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+[    {        "title": "1609.05603v1.Effective_indirect_multi_site_spin_spin_interactions_in_the_s_d_f__model.pdf",        "content": "arXiv:1609.05603v1  [cond-mat.str-el]  19 Sep 2016Effective indirect multi-site spin-spin interactions\nin the s-d(f) model\nK.K.Komarova), D.M.Dzebisashvilia,b)\naKirensky Institute of Physics, Federal Research Center KSC SB RAS, 660036, Krasnoyarsk,\nRussian Federation\nbSiberian State Aerospace University, Krasnoyarsk, 660014 , Russian Federation\nUsing the diagram technique for Matsubara Green’s function s it is shown that the dynamics\nof the localized spin subsystem in the s-d(f) model can be des cribed in terms of an effective spin\nmodel with multi-site spin-spin interactions. An exact rep resentation of the action for the effective\npurely spin model is derived as an infinite series in powers of s-d(f) exchange interaction J. The\nindirect interactions of the 2-nd, 3-rd and 4-th order are di scussed.\n1. Introduction\nKondo lattice model or s-d(f) exchange model is widely used to desc ribe the correlation\neffects in metals (and their compounds) with unfilled d- andf-shells.\nThe nature of the ground state in these systems is largely determin ed by the result of\ncompetition between two interactions. On the one hand, s-d(f) ex change coupling between\nspins of itinerant s- and localized d(f)- electrons due to Kondo fluct uations screens the\nlocalized spins and tends to form a non-magnetic ground state [1]. I n the opposite direction\nacts indirect exchange (RKKY) interaction between spins of d(f)- electrons [2], trying to set\na long-rang magnetic order which is not necessarily a ferromagnetic or antiferromagnetic.\nCorrelation effects can lead to, for example, helical magnetic struc tures [3], with period that\nis determined by the Fermi surface singularities [5, 6].\nIt is clear that the study of competition between different effective interactions, that\noccur in the localized spin subsystem and are caused by the same itine rant electrons, must\nbe carried out within the same unified approach.\nThe purpose of this paper is to derive such an effective Hamiltonian (o r rather action),\nwhich will allow to study the localized spin subsystem in the s-d(f) mode l within the model\nwith only spin-spin interactions. At the same time all possible spin-spin interactions that\noccur in different orders of perturbation theory in J, should be determined by the same\nmodel parameters: 1) the s-d(f)-exchange coupling constant J; 2) the dispersion law of\nconduction electrons; 3) the filling degree of the conduction band. Such an approach should\nallow to analyze all types of exchange spin-spin interactions from a u nified point of view.\nThis problem is solved by integrating charge degrees of freedom usin g the diagram tech-\nnique for Matsubara Green’s functions. It is shown that in addition t o the two-spin indirect\nexchange interaction, which occurs in the second order in s-d(f)- parameter J, in the follow-\ning orders also appear terms describing, in particular, the ring and b iquadratic exchange\ninteractions. The essential point of all these interactions is the ac count for retardation ef-\nfects provided by the imaginary time dependence of all the effective multi-site exchange\nparameters.\n12. The Hamiltonian of the s-d(f) exchange model\nThe Hamiltonian of the Kondo lattice model (or s-d(f) exchange mod el) can be written as a\nsum of two terms:\nˆH=ˆH0+ˆHint, (1)\nwhere\nˆH0=/summationdisplay\nkα(εk−µ)c+\nkαckα,ˆHint=J\n2/summationdisplay\nfc+\nf˜Sfcf. (2)\nOperator ˆH0stands for the energy of noninteracting current carriers (elect rons or holes) with\ndispersion εk,µ– chemical potential. Operator c+\nkα(ckα) creates (annihilates) a particle in\nthe state with quasimomentum kand spin projection α=±1/2.\nThe second term in (1) describes Kondo exchange interaction betw een localized spins\nand itinerant quasiparticles. The intensity of this interaction is defin ed by the constant\nJ. Operator ˜Sf=/vectorSf/vector σis a product of a localized spin vector operator /vectorSf= (Sx\nf,Sy\nf,Sz\nf)\nand a vector /vector σ= (σx,σy,σz) which is formed of Pauli matrices. In definition (2) the spinor\nnotations:\nc+\nf=/parenleftbigc+\nf↑c+\nf↓/parenrightbig\n, cf=/parenleftbiggcf↑\ncf↓/parenrightbigg\n,˜Sf=/parenleftbiggSz\nfS−\nf\nS+\nf−Sz\nf/parenrightbigg\n, S±\nf=Sx\nf±iSy\nf, (3)\nare used. The operators cfandckare related to each other by Fourier transformation:\ncf=N−1/2/summationtext\nkeikfck.\n3. Spin Green’s functions and effective action\nThe thermodynamic properties of localized spin subsystem convenie ntly studied on the basis\nof the diagram technique for Matsubara Green’s functions [7]:\nGjj′(f−f′,τ−τ′) =−/angbracketleftBig\nTτ¯Sj\nf(τ)¯Sj′\nf′(τ′)/angbracketrightBig\n, j,j′={x,y,z}. (4)\nIn this expression the spin operators are written down in the Heisen berg representation:\n¯Sj\nf(τ) =eτˆHSj\nfe−τˆH, where ˆHis the s-d(f) exchange Hamiltonian (1), and τ– imaginary\ntime varying within interval (0 ,1/T) (T– is the temperature). The imaginary time ordering\noperator Tτarranges all operators on the right side of it in the descending orde r of the\nimaginary time τfrom left to right. Angle brackets in (4) denote a thermodynamic av erage\nover the grand canonical ensemble described by the Hamiltonian ˆH.\nAs is known [8], in the interaction representation: Sj\nf(τ) =eτˆH0Sj\nfe−τˆH0, the expression\n(4) transforms into:\nGjj′(f−f′,τ−τ′) =−/angbracketleftBig\nTτSj\nf(τ)Sj′\nf′(τ′)S(β)/angbracketrightBig\n0. (5)\nHere the scattering matrix S(β) is defined by the formula:\nS(β) =Tτexp/braceleftbigg\n−/integraldisplayβ\n0dτˆHint(τ)/bracerightbigg\n, (6)\n2whereintheexponentundertheintegralstandstheinteractiono perator(2)intheinteraction\nrepresentation. Lower index ”0”, on the right of the angle bracke ts in (5) indicates that the\nthermodynamic averaging is over the ensemble of non-interacting s pin and fermion systems.\nBesides, when expanding the thermodynamic averages using Wick’s t heorem only connected\ndiagrams should be considered.\nNote that for calculating the Green’s function (5) along with the usu al diagramtechnique\n[8] it is necessary to use the diagram technique for spin operators [9 ] as well. The expanding\nof aTτ-ordered average of a product of spin S- and fermion c-operators can be divided into\ntwo stages: first, only c-operators are paired, and then the Wick’s theorem is applied to the\nremaining spin operators. Formally this can be written down as follows :\n/angbracketleftbig\nTτS1...Slc1...c+\nm/angbracketrightbig\n0=/angbracketleftBig\nTτS1...Sl/angbracketleftbig\nTτc1...c+\nm/angbracketrightbig\nc0/angbracketrightBig\nS0. (7)\nInternal thermodynamic averageontherightsideoftheequation (7)withindex ” c0”denotes\naveraging only over an ensemble of non-interacting fermions. Exte rnalTτ-ordered averaging,\nindicated by the index ” S0”, should be done using the ensemble of non-interacting spin\nsubsystems. Applying equation (7) to the definition of the Green’s f unction (5) we can\nwrite:\nGjj′(f−f′,τ−τ′) =−/angbracketleftBig\nTτSj\nf(τ)Sj′\nf′(τ′)SS(β)/angbracketrightBig\nS0, (8)\nwhere the effective scattering matrix SS(β) is defined by the expression:\nSS(β) =/an}bracketle{tS(β)/an}bracketri}htc0. (9)\nTo obtaintheeffective purely spin model first weshould pair all the c-operatorsaccording\nto the Wick’s theorem in each order of the S-matrix expansion in powers of the coupling\nconstants J. Afterthattheresultmustberewrittenasa Tτ-orderedexponent. Theargument\nof the exponent will determine the required effective interaction in t he subsystem of localized\nspins. Note that the described here scheme of integrating over th e charge degrees of freedom\nin the s-d(f) model in some sense is similar to the proof of equivalence between diagrammatic\nexpansion for the Green’s function of localized f-electrons in the periodic Anderson model\nand diagrammatic expansion of the fermion Green’s function in the Hu bbard model [10, 11].\nLet us consider for the average /an}bracketle{tS(β)/an}bracketri}htc0the diagrams of n-th order with respect to\nthe constant J. Since each term in ˆHintconsists of one spin operator and two second-\nquantization operators ( c+andc), the structure of diagrams emerging after all possible\npairings of c-operators is characterized by a some number of fermionic loops (s ee figure\n1). The order of each loop is determined by the number of fermionic lin es, corresponding\nto the fermion Green’s functions G(0)\nk(τ−τ′) =−/an}bracketle{tTτck(τ)c+\nk(τ′)/an}bracketri}ht0, and the same number of\nvertices. In contrast to the conventional diagram technique in th is case each vertex is related\nnot to a function (or a number) but to the spin operator. Therefo re then-th order diagram\ncorresponds to a product of nspin operators entering to the Tτ-ordered thermodynamic\naverage/an}bracketle{tTτ.../an}bracketri}htS0.\nLet us denote by pjthe number of loops of the same order mj, where the index j\nenumerates all loops of different order. Then obviously n=p1m1+p2m2+.... Until the\nspin operators are not paired the analytical expression correspo nding to each n-th order\ndiagram can be represented as a product of some multipliers. Each m ultiplier is related to\na single fermionic loop. Note that in the case of further application of the spin diagram\n3L\n1pjp121m3\n121m3\n12jm3\n12jm3\nLL L\nFigure 1: The n-th order diagrams, arising due to pairing of all c-operators in the expansion\nof/an}bracketle{tS(β)/an}bracketri}htc0, can be represented as a set of fermionic loops. The order of each loopmjis\ndetermined by the number of vertices represented by points and t he same number of lines\nwith arrows denoting the fermion propagator. Each vertex corre sponds to the spin operator.\nIf the number of vertices of the same order mjis denoted by pj, then:p1m1+p2m2+···=n.\ntechnique [9] there should be taken into account only such pairings o f spin operators when\nall fermionic loops are connected graphically with elements of the spin diagram technique.\nThe fermion loop of the mj-th order is formed due to mjoperators ˆHint. Let’s use the\nindex ”L” in angle brackets /an}bracketle{t.../an}bracketri}htLto indicate the fact that when pairing all the c-operators\nin theTτ-ordered average of mjoperators ˆHintthere should be taken into account only one-\nloop diagrams of mj-th order. Then the analytical contribution to the one loop of mj-th\norder will be determined by the average: /an}bracketle{tTτˆHint(τ1)...ˆHint(τmj)/an}bracketri}htLc0. Strictly speaking this\naverage gives rise to ( mj−1)! equivalent fermionic loops of mj-th order. Using the definition\nof the average /an}bracketle{tTτ.../an}bracketri}htLc0the expansion of /an}bracketle{tS(β)/an}bracketri}htc0can be written as:\n/angbracketleftbig\nS(β)/angbracketrightbig\nc0=Tτ/summationdisplay\nn(−1)n\nn!/summationdisplay\np1,p2...\nm1,m2...\n(p1m1+···=n)A(p1m1;p2m2;...)×\n×/productdisplay\nj/integraldisplay\ndτ(j)\n1...dτ(j)\nm1/angbracketleftbig\nTτˆHint(τ(j)\n1)...ˆHint(τ(j)\nm1)/angbracketrightbig\nLc0. (10)\nInderivationtheformula(10)ineach n-thorderof ˆHintthethermodynamicaverage /an}bracketle{tTτ.../an}bracketri}htc0\nwas represented as a sum of products of averages /an}bracketle{tTτ.../an}bracketri}htLc0. The second sum in (10) takes\ninto account all possible partitions of noperators ˆHintintop1+p2+...groups. Each\ngroup corresponds to a single fermionic loop and the equality: p1m1+p2m2+···=nmust\nbe satisfied. The index jin the product/producttext\njruns all groups (loops) at a fixed partition.\nBecause the averaging in the /an}bracketle{tS(β)/an}bracketri}htc0is carried out only over the charge degrees of freedom,\nthe remaining spin operators must be still ordered by the imaginary t ime. This justifies the\npresence of operator Tτin the right side of (10). Combinatorial factor A(p1m1;p2m2;...)\ntakes into account the equivalent contributions arising at each pos sible partitioning of n\noperators ˆHintin groups.\nIt can be shown (cf. [8]) that:\nA(p1m1;p2m2;...) =n!\np1!(m1!)p1p2!(m2!)p2.... (11)\nIntroducing notations:\nΞm=(−1)m+1\nm!/integraldisplayβ\n0dτ1...dτm/angbracketleftBig\nTτˆHint(τ1)...ˆHint(τm)/angbracketrightBig\nLc0, (12)\n4Figure 2: The effective action Ξ can be expressed as an infinite series of terms Ξ n. Each\nterm Ξ nin the diagrammatic representation corresponds to a single loop of n-th order in the\nexchange coupling constant Jand describes the effective n-site spin-spin interaction. The\nlines with arrows represent bare fermion Green’s functions and eac h vertex, shown with a\nbold circle, corresponds to a spin operator ˜S.\nthe expression (10) can be rewritten as:\n/angbracketleftbig\nS(β)/angbracketrightbig\nc0≡SS(β) =Tτ/summationdisplay\np1,p2...1\np1!(−Ξ1)p11\np2!(−Ξ2)p2···=Tτexp{−Ξ1−Ξ2−...}.(13)\nUsing the explicit form of ˆHintcarry out in (12) pairings of all the c-operators. Then\nintroducing the variable xj= (/vectorRfj,τj) we obtain expression for the operator Ξ min the\nWannier representation:\nΞm=1\nm/parenleftbiggJ\n2/parenrightbiggm/integraldisplay\ndx1...dxmG(0)(x1−x2)...G(0)(xm−x1) Sp/braceleftBig\n˜S(x1)...˜S(xm)/bracerightBig\n,(14)\nwhere the integral over d xdenotes the operation/integraltextβ\n0dτ/summationtext\nfand the trace is calculated over\nthe Pauli matrix indices.\nFrom the formula (13) it follows that the effective interactions in the localized spin\nsubsystem are determined by the structure of the series:\nΞ =∞/summationdisplay\nm=1Ξm. (15)\nThe effective action Ξ describes all possible multi-site spin-spin intera ctions in the localized\nspin subsystem in the arbitrarily order of the coupling constant J. The partial action Ξ m\ndetermines the m-th order interactions and diagrammatically can be represented as a loop\nwithmlines, corresponding to propagators G(0), andmvertices related to spin operators ˜S\n(see figure 2). It can be seen that all effective interactions in Ξ tak e into account retardation\neffects. This is due to imaginary time dependence of all the effective in teractions in each\nterm (14) of the series for Ξ.\n4. Effective interactions of the 2nd, 3rd and 4th order\ninJ\nLet us consider the first several terms of the series for Ξ. The fir st term with n= 1 is zero,\nbecause Sp {˜S(x1)}=/summationtext\njSj(x1)Sp{σj}, and Sp{σj}= 0 at any j=x,y,z.\n5To calculate the operator Ξ 2one should to derive the trace Sp/braceleftBig\n˜S(x1)˜S(x2)/bracerightBig\n. Using the\nidentity: σiσj=δij+iεijlσl, whereεijl– Levi-Civita antisymmetric tensor we find:\nSp/braceleftBig\n˜S(x1)˜S(x2)/bracerightBig\n= 2/parenleftBig\n/vectorS(x1)/vectorS(x2)/parenrightBig\n. (16)\nThen the operator Ξ 2takes the form:\nΞ2=/integraldisplay\ndx1dx2V2(x1−x2)/vectorS(x1)/vectorS(x2), (17)\nwhere the effective interaction between spins is defined via a polariza tion loop (see figure 3):\nV2(x1−x2) =/parenleftbiggJ\n2/parenrightbigg2\nG(0)(x1−x2)G(0)(x2−x1). (18)\nFigure 3: The action Ξ 2can be represented by a loop of the second order in the exchange\ncoupling constant J.\nFrom equation (17) it follows, that at f1/ne}ationslash=f2the second order effective action Ξ 2de-\nscribesindirect exchangeinteractionoftwo localizedspinsthrough thesubsystem ofitinerant\nelectrons. However, in contrast to the usual RKKY-interaction h ere the retardation effects,\ncaused by τ-dependence of V2, are taken into account. Note that in the Ξ 2there is also a\nterm with f1=f2. The Fourier transform of the indirect exchange interaction (18) has the\nform:\nV2(k,iωm) =/parenleftbiggJ\n2/parenrightbigg2\nχ0(k,iωm), (19)\nwhere\nχ0(k,iωm) =1\nN/summationdisplay\nqfq−fq+k\niωm+εq−εq+k(20)\nis the Lindhard susceptibility in the Matsubara representation, ωm= 2mπTwithm∈Z,\nandfq= (exp{(εq−µ)/T}+1)−1is the Fermi-Dirac distribution function. The intensity of\nthe exchange interaction is largely determined by the properties of the itinerant subsystem.\nThe well known RKKY-interaction follows from (20) at ωm≡0.\nCalculating the 3rd order effective action Ξ 3we obtain (see also figure 4):\nΞ3=/integraldisplay\ndx1dx2dx3V3(x1,x2,x3)/vectorS(x1)·/parenleftBig\n/vectorS(x2)×/vectorS(x3)/parenrightBig\n, (21)\nwhere\n6Figure 4: Diagrammatic representation of the third order effective action Ξ 3.\nFigure 5: Fourth order effective action Ξ 4in the diagrammatic representation.\nV3(x1,x2,x3) =i2\n3/parenleftbiggJ\n2/parenrightbigg3\nG(0)(x1−x2)G(0)(x2−x3)G(0)(x3−x1). (22)\nThe formula (21) describes the three-spin interactions in the form of a mixed product of\nthree spin operators. This type of interaction was first considere d in [12] but without taking\ninto account the retardation effects.\nItisevidentthattheinteraction(21)favorsthechiralorderinth emagneticsubsystem. In\nthis regard we note that in the paper [13] the third order correctio ns to the Hall conductivity\ndue to s-d(f) exchange interaction were shown to give rise the ano malous Hall effect provided\nthat non trivial spin configuration (chirality) is formed in the spin sub system. Interestingly\nthe structure of the expression for the Hall conductivity obtaine d in [13] is similar to that\nof (21).\nThe operator Ξ 4, which determines the effective spin-spin interactions in the fourth order\nin the coupling constant J, after calculating the trace Sp {˜S(x1)˜S(x2)˜S(x3)˜S(x4)}takes the\nform (see also figure 5):\nΞ4=/integraldisplay\ndx1dx2dx3dx4V4(x1,x2,x3,x4)/parenleftBig\n/vectorS(x1)/vectorS(x2)/parenrightBig\n·/parenleftBig\n/vectorS(x3)/vectorS(x4)/parenrightBig\n, (23)\nwhere\nV4(x1,x2,x3,x4) =1\n2/parenleftbiggJ\n2/parenrightbigg4\n[G(0)(x1−x2)G(0)(x2−x3)G(0)(x3−x4)G(0)(x4−x1)−\n−G(0)(x1−x3)G(0)(x3−x2)G(0)(x2−x4)G(0)(x4−x1)+\n+G(0)(x1−x4)G(0)(x4−x3)G(0)(x3−x2)G(0)(x2−x1) ].(24)\nThe terms ofthe expression (23) with unequal indices of sites fj(j= 1,...,4) correspond\nto the four-spin exchange interactions. Among them there are, in particular, interactions\ndescribing ring exchange of four spins located, for example, at the square plaquette vertices.\nFor the first time the ring exchange (without retardation effects) was obtained from the\n7Hubbard model at half filling in the fourth order of the perturbation theory in the parameter\nt/U, wheretisthetunneling integral, and UistheCoulombrepulsion energyoftwo electrons\nat the same site [14]. The four-spin ring exchange interaction was inv olved to explain the\nmagnetic ordering features in the quantum crystal3He [15]. In the paper [16] it was argued\nthat ring exchange is important to describe magnetic properties of cuprate high-temperature\nsuperconductors. Effect of ring exchange interaction on the sup erconductivity in cuprates\nwas investigated in [17].\nAt pairwise coincident site indices: f3=f1andf4=f2, in the sum (23) there are terms\nthat are responsible for biquadratic exchange interaction. These interactions were first used\nin[18]forexplaining theparamagneticresonanceonMnionsinthecom poundMnO.Besides,\nthe biquadratic exchange interaction is essential in multilayer magne tic systems [19].\n5. Conclusion\nThe paper presents a method of deriving all possible kinds of effectiv e indirect interactions\nbetween the localizedspins dueto s-d(f) exchange coupling of thes e spins withthe subsystem\nof itinerant electrons. After integrating over charge degrees of freedom in the s-d(f) exchange\nmodel an exact representation for an action of a purely spin model is obtained. Using this\naction allows to study the spin subsystem in the s-d(f) model in the f ramework of effective\npurely spin model. The important point of this model is that all effectiv e interactions take\ninto account retardation effects. Although explicit expressions ar e written only for two-,\nthree- and four-spin interactions, the formula (14) allows to gene rate multi-site spin-spin\ninteractions in the arbitrary order in the s-d(f) exchange coupling constant J.\nThe work was supported by the Russian Foundation for Basic Resea rch (project nos.\n16-42-240435 and 15-42-04372).\nReferences\n[1] V.V. Moschalkov, N.B. Brandt, UFN 149(1986) 585.\n[2] M.A. Ruderman, C. Kittel, Phys. Rev. 96(1954) 99; K. Yosida, Phys. Rev. 106(1957)\n893; T. Kasuya, Prog. Theor. Phys. 16(1956) 45.\n[3] M. Hamada, H. Shimahara, Phys. Rev. B. 51(1995) 3027.\n[4] N. de C. Costa, J.P. de Lima, R.R. dos Santos, arXiv: 1506.00890v 1 [cond-mat.str-el]\n(2015).\n[5] K. Yosida, A. Watabe, Prog. Theor. Phys. 28(1961) 361.\n[6] I.E. Dzyaloshinskii, ZhETF 47(1964) 336.\n[7] T. Matsubara, Prog. Theor. Phys. 14(1955) 351.\n[8] A.A. Abrikosov, L.P. Gorkov, I.E. Dzyaloshinskiy, Methods of qua ntum field theory\nin statistical physics. – Moscow: Fizmatgiz, 1962. – 444 p.\n[9] Yu.A. Izyumov, F.A. Kassan-ogly, Yu.N. Skryabin, Field methods in the theory of\nferromagnetism. – Moscow: Nauka, 1974. S. – 224\n8[10] V.A. Moskalenko, TMF 110(1997) 308.\n[11] V.V. Val’kov, D.M. Dzebisashvili, Pis’ma v ZhETF 84(2006) 251.\n[12] H.A. Brown, JMMM 43(1984) L1-L2.\n[13] G. Tatara, H. Kawamura, JPSJ 71(2002) 2613.\n[14] M. Takahashi, J. Phys. C: Solid State Phys. 10(1977) 1289.\n[15] M. Roger, J.H. Hetherington, J.M. Delrieu, Rev. Mod. Phys. 55(1983) 1-64.\n[16] M. Roger, J.M. Delrieu, Phys. Rev. B 39(1989) 2299.\n[17] E.I. Shneyder, S.G. Ovchinnikov, A.V. Shnurenko, Pis’ma v ZhETF 95(2012) 211.\n[18] E.A.Harris, J.Owen, Phys. Rev. Lett. 11(1963) 9.\n[19] J.C. Slonczewski, J. Appl. Phys. 73(1993) 5957.\n9"    },    {        "title": "0906.5111v1.Steady_state_spin_densities_and_currents.pdf",        "content": "arXiv:0906.5111v1  [cond-mat.mes-hall]  27 Jun 2009Steady-state spin densities and currents\nDimitrie Culcer1,2,3\n1Condensed Matter Theory Center, Department of Physics,\nUniversity of Maryland, College Park MD 20742.\n2Advanced Photon Source, Argonne National Laboratory, Argo nne, IL 60439.\n3Northern Illinois University, De Kalb, IL 60115.\n(Dated: December 4, 2018)\nThis article reviews steady-state spin densities and spin c urrents in materials with strong spin-\norbit interactions. These phenomena are intimately relate d to spin precession due to spin-orbit\ncoupling which has no equivalent in the steady state of charg e distributions. The focus will be\ninitially on effects originating from the band structure. In this case spin densities arise in an\nelectric field because a component of each spin is conserved d uring precession. Spin currents arise\nbecause a component of each spin is continually precessing. These two phenomena are due to\nindependent contributions to the steady-state density mat rix, and scattering between the conserved\nand precessing spin distributions has important consequen ces for spin dynamics and spin-related\neffects in general. In the latter part of the article extrinsi c effects such as skew scattering and side\njump will be discussed, and it will be shown that these effects are also modified considerably by\nspin precession. Theoretical and experimental progress in all areas will be reviewed.\nI. INTRODUCTION\nSpin electronics seeks to harness the spin degree of freedom of th e electron, aiming to generate and maintain a\nspin polarization for possible use in information storage through the manufacture of magnetic memory devices. For\nspin-based quantum computing the manipulation of a spin polarization is also crucial. A spin polarization can be\ngenegrated by optical, magnetic and electrical means. Optically, on e uses light to transfer angular momentum to\ncharge carriers during interband transitions.[1] This reliable proced ure has been tested successfully decades ago, yet\ndevices that require an input of light are currently impractical. Magn etically, one uses the Zeeman effect, to which\nsimilarobservationsapply. Electrically, onedrivesacurrentfromas pin-polarizedmaterialintoanunpolarizedoneand\nthus carry spin-polarized electrons across the interface. This wo rks in the case of metals but modern technology relies\noverwhelmingly on semiconductors, and spin injection from ferroma gnetic metals into semiconductors is hampered by\nthe resistivity mismatch [2] at the interface between the metal and the semiconductor, which causes most of the spin\npolarization to be lost at the interface. Another avenue explores t he use of ferromagnetic semiconductors[3] as spin\ninjection devices, but room temperature ferromagnetism in semico nductors remains a long-term goal.\nAs a result of these challenges, purely electrical means of spin gene ration have been sought. The past decade\nhas witnessed an explosion of interest in electrically-induced spin phe nomena in semiconductors and metals, which\nhas accompanied significant experimental and theoretical progre ss. It has been found that the application of an\nelectric field gives rise to a nonequilibrium spin density in the bulk of the s ample, discussed in Refs. [[4]-[12]], and a\nnonequilibrium spin current, discussed in Refs. [[13]-[85]]. The appeara nce of a steady-state spin density in an electric\nfield was predicted several decades ago [4] and subsequently obse rved in tellurium. [5] This work was followed by a\nnumber of theoretical [6, 7, 8, 9] and experimental [10, 11, 12] st udies. Spin currents also have a long history, their\ntheoretical prediction stretching back almost four decades to th e work of Dyakonov.[13] The theoretical work was,\nhowever, not followed by experiments, and the topic was silent for o ver a quarter of a century, until the theory picked\nup again, led by Hirsch [14], Murakami et al.[15] and Sinova et al.[16] During the recent surge ofinterest in electrically-\ninduced spin phenomena the focus has been principally on spin curren ts, with the ultimate goal of measuring spin\nflow in a given direction. In the absence of a spin-ammeter , initially the only way of detecting a spin current relied on\nmeasuring the spin accumulation it produced at the edge of the samp le. This procedure, as will be discussed below,\nis complicated by the fact that the spin current is often not well defi ned and its relationship to spin accumulation\nis not clear. Thus one can detect a spin accumulation at the edge of t he sample as a result of the spin current, but\ninferring the size of the spin current from such a measurement is be yond current understanding. Nevertheless, in just\na few years experiment has been surging ahead. First, in a recent e xperiment, Crooker and Smith [17] imaged the\nflow of spin in semiconductors in the presence of electric and magnet ic fields, as well as strain. Following that, spin\naccumulations resulting from spin currents were measured by seve ral groups. [18, 19, 20, 21, 22, 23, 24] In what is\nsurely a new generation of experiments, Valenzuela and Tinkham[25 ] made use of the fact that a spin current gives\nrise to a transversecharge current. Relying on a nonlocal techniq ue, their group successfully detected a charge current\nflowing as a result of a spin current in Al metal. Cui et al.[26] also observed an electrical current induced by a spin\ncurrent, though in that case the spin current was generated opt ically. This technique was used by several groups2\n[27, 28, 29, 30, 31] shortly afterwards in the study of platinum- an d gold-based structures. In gold an unusually large\nspin current was observed[30] even at room temperature. Room t emperature spin currents were also reported by\nSternet al[21] in ZnSe. Thus, from their initial observation four years ago, sp in currents have begun an experimental\nrevolution, and the field has matured to the point where experiment and theory interact constructively.\nThe phenomena discussed in this review are all related to spin-orbit in teractions, which are present in the band\nstructure and in potentials due to impurity distributions. Band stru cture spin-orbit interactions are frequently the\nmost important factor determining spin dynamics in solids, and are pa rticularly important in semiconductors, where\ncarriersinvolvedin steady-stateprocesseshavewavevectorsw ell within the first Brillouinzone. At these wavevectors,\nwhile the spin-orbit contribution to the carrierenergy is still smaller t han the kinetic energy, it plays an important role\nin determining the energy spectrum. Band structure spin-orbit co upling may arise from the inversion asymmetry of\nthe underlying crystal lattice [86], from the inversion asymmetry of the confining potential in two dimensions [87], and\nmay be present also in inversion symmetric systems. [88] Spin-orbit c oupling is also in principle present in impurity\npotentials. Spin-orbit interactionsin the impurity potentialscauses kewscattering, orasymmetricscatteringofup and\ndown spins, giving scattering-dependent or extrinsic contributions to spin currents. Finallly, spin-orbit interactions\ncause a modification ofthe position and velocity operators, which aff ects the interactionwith an electric field as well as\nthe scattering term. This last mechanism is referred to as side jump and is typically classified as extrinsic although its\ncontributions are manifold. These extrinsic mechanisms were discov ered in the anomalous Hall effect [89, 90, 91, 92]\nbut have acquired a new relevance in recent years in electrically-indu cedspinphenomena.\nIn charge transport, the steady state is characterized by a non equilibrium density matrix that is divergent in the\nclean limit, indicating a competition between the electric field, accelera ting charge carriers, and scattering, which\ninhibits their forward motion. The density matrix is a scalar and the co rrection to it caused by the electric field is\n∝n−1\ni, whereniis the impurity density. When considering the steady state of syste ms with spin-orbit interactions\nthree different situations are distinguished: the case when only ext rinsic mechanisms are important, the case when\nonly band-structure mechanisms are important, and the case whe n both extrinsic and band-structure mechanisms are\nimportant. The differences between these cases are significant an d nontrivial.\nIf band structure spin-orbit interactions are negligible and only ext rinsic mechanisms are important, there is no\nspin precession and the correction to the spin density matrix due to an electric field is very similar to the correction to\nthe scalar density matrix in charge transport. This correction is ∝n−1\niand it gives rise to a steady-state spin current\nbut not to a steady-state spin density. There have been many stu dies of extrinsic spin currents in semiconductors\n(Refs. [[77]-[79]]) and metals (Refs. [[14],[84]].)\nNonequilibrium corrections that arise as a result of band structure spin-orbit coupling in crystal Hamiltonians\nrepresent a different kind of interplay between the electric field and scattering processes. The spin-orbit splitting\nof the bands gives rise to spin-dependent scattering even from sp in-independent potentials, and spin currents and\nspin densities in an electric field arise from linearly independent contrib utions to the density matrix. The steady-\nstate density matrix contains a contribution due to precessing spin s and one due to conserved spins. Steady-state\ncorrections ∝n−1\niare associated with the absenceof spin precession and steady-state corrections independent of ni\nare associated with spin precession. Steady-state corrections ∝n−1\nigive rise to spin densities in external fields while\nsteady-state corrections independent of nigive rise to spin currents in external fields. Scattering between th ese two\ndistributions induces significant corrections to steady-state spin currents, and may cause spin currents to vanish.\nThe phrase spin current refers to the flow of spins across a sample and, if spin were conserv ed, one could distinguish\nbetween spin-up and spin-down charge currents. In systems with band structure spin-orbit interactions the spin\ncurrent is not well defined [32, 33, 34] and its relationship to spin acc umulation is unclear. Spin transport in these\nsystems usually does not involve charge transport as the charge c urrents in the direction of spin flow cancel out.\nIn addition, since spin currents may be accompanied by steady-sta te spin densities, this makes it more difficult to\nseparate experimentally the signals due to these contributions. Th e study of spin currents in systems with band\nstructure spin-orbit interactions has thus encountered a numbe r of profound physical issues on which no consensus\nexists at present. There has been considerable research has on d efinitions of spin currents in such systems (Refs. [[32]-\n[35]]), the role played by the symmetry of the underlying crystal latt ice,[36] whether spin currents as a result of\nband structure spin-orbit coupling are transport or background currents, [37] the relationship between band structure\nspin currents and spin accumulation, (Refs. [[38]-[45]]) and the form o f the Maxwell equations in systems with band\nstructure spin-orbit coupling.[46, 47] Most theoretical studies ha ve focused on metals, common semiconductors and\nasymmetric quantum wells with band structure spin-orbit interactio ns, such as Refs. [[16] -[76]].\nWhen both band structure and extrinsic mechanisms are present, the situation becomes considerably more compli-\ncated. There is no simple association between spin conservation and spin densities, or between spin precession and\nspin currents. Both spin densities and currents may contain terms ∝n−1\niand independent of ni. A careful analysis\nshows that the presence of band structure spin-orbit interactio ns causes skew scattering and side jump to behave very\ndifferently in the steady state, and under certain circumstances, the contributions of these extrinsic mechanisms to the\nspin current vanish. The interplay of band structure and extrinsic mechanisms has been considered in Refs. [[80]-[83]].3\nThis review will cover steady-state spin densities and currents with in the framework of a density matrix theory.\nSuch a framework is important in order to demonstrate the unity be hind all observed phenomena. I will consider\nlarge, uniform systems, and work in momentum space. Although man y observations in this entry are general, the\ndiscussion will focus on non-interacting spin-1/2electron systems , which are pedagogicallyeasier. This review is based\non a kinetic-equation formalism equivalent to linear response theory and correspondences with other methods will be\nidentified, such as linear response theories based on Green’s funct ions and semiclassical wave packet dynamics.\nThisreviewarticleisstructuredasfollows. Section2will introducead ensitymatrixpictureofspindynamics, briefly\nsketching the way different theories stem from the quantum Liouville equation. In section 3 a brief introduction to\nthe spin-orbit interaction in spin-1/2 systems is given, after which s pin densities and spin currents as a result of band\nstructure spin-orbit interactions are discussed, as well as their r elationship to spin precession. Section 4 is devoted\nto spin densities and spin currents that arise as a result of extrinsic mechanisms, together with the case when both\nextrinsic and band-structure mechanisms are present. Section 5 discusses briefly various definitions of spin currents\nand the relationship between spin currents and spin acumulation. Se ction 6 outlines the role of the underlying crystal\nlattice in the establishment of spin densities and currents and sectio n 5 is concerned with open issues such as the\ndefinition and nature of spin currents, as well as the relationship be tween spin currents and spin accumulation. In\nsection 7 the experimental situation is summarized and in closing futu re directions are discussed. Related topics such\nas the quantum spin-Hall effect [93] and the spin-Hall insulator [94] a re beyond the scope of this review.\nII. DENSITY MATRIX PICTURE OF SPIN DYNAMICS\nElectrically-induced spin phenomena encompass a wide variety of pro cesses that have been studied using different\nmethods. In order to bring out the unity behind the processes invo lved, as well as behind the theoretical approaches\nemployed, it will be useful to have a unified frameworkin which electric ally-induced spin phenomena can be discussed.\nIn this section such a framework will be constructed beginning with t he quantum Liouville equation. This equation,\nin one form or another, is the starting point of most theories of spin dynamics, and this section will outline the way\nvarious approaches are related to each other. The focus will be on systems with large homogeneous systems with long\nmean free paths, and diffusion will not be considered explicitly. Electr ic fields will be assumed uniform.\nA. Quantum Liouville equation\nA system of non-interacting spin-1/2 electrons is represented by a one-particle density operator ˆ ρ. The expectation\nvalue of an observable represented by a Hermitian operator ˆOis given by Tr(ˆ ρˆO), where Tr denotes the most general\noperator trace, which involves summation over discrete degrees o f freedom and integration over continuous ones. The\nusual matrix trace will be denoted by tr. The dynamics of ˆ ρare described by the quantum Liouville equation,\ndˆρ\ndt+i\n/planckover2pi1[ˆH+ˆUdis+eE·ˆr,ˆρ] = 0. (1)\nThe Hamiltonian ˆHcontains contributions due to the kinetic energy and spin-orbit cou pling, while ˆris the position\noperator. The effect of the lattice-periodic potential of the ions is taken into account through a replacement of\nthe carrier mass by the effective mass. The potential ˆUdisaccounts for scattering processes, which may be due to\nimpurities, phonons, surface roughness, or other perturbation s. This review focuses on impurity scattering, as the\neffects discussed are frequently observed at low temperatures, where scattering due to phonons may be neglected. It\nis assumed that the electric field is small and a solution is sought to firs t order in the electric field. The solution of the\nLiouville equation (1) in the absence of the external field is provided b y ˆρ0, the Fermi-Dirac function. The correction\ndue to the external field, ˆ ρE, satisfies\ndˆρE\ndt+i\n/planckover2pi1[ˆH+ˆUdis,ˆρE] =−i\n/planckover2pi1[eE·ˆr,ˆρ0]. (2)\nEquation(2)isprojectedontoacompletesetoftime-independent statesofdefinitewavevector {|ks∝an}bracketri}ht}. Thesestatesare\nnot assumed to be eigenstates of ˆH. The matrix elements of ˆ ρin this basis will be written as ρkk′≡ρss′\nkk′=∝an}bracketle{tks|ˆρ|k′s′∝an}bracketri}ht,\nwith corresponding notations for the matrix elements of ˆHandˆUdis. Spin indices will not be shown explicitly in the\nsubsequent discussion, and it will be understood that the quantitie sρkk′,Hkk′, andUdis\nkk′are matrices in spin space.\nρkk′is referred to as the density matrix. With our choice of basis functio ns of definite wave vector, matrix elements of\nthe Hamiltonian Hkk′=Hkδkk′arediagonalin k. However,sincethe Hamiltoniancontainsspin-orbitcouplingterms,\nmatrix elements Hkare generally off-diagonal in spin space. Matrix elements of the scat tering potential Udis\nkk′are4\noff-diagonal in k. Matrix elements diagonal in kin the scattering potential would lead to a redefinition of Hk, which\nis analogous, in Green’s function formalisms, to the offset introduce d by the real part of the self energy. Scattering\nis assumed elastic and impurities uncorrelated, and the normalization is such that the configurational average of\n∝an}bracketle{tks|ˆUdis|k′s′∝an}bracketri}ht∝an}bracketle{tk′s′|ˆUdis|ks∝an}bracketri}htis (ni|Ukk′|2δss′)/V, whereniis the impurity density, Vthe crystal volume and Ukk′the\nmatrix element of the potential of a single impurity. Configurational averages over terms of higher order in ˆUdisare\nperformed in a similar fashion. [96] It is assumed that εFτ//planckover2pi1≫1, where εFis the Fermi energy and τa characteristic\nscattering time. This is equivalent to the assumption that the carrie r mean free path is much larger that the de\nBroglie wavelength. None of the methods presented below are valid o nceεFτp//planckover2pi1becomes comparable to unity.\nSpin densities and spin currents are obtained as expectation values of spin and spin current operators. The spin\noperator is given by sσ= (/planckover2pi1/2)σσ, whereσσis a Pauli spin matrix. The spin current operator will be taken to be\nˆJσ\ni= (1/2){sσ,vi}, where the velocity operator is vi= (1//planckover2pi1)∂Hk/∂ki. Both operators are diagonal in k. One is\ntherefore primarily interested in the part of the density matrix diag onal ink. With this observation in mind, ρkk′is\ndivided into a part diagonal in kand a part off-diagonal in kasρkk′=fkδkk′+gkk′, where, in gkk′, it is understood\nthatk∝ne}ationslash=k′. Further, fkis decomposed into a scalar part and a spin-dependent part, fk=nk11 +Sk, with11 the\nidentity matrix. In an electric field, fk=f0k+fEk, wherefEkis a correction linear in Eand has a corresponding\ndecomposition into a scalar part and a spin-dependent part. At the end one takes the trace of the spin and spin\ncurrent operators with fEkgiven by Eq. (4). The final result is usually expressed in many ways.\nIt should be noted that the electric field induces coherence betwee n bands. In equilibrium the Fermi-Dirac function\nf0kcommutes with the Hamiltonian and is stationary. However the sourc e term due to Ein Eq. (2) does not in\ngeneral commute with the Hamiltonian. If the problem is considered in the basis of eigenstates of Hk, the source\nterm couples different energy bands. If a basis is used in which one sp in component is a good quantum number, then\nthe source term couples up and down spins.\nB. Linear response Green’s function formalism\nThe linear response Green’s function formalism is closely related to th e density matrix formalism but relies on a\nsomewhat different way of regarding the problem [96]. The electric fi eld-induced correction to the density matrix ˆ ρE\nis found by applying the time evolution operator corresponding to th etotalHamiltonian ˆH+ˆUdis, incuding disorder,\nin Eq. (2). The matrix elements of the time evolution operator in the b asis{|ks∝an}bracketri}ht}constitute the Green’s function\nGss′\nkk′(t), defined by\nGss′\nkk′(t) =∝an}bracketle{tks|e−i(ˆH+ˆUdis)t//planckover2pi1|k′s′∝an}bracketri}ht. (3)\nSpin indices are again suppressed. All the information about the sys tem is contained in the Green’s function, which\nis also referred to as the propagator, since it gives the amplitude fo r a particle to propagate in time tfrom state\n|ks∝an}bracketri}htto state|k′s′∝an}bracketri}ht. The Green’s function is also the kernel of the Schrodinger equatio n and represents the response\nof the system to a perturbation that is localized in real space or mom entum space. The total perturbation is built\nup as a sum of localized ones. The problem can be formulated in any bas is but for our purposes it is still the most\nconvenient to work in the basis of eigenstates of Hsok. In general the Green’s function is a matrix in spin space. For\npractical purposes one also defines the retarded and advanced G reen’s functions, GRandGA, asGR=−iGθ(t) and\nGA=iGθ(−t), withθthe Heaviside step function. The electric field-induced correction t o the part of the density\nmatrix diagonal in kis\nfEk=−ieE\n/planckover2pi1·/summationdisplay\nk′/integraldisplay∞\n−∞dt′GR\nkk′(t′)[ˆr,ˆρ0]k′GA\nk′k(−t′) (4)\nAt this stage it is easiest to carry out a Fourier transformation with respect to time. The integration in Eq. (4) has\nthe same form except the time variable is replaced by the frequency variable ω\nfEk=−ieE\n/planckover2pi1·/summationdisplay\nk′/integraldisplay∞\n−∞dωGR\nkk′(ω)[ˆr,ˆρ0]k′GA\nk′k(−ω). (5)\nThe correction fEkto the density matrix needs to be averaged over impurities, and it is a ssumed henceforth that\nonly the impurity averaged density matrix is of interest. We denote t his impurity average by the symbol ∝an}bracketle{t∝an}bracketri}ht, but the\nimpurity-averaged density matrix will be abbreviated as ∝an}bracketle{tfEk∝an}bracketri}ht ≡fEk. The full Green’s function is determined by the\ntotal Hamiltonian ˆH+ˆUdisincluding the scattering potential. The Green’s function is expanded in the scattering5\npotential, and one is only interested in the Green’s function averaged over impurity configurations. For uncorrelated\nimpurities the impurity-averaged Green’s function, denoted by ∝an}bracketle{tG∝an}bracketri}ht, obeys the recursion relation\n∝an}bracketle{tGR/A(k,ω)∝an}bracketri}ht=GR/A\n0(k,ω)+GR/A\n0(k,ω)ΣR/A(k,ω)GR/A(k,ω) (6)\nin which G0is the equilibrium Green’s function, corresponding to the Hamiltonian ˆHwithout disorder, and the self\nenergy Σ is typically evaluated in the first Born approximation\nΣR/A(k,ω) =ni/integraldisplayddk′\n(2π)d|Ukk′|2GR/A\n0(k′,ω) (7)\nwithdthe dimensionality of the system. The correction fEkto the density matrix depends on the product of two\nGreen’s functions. After the average over impurity configuration s we obtain terms which are expressible as a product\nof two individual impurity-averagedGreen’s functions and cross te rms, which connect different Green’s functions, and\nwhich are not expressible as a product. These form a vertex Γ kk′defined by\n∝an}bracketle{tGR(k,ω)GA(k′,ω)∝an}bracketri}ht=∝an}bracketle{tGR(k,ω)∝an}bracketri}ht∝an}bracketle{tGA(k,ω)∝an}bracketri}ht+Γkk′(ω)∝an}bracketle{tGR(k′,ω)∝an}bracketri}ht∝an}bracketle{tGA(k′,ω)∝an}bracketri}ht. (8)\nThe vertex is in general expressed as a sum of different classes of d iagrams, of which the most relevant to transport\nare the ladder diagrams and the maximally crossed diagrams. It also s atisfies the recursion relation\nΓkk′(ω) =Mkk′(ω)+/integraldisplayddk′′\n(2π)dMkk′′(ω)∝an}bracketle{tGR(k′′,ω)∝an}bracketri}ht∝an}bracketle{tGA(k′′,ω)Γk′′k′(ω)∝an}bracketri}ht (9)\nwhereMkk′(ω) can alsobe expressed in terms ofdiagramsas shown in Ch. 8 ofRef. [[96]]. Once the impurity-averaged\nGreen’sfunction ∝an}bracketle{tG∝an}bracketri}htandthe vertexareknown, fEkcan be found. It will be noticed that in the linearresponseGreen’s\nfunction approach disorder is contained in the self energy Σ( k) and in the vertex Γ kk′. The response function contains\nall the disorder up to the desired approximation. Once the Green’s f unction and the vertex function are found, the\ntrace with the spin and spin current operators is taken.\nC. Kinetic equation formalism\nThe kinetic equation formalism can be derived from the quantum Liouv ille equation, as will be done below, or using\na Keldysh Green’s function formalism.[57] The kinetic equation obtaine d is naturally the same. Starting from the\nquantum Liouville equation, this can be broken down into equations fo rfkandgkk′\ndfEk\ndt+i\n/planckover2pi1[Hk,fEk] =−i\n/planckover2pi1[ˆU,ˆgE]kk−i\n/planckover2pi1[eE·ˆr,ˆρ0]kk, (10a)\ndgEkk′\ndt+i\n/planckover2pi1[ˆH,ˆgE]kk′=−i\n/planckover2pi1[ˆU,ˆfE+ ˆgE]kk′. (10b)\nIn the first Born approximation the solution to Eq. (10b) can be writ ten as\ngEkk′=−i\n/planckover2pi1/integraldisplay∞\n0dt′e−iˆHt′//planckover2pi1/bracketleftBig\nˆU,ˆfE(t−t′)/bracketrightBig\neiˆHt′//planckover2pi1|kk′. (11)\nSinceεFτp//planckover2pi1≫1, we shall expand ˆf(t−t′) in the time integral around tand, noting that terms beyond ˆf(t) are of\nhigher order in the scattering potential, we shall only retain the firs t term,ˆf(t). The equation for fkthen becomes\ndfEk\ndt+i\n/planckover2pi1[Hk,fEk]+ˆJ(fEk) =eE\n/planckover2pi1·/parenleftbigg∂f0k\n∂k−i[R,f0k]/parenrightbigg\n(12a)\nin which the scattering term ˆJ(fk) is given by\nˆJ(fEk) =1\n/planckover2pi12/integraldisplay∞\n0dt′/bracketleftBig\nˆU,e−iˆHt′//planckover2pi1/bracketleftBig\nˆU,ˆfE(t)/bracketrightBig\neiˆHt′//planckover2pi1/bracketrightBig\nkk(12b)\nand the source term on the RHS contains the covariant derivative with respect to k, which takes into account the\nfact that the basis functions themselves may depend on k. It includes the gauge connection matrix R≡Rss′=6\n∝an}bracketle{tks|i∂/∂k|ks′∝an}bracketri}ht. Theresponsefunction, aswellasthescatteringtermneedtobe averagedoverimpurityconfigurations,\nand the notation fEkis used henceforth as above to denote the impurity-averaged den sity matrix. The scattering\nterm can be expanded further in the basis in spin space spanned by s pin eigenstates | ↑∝an}bracketri}htand| ↓∝an}bracketri}ht(the Pauli basis.)\nIn this basis the connection matrix Rvanishes and the scattering potential Ukk′=Ukk′11 is diagonal in spin space.\nConverting sums over wave vector into integrals/summationtext\nk′→V/integraltextddk′\n(2π)d, and performing the time integral the scattering\nterm can be expressed in the form ( ˆJ0+ˆJb)(nk)+ˆJ0(Sk), with\nˆJ0(Sk) =2πni\n/planckover2pi1/integraldisplayddk′\n(2π)d|Ukk′|2(Sk−Sk′)δ(ε0k−ε0k′) (13a)\nˆJb(nk) =2πni\n/planckover2pi1/integraldisplayddk′\n(2π)d|Ukk′|2(nk−nk′)1\n2σ·(Ωk−Ωk′)∂\n∂ε0kδ(ε0k−ε0k′).\nThe term ˆJb(nk) in Eq. (13b) illustrates the fact that, when spin-orbit interaction s are present in the band structure,\neven spin-independent scattering potentials give rise to spin-depe ndent terms in the scattering integral. For potentials\ndiagonalin spinspaceandspin-degeneratebands, Eq.(12b) simplifi estothe customaryFermi’sgoldenrule. Therefore,\nEq. (12) is a generalization of Fermi’s golden rule that explicitly takes in to account the spin degree of freedom.\nD. Boltzmann-wave packet formalism\nThe density matrix is the most complete description of a physical sys tem. It contains all the relevant physics,\nincluding interband coherence due to the electric field and scatterin g. It represents the system as a whole and\naccounts for the individual particle motion, which occurs between c ollisions, as well as for the distribution of electrons\nin phase space and the way it is altered by collisions. As was shown abov e, linear response theories based on Green’s\nfunctions and theories based on kinetic equations originate directly from the Liouville equation for the density matrix.\nA third approach, which may be called a semiclassical or Boltzmann-wa ve packet approach, separates the motion\nof the charge carriers from their distribution in phase space. To de termine the dynamics of the charge carriers one\nenvisages them as being represented by wave packets, and the ph ase space distribution of wave packets is given by a\nfunction of the Boltzmann form. One use of the word semiclassical , which will be adopted in this work, is to refer to\ntheories which consider the position and momentum of a particle simult aneously. Semiclassical approaches exploit the\nsmooth variation of transport fields on atomic length scales in order to provide intuitive descriptions of steady-state\nprocesses, which can be easily extended to cover inhomogeneous s ystems and spatially dependent fields.\nThe semiclassical approach is also closely related to the density matr ix and the Liouville equation. Firstly, in order\nto separate the particle dynamics and the phase space distribution one must be able to label the particles by a band\nindex, therefore the theory must be formulated in the basis of eige nstates of Hk. Since these eigenstates are usually\nk-dependent it is the covariant derivative that enters the source t erm in the kinetic equation. The point at which\nsemiclassical theory branches out of the kinetic equation is Eq. (12 a). In order to turn the density matrix into a\nBoltzmann distribution function one takes the diagonal elements of this equation. If we consider for simplicity a spin-\n1/2system, which ismade up oftwobandsthat will be called 1and2, we canlabel the diagonalelements ofthe density\nmatrix as f1kandf2k. These become the Boltzmann distributions for bands 1 and 2. The c ommutator [ Hk,fk] has\nno diagonal elements and ˆJ(fk) reduces to one scattering term for each band, which is given by Fe rmi’s Golden Rule,\nplus interband scattering terms which couple f1kandf2k. In this way Eq. (12a) reduces to two coupled Boltzmann\nequations for the two bands and the phase space distribution has b een separated out of the kinetic equation.\nIn reducing the kinetic equation to a series of Boltzmann equations o ne neglects interband coherence due to the\nelectric field, and this must be recovered elsewhere. In the semiclas sical approach, this occurs in several ways. Firstly,\none needs to consider the nature of the carriers in band sand invoke explicitly the fact that carriers are described\nby wave packets. This is the point at which the formalism turns into a t rue semiclassical theory. The construction\nof a wave-packet representing a charge and spin carrier in band s, which has real and k-space coordinates ( rcs,kcs),\nhas been thoroughly treated by Sundaram and Niu [97] and its exten sion to multiple bands was carried out in Ref.\n[[98]]. The macroscopic distribution associated with a quantum mechan ical operator ˆOis no longer given simply by\nthe expectation value of the operator ˆO, but involves explicitly a sum over wave packets centered at ( rcs,kcs). For\nexample the spin density distribution is\nSσ(r,t) =/summationdisplay\ns/integraldisplay\nd3kcs/integraldisplay\nd3rcsfs(kcs,t)∝an}bracketle{tδ(r−ˆ r)ˆsσ∝an}bracketri}hts, (14)\nwhere the bracket indicates quantum mechanical average over th e wavepacket with charge centroid ( rcs,kcs). An\nanalogous expression exists for the spin current distribution. The wave-packet expectation values ∝an}bracketle{t∝an}bracketri}htsin Eq. (14)7\nr rCharge Spin\nc s\nFIG. 1: For a particle of finite extent the charge and spin dist ributions in real space are in general do not coincide. The sa me\nis true of the charge and spin distributions in reciprocal sp ace.\neventually yield functions of ( rcs,kcs), the dynamics of which are discussed below. It may appear contra dictory that\none has to integrate over the real-space coordinates of the wave packetsrcseven in the case of homogeneous systems\nstudied in this work, where the distribution function is not a function ofrcs. This is because even in a homogeneous\nsystem of the center of charge and the center of spin are not the same, so variations of the spin distribution on the\nspatial scales comparable to that of the wave packet need to be ta ken into account, as shown in Fig. 1. All these\nterms must be considered in order to reach agreement with approa ches based directly on the Liouville equation.\nIn a constant uniform electric field Ethe coordinates of the wave packet center ( rcs,kcs) drift according to the\nsemiclassical equations of motion [97]\n/planckover2pi1˙kcs=−eE\n/planckover2pi1˙rcs=∂εs\n∂kc+eE×Fs,(15)\nwhereFs=∇k×RssrepresentstheBerry,orgeometricalcurvature[97], with ∇kthegradientoperatorinmomentum\nspace. A careful analysis reveals that the Berry curvature also r epresents coherence between bands and must be taken\ninto account in order to obtain agreement with approaches based d irectly on the Liouville equation. It is emphasized\nthat the Berry curvature is not a result of the particles being desc ribed by wave packets, which have a finite extent in\nreal and momentum space. Rather, it emerges from the phase of t he Bloch wave functions involved in the construction\nof the wave packets, implying that this formulation of semiclassical t ransport theory is a useful tool for capturing\nBerry-phase effects.\nThe electric field and disorder potential also give rise to a nonadiabat ic mixing of the bands such that the basis\nstates|ks∝an}bracketri}htbecome|˜ks∝an}bracketri}ht, given by\n|˜ks∝an}bracketri}ht=|ks∝an}bracketri}ht+/summationdisplay\nk′,s′/negationslash=s[eEδkk′−(∇Udis)kk′]·Rs′s\nεks−εks′|ks′∝an}bracketri}ht, (16)\nwhere the |ks∝an}bracketri}htare the unperturbed eigenstates. The |˜ks∝an}bracketri}htalso form a complete set. The distribution functions\nfs(kcs,t) are made up of an equilibrium part and a part linear in the electric field, fs(kcs,t) =f0s(kcs,t)+fEs(kcs,t),\nand the expectation values ∝an}bracketle{t∝an}bracketri}htsin Eq. (14) also have terms of zeroth and linear orders in the electric field. Therefore\nto first order in the field the expectation value of operator ˆOcontains terms arising from the product of fEs(kcs,t)\nwith the zeroth order term in ∝an}bracketle{t∝an}bracketri}htsas well as from the equilibrium f0s(kcs,t) multiplied by the term in ∝an}bracketle{t∝an}bracketri}htslinear in the\nelectric field.\nThe above exposition shows that interband coherence lost in passin g from the density matrix to the Boltzmann\napproach is thus recovered in many ways. It is present in the Berry curvature, in the nonadiabatic correction to\nthe wave functions, and in the interband scattering terms coupling the two distribution functions. The construction\nof a correct semiclassical theory that contains all the physics rele vant to transport is seen to entail a sizable effort,\nparticularly in the case of transport of non-conserved quantities such as spin.8\nIII. BAND-STRUCTURE SPIN DENSITIES AND CURRENTS\nThis work will now focus on the kinetic equation formalism and determin e the spin densities and spin currents\ninduced by external electric fields under a variety of circumstance s. It is most straightforward to begin with effects\noriginating in the band structure. The Hamiltonian of spin-1/2 electr on systems typically contains a kinetic energy\nterm and a spin-orbit coupling term, Hk=/planckover2pi12k2\n2m∗+Hso\nk, wherem∗is the electron effective mass. In spin-1/2 electron\nsystems, band structure spin-orbit coupling can always be repres ented as a Zeeman-like interaction of the spin with\na wave vector-dependent effective magnetic field Ωk, thusHso\nk= (/planckover2pi1/2)σ·Ωk. Common examples of effective fields\nare the Rashba spin-orbit interaction, [87] which is often dominant in quantum wells with inversion asymmetry, and\nthe Dresselhaus spin-orbit interaction, [86] which is due to bulk inver sion asymmetry.\nAn electron spin at wave vector kprecesses about the effective field Ωkwith frequency Ω k//planckover2pi1≡ |Ωk|//planckover2pi1and is\nscattered to a different wavevectorwithin a characteristicmomen tum scatteringtime τ. Within the range εFτ//planckover2pi1≫1,\nthe relative magnitude of the spin precession frequency Ω kand inverse scattering time 1 /τdefine three qualitatively\ndifferent regimes. In the ballistic or clean regime no scattering occur s and the temperature tends to absolute zero,\nso thatεFτ//planckover2pi1→ ∞and Ω kτ//planckover2pi1→ ∞. The weak scattering regime is characterized by fast spin precess ion and little\nmomentum scattering due to, e.g., a slight increase in temperature, yielding εFτ//planckover2pi1≫Ωkτ//planckover2pi1≫1. In the strong\nmomentum scattering regime εFτ//planckover2pi1≫1≫Ωkτ//planckover2pi1.\nThe kinetic equation (12a) has a scalar part which determines nEk\n∂nEk\n∂t+ˆJ0(nEk) =eE\n/planckover2pi1·∂n0k\n∂k. (17)\nThe solution of this equation is given by the well-known expression\nnEk=eEτp\n/planckover2pi1·∂n0k\n∂k, (18)\nin other words, nEkdescribes the shift of the Fermi sphere in the presence of the elec tric field E. The expression\nfor the momentum relaxation time τpis a little different depending on the dimensionality of the system. Using γto\ndenote the relative angle between kandk′,\n1\nτp=nimk\n2π/planckover2pi13/integraldisplayπ\n0dγ|Ukk′|2sinγ(1−cosγ) (3D) (19a)\nnim\n2π/planckover2pi13/integraldisplay2π\n0dγ|Ukk′|2(1−cosγ) (2 D). (19b)\nThe spin-dependent part of the nonequilibrium correction to the de nsity matrix SEkis the spin density induced by\nE. Its time evolution is governed by SEkis\n∂SEk\n∂t+i\n/planckover2pi1[Hk,SEk]+ˆJ0(SEk) =eE\n/planckover2pi1·∂S0k\n∂k−ˆJb(nEk). (20)\nSpin-dependent scattering gives rise to a renormalization of the dr iving term in the equation for SEkwith no analog\nin charge transport, see Eq. (17).\nSince an electron spin at wave vector kprecesses about Ωk, the spin can be resolved into components parallel and\nperpendicular to Ωk. In the course of spin precession the component of the spin paralle l toΩkis conserved, while the\nperpendicular component is continually changing. It will prove usefu l in our analysis to divide the spin distribution\ninto a part representing conserved spin and a part representing p recessing spin. The effective source term, which\nenters the RHS of Eq. (20), is divided into two parts, ( eE//planckover2pi1)·∂S0k/∂k−ˆJb(nEk) = ΣEk/bardbl+ΣEk⊥. ΣEk/bardblcommutes\nwithHso\nkwhile Σ Ek⊥isorthogonal to it and tr(Σ Ek⊥Hso\nk) = 0. Projections onto and orthogonal to Hso\nkare most\neasily carried out by defining projectors P/bardblandP⊥by their actions on the basis matrices σias described in Ref. [[36]].\nSEkis likewise divided into two terms: SEk/bardbl, commuting with the spin-orbit Hamiltonian and SEk⊥, orthogonal to\nit.SEk/bardblis the distribution of conserved spins while SEk⊥is the distribution of precessing spins. Equation (20) is\ndivided into separate equations for SEk/bardblandSEk⊥\n∂SEk/bardbl\n∂t+P/bardblˆJ0(SEk) = ΣEk/bardbl, (21a)\n∂SEk⊥\n∂t+i\n/planckover2pi1[Hk,SEk⊥]+P⊥ˆJ0(SEk) = ΣEk⊥. (21b)9\nThe absence of the commutator [ Hk,SEk/bardbl] = 0 in Eq. (21a) indicates the absence of spin precession, while the\ncommutator [ Hk,SEk⊥] in Eq. (21b) represents spin precession. In order to solve Eqs. ( 21a) and (21b) for arbitrary\nscattering, it is necessaryto expand SEk/bardblandSEk⊥inni, asSEk/bardbl=S(−1)\nEk/bardbl+S(0)\nEk/bardbl+O(ni) andSEk⊥=S(0)\nEk⊥+O(ni).\nThis is an expansion in the parameter /planckover2pi1/(Ωkτp) and is most suited to systems in the weak scattering regime. The\nexpansion of SEk/bardblstarts at order −1, a fact which can be understood by inspecting Eq. (21a). In the s teady state\nthe time derivative drops out, and the operator ˆJ0is first order in ni, while the right-hand side is independent of ni.\nAs a result, the expansion of the solution must start at order −1. Equation (21b) for SEk⊥tells us that, since Hkis\nindependent of niand the right hand side is also independent of ni, the expansion of SEk⊥must start at order zero.\nEquation (21a) can be solved iteratively for any scattering\nS(−1)\nEk/bardbl= ΣEk/bardblτ0+P/bardbl/parenleftbiggm∗\n2π/planckover2pi13/integraldisplay\ndθ′|Ukk′|2ΣEk/bardbl/parenrightbigg\nτ2\n0+... (22)\nwhereτ0=nim∗/integraltext\ndθ′|Ukk′|2/(2π/planckover2pi13) is the quantum lifetime of the carriers. Above and henceforth inte grals over\nwave vectors will be represented as two-dimensional, and θ′will refer to the polar angle of k′. The extension to three\ndimensions is straightforward. The equations for higher orders in niare easily deduced. However, the term of order\n−1 is by far the dominant one in the weak momentum scattering regime a nd is expected to be dominant over a wide\nrange of strengths of the scattering potential.\nIt is evident that the steady state for conserved spins involves no spin precession, and that the correction SEk/bardbl\ndepends explicitly on the nonequilibrium shift in the Fermi surface. In addition, scattering terms contain only the\neven function |Ukk′|2. As a result, SEk/bardbldoes not give rise to a spin current. Inspection of Eq. (22) shows t hat integrals\nof the form/integraltext\ndθˆJσ\niSEk/bardblcontain an odd number of powers of kand are therefore zero. In the absence of impurity\nspin-orbit interactions, the distribution of conserved spins can giv e no spin current. It can, however, give rise to a\nnonequilibrium spin density since integrals of the form/integraltext\ndθˆsσSEk/bardblcontain an even number of powers of kand may\nbe nonzero.\nThe leading term S(0)\nEk⊥is found to be\nS(0)\nEk⊥=/planckover2pi1\n2σ·ˆΩk×[ΣEk⊥−P⊥ˆJ0(SEk/bardbl)]\nΩk, (23)\nwhere we have written Σ Ek⊥= (1/2)ΣEk⊥·σandSEk/bardbl= (1/2)SEk/bardbl·σ. The result expressed by Eq. (23) is valid\nfor any elastic scattering. An argument similar to that given above s hows that S(0)\nEk⊥cannot lead to a nonequilibrium\nspin density (although, as will be shown below, higher-order terms in SEk⊥can contribute to the spin density). For,\ntaking the expectation value of the spin operator, one arrives at in tegrals of the form/integraltext\ndθˆsσS(0)\nEk⊥, which involve odd\nnumbers of powers of kand are therefore zero. This term in the distribution of precessing spin does, however, give\nrise to nonzero spin currents, since integrals if the form/integraltext\ndθˆJσ\niSEk⊥contain an even numbers of powers of kand\nmay be nonzero. Consequently, in the absence of spin-orbit couplin g in the scattering potential, nonequilibrium spin\ncurrents arise from spin precession.\nThe dominant contribution to the nonequilibrium spin density in an elect ric field exists because in the course of spin\nprecession a component of each individual spin is preserved. For an electron with wave vector k, this spin component\nis parallel to Ωk. In equilibrium the average of these conserved components is zero . However, when an electric field is\napplied, the Fermi surface is shifted, and the average of the cons erved spin components may be nonzero, as illustrated\nin Fig. 2. This intuitive physicalargumentexplains why the nonequilibriu m spin density ∝τ−1\npandrequiresscattering\nto balance the drift of the Fermi surface. It is interesting to note , also, that, although spin densities in electric fields\nrequire the presence of band structure spin-orbit interactions a nd therefore spin precession, the dominant contribution\narises as a result of the absence of spin precession. Band structu re spin currents on the other hand are associated\nwith displacement of spins. The relation between spin currents and s pin precession was made explicit by Sinova et al.\n[16] Both SEk/bardblandSEk⊥are invariant under time-reversal. As a result, the tensor charac terizing the response of\nspin currents to electric fields is invariant under time reversal, wher eas the tensor characterizing the response of spin\ndensities to electric fields changes sign under time reversal.\nIt will be noticed that the driving term in the equation for the nonequ ilibrium spin distribution SEkis renormalized\nby the term ˆJs(nEk), which accounts for spin-dependent scattering. In addition, Eq . (21b) shows that scattering\nmixes the distributions of conserved and precessing spins. When on e spin at wave vector kand precessing about\nΩkis scattered to wave vector k′and precesses about Ωk′, its conserved component changes, a process which\nalters the distributions of conserved and precessing spin. Conseq uently, scattering processes in systems with spin-\norbit interactions cause a renormalization of the driving term for th e spin distribution as well as scattering between\nthe conserved and precessing spin distributions. Contributions du e to these two processes are contained in vertex10\nkxky\nkxky\n0 00 0E=0 E>0 (b) (a)\nFIG. 2: Effective field Ωkat the Fermi energy in the Rashba model (a) without ( E= 0) and (b) with ( E >0) an external\nelectric field.\ncorrections to spin-dependent quantities found in Green’s functio ns formalisms. For short-range impurities it is easy\nto show [36, 75] that the scattering correction P⊥ˆJ0(SEk/bardbl) to Eq. (23) depends on the steady state spin density, and\nit becomes evident that the existence of a nonzero nonequilibrium sp in density does affect the spin current. This fact\nwas first pointed out by Raimondi [75] for the Rashba model.\nThe spin-Hall current was initially determined for spin-3/2 holes in GaA s [15] and for an asymmetric quantum well\n[16] in which the spin-orbit interaction is described by the Rashba mod el. The latter calculation yielded a spin-Hall\nconductivity σz\nxy=e/(8π), in which σi\njkis understood as referring to spin component iflowing in direction jin\nresponse to an electric field applied along k. However, it was subsequently shown that a more careful treatm ent of\ndisorder renormalizes this result to zero. In fact in two dimensions, for Hamiltonians linear in wave vector, the spin\ncurrent vanishes for short-range impurities (Refs. [[50]-[55]]), for small-angle scattering [56, 57] and in general for any\nelastic scattering. [33] However, it does not vanish in a generalized R ashba model as was shown by Krotkov and Das\nSarma. [68] For spin-orbit Hamiltonians characterized solely by one a ngular Fourier component Nthe spin current\n∝N. [57]\nAb initio calculation of band structure spin currents were also perfo rmed by Guo et al.[72] in semiconductors and\nby Yaoet al.[73] in semiconductors and simple metals such as tungsten, platinum a nd gold. Spin transport in metals\nusually requires complex band structure calculations, which are typ ically done numerically. Yao et al.[73] found that\nthe band structure spin-Hall effect in metals can be significantly larg er than in semiconductors and that the spin-Hall\nconductivity can even undergo sign changes under certain circums tances.\nA. Spin adiabatically following a magnetic field\nThe existence of a spin current in an electric field and its association w ith spin precession, can be understood from\na straightforward argument due to Sinova.[16] A spin precesses ab out an effective magnetic field Ωkwhich depends\nonk. The electric field changes the wave vector kand in the process changes Ωk, so the spin is now precessing\nabout magnetic field that is changing adiabatically. In the adiabatic limit the spin follows the magnetic field. [99] In\nparticular, in 2D if the magnetic field is in the plane of the spin, the spin n ever acquires a significant out-of plane\ncomponent. However the spin in general acquires a small out-of-p lane component. Consider a generalized magnetic\nfieldΩand take the simple example in which the magnetic field ∝bardblˆxand has a small y-component which increases\nlinearly with time, such that Ω y=ǫt. The spin starts out along x, initially parallel to the field. In both cases, sxwill\nbe considered large compared to sy,szand in this section /planckover2pi1= 1. The adiabatic condition means that the magnitude\nof the magnetic field Ω changes little over one revolution of the electr on spin about it. This means that, if we start\natt= 0 and consider a small time increment τ, Ω(τ)−Ω(0)<<Ω(0).\nΩ(τ) = Ω(0)+dΩ\ndt|0τ\nΩ(τ)−Ω(0) =dΩ\ndt|0τ→dΩ\ndt|0τ <<Ω(0).(24)\nThe adiabatic limit in this case is the limit of fast precession so one can ch oose as the small time τthe (approximate)\nprecession period around the magnetic field, τ=2π\nΩ. The condition becomes\n2π\nΩdΩ\ndt<<Ω. (25)11\nFIG. 3: Spin splitting of energy spectrum in the Rashba model . The green arrows represent the direction of the momentum\nand the red arrows represent the direction of the spin.\nConsider the case in which a small damping term exists α(ds\ndt×s). The equations of motion for the three spin\ncomponents are\ndsx\ndt=−szΩy+α(dsy\ndtsz−dsz\ndtsy)\ndsy\ndt=szΩx+α(dsz\ndtsx−dsx\ndtsz)\ndsz\ndt=sxΩy−syΩx+α(dsx\ndtsy−dsy\ndtsx).(26)\nIt is clear that all the changes in sxare at least of second order in small quantities, so sxcan be treated as a constant.\nThe solution is approximately\nsy(t) =ǫsx\nΩ2x(Ωxt−e−t/τsinΩxt)\nsz(t) =ǫsx\nΩ2[1−e−t/τ(cosΩt−1\nΩτsinΩt)].(27)\nwhereτ=1\nαΩxsx. After the oscillations in sy,szare damped, what remains is\nsy(t >> τ) =ǫsx\nΩxt\nsz(t >> τ) =ǫsx\nΩ2.(28)\nThe spin, which was originally in the plane, acquires a small steady-sta te out-of-plane component. This component\ndepends on sx, andsxhas the opposite sign on different sides of the Fermi surface. As a r esult,szup and down spins\ntravel in opposite directions and a spin-Hall current is established. This derivation, illustrated by Fig. 3, is another\nway to clarify the role of spin precession in steady-state band stru cture spin currents.\nIV. EXTRINSIC SPIN DENSITIES AND CURRENTS\nUntil now the scattering potential has been treated as a scalar. H owever, in general the spin-orbit interaction makes\na contribution to the disorder potential. The disorder potential inc luding the scalar and spin-orbit parts takes the\nform\nUkk′= [1+iλσ·(k×k′)]Ukk′\nV. (29)12\nFIG. 4: An electric field displaces the Fermi surface and the e lectron spins are tilted up for py>0 and down for py<0.\nwithUkk′the matrix element of the scalarpart of the potential. The second term comes from spin-orbit coup ling\ninUkk′andλis a material-specific constant. Configurational averages of term s of the form Uimp\nkk′Uimp\nk′k′′Uimp\nk′′kare also\nlinear in nibut are rather lengthy to be displayed explicitly.\nThe spin-orbit interaction also produces a correction[92] to the po sition operator ˆr=ˆrord+2λσ×kwhereˆrord\nis the ordinary position operator. Since the external electric field e nters through the Hamiltonian HE=eE·ˆrthe\ncorrectionto ˆrgivesaside jump termHj\nE= 2eλσ·k×Eand this side jump term makes an additional contribution[92]\nto the velocity operator vj\nE=−2eλ\n/planckover2pi1σ×E. In the kinetic equation extra spin-dependent driving term −(i//planckover2pi1)[Hj\nE,f0k]\nmust be taken into account due to the change in the position operat or. This term is nonzero if the equilibrium density\nmatrixf0kis spin-dependent. The side jump mechanism is typically classified as ex trinsic although its contributions\nare manifold. These mechanisms were studied extensively in the anom alous Hall effect [89, 90, 91, 92] and recently\ntheir role in steady-state spin densities and currents have been hig hlighted in semiconductors[77, 78, 79, 83] and\nmetals.[84]\nThe solution for nEkis found as above and the spin-dependent scattering term ˆJsacts onnEkand produces an\nadditional source terms for SEk, such that the RHS of Eq. (20) becomes\nΣEk−ˆJs(nEk)−i\n/planckover2pi1[Hj\nE,S0k]−ˆJj(f0k). (30)\nSkew scattering emerges in the second Born approximation as a thir d-order term in the potential U. Substituting for\nnEkand introducing the angles γ1=θ′−θ,γ2=θ′′−θ′,γ3=θ−θ′′and the solid angles ω′,ω′′,\nˆJs(nk) =−6π2λnieτpm∗2k2d−2\n/planckover2pi16δ(k−kF)σ·ˆk×I\nI=/integraldisplaydω′\n(2π)d/integraldisplaydω′′\n(2π)dU(γ1)U(γ2)U(γ3)(ˆk′)E·(ˆk′−ˆk′′).(31)\nThe integraloversolidangles Iis independent of ˆk. In twodimensions the integrandis expanded in Fourierharmonics\nand the integration is straightforward. In three dimensions the int egrand is expanded in Legendre polynomials of γ1,\nγ2andγ3, and the integral over the two solid angles of Pl(γ1)Pm(γ2)Pn(γ3) is independent of ˆk. The modification of\nthe position operator produces an additional side jump scattering term\n−ˆJj(f0k) =−Hj\nEδ(εk−εF)\nτp. (32)\nThis term is related quite literally to transverse jumps undergone by a carrier during scattering [92] and its form\nreflects the conservation of momentum during such a side jump sca ttering event. [79, 92]13\nThe intrinsic source Σ Ekwas discussed in the previous section. The extrinsic source is hence forth denoted by\nTEk=−i\n/planckover2pi1[Hj\nE,S0k]−��Jj(f0k)−ˆJs(nEk). If the band structure spin-orbit interactions are zero one find s easily\nSext\nEk=Hj\nEδ(εk−εF)−ˆJs(nEk)τp, (33)\nwhere the superscript extindicates that only extrinsic contributions are considered. This exp ression averages to zero\nover directions in momentum space and does not give a spin density. I t does however contribute to the spin current,\ngiving a spin-Hall conductivity ∝n−1\nidue to skew scattering and a spin-Hall conductivity neλindependent of nidue\nto side jump. These terms were discussed by Engel et al.[77] and by Hankiewicz and Vignale [79].\nIf band structure spin-orbit interactions are nonzero then, as w as pointed out by several groups,[80, 81, 82, 83]\nspin precession is crucial in establishing the steady state. To illustra te this consider once again the decomposition\nSext\nEk=Sext\nEk/bardbl+Sext\nEk⊥into linearly independent components. In two-dimensional systems grown along (001) the entire\nextrinsic source term is orthogonal to Hso, therefore TEk/bardbl= 0 and there is no term in Sext\nEkthat is∝n−1\ni. The\nsolution is\nSext\nEk⊥=σ·ˆΩk×TEk\n2Ωk. (34)\nThe contributions to Sext\nEk⊥due to skew scattering and side jump contain kto an even power, therefore skew scattering\nand side jump do not contribute to the spin current in the presence of spin precession (they can be restored by an\nexternal magnetic field B∝bardblˆzas found in Ref. [[80]].) The only contribution to the spin current comes from the\nspin-dependent driving term. The spin-Hall conductivity originating from this term is neλregardless of the form of\nthe band structure spin-orbit interaction. Skew scattering in the presence of band structure spin-orbit interactions\ndoes give a steady-state spin density, which was pointed out in [[83]], a nd a similar spin density arises from side jump.\nIt is therefore seen that skew scattering and side jump contribut e very differently to the steady-state density matrix\nwhen band structure spin-orbit interactions are present. This ca n be understood from the following argument. In\norder to get a sizable spin current from skew scattering, spins mus t be conserved as they travel. Scattering processes\nproduce a separation between spin-up and spin-down, while preces sion tries to destroy the orientation of the spins.\nThus once the spins are scattered it is the conserved spin fraction that carries the spin current. Evidently the same\nargument applies for side jumps undergone during scattering.\nV. DEFINITIONS OF THE SPIN CURRENT AND BOUNDARY CONDITIONS\nThe spin current is defined in an intuitive manner by ˆJσ\ni= (1/2){sσ,vi}, and this definition is used by the majority\nof researchers working on this topic. Nevertheless, in the presen ce of band structure spin-orbit interactions this spin\ncurrent is not conserved. The equation of continuity satisfied by t he spin density and current was shown by Shi et al.\nto be different [32] from the usual equation of continuity for the ch arge density and current. A source term exists in\nthis equation which reflects spin non-conservation. As a result, Sh iet al.as well as Bryksin and Kleinert [34] have\nproposed an alternative definition of the spin current accordingto whichˆJσ\ni=d/dt(ˆriˆsσ). The equation of continuity\nsatisfied by this current still contains a source term, but this sour ce term vanishes in many commonly used models.\nThis definition was used in Refs. [[33, 69]]. Recently, Tokatly [35] has a rgued that the intuitive definition of the spin\ncurrent represents a dissipative current which is conjugate to an effective SU(2) electric field.\nIn order to obtain the spin accumulation the kinetic equation needs t o be supplemented by boundary conditions.\nThis was done by a number of groups. [38, 39, 40, 41, 42, 43, 44, 45 ] Unfortunately, it was shown that the form of\nthe spin accumulation depends on the theoretical model of the bou ndary. [43, 44] The nontrivial physics associated\nwith boundary conditions and the implications of various formulations of boundary conditions for spin transport were\ndiscussed by Bleibaum [39] and Tserkovnyak et al.[38] An interesting argument due to Adagideli and Bauer [45] points\nout that, in systems in which the band structure spin current is exp ected to vanish in the bulk (i.e. for Rashba and\nDresselhaus spin-orbit coupling), the situation is not the same near the edges and near metal contacts, where a finite\nspin current exists which can be detected.\nVI. CRYSTAL SYMMETRY\nSpin densities and spin currents in a crystal are closely tied to the sy mmetry of the underlying lattice. An analysis\nrelating the response tensor to the symmetry of the underlying cr ystal lattice has been enlightening in the context of\nnonequilibrium spin densities excited by an electric field. If the respon se of the spin density sto an electric field Eis14\ngiven by sσ=Qσ\njEj, nonzero components for the material-specific spin density respo nse tensor Qσ\njare permitted only\nin gyrotropiccrystals [5]. For spin transport such an analysis was pe rformed in Ref.[[36]], determining the components\nof the spin-current response tensor allowed by symmetry in an elec tric field and providing systematic proof that spin\ncurrents in response to an electric field can be much more complex th an the spin-Hall effect [49]. This result is\ncompletely general and is not sensitive to the definition of the spin cu rrent or to whether the electric field is constant\nor time-dependent.\nThe spin current operator can be defined as ˆJσ\ni=1\n2(ˆsσˆvi+ ˆviˆsσ) orˆJσ\ni=d/dt(ˆriˆsσ). From a symmetry point of\nview these two definitions are equivalent. The spin current ˆJis a second rank tensor that can be decomposed into a\npseudoscalar part, an antisymmetric (spin-Hall) part, and a symme tric part. The pseudoscalar part is tr( ˆJ) =1\n3ˆs·ˆv\nand represents a spin flowing in the direction in which it is oriented. The symmetric and antisymmetric parts are\ngiven, respectively, by1\n2(ˆsσˆvi±ˆvσˆsi). The pseudoscalar and symmetric parts will be referred to as non-spin-Hall\ncurrents. Under the full orthogonalgroup only the antisymmetr ic (spin-Hall) components of ˆJare allowed, indicating\nthat these components are always permitted by symmetry.\nIn general, the spin current responseof a crystal to an electric fi eldEis characterizedby a materialtensor Tdefined\nbyJσ\ni=Tσ\nijEj. Forthe 32crystallographicpoint groupsthe symmetry analysis[10 0] for the tensor Tis establishedby\nmeans of standard compatibility relations [101]. One is particularly inte rested in those groups in which non-spin-Hall\ncomponents may be present. The pseudoscalar part of the spin cu rrent is only allowed by 13 point groups, while the\nsymmetric part is allowed in all systems except those with point group sO,Td(zinc blende), or Oh(diamond). Lower\nsymmetries, allowing non-spin-Hall currents, are characteristic o f systems of reduced dimensionality. In such systems\nit was shown that non-spin Hall currents exist.[36] For a quantum we ll grown along (113), the effective magnetic field\ncorresponding to the Rashba spin-orbit interaction has a compone nt in the ˆz-direction as well, and Ref. [[36]] showed\nthat the spin conductivities σx\nxx,σx\nyy,σz\nyxandσz\nxyare nonzero.\nSkew scattering and side jump contributions to the spin current in t he presence of band structure spin-orbit\ninteractions can also be restored by growing the structure along a direction that is not one of the main crystal axes.\nThe contribution to the spin density matrix due to extrinsic mechanis ms has two parts,\nSext\nEk/bardbl=1\n2/parenleftbiggTEkzΩkz\nΩ2\nk/parenrightbigg\nσ·Ωkτ (35)\nandSext\nEk⊥, which is given by Eq. (34). Taking again a quantum well grown along (1 13),Sext\nEk/bardblgives no spin density\nbut gives a spin current from skew scattering and side jump. Skew s cattering and side jump give the same nonzero\ncomponents of the spin conductivity tensor, σx\nxx,σx\nyy,σz\nyxandσz\nxy.\nVII. EXPERIMENTAL SITUATION\nSpin densities in semiconductors were first predicted by Ivchenko [4 ] and observed the following year in tellurium\nby Vorob’ev. [5] Recent experiments[10, 11, 12] have also reporte d the observation of a steady-state spin density in\nsemiconductors. Experimentally the spin density can be found by Ke rr rotation, in which a beam of linearly-polarized\nlight is sent into the sample and its polarizationvectorrotatesby an a nglethat is proportionalto the spin polarization.\nIn so far as measuring a spin current, the situation is more complicat ed. Initially the approach adopted was to\ngenerate a spin current which would flow to the edges of the sample w here it would produce a spin accumulation.\nThis spin density at the edge of the sample could then be detected by Kerr rotation or an equivalent technique.\nJ. Wunderlich et al.[18] used photoluminescence to detect a spin accumulation due to a s pin-Hall current in a two\ndimensional hole gas, which is believed to be due to spin-orbit interact ions in the band structure. Kato et al.[19]\nused Kerr rotation to detect a spin accumulation due to a spin-Hall c urrent in n-GaAs, in which the spin current is\nbelieved to be due to extrinsic mechanisms [77]. A similar experiment was carried out shortly thereafter by Sih et al.\n[20]. Stern et al.[21] used the same technique to detect a spin-Hall current in ZnSe a t room temperature. Following\nthis work, Sih et al.[22] performed an experiment designed to demonstrate explicitly th at the observed edge spin\naccumulation was due to the spin-Hall effect, as opposed to edge eff ects associated with the electric field. The group\nmanufactured samples with a series of transverse channels, such that the spin-Hall current generated by the electric\nfield was allowed to drift into regions where the electric field was effect ively zero. Since a spin accumulation was still\nmeasured at the edge, this showed unambiguously that the effect w as due to the spin current. Further experiments\nby Stern et al.[23] imaged the spatial distribution of the spin accumulation genera ted by the spin-Hall effect as well\nas its behavior in a magnetic field. Chang et al.[24] also reported, using photoluminescence, a spin accumulation du e\nto the spin-Hall effect in InGaN/GaN superlattices.\nA second type of experiments relies on an effect referred to as the inverse spin-Hall effect. Briefly, a spin current in\nturn generates a charge current, which can be detected by conv entional means. This was shown by Hirsch [14] and its15\nextension to a Landauer-Buttiker type multi-terminal nonlocal me asurement was presented by Hankiewicz et al.[74]\nThis technique was used by Valenzuela and Tinkham [25] to detect a c harge current as a result of the spin current\nin aluminium, and by Saitoh et al.[27] in platinum. Kimura et al.[28] observed the spin-Hall effect in platinum at\nroom temperature and their findings were explained theoretically by Guoet al., [76] who demonstrated that the effect\nis due to band structure spin-orbit interactions. Vila et al.also obtained results for platinum nanowires [29] while\nSekiet al.[30] reported a giant spin Hall effect in FePt/Au devices. The group u sed a multi-terminal device with a\ngold cross and FePt acting as a spin injector. The spin-Hall resistan ce was measured to be 2.9 mΩ and is attributed\nto the large skew scattering in gold, being thus extrinsic in nature. W enget al.[31] carried out similar experiments\non platinum, aluminium and gold and obtained results in agreement with t hose found to date. It should be noted\nthat Cui et al.[26] also observed an electrical current induced by an optically-gen erated spin current.\nVIII. FUTURE DIRECTIONS\nWhereas the scientific community working on steady-state spin den sities and currents appears to be in agreement\nthat spin currents exist and are experimentally measurable, a numb er of questions remain to be addressed in the\nfuture. For example, the relative magnitude of band structure sp in currents and spin currents due to extrinsic\nmechanisms such as skew scattering and side jump remains to be det ermined for a general band structure spin-\norbit interaction. In addition, the fact that no unique definition of t he spin current exists causes difficulties in the\ncomparison of experimental data with theoretical predictions. Th is ambiguity is exacerbated by the fact that different\ndefinitions of the spin current give results that often differ by a sign . [32, 33] Thanks to the non-conservation of\nspin, the relationship between spin current and spin accumulation at the boundary remains to be clarified. It appears\nthat what happens at the boundary is sensitive to the type of boun dary conditions assumed.[43, 44] Thus so far as\nquantitative interpretation of experimental data is concerned, t heory still has some way to go.\nOn the experimental side, despite tremendous progress, the com munity is still searching for a reliable way to\nmeasure, as opposed to detect, spin currents directly. A possible new experimental avenue relies o n magnetoresistance\ncaused by an edge spin accumulation, proposed by Dyakonov. [85] A spin current produces a spin accumulation near\nthe sample edges, which in turn causes the sample resistance to dec rease by a small amount, whereas an external\nmagnetic field can destroy the edge spin polarization and yield a positiv e magnetoresistance. An alternative path was\nfollowed by Wang et al.,[46] who started from the Dirac equation and obtained in the weakly relativistic limit a set of\nMaxwell equations in the presence of spin-orbit interactions. Altho ugh the spin current does not appear explicitly, it\nis contained in the Maxwell equations and the authors demonstrate that the relativistic conservation laws imply that\nthe spin current yields an electrical polarization, which could be dete cted directly. Similarly, a set of Maxwell-like\nequations for spin was formulated by Bernevig et al.[47] These methods remains to be attempted experimentally.\nPractically, the question of what to do with the electron spin once it h as been generated or transported to the\nedge of the sample remains. The revolutionary electronic device tha t harnesses spin currents for a practical purpose\nremains to be made, and the challenge of its design confronts exper imentalists and theorists alike.\nThe research at Argonne National Laboratory was supported by the US Department of Energy, Office of Science,\nOffice of Basic Energy Sciences, under Contract No. DE-AC02-06C H11357.\n[1] F. Meier and B. P. Zakharchenya, Optical Orientation (Elsevier, Amsterdam, 1984.)\n[2] G. Schmidt et al., Phys. Rev. B 62, R4790(2000).\n[3] T. Jungwirth, J. Sinova, J. Masek, J. Kucera, and A. H. Mac Donald, Rev. Mod. Phys. 78, 809 (2006).\n[4] E. L. Ivchenko and G. E. Pikus, JETP Lett. 27, 604 (1978).\n[5] L. E. Vorob’ev, E. L. Ivchenko, G. E. Pikus, I. I. Farbstei n, V. A. Shalygin, and A. V. Sturbin, JETP Lett. 29, 441\n(1979).\n[6] L. S. Levitov, Y. V. Nazarov, and G. M. Eliashberg, Sov. Ph ys. JETP 61(1), 133 (1985).\n[7] V. M. Edelstein, Solid State Comm. 73, 233 (1990).\n[8] A. G. Aronov, Y. B. Lyanda-Geller, and G. E. Pikus, Sov. Ph ys. JETP 73, 537 (1991).\n[9] L. I. Magarill, A. V. Chaplik, and M. V. Entin, Semiconduc tors35(9), 1081 (2001).\n[10] Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom , Phys. Rev. Lett. 93, 176601 (2004).\n[11] S. Ganichev et al, J. Magn. Magn. Mater. 300, 127 (2006).\n[12] A. Y. Silov, P. A. Blajnov, J. H. Wolter, R. Hey, K. H. Ploo g, and N. S. Averkiev, Appl. Phys. Lett. 85, 5929 (2004).\n[13] M. I. D’yakonov and V. I. Perel’, Zh. Eksp. Teor. Fiz 60, 1954 (1971).\n[14] J. E. Hirsch, Phys. Rev. Lett. 83, 1834 (1999).\n[15] S. Murakami, N. Nagaosa, and S.-C. Zhang, Science 301, 1348 (2003).\n[16] J. Sinova, D. Culcer, Q. Niu, N. A. Sinitsyn, T. Jungwirt h, and A. H. MacDonald, Phys. Rev. Lett. 92, 126603 (2004).16\n[17] S. A. Crooker and D. L. Smith, Phys. Rev. Lett. 94, 236601 (2005).\n[18] J. Wunderlich, B. Kaestner, J. Sinova, and T. Jungwirth , Phys. Rev. Lett. 94, 047204 (2005).\n[19] Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom , Science 306, 1910 (2004).\n[20] V. Sih, R. C. Myers, Y. K. Kato, W. H. Lau, A. C. Gossard, an d D. D. Awschalom, Nature Phys. 1, 31-35 (2005).\n[21] N. P. Stern, S. Ghosh, G. Xiang, M. Zhu, N. Samarth, and D. D. Awschalom, Phys. Rev. Lett. 97, 126603 (2006).\n[22] V. Sih, W.H. Lau, R.C. Myers, V.R. Horowitz, A.C. Gossar d, D.D. Awschalom, Phys. Rev. Lett. 97, 096605 (2006).\n[23] N. P. Stern, D. W. Steuerman, S. Mack, A. C. Gossard, D. D. Awschalom, Appl. Phys. Lett. 91, 062109 (2007).\n[24] H. J. Chang et al, Phys. Rev. Lett. 98, 136403 (2007).\n[25] S. O. Valenzuela and M. Tinkham, Nature 442, 176 (2006).\n[26] X.-D. Cui, S.-Q. Shen, J. Li, W. Ge, and F.-C. Zhang, Appl . Phys. Lett. 90, 242115 (2007).\n[27] E. Saitoh et al, Appl. Phys. Lett. 88, 182509 (2006).\n[28] T. Kimura et al, Phys. Rev. Lett. 98, 156601 (2007).\n[29] L. Vila et al, Phys. Rev. Lett. 99, 226604 (2007).\n[30] T. Seki, Y. Hasegawa, S. Mitani, S. Takahashi, H. Imamur a, S. Maekawa, J. Nitta, and K. Takanashi, Nature Materials\n7, 125 - 129 (2008).\n[31] K. C. Weng, N. Chandrasekhar, C. Miniatura, B.-G. Engle rt, arXiv:0804.0096.\n[32] J. Shi, P. Zhang, D. Xiao, and Q. Niu, Phys. Rev. Lett. 96, 076604 (2006).\n[33] N. Sugimoto, S. Onoda, S. Murakami, and N. Nagaosa, Phys . Rev. B 73, 113305 (2006).\n[34] V. V. Bryksin and P. Kleinert, Phys. Rev. B 73, 165313 (2006).\n[35] I. V. Tokatly, Phys. Rev. Lett. 101, 106601 (2008).\n[36] D. Culcer and R. Winkler, Phys. Rev. Lett. 99, 226601 (2007).\n[37] E. I. Rashba, Phys. Rev. B 70, 201309 (2004).\n[38] Y. Tserkovnyak, B. I. Halperin, A. A. Kovalev, and A. Bra taas, Phys. Rev. B 76, 085319 (2007).\n[39] O. Bleibaum, Phys. Rev. B 74, 113309 (2006).\n[40] G. Usaj and C. A. Balseiro, Europhys. Lett. 72, 631 (2005).\n[41] A. G. Mal’shukov, L. Y. Wang, C. S. Chu, and K. A. Chao, Phy s. Rev. Lett. 95, 146601 (2005).\n[42] K. Nomura, J. Wunderlich, J. Sinova, B. Kaestner, A. H. M acDonald, and T. Jungwirth, Phys. Rev. B 72, 245330 (2005).\n[43] Tse, W.-K., Fabian, J., Zutic, I. and Das Sarma, S. (2005 ) Spin accumulation in the extrinsic spin-Hall effect, Phys. Rev.\nB72, 241303.\n[44] V. M. Galitski, A. A. Burkov, and S. Das Sarma, Phys. Rev. B74, 115331 (2006).\n[45] I. Adagideli and G. E. W. Bauer, Phys. Rev. Lett. 95, 256602 (2005).\n[46] Y. Wang, K. Xia, Z.-B. Su, and Z. Ma, Phys. Rev. Lett. 96, 066601 (2006).\n[47] B. A. Bernevig, X. Yu, and S.-C. Zhang, Phys. Rev. Lett. 95, 076602 (2005).\n[48] D. Culcer and R. Winkler, Phys. Rev. B 76, 245322 (2007).\n[49] N. A. Sinitsyn, E. M. Hankiewicz, W. Teizer, and J. Sinov a, Phys. Rev. B 70, 081312 (2004).\n[50] J. Inoue, G. E. W. Bauer, and L. W. Molenkamp, Phys. Rev. B 70, 041303 (2004).\n[51] O. V. Dimitrova, Phys. Rev. B 71, 245327 (2005).\n[52] P. Schwab and R. Raimondi, Eur. Phys. J. B 25, 483 (2002).\n[53] E. G. Mishchenko, A. V. Shytov, and B. I. Halperin, Phys. Rev. Lett. 93, 226602 (2004).\n[54] C. Grimaldi, E. Cappelluti, and F. Marsiglio, Phys. Rev . B.73, 081303 (2006).\n[55] S. Y. Liu, X. L. Lei, and N. J. M. Horing, Phys. Rev. B 73, 035323 (2006).\n[56] A. Khaetskii, Phys. Rev. Lett. 96, 056602 (2006).\n[57] A. V. Shytov, E. G. Mishchenko, H.-A. Engel, and B. I. Hal perin, Phys. Rev. B 73, 075316 (2006).\n[58] A. G. Mal’shukov and K. A. Chao, Phys. Rev. B 71, 121308 (2005).\n[59] S. Zhang and Z. Yang, Phys. Rev. Lett. 94, 066602 (2005).\n[60] L. Sheng, D. N. Sheng, C. S. Ting, and F. D. M. Haldane, Phy s. Rev. Lett. 95, 136602 (2005).\n[61] T. Damker, H. Bottger, and V. Bryksin, Phys. Rev. B 69, 205327 (2004).\n[62] B. K. Nikolic, L. P. Zarbo, and S. Souma, Phys. Rev. B 73, 075303 (2006).\n[63] O. Bleibaum and S. Wachsmuth, Phys. Rev. B 74, 195330 (2006).\n[64] J. Schliemann and D. Loss, Phys. Rev. B 71, 085308 (2005).\n[65] P. -Q. Jin and Y. -Q. Li, Phys. Rev. B 74, 085315 (2006).\n[66] J. Li, L. Hu, and S. Q. Shen, Phys. Rev. B 71, 241305 (2005).\n[67] S. Y. Liu and X. L. Lei, Phys. Rev. B 72, 155314 (2005).\n[68] P. L. Krotkov and S. Das Sarma, Phys. Rev. B 73, 195307 (2006).\n[69] M. Zarea. and S. Ulloa, Phys. Rev. B. 73, 165306 (2006).\n[70] N. A. Sinitsyn, A. H. MacDonald, T. Jungwirth, V. K. Duga ev, and J. Sinova, Phys. Rev. B 75, 045315 (2007).\n[71] D. N. Sheng, L. Sheng, Z. Y. Weng, and F. D. M. Haldane, Phy s. Rev. B 72, 153307 (2005).\n[72] G. Y. Guo, Y. Yao, and Q. Niu, Phys. Rev. Lett. 94, 226601 (2005).\n[73] Y. Yao and Z. Fang, Phys. Rev. Lett. 95, 156601 (2005).\n[74] E. M. Hankiewicz, L. W. Molenkamp, T. Jungwirth, and J. S inova, Phys. Rev. B 70, 241301 (2004).\n[75] R. Raimondi et al., Phys. Rev. B 74, 035340 (2006).\n[76] G. Y. Guo, S. Murakami, T.-W. Chen, and N. Nagaosa, Phys. Rev. Lett. 100, 096401 (2008).\n[77] H. -A. Engel, B. I. Halperin, and E. I. Rashba, Phys. Rev. Lett.95, 166605 (2005).\n[78] S. Onoda, N. Sugimoto, and N. Nagaosa, Prog. Theor. Phys .116, 61 (2006).17\n[79] E. M. Hankiewicz and G. Vignale, Phys. Rev. B 73, 115339 (2006).\n[80] E. M. Hankiewicz and G. Vignale, Phys. Rev. Lett. 100, 026602 (2008).\n[81] W.-K. Tse and S. Das Sarma, Phys. Rev. B 74, 245309 (2006).\n[82] L. Hu, Z. Huang, and S. Hu, Phys. Rev. B 73, 235314 (2006).\n[83] J. L. Cheng and M. W. Wu, J. Phys. Cond. Mat. 20, 085209 (2008).\n[84] R. V. Shchelushkin and A. Brataas, Phys. Rev. B 71, 045123 (2005).\n[85] M. I. Dyakonov, Phys. Rev. Lett. 99, 126601 (2007).\n[86] G. Dresselhaus, Phys. Rev. 100, 580 (1955).\n[87] Y. A. Bychkov and E. I. Rashba, JETP Lett. 39, 66 (1984).\n[88] J. M. Luttinger, Phys. Rev. 102, 1030 (1956).\n[89] Q. Kohn and J. M. Luttinger Phys. Rev. 108, 590 (1957).\n[90] J. Smit, Physica 21, 877 (1955); ibid. 23, 39 (1958).\n[91] L. Berger, Phys. Rev. B 2, 4559 (1970).\n[92] P. Nozieres and C. Lewiner, Le Journal de Physique 34, 901 (1973).\n[93] C. L. Kane and E J. Mele, Phys. Rev. Lett. 95, 146802 (2005).\n[94] S. Murakami, N. Nagaosa, and S.-C. Zhang, Phys. Rev. Let t.93, 156804 (2004).\n[95] R. Winkler, Phys. Rev. B 69, 045317 (2004).\n[96] Out of the many texts on this topic, the excellent exposi tion by J. Rammer, Quantum Transport Theory (Perseus Books,\nReading MA, 1998) stands out. Chs. 2, 3, 7 and 8 discuss the fun damentals needed to understand linear response theory\nin systems with disorder.\n[97] G. Sundaram and Q. Niu, Phys. Rev. B 5914915 (1999).\n[98] D. Culcer, Y. Yao, A. H. MacDonald, and Q. Niu, Phys. Rev. B 72, 045215 (2005).\n[99] J. J. Sakurai, Modern Quantum Mechanics (Addison-Wesley, Reading MA, 1994).\n[100] G. L. Bir and G. E. Pikus, Symmetry and Strain-Induced Effects in Semiconductors (Wiley, New York, 1974).\n[101] G. F. Koster, J. O. Dimmock, R. G Wheeler, and H. Statz, Properties of the Thirty-two Point Groups (MIT Press,\nCambridge MA, 1963)."    },    {        "title": "1305.2443v3.Spin_dynamics_in_a_spin_orbit_coupled_Fermi_gas.pdf",        "content": "arXiv:1305.2443v3  [cond-mat.quant-gas]  26 Sep 2013Spin dynamics in a spin-orbit coupled Fermi gas\nStefan S. Natu∗and S. Das Sarma\nCondensed Matter Theory Center and Joint Quantum Institute , Department of Physics,\nUniversity of Maryland, College Park, Maryland 20742-4111 USA\nWe study the dynamics of a non-degenerate, harmonically tra pped Fermi gas following a sudden\nramp of the spin-orbit coupling strength using a Boltzmann e quation approach. In the absence of\ninteractions and a Zeeman field, we solve the spin-orbit coup led Boltzmann equation analytically ,\nand derive expressions for the phase-space and temporal dyn amics of an arbitrary initial spin state.\nFor a fully spin polarized initial state, the total magnetiz ation exhibits collapse and revival dynamics\nin time with a period set by the trapping potential. In real sp ace, this corresponds to oscillations\nbetween a fully polarized state and a spin helix. To make pred ictions relevant to current experiments\non spin-orbit coupled Fermi gases, we then numerically stud y the dynamics in the presence of an\nadditional momentum independent Zeeman field. We find that th e spin helix is robust for weak\nmagnetic fields but disappears for stronger field strengths. Finally, we explore the spin dynamics\nin the presence of interactions and find that weak interactio nsenhance the amplitude of the spin\nhelix.\nI. INTRODUCTION\nThephysicsofspin-orbitcouplingisattheheartoffun-\ndamental phenomena such as the spin Hall effect [1, 2]\nas well as practical devices such as the spin transistor\n[3]. Key to these developments in the field of spintron-\nics is the understanding of how parameters such as the\nspin-orbit coupling, interaction, disorder, and geometry\nseparately, as well as collectively influence spin dynamics\n[4,5]. Thecreationoflowtemperatureatomicandmolec-\nular spin-orbit coupled Bose and Fermi gases has paved\nthe way for studying this physics in a setting where these\nparameters are well characterized [6–10]. Furthermore, a\ntool largely unique to ultra-cold gases is the ability to\ninduce the spin-orbit coupling dynamically , thereby en-\nabling the study of out of equilibrium physics in these\nsystems. In this paper, we solve the dynamics of a non-\ninteracting non-degenerate Fermi gas following a sudden\nramp of the spin-orbit coupling strength. The resulting\nout-of-equilibriumdynamicsisrich[11–13]: coherencein-\nherent in cold atomic gases, but almost always absent in\nsolid state systems leads to collapse and revival of the\ntotal magnetization [12]. In real space, this is manifested\nby the spontaneous appearance of a helical spin texture\nwhichistheanalogueofthepersistentspinhelixobserved\nin two-dimensional electron gases [14, 15].\nA further advantage of cold atomic systems is the abil-\nity to use magnetic fields to control the interactions be-\ntween the atomic states [16, 17]. This, particularly in\nthe presenceofspin-orbitcouplingands-wavesuperfluid-\nity, may enable experimentalists to realize novel states of\nmatter with topological properties whose excitations ex-\nhibit non-Abelian statistics [18, 19]. In addition to their\nfundamental importance, such topological states of mat-\nter may serve as a platform for fault tolerant quantum\ncomputation [20]. However a key challenge in observing\n∗Electronic address: snatu@umd.eduthis physicsin ultra-coldgasesisthe inherent difficulty in\nattaining sufficiently low temperatures needed to realize\nthese topological states. In contrast, the non-equilibrium\ndynamics we study here occurs at high temperatures and\ncan be observed in currentexperiments. We numerically\nstudy the spin dynamics of a fully polarized, weaklyin-\nteracting Fermi gas using a collisionlessBoltzmann equa-\ntion. We find large amplitude spin waves analogous to\nthose previously observed in dilute spin polarized Hy-\ndrogen [21–25] and more recently in ultra-cold Bose and\nFermi gases [26–31]. Our numerical study of spin dy-\nnamics in the collisionless spin-orbit coupled Fermi gas\ncomplements a recent, linear response study on a homo-\ngeneous system by Tokatly and Sherman [11]. We point\nout that the ultra-low temperature degenerate version of\nour system (with spin-orbit coupling and large Zeeman\nsplitting) would manifest topological superfluidity in the\npresenceofordinarys-wavesuperfluidityinducedbysuit-\nable Feshbach resonance [19].\nWeshowthatspin-orbitcoupling, whencombinedwith\nthe long coherence times inherent to cold atomic and\nmolecular gases, leads to surprising dynamical phenom-\nena even at high temperatures where the cold gas is not\nnecessarily quantum degenerate. Furthermore the non-\ndegenerate gas is an ideal conceptual starting point for\nstudying how interactions influence the spin dynamics in\nthe presence of spin-orbit coupling.\nThe effects predicted in our work are unlikely to be of\nmuch experimental significance in solid state spin-orbit\ncoupled systems because of strong disorder and decoher-\nence intrinsically present in these systems as well as the\nultra-fast time scales for spin relaxation [32]. But in\ncold atomic and molecular systems, our proposed physics\ncould be studied in existinglaboratory systems. In prin-\nciple, the physics we predict should be present in both\natomic/molecular Fermi/Bose gases since it is an intrin-\nsically high temperature phenomenon, but in light of re-\ncent experiments [8–10], we will discuss our theoretical\ndetails using the atomic Fermi gas as the representative\nsystem of study.2\nThis paper is organized as follows: In Section II, we\ndescribe our system and derive the collisionless Boltz-\nmann equation for a two-component Fermi gas in a 2D\nharmonic trap in the presence of spin-orbit coupling. In\nthe subsequent sections, we choose an initial state which\nis a stationary state of the Hamiltonian in the absence\nof spin-orbit coupling. We then drive the system out-\nof-equilibrum by suddenly turning on the spin-orbit cou-\npling and study the resulting dynamics. In Section III,\nwe solve the Boltzmann equation in the absence of in-\nteractions, and in Section IV, we consider the effect of\ninteractions on the spin dynamics. We summarize our\nresults in Section V.\nII. SETUP\nIn the absence of spin-orbit coupling, the Hamiltonian\nfortwohyperfinestatesofaFermigascanbeexpressedas\na sum of single particle and two-body interaction terms:\nH=Hs+Hint (1)\nwhere the single particle Hamiltonian is composed of a\nkinetic term and a potential term arising from the exter-\nnal trapping potential\nHs=/summationdisplay\ni=↑,↓/integraldisplay\nd3rψ†\ni/parenleftBig\n−/planckover2pi12∇2\nr\n2m+U(r)/parenrightBig\nψi\nwhereψidenotes the fermionic annihilation operator for\na particle in hyperfine state {↑,↓}and massm. Here\nU(r) refers to the external trapping potential, which we\nassume to be cylindrically symmetric U(r) =1\n2mω2\nr(x2+\ny2)+1\n2mω2\nzz2, whereωrandωzare the trapping frequen-\ncies in the radial and longitudinal directions respectively.\nAs the relevant spin-orbit physics is two-dimensional, we\nassume a quasi-two dimensional, pancake geometry, ob-\ntained by tight confinement in the longitudinal ( z) di-\nrection (ωz≫ωr). For simplicity, we assume that both\nthe atomic states experience identical trapping poten-\ntials, but all our calculations can be readily extended to\ninclude more general trapping potentials realized in ex-\nperiments.\nAt the ultra-cold temperatures realized in these exper-\niments, the dominant contribution to scattering comes\nfrom the s-wave channel. The interaction Hamiltonian\ntherefore takes the simple form of a contact interaction:\nHint=g\n2/integraldisplay\ndrψ†\n↑(r)ψ†\n↓(r)ψ↓(r)ψ↑(r) (2)\nwith interaction strength g= 4π/planckover2pi12a/mwhereadenotes\nthe s-wave scattering length. A key advantage of the\nfermonicexperiments[8–10] istheabilityto usemagnetic\nfields to tune the interactions between different internal\nstates, viaaFeshbachresonance[16,17]. Hereweworkin\nthe weakly interacting regime, which can be realized by\nworkingnearthe zerocrossingofthe Feshbachresonance.The spin-orbit Hamiltonian containing terms linear in\nmomentum takes the form:\nHSOC=α(σxpy−σypx)+β(σxpx−σypy) (3)\nwhere the first term is the Rashba contribution\nand the second term is the Dresselhaus contribution,\nparametrized by the coupling constants αandβrespec-\ntively. Here σx,σyandσzdenote Pauli matrices.\nIn the ultra-coldgassetting, spin-orbit couplingis gen-\nerated by using a pair of Raman beams to drive tran-\nsitions between two hyperfine states of an atom, while\nsimultaneously imparting a momentum kick [6–9]. The\nresulting spin-orbit Hamiltonian takes the Rashba equal\nDresshelhaus form α=β, where the magnitude of αis\ndetermined by the wave-length of the Raman beams and\nthe angle at which they intersect. The Raman beams\nalso produce a momentum independent Zeeman field,\nHZ=−/planckover2pi1ΩRσz, where Ω Ris the strength of the Ra-\nman coupling, and is proportional to the intensity of the\nRaman lasers. As we show below, this term plays an\nimportant role in the dynamics. The scheme described\nhere was first successfully demonstrated in experiments\nat NIST using bosonic87Rb [6, 7]. Recently, a simi-\nlar scheme was used to generate spin-orbit coupling in a\nFermi gas of6Li and40K [8–10].\nAs we are primarily motivated by current experiments,\nwe limit ourselves to the case α=βin Eq. 3. In this\nlimit, the spin-orbit Hamiltonian can be diagonalized by\nindependently rotating the momentum co-ordinate and\nperforming a global spin rotation. We will denote the di-\nagonal basis as{ψ+,ψ−}. Throughout, we will use both\nthe diagonal basis ( {+,−}) and the pseudo-spin basis\n({↑,↓}) corresponding to the original hyperfine states,\ndepending on the context. The generalization to α/ne}ationslash=β\nis straightforward within our formalism, and will be the\nsubject of a future work [33].\nAll of the work described here is in the non-degenerate\nlimit. The onlyrequirementonthe temperature Tis that\nit should be smaller than the detuning energy between\nthe magnetic sublevels, so that the gas can be described\nby a two-level (pseudo-spin1\n2) system. This is readily\naccomplished as the splitting between hyperfine levels is\nmuch larger than the Fermi energy for typical densities\n[8].\nThe physics of the weakly interacting gas is domi-\nnated by coherent mean-field dynamics which occurs on\na timescale τmf∼(an0/m)−1which is much faster than\nthe timescale for energy exchanging collisions τcoll∼\n(4πa2n0v)−1(vis the characteristic velocity of the parti-\ncles andn0is the density). For a trapped gas of N∼105\n6Li atoms at a temperature T∼10−6K, the collision-\nless limit (τmf≪τcoll) corresponds to scattering lengths\na∼10aB, whereaBis the Bohr radius. The recent ex-\nperiment of Cheuk et al.is already in this regime [9],\nwhile the experiment of Wang et al.[8] has stronger in-\nteractions of a∼200aB, which can be tuned near zero\nusing a Feshbach resonance [17].3\nMathematically, the weakly interacting gas can be de-\nscribed using a collisionlessBoltzmann equation. Follow-\ningRef. [34], weusetheHeisenbergequationsfor ψσ(r, t)\nto derive the equations of motion for the spin dependent\nWigner function\n← →F=/parenleftbigg\nf↑↑(p,R,t)f↑↓(p,R,t)\nf↓↑(p,R,t)f↓↓(p,R,t)/parenrightbigg\n(4)\nfσσ′(p,R,t) =/integraldisplay\ndreip·r/an}b∇acketle{tψ†\nσ(R−r\n2,t)ψσ′(R+r\n2,t)/an}b∇acket∇i}ht,\nwhich is the quantum analogue of the classical distri-\nbution function. Here prepresents the momentum,\nr=r1−r2is the relative coordinate and R=r1+r2\n2\nis the center of mass coordinate.\nThe diagonal components of← →Fcan be integrated\nin momentum to give the respective spin densities\nnσσ(R,t) =/an}b∇acketle{tψ†\nσ(R,t)ψσ(R,t)/an}b∇acket∇i}ht=/integraltextdp\n(2π)3fσσ(p,R,t),\nwhile the off-diagonal components correspond to quan-\ntum coherences that are absent in a classical model of a\nspin-1\n2gas.\nFor the pancake geometry considered here, all the\nrelevant dynamics is two dimensional. Assuming ther-\nmal equilibrium in the longitudinal direction, we de-\ncompose the Wigner function in the radial and axial\ndirections as fσσ′(p,R,t) =fσσ′(pr,r,t)f(pz,z) where\nf(pz,z) =e−β(p2\nz/2m+1\n2mω2\nzz2)whereβ= 1/kBT.\nWe obtain a two dimensional density n2D\nσσ′(r,t) =/integraltext\ndprfσσ′(pr,r,t)/integraltext\ndpzf(pz,z) by integrating out\nthe longitudinal co-ordinate. Averaging over the\nz−direction, we obtain an effective quasi 2D Boltzmann\nequation:\n∂t← →F+p\nm·∇r← →F−∇rU∇p← →F=i[← →V,← →F]+ (5)\n1\n2{∇r← →V,∇p← →F}+iαp+[σ+,← →F]+α\n2{σ+,∇r+← →F}\nwherep+=px+py,σ+=σx+σy, and the interaction\npotential← →Vis [28]:\n← →V=/parenleftbigg\ng2Dn↓↓−/planckover2pi1ΩR−gn↑↓\n−gn↓↑gn↑↑+/planckover2pi1ΩR/parenrightbigg\n(6)\nwhereg2Dis an effective two-dimensional interaction\nstrength given by g2D= 2√π/planckover2pi12a/(mΛth), where Λ th=/radicalbig\n2π/planckover2pi12/mkBT. The diagonal components in the interac-\ntion matrix arise from forward scattering (Hartree) while\nthe off-diagonal terms arise from exchange interactions\n(Fock). Commutatorsandanti-commutatorsaredenoted\nby [,] and{,}respectively.\nThe single particle limit of Eq. 5 was derived by\nMishchenko and Halperin [5], who used this approach to\nstudy the transport properties of a 2D electron gas. Here\nwe generalize the Boltzmann equation to the include ef-\nfect of← →Von the phase space and spin space evolution of\nthe Wigner function.\nIn general, Eq. 5 is a non-linear matrix equation which\nhas to be solved numerically. Below we assume an initialstatewhichisastationarystateoftheHamiltonianinthe\nabsence of spin-orbit coupling ( α= ΩR= 0). We then\nsuddenly turn on the Raman coupling, and investigate\nthe resulting out-of-equilibrium dynamics.\nIII. NON-INTERACTING LIMIT\nA.ΩR= 0\nThe non-interacting limit in ultra-cold Fermi gases is\nachieved by working at the zero crossing of a Feshbach\nresonance. As the magnetic fields corresponding to the\nzerocrossingofthe interactionsaretypicallysmall[9], we\ndo not expect the Raman couplings to deviate apprecia-\nbly from their zero field values [35]. We first consider the\ncase where upon switching on the Raman coupling, the\nspin-orbit coupling ( α) is non-zero, but the Zeeman term\nΩR= 0. While this scenariodoes not correctlymodel the\npresent experiments, in this limit the Boltzmann equa-\ntion is exactly soluble for an arbitrary initial spin state,\nthus serving as a conceptual starting point. We remark\nthat although we only consider the non-degenerate limit\nhere, our results can be readily generalized to tempera-\ntures below the Fermi temperature.\nIn order to proceed, we introduce dimensionless po-\nsition and momentum coordinates ˜ r=r/rtrapand\n˜p=p√\n2π/planckover2pi1/Λth, wherertrap=/radicalbig\n/planckover2pi1/mωris the char-\nacteristic length scale of motion in the trap and and\nΛth=/radicalbig\n2π/planckover2pi12/mkBTis the thermal deBroglie wave-\nlength. We normalize time in units of the radial trap-\nping frequency ˜t=t/ωr. We also introduce a parameter\nη=/radicalbig\n/planckover2pi1ωr/kBT. As the spin-orbit Hamiltonian only\ncouples to momentum in the px+pydirection, it suf-\nfices to consider the evolution of the distribution in the\np+=px+pyandr+=x+ydirections of phase space.\nConsider an arbitrary initial spin state given by the\nWigner distribution function:← →F(˜p+,˜r+,t= 0) =\ne−1\n4(˜p2\n++η2r2\n+)← →fwhere← →fis a 2×2matrix corresponding\nto the initial spin state. We omit the p−,r−directions\nfor now as they have no dynamics. In the absence of\nspin-orbit coupling or interactions ( α= ΩR=g2D= 0),\nthe initial state is stationary.\nNext, we expressthe Boltzmann equation in the diago-\nnal basis by performing the global transformation← →F→\nB†← →f Bwhere the unitary matrix B=/parenleftBigg1√\n2−1+i\n2\n1+i\n21√\n2/parenrightBigg\nrotatesσx+σyto√\n2σz. In the rotated basis, the spin +\nand−components evolve independently, and the prob-\nlem reduces to a single particle problem in the presence\nof amomentum dependent magnetic field.\nThe dynamics in phase space can now be solved by\nmaking the following ansatz for the diagonal components\nof the rotated spin density matrix:\nF++/−−(˜p+,˜r+,˜t) =A+/−e−1\n4{(˜p+∓a(˜t))2+η2(˜r+∓b(˜t))2}(7)4\n/Minus10/Minus505100Π\n2Π3Π\n22Π\nr/Slash1rtrapt/LParen11/Slash1Ωtrap/RParen1\nFIG. 1: Time evolution of the longitudinal (density plot) an d\ntransverse magnetization (arrows) in the r+=x+ydirection,\nfollowing a sudden ramp of the spin-orbit coupling strength .\nThe Zeeman term is set to zero here (Ω R= 0). Brighter\ncolors indicate positive magnetization ( ↑) and darker colors\nindicate negative magnetization ( ↓). The arrows indicate the\ndirection of the magnetization in the x−yplane, and the\nlength of the arrows indicate the magnitude of the transvers e\nspin normalized to the total spin. At t= 0 all the spins are\npointing in the zdirection (all atoms in the ↑state). spin-\norbit coupling causes the atoms to precess in time, and at\nhalf the trapping period a spin helix is produced. The wave-\nvector of the helix at t=π/ωrdepends only on the spin-orbit\ninteraction and is λsh=π rtrap/2˜α, wherertrap=/radicalbig\n/planckover2pi1/mωr.\nTo clearly illustrate this effect, we choose a weak spin-orbi t\ncoupling of ˜ α=√\n2α/radicalbig\nm//planckover2pi1ωr= 0.125 and η=/radicalbig\n/planckover2pi1ω/kBT=\n0.25 in these simulations. The initial state is recovered afte r\nt= 2π/ωr.\nwherea(0) =b(0) = 0. The coefficients A+/−are the\ndiagonal matrix elements of the spin density matrix← →f\nafter rotation into the {+,−}basis.\nSubstituting the ansatz of Eq. 7 into Eq. 5 we find\na(˜t) = ˜αη(cos(˜t)−1) andb(˜t) = ˜αsin(˜t), where we have\nintroduced a dimensionless, spin-orbit coupling constant\n˜α=√\n2α/radicalbig\nm//planckover2pi1ωr, which parametrizes the strength of\nthe spin-orbit coupling relative to the trapping potential.\nThus the rotated spin densities simply perform oscilla-\ntions in real and momentum space with an amplitude set\nby the strength of the spin-orbit interaction and period\nset by the trap frequency.\nSimilarly, one can solve for the dynamics of the off-\ndiagonal components to find:\nF+−(˜p,+˜r+,˜t) =A+−e−4˜α2(1−cos(˜t))\nη2×(8)\ne−1\n2{(˜p+−2i˜α/ηsin(˜t))2+η2(˜r+−2i˜α/η2(1−cos(˜t)))2}\nwhereA+−is the off-diagonal matrix element of the spin\ndensity matrix after rotation into the ±basis. The dy-\nnamics ofF−+is obtained by replacing ˜ α→−˜αin Eq. 8.\nUnlike the diagonal components, the magnitudes of theoff-diagonal components are notconserved and oscillate\nin time.\nThe corresponding spin densities are found by inte-\ngrating the above expressions for the Wigner functions\nin momentum space. By rotating the diagonalbasis back\ninto the hyperfine basis {↑,↓}, one obtains the dynamics\nof an arbitrary initial spin state.\nTo illustrate the role of quantum coherence, we con-\nsider the dynamics of a fully polarized initial state cor-\nresponding to all particles in the ↑state. We study the\ndynamics of the longitudinal magnetization density and\nthe total magnetization:\nmz(r,t) =/integraldisplay\ndp/parenleftBig\nf↑↑(p,r,t)−f↓↓(p,r,t)/parenrightBig\n(9)\nM(t) =/integraldisplay\ndrmz(r,t)\nThe zero temperature dynamics of the total magneti-\nzation for this initial state was considered previously by\nStanescu, Zhang and Galitski [12]. By exactly solving\nfor the quantum dynamics in a trap, they demonstrated\nthat the total magnetization exhibits collapse and revival\ndynamics, and produced analytic formulas for the total\nmagnetization in weak spin-orbit coupling limit.\nHere we show that similar dynamics also occurs in the\nnon-degenerate gas, which is much more readily acces-\nsible in experiments. Furthermore, we obtain analytic\nexpressions for the total magnetization for arbitrary val-\nues of the spin-orbit coupling.\nRotating the spin polarized state to the diagonal ba-\nsis, one finds that the density matrix has both diagonal\nand off-diagonal matrix elements ( A+=A−= 1/2 and\nA+−=−1−i\n2√\n2), whose dynamics is given by Eqns. (7, 8).\nTransforming back to the hyperfine basis and integrating\nover momentum, the longitudinal magnetization density\ntakes the form:\nmz(˜r+,˜t)∼e−1\n2η2˜r2\n+−˜α2\nη2(1−cos(2˜t))× (10)\ncos(2˜r+˜α(1−cos(˜t))\nand the total magnetization is:\nM(˜t)∼/radicalbigg2π\nηe−4˜α2\nη2(1−cos(˜t))(11)\nwhere we haveignored an overallnormalizationfactor re-\nsultingfromintegrationovermomentum. The expression\nfor the transverse magnetization density is rather cum-\nbersome, but the total transverse magnetization remains\nzero at all times.\nFromEq.11, it isclearthatthe totallongitudinalmag-\nnetization exhibits collapse and revival dynamics in time\nwith a period which depends onlyon the trapping poten-\ntial, and is completely independent of the temperature\nor the spin-orbit coupling strength. At fixed tempera-\nture, for weak spin-orbit coupling ˜ α≪1, our expression5\nreadsM∼1−4˜α2/η2(1−cos(˜t)) [12]. In this limit,\nthe magnetization exhibits sinusoidaloscillations with an\namplitude which is proportional to 8˜ α2/η2. For strong\nspin-orbit coupling, ˜ α≫1, the magnetization becomes\nstrongly peaked near t= 2πn/ωrwherenis an integer,\nand decays exponentially, away from these points.\nThe collapse and revival of the total magnetization is\na trap effect. In a homogeneous system, the momentum\ndependent spin-orbit magnetic field will simply cause the\nspins to dephase, particles with different momenta will\nprecessatdifferentrates,andthetotalmagnetizationwill\ngo to zeroirreversiblyon a timescale set by the spin-orbit\ncoupling strength [11]. It is also important to emphasize\nthat the collapse and revival in the total magnetization\ndescribed above has a different origin from what is ob-\nserved in the experiments of Wang et al.[8]. We will\ndiscuss this in more detail later.\nTo understand the origin of the magnetization oscil-\nlations, we now turn to the dynamics of the longitudi-\nnal and transverse magnetization density following the\nramp. In a trapped geometry, from Eq. 10, we find that\nthe longitudinal magnetization density exhibits periodic\noscillations in space and in time. The temporal oscilla-\ntions have a period of t= 2π/ωr, while att=π/ωrthe\nspatial oscillations have a characteristic wave-length of\nλsh=π rtrap/(2˜α) wherertrap=/radicalbig\n/planckover2pi1/mωr.\nIn Fig. 1 we plot the magnetization density normal-\nized to the initial magnetization at the center ( m(r=\n0,t= 0)) as a function of time for the parameters\nabove. Brighter colors indicate positive magnetization\nwhile darker colors indicate negative values of mz. We\nchoose a rather weak spin-orbit coupling strength in or-\nder to enhance the wavelength of the spatial oscillations\natt=π/ωr. The transverse components of the spin\nare indicated by arrows whose length corresponds to the\nmagnitude of the spin vector in the x−yplane. At\nt= 0, all spins are pointing in the ↑direction indicated\nby the bright region in the density plot. Over time a\ntransverse component develops and a spin helix emerges.\nAtt=π/ωr, the spin oscillations reach the maximum\namplitude proportional to e−1\n2η2˜r2\n+, with a wave-length\nof Λsh/rtrap=π/2˜α.\nEnergy conserving dynamics in phase space implies\nthat the momentum of a particle is linked to its position.\nMoreover, the spin of an atom is linked to its momentum\nvia the spin-orbit coupling. The spin-orbit Hamiltonian\nshifts the minimum of the dispersion to finite momenta.\nIn a harmonically confined system, the iso-energy con-\ntours are circles in phase space, that are now shifted to\nfinite momenta due to the spin-orbit coupling. A wave-\npacket polarized in the ↑direction centered at r=p= 0\nwill follow the iso-energy contours in phase space, while\nsimultaneously rotating in spin space. At t=π/ωrthe\natomic wave-packet is centered around r= 0 in real\nspace, and in order to conserve the total energy, the dis-\ntribution will be centered around ˜ p+=±2˜αηin momen-\ntum space. As a result, atoms with opposite momenta\nprecess in opposite directions in spin space, producing aspin helix. At t= 2π/ωr, the atoms return to their orig-\ninal distribution in real and momentum space, and the\ninitial state is recovered. The spin helix has a smaller net\nmagnetization as compared to the fully polarized initial\nstate, thus explaining the oscillations in the net magne-\ntization.\nExperimentally, the spin density can be imaged in situ\nusing phase contrast imaging [36, 37]. The wave-length\nof the spin helix is set by the spin-orbit interaction which\nis determined by the wave-length of the Raman beams.\nFor the parameters used in the experiment of Cheuk et\nal., Λsh∼0.5µm [9], which may be below the experi-\nmental resolution. However, the wave-length can be in-\ncreased by decreasing the spin-orbit coupling strength.\nAt present, experiments on spin-orbit coupled cold gases\nsuffer from extremely short lifetimes ( <500ms) due to\nthe large heating rates resulting from inelastic light scat-\ntering from the Raman beams [38]. For typical trapping\npotentials, the timescale for the appearance of the spin\nhelix ist=π/ωr∼50ms, so the short lifetimes may not\nbe a major limitation in observing the spin helix.\nB.ΩR/negationslash= 0\nWe now turn to the experimentally relevantcase where\nuponsuddenlyswitchingontheRamanbeams, theatoms\nalsoexperienceaconstant, momentumindependent mag-\nnetic field, parametrized by a dimensionless parameter\n˜B= ΩR/ωr. In this limit, the problem cannot be solved\nanalytically as the Zeeman and spin-orbit terms in Eq. 5\ndonotcommute. Instead we numerically integrate the\nBoltzmann equation on a 4D grid in phase space. We\nchoose a 20×20 lattice in Randpwith a spatial res-\nolution ofδr= 2rtrap, wherertrap=/radicalbig\n/planckover2pi1/mωris the\ncharacteristic length scale of motion in the trap, and mo-\nmentum resolution of δp= 0.6 2π/Λth. The integration\nis done using a split-step method that conserves total\nparticle number and the total energy to high accuracy\nfor sufficiently small time steps.\nFor simplicity we choose a fully polarized initial state,\nwhich is stationary in the absence of interactions or spin-\norbitcoupling. Wethenconsidertwolimits, uponswitch-\ning on the Raman beams: ˜B∼˜αand˜B >˜α. The\ntwo parameters can be controlled independently as the\nmagnitude of ˜ αis set by the wave-length of the Raman\nbeams, and Ω Ris set by the laser intensity. The result-\ning dynamics of the total magnetization as well as the\nmagnetization density is plotted in Fig. 2.\nAs shown in Fig. 2, the addition of a Zeeman term\ncauses the magnetic field oscillations to decay over time.\nAt long times, the total magnetization acquires a new\nsteady state value which is smaller than 1. As the\nstrength of the Zeeman field is increased, the fully po-\nlarized initial state becomes increasingly stable, and the\ntotal magnetization remains close to 1 at long times with\nsmall oscillations.\nThe dynamics can be understood as follows: in6\n02Π4Π6Π8Π1\n0.5\n1\n0.5\n0\n1\nt/LParen11/Slash1Ωr/RParen1M/LParen1t/RParen1/Slash1N/Minus20/Minus1001020/Minus0.400.40.8\nr/Slash1rtrapS/LParen1r/RParen1/Slash1n\n/Minus20/Minus100102000.40.8\nr/Slash1rtrapS/LParen1r/RParen1/Slash1n\n/Minus20/Minus100102000.40.8\nr/Slash1rtrapS/LParen1r/RParen1/Slash1n\nFIG. 2: (Left) Time Evolution of the total magnetization\nin the system (see Eq. 9) normalized to the total parti-\ncle number for three different values of the dimensionless\nZeeman coupling ˜B= ΩR/ωr. In each case, we fix ˜ α=√\n2α/radicalbig\nm//planckover2pi1ωr= 0.25. From top to bottom: (Black) ˜B= 0;\n(Blue)˜B= 0.25; (Red) ˜B= 1. (Right) Magnetization\ndensity along the r+=x+ydirection for the same val-\nues of the spin-orbit coupling strength and Zeeman field as\nin the left figure at fixed time t=π/ωr. The magnetiza-\ntion densities are normalized to the total central density d e-\nnotedn=n(r= 0,t= 0) =/integraltext\ndp[f↑↑(p,0,0) +f↓↓(p,0,0)].\nSolid curves represent the magnetization density in the zdi-\nrection (Eq. 9), while the dashed curves indicate the mag-\nnetization density in the xdirection: mx(r,t=π/ωr) =/integraltext\ndp[f↑↓(p,r,π/ωr)+f↓↑(p,r,π/ωr)].\nthe presence of a Zeeman field, each spin precesses\nabout a new magnetic field, which is the sum of the\nspin-orbit magnetic field and the Zeeman field, and is\ntilted away from the x-y plane by an angle sin( θp) =\n˜B//radicalBig\n˜B2+(˜αη˜p+)2. As the initial state has a Gaussian\ndistribution of atoms with different momenta with spins\npointinginthe z−direction,atomswith momentagreater\nthanp>˜pcrit=˜B/˜αηwill primarilyexperience the spin-\norbit magnetic field and precess about the x−yplane,\nwhile atoms with momenta p<pcritwill predominantly\nsee the Zeeman field, and precess about the z−axis.\nIn a thermal gas, the characteristic width of the mo-\nmentum distribution is set by pth=√\n2π/planckover2pi1/Λth. If the\nZeeman field is weak compared to the spin-orbit strength\n(αpth>>/planckover2pi1ΩR), the majority of the atoms still expe-\nrience the spin-orbit magnetic field, and the spin den-\nsity wave is preserved (as shown in the blue curves in\nFig. 2) for the first few oscillations. On longer timescales\nthe magnetic field affects the spin precession of even the\natoms with p >> p critand the spin density wave disap-\npears. On very long times the system settles into a new\nsteady state with a lower net magnetization.\nOn the other hand, if the Zeeman field is strong(αpth<</planckover2pi1ΩR), the majority of the atoms experience\na net magnetic field which is only slightly tilted away\nfrom thez-axis. As the initial state is fully polarized,\nthe total magnetization remains close to 1 at all times\nwith rapid oscillations whose frequency is given by Ω R.\nThe transverse spin density remains small at all times\n(see the dashed red curve in Fig. 2) and no spin helix is\nfound.\nThe spin helix is the result of the interplay be-\ntween dynamics in spin space and dynamics in phase\nspace. Observing the spin helix therefore requires that\nthe timescales for spin dynamics be comparable to the\ntimescales for dynamics in phase space.\nIn the presence of a large Zeeman field, as in the ex-\nperiments of Wang et al.[8] (EZ∼kHz≫/planckover2pi1ωr∼50Hz),\nthe timescale for spin dynamics is set by the Zeeman\nterm, which is much too short for any dynamics to oc-\ncur in phase space. Thus while the total magnetiza-\ntion shows collapse and revivals (on a timescale set by\nt∼2π/EZ∼0.1ms), the magnetization density in real\nspace is unaffected. By contrast the collapse and revival\ndynamics discussed in Sec. III A occurs because the ini-\ntial state is out of equilibrium in phase space , when the\nRamancouplingis switched on. As a result, the magneti-\nzation oscillates on a timescale set by t∼2π/ωr∼10ms,\nand a spin spiral appears.\nIV. INTERACTIONS\nWe now consider the effect of weak interactions on the\ndynamics discussed above. As discussed previously, we\nwork in the the collisionless limit (sometimes referred to\nas the Knudsen regime), which is valid as long as τmf≪\nτcoll.\nSpin waves in dilute, quantum gases were studied the-\noretically by Bashkin and others [21–24] and observed in\nexperiments on spin polarized hydrogen [25]. More re-\ncently, Du et al.explored similar physics in a weakly in-\nteracting, thermal gas of6Li atoms [26]. Here we explore\nspin waves in a collisionless thermal gas with spin-orbit\ncoupling.\nTo compare with the results of Sec III, we consider a\nfullypolarizedinitialstatewhichisthestationarystateof\nthe Boltzmann equation in the absence of spin-orbit cou-\npling or interactions. We then simultaneously switch on\nthe interactions, and the spin-orbit coupling. We define a\ndimensionless interaction strength ˜ g=g2Dn//planckover2pi1ωr, where\nnis the total initial density at the trap center n=n(r=\n0,t= 0) =/integraltext\ndp[f↑↑(p,0,0) +f↓↓(p,0,0)]. For typical\ntrapping potentials used in experiments, ωr∼2π×100\nHz, the Knudsen regime corresponds to ˜ g∼0.1 [26]. We\nchoose ��g= 0.25. Furthermore, as shown in Sec III B,\na large Zeeman field ( ˜B≫˜α) stabilizes the spin polar-\nized state and produces little spin dynamics. Hence we\nconsider ˜B= ˜α= 0.25.\nIn Fig. 3 we plot the total magnetization density along\nther+direction and the total net magnetization as a7\n/Minus20/Minus100102002Π4Π6Π8Π\nr/Slash1rtrapt/LParen11/Slash1Ωtrap/RParen1t/EqualΠ/Slash1Ωr\n/Minus20/Minus100102000.40.8\nr/Slash1rtrapS/LParen1r/RParen1/Slash1n\nt/Equal2.5Π/Slash1Ωr\n/Minus20/Minus1001020/Minus1/Minus0.500.5\nr/Slash1rtrapS/LParen1r/RParen1/Slash1n\nt/Equal5Π/Slash1Ωr\n/Minus20/Minus100102000.40.8\nr/Slash1rtrapS/LParen1r/RParen1/Slash1n\n02Π4Π6Π8Π1\n0\n1\n/Minus0.5\n1\n/Minus0.6\nt/LParen11/Slash1Ωr/RParen1M/LParen1t/RParen1/Slash1N\nFIG. 3: Top Left: Density plot showing the magnetization\ndensity along r+(see Eq. 9), normalized to the total initial\ncentral density as a function of time. The parameters are\n˜g=˜B= ˜α= 0.25 (Recall that ˜ g=gn//planckover2pi1ωr,˜B= ΩR/ωrand\n˜α=√\n2α/radicalbig\nm//planckover2pi1ωr. Brighter colors indicate positive magne-\ntization while darker colors indicate negative magnetizat ion.\nBottom Left: Total magnetization in the entire system as a\nfunction of time for three different values of ˜ g. In each plot,\n˜B= ˜α= 0.25. From top to bottom ˜ g= 0 (dashed), ˜ g= 0.25\n(solid) and ˜ g= 0.5 (dotted). Panels on Right: Longitudinal\n(solid) and transverse (dashed) magnetization densities i n the\nr+direction at different times. The top panel shows the spin\nhelix without interactions for ˜B= ˜α= 0.25 (same as the cen-\ntral panel in Fig. 2) for comparison. The central and bottom\npanels are snapshots of the longitudinal and transverse mag -\nnetization at time t= 2.5π/ωrandt= 5π/ωrfor the same\nparameters as in the Top Left.\nfunction of time. As is apparent from the figure, the\ninclusion of interaction drives large magnetization oscil-\nlations on a characteristic timescale much larger than\nt=π/ωr. This timescale grows as the interaction\nstrength is increased. In real space, the magnetization\nminimum is manifested as a large amplitude longitudi-\nnal spin wave. The panels on the right show the mag-\nnetization density in the longitudinal and transverse di-\nrections (solid and dashed respectively) at two different\ntimest= 2.5π/ωr(red) andt= 5π/ωr(blue) for pa-\nrameters corresponding to the density plot. The spin\ndensities for the spin helix (c.f Fig. 2(right) center panel)\nfor the same values of ˜Band ˜α, but ˜g= 0 are shown in\nthe top panel for comparison. The amplitude of the spin\nwave seen here is much larger than what was observed\nin previous studied of the spin-1\n2gas in the absence of\nspin-orbit coupling [26].\nInteractionsalterthephysicsofthenon-interactinggas\nin two crucial ways. Forward scattering modifies the iso-\nenergycontoursinphasespace: spin ↑atomsexperiencea\nmean-field proportionalto the density ofthe ↓atoms and\nvice versa. More importantly, due to the exchange inter-action, when two atoms collide, they precess about the\ncommon axis of their total spin. As argued by Lhuillier\nand Lalo¨ e, it is this effect that gives rise to spin waves in\nthe collisionless gas [22]. In the experiments of Du et al.\n[26], theinitialstatewaspolarizedalongthe x−direction,\nand spin dynamics was the result of the negligible differ-\nence in the trapping potentials experienced by the spin ↑\nand↓atoms. Consequently the amplitude of the result-\ning spin wave was very small, mz(r)/n0≪1 [26].\nBy contrast in the present case the initial state is po-\nlarized along z. Absent spin-orbit coupling, this state\nhasnodynamics as it is stationary with respect to the\nZeeman field and the interaction term. spin-orbit cou-\npling however causes the spins to precess about the x−y\nplane with a precession rate proportional to the momen-\ntum. Over time, atoms with different momenta precess\nat different rates and can collide with one another. The\natomic collisions subsequently lead to a dynamical spin\nsegregation in real space [22]. Unlike the experiments of\nDuet al., the initial spin precession in our system occurs\ndue to spin-orbit coupling, which is not a small effect.\nConsequently the amplitude of the resulting spin wave is\nalso larger than what was found in Ref. [26].\nWe remark that the amplitude of the total magneti-\nzation oscillations, and the associated spin texture in\nreal space, depends non-monotonically on the strength\nof the Zeeman field. For ˜B≪˜α, and weak interactions\n˜g≪1, the magnetization oscillations are only slightly\nlarger than those studied in Sec III A, and the real space\nmagnetization density resembles the spin helix shown in\nFig. 3 (top right). The magnetization oscillations reach\na maximum at some ˜Bcrit, which depends on ˜ gand ˜α.\nUpon further increase of ˜B, the spin↑state becomes\nstable and there is no spin dynamics.\nV. SUMMARY AND CONCLUSIONS\nIn summary, we studied the spin dynamics in a weakly\ninteracting, non-degenerate Fermi gas following a sudden\nramp of the spin-orbit coupling strength. In the absence\nof interactions and a Zeeman field, we produced analytic\nexpressions for the total magnetization and the magneti-\nzation density for arbitrary values of the spin-orbit cou-\npling, trap frequency and the temperature. We argued\nthat a fully polarized initial state will give rise to a spin\nhelix on a timescale set by the trapping period with a\nwave-length that depends on the ratio of the spin-orbit\ninteraction to the trap frequency. For the high temper-\nature gas, we generalized the analytic results obtained\nby Stanescu et al.[12] to arbitrary spin-orbit coupling\nstrength.\nWe then numerically studied the spin dynamics in the\npresence of interactions and a Zeeman field, highlighting\nthe role played both separately and collectively by these\nterms. For weak Zeeman fields the spin helix is preserved\nbut it disappears for very large Zeeman fields, as the\nfully polarized initial state becomes increasingly stable.8\nIn the presence of interactions however the dynamics is\nmore complicated. Interactions tend to enhance the am-\nplitude of the spin oscillations. We explain this effect as\na dynamical spin segregation driven by exchange.\nFinally, we briefly comment on the role of Fermi statis-\ntics in our calculations. In principle, all the results dis-\ncussed in this paper apply equally well to bosons. The\nspin helix discussed in Sec III is a single-particle effect,\nand should also occur in a bosonic system. Importantly,\nit is adynamical effect and should be contrasted with the\nspin stripe phase predicted by Ho and Zhang, which oc-\ncurs in equilibrium spin-orbit coupled Bose-Einstein con-\ndensates [39].\nTo conclude, our work shows that a seemingly one-\nparticle term in the Hamiltonian (i.e. spin-orbit cou-\npling), when coupled with coherent non-equilibrium dy-\nnamics of the Fermi gas, could manifest rather non-\nobvious and complex (and more importantly, observable)behavior in cold atoms and molecules even at relatively\nhigh temperature where the quantum degeneracy of the\nsystem is not of any key importance. The dynamical\nphysics we predict here is unlikely to manifest itself in\nany solid state systems (and can only be seen in atomic\ngases) although the basic concept of spin-orbit coupling\noriginates in solids as discussed in detail in Refs. [1, 2].\nVI. ACKNOWLEDGEMENTS\nIt is a pleasure to thank Juraj Radic for his insights,\nand his careful reading of the manuscript. We also\ngratefully acknowledge discussions with Victor Galitski,\nI.V.Tokatly and E. Y. Sherman. This work is supported\nby JQI-NSF-PFC, AFOSR-MURI, and ARO-MURI.\n[1] J. E. Hirsch, Phys. Rev. Lett. 831834 (1999).\n[2] I. Zutic, J. Fabian and S. Das Sarma, Rev. Mod. Phys.\n76323 (2004).\n[3] S. Datta and B. Das, Appl. Phys. Lett. 56665 (1990); J.\nWunderlich et al.Science3301801 (2010).\n[4] M. I. Dyakonov et al.Sov. Phys. JETP 63655 (1986); E.\nG. Mishchenko, A. V. Shytov, and B. I. Halperin, Phys.\nRev. Lett. 93226602 (2004); A. A. Burkov, A. S. Nunez\n, A. H. MacDonald, Phys. Rev. B 70155308 (2004); D.\nD. Awschwalom and M. E. Flatt´ e, Nature Physics 3153\n(2007); B. A. Bernevig, J. Hu, Phys. Rev. B 78245123\n(2008).\n[5] E. G. Mishchenko and B. I. Halperin, Phys. Rev. B 68\n045317 (2003).\n[6] Y. K Lin, K. Jimenez-Garcia and I. B. Spielman, Nature,\n47183 (2011).\n[7] Y.-J. Lin, R. L. Compton, A. R. Perry, W. D. Phillips,\nJ. V. Porto, and I. B. Spielman, Phys. Rev. Lett. 102\n130401 (2009).\n[8] P. Wang, Z-Q. Yu, Z. Fu, J. Miao, L. Huang, S. Chai, H.\nZhai and J. Zhang Phys. Rev. Lett. 109095301 (2012).\n[9] L. W. Cheuk, A. T. Sommer, Z. Hadzibabic, T. Yefsah,\nW. S. Bakr and M. W. Zwierlein Phys. Rev. Lett. 109\n095302 (2012).\n[10] R. A. Williams, M. C. Beeler, L. J. LeBlanc, K. Jimenez-\nGarcia, I. B. Spielman eprint.arXiv:1306.1965.\n[11] I.V.TokatlyandE. Y.Sherman, Phys.Rev.A 87041602\n(R) (2013).\n[12] T. D. Stanescu, C. Zhang and V. Galitski Phys. Rev.\nLett.99110403 (2007).\n[13] J. Larson, B. M. Anderson and A. Altland, Phys. Rev.\nA87013624 (2013).\n[14] B. A. Bernevig, J Orenstein, and S- C. Zhang, Phys. Rev.\nLett.97236601 (2006).\n[15] J. D. Koralek et al.Nature458610 (2009).\n[16] I. Bloch, J. Dalibard and W. Zwerger Rev. Mod. Phys.\n80885 (2008).\n[17] W. Ketterle and M. W. Zwierlein, Riv. Nuovo Cimento\n31247 (2008).\n[18] J. D. Sau, R. Sensarma, S. Powell, I.B. Spielman and S.Das Sarma Phys. Rev. B 83140510 (R) (2011).\n[19] C. Zhang, S. Tewari, R. Lutchynand S. Das Sarma Phys.\nRev. Lett. 101160401 (2008).\n[20] A. Kitaev, Ann. Phys. 3032 (2003).\n[21] E. P. Bashkin, JETP Lett. 338 (1981).\n[22] C. Lhuillier and F. Lalo¨ e, J. Phys (Paris) 43197 (1982);\nC. Lhuillier and F. Lalo¨ e, J. Phys (Paris) 43225 (1982).\n[23] W. J. Gully and W. J. Mullin, Phys. Rev. Lett. 521810\n(1984).\n[24] B. Castaing, Physica B 126(1984).\n[25] B. R. Johnson, J. S. Denker, N. Bigelow, L. P. Levy, J. H.\nFreed and D. M. Lee, Phys. Rev. Lett. 521508 (1984).\n[26] X. Du, L. Luo, B. Clancy and J. E. Thomas, Phys. Rev.\nLett.101150401 (2008).\n[27] H. J. Lewandowski, D. M. Harber, D. L. Whitaker and\nE. A. Cornell, Phys. Rev. Lett. 88070403 (2002).\n[28] S. S. Natu and E. J. Mueller, Phys. Rev. A 79051601(R)\n(2009).\n[29] U. Ebling, A. Eckardt and M. Lewenstein, Phys. Rev. A\n84063607 (2011).\n[30] F. Pi´ echon, J. N. Fuchs and F. Lalo¨ e, Phys. Rev. Lett.\n102215301 (2009).\n[31] M. ¨O. Oktel and L. S. Levitov, Phys. Rev. Lett. 88\n230403 (2002).\n[32] M. I. Dyakonov and V. I. Perel, Zh. Eksp. Teor. Fiz 60\n1954 (1971) [Sov. Phys. JETP 331053 (1971)].\n[33] J. Radic, S. S. Natu and V. Galitski, in preparation .\n[34]Quantum Statistical Mechanics , L. P. Kadanoff, and G.\nBaym, (W. A. Benjamin, Inc. New York 1962).\n[35] R. Wei and E. J. Mueller Phys. Rev. A 87042514 (2013).\n[36] Y. Shin, M. W. Zwierlein, C. H. Schunck, A. Schirotzek\nand W. Ketterle, Phys. Rev. Lett. 97030401 (2006).\n[37] L. E. Sadler, J. M. Higbie, S. R. Leslie, M. Vengalattore\nand D. M. Stamper-Kurn, Phys. Rev. Lett. 98110401\n(2007).\n[38] I. B. Spielman, Phys. Rev. A 79063613 (2009).\n[39] T-L. Ho, and S. Zhang, Phys. Rev. Lett. 107150403\n(2011)."    },    {        "title": "1008.2784v1.Transient_entanglement_in_a_spin_chain_stimulated_by_phase_pulses.pdf",        "content": "arXiv:1008.2784v1  [quant-ph]  16 Aug 2010Transient entanglement in a spin chain stimulated by phase p ulses.\nIsabel Sainz1,Gennadiy Burlak2, and Andrei B. Klimov1\n1Departamento de F´ ısica, Universidad de Guadalajara,\nRevoluci´ on 1500, Guadalajara, Jalisco, 44420, M´ exico. a nd\n2Centro de Investigaci´ on en Ingenier´ ıa y Ciencias Aplicad as,\nUniversidad Aut´ onoma del Estado de Morelos, Cuernavaca, M orelos, M´ exico\nDynamics of the one-dimensional open Ising chain under influ ence ofπ-pulses is studied. It is\nshown that the application of a specific sequence of such inst ant kicks to selective spins stimulates\narisingofperfect dynamicalpairwise entanglementbetwee nendsofthespinchain. Analyticformulas\nfor the concurrence dynamics are derived. It is also shown th at the time required to perfectly\nentangle the ends of the chains grows linearly with the numbe r of spins in the chain. The final\nentangled state of the ending spins is always the same and doe s not depend on length the chain.\nI. INTRODUCTION\nGeneration and distribution of entanglement between\nremote particles is a necessary requirement for quantum\ninformation transfer [1], [2]. It turns out that spin chains\nare one of the most promising quantum channels for cre-\nation of spin-spin entanglement [3]. It appears, that\nin spite of existence of correlations in spin chains, the\namount of entanglement generated between two particu-\nlar spins rapidly decays with the distance between them\n[3], [4] even near the point of quantum phase transition\n[5]. So that, the implementation of efficient mechanisms\nfor entangling spatially separated subsystems is still a\nchallenging problem, especially for massive particles.\nDuring the last years different types of spin-spin in-\nteractions have been extensively studied as possible can-\ndidates to entangle remote particles. It has been proved\nthat forcertain typesofspin-1 /2open Heisenbergchains,\nit is possible to entangle the ends of the chain by action\nofstationaryHamiltonians[6]andimplementingdifferent\nmechanisms like mirror symmetry [7], dimerized models\nor weak couplings at the ends of the chain [8], perform-\ningaquench[9], [10], orconsideringalternatingcouplings\nbetween the spins [11]. Also, several dynamical schemes,\nsuch as application of optimized time-dependent mag-\nnetic field [12] and periodically time-dependent nearest-\nneighbor coupling [13] have been proposed to faithfully\nentangle the ends of an open spin chain. Unfortunately,\nonly a partial (although a quite large) entanglement can\nbe achieved with these schemes. Besides, owing math-\nematical complexity of the problem, the main tool for\nstudying such systems remains numerical simulations.\nThe simplest type of interaction between massive par-\nticles is the homogeneous Ising chain. Being the simplest\ntype of interaction, it has been a candidate for imple-\nmentation of quantum information algorithms since the\nvery beginning [14]. It worth noting a recent proposal\nfor measurement-based quantum computation [15] using\nhomogeneous Ising-like interaction. Also, this kind of in-\nteraction accompanied by the global πpulses allows a\nperfect qubit transport, quantum mirrors and universal\nquantum computation [16].\nOn the other hand, the possibility of application ofpulses to quantum systems in order to preserve the co-\nherencewasdiscussedseveralyearsago[17], andrecently,\nthe idea of applying optimized πpulses at irregularinter-\nvals of time was proposed and studied by Uhrig [18] for\nefficient control of dephasing in spin systems and later\napplied for preserving entanglement in dephasing envi-\nronments [19].\nIn this article we propose a simple analytical scheme\nwhich allows to create a transient entanglement between\ntheendsofanhomogeneousspinchainbymeansofappli-\ncation of a sequence of πpulses on some particular spins.\nWe apply πpulses not to all qubits (as it was proposed\nin [16] to control a qubit transport) but only to a part of\nthem, which results in a perfect transient entanglement\nbetween the ends the chain.\nII. THE MODEL\nLet us consider an open chain of Nspins 1/2 with ho-\nmogeneous Ising-like interaction (the coupling constant\nis taken to be unity), governed by the Hamiltonian\nH=N/summationdisplay\nj=1ˆszjˆszj+1, (1)\nand initially prepared in the coherent superposition\n|Ψ0/angbracketright= ΠN\nj=1|+/angbracketrightj,|+/angbracketrightj=1√\n2/parenleftBig\n|0/angbracketrightj+|1/angbracketrightj/parenrightBig\n.\nSince|Ψ0/angbracketrightis not an eigenstate of the Hamiltonian\nEq.(1), some specific spin-spin correlations arise during\nthe Hamiltonian evolution, ˆU(t) =e−itH. The evolution\nof the density matrix ˆ ρcan be found in the following2\nclosed form,\nˆρ(t) =ˆU(t)|Ψ0/angbracketright/angbracketleftΨ0|ˆU†(t)\n=/parenleftbigg1\n2+cost\n2ˆsx1+2sint\n2ˆsy1ˆsz2/parenrightbigg\nN−1/productdisplay\nj=2/bracketleftbigg1\n2+cos2t\n2ˆsxj−4sin2t\n2ˆsxjˆszj−1ˆszj+1\n+2sint\n2cost\n2ˆsyj(ˆszj−1+ ˆszj+1)/bracketrightbigg\n/parenleftbigg1\n2+cost\n2ˆsxN+2sint\n2ˆsyNˆszN−1/parenrightbigg\n,(2)\nIn the Hamiltonian (1) only nearest neighbors interact,\nand a dynamical pairwise entanglement is generated be-\ntween them. As it can easily be seen from (2), the entan-\nglement dynamics between all the pairs in the middle of\nthe chain is the same, the concurrence [20], is given by\nCjj+1= max/bracketleftBigg\n0,|sint|\n2−sin2t\n2\n2/bracketrightBigg\n,(3)\nwherej= 2,...,N−2 and its maximum is ∼0.31. Since\nthe spin chain is open, the concurrences for the spins 1\nand 2, and the spins N−1 andNare different that given\nin Eq. (3),\nC12=CN−1N=|sint|\n2, (4)\nreaching the maximum of 1 /2 att= (2n+ 1)π/2,with\nn= 0,1,....\nIt is important to stress that non neighboring pairs\nof spins will never get entangled under the Hamiltonian\nevolution.\nIII. ENTANGLING THE ENDS OF THE CHAIN\nWITH π-PULSES\nIn order to generate entanglement between the ends\nof the chain let us apply instant πpulses to the first\nN−1 spins at certain times tj. Such kicks correspond\nto rotations around the y-axis, and for the j-th spin are\nrepresented by the operator ˆRj=e−iπˆsyj/2. The main\nidea is as follows: initially all the spins are in eigenstates\nof ˆsxjso that, in the course of the evolution governed\nby Eq.(1), some spins become correlated. At the instant\nt1=πwe apply a π-rotation to the first N−1 spins,\nproducing flip of those spins. Afterwards, the evolution\nundertheHamiltonianEq.(1)allowedtocontinueleading\nto the recoupling of all the spins. Then, at the instant\nt2= 2πwe apply the inverse π-rotation, allowing the\nsystem to evolve further and repeat this cycle until N−2\nspin flips areperformed at times tj=jπ,j= 1,2,...,N−\n2, so that\n|Ψ(t)/angbracketright=ˆU(t−π(N−2))ˆR(−1)N−1ˆU(π)...\n...ˆU(π)ˆR−1ˆU(π)ˆRˆU(π)|Ψ0/angbracketright,whereˆR=⊗ΠN−1\nj=1ˆRj. Using (5) we now proceed\nto compute the reduced density matrix and the (time-\ndependent) concurrence C(t) [20].\nThe three-spin case is special, since the middle spin is\ncoupled to both ends. For a three-spin chain a single π\npulse at t1=π, applied to the first and second spins,\nproduces entanglement between the ends. It is straight-\nforward to obtain the concurrence for the first and third\nspins:\nC13(t) =/braceleftbigg\ncos2t\n2, t≥π\n0, t < π,\ni.e. the first and third spins, which were not entangled\nbefore the pulse ( t < π), become completely entangled\natt= 2πk, k= 1,2,..., whileC12(t) =C23(t) = 0 for\nt≥π.\nFor longer chains the dynamical behavior of the pair-\nwise entanglement is rather unusual. After the first kick\nand before the ( N−2)-nd kick the pairwise entanglement\nbetween any pair of spins disappears. Since the system is\nclosed, the pairwise entanglement transforms into a mul-\ntipartite entanglement during that time. Moreover, after\nthe (N−2)−nd kickit is possibleto obtain the following\nanalytic expression for the concurrence between the ends\nof the chain:\na) for the even number of spins\nC(t) =/braceleftBigg\nmax/bracketleftBig\n0,|sint\n2|−cos2t\n2\n2/bracketrightBig\n, t≥(N−2)π\n0, t < (N−2)π,(5)\nand\nb) for the odd number of spins\nC(t) =/braceleftBigg\nmax/bracketleftBig\n0,|cost\n2|−sin2t\n2\n2/bracketrightBig\n, t≥(N−2)π\n0, t < (N−2)π,\n(6)\nwhile the nearest neighbors in the middle of the chain\nrecover their original transient entanglement Cjj+1(t) =\nmax[0,|sint|−sin2t/2]/2 for times t >(N−2)π.\nThus, the subsystem containing the first and the last\nspins of the chain stays in an unentangled state un-\ntil the last kick is applied. As a result the subsys-\ntem reaches a maximally entangled state at the mo-\nmentst= (N+2k−1)π, k= 0,1,..., being unen-\ntangled with the rest of the chain, in accordance with\nthe monogamy property [21] .So that, the time needed\nto completely entangle the ends of the chain linearly de-\npends on the length of the chain. In particular, at the\ninstantt= (N−1)πthe state of the chain is as follows:\n|Ψ/angbracketright=1\n2(N−1)/2N−1/productdisplay\nj=2/bracketleftBig\n|1/angbracketrightj+(−1)j|0/angbracketrightj/bracketrightBig\n[|00/angbracketright1N+|11/angbracketright1N+i(|01/angbracketright1N+|10/angbracketright1N)].\nOn the other hand, the concurrence between the spins\n1 and 2, and the spins N−1 andN, remains zero for any\nt > π.3\n0 5 10 15 20 25 3000.20.40.60.81\n(a)\n0 5 10 15 20 25 3000.20.40.60.81\n(b)t1t2\nt1t2t3t4t5\nt\nFIG. 1: (Color online) Dynamics of concurrence C(t) for the\nchain with (a) 4 spins for t1=π,t2= 2π; and (b) 7 spins\nfort1=π,t2= 2π,t3= 3π,t4= 4π,t5= 5π. Dashed line\nshowsTr(ρ2\n1N), arrows indicate when the kicked pulsed are\napplpied.\nIn Fig.(1) we show the dynamics of the concurrence\nC(t) for the chain with (a) 4 spins and π-pulses applied\nto the first 3 spins at tj=jπ,j= 1,2, and (b) 7 spins\nandπ-pulses applied to the first 6 spins at tj=jπ,\nj= 1,...5. The dashed line shows the purity of the of\nsubsystem composed by the first and the last spins of\nthe corresponding chain, Tr(ρ2\n1N). Observe that, due to\nthe Hamiltonian evolution, the concurrence and purity\nkeeps oscillating after application of the last kick.\nIn order to study the behavior of concurrence C(t)\nwhen the times tjof application of the π-pulses are not\ncommensurable, we have numerically calculated C(t) =\nC(t,t1,t2) as a function of t1andt2> t1at different\nfixed times. In Fig.2 we plot the concurrence C(t) for 4\nspins at time t= 3π(compare with Fig.1 (a)) when t1\n∈[0.1,5] andt2∈[5.1,9] . We can observe that C(t1,t2)\nhas a pronounced maximum, C= 1,att1=π, and\nt2= 2πand smoothly decreases when t1,2deviate from\nthese optimal values.\nIV. DISCUSSION\nWe have studied the dynamics a one-dimensional open\nIsing chain of Nspins assisted by phase kicks at times051015202530\n05101520253000.20.40.60.81 \nt2t1 \nFIG. 2: (Color online) The concurrence Cfor chain of 4 spins\nas a function of the moments of applied kicks at fixed time\nt= 3π.\ntj=jπ,j= 1,2,...,N−2. It is found that the ap-\nplication of N−2 instant pulses to N−1 spins leads\nto arising of transient perfect entanglement between the\nfirst and the last spins of the chain. One interesting re-\nsults is that, if the number of pulses is less than required,\nthen every pairwise concurrence would be zero. The pe-\nriodic behavior of the concurrence between the ends of\nthe chain, C1N(t) reaches the maximum value C= 1 at\nsome specific times even when the pulses are stopped be-\ning applied and the chain evolves under the Hamiltonian\n(1) only. This effect substantially enlarges possible ap-\nplications of the low-dimensional spin chains in quantum\ninformation technology.\nOn the other hand, in this particular model we may\nobserve an important example of a non-trivial effect of\nlocal transformations on the entanglement production in\nnon-linear systems.\nIt is easy to see that combining the present scheme\nwith quench one can entangle any two spins rands,\n1< r < s < N , in the chain, i.e. to model a quantum\nrouter. Really, we just have to “disconnect” the chain\n[10] from 1 to r−1 and from s+1 toNand then apply\nthe above discussed sequence of pulses.\nIt worth noting that similar local transformations in\nthe form of instant pulses were also used for another pur-\nposes: for atomic squeezing enhancement in the Dicke\nstates [22], for intensification of the entanglement in\ncontinuous-variables [23] and in two-spin [24] systems.\nThis work is partially supported by the Grant 106525\nof CONACyT (Mexico)4\n[1] M. Horodecki, P. Horodecki, and R. Horodecki, Phys.\nRev. A60, 1888 (1999).\n[2] A. Bayat and S. Bose, Phys. Rev. A 81, 012304 (2010).\n[3] L. Amico, R. Fazio, A. Osterloh and V. Vedral,\nRev.Mod.Phys. 80, 517 (2008).\n[4] S. Bose, Contemporary Physics 48, 13 (2007).\n[5] A. Osterloh, L. Amico, G. Falci, and R. Facio, Nature\n416, 608 (2002); T. J. Osborne, and M. A. Nielsen, Phys.\nRev. A66, 032110 (2002).\n[6] S. Bose, Phys. Rev. Lett. 91, 207901 (2003)\n[7] M. H. Yungand S.Bose, Phys.Rev. A 71, 032310 (2005).\n[8] L. CamposVenuti, C. DegliEspostiBoschi, and M.\nRoncaglia, Phys. Rev. Lett. 96, 247206 (2006); L. Cam-\nposVenuti, S. M. Giampaolo, F. Illuminati, and P. Za-\nnardi, Phys. Rev. A 76, 052328 (2007); A. Ferreira, J.\nM. B. L. dosSantos, Phys. Rev. A 77, 034301 (2008).\n[9] H. Wichterich and S. Bose, Phys. Rev. A 79, 060302(R)\n(2009).\n[10] A. Bayat, S. Bose, and P. Sodano, Preprint\narXiv:1007.4516 (2010).\n[11] C. Di Franco, M. Paternostro, and M. S. Kim, Phys.\nRev.A77, 020303(R) (2008).\n[12] X. Wang, A. Bayat, S. G. Schirmer, and S. Bose, Phys.\nRev. A81, 032312 (2010).\n[13] F. Galve, D. Zueco, S. Kohler, E. Lutz, P. H¨ anggi,\nPhys. Rev. A 79, 032332 (2009); F. Galve, D. Zueco,\nG. M. Reuther, S. Kohler, and P. H¨ anggi, Europ. Phys.\nJ: Special Topics 180, 237 (2009).\n[14] G. P. Berman, G. D. Doolen, G. D. Holm, andV. I. Tsifrinovich, Phys. Lett. A 193, 444 (1994);\nI. L. Chuang, N. Gershenfeld, and M. Kubinec,\nPhys. Rev. Lett. 80, 3408 (1998).\n[15] H. J. Briegel et al, Nature Physics 5, 19 (2009).\n[16] J. Fitzsimons and J. Twamley, Phys. Rev. Lett. 97,\n090502 (2006)\n[17] L. Viola, and S. Lloyd, Phys. Rev. A 58, 2733 (1998); M.\nBan, J. Mod. Opt. 45, 2315 (1998).\n[18] G. S. Uhrig, Phys. Rev. Lett. 98, 100504 (2007); K.\nKhodjasteh and D. A. Lidar, Phys. Rev. Lett. 95, 180501\n(2005); W. Yang and R. B. Liu, Phys. Rev. Lett. 101,\n180403 (2008); G. S. Uhrig, New J. Phys. 10, 083024\n(2008).\n[19] G.S. Agarwal, arXiv:0911.4158 (2009).\n[20] W. K. Wootters, Phys. Rev. Lett. 80, 2245 (1998).\n[21] V. Coffman, J. Kundu, and W. K. Wootters, Phys. Rev.\nA61, 052306 (2000); T. J. Osborne and F. Verstraete,\nPhys. Rev. Lett. 96, 220503 (2006).\n[22] D. Shindo, A. Chavez,S. M. Chumakov, and\nA. B. Klimov, J. Opt. B: Quamtum Semiclassical\nOpt.6, 34 (2004).\n[23] B. Kraus, K. Hammerer, G. Giedke, and J. I. Cirac,\nPhys. Rev. A 67, 042314 (2003).; K. Hammerer,\nK. Mølmer, E. S. Polzik, and J. I. Cirac, Phys. Rev. A\n70, 044304 (2004).\n[24] G. Burlak, I. Sainz and A.B. Klimov, Phys. Rev. A 80,\n024301 (2009)."    },    {        "title": "0909.3711v3.Engineering_ultralong_spin_coherence_in_two_dimensional_hole_systems_at_low_temperatures.pdf",        "content": "arXiv:0909.3711v3  [cond-mat.mes-hall]  12 Feb 2010Engineering ultralong spin coherence in\ntwo-dimensional hole systems at low temperatures\nT Korn, M Kugler, M Griesbeck, R Schulz, A Wagner, M\nHirmer, C Gerl, D Schuh, W Wegscheider ‡and C Sch¨ uller\nE-mail:tobias.korn@physik.uni-regensburg.de\nInstitut f¨ ur Experimentelle und Angewandte Physik, Universit¨ at Regensburg,\nD-93040 Regensburg, Germany\nAbstract. For the realisation of scalable solid-state quantum-bit systems, sp ins\nin semiconductor quantum dots are promising candidates. A key req uirement for\nquantum logic operations is a sufficiently long coherence time of the sp in system.\nRecently, hole spins in III-V-based quantum dots were discussed a s alternatives to\nelectron spins, since the hole spin, in contrast to the electron spin, is not affected by\ncontacthyperfine interactionwith the nuclearspins. Here, we rep orta breakthroughin\nthe spin coherence times of hole ensembles, confined in so called natu ral quantum dots,\nin narrow GaAs/AlGaAs quantum wells at temperatures below 500 mK. Consistently,\ntime-resolvedFaradayrotationand resonantspin amplification tec hniquesdeliver hole-\nspincoherencetimes, whichapproachinthe lowmagneticfieldlimit value sabove70ns.\nThe optical initialisation of the hole spin polarisation, as well as the inte rconnected\nelectron and hole spin dynamics in our samples are well reproduced us ing a rate\nequation model.\nPACS numbers: 78.67.De, 78.55.Cr\nSubmitted to: New J. Phys.\n‡present address: Solid State Physics Laboratory, ETH Zurich, 80 93 Zurich, SwitzerlandEngineering ultralong spin coherence in two-dimensional h ole systems 2\n1. Introduction\nAmongthemostpromisingsystems fortherealisationofquantumco mputingdevices are\nspins in semiconductor quantum dots (QDs) [1]. Using electrostatic g ating techniques,\nthese dots may be defined within a two-dimensional electron system (2DES) by local\ndepletion. This approach has the advantage that it allows for the fa brication of\nscalable arrays of quantum bits, as it is based on high-resolution litho graphy techniques\ninstead of self-organized growth of QDs. While the spin dephasing of electrons in high-\nmobility GaAs/AlGaAs-based 2DES is extremely fast - on the order of only a few\ntens of picoseconds [2, 3, 4] in the low-excitation limit - for electron s confined in QDs,\nspin dephasing is strongly reduced and the main remaining spin dephas ing channel is\ncontact hyperfine interaction with the nuclei [5]. The latter may be suppressed by using\nelaborate spin echo techniques [6]. Several alternative material sy stems like silicon [7]\nand graphene [8] have been suggested to overcome the problem of hyperfine interaction\nof electrons and nuclei. Recently, hole spins confined in QDs have bee n considered for\nquantum computing, as long hole spin dephasing times (SDT) were obs erved in p-doped\nself-organized QDs [9, 10]. Additionally, efficient electrical tuning of t he effective hole g\nfactorinlow-dimensional structureswaspredicted[11]andexper imentally demonstrated\n[12]. In the present work we demonstrate that very long hole SDTs c an be observed in\na two-dimensional hole system (2DHS), residing in a narrow quantum well (QW) in a\nGaAs/AlGaAs-based heterostructure. We show that the hole SDT strongly depends on\nthe energy splitting between the quantized heavy-hole (HH) and ligh t-hole (LH) states\nwithin the QW, which is controlled by the QW width. Specifically, the hole S DT can\nreach values above 70 ns at low temperatures and in small magnetic fi elds, which is\nabout two orders of magnitude longer than previously reported re sults on wider GaAs\nQWs [12, 14]. In particular, these long SDTs allow us to use resonant s pin amplification\n(RSA)techniques forprecisemeasurementsoftheholespindynam ics. Theresultsshown\nhere suggest that GaAs-based 2DHS may be a viable alternative to 2 DES for quantum\ncomputing applications, which rely on electrostatically confined char ged carriers.\n2. Sample design and experimental methods\nOur samples are single-side p-modulation-doped GaAs/Al 0.3Ga0.7As QWs containing\na 2DHS with relatively low hole density (see Table 1). Previous investiga tions on\nsimilar samples with relatively wide wells (15 nm and 10 nm) [12] showed th at at\nthese low densities, the optical recombination spectra at liquid-Heliu m temperatures\nare governed by recombinations of neutral and positively-charge d excitons, i.e., bound\nexcitonic complexes, consisting of one electron and two holes. Most importantly, even\nfor the wider QWs, at very low temperatures, the resident holes be come localised in\npotential fluctuations in the plane of the QW [12]. It was shown by Syp erek et al. [14]\nand Kugler et al. [12] that localisation of the holes is crucial for the ob servation of long\nhole SDTs on the order of a few-hundred picoseconds in those samp les. Here, we reportEngineering ultralong spin coherence in two-dimensional h ole systems 3\nTable 1. Sample data. Density and mobility were determined from\nmagnetotransport measurements at 1.3 K.\nSample QW width hole density phole mobility µelectron g\n(nm) (1011cm−2) (105cm2/Vs) factor |ge|\nA 15 0.90 5.0 0 .280±0.005\nB 9 1.03 3.6 0 .133±0.01\nC 7.5 1.10 5.3 0 .106±0.01\nD 4 1.10 0.13 0 .266±0.003\non hole SDTs in narrow QWs, which are up to two orders of magnitude lo nger.\nThe structures are grown by molecular-beam epitaxy (MBE) on [001 ] substrates.\nSome characteristic properties are listed in Table 1. The table clearly shows how the\nhole mobility is significantly reduced for the thinnest sample D, most like ly due to\nthe increased influence of monolayer fluctuations at the AlGaAs/Ga As interfaces on\nthe hole wave function within the QW. For time-resolved Faraday rot ation (TRFR)\nmeasurements, the samples are first glued onto a sapphire substr ate with optically\ntransparent glue, then the semiconductor substrate is removed by grinding and selective\nwet etching. All samples contain a short-period GaAs/AlGaAs super lattice, which\nserves as an etch stop, leaving only the MBE-grown layers. The TRF R and resonant\nspin amplification (RSA) measurements are performed in an optical c ryostat with3He\ninsert, allowing for sample temperatures below 400 mK and magnetic fi elds of up to\n11.5 Tesla. A pulsed Ti-Sapphire laser system generating pulses with a length of 600 fs\nand a spectral width of 3-4 meV is used for the optical measuremen ts. The repetition\nrate of the laser system is 82 MHz. The laser pulses are split into a circ ularly-polarised\npump beam and a linearly-polarised probe beam by a beam splitter. A me chanical\ndelay line is used to create a variable time delay between pump and prob e. Both\nbeams are focussed to a diameter of about 80 µm on the sample using an achromat,\nresulting in an excitation density of about 2 Wcm−2. The center wavelength of the\nlaser system is tuned to near-resonance with the HH excitonic abso rption lines of the\nsamples. For this, photoluminescence spectra, taken using nonre sonant excitation, are\nusedtodetermine thetransitionenergiesoftheneutralandchar gedexcitons[12]. Asthe\nspectral linewidth of our laser system exceeds the energy splitting between the charged\nandneutral exciton transitions, bothareexcited by the pump lase r pulses. In theTRFR\nand RSA experiments, the circularly-polarised pump beam is generat ing electron-hole\npairs in the QW, with spins aligned parallel or antiparallel to the beam dir ection, i.e.,\nthe QW normal. Due to the spectral linewidth of the laser, typically, b oth, neutral and\npositively chargedexcitons areexcited near-resonantly. IntheT RFRmeasurements, the\nspinpolarisationcreatedperpendiculartothesampleplanebythepu mpbeam, isprobed\nby the time-delayed probe beam via the Faraday effect: the axis of lin ear polarisation\nof the probe beam is tilted by a small angle [13], which is proportional to the out-of-\nplane component of the spin polarisation. This small angle is detected using an opticalEngineering ultralong spin coherence in two-dimensional h ole systems 4\nbridge. A lock-in scheme is used to increase sensitivity. In the RSA me asurements, the\nFaraday rotation angle is measured for a fixed time delay as a functio n of an applied\nin-plane magnetic field. In our investigations, we exploit the strengt hs of both methods\nin order to explore the limits of hole spin dynamics in GaAs-based hole sy stems: In the\nTRFR experiments, one measures in the time domain via a pump-probe scheme, and\nthe SDT as well as photocarrier lifetimes can be extracted from the measurements. The\ndisadvantage of this method, however, is the limitation of the acces sible time range by\nthe travel length of the optical delay line. In our setup, this limits th e time range to\nabout 2 ns. This hinders an accurate determination of SDTs, which a re significantly\nlonger than the measurement range. The TRFR method encounter s severe problems, if\nthe SDT is even longer than the time interval between two subseque nt laser pulses in\nthe pulse train of the mode-locked laser (in our case about 12 ns). F ortunately, with\nthe RSA technique [15], one can overcome these problems, since her e one works with a\nfixed time delay between pump and probe pulses. The method is based on the resonant\namplification of the Faraday signal, when integer multiples of the prec ession period of\nthe spins in an inplane magnetic field coincide with the inverse laser repe tition rate, i.e.,\nthe time interval between subsequent pulses. However, here the extraction of the SDT\nfrom the experimental data is more involved, since it is hidden in the line widths of the\nRSA maxima. As we will show below, in particular in our case, where elect ron and hole\nrecombination and spin dynamics are interconnected, this is rather complex.\n3. Quantum-well width dependence of hole spin-dephasing ti me\nWe start our discussion by presenting TRFR experiments on our sam ples as a function\nof an in-plane magnetic field. As discussed above, in these experimen ts we are detecting\nthe charge carrier and spin dynamics of photoexcited electrons an d holes within a time\nrange<2 ns. This will allow us to gain important information on the interconnec ted\nelectron and hole photocarrier and spin dynamics. Figure 1(a) show s a series of TRFR\ntraces measured on sample D, the sample with the thinnest QW, at 1.2 K at three\ndifferent magnetic fields, applied in the sample plane. The trace at B= 0 shows a\nsingle exponential decay of the Faraday signal after pulsed excita tion with a decay time\nofτR= 70 ps. Each of the other two traces, however, exhibits the supe rposition of\ntwo damped oscillations with markedly different frequencies and damp ing constants: A\nfast oscillation, which is observable within about the decay time of the B= 0 trace,\nonly, and a slower oscillation, which, in the B= 2 T trace, exceeds the measurement\nrange by far. The sum of two damped cosine functions is fit to the da ta in order to\nextract the precession frequencies and decay constants. As a r esult, we can identify\nthe high-frequency oscillations, which decay significantly faster, a s the precession of\nphotogeneratedelectrons, andthelow-frequency oscillationsas theprecession ofresident\nholes within the sample: In figure 1(b), the dispersions of the two pr ecession frequencies\nare plotted, and the electron and hole gfactors are extracted from the data. The\nelectron gfactor|ge|= 0.266 is in good agreement with values measured for QWs ofEngineering ultralong spin coherence in two-dimensional h ole systems 5\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s48 /s50 /s52 /s54 /s56 /s49/s48/s48/s53/s49/s48/s49/s53/s50/s48/s50/s53/s51/s48/s51/s53/s52/s48\n/s66/s61/s56/s32/s84\n/s66/s61/s50/s32/s84\n/s32/s32/s70/s97/s114/s97/s100/s97/s121/s32/s115/s105/s103/s110/s97/s108/s32/s40/s110/s111/s114/s109/s97/s108/s105/s122/s101/s100/s41\n/s116/s105/s109/s101/s32/s116/s32/s40/s110/s115/s41/s66/s61/s48/s32/s84/s40/s97/s41\n/s84/s61/s49/s46/s50/s32/s75/s40/s98/s41\n/s32/s32/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s71/s72/s122/s41\n/s77/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32/s40/s84/s41/s124/s103\n/s104/s124/s61/s48/s46/s48/s54/s53/s53/s124/s103\n/s101/s124/s61/s48/s46/s50/s54/s54\nFigure 1. TRFR for different in-plane magnetic fields and disp ersion of\nprecession frequencies. (a) TRFR traces for sample D, measured at 1.2 K for\ndifferent magnetic fields. (b) Dispersion of electron (black dots) an d hole (red stars)\nspin precession determined for sample D from TRFR measurements. The solid lines\nare linear fits to the data.\nsimilar widths [16, 17], while the hole gfactor|gh|= 0.0655 is close to zero, as was\ntheoretically predicted for GaAs-based QWs grown along the [001] d irection [18] and\nexperimentally observed [12, 14, 19].\nWe note that the decay constant of the electron spin precession, τe= 75±5 ps,\nremains almost constant throughout the investigated magnetic fie ld range. This may be\nexplained as follows: as our samples are p-doped, the optically orient ed electron spins\ncan only be observed during the photocarrier lifetime. The measure d decay constant\nτe= 75±5 ps therefore corresponds to the photocarrier lifetime in sample D ,τR, as\nthe electron SDT is typically longer. In stark contrast, the hole SDT decreases strongly\nwith increasing magnetic field, as can be seen quite drastically from th eB= 2 T and\nB= 8 T traces in figure 1(a). However, at this stage the important qu estion arises,\nwhy the trace at B= 0 in figure 1(a) obviously does not show a long lasting hole spin\nsignal but a spin signal, which is governed by the photocarrier lifetime , only? We will\naddress this question further below in the next section.\nTo investigate the strong magnetic field dependence of the hole SDT in more detail,\nwe study the hole spin dynamics as a function of the applied in-plane ma gnetic field for\nsamples with different QW widths. Figure 2(a) shows TRFR traces for all 4 samples,\nmeasured at 1.2 K with an applied in-plane magnetic field of 6 T. The trac es haveEngineering ultralong spin coherence in two-dimensional h ole systems 6\n0.0 0.2 0.4 0.6 0.8 1.0 0 2 4 6 8 100.1110\nSample D\nSample C\nSample B\n  Faraday signal (norm.)\ntime t (ns)(a)\nSample AT=1.2 K\n D\n C\n B\n A(b)\n  Hole SDT (ns)\nMagnetic field  (T)Sample\nFigure 2. TRFR for different well widths and magnetic-field de pendence\nof SDTs. (a) TRFR traces for samples A, B, C and D at 1.2 K. A 6 Tesla in-plane\nmagnetic field was applied during the measurements. (b) Hole spin dep hasing times\nfor samples A, B, C and D as a function of magnetic field. The lines repr esent fits of\na 1/B dependence.\nbeen scaled and vertically shifted in order to allow easy comparison of the hole spin\ndynamics. In all traces, again, a fast, rapidly decaying electron sp in precession can be\nobserved within the first ∼100 ps. After photocarrier recombination, only larger-period\nhole spin precession is visible in the traces. Even though the applied in- plane magnetic\nfield was fixed at 6 T, in all measurements shown in figure 2(a), it can b e seen clearly\nthat both, the electron and the hole spin precession frequencies, are different for all 4\nsamples. For electrons, it is well-known that the gfactor in a QW depends on the QW\nwidth [16, 17]: due to the nonparabolicity of the conduction band in Ga As, the electron\ng factor changes depending on the electron energy above the con duction band edge, and\nin good approximation, the QW confinement energy of an electron will lead to a similar\nchange of the g factor. Additionally, as the electron wave function has a nonvanishing\namplitude within the QW barriers for thin GaAs QWs, the different elect rongfactors\nof pure GaAs and the barrier material are admixed. The values of th e electron gfactors\nextracted from TRFR measurements on all four samples are listed in Table 1. We note\nthat the sign of the g factor cannot be directly determined from TR FR measurements,\nhowever, by comparing our measured values to literature data [16, 17], we conclude that\nthe electron g factor for samples A-C is negative, while it is positive fo r sample D.\nOn the other hand, the different hole gfactors observed in the TRFR traces stemEngineering ultralong spin coherence in two-dimensional h ole systems 7\nfrom the strong anisotropy of the hole gtensor, which was predicted by Winkler et al.\n[18] for [001]-grown GaAs QWs. The effective hole gfactor,g∗\nh, measured in TRFR is\ngiven by the geometric sum of the in-plane ( g⊥) and out-of-plane ( g||) components of\nthe hole gtensor:\ng∗\nh=/radicalBig\ng2\n⊥cos2α+g2\n||sin2α. (1)\nHere,αis the tilt angle of the magnetic field with respect to the QW plane. While\nthe in-plane component of the hole gfactor is close to zero, g⊥∼0, the out-of-plane\ncomponent is g||∼ −0.7 for bulk GaAs [20]. Therefore, even small tilt angles αresult\nin markedly different g∗\nh.\nIn figure2(b), thehole SDTs forall 4 samples areshown asa functio n of theapplied\nmagnetic field. Two effects are clearly visible here:\n(i) For all samples, the hole SDT decreases as the magnetic field is incr eased, following\napproximately a 1 /Bdependence.\n(ii) The hole SDT increases systematically as the QW width is reduced. I t is smallest\nfor sample A with the widest QW and largest for sample D with the thinne st QW.\nThe decrease of the hole SDT with magnetic field is a well-known phenom enon. It\nis believed to be caused by the inhomogeneity of the hole gfactor, ∆ g∗\nh, which leads to\na dephasing of the hole spin polarisation due to different precession f requencies. The\ndephasing rate due to this inhomogeneity is proportional to the app lied magnetic field.\nTherefore, the hole spin dephasing time observed in the experiment , which is the spin\ndephasing timeofaninhomogeneously broadenedensemble, T∗\n2, isinfirstapproximation\ngiven by [21]\nT∗\n2=/parenleftbigg1\nT2+∆g∗\nhµBB\n¯h/parenrightbigg−1\n, (2)\nif ∆g∗\nhis considered as the only source of inhomogeneity. Here, T2is the hole spin\ndephasing time in the absence of inhomogeneous broadening.\nIn order to understand the influence of the QW width on the hole SDT , we have\nto think about the main mechanisms for hole spin dephasing. The HH an d LH states\nhave different angular momenta, transitions between these state s will therefore destroy\nhole spin orientation. In bulk GaAs, where HH and LH valence bands ar e degenerate at\nk= 0, any momentum scattering may lead to a transition between HH an d LH states,\nwhich leads to hole SDTs on the order of the momentum scattering tim e [22]. In QWs,\nthis degeneracy is lifted. However, for k >0, the valence bands have a mixed HH/LH\ncharacter [23], which may also lead to rapid hole spin dephasing due to s cattering\n[24]. At low temperatures, resident holes in QWs may become localised a t potential\nfluctuations within the QW, arising from QW thickness fluctuations du e to monolayer\nsteps at the interfaces, as well as from the granular distribution o f the remote donors.\nLocalisationsignificantlyreduces theholequasimomentum, keeping r esident holesinHH\nstates with k∼0. However, even for k∼0, there is a finite admixture of the LH states\nto the first HH subband [25]. The significant increase in hole SDT with de creasing QWEngineering ultralong spin coherence in two-dimensional h ole systems 8\nwidth can therefore be attributed to an increased HH/LH splitting, which reduces the\nLH contribution to the HH ground state [26]. In a first approximatio n, if we consider\ninfinitely high square-well potentials, this energy splitting ∆ Eis proportional to 1 over\nthe well-width Lsquared:\n∆E∼/parenleftbigg1\nmLH−1\nmHH/parenrightbigg¯h2π2\n2L2. (3)\nFor narrower QWs of finite potential height, however, the hole wav e function penetrates\nstrongly into the barrier, and the energy splitting is reduced again. For a QW width of\n4 nm, the maximum HH/LH splitting was theoretically predicted and exp erimentally\nobserved [27]. As figure 2(b) shows, we indeed observe the longest hole SDT in sample\nD with a well width of 4 nm. This suggests that the maximum HH/LH splitt ing in\nthis sample is responsible for the long SDT. We will discuss the possible a lternative\nmechanisms, which might govern hole spin dephasing in the last section . Before, we will\nin the following explore the limits of hole spin dephasing in our samples, st arting with\nthe optical initialisation process of hole spin polarisation in the next se ction.\n4. Buildup of a resident hole spin polarisation\nIn this section, we present experimental results and simulation dat a concerning the\ninitialization process of the hole spin polarisation, which was described by Syperek et\nal. [14]: excitation of the sample with circularly-polarised light will creat e spin-polarised\nelectron-hole pairs according to the optical selection rules. If the re is no significant\nelectron spin dephasing during the photocarrier lifetime, the optica lly oriented electrons\nwill recombine with holes which have matching spin orientation. Theref ore, no spin\npolarisation will remain within the sample after photocarrier recombin ation. This\nprocess is sketched schematically in the left panel of figure 3(c).\nIn an applied in-plane magnetic field, however, electrons and holes pr ecess with\ndifferent precession frequencies due to their strongly different gfactors. Therefore,\nduring their photocarrier lifetime, the electron spins will recombine w ith holes with\narbitrary spin orientation. A part of the optically oriented holes may therefore remain\nwithin the sample after photocarrier recombination, as depicted in t he right panel of\nfigure 3(c).\nIn order to gain deeper insight into the combined dynamics of electro n and hole\nspins, and to analyse our experimental results quantitatively, we s et up a model in the\nfollowing. The combined dynamics of the electron and hole spins can be described via\ncoupled differential equations for the electron ( e) and hole ( h) spin polarisation vectors:\nde\ndt=−e\nτR+geµB\n¯h(B×e) (4)\ndh\ndt=−h\nτh+ghµB\n¯h(B×h)+ezz\nτR(5)\nIn our model we assume the following: the electron spin polarisation is reduced at a\nrate, given by thephotocarrier recombination andprecesses abo utthe in-planemagneticEngineering ultralong spin coherence in two-dimensional h ole systems 9\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48\n/s32/s32/s70/s97/s114/s97/s100/s97/s121/s32/s115/s105/s103/s110/s97/s108/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s116/s105/s109/s101/s32/s40/s110/s115/s41/s32/s32/s66/s61/s54/s32/s84\n/s32/s32/s66/s61/s51/s32/s84\n/s32/s32/s66/s61/s48/s32/s84\n/s102/s105/s101/s108/s100/s32/s115/s119/s101/s101/s112/s84/s61/s49/s46/s50/s32/s75\n/s48 /s50 /s52 /s54/s48/s49/s40/s98/s41/s32\n/s32/s72/s111/s108/s101/s32/s115/s112/s105/s110/s32/s115/s105/s103/s110/s97/s108/s32/s64/s32/s48/s46/s52/s32/s110/s115/s32/s40/s110/s111/s114/s109/s46/s41/s32\n/s77/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32/s40/s84/s101/s115/s108/s97/s41/s32/s70/s97/s114/s97/s100/s97/s121/s32/s115/s105/s103/s110/s97/s108\n/s32/s82/s97/s116/s101/s32/s101/s113/s110/s46/s32/s109/s111/s100/s101/s108/s40/s97/s41\nh+e-Optically□oriented\nelectron-hole□pairsResident\nholesNo□magnetic□field In-plane□magnetic□field\nh+e-\nh+e-\nh+e-recombinationprecessiontime(c)\nFigure 3. TRFR for different magnetic fields and buildup of hol e spin\npolarisation. (a) TRFR traces for sample C taken at 1.2 K for different magnetic\nfields. The arrow indicates the delay position for the magnetic field sw eep shown in\n(b). (b) Kerr signal (black dots) as a function of magnetic field tak en at a fixed time\ndelay (400 ps) between pump and probe pulses. The orange line show s the buildup\nof the hole spin polarisation as a function of magnetic field as calculate d by the rate\nequations model. (c) Diagram of the combined spin and recombination dynamics with\n(right panel) and without an applied magnetic field (left panel).Engineering ultralong spin coherence in two-dimensional h ole systems 10\n-1.0 -0.5 0.0 0.5 1.0-0.1 0.0 0.1 0.2 0.3 0.4\n0.1 1 1011010020x\nSimulationT=0.4 KT=1.2 K\n  RSA signal (arb. units)\nMagnetic field (T)T=4.5 K(a)   0.4 K Experiment\n  Simulation\n  RSA signal (arb. units)\nMagnetic field (T)(b)n = 1234\n 0.4 K\n 1.2 K\n 4.5 KTRFR data\n  Hole SDT (ns)\nMagnetic field (T)(c)RSA data\nFigure 4. RSA measurements for different temperatures and ma gnetic field\ndependence of hole SDT for different temperatures. (a) RSA traces for sample\nD, measured at different temperatures, compared to simulation da ta. (b) RSA trace\nmeasured at 400 mK (open circles) compared to simulation data (ora nge line). (c)\nHole SDT, determined from RSA data (left of the dotted line and TRFR data (right\nof the dotted line), for different temperatures as a function of ma gnetic field.\nfield vector B. Electron spin dephasing has been neglected, as the electron SDT is\nexpected to be significantly longer than the photocarrier recombin ation time τR, which\nis a reasonable assumption. The hole spin polarisation is reduced at a r ate given by the\nhole SDT τhand precesses about the in-plane magnetic field vector B. The last term in\nthe second equation describes the change of hole spin polarisation d ue to recombination\nof spin-polarised electrons with holes with matching spin orientation. Here,ezis thez\ncomponent of the electron spin polarisation, and zis the unit vector along the growth\ndirection. Asimilar approachwasused by Yugovaet al. [28] todescrib e theinitialization\nof a resident electronspin polarisation in n-doped QWs. However, we need to include,\nboth, electron and hole spin precession (second term in equation (5 )) in our differential\nequations to correctly model the resonant spin amplification spect ra we observe, as\ndescribed in the section below. In order to test the validity of this mo del, we compare it\nto experimental results. Figure 3(a) shows TRFR traces of sample C, taken at different\napplied in-plane fields. The sample has been carefully aligned so that th e magnetic field\nis applied along the QW plane, i.e., α∼0, resulting in a very low effective hole gfactor\ng∗\nh. For zero magnetic field, the TRFR signal decays monoexponentially with a decay\nconstant of τR= 48 ps, which reflects the photocarrier recombination time in sample C.\nFor larger magnetic fields, pronounced electron spin precession ca n be observed during\nτR, and a significant nonzero TRFR signal remains after photocarrier recombination.Engineering ultralong spin coherence in two-dimensional h ole systems 11\nThis signal is due to the buildup of a resident hole spin polarisation, as e xplained above.\nIts amplitude increases as the magnetic field is increased. Due to the very small hole\ngfactor (|g∗\nh|<0.005), no hole spin precession is observed in the time range shown\nin figure 3(a). This allows us to study the buildup of the hole spin polaris ation in\nmore detail using the following experimental technique: the TRFR sig nal is recorded\nfor a fixed time delay between pump and probe. The time delay is chose n such that\nphotocarrier recombination is complete, thereby leaving a TRFR sign al from resident\nholes, only. The magnetic field is ramped up from zero in small incremen ts. Figure 3(b)\nshows such a measurement for a time delay of ∆ t= 400 ps. It is clearly visible how\nthe resident hole spin polarisation increases from zero to a maximum v alue, at which\nit saturates. This buildup can easily be modeled by the differential equ ation system\ndescribed above. For this, we used the parameters extracted fr om the experiment:\n|ge|= 0.118,τR= 48 ps, g∗\nh= 0. Hole spin dephasing was neglected to model the\ndataset. The resulting hole spin polarisation as a function of magnet ic field is shown as\nan orange solid line in figure 3(b). It is in excellent agreement with the e xperimental\ndata. These measurements clearly demonstrate the crucial role o f an applied in-plane\nmagnetic field for transferring spin polarisation from optically orient ed photocarriers to\nresident holes.\n5. Resonant spin amplification measurements of hole spin dyn amics\nTo explore the hole SDT in very small applied magnetic fields, and to app roach the\nultimate limit of hole SDT in our samples, we employ the RSA technique [15 ], which\nhasbeenusedinrecentyears, e.g.,tostudyelectronspindynamics inn-dopedbulkGaAs\n[29] and 2DES in CdTe-based QWs [30]. RSA is based on the constructiv e interference\nof spin polarisations created by subsequent laser pulses in a time-re solved Faraday or\nKerr rotation measurement. The Faraday/Kerr signal is measure d as a function of the\nin-plane magnetic field for a fixed time delay ∆ t, typically before the arrival of a pump\npulse. If the optically oriented spins precess by an integer multiple of 2πwithin the\ntime delay between two pump pulses, a maximum in the RSA signal is obse rved. The\nmaxima are typically periodic in B. The SDT can be determined from the half-width\nof the maxima, and their spacing yields the gfactor. In systems where the buildup\nof a resident spin polarisation is governed by the interplay of electro n and hole spin\ndynamics, however, a more complex shape of the RSA signal can be e xpected, as will\nbe demonstrated now.\nFigure 4(a) shows a series of RSA traces, measured on sample D for different\ntemperatures, compared to a numeric simulation, using the coupled differential equation\nmodel introduced above. The unusual ’butterfly’ shape [28] of t he RSA signals is\nclearly visible for the traces measured at 1.2 K and 0.4 K. It is well-repr oduced by\nthe simulation data. The RSA signal at 4.5 K is about 20 times weaker th an the\nlower-temperature signals. In figure 4(b), a closeup of the first f ew RSA maxima\nmeasured at 0.4 K is compared to simulation data. There is no RSA maxim um inEngineering ultralong spin coherence in two-dimensional h ole systems 12\nthe measurement and the simulation for B= 0, in contrast to typical RSA curves\nmeasured, e.g., in n-doped GaAs bulk. This is due to the peculiar trans fer process\nthat leads to a resident hole spin polarisation. As demonstrated abo ve, a finite in-plane\nmagneticfieldisnecessarytocreatearesident holespinpolarisation . Thefirstmaximum\nat finite field (numbered as n= 1) has a distinct shape resembling the derivative of a\nLorentzian, while subsequent maxima resemble slightly asymmetrical Lorentzian curves.\nThe amplitude of the maxima first increases with n, then decreases again for n >5 (see\nfigure 4(a)), while the FWHM of the maxima increases. These featur es are clearly\nreproduced in the simulation. They can be explained as follows: for sm all magnetic\nfields, there is only a partial transfer of spin polarisation to the res ident holes, a process\nwhich saturates in our experiment for B∼0.5 T. For larger magnetic fields, the RSA\nFWHM increases due to inhomogeneous broadening, and the amplitud es of the RSA\nmaxima decrease accordingly, until the RSA signal drops to the nois e level at about\n1.6 T in the low-temperature measurements. Interestingly, direct comparison of the\nRSA traces measured at 1.2 K and 4.5 K reveals that the effective hole g factor |g∗\nh|\ndecreases from 0.067 to 0.050 as the temperature is increased, in g ood agreement with\nprevious observations [14].\nFigure 4(c) shows the hole spin lifetime insample D for different temper atures. The\ndata for low magnetic fields (below 1 T) have been determined from RS A data, the data\nfor higher magnetic fields (above 2 T) were determined from TRFR da ta. Interestingly,\nwhile at 4.5 K, the hole SDT saturates at about 2.5 ns, even in low magne tic fields,\nit follows a clear 1 /B-like dependence down to very small magnetic fields in the data\nmeasured at lower temperatures, yielding values of T∗\n2= 74±15 ns at 0.4 K and\nT∗\n2= 61±11 ns at 1.2 K (determined from the FWHM of the second RSA maximum a t\nB∼0.2 T) [36]. The power of the RSA technique is evident if one compares th e hole\nSDT measured at higher fields: in the magnetic field range where TRFR measurements\nyield precise results of the hole SDT, very little difference is observed in the values of\nthe hole SDT in the temperature range from 0.4 K to 4.5 K. From the ma gnetic field\ndependence oftheholeSDT, wemayinferthatthe T2timeoftheholespinisabove80ns\nat temperatures below 500 mK. By fitting equation 2 to the magnetic field dependence,\nwe determine the g factor inhomogeneity ∆ g∗\nh= 0.003±0.0002.\nFurthermore, the RSA measurements allow for a precise determina tion of the\neffective hole gfactor, which is given by g∗\nh= 2πfrep¯h/(µB∆B). Here, frepis the laser\npulse repetition frequency, and ∆ Bis the magnetic field spacing between two adjacent\nRSA maxima. Figure 5 shows RSA traces measured at 1.2 K for differen t angles α\nof the magnetic field with respect to the QW plane [35]. The geometry is sketched in\nthe figure. The spacing of the maxima is significantly reduced as the m agnetic field\nacquires an out-of-plane component. The effective hole gfactor|g∗\nh|is extracted from\nthis spacing, the results are shown in the inset. The increase of the effective hole gfactor\nwith the magnetic field angle is due to an admixture of the out-of-plan e component of\nthe hole gfactorg||, which is typically far larger than the in-plane hole gfactorg⊥,\nas described by equation 1. By fitting the results with this equation a s indicated byEngineering ultralong spin coherence in two-dimensional h ole systems 13\n0.0 0.1 0.2 0.3 0.4 0.5-10 -5 0 5 100.050.100.15\nα\n  RSA signal (norm.)\nMagnetic field (T)0o4.5o\n3oB\nT=1.2 K\n  eff. hole g factorTilt angle α (deg.)\nFigure 5. RSA measurements for different tilt angles of the ma gnetic field\nand effective hole g factor dependence on tilt angle. RSA traces for sample D,\nmeasured at 1.2 K, for different tilt angles of the sample with respect to the external\nmagnetic field. The inset shows the effective hole g factor |g∗\nh|determined from the\nspacing of the RSA maxima (black dots). The solid red line represents a fit of the\nresults using equation 1.\nthe solid red line in the inset, both components of the hole g factor at 1.2 K can be\ndetermined with high accuracy: |g⊥|= 0.059±0.003,|g|||= 0.89±0.03.\nFinally, we will discuss the possible mechanisms, which might limit the hole S DT\nin our experiments. At low temperatures and in low magnetic fields, wh ere thegfactor\ninhomogeneity may be neglected, in principle several mechanisms migh t be thought\nof to pose the ultimate limit of the hole SDT. In the following, we will discu ss their\ndependence on QW thickness in the light of our experiments, and iden tify the most\nimportant mechanism.\nThermal activation of holes from localized states: The thermal activation rate is\nproportional to the Boltzmann factor exp/parenleftBig\n(−ELoc)/(kBT)/parenrightBig\n. For localization of holes at\nQW monolayer thickness fluctuations, the localization energy increa ses drastically with\nareductionoftheQWwidth[25]. Fora15nmwideQW,itislessthan1meV, whilefora\n4 nm wide QW, it is about 10 meV. In both cases, the localization energy is significantly\nlargerthanthethermalenergy, evenatthehighestmeasuremen ttemperatureusedinour\nexperiments: Eth(4.2 K)=0.36 meV. This indicates that thermal activation of holes outEngineering ultralong spin coherence in two-dimensional h ole systems 14\nof localized states is unlikely to affect hole spin dynamics at liquid-helium t emperatures\nand below.\nDipole-Dipole interaction of holes with nuclei: Even though the p-like symmetry of\nvalence band states does not allow for a contacthyperfine interaction between holes\nand nuclei, a coupling is possible via dipolar interactions [31, 10]. The nuc lei act as\nslowly varying random magnetic fields, which lead to both, single spin de coherence\nand ensemble spin dephasing. The variance of the nuclear magnetic fi eld is inversely\nproportional to thenumber of nuclei that interact with the hole co nfined in the quantum\ndot, which maybe estimated fromthe dot size. Wenotethat thetyp ical size of quantum\ndots which arise from monolayer thickness fluctuations in high-qualit y GaAs QWs (TF-\nQDs) is on the order of (100 nm)2, significantly larger than that of self-assembled InAs\ndots. One may estimate that a hole interacts with about 106nuclei in TF-QDs or\nelectrostatically controlled QDs [6], whereas self-assembled dots ty pically contain only\nabout 5·104nuclei [10]. Additionally, the random magnetic fields associated with th e\nnuclei may be suppressed very efficiently by a small, external magne tic field applied\nwithin the sample plane [31]. It is shown above that the application of an in-plane\nmagnetic field is necessary for the transfer of spin polarisation of o ptically oriented\ncarriers to resident holes. Therefore, we may neglect hole spin dec oherence due to\ninteractionwithnucleiasthedominantprocess, limitingtheholeSDTin ourexperiment.\nFinite admixture of LH states to HH states: Even localized holes have a finite\nquasimomentum, which is given by the hole temperature. In an analog on to the Elliott-\nYafet [32, 33] mechanism, which has been studied in detail for electr ons, any momentum\nscattering may therefore destroy hole spin orientation via hole spin flip due to the slight\nadmixtureofLHstatestoHHstates. Thisadmixtureisontheorder of2percent forthin\nQWs [25]. It increases significantly for wider QWs due to the decreasin g HH/LH energy\nsplitting, increasing the probability of a hole spin flip during momentum s cattering [34].\nThespin dephasing rateisdirectly proportionaltothemomentum sc attering rate, which\nin turn increases with the sample temperature.\nSummarizing this part: We believe that the increased HH/LH splitting is\nresponsible for the increased hole SDT with decreasing QW width in our experiments.\nOn the other hand, the still finite LH admixture to the HH ground sta te and, hence,\nthe possibility of spinflip scattering of Elliott-Yafet type sets the limit for hole spin\ndephasing.\nConclusions\nIn conclusion, we have investigated hole spin dynamics in two-dimensio nal hole systems\nembedded in quantum wells of different width. Due to the increased en ergy splitting\nbetween heavy- and light-hole states, the hole spin dephasing time in creases drastically\nin narrow quantum wells. The inhomogeneity ∆ g∗\nhof the hole gfactor leads to aEngineering ultralong spin coherence in two-dimensional h ole systems 15\ntypical 1/B-like dependence of the hole SDT. In order to transfer spin polarisa tion from\nthe optically oriented photocarriers to the 2DHS, a finite magnetic fi eld is necessary.\nThis also leads to a peculiar shape of resonant spin amplification signals measured on\n2DHS. From these RSA signals, the hole SDT in low magnetic fields can be extracted.\nIt is on the order of 80 ns, one order of magnitude larger than for e lectrons in\nquantum dots of similar dimensions defined in 2DES by external gates [6]. This makes\n2DHS an interesting material system for scalable quantum computin g devices based\non electrostatically confined charge carriers. Additionally, RSA mea surements in tilted\nmagnetic fields allow us to accurately determine both, the in-plane an d out-of plane\ncomponents of the hole g factor.\nAcknowledgments\nThe authors would like to thank E.L. Ivchenko, M.M. Glazov and M.W. Wu for fruitful\ndiscussion. Financial support by the DFG via SPP 1285 and SFB 689 is g ratefully\nacknowledged.\nReferences\n[1] Awschalom D D, Loss D and Samarth N 2002 Semiconductor Spintronics and Quantum\nComputation (Berlin: Springer)\n[2] Brand M A, Malinowski A, Karimov O Z, Marsden P A, Harley R T, Shield s A J, Sanvitto D,\nRitchie D A and Simmons M Y 2002 Phys. Rev. Lett. 89236601\n[3] Stich D, Zhou J, Korn T, Schulz R, Schuh D, Wegscheider W, Wu M W a nd Sch¨ uller C 2007 Phys.\nRev. Lett. 98176401\n[4] Stich D, Zhou J, Korn T, Schulz R, Schuh D, Wegscheider W, Wu M W a nd Sch¨ uller C 2007 Phys.\nRev.B76073309\n[5] Khaetskii A V, Loss D and Glazman L 2002 Phys. Rev. Lett. 88186802\n[6] Petta J R, Johnson A C, Taylor J M, Laird E A, Yacoby A, Lukin M D, M arcus C M, Hanson M\nP and Gossard A C 2005 Science 3092180\n[7] Eriksson M A, Friesen M, Coppersmith S N, Joynt R, Klein L J, Slinker K, Tahan C, Mooney P\nM, Chu J O and Koester S J 2004 Quantum Information Processing 3133\n[8] Trauzettel B, Bulaev D V, Loss D and Burkard G 2007 Nature Phy s.3192\n[9] Heiss D, Schaeck S, Huebl H, Bichler M, Abstreiter G, Finley J, Bula ev D V and Loss D 2007 Phys.\nRev.B76241306\n[10] Eble B, Testelin C, Desfonds P, Bernardot F, Balocchi A, Amand T, Miard A, Lemaˆ ıtre A, Marie\nX and Chamarro M 2009 Phys. Rev. Lett. 102146601\n[11] Andlauer T and Vogl P 2009 Phys. Rev. B79045307\n[12] Kugler M, Andlauer T, Korn T, Wagner A, Fehringer S, Schulz R, K ubova M, Gerl C, Schuh D,\nWegscheider W, Vogl P and Sch¨ uller C 2009 Phys. Rev. B80035325\n[13] forthesamplesinvestigatedhere, themaximumTRFRsignalcor respondstoatilt angleof70 µRad\n[14] Syperek M, Yakovlev D R, Greilich A, Misiewicz J, Bayer M, Reuter D and Wieck A D 2007 Phys.\nRev. Lett. 99187401\n[15] Kikkawa J M and Awschalom D D, Phys. Rev. Lett. 801998 4313\n[16] Snelling M J, Flinn G P, Plaut A S, Harley R T, Tropper A C, Eccleston R and Phillips C C 1991\nPhys. Rev. B4411345\n[17] Yugova I A, Greilich A, Yakovlev D R,Kiselev A A,Bayer M,Petrov V V,Do lgikh Yu K,Reuter D\nand Wieck A D 2007 Phys. Rev. B75245302Engineering ultralong spin coherence in two-dimensional h ole systems 16\n[18] Winkler R, Papadakis S J, De Poortere E P and Shayegan M 2001 Phys. Rev. Lett. 854574\n[19] Marie X, Amand T, Le Jeune P, Paillard M, Renucci P, Golub L E, Dym nikov V D and Ivchenko\nE L 1991 Phys. Rev. B605811\n[20] Landolt-B¨ ornstein 1987 Semiconductors: Intrinsic Properties of Group IV Elements and III-V, II-\nVI and I-VII Compounds New Series Group III Vol. 22Pt. A ed. O. Madelung (Berlin: Springer)\n[21] YakovlevD R and BayerM 2008 Coherent Spin Dynamics of carriers, chapter 6 in Spin Physic s in\nSemiconductors Springer Series in Solid-State Sciences 157ed. Dyakonov M I (Berlin: Springer)\n[22] Hilton D J and Tang C L 2002 Phys. Rev. Lett. 89146601\n[23] Pfalz S, Winkler R, Nowitzki T, Reuter D, Wieck A D, H¨ agele D and O estreich M 2005 Phys. Rev.\nB71165305\n[24] Damen T C, Vi˜ na L, Cunningham J E, Shah J E and Sham L J 1991 Phys. Rev. Lett. 673432\n[25] Luo J-W, Bester G and Zunger A 2009 Phys. Rev. B79125329\n[26] Winkler R 2003 Spin-orbit Coupling Effects in 2D Electron and Hole Systems (Berlin: Springer)\n[27] El Khalifi Y, Gil B, Mathieu H, Fukunaga T and Nakashima H 1989 Phys. Rev. B3913533\n[28] Yugova I A, Sokolova A A, Yakovlev D R, Greilich A, Reuter D, Wieck A D and Bayer M 2009\nPhys. Rev. Lett. 102167402\n[29] Kikkawa J J and Awschalom D D 1999 Nature 397, 139\n[30] Zhukov E A, Yakovlev D R, Bayer M, Karczewski G, Wojtowicz T a nd Kossut J 2006 Phys. Status\nSolidi(b)243878\n[31] Fischer J, Coish W A, Bulaev D V and Loss D 2008 Phys. Rev. B78155329\n[32] Elliott R J 1954 Phys. Rev. 96266\n[33] Yafet Y 1963 Solid State Phys. 141\n[34] L¨ u C, Cheng J L and Wu M W 2005 Phys. Rev. B71075308\n[35] The sample tilt angle is determined by measuring the angle of the pu mp beam which is reflected\nfrom the sample surface.\n[36] The hole SDT is proportional to 1 /(δB2) for all RSA maxima with n >1. Here, δBis the full\nwidth at half maximum (FWHM) of the RSA maximum."    },    {        "title": "2402.04719v2.Quantum_Theory_of_Spin_Transfer_and_Spin_Pumping_in_Collinear_Antiferromagnets_and_Ferrimagnets.pdf",        "content": "Quantum Theory of Spin-Transfer and Spin-Pumping in Collinear Antiferromagnets\nand Ferrimagnets\nHans Gløckner Giil and Arne Brataas\nCenter for Quantum Spintronics, Department of Physics,\nNorwegian University of Science and Technology, NO-7491 Trondheim, Norway\n(Dated: April 12, 2024)\nAntiferromagnets are promising candidates as active components in spintronic applications. They\nshare features with ferrimagnets in that opposing spin orientations exist in two or more sublattices.\nSpin transfer torque and spin pumping are essential ingredients in antiferromagnetic and ferrimag-\nnet spintronics. This paper develops an out-of-equilibrium quantum theory of the spin dynamics of\ncollinear magnets containing many spins coupled to normal metal reservoirs. At equilibrium, the\nspins are parallel or antiparallel to the easy axis. The theory, therefore, covers collinear antiferro-\nmagnets and ferrimagnets. We focus on the resulting semi-classical spin dynamics. The dissipation\nin the spin dynamics is enhanced due to spin-pumping. Spin accumulations in the normal metals\ninduce deterministic spin-transfer torques on the magnet. Additionally, each electron’s discrete spin\nangular momentum causes stochastic fluctuating torques on the antiferromagnet or ferrimagnet. We\nderive these fluctuating torques. The fluctuation-dissipation theorem holds at high temperatures,\nincluding the effects of spin-pumping. At low temperatures, we derive shot noise contributions to\nthe fluctuations.\nI. INTRODUCTION\nSpin transfer torque (STT) and spin pumping (SP)\nare essential ingredients in the generation and detection\nof spin currents and are central components in modern\nspintronics research and devices [1]. The use of mag-\nnetic insulators enables signal propagation without mov-\ning charges and could provide low-dissipation and ultra-\nfast memory devices [2]. Initially, much of spintronic\nresearch focused on the study of STT [3–5] and SP [6–\n8] in ferromagnets (FMs). Subsequently, this included\nalso works on fluctuations [9–11] and pumped magnon\ncondensates [12–14].\nUnlike FMs, whose macroscopically apparent magnetic\nproperties have been known for thousands of years, anti-\nferromagnets (AFMs) carry zero net magnetic moments\nand were elusive for some time. Even after their dis-\ncovery, AFMs were believed to have few potential ap-\nplications [15] and were disregarded in the early days\nof spintronics research. Recent theoretical and experi-\nmental findings have highlighted the potential of using\nAFMs in spintronics applications, thus starting the field\nof antiferromagnetic spintronics. Key discoveries were\nthe robustness of AFMs to external magnetic perturba-\ntions and the high resonance frequency of antiferromag-\nnetic material [16, 17]. The prediction [18] and subse-\nquent experimental detection [19] of an STT in AFMs\nsparked a massive interest in using AFMs as the active\ncomponent in spintronics devices [16, 20]. Moreover, it\nwas predicted that contrary to what was believed, anti-\nferromagnets are as efficient in pumping spin currents as\nFMs [21]. This effect was later experimentally detected in\nthe easy-axis AFM MnF 2[22]. These discoveries opened\nup the possibility of utilizing AFMs in spintronic applica-\ntions, enabling the possible fabrication of stray-field-free\ndevices operating in the THz-regime [16, 23], allowing for\nmuch faster device operation than in FMs.In recent years, the spin dynamics in AFMs have been\nexplored extensively, including the effects of disorder [24],\ngeneration of spin-Hall voltages [25], and the proper-\nties of antiferromagnetic skyrmions [26]. The spin dy-\nnamics of ferrimagnetic materials have also been stud-\nied [27]. Phenomenological models of intra- and cross-\nlattice torques were introduced in [28]. Ref. 29 fur-\nther discusses the competition between intra and cross-\nsublattice spin pumping in specific models of antiferro-\nmagnets.\nAs in antiferromagnets, ferrimagnets have opposing\nmagnetic moments. However, these moments have differ-\nent magnitudes, resulting in a net magnetization. These\nfeatures result in rich spin dynamics ranging from be-\nhavior reminiscent of antiferromagnets to ferromagnets.\nA prime example of a ferrimagnet is yttrium-iron-garnet\n(YIG). The low-energy magnon modes in YIG resemble\nmodes in ferromagnets.\nIn the study of non-equilibrium effects, the Keldysh\npath integral approach to non-equilibrium quantum field\ntheory is a powerful tool in the study of non-equilibrium\nsystems beyond linear response [30, 31]. Although most\nof the research on STT and SP utilized a semiclassical\napproach, some works have used the Keldysh framework\nin the study of spin dynamics in FMs out of equilib-\nrium [10, 11, 32–34]. Moreover, the Keldysh method was\nrecently used to formalize a fully quantum mechanical\ntheory of STT and SP, including the effects of quan-\ntum fluctuations [35]. These fluctuations have become\nincreasingly relevant with the development of new devices\noperating in the low-temperature regime. Nevertheless,\napplying the Keldysh method to derive microscopic re-\nlations for SP, STT, and fluctuating torques in an AFM\nsystem is lacking.\nIn this paper, we extend the approach of Ref. 35,\nwhich examined a ferromagnet in the macrospin approx-\nimation coupled to normal metals featuring spin andarXiv:2402.04719v2  [cond-mat.mes-hall]  11 Apr 20242\ncharge accumulation, to a similar system but instead\nfeaturing a collinear magnet with many individual spins\ncoupled at different sublattices to normal metals. Our\nstudy thus covers antiferromagnets, ferrimagnets, and\nferromagnets. We derive the spin dynamics using a\nfully quantum mechanical Keldysh non-equilibrium ap-\nproach. We find expressions for the spin transfer torque,\nspin-pumping-induced Gilbert damping, and fluctuat-\ning fields, including low-temperature shot-noise contri-\nbutions. The Gilbert damping and fluctuations contain\nboth inter-lattice and intra-lattice terms. Using Onsager\nreciprocal relations, we relate the spin pumping and spin\ntransfer coefficients. Our results enhance the knowledge\nof the microscopic expressions of STT and SP and fluctu-\nating torques in antiferromagnets and ferrimagnets cou-\npled to normal metals in the low-energy regime, where\nquantum fluctuations become essential.\nThe subsequent sections of this paper are structured\nas follows. In Sec. II, we introduce the model employed\nfor the itinerant electrons in the normal metals, the lo-\ncalized magnetic moments in the antiferromagnetic or\nferrimagnet, and the electron-magnon coupling between\nthem. We then present the key findings of this paper\nin Sec. III, including microscopic definitions of the spin\ntransfer torque, spin pumping, and fluctuating torques\nin many spin magnets, being an antiferromagnetic, fer-\nrimagnet, or ferromagnet. The derivation of an effective\nmagnon action, achieved by integrating fermionic degrees\nof freedom resulting from the interaction with normal\nmetals, is detailed in Sec. IV. The evaluation of this ef-\nfective action is then provided in Sec. V. Finally, Sec. VI\nconcludes the paper.\nII. MODEL\nWe consider a bipartite collinear magnet coupled to\nan arbitrary number of normal metal reservoirs. The\nmagnet can represent an antiferromagnet, a ferrimagnet,\nor a ferromagnet. The total Hamiltonian is\nˆH=ˆHe+ˆHem+ˆHm (1)\nin terms of the Hamiltonian describing the electrons in\nthe normal metal ˆHe, the Hamiltonian describing the in-\nteraction between the electrons and the magnet ˆHem, and\nthe Hamiltonian of the magnet ˆHm.\nThe Hamiltonian of the electrons combined with the\nHamiltonian representing the interaction between the\nelectrons and the magnet is\nˆHe+ˆHem=Z\ndrˆψ†\"\nHe+ℏ−1X\niuiσ·ˆSi#\nˆψ , (2)\nwhere ˆψ†= ( ˆψ†\n↑,ˆψ†\n↓) is the spatially dependent 2-\ncomponent itinerant electron field operator, and σis the\nvector of Pauli matrices in the 2 ×2 spin space. In the\nHamiltonian (2), ui(r) represents the spatially depen-\ndent exchange interaction between the localized spin atsiteiand the itinerant electrons. This interaction is lo-\ncalized around spin iinside the magnet. The sum over\nthe localized spins iconsists of a sum over sites in sublat-\nticeAand sublattice B, i.e.,P\ni. . .→P\na. . .+P\nb. . ..\nThe localized spin operator ˆSihas a total spin angular\nmomentum Si=ℏp\nsi(si+ 1) where siis the (unitless)\nspin quantum number of the localized spin, such that\nˆSi2=ℏ2si(si+ 1). For large si, the difference between\nSi/ℏandsiis a first-order correction, and we can ap-\nproximate Si≈ℏsi.\nThe spin-independent part of the single-particle elec-\ntron Hamiltonian is\nHe=−ℏ2\n2m∇2+Vc, (3)\nwhere Vcis the spatially dependent charge potential.\nIn the classical limit of the magnet, the spins at sublat-\nticeAare along a certain direction and the spins at sub-\nlattice Bare along the opposite direction in the ground\nstate. We will consider the semiclassical spin dynam-\nics near the instantaneous classical direction of the spins\nthat we let be along the zdirection and adiabatically\nadjust the evolution of the small deviation [10, 11, 35].\nIn the following, it is constructive to expand the in-\nteraction term to the second order in the magnet cre-\nation/annihilation operators using a Holstein-Primakoff\ntransformation,\nˆHem=ˆH0+ˆH1+ˆH2, (4)\nwhere ˆH0is the interaction with the classical magnetic\nground state and ˆH1(ˆH2) is the interaction term to the\nfirst (second) order. The classical ground state contribu-\ntion to the interaction is then\nˆH0=Z\ndrˆψ†Vsσzˆψ , (5)\nwhere the magnitude of the spatially dependent spin po-\ntential experienced by the itinerant electrons is\nVs(r) =X\nasaua(r)−X\nbsbub(r), (6)\nand oscillates rapidly with the staggered field.\nIn the macrospin approximation,P\niSican be treated\nas a giant spin in ferromagnets. Then, ui(r) becomes\nthe effective exchange interaction. Ref. 35 shows how the\nelectronic Hamiltonian ˆHecombined with the electron-\nmagnon Hamiltonian to zeroth order ˆH0become partic-\nularly transparent in ferromagnet-normal metal systems\nin terms of the scattering states of the itinerant electrons\nfor the macrospin dynamics. We generalize this approach\nto magnet-normal metal systems with individual local-\nized spins. In this picture, the electronic Hamiltonian\nremains simple, as in Ref. 35:\nˆHe+ˆH0=X\nsαϵαˆc†\nsαˆcsα, (7)3\nwhere ˆ csαannihilates an electron with spin s(s=↑or\ns=↓). The quantum number α=κnϵcaptures the\nleadκ, the transverse waveguide mode n, and the elec-\ntron energy ϵ. The electron energy consists of a trans-\nverse contribution ϵnand a longitudinal contribution\nϵ(k) =k2/2m, where kis the longitudinal momentum,\nsuch that ϵ=ϵn+ϵ(k). The eigenenergy is spin degener-\nate, since the leads are paramagnetic. Furthermore, we\nconsider identical leads such that the eigenenergy is in-\ndependent of the lead index. The system setup is shown\nL R N N AF\nFIG. 1. An antiferromagnet (AF) with conductors (N) on\neither side connected to a right (R) and a left (L) lead.\nin Fig. 1 for the case of two leads. In Eq. (7) and similar\nexpressions to follow, the sum over the scattering states\nimplies thatP\nαXsα=P\nκnR∞\nϵndϵXsκn(ϵ). In the scat-\ntering approach, the field operator is\nˆψs=X\nαˆcsαψsα, (8)\nwhere ψsα(r) is the wave function of a scattering state\nof spin sand quantum number α.\nThe Hamiltonian of the antiferromagnet is [ idenotes\na site at sublattice A(i=a) orB(i=b)]\nHm=ℏ−2X\nijJijˆSi·ˆSj−Kℏ−2X\ni\u0010\nˆSi·z\u00112\n+γµ0X\naHA\na·ˆSa+γµ0X\nbHB\nb·ˆSb, (9)\nwhere Jijis the symmetric exchange interaction, K > 0\nis the easy-axis anisotropy energy, and γ=g∗µB/ℏis\nthe (absolute value of) the effective gyromagnetic ratio,\nwhere g∗is the effective Land´ e g-factor and µBis the\nBohr magneton. In Eq. (9), HA,B\niis the external mag-\nnetic field in units of Am−1at lattice site i={a, b}, and\nµ0is the vacuum permeability, which appears because\nwe are employing SI units. In reality, HA=HBin the\npresence of a uniform external magnetic field. However,\nto illustrate and understand the physics, we allow the\nexternal fields at sublattices AandBto differ, and to\ndepend on the lattice site.\nWe consider the low-energy excitations from the semi-\nclassical ground state of the staggered spin orientation.\nTo this end, we carry out a Holstein-Primakoff expansion\nto the second order in magnon excitations at each sub-\nlattice AandBdescribed via the annihilation operators\nˆaaandˆbbas detailed in Appendix A. Introducing theraising/lowering fields as H±=Hx±iHy, the magnon\nHamiltonian becomes\nHm=E0+X\naEA\naˆa†\naˆaa+X\nbEB\nbˆb†\nbˆbb\n+ 2X\naa′Jaa′√sasa′ˆa†\naˆaa′+ 2X\nbb′Jbb′√sbsb′ˆb†\nbˆbb′\n+ 2X\nabJab√sasb[ˆaaˆbb+ ˆa†\naˆb†\nb]\n+γµ0ℏX\narsa\n2[HA\na−ˆaa+HA\na+ˆa†\na]\n+γµ0ℏX\nbrsb\n2[HB\nb−ˆb†\nb+HB\nb+ˆbb], (10)\nwhere the classical ground state energy E0is\nE0=X\naa′sasa′Jaa′+X\nbb′sbsb′Jbb′−2X\nabsasbJab\n−2KX\nis2\ni+ℏµ0X\nasaHA\naz−ℏµ0X\nbsbHB\nbz,(11)\nand is disregarded in the following.\nEA(B)\na(b)= 2X\nb(a)sb(a)Jab−X\na′(b′)sa′(b′)Ja(b)a′(b′)\n+ 2sa(b)K∓ℏγµ0HA(B)\na(b)z(12)\nis the energy of a local excitation, where the upper sign\nholds for sites on sublattice Aand the lower sign holds\nfor sites on sublattice B.\nIn the scattering basis of the electronic states, the cor-\nrections to the antiferromagnetic ground state electron-\nmagnon interaction to quadratic order in the magnet op-\nerators becomes ˆHem−ˆH0=ˆH1+ˆH2. The first-order\ncontribution of electron-magnon interaction is\nˆH1=X\naαβr\n2\nsah\nˆaaˆc†\n↓αWαβ\na↓↑ˆc↑β+ ˆa†\naˆc†\n↑αWαβ\na↑↓ˆc↓βi\n+X\nbαβr\n2\nsbh\nˆb†\nbˆc†\n↓αWαβ\nb↓↑ˆc↑β+ˆbbˆc†\n↑αWαβ\nb↑↓ˆc↓βi\n,(13)\nand describes the spin-flip scattering of the itinerant elec-\ntrons associated with creating or annihilating localized\nmagnons. The dimensionless matrix Wiis governed by\nthe exchange potential ui(r) and the scattering states\nwave functions ψsα:\nWαβ\niss‘=Z\ndrψ∗\nsα(r)siui(r)ψs‘β(r), (14)\nand is Hermitian, Wαβ\ni↑↓= [Wβα\ni↓↑]∗. The electron-magnon\ninteraction that is second order in the magnon operators4\nis\nˆH2=−X\naαβˆa†\naˆaa\nsah\nˆc†\n↑αWαβ\na↑↑ˆc↑β−ˆc†\n↓αWαβ\na↓↓ˆc↓βi\n+X\nbαβˆb†\nbˆbb\nsbh\nˆc†\n↑αWαβ\nb↑↑ˆc↑β−ˆc†\n↓αWαβ\nb↓↓ˆc↓βi\n,(15)\nwhere the matrix elements are defined in Eq. (14). We\nnote that our electron-magnon-interaction is isotropic in\nspin space and will give rise to zeroth-, first-, and second-\norder magnon terms in the Hamiltonian, i.e. ˆH0,ˆH1and\nˆH2, respectively. This is in contrast to the model used\nin Refs. 12 and 32, where only the first-order term ˆH1is\nconsidered.\nFinally, in normal metal reservoirs, the occupation of\nthe state is\n⟨c†\ns′αcsβ⟩=δαβnss′α, (16)\nwhere the 2 ×2 out-of-equilibrium distribution is\nnss′α=1\n2[fκ↑(ϵα)) +fκ↓(ϵα))]δss′\n+1\n2[fκ↑(ϵα))−fκ↓(ϵα))]uκ·σss′, (17)\nallowing for a (lead-dependent) spin accumulation in the\ndirection of the unit vector uk.f↑andf↓are general\ndistribution functions for spin-up and spin-down parti-\ncles, which generally differ for elastic or inelastic trans-\nport [35]. In equilibrium, the distribution function only\ndepends on energy,\nfeq\nκ↑(ϵ) =feq\nκ↓(ϵ) =f(ϵ−µ0), (18)\nwhere fis the equilibrium Fermi-Dirac distribution and\nµ0is the equilibrium chemical potential.\nIn inelastic transport, the spin- and charge accumula-\ntions µCandµScorrespond to chemical potential in a\n(spin-dependent) Fermi-Dirac function,\nfin\nκ↑(ϵ) =f(ϵ−µ0−µC\nκ−µS\nκ/2) (19a)\nfin\nκ↓(ϵ) =f(ϵ−µ0−µC\nκ+µS\nκ/2). (19b)\nFor notational simplicity, we define the chemical poten-\ntials\nµκ↑=µ0+µC\nκ+µS\nκ/2 (20a)\nµκ↓=µ0+µC\nκ−µS\nκ/2. (20b)\nIn the limit of small charge and spin accumulations com-\npared to the Fermi level, it can be derived that\nµC\nκ+µS\nκ\n2=Z\ndϵ\u0002\nfin\nκ↑(ϵ)−f(ϵ)\u0003\n. (21)\nIn the elastic regime, the distribution function cannot\ngenerally be described as a Fermi-Dirac function. Thedistribution function is instead given as a linear combi-\nnation of Fermi-Dirac functions in the connected reser-\nvoirs [35],\nfel\nsκ(ϵ) =X\nlRsκlf(ϵ−µl), (22)\nwhere the index lruns over the reservoirs, and Rsκlis the\nlead and spin-dependent transport coefficient for reser-\nvoirl. The transport coefficients satisfy\nX\nlRsκl= 1. (23)\nIn the elastic transport regime, it is advantageous to de-\nfine the effective charge and spin accumulations through\nµC\nκ+µS\nκ\n2=Z\ndϵ\u0002\nfel\nκ↑(ϵ)−f(ϵ)\u0003\n. (24)\nThe elastic and inelastic transport regime results in dif-\nferent results for the fluctuations in the magnetization\ndynamics of the magnet.\nHaving specified the model for the system in consider-\nation, we proceed by presenting the main results of the\npaper.\nIII. MAIN RESULTS: EQUATIONS OF MOTION\nThis section presents the main results of our work.\nOur primary result is the derivation of a Lan-\ndau–Lifshitz–Gilbert–Slonczewski (LLGS) equation for\nthe localized (normalized) spins mi=Si/Siin a gen-\neral magnet coupled to normal metal reservoirs,\n∂tmi=τb\ni+τf\ni+τsp\ni+τstt\ni (25)\nvalid for low-energy excitations when the equilibrium\nmagnetization is parallel (antiparallel) to the z-axis. The\nbulk antiferromagnet torque τb\nifor a site i={a, b}arises\nfrom contributions of anisotropy, exchange coupling, and\nexternal fields, and reads\nτb\ni=−z×\u0000\nℏ−1Eimi+γµ0Hi\u0001\n. (26)\nwhere Eiis the energy of a local excitation and Hiis\nthe applied field. Hence, the bulk torque remains unaf-\nfected by the presence of normal metal reservoirs and the\nassociated spin- and charge accumulations.\nThe spin transfer torque τstt\niis induced by spin accu-\nmulation in the normal metals, and can be expressed as\nfollows:\nτstt\ni=ℏ−1X\nκ\u0002\nβI\niκz×µS\nκ−βR\niκz×(z×µS\nκ)\u0003\n.(27)\nIn Eq. (27), the superscripts ” R,I” denote the real and\nimaginary part. The site and lead-dependent coefficients5\nβiκare expressed in terms of the microscopic scatter-\ning matrix elements defined in Eq. (14) evaluated at the\nFermi energy:\nβiκ=−2i\nsiX\nnWκnκn\ni↑↓, (28)\nand can be calculated numerically for any particular sys-\ntem configuration.\nThe spin pumping torque τsp\nicontains contributions\nfrom both sublattices and is given by\nτsp\ni=X\nj\u0002\nαR\nijz×∂tmj+αI\nijz×(z×∂tmj)\u0003\n,(29)\nwhere jruns over all sites and αijis expressed in the low-\nenergy limit using the scattering matrix elements evalu-\nated at the Fermi energy,\nαij=2π√sisjX\nκλnmWκnλm\ni↓↑Wλmκn\nj↑↓, (30)\nandαR(I)denotes the real (imaginary) part of the matrix.\nUsing the Onsager reciprocal relations in Appendix B, we\nfind that the spin transfer torque and spin pumping are\nrelated in the case of the most relevant case of a single\nreservoir,\nX\njαij=βi. (31)\nFinally, the fluctuating torque τf\niis expressed in terms\nof a fluctuating transverse field Hf\ni,\nτf\ni=−γµ0z×Hf\ni. (32)\nThe fluctuating field exhibit interlattice and intralattice\ncorrelators ⟨HµiHνj⟩, where µ, ν={x, y}:\n2√sisjγ2µ2\n0⟨Hf\nxiHf\nxj⟩= ImΣK\nij+ 4Im ˜Σ↑↓ij (33a)\n2√sisjγ2µ2\n0⟨Hf\nxiHf\nyj⟩=−ReΣK\nij−4Re˜Σ↑↓ij (33b)\n2√sisjγ2µ2\n0⟨Hf\nyiHf\nyj⟩= ImΣK\nij−4Im˜Σ↑↓ij, (33c)\nwhere the time arguments tandt′of the fields and the\nrelative time argument ( t−t′) of the self energies are\nomitted for simplicity. The self-energy Σ is due to charge\nand longitudinal spin accumulations in the normal metals\nand is nonzero even in equilibrium. It is conveniently\nwritten as a product of a frequency-dependent quantity\nπ(ω) and a site and scattering states dependent quantity\nσij[35]:\nΣK\nij(ω) =i\nℏX\nκλσijκλπκλ(ω), (34)\nwith\nπκλ(ω) =−2Z\ndϵ[2n↑↑κ(ϵ)n↓↓λ(ϵ+ℏω)\n−n↑↑κ(ϵ)−n↓↓λ(ϵ+ℏω)] (35a)\nσijκλ=2π√sisjX\nnmWκnλm\ni↓↑Wλmκn\nj↑↓. (35b)Conversely, the self-energy matrices ˜Σ↑↓are due to trans-\nverse spin accumulation in the normal metals and, as a\nresult, vanish in equilibrium. Analogous to the decom-\nposition in Eq. (34), we write\n˜ΣK\n↑↓ij=−i\nℏX\nκλ˜σ↑↓ijκλ˜πκλ(ω), (36)\nwhere\n˜π↑↓(ω) =−4Z\ndϵn↑↓κ(ϵ)n↑↓λ(ϵ+ℏω) (37a)\n˜σ↑↓ijκλ=−π√sisjX\nnmWκnλm\ni↓↑Wλmκn\nj↓↑, (37b)\nThe noise matrices π(ω) and ˜ π(ω) are similar to what\nwas found in Ref. 35, and are calculated in the equi-\nlibrium, elastic, and inelastic transport regime in Sec.\nV B. Crucially, the shot noise differs on various sites, due\nto the site-dependence of σand ˜σ. At equilibrium, the\nfluctuation-dissipation theorem holds, e.g.\n2siγ2µ2\n0⟨Hf\nµiHf\nνi⟩=δµναii4kBTξ\u0012ℏω\n2kBT\u0013\n, (38)\nwhere ξ(x) =xcothx.\nIn the next section, we discuss the Keldysh action of\nthe model presented in this section and derive an effec-\ntive action by integrating out the fermionic degrees of\nfreedom.\nIV. KELDYSH THEORY AND EFFECTIVE\nACTION\nIn this section, we derive the semiclassical spin dynam-\nics by using an out-of-equilibrium path integral formal-\nism [30]. We introduce the closed contour action Sand\nthe partition function Z,\nZ=Z\nD[¯aa¯bb¯c↑c↑¯c↓c↓]eiS/ℏ. (39)\nThe action Sconsists of contributions from the localized\nmagnetic excitations aandb, and the spin-up c↑and\nspin-down electrons c↓from the scattering states. We\nwill integrate out the fermion operators and get an effec-\ntive action for the magnetic excitations aandb, which\nincludes effective transverse and longitudinal fields that\narise from the charge and spin accumulations in the nor-\nmal metals.\nWe follow Ref. 30 and replace the fields in Eq. (39)\nwith ” ±” fields residing on the forward and backward\npart of the Schwinger-Keldysh contour. The action in\nthe±basis is given in Appendix C. These fields are not\nindependent of each other and can be Keldysh rotated\ninto a new basis that takes into account the coupling\nbetween them. The rotated fields have the advantage\nof suggesting a transparent physical interpretation, cor-\nresponding to the semiclassical equations and quantum\ncorrections.6\nA. Keldysh action\nFor magnons, the classical ( cl) and quantum ( q) fields\nare defined linear combinations of the ±-fields, as de-\nscribed in detail in Appendix C. In Keldysh space, it is\nconvenient to also introduce the matrices\nγq=\u0012\n0 1\n1 0\u0013\nγcl=\u0012\n1 0\n0 1\u0013\n. (40)\nThe Keldysh rotated magnon action becomes\nSm=X\nat¯aq\na(iℏ∂t−EA\na)acl\na+X\nbt¯bq\nb(iℏ∂t−EB\nb)bcl\nb\n+X\nataq\na(iℏ∂t−EA\na)¯acl\na+X\nbtbq\nb(iℏ∂t−EB\nb)¯bcl\nb\n−2X\naa′t√sasa′Jaa′\u0002\n¯aq\naacl\na′+h.c.\u0003\n−2X\nbb′t√sbsb′Jbb′\u0002¯bq\nbbcl\nb′+h.c\u0003\n−2X\nabt√sasbJab\u0002\n¯aq\na¯bcl\nb+ ¯acl\na¯bq\nb+h.c.\u0003\n−γµ0ℏX\nat√sa\u0002\nHA\na−aq\na+HA\na+¯aq\na\u0003\n−γµ0ℏX\nbt√sb\u0002\nHB\nb−¯bq\nb+HB\nb+bq\nb\u0003\n, (41)where we wrote the time integral as a sum for compact\nnotation. In Eq. (41), h.c.denotes the hermitian conju-\ngate of the previous term. The fermion action becomes\nSe+S0=X\nst¯Csγcl(iℏ∂t−ϵ)Cs, (42)\nwhere we introduced vector notation for the 1 /2-fields,\n¯Cs= (¯c1\nsα,¯c2\nsα), and ϵis a diagonal matrix containing\nthe single-particle energies of the electrons. The Keldysh\nrotated first-order electron-magnon interaction is\nS1=−X\nat\nαβ1√sah\nacl\naWαβ\na↓↑¯C↓αγclC↑β+aq\naWαβ\na↓↑¯C↓αγqC↑β+h.c.i\n−X\nbt\nαβ1√sbh\n¯bcl\naWαβ\nb↓↑¯C↓αγclC↑β+¯bq\nbWαβ\nb↓↑¯C↓αγqC↑β+h.c.i\n, (43)\nand, finally, the second-order term reads\nS2=X\nat\nαβ1\n2sah\nWαβ\na↑↑\u0000¯AaγclAa¯C↑αγclC↑β+¯AaγqAa¯C↑αγqC↑β\u0001\n−Wαβ\na↓↓\u0000¯AaγclAa¯C↓αγclC↓β+¯AaγqAa¯C↓αγqC↓β\u0001i\n−X\nbt\nαβ1\n2sbh\nWαβ\nb↑↑\u0000¯BbγclBb¯C↑αγclC↑β+¯BbγqBb¯C↑αγqC↑β\u0001\n−Wαβ\nb↓↓\u0000¯BbγclBb¯C↓αγclC↓β+¯BbγqBb¯C↓αγqC↓β\u0001i\n,\n(44)\nwhere the magnon q/cloperators are consolidated in vec-\ntors ¯Aaand ¯Bb. The Keldysh rotated action proves to\nbe well-suited for the computation of an effective magnon\naction, a topic we delve into in the following section.B. Integrating out the fermionic degrees of\nfreedom\nFor the itinerant electrons in the normal metal and\nantiferromagnet, the total effective electron action is7\nSe,tot=Se+Sem, and can be expressed as\nSe,tot=X\nss′tt′¯Cs,tG−1\nss′,tt′Cs′,t′, (45)\nwhere the interacting Green function Gis given in terms\nof the noninteracting Green function G0and interaction\nterms as\nG−1=G−1\n0+˜W1+˜W2. (46)\nHere, ˜W1contains the first-order magnon operators on\nboth sublattices,\n˜W1=δ(t−t′)\u0002\nWA\n1+WB\n1\u0003\n, (47)\nwhere WA\n1andWB\n1are spin flip operators:\nWA\n1=−X\nxa1√saγx\u0002\nWa↑↓¯ax\naσ++Wa↓↑ax\naσ−\u0003\n(48a)\nWB\n1=−X\nxb1√sbγx\u0002\nWb↑↓bx\nbσ++Wb↓↑¯bx\nbσ−\u0003\n.(48b)\nIn Eq. (48), the variable x={cl, q}represents a Keldysh\nspace index, and σ±are the usual raising and lowering\nPauli matrices. Similarly, ˜W2contains the magnon oper-\nators to quadratic order for both sublattices,\n˜W2=δ(t−t′)\u0002\nWA\n2−WB\n2\u0003\n, (49)\nwith WA\n2andWB\n2given by\nWA\n2=X\naxy1\n2sa¯ax\naγxay\naγy\u0012\nWa↑↑ 0\n0−Wa↓↓\u0013\n(50a)\nWB\n2=X\nbxy1\n2sb¯bx\nbγxby\nbγy\u0012\nWb↑↑ 0\n0−Wb↓↓\u0013\n, (50b)\nwhere the spin structure is explicitly written out as a ma-\ntrix. The matrices WA(B)\n1 andWA(B)\n2 have a structure\nin the scattering states space from Wa(b), spin space from\nthe Pauli matrices, and Keldysh space from γx. The in-\nverse free electron Green function G−1\n0from Eq. (46) has\nthe conventional causality structure in Keldysh space,\nwith a retarded ( R), advanced ( A), and Keldysh ( K)\ncomponent:\nG−1\n0=\u0012\n[GR\n0]−1[GK\n0]−1\n0 [ GA\n0]−1\u0013\n, (51)\nand has equilibrium components that are diagonal in\nboth spin space and in the scattering states space,\n[G−1\n0]R(A)\nαβ,ss′=δαβδss′δ(t−t′)(iℏ∂t−ϵα±iδ),(52)\nwhere the upper sign corresponds to the retarded compo-\nnent, while the lower sign is applicable to the advanced\ncomponent. The Keldysh component includes informa-\ntion about the distribution function, and will be dis-\ncussed below, when we Fourier transform the Green func-\ntions.From the effective electron action in Eq. (45), it is\nevident that the partition function of Se,tottakes on a\nGaussian form with respect to the fermionic operators.\nHence, the fermionic integral in the partition function\ncan be evaluated exactly, with an inconsequential pro-\nportionality constant being disregarded:\nZ\nD[C]eiSe,tot/ℏ= eTr[ln[1+G0˜W1+G0˜W2]]. (53)\nIn Eq. (53), we have used the short-hand notation for\nthe functional integral measure of all fermionic states,\nD[C] =D[¯C↑C↑¯C↓C↓]. We have absorbed a normaliza-\ntion constant into the functional integral measure for sim-\nplicity. We note that as a consequence of the continuity\nof the time coordinate and scattering states energy that\nwe are employing, the unit matrix is a delta function in\ntime and energy, 1 ≡δ(t−t′)δ(ϵα−ϵβ), and thus quanti-\nties inside the logarithm carries dimension J−1s−1. The\ntrace, on the other hand, is an integral operator with unit\nJ s. As long as one interprets the logarithm in terms of\nits Taylor expansion, this does not lead to any problems,\nas the exponent of Eq. (53) becomes dimensionless for all\nterms in the expansion. The exponent is interpreted as\nan additional contribution to the magnon action,\ni\nℏSeff= Trh\nln[1 + G0˜W1+G0˜W2]i\n. (54)\nThe way forward is to treat this interaction as a pertur-\nbation, expanding the logarithm in first and second-order\ncontributions and disregarding higher-order terms,\nSeff≈ −iℏTrh\nG0˜W1i\n−iℏTrh\nG0˜W2i\n+iℏ\n2Trh\nG0˜W1G0˜W1i\n. (55)\nTo evaluate the trace in these terms, it is convenient to\nFourier transform all quantities from the time domain\nto the energy domain. This diagonalizes the noninter-\nacting Green functions, making calculations much more\nstraightforward.\nC. Fourier representation\nThe paper employs the Fourier transform convention\ndefined in Appendix D. In Fourier space, the fermion\nequilibrium Green function components are particularly\nsimple:\n[G0]R(A)\nαβ,ss′(ω) =δαβδss′(ℏω−ϵ±iδ)−1, (56)\nwhere δ > 0 is an infinitesimal quantity ensuring con-\nvergence. The Keldysh component accounts for non-\nequilibrium phenomena though the spin-dependent dis-\ntribution nss′αdefined in Eq. (17),\n[G0]K\nαβ,ss′(ω) =−2πiδαβδ(ℏω−ϵα) [δss′−2nss′α].(57)8\nThe Keldysh component has off-diagonal terms in spin\nspace if the distribution function nss′αhas off-diagonal\nelements, i.e. if there is a transverse spin accumulation\nin the normal metals.\nV. NON-EQUILIBRIUM SPIN DYNAMICS\nHaving derived the effective action as expressed in Eq.\n(55), we proceed by evaluating the traces and delving\ninto the resultant terms. The discussion unveils effective\nlongitudinal and transverse fields, which we ascribe to\nspin transfer torque and spin pumping originating from\nthe normal metal reservoirs.\nA. First-order contribution\nEvaluating the trace in the first order term in Eq.\n(55) corresponds to summing over the diagonal elements\nin spin space and Keldysh space, integrating over both\ntime variables, and summing over the space of scattering\nstates, we find\n−iℏTrh\nG0˜W1i\n=−X\naα2√saWαα\na↑↓n↓↑αZ\ndt¯aq\na(t)\n−X\naα2√saWαα\na↓↑n↑↓αZ\ndtaq\na(t)\n−X\nbα2√saWαα\nb↑↓n↓↑αZ\ndtbq\nb(t)\n−X\nbα2√saWαα\nb↓↑n↑↓αZ\ndt¯bq\nb(t).(58)\nHere, we have used the general Green function identity\nGR(t, t)+GA(t, t) = 0 [30], and written the time integra-\ntion explicitly. Comparing the first-order contribution in\nEq. (58) with the magnon action in Eq. (41), we observe\nthat the first-order effect of the spin accumulation in the\nnormal metal is equivalent to an effective deterministic\ntransverse magnetic field Hstt\ni, which act on a localized\nspin at site i={a, b}in the antiferromagnet. The ”stt”\nsuperscript indicates that this field will take the form of\na spin transfer torque, which will be elaborated on be-\nlow. The magnitudes of these effective transverse fields\nare given by\nγµ0Hstt\ni−=2\nsiℏX\nαWαα\ni↓↑n↑↓α (59a)\nγµ0Hstt\ni+=2\nsiℏX\nαWαα\ni↑↓n↓↑α, (59b)\nwhich implies that the Cartesian components read\nγµ0Hstt\nix=2\nsiℏX\nαRe\u0002\nWαα\ni↑↓n↓↑α\u0003\n(60a)\nγµ0Hstt\niy=2\nsiℏX\nαIm\u0002\nWαα\ni↑↓n↓↑α\u0003\n. (60b)Recalling that the spin accumulation is given by Eq. (24)\nand Eq. (21), we write the effective fields from Eq. (60)\nin the conventional spin transfer torque form:\nγµ0Hstt\ni=1\nℏX\nκ\u0002\nβR\niκz×µS\nκ+βI\niκz×(z×µS\nκ)\u0003\n,(61)\nwhere the appearance of zis a consequence of our the-\nory being restricted to small deviations for the equilib-\nrium magnetization ±z. This results in the spin transfer\ntorque given in Eq. (27). In Eq. (61), the superscripts\n”R” and ” I” denote the real and imaginary parts and the\nlead- and site-dependent constants βiκhave been intro-\nduced as sums over the transverse modes of the scattering\nmatrix elements,\nβiκ=−2i\nsiX\nnWκnκn\ni↑↓, (62)\nand where we have assumed that the transverse spin dis-\ntribution functions n↑↓andn↓↑are only significant close\nto the Fermi surface, such that the scattering states ma-\ntrix elements are well approximated by their value at\nthe Fermi surface. The expression for the spin transfer\nfield in Eq. (61) is valid in both the elastic and inelastic\nregime, and vanishes in equilibrium. We note that the\ncoefficient βiκ, for i={a, b}, depends not only on the\npotential at lattice site ibut also indirectly of all lattice\nsites on both sublattices through the scattering states.\nTo the lowest order, the sublattice magnetizations are\nparallel and antiparallel to the z-axis, mA≈zand\nmB≈ −z. Thus, to the lowest order in the magnon\noperators, the expressions for the transverse fields are\nambiguous, and we can write the transverse field in Eq.\n(61) in terms of mAormB. To the lowest order in the\nmagnon operators, the Keldysh technique cannot be used\nto identify which sublattice the transverse fields in Eq.\n(58) originate from.\nB. Second order contribution\nThe second order contribution in Eq. (55) has contri-\nbutions from ˜W2,\nS21=−iℏTrh\nG0˜W2i\n, (63)\nas well as a contribution from ˜W1,\nS22=iℏ\n2Trh\nG0˜W1G0˜W1i\n. (64)\nProceeding in a manner analogous to the treatment of\nthe first-order term, the trace in S21is evaluated:9\nS21=−X\naαπ\nsa\u0002\nWαα\na↑↑(1−2n↑↑α)−Wαα\na↓↓(1−2n↓↓α)\u0003Z\ndt¯Aa(t)γqAa(t)\n+X\nbαπ\nsb\u0002\nWαα\nb↑↑(1−2n↑↑α)−Wαα\nb↓↓(1−2n↓↓α)\u0003Z\ndt¯Bb(t)γqBb(t). (65)\nFrom Eq. (41), it is apparent that the second-order terms in S21are equivalent with a longitudinal magnetic field,\nwith magnitude\nγµ0HA21\niz=−π\nℏsiX\nα\u0002\nWαα\ni↑↑(1−2n↑↑α)−Wαα\ni↓↓(1−2n↓↓α)\u0003\n, (66)\nwhich, in this reference frame, renormalizes the energies of localized magnon excitations. However, such longitudinal\nmagnetic fields should not affect the spin dynamics since they, in the instantaneous reference field, correspond to\ncontributions to the total free energy proportional to S2\ni.\nThe final contribution S22to the effective action contains inter-lattice and intra-lattice terms and can be written\ncompactly by introducing a field di={aa,¯bb}and summing over the two field components, i.e.P\nidi=P\naaa+P\nb¯bb:\nS22=Z\ndtdt′X\nxx′ijiℏ\n2√sisjTrh\nG0,↑↓(t′, t)Wi↓↑γxG0,↑↓(t, t′)γx′Wj↓↑i\ndx\ni(t)dx′\nj(t′)\n+Z\ndtdt′X\nxx′ijiℏ\n2√sisjTrh\nG0,↓↑(t′, t)Wi↑↓γxG0,↓↑(t, t′)γx′Wj↑↓i\n¯dx\ni(t)¯dx′\nj(t′)\n+Z\ndtdt′X\nxx′ijiℏ\n2√sisjTrh\nG0,↓↓(t′, t)Wi↓↑γxG0,↑↑(t, t′)γx′Wj↑↓i\ndx\ni(t)¯dx′\nj(t′)\n+Z\ndtdt′X\nxx′ijiℏ\n2√sisjTrh\nG0,↑↑(t′, t)Wi↑↓γxG0,↓↓(t, t′)γx′Wj↓↑i\n¯dx\ni(t)dx′\nj(t′), (67)\nwhere the trace is taken only over the 2 ×2 Keldysh space and the space of scattering states α. The interlattice terms,\ni.e.d=d′, are discussed in Ref. 35 for a macrospin ferromagnet. Here, we summarize this discussion and highlight\nthe addition of the inter-lattice terms not present in the macrospin ferromagnet.\nEvaluating the trace in the first and second line of Eq. (67), we note that only the Keldysh component has off-\ndiagonal elements in spin space, and find a contribution only from x=x′=q,\n˜Sqq\n22=ℏZ\ndtdt′X\nijh\ndq\ni(t)˜ΣK\n↑↓ij(t, t′)dq\nj(t′)i\n(68a)\n˜S¯q¯q\n22=ℏZ\ndtdt′X\nijh\n¯dq\ni(t)˜ΣK\n↓↑ij(t, t′)¯dq\nj(t′)i\n, (68b)\nwhere the self-energies are\n˜ΣK\n↑↓ij(t−t′) =−2i\nℏ2√sisjX\nαβn↑↓αn↑↓βWαβ\ni↓↑Wβα\nj↓↑ei(ϵα−ϵβ)(t−t′)/ℏ(69a)\n˜ΣK\n↓↑ij(t−t′) =−2i\nℏ2√sisjX\nαβn↓↑αn↓↑βWαβ\ni↑↓Wβα\nj↑↓ei(ϵα−ϵβ)(t−t′)/ℏ. (69b)\nThe reasoning behind identifying this self-energy as a Keldysh component is that it couples the quantum components\nof the fields, see Eq. (68). The terms in Eq. (68) do not have a direct analog in the magnon action in Eq. (41), and\ninterpreting these will be the subject of Sec. V C. The self-energies in Eq. (69) are invariant under a joint time and\nlattice site reversal, i.e. ˜Σij(t−t′) =˜Σji(t′−t). Moreover, due to the properties n↑↓=n∗\n↓↑andWαβ\ni↑↓= [Wβα\ni↓↑]∗, we\nsee that the self-energies are related by ˜ΣK\n↑↓ij(t−t′) =−[˜ΣK\n↓↑ij(t−t′)]∗, which will be important later.\nDisregarding terms of the order kBT/ϵFandµs/ϵF[35], we find that the Fourier-transformed self-energy becomes\n˜ΣK\n↑↓ij(ω) =−i\nℏX\nκλ˜σ↑↓ijκλ˜πκλ(ω), (70)10\nwhere\n˜πκλ(ω) =−4Z\ndϵn↑↓κ(ϵ)n↑↓λ(ϵ+ω) ˜ σ↑↓ijκλ=−π√sisjX\nnmWκnλm\ni↓↑Wλmκn\nj↓↑, (71)\nand where the matrix elements Ware evaluated at the Fermi surface. This is a straightforward generalization of\nthe macrospin ferromagnet case, with the addition of shot-noise contributions from inter-lattice and intra-lattice\ninteractions between different lattice sites. We can evaluate the quantity ˜ πκλ↑↓(ω) by using Eq. (17):\n˜π↑↓κλ(ω) =−uκ−uλ−Z\ndϵ[f↑κ(ϵ)−f↓κ(ϵ)] [f↑λ(ϵ+ℏω)−f↓λ(ϵ+ℏω)] (72)\nwhere we introduced the conventional ”lowering” vector u−=ux−iuy. This can be computed in equilibrium, elastic,\nand inelastic scattering cases, and results exactly similar to those in Ref. 35.\nWe now turn our attention to the third and fourth lines of the second-order action in Eq. (67). The contributions\nfrom the two lines are equal, which is evident from interchanging summation indices and rearranging terms. Their\ntotal contribution to the action S22can be split into contributions S¯qq\n22,S¯qcl\n22, and S¯clq\n22. The contribution Sclclvanishes,\ndue to the quantity GR(t′−t)GR(t−t′) being nonzero only for t=t′, which has measure zero, and similarly for\nGA. This ensures that the action satisfies the general requirement S[ϕcl, ϕq= 0] = 0 [30]. Introducing, for notational\nconvenience, the vector ¯Di=\u0000¯dcl¯dq\u0001\n, we find\nS¯qq\n22+S¯qcl\n22+S¯clq\n22=ℏZ\ndtdt′X\nij¯Di(t)ˆΣij(t−t′)Dj(t′), (73)\nwhere the self-energy matrix has structure in Keldysh space and in the sublattice space,\nˆΣij(t−t′) =\u0012\n0 ΣA(t−t′)\nΣR(t−t′) ΣK(t−t′)\u0013\nij, (74)\nand its components are given by\nΣK\nij(t−t′) =2i√sisjℏ2X\nαβ(n↑↑α+n↓↓β−2n↑↑αn↓↓β)Wαβ\ni↓↑Wβα\nj↑↓ei(ϵα−ϵβ)(t−t′)/ℏ(75a)\nΣR\nij(t−t′) =2i√sisjℏ2θ(t−t′)X\nαβ(n↑↑α−n↓↓β)Wαβ\ni↓↑Wβα\nj↑↓ei(ϵα−ϵβ)(t−t′)/ℏ(75b)\nΣA\nij(t′−t) =−2i√sisjℏ2θ(t−t′)X\nαβ(n↑↑α−n↓↓β)Wαβ\ni↓↑Wβα\nj↑↓ei(ϵα−ϵβ)(t−t′)/ℏ. (75c)\nThe Keldysh component of this self-energy has the symmetry [ΣK\nij(t−t′)]∗=−ΣK\nji(t′−t). Imperatively, as a\nconsequence of this symmetry, the quantities\nΣK\nij(t−t′)−ΣK\nji(t′−t) = 2Re\u0002\nΣK\nij(t−t′)\u0003\n(76)\niΣK\nij(t−t′) + iΣK\nji(t′−t) =−2Im\u0002\nΣK\nij(t−t′)\u0003\n, (77)\nare real numbers, which will be important in the next section. We proceed by a similar analysis to what was done\nwith ˜Σ, writing it in terms of a shot-noise matrix. We assume that the matrices Wcan be approximated by their\nvalue on the Fermi surface, and write\nΣK\nij(ω) =i\nℏX\nκλσijκλπκλ(ω), (78)\nwhere we introduced the matrices\nπκλ(ω) =−2Z\ndϵ[2n↑↑κ(ϵ)n↓↓λ(ϵ+ℏω)−n↑↑κ(ϵ)−n↓↓λ(ϵ+ℏω)] σijκλ=2π√sisjX\nnmWκnλm\ni↓↑Wλmκn\nj↑↓.(79)\nThe matrix π(ω) can also be evaluated in equilibrium, and for elastic and inelastic scattering, and the results are\nagain exactly similar to those in Ref. 35.\nwhere ξ(x) =xcothxis an asymptotically linear function for high xandpss′κ= (1−uzκ)/2+uzκδssis a projection11\nfactor introduced for notational convenience.\nComparing with the magnetic action in Eq. (41), we\nnotice that the terms with the retarded and advanced\nself-energies are equivalent with longitudinal fields, which\nwe in the following will show consists of dissipative\nGilbert-like terms and non-dissipative field-like terms.\nFourier transforming and applying the identity (D5), the\nretarded and advanced self-energies from Eq. (75b) and\nEq. (75c) become\nΣR,A\nij=−2√sisjℏX\nαβn↑↑α−n↓↓β\nℏω+ϵα−ϵβ±iδWαβ\ni↓↑Wβα\nj↑↓.(80)\nThis self-energy has equilibrium contributions as well\nas non-equilibrium contributions, however, the non-\nequilibrium contributions scale as µ↑↑/ϵFandµ↓↓/ϵFand\nare disregarded in the following. The equilibrium part of\nEq. (80) becomes particularly transparent when expand-\ning to first order in the frequency ω:\nΣR/A\n↑↓ij(ω)≈ΣR/A\n↑↓ij(ω= 0)±iωαij, (81)\nwhere we introduced the frequency-independent matrix\nelement\nαij=2π√sisjX\nαβ[−f′(ϵα)]δ(ϵα−ϵβ)Wαβ\ni↓↑Wβα\nj↑↓,(82)\nwhich can be approximated to\nαij=2π√sisjX\nκλnmWκnλm\ni↓↑Wλmκn\nj↑↓, (83)\nwhere the scattering states matrix elements are evalu-\nated at the Fermi surface. We note from the identity\n[Wαβ\ni↑↓]∗=Wβα\ni↓↑thatαis a Hermitian matrix in the space\nof lattice sites, i.e. [ αij]∗=αji. The zeroth order term\nin frequency is\n[S¯qcl\n22+S¯clq\n22]0=ℏX\nijZ\ndω¯dq\ni(ω)ΣR\n↑↓ij(0)dcl\nj(ω)\n+ℏX\nijZ\ndω¯dcl\ni(ω)ΣA\n↑↓ij(0)dq\nj(ω),(84)which is a constant longitudinal field that plays no role\nin the instantaneous reference frame, as discussed above.\nThe first-order term in frequency is finite even in equi-\nlibrium,\n[S¯qcl\n22+S¯clq\n22]1=ℏX\nijαijZ\ndt¯Diγq∂tDj, (85)\nand takes the form of a Gilbert damping term, includ-\ning both inter-lattice and intra-lattice contributions. The\nspin transfer torque coefficient αand the spin pumping\ncoefficient βare related to each other as a consequence of\nthe Onsager reciprocal relations [36]. In Appendix B we\nderive this relation, which is given in Eq. (B11), and de-\nrive an optical theorem relating the scattering matrices,\ngiven in Eq. (B12).\nSummarizing this section, we have found that the cor-\nrections to the magnon action Smin the presence of spin\nand charge accumulations in surrounding normal metals\nisS1+S21+S¯qcl\n22+S¯clq\n22+˜Sqq\n22+S¯qq\n22, and found that the\nfirst three of these contributions appear like magnetic\nfields and (in the low-frequency limit) like Gilbert-like\ndamping terms in the effective magnon action. Impor-\ntantly, we find both longitudinal and transverse fields in\nthe general case. The last two contributions to the ac-\ntion consist of coupled quantum fields and are the result\nof purely quantum effects. These terms are the subject\nof the next section.\nC. Fluctuating fields\nFrom the effective action in the last section, we were\nable to associate the ( q, cl) and ( cl, q) terms with longi-\ntudinal fields by comparing them with the magnon ac-\ntion in Eq. (41). Now, we must address the issue of\nhow to interpret the ( q, q) terms, which lack an ana-\nlog in the action described in Eq. (41). In this section,\nwe derive fluctuating forces from these terms by employ-\ning a Hubbard-Stratonovich (HS) transformation on the\nquadratic fields in the effective action, introducing aux-\niliary fields in the process. Commencing with the con-\ntribution from the term S¯qq\n22, we introduce the complex\nauxiliary field h¯qq\ni(in units of inverse second) via a con-\nventional Hubbard–Stratonovich transformation:\neiS¯qq\n22/ℏ= exp\u0014Z\ndtdt′X\nij¯dq\ni(t)iΣK\nij(t−t′)dq\nj(t′)\u0015\n=1\ndet [−iΣK]ZY\niD[h¯qq\ni] exp\u0014\niZ\ndtX\nih¯qq\ni(t)¯dq\ni(t) +h.c.−Z\ndtdt′X\nij¯h¯qq\ni(t)[−iΣK\nij(t−t′)]−1hqq\nj(t′)\u0015\n,\n(86)\nwhere a shorthand notation for the measure was in-\ntroduced as D[h¯qq\ni] = Π k\b\nd[Imh¯qq\ni(tk)]d[Reh¯qq\ni(tk)]/π\t\n,where kis the index used to order the discretization of12\nthe time coordinate. From the Gaussian form of Eq. (86),\nthe correlators of the auxiliary field can be identified as\n⟨h¯qq\ni(t)⟩= 0 (87a)\n⟨h¯qq\ni(t)h¯qq\nj(t′)⟩= 0 (87b)\n⟨¯h¯qq\ni(t)h¯qq\nj(t′)⟩=−iΣK\nij(t−t′). (87c)\nThe second term in the exponent is quadratic in the newfields, and gives no contribution to the magnon action,\nwhile the first term is linear in the magnon field diand is\ninterpreted as an effective transverse field in the magnon\naction.\nThe contribution from the terms ˜Sqq\n22+˜S¯q¯q\n22is HS trans-\nformed by performing an unconventional transformation\nin the two complex fields ˜hqq\niand˜hqq\niseparately:\nei˜S22/ℏ=1q\ndet [−2i˜ΣK\n↓↑]q\ndet [−2i˜ΣK\n↑↓]ZY\niD[˜hqq\ni] exp\"\niZ\ndtX\ni\u0010\n˜hqq(t)˜hqq(t)\u0011\ni\u0012\nd(t)\n¯d(t)\u0013\ni+h.c.\n−Z\ndtdt′X\nij\u0010\n˜hqq(t)˜hqq(t)\u0011\ni \n0 −i˜ΣK\n↓↑(t−t′)\n−i˜ΣK\n↑↓(t−t′) 0!−1\nij ˜hqq(t′)\n˜hqq(t′)!\nj#\n,(88)\nagain interpreting the exponent as an effective action in-\ncluding the field ˜hqq\ni, which has the correlators\n⟨˜hqq\ni(t)⟩= 0 (89a)\n⟨˜hqq\ni(t)˜hqq\nj(t′)⟩=−i˜ΣK\n↑↓ij(t−t′) (89b)\n⟨˜hqq\ni(t)˜hqq\nj(t′)⟩=−i˜ΣK\n↓↑ij(t−t′) (89c)\n⟨˜hqq\ni(t)˜hqq\nj(t′)⟩= 0. (89d)\nWe remark that the unconventional form of the Hubbard-\nStratonovich decoupling leads to non-zero correlators for\nequal fields, as opposed to the conventional approach\nwhere the non-zero correlators involve one field being\nthe complex conjugate of the other. The fields h¯qqand\n˜hqqare interpreted as fluctuating transverse fields with,\nin general, different amplitudes depending on the lattice\nsite, but with correlators between lattice sites. Compar-\ning the effective action in Eq. (86) and Eq. (88) with the\nmagnon action in Eq. (41), the components of the total\nfluctuating field Hfcan be identified as\nγµ0Hf\n+,i=−1√sih\n2˜hqq\ni+h¯qq\nii\n(90a)\nγµ0Hf\n−,i=−1√sih\n2˜hqq\ni+¯h¯qq\nii\n. (90b)In this expression, the factor of 2 arises from the uncon-\nventional nature of the Hubbard-Stratonovich transfor-\nmation in Eq. (88). The correlators between the Carte-\nsian components of the fluctuating field can be calculated\nusing Eq. (89) and Eq. (90),\n2√sisjγ2µ2\n0⟨Hf\nxiHf\nxj⟩= ImΣK\nij+ 4Im ˜Σ↑↓ij (91a)\n2√sisjγ2µ2\n0⟨Hf\nxiHf\nyj⟩=−ReΣK\nij−4Re˜Σ↑↓ij (91b)\n2√sisjγ2µ2\n0⟨Hf\nyiHf\nyj⟩= ImΣK\nij−4Im˜Σ↑↓ij, (91c)\nfrom which we conclude that the correlators in the fluc-\ntuating field Hfare real numbers. In Eq. (91), we omit-\nted the time arguments for notational simplicity. Fur-\nthermore, it is evident that for i=jandt=t′, the\ncorrelators in Eq. (91a) and (91c) are positive, aligning\nwith the conditions expected for representing the vari-\nance of a real field.13\nD. Equations of motion\nAfter HS decoupling the qqcomponents, the effective action reads\nSeff=−γℏµ0Z\ndt\"X\na√sa\u0000\nHstt\na++Hf\na+\u0001\n¯aq\na(t) +X\nb√sb\u0000\nHstt\nb−+Hf\nb−\u0001¯bq\nb(t) +h.c.#\n+ℏZ\ndt\"X\naa′βaa′¯aq\na∂tacl\na+X\nabβabaq\na∂tbcl\nb+X\nbaβba¯bq\nb∂t¯acl\na+X\nbb′βbb′¯bq\nb∂tbcl\nb′+h.c.#\n. (92)\nHaving cast the total action Sm+Seffin a form that is linear in the quantum fields ¯ aqand¯bqand their complex\nconjugates, we can integrate over these fields in the partition function, producing the functional delta function imposing\nthe semiclassical equations of motion for the fields aclandbcl[30]. Using acl\na=S+a/(ℏ√sa) and ¯bcl\nb=S+b/(ℏ√sb) in\nthe semiclassical limit, we find the coupled equations of motion:\ni∂tSi+=ℏ−1EiSi++siℏµ0γ\u0000\nHi++Hf\ni++Hstt\ni+\u0001\n−X\njβij∂tSj+, (93)\nas well as its complex conjugated counterpart. Both in the definition of this field and in Eq. (93), the upper sign holds\nfor sublattice A, and the lower sign holds for sublattice B. We find the Cartesian components by taking the real and\nimaginary parts and divide with ℏsito find an equation for the vector mi=Si/(ℏsi),\n∂tmi=τb\ni+τf\ni+τsp\ni+τstt\ni, (94)\nwhere\nτb\ni=−z×\u0000\nℏ−1Eimi+γµ0Hi\u0001\n(95a)\nτf\ni=−γµ0z×Hf\ni (95b)\nτstt\ni=−γµ0z×Hstt\ni (95c)\nτsp\ni=X\njReβijz×∂tmj+X\njImβijz×(z×∂tmj) (95d)\nis microscopic expressions for the bulk torque τb, the fluctuating torque τf, the spin pumping torque τsp, and the\nspin transfer torque τstt.\nVI. CONCLUSION\nIn this paper, we have presented a general quantum\ntheory of spin dynamics in magnet-normal metal systems,\ngeneralizing earlier results to a general antiferromagnetic\nor ferrimagnetic bipartite lattice. Spin and charge ac-\ncumulations in the normal metals influence the magne-\ntization dynamics in the magnet through spin transfer\ntorque, and the damping is enhanced due to spin pump-\ning, including both inter- and intra-lattice contributions.\nWe derived expressions for transverse fluctuating fields\narising due to the electron magnon interactions. These\nfields have contributions from equilibrium terms as well\nas charge and spin accumulation in the normal metals.\nWe found site-dependent shot noise contributions that\nare non-negligible at low temperatures.ACKNOWLEDGMENTS\nThis work was supported by the Research Council\nof Norway through its Centers of Excellence funding\nscheme, Project No. 262633, ”QuSpin”.\nAppendix A: Holstein-Primakoff transformation\nIn this Appendix, we discuss the transformations used\nto diagonalize the magnon Hamiltonian of Eq. (9). To\ngo from the SU(2) spin operators to bosonic annihila-\ntion and creation operators, we employ the Holstein-\nPrimakoff transformation [37, 38] at sublattices AandB\nand expand to the lowest order in the bosonic operators,\nassuming the antiferromagnet is close to the N´ eel state,\ni.e. that all spins on sublattice A(B) is close to being\nparallel (antiparallel) to the z-direction. At sublattice A,14\nwe expand\nˆSa+=ℏ√\n2sa\u0012\n1−ˆa†\naˆaa\n2sa\u00131/2\nˆaa≈ℏ√\n2saˆaa (A1)\nˆSa−=ℏ√\n2saˆa†\na\u0012\n1−ˆa†\naˆaa\n2sa\u00131/2\n≈ℏ√\n2saˆa†(A2)\nˆSaz=ℏ(sa−ˆa†\naˆaa), (A3)\nwhere aaannihilates a localized magnon and sais the\ntotal spin at lattice site a. In the expansion of the square\nroots in Eq. (A1) and Eq. (A2), we assumed sa≫1 and\nexpanded the square root to lowest order in 1 /sA. We\nhave employed the standard raising and lowering spin\noperators, defined as S±=Sx±iSy.\nSimilarly, at sublattice B, we expand\nˆSb+=ℏ√\n2sbˆb†\nb \n1−ˆb†\nbˆbb\n2sb!\n≈√\n2sbˆb†\nb, (A4)\nˆSb−=ℏ√\n2sb \n1−ˆb†\nbˆbb\n2sb!\nˆbb≈√\n2sbˆbb, (A5)\nˆSbz=ℏ\u0010\n−sb+ˆb†\nbˆbb\u0011\n, (A6)\nwhere ˆbannihilates a localized spin-up magnon.\nAppendix B: Relating spin transfer torque and spin\npumping coefficients\nWe relate the spin transfer pumping coefficients de-\nfined in Eq. (82) to the spin transfer coefficients found\nin Eq. (62) in the case of one normal metal reservoir us-\ning the Onsager reciprocal relations [36]. We start by\ndefining the pumped spin current (in units of electrical\ncurrent, i.e. Ampere) into normal metal as the change in\ntotal spin inside the antiferromagnetic due to spin pump-\ning, i.e.\nIS=−e\nℏX\njSjτsp\nj. (B1)\nThe appearance of Sj=ℏp\nsj(sj+ 1) is due to the way\nwe have defined the torques in the main text, causing\nthem to have the dimension of inverse time. The dynam-\nics of the localized magnetic moment µj=−γSjmjand\nthe spin current are driven by the external effective field\nHeffand the spin accumulation µS, which are the ther-\nmodynamic forces in our system. In linear response, we\ncan then write the equations for the spin dynamics and\nthe spin current in matrix form:\n\u0012−γSi∂tmi\nIS\u0013\n=\u0012Lmm\nijLms\ni\nLsm\njLss\u0013\u0012µ0Heff\nj\nµS/e\u0013\n, (B2)\nwhere the matrix elements 3 ×3 tensors that effectively\napply the relevant cross products to make Eq. (B2) con-\nsistent with the Landau-Lifshitz equation, and where\nwe use the Einstein summation convention for repeated\nLatin indices.1. Identifying Lsm\nInserting the spin pumping torque from Eq. (29), the\nspin current becomes\nIS=−Xj∂tmj, (B3)\nwhere we defined the 3 ×3 matrix Xjas\nXj=e\nℏSjX\nih\nαR\nij˜O+αI\nij˜O2i\n, (B4)\nand the 3 ×3 matrix ˜Oimplements the cross product\nz×v=˜Ovand can be defined in terms of the Levi-\nCivita tensor. The LLG equation in the absence of spin\naccumulation (causing the spin transfer torque to vanish)\nreads\n(1−αb˜O)∂tmi=˜O(−γµ0Heff\ni), (B5)\nwhere αbis the (bulk) Gilbert damping constant. Hence,\nwe identify\nLsm\nj=γXj˜O(1−αb˜O)−1. (B6)\n2. Identifying Lms\nInserting the spin transfer torque from Eq. (27) into\nthe LLGS equation in the absence of an effective field,\nwe find\n∂tmi=ℏ−1(1−αb˜O)−1h\nβI\ni˜O−βR\ni˜O2i\nµS,(B7)\nmeaning that we can identify the linear response coeffi-\ncient Lmsas (no Einstein summation)\nLms\ni=−Siγe\nℏ(1−αb˜O)−1h\nβI\ni˜O−βR\ni˜O2i\n. (B8)\n3. Deriving relations from the Onsager reciprocal\nrelations\nWe are now looking to employ Onsager’s reciprocal\nrelation:\n[Lsm\ni({−mj})]T=Lms\ni({mj}), (B9)\nwhere the superscript Tindicates a matrix transpose in\nthe 3 ×3 Cartesian space. Using the matrix identity\n˜O3=−˜O, we find that Eq. (B9) implies that\nβI\nj˜O−βR\nj˜O2=X\nih\nαI\nij˜O−αR\nij˜O2i\n(B10)\nThis equality is satisfied if\nβj=X\niαij, (B11)15\nwhich generalizes the result from Ref. 35. Inserting the\ndefinitions of these coefficients in the low-temperature\nlimit, we find that\nX\nnWnn\nj↑↓= iπX\ninmWnm\ni↓↑Wmn\nj↑↓, (B12)\nwhich we classify as a generalized optical theorem, since\nin the diagonal case i=j, we can rewrite the imaginary\npart of this to\nIm\"X\nnWnn\ni↑↓#\n=πX\ninm|Wnm\ni↓↑|2, (B13)\nwhich is reminiscent of the optical theorem in wave scat-\ntering theory.\nAppendix C: Contour fields and Keldysh rotations\nIn this Appendix, we show how the action can be writ-\nten in the ±basis, and introduce the Keldysh rotated\nfields, which differ in the case of fermionic and bosonic\nfields. In the ±field basis, the action of the scattering\n(electron) states, corresponding to the Hamiltonian in\nEq. (7), reads\nSe+S0=X\nsZ∞\n−∞dt¯c+\ns(iℏ∂t−ϵ)c+\ns\n−X\nsZ∞\n−∞dt¯c−\ns(iℏ∂t−ϵ)cs−\n=X\nsξt¯cξ\ns(iℏ∂t−ϵ)cξ\ns, (C1)\nwhere now csis a vector containing the scattering fields,\n¯csdenotes its complex conjugate, and ϵis a diagonal\nmatrix containing all energy eigenvalues of the scatter-\ning states. In the final line, we have written the time\nintegration as a sum for concise notation. Additionally,\nwe introduced the sum over ” ±” fields as a sum over\nξ={+,−}, with an implicit negative sign before the ”-”\nfield, i.e.P\nξ. . .ξ=. . .+−. . .−. A similar notation will\nalso be used for the magnon fields below. The negative\nsign ( ξ=−) in the integral in Eq. (C1) and in the other\nactions below originates from reversing the integrationlimits on the backward contour. The magnon action is\nSm=X\nξabt[¯aξ\na(iℏ∂t−EA\nab)aξ\na+¯bξ\nb(iℏ∂t−EB\nab)bξ\nb]\n−2X\naa′Jaa′√sasa′¯aξ\naaξ\na′\n−2X\nbb′Jbb′√sbsb′¯bξ\nbbξ\nb′\n−2X\nξabtJab√sasb[aξ\nabξ\nb+ ¯aξ\na¯bξ\nb]\n−γµ0ℏX\nξatrsa\n2[HA\na−aξ\na+HA\na+¯aξ\na]\n−γµ0ℏX\nξbtrsb\n2[HB\nb−¯bξ\nb+HB\nb+bξ\nb]. (C2)\nThe first-order electron-magnon interaction is\nS1=−X\nξat\nαβr\n2\nsah\naξ\na¯cξ\n↓αWαβ\na↓↑cξ\n↑β+ ¯aξ\na¯cξ\n↑αWαβ\na↑↓cξ\n↓βi\n−X\nξbt\nαβr\n2\nsbh\n¯bξ\nb¯cξ\n↓αWαβ\nb↓↑cξ\n↑β+bξ\nb¯cξ\n↑αWαβ\nb↑↓cξ\n↓βi\n,\n(C3)\nand the second-order term is\nS2=X\nξat\nαβ1\nsa¯aξ\naaξ\nah\n¯cξ\n↑αWαβ\na↑↑cξ\n↑β−¯cξ\n↓αWαβ\na↓↓cξ\n↓βi\n−X\nξbt\nαβ1\nsb¯bξ\nbbξ\nbh\n¯cξ\n↑αWαβ\nb↑↑cξ\n↑β−¯cξ\n↓αWαβ\nb↓↓cξ\n↓βi\n.(C4)\nFor a general bosonic field ϕ, the classical ( cl) and\nquantum ( q) fields are defined as [30]:\nϕcl/q=1√\n2(ϕ+±ϕ−)¯ϕcl/q=1√\n2(¯ϕ+±¯ϕ−).(C5)\nIn our case, we have ϕ={a, b}. The upper (lower) sign\nholds for the classical (quantum) fields. For a fermionic\nfield c, the rotated fields are denoted by 1 and 2, and\ndefined as\nc1/2=1√\n2(c+±c−) ¯c1/2=1√\n2(¯c+∓¯c−).(C6)\nFor fermions, ¯ candcare independent variables, not re-\nlated by complex conjugation.\nAppendix D: Fourier transform\nFor a general function of relative time t−t′, we define\nthe Fourier transform between the relative time domain16\nand the energy domain as\nf(ω) =Z∞\n−∞d(t−t′)eiω(t−t′)f(t−t′), (D1)\nf(t−t′) =Z∞\n−∞dω\n2πe−iω(t−t′)f(ω). (D2)\nThe delta function can be represented as\nδ(t−t′) =Z∞\n−∞dω\n2πe−iω(t−t′), (D3)\nδ(ω) =1\n2πZ∞\n−∞d(t−t′)eiω(t−t′). (D4)Finally, we note a frequently employed identity,\n−iZ∞\n−∞d(t−t′)eiω(t−t′)θ(t−t′) = (ω+ iδ)−1,(D5)\niZ∞\n−∞d(t−t′)eiω(t−t′)θ(t′−t) = (ω−iδ)−1,(D6)\nwhere δis an infinitesimal positive quantity.\n[1] A. Hirohata, K. Yamada, Y. Nakatani, I.-L. Prejbeanu,\nB. Di´ eny, P. Pirro, and B. Hillebrands, Journal of Mag-\nnetism and Magnetic Materials 509, 166711 (2020).\n[2] A. Brataas, B. van Wees, O. Klein, G. de Loubens, and\nM. Viret, Phys. Rep. 885, 1 (2020).\n[3] L. Berger, Phys. Rev. B 54, 9353 (1996).\n[4] J. Slonczewski, Journal of Magnetism and Magnetic Ma-\nterials 159, L1 (1996).\n[5] A. Brataas, Yu. V. Nazarov, and G. E. W. Bauer, Phys-\nical Review Letters 84, 2481 (2000).\n[6] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys.\nRev. Lett. 88, 117601 (2002).\n[7] B. Heinrich, Y. Tserkovnyak, G. Woltersdorf, A. Brataas,\nR. Urban, and G. E. W. Bauer, Phys. Rev. Lett. 90,\n187601 (2003).\n[8] Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I.\nHalperin, Rev. Mod. Phys. 77, 1375 (2005).\n[9] J. Foros, A. Brataas, Y. Tserkovnyak, and G. E. W.\nBauer, Phys. Rev. Lett. 95, 016601 (2005).\n[10] A. L. Chudnovskiy, J. Swiebodzinski, and A. Kamenev,\nPhys. Rev. Lett. 101, 066601 (2008).\n[11] J. Swiebodzinski, A. Chudnovskiy, T. Dunn, and\nA. Kamenev, Phys. Rev. B 82, 144404 (2010).\n[12] S. A. Bender, R. A. Duine, and Y. Tserkovnyak, Phys.\nRev. Lett. 108, 246601 (2012).\n[13] S. A. Bender, R. A. Duine, A. Brataas, and\nY. Tserkovnyak, Phys. Rev. B 90, 094409 (2014).\n[14] B. Divinskiy, H. Merbouche, V. E. Demidov, K. O. Niko-\nlaev, L. Soumah, D. Gou´ er´ e, R. Lebrun, V. Cros, J. B.\nYoussef, P. Bortolotti, A. Anane, and S. O. Demokritov,\nNature Communications 12, 6541 (2021).\n[15] L. N´ eel, Magnetism and the local molecular field, Nobel\nLecture (1970).\n[16] T. Jungwirth, X. Marti, P. Wadley, and J. Wunderlich,\nNat. Nanotechnol. 11, 231 (2016).\n[17] S. M. Rezende, A. Azevedo, and R. L. Rodr´ ıguez-Su´ arez,\nJ. Appl. Phys. 126, 151101 (2019).\n[18] A. S. N´ u˜ nez, R. A. Duine, P. Haney, and A. H. MacDon-\nald, Phys. Rev. B 73, 214426 (2006).[19] Z. Wei, A. Sharma, A. S. Nunez, P. M. Haney, R. A.\nDuine, J. Bass, A. H. MacDonald, and M. Tsoi, Phys.\nRev. Lett. 98, 116603 (2007).\n[20] V. Baltz, A. Manchon, M. Tsoi, T. Moriyama, T. Ono,\nand Y. Tserkovnyak, Reviews of Modern Physics 90,\n015005 (2018).\n[21] R. Cheng, J. Xiao, Q. Niu, and A. Brataas, Phys. Rev.\nLett. 113, 057601 (2014).\n[22] P. Vaidya, S. A. Morley, J. van Tol, Y. Liu, R. Cheng,\nA. Brataas, D. Lederman, and E. del Barco, Science 368,\n160 (2020).\n[23] F. Keffer and C. Kittel, Phys. Rev. 85, 329 (1952).\n[24] S. A. Gulbrandsen and A. Brataas, Phys. Rev. B 97,\n054409 (2018).\n[25] Ø. Johansen and A. Brataas, Phys. Rev. B 95, 220408\n(2017).\n[26] J. Barker and O. A. Tretiakov, Physical Review Letters\n116, 147203 (2016).\n[27] A. Kamra and W. Belzig, Phys. Rev. Lett. 119, 197201\n(2017).\n[28] A. Kamra, R. E. Troncoso, W. Belzig, and A. Brataas,\nPhys. Rev. B 98, 184402 (2018).\n[29] J. Tang and R. Cheng, APL Materials 11, 111117 (2023).\n[30] A. Kamenev, Field Theory of Non-Equilibrium Systems\n(Cambridge University Press, Cambridge ; New York,\n2011).\n[31] L. V. Keldysh, Zh. Eksp. Teor. Fiz. (1964).\n[32] S. Takei, Phys. Rev. B 100, 134440 (2019).\n[33] R. A. Duine, A. S. N´ u˜ nez, J. Sinova, and A. H. MacDon-\nald, Physical Review B 75, 214420 (2007).\n[34] C. Tassi, M. Barbieri, and R. Raimondi, Physical Review\nB100, 184418 (2019).\n[35] A. Brataas, Phys. Rev. B 106, 064402 (2022).\n[36] A. Brataas, Y. Tserkovnyak, G. E. W. Bauer, and\nP. J. Kelly, Spin Pumping and Spin Transfer (2012),\narxiv:1108.0385.\n[37] T. Holstein and H. Primakoff, Phys. Rev. 58, 1098\n(1940).\n[38] S. M. Rezende and R. M. White, Physical Review B 14,\n2939 (1976)."    },    {        "title": "2110.11152v1.The_spin_spin_problem_in_Celestial_Mechanics.pdf",        "content": "THE SPIN-SPIN PROBLEM IN CELESTIAL MECHANICS\nALESSANDRA CELLETTI, JOAN GIMENO, AND MAURICIO MISQUERO\nAbstract. We study the dynamics of two homogeneous rigid ellipsoids subject to\ntheir mutual gravitational in\ruence. We assume that the spin axis of each ellipsoid\ncoincides with its shortest physical axis and is perpendicular to the orbital plane. Due\nto such assumptions, the problem is planar and depends on particular parameters of\nthe ellipsoids, most notably, the equatorial oblateness and the \rattening with respect\nto the shortest physical axes. We consider two models for such con\fguration: while\nin the full model, there is a coupling between the orbital and rotational motions, in\nthe Keplerian model, the centers of mass of the bodies are constrained to move on\ncoplanar Keplerian ellipses. The Keplerian case, in the approximation that includes\nthe coupling between the spins of the two ellipsoids, is what we call spin-spin problem,\nthat is a generalization of the classical spin-orbit problem. In this paper we continue the\ninvestigations of [Mis21] on the spin-spin problem by comparing it with the spin-orbit\nproblem and also with the full model.\nBeside detailing the models associated to the spin-orbit and spin-spin problems, we\nintroduce the notions of standard and balanced resonances, which lead us to investi-\ngate the existence of periodic and quasi-periodic solutions. We also give a qualitative\ndescription of the phase space and provide results on the linear stability of solutions for\nthe spin-orbit and spin-spin problems. We conclude by providing a comparison between\nthe full and the Keplerian models with particular reference to the interaction between\nthe rotational and orbital motions.\n1.Introduction\nThe dynamics of two rigid bodies orbiting under their mutual gravitational attraction is\na classical problem of Celestial Mechanics known as the Full Two-Body problem . In this\ncontext, Kinoshita investigated the problem by using Hori-Deprit perturbation theory\n[Kin72], assuming that one of the bodies is spherical and the other body is triaxial. Later,\nthe problem of two extended rigid bodies was studied in [Mac95] as a Hamiltonian system\nwith respect to a non-canonical structure, which is used to characterize the relative\nequilibria. A seminal work was performed in [Bou17] to which we refer for an alternative\ndescription of the model of the full two rigid body problem using spherical harmonics\nand Wigner D-matrices. In [Sch09], the problem is restricted to a planar con\fguration\nwith the potential expanded to order 1 =r3, whereris the relative distance between the\ntwo rigid bodies; under this condition, [Sch09] describes the relative equilibria and their\nstability properties.\nKey words and phrases. Spin-spin model jSpin-orbit model jTwo-body problem jResonancesj\nPeriodic orbitsjQuasi-periodic solutions.\nA.C. and M.M. thank MSCA-ITN-ETN Stardust-R, Grant Agreement 813644 and J.G. thanks MIUR-\nPRIN 20178CJA2B \\New Frontiers of Celestial Mechanics: theory and Applications\". J.G. has also been\nsupported by the Spanish grants PGC2018-100699-B-I00 (MCIU/AEI/FEDER, UE) and the Catalan\ngrant 2017 SGR 1374.\n1arXiv:2110.11152v1  [math.DS]  21 Oct 20212 A. CELLETTI, J. GIMENO, AND M. MISQUERO\nIn this paper, we investigate di\u000berent simpli\fed models of rotational dynamics of\ncelestial bodies, subject to the mutual gravitational attraction. The spin-spin problem\nwas introduced in [Mis21] as a planar version of the Full Two-Body problem for ellipsoids\n(compare with [BM15, BL09]), by using the expansion of the potential up to order 1 =r5,\nwhich results in the coupling of the spins of both bodies. An equivalent model was\nstudied in [JA16] (see also [HX17]).\nIndeed, we consider a hierarchy of models with di\u000berent complexity. In particular, we\nstart by considering two homogeneous rigid ellipsoidal bodies subject to the following\nassumptions:\nAssumption 1. The spin axis of each ellipsoid is perpendicular to the orbital plane.\nAssumption 2. The spin axis of each ellipsoid is aligned with the shortest physical axis\nof the satellite.\nAssumptions 1 and 2 imply that the motion takes place on a plane. Following [Mis21],\nwe introduce a Hamiltonian function that includes both the orbital and rotational mo-\ntions. Using the conservation of the angular momentum, the system is described by a\nHamiltonian with 3 degrees of freedom that depends on several parameters of each ellip-\nsoid, among which there are the equatorial oblateness and the \rattening with respect to\nthe shortest physical axis. Such parameters are typically small for natural bodies of the\nsolar system. We refer to this model as the fullproblem, since it includes the coupling\nbetween the orbital and rotational motions.\nThe potential of the problem can be written as V=V0+P1\nl=1V2l, whereV0denotes\nthe Keplerian potential and the terms V2lare proportional to 1 =r2l+1, whereris the\ninstantaneous distance of the two centers of mass. If we consider the expansion up to\norderl= 2, sayV=V0+V2+V4, we obtain that the model includes the coupling of\nthe spins of the two ellipsoids through the term V4, which contains combinations of the\nrotation angles of the two satellites. When the two spins interact, we refer to the problem\nas the full spin-spin model. If, instead, we limit the potential to V=V0+V2, then we\nobtain two decoupled systems, the full spin-orbit models (see, e.g., [Bel66, Cel90, GP66]).\nWhen one of the bodies is spherical, its spin is uniform and the dynamics of the spin-spin\nmodel becomes very similar to that of the spin-orbit model, but including the terms in\nV4.\nNext, we introduce another assumption, namely:\nAssumption 3. The orbital motion of the ellipsoids coincides with that of two point\nmasses, so that both centers of mass move on coplanar Keplerian orbits with eccentricity\ne2[0;1) and with a common focus at the barycenter of the system.\nAssumption 3 implies that the orbit is not a\u000bected by the rotational motion. To\nthefullspin-spin and spin-orbit problems, it corresponds the Keplerian spin-spin and\nspin-orbit models, described by a Hamiltonian function with an explicit periodic time\ndependence.\nNote that we are considering rigid bodies only, which means that dissipative e\u000bects\ndue to tidal torques are not considered. We refer to [CC08, CC09, MO20, Mis21, GP66]\nfor a description of the dissipative spin-orbit and spin-spin problems.THE SPIN-SPIN PROBLEM IN CELESTIAL MECHANICS 3\nThe previously described models have some symmetries that are a direct consequence\nof the mirror symmetries of the ellipsoids. This fact leads us to introduce the following\ntwo types of resonances within the spin-orbit problem of a single ellipsoid:\n(R1) We call standardm:nspin-orbit resonance, for some integers m,n, when the\nspinning body makes mrotations during norbital revolutions;\n(R2) We call balancedm: 2 spin-orbit resonance, for some integers m, when the\nspinning body makes m=2 of a rotation during one orbital revolution.\nWe remark that a balanced spin-orbit resonance is also a standard resonance, but the\nconverse is not always true. For example, a balanced 2 k: 2 resonance for k2Zis\nequivalent to a k: 1 spin-orbit resonance, but there is not such an equivalence for order\n(2k+ 1) : 2. Both de\fnitions ( R1) and ( R2) extend to the spin-spin problem of type\n(m1:n1;m2:n2) for integers m1,m2,n1,n2, when the \frst ellipsoid is in a m1:n1\nspin-orbit resonance and the second ellipsoid in a m2:n2spin-orbit resonance.\nWe also stress that spin-orbit resonances \fnd many applications in the solar system;\nin fact, the Moon is an example of a 1 : 1 spin-orbit resonance1, since it makes a rotation\nin the same period it takes to make an orbit around the Earth. This is also called a\nsynchronous spin-orbit resonance, which is common to many satellites of other planets,\nincluding Mars, Jupiter, Saturn, Uranus, Neptune. Among the planets, Mercury is locked\nin a 3 : 2 spin-orbit resonance around the Sun. On the other hand, the Pluto-Charon\nsystem is locked in the double synchronous spin-spin resonance (1 : 1 ;1 : 1).\nIn this work, we study the behavior of the solutions of the spin-orbit and spin-spin\nproblems as the parameters and the initial conditions are varied. In particular, we\ninvestigate the boundary conditions that lead to the existence of symmetric periodic\norbits. Such results (see Propositions 5 and 10) use some symmetry properties of the\nequations of motion. We remark that these symmetries are lost if we include dissipation;\nhowever, such periodic solutions might be continued to the dissipative setting as shown\nin [MO20, Mis21]. Beside the study of the periodic orbits, we provide the conditions for\nthe existence of quasi-periodic solutions of the Keplerian version of the spin-spin model.\nWe also give a qualitative study of the spin-orbit problem as well as the spin-spin\nproblem with spherical and non-spherical companion. Within such investigation, we\ndiscover some new features, like the measure synchronization (see, e.g., [HZ99]) for the\nspin-spin problem with identical bodies. Our study leads to analyze the multiplicity\nof solutions and the linear stability of the periodic orbits (compare with [CC00]). In\ngeneral, we \fnd that there is not a unique solution associated to a particular resonance;\nhowever, for some values of the parameters such a uniqueness exists. Finally, we provide\nsome results on the comparison between the full and Keplerian models, as well as on\nthe interaction between the spin and the orbital motion, motivated by the fact that the\ncoupling between the rotational and orbital motions has not been much explored in the\nliterature.\nThis work is organized as follows. In Section 2 we present the spin-orbit and spin-spin\nmodels. The de\fnition of resonances and the existence of periodic and quasi-periodic\norbits are given in Section 3. A qualitative description of the phase space is given in\n1A 2 : 2 balanced spin-orbit resonance is equivalent to a 1 : 1 standard spin-orbit resonance.4 A. CELLETTI, J. GIMENO, AND M. MISQUERO\nSection 4. The linear stability of symmetric periodic orbits is investigated in Section 5.\nFinally, a comparison between the full and Keplerian models is presented in Section 6.\n2.The models for the spin-orbit and spin-spin coupling\nThe aim of this section is to present the so-called spin-orbit andspin-spin models, that\nwe are going to introduce as follows. The assumptions and the notations are given in\nSection 2.1; di\u000berent models, subject to some or all the assumptions listed in Section 2.1,\nare presented in Sections 2.2 and 2.3.\n2.1.General assumptions. Consider two homogeneous rigid ellipsoids, say E1andE2,\nwith masses M1andM2, respectively. Let Aj<Bj<Cj,j= 1;2, be their principal\nmoments of inertia with corresponding principal semi-axes aj>bj>cj. We refer to the\nFull Two-Body Problem (hereafter F2BP) as the problem of two rigid bodies interacting\ngravitationally (see, e.g., [Sch02, Sch09]). When the bodies have ellipsoidal shape, we\nspeak of the ellipsoidal F2BP , where we make the assumptions 1, 2, 3 of Section 1:\norbit of\norbit of\nFigure 1. The planar spin-spin problem considering Assumptions 1 to 3.\nAssumptions 1 and 2 guarantee that the problem we deal with is a planar problem.\nAdditionally, Assumption 3 restricts the problem so that we obtain a model with two\ndegrees of freedom and a periodic time dependence. This assumption is equivalent to\nsay that the spin motion will not in\ruence the orbital motion. Besides, note that we\nare neglecting the gravitational contribution of other bodies, we are not considering any\ndissipative e\u000bect that might arise, for example, from the non-rigidity of the ellipsoidal\nbodies and we do not take into account the obliquity, namely the inclination of the\nspin-axes with respect to the orbital plane.\nWe are going to work with units adapted to the system. If we call \u001cthe orbital period\nof the Keplerian orbit, then we will use units such that\nM1+M2= 1;C1+C2= 1; \u001c = 2\u0019 :\nRecall Kepler's third law for the Two-Body Problem\nG(M1+M2)\u0010\u001c\n2\u0019\u00112\n=a3;THE SPIN-SPIN PROBLEM IN CELESTIAL MECHANICS 5\nwhereGis the gravitational constant, ais the semi-major axis associated to the motion\nof the reduced mass of the system, say \u0016=M1M2in our units. In consequence, G=a3\nin our units.\nLet us now de\fne the parameters for each ellipsoid\ndj=Bj\u0000Aj; qj= 2Cj\u0000Bj\u0000Aj;\nthe quantity dj=Cjmeasures the equatorial oblateness of each ellipsoid with respect to\nthe plane formed by the directions of ajand bj, whereasqj=Cjmeasures the \rattening\nwith respect to the direction corresponding to the cj-axis.\nNote that ifAj\u0014Bj\u0014Cj, then, in our units there are some bounds for the parameters\nof the system given by\n0\u0014dj\u0014Cj\u00141; dj\u0014qj\u00142Cj\u00142; Mja2\nj=5\n2(Cj+dj)\u00145Cj\u00145: (1)\nThe last relation in (1) comes from the fact that the moments of inertia of an ellipsoid\nhold the identities\nAj=1\n5Mj(b2\nj+c2\nj);Bj=1\n5Mj(a2\nj+c2\nj);Cj=1\n5Mj(a2\nj+b2\nj):\n2.2.The full models. First, let us derive the equations of the full models of spin-orbit\nand spin-spin coupling, for which only Assumptions 1 and 2 hold, namely we do not\nconstrain the centers of mass of E1andE2to move on Keplerian ellipses.\nThe equations of motion are obtained by computing the Hamiltonian function through\na Legendre transformation of the Lagrangian, say L=T\u0000V, whereTis the kinetic\nenergy and Vthe potential energy of the system. We split Tin two parts, associated\nrespectively to the orbital and rotational motions, say T=Torb+Trot.\nLet us identify the orbital plane with the complex plane C, consider the center of mass\nof the system \fxed in the origin and let the position of each ellipsoid be rj2C. Then,\nby de\fnition of the barycenter we have\nM1r1+M2r2=0:\nIf we de\fne the relative position vector r=r2\u0000r1, since in our units M1+M2= 1, then\nwe have\nr1=\u0000M2r;r2=M1r: (2)\nFigure 2. Generalized coordinates of the full planar model.6 A. CELLETTI, J. GIMENO, AND M. MISQUERO\nWe introduce the Lagrangian generalized coordinates ( r;f;\u0012 1;\u00122), illustrated in Fig-\nure 2, in the following way. The distance r > 0 and the angle fde\fne the relative\nposition vector r=rexp(if)2Cwith respect to an inertial reference frame with a \fxed\nx-axis. The angles \u00121,\u00122provide the orientation of the axes a1,a2with respect to the\nx-axis. The variables randfde\fne the orbital motion of the system and \u0012jthe spin\nmotion of the ellipsoid Ej.\nTo write the kinetic energy, we notice that we can express rin components as r=\n(rcosf;rsinf), which gives _r= ( _rcosf\u0000r_fsinf;_rsinf+r_fcosf). Then, using (2),\nthe total orbital and the rotational kinetic energies are given by\nTorb=1\n2(M1_r2\n1+M2_r2\n2) =\u0016\n2_r2=\u0016\n2( _r2+r2_f2); Trot=1\n2C1_\u00122\n1+1\n2C2_\u00122\n2;\nwhere\u0016=M1M2in our units. The Lagrangian is given by\nL(r;f;\u0012 1;\u00122;_r;_f;_\u00121;_\u00122) =Torb(r;_r;_f) +Trot(_\u00121;_\u00122)\u0000V(r;f;\u0012 1;\u00122);\nwhere, according to [Mis21, Bou17], the full expansion of the potential energy of the\nsystemV=V(r;f;\u0012 1;\u00122) takes the form\nV(r;f;\u0012 1;\u00122) =\u0000GM 1M2\nrX\n(l1;m1)2\u0007\n(l2;m2)2\u0007\u0003l1;m1\nl2;m2\nr2(l1+l2)cos(2m1(\u00121\u0000f) + 2m2(\u00122\u0000f)); (3)\nwhere\n\u0007 =f(l;m)2Z2: 0\u0014jmj\u0014lg;\nand the constants \u0003l1;m1\nl2;m2are de\fned in Appendix A; we refer to [Mis21] for full details.\nLet us de\fne the momenta as\npr=@_rL=\u0016_r; pf=@_fL=\u0016r2_f; pj=@_\u0012jL=Cj_\u0012j;\nthen, the Hamiltonian of the system becomes\nH(r;f;\u0012 1;\u00122;pr;pf;p1;p2) =p2\nr\n2\u0016+p2\nf\n2\u0016r2+p2\n1\n2C1+p2\n2\n2C2+V(r;f;\u0012 1;\u00122): (4)\nConsequently, the equations of motion are\n_r=pr\n\u0016;_f=pf\n\u0016r2;_pr=p2\nf\n\u0016r3\u0000@rV; _pf=\u0000@fV (5)\nand\n_\u0012j=pj\nCj;_pj=\u0000@\u0012jV : (6)\nWe remark that with these variables, the problem splits in two parts: equations (5)\ndescribe the orbital motion, while equations (6) describe the rotational motion. The\nevolution of both parts is coupled through the potential V.\nThe potential V, expanded in (3), can be written in a perturbative way as\nV(r;f;\u0012 1;\u00122) =V0(r) +Vper(r;f;\u0012 1;\u00122); V 0(r) =\u0000GM 1M2\nr: (7)\nWe remark that the term V0corresponds to the classical Keplerian form for the poten-\ntial and the term Vperprovides the coupling between the spin and the orbital motions.THE SPIN-SPIN PROBLEM IN CELESTIAL MECHANICS 7\nMoreover, we can expand VperasVper=P1\nl=1V2l;whereV2lare suitable terms propor-\ntional to 1=r2l+1. A truncation of the expansion of Vperwill result in an approximated\ndynamics of our system. The explicit expressions of the \frst two terms of such a expan-\nsion are given in [Mis21] and we report them here:\nV2=\u0000GM 2\n4r3(q1+ 3d1cos(2\u00121\u00002f))\u0000GM 1\n4r3(q2+ 3d2cos(2\u00122\u00002f));\nV4=\u00003G\n43r5(\n12q1q2+15\n7[M2\nM1d2\n1+ 2M2\nM1q2\n1+M1\nM2d2\n2+ 2M1\nM2q2\n2]\n+d1M2\u001a\n[20q2\nM2+100\n7q1\nM1] cos(2\u00121\u00002f) + 25d1\nM1cos(4\u00121\u00004f)\u001b\n+d2M1\u001a\n[20q1\nM1+100\n7q2\nM2] cos(2\u00122\u00002f) + 25d2\nM2cos(4\u00122\u00004f)\u001b\n+6d1d2cos(2\u00121\u00002\u00122) + 70d1d2cos(2\u00121+ 2\u00122\u00004f))\n: (8)\nFrom now on, we will refer to (5) and (6) as the full models: full spin-orbit model if we\ntakeVper=V2, in which the angles \u00121,\u00122appear in di\u000berent trigonometric terms, and full\nspin-spin model ifVper=V2+V4, which contains trigonometric terms with combinations\nof the rotation angles \u00121,\u00122. These names are motivated by the well-known spin-orbit\nmodel, [Cel10], and the spin-spin model from [Mis21]. If we consider the models under\nAssumption 3 which gives a constraint on the orbit, we speak of Keplerian spin-orbit\nmodel andKeplerian spin-spin model (compare with Section 2.3).\n2.2.1. Conservation of the angular momentum. Note that the Hamiltonian Hin (4) is\ninvariant under the transformation ( r;f;\u0012 1;\u00122)7!(r;f+\u000ef;\u0012 1+\u000ef;\u0012 2+\u000ef), where\u000ef\nis an in\fnitesimal angular increase, because the angular arguments of V(r;f;\u0012 1;\u00122) only\ndepend on the di\u000berences \u00121\u0000fand\u00122\u0000f. This symmetry is related, by Noether's\ntheorem [Gol80], with a conserved quantity, say, the total angular momentum pf+p1+p2.\nThis can be proved through the following change of variables\n(r;f;\u0012 1;\u00122;pr;pf;p1;p2)7!(r;f;\u001e 1;\u001e2;pr;Pf;p1;p2);\nwhere\n\u001ej=\u001ej(f;\u0012j) =\u0012j\u0000f; Pf=Pf(pf;p1;p2) =pf+p1+p2: (9)\nThe transformation of coordinates (9) is canonical, since\ndr^dpr+ df^dpf+2X\nj=1d\u0012j^dpj= dr^dpr+ df^dPf+2X\nj=1d\u001ej^dpj:\nThen, the Hamiltonian (4) in this new set of variables is given by\nH(r;f;\u001e 1;\u001e2;pr;Pf;p1;p2) =p2\nr\n2\u0016+(Pf\u0000p1\u0000p2)2\n2\u0016r2+p2\n1\n2C1+p2\n2\n2C2+V(r;\u001e1;\u001e2);(10)8 A. CELLETTI, J. GIMENO, AND M. MISQUERO\nwhereV(r;\u001e1;\u001e2) =V(r;f;\u001e 1+f;\u001e 2+f). Now it is clear that fis an ignorable variable\nin (10) and that Pfis a constant of motion, corresponding to the total angular momentum\nof the system.\nIn summary, in the evolution of the system, there is a transfer of angular momentum\nbetween the spin part, given for each body by pj=Cj_\u0012j, and the orbital part, given by\npf=\u0016r2_f.\n2.3.The Keplerian models. In this section, we introduce the Assumption 3 to the\nmodel of Section 2.2. From (5) and (6), it is straightforward to constrain the orbit\nto be Keplerian; only in the orbital part we retain just the term V0of the potential\nV=V0+Vperin (7), thus obtaining the following equations:\n_r=pr\n\u0016;_f=pf\n\u0016r2;_pr=p2\nf\n\u0016r3\u0000@rV0;_pf=\u0000@fV0= 0; (11)\nand\n_\u0012j=pj\nCj;_pj=\u0000@\u0012jV=\u0000@\u0012jVper: (12)\nA convenient procedure to numerically integrate the equations of motion (12) is pre-\nsented in Appendix B.\n2.3.1. Orbital motion. Note that since @\u0012jV0= 0, the system (11) is now decoupled from\n(12). Moreover, (11) is the Kepler problem, whose solutions depend on the eccentricity\neand the semi-major axis aof the orbit. Here we assume for simplicity that the orbit\nis a 2\u0019-periodic Keplerian ellipse of eccentricity e2[0;1) with focus at the origin and\nwith the periapsis on the positive x-axis.\nSince the orbital period is 2 \u0019, then, we can take the time tto coincide with the mean\nanomaly . We denote by utheeccentric anomaly , which, in our units, is related to the\nmean anomaly by Kepler's equation\nt=u\u0000esinu : (13)\nThe orbital radius is related to uby\nr=a(1\u0000ecosu): (14)\nWe can write the vector r2Cin terms of the eccentric anomaly also as\nrexp(if) =a(cosu\u0000e+ip\n1\u0000e2sinu): (15)\nNote that for t= 0 we assumed, without loss of generality, that f=u= 0, and\nconsequently, f=u=\u0019whent=\u0019. From (15) we obtain the following useful relations\nbetweenfandu\ncosf=cosu\u0000e\n1\u0000ecosu; sinf=p\n1\u0000e2sinu\n1\u0000ecosu: (16)\nWith the previous de\fnitions, the Keplerian orbit of eccentricity eand semi-major\naxisais given by the functions\nr=r(t;a;e); f =f(t;e); pr=pr(t;a;e); pf=pf(a;e) =\u0016a2p\n1\u0000e2; (17)\nthat correspond to the solution of equations (11) generated by the initial conditions\nr(0) =a(1\u0000e); pr(0) = 0; f(0) = 0; pf(0) =\u0016a2p\n1\u0000e2: (18)THE SPIN-SPIN PROBLEM IN CELESTIAL MECHANICS 9\n2.3.2. Spin motion. The spin motion is described by equations (12) with the Keplerian\nperiodic input (17) given implicitly by eqs. (13) to (15). This motion can be described\nby the non-autonomous Hamiltonian\nHK(t;\u00121;\u00122;p1;p2) =H(r(t;a;e);f(t;e);\u00121;\u00122;pr(t;a;e);pf(a;e);p1;p2); (19)\nwhereH(r;f;\u0012 1;\u00122;pr;pf;p1;p2) is the Hamiltonian of the full model de\fned in (4). The\nHamiltonian (19) is hence of the form\nHK(t;\u00121;\u00122;p1;p2) =p2\n1\n2C1+p2\n2\n2C2+W(t;\u00121;\u00122); (20)\nwhere the potential Wis 2\u0019-periodic in tand\u0019-periodic in \u00121and\u00122. The equations of\nmotion (12) take the form\n_\u0012j=pj\nCj;_pj=\u0000@\u0012jW(t;\u00121;\u00122): (21)\nLet us de\fne the non-dimensional parameters of the model:\n\u0015j= 3\u0016\nMjdj\nCj; \u001bj=1\n3Cj\n\u0016a2; ^qj=qj\nMja2; (22)\nwhere\u0015jrepresents the equatorial oblateness of Ej;\u001bjis the ratio between the moment\nof inertia ofEjand the orbital one; and ^ qjmeasures the \rattening of Ejwith respect\nto the size of the orbit. Note that the parameters in (22) are small for bodies that are\nclose to spherical. Besides, not all the parameters de\fned previously are free, because\nwe have the constraint C1\u001b2=C2\u001b1.\nIf we takeVper=V2, then the system (12) becomes\n\u0012j+\u0015j\n2\u0012a\nr(t;e)\u00133\nsin(2\u0012j\u00002f(t;e)) = 0; j = 1;2; (23)\nthat is a system of two uncoupled spin-orbit problems. Each of these problems depends\njust on two parameters: ( e;\u0015j). On the other hand, if Vper=V2+V4, from (12) and (8)\nwe obtain the following system for j= 1;2,\n0 =\u0012j+\u0015j\n2(\u0012a\nr(t;e)\u00133\nsin(2\u0012j\u00002f(t;e))+\n+\u0012a\nr(t;e)\u00135\"\n5\n4\u0012\n^q3\u0000j+5\n7^qj\u0013\nsin(2\u0012j\u00002f(t;e)) +25\n8\u0015j\u001bjsin(4\u0012j\u00004f(t;e))\n+\u00153\u0000j\u001b3\u0000j\u00123\n8sin(2\u0012j\u00002\u00123\u0000j) +35\n8sin(2\u00123\u0000j+ 2\u0012j\u00004f(t;e))\u0013#)\n;(24)\nthat we call spin-spin problem. From the previous discussion, this model depends on\nseven independent parameters2(e;C1;\u00151;\u00152;\u001b1;^q1;^q2). Note that in (24), the coupling\nbetween the dynamics of \u00121and\u00122is given by \u001b1and\u001b2. Moreover, if ^ qj=\u001bj= 0, the\nspin-spin problem (24) is reduced to a pair of spin-orbit problems (23).\n2In [Mis21] there was an error because ^ q1and ^q2are actually independent, it is not always true that\nC1\u00151^q2=C2\u00152^q1, but it can be regarded as an additional constraint.10 A. CELLETTI, J. GIMENO, AND M. MISQUERO\nLet us now consider (24) in the case that E2is a sphere, that is, d2=q2= 0. Then,\n\u00152=\u001b2= ^q2= 0, which implies that E2is in uniform rotation \u00122(t) = _\u00122(0)t+\u00122(0).\nThe dynamics of \u00121is uncoupled from \u00122and is given by\n0 =\u00121+\u00151\n2n\u0012a\nr(t;e)\u00133\nsin(2\u00121\u00002f(t;e))+\n+\u0012a\nr(t;e)\u00135h25^q1\n28sin(2\u00121\u00002f(t;e)) +25\u00151\u001b1\n8sin(4\u00121\u00004f(t;e))io\n;(25)\nthat is a spin-orbit problem up to order 1 =r5. An equivalent system was studied previ-\nously in [JNA16]. Here the parameters \u001b1and ^q1perturb the framework of the spin-orbit\nproblem (23).\n2.3.3. Reversing symmetries. The equations for the spin motion in the Keplerian mod-\nels have some reversing symmetries, i.e., transformations in the phase space that keep\ninvariant the equations of motion with a time reversal.\nDe\fnition 1. Consider the di\u000berential equation\ndx\ndt=F(x);x2Rn; (26)\nand let x(t;x0) be the solution of (26) with initial condition x(0;x0) =x0.\n(1) A transformation R:Rn!Rnis called a reversing symmetry of (26) if\ndR(x)\ndt=\u0000F(R(x)):\n(2) The \fxed point set of Ris given by Fix( R) =fx2Rn:R(x) =xg.\n(3) An orbit o(x0) =fx(t;x0):t2RgisR-symmetric if R(o(x0)) =o(x0).\nThe system (23) with j= 1 can be written in the autonomous form (26) with\nx= (t;\u00121;_\u00121);F(x) = \n1;_\u00121;\u0000\u00151\n2\u0012a\nr(t;e)\u00133\nsin(2\u00121\u00002f(t;e))!\n: (27)\nOne can easily check that each transformation de\fned by\nR\u000b;\f(x) = (2\u000b\u0000t;2\f\u0000\u00121;_\u00121);with (\u000b;\f)2\u0019Z\u0002\u0019\n2Z; (28)\nis a reversing symmetry of equation (26) with Fas in (27) because\nf(2\u000b\u0000t;e) = 2\u000b\u0000f(t;e); r(2\u000b\u0000t;e) =r(t;e):\nThe same is true replacing in (28) the quantities \u00121,_\u00121by\u00122,_\u00122.\nOn the other hand, the spin-spin problem in its Hamiltonian formulation is given by\n(21), that can be written as (26) with\nx= (t;\u00121;\u00122;p1;p2);F(x) =\u0012\n1;p1\nC1;p2\nC2;\u0000@\u00121W(t;\u00121;\u00122);\u0000@\u00122W(t;\u00121;\u00122)\u0013\n:(29)\nEach transformation de\fned by\nR\u000b;\f1;\f2(x) = (2\u000b\u0000t;2\f1\u0000\u00121;2\f2\u0000\u00122;p1;p2);with (\u000b;\f 1;\f2)2\u0019Z\u0002\u0019\n2Z\u0002\u0019\n2Z;(30)\nis a reversing symmetry of equation (26).THE SPIN-SPIN PROBLEM IN CELESTIAL MECHANICS 11\nIn the following sections we are going to emphasize the study of orbits that are sym-\nmetric with respect to the mentioned reversing symmetries using the following lemma.\nLemma 2 (from Theorem 4.1 in [LR98]) .An orbit of (26) is R-symmetric if, and only\nif, it intersects the \fxed point set Fix(R).\n3.Periodic and quasi-periodic solutions of the Keplerian models\nIn this section we provide the de\fnitions of spin-orbit and spin-spin resonances, giving\nsome results on the existence of periodic orbits (Section 3.1) and KAM tori (Section 3.2).\n3.1.The spin-orbit and spin-spin resonances. The periodic solutions of the Kep-\nlerian models presented in the Section 2.3 correspond to resonances between the orbital\nmotion and the spin motion. The expansion of the potential in (8) contains much inter-\nesting information concerning such resonances. First, let us introduce the de\fnition of\nspin-orbit resonances.\nDe\fnition 3. We say that the ellipsoid E1is in a standard spin-orbit resonance of order\nm:nwithm2Z;n2Znf0g, if\n\u00121(t+ 2\u0019n) =\u00121(t) + 2\u0019m : (31)\nThe associated resonant angle  m:n\n1(t) =mt\u0000n\u00121(t) is a periodic function of period 2 \u0019n.\nThe same de\fnition holds for E2.\nRecalling that we have normalized the mean motion to unity, we remark that De\fni-\ntion 3 states that the ratio of the orbital period of Ejover its period of rotation is m=n,\nn6= 0. Additionally, according to De\fnition 3, the resonance m:n,n6= 0, is also of\norderkm:kn,k2Znf0g, but the converse is not true in general. For example, with\n(31), the resonance 1 : 1 is also of order 2 : 2, but a resonance 2 : 2 may not be of order\n1 : 1. We can say that the resonance m:nis of higher order than the resonance m0:n0\nifm=n>m0=n0. This will be denoted using the notation m:n>m0:n0.\nNext, we introduce a di\u000berent de\fnition of spin-orbit resonance.\nDe\fnition 4. We say that the ellipsoid E1is in a balanced spin-orbit resonance of order\nm: 2,m2Z, if\n\u00121(t+ 2\u0019) =\u00121(t) +m\u0019 : (32)\nIn this case, the resonant angle  m:2\n1(t) is 2\u0019-periodic. The same de\fnition holds for E2.\nNotice that the two notions (31) and (32) are not equivalent: (32) implies (31) for\nm: 2, but the converse is not true. Actually, note that a balanced 2 k: 2 resonance, with\nk2Z, is a spin-orbit resonance of order k: 1. This new de\fnition was motivated by\n[BL75], where the solutions associated to the resonance 3 : 2 of the spin-orbit problem\n(say, (23) with j= 1) were studied numerically. Basically, they found out that the\nsolutions satisfying (31), but not (32), appear only for large values of \u00151(&1), also, for\na given point ( e;\u00151) the solutions appear in multiplets, and \fnally, the corresponding\nresonant angles have large amplitudes ( j 3:2\n1(t)j&0:75), see Table 1 in [BL75]. On the\nother hand, solutions that obey (32) exist for any point in the ( e;\u00151)-plane, including\nlarge regions of uniqueness of solution and resonant angle with small amplitude. Note\nthat such amplitude is a measure of the deviation of the solution with respect to the\nuniform rotation of angular velocity3\n2t. In De\fnition 4 we generalize these two types of12 A. CELLETTI, J. GIMENO, AND M. MISQUERO\nresonances for any order m: 2, since this let us determine the main resonances in the\n\frst orbital revolution.\nFrom [DeV58], for instance, we know that a good tool to study a di\u000berential equation\nthat has some reversing symmetries is by studying the periodic orbits that are invariant\nunder such transformations.\nIn the next proposition we provide some boundary conditions that characterize the\nsymmetric orbits in spin-orbit resonances.\nProposition 5. The following statements hold for the spin-orbit problem (23), with\nj= 1 (ellipsoidE1):\n(1)AnyR\u000b;\f-symmetric orbit, with R\u000b;\fde\fned in (28), associated to a m:nspin-\norbit resonance is equivalent to a solution that satis\fes the following Dirichlet\nconditions:\n\u00121(\u000b) =\f; \u0012 1(\u000b+n\u0019) =\f+m\u0019 ; (33)\nwith\u000b2f0;\u0019gand\f2f0;\u0019\n2g(four combinations). Moreover, such solution\nsatis\fes the following symmetry property \u00121(t) = 2\f\u0000\u00121(2\u000b\u0000t).\n(2)There are two independent types of R\u000b;\f-symmetric orbits representing a balanced\nm: 2spin-orbit resonance and are given by:\nType 0 :\u00121(0) = 0; \u0012 1(\u0019) =m\u0019\n2; (34)\nType\u0019\n2:\u00121(0) =\u0019\n2; \u0012 1(\u0019) =(m+ 1)\u0019\n2: (35)\nMoreover, the corresponding symmetry relations are: \u00121(t) =\u0000\u00121(\u0000t)for type I\nand\u00121(t) =\u0019\u0000\u00121(\u0000t)for type II.\nThe same is true for (23), with j= 2 (ellipsoidE2), and also for (25).\nProof. Let us apply Lemma 2 to (26) with F(x) given by (27). The \fxed point set of each\nreversing symmetry R\u000b;\fis Fix(R\u000b;\f) =f(t;\u00121;_\u00121):t=\u000b;\u0012 1=\fg, then, the symmetric\norbits can be found with initial conditions \u00121(\u000b) =\f. Since F(x) is 2\u0019-periodic in tand\n\u0019-periodic in \u00121, it is enough to consider \u000b2f0;\u0019g(the periapsis and the apoapsis) and\n\f2f0;\u0019\n2g.\nNow, since R\u000b;\fis a reversing symmetry, if \u00121(t) is a solution of (23) with j= 1, so it\nis (t) = 2\f\u0000\u00121(2\u000b\u0000t). Additionally, if \u00121(\u000b) =\f, then, both solutions coincide, so\nthe symmetry relation \u00121(t) = 2\f\u0000\u00121(2\u000b\u0000t) holds for it. Now, if \u00121(t) is in am:n\nspin-orbit resonance, then, replacing t=\u000b\u0000n\u0019in (31), we get\n\u00121(\u000b+n\u0019) =\u00121(\u000b\u0000n\u0019) + 2\u0019m :\nFrom the symmetry relation we get additionally that\n\u00121(\u000b\u0000n\u0019) = 2\f\u0000\u00121(\u000b+n\u0019):\nCombining the two expressions we prove (33).\nNow let us prove the converse, that a solution \u00121(t) satisfying the conditions (33) is in\nm:nspin-orbit resonance. Let the initial conditions of such solution be\n\u00121(\u000b+n\u0019) =\f+m\u0019; _\u00121(\u000b+n\u0019) =~\f ; (36)THE SPIN-SPIN PROBLEM IN CELESTIAL MECHANICS 13\nfor some real constant ~\f. Since\u00121(\u000b) =\f, the solution has the symmetry relations\n\u00121(t) = 2\f\u0000\u00121(2\u000b\u0000t) and _\u00121(t) =_\u00121(2\u000b\u0000t). Using such relations we get that\n\u00121(\u000b\u0000n\u0019) =\f\u0000m\u0019; _\u00121(\u000b\u0000n\u0019) =~\f : (37)\nThe initial conditions (36) and (37) show that, while the time increases 2 \u0019n, the angle\u00121\nincreases in 2 \u0019m, and the angular velocity is the same for both cases, then, the solution\nis periodic. With this, we have proved item 1.\nLet us now consider the balanced m: 2 case in item 2. Note that, using the de\fnition\n(32) instead of (31), we can follow the same procedure as before to prove that a balanced\nm: 2 solution satis\fes the following boundary conditions\n\u00121(\u000b) =\f; \u0012 1(\u000b\u0006\u0019) =\f\u0006m\u0019\n2; (38)\nand the symmetry relation \u00121(t) = 2\f\u0000\u00121(2\u000b\u0000t). Then, we get that the four types are\nin this case (34), with \u000b= 0;\f= 0; (35), with \u000b= 0;\f=\u0019=2;\nType 00:\u00121(0) =\u0000m\u0019\n2; \u0012 1(\u0019) = 0;\nwith\u000b=\u0019;\f= 0, and\nType\u0019\n20\n:\u00121(0) =(1\u0000m)\u0019\n2; \u0012 1(\u0019) =\u0019\n2;\nwith\u000b=\u0019;\f =\u0019=2. Since an ellipsoid has a mirror symmetry with respect to any\nplane containing a pair of semi-axes, the angle \u00121is equivalent to \u00121+k\u0019,k2Z. In\nconsequence, if m= 2k1;k12Z, type 00is equivalent to type 0 and type\u0019\n20is equivalent\nto type\u0019\n2. Likewise, for m= 2k2+ 1,k22Z, type 00is equivalent to type\u0019\n2and type\u0019\n20\nis equivalent to type 0. Then, for resonances of order m: 2,m2Z, we will take types 0\nand\u0019\n2as representatives. With this we have proved item 2.\nThe previous facts rely only on the symmetries and the periodicity of equation (23),\nincluding the discussion in [CC00]. Since equation (25) for the spin motion of E1, with\nE2spherical, has exactly the same properties, then, the proof above is also valid in such\ncase. \u0003\nProposition 5 let us characterize all the balanced spin-orbit resonances in the \frst\nhalf of an orbital revolution, additionally, the corresponding solutions have a certain\nsymmetry relation. In the generalization to spin-spin resonances we want to combine\ndi\u000berent spin-orbit resonances and we will use the boundary conditions in the same time\ninterval.\nRemark 6.Note that solutions of type\u0019\n2can be recovered with the conditions of type 0\nby considering negative values of \u00151. More precisely, solutions of type\u0019\n2of equation (23)\nwith\u00151=\u0015\u0003>0 are equivalent to solutions of (23) with \u00151=\u0000\u0015\u0003satisfying conditions\nof type 0. The same holds for (25).\nRemark 7.Proposition 5 gives a way to numerically search for balanced resonances.\nIndeed, eqs. (34) and (35) can be used to apply a Newton method, that is, to \fnd the\ninitial condition _\u00121(0) such that the conditions for either Type 0 or Type\u0019\n2are satis\fed.\nNext we introduce the following de\fnition, which deals with the spins of both objects.14 A. CELLETTI, J. GIMENO, AND M. MISQUERO\nDe\fnition 8. We say that the ellipsoids E1,E2are in a standard spin-spin resonance3of\norder (m1:n1;m2:n2) withm1;m22Z;n1;n22Znf0g, if the ellipsoidEjis in amj:nj\nspin-orbit resonance. In such case, the resonant angles  m1:n1m2:n2(t) = m1:n1\n1(t)\u0006 m2:n2\n2(t)\nare 2\u0019n-periodic functions, where nis the least common multiple of n1andn2.\nAn analogous de\fnition holds for resonances of the balanced type.\nDe\fnition 9. We say that the ellipsoids E1,E2are in a balanced spin-spin resonance\nof order (m1: 2;m2: 2) withm1;m22Z, if the ellipsoidEjis in amj: 2 spin-orbit\nresonance for j= 1;2.\nNote that a spin-spin resonance of order ( m1:n1;m2:n2) is also of order ( \u00141m1:\nn;\u0014 2m2:n), wheren=\u00141n1=\u00142n2is the least common multiple of n1andn2. Again,\nthe converse is not true in general. For example, the resonance (1 : 1 ;3 : 2) is of order\n(2 : 2;3 : 2), but not the opposite. However, a balanced resonance (2 : 2 ;3 : 2) is a\nspin-spin resonance of order (1 : 1 ;3 : 2).\nThe following proposition generalizes Proposition 5 to the spin-spin problem.\nProposition 10. The following statements hold for the spin-spin problem (24):\n(i)AnyR\u000b;\f1;\f2-symmetric orbit, with R\u000b;\f1;\f2de\fned in (30), associated to a (m1:\nn;m 2:n)spin-orbit resonance is equivalent to a solution that satis\fes the fol-\nlowing Dirichlet conditions:\n\u0012j(\u000b) =\fj; \u0012j(\u000b+n\u0019) =\fj+mj\u0019 ;\nwith\u000b2f0;\u0019gand\fj2f0;\u0019\n2g(eight combinations). Moreover, such solution\nsatis\fes the following symmetry property \u0012j(t) = 2\fj\u0000\u0012j(2\u000b\u0000t).\n(ii)There are four independent types of R\u000b;\f1;\f2-symmetric orbits representing a bal-\nanced (m1: 2;m2: 2)spin-spin resonance and are given by:\nType (\f1;\f2) :\u0012j(0) =\fj; \u0012j(\u0019) =\fj+mj\u0019\n2; (39)\nwith\fj2 f0;\u0019\n2g. Moreover, the corresponding symmetry relation is \u0012j(t) =\n2\fj\u0000\u0012j(\u0000t).\nProof. This proof is based on the fact that the same arguments used to prove Proposi-\ntion 5 can be generalized in a straightforward way for the spin-spin problem (24).\nNow we apply Lemma 2 to (26) with F(x) given by (29). The \fxed point set of each\nreversing symmetry R\u000b;\f1;\f2is\nFix(R\u000b;\f1;\f2) =f(t;\u00121;\u00122;p1;p2):t=\u000b;\u0012 1=\f1;\u00122=\f2g:\nThen, due to the periodicity of Win (20), the periodic orbits can be found at \u0012j(\u000b) =\fj,\nwhere we can take any combination between \u000b2f0;\u0019gand\fj2f0;\u0019\n2g. A similar\nmethod was used in [Gre79] for the standard map and was generalized for the spin-orbit\nproblem in [CC00] and for a standard map of two degrees of freedom in [CFL04].\nThe rest of the proof of item i follows analogously to the proof of item (1) of Proposi-\ntion 5 using the reversing symmetries.\n3We remark that this is a practical de\fnition because it is useful for the physical interpretation.\nHowever, we notice that there is a more general de\fnition given by the resonant combination n0t\u0000\nn1\u00121\u0000n2\u00122, withn0;n1;n2integers.THE SPIN-SPIN PROBLEM IN CELESTIAL MECHANICS 15\nThe proof of item ii needs more detail. In the same way as (38), we obtain easily that\nthe balanced resonances ( m1: 2;m2: 2) are given by\n\u0012j(\u000b) =\fj; \u0012j(\u000b\u0006\u0019) =\fj\u0006mj\u0019\n2: (40)\nWe see that (39) corresponds to (40) with \u000b= 0 and the positive sign. From the case\n\u000b=\u0019and the negative sign we have that\n\u0012j(0) =\fj\u0000mj\u0019\n2; \u0012j(\u0019) =\fj: (41)\nNote that if, for example, we take j= 1 andm1= 2k1, withk12Z, then, the conditions\n(39) and (41) are equivalent. On the other hand, if m1= 2k1+ 1, then, the conditions\n(39) with\f1= 0 and\u0019\n2are equivalent respectively to (41) with \f1=\u0019\n2and 0. The same\nis true forj= 2. Then, with (39) all the possibilities are covered. \u0003\nRemark 11.Results in Proposition 10 allow to apply a Newton method to compute\nresonances for the spin-spin problem just considering as unknowns _\u0012j(0) and correct\nthem by imposing the conditions in (39) or (41).\nFor circular orbits ( e= 0), each of the spin-orbit models (23) is a classical pendulum\nwhose only stable equilibrium point corresponds to a 1 : 1 resonance that is given by\n\u0012j(t) =t. Similarly, the spin-spin model (24) consists of two coupled penduli whose only\nstable solution is \u00121(t) =\u00122(t) =tthat is a (1 : 1 ;1 : 1) resonance. However, for e6= 0,\nf(t;e) does not coincide with tand more stable spin-orbit resonances may appear.\nIn order to study the spin-orbit and spin-spin resonances, it is useful to compute the\nexpansion of V2andV4up to some power of the eccentricity. This expansion is obtained\nsolving Kepler's equation (13) up to a \fnite order in the eccentricity, then inserting the\nsolutionu=u(t) in (14), (16), expand them in series of the eccentricity and \fnally\nexpanding the trigonometric terms appearing in (8).\nThis procedure leads to the expansions of V2andV4that, for simplicity, we give up to\nthe order 2 in the eccentricity in the Appendix C. In those expressions, the trigonometric\nterms with arguments  mj:nj\nj(t) =mjt\u0000nj\u0012jare associated to mj:njspin-orbit reso-\nnances forEj, whereas the terms with argument  m1:n1m2:n2(t) = (m1\u0006m2)t\u0000n1\u00121(t)\u0007n2\u00122(t)\ncorrespond to spin-spin resonances by combining spin-orbit resonances of orders m1:n1\nandm2:n2. For each order of the expansion in the eccentricity e, there are some reso-\nnances appearing. They are shown in Table 1, where we can recognize a hierarchy: the\nmost important spin-orbit resonance is 1 : 1, then we have 3 : 2, 1 : 2 and so on, because\nthey appear for low orders of the eccentricity. Resonances of further orders are relevant\nonly for large eccentricities.\nNote that the most relevant spin-orbit resonances are of order m: 2,m2Z. Ad-\nditionally, spin-spin resonances appearing at order e\u000binV4are obtained by combining\nspin-orbit resonances appearing at order e\u000b1ande\u000b2inV2such that\u000b1+\u000b2=\u000b.\nFinally, note that for V2, the coe\u000ecients associated to spin-orbit resonances are of order\none ind1;d2, whereas for V4, the spin-orbit coe\u000ecients are of order two in d1;d2;q1;q2,\nand the spin-spin coupling coe\u000ecients are of order two in d1;d2.\n3.2.KAM tori in the spin-spin problem. Now we deal with quasi-periodic solutions\nof the Keplerian version of the spin-spin model. We denote by\n!= (1;!1;!2) (42)16 A. CELLETTI, J. GIMENO, AND M. MISQUERO\nspin-orbit resonances spin-spin resonances\norder V2 V4 V4\ne01 : 1 1 : 1 combine 1 : 1 with 1 : 1\ne13 : 2, 1 : 2 3 : 2, 1 : 2, 3 : 4, 5 : 4 combine 1 : 1 with 3 : 2 and 1 : 2\ne21 : 1, 2 : 1, 0 : 11 : 1, 2 : 1, 0 : 1,\n1 : 2, 3 : 2combine 1 : 1 with 1 : 1, 2 : 1 and 0 : 1\nall combinations between 3 : 2 and 1 : 2\nTable 1. Resonances appearing in the expansion of the potential V2+V4\nfor each order of the eccentricity.\nthe frequency vector with\n!1=p1\nC1; ! 2=p2\nC2:\nThe Hessian matrix associated to (20) has determinant di\u000berent from zero, whenever\np16= 0,p26= 0. This implies that (20) satis\fes the Kolmogorov non-degeneracy condition,\nwhich is a requirement for the applicability of KAM theorem [Kol54, Arn63, Mos62]. The\nother essential requirement in KAM theory is the assumption that the frequency satis\fes\na Diophantine inequality, namely there exist C > 0 and\u0018\u00152, such that\njk\u0001!j\u00001\u0014Cjkj\u0018(43)\nfork2Z3nf0g.\nWe remark that a possible choice of !satisfying (43) can be obtained as follows. Let\n\u000bbe an algebraic number of degree 3, namely the solution of a polynomial equation of\ndegree 3 with integer coe\u000ecients, not being the root of polynomial equations of lower\ndegree. Let us consider the vector != (1;!1;!2) obtained as\n0\nBBB@1\n!1\n!21\nCCCA=0\nBBB@1 0 0\nb1a11a12\nb2a21a221\nCCCA0\nBBB@1\n\u000b\n\u000b21\nCCCA; (44)\nwhere (b1;b2) and the matrix A\u0011(amn) have rational coe\u000ecients and det A6= 0. By\nnumber theory results (see, e.g., [CFL04]), a vector !as in (44) satis\fes (43) with\n\u0018= 2. Under smallness conditions of the parameters, say \u0015jin (22), KAM theory ensures\nthe existence of a quasi-periodic torus with Diophantine frequency. We remark that\nthe theory presented in [CCGdlL21c] for the spin-orbit problem (see also [CCGdlL21b,\nCCGdlL21a]) can be extended to provide explicit estimates for (20) and an explicit\nalgorithm to construct quasi-periodic solutions.\n4.Qualitative description of the spin models\nIn this section, we give a qualitative description of the phase space associated to\nthe spin-orbit problem (Section 4.1), the spin-spin problem with spherical companion\n(Section 4.2) and with non-spherical companion (Section 4.3).THE SPIN-SPIN PROBLEM IN CELESTIAL MECHANICS 17\n4.1.Spin-orbit problem ( Vper=V2).Since the system (23) corresponds to two uncou-\npled spin-orbit problems, then the Poincar\u0013 e map of the whole system can be understood\nas the direct product of the Poincar\u0013 e maps for each of the bodies. Consider for exam-\nple the dynamics of E1. Figure 3 is a typical Poincar\u0013 e map of the spin-orbit problem\nobtained using a Taylor integrator [JZ05] and using a similar approach as the one ex-\nplained in Appendix B, in this case with parameters ( e;\u00151) = (0:06;0:05); it represents\nsolutions at t= 2\u0019k,k2Z, in the plane ( \u00121;_\u00121) restricted to \u001212[\u0000\u0019;\u0019]. The main\nstable resonances are tagged with their corresponding order m:n. The Poincar\u0013 e map\nfor su\u000eciently small parameters ( e;\u00151) has the following features:\n1) The main stable spin-orbit resonances are represented by \fxed points in the plane\n(\u00121;_\u00121) surrounded by islands of invariant librational tori. High order resonances\nappear above low order resonances.\n2) It looks that the spin-orbit resonances of order m: 2\u00151 : 1 are balanced:\nsolutions of type 0 (34) are stable and those of type\u0019\n2(35) are unstable. On\nthe contrary, for the 1 : 2 resonance, type 0 is unstable and type\u0019\n2is stable.\nConcerning other resonances (33), for instance in the 3 : 4 resonance, types with\n\u000b= 0 are unstable and types with \u000b=\u0019are stable. Exactly the opposite occurs\nfor the case 5 : 4.\n3) It is possible to have stable secondary resonances, namely small resonances sur-\nrounding other resonances. This is clear for the 1 : 1 resonance. Beyond the\nlibrational islands associated to the resonance 1 : 1 there is a chaotic region that\nincludes the unstable resonances and that is larger for large parameters ( e;\u00151).\nThe chaotic region can appear for other resonances and is limited by rotational\ntori that also distinguish the domains of resonances of di\u000berent orders.\nFigure 3. Poincar\u0013 e map for the spin-orbit problem.\n4.2.Spin-spin problem ( Vper=V2+V4) with spherical companion. Let us now\nconsider the case when E2is a sphere. Then, \u00122(t) =\u00122(0) + _\u00122(0)tand the dynamics of\n\u00121is given by (25), that depends on the parameters ( e;\u00151;^q1;\u001b1). Here the parameters18 A. CELLETTI, J. GIMENO, AND M. MISQUERO\n\u001b1and ^q1perturb the previous framework of the spin-orbit problem, see Section 4.1.\nWe can see a comparison between both problems in Figure 4: we see that the only\nqualitative di\u000berence between both cases is that the Poincar\u0013 e map associated to equation\n(25) is slightly more chaotic. This minor di\u000berence is due to the fact that we take\n\u001b1= ^q1= 0:01, that are small parameters. As we will see in Section 6, the spin-spin\nmodel is a good approximation for the dynamics of two ellipsoids for \u001bjand ^q2up to\nthe order of magnitude of about 10\u00002, because larger values could lead to a collision, see\nSection 6.2.\nFigure 4. Poincar\u0013 e maps. Left: usual spin-orbit problem with\n(e;\u00151) = (0:06;0:05). Right: spin-orbit problem up to order 1 =r5with\n(e;\u00151;\u001b1;^q1) = (0:06;0:05;0:01;0:01).\n4.3.Spin-spin problem ( Vper=V2+V4) with non-spherical companion. Now we\ndeal with the general system (24). Let \t( t) = (\u00121(t);\u00122(t);_\u00121(t);_\u00122(t)) be a solution of\n(24) and its respective projections \t 1(t) = (\u00121(t);_\u00121(t)) and \t 2(t) = (\u00122(t);_\u00122(t)). From\nnow on we restrict \u0012j(t) to [\u0000\u0019;\u0019]. Let the Poincar\u0013 e map associated to such solution\nbe de\fned byP(\t(0)) = \t(2 \u0019), and its projections by Pj(\tj(0)) = \t j(2\u0019). It is\nnot possible to represent the iteration of the map Pin a single plot, because it is 4-\ndimensional, so we will represent the projections Pj. That is to say, for a solution \t( t),\nwe are interested in the behavior of the two families of points \t 1(2\u0019k) and \t 2(2\u0019k),\nk2Z, in a single 2-dimensional strip \u0005 = f(x;y)2R2:jxj\u0014\u0019g. We recognize the\nfollowing features:\n1) A solution in spin-spin resonance corresponds to a family of isolated recurrent\npoints ofP(namely, the successive points on the Poincar\u0013 e map), whose projec-\ntions are represented in \u0005 as a pair of families of recurrent points.\n2) If the spin-spin resonance is stable, then nearby solutions would librate around\nsuch points. While in the uncoupled system, librating solutions belong to 2-\ndimensional tori, here tori can be higher dimensional. As a result, the projected\npoints represented in \u0005 are distributed in two clouds of points surrounding each\nrecurrent point. A cloud of this kind covers an annulus-like region of a certain\nthickness that is usually thicker for stronger couplings. A similar behavior occursTHE SPIN-SPIN PROBLEM IN CELESTIAL MECHANICS 19\nfor rotational solutions, whose corresponding clouds are distributed in strips of\ncertain thickness, see Figure 5.\n 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7\n-1 -0.8 -0.6 -0.4 -0.2  0  0.2  0.4  0.6  0.8  1dθ/dt(t)\nθ(t)/πσ=0.2 θ 1(0)=0 dθ 1/dt(0)=0.9 θ 2(0)=0 dθ 2/dt(0)=1.45\nbody 1\nbody 2\n 0.4 0.6 0.8 1 1.2 1.4 1.6\n-1 -0.8 -0.6 -0.4 -0.2  0  0.2  0.4  0.6  0.8  1dθ/dt(t)\nθ(t)/πσ=0.25 θ 1(0)=0 dθ 1/dt(0)=0.5 θ 2(0)=0 dθ 2/dt(0)=1.51\nbody 1\nbody 2\nFigure 5. Left: Projections \t 1(2\u0019k) and \t 2(2\u0019k),k2Nof a solution\n\t(2\u0019k) librating around a stable (1 : 1 ;3 : 2) spin-spin resonance. Right:\nSolution for which one of the bodies librates around a stable spin-orbit\nresonance and the other one has a rotational behavior. The common pa-\nrameter values are e= 0:06,\u00151=\u00152= 0:05, and ^q1= ^q2= 0:001.\n3) We expect that, for small enough coupling parameters \u001bj, the spin-spin resonances\nare located close to the corresponding spin-orbit resonances for each ellipsoid, and\nwould keep the same stability as for the uncoupled problem. However, we have\nfound that, for larger \u001bj, the stability may change with respect to the uncoupled\nsystem, see Figure 6.\n4) The coupled system is 5-dimensional, then, invariant tori, if there exist, would\nnot con\fne solutions in determined regions (as in the uncoupled system), but\nArnold di\u000busion is expected to take place.\nA particular behavior occurs only when both bodies are identical ( C1=C2= 0:5,\n\u0015=\u00151=\u00152,\u001b=\u001b1=\u001b2and ^q= ^q1= ^q2), the so-called measure synchronization . This\nphenomenon was observed numerically in [HZ99] for an autonomous Hamiltonian system\nof two degrees of freedom (a pair of identical coupled oscillators): the system librates\naround a stable periodic solution in a very particular way described as follows for our\nsystem (see the phenomenon illustrated in Figure 7). Take a solution \t( t) librating\naround a stable spin-spin resonance of order ( m:n;m :n). Consider the two families\nof points \t 1(2\u0019k) and \t 2(2\u0019k) and their corresponding annulus-like region in the plane\n\u0005. There are two possibilities: either both clouds of points are distributed in separated\nannulus-like regions or both regions coincide. That is to say, either the overlap is empty\nor there is a complete overlap. Moreover, if we start with a solution with separated\nregions, we can obtain the complete overlap by increasing the coupling parameter \u001b\n(keeping the same \t(0)). The phenomenon takes place suddenly for a speci\fc \u001b=\u001b\u0003\nwhen the outer boundary of the inner ring touches the inner boundary of the outer ring.\nAt that moment, there is a concentration of density of points in the contact region.20 A. CELLETTI, J. GIMENO, AND M. MISQUERO\n 0.4 0.6 0.8 1 1.2 1.4 1.6\n-1 -0.8 -0.6 -0.4 -0.2  0  0.2  0.4  0.6  0.8  1dθ/dt(t)\nθ(t)/πσ=0.15 θ 1(0)=1.571 dθ 1/dt(0)=0.48 θ 2(0)=0 dθ 2/dt(0)=1.48\nbody 1\nbody 2\n 0.4 0.6 0.8 1 1.2 1.4 1.6\n-1 -0.8 -0.6 -0.4 -0.2  0  0.2  0.4  0.6  0.8  1dθ/dt(t)\nθ(t)/πσ=0.55 θ 1(0)=0 dθ 1/dt(0)=0.48 θ 2(0)=0 dθ 2/dt(0)=1.48\nbody 1\nbody 2\nFigure 6. Left: Projections \t 1(2\u0019k) and \t 2(2\u0019k),k2Zof a solution\n\t(t) close to an unstable (1 : 2 ;3 : 2) spin-spin resonance. Right: Rep-\nresentation of a solution with identical \t(0) for larger coupling, now the\nnearby spin-spin resonance has become stable. The common parameter\nvalues aree= 0:06,\u00151=\u00152= 0:05, and ^q1= ^q2= 0:001.\nThe phenomenon of measure synchronization disappears when the bodies are not\nequal. See in Figure 8 how the domains of both ellipsoids can overlap without merging\ninto a single ring.\n5.Linear stability of resonances\nIn this section we analyze the stability (in the linear approximation) of solutions asso-\nciated to the resonances in di\u000berent models, namely the spin-orbit problem (Section 5.1)\nand the spin-spin problem with spherical (Section 5.2) and non-spherical (Section 5.3)\ncompanion.\nIn all cases, we will only deal with balanced resonances (32), because they appear to\nbe simpler and more relevant (see Section 4) than resonances of the general type (31).\nActually, we can establish regions in the space of parameters where solutions associated\nto balanced resonances are unique or have some low multiplicity. In the case of the spin-\nspin problem, we restrict ourselves to regions of uniqueness. The study of linear stability\nof such periodic solutions in the space of parameters complements the understanding of\nthe dynamics that we presented in previous sections, especially Section 4.\n5.1.Spin-orbit problem ( Vper=V2).Consider the spin-orbit problem (23) with j= 1,\nthat is, the motion of the ellipsoid E1. Let\u00121=\u0012\u0003(t) be a solution in a balanced m: 2\nspin-orbit resonance, whose associated variational equation is\ny+\u00151\u0012a\nr(t;e)\u00133\ncos(2\u0012\u0003(t)\u00002f(t;e))y= 0; y2R: (45)\nFore6= 0, (45) is a linear equation with a 2 \u0019-periodic coe\u000ecient. Particularly, this is a\nHill's equation [MW79]. Assume that \b( t) is a matrix solution of (45) with \b( t0) =12,\nthe identity matrix 2 \u00022. The stability of (45) is determined by the structure of the\nmatrixM= \b(t0+ 2\u0019), called monodromy matrix. If jTr(M)j<2, we have ellipticTHE SPIN-SPIN PROBLEM IN CELESTIAL MECHANICS 21\n 0.9 0.95 1 1.05 1.1\n-0.15 -0.1 -0.05  0  0.05  0.1  0.15dθ/dt(t)σ=0\nbody 1\nbody 2\n 0.9 0.95 1 1.05 1.1\n-0.15 -0.1 -0.05  0  0.05  0.1  0.15σ=(0.1,0.1)\nbody 1\nbody 2\n 0.9 0.95 1 1.05 1.1\n-0.15 -0.1 -0.05  0  0.05  0.1  0.15dθ/dt(t)\nθ(t)/πσ=0.268\nbody 1\nbody 2\n 0.9 0.95 1 1.05 1.1\n-0.15 -0.1 -0.05  0  0.05  0.1  0.15\nθ(t)/πσ=0.268884\nbody 1\nbody 2\nFigure 7. Projections \t 1(2\u0019k) and \t 2(2\u0019k),k2Zof a solution \t( t)\nclose to a stable (1 : 1 ;1 : 1) spin-spin resonance for di\u000berent values of\n\u001b1and\u001b2. Keeping the same parameters e= 0:06,\u00151=\u00152= 0:05,\n^q1= ^q2= 0:001,\u00121(0) =\u00122(0) = 0, _\u00121(0) = 0:92 and _\u00122(0) = 1:05. The\nexternal ring (body 1) keeps similar thickness, the internal ring changes\nfrom a thin one to another that occupies values close to (0,0) to thin one\nto \fnaly collapse with the exterior one.\nstability, whereas if jTr(M)j>2 we have hyperbolic instability. In the parabolic case,\nwhenjTr(M)j= 2, the system is stable if the Jordan canonical form of Mis12or\n\u000012, otherwise the system is unstable. Actually, if the system is parabolic unstable, the\ninstability of the linear system is linear in time, but hyperbolic instability is associated to\nan exponential divergence in time. In our case we want to distinguish regions of stability\nand instability in the ( e;\u00151)-plane for a given solution, which is continuous in ( e;\u00151).\nFrom properties of Hill's equations, regions of elliptic stability are separated from regions\nof hyperbolic instability by parabolic curves ( jTr(M)j= 2). These curves are made of\nunstable points, except if there are intersections of parabolic curves, because points of\nintersection become stable. This phenomenon is called coexistence , [MW79].22 A. CELLETTI, J. GIMENO, AND M. MISQUERO\n 0.9 0.92 0.94 0.96 0.98 1 1.02 1.04 1.06 1.08 1.1\n-0.4 -0.3 -0.2 -0.1  0  0.1  0.2  0.3  0.4dθ/dt(t)\nθ(t)/πbody 1\nbody 2\nFigure 8. Partial overlapping of the domains of \t 1(2\u0019k) and \t 2(2\u0019k),\nk2Zof a solution \t( t) close to a stable (1 : 1 ;1 : 1) spin-spin resonance in\nthe case of di\u000berent bodies: no measure synchronization. The parameter\nvalues aree= 0:06,\u00151= 0:009,\u00152= 0:05, ^q1= ^q2= 0:001,\u001b1=\u001b2= 0:3,\n\u00121(0) =\u00122(0) = 0, _\u00121(0) = 0:92, and _\u00122(0) = 1:064.\nFor a given point ( e;\u00151) and a given balanced m: 2 resonance, we want to know how\nmany solutions there are of each type (34) or (35), and their linear stability. First, recall\nRemark 6: solutions of type\u0019\n2satisfy conditions of type 0 for the equation (23) with j= 1,\ntaking\u0000\u00151instead of\u00151. Consequently, for each ( e;\u00151), withe2[0;1) and\u001512(\u00003;3),\nwe can obtain all the solutions corresponding to a balanced resonance by applying the\nshooting method: take a solution \u00121(t) with initial conditions4\u00121(\u0019) =m\u0019=2,_\u00121(\u0019) =\n\r2Rand let\rvary until the boundary condition \u00121(0) = 0 is reached. Finally, we obtain\nthe linear stability of the solution by computing \b( t) such that \b( \u0019) =12. Actually,\nfor this procedure, we can take any of the boundary conditions in eqs. (34) and (35), we\njust need one type to generate all solutions.\nThe results of this method for the main balanced spin-orbit resonances are shown in\nFigure 9. For these computations we used a Runge-Kutta Verner 8(9) integrator [Ver78],\ninstead of a Taylor integrator [JZ05]; the reason is that, for some parameter values, the\nsolutions are constant or polynomials and the Taylor method su\u000bers in choosing a good\nstep size. Thus, Figure 9 required around 3.5 days with 34 CPUs to be generated with\na discretization mesh size of 2000 \u00022000\u00022000 for (e;\u00151;\r).\nWe can recognize the following characteristics5:\n1) Each of the balanced resonances is represented in the ( e;\u00151;\r)-space by a contin-\nuous surface. In the case 1 : 1, the surface is made of two sheets connected only\nin one point ( e;\u00151) = (0;1).\n4We choose to take initial conditions at t=\u0019and nott= 0 because the values of _\u00121(0) producing\nspin-orbit resonances for large eand\u00151are too large to be represented in a 3-dimensional plot as in\nFigure 9.\n5The linear stability of the multiple solutions associated to the resonances 1 : 1 and 3 : 2 was already\nstudied in [ZOST66, Bel66, BL75], but we include them here in order to have a more complete view.THE SPIN-SPIN PROBLEM IN CELESTIAL MECHANICS 23\nFigure 9. Diagrams of linear stability for the balanced resonances of\norder 1 : 2 (left), 1 : 1 (middle left), 3 : 2 (middle right) and 2 : 1 (right).\nBlue-instability and yellow-stability. The upper diagrams are projections\nof the lower diagrams in the ( e;\u00151)-plane with some transparency in order\nto identify regions with multiplicity of solutions.\n2) The region of uniqueness in the ( e;\u00151)-plane is quite large. The multiplicity\nis generated by bifurcation of solutions: the surface folds generating multiple\nsolutions (from one to three, as far as we see). Particularly, the bifurcation in the\ncase 1 : 1 occurs at ( e;\u00151) = (0;1) producing a secondary sheet behind the main\none. In general, the multiplicity takes place for some regions with j\u00151j>1. The\nresonances of order m: 2 withm > 2, have two characteristic folds, one with a\nV-like shape in solutions of type 0 for \u00151>1, and another small one in solutions\nof type\u0019\n2for\u00151\u0018\u00002 and very large eccentricities.\n3) Instability is predominant in the diagrams, especially in solutions of type 0 for\nthe resonance 1:2 and of type\u0019\n2for the rest of the resonances. We see that the\nmain regions of linear stability are continuation of stable solutions from e= 0,\nmuch of which are close to small j\u00151j. Except for the resonance 1:2, the stability\nregion for large eccentricities ( e>0:6) of the other resonances has a similar shape,\ncharacterized by a bifurcation with an interchange of stability. It is interesting\nto note that the folds producing the multiplicity are associated to some stable\nregions with peculiar shapes. In the resonance 1 : 1 we \fnd two unstable regions\nbifurcating from the exact solution \u00121(t) =tfore= 0: one at \u00151= 1=4 = 0:25\n(main sheet) and the other one at \u00151= 9=4 = 2:25 (secondary sheet).\nNow let us consider both bodies. Since the system (23) is uncoupled, then, the multi-\nplicity and stability associated to a spin-spin resonance is given by each of the separated\nproblems. For example, take the (1 : 1 ;3 : 2) balanced spin-spin resonance of type (0 ;\u0019\n2)24 A. CELLETTI, J. GIMENO, AND M. MISQUERO\nfor (e;\u00151;\u00152) = (0:3;0:1;0:5), that is, the red points shown in Figure 9. It has associated\na unique solution and it is unstable because it is so for \u00122.\n5.2.Spin-spin problem ( Vper=V2+V4) with spherical companion. In this case\nwe know thatE2is in uniform rotation \u00122(t) =\u00122(0) + _\u00122(0)t, while the dynamics of\n\u00121is given by (25), that depends on ( e;\u00151;^q1;\u001b1). For this problem, we can proceed in\nthe same way as for the spin-orbit problem of Section 5.1. On one hand, the variational\nequation associated to a solution in a spin-orbit resonance is a Hill's equation like (45),\nso the linear stability of the solution is characterized by the corresponding monodromy\nmatrix. On the other hand, we can \fnd all the solutions associated to a balanced spin-\norbit resonance using the shooting method for only one type of boundary conditions by\nincluding negative values of \u00151, see Remark 6.\nComparing Figures 9 and 10 we can see, for example, how the balanced 3 : 2 resonance\nis perturbed when we turn on the parameters (^ q1;\u001b1):\nFigure 10. Diagrams of linear stability for the 3 : 2 balanced resonance of\n(25) for di\u000berent values of the indicated parameters (^ q1;\u001b1). Blue denotes\ninstability and yellow denotes stability. Each of the columns have required\naround 15h using 24 CPUs and a mesh size of 512 \u0002512\u0002512.\n1) The e\u000bect of the new parameters is remarkable for large eandj\u00151j. Especially\nwhen ^q1and\u001b1are large.\n2) At very large eccentricities, the surface has a complicated structure resulting in\nmultiple solutions. The e\u000bect of ^ qis mainly to alter the stability for large e:\nsolutions of type\u0019\n2become always unstable, while stable regions of solutions of\ntype 0 are more concentrated. The growth of ^ qalso increases the multiplicity of\nsolutions of type 0 and very large e. On the other hand, increasing \u001b1has a more\ndramatic e\u000bect on the complexity of the surface and also modi\fes the stability for\nlargee. Actually, for some values of \u001b1, the existing V-shaped fold connects with\nthe complex structure of large eccentricities in the upper part of the diagram.THE SPIN-SPIN PROBLEM IN CELESTIAL MECHANICS 25\n5.3.Spin-spin problem ( Vper=V2+V4) with non-spherical companion. We study\nthe linear stability of the resonances of the full coupled spin-spin model in (24). In this\ncase linear stability is determined by a more general theory and we will restrict ourselves\nto zones in the space of parameters where there is uniqueness.\nSince we have two degrees of freedom, the \frst variation at a particular resonance is\nnot a Hill's equation anymore, but a coupled system of second order. A Hill's equation is\na particular case of linear Hamiltonian system with periodic coe\u000ecients, hereafter LPH\nsystems.\nIf we de\fne z= (\u00121;\u00122;p1;p2)T, wherepj=Cj_\u0012j, the spin-spin problem in the Hamil-\ntonian form (21) can be written as\n_z=J2@zHK(t;z); (46)\nwhereJlis the square matrix of order 2 lgiven by\nJl=0\n@01l\n\u00001l01\nA; l= 1;2;::: (47)\nwith 1lthe unit matrix of order l. The non-autonomous Hamiltonian HK(t;z) is given\nin (19) for Vper=V2+V4. Letz=z\u0003(t) be a solution of (46) that is in a balanced\nspin-spin resonance of type ( m1: 2;m2: 2). Then, the \frst variation at such solution\nhas the form\n_y=J2@z;zHK(t;z\u0003(t))y; y2R4; (48)\nwhere@z;zHKdenotes the Hessian matrix of the Hamiltonian HKin the 4-dimensional\nvariablez. The system (48) is an LPH system of period 2 \u0019. Assume that \b( t) is a\nmatrix solution of (48) with \b( t0) =14. The stability of (48) is determined by the\nFloquet multipliers, that are the eigenvalues of the monodromy matrix \b( t0+ 2\u0019).\nAn LPH system has particular stability properties, see the general theory in [YS75,\nEke90] and an application to the double synchronous spin-spin resonance in [Mis21].\nFor example, assume that '2Cis a Floquet multiplier of an LPH system. Then, its\ninverse'\u00001, its complex conjugates \u0016 'and \u0016'\u00001are also multipliers and have the same\nmultiplicity as '. That is, the Floquet multipliers have a symmetric distribution with\nrespect to the real line and the unit circle of the complex plane. In consequence, a\nnecessary condition for stability is that all multipliers have modulus 1. Moreover, if all\nmultipliers have modulus 1 and they are all di\u000berent, then the system is stable. When all\nthe multipliers have modulus 1 and some of them coincide, the situation is not trivial and\nthe stability depends on further algebraic properties of the multipliers (Krein's theory,\n[Kre50]). Then, unlike for Hill's equations, here we do not have a quantity like the trace\nof the monodromy matrix in order to characterize the boundary of stability/instability\nregions. Instead, we will use the following de\fnition.\nDe\fnition 12. We will say that an LPH system is hyperbolic unstable if\nmax\nkj'kj>1;\nwhere'k,k= 1;2;:::, are all the Floquet multipliers of the system.\nAssume that the solution z\u0003(t) is continuous in some domain of the parameters of the\nmodel (e;C1;\u00151;\u00152;\u001b1;^q1;^q2). The equation max\nk=1;:::;4j'kj= 1 +\", with\" >0, de\fnes a26 A. CELLETTI, J. GIMENO, AND M. MISQUERO\n1-parametric family of hyperbolic unstable manifolds in the space of parameters; then,\nthe boundary of hyperbolic instability will be found if we take the limit of such manifolds\nas\"!0.\nOn the other hand, it is possible to \fnd all the solutions of a given type of a balanced\nspin-spin resonance using the shooting method as in the previous section. However, since\nthe phase space and the space of parameters have large dimensions, our approach will be\nto obtain the solutions by continuation. Note that the solutions of (24) for \u0015j= 0 and\nanye2[0;1) are exactly given by \u0012j(t) =\u0012j(0) + _\u0012j(0)t. Then, the unique solution of\ntype (\f1;\f2) of a balanced spin-spin resonance ( m1: 2;m2: 2) is just \u0012j(t) =\fj+mj\n2t.\nSuch solution can be continued for j\u0015jj>0. For small enough j\u0015jj, the systems in\neqs. (23) to (25) can be regarded as di\u000berent perturbations of the system \u0012j= 0, then,\nFigures 9 and 10 give us a quite clear idea of a region of uniqueness of a given type\nassociated to a balanced spin-spin resonance of (24). Moreover, such solution can be\nfound by continuation in the space of parameters.\nFigure 11. Regions of (hyperbolic) linear instability of solutions asso-\nciated to di\u000berent spin-spin resonances of the problem (24) (case: equal\nbodies) for di\u000berent values of the parameters (^ q;\u001b). Blue denotes instabil-\nity and yellow denotes stability. Each plot has required around 10 minutes\nusing 15 CPUs and a mesh size 750 \u0002750.THE SPIN-SPIN PROBLEM IN CELESTIAL MECHANICS 27\nCase of equal bodies E1=E2.In this case, the equation (24) depends on four parameters\n(e;\u0015;^q;\u001b), where\u0015=\u00151=\u00152, ^q= ^q1= ^q2and\u001b=\u001b1=\u001b2. From Figure 10, it looks\nthat a good choice for doing the continuation is in the range 0 \u0014e.0:8, 0\u0014\u0015.1 and\n0\u0014^q;\u001b.0:01. In this range, the linear stability of the solution of type (0 ;0) associated\nto the resonance of order (1 : 1 ;1 : 1) was investigated in [Mis21]. Figure 11 shows\nthe stability diagrams of the resonance (1 : 1 ;3 : 2) of type (0 ;0) and the resonance\n(1 : 2;3 : 2) of type (\u0019\n2;0) and (0;0) for di\u000berent (^ q;\u001b). Let us point out some properties:\n1) Note that a plot with (^ q;\u001b) = (0;0) is just the superposition of diagrams in\nFigure 9, while a plot with (^ q;\u001b) = (0:01;0) is the superposition of diagrams\nsimilar to those in Figure 10. Then, the e\u000bect of the coupling can be seen in the\nplots with\u001b6= 0.\n2) The e\u000bect of the coupling is di\u000berent in each case: for the resonance (1 : 2 ;3 : 2)\nof type (0;0), we only see an additional thin unstable region for e < 0:1 and\n0:25<\u0015< 1. For the resonance (1 : 2 ;3 : 2) of type (\u0019\n2;0), we see that the region\nwith small ebecomes unstable, whereas for large ethe stability is somewhat\naltered. Finally, without coupling, the resonance (1 : 2 ;3 : 2) of type (0 ;0) is\nunstable for almost all the points, but the coupling introduces the stability for\nsmalle. Note that this in agreement with Figure 6.\n6.Comparison between the full and the Keplerian models\nIn this section, we provide some results on the comparison between the full and Ke-\nplerian problems (Section 6.1), and we give some numerical results on the interaction\nbetween the spin and orbital motions (Section 6.2).\n6.1.Hamiltonian approach. It will be useful to write the dynamical equations of the\nHamiltonian of the full model (4) in the compact form\n_z=J4@zH(z); (49)\nwherez= (r;f;\u0012 1;\u00122;pr;pf;p1;p2)TandJ4is de\fned by (47).\nLet us now formulate the Keplerian models (spin-orbit and spin-spin, see Section 2.3)\nas perturbations of the full model, so we can compare both families of models. Consider\na function\u0010(t) that satis\fes the equation\n_\u0010(t) =J4@zH(\u0010(t))\u0000h(\u0010(t)); (50)\nwhere the vector function\nh(z) = (0;0;0;0;\u0000@rVper(z);\u0000@fVper(z);0;0)T\nis responsible of subtracting the perturbative part of the potential (see equation (7))\nonly in the equations for _ prand _pf. Thus, equation (50) represents the Keplerian models\nincluding the orbital part: the spin-orbit model corresponds to (50) with Vper=V2and\nthe spin-spin model with Vper=V2+V4. The corresponding equations of motion are (11)\nand (12), so we can write the solution in the form\n\u0010(t) = (r(t;a;e);f(t;e);\u00121(t);\u00122(t);pr(t;a;e);pf(a;e);p1(t);p2(t));\nwhereaandeare the semimajor axis and the eccentricity of the Keplerian orbit, see\n(17). Now it is clear that the Keplerian model (50) is not Hamiltonian, even though we\ncan split it into one autonomous Hamiltonian system (orbital part with V=V0) and28 A. CELLETTI, J. GIMENO, AND M. MISQUERO\nanother non-autonomous one (spin part with V=V0+Vper). Moreover, unlike for the\nfull model, in a Keplerian model neither the total angular momentum Pf=pf+p1+p2\nis conserved6, since we can compute that\n_Pf(t) =\u00002X\nj=1@\u0012jVper(r(t;a;e);f(t;e);\u00121(t);\u00122(t)):\nWe want to know if \u0010(t) is a good approximation for a solution of the full problem (49)\nwith the same initial conditions. Consider the solution z=z(t) =\u0010(t)\u0000\u000ez(t) of (49)\nsuch thatz(0) =\u0010(0). Then, expanding the equation (49) up to \frst order in \u000ez, we\nobtain that the function \u000ez=\u000ez(t) satis\fes the equation\nd\ndt\u000ez=J4@2\nz;zH(\u0010(t))\u000ez\u0000h(\u0010(t)) (51)\nwith initial condition \u000ez(0) = 0. In equation (51), @2\nz;zH(z) is the Hessian matrix associ-\nated to the Hamiltonian in (4). If the system (51) is stable, then the norm jj\u0010(t)\u0000z(t)jj\nis bounded. Additionally, it is easy to see that the system (51) is Lyapunov stable if and\nonly if the trivial solution of the homogeneous part\n_y=J4@2\nz;zH(\u0010(t))y; y2R8; (52)\nis Lyapunov stable. A general form of \u0010(t) is unknown because, although the orbital\npart is given by the classical Kepler problem, the spin part is given by a non-autonomous\nperiodic system of nonlinear equations. However, \u0010(t) is a periodic solution at spin-spin\nresonances, with period 2 \u0019for balanced resonances. In such cases, equation (52) is an\nLPH system, some of whose properties were mentioned in Section 5.3.\nAt this point, we wonder if it is possible for the periodic system (52) to be stable. Let us\npoint out an argument supporting a negative answer, even for stable spin-spin resonances.\nIt is known that the periodic solutions of the planar Kepler problem are not linearly\nstable. This can be easily checked using Poincar\u0013 e variables (action-angle variables): we\nobtain a positive eigenvalue for a linear system of constant coe\u000ecients. See further\nrelated discussions in [BO16, Sch72]. We can see that 1 is the only associated Floquet\nmultiplier and has multiplicity four; then, the instability of the periodic solutions of the\nKepler problem is not hyperbolic, according to De\fnition 12, but rather of parabolic\nkind. From this discussion, we expect that \u0010(t) andz(t) are divergent functions, but we\nwant to know if such divergence is exponential in time, that is, when (52) is hyperbolic\nunstable.\nSuppose that the function \u0010(t) is continuous on a domain of the space of parameters\n(a;e;C1;\u00151;\u00152;\u001b1;^q1;^q2). Note that we add ato the parameters of the spin-spin model\nbecause it varies with the orbital initial conditions. On the other hand, the Hamil-\ntonianHdepends on the parameters of the full model, we take the independent set\n(\u0016;C1;d1;d2;q1;q2), where the value of \u0016informs us about the disparity in the masses of\nthe bodies because \u0016=M1M2=M1(1\u0000M1). We can obtain ( \u0016;C1;d1;d2;q1;q2) from\n(a;e;C1;\u00151;\u00152;\u001b1;^q1;^q2) as follows: First, from (22), we see that, assuming \u001b1>0,\n\u0016=C1\n3\u001b1a2; (53)\n6Instead, the Keplerian assumption results in the conservation of the orbital angular momentum\npf(t) =\u0016r(t;a;e)2_f(t;e) =\u0016a2p\n1\u0000e2.THE SPIN-SPIN PROBLEM IN CELESTIAL MECHANICS 29\nfrom this value we can obtain M1because\u0016=M1(1\u0000M1) withM12(0;1). This is\nenough to write M2= 1\u0000M1,C2= 1\u0000C 1and, from the de\fnitions (22), it holds that\nd1=\u00151C1\n3M2; d 2=\u00152C2\n3M1; q 1= ^q1M1a2; q 2= ^q2M2a2:\nFrom these relations, it is obvious that, in order to compare the full and the Keplerian\nmodels, the spin-orbit versions are not enough, and we need to take Vper=V2+V4, say,\nthe spin-spin models.\nAdditionally, the initial conditions associated to \u0010(t) are given by (18), for the orbital\npart, and the values \u0012j(0);_\u0012j(0) are such that the spin part satis\fes the boundary con-\nditions for the spin-spin problem in a spin-spin resonance of a certain type as in (39).\nRecall also that, in our units, the Keplerian orbital period is T= 2\u0019andG=a3.\nAs in Section 5.3, we can \fnd the region (in the parameters space) of hyperbolic\ninstability and its boundary, that is given by the limit as \"!0 of the manifolds that\nsatisfy max\nk=1;:::;8j'kj= 1 +\", where'kare the Floquet multipliers of (52). It is important\nthat we focus on a region of the parameters where the corresponding solution associated\nto a spin-spin resonance is linearly stable, so we can see what is the e\u000bect of putting\nthe orbital and the spin parts altogether. Actually, not even the spin part of (52) is\ncompletely equivalent to (48), because all the derivatives of Vper(r;f;\u0012 1;\u00122), evaluated at\n\u0010(t), appear in (52) for both orbital and spin variables.\nFigure 12. Regions of (hyperbolic) linear instability of (1 : 2 ;3 : 2) of\ntype (\u0019=2;0) of the equation (52) for equal bodies and di\u000berent values\nof the parameters. Blue denotes instability and yellow denotes stability.\nEach plot has required around 10 minutes using 15 CPUs and a mesh size\nof 750\u0002750.\nFigure 12 shows the regions of hyperbolic instability of (52) in the case of equal bodies\nfor a particular spin-spin resonance with di\u000berent values of the parameters. We observe\nthe following:30 A. CELLETTI, J. GIMENO, AND M. MISQUERO\n(1) We can compare Figure 12 with Figure 11 for similar values of ^ qand\u001b. In the\ncase of equal bodies ( \u0016= 0:25 andC1= 0:5), the value of ais determined by\n(53): smaller \u001balso implies further bodies. Then, the plots with \u001b= 0:0001 in\nFigure 12 and be compared approximately with those with \u001b= 0 in Figure 11.\n(2) From the comparison, we see that the plots in Figure 12 with Figure 11 with\n^q= 0 are similar for small eccentricities, but, for large e, the unstable regions\nof plots in Figure 12 are larger. On the other hand, the plots in Figure 12 with\n^q= 0:01 include unstable stripes invading the stable regions for small e(they\ngrow with\u001b), whereas for large ethe diagrams become totally unstable.\n(3) As we increase \u001b(closer bodies), the plots become more and more unstable.\nWe conclude that the spin-spin model should be a reliable simpli\fcation of the full\nmodel in the regions of the parameters with small e, largeaand close to regions with\nstable spin-spin resonances. We remark also that instability of spin-spin resonances not\nnecessarily implies hyperbolic instability of (48): there are some regions for which the\nFloquet multipliers of the spin-spin resonance are not too far from the unit circle, so\nthat when we consider the corresponding system (48), all its Floquet multipliers belong\nto the unit circle. It is also remarkable that, although the comparison makes sense only\nbetween the full and Keplerian spin-spin models, we obtain almost identical plots as in\nFigure 12 independently if we use Vper=V2orVper=V2+V4in the Hamiltonian in (52).\n6.2.Quantitative numerical approach. In this section, we want to investigate, from\na numerical point of view, two questions that the Hamiltonian approach left unanswered:\n(Q1) Can we quantify the in\ruence of the spin motion on the orbital one? (Q2) Is it\npossible that the bodies end up colliding, even though (52) is not hyperbolic unstable?\nIn Section 6.1, we saw that the solution of a Keplerian model \u0010(t) corresponding to a\nspin-spin resonance should diverge from a solution of the full model z(t) with identical\ninitial conditions. In the region of parameters of hyperbolic instability the divergence\nshould be exponential; however, in the rest of the parametric space, we want to quantify\nhow di\u000berent both motions are. So, let us take the point ( a;e;C1;\u00151;\u00152;\u001b1;^q1;^q2) and\ncompute the functions \u0010(t) andz(t) in order to compare them.\nThe orbital motion of \u0010(t) is characterized unambiguously by the set of Keplerian\nelements (a;e;! ), where!= 0 is the argument of the periapsis, together with t, the\nmean anomaly. On the other hand, let the orbital part of the solution z(t) of the full\nmodel be given by ( rF(t);fF(t)). Then, we can transform the orbital position rF(t) =\nrF(t) exp(ifF(t)) to the osculating Keplerian elements ( aF(t);eF(t);!F(t)) of the two-\nbody problem using the following expressions. From the geometrical identity j_rFj2=\nG\u0010\n2\nrF\u00001\naF\u0011\n, we obtain\naF(t) = \n2\nrF\u0000_r2\nF+_f2\nFr2\nF\nG!\u00001\n:\nNow let us de\fne the orbital angular momentum per unit mass hF=rF^_rF, where^\nis the vector product, and the eccentricity vector\neF(t) =_rF^hF\nG\u0000rF\nrF;THE SPIN-SPIN PROBLEM IN CELESTIAL MECHANICS 31\nwhose modulus is given by\neF(t) =s\n1\u0000r4\nF_f2\nF\nGaF:\nWhenevereF6= 0, we can de\fne !F2[\u0000\u0019;\u0019) as the polar angle of eF. The mean\nanomaly associated to the full model can be de\fned too, but we will not use it in our\nstudy. The balanced spin-spin resonance of order ( m1: 2;m2: 2) in the function \u0010(t) is\ncharacterized by the modi\fed resonant angles  mj:2\nj(t) =mjf(t;e)\u00002\u0012j(t). Seemingly,\nthe spin part of the solution z(t) of the full model is given by ( \u00121;F(t);\u00122;F(t)), so, in\norder to compare with \u0010(t), let us de\fne  mj:2\nj;F(t) =mjfF(t)\u00002\u0012j;F(t). Now de\fne the\nfunctions\n\u000ea(t) =aF(t)\u0000a\na; \u000ee(t) =eF(t)\u0000e; \u000eres;j(t) = mj:2\nj(t)\u0000 mj:2\nj;F(t); (54)\nwhere\u000eais the relative deviation in semi-major axis, while \u000eeand\u000eres;jare the absolute\ndeviations in eccentricity and resonant angles, respectively.\nFinally, recall from (1) that we have an expression for ajin terms of the parameters of\nthe model. Then, for our purpose, we will say that there is a collision if rF(t)\u0014a1+a2.\nNow we are in a position to compare the solutions of the full model with respect with\nthose of the Keplerian models.\nIn Figure 13 we see, for di\u000berent values of the parameters and the case of identical\nbodies, the comparison between z(t) and\u0010(t) in a resonance (1 : 1 ;3 : 2) of type (0 ;0).\nParticularly, we see the evolution of the \u000e-functions in (54) and some Kepler elements of\nz(t) in 100 revolutions. In addition, Table 2 informs us about the corresponding Floquet\nmultipliers of both z(t) and\u0010(t). We summarize the following observations:\n(1) We chose two cases for the comparison. In the \frst case, e= 0 and\u001b= 10\u00003, so\nthe bodies are relatively close to each other ( a\u001939) in circular orbits, whereas in\nthe second case, e= 0:1 and\u001b= 10\u00007, so the bodies are much further ( a\u00193900)\nin moderately elliptic orbits. For both cases, we took values of parameters whose\ncorresponding resonance is not hyperbolic unstable, say, \u0015= 0:05 and ^q= 0:01.\n(2) From Figure 13, we see that the \frst case has a more regular behavior than the\nsecond one. In the \frst case, the orbit of the full model oscillates regularly very\nclose to the circular orbit of the Keplerian model, with a precession that increases\nuniformly in average. Actually, in a shorter time scale, the eccentricity vector\ndescribes a circles passing through the origin. On the other hand, the orbit of\nthe second case is greatly irregular, with variations of size up to \u001825%. Even its\nprecession changes direction at some moment and the eccentricity vector swings\nchaotically in short time scales.\n(3) The variation of the angles is quite regular in both cases. Except for the reso-\nnant angle of the resonance 1:1 of the \frst case, which oscillates regularly with\nsmall amplitude, the rest of the resonant angles increase/decrease more or less\nconstantly, with larger variations in the second case.\n(4) The Floquet multipliers in Table 2 gives us more information about both cases.\nWe remark that we display the multipliers of \u0010(t) corresponding to spin and orbit\nseparately, say, the \frst two rows are the four multipliers of the spin-spin model\n(24), whereas the third one shows the four coincident multipliers of the Kepler\nproblem. We con\frm that the spin of \u0010(t) is elliptic stable in the two cases.32 A. CELLETTI, J. GIMENO, AND M. MISQUERO\n-0.0004-0.0002 0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016\n 0  20  40  60  80  100δa(u)\nu/(2π) 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035\n 0  20  40  60  80  100δe(u)\nu/(2π)\n 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035\n 0 1 2 3 4 5 6 7 8eF(u)\nu/(2π)-1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1\n 0  20  40  60  80  100ω(u)/(2π)\nu/(2π) 0  0.01  0.02  0.03eF(ω)\n-0.4-0.3-0.2-0.1 0 0.1 0.2 0.3 0.4\n 0  20  40  60  80  100δres,1(u)\nu/(2π)-90-80-70-60-50-40-30-20-10 0\n 0  20  40  60  80  100δres,2(u)\nu/(2π)-15-10-5 0 5 10 15 20 25\n 0  20  40  60  80  100\nu/(2π)fF(u) - f(u)\nθ1,F(u) - θ 1(u)\nθ2,F(u) - θ 2(u)\n-0.3-0.25-0.2-0.15-0.1-0.05 0 0.05 0.1 0.15 0.2 0.25\n 0  20  40  60  80  100δa(u)\nu/(2π)-0.03-0.02-0.01 0 0.01 0.02 0.03\n 0  20  40  60  80  100δe(u)\nu/(2π)\n 0.075 0.08 0.085 0.09 0.095 0.1 0.105 0.11\n 0 1 2 3 4 5 6 7 8eF(u)\nu/(2π)-0.7-0.6-0.5-0.4-0.3-0.2-0.1 0 0.1 0.2 0.3\n 0  20  40  60  80  100ω(u)/(2π)\nu/(2π) 0  0.05  0.1eF(ω)\n-100 0 100 200 300 400 500 600 700 800 900\n 0  20  40  60  80  100δres,1(u)\nu/(2π)-900-800-700-600-500-400-300-200-100 0 100\n 0  20  40  60  80  100δres,2(u)\nu/(2π)-400-300-200-100 0 100 200 300 400 500\n 0  20  40  60  80  100\nu/(2π)fF(u) - f(u)\nθ1,F(u) - θ 1(u)\nθ2,F(u) - θ 2(u)\nFigure 13. Comparison between the full and the Keplerian solutions in\na resonance (1 : 1 ;3 : 2) of type (0 ;0) for di\u000berent parameter values in the\ncase of equal bodies. Top plots: e= 0,\u0015= 0:05, ^q= 0:01,\u001b= 10\u00003.\nBottom plots: e= 0:1,\u0015= 0:05, ^q= 0,\u001b= 10\u00007.THE SPIN-SPIN PROBLEM IN CELESTIAL MECHANICS 33\nOn the other hand, the multipliers of z(t) are computed with (49). We \fnd out\nthat, although the \frst case is more regular, the solution is actually hyperbolic\nunstable, whereas the opposite occurs for the second case: it is irregular but not\nhyperbolic. This is consistent with Figure 12.\ne= 0;\u0015= 0:05;^q= 0:01;\u001b= 10\u00003e= 0:1;\u0015= 0:05;^q= 0:01;\u001b= 10\u00007\nArgument Modulus Argument Modulus\n\u0010(t)\u00061:42 1\u00061:39 1\n\u00067:36\u000110\u000081\u00068:34\u000110\u000011\n0 1 (quadruple) 0 1 (quadruple)\nz(t)\u00061:41 1\u00061:39 1\n\u00067:64\u000110\u000091\u00068:34\u000110\u000011\n\u00061:65\u000110\u000011:09\u00063:77\u000110\u000041\n\u00061:65\u000110\u000010:915 0 1 (double)\nTable 2. Modulus and argument of the Floquet multipliers of the func-\ntionsz(t) and\u0010(t) compared in Figure 13.\nWe conclude that, even when there is hyperbolic instability, the solution of full system\nremains quite close to the solution of the Keplerian system with circular orbit. However,\neven with no hyperbolic instability, for e6= 0, the behavior of both solutions is remarkably\ndi\u000berent. This fact should be attenuated with very small eccentricities.\nWe can make a step further in the comparison when e= 0. Figure 14 shows a\nquantitative comparison in the orbits of the full and Keplerian models. We vary the\nparameters \u001band\u0015keeping ^q= 0 constant. For large enough \u001b(smalla), there is a\nregion of parameters leading to collision. The rest of the diagrams show di\u000berent orders\nof magnitude of \u000e-functions corresponding to aande. We check that, for very small \u0015,\nthe orbit of the full model is very close to the Keplerian one.\n7.Conclusions\nThis research is, primarily, a careful numerical study of the spin-spin model, presented\nas such in [Mis21]. For this purpose, we have considered several parallel models describing\nthe planar gravitational dynamics of two ellipsoids under the Assumptions 1 to 3. We\ndistinguish between full and Keplerian models, and also, between spin-orbit and spin-spin\nmodels. The known dynamics of the Keplerian spin-orbit problem was used as reference\nto develop our results.\nIn \frst place, we realize that the symmetries of the ellipsoids lead to certain symmetries\nin the equations of the Keplerian models, and so, to symmetric periodic solutions that\nstructure the overall dynamics. We complete this general view by providing conditions\nfor existence of quasi-periodic solutions. Between the periodic solutions, we focus on\na special class that comprises the simplest solutions from the physical interpretation,34 A. CELLETTI, J. GIMENO, AND M. MISQUERO\nFigure 14. Contour plots of max flog10j\u000ee(u)j:u2[0;4\u0019]gon the left\nand maxflog10j\u000ea(u)j:u2[0;4\u0019]gon the right for the resonance (1 : 1 ;3 :\n2) of type (0 ;0) and parameter values e= 0 and ^q= 0 for equal bodies.\nThe black region denotes the collision during the integration of the full\nspin-spin model, i.e. when rF(u)\u0014a1+a2with ajde\fned in (1). Using\n80 CPUs the plots required around 1 hour with a mesh of 10242and 2048\nstopping points in [0 ;4\u0019].\nthe balanced spin-orbit and spin-spin resonances. We study the high-dimensional phase\nspace of the spin-spin problem by means of projections of Poincar\u0013 e maps, taking those\nof the spin-orbit problem as reference.\nFirst, we see that when one body is spherical, the dynamics of the spin-orbit problem\nis reproduced with small variations. For two ellipsoids, the projections of the Poincar\u0013 e\nmaps show structures similar to those of the spin-orbit ones. There is a particular\nbehavior when both bodies are identical, the so-called measure synchronization.\nAfter that, we study existence, multiplicity and linear stability of the solutions in bal-\nanced resonances as we vary the parameters of the problem. We focus on the qualitative\nchanges in the stability diagrams produced by the variation of each parameter. We use\nthe case of identical bodies to illustrate our observations most of the time.\nIn the last part of the paper, we compare the solutions of the full and the Keplerian\nmodels by means of rewriting the equations in a Hamiltonian-like form. We produce\nstability diagrams similar to the previous ones to show such a comparison. Representing\ndi\u000berent aspects with similar diagrams allows one a much more global understanding\nof a problem with so many variables and parameters. We end up by comparing the\nevolution of particular solutions for certain parameters. The orbit of the full problem is\ncharacterized by Kepler's elements to facilitate the comparison. From the parameters,\nwe observe that the comparison only makes sense for the spin-spin models. Here we \fnd\nout that, beside the case of circular orbits, for which full and Keplerian models show\na close behavior, for eccentric orbits, the trajectories are very dissimilar. We \fnish the\ncomparison in the case of circular orbits by showing that the space of parameters isTHE SPIN-SPIN PROBLEM IN CELESTIAL MECHANICS 35\ndivided into regions leading to collision or regions with orbits di\u000bering speci\fc orders of\nmagnitude.\nThe topics and results presented in this work are just a taster of some features of the\nrich spin-spin dynamics. We also include a comparison with the full models, which is not\nwidely studied in the literature; however, this is crucial, because it shows us the limits\nof validity of the models for applications. We believe that here is still a lot to explore\nin this model, which can be used to investigate problems like Arnold's di\u000busion or the\nexistence of normally hyperbolic manifolds.\nAppendix A.Coefficients of the expansion of the potential\nThe full expansion of the potential energy of the Full Two-Body Problem was derived\nin [Bou17]. Later, [Mis21] detailed that expression to the case of planar motion of two\nellipsoids, which is given by\nV=\u0000GM 1M2\nrX\n(l1;m1)2\u0007\n(l2;m2)2\u0007\u0000l1;m1\nl2;m2\u0012R1\nr\u00132l1\u0012R2\nr\u00132l2\n\u0002\n\u0002Z1)\n2l1;2m1Z2)\n2l2;2m2cos(2m1(\u00122\u0000f) + 2m2(\u00122\u0000f));\nwhere\n\u0007 =f(l;m)2Z2: 0\u0014jmj\u0014lg;\nand, if we take L=l1+l2andM=m1+m2, the constants\n\u0000l1;m1\nl2;m2=(\u00001)L\u0000M\n4Lp\n(2l1\u00002m1)!(2l1+ 2m1)!(2l2\u00002m2)!(2l2+ 2m2)!(2L\u00002M)!(2L+ 2M)!\n(L\u0000M)!(L+M)!\nare numbers dealing with the interaction between the extended bodies. On the other\nhand,RjandZj)\n2lj;2mjare, respectively, the mean radius and the Stokes coe\u000ecients of\neachEj. The quantitiesZj)\nl;mprovide the expansion of the potential created for the body\nEj. They are related to the usual parameters Cj)\nl;mandSj)\nl;mby\nCj)\nl;m+iSj)\nl;m= (\u00001)m 2\n1 +\u000em;0s\n(l\u0000m)!\n(l+m)!\u0016Zj)\nl;m; m\u00150;\nwhere\u000em;nis the Kronecker delta.\nAppendix B.A different formulation of the equations of motion\nTo perform the integration of (12), it is convenient to adopt the eccentric anomaly u\nas independent variable, according to the following procedure.\nLet us write the equations (12) as\nCjd2\u0012j\ndt2(t) +\u0010a\nr\u00115\n\"jFj(t;\u0012) = 0; j = 1;2; (55)36 A. CELLETTI, J. GIMENO, AND M. MISQUERO\nwhere\u0012= (\u00121;\u00122),C1+C2= 1 andFj=@\u0012j(V2+V4),j= 1;2. The explicit expression\nforF1andF2is given by\nF1(t;\u0012) =\u0012\u0010r\na\u00112\n+5\n4\u0000\n^q2+5\n7^q1\u0001\u0013\nsin(2\u00121\u00002f)\n+25^d1\n8sin(4\u00121\u00004f) +3^d2\n8sin(2\u00121\u00002\u00122) +35^d2\n8sin(2\u00122+ 2\u00121\u00004f)\nF2(t;\u0012) =\u0012\u0010r\na\u00112\n+5\n4\u0000\n^q1+5\n7^q2\u0001\u0013\nsin(2\u00122\u00002f) +25^d2\n8sin(4\u00122\u00004f)\n\u00003^d1\n8sin(2\u00121\u00002\u00122) +35^d1\n8sin(2\u00122+ 2\u00121\u00004f): (56)\nLet us consider the change of variables given by the Kepler's equation\nxj(u) =\u0012j(u\u0000esinu); j = 1;2:\nThen, we have\nd2\u0012j\ndt2(t) =\u0012a\nr(u)\u00132d2xj\ndu2(u)\u0000\u0012a\nr(u)\u00133dxj\ndu(u)esinu; j = 1;2;\nso that, assuming Cj6= 0, (55) becomes\nd2xj\ndu2(u)\u0000a\nr(u)dxj\ndu(u)esinu+\u0012a\nr(u)\u00133\"j\nCjFj(u;x) = 0; j = 1;2: (57)\nWe can write the system (57) as\ndxj\ndu(u) =yj(u)\ndyj\ndu(u) =a\nr(u)yj(u)esinu\u0000\u0012a\nr(u)\u00133\"j\nCjFj(u;x);\nforj= 1;2 andx= (x1;x2). From the well-known relations used in the study of Kepler's\nproblem\ncosf=cosu\u0000e\n1\u0000ecosu; sinf=p\n1\u0000e2sinu\n1\u0000ecosu;\nwe can de\fne the functions s=s(xj),c=c(xj) as\ns(xj) = sin(2xj)(2 cos2f\u00001)\u0000cos(2xj)2 cosfsinf\nc(xj) = cos(2xj)(2 cos2f\u00001) + sin(2xj)2 cosfsinf ;THE SPIN-SPIN PROBLEM IN CELESTIAL MECHANICS 37\nso thatF1andF2in (56) take the form\nF1(u;x) =\u0012\u0012r(u)\na\u00132\n+5\n4\u0000\n^q2+5\n7^q1\u0001\u0013\ns(x1)\n+25^d1\n4s(x1)c(x1) +3^d2\n8sin(2x1\u00002x2) +35^d2\n4s(x1+x2)c(x1+x2)\nF2(u;x) =\u0012\u0012r(u)\na\u00132\n+5\n4\u0000\n^q1+5\n7^q2\u0001\u0013\ns(x2) +25^d2\n4s(x1)c(x1)\n\u00003^d1\n8sin(2x1\u00002x2) +35^d1\n4s(x1+x2)c(x1+x2):\nAppendix C.Expansion of V2andV4\nWe give below the expansion of V2andV4up to second order in the eccentricity:\nV2=\u00003d1e2GM 2cos(2\u00121)\n8a3\u00003d2e2GM 1cos(2\u00122)\n8a3\u000027d1e2GM 2cos(2t\u00002\u00121)\n8a3\n\u00003d1e2GM 2cos(4t\u00002\u00121)\n4a3\u000027d2e2GM 1cos(2t\u00002\u00122)\n8a3\u00003d2e2GM 1cos(4t\u00002\u00122)\n4a3\n\u00009d1eGM 2cos(t\u00002\u00121)\n8a3\u00009d1eGM 2cos(3t\u00002\u00121)\n8a3\u00009d2eGM 1cos(t\u00002\u00122)\n8a3\n\u00009d2eGM 1cos(3t\u00002\u00122)\n8a3\u00003d1GM 2cos(2t\u00002\u00121)\n4a3\u00003d2GM 1cos(2t\u00002\u00122)\n4a3\n\u00003e2GM 2q1cos(2t)\n8a3\u00003e2GM 1q2cos(2t)\n8a3\u00009e2GM 2q1\n8a3\u00009e2GM 1q2\n8a3\u00003eGM 2q1cos(t)\n4a3\n\u00003eGM 1q2cos(t)\n4a3\u0000GM 2q1\n4a3\u0000GM 1q2\n4a3;\nV4=\u0000225GM 2q2\n1e2\n112a5M1\u0000225Gcos(2t)M2q2\n1e2\n224a5M1\u0000225GM 1q2\n2e2\n112a5M2\u0000225Gcos(2t)M1q2\n2e2\n224a5M2\n\u0000105Gcos(2t\u00002\u00121\u00002\u00122)d1d2e2\n16a5\u0000525Gcos(4t\u00002\u00121\u00002\u00122)d1d2e2\n16a5\n\u0000315Gcos(6t\u00002\u00121\u00002\u00122)d1d2e2\n32a5\u000045Gcos(2\u00121\u00002\u00122)d1d2e2\n16a5\n\u000045Gcos(2t+ 2\u00121\u00002\u00122)d1d2e2\n64a5\u000045Gcos(2t\u00002\u00121+ 2\u00122)d1d2e2\n64a5\u0000225Gd2\n1M2e2\n224a5M1\n\u0000225Gcos(2t)d2\n1M2e2\n448a5M1\u000075Gcos(2t\u00004\u00121)d2\n1M2e2\n32a5M1\u0000375Gcos(4t\u00004\u00121)d2\n1M2e2\n32a5M1\n\u0000225Gcos(6t\u00004\u00121)d2\n1M2e2\n64a5M1\u000075Gcos(2t\u00002\u00122)d2q1e2\n8a5\u0000165Gcos(4t\u00002\u00122)d2q1e2\n64a5\n\u0000135Gcos(2\u00122)d2q1e2\n64a5\u0000375Gcos(2t\u00002\u00121)d1M2q1e2\n56a5M1\u0000825Gcos(4t\u00002\u00121)d1M2q1e2\n448a5M1\n\u0000675Gcos(2\u00121)d1M2q1e2\n448a5M1\u000075Gcos(2t\u00002\u00121)d1q2e2\n8a5\u0000165Gcos(4t\u00002\u00121)d1q2e2\n64a538 A. CELLETTI, J. GIMENO, AND M. MISQUERO\n\u0000135Gcos(2\u00121)d1q2e2\n64a5\u000045Gq1q2e2\n8a5\u000045Gcos(2t)q1q2e2\n16a5\u0000375Gcos(2t\u00002\u00122)d2M1q2e2\n56a5M2\n\u0000825Gcos(4t\u00002\u00122)d2M1q2e2\n448a5M2\u0000675Gcos(2\u00122)d2M1q2e2\n448a5M2\u0000225Gd2\n2M1e2\n224a5M2\u0000225Gcos(2t)d2\n2M1e2\n448a5M2\n\u000075Gcos(2t\u00004\u00122)d2\n2M1e2\n32a5M2\u0000375Gcos(4t\u00004\u00122)d2\n2M1e2\n32a5M2\u0000225Gcos(6t\u00004\u00122)d2\n2M1e2\n64a5M2\n\u0000225Gcos(t)M2q2\n1e\n224a5M1\u0000225Gcos(t)M1q2\n2e\n224a5M2\u0000525Gcos(3t\u00002\u00121\u00002\u00122)d1d2e\n64a5\n\u0000525Gcos(5t\u00002\u00121\u00002\u00122)d1d2e\n64a5\u000045Gcos(t+ 2\u00121\u00002\u00122)d1d2e\n64a5\n\u000045Gcos(t\u00002\u00121+ 2\u00122)d1d2e\n64a5\u0000225Gcos(t)d2\n1M2e\n448a5M1\u0000375Gcos(3t\u00004\u00121)d2\n1M2e\n128a5M1\n\u0000375Gcos(5t\u00004\u00121)d2\n1M2e\n128a5M1\u000075Gcos(t\u00002\u00122)d2q1e\n32a5\u000075Gcos(3t\u00002\u00122)d2q1e\n32a5\n\u0000375Gcos(t\u00002\u00121)d1M2q1e\n224a5M1\u0000375Gcos(3t\u00002\u00121)d1M2q1e\n224a5M1\u000075Gcos(t\u00002\u00121)d1q2e\n32a5\n\u000075Gcos(3t\u00002\u00121)d1q2e\n32a5\u000045Gcos(t)q1q2e\n16a5\u0000375Gcos(t\u00002\u00122)d2M1q2e\n224a5M2\n\u0000375Gcos(3t\u00002\u00122)d2M1q2e\n224a5M2\u0000225Gcos(t)d2\n2M1e\n448a5M2\u0000375Gcos(3t\u00004\u00122)d2\n2M1e\n128a5M2\n\u0000375Gcos(5t\u00004\u00122)d2\n2M1e\n128a5M2\u000045GM 2q2\n1\n224a5M1\u000045GM 1q2\n2\n224a5M2\u0000105Gcos(4t\u00002\u00121\u00002\u00122)d1d2\n32a5\n\u00009Gcos(2\u00121\u00002\u00122)d1d2\n32a5\u000045Gd2\n1M2\n448a5M1\u000075Gcos(4t\u00004\u00121)d2\n1M2\n64a5M1\n\u000015Gcos(2t\u00002\u00122)d2q1\n16a5\u000075Gcos(2t\u00002\u00121)d1M2q1\n112a5M1\n\u000015Gcos(2t\u00002\u00121)d1q2\n16a5\u00009Gq1q2\n16a5\u000075Gcos(2t\u00002\u00122)d2M1q2\n112a5M2\u000045Gd2\n2M1\n448a5M2\n\u000075Gcos(4t\u00004\u00122)d2\n2M1\n64a5M2:\nReferences\n[Arn63] V. I. Arnol'd. Proof of a theorem of A. N. Kolmogorov on the invariance of quasi-periodic\nmotions under small perturbations. Russian Math. Surveys , 18(5):9{36, 1963.\n[Bel66] V. V. Beletskii. Motion of an arti\fcial satellite about its center of mass . Mechanics of\nSpace Flight. Israel Program for Scienti\fc Translations; [available from the U.S. Dept. of\nCommerce, Clearinghouse for Federal Scienti\fc and Technical Information, Spring\feld,\nVa.], Jerusalem, 1966.\n[BL75] V. V. Beletskii and E. K. Lavrovskii. On the theory of the resonance rotation of Mercury.\nAstronomicheskii Zhurnal , 52:1299{1308, December 1975.\n[BL09] Gwena el Bou\u0013 e and Jacques Laskar. Spin axis evolution of two interacting bodies. Icarus ,\n201(2):750 { 767, 2009.THE SPIN-SPIN PROBLEM IN CELESTIAL MECHANICS 39\n[BM15] Konstantin Batygin and Alessandro Morbidelli. Spin-Spin coupling in the Solar System.\nThe Astrophysical Journal , 810(2):110, sep 2015.\n[BO16] Alberto Boscaggin and Rafael Ortega. Periodic solutions of a perturbed kepler problem\nin the plane: From existence to stability. Journal of Di\u000berential Equations , 261(4):2528{\n2551, 2016.\n[Bou17] Gwena el Bou\u0013 e. The two rigid body interaction using angular momentum theory formulae.\nCelestial Mechanics and Dynamical Astronomy , 128(2):261{273, Jun 2017.\n[CC00] Alessandra Celletti and Luigi Chierchia. Hamiltonian stability of spin{orbit resonances in\ncelestial mechanics. Celestial Mechanics and Dynamical Astronomy , 76:229{240, 05 2000.\n[CC08] A. Celletti and L. Chierchia. Measures of basins of attraction in spin-orbit dynamics.\nCelestial Mechanics and Dynamical Astronomy , 101:159{170, May 2008.\n[CC09] A. Celletti and L. Chierchia. Quasi-Periodic Attractors in Celestial Mechanics. Archive\nfor Rational Mechanics and Analysis , 191:311{345, February 2009.\n[CCGdlL21a] Renato C. Calleja, Alessandra Celletti, Joan Gimeno, and Rafael de la Llave. Break-down\nthreshold of invariant attractors in the dissipative spin-orbit problem. Preprint , 2021.\n[CCGdlL21b] Renato C. Calleja, Alessandra Celletti, Joan Gimeno, and Rafael de la Llave. E\u000ecient\nand accurate KAM tori construction for the dissipative spin-orbit problem using a map\nreduction. Preprint , 2021.\n[CCGdlL21c] Renato C. Calleja, Alessandra Celletti, Joan Gimeno, and Rafael de la Llave. KAM quasi-\nperiodic tori for the dissipative spin-orbit problem. Preprint , 2021.\n[Cel90] A. Celletti. Analysis of resonances in the spin-orbit problem in celestial mechanics: The\nsynchronous resonance (Part I). Zeitschrift Angewandte Mathematik und Physik , 41:174{\n204, March 1990.\n[Cel10] Alessandra Celletti. Stability and Chaos in Celestial Mechanics . Springer, Berlin, Heidel-\nberg, 01 2010.\n[CFL04] A. Celletti, C. Falcolini, and U. Locatelli. On the break-down threshold of invariant tori\nin four dimensional maps. Regul. Chaotic Dyn. , 9(3):227{253, 2004.\n[DeV58] Ren\u0013 e DeVogelaere. On the Structure of Symmetric Periodic Solutions of Conservative\nSystems, with Applications , pages 53{84. Princeton University Press, 1958.\n[Eke90] I. Ekeland. Convexity Methods in Hamiltonian Mechanics , volume 19 of Ergebnisse der\nMathematik und ihrer Grenzgebiete : a series of modern surveys in mathematics. Folge\n3. Springer-Verlag, Berlin Heidelberg, 1990.\n[Gol80] Herbert Goldstein. Classical Mechanics . Addison-Wesley, 1980.\n[GP66] Peter Goldreich and Stanton Peale. Spin orbit coupling in the Solar System. The Astro-\nnomical Journal , 71:425, 07 1966.\n[Gre79] John M. Greene. A method for determining a stochastic transition. J. Math. Phys. ,\n20(6):1183{1201, 1979.\n[HX17] Xiyun Hou and Xiaosheng Xin. A note on the spin{orbit, spin{spin, and spin{orbit{spin\nresonances in the binary minor planet system. The Astronomical Journal , 154(6):257, nov\n2017.\n[HZ99] Alan Hampton and Dami\u0013 an H. Zanette. Measure synchronization in coupled hamiltonian\nsystems. Phys. Rev. Lett. , 83:2179{2182, Sep 1999.\n[JA16] Mahdi Jafari Nadoushan and Nima Assadian. Geography of the rotational resonances and\ntheir stability in the ellipsoidal full two body problem. Icarus , 265:175 { 186, 2016.\n[JNA16] Mahdi Jafari Nadoushan and Nima Assadian. Chirikov di\u000busion in the sphere-ellipsoid\nbinary asteroids. Nonlinear Dynamics , 85, 08 2016.\n[JZ05] \u0012A. Jorba and M. Zou. A software package for the numerical integration of ODEs by means\nof high-order Taylor methods. Experiment. Math. , 14(1):99{117, 2005.\n[Kin72] H. Kinoshita. First-order perturbations of the two \fnite body problem. Publ. Astron. Soc.\nJapan , 24:423, January 1972.\n[Kol54] Andrej N. Kolmogorov. On the conservation of conditionally periodic motions under small\nperturbation of the hamiltonian. In Dokl. Akad. Nauk. SSR , volume 98, pages 2{3, 1954.\n[Kre50] M.G. Krein. Generalization of certain investigations of A.M. Lyapunov on linear di\u000ber-\nential equations with periodic coe\u000ecients. Dokl. Akad. Nauk USSR , 73:445{448, 1950.40 A. CELLETTI, J. GIMENO, AND M. MISQUERO\n[LR98] Jeroen S.W. Lamb and John A.G. Roberts. Time-reversal symmetry in dynamical sys-\ntems: A survey. Physica D: Nonlinear Phenomena , 112(1):1{39, 1998. Proceedings of the\nWorkshop on Time-Reversal Symmetry in Dynamical Systems.\n[Mac95] Andrzej J. Maciejewski. Reduction, relative equilibria and potential in the two rigid bodies\nproblem. Celestial Mechanics and Dynamical Astronomy , 63(1):1{28, Mar 1995.\n[Mis21] Mauricio Misquero. The spin-spin model and the capture into the double synchronous\nresonance. Nonlinearity , 34:2191{2219, 2021.\n[MO20] Mauricio Misquero and Rafael Ortega. Some rigorous results on the 1:1 resonance of the\nspin-orbit problem. SIAM J. Applied Dynamical Systems , 19(4):2233{2267, 2020.\n[Mos62] J. Moser. On invariant curves of area-preserving mappings of an annulus. Nachr. Akad.\nWiss. G ottingen Math.-Phys. Kl. II , 1962:1{20, 1962.\n[MW79] W. Magnus and S. Winkler. Hill's Equation . Dover, New York, 1979.\n[Sch72] H.R. Schwarz. Stability of Kepler motion. Computer Methods in Applied Mechanics and\nEngineering , 1(3):279{299, 1972.\n[Sch02] D. J. Scheeres. Stability in the Full Two-Body Problem. Celestial Mechanics and Dynam-\nical Astronomy , 83(1):155{169, 2002.\n[Sch09] D. J. Scheeres. Stability of the planar full 2-body problem. Celestial Mechanics and Dy-\nnamical Astronomy , 104(1):103{128, Jun 2009.\n[Ver78] J.H. Verner. Explicit Runge-Kutta methods with estimates of the local truncation error.\nSIAM J. Numer. Anal. , 15(4):772{790, 1978.\n[YS75] V. A. Yakubovich and V. M. Starzhinskii. Linear Di\u000berential Equations with Periodic\nCoe\u000ecients . Wiley, New York, 1975.\n[ZOST66] V.A. Zlatoustov, D.E. Ohotzimsky, V.A. Sarychev, and A.P. Torzhevsky. Investigation of\na satellite oscillations in the plane of an elliptic orbit. In G ortler H., editor, Applied Me-\nchanics. Proceedings of the Eleventh International Congress of Applied Mechanics Munich\n(Germany) 1964 , pages 436{439. Springer, Berlin, Heidelberg, 1966.\nDepartment of Mathematics, University of Rome Tor Vergata, Via della Ricerca\nScientifica 1, 00133 Rome (Italy)\nEmail address :celletti@mat.uniroma2.it\nDepartment of Mathematics, University of Rome Tor Vergata, Via della Ricerca\nScientifica 1, 00133 Rome (Italy)\nEmail address :gimeno@mat.uniroma2.it\nDepartment of Mathematics, University of Rome Tor Vergata, Via della Ricerca\nScientifica 1, 00133 Rome (Italy)\nEmail address :misquero@mat.uniroma2.it"    },    {        "title": "1211.4926v1.The_quantum_dynamics_of_two_coupled_large_spins.pdf",        "content": "The quantum dynamics of two coupled large spins  \nV.E. Zobov \nL.V.Kirensky Institu te of Physics, Ru ssian Academy of Sciences, Siberian Branch, \n660036,Krasnoyarsk, R ussia; rsa@iph.krasn.ru \n \nWe calculate the tim e evolution of m ean spin com ponents and the squared I-concurrence \nof two coupled large spins S. As the initial conditions we take tw o cases: spin coherent states and \nuniform  superposition states. For the spin cohere nt states we have obtained the asym ptotic for-\nmulas at S>>1 and t<<T  (T is period ). We draw a conclusion that  quantum com putation on \nqudits, represented by large spins, do not get th e principal lim itations on th e spin num ber S. \n \nIntroductio n \nIn recent years, quantum com putation not  only on quantum  system s with two levels \n(qubits) but on d-level system s (qudits) [1-3] is investigated. Qudit quant um infor mation proc-\nessing em ploys fewer coupled quantum  system s: a considerable advantage for the experim ental \nrealization of quantum  computing:  \n2ln/ ln ln/ ln N n d N n =⇔ = . \nA particle w ith large spin S, for example a qua drupole nucleus or a single-m olecule m agnet, can \nbe treated as a qudit with d=2S+1. Interactions of  the spin with a sta tic m agnetic f ield and with \nan electric field gradient result in form ation non-equidistant energy levels,  allowing to selective \ncontro l of the sta tes. W e can ask: is there li mitation on the va lue of  S? W ill quan tum behaviour \n(superposition and entanglem ent) be retained with incre asing in S?  \nWe can f ound papers ( see e.g. [ 4-5]), in which it is expected that with incr easing of  S will \nbe the transition from  quantum  to classical sp in angular m omentum . However, another authors \n[7-8] do not share this expectation, because the entanglem ent [8-9] and the boundary of Bell' s \ninequalities [7] increase with increas ing S. To st udy this problem , we now  consider the quantum  \ndynam ics of a sim ple system : two coupled large sp ins. W e calculate the tim e evolution of spin \ncomponents and the squared I-concurrence [ 10], which is an entanglem ent m easure. \n \nThe system \n The spin-sp in inte raction  between tw o spins S is given by the  Ising- like Ham iltonian  \nZZ\nSS SSSJH21−= ,     ( 1) \nwhere  is Z com ponent of spin-S angular-m omentum  operatorZ\niSiSr\n (i=1, 2). To study the lim it S \n→ ∞, we have scaled a constant  interaction inver sely with S, f ollow B.C. Sanders [ 4]. \nLet us use the basis of eigenstates of  (with certain value at the projection on Z-axis \nfor each spin): Z\niS\n2 1 m m⊗ , where m 1 and m 2 are 2S +1 values:              -S, -S+1,  .  .  .  , S-1, S. \nSuppose that the two spins are init ially in a pure product state: \n 12 1Ψ⊗Ψ=Ψ ,  ∑=\n−==ΨSm\nS mm i mC , \nwhere iΨ is an arbitrary sup erposition s tate. This  state under th e action of the Ha miltonian (1) \nbecom es time dependence  \n∑∑=\n−==\n−=⊗\n⎭⎬⎫\n⎩⎨⎧=ΨSm\nSmmnSn\nSnn mSJitmn CC t exp )( .   (2) \nNote that du e to the d iscreten ess of the leve ls the  quantum  dynam ics of  the isolated sy stem  is \nperiod ic with a recu rrence tim e of  JSTπ4= . \nFor concrete calcu lations , we choose two im portant variants o f the initial conditions :  \n1) the spin coherent states - eigenstates of with maxim al X-projection [9,11]:  X\niS\n∑=\n−=⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n+=Sm\nS mZ S XmmSSS2/12\n21;     ( 3) \n2) the uniform superpos ition states [12]:  \n∑=\n−==ΨSm\nS mS m\nd2/11.      ( 4) \nThese states can be prepared from  the ground state Z ZS S⊗=Ψ0  by following op-\nerations to b oth spins for each of  the two varian ts, respectively:  \n1) π/2-rotation around the Y-axis (t he nonselective operation);  \n2) the operator of the quant um Fourier transform  QFT d (we found the sequence of selective rota-\ntions for d=3 ÷10 [13]).  \n \nThe motion  of spin co mponents  \nCalculate th e tim e evolution of the m ean value of the first spin X-projection )(1tSX. \nAs usually, the spin oper ator can be decom posed into th e lad der oper ators:  \n21) (−++= S S SX, \nwhich have only nonzero m atrix elem ents between neighboring states  \n()()2/1 2/11 1 1 mS mS mS m mSm ++−= −=−− +. \nFor function norm alized to 1 at t = 0, we obtain tim e depende nce  \n∑=\n−=++\n⎭⎬⎫\n⎩⎨⎧=\nΨΨΨΨ=Sn\nSnnSJitn C\nSt SttF exp\n)0( )0()( )()(2\n11,    (5) \n 2() ( ∑=\n+−=−∗ +++− =ΨΨSm\nS mm m mS mS CC S\n12/1 2/1\n1 1 1 )0( )0( ). \nFor the two varian ts, we  have, respectively:  \n1)  S SSJttFS\ncoh =ΨΨ⎟\n⎠⎞⎜\n⎝⎛=+)0( )0( ,2cos)(12\n;           (6) \n2) ()() (∑=\n+−=+++−+=ΨΨ++=Sm\nS mmS mSSSS Jt SS JtF\n12/1 2/1\n1 sup 1121)0( )0( ,))2/( sin()12())2/(11( sin). \nThe dependencies are shown in Fig. 1 fo r small (a) and large (b) spin num ber.  \nThe period o f oscilla tion is inc reasing with the g rowth of S. When S>> 1, t <<T we can  \napproxim ate the functions as follows: by Gaussian f unctions at first and by sine divided by Jt at \nsecond -  \n(),sin\nsupJtJtF=  ⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛−=≅SJttFtFg coh4)(exp)( )(2\n.   (7) \nIn the lim it S → ∞ for the spin coherent s tates, we get a con stant, i.e. th e result coin cides \nwith the res ult for two c lassical m agnetic m oments, lying in the plan e, be cause th e interac tion (1 ) \nturns to zero  as . The second state keeps a supe rposition wh ich non-repres ented \nclass ically.  0 ,02 1 ==Z ZS S\n \nSquared I-concurrence  \nAs an entan glement measure we will use th e squared I-concurrenc e [10], that for pure \nstates tak es the for m,  \n()2\n112ˆ 11ρTrddC −−= ,     ( 8) \nwhere )()( ˆ2 1 t t Tr ΨΨ=ρ  is th e reduced den sity m atrix o f the f irst sp in (tra ced by the sec -\nond spin), w hich we find  \n∑∑∑=\n−==\n−=∗=\n−= ⎭⎬⎫\n⎩⎨⎧− =Sm\nS mSn\nSnn m mSm\nS m SJmmitn C mmCC2\n22 11\n1) ( exp ˆ2 12\n2 1 1ρ . \nLet us com pare th is formula with the previous  case (5). There trans itions b etween near \nlevels  12 1=−m m  take part only. Now all transitions take part and result in double sum and \nlarge coefficient S m m≈−2 1  in expo nent. That is  mathem atics basis of differences in tim e \nbehaviors of  the m ean spin com pone nt and the entanglem ent.  \nCalculation gives:  \n 31) The coherent state:  \n) ( ,24\n21\n2cos24\n22ˆ2 1 442\n142\n11 mm MSS\nSJtMMSSTrSSS M\nMS−=⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛+⎥⎦⎤\n⎢⎣⎡\n⎭⎬⎫\n⎩⎨⎧\n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n+=∑=\n=ρ . (9) \nAt sm all times    () )4(2\n412 2tO JtSS\ncohC ++= . \nThe tim e average for th e period  2 2\n4 2\n21121 2\n24)2(121 2⎟\n⎠⎞⎜\n⎝⎛−+≅⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛−+=−\nS SS\nSS\nSSCS\ncohπ \n2) The uniform  superposition state:  \nd S JtMS JtMdMd\ndTrd\nM1\n)/ cos(1)/ cos(1) (2ˆ1\n142\n11 +−−− =∑−\n=ρ .   (10) \nAt sm all times  () ()22\n4 2\n23\n2\nsup9)(\n)12()1(JtStO Jt\nSd dC ≈++= .  \nThe tim e average for th e period  1 22 112\nsup+=−=SS\ndC .  \nIn Refs. [9, 11, 12] have been studied the entang lement for such states. T he results are sim ilar. \nThe tim e dependences of the squared I-concurren ce in som e values of S, calcu lated b y the \nobtained formulas, are shown in Fig. 2. W e may see that when the spin n umbers increase, th e \nsquared I-concurrence grows and is closed in 1. This deviation from  1 for the uniform  superposi-\ntion state is less than for the cohe rent state. There are th e tim e oscillations.  Starting with the tim e \nT/4 the en tanglem ent will be de creased. G. Burlak , I. Sainz, a nd A.B. Klimov [ 9] found that in-\nstead of decrease can b e obtained a further in crease in th e con curren ce if to apply ± π / 2 rota tions \naround the Y-axis to both spins at some appropriate tim e moments: t 1 and t2.  \n \nAsymptotic formula for the squared I-concurrence  of co herent sta tes \nIn the case o f coherent s tates with S>>1 for the squared I-concu rrence can be obtained a \nsimple asymptotic form ula. In Eq. (9 ) for  when t<<T, we rep lace app roxim ately the bi-\nnominal coefficients and  the cos ine to the high power 4S by Gaussian functions:  2\n11ˆρTr\n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛− ≈⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n+ SM\nS MSS\nS2exp\n21\n24\n212\n4π, \n⎭⎬⎫\n⎩⎨⎧−≈⎥⎦⎤\n⎢⎣⎡\n⎭⎬⎫\n⎩⎨⎧\nSJtMSJtMS\n2)(exp2cos2\n24\n, \nand summ ation by integration. Then we obtain  \n 4S StJerf\nJtdM JtSM\nSTrS\nSπ πρ\n21\n81\n1)(1))(1(2exp\n21ˆ22\n222 2\n22\n11 ++\n+−\n⎭⎬⎫\n⎩⎨⎧+− ≈∫\n−. (11) \nHere we too k into ac count the sp ecial role of  summ ands 0) (2 1=−= m mМ , which do not \nvary over tim e and, therefore, does not decay. T heir con tribution is \nS SSTrS\nπρ\n21\n24)2( ˆ4 2\n11 ≅⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛=−. \nLet us replace approximately 2S by ∞ in th e inte gration lim its, and then we f ind  \n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n−+−\n+−+=\nS StJerf\ntJ SSCcohπ21)811(\n11121 222\n222.  (12) \nConsider ag ain the exact tim e dependence of squared I-concurrence in Fig.2. W ith in-\ncreased tim e on the background of monotonous growth we see a sequence of local m inima in the \npoints πnSJMt=2 in which the cosine is drawn to ± 1.  When S>>1, thes e minima can be de-\nscribed by G aussian functions as:  \n∑=\n= ⎪⎭⎪⎬⎫\n⎪⎩⎪⎨⎧\n⎟⎠⎞⎜⎝⎛− −≈⎥⎦⎤\n⎢⎣⎡\n⎭⎬⎫\n⎩⎨⎧Mn\nnS\nMnTtJSM\nSJtM\n02\n22 4\n2 2exp2cos . \nThen to get a squared I-concurrence  \n⎟⎟\n⎠⎞\n⎪⎭⎪⎬⎫\n⎪⎩⎪⎨⎧\n⎥⎥\n⎦⎤\n⎢⎢\n⎣⎡\n⎟\n⎠⎞⎜\n⎝⎛−+− −⎜⎜\n⎝⎛\n−−+−\n+−+=\n∑∑=\n==\n=Mn\nnS M\nMcoh\nMnTtJSM\nSS StJerf\ntJ SSC\n12\n22 2\n122\n222\n212exp\n2221)811(\n11121 2\nππ\n.   (13) \nIn Fig. 3 perfor med a comparison of the de pendences, calculated on the exact and asymp-\ntotic f ormulas. In the las t (13), due to the rap id decrease of amplitude with increasing M, we are \nrestricted by the first three i ngredients with M = 2, 3 and 4. We see good agreem ent the asym p-\ntotic form ula with exac t one. W hen S increas es its accu racy will in crease, and the dep th of m in-\nima will be r educed. \nThe changes in behaviors  and  with the growth of S shown in Fig. 4. The \ntime evolution of  slows down and thus closes in th e motion of classical magnetic m o-\nments. W hile the squared I-conc urrence rea ches m aximum value C = 1, indicating preservation \nof the quantum  properties. Previously, such a dependence of entanglem ent has been dem on-)(t Fcoh )(2t Ccoh\n)(t Fcoh\n 5strated in another m odel system : a particle with  spin, oscillating in a non-uniform  magnetic field \n[8].  \n \nConclusions \nIf the two sp ins are p repared in the sp in coherent states, the m otion of their m ean projec-\ntions in the lim it S → ∞ will conc ede with the m otion of  the p rojections of  two class ical mag-\nnetic m oments. Neverth eless, the sq uared I -conc urrenc e, and  thus the entanglem ent will grow \nwith tim e to values app roaching to th e maximum C = 1 when S → ∞.  \nIf the two sp ins are p repared in the un iform super position states, we observe the quantum  \ndynam ics in any S, so far as classical m agnetic mo ments can not be in supe rposition state (like of \nthe fam ous «Schrodinger cat» [14] ). (Although the preparation of th e superposition state at large \nS can be technically very difficult).  \nThus, the large spins m ay show both the quant um and classical prop erties, accord ing to \nprepared states and observe conditions. It is im portant for us (and this is the m ain goal of this \nstudy) that quantum  computation on qudits, represented by large sp ins, do not get the principal \nlimitations o n the spin n umber S.  \n \nThis work was supported by the R ussian Founda tion for Basic Research (project. no. 09-07-\n00138-a). \n \n 6References  \n1. D. Gottesm an,.Lect. Not. Co mp. Sci. 1509,  302 (1999). \n2. A.R. Kessel, V.L. Er makov, JETP Letters 70, 61 (1999); 71, 307 (2000).  \n3. A. Muthukrishnan, C. R. Stroud  (Jr.), Phys. Rev. A 62, 052309 (2000). \n4. B.C. Sanders, Phys. Rev. A 40, 2417 (1989). \n5. J.L. Garcia-Palacios, S Dattagupta, Phys. Rev. Lett. 95, 190401 (2005).  \n6. A. Polkovnikov, “Representation of quantum  dynamics of interacting system s through clas-\nsical trajectories ”, cond-m at/0905.3384. \n7. A Cabello, Phys. Rev. A 65, 062105 (2002). \n8. S. Ghose, P.M. Alsing, B.C. Sanders, I.H. Deutsch, Phys. Rev. A 72, 014102 (2005).  \n9. G. Burlak, I. Sainz, A.B. Klim ov, Phy s. Rev. A 80, 024301 (2009). \n10. P. Rungta, V. Buzek, C.V. Caves et al., Phys. Rev. A 64, 042315 (2001); P. Rungta, C.V. \nCaves, Phys. Rev. A 67, 012307 (2003). \n11. Y.Yang, W .Liu, Z. Sun, X. W ang, Phys. Rev. A 79, 054104 (2009). \n12. X. Lu, X. Wang, Y.Yang, J. Chen, Quantum  Inf. Co mput. 8, 671 (2008); quant-\nph/0803.1032. \n13. A.S. Er milov, V.E. Zob ov, Opt. Spectrosc. 103, 969 (2007).  \n14. R. Penrose, «The Em peror' s New Mind»,  Oxford University Press, 1989.  \n 7Квантовая  динамика  двух  взаимодействующих  больших  спинов \nВладимир  Евгеньевич  Зобов  \nИнститут  физики  им. Л. В. Киренского  СО РАН , \n660036, Красноярск , Академгород ок,50, стр.38. \nrsa@iph.krasn.ru\n \n Рассчитывается  временная эволюция  средних  спиновых  компонент  и квадратичной  \nI-согласованности  двух взаимодейству ющих больших  спинов  S. В качестве  начальных  ус-\nловий  взяты два случая : спиновые  когерентные  состояния  и равномерные  суперпозицион -\nные состояния . Для спиновых  когерентных  состояний  получено  асимптотическое  выра -\nжение  при S>>1 и t<<T (Т-период ). Сделан  вывод , что квантовые  вычисления  на кудитах, \nпредставленных  большими  спинами , не встречают  принципиальных  ограничений  на спи-\nновое  число S.  \n 8-101\n05 10\nJtF\ncoh; S=1\nsup; S=1\ncoh; S=3/2\nsup; S=3/2(а) \n-0,4-0,200,20,40,60,81\n02 468 10 12\nJtFcoh; S=9 /2\ng;    S=9/2\nsup; S=9/2\ncoh; S=1 0\ng;     S=10\nsup; S=10\ncoh; S=1000\ng;     S=1000\nsup; S=1000cohF\nsupF\nsupFcohF (b) \n \nFig. 1. Tim e evolution of the norm alized m ean va lue of  the f irst spin X-p rojection f or small S (a) \nand large S (b) (the Gaussian functions F g(t) (7) f or correspondence S are shown by open cir-\ncles).  \n 900,20,40,60,81\n01 23 45 67\nJt S=1/2\ncoh; S=1\nsup; S=1\ncoh; S=3/2\nsup; S=3/2(a) \n2\nsupC\n2\ncohC\n \n00,20,40,60,81\n05 10 Jt 15\n 2\nsupC (b) \n2\ncohC\n00,20,40,60,81\n0 5 10 15 20 25 30 Jt\nFig. 2. Tim e evolution of the s quared I-concurrence for different valu es of S: ½, 1, 3/2 (a), and \n9/2 (b) where inset shows this dependence up to tim e T/2. \n 100,40,50,60,70,80,91\n02468 10 12 14Jtasymptotic (12)\nasymptotic (13)\nexact (8),(9)2\ncohC\n \nFig. 3. Tim e evolution of the s quared I-concurrence in the spin cohe rent states for S = 9/2, calcu -\nlated on the exact form ula (8) with (9 ) and the asymptotic formulas (12) and (13).  \n \n 1100,20,40,60,81\n0246 8 10 12Jt\n 1000 \n9/2 cohF\n1000 \n100 \n2\ncohC\n10 \n9/2\nFig. 4. Tim e evolution of  and , for different values of S (the num bers on the \ncurves), calculated on th e asym ptotic fo rmulas (7) and (12), respectively.  )(t Fcoh)(2t Ccoh\n 12"    },    {        "title": "0910.3776v1.Bifurcation_and_chaos_in_spin_valve_pillars_in_a_periodic_applied_magnetic_field.pdf",        "content": "arXiv:0910.3776v1  [cond-mat.other]  20 Oct 2009APS/123-QED\nBifurcation and chaos in spin-valve pillars in a periodic ap plied magnetic field\nS. Murugesh1∗and M. Lakshmanan2†\n1Department of Physics & Meteorology, IIT-Kharagpur, Khara gpur 721 302, India and\n2Centre for Nonlinear Dynamics, School of Physics, Bharathi dasan University, Tiruchirapalli 620 024, India\n(Dated: November 10, 2018)\nWe study the bifurcation and chaos scenario of the macro-mag netization vector in a homogeneous\nnanoscale-ferromagnetic thin film of the type used in spin-v alve pillars. The underlying dynamics\nis described by a generalized Landau-Lifshitz-Gilbert (LL G) equation. The LLG equation has\nan especially appealing form under a complex stereographic projection, wherein the qualitative\nequivalence of an applied field and a spin-current induced to rque is transparent. Recently chaotic\nbehavior of such a spin vector has been identified by Zhang and Li using a spin polarized current\npassing through the pillar of constant polarization direct ion and periodically varying magnitude,\nowing to the spin-transfer torque effect. In this paper we sho w that the same dynamical behavior\ncan be achieved using a periodically varying applied magnet ic field, in the presence of a constant\nDC magnetic field and constant spin current, which is technic ally much more feasible, and demon-\nstrate numerically the chaotic dynamics in the system for an infinitely thin film. Further, it is\nnoted that in the presence of a nonzero crystal anisotropy fie ld chaotic dynamics occurs at much\nlower magnitudes of the spin-current and DC applied field.\nBolstered by the importance of Giant Magneto\nResistance (GMR), a sequence of experimental\nand theoretical developments in the last few years\non current induced switching of magnetization in\nnanoscale ferromagnets has thrown open several\nprospects in next generation magnetic memory\ndevices. The direct role of spin polarized current,\nas against the traditional applied field, in control-\nling spin dynamics has brought in the possibility\nof new types of current-controlled memory de-\nvices and microwave resonators. The system un-\nder consideration is primarily a nanoscale spin-\nvalve pillar structure, with one freeferromagnetic\nlayer and another pinnedlayer separated by a non-\nferromagnetic conducting layer. The behavior of\nthe dynamical quantity of interest, the magneti-\nzation field in the free layer, is well modeled by an\nextended Landau-Lifshitz equation with Gilbert\ndamping, which is a fascinating nonlinear dynam-\nical system. The free layer is usually assumed\nto be of single magnetic domain. Owing to the\nhighly nonlinear nature of the LLG equation it is\nimperative to study the chaotic dynamical regime\nof the magnetization field. Indeed, several recent\nexperiments have exclusively focused on chaos as-\npect of the system. In this article we have shown\nthat a small applied periodically varying (AC)\nmagnetic field, in the presence of a constant spin-\ncurrent and a steady applied magnetic field, can\ninduce parametric regimes displaying a broad va-\nriety of dynamics and period doubling route to\nchaos. A numerical study of the effects of a non-\n∗Electronic address: murugesh@phy.iitkgp.ernet.in\n†Electronic address: lakshman@cnld.bdu.ac.inzero anisotropy field reveals chaotic dynamics at\nmuch lower magnitudes of the spin-current and\napplied DC field. This could be an important\nfactor to consider in microwave resonator appli-\ncations of spin-valve pillars.\nI. INTRODUCTION\nFollowing the success of GMR, the next major devel-\nopment in classical computer technology is expected to\nbe through MRAMs (1; 2; 3). Apart from a manifold\nreduction in power consumption, being inherently non-\nvolatile in nature, they also bring in the prospect of\ncomputers that need not be rebooted. Understandably,\nfabrication of MRAMs has been a major thrust area of\nresearch in the last two decades. Typically, the mem-\nory unit consists of two nanoscale magnetic films sepa-\nrated by a spacer conductor/semiconductor medium and\nworks on the principle of GMR. The imminent prospects\nin the recording media industry has prompted a breadth\nof development in the field. Besides the eminent role\nas a memory unit, the possibility of a single MRAM as\na logical unit has also been proposed (4; 5). One no-\ntable technological hiccup in fabricating large MRAM\ngrids is the extent to which the applied magnetic field\ncan be localized. This imposes limitations on the ef-\nficiency with which an individual unit can be manipu-\nlated. The applied field required for the purpose is in\nmost cases the Oersted field generated through electrical\ncurrents. Asignificantstepforward,inbypassingthelim-\nitation on field localization, occurred when Slonczewski\nandBergerindependentlyenvisagedamoredirectrolefor\n(spin polarized) current on magnetization(6; 7). They\npredicted that the angular momentum acquired by the\nspin polarized current can interact with the magnetiza-\ntion vector, and thus a suitable spin polarized current2\ncan possibly flip its direction. This phenomenon has\nbeen well established in a series of experiments in the\nlast decade and referred to as the spin-torque effect in\nthe literature(8; 9; 10; 11; 12; 13; 14; 15). Interest-\ningly, the effect has a simple semiclassical description in\nthe form of an extended Landau-Lifshitz equation with\nGilbert damping (6; 7; 16). Several proposals have ap-\npeared in the last few years suggesting an increased role\nof the spin current and the associated torque in the even-\ntually expected version of the MRAM.\nOneimportantassumptionoftenmadein mostofthese\nstudies is that the magnetization in the film is homoge-\nneous, or at least enough well defined that one can con-\nsider the film to be a mono-domain layer. This homo-\ngeneity assumption effectively nulls the Heisenberg in-\nteraction between neighboring spins, and allows one to\ntreat the system as a single spin unit. As the size of\nthe magnetic film is increased, this approximation is ex-\npected to fail. Indeed, chaotic behavior has been ob-\nserved due to spin field inhomogeneity at lateral sizes of\norder 60 −130nm(17; 18). Besides, it is well realized\nexperimentally that spin-transfer can induce microwave\noscillations(17; 19; 20; 21). The possible role of such\ncurrent controlled microwave oscillators in the nanoscale\nhas been well realized (17). For higher power levels that\nare practically desired it is natural to look at a series of\nsuch coupled spin-valve oscillators. Studies on synchro-\nnization of networks of such coupled nano-spin trans-\nfer oscillators, each one modeled on the extended LLG\nequation have been carried out both experimentally and\nnumerically in an effort to enhance the emitted power\n(22; 23; 24; 25).\nSince theextended LLGequationis highlynonlinearin\nnature, a detailed study of the underlying nonlinear dy-\nnamics in spin-valve structures becomes inevitable. The\nstability diagram based on Melnikov theory in the space\nof the external field along the direction of anisotropy and\nthe strength of spin torque due to a DC current was ob-\ntained in detail by Bertotti et. al.,in (26). Following\nthis, it was shown by Z. Li, Y. C. Li and S. Zhang that\nan applied alternating current can induce three broad\ntypes of dynamics, vis−a−visChaos, Multiply periodic\nand Periodic (27). Qualitatively similar features are also\nnoted when the effects of nearest neighbor interactions\nare included in a long one dimensional ferromagnet with\na spin torque term. Indeed the ferromagnetic chain ex-\nhibits both periodic and chaotic behavior in the presence\nof an applied AC spin current. (28). The multiply pe-\nriodic behavior refers to the case where, with change in\nparameter, the system moves from periodic to multiply\nperiodic behavior but not leading to chaos. It must be\nmentioned at this point that the last two types ‘multiply\nperiodic’ and ‘periodic’ are also referred to in the litera-\nture as ‘modification’ and ‘synchronization’(27; 29). The\nusage here is deliberate as the periods do not have a di-\nrect relation to the period of the applied magnetic field.\nIt was also shown that in the space of the applied exter-\nnal field and the strength of the spin current the systemexhibits chaotic behavior for a range of values within the\nboundary predicted by Melnikov theory (29).\nThe possibility of chaotic oscillations in monodomain\nsingle nanoscale spin-valve requires further in depth\nstudy of underlying bifurcation and chaos scenarios, in-\ncluding the detailed phase diagrams, at least for two rea-\nsons: (i) Chaotic oscillations in spin transfer oscillators\nmay be unfavorable from a practical/technological point\nof view and such oscillations need to be suppressed by\nminimal intervention using one of the several control-\nling techniques developed in the field of chaotic dynam-\nics (30; 31), where already such a study has been made\n(32). (ii) The previously mentioned possibility of syn-\nchronization of coupled spin-valve oscillators raises com-\nplex problems which are new in spintronics and is related\nto the topic of chaos synchronization in dynamical sys-\ntems (30; 31; 33), similar to synchronization of chaoti-\ncally evolving networks of Josephson junctions, laser sys-\ntems and nonlinear oscillators. As synchronization of\nchaotic oscillators is considered as a potential technique\nfor secure communication including cryptography, there\nexistspossibleapplicationsofnetworksofnanoscalespin-\nvalve structures along these lines, although this may be\ncomplicated due to the presence of several parameters in\nthe system. Consequently the study of the full nonlinear\ndynamics of spin transfer oscillators will be of consider-\nable significance. It may be noted that chaosin magnetic\nsystems, such as yttrium garnet, driven by external fields\nhave been extensively studied in the past (34; 35; 36).\nNumerical experiments on a model of thin magnetic film\nwherein spin wave excitations induced by spin-current\nlead to chaos have also been studied in (37). However,\nchaosinnano-spinvalvegeometryisreasonablynew with\npotential new applications as discussed above.\nIn this paper, we study the chaotic dynamics of the\nmagnetization vector in a single domain current driven\nspin-valve pillar, induced by a periodically varying (AC)\napplied magnetic field in the presence of a constant spin\ncurrent and steady (DC) magnetic field, using the ex-\ntended LLG equation as the model for the system. Mak-\ning use of a complex stereographic variable, we observe\nthat the spin current induced torque is effectively equiv-\nalent to an applied magnetic field. Following this ob-\nservation, we show numerically that a periodically alter-\nnating field can lead to chaotic behavior of the magne-\ntization vector, which is similar to that of an alternat-\ning spin-current induced torque, studied in Li, Li, and\nZhang (27). It will be shown that the order of magni-\ntude of the applied alternating field required for chaotic\nmotion is substantially lower, within practically achiev-\nable limits, compared to the alternating current magni-\ntudes shown in (27; 29). This is expected to be helpful\nin applications such as resonators, as an AC magnetic\nfield is much more feasible practically than a AC spin-\npolarized current (although, when it comes to DC fields,\nthe current induced DC Oersted fields are more cumber-\nsome to produce in spin valve geometries compared to\nDC spin currents). We study the dynamics in an infi-3\nnite thin film, both with a vanishing and non-vanishing\nanisotropy field. In the setup we shall consider the mon-\nodomain spin layer influenced by a DC spin-current, and\nboth a DC and AC applied magnetic field. Although\nthe applied magnetic field is qualitatively equivalent to\na suitable spin-current(38), we employ both a DC spin-\ncurrent and applied magnetic field as the magnitudes at\nwhich chaos is observed is quite high, and hence difficult\nto achieve exclusively using either of the two. It may be\nnoted that the chaotic dynamics studied here is induced\nby an AC applied magnetic field, and is phenomenologi-\ncally different from the spin field inhomogeneity induced\nchaotic behavior that is observed in (17). Periodic dy-\nnamics is possible even in the absence of an alternating\ncurrent, or field, as has been noticed in ferromagnetic\nfilms induced by a spin-current (19). Futher it has been\nreported therein that the current magnitudes at which\nperiodic behavior is seen share a linear relation of nega-\ntive slope with the oscillationfrequencies. Our numerical\nresults based on the extended LLG model are shown to\nbe in agrement with these observations. Interestingly,\nin the presence of a non-zero in plane easy-axis crystal\nanisotropy field (taken along the zdirection), the chaotic\ndynamical regime is observed at much lower magnitudes\nof the DC applied field and spin-current. This could be\nan aspect to factor in while designing spin-valve based\nmicrowave resonators.\nThe paper is organized as follows. In Section 2, we\ndetail the various interactions, including spin-transfer\ntorque, that make up the extended LLG equation.\nFurther, we introduce a complex stereographic vari-\nable equivalent to the macro-magnetization vector, and\nrewrite the LLG equation in this variable. As will be\nshown, this elucidates the role of the spin transfer torque\nas equivalent to an applied magnetic field. In Section\n3, we present our numerical results which demonstrate\nchaoticdynamics ofthe magnetizationvectorin the pres-\nence of a periodic applied field. Here we shall consider\ntwocases,namelyresponseinthepresenceandabsenceof\na crystal anisotropy field. As will be noticed, the chaotic\nregimes in the two cases are significantly different. In\nSection 4 we conclude with a discussion on our results.\nII. THE EXTENDED LANDAU-LIFSHITZ EQUATION\nAND COMPLEX REPRESENTATION\nA typical spin-valve pillar used in most experiments is\nschematicallyshowninFIG 1. Acurrentpassingthrough\nthis ferromagnet acquires a spin polarization in the ˆ zdi-\nrection. The thickness of the spacer conductor medium\nshould be less than the spin diffusion length of the polar-\nized current. The polarized current then passing through\nthe free layer causes a change in the magnetization vec-\ntorˆS, an effective torque referred to as the spin transfer\ntorque. Interestingly, it has been realized that semiclas-\nsically the phenomenon can be described by an extensionj\nPinned layer Conductor Thin film ConductorS S p\nxz\ny~100 nm 2−10 nm\nFIG. 1 A schematic figure of a spin-valve pillar. The cross\nsection of the free layer is roughly 5000 nm2.ˆSis the magne-\ntization vector in the free layer, and is the dynamical quant ity\nof interest. ˆSPis the direction of polarization of the spin cur-\nrent.\nto the LLG equation, (6; 7)\ndˆS\ndt=−γˆS×/vectorHeff+λˆS×dˆS\ndt−γ aˆS×(ˆS׈Sp),(1)\nHere,ˆS={S1,S2,S3}is the unit vector along the di-\nrection of the magnetization vector in the ferromagnetic\nfilm, which isthe dynamicalvariableofinterest, γthe gy-\nromagnetic ratio, and λthe dissipation coefficient. /vectorHeff\nin Eq. (1) is the effective field due to exchange inter-\naction, anisotropy, demagnetization and an applied field\n(see (38) for details):\n/vectorHeff=DS0∇2ˆS+κ(ˆS·ˆ e/bardbl)ˆ e/bardbl+/vectorHdemag+/vectorBapplied,(2)\nˆ e/bardblbeing the unit vector along the direction of the\nanisotropy axis. The demagnetization field is obtained\nas a solution of the differential equation\n∇·/vectorHdemag=−4πS0∇·ˆS, (3)\nand/vectorBappliedis the applied magnetic field on the sample.\nThe last term in Eq. (1) is the additional term describing\nthe spin-transfer torque, and the parameter ‘ a’ depends\nlinearly on the current density j.ˆSpis the direction of\nmagnetization of the polarizer, i.e., the polarization of\nthe spin current.\nIn this study we shall assume the magnetization to be\nhomogeneous. This effectively nulls the exchange inter-\naction, while Eq. (3) can be immediately solved to give\nthe demagnetization field as\n/vectorHdemagnetization =−4πS0(N1S1ˆ x+N2S2ˆ y+N3S3ˆ z),\n(4)\nwhereNi,i= 1,2,3 are conveniently chosen such that\nN1+N2+N3= 1 (after suitable rescaling of the magni-\ntude of the spin). For a spherical sample N1=N2=N3,\nand the demagnetization term is effectively null in Eq.\n(1). In the next sections we study chaotic dynamics\nshown by the system when an alternating magnetic field\nis applied.\nRewriting Eq. (1) using the complex stereographic\nvariable (39; 40)\nΩ≡S1+iS2\n1+S3, (5)4\nprovidesamorecomprehensiblepictureoftheroleofspin\ntransfer torque. For the spin valve system, the direction\nof polarization of the spin-polarized current ˆSpremains\na constant, and lies in the plane of the film. Without loss\nof generality, we call this the direction ˆ zin the internal\nspin space, i.e., ˆSp=ˆ z. Asmentioned in Sec. 2, wedisre-\ngard the exchange term. We choose the applied externalfield also in the ˆ zdirection, i.e., /vectorBapplied={0,0,ha3}.\nDefining\nˆ e/bardbl={sinθ/bardblcosφ/bardbl,sinθ/bardblsinφ/bardbl,cosθ/bardbl}(6)\nand upon using Eq. (5) in Eq. (1), we get\n(1−iλ)˙Ω =−γ(a−iha3)Ω+iS/bardblκγ/bracketleftBig\ncosθ/bardblΩ−1\n2sinθ/bardbl(eiφ/bardbl−Ω2e−iφ/bardbl)/bracketrightBig\n−iγ4π S0\n(1+|Ω|2)/bracketleftBig\nN3(1−|Ω|2)Ω\n−N1\n2(1−Ω2−|Ω|2)Ω−N2\n2(1+Ω2−|Ω|2)Ω−(N1−N2)\n2¯Ω/bracketrightBig\n,(7)\nwhereS/bardbl=ˆS·ˆ e/bardbl. Using Eq. (5) and Eq. (6), S/bardbl, and\nthus Eq. (7), can be written entirely in terms of Ω. For\nfurther details of derivation of Eq. (7) see Ref. (38).\nIt is interesting to note that in this representation the\nspin-transfertorque,proportionaltotheparameter a, ap-\npears only in the first term in the right hand side of Eq.\n(7) as an addition to the applied magnetic field ha3but\nwith a prefactor i. Thus the spin torque term can be\nconsidered as an effective applied magnetic field (38). It\nwas further explicitly shown in (38), in the absence of the\ncrystal anisotropy field, that the switching time due to\nthe spin-torque will be shorter by a factor λ, compared\nto that of magnetic field induced switching. Further, the\nspin-torque produced the dual effect of precession and\ndissipation even in the presence of the external applied\nfield. The nature of switching of magnetization for other\ntypes of materials can be investigated by analyzing Eq.\n(7), and for typical materials this has been carried out in\n(38).\nIII. CHAOTIC DYNAMICS\nMagnetization reversal in a spin mono-domain layer in\nthepresenceofbothasteadyappliedmagneticfieldanda\nsteady polarized current corresponds to a rather compli-\ncated phasediagram,as revealedin (26). Using Melnikov\ntheory, it was also shown therein that the magnetization\nvector also has limit cycles for a range of values of the\nparameters, with frequency in the microwave range. The\ndynamical quantity in question, the magnetization vec-\ntorˆS, is two dimensional, owing to its constant (unit)\nmagnitude. Hence, chaotic behavior is ruled out. How-\never, making the applied field, or current, time depen-\ndent is one way of increasing the effective dimensionality\nof the system to three and hence introduce a possibility\nof chaotic dynamics. Following (26) it has been shown in\n(27) that a small alternating current can produce a vari-\nety of dynamics, namely, Multiply periodic, Periodic and\nChaos. It is also noticed that, along with a steady spincurrent of order 250 Oeand a steady applied magnetic\nfield of the same order, inclusion of a small alternating\nspin polarized current leads to chaotic dynamics (29).\nThe dynamical similarity of the applied field and the\nspin-torque was noted in Section II. In this section we\nshow numerically that an applied AC field can also pro-\nduce diverse dynamical scenarios and point out the ad-\nvantagesin using a periodically varyingapplied magnetic\nfield instead of an alternating current. i.e., in Eq. (1) (or\nequivalently Eq. (7)) we take\n/vectorBapplied={0,0,ha3+haccosωt}. (8)\nIt will be notedthat chaoticdynamicsis possibleatmuch\nlower DC applied field strengths and spin current in the\npresence of an anisotropy field.\nFor the film geometry, the film is taken to be in the\ny−zplane, and the demagnetization field perpendicular\nto the film, the ˆ xdirection, i.e.,\nHdemagnetization =−4πS0S1ˆ x. (9)\nThe saturation magnetization is taken to be that of\npermalloy, so that 4 πS0= 8400Oe. Two different scenar-\nios, one without anisotropy, and anotherwith an in-plane\neasy axis anisotropy of strength κ= 45Oealong the z\ndirection, are investigated. All the numerical results that\nfollow have been obtained by directly simulating Eq. (1)\nfor the vector ˆS. The same results were confirmed using\nEq. (7) as well.\nA. Regions of Chaos in the presence of an applied\nalternating field\nAs a first step we show below (FIG 2) the regions\nof chaos, or regions of positive Lyapunov exponent, in\nthe space of the DC magnetic field and the DC current\nfor an alternating applied magnetic field of magnitude\n10Oeand frequency 15 ns−1, first without an anisotropy\nfield,κ= 0, (FIG 2a), and then with anisotropy field,5\n(a)\nha3(Oe)a (Oe)\n800700600500400300200100280\n270\n260\n250\n240\n230\n220\n210\n(b)\nha3(Oe)a (Oe)\n1800 1400 1000 600 200400\n350\n300\n250\n200\n150\n100\n50\n0\nFIG. 2 Regions of chaos in the a−ha3space, for an applied\nalternating magnetic field of amplitude hac= 10Oeand fre-\nquencyω= 15ns−1, a) without anisotropy field, κ= 0,\nand b) with anisotropy field of strength κ= 45Oealong the\nzdirection. The dark regions indicate values for which the\ndynamics is chaotic, i.e, regions where the largest Lyapuno v\nexponentispositive. Ina)chaosis rarelynoticedforlower val-\nues ofha3. The other parameters are N1= 1,N2=N3= 0,\n4πS0= 8400Oe. The points are plotted at intervals of 5 Oe\nalong both axes, and hence the figure offers only limited res-\nolution in the dark(chaotic) regions.\nκ= 45Oe, (FIG 2b). The regions are obtained by di-\nrect numerical simulation using the model in Eq. (1), or\nequivalentlyEq. (7). ThedarkregionsinFIG2represent\nparameter values when the largest Lyapunov exponent is\npositive.\n1. Case a: No anisotropy ( κ= 0)\nThe similarity of FIG 2a with that of regions of chaos\nfor an alternating spin current must be noted (see FIG 1\nin (29)). The figure is a demonstration of the qualitative\nequivalence of the applied field and current induced spin-\ntorque in describing the gross dynamical scenario. From\nFIG 2a, chaotic behavior of spins is observed for spin-\ncurrent magnitudes in the range of a= 200−300Oe, and\nDC magneticfieldsabove100 Oe. These valuesof‘ a’ cor-\nrespond to spin-current magnitudes of over 1012A/cm2,\nwhich is at the higher end of the presently achievable\nlevels.\nIn FIG 3 we present the bifurcation diagram showingthe period doubling route to chaos as the DC current\nis varied. These diagrams are obtained by plotting the\nminimum values for one of the components of the spin,\nin this case S1, over severalperiods of time for each value\nofain the given range. From the data in FIG 3a, the\nfirst five period doubling bifurcations are seen to occur\natan= 268.4845,267.7723,267.6055,267.5685,267.5605.\nConsequently, the ratio δn= (an−an−1)/(an−1−an+1)\ntakesvalues4 .2698,4.5081,4.625,clearlyapproachingthe\nFeigenbaum constant. The Lyapunov spectrum for this\nrange of ‘ a′is shown as inset. On comparison, we notice\nthe largest Lyapunov exponent is positive for values of\nawhere the dynamics is chaotic in FIG 3a. A similar\ncheck has been made for FIG 3b, which again shows the\nperiod doubling route in the presence of an anisotropy\nfield. Period doubling route to chaos is also noticed as\nthe strength of the steady applied magnetic field is var-\nied(keeping the current and frequency fixed), and when\na (Oe)λ\n268.6268267.40\n-4\n-8\n-12(a)\na (Oe)S1minimum\n268.6268.4268.2268267.8267.6267.4-0.994\n-0.995\n-0.996\n-0.997\n-0.998\n(b)\na (Oe)S1minimum\n250.5250 249.5249 248.5-0.984\n-0.986\n-0.988\n-0.99\n-0.992\n-0.994\n-0.996\nFIG. 3 Period doubling route to chaos as ais varied. The\nfigure is a plot of the minimum values of S1over several pe-\nriods for the given parameter values a) without anisotropy,\nand b) with anisotropy of κ= 45Oealong the zdirection.\nThe applied DC field ha3= 200Oe. All the other parameters\nremain the same as in FIG 2. The corresponding Lyapunov\nspectrum is shown as an inset.6\nthe frequency of the applied field is varied (keeping cur-\nrent and magnetic field strength steady), respectively.\n(a)\na (Oe)ω(ns−1)\n60055050045040035030025045\n40\n35\n30\n25\n20\n15\n10\n5\n(b)\na (Oe)ω(ns−1)\n60055050045040035030025045\n40\n35\n30\n25\n20\n15\n10\n5\nFIG. 4 Regions of chaos(red dots) and periodicity (blue\nwings) in theparameter space of DCcurrent‘ a′andfrequency\n‘ω′, a) without anisotropy and b)with anisotropy ( κ= 45Oe)\nalong the zdirection. The left over regions show multiply pe-\nriodicbehavior. All other parameters remain the same as in\nFIG 3. The power spectrum at the two dark points in (a)\n(255,25) and (280 ,25) are shown in FIG 5 (a) and (b), re-\nspectively.\n2. Case b: Non-zero anisotropy( κ= 45Oe)\nContrary to the observation in FIG 2a, in the pres-\nence of a non-zero easy axis anisotropy, chaos is noted\nat much lower values of the spin current and DC applied\nfield (FIG 2b). FIGs 3b shows period doubling bifur-\ncation scenario in the presence of anisotropy. In FIG\n3b the current magnitude is varied, keeping the mag-\nnetic field strength and frequency of the AC component\nfixed. Similar period doubling route to chaos is also no-\nticed as a) the magnitude of the steady magnetic field is\nvaried (keeping current and frequency fixed), and when\nthe frequency of the AC component of the applied mag-netic field is changed (keeping the current and magnetic\nfield strengths constant). In either case, an easy axis\nanisotropy of magnitude κ= 45Oeis chosen along the z\ndirection, which is also the polarization direction of the\nspin current. The figures corresponding to these results\nare, however, not presented in here.\nB. Periodic, Multiply periodic and Chaotic dynamics\nIn the presence of an AC spin-current induced torque,\nit is known that as the frequency of the spin-current is\nvaried the system exhibits three distinct phases wherein\nthe dynamicsispredominantlyeither- Periodic, Multiply\nperiodicorChaotic(27). Herein we show that an applied\nAC magnetic field of magnitude 10 Oe, instead of a AC\ncurrent, results in the three dynamical phases as the fre-\nquency of the applied field is varied for a constant value\nof the applied DC field.\nFIG 4a shows regions of periodic (blue wings), and\nchaotic (red stem) dynamics in the parameter space of\nthe spin current magnitude and the frequency of the ap-\npliedmagneticfield. Multiplyperiodicbehaviorisseenin\ntheunshadedregions. Hereagainthesimilaritywith that\nofFIG1in(27)maybenoted. Thisfurtherillustratesthe\nqualitative similarity of the spin current induced torque\nwith that of the applied field. Our numerical results fur-\nther show a number of wing like bands in the ω−aspace\nwhere periodic behavior is noticed. This is clearly ab-\nsent with a periodic spin-current as seen in (27). The\npowerspectrum correspondingto two specific points, one\nchaotic and the other periodic (indicated with dark dots\nin FIG 4a), are shown in FIG 5. Periodic behavior is no-\nticed even in the limit ω→0 forcertain values of a, while\nmultiply periodic behavior is noticed for the other inter-\nmediate values. The power spectrum corresponding to\na = 280Oe\n20010008\n4\n0\n-4\n-8\na = 255Oe\nω(ns−1)log(Power)\n2502001501005008\n6\n4\n2\n0\n-2\n-4\n-6\n-8\nFIG. 5 The power spectrum distribution corresponding tope-\nriodic,a= 280Oe(inset), and chaotic, a= 255Oe, scenarios\nin FIG 4a. The first peak in the inset is seen at ω= 25ns−1.\nThe anisotropy is taken zero, and all other parameters are th e\nsame as in FIG 4a.7\nthese current values are shown in FIG. 6. Such a behav-\nior, induced by a spin-currentof constantmagnitude, has\nbeen noticed in (19). Indeed, the current magnitudes, a,\nwhere periodic behavior is seen to vary linearly (with a\nnegative slope) with the corresponding periods (see FIG.\n6 inset), in further agreement with (19).\nHowever,in the presenceof the anisotropyfield chaotic\nregime is much more pronounced and wider, as seen in\nFIG 4b. For a much lower value of the DC applied field\n(ha3= 20Oe) there appear wide bands in the ω−aspace\nwhere periodic behavior is exclusively noted at low fre-\nquencies (FIG 7). As the frequency is increased multiple\nperiodicity islands appear in between periodicbands, and\nfor even higher frequencies the dynamics is largely one of\nmultiply periodic type. No chaotic dynamics is noted\nin the parameter range chosen. These thick periodicity\nbands present better regions to choose in applications\nsuch as the microwave resonator discussed earlier, while\nchaos synchronization studies can be carried out in the\nchaos regimes.\nIV. DISCUSSION AND CONCLUSION\nUsing the complex stereographic variable to represent\nthe spin vector, and rewriting the modified Landau-\nLifshitz-Gilbert equation, we have shown that the spin-\ncurrent induced torque is qualitatively equivalent to an\napplied field. Using this equivalence we have shown that\nan applied AC magnetic field in the presence of con-\nstant spin current and DC applied magnetic field can\nlead to varied dynamical scenarios including chaos. We\nhave explicitly demonstrated numerically the chaotic be-\nhavior for a range of values of the parameters. The sys-\n182Oe190Oe198Oe199Oe\n207Oe\n215Oe 241OeAng.freq. (ns−1)a(Oe)\n2.31.81.3240\n220\n200\n180\nAng.freq. (ns−1)Power\n2.42.221.81.61.41.210.25\n0.2\n0.15\n0.1\n0.05\n0\nFIG. 6 The power spectrum distribution in the limit ω= 0,\nat certain values of a(indicated on each spectrum) where\nperiodic behavior is noted. Multiply periodic behavior is n o-\nticed for other values of ain the range shown. The current\nmagnitudes vary linearly and decrease with the frequency of\noscillation ( inset).hac= 0, while all other parameters are the\nsame as in FIG. 3.a (Oe)ω(GHz)\n4003503002502001501005045\n40\n35\n30\n25\n20\n15\n10\n5\nFIG. 7 Regions of multiply periodic dynamics for the system\nwith the DC applied field fixed at ha3= 20Oe, and non-zero\nanisotropy. All the other parameters remain the same as in\nFIG 4b. Synchronization is noted in the unshaded regions,\nwhile chaotic dynamics is not noticed in the parameter range\nshown in the figure. Islands of multiply periodic behavior ap -\npear between regions of periodic behavior for low frequenci es.\nFor higher frequencies, the dynamics is exclusively multip ly\nperiodic.\ntem also exhibits regular periodic behavior for a different\nrange of values. It is now realized that such nanoscale\nmonodomain layers can find application as resonators,\nthrough periodic oscillations induced by an alternating\nspin-current. The resultspresentedhereprovideanalter-\nnativemethod throughoscillationsinduced byanapplied\nmagnetic field. It is further noticed that the range of\nthe chaotic regime strongly depends on the presence of a\ncrystal anisotropy field. In the presence of an anisotropy\nfield chaotic behavior is noticed for much lower values\nof the DC field and spin-current, which are more suited\nfor chaos synchronization studies. However, there are re-\ngions in the ω−aspace where regular periodic motion is\nmore robust and presents a better alternative in applica-\ntions. In a future study we will present the possibility of\nchaos synchronization in spin-valve structures.\nACKNOWLEDGMENTS\nS.M. wishes to thank DST, India, for funding through\nthe FASTTRACK scheme. The work of M.L. forms part\nof a DST-IRHPA research project and is supported by a\nDST-Ramanna fellowship.\nReferences\n[1] S. A. Wolf, A. Y. Chtchelkanova and D. M. Treger, IBM\nJ. Res. Dev. 50, 101 (2006).\n[2] R. K. Nesbet, IBM J. Res. Dev. 42, 53 (1998).8\n[3] C. H. Tsang, R. E. Fontana, Jr., T. Lin, D. E. Heim,\nB. A. Gurnet and M. L. Williams, IBM J. Res. Dev. 42,\n103 (1998).\n[4] W. C. Black, Jr. and B. Das, J. Appl. Ph. 87, 6674\n(2000).\n[5] A. Ney, C. Pampuch, R. Koch and K. H. Ploog, Nature\n425, 485 (2003).\n[6] L. Berger, Phys. Rev. B 54, 9353 (1996).\n[7] J. C. Slonczewski, J. Mag. Mag. Mat. 159, L1 (1996).\n[8] D. Berkov and J. Miltat, J. Mag. Mag. Mat. 320, 1238\n(2008).\n[9] M. D. Stiles and J. Miltat, Topics Appl. Physics 101, 225\n(2006).\n[10] E. B. Myers, D. C. Ralph, J. A. Katine, R. N. Louie and\nR. A. Buhrman, Science 185, 867 (1999).\n[11] J.-E. Wegrowe, D. Kelly, X. Hoffer, Ph. Guittienne and\nJ.-Ph. Ansermet, J. Appl. Phy. 89, 7127 (2001).\n[12] J. A. Katine, F. J. Albert, R. A. Buhrman, E. B. Myers\nand D. C. Ralph, Phys. Rev. Lett. 84, 3149 (2000).\n[13] J. Grollier, V. Cros, A. Hamzic, J. M. George and H.\nJarrf´ es, Appl. Phys. Lett. 78, 3663 (2001).\n[14] B.¨Ozyilmaz, A. Kent, D. Monsma, J. Z. Sun, M. J.\nRooks and R. H. Koch, Phys. Rev. Lett. 91, 067203\n(2003).\n[15] S. Urazhdin, Norman O. Birge, W. P. Pratt Jr. and J.\nBass, Phys. Rev. Lett. 91, 146803 (2003).\n[16] Y. B. Bazaliy, B. A. Jones and S.-C. Zhang, Phys. Rev.\nB57, R3213 (1998).\n[17] S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Em-\nley, R. J. Schoelkopf, R. A. Buhrman, and D. C. Ralph,\nNature425, 380 (2003).\n[18] K-J. Lee, A. Deac, O. Redon, J-P. Nozi´ eres and B. Dieny,\nNature Materials 3, 877 (2004).\n[19] W. H. Rippard, M. R. Pufall, S. Kaka, S. E. Russek, and\nT. J. Silva, Phys. Rev. Lett. 92, 027201 (2004).\n[20] W. Weber, S. Riesen, and H. G. Siegmann, Science 291,\n1015 (2001).\n[21] H. Xi, K. Z. Gao, and Y. Shi, Appl. Phys. Lett. 84, 4977\n(2004).\n[22] F. B. Mancoff, N. D. Rizzo, B. N. Engel and S. Tehrani,Nature437, 393 (2005).\n[23] S. Kaka, M. R. Pufall, W. H. Rippard, T. J. Silva, S. E.\nRussek and J. A. Katine, Nature 437, 389 (2005).\n[24] J. Grollier, V. Cros and A. Fert, Phys. Rev. B 73,\n060409(R) (2006).\n[25] J. Persson, Y. Zhou and J. Akerman, J. Appl. Phys. 101,\n09A503 (2007).\n[26] G. Bertotti, C. Serpico, I. D. Mayergoyz, A. Magni,\nM. d’Aquino, and R. Bonin, Phys. Rev. Lett. 94, 127206\n(2005).\n[27] Z. Li, Y. C. Li, and S. Zhang, Phys. Rev. B 74, 054417\n(2006).\n[28] Y. Lan, and M. C. Li, Nonlinearity 21, 2801 (2008).\n[29] Z. Yang, S. Zhang, and Y. C. Li, Phys. Rev. Lett. 99,\n134101 (2007).\n[30] M. Lakshmanan and K. Muruali Chaos in Nonlinear\nOscillators: Controlling and Synchronization (World-\nScientific, Singapore, 1995).\n[31] M. Lakshmanan and S. Rajasekhar, Nonlinear Dynamics\n(Springer-Verlag, Berlin, 2003).\n[32] H. Z. Xu, X. Chen and J.-M. Liu, J. Appl. Phys. 104,\n093919 (2008).\n[33] A. Pikovsky, M. Rosenblum and J. Kurths, Synchroniza-\ntion: A Universal Concept in Nonlinear Sciences (Cam-\nbridge University Press, 2008).\n[34] S. M. Rezende and F. M. de Aguiar, Pr. of IEEE. 78,\n893 (1990).\n[35] T. L. Carroll, M. L. Pecora and F. J. Rachford, J. Appl.\nPhys.70, 3197 (1991).\n[36] N. Y. Pisken and P. E. Wigen, J. Appl. Phys. 85, 4521\n(1999).\n[37] F. M. de Aguiar, A. Azevadoand S. M. Rezende, J. Appl.\nPhys.91, 8046 (2002).\n[38] S. Murugesh, and M. Lakshmanan, Chaos, Sol. & Frac.\n41, 2773 (2009).\n[39] C. Kosaka, K. Nakamura, S. Murugesh, and M. Laksh-\nmanan, Physica D 203, 233 (2005).\n[40] M. Lakshmanan and K. Nakamura, Phys. Rev. Lett. 53,\n2497 (1984)."    },    {        "title": "2208.02216v2.Tunable_itinerant_spin_dynamics_with_polar_molecules.pdf",        "content": " 1 Tunable itinerant spin dynamics with polar molecules Jun-Ru Li, Kyle Matsuda, Calder Miller, Annette N. Carroll, William G. Tobias, Jacob S. Higgins, Jun Ye 1JILA, National Institute of Standards and Technology and Department of Physics, University of Colorado, Boulder, CO 80309. *Corresponding authors. Emails: junru.li@colorado.edu, ye@jila.colorado.edu   Strongly interacting spins underlie many intriguing phenomena and applications1-4 ranging from magnetism to quantum information processing. Interacting spins combined with motion display exotic spin transport phenomena, such as superfluidity arising from pairing of spins induced by spin attraction5,6. To understand these complex phenomena, an interacting spin system with high controllability is desired. Quantum spin dynamics have been studied on different platforms with varying capabilities7-13. Here we demonstrate tunable itinerant spin dynamics enabled by dipolar interactions using a gas of potassium-rubidium molecules confined to two-dimensional planes, where a spin-1/2 system is encoded into the molecular rotational levels. The dipolar interaction gives rise to a shift of the rotational transition frequency and a collision-limited Ramsey contrast decay that emerges from the coupled spin and motion. Both the Ising and spin exchange interactions are precisely tuned by varying the strength and orientation of an electric field, as well as the internal molecular state. This full tunability enables both static and dynamical control of the spin Hamiltonian, allowing reversal of the coherent spin dynamics. Our work establishes an interacting spin platform that allows for exploration of many-body spin dynamics and spin-motion physics utilizing the strong, tunable dipolar interaction.    2 Ultracold polar molecules, with (pseudo-)spins encoded in the molecules’ internal states, provide a unique platform for investigating many-body spin dynamics with high tunability13-20. The effective spin coupling is realized via long-range, anisotropic dipolar interactions mediated by the molecule's electric dipole moment, which is tunable with external electric fields and microwaves. Pioneering proposals14-16, 19 and experiment13 used polar molecules localized in optical lattices to realize lattice-spin models. Even with modest lattice fillings, the strong dipolar interaction allowed the observation of spin-exchange dynamics13. Recently, spin correlations have been directly imaged under a microscope12. The strong tunability of this lattice-spin system enables the realization of a variety of spin Hamiltonians displaying exotic correlations and phase transitions16,17,19-21. In parallel, molecules loaded in a two-dimensional (2D) layer are allowed to move freely within the plane, and dipolar interactions control the collisional process. The repulsion from the dipolar interaction enhances elastic collisions while suppressing reactive losses, allowing evaporative cooling to quantum degeneracy22. The engineered dipolar interactions create a strong repulsive energy barrier that allows control of the molecular reaction with only a small change of the electric field23,24 or microwaves25,26. In similar systems with magnetic atoms, magnetic dipolar interactions, albeit relatively weak, yield a variety of quantum phases and dynamics11. Combining coherent spin interactions and motional physics in a single platform is expected to give rise to rich dynamics. In this article, we report the realization of an itinerant spin system that evolves from a fully controlled spin Hamiltonian to a more complex spin-motion coupled system. Isolated 2D layers of dipolar molecules are addressed by a controlled external electric field (E) to realize both Ising and exchange interactions with high tunability (Fig. 1a). At short evolution times, the system’s dynamics are governed by dipolar interactions in the spin degrees of freedom, realizing a spin Hamiltonian that can generate highly entangled spin states27. At longer times where collisions mediated by the molecular thermal motion and dipolar interaction manifest, we observe the onset of spin decoherence that is not reversible by engineered echo pulse sequences. Our work establishes a tunable dipolar system where the spin and motional dynamics are coupled by strong dipolar interactions.  We create gases of fermionic 40K87Rb (KRb) loaded into 2D harmonic traps formed by a one- 3 dimensional optical lattice22,28 (see Methods). The spin-1/2 degree of freedom is mapped onto molecular rotations as |¯ñ = |0, 0ñ and |1ñ = |1, 0ñ or |2ñ = |1, -1ñ (Fig.1b). Here, |N, mNñ denotes a rotational state with N being the field-dressed rotational quantum number and mN its projection on the quantization axis set by E (or the magnetic field along 𝒚\" at |E| = 0). Spin states are prepared using a microwave field to drive the molecules’ rotational transitions. The magnitude and orientation of E and the molecular spin states are used to tune the dipolar interaction, and consequently, the spin and collision dynamics. At short times prior to the onset of collisions, the system evolves primarily through the spin-1/2 degree of freedom, as molecular motion is effectively frozen. Dipolar interactions couple individual molecular spins 𝒔$={𝑠$(,𝑠$*,𝑠$+} occupying the harmonic oscillator mode i, leading to spin dynamics described by an XXZ Heisenberg Hamiltonian27: 𝐻=\t/0∑2𝐽$4+𝑠$+𝑠4++𝐽$467𝑠$(𝑠4(+$4𝑠$*𝑠4*89+∑𝑠$+ℎ$+.$Here,\tℎ$+ is the effective field. 𝐽$4+\tand 𝐽$46 are the Ising and the exchange interactions arising from the E-induced dipole moments d¯ and d and the transition dipole moment 𝑑↓↑, respectively.  In contrast to lattice spins13, spins delocalized in the 2D plane have large and nearly uniform spatial wavefunction overlap between any pairs of molecules. As a result, 𝐽$46, 𝐽$4+, and ℎ$+ have only weak dependence on {i, j}, leading to an effective description of dynamics in terms of the ensemble averaged values 𝐽6, 𝐽+, ℎ+\tand collective spin27 𝑆@=∑𝑠$@$\t(𝛾\t=𝑋,𝑌,𝑍): 𝐻=𝐽6𝑆0+𝜒H𝑆+0+ℎ+𝑆+ .    (1) 𝐽6 enforces an energy cost to change |S| and thus protects the collective spin from decoherence. ℎ+ represents an effective magnetic field. The strength of spin-spin interaction is characterized by 𝜒H=𝐽+−𝐽6, which is the difference between the average Ising and exchange interaction. This term produces a one-axis twisting that can generate highly entangled spin states27,29,30. All these terms are tunable with E or the choice of spin states. In particular, both the magnitude and sign of the interaction parameter 𝜒H∝(3cos0𝛼−1)[(𝑑↓−𝑑↑)0−2𝑑↓↑𝑑↑↓] (see Methods) can be varied by changing the angle a between E and 𝒚\" (Fig. 1a) or |E|, as exemplified by the spin manifold {|¯ñ,|1ñ} where 𝑑↓↑=𝑑↑↓=⟨↓|𝑑|↑/⟩ (Fig. 1c). At long times, dipolar interactions lead to elastic collisions that change the motional modes of the  4 molecules at a rate depending on the molecular spins and coherence. In the motional degree of freedom, these mode-changing collisions lead to dipolar thermalization22. In the spin degree of freedom, they cause decoherence of the collective spin at the collision rate because each of the collisional events introduces random spin rotations on the two colliding spins. This decoherence cannot be reversed due to scrambled spin-motion coupling.  Our system exhibits unique dynamics arising from the molecular spin and motion intertwined by dipolar interactions: Delocalized molecules facilitate the couplings between molecular spins, and spin dynamics affect the molecular collisions through spin coherence. Using dynamical decoupling techniques, we characterize both effects with Ramsey spectroscopy. The former manifests as a frequency shift on the spin transition, the latter as an interaction dependent fringe contrast decay.  We first characterize the spin dynamics by measuring the frequency shift arising from 𝜒H. At the mean-field level, 𝜒H𝑆+0≈2𝜒H𝑆+⟨𝑆+⟩ produces a shift to the spin transition frequency that is proportional to áSZñ,  representing a population imbalance of (N - N¯)/2 in a coherent superposition of the two spin states. Here, N and N¯ are the numbers of molecules in each spin state. For a Ramsey experiment with an interrogation time T and an initial pulse area q that prepares the system with áSZñ = - [(N + N¯)/2] cosq, this leads to a dipolar interaction-induced fringe phase shift  Δ𝜙/2𝜋=−𝑛𝜒𝑇\t𝑐𝑜𝑠𝜃.    (2) Here, 𝜒=(𝜒H/ℎ)[(𝑁↑+𝑁↓)/𝑛], where n is the average 2D molecular density and h is the Planck constant. The strength of the mean-field interaction Uc is thus defined as Uc = nc. For a typical density n = 1 ´ 107 cm-2, the frequency shift (Δ𝜙/2𝜋𝑇) is expected to be about 100 Hz. Measuring such shifts requires an appropriately long spin coherence time. However, the experimentally achieved coherence time T2* between molecular rotations is limited by single particle dephasing mechanisms28,31. Grey circles in Fig. 2b show decay of the Ramsey contrast for |E| = 0 under typical trapping conditions. A Gaussian fit gives T2* of only 0.24(1) ms. Extending T2* to milliseconds has been demonstrated by orienting the bias electric field to the magic angle28,31-33. However, achieving long T2* in an arbitrary configuration of E, which is required for tuning of the dipolar interaction, has been challenging.  5 We perform dynamical decoupling with an XY8 multi-pulse sequence34, consisting of eight Rabi-p rotations along 𝑋b and 𝑌b spaced by time t  (Fig. 2a), to suppress single particle dephasing. Multiple XY8 sequences can be concatenated (denoted XY8´M) to achieve longer T (see Methods). Implementing XY8´3 leads to a factor of 71(5) improvement in T2* from 0.24(1) ms to 17(1) ms for the same field and trapping condition (orange squares in Fig. 2b), sufficient for studying the dipolar density shift as a function of |E| and a. To measure the frequency shift and thus c , a microwave pulse with an area q first prepares KRb in a superposition state cos(q/2)|¯ñ+ sin(q/2)|1ñ. After a sequence of XY8 with T = 1.2 ms (Fig. 2a), the accumulated phase is mapped onto population by a second Ramsey pulse with an area of p-q and a variable phase f. The fraction in |1ñ as a function of f are subsequently measured by imaging molecules in both states28, from which we obtain Ramsey fringes like those shown in Fig. 2c for |E| = 0. By fitting the fringes to a sinusoidal function, we extract the corresponding phase shifts Df. At a fixed n, we measure Df at several different values of q  and obtain Uc as the slope of the linear fit to Df versus cosq (solid black and solid grey lines in Fig. 2d). We repeat the measurement of Uc at several values of n to measure c. The results for |E| = 0 are shown with grey circles in Fig. 2e. The measured Uc as a function of n is well captured by a linear fit, consistent with the mean-field model. From the slope of the fit, we extract 𝜒/ = -4.9(3) ´ 10-6 Hz/cm-2 at |E| = 0 between |¯ñ and |1ñ. Leveraging a configurable |E| that mixes the molecular rotational states allows us to vary the relative strength between the Ising and exchange interactions in the spin dynamics. We demonstrate this tunability by measuring c over a wide range of |E| at a = 0° for the manifold {|¯ñ,|1ñ}. The results, shown in Fig. 2f, are well captured by scaling 𝜒∝[(𝑑↓−𝑑↑)0−2𝑑↓↑𝑑↑↓] (black solid line) to fit the experimental data. At low |E|, dipolar exchange dominates and c < 0. As |E| increases, the increasing strength of the Ising interactions from the induced dipoles competes with the spin exchange eventually leading to c > 0 (Fig. 1e) at high |E|. Tuning |E| to approximately 7 kV/cm where c = 0 would realize the Heisenberg XXX model. In future studies, such tunability will allow exploration of qualitatively different spin dynamics hosted by a spin-1/2 system16.  In 2D, the average dipolar interaction can be controlled by orienting the dipole moments relative  6 to the 2D plane, as a direct consequence of the anisotropy of the dipolar interaction. We measure c between |¯ñ and |1ñ as a function of a at |E| = 1.02 kV/cm. By rotating the dipoles, c is varied continuously from negative to positive (Fig. 2g). At a = 0°, molecules interact repulsively as the dipole moments are aligned perpendicular to the 2D plane, leading to c < 0. At a  = 90°, dipoles are aligned within the plane and the average interaction is attractive, leading to c > 0. Assuming symmetric transverse trapping, c µ -(3cos2a - 1). Notably, this tunability of c by rotating E arises from a common geometric factor on the Ising and exchange interactions, in contrast to tuning with |E|, which controls the relative strength of the two interactions. The choice of the internal states used for the spin-1/2 system offers additional control over the interaction parameters. The green squares in Fig. 2e show a measurement of Uc between |¯ñ and |2ñ = |1, -1ñ at |E| = 0. Using |2ñ instead of |1ñ changes the magnitude of c and reverses its sign at |E| = 0, as a result of the different transition dipole moments in {|¯ñ, |2ñ} where 𝑑↑↓=−𝑑↓↑=⟨↓|𝑑|↑/⟩/√2. We measure c between |¯ñ and |2ñ to be 𝜒0 = 2.3(8)´10-6 Hz/cm-2, consistent with -1/2 of the interaction between |¯ñ and |1ñ at this field.  The ability to manipulate dipolar interactions through the internal state allows us to dynamically change the interaction and the evolution of the many-body state. In contrast to tuning with the dc electric field, internal state control of the interaction can be achieved rapidly and coherently with microwave pulses. Such a capability is essential for applications such as dynamic Hamiltonian engineering35, high fidelity quantum gate operations36, and studying quantum quenched dynamics with dipolar spin systems37,38. By implementing a dynamical control pulse sequence, we demonstrate reversal of the coherent many-body spin dynamics. In particular, we coherently swap the excited state from |2ñ to |1ñ in the middle of the Ramsey evolution, which instantaneously changes the interaction parameter c by a factor of -2, and consequently reverses the associated phase progress in a coherent evolution.  Our sequence consists of two phase accumulation stages (Fig. 3a). After preparation in a superposition of |¯ñ, |2ñ, molecules evolve with 𝜒0 (stage I) until a composite microwave pulse R (Inset of Fig. 3a) coherently transfers the excited state population from |2ñ to |1ñ while  7 maintaining the phase relative to |¯ñ. Molecules then continue to evolve in {|¯ñ, |1ñ} (stage II) with 𝜒/=−2𝜒0 before being detected after the final Ramsey pulse. The XY8 decoupling sequence is used during both stages. The duration of R is 70 µs, which is short compared to the interaction timescales. We then extract the phase shift DF  = Df(q = p/4) -Df(q = 3p/4) as a function of T, plotted in Fig. 3b. (see Methods) We observe a sign change in the rate of phase accumulation d(DF)/dT at ts = 1.2 ms when R is implemented, indicating a reversal of the mean-field interaction sign. The data is well described by a piecewise linear fit with the ratio of d(DF)/dT before and after the reversal constrained to be −2, consistent with the measurement in Fig. 2e. At T » 1.6 ms, DF  returns to the initial value, completing the phase reversal.  Our protocol realizes a complete reversal of the spin Hamiltonian at |E| = 0. Such a capability is essential for studying out-of-time-ordered correlators which are used to understand dynamics of interacting quantum many-body systems such as quantum information scrambling39-42. Similar many-body echo processes also allow applications such as robust Heisenberg-limited phase sensitivity without single-particle-resolved state detection27,43-45.  At long evolution times, motional and spin dynamics are coupled by dipolar interactions, resulting in dynamical evolution beyond the spin model. Mode-changing dipolar collisions lead to loss of coherence, manifesting as an exponential decay of the Ramsey contrast, exp(-G t). G is proportional to the rate of elastic collisions ns vT, where vT is the thermal velocity and s is the cross section for dipolar elastic collisions. For spin-polarized fermionic molecules with dipole moment d induced by E, previous theories46,47 and experiments22,24 have shown s µ d4. By comparison, two molecules in a coherent superposition carry oscillating dipoles with amplitudes of 𝑑↓↑ in the lab frame, leading to dipolar collisions even at low |E| where induced dipole moments are small26,48. When the oscillations are fully coherent, the molecules collide with maximum effective dipole moment 𝑑↓↑/√2 and s (Ref. 48). These collisions couple the spin and motion, resulting in spin decoherence that cannot be removed using multi-pulse sequences, in contrast to interaction-induced dephasing in a lattice13. We study the effect of dipolar collisions by measuring contrast decay as a function of n. To  8 investigate collisional decoherence, the single particle dephasing rate must be below the collisional rate, which was achieved by increasing the number of decoupling pulses.  We observe the characteristic exponential decay of the contrast for a range of n (Fig. 4a). The extracted G increases with density (Fig. 4b), indicating collision-limited coherence time. At n = 1.1(1) ´ 107 cm-2 and |E| = 0, we measure G = 130(5) s-1, consistent with dipolar elastic collision rates measured in Ref. 22 with d being 𝑑↓↑/√2  (see Methods). This gives a collisional timescale shorter than the trap oscillation period of ~22 ms (see Methods). Rotating the electric field to a = 36° at |E| = 1.02 kV/cm reduces the strength of the dipolar interaction. We observe longer coherence times at the same density (purple squares in Fig. 4b), indicating reduced s. By extracting the average s = G/nvT for each E, we found s(a = 0°, |E| = 0)/s(36°, 1.02 kV/cm) = 3.1(4). This variation of s is expected to arise from the changing dipolar cross section as a function of a, similar to scattering with induced dipole moments24,47. In analogy to the relationship in atomic systems between the mean-field shift and elastic cross section, the variation of s could also be explained by a scaling s µ c2. The measured ratio of s is consistent with [c(0°, 0)/c(36°, 1.02 kV/cm)]2 = 3.5(9) from Fig. 2f, 2g. These results suggest that dipolar effects dominate the collisional decoherence process. The high controllability of our system allows tuning the motional dynamics relative to the coherent spin dynamics by changing parameters such as the strength of optical trapping49, temperature, and dipolar interaction strength. These capabilities would allow the investigation of other many-body spin-motion effects such as unconventional paired superfluidity50,51, spin-wave and spin transport physics52,53, and dipole-mediated spin-orbital dynamics54. In conclusion, we have demonstrated tunable dynamics of the collective spin and motion in a 2D itinerant spin system formed from a gas of fermionic polar molecules. We have shown static and dynamical control over the spin Hamiltonian using external fields, microwaves, and internal states of the molecules. Our results establish a highly controllable spin system that can be used to investigate a broad range of many-body phenomena.      9 Methods Molecular spin system and preparation We create ultracold gases of KRb in 2D with the procedure detailed in Ref. 22 and Ref. 28. In brief, degenerate gases of 40K and 87Rb are loaded into a stack of 2D harmonic traps formed by a one-dimensional optical lattice. Molecules in the ground rotational state |¯ñ are subsequently produced by magneto-association and stimulated Raman adiabatic passage (STIRAP) directly at the electric field E. We typically produce 15´103 molecules at a temperature T0 » 450 nK, occupying 19(1) layers with a peak number of about 1000 molecules in a single layer. The harmonic trapping frequencies within each layer for molecules in |¯ñ at |E| = 0 are (wx, wy, wz) = 2p ´ (45, 17000, 45) Hz. Since the harmonic level spacing ħwy along the tight confinement direction is larger than kBT0 and the interaction energy scale, our molecules predominantly remain in the lowest harmonic level, forming a quasi-2D system throughout the experiments. Here, kB is the Boltzmann constant.   Couplings between these 2D layers are weak compared to the intralayer interaction28 due to the large spatial extension of the fermionic molecular gases in each layer that averages out the interlayer interaction. This weak interlayer interaction is expected to manifest as only a small correction to c for the timescales considered in this work. We therefore approximate the system as a stack of isolated layers. At |E| = 0 and typical trapping condition, the resonant transition frequency between |¯ñ and |1ñ (|¯ñ and |2ñ ) is measured to be around 2228.138 MHz (2227.742 MHz). Compared to our typical Rabi frequency of 100 kHz, the large difference between resonant frequencies allows us to resolve these states by changing the microwave frequency. The degeneracy of |1,1ñ  and |2ñ = |1, -1ñ is broken by a weak coupling of the nuclear and rotational degrees of freedom, resulting in an energy difference of about 100 kHz between the two states. To avoid off-resonant transfer to |1,1ñ when driving the transition between |¯ñ and |2ñ,  we reduce the microwave Rabi frequency to about 60 kHz.  Besides rotational structure, molecules possess hyperfine states. In our experiment, KRb is prepared in |¯ñ = |N = 0, mN = 0, mK = -4, mRb = 1/2ñ. Here, mK and mRb are the projection of the  10 nuclear spins on the quantization axis. For states with N = 1 (|1ñ and |2ñ), electric quadrupole interaction slightly mixes the hyperfine states and the rotational states. Since hyperfine changing transitions are weak and their energy spectrum is well resolved, we use states with the largest projection on |1, 0, -4, 1/2> and |1, -1, -4, 1/2> as |1ñ and |2ñ, respectively. In this work, we avoid driving hyperfine-changing transitions, and thus neglect the hyperfine degree of freedom.   The density of the molecular gas is varied by adding an extra cleaning procedure after the molecules are produced and before the Ramsey interrogation. Specifically, a microwave pulse with an area of qc prepares molecules in a superposition of |¯ñ and |1ñ. The STIRAP beam that is resonant with |¯ñ and the electronically-excited intermediate state of the molecules is switched on for 50 µs to eliminate the |¯ñ component of the superposition. The remaining molecules are then flipped back to |¯ñ with a p pulse. The density is reduced to sin2(qc /2) of the initial density. At certain values of E, the cleaning STIRAP light pulse can also off-resonantly excite molecules in |1ñ. In that case, molecules in |1ñ are shelved in |2,0ñ during the optical blast using an additional microwave pulse, and no loss of |1ñ molecules is observed.  Since the size of the STIRAP beam used for cleaning is larger than the size of the sample, we do not observe significant temperature change from the process. This method therefore allows us to control the sample density without changing the temperature or the spatial distribution in the regime where the experiments are conducted. Timescales of the dynamics Multiple timescales are involved in the dynamics presented in this work: 1) interaction timescale set by Uc, 2) collision timescale set by G , and 3) single-particle motion timescale set by the period of harmonic trapping.   The spin model is valid in the regime t < 1/G (short time limit). For n = 1.1(1) ´ 107 cm-2 (where the strongest collisional effects are explored in the current work), 1/G  » 7.4 ms. At this density, Uc  » 60 Hz, giving a timescale of 1/ Uc » 17 ms. Since Uc µ 𝜒 while s, therefore G, scales with c2, it is possible to tune the relative timescale of these two processes. Spin dynamics are favored for small c, while collisional dynamics dominate at large c.   11 The period of trap oscillation is ~22 ms. Therefore, for the highest density (1/G » 7.4 ms), molecules collide before they complete one trap oscillation (akin to the hydrodynamic regime). For the lowest collision rate we measured when we lowered the density or reduced the strength of dipolar interaction, molecules would undergo several oscillations before a collision occurs (collisionless regime). The relative rates of the two processes can be controlled by changing the strength of the optical trapping, temperature, density, or c. The capability of tuning the system across different regimes highlights the controllability of our system.  2D density calibration We use the following procedure to obtain the average 2D densities in the layers from the total molecule number Ntot. The molecules occupy multiple layers formed by an optical lattice. We first measure the molecule distribution using layer-resolved spectroscopy28. In brief, an electric field gradient along the longitudinal direction of the optical lattice shifts the transition frequency between |0,0ñ and |1,0ñ on each layer. When the differential shift for molecules in adjacent layers is large compared to the linewidth of the transition in each layer, molecules in different layers can be addressed, resolved, and measured spectroscopically using microwaves. A typical measurement (Extended data figure 1) reveals a Gaussian distribution with a full-width-half-maximum of 11(1) layers.  \n Extended data figure 1. A typical molecular number distribution measured via layer-resolved spectroscopy. \n 12 For two-body processes considered in this work, such as the mean-field frequency shift and collisional decoherence, we calculate the average number of molecules per layer using an effective number of layers L, defined as\t𝐿=\tef∑egfg, where Nk is the number for molecules in the k-th layer. From the layer-resolved number distribution, we extract L = 19(1) and use N2D = Ntot/L as the average number of molecules in a single layer. The average 2D density n is calculated from N2D using temperatures measured with time-of-flight thermometry and the transverse trapping frequencies for |¯ñ at E.  Our procedure for varying the density is not expected to change the number distribution across layers. We therefore use L = 19 for all our measurements.  Geometric factor of c The dipolar interaction between two dipoles has a geometric factor (1-3cos2Qij), where Qij is the angle between the dipole moment and the vector between the two particles. When the molecular motion is confined to 2D, cos2Qij can be decomposed into a and the azimuth angle b as cos2Qij = sin2a cos2b. This geometric factor can be expressed in terms of a by taking the average over the harmonic oscillator states in the 2D plane. In the mean-field regime and with approximately symmetric transverse trapping, this averaging reduces to an average over b, yielding ácos2bñ = 1/2 and thus á1-3cos2Qijñ = (3cos2a -1)/2. Measurement of Ramsey coherence time For the coherence time measurements, we measure the contrast at different Ramsey time T with the number of XY8 echo pulses fixed. To extract the contrast at each T, we perform 8 to 16 measurements of the fraction of |1ñ with f equally spaced between 0 and 360 degrees. The contrast and its standard deviation (SD) at time T are extracted from the measured fractions via bootstrapping following the procedure described in Ref. 28. Dynamical decoupling and calibration of the pulse sequence Grey circles are the experimental measurements. Black solid line is a fit to a summation of equally spaced Gaussian functions with a global Gaussian envelope. The width of each narrow Gaussian peak is assumed to be the same. The data is taken at |E| = 1.02 kV/cm and a  = 36° (magic condition) to reduce broadening of the single-layer transition linewidth due to differential polarizability.   13 We perform dynamical decoupling with an XY8 multi-pulse sequence to suppress single particle dephasing. The consecutive spin echo pulses spaced by time t form a bandpass filter that rejects noise outside of a window with center frequency f0 = 1/(2t) and width ~1/T, suppressing the Ramsey phase fluctuations caused by noise in E and the nonzero differential ac polarizability. The XY8 sequence’s time-reversal symmetry and alternation between rotation axes further improve robustness against pulse area error and finite pulse duration. Multiple XY8 sequences are concatenated to achieve longer Ramsey time T without affecting f0. The efficacy of the dephasing suppression depends on t, pulse duration tp, and the timing of the sequence, which we optimize with the following procedure: An ideal XY8 sequence consists of infinitely fast echo pulses. However, if the microwave Rabi frequency is too high, molecular hyperfine states that are not in the spin-1/2 manifold can be coupled off-resonantly by the microwave field. To avoid this effect, we limit the Rabi frequency such that the p pulses time is around 10 µs for measurements in {|¯ñ, |1ñ}, and around 16 µs for measurements in {|¯ñ, |2ñ}.  At fixed microwave power, we precisely determine the pulse duration in order to reduce the rotation error of the pulses. To do this, we perform a p/2 rotation followed immediately by up to eight closely spaced echo pulses. We fine tune the pulse duration tp until equal population of the two spin states are obtained regardless of the number of echo pulses applied.  The pulse spacing t determines the passband frequency of the noise filter. Shorter t gives higher passband frequencies and less sensitivity to low frequency electric field noise, which is the dominant noise source in our system. We characterize noise rejection as a function of t by measuring the Ramsey phase fluctuations after a fixed T with {|¯ñ, |1ñ}. We vary t by changing the number of equally spaced Rabi-p pulses inserted between the two Ramsey pulses. We choose the phase of the second Ramsey pulse to maximize the sensitivity of the fraction of |1ñ to the Ramsey phase. For each t, we repeat the measurement 10 times to extract the SD of Ramsey phase, as computed from the measured fraction of |1ñ. We use echoes along 𝑋b only for this experiment, which provides the same electric field noise rejection performance as XY8.  14 Extended data figure 2 shows the SD of Ramsey phase as a function of t for T = 1.5 ms. We observe reduction of the phase fluctuations as t is reduced. To balance between decoupling efficacy and a hight/tp ratio to minimize system evolution during the pulses, we chose t = 140 µs for the mean-field frequency shift measurements. To account for the finite widths of the Ramsey pulses, we adjusted the delay (t/2 for infinitely short pulses) after the first Ramsey pulse and before the final detection pulse to minimize the sensitivity of the Ramsey phase shift to detuning D at resonance (𝜕Df/𝜕D)D\tij and maximize the range of D within which the phase shift is insensitive to D at first order. The amount of the adjustment is determined by simulating the XY8 sequence on a two-level system, and depends on the rotation angle q, which is accounted for in the measurements. Extracting phase shifts for the reversal measurements Switching between two spin manifolds requires changing the frequency of the microwave source. We use two procedures to reduce extra phase shift introduced by this process: (1) we measure the  Extended data figure 2. Noise suppression of the dynamical decoupling sequence. Data is taken at |E| = 1.02 kV/cm and a = 0°. We use X-echo only, which provides similar performance to XY8 in terms of rejecting noise in D. Each data point is extracted from 10 repetitions. The dashed line is the noise floor of our measurement. \n 15 phase difference DF =Df(q = p/4) - Df(q = 3p/4) between the Ramsey fringes obtained for q =p /4 and q =3p /4; (2) For q =3p /4 measurement, we first prepare the molecules in |2ñ instead of |¯ñ and then apply a q =p /4 pulse. This allows us to keep the timings of the sequences for q =p /4 and q =3p /4 identical. Dipolar collisions with transition dipole moment In Ref. 22, the dipolar elastic collision rate between spin polarized KRb in 2D was measured. With n » 5.0´107 cm-2, T0 » 250 nK, d = 0.2 D, Ref. 22 reported G 0 = 168(48) s-1. Scaling G 0 as 𝛤∝𝑛\tl𝑇j\t𝑑m with n = 1.1(1)´107 cm-2, T0 = 463(9) nK, and 𝑑=\t𝑑↓↑/√2\t(conditions for the grey circles in Fig. 4a), we obtain G = 95(29) s-1.  Acknowledgement We thank Thomas Bilitewski and Ana Maria Rey for inspirational discussions and critical reading of the manuscript. We acknowledge funding from the DOE Quantum System Accelerator, the National Science Foundation QLCI OMA-2016244, the National Science Foundation Phys-1734006, and the National Institute of Standards and Technology. J. S. H. acknowledges support from the National Research Council postdoctoral fellowship. C.M. acknowledges the NDSEG Graduate Fellowship.  Author contributions All authors performed the experiments, analyzed the data, and contributed to interpreting the results and writing the manuscript. Data availability The datasets generated and analyzed during the current study are available from the corresponding authors on reasonable request. Competing interests The authors declare no competing interests.    16 References 1. Manousakis, E. The spin-1/2 Heisenberg antiferromagnet on a square lattice and its application to the cuprous oxides. Rev. Mod. Phys. 63, 1–62 (1991). 2. Žutić, I., Fabian, J. & Das Sarma, S. Spintronics: Fundamentals and applications. Rev. Mod. Phys. 76, 323–410 (2004). 3. Wineland, D. J., Bollinger, J. J., Itano, W. M., Moore, F. L. & Heinzen, D. J. Spin squeezing and reduced quantum noise in spectroscopy. Phys. Rev. A 46, R6797--R6800 (1992). 4. Nielsen, M. A. & Chuang, I. Quantum Computation and Quantum Information. Am. J. Phys. 70, 558–559 (2002). 5. Levin, K. & Hulet, R. G. Chapter 3 - The Fermi Gases and Superfluids: Experiment and Theory. in Ultracold Bosonic and Fermionic Gases (eds. Levin, K., Fetter, A. L. & Stamper-Kurn, D. M.) vol. 5 69–94 (Elsevier, 2012). 6. Sobirey, L. et al. Observation of superfluidity in a strongly correlated two-dimensional Fermi gas. Science. 372, 844–846 (2021). 7. Bloch, I., Dalibard, J. & Nascimbène, S. Quantum simulations with ultracold quantum gases. Nat. Phys. 8, 267–276 (2012). 8. Labuhn, H. et al. Tunable two-dimensional arrays of single Rydberg atoms for realizing quantum Ising models. Nature 534, 667–670 (2016). 9. Monroe, C. et al. Programmable quantum simulations of spin systems with trapped ions. Rev. Mod. Phys. 93, 25001 (2021). 10. Zu, C. et al. Emergent hydrodynamics in a strongly interacting dipolar spin ensemble. Nature 597, 45–50 (2021). 11. Chomaz, L. et al. Dipolar physics: A review of experiments with magnetic quantum gases. arXiv (2022) doi:10.48550/ARXIV.2201.02672. 12. Christakis, L. et al. Probing site-resolved correlations in a spin system of ultracold molecules. arXiv (2022) doi:10.48550/ARXIV.2207.09328. 13. Yan, B. et al. Observation of dipolar spin-exchange interactions with lattice-confined polar molecules. Nature 501, 521–525 (2013). 14. Micheli, A., Brennen, G. K. & Zoller, P. A toolbox for lattice-spin models with polar molecules. Nat. Phys. 2, 341–347 (2006). 15. Barnett, R., Petrov, D., Lukin, M. & Demler, E. Quantum Magnetism with Multicomponent  17 Dipolar Molecules in an Optical Lattice. Phys. Rev. Lett. 96, 190401 (2006). 16. Hazzard, K. R. A., Manmana, S. R., Foss-Feig, M. & Rey, A. M. Far-from-Equilibrium Quantum Magnetism with Ultracold Polar Molecules. Phys. Rev. Lett. 110, 75301 (2013). 17. Mishra, T., Greschner, S. & Santos, L. Polar molecules in frustrated triangular ladders. Phys. Rev. A 91, 43614 (2015). 18. Bohn, J. L., Rey, A. M. & Ye, J. Cold molecules: Progress in quantum engineering of chemistry and quantum matter. Science 357, 1002–1010 (2017). 19. Gorshkov, A. V et al. Quantum magnetism with polar alkali-metal dimers. Phys. Rev. A 84, 33619 (2011). 20. Yao, N. Y., Zaletel, M. P., Stamper-Kurn, D. M. & Vishwanath, A. A quantum dipolar spin liquid. Nat. Phys. 14, 405–410 (2018). 21. Kwasigroch, M. P. & Cooper, N. R. Bose-Einstein condensation and many-body localization of rotational excitations of polar molecules following a microwave pulse. Phys. Rev. A 90, 21605 (2014). 22. Valtolina, G. et al. Dipolar evaporation of reactive molecules to below the Fermi temperature. Nature 588, 239–243 (2020). 23. Matsuda, K. et al. Resonant collisional shielding of reactive molecules using electric fields. Science 370, 1324–1327 (2020). 24. Li, J.-R. et al. Tuning of dipolar interactions and evaporative cooling in a three-dimensional molecular quantum gas. Nat. Phys. 17, 1144–1148 (2021). 25. Anderegg, L. et al. Observation of microwave shielding of ultracold molecules. Science 373, 779–782 (2021). 26. Schindewolf, A. et al. Evaporation of microwave-shielded polar molecules to quantum degeneracy. Nature 607, 677–681 (2022). 27. Bilitewski, T. et al. Dynamical Generation of Spin Squeezing in Ultracold Dipolar Molecules. Phys. Rev. Lett. 126, 113401 (2021). 28. Tobias, W. G. et al. Reactions between layer-resolved molecules mediated by dipolar spin exchange. Science 375, 1299–1303 (2022). 29. Kitagawa, M. & Ueda, M. Squeezed spin states. Phys. Rev. A 47, 5138–5143 (1993). 30. Leibfried, D. et al. Creation of a six-atom ‘Schrödinger cat’ state. Nature 438, 639–642 (2005). 31. Neyenhuis, B. et al. Anisotropic Polarizability of Ultracold Polar 40K87Rb Molecules. Phys. Rev.  18 Lett. 109, 230403 (2012). 32. Seebelberg, F. et al. Extending Rotational Coherence of Interacting Polar Molecules in a Spin-Decoupled Magic Trap. Phys. Rev. Lett. 121, 253401 (2018). 33. Burchesky, S. et al. Rotational Coherence Times of Polar Molecules in Optical Tweezers. Phys. Rev. Lett. 127, 123202 (2021). 34. Gullion, T., Baker, D. B. & Conradi, M. S. New, compensated Carr-Purcell sequences. J. Magn. Reson. 89, 479–484 (1990). 35. Geier, S. et al. Floquet Hamiltonian engineering of an isolated many-body spin system. Science  374, 1149–1152 (2021). 36. Ni, K.-K., Rosenband, T. & Grimes, D. D. Dipolar exchange quantum logic gate with polar molecules. Chem. Sci. 9, 6830–6838 (2018). 37. Eisert, J., Friesdorf, M. & Gogolin, C. Quantum many-body systems out of equilibrium. Nat. Phys. 11, 124–130 (2015). 38. Guardado-Sanchez, E. et al. Probing the Quench Dynamics of Antiferromagnetic Correlations in a 2D Quantum Ising Spin System. Phys. Rev. X 8, 21069 (2018). 39. Shenker, S. H. & Stanford, D. Black holes and the butterfly effect. J. High Energy Phys. 2014, 67 (2014). 40. Maldacena, J., Shenker, S. H. & Stanford, D. A bound on chaos. J. High Energy Phys. 2016, 106 (2016). 41. Braumüller, J. et al. Probing quantum information propagation with out-of-time-ordered correlators. Nat. Phys. 18, 172–178 (2022). 42. Green, A. M. et al. Experimental Measurement of Out-of-Time-Ordered Correlators at Finite Temperature. Phys. Rev. Lett. 128, 140601 (2022). 43. Davis, E., Bentsen, G. & Schleier-Smith, M. Approaching the Heisenberg Limit without Single-Particle Detection. Phys. Rev. Lett. 116, 53601 (2016). 44. Colombo, S. et al. Time-Reversal-Based Quantum Metrology with Many-Body Entangled States. Nat. Phys. 18, 925-930 (2022). 45. Gilmore, K. A. et al. Quantum-enhanced sensing of displacements and electric fields with two-dimensional trapped-ion crystals. Science 373, 673–678 (2021). 46. Bohn, J. L., Cavagnero, M. & Ticknor, C. Quasi-universal dipolar scattering in cold and ultracold gases. New J. Phys. 11, 55039 (2009).  19 47. Ticknor, C. Quasi-two-dimensional dipolar scattering. Phys. Rev. A 81, 42708 (2010). 48. Karman, T., Yan, Z. Z. & Zwierlein, M. Resonant and first-order dipolar interactions between ultracold 1S molecules in static and microwave electric fields. Phys. Rev. A 105, 13321 (2022). 49. Martin, M. J. et al. A Quantum Many-Body Spin System in an Optical Lattice Clock. Science 341, 632–636 (2013). 50. Bruun, G. M. & Taylor, E. Quantum Phases of a Two-Dimensional Dipolar Fermi Gas. Phys. Rev. Lett. 101, 245301 (2008). 51. Lee, H., Matveenko, S. I., Wang, D.-W. & Shlyapnikov, G. V. Fulde-Ferrell-Larkin-Ovchinnikov state in bilayer dipolar systems. Phys. Rev. A 96, 61602 (2017). 52. Peter, D., Müller, S., Wessel, S. & Büchler, H. P. Anomalous Behavior of Spin Systems with Dipolar Interactions. Phys. Rev. Lett. 109, 25303 (2012). 53. Koschorreck, M., Pertot, D., Vogt, E. & Köhl, M. Universal spin dynamics in two-dimensional Fermi gases. Nat. Phys. 9, 405–409 (2013). 54. Syzranov, S. V, Wall, M. L., Gurarie, V. & Rey, A. M. Spin–orbital dynamics in a system of polar molecules. Nat. Commun. 5, 5391 (2014).   20  Figure 1. A 2D itinerant spin system with polar molecules. a. Molecular spins are free to move within a 2D layer. Their dipolar interactions are tuned by a bias electric field E with configurable magnitude and orientation in the x-y plane. Yellow, blue, green, and grey curves illustrate interaction processes detailed in c. Red arrows represent dipolar elastic collisions. b. Energy diagram for the ground and first excited rotational states in which we encode the spin-1/2 degree of freedom. The transitions are driven by microwaves. c. Calculated dipolar interaction strength as a function of |E| for the spin manifold {|¯ñ, |1ñ}. The dipolar coupling strength is given in units of the permanent dipole moment of KRb dp = 0.574 Debye. The molecules interact via their induced dipole moments d¯ and d, as well as the transition dipole moment 𝑑↓↑. The black line shows the field dependence of 𝜒H.    \n 21     Figure 2. Dynamical decoupling and tunable dipolar interactions between molecules. a. The standard Ramsey sequence consists of an initial pulse of area q, followed by an evolution time T, and finally a pulse of area p - q about an axis 𝑛n=\t𝑋b𝑐𝑜𝑠𝜙+\t𝑌b𝑠𝑖𝑛𝜙. We insert one or more XY8 sequences during T, denoted as XY8´M. One XY8 sequence (zoomed region) consists of 8 Rabi-p pulses spaced by time t, for a total free precession time of 8t (see Methods). b. Decay of Ramsey fringe contrast between |¯ñ and |1ñ without dynamical decoupling (grey circles) and with XY8´3 at a density of n = 0.14(2) ´ 107 cm-2 (orange squares). We use a low density to reduce interaction effects. Error bars are 1 standard deviation from bootstrapping (see Methods). c. Measured Ramsey fringes with dynamical decoupling for an average 2D density n = 1.4(1) ´ 107 cm-2 for initial population imbalance q = p/4 (blue diamonds), p/2 (black circles) and 3p/4 (red squares) for T = 1.2 ms at |E| = 0. Solid lines are fits to Acos[(p/180f) - Df] plus an offset, from which we extract the measured shift Df.d. Measured Df versus cosq for n =1.4(1) ´ 107 cm-2 (black circles) and 0.65(7) ´ 107 cm-2 (grey squares). The strength of the mean-field interaction Uc is extracted from the slope of the linear fit (solid lines). Error bars are 1 s.e.. e. Density dependence of Uc for the {|¯ñ, |1ñ} manifold (grey) and the {|¯ñ, |2ñ} manifold (green). Black and green solid lines are linear fits to the data. The slopes of the fits are direct measurements of c/ and c0 respectively.  Error bars are 1 s.e. from linear fits. f. Dependence of c for the {|¯ñ, |1ñ} manifold on |E| at a = 0°. The solid line is a one parameter fit to 𝐴[(𝑑↓−𝑑↑)0−2𝑑↓↑𝑑↑↓] calculated for KRb at the experimental magnetic field and trapping conditions which includes modifications to the dipole moments from the mixing of hyperfine and rotational states at |E| below ~500 V/cm (shaded area). Grey line indicates zero.  Error bars are 1 s.e. from linear fits. g. Angular dependence of c for the {|¯ñ, |1ñ} manifold at |E| = 1.02 kV/cm. Solid line is a one parameter fit to -A(3cos2a - 1). Grey line indicates zero.  Error bars are 1 s.e. from linear fits. \n 22     Figure 3. Reversal of the spin dynamics. a. Measurement sequence. Molecules are initially prepared and allowed to evolve in a superposition of |¯ñ and |2ñ. A composite pulse R coherently transfers the population in |2ñ to |1ñ while preserving its relative phase to |¯ñ. Molecules then evolve in {|¯ñ ,|1ñ}. Inset: Composition of R. b. Measured time evolution of DF  = Df(q = p/4) -Df(q = 3p/4) at n = 1.1(1) ´ 107 cm-2. The phase accumulation in stage I (green squares) and stage II (black circles) are plotted over time. For T <1.2 ms, we fix the duration of stage II to be 80 µs and scan the time of stage I. For T >1.2 ms, we fix the duration of stage I to be 1.2 ms and scan the time of stage II. The total time plotted excludes the widths of the microwave pulses. A piecewise linear fit to the time evolution of DF is shown as a solid line. The ratio between the two slopes is constrained to -2. Error bars are 1 s.e.. \n 23  \n Figure 4. Dipolar collisional decoherence. a. Decay of Ramsey contrast between |¯ñ and |1ñ for density n = 1.1(1) ´ 107 cm-2 (grey circles), 0.57(4) ´ 107 cm-2 (blue diamonds), 0.28(3) ´ 107 cm-2 (orange squares) at |E| = 0. Decoupling sequences used are XY8´3 for the grey and blue traces and XY8´7 for the orange trace.  Error bars are 1 standard deviation from bootstrapping (see Methods). b. Contrast decay rate G as a function of n for |E| = 0, a = 0° (grey circles), and |E| = 1.02 kV/cm, a = 36° (purple squares).  Error bars are 1 s.e. from exponential fits. \n"    },    {        "title": "1311.0965v1.Spin_accumulation_detection_of_FMR_driven_spin_pumping_in_silicon_based_metal_oxide_semiconductor_heterostructures.pdf",        "content": "1 \n Spin accumulation detection of FMR driven  spin pumping  in silicon -based \nmetal -oxide -semiconductor hetero structures  \n \nY. P u1, P. M. Odenthal2, R. Adur1, J. Beardsley1, A. G. Swartz2, D. V. Pelekhov1, R. K. \nKawakami2, J. Pelz1, P. C. Hammel1, E. Johnston -Halperin1  \n \n1Department of Physics, The Ohio State University , Columbus, Ohio 43210  \n2Department of Physics and Astronomy, University of California, Riverside, California 92521  \n \nThe use of the spin  Hall effect and its inverse  to electrically detect and manipulate dynamic \nspin currents generated via ferromagnetic resonance (FMR) driven spin pumping has \nenabled the investigation of  these dynamically injected currents across a wide variety  of \nferromagnetic materials.  However, while this approach has proven to be an invaluable \ndiagnostic for exploring the spin pumping process it requires  strong spin -orbit coupling , \nthus substantially  limit ing the materials basis available for  the detector/ch annel material \n(primarily Pt, W and Ta) . Here , we re port FMR driven  spin pumping into a weak spin -\norbit channel through the measurement of a spin  accumulation voltage in a Si-based  metal -\noxide -semiconductor (MOS)  heterostructure . This alternate experimental approach \nenables the investigation of dynamic  spin pumping in a broad  class of materials with weak \nspin-orbit coupling and long spin  lifetime  while providing additional information regarding \nthe phase evolution of the injected s pin ensemble  via Hanle -based measurements of the \neffective spin lifetime .  \n 2 \n The creation and manipulation of non -equilibrium spin populations in non -magnetic \nmaterials (NM) is one of the cornerstones of modern spintronics. These excitations have to date \nrelied primarily on charge based phenomena, either via direct electrical injection from a \nferromagnet (FM)1-7 or through the exploitation of the spin -orbit interaction8-10. Ferromagnetic \nresonance (FMR) driven spin pumping11-22 is an emerging method to dynam ically inject pure \nspin current  into a NM with no need for an accompanying charge current, implying substantial \npotential impacts on low energy cost , high efficiency spintronics . However, while the creation  of \nthese non -equilibrium spin currents does not require a charge current , previous studies of \ntransport -detected spin pumping do rely on a strong spin -orbit interaction in the NM to convert \nthe spin current into a charge current  in the detector via the  inverse spin -Hall effect (ISHE) 12-20. \nThis approach has proven  to be an  instrumental diagnostic , but it does carry with it several \nlimitations; specifically, the ISHE measures spin current  not spin density , is only sensitive to a \nsingle component of the ful l spin vector  and is only effective in materials with strong spin -orbit \ncoupling . Here we demonstrate an alternate detection geometry relying on the measurement of a \nspin accumulation voltage using a ferromagnetic electrode, similar to the three -terminal  \ngeometry pioneered for electrically -driven spin injection4,23-28. This approach dramatically \nexpands the materials basis for FMR  driven spin pumping, allows for the direct measurement of \nspin accumulation in the channel and enables the phase -sensitive meas urement of the injected \nspin population.  \n Our study is performed in  a silicon -based metal -oxide -semiconductor (MOS) structure \ncompatible with current semiconductor logic technologies.  Using  a Fe/MgO/p -Si tunnel diode \nwe achieve spin pumping into  a semiconductor across an insulat ing dielectric. This approach \nallows voltage -based detection of the spin accumulation under the electrode4-6,23-29. Further, we 3 \n demonstrate sensitivity to the phase of the injected spin via the observation of Hanle dephasing  \nin the presence of an out -of-plane magnetic field . These results establish a bridge between the \npure spin currents generated by FMR driven spin pumping and traditional charge -based spin \ninjection,  laying the foundation for a new class of experimental probes and promising the \ndevelopment of novel spin -based devices compatible with current CMOS technologies.  \n Tunne l diodes are fabricated from  Fe(10nm)/ MgO( 1.3nm)/ Si(100) heterostructures  \ngrown by mo lecular beam epitaxy (MBE). The p -type Si substrates are semiconductor on \ninsulator (SOI) wafers with a 3 μm thick Si device layer containing 5×1018 cm-3 boron dopants , \nproducing a room temperature resistivity of 2× 10-2 Ωcm. The device is  patterned by conv entional \nphotolithography technique s into  a Fe/MgO/Si tunnel contact  of 500  μm × 500 μm lateral size, \nplaced 1 mm away from Au reference contacts for voltage measurements.  Spin pumping and \nFMR measurements are performed in the center of a radio frequency (RF) microwave cavity \nwith f = 9.85GHz  with a DC magnetic field , \nDCH , applied along the x-axis, as sketched in Fig.  \n1a.   On resonance , a pure sp in current is injected into the silicon channel via coupling between  \nthe precessing magnetization  of the ferromagnet,  M, and the conduction electrons in the silicon. \nThis spin current  induces an imbalance in the spin -resolved electrochemical potential and \nconsequent  spin accumulation  given by\n S , where  \nand \n are the chemical \npotentials of up and down spin s, respectively. Using a standard electrical  spin detection \ntechnique4-6,23-29 the spin accumulation can be detected electrically via the relationship : \n)1(2eP VS\nS\n \nwhere \nSV  is the spin -resolved voltage between Fe and Si, \nP is the spin polarization of Fe , \n is \nthe spin detection efficiency , and \nS  is assumed to be proportional to the component of the net 4 \n spin polarization parallel to M30. Figure 1c shows the magnetic field dependence of the FMR \nintensity (upper panel) and the spin accumulation induced voltage VS (lower panel), clearly \ndemonstrating  spin accumulation at the ferromagnetic resonance . \n Figure 2a show s the RF power , PRF, dependence of the FMR intensity  (upper panel)  and \nSV\n (lower panel) on resonance ; the former is proportional to the square root of  \nRFP  and latter is \nlinear with\nRFP , consistent with ISHE detected spin pumping18-20. As shown in Fig.2b, \nSV  is \nconstant  when M reverses, consistent with our local detection geometry wherein the injected spin \nis always parallel to the magnetization of the FM electrode . Note that this is in contrast to the \nmagnetization dependence of ISHE detection, wherein the sign of the ISHE gives a  measure of \nthe spin orientation relative to the  detection electrode . As a result, o ur technique distinguishes the \nspin accumulation signal from  artifacts due to magneto -transport or spin transport, such as the \nanomalous Hall effect, ISHE or spin Seebeck effect , which depend on the direction of M. In \naddition, the current -voltage characteristic ( I-V) of the  tunnel contact is linear at room \ntemperature (see Supplementary Information ), implying at best a weak rectification of any RF -\ninduced pickup currents. This expectation is confirmed by the small offset (below 10 V) \nobserved in spin accumulation voltage measurement s.  Our technique  also rules out potential  \nspurious signal s due to magneto -electric transport such as tunneling anisotropic magneto -\nresistance (TAMR)  that would  require  a rectified  bias voltage of at least mV scale to give the  \nobserved  V scale signal  observed  at resonance.      \n In order to probe the dynamics of the  observed spin accumulation, the applied magnetic \nfield is rotated towards the sample normal (within the xz-plane) by an angl e \n, introducing an \nout-of-plane component,  \nzH. Due to the strong  demagnetization field of our thin -film geometry \n(~2.2 T) the orientation of the magnetization lags the orientation of the applied field,  remaining 5 \n almost entirely in -plane ( the maximum estimated deviation is 2°). As a result , the injected spins \n(parallel to M) precess due to the applied field.  As the magnitude of \nzH  increases this precession \nwill lead to a dephasing of the spin ensemble  and consequent decrease in its net magnetization , \n(the Hanle effect3-6,23-29). Figure 3a shows the FMR spectrum for different angles \n ; the \nresonant field \nFMRH  changes from 250G to 400G as \n  changes from 0 to 40 degrees. This \nincrease is consistent with the fact that the in -plane component of H primarily determines the \nresonance condition, so as \n  increases a larger total applied field is therefore required to drive \nFMR  (see Supplementa ry Information ). Figure 3b shows \nSV  vs. \nDCH  over the same angular \nrange. The peak position of \nSV  shifts in parallel with the FMR spectrum, but the peak value \ndecreases with increasing  \nzH , as expected for Hanle -induced dephasing in an ensemble of \ninjected spins3-6,23-29.  \nFor an isotropic ensemble of spins precessing in a uniform field perpendicular to M the \neffect of this dephasing on \nSV  can be described by a simple Lorentzian function:  \n)2()(1)(20\n SHS Vz x S S\n \nwhere  \n  is the Larmor frequency given by \n/z B effH g , \neffg  is the effective Land é g-factor, \nB\n is the Bohr magneton ,\n is the Plank’s constant , and \n  is the spin lifetime. In the more \ngeneral case that H is not perpendicular to M, as is the case here, then Eq.  (2) should be replaced \nby the more general function:  \n)3() (11\n2 22 2\n22\n0\n\n  \n\ntotal totalz y\ntotalx\nx S S S V\n \nwhere \n),, (/ zyxi gHB effi i    23,27.  If H is in the xz-plane, this reduces to  6 \n \n)4() (11sin cos22 2\n0\n\n\n \ntotalx S S S V \n This general behavior has been observed in previous studies of three terminal  electrical \nspin injection4,23-28 (Fig. 4a). However, it has been widely reported in electrically detected spin \ninjection experiments  that spatially varying local fields due to the magnetic electrodes, coupling \nto interfacial spin states and other non -idealities generate spin dynamics that are not well \ndescribed by this simple model. As a result \n  is generally understo od to represent an  effective \nspin lifetime, \neff , and while there are some initial efforts to more quantitatively account for the \nreal sample environment, such as the so -called “inver ted-Hanle” measurement23,26,27, a detailed \nmodel of these interactions is currently lacking.  \nWe explore the functional dependence of the dephasing of our FMR driven spin current \nby plotting the peak spin accumulation voltage, \npeak\nSV , as a function of \nzH  (Fig. 4b, solid \ncircles). The suppression of the spin accumulation at high magnetic fields seen in Fig. 3 is a clear \nindication of the dephasing of the injected spin ensemble; however, attempts to fit this behavior \nto Eq. (4) reveal that this simple, isotropic model fails to accurately reproduce our data (Fig. 4b, \nblack dashed  lines). In particular, it is a feature of the isotropic model that for a magnetic field \nthat is not parallel to z that the spin polarization along M, and therefore the measured \naccumulation voltage, does not go to zero at high field even for infinite spin lifetime.  This \ndiscrepancy likely arise s from  contributions due to the various non -idealities discussed above ; in \nparticular, as we discuss below, we believe that the coupling to localized states and the impact of \nbulk spin diffusion may play a more central role in this experimental geometry.  \nWe note that our data is well described by  a simple Lorentzian  (Eq. 2), though the \nrelationship between the effective spin lifetime extracted from this fit  (0.6 ns) , which we label 7 \n \nFMR, to the \neff  defined in Eq. (4) is not clear. For comparison, the intensity of the FMR signal \nis found to be constant  to within roughly 10% (solid red triangles) , suggesting  that the spin \ncurrent is roughly  constant and indicating that FMR -driven heating19,20, if present, does not \ncontribute significantly to the field -induced suppression of  \npeak\nSV  seen i n Fig. 4b.  \nA key advantage of our experimental geometry is that it allows direct comparison of this \ndephasing with the more traditional three -terminal electrical injectio n within the same device26. \nFigure 4b (open purple circles) shows the spin accumulation voltage measured in the three -\nterminal geometry as a function of a perpendicular applied field, \nzH . The dephasing in this \ngeometry is clearly much slower than in FMR driven s pin pumping. This observation is \nsupported by the Lorentzian fit to Eq. (2) indicated by the solid purple line, yielding an effective \nlifetime of 0.11 ns, consistent with previous reports by our group and others4,23-27. In considering \nthe origin of this discrepancy in observed lifetime a natural suspicion falls on the different \nexperimental methodologies . Specifically , for the spin pumping case the field is applied at an \nangle \n , resulting in both in -plane and perpendicular components to the field, while for the DC \ncurrent injection only the perpendicular component is present.  \nThe consequence of this vector magnetic field is twofold: first, it will rotate the \nprecession axis of the injected spins away from the perpendicular case implicit in the simple \nHanle model as described above, and second, it will generate an “inver ted” Hanle effect that has \nbeen proposed to derive from the interplay between an in -plane applied magnetic fiel d and some \nfinite inhomogeneity in the local fields due to the magnetization of the electrode.  In Fig. 4c w e \nexplore this behavior in a control sample wherein we perform both traditional Hanle (i.e. wherein \nthe only applied magnetic field is\nzH ) and rotating Hanle measurements (i.e. wherein the \nmagnetic field is rotated by an angle \n , yielding both in -plane and perpendicular components to  8 \n \nH), see Supplementary Information.  The field values for the rotating Hanle experiment are \nchosen to correspond to the values o f \nzH  from Fig. 4b. As expected, the traditional Hanle \nmeasurement again yields an effective lifetime of roughly 0.1 4 ns (open purple circles) . While \nthe rotating Hanle geometry does yield a slightly shorter lifetime of  0.09 ns (solid blue squares) , \nthis variation is too small  (and in the wrong direction)  to account for  the longer effective lifetime  \nobserved for FMR driven spin pumping. We therefore conclude that th e enhanced dephasing rate \nobserved in Fig. 4b indicates the FMR -driven  and electrically -driven spin injection  processes \ndiffer . \nWhile the origin of this discrepancy  is still an active area of investigation, we note that \nthe observed  \nFMR  of 0.6 ns is consistent with previous measurements of the spin lifetime in the \nbulk silicon channel at these doping level4,23-25. Further, the requirement that there be no net \ncharge flow during FMR driven spin injection implies that any forward propagati ng tunneling \nprocess be balanced by an equal and opposite back tunneling process. As shown by the band \ndiagram of spin pumping  in Fig. 4a this opens up a potential pathway for coupling from  the bulk \nSi states into the intermediate states that dominate the three -terminal accumulation voltage26,28. \nThe band diagram of electrical spin injection (upper panel of Fig. 4a) shows that  this process is \nstrongly suppressed in electrical spin injection due to the finite bias ( 5 mV in this case) present \nacross the tunnel junction . If we assume for the sake of argument that  the spin pumping \nmeasurement is in fact sensitive to the bulk spin polarization in silicon, we can calculate the spin \ncurrent  using the relation \nsf S SeJ ; according to Eq . (1) with  \nP=0.4 for Fe and assuming  \n\n=0.5, the spin current density is \nSJ  ~ 2 ×  105 Am-2. This spin current is roughly one order of \nmagnitude smaller than previous reports of spin pumping from conventional ferromagnets into \nmetals.  9 \n  In summary, we demonstrate spin pumping into  a semiconductor through an insulat ing \ndielectric . This approach allows observation of  precession  of the dynamically injected spins an d \ncharacteriz ation of the  effective spin lifetime . Our results directly probe the coherence and phase \nof the dynamically injected spins and the spin manipulation  of that spin ense mble via spin \nprecession, lay ing the foundation for novel spin pumping based spintronic applications . \n \nAcknowledgements  \nThis work is supported by the Center for Emergent Materials at the Ohio State University, \na NSF Materials Research Science and Engineering Center (DMR -0820414) (YP, PO, JB, AS, \nRK, JP and EJH) and by the Department of Energy through grant DE -FG02 -03ER46054 (RA \nand PCH). Technical  support is provided by the NanoSystems Laboratory at the Ohio State \nUniversity.  The aut hors thank Andrew Berger  and Steven Tjung  for discussion s and assistance.  \n \nReference and notes   \n1. Žutić, I. , Fabian , J. & Das Sarma , S. Spintronics: Fundamentals and applications. Rev. Mod. \nPhys.  76, 323 (2004).  \n2. Awschalom , D. D.  & Flatté , M. E. Challenges for semiconductor spintronics. Nature Phys. \n3, 153 (2007).  \n3. Appelbaum , I., Huang , B., & Monsma, D. J. Electronic measurement and control of spin \ntransport in silicon . Nature  447, 295 (2007).  \n4. Dash, S. P. , Sharma,  S., Patel, R. S. , de Jong, M. P. & Jansen, R. Electrical creation of spin \npolarization in silicon at room temperature . Nature  462, 491 (2009).  10 \n 5. Jedema, F. J., Heersche, H. B., Filip, A. T., Baselmans, J. J. A. & van Wees, B. J. Electrical \ndetection of spin precession in a metallic mesoscopic spin valve. Nature  416, 713 (2002).  \n6. Lou, X. et al.  Electrical detection of spin transport in lateral  ferromagnet  semiconductor \ndevices. Nature Phys.  3, 197  (2007).  \n7. Jonker,  B. T. , Kioscog lou, G., Hanbicki, A. T., Li, C. H., & Thompson, P. E.  Electrical spin -\ninjection into silicon from a ferromagnetic metal /tunnel barrier contact . Nature Phys.  3, 542  \n(2007).  \n8. Kato , Y. K., Myers , R. C., Gossard , A. C.  & Awschalom , D. D. Observation of the Spin Hall \nEffect in Semiconductors . Science  306, 1910 (2004).  \n9. Valenzuela , S. O.  & Tinkham , M. Direct electronic measurement of the spin Hall  effect . \nNature  442, 176 (2006).  \n10. Liu, L. et al.  Spin-Torque Switching with the Giant Spin Hall Effect of Tantalum. Science  \n336, 555 (2012).  \n11. Tserkovnyak Y., Brataas, A. & Bauer, G. E.  W. Enhanced Gilbert damping in thin \nferromagnetic Films. Phys. Rev. Lett . 88, 117601 (2002).  \n12. Saitoh, E., Ueda, M., Miyajima, H. & Tatara, G. Conversion of spin current into charge \ncurrent at room temperature: Inverse spin -Hall effect. Appl. Phys. Lett.  88, 182509 (2006).  \n13. Tserkovnyak Y., Brataas, A. & Bauer, G. E.W. & Halperin, B. I. Nonlocal magnetization \ndynamics in ferromagnetic heterostructures. Rev. Mod. Phys.  77, 1375 (2005).  \n14. Kajiwara , Y. et al . Transmission of electrical signals by spin -wave interco nversion in a \nmagnetic insulator . Nature  464, 262 (2010).  \n15. Kurebayashi , H. et al . Controlled enhancement of spin -current emission  by three -magnon \nsplitting . Nature Materials  10, 660 (2011).  11 \n 16. Sandweg, C. W . et al. Spin Pumping by Parametrically Excited Exchange Magnons . Phys. \nRev. Lett.  106, 216601 (2011).  \n17. Czeschka , F. D. et al. Scaling behavior of the spin pumping effect in ferromagnet -platinum \nbilayers . Phys. Rev. Lett.  107, 046601 (2011).  \n18. Ando , K. et al.  Inverse spin -Hall effect induced by spin pumping in metallic system . J. Appl. \nPhys.  109, 103913 (2011).  \n19. Ando , K. et al . Electrically tunable spin injector free from the  impedance mismatch \nproblem . Nature Materials  10, 655 (2011).  \n20. Ando , K. & Saitoh, E.  Observation of the inverse spin Hall effect i n silicon. Nature Comm.  \n3, 629 (2012 ). \n21. Costache, M. V. et al.  Electrical detection of spin pumping due to the precessing \nmagnetization of a single ferromagnet. Phys. Rev. Lett.  97, 216603 (2006).  \n22. Heinrich , B. et al . Spin pumping at the magnetic insulator (YIG)/normal metal (Au) \ninterfaces . Phys. Rev. Lett.  107, 066604 (2011) . \n23. Dash, S. P. et al. Spin precession and inverted Hanle effect in a semiconductor near a finite -\nroughness ferromagnetic interface . Phys. Rev. B  84, 054410  (2011).  \n24. Li, C. H., van‘t Erve , O. & Jonker , B. T. Electrical injection and detection of spin  \naccumulation in silicon at 500K with magnetic metal/silicon dioxide contacts . Nature \nComm.  2, 245 (2011 ). \n25. Gray , N. W. & Tiwaria , A. Room temperature electrical injection and detection of spin \npolarized  carriers in silicon using MgO tunnel barrier . Appl. Phys. Lett.  98, 102112 (2011 ). \n26. Pu, Y . et al. Correlation of electrical spin injection and non -linear charge -transport in \nFe/MgO/Si . Appl. Phys. Lett.  103, 012402 (2013 ). 12 \n 27. Jeon, K. et al. Electrical investigation of the oblique Hanle effect in ferromagnet / oxide / \nsemiconductor c ontacts . arXiv  1211.3486 (2013).  \n28. Tran, M. et al. Enhancement of the spin accumulation at the i nterface  between a spin-\npolarized tunnel junction and a semiconductor . Phys. Rev. Lett. 102, 036601 (2009 ). \n29. Sasaki, T., Oikawa,  T., Suzuki, T., Shiraishi,  M., Suzuki, Y. & Noguchi,  K. Comparison of \nspin signals in silicon between nonlocal four -terminal  and three -terminal methods. Appl. \nPhys. Lett.  98, 012508 (2011).  \n30. As discussed in Ref. 26, Eq. (1) is derived assuming a linear tunneling model, and might \nunderestimate the actual value of \nS  at higher  bias if the current depends super -linearly on \napplied bias.   \n  13 \n Figure legends  \n \nFigure 1 | Experimental setup  \n(a) Schematic of experimental setup. ( b) Diagram of spin accumulation and spin -resolved  \nvoltage  \nSV. (c) FMR intensity (upper panel; arrows indicate state of the Fe magnetiza tion) and \nspin-resolved voltage  \nSV (lower panel) as a function of \nDCH .  \n \nFigure 2  | RF power - and magnetic field - dependence  \n(a) FMR intensity (upper panel) and spin-resolved voltage (lower panel) as a function of RF \npower; ( b) Solid symbols: \nSV  vs. \nFMR DC H H  when \nDCH  is parallel or anti -parallel with the x-\naxis; open symbols indicate the voltage between two Au/Si reference contacts; all data is \nmeasured under the same experimental conditions.  A background offset of ~2 V has been \nsubtracted from all data.  \n \nFigure 3 | Experiments with increasing Hz \n(a) FMR intensity spectra at various magnetic field orientations \n  as described in the text; ( b) \nSV\n vs. \nDCH measured at the same set of magnetic field orientations. The shift in FMR center \nfrequency tracks the expected magnetization anisotropy of the Fe thin film, see text.  \n \nFigure 4 | Hanle effect measurements  14 \n (a) Schematics of experimental setup and band diagram for three -terminal electrical spin \ninjection (upper panel) and spin pumping (lower panel); ( b) Hanle effect as a function of \nzH  for \nthree -terminal (open circles) and spin pumping (solid circles), solid lines are Lorentzian fits \nyielding 0.11 ns and 0.6 ns, respectively; solid triangles are FMR absorption as a function  of H z, \nthe red dashed line is a guide to the eye and the scale bar represents 10% variation; Black  dashed \nlines are simulated using Eq. (4) with \n , \nxH = 248G ( dash dot ) and 310G ( dash), \nrespectively , see Supplementary Information . (c) Hanle effect measured in a control sample by \nthe three -terminal method, open circles are measured with magnetic field applied out of plane, \nsolid squares are obtained using same magnetic field configuration as for FMR driven spin \npumping;  lines are Lorentzian fits yielding 0.14 ns and 0.09 ns, respectively.   15 \n Figures  \n \n \n \nFigure 1  Y. Pu et al.   \n16 \n  \n \n \n \nFigure 2   Y. Pu et al.  \n  \n17 \n  \n \n \n \nFigure 3  Y. Pu et al.  \n  \n18 \n  \n \n \n \nFigure 4   Y. Pu et al.   \n19 \n Supplementary Information  \n \nSpin accumulation detection of FMR driven spin pumping in silicon -based \nmetal -oxide -semiconductor heterostructures  \n \nY. Pu1, P. M. Odenthal2, R. Adur1, J. Beardsley1, A. G. Swartz2, D. V. Pelekhov1, R. K. \nKawakami2, J. Pelz1, P. C. Hammel1, E. Johnston -Halperin1  \n \n1Department of Physics, The Ohio State University, Columbus, Ohio 43210  \n2Department of Physics and Astronomy, University of California, Riverside, California 92521  \n \nA. Linear I -V of Fe/MgO/Si contact at room temperature  \n The I-V characteristic of the Fe/MgO/p -Si contact is linear at room temperature , as \nshown in Fig. S1, indicating that in this regime  the contact resistance is domin ated by the MgO \ninsulating barrier and contribution from the Schottky barrier is negligible. The linear fit gives \n40.9 resistance with about 25m  uncertainty.   \n \nB.  Impact of interface roughness and in -plane external magnetic field  \n  In contrast to traditional three -terminal spin accumulation measurements, for the FMR \ndriven measurements described in Figs. 3 and 4 it is necessary to apply an external magnetic \nfield both parallel and perpendicular to the magnetization. As described in the main text, the \nparallel component of the field is necessary to satisfy the conditions of magnetic resonance and 20 \n the perpendicular component contributes to  the decay of spin accumulation, allowing for a \nHanle -style measurement of the effective spin life time. However, in consulting the literature23,27 \nit becomes evident that this geometry potentially raises an additional concern regarding the \ninterpretation of this data. Specifically, the in plane component of the magnetic field will itself \ninduce an “inv erted -Hanle” effect wherein the measured spin accumulation rises  with the \nmagnitude of the parallel component of the magnetic field (Fig. S2a). The accepted interpretation \nof this effect is an annealing of fluctuations in the magnetization induced by surfa ce roughness of \nthe magnetic layer in large external fields23,27.   \n We measure the spin accumulation on a control sample via 3T electrical spin injection, as \nshown in Fig. S2a, where magnetic field is applied in xz -plane with orientation ranging from in -\nplane ( H//x, 0 degree) to out of plane (H//z, 90 degree). Using the 3T data of \n) (FMR S H HV , \nwhere \nFMRH  for given magnetic field orientation is obtained by spin pumping experiment as \nshown in Figure 3, we can first directly compare electrical spin injection and spin pumping under \nthe same experimental configuration, as discussed in main text.  \nTo get a quantitative understanding of the angular dependence shown in Fig. S2a, one can \nstart with a general formula:  \nSSωSSDt\n      (S1) \nwhere  \n  is the Larmor frequency , \nD is the spin diffusion constant and \n  is the spin lifetime. In \nthe 3T geometry the impact of spin diffusion is usually believed to be negligible23,27, from this \none can obtain an analytical solution under arbitrary applied magnetic field:  21 \n  \n\n\n 2 22 2\n22\n0) (11\n \n\ntotal totalz y\ntotalx\nxS S       (S2) \nwhere \n),, (/ zyxi gHB effi i     , representing each component of the applied magnetic field.  \nFigure S2b shows the simulation according to Equation (S2) with \n = 0.14ns as obtained by \nLorentzian  fit. Clearly the model fails to explain the data shown in Fig. S2a, especially  the \nobservation that at certain orientations the measured spin accumulation rises  with the applied \nmagnetic field increasing.  \n As pointed out by previous studies23,27 stray magnetic fields from the injector due to \ninterface roughness should strongly impact on the Hanle -style measurements. The total magnetic \nfield should be taken as\n),, ( zyxi H H Hms\niext\ni i  , which re presents the contribution from \nexternal and magnetic -stray fields. The stray field strength is taken to have a spatial variation \n)/2cos()0( )( x Hx Hms\nims\ni \n, where \n  is the typical length scale (~20nm) of the surface \nroughness23,27.  Assuming the spin diffusion length is much longer than \n  we average the total \nmagnetic field over a full period of \n , i.e. \n2 2 2) () (ms\niext\ni i H H H  , we therefore have a formula:  \n\n\n\n2 2 2 22 2\n22 2\n0)/ (11 ) () () () (\n total B totalext\nzms\ntotalext\nxms\nx\nxH g HH H\nHH HS S\n      (S3) \nwhere \nms\nxH  and\nmsH  represent the averaged stray field parallel or perpendicular to the injected \nspins, respectively.  Figure S2c shows the simulation according to Equation (S3) with parameter s \n\n= 0.9ns, \nms\nxH 270G and \nmsH 440G.  The simulation qualitatively agrees with the \nexperiment, but shows some systematic deviations especially in the low -field regime.  22 \n  Although the Equation (S1) is generally accepted, and in principle rigorous analysis can \nbe done with spin precessi on, spin diffusion and spin flip involved, a well -established approach \nto determine the intrinsic spin lifetime using the local spin detection geometry is still lacking. A \nprecise determination of the intrinsic spin lifetime in our sample is beyond the sco pe of this \nreport; we treat the lifetime obtained by the simple Lorentzian fit as an effective spin lifetime or \nspin dephasing time, which represents the decay rate of average spin accumulation under applied \nperpendicular magnetic field.  \n \nC. Simulation on the Hanle effect under FMR condition  \nAs indicated by the FMR spectrum, at FMR there are varying x- and z- components of the \napplied magnetic field with different field orientation \n . The table below is a summary:  \n (deg.)  0 10 20 25 30 40 60 \nHx (G) 250 276 287 295 310 302 248 \nHz (G) 0 49 105 138 179 253 430 \n \nAs shown in the table, \nzH   increases monotonically with \n  and \nxH  is in range of 248 – 310 G \n(roughly constant to maintain the conditions for magnetic resonance), both should impact on the \nHanle effect. The black dashed curves shown in Fig. 4(b) are simulated using Eq. (4) with \n\nand \nxH = 248G, 310G respectively. In the situation that Eq. (4) is valid, finite values of the spin \nlifetime should give a weaker H z-dependence than the simulation curve.  23 \n  \n \nFigure S1: Plot of current vs. voltage of the Fe/MgO/p -Si contact at room temperature, symbol s \nare data and the solid line is a linear fit.  \n \n \n \n  \n24 \n  \n \nFigure S 2: (a) Spin accumulation from 3T electrical spin injection, magnetic field is applied in \ndifferent orientations, ranging from in -plane ( H//x, 0 degree, top curve) to out of plane ( H//z, 90 \ndegree, bottom curve); (b) Simu lations according to Equation (S2)  with \n= 0.14ns, assuming no \nstray field; ( c) Simulations  using Equation (S3 ) with parameters  \n= 0.9ns, \nms\nxH 270G and \nmsH\n440G . \n \n  \n \n"    },    {        "title": "1704.02707v1.Majorana_dynamical_mean_field_study_of_spin_dynamics_at_finite_temperatures_in_the_honeycomb_Kitaev_model.pdf",        "content": "APS/123-QED\nMajorana dynamical mean-\feld study of spin dynamics at \fnite temperatures in the\nhoneycomb Kitaev model\nJunki Yoshitake,1Joji Nasu,2Yasuyuki Kato,1and Yukitoshi Motome1\n1Department of Applied Physics, University of Tokyo, Bunkyo, Tokyo 113-8656, Japan\n2Department of Physics, Tokyo Institute of Technology, Meguro, Tokyo 152-8551, Japan\n(Dated: April 11, 2017)\nA prominent feature of quantum spin liquids is fractionalization of the spin degree of freedom.\nFractionalized excitations have their own dynamics in di\u000berent energy scales, and hence, a\u000bect\n\fnite-temperature ( T) properties in a peculiar manner even in the paramagnetic state harboring\nthe quantum spin liquid state. We here present a comprehensive theoretical study of the spin\ndynamics in a wide Trange for the Kitaev model on a honeycomb lattice, whose ground state is\nsuch a quantum spin liquid. In this model, the fractionalization occurs to break up quantum spins\ninto itinerant matter fermions and localized gauge \ruxes, which results in two crossovers at very\ndi\u000berentTscales. Extending the previous study for the isotropic coupling case [J. Yoshitake, J. Nasu,\nand Y. Motome, Phys. Rev. Lett. 117, 157203 (2016)], we calculate the dynamical spin structure\nfactorS(q;!), the NMR relaxation rate 1 =T1, and the magnetic susceptibility \u001fwhile changing\nthe anisotropy in the exchange coupling constants, by using the dynamical mean-\feld theory based\non a Majorana fermion representation. We describe the details of the methodology including the\ncontinuous-time quantum Monte Carlo method for computing dynamical spin correlations and the\nmaximum entropy method for analytic continuation. We con\frm that the combined method provides\naccurate results in a wide Trange including the region where the spins are fractionalized. We\n\fnd that also in the anisotropic cases the system exhibits peculiar behaviors below the high- T\ncrossover whose temperature is comparable to the average of the exchange constants: S(q;!) shows\nan inelastic response at the energy scale of the averaged exchange constant, 1 =T1continues to\ngrow even though the equal-time spin correlations are saturated and almost Tindependent, and\n\u001fdeviates from the Curie-Weiss behavior. In particular, when the exchange interaction in one\ndirection is stronger than the other two, the dynamical quantities exhibit qualitatively di\u000berent T\ndependences from the isotropic case at low T, re\recting the opposite parity between the \rux-free\nground state and the \rux-excited state, and a larger energy cost for \ripping a spin in the strong\ninteraction direction. On the other hand, when the exchange anisotropy is in the opposite way,\nthe results are qualitatively similar to those in the isotropic case. All these behaviors manifest\nthe spin fractionalization in the paramagnetic region. Among them, the dichotomy between the\nstatic and dynamical spin correlations is unusual behavior hardly seen in conventional magnets.\nWe discuss the relation between the dichotomy and the spatial con\fguration of gauge \ruxes. Our\nresults could stimulate further experimental and theoretical analyses of candidate materials for the\nKitaev quantum spin liquids.\nI. INTRODUCTION\nQuantum many-body systems show various intriguing\nphenomena which cannot be understood as an assembly\nof independent particles. One of such phenomena is frac-\ntionalization, in which the fundamental degree of freedom\nin the system is fractionalized into several quasiparticles.\nA well-known example of such fractionalization is the\nfractional quantum Hall e\u000bect, in which the Hall conduc-\ntance shows plateaus at fractional values of e2=h(eis the\nelementary charge and his the Planck constant) [1, 2].\nIn this case, the quasiparticles carry a fractional value of\nthe elementary charge, as a collective excitation of the\nelementary particle, electron. This is fractionalization of\ncharge degree of freedom. On the other hand, another\ndegree of freedom of electrons, spin, can also be fraction-\nalized. Such a peculiar phenomenon has been argued for\nquantum many-body states in insulating magnets, e.g., a\nquantum spin liquid (QSL) state.\nQSLs are the magnetic states which preserve all the\nsymmetries in the high-temperature( T) paramagnet evenin the ground state and evade a description by conven-\ntional local order parameters. A typical example of QSLs\nis the resonating valence bond (RVB) state, proposed by\nP. W. Anderson [3]. The RVB state is a superposition\nof valence bond states (direct products of spin singlet\ndimers), which does not break either time reversal or\ntranslational symmetry. In the RVB state, the spin de-\ngree of freedom is fractionalized: the system exhibits two\ndi\u000berent types of elementary excitations called spinon\nand vison [4, 5]. Spinon is a particlelike excitation car-\nrying no charge but spin S= 1=2. Meanwhile, vison\nis a topological excitation characterized by the parity\nof crossing singlet pairs with its trace. Another exam-\nple of QSLs is found in quantum spin ice systems, in\nwhich peculiar excitations are assumed to be magnetic\nmonopoles, electric gauge charges, and arti\fcial photons\nresulting from fractionalization of the spin degree of free-\ndom [6, 7].\nAmong theoretical models for QSLs, the Kitaev model\nhas attracted growing interest, as it realizes the frac-\ntionalization of quantum spins in a canonical form [8].arXiv:1704.02707v1  [cond-mat.str-el]  10 Apr 20172\nThe Kitaev model is a localized spin S= 1=2 model de-\n\fned on a two-dimensional honeycomb lattice with bond-\ndependent anisotropic interactions (see Sec. II A). In this\nmodel, the ground state is exactly obtained as a QSL,\nin which quantum spins S= 1=2 are fractionalized into\nitinerant Majorana fermions and localized gauge \ruxes.\nThe fractionalization a\u000bects the thermal and dynami-\ncal properties in this model. For instance, the di\u000berent\nenergy scales between the fractionalized excitations ap-\npear as two crossovers at largely di\u000berent Tscales; in\neach crossover, itinerant Majorana fermions and local-\nized gauge \ruxes release their entropy, a half of log 2\nper site [9, 10]. Also in the ground state, the dynamical\nspin structure factor S(q;!) shows a gap due to the \rux\nexcitation and strong incoherent spectra from the com-\nposite excitations between itinerant Majorana fermions\nand localized gauge \ruxes [11]. Such incoherent spectra\nwere indeed observed in recent inelestic neutron scatter-\ning experiments for a candidate for the Kitaev QSL, \u000b-\nRuCl 3[12, 13]. The magnetic Raman scattering spectra\nalso shows a broad continuum dominated by the itinerant\nMajorana fermions, in marked contrast to conventional\ninsulating magnets [14]. Such a broad continuum was ex-\nperimentally observed also in \u000b-RuCl 3[15]. Furthermore,\ntheTdependence of the incoherent response was theoret-\nically analyzed and identi\fed as the fermionic excitations\nemergent from the spin fractionalization [16, 17].\nIn the previous study, the authors have studied dy-\nnamical properties of the Kitaev model at \fnite Tby a\nnewly developed numerical technique, the Majorana dy-\nnamical mean-\feld method [18]. Indications of the spin\nfractionalization were identi\fed in the Tdependences of\nS(q;!), the relaxation rate in the nuclear magnetic res-\nonance (NMR), 1 =T1, and the magnetic susceptibility \u001f.\nIn the previous study, however, the results were limited\nto the case with the isotropic exchange constants, despite\nthe anisotropy existing in the Kitaev candidate materi-\nals [19, 20]. In the present paper, to complete the anal-\nysis, we present the numerical results of the dynamical\nquantities for anisotropic cases. We also provide the com-\nprehensive description of the theoretical method, includ-\ning the details of the cluster dynamical mean-\feld theory\n(CDMFT), the continuous-time quantum Monte Carlo\n(CTQMC) as a solver of the impurity problem to calcu-\nlate the dynamical spin correlations, and the maximum\nentropy method (MEM) for the analytic continuation.\nWe discuss a prominent feature proximate to the QSL,\ni.e., the dichotomy between static and dynamical spin\ncorrelations, from the viewpoint of the fractionalization\nof spins.\nThe paper is organized as follows. In Sec. II, after in-\ntroducing the Kitaev model and its Majorana fermion\nrepresentation, we describe the details of the CDMFT,\nCTQMC, and MEM. In Sec. III, we show the numeri-\ncal results for S(q;!), 1=T1, and\u001fwhile changing the\nanisotropy in the exchange constants. In Sec. IV, we\ndiscuss the dichotomy between the static and dynamical\nspin correlations by comparing the Tdependences for the\nK2 ΓM1\nM2K1(a) (b)FIG. 1. (a) Schematic picture of the Kitaev model on the\nhoneycomb lattice. The blue, green, and red bonds represent\nthep=x;y, andzbonds in Eq. (1), respectively. The dashed\noval represents the 26-site cluster used in the CDMFT calcu-\nlations. (b) The \frst Brillouin zone (black hexagon) and the\nsymmetric lines (red lines) used in Figs. 3 and 4.\nuniform and random \rux con\fgurations. The cluster-size\ndependence in the CDMFT is examined in Appendix A.\nThe accuracy of MEM is also examined in Appendix B in\nthe one-dimensional limit where the dynamical properties\ncan be calculated without analytic continuation. We also\nshow theTand!dependence of spin correlations and the\nTdependence of the Korringa ratio in Appendix C and\nD, respectively.\nII. MODEL AND METHOD\nIn this section, we describe the details of the meth-\nods used in the present study, the Majorana CDMFT\nand CTQMC methods. After introducing the Majorana\nfermion representation of the Kitaev model in Sec. II A,\nwe describe the framework of the Majorana CDMFT in\nSec. II B, in which the impurity problem is solved exactly.\nIn Sec. II C, we introduce the CTQMC method which is\napplied to the converged solutions obtained by the Majo-\nrana CDMFT for calculating dynamical spin correlations.\nWe also touch on the MEM used for obtaining the dy-\nnamical spin correlations as functions of real frequency\nfrom those of imaginary time in Sec. II D.\nA. Kitaev model and the Majorana fermion\nrepresentation\nWe consider the Kitaev model on a honeycomb lattice,\nwhose Hamiltonian is given by [8]\nH=\u0000X\npJpX\nhj;j0ipSp\njSp\nj0; (1)\nwherep=x,y, andz, and the sum of hj;j0ipis taken\nfor the nearest-neighbor (NN) sites on three inequiv-\nalent bonds of the honeycomb lattice, as indicated in\nFig. 1(a);Sp\njis thepcomponent of the S= 1=2 spin\nat sitej. Hereafter, we denote the average of JpasJ3\nand set the energy scale asP\npjJpj= 3, i.e.,J= 1, and\nparametrize the anisotropy of the exchange coupling con-\nstants asJx=Jy=\u0006\u000bandJz=\u0006(3\u00002\u000b), where +\nand\u0000correspond to the ferromagnetic (FM) and anti-\nferromagnetic (AFM) cases, respectively. We note that\nthe FM and AFM cases are connected through unitary\ntransformations [8].\nAs shown by Kitaev [8], the model is soluble and the\nexact ground state is obtained as a QSL. The spin corre-\nlations are extremely short-ranged: hSp\njSp\nj0iare nonzero\nonly for the NN sites j;j0on thepbonds as well as the\nsame sitej=j0[23]. Hereafter, we denote the NN corre-\nlations ashSp\njSp\nj0iNN. There are two types of QSL phases\ndepending on the anisotropy in Jp: one is a gapless QSL\nrealized in the region with 0 :75\u0014\u000b\u00141:5 including the\nisotropic point \u000b= 1 (Jx=Jy=Jz=\u00061), while the\nother is gapful for 0 \u0014\u000b < 0:75. The ground state\nhas nontrivial fourfold degeneracy in the thermodynamic\nlimit [24].\nThe exact solution for the ground state was originally\nobtained by introducing four types of Majorana fermions\nfor eachS= 1=2 spin [8]. In this method, the Hilbert\nspace in the original spin representation, 2N, is extended\nto 4Nin the Majorana fermion representation ( Nis the\nnumber of spins). Thus, to calculate physical quantities,\nsuch as spin correlations, it is necessary to make a pro-\njection from the extended Hilbert space to the original\none.\nSoon later, however, another way of solving the model\nwas introduced by using only two types of Majorana\nfermions [25{27], in which the projection is avoided as\nthe Hilbert space is not extended. In this method, the\nspin operators are written by spinless fermions by ap-\nplying the Jordan-Wigner transformation to the one-\ndimensional chains composed of two types of bonds, say,\ntheJxandJybonds. Then, by introducing two Ma-\njorana fermions cjand \u0016cjfor the spinless fermions, the\nHamiltonian in Eq. (1) is rewritten as\nH=iJx\n4X\n(j;j0)xcjcj0\u0000iJy\n4X\n(j;j0)ycjcj0\u0000iJz\n4X\n(j;j0)z\u0011rcjcj0;\n(2)\nwhere the sum over ( j;j0)pis taken for the NN sites on a\npbond withj <j0.\u0011r=i\u0016cj\u0016cj0is de\fned on each zbond\nconnecting jandj0sites (ris the index of the zbond).\nHere,\u0011ris considered as a Z2variable taking\u00061, as\n\u0011rcommutes with the total Hamiltonian as well as with\nother\u0011r0and as\u00112\nr= 1. Thus, the model in Eq. (2) de-\nscribes itinerant Majorana fermions fcjg(called matter\nfermions) coupled to the Z2variablesf\u0011rg(called gauge\n\ruxes). The ground state is given by all \u0011r= 1, giving\nQSLs with gapless or gapful excitations depending on \u000b,\nas in the original Kitaev's solution.\nIn the present numerical study at \fnite T, we adopt\nthe Majorana representation used in Eq. (2). This is\nbecause the form of Eq. (2) is suitable for the CDMFT\ncalculations (see Sec. II B), as the interaction term, thethird term in Eq. (2), only lies on zbonds. In this study,\nwe apply the CDMFT to deal with thermal \ructuations\nand compute the static quantities. For calculating dy-\nnamical quantities, we apply the CTQMC method to\nthe converged solutions obtained the CDMFT. While the\nframework was brie\ry introduced in Ref. [18], we describe\nfurther details in the following sections.\nB. Cluster dynamical mean-\feld theory in the\nMajorana fermion representation\nAs presented in the previous study by the real-space\nQMC simulation [10], spacial correlations between \u0011rde-\nvelop at low T. To take into account such spacial corre-\nlations, we adopt a cluster extension of DMFT [28]. As\nthe Majorana Hamiltonian in Eq. (2) is formally simi-\nlar to the Falicov-Kimball model or the double-exchange\nmodel with Ising localized moments, we follow the DMFT\nframework for the double-exchange model [29].\nIn the CDMFT, we regard the whole lattice as a pe-\nriodic array of clusters. The Hamiltonian in Eq. (2) is\nrewritten into the matrix form of\nH=X\n\r;\r0;j;j01\n2H0\n\r;j;\r0;j0c\r;jc\r0;j0+X\n\r;j;j01\n2Hf\u0011g\nj;j0c\r;jc\r;j0;\n(3)\nwhere\rand\r0are the indices for the clusters, and jand\nj0denote the sites in each Nc-site cluster. The coe\u000ecient\n1=2 in Eq. (3) is introduced to follow the notation in\nRef. [30]. In Eq. (3), the \frst term corresponds to the\n\frst and second terms in Eq. (2), while the second term\nis for the third term. Green's function for Eq. (3) is\nformally written as\nG(k;i!n) = (i!n\u00002H0(k)\u0000\u0006(k;i!n))\u00001;(4)\nwhere!n= (2n+ 1)\u0019Tis the Matsubara frequency ( n\nis an integer, and the Boltzmann constant kBand the\nreduced Planck constant ~are set to unity), \u0006( k;i!n)\nis the self-energy, and H0(k) is the Fourier transform of\nH0\n\r;j;\r0;j0in Eq. (3) given by the matrix:\nH0\nj;j0(k) =X\n\rH0\n\r;j;0;j0e\u0000ik\u0001r\r; (5)\nwhere r\ris the coordinate of the cluster \r.\nFollowing the spirit of the DMFT [31, 32], we omit the\nkdependence of the self-energy: \u0006( k;i!n) = \u0006(i!n).\nIn this approximation, local Green's function is de\fned\nwithin a cluster as\nGj;j0(i!n) =1\nN0X\nk\u0002\n(i!n\u00002H0(k)\u0000\u0006(i!n))\u00001\u0003\nj;j0;\n(6)\nwhereN0is the number of clusters in the whole lattice\n(N=NcN0), andjandj0denotes the sites in the cluster.4\nThe Weiss function is introduced to take into account the\ncorrelation e\u000bects in other clusters as\nG0\nj;j0(i!n)\u00001=Gj;j0(i!n)\u00001+ \u0006j;j0(i!n): (7)\nIn order to take into account the interaction Hf\u0011gin\nEq. (3) within the cluster that we focus on, we consider\nthe impurity problem for the cluster described by the ef-\nfective action in the path-integral representation for Ma-\njorana fermions [30]. The partition function is given by\nZ=X\nf\u0011gZf\u0011g; (8)\nwhere\nZf\u0011g=Z\nD\u001fexp(\u0000Sf\u0011g\ne\u000b): (9)\nHere, the sum of f\u0011gin Eq. (8) runs over all possible\ncon\fgurations of f\u0011g, andD\u001f=Q\nj;nd\u001fj;!nin Eq. (9);\n\u001fj;!nis the Grassmann number corresponding to the Ma-\njorana operator cj(more precisely, cj=p\n2 following the\nnotation in Ref. [30]). The e\u000bective action is given by\nSf\u0011g\ne\u000b=\u0000TX\nj;j0;n\u00150\u001fj;\u0000!n(G0(i!n))\u00001\nj;j0\u001fj0;!n\n+ 2TX\nj;j0;n\u00150\u001fj;\u0000!nHf\u0011g\nj;j0\u001fj0;!n: (10)\nFor a given con\fguration of f\u0011g, the impurity problem\nde\fned by Eq. (9) is exactly solvable because it is nothing\nbut a free fermion problem. Green's function is obtained\nas\nh\n(Gf\u0011g(i!n))\u00001i\nj;j0=\u0002\n(G0(i!n))\u00001\u0003\nj;j0\u00002Hf\u0011g\nj;j0:(11)\nNote that we slightly modi\fed the notation from the pre-\nvious study in Ref. [18]. Then, local Green's function for\nthe impurity problem is calculated by\nGimp\nj;j0(i!n) =X\nf\u0011gP(f\u0011g)Gf\u0011g\nj;j0(i!n); (12)\nwhereP(f\u0011g) is the statistical weight for the con\fgura-\ntionf\u0011ggiven by\nP(f\u0011g) =Zf\u0011g=X\nf\u0011gZf\u0011g: (13)\nZf\u0011gis obtained from Green's functions as\nZf\u0011g=Y\nn\u00150det[\u0000Gf\u0011g(i!n)]: (14)\nWe note that Gimp(i!n) is obtained exactly by comput-\ningGf\u0011g(i!n) andP(f\u0011g) for all 2Nc=2con\fgurations off\u0011gin theNc-site cluster [33]. The self-energy for the\nimpurity problem is obtained as\n\u0006j;j0(i!n) =\u0002\n(G0(i!n))\u00001\u0003\nj;j0\u0000\u0002\n(Gimp(i!n))\u00001\u0003\nj;j0:\n(15)\nIn the CDMFT, the above equations, Eqs. (6), (7),\n(12), and (15), are solved in a self-consistent way. The\nself-consistent condition is given by\nG(i!n) =Gimp(i!n); (16)\nnamely, the calculation is repeated until local Green's\nfunction in Eq. (6) agrees with Green's function calcu-\nlated for the impurity problem in Eq. (12).\nThe Majorana CDMFT framework provides a concise\ncalculation method for Tdependences of static quanti-\nties of the Kitaev model, such as the speci\fc heat and the\nequal-time spin correlations hSp\njSp\nj0i. It is worth noting\nthat the CDMFT calculations can be performed without\nany biased approximation except for the cluster approxi-\nmation: the exact enumeration for all the 2Nc=2con\fgu-\nrations in Eq. (12) enables the exact calculations for the\ngiven cluster. Furthermore, the cluster-size dependence\nis su\u000eciently small at all the Trange above the critical\ntemperature for the arti\fcial phase transition due to the\nmean-\feld nature of the CDMFT, as demonstrated for\nthe isotropic case with \u000b= 1:0 in the previous study [18]\n(see also Sec. III A and Appendix A for the anisotropic\ncases). On the other hand, for obtaining dynamical quan-\ntities, such as the dynamical spin correlations hSp\nj(\u001c)Sp\nj0i\n(\u001cis the imaginary time), we need to make an additional\ne\u000bort beyond the exact enumeration in the CDMFT, as\ndiscussed in the next subsection.\nIn the CDMFT+CTQMC calculations in Sec. III, we\nuse the 26-site cluster shown in Fig. 1(a). In Appendix A,\nwe examine the dependence on the cluster size as well as\nshape.\nC. Continuous-time quantum Monte Carlo method\nIn order to calculate the dynamical spin correlations\nhSp\nj(\u001c)Sp\nj0i, we need to take into account the imaginary-\ntime evolution of f\u0016cgthat compose the conserved quan-\ntitiesf\u0011g, e.g.,Sz\nj(\u001c) =\u0006i\u001fj(\u001c)\u0016cj(\u001c)=p\n2; the sign de-\npends on the sublattice on the honeycomb structure.\nFor this purpose, we adopt the CTQMC method based\non the strong coupling expansion [34]. In this method,\nhSz\nj(\u001c)Sz\nj0ion anr0bond is calculated as\nhSz\nj(\u001c)Sz\nj0i=X\nf\u0011g0;\u0011r0=\u00061P(f\u0011g0;\u0011r0)hSz\nj(\u001c)Sz\nj0if\u0011g0;\n(17)\nwheref\u0011g0represents the con\fgurations of \u0011rexcept for\n\u0011r0on ther0bond.P(f\u0011g0;\u0011r0) is obtained from the5\nconverged solution of the Majorana CDMFT in Sec. II B.\nhSz\nj(\u001c)Sz\nj0if\u0011g0is the dynamical spin correlation on the r0\nbond calculated by the CTQMC method for each con\fg-\nurationf\u0011g0. The sum off\u0011g0runs over all possible con-\n\fgurations off\u0011g0within the cluster. Note that Eq. (17)\nis derived from the fact that Sz\njcommutes with \u0011rin\nf\u0011g0, whereas it does not commute with \u0011r0. Thus, for a\ngivenf\u0011g0, the interaction lies only on the r0bond, and\nhence, it is su\u000ecient to solve the two-site impurity prob-\nlem in the CTQMC calculations. The two-site impurity\nproblem is de\fned by the integration in Eq. (9) on \u001fj;!n\nwhosejdoes not belong to the r0bond. Then, we obtain\nSf\u0011g0\ne\u000b=Sf\u0011g0\nhyb+Slocal; (18)\nwhere\nSf\u0011g0\nhyb=\u0000X\nj;j0Z\f\n0d\u001cZ\f\n0d\u001c0\u001fj(\u001c)\u0001f\u0011g0\nj;j0(\u001c\u0000\u001c0)\u001fj0(\u001c0);\n(19)\nSlocal=X\nj;j0Z\f\n0d\u001c\u001fj(\u001c)\u0012\u000ej;j0\n2@\n@\u001c+Hf\u0011g\nj;j0\u0013\n\u001fj0(\u001c);(20)andj;j0in Eqs. (19) and (20) are the sites on the r0\nbond;\f= 1=Tis the inverse temperature. In Eq. (19),\nthe hybridization function \u0001f\u0011g0\nj;j0(\u001c) is calculated from\nGf\u0011g\nj;j0(i!n) in the converged solution of CDMFT as fol-\nlows. Let us de\fne the matrix ~Gf\u0011g(i!n) as a 2\u00022 sub-\nmatrix ofGf\u0011g(i!n), as\n~Gf\u0011g\nj;j0(i!n) =Gf\u0011g\nj;j0(i!n): (21)\nThen, the hybridization function is given as a function of\nthe Matsubara frequency in the form\n\u0001f\u0011g0\nj;j0(i!n) = [ ~Gf\u0011g(i!n)]\u00001\nj;j0\u0000(i!n\u00002Hf\u0011g\nj;j0):(22)\nNote that \u0001f\u0011g0\nj;j0(i!n) does not depend on f\u0011g, which is\nstraightforwardly shown by the matrix operations in the\nright hand side. Converting Eq. (22) to the imaginary-\ntime representation, we obtain\n\u0001f\u0011g0\nj;j0(\u001c) =T\n2X\nne\u0000i!n\u001c\u0001f\u0011g0\nj;j0(i!n): (23)\nGiven Eqs. (18)-(20), the partition function of the system\nis expanded in terms of Sf\u0011g0\nhybas\nZ\nZlocal=R\nD\u001fe\u0000Sf\u0011g0\nhybe\u0000Slocal\nR\nD\u001fe\u0000Slocal=he\u0000Sf\u0011g0\nhybilocal\n=X\nd;i0;:::;i 2d\u00001Z\f\n0d\u001c0:::Z\f\n0d\u001c2d\u000011\nd!h\u001fi0(\u001c0):::\u001fi2d\u00001(\u001c2d\u00001)ilocalPf(^\u0001f\u0011g0(d;i0;\u001c0;:::;i 2d\u00001;\u001c2d\u00001)); (24)\nwhereZlocal=R\nD\u001fe\u0000Slocalis the partition function for\nthe two sites described by Slocal, andhAilocalrepresents\nthe expectation value in the two-site problem as\nhAilocal=R\nD\u001fAe\u0000Slocal\nR\nD\u001fe\u0000Slocal: (25)\nIn the second line of Eq. (24), dis the order ofSf\u0011g0\nhybin\nthe expansion of e\u0000Sf\u0011g0\nhyb, Pf(M) is the Pfa\u000ean of skew-\nsymmetric matrix M, and ^\u0001f\u0011g0(d;i0;\u001c0;:::;i 2d\u00001;\u001c2d\u00001)\nis a 2d\u00022dmatrix, whose ( m;n) element is given by\n^\u0001f\u0011g0(d;i0;\u001c0:::;i2d\u00001;\u001c2d\u00001)m;n= \u0001f\u0011g0\nim;in(\u001cm\u0000\u001cn):\n(26)\nWe note that this is the \frst formulation of the CTQMC\nmethod with using the Pfa\u000ean in the weight function to\nour knowledge, whereas a QMC simulation in the Ma-\njorana representation has been introduced for itinerant\nfermion models [35].In the CTQMC calculation, we perform MC sam-\npling over the con\fgurations ( d;i0;\u001c0;:::;i 2d\u00001;\u001c2d\u00001)\nby using the integrand in Eq. (24) as the statis-\ntical weight for each con\fguration. In each MC\nstep, we perform an update from one con\fguration\nto another; for instance, an increase of the order\nof expansion das (d;i0;\u001c0;:::;i 2d\u00001;\u001c2d\u00001) to (d+\n1;i0;\u001c0;:::;i 2d\u00001;\u001c2d\u00001;i2d;\u001c2d;i2d+1;\u001c2d+1) by adding\n(i2d;\u001c2d);(i2d+1;\u001c2d+1). To judge the acceptance of such\nan update, we need to calculate the ratio of the Pfaf-\n\fan. This is e\u000eciently done by using the fast update\nalgorithm, as in the hybridization expansion scheme for\nusual fermion problems (for example, see Ref. [36]). For\nthe above example of increasing d, the ratio is calculated\nby adding two rows and columns in the matrix ^\u0001f\u0011g0as\nPf(^\u0001f\u0011g0(d;i0;\u001c0;:::;i 2d\u00001;\u001c2d\u00001))\nPf(^\u0001f\u0011g0(d+ 1;i0;\u001c0;:::;i 2d+1;\u001c2d+1)); (27)\nwhose calculation cost is in the order of d2by using the\nfast update algorithm. On the other hand, in Eq. (24),6\nhT\u001c\u001fi0(\u001c0):::\u001fi2d\u00001(\u001c2d\u00001)ilocal is obtained as the aver-\nage in the two-site problem, which can be calculated by\nconsidering the imaginary-time evolution of all the fourstates in the two-site problem.\nThen, the dynamical spin correlation for the con\fgu-\nrationf\u0011g0,hSz\nj(\u001c)Sz\nj0if\u0011g0in Eq. (17), is calculated as\nhSz\nj(\u001c)Sz\nj0if\u0011g0=Zlocal\nZX\nd;i0;:::;i 2d\u00001Z\f\n0d\u001c0:::Z\f\n0d\u001c2d\u000011\nd!h\u001fi0(\u001c0):::\u001fi2d\u00001(\u001c2d\u00001)Sz\nj(\u001c)Sz\nj0ilocal\n\u0002Pf(^\u0001f\u0011g0(d;i0;:::;i 2d\u00001;\u001c0;:::;\u001c 2d\u00001)): (28)\nFor the MC sampling, we need to evaluate\nhT\u001c\u001fi0(\u001c0):::\u001fi2d\u00001(\u001c2d\u00001)Sz\nj(\u001c)Sz\nj0ilocal\nhT\u001c\u001fi0(\u001c0):::\u001fi2d\u00001(\u001c2d\u00001)ilocal: (29)\nThis is again calculated by considering the imaginary-\ntime evolution of all the four states in the two-site prob-\nlem. In the isotropic case with \u000b= 1:0,hSp\nj(\u001c)Sp\nj0ifor\np=x;yare equivalent to hSz\nj(\u001c)Sz\nj0i. Meanwhile, for the\nanisotropic case, we compute hSp\nj(\u001c)Sp\nj0iforp=x;yby\nthe same technique described above with using the spin\nrotationsfSx;Sy;Szg!fSy;Sz;SxgorfSx;Sy;Szg!\nfSz;Sx;Syg.\nIn the CTQMC calculations in Sec. III, for each con\fg-\nurationf\u0011g0, we typically run 107MC steps and perform\nthe measurements at every 20 steps, after 105MC steps\nfor the initial relaxation.\nD. Maximum entropy method\nBy using the CTQMC method as the impurity solver\nin the CDMFT, which we call the CDMFT+CTQMC\nmethod, we can numerically estimate the dynamical\nspin correlation as a function of the imaginary time,\nhSp\nj(\u001c)Sp\nj0i. To obtain the physical observables, such as\nthe dynamical spin structure factor and the NMR re-\nlaxation rate, which are given by the dynamical spin\ncorrelations as functions of frequency !, we need to in-\nversely solve the equation given by the generic form\ng(\u001c) =R\nd!\u001a(!)e\u0000!\u001c. In our problem, g(\u001c) and\u001a(!)\ncorrespond to the dynamical spin correlations as func-\ntions of imaginary time \u001cand real frequency !:g(\u001c) =\nhSp\nj(\u001c)Sp\nj0iand\u001a(!) =Sp\nj;j0(!). In the following cal-\nculations, we utilize the Legendre polynomial expansion\nfollowing Refs. [37, 38]:\ngm=p\n2m+ 1Z\f\n0d\u001cPm(x(\u001c))g(\u001c); (30)\nwherePm(x) is themth Legendre polynomials and\nx(\u001c) = 2\u001c=\f\u00001. Then, the inverse problem is given\nby\ngm=Z\nd!\u001a(!)Km(!); (31)where\nKm(!) =p\n2m+ 1Z\f\n0d\u001cPm(x(\u001c))e\u0000!\u001c: (32)\nFor solving the inverse problem, we adopt the maxi-\nmum entropy method (MEM) [39]. The following pro-\ncedure is the standard one, but we brie\ry introduce it\nto make the paper self-contained. In the MEM, we dis-\ncretize\u001a(!) to\u001al=\u001a(!l), and determine \u001alto minimize\nthe function\nF=1\n2X\nm;n(gm\u0000~gm)\u0010\u00001C\u00001\nm;n(gn\u0000~gn)\n\u0000\u000eX\nl\"\n\u001al\u0000\u001a(0)\nl\u0000\u001alln \n\u001al\n\u001a(0)\nl!#\n; (33)\nwhere\u0010and\u000eare the coe\u000ecients described below,\nandCis a variance-covariance matrix of gm; ~gm=P\nl\u0001!\u001alKm(!l). We take the Legendre expansion up\nto 50th order and \u0001 != 0:01125 in the following calcu-\nlations. In Eq. (33), \u001a(0)\nlis the advance estimate of \u001al,\nwhich we set to be a constant in this study.\nOnce neglecting the second term in the right hand side\nof Eq. (33), the minimization of Fis equivalent to the\nleast squares method. The least squares method is un-\nstable, asgmis rather insensitive to a change of \u001al. The\nsecond term, called the entropy term, stabilizes the min-\nimization process. In the following calculations, we set\n\u0010= 625 to su\u000eciently take into account the e\u000bect of the\nentropy term, where the value of \u000eis determined self-\nconsistently in each MEM calculation based on the max-\nimum likelihood estimation, called the classical MEM [39]\n(typically,\u000e'1-10). We note that the deviations of ~ gm\nfromgmare typically comparable to the statistical errors\nin the CTQMC calculations. In the following results, we\nestimate the errors of \u001a(!) by the standard deviation be-\ntween the data for \u0010= 100, 625 and 10000 in the range\nwhere the MEM retains the precision.\nIn the MEM, \u001a(!) should be positive for all !. In our\nproblem, the onsite correlation Sp\nj;j(!) satis\fes this con-\ndition automatically, whereas Sp\nj;j0(!) for the NN sites\nj;j0on thepbond, which is denoted by Sp\nNN(!) hereafter,\ncan be negative. (Note that all the further-neighbor7\ncorrelations beyond the NN sites vanish in the Kitaev\nmodel [23].) To obtain Sp\nNN(!) properly, we calculate\nSp\nj;j(!) + 2Sp\nj;j0(!) +Sp\nj0;j0(!), which is positive de\fnite\nfor all!, and subtract the onsite contributions [40]. The\naccuracy of Sp\nj;j0(!) obtained by the MEM are examined\nin Appendix B in the one-dimensional limit with \u000b= 1:5,\nwhereSp\nj;j0(!) can be calculated without using the MEM.\nIII. RESULT\nIn this section, we present the results obtained by\nthe CDMFT and the CDMFT+CTQMC methods. In\nSec. III A, we present the speci\fc heat and equal-time\nspin correlations for the NN sites obtained by the\nCDMFT for the cases with anisotropic Jx,Jy, and\nJz. By comparing the results with those by the QMC\nmethod [10], we con\frm that the CDMFT is valid in the\nTrange above the arti\fcial critical temperature close\nto the low- Tcrossover. In Sec. III B, III C, and III D,\nwe present the CDMFT+CTQMC results for dynami-\ncal quantities, i.e., the dynamical spin structure factor,\nthe NMR relaxation rate, and the magnetic susceptibil-\nity, respectively, in the quali\fed Trange. We discuss the\nresults in comparison with the isotropic case reported\npreviously in Ref. [18].\nA. Static quantities: comparison to the previous\nQMC results\nFigure 2 shows the benchmark of the Majorana\nCDMFT. We compare the speci\fc heat Cvand equal-\ntime spin correlations for NN pairs on the pbonds,\nhSp\njSp\nj0iNN, obtained by the Majorana CDMFT, with\nthose by QMC in Ref. [10]. The data are calculated for\nthe FM case with bond asymmetry: \u000b= 0:8 (Jx=Jy=\n0:8 andJz= 1:4) and\u000b= 1:2 (Jx=Jy= 1:2 and\nJz= 0:6). While the data of Cvare common to the\nFM and AFM cases, the sign of hSp\njSp\nj0iNNis reversed for\nthe AFM case. Note that similar comparison was made\nfor the isotropic case \u000b= 1:0 (Jx=Jy=Jz= 1) in\nRef. [18].\nAs indicated by two broad peaks in the speci\fc heat\nin the QMC results, the system exhibits two crossovers\nowing to thermal fractionalization of quantum spins [10];\nthe crossover temperatures were estimated as TL'0:012\nandTH'0:375 in the isotropic case. In the anisotropic\ncases, the low- Tcrossover takes place at a lower T,\ni.e.,TL'0:0052 for\u000b= 0:8 andTL'0:0075 for\n\u000b= 1:2, while the high- Tone is almost unchanged, i.e.,\nTH'0:375. These behaviors are excellently reproduced\nby the Majorana CDMFT, except for the low- Tpeak; the\nCDMFT results show a sharp anomaly at ~Tc'0:0063 for\n\u000b= 0:8 and ~Tc'0:013 for\u000b= 1:2. This is due to a\nphase transition by ordering of \u0011, which is an artifact of\nthe mean-\feld nature of CDMFT.\n0.00.20.40.60.81.0\n0.00.20.40.60.8\n10-210-1100\nT(a)\n(b)CDMFT QMC\nCDMFT QMCFIG. 2. The speci\fc heat Cvand equal-time spin correla-\ntions for the NN sites, hSz\njSz\nj0iNNandhSx\njSx\nj0iNN, obtained by\nthe Majorana CDMFT for the FM case at (a) \u000b= 0:8 and (b)\n\u000b= 1:2. Note thathSx\njSx\nj0iNN=hSy\njSy\nj0iNNfrom the symme-\ntry. QMC data in Ref. [10] are plotted by gray symbols for\ncomparison.\nOn the other hand, the QMC results for the NN spin\ncorrelations are also precisely reproduced by the Majo-\nrana CDMFT in the wide Trange above the arti\fcial\nphase transition temperature ~Tc. Although they appear\nto be reproduced even below ~Tc, there is a small anomaly\nat~Tcassociated with the arti\fcial transition, while the\nQMC data smoothly change around TL. (Note that the\nappropriate sum of the NN spin correlations is nothing\nbut the internal energy, and hence, the Tderivative cor-\nresponds to the speci\fc heat.)\nThus, the comparison indicates that the Majorana\nCDMFT gives quantitatively precise results in the wide\nTrange above the arti\fcial transition temperature ~Tc: in\nthe present cases with \u000b= 0:8 and 1:2, the CDMFT is\nreliable for T&0:007 andT&0:014, respectively. As\ndiscussed in the previous study [10], the thermal fraction-\nalization of quantum spins sets in below T'TH, which\nis well above ~Tc. Thus, the Tranges quali\fed for the\nCDMFT include the peculiar paramagnetic state show-\ning the thermal fractionalization. In the following sec-\ntions, we apply the CDMFT+CTQMC method in these\nquali\fedTranges to the study of spin dynamics, which\nwas not obtained by the previous QMC method [10].8\n(a)\n(d)\n(g)\n(j)\nM2 ΓK1 M1K1 K2 Γ(b)\n(e)\n(h)\n(k)\nM2 ΓK1 M1K1 K2 Γ(c)\n(f)\n(i)\n(l)\nM2 ΓK1 M1K1 K2 Γ M2 M2 M2\nFIG. 3. Dynamical spin structure factor S(q;!) obtained by the Majorana CDMFT+CTQMC method for the FM case\nwith (a)(d)(g)(j) \u000b= 1:0, (b)(e)(h)(k) \u000b= 0:8, and (c)(f)(i)(l) \u000b= 1:2: (a)(b)(c) T'2TL, (d)(e)(f)T'pTLTH, (g)(h)(i)\nT'0:64TH, and (j)(k)(l) T'6:4TH. Here,TL'0:012, 0:0052, and 0 :0075 for\u000b= 1:0, 0:8, and 1:2, respectively, while\nTH'0:375 for all the cases.\nB. Dynamical spin structure factor\nFigure 3 shows the CDMFT+CTQMC results for the\ndynamical spin structure factor S(q;!) at several Tfor\nthe FM case with \u000b= 1:0, 0:8, and 1:2.S(q;!) is calcu-\nlated as\nS(q;!) =X\npSp(q;!); (34)\nSp(q;!) =1\n3NX\nj;j0eiq\u0001(rj\u0000rj0)Sp\nj;j0(!); (35)whereSp\nj;j0(!) is obtained by the MEM described in\nSec. II D from the imaginary-time correlations hSp\nj(\u001c)Sp\nj0i\nby CDMFT+CTQMC. As mentioned above, nonzero\ncontributions in Eq. (35) come from only the onsite and\nNN-site components of Sp\nj;j0(!); we present their Tde-\npendences in Appendix C. The Brillouin zone and sym-\nmetric lines on which S(q;!) is plotted are presented in\nFig. 1(b). Although the results at \u000b= 1:0 were shown\nin the previous study [18], we present them (for a sightly\ndi\u000berentTset) for comparison. We show the data at\nfour temperatures: T'2TL,pTLTH, 0:64TH, and 6:4TH.\nNote thatTL'0:012, 0:0052, and 0 :0075 for\u000b= 1:0,9\n0:8, and 1:2, respectively, while TH'0:375 for all the\ncases.\nAs shown in Fig. 3, at su\u000eciently high TthanTH,\nS(q;!) does not show any signi\fcant qdependence for\nall\u000bstudied here; S(q;!) shows only a di\u000busive re-\nsponse centered at !\u00180, as shown in Figs. 3(j)-3(l).\nWhen lowering TbelowTH, the di\u000busive weight is shifted\nto the positive !region ranging up to above !\u0018J\nfor all the cases, as shown in Figs. 3(g)-3(i). Simulta-\nneously, a quasi-elastic component grows gradually at\n!\u00180. Both the inelastic and the quasi-elastic com-\nponents show a discernible qdependence; in particular,\nthe latter increases the intensity around the \u0000 point re-\n\recting the FM interactions. While S(K1;!) =S(K2;!)\nandS(M1;!) =S(M2;!) for\u000b= 1:0 from the symme-\ntry, the quasi-elastic response is small (large) around the\nM1-K1line compared to that around the M 2-K2line for\n\u000b= 0:8 (1:2) because of the anisotropy.\nWhen further lowering Tand approaching TL, the\nquasi-elastic component increases its intensity, while the\ninelastic response at !\u0018Jdoes not change substantially.\nIn particular, in the case of \u000b= 0:8, the quasi-elastic\ncomponent is sharpened and develops to a \u000e-function like\npeak as shown in Figs. 3(e) and 3(b). In addition, the\nbroad incoherent weight splits from the coherent peak.\nThese behaviors appear to asymptotically converge onto\nthe result at T= 0, where the \u000e-function peak appears\ndue to the change of the parity between the ground state\nand the \rux-excited state [11] (for the \u000e-function peak,\nsee also Fig. 16 in Appendix B). On the other hand,\nS(q;!) at\u000b= 1:2 does not show such a drastic change,\nand the quasi-elastic component grows continuously, as\nshown in Figs. 3(f) and 3(c). We note that the results\nfor\u000b= 1:2 are qualitatively similar to those for \u000b= 1:0\nin Figs. 3(d) and 3(a), except for di\u000berent qdependence\nmentioned above.\nFigure 4 shows the results for the AFM case. The over-\nall!dependence of S(q;!) is similar to that for the FM\ncase at allT: the di\u000busive response centered at !\u00180 for\nT&TH[Figs. 4(j)-4(l)], the shift of the di\u000busive weight\nto the region of !\u0018Jand the growth of a quasi-elastic\ncomponent at !\u00180 belowTH[Figs. 4(g)-4(i)], and the\n\u000e-function like peak for \u000b= 0:8 while approaching to\nTL[Figs. 4(e) and 4(b)]. The similarity of the !depen-\ndences ofS(q;!) between FM and AFM cases is partly\nunderstood by the relation 2 S(K1;!)FM+S(K2;!)FM=\n2S(K1;!)AFM +S(K2;!)AFM, which holds for Jx=Jy\n[S(q;!)FMandS(q;!)AFM areS(q;!) for the FM and\nAFM cases, respectively]. On the other hand, the qde-\npendence is in contrast to the FM case: while the weight\nof the quasi-elastic response almost vanishes around the\n\u0000 point, those on the zone boundary are enhanced in\nan almost opposite manner to the FM cases. In addi-\ntion, the incoherent weight at !\u0018Jalso shows the\nopposite qdependence to the FM case: the weight\nis stronger around the \u0000 point than that on the zone\nboundary. The opposite qdependences between the\nFM and AFM cases directly follow from the relationS(q;!)AFM =\u0000S(q;!)FM+ (2=3)P\npSp\nj;j(!).\nIn order to show the Tdependences of S(q;!) more\nexplicitly, we present in Figs. 5-8 the T-!plot ofS(q;!)\natq= \u0000, K 1, and K 2with the intensity pro\fles for the\nsame set of Tused in Figs. 3 and 4. Figure 5 shows the\nresult for the FM case at \u000b= 0:8. The overall weight\nofS(q;!) shifts from !\u00180 to a large- !region when\nthe system is cooled down below T\u0018TH. BelowTH,\nquasi-elastic response gradually grows and develops to\nthe\u000e-function like peak. The peak intensity in S(\u0000;!)\nandS(K2;!) is larger than that for S(K1;!), re\recting\nthe anisotropy of the interaction.\nFigure 6 shows the corresponding plot for the AFM\ncase at\u000b= 0:8. In contrast to the FM case, the strong\nquasi-elastic response is seen for q= K 1, which develops\nto the\u000e-function like peak at low T. We note that the dip\nand shoulder like structures around != 0 in the interme-\ndiateTfor the result at q= K 2may be an artifact origi-\nnating from low precision in the MEM for this AFM case\nbecause of the following reason. As described in Sec. II D,\nwe calculate Sp\nj;j0(!) for the NN bonds by subtracting the\nonsite component Sp\nj;j(!) fromSp\nj;j(!)+Sp\nj;j0(!), both of\nwhich are obtained by the MEM. In the present case, as\nboth ofSz\nj;j(!) andSz\nj;j(!)+Sz\nj;j0(!) become large around\n!= 0 due to the development of the \u000e-function like peak,\nthe relative error becomes large for S(K2;!\u00180), which\nmay lead to arti\fcial structures.\nFigures 7 and 8 show the results at \u000b= 1:2. As ob-\nserved in Figs. 3 and 4, S(q;!) for both the FM and\nAFM cases behave similarly to those at \u000b= 1:0 [18].\nIn the anisotropic cases, however, the di\u000berence between\nS(K1;!) andS(K2;!) is obvious: the quasi-elastic peak\nforS(K1;!) is larger (smaller) than that for S(K2;!) in\nthe FM (AFM) case.\nAs discussed in the previous study [18], there is a rela-\ntion between the static spin correlation and the average\nfrequency of S(\u0000;!), \u0016!\u0011R\n!S(\u0000;!)d!=R\nS(\u0000;!)d!,\noriginating from the sum rule for S(q;!).Tdependences\nof \u0016!are shown by white dashed curves in Figs. 5(b),\n6(b), 7(b), and 8(b). In all cases, \u0016 !is nearly zero for\nsu\u000eciently high T, but it grows at T\u0018THand becomes\nalmost independent of TforT.TH. TheseTdepen-\ndences are similar to those of the static spin correlation\nbetween the NN sites shown in Fig. 2.\nC. NMR relaxation rate\nFigure 9 shows the NMR relaxation rate 1 =T1obtained\nby the CDMFT+CTQMC method. While the results at\n\u000b= 1:0 were presented in the previous study [18], we\npresent them for comparison in Fig. 9(a). 1 =T1in the\nmagnetic \feld applied to the zdirection, which is denoted\nby 1=Tz\n1, is given by\n1=Tz\n1/TX\nqjAqj2Im\u001f?(q;!0)\n!0; (36)10\n(a)\n(d)\n(g)\n(j)\nM2 ΓK1 M1K1 K2 Γ(b)\n(e)\n(h)\n(k)\nM2 ΓK1 M1K1 K2 Γ(c)\n(f)\n(i)\n(l)\nM2 ΓK1 M1K1 K2 Γ M2 M2 M2\nFIG. 4. Dynamical spin structure factor S(q;!) obtained by the Majorana CDMFT+CTQMC method for the AFM case.\nThe values of \u000bandTare common to Fig. 3.\nwhereAqis the hyper\fne coupling constant, \u001f?(q;!)\nis the dynamical susceptibility for the spin component\nperpendicular to the magnetic \feld direction, and !0\nis the resonance frequency in the NMR measurement.\nThe dynamical susceptibility \u001f(q;!) is related with the\ndynamical spin structure factor through the \ructuation-\ndissipation theorem, as\nS(q;!) =1\n\u0019(1\u0000e\u0000\f!)Im\u001f(q;!): (37)\nIn the NMR experiments, !0is in general negligibly small\ncompared to the typical energy scale of the system, J.\nThus, by taking the limit of !0!0 in Eq. (36) andusing Eq. (37), we obtain\n1=Tz\n1=a0Sx\nj;j(!= 0) +a1Sy\nj;j(!= 0)\n+a2Sx\nNN(!= 0) +a3Sy\nNN(!= 0); (38)\nwhere the coe\u000ecients a0,a1,a2, anda3are determined\nbyAq. The similar equations are obtained for 1 =Tx\n1and\n1=Ty\n1by the cyclic permutation of x;y;z (1=Tx\n1= 1=Ty\n1\nfor the present cases from the symmetry). Because Aq\ndepends on the details of the system, we here compute\nthe onsite and NN-site components of 1 =T1separately\nwith omitting the coe\u000ecients: the onsite components are11\n(a)\n(c) (d)(b)\n(e)(f)0.00.20.40.60.81.01.2\n-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0\nω0.00.20.40.60.81.01.2\n-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0\nω\n0.00.20.40.60.81.01.2\n-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0\nω0.00.20.40.60.81.01.2\n-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0\nω0.00.20.40.60.81.01.2\n-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0\nω0.00.20.40.60.81.01.2\n-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0\nω\nFIG. 5. (a) S(\u0000;!), (c)S(K1;!), and (e)S(K2;!) for the\nFM case with \u000b= 0:8 at several T. The corresponding con-\ntour plots in the T-!plane are shown in (b)(d)(f). The arrows\nindicate the temperatures used for the data in (a)(c)(e), while\nthe white and gray dotted lines indicate THandTL, respec-\ntively. Note that the Tset is common to that used in Figs. 3\nand 4. The dashed curve in (b) represent the average fre-\nquency ofS(\u0000;!) (see the text for details). In (a)(c)(e), the\nerrorbars are shown for every ten data along the !axis.\ncalculated as\n1=Tz\n1=Sx\nj;j(!= 0) +Sy\nj;j(!= 0); (39)\n1=Tx\n1=Sy\nj;j(!= 0) +Sz\nj;j(!= 0); (40)\nwhile the NN-site ones are\n1=Tz\n1=\u0006(Sx\nNN(!= 0) +Sy\nNN(!= 0)); (41)\n1=Tx\n1=\u0006(Sy\nNN(!= 0) +Sz\nNN(!= 0)); (42)\nwhere the sign is +( \u0000) for the FM (AFM) case. We note\nthat, in the anisotropic cases \u000b6= 1:0, the NN-site 1 =Tx\n1\nis not simply given by the sum in Eq. (42): it will be given\nby a linear combination of Sy\nNN(!= 0) andSz\nNN(!= 0)\nwith appropriate coe\u000ecients determined by Aq. Such\na linear combination, however, can be constructed from\nour data for Eqs. (41) and (42) by noting that 1 =Tz\n1=\n2Sy\nNN(!= 0) forJx=Jy. Hence, we present the results\nby Eqs. (41) and (42) in Fig. 9 for simplicity.\nAs shown in Fig. 9, for all cases, the onsite component\nof 1=Tp\n1is nonzero and almost Tindependent above T\u0018\nTH, as expected for the conventional paramagnets [41].\nOn the other hand, the NN-site component is zero in the\nhigh-Tlimit and increases as decreasing T. This behavior\n(a)\n(c) (d)(b)\n(e)(f)\n0.00.20.40.60.81.01.2\n-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0\nω0.00.20.40.60.81.01.2\n-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0\nω0.00.20.40.60.81.01.2\n-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0\nω0.00.20.40.60.81.01.2\n-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0\nω0.00.20.40.60.81.01.2\n-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0\nω0.00.20.40.60.81.01.2\n-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0\nωFIG. 6. (a) S(\u0000;!), (c)S(K1;!), and (e)S(K2;!) for the\nAFM case with \u000b= 0:8 at several T. The corresponding\ncontour plots in the T-!plane are shown in (b)(d)(f). The\nnotations are common to those in Fig. 5.\ncorresponds to the development of NN-site static spin\ncorrelations shown in Sec. III A, as they have a relation\nthrough the sum rule,R\nSp\nj;j0(!)d!=hSp\njSp\nj0i.\nWhen lowering TbelowTH, 1=Tx\n1for\u000b= 0:8 substan-\ntially increases, as shown in Fig. 9(b). The enhancement\nis much larger than the case of \u000b= 1:0 in Fig. 9(a).\nThis is due to the evolution of the \u000e-function like peak in\nSz(q;!) discussed in Sec. III B. In contrast, Sx(q;!) and\nSy(q;!) do not develop such \u000e-function like peaks, and\nhence, 1=Tz\n1does not show enhancement unlike 1 =Tx\n1.\nWhile further decreasing T, 1=Tx\n1shows a peak slightly\naboveTL. The decrease at low Tre\rects a spin gap orig-\ninating from the nonzero \rux gap in the ground state [8].\nOn the other hand, the onsite and NN-site components of\n1=Tz\n1are both suppressed below T\u0018TH, after showing\na plateau and broad peak, respectively. The suppression\nof 1=Tz\n1is due to an increase of energy cost for a spin \rip\non the strong zbond under the well-developed static spin\ncorrelations between NN sites in this Trange. Actually,\nthe energy cost is represented by the average frequency\nofSx\nj;j(!) as there is a relation\n\u0016!x\nonsite =R\n!Sx\nj;j(!)d!R\nSx\nj;j(!)d!\n=P\nm;ne\u0000\fEn(Em\u0000En)jhmjSx\njjnij2\n1\n4P\nne\u0000\fEn:(43)12\n(a)\n(c) (d)(b)\n(e)(f)\n0.00.20.40.60.81.01.2\n-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0\nω0.00.20.40.60.81.01.2\n-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0\nω0.00.20.40.60.81.01.2\n-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0\nω0.00.20.40.60.81.01.2\n-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0\nω0.00.20.40.60.81.01.2\n-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0\nω0.00.20.40.60.81.01.2\n-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0\nω\nFIG. 7. (a) S(\u0000;!), (c)S(K1;!), and (e) S(K2;!) for\nthe FM case with \u000b= 1:2 at several T. The corresponding\ncontour plots in the T-!plane are shown in (b)(d)(f). The\nnotations are common to those in Fig. 5.\nOn the other hand, \u0016 !x\nonsite is also written as \u0016 !x\nonsite =\n(JyhSy\njSy\nj0iNN+JzhSz\njSz\nj0iNN)=2 by the sum rule [18].\nThus, the energy cost becomes large below THaccord-\ning to the growth of hSz\njSz\nj0iNN.\nIn contrast, as shown in Fig. 9(c), Tdependence of\n1=Tp\n1at\u000b= 1:2 is similar to that at \u000b= 1:0 in Fig. 9(a).\nBoth 1=Tx\n1and 1=Tz\n1increase below THwhile decreasing\nT, in contrast to the case with \u000b= 0:8. For\u000b= 1:2,\nhowever, 1=Tz\n1is larger than 1 =Tx\n1, re\recting the stronger\ninteractions on the xandybonds than the zbond. On\nfurther decreasing T, 1=Tp\n1at\u000b= 1:2 also shows the\npeak structure slightly above TLand then decreases, as\nexpected from the \fnite \rux gap in the ground state.\nAlthough the system is described by free Majorana\nfermions coupled to localized gauge \ruxes, the NMR re-\nlaxation rate does not obey the Korringa law, 1 =(T1T)\u0018\nconstant, which is expected for free fermion systems.\nThis is natural because the spin-\rip excitation in the\nNMR process is a composite of both itinerant matter\nfermions and localized gauge \ruxes. Nonetheless, for\ncomparison to forth-coming experiments, we plot the Ko-\nrringa ratio as a function of Tin Appendix D.\n(a)\n(c) (d)(b)\n(e)(f)\n0.00.20.40.60.81.01.2\n-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0\nω0.00.20.40.60.81.01.2\n-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0\nω0.00.20.40.60.81.01.2\n-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0\nω0.00.20.40.60.81.01.2\n-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0\nω0.00.20.40.60.81.01.2\n-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0\nω0.00.20.40.60.81.01.2\n-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0\nωFIG. 8. (a) S(\u0000;!), (c)S(K1;!), and (e)S(K2;!) for the\nAFM case with \u000b= 1:2 at several T. The corresponding\ncontour plots in the T-!plane are shown in (b)(d)(f). The\nnotations are common to those in Fig. 5.\nD. Magnetic susceptibility\nFigures 10 and 11 show the Tdependences of the mag-\nnetic susceptibility \u001fpfor the FM and AFM cases, respec-\ntively.\u001fpat\u000b= 1:0 presented in the previous study [18],\nare also presented in Fig. 10(a) and Fig. 11(a) for com-\nparison.\u001fpis calculated from the imaginary-time spin\ncorrelations as\n\u001fp=1\nNX\nj;j0Z\f\n0d\u001chSp\nj(\u001c)Sp\nj0i: (44)\nNote that this is obtained without the MEM. In all the\ncases, at su\u000eciently high Tcompared to the dominant\nJp,Jmax\np= maxfJpg,\u001fpobeys the Curie-Weiss law,\n\u001fp\nCW=1\n4T\u0000Jmaxp; (45)\nwhich is obtained by the standard mean-\feld approxima-\ntion in the original spin representation. While decreasing\nT,\u001fpshows a deviation from \u001fp\nCWbelowT\u0018Jmax\np.\nAmong the results, \u001fxfor the FM case and \u001fzfor the\nAFM case at \u000b= 0:8 show peculiar Tdependences at low\nT. The former largely deviates from the Curie-Weiss be-\nhavior and saturates to a small nonzero value, as shown\nin Fig. 10(b) [42]. Meanwhile, the latter shows a broad\nhump atT\u0018THand decreases as lowering T, as shown13\n(a)\n(b)\n(c)\n10-210-1100\n10-210-110010-310-210-11001011 /T1p10-210-11001 /T1ponsite\nNN-site\nzx\nonsite\nNN-site\nzx\nonsite\nNN-site1 /T1p\nT\nFIG. 9.Tdependences of the NMR relaxation rate 1 =Tp\n1\n(p=z;x) at (a)\u000b= 1:0, (b)\u000b= 0:8, and (c)\u000b= 1:2. Note\nthat 1=Tz\n1= 1=Tx\n1for\u000b= 1:0 and 1=Tx\n1= 1=Ty\n1for all the\ncases from the symmetry. The vertical dotted lines indicate\nTLandTHfor each\u000b.\nin Fig. 11(b). These Tdependences are qualitatively un-\nderstood by considering a two-site dimer model on the z\nbond obtained by setting Jx=Jy= 0. The dimer model\ngives the analytical forms for the magnetic susceptibility\nas\n\u001fz\ndimer =\f\n2exp(\fJz=4)\nexp(\fJz=4) + exp(\u0000\fJz=4); (46)\n\u001fx\ndimer =1\nJztanh\u0012\n\fJz\n4\u0013\n: (47)\nThe results are plotted by the dashed-dotted curves in\nFigs. 10(b) and 11(b). \u001fx\ndimer for the FM case almost\nsaturates around T\u0018Jz=4, as the dominant Jzinter-\naction suppresses the magnetization in the xdirection.\nThis accounts for the behavior of \u001fxin Fig. 10(b) quali-\ntatively. Meanwhile, \u001fz\ndimer also well reproduces a hump\nzx(a)\n(b)\nzx (c)01234567\n010203040\n0.00.51.01.52.0\n0246810\n10-210-1100\nTxpxpxpFIG. 10. Tdependences of the magnetic susceptibility \u001fp\n(p=z;x) at (a)\u000b= 1:0, (b)\u000b= 0:8, and (c)\u000b= 1:2 for the\nFM case. The dashed curves represent \u001fp\nCWin Eq. (45). The\nred and black dashed-dotted curves in (b) represent \u001fp\ndimer for\np=z[Eq. (46)] and p=x[Eq. (47)], respectively. Note that\n\u001fz=\u001fxfor\u000b= 1:0 and\u001fx=\u001fyfor all the cases from the\nsymmetry. The vertical dotted lines indicate TLandTHfor\neach\u000b.\natT\u00180:5 in\u001fzfor the AFM case in Fig. 11(b); \u001fz\nremains nonzero down to low Tas nonzero JxandJy\nsmear out the dimer gap.\nIn the case of \u000b= 1:2,Tdependences of \u001fpshown\nin Figs. 10(c) and 11(c) are similar to those for \u000b=\n1:0 in the previous study [18] replotted in Figs. 10(a)\nand 11(a), respectively; on decreasing T,\u001fpcontinues to\nincrease down to T\u0018TLin the FM case, whereas \u001fp\nshows broad peak at a higher Tin the AFM case. The\ne\u000bect of anisotropic Jp, however, is clearly observed: the\nstronger interactions on the x;ybonds than the zbond\nresult in larger (smaller) \u001fxthan\u001fzin the FM (AFM)\ncase. In addition, the temperature of the broad peak of\n\u001fz(\u001fx) in the AFM case shifts to a lower (higher) Tthan14\nzx(a)\n(c) zx(b)0.00.10.20.30.40.5\n0.00.10.20.30.40.50.6\n0.00.10.20.30.40.50.6\n10-210-1100\nTxpxpxp\nFIG. 11. Tdependences of the magnetic susceptibility \u001fp\n(p=z;x) at (a)\u000b= 1:0, (b)\u000b= 0:8, and (c)\u000b= 1:2 for the\nAFM case. The notations are common to those in Fig. 10.\nthat for\u000b= 1:0.\nIV. DISCUSSION\nAs pointed out in the previous study for the isotropic\ncase by the authors [18] and con\frmed also for the\nanisotropic cases in the present study, a remarkable fea-\nture in the Kitaev model is the dichotomy between the\ndynamical and static spin correlations; namely, the NMR\nrelaxation rate 1 =Tp\n1and the magnetic susceptibility,\nboth of which re\rect the dynamical spin correlations,\nshow substantial Tdependences below TH(Figs. 9-11),\neven though the static spin correlations hSp\njSp\nj0iNNalmost\nsaturate to the T= 0 values (Fig. 2). The dichotomy\nis unconventional behavior hardly seen in conventional\ninsulating magnets. This might be a signature of thefractionalization of quantum spins, as THis the temper-\nature where the fractionalization sets in as indicated in\nthe speci\fc heat and entropy [10].\nTo examine the dichotomy in more detail, we calcu-\nlate theTdependences of 1 =Tp\n1andhSp\njSp\nj0iNNfor two\nextreme cases by assuming the con\fguration of f\u0011gby\nhand. One is the \rux-free state with all \u0011r= +1, which\nis realized in the ground state. The other is the state\nwith completely random f\u0011g, corresponding to the high-\nTlimit. For this purpose, we regard a single zbondr0\nas the cluster in CDMFT, and take P(\u0011r0= 1) = 1 and\nP(\u0011r0=\u00001) = 0 for the former uniform state, while\nP(\u0011r0= 1) =P(\u0011r0=\u00001) = 1=2 for the latter ran-\ndom state, in Eq. (12) of the self-consistent equation of\nCDMFT [43].\nFigure 12 shows the results. In all cases, hSp\njSp\nj0iNN\nfor both uniform and random f\u0011gshows almost simi-\nlarTdependence to the CDMFT results, as shown in\nFigs. 12(a)-12(c). However, 1 =Tp\n1exhibits considerably\ndi\u000berentTdependence. For instance, in the isotropic\ncase with\u000b= 1:0, although 1 =T1is almostTindepen-\ndent forT >T Hfor both uniform and random f\u0011gsim-\nilar to the result by the CDMFT+CTQMC method in\nRef. [18], it shows di\u000berent behavior below THbetween\nthe two cases, as shown in Fig. 12(d). For the case with\nuniformf\u0011g, 1=T1decreases to zero after showing a small\nhump. The suppression at low Tre\rects the \rux gap\n\u0001'0:065Jin the \rux-free state [8, 11]. On the other\nhand, for the case with random f\u0011g, 1=T1monotonically\nincreases while decreasing Tin the calculated Trange.\nSimilarTdependences of 1 =Tp\n1are obtained for 1 =Tx\n1at\n\u000b= 0:8 and 1=Tx;z\n1at\u000b= 1:2, as shown in Figs. 12(e)\nand 12(f), respectively. We note that 1 =Tzfor\u000b= 0:8\nbehaves di\u000berently; we will comment on this point in the\nend of this section.\nThe results clearly indicate that the peculiar Tde-\npendences of 1 =T1found in the CDMFT+CTQMC re-\nsults are closely related with \ructuations of the gauge\n\ruxesf\u0011gcomposed of localized Majorana fermions f\u0016cg\nemergent from the spin fractionalization. As seen in\nthe equal-time spin correlations shown in Figs. 12(a)-\n12(c), itinerant matter fermions develop their kinetic en-\nergy to the saturation at T\u0018TH(the equal-time spin\ncorrelations correspond to the kinetic energy of mat-\nter fermions). Due to the fractionalization, however,\nthe localized gauge \ruxes are still disordered even below\nTH[10], which results in the enhancement of 1 =T1, as in-\ndicated in Figs. 12(d)-12(f). When approaching TL,f\u0011g\nare aligned in a coherent manner [10], and hence, 1 =T1is\nrapidly suppressed at T\u0018TL. Thus, the Tdependence\nof 1=T1is qualitatively explained by the crossover from\nthat for the random f\u0011gto the fully-aligned f\u0011gwhile\ndecreasing T. The crossover occurs well below THand\nclose toTL. Of course, as the original quantum spin is\na composite of itinerant matter fermions and localized\ngauge \ruxes, the spin-\rip dynamics is a composite exci-\ntation. Nevertheless, our results indicate that the pecu-\nliarTdependence of the NMR relaxation rate as well as15\nT T T(a) (b) (c)\n(d) (e) (f)z x\nuniform\nrandom\nCDMFT\n10-210-110010-210-11000.00.20.40.60.81.0\n10-310-210-1100101\n10-210-1100uniform\nrandom\nCDMFTz x\nuniform\nrandom\nCDMFT\nuniform\nrandom\nCDMFT+CTQMCuniform\nrandom\nCDMFT+CTQMCz x\nuniform\nrandom\nCDMFT+CTQMCz x\nFIG. 12. (a)(b)(c) 4 hSp\njSp\nj0iNNfor the FM case and (d)(e)(f) the onsite components of 1 =Tp\n1(p=z;x) calculated by setting\nall\u0011= 1 (uniform) and all \u0011being random (random) in the CDMFT calculations: (a)(d) \u000b= 1:0, (b)(e)\u000b= 0:8, and (c)(f)\n\u000b= 1:2. In (a) and (d), hSp\njSp\nj0iNNand 1=Tp\n1are equivalent for p=x;z. For comparison, we plot the data in Figs. 2 (CDMFT)\nand 9 (CDMFT+CTQMC). The vertical dotted lines represent TLandTHfor each\u000b.\nthe magnetic susceptibility is dominated by the emergent\ngauge \ruxes from the fractionalization.\nAs noted above, 1 =Tz\n1for\u000b= 0:8 behaves di\u000berently\nfrom others: 1 =Tz\n1for the randomf\u0011gis smaller than\nthat for the uniform f\u0011gat lowT, as shown in Fig. 12(e).\nThis is presumably because of the peculiar Tdependence\nof the density of states (DOS) for the itinerant matter\nfermions at \u000b= 0:8. In the gapless QSL region for\n0:75\u0014\u000b\u00141:5 but close to the gapless-gapful boundary\nat\u000b= 0:75, the DOS opens a gap as f\u0011gare thermally\ndisordered by raising T[10]. Thus, the DOS for matter\nfermions is gapless for the uniform f\u0011g, while gapped for\nthe randomf\u0011g. As spin excitations by Sx\njandSy\njare\ncomposite excitations of both itinerant matter fermions\nand localized gauge \ruxes, the gap in the DOS for matter\nfermions suppresses 1 =Tz\n1for the random case compared\nto the uniform one. Since f\u0011gare aligned uniformly be-\nlowTL, we expect that 1 =Tz\n1shows an abrupt increase\nwhile decreasing TthroughTL. This indicates that while\na rapid change of 1 =T1when approaching TLis yielded\nby the coherent alignment of f\u0011g, either increase or de-\ncrease of 1=T1atTLmay be a\u000bected by the itinerant\nmatter fermions.\nV. SUMMARY\nTo summarize, we have presented numerical results for\nspin dynamics of the Kitaev model with the anisotropyin the bond-dependent coupling constants. We calculated\nthe experimentally-measurable quantities, the dynamical\nspin structure factor S(q;!), the NMR relaxation rate\n1=T1, and the magnetic susceptibility \u001f, in the wide T\nrange including the peculiar paramagnetic region where\nquantum spins are fractionalized. The results have been\nobtained by the Majorana CDMFT+CTQMC method,\nwhich were developed by the authors previously [18]; we\ngave detailed descriptions of the method, including the\nMEM for analytical continuation. We also con\frmed the\nMajorana CDMFT is precise enough in the range of T\nand anisotropy that we investigated in the present study.\nWe found that the Kitaev model exhibits unconven-\ntional behaviors in spin dynamics in the \fnite- Tparam-\nagnetic state in proximity to the QSL ground state. The\nprominent feature is the dichotomy between static and\ndynamical spin correlations as a consequence of the spin\nfractionalization. The dichotomy appears clearly in the\nincrease of 1 =T1belowTHwhere the fractionalization sets\nin, despite the saturation of static correlations. Similar\nbehavior was also seen in the isotropic case in the previ-\nous study [18]. Our results suggest that the dichotomy\nis found universally in the fractionalized paramagnetic\nregion irrespective of the anisotropy in the system.\nOn the other hand, we also clari\fed interesting behav-\niors that depend on the anisotropy at low T. When one\nof the three bond-dependent interactions is stronger than\nthe other two, the spin dynamics shows peculiar Tand\nenergy dependences distinct from those in the isotropic16\ncoupling case as follows. As lowering T,S(q;!) develops\na\u000e-function like peak, which is well separated from the\nincoherent continuum. 1 =T1monotonically decreases in\nthe spin component for the stronger bond. \u001fincreases\nand saturates to a nonzero value for the spin component\nfor the weaker bonds, while it shows hump and then de-\ncreases for the stronger-bond component in the antifer-\nromagnetic case. We also showed that the peculiar Tde-\npendences of \u001fare qualitatively explained by the two-site\ndimer model. In contrast, when the anisotropy is oppo-\nsite, i.e., when the two types of bonds become stronger,\nthe results are qualitatively unchanged from those for the\nisotropic case, while the e\u000bect of anisotropy is obvious in\ntheqdependence in S(q;!) and the di\u000berent components\nin 1=T1and\u001f.\nOur results will stimulate further experimental and\ntheoretical analyses of candidate materials for the Ki-\ntaev QSLs. As most of the materials are assumed to be\nanisotropic in the exchange constants [19{22], our results\nwill be helpful for understanding of unusual behaviors\nin the real compounds. We emphasize that our numer-\nical data obtained by the Majorana CDMFT+CTQMC\nmethod are quantitatively reliable in the calculated para-\nmagnetic regime, as the cluster approximation and the\nanalytic continuation are both well controlled. Although\nthere are residual interactions in addition to the Kitaev-\ntype ones in real materials, our results provide good ref-\nerences in the limit of the pure Kitaev model for inter-\npreting the role of the additional interactions.\nWhile we have calculated dynamical quantities of the\nKitaev model in the wide Trange, the calculations were\nlimited above TLdue to the phase transition which is\nartifact of the mean-\feld nature of CDMFT. It is nec-\nessary to develop more sophisticated method to study\nthe dynamical properties below TL. The low-Tspin dy-\nnamics will be interesting, in particular, for extensions of\nthe Kitaev model to three-dimensional lattices, such as\nhyperhoneycomb and hyperoctagon lattices [44]. In the\nthree-dimensional cases, in general, the Kitaev models\nmay cause a \fnite- Tphase transition between the para-\nmagnetic and QSL phases. Indeed, such an exotic transi-\ntion was found for the hyperhoneycomb Kitaev model [9].\nThe phase transition is triggered by the con\fnement and\ndecon\fnement of emergent loops composed of excited\n\ruxes [9]. This is a topological phase transition that can-\nnot be described by a local order parameter. Although it\nis expected that dynamical quantities exhibit peculiar be-\nhavior associated with the topological phase transition,\nthe CDMFT is not able to describe such a transition.\nThus, with bearing the fact in mind that there are some\ncandidates for the three-dimensional Kitaev QSLs [45{\n49] the calculation of dynamical quantities in all Trange\nbeyond the CDMFT is an interesting challenge left for\nfuture works.\n(a) (b)FIG. 13. Schematic pictures of the di\u000berent types of clusters\nused in the benchmark of CDMFT. The color of the bonds\nare common to Fig. 1(a).\nAppendix A: Cluster size dependence\nIn the CDMFT, we replace the lattice model to the\nimpurity model with a \fnite-size cluster. The CDMFT\nbecomes exact in the limit of in\fnite size cluster. Al-\nthough the cluster size dependence was examined for the\nisotropic case with \u000b= 1:0 in Supplemental Material\nfor the previous study [18], here we present the cluster\nsize dependences of \u001fpand 1=Tp\n1for\u000b= 0:8 and 1:2 in\ncomparison with the \u000b= 1:0 case. As the onsite and\nNN-site components of 1 =Tp\n1shows almost the same T\ndependences below TH(see Fig. 9), we present only the\nonsite one.\nFigure 14 shows the cluster size dependence of \u001fpand\n1=Tp\n1obtained by the CDMFT+CTQMC calculations for\nthree di\u000berent types of clusters shown in Figs. 1(a), 13(a),\nand 13(b). In each type, we change the cluster sizes in\nthe width in the xy-chain direction while keeping that in\nthez-bond direction. This is because the width in the\nxy-chain direction is rather relevant compared to that in\nthez-bond direction in the present CDMFT, presumably\ndue to the Majorana representation based on the Jordan-\nWigner transformation along the xychains. Hereafter,\nwe de\fne the size of the cluster by the average width in\nthexy-chain direction: for instance, 4 :3 for the cluster\nin Fig. 1(a), while 4 and 5 for Figs. 13(a) and 13(b),\nrespectively.\nAs shown in Figs. 14(a)-14(j), the CDMFT+CTQMC\nresults for\u001fpshow quick convergence with respect to the\ncluster width for all the cluster types. Even close to the\narti\fcial critical temperature ~Tc, the results for the width\nlarger than 4 are almost convergent to the large width\nlimit for all types of the clusters: the remnant relative\nerrors are .5%. Note that ~Tc\u00180:014 for\u000b= 1:0,~Tc\u0018\n0:0063 for\u000b= 0:8, and ~Tc\u00180:013 for\u000b= 1:2 (for the\nrotated lattice coordinate used to calculate hSp\nj(\u001c)Sp\nj0i\nforp=x;y,~Tcbecomes slightly lower: ~Tc\u00180:0052 for\n\u000b= 0:8 and ~Tc\u00180:0094 for\u000b= 1:2).\nOn the other hand, as shown in Figs. 14(k)-14(o), the\ncluster-size dependences of 1 =T1remains up to relatively\nhigherTthan\u001fp. But the remnant relative errors are\n.10% for the cluster width larger than 4, which are su\u000e-\nciently small to observe the characteristic Tdependences\nof 1=T1as shown in Figs. 9.17\n(A)(B)(C)(a) α= 1.0\n(b) α= 0.8, p= z\n(c) α= 0.8, p= x\n(d) α= 1.2, p= z\n(e) α= 1.2, p= x\n(f) (g) (h) (i) (j) \n(k) (l) (m) (n) (o) 4.55.05.56.06.57.0\n0.270.280.290.300.31\n0.30.40.50.60.70.80.91.0\n234567891011121 / T1p\ncluster width1.51.71.92.12.32.5\n0.400.420.440.460.482.02.53.03.5\n0.500.520.540.560.58\n1.01.52.02.5\n23456789101112\ncluster width56789101112\n0.200.210.220.230.24\n0.70.80.91.01.11.21.3\n23456789101112\ncluster width(A)(B)(C)(A)(B)(C)(A)(B)(C)χpχp\n0.000.020.040.060.08\n23456789101112\ncluster width0.050.060.070.080.092025303540455055\n357911\n23456789101112\ncluster width(A)(B)(C)\nFIG. 14. Cluster-size dependences of the magnetic susceptibility \u001fpfor (a)-(e) the FM case and (f)-(j) the AFM case, and (k)-\n(o) the onsite component of the NMR relaxation rate 1 =Tp\n1: (a)(f)(k)\u000b= 1:0, (b)(c)(g)(h)(l)(m) \u000b= 0:8, and (d)(e)(i)(j)(n)(o)\n\u000b= 1:2. The data for two di\u000berent Tare plotted in each case. In (a)(f)(k), the data are common to p=zandx. Calculations\nare performed for the cluster series denoted in (A) Fig. 1(a), (B) Fig. 13(a), and (C) Fig. 13(b). Symbols in (a)-(e) are common\nfor the same parameters in (f)-(o).\nAppendix B: Accuracy of the maximum entropy\nmethod\nIn the CDMFT+CTQMC calculations, we calculate\nSp\nj;j0(!) fromhSp\nj(\u001c)Sp\nj0iby the MEM as described in\nSec. II D. In this Appendix, we examine the accuracy\nof the MEM in the limit of decoupled one-dimensional\nchains, i.e., \u000b= 1:5 (Jz= 0), where Sp\nj;j0(!) can be\nobtained directly without the MEM. We also examine\nthe accuracy by comparing Sp\nj;j0(!) at su\u000eciently low- T\nwith the analytical solution in the ground state.\nFirst, we show the comparison in the limit of decoupled\none-dimensional chains, i.e., \u000b= 1:5 (Jz= 0). In this\nlimit, the Kitaev Hamiltonian in Eq. (1) is written only\nby itinerant matter fermions fcg, in the form of Eq. (2)\nwithJz= 0. In this noninteracting problem, following\nRef. [50], we can calculate Sx\nj;j0(!) by considering the\nreal-time evolution (RTE) of hSx\nj(t)Sx\nj0i, instead of the\nimaginary-time correlation hSx\nj(\u001c)Sx\nj0i, as\nSx\nj;j0(!) =Z1\n\u00001dtei!t\u0000\u000fjtjhSx\nj(t)Sx\nj0i: (B1)\nWe call this method as the RTE in the following. In the\nRTE calculations, we consider an xychain with 600 sites\nunder the open boundary condition and take a su\u000eciently\nsmall\u000f= 0:04 in Eq. (B1).\nOn the other hand, Sz\nj;j0(!) has a nonzero value only\nfor the onsite component, which is given by 4 hSz\nj(\u001c)Sz\nji=\nhcj(\u001c)cji. Hence,Sz\nj;j(!) is obtained as\nSz\nj;j(!) =1\n2(1 +e\u0000\f!)D(!); (B2)whereD(!) is the DOS for itinerant matter fermions in\nthe one-dimensional limit:\nD(!) =1\n\u0019p\n1:52\u0000!2: (B3)\nWe call this method to estimate Sz\nj;j(!) the exact-DOS\nin the following.\nFigure 15 shows the results of Sp\nj;j0(!) obtained by the\nMEM, RTE, and exact-DOS methods for the FM case\nwith\u000b= 1:5 (Jx=Jy= 1:5 andJz= 0). We present\nboth onsite and NN-site components for Sx\nj;j0(!), while\nonly the onsite one for Sz\nj;j0(!). We \fnd that overall !\ndependence of Sp\nj;j0(!) is well reproduced by the MEM.\nIn particular, the agreement is excellent in the low !\nregion; the growth of Sx\nj;j0(!= 0) on decreasing T, which\ncontributes to 1 =T1, is well reproduced by the MEM. On\nthe other hand, the relatively sharp structures at !\u0018\n1:5 are blurred in the MEM results for both p=xand\nz, presumably because hSp\nj(\u001c)Sp\nj0iis more insensitive to\nSp\nj;j0(!) in the larger !region. Nevertheless, as shown\nin Figs. 15(e) and 15(f), the MEM results reproduce the\nbroad incoherent peak of Sx\nj;j(!)\u0000Sx\nNN(!).\nNext, we examine the accuracy of the MEM for the\ndata at su\u000eciently low Twith the analytical solution in\nthe ground state [11]. Figure 16 shows Sz(\u0000;!) obtained\nby the Majorana CDMFT+CTQMC method for the FM\ncase with\u000b= 0:8 atT= 0:003. In the ground state,\nthe energy required to \rip a single \u0011ris \u0001'0:042 at\n\u000b= 0:8. Re\recting the \rux gap, Sz(\u0000;!) at lowThas\na\u000e-function like peak at \u0001 '0:042 [11]. As shown in\nFig. 16, our CDMFT+CTQMC result shows a peak at\nthis energy, which is considered to precisely reproduce18\nSx\njj(ω)\n-0.5 Sx\nNN(ω) Sz\nj,j(ω)\nω ω(a) (b)\n(c) (d)\n(e) (f)\n(g) (h)0.00.51.0\n0.00.51.0\n0.00.10.20.30.4\n0.00.20.40.60.8\n-1 0 1 -1 0 1 2MEM\nRTEMEM\nRTE\nMEM\nRTEMEM\nRTE\nMEM\nRTEMEM\nRTE\nMEM\nexactMEM\nexactS(ω) -Sx\nNN(ω)x\nj,j\nFIG. 15. Comparison between the MEM, RTE, and exact-\nDOS results for (a)(b) Sx\nj;j(!), (c)(d)Sx\nNN(!), (e)(f)Sx\nj;j(!)\u0000\nSx\nNN(!), and (g)(h) Sz\nj;j(!) at (a)(c)(e)(g) T= 0:375 and\n(b)(d)(f)(h) T= 0:0375.\nthe low-energy structure of the dynamical spin structure\nfactor.\nFrom these observations, we consider that the MEM\nresults for S(q;!) and 1=T1in Sec. III B and III C are\naccurate enough to discuss the Tand!dependences.\nAppendix C: Spin correlations as functions of Tand\n!\nIn this Appendix, we present the spin correlations\nas functions of Tand!, which are obtained by the\nMEM. Figure 17 shows the results for onsite and NN-site\ncomponents for \u000b= 1:0, 0:8, and 1:2. The data are used\nto obtain the dynamical quantities in Sec. III B and III C.\n0123456\n0.000.020.040.060.080.10Sz(Γ, ω)\nωFIG. 16. Sz(\u0000;!) obtained by the Majorana\nCDMFT+CTQMC method for the FM case for \u000b= 0:8 at\nT= 0:003. Vertical line at !\u00180:042 represents the value of\n\rux gap of the ground state calculated exactly.\nAppendix D: Tdependence of the Korringa ratio\nFigures 18 and 19 display the Tdependences of the\nKorringa ratio de\fned as\nKp=1\nTp\n1T(\u001fp)2; (D1)\nwhich is computed by using the NMR relaxation rate\n1=Tp\n1and the magnetic susceptibility \u001fpobtained in\nSec. III C and III D. Interestingly, as shown in Fig. 18(a),\nKpfor the isotropic FM case is almost constant close to\n1 forTL.T.TH, which is apparently consistent with\nthe behavior expected for free electron systems. This is\nalso the case for the xcomponent for the FM case with\n\u000b= 1:2, as shown in Fig. 18(c). However, the suggestive\nbehavior is presumably super\fcial, as the results for the\nAFM cases as well as for \u000b= 0:8 behave di\u000berently with\nsubstantial Tdependence.\nACKNOWLEDGMENTS\nThe authors thank M. Imada, Y. Kamiya, K. Oh-\ngushi, S. Takagi, M. Udagawa, and Y. Yamaji for\nfruitful discussions. Y. M. thanks A. Banerjee, C.\nD. Batista, K.-Y. Choi, S. Ji, S. Naglar, and J.-H.\nPark for constructive suggestions. This research was\nsupported by Grants-in-Aid for Scienti\fc Research un-\nder Grants No. JP15K13533, No. JP16K17747, and\nNo. JP16H02206. Parts of the numerical calculations\nwere performed in the supercomputing systems in ISSP,\nthe University of Tokyo.\n[1] D. C. Tsui, H. L. Stormer, and A. C. Gossard, Two-\nDimensional Magnetotransport in the Extreme QuantumLimit, Phys. Rev. Lett. 48, 1559 (1982).19\n(a) (b)\n(c) (d)\n(e) (f)\n(g) (h)\n(i) (j)\nFIG. 17. Spin correlations as functions of Tand!for\nthe FM case: (a)(c)(e)(g)(i) onsite components Sj;j(!) and\n(b)(d)(f)(h)(j) NN-site components SNN(!). (a) and (b) are\nforSx\nj;j(!) =Sz\nj;j(!) andSx\nNN(!) =Sz\nNN(!), respectively, at\n\u000b= 1:0. (c), (d), (e), and (f) [(g), (h), (i), and (j)] are for\nSx\nj;j(!),Sz\nNN(!),Sz\nj;j(!), andSz\nNN(!), respectively, at \u000b= 0:8\n(1:2). The white and gray dotted lines indicate THandTL,\nrespectively, for each \u000b.\n[2] H. L. Stormer, D. C. Tsui, and A. C. Gossard, The frac-\ntional quantum Hall e\u000bect, Phys. Mod. Phys. 71, S298\n(1999).\n[3] P. W. Anderson, Resonating valence bonds: A new kind\nof insulator?, Mater. Res. Bull. 8, 153 (1973).\n[4] S. A. Kivelson, D. S. Rokhsar, and J. P. Sethna, Topology\nof the resonating valence-bond state: Solitons and high-\nTcsuperconductivity, Phys. Rev. B 35, 8865 (1987).\n[5] T. Senthil and M. P. A. Fisher, Z2gauge theory of\nelectron fractionalization in strongly correlated systems,\nPhys. Rev. B 62, 7850 (2000).\n[6] M. Hermele, M. P. A. Fisher, and L. Balents, Pyrochlore\nphotons: The U(1) spin liquid in a S=1\n2three-\ndimensional frustrated magnet, Phys. Rev. B, 69, 064404\nT(a)\n(b)\n(c)zx\nzx\n10-1100101\n10-210-110010-1100101\n10-410-310-210-1100101102onsite\nNN-site\nonsite\nNN-site\nonsite\nNN-siteFIG. 18. Tdependences of the Korringa ratio Kp=\n1=(Tp\n1T(\u001fp)2) for the FM case at (a) \u000b= 1:0, (b)\u000b= 0:8, and\n(c)\u000b= 1:2 (p=z;x). Note thatKz=Kxfor\u000b= 1:0 and\nKx=Kyfor all the cases from the symmetry. The vertical\ndotted lines indicates TLandTHfor each\u000b.\n(2004).\n[7] C. Castelnovo, R. Moessner, and S. L. Sondhi, Magnetic\nmonopoles in spin ice, Nature 451, 42 (2008).\n[8] A. Kitaev, Anyons in an exactly solved model and be-\nyond, Ann. Phys. 321, 2 (2006).\n[9] J. Nasu, M. Udagawa, and Y. Motome, Vaporization\nof Kitaev Spin Liquids, Phys. Rev. Lett. 113, 197205\n(2014).\n[10] J. Nasu, M. Udagawa, and Y. Motome, Thermal frac-\ntionalization of quantum spins in a Kitaev model:\nTemperature-linear speci\fc heat and coherent transport\nof Majorana fermions, Phys. Rev. B 92, 115122 (2015).\n[11] J. Knolle, D.L. Kovrizhin, J.T. Chalker, and R. Moess-\nner, Dynamics of a Two-Dimensional Quantum Spin Liq-\nuid: Signatures of Emergent Majorana Fermions and\nFluxes, Phys. Rev. Lett. 112, 207203 (2014).\n[12] A. Banerjee, C. A. Bridges, J-Q. Yan, A. A. Aczel, L. Li,\nM. B. Stone, G. E. Granroth, M. D. Lumsden, Y. Yiu, J.20\nT(a)\n(b)\n(c)zx\nzx\n10-1100101102103\n10-210-110010-110010110210310-1100101102103104\nonsite\nNN-site\nonsite\nNN-site\nonsite\nNN-site\nFIG. 19. Tdependences of the Korringa ratio Kp=\n1=(Tp\n1T(\u001fp)2) for the AFM case at (a) \u000b= 1:0, (b)\u000b= 0:8,\nand (c)\u000b= 1:2 (p=z;x). The notations are common to\nthose in Fig. 18.\nKnolle, D. L. Kovrizhin, S. Bhattacharjee, R. Moessner,\nD. A. Tennant, D. G. Mandrus, S. E. Nagler, Proximate\nKitaev quantum spin liquid behaviour in a honeycomb\nmagnet, Nature Mater. 15733 (2016).\n[13] S.-H. Do, S.-Y. Park, J. Yoshitake, J. Nasu, Y. Motome,\nY. S. Kwon, D. T. Adroja, D. J. Voneshen, K. Kim,\nT.-H. Jang, J.-H. Park, K.-Y. Choi, S. Ji, Incarnation\nof Majorana Fermions in Kitaev Quantum Spin Lattice,\npreprint (arXiv:1703.01081).\n[14] J. Knolle, G.-W. Chern, D. L. Kovrizhin, R. Moessner,\nand N. B. Perkins, Raman Scattering Signatures of Ki-\ntaev Spin Liquids in A2IrO3Iridates with A=Na or Li,\nPhys. Rev. Lett. 113, 187201 (2014).\n[15] L. J. Sandilands, Y. Tian, K. W. Plumb, Y.-J. Kim, and\nK. S. Burch, Scattering Continuum and Possible Frac-\ntionalized Excitations in \u000b-RuCl 3, Phys. Rev. Lett. 114,\n147201 (2015).\n[16] J. Nasu, J. Knolle, D. L. Kovrizhin, Y. Motome, and R.\nMoessner, Fermionic response from fractionalization inan insulatin two-dimensional magnet, Nature Phys. 12,\n912 (2016).\n[17] A. Glamazda, P. Lemmens, S.-H. Do, Y. S. Choi, and K.-\nY. Choi, Raman spectroscopic signature of fractionalized\nexcitations in the harmonic-honeycomb iridates \fand\r-\nLi2IrO3, Nat. Commun. 7, 12286 (2016).\n[18] J. Yoshitake, J. Nasu, and Y. Motome, Fractional Spin\nFluctuation as a Precursor of Quantum Spin Liquids:\nMajorana Dynamical Mean-Field Study for the Kitaev\nModel, Phys. Rev. Lett. 117, 157203 (2016).\n[19] R. D. Johnson, S. C. Williams, A. A. Haghighirad, J.\nSingleton, V. Zapf, P. Manuel, I. I. Mazin, Y. Li, H. O.\nJeschke, R. Valent\u0013 \u0010, and R. Coldea, Monoclinic crystal\nstructure of \u000b\u0000RuCl 3and the zigzag antiferromagnetic\nground state, Phys. Rev. B, 92, 235119 (2015).\n[20] Y. Yamaji, Y. Nomura, M. Kurita, R. Arita, and M.\nImada, First-Principles Study of the Honeycomb-Lattice\nIridates Na 2IrO3in the Presence of Strong Spin-Orbit\nInteraction and Electron Correlations, Phys. Rev. Lett.\n113, 107201 (2014).\n[21] S. K. Choi, R. Coldea, A. N. Kolmogorov, T. Lancaster,\nI. I. Mazin, S. J. Blundell, P. G. Radaelli, Yogesh Singh,\nP. Gegenwart, K. R. Choi, S.-W. Cheong, P. J. Baker,\nC. Stock, and J. Taylor, Spin Waves and Revised Crys-\ntal Structure of Honeycomb Iridate Na 2IrO3, Phys. Rev.\nLett. 108, 127204 (2012).\n[22] S. M. Winter, Y. Li, H. O. Jeschke, and R. Valenti, Chal-\nlenges in design of Kitaev materials: Magnetic interac-\ntions from competing energy scales, Phys. Rev. B 93,\n214431 (2016).\n[23] G. Baskaran, Saptarshi Mandal, and R. Shankar, Exact\nResults for Spin Dynamics and Fractionalization in the\nKitaev Model, Phys. Rev. Lett. 98, 247201 (2007).\n[24] S. Mandal, R. Shankar, and G. Baskaran, RVB gauge the-\nory and the topological degeneracy in the honeycomb Ki-\ntaev model, J. Phys. A Math. Theor. 45, 335304 (2012).\n[25] H.-D. Chen and J. Hu, Exact mapping between classical\nand topological orders in two-dimensional spin systems,\nPhys. Rev. B 76, 193101 (2007).\n[26] X.-Y. Feng, G.-M. Zhang, and T. Xiang, Topological\nCharacterization of Quantum Phase Transitions in a\nSpin-1=2 Model, Phys. Rev. Lett. 98, 087204 (2007).\n[27] H.-D. Chen, and Z. Nussinov, Exact results of the Ki-\ntaev model on a hexagonal lattice: spin states, string\nand brane correlators, and anyonic excitations, J. Phys.\nA Math. Theor. 41, 075001 (2008).\n[28] G. Kotliar, S. Y. Savrasov, G. P\u0013 alsson, and G. Biroli,\nCellular Dynamical Mean Field Approach to Strongly\nCorrelated Systems, Phys. Rev. Lett. 87, 186401 (2001).\n[29] N. Furukawa, Transport Properties of the Kondo Lattice\nModel in the Limit S=1andD=1, J. Phys. Soc.\nJpn.63, 3214 (1994).\n[30] J. Nilsson and M. Bazzanella, Majorana fermion descrip-\ntion of the Kondo lattice: Variational and path integral\napproach, Phys. Rev. B 88, 045112 (2013).\n[31] W. Metzner and D. Vollhardt, Correlated Lattice\nFermions in d=1Dimensions, Phys. Rev. Lett. 62,\n324 (1989).\n[32] A. Georges, G. Kotliar, W. Krauth, and M. J. Rozen-\nberg, Dynamical mean-\feld theory of strongly correlated\nfermion systems and the limit of in\fnite dimensions, Rev.\nMod. Phys. 68, 13 (1996).\n[33] M. Udagawa, H. Ishizuka, and Y. Motome, Non-Kondo\nMechanism for Resistivity Minimum in Spin Ice Conduc-21\ntion Systems, Phys. Rev. Lett. 108, 066406 (2012).\n[34] P. Werner, A. Comanac, L. de' Medici, M. Troyer, and A.\nJ. Mills, Continuous-Time Solver for Quantum Impurity\nModels, Phys. Rev. Lett. 97, 076405 (2006).\n[35] Z.-X. Li, Y.-F. Jiang, and H. Yao, Solving the fermion\nsign problem in quantum Monte Carlo simulations by\nMajorana representation, Phys. Rev. B 91, 241117(R)\n(2015).\n[36] A. N. Rubtsov, V. V. Savkin, and A. I. Lichten-\nstein, Continuous-time quantum Monte Carlo method for\nfermions, Phys. Rev. B 72, 035122 (2005).\n[37] L. Boehnke, H. Hafermann, M. Ferrero, F. Lechermann,\nO. Parcollet, Orthogonal polynomial representation of\nimaginary-time Green's functions, Phys. Rev. B 84,\n075145 (2011).\n[38] R. Levy, J. P. F. LeBlanc, and E. Gull, Implementation of\nthe maximum entropy method for analytic continuation,\nComp. Phys. Commun. 215, 149 (2017).\n[39] M. Jarrell and J. E. Gubernatis, Bayesian inference and\nthe analytic continuation of imaginary-time quantum\nMonte Carlo data, Phys. Rep. 269, 133 (1996).\n[40]Sp\nNN(!) for the FM case ( Sp\nNN(!)FM) and the AFM\ncase (Sp\nNN(!)AFM) satisfy the relation Sp\nNN(!)FM =\n\u0000Sp\nNN(!)AFM. Nevertheless, Sp\nNN(!)FMandSp\nNN(!)AFM\nobtained by the present numerical procedure violates the\nrelation because of the precision of the MEM. In the\ndata presented in Sec. III, we calculate Sp\nNN(!)FMby\nthe procedure, and Sp\nj;j0(!)AFM are set asSp\nj;j0(!)AFM =\n\u0000Sp\nj;j0(!)FM.\n[41] T. Moriya, Nuclear Magnetic Relaxation in Antiferro-\nmagnetics, II, Prog. Theor. Phys., 16, 641 (1956).\n[42]\u001fxfor the AFM case at \u000b= 0:8 also shows similar behav-\nior, but it is qualitatively explained by the Curie-Weiss\nlaw in Eq. (45) for this case.\n[43] Thermal averages h\u0011r0iobtained by the CTQMC calcu-\nlations deviate from 1 even in the uniform case, althoughwe \fxP(\u0011r0= 1) = 1 in the CDMFT. This is because\nthe statistical weight of \u0011r0=\u00001 is nonzero in the two-\nsite impurity problem in the CTQMC calculations. Such\na discrepancy ofh\u0011r0ibetween the CDMFT and CTQMC\nsolutions never occurs when we do not \fx P(\u0011r0) by hand.\n[44] M. Hermanns and S. Trebst, Quantum spin liquid with a\nMajorana Fermi surface on the three-dimensional hyper-\noctagon lattice, Phys. Rev. B 89, 235102, (2014).\n[45] Y. Okamoto, M. Nohara, H. Aruga-Katori, and H. Tak-\nagi, Spin-Liquid State in the S= 1=2 Hyperkagome\nAntiferromagnet Na 4Ir3O8, Phys. Rev. Lett. 99, 137207\n(2007).\n[46] H. Kuriyama, J. Matsuno, S. Niitaka, M. Uchida, D.\nHashizume, A. Nakao, K. Sugimoto, H. Ohsumi, M.\nTakata, and H. Takagi, Epitaxially stabilized iridium\nspinel oxide without cations in the tetrahedral site, Appl.\nPhys. Lett. 96182103 (2010).\n[47] K. A. Modic, T. E. Smidt, I. Kimchi, N. P. Breznay, A.\nBi\u000en, S. Choi, R. D. Johnson, R. Coldea, P. Watkins-\nCurry, G. T. McCandless, J. Y. Chan, F. Gandara, Z.\nIslam, A. Vishwanath, A. Shekhter, R. D. McDonald, and\nJ. G. Analytis, Realization of a three-dimensional spin-\nanisotropic harmonic honeycomb iridate, Nat. Comm. 5,\n4203 (2014).\n[48] T. Takayama, A. Kato, R. Dinnebier, J. Nuss, H. Kono,\nL. S. I. Veiga, G. Fabbris, D. Haskel, and H. Takagi, Hy-\nperhoneycomb Iridate \f-Li2IrO3as a Platform for Kitaev\nMagnetism, Phys. Rev. Lett. 114, 077202 (2015).\n[49] I. Kimchi and A. Vishwanath, Kitaev-Heisenberg models\nfor iridates on the triangular, hyperkagome, kagome, fcc,\nand pyrochlore lattices, Phys. Rev. B 89, 014414 (2014).\n[50] O. Derzhko, T. Krokhmalskii, and J. Stolze, Dynamics\nof the spin-1\n2isotropicXYchain in a transverse \feld, J.\nPhys. A: Math. Gen., 33, 3063 (2000)."    },    {        "title": "1609.02319v1.Pumping_dynamics_of_nuclear_spins_in_GaAs_quantum_wells.pdf",        "content": "arXiv:1609.02319v1  [cond-mat.mes-hall]  8 Sep 2016Pumping dynamics of nuclear spins in GaAs quantum wells\nRaphael W. Mocek,∗Danil O. Tolmachev, Giovanni Cascio, and Dieter Suter†\nExperimental Physics III, TU Dortmund University, Otto-Ha hn-Str. 4a, 44227 Dortmund,Germany\n(ΩDated: August 3, 2018)\nIrradiating a semiconductor with circularly polarized lig ht creates spin-polarized charge carriers.\nIf the material contains atoms with non-zero nuclear spin, t hey interact with the electron spins via\nthe hyperfinecoupling. Here, we consider GaAs/AlGaAs quant um wells, where the conduction-band\nelectron spins interact with three different types of nuclea r spins. The hyperfine interaction drives\na transfer of spin polarization to the nuclear spins, which t herefore acquire a polarization that is\ncomparable to that of the electron spins. In this paper, we an alyze the dynamics of the optical\npumping process in the presence of an external magnetic field while irradiating a single quantum\nwell with a circularly polarized laser. We measure the time d ependence of the photoluminescence\npolarization to monitor the buildup of the nuclear spin pola rization and thus the average hyperfine\ninteraction acting on the electron spins. We present a simpl e model that adequately describes the\ndynamics of this process and is in good agreement with the exp erimental data.\nPACS numbers: 78.67.De, 78.55.Cr, 78.66.Fd, 76.60.-k\nKeywords: Optical Orientation, Semiconductors, Optical P umping, Spin Dynamics\nI. INTRODUCTION\nOptical pumping creates electrons and holes in semi-\nconductor samples with spin polarizations far from equi-\nlibrium, as shown by the level scheme and transition dia-\ngramoffig. 1. Dependingontheconduction-andvalence-\nband states involved in the optical transition, the polar-\nization of the electron spins can reach almost 100%[ 1].\nFIG. 1. Selection rules for the optical transitions between\nthe valence-band (VB) and the conduction-band (CB) of a\nsemiconductor quantum well. The confinement lifts the de-\ngeneracy of the hole states. (modified after[ 2])\nThe spin-polarizationofthe conduction-bandelectrons\ncan be monitored through the photoluminescence (PL)\npolarization[ 3–5]\nDOP=I(σ+)−I(σ−)\nI(σ+)+I(σ−), (1)\nwhereI(σ±) is the intensity of right- or left-circularly\npolarized PL.\n∗raphael.mocek@tu-dortmund.de\n†dieter.suter@tu-dortmund.deInmanymaterials,andinparticularintheGaAsquan-\ntum wells that we consider in this work, the electrons are\ncoupled to different nuclear spins by the hyperfine inter-\naction. Accordingly, the electron spin orientation can be\ntransferred to the nuclear spins[ 6–9] in a process known\nas dynamic nuclear polarization (DNP)[ 2,10,11]. Con-\nversely, the ensemble of polarized nuclear spins affect the\nevolution of the electron spins. The overall effect can be\nsummarizedbyaneffectivenuclearmagneticfield[ 10,12–\n17]. This is used, e.g., in optically detected nuclear mag-\nnetic resonance[ 6,13,18–22].\nThe goal of this study is a detailed understanding of\nthe buildup of the nuclear spin polarization during opti-\ncal pumping. For this purpose, we rely mostly on mea-\nsurements ofthe time dependence of the PL polarization,\nfrom which we determine the buildup of the nuclearmag-\nnetic field. The paper is organized as follows. In Sec. II,\nwe describe a simple model of the spin dynamics during\noptical pumping. In Sec. IIIwe present the experimental\nsetup and the sample under investigation. Section IVA\ncontainstheexperimentalresultsforthetimedependence\nofthe optical pumping processandcomparesthem to the\ntheoretical prediction. Section IVBsummarizes the ef-\nfect of the control parameters laser intensity and optical\ndetuning on the optical pumping dynamics. The paper\nends with a short discussion and conclusions.\nII. THEORY\nA. Electron spin polarization\nThe optical pumping process, as well as the optical\ndetection couple the photon angular momentum directly\nto the spin ofthe chargecarriers. We therefore start with\nthe equation of motion for the spin density operator ρ:2\n∂ρ\n∂t=−i\n/planckover2pi1[H,ρ]−ΓRρ−ΓS/parenleftbigg\nρ−Tr{ρ}\n2\n/BD/parenrightbigg\n+˜P(2)\nH=/planckover2pi1γe/vectorB·/vectorS\nWe use the spin operators defined as Si=1\n2σiwithi∈\n[x,y,z],γeis the gyromagnetic ratio of the conduction\nelectrons and /BDis the two-dimensional unity matrix. /vectorB\nis the magnetic field, Γ Rdescribes the recombination of\nthe electrons from the conduction- to the valence-band\nand ΓSthe spin relaxation rate. The matrix ˜Pdescribes\nthe buildup of electron spin density by the absorption of\ncircularly polarized photons. In the coordinate system\ndefined in fig. 2, this matrix is\n˜P=P/parenleftbigg\ncos2ΘL\n21\n2sinΘL\n1\n2sinΘLsin2ΘL\n2/parenrightbigg\n. (3)\nPis the rate at which the optical pumping process gener-\nates electron spin density in the conduction band. Θ Lis\nthe angle between the incident laser beam and the z-axis,\nas defined in fig. 2.\nFIG. 2. Schematic overview of the chosen coordinate system\nand the relevant angles. The laser light hits the sample per-\npendicularly.\nIf the nuclear spins are polarized, they collectively\nmodify the effective magnetic field acting on the electron\nspin. We therefore write the total field /vectorBas the sum\nof the external field /vectorBextand the nuclear field /vectorBnuc. In\nour coordinate system, both are oriented along the z-axis\nand we therefore write Bext+Bnucfor thez−component\nof the effective magnetic field. Under stationary condi-\ntions, the expectation values of the three components of\nthe electron spin are\n/an}bracketle{tSx/an}bracketri}ht=Tr{Sxρ}=P\n2γe∆BsinΘL\n(Bext+Bnuc)2+∆B2(4)\n/an}bracketle{tSy/an}bracketri}ht=Tr{Syρ}=P\n2γe(Bext+Bnuc)sinΘ L\n(Bext+Bnuc)2+∆B2\n/an}bracketle{tSz/an}bracketri}ht=Tr{Szρ}=P\n2γecosΘL\n∆B.\nHere the parameter ∆ Bis defined as ∆ B=/planckover2pi1(ΓR+ΓS)\n|g∗|µB\nwithg∗asthe g-factorofthe conduction electrons[ 23–25]andTr{ρ}=P\nΓR. In our case, the direction of detection\n/vector eis close to the x-axis\n/vector e=/parenleftbigsinΘD,0,cosΘD/parenrightbig\n, (5)\nwhere the angle θDis defined in fig. 2.\nExperimentally, we measure the degree of photon po-\nlarization (see eq.( 1)) in the direction /vector eby dividing the\ndifferencebetweenthetwointensities I(σ±)bytheirsum.\nThe result is\nSD=1\nTr{ρ}(sinΘD/an}bracketle{tSx/an}bracketri}ht+cosΘ D/an}bracketle{tSz/an}bracketri}ht)\n=S0/parenleftBigg\ncosΘLcosΘD+∆B2sinΘLsinΘD\n∆B2+(Bext+Bnuc)2/parenrightBigg\n.(6)\nHereS0=1\n2ΓR\nΓR+ΓSdescribes the equilibrium spin polar-\nization in the absence of a magnetic field. This signal,\nmeasured as a function of the external magnetic field,\nis known as a (shifted) Hanle curve. It represents a\nLorentzian, with a maximum at Bext=−Bnucand a\nwidth ∆B.\nB. Nuclear spin polarization\nIn this study, we concentrate on the evolution of the\nnuclear spin polarization. The equation of motion for the\nnuclear spin populations can be written as\nd\ndt/parenleftbigg\np↑\np↓/parenrightbigg\n=/parenleftbigg−κs↓−1\n2T1κs↑+1\n2T1\nκs↓+1\n2T1−κs↑−1\n2T1/parenrightbigg/parenleftbigg\np↑\np↓/parenrightbigg\n,(7)\nwhereκis the transfer rate at which electronic and nu-\nclear spins undergo mutual flip-flop transitions and s↑↓\nare the densities of the electron spins generated by the\noptical pumping process. According to eq.( 4) they are\ns↑↓=P\n2ΓR±/an}bracketle{tSz/an}bracketri}ht.\nFor quantum wells with dimensions of ≈20 nm, the\nrecombination rate of the electrons is Γ R≈109s−1[26].\nFor time-independent parameters, eq.( 7) can be solved\nanalytically. If the system is initially in thermal equilib-\nrium,p↑↓(0) = 0.5, the solution is\np↑↓(t) =1±∆p(t)\n2, (8)\nwhere the population difference is\n∆p(t) = ∆p∞/parenleftbigg\n1−e−/parenleftBig\n1\nT1+κP\nΓR/parenrightBig\nt/parenrightbigg\n(9)\nand it’s equilibrium value\n∆p∞=/an}bracketle{tSz/an}bracketri}ht2κT1ΓR\nΓR+κT1P. (10)3\nThe nuclear spin polarization can be measured through\nits effect on the electron spin, via the effective nuclear\nfield\nBnuc(t) =Bmax∆p(t)\n=Bmax∆p∞/parenleftbigg\n1−e−/parenleftBig\n1\nT1+κP\nΓR/parenrightBig\nt/parenrightbigg\n.(11)\nAccording to eq.( 6),Bnucis given by the maximum of\nthe Hanle curve. To measure the time dependence of the\npopulations of the nuclear spin, we therefore measure the\nHanle curves for different pumping times.\nIII. EXPERIMENTAL\nThe sample used for this investigation was grown by\nmolecular beam epitaxy on a Te-doped GaAs substrate.\nIt consists of 13 undoped GaAs/Al0.35Ga0.65As quantum\nwells with thicknesses dranging from 2.8 to 39 .3nm[27].\nFIG. 3. Experimental setup of the optical pumping exper-\niment. L1, L2 and L3: lenses, LP: linear polarizer, PEM:\nphoto elastic modulator, APD: avalanche photo diode, BS:\nbeam splitter, λ/2,λ/4 : retardation plates.\nFigure3shows a schematic representation of the experi-\nmental setup. For the optical excitation we use a semi-\nconductor laser (Toptica DLC DL PRO), which covers\nthe wavelength range of λexc= 799−812nm. The mag-\nneticfieldiscreatedbyaresistiveelectromagnet(Bruker)\nwith a range of Bext= 0T−1.4T. The sample is\nmounted on the cold finger of a home-built flow-cryostat\nand kept at temperatures of T≈4.7±0.3K. The laser\nbeam analysis includes a spectrometer (APE waveScan\nUSB) for monitoring the laser wavelength and a pho-\ntodiode to monitor the laser power. The PL is passed\nthrough a monochromator (Spex 1704) and a photo-\nelastic modulator (Hinds Instruments PEM 90) and de-\ntected with an avalanche photodiode (APD, Hamamatsu\n5640). Two lock-in amplifiers (Stanford Research SR830\nDSP) are then used to measure the total PL powerIΣ=I(σ+) +I(σ−) and the difference between right-\nand left-circularly polarized light I∆=I(σ+)−I(σ−).\nThe optical pumping process took place in constant\nexternal magnetic fields of Bext= 0.3T or 1T. We\nmonitored the buildup of the nuclear spin polarization\nby measuring Hanle curves[ 2,28] as a function of the\npumping time. For the Hanle curves, the magnetic field\nwas scanned from Bexteither upward or downward, de-\npending on the displacement of the Hanle curve. The\ntime for measuring a Hanle curve was about 10s, short\ncompared to the duration of the optical pumping. Dur-\ningtheHanlemeasurements,the laserpowerwasreduced\ntoPL≈2mW, to minimize the optical pumping effects.\nThe angles defined in fig. 2were Θ D= 81◦and Θ L= 78◦\nfor all experiments.\nIV. RESULTS\nA. Nuclear field buildup\nThe main goal of these experiments was a quantita-\ntive understanding of the process that generates the nu-\nclear spin polarization. For this purpose, we performed\na set of measurements that consisted of a pumping pe-\nriodTpumpduring which the sample was irradiated with\ncircularly polarized light in a constant magnetic field.\nImmediately after this pumping period, we performed\na rapid scan of the magnetic field to measure a Hanle\ncurve. According to eq.( 4), the maxima of these curves\ncorrespond to the effective nuclear field |Bnuc|and can\ntherefore be used as a probe of the nuclear spin polar-\nization. Measured Hanle curves after two different times\nTpumpare shown in fig. 4(a). The experimental param-\neters for these measurements are Bext= 1T and laser\npowerPL= 49mW. The monochromator was set to\nthe maximum of the PL line of the d= 19.7nm quan-\ntum well and the optical detuning of the laser beam was\n∆λ=λdet−λexc= 811.6nm−811.3nm = 0 .3nm.\nFigure4(b) shows the buildup of the effective nuclear\nfield|Bnuc(t)|. It compares the experimental data with\nthe theoretical curve calculated from eq.( 11). For the\nparameters,weusedaspin-latticerelaxationtimeof T1=\n596±160s, which we measured independently, and is\ncomparable to literature values for similar systems [ 29].\nFrom the fitted curve and our experimental parameters,\nwe calculated P= 7.7·1013#e−\ns·µm3. Table Ishows the\nother relevant parameters determined from these curves.\nBmax[T] ∆ p∞[%]κ/bracketleftBig\nµm3\ns·#e−/bracketrightBig\n29 3 .4 6.8·10−8\nTABLE I. Fit parameters for the data shown in fig. 4(b).\nThe fit-result for the maximum field Bmaxis signifi-\ncantly larger than some values from the literature [ 2,12].\nThebuildupofthenuclearspinpolarizationcanalsobe4\n(b)\nFIG. 4. Time dependence of the effective nuclear field |Bnuc|:\n(a) Hanle curves measured after optical pumping periods of\nTpump= 120s and 900s. (b) Evolution of the nuclear field\nduring the optical pumping. The solid red line is the fit resul t\nusing eq.( 11) and the filled circles represent the experimental\nvalues.\nmonitored through the time dependence of the PL polar-\nization during the optical pumping process, as shown in\nfig.5. The theoretical curve was calculated from eqs.( 6)\nand (11) with the parameters of tab. I.\nB. Dependence on the laser intensity\nThe parameter Pintroduced in eq.( 3) describes the\nrateat which electron spin density is createdin the mate-\nrial. Over some range, we therefore expect that Pshould\nbe proportional to the laser power PL. Since only ab-\nsorbed photons generate electron spins, the rate should\nalso depend on the absorption probability of the photons\nand reach a maximum at the optical resonance. The in-\nfluence of optical detuning is discussed in Sec. IVC.\nWe examined the dependence of the pumping dy-\nnamics on the laser power by performing a series of\nmeasurements with increasing laser intensity. After\nFIG. 5. Time dependence of the optical pumping process:\nmeasured PL polarization under optical pumping conditions\nwithBext= 1T,PL= 49mW and ∆ λ= 0.3nm. The solid\nred line was calculated with the parameters of table Iusing\neq.(6) and (11).\npumping times Tpump= [30s,60s,180s,300s,600s],\nwe measured Hanle curves to monitor the evolution\nof the effective nuclear magnetic field |Bnuc|. We\nrepeated this procedure with laser powers of PL=\n[20mW,25mW,30mW,35mW,40mW]. Further ex-\nperimental parameters were Bext= 0.3T and ∆ λ=\n0.5nm. The detection wavelength was λdet= 811.6nm,\nwhich corresponds to the maximum of the PL line of the\nd= 19.7nm quantum well.\nFIG. 6. Evolution of the effective nuclear magnetic field\n|Bnuc|for different laser powers PL. The circles mark the\nexperimental data while the solid lines are the result of the\nfit using eq.( 11) andBmax= 5.3T,κ= 6.3·10−8µm3\ns·#e−as\nfixed parameters.\nFigure6showsthebuildupoftheeffectivenuclearmag-5\nnetic field |Bnuc(t)|for different laser intensities. The\nexperimental data, which are shown as circles, are the\nmaxima of the Hanle curves taken after each optical\npumping period Tpump. We used Bmax= 29T and\nκ= 6.8·10−8µm3\ns·#e−as fixed parameters obtained by\nthe measurement presented in Sec. IVAand eq.( 11) to\nfit the experimental data shown in fig. 6. The resulting\n|Bnuc(t)|are shown as solid curves.\nFIG. 7. Rate of change in electron spin density Ptogether\nwith the PL light power as a function of the laser power PL.\nThe solid line is the result of the linear fit.\nFrom the observed buildup-rate of |Bnuc|, we calcu-\nlated the rate Pat which electron-spin density is gen-\nerated by the pumping process. Figure 7shows the re-\nsulting rates Pas a function of the laser power PL. The\nrate at which electrons are generated in the conduction-\nband should also be reflected in the PL power, which we\nmeasured independently. As shown in fig. 7, both quan-\ntities are roughly proportional to the laser power, with\nproportionality factors mP= (1.7±0.5)·1012#e−\ns·µm3·mW\nandmPLpower= 0.4±0.1nW\nmW, respectively.\nC. Optical detuning\nThe rate Pat which electron-spins are generated de-\npends also on the frequency of the laser with respect to\nthe resonance frequency of the quantum well. We mea-\nsured the time dependence of the PL polarization un-\nder optical pumping conditions for different optical de-\ntunings ∆ λ= [0.7...2.1]nm relative to the peak wave-\nlengthλdet= 811.6nm of the d= 19.7nm quantum\nwell. The experimental parameters were Bext= 0.3T\nandPL= 47mW.\nFigure8shows some of the measured curves. The ex-\nperimental data are compared to the theoretical expec-\ntations calculated from eq.( 6) and (11), using the exper-\nimental parameters of our system and the fit parameters\nFIG. 8. Time dependence of the optical pumping process:\nmeasured PL polarization under optical pumping conditions\nwithBext= 0.3T,PL= 47mW and Tpump= 600s for differ-\nent optical detunings ∆ λ. The solid lines are the fit results\nbased on eq.( 11),(6).\ngiven in tab. I. The electron-spin density generation rate\nPwas adjusted for the curves to fit the experimental\ndata.\nFIG. 9. Detuningdependence of the rate of change in electron\nspin density P: the circles mark the fit-parameters Pand\nthe solid red line is the result of the Lorentzian fit function\neq.(12). Detuning dependence of the PL light power: the\ndiamonds mark the measured PL light power and the solid\nblack line is given by eq.( 13).\nFigure9shows the resulting fit-parameters Pas red cir-\ncles for the complete set of measurements as a function\nof the laser detuning ∆ λ.We compare the experimental6\ndata points to a Lorentzian,\nP(∆λ) =a1/parenleftbig∆λ\nHWHM/parenrightbig2+1. (12)\nFitting the data marked as circles in fig. 9to eq.(12) re-\nsults ina1= (5.5±0.9)·1015#e−\ns·µm3·nmand a half width\nat half maximum HWHM = (0.5±0.1) nm.\nThe number of photons absorbed by the QW should\nalso be reflected in the rate of emitted PL. We there-\nfore also measured the average PL power as a function\nof the detuning ∆ λ. The results are shown in fig. 9as\nblack diamonds. We compare them to the theoretically\nexpected behavior of a Lorentzian line similar to eq.( 12),\nbut with an additional offset PLoffreflecting the effect\nof non-resonant excitation:\nPLpower(∆λ) =a2/parenleftbig∆λ\n0.5nm/parenrightbig2+1+PLoff.(13)\nThe fit result is shown in fig. 9as a solid black line using\nthe fit-parameters PLpower(∆λ= 0)≈40.8nW,a2=\n31.7±2.1nW and PLoff= 9.2±0.8nW. The value of\nPLoffwas also measured independently by exciting the\nsystem with a blue laser with λexc= 406nm and a laser\npower of PL= 10mW. We obtained a PL power that\nwas compatible with the value given above and found no\nsignificant PL polarization, which is consistent with the\naboveassumptionthatthenon-resonantprocessesshould\nnot contribute to the spin density[ 21].\nD. Influence of the laser beam cross section\nFigure10shows a representative series of Hanle curves\nmeasured for different pumping times Tpumpin an exter-\nnal fieldBext= 0.3T with a laser power of PL= 39mW.\nSimilar to the data shown above, they show a buildup\nof nuclear spin polarization, manifested as a shift of the\nmaximum towards higher fields. Compared to the data\nshownabove,the monochromatorwassettothe PLmax-\nimum of the d= 39.1nm quantum well, and the optical\ndetuning of the laser was ∆ λ= 7nm, which resulted in\ncorrespondingly lower PL polarization. We used smaller\nincrements for Tpumpin order to achieve higher temporal\nresolution of the pumping process. Using eq.( 6), we fit-\ntedeachcurveseparatelytothe theoreticalshape, adding\nan offset of 0 .3%, which may be due to stray light. As\nshown in fig.( 10), we find the expected increase of the\nnuclear field with the pumping time. In addition, the\ncurves broaden and the maximum of the polarization de-\ncreases with increasing pumping time. To understand\nthese additional effects, we consider the intensity distri-\nbution over the laser beam cross section, which we as-\nsume to be Gaussian. As a result of this distribution,\ndifferent sample regions experience different intensities\nresulting in different pumping rates. To describe this\neffect, we write the laser intensity as a function of the\nFIG. 10. Shifted Hanle curves: measured data and calculated\nHanle curves based on the fit-parameters shown in fig. 12.\ndistance rfrom the center of the beam as\nI(r) =I0·e−r\n2σ2\n0, (14)\nwith the maximum intensity\nI0=PL\n2πσ2\n0. (15)\nFor a laser power PL= 39mW and beam width\nσ0= 88µm, we obtain I0= 0.8µW\nµm2. For a numeri-\ncal simulation of the observed Hanle curves, we calcu-\nlated the pumping dynamics and resulting Hanle curves\nh(Bext,I(r)) for annular segments of the laser beam us-\ning eq.(6). The individual curves were then weighted\nwith the PL power I(r)rdremitted by each ring. The\nresulting calculated Hanle curve is\nSC=´\ndrh(Bext,I(r))rI(r)´\ndrrI(r). (16)\nFigure11shows the calculated Hanle curves SCfor in-\ncreasing times Tpump= [0...2000]s. The saturation of\nthe nuclear field Bnucas well as the broadening of the\ncurves and the decrease in the maximal degree of PL\npolarization are clearly visible. Figure 12compares the\npredictions from this simple model with the experimen-\ntal Hanle curves. The decreasing maximal degree of PL\npolarization and the broadening of the curves, repre-\nsented by ∆ B, as well as the shift of the maxima, |Bnuc|,\nare qualitatively well described by the theoretical model.\nFigure13summarizes this model in a different way: it\nshows the variation of the effective field Bext+Bnucover\nthe laser beam cross section. The dependence of the ef-\nfective field on the position ris calculated for two dif-\nferent pumping times, Tpump= 36s (solid red line) and\nTpump= 271s (dashed red line) with an external field of\nBext= 0.5T, and a laser power of PL= 49mW.7\nFIG. 11. Calculated Hanle curves SCfor increasing times\nTpump= [0...2000]s.\nFIG. 12. Comparison between the experimental parameters\n|Bnuc|, ∆BandS0obtained byfittingthe Hanle curvesshown\nin fig.10(a) using eq.( 6) with the result of the simulation.\nV. DISCUSSION AND CONCLUSION\nAs we have shown, our experimental setup allows one\nto measure the optical pumping dynamics of nuclear\nspins in GaAs quantum wells. We presented a simple\ntheoretical model that describes the buildup of the nu-\nclear spin polarization and therefore of the nuclear field\nin a quantitative way and agrees with the experimental\ndata, within the experimental uncertainties. This model\nstarts with the generation of electron spins by the optical\npumping process. The relevant rate of electron spin den-\nsity production is proportional to the laser power PLand\ndecreases with increasing optical detuning ∆ λ. The spin\npolarizationis then transferredfrom the electron spins to\nthe nuclear spins, and we also determined the rate con-\nstant for this process. The experimental data show some\nFIG. 13. Laser intensity and effective field Bext+Bnucwith\nBext= 0.5T as a function of the position in the laser beam\nin units of σ0= 88µm. The solid red line is calculated for\na pumping time, Tpump= 36s and the dashed red line for\nTpump= 271s.\nsignificantdeviationsfromthe simple model, whichcould\nbe explained quantitatively by taking into account that\nthe laser beam does not illuminate the sample homo-\ngeneously, but with a roughly Gaussian profile. These\nresults can be used to prepare the nuclear spin system\ne.g. for optically detected nuclear magnetic resonance\nexperiments.\nACKNOWLEDGMENTS\nWe gratefully acknowledge the support by the In-\nternational Collaborative Research Centre TRR 160\n“Coherent manipulation of interacting spin excitations\nin tailored semiconductors,” funded by the Deutsche\nForschungsgemeinschaft.8\n[1] S. Pfalz, R. Winkler, T. Nowitzki, D. Reuter, A. D.\nWieck, D. H¨ agele, and M. Oestreich, Phys. Rev. B 71,\n165305 (2005).\n[2] M. DyakonovandV. Perel, in Optical Orientation , edited\nby F. Meier and B. Zakharchenya (Elsevier, 1984), vol. 8\nofModern Problems in Condensed Matter Sciences , pp.\n11 – 71.\n[3] A. Ekimov and V. Safarov, Soviet Journal of Experimen-\ntal and Theoretical Physics 12, 1 (1970).\n[4] R. R. Parsons, Phys. Rev. Lett. 23, 1152 (1969).\n[5] R. I. Dzhioev, B. P. Zakharchenya, V. L. Korenev, P. E.\nPak, D. A. Vinokurov, O. V. Kovalenkov, and I. S.\nTarasov, Physics of the Solid State 40, 1587 (1998).\n[6] S. E. Barrett, R. Tycko, L. N. Pfeiffer, and K. W. West,\nPhys. Rev. Lett. 72, 1368 (1994).\n[7] T. Pietraß, A. Bifone, T. R˜ o˜ om, and E. L. Hahn, Phys.\nRev. B53, 4428 (1996).\n[8] A. I. Ekimov and V. I. Safarov, Soviet Journal of Exper-\nimental and Theoretical Physics Letters 15, 179 (1972).\n[9] O. S. Leifson and C. D. Jeffries, Phys. Rev. 122, 1781\n(1961).\n[10] G. Lampel, Phys. Rev. Lett. 20, 491 (1968).\n[11] A. Abragam, The Principles of Nuclear Magnetism (Ox-\nford University Press, 1967).\n[12] D.Paget, G.Lampel, B.Sapoval, andV.I.Safarov, Phys.\nRev. B15, 5780 (1977).\n[13] G. Salis, D. D. Awschalom, Y. Ohno, andH. Ohno, Phys.\nRev. B64, 195304 (2001).\n[14] M.J. Snelling, G.P. Flinn, A.S. Plaut, R.T. Harley,\nA.C. Tropper, R. Eccleston, and C.C. Phillips, Physical\nReview B 44, 11345 (1991).\n[15] V. Kalevich, V. Korenev, and O. Fedorova, Soviet Jour-\nnal of Experimental and Theoretical Physics Letters 52,349 (1990).\n[16] M. Dyakonov and V. Perel, Soviet Journal of Experimen-\ntal and Theoretical Physics 38, 177 (1974).\n[17] M. Dyakonov, V. Perel, V. Berkovits, and V. Safarov,\nSoviet Journal of Experimental and Theoretical Physics\n40, 950 (1975).\n[18] B. Cavenett, Advances in Physics 30, 475 (1981).\n[19] M. Eickhoff, B. Lenzmann, G. Flinn, and D. Suter, Phys.\nRev. B65, 125301 (2002).\n[20] M. Eickhoff, B. Lenzmann, D. Suter, S. E. Hayes, and\nA. D. Wieck, Phys. Rev. B 67, 085308 (2003).\n[21] M. Eickhoff and D. Suter, J. Magn. Reson. 166, 69\n(2004).\n[22] M. Eickhoff, S. Fustmann, and D. Suter, Phys. Rev. B\n71, 195332 (2005).\n[23] C. Weisbuch and C. Hermann, Phys. Rev. B 15, 816\n(1977).\n[24] R. Hannak, M. Oestreich, A. Heberle, W. R¨ uhle, and\nK. K¨ ohler, Solid State Communications 93, 313 (1995).\n[25] M. J. Snelling, E. Blackwood, C. J. McDonagh, R. T.\nHarley, and C. T. B. Foxon, Phys. Rev. B 45, 3922\n(1992).\n[26] J. Shi-Rong, X. Zhong-ying, L. Jin-sheng, L. Chang-pin g,\nX. Ji-zong, and Z. Bao-zhen, Acta Physica Sinica (Over-\nseas Edition) 3, 384 (1994).\n[27] S. Eshlaghi, W. Worthoff, A. D. Wieck, and D. Suter,\nPhys. Rev. B 77, 245317 (2008).\n[28] W. Hanle, Zeitschrift fuer Physik 30, 93 (1924).\n[29] A. Berg, M. Dobers, R. R. Gerhardts, and K. v. Klitzing,\nPhys. Rev. Lett. 64, 2563 (1990)."    },    {        "title": "1003.1812v1.Laser_catalyzed_spin_exchange_process_in_a_Bose_Einstein_condensate.pdf",        "content": "arXiv:1003.1812v1  [quant-ph]  9 Mar 2010Laser-catalyzed spin-exchange process in a Bose-Einstein condensate\nH. Jing1,2, Y. Jiang1, Weiping Zhang2, and P. Meystre3\n1Department of Physics, Henan Normal University, Xinxiang 4 53007, People’s Republic of China\n2State Key Laboratory of Precision Spectroscopy, Departmen t of Physics,\nEast China Normal University, Shanghai 200062, People’s Re public of China\n3B2 Institute and Department of Physics, The University of Ar izona, Tucson, Arizona 85721, USA\n(Dated: November 5, 2018)\nWe show theoretically that it is possible to optically contr ol collective spin-exchange processes\nin spinor Bose condensates through virtual photoassociati on. The interplay between optically in-\nduced spin exchange and spin-dependent collisions provide s a flexible tool for the control of atomic\nspin dynamics, including enhanced or inhibited quantum spi n oscillations, the optically-induced\nferromagnetic-to-antiferromagnetic transition, and coh erent matter-wave spin conversion.\nPACS numbers: 42.50.-p, 03.75.Pp, 03.70.+k\nSpinor Bose-Einstein condensates, consisting of atoms\nwith internal spin states, provide a promising bridge\nbetween atomic, molecular and optical physics (AMO),\nmatter-wave optics, many-body physics, and quantum\ninformation science [1]. Beside the study of their rich\nground-sate properties [2] and spatial spin structures [3–\n5], the magneticcontrolofquantum spin mixinghasbeen\na central topic of investigations for atomic spin systems\n[6–13]. Also, by steering the spin degrees of freedom\nin ultracold quantum gases, the creation of topological\nskyrmions, Dirac monopoles, dark solitons, and quantum\nentanglement have been investigated [14–16].\nIn parallel to these works, which concentrate largely\non the role of external magnetic fields on spinor con-\ndensates, there have also been important developments\non their magneto-optical manipulation. For example,\nZhangetal.[17] studied the spin waves induced by light-\ninduceddipole-dipoleinteractionsinanatomicspinchain\ntrapped in a lattice potential. More recently, advances\nin coherent photoassociation (PA) at ultracold tempera-\ntures [18] were exploited in theoretical and experimental\nstudies of spin mixing and spin-dependent PA in spin-1\ncondensates [19, 20]. In very recent work, Kobayashi\net al.observed the spin-selective formation of spinor\nmolecules in ferromagnetic atoms87Rb [21].\nIn this Rapid Communication, we demonstrate theo-\nreticallytheopticalcontrolofatomicspinmixingin afer-\nromagnetic spin-1 Bose gas. In the proposed method the\nopticalfieldsinducevirtualPAprocessesintheatoms,for\nexample via a dark molecular state. We show that this\nstep, which can be intuitively coined a laser-catalyzed\nspin-exchange process (LCSE), opens up an efficient and\nwell-controlled optical channel for coherent atomic spin\nmixing. By tuning the the strength ratio of these two\nchannels of LCSE and spin-dependent collisions, three\ndifferent regimes can be identified in the laser-controlled\nquantum spin dynamics, i.e., going from the collision-\ndominated regime to the no-spin-mixing regime, and to\nthelaser-inducedantiferromagneticregime,whichisrem-\niniscent of the prominent role of long-range dipole-dipole\ninteraction in a dipolar spin gas (by changing the ratio of\nspin-dependent collision and dipolar interaction). As aresult, and in contrast to the collision-dominated ”single-\nchannel” case, a wealth of important new effects arise in\nthe spin dynamics of the atoms.\nFor concreteness we concentrate on two limiting situ-\nations, the adiabatic far off-resonant regime and the res-\nonant case [22]. Our purpose here is to show that the\ninterplay of two channels, the spin-dependent collisions\nand LCSE, provides a flexible tool for the control of the\natomic spin dynamics, including enhanced and inhibited\nquantum spin oscillations, laser-induced ferromagnetic-\nto-antiferromagnetic (F-AF) transitions, as well as effi-\ncient coherent spin transfer triggered even by quantum\nvacuum noise. This method can be also extended to\nstudy e.g. the optical control of domain formation and of\nspin textures in a spinor gas [4]. As such, optical LCSE-\ncontrolled spinor condensates provide a promising new\ntool for the study of collective chemically-driven quan-\ntum spin dynamics.\nFigure 1 illustrates the process under consideration,\nthe dynamics of a spin-1 atomic condensate resulting\nfrom the virtual PA of two mF= 0 atoms into a molecu-\nlar state, followed by dissociation into a pair of mF=−1\nandmF= +1 atoms. Accounting in addition for spin-\ndependent collisions between atoms, this system is de-\nscribed by the Hamiltonian ( /planckover2pi1= 1)\nH=Hcoll+Hpa, (1)\nwhereHcollandHparefer to spin-dependent collisions\nand the controllable light-assisted interactions, respec-\ntively, with\nHcoll=/integraldisplay\ndr/bracketleftBig\nˆψ†\ni(−/planckover2pi12\n2m∇2+V+Ei)ˆψi+c′\n0\n2ˆψ†\niˆψ†\njˆψjˆψi\n+c′\n2\n2ˆψ†\nkˆψ†\ni(Fγ)ij(Fγ)klˆψjˆψl/bracketrightBig\n, (2)\nHpa=/integraldisplay\ndr/bracketleftBig\n∆′ˆψ†\nmˆψm+Ωp(ˆψ†2\n0ˆψm+ˆψ†\nmˆψ2\n0)\n−Ωd(ˆψ†\nmˆψ+ˆψ−+ˆψ†\n−ˆψ†\n+ˆψm)/bracketrightBig\n. (3)\nHereˆψi(r) is the annihilation operatorof the i-th compo-\nnent atom ( i= 0,±1) andˆψm(r) the corresponding op-2\nFIG. 1: (Color online) Schematic of coherent two-channel\nspin-exchange interactions in an optically-controlled sp in-1\nBose gas. In addition to the familiar collisional channel, t he\noptical channel proceeds via the virtual PA of two |mF= 0/angbracketright\natoms into an intermediate molecular state |m/angbracketright, which disso-\nciates into a pair of |mF= +/angbracketrightand|mF=−/angbracketrightatoms.\nerator for the intermediate molecular field, Vis the trap\npotential,Eiis the Zeeman shift and Fγ=x,y,zis the spin-\n1 matrix [3]. The coefficients c′\n0= 4π/planckover2pi12(a0+ 2a2)/3m\nandc′\n2= 4π/planckover2pi12(a2−a0)/3mgive the strength of the spin-\nconservingtwo-body collisions, with a0,a2the scattering\nlengths of the accessible collision channels, and mis the\natomic mass. The Rabi frequencies Ω p,ddescribe the\nstrength of the photoassociation of mF= 0 atoms and\ndissociation into mF=±1 atoms, and ∆′is the detuning\nbetween the molecular and atomic states.\nWe first consider the adiabatic off-resonant regime by\nassuming that ∆′is the largest parameter in the system.\nFrom∂ˆψm/∆′∂t≃0, we have ˆψm≃Ωdˆψ+ˆψ−−Ωpˆψ2\n0\n∆′, and\nsubstitute this form into the Heisenberg equations of mo-\ntion derived from Eq. (1) [23]. It is easily seen that the\nresulting equations can also be derived from the effective\nHamiltonian\nHeff=Hcoll+/integraldisplay\ndr/bracketleftBig\nΩ′(ˆψ†\n+ˆψ†\n−ˆψ2\n0+ˆψ†2\n0ˆψ+ˆψ−)+O/bracketrightBig\n,(4)\nwith\nΩ′= ΩpΩd/∆′,O=−Ω2\np\n∆′ˆψ†2\n0ˆψ2\n0−Ω2\nd\n∆′ˆψ†\n−ˆψ†\n+ˆψ+ˆψ−.\nIn addition to the familiar spin-dependent collisions, spin\ncoupling now also results from optical LCSE, resulting in\nthe two-channel spin-exchange Hamiltonian\nH=C′/integraldisplay\ndr(ˆψ†\n+(r,t)ˆψ†\n−(r,t)ˆψ0(r,t)ˆψ0(r,t)+h.c.).\nHereC′= Ω′+c′\n2describes the combined effect of spin-\ndependent collisionsand optical LCSE.Clearly, the effec-\ntive spin-coupling strength C′can be negative or positivewith suitable optical parameters, leading to significantly\ndifferent spin dynamics.\nIn the limit where the spatial degrees of freedom de-\ncouple from the spinor dynamics, that is, when the spin\nhealing length is larger than the condensate size, it is\npossible to invoke the single-mode approximation [9, 13]\nwhereˆψi(r,t)→φ(r)ˆai(t),ˆψm(r,t)→φ(r)ˆm(t), where\nφ(r) is the spatial wave function of the condensate with\nˆaiand ˆmbeing the atomic or molecular annihilation\noperators. The rest of this paper presents results ob-\ntained in this approximation. For convenience we also\nintroduce the scaled parameters c0,2=c′\n0,2/integraltext\ndr|φ(r)|4,\nΩ = Ω′/integraltext\ndr|φ(r)|4, ∆ =Ω2\nd\n∆′/integraltext\ndr|φ(r)|4, andτ=c0nt,\nwherenis the initial atomic density.\nThe mean-field evolution of the spin-0 atomic popu-\nlationn0is illustrated in Fig. 2 for several values of\nC= Ω +c2and for the initial state ( n+, n0, n−) =\n(0.05,0.9,0.05). The specific example of atoms87Rb\nis considered here, with a0= (101.8±2)aBanda2=\n(100.4±1)aB[24], where aBis the Bohr radius. In our\ncalculations, we assume the optical detuning ∆′= 100Ωp\nand Ωd= 10Ωp, which is reasonable for the adiabatic\noff-resonant case. The typical initial atomic density\nn∼1014cm−3,corresponding to c0n∼9700Hz.\nSignificantly different regimes are reached by varying\nthe optical detuning ∆′: (i) In the collision-dominated\nregime|c2|>|Ω|, the population of spin-0 atoms is al-\nwayslargerthan its initial value 0 .9. In this perturbed\nregime, the spin coupling is still ferromagnetic ( C<0);\n(ii) ForC= 0, i.e. Ω = −c2(for ∆′>0) the two channels\n(spin-dependent collisions and LCSE) interfere destruc-\ntively, leadingtofrozenorinhibitedspinmixing; (iii)The\nsignofCcan be reversedby tuning the laserfields, result-\ning in an effective antiferromagnetic regime C>0. This\nis reminiscent of the F-AF transition induced by long-\nrange dipolar interactions in a spinor gas [10]. In this\nreversed regime, the atomic spin oscillations ( |Ω|>|c2|)\nturn out to be belowthe initial value 0 .9.\nThe mean-field atomic spinor dynamics is well de-\nscribed by a nonrigid pendulum model [3]. By expressing\nthec-number amplitudes aiin terms of real amplitudes\nand phases, i.e. a±,0=√n±,0e−iθ±,0, the energy func-\ntional of the optically-controlled spinor system can be\nderived as\nE=q(1−n0)+C/radicalbig\n(1−n0)2−m2cosθ\n+c2n0(1−n0)+∆\n4n0(2−n0)−Ω2\n∆n2\n0(5)\nwhereqdenotes the quadratic Zeeman effect, θ=θ++\nθ−−2θ0is the relative phase of the spin components and\nm=n+−n−is the atomic magnetization. The last two\nterms in Eq. (5) account for the optical energy shift.\nThe effective spin-coupling parameter Chas an impor-\ntant impact on the properties of the system, as already\nmentioned. Figure 3 plots equal-energy contours in the\nphase space ( θ,n0), for several values of Cform= 0 and\nq= 0.01, corresponding to the fixed magnetic field about3\n460mGin our specific example, which is larger than the\nresonance magnetic field Bres∼330 mG, see [27]. We\nhave also carried out numerical simulations for values of\nqcorrespondingto B <B resand found similar results for\nappropriate values of C. For our parameters only open\ntrajectories exist in the absence of laser fields ( C=c2) as\nwell as for perturbed case C= 0.5c2. In contrast, the ”re-\nversed” cases C=−0.5c2andC=−c2are characterized\nby the coexistence of closed and open trajectories.\n0 500 1000 1500 20000.6 0.70.80.91\ntimePop ulationsC=0.5c2\nC=−c2C=c2 C=0\nC=−10c2\nFIG. 2: (color online) Population of spin-zero atoms with\n(solid lines) or without (dashed lines) laser fields. In all c ases\nthe atomic density is 2 ×1014cm−3[25], and the time is scaled\ntoτ=c0nt, corresponding to the time unit 0 .1ms.\nWe remark that Eq. (5) also permits to study the in-\nstabilities and domain formation in spin-1 atoms [3, 5].\nWe actually have done this and found that the LCSE-\ndominated case is again different from the collision-\ndominated case [26].\nWe now turn to the resonant situation, where one can\nexploit the existence of an atom-molecule dark state to\npreventthebuild-upofasignificantmolecularpopulation\nthroughout the process [18]. This two-photon resonant\ncontrol of atomic spinor dynamics represents a promis-\ning new step in developing the field of all-optical manip-\nulation of matter-wave spins. In a mean-field approach\nwhereˆψi→√nφi[19] we find\ndφ+\ndτ=−i[c2(|φ+|2+|φ0|2−|φ−|2)]φ+\n−ic2φ2\n0φ∗\n−+iΩ′\ndφmφ∗\n−−i(Θ+δ)φ+,\ndφ0\ndτ=−i[c2(|φ+|2+|φ−|2)]φ0−2ic2φ+φ−φ∗\n0\n−2iΩ′\npφmφ∗\n0\ndφ−\ndτ=−i[c2(|φ−|2+|φ0|2−|φ+|2)]φ−,\n+iΩ′\ndφmφ∗\n+,\ndφm\ndτ=iΩ′\ndφ+φ−−iΩ′\npφ2\n0−(iδ+γ)φm, (6)\nwhere Ω′\np= Ωp/c0√n, Ω′\nd= Ωd/c0√n, Θ = ∆′/c0n,\nδ=δ′/c0n,δis the frequency difference between the(a) (b)\n(c) (d)C=c2\nθC=0.5c2\n0n0n\nθC=c2− C= c20.5−\nFIG. 3: (color online) Equal energy contours of Ewithout (a)\n(C=c2) and with (b) ( C= 0.5c2), (c) (C=−0.5c2, and (d)\n(C=−c2)) optically-controlled SER for q= 0.01 andm= 0.\nmolecular state and the atomic hyperfine states, and the\nphenomenologicaldecayrate γaccountsfor the lossofin-\ntermediate molecules. Following the method of Ref. [22],\nwe find that Eqs. (6) admit a steady-state coherent pop-\nulation trapping (CPT) solution ( nm= 0),\nn±,s=1\n2+Ω′\nd/Ω′p, n0,s=1\n1+2Ω′p/Ω′\nd,\nunder the generalized two-photon resonance condition\nΘ(t) =−δ+c2/bracketleftbig\n2(n+,s+n−,s)+2√n+,sn0,s−4n0,s/bracketrightbig\n.\nThis time-dependent resonance condition is determined\nby the CPT steady-state values of the atomic density,\nwhich thereby can be tuned by choosing suitable pump-\ning and (time-dependent) dumping laser fields.\nFigure 4 shows the populations of the atomic sublevels\nfor Ω′\np= 1 (corresponding to 9700 Hz) and\nΩ′\nd(t) = Ω′\nd,0sech(t/t0),\nwith Ω′\nd,0= 40,t0= 20, the other parameters being as in\nFig. 2. In contrast to the off-resonant spin oscillations,\nwe have now a full transfer of population from the initial\nstate, say |mF= 0/angb∇acket∇ightto a final coherent superposition\nof|mF=±1/angb∇acket∇ighthyperfine states. We have carried out\nnumerical simulations for a large set of initial seedsof\nspin-±atoms and found that the stable spin conversion\nis always possible for e.g. δ= 3. The departure of the\nspin-±populations from the ideal CPT value is due to\nthe fact that only an approximate adiabatic condition\nexists for the CPT state [22].4\n0 50 100 150 200 250 30000.20.40.60.81\ntimePop ulationsn± (10−5)n± (10−4)\nn± (10−6)n± (0.05)n0(0.9)\nCPTn0 (CPT)\nFIG. 4: (color online) Atomic populations in the resonant\nregime for87Rb, with detuning δ= 3 and γ= 1. To illustrate\ntherole of small seeds ofatoms n±, several values of theinitial\npopulations of the corresponding spin states have been con-\nsidered, as denoted in the brackets. The line labeled ”CPT”\nshows the ideal population of the target states |mF=±1/angbracketright.\nFollowing the preparation of all atoms in the spin-0\nstate, Klempt et. al[28] have recently studied the mag-\nnetic field dependence of the fraction of atoms trans-\nferred to the spin- ±states. This transfer process is of\nquantum nature as the mean-field equations break down\nfor an initial vacuum of spin- ±atoms. Our system also\npermits to study quantum spin transfer with quantum-\nnoise-triggered seed. Followingthestrategyfamiliarfrom\nquantum optics, seee.g. [1, 29, 30], we separatethe prob-lem into an initial stage dominated by quantum noise\nfollowed by a classical stage that arises once the target\nstate has acquired a macroscopic population [31]. We\nhave performed an analysis of the present system along\nthese lines, and it confirms that its evolution from quan-\ntum noise is also efficient, with dynamics similar to those\nwith the classical seedn ±= 10−5in Fig. 5 [26, 31].\nIn conclusion, we havedemonstrated theoretically that\ntheinterplayofLCSEandspin-dependentcollisionsleads\nto a variety of remarkable effects such as inhibited spin\noscillations, laser-induced F-AF transition, and quantum\nspin transfer between different components. The optical\ntuning or even elimination of spin-exchange interactions\nmay be used to e.g. probe the weak signal of dipolar\ninteractions which is generally far smaller than the spin-\ndependent collisions [10]. The optical control of atomic\nspin interactions is somewhat reminiscent of the role of\nmagnetic Feshbach resonances in the study of ultracold\ngases. This work hints at promising possibilities to carry\nout an all-optical control of atomic spins [32]. When\ncompared to the magnetic control of spinor atoms [1-\n16], LCSE offers an exciting new route to the study of\natomic spin coupling independent of the collisions. Ulti-\nmately, magneto-optical methods will likely combine the\nbest ofboth opticaland purelymagneticapproaches. Fu-\nture work will study the magneto-optical control of spa-\ntial spin structures and will explore possibilities of spin-\ndependent ultracold chemistry or atom-molecule hybrid\nspin mixing.\nP.M.is supportedby the U.S. OfficeofNavalResearch,\nbytheU.S. NationalScienceFoundation, andbythe U.S.\nArmy Research Office. W.Z. is supported by the NSFC\nand by the 973 project. H.J. is supported by the NSFC.\n[1] P.Meystre, Atom Optics , (Springer-Verlag, Berlin, 2001).\n[2] T.-L. Ho, Phys. Rev. Lett. 81, 742 (1998); T. Ohmi and\nK. Machida, J. Phys. Soc. Jpn. 67, 1822 (1998).\n[3] W. Zhang, et al., Phys. Rev. Lett. 95, 180403 (2005).\n[4] L. E. Sadler, et al., Nature (London) 443, 312 (2006).\n[5] Q.GuandH.-B.Qiu, Phys.Rev.Lett. 98, 200401 (2007).\n[6] C. K. Law, H. Pu, and N. P. Bigelow, Phys. Rev. Lett.\n81, 5257 (1998).\n[7] L. I. Plimak, et al., Opt. Commun. 264, 311 (2006).\n[8] L. Chang, et al., Phys. Rev. Lett. 99, 080402 (2007).\n[9] M. S. Chang, et al., Nature Phys, 1, 111 (2005).\n[10] S. Yi, L. You, and H. Pu, Phys. Rev. Lett. 93, 040403\n(2004).\n[11] J. Stenger, et al., Nature (London) 396, 345 (1998).\n[12] H.-J. Miesner, et al., Phys. Rev. Lett. 82, 2228 (1999).\n[13] A. T. Black, et al., Phys. Rev. Lett. 99, 070403 (2007);\nY. Liu,et al.,ibid.102, 125301 (2009).\n[14] V. Pietil¨ a and M. M¨ ott¨ onen, Phys. Rev. Lett. 103,\n030401 (2009).\n[15] H. Pu and P. Meystre, Phys. Rev. Lett. 85, 3987 (2000);\nL.-M. Duan, et al.,ibid.85, 3991 (2000).\n[16] A. Widera, et al., Phys. Rev. Lett. 92,160406 (2004).\n[17] W. Zhang, et al., Phys. Rev. Lett. 88, 060401 (2002).[18] K. Winkler, et al., Phys. Rev. Lett. 95, 063202 (2005).\n[19] J. Cheng, H. Jing, and Y.-J. Yan, Phys. Rev. A 77,\n061604(R) (2008).\n[20] C. D. Hamley, et al., Phys. Rev. A 79, 023401 (2009).\n[21] J. Kobayashi et al., Appl. Phys. B, 95, 37 (2009).\n[22] H.-Y. Ling, et al., Phys. Rev. Lett. 93, 250403 (2004).\n[23] J. Calsamiglia, M. Mackie, and K-A. Suominen, Phys.\nRev. Lett. 87, 160403 (2001).\n[24] E. G. M. van Kempen et al.,Phys. Rev. Lett. 88, 093201\n(2002).\n[25] Y. Liu, et al., Phys. Rev. Lett. 102, 225301 (2009).\n[26] The details of the results will be published elsewhere.\n[27] J. Kronj¨ ager, et al., Phys. Rev. Lett. 97, 110404 (2006);\nJ. M.-Petit, Phys. Rev. A 79, 063603 (2009).\n[28] C. Klempt, et al.,Phys. Rev. Lett. 103, 195302 (2009).\n[29] M. G. Moore and P. Meystre, Phys. Rev. Lett. 83, 5202\n(1999); H. Pu and P. Meystre, ibid. 85, 3987 (2000).\n[30] C. P. Search and P. Meystre, Phys. Rev. Lett. 93, 140405\n(2004).\n[31] X. Li, et al., Phys. Rev. Lett. 101, 043003 (2008); H.\nJing, J. Cheng, and P. Meystre, ibid.101, 073603 (2008).\n[32] T. Moriyasu, et al., Phys. Rev. Lett. 103, 213602 (2009)."    },    {        "title": "1101.5301v1.Magnetic_excitations_in_one_dimensional_spin_orbital_models.pdf",        "content": "arXiv:1101.5301v1  [cond-mat.str-el]  27 Jan 2011Magnetic excitations in one-dimensional spin-orbital mod els\nAlexander Herzog and Peter Horsch\nMax-Planck-Institut f¨ ur Festk¨ orperforschung, Heisenb ergstrasse 1, D-70569 Stuttgart, Germany\nAndrzej M. Ole´ s\nMax-Planck-Institut f¨ ur Festk¨ orperforschung, Heisenb ergstrasse 1, D-70569 Stuttgart, Germany and\nMarian Smoluchowski Institute of Physics, Jagellonian Uni versity, Reymonta 4, PL-30059 Krak´ ow, Poland\nJesko Sirker∗\nDepartment of Physics and Research Center OPTIMAS,\nUniversity of Kaiserslautern, D-67663 Kaiserslautern, Ge rmany and\nMax-Planck-Institut f¨ ur Festk¨ orperforschung, Heisenb ergstrasse 1, D-70569 Stuttgart, Germany\n(Dated: November 6, 2018)\nWe study the dynamics and thermodynamics of one-dimensiona l spin-orbital models relevant for\ntransition metal oxides. We show that collective spin, orbi tal, and combined spin-orbital excitations\nwith infinite lifetime can exist, if the ground state of both s ectors is ferromagnetic. Our main focus\nis the case of effectively ferromagnetic (antiferromagneti c) exchange for the spin (orbital) sector,\nrespectively, and we investigate the renormalization of sp in excitations via spin-orbital fluctuations\nusing a boson-fermion representation. We contrast a mean-fi eld decoupling approach with results\nobtained by treating the spin-orbital coupling perturbati vely. Within the latter self-consistent ap-\nproach we find a significant increase of the linewidth and addi tional structures in the dynamical\nspin structure factor as well as Kohn anomalies in the spin-w ave dispersion caused by the scattering\nof spin excitations from orbital fluctuations. Finally, we a nalyze the specific heat c(T) by compar-\ning a numerical solution of the model obtained by the density -matrix renormalization group with\nperturbative results. At low temperatures Twe find numerically c(T)∼Tpointing to a low-energy\neffective theory with dynamical critical exponent z= 1.\nPACS numbers: 75.10.Pq, 75.30.Et, 05.10.Cc, 05.70.Fh\nI. INTRODUCTION\nIn condensed matter systems the coupling between\ndifferent degrees of freedom often plays an important\nrole. The electron-phonon coupling, for example, can\nlead to the formation of renormalized quasiparticles, so-\ncalled polarons,1,2as well as to phase transitions like the\nPeierls instability.3In recent years, the coupling between\nfermionic and bosonic degrees of freedom has also been\nintensely studied in Bose-Fermi (BF) mixtures of ultra-\ncold quantum gases.4–8\nCoupled degrees of freedom also seem to be important\nin certain transition metal oxides where the low-lying\nelectronic states (termed “orbitals”) are not completely\nquenchedsothattemperatureordopingcanleadto asig-\nnificant redistribution of the valence electron density. In\ninsulating materials with partly filled degenerate orbitals\nthe superexchange between the magnetic degrees of free-\ndom then becomes a function of the orbital occupation.\nThis leads to models of coupled spin and orbital degrees\nof freedom as, for example, the Kugel-Khomskii model\nwhere the orbitals are represented by a pseudospin.9\nLaMnO 3,10–13LaTiO 3,14–17and LaVO 3or YVO 3,18–25\nare well-known examples for compounds believed to be\ndescribed by effective spin-orbital models. They exhibit\na wide range of fascinating effects ranging from colossal\nmagnetoresistance12to temperature-induced magnetiza-\ntion reversals.18,19Common to all these transistion metal oxides is a lift-\ning of the fivefold degeneracy of the dorbitals into two eg\norbitals ( x2−y2and 3z2−r2) and three t2gorbitals ( xy,\nyz, andxz). This splitting is due to the perovskite struc-\nture where oxygen ions, O2−, form octahedra around the\ntransition metal ions which are therefore exposed to an\napproximately cubic crystal field. As a consequence, the\norbitals pointing towards the oxygen ions are energeti-\ncally unfavorable.\nIn YVO 3thet2gorbitals are occupied by two electrons\nforming an effective spin S= 1 due to large Hund’s rule\ncoupling. The material is an insulator with an interest-\ning phase diagram.18–20,22At temperatures below 77 K\nthe system is in a G-type antiferromagnetic (AF) phase,\ni.e., AF in all three directions. In a range of higher tem-\nperatures, 77 K < T <116 K, the magnetic structure is\nC-type with spins ordering antiferromagnetically in the\n(a,b) plane and ferromagnetically along the caxis. The\nsurprising fact that the ferromagnetic (FM) exchange in-\ntegral in this phase is much larger than the AF exchange\ninteractions in the ( a,b) plane22was explained by strong\norbital fluctuations along the c-axis chains that trigger\nferromagnetism.21In theC-type phase a neutron scat-\ntering study revealed that the magnon dispersion along\nthe FMc-axis chainsconsists of twobranches. This split-\nting has been interpreted as due to a periodic modula-\ntion of the FM exchange along these chains caused by\nan entropy gain of fluctuating orbital occupations.22,23\nSupport for an orbital Peierls effect in this material was2\n. . .j−1jj+ 1. . .|FS,Fτ/angbracketright=(a)\n. . .j−1jj+ 1. . .S−\nj|FS,Fτ/angbracketright=(b)\n. . .j−1jj+ 1. . .S−\njτ−\nj|FS,Fτ/angbracketright=(c)\nFIG. 1: (a) Ground state of the FM spin-orbital model,\nEq. (1.1), (b) a spin excitation, and (c) a coupled spin-orbi tal\nexcitation. The two orbitals per site are assumed to be de-\ngenerate (the splitting is only for clarity of presentation ).\ngiven by numerical investigations23,24and a mean-field\n(MF) decoupling approach.26\nHowever, the dynamics in such systems cannot easily\nbe studied numerically and a MF decoupling is unable\nto explain important features of coupled spin-orbital de-\ngrees of freedom27as can be seen in the following ex-\nample. Consider the one-dimensional (1D) spin-orbital\nHamiltonian28–30\nH=J/summationdisplay\nj(Sj·Sj+1+x)(τj·τj+1+y),(1.1)\nwithferromagnetic superexchange interaction J <0,\nwhereSjandτjare spin Sand pseudospin τoperators\nat sitej, respectively, and xandyare constants. For\ngeneralx,ythe model has an SU(2) ⊗SU(2) symmetry\nand exhibits an additional Z2symmetry, interchanging\nspin and orbital sectors, if x=y. ForS=τ= 1/2 and\nx=y= 1/4 the symmetry is enlarged to SU(4).31\nIn the following we discuss the case S=τ= 1/2\nand we choose xandysuch that the ground state/vextendsingle/vextendsingleFS,Fτ/angbracketrightbig\nis given by fully polarized spin and orbital sec-\ntors as illustrated in Fig. 1(a). Using the equation of\nmotion method we find that the state S−\nj/vextendsingle/vextendsingleFs,Fτ/angbracketrightbig\nshown\nin Fig.1(b) is always an elementary excitation with dis-\npersionωS(q) =|J|(1 + 4y)(1−cosq)/4. Analogously,\nthe orbital flip is also an elementary excitation with\nωτ(q) =|J|(1+4x)(1−cosq)/4. However,thesecollective\nexcitations are not the only undamped elementary exci-\ntations of the Hamiltonian, Eq. ( 1.1). In addition, a cou-\npled spin-orbital excitation S−\njτ−\nj, as shown in Fig. 1(c)\nmay exist. To investigate this issue we again apply theequation of motion method leading to\n/braceleftbig/bracketleftbig\nH,S−\njτ−\nj/bracketrightbig\n−|J|[Cj(x,y)+Dj(x,y)]/bracerightbig\n|FS,Fτ/an}bracketri}ht= 0,\n(1.2)\nwhere\nCj(x,y) = (x+y)S−\njτ−\nj−1\n4/parenleftbig\nS−\nj−1τ−\nj−1+S−\nj+1τ−\nj+1/parenrightbig\nis the coherent part, and\nDj(x,y) =−1\n2/bracketleftbigg/parenleftbigg\nx−1\n4/parenrightbigg/parenleftbig\nS−\njτ−\nj−1+S−\njτ−\nj+1/parenrightbig\n+/parenleftbigg\ny−1\n4/parenrightbigg/parenleftbig\nS−\nj−1τ−\nj+S−\nj+1τ−\nj/parenrightbig/bracketrightbigg\n(1.3)\ncontains terms which lead to a spatial decoherence of\nthe excitation. Hence in order to have a fully confined\nspin-orbital excitation, Dj(x,y) has to vanish, which ob-\nviously is the case if x=y= 1/4. From this we draw\nthe conclusion that confined spin-orbital excitations are\nrather the exception than the rule, relying on the par-\nticular value of the constants. If we introduce the Bloch\nstates\nΨSτ(q) =1\nN/summationdisplay\njeijqS−\njτ−\nj|FS,Fτ/an}bracketri}ht,(1.4)\nthe dispersion of the coupled spin-orbital excitation for\nx=y= 1/4 is given by\nωSτ(q) =|J|(1−cosq)/2. (1.5)\nThus for x=y= 1/4 we have ωS(q) =ωτ(q) =ωSτ(q),\ni.e., the dispersions of all three elementary excitations\nare degenerate.32Interestingly, they all lie withinthe\ncontinuum of spin-orbital excitations given by γ(q,p) =\nωS(q/2+p) +ωτ(q/2−p). The Hamiltonian, however,\ndoes not allow for a decay of these three elementary ex-\ncitations in the ferromagnetic case. For the case of the\ncoupled spin-orbitalexcitation we see from Eq. ( 1.3) that\nsuch a decay becomes possible once we move away from\nthe special point x=y= 1/4. Our conclusions partly\ndiffer from the ones presented in Ref. 30, where the cou-\npledspin-orbitalexcitationisconsideredasaboundstate\nbelow the spin-orbital continuum.33\nThere are several ways to generalize the S=τ= 1/2\ncasetoarbitraryspin-andpseudospinquantumnumbers.\nIfwe startagainfrom fully polarizedspin and orbitalsec-\ntors and only demand that S−\njτ−\nj/vextendsingle/vextendsingleFS,Fτ/angbracketrightbig\nstays confined,\nwe find the condition x=S(1−S) andy=τ(1−τ).\nAnother way of generalizing the S=τ= 1/2 case to ar-\nbitrarySandτrelies on the fact that the Hamiltonian,\nEq. (1.1), withx=y= 1/4 is equivalent to\nH=J\n4/summationdisplay\njDS=1\n2\nj,j+1Dτ=1\n2\nj,j+1. (1.6)\nwhereDσ=1\n2\nj,lwithσ∈ {S,τ}is Dirac’sexchangeoperator\nforσ= 1/2.34A generalization of this exchange operator3\nto arbitrary spin has been discussed by Schr¨ odinger.35\nFor instance for σ= 1 the spin (pseudospin) exchange\noperator is given by Dσ=1\nj,l= (σj·σl)2+σj·σl−1.36\nThe Hamiltonian in Eq. ( 1.6) with arbitrary spin (pseu-\ndospin) quantum number does not only keep the sin-\ngle spin-orbital flip confined as in the generalization dis-\ncussed above but rather all spin-orbital excitations of the\ntype (S−\nj)mS(τ−\nj)mτ/vextendsingle/vextendsingleFS,Fτ/angbracketrightbig\nwheremSandmτare the\nmultiplicities forspin and pseudospin, respectively. Since\na MF decoupling solution treats the spin-orbital chain\nas two separate chains with effective exchange parame-\nters determined self-consistently, the physics of coupled\nspin-orbital excitations cannot be captured within this\napproach.\nThe purpose of this paper is to study the importance\nofcoupled spin-orbitalexcitations in aspin-orbitalmodel\nwithantiferromagnetic superexchange and anisotropic\norbital exchange. This case is intriguing as spin and or-\nbital degrees of freedom may be expected to be strongly\nentangled.27In fact, it has been shown that compos-\nite spin-orbital excitations have to be analyzed together\nwith spin waves in systems with active egorbitals, such\nas for instance KCuF 3.37,38This follows from the non-\nconservation of the orbital flavor in hopping processes\nwhich implies that spin excitations are not independent\nand may occur in general together with an orbital flip.\nHere we will consider an anisotropic generalization of the\nspin-orbital model ( 1.1) with parameters x,ysuch that\nthespinsstillorderferromagneticallyinthegroundstate.\nThe orbital sector, however, will no longer be in a fully\npolarized state due to the AF superexchange which fa-\nvors orbital alternation. Independent spin, orbital, and\ncoupled spin-orbital excitations of collective type, as dis-\ncussed above, therefore can no longer exist. We will fo-\ncus, in particular, on the question how spin excitations\nare modified by the presence of orbitals in this case.\nThe paper is organizedas follows: In Sec. IIwe present\na generalization of the spin-orbital model, Eq. ( 1.1), to a\nmodelwithanisotropicorbitalexchange. Fortheextreme\nquantumlimit oftheorbitalsectorinteractingviaanXY-\ntypecouplingwethen deriveaneffective BFmodelwhich\nresembles models considered in the context of ultracold\nBF gases. By using a density matrix renormalization\ngroupalgorithmapplied totransfermatrices(TMRG) we\nexemplarily investigate numerically the crossover from\nAF to FM correlations. In Sec. IIIwe discuss the MF\ndecoupling approach. We allow for a dimerization in\nboth sectorsanddiscusstheobtainedMF phasediagram.\nIn Sec.IVwe summarize the results for the dynamical\nspin structure factor S(q,ω) obtained within the modi-\nfied spin-wave theory (MSWT) for the uniform FM spin\nchain.39,40In Sec.Vwe formulate an approach where\nthe coupling between spins and orbitals is treated per-\nturbatively. The approach is based on representing the\nspins by bosons using the MSWT41,42and the orbitals by\nJordan-Wigner fermions. Finally, in Sec. VI, we consider\nthe effects of coupled spin-orbital degrees of freedom on\nthe thermodynamics of the system. We focus, in partic-ular, on the specific heat as a function of temperature\nand compare perturbative results with numerical data\nobtained by TMRG. In Sec. VIIwe summarize and dis-\ncuss our results. The Appendix provides details of the\nperturbative approach.\nII. SPIN-ORBITAL MODEL AND MAPPING\nONTO A BOSON-FERMION MODEL\nA. One-dimensional spin-orbital model\nWe will focus here on the physical situation realized\nin the vanadium perovskites, such as YVO 3, where the\nsuperexchange interactions are antiferromagnetic. In\nYVO3the twodelectrons occupy the lower lying t2gor-\nbitals while the egorbitals are empty. From electronic\nstructure calculations43–45it is concluded that the t2g\norbitals are split into a lower-lying xyorbital level and\na higher-lying doublet of xzandyzorbitals. Therefore\nthexyorbital will always be occupied by one electron,\ncontrolling the AF correlations in the ( a,b) planes. The\nlarge Hund’s coupling JH(normalized to the interatomic\nColoumb interaction U) present in YVO 3will support\nparallel alignment of electronic spins at V3+ions ind2\nconfigurations,46leading to S= 1 spins. Therefore, the\nremaining electron will be placed in one of the two other\norbitals{xz,yz}which constitutes the τ= 1/2 orbital\ndegree of freedom. On this basis, an effective spin-orbital\nsuperexchange model for YVO 3with spins S= 1 has\nbeen derived.21Here we shall study the 1D spin-orbital\nmodel extracted from it for the c-axis.26\nThe simplest Hamiltonian for the c-axis FM chains in\nYVO3takingJHinto account is given by Eq. ( 1.1) with\nJ >0,x= 1, and y= 1/4−γH. HereγHis proportional\nto Hund’s coupling JH, supporting FM correlations in\nthe spin sector. For S= 1 and realistic values of Hund’s\ncoupling for vanadates, γH∼0.1, numerical investiga-\ntions ofthis model showedstrongbut short-rangeddimer\ncorrelations in a certain finite temperature range caused\nby the related entropy gain although the ground state is\nuniformly FM.24The same Hamiltonian was also studied\nusing a MF decoupling scheme.26Within this approach a\nfinite temperature phasewith dimer orderin both sectors\nwasfound. However,asdiscussedintheintroduction, the\nMF decoupling approach has severe limitations as it does\nnot take the coupled spin-orbital dynamics into account.\nWe start from a generalizationof Eq. ( 1.1) which reads\nHSτ(Γ) =J/summationdisplay\nj(Sj·Sj+1+x)/parenleftbig\n[τj·τj+1]Γ+y/parenrightbig\n,(2.1)\nwith\n[τj·τj+1]Γ≡τj·τj+1−Γτz\njτz\nj+1.(2.2)\nThe calculations presented here are valid for general S\nandx, but we will, unless stated otherwise, only address4\n-1.5-1-0.500.51<SiSi+1>\n0 1\nT/J-0.4-0.200.20.4<τx\niτx\ni+1+τy\niτy\ni+1>y=0.4y=-0.3\ny=0.4\ny=-0.3\nFIG. 2: (Color online) Nearest-neighbor spin and orbital co r-\nrelation functions for the spin-orbital model (2.1) with Γ = 1\nas a function of temperature Tin units of J(we setkB= 1).\nIn both panels y= 0.4, 0.3, 0.2, 0.15, 0.1, 0.05, 0.0, −0.1,\n−0.2,−0.3 in arrow direction. The spin correlations switch\nfrom AF to FM at y= 0.1 (dashed lines). The dotted lines\nin the upper (lower) panel correspond to the limiting values\n1 and−1.4015 (−1/πand 1/π), respectively.\nthe caseS= 1,x= 1 relevant for YVO 3in the following.\nFor Γ = 1 the pseudospin sectorreduces to an XY model.\nIn Fig.2the nearest-neigbor spin and orbital corre-\nlation functions for Γ = 1 as a function of temperature\nfor various parameters yobtained by TMRG are shown.\nThis method allows us to obtain thermodynamic quan-\ntities for 1D quantum systems directly in the thermody-\nnamic limit.47–49Fory/lessorsimilar0.1 the ground state has ferro-\nmagneticallyalignedspins. Thephasetransitionbetween\nthefullypolarizedFMstateandastatewithAFspincor-\nrelations at y≈0.1 is first order. In the limit y≫1 the\nvalue/an}bracketle{tSjSj+1/an}bracketri}ht ≃ −1.4015 for a Haldane S= 1 Heisen-\nberg chain is reached,50while the orbital correlations ap-\nproach/an}bracketle{tτx\njτx\nj+1+τy\njτy\nj+1/an}bracketri}ht →1/π. We note that the FM\nground state is lost at y≃0.1 both in the model with\nΓ = 1 investigated here as well as in the model with an\nisotropic pseudospin sector (Γ = 0).24,51For the model\nwith Γ = 1, however, we have a direct phase transition\nfrom the FM to the Haldane phase while for the isotropic\nmodel an orbital valence bond phase is intervening be-\ntween these two phases.51\nThe isotropic model ( 2.1), Γ = 0, has also been in-\ntensely studied for S=τ= 1/2. Here the phase diagram\nis more complex than in the S= 1,τ= 1/2 case.52,53Forx= 1 the FM spin state is again found to be stable\nfory/lessorsimilar0.1. However, now the transition at y∼0.1 is to\na gapless “renormalized SU(4)” phase followed by a fur-\nther phase transition at larger yinto a dimer phase. The\nphase with ferromagnetically polarized orbitals is absent\nbecause/an}bracketle{tSj·Sj+1+x/an}bracketri}ht>0 forx= 1 but is again present\nfor large yifx/lessorsimilarln2−1/4.\nB. Boson-fermion model\nFor the 1D spin-orbital model, Eq. ( 2.1), at the point\nΓ = 1, we will now derive an effective BF model which\nwill be used as a starting point for the perturbative\napproach. First, applying the Jordan-Wigner transfor-\nmation the orbital part is mapped onto a free fermion\nmodel (for Γ /ne}ationslash= 1 the pseudospins map onto interacting\nfermions). The spin part of the spin-orbital Hamiltonian\n(2.1) will be represented by bosons. Concentrating on\nthe case where the spin part is ferromagnetically polar-\nized in the ground state, we can treat the spin sector by\nthe MSWT.41,42To this end, we introduce bosonic op-\nerators by a Dyson-Maleev transformation. If we retain\nbosonic operators only up to quadratic order we end up\nwith\nH≡ HSτ(1)−JN(S2+x)y≃H0+H1.(2.3)\nHereH0is already diagonal\nH0=/summationdisplay\nkωB(k)b†\nkbk+/summationdisplay\nqωF(q)f†\nqfq,(2.4)\nwithf†\nqandfq(b†\nkandbk) being the fermionic (bosonic)\ncreation and annihilation operators, respectively. The\nmagnon dispersion is given by\nωB(k) = 2JS|y|(1−cosk), (2.5)\nand the fermion dispersion reads\nωF(q) =J(S2+x)cosq. (2.6)\nThe spinless fermions fill up the Fermi sea between the\nFermi points at kF=±π/2.\nFor FM spin chains usual spin-wave theory has to be\nmodified by a Lagrange multiplier µacting as a chemical\npotential which enforces the Mermin-Wagner theorem of\nvanishing magnetization at finite temperature41\nS=1\nN/summationdisplay\nk/angbracketleftBig\nb†\nkbk/angbracketrightBig\n. (2.7)\nThermodynamic quantities calculated with this method\nare in excellent agreement with the exact Bethe ansatz\nsolution for the uniform chain as well as with numerical\nTMRGdataforthedimerizedFMchainfortemperatures\nup toT∼ |Jeff|S2, withJeffbeing the effective exchange\nconstant of the model under consideration.26,41,42,545\nThe interacting part couples bosons and fermions and\nreads\nH1=1\nN/summationdisplay\nk1,k2,qωBF(k1,k2,q)b†\nk1bk2f†\nqfk1−k2+q,(2.8)\nwith the vertex\nωBF(k1,k2,q)≡JS[cos(k2−q)+cos(k1+q)\n−cos(k1−k2+q)−cosq].(2.9)\nThe Hamiltonian H=H0+H1withH0andH1given\nby Eqs. ( 2.4) and (2.8), supplemented by the constraint\n(2.7), is an effective BF representation valid at low tem-\nperatures. We will investigate this model in Sec. Vtreat-\ning the BF coupling perturbatively.\nIII. MEAN-FIELD DECOUPLING\nThe spin-orbital model ( 2.1) contains rich and inter-\nesting physics. A first attempt to understand the prop-\nerties of the model is to apply a MF decoupling which\nneglects the coupled spin-orbital degrees of freedom and\ntreats the spin-orbital chain as two separate chains with\neffective coupling constants which have to be determined\nself-consistently. Note, however,that this treatment does\nnot involve site variables as in the classical Weiss-MF\ntheory but takes the correlations on a bond as relevant\nvariables. Interestingly, these expectation values never\nvanish, which makes them useful particularly in cases\nwithout long-range order.\nA. Decoupling into spin and orbital chain\nApplying a MF decoupling and allowing for a dimer-\nization in both sectors26,27we obtain from Eq. ( 2.1)\nHSτ(Γ)≃HMF\nS+HMF\nτ(Γ), (3.1)\nwith the spin and orbital Hamiltonians\nHMF\nS=JSN/summationdisplay\nj=1{1+(−1)jδS}Sj·Sj+1,\nHMF\nτ(Γ) =JτN/summationdisplay\nj=1{1+(−1)jδτ}[τj·τj+1]Γ.(3.2)\nWithin this approximation the effective superexchange\nconstants and dimerization parameters are given by\nJτ=J∆+\nSS+2x\n2, δ τ=∆−\nSS\n∆+\nSS+2x,\nJS=J∆+\nττ+2y\n2, δ S=∆−\nττ\n∆+ττ+2y,(3.3)where we have defined\n∆±\nSS=/an}bracketle{tS2j·S2j+1/an}bracketri}ht±/an}bracketle{tS2j·S2j−1/an}bracketri}ht,\n∆±\nττ=/angbracketleftbig\n[τ2j·τ2j+1]Γ/angbracketrightbig\n±/angbracketleftbig\n[τ2j·τ2j−1]Γ/angbracketrightbig\n.(3.4)\nHere ∆−\nσσwithσ=S(σ=τ) is an order parameter for\nthe spin (orbital) dimerization, respectively. Thus, the\nexchangeconstantsand dimerization parametersfor each\nsectoraredeterminedbythe nearest-neighborcorrelation\nfunctions in the other sector, making a self-consistent\ncalculation necessary. In the following we want to solve\nEqs. (3.2)-(3.4) for the spin exchange being effectively\nFM, i.e. JS<0.\nB. Dimerized orbital correlations\nNumerical investigations of the model with isotropic\norbital exchange, HSτ(0), have shown orbital-singlet for-\nmation in the ground state51fory/greaterorsimilar0.1. Moreover,\nalthough the ground state consists of a fully spin polar-\nized FM state for y/lessorsimilar0.1 with AF orbital correlations,\nit has been shown that a tendency towards orbital sin-\nglet formation is still present but has to be activated by\nthermal fluctuations.24In Ref. 26 the model ( 3.1) was\nstudied in the FM regime with x= 1,y=1\n4−γH,\nΓ = 0 and γH= 0.1 in order to address the ques-\ntion whether this orbital-Peierls effect can be captured\nwithin a MF decoupling approach. A dimerized phase\nfor 0.10/lessorsimilarT/J/lessorsimilar0.49 (we set kB=/planckover2pi1= 1) was found\nwith the dimerization amplitude in the spin sector being\nmuch larger than in the orbital sector.\nWe now want to compare this result with the case\nwhere we set Γ = 1 in Eq. ( 3.1) so that the self-\nconsistent Eqs. ( 3.3) can be solved analytically by ap-\nplying a Jordan-Wigner transformation and MSWT. In-\ntroducing fermionic operators f(†)\nj,eif the index jis even\nandf(†)\nj,oifjis odd for the pseudospins, we rewrite\nHMF\nτ≡HMF\nτ(Γ = 1) in Fourier representation. Finally\nintroducing new fermionic operators φ(†)\nqandϕ(†)\nqwhich\ndiagonalize the Hamiltonian HMF\nτ, we find\nHMF\nτ=/summationdisplay\nqωMF\nF(q,δτ)(φ†\nqφq+ϕ†\nqϕq),(3.5)\nwith the fermionic dispersion55\nωMF\nF(q,δτ)≡ Jτ/radicalBig\ncos2q+δ2τsin2q.(3.6)\nWe can now calculate ∆±\nττ, as given in Eq. ( 3.4)\nstraightforwardly and obtain\n∆−\nττ=2δτ\nN/summationdisplay\nq/braceleftbig\n2nF[ωMF\nF(q,δτ)]−1/bracerightbig\nsin2q/radicalBig\ncos2q+δ2τsin2q,\n∆+\nττ=2\nN/summationdisplay\nq/braceleftbig\n2nF[ωMF\nF(q,δτ)]−1/bracerightbig\ncos2q/radicalBig\ncos2q+δ2τsin2q,(3.7)6\nwherenF(x) ={exp(βx)+1}−1is the Fermi function\nandβ= 1/T.\nC. Dimerized spin correlations\nNext we turn to the spin part of Eq. ( 3.1) to which\nwe apply the MSWT.41,42We introduce two bosonic op-\neratorsb(†)\nj,e[b(†)\nj,o] forjeven [odd] by means of a Dyson-\nMaleev transformation. Retaining only terms bilinear in\nthe bosonic operators we can diagonalize the resulting\nHamiltonian by a Bogoliubov transformation leading to\nHMF\nS=/summationdisplay\nk/braceleftBig\nωMF\nB,−(k,δS)α†\nkαk+ωMF\nB,+(k,δS)β†\nkβk/bracerightBig\n+JSNS2,\n(3.8)\nwith the two magnon branches\nωMF\nB,±(k,δS) = 2|JS|S/parenleftbigg\n1±/radicalBig\ncos2k+δ2\nSsin2k/parenrightbigg\n.(3.9)\nThe constraint of vanishing magnetization at finite tem-\nperature ( 2.7) now reads\nS=1\nN/summationdisplay\nk/braceleftbig\nnB[ζ−(k,δS)]+nB[ζ+(k,δS)]/bracerightbig\n,(3.10)\nwherenB(x) ={exp{βx}−1}−1is the Bose function and\nζ±(k,δS) =ωMF\nB,±(k,δS)−µ(δS).\nTo calculate the nearest-neighborcorrelationfunctions\nB±≡/angbracketleftbig\nSj·Sj±1/angbracketrightbig\nit is necessaryto go beyond linear spin-\nwave theory. Taking terms of quartic order into account\nand using Eq. ( 3.10) we obtain26\nB±=\n1\nN/summationdisplay\nkf±(k,δS)/summationdisplay\nσ∈{±}σnB[ζσ\nB(k,δS)]\n2\n.\n(3.11)\nHere we have defined\nf±(k,δS)≡cos2k±δSsin2k/radicalBig\ncos2k+δ2\nSsin2k.(3.12)\nFrom these expressions we can obtain ∆±\nSSwhich, com-\nbined with Eq. ( 3.7), allows us to solve Eqs. ( 3.1)-(3.4)\nself-consistently.\nD. Mean-field phase diagram\nWe first discuss the ground state phase diagram of the\nHamiltonian ( 3.1) for Γ = 1. Depending on the sign of\nthe effective coupling constant JSwe find/an}bracketle{tSjSj+1/an}bracketri}ht=\n1,−1.4015 with the latter value being the approximate\nresult for the S= 1 AF Haldane chain. In the following,\nwe restrict our discussion to −1< x <1.4015 so that0.1 0.15 0.2y00.10.20.30.4 T/Jdimerizeduniform\nuniform(a)\nytpT2T1\ntricritical \npoint\n0.2 0.3 0.4\nT/J00.51Dimerization parameters(b)δS\nδτ\nT2/J T1/Jdimerized\nFIG. 3: (a) Phase diagram of the Hamiltonian (3.1) with\nΓ = 1 and x= 1 in mean-field decoupling. The shaded area\nrepresents the dimerized phase. The phase transition at T2\nis first order whereas the transition at T1is of second order.\nThe two transition lines merge at the tricritical point ytp. (b)\nDimerization parameters δSandδτforx= 1 and y= 0.14.\nThe lines are guides to the eye. The shaded area marks the\ntemperature range where the dimerization is nonzero.\n/an}bracketle{tSjSj+1/an}bracketri}htandJτ=J(/an}bracketle{tSjSj+1/an}bracketri}ht+x) always have the\nsame sign. For the orbital sector we obtain, on the other\nhand,/an}bracketle{tτx\njτx\nj+1+τy\njτy\nj+1/an}bracketri}ht=±1/π.y >1/πimpliesJS>0\nand the ground state is therefore certainly AF (Haldane\nphase) whereas JS<0 fory <−1/πleading to a FM\nstate. In the regime −1/π < y < 1/πthe self-consistent\nequationshave twosolutions with energies EAF\n0≈(1/π+\ny)(−1.4015+x) andEFM\n0= (−1/π+y)(1+x) and a first\norder phase transition between the FM and AF states\noccurs where the energies cross. For the case x= 1 we\nare focussing on here, this happens at yc≈0.212 and the\nFM stateis stablefor y < yc. Comparedto the numerical\nsolution where yc≈0.1 (see Fig. 2) the range of stability\nof the FM state is therefore increased in the MF solution.\nNext, we investigate the possibility of a finite tem-\nperature dimerization for x= 1 in that part of the\nphase diagram where the ground state is FM. As shown7\nin Fig.3(a) we find that a dimerized phase at finite\ntemperatures does indeed exist in MF decoupling for\nytp≈0.128/lessorsimilary/lessorsimilaryc≈0.212 where ytpdenotes the\ntricrictal point. As in the model with an isotropic pseu-\ndospin sector,26the temperature range where the dimer-\nized phase is stable depends on y. At the onset temper-\natureT1the phase transition is of second order whereas\nat the reentrance temperature T2it is of first order, see\nFig.3(b). As in the case Γ = 0, the dimerization in\nthe spin sector is always much larger than in the orbital\nsector.\nAs pointed out before, the MF decoupling suffers from\nsevere limitations and it is expected to be an even worse\napproximation in the extreme quantum case Γ = 1 than\nin the case Γ = 0 studied previously.26In particular the\ncoupling between spin and orbital degrees of freedom is\ncompletely lost within this approach. In the following\nsections we will therefore develop an alternative pertur-\nbative treatment of the spin-orbital coupling.\nIV. DYNAMICAL SPIN STRUCTURE FACTOR\nFOR THE UNIFORM FERROMAGNETIC CHAIN\nIn order to investigate coupled spin-orbital degrees of\nfreedom and, in particular, their implications on the spin\ndynamics of the spin-orbital chain, a detailed under-\nstanding of the spin dynamics of a FM chain is useful.\nWe shall avoid the complications of the dimerized chain\nand focus our study on the uniform 1D ferromagnet.56\nIn doing so we neglect the coupling between spin and\npseudospin operators for a moment and consider\nHS=JS/summationdisplay\njSj·Sj+1, (4.1)\nwithJS<0. It is well-known that MSWT does not\nrespect the SU(2) symmetry of the FM Heisenberg chain\nEq. (4.1). We therefore directly calculate the full spin\ncorrelation function39,40\nG(r,τ)≡ −/an}bracketle{tT[Sj(0)·Sj+r(τ)]/an}bracketri}ht.(4.2)\nIn Fourier space we obtain\nG(q,ων,B)=1\nN/summationdisplay\nk(1+nB[ζ(k)])nB[ζ(q−k)]1−e−βǫq(k)\niων,B−ǫq(k),\n(4.3)\nwhere we have used the bosonic Matsubara frequencies\nων,Bandǫq(k)≡ζ(k)−ζ(q−k) withk∈[−π,π]. The\nreduced magnon dispersion reads\nζ(k) = 2JSS(1−cosk)−µ. (4.4)\nIn Fig.4(a) the dynamical spin structure factor,\nS(q,ω) = 2nB(−ω)ImGret(q,ω),(4.5)0 1 2 3 4\nω/|JS|050100 |JS|S(q,ω)\n3 3.5\nω/|JS|02040|JS|S(q,ω)\n3 3.5 3.8025 4 4048\n|JS|ρ(ω)q=0q=π/5q=2π/5q=3π/5q=4π/5q=π(a)\n(b)\nω4π/5max\nFIG. 4: (Color online) (a) Dynamical spin structure factor\nS(q,ω) as obtained for T/|JS|= 0.1 and 0 ≤q≤π. The\ndashed line indicates the upper boundary of the two magnon\ncontinuum. The dots are projections of the peak positions\nonto the ( q,ω) plane. They are connected by the dotted line\nwhich is a guide to the eye. (b) Dynamical spin structure\nfactorS(q,ω) for the same parameters at q= 4π/5 (solid\nline) and the corresponding density of states (dashed line) .\nis shown for the uniform FM chain at ω >0, where\nImGret(q,ω) =π\nN/summationdisplay\nk(1+nB[ζ(k)])nB[ζ(q−k)]\n×/parenleftBig\ne−βǫq(k)−1/parenrightBig\nδ(ω−ǫq(k))(4.6)\nis the imaginary part of the retarded Green’s function\nobtained from Eq. ( 4.3) by analytical continuation. Up\nto a factor of 2 π, as a matter of definition, we obtain\nthe result previouslygivenbyTakahashi.39The structure\nfactor fulfills detailed balance, S(q,ω) = eβωS(q,−ω).8\nThe symbols in Fig. 4show the peak positions projected\nonto the ( q,ω) plane. They follow the reduced disper-\nsion Eq. ( 4.4). Also shown in Fig. 4(a) as a dashed\ncurve is ωmax\nq= 4|JS|Ssinq\n2corresponding to the up-\nper boundary of the two magnon continuum ǫq(k) above\nwhichS(q,ω) is zero in this approximation.\nAt the edge of the two magnon continuum S(q,ω) has\na singularity. In Fig. 4(b) the dynamical spin structure\nfactor for the same parameters as used in Fig. 4(a) is\nshown at q= 4π/5 together with the density of states\nwhich is given by ρq(ω) = 1//radicalBig\n(ωmaxq)2−ω2. Right be-\nlow the singularityat ωmax\nqthe density of states to lowest\norder reads ρq(ωmax\nq−δω)∼1/√\nδω, i.e.,S(q,ω) shows\na square root divergence at the upper threshold. If the\nedge singularity and the central peak are well separated\nthen the spectral weight of the edge singularity is much\nsmaller than the spectral weight of the central peak. If,\non the other hand, the edge singularity is close to the\ncentral peak then the shape of the latter is strongly af-\nfected by the occurence of the edge singularity. In this\ncase the edge singularity gives a significant contribution.\nIt is instructive to analyze S(q,ω) in the limit of small\nq. If the edge singularity and the peak of the struc-\nture factor are well separated, the lineshape of the peak\ncan be obtained approximately. To this end, for small\nqbut|JS|S2q/T≫1 we only retain the leading terms\nof Eq. (4.3). Performing a saddle point approximation\nto lowest order we find S(q,ω)∼nB(−ω)(a(q,ω)−\na(q,−ω)), with\na(q,ω)≈2S|JS|Sq\nξ\n(ω−JSSq2)2+/parenleftBig\nJSSq\nξ/parenrightBig2.(4.7)\nThis Lorentzian lineshape is only valid for low temper-\natures. Here ξ≈ |JS|S2/Tis the correlation length in\nthe low-temperaturelimit.39–42Finally, we want to stress\nthat forT→0 the peaks will reduce to δ-functions, i.e.,\nonly thermal broadening is included in this approxima-\ntion.\nV. PERTURBATION THEORY\nIn this section we intent to go beyond the MF decou-\npling approach treating the influence of the BF inter-\naction, Eq ( 2.8), on the spin-wave dispersion perturba-\ntively. Naively one would expect that the magnon should\nbe able to couple to the fermionic degrees of freedom if\nit lies inside the fermionic two-particle continuum. The\nupper and lower boundary of the latter are given by\nǫmax\nF(q) = 2J(S2+x)sin(q/2),\nǫmin\nF(q) =J(S2+x)sinq, (5.1)\nrespectively. The continuum and the magnon dispersion\nωB(q) fory=−1 are shown in Fig. 5. One would there-\nfore expect that in this case ωB(q) is unaffected by the0 0.5 1\nq/π01234 ω/JωB(q)\nFIG. 5: (Color online) Magnon dispersion ωB(q) (dashed line)\nfory= 1 andfermionic two-particle continuum(shaded area).\npresenceofthe fermionsfor q < π/2, sinceit can not cou-\nple to these degrees of freedom. However for higher mo-\nmenta the spin wave may couple to the fermionic degrees\noffreedomandthus abroadeningof S(q,ω)shouldoccur.\nMoreover by choosing different values for ythe point at\nwhich the magnon enters the fermionic two-particle con-\ntinuum is changed. Thus the momentum at which the\nspin wave is affected by the coupling to the fermionic de-\ngrees of freedom depends directly on the parameter y.\nThese argumentsgive the qualitatively correct picture,\ni.e., we find indeed that the coupling of the magnon to\nthe fermionic degrees of freedom has strong effects on\nthe dynamical spin structure factor at intermediate and\nhigh momenta and that the onset of these effects can be\nwell estimated by our simple argument. However, there\narealsocertain aspects which can not be captured within\nthis picture. Forinstance, for 2 S|y|>(S2+x) it suggests\nthat the spin wave may leave the fermionic two-particle\ncontinuum at a certain momentum qland thus should be\nunaffected by the BF coupling for q > ql. The detailed\ncalculation, however, revealsthat this is not true because\nthe spin wave decays into a fermionic particle-hole anda\nremainingspin waveaswill becomeclearinthe following.\nA. General formulation\nHere we want to study the Hamiltonian Hgiven by\nEq. (2.3), with its noninteracting part H0and interact-\ning part H1defined by Eqs. ( 2.4) and (2.8), by treat-\ning the BF interaction perturbatively, i.e., in the limit\n|x|,|y| ≫1. As explained in the appendix, we start by\nperforming a MF decoupling for the interaction H1. Cor-\nrections to this solution are then taken into account per-\nturbatively. Here we adress the bosonic Green’s function\nat zero temperature\nGB(q,t) =−i/angbracketleftbig\nTt/bracketleftbig\nbq(t)b†\nq(0)/bracketrightbig/angbracketrightbig\n. (5.2)9\nq qk1 k1k2\nq qq−k1+k2\nk2k1q\nqq−k1+k2 k2 k1(c) (a) (b)\nFIG. 6: Diagrams which contribute to a renormalization of th e magnon in a perturbation theory: (a,b) Diagrams with momen -\ntum exchange between the magnon and the fermions, and (c) dia gram without momentum exchange. All diagrams are second\norder. Fermionic propagators are shown by solid lines, wher eas bosonic propagators are shown as dashed lines.\nq qq−k1+k2\nk2k1\nFIG. 7: Second order diagram for a system of interacting\nfermions with momentum exchange.\nIn Fig.6all distinct, connected diagrams beyond the MF\ndecoupling up to second order are shown.\nWe calculate the Green’s function from the Dyson\nequation\nGB(q,ω) =1\n/braceleftBig\nG(0)\nB(q,ω)/bracerightBig−1\n−Σ(q,ω),(5.3)\nwith\n/braceleftBig\nG(0)\nB(q,ω)/bracerightBig−1\n=ω−ζ(q), (5.4)\nwhereζ(q) is the reduced magnon dispersion defined\nin Eq. ( 4.4) withJS=J(y−1/π), and the self-\nenergy Σ( q,ω) is approximated by the proper self-energy\nΣ2(q,ω) obtained by summing up the diagrams which\ncan be composed of the diagrams shown in Fig. 6.\nThe diagrams shown in Figs. 6(a) and6(b) are of par-\nticular interest because they are the lowest order dia-\ngrams where bosons and fermions exchange momentum.\nThey describe the part of spin-orbital dynamics which\ncannot be captured within the MF decoupling approach\ndiscussed in section III. The diagram shown in Fig. 6(b)\nhas to be thermally activated, i.e., it does not give any\ncontribution at zero temperature. The same is true for\nthe diagram shown in Fig. 6(c). Thus at T= 0 the only\nsecondorderdiagramwhichcontributestotheselfenergyis the one shown in Fig. 6(a) leading to\nΣ+\nBF(q,ω) =−1\nN2/summationdisplay\nk1,k2ω2\nBF(q,k1,k2)\n×Θ[ωF(q−k1+k2)]Θ[−ωF(k2)]\nω−Ω+q(k1,k2)+i0+,(5.5)\nwhere we have abbreviated57\nΩ±\nq(k1,k2)≡ ±ζ(k1)+ωF(q−k1+k2)−ωF(k2),(5.6)\nwithk1,k2∈[−π,π].\nFor systems of interacting fermions we know that per-\nturbation theory in one dimension often leads to infrared\ndivergencies.58,59Such divergencies occur, for example,\nfor the fermionic analogon of the diagram with momen-\ntum exchange, see Fig. 7. These problems can be over-\ncome by the Dzyaloshinski-Larkin solution or bosoniza-\ntion techniques. For the model considered here, however,\nwe find no divergencies within the considered diagrams.\nOne reason for this behavior is a lack of nesting. While\nfor a fermionic interaction as shown in Fig. 7all the\ndispersions in the denominator of Eq. ( 5.5) are approxi-\nmately linear at low energies here one of the dispersions\nis approximately quadratic so that nesting only occurs\nfor singular points. As a further check, we have evalu-\nated the integrals in Eq. ( 5.5) for a constant vertex at\nsmallqandωand did not find any infrared divergencies.\nFor finite temperatures the Matsubara formalism can\nbe applied straightforwardly. The self-energy Eq. ( 5.5)\nnow reads\nΣ+\nBF(q,ων,B) =−1\nN2/summationdisplay\nk1,k2ω2\nBF(q,k1,k2)\niων,B−Ω+q(k1,k2)\n×N+\nF,B(k1,k2,ων,B,T)NF,F(q,k1,k2,ων,B,T),\n(5.7)\nwhere we have abbreviated\nN±\nF,B(k1,k2,ων,B,T)≡nB[ζ(k1)]±nF[ωF(k2)],\nNF,F(q,k1,k2,ων,B,T)≡nF[ωF(q−k1+k2)]\n−nF[ωF(k2)−ζ(k1)].(5.8)10\n0102030JS(q,ω)\n0 2 4 68\nω/J10-810-4100-ImΣ(q,ω)q=3π/10\nq=5π/10\nq=7π/10\nq=9π/10-1-0.500.51\nq/π051015ω/J(a)\n(b)0204060JS(q,ω)\n0 5 10 15\nω/J10-810-4100-ImΣ(q,ω)q=3π/10\nq=5π/10\nq=7π/10\nq=9π/10-1-0.500.51\nq/π051015ω/J(c)\n(d)\nFIG. 8: (Color online) Perturbative results for the BF model at zero temperature with x= 1. In the left (right) panel y=−1\n(y=−2), respectively. (a),(c) S(q,ω) with the inset showing the region for which ImΣ+\nBF(q,ω) is nonzero as a shaded area\nand the renormalized spin-wave dispersion as a dotted line. WhileS(q,ω) is sharply peaked at low momenta, a significant\nbroadening occurs at higher q. Moreover we find additional structures which, as explained in the text, are due to coupled\nspin-orbital excitations. (b),(d) −ImΣ+\nBF(q,ω) as given in Eq. (5.5) for the corresponding values of qshown in (a) and (c),\nrespectively. The coupled spin-orbital excitations show u p as peaks and edges in ImΣ+\nBF(q,ω) (notice the logarithmic scale).\nNote that at finite temperatures both, the reduced spin-\nwave dispersion ζ(q) as well as the fermionic dispersion\nis renormalized due to the MF decoupling applied to\nEqs. (2.8). The respective expressions are given in the\nappendix, see Eqs. ( A.1) and (A.2).\nAt finite temperatures also the diagram shown in\nFig.6(b) contributes and is given by\nΣ−\nBF(q,ων,B) =−1\nN2/summationdisplay\nk1,k2ω2\nBF(q,k1,k2)\niων,B−Ω−q(k1,k2)\n×(1+N−\nF,B(k1,k2,ων,B,T))NF,F(q,k1,k2,ων,B,T).\n(5.9)B. Dynamical spin structure factor\nBelow we present the results obtained by summing up\nthe diagrams shown in Fig. 6in a Dyson series, but re-\nplacingtheexternallegsbytheSU(2)symmetricfunction\ngiven in Eq. ( 4.3). While the perturbative results can,\nstrictly speaking, only be valid for |x|,|y| ≫1 we extend\nthe results here to more physical values |x|,|y| ∼ O(1)\nwhere we still expect perturbation theory to give at least\na qualitatively correct picture. Numerical results ob-\ntained for the dynamical spin structure factor within this\nperturbative approach are shown in Fig. 8forT= 0 and\ny=−1 andy=−2. In both cases S(q,ω) is sharply\npeaked at small momenta whereas a significant broad-\nening occurs at higher momenta. Note, that within the11\n0 0.2 0.4 0.60.8\nq/π0246810 ω/J\n0.5 0.52 0.54\nq/π22.22.42.6ω/J\ny=-0.75y=-1y=-2\nFIG. 9: (Color online) Renormalized magnon dispersions ωq\nin the FM chain for x= 1 and selected values of y. Inset:\nThe most pronounced Kohn anomalies occur at qnearπ/2.\nMSWTS(q,ω) is always a δ-function for the pure spin\nmodel at T= 0, i.e., the broadening here is solely due to\nthe coupling to orbital excitations.\nBy extracting the central peaks of the dynamical spin\nstructurefactoratvariousmomenta, weobtaintherenor-\nmalized spin-wave dispersion ωqwithin the perturbative\napproach. The result of this is shown in the insets of\nFig.8(a,c) and in more detail in Fig. 9. The magnon dis-\npersion is renormalized and small kinks are visible close\ntoq=π/2, which may be interpreted as Kohn anoma-\nlies (see below). The inset of Fig. 9shows the Kohn\nanomalies with a higher resolution. For itinerant ferro-\nmagnets Kohn anomalies are well-known. Here the inter-\naction between the spins of localized ions is mediated by\nan exchange with the conduction electrons.60–66These\nKohn anomalies can thus be used to gain information\nabout the Fermi surface of the conduction electrons.60–62\nHowever to the best of our knowledge Kohn anomalies in\nthe spin-wavedispersionforinsulatingmaterialshavenot\nbeen adressed so far. As we will show below, the Kohn\nanomalies in our case are caused by coupled spin-orbital\ndegrees of freedom.\nApart from extracting the effective spin-wave disper-\nsion from S(q,ω), we also want to discuss the magnon\nbandwidth (full width at half maximum (FWHM)) Γ q\nof the central peaks. A broadening of the zero temper-\nature peaks occurs whenever the imaginary part of the\nself-energy,\nIm Σ+\nBF(q,ω) =−π\nN2/summationdisplay\nk1,k2ω2\nBF(q,k1,k2)Θ[−ωF(k2)]\n×Θ[ωF(q−k1+k2)]δ(ω−Ω+\nq(k1,k2)),\n(5.10)\nis non-zero at ω= Ω+\nq(k1,k2). The contributions within\nthe sums are now determined by the argument of the δ-\nfunction as well as by the constraints given by the Heav-\niside functions. This procedure, for a given set of param-0 0.2 0.4 0.60.8 1\nq/π00.10.20.30.40.5Γq/Jy=-0.75\ny=-1\ny=-2\nFIG. 10: (Color online) Magnon linewidth Γ q(FWHM of\nS(q,ω)) at zero temperature (data points), as obtained for\nx= 1 and the representative values of yindicated in the plot.\nThe lines are guides to the eye.\netersx,y,andS, effectively yields a region within which\nthe spin wave may scatter on fermion pairs. Henceforth\nwecallthisregiontheBFcontinuum. TheBFcontinuum\nis shown in the insets of Figs. 8(a) and8(c) as shaded ar-\neas. The upper boundary of the BF continuum (solid\nlines), which is periodic with a period of 2 π, is given by\n−4JS(y−1/π) + 2J(S2+x) forq= 0, and decreases\nmonotonously from this value with increasing |q|. The\nlower boundary (dashed lines) is periodic with a period\nofπ.\nTo obtain the FWHM of the structure factor, the mag-\nnitude of the contributions to the sums given in Eq.\n(5.10) are essential. Here not only the ( k1,k2)-region\nwhich contributes to the summation but also the magni-\ntude of the vertex ωBFis of importance. We observe that\nthe vertex is small at small momenta but increases at in-\ntermediate and high momenta. This leads to a strong\nincrease of the magnitude of the imaginary part of the\nself-energyas shownin panels (b) and (d) ofFig. 8. From\nthe insets of Fig. 8it becomes clear that the spin-wave\ndispersion enters the BF continuum depending on y. For\nhigher values of |y|the spin wave enters at lower mo-\nmenta. However, since the vertex gives smaller contribu-\ntions at smaller momenta, the broadening of the central\npeaks of the dynamical spin structure factor turns out\nto be smaller the smaller the momenta are at which the\nspin wave enters the BF continuum. This can be seen in\nFig.10where the magnon linewidth Γ qis shown. The\nonset of a finite Γ qsignals the entrance of the spin-wave\ndispersion into the BF continuum and depending on the\nmomentum at which the entrance occurs the increase of\nΓqis either smooth (entranceat lowmomentum) orsteep\n(entrance at high momentum). In addition, we observe\nthat Γ qhas a maximum at the boundary of the Brillouin\nzone for stronger interactions ( y=−0.75 andy=−1\nin Fig.10respectively), whereas for smaller interactions\nwe observe the maximum at smaller momenta followed12\n0 0.2 0.4 0.60.8 1\nq/π0246810 ω/Jωq\nFIG. 11: (Color online) Effective dispersion of the spin\nwaveωqfory=−1 (red solid line) together with coupled\nspin-orbital excitations deduced from −ImΣ2(q,ω), shown\nby (green) dots within the Bose-Fermi continuum (shadded\narea).\nby a decrease of the FWHM towards the zone boundary\n(y=−2 in Fig. 10).\nInterestingly, the coupling to the orbital degrees of\nfreedom does not only give rise to a featureless broaden-\ning ofS(q,ω) but produces additional structures. These\nadditional structures aremost obviousin Fig. 8(a). From\nFig.8(b) it becomes clear that these structures are dom-\ninated by local extrema as well as edges in the imagi-\nnary part of the self-energy. Eq. ( 5.10) shows that such\nextrema can occur if Ω+\nq(k1,k2), Eq. (5.6), becomes sta-\ntionaryas a function ofthe momenta k1,k2as long as the\nHeaviside functions in Eq. ( 5.10) for these momenta are\nnon-zero. The position of the local maxima in the imag-\ninary part of the self-energy is therefore approximately\ngiven by the values Ω+\nq(k1,k2) at these stationary points.\nThese values correspond to the energy of spin-orbital ex-\ncitations into which the initial spin wave can decay, see\nFig.6(a) and which are stable against small redistribu-\ntions of momenta.\nWe conclude that while we do not have completely\nsharp spin-orbital excitations any more as in the Hamil-\ntonian (1.6) with FM exchange considered in the intro-\nduction, there are still characteristic spin-orbital excita-\ntions of finite width within the spin-orbital continuum.\nAsshowninFig. 11wecanextractthe dispersionofthese\ncharacteristic excitations and find that the coupled spin-\norbital excitations are gapless for the parameters con-\nsidered here. However, as can be clearly seen in Figs.\n8(b) and8(d), the weights of the low-energy excitations\nare orders of magnitude smaller than the excitations lo-\ncated at higher energies. Hence the excitations at high\nenergies give the most dominant contribution to the dy-\nnamical spin structure factor. We therefore expect that\nthese excitations will generate additional entropy in the\ncorresponding temperature range which should show up,0 2 4 6\nω/J0102030JS(q,ω)q=3π/10\nq=5π/10\nq=7π/10\nq=9π/10\nFIG. 12: (Color online) Dynamical spin structure factor\nS(q,ω) calculated perturbatively for the spin-orbital chain at\ntemperature T/J= 0.1 withy=−1.\nfor example, in the specific heat which will be studied in\nthe next section.\nMoreover, we find that these coupled spin-orbtial exci-\ntationsareresponsiblefortheKohnanomaliesmentioned\nabove. We observe that the Kohn anomalies at interme-\ndiate momenta occur when the energy of the spin wave\ncoincides with that of a characteristic spin-orbital exci-\ntation. This is different from the Kohn anomaly in the\nspin-wave dispersion of itinerant ferromagnets. In this\ncase the interaction between the localized spins given by\nthe lattice ions is induced by scattering with conduction\nelectrons and hence the Kohn anomaly is determined by\nthe shape of the Fermi surface.60–66The Kohn anomaly\nwe find within the present context is also due to interac-\ntioneffects, wherethenatureoftheinteractions-coupled\nspin-orbital degrees of freedom - is distinct from the ones\nof the itinerant ferromagnets. For the case x=−y= 1\nthe spin-wave dispersion has a discontinuity of the order\n∆ω≃0.01Jat the point q≃0.509π(see inset of Fig. 9).\nHowever, for the crossing points of the magnon and the\ncoupled spin-orbital excitation located at q≃0.37πand\nq≃0.76π(see Fig. 11) no Kohn-anomaly could be re-\nsolved. We believe that this is a consequence of the\nweight of the coupled spin-orbital excitations: Whereas\natq≃0.509πthe imaginary part of the self-energy dis-\nplays a steep increase of several magnitudes, at the other\ncrossing points the slope towards the local maxima is far\nmore moderate.\nAt finite temperatures two effects contribute to the\nbroadening of the central peaks of the dynamical spin\nstructure factor. First, there is a broadening due to ther-\nmally excited magnons which is already present in the\n1D Heisenberg chain discussed in Sec. IV. This is com-\nbined with the broadening due to the interaction with\nthe orbital degrees of freedom. Here the BF continuum\nis smeared out by thermal fluctuations compared to the\nzero temperature case. Results for the structure factor13\n0 0.2 0.4 0.60.8 1\nq/π00.10.20.30.40.5Γq/JT/J=0.2\nT/J=0.1\nT/J=0\nFIG. 13: (Color online) Magnon linewidth Γ q(FWHM of\nS(q,ω)) as a function of q, as obtained at y=−1 and tem-\nperatures T/J= 0, 0.1 and 0.2. The lines are guides to the\neye.\nat finite temperatures are shown in Fig. 12.\nThe broadening at small momenta is dominated by\nthermal fluctuations and the lineshape is very similar to\nthat of the pure spin model discussed in Sec. IV. A\nfurther strong broadening in going from q= 0.3πto\nq= 0.5πsignals the relevance of coupled spin-orbital\ndegrees of freedom on the spin dynamics at intermedi-\nate and high momenta. Again, additional structures in\nS(q,ω) are visible related to the spin-orbital excitations\ndiscussed above.\nFinally, we analyze the variation of the FWHM with\nincreasing temperature for a representative value of y=\n−1, see Fig. 13. AtT >0 the thermal broadening at\nsmall momenta is clearly visible. As in the zero tempera-\nture case another strong increase of Γ qbetween q= 2π/5\nandq=π/2 is observed due to coupled spin-orbital de-\ngrees of freedom. For T/J= 0.1 andT/J= 0.2 we\nobserve that Γ qhas a temperature dependent maximum\nfrom where Γ qdecreases towards the boundary of the\nBrillouin zone. This is due to the fact that the ther-\nmal broadening of the central peaks of the dynamic spin\nstructure factor decreases from intermediate to high mo-\nmenta (see Fig. 4). Actually without coupling to any or-\nbital degrees of freedom we expect Γ qto be very small at\nthe boundary of the Brillouin zone. Thus a large band-\nwidth at q=πmakes the spin-orbital model distinct\nfrom a pure 1D Heisenberg ferromagnet.\nVI. THERMODYNAMICS\nCoupled spin-orbital degrees of freedom will not only\ninfluence the spin dynamics but also the thermodynam-\nics of the system. We expect, in particular, that the\nspin-orbital excitations which were shown to affect the/Bullet /Bullet /Bullet /Bullet\nFIG.14: Diagramatic representations ofthesecond orderco n-\ntributions to the free energy as given by Eq. (6.6).\ndynamical spin structure factor in the previous section\nwill also become observable in thermodynamic quanti-\nties when comparing the MF decoupling and the per-\nturbative solution. In order to investigate this issue we\nrewrite Eq. ( 2.1) as\nHSτ(1) =HMF+δH, (6.1)\nwithδH=HSτ(1)−HMF. The MF part reads\nHMF=N/summationdisplay\nj=1/braceleftbig\nJτ[τj·τj+1]1+JSSj·Sj+1\n−/an}bracketle{tSj·Sj+1/an}bracketri}htMF/angbracketleftbig\n[τj·τj+1]1/angbracketrightbig\nMF/bracerightbig\n.(6.2)\nThe exchange constants JS,τare defined in Eqs. ( 3.3).67\nWe use the Hamiltonian ( 6.1) to determine the free en-\nergy per site perturbatively, following the expansion,\nf=fS\nMF+fτ\nMF+1\nN/an}bracketle{tδH/an}bracketri}htc\nMF−1\n2NT/angbracketleftbig\nδH2/angbracketrightbigc\nMF+...,\n(6.3)\nwherefS\nMF(fτ\nMF) is the expression for the free energy per\nsite stemming from the spin (pseudospin) sector within\nthe MFdecouplingsolution. The subscriptindicatesthat\nthe respective correlation functions are calculated with\nHMF. Moreover the superscript cmeans that the above\nexpansion of the free energy is restricted to connected\ndiagrams. We note that Eq. ( 6.3) is a high temperature\nexpansion valid if |x|T/J≫1 and|y|T/J≫1.\nA straightforward calculation shows that the first or-\nder contribution only shifts the free energy, Eq. ( 6.3),\nand will not show up in thermodynamic observables ob-\ntained by taking derivatives of the free energy. For the\nsecond order contribution we have to evaluate two- and\nfour-point correlationfunctions both for the spin and the\npseudospin part. We use the abbreviations\n/an}bracketle{t(Sj·Sj+1)(Sl·Sl+1)/an}bracketri}ht=a+b(j,l) (6.4)\nfor the spins and\n/angbracketleftbig\n[τj·τj+1]1[τl·τl+1]1/angbracketrightbig\n=c+d(j,l) (6.5)\nfor the pseudospins. Here the site-independent quanti-\ntiesaandcstand for the disconnected parts of the four-\npoint correlation functions whereas b(j,l) andd(j,l) fol-\nlow from the connected ones. One finds after a straight-\nforward calculation that only the product of the con-\nnected parts contributes to the second order correction,14\nleading to\n/angbracketleftbig\nδH2/angbracketrightbig\nMF=J2N/summationdisplay\nj,l=1b(j,l)d(j,l).(6.6)\nTo proceed further, we again apply the MSWT to the\nspin and a Jordan-Wigner transformation to the pseu-\ndospin part. The evaluation of d(j,l) is again straight-\nforward and yields\nd(j,l) =1\n2N2/summationdisplay\nk1,k2nF[ωMF\nF(k1,0)]{1−nF[ωMF\nF(k2,0)]}\n×ei(k1−k2)(j−l){1+cos(k1+k2)}\n(6.7)\nwithωMF\nF(q,δ) as given in Eq. ( 3.6).\nThe evaluation of Eq. ( 6.4) is more involved. We first\napply a Dyson-Maleev transformation and treat the ob-\ntained expressionsusing Wick’s theorem. In addition, we\nalso have to account for the constraint of nonzero mag-\nnetization at T >0 imposed by the MSWT. The cor-\nresponding diagrams are shown in Fig. 14. Within this\napproximation the specific heat per site reads\nc=cMF+c2+... , (6.8)\nwith\nc2=J2T∂2\n∂T2N/summationtext\nj,l=1b(j,l)d(j,l)\n2NT.(6.9)\nWe calculate the first term in Eq. ( 6.8) within the MF\ndecoupling, cMF=cS\nMF+cτ\nMF, from the internal energy\nwhich is determined by the respective nearest-neighbor\ncorrelation functions allowing us to keep terms up to\nquartic order in the bosonic operators.54This strategy\nmakes it possible to obtain reliable results for cS\nMFup to\nT/(|JS|S2)≤1. The second order correction c2given in\nEq. (6.9) is obtained using the Dyson-Maleev transfor-\nmation so that quartic terms are also included and the\norder of approximation is the same. Since we are using\na high temperature expansion, Eq. ( 6.3), in combination\nwith the MSWT to evaluate the diagrams, our results\nare only valid in an intermediate temperature regime. If\nwe restrict ourselves to parameters x=−y >0 then this\ntemperature range is given by 1 /x≪T/J≪xS2. In\nthe following we therefore only consider the case x≫1\nand compare the results from perturbation theory with\nnumerical data obtained by TMRG.\nAs shown in Fig. 15, the specific heat c/(Jx)2exhibits\na broad maximum which corresponds to the character-\nistic energies of spin and fermionic particle-hole excita-\ntions. In the temperature range where the perturbative\napproach is valid we find excellent agreement with the\nnumerical solution. In particular, the perturbative cor-\nrectionc2(see Fig. 16) correctlycapturesthe weight shift\nfrom low to intermediate temperatures visible when com-\nparing the numerical and the MF decoupling solution. In0 0.2 0.4 0.60.8 1\nT/(Jx)00.20.40.6c/(4J2)00.20.40.6c/(100J2)\n0 0.2 0.4 0.60.8 1\nT/(10J)00.20.40.6c/(100J2)\n0 0.2 0.4 0.60.8 1\nT/(2J)00.20.40.6c/(4J2)(a)\n(b)cMFScMFτcMF\ncMFScMFτcMF\nFIG. 15: Specific heat per site c/(Jx)2as a function of T/Jx\nfor (a)x= 10 and (b) x= 2. The circles denote the nu-\nmerical data from TMRG with the solid line obtained by\na low-temperature fit of the TMRG data for the inner en-\nergy. The dashed lines correspond to MF decoupling while\nthe dashed-dotted lines are the results obtained by perturb a-\ntion theory. The perturbative results are expected to be val id\nfor 1/x2≪T/(Jx)≪1. Insets: Specific heat cMFwithin the\nMF solution (dashed-dotted line) with the contributions fr om\nthe spin ( cS\nMFsolid line) and the orbital ( cτ\nMFdashed line)\nsector shown separately.\nspiteofthisweightshift, theMFdecouplingyieldsoverall\na very reasonabledescription of the specific heat for both\ncases shown in Fig. 15. Forx= 10, (Fig. 15(a)) the spe-\ncific heat chas a broad maximum at T/(Jx)≃0.4. This\nmaximum results from a distinct maximum in the orbital\ncontribution cτ\nMF, see insets in Fig. 15. In contrast the\nspin contribution cS\nMFincreases steadily with increasing\ntemperature, in agreement with the higher energy scale\nfor spin excitations. As a result, the total specific heat\nhas only a weaker and broader maximum than suggested\nby the orbital part.\nThe MF decoupling does seem to fail, however, at very\nlow temperatures. Here the MF solution predicts that\nspin excitations give the dominant contribution leading15\n0 0.2 0.4 0.60.8 1\nT/(Jx)-0.0300.03c2/(Jx)2\nFIG. 16: (Color online) Perturbative contribution to the sp e-\ncific heat per site c2/(Jx)2, Eq. (6.9), as a function of T/Jx\nforx= 10 (solid line) and x= 2 (dashed line).\nto ac(T)∼√\nTbehavior. This is in contrast to an ex-\ntrapolation of the numerical data shown as dot-dashed\nlines in Fig. 15which suggests an approximately linear\ndependenceontemperature. Thisdiscrepancycomesasa\nsurprise because our perturbative calculations of the dy-\nnamical spin-structure factor in Sec. VBlead us to the\nconclusion that the magnons survive as sharp quasipar-\nticles at low energies. More generally, one might argue\nthat the ground state does not show any entanglement\nbetween the two sectors because of the classical nature\nof the FM state thus allowing for spin-wave excitations\nat low energies. From the point of view of a low-energy\neffective field theory, however, the situation is much less\nclear. While a FM chain is described by a low-energy\neffective theory with dynamical critical exponent z= 2,\nthe fermionic orbital chain has z= 1. A coupling of\nspatial spin deviations to time-dependent orbital fluc-\ntuations then seems to require that the low-energy ef-\nfective theory for the coupled system has z= 1. As a\nconsequence the temperature dependence of c(T) would\nindeed be linear. However, such an approach leaves open\nthe role and treatment of the Berry phase terms.\nWithin the perturbative approach the open question\nis whether or not higher order contributions to the self-\nenergy might induce a significant broadening of the dy-\nnamical spin-structure factor also at low energies. In this\nregard we note that the vertex responsible for the broad-\nening of S(q,ω) studied in Sec. VBdoes not play any\nrolefor the thermodynamics ofthe system. Here the con-\nstraint of vanishing magnetization means that such dia-\ngrams do not contribute to staticcorrelation functions so\nthat the lowestordercorrectionsarecaused by the vertex\nshown in Fig. 14.VII. CONCLUSIONS\nIn summary, we have investigated coupled spin-orbital\ndegrees of freedom in a one-dimensional model. For fer-\nromagnetic exchange we have shown that the considered\nmodelatspecialpointsinparameterspacecanbewritten\nin terms ofDiracexchangeoperatorsfor spin Sand pseu-\ndospinτ. As a consequence, three collective excitations\nof spin, orbital and coupled spin-orbital type do exist. In\nparticular, we discussed the case of Dirac exchange oper-\nators for S=τ= 1/2 where the dispersions of all three\nelementary excitations are degenerate and lie within the\nspin-orbital continuum. While the spin-orbital excita-\ntions stay confined in this case, a decay becomes possible\nonce we move away from this special point.\nFor antiferromagnetic exchange the one-dimensional\nspin-orbital model captures fundamental aspects of\nphysics relevant for transition metal oxides and, as we\nhave shown, sharp excitations do not exist. To address\nthe question how the spin dynamics is influenced by fluc-\ntuatingorbitalsweconsideredtheextremequantumlimit\nof orbitals interacting via an XY-type coupling. This al-\nlowed us to map the orbital sector onto free fermions\nusing the Jordan-Wigner transformation. In spin-wave\ntheory the spin sector is described by bosons so that our\nmodel corresponds to an effective boson-fermion model\nwhich applies for low temperatures. An analytic calcu-\nlation of the properties within a mean-field decoupling\napproachis then straightforward. Compared to a numer-\nical phase diagram based on the density-matrix renor-\nmalization group we find that the regime with ferromag-\nnetically polarized spins is stabilized by the decoupling\nprocedure. Furthermore, the mean-field decoupling gives\nrise to a finite temperature dimerized phase for certain\nparameterswhen starting from the ferromagnetic ground\nstate. Whileaphasewithlong-rangedimerorderatfinite\ntemperatures is not possible in a purely one-dimensional\nmodel, the mean-field approach also completely ignores\nany kind of coupled spin-orbital excitations.\nThus we developed a self-consistent perturbative\nscheme to explore the role played by spin-orbital cou-\npling. In perturbation theory the boson-fermion inter-\naction does not produce any infrared divergencies in the\none-dimensional model due to the lack of nesting. This\nmakes a perturbative calculation of the spin structure\nfactorS(q,ω) possible. At large momenta q, we find that\nS(q,ω) shows a significant broadening due to scattering\nof magnons by orbital excitations. For small momenta,\non the other hand, no broadening in this lowest order\nperturbative approach is observed because the magnon\ncannot scatter on these excitations. The onset of the\nbroadening occurs at momenta where the magnon enters\nthe boson-fermion spectrum. This point, as well as de-\ntails of the full width at half maximum is determined by\nthe strength of interaction. Most interestingly, S(q,ω)\ndoes show additional peaks and shoulders corresponding\ntocharacteristicspin-orbitalexcitations. Atpointswhere\nthe renormalized spin-wave dispersion and the dispersion16\nof these excitations cross, Kohn anomalies do occur.\nFurthermore, we compared numerical data for the spe-\ncific heat of the spin-orbital model with the mean-field\ndecoupling solution and an approach where we also took\nthe second order correction to the mean-field result into\naccount. Overall, we found that the mean-field decou-\npling does describe the specific heat reasonably well. A\nredistribution of entropic weight from low to intermedi-\nate temperatures observed when comparing the numeri-\ncal data and the mean-field solution is very well captured\nby the second order perturbative correction. An interest-\ning open point is the behavior of the specific heat c(T) at\nlow temperatures. While the mean-field solution predicts\nc(T)∼√\nTdue to spin-wave excitations, the numerical\ndata suggest instead that c(T)∼T. We have speculated\nthat a coupling of the two sectors might indeed lead to a\nlow-energy effective theory with dynamical critical expo-\nnentz= 1 but details of such a theory need to be worked\nout in the future.\nIn conclusion, we have shown that while collective\nspin-orbital excitations with infinite lifetime do not ex-\nist forantiferromagneticsuperexchangethe coupled spin-\norbital degrees of freedom have a strong influence on the\nspin excitation spectrum as well as on the thermody-\nnamic properties of the system. However, the treatment\nof spin-orbital systems beyond the range of validity of\nmean-field decoupling and perturbative schemes remains\nan open problem in theory.\nAcknowledgments\nThe authors thank O. P. Sushkov for valuable dis-\ncussions. A.M.O. acknowledges support by the Foun-\ndation for Polish Science (FNP) and by the Polish Min-\nistry of Science and Higher Education under Project No.\nN202069639. J.S. acknowledgessupport by the graduate\nschool of excellence MAINZ (MATCOR).\nAppendix: Details of the perturbative approach for\nthe boson-fermion model\nWe start bya MF decoupling, rewritingthe interaction\nas\nH1=H1,MF+(H1−H1,MF)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nδH,\nwith\nH1,MF=1\nN2/summationdisplay\nkωBF(k,k,q)2/bracketleftBig\n˜nb,kf†\nqfq+ ˜nf,qb†\nkbk/bracketrightBig\n.\nWe treat δHas perturbation. The averages ˜ na,p=/angbracketleftbig\na†\npap/angbracketrightbig\nare determined self-consistently within this MF\nscheme. This leads to a renormalization of the magnon\nand fermion dispersions. We find\nζ(q) = 2JS/parenleftBigg\n|y|−1\nN/summationdisplay\nkcosk˜nf,k/parenrightBigg\n(1−cosq)−µ,\n(A.1)and\nωF(q) =/parenleftBigg\nS2+x+2\nN/summationdisplay\nk(1−cosk)˜nb,k/parenrightBigg\ncosq.(A.2)\nForT= 0 only the magnon dispersion is renormalized to\nζ(q) = 2JS(|y|+1/π)(1−cosq).\nBy virtue of Dyson’s equation we may calculate the\nbosonic Green’s function at T= 0.68Since the MF de-\ncoupling already takes the first order contributions into\naccount, the lowest order diagrams we obtain are of sec-\nond order. The self energy is thus approximated by\nthe proper self-energy obtained by summing up those\ndiagrams which may be composed by the second order\ndiagrams. From this we have Σ( q,ω)≈Σ2(q,ω)≡\nΣ+\nBF(q,ω)+Σ−\nBF(q,ω)+Σ2,1(q) where the diagrams are\ngivenby Σ+\nBF(q,ω) (Fig.6(a)), Σ−\nBF(q,ω) (Fig.6(b)), and\nΣ2,1(q) (Fig.6(c)). A straigthforward calculation reveals\nthat at zero temperature Σ 2,1(q) and Σ−\nBF(q,ω) vanish.\nFor Σ+\nBF(q,ω) we find the expression given in Eq. ( 5.5).\nFor finite temperatures we calculate the imaginary\ntime Green’s function. We find\nΣ±\nBF(q,ων,B) =−T2\nN2/summationdisplay\nk1,k2ω2\nBF(q,k1,k2)\n×/summationdisplay\na,bG(0)\nF(k2,ωa,F)G(0)\nF(q−k1+k2,ωb,F)\n×G(0)\nB(k1,±(ων,B−ωb,F+ωa,F))\n(A.3)\nfor the diagrams in Fig. 6(a,b), and\nΣ2,1(q) =−T2\nN2/summationdisplay\nk1,k2ωBF(q,q,k1)ωBF(k2,k2,k1)\n×/summationdisplay\na,b/bracketleftBig\nG(0)\nF(k1,ωa,F)/bracketrightBig2\nG(0)\nB(k2,ωb,B)(A.4)\nfor the diagram given in Fig. 6(c). Here we have used\nG(0)\nF(q,ωµ,F) = (iωµ,F−ωF(q))−1andG(0)\nB(q,ων,B) =\n(iων,B−ζ(q))−1as the fermionic and bosonic Matsub-\nara Green’s function for the noninteracting Hamiltonian,\nrespectively. ων,Fare the fermionic Matsubara frequen-\ncies. After performing the frequency sums we end up\nwith\nΣ2,1(q) =−1\nTN2/summationdisplay\nk1,k2ωBF(q,q,k1)ωBF(k2,k2,k1)\n×nB[ζ(k2)](1−nF[ωF(k1)])nF[ωF(k1)],\n(A.5)\nand the self-energies Σ±\nBF(q,ων,B) as given in Eqs. ( 5.7)\nand (5.9).17\n∗j.sirker@fkf.mpg.de\n1L. D. Landau, Phys. Z. Sowjetunion 3, 664 (1933).\n2H. Fr¨ ohlich, Advances in Physics 3, 325 (1954).\n3R. E. Peierls, Quantum Theory of Solids (Oxford Univer-\nsity Press, Oxford, 1955).\n4K. K. Das, Phys. Rev. Lett. 90, 170403 (2003).\n5A. Albus, F. Illuminati, and J. Eisert, Phys. Rev. A 68,\n023606 (2003).\n6M. A. Cazalilla and A. F. Ho, Phys. Rev. Lett. 91, 150403\n(2003).\n7L. Mathey, D.-W. Wang, W. Hofstetter, M. D. Lukin, and\nE. Demler, Phys. Rev. Lett. 93, 120404 (2004).\n8M. Lewenstein, L. Santos, M. A. Baranov, and\nH. Fehrmann, Phys. Rev. Lett. 92, 050401 (2004).\n9K. I. Kugel and D. I. Khomskii, Sov. Phys. JETP 52, 501\n(1980).\n10L. F. Feiner and A. M. Ole´ s, Phys. Rev. B 59, 3295 (1999).\n11Y. Tokura and N. Nagaosa, Science 288, 462 (2000).\n12Colossal Magnetoresistive Oxides , edited by Y. Tokura,\n(Gordon and Breach, Amsterdam, 2000).\n13N. N. Kovaleva, A. M. Ole´ s, A. M. Balbashov, A. Maljuk,\nD. N. Argyriou, G. Khaliullin, and B. Keimer, Phys. Rev.\nB81, 235130 (2010).\n14G. Khaliullin and S. Maekawa, Phys. Rev. Lett. 85, 3950\n(2000).\n15B. Keimer, D. Casa, A. Ivanov, J. W. Lynn, M. v. Zim-\nmermann, J. P. Hill, D. Gibbs, Y. Taguchi, and Y. Tokura,\nPhys. Rev. Lett. 85, 3946 (2000).\n16J. Hemberger, H.-A. K. von Nidda, V. Fritsch, J. Deisen-\nhofer, S. Lobina, T. Rudolf, P. Lunkenheimer, F. Lichten-\nberg, A. Loidl, D. Bruns, and B. B¨ uchner, Phys. Rev. Lett.\n91, 066403 (2003).\n17M. Cwik, T. Lorenz, J. Baier, R. M¨ uller, G. Andr´ e,\nF. Bour´ ee, F. Lichtenberg, A. Freimuth, R. Schmitz,\nE. M¨ uller-Hartmann, and M. Braden, Phys. Rev. B 68,\n060401 (2003).\n18Y. Ren, T. T. M. Palstra, D. I. Khomskii, A. A. Nugroho,\nA. A. Menovsky, and G. A. Sawatzky, Nature (London)\n396, 441 (1998).\n19Y. Ren, T. T. M. Palstra, D. I. Khomskii, A. A. Nugroho,\nA. A. Menovsky, and G. A. Sawatzky, Phys. Rev. B 62,\n6577 (2000).\n20M. Noguchi, A. Nakazawa, S. Oka, T. Arima, Y. Wak-\nabayashi, H. Nakao, and Y. Murakami, Phys. Rev. B 62,\nR9271 (2000).\n21G. Khaliullin, P. Horsch, and A. M. Ole´ s, Phys. Rev. Lett.\n86, 3879 (2001).\n22C. Ulrich, G. Khaliullin, J. Sirker, M. Reehuis, M. Ohl,\nS. Miyasaka, Y. Tokura, and B. Keimer, Phys. Rev. Lett.\n91, 257202 (2003).\n23P. Horsch, G. Khaliullin, and A. M. Ole´ s, Phys. Rev. Lett.\n91, 257203 (2003).\n24J. Sirker and G. Khaliullin, Phys. Rev. B 67, 100408\n(2003).\n25P. Horsch, G. Khaliullin, A. M. Ole´ s, and L. F. Feiner,\nPhys. Rev. Lett. 100, 167205 (2008).\n26J. Sirker, A. Herzog, A.M. Ole´ s, and P. Horsch, Phys. Rev.\nLett.101, 157204 (2008).\n27A. M. Ole´ s, P. Horsch, L. F. Feiner, and G. Khaliullin,\nPhys. Rev. Lett. 96, 147205 (2006).\n28C. Itoi, S. Qin, and I. Affleck, Phys. Rev. B 61, 6747(2000).\n29J. Sirker, Phys. Rev. B 69, 104428 (2004).\n30J. van den Brink, W. Stekelenburg, D. I. Khomskii, G. A.\nSawatzky, and K. I. Kugel, Phys. Rev. B 58, 10276 (1998).\n31Y. Q. Li, M. Ma, D. N. Shi, and F. C. Zhang, Phys Rev\nLett81, 3527 (1998).\n32This is reminiscent to the SU(4) model with AF exchange\nwhere all three elementary excitations are degenerate.\nHowever in the latter case the elementary excitations are\nof coupled spin-orbital type, whereas in the present case\nwe find elementary spin, orbital and combined spin-orbital\nexcitations.\n33This, we believe, is a consequence of a factor of 2 missed\nin the analysis of the elementary dispersions in this work.\n34P. A. M. Dirac, Principles of Quantum Mechanics (Oxford\nUniversity Press, Oxford, 1958).\n35E. Schr¨ odinger, Proceedings of the Royal Irish Academy\nSection A 47, 39 (1941).\n36H. A. Brown, Phys. Rev. B 31, 3118 (1985).\n37L.F. Feiner, A.M.Ole´ s, andJ. Zaanen, J. Phys.: Condens.\nMatter10, L555 (1998).\n38A. M. Ole´ s, L. F. Feiner, and J. Zaanen, Phys. Rev. B 61,\n6257 (2000).\n39M. Takahashi, Phys. Rev. B 42, 766 (1990).\n40M. Takahashi, Phys. Rev. B 47, 8336 (1993).\n41M. Takahashi, Phys. Rev. Lett 58, 168 (1987).\n42M. Takahashi, Prog. Theor. Phys. Suppl. 87, 233 (1986).\n43T. Mizokawa and A. Fujimori, Phys. Rev. B 54, 5368\n(1996).\n44H. Sawada and K. Terakura, Phys. Rev. B 58, 6831 (1998).\n45T. Mizokawa, D. I. Khomskii, and G. A. Sawatzky, Phys.\nRev. B60, 7309 (1999).\n46J.-H. Park, L. H. Tjeng, A. Tanaka, J. W. Allen, C. T.\nChen, P. Metcalf, J. M. Honig, F. M. F. de Groot, and\nG. A. Sawatzky, Phys. Rev. B 61, 11506 (2000).\n47R. J. Bursill, T. Xiang, and G. A. Gehring, J. Phys. Cond.\nMat.8, L583 (1996).\n48X. Wang and T. Xiang, Phys Rev B 56, 5061 (1997).\n49J. Sirker and A. Kl¨ umper, Europhys. Lett. 60, 262 (2002).\n50S. R. White and D. S. Huse, Phys. Rev. B 48, 3844 (1993).\n51S. Miyashita and N. Kawakami, J. Phys. Soc. Japan 74,\n758 (2005).\n52C. Itoi, S. Qin, and I. Affleck, Phys. Rev. B 61, 6747\n(2000).\n53Y. Chen, Z. D. Wang, Y. Q. Li, and F. C. Zhang, Phys.\nRev. B75, 195113 (2007).\n54A. Herzog, P. Horsch, A. M. Ole´ s, and J. Sirker, Journal\nof Physics Conference Series 200, 022017 (2010).\n55P. Pincus, Solid State Communications 9, 1971 (1971).\n56The discussion of the dynamical spin structure factor for\nthe dimerized FM chain will be given elsewhere.\n57We note that the MF decoupling applied to the interaction\ngiven in Eq. ( 2.8) does not renormalize the dispersion of\nthe fermions at T= 0.\n58T. Giamarchi, Quantum Physics in One Dimension\n(Clarendon Press, Oxford, 2003).\n59W. Metzner, C. Castellani, and C. D. Castro, Advances in\nPhysics47, 317 (1998).\n60W. Kohn, Phys. Rev. Lett. 2, 393 (1959).\n61E. J. Woll and S. J. Nettel, Phys. Rev. 123, 769 (1961).\n62G. Barnea and G. Horwitz, J. Phys. C 8, 2124 (1975).18\n63H. B. Møller and J. C. G. Houmann, Phys. Rev. Lett. 16,\n737 (1966).\n64S. V. Halilov, H. Eschrig, A. Y. Perlov, and P. M. Oppe-\nneer, Phys. Rev. B 58, 293 (1998).\n65M. Pajda, J. Kudrnovsk´ y, I. Turek, V. Drchal, and\nP. Bruno, Phys. Rev. B 64, 174402 (2001).\n66S. Mor´ an, C. Ederer, and M. F¨ ahnle, Phys. Rev. B 67,\n012407 (2003).\n67Performing the MF decoupling with the Hamiltonian ( 2.1)also takes the quartic orders of the Dyson-Maleev transfor-\nmation into account. These higher orders are neglected in\nthe MF decoupling performed in Sec. V. For low temper-\natures, i.e. the temperature region we have used in Sec. V\nthe contributions from these higher order terms are small.\n68A. L. Fetter and J. D. Walecka, Quantum Theory of Many-\nParticle Sytems (McGraw-Hill, Inc., New York, 1971)."    },    {        "title": "1912.07728v1.Spin_current_manipulation_of_photoinduced_magnetization_dynamics_in_heavy_metal___ferromagnet_double_layer_based_nanostructures.pdf",        "content": "IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 11, NOVEMBER 2017 1\nSpin-current manipulation of photoinduced magnetization dynamics\nin heavy metal / ferromagnet double layer based nanostructures\nSteffen Wittrock1, Dennis Meyer2, Markus M ¨uller2, Henning Ulrichs2, Jakob Walowski3, Maria Mansurova3,\nUlrike Martens3, and Markus M ¨unzenberg3\n1Unit´e Mixte de Physique CNRS/Thales, Universit ´e Paris Sud, 1 Avenue Augustin Fresnel, 91767 Palaiseau, France\n2I. Physikalisches Institut, Georg-August-Universit ¨at G ¨ottingen, Friedrich-Hund-Platz 1, 37077 G ¨ottingen, Germany\n3Institut f ¨ur Physik, Ernst-Moritz-Arndt-Universit ¨at Greifswald, Felix-Hausdorff-Str. 6, 17489 Greifswald, Germany\nSpin currents offer a way to control static and dynamic magnetic properties, and therefore they are crucial for next-generation\nMRAM devices or spin-torque oscillators. Manipulating the dynamics is especially interesting within the context of photo-magnonics.\nIn typical 3dtransition metal ferromagnets like CoFeB, the lifetime of light-induced magnetization dynamics is restricted to about 1\nns, which e.g. strongly limits the opportunities to exploit the wave nature in a magnonic crystal filtering device. Here, we investigate\nthe potential of spin-currents to increase the spin wave lifetime in a functional bilayer system, consisting of a heavy metal (8 nm\nof\f-Tantalum (Platinum)) and 5 nm CoFeB. Due to the spin Hall effect, the heavy metal layer generates a transverse spin current\nwhen a lateral charge current passes through the strip. Using time-resolved all-optical pump-probe spectroscopy, we investigate how\nthis spin current affects the magnetization dynamics in the adjacent CoFeB layer. We observed a linear spin current manipulation\nof the effective Gilbert damping parameter for the Kittel mode from which we were able to determine the system’s spin Hall angles.\nFurthermore, we measured a strong influence of the spin current on a high-frequency mode. We interpret this mode an an exchange\ndominated higher order spin-wave resonance. Thus we infer a strong dependence of the exchange constant on the spin current.\nIndex Terms —Spin Hall effect, spin current, magnetization dynamics, magnetooptical Kerr-effect.\nI. I NTRODUCTION\nTHE spin-transfer effect describes the transfer of spin an-\ngular momentum to a ferromagnet’s magnetization from\nan injected spin polarized current. Since its prediction by L.\nBerger [1], this effect has experienced a high research interest\nas it permits to manipulate and control the magnetization of a\nthin ferromagnetic (FM) layer. Especially when the resulting\nspin transfer torque is collinear with the damping torque,\nmagnetic dissipation can be controlled. Thereby the life time\nof spin wave dynamics can be drastically enhanced.\nAmong the possible methods of creating the necessary spin\ncurrent, the exploitation of the spin Hall effect has become a\npowerful mean since its first observation only a decade ago\n[2], [3], [4], [5]. Governed by spin-orbit coupling phenomena,\nit generates a spin current jsfrom a transverse charge current\njewithout any need for neither a ferromagnet nor an external\nmagnetic field. The efficiency of the conversion process can\nbe described by the spin Hall angle (SHA) \u0002SH=js=je.\nHere, we investigate the photo-induced magnetization dy-\nnamics in a few nanometer thin soft magnetic layer consisting\nof amorphous metallic cobalt iron boron alloy (Co 20Fe60B20)\nunder influence of a strong spin current generated by the SHE\nin an adjacent heavy metal film made of platinum (Pt) or\ntantalum (Ta). The sputter conditions for Ta were chosen such\nthat the film has grown in the high resistive \f-phase for which\na high SHA has recently been reported [6].\nII. E XPERIMENTAL\nThe samples consist of two functional thin layers (fig. 1a):\n8nm Pt or\f-Ta as a SHE-material generating the spin current\nManuscript received April 21, 2017; revised April 21, 2017. Corresponding\nauthor: S. Wittrock (email: steffen.wittrock@u-psud.fr).and5nm of ferromagnetic amorphous CoFeB, into which\nthe spin current is being injected in order to manipulate its\nmagnetization dynamics. The layer stack is complemented by\n3nm of Ru as a capping layer. CoFeB and Ta are deposited by\nargon ion sputtering and Pt and Ru by e-beam evaporation. All\npreparation steps are conducted in situ in ultra-high vacuum.\nSubsequent structuring of the samples by e-beam lithography\nenables the electrical contacting and generation of a high\ncharge current density (fig. 1b).\n(a)\n(b)\n (c)\nFig. 1: Experimental characteristics. (a) Schematic processes\nin the functional layer stack of a SHE material (blue) and the\nferromagnetic CoFeB (yellow). (b) Patterned sample structure,\nthe widthxwas12\u0016m for the results shown here. The red\nmarked area was excited by the laser spot. (c) Pump-probe\nsetup with ratio of powers Ppump :Pprobe = 95 : 5 , time\nresolution is realized by a delay stage, a double modulation\ntechnique of photoelastic modulator (PEM) and chopper fre-\nquency was used [7].\nWe used time resolved pump-probe spectroscopy exploitingIEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 11, NOVEMBER 2017 2\nthe magnetooptical Kerr effect to excite and measure magne-\ntization dynamics (schematical setup in 1c). The laser pulses\nwith central wave length \u0015= 800 nm have an autocorrelation\nlength of \u0001\u001c\u001980fs and a repetition rate of 250kHz. The\npump spot size was \u001960\u0016m providing an optical fluence of\nF= 15 mJ=cm2.\nThe incoming pump pulse induces the dynamics at \u001c= 0\nby firstly generating hot electrons, which thermalize on a\ntimescale of \u001c\u0018100fs due to electron-electron-scattering.\nFurther scattering events with phonons and spins lead to\nenergy transfer into the phonon and spin system giving rise to\nultrafast demagnetization [8]. Caused by the high change in\ntemperature on the time scale of \u00181ps, the local anisotropy\nconstant of the ferromagnetic material and thus the effective\nmagnetic field ~Heffis changed. With ~Heffrereaching its\nequilibrium position after a few picoseconds, the magnetiza-\ntion dynamics corresponding to the Landau-Lifshitz-Gilbert-\nequation (LLG) is excited. In order to subsequently trigger\na coherent precession of the magnetization, an external field\n(145mT) was applied at an out-of-plane angle of '= 35\u000e,\ntransversal to ~je.\nIII. R ESULTS -NANOSECOND TIMESCALE\n-0.5-0.4-0.3-0.2-0.10.00.10\n2 004 006 008 001 000-0.04-0.020.000.020.040.06T\nime delay τ [ps]TRMOKE signal [a.u.] \nData \nBackroundµ0H = 145 mT ;  ϕ = 35° ;  j = 6.2e10 A/m2 \nbackround-cleaned, fitted data \nDamping ∼ exp(- τ/τα)\n(a)\n-10-8-6-4-2024681012.012.112.212.312.412.512.612.712.812.9Kittel-Frequency [GHz]C\nurrent density in Ta-layer [1010 A/m2]µ0H = 145 mT ;  ϕ = 35° \n(b)\n-10-8-6-4-202468101.301.351.401.451.501.551.601.65µ0MS [T]C\nurrent density in Ta-layer [1010 A/m2]µ0H = 145 mT ;  ϕ = 35° (c)\nFig. 2: (a) Typical measurement data and Kittel mode after\nsubstraction of exponential background. The TRMOKE-signal\nis proportional to the magnetization. Dependence of (b) Kittel\nfrequency and (c) saturation magnetization on the current\ndensity in the Ta-layer.\nBesides the dominating coherent, in spatially homogeneous\ngeometries called Kittel mode oscillations, also incoherent\nphonons and magnons contribute to the signal and give rise\nto a certain background [9], [8], which can be modelled by\nexponential functions. Substracting it from the raw data (fig.\n2) highlights the magnetic oscillations. These are analysedwith respect to frequency and damping for different current\ndensities passing through the SHE-material.\nA. Kittel frequencies, magnetization, and Oersted field\nThe dependence of the Kittel frequency on jis shown in\nfig. 2b. The mainly parabolic behaviour can be attributed to\nthe reduction of the saturation magnetization due to Joule-\nheating. The observable asymmetries for opposite current\ndirections can be related to the Oersted field produced by the\ncurrent, or to the presence of a field-like torque. Analysing\nthe asymmetries, an Oersted field of HOe=ajwitha=\n(1:23\u00060:04)\u000110\u00008m is determined which agrees well with\ntheoretical estimations. Even if present, the field-like torque\nmust be much smaller than the effect arising from the Oersted\nfield. Taking also into account the in-plane applied magnetic\nfield component Hx, the saturation magnetization can be deter-\nmined from the Kittel formula !=\r\u00160p\nHx(Hx+MS)(fig.\n2c). Note that for the given field geometry, the magnetization’s\nout-of-plane component is only around 2%, as is estimated\nby micromagnetic simulations; therefore this approximation\nis valid. Besides Joule-heating, also the energy deposition of\nthe pump pulse heats up the sample locally to around 400-\n450K at a time scale of up to 1ns. Thus the saturation\nmagnetization at j= 0 A/m2is slightly lower than the\nroom temperature value of \u00160MS= 1:63T, determined by a\nvibrating sample magnetometer [10]. The Joule-heating effect\nleads to a temperature increase especially in Ta of up to 200K\nat a maximum current density of jTa= 9:3\u00011010A/m2.\nB. Effective Gilbert damping parameter of the Kittel mode\nFrom the exponential decay time \u001c\u000bof the Kittel mode (fig.\n2a) and taking into account the full in-plane magnetic field\nHx=Hextcos(') +HOe, and the current dependence of\nthe magnetization, we calculate the effective Gilbert damping\nparameter using:\n\u000bKittel =\u0012\n\u001c\u000b\r\u00160\u0012\nHx+MS\n2\u0013\u0013\u00001\n: (1)\n-10-8-6-4-202468100.0050.0060.0070.0080.0090.0100.011Effective Gilbert dampingC\nurrent density in Ta-layer [1010 A/m2] Calculated data \nLinear fit\n(a)\n-30- 20- 100 1 02 00.000.010.020.030.04C\nurrent density in Pt-layer [1010 A/m2]Effective Gilbert damping Calculated data \nLinear fit (b)\nFig. 3: Current dependence of the effective Gilbert damping\nparameter for (a) the Ta based sample and (b) the Pt based\nsample. From the linear fit, the SHE efficiency can be ex-\ntracted.\nResults for the two SHE-materials Ta and Pt are shown in\nfig. 3. We determined the SHA \u0002SHby fitting the formula:\n~\u000b=\u000b+j\u0001\u0002SH\u0001~\n2ed\u00160MSHx: (2)IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 11, NOVEMBER 2017 3\nNote that this expression was derived from linearizing the\nLandau-Lifshitz-Gilbert-Slonczewski-equation (LLGS).\nAveraging a few datasets gives the SHAs of:\n\u0002Ta\nSH=\u00000:043\u00060:011;\nand\n\u0002Pt\nSH= 0:086\u00060:012:\nThese results lie within the values obtained by other groups\n[11] and in particular confirm the different sign to be expected\nfor Pt [12], [13] and Ta [14], [6].\nC. Spin-pumping effect\nThe spin mixing conductance g\"#of a layer combination\ncan be estimated through comparison of the effective damping\nparameters ~\u000bof two layer stacks involving the same ferromag-\nnetic material. Using Ta and Pt as SHE-materials and CoFeB\nas the ferromagnet, the spin mixing conductance of Ta can be\ncalculated using:\ng\"#\nTa=\u00004\u0019MSd\u0001\u0001~\u000b\n~\r+g\"#\nPt (3)\n\u0001~\u000b= ~\u000bPt\u0000~\u000bTadenotes the difference of the effective\nmagnetic damping constants of the d= 5 nm thick CoFeB\nlayer for the two different adjacent SHE-materials. Average\nvalues of a few measurements give ~\u000bPt= (1:20\u00060:15)\u000110\u00002\nand~\u000bTa= (0:69\u00060:05)\u000110\u00002forj= 0. The Pt/CoFeB spin\nmixing conductance of g\"#\nPt= (4\u00061)\u00011019m\u00002[18] was used\nas a reference value.\nWe evaluate the spin mixing conductance of Ta/CoFeB to\ng\"#\nTa= (1:9\u00061:2)\u00011019m\u00002. The smaller value for Ta/CoFeB\ncompared to Pt/CoFeB is expected as similar values were\nalready obtained in Ta/NiFe [15] resp. Pt/NiFe [16] bilayers.\nIn conclusion, this analysis shows that TRMOKE is also a\nsuitable method to determine the spin mixing conductance\nwhich is an important parameter for heavy metal/FM based\nbilayer systems.\nIV. R ESULTS -PICOSECOND TIMESCALE\nOn the timescale of a few picoseconds, when the magneti-\nzation starts relaxating again, we observed a strongly damped,\nultrafast oscillation in the terahertz regime (fig. 4c), usually\nexistent for one or two periods (fig. 4). We identified this os-\ncillation as the well-known perpendicular standing spin-wave\n(PSSW) mode of first order n= 1. Analysing its frequencies\n(fig. 4c) and taking into account the saturation magnetization\ngathered on the nanosecond timescale, especially the exchange\nstiffnessA(fig. 4d) can be obtained from:\n!=\r\u00160s\u0012\nHx+2A\n\u00160MSk2\u0013\u0012\nHx+2A\n\u00160MSk2+MS\u0013\n(4)\nwithk2=k2\nz= (n\u0019=d)2,n2Nthe quantized wavevector\nnormal to the plane. The exchange stiffness is shown in fig.\n4d and found to depend on the spin current.Note that pinning at interfaces can alter the effective wave\nlength of the exchange mode. Without further experimental\nevidence, assuming zero pinning is the simplest model. Since\nthe exchange constant fits to known values determined from\nTRMOKE experiments on thicker films [17], we stick to this\nmodel.\n-2-10123456789-0.5-0.4-0.3-0.2-0.10.00.1 \nsmoothed data \n    data pointsTRMOKE signal [a.u.]T\nime delay τ [ps]Demagnetizationµ0H = 145 mT ;  ϕ = 35° ;  j = 6.2e10 A/m2\n(a)\n34 5 6 7 8 9 -0.06-0.04-0.020.000.020.040.06 original data \nsmoothed data \ndamped sinus fitTRMOKE signal [a.u.]T\nime delay τ [ps]\n(b)\n-10-8-6-4-20246810520540560580600620640Frequency [GHz]C\nurrent density in Ta-layer [1010 A/m2] (c)\n-8-6-4-2024682.32.42.52.62.72.82.93.03.13.23.3Exchange stiffness A [10-11 J/m]C\nurrent density in Ta-layer [1010 A/m2]\n(d)\n-8-6-4-202 4 6 8 0.020.040.060.080.100.120.140.16C\nurrent density in Ta-layer [1010 A/m2]αPSSW (e)\nFig. 4: (a) The first ten picoseconds of the excited dynamics\nincluding an ultrafast, highly damped oscillation during re-\nmagnetization (b). (c-e) Current dependence of certain derived\nparameters.\nThe Gilbert damping parameter of the PSSW mode (fig.\n4e) can again be determined from the oscillation’s exponential\ndecay time\u001c\u000b:\n\u000bPSSW =\u0012\n\u001c\u000b\r\u00160\u0012\nHx+2Ak2\n\u00160MS+MS\n2\u0013\u0013\u00001\n:(5)\nEspecially the exchange term \u0018Ak2=MSproves to be dom-\ninant due to the small CoFeB thickness. We attribute the (at\nleast for positive jTa) linear behaviour of \u000bPSSW to the SHE.\nThe curve’s slope (fig. 4e, jTa\u00150) possesses the same sign\nas the Kittel mode damping \u000bKittel . Furthermore, \u000bPSSW\nis found to be one order of magnitude higher than \u000bKittel .\nThe relative change from \u000bPSSW\u00190:04for a high positive\ncurrent density up to \u000bPSSW\u00190:13forj= 0shows a strong\ndependence on the spin current which is around three times\nhigher than for the Kittel mode.\nV. D ISCUSSION\nAccording to our experiments, the SHA for a Pt-based layer\nstructure is almost double the SHA of the Ta-system. A strongIEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 11, NOVEMBER 2017 4\nJoule-heating effect occurred especially in the Ta based struc-\nture because of its high resistivity. The latter also contributes\nto an eventual systematic reduction of the tantalum’s SHA by\nup to 10% assuming the capped Ru displays a small negative\nSHA of \u0002SH\u0019\u00000:001[19]. This would lead to a spin current\nwith opposing sign flowing into the CoFeB from above.\nThe strong spin pumping mechanism and thus increase of\nthe magnetic damping constant especially in the Pt sample\nis theoretically expected due to platinum’s high spin flip\nprobability. Adding an interlayer with high spin diffusion\nlength (such as Cu [20], [21], [22]) could solve this issue.\nThe interesting discovery of the fast, highly damped mag-\nnetic oscillation within the first 10ps of the remagnetization\nprocess is identified as the PSSW mode of first order. It\nshows a strong dependence on the injected spin current which\ninfluences its frequency, the exchange stiffness and the mode’s\ndamping. Note that usually, the PSSW mode is optically\nexcitable and observable in samples, whose layer thickness\nis higher than the optical penetration depth which is \u001830nm\nin our case. To obtain clear evidence of this mode’s properties,\nfurther investigations with enhanced signal to noise ratio are\nnecessary.\nVI. C ONCLUSION AND OUTLOOK\nIn this article we discussed the photoinduced magnetization\ndynamics under influence of an injected spin current, generated\nby the SHE. The powerful tool of time resolved magnetoop-\ntical Kerr effect, compared to more established methods like\nBLS or ST-FMR, allowed to investigate nanosecond as well\nas (sub-)picosecond dynamics and to get insights into high-\nfrequency and non-equilibrium dynamics so far not in the\nfocus within this context.\nWe discussed the influence of Joule-heating on our samples\nand described the linear manipulation of the Kittel mode’s\nGilbert damping through a spin current. The spin Hall angles\nwhich could be determined for the two different, commonly\nused SHE materials Ta and Pt, lie within other reported values\n[11].\nWe reported a magnetic oscillation at the picosecond\ntimescale which is found to be highly dependent on the spin\ncurrent. The exact behaviour of this mode has to be further\ninvestigated in future experiments with enhanced signal to\nnoise ratio. Furthermore, the interesting timescales of the de-\nmagnetization process will be adressed in future publications.\nStandard methods such as ST-FMR do not allow to address\nsuch high-frequency dynamics easily. Our approach enables\nus to enter this temporal regime, which provides new insights\non the action of spin torques on picosecond time scales.\nACKNOWLEDGMENT\nThe authors acknowledge financial support by the DFG,\nwithin the CRC 1073 ’Atomic scale control of energy con-\nversion’.\nREFERENCES\n[1] B ERGER , L.: Exchange interaction between ferromagnetic domain wall\nand electric current in very thin metallic films. Journal of Applied Physics\n55, 6, p. 1954-1956, 1984.[2] K ATO, Y.K.; M YERS , R.C.; G OSSARD , A.C.; A WSCHALOM , D.D.:\nObservation of the Spin Hall Effect in Semiconductors. Science 306, 5703,\np. 1910-1913, 2004.\n[3] W UNDERLICH , J.; K ¨ASTNER , B.; S INOVA , J.; J UNGWIRTH , T.: Exper-\nimental discovery of the spin-Hall effect in Rashba spin-orbit coupled\nsemiconductor systems. condmat/0410295.\n[4] D AY, C.: Two Groups Observe the Spin Hall Effect in Semiconductors.\nPhys. Today 58, 2, 17, 2005.\n[5] W UNDERLICH , J.; K ¨ASTNER , B.; S INOVA , J.; J UNGWIRTH , T.: Experi-\nmental observation of the spin-Hall effect in a two-dimensional spin-orbit\ncoupled semiconductor system. Physical Review Letters, 94(4), 047204,\n2005.\n[6] L IU, L.; P AI, C.-F.; L I, Y.; T SENG , H. W.; R ALPH , D. C.; B UHRMAN ,\nR. A.: Spin-torque switching with the giant spin Hall effect of tantalum.\nScience 336, 6081, 2012.\n[7] K OOPMANS , B.: Laser-Induced Magnetization Dy-\nnamics. In: H ILLEBRANDS , B.; O UNADJELA , K.:\nSpin Dynamics in Confined Magnetic Structures II. , 2003.\n[8] B EAUREPAIRE , E.; M ERLE , J.-C.; D AUNOIS , A.; B IGOT , J.-Y.: Ultra-\nfast Spin Dynamics in Ferromagnetic Nickel. Phys. Rev. Lett. 76, p. 4250-\n4253, 1996.\n[9] D JORDJEVIC , M.; L ¨UTTICH , M.; M OSCHKAU , P.; G UDERIAN , P.;\nKAMPFRATH , T.; U LBRICH , R. G.; M ¨UNZENBERG , M.; F ELSCH , W.;\nMOODERA , J. S.: Comprehensive view on ultrafast dynamics of ferro-\nmagnetic films. Physica status solidi (c) 3, 5, p. 1347-1358, 2006.\n[10] M ANSUROVA , M.: Magnetization and elastic dynamics in nanostruc-\ntured metamaterials. I. Physikalisches Institut, Georg-August-Universit ¨at\nG¨ottingen, 2016.\n[11] S INOVA , J.; V ALENZUELA , S.O.; W UNDERLICH , J.; B ACK, C.H.;\nJUNGWIRTH , T.: Spin Hall effects. Reviews of Modern Physics 87, 4,\np. 1213-1260, 2015.\n[12] L IU, L.; M ORIYAMA , T.; R ALPH , D.C.; B UHRMAN , R.A.: Spin-Torque\nFerromagnetic Resonance Induced by the Spin Hall Effect. Phys. Rev.\nLett. 106, 036601, 2011.\n[13] A NDO , K.; T AKAHASHI , S.; H ARII, K.; S ASAGE , K.; I EDA, J.;\nMAEKAWA , S.; S AITOH , E.: Electric Manipulation of Spin Relaxation\nUsing the Spin Hall Effect. Phys. Rev. Lett. 101, 036601, 2008.\n[14] M OROTA , M.; N IIMI, Y.; O HNISHI , K.; W EI, D.H.; T ANAKA , T.;\nKONTANI , H.; K IMURA , T.; O TANI , Y.: Indication of intrinsic spin Hall\neffect in 4 dand 5 dtransition metals. Phys. Rev. B 83, 174405, 2011.\n[15] M ONTOYA , E.; O MELCHENKO , P.; C OUTTS , C.; L EE-HONE, N.R.;\nH¨UBNER , R.; B ROUN , D.; H EINRICH , B.; G IRT, E.: Spin transport in\ntantalum studied using magnetic single and double layers. Phys. Rev. B\n94, 054416, 2016.\n[16] C ZESCHKA , F.D.; D REHER , L.; B RANDT , M.S.; W EILER , M.; A L-\nTHAMMER , M.; I MORT , I.-M.; R EISS, G.; T HOMAS , A.; S CHOCH , W.;\nLIMMER , W.; H UEBL , H.; G ROSS , R.; G OENNENWEIN , S.T.B.: Scaling\nbehavior of the spin pumping effect in ferromagnet/platinum bilayers.\narXiv:1012.3017 , 2011.\n[17] U LRICHS , H; L ENK, B.; M ¨UNZENBERG , M.: Magnonic spin-wave\nmodes in CoFeB antidot lattices. Appl. Phys. Lett. 97, 092506, 2010.\n[18] R UIZ-CALAFORRA , A.; B R¨ACHER , T.; L AUER , V.; P IRRO , P.; H EINZ ,\nB.; G EILEN , M.; C HUMAK , A.V.; C ONCA , A.; L EVEN , B.; H ILLE -\nBRANDS , B.: The role of the non-magnetic material in spin pumping and\nmagnetization dynamics in NiFe and CoFeB multilayer systems. Journal\nof Applied Physics 117, 16, 2015.\n[19] K AMPFRATH , T.; B ATTIATO , M.; M ALDONADO , P.; E ILERS , G.;\nN¨OTZOLD , J.; M ¨AHRLEIN , S.; Z BARSKY , V.; F REIMUTH , F.;\nMOKROUSOV , Y. ; B L¨UGEL , S.; W OLF, M.; R ADU , I.; O PPENEER ,\nP.M. ; M ¨UNZENBERG , M.: Terahertz spin current pulses controlled by\nmagnetic heterostructures. Nature nanotechnology 8, 4, p. 256260, 2013.\n[20] M IZUKAMI , S.; A NDO , Y.; M IYAZAKI , T.: Effect of spin diffusion on\nGilbert damping for a very thin permalloy layer in Cu/permalloy/Cu/Pt\nfilms. Phys. Rev. B 66, 104413, 2002.\n[21] S UN, Y.; C HANG , H.; K ABATEK , M.; S ONG , Y.-Y.; W ANG , Z.;\nJANTZ , M.; S CHNEIDER , W.; W U, M.; M ONTOYA , E.; K ARDASZ , B.;\nHEINRICH , B.; V ELTHUIS , S.G.E.; S CHULTHEISS , H.; H OFFMANN , A.:\nDamping in Yttrium Iron Garnet Nanoscale Films Capped by Platinum.\nPhys. Rev. Lett. 111, 106601, 2013.\n[22] R OJAS -S´ANCHEZ , J.-C.; R EYREN , N.; L ACZKOWSKI , P.; S AVERO ,\nW.; A TTAN ´E, J.-P.; D ERANLOT , C.; J AMET , M.; G EORGE , J.-M.; V ILA,\nL.; J AFFR `ES, H.: Spin Pumping and Inverse Spin Hall Effect in Platinum:\nThe Essential Role of Spin-Memory Loss at Metallic Interfaces. Phys. Rev.\nLett. 112, 106602, 2014."    },    {        "title": "1608.08153v1.Spin_pumping_and_measurement_of_spin_currents_in_optical_superlattices.pdf",        "content": "Spin pumping and measurement of spin currents in optical superlattices\nC. Schweizer1;2, M. Lohse1;2, R. Citro3;4, I. Bloch1;2\n1Fakultät für Physik, Ludwig-Maximilians-Universität, Schellingstrasse 4, D-80799 München, Germany\n2Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Strasse 1, D-85748 Garching, Germany\n3Dipartimento di Fisica “E. R. Caianiello”, Università degli Studi di Salerno,\nVia Giovanni Paolo II 132, I-84084 Fisciano (Salerno), Italy\n4SPIN-CNR Salerno, Via Giovanni Paolo II 132, I-84084 Fisciano (Salerno), Italy\n(Dated: September 11, 2021)\nWe report on the experimental implementation of a spin pump with ultracold bosonic atoms in\nan optical superlattice. In the limit of isolated double wells it represents a 1D dynamical version\nof the quantum spin Hall effect. Starting from an antiferromagnetically ordered spin chain, we\nperiodically vary the underlying spin-dependent Hamiltonian and observe a spin current without\ncharge transport. We demonstrate a novel detection method to measure spin currents in optical\nlattices via superexchange oscillations emerging after a projection onto static double wells. Further-\nmore, we directly verify spin transport through in-situ measurements of the spins’ center of mass\ndisplacement.\nExposing materials to strong magnetic fields has led to\nremarkable discoveries, most prominently the pioneering\nobservation of the integer and fractional quantum Hall\neffect [1, 2]. These quantum phenomena surprise due\nto their robustness and independence of material prop-\nerties, arising from their topological nature [3]. In this\ncontext, Thouless recognized that 1D dynamical systems\ncan share the same topological character as the 2D in-\nteger quantum Hall (IQH) effect [4, 5]. Such topological\nchargepumpsexhibitaquantizedtransportperpumpcy-\ncle in a gapped filled band of an adiabatically and period-\nically evolving potential. More recently, a fundamentally\ndifferent quantum state was observed [6], the topologi-\ncal insulator (TI) [7, 8], which preserves in addition to\ncharge time-reversal symmetry. In 2D systems with spin\nconservation, it exhibits the quantum spin Hall (QSH)\neffect, characterized by a quantized spin but vanishing\ncharge conductance. Analogous to the Thouless pump, a\ndynamical version of a TI can be designed – a quantum\nspin pump [9–11].\nSpin pumps could serve as spin current sources, e.g. for\nspintronic applications [12]. Such spin current generators\nhave been proposed based on the spin Hall effect [13, 14],\nthe cyclic variation of two system parameters in inter-\nacting quantum wires [15, 16], and topological insula-\ntors [17]. However, spin pumps have been realized only\nin few experiments, e.g. in quantum dot structures [18]\nand by parametrically excited exchange magnons [19].\nHere, we demonstrate the first implementation of a spin\npump with ultracold bosonic atoms in optical superlat-\ntices and present a direct measurement of the arising spin\ncurrent.\nIn1983,Thoulessinvestigatedparticletransportintwo\nsuperimposed1Dperiodicpotentialsadiabaticallymoved\nrelative to each other . This sliding motion periodically\nvaries the combined potential and thereby the underlying\nsingle-particle Hamiltonian, which can be parametrized\nby the cyclic pump parameter \u001e. During the pumping, aparticle acquires an anomalous velocity proportional to\ntheBerrycurvaturedefinedontheclosedsurfacespanned\nby\u001eand the quasi-momentum k. The resulting displace-\nmentafteronepumpcycleonlydependsonthegeometric\nproperties of the pump cycle and is not quantized unless\nall quasi-momenta of a band are occupied equally. In\nthis case, the pump is topological with a displacement\nproportional to the Chern number, the integral of the\nBerry curvature over the entire surface. Thus, transport\nis quantized and robust against perturbations [4, 5]. Re-\ncently, such geometric and quantized, topological pumps\nhave been realized with ultracold bosonic [20, 21] and\nfermionic atoms [22].\nIn analogy to the Thouless pump, a spin pump can\nbe thought of as a dynamical version of a QSH system\n[9], which is characterized by a bulk excitation gap and\ngapless edge excitations. In general, the electron spin\nis not conserved, e.g. in the presence of spin-orbit cou-\npling, and therefore unconventional topological invari-\nants, like the Z2index [7], are needed for classification.\nNon-interacting QSH systems with spin conservation can\nbe interpreted as two independent IQH systems. There-\nfore, a quantum spin pump can be composed of two in-\ndependent pumps, where the up and down spins have\ninverted Berry curvature and are therefore transported\nin opposite directions.\nA quantum spin pump can be implemented with ultra-\ncold atoms in two hyperfine states in a spin-dependent\ndynamically controlled optical superlattice, which can be\nformed by superimposing two lattices with periods ds\nanddl= 2ds. In the tight binding limit, a spin in this\nsuperlattice is described by the Rice-Mele model [23],\nwhich comprises staggered on-site energies \u0006\u0001=2be-\ntween neighboring sites and alternating tunnel cou-\nplings1\n2(J\u0006\u000eJ)with the dimerization parameter \u000eJ.\nPumping can be induced by an adiabatic modulation\nof the potential and corresponds to a loop in parame-\nter space (\u000eJ,\u0001) around the degeneracy point ( \u000eJ= 0,arXiv:1608.08153v1  [cond-mat.quant-gas]  29 Aug 20162\n\u0001 = 0). If the two spin components do not interact\nwith each other, their pumping motion is independent\nand a spin pump can be realized by a spin-dependent\ndeformation of the potential, so that time-reversal sym-\nmetry is retained and their Berry curvature is reversed.\nThe associated spin transport is quantized only for equal\noccupation of all quasi-momenta, which can be realized\nwith non-interacting fermions by placing the Fermi en-\nergy in the band gap and bosons by localizing each spin\ncomponent to a Mott insulator with negligible inter-spin\ninteraction.\nIn addition, the two spin components can be coupled\nby introducing on-site interactions Ubetween the atoms.\nFor hardcore interactions and unit filling, the bare tun-\nneling is suppressed and the system can be described by\na 1D spin chain\n^H=\u00001\n4X\nm\u0000\nJex+ (\u00001)m\u000eJex\u0001\u0010\n^S+\nm^S\u0000\nm+1+h.c.\u0011\n+\u0001\n2X\nm(\u00001)m^Sz\nm(1)\nwithspin-dependenttilt \u0001andalternatingexchangecou-\npling1\n2(Jex\u0006\u000eJex)'(J\u0006\u000eJ)2=U. For large tilts \u0001\u001d\n1\n2(Jex+\u000eJex)the many-body ground state are locked\nspins in an antiferromagnetic order, while for strong\nexchange coupling1\n2(Jex+\u000eJex)\u001d\u0001dimerized en-\ntangled pairs are favored. Implementing this Hamilto-\nnian requires a dynamically controllable spin-dependent\nsuperlattice [24, 25]. In the limit of isolated double\nwells\u000eJex\u0019Jex, applying a global gradient to a spin-\nindependent superlattice can locally reproduce the stag-\ngered tilts (Fig.1(a)). In this situation, a variation of\nthe parameters ( \u000eJ,\u0001) during the spin pump cycle cor-\nresponds to a modulation of ( Jex,\u0001) in the interacting\n1D spin chain. This cycle needs to be performed adiabat-\nically with respect to the intra double well exchange cou-\npling1\n2(Jex+\u000eJex). The pump cycle encircles the degen-\neracy point ( \u000eJex= 0,\u0001 = 0) as illustrated in Fig.1(a).\nDespite the antiferromagnetically ordered state being an\nexcitedmany-bodystateinthegloballytiltedsystem, the\npump can still be described by a topological invariant as\nthe pump of Eq.1 with local tilts, if the pumping is fast\ncompared to1\n2(Jex\u0000\u000eJex), which determines the gaps of\nadditional level crossings in the spectrum [26]. This can\nbe readily achieved in the experiment by choosing lattice\ndepths, for which inter-double well coupling is quenched.\nThe experimental setup consists of a 3D optical lat-\ntice with a superlattice along the x-axis and deep trans-\nverse lattices along yandzto create an array of decou-\npled 1D systems. Each system is initially occupied by\nan antiferromagnetically ordered spin chain of up j\"i=\njF= 1;mF=\u00001iand downj#i=jF= 1;mF= +1i\n87Rb atoms [27], localized on individual sites with Jex\u0019\n\u000eJex\u00190[26]. To start the pump cycle, every second\nbarrier is decreased to transfer two neighboring spins tothe ground state of a double well. Since a large mag-\nnetic gradient \u0001\u001dJexis present along x, the up (down)\nspin stays localized on the left (right) side, shortly de-\nnoted asj\";#i. The gradient is then reversed adiabati-\ncally compared to Jex. Thereby the wavefunction follows\nthe instantaneous eigenstate and a spin current occurs\nas the two spins exchange their positions via the delocal-\nized triplet state1p\n2\u0000\nj\";#i+j#;\"i\u0001\nat\u0001 = 0(Fig.1(b)).\nAt the end of the first half cycle, individual sites are\ndecoupled by increasing the short lattice depth. Subse-\nquently,\u000eJexis inverted by flipping the dimerization and\nalso the magnetic gradient to maintain the correct local\ntilts (insets Fig.1(a)). This corresponds to a projection\non double wells shifted by one lattice site. The state of\nthe system remains unchanged during this sudden switch\nand the spins in the new shifted double well are in the\nground state. After a full cycle, the two spin components\nhave each moved by 2dsin opposite directions; therefore\nthe total particle current vanishes as the contributions\nRLRφ = π\nB´x B´xU∆Jex J\nspin transportφδJex\n∆\nRLRφ = 0\nba\nds\nFigure 1. Spin pump cycle. (a) Spin pump cycle in parameter\nspace (green) of spin-dependent tilt \u0001and exchange coupling\ndimerization \u000eJex. The path can be parametrized by the an-\ngle\u001e, the pump parameter. The insets in the quadrants show\nthe local mapping of globally tilted double wells to the cor-\nresponding local superlattice tilts with the black rectangles\nindicating the decoupled double wells. Between \u001e= 0and\n\u0019,j\"iandj#ispins exchange their position, which can be\nobserved by site-resolved band mapping images detecting the\nspin occupation on the left (L) and right (R) sites, respec-\ntively. (b) Evolution of the two-particle ground state in a\ndouble well around \u0001 = 0with tunnel coupling1\n2(J+\u000eJ),\non-site interaction energy U, and spin-dependent tilt \u0001as\nwell as the exchange coupling Jex'(J+\u000eJ)2=Uand the\nlattice constant ds.3\n-500 -250 0 250 500\n∆s/h (Hz)0.000.060.120.18A\nII\nIII\nI\nImbalance   \n0.60.70.8\n0 1 2\nt´ (ms)III-0.10.00.10 2.5 5\nII\n0 3 6-0.5-0.4-0.3 I\n-10 0 10-0.800.8∆/h (kHz)\n-10 0 1000.060.12a2\n-10 0 10\nt (ms)-101b a\nFigure 2. Spin current measurement. (a) Illustration of the measurement scheme. \u0001is ramped with a rate of _\u0001 = 82(2) kHz/s\nat\u0001 = 0and the ramp is stopped abruptly at different points in the cycle \u0001s(upper graph). During this ramp, the two-particle\nwavefunction initially in the ground state has a small admixture a2of the first excited state around \u0001 = 0(middle graph).\nAfter the stop of the ramp, this admixture leads to spin imbalance oscillations with an amplitude proportional to a2and thus to\nthe instantaneous spin current at ts. The lower graph shows a numerical simulation of the spin imbalance time traces assuming\nperfect adiabaticity. (b) Spin imbalance oscillation amplitude Aat different points in the pump cycle for Jex=h= 342(2) Hz\n(blue) and Jex=h= 467(3) Hz (orange). Each point is the amplitude obtained by fitting Eq.3 to the spin imbalance that was\nmeasured as a function of the holdtime t0; error bars are the fit uncertainty. Three traces are shown on the right hand side\nfor\u0001s=h=\u0000144(7)Hz (I, dark blue), \u0001s=h= 18(5)Hz (II, blue), and \u0001s=h= 530(30) Hz (III, light blue) corresponding to\nthe illustrations in (a). Each trace consists of 26 points, which were averaged five times. The light solid lines in the main\nplot show the numerical calculation for the oscillation amplitude taking into account the reduced detection efficiency due to a\nresidual exponential decay of \u0001. The dark solid lines include additionally a finite ground state occupation of 97(1) %and a\npump efficiency of 89(1) %, which were measured separately by band mapping.\nfrom the two spin components cancel each other exactly.\nThus, pumping leads to a spin transport without induc-\ning a particle current.\nThe spin current jbetween the left (L) and the right\n(R) siteof adouble well isrelated tothe change in theex-\npectation value of the spin imbalance I=1\n2(nL#\u0000nL\"\u0000\nnR#+nR\")given by the integral form of the continuity\nequation 2j=@tI, withni\u001bthe occupation of spin \u001b\non sitei. To understand how this spin current arises\nand how it can be detected, it is useful to examine the\nevolution of the eigenstates during the adiabatic change\nof\u0001(t). Two spins initially at time tiin the eigenstate\njntiiofthetwo-particledoublewellHamiltonian ^HDW(ti)\n[26] follow the instantaneous eigenstate jnti, but acquire\n– even for a perfect adiabatic evolution – a small imag-\ninary contribution iam(t)from other eigenstates jmti.\nThis admixture occurs only temporarily during the ramp\nand induces an anomalous spin velocity (Fig.2(a)). The\ncoefficients amcan be calculated in first-order perturba-\ntion theory am(t) =\u0000_\u0001hmtj~@\u0001jnti\nEn(t)\u0000Em(t)with _\u0001 =@t\u0001(t)\nthe ramp speed and El(t)the eigenenergy of jlti[28].\nWhen starting from the ground state j1ti, the wavefunc-\ntion is well approximated by only considering contribu-\ntions from the first excited state j ti\u0019j1ti+ ia2(t)j2ti.\nTheam-coefficients of higher lying states are stronglysuppressed as the corresponding wavefunctions depend\nweakly on \u0001andEm\u0000E1\u001dJex. The wavefunction j ti\ncan be probed by a sudden stop of the pump cycle at\ntimetsby projecting it onto ^HDW(ts). During the subse-\nquent time evolution, the two states j1tiandj2tiacquire\na relative phase leading to oscillations of the spin im-\nbalanceI(t) =Is+Asin[(E2(ts)\u0000E1(ts))=~\u0001(t\u0000ts)]\nwithIsthe imbalance at time ts. The oscillation am-\nplitudeA=\u00002a2(ts)h1tsj^Ij2tsiis proportional to the\nadmixture of the second eigenstate and can be related to\nthespincurrent j(ts)throughthecontinuityequation[26]\nj(ts) =AE2(ts)\u0000E1(ts)\n2~: (2)\nExperimentally, the gradient ramp was abruptly stopped\nat\u0001s= \u0001(ts)and a time trace of the resulting double\nwell superexchange oscillation [29] was recorded by a si-\nmultaneous measurement of nL\",nL#,nR\"andnR#with\nStern-Gerlach separated site-resolved band mapping im-\nages (Fig.2(a)). The amplitude Awas found by fitting\nIfit(t0) =Ae\u0000t0=\u001cexsin (!ext0+\u0012) +Is+Ide\u0000t0=\u001cd(3)\nto the oscillation data, where t0=t\u0000tsand\u0012\u00190\na phase shift induced by a finite freezing ramp speed.\nCompared to the ideal evolution, two additional effects4\nj\n250 350 450 550\nJex/h (Hz)0.00.10.2A250 350 450 550\nJex/h (Hz)0.00.51.01.5int.  \nFigure 3. Imbalance oscillation amplitude Aat\u0001s= 0as a\nfunction of Jex. Each point is an average of the fitted am-\nplitudes of three time traces; the error bars are the standard\ndeviations. The blue lines are a numerical calculation taking\ninto account the reduced detection efficiency as well as the\nmeasured initial state occupation and finite pump efficiency.\nAssuming a constant scaling factor for each Jexas indicated\nby the measurement in Fig.2, the integrated current per cycle\ncan be estimated and is shown in the inset. In the ideal case,\nthe integrated current is equal to one (gray line).\nare taken into account. First, an exponential decay of\nthe amplitude with a time constant \u001cexaccounts for de-\nphasing between individual double wells. Both, \u001cexand\nthe oscillation frequency !ex'(E2\u0000E1)=~were deter-\nmined for each \u0001swith an independent superexchange\noscillation measurement. Second, an additional decay of\nthe imbalance offset Idis caused by an exponential re-\nlaxation of a small residual magnetic field gradient after\nthe abrupt stop of B0=\u000024:3(6)Hz=ds, with a decay\nconstant\u001cd= 1:05(5)ms. The resulting oscillation am-\nplitudes during the pump cycle for two different Jexas\nwell as exemplary spin imbalance traces are summarized\ninFig.2(b). Thespincurrentpeaksaround \u0001 = 0, where\nthe ground state is delocalized and spins move. For large\ngradients the eigenstates are independent of \u0001and the\nspin current vanishes. Note that the residual gradient\n\u0001dslightly shifts the peak towards higher \u0001. Further-\nmore, the spin current strongly depends on the exchange\ncouplingJex. With increasing Jex, the wavefunction de-\nlocalizes and depends less on \u0001, so the peak width in-\ncreases while the maximum amplitude decreases. How-\never, unlike the instantaneous current, the transported\nspin during one pump cycle, is independent of the pump\nparameters.\nTo compare the data with theoretical expectations, we\nperformed a numerical calculation including a reduced\ndetection efficiency caused by the residual gradient de-\ncay. Imbalance time traces for  tswere evaluated using\na two spin, two-site extended Bose-Hubbard Hamiltonian\n^HDW(t0)with \u0001(t0) = \u0001de\u0000t0=\u001cd+ \u0001s[26, 30]. The cal-\nculated time traces were also fitted with Eq.3; the re-\nsulting oscillation amplitude describes ideal transport of\n0 0.5 1 1.5 2\nφ (2π )-3-2-10123x (ds )\nFigure4. Center-of-masspositionofup(red)anddown(blue)\nspins as a function of the pump parameter \u001e. The points\nshow the center-of-mass position averaged over ten data sets\nof a spin-selective imaged atom cloud; the error bars show\nthe error of the mean. Each data set consists of an average\nof ten pairs, which contain an image obtained by a sequence\nwith pumping and one using a reference sequence with the\nsame length but constant pump parameter \u001e= 0. Difference\nimages of both sequences for up and down spin are shown on\nthe right side. The solid lines depict the calculated motion\nof a localized spin for the ideal case (light gray) and for a\nreduced ground state occupation and a pump efficiency per\nhalf pump cycle that was determined independently through\na band mapping sequence (gray).\nground state spins (light lines in Fig.2). This curve can\nbe fitted to the data by rescaling the amplitude with a\nfactor of 0:84(6), which directly determines the reduction\nof the integrated measured current compared to the ideal\none and thereby the transported spin polarization. The\ndeviation from ideal transport can be attributed to an\nimperfect initial state preparation with 97(1) %ground\nstate occupation and a pump efficiency per half a pump\ncycle of 89(1) %, which describes the fraction of double\nwells that remain in the ground state after half a cycle.\nConsidering this additional occupation of the first ex-\ncited state, which creates an opposite current, the total\nspin current can be calculated (dark lines in Fig.2(b)).\nFitting these expected oscillation amplitudes to the mea-\nsured data by rescaling with a global factor results in a\nfit value of 1:05(8)forJex=h= 342(2) Hz and 1:06(8)\nforJex=h= 467(3) Hz. This shows that even though\nthe shape and amplitude of the curve changes, the in-\ntegrated current is only defined by the pumps’ topology\nnot by the specific tunneling parameters. Furthermore,\nwe note that the deviation to the theoretically expected\nintegrated current can not be attributed to edge effects\nas they are negligibly small for the present trapping po-\ntential.\nTo study the dependence of the maximum current\non the exchange coupling, the oscillation amplitude was\nmeasured at \u0001s= 0for various Jex(Fig.3). The max-\nimum amplitude decreases with rising Jex, as the spins’\nwavefunction is more delocalized for the same \u0001and5\ntherefore the current flow is spread over a larger sector\nin the pump cycle. The observed peak amplitude agrees\nwith the numerical model including the initial ground\nstate occupation and pump efficiency. As suggested by\nthe measurements in Fig.2, the integrated spin current\ncan be extracted by rescaling the ideal amplitude with a\nglobal factor and is found to be constant for all exchange\ncouplings (inset Fig.3).\nIndependent evidence for the spin separation and a\nquantitative comparison with the total spin current can\nbe obtained by measuring the center-of-mass position of\nthe two spin components from in-situ absorption images\nafter removing one of the spin components. When vary-\ning\u001e, the up and down spins clearly separate (Fig.4).\nThis independently verifies the spin transport and shows\nquantitative agreement with the results of the spin cur-\nrent measurement [26].\nIn conclusion, we have demonstrated the implemen-\ntation of a spin pump and introduced a new method for\ndirectly measuring instantaneous spin currents. Compar-\ning the measured spin imbalance oscillation amplitudes\nwith the adiabatic theory shows that the integrated cur-\nrent is independent of the specific pump parameters and\ngives evidence for the utility of the developed current\nmeasurement method. The method can also be extended\nto more general systems by performing an instantaneous\nprojection onto double wells. Investigating such spin\npumps on a single site level would allow for local obser-\nvation of spin currents and the direct observation of edge\nexcitations in finite systems [31]. A system described by\nthe non-trivial Z2-invariant can be realized with time-\nreversal invariant spin orbit interaction [9, 11]. When\nbreaking time-reversal symmetry, the topological prop-\nerties of the QSH system remain but spin-Chern num-\nbers are required for the description [11]. Furthermore,\na topological, interaction-driven quantum motor [32, 33]\ncan be accomplished by only pumping one of the compo-\nnents while the other is coupled by interaction. For spin\npumps with highly degenerate many-body ground states,\nfractional transport is predicted [34].\nWe acknowledge insightful discussions with M. Aidels-\nburger. This work was supported by NIM, the EU\n(UQUAM, SIQS), and the DFG (DIP & FOR2414).\nM.L. was additionally supported by ExQM and R.C. by\nFIRB-2012-HybridNanoDev(GrantNo. RBFR1236VV).\n[1] K. v. Klitzing, G. Dorda, and M. Pepper, Phys. Rev.\nLett.45, 494 (1980).[2] D. C. Tsui, H. L. Stormer, and A. C. Gossard, Phys. Rev.\nLett.48, 1559 (1982).\n[3] D. J. Thouless, M. Kohmoto, M. P. Nightingale, and\nM. den Nijs, Phys. Rev. Lett. 49, 405 (1982).\n[4] D. J. Thouless, Phys. Rev. B 27, 6083 (1983).\n[5] Q. Niu and D. J. Thouless, J. Phys. A 17, 2453 (1984).\n[6] M. König, S. Wiedmann, C. Brüne, et al., Science 318,\n766 (2007).\n[7] C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801\n(2005).\n[8] B. A. Bernevig and S.-C. Zhang, Phys. Rev. Lett. 96,\n106802 (2006).\n[9] R. Shindou, J. Phys. Soc. Jpn. 74, 1214 (2005).\n[10] L. Fu and C. L. Kane, Phys. Rev. B 74, 195312 (2006).\n[11] C. Q. Zhou, Y. F. Zhang, L. Sheng, et al., Phys. Rev. B\n90, 085133 (2014).\n[12] G. A. Prinz, Science 282, 1660 (1998).\n[13] S. Murakami, N. Nagaosa, and S.-C. Zhang, Science 301,\n1348 (2003).\n[14] Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D.\nAwschalom, Science 306, 1910 (2004).\n[15] P. Sharma and C. Chamon, Phys. Rev. Lett. 87, 096401\n(2001).\n[16] R.Citro, N.Andrei,andQ.Niu,Phys.Rev.B 68,165312\n(2003).\n[17] R. Citro, F. Romeo, and N. Andrei, Phys. Rev. B 84,\n161301 (2011).\n[18] S. K. Watson, R. M. Potok, C. M. Marcus, and V. Uman-\nsky, Phys. Rev. Lett. 91, 258301 (2003).\n[19] C.W.Sandweg,Y.Kajiwara,A.V.Chumak,etal.,Phys.\nRev. Lett. 106, 216601 (2011).\n[20] M. Lohse, C. Schweizer, O. Zilberberg, M. Aidelsburger,\nand I. Bloch, Nat. Phys. 12, 350 (2016).\n[21] H.-I. Lu, M. Schemmer, L. M. Aycock, et al., Phys. Rev.\nLett.116, 200402 (2016).\n[22] S. Nakajima, T. Tomita, S. Taie, et al., Nat Phys 12, 296\n(2016).\n[23] M. J. Rice and E. J. Mele, Phys. Rev. Lett. 49, 1455\n(1982).\n[24] P. J. Lee, M. Anderlini, B. L. Brown, et al., Phys. Rev.\nLett.99, 020402 (2007).\n[25] H.-N. Dai, B. Yang, A. Reingruber, et al., Nat. Phys. 12,\n783 (2016), article.\n[26] See Supplemental Material.\n[27] A. Widera, F. Gerbier, S. Fölling, et al., Phys. Rev. Lett.\n95, 190405 (2005).\n[28] D. Xiao, M.-C. Chang, and Q. Niu, Rev. Mod. Phys. 82,\n1959 (2010).\n[29] S. Trotzky, P. Cheinet, S. Fölling, et al., Science 319, 295\n(2008).\n[30] V. W. Scarola and S. Das Sarma, Phys. Rev. Lett. 95,\n033003 (2005).\n[31] Y. Hatsugai and T. Fukui, Phys. Rev. B 94, 041102\n(2016).\n[32] F. Zhou, Phys. Rev. B 70, 125321 (2004).\n[33] R. Bustos-Marún, G. Refael, and F. von Oppen, Phys.\nRev. Lett. 111, 060802 (2013).\n[34] D. Meidan, T. Micklitz, and P. W. Brouwer, Phys. Rev.\nB84, 075325 (2011).6\nSupplemental Material for:\nSpin pumping and measurement of spin currents in optical superlattices\nC. Schweizer1;2, M. Lohse1;2, R. Citro3;4, I. Bloch1;2\n1Fakultät für Physik, Ludwig-Maximilians-Universität, Schellingstrasse 4, D-80799 München, Germany\n2Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Strasse 1, D-85748 Garching, Germany\n3Dipartimento di Fisica “E. R. Caianiello”, Università degli Studi di Salerno,\nVia Giovanni Paolo II 132, I-84084 Fisciano (Salerno), Italy\n4SPIN-CNR Salerno, Via Giovanni Paolo II 132, I-84084 Fisciano (Salerno), Italy\nSPIN CURRENT MEASUREMENT\nTwo-site extended Bose-Hubbard model\nIn the tight binding limit, two spins on an isolated\ndoublewell( \u000eJ=J)canbedescribedbyatwo-siteBose-\nHubbard model, where we denote the spins \u001b=f\";#g\nand the left (right) site with L (R). The Hamiltonian is\ngiven by\n^HBH=\u0000JX\n\u001b=f\";#g\u0010\n^ay\nL;\u001b^aR;\u001b+h.c.\u0011\n\u0000\u0001\n2(^nL;#\u0000^nL;\"\u0000^nR;#+ ^nR;\")\n+U(^nL;\"^nL;#+ ^nR;\"^nR;#)(S.1)\nwith ^ay\nR/L;\u001b(^aR/L;\u001b) the creation (annihilation) operator\nof spin\u001bon the left or right site, ^nR/L;\u001bthe number op-\nerator counting the spins, Jthe tunneling rate, \u0001the\nspin-dependent energy offset between left and right site,\nandUthe on-site interaction energy. The accuracy of\nthe exchange coupling, especially at large J=U, can be in-\ncreased by including corrections from density-dependent\nhoppingJddh=\u0000gR\nw3\nL(r)wR(r)d3rand nearest neigh-\nbor interaction ULR=gR\nw2\nL(r)w2\nR(r)d3r[S1, S2]. Here,\nwL/R(r)denotes the Wannier function on the left/right\nsite. The Hamiltonian of this extended Bose-Hubbard\nmodel ^HDWis in the basisj\"#;0i,j\";#i,j#;\"i,j0;\"#i\nwithJ0=J+Jddh:\n^HDW=0\nBB@U\u0000J0\u0000J0ULR\n\u0000J0ULR+ \u0001ULR\u0000J0\n\u0000J0ULRULR\u0000\u0001\u0000J0\nULR\u0000J0\u0000J0U1\nCCA:(S.2)\nThe dependence of the energy spectrum Enof the ex-\ntended Bose-Hubbard model on \u0001is shown in Fig.S1(a).\nWe define the exchange coupling Jexas the gap between\nthe first and second eigenenergy at \u0001 = 0. The ground\nstate in the limitj\u0001j\u001dJexisj1i\u0019j#;\"ifor positive and\nj\";#ifor negative \u0001. At \u0001 = 0andU\u001dJthe ground\nstate is approximately a triplet state1p\n2\u0000\nj\";#i+j#;\"i\u0001\n.\nNote that the third eigenstate is independent of \u0001.\n-500 -250 0 250 500\n∆/h (Hz)-0.100.1\n-500 -250 0 250 500\n∆/h (Hz)04080120\nam (10-3)am (10-3)\na2a4\na3-4-2E/h (kHz)024\n2 1\n∆/h (kHz)0 -1 -2 ba\n|1/angbracketright|2/angbracketright|3/angbracketright|4/angbracketrightFigure S1. Energy spectrum and am-coefficients. (a) Energy\nspectrum of the extended Bose-Hubbard model for two spins\non a double well. (b) Magnitude of the am-coefficients at\ndifferent spin-dependent tilt values \u0001during the magnetic\nfield ramp with a speed of _\u0001 = 82(2) kHz/s and exchange\ncoupling of Jex=h= 342(2) Hz.\nAn adiabatic change of the tilt \u0001with rate _\u0001leads in\nfirst-order approximation to the temporary wavefunction\nj ti=jnti+ iX\nm6=nam(t)jmti (S.3)\nusing parallel transport conditions [S3]. Here, jltiis an\ninstantaneous eigenstate of ^HDWand the admixture co-\nefficients are\nam(t) =\u0000hmtj~@tjnti\nEn(t)\u0000Em(t)=\u0000_\u0001hmtj~@\u0001jnti\nEn(t)\u0000Em(t):(S.4)\nA detailed derivation can be e.g. found in Ref.[S3]. The\nam-coefficients forjnti=j1ti, i.e. starting from the\nground state, are depicted in Fig.S1(b); the contribu-\ntiona2clearly dominates and other coefficients can be7\nneglected because the dependence of the eigenstates j4i\nandj3ion\u0001is weak or even vanishing because of the\nlarge gapjEm\u0000E1j\u0018U\u001dJex, form > 2. Then the\ntemporary wavefunction Eq.S.3 reduces to\nj ti\u0019j1ti+ ia2(t)j2ti: (S.5)\nConnection Between Amplitude and Current\nFor the current measurement, the pump cycle is\nabruptly stopped at \u0001s. In the ideal case this hap-\npensinstantaneously, suchthatthewavefunction j (\u0001s)i\nsubsequently evolves according to the static Hamilto-\nnian ^HDW(\u0001s). Sincej (\u0001s)iis in general not an eigen-\nstateof ^HDW(\u0001s), theexpectationvalueoftheimbalance\noperatorIoscillates in time\nI(t0) =Asin\u0012E2(ts)\u0000E1(ts)\n~t0\u0013\n+Is(S.6)\nwitht0=t\u0000ts. The oscillation amplitude is directly\nproportional to the a2-coefficient\nA=\u00002a2(ts)h1tsj^Ij2tsi: (S.7)\nWhen comparing this result to the the integral form of\nthe continuity equation, the spin current at tsis directly\nconnected to the oscillation amplitude Avia\nj(ts) =1\n2@tI(t0)\f\f\f\f\nt=ts\n=AE2(ts)\u0000E1(ts)\n2~cos\u0012E2\u0000E1\n~t0\u0013\f\f\f\f\nt=ts\n=AE2(ts)\u0000E1(ts)\n2~:(S.8)\nOscillation Amplitude – Model and Corrections\nFor two spins occupying the ground state of an isolated\ndoublewell, theidealoscillationamplitudefora perfectly\nabrupt stop of the ramp can be evaluated from Eq.S.7.\nThis ideal amplitude, within ^HDW, is shown as a func-\ntion of \u0001sin Fig.S2 (green solid lines). As the abrupt\nstop is smoothed by a decaying residual magnetic gra-\ndient, the measured amplitude for a given spin current\nis slightly reduced. This amplitude can be calculated\nnumerically by solving the time-dependent Schrödinger\nequation including the residual magnetic gradient de-\ncay. The model assumes an initial state after the adia-\nbatic evolution (Eq.S.5) stopped at \u0001s+ \u0001dand a time-\ndependent Hamiltonian ^HDW(\u0001de\u0000t0=\u001cd+ \u0001s). For long\ntimest0\u001d\u001cd, theHamiltonianapproachestheidealcase.\nWith a numerical time propagation, the evolution of the\nwavefunction and thereby I(t0)can be calculated. From\nthis time trace the amplitude is extracted with a fit of\n0.070.140.21AJex/h=467Hz\n0.00\n-500 -250 0 250 500\n∆/h (Hz)0.000.070.140.21AJex/h=342HzFigure S2. Current measurement. The figure summarizes dif-\nferent theory curves for both data sets at Jex=h= 467(3) Hz\n(orange) and Jex=h= 342(2) Hz (blue) from 2 in the main\ntext. The green solid line shows the ideally expected oscilla-\ntion amplitude. A corrected description taking into account\nthe reduced detection efficiency due to a residual exponential\ndecay of the magnetic field gradient (orange solid line), and\na description including additionally a finite ground state oc-\ncupation of 97(1) %and a pump efficiency of 89(1) %(blue\nsolid line) are shown. Furthermore, a measurement of the\nparticle current is shown (gray), obtained by analyzing the\nparticle imbalance oscillations between the left and right side\nof the double well based on the same data sets as for the spin\nimbalance oscillations.\n3 from the main text as for the experimental data. The\neffect of this detection correction is visualized in Fig.S2\nas orange solid lines.\nPreparing only ground state double wells is extremely\nchallenging and therefore a small residual amount of ex-\ncited state double wells is expected. When considering\nonly ground and first excited state, the spin imbalance is\na good measure for the state occupation at j\u0001j\u001dJex,\nwhere the spins are perfectly localized. By measuring\nthe spin imbalance before ( Ii) and after half a pump cy-\ncle (If), the initial state preparation and the pump ef-\nficiency can be characterized. The pump efficiency \f1\nis a measure of the fraction of double wells that remain\nin the ground state during half a pump cycle and can\nbe deduced from the spin imbalance: \f1=Ii\u0000If\n2Ii. The\ninitial fraction of ground and excited state double wells\nisn(i)\n1=I1+Ii\n2I1andn(i)\n2=I2+Ii\n2I2=I1\u0000Ii\n2I1, assuming\nI1=\u0000I2being the ideal spin imbalance at the initial\nconditions.8\nThe integrated spin current j=Rtf\ntij(t)dtis the sum of\nthe ideal integrated spin currents j1=2,idealof both states\nweighted by the initial band occupation and the pump\nefficiency. The ideal spin currents are oppositely directed\nj1,ideal =\u0000j2,ideal =I1and given by the ideal initial spin\nimbalance. Furthermore, equal pump efficiency \f2=\f1\nfor ground and first excited state are assumed.\nj=n(i)\n1\f1j1,ideal +n(i)\n2\f2j2,ideal\n=\u0010\nn(i)\n1\u0000n(i)\n2\u0011\n\f1j1,ideal\n=Ii\u0000If\n2(S.9)\nThe result can be verified by a comparison with the in-\ntegral form of the continuity equation 2j=@tI.\nThe reduction of the integrated current as a result of\na finite excited state occupation and a reduced pump ef-\nficiency can be approximately captured in the data anal-\nysis by rescaling of the current j(t)with a global factor.\nSuch a rescaling corresponds to a description by an aver-\nage state occupation with perfect pump efficiency. The\nblue line in Fig.S2 shows the spin current taking into\naccount the detection efficiency, the initial ground state\noccupation as well as the pump efficiency.\nENERGY SPECTRUM\nIn the experiment, the spin chain with local tilts is re-\nalized in the limit of isolated double wells with a global\nmagnetic gradient. In this system unlike for ^Hin 1\nof the main text, the prepared initial state is not the\nground state but an excited state. During the pump cy-\ncle, however, it evolves in the same way as the ground\nstate of a spin-dependent superlattice with the excep-\ntion of a number of additional crossings that occur in\nthe energy spectrum (see Fig.S3). However, the gaps are\nvery small – on the order of the inter double well ex-\nchange coupling1\n2(Jex\u0000\u000eJex)– and can be crossed non-\nadiabatically. In conclusion the gradient model requires\nnot only adiabaticity with respect to the intra double\nwell exchange tunneling1\n2(Jex+\u000eJex)but also pure non-\nadiabatictransferwithrespecttogapsontheorderofthe\ninter double well exchange coupling1\n2(Jex\u0000\u000eJex). As the\nlatter can be suppressed exponentially with the long lat-\ntice depth, and adiabaticity on the double well scale can\nbe reached by slower ramp speed, the transport in these\nmodels can be described by the same topological invari-\nant. An energy spectrum for two up and two down spins\non two double wells is shown in Fig.S3 for the experi-\nmental parameter set for Jex=h= 342(2) Hz. The pump\ncycle follows the thicker darker depicted state, which is\neither strongly gapped around \u0001 = 0or crosses states\nwith negligibly small gaps. Additionally, the state over-\nlap between the groundstate in the staggered model and\n1\n2(J\n+δJ\n)|/angbracketleft1\n|Ψ\n /angbracketright|2\n01\n-3 -2 -1 0 1 2 3\n∆/h (kHz)-4-3-2-10123E/h (kHz)\n1\n2(J\n−δJ\n)≈0Figure S3. Energy spectrum and state overlap for a superlat-\ntice with staggered tilt and a global gradient, respectively. In\nthe upper panel the energy spectrum of two up and two down\nspins on two double wells with local, staggered tilt (red) and\nglobal gradient (blue) is shown. The thicker darker lines rep-\nresent the ground state of the staggered superlattice j1stiand\nthe corresponding state in the globally tilted lattice j\u001egradi,\nwhich is used for pumping in the experiment. In the lower\npanel the state overlap between these two states is depicted.\nthe pumped state for the model with a global gradient is\ncalculated and found to be one, apart from the vicinity\nof the tiny gaps. This shows that the pumped state in\nthe gradient model is very similar to the one in the model\nwith staggered tilts.\nEXPERIMENTAL SEQUENCE\nThe experimental sequence starts with a Mott insu-\nlator of (F= 1;mF=\u00001)87Rb atoms in a 3D opti-\ncal lattice of three mutually orthogonal standing waves\nwith wavelengths \u0015x=\u0015y= 767nm and\u0015z= 844nm.\nWith a sequence of microwave driven adiabatic passages,\nthe atoms are transferred to the (F= 1;mF= 0)via\nthe(F= 2;mF= +1)state (seeft1,ft2in Fig.S4).\nThen two neighboring lattice sites are merged along the\nx-direction into decoupled sites with twice the period\nby ramping up a long lattice with period dl= 2ds,\nwhereds=\u0015x=2, and simultaneously turning the short\nperiod lattice off. With coherent microwave-mediated\nspin changing collisions [S4] each atom pair is trans-\nferred to a pair of j\"i=jF= 1;mF=\u00001iandj#i=\njF= 1;mF= +1iatoms. Subsequently, a magnetic field\ngradient is turned on and the long lattice sites are split\nadiabatically into two decoupled sites by turning on the\nshort period lattice to a depth of Vs= 40Er,sin50ms\nwithEr,i=h2\n2mRb\u00152\ni, wheremRbis the mass of a Rubid-9\n20ms60ms38.5ms38.5ms\n537 540 460 460100\nπ50\n0MWBQuad\nfsc ft2 ft1Bx\nMWBQuadBxVi (Er,i )Vz Vy\nVx,lVx,s150ms50ms50ms150ms90ms\nt (ms)\nt (ms)460 310 150 0100\n50\n0Vi (Er,i )a Initial state preparation\nb Current measurement Pump cyclec\nFigure S4. Summary of the experimental sequences. The lat-\ntice depths are shown in each upper panel in their respective\nrecoil energies: along the x-direction with period ds(blue)\nanddl(red), along the y-direction with period ds(green)\nand along the z-direction with period dz(yellow). In the\nlower panel the two contributions to the magnetic field gra-\ndient as well as the mircowave pulses are sketched. (a) A\ntime sequence for the initial state preparation is depicted. It\nends with decoupled sites occupied by antiferromagnetically\norderedj\"i=jF= 1;mF=\u00001iandj#i=jF= 1;mF= +1i\nspins. (b) Example sequence for the current measurement\nat\u0001s= 0which directly follows the initial state prepara-\ntion. (c) The spin pump sequences for the in-situ and band\nmapping measurement starting directly after the initial state\npreparation. Noteafterhalfacyclethedimerizationisflipped\nby swapping to a second laser creating \u0019shifted double wells,\nand the magnetic field gradient is reversed in 2ms.\nium atom. Due to the present magnetic field gradient\nof\u0001=h= 2:7(2)kHz during the splitting, the up and\ndown spins order antiferromagnetically in individual, de-\ncoupled sites with Jex\u00190. A detailed time sequence\nof the individual experimental parameters is shown in\nFig.S4(a). This initial state preparation is then followed\nby the pumping sequence.The spin pump sequence for multiple cycles, which\nwas used for the in-situ measurements, starts by cou-\npling neighboring lattice sites of each double well by de-\ncreasing the short period lattice to the final value for the\ncorresponding superexchange coupling. Then, the spin-\ndependent gradient is inverted adiabatically by changing\nthe bias field along the x-direction (see gradient calibra-\ntion), and at the end of the first half pump cycle the\ndistribution is frozen again by increasing the short lat-\ntice depth such that Jex\u00190. To continue the cycle, the\ndimerization is changed by swapping to a second laser\nwhich creates a \u0019-shifted long lattice, and simultane-\nously setting the magnetic bias field back to its initial\nvalue in 2ms. Then the previously described cycle is re-\npeated multiple times. Before in-situ imaging, residual\natoms in the ( F= 1,mF= 0) state are removed by ap-\nplying simultaneously a microwave field on the ( F= 1,\nmF= 0)!(F= 2,mF= 0) transition and a resonant\nimaging light pulse to remove the atoms in the F= 2\nmanifold. Subsequently, one of the spin states is selected\nand transferred to the ( F= 2,mF= 0) state by a mi-\ncrowave driven adiabatic passage and imaged by absorp-\ntion imaging.\nThe current measurement sequence is limited to the\nfirst half pump cycle. Neighboring sites are coupled\nby decreasing the short lattice depth Vsand the bias\nfieldBxis changed with a constant rate to different fi-\nnal values of the magnetic tilt \u0001s, where it is abruptly\nstopped. After a variable holdtime t, the spin distribu-\ntionisfrozenbyrampinguptheshortperiodlatticeanda\nsite-resolved band mapping was performed. During time-\nof-flight, the spin-components are separated spatially by\na Stern-Gerlach field and the four site occupations nL#,\nnL\",nR#, andnR\"can be simultaneously extracted from\neach absorption image.\nCALIBRATION OF LATTICE, GRADIENT\nFIELD AND HUBBARD PARAMETERS\nCalibration of bare tunneling rate: The bare tunnel-\ning matrix element Jcouples two neighboring sites and\ncan be calibrated from left-right oscillations of a single\natom on a double well with degenerate on-site energies\n(\u0001 = 0). At the beginning, a single atom is prepared\non the left site of each double well. To this end, an\nn= 1Mott insulator in the long lattice is prepared and\nadiabatically split with the short lattice in the presence\nof a large potential tilt. The short lattice depth is in-\ncreased such that the atoms localize on the lower-lying\nleft site (J\u00190). Then, the tilt is removed and subse-\nquently the short period lattice is ramped down in 200\u0016s\nto the lattice depth at which Jneeds to be calibrated.\nThe localized wavefunction is not an eigenstate of the\nsymmetric double well, but an equal superposition of the\ngroundj1iand first excited state j2iand thus the left-10\n-0.8 -0.4 0.0 0.4 0.8\nBx (G)-1.2-0.60.00.61.2∆/h (kHz)\nFigure S5. Calibration of the magnetic gradient. The data\npoints show a measurement of the magnetic tilt \u0001by laser-\nassisted tunneling spectroscopy in a tilted double well as a\nfunction of the magnetic bias field Bx. From the measured\nfrequency, the tilt can be directly inferred by 2\u0001 =hfr\u0000\u000e0.\nThe error bars show the standard deviation of an average of\neight points, four for ( F= 2,mF=\u00062) each.\nrightoccupationoscillatesintime. Theleft-rightfraction\ncan be measured with site-resolved band mapping and its\noscillation frequency fbareis equal to the difference of the\nsingle-particle eigenenergies hfbare=\u000f2\u0000\u000f1= 2J.\nCalibration of on-site interaction energy: The on-site\ninteraction energy Uis the extra energy for placing a\nsecond particle on the same site and can be calibrated\nvia the superexchange oscillation frequency. An up and\na down spin are localized on the left and right site\nof a double well prepared in the presence of a large\nspin-dependent tilt \u0001as for the current measurement.\nThen, the magnetic fields, except for a small bias field to\nmaintain the quantization axis, are switched off to non-\nadiabatically remove the tilt while Jex\u00190and after-\nwards the short period lattice is decreased within 200\u0016s\nto the final value, at which Uis calibrated. The local-\nized state is not an eigenstate anymore and evolves in\ntime. The time evolution leads to superexchange oscilla-\ntions [S2], which for the experimentally used parameters\nare dominated by the contributions from the ground and\nfirst excited state ~!ex'E2\u0000E1=Jex. The exchange\ncouplingJexcan be calculated from an extended Bose-\nHubbard model and depends strongly on U. Thus,Ucan\nbe inferred from !ex.\nGradient Calibration: The spin-dependent gradient\nfield is generated by two pairs of coils: an anti-Helmholtz\npair along the z-direction, which creates a quadrupole\nfieldBquad(xex+yey\u00002zez), and a Helmholtz pair\nalong thex-direction, which creates a homogenous bias\nfieldBxexwitheithe unit vector in i-direction. At\nthe atom cloud ( x\u00190), the total magnetic field is\nB=p\n(Bquadx+Bx)2+B2\nofs(y;z)and the magnetic\nfield gradient B0=@B=@xdepends linearly on the bias\nfieldBxforBx\u001cBofs. A precise knowledge of the\n0 2 4 6\nt´ (ms)∆/h (Hz)\n05101520\n-2 0 2 4 6 8 10\nt´ (ms)∆/h (Hz)\n050100150200Figure S6. Calibration of the residual exponential decay of \u0001\nafter the ramp stop t0=t\u0000ts. The data points show the\nmagnetic gradient determined by microwave spectroscopy on\nthe (F= 1,mF=\u00001)!(F= 2,mF=\u00002) transition when\nstopping the gradient ramp at \u0001s= 0. The linear ramp be-\nfore and the exponential decay after the stop time tsis clearly\nvisible. By fitting a linear ( t <0) and an exponential func-\ntion (t >0) to the data, the decay constant \u001cd= 1:05(5)ms\nand\u0001d, which depends on the double well site distance, can\nbe determined.\nspin-dependent double well tilt is required for the current\nmeasurementmethod. Tothisend, acalibrationwasper-\nformed using laser-assisted tunneling spectroscopy. The\nsetup comprises two beams interfered under an angle of\n90\u000ewith a frequency difference of \u000e!forming a running\nwave lattice oriented at 45\u000eto the physical lattice. A\nsingle spin in the ( F= 2,mF=\u00062)-state is loaded in\nthe ground state of a tilted double well potential with\n\u000e0=h= 4:91(2)kHz and additional magnetic tilt 2\u0001,\nwhich needs to be calibrated. The running wave lattice\nmodulates neighboring lattice sites relative to each other\nand will induce tunneling if ~\u000e!= 2\u0001 +\u000e0. A series of\nspectroscopy scans varying \u000e!are performed for various\nmagnetic field values Bx. The resulting data is shown in\nFig.S5. Forlarge Bxa deviationfrom thelinear behavior\nis visible, which can be captured by fitting the magnetic\nfield distribution of a quadrupole field\n\u0001/BxBquadp\nB2x+B2\nofs; (S.10)\nwhere both BquadandBofsare fit variables.\nDecay Calibration: During the current measurement\nsequence, the magnetic field gradient is ramped to a final\nvalue \u0001sand abruptly stopped there. Experimentally,\nan instantaneous stop is not realizable but a small resid-\nual gradient remains which decays slowly during the cur-\nrent measurement. The calibration of the decay time \u001cd\nas well as the residual gradient \u0001dattsis essential for\nthe current measurement and is realized with microwave\nspectroscopy. The sequence starts with a single spin in\nthe (F= 1,mF=\u00001) state localized on the left site by a\nstrong magnetic gradient. The gradient is reduced with11\nthe identical rate as for the current measurement and\nstopped at \u0001s= 0but individual sites are decoupled by\na large short lattice depth. Now, resonance frequency\nscans on the ( F= 1,mF=\u00001)!(F= 2,mF=\u00002)\nmicrowave transition with a pulse duration of 44\u0016s are\nrecorded at various times around ts; the gradient \u0001(t0)\ndetermined from the center frequencies is summarized in\nFig.S6. A clear exponential decay \u0001de\u0000t0=\u001cdcan be fit-\nted fort0>0with a decay constant \u001cd= 1:05(5)ms. The\namplitude \u0001dcan be calibrated by comparing the linear\nincrease for t0<0with the gradient calibration (Fig.S5)\nand leads to \u0001d=h= 24:3(6)Hz for sites separated by ds.\nThis corresponds to a tilt in the superlattice for the ex-\nperimental parameters of \u0001d=h= 19:8(5)Hz atJex=h=\n342(2)Hz and \u0001d=h= 19:4(5)Hz atJex=h= 467(3) Hz.\nThe difference originates mainly in the slightly different\ndistance between the sites depending on the double well\nparameters.\nSimultaneous band mapping of two spins: The simul-\ntaneous site-resolved detection of two spins per double\nwell suffers from an additional reduction of the detected\nimbalance. This reduction occurs during the merging of\nthe left and right site of each double well in the pres-\nence of a spin-independent tilt into a single site of the\nlong lattice. This process transfers atoms from the left\nsite to the ground band and spins from the right site to\nthe second excited band of the long lattice. The subse-\nquent band mapping measures the band occupation and\nhence also the site occupations. However, spins in these\nbands undergo singlet-triplet oscillations after and also\nduring the merging [S2, S5]. A calibrated holdtime be-\nfore the release is chosen such that correct imbalances\nare detected. Nevertheless, the detected imbalance is re-\nduced most likely due to dephasing during the merging\nramps. The value of this reduction can be calibrated to\nrescale the measured imbalances with a constant factor\nto correctly determine the imbalance. The calibration\nmeasurement compares the two spin site-resolved band\nmapping with a single spin band mapping, where shortly\nbeforethemergingoneofthespincomponentsisremoved\nby an adiabatic spin transfer and a subsequent resonant\nlight pulse.\nMULTIPLE PUMP CYCLES\nBand mapping data\nThe in-situ data show a separation and opposite trans-\nport of the two spin components for multiple pump cy-\ncles. In Fig.S7 the spin imbalance Isduring the pump\ncycle is depicted versus the pump parameter \u001e, which is\ndefined as the angle of the pump path in parameter space\n(\u000eJex=\u000eJex,max;\u0001=\u0001max). The imbalance starts with a\nnegative value, the state is predominantly j\"#i, and in-\n0.0 0.5 1.0 1.5\nφ (2π)-1.0-0.50.00.51.0Figure S7. Static spin imbalance Iduring the pump cycle.\nThe data is shown with respect to the pump parameter \u001e\ndefined as the angle of the pump path in parameter space\n(\u000eJex=\u000eJex,max;\u0001=\u0001max). The error bars show the error of\nthe mean of five repetitions each.\nverts during the first half pump cycle. After switching\nthe dimerization, the pump cycle continues by inverting\nthe spin-imbalance each half pump cycle. At \u001e= 2\u0019a\nsmall step inIis visible, which originates mainly from\nsingly occupied sites created at the surface of the atom\ncloud during pumping.\nInitial state and pump efficiency\nFor the in-situ measurement the calculated motion of a\nlocalized spin is shown, which takes into account the ini-\ntial ground state occupation n(i)\n1and pump efficiency \fi\nperi-th half pump cycle. The model is analogous to the\none for the spin current, where the corrections are ex-\ntracted from the spin-imbalance measured at each half\npump cycle. The step height is then given by\nsi=\fi\u0010\n2n(i)\n1i\u00001Y\nj=0\fj\u00001\u0011\n(S.11)\nwithn(i)\n1= 0:94and\f0:::4=f1;0:97;0:91;0:96;0:90g. In\ntotal, the displacement after the i-th half pump cycle is\nx=X\nj=1sj: (S.12)\nPUMP SCHEME IN A TIGHT-BINDING MODEL\nWITH MAGNETIC FIELD GRADIENT\nIn the tight-binding approximation the dynamics of\nnon-interacting atoms in an optical superlattice poten-\ntial in the presence of a field gradient is described by a\ngeneralized Harper model with a site-dependent Zeeman-12\nlike term\n^Hs=^HJ+^H\u0001\n=\u0000X\nm;\u001b1\n2(J+\u000eJm) (^ay\nm+1;\u001b^am;\u001b+h.c.)\n+X\nmm\u0001\u0010\n^ay\nm;\"^am;\"\u0000^ay\nm;#^am;#\u0011(S.13)\nwith\u000eJm= (\u00001)m\u000eJ. The pumping scheme is imple-\nmented with a cycle, in which (\u000eJ;\u0001)!(\u000eJ(\u001e);\u0001(\u001e))\nand where the pump parameter changes constantly in\ntime\u001e= 2\u0019t=T. In the deep tight-binding regime the\nparameter space describes an ellipse (\u000eJ(t);\u0001(t)) =\n(\u000eJsin(2\u0019t=T );\u0001 cos(2\u0019t=T )). At \u0001 = 0, the spectrum\nhas an energy gap \u0001E= 2\u000eJ, and thus the adiabatic\ncondition is met if T\u001d~=\u000eJ.\nThe part of the Hamiltonian ^H\u0001is odd while the part\n^HJis even under time-reversal symmetry and therefore\nEq.S.13 belongs to the class that satisfy the condition\n^H[\u0000t] =^\u0002^H[t]^\u0002\u00001, where ^\u0002is the time-reversal opera-\ntor. Moreover, the Hamiltonian is time-reversal invariant\nat two points t1=T\n4andt2=3T\n4, where ^HJdominates.\nThe existence of these two points plays a crucial role in\nthe classification of the pump cycle. In particular, pump\ncycles in which ^H[t1]and ^H[t2]have different time re-\nversal polarization are topologically distinct from trivial\ncycles and define a Z2spin-pump. In a single double well\nat timet= 0,^H\u0001dominates and locks the up (down)\nspins on the left (right) well, denoted by j\";#i. This\nstate evolves into the j#;\"istate after half a pump cycle\natt=T\n2, where the two spins have exchanged their po-\nsitions. In contrast, at t=T\n4andt=3T\n4, the term ^HJ\ndominates and the spins are delocalized over the double\nwells; then the system is dimerized.\nCENTER OF MASS SHIFT AND\nTIME-REVERSAL POLARIZATION\nConsider a Hamiltonian Eq.S.13 with lattice constant\nds=dl=2 = 1and periodic boundary conditions. Then,\nin absence of spin-orbit type of interaction that means\nindependent spin components without inter-spin inter-\nactions, the spin transport for a homogeneously popu-\nlated band is characterized by the spin Chern number\nCsc=\u0017\"\u0000\u0017#. Since the spin components are decoupled\neven in the presence of a field gradient, the Chern num-\nbers\u0017\u001bcan be evaluated using the Thouless-Kohmoto-\nNightingale-Nijs expression [S6]:\n\u0017\u001b=1\n2\u0019ZT\n0dtZ\u0019\n\u0000\u0019dk\n\u001b(t;k);(S.14)\nwhere \n\u001bis the Berry curvature associated to the single-\nparticle wavefunction\n\n\u001b(t;k) = i (h@tu\u001bj@ku\u001bi\u0000h.c.): (S.15)The spin Chern number can be furthermore related for\nthe non-interacting case to the Z2topological invariant\nI=mod 2(Csc=2)thatdistinguishesanontrivial Z2pump\nfrom a trivial one and is related to the change in time\nreversal polarization.\nAs known from polarization theory [S7], the charge\npolarization is the center of mass of a localized Wan-\nnier state and is in turn related to Berry’s phase of the\ncorresponding Bloch functions. In the same way, the po-\nlarization of a single spin component is given by:\nP\u001b=1\n2\u0019Z\u0019\n\u0000\u0019dkA\u001b(k); (S.16)\nwhereA\u001b(k) = iPhu\u001bj@ku\u001biis the Berry connection.\nThe change in polarization induced by changing the\npump parameter \u001eby2\u0019, or the time variable, corre-\nsponds to the Chern number [S7]\n\u0017\u001b=Z2\u0019\n0d\u001e @\u001eP\u001b(\u001e): (S.17)\nThe spin-density can be directly measured by in-situ\nabsorption imaging for a single spin component. The\nchange of spin-polarization \u0001P\u001b=P\u001b(\u001e1)\u0000P\u001b(\u001e2)at\ntwo different times t1andt2, coincides with the spatial\nshift of the Wannier function. Measuring the center of\nmass shift for a single spin component thus gives the\nChern number of this component.\nFor time-reversal invariant systems, taking into ac-\ncount the role of Kramer’s degeneracy, one can define\na corresponding time-reversal polarization in terms of\nthe difference of the individual spin polarizations Ps=\nP\"\u0000P#. Hence, the change in time-reversal polarization\nduringacyclegivesthe Z2topologicalinvariant. Further-\nmore, it is equal to the integration of the instantaneous\nspin-current jover the pump cycle as,RT\n0dtj(t).\nSPIN PUMPING WITH INTERACTIONS\nWhen in addition hardcore interactions between the\nspin components are assumed, spin pumping can be un-\nderstood in a similar way as in the non-interacting case.\nFor half filling a representation in terms of spin operators\ncan be introduced in this limit:\n^S+\nm= ^ay\nm\"^am#;\n^S\u0000\nm= ^ay\nm#^am\";\n^Sz\nm= ^ay\nm\"^am\"\u0000^ay\nm#^am#:(S.18)13\nModel Eq.S.13 can thus be mapped to a model of an\nantiferromagnetic spin chain with two perturbations\n^Heff=^Hxy+^Hdim+^H\u0001\n=\u0000Jex\n4X\nm(^S+\nm^S\u0000\nm+1+h.c.)\n\u0000\u000eJex\n4X\nm(\u00001)m(^S+\nm^S\u0000\nm+1+h.c.)\n+ \u0001X\nmm^Sz\nm;(S.19)\nwhere the second term describes a staggered compo-\nnent of the exchange interaction, while the last one is\na Zeeman-like term, which controls the on-site energies.\nA cycle, in which (\u000eJex;\u0001)are adiabatically var-\nied defines a topological spin pump [S8]. Such a spin\npump transfers Sz=~per cycle, which can be still\ndescribed by the Z2topological invariant. Note that\nthe Hamiltonian ^Hxyin Eq.S.19 corresponds to a co-\nsine band \u000f(k) =\u0000Jex=2 cos(k). The staggered ex-\nchange interaction \u000eJexopens a gap at k=\u0006\u0019\n2and for\nhalf filling, only the lowest subband is occupied. Due\nto the\u0019periodicity in k-space, the double degenerate\npoint (k; \u000eJex;\u0001) = (\u0019\n2;0;0)is identical to that at\n(\u0000\u0019\n2;0;0)and becomes the source and sink for a vector\nfieldB+1andB\u00001defined in the k-\u001e-parameter space.\nIf a pump path \rencloses the origin (\u000eJex;\u0001) = (0;0),\ne.g. (\u000eJex;\u0001) = (\u000eJex,max cos\u001e;\u0001maxsin\u001e), where\n\u001e: 0!2\u0019, the number of lattice sites that a spin is\ntransported, is given by the flux B+1enclosed by the\npath\nI\n\rZ\u0019\nk=\u0000\u0019dS\u0001B+1= 1: (S.20)\nThis corresponds to a quantized spin transport. The to-\ntalSzat one end of this system increases while that at\nthe other end decreases by one during the entire cycle as\nlong as the gap is maintained open and the point (0;0)\nis not outside the 2D closed surface.\nAway from the hard-core constraints for the bosons,\nthe effect of a finite interaction can be taken into account\nvia a bosonization approach. When applying Haldane’s\nbosonization of interacting bosons [S9] to the Hamilto-\nnian Eq.S.13 and \u000eJex;\u0001 = 0, the Hamiltonian of the\nbosons can be written as:\n^H0=X\n\u001bZdx\n2\u0019\u0014\nv\u001bK\u001b(\u0019\u0005\u001b)2+v\u001b\nK\u001b(@x\b\u001b)2\u0015\n;(S.21)where the two canonical fields fulfill [\b\u000b(x);\u0005\f(x0)] =\ni\u000e\u000b\f\u000e(x\u0000x0),v\u001bis the velocity of excitations, and K\u001bis\nthe Tomonaga-Luttinger exponent. In the case of hard-\ncore bosons, v\u001b=Jexsin(\u0019\u001a0\n\u001b)andK\u001b= 1, while\u001a0is\nthe boson density.\nIntroducing the fields \u0012\u000b=\u0019Rx\u0005\u000b, the boson annihi-\nlation operators can be represented as [S9]:\naj\u001b= \u001b(x) (S.22)\n=ei\u0012\u001b(x)+1X\nm=0c\u001b\nmcos(2m\b\u001b(x)\u00002m\u0019\u001a(0)\n\u001bx);\nwherec\u001b\nmare non-universal coefficients. For hardcore\nbosons at half filling, these coefficients have been found\nanalytically [S10]. From Eq.S.22, the bosonized expres-\nsion of the staggered hopping term can be deduced:\n^Hhop./\u000eJZ\ndxsin(2\bc) cos(2\bs);(S.23)\nwith only the most relevant term in the renormalization\ngroup sense and the charge and spin variables \b\"=#=\n(\bc\u0006\bs). When \bcis pinned (e.g. at commensurate\nfillings) in the gapped spin phase also the field \bsis\npinnedh\bsi\u0011\u0019\n4(1 +sign(\u000eJ)). The excitations above\nthe ground state are solitons and antisolitons, which are\ntopological excitations of the field \bsthat carry a spin\n1=2.\nThe time reversal-polarization is identified as Ps=\nmod 2(2\bs\n\u0019)and because under time reversal ^\u0002\bs^\u0002\u00001=\n\u0000\bs, time reversal polarization is either 0or1. Thus,\nthe topological classification of the spin pump remains\nalso away from the hard-core bosons limit.\n[S1] V. W. Scarola and S. Das Sarma, Phys. Rev. Lett. 95,\n033003 (2005).\n[S2] S. Trotzky, P. Cheinet, S. Fölling, et al., Science 319,\n295 (2008).\n[S3] D. Xiao, M.-C. Chang, and Q. Niu, Rev. Mod. Phys.\n82, 1959 (2010).\n[S4] A. Widera, F. Gerbier, S. Fölling, et al., Phys. Rev.\nLett.95, 190405 (2005).\n[S5] M. Anderlini, P. J. Lee, B. L. Brown, et al., Nature 448,\n452 (2007).\n[S6] D. J. Thouless, M. Kohmoto, M. P. Nightingale, and\nM. den Nijs, Phys. Rev. Lett. 49, 405 (1982).\n[S7] N. Marzari and D. Vanderbilt, Phys. Rev. B 56, 12847\n(1997).\n[S8] R. Shindou, J. Phys. Soc. Jpn. 74, 1214 (2005).\n[S9] F. D. M. Haldane, Phys. Rev. Lett. 47, 1840 (1981).\n[S10] A. A. Ovchinnikov, J. Phys. Condens. Matter 16, 3147\n(2004)."    },    {        "title": "1005.3853v2.Cavity_spin_optodynamics.pdf",        "content": "A spin optodynamics analogue of cavity optomechanics\nN. Brahms1and D.M. Stamper-Kurn1;2\u0003\n1Department of Physics, University of California, Berkeley CA 94720, USA\n2Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA\n(Dated: October 30, 2018)\nThe dynamics of a large quantum spin coupled parametrically to an optical resonator is treated in\nanalogy with the motion of a cantilever in cavity optomechanics. New spin optodynamic phenomena\nare predicted, such as cavity-spin bistability, optodynamic spin-precession frequency shifts, coherent\nampli\fcation and damping of spin, and the spin optodynamic squeezing of light.\nCavity optomechanical systems are currently being ex-\nplored with the goal of measuring and controlling me-\nchanical objects at the quantum limit, using interactions\nwith light [1]. In such systems, the position of a mechan-\nical oscillator is coupled parametrically to the frequency\nof cavity photons. A wealth of phenomena result, in-\ncluding quantum-limited measurements [2], mechanical\nresponse to photon shot noise [3], cavity cooling [4], and\nponderomotive optical squeezing [5].\nConcurrently, spins and psuedospins coupled to elec-\ntromagnetic cavities are being researched in atomic [6],\nionic [7], and nanofabricated systems [8, 9], with appli-\ncations including magnetometry [10], atomic clocks [11],\nand quantum information processing [6, 9, 12]. In con-\ntrast to mechanical objects, spin systems are more eas-\nily disconnected from their environment and prepared in\nquantum states, including squeezed states [11].\nIn this Rapid Communication, we seek to link these\ntwo \felds by exploiting the similarities between large-\nspin systems and harmonic oscillators [13] to construct a\ncavity spin optodynamics system in analogy to cavity op-\ntomechanics. Optomechanical phenomena map directly\nto our proposed system, resulting in spin cooling and\nampli\fcation [14, 15], nonlinear spin sensitivity and spin-\ncavity bistability [16, 17], and spin opto-dynamic squeez-\ning of light [14, 18]. Such a system may \fnd application\nas a quantum-limited spin ampli\fer or as a latching spin\ndetector. We detail these phenomena using currently ac-\ncessible parameters, and we propose realizations either\nusing cold atoms and visible light or using cryogenic solid\nstate systems and microwaves.\nAn ideal cavity optomechanics system, consisting of\na harmonic oscillator coupled linearly to a single-mode\ncavity \feld, obeys the Hamiltonian\nH=~!c^n+~!z^ay^a\u0000fzHO\u0000\n^ay+ ^a\u0001\n^n+Hin=out:(1)\nHere ^ais the oscillator's phonon annihilation operator,\n^nis the photon number operator, !zis the natural fre-\nquency of the oscillator in the dark, and !cis the bare\ncavity resonance frequency. fis the radiation-pressure\nforce applied by a single photon, while zHO=p\n~=2m!z\nis the harmonic oscillator length for oscillator mass m.\nHin=outdescribes the coupling of the cavity \feld to exter-\nnal light modes. Under this Hamiltonian, the cantileverposition ^zand momentum ^ pevolve asd^z=dt = ^p=m and\nd^p=dt =\u0000m!2\nz^z+f^n.\nTo construct a spin analogue of this system, we con-\nsider a Fabry-Perot cavity with its axis along k(Fig.\n1). For the collective spin, we \frst consider a gas of\nNhydrogenlike atoms in a single hyper\fne manifold of\ntheir electronic ground state, each with dimensionless\nspinsand gyromagnetic ratio \r. The atoms are op-\ntically con\fned at an antinode of the cavity \feld. An\nexternal magnetic \feld B=Bbis applied to the atoms.\nThe detuning \u0001 cabetween the cavity resonance is cho-\nsen to be large compared to both the natural linewidth\nand the hyper\fne splitting of the atoms' excited state.\nIn this limit, spontaneous emission may be ignored and\nthe single-atom cavity-\feld interaction energy, HStark =\n(~g2\n0=\u0001ca)^n(1\u0006\u001dk\u0001^ s), comprises the scalar and vector\nac Stark shifts [19], where g0quanti\fes the atom-cavity\ncoupling and \u001dthe vector shift.\nSumming over all atoms q, we obtain the system\nHamiltonian:\nH=~!c(^n++ ^n\u0000) +Hin=out +X\nq\u0012\n\u0000~\rB\u0001^ sq\n+~g2\n0\n\u0001ca[(^n++ ^n\u0000) +\u001d(^n+\u0000^n\u0000)k\u0001^ sq]\u0013\n;(2)\nwith number operators ^ n\u0006for the\u001b\u0006polarized optical\nmodes.\nThe above Hamiltonian can be rewritten as the inter-\naction of the collective spin operator ^S\u0011P\nq^ sqwith an\ne\u000bective total magnetic \feld Be\u000b\u0011\ne\u000b=\r, giving [20]\n\ne\u000b= \nLb+ \nc(^n+\u0000^n\u0000)k: (3)\nHere \nL=\rBand \nc=\u0000\u001dg2\n0=\u0001ca. Altogether, the\ncavity spin optodynamical Hamiltonian is\nH=~\u0012\n!c+Ng2\n0\n\u0001ca\u0013\n(^n++ ^n\u0000)+Hin=out\u0000~\ne\u000b\u0001^S:(4)\nNow consider the external magnetic \feld to be static\nand oriented along i, orthogonal to the cavity axis. In\nthe limith^Si'Si, the spin dynamics become\nd^Sj\ndt= \nL^Sk\u0000\ncS(^n+\u0000^n\u0000);d^Sk\ndt=\u0000\nL^Sj:(5)arXiv:1005.3853v2  [quant-ph]  3 Jan 20112\nS\nΩLiΩc (n+-n-)k\nσ+ σ -\nFIG. 1. (Color) An ensemble of atoms trapped within a driven\noptical resonator experiences an externally imposed magnetic\n\feld along iand a light-induced e\u000bective magnetic \feld along\nthe cavity axis k. The evolution of the collective spin ^Sre-\nsembles that of a cantilever in cavity optomechanics.\nThe analogy between cavity optomechanics and\nspin optodynamics is established by assigning\n^z! \u0000zHO^Sk=\u0001SSQL and ^p!pHO^Sj=\u0001SSQL,\nwherezHOandpHO =~=(2zHO) are de\fned with\n!z!\nL[13] and \u0001 SSQL =p\nS=2 is the standard\nquantum limit for transverse spin \ructuations. Eqs. (5)\nnow match the optomechanical equations of motion\nwith the optomechanical coupling de\fned through\nfzHO^n!\u0000~\nc\u0001SSQL(^n+\u0000^n\u0000). The main result of this\nwork, that various cavity optomechanical phenomena\nare manifest also in cavity spin optodynamical systems,\nis immediately established.\nLet us now elaborate on these phenomena. To obtain\ngeneral results, we will proceed without assuming ^S'Si,\nexcept in certain cases, noted in the text, where some\nphysical insight is gained. We begin with e\u000bects for which\nboth the light \feld and the ensemble spin may be treated\nclassically, i.e. by letting S=h^Siand \u0016n\u0006=h^n\u0006i.\nCavity-spin bistability: We start with the static\nbehavior of the system by \fnding the \fxed points of the\nsystem. The collective spin vector is static when Sis\nparallel to \ne\u000b. Writing S=S(isin\u00120+kcos\u00120), this\ncondition requires \u0016 n+\u0000\u0016n\u0000= (\nL=\nc) cot\u00120. The intra-\ncavity photon numbers are determined also by the stan-\ndard expression for a driven cavity of half line-width \u0014,\ni.e. \u0016n\u0006= \u0016nmax;\u0006[1 + (\u0001p;\u0006\u0006\ncScos\u00120)2=\u00142]\u00001with\n!\u0006= (!c+Ng2\n0=\u0001ca) + \u0001p;\u0006being the frequency of\nlaser light of polarization \u001b\u0006driving the cavity and\n\u0016nmax;\u0006characterizing its power. These two expressions\nfor \u0016n+\u0000\u0016n\u0000may admit several solutions (Fig. 2).\nAs typical in instances of cavity bistability [22], sev-\neral of the static solutions for the intracavity intensities\nmay be unstable. To identify such instabilities, we con-\nsider the torque on the collective spin when it is dis-\nplaced slightly toward + kfrom its static orientation.\nStable dynamics result when such displacement yields a\ntorque N\u0001jwith the sign \u000b= sgn(sin\u00120). Geometri-\ncally, this stability requires that the spin vector be dis-\nplaced further in the + kdirection than the vector \u000b\ne\u000b.\nQuantifying the linear response of the intracavity e\u000bec-\nS\n15\n0\n-10 -5 0 5 1015\n0(a)(b)\nΔp/κ Δp/κn -n+\nθ0π\n-π\n-10 -5 0 5 10FIG. 2. (Color) Cavity-spin bistability in a cavity driven\nwith linearly polarized light. We consider N= 5000 spin-\n287Rb atoms, \n c=\u0014= 1:25\u000210\u00003, \nL=\u0014= 3:3\u000210\u00002, and\n\u0016nmax;\u0006= 15 (similar to Ref. [21]). (a) As \u0001 pis varied, several\nstable (black) and unstable (gray) static spin con\fgurations\nare found. Con\fgurations for \u0001 p=\u0014=\u00004:8 are depicted. (b)\nThe cavity exhibits hysteresis as the probe is swept with posi-\ntive (dashed blue) or negative (red) frequency chirps, with the\nspin initially along i. Rapid transitions as \u0001 p=\u0014is swept up-\nward from -2.8 or downward from 0 involve symmetry break-\ning as the cavity becomes birefringent; we display \u0016 n+and \u0016n\u0000\nassuming the stable branch closer to \u00120= 0 is selected. Here,\n\u0001ca=2\u0019= 20 GHz from the D2 transition, g0=2\u0019= 15 MHz,\n\u0014=2\u0019= 1:5 MHz.\ntive magnetic \feld to variations of the collective spin via\n\u0015= \ncd(\u0016n+\u0000\u0016n\u0000)=dSk, the static spin orientations are\nfound to be unstable when \u000b\u0015> \nLjcsc3\u00120j=S.\nOpto-dynamical Larmor frequency shift: The\ndynamics of the spin precessing about one of the stable\ncon\fgurations can be parameterized by the precession\nfrequency, which is shifted from \n Lby two e\u000bects. First,\nthere is an upward frequency shift from the static modi-\n\fcation of the e\u000bective magnetic \feld, leading to preces-\nsion at the frequency \n L0= \nLjcsc\u00120jwhen\u0015= 0. A\nsecond shift occurs when the spin dynamics are slow com-\npared to the response time of the cavity \feld (\n L0\u001c\u0014).\nHere, the precessing spin modulates the cavity \feld,\nwhich, in turn, acts back upon the spin to modify its\nprecession frequency, When the precession amplitude is\nsmall, a solution of the spin equations of motion derived\nfrom the Hamiltonian in Eq. (4) yields an overall preces-\nsion frequency \n L00, where\n\nL002= \nL02\u0000\u0015\nLSsin\u00120: (6)\nThe quantity kS\u0011\u0000\u0015\nLSsin\u00120serves as the analogue\nof the optical spring constant [23], and leads to shifts of\nthe Larmor precession frequency with a sign and mag-\nnitude that depend on the spin orientation, \u0015and the\nfrequency, intensity, and polarization of the cavity probe\n\felds. When the precession amplitude is large, the dy-\nnamics become essentially nonlinear. In this case the dy-\nnamics can be described by numerical simulation (Fig. 3).\nCoherent ampli\fcation and damping of spin:3\nNow we consider the e\u000bects of the \fnite cavity response\ntime\u0014\u00001on the spin dynamics. To develop an intu-\nitive picture, we consider the unresolved sideband regime\n\nL<\u0014, in a frame (indicated by the index \\r\") corotat-\ning with the collective spin, with iraligned to the \fxed\npoint. We assume the spin to be precessing at a near\nconstant rate, and the cavity \feld response to this pre-\ncession to be simply delayed by \u0014\u00001. Employing the\nrotating-wave approximation, the delay causes the ef-\nfective \feld \ne\u000b,rto point out of the ir-krplane, with\n\ne\u000b,r\u0001jr=\u0000(\u000b\u0015Sk;rsin2\u00120sin\u001e)=2, where\u001e= \nL00=\u0014.\nThe collective spin now experiences a torque in the kr\ndirection, giving\ndSk;r\ndt=\u0000\u000b\u0015sin2\u00120sin\u001eSi;r\n2Sk;r (7)\nFor positive (negative) values of \u000b\u0015, the Larmor pre-\ncession frequency is shifted down (up) and the spin is\ndamped toward (ampli\fed away from) its stable point.\nSimilar relations apply to cavity optomechanics [24]. The\nde\rection of the spin toward or away from the stable\npoints persists for large precession amplitudes (Fig. 3).\nThis cavity-induced spin ampli\fcation or damping dif-\nfers from conventional optical pumping in two important\nrespects. First, while the spin polarization generated by\noptical pumping relies on the polarization of the pump\nlight, the target state for cavity-induced spin damping\nis selected energetically. Similar to cavity optomechani-\ncal cooling [4], cavity enhancement of Raman scattered\nlight drives spins to the high- or low-energy spin state\naccording to the detuning of probe light from the cavity\nresonance, independent of the polarization. Second, this\nampli\fcation or damping of the intracavity spin is coher-\nent, preserving the phase of Larmor precession, at least\nwithin the limits of a quantum ampli\fer.\nSpin optodynamical squeezing of light: We now\nconsider quantum optical e\u000bects of cavity spin optody-\nnamics. One such e\u000bect is the disturbance of the collec-\ntive spin due to quantum optical \ructuations of the cav-\nity \felds. In cavity optomechanics, intracavity photon\nnumber \ructuations disturb the motion of a cantilever,\nproviding the necessary backaction of a quantum mea-\nsurement of position [25]. The analogous disturbance of\noptically probed atomic spins (or pseudo-spins) has been\nstudied both in free-space [26] and intracavity [11, 27]\ncon\fgurations. In an optomechanics-like con\fguration,\ne.g. with B/i, backaction heating of the atomic spin\nenforces quantum limits to measurement of the precess-\ning ensemble and also set limits on optodynamical cool-\ning. In contrast with optomechanical systems, optically\nprobed spin ensembles readily present the opportunity to\nperform quantum-non-demolition (QND) measurements;\nwithB/k, the detected spin component Skis a QND\nvariable representing the energy of the spin system.\nThe noise-perturbed spin acts back upon the cavity\noptical \feld, mediating a self-interaction of the light \feld\n00 . 1-500005000\n246\nt (ms)8.2 8.3 8.4Si and S kSk\nSi\n0\nf (kHz)100 200 300 400-π/201020\n-10π/2(a)\n(b)\nS\nk\nS\ni\n100\n 200\n 300\n 400dBFIG. 3. (Color) Simulations of spin dynamics for S= 5000,\n\nL=2\u0019= 200 kHz, \n c=2\u0019=\u00002:3 kHz,\u0014=2\u0019= 1:8 MHz,\n^n+= 10, and \u0001 p;+= 0:37\u0014. (a) Time evolution of Si(black)\nandSk(blue), following spin preparation near i, shows ampli-\n\fcation, reorientation, and damping toward the high-energy\nstable orientation near \u0000i. Note the di\u000berent scales on the\nhorizontal axis. (b) Logarithmic optical spectral noise power\nrelative to that of coherent light, plotted vs. quadrature an-\ngle\u001e(amplitude quadrature at \u001e= 0), shows inhomogeneous\noptical squeezing. Simulation results shown in color, and lin-\nearized theory (Eq. 10) as contour lines every 5 dB.\nthat can result in optical squeezing. To exhibit this ef-\nfect, we consider a cavity illumined with \u001b+circular po-\nlarized probe light with detuning \u0001 p. The dynamics of\nthe cavity \feld are given by\nd^c+\ndt= (i\u0001p\u0000\u0014+i\nc^Sk)^c++\u0014\u0010\n\u0011+^\u0018+\u0011\n:(8)\nHere,\u0011gives the coherent-state amplitude of the drive\n\feld and the noise operator ^\u0018+represents its \ructua-\ntions. When evaluating the dynamics numerically, we\nconsider a semiclassical Langevin equation, converting\n^\u0018+into a Gaussian stochastic variable with statistics re-\nlated to those of the noise operator, and replacing the\noperators ^c+and^Swithc-numbers. This substitution is\nappropriate for moderately large values of \u0016 nandS.\nFig. 3 portrays the simulated evolution of a spin pre-\npared initially in a low-energy spin orientation (close\ntoi), driven by a blue-detuned cavity probe. Coher-\nent spin ampli\fcation directs the spin toward the stable\nhigh-energy con\fguration (near \u0000i), yielding a dynami-\ncal steady state characterized by a negative temperature.\nTo obtain analytical expressions for the evolution dy-\nnamics, we follow the example of cavity optomechanics [5]4\nby linearizing the Langevin equations for spin and opti-\ncal \ructuations about their steady-state value. The spin\nprojectionSkresponds to amplitude-quadrature \ructua-\ntions of the cavity \feld \u0018A(!) with susceptibility\n\u001f(!)\u0011Sk(!)\n\u0018A(!)=\u0000\nL0\ncp\u0016n+\n\nL02+kSR(!)\u0000!2+i!\u0000o(!);(9)\nwhere \u0000o(!) = 2\u0014\nL02\u0000!2\n\u00142+\u00012\np\u0000!2is the cavity optodynamic\nspin damping, and R(!) =\u00142+\u00012\np\n\u00142+\u00012p\u0000!2. The susceptibil-\nity is largest for !'\nL00. The driven spin feeds the\n\ructuations back onto the cavity \feld, yielding the in-\ntracavity \feld \ructuation spectrum\nc+(!) =\nL02+i\u0014+!\n\u0001pkSR(!)\u0000!2+i!\u0000o(!)\n\nL02+kSR(!)\u0000!2+i!\u0000o(!)\u0018A+i\u0018P;\n(10)\nwhere\u0018P(!) is the input spectrum of phase \ructuations.\nThis \ructuation spectrum exhibits inhomogeneous opti-\ncal squeezing (Fig. 3b).\nApplications: The analogy of cavity optodynamics\nwidens the range of phenomena accessed through the ma-\nnipulation and detection of quantum spins within opti-\ncal cavities, enabling several applications. For example,\nbistability in cavity-coupled single-spin systems serves to\nincrease the readout \fdelity of cavity-coupled qubits [28].\nSimilarly here, cavity-spin bistability could be used as a\nSchmitt trigger for the collective spin: if the probe power\nis turned on diabatically in the bistable regime, the cav-\nity transmission will latch into either a bright or dark\nstate, depending on whether the initial spin state is be-\nlow or above a parametrically chosen threshold value.\nThe cavity-spin system may also be used as a phase-\npreserving ampli\fer for spin dynamics occurring near\nthe shifted precession frequency \n L00, with ampli\fcation\nnoise given in Eq. (10). Both applications may aid mea-\nsurements of ac magnetic \felds, amplifying weak signals\nabove technical sensitivity limits.\nConversely, cavity spin optodynamics may be applied\nas a powerful simulator of cavity optomechanics, with the\nspin system allowing for new means of control. For exam-\nple, precession frequencies may be tuned rapidly by vary-\ning the applied magnetic \feld, simulating optomechanics\nwith a dynamically variable mechanical spring constant.\nAlternately, spatial control of inhomogeneous magnetic\n\felds may be used to divide a spin ensemble into sev-\neral independent subensembles, simulating optomechan-\nics with several mechanical modes.\nIn addition to the dilute gas implementation discussed\nso far, a similar system could be constructed using solid-\nstate spin ensembles and microwave resonators. For ex-\nample, using ensembles of nitrogen-vacancy defects in di-\namond coupled to the circular polarized evanescent radi-\nation of a crossed microwave resonator [29], the Hamilto-\nnian of Eq. (4) is obtained by the ground ms=\u00061 elec-tronic states as the pseudospin and replacing the ac Stark\nshift with an ac Zeeman shift from microwave radiation\nnear the 2.8-GHz crystal-\feld-split Zeeman transition.\nWe thank H. Mabuchi and K.B. Whaley for inspiring\ndiscussions. This work was supported by the NSF and\nthe AFOSR. D.M.S.-K. acknowledges support from the\nMiller Institute for Basic Research in Science.\n\u0003dmsk@berkeley.edu\n[1] T. Kippenberg and K. Vahala, Science 321, 1172 (2008).\n[2] V. Braginskii and F. Y. Khalili, Quantum Measurement\n(Cambridge University Press, Cambridge, 1995).\n[3] K. W. Murch, K. L. Moore, S. Gupta, and D. M.\nStamper-Kurn, Nature Physics 4, 561 (2008).\n[4] V. Vuleti\u0013 c and S. Chu, Phys. Rev. Lett. 84, 3787 (2000).\n[5] C. Fabre et al., Phys. Rev. A 49, 1337 (1994); S. Mancini\nand P. Tombesi, Phys. Rev. A 49, 4055 (1994).\n[6] S. Haroche and J.-M. Raimond, Exploring the Quantum:\nAtoms, Cavities, and Photons (Oxford University Press,\n2006).\n[7] P. F. Herskind et al. , Nature Physics 5, 494 (2009).\n[8] P. E. Barclay, K.-M. C. Fu, C. Santori, and R. G. Beau-\nsoleil, App. Phys. Lett. 95, 191115 (2009).\n[9] L. DiCarlo et al. , Nature 460, 240 (2009).\n[10] J. M. Geremia, J. K. Stockton, A. C. Doherty, and\nH. Mabuchi, Phys. Rev. Lett. 91, 250801 (2003).\n[11] I. D. Leroux, M. H. Schleier-Smith, and V. Vuletic, Phys.\nRev. Lett. 104, 073602 (2010).\n[12] Q. Turchette et al. , Phys. Rev. Lett. 75, 4710 (1995).\n[13] T. Holstein and H. Primako\u000b, Phys. Rev. 58, 1098\n(1940).\n[14] V. B. Braginsky and A. B. Manukin, Sov. Phys. JETP\n25, 653 (1967).\n[15] T.J. Kippenberg et al., Phys. Rev. Lett. 95, 033901\n(2005); S. Gigan et al., Nature 444, 67 (2006); O. Ar-\ncizet et al., Nature 444, 71 (2006).\n[16] A. Dorsel et al. , Phys. Rev. Lett. 51, 1550 (1983).\n[17] S. Gupta, K. L. Moore, K. W. Murch, and D. M.\nStamper-Kurn, Phys. Rev. Lett. 99, 213601 (2007).\n[18] H. J. Kimble et al. , Phys. Rev. D 65, 022002 (2001).\n[19] W. Happer and B. S. Mathur, Phys. Rev. Lett. 18, 577\n(1967).\n[20] C. Cohen-Tannoudji and J. Dupont-Roc, Phys. Rev. A\n5, 968 (1972).\n[21] T. Purdy et al. , preprint, arXiv:1005.4085.\n[22] H. M. Gibbs, Optical bistability: Controlling light with\nlight (Academic Press, New York, 1985).\n[23] A. Buonanno and Y. B. Chen, Phys. Rev. D 65, 042001\n(2002).\n[24] T. Corbitt et al. , Phys. Rev. Lett. 98, 150802 (2007).\n[25] C. M. Caves, Phys. Rev. Lett. 45, 75 (1980).\n[26] C. Schori, B. Julsgaard, J. L. S\u001crensen, and E. S. Polzik,\nPhys. Rev. Lett. 89, 057903 (2002).\n[27] M. H. Schleier-Smith, I. D. Leroux, and V. Vuletic, Phys.\nRev. Lett. 104, 073604 (2010).\n[28] I. Siddiqi et al. , Phys. Rev. B 73, 054510 (2006).\n[29] J. J. Henderson, C. M. Ramsey, H. M. Quddusi, and E. d.\nBarco, Rev. Sci. Instrum. 79, 074704 (2008)."    },    {        "title": "1608.07646v1.Tunneling_induced_spin_dynamics_in_a_quantum_dot_lead_hybrid_system.pdf",        "content": "arXiv:1608.07646v1  [cond-mat.mes-hall]  27 Aug 2016Tunneling induced spindynamics inaquantum dot-leadhybri d system\nTomohiro Otsuka,1,2,∗Takashi Nakajima,1,2Matthieu R. Delbecq,1Shinichi Amaha,1\nJun Yoneda,1,2Kenta Takeda,1Giles Allison,1Peter Stano,1,3Akito Noiri,1,2Takumi\nIto,1,2Daniel Loss,1,4Arne Ludwig,5Andreas D. Wieck,5and Seigo Tarucha1,2,6,7\n1Center for Emergent Matter Science, RIKEN, 2-1 Hirosawa, Wa ko, Saitama 351-0198, Japan\n2Department of Applied Physics, University of Tokyo, Bunkyo , Tokyo 113-8656, Japan\n3Institute of Physics, Slovak Academy of Sciences, 845 11 Bra tislava, Slovakia\n4Department of Physics, University of Basel, Klingelbergst rasse 82, 4056 Basel, Switzerland\n5Angewandte Festk¨ orperphysik, Ruhr-Universit¨ at Bochum , D-44780 Bochum, Germany\n6Quantum-Phase Electronics Center, University of Tokyo, Bu nkyo, Tokyo 113-8656, Japan\n7Institute for Nano Quantum Information Electronics, Unive rsity of Tokyo, 4-6-1 Komaba, Meguro, Tokyo 153-8505, Japan\n(Dated: September 2, 2021)\nSemiconductor quantum dots (QDs)offer a platform toexplor e the physics of quantum electronics including\nspins. Electron spins in QDs are considered good candidates for quantum bits [1] in quantum information\nprocessing [2, 3], and spin control and readout have been est ablished down to a single electron level [4]. We\nuse these techniques to explore spin dynamics in a hybrid sys tem, namely a QD coupled to a two dimensional\nelectronic reservoir. The proximity of the lead results in r elaxation dynamics of both charge and spin, the\nmechanism of which is revealed by comparing the charge and sp in signal. For example, higher order charge\ntunnelingeventscanbemonitoredbyobservingthespin. Wee xpecttheseresultstostimulatefurtherexploration\nof spindynamics inQD-leadhybridsystems and expand the pos sibilities for controlledspin manipulations.\nElectron spins in semiconductor QDs have relatively long\ncoherence times [5–8], while the solid state structures hav e\npotential scalability by utilizing current extensive semi con-\nductor fabricationtechniques. They are also consideredgo od\ncandidates for quantum bits [1] in quantum information pro-\ncessing [2, 3]. The spin states can be initialized using a lar ge\nexchangeenergyofasingleQD[9],andtherelatedPaulispin\nblockadeinadoubleQD(DQD)[10]. Thedemonstratedways\nofmanipulationincludethespin-spinexchangeinteractio nbe-\ntween neighboringQDs [9], and electron spin resonances in-\nducedbymicrocoils[11],nuclearspins[12],spinorbitint er-\naction [13], or micro magnets [14–16]. Finally, the spin can\nbereadout byPaulispin blockade,or a tunnelingsensitive t o\ntheZeemanenergy[17].\nIn experiments aimed at the minimization of the dissipa-\ntion, the QDs have been typically isolated from their envi-\nronment, including the leads, as much as possible. How-\never, it is worth to explore physics in hybrid systems, where\nthe dot-environmentcoupling is stronger, since this coupl ing\ncan be tuned straightforwardly and precisely tuned by gate\nvoltages. In addition, the electronic reservoirs can be tai -\nlored themselves, by applying bias voltages or using specifi c\nstatessuch asferromagnets[18],superconductors[19],qu an-\ntum Hall states [20], and others. This variability gives ris e\nto attractive physics like Fano interference [21, 22], Kond o\nstates[23,24],orgeneralphysicsofopenandnonequilibri um\nsystems,andpossiblyleadtonewwaysofspinmanipulations\nutilizinginteractionsinducedbytheenvironment[25].\nInthiswork,weexplorespindynamicsinaQD-leadhybrid\nsystemutilizingthespinmanipulationandreadouttechniq ues\ndeveloped in previous spin qubit experiments. Specifically ,\nwe monitor changes of the spin and charge states induced by\ncoupling of the QD to an electronic reservoir. With the QD\nbeingclosetoachargetransition,weobservespinandcharg erelaxation,correspondingtofirst-ordertunnelingevents . With\nthedotinaCoulombblockadeconfiguration,weobserveonly\nthe relaxation of spin, corresponding to second-order tunn el-\ningevents.\nFigure 1(a) shows a scanning electron micrograph of the\ndevice. By applyingnegativevoltages on the gate electrode s,\na DQD and a QD charge sensor [26] are formed at the lower\nanduppersides,respectively. TheleftQDintheDQDcouples\nto a lead andthe couplingstrengthis tunedbythe voltage VT\nappliedongateT.TheQDchargesensorisconnectedtoaRF\nresonator formedby the inductor Land the stray capacitance\nCpforRF reflectometry[26–28]. The numberof electronsin\neachQD(n1,n2)ismonitoredbytheintensityofthereflected\nRF signal Vsensor. A change in the electrostatic environment\naround the sensing dot changes its conductance, which shift s\nthe tank circuit resonance and modifies Vsensormeasured at\nfres=297MHz, thecircuitresonancefrequency.\nFigure1(b)showsthechargestabilitydiagramoftheDQD.\nWe measured the sensor signal Vsensoras a function of the\nplunger gate voltages of QD 2(VP2), and QD 1(VP1). We\nobserve a change ∆Vsensoreach time the DQD charge con-\nfiguration (n1,n2)changes. Depicted in Fig. 1(b), the values\n(n1,n2)areassignedbycountingthenumberofchargetransi-\ntionlinesfromthefullydepletedconfiguration (n1,n2)=(0,0)\n[the latter not shown on Fig. 1(b)]. Around the charge state\ntransition (1,1)↔(0,2), we observe a suppression of the\n(0,2)charge signal due to the Pauli spin blockade [in the re-\ngion indicated by the triangle in Fig. 1(b)]. In this specific\nmeasurementofthestability diagram,unlikeelsewhere,up on\npulsing(2,0)→(1,1)we move through the singlet-triplet\nT+anti-crossing very slowly (adiabatically), to induce a siz -\nable triplet component of the (1,1)state even at a zero inter-\naction time. Pulsing quickly back (1,1)→(0,2)results in a\nPauli blocked signal inside the denoted triangular area. Th is2\n(a) (b)\n500 nmP1P2SRF\nL\nCpLead\n-800-790-780-770VP1 (mV)\n-180-170-160-150\nVP2 (mV)-200 200ΔVsensor (mV)\n(1,1) (1,2)\n(0,2) (0,1)SB\nMO1\nIO2\n(c)\nInitialize\nLeadQD1 QD2Operation Measurement\n?\nInteraction SBT z\nyxδ\nFIG. 1: (a) Scanning electron micrograph of the device and th e\nschematic of the measurement setup. A DQD is formed at the low er\nside and the charge states are monitored by the charge sensor QD at\nthe upper side. The charge sensor is connected to resonators formed\nbytheinductor Landthestraycapacitance CpfortheRFreflectom-\netry. The external magnetic field of 0.5 T is applied in plane a long\nthezaxis. (b) ∆Vsensoras a function of VP2andVP1. Changes of\nthe charge states are observed. The number of electrons in ea ch QD\nisgiven as (n1,n2). Thetriangle shows the regionof spinblockade.\nThepositionscorrespondingtostepsofpulsesequences(O, I,M)are\nindicated. (c) Schematic of the measurement scheme. The spi nstate\nis initializedtoa (0,2) singlet at I.Next, the we move intoO in(1,1)\nwherethespincouplestothelead. Finally,thespinstateis measured\nusing spinblockade at M.\nshowsuswherewecanutilizethePaulispinblockadetoread-\nout the spin state in the followingmeasurements,probingth e\ndotspinandchargetunneling-induceddynamics.\nThe operation scheme to measure the effect of the lead on\nthe spin is depicted in Fig. 1(c). We initialize the state to a\n(0,2)singlet by waiting at the initialization point I denot edin\nFig. 1(b). Next, we move to the operation point O. In this\nstep, the electron in QD 1interacts with the lead and the dot\nstate might be changed by electron tunneling. The tunneling\nrate can be modified by tuning VT, which changes the tunnel\ncoupling,andthepositionofO,whichchangesthedotpoten-\ntial with respect to the Fermi energyof the lead (O 1: close to\na charge transition, O 2: deep in the Coulomb blockade). At\nthe next step, the spin state is measured using spin blockade\nby pulsingthe dot to the pointdenotedbyM. If the spin state\ndid notchange,we observethe (0,2)singlet again. If the spi n\nstate changed,a polarized triplet componentis measuredas a\nblocked(1,1)→(0,2)charge transition. From the charge\nsignal,wecanthereforededucethespinstate.\nIn this way, we first measure the spin relaxation using\nthe operation point O 1close to a charge transition line, see\nFig.1(b),wheretheQDlevelisclosetotheFermilevelofthe\nlead. Thetunnelinggatevoltageissetto VT=−660mV.The(b)(a) (c)\n1.0\n0.9\n0.8\n0.7\n0.6\n0.5Singlet Probability\n12840\nTime (µs)-920-900-880-860\nCharge Signal (mV)\n12\n8\n4\n0Time (µs)\n-1200 -800 -400\nVsensor (mV)12x10-3\n8\n4\n0(1,1) (0,1)Lead QD1\nEmptyN(Vsensor)/Ntot\nFIG.2: (a) Observed spin and charge signals (the singlet pro bability\nand the average of the sensor signal /angbracketleftVsensor/angbracketright) as a function of the\ninteraction time. Red circles show the spin signal (left axi s). The\nblue trace shows the charge signal (right axis). The smooth l ines are\nexponential fitsresultingintherelaxationtimeof3.0 µsforthespin,\nand 1.8µs for the charge. (b) Statistics of the charge signal at the\noperation point. Histogram of observed values of the charge sensor\nVsensor(on the x axis), N(Vsensor)/Ntotis plotted as a function of\nthe interactiontime(yaxis). The twopeaks, at Vsensor=−960mV,\nand−780mV, correspond to the (1,1) and the (0,1) charge states,\nrespectively. The weight of the (0,1) component increases w ith the\nlonger interaction time. (c) Schematic of the spin relaxati on by a\nfirst-order tunneling process. An electron escapes from the QD and\nthe QDbecomes empty. Another electroncomes inafterthat.\nred circles in Fig. 2(a) show the measured singlet probabili ty\nas a function of the interaction time at O 1. Initially at 1, the\nsinglet probability decreases upon increasing the interac tion\ntime from zero. This decrease indicates that a triplet compo -\nnentisformedbytheinteractionwiththelead. Fittingwith an\nexponentialrevealsa relaxationtimeof3.0 µs.\nSimilarly to spin, we also measure the lifetime of charge\nin this configuration. The blue trace in Fig. 2(a) shows the\naveraged Vsensorasafunctionoftheinteractiontime. Asseen\nthere,∝an}bracketle{tVsensor∝an}bracketri}htchanges exponentially, with the fitted charge\nrelaxationtimeof1.8 µs. Toexaminethechargerelaxationin\nmore detail, we plot in Fig. 2b histograms (the x axis) of the\nvalues of Vsensorfor a varying interaction time (the y axis).\nThe two peaks along a horizontal cut correspond to the (1,1)\nand the (0,1)chargestates, respectively. At a zero interac tion\ntime, only the (1,1) state signal is present, while (0,1) sta te\nappearsforfiniteinteractiontimes.\nIn this configuration, the mechanism of the relaxation for\nboth spin and charge is a first-order tunneling process [29].3\n12\n8\n4\n0Time (µs)\n-1000 -800 -600\nVsensor (mV)20x10-3\n15\n10\n5\n00.9\n0.8\n0.7Singlet Probability\n12840\nTime (µs)-800-780-760-740-720\nCharge Signal (mV)\n(b)(a) (c)\n(1,1)Lead QD1N(Vsensor)/Ntot\nFIG.3: (a) The observed singlet probability and /angbracketleftVsensor/angbracketrightas a func-\ntion of the interaction time at O 2[see Fig. 1(b)]. Red circles show\nthe spin signal (left axis). The blue trace shows the charge s ignal\n(right axis). The red smooth curve is an exponential fit resul ting in\ntherelaxationtimeof4.5 µs. Thecharge signalshowsnorelaxation.\n(b) Histogram of observed values of the charge sensor Vsensor(on\nthe x axis), N(Vsensor)/Ntot, is plotted as a function of the interac-\ntion time (y axis). The peak corresponds to the (1,1) charge s tate.\n(c)Schematicofthespinrelaxationbyasecond-order tunne lingpro-\ncess. An electron of the QD 1is swapped with a one in the lead in a\nsingle step. The spinstateis changed even though the charge state is\nstable.\nNamely, the electron tunnels out of the QD 1into the lead,\nafter which the dot is refilled from the lead, and the initial\ninformationislost. Thespinandchargerelaxationhappens i-\nmultaneously,theinformationlossofthespindemonstrate din\nFig.2(a),andofthechargeinFig.2(a-b). Wenotethatthoug h\nthe relaxation timescales are similar, they are not identic al.\nThedifferencecomesfromadifferenceintheratedependenc e\nonFermioccupationofthelead(seetheSupplementaryInfor -\nmation).\nWenowinvestigatethespindynamicsinaCoulombblock-\nadeddot. Tothisend,werepeatthepreviouslydescribedmea -\nsurement using the operation point O 2, deep in the (1,1) re-\ngion, see Fig. 1(b). Here the QD level is far below the Fermi\nlevel of the lead. To increase the speed of the lead induced\nspindynamicsonthedot,weincreasethedot-leadtunnelcou -\npling by setting VT=−560mV. As can be seen in Fig. 3(a),\nsimilarly as before, the spin state displays an exponential de-\ncay, with the relaxation time of 4.5 µs. However, now the\ncharge signal barely changes, indicating that the charge st ate\nis not affected. (The slight change of the charge signal in\nFig. 3(a) is caused by the distorted voltage pulses applied o n2.5\n2.0\n1.5\n1.0\n0.5\n0.0Relaxation Rate (MHz)\n543210\nδ (mV)\n0.9\n0.8\n0.7Singlet Probability\n543210\nTime (µs)(a)\n(b)\nFIG. 4: (a) The spin relaxation rate as a function of δ. Circles show\nthe experimental data and the line shows a theoretical curve consid-\neringthesecond-ordertunnelingprocess. (b)Thespinrela xationsig-\nnal as a function of the interaction time at O2for different values of\nthe gate voltage applied on gate T VT. Circles,triangles and squares\nshow the result at VT=−560,−565,−570mV, respectively. The\nlines are exponential fits.\nP1 and P2. Due to a cross-talk between the plunger gates\nandthesensor,thepulsedistortionslightlyaffecttheobs erved\nchargesignal.) This is confirmedby Fig. 3(b), where the his-\ntograms of the values of Vsensordisplay a single peak corre-\nsponding to the (1,1) charge state. The spin therefore decay s\nata fixedQDchargeconfiguration.\nWe therefore interpret this as the observation of a spin re-\nlaxation induced by a second-order tunneling process [30],\nwhere the electron in QD 1swaps with a random one from\ntheleadinasinglestep. Figure4(a)showsthespinrelaxati on\nrate as we change the operation point from O 2toward O 1,\nparametrizingthedisplacementbyvoltage δ. Uponincreasing\nδ(moving towards the charge transition line) the spin relax-\nationrateisenhanced. Themeasureddependenceisverywell\nfitted by an analytical expression for an inelastic cotunnel ing\nrate,giving ∝(1/(µ(2)−µF)+1/(µF−µ(1)))2,withµ(N)\nandµFbeing the electrochemical potential at the dot with N\nelectrons [4] and the Fermi energy of the lead, respectively\n(see the SupplementalInformationfor details). In additio n to\nplungervoltages,wecantunethe spindecaytimescalebythe\nvoltage applied on gate T, VT. Indeed, as seen in Fig. 4, ap-\nplyingmore negativevoltage VTprolongsthe spin relaxation\ntime,bydecreasingthetunnelcouplingtothelead,from0.7 to\n1.7 to 5.0 µs, forVT=−560,−565,−570mV, respectively.\n(We notethatthe relaxationtime at VT=−560mV isdiffer-4\nentfromthecorrespondingvalueof VTgiveninFig.3(a)due\nto a shift of the QD conditions between experiments.) This\ndemonstratesthetwohandlesonthespeedoftheleadinduced\ndynamicsoftheQD spin.\nTo sum up the results observed in the Coulomb blockade\nregime, here the interaction with the lead influences only th e\ndot spin, but not charge. The spin relaxationthusdirectly u n-\ncoversthe second ordertunnelingprocesses. This interact ion\ncanbeutilizedforthespininitialization,thoughalsothe mea-\nsurement and manipulation might be envisioned, considerin g\nleads with special properties. We note that even though the\ntimescale of the dot-lead interaction realized in this expe ri-\nment was tuned to ∼µs, it is straightforward to enhance it\nby increasing the tunnel coupling, and/or utilizing the Kon do\neffect,whichenhancesthesecond-ordertunnelingatlowte m-\nperatures.\nIn conclusion, we have measured spin dynamics in a QD-\nleadhybridsystem. Close toa chargetransition,we observe d\nspin and charge relaxation signals corresponding to the firs t-\norder tunneling process. In the Coulomb blockade, we ob-\nserved spin relaxation at a fixed charge configuration, corre -\nsponding to the second-ordertunneling process. The demon-\nstrated dot-lead spin exchange can be useful as a general re-\nsource for spin manipulations, and simulations of open sys-\ntemsundernon-equilibriumconditions.\nMETHODS\nThe device was fabricated from a GaAs/AlGaAs het-\nerostructure wafer with an electron sheet carrier density o f\n2.0×1015m−2and a mobility of 110 m2/Vs at 4.2 K, mea-\nsured by Hall-effect in the van der Pauw geometry. The two-\ndimensional electron gas is formed 90 nm under the wafer\nsurface. We patterned a mesa by wet-etching and formed\nTi/Au Schottky surface gates by metal deposition, which ap-\npear white in Fig. 1(a). All measurementswere conductedin\na dilutionfridgecryostatat a temperatureof13mK.\nACKNOWLEDGEMENTS\nWe thank J. Beil, J. Medford, F. Kuemmeth, C. M. Mar-\ncus, D. J. Reilly, K. Ono, RIKEN CEMS Emergent Mat-\nter Science ResearchSupportTeamandMicrowaveResearch\nGroup in Caltech for fruitful discussions and technical sup -\nports. Part of this work is supported by the Grant-in-Aid for\nScientific Research (No. 25800173, 26220710, 26709023,\n26630151, 16H00817), CREST, JST, ImPACT Program of\nCouncilforScience,TechnologyandInnovation(CabinetOf -\nfice, Government of Japan), Strategic Information and Com-\nmunications R&D Promotion Programme, RIKEN Incentive\nResearch Project, Yazaki Memorial Foundation for Science\nand TechnologyResearch Grant, Japan Prize FoundationRe-\nsearchGrant,AdvancedTechnologyInstituteResearchGran t,the Murata Science Foundation Research Grant, Izumi Sci-\nence and Technology Foundation Research Grant, TEPCO\nMemorialFoundationResearchGrant,IARPAproject“Multi-\nQubit Coherent Operations” throughCopenhagenUniversity ,\nDFG-TRR160,andtheBMBF - Q.com-H16KIS0109.\nAUTHOR CONTRIBUTIONS\nT. O., T. N., M. D., S. A., J. Y., K. T., G. A. and S. T.\nplannedtheproject;T.O.,T.N.,M.D.,S.A.,A.L.andA.W.\nperformeddevicefabrication;T. O., T.N., M. D., S. A., J. Y. ,\nK. T., G. A., P. S., A. N., T. I., D. L. and S. T. conductedex-\nperimentsand data analysis; all authorsdiscussed the resu lts;\nT. O., T. N., M. D., S. A., J. Y., K. T., G. A., P. S. and S. T.\nwrotethemanuscript.\n∗tomohiro.otsuka@riken.jp\n[1] Loss, D., and DiVincenzo, D. P. Quantum computation with\nquantum dots. Phys.Rev. A 57, 120(1998).\n[2] Nielsen, M. A., and Chuang, I. L. Quantum Computation and\nQuantum Information. (Cambridge UniversityPress,2000).\n[3] Ladd, T. D. et al. Quantum computers. Nature 464, 45-53\n(2010).\n[4] Hanson, R. et al. Spins in few-electron quantum dots. Rev.\nMod. Phys. 79, 1217 (2007).\n[5] Bluhm, H. et al. Dephasing time of GaAs electron-spin qubits\ncoupled to a nuclear bath exceeding 200 µs. Nat. Phys. 7, 109-\n113 (2011).\n[6] Shulman, M. D. et al. Suppressing qubit dephasing using real-\ntime Hamiltonianestimation. Nat.Commun. 5,5156 (2014).\n[7] Kawakami, E. etal.Electricalcontrolofalong-livedspinqubit\nina Si/SiGequantum dot.Nat. Nano. 9, 666-670 (2014).\n[8] Veldhorst, M. et al. An addressable quantum dot qubit with\nfault-tolerant control-fidelity.Nat. Nano. 9, 981-985 (2014).\n[9] Petta, J. R. et al. Coherent Manipulation of Coupled Electron\nSpins in Semiconductor Quantum Dots. Science 309, 2180-\n2184 (2005).\n[10] Ono, K., Austing, D. G., Tokura, Y., and Tarucha, S. Curr ent\nRectification by Pauli Exclusion in a Weakly Coupled Double\nQuantum Dot System.Science 297, 1313-1317 (2002).\n[11] Koppens, F.H.L. etal. Drivencoherent oscillations of a single\nelectron spinina quantum dot. Nature 442, 766-771 (2006).\n[12] Laird, E. A. et al. Hyperfine-Mediated Gate-Driven Electron\nSpinResonance. Phys. Rev. Lett. 99, 246601 (2007).\n[13] Nowack, K. C., Koppens, F. H. L., Nazarov, Y. V., and Van-\ndersypen, L.M. K. Coherent Control of a Single Electron Spin\nwithElectricFields.Science 318, 1430-1433 (2007).\n[14] Tokura, Y., van der Wiel, W. G., Obata, T., and Tarucha, S .\nCoherent Single Electron Spin Control in a Slanting Zeeman\nField.Phys. Rev. Lett. 96, 047202 (2006).\n[15] Pioro-Ladriere,M. etal.Electricallydrivensingle-electronspin\nresonance in a slanting Zeeman field. Nat. Phys. 4, 776-779\n(2008).\n[16] Yoneda,J. etal.FastElectricalControlofSingleElectronSpins\ninQuantumDotswithVanishingInfluencefromNuclearSpins.\nPhys. Rev. Lett. 113, 267601 (2014).5\n[17] Elzerman,J.M. etal.Single-shotread-outofanindividualelec-\ntronspinina quantum dot. Nature 430, 431-435 (2004).\n[18] Hamaya, K. et al. Spin transport through a single self-\nassembled InAs quantum dot with ferromagnetic leads. Appl.\nPhys.Lett. 90, 053108 (2007).\n[19] Deacon,R.S. etal.TunnelingSpectroscopyofAndreevEnergy\nLevels in a Quantum Dot Coupled to a Superconductor. Phys.\nRev. Lett. 104, 076805 (2010).\n[20] Altimiras,C. etal.Non-equilibriumedge-channelspectroscopy\nintheintegerquantum Hallregime.Nat.Phys. 6,34-39(2010).\n[21] Fano,U.Effectsof ConfigurationInteractiononIntens itiesand\nPhase Shifts.Phys. Rev. 124, 1866-1878 (1961).\n[22] Kobayashi, K., Aikawa, H., Katsumoto, S., and Iye, Y. Tu ning\nof the Fano Effect through a Quantum Dot in an Aharonov-\nBohm Interferometer. Phys.Rev. Lett. 88, 256806 (2002).\n[23] Goldhaber-Gordon, D. et al. Kondo effect in a single-electron\ntransistor.Nature 391, 156-159 (1998).\n[24] van der Wiel, W. G. et al. The Kondo Effect in the Unitary\nLimit.Science 289, 2105-2108 (2000).\n[25] Anders, F. B., and Schiller, A. Spin precession and real -time\ndynamics in the Kondo model: Time-dependent numerical\nrenormalization-group study. Phys.Rev. B 74, 245113 (2006).\n[26] Barthel,C. etal.Fastsensingofdouble-dotchargearrangement\nandspinstatewitharadio-frequencysensorquantumdot.Ph ys.\nRev. B81, 161308 (2010).\n[27] Schoelkopf, R. J. et al. The Radio-Frequency Single-Electron\nTransistor (RF-SET): A Fast and Ultrasensitive Electromet er.\nScience280, 1238-1242 (1998).\n[28] Reilly, D. J., Marcus, C. M., Hanson, M. P., and Gossard, A.\nC. Fast single-charge sensing with a rf quantum point contac t.\nAppl.Phys. Lett. 91, 162101 (2007).\n[29] Biesinger, D. E. F. et al. Intrinsic Metastabilities in the Charge\nConfiguration of a Double Quantum Dot Phys. Rev. Lett. 115,\n106804 (2015).\n[30] Schleser,R. etal.Cotunneling-MediatedTransportthroughEx-\ncitedStatesintheCoulomb-Blockade Regime.Phys.Rev.Let t.\n94, 206805 (2005).1\nSupplementalMaterialto`Tunneling induced spin dynamics in a quantumdot-leadhybridsystem'\nCHARGEAND SPINRELAXATIONSIGNALS\nWe describethedynamicsofthechargeandspinontheQD bycon sideringtherateequation\n∂tPσ=−Γσ(1−fσ)Pσ+ΓσfσPe, (S1)\nfor the probability Pσthat the dot is occupied by a single electron with spin σ∈ {↑,↓}(we alternative use σ=±1), where\nPeis the probability that the dot is empty. We do not consider an y other states, which gives the normalization condition\nP↑+P↓+Pe= 1. Eq. (S1) includesthe processof an electron with a given spi n leaving the dot into the lead where an empty\nstateexistswiththeprobability (1−fσ)andenteringanemptydotfromtheleadstateoccupiedwithpr obability fσ. ThisFermi\nfactor is given by fσ=fFD(µ(1)−σEz), withµ(1) =eVg+ǫ1being the energycost to add an electron into the dot, which\nincludesthe electrostatic potentialenergy eVg, the orbital (quantization)energy ǫ1. The Zeemanenergyis Ez=gµBB/2, and\ntheFermi-Diracdistribution\nfFD(ǫ) =/braceleftbigg\nexp/bracketleftbiggǫ−µF\nkBT/bracketrightbigg\n+1/bracerightbigg−1\n, (S2)\ndependson the temperature T, and the lead Fermi energy µF. Apart fromthe Fermi factors fσ, the tunnelingrates for hopping\nonandoffthe dotare identical, Γσ. We, however,allowfora spin dependenceofthe tunnelingra tewhichhasbeenfoundto be\nanappreciableeffect(theasymmetryoftheratescanbeofth eorderoftheratesthemselves),mostprobablyduetotheexc hange\ninteractioninthelead[1–3].\nTo expose the spin and charge dynamics, we introduce new vari ables, the probability of charge occupation, Po=P↑+P↓\nand the spin polarization, s=P↑−P↓, and new parameters, for the average, Γ, and the dimensionless asymmetry α, in the\ntunnelingrates, bywriting Γσ= Γ(1+ σα), andsimilarly fortheFermi factors, fσ=f+σfδ. Equation(S1) canbe nowcast\nintothematrixformforthe vectorofunknowns, v= (Po,s)T,namely\n∂tv=−M(v−v0), (S3)\nwiththe matrixdefiningthesystem propagator\nM= Γ/parenleftbigg\n1+f+fδα−fδ+(1−f)α\nfδ+(1+f)α1−f−fδα/parenrightbigg\n, (S4)\nandthesteadystate solution\nv0=2\n1−f2+f2\nδ/parenleftbigg\nf(1−f)−f2\nδ\nfδ/parenrightbigg\n. (S5)\nThesteadystateisindependentonthetunnelingrates,andd ependsonlyontheleadFermifactorsforthetwospins,asit s hould\nbe,whilethepropagatormatrixdependsonallparametersof theproblem. Eventhoughitisstraightforwardtosolvethep roblem\ninthe mostgeneralcase,it isusefultoconsider Mforα= 0(spinindependenttunnelingrates),whichgives\nM= Γ/parenleftbigg\n1+f−fδ\nfδ1−f/parenrightbigg\n. (S6)\nForanegligibledifferenceoftheFermifunctionvaluesfor thetwospins, fδ→0,thechargeandspindecaytotheirsteadystate\nvalues independently,with the rates Γ(1+f), andΓ(1−f), respectively. The steady states are also markedly differe ntin this\nlimit, asPo(t=∞) = 2f/(1+f)depends on the Fermi factors, while s(t=∞) = 0does not. This is then the reason for\ndifferenceinthedecayscales: whilethechargedecaystowa rdsthesteadystate witheffectivelythesumoftheratesfor leaving,\n(1−f)Γ,andentering, 2fΓ,thedot,onlytheeventsofelectronsleavingthedotscanre laxthespinpolarization s. Thespinand\nchargerelaxationscaleswill thenbemostdifferentif f≈1,wherethechargeequilibratesmuchfaster thanthespin.\nTodemonstratethisdifference,seenalsoexperimentally, weplotthechargeandspinsignalsinFig.S1(a)and(b),resp ectively.\nTheparametersare set as T= 200mK,B= 0.5T,g=−0.37[4]. Thetracesshowtheresultswith f= 0.2,0.4,0.6,0.8from\nthebottomto thetop.2\n1.0\n0.8\n0.6\n0.4Charge Signal\n543210(a)\nt(1/Γ)1.0\n0.9\n0.8\n0.7\n0.6Spin Signal\n543210(b)\nt(1/Γ)\nFIG.S1: (a) Calculatedcharge signal asa functionof t. Thetraces show the resultswith f= 0.2,0.4,0.6,0.8from thebottom tothe top. (b)\nCalculatedspinsignal as afunction of t.\nCO-TUNNELINGRATE\nTo derive the formula for the spin relaxation by cotunneling , which was used in the main text to fit the data on Fig. 4(a), we\nconsidertheHamiltonianofa QDcoupledtoa lead, H=HD+HL+HT. HerethedotHamiltonianis\nHD=/summationdisplay\nα∈{0,σ,S}ǫα|α∝an}bracketri}ht∝an}bracketle{tα|, (S7)\nwheretheindex αlabelsthestatesofthedot |α∝an}bracketri}htwithenergies ǫα,and|0∝an}bracketri}htdenotesanemptydot, |σ∝an}bracketri}ht=d†\nσ|0∝an}bracketri}htadotwithasingle\nelectronwith spin σ, and|S∝an}bracketri}ht=d†\n↑d†\n↓|0∝an}bracketri}hta dot with a two electronsinglet state, and d†\nσis the creation operatorof a dot electron\nwithspin σ. Theleadisdescribedby\nHL=/summationdisplay\nkσǫkσc†\nkσckσ, (S8)\nwherekisawave-vector(forsimplicity,weconsideraonedimensio nallead,sothat kisascalar). Finally,thedot-leadcoupling\nis\nHT=/summationdisplay\nkσtkσc†\nkσdσ+t∗\nkσd†\nσckσ, (S9)\nwhich desctribes a spin-preserving lead-dot tunneling wit h, in general complex and spin and energy dependent, tunneli ng am-\nplitudestkσ.\nWe now repeat the standard calculation [5–9] with minor adju stments to to arrive at the inelastic spin decay rather than t he\nco-tunnelingcurrent. Tothisend,wedefinethetransitionr ate bythe Fermi’sGoldenruleformula\nΓi→f=/summationdisplay\nileadpilead/summationdisplay\nflead2π\n/planckover2pi1|∝an}bracketle{tfdot⊗flead|G|idot⊗ilead∝an}bracketri}ht|2δ(Ei−Ef), (S10)\nwhereiandfaretheinitialandfinalstateswithenergies EiandEf,respectively,consideredtobeseparable(totheleadandd ot\ncomponents) eigenstates of the unperturbed system describ ed byH0=HD+HL. As we are not conditioning the transitions\non the states of the lead, the rate is summed over all possible initial lead states, with the correspondingprobabilities pilead, and\nall lead final states. The former gives the prescription for a replacement/summationtext\nileadpilead|ilead∝an}bracketri}ht∝an}bracketle{tilead| →ρthermal\nlead, with the latter\nthe equilibriumdensity matrixcorrespondingto a system wi th Hamiltonian HL, at a temperature T. Finally, Gis the transition\noperatorwhichcanbeexpandedinpowersofthetunnelingter m\nG=HT+HT1\nE−H0−iγHT+... (S11)\nwithE=Ei=Ef. The two terms describe, respectively, the direct tunnelin g and the co-tunneling, and γis a regularization\nfactor[10].3\nSimple results can be derived in the well justified case of a ne gligible dependence of the tunneling amplitudes on the wave\nvector,tkσ≈tσ. Using the first term in Eq. (S11) gives in this limit the follo wing expression for the direct tunneling rates\ndefinedin Eq.(S1)\nΓσ=2π\n/planckover2pi1|tσ|2gF, (S12)\nwhere we denoted Γσ≡Γσ→0= Γ0→σandgFis the density of states in the lead at the Fermi energy. Simil arly, keeping\nonlythe secondterminEq.(S11)givestheco-tunnelingrate fora spin-flip(from σto theoppositevalue σ)ofa singleelectron\noccupyingthedot,\nΓσ→σ=/planckover2pi1\n2πΓσΓσ/integraldisplay\ndǫfFD(ǫ+σEz)[1−fFD(ǫ+σEz)]/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nǫ−µ(2)+iγ−1\nǫ−µ(1)−iγ/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n, (S13)\nwhereµ(2) =ǫS−ǫ1isthe(spinindependentpartofthe)energycosttoaddaseco ndelectronintothedot. Theexpressioncan\nbe furthersimplified if the dot is deep in the Coulombblockad e,so that the chargeexcitation energiesare muchlarger tha n the\ntemperature,namely µ(1)≪µF≪µ(2)arewellfulfilledontheenergyscaleofthetemperature, kBT. Theenergydependence\nofthelast terminEq.(S13)canbe thenneglected,replacing ǫ→µF, andtheremainingintegralcanbeevaluatedresultingin\nΓσ→σ=/planckover2pi1\n2πΓσΓσ2σEz\nexp/parenleftBig\n2σEz\nkBT−1/parenrightBig/parenleftbigg1\nµ(2)−µF+1\nµF−µ(1)/parenrightbigg2\n, (S14)\nwherewealsoneglectedtheregularizationfactors. Inthel argetemperaturelimit, kBT≫Ez,thetemperaturedependentfactor\nbecomeskBT,whileintheoppositelimit, kBT≫Ez,itgives2Ezforσ=↓,and0forσ=↑. However,Eq.(S14)isalreadyin\ntheformwhichwasusedtofit the dataandisthusthefinalresul tofthissection.\n∗tomohiro.otsuka@riken.jp\n[S1] S. Amasha, K. MacLean, Iuliana P. Radu, D. M. Zumb¨ ul, M. A. Kastner, M. P. Hanson, and A. C. Gossard, Spin-dependent tunneling\nof single electrons intoan empty quantum dot , Phys.Rev. B 78, 041306(R) (2008).\n[S2] P.StanoandPh. Jacquod, Spin-dependent tunneling intoan empty lateral quantum dot , Phys.Rev. B 82, 125309 (2010).\n[S3] M.Yamagishi,N.Watase,M.Hashisaka,K.Muraki,andT. Fujisawa, Spin-dependent tunnelingratesforelectrostaticallydefi nedGaAs\nquantum dots , Phys. Rev. B 90, 035306 (2014).\n[S4] T. Otsuka, T. Nakajima, M. R. Delbecq, S. Amaha, J. Yoned a, K. Takeda, G. Allison, T. Ito, R. Sugawara, A. Noiri, A. Lud wig, A. D.\nWieckand S.Tarucha, Single-electronSpin Resonance inaQuadruple Quantum Dot , Sci.Rep. 6, 31820 (2016).\n[S5] D. V. Averin and Yu. V. Nazarov, Virtual electron diffusion during quantum tunneling of the electric charge , Phys. Rev. Lett. 65, 2446\n(1990).\n[S6] M. R.Wegewijs,and Yu.V. Nazarov, Inelastic co-tunneling through anexcitedstate of aquantu m dot, arXiv:cond-mat/0103579.\n[S7] V. N. Golovach and D. Loss, Transport through a double quantum dot in the sequential tun neling and cotunneling regimes , Phys. Rev.\nB69, 245327 (2004).\n[S8] J.Lehmann andD.Loss, Cotunneling current throughquantum dots withphonon-assi sted spin-flipprocesses , Phys.Rev. B73, 045328\n(2006)\n[S9] P. Stano, J. Klinovaja, F. R. Braakman, L. M. K. Vandersy pen, and D. Loss, Fast Long-Distance Control of Spin Qubits by Photon\nAssistedCotunneling , Phys.Rev. B 92, 075302 (2015).\n[S10] G. Begemann, S.Koller, M. Grifoni,andJ. Paaske, Inelastic cotunneling inquantum dots and molecules withwe akly broken degenera-\ncies, Phys.Rev. B 82, 045316 (2010)."    },    {        "title": "1012.3575v1.Spin_Relaxation_in_Quantum_Wires.pdf",        "content": "arXiv:1012.3575v1  [cond-mat.mes-hall]  16 Dec 2010Spin Relaxation in Quantum Wires\nP. Wenk∗\nSchool of Engineering and Science, Jacobs University Breme n, Bremen 28759, Germany\nS. Kettemann†\nSchool of Engineering and Science, Jacobs University Breme n, Bremen 28759,\nGermany, and Asia Pacific Center for Theoretical Physics and Division of Advanced\nMaterials Science Pohang University of Science and Technol ogy (POSTECH) San31,\nHyoja-dong, Nam-gu, Pohang 790-784, South Korea\nThe spin dynamics and spin relaxation of itinerant electron s in quantum wires with spin-orbit\ncoupling is reviewed. We give an introduction to spin dynami cs, and review spin-orbit coupling\nmechanisms in semiconductors. The spin diffusion equation w ith spin-orbit coupling is derived,\nusing only intuitive, classical random walk arguments. We g ive an overview of all spin relaxation\nmechanisms, with particular emphasis on the motional narro wing mechanism in disordered conduc-\ntors, the D’yakonov-Perel’-Spin relaxation (DPS). Here, w e discuss in particular, the existence of\npersistent spin helix solutions of the spin diffusion equati on, with vanishing spin relaxation rates.\nWe then, derive solutions of the spin diffusion equation in qu antum wires, and show that there is an\neffective alignment of the spin-orbit field in wires whose wid th is smaller than the spin precession\nlengthLSO. We show that the resulting reduction in the spin relaxation rate results in a change\nin the sign of the quantum corrections to the conductivity. F inally, we present recent experimental\nresults which confirm the decrease of the spin relaxation rat e in wires whose width is smaller than\nLSO: the direct optical measurement of the spin relaxation rate , as well as transport measurements,\nwhich show a dimensional crossover from weak antilocalizat ion to weak localization as the wire\nwidth is reduced. Open problems remain, in particular in nar rower, ballistic wires, were optical and\ntransport measurements seem to find opposite behavior of the spin relaxation rate: enhancement,\nsuppression, respectively. We conclude with a review of the se and other open problems which still\nchallenge the theoretical understanding and modeling of th e experimental results.2\nContents\nI. Introduction 3\nII. Spin Dynamics 3\nA. Dynamics of a Localized Spin 3\nB. Spin Dynamics of Itinerant Electrons 4\n1. Ballistic Spin Dynamics 4\n2. Spin Diffusion Equation 4\n3. Spin-Orbit Interaction in Semiconductors 5\n4. Spin Diffusion in the Presence of Spin-Orbit Interaction 7\nIII. Spin Relaxation Mechanisms 9\nA. D’yakonov-Perel’ Spin Relaxation 10\nB. DP Spin Relaxation with Electron-Electron and Electron- Phonon Scattering 11\nC. Elliott-Yafet Spin Relaxation 12\nD. Spin Relaxation due to Spin-Orbit Interaction with Impur ities 12\nE. Bir-Aronov-Pikus Spin Relaxation 13\nF. Magnetic Impurities 13\nG. Nuclear Spins 13\nH. Magnetic Field Dependence of Spin Relaxation 13\nI. Dimensional Reduction of Spin Relaxation 14\nIV. Spin-Dynamics in Quantum Wires 14\nA. One-Dimensional Wires 14\nB. Spin-Diffusion in Quantum Wires 14\nC. Weak Localization Corrections 17\nV. Experimental Results on Spin Relaxation Rate in Semicond uctor Quantum Wires 19\nA. Optical Measurements 19\nB. Transport Measurements 20\nVI. Critical Discussion and Future Perspective 20\nVII. Summary 21\nSymbols 22\nAcknowledgments 22\nReferences 223\nI. INTRODUCTION\nThe emerging technology of spintronics intends to use the ma nipulation of the spin degree of freedom of individual\nelectrons for energy efficient storage and transport of infor mation.1In contrast to classical electronics, which relies on\nthe steering of charge carriers through semiconductors, sp intronics uses the spin carried by electrons, resembling ti ny\nspinning tops. The difference to a classical top is that its an gular momentum is quantized, it can only take two discrete\nvalues, up or down. To control the spin of electrons, a detail ed understanding of the interaction between the spin and\norbital degrees of freedom of electrons and other mechanism s which do not conserve its spin, is necessary. These are\ntypically weak perturbations, compared to the kinetic ener gy of conduction electrons, so that their spin relaxes slowl y\nto the advantage of spintronic applications. The relaxatio n, or depolarization of the electron spin can occur due to\nthe randomization of the electron momentum by scattering fr om impurities, and dislocations in the material, and due\nto scattering with elementary excitations of the solid such as phonons and other electrons, when it is transferred to\nthe randomization of the electron spin due to the spin-orbit interaction. In addition, scattering from localized spins ,\nsuch as nuclear spins and magnetic impurities are sources of electron spin relaxation. The electron spin relaxation can\nbe reduced by constraining the electrons in low dimensional structures, quantum wells (confined in one direction, free\nin two dimensions), quantum wires ( confined in two direction s, free in one direction), or quantum dots ( confined in\nall three directions). Although spin relaxation is typical ly smallest in quantum dots due to their discrete energy leve l\nspectrum, the necessity to transfer the spin in spintronic d evices, recently lead to intense research efforts to reduce t he\nspin relaxation in quantum wires, where the energy spectrum is continuous. In the following we will review the theory\nof spin dynamics and relaxation in quantum wires, and compar e it with recent experimental results. After a general\nintroduction to spin dynamics in Section II, we discuss all relevant spin relaxation mechanisms and how they depend\non dimension, temperature, mobility, charge carrier densi ty and magnetic field in Section III. In particular, we review\nrecent results on spin relaxation in semiconducting quantu m wires, and its influence on the quantum corrections to\ntheir conductance in Section IV. These weak localization corrections are thereby a very sen sitive measure of spin\nrelaxation in quantum wires, in addition to optical methods as we review in Section V. We set /planckover2pi1= 1in the following.\nII. SPIN DYNAMICS\nBefore we review the spin dynamics of conduction electrons a nd holes in semiconductors and metals, let us first\nreconsider the spin dynamics of a localized spin, as governe d by the Bloch equations.\nA. Dynamics of a Localized Spin\nA localized spin ˆ s, like a nuclear spin, or the spin of a magnetic impurity in a so lid, precesses in an external\nmagnetic field Bdue to the Zeeman interaction with Hamiltonian HZ=−γgˆ sB, whereγgis the corresponding\ngyromagnetic ratio of the nuclear spin or magnetic impurity spin, respectively, which we will set equal to one, unless\nneeded explicitly. This spin dynamics is governed by the Blo ch equation of a localized spin,\n∂tˆ s=γgˆ s×B. (1)\nThis equation is identical to the Heisenberg equation ∂tˆ s=−i[ˆ s,HZ]for the quantum mechanical spin operator ˆ sof\nanS= 1/2-spin, interacting with the external magnetic field Bdue to the Zeeman interaction with Hamiltonian HZ.\nThe solution of the Bloch equation for a magnetic field pointi ng in the z-direction is ˆsz(t) = ˆsz(0), while the x- and y-\ncomponents of the spin are precessing with frequency ω0=γgBaround the z-axis, ˆsx(t) = ˆsx(0)cosω0t+ˆsy(0)sinω0t,\nˆsy(t) =−ˆsx(0)sinω0t+ ˆsy(0)cosω0t. Since a localized spin interacts with its environment by ex change interaction\nand magnetic dipole interaction, the precession will depha se after a time τ2, and the z-component of the spin relaxes\nto its equilibrium value sz0within a relaxation time τ1. This modifies the Bloch equations to the phenomenological\nequations,\n∂tˆsx=γg(ˆsyBz−ˆszBy)−1\nτ2ˆsx\n∂tˆsy=γg(ˆszBx−ˆsxBz)−1\nτ2ˆsy\n∂tˆsz=γg(ˆsxBy−ˆsyBx)−1\nτ1(ˆsz−sz0). (2)4\nB. Spin Dynamics of Itinerant Electrons\n1. Ballistic Spin Dynamics\nThe intrinsic degree of freedom spin is a direct consequence of the Lorentz invariant formulation of quantum\nmechanics. Expanding the relativistic Dirac equation in th e ratio of the electron velocity and the constant velocity\nof lightc, one obtains in addition to the Zeeman term, a term which coup les the spin swith the momentum pof the\nelectrons, the spin-orbit coupling\nHSO=−µB\n2mc2ˆ s p×E=−ˆ sBSO(p), (3)\nwhere we set the gyromagnetic ratio γg= 1.E=−∇V, is an electrical field, and BSO(p) =µB/(2mc2)p×E.\nSubstitution into the Heisenberg equation yields the Bloch equation in the presence of spin-orbit interaction:\n∂tˆ s=ˆ s×BSO(p), (4)\nso that the spin performs a precession around the momentum de pendent spin-orbit field BSO(p). It is important to\nnote, that the spin-orbit field does not break the invariance under time reversal ( ˆ s→ −ˆ s,p→ −p), in contrast to\nan external magnetic field B. Therefore, averaging over all directions of momentum, the re is no spin polarization of\nthe conduction electrons. However, injecting a spin-polar ized electron with given momentum pinto a translationally\ninvariant wire, its spin precesses in the spin-orbit field as the electron moves through the wire. The spin will be\noriented again in the initial direction after it moved a leng thLSO, the spin precession length. The precise magnitude\nofLSOdoes not only depend on the strength of the spin-orbit intera ction but may also depend on the direction of its\nmovement in the crystal, as we will discuss below.\n2. Spin Diffusion Equation\nTranslational invariance is broken by the presence of disor der due to impurities and lattice imperfections in the\nconductor. As the electrons scatter from the disorder poten tial elastically, their momentum changes in a stochastic\nway, resulting in diffusive motion. That results in a change o f the the local electron density ρ(r,t) =/summationtext\nα=±|ψα(r,t)|2,\nwhereα=±denotes the orientation of the electron spin, and ψα(r,t)is the position and time dependent electron\nwave function amplitude. On length scales exceeding the ela stic mean free path le, that density is governed by the\ndiffusion equation\n∂ρ\n∂t=De∇2ρ, (5)\nwhere the diffusion constant Deis related to the elastic scattering time τbyDe=v2\nFτ/dD, wherevFis the Fermi\nvelocity, and dDthe Diffusion dimension of the electron system. That diffusio n constant is related to the mobility\nof the electrons, µe=eτ/m∗by the Einstein relation µeρ=e2νDe, whereνis the density of states per spin\nat the Fermi energy EF. Injecting an electron at position r0into a conductor with previously constant electron\ndensityρ0, the solution of the diffusion equation yields that the elect ron density spreads in space according to\nρ(r,t) =ρ0+exp(−(r−r0)2/4Det)/(4πDet)dD/2, wheredDis the dimension of diffusion. That dimension is equal to\nthe kinetic dimension d,dD=d, if the elastic mean free path leis smaller than the size of the sample in all directions.\nIf the elastic mean free path is larger than the sample size in one direction the diffusion dimension reduces by one,\naccordingly. Thus, on average the variance of the distance t he electron moves after time tis/angb∇acketleft(r−r0)2/angb∇acket∇ight= 2dDDet. This\nintroduces a new length scale, the diffusion length LD(t) =√Det. We can rewrite the density as ρ=/angb∇acketleftψ†(r,t)ψ(r,t)/angb∇acket∇ight,\nwhereψ†= (ψ†\n+,ψ†\n−)is the two-component vector of the up (+), and down (-) spin fe rmionic creation operators, and\nψthe 2-component vector of annihilation operators, respect ively,/angb∇acketleft.../angb∇acket∇ightdenotes the expectation value. Accordingly,\nthe spin density s(r,t)is expected to satisfy a diffusion equation, as well. The spin density is defined by\ns(r,t) =1\n2/angb∇acketleftψ†(r,t)σψ(r,t)/angb∇acket∇ight, (6)\nwhereσis the vector of Pauli matrices,\nσx=/parenleftbigg\n0 1\n1 0/parenrightbigg\n,σy=/parenleftbigg\n0−i\ni0/parenrightbigg\n,andσz=/parenleftbigg\n1 0\n0−1/parenrightbigg\n.5\nThus the z-component of the spin density is half the differenc e between the density of spin up and down electrons,\nsz= (ρ+−ρ−)/2, which is the local spin polarization of the electron system . Thus, we can directly infer the diffusion\nequation for sz, and, similarly, for the other components of the spin densit y, yielding, without magnetic field and\nspin-orbit interaction,2\n∂s\n∂t=De∇2s−s\nˆτs. (7)\nHere, in the spin relaxation term we introduced the tensor ˆτs, which can have non-diagonal matrix elements. In the\ncase of a diagonal matrix, τsxx=τsyy=τ2, is the spin dephasing time, and τszz=τ1the spin relaxation time.\nThe spin diffusion equation can be written as a continuity equ ation for the spin density vector, by defining the spin\ndiffusion current of the spin components si,\nJsi=−De∇si. (8)\nThus, we get the continuity equation for the spin density com ponentssi,\n∂si\n∂t+∇Jsi=−/summationdisplay\njsj\nτsij. (9)\n3. Spin-Orbit Interaction in Semiconductors\nWhile silicon and germanium have in their diamond structure an inversion symmetry around every midpoint on each\nline connecting nearest neighbor atoms, this is not the case for III-V-semiconductors like GaAs, InAs, InSb, or ZnS.\nThese have a zinc-blende structure which can be obtained fro m a diamond structure with neighbored sites occupied by\nthe two different elements. Therefore the inversion symmetr y is broken, which results in spin-orbit coupling. Similarl y,\nthat symmetry is broken in II-VI-semiconductors. This bulk inversion asymmetry (BIA) coupling, or often so called\nDresselhaus-coupling, is anisotropic, as given by3\nHD=γD/bracketleftbig\nσxkx(k2\ny−k2\nz)+σyky(k2\nz−k2\nx)+σzkz(k2\nx−k2\ny)/bracketrightbig\n, (10)\nwhereγDis the Dresselhaus-spin-orbit coefficient. Confinement in qu antum wells with width aon the order of the\nFermi wave length λFyields accordingly a spin-orbit interaction where the mome ntum in growth direction is of the\norder of 1/a. Because of the anisotropy of the Dresselhaus term, the spin -orbit interaction depends strongly on the\ngrowth direction of the quantum well. Grown in [001]direction, one gets, taking the expectation value of Eq. ( 10) in\nthe direction normal to the plane, noting that /angb∇acketleftkz/angb∇acket∇ight=/angb∇acketleftk3\nz/angb∇acket∇ight= 0,3\nHD[001]=α1(−σxkx+σyky)+γD(σxkxk2\ny−σykyk2\nx). (11)\nwhereα1=γD/angb∇acketleftk2\nz/angb∇acket∇ightis the linear Dresselhaus parameter. Thus, inserting an ele ctron with momentum along the x-\ndirection, with its spin initially polarized in z-directio n, it will precess around the x-axis as it moves along. For nar row\nquantum wells, where /angb∇acketleftk2\nz/angb∇acket∇ight ∼1/a2≥k2\nFthe linear term exceeds the cubic Dresselhaus terms. A speci al situation\narises for quantum wells grown in the [110]- direction, where it turns out that the spin-orbit field is po inting normal\nto the quantum well, as shown in Fig. 1, so that an electron whose spin is initially polarized along the normal of the\nplane, remains polarized as it moves in the quantum well. In q uantum wells with asymmetric electrical confinement\nthe inversion symmetry is broken as well. This structural in version asymmetry (SIA) can be deliberately modified\nby changing the confinement potential by application of a gat e voltage. The resulting spin-orbit coupling, the SIA\ncoupling, also called Rashba-spin-orbit interaction4is given by\nHR=α2(σxky−σykx), (12)\nwhereα2depends on the asymmetry of the confinement potential V(z)in the direction z, the growth direction of the\nquantum well, and can thus be deliberately changed by applic ation of a gate potential. At first sight it looks as if the\nexpectation value of the electrical field Ec=−∂zV(z)in the conduction band state vanishes, since the ground stat e of\nthe quantum well must be symmetric in z. Taking into account the coupling to the valence band,5,6the discontinuities\nin the effective mass,7and corrections due to the coupling to odd excited states,8yields a sizable coupling parameter\ndepending on the asymmetry of the confinement potential6,9.\nThis dependence allows one, in principle, to control the ele ctron spin with a gate potential, which can therefore be\nused as the basis of a spin transistor.106\nSIA\n00 BIA/LBracket1111/RBracket1\n00\nBIA/LBracket1001/RBracket1\n00\n/LBracket1100/RBracket1/LBracket1010/RBracket1\nBIA/LBracket1110/RBracket1\n0\n/LBracket11/OverBar/OverBar10/RBracket10/LBracket1001/RBracket10/LBracket1110/RBracket1\nFigure 1: The spin-orbit vector fields for linear structure i nversion asymmetry (Rashba) coupling, and for linear bulk i nversion\nasymmetry (BIA) spin orbit coupling for quantum wells grown in [111], [001] and [110] direction, respectively.\nWe can combine all spin-orbit couplings by introducing the s pin-orbit field such that the Hamiltonian has the form\nof a Zeeman term:\nHSO=−sBSO(k), (13)\nwhere the spin vector is s=σ/2. But we stress again that since BSO(k)→BSO(−k) =−BSO(k)under the time\nreversal operation, spin-orbit coupling does not break tim e reversal symmetry, since the time reversal operation also\nchanges the sign of the spin, s→ −s. Only an external magnetic field Bbreaks the time reversal symmetry. Thus,\nthe electron spin operator ˆ sis for fixed electron momentum kgoverned by the Bloch equations with the spin-orbit\nfield,\n∂ˆ s\n∂t=ˆ s×(B+BSO(k))−1\nˆτsˆ s. (14)\nThe spin relaxation tensor is no longer necessarily diagona l in the presence of spin-orbit interaction.\nIn narrow quantum wells where the cubic Dresselhaus couplin g is weak compared to the linear Dresselhaus and Rashba\ncouplings, the spin-orbit field is given by\nBSO(k) =−2\n−α1kx+α2ky\nα1ky−α2kx\n0\n, (15)\nwhich changes both its direction and its amplitude |BSO(k)|= 2/radicalbig\n(α2\n1+α2\n2)k2−4α1α2kxky, as the direction of\nthe momentum kis changed. Accordingly, the electron energy dispersion cl ose to the Fermi energy is in general7\nanisotropic as given by\nE±=1\n2m∗k2±αk/radicalbigg\n1−4α1α2\nα2cosθsinθ, (16)\nwherek=|k|,α=/radicalbig\nα2\n1+α2\n2, andkx=kcosθ. Thus, when an electron is injected with energy E, with momentum\nkalong the [100]-direction, kx=k,ky= 0, its wave function is a superposition of plain waves with the positive\nmomentak±=∓αm∗+m∗(α2+2E/m∗)1/2. The momentum difference k−−k+= 2m∗αcauses a rotation of the\nelectron eigenstate in the spin subspace. When at x= 0the electron spin was polarized up spin, with the Eigenvecto r\nψ(x= 0) =/parenleftbigg\n1\n0/parenrightbigg\n,\nthen, when its momentum points in x-direction, at a distance x, it will have rotated the spin as described by the\nEigenvector\nψ(x) =1\n2/parenleftbigg1\nα1+iα2\nα/parenrightbigg\neik+x+1\n2/parenleftbigg1\n−α1+iα2\nα/parenrightbigg\neik−x. (17)\nIn Fig. 2we plot the corresponding spin density as defined in Eq. ( 6) for pure Rashba coupling, α1= 0. The spin\n0 LSO/Slash12 LSO/Minus101\nxsz\nFigure 2: Precession of a spin injected at x= 0, polarized in z-direction, as it moves by one spin precessio n length LSO=π/m∗α\nthrough the wire with linear Rashba spin orbit coupling α2.\nwill point again in the initial direction, when the phase diff erence between the two plain waves is 2π, which gives the\ncondition for spin precession length as 2π= (k−−k+)LSO, yielding for linear Rashba and Dresselhaus coupling, and\nthe electron moving in [100]- direction,\nLSO=π/m∗α. (18)\nWe note that the period of spin precession changes with the di rection of the electron momentum since the spin-orbit\nfield, Eq. ( 15), is anisotropic.\n4. Spin Diffusion in the Presence of Spin-Orbit Interaction\nAs the electrons are scattered by imperfections like impuri ties and dislocations, their momentum is changed ran-\ndomly. Accordingly, the direction of the spin-orbit field BSO(k)changes randomly as the electron moves through the\nsample. This has two consequences: the electron spin direct ion becomes randomized, dephasing the spin precession\nand relaxing the spin polarization. In addition, the spin pr ecession term is modified, as the momentum kchanges\nrandomly, and has no longer the form given in the ballistic Bl och-like equation, Eq. ( 14). One can derive the diffu-\nsion equation for the expectation value of the spin, the spin density Eq. ( 6) semiclassically,11,12or by diagrammatic8\nexpansion.13In order to get a better understanding on the meaning of this e quation, we will give a simplified classical\nderivation, in the following. The spin density at time t+∆tcan be related to the one at the earlier time t. Note that\nfor ballistic times ∆t≤τ, the distance the electron has moved with a probability p∆x,∆x, is related to that time\nby the ballistic equation, ∆x=k(t)∆t/mwhen the electron moves with the momentum k(t). On this time scale the\nspin evolution is still governed by the ballistic Bloch equa tion Eq. ( 14). Thus, we can relate the spin density at the\nposition xat the time t+∆t, to the one at the earlier time tat position x−∆x:\ns(x,t+∆t) =/summationdisplay\n∆xp∆x/parenleftbigg/parenleftbigg\n1−1\nˆτs∆t/parenrightbigg\ns(x−∆x,t)−∆t[B+BSO(k(t))]×s(x−∆x,t)/parenrightbigg\n. (19)\nNow, we can expand in ∆tto first order and in ∆xto second order. Next, we average over the disorder potentia l,\nassuming that the electrons are scattered isotropically, a nd substitute/summationtext\n∆xp∆x...=/integraltext\n(dΩ/Ω)...whereΩis the total\nangle, and/integraltextdΩdenotes the integral over all angles with/integraltext(dΩ/Ω) = 1 . Also, we get (s(x,t+∆t)−s(x,t))/∆t→\n∂ts(x,t)for∆t→0, and/angb∇acketleft∆x2\ni/angb∇acket∇ight= 2De∆t, whereDeis the diffusion constant. While the disorder average yields\n/angb∇acketleft∆x/angb∇acket∇ight= 0, and/angb∇acketleftBSO(k(t))/angb∇acket∇ight= 0, separately, for isotropic impurity scattering, averagin g their product yields a finite\nvalue, since ∆xdepends on the momentum at time t,k(t), yielding /angb∇acketleft∆xBSOi(k(t))/angb∇acket∇ight= 2∆t/angb∇acketleftvFBSOi(k(t))/angb∇acket∇ight, where\n/angb∇acketleft.../angb∇acket∇ightdenotes the average over the Fermi surface. This way, we can a lso evaluate the average of the spin-orbit term in\nEq. (19), expanded to first order in ∆x, and get, substituting ∆t→τthe spin diffusion equation,\n∂s\n∂t=−B×s+De∇2s+2τ/angb∇acketleft(∇vF)BSO(p)/angb∇acket∇ight×s−1\nˆτss, (20)\nwhere/angb∇acketleft.../angb∇acket∇ightdenotes the average over the Fermi surface. Spin polarized e lectrons injected into the sample spread\ndiffusively, and their spin polarization, while spreading d iffusively as well, decays in amplitude exponentially in tim e.\nSince, between scattering events the spins precess around t he spin-orbit fields, one expects also an oscillation of the\npolarization amplitude in space. One can find the spatial dis tribution of the spin density which is the solution of\nEq. (20) with the smallest decay rate Γs. As an example, the solution for linear Rashba coupling is,12\ns(x,t) = (ˆeqcosqx+Aˆezsinqx)e−t/τs, (21)\nwith1/τs= 7/16τs0where1/τs0= 2τk2\nFα2\n2and where the amplitude of the momentum qis determined by Deq2=\n15/16τs0, andA= 3/√\n15, andˆeq=q/q. This solution is plotted in Fig. 3forˆeq= (1,1,0)/√\n2. In Fig. 4we plot the\nlinearly independent solution obtained by interchanging cosandsinin Eq. ( 21), with the spin pointing in z-direction,\ninitially. We choose ˆeq= ˆex. Comparison with the ballistic precession of the spin, Fig 4shows that the period\nof precession is enhanced by the factor 4/√\n15in the diffusive wire, and that the amplitude of the spin densi ty is\nmodulated, changing from 1toA= 3/√\n15.\n0\n/LParen115/Slash14/RParen1LSO/Slash12\n/LParen115/Slash14/RParen1LSOx/Minus0.500.5Sy\n/Minus0.500.5\nSz\nFigure 3: The spin density for linear Rashba coupling which i s a solution of the spin diffusion equation with the relaxatio n\nrate7/16τs. The spin points initially in the x−y-plane in the direction (1,1,0).9\n0 Lso/Slash12 Lso/Minus/FractionBarExt1\n20/FractionBarExt1\n2\nx/LessS/Greaterz\n0.770.891/VertBar1S/VertBar1\nFigure 4: The spin density for linear Rashba coupling which i s a solution of the spin diffusion equation with the relaxatio n rate\n1/τs= 7/16τs0. Note that, compared to the ballistic spin density, Fig. 2, the period is slightly enhanced by a factor 4/√\n15.\nAlso, the amplitude of the spin density changes with the posi tionx, in contrast to the ballistic case. The color is changing in\nproportion to the spin density amplitude.\nInjecting a spin-polarized electron at one point, say x= 0, its density spreads the same way it does without spin-\norbit interaction, ρ(r,t) = exp(−r2/4Det)/(4πDet)dD/2, whereris the distance to the injection point. However, the\ndecay of the spin density is periodically modulated as a func tion of2π/radicalbig\n15/16r/LSO.14The spin-orbit interaction\ntogether with the scattering from impurities is also a sourc e of spin relaxation, as we discuss in the next Section\ntogether with other mechanisms of spin relaxation. We can fin d the classical spin diffusion current in the presence of\nspin-orbit interaction, in a similar way as one can derive th e classical diffusion current: The current at the position\nris a sum over all currents in its vicinity which are directed t owards that position. Thus, j(r,t) =/angb∇acketleftvρ(r−∆x)/angb∇acket∇ight\nwhere an angular average over all possible directions of the velocity vis taken. Expanding in ∆x=lev/v, and\nnoting that /angb∇acketleftvρ(r)/angb∇acket∇ight= 0, one gets j(r,t) =/angb∇acketleftv(−∆x)∇ρ(r)/angb∇acket∇ight=−(vFle/2)∇ρ(r) =−De∇ρ(r). For the classical spin\ndiffusion current of spin component Si, as defined by jSi(r,t) =vSi(r,t), there is the complication that the spin\nkeeps precessing as it moves from r−∆xtor, and that the spin-orbit field changes its direction with the direction\nof the electron velocity v. Therefore, the 0-th order term in the expansion in ∆xdoes not vanish, rather, we get\njSi(r,t) =/angb∇acketleftvSk\ni(r,t)/angb∇acket∇ight −De∇Si(r,t), whereSk\niis the part of the spin density which evolved from the spin den sity\natr−∆xmoving with velocity vand momentum k. Noting that the spin precession on ballistic scales t≤τis\ngoverned by the Bloch equation, Eq. ( 14), we find by integration of Eq. ( 14), thatSk\ni=−τ(BSO(k)×S)iso that we\ncan rewrite the first term yielding the total spin diffusion cu rrent as\njSi=−τ/angb∇acketleftvF(BSO(k)×S)i/angb∇acket∇ight−De∇Si. (22)\nThus, we can rewrite the spin diffusion equation in terms of th is spin diffusion current and get the continuity equation\n∂si\n∂t=−De∇jSi+τ/angb∇acketleft∇vF(BSO(k)×S)i/angb∇acket∇ight−1\nˆτsijsj. (23)\nIt is important to note that in contrast to the continuity equ ation for the density, there are two additional terms, due\nto the spin orbit interaction. The last one is the spin relaxa tion tensor which will be considered in detail in the next\nsection. The other term arises due to the fact that Eq. ( 20) contains a factor 2in front of the spin-orbit precession\nterm, while the spin diffusion current Eq. ( 22) does not contain that factor. This has important physical c onsequences,\nresulting in the suppression of the spin relaxation rate in q uantum wires and quantum dots as soon as their lateral\nextension is smaller than the spin precession length LSO, as we will see in the subsequent Sections.\nIII. SPIN RELAXATION MECHANISMS\nThe intrinsic spin-orbit interaction itself causes the spi n of the electrons to precess coherently, as the electrons\nmove through a conductor, defining the spin precession lengt hLSO, Eq. ( 18). Since impurities and dislocations in\nthe conductor randomize the electron momentum, the impurit y scattering is transferred into a randomization of the10\nelectron spin by the spin-orbit interaction, which thereby results in spin dephasing and spin relaxation. This results\nin a new length scale, the spin relaxation length, Ls, which is related to the spin relaxation rate 1/τsby\nLs=/radicalbig\nDeτs. (24)\nA. D’yakonov-Perel’ Spin Relaxation\nD’yakonov-Perel’ spin relaxation (DPS) can be understood q ualitatively in the following way: The spin-orbit field\nBSO(k)changes its direction randomly after each elastic scatteri ng event from an impurity, that is, after a time of\nthe order of the elastic scattering time τ, when the momentum is changed randomly as sketched in Fig. 5. Thus, the\nFigure 5: Elastic scattering from impurities changes the di rection of the spin-orbit field around which the electron spi n is\nprecessing.\nspin has the time τto perform a precession around the present direction of the s pin-orbit field, and can thus change\nits direction only by an angle of the order of BSOτby precession. After a time twithNt=t/τscattering events,\nthe direction of the spin will therefore have changed by an an gle of the order of |BSO|τ√Nt=|BSO|√\nτt. Defining\nthe spin relaxation time τsas the time by which the spin direction has changed by an angle of order one, we thus\nfind that 1/τs∼τ/angb∇acketleftBSO(k)2/angb∇acket∇ight, where the angular brackets denote integration over all ang les. Remarkably, this spin\nrelaxation rate becomes smaller, the more scattering event s take place, because the smaller the elastic scattering tim e\nτis, the less time the spin has to change its direction by prece ssion. Such a behavior is also well known as motional ,\nordynamic narrowing of magnetic resonance lines.15A more rigorous derivation for the kinetic equation of the sp in\ndensity matrix yields additional interference terms, not t aken into account in the above argument. It can be obtained\nby iterating the expansion of the spin density Eq. ( 19) once in the spin precession term, which yields the term\n/angbracketleftBigg\ns(x,t)×/integraldisplay∆t\n0dt′BSO(k(t′))×/integraldisplay∆t\n0dt′′BSO(k(t′′))/angbracketrightBigg\n, (25)\nwhere/angb∇acketleft.../angb∇acket∇ightdenotes the average over all angles due to the scattering fro m impurities. Since the electrons move\nballistically at times smaller than the elastic scattering time, the momenta are correlated only on time scales smaller\nthanτ, yielding /angb∇acketleftki(t′)kj(t′′)/angb∇acket∇ight= (1/2)k2δijτδ(t′−t′′).\nNoting that (A×B×C)m=ǫijkǫklmAiBjCland/summationtextǫijkǫklm=δilδjm−δimδjlwe find that Eq. ( 25) simplifies to\n−/summationtext\ni(1/τsij)Sj, where the matrix elements of the spin relaxation terms are g iven by,16\n1\nτsij=τ/parenleftbig\n/angb∇acketleftBSO(k)2/angb∇acket∇ightδij−/angb∇acketleftBSO(k)iBSO(k)j/angb∇acket∇ight/parenrightbig\n, (26)\nwhere/angb∇acketleft.../angb∇acket∇ightdenotes the average over the direction of the momentum k. These non-diagonal terms can diminish the\nspin relaxation and even result in vanishing spin relaxatio n. As an example, we consider a quantum well where the\nlinear Dresselhaus coupling for quantum wells grown in [001]direction, Eq. ( 11), and linear Rashba-coupling, Eq. ( 12),\nare the dominant spin-orbit couplings. The energy dispersi on is anisotropic, as given by Eq. ( 16), and the spin-orbit\nfieldBSO(k)changes its direction and its amplitude with the direction o f the momentum k:\nBSO(k) =−2\n−α1kx+α2ky\nα1ky−α2kx\n0\n, (27)11\nwith|BSO(k)|= 2/radicalbig\n(α2\n1+α2\n2)k2−4α1α2kxky. Thus we find the spin relaxation tensor as,\n1\nˆτs(k) = 4τk2\n1\n2α2−α1α20\n−α1α21\n2α20\n0 0 α2\n. (28)\nDiagonalizing this matrix, one finds the three eigenvalues (1/τs)(α1±α2)2/α2and2/τswhereα2=α2\n1+α2\n2, and\n1/τs= 2k2τα2. Note, that one of these eigenvalues of the spin relaxation t ensor vanishes when α1=α2=α0. In\nfact, this is a special case, when the spin-orbit field does no t change its direction with the momentum:\nBSO(k)|α1=α2=α0=2α0(kx−ky)\n1\n1\n0\n. (29)\nIn this case the constant spin density given by\nS=S0\n1\n1\n0\n, (30)\ndoes not decay in time, since the spin density vector is paral lel to the spin orbit field BSO(k), Eq. ( 29), and cannot\nprecess, as has been noted in Ref. [ 17]. It turns out, however, that there are two more modes which d o not decay in\ntime, whose spin relaxation rate vanishes for α1=α2. These modes are not homogeneous in space, and correspond to\nprecessing spin densities. They were found previously in a n umerical Monte Carlo simulation and found not to decay in\ntime, being called therefore persistent spin helix .18,19Recently, a long living inhomogeneous spin density distrib ution\nhas been detected experimentally in Ref. [ 20]. We can now get these persistent spin helix modes analytically, by solving\nthe full spin diffusion equation Eq. ( 20) with the spin relaxation tensor given by Eq. ( 28). We can diagonalize that\nequation, noting that its eigenfunctions are plain waves S(x)∼exp(iQx−Et). Thereby one finds, first of all, the\nmode with Eigenvalue E1=DeQ2, with the spin density\nS=S0\n1\n1\n0\nexp(iQx−DeQ2t). (31)\nIndeed for Q= 0, the homogeneous solution, it does not decay in time, in agre ement with the solution we found\nabove, Eq. ( 31). There are, however, two more modes with the eigenvalues\nE±=1\nτs(˜Q2+2±2|˜Qx−˜Qy|), (32)\nwhere˜Q=LSOQ/2π. At˜Qx=−˜Qy=±1, these modes do not decay in time. These two stationary solut ions, are\nS=S0\n1\n−1\n0\nsin/parenleftbigg2π\nLSO(x−y)/parenrightbigg\n+S0√\n2\n0\n0\n1\ncos/parenleftbigg2π\nLSO(x−y)/parenrightbigg\n, (33)\nand the linearly independent solution, obtained by interch angingcosandsinin Eq. ( 33). The spin precesses as the\nelectrons diffuse along the quantum wire with the period LSO, the spin precession length, forming a persistent spin\nhelix, as shown in Fig. 6.\nB. DP Spin Relaxation with Electron-Electron and Electron- Phonon Scattering\nIt has been noted, that the momentum scattering which limits the D’yakonov-Perel’ mechanism of spin relaxation\nis not restricted to impurity scattering, but can also be due to electron-phonon or electron-electron interactions.21–24\nThus the scattering time, τis the total scattering time as defined by,21,221/τ= 1/τ0+1/τee+1/τep, where1/τ0is the\nelastic scattering rate due to scattering from impurities w ith potential V, given by 1/τ0= 2πνni/integraltext\n(dθ/2π)(1−cosθ)|\nV(k,k′)|2, whereνis the density of states per spin at the Fermi energy, niis the concentration of impurities with\npotentialV, andkk′=kk′cos(θ). In degenerate semiconductors and in metals, the electron- electron scattering rate\nis given by the Fermi liquid inelastic electron scattering r ate1/τee∼T2/ǫF. The electron-phonon scattering time\n1/τep∼T5decays faster with temperature. Thus, at low temperatures t he DP spin relaxation is dominated by elastic\nimpurity scattering τ0.12\n0\nLSO/Slash12\nLSOx\n/MinusS00S0\n/LessS/Greatery/Minus/Radical1/Radical1Extens/Radical1Extens/Radical1Extens/Radical1Extens/Radical1Extens2S00/Radical1/Radical1Extens/Radical1Extens/Radical1Extens/Radical1Extens/Radical1Extens2S0\n/LessS/Greaterz\nFigure 6: Persistent spin helix solution of the spin diffusio n equation for equal magnitude of linear Rashba and linear Dr esselhaus\ncoupling, Eq.( 33).\nC. Elliott-Yafet Spin Relaxation\nBecause of the spin-orbit interaction the conduction elect ron wave functions are not Eigenstates of the electron\nspin, but have an admixture of both spin up and spin down wave f unctions. Thus, a nonmagnetic impurity potential\nVcan change the electron spin, by changing their momentum due to the spin-orbit coupling. This results in another\nsource of spin relaxation which is stronger, the more often t he electrons are scattered, and is thus proportional to the\nmomentum scattering rate 1/τ.25,26For degenerate III-V semiconductors one finds27,28\n1\nτs∼∆2\nSO\n(EG+∆SO)2E2\nk\nE2\nG1\nτ(k), (34)\nwhereEGis the gap between the valence and the conduction band of the s emiconductor, Ekthe energy of the\nconduction electron, and ∆SOis the spin-orbit splitting of the valence band. Thus, the El liott-Yafet spin relaxation\n(EYS) can be distinguished, being proportional to 1/τ, and thereby to the resistivity, in contrast to the DP spin\nscattering rate, Eq. ( 26), which is proportional to the conductivity. Since the EYS d ecays in proportion to the inverse\nof the band gap, it is negligible in large band gap semiconduc tors likeSiandGaAs . The scattering rate 1/τis again\nthe sum of the impurity scattering rate,25the electron-phonon scattering rate,26,29and electron-electron interaction,30\nso that all these scattering processes result in EYS. In non- degenerate semiconductors, where the Fermi energy is\nbelow the conduction band edge, 1/τs∼τT3/EGattains a stronger temperature dependence.\nD. Spin Relaxation due to Spin-Orbit Interaction with Impur ities\nThe spin-orbit interaction, as defined in Eq. ( 3), arises whenever there is a gradient in an electrostatic po tential.\nThus, the impurity potential gives rise to the spin-orbit in teraction\nVSO=1\n2m2c2∇V×k s. (35)\nPerturbation theory yields then directly the correspondin g spin relaxation rate\n1\nτs=πνni/summationdisplay\nα,β/integraldisplaydθ\n2π(1−cosθ)|VSO(k,k′)αβ|2, (36)\nproportional to the concentration of impurities ni. Hereα,β=±denotes the spin indices. Since the spin-orbit\ninteraction increases with the atomic number Zof the impurity element, this spin relaxation increases as Z2, being\nstronger for heavier element impurities.13\nE. Bir-Aronov-Pikus Spin Relaxation\nThe exchange interaction Jbetween electrons and holes in p-doped semiconductors resu lts in spin relaxation, as\nwell.31,32Its strength is proportional to the density of holes pand depends on their itinerancy. If the holes are\nlocalized they act like magnetic impurities. If they are iti nerant, the spin of the conduction electrons is transferred by\nthe exchange interaction to the holes, where the spin-orbit splitting of the valence bands results in fast spin relaxati on\nof the hole spin due to the Elliott-Yafet, or the D’yakonov-P erel’ mechanism.\nF. Magnetic Impurities\nMagnetic impurities have a spin Swhich interacts with the spin of the conduction electrons by the exchange\ninteraction J, resulting in a spatially and temporarily fluctuating local magnetic field\nBMI(r) =−/summationdisplay\niJδ(r−Ri)S, (37)\nwhere the sum is over the position of the magnetic impurities Ri. This gives rise to spin relaxation of the conduction\nelectrons, with a rate given by\n1\nτMs= 2πnMνJ2S(S+1), (38)\nwherenMis the density of magnetic impurities, and νis the density of states at the Fermi energy. Here, Sis the\nspin quantum number of the magnetic impurity, which can take the valuesS= 1/2,1,3/2,2.... Antiferromagnetic\nexchange interaction between the magnetic impurity spin an d the conduction electrons results in a competition\nbetween the conduction electrons to form a singlet with the i mpurity spin, which results in enhanced nonmagnetic\nand magnetic scattering. At low temperatures the magnetic i mpurity spin is screened by the conduction electrons\nresulting in a vanishing of the magnetic scattering rate. Th us, the spin scattering from magnetic impurities has a\nmaximum at a temperature of the order of the Kondo temperatur eTK∼EFexp(−1/νJ), whereνis the density of\nstates at the Fermi energy.33–35In semiconductors TKis exponentially small due to the small effective mass and the\nresulting small density of states ν. Therefore, the magnetic moments remain free at the experim entally achievable\ntemperatures. At large concentration of magnetic impuriti es, the RKKY-exchange interaction between the magnetic\nimpurities quenches however the spin quantum dynamics, so t hatS(S+1)is replaced by its classical value S2. In\nMn-p-doped GaAs, the exchange interaction between the Mn do pants and the holes can result in compensation of the\nhole spins and therefore a suppression of the Bir-Aronov-Pi kus (BAP) spin relaxation.36\nG. Nuclear Spins\nNuclear spins interact by the hyperfine interaction with con duction electrons. The hyperfine interaction between\nnuclear spins ˆIand the conduction electron spin, ˆs, results in a local Zeeman field given by37\nˆBN(r) =−8π\n3g0µB\nγg/summationdisplay\nnγnˆI,δ(r−Rn), (39)\nwhereγnis the gyromagnetic ratio of the nuclear spin. The spatial an d temporal fluctuations of this hyperfine\ninteraction field result in spin relaxation proportional to its variance, similar to the spin relaxation by magnetic\nimpurities.\nH. Magnetic Field Dependence of Spin Relaxation\nThe magnetic field changes the electron momentum due to the Lo rentz force, resulting in a continuous change of\nthe spin-orbit field, which similar to the momentum scatteri ng results in motional narrowing and thereby a reduction\nof DPS:28,38–40\n1\nτs∼τ\n1+ω2cτ2. (40)14\nAnother source of a magnetic field dependence is the precessi on around the external magnetic field. In bulk semi-\nconductors and for magnetic fields perpendicular to a quantu m well, the orbital mechanism is dominating, however.\nThis magnetic field dependence can be used to identify the spi n relaxation mechanism, since the EYS does have only\na weak magnetic field dependence due to the weak Pauli-parama gnetism.\nI. Dimensional Reduction of Spin Relaxation\nElectrostatic confinement of conduction electrons can redu ce the effective dimension of their motion. In quantum\ndots, the electrons are confined in all three directions, and the e nergy spectrum consists of discrete levels like in\natoms. Therefore, the energy conservation restricts relax ation processes severely, resulting in strongly enhanced s pin\nrelaxation times in quantum dots.41,42Then, spin relaxation can only occur due to absorption or emi ssion of phonons,\nyielding spin relaxation rates proportional to the inelast ic electron-phonon scattering rate.41Quantitative comparison\nof the various spin relaxation mechanisms in GaAs quantum do ts resulted in the conclusion that the spin relaxation is\ndominated by the hyperfine interaction.43–45A similar conclusion can be drawn from experiments on low tem perature\nspin relaxation in low density n-type GaAs, where the locali zation of the electrons in the impurity band results in\nspin relaxation dominated by hyperfine interaction as well.46,47For linear Rashba and linear Dresselhaus spin-orbit\ncoupling we can see from the spin diffusion equation Eq. ( 20) with the DP spin relaxation tensor Eq. ( 28) that the\nspin relaxation vanishes, when the spin current Eq. ( 22) vanishes, in which case the last two terms of Eq. ( 20)\ncancel exactly. The vanishing of the spin current is imposed by hard wall boundary condition for which the spin\ndiffusion current vanishes at the boundaries of the sample, jSin|Boundary= 0, wherenis the normal to the boundary.\nWhen the quantum dot is smaller than the spin precession leng thLSOthe lowest energy mode thus corresponds to a\nhomogeneous solution with vanishing spin relaxation rate. Cubic spin-orbit coupling does not yield such a vanishing\nof the DP spin relaxation rate. Only in quantum dots whose siz e does not exceed the elastic mean free path lethe\nDP spin relaxation from cubic spin relaxation becomes dimin ished. In quantum wires , the electrons have a continuous\nspectrum of delocalized states. Still, transverse confinem ent can reduce the DP spin relaxation as we review in the\nnext section.\nIV. SPIN-DYNAMICS IN QUANTUM WIRES\nA. One-Dimensional Wires\nIn one dimensional wires, whose width Wis of the order of the Fermi wave length λF, impurities can only reverse the\nmomentum p→ −p. Therefore, the spin-orbit field can only change its sign, wh en a scattering from impurities occurs.\nBSO(p)→BSO(−p) =−BSO(p). Therefore, the precession axis and the amplitude of the spi n orbit field does not\nchange, reversing only the spin precession, so that the D’ya konov-Perel’-spin relaxation is absent in one dimensional\nwires.48In an external magnetic field, the precession around the magn etic field axis, due to the Zeeman-interaction is\ncompeting with the spin-orbit field, however. Then, as the el ectrons are scattered from impurities, both the precession\naxis and the amplitude of the total precession field is changi ng, since\n|B+BSO(−p)|=|B−BSO(p)|/negationslash=|B+BSO(p)|,\nresulting in spin dephasing and relaxation, as the sign of th e momentum changes randomly.\nB. Spin-Diffusion in Quantum Wires\nHow does the spin relaxation rate depend on the wire width Wwhen the quantum wire has more than one channel\noccupied,W >λ F? Clearly, for large wire widths, the spin relaxation rate sh ould converge to a finite value, while it\nvanishes for W→λF. It is both of practical importance for spintronic applicat ions and of fundamental interest to\nknow on which length scales this crossover occurs. Basicall y, there are three intrinsic length scales characterizing t he\nquantum wire relative to its width W. The Fermi wave length λF, the elastic mean free path leand the spin precession\nlengthLSO, Eq. ( 18). Suppression of spin relaxation for wire widths not exceed ing the elastic mean free path le, has\nbeen predicted and obtained numerically in Refs. [ 11,49–53]. Is the spin relaxation rate also suppressed in diffusive\nwires in which the elastic mean free path is smaller than the wire wi dth as in the wire shown schematically in Fig. 7?\nWe will answer this question by means of an analytical deriva tion in the following. The transversal confinement\nimposes that the spin current vanishes normal to the boundar y,jSin|Boundary= 0. For a wire grown along the [010]15\nFigure 7: Elastic scatterings from impurities and from the b oundary of the wire change the direction of the spin-orbit fie ld\naround which the electron spin is precessing.\ndirection, n= ˆexis the unit vector in the x-direction. For wire widths Wsmaller than the spin precession length LSO,\nthe solutions with the lowest energy have thus a vanishing tr ansverse spin current, and the spin diffusion equation\nEq. (20) becomes\n∂si\n∂t=−De∂yjSiy+τ/angb∇acketleft∇vF(BSO(k)×S)i/angb∇acket∇ight−/summationdisplay\nj1\nˆτsijsj. (41)\nwith\njSix|x=±W/2= (−τ/angb∇acketleftvx(BSO(k)×S)i/angb∇acket∇ight−De∂xSi)|x=±W/2= 0, (42)\nwhereWis the width of the wire. One sees that this equation has a pers istent solution, which does not decay in time\nand is homogeneous along the wire, ∂yS= 0. In this special case the spin diffusion equation simplifies t o12\n∂tS=−1\nτsα2\nα2\n1−α1α20\n−α1α2α2\n20\n0 0 α2\nS. (43)\nIndeed this has one persistent solution given by\nS=S0\nα2\nα1\n0\n, (44)\nThus, we can conclude that the boundary conditions impose an effective alignment of all spin-orbit fields, in a direction\nidentical to the one it would attain in a one-dimensional wir e, along the [010]-direction, setting kx= 0in Eq. ( 27),\nBSO(k) =−2ky\nα2\nα1\n0\n, (45)\nwhich therefore does not change its direction when the elect rons are scattered. This is remarkable, since this alignmen t\nalready occurs in wires with many channels, where the impuri ty scattering is two-dimensional, and the transverse\nmomentum kxactually can be finite. Rather, the alignment of the spin-orb it field, accompanied by a suppression of\nthe DP spin relaxation rate occurs due to the constraint on th e spin-dynamics imposed by the boundary conditions\nas soon as the wire width Wis smaller than the length scale which governs the spin dynam ics, namely, the spin\nprecession length LSO. It turns out that the spin diffusion equation Eq. ( 41) has also two long persisting spin helix\nsolutions in narrow wires13,54which oscillate periodically with the period LSO=π/m∗α. In contrast to the situation\nin 2D systems we reviewed in the previous Section, in quantum wires of width W < L SOthese solutions are long\npersisting even for α1/negationslash=α2. These two stationary solutions, are\nS=S0\nα1\nα\n−α2\nα\n0\nsin/parenleftbigg2π\nLSOy/parenrightbigg\n+S0\n0\n0\n1\ncos/parenleftbigg2π\nLSOy/parenrightbigg\n, (46)16\n0\n/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExtLSO\n2\nLSOy\n/Minus101\nSy/Slash1S0/Minus101\nSz/Slash1S0\n0/FractionBarExt/FractionBarExt/FractionBarExtΠ\n4/FractionBarExt/FractionBarExt/FractionBarExtΠ\n2/CurlyPhi\nFigure 8: Persistent spin helix solution of the spin diffusio n equation in a quantum wire whose width Wis smaller than the spin\nprecession length LSOfor varying ratio of linear Rashba α2=αsinϕand linear Dresselhaus coupling, α1=αcosϕ, Eq.( 46),\nfor fixed αandLSO=π/m∗α.\nand the linearly independent solution, obtained by interch angingcosandsinin Eq. ( 46). The spin precesses as the\nelectrons diffuse along the quantum wire with the period LSO, the spin precession length, forming a persistent spin\nhelix, whose x-component is proportional to the linear Dresselha us-coupling α1while its y-component is proportional\nto the Rashba-coupling α2as seen in Fig. 8. A similar reduction of the spin relaxation rate is not effect ive for cubic\nspin-orbit coupling for wire widths exceeding the elastic m ean free path le. One can derive the spin relaxation rate as\nfunction of the wire width for diffusive wires le<W <L SO. The total spin relaxation rate, in the presence of both\nlinear Rashba spin-orbit coupling α2and linear and cubic Dresselhaus coupling α1, andγD, is as function of wire\nwidthWgiven by,54\n1\nτs(W) =1\n12/parenleftbiggW\nLSO/parenrightbigg2\nδ2\nSO1\nτs+De(m∗2ǫFγD)2, (47)\nwhere1/τs= 2p2\nF(α2\n2+(α1−m∗γDǫF/2)2)τ. We introduced the dimensionless factor, δSO= (Q2\nR−Q2\nD)/Q2\nSOwith\nQ2\nSO=Q2\nD+Q2\nRwhereQDdepends on Dresselhaus spin-orbit coupling, QD=m∗(2α1−m∗ǫFγ).QRdepends on\nRashba coupling: QR= 2m∗α2. Thus, for negligible cubic Dresselhaus spin-orbit coupli ng the the spin relaxation\nlength increases when decreasing the wire width Was,\nLs(W) =/radicalbig\nDeτs(W)∼L2\nSO\nW. (48)\nThis can be understood as follows:54–56In a wire whose width exceeds the spin precession length LSO, the area\nan electron covers by diffusion in time τsisWLs. To achieve spin relaxation, this area should be equal to the\ncorresponding 2D spin relaxation area Ls(2D)2, whereLs(2D) =LSO/(2π). Thus, the smaller the wire width, the\nlarger the spin relaxation length becomes, Ls∼(LSO)2/Win agreement with Eq. ( 48). For larger wire widths, the spin\ndiffusion equation can be solved as well, and one finds that the spin relaxation rate does not increase monotonously\nto the 2D limiting value but shows oscillations on the scale LSO, which can be understood in analogy to Fabry-Pérot\nresonances.54For pure linear Rashba coupling that behavior can be derived analytically, in the approximation of a\nhomogeneous spin density in transverse direction, yieldin g a relaxation rate given by\n1\nτs(W) =De\n2Q2\nSO/parenleftbigg\n1−sin(QSOW)\nQSOW/parenrightbigg\n, (49)\nwhereQSO= 2π/LSO. Furthermore, taking into account the transverse modulati on of the spin density by performing\nan exact diagonalization of the spin diffusion equation with the transverse boundary conditions, Eq. ( 42), one finds17\nforW >L SOmodes which are localized at the boundaries and have a lower r elaxation rate than the bulk modes.12,13\nFor pure Rashba spin relaxation we find that there is a spin-he lix solution located at the edge whose relaxation rate\n1/τs=.31/τs0is smaller than the spin relaxation rate of bulk modes 1/τs= 7/16τs0.\nC. Weak Localization Corrections\nQuantum interference of electrons in low-dimensional, dis ordered conductors results in corrections to the electrica l\nconductivity ∆σ. This quantum correction, the weak localization effect, is k nown to be a very sensitive tool to study\ndephasing and symmetry breaking mechanisms in conductors.57–59The entanglement of spin and charge by spin-orbit\ninteraction reverses the effect of weak localization and the reby enhances the conductivity, the weak antilocalization\neffect. The quantum correction to the conductivity ∆σarises from the fact, that the quantum return probability\nto a given point x0after a time t,P(t), differs from the classical return probability, due to quant um interference.\nAs the electrons scatter from impurities, there is a finite pr obability that they diffuse on closed paths, which does\nincrease the lower the dimension of the conductor. Since an e lectron can move on such a closed orbit clockwise or\nanticlockwise as shown in light and dark blue in Fig. 9, with equal probability, the probability amplitudes of bot h\npaths add coherently, if their length is smaller than the dep hasing length Lϕ. In a magnetic field, indicated by the\nFigure 9: Electrons can diffuse on closed paths, orbit clockw ise or anticlockwise as indicated by the light and dark blue a rrows,\nrespectively. Middle figure: Closed electron paths enclose a magnetic flux from an external magnetic field, indicated as t he red\narrow, breaking time reversal symmetry. Right figure: The sc attering from a magnetic impurity spin, breaks the time reve rsal\nsymmetry between the clock- and anticlockwise electron pat hs.\nred arrow in the middle Fig. 9, the electrons acquire a magnetic flux phase. This phase depe nds on the direction in\nwhich the electron moves on the closed path. Thus, the quantu m interference is diminished in an external magnetic\nfield since the area of closed paths and thereby the flux phases are randomly distributed in a disordered wire, even\nthough the magnetic field can be constant. Similarly, the sca ttering from magnetic impurities breaks the time reversal\ninvariance between the two directions in which the closed pa th can be transversed. Therefore magnetic impurities\ndiminish the quantum corrections in proportion to the rate w ith which the electron spins scatter from them due to\nthe exchange interaction, 1/τMs, Eq. ( 38).\nThus, the quantum correction to the conductivity, ∆σis proportional to the integral over all times smaller than t he\ndephasing time τϕof the quantum mechanical return probability P(t) =λd\nFρ(x,t), wheredis dimension of diffusion,\nandρis the electron density. In the presence of spin-orbit scatt ering, the sign of the quantum correction changes to\nweak antilocalization as was as predicted by Hikami, Larkin , and Nagaoka60for conductors with impurities of heavy\nelements. As conduction electrons scatter from such impuri ties, the spin-orbit interaction randomizes their spin,\nFig.10. The resulting spin relaxation suppresses interference of time reversed paths in spin triplet configurations,\nwhile interference in singlet configuration remains unaffec ted as indicated in Fig. 10. Since singlet interference reduces\nthe electron’s return probability it enhances the conducti vity, the weak antilocalization effect. Weak magnetic fields\nsuppress also these singlet contributions, reducing the co nductivity and resulting in negative magnetoconductivity .\nIf the host lattice of the electrons provides spin-orbit int eraction, the spin relaxation of DP or EY type does have\nthe same effect of diminishing the quantum corrections in the triplet configuration. When the dephasing length\nLϕis smaller than the wire width W, the quantum corrections are determined by the interferenc e of 2-dimensional\nclosed diffusion paths, and as a result, the conductivity inc reases logarithmically with Lϕwhich increases itself as\nthe temperature is lowered. At low temperatures, the electr on-electron scattering is the dominating mechanism of\nspin dephasing, yielding Lϕ∼T−1/2. One can derive the magnetic field dependence of that quantum correction18\nFigure 10: As electrons diffuse, their spin precesses around the spin-orbit field, which changes its orientation, when th e electron\nis scattered. Electrons which enter closed paths with the sa me spin leave it therefore with a different spin if they choose the\npath in the opposite sense, as indicated by the light and dark blue arrows. However, electrons which enter the closed path with\nopposite spin, and move through the closed path in opposite s ense, attain the same quantum phase. This is a consequence of\ntime reversal invariance.\nnonperturbatively.42,60–64An approximate expression showing the logarithmic depende nce explicitly is given by\n∆σ=−1\n2πlnB+4\n3HMs+Hϕ\nHτ+1\n2πlnB+Hϕ+Hs+2\n3HMs\nHτ+1\nπlnB+Hϕ+cHs+2\n3HMs\nHτ, (50)\nin units of e2/h. All parameters are rescaled to dimensions of magnetic field s:Hϕ= 1/(4eDeτϕ) = 1/(4eL2\nϕ),\nHτ=/planckover2pi1/(4eDeτ), the spin relaxation field due to spin orbit relaxation, Hs=/planckover2pi1/(4eDeτs),61and the spin relaxation\nfield due to magnetic impurities HMs=/planckover2pi1/(4eDeτMs). Here1/τsis the DP relaxation rate in the 2D limit derived in\nthe previous section.61,65The prefactor cdepends on the particular spin-orbit interaction. For line ar Rashba-coupling,\nc= 7/16. Note that 7/16τsis the smallest spin relaxation rate of an inhomogeneous spi n density distribution13as\nderived in the section II B 4.1/τMsis the magnetic scattering rate from magnetic impurities, E q. (38). Indeed we\nsee that the first term does not depend on the DP spin relaxatio n rate. This term originates from the interference\nof time reversed paths, indicated in Fig. 10, which contributes to the quantum conductance in the single t state,\n|S= 0;m= 0/angb∇acket∇ight= (| ↑↓/angb∇acket∇ight− | ↓↑/angb∇acket∇ight )/√\n2, the minus sign in front of the second term is the origin of the change in sign\nin the weak localization correction. The other three terms a re suppressed by the spin relaxation rate, since they\noriginate from interference in triplet states, |S= 1;m= 0/angb∇acket∇ight= (| ↑↓/angb∇acket∇ight+| ↓↑/angb∇acket∇ight)/√\n2,|S= 1;m= 1/angb∇acket∇ight,|S= 1;m=−1/angb∇acket∇ight\nwhich do not conserve the spin symmetry. Thus, at strong spin -orbit induced spin relaxation the last three terms are\nsuppressed and the sign of the quantum correction switches t o weak antilocalization. In quasi-1-dimensional quantum\nwires which are coherent in transverse direction, W < L ϕthe weak localization correction is further enhanced, and\nincreases linearly with the dephasing length Lϕ. Thus, for WQSO≪1the weak localization correction is54\n∆σ=√HW/radicalBig\nHϕ+1\n4B∗(W)+2\n3HMs−√HW/radicalBig\nHϕ+1\n4B∗(W)+Hs(W)+2\n3HMs\n−2√HW/radicalBig\nHϕ+1\n4B∗(W)+1\n2Hs(W)+4\n3HMs, (51)\nin units ofe2/h. We defined HW=/planckover2pi1/(4eW2), and the effective external magnetic field,\nB∗(W) =/parenleftbigg\n1−1//parenleftbigg\n1+W2\n3l2\nB/parenrightbigg/parenrightbigg\nB. (52)19\nThe spin relaxation field Hs(W)is forW <L SO,\nHs(W) =1\n12/parenleftbiggW\nLSO/parenrightbigg2\nδ2\nSOHs, (53)\nsuppressed in proportion to (W/LSO)2. Taking one transverse mode into account the quantum conduc tivity correction\nis plotted in Fig. 11for different wire withs for pure Rashba SOC, showing the cros sover from weak localization (positive\nmagnetoconductivity) to weak antilocalization (negative magnetoconductivity). In analogy to the effective magnetic\nFigure 11: The quantum conductivity correction in units of 2e2/has function of magnetic field B(scaled with bulk relaxation\nfieldHs), and the wire width W(scaled with LSO/2π), for pure Rashba coupling, δSO= 1.\nfield, Eq. ( 52), the spin orbit coupling acts in quantum wires like an effect ive magnetic vector potential.55One can\nexpect that in ballistic wires, le>W, the spin relaxation rate is suppressed in analogy to the flux cancellation effect,\nwhich yields the weaker rate, 1/τs= (W/Cle)(DeW2/12L4\nSO), whereC= 10.8.66–68A dimensional crossover from\nweak antilocalization to weak localization and a reduction of spin relaxation has recently been observed experimental ly\nin quantum wires as we will review in the next Section.\nV. EXPERIMENTAL RESULTS ON SPIN RELAXATION RATE IN SEMICOND UCTOR QUANTUM\nWIRES\nA. Optical Measurements\nOptical time-resolved Faraday rotation (TRFR) spectrosco py69has been used to probe the spin dynamics in an\narray of n-doped InGaAs wires by Holleitner et al. in Ref. [ 70,71]. The wires were dry etched from a quantum well\ngrown in the [001]-direction with a distance of 1 µm between the wires. Spin aligned charge carriers were creat ed\nby absorption of circularly-polarized light. For normal in cidence, the spins point then perpendicular to the quantum\nwell plane, in the growth direction [001]. The time evolutio n of the spin polarization was then measured with a\nlinearly polarized pulse, see inset of Fig. 1c of Ref. [ 70]. The time dependence fits well with an exponential decay\n∼exp(−∆t/τs). As seen in Fig. 2a of Ref. [ 70], the thus measured lifetime τsat fixed temperature T= 5K of the\nspin polarization is enhanced when the wire width Wis reduced70: While for W >15µm it isτs= (12±1)ps,\nit increases for channels grown along the [100]- direction t o almostτs= 30 ps, and in the [110]- direction to about\nτs= 20ps. Thus, the experimental results show that the spin relaxa tion depends on the patterning direction of the\nwires: wires aligned along [100] and [010] show equivalent s pin relaxation times, which are generally longer than the\nspin relaxation times of wires patterned along [110] and [ 110]. The dimensional reduction could be seen already for\nwire widths as wide as 10µm, which is much wider than both the Fermi wave length and the e lastic mean free path\nlein the wires. This agrees well with the predicted reduction o f the DP scattering rate, Eq. ( 47) for wire widths\nsmaller than the spin precession length LSO. From the measured 2D spin diffusion length Ls(2D) = (0.9−1.1)µm,20\nand its relation to the spin precession length Eq. ( 18),LSO= 2πLs(2D), we expect the crossover to occur on a scale\nofLSO= (5.7−6.9)µm as observed in Fig. 2a of Ref. [ 70]. FromLSO=π/m∗αwe get with m∗= 0.064mea\nspin-orbit coupling α= (5−6)meVÅ. According to Ls=√Deτs, the spin relaxation length increases by a factor of/radicalbig\n30/12 = 1.6in the [100]-, and by/radicalbig\n20/12= 1.3in the [110]- direction.\nThe spin relaxation time has been found to attain a maximum, h owever, at about W= 1µm≈Ls(2D), decaying\nappreciably for smaller widths. While a saturation of τscould be expected according to Eq. ( 47) for diffusive wires,\ndue to cubic Dresselhaus-coupling, a decrease is unexpecte d. Schwab et al., Ref. [ 12], noted that with wire boundary\nconditions which do not conserve the spin of the conduction e lectrons one can obtain such a reduction. A mechanism\nfor such spin-flip processes at the edges of the wire has not ye t been identified, however. The magnetic field dependence\nof the spin relaxation rate yields further confirmation that the dominant spin relaxation mechanism in these wires is\nDPS: It follows the predicted behavior Eq. ( 40), as seen in Fig. 3a of Ref. [ 70], and the spin relaxation rate is enhanced\ntoτs(B= 1T) = 100 ps for all wire growth directions, at T= 5K and wire widths of W= 1.25µm.\nB. Transport Measurements\nA dimensional crossover from weak antilocalization to weak localization and a reduction of spin relaxation has\nrecently been observed experimentally in n-doped InGaAs qu antum wires,72,73in GaAs wires,74as well as in Al-\nGaN/GaN wires.75The crossover indeed occurred in all experiments on the leng th scale of the spin precession length\nLSO. We summarize in the following the main results of these expe riments.\nWirthmann et al., Ref. [ 72], measured the magnetoconductivity of inversion-doped In As quantum wells with a density\nofn= 9.7×1011/cm2, and a measured effective mass of m∗= 0.04me. In the wide wires the magnetoconductivity\nshowed a pronounced weak antilocalization peak, which agre ed well with the 2D theory,61,65with a spin-orbit-coupling\nparameter of α= 9.3meVÅ. They observed a diminishment of the antilocalization peak which occurred for wire widths\nW <0.6µm, atT= 2K, indicating a dimensional reduction of the DP spin relaxat ion rate.\nSchäpers et al. observed in Ga xIn1−xAs/InP quantum wires a complete crossover from weak antiloc alization to weak\nlocalization for wire widths below W= 500 nm. Such a crossover has also been observed in GaAs-quantum w ires by\nDinter et al., Ref. [ 74].\nVery recently, Kunihashi et al., Ref. [ 76] observed the crossover from weak antilocalization to weak localization in\ngate controlled InGaAs quantum wires. The asymmetric poten tial normal to the quantum well could be enhanced by\napplication of a negative gate voltage, yielding an increas e of the SIA-coupling parameter α, with decreasing carrier\ndensity, as was obtained by fitting the magnetoconductivity of the quantum wells to the theory of 2D weak localization\ncorrections of Iordanskii et al., Ref. [ 65]. Thereby, the spin relaxation length Ls=LSO/2πwas found to decrease from\n0.5µm to0.15µm, which according to LSO=π/m∗αcorresponds to an increase of αfrom(20±1)meVÅ at electron\nconcentrations of n= 1.4×1012/cm2toα= (60±1)meVÅ at electron concentrations of n= 0.3×1012/cm2. The\nmagnetoconductivity of a sample with 95 quantum wires in par allel showed a clear crossover from weak antilocaliza-\ntion to localization. Fitting the data to Eq. ( 51) a corresponding decrease of the spin relaxation rate was ob tained,\nwhich was observable already at large widths of the order of t he spin precession length LSOin agreement with the\ntheory Eq. ( 47). However, a saturation as obtained theoretically in diffus ive wires, due to cubic BIA-coupling was not\nobserved. This might be due to the limitation of Eq. ( 47), to diffusive wire widths, le< W, while in ballistic wires\na suppression also of the spin relaxation due to cubic BIA-co upling can be expected, since it vanishes identically in\n1-D wires, see section IVA. Also, an increase of the spin scattering rate in narrower wi res,W < L s(2D), was not\nobserved in contrast to the results of the optical experimen ts, Ref. [ 70], reviewed above.\nThe dimensional crossover has also been observed in the hete rostructures of the wide gap semiconductor GaN.75The\nmagnetoconductivity of 160 AlGaN/GaN- quantum wires were m easured. The effective mass is m∗= 0.22me, all\nwires were diffusive with le< W. For electron densities of n≈5×1012/cm2an increase from Ls(2D)≈550nm\ntoLs(W≈130nm)>1.8µm, and for densities n≈2×1012/cm2an increase from Ls(2D)≈500nm to\nLs(W≈120nm)>1. µm was observed. Using Ls(2D) = 1/2m∗α, one obtains for both densities n, the spin-\norbit coupling α≈5.8meVÅ. A saturation of the spin relaxation rate could not be ob served, suggesting that the\ncubic BIA-coupling is negligible in these structures.\nWe note, that an enhancement of the spin relaxation rate as in the optical experiments of narrow InGaAs quantum\nwires, Ref. [ 70], was not observed in these AlGaN/GaN-wires.\nVI. CRITICAL DISCUSSION AND FUTURE PERSPECTIVE\nThe fact that optical and transport measurements seem to find opposite behavior, enhancement and suppression\nof the spin relaxation rate, respectively, in narrow wires, calls for an extension of the theory to describe the crossove r21\nto ballistic quantum wires. This can be done, using the kinet ic equation approach to the spin-diffusion equation,12a\nsemiclassical approach,77,78or an extension of the diagrammatic approach.13In particular, the dimensional crossover\nof DPS due to cubic Dresselhaus coupling, which we found not t o be suppressed in diffusive wires, needs to be studied\nfor ballistic wires, le> W, as many of the experimentally studied quantum wires are in t his regime. Furthermore,\nusing the spin diffusion equation, one can study the dependen ce on the growth direction of quantum wires, and find\nmore information on the magnitude of the various spin-orbit coupling parameters, α1,α2,γD, by comparison with the\ndirectional dependence found in both the optical measureme nts70of the spin relaxation rate, as well as in recent gate\ncontrolled transport experiments.76\nIn narrow wires, corrections due to electron-electron inte raction can become more important and influence especially\nthe temperature dependence. Ref. [ 71] reports a strong temperature dependence of the spin relaxa tion rate in narrow\nquantum wires. As shown in Ref. [ 23], the spin relaxation rates obtained from the spin diffusion equation and the\nquantum corrections to the magnetoconductivity can be diffe rent, when corrections due to electron-electron interacti on\nbecome important. As the DPS becomes suppressed in quantum w ires other spin relaxation mechanisms like the EYS\nmay become dominant, since it is expected that the dimension al dependence of EYS is less strong. In more narrow\nwires, disorder can also result in Anderson localization. S imilar as in quantum dots,41,45this can yield enhanced spin\nrelaxation due to hyperfine coupling, Eq. ( 39). The spin relaxation in metal wires is believed to be domina ted by\nthe EYS mechanism, which is not expected to show such a strong wire width dependence, although this needs to be\nexplored in more detail. Even dilute concentrations of magn etic impurities of less than 1 ppm, do yield measurable\nspin relaxation rates in metals and allow the study of the Kon do effect with unprecedented accuracy.34,35\nVII. SUMMARY\nThe spin dynamics and spin relaxation of itinerant electron s in disordered quantum wires with spin-orbit coupling\nis governed by the spin diffusion equation Eq. ( 20). We have shown that it can be derived by using classical rand om\nwalk arguments, in agreement with more elaborate derivatio ns.12,13In semiconductor quantum wires all available\nexperiments show that the motional narrowing mechanism of s pin relaxation, the D’yakonov-Perel’-Spin relaxation\n(DPS) is the dominant mechanism in quantum wires whose width exceeds the spin precession length LSO. The solution\nof the spin diffusion equation reveals existence of persiste nt spin helix modes when the linear BIA- and the SIA-spin-\norbit coupling are of equal magnitude. In quantum wires whic h are more narrow than the spin precession length LSO\nthere is an effective alignment of the spin-orbit fields givin g rise to long living spin density modes for arbitrary ratio\nof the linear BIA- and the SIA-spin-orbit coupling. The resu lting reduction in the spin relaxation rate results in a\nchange in the sign of the quantum corrections to the conducti vity. Recent experimental results confirm the increase\nof the spin relaxation rate in wires whose width is smaller th anLSO, both the direct optical measurement of the spin\nrelaxation rate, as well as transport measurements. These s how a dimensional crossover from weak antilocalization\nto weak localization as the wire width is reduced. Open probl ems remain, in particular in narrower, ballistic wires,\nwere optical and transport measurements seem to find opposit e behavior of the spin relaxation rate: enhancement,\nsuppression, respectively. The experimentally observed r eduction of spin relaxation in quantum wires opens new\nperspectives for spintronic applications, since the spin- orbit coupling and therefore the spin precession length rem ains\nunaffected, allowing a better control of the itinerant elect ron spin. The observed directional dependence moreover\ncan yield more detailed information about the spin-orbit co upling, enhancing the spin control for future spintronic\ndevices further.22\nSymbols\nτ0elastic scattering time\nτeescattering time due to electron-electron interaction\nτepscattering time due to electron-phonon interaction\nτtotal scattering time 1/τ= 1/τ0+1/τee+1/τep.\nˆτsspin relaxation tensor\nDediffusion constant, De=v2\nFτ/dD, wheredDis the dimension of diffusion.\nleelastic mean free path\nLSOspin precession length in 2D. The spin will be oriented again in the initial direction after it moved ballistically\nthe lengthLSO.\nQSO= 2π/LSO\nLsspin relaxation length, Ls(W) =/radicalbig\nDeτs(W)withLs(W)|W→∞=Ls(2D) =LSO/2π\nLϕdephasing length\nα1linear (Bulk inversion Asymmetry (BIA) = Dresselhaus)-par ameter\nα2linear (Structural inversion Asymmetry (SIA) =Bychkov-Ra shba)-parameter\nγDcubic (Bulk inversion Asymmetry (BIA) = Dresselhaus)-para meter\nγggyromagnetic ratio\nAcknowledgments\nWe thank V. L. Fal’ko, F. E. Meijer, E. Mucciolo, I. Aleiner, C . Marcus, T. Ohtsuki, K. Slevin, J. Ohe, and A.\nWirthmann for helpful discussions. This research was suppo rted by DFG-SFB508 B9 and by WCU ( World Class\nUniversity ) program through the Korea Science and Engineer ing Foundation funded by the Ministry of Education,\nScience and Technology (Project No. R31-2008-000-10059-0 ).\n∗p.wenk@jacobs-university.de ;www.physnet.uni-hamburg.de/hp/pwenk/\n†s.kettemann@jacobs-university.de ;www.jacobs-university.de/ses/skettemann\n1I. Zutic, J. Fabian, and S. S. Das, Rev. Mod. Phys. 76, 323 (2004).\n2H. C. Torrey, Phys. Rev. 104, 563 (1956).\n3G. Dresselhaus, Phys. Rev. 100, 580 (1955).\n4E. Rashba, Soviet Physics-Solid State 2, 1109 (1960).\n5R. Lassnig, Phys. Rev. B 31, 8076 (1985).\n6R. Winkler, Spin-Orbit Coupling Effects in Two-Dimensional Electron an d Hole Systems , vol. 191 of Springer Tracts in\nModern Physics (Springer-Verlag, Berlin, 2003).\n7F. Malcher, G. Lommer, and U. Rössler, Superlattices and Mic rostructures 2, 267 (1986).\n8E. Bernardes, J. Schliemann, M. Lee, J. C. Egues, and D. Loss, Phys. Rev. Lett. 99, 076603 (2007).\n9J. Fabian, A. Matos-Abiague, C. Ertler, P. Stano, and I. Zuti c, Acta Physica Slovaca 57, 565 (2007).\n10S. Datta and B. Das, App. Phys. Lett. 56, 665 (1990).\n11A. G. Mal’shukov and K. A. Chao, Phys. Rev. B 61, R2413 (2000).\n12P. Schwab, M. Dzierzawa, C. Gorini, and R. Raimondi, Phys. Re v. B74, 155316 (2006).\n13P. Wenk and S. Kettemann, Phys. Rev. B 81, 18 (2010).\n14V. A. Froltsov, Phys. Rev. B 64, 045311 (2001).\n15N. Bloembergen, E. M. Purcell, and R. V. Pound, Phys. Rev. 73, 679 (1948).\n16M. I. D’yakonov and V. I. Perel’, Sov. Phys. Solid State 13(1972), [ Fiz. Tverd. Tela , 13:3581, 1971].\n17N. S. Averkiev and L. E. Golub, Phys. Rev. B 60, 15582 (1999).23\n18B. A. Bernevig, J. Orenstein, and S.-C. Zhang, Phys. Rev. Let t.97, 236601 (2006).\n19Y. Ohno, R. Terauchi, T. Adachi, F. Matsukura, and H. Ohno, Ph ys. Rev. Lett. 83, 4196 (1999).\n20C. P. Weber, J. Orenstein, B. A. Bernevig, S.-C. Zhang, J. Ste phens, and D. D. Awschalom, Phys. Rev. Lett. 98, 076604\n(2007).\n21M. M. Glazov and E. L. Ivchenko, Jetp Lett. 75, 403 (2002).\n22M. M. Glazov and E. L. Ivchenko, J. Exp. and Theoret. Phys. 99, 1279 (2004).\n23A. Punnoose and A. M. Finkel’stein, Phys. Rev. Lett. 96, 057202 (2006).\n24A. Dyson and B. K. Ridley, Phys. Rev. B 69, 125211 (2004).\n25R. J. Elliott, Phys. Rev. 96, 266 (1954).\n26Y. Yafet, in Solid State Physics , edited by F. Seitz and D. Turnbull (Academic, New York, 1963 ), vol. 14.\n27J. N. Chazalviel, Phys. Rev. B 11, 1555 (1975).\n28G. E. Pikus and A. N. Titkov, in Optical Orientation , edited by F. Meier and B. P. Zakharchenya (North-Holland, A mster-\ndam, 1984), vol. 8 of Modern Problems in Condensed Matter Sciences , chap. 3.\n29C. Grimaldi and P. Fulde, Phys. Rev. B 55, 15523 (1997).\n30P. Boguslawski, Solid State Commun. 33, 389 (1980).\n31G. L. Bir, A. G. Aronov, and G. E. Pikus, Sov. Phys.-JETP 42, 705 (1976), [ Zh. Eksp. Teor. Fiz. , 69:1382, 1975].\n32G. E. Pikus and G. L. Bir, Sov. Phys.-JETP 33, 108 (1971), [ Zh. Eksp. Teor. Fiz. , 60:195, 1971].\n33Müller-Hartmann, E., and J. Zittartz, Phys. Rev. Lett. 26, 428 (1971).\n34T. Micklitz, A. Altland, T. A. Costi, and A. Rosch, Phys. Rev. Lett.96, 226601 (2006).\n35G. Zaránd, L. Borda, J. von Delft, and N. Andrei, Phys. Rev. Le tt.93, 107204 (2004).\n36G. V. Astakhov, R. I. Dzhioev, K. V. Kavokin, V. L. Korenev, M. V. Lazarev, M. N. Tkachuk, Y. G. Kusrayev, T. Kiessling,\nW. Ossau, and L. W. Molenkamp, Phys. Rev. Lett. 101, 076602 (2008).\n37A. W. Overhauser, Phys. Rev. 89, 689 (1953).\n38E. L. Ivchenko, Sov. Phys. Solid State 15, 1048 (1973).\n39E. L. Ivchenko, Fiz. Tverd. Tela 15, 1566 (1973).\n40A. A. Burkov and L. Balents, Phys. Rev. B 69, 245312 (2004).\n41A. V. Khaetskii and Y. V. Nazarov, Phys. Rev. B 61, 12639 (2000).\n42I. L. Aleiner and V. I. Fal’ko, Phys. Rev. Lett. 87, 256801 (2001).\n43S. I. Erlingsson, Y. V. Nazarov, and V. I. Fal’ko, Phys. Rev. B 64, 195306 (2001).\n44A. Khaetskii, D. Loss, and L. Glazman, Phys. Rev. B 67, 195329 (2003).\n45A. V. Khaetskii, D. Loss, and L. Glazman, Phys. Rev. Lett. 88, 186802 (2002).\n46R. I. Dzhioev, K. V. Kavokin, V. L. Korenev, M. V. Lazarev, B. Y . Meltser, M. N. Stepanova, B. P. Zakharchenya,\nD. Gammon, and D. S. Katzer, Phys. Rev. B 66, 245204 (2002).\n47P. I. Tamborenea, D. Weinmann, and R. A. Jalabert, Phys. Rev. B76, 085209 (2007).\n48J. S. Meyer, V. I. Fal’ko, and B. L. Altshuler (Kluwer Academi c Publishers, Dordrecht, 2002), vol. 72 of Nato Science Series\nII, p. 117.\n49A. Bournel, P. Dollfus, P. Bruno, and P. Hesto, The European P hysical Journal Applied Physics 4, 1 (1998).\n50A. A. Kiselev and K. W. Kim, Phys. Rev. B 61, 13115 (2000).\n51T. P. Pareek and P. Bruno, Phys. Rev. B 65, 241305 (2002).\n52T. Kaneko, M. Koshino, and T. Ando, Phys. Rev. B 78, 245303 (2008).\n53R. L. Dragomirova and B. K. Nikolic, Phys. Rev. B 75, 085328 (2007).\n54S. Kettemann, Phys. Rev. Lett. 98, 176808 (2007).\n55V. L. Fal’ko (2003), private communication.\n56T. Schäpers, V. A. Guzenko, M. G. Pala, U. Zülicke, M. Governa le, J. Knobbe, and H. Hardtdegen, Phys. Rev. B 74, 081301\n(2006).\n57B. L. Altshuler, A. G. Aronov, D. E. Khmelnitskii, and A. I. La rkin (Mir, Moscow, 1982), Quantum Theory of Solids.\n58G. Bergmann, Physics Reports 107, 1 (1984).\n59S. Chakravarty and A. Schmid, Physics Reports 140, 193 (1986).\n60S. Hikami, A. I. Larkin, and Y. Nagaoka, Progress of Theoreti cal Physics 63, 707 (1980).\n61W. Knap, C. Skierbiszewski, A. Zduniak, E. Litwin-Staszews ka, D. Bertho, F. Kobbi, J. L. Robert, G. E. Pikus, F. G. Pikus,\nS. V. Iordanskii, et al., Phys. Rev. B 53, 3912 (1996).\n62J. B. Miller, D. M. Zumbühl, C. M. Marcus, Y. B. Lyanda-Geller , D. Goldhaber-Gordon, K. Campman, and A. C. Gossard,\nPhys. Rev. Lett. 90, 076807 (2003).\n63Y. Lyanda-Geller, Phys. Rev. Lett. 80, 4273 (1998).\n64L. E. Golub, Phys. Rev. B 71, 235310 (2005).\n65S. V. Iordanskii, Y. Lyanda-Geller, and G. E. Pikus, JETP Let t.60, 206 (1994).\n66C. W. J. Beenakker and H. van Houten, Phys. Rev. B 37, 6544 (1988).\n67V. K. Dugaev and D. E. Khmel’nitskii, Soviet Physics - JETP 59, 1038 (1984).\n68S. Kettemann and R. Mazzarello, Phys. Rev. B 65, 085318 (2002).\n69D. Stich, J. H. Jiang, T. Korn, R. Schulz, D. Schuh, W. Wegsche ider, M. W. Wu, and C. Schüller, Phys. Rev. B 76, 073309\n(2007).\n70A. W. Holleitner, V. Sih, R. C. Myers, A. C. Gossard, and D. D. A wschalom, Phys. Rev. Lett. 97, 036805 (2006).\n71A. W. Holleitner, V. Sih, R. C. Myers, A. C. Gossard, and D. D. A wschalom, New Jounral of Physics 9, 342 (2007).\n72A. Wirthmann, Y. Gui, C. Zehnder, D. Heitmann, C.-M. Hu, and S . Kettemann, Physica E: Low-dimensional Systems24\nand Nanostructures 34, 493 (2006), proceedings of the 16th International Confere nce on Electronic Properties of Two-\nDimensional Systems (EP2DS-16).\n73F. E. Meijer (2005), private communication.\n74R. Dinter, S. Löhr, S. Schulz, C. Heyn, and W. Hansen (2005), u npublished.\n75P. Lehnen, T. Schäpers, N. Kaluza, N. Thillosen, and H. Hardt degen, Phys. Rev. B 76, 205307 (2007).\n76Y. Kunihashi, M. Kohda, and J. Nitta, Phys. Rev. Lett. 102, 226601 (2009).\n77O. Zaitsev, D. Frustaglia, and R. K., Phys. Rev. Lett. 94, 026809 (2005).\n78O. Zaitsev, D. Frustaglia, and R. K., Phys. Rev. B 72, 155325 (2005)."    },    {        "title": "2011.02369v2.Dynamical_spin_correlations_of_the_kagome_antiferromagnet.pdf",        "content": "Dynamical spin correlations of the kagome antiferromagnet\nP. Prelov\u0014 sek,1, 2M. Gomil\u0014 sek,1T. Arh,1, 2and A. Zorko1, 2,\u0003\n1Jo\u0014 zef Stefan Institute, Jamova c. 39, SI-1000 Ljubljana, Slovenia\n2Faculty of Mathematics and Physics, University of Ljubljana, Jadranska u. 19, SI-1000 Ljubljana, Slovenia\nTemperature-dependent dynamical spin correlations, which can be readily accessed via a variety of\nexperimental techniques, hold the potential of o\u000bering a unique \fngerprint of quantum spin liquids\nand other intriguing dynamical states. In this work we present an in-depth study of the temperature-\ndependent dynamical spin structure factor S(q;!) of the antiferromagnetic (AFM) Heisenberg spin-\n1/2 model on the kagome lattice with additional Dzyaloshinskii{Moriya (DM) interactions. Using\nthe \fnite-temperature Lanczos method on lattices with up to N= 30 sites we \fnd that even without\nDM interactions, chiral low-energy spin \ructuations of the 120 °AFM order parameter dominate the\ndynamical response. This leads to a nontrivial frequency dependence of S(q;!) and the appearance\nof a pronounced low-frequency mode at the M point of the extended Brillouin zone. Adding an\nout-of-plane DM interactions Dzgives rise to an anisotropic dynamical response, a softening of\nin-plane spin \ructuations, and, ultimately, the onset of a coplanar AFM ground-state order at\nDz>0:1J. Our results are in very good agreement with existing inelastic neutron scattering and\ntemperature-dependent NMR spin-lattice relaxation rate (1 =T1) data on the paradigmatic kagome\nAFM herbertsmithite, where the e\u000bect of its small Dzon the dynamical spin correlations is shown\nto be rather small, as well as with 1 =T1data on the novel kagome AFM YCu 3(OH) 6Cl3, where its\nsubstantial Dz\u00190:25Jinteraction is found to strongly a\u000bect the spin dynamics.\nI. INTRODUCTION\nThe antiferromagnetic (AFM) Heisenberg spin-1/2\nmodel on the kagome lattice (KLHM) is one of the\nmost intensively studied quantum spin models, owing\nto its unique ground state (GS) and low- Tproperties\n[1{4]. Various theoretical and numerical investigations\nhave established KLHM as the most promising candidate\namongst isotropic spin models to feature a quantum spin\nliquid (SL) GS, where the absence of low- Tlong-range\norder is accompanied by strong quantum entanglement\nbetween constituent spins. However, the nature of the\nSL GS, including the presence of either a \fnite [5{15] or\na vanishing [16{21] energy gap \u0001 tto spin-triplet excita-\ntions, remains controversial. Properties of the KLHM at\n\fnite temperatures may provide important insights into\nthis long-standing issue.\nThermodynamic quantities such as the uniform suscep-\ntibility\u001f0(T), magnetic speci\fc heat c(T), and the re-\nlated entropy density s(T) of the KLHM have previously\nbeen studied by high- Tseries expansion [22, 23], via nu-\nmerical linked cluster methods [24, 25], and more recently\nwith the \fnite-temperature Lanczos method (FTLM)\n[13, 15, 26] on \fnite spin systems with up to N= 42 sites.\nApart from an evidence of \fnite spin triplet gap \u0001 t>0,\nFTLM results indicate that there is a substantial rem-\nnant entropy s(T)>0 at very low T, which is a signature\nof a large density of low-energy singlet excitations with a\n(nearly) vanishing spin singlet energy gap \u0001 s\u001c\u0001t. The\nstatic (equal-time) spin correlation function S\u000b\u000b(q) has\nalso been studied both at T= 0 [17] and at \fnite tem-\nperatures [27, 28]. However, dynamical spin properties\n\u0003andrej.zorko@ijs.siof the KLHM, in particular the dynamical spin struc-\nture factor (DSF) S\u000b\u000b(q;!), are theoretically poorly un-\nderstood even though the temperature-dependent DSF\nis potentially a unique \fngerprint of SL states, and is\nexperimentally directly accessible via inelastic neutron-\nscattering (INS) and nuclear magnetic resonance (NMR)\nrelaxation [29]. Because of its fundamental importance\nvarious analytical concepts and methods [30{32], as well\nas numerical approaches [28, 33], have been employed to\nstudy it, though they have mostly led to inconclusive re-\nsults.\nOne reason for the theoretical di\u000eculties lies in the\nlarge density of low-energy spin-singlet states of the\nKLHM [15], which implies that a meaningful evalua-\ntion of the DSF would require a more challenging \fnite-\ntemperature instead of GS treatment. Another reason is\nthat the DSF of a SL, like the one in KLHM, is usually\n(implicitly) assumed to be rather featureless due to the\nfractionalization of spin excitations. We show that the\nKLHM DSF instead has some quite pronounced spectral\nfeatures.\nOn the experimental front, investigations of the KLHM\nhave been boosted in the last couple of decades by\nthe discovery of several promising kagome-lattice (KL)\nmaterials exhibiting SL properties at low tempera-\ntures. The most prominent example is herbertsmithite,\nZnCu 3(OH) 6Cl2[34{36], where the availability of single\ncrystals allows for full access to the DSF S\u000b\u000b(q;!) [37{\n39]. While several other KL materials have been discov-\nered in recent years [40{45], we will mostly focus on the\nrecently synthesized [46] and investigated [26, 45, 47, 48]\ncompound YCu 3(OH) 6Cl3, which has the distinct ad-\nvantage of having a structurally-perfect kagome lattice\nwithout any substitutional disorder, in contrast to most\nother KL materials including herbertsmithite [35, 36, 49].\nBesides potential imperfections the relation of KL ma-arXiv:2011.02369v2  [cond-mat.str-el]  20 Jan 20212\nterials to the ideal KLHM is often further complicated\nby additional Dzyaloshinskii{Moriya (DM) interactions,\nwhich are usually allowed in these systems as most lack\nlocal inversion symmetry on superexchange bonds Jbe-\ntween nearest-neighbor magnetic ions. While weak DM\ninteractions are expected to lead to mostly quantitative\ncorrections of observables at low T[23, 25], as in the\ncase of herbertsmithite [50, 51], strong DM interactions\ncan lead to a quantum phase transition from a SL to\na long-range ordered (LRO) GS [52{54], as in the case\nof YCu 3(OH) 6Cl3where an out-of-plane Dz\u00190:25Jin-\nduces chiral 120 °AFM LRO [26, 48]. The addition of\nDM interactions to the KLHM is therefore crucial for ex-\nplaining the observed properties of many KL materials,\nespecially low- Tordered ones like YCu 3(OH) 6Cl3.\nIn this paper we present a comprehensive numerical\nstudy of the DSF S\u000b\u000b(q;!) of the KLHM with additional\nout-of-plane DM interactions D=Dzat \fnite tempera-\ntures. To this end we employ the FTLM on systems with\nup toN= 30 sites under periodic boundary conditions.\nThis method is particularly suitable for frustrated spin\nsystems (and in general strongly-correlated systems) that\ndo not possess long-range correlations down to T\u001cJ,\nwhich allows us to obtain static and dynamical properties\nof macroscopic validity down to temperatures many times\nlower [15, 26] than in systems with GS LRO [15, 55]. In\ncontrast to previous investigations of the KLHM DSF [28]\nwe \fnd that it is in fact not featureless. Even at D= 0 we\n\fnd particularly pronounced low-energy chiral 120 °AFM\n\ructuations corresponding to the wavevector q= 0 in the\nreduced Brillouin zone (BZ), or, equivalently, the M point\nof the extended BZ. Furthermore, we \fnd that the low- T,\nlow-energy DSF of the KLHM seems to be governed by a\n\fnite spin triplet gap \u0001 t>0. Adding \fnite DM interac-\ntionsD> 0 results in an anisotropic DSF and a softening\nof the in-plane spin triplet gap \u0001x\ntthat ultimately leads\nto GS LRO for D > Dc\u00190:1J. The calculated DSF is\nalso used to evaluate temperature-dependent local spin\n\ructuation (LSF) spectra S\u000b\u000b\nL(!), which are directly re-\nlated to experimental NMR spin-lattice relaxation rates\n1=T1. Finally, the obtained numerical DSF and LSF re-\nsults are compared with experimental INS [37] and NMR\nresults [38] on herbertsmithite and on the impurity-free\nYCu 3(OH) 6Cl3[56].\nII. MODEL, NUMERICAL METHOD AND\nCONSIDERED QUANTITIES\nWe consider the KLHM with AFM isotropic Heisen-\nberg nearest-neighbor exchange interactions Jbetween\nS= 1=2 spins on a KL with additional out-of-plane DM\ninteractions D,\nH=X\nhijih\nJSi\u0001Sj+D(Si\u0002Sj)zi\n; (1)\nwherehijiis a sum over nearest-neighbor spin pairs and\nthe spins in the DM term appear in the clockwise direc-\nFigure 1. Finite size kagome lattices with N= 24, 27 and 30\nsites used in our FTLM calculations. The primitive vectors\nof the underlying hexagonal Bravais lattice are denoted by a1\nanda2, while the three basis vectors of the kagome lattice are\ndenoted by r0,r1, and r2.\ntion around each lattice triangle (see Fig. 1). Except in\nSec. V where we compare our numerical results with ex-\nperiment, we use ~=kB= 1 units as well as J= 1. All\nenergies, frequencies and temperatures are thus implicitly\ngiven relative to J. In KL materials the DM interaction\nhas the general form Dij\u0001(Si\u0002Sj) where Dijis a vec-\ntor with an out-of-plane component Dz\nijand an in-plane\ncomponent Dp\nij. In this paper we consider only the e\u000bect\nof a non-zero Dz\nij=Dfor three reasons. Firstly, an in-\nplaneDp\nijis symmetry-allowed only when the kagome\nplane is not also a crystallographic mirror plane [53],\nand is thus present less often. Secondly, the e\u000bect of\nDp\nij6= 0 appears to be weaker and qualitatively less im-\nportant than that of Dz\nij6= 0 in the KLHM, as con\frmed\nboth theoretically and experimentally [23, 25, 26, 52].\nAnd thirdly, as a practical bene\ft, when Dp\nij= 0 the\nhamiltonianHremains uniaxially symmetric about the\nzaxis, conserving the zcomponent of total magnetiza-\ntionSz\ntot=P\niSz\ni, which signi\fcantly reduces the di-\nmensionality of invariant Hilbert subspaces, and hence\nthe memory requirements, of the FTLM.\nThe standard de\fnition of the DSF is\nS\u000b\f(q;!) =1\n2\u0019Z1\n\u00001dt ei!thS\u000b\n\u0000q(t)S\f\nq(0)i;(2)\nwhereh:::idenotes the canonical thermal average, \u000b\nand\fare components of q-space spin operators Sq=\n(1=p\nN)P\nieiq\u0001RiSide\fned via the positions Riof spins\nin the KL, and Nis the total number of KL sites. As\nthe KL is formed of three basis vectors rk(k= 0;1;2) on\nan underlying hexagonal Bravais lattice of down-pointing\ntriangle centers eRn(see Fig. 1) one has Ri=eRn+rk\nwherei\u0011(n;k). Due to the three KL basis vectors\nthe DSF is only q-periodic over an extended BZ that\nis 4-times larger than the reduced BZ of the underlying\nhexagonal Bravais lattice (see Fig. 5).\nA more insightful de\fnition of the DSF for the KL,3\nwhich explicitly takes into account its threefold rota-\ntional symmetry, and which is bene\fcial both numerically\nas well as for theoretical understanding, instead involves\nchiral spin operators in q-space,\neScq=1p\nNX\nneiq\u0001eRn\u0002\nS(n;0)+\u0010cS(n;1)+\u0010\u0000cS(n;2)\u0003\n;\n(3)\nwherenin (n;k) runs over all down-pointing triangles of\nthe KL,kruns over the three spins inside these triangles,\n\u0010=e2\u0019i=3, andc=\u00001;0;1 denotes the vector spin chi-\nrality of the KL triangles. Note that the standard 120 °\nAFM LRO on the KL involves only the chiral spin oper-\nators withc=\u00061, while ferromagnetic LRO involves the\nc= 0 chiral spin operators. Using Eq. (3) we de\fne the\n(diagonal) chiral DSF on the KL as\neS\u000b\f\nc(q;!) =1\n2\u0019Z1\n\u00001dt ei!theS\u000by\ncq(t)eS\f\ncq(0)i;(4)\nwhich is q-periodic over the reduced BZ, not just over\nthe larger extended BZ of the standard DSF of Eq. (2)\n(see Fig. 5). Note that at a generic qone could also\nexpect non-vanishing o\u000b-diagonal terms heS\u000by\ncq(t)eS\f\nc0q(0)i\nwithc6=c0. Nevertheless, these terms are expected to\nbe less important than the diagonal ones, and are much\nmore di\u000ecult to handle within the FTLM, so we neglect\nthem. We can then express the standard DSF of Eq. (2)\nfrom the chiral DSF of Eq. (4) as\nS\u000b\u000b(q;!) =1X\nc=\u00001j\u0018c(q)j2eS\u000b\u000b\nc(q;!);\n\u0018c(q) =1\n32X\nk=0eiq\u0001rk\u0010\u0000ck;(5)\nwhereeS\u000b\f\nc(q;!) = 0 for \u000b6=\fandSxx(q;!) =\nSyy(q;!) sinceSz\ntotis a conserved quantity.\nWe evaluate the chiral DSF at T >0 using the FTLM,\nintroduced in Refs. [55, 57] and used in numerous stud-\nies of static and dynamical properties of various corre-\nlated systems [58]. In the case of the KLHM, the FTLM\nhas previously been employed only for the calculation of\nthermodynamic quantities, such as the uniform suscepti-\nbility\u001f0(T), entropy density s(T) and speci\fc heat c(T)\n[13, 15, 26], that involve only the conserved quantities\nof energy and total magnetization Sz\ntot. In contrast, the\nevaluation of the chiral DSF (here given in the Lehmann\nrepresentation) is more involved,\neS\u000b\u000b\nc(q;!) =1\nZX\nne\u0000\u000fn=Th njeS\u000by\ncq\u000e(!+\u000fn\u0000H)eS\u000b\ncqj ni;\n(6)\nwhereZ=P\nne\u0000\u000fn=Tis the canonical partition function,\nj niare eigenfunctions of Hand\u000fnare their eigenener-\ngies. As the chiral DSF already takes into account both\ntranslation symmetry and the conservation of Sz\ntot, the\nneeded Hilbert subspaces remain the same as for staticquantities; e.g., the largest subspace for N= 30 sites\ncontainsNst\u0018107states. In FTLM we replaceP\nnover\nall eigenfunctions with a trace over R> 1 random initial\nwavefunctionsjriand the expectation value with a dou-\nble sum over the emerging Lanczos (eigen)functions j\u001er\nii,\nje\u001er\njiin di\u000berent qwavevector sectors [55, 57, 58], with\ni;j\u0014NLwhereNLis the number of performed Lanczos\nsteps. This requires additional storage of 2 NLwavefunc-\ntions meaning that the total memory requirements for\nthe dynamical FTLM are O(NLNst). To achieve satis-\nfactory!resolution in the DSF NL>100 is typically\nrequired.\nIn the following we evaluate the chiral DSF on several\n\fnite-sized lattices with N= 24, 27 and 30 sites (Fig. 1).\nWhile theN= 24 and 30 lattices break the rotational\nsymmetry of the in\fnite KL the N= 27 lattice preserves\nit, but is less convenient because of its Stot= 1=2 GS,\nwhereas the in\fnite KLHM should have a Stot= 0 GS\n[1{4]. While for N= 24 and 27 we can a\u000bord NL\u0018200\nandR > 10, most of the present results are for N= 30\nsites where we used NL= 120 and R= 3 within each\nsymmetry sector. We note that the main criterion for\n(even macroscopic) validity of FTLM results (in the given\nmodel and system size) is that the modi\fed thermody-\nnamic sumeZ(T) =RTr[exp(\u0000(H\u0000E0)=T)]>eZ(Tfs)\u001d\n1 [55, 57], where E0is the ground-state energy and trace\nTr involves the sum over all wave vector and Sz\ntotsectors.\nDue to very large density of low-lying states in SL sys-\ntems (and directly related large entropy even at low T),\neven modest R= 3 is enough to reach valid results down\nto temperatures T >T fs\u00180:1J+D[15, 26] below which\nthey are limited by \fnite-size e\u000bects, i.e., by the onset of\nlonger-range correlations for D> 0.\nFinally, while S\u000b\u000b(q;!) contains all of the dynamical\ninformation, it is also useful to extract the equal-time\nspin correlation function S\u000b\u000b(q) and the d.c. spin sus-\nceptibility\u001f\u000b\u000b\n0(q), de\fned from the DSF as\nS\u000b\u000b(q) =Z1\n\u00001d! S\u000b\u000b(q;!) =hS\u000b\n\u0000qS\u000b\nqi;\n\u001f\u000b\u000b\n0(q) =PZ1\n\u00001d!1\u0000e\u0000!=T\n!S\u000b\u000b(q;!);(7)\nwherePdenotes the Cauchy principal value. Note that\n!\u000b(q) =p\nS\u000b\u000b(q)=\u001f\u000b\u000b\n0(q) can be interpreted as the\ncharacteristic spin-\ructuation frequency at a given qand\nfor a given direction \u000b.\nIII. HEISENBERG MODEL ON KAGOME\nLATTICE\nIn this section we consider the D= 0 KLHM. Since\nthis model is isotropic in spin space it has an isotropic\nchiral DSFeS\u000b\u000b\nc(q;!) =eSc(q;!) and hence also isotropic\nderived quantities in Eqs. (7). In the following we present\nnumerical results for the standard choice \u000b=z, which4\n-10 1 2 3 4 5 6 0.00.10.20.3  \nSc(q, ω)ω\n Γ, c = ±1 (N = 24) \nΓ, c = ±1 (N = 27) \nΓ, c = ±1 (N = 30) \nM, c = 0 (N = 24) \nM, c = 0 (N = 30)\nFigure 2. Chiral DSF's eS\u00061(q= 0;!) andeS0(q= M;!) at\nT= 0:2 calculated using the FTLM for di\u000berent lattice sizes\nN= 24{30 (Fig. 1). Note that the N= 27 lattice does not\ncontain the M point of the reduced BZ. The vertical dotted\nline at!= 1:5 corresponds to triplet excitations within an\nisolated Heisenberg spin triangle.\nis numerically less costly to evaluate since the relevant\noperators are diagonal in the Sz\ntotbasis.\nA. Dynamical spin structure factor\nIn Fig. 2 we present a comparison of chiral DSF's cal-\nculated at T= 0:2 using the FTLM on lattices with\nN= 24, 27 and 30 sites (see Fig. 1). We choose two\nrather extreme cases of the \u0000 and M points of the re-\nduced BZ [see inset in Fig. 3(b)]. The largest dynamical\nresponse is at the \u0000 point ( q= 0) with chirality c=\u00061,\nwhich represents uniform \ructuations of the AFM order\nparameter for 120 °ordered spins on each KL triangle.\nWe see that these results are quite independent of lattice\nsizeN, which con\frms that the spin correlation length\nis quite short even at this low temperature due to strong\ngeometric frustration. The spectra are not featureless, as\nthey exhibit two distinct frequency maxima, which seem\nquite robust. These were already tentatively observed via\nthe numerical linked cluster method [29] by assuming an\nad hoc Lorentzian line shape. Our FTLM calculations,\non the other hand, do not require any a priori assump-\ntions on the line shape. The higher-energy maximum can\nbe traced back to transitions within individual spin tri-\nangles, for which the energy gap between the S= 1=2\nGS and excited S= 3=2 spin states is != 1:5 (dashed\nline in Fig. 2).\nIn Fig. 3 we show the full chiral DSF eSc(q;!) atT=\n0:2 for all inequivalent q's in the reduced BZ for both\nchirality branches c=\u00061 andc= 0, calculated on the\nlargestN= 30 site lattice. Since the c= +1 andc=\u00001\nchiral DSF's are in general not equal at generic q6= 0 we\nplot in Fig. 3(a) the averaged chiral DSF\nS1(q;!) =1\n2h\neS1(q;!) +eS\u00001(q;!)i\n; (8)\n0.00.10.20.3\n-1 0 1 2 3 4 5 60.000.020.040.060.08\n(a)S1(q, ω)\n Γ\n q1\n q2\n q3\n M\n q5S0(q, ω)\nω(b)\nΓM\nq-1q1\nq2q-2\nq3q-3\nq-5\nq5Figure 3. (a) Average chiral DSF S1(q;!) [Eq. (8)], and (b)\nchiral DSF eS0(q;!) atT= 0:2 for all inequivalent numerical\nq's within the reduced BZ calculated on the N= 30 lattice.\nNote the very di\u000berent vertical scales of both panels. Inset\nin (b) shows the numerical q-cells in the reduced BZ of the\nN= 30 lattice (Fig. 1).\nwhile atq= 0 both chiralities c=\u00061 match and we have\nS1(q= 0;!) =eS\u00061(q= 0;!). We see that chiral c=\u00061\n\ructuations indeed dominate the response (Fig. 3), with\nthe largest intensity found at the q= 0 (\u0000) point and a\nslightly reduced intensity found at the smallest nonzero\nq=q1. Chiral DSF spectra at these low qshow the char-\nacteristic double-maximum frequency dependence with\nmaxima near !\u00190:3 and!\u00191:5 [Fig. 3(a)]. This struc-\nture is reproduced even in the considerably weaker c= 0\nresponse at q=q1[Fig. 3(b)]. At larger q, nearer the\nBZ boundary, all spectra are broad ( \u000e!&3), weak and\nfeatureless. The observed qandcdependence thus in-\ndicates that longer-ranged chiral 120 °AFM correlations\ndominate the dynamical response of the KLHM at low\nfrequencies, with a correlation length \u0018 > 1 extending\nfurther than a single spin triangle.\nIn Fig. 4 we present the temperature evolution of the\ndominantq= 0,c=\u00061 chiral DSF. It is evident that\nthe double-maximum frequency structure is not just a\nlow-Tfeature as it persists to temperatures as large as\nT\u00181. Partly, the low-energy peak at !\u00180:3 is simply\na consequence of the detailed balance relation for DSF's,\nS(\u0000!) = exp(\u0000!=T)S(!), which implies d S=d!j!=0=\nS(0)=(2T)>0 and thus always leads to a maximum at\n! > 0. On the other hand, at the lowest T= 0:1\u0018Tfs\nwe \fnd a further reduced != 0 response, which could\nindicate a \fnite spin triplet gap \u0001 t>0, at least on our\n\fnite-sized N= 30 lattice.\nFinally, we note that the q= 0,c= 0 chiral DSF had to5\nbe explicitly excluded from our FTLM calculations since\nit is singular in \fnite systems, eS0(q= 0;!)/\u000e(!), due\nto the conservation of Sz\ntot. Nevertheless, in the macro-\nscopic limit N!1 atT > 0 this singular DSF should\nevolve into the q\u00190 spin di\u000busion peak with a spectral\nwidth that is expected to scale as \u000e!/Ddi\u000bq2where\nDdi\u000b(T) is the temperature-dependent spin di\u000busion con-\nstant [59]. We discuss the experimental relevance of this\ncontribution in more detail in Sec. V B.\nB. Equal-time correlations and static response\nFor comparison with experimental INS data as well as\nwith previous theoretical calculations, it is informative\nto also look at the standard INS DSF S(q;!) [Eq. (2)] in\nthe extended BZ, which is calculated from the chiral DSF\neSc(q;!) via Eq. (5). Firstly, we consider the q-dependent\nequal-time spin correlation function S(q) [Eqs. (7)] over a\nbroad range of temperatures T= 0:1{2:0 on theN= 30\nlattice (Fig. 5). Consistent with several previous numer-\nical studies of this quantity [17, 27, 28], S(q) has a pro-\nnounced but spread-out region of high intensity around\nthe whole extended BZ boundary that remains visible\neven at very high T\u00182. This can be understood by\nconsidering the q-dependence of the chiral weighing fac-\ntorj\u0018c(q)j2in Eq. (5) that suppresses the contribution of\nthe dominant chiral c=\u00061 \ructuations to the standard\nDSFS(q;!) near the \u0000 point of the extended BZ, but\nnot at the extended BZ boundary. Weak global maxima\nofS(q) appear for T >0:1 at corner K points of the ex-\ntended BZ (note that our N= 30 lattice does not contain\nthis point), qualitatively consistent with previous stud-\nies, but appreciable intensity can also be found at the M\npoints (corresponding to periodic images of the \u0000 point\nof the reduced BZ).\nA complementary quantity, which is more sensitive to\nlow-energy \ructuations as is obvious from Eqs. (7), is the\nq-dependent d.c. susceptibility \u001f0(q), which we present\nin the extended BZ over a broad range of temperatures\n-2-101234560.00.10.20.3 T = 0.1 \nT = 0.2 \nT = 0.4 \nT = 0.8 \nT = 2 \nS±1(q = 0, ω)ω\nFigure 4. The chiral DSF eS\u00061(q= 0;!) at di\u000berent T= 0:1{\n2:0 on theN= 30 lattice.\nFigure 5. The equal-time spin correlation function S(q) in\nthe extended BZ (large hexagon) at di\u000berent T= 0:1{2:0 on\ntheN= 30 lattice. The reduced BZ is shown by the smaller\nhexagon. The left halves of the panels show raw FTLM results\nonq-cells shown in the inset in Fig. 3(b), while the right\nhalves are rotationally symmetrized (to recover the threefold\nsymmetry of the KL) and smoothed via interpolation.\nFigure 6. The d.c. spin susceptibility \u001f0(q) in the extended\nBZ (large hexagon) at di\u000berent T= 0:1{2:0 on theN= 30\nlattice. The reduced BZ is shown by the smaller hexagon.\nThe left halves of the panels show raw FTLM results on q-\ncells shown in the inset in Fig. 3(b), while the right halves are\nrotationally symmetrized as in Fig. 5.\nin Fig. 6. A striking di\u000berence to S(q) (Fig. 5) is a very\npronounced maximum of \u001f0(q) at the M point of the\nextended BZ, which is directly related to the dominant6\n-2-101234560.000.020.040.06 T = 0.1 \nT = 0.2 \nT = 0.4 \nT = 0.8 \nT = 2 \nSL(ω)ω\nFigure 7. The LSF SL(!) at di\u000berent T= 0:1{2:0 on the\nN= 30 lattice.\nlow-energy q= 0,c=\u00061 chiral \ructuations seen in\nthe chiral DSF (Fig. 3). This maximum is much more\nsensitive to temperature than the maximum in S(q) and\ndisappears for T > 1, consistent with the broadening of\nthe chiral response visible in Fig. 4. It should be stressed\nthat the same maximum is directly related to the one\nobserved by low-energy INS in herbertsmithite [37], as\nwill be discussed in more detail in Sec. V A.\nC. Local spin \ructuations\nLSF can be expressed from the chiral DSF as\nS\u000b\u000b\nL(!) =Z1\n\u00001dt\n2\u0019ei!thS\u000b\ni(t)S\u000b\ni(0)i=1\nNX\ncqeS\u000b\u000b\nc(q;!);\n(9)\nand are likewise isotropic in the D= 0 KLHM, i.e.\nS\u000b\u000b\nL(!) =SL(!). Their value at !\u00190 is experimen-\ntally highly relevant, as it is directly proportional to the\nexperimental NMR spin-lattice relaxation rate 1 =T1, pro-\nvided that hyper\fne form factors do not play an essential\nrole, as discussed in detail in Sec. V B. As mentioned pre-\nviously, we omit the singular q= 0,c= 0 spin di\u000busion\ncontribution, discussed further in Sec. V B.\nIn Fig. 7 we show the temperature evolution of the\nLSF over a broad range of T= 0:1{2:0 on theN= 30\nlattice. Apart from a pronounced low-energy peak arising\nfrom the dominant q= 0,c=\u00061 chiral \ructuations at\nT < 0:2 the LSF are quite temperature independent for\nT > 0:2, even at the relevant !\u00190 energy scale of\nNMR experiments. At T < 0:2, a drop of SL(!= 0) is\nobserved, which is again a signature of a \fnite spin triplet\ngap \u0001t>0, at least on \fnite-sized lattices [13{15].\nIt is instructive to compare the calculated LSF SL(!=\n0) to Moriya's Gaussian approximation [60] frequently\nused at high T\u001d1, but also extended to lower Tvia\nhigher-order corrections in the case of the D= 0 KLHM\n[29]. In a uniform Heisenberg spin-1 =2 model the LSF\nfrequency moments \u0016k=R\nd! !kSL(!), are exactly\nknown atT!1 , with the LSF sum rule \u00160= 1=4 and\u00162=z=8, wherez= 4 is the number of nearest-neighbors\nin the KL. These yield the expected != 0 value of the\nKLHM LSF under Gaussian line shape approximation\nSMoriya\nL (0) =\u00160p8\u0019\u00162\u00190:071; (10)\nwhich is reasonably close to the actual KLHM value\nSL(0)\u00190:055 atT= 2 calculated with the FTLM. We\nnote, though, that the frequency-dependent LSF SL(!)\nare not, in fact, Gaussian in shape, as is obvious from\nFig. 7, and T= 2 is not yet\u001d1.\nIV. DZYALOSHINSKII-MORIYA\nINTERACTIONS\nIn this section we consider an extension of the KLHM\nwith out-of-plane DM interactions 0 \u0014D\u00140:25 [Eq. (1)]\n(note that the dynamical response is not sensitive to the\nsign ofD) [25], which are relevant in many KL materials\n[26, 48, 50, 51, 54]. The out-of-plane Dleads to a uniax-\nially anisotropic chiral DSF eS\u000b\u000b\nc(q;!) with equal \u000b=x\nand\u000b=ycomponents that di\u000ber from the \u000b=zcom-\nponent, which has to be calculated separately. The same\nalso holds for all derived quantities, including the stan-\ndard DSFS\u000b\u000b(q;!) [Eq. (2)] and quantities in Eqs. (7).\nWe note that chiral spin operators eS\u000b\ncqwith\u000b=x;y\nare o\u000b-diagonal in the Sz\ntotbasis, which substantially\nincreases the overall computational complexity and re-\nquirements of FTLM compared to the \u000b=zcase, where\nthey are diagonal in the subspace. In particular, the em-\nployed reduced summation over Sz\ntotsubspaces (having\nlesser e\u000bect on diagonal correlations) appears to in\ruence\nmore the calculation of o\u000b-diagonal \u000b=x;ycorrelations.\nTo reduce di\u000berences we normalize \u000b=x;yresults by a\nscaling factor of 1 :15 to reproduce the \u000b=zsum rules\natD= 0.\nA. Dynamical spin structure factor\nIt is known that at low temperatures KLHM systems\ncan be signi\fcantly a\u000bected by the presence of additional\nDM interactions, with a quantum phase transition from\na SL GS to a 120 °AFM LRO GS with nonzero vector\nspin chirality when D >Dc\u00190:1 [52, 53]. This mainly\ncorresponds to a gradual softening of the dominant q= 0,\nc=\u00061 chiral \ructuations as Dincreases towards the\nquantum critical point Dc, beyond which these emerge\nas in-plane chiral 120 °AFM LRO.\nIn Fig. 8 we present the dominant q= 0,c=\u00061 chi-\nral DSF at a temperature T= 0:3 high enough to avoid\nlonger-ranged AFM correlations leading to strong \fnite-\nsize e\u000bects in our FTLM calculations. A \fnite D > 0\nsubstantially decreases the \u000b=zcomponent of the chiral\nDSF at low !, consistent with an increase of the e\u000bec-\ntive out-of-plane spin triplet gap \u0001z\nt. At the same time,\nthe\u000b=zspectra become sharper (more coherent) for7\n0.00.10.20.30.40.5-\n10 1 2 3 4 0.00.20.40.60.81.00.00.10.20.00.5(a) \nSzz±\n1(q = 0, ω) \nD = 0 \nD = 0.05 \nD = 0.1 \nD = 0.15 \nD = 0.2 \nD = 0.25 \nSxx±\n1(q = 0, ω)ω\n(b) \nωzzm\naxD\nModelH yperbola (User)E\nquation( k*dx) * sqrt(1 + ((x-x0)/dx)^2)P\nloto mega_maxx\n00  ± 0d\nx0 .05664 ± 0.0026k\n3 .04742 ± 0.02872R\neduced Chi-Sqr8 .28055E-5R\n-Square (COD)0 .99883A\ndj. R-Square0 .99853\nFigure 8. Chiral DSF's (a) eSzz\n\u00061(q= 0;!) and (b) eSxx\n\u00061(q=\n0;!) at a \fxed T= 0:3 and di\u000berent D= 0{0:25 on theN=\n30 lattice. Inset in (a) shows the frequency of the lower-energy\nmaximum of eSzz\n\u00061(q= 0;!) (symbols) with curves serving as\nguides to the eye.\nD > 0:1, i.e. beyond the quantum critical point, with\nthe energy of the spectral peak scaling nearly linearly as\n!max\u00193:0D[see inset in Fig. 8(a)]. This is consistent\nwith the linear scaling of the lower speci\fc heat peak\nTmax\u00190:91Dfound via the FTLM in Ref. [26], below\nwhich the spin correlation length \u0018increases substan-\ntially. The \u000b=x;ycomponents of the chiral DSF show\nthe latter e\u000bect quite clearly [Fig. 8(b)] as low-energy\noscillations due to \fnite-size magnon-like excitations be-\ncome visible at D&0:1 and increase in prominence as\nDincreases further. This indicates a considerable in-\ncrease in the spin correlation length \u0018 > 1 already at\nT&Tmaxwith increasing D>Dc. Concomitantly, there\nis a substantial increase of low- !intensity in \u000b=x;y\ncomponents of the chiral DSF, in contrast to a decrease\nin the\u000b=zcomponent, consistent with a softening of\nchiral \ructuations above a 120 °AFM GS with in-plane\nLRO spins [26, 48, 50, 52{54].\nB. Equal-time correlations and local spin\n\ructuations\nSimilar conclusions can be drawn from the tempera-\nture dependence of the chiral equal-time correlation func-\ntioneS\u000b\u000b\nc(q), which is de\fned by replacing the standard\nDSFS\u000b\u000b(q;!) in Eqs. (7) by the chiral DSF eS\u000b\u000b\nc(q;!).\nWe focus on the dominant q= 0,c=\u00061 correla-\ntions, which are shown in Fig. 9. We see that they\nare weakly T-dependent over the whole T > T fsrange\n0.30.40.50\n.00 .51 .01 .50.30.40.5 D = 0 \nD = 0.05 \nD = 0.1 \nD = 0.15 \nD = 0.2 \nD = 0.25(a)Szz±\n1(q = 0)T\n(b)Sxx±\n1(q = 0)Figure 9. The temperature dependence of chiral equal-time\nspin correlation functions (a) eSzz\n\u00061(q= 0) and (b) eSxx\n\u00061(q= 0)\nfor di\u000berent D= 0{0:25 on theN= 27 lattice.\nwhenD < Dc. The behavior changes qualitatively for\nD > Dc. While the \u000b=zcomponent remains rela-\ntively una\u000bected [Fig. 9(a)], the \u000b=x;ycomponents\nshow a strong increase below T.2D[Fig. 9(b)] consis-\ntent with the gradual onset of longer-range correlations\naroundT\u0018Tmax[26] and ultimate chiral AFM LRO at\nT= 0.\nFinally, we consider the temperature dependence of\nthe!= 0 LSF, which are directly relevant for NMR\nspin-lattice relaxation rate (1 =T1) experiments that we\ndiscuss in Sec. V B. Here we \fnd it useful to separately\nconsider the individual chiral LSF contributions\neS\u000b\u000b\nLc(!) =3\nNX\nqeS\u000b\u000b\nc(q;!); (11)\nto the full LSF S\u000b\u000b\nL(!) = (1=3)P\nceS\u000b\u000b\nLc(!). Note that\nthec= +1 andc=\u00001 chiral LSF are equal.\nIn Fig. 10 we present the calculated temperature de-\npendence of the != 0 chiral LSF for a range of D= 0{\n0:25 on theN= 30 lattice. We \fnd that the c=\u00061 chi-\nral LSF are highly sensitive to D, especially at T.2D\nwhere the\u000b=zcomponent is suppressed [Fig. 10(a)]\ndue to a shift of spectral intensity to higher !\u0018!max\n[Fig. 8(a)], while the \u000b=x;ycomponents are strongly\nenhanced [Fig. 10(c)] due to the gradual onset of longer-\nrange correlations at T\u0018Tmax[Fig. 8(b)]. On the other\nhand, components of the c= 0 chiral LSF are nearly\nequal and mostly insensitive to D, showing just a steady\nincrease with increasing temperature due to increasingly\nincoherent spin dynamics at T&1 [Fig. 10(b,d)].8\n0.000.020.040.06\n0.0 0.5 1.0 1.50.000.020.040.060.080.000.020.040.06\n0.0 0.5 1.0 1.50.000.020.040.06\n(a)\n D = 0\n D = 0.05\n D = 0.1\n D = 0.15\n D = 0.2\n D = 0.25Szz\nL ±1(ω = 0)\n(c)Sxx\nL ±1(ω = 0)\nT(b)\nSzz\nL0(ω = 0)\n(d)\nSxx\nL0(ω = 0)\nT\nFigure 10. The temperature dependence of d.c. chiral LSF (a) eSzz\nL\u00061(!= 0), (b) eSzz\nL0(!= 0), (c) eSxx\nL\u00061(!= 0), and (d)\neSxx\nL0(!= 0) for di\u000berent D= 0{0:25 on theN= 30 lattice.\nV. COMPARISON WITH EXPERIMENT\nIn this section we reinstate J6= 1 and SI units.\nA. Inelastic neutron scattering\nINS is a very powerful experimental technique as it\ndirectly probes the magnetic DSF S\u000b\u000b(q;!), with typi-\ncal interaction energies in KLHM materials J\u0018kB(60{\n230 K) = 5{20 meV in a convenient energy range for this\ntechnique. Unfortunately, most KL materials are not\nyet available in single-crystal form, therefore the intrinsic\nDSF anisotropy and q-dependence is often averaged out\nin experiment. To avoid this issue we concentrate on INS\nresults on single-crystal herbertsmithite [37], a material\nthat remains in a SL state down to the lowest measured\nkBT\u001910\u00004J. As the experimentally determined D=\n(0:04{0:08)J[50, 51] plays only a modest role against a\nmuch stronger J=kB\u0019190 K (16:4 meV) in equal-time\nproperties relevant for INS (Fig. 9), we compare INS re-\nsults with model calculations for D= 0. Moreover, low-\n!results may be strongly in\ruenced by structural and\nchemical disorder, especially at low kBT\u001cJ. We there-\nfore restrict ourselves to INS energies ~!>1 meV, above\nthe energy scale of impurity contributions, which mostly\ncontribute to a low- !quasielastic INS peak [37, 61].\nFirstly, we note that experimental frequency-\ndependent INS spectra show a broad maximum at ~!\u0018\n6 meV [37], which is consistent with the calculated low-\nenergy peak at ~!\u00190:3J= 5 meV (Figs. 3 and 4).\nSecondly, our calculations also nicely reproduce the q-dependence of the DSF integrated over a broad frequency\nwindow 1 meV <~!<11 meV (i.e. 0 :06J <~!<0:67J)\nalong the (\u00002;1 +K;0) cut in q-space, which shows the\nmost pronounced variation (Fig. 11). The position of the\nexperimental INS peak at ( \u00002;1;0) corresponds to the\nM point of the extended BZ and is well accounted for by\nour FTLM results. Having separated chiral contributions\n-1.0- 0.50 .00 .51 .00.00.10.20.3 \nHan et al. (2012) \nDimer model \nFTLM \nIntegrated S(q, ω) (arb. units)(\n-2, 1 + K, 0)/s49/s8211/s49/s49/s32/s109/s101/s86\nFigure 11. Low- Therbertsmithite INS measurements of the\nmagnetic DSF S(q;!) (symbols) integrated over 1 meV <\n~! < 11 meV along the ( \u00002;1 +K;0) cut in q-space from\nRef. [37]. The presented magnetic DSF was obtained from\nraw experimental data by dividing INS intensities by the free-\nCu2+magnetic form factor jF(q)j2[37, 62]. The experimental\nmagnetic DSF agrees well with q-interpolated FTLM calcu-\nlations atT= 0:1 on theN= 30 lattice (blue line), but\nsigni\fcantly worse with a toy model of independent singlet\ndimers [37] (dashed line).9\nat di\u000berent wave vectors, we can attribute this peak to\nthe dominant low-energy q= 0,c=\u00061 chiral \ructua-\ntions [Fig. 3 and Eq. (5)]. The position of the peak is\nalso consistent with expectations from the q-dependent\nd.c. susceptibility \u001f0(q), which is also sensitive mainly to\nlow-energy \ructuations, and which also has a very pro-\nnounced peak at the same wavevector (Fig. 6 and discus-\nsion in Sec. III B). Finally, we stress that not only the\nposition but also the width of the experimental INS peak\nis well reproduced by model calculations (Fig. 11), and is\nconsiderably smaller than the width predicted by a sim-\nple independent singlet dimer model [37]. This indicates\nthat the chiral AFM \ructuations in the KLHM have a\nnontrivial low- Tcorrelation length \u0018 > 1 that extends\nbeyond nearest KL neighbors.\nB. NMR spin-lattice relaxation rate\n1. Theory\nNMR spin relaxation experiments probe low-energy\nelectron spin \ructuations via the hyper\fne coupling be-\ntween nuclear and electron spins. In a crystal, the spin-\nlattice relaxation rate of a given nucleus is given by\n[63, 64]\n1\nT1=\r2\nn\n2Z1\n\u00001dt ei!0tX\nij\u000b\f(\u000e\u000b\f\u0000^B\u000b^B\f)h\u000eb\u000b\ni(t)\u000eb\f\nj(0)i;\n(12)\nwhere\rnis the nuclear gyromagnetic ratio, !0=\rnB\u001c\nJ=~is the nuclear Larmor angular frequency in an exter-\nnal \feld B,^B=B=jBjis a unit vector pointing along\nB, and\u000eb\u000b\ni=b\u000b\ni\u0000hb\u000b\nii=\u0000P\n\u0016A\u000b\u0016\niS\u0016\niis the e\u000bective\n\ructuating local \feld at the position of the nucleus due\nto hyper\fne coupling with the electron spin S\u0016\nivia the\nspeci\fc hyper\fne coupling tensor A\u000b\u0016\ni. De\fning the chi-\nral hyper\fne coupling tensor in q-space in analogy with\n[Eq. (3)] as\neA\u000b\u0016\ncq=1p\nNX\nneiq\u0001eRnh\nA\u000b\u0016\n(n;0)+\u0010cA\u000b\u0016\n(n;1)+\u0010\u0000cA\u000b\u0016\n(n;2)i\n;\n(13)\nwe can further succinctly express the NMR spin-lattice\nrelaxation rate in terms of the chiral DSF eS\u000b\f\nc(q;!)\n[Eq. (4)] as\n1\nT1=\u0019\r2\nnX\ncqtrn\neAy\ncq\u0001P?\u0001eAcq\u0001eSc(q;!0)o\n;(14)\nwhere the tensor P?= I\u0000^B\n^Bprojects onto a plane\northogonal to B, whileeAcqandeSc(q;!0) are 3\u00023 tensors\nwith components eA\u000b\u0016\ncqandeS\u000b\f\nc(q;!0), respectively.\nIn the simplest, yet experimentally highly relevant,\ncase of a nucleus coupled to z1spins of the KL triangle\nwith hyper\fne eigenaxes along the crystallographic axes\n[i.e. forA\u000b\u0016\n(n;k)=A\u000b\u000e\u000b\u0016\u000en;n0\u000ek\u0014z1], this further simpli\festo an expression involving only the chiral LSF eS\u000b\u000b\nLc(!0)\n[Eq. (11) and Fig. 10]\n1\nT1=\u0019\r2\nnX\n\u000bA2\n\u000b(1\u0000^B2\n\u000b)1\neT\u000b\u000b\n1;\n1\neT\u000b\u000b\n1=1\n31X\nc=\u00001fceS\u000b\u000b\nLc(!0);(15)\nwhere 1=eT\u000b\u000b\n1are directional contributions to the spin-\nlattice relaxation rate 1 =T1that depend on the number\nof spinsz1the nucleus is coupled to via the chiral form\nfactorsfcsummarized in Table I.\nIn Figs. 12(a{c) we show the impact of di\u000berent z1on\n1=eT\u000b\u000b\n1in more detail. We consider the \u000b=x;ycompo-\nnent due to spin \ructuations within the kagome plane,\nthe\u000b=zcomponent due to out-of-plane spin \ructua-\ntions, and a powder average of both, for both zero and\nlargeD= 0:25Jon theN= 30 lattice. We present our\nresults normalized to Moriya's Gaussian approximation\n[60] for the KLHM where eS\u000b\u000b;Moriya\nLc (0) =SMoriya\nL (0) for\nallcand\u000b[Eq. (10)], yielding\n1\neTMoriya\n1=~z1\n8p\u0019J: (16)\nFirstly, in the z1= 1 case [Fig. 12(a)], where each nucleus\nis coupled to a single magnetic ion, we have f0=f\u00061= 1\n(Table I), i.e. all chiralities contribute equally, and\n1=eT\u000b\u000b\n1=S\u000b\u000b\nL(!0\u00190) [Eq. (9)]. In the intermediate case\nofz1= 2 [e.g. when each nucleus is coupled equally to\ntwo spins on an exchange bond; see inset in Fig. 12(a)] we\nhavef0>f\u00061>0, where again all chiralities contribute\nto 1=eT\u000b\u000b\n1but chiral the c=\u00061 contributions are sup-\npressed compared to the c= 0 contribution [Fig. 12(b)].\nFinally, in the case of z1= 3 [e.g. when each nucleus\nis positioned symmetrically at or above the center of a\nKL triangle; see inset in Fig. 12(a)] we have f\u00061= 0, so\nthat thec=\u00061 \ructuations are completely \fltered out,\nand just the c= 0 chiral LSF [Figs. 10(b,d)] contribute,\nresulting in a nearly isotropic 1 =eT\u000b\u000b\n1= 3eS\u000b\u000b\nL0(!0\u00190)\nsteadily increasing with T[Fig. 12(c)].\nTable I. NMR chiral form factors fcin Eqs. (15) for coupling\nto di\u000berent numbers of spins z1in a single KL triangle and\nexamples of relevant nuclei in herbertsmithite, YCu 3(OH) 6Cl3\nand other compounds [see inset in Fig. 12(a)]. Note thatP\ncfc= 3z1.\nz1Nuclear position Nucleus Non-chiral f0Chiralf\u00061\n1 Magnetic ion (on-site)63,65Cu 1 1\n2 Exchange bond (NN)17O,1H 4 1\n3 Center of spin triangle35Cl 9 010\n0.0 0.5 1.0 1.50.00.20.40.60.80.00.20.40.60.8\n0.00.20.40.60.8\n0.0 0.5 1.0 1.50.00.20.40.60.8\n(d)35Cl in YCu3(OH)6Cl3 (×0.40)\n 4.7 T // c\n 4.7 T // a\n17O in herbertsmithite (×0.57)\n 9 T // c\n 9 T // a*\n 3.2 T // a* \nkBT/J\nTMoriya\n1/T1D = 0.25J\n Powder\n zz\n xx(a) z1 = 1TMoriya\n1/Tαα\n1\nD = 0\n(b) z1 = 2\nTMoriya\n1/Tαα\n1\nD = 0.25J\n Powder\n zz\n xxD = 0\n(c) z1 = 3TMoriya\n1/Tαα\n1\nkBT/JD = 0\nD = 0.25J\n Powder\n zz\n xxO2–\nCu2+\nCl–\nH+\nFigure 12. Directional spin-lattice relaxation rate contributions 1 =eT\u000b\u000b\n1[Eqs. (15) and Table I] normalized by Moriya's Gaussian\napproximation 1 =eTMoriya\n1 [Eq. (16)] for nuclei coupled to z1electron spins where (a) z1= 1 (63,65Cu-type), (b) z1= 2 (17O-type),\nand (c)z1= 3 (35Cl-type nuclei) calculated for D= 0 andD= 0:25Jon theN= 30 lattice. Shown are the \u000b=x;ycomponent\n(xx), the\u000b=zcomponent ( zz), and a powder average of both given by 1 =eTpowder\n1 = (1=3)P\n\u000b1=eT\u000b\u000b\n1. Representative nuclei\nin herbertsmithite and similar materials are shown on the inset in panel (a). (d) Symbols show the17O NMR spin-lattice\nrelaxation rate 1 =T1of herbertsmithite with J=kB= 190 K and z1= 2 from Ref. [38] (green) and the35Cl NMR spin-lattice\nrelaxation rate of YCu 3(OH) 6Cl3withJ=kB= 82 K [26] and z1= 3 from Ref. [56] (red), both normalized by Eq. (17). These\nvalues are further uniformly rescaled by a factor 0 :57 in the case of herbertsmithite and 0 :40 in the case of YCu 3(OH) 6Cl3.\nThese are compared to FTLM results at D= 0 andD= 0:25Jon theN= 30 lattice, with the curves taking appropriate\naverages over relevant directions \u000band chiral form factors fcin Eqs. (15) and Eq. (17).\n2. Experiment\nFirst we compare our FTLM model results with NMR\nexperiments on herbertsmithite, ZnCu 3(OH) 6Cl2. Even\nthough several experimental NMR spin relaxation stud-\nies have been carried out on this material over the years,\nsingle-crystal studies at kBT >0:1Jrelevant for compar-\nison with our model calculations are rare. In Fig. 12(d)\nwe summarize the17O NMR 1=T1results from Ref. [38]\nmeasured in the direction orthogonal to the kagome\nplanes (c-axis) and within the kagome planes ( a\u0003-axis).\nThe appropriate components of the hyper\fne coupling\ntensors (Aa;Aa\u0003;Ac) = (3:3ga;4:3ga;3:6gc) T are taken\nfrom Ref. [29]. Here ga= 2:14 andgc= 2:25 are in-\nplane and out-of-plane components, respectively, of the\nCu2+g-factor tensor at high- T[49]. As the oxygen nuclei\nare positioned symmetrically with respect to two neigh-\nboring magnetic Cu2+ions [inset in Fig. 12(a)], we have\nz1= 2 and the corresponding chiral form factors are\nf0= 4 andf\u00061= 1 (Table I). To compare our calcula-\ntions with experiment, we normalize all 1 =T1values toGaussian approximation [60]\n1\nTMoriya\n1=p\u0019\r2\nn~z1\n8JX\n\u000bA2\n\u000b(1\u0000^B2\n\u000b); (17)\nwhich can be obtained by inserting the directional\n1=eTMoriya\n1 [Eq. (16)] into the full 1 =T1[Eqs. (15)]. Like\nin the INS analysis in Sec. V A we compare experimental\nresults with FTLM calculations for D= 0, as the e\u000bect\nof the DM interaction on the chiral LSF at the experi-\nmentally determined value of D= (0:04{0:08)J[50, 51]\nis very small for all directions and chiralities (see the\nD= 0:05 curves in Fig. 10). The experimental 1 =T1\nalong the two crystallographic directions indeed coincide\nwhen normalized by Eq. (17), and their graduate decrease\nwith lowering Tnicely follows the theoretical prediction\ndown tokBT\u00190:3J. The downturn of the experimental\n17O NMR 1=T1below this temperature, which ultimately\nleads to 1=T1/T0:84belowkBT\u00190:05J[39], also seems\nto be qualitatively supported by our model calculations,\nwhere in the latter the downturn is the signature of a\nquite robust triplet gap \u0001 t>0 in the considered D= 0\nmodel system.\nThe second experimental example is the novel KL ma-11\nterial YCu 3(OH) 6Cl3, which, like herbertsmithite, has a\nnearest-neighbor Heisenberg exchange coupling J=kB=\n82 K that is by far the dominant isotropic magnetic inter-\naction [26]. However, unlike herbertsmithite, this mate-\nrial enters a chiral 120 °AFM LRO GS at kBTN= 0:15J\n[45, 47, 48], which is attributed to a sizable out-of-plane\nDM interaction D= 0:25J[26].35Cl NMR 1=T1results\nfrom Ref. [56] measured in the direction orthogonal to the\nkagome planes ( c-axis) and within the kagome planes ( a-\naxis), on one of the two chlorine crystallographic sites,\nare shown in Fig. 12(d). The chosen35Cl site is cou-\npled symmetrically with all three Cu2+spins on a given\nKL spin triangle [inset in Fig. 12(a)], similar to chlorine\nsites in herbertsmithite. The appropriate components of\nthe hyper\fne coupling tensor to a single electron spin are\nAa=Ac= 0:28 T. As evident by model calculations for a\nnucleus in such a symmetric z1= 3 position [Fig. 12(c)],\nthe anisotropy of the measured 1 =T1is minimal, sug-\ngesting that highly anisotropic chiral c=\u00061 local spin\n\ructuations [Figs. 10(a,c)] are indeed highly suppressed\nat the35Cl site, broadly consistent with the expected\nchiral form factors (Table I). Nevertheless, even though\nthe theoretically predicted trend of decreasing 1 =T1with\nloweringTis followed by experiment, the experimentally-\nobserved decrease is less pronounced [Fig. 12(d)]. As the\nc= 0 chiral LSF, which should represent the only contri-\nbution to 1=T1according to Table I, is expected to nearly\nvanish at low T[Figs. 10(b,d)], while the experimental\n1=T1does not, this suggests that a remnant c=\u00061 con-\ntribution must still a\u000bect the experimental 1 =T1to a cer-\ntain extent. This could be a telltale sign of reduced local\nthreefold rotational symmetry in YCu 3(OH) 6Cl3, similar\nto the recently discovered symmetry reduction in her-\nbertsmithite [49].\nFinally, we note that the experimental 1 =T1results\non both herbertsmithite and YCu 3(OH) 6Cl3need to be\nrescaled by factors of 0 :57 and 0:40, respectively, to\nachieve a quantitative match with model calculations\n[Fig. 12(d)]. This might be partly attributed to uncer-\ntainty in experimental parameters such as the hyper\fne\ncoupling constants. A further source of uncertainty is\nalso the spin di\u000busion contribution, i.e. the contribution\nfrom theq\u00190,c= 0 spin \ructuations, which we omit as\nmentioned in Sec. III A. In generic 2D systems at !0!0\nthis contribution might even be singular [59]. Never-\ntheless, in systems with strong AFM \ructuations, like\ncuprates [65] and KLHM materials, the spin di\u000busion\ncontribution to 1 =T1is generally considered to be rela-\ntively small and sizable only at high T. Still, it might\ncontribute a relevant quantitative correction to the cal-\nculated 1=T1.\nVI. SUMMARY AND OUTLOOK\nOur comprehensive numerical study of the dynamical\nspin correlations of the KL AFM via FTLM calculations\nhas led to several pertinent \fndings. By separating thechiral correlations ( c=\u00061) from non-chiral ones ( c= 0),\nwe have shown that former dominate the low-energy dy-\nnamics even of the isotropic D= 0 KLHM (Figs. 2\nand 3). These corresponds to \ructuations of the uni-\nform (q= 0) 120 °AFM order parameter, leading to a\npronounced low-energy response in the DSF S(q;!) at\nthe M point of the extended BZ. The dominant chiral\nDSF features a nontrivial frequency dependence charac-\nterized by a double-maximum structure that persists up\ntokBT\u0019J(Fig. 4), even though the lower-energy peak\ncorresponds to energies of only around 0 :3J. As a di-\nrect consequence of this low-energy peak, the d.c. sus-\nceptibility\u001f0(q) exhibits a pronounced peak at the M\npoint at low T(Fig. 6). In clear contrast, the equal-time\nS(q), which sums over all energies, exhibit a pronounced\nregion of high intensity that is spread out around the\nwhole extended BZ boundary, remains stable up to high\nkBT\u00182J, and has apparent weak maxima in the corner\nK points of the extended BZ (Fig. 5).\nAllowing for \fnite DM interactions perpendicular to\nthe kagome plane makes the chiral DSF anisotropic\n(Fig. 8). Such magnetic anisotropy mainly a\u000bects the\nq= 0,c=\u00061 chiral 120 °AFM \ructuations, which\nsoften at the quantum critical point Dc\u00190:1J. The\ncorresponding out-of-plane chiral DSF response spec-\ntraeSzz\n\u00061(q= 0;!) become more coherent with increas-\ningDwith an increase in an e\u000bective out-of-plane spin\ntriplet gap [Fig. 8(a)], while in-plane chiral DSF spectra\neSxx\n\u00061(q= 0;!) show enhanced low-energy \ructuations and\nlonger-range correlations [Figs. 8(b) and 9]. The change\nin local (i.e. integrated over q) spin \ructuations, which\nare highly relevant for local-probe experiments like NMR,\nfrom the isotropic D= 0 case is also dominated by chiral\nc=\u00061 \ructuations (Fig. 10).\nAll of the observed characteristic features of the KL\nantiferromagnet DSF can also be probed experimen-\ntally via INS and NMR spin-lattice relaxation measure-\nments. We critically compare our results to two most\nrelevant examples of the nearest-neighbor KL materials,\nthe archetypal herbertsmithite and the novel KL mate-\nrial YCu 3(OH) 6Cl3. The former possesses rather small\nDM magnetic anisotropy and lacks LRO down to the\nlowest experimentally accessible temperatures, while the\nlatter is characterized by a much larger DM anisotropy\nand chiral 120 °LRO at low T. Single-crystal KL INS\nmeasurements with the required q-space resolution are\nso far only available for herbertsmithite. These measure-\nments indeed show a broad low-energy peak [37] at en-\nergies that are entirely consistent with the lower-energy,\n0:3Jpeak that we \fnd numerically. Furthermore, our\nmodel calculations also convincingly reproduce the vari-\nation ofS(q;!) measured along the ( \u00002;1 +K;0)q-cut\nin Ref. [37] (Fig. 11). We \fnd that the peak is consider-\nably narrower than predicted by a simple singlet-dimer\ntoy model [37], which indicates that chiral AFM \ructua-\ntions in the KLHM have a \fnite low- Tcorrelation length\n\u0018>1.\nFurthermore,17O NMR spin-lattice relaxation rate12\n1=T1measurements on herbertsmithite [38] are reason-\nably reproduced by model calculations [Fig. 12(d)], show-\ning that the e\u000bect of small DM interactions that are\npresent in this compound on dynamical spin correla-\ntions is indeed small. The experimental T-dependence\nis well consistent with numerical result, in particular at\nkBT > 0:3J. The observed variation predominantly re-\n\rects the evolution of the non-chiral ( c= 0) \ructuations,\nas the contribution of the chiral ( c=\u00061) \ructuations is\npartly \fltered out on the symmetric position of the17O\nnuclei. The situation is even more extreme in the case\nof35Cl NMR spin-lattice relaxation rate 1 =T1measure-\nments on YCu 3(OH) 6Cl3[56], where chiral c=\u00061 \ructu-\nations should be completely \fltered out due to rotational\nsymmetry at the nuclear site. Indeed, we observe almost\nno anisotropy in experimental 1 =T1, however the experi-\nment notably deviates from theory at low T, suggesting\nthat the chiral contribution might still contribute, likely\ndue to reduced local rotational symmetry.\nOur study has demonstrated that detailed knowledge\nof the dynamical spin structure factor of KL AFM atT >0 can indeed provide invaluable insight into the na-\nture of its low-energy spin excitations and represent a\nlink to numerous theoretical studies of the ground state of\nthis enigmatic model. Especially intriguing is the robust,\nyet hitherto underappreciated, chiral nature of the dom-\ninant spin \ructuations. The scope of our results is fur-\nther extended by the inclusion of experimentally highly-\nrelevant DM interactions with nontrivial consequences.\nWe have furthermore demonstrated that our unbiased\nstate-of-the-art numerical calculations provide a reliable\nbasis upon which past and future experiments on kagome\nmaterials can be judged and interpreted.\nACKNOWLEDGMENTS\nWe acknowledge the \fnancial support of the Slove-\nnian Research Agency through programs P1-0044 and\nP1-0125, and projects N1-0088, Z1-1852, N1-0148, and\nJ1-2461.\n[1] P. A. Lee, An end to the drought of quantum spin liquids,\nScience 321, 1306 (2008).\n[2] L. Balents, Spin liquids in frustrated magnets, Nature\n464, 199 (2010).\n[3] L. Savary and L. Balents, Quantum spin liquids: A re-\nview, Rep. Prog. Phys. 80, 016502 (2017).\n[4] C. Broholm, R. J. Cava, S. A. Kivelson, D. G. Nocera,\nM. R. Norman, and T. Senthil, Quantum Spin Liquids,\nScience 367, 263 (2020).\n[5] F. Mila, Low-energy sector of the kagome antiferromag-\nnet, Phys. Rev. Lett. 81, 2356 (1998).\n[6] C. Waldtmann, H. U. Everts, B. Bernu, C. Lhuillier,\nP. Sindzingre, P. Lecheminant, and L. Pierre, First ex-\ncitations of the spin 1/2 Heisenberg antiferromagnet on\nthe kagom\u0013 e lattice, Eur. Phys. J. B 2, 501 (1998).\n[7] R. R. P. Singh and D. A. Huse, Ground state of the spin-\n1/2 kagome-lattice Heisenberg antiferromagnet, Phys.\nRev. B 76, 180407(R) (2007).\n[8] R. R. P. Singh and D. A. Huse, Triplet and singlet excita-\ntions in the valence bond crystal phase of the kagome lat-\ntice Heisenberg model, Phys. Rev. B 77, 144415 (2008).\n[9] S. Yan, D. A. Huse, and S. R. White, Spin-liquid ground\nstate of the S=1/2 kagome Heisenberg antiferromagnet,\nScience 322, 1173 (2011).\n[10] A. M. L auchli, J. Sudan, and E. S. S\u001crensen, Ground-\nstate energy and spin gap of spin-1/2 Kagom\u0013 e-Heisenberg\nantiferromagnetic clusters: Large-scale exact diagonal-\nization results, Phys. Rev. B 83, 212401 (2011).\n[11] Y. Iqbal, F. Becca, and D. Poilblanc, Valence-bond crys-\ntal in the extended kagome spin-1/2 quantum Heisenberg\nantiferromagnet: A variational Monte Carlo approach,\nPhys. Rev. B 83, 100404(R) (2011).\n[12] S. Depenbrock, I. P. McCulloch, and U. Schollw ock, Na-\nture of the Spin-Liquid Ground State of the S= 1=2\nHeisenberg Model on the Kagome Lattice, Phys. Rev.\nLett. 109, 067201 (2012).[13] J. Schnack, J. Schulenburg, and J. Richter, Magnetism of\ntheN= 42 kagome lattice antiferromagnet, Phys. Rev.\nB98, 094423 (2018).\n[14] A. M. L auchli, J. Sudan, and R. Moessner, S=1/2 kagome\nHeisenberg antiferromagnet revisited, Phys. Rev. B 100,\n155142 (2019).\n[15] P. Prelov\u0014 sek, K. Morita, T. Tohyama, and J. Herbrych,\nVanishing Wilson ratio as the hallmark of quantum spin-\nliquid models, Phys. Rev. Research 2, 023024 (2020).\n[16] Y. Ran, M. Hermele, P. A. Lee, and X.-G. Wen,\nProjected-Wave-Function Study of the Spin-1 =2 Heisen-\nberg Model on the Kagom\u0013 e Lattice, Phys. Rev. Lett. 98,\n117205 (2007).\n[17] Y. Iqbal, F. Becca, S. Sorella, and D. Poilblanc, Gap-\nless spin-liquid phase in the kagome spin-1\n2Heisenberg\nantiferromagnet, Phys. Rev. B 87, 060405(R) (2013).\n[18] Z. Y. Xie, J. Chen, J. F. Yu, X. Kong, B. Normand,\nand T. Xiang, Tensor renormalization of quantum many-\nbody systems using projected entangled simplex states,\nPhys. Rev. X 4, 011025 (2014).\n[19] Y. Iqbal, D. Poilblanc, and F. Becca, Vanishing spin\ngap in a competing spin-liquid phase in the kagome\nHeisenberg antiferromagnet, Phys. Rev. B 89, 020407(R)\n(2014).\n[20] Y.-C. He, M. P. Zaletel, M. Oshikawa, and F. Poll-\nmann, Signatures of Dirac Cones in a DMRG Study of\nthe Kagome Heisenberg Model, Phys. Rev. X 7, 031020\n(2017).\n[21] H. J. Liao, Z. Y. Xie, J. Chen, Z. Y. Liu, H. D. Xie, R. Z.\nHuang, B. Normand, and T. Xiang, Gapless spin-liquid\nground state in the s= 1=2 kagome antiferromagnet,\nPhys. Rev. Lett. 118, 137202 (2017).\n[22] G. Misguich and P. Sindzingre, Magnetic suscepti-\nbility and speci\fc heat of the spin-1/2 Heisenberg\nmodel on the kagome lattice and experimental data on\nZnCu 3(OH) 6Cl2, Eur. Phys. J. B 59, 305 (2007).13\n[23] B. Bernu, L. Pierre, K. Essa\f, and L. Messio, E\u000bect of\nperturbations on the kagome s=1\n2antiferromagnet at\nall temperatures, Phys. Rev. B 101, 140403 (2020).\n[24] M. Rigol and R. R. P. Singh, Magnetic Susceptibility\nof the Kagome Antiferromagnet ZnCu 3(OH)6Cl2, Phys.\nRev. Lett. 98, 207204 (2007).\n[25] M. Rigol and R. R. P. Singh, Kagome lattice antiferro-\nmagnets and Dzyaloshinsky-Moriya interactions, Phys.\nRev. B 76, 184403 (2007).\n[26] T. Arh, M. Gomil\u0014 sek, P. Prelov\u0014 sek, M. Pregelj,\nM. Klanj\u0014 sek, A. Ozarowski, S. J. Clark, T. Lancaster,\nW. Sun, J.-X. Mi, and A. Zorko, Origin of magnetic or-\ndering in a structurally perfect quantum kagome antifer-\nromagnet, Phys. Rev. Lett. 125, 027203 (2020).\n[27] T. Shimokawa and H. Kawamura, Finite-temperature\ncrossover phenomenon in the S= 1=2 antiferromagnetic\nHeisenberg model on the kagome lattice, J. Phys. Soc.\nJpn.85, 6 (2016).\n[28] N. E. Sherman and R. R. P. Singh, Structure factors of\nthe kagome-lattice Heisenberg antiferromagnets at \fnite\ntemperatures, Phys. Rev. B 97, 014423 (2018).\n[29] N. E. Sherman, T. Imai, and R. R. P. Singh, Nuclear\nrelaxation rates in the herbertsmithite kagome antifer-\nromagnets ZnCu 3(OH)6Cl2, Phys. Rev. B 94, 140415\n(2016).\n[30] Z. Hao and O. Tchernyshyov, Structure factor of low-\nenergy spin excitations in a S= 1=2 kagome antiferro-\nmagnet, Phys. Rev. B 81, 1 (2010).\n[31] M. Punk, D. Chowdhury, and S. Sachdev, Topological ex-\ncitations and the dynamic structure factor of spin liquids\non the kagome lattice, Nat. Phys. 10, 289 (2014).\n[32] W. Zhu, S.-s. Gong, and D. N. Sheng, Identifying spinon\nexcitations from dynamic structure factor of spin-1/2\nHeisenberg antiferromagnet on the Kagome lattice, Proc.\nNatl. Acad. Sci. 116, 5437 (2019).\n[33] T. Shimokawa, K. Watanabe, and H. Kawamura, Static\nand dynamical spin correlations of the S=1\n2random-\nbond antiferromagnetic Heisenberg model on the trian-\ngular and kagome lattices, Phys. Rev. B 92, 1 (2015).\n[34] M. P. Shores, E. A. Nytko, B. M. Bartlett, and D. G.\nNocera, A structurally perfect s = 1/2 kagom\u0013 e antiferro-\nmagnet, J. Am. Chem. Soc. 127, 13462 (2005).\n[35] P. Mendels and F. Bert, Quantum kagome antiferro-\nmagnet ZnCu 3(OH) 6Cl2, J. Phys. Soc. Jpn. 79, 011001\n(2010).\n[36] M. R. Norman, Colloquium: Herbertsmithite and the\nsearch for the quantum spin liquid, Rev. Mod. Phys. 88,\n041002 (2016).\n[37] T. H. Han, J. S. Helton, S. Chu, D. G. Nocera, J. A.\nRodriguez-Rivera, C. Broholm, and Y. S. Lee, Fraction-\nalized excitations in the spin-liquid state of a kagome-\nlattice antiferromagnet, Nature 492, 406 (2012).\n[38] M. Fu, T. Imai, T.-H. Han, and Y. S. Lee, Evidence for a\ngapped spin-liquid ground state in a kagome Heisenberg\nantiferromagnet, Science 350, 655 (2015).\n[39] P. Khuntia, M. Velazquez, Q. Barth\u0013 elemy, F. Bert,\nE. Kermarrec, A. Legros, B. Bernu, L. Messio, A. Zorko,\nand P. Mendels, Gapless ground state in the archetypal\nquantum kagome antiferromagnet ZnCu 3(OH) 6Cl2, Nat.\nPhys. 16, 469 (2020).\n[40] Z. Hiroi, M. Hanawa, N. Kobayashi, M. Nohara, H. Tak-\nagi, Y. Kato, and M. Takigawa, Spin-1/2 Kagome-Like\nLattice in Volborthite Cu 3V2O7(OH) 2·2H2O, J. Phys.\nSoc. Jpn. 70, 3377 (2001).[41] B. F\u0017 ak, E. Kermarrec, L. Messio, B. Bernu, C. Lhuil-\nlier, F. Bert, P. Mendels, B. Koteswararao, F. Bouquet,\nJ. Ollivier, A. D. Hillier, A. Amato, R. H. Colman, and\nA. S. Wills, Kapellasite: A kagome quantum spin liq-\nuid with competing interactions, Phys. Rev. Lett. 109,\n037208 (2012).\n[42] Y. Li, B. Pan, S. Li, W. Tong, L. L., Z. Yang, J. Wang,\nZ. Chen, Z. Wu, and Q. Zhang, Gapless quantum spin\nliquid in the S = 1/2 anisotropic kagome antiferromagnet\nZnCu 3(OH) 6SO4, New J. Phys. 16, 093011 (2014).\n[43] M. Gomil\u0014 sek, M. Klanj\u0014 sek, M. Pregelj, F. C. Coomer,\nH. Luetkens, O. Zaharko, T. Fennell, Y. Li, Q. M. Zhang,\nand A. Zorko, Instabilities of spin-liquid states in a quan-\ntum kagome antiferromagnet, Phys. Rev. B 93, 060405\n(2016).\n[44] Z. Feng, Z. Li, X. Meng, W. Yi, Y. Wei, J. Zhang, Y. C.\nWang, W. Jiang, Z. Liu, S. Li, F. Liu, J. Luo, S. Li, G. Q.\nZheng, Z. Y. Meng, J. W. Mei, and Y. Shi, Gapped Spin-\n1/2 Spinon Excitations in a New Kagome Quantum Spin\nLiquid Compound Cu 3Zn(OH) 6FBr, Chin. Phys. Lett.\n34(2017).\n[45] A. Zorko, M. Pregelj, M. Klanj\u0014 sek, M. Gomil\u0014 sek,\nZ. Jagli\u0014 ci\u0013 c, J. S. Lord, J. A. T. Verezhak, T. Shang,\nW. Sun, and J.-X. Mi, Coexistence of magnetic order\nand persistent spin dynamics in a quantum kagome anti-\nferromagnet with no intersite mixing, Phys. Rev. B 99,\n214441 (2019).\n[46] W. Sun, Y.-X. Huang, S. Nokhrin, Y. Pan, and J.-X.\nMi, Perfect Kagom\u0013 e lattices in YCu 3(OH) 6Cl3: a new\ncandidate for the quantum spin liquid state, J. Mater.\nChem. C 4, 8772 (2016).\n[47] Q. Barth\u0013 elemy, P. Puphal, K. M. Zoch, C. Krell-\nner, H. Luetkens, C. Baines, D. Sheptyakov, E. Ker-\nmarrec, P. Mendels, and F. Bert, Local study\nof the insulating quantum kagome antiferromagnets\nYCu 3(OH)6OxCl3\u0000x(x= 0;1=3), Phys. Rev. Materials\n3, 074401 (2019).\n[48] A. Zorko, M. Pregelj, M. Gomil\u0014 sek, M. Klanj\u0014 sek, O. Za-\nharko, W. Sun, and J.-X. Mi, Negative-vector-chirality\n120\u000espin structure in the defect- and distortion-free\nquantum kagome antiferromagnet YCu 3(OH)6Cl3, Phys.\nRev. B 100, 144420 (2019).\n[49] A. Zorko, M. Herak, M. Gomil\u0014 sek, J. van Tol,\nM. Vel\u0013 azquez, P. Khuntia, F. Bert, and P. Mendels,\nSymmetry Reduction in the Quantum Kagome Antifer-\nromagnet Herbertsmithite, Phys. Rev. Lett. 118, 017202\n(2017).\n[50] A. Zorko, S. Nellutla, J. van Tol, L. C. Brunel, F. Bert,\nF. Duc, J. C. Trombe, M. A. de Vries, A. Harrison,\nand P. Mendels, Dzyaloshinsky-Moriya anisotropy in the\nspin-1/2 kagome compound ZnCu 3(OH) 6Cl2, Phys. Rev.\nLett. 101, 026405 (2008).\n[51] S. El Shawish, O. C\u0013 epas, and S. Miyashita, Electron spin\nresonance in S= 1=2 antiferromagnets at high tempera-\nture, Phys. Rev. B 81, 224421 (2010).\n[52] O. C\u0013 epas, C. M. Fong, P. W. Leung, and C. Lhuillier,\nQuantum phase transition induced by Dzyaloshinskii-\nMoriya interactions in the kagome antiferromagnet,\nPhys. Rev. B 78, 140405(R) (2008).\n[53] M. Elhajal, B. Canals, and C. Lacroix, Symmetry break-\ning due to Dzyaloshinsky-Moriya interactions in the\nkagom\u0013 e lattice, Phys. Rev. B 66, 014422 (2002).\n[54] A. Zorko, F. Bert, A. Ozarowski, J. van Tol, D. Boldrin,\nA. S. Wills, and P. Mendels, Dzyaloshinsky-Moriya in-14\nteraction in vesignieite: A route to freezing in a quan-\ntum kagome antiferromagnet, Phys. Rev. B 88, 144419\n(2013).\n[55] J. Jakli\u0014 c and P. Prelov\u0014 sek, Finite-temperature properties\nof doped antiferromagnets, Adv. Phys. 49, 1 (2000).\n[56] T. Arh, M. Klanj\u0014 sek, M. Pregelj, M. Gomil\u0014 sek, and\nA. Zorko, Local insight into spin correlations in the\nkagome antiferromagnet YCu 3(OH) 6Cl3, manuscript in\npreparation.\n[57] J. Jakli\u0014 c and P. Prelov\u0014 sek, Lanczos method for the calcu-\nlation of \fnite-temperature quantities in correlated sys-\ntems, Phys. Rev. B 49, 5065 (1994).\n[58] P. Prelov\u0014 sek and J. Bon\u0014 ca, Ground State and Finite\nTemperature Lanczos Methods, in Strongly Correlated\nSystems|Numerical Methods , edited by A. Avella and\nF. Mancini (Springer, Berlin, 2013).\n[59] A. Sokol, E. Gagliano, and S. Bacci, Theory of nuclear\nspin{lattice relaxation in La 2CuO 4at high temperatures,\nPhys. Rev. B 47, 14646 (1993).\n[60] T. Moriya, Nuclear magnetic relaxation in antiferromag-\nnets, Prog. Theor. Phys. 16, 23 (1956).\n[61] T.-H. Han, M. R. Norman, J.-J. Wen, J. A. Rodriguez-Rivera, J. S. Helton, C. Broholm, and Y. S. Lee, Cor-\nrelated impurities and intrinsic spin-liquid physics in\nthe kagome material herbertsmithite, Phys. Rev. B 94,\n060409 (2016).\n[62] G. Shirane, S. Shapiro, and J. Tranquada, Neutron Scat-\ntering with a Triple-Axis Spectrometer: Basic Techniques\n(Cambridge University Press, Cambridge, 2006).\n[63] M. Horvati\u0013 c, Magnetic nuclear spin-lattice relaxation in\nNMR of orthorhombic crystals in the presence of strong\nquadrupole coupling, J. Phys. Condens. Matter. 4, 5811\n(1992).\n[64] M. Horvati\u0013 c and C. Berthier, NMR Studies of Low-\nDimensional Quantum Antiferromagnets, in High Mag-\nnetic Fields: Applications in Condensed Matter Physics\nand Spectroscopy , edited by C. Berthier, L. P. L\u0013 evy, and\nG. Martinez (Springer Berlin Heidelberg, Berlin, Heidel-\nberg, 2001) pp. 191{210.\n[65] T. Imai, C. P. Slichter, K. Yoshimura, and K. Kosuge,\nLow frequency spin dynamics in undoped and Sr-doped\nLa2CuO 4, Phys. Rev. Lett. 70, 1002 (1993)."    },    {        "title": "1505.00106v1.Spin_charge_transport_driven_by_magnetization_dynamics_on_disordered_surface_of_doped_topological_insulators.pdf",        "content": "arXiv:1505.00106v1  [cond-mat.mes-hall]  1 May 2015Spin-charge transport driven by magnetization dynamics on\ndisordered surface of doped topological insulators\nK. Taguchi, K. Shintani, and Y. Tanaka\nDepartment of Applied Physics, Nagoya University, Nagoya, 464-8603, Japan\nand CREST, Japan Science and Technology Corporation (JST), Nagoya 464-8603, Japan\n(Dated: June 4, 2021)\nAbstract\nWe theoretically study the spin and charge generation along with the electron transport on a\ndisordered surface of a doped three-dimensional topologic al insulator/magnetic insulator junction\nby using Green’s function techniques. We find that the spin an d charge current are induced by not\nonly local but also by nonlocal magnetization dynamics thro ugh nonmagnetic impurity scattering\non the disordered surface of the doped topological insulato r. We also clarify that the spin current\nas well as charge density are induced by spatially inhomogen eous magnetization dynamics, and\nthe spin current diffusively propagates on the disordered sur face. Using these results, we discuss\nboth local and nonlocal spin torques before and after the spi n and spin current generation on the\nsurface, and provide a procedure to detect the spin current.\n1I. INTRODUCTION\nIn spintronics, the mutual control of the direction and the flow of spin is a central issue\nfor wide applications. The flow of spin, i.e., spin current, is the differen ce between the\ncurrents of up and down-spin conduction electrons. It is known th at the spin current is\ninduced in the setup of the ferromagnetic metal (FM)/normal met al (NM) junction1–3. Its\norigin is due to the magnetization dynamics in ferromagnet, which tra nsfers the spin angular\nmomentum of the magnetization into that of the conduction electro ns. The transfer of the\nspin angular momentum is called spin-pumping. Here, the spin-pumping of magnetization\ndynamics generates the spin-current in the NM, and the spin-curr ent can be converted into\ncharge current through spin-orbit interactions1–5.\nTopological insulator (TI) is a new class of materials which has a gaples s surface state,\ndubbed as the helical surface state, in which the spin and momentum are locked by the spin-\norbit interactions6–8. On the surface of the TI, the direction of charge current and th at of\nthe spin of conduction electrons can be mutually manipulated by an ap plied electromagnetic\nfield through the spin-momentum locking. There have been many the oretical works in\nhybrid systems including superconducting junctions on the surfac e of TI stimulated by the\nexotic surface state7,9–12. In the TI/FM junctions, the anomalous charge-spin transport12–15,\nthe anomalous tunnel conductance16–20, the giant magneto resistance21–23, and the current-\ninduced spin-transfer torque24,25have been studied up to now. The exotic phenomena are\ntriggered in the presence of static magnetization and an applied elec tromagnetic field. The\nmagnetizationoftheferromagnetplaystheroleofaneffectivevec torpotentialforconduction\nelectrons, which is like a vector potential of electromagnetic fields. Owing to the effective\nvector potential, the time-derivative of the magnetization can be r egarded as an effective\nelectric field, and the magnetization dynamics generates charge cu rrent on the surface of the\nTI/FM junction even in the absence of electromagnetic fields26–29. This is called as the spin-\ncharge conversion. The direction of the induced charge current is perfectly perpendicular\nto the magnetization dynamics due to the spin-momentum locking. Th e relation between\nthe direction of the magnetization and the induced charge current can be a characteristic\nproperty on the surface of the TI. The property of the spin-pum ping on the surface of TI\ncan be applicable for spintronics devices.\nExisting works of the spin-charge conversion have been done in the case of a clean sur-\n2face of the TI, namely the ballistic transport regime. However, the actual charge trans-\nport on the surface of the TI is in the diffusive regime due to the nonm agnetic impurity\nscattering21–25,30–32. Since Burkov et al., have predicted not only the local but also the no n-\nlocal current onthe disordered surface of the TI inthe presence of the applied electric field21;\nwe can naturally expect nonlocal current is driven by the magnetiza tion dynamics even on\nthe disordered surface of the TI in the presence of the magnetiza tion dynamics.\nIn this article, we study the charge-spin transport due to the mag netization dynamics on\nthe disordered surface of the three-dimensional doped TI/magn etic insulator (MI) junction,\nas shown in Fig. 1, where we show that charge current and spin polar ization on the surface\nof the TI are induced not only by a local, but also by a nonlocal magnet ization dynamics.\nBesides this, we clarify that the spin current is driven by the dynamic s of the spatially\ninhomogenous magnetization, and the spin current diffusively propa gates on the surface.\nThe magnitude of the spin current reflects the spatially inhomogeno us spin structure of the\nMI. The directions of the spin flow and the spin projection of the spin current are perfectly\nlinked by the spin-momentum locking on the surface of TIs. The pres ent features may serve\nas a guide to fabricate future spintronics devices based on the sur face of TIs with magnetic\nsubstance.\nThe merit of the choice of MI on the surface of the TI instead of met allic ferromagnet is\nto prevent the induced charge current going through the bulk of t he MI. Then, we can focus\non the charge transport on the surface of TI. Besides, the Gilber t damping constant in MIs\ntends to be smaller than that in ferromagnetic metals. The small valu e of the damping in\nMIs can be useful for the detection of the spin current on the diso rdered surface of the TI,\nas discussed in sec. VC\nII. MODEL\nWe consider conduction electrons coupled to an effective localized sp in on the disordered\nsurface of the three-dimensional doped TI attached with the MI, as shown in Fig. 1. The\nsetup in Fig. 1 is similar to a system, where conduction electrons coup le with the magnetic\nmoments of ferromagnetic metals deposited on the surface of the TI12,16,17. We expect\nthat on the surface of the TI, the effective localized spin ( S) can be produced from the\nmagnetization in the MI through magnetic proximity effects. In the f ollowing, we use the\n3Hamiltonian, describing the surface of the TI with MI, given by\nH=HTI+Hsd+Vimp, (1)\nwhere the first term in Eq. (1), HTIis Hamiltonian of the conduction electrons on one of\nthe surface of the doped TI without Sas\nHTI=/integraldisplay\ndxψ†[−i/planckover2pi1vF(ˆσ×∇)z−ǫF]ψ, (2)\nHere,ψ†≡ψ†(x,t) = (ψ†\n↑ψ†\n↓), andψare the creation and annihilation operators of the\nconduction electron, respectively (where indices ↑and↓represent spin), ǫFis the Fermi\nenergy, and vFis the Fermi velocity of the bare electron on the surface of the dop ed TI. The\nˆσis the Pauli matrices in spin space. The second term of Eq. (1), Hsd, shows the exchange\ninteraction between the conduction electron spin s=1\n2ψ†σψand the localized spin Son\nthe disordered surface of the doped TI, as described by\nHsd=−/integraldisplay\ndxJsdψ†S·ˆσψ, (3)\nwhereJsd>0 is the exchange coupling constant. The localized spin Scan be described\nby the magnetization of the MI as S=−(S/M)M, whereSandMare the magnitude of\nthe localized spin and of the magnetization, respectively. We conside r that in general, the\nlocalized spin S≡S(x,t) depends on the time and position on the surface of the TI. The\nS(x,t) changes slowly compared with the electron transport relaxation t ime (τ) and varies\nin space compared with the electron mean-free path ( ℓ). We expect that from the Eqs.\n(2)-(3), the in-plane component of the localized spin, S/bardbl≡S−Szz, can play the role of\nthe effective vector potential for the conduction electrons on th e surface. The out-of plane\ncomponent of the localized spin Szplays a role to open the energy gap of the dispersion on\nthe surface of the doped TI. We assume that the band gap opened bySzis smaller than the\nFermi energy on the surface of the doped TI, i.e.,ǫF−JsdSz>0. The third term of Eq. (1),\nVimp=Ni/summationdisplay\nj=1/integraldisplay\ndxUiψ†ψ, (4)\nrepresents nonmagnetic impurity scattering on the disordered su rface of the doped TI. The\nimpurity scattering causes the relaxation time τof the transport of conduction electrons on\nthesurfaceoftheTI. Here Ui=uiδ(x−rj)isadelta-functiontype potential, uiisapotential\n4FIG. 1: (Color Online) A setup of spin-charge current genera tion due to magnetization dynamics\nin MIs deposited on disordered surface of a doped TI.\nenergy density, rjis the position of impurities, and Nishows the number of impurities. The\ncontribution of Vican be treated by the impurity average.\nWe will calculate the spin current and charge current due to the spin -pumping in the\nlinear response to Sunder the condition Jsd≪/planckover2pi1/τ. We expect that the condition could\nbe realized on a metallic disordered surface of the TI, which satisfies /planckover2pi1/(ǫFτ)≪1. For\nexample, the exchange coupling Jsdcan be estimated by Jsd≃6meV28for Ni81Fe19/Bi2Te3\njunction. If/planckover2pi1\nǫFτ<10−2is satisfied, the perturbation can be accessible.\nA. Renormalization of the Fermi velocity\nGreen’s function on disordered surface of doped TI can be describ ed by using HTIin-\ncludingVimpwithin the self-consistent Born approximation of Vimpas\nˆgk,ω= [/planckover2pi1ω−{/planckover2pi1vF(ˆσ×k)z−ǫF}−ˆΣk,ω]−1, (5)\nwhereˆΣk,ωis the self-energy within the Born approximation given by\nˆΣk,ω=ni/summationdisplay\nk′|uk−k′|2ˆgk′,ω. (6)\nHere,ˆΣk,ωsatisfiestheWard-Takahashiidentity33. Toestimatethevalueof ˆΣk,ω, weconsider\nthek′-dependence of uk−k′, which plays the role to prevent the ultraviolet-divergence over\na large momentum in the k′-integration25,34. When ˆΣk,ωcan be described by 2 ×2 matrix\nˆΣk,ω= Σ0+ˆΣ/bardbl+ Σzˆσz,ˆΣ is estimated, where Σ 0and Σzare independent of k, while\n5Σ/bardbl≡Σxˆσx+Σyˆσydepends on k. Then, the Green’s function ˆ gk,ωis described by Σ within\nthe Born approximation as25,34\nˆgr\nk,ω=/bracketleftbigg\n/planckover2pi1ω−{/planckover2pi1˜vF(ˆσ×k)z−ǫF}+i/planckover2pi1\n2τ/bracketrightbigg−1\n, (7)\nwhere ˆgr\nk,ωrepresents the retarded Green’s function. From Eq. (7), the Fe rmi velocity is\nrenormalized by Σ/bardbl, and the renormalized Fermi velocity is represented by ˜ vF=vF/(1+ξ),\nwhereξ=niu2\n0/(4π/planckover2pi12v2\nF) is a small value depending on the relaxation time35. The last term\nin Eq. (7) is caused by the retarded component of Im[Σ 0].\nEquation (7) indicates the Green’s function on the disordered surf ace of the doped TI\nestimated within the Born approximation of Vimp. Therefore, we could expect that /planckover2pi1˜vF(ˆσ×\nk)z−ǫFin the Eq. (7) corresponds to the dispersion on the disordered sur face of the TI.\nThe dispersion is different from that of HTIwithoutVimp. In the following work, we will use\nan effective Hamiltonian ˜HTIobtained by replacing vFwith ˜vFin Eq. (2). This replacement\nis needed for satisfying the charge conservation law on the disorde red surface of the doped\nTI36.\nIII. SPIN CURRENT DUE TO MAGNETIZATION DYNAMICS\nIn this section, we show spin current driven by magnetization dynam ics on the disordered\nsurface of the doped TI/MI junction. Here, the spin current and charge density are mutually\nrelated each other, because of the spin-momentum locking on the s urface of the TI.\nA. Definition of spin current on the surface of topological insulators\nIn order to derive the spin current on the disordered surface of t he doped TI, we demon-\nstrate the definition of the spin current. The spin current jα\niis defined from\n∂tsα+∇ijα\ni=Tα, (8)\nwheresα=1\n2∝an}b∇acketle{tψ†ˆσαψ∝an}b∇acket∇i}htis the spin density, jα\nishows the spin current density, and Tαis the\nspin relaxation torque on the surface. Here the subscript and sup erscript ofjα\nirepresent the\ndirection of flow and spin of the spin current, respectively. From Eq s. (1)-(5),jα\niandTα\nare given by\njα\ni=˜vF\n2ǫzαi∝an}b∇acketle{tψ†ψ∝an}b∇acket∇i}ht=˜vF\n2eǫzαiρe. (9)\n6with the Levi-Civita symbol ǫzαi. In the above equation, we used the commutation relation:\n[ψ†,H] =−i/planckover2pi1˜vFǫzαℓ(∇ℓψ†)σα+Jsdψ†Sασα,\n[ψ,H] =−i/planckover2pi1˜vFǫzαℓσα(∇ℓψ)−JsdSασαψ.\nFrom Eq. (9), the spin current is proportional to the charge dens ityρe=e∝an}b∇acketle{tψ†ψ∝an}b∇acket∇i}ht, where\ne <0 is the charge of electrons. Moreover, the directions of the spin a nd flow of the spin\ncurrent are perpendicular to each other because of the spin-mom entum locking. The spin\nrelaxation torque is derived from Eqs. (8)-(9). The torque can be separated as\nTα=Tα\nTI+Tα\nsd, (10)\nwhereTα\nTIandTα\nsdare spin relaxation torque caused by HTIandHsd, respectively. Here, Tα\nTI\nandTα\nsdare given by\nTα\nTI=i˜vF\n2ǫβzℓǫβαν∝an}b∇acketle{t(∇ℓψ†)ˆσνψ−ψ†ˆσν∇ℓψ∝an}b∇acket∇i}ht, (11)\nTα\nsd=2Jsd\n/planckover2pi1ǫνβαsνSβ. (12)\nWenotethat thedefinition ofspincurrent depends onthat ofthes pin relaxationtorque37–39.\nFor example, we consider the case when the spin relaxation torque c an be described by\nτα=Tα+∇iPα\ni, where the polarization Pα\niis an arbitrary vector with ∇iPα\ni= 0, whose\nindexiandαrepresent the direction of the polarization in the real space and th at of the\nspin in the spin space, respectively. Then, the spin current Jα\nican be also represented by\nJα\ni=jα\ni+Pα\niandJα\nisatisfies the conservation law as ∂tsα+∇iJα\ni=τα. We discuss the\nspin current defined in Eq. (9). To consider the spin current and th e spin relaxation torque,\nwe calculate the charge density and spin density in the following subse ctions.\nB. Charge density\nFirst, we will calculate the charge density ρein the linear response to S.ρeis de-\nscribed by using the lesser component of the Keldysh-Green’s func tion,−i/planckover2pi1G<(x,t,x,t) =\n∝an}b∇acketle{tψ†(x,t)ψ(x,t)∝an}b∇acket∇i}htin the same position and time as\nρe=−i/planckover2pi1etr/bracketleftbigˆG<(x,t,x,t)/bracketrightbig\n. (13)\n7Hence,ρeis given by\nρe=i/planckover2pi1eJsd\nL2/summationdisplay\nq,Ωei(Ωt−q·x)tr[ˆΠ0ν(q,Ω)Sν\nq,Ω], (14)\nwhereL2is the area of the disordered surface of the TI, and q= (qx,qy) and Ω indicate\nthe momentum and frequency of the localized spin Sν\nq,Ω(ν=x,y,z), respectively. Here, the\ncharge-spin correlation function ˆΠ0νis given by\nˆΠ0ν(q,Ω) =/summationdisplay\nk,ω[ˆgk−q\n2,ω−Ω\n2ˆΛν(q,Ω)ˆgk+q\n2,ω+Ω\n2]<, (15)\nwhere ˆgk±q\n2,ω±Ω\n2is the non-perturbative Green’s function of HTIincludingVi, which is taken\ninto account within the Born approximation. The retarded (advanc ed) Green’s function ˆ gr\nk,ω\n(ˆga\nk,ω= [ˆgr\nk,ω]†) is given by\nˆgr\nk,ω= [/planckover2pi1ω+ǫF−/planckover2pi1˜vFˆσ·(k×z)+i/planckover2pi1/(2τ)]−1,\nwhere/planckover2pi1/(2τ) =πniu2\niνe/2 represents the self-energy due to Viwithin the Born approxima-\ntion. The vector ˆΛνin Eq. (15) is the vertex function, which is described by\nˆΛγ(q,Ω) = ˆσγ+/summationdisplay\nζ=0,x,y,z[˜Γ(q,Ω)+[˜Γ(q,Ω)]2+···]γζˆσζ. (16)\nHere ˆσ0=ˆ12×2is the identity matrix and ˜Γγζis given by ˆΓγ:\nˆΓγ(q,Ω)≡niu2\ni/summationdisplay\nkˆgk−q\n2,ω−Ω\n2ˆσγˆgk+q\n2,ω+Ω\n2\n=˜Γγζˆσζ. (17)\nThe correlation function ˆΠ0νcan be decomposed into the retarded and advanced Green’s\nfunction by using the formula ˆ g<\nk,ω=fω(ˆga\nk,ω−ˆgr\nk,ω)40, wherefωis the Fermi distribution\nfunction. Using the formula, we can estimate the correlation funct ionˆΠ0νon the surface of\nthe doped TI, i.e.,/planckover2pi1/(ǫFτ)≪1 regime as ˆΠ0γ=ˆΠra\n0γ+o(/planckover2pi1/(ǫFτ)), where ˆΠra\n0γis represented\nby\nˆΠra\n0γ=/summationdisplay\nk,ω(fω+Ω\n2−fω−Ω\n2)ˆgr\nk−q\n2,ω−Ω\n2ˆΛra\nγˆga\nk+q\n2,ω+Ω\n2. (18)\nHereˆΛra\nγis given by the Pauli matrix as\nˆΛra\nγ=/summationdisplay\nζ=0,x,y,z/bracketleftbigˆ1+˜Γra+(˜Γra)2+···/bracketrightbig\nγζˆσζ≡/summationdisplay\nζ˜Λra\nγζˆσζ, (19)\nˆΓra\nγ=niu2\ni/summationdisplay\nkˆgr\nk−q\n2,ω−Ω\n2ˆσγˆga\nk+q\n2,ω+Ω\n2. (20)\n8Using Eqs. (19)-(20) under the condition Ω τ≪1, we can calculate ˆΠra\n0γin the low-\ntemperature limit. Besides this, we can calculate the response func tionˆΠra\n0γby postuating\nΩτ≪1,qℓ≪1, andq2\nx=q2\ny=q2/2.ˆΠra\n0γis given by\nˆΠra\n0γ=−Ω\n2π/summationdisplay\nζ=0,x,y,zˆΓra\nζ˜Λra\nγζ, (21)\nwhereˆΓra\nζ≡/summationtext\nν=0,x,y,z˜Γra\nζνσνcan be expressed by 4 ×4 matrix ˜Γraas\n˜Γra=\n1−iΩτ−1\n2ℓ2q2 i\n2ℓqy −i\n2ℓqx 0\ni\n2ℓqy1\n2(1−iΩτ−1\n2ℓ2q2)1\n4ℓ2qxqy−i\n4qxℓ/planckover2pi1\nǫFτ\n−i\n2ℓqx1\n4ℓ2qxqy1\n2(1−iΩτ−1\n2ℓ2q2)−i\n4qyℓ/planckover2pi1\nǫFτ\n0i\n4qxℓ/planckover2pi1\nǫFτi\n4qyℓ/planckover2pi1\nǫFτo(/planckover2pi1\nǫFτ)2\n.(22)\nIn the above equation, we have used niu2\niπνe/(/planckover2pi1/2τ) = 1/2. Here,νe=ǫF/(2π/planckover2pi12˜v2\nF) is\nthe density of states at the Fermi energy on the surface of the d oped TI. From the above\nequation, themagnitudesof ˜Γζzand˜Γzζarenegligiblysmaller thanthatof ˜Γνµ(ν,µ= 0,x,y)\nfor/planckover2pi1/(ǫFτ)≪1. As a result, ˆΓµ=/summationtext\nν=0,x,y[˜Γ]µνˆσν+o(/planckover2pi1/(ǫFτ)) is obtained by\nˆΓra\n0=/parenleftbigg\n1−iΩτ−1\n2ℓ2q2/parenrightbigg\nˆσ0+i\n2ℓˆσaqbǫabz, (23)\nˆΓra\nµ=x,y=/bracketleftbigg1\n2/parenleftbigg\n1−iΩτ−3\n4ℓ2q2/parenrightbigg\nδµν+1\n4ℓ2qµqν/bracketrightbigg\nˆσν+i\n2ℓqaǫµazˆσ0. (24)\nThen,˜Λra\nζγcan also be estimated by using ˜Γraas41\n˜Λra\nγζ= [(1−˜Γra)−1]γζ. (25)\nTherefore, from Eqs. (13) and (21)-(24), the charge density ρeis obtained by\nρe=eνeJsdτ\nL2/summationdisplay\nq,Ωei(Ωt−q·x)ℓΩ\nq2ℓ2+iΩτ(qySx\nq,Ω−qxSy\nq,Ω)\n=−eνeJsdτℓ[∇×∂t∝an}b∇acketle{tS/bardbl∝an}b∇acket∇i}htD]z, (26)\nFrom Eq. (26), we find that the charge density ρeis induced by ∂t[∇× ∝an}b∇acketle{tS/bardbl∝an}b∇acket∇i}htD]z. Here\n∝an}b∇acketle{tS/bardbl∝an}b∇acket∇i}htDis defined by the convolution of the in-plane of the localized spin S/bardbland a diffusion\npropagator function Don the disordered surface of the TI as\n∝an}b∇acketle{tS/bardbl∝an}b∇acket∇i}htD(x,t)≡1\nτ/integraldisplay\ndt′/integraldisplay\ndx′D(x−x′,t−t′)S/bardbl(x′,t′), (27)\nD(x,t) =1\nL2/summationdisplay\nq,Ωei(Ωt−q·x)1\n2Dq2+iΩ, (28)\n9where,D≡˜v2\nFτ/2 is a diffusion constant and ∝an}b∇acketle{tS/bardbl∝an}b∇acket∇i}htDdenotes the nonlocal spin, which\ndiffusively propagates by the diffusion propagator D. TheDresults because of nonmagnetic\nimpurity scattering on the disordered surface of the doped TI. We also find that the charge\ndensity due to the out-of plane of the localized spin, Sz, is negligible smaller than that due\ntoS/bardbl.\nThe diffusion propagator Dsatisfies the differential equation\n(∂t−2D∇2)D(x−x′,t−t′) =δ(x−x′)δ(t−t′). (29)\nWe find that from Eqs. (26) and (27), the diffusive motion of the cha rge density obeys the\ndiffusion equation:\n(∂t−2D∇2)ρe=−eνeJsdℓ(∇×∂tS/bardbl)z. (30)\nThe above equation means that the diffusion propagator of the cha rge density is caused\nby the spatial and time derivative of the localized spin, ( ∇×∂tS/bardbl)z, on the surface of the\ndoped TI. When the localized spin is spatially uniform, ρeis not driven by the magnetization\ndynamics.\nC. Spin current\nWe will now consider the spin current due to the magnetization dynam ics on the disor-\ndered surface of the doped TI. The spin current is proportional t o the charge density [see\nEq. (9)]. From the result of the charge density due to magnetizatio n dynamics [see Eq.\n(26)], the spin current is given by\njα\ni=−1\n2ǫzαiνeJsdℓ2[∇×∂t∝an}b∇acketle{tS/bardbl∝an}b∇acket∇i}htD]z. (31)\nThis is one of the main results of this paper. From Eq. (31), the direc tion of spin of the\nspin current is perfectly locked and is perpendicular to the direction of the flow of the spin\ncurrent. The origin lies on the spin-momentum locking on the surface of the TI. The spin\ncurrent is proportional to the coefficients, which are the density o f states at Fermi energy\nνe, thes-dexchange coupling Jsd, and the square of the mean-free path ℓ2. Herejα\niis\nproportional to the spatial and time derivative of the nonlocal spin as [∇×∂t∝an}b∇acketle{tS/bardbl∝an}b∇acket∇i}htD]z. We\nfind that the local spin does not contribute to the spin current gen eration. In the case\n10when the spin structure of the MI is spatially uniform, the spin curre nt vanishes. Since the\nspin current is proportional to the charge density, we expect tha t the spin current can be\narising from the accumulation of the diffusive charge density, which is given by Eq. (30).\nAdditionally, Eq. (31) indicates that the spin current is an even-fun ction of ˜vF, the sign of\nwhich depends on the helicity of electron on the surface of the TI. T herefore, the direction\nof the spin and flow of the spin current on top surface ( jα\ni,top) and that on bottom surface of\nthe TI (jα\ni,bottom) are equal as jα\ni,top=jα\ni,bottom. We find that the in-plane component of the\nlocalized spin S/bardbl≡S−Szzcontributes to the spin current, but the out-of plane component\nof the localized spin Szzdoes not. We expect that its origin lies on the spin-orbit coupling\nofHTI. From ( ˆσ×p)z=ˆσ·(p×z) andS/bardbl= (z×S/bardbl)×z, the Hamiltonian HTI+Hsd\ncan be described by\nHTI+Hsd=/integraldisplay\ndxψ†/braceleftbigg\n˜vFˆσ·/bracketleftbigg/parenleftbigg\np−Jsd\n˜vF(z×S/bardbl)/parenrightbigg\n×z/bracketrightbigg\n−JsdSzˆσz−ǫF/bracerightbigg\nψ(32)\nFrom the above equation, we can regard that the conduction elect rons momentum pis\nshifted by the in-plane localized spin S/bardbl:p→p−Jsd\n˜vF(z×S/bardbl). The in-plane localized spin\nz×S/bardblplays a role like an electromagnetic vector potential A=Jsd\ne˜vF(z×S/bardbl)13,14. Then, the\nobservable quantity should be proportional to the gauge invariant form: an effective electric\nfieldE≡ −∂tAor an effective magnetic field B=∇×A, as represented by\nE=−Jsd\ne˜vF(z×∂tS/bardbl), (33)\nB=Jsd\ne˜vF∇×(z×S/bardbl). (34)\nThedynamics ofthein-planecomponent ofthelocalizedspincanbere gardedastheeffective\nelectromagnetic field, which acts as a driving force to trigger the mo tion of conduction\nelectrons. While, theout-ofplaneone Szzplays arolelike magnetic fields fortheconduction\nelectronsanddoesnotdirectlyshift pinthemomentumspace. Weexpectfromthedifference\nof these properties of the localized spin, the contribution from Szzcould be smaller than\nthat from S/bardbl.\nThe spin current can be represented by using the effective electric fieldEand{∇×\n[∝an}b∇acketle{tE∝an}b∇acket∇i}htD×z]}z=−∇·∝an}b∇acketle{tE∝an}b∇acket∇i}htDas\njα\ni=−1\n2ǫzαiνee˜vFℓ2∇·∝an}b∇acketle{tE∝an}b∇acket∇i}htD. (35)\n11From the above equation, we find that the spin current is proportio nal to∇·∝an}b∇acketle{tE∝an}b∇acket∇i}htDstemming\nfrom charge density. In fact, the charge density can be represe nted byρe∝∇· ∝an}b∇acketle{tE∝an}b∇acket∇i}htD, as\nshown in Eq. (26). Here, the charge density is also proportional to ∇·∝an}b∇acketle{tE∝an}b∇acket∇i}htDand is similar to\nthe Gauss’s law in Maxwell equations as ρe=ǫ∇·E, whereǫis a permittivity and Eis an\napplied electric field. Thus, we can interpret that the charge densit y and the spin current on\nthe surface of the TI are generated by the divergence of the effe ctive electric field. Equations\n(26), (31), and (35) are the main results of this section.\nIV. CHARGE CURRENT DUE TO MAGNETIZATION DYNAMICS\nIn this section, we show charge current due to magnetization dyna mics on the disordered\nsurface of the doped TI. Because of the spin-momentum locking, t he charge current is\nproportional to the density of the spin polarization on the surface of the doped TI. We\ncalculate the spin density, the charge current and the resulting sp in-relaxation torque.\nA. Spin density\nTo discuss the charge current, we calculate the spin density due to the magnetization\ndynamics in the linear response to the localized spin. The spin density s=1\n2∝an}b∇acketle{tψ†ˆσψ∝an}b∇acket∇i}htis\ngiven by\nsµ=i/planckover2pi1Jsd\n2L2/summationdisplay\nq,Ωei(Ωt−q·x)tr[ˆΠµν(q,Ω)Sν\nq,Ω], (36)\nwhere, Π µν(µ,ν=x,y,z) is the spin-spin correlation function. Π µνcan be calculated within\nthesameformalismasinthesection3.2,andisrepresentedby ˆΠµν= ˆσµˆΠ0ν. Fromtheresult,\nwe can obtain the spin density s. Here,scan be decomposed into two terms: s=s/bardbl+szz,\nwheres/bardbl=s−szzandszzshow the in-plane and out-of plane component of the spin\non the disordered surface of the doped TI, respectively. We find t hatszzis proportional\nto∂tSz, and its magnitude is negligibly smaller than that of the magnitude of s/bardblwithin\nthe approximation |sz|/|s/bardbl| ∼o[/planckover2pi1/(ǫFτ)≪1]. Thus, the spin density can be estimated\nbys=s/bardbl+o[/planckover2pi1/(ǫFτ)] andSzdoes not contribute to the generation of s. The dominant\ncontribution of scan also be decomposed into two terms:\ns=sL+sD, (37)\n12wheresLis the local spin density due to S/bardbl, and is given by\nsL=−1\n2νeJsdτ∂tS/bardbl. (38)\nThe local spin density sLis induced by the time-derivative of the in-plane component of the\nlocalized spin ∂tS/bardbl. On the other hand, the second term of Eq. (37), sD, is the diffusive\nspin density and is given by\nsD=−1\n2νeJsdτℓ2(z×∇)[∇×∂t∝an}b∇acketle{tS/bardbl∝an}b∇acket∇i}htD]z (39)\n=ℓ\n2e(z×∇)ρe. (40)\nFrom Eq. (40), sDis generated by the driving field ( z×∇)(∇×∂t∝an}b∇acketle{tS∝an}b∇acket∇i}htD)z, which is the\nspatial gradient and the time-derivative of the nonlocal localized sp in∝an}b∇acketle{tS/bardbl∝an}b∇acket∇i}htD. The driving\nfield is also described by ( z×∇)(∇×∂t∝an}b∇acketle{tS∝an}b∇acket∇i}htD)z=∂t[∇2∝an}b∇acketle{tS∝an}b∇acket∇i}htD−∇(∇· ∝an}b∇acketle{tS∝an}b∇acket∇i}htD)]. Here, sD\nis described by the spatial gradient of the charge density, which is c aused by the electron\ndiffusion on the surface of the TI, as shown in Eq. (26). In addition, sDis also represented\nby the spin current: The charge density is proportional to the spin current,ρe=e\n˜vF(jx\ny−jy\nx),\nandsDbecomes\nsD=1\n2τǫzαi(z×∇)jα\ni. (41)\nFrom Eqs. (38)-(39), we find that sis an even-function of ˜ vFand is independent of the\nhelicity on the surface of the TI. Therefore, the direction of sdoes not depend on whether\nwe are focusing on the top or bottom surface of the TI.\nWe find that from Eqs. (38)-(39), the spin is polarized not by a stat ic magnetization\nbut by magnetization dynamics. Therefore, we expect that static magnetization does not\ninduce spin polarization on the surface of the TI. This seems to be an omalous property on\nthe surface. The response between the spin polarization and the s tatic magnetization on\nthe surface of the TI is different from that in conventional metals: In the metals, a spin is\npolarized even by static magnetization. We will consider shortly why s tatic magnetization\ndoes not generate spin polarization on the surface of the TI. The m agnetization on the\nsurface of the TI plays the role to shift the momentum of conductio n electrons from pinto\np−Jsd\n˜vF(z×S/bardbl) in momentum space. As a result, the center of Fermi sphere is also shifted\nfromp= 0 into p=Jsd\n˜vF(z×S/bardbl). Then, the direction of the spin at each momentum\nare perfectly perpendicular to that of the momentum. Besides, th e spin configuration in\n13momentum space does not change before and after the shift, bec ause the direction of the\nspin at each momentum are independent of the shift. Therefore, n o spin polarization is\ndriven by momentum shift due to a static magnetization.\nB. Charge current\nWe note that on the surface of the TI, the charge current jis proportional to the spin\ndensity on the surface of the TI. The charge current is represen ted by using the renormarized\nvelocity operator as j= 2e˜vF(z×s)36. From Eqs. (38)-(40), the charge current is also\ndecomposed into two terms: j=jL+jDas\njL=−eνeJsdℓ(z×∂tS/bardbl), (42)\njD=eνeJsdℓ3∇[∇×∂t∝an}b∇acketle{tS/bardbl∝an}b∇acket∇i}htD]z. (43)\nThejLis the local charge current and is induced by the time-derivative of t he localized spin\nS/bardbl. The direction of jLis parallel to z×S/bardbl. On the other hand, jDis the diffusive charge\ncurrent caused by impurity scattering on the disordered surface of the TI. In fact, jDcan be\nrepresented by the spatial gradient of the charge density as jD=−2D∇ρe. This means that\ndiffusive current is generated by the spatial gradient of the charg e density on the surface of\nthe TI. We note that the charge current is an odd-function of ˜ vF, so that the direction of\nthe charge current on the top surface jtopis opposite to that on the bottom surface jbottom,\nifSis same on the top and bottom surface. It is noted that, from Eqs. (26) and (42)-(43),\nthe charge density ρeand charge current jsatisfy the conservation law ˙ ρe+∇·jγ= 0. The\ndetail is shown in Appendix B.\nNext, we comment on the relationship between the spin current and the charge current\non the disordered surface of the doped TI. Substituting ǫzαijα\ni= ˜vFρe/einto Eq. (43), we\nfind that the diffusive charge current can be described by the spin c urrent as\njD=−eℓǫzαi∇jα\ni. (44)\nThis is also the main result of this paper. We expect that the above eq uation displays\nthe conversion between the spin current into the diffusive charge c urrent on the disordered\nsurface of the doped TI by using the spatial gradient of the spin cu rrent. The spin current\ncan be converted into the diffusive charge current when the spin cu rrent depends on the\n14space on the disordered surface. The relation in Eq. (44) is plausible on the disordered\nsurface of the doped TI, because the charge density ρeis proportional to the spin current,\nand a diffusive particle current generally proportional to a spatial g radient of particles. We\nnote that there is no relation between the spin current and the loca l charge current jL.\nC. Effective conductivity\nThechargecurrent duetomagnetizationdynamics jcanbealsodescribed bytheeffective\nelectric field E:\nj=e2˜v2\nFνeτE+e2˜vFνeℓ3∇[∇·∝an}b∇acketle{tE∝an}b∇acket∇i}htD]. (45)\nThe first term and the second term are corresponding to the local and diffusive charge cur-\nrent, respectively. From the above results, we will consider an effe ctive conductivity: an\nefficiency of the charge flow due to the applied effective electric field E. This is similar\nto the conventional electric conductivity: In general, the longitud inal electrical conductiv-\nity is defined from dividing the charge current by an applied electric fie ld42. We expect\nthat the current corresponds to the local current. Then, an eff ective longitudinal electrical\nconductivity can be defined by jL=σE, and is given by the first term in Eq. (45) as\nσ=e2˜v2\nFνeτ. (46)\nTheconductivityonlydependsontheparametersonthesurfaceo ftheTI,andisindependent\nof parameters attached to the MI.\nD. Spin relaxation torque\nWe will consider the spin relaxation due to the magnetization dynamics on the surface\nof the TI. Using Eqs. (31) and (37)-(39), we can describe ∂tsαand∇ijα\nias\n∂tsα=−1\n2νeJsdτ∂2\ntS/bardbl,α+1\n2νeJsdℓ2τǫαiz∇i[∇×∂2\nt∝an}b∇acketle{tS/bardbl∝an}b∇acket∇i}htD]z, (47)\n∇ijα\ni=−1\n2νeJsdℓ2ǫαiz∇i[∇×∂t∝an}b∇acketle{tS/bardbl∝an}b∇acket∇i}htD]z. (48)\nTherefore, the spin relaxation torque Tα=∂tsα+∇ijα\niin the linear response to S/bardblis\nobtained by\nT=−1\n2νeJsdτ∂2\ntS/bardbl+1\n2νeJsdℓ2(1−τ∂t)(z×∇)[∇×∂t∝an}b∇acketle{tS/bardbl∝an}b∇acket∇i}htD]z+O(J2\nsd).(49)\n15The first term in Eq. (47) shows a local spin relaxation torque and is in duced by∂2\ntS/bardbl. The\nsecond term is a nonlocal spin relaxation torque and is driven by ( z×∇)(∇×∂t∝an}b∇acketle{tS/bardbl∝an}b∇acket∇i}htD)z.\nThe nonlocal torque vanishes when the magnetization is spatially unif orm. The third term\nin Eq. (47) represents the higher order of JsdandS. From Eq. (12), the spin relaxation\ntorqueTsdis proportional to Sands, which is proportional to S/bardbl. Therefore, the third\nterm in Eq. (47) corresponds to Tsdwithin the linear response to S. We expect that the\nthird term Tsd≃o(Jsdτ\n/planckover2pi1)2can be negligibly small and be ignored in comparison with the\nfirst and the second terms of Tin the regimeJsdτ\n/planckover2pi1≪1.\nThe spin relaxation torque Tis also represented by the effective electric field as\nT=1\n2eνe˜vFτ(∂tE×z)+1\n2eνe˜vFℓ2(1−τ∂t)(z×∇)[∇·∝an}b∇acketle{tE∝an}b∇acket∇i}htD]. (50)\nThe local spin relaxation is written as the time-derivative of the effec tive electric field and\nthe diffusive one is induced by the spatial gradient of the nonlocal eff ective electric field. The\nelectric field dependence of the spin relaxation torque on the surfa ce of the TI is different\nfrom that in NM with spin-orbit interactions: The spin relaxation torq ue in the NM, TNM,\nis proportional to the spatial gradient of the applied electric field39. We expect that the\ndifferencecanbecausedbythe k-dependenceoftheenergydispersion: Theenergydispersion\non the surface of the TI is a linear function of k, while that in the NM proportional to the\nsquare of k. Equations (39), (44), and (46) are the main results of this sectio n.\nV. DISCUSSION\nA. Spin torque\nWe will phenomenologically study the change of the magnetization dyn amics in before\nand after the spin-charge generation due to the ferromagnetic r esonance (FMR). Now, we\nconsider a disordered surface of the MI/TI junction, as shown in F ig.1. Further, in the\njunction, a static magnetic field and ac magnetic field are additionally a pplied. Here ac\nmagnetic field is given by a microwave irradiation, and is needed for FMR in the MI. When\nwe apply this magnetic field, the magnetization dynamics is triggered in the MI, and the\nmagnetization dynamics induces the spin polarization on the surface of the TI [see Eqs.\n(37)-(39)]. Then, the induced spin polarization splays the role of an exchange field acting\non the magnetization in the MI. As a result, the magnetization dynam ics is affected from\n16the generated spin s. Here, the mutual interaction between the magnetization dynamic s\nand the induced spin are called as the feedback effect25.\nThe magnetization dynamics on the disordered surface of the dope d TI is described from\nthe Landau–Lifshitz–Gilbert (LLG) equation as43\n∂tM=−γµ(M×H)+α\nMM×∂tM+Te, (51)\nwhere,M=−M\nSSis the magnetization of the MI, γis the gyromagnetic ratio, µis a\npermeability, αis a Gilbert damping constant, H=H0+Hacis an applied magnetic field\non the disordered surface of the doped TI. H0andHacdenote a static and ac magnetic\nfield, respectively. The spin torque on the disordered surface of t he doped TI is given by\nTe=2Jsda2\n/planckover2pi1M×s. The torque can be decomposed into two terms: Te=TL\ne+TD\ne,\nwhereTL\ne=2Jsda2\n/planckover2pi1M×sLandTD\ne=2Jsda2\n/planckover2pi1M×sDare the local and diffusive spin torque,\nrespectively. Here, ais a lattice constant on the surface of the TI. These spin torques a re\nobtained from Eqs. (37)-(39) and S/bardbl=−S/bardbl\nM/bardblM/bardblas\nTL\ne=κ\nM/bardblM×∂tM/bardbl, (52)\nTD\ne=κ\nM/bardblℓ2M×(z×∇)(∇×∂t∝an}b∇acketle{tM/bardbl∝an}b∇acket∇i}htD)z, (53)\nwhere,M/bardblis the magnitude of the in-plane magnetization M/bardbl≡M−Mzz, andκ=\nνea2J2\nsdτS/bardbl//planckover2pi1is dimensionless coefficient proportional to J2\nsdandτ. We find that TL\ne∝\nM×∂tM/bardblis slightly different from the damping torqueα\nMM×∂tM; which is a damping\nof the magnetization. We could expect that the contribution from t he local spin torque TL\ne\ncan be observed in the experiments on the surface of FM/TI junct ion27,28. Here,TL\neplays\nthe role of an anisotropic damping torque unless the static magnetic field and microwave\nare parallel to the zdirection. The anisotropic damping affects the magnetic permeability :\nFor example, when the static magnetic field and microwave are paralle l to they-direction,\nthen, the longitudinal magnetic permeability χxxandχzzare not equal each other. Here,\nTD\neseems to be a new type of spin torque TD\neon the surface of the TI. TD\nein Eq. (53) is\ninduced by the spatial gradient of the magnetization, M×(z×∇)(∇×∂t∝an}b∇acketle{tM/bardbl∝an}b∇acket∇i}htD)z. When\nthe magnetization is spatially uniform, TD\neis zero and TL\neis nonzero.\nSince,jα\niandTD\neare proportional to the charge density on the disordered surfac e of the\ndoped TI, TD\necan be described by the spin current jα\ni:\nTD\ne=Jsdτa2\n/planckover2pi1M×[(z×∇)ǫzαijα\ni]. (54)\n17The spatial gradient of the spin current induces the diffusive spin to rque on the disordered\nsurface of the doped TI. We expect that the contribution of jα\niis detected from the change\nof the half-width value, as well as the change of a shift of the magne tic resonance frequency\nthroughTD\ne[discussed in Sec.VC]. Equations (52)-(54) are the main results of th is section.\nB. Magnetic permeability without diffusion\nUsing Eqs. (51)-(53), we discuss the magnetic permeability in FMR wh en the magneti-\nzation is spatially uniform on the surface of the MI/TI junction. We c onsider that in the\njunction, the applied static magnetic field H0and the microwave of the ac magnetic field\nHacare applied along the ydirection: H0= (0,H,0) andHac= (hx,0,hz). For an uniform\nmagnetization case, the spin becomes sL∝ne}ationslash= 0 and sD= 0, and the spin torque are TL\ne∝ne}ationslash= 0,\nbutTD\ne= 0.\nThen, the LLG equation on the surface of the MI/TI junction can b e described by\n∂tM=γµ(H×M)+α\nM(M×∂tM)+κ\nM/bardbl(M×∂tM/bardbl). (55)\nTo estimate the magnetic permeability on the surface of the TI, we a ssume that the |H0|is\nlarger than the |Hac|,i.e.,|hx|≪Hand|hz|≪H. Then, from the applied magnetic field,\nwe expect that the local magnetization on the surface M= (mx,My,mz) can be satisfied\nmx≪Myandmz≪My. Moreover we assume that the time-dependence of the precessio n\nofMis given by mx∝mz∝eiΩtand∂tMy∼0. In order to solve the LLG equation, we\ntake a linear approximation of mi:mimj∼0,My∼M(=|M|), andM/bardbl∼M. Then, the\nLLG equation becomes\n∂tmx=γµ(Hmz−hzM)+α∂tmz,\n∂tmz=γµ(hxM−Hmx)−(α+κ)∂tmx,(56)\nand the magnetic permeability is given by\n/parenleftbiggmx\nmz/parenrightbigg\n=/parenleftbiggχxxχxz\nχzxχzz/parenrightbigg/parenleftbigghx\nhz/parenrightbigg\n. (57)\nThe frequency dependence of the longitudinal magnetic permeabilit yχxxandχzzare de-\n18scribed by\nχxx=ωM(ωH+iαΩ)\n(ωH+iαΩ)[ωH+i(α+κ)Ω]−Ω2, (58)\nχzz=ωH+i(α+κ)Ω\nωH+iαΩχxx, (59)\nwhere.ωH=γµHandωM=γµMare the angular frequency of the applied static magnetic\nfield and that of the magnetization, respectively. The permeability χxxandχzzare different\neach other originating from the anisotropic spin-transfer torque TL\ne. Transverse magnetic\npermeability has the relation χxz=−χzxand is given by\nχxz=−iΩ\nωH+iαΩχxx. (60)\nThe real part of the permeability Re[ χxx] shows a Lorentzian profile due to the magnetic\nresonance around the resonant frequency Ω r, which is proportional to H. The Im[χxx]\nindicates the energy absorption of the applied microwave around Ω r. Half-width value of\nIm[χxx] expresses the damping of the precessional motion of the magnet ization. From Eq.\n(58), the half-width value ∆Ω is caused by the Gilbert damping ( αM×∂tM) and the local\nspin torque TL\negiven by\n∆Ω≃(2α+κ)Ωr. (61)\nWe expect that in the MI without TI, the half-width value of the magn etic permeability\nestimates ∆Ω = 2 αΩr. Eq. (61) indicates that on the surface of the doped TI, ∆Ω is\nenhanced from 2 αΩrinto (2α+κ)Ωr. The origin lies on TL\ne, which is triggered by the\ninduced spin, where the spin is induced by the magnetization dynamics on the surface of\nthe doped TI. This enhancement of ∆Ω has been verified in the recen t experiment28.\nWe compare ∆Ω in MI/TI junction with ∆Ω in FM/NM junction. In the FM/ NM\njunction, it has been demonstrated that the enhancement of ∆Ω is triggered by the spin\ncurrent in the NM, which is generated by the magnetization dynamics of the FM44. On the\nsurface of the MI/TI junction with uniform spin structure, on the other hand, spin current\nis not generated by magnetization dynamics, and the spin current d oes not contribute to\n∆Ω. The enhancement of ∆Ω is caused by the spin polarization due to t he magnetization\ndynamics on the surface.\n19C. Magnetic permeability with diffusion\nWe will consider the surface of the TI/MI junction, where the localiz ed spin in the MI is\nspatially inhomogeneous. The spin structure of the localized spin we c onsider is a spin-wave\nor longitudinal conical spin structure, which are realized in ferrimag netic insulator yttrium\niron garnets (YIG) or multiferroics45, respectively. We expect that even on the surface of\nthe TI, the localized spin depends on the position through proximity e ffects from the MI.\nThen, if we apply ac magnetic field of microwave in the junction along th eydirection, we\nassume that the localized spin becomes precessional motion by the a pplied magnetic field.\nThe localized spin on the surface of the TI can be described by\nS(x,t) = [Scos(q·x−Ωt),Sy,Ssin(q·x−Ωt)], (62)\nwhere,SandSyare a constant coefficient independent of space. We assume |S|≪|Sy|\nand|S|∼Sy. Hereq= (qx,qy) is the momentum of the localized spin and is assumed to\nbe monochromatic. The direction and the magnitude of qdepends on materials of the MI.\nIn the spin structure, nonlocal diffusive spin ∝an}b∇acketle{tS∝an}b∇acket∇i}htDis given by Eqs. (27) and (62) as\n∝an}b∇acketle{tSx∝an}b∇acket∇i}htD=Aq,ΩScos[q·x−Ωt]−Bq,ΩSsin[q·x−Ωt],\n∝an}b∇acketle{tSz∝an}b∇acket∇i}htD=Bq,ΩScos[q·x−Ωt]+Aq,ΩSsin[q·x−Ωt].(63)\nThe component of ∝an}b∇acketle{tS∝an}b∇acket∇i}htDis different from that of S:∝an}b∇acketle{tSx∝an}b∇acket∇i}htDhas cos[q·x−Ωt] components,\nas well as sin[ q·x−Ωt] components. Here, the coefficients AandBare obtained (see\nAppendix A) as\nAq,Ω=q2ℓ2\n(Ωτ)2+(q2ℓ2)2, (64)\nBq,Ω=Ωτ\n(Ωτ)2+(q2ℓ2)2. (65)\nThecoefficients Aq,ΩandBq,ΩdependonqℓandΩτ, whereparameters ℓandτaredetermined\nby the TI, and qand Ω are chosen as characteristic values of the MI. Then, the diffu sive\nspinsDis given from Eqs. (39) as\nsD=νeJsdτℓ2[q2∂t∝an}b∇acketle{tS/bardbl∝an}b∇acket∇i}htD−q(q·∂t∝an}b∇acketle{tS/bardbl∝an}b∇acket∇i}htD)]. (66)\nThen, nonlocal diffusive spin torque TD\neis obtained by using ∝an}b∇acketle{tM/bardbl∝an}b∇acket∇i}htD=−(M/bardbl/S/bardbl)∝an}b∇acketle{tS/bardbl∝an}b∇acket∇i}htDas\nTD\ne=−κℓ2\nM/bardbl/bracketleftbig\nq2M×∂t∝an}b∇acketle{tM/bardbl∝an}b∇acket∇i}htD−(M×q)(q·∂t∝an}b∇acketle{tM/bardbl∝an}b∇acket∇i}htD)/bracketrightbig\n. (67)\n20The first term of Eq. (67) plays the role of an anisotropic damping to rque, which is similar\ntoTL\ne. The direction of this torque is independent of the direction of q. On the other hand,\nthe second term of Eq. (67) is proportional to ( M×q)(q·∂t∝an}b∇acketle{tM/bardbl∝an}b∇acket∇i}htD), and its direction\ndepends on q.\nUsing Eqs. (62)-(67), we consider the magnetic permeability affect ed byTD\ne.Then, the\nLLG equation is given within the linear order of the magnetization as\n∂tmx=ωHmz−hzωM+α∂tmz\n∂tmz=hxωM−ωHmx−(α+κ)∂tmx+κq2\nyℓ2∂t∝an}b∇acketle{tM/bardbl,x∝an}b∇acket∇i}htD.\nThelasttermoftheaboveequationiscausedby TD\ne. Inordertodiscuss thepermeabilitydue\ntoTD\ne, we consider when the momentum has the y-component ( q=qy), whose direction\nis parallel to the applied static magnetic field. That means that the sp in structure we\nconsider is a longitudinal conical spin structure. From the above eq uation, using ∝an}b∇acketle{tM/bardbl,x∝an}b∇acket∇i}htD=\nAq,Ωmx−Bq,Ωmz, we obtain the magnetic permeability as\n/parenleftbiggmx\nmz/parenrightbigg\n=/parenleftbiggχD\nxxχD\nxz\nχD\nzxχD\nzz/parenrightbigg/parenleftbigghx\nhz/parenrightbigg\n. (68)\nHere the longitudinal and transverse magnetic permeability are give n by\nχD\nxx(q,Ω) =(ωH+iαΩ)ωM\n[ωH+iαΩ][ωH+i(α+ ˜κq,Ω)Ω]−ζq,ΩΩ2, (69)\nχD\nzz(q,Ω) =ωH+i(α+ ˜κq,Ω)Ω\nωH+iαΩχD\nxx, (70)\nχD\nxz(q,Ω) =−ζq,ΩχD\nzx. (71)\nThe obtained permeability is different from that in Eqs. (58)-(60). T he difference is caused\nby coefficients ˜ κq,Ωandζq,Ω:\n˜κq,Ω=κ(1−q2ℓ2Aq,Ω), (72)\nζq,Ω= 1+κq2ℓ2Bq,Ω. (73)\nThe ˜κq,Ωandζq,Ωdepend on qℓand Ωτ. Ifq= 0, one can demonstrate ˜ κq,Ω=κand\nζq,Ω= 1, and the magnetic permeability in Eqs. (69)-(71) equal to that in Eqs. (58)-(60),\nrespectively. Figure 2(a) indicates the Ω τdependence of ˜ κ/κfor several momentum qℓ. The\nparameter ˜κ/κapproaches to 0 from ˜ κ/κ= 1 with increasing Ω τ: In the case for ˜ κ/κ= 0,\n21FIG. 2: (Color online) (a) The Ω τdependence of ˜ κ/κfor several qℓ. (b) The qℓdependence of the\nζfor several frequency (2 πf= 1,2,5,10 GHz) in the fixed relaxation time ( τ= 0.087 ps).\nTL\neandTD\neare canceled out each other, and the spin torque Tevanishes. On the other\nhand, ˜κ/κ= 1 means that TD\neis zero and TL\neis nozero. The relation ˜ κ/κsignificantly\nchanged when ( qℓ)2∼Ωτis satisfied. Figure 2(b) shows the qℓdependence of ζfor several\nangular frequency of the applied ac magnetic field Ω, where we take a realistic relaxation\ntime (τ= 0.087 ps46). The magnitude of ζchanges around qℓ∼√\nΩτand approaches to\nζ= 1 with increasing qℓ.\nFigure 3 (a) and (b) show the Ω τdependence of the real and imaginary part of the\nlongitudinal magnetic permeability, Re[ χD\nxx] and−Im[χD\nxx], respectively. In Figs. 3(a) and\n(b), we choose realistic parameters of the TI: ℓ= 40nm,vF= 4.6×105m/s46,τ= 0.087 ps,\nξ= 0.003,and ˜vF/vF= 0.997. Besides, wechoosematerialparametersoftheferromagne ts28:\nα= 0.015,Jsd∼6meV,S/bardbl∼0.3, and Fermi wavenumber kF= 3.9×108m−1. Then,\nκ= 0.008 is obtained. The frequency of the magnetization ωM/(2π) =fM= 0.28 GHz\nin the MI. The magnitude of the frequency is evaluated by the mater ial parameters of the\npermalloy28.\n22FIG. 3: (Color online) (a)-(b) The frequency dependence of t he real part and imaginary part of the\nlongitudinal permeability, Re[ χD\nxx] and−Im[χD\nxx], for a fixed Gilbert dampingconstant ( α= 0.015),\nthe relaxation time [ τ= 0.087ps], the anisotropic damping constant [ κ= 0.008], and the frequency\n[fH= 1.6 GHz and fM= 0.28 GHz] for several momentum qℓ.\nWe plot the Re[ χD\nxx] and−Im[χD\nxx] functions as the frequency of the ac magnetic field for\nseveral momentum of the localized spin, when we take the frequenc y of the static magnetic\nfieldωH/(2π) =fH= 1.6 GHz. The magnitude of Ω τcan be estimated as Ω τ∼ωHτ=\n8.6×10−3. Forqℓ <0.03, the permeability Re[ χD\nxx] and−Im[χD\nxx] are not dramatically\nchanged from that without diffusion, Re[ χxx] and−Im[χxx], respectively. The reason is due\nto the profile of ˜ κ/κandζ: Both ˜κ/κandζare about 1 within qℓ<0.03 in Ωτ∼8.6×10−3.\nWhileqℓis nearqℓ= 0.03,χD\nxxchanges: The magnitude of χD\nxxincreases from that of χxx.\nBesides, the resonant frequency of χD\nxxincreases from that of χxx[seeqℓ= 0.03 in Figs. 3\n(a)-(b)]. The change of the resonant frequency is also shown in Fig . 4 (a). We find that\nwhen (qℓ)2∼Ωτis satisfied, ˜ κ/κandζdeviate from 1, as shown in Figs. 2(a)-(b). After\nincreasingqℓfromqℓ= 0.03, the shifted resonant frequency gets back again. The magnitu de\nof the permeability increases with increasing qℓ.\nThe half-width value is also changed by the magnitude of Im[ χD\nxx] for several momentum\n23FIG. 4: (Coloronline) The qℓdependenceoftheresonant frequency(a) andthehalf-width value(b)\nnormalized by the angular frequency of the applied magnetic field for several frequencies (2 πfH=1,\n2, 5, and 10 GHz) in 2 α+κ= 0.038 and 2 α= 0.030.\nqℓ. This trade off between the half-width value and qℓfor several frequency (2 πfH= 1,\n2, 5, and 10 GHz ) is shown in Fig. 4(b). In the case for τ= 0.087 ps, the half-width\nvalue ∆Ω significantly changed around the qℓ, which satisfies qℓ∼√\nΩτ(e.g.,qℓ= 0.03 in\nωH= 10 GHz). Figure 4(a) indicates qℓdependence of the resonant frequency rate Ω r/ωH.\nThe Ω r/ωHchanged from Ω r/ωH∼1 into Ω r/ωH= 0.998 around the qℓ, which satisfies\nqℓ∼√\nΩτ.\nWe discuss the role of diffusive spin torque TD\nefrom Figs. 4(a) and (b). Figure 4(a)\nshows that the resonant angular frequency Ω rtends to decrease with increasing qℓ. The\ndecrease of Ω rcan be caused by the increase of ζq,Ω, and the change of ζq,Ωis caused by TD\ne.\nTherefore, we expect that TD\neplays a role as a field-like torque to shift the resonant fre-\nquency. Then, Fig.4 (b) indicates that the half-width value tends to decrease with increasing\n˜κq,Ωin Eq. (73), which is caused by TD\ne. It means that the damping of the magnetization\ndynamics is reduced by TD\neon the disordered surface of the doped TI. Thus, TD\nebehaves\n24TABLE I: A brief summary of the role of the spin torques and the half-width value in several\nspin structures on the surface of the TI, when the direction o fH0and the propagation of Hacare\nparallel to y-axis.\nUniform Transverse conical ( qy= 0) Longitudinal conical ( qy∝ne}ationslash= 0)\nTL\neDamping torque Damping torque Damping torque\nTD\ne – – Damping torque and field-like torque\n∆Ω/Ωr2α+κ 2α+κ 2α+ ˜κ\nboth the damping torque and the field-like torque. The role of the sp in torque in several\nspin structures are summarized in the Table I. Fromthe table I, we c an expect to distinguish\nthe half-width value contributed from TL\neandTD\neby tuning of the magnitude of the applied\nmagnetic field H0. The reason is why the longitudinal spin structure changes an spat ial\nuniform ferromagnetic structure if when we apply a strong magnet ic field, which broke the\nlongitudinal spin structure. Then, qℓof the spatial uniform spin structure can be regarded\nasqℓ= 0. Therefore, we expect that if when we apply the strong applied m agnetic field\nin the longitudinal spin structure, the half-width value with a finite qℓchanges into the\nhalf-width value with qℓ= 0 as ∆Ω/Ωr= 2α+ ˜κ→2α+κ[see Fig.4 (b)].\nFor magnetization dynamics due to the magnetic resonance, we nee d to apply magnetic\nfields in the MI/TI junction. Then, the spin-charge generation and transport are triggered\nnot only by the magnetization dynamics, but also by the applied magne tic field. We will\nestimate when the contribution due to the magnetic field can be relev ant. The contribution\nfromtheappliedmagneticfieldcanbedescribedbytheZeemaneffect ,HZ=−2/planckover2pi1γ/integraltext\ndxB·s,\nwhereBis the applied magnetic field and couples with conduction electrons spin on the\nsurface of the TI. The contribution from Bcan be treated within the same formalism in\nsections 3 and 4 by replacing S→S+2(/planckover2pi1γ/Jsd)Bin Eq. (3). As a result, the spin-charge\ngeneration and transport due to HZandHsdare obtained by replacing S→S+2(/planckover2pi1γ/Jsd)B\nin Eqs. (26), (31), (38)-(39), and (42)-(43). We expect that t he contribution from HZcan be\nignored compared with that from Hsd, when the energy scale of the Zeeman effect is smaller\nthan that of the exchange energy on the surface of the TI as 2 /planckover2pi1γ|B|/(Jsd|S|)≪1. The\n|B|/Jsdvalue can be estimated by |B|/Jsd≪1/(2/planckover2pi1γ)∼4×104T·eV−1inS∼1. Then, the\n25magnitude of the exchange coupling Jsdcan be estimated. Using realistic parameters, the\nmean-free path ℓ= 40nm, Fermi velocity vF= 4.6×105m/s46, we obtain the /planckover2pi1/τ∼15meV\nandJsd≪15 meV, which is requested for the perturbation condition Jsdτ//planckover2pi1≪1. Then, if\n|B| ≫10T is satisfied, we need to consider the contribution from HZ.\nAt the end of this section, we estimate the spin density due to the dy namics of the\nlongitudinal spin structure S=S/bardbl(sinθcosΩt,1,sinθsinΩt) in the case of θ≪1 around\nresonance angular frequency Ω ∼1×1010s−1. The magnitude of the spin depends on the\nregime ofq2ℓ2≪Ωτorq2ℓ2≫Ωτ. Inq2ℓ2≪Ωτregime, the magnitude of the nonlocal spin\ncan be negligible small compared with that of the local term, and the s pin is estimated by\n|s| ∼1.6×10−10˚A−2atθ∼0.1radintheFMR.Then, wefindthatthemagnitudeofthespin\ndue to the spin-pumping is smaller than that of the spin due to the app lied electric field47.\nOn the other hand, in q2ℓ2≫Ωτregime, the local and nonlocal spin vanishes each other\neven in the presence of FMR. From the results, we expect that the change of the magnitude\nof the spin dependent on q2ℓ2/Ωτcan be measurable for several applied magnetic fields,\nbecause the inhomogenous spin structure ( qℓ∝ne}ationslash= 0) changes into an uniform spin structure\n(qℓ∼0) by using an applied strong magnetic field.\nD. Spin current and charge current\nWewill discuss thespincurrent onthesurfaceofthedisorderedMI /TIjunctioncompared\nwith that in the FM/NM junction. The spin current due to the spin-pu mpingjα\ni,FM/NMin\nthe FM/NM junction is triggered by the magnetization dynamics as1,2,5\njα\ni,FM/NM=b∇i∂tSα+O(S2), (74)\nwherebis a coefficient dependent on materials. It is similar to the spin current in Eq. (74),\nthat thespin current isproportional to a time-dependent magnet izationand thespin current\nvanishes when the magnetization is spatially uniform. The direction of the spin (α) and the\nflow (i) of the spin current in Eq. (74) are not related each other. On the other hand, spin\ncurrent on the surface of the TI, whose direction of spin and flow a re perfectly perpendicular\nto each other. The difference lies on the spin-orbit interaction. The jα\niin Eq. (31) includes\nthe contribution of the spin-orbit interaction, which is absent in Eq. (74) does not.\nCharge current due to the spin-pumping in the FM/NM junction is also given by the\n26magnetization dynamics and Rashba type spin-orbit interactions. W hen the magnetization\nis spatially uniform, the charge current jFM/NMbecomes2,5\njFM/NM=α×(S×∂tS), (75)\nwhereαis a constant vector including the contribution from the spin-orbit in teraction. The\nchargecurrent inEq. (75)isproportionaltothesecond-order o fthelocalizedspin S. That is\ndifferent from the charge current on the surface of TIs. The cha rge current on the surface of\nthe TI is proportional to the localized spin ∂tS/bardbl, as shown in Eq. (43). The difference is due\nto the property of the localized spin: the localized spin plays the role o f the effective vector\npotential on the surface of the TI. We note that the frequency d ependence of these charge\ncurrent is also different; jTI∝∂tS/bardbloscillates with time of the localized spin in the FMR,\nbutjFM/NM∝S×∂tSdoes not. For example, the ac current jTI∝(cosΩt,sinΩt,0) is given\nwhen we apply the magnetic field parallel to the zdirection on the TI/MI junction. On\nthe other hand, the dc current jFM/NM∝Ω(0,0,1) is obtained when we apply the magnetic\nfield parallel to the zdirection in the NM/FM junction.\nVI. SUMMARY\nWe have studied the spin-charge generation and transport due to the magnetization dy-\nnamics on the disordered surface of the doped TI/MI junction. Th e spin current jα\ns,iis\nproportional to the charge density ρeand the direction of its spin and its flow are perfectly\nperpendicular to each other, because of the spin-momentum lockin g on the surface of the\nTI. We have found that jα\niandρeare induced by the time- and spatial-dependent of nonlo-\ncal magnetization dynamics, which is affected by nonmagnetic impurit y scatterings on the\ndisordered surface of the doped TI. These results of jα\niandρeare shown in Eqs. (31) and\n(26), respectively. jα\niandρeis induced except when the magnetization dynamics is spatially\nuniform. We have also shown the induced spin sand charge current density jdue to the\nmagnetization dynamics. Because of the spin-momentum locking, th e spinsand charge\ncurrent density jare proportional to each other. The sandjare generated not only by\nthe local magnetization dynamics, but also by the nonlocal magnetiz ation dynamics with\nthe diffusion on the disordered surface of the doped TI. These res ults ofsandjare shown\nin Eqs. (38)-(39) and (42)-(43), respectively. A brief summary o f the local and nonlocal ρe,\n27TABLE II: A brief summary of the charge, charge current, spin , and spin current density due to\nthe magnetization dynamics on the disordered surface of the TI. These terms are driven by the\neffective electric field E.\ncharge density ρecurrent density jispin density sαspin current density jα\ni\nLocal term – z×∂tS/bardbl∂tS/bardbl–\nNonlocal term [ ∇×∂t∝an}b∇acketle{tS/bardbl∝an}b∇acket∇i}htD]z∇[∇×∂t∝an}b∇acketle{tS/bardbl∝an}b∇acket∇i}htD]z(z×∇)[∇×∂t∝an}b∇acketle{tS/bardbl∝an}b∇acket∇i}htD]z[∇×∂t∝an}b∇acketle{tS/bardbl∝an}b∇acket∇i}htD]z\nDriving field ∇·∝an}b∇acketle{tE∝an}b∇acket∇i}htD E,∇2∝an}b∇acketle{tE∝an}b∇acket∇i}htDz×E, (z×∇)[∇·∝an}b∇acketle{tE∝an}b∇acket∇i}htD] ∇·∝an}b∇acketle{tE∝an}b∇acket∇i}htD\njα\ni,s, andjdue to the magnetization dynamics are represented in Table. II. As a result,\nwe have discussed the modification of the magnetization dynamics be fore and after these\nspin-charge generation and transport on the disordered surfac e of the doped TI. These spin-\ncharge generation and transport can be detected from the half- width value of the magnetic\npermeability in the magnetic resonance in the MI/TI junction, as disc ussed in section V.\nThe magnitude of a Gilbert damping constant αin ferromagnetic insulator is smaller than\nthat in ferromagnetic metals. Then, we can easily detect the chang e of thefHdependence\nof the resonant frequency and half-width value, which are shown in Figs. 4 (a)-(b).\nThe preparation of the hybrid system with the ferromagnetic insula tor deposited on the\nsurface of the TI, EuS/Bi 2Se3, has been reported50, where the magnetic moment of Eu\nlocates at the interface between the EuS and Bi 2Se3. If the magnetization dynamics of\nthe magnetic moment of Eu is triggered by an applied magnetic field, th e spin density\nand charge current can be induced on the surface of the TI. Addit ionally, the magnetic\ndistribution of Eu has a magnetic domain, which is spatially dependent o n the position on\nthe surface. Therefore, when we move the magnetic domain by usin g an applied magnetic\nfield, the charge density and the spin current are also triggered on ly around the magnetic\ndomain. Recently, magnetic insulator with noncoplanar spin structu re has been reported.\nFor example, magnetoelectric insulator Cu 2OSeO3has spatially dependent spin structure,\nand is called skyrmion, which is topologically protected magnetic spin vo rtex-like object51,52.\nIf one can prepare the vortex-like spin structure deposited on th e surface of the TI and can\ntrigger the magnetization dynamics of the skyrmion, we expect tha t the charge and spin\n28currents are driven by the magnetization dynamics of the skyrmion . Moreover, the spatial\ndistributions of the charge density and the magnitude of the spin cu rrent depends on the\nposition of the skyrmion, because the spatial derivative of the loca lized spin depends on the\npositions in the skyrmion. We comment that the induced spin and char ge currents could be\nindependent of the polarity of the skyrmion, since these current a re triggered by the in-plane\ncomponent of the localized spin [see Eqs. (31) and (42)-(43)]. Then , we expect that in the\nMIs deposited on the surface of the TI, the magnetization dynamic s induces not only the\nlocal spin-charge generation and transport, but also the diffusive one. Our obtained results\nwill enable the applications of TI nanomembrane in spintronics devices .\nACKNOWLEDGEMENTS\nThe authors would like to thank A. A. Golubov, A. Dutt, and A. Yamak aga for valu-\nable discussions. This work was supported by Grants-in-Aid for You ng Scientists (B) (No.\n22740222 and No. 23740236), by Grants-in-Aid for Scientific Rese arch on Innovative Areas\n“Topological Quantum Phenomena” (No. 22103005 and No. 251037 09) from the Ministry\nof Education, Culture, Sports, Science, and Technology, Japan ( MEXT), and by the Core\nResearch for Evolutional Science and Technology (CREST) of the J apan Science. K.T. ac-\nknowledges support from a Grant-in-Aid for Japan Society for the Promotion of Science\n(JSPS) Fellows.\nAppendix A: Derivation of Eqs. (65)-(66)\nWe show details of calculation of coefficient Aq,ΩandBq,Ωof nonlocal localized spin ∝an}b∇acketle{tS∝an}b∇acket∇i}htD.\nThe∝an}b∇acketle{tS∝an}b∇acket∇i}htDis given by\n∝an}b∇acketle{tS∝an}b∇acket∇i}htD(x,t)≡/integraldisplay/integraldisplay\ndt′dx′D(x−x′,t−t′)S(x′,t′) (A1)\nTo substitute S=S(1,0,−i)ei(q·x−Ωt)∝ei(q·x−Ωt)into the above equation, we calculate\n∝an}b∇acketle{tn∝an}b∇acket∇i}htD:\n∝an}b∇acketle{tn∝an}b∇acket∇i}htD=1\nL2/integraldisplay/integraldisplay\ndt′dx′/summationdisplay\nQ,ωei[ω(t−t′)−Q·(x−x′)]ei(q·x′−Ωt′)\nQ2ℓ2+iωτ\n=ei(q·x−ωt)\nq2ℓ2−iΩτ(A2)\n29The resulting ∝an}b∇acketle{tSx∝an}b∇acket∇i}htDis obtained from the real part of S∝an}b∇acketle{tn∝an}b∇acket∇i}htDin the above equation. In the\nsame way, ∝an}b∇acketle{tSz∝an}b∇acket∇i}htDis obtained from the real part of −iS∝an}b∇acketle{tn∝an}b∇acket∇i}htD. Thus, from the above equation,\nthe coefficient of Aq,ΩandBq,Ωare derived as Eqs. (65)-(66).\nAppendix B: Charge conservation\nTo check the validity of the spin current and the charge current we calculate, we use the\ncharge conservation law ∂tρe+∇·j= 0. The charge density ∂tρeis given by\n∂tρe=eνeJsdτ\nL2/bracketleftbigg/summationdisplay\nq,Ωei[Ωt−q·x]iℓΩ2\nq2ℓ2+iΩτ(qySx\nq,Ω−qxSy\nq,Ω)/bracketrightbigg\n, (B1)\nwhereℓ= ˜vFτis the mean-free path. The resulting ∇·jbecomes\n∇xjx=e˜vFνeJsdτ\nL2/summationdisplay\nq,Ωei[Ωt−q·x]Ω/bracketleftbigg/braceleftbigg\n1−1\n2q2ℓ2\nq2ℓ2+iΩτ/bracerightbigg\nqxSy\nq,Ω+1\n2ℓ2q2\nℓ2q2+iΩτqySx\nq,Ω/bracketrightbigg\n,\n∇yjy=−e˜vFνeJsdτ\nL2/summationdisplay\nq,Ωei[Ωt−q·x]Ω/bracketleftbigg/braceleftbigg\n1−1\n2q2ℓ2\nq2ℓ2+iΩτ/bracerightbigg\nqySx\nq,Ω+1\n2ℓ2q2\nℓ2q2+iΩτqxSy\nq,Ω/bracketrightbigg\n,\nand\n∇xjx+∇yjy=e˜vFνeJsdτ\nL2/summationdisplay\nq,Ωei[Ωt−q·x]Ω/bracketleftbigg/braceleftbigg\n1−q2ℓ2\nq2ℓ2+iΩτ/bracerightbigg\n(qxSy\nq,Ω−qySx\nq,Ω)/bracketrightbigg\n.\n=eνeJsdτ\nL2/summationdisplay\nq,Ωei[Ωt−q·x]/bracketleftbiggiΩ2˜vFτ\nq2ℓ2+iΩτ(qxSy\nq,Ω−qySx\nq,Ω)/bracketrightbigg\n=−∂tρe.(B2)\nTherefore, the ρeandjsatisfy∂tρe+∇·j= 0.\n1Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Spin pumping and magnetization dynamics\nin metallic multilayers , Phys. Rev. Lett. 88, 117601 (2002).\n2J.-I. Ohe, A. Takeuchi, and G. Tatara, Charge Current Driven by Spin Dynamics in Disordered\nRashba Spin-Orbit System , Phys. Rev. Lett. 99, 266603 (2007).\n3E.Saitoh, M. Ueda, H. Miyajima, andG. Tatara, Conversion of spin current into charge current\nat room temperature: Inverse spin-Hall effect , Appl. Phys. Lett. 88, 182509 (2006).\n4A. Stern, Berry’s phase, motive forces, and mesoscopic conductivity , Phys. Rev. Lett. 681022\n(1992).\n305A. Takeuchi and G. Tatara, Charge and Spin Currents Generated by Dynamical Spins , J. Phys.\nSoc. Jpn. 77, 074701 (2008).\n6M. Z. Hasan and C. L. Kane, Colloquium: Topological insulators , Rev. Mod. Phys. 82, 3045\n(2010).\n7X.-L. Qi and S.-C. Zhang, Topological insulators and superconductors , Rev. Mod. Phys. 83,\n1057 (2011).\n8Y. Ando, Topological Insulator Materials , J. Phys. Soc. Jpn. 82, 102001 (2013).\n9L. Fu and C. L. Kane, Superconducting Proximity Effect and Majorana Fermions at t he Surface\nof a Topological Insulator , Phys. Rev. Lett. 100, 096407 (2008).\n10A. R. Akhmerov, J. Nilsson, and C. W. J. Beenakker, Electrically Detected Interferometry of\nMajorana Fermions in a Topological Insulator , Phys. Rev. Lett. 102, 216404 (2009).\n11Y. Tanaka, T. Yokoyama and N. Nagaosa, Manipulation of the Majorana Fermion, Andreev\nReflection, and Josephson Current on Topological Insulators , Phys. Rev. Lett. 103, 107002\n(2009).\n12X.-L. Qi, T. L. Hughes, and S.-C. Zhang, Topological field theory of time-reversal invariant\ninsulators , Phys. Rev. B 78, 195424 (2008).\n13K.NomuraandN. Nagaosa, Electric charging of magnetic textures on the surface of a to pological\ninsulator , Phys. Rev. B 82, 161401 (2010).\n14H. T. Ueda, A. Takeuchi, G. Tatara, and T. Yokoyama, Topological charge pumping effect by the\nmagnetization dynamics on the surface of three-dimensiona l topological insulators , Phys. Rev.\nB85, 115110 (2012).\n15I. Garate and M. Franz, Magnetoelectric response of the time-reversal invariant h elical metal ,\nPhys. Rev. B 81, 172408 (2010).\n16T. Yokoyama, Y. Tanaka, and N. Nagaosa, Anomalous magnetoresistance of a two-dimensional\nferromagnet/ferromagnet junction on the surface of a topol ogical insulator , Phys. Rev. B, 81,\n121401(R) (2010).\n17S. Mondal, D. Sen, K. Sengupta, and R. Shankar, Magnetotransport of Dirac fermions on the\nsurface of a topological insulator , Phys. Rev. B. 82, 045120 (2010).\n18B. D. Kong, Y. G. Semenov, C. M. Krowne, and K. W. Kim, Unusual magnetoresistance in a\ntopological insulator with a single ferromagnetic barrier , App. Phys. Lett. 98, 243112 (2011).\n19D. Culcer, Transport in three-dimensional topological insulators: Th eory and experiment , Phys-\n31ica E,44, 860, (2012).\n20M. J. Ma, M. B. A. Jalil, S. G. Tan, Y. Li, and Z. B. Siu, Spin current generator based on\ntopological insulator coupled to ferromagnetic insulator s, AIP Advances 2, 032162 (2012)\n21A. A. Burkov and D. G. Hawthorn, Spin and Charge Transport on the Surface of a Topological\nInsulator , Phys. Rev. Lett. 105, 066802 (2010).\n22P. Schwab, R. Raimondi, and C. Gorini, Spin-charge locking and tunneling into a helical metal ,\nEPL (Europhysics Letters), 93, 67004 (2011).\n23K. Taguchi, T. Yokoyama and Y. Tanaka, Giant magnetoresistance in the junction of two\nferromagnets on the surface of diffusive topological insula tors, Phys. Rev. B 89, 085407 (2014).\n24T. Yokoyama, J. Zang, and N. Nagaosa, Theoretical study of the dynamics of magnetization on\nthe topological surface , Phys. Rev. B 81, 241410 (2010).\n25A. Sakai and H. Kohno, Spin torques and charge transport on the surface of topologi cal insulator ,\nPhys. Rev. B 89, 165307 (2014).\n26A. R. Mellnik et al., Spin-transfer torque generated by a topological insulator , Nature (London)\n511, 449 (2014).\n27P. Deorani, J. Son, K. Banerjee, N. Koirala, M. Brahlek, S. Oh , and H. Yang, Observation of\ninverse spin Hall effect in bismuth selenide , Phys. Rev. B 90, 094403 (2014).\n28Y. Shiomi, K. Nomura, Y. Kajiwara, K. Eto, M. Novak, Kouji Seg awa, Yoichi Ando, and E.\nSaitoh,Spin-Electricity Conversion Induced by Spin Injection into To pological Insulators , Phys.\nRev. Lett. 113196601 (2014).\n29M. Jamali, J.-S. Lee, Y. Lv, Z. Zhao, N. Samarth, and J.-P. Wan g,Room Temperature Spin\nPumping in Topological Insulator Bi2Se3, arXiv:1407.7940\n30X. Liu and J. Sinova, Reading Charge Transport from the Spin Dynamics on the Surfac e of a\nTopological Insulator , Phys. Rev. Lett. 111, 166801 (2013).\n31M. H. Fischer, A. Vaezi, A. Manchon, and E.-Ah Kim, Large Spin Torque in Topological Insu-\nlator/Ferromagnetic Metal Bilayers , arXiv:1305.1328.\n32T. Yokoyama and Y. Tserkovnyak, Spin diffusion and magnetoresistance in\nferromagnet/topological-insulator junctions , Phys. Rev. B 89, 035408 (2014).\n33G. D. Mahan, Many-Particle Physics , 3rd ed. (Springer, 2000), p.514.\n34J. Fujimoto, A. Sakai, H. Kohno, Ultraviolet divergence and Ward-Takahashi identity in a tw o-\ndimensional Dirac electron system with short-range impuri ties, Phys. Rev. B 87, 085437 (2013).\n3235Using/planckover2pi1\n2τ=1\n2πνeniu2\n0,νe=ǫF\n2π/planckover2pi12˜v2\nF, and ˜vF/vF= (1+ξ)−1, wecanobtaintherelation: ξ(1+ξ)2=\n/planckover2pi1\n4πǫFτ. Because of/planckover2pi1\nǫFτ≪1,ξ </planckover2pi1\n4πǫFτbecomes negligible small as ˜ vF/vF= 1+o(/planckover2pi1\nǫFτ). Streakily\nspeaking, there is no problem to take ˜ vF≃vF.\n36If we do not take the replacement vF→˜vFin Eq. (2), the velocity operator on the surface of\nthe TI becomes ˆ v=vF(z׈σ). Then, the charge current is given by the multiple of vF/˜vFin\nEqs. (42)-(43). The charge current does not satisfy the char ge conservation law, which is shown\nin Appendix B.\n37D. Culcer, J. Sinova, N. A. Sinitsyn, T. Jungwirth, A. H. MacD onald, and Q. Niu, Semiclassical\nSpin Transport in Spin-Orbit-Coupled Bands , Phys. Rev. Lett. 93, 046602 (2004).\n38J. Shi, P. Zhang, D. Xiao, and Q. Niu, Proper Definition of Spin Current in Spin-Orbit Coupled\nSystems, Phys. Rev. Lett. 96, 076604 (2006).\n39N. Nakabayashi, A. Takeuchi, K. Hosono, K. Taguchi, and G. Ta tara,Theory of spin relaxation\ntorque in metallic ferromagnets , Phys. Rev. B. 82, 014403 (2010).\n40H. Haug and A. P. Jauho, Quantum Kinetics in Transport and Optics of Semiconductors , 2nd\ned. (Springer, New York, 2007), pp. 45–46.\n41ˆΛra= [1−˜Γra]−1is given when max{/summationtext\nν[˜Γra]µν}<1 is satisfied.\n42C. Kittel, Introduction to Solid State Physics , 8th ed. (Wiley, 2004), p.148.\n43T. L. Gilbert, A phenomenological theory of damping in ferromagnetic mate rials, IEEE Trans.\nMagn.40, 3443 (2004).\n44S. Mizukami, Y. Ando, and T. Miyazaki, Effect of spin diffusion on Gilbert damping for a very\nthin permalloy layer in Cu/permalloy/Cu/Pt films , Phys. Rev. B. 66, 104413 (2002).\n45Y. Tokura and N. Kida Dynamical magnetoelectric effects in multiferroic oxides , Phil. Trans.\nR. Soc. A 369, 3679 (2011).\n46A. A. Taskin, S. Sasaki, K. Segawa, and Y. Ando, Manifestation of Topological Protection in\nTransport Properties of Epitaxial Bi 2Se3Thin Films , Phys. Rev. Lett. 109, 066803 (2012).\n47T. Misawa, T. Yokoyama, and S. Murakami, , Phys. Rev. B. 84.165407 (2011).\n48Y.Kajiwaraet. al., Transmission of electrical signals by spin-wave interconv ersion in a magnetic\ninsulator , Nature 464, 262 (2010).\n49I. Vobornik, U. Manju, J. Fujii, F. Borgatti, P. Torelli, D. K rizmancic, Y. S. Hor, R. J. Cava,\nandG. Panaccione, Magnetic Proximity Effect as a Pathway to Spintronic Applica tions of Topo-\nlogical Insulators , Nano Lett. 11, 4079 (2011).\n3350P. Wei, F. Katmis, B.A. Assaf, H. Steinberg, P. Jarillo-Herr ero, D. Heiman, J.S. Moodera,\nExchange-Coupling-Induced Symmetry Breaking in Topologica l Insulators , Phys. Rev. Lett. 110,\n186807 (2013).\n51S. Seki, X. Z. Yu, S. Ishiwata, Y. Tokura, bservation of Skyrmions in a Multiferroic Material ,\nscience,336, 198 (2012).\n52J. S. White, et. al., Electric-Field-Induced Skyrmion Distortion and Giant Latt ice Rotation in\nthe Magnetoelectric Insulator Cu3OSeO3, Phys. Rev. Lett. 113, 107203 (2014).\n34"    },    {        "title": "1412.4551v1.Characterization_of_S_T_____Transition_Dynamics_via_Correlation_Measurements.pdf",        "content": "arXiv:1412.4551v1  [cond-mat.mes-hall]  15 Dec 2014Characterization of S- T+Transition Dynamics via Correlation Measurements\nChristian Dickel,1,∗Sandra Foletti,2Vladimir Umansky,3and Hendrik Bluhm1\n1JARA-Institute for Quantum Information, RWTH Aachen Unive rsity, D-52074 Aachen, Germany\n2Department of Physics, Harvard University, Cambridge, Mas sachusetts 02138, USA\n3Braun Center for Submicron Research, Department of Condens ed Matter Physics,\nWeizmann Institute of Science, Rehovot 76100, Israel\nNuclear spins are an important source of dephasing for elect ron spin qubits in GaAs quantum\ndots. Most studies of their dynamics have focused on the rela tively slow longitudinal polarization.\nWe show, based on a semiclassical model and experimentally, that the dynamics of the transverse\nhyperfine field can be probed by correlating individual Landa u-Zener sweeps across the S- T+tran-\nsition of a two-electron spin qubit. The relative Larmor pre cession of different nuclear spin species\nleads to oscillations in these correlations, whose decay ar ises from dephasing of the nuclei. In the\npresence of spin orbit coupling, oscillations with the abso lute Larmor frequencies whose amplitude\nreflects the spin orbit coupling strength are expected.\nPACS numbers: 73.63.Kv, 03.65.Yz, 76.70.Dx\nElectron spins in semiconductor quantum dots have\nbeen studied much as a platform for quantum informa-\ntion processing. While at first it was envisioned to use\nsingle electron spins as qubits [1], it is also possible to\nencode a qubit in several electron spins. The experi-\nmentally most established approach is to use the singlet\nstate S and the m= 0triplet state, T0, of two elec-\ntrons in a double quantum dot (DQD). Alternatively, it\nwas proposed [2] to replace T0with them= 1state,\nT+. Coherent oscillations between these two states were\nobserved using Landau-Zener-Stückelberg interferometry\n[2], and pulse sequences to control this type of qubit have\nbeen constructed [2–4]. The transition between S and T0\nis a narrow avoided crossing mediated by perpendicular\neffective magnetic fields arising from nuclear spins and\nspin orbit (SO) coupling [5]. It is thus important to un-\nderstand the quantum dynamics of transitions between\nthese two states, which, as we show here, are strongly\naffected by the precession of the nuclear spins in the ex-\nternal field. Nuclear spin effects are especially prominent\nin GaAs-based quantum dots, currently the most devel-\noped material system available for gated semiconductor\nquantum dots.\nIn addition to qubit control, the S- T+transition can be\nused to transfer angular momentum from the electronic\nsystem to the nuclear spins. The resulting dynamic nu-\nclear polarization (DNP) has been instrumental in re-\nducing dephasing [6] and achieving universal control [7]\nof S-T0qubits. Periodically traversing the crossing can\nbe used to polarize the nuclear spins and even to sup-\npress fluctuations via a feedback loop. Such nuclear spin\ncontrol is useful for both gate defined and self assembled\nquantum dots [8–10]. In gated dots, the performance\nof such DNP schemes is currently limited by a low spin\ntransfer probability from electrons to nuclei [6], which\nis poorly understood. It was observed that the electron\nspin flip rate exceeds the nuclear polarization rate, which\nmay result from an additional SO spin flip channel [11–14]. Probing the transitions between S and T+states\nand the role of SO coupling is thus also relevant for un-\nderstanding the limitations of DNP methods.\nHere, we present a measurement technique that probes\nthe dynamics of the matrix element driving transitions\nbetween S and T+. Direct measurements of the Over-\nhauser field have previously been used to probe fluctua-\ntions of its component parallel to the external magnetic\nfield [6, 15]. However, the transverse part that drives the\noscillations between S and T+exhibits faster dynamics\nthat are harder to access. Our approach is based on\ncorrelating the outcomes of subsequent individual single\nshot measurements of the qubit state after Landau-Zener\n(LZ) transitions similar to [16]. After each LZ sweep, the\nstate of the qubit is measured. From this data, the auto-\ncorrelation of single shot measurements is computed as a\nfunction of the time delay between them. The time res-\nolution is only limited by the accuracy of pulse timing,\nbut not by the required averaging time.\nWe will show in the following that these correlations\nreveal the relative nuclear spin Larmor precession and de-\nphasing of these precessions on a µs scale. In the presence\nof SO coupling, the oscillations would exhibit additional\nfrequency components. Therefore, the method can pro-\nvide direct, qualitative and quantitative evidence for SO\ncoupling. We also present proof of principle experimental\nresults. They show nuclear spin effects but no SO cou-\npling. We also point out that averaging the LZ transition\nprobability over nuclear spin configurations leads to an\nalgebraic instead of exponential dependence on the sweep\nrate. A SO component of the transition matrix element\nis not subject to this averaging and accordingly leads to\nmore exponential behavior.\nThe two-electron spin qubits considered here consist\nof two electrons trapped in a double-well potential as\nseen in Fig. 1(a) formed by electrostatic gating of a two-\ndimensional electron gas in a GaAs/AlGaAs heterostruc-\nture. A comprehensive derivation of the relevant DQD2\nE\nxεBext Bso (a)\nE\nInitialize Sweep t\nMeasure Bext \nS(0,2)S(1,1)T0T-\nT+S(1,1)\nΔε Δtε\nδt2τ (b) \n2Δ \n(c) \n/g1636∆Bnuc \n69 Ga 71 Ga 75 As ⊙Bext \nω69 ω71 ω75 Bnuc L\nBnuc R\nΩΔ\nFigure 1. (a) DQD potential landscape along the dot-dot axis .\nVectors indicate effective magnetic fields from SO due to tun-\nneling and nuclear spins via the Fermi contact interaction.\nThe detuning εis the energy difference between the single\nelectron levels in the dots. (b) DQD energy diagram and\npulse forms for LZ correlation measurements. Pulse cycles\nare concatenated with a time delay δt. (c)∆B⊥\nnuchas contri-\nbutions from the three different nuclear species that preces s in\nBextwith different Larmor frequencies. The S-T +transition\nmatrix element ∆is a vector sum of the static SO field Ωand\n∆B⊥\nnuc\nelectronic Hamiltonian can be found in [5]. In the fol-\nlowing we provide a qualitative summary. The qubit is\ncontrolled via the detuning, ε, defined as the energy dif-\nference between the lowest (1, 1) and (0, 2) charge con-\nfiguration, where the numbers indicate the occupancy of\neach dot. Positive detuning favors double occupation of\none dot in the singlet state, while at negative detuning\nthe electrons are separated. The resulting level diagram\nas a function of εis shown in Fig. 1(b). The singlet states\nexhibit an avoided crossing with tunnel coupling τ. Due\nto the Pauli principle, the triplet states |T+/an}bracketri}ht,|T−/an}bracketri}htand\n|T0/an}bracketri}htwith magnetic quantum numbers m=±1,0always\nstay in the (1, 1) configuration and are energetically sep-\narated by the Zeeman energy in an external magnetic\nfieldBext. This setting allows fast electric singlet initial-\nization and single shot readout that distinguishes singlet\nand triplet states via spin to charge conversion [15], in-\ndependent of which triplet is used.\nThe focus of this Letter is the S-T +degeneracy point\n(called S-T +transition), where transitions between these\nstates are possible. A convenient way to probe this nar-\nrow crossing is to employ LZ sweeps, i.e., linear sweeps\nacross the transition as illustrated in Fig. 1(b) that do\nnot require precise knowledge of the position of the tran-\nsition. For a sufficiently small sweep range, the other\nstates can be neglected. We use an effective Hamil-\ntonian in the basis {|S∗(ε)/an}bracketri}ht,|T+/an}bracketri}ht}, where |S∗(ε)/an}bracketri}ht=cosψ|S(1,1)/an}bracketri}ht+ sinψ|S(0,2)/an}bracketri}htis the hybridized singlet\nclose to the crossing:\nH=Bext−J(ε)\n2σz+∆σx, (1)\nwhereσxandσzare Pauli matrices. We use units\nof energy for all electronic Zeeman fields, thus setting\ngµB= 1. The parameters are the exchange splitting\nJ=ES∗−ET0and the coupling\n∆ =1\n2/vextendsingle/vextendsingle/vextendsingleBSOsinψ+∆B⊥\nnuccosψ/vextendsingle/vextendsingle/vextendsingle. (2)\nThe singlet mixing angle, ψ, at the transition sets the\nweight of the effective SO field BSO, which couples\n|S(0,2)/an}bracketri}htand|T+/an}bracketri}ht, and the difference in transverse hyper-\nfine fields ∆B⊥\nnuc=/parenleftBig\nB⊥,R\nnuc−B⊥,L\nnuc/parenrightBig\nbetween the dots,\nwhich couples |S(1,1)/an}bracketri}htand|T+/an}bracketri}ht[5]. The mixing angle\ndepends on τand indirectly on Bextvia the position of\nthe transition. For J=αt, whereαis the sweep rate and\na sweep from −∞to∞, the probability that an initial\nsinglet stays in a singlet is given by PS= exp/parenleftBig\n−2π∆2\n/planckover2pi1α/parenrightBig\n[17, 18].\nIn this work, we assume Bext≪τso thatcosψ≈1\nandsinψ≪1at the transition. BSO=4τ\n3l\nlSOdepends\nthe inter-dot distance l≈200nm and the SO length\nlSO[5]. Therefore the effective field from SO coupling\nis expected to be static with a value on the order of\nBSO∼100 mT. For two million unit cells in each dot\n[19],/angbracketleftbigg/parenleftBig\n∆B⊥\nnuc/parenrightBig2/angbracketrightbigg1/2\nis on the order of 4 mT. Thus even\na suppressed SO coupling could approach the expected\nOverhauser field. We introduce the overall SO contribu-\ntion to∆,Ω=BSOsinψ. In the regime of interest here,\nΩ/lessorsimilar/angbracketleftbigg/parenleftBig\n∆B⊥\nnuc/parenrightBig2/angbracketrightbigg1/2\n.\nThe dynamical effects considered here originate in the\ntime dependence of ∆via∆B⊥\nnuc, which arises from the\ninteractions of each electron spin with on the order of\n106nuclear spins. As illustrated in Fig. 1(c), ∆B⊥\nnuc\nis a vector sum of contributions from the three isotopes\npresent in GaAs,69Ga,71Ga and75As, all with nuclear\nspin3/2. Following earlier work, we treat their fields as\nclassical random variables [20].\nIn addition to the Larmor precession of the nuclei, we\nalso account for dephasing of this precession due to mi-\ncroscopic field fluctuations originating from dipole-dipol e\ninteractions and quadrupole splittings. To this end, we\nsubdivide each species into subspecies with slightly differ -\nent precession frequencies ωi=γi(Bext+δBi), whereγi\nis the nuclear gyromagnetic ratio. The field fluctuations\nδBiare taken from a discretized normal distribution with\nstandard deviation δBloc, and the index idenotes both\nthe nuclear species and the local field value. We find that\ngood convergence is obtained for five bins per species.3\nWe incorporate the above dynamics by writing\n∆(t)2=1\n4/parenleftBig/summationdisplay\nijB+\niB−\njei(ωi−ωj)t\n+/summationdisplay\niΩ/parenleftbig\nB+\nieiωit+B−\nie−iωit/parenrightbig\n+Ω2/parenrightBig\n,(3)\nwhereB±\ni=Bx\ni±iBy\ni. Note that the contribution from\nSO coupling behaves like an additional, static nuclear\nspin species. During a single LZ sweep, we approximate\n∆as constant because the nuclear spin dynamics are slow\ncompared to the time spent at the crossing. This approx-\nimation is justified as long as γN/γeBext≪∆, whereγN\nandγeare nuclear and electronic gyromagnetic ratios.\nThe expectation values over nuclear states of the\nsingle shot measurement correlations are given by\n/an}bracketle{tPS(t)PS(t+δt)/an}bracketri}ht=/angbracketleftBig\nexp/parenleftBig\n−2π\nα/planckover2pi1(∆(t)2+∆(t+δt)2/parenrightBig/angbracketrightBig\n[16]. To evaluate them one has to integrate over all pos-\nsible initial conditions B±\ni. In the homogeneous coupling\napproximation used here, each component of the effec-\ntive field of each species is approximated as normally\ndistributed with standard deviation Bi,rms=Ai\nN/radicalbig\n5Ni/2,\nwhich corresponds to averaging over an infinite temper-\nature nuclear ensemble. Niis the number of spins in\ntheith bin,Aithe species-dependent hyperfine coupling\nstrength and Nthe number of unit cells occupied by each\nelectron.\nTo perform this integration, it is useful to decouple the\nterms via the T-matrix method [20]. The exponent of the\nLZ correlations is rewritten as\n2π\n/planckover2pi1α/parenleftBig\n∆(t)2+∆(t+δt)2/parenrightBig\n=π\nα/planckover2pi1Ω2+/summationdisplay\nijTij(δt)z+\niz−\nj\n+/summationdisplay\niVi(δt)∗z−\ni+/summationdisplay\niVi(δt)z+\ni\nwithzi=xi+iyi, wherexandyare normally distributed\nwith unit variance. Diagonalizing the T-matrix, applying\nthe same basis change to the linear terms and integrating\nover thezidistributions results in\n/an}bracketle{tPS(t)PS(t+δt)/an}bracketri}ht= exp/parenleftbigg\n−πΩ2\nα/planckover2pi1/parenrightbigg\n×/productdisplay\ni1\n1+2λi·exp/parenleftBigg\n2˜Vi2\n(1+2λi)/parenrightBigg\n,\nwhereλiare the eigenvalues of the T-matrix and the ˜Vi\nare the components of the vector Viin the eigenbasis of\nthe T-matrix. A similar, straightforward calculation also\nreveals a modification of the average flip probability due\nto averaging over nuclear spin configurations:\n/an}bracketle{tPS/an}bracketri}ht=α/planckover2pi1\nα/planckover2pi1+π/summationtext\niB2\ni,rms·exp/parenleftBigg\n−πΩ2\n2/parenleftbig\nα/planckover2pi1+π/summationtext\niB2\ni,rms/parenrightbig/parenrightBigg\n.\n(4)0 20 40 60 80 100 12 0 14 00.250.30.350.40.45\n Ω = 0.35   ( ΔBnuc )²    ½ Ω = 0.70   ( ΔBnuc )²    ½\nT75 \n/g1636/g1636ω71 ω69 ω75 0\nω71 \nω71-75 \nω71-69 ≈ ω 69-75 \n Ω = 0 \nδt (μs) T71-75 ωPS(t)PS(t+δt)Bext  = 20 mT \nδBloc  = 0.3 mT \nNunit cells  = 2.3 ·10 6\nα = 20 µeV/µs \nT71-69 ≈ T69-75 \nFigure 2. Simulations of correlations for different SO cou-\npling. Above the nuclear spin frequencies are given. For\nΩ = 0 , correlations oscillate with the relative Larmor fre-\nquencies. For Ω>0additional frequencies appear, and the\ncorresponding oscillations extend to longer times while th e\noriginal peaks become smaller and some split into two.\nForΩ = 0 , the behavior changes from exponential to\nalgebraic.\nFig. 2 shows the correlations for typical experimental\nparameters and different SO coupling constants Ω = 0 .\nAs also seen in equation 3, ∆oscillates with the Larmor\nfrequency differences ωi−ωjforΩ = 0 . In GaAs, the\nthreeγiare approximately equidistant. Therefore, the\nrelative orientations of the three contributions to ∆B⊥\nnuc\nand thus ∆return to nearly the same value after one\nperiodTij= 2π(ωi−ωj)−1of the slower relative pre-\ncession,T75−69≈T71−69. Hence, large revivals in the\ncorrelations occur at this value of δt. The smaller peaks\natδt=T75−71result from a partial realignment of these\ntwo species only. The envelope decay of the correlations\non a time scale of approximately 40 µs arises from nuclear\nspin dephasing.\nAsΩis increased, oscillations at the bare frequencies\nωiextending to larger δtstart to appear as well. In addi-\ntion, the large peak around δt= 17µssplits. Observing\nthese features would represent qualitative evidence for\nSO coupling. At the same time, the magnitude of the\nvariation in the correlations is reduced.\nWe now turn to proof of principle experimental results,\nobtained from the same DQD sample as used in [19]. In\nall measurements, the sweep range was constrained to\nkeepJ(ε)≫Bnucto prevent mixing of S and T 0. As\nBext≫∆, the nonlinear sweep and the finite sweep range4\ndo not introduce a large error. A measured background\nreflecting the direct coupling of the gate pulses to the\ncharge sensor used for single shot readout was subtracted\nfrom each dataset.\nTo verify the expected behavior of flip the probability,\nwe measured its dependence on sweep rate and magnetic\nfield, as shown in Fig. 3(a). The scaling factor used\nto convert the readout signal to triplet probability was\nchosen in accordance with the contrast of the (1,1)-(0,2)\ncharge transition, taking relaxation during the readout\nstage into account. The scaling of the x-axis is motivated\nby the fact that in the experiment the sweeps result in a\nlinear time dependence of εrather than J(ε). To com-\npute the sweep rate α=dJ/dt=dJ/dε·dε/dt, we employ the\nphenomenological relation J(ε) =J0exp/parenleftBig\n−ε\nε0/parenrightBig\n[21]. At\nthe S-T+ transition, J(ε) =Bext, so thatα=Bext1\nε0∆ε\n∆t,\nwhere∆tis time taken to sweep detuning by ∆ε. Hence,\nPSis expected to be a function of ∆t/Bext, independent\nofBext. Indeed, the curves for different Bextin Fig. 3(a)\ncollapse reasonably well onto each other. The deviations\nfor slow sweeps may be a result of relaxation. The solid\nline was obtained from Eq. 4 using Ω = 0 and a value\nof∆2/αthat is a factor 6 smaller than expected based\non measurements of T∗\n2, from which N= 2.3·106can be\nextracted [19]. This adjustment was required to obtain\nadequate agreement with the model, which contains no\nfurther free parameters. As can be seen from the rela-\ntively large slope at large ∆t, the algebraic dependence\nobtained from nuclear averaging in equation 4 gives a bet-\nter overall fit than the exponential LZ formula (dashed\nline), even if adjusting the scaling factor of the latter.\nGuided by these results, we fixed ∆tat 0.3µs for the\ncorrelation measurements discussed next. Initialization –\nLZ-sweep–readout cycles as shown in Fig. 1(b) were re-\npeated at a rate of 0.5 MHz. Initialization was carried\nout via a standard procedure through electron exchange\nwith the leads [2]. Readout was accomplished via RF re-\nflectometry of a quantum point contact [15]. Instead of\nsingle shot discrimination, we computed the correlations\ndirectly from the readout signal, which was averaged over\n1µsin each cycle. Assuming the readout signal is the\nsum of a contribution from the qubit measurement and\nnoise uncorrelated with the qubit state, this procedure is\nequivalent to correlating single shot readout values apart\nfrom additional noise. Due to the fixed sampling rate,\nthe time resolution was limited to 2 µs.\nThe data shown in Fig. 3(b) clearly shows the expected\noscillations. For fitting our model to the experiment,\nonly an offset of the QPC signal was adjusted individ-\nually for each data set. The y-scale was set analogous\nto Fig. 3(a). Obtaining good agreement of the mag-\nnitude required ∆2/αto be chosen a factor 18 smaller\nthan expected for N= 2.3·106unit cells and experi-\nmental sweep rates. The δt-dependence is unaffected by\nthis adjustment. Nuclear spin Larmor frequencies were  \nΔt / Bext   ( μs / T) \n0 20 40 60 80 10000.050.10.150.20.250.3 Bext  = 40 mT \nBext  = 35 mT \nBext  = 30 mT \nBext  = 25 mT \nBext  = 20 mT \nBext  = 15 mT 0 10 20 30 40 50 60 70 80 00.250.50.751\nBext  = 75 mT Bext  = 15 mT \nδt (μs) PSPS(t)PS(t+δt)(a)\n(b) 1 - \nFigure 3. (a) Triplet probabilities as a function of sweep-t ime\n∆tfrom 0µs to 1µs with a constant ∆εfor different Bext.\nThe scaling of the time axis with magnetic field leads to a\nreasonable collapse of the curves. Data in color. The black\nlines shows the LZ-model (dashed) with a fixed transition\nmatrix element, and the nuclear spin averaged LZ behavior\n(solid). With N= 2.3·106fixed,∆2/αrequires a scaling\nfactor of 6 to match experiment. (b) Correlation of triplet\nprobabilities for different Bext, data in black and model in\nred. Except for the magnitude, the correlations behave as\nexpected as a function of Bext.\ntaken from [22]. δBlocdetermines the time scale of the\noverall decay and was fixed to 0.33 mT, consistent with\nearlier measurements [19]. As expected for the low mag-\nnetic fields considered here, the effect of SO coupling is\nnegligible. At higher fields, contrast is lost faster than\nexpected from our model, but the time resolution used\nhere also becomes too coarse to resolve the Larmor pre-\ncession.\nThe experiments thus confirm the overall picture ob-\ntained from our model and the usefulness of single shot\ncorrelation measurements. The significant quantitative\ndiscrepancy in ∆2/αis not understood, but may at least\npartly arise from noise and non-uniformity of the sweep5\nresulting from the discrete 1 ns sampling interval of the\nwave form generator. More detailed measurements would\nbe required to explore these effects. By varying the de-\nlay between subsequent pulses instead of using a fixed\nrepetition rate, the time resolution could be improved\nsignificantly. This would enable an extension to larger\nmagnetic fields where SO coupling might become impor-\ntant.\nIn conclusion, we have shown that correlation measure-\nments of Landau-Zener sweeps through the S-T+ transi-\ntion reveal nuclear spin precession and dephasing. These\ndynamics are seen in an experimental implementation of\nthe scheme. They imply that the perpendicular compo-\nnent of the Overhauser field cannot be controlled as well\nas thez-component. Our model indicates that the same\nmeasurement technique should be able to qualitatively\nreveal SO coupling at higher magnetic fields.\nThis work was supported by the Alfried Krupp von\nBohlen und Halbach Foundation and DFG under Grant\nNo. BL 1197/2-1. We would like to thank Amir Yacoby,\nMark Rudner and Izhar Neder for discussions.\n∗Present address: QuTech and Kavli Institute of\nNanoscience, Delft University of Technology, 2600 GA\nDelft, The Netherlands\n[1] D. Loss and D. P. DiVincenzo,\nPhys. Rev. A 57, 120 (1998).\n[2] J. R. Petta, H. Lu, and A. C. Gossard,\nScience 327, 669 (2010).\n[3] H. Ribeiro, J. R. Petta, and G. Burkard,\nPhys. Rev. B 87, 235318 (2013).\n[4] H. Ribeiro, G. Burkard, J. R. Petta, H. Lu, and A. C.\nGossard, Phys. Rev. Lett. 110, 086804 (2013).\n[5] D. Stepanenko, M. Rudner, B. I. Halperin, and D. Loss,Phys. Rev. B 85, 075416 (2012).\n[6] H. Bluhm, S. Foletti, D. Mahalu, V. Umansky, and\nA. Yacoby, Phys. Rev. Lett. 105, 216803 (2010).\n[7] S. Foletti, H. Bluhm, D. Mahalu, V. Umansky, and\nA. Yacoby, Nature Physics 5, 903 (2009).\n[8] C. Kloeffel, P. A. Dalgarno, B. Urbaszek, B. D. Gerardot,\nD. Brunner, P. M. Petroff, D. Loss, and R. J. Warburton,\nPhys. Rev. Lett. 106, 046802 (2011).\n[9] C. Latta, A. Hoegele, Y. Zhao, A. N. Vamivakas,\nP. Maletinsky, M. Kroner, J. Dreiser, I. Carusotto,\nA. Badolato, D. Schuh, W. Wegscheider, M. Atature,\nand A. Imamoglu, Nature Physics 5, 758 (2009).\n[10] T. D. Ladd, D. Press, K. De Greve, P. L. McMahon,\nB. Friess, C. Schneider, M. Kamp, S. Höfling, A. Forchel,\nand Y. Yamamoto, Phys. Rev. Lett. 105, 107401 (2010).\n[11] M. S. Rudner, I. Neder, L. S. Levitov, and B. I. Halperin,\nPhys. Rev. B 82, 041311 (2010).\n[12] A. Brataas and E. I. Rashba,\nPhys. Rev. B 84, 045301 (2011).\n[13] A. Brataas and E. I. Rashba,\nPhys. Rev. Lett. 109, 236803 (2012).\n[14] I. Neder, M. S. Rudner, and B. I. Halperin,\nPhys. Rev. B 89, 085403 (2014).\n[15] C. Barthel, D. J. Reilly, C. M. Marcus, M. P. Hanson,\nand A. C. Gossard, Phys. Rev. Lett. 103, 160503 (2009).\n[16] T. Fink and H. Bluhm,\nPhys. Rev. Lett. 110, 010403 (2013).\n[17] L. Landau, Physics of the Soviet Union 2 , 46 (1932).\n[18] C. Zener, Proc. Roy. Soc. London, Ser. A 137, 696 (1932).\n[19] H. Bluhm, S. Foletti, I. Neder, M. Rudner,\nD. Mahalu, V. Umansky, and A. Yacoby,\nNature Physics 7, 109 (2011).\n[20] I. Neder, M. S. Rudner, H. Bluhm, S. Fo-\nletti, B. I. Halperin, and A. Yacoby,\nPhys. Rev. B 84, 035441 (2011).\n[21] O. E. Dial, M. D. Shulman, S. P. Harvey,\nH. Bluhm, V. Umansky, and A. Yacoby,\nPhys. Rev. Lett. 110, 146804 (2013).\n[22] D. Paget, G. Lampel, B. Sapoval, and V. I. Safarov,\nPhys. Rev. B 15, 5780 (1977)."    },    {        "title": "0801.4132v2.Spin_dynamics_of_two_dimensional_electrons_with_Rashba_spin_orbit_coupling_and_electron_electron_interactions.pdf",        "content": "arXiv:0801.4132v2  [cond-mat.mes-hall]  3 Nov 2008Spin dynamics of two-dimensional electrons with Rashba spi n-orbit coupling and\nelectron-electron interactions\nYuan Li1,2and You-Quan Li1\n1Department of Physics, Zhejiang University, Hangzhou 310027, P . R. China\n2Institute of Materials Physics, Hangzhou Dianzi University, Hangz hou 310018, P. R. China\n(Dated: October 29, 2018)\nWe study the spin dynamics of two dimensional electron gases (2DEGs) with Rashba spin-orbit\ncoupling by taking account of electron-electron interacti ons. The diffusion equations for charge\nand spin densities are derived by making use of the path-inte gral approach and the quasiclassical\nGreen’s function. Analyzing the effect of the interactions, we show that the spin-relaxation time\ncan be enhanced by the electron-electron interaction in the ballistic regime.\nPACS numbers: 72.25.Rb, 71.70.Ej, 71.10.Ca\nI. INTRODUCTION\nSpin based electronics [1] or spintronics [2] has been\nan active research area in the past decade. The effort\nfor effectively manipulating electron spin by means of an\napplied electric field [3, 4, 5] is an important issue there.\nThe system with spin-orbit (SO) couplings makes those\nefforts possible and thus brings great interests from both\nacademicandpracticalaspectsrecently. Thus,itisessen-\ntial to study the spin relaxation for further development\nof spintronics.\nThere are four main mechanisms of spin relaxation in\nsemiconductor systems [2, 6, 7, 8, 9, 10]. In Elliott-Yafet\nmechanism, the spin-orbit coupling induces a mixing of\nwave functions for valence-band states and conduction-\nband states. The mixing that results in the spin relax-\nation ofelectrons is due to the scattering by impurities or\nphonons. The Elliott-Yafet mechanism operates in semi-\nconductors with and without a center of inversion sym-\nmetry, while it is most prominent in the centrosymmetric\nones (such as silicon). The Bir-Aronov-Pikus mechanism\nis applicable for p-doped semiconductors in which the\nelectron spin flipping is induced by exchange interaction\nwith holes. The hyperfine interaction provides another\nimportant mechanism [11] for ensemble spin dephasing\nand single spin decoherence of localized electrons. The\nD’yakonov-Perel mechanism depicts that electrons can\nfeel an effective random magnetic field arising from the\nspin-orbit coupling in systems with inversion asymme-\ntry such that spin relaxation occurs. This mechanics\ncan interpret the spin dephasing in crystals without in-\nversion center and is particularly applicable for n-type\nsamples. For two dimensional n-type semiconductor sys-\ntems without inversion symmetry, the D’yakonov-Perel\nmechanism is believed to be most important in wide\nranges of carrier temperature and concentration. Under\ncertain conditions, the Elliott-Yafet mechanism may af-\nfect spin dynamics of two-dimensional electrons in these\nsystems. The Bir-Aronov-Pikus mechanism is impor-\ntant forp-type semiconductor systems and the hyperfine-\ninteraction mechanism dominates for localized electrons.\nMost studies of spin relaxation in semiconductors havefocused on impurity (somewhat less phonon) mediated\nspin flips while neglecting the effect of electron-electron\ninteractions for a long time. It has been noticed re-\ncentlythatelectron-electroninteractionsplaycertainrole\nin spin relaxation and dephasing in semiconductor sys-\ntems. The electron-electron interaction is known to play\na crucial role in determining the transport and thermo-\ndynamic properties near the metal-insulatortransition in\ntwo-dimensionalelectronsystems[12], whichissuspected\nto affect the spin relaxation for the spin susceptibility\nbehaving critically when the metal-insulator transition\noccur [13, 14, 15]. There are several experimental and\ntheoretical studies on the effect of electron-electron in-\nteractions on spin relaxation. The electron-electron scat-\ntering results in additional momentum relaxation which\ninduces spin dephasing of electrons through the motional\nnarrowing of the D’yakonov-Perel type [16] as measured\ninn-GaAs/AlGaAsquantumwells[17, 18]. Theelectron-\nelectron scattering effect on the spin dephasing has been\nconsidered[19] in a magnetic field, and a momentum de-\npendent effective random magnetic field induced by the\nelectron-electron exchange interaction can lead to spin\ndephasing of electrons [20, 21, 22]. It is also observed\nthat the spin relaxation caused by the D’yakonov-Perel\nmechanism gives considerably different rates depending\non the technique employed [23].\nHowever, as we are aware, the explicit form of the\ndiffusion equation for two-dimensional electron gases\n(2DEGs) with spin-orbit couplings has not been derived\nby taking account of electron-electron interactions. It is\nthus obligatory to develop the explicit form of the dif-\nfusion equation to study the spin dynamics for 2DEGs\nwith spin-orbit couplings as well as electron-electron in-\nteractions. In this paper, we focus attention on the\nD’yakonov-Perel spin-relaxation mechanism. We inves-\ntigate the spin dynamics of electrons in two-dimensional\nn-type semiconductor systems with electron-electron in-\nteractions and Rashba spin-orbit coupling.\nThe paper is organized as follows. In Sec. II, we\ntake account of the electron-electron interaction for the\n2DEGs with the Rashba spin-orbit coupling. Applying\nthe path integral formulation, we decoupled the interac-2\nFIG. 1: The Keldysh contour.\ntion in terms of an auxiliary Bose field. In Sec. III, we\nemploy the quasiclassical Green’s function to investigate\nthe spin dynamics of electrons. In Sec. IV, the diffusion\nequations for spin and charge densities as well as the ex-\nplicit expression of spin-relaxation time are derived. A\nsummary is given in Sec. V and some complicated for-\nmulae are given in the Appendix.\nII. AUXILIARY FIELDS DESCRIBING THE\nELECTRON-ELECTRON INTERACTION\nTaking the electron-electron interaction into account,\nwe study the spin dynamics of electrons in two-\ndimensionalsystemswith structureinversionasymmetry.\nAs the Fourier transform of the Coulomb repulsion be-\ntween electrons reads V(q) = 2πe2/|q|, the Hamiltonian\nof such a system is given by\nˆH=/integraldisplay/braceleftig/summationdisplay\nλ,λ′ˆψλ†(r)/bracketleftbig/parenleftbig\n−/planckover2pi12\n2m∇2+U(r)−µ/parenrightbig\nδλ,λ′\n+b·/vector σλλ′/bracketrightbigˆψλ′(r)/bracerightig\nd2r (1)\n+1\nA/summationdisplay\nq/negationslash=0πe2\n|q|ˆρ(q)ˆρ(−q),\nwhereˆψλ†(r) andˆψλ(r) representthefieldoperatorswith\nλ=↑,↓labelling the spin state of the electron, ˆ ρ(q)\nrepresents the Fourier transform of the density operator\nˆρ(r) =/summationtext\nλˆψλ†(r)ˆψλ(r) and/vector σ= (σx,σy,σz) the Pauli\nmatrices in spin space, U(r) a random disorder poten-\ntial andµthe chemical potential. The other notions in\nEq. (1) are A=L2withLreferring to the size of the\nsample and b=αp×ezwithαreferring to the Rashba\nspin-orbit coupling strength. In holonomyrepresentation\n(or the called coherent state representation), the Green’s\nfunction can be expressed as a functional integral over\ntheGrassmannfields ψλand¯ψλthatreflectthefermionic\nnature of electrons,\nGλλ′(r,t;r′,t′) =∝an}b∇acketle{tψλ(r,t)¯ψλ′(r′,t′)∝an}b∇acket∇i}ht\n=/integraldisplay\nD¯ψDψψλ(r,t)¯ψλ′(r′,t′)e−iS[ψ,¯ψ]\n/integraldisplay\nDψD¯ψe−iS[ψ,¯ψ].(2)Here we adopted the unit /planckover2pi1= 1 and the simplified no-\ntationDψ=Dψ↑Dψ↓. The action S[ψ,¯ψ] in the above\nequation is given by\nS[ψ,¯ψ] =/integraldisplay\ndt/braceleftig/integraldisplay\nd2r/summationdisplay\nλλ′¯ψλ(r,t)Wλλ′ψλ′(r,t)\n+1\nA/summationdisplay\nq/negationslash=0πe2\n|q|/parenleftbig\nρ(q,t)ρ(−q,t)/parenrightbig/bracerightig\n,(3)\nwhereWλλ′=/parenleftbig\n−i∂/∂t−∇2\nr/2m+U(r)−µ/parenrightbig\nδλλ′+b·/vector σλλ′.\nWe divide the fermionic field ψλ(r,t) into two compo-\nnentsψλ\n1(r,t) andψλ\n2(r,t) which reside, respectively, on\nthe upper and lower branches of the Keldysh time con-\ntour shown in Fig. (1). Hence, the second line of Eq. (3),\nwhich refers to the interaction part, can be written as\nSint[ψ1,¯ψ1]−Sint[ψ2,¯ψ2] with\nSint[ψi,¯ψi] =/integraldisplay\ndt/summationdisplay\nq/negationslash=0πe2\nA|q|/parenleftbig\nρi(q,t)ρi(−q,t)/parenrightbig\n,\nwherei= 1,2. With the help of two auxiliary bosonic\nfields˜φi(r,t), we can decouple those two terms rele-\nvant to electron-electron interactions via the Hubbard-\nStratonovich transformation [24], namely,\nexp/bracketleftbig\n−i/integraldisplay\ndt/summationdisplay\nq/negationslash=0πe2\nA|q|ρi(q,t)ρi(−q,t)/bracketrightbig\n=/integraldisplay\nD˜φi(q,t)exp/bracketleftbig\ni/integraldisplay\ndt/summationdisplay\nq/negationslash=0|q|\n4π˜φi(q,t)˜φi(−q,t)/bracketrightbig\n×exp/bracketleftig\ni/integraldisplay\ndte\n2√\nA/summationdisplay\nq/negationslash=0/braceleftbig˜φi(q,t)ρi(−q,t)\n+ρi(q,t)˜φi(−q,t)/bracerightbig/bracketrightig\n.\nThen we can write the Green’s function as follows,\nˆGλλ′(r,t;r′,t′)\n=/integraldisplay\nD¯ΨDΨDΦΨλ(r,t)¯Ψλ′(r′,t′)e−iS[Ψ,¯Ψ,Φ]\n/integraldisplay\nDΨD¯ΨDΦe−iS[Ψ,¯Ψ,Φ],\n(4)\nwhere the action in real space is given by3\nS[Ψ,¯Ψ,Φ] =/integraldisplay\ndtd2r/braceleftig/summationdisplay\nλλ′¯Ψλ(r,t)/bracketleftbig\nWλλ′σ3−e˜φi˜γiδλλ′/bracketrightbig\nΨλ′(r,t)/bracerightig\n+/integraldisplay\ndt/integraldisplay\nd2rd2r′/braceleftig\nΦT(r,t)−e2\n2V−1\n0(r−r′)σ3Φ(r′,t)/bracerightig\n, (5)\nin which the Pauli matrix σ3= diag(1,−1) is defined on\nthe Keldysh space, and V−1\n0is defined via the following\nrelation\n/integraldisplay\nd2r1V0(r−r1)V−1\n0(r1−r′) =δ(r−r′).\nThe other notions appeared in Eq. (5) are fermionic dou-\nblet Ψ, bosonic doublet Φ, and vertex matrices ˜ γi, they\nare defined as\nΨλ=/parenleftbigg\nψλ\n1\nψλ\n2/parenrightbigg\n,Φ =/parenleftbigg˜φ1\n˜φ2/parenrightbigg\n,\n˜γ1=/parenleftbigg\n1 0\n0 0/parenrightbigg\n,˜γ2=/parenleftbigg\n0 0\n0−1/parenrightbigg\n.\nFor calculation convenience, one can introduce a parti-\ntion function for the coupling between the fermionic and\nbosonic doublets,\nZ[Φ] =∝an}b∇acketle{tTCe−iSR[Φ,Ψ]∝an}b∇acket∇i}htΨ,\nSR[Φ,Ψ] =/integraldisplay\ndt d2r/braceleftig/summationdisplay\nλ¯Ψλ/parenleftbig\n−e˜φi˜γi/parenrightbig\nΨλ/bracerightig\n,\nwhere T Cstands for time ordering along the contour C\nand∝an}b∇acketle{t ··· ∝an}b∇acket∇i}ht Ψmeans functional integration over Ψ field\nwith the action\nS[Ψ] =/integraldisplay\ndtd2r/braceleftig/summationdisplay\nλλ′Wλλ′¯Ψλ(r,t)σ3Ψλ′(r,t)/bracerightig\n.\nThen the Green’s function in Eq. (4) can be formally\nexpressed as a functional integration over the bosonic\nfields,\nˆGλλ′(r,t;r′,t′) =N/integraldisplay\nDΦˆGλλ′(r,t;r′,t′|Φ)\n×exp/braceleftbig\n−iSe[Φ]/bracerightbig\n, (6)\nwhere the normalization coefficient is denoted by Nand\nthe actionSe[Φ] is defined by\nSe[Φ] =ilnZ[Φ]+/integraldisplay\ndt d2rd2r′\n×/braceleftbig\nΦT(r,t)−e2\n2V−1\n0(r−r′)σ3Φ(r′,t)/bracerightbig\n,(7)\nand the kernel ˆG(r,t;r′,t′|Φ) is given by\nˆGλλ′(r,t;r′,t′|Φ)\n=1\nZ[Φ]∝an}b∇acketle{tTCΨλ(r,t)¯Ψλ′(r′,t′)e−iSR[Φ,Ψ]∝an}b∇acket∇i}htΨ.(8)We can averagethe Green’s function ˆGλλ′(r,t;r′,t′) over\ndisorder as follows [25]\n∝an}b∇acketle{tˆGλλ′(r,t;r′,t′)∝an}b∇acket∇i}htdis=N/integraldisplay\nDΦ∝an}b∇acketle{tˆGλλ′(r,t;r′,t′|Φ)∝an}b∇acket∇i}htdis\n×exp/braceleftbig\n−i∝an}b∇acketle{tSe[Φ]∝an}b∇acket∇i}htdis/bracerightbig\n,(9)\nwhere∝an}b∇acketle{t···∝an}b∇acket∇i}htdisrefers to the average over disorder. The\nrandom disorder potential U(r) is assumed to be charac-\nterized by a correlation function\n∝an}b∇acketle{tU(r)U(r′)∝an}b∇acket∇i}htdis=1\n2πντδ(r−r′),\nwhereν=m/π/planckover2pi12stands for the density of states. The\naverage of the Green’s function over disorder introduces\nthe elastic scattering time τwhich is relevant to the ran-\ndom disorder. We neglect correlationsbetween the meso-\nscopic fluctuations of ∝an}b∇acketle{tSe[Φ]∝an}b∇acket∇i}htdisand the fermionic opera-\ntors in Eq. (8) so that the averageofthe Green’s function\nˆGλλ′(r,t;r′,t′|Φ) can be separated from the bosonic ac-\ntionSe[Φ]. This approximation is valid since the meso-\nscopic fluctuation is smaller than average quantities.\nAfter averaging over disorder, we rotate the Keldysh\nbasesˆG→Lσ3ˆGL†through a unitary matrix L\nL=1√\n2/parenleftbigg\n1−1\n1 1/parenrightbigg\n,\nso that the Green’s function takes the following shape,\nˆG(r,t;r′,t′|Φ)\n=/parenleftbigg\nGR(r,t;r′,t′|Φ)GK(r,t;r′,t′|Φ)\nGZ(r,t;r′,t′|Φ)GA(r,t;r′,t′|Φ)/parenrightbigg\n.(10)\nNote that the Green’s function ˆG(r,t;r′,t′|Φ) is a 2 ×2\nmatrix defined in the Keldysh space, of which the matrix\nentities are again 2 ×2 matrices defined in spin space.\nThe bosonic fields after rotation take the following two\ncomponents φ1=e\n2(˜φ1+˜φ2) andφ2=e\n2(˜φ1−˜φ2) that\nreside on the upper and lower branches of the contour\nC, respectively. Then the corresponding vertex matrices\nturn toγ1(2)=L(˜γ1±˜γ2)σ3L†, namely,\nγ1=/parenleftbigg\n1 0\n0 1/parenrightbigg\n, γ2=/parenleftbigg\n0 1\n1 0/parenrightbigg\n,\nand the interaction term ¯Ψλ(−e˜φi˜γi)Ψλbecomes\n−¯Ψλ/parenleftbigg\nφ1φ2\nφ2φ1/parenrightbigg\nΨλ. Asγ1andγ2constitute a repre-\nsentation of Z2group, the interaction can be regarded as\nthe coupling between Fermi field and Z2Bose field.4\nThese Bose fields define the following propagators:\nDK(r1,r2;t1,t2) =−2i∝an}b∇acketle{tφ1(r1,t1)φ1(r2,t2)∝an}b∇acket∇i}ht,\nDR(r1,r2;t1,t2) =−2i∝an}b∇acketle{tφ1(r1,t1)φ2(r2,t2)∝an}b∇acket∇i}ht,\nDA(r1,r2;t1,t2) =−2i∝an}b∇acketle{tφ2(r1,t1)φ1(r2,t2)∝an}b∇acket∇i}ht,\n∝an}b∇acketle{tφ2(r1,t1)φ2(r2,t2)∝an}b∇acket∇i}ht= 0. (11)\nOne can show that those propagators obey the Dysonequations in the saddle point approximation, namely,\nˆD(x1,x2) =ˆD0+/integraldisplay\ndx3dx4ˆD0(x1,x3)ˆΠ(x3,x4)ˆD(x3,x2),\n(12)\nwith notations x≡(r,t),DR\n0(q) =DA\n0(q) =−2πe2/|q|,\nand\nˆD=/parenleftbigg\nDRDK\n0DA/parenrightbigg\n,ˆD0=/parenleftbigg\nDR\n00\n0DA\n0/parenrightbigg\n,ˆΠ =/parenleftbigg\nΠRΠK\n0 ΠA/parenrightbigg\n,\nΠR(x1,x2) = ΠA(x2,x1) =δTrsGK(r1;t1,t1|Φ)\n−2iδφ1(r2,t2),\nΠK(x1,x2) =δTrs/bracketleftbig\nGK(r1;t1,t1|Φ)+GZ(r1;t1,t1|Φ)/bracketrightbig\n−2i δφ2(r2,t2),\n(13)\nwhere Tr sstands for the trace in the spin space.\nIII. THE KINETIC EQUATION\nIn previous section, the electron-electron interaction\nhas been decoupled with the help of auxiliary bosonic\nfieldsφ1(2). This means that the influence of the interac-\ntion can be described by a Z2Bose field,\nˆϕ(r,t) =/parenleftbigg\nφ1(r,t)φ2(r,t)\nφ2(r,t)φ1(r,t)/parenrightbigg\n.\nNow we are able to apply the quasiclassicalGreen’s func-\ntion[26, 27, 28]approachtostudythespindynamics. We\nderive the Eilenberger equation from the right-hand and\nleft-hand Dyson equations obeyed by the Green’s func-\ntionˆG(r,t;r′,t′|Φ) in Eq. (10):\n˜∂tˆg+vF·/vector∇ˆg+i/bracketleftbig\nb·/vector σ,ˆg/bracketrightbig\n=ˆg∝an}b∇acketle{tˆg∝an}b∇acket∇i}htn−∝an}b∇acketle{tˆg∝an}b∇acket∇i}htnˆg\n2τ,(14)\nwherevFdenotes the Fermi velocity, τis the elastic scat-\ntering time arising from the adoption of the standard\nself-consistent Born approximation; ∝an}b∇acketle{t···∝an}b∇acket∇i}htnmeans tak-\ning average over the direction of the electron momentum\nn=p/|p| ≡(cosθ,sinθ), and the covariant derivative\nis defined by\n˜∂tˆg=∂t1ˆg+∂t2ˆg+iˆϕ(r,t1)ˆg−iˆgˆϕ(r,t2).(15)\nThe quasiclassical Green’s function in Keldysh and spin\nspaces,\nˆg=/parenleftbigg\ngRgK\ngZgA/parenrightbigg\n, (16)can be derived by integrating the Fourier transform of\nthe Green’s function in Eq. (10) over energy variables,\ni.e.,\nˆg(t1,t2;n,r) =i\nπ/integraldisplay\ndξˆG(t1,t2;p,r),\nˆG(t1,t2;p,r) =/integraldisplay\nd2r′eip·r′ˆG(r1,t1;r2,t2|Φ),(17)\nwhereξ=p2/2m−µ,r′=r1−r2,r= (r1+r2)/2. The\nelectron polarization operators can be obtained in terms\nof Eq. (13) and Eq. (17), i.e.,\nΠR(x1,x2) = ΠA(x2,x1)\n=ν/integraldisplaydθ\n2π/bracketleftbig\nδ(x1−x2)+π δTrsgK(t1,t1;n,r1)\n2δφ1(r2,t2)/bracketrightbig\n,\nΠK(x1,x2)\n=πν/integraldisplaydθ\n2πδTrs/bracketleftbig\ngK(t1,t1;n,r1)+gZ(t1,t1;n,r1)/bracketrightbig\n2δφ2(r2,t2).\n(18)\nSince physical observables are determined by the\nKeldysh component of the quasiclassical Green’s func-\ntion, namely ∝an}b∇acketle{tgK(t1,t2;n,r)∝an}b∇acket∇i}htΦ(here the subscript Φ\nrefers that the functional average [29] is taken over the\nfield Φ), we need to solve this component from the\nEilenberger equation. Decomposing the Green’s func-\ntion in chargeand spin components ∝an}b∇acketle{tgK(t1,t2;n,r)∝an}b∇acket∇i}htΦ=\ngK\n0+gK·/vector σ, one can obtain the charge and spin densities,5\nrespectively,\nρ(r,t) =−1\n4eν/integraldisplay\ndǫ∝an}b∇acketle{tgK\n0(t,ǫ;n,r)∝an}b∇acket∇i}htn,\nS(r,t) =−1\n4ν/integraldisplay\ndǫ∝an}b∇acketle{tgK(t,ǫ;n,r)∝an}b∇acket∇i}htn.(19)\nNow we turn to the kinetic equations for the two inde-\npendent components gKandgZ. For∝an}b∇acketle{tgZ∝an}b∇acket∇i}htΦ= 0 in all\norders of the perturbation theory, we have\ngK=∝an}b∇acketle{tgK∝an}b∇acket∇i}htΦ+δgK, gZ=δgZ,\nwhere the fluctuation parts δgimply the effects con-\ntributed by the auxiliary bosonic fields. One can obtain\nfrom Eq. (14) that δgZobeys the following equation,\n(˜∂t+vF·/vector∇)δgZ+i/bracketleftbig\nb·/vector σ,δgZ/bracketrightbig\n−1\nτ/bracketleftbig\nδgZ−∝an}b∇acketle{tδgZ∝an}b∇acket∇i}htn/bracketrightbig\n=−2iφ2(r,t1)δ(t1−t2)Is, (20)\nwhere I sdenotes the unit matrix in spin space. When\nderiving the above equation, we have used the conditions\ngR=δ(t1−t2)Is−gKδgZ/2 andgA=−δ(t1−t2)Is+\nδgZgK/2. Equation (20) gives rise to\nδgZ(t1,t2;n,r) = 2iδ(t1−t2)/integraldisplay\ndr1dt3/integraldisplaydθ′\n2π\n×φ2(r1,t3)Γρ(t3−t1,n′,n;r1,r),\nΓρ(t,n′,n;r1,r2) =/integraldisplaydωd2q\n(2π)3eiq·(r1−r2)−iωt\n×Γρ(n′,n;ω,q), (21)where the diffusion propagator Γ ρis defined by\n(−iω+ivFn·q)Γρ(n,n′;ω,q)+1\nτ/bracketleftbig\nΓρ(n,n′;ω,q)\n−∝an}b∇acketle{tΓρ(n,n′;ω,q)∝an}b∇acket∇i}htn/bracketrightbig\n= 2πδ(n−n′). (22)\nAfter obtaining the explicit form of δgZ, we can further\nsolve theδgKfrom the following relation\n(˜∂t+vF·/vector∇)δgK+i/bracketleftbig\nb·/vector σ,δgK/bracketrightbig\n+1\nτ/bracketleftbig\nδgK−∝an}b∇acketle{tδgK∝an}b∇acket∇i}htn/bracketrightbig\n= 2iφ2(r,t1)δ(t1−t2)Is−i/bracketleftbig\nφ1(r,t1)−φ1(r,t2)/bracketrightbig\n∝an}b∇acketle{tgK∝an}b∇acket∇i}htΦ\n+1\n4τ/bracketleftig\n∝an}b∇acketle{tgK∝an}b∇acket∇i}htΦ∝an}b∇acketle{tδgZ∝an}b∇acketle{tgK∝an}b∇acket∇i}htΦ∝an}b∇acket∇i}htn−∝an}b∇acketle{t ∝an}b∇acketle{tgK∝an}b∇acket∇i}htΦ∝an}b∇acket∇i}htnδgZ∝an}b∇acketle{tgK∝an}b∇acket∇i}htΦ\n−∝an}b∇acketle{tgK∝an}b∇acket∇i}htΦδgZ∝an}b∇acketle{t ∝an}b∇acketle{tgK∝an}b∇acket∇i}htΦ∝an}b∇acket∇i}htn+∝an}b∇acketle{t ∝an}b∇acketle{tgK∝an}b∇acket∇i}htΦδgZ∝an}b∇acket∇i}htn∝an}b∇acketle{tgK∝an}b∇acket∇i}htΦ/bracketrightig\n.(23)\nWe take only the zeroth and first angular harmonics\ninto account in the Keldysh component assumed spatial\nsmoothness,\n∝an}b∇acketle{tgK(t1,t2;n,r)∝an}b∇acket∇i}htΦ≈ ∝an}b∇acketle{tgK(t1,t2;n′,r)∝an}b∇acket∇i}htΦ,n′\n+2n·∝an}b∇acketle{tn′gK(t1,t2;n′,r)∝an}b∇acket∇i}htΦ,n′. (24)\nDecomposing the fluctuating term in charge and spin\ncomponents δgK=δgK\n0+δgK·/vector σ, one can easily ob-\ntain the explicit expression of the δgK\n0which is given in\nEq. (A1). The fluctuation part δgKrelated to the spin\ncomponents fulfils the following equation,\n(˜∂t+vF·/vector∇)δgK−2b×δgK+1\nτ/bracketleftbig\nδgK−∝an}b∇acketle{tδgK∝an}b∇acket∇i}htn/bracketrightbig\n=−i/bracketleftbig\nφ1(r,t1)−φ1(r,t2)/bracketrightbig/parenleftig\n∝an}b∇acketle{tgK(t1,t2;n,r)∝an}b∇acket∇i}htΦ,n\n+2n·∝an}b∇acketle{tn′gK(t1,t2;n′,r)∝an}b∇acket∇i}htΦ,n′/parenrightig\n+1\n2τ/braceleftig\n∝an}b∇acketle{tgK\n0∝an}b∇acket∇i}htΦ,n/parenleftbig\n∝an}b∇acketle{tδgZ∝an}b∇acket∇i}htn−δgZ/parenrightbig\n∝an}b∇acketle{tgK∝an}b∇acket∇i}htΦ,n+∝an}b∇acketle{tgK∝an}b∇acket∇i}htΦ,n/parenleftbig\n∝an}b∇acketle{tδgZ∝an}b∇acket∇i}htn−δgZ/parenrightbig\n∝an}b∇acketle{tgK\n0∝an}b∇acket∇i}htΦ,n/bracerightig\n.(25)\nIf denoting\nQ=\nδgK\nx\nδgK\ny\nδgK\nz\n, Lk=\ngK\nx\ngK\ny\ngK\nz\n,\nwe can write Eq. (25) in the following matrix equation,\n(˜∂t+vF·/vector∇)Q−2ζQ+1\nτ/parenleftbig\nQ−∝an}b∇acketle{tQ∝an}b∇acket∇i}htn/parenrightbig\n=−i/bracketleftbig\nφ1(r,t1)−φ1(r,t2)/bracketrightbig/parenleftbig\n∝an}b∇acketle{tLk∝an}b∇acket∇i}htΦ,n+2n·∝an}b∇acketle{tn′Lk∝an}b∇acket∇i}htΦ,n′/parenrightbig\n+1\n2τ/braceleftig\n∝an}b∇acketle{tgK\n0∝an}b∇acket∇i}htΦ,n/parenleftbig\n∝an}b∇acketle{tδgZ∝an}b∇acket∇i}htn−δgZ/parenrightbig\n∝an}b∇acketle{tLk∝an}b∇acket∇i}htΦ,n\n+∝an}b∇acketle{tLk∝an}b∇acket∇i}htΦ,n/parenleftbig\n∝an}b∇acketle{tδgZ∝an}b∇acket∇i}htn−δgZ/parenrightbig\n∝an}b∇acketle{tgK\n0∝an}b∇acket∇i}htΦ,n/bracerightig\n, (26)where the matrix ζis given by\nζ=\n0 0 −αpFcosθ\n0 0 −αpFsinθ\nαpFcosθ αpFsinθ0\n.\nThen equation (26) can be solved by utilizing the follow-\ning expression\n(−iω+ivFn·q−2ζ)Γs(n,n′;ω,q)+1\nτ/bracketleftbig\nΓs(n,n′;ω,q)\n−∝an}b∇acketle{tΓs(n,n′;ω,q)∝an}b∇acket∇i}htn/bracketrightbig\n= 2πδ(n−n′). (27)\nWe give the explicit expression of δgKin Eq. (A2) in the\nappendix A.\nSince the concrete forms of δgZandδgKhave been\nobtained, we can write down the kinetic equation satis-6\nfied by the Keldysh function through averaging the K\ncomponent of Eq. (14) over the auxiliary bosonic fields,\n(˜∂t+vF·/vector∇)∝an}b∇acketle{tgK∝an}b∇acket∇i}htΦ+i/bracketleftbig\nb·/vector σ,∝an}b∇acketle{tgK∝an}b∇acket∇i}htΦ/bracketrightbig\n=Cel/braceleftbig\n∝an}b∇acketle{tgK∝an}b∇acket∇i}htΦ/bracerightbig\n+Cin/braceleftbig\n∝an}b∇acketle{tgK∝an}b∇acket∇i}htΦ/bracerightbig\n, (28)\nwhere the inelastic collision integral reads\nCin{∝an}b∇acketle{tgK∝an}b∇acket∇i}htΦ}(t1,t2;n,r)\n=−i∝an}b∇acketle{t/bracketleftbig\nφ1(r,t1)−φ1(r,t2)/bracketrightbig\nδgK∝an}b∇acket∇i}htΦ,(29)and the elastic collision integral is given by\nCel/braceleftbig\n∝an}b∇acketle{tgK∝an}b∇acket∇i}htΦ/bracerightbig\n(t1,t2;n,r) =1\nτ/bracketleftbig\n∝an}b∇acketle{t ∝an}b∇acketle{tgK(t1,t2;n,r)∝an}b∇acket∇i}htΦ∝an}b∇acket∇i}htn−∝an}b∇acketle{tgK(t1,t2;n,r)∝an}b∇acket∇i}htΦ/bracketrightbig\n+/integraldisplay\ndt3dθ1\n2π/bracketleftbig\n∝an}b∇acketle{tgK(t1,t3;n1,r)∝an}b∇acket∇i}htΦΛA(t3,t2;n1,n;r)−∝an}b∇acketle{tgK(t1,t3;n,r)∝an}b∇acket∇i}htΦΛA(t3,t2;n,n1;r)/bracketrightbig\n+/integraldisplay\ndt3dθ1\n2π/bracketleftbig\nΛR(t1,t3;n,n1;r)∝an}b∇acketle{tgK(t3,t2;n1,r)∝an}b∇acket∇i}htΦ−ΛR(t1,t3;n1,n;r)∝an}b∇acketle{tgK(t3,t2;n,r)∝an}b∇acket∇i}htΦ/bracketrightbig\n, (30)\nwhere\nΛA(t1,t2;n,n1;r) =1\n4τ/integraldisplay\ndt3∝an}b∇acketle{t/bracketleftbig\nδgZ(t1,t3;n1,r)−δgZ(t1,t3;n,r)/bracketrightbig\nδgK(t3,t2;n,r)∝an}b∇acket∇i}htΦ,\nΛR(t1,t2;n,n1;r) =1\n4τ/integraldisplay\ndt3∝an}b∇acketle{tδgK(t1,t3;n,r)/bracketleftbig\nδgZ(t3,t2;n1,r)−δgZ(t3,t2;n,r)/bracketrightbig\n∝an}b∇acket∇i}htΦ. (31)\nSubstituting the explicit forms of δgZandδgKgiven in\nEq. (21), (A1) and (A2) into Eq. (28), one get the ki-\nnetic equation which can be used to study the influence\nof electron-electron interaction on the spin dynamics of\n2DEGs with Rashba spin-orbit coupling. After some te-\ndious calculation, we obtain the explicit expressions of\nthe inelastic and elastic collision integrals, respectively,\nwhich are given in Eq. (A3) and (A6) in the appendix A.\nIV. SPIN DYNAMICS\nAfter taking average over the direction of the momen-\ntumn, one can see from Eq. (29-30) that the elastic col-\nlision integral vanishes,\n/integraldisplaydθ\n2πCel/braceleftbig\n∝an}b∇acketle{tgK∝an}b∇acket∇i}htΦ/bracerightbig\n(t,ǫ;n,r) = 0, (32)\nbut the average of the inelastic collision integral over the\ndirection does not vanish. This means that the elastic\ncollision integral preserves the number of electrons on a\ngivenenergyshell defined by Eq. (A5), while the inelastic\ncollision integral does not preserve it. When t1=t2,\nequation (A4) gives rise to\n/integraldisplay\ndǫCin/braceleftbig\n∝an}b∇acketle{tgK∝an}b∇acket∇i}htΦ/bracerightbig\n(t1,ǫ;n,r) =Cin/braceleftbig\n∝an}b∇acketle{tgK∝an}b∇acket∇i}htΦ/bracerightbig\n(t1,t1;n,r).One can see from Eq. (29) that the right-hand side is\nalways zero. Thus we obtain\n/integraldisplay\ndǫCin/braceleftbig\n∝an}b∇acketle{tgK∝an}b∇acket∇i}htΦ/bracerightbig\n(t,ǫ;n,r) = 0. (33)\nThis implies that not only the total number of electrons\nis conserved, but also the number of electrons moving\nalong a concrete direction nis conserved.\nDecomposing the Green’s function in charge and spin\ncomponents in the approximation of Eq. (24), separat-\ning the zeroth and first angular harmonics and utilizing\nEq. (32) and Eq. (33), we obtain from Eq. (28) that\nvFn·/vector∇∝an}b∇acketle{tgK(t,ǫ;n,r)∝an}b∇acket∇i}htΦ,n+i/bracketleftbig\nb·/vector σ,\n∝an}b∇acketle{tgK(t,ǫ;n,r)∝an}b∇acket∇i}htΦ,n/bracketrightbig\n=Cel/braceleftbig\n∝an}b∇acketle{tgK∝an}b∇acket∇i}htΦ/bracerightbig\n(t,ǫ;n,r),(34)\n∂t∝an}b∇acketle{tgK(t,ǫ;n,r)∝an}b∇acket∇i}htΦ,n+i/bracketleftbig\nb·/vector σ,2n·∝an}b∇acketle{tn′gK(t,ǫ;n′,r)∝an}b∇acket∇i}htΦ,n′/bracketrightbig\n+vFn·/vector∇/parenleftbig\n2n·∝an}b∇acketle{tn′gK(t,ǫ;n′,r)∝an}b∇acket∇i}htΦ,n′/parenrightbig\n= 0. (35)\nThere is no contribution of the inelastic collision integral\nto the spin dynamics due to the condition Eq. (33). Solv-\ning Eq. (34) and substituting ∝an}b∇acketle{tn′gK(t,ǫ;n′,r)∝an}b∇acket∇i}htΦ,n′into\nEq. (35), we can obtain the spin and charge diffusion\nequations. The diffusion equation for the charge density\nreads,\n∂tρ−CD∂2\nXρ= 0, (36)7\nwhere∂X= (∂x,∂y) andCD=v2\nFτ/2. We introduce\nthe distribution function fwhich reduces to the Fermi\ndistribution in equilibrium,\nf=f0+/vector σ·fk=1\n2(1−1\n2gK). (37)\nIn the time τ, the charge density becomes isotropic but\nthe spin relaxation process does not start yet, hence [30]\ngK\n0(ǫ) = 2(1−2f0(ǫ)),\ngK(t,ǫ;r) =−4fk(t,ǫ;r), (38)\nwhere\nf0(ǫ) =/parenleftbig\nf+(ǫ)+f−(ǫ)/parenrightbig\n/2,\nfk(t,ǫ;r) =/parenleftbig\nf+(ǫ)−f−(ǫ)/parenrightbig\ns(t,r),\nf±(ǫ) = [exp(ǫ∓∆µ/2\nkBT)+1]−1,(39)\nwheres= (sx,sy,sz) denotes the unit vector along the\nspin,f±(ǫ) represent the distribution functions projected\nalongthe directionparallelorantiparalleltothe unit vec-\ntors(all the energiesare counted from the Fermi energy)\nand ∆µ= (µ+−µ−) refers to the difference between the\nchemical potentials µ±of the electron spin subsystems.\nThediffusionequationsforthespincomponentsaregiven\nby\n∂tSx−CD∂2\nXSx−2CE∂xSz+1\nτ′sSx\n=1\nτexxSx+Fx(Sx,Sy,Sz),\n∂tSy−CD∂2\nXSy−2CE∂ySz+1\nτ′sSy\n=1\nτeyySy+Fy(Sx,Sy,Sz),\n∂tSz−CD∂2\nXSz+2CE∂xSx+2CE∂ySy+2\nτ′sSz\n=1\nτezzSz+Fz(Sx,Sy,Sz), (40)\nwhereCE=αvFpFτ,τ′\ns= 1/[2(αpF)2τ] and\nFℓ(Sx,Sy,Sz) is a quadratic form of ( Sx,Sy,Sz) lack-\ning of theS2\nℓ(ℓ=x,y,z) term. The characteristic times\nτe\nℓℓdescribe the effect of the electron-electron interaction\non the spin relaxation, and their explicit expressions aregiven by\n1\nτexx=2(αpF)2τ\nM/braceleftig/integraldisplay\ndǫ/integraldisplaydω\n2π/bracketleftbig/parenleftbig\nf+(ǫ−ω)−f−(ǫ−ω)/parenrightbig\n×Im(Rxx\n2)gK\n0(ǫ)+/parenleftbig\nf+(ǫ)−f−(ǫ)/parenrightbig\nRxx\n1gK\n0(ǫ−ω)/bracketrightbig/bracerightig\n,\n1\nτeyy=2(αpF)2τ\nM/braceleftig/integraldisplay\ndǫ/integraldisplaydω\n2π/bracketleftbig/parenleftbig\nf+(ǫ−ω)−f−(ǫ−ω)/parenrightbig\n×Im(Ryy\n2)gK\n0(ǫ)+/parenleftbig\nf+(ǫ)−f−(ǫ)/parenrightbig\nRyy\n1gK\n0(ǫ−ω)/bracketrightbig/bracerightig\n,\n1\nτezz=1\nτexx+1\nτeyy\n=2(αpF)2τ\nM/braceleftig/integraldisplay\ndǫ/integraldisplaydω\n2π/bracketleftbig/parenleftbig\nf+(ǫ−ω)−f−(ǫ−ω)/parenrightbig\n×/parenleftbig\nIm(Rxx\n2)+Im(Ryy\n2)/parenrightbig\ngK\n0(ǫ)\n+/parenleftbig\nf+(ǫ)−f−(ǫ)/parenrightbig/parenleftbig\nRxx\n1+Ryy\n1/parenrightbig\ngK\n0(ǫ−ω)/bracketrightbig/bracerightig\n,\n(41)\nwhereM=/integraldisplay\ndǫ/parenleftbig\nf+(ǫ)−f−(ǫ)/parenrightbig\n. In order to obtain the\nconcrete expressions of the characteristic times τe\nℓℓ, we\nfirstly take the energy integration in Eq. (41). Since the\nspin splitting is small, i.e.,\n|µ+−µ−|≪|µ+|,|µ−|,\nthe energy integration can be taken as follows,\n1\nM/integraldisplay∞\n−∞dǫ/parenleftbig\nf+(ǫ−ω)−f−(ǫ−ω)/parenrightbig/parenleftbig\nf+(ǫ)+f−(ǫ)/parenrightbig\n≈2/integraldisplay∞\n−∞dǫ∂f0(ǫ−ω)\n∂ǫf0(ǫ)\n/integraldisplay∞\n−∞dǫ∂f0(ǫ)\n∂ǫ\n= 1−∂\n∂ω(ωcothω\n2kBT). (42)\nAfter the energy integration, the characteristic times τe\nℓℓ\nhave the forms\n1\nτexx=1\nτeyy= 8(αpF)2τ/integraldisplay∞\n0dω\n2π/bracketleftbig∂\n∂ω(ωcothω\n2kBT)/bracketrightbig\n×/bracketleftbig\nIm(Rxx\n2)−Rxx\n1/bracketrightbig\n, (43)\nthe detail of the calculationsof the kernels Rı\n1and ImRı\n2\nare given in appendix B. Now we discuss the influence of\nthe electron-electron interaction on the spin-relaxation\ntime in the ballistic regime Tτ≫1.\nWe can obtain the characteristic time τe\nxxin the ballis-8\ntic regime utilizing the kernels Rı\n1and ImRı\n2in Eq. (B7)\n1\nτexx(Tτ≫1) = 8(αpF)2τ/integraldisplay∞\n0dω\n2π/bracketleftbig∂\n∂ω(ωcothω\n2kBT)/bracketrightbig\n×/bracketleftbig−1\n4πνv2\nF(3π\n2+tan−1ωτ−2ωτ\n1+ω2τ2)/bracketrightbig\n≈−4(αpF)2τ\nνv2\nF/integraldisplay∞\n0dω\n2π/bracketleftbig∂\n∂ω(ωcothω\n2kBT)/bracketrightbig\n=2(αpF)2τ\nπνv2\nF(2kBT−EFcothEF\n2kBT), (44)\nwheretan−1(ωτ)isreplacedby π/2forωτ≫1inthebal-\nlistic regime and EFis in the place of the upper limit of\nthe integral. In the low temperature regime kBT≪EF,\nthe second term approaches a constant independent of\nthe temperature, so the first term manifests the temper-\nature effect in the contribution of the electron-electron\ninteraction to the spin-relaxation time.\nWhen the total spin density Sis spatially homo-\ngeneous and parallel to the ℓth-axis of the coordi-\nnate frame, the contribution of Fℓ(Sx,Sy,Sz) vanishes,\nnamelyFℓ(Sx,Sy,Sz) = 0. The diffusion equations for\nspin components Sℓcan be simplified, for example,\n∂tSx=−1\nτ′sSx+1\nτexxSx=−1\nτsxxSx,(45)\nwhereτs\nxx=τ′\ns/(1−τ′\ns\nτexx). Therefore, the spin-relaxation\ntimes can be determined by τ′\nsandτe\nℓℓ, consequently,\nτs\nxx=τ′\ns\n1−τ′\ns\nτexx, τs\nyy=τ′\ns\n1−τ′\ns\nτeyy,\n(τs\nzz)−1= (τs\nxx)−1+(τs\nyy)−1. (46)\nWe can see that the total spin decays exponentially when\n0< τ′\ns/τe\nℓℓ<1. In terms of the explicit forms of\nthe characteristic times τe\nℓℓin the ballistic regime, the\nspin-relaxation times involving the effect of the electron-\nelectron interaction take the following forms\nτs\nxx=τs\nyy= 2τs\nzz=τ′\ns\n1−(T\nTF−1\n2), Tτ≫1,(47)\nwhereTF=EF/kBis the Fermi temperature. It is\nworthwhile to indicate that there exists an obvious en-\nhancement of the spin-relaxation time with increment of\nthe temperature in the ballistic regime. The increasingamplitude of the spin-relaxation time depends on the ra-\ntio of the temperature to the Fermi temperature. In con-\nclusion, an obvious enhancement of the spin-relaxation\ntime can be induced by the electron-electron interaction\nin the ballistic regime for systems under consideration.\nV. SUMMARY\nIn the above, we presented a theoretical study of the\ninfluence of electron-electron interactions on the spin dy-\nnamics for 2DEGs with Rashba spin-orbit coupling. We\nemployed the path-integral approach and the quasiclas-\nsical Green’s function to deal with the electron-electron\ninteraction. With the help of the auxiliary Bose field,\nthe electron-electron interaction was decoupled via the\nHubbard-Stratonovich transformation. Then one is able\nto derive the Elienberger equation by using the Green’s\nfunction after the transformation. Through tedious cal-\nculation, we further derived the spin and charge diffu-\nsion equations, from which the spin-relaxation time can\nbe given explicitly. We analyzed the influence of the\nelectron-electron interaction on the spin-relaxation time\nin the ballistic regime and found an obvious enhance-\nment of the spin-relaxation time with the increment of\nthe temperature T. The increasing amplitude of the\nspin-relaxation time depends on the ratio of the temper-\nature to the Fermi temperature. The electron-electron\ninteraction changes the wave vector kand hence results\nin the variation of the spin precession vector. This ex-\nhibits that the electron-electron interaction plays an im-\nportant role in the spin relaxation of electrons when\nthe D’yakonov-Perel spin relaxation mechanism domi-\nnates. It is expected to be helpful for understanding\nthe spin dynamics of 2DEGs with spin-orbit couplings\nand electron-electron interactions. Our formulation can\nalso be extend to the case of bulk inversion asymme-\ntry, namely the additional Dresselhaus term [31] with\nb=β(px,−py)+γ(pxp2\ny−pyp2\nx).\nAcknowledgments\nTheworkwassupportedbyNSFCGrantNo. 10674117\nand partially by PCSIRT Grant No. IRT0754.\nAPPENDIX A: EXPLICIT FORMS\nThe explicit form of δgK\n0is9\nδgK\n0(t1,t2;n,r) =−i/integraldisplay\ndtθ/bracketleftbig\nφ1(r1,t1−tθ)−φ1(r1,t2−tθ)/bracketrightbig/integraldisplaydθ′\n2πΓρ(tθ,n,n′;r,r1)\n×/braceleftig\n∝an}b∇acketle{tgK\n0(t1−tθ,t2−tθ;n1,r)∝an}b∇acket∇i}htΦ,n1+2n′·∝an}b∇acketle{tn1gK\n0(t1−tθ,t2−tθ;n1,r)∝an}b∇acket∇i}htΦ,n1/bracerightig\n+/integraldisplaydθ′\n2πdθ′′\n2π/integraldisplay\nd2r1dtθΓρ(tθ,n,n′;r,r1)/braceleftig\n2iφ2(r1,t1−tθ)δ(t1−t2)\n+i\nτ/bracketleftbig\n∝an}b∇acketle{tΓρ(t4−t3,n′′,n1;r2,r1)∝an}b∇acket∇i}htn1−Γρ(t4−t3,n′′,n;r2,r1)/bracketrightbig\nφ2(r2,t4)\n×/bracketleftbig\n∝an}b∇acketle{tgK\n0(t1−tθ,t3;n1,r)∝an}b∇acket∇i}htΦ,n1∝an}b∇acketle{tgK\n0(t3,t2−tθ;n1,r)∝an}b∇acket∇i}htΦ,n1\n+∝an}b∇acketle{tgK(t1−tθ,t3;n1,r)∝an}b∇acket∇i}htΦ,n1·∝an}b∇acketle{tgK(t3,t2−tθ;n1,r)∝an}b∇acket∇i}htΦ,n1/bracketrightbig/bracerightig\n. (A1)\nThe explicit expression of δgKis\nQ(t1,t2;n,r) =−i/integraldisplay\ndtθ/bracketleftbig\nφ1(r1,t1−tθ)−φ1(r1,t2−tθ)/bracketrightbig/integraldisplaydθ′\n2πΓs(tθ,n,n′;r,r1)\n×/braceleftig\n∝an}b∇acketle{tLk(t1−tθ,t2−tθ;n1,r)∝an}b∇acket∇i}htΦ,n1+2n′·∝an}b∇acketle{tn1Lk(t1−tθ,t2−tθ;n1,r)∝an}b∇acket∇i}htΦ,n1/bracerightig\n+2i\nτ/integraldisplaydθ′\n2πdθ′′\n2π/integraldisplay\nd2r1dtθΓs(tθ,n,n′;r,r1)∝an}b∇acketle{tgK\n0(t1−tθ,t3;n1,r)∝an}b∇acket∇i}htΦ,n1\n×/braceleftig/bracketleftbig\n∝an}b∇acketle{tΓρ(t4−t3,n′′,n1;r2,r1)∝an}b∇acket∇i}htn1−Γρ(t4−t3,n′′,n1;r2,r1)/bracketrightbig\n×φ2(r4,t2)∝an}b∇acketle{tLk(t3,t2−tθ;n1,r)∝an}b∇acket∇i}htΦ,n1/bracerightig\n. (A2)\nThe inelastic collision integral reads\nCin/braceleftbig\n∝an}b∇acketle{tgK∝an}b∇acket∇i}htΦ/bracerightbig\n(t,ǫ;n,r)\n=i\n2τ/integraldisplay\nd2r1d2r2/integraldisplaydω\n2π/bracketleftbig\nDR(ω;r,r2)−DA(ω;r2,r)/bracketrightbig\n×/bracketleftbig\n∝an}b∇acketle{tΓρ(ω;r2,r1)∝an}b∇acket∇i}ht∝an}b∇acketle{tΓρ(−ω;r,r1)∝an}b∇acket∇i}ht−∝an}b∇acketle{tΓρ(−ω;r,r1)Γρ(ω;r2,r1)∝an}b∇acket∇i}ht/bracketrightbig\n×/bracketleftbig\n∝an}b∇acketle{tgK\n0(t,ǫ−ω;n,r)∝an}b∇acket∇i}htΦ,n∝an}b∇acketle{tgK\n0(t,ǫ;n,r)∝an}b∇acket∇i}htΦ,n+∝an}b∇acketle{tgK(t,ǫ−ω;n,r)∝an}b∇acket∇i}htΦ,n·∝an}b∇acketle{tgK(t,ǫ;n,r)∝an}b∇acket∇i}htΦ,n/bracketrightbig\n−i\n2/integraldisplay\nd2r1/integraldisplaydω\n2πDK(ω;r,r1)σm/bracketleftbig\n∝an}b∇acketle{tΓs(−ω;r,r1)∝an}b∇acket∇i}ht+∝an}b∇acketle{tΓs(ω;r1,r)∝an}b∇acket∇i}ht/bracketrightbig/bracketleftbig\n∝an}b∇acketle{tLk(t,ǫ;n,r)∝an}b∇acket∇i}htΦ,n−∝an}b∇acketle{tLk(t,ǫ−ω;n,r)∝an}b∇acket∇i}htΦ,n/bracketrightbig\n+i\nτ/integraldisplay\nd2r1d2r2/integraldisplaydω\n2π/bracketleftbig\nDR(ω;r,r2)−DA(ω;r2,r)/bracketrightbig\nσm\n×/bracketleftbig\n∝an}b∇acketle{tΓs(−ω;r2,r1)∝an}b∇acket∇i}ht∝an}b∇acketle{tΓρ(−ω;r,r1)∝an}b∇acket∇i}ht−∝an}b∇acketle{tΓs(−ω;r,r1)Γρ(ω;r2,r1)∝an}b∇acket∇i}ht/bracketrightbig\n∝an}b∇acketle{tgK\n0(t,ǫ−ω;n,r)∝an}b∇acket∇i}htΦ,n∝an}b∇acketle{tLk(t,ǫ;n,r)∝an}b∇acket∇i}htΦ,n,(A3)\nwhere\nCin/braceleftbig\n∝an}b∇acketle{tgK∝an}b∇acket∇i}htΦ/bracerightbig\n(t1,t2;n,r)\n=/integraldisplaydǫ\n2πCin/braceleftbig\n∝an}b∇acketle{tgK∝an}b∇acket∇i}htΦ/bracerightbig/parenleftbigt1+t2\n2,ǫ;n,r/parenrightbig\neiǫ(t2−t1),(A4)\nand∝an}b∇acketle{tΓρ(s)∝an}b∇acket∇i}htmeans angular averaging defined in\nEq. (A10), the matrix σm= (σx,σy,σz) and the tem-\nporal transformation of the Green’s function has beenused due to a much faster dependence on the difference\nt1−t2than on the t1+t2\ngK(t1,t2;n,r) =/integraldisplaydǫ\n2πgK/parenleftbigt1+t2\n2,ǫ;n,r/parenrightbig\neiǫ(t2−t1),(A5)\nthe propagators of auxiliary fields have the same trans-\nformation. The elastic collision integral can be written\nas10\nCel/braceleftbig\n∝an}b∇acketle{tgK∝an}b∇acket∇i}htΦ/bracerightbig\n(t,ǫ;n,r) =−2\nτnı∝an}b∇acketle{tnıgK(t,ǫ;n,r)∝an}b∇acket∇i}htΦ,n−2\nτ/integraldisplaydω\n2πnıRı\n1(ω)∝an}b∇acketle{tgK\n0(t,ǫ−ω;n,r)∝an}b∇acket∇i}htΦ,n\n×∝an}b∇acketle{tngK(t,ǫ;n,r)∝an}b∇acket∇i}htΦ,n+i\nτ/integraldisplaydω\n2πnıRı\n2(ω)∝an}b∇acketle{tngK(t,ǫ−ω;n,r)∝an}b∇acket∇i}htΦ,n∝an}b∇acketle{tgK(t,ǫ;n,r)∝an}b∇acket∇i}htΦ,n\n−i\nτ/integraldisplaydω\n2πnı/parenleftbig\nRı\n2(ω)/parenrightbig⋆∝an}b∇acketle{tgK(t,ǫ;n,r)∝an}b∇acket∇i}htΦ,n∝an}b∇acketle{tngK(t,ǫ−ω;n,r)∝an}b∇acket∇i}htΦ,n\n+i\nτ/integraldisplaydω\n2πnı/bracketleftig\nσmRı\n3(ω)∝an}b∇acketle{tLk(t,ǫ−ω;n,r)∝an}b∇acket∇i}htΦ,n∝an}b∇acketle{tngK(t,ǫ;n,r)∝an}b∇acket∇i}htΦ,n\n−∝an}b∇acketle{tngK(t,ǫ;n,r)∝an}b∇acket∇i}htΦ,nσm/parenleftbig\nRı\n3(ω)/parenrightbig⋆∝an}b∇acketle{tLk(t,ǫ−ω;n,r)∝an}b∇acket∇i}htΦ,n/bracketrightig\n. (A6)\nWherenı()referstothe ı()componentoftheunitvector\nn, withı,=x,yand the kernels Rı\n1(ω)−Rı\n3(ω) in\nEq. (A6) are defined by\nRı\n1(ω) = Im/integraldisplayd2q\n(2π)2DR(ω;q)/braceleftbig\n∝an}b∇acketle{tΓρ(n,ω;q)n∝an}b∇acket∇i}ht\n×∝an}b∇acketle{tnıΓρ∝an}b∇acket∇i}ht−1\n2δı,/parenleftbig\n∝an}b∇acketle{tΓρ∝an}b∇acket∇i}ht∝an}b∇acketle{tΓρ∝an}b∇acket∇i}ht−∝an}b∇acketle{tΓρΓρ∝an}b∇acket∇i}ht/parenrightbig/bracerightbig\n,(A7)\nRı\n2(ω) =/integraldisplayd2q\n(2π)2DR(ω;q)/braceleftbig\n∝an}b∇acketle{tΓρ∝an}b∇acket∇i}ht∝an}b∇acketle{tnıΓρn∝an}b∇acket∇i}ht\n−∝an}b∇acketle{tΓρnıΓρn∝an}b∇acket∇i}ht−∝an}b∇acketle{tΓρnı∝an}b∇acket∇i}ht∝an}b∇acketle{tΓρn∝an}b∇acket∇i}ht/bracerightbig\n, (A8)\nRı\n3(ω) =/integraldisplayd2q\n(2π)2DR(ω;q)/braceleftbig\n∝an}b∇acketle{tΓρn∝an}b∇acket∇i}ht∝an}b∇acketle{tnıΓs∝an}b∇acket∇i}ht\n−1\n2δı,/parenleftbig\n∝an}b∇acketle{tΓρ∝an}b∇acket∇i}ht∝an}b∇acketle{tΓs∝an}b∇acket∇i}ht−∝an}b∇acketle{tΓρΓs∝an}b∇acket∇i}ht/parenrightbig/bracerightbig\n, (A9)\nwhere we have introduced the notation\n∝an}b∇acketle{tfΓρ(s)h∝an}b∇acket∇i}ht=/integraldisplaydθdθ′\n(2π)2f(n)Γρ(s)(n,n′;ω,q)h(n′).(A10)\nAPPENDIX B: CALCULATION OF THE\nKERNELS Rı\ni\nAccording to the definition of the diffusion propagator\nΓρin Eq. (22), we can obtain\nΓρ(n,n′;ω,q) = 2πδ(n−n′)Γρ0(n;ω,q)\n+ Γρ0(n;ω,q)Γρ0(n′;ω,q)1\nτ−1\nΥ,\n(B1)\nwhere\nΓρ0(n;ω,q) =1\n−iω+ivFn·q+1\nτ\n=1\n−iω+ivFqcos(φ−φq)+1\nτ,\nΥ =/radicalbigg\n(−iω+1\nτ)2+v2\nFq2, (B2)withφqbeing the angle between the wave vector qand\nthex-axis. In terms of the explicit form of the diffusion\npropagator, one can obtain\n∝an}b∇acketle{tΓρ∝an}b∇acket∇i}ht=τ\nΥτ−1,\n∝an}b∇acketle{tΓρnx∝an}b∇acket∇i}ht=∝an}b∇acketle{tnxΓρ∝an}b∇acket∇i}ht=τ\nivFq(Υτ−1)(Υ+iω−1\nτ)cosφq,\n∝an}b∇acketle{tΓρny∝an}b∇acket∇i}ht=∝an}b∇acketle{tnyΓρ∝an}b∇acket∇i}ht=τ\nivFq(Υτ−1)(Υ+iω−1\nτ)sinφq,\n∝an}b∇acketle{tΓρΓρ∝an}b∇acket∇i}ht=−iω+1\nτ\nΥ(Υ−1\nτ)2,\n∝an}b∇acketle{tΓρnxΓρ∝an}b∇acket∇i}ht=1\nΥsin2φq\n−Υ\nv2\nFq2(Υτ−1)(1−−iω+1\nτ\nΥ)2cos2φq,\n∝an}b∇acketle{tΓρnyΓρ∝an}b∇acket∇i}ht=1\nΥcos2φq\n−Υ\nv2\nFq2(Υτ−1)(1−−iω+1\nτ\nΥ)2sin2φq,\n∝an}b∇acketle{tΓρnxΓρnx∝an}b∇acket∇i}ht=τ\nΥτ−1/parenleftbig−iω+1\nτ\nΥ2sin2φq\n−Υ−(iω+1\nτ)\nΥ2(Υτ−1)cos2φq/parenrightbig\n,\n∝an}b∇acketle{tΓρnyΓρny∝an}b∇acket∇i}ht=τ\nΥτ−1/parenleftbig−iω+1\nτ\nΥ2cos2φq\n−Υ−(iω+1\nτ)\nΥ2(Υτ−1)sin2φq/parenrightbig\n.\n(B3)\nUtilizing above formulas, we find that the kernels Rı\n1(ω)\nandRı\n2(ω) are diagonal, Rı\ni=δıRi, which can be writ-11\nten as\nR1(ω) =−Im/integraldisplay∞\n0qdq\n4πDR(ω;q)/braceleftig1\nv2\nFq2/parenleftbigΥ+iω−1\nτ\nΥ−1\nτ/parenrightbig2\n+Υ+iω−1\nτ\nΥ(Υ−1\nτ)2/bracerightig\n,\nImR2(ω) = Im/integraldisplay∞\n0qdq\n4πDR(ω;q)/braceleftig1\nv2\nFq2[Υ+iω−1\nτ]2\nΥ(Υ−1\nτ)\n+Υ+iω−1\nτ\nΥ(Υ−1\nτ)2/bracerightig\n.(B4)\nIt is not difficult to calculate the concrete forms of the\nelectron polarization operators from Eq. (18), for exam-\nple,\nΠR(ω,q) =ν[1+iω\nΥ−1\nτ]. (B5)\nSubstituting the polarization operator into Eq. (12), weobtain the propagator of the Bose fields, i.e.,\nDR(ω,q) =DR\n0\n1−DR\n0ΠR=−2πe2/q\n1+2πe2\nqν[1+iω\nΥ−1\nτ]\n≈ −1\nΠR=−1\nνΥ−1\nτ\nΥ−1\nτ−iω, (B6)\nwhere the approximation in the second line corresponds\ntotheunitarylimitassociatingwithlargerdistancesthan\nthe screening radius.\nWe obtain the concrete expressions of the kernels R1\nandR2in the ballistic regime Tτ≫1,\nR1(Tτ≫1)∝1\n8πνv2\nF(3π\n2+tan−1ωτ−2ωτ\n1+ω2τ2),\nImR2(Tτ≫1)∝ −1\n8πνv2\nF(3π\n2+tan−1ωτ−2ωτ\n1+ω2τ2).\n(B7)\n[1] S. A. Wolf, D. D. Awswchalom, R. A. Buhrman, J.\nM. Daughton, S. Von. Molnar, M. L. Roukes, A. Y.\nChtchelkanova, D. M. Tresger, Science 294, 1488 (2001),\nand references therein.\n[2] I. Zutic, J. Fabian and S. Das Sarma, Rev. Mod. Phys.\n76, 323 (2004), and references therein.\n[3] S. Datta and B. Das, Appl. Phys. Lett. 56, 665 (1990).\n[4] J. Schliemann, J. C. Egues and Daniel Loss, Phys. Rev.\nLett.90, 146801 (2003).\n[5] S. Murakami, N. Nagaosa and S. C. Zhang, Science 301,\n1348 (2003).\n[6] G. E. Pikus and A. N. Tikov, in Optical Orientation,\nModern Problems in Condensed Matter Science, Vol.8\nedited by F. Meier and B. P. Zakharchenya (North-\nHolland, Amsterdam), p109.\n[7] N. S. Averkiev and L. E. Golub, Phys. Rev. B 60, 15582\n(1999).\n[8] N. S. Averkiev, L. E. Golub and M. Willander, J. Phys.\nCondens. Matter 14, R271 (2002).\n[9] M. E. Flatt´ e, J. M. Byers and W. H. Lau, Spin Dynam-\nics in Semiconductors in Semiconductor spintronics and\nQuantum Computation, edited by D. D. Awschalom et\nal. (Springer, Berlin, 2002), chapter 4.\n[10] Y. Li and Y. Q. Li, J. Phys. Condens. Matter 19, 346231\n(2007).\n[11] M. I. D’yakonov and V. I. Perel, Zh. Eksp. Teor. Fiz, 38,\n362 (1973) [Sov. Phys. JETP. 38, 177 (1973)].\n[12] S. V. Kravchenko and M. P. Sarachik, Rep. Prog. Phys.\n67, 1 (2004).\n[13] O. Prus, Y. Yaish, M. Reznikov, U. Sivan and V. Pu-\ndalov, Phys. Rev. B 67, 205407 (2003).\n[14] A. A. Shashkin, S. Anissimova, M. R. Sakr, S. V.\nKravchenko,V.T. Dolgopolov andT. M.Klapwijk, Phys.\nRev. Lett. 96, 036403 (2006).\n[15] A. Punnoose and A. M. Finkel’stein, Science 310, 289(2005).\n[16] M. M. Glazov and E. L. Ivchenko, JETP Lett. 75, 403\n(2002).\n[17] D. Stich, J. Zhou, T. Korn, R. Schulz, D. Schuh, W.\nWegscheider, M.W.WuandC. Sch¨ uller, Phys.Rev.Lett.\n98, 176401 (2007).\n[18] W. J. H. Leyland, G. H. John, R. T. Harley, M. M.\nGlazov, E. L. Ivchenko, D. A. Ritchie, I. Farrer, A. J.\nShields and M. Henini, Phys. Rev. B 75, 165309 (2007).\n[19] M. W. Wu and C. Z. Ning, Eur. Phys. J. B 18, 373\n(2002).\n[20] M. Q. Weng and M. W. Wu, Phys. Rev. B 68, 075312\n(2003).\n[21] M. Q. Weng, M. W. Wu and L. Jiang, Phys. Rev. B 69,\n245320 (2004).\n[22] M. M. Glazov and E. L. Ivchenko, JETP 99, 1279 (2004).\n[23] A. Punnoose and A. M. Finkel’stein, Phys. Rev. Lett.\n96, 057202 (2006).\n[24] N. Nagaosa, Quantum Field Theory in Condensed Mat-\nter Physics, translated by S. Heusler (Springer, Berlin,\n1999), p95.\n[25] G. Zala, B. N. Narozhny and I. L. Aleiner, Phys. Rev. B\n64, 214204 (2001).\n[26] G. Eilenberger, Z. Phys. B 214,195 (1968).\n[27] J. Rammer and H. Smith, Rev. Mod. Phys. 58, 323\n(1986).\n[28] P. Schwab and R. Raimondi, Ann. Phys. 12, 471 (2003).\n[29] Note that the average over a variable means to ingtegrat e\nthe variable out simply, while the average over a func-\ntionalφalwayimplies anadditionalmeassureofFeymann\npath-integral, e−iS[φ], in the functional intergration.\n[30] M. I. Dyakonovand V. I. Perel, Fiz. Tverd. Tela 13, 3581\n(1971) [Sov. Phys. Solid State 13, 3023 (1972)].\n[31] G. Dresselhaus, Phys. Rev. 100, 580 (1955)."    },    {        "title": "1007.0853v1.Radial_Spin_Helix_in_Two_Dimensional_Electron_Systems_with_Rashba_Spin_Orbit_Coupling.pdf",        "content": "arXiv:1007.0853v1  [cond-mat.mes-hall]  6 Jul 2010Radial Spin Helix in Two-Dimensional\nElectron Systems with Rashba Spin-Orbit Coupling\nYuriy V. Pershin∗\nDepartment of Physics and Astronomy and USC Nanocenter,\nUniversity of South Carolina, Columbia, SC 29208, USA\nValeriy A. Slipko\nDepartment of Physics and Technology, V. N. Karazin Kharkov National University, Kharkov, Ukraine\nWe suggest a long-lived spin polarization structure, a radi al spin helix, and study its relaxation\ndynamics. For this purpose, starting with a simple and physi cally clear consideration of spin trans-\nport, we derive a system of equations for spin polarization d ensity and find its general solution in\nthe axially symmetric case. It is demonstrated that the radi al spin helix of a certain period relaxes\nslower than homogeneous spin polarization and plain spin he lix. Importantly, the spin polarization\nat the center of the radial spin helix stays almost unchanged at short times. At longer times, when\nthe initial non-exponential relaxation region ends, the re laxation of the radial spin helix occurs with\nthe same time constant as that describing the relaxation of t he plain spin helix.\nI. INTRODUCTION\nAt the present time, there is a significant inter-\nest in the field of electron spin relaxation in semicon-\nductors stimulated by possible future applications of\nspins in electronics and computing1,2. In many two-\ndimensional (2D) electron systems the leading mecha-\nnism of spin relaxation is the D’yakonov-Perel’ spin re-\nlaxation mechanism3,4. Within this mechanism, electron\nspins feel an effective momentum-dependent magnetic\nfield randomized by electron scattering events resulting\nin relaxation of electron spin polarization. A number\nof theoretical and experimental studies on peculiarities\nof D’yakonov-Perel’ spin relaxation were reported in the\nlast decade5–16.\nIt wasshownin Ref. 9that the spin relaxationtime for\n2D electrons depends not only on material parameters\n(e.g., strength of spin-orbit interaction, electron mean\nfree path, etc.) but also on the initial spin polarization\nprofile. In particular, it was demonstrated that a plain\nspin helix in a 2D electron system with Rashba spin-\norbit interaction has a longer spin relaxation time than a\nhomogeneousspinpolarization9. Alaterstudy11revealed\nthat in a system with both Rashba17and Dresselhaus18\ninteractions such an increase in spin relaxation time can\nbe even more dramatic. This effect was also observed\nexperimentally13,19.\nIn this paper, we consider spin relaxation of a radial\nspin helix in a 2D electron system with Rashba spin-orbit\n(SO) interaction. This structure is interesting because it\nprovides, to the best of our knowledge, the longest spin\nrelaxation time of 2D electrons subjected to Rashba SO\ninteraction. Such a property is related to spin polariza-\ntion in the vicinity ofthe special point of radialspin helix\nr= 0. Physically, at short times, when the period of ra-\ndial spin helix is equal to the period of spin precession,\nan electron diffusing from any direction to a point in the\nvicinity of r= 0 has the same direction of spin polariza-\ntion as the initial spin polarization at this point. Thus,D’yakonov-Perel’ spin relaxation becomes inefficient at\nshort times for spin polarization in the vicinity of r= 0.\nIn the radial spin helix, the initial distribution of spin\npolarization has a cylindrical (axial) symmetry and is\ngiven by\nSr(r,t= 0) =−S0sin(kr), (1)\nSϕ(r,t= 0) = 0 , (2)\nSz(r,t= 0) =S0cos(kr), (3)\nwherekisthe wavevector, and S0istheinitial amplitude\nof spin polarization. In Fig. 1(a) we show schematically\nspin polarizationdistribution in the radialdirection. The\noverall distribution of z-component of the spin polariza-\nz (a) z\nr(a)\nFIG. 1. (Color online) (a) Schematic of initial spin polariz a-\ntion distribution in a radial spin helix. (b) Initial distri bution\nofz-component of spin polarization ( Sz(r,0)/S0) in the radial\nspinhelix offinite radius usedin ourMonte Carlo simulation s.2\ntion in radial spin helix of finite radius is shown in Fig.\n1(b). Moreover, in this paper we will often refer to a\nplain spin helix suggested in Ref. 9. The initial distri-\nbution of spin polarization components in the plain spin\nhelix is\nSx(x,t= 0) =−S0sin(kx), (4)\nSy(x,t= 0) = 0 , (5)\nSz(x,t= 0) =S0cos(kx). (6)\nThis paper is organized as follows. In Sec. II, a drift-\ndiffusion equation approach is used to study relaxation\ndynamics of the radial spin helix. We obtain general ex-\npressions for spin polarization as a function of time and\nstudy its short and long time behavior. Our analytical\nstudies are supported by Monte Carlo simulations pre-\nsented in Sec. III. Our main results and conclusions\nare summarized in Sec. IV. Moreover, in several Ap-\npendices following the main text, we provide additional\ncalculation details. Specifically, electron spin rotations\ninduced by Rashba SO interaction are considered in the\nAppendix A, spin drift-diffusion equations are derived\nin the Appendix B, a general analytical solution of spin\ndrift-diffusion equation in the axially symmetric case is\npresentedin the Appendix C, andrelaxationofplainspin\nhelix is discussed in the Appendix D.\nII. DRIFT-DIFFUSION DESCRIPTION OF\nRADIAL SPIN HELIX\nIn this section we consider the relaxation dynamics of\nthe radial spin helix analytically. The initial spin polar-\nization in the radial spin helix is of cylindrical symmetry\nand described by Eqs. (1-3). Intuitively, a special point\nin the radial spin helix is r= 0, because the electrons\nmotion through this point along straight trajectories in\nalldirections should not lead to spin relaxation at short\ntimes for a specific value of the wave vector k. Corre-\nspondingly, the spin lifetime of electrons located in a re-\ngion within r= 0 should be longer than the spin lifetime\nof homogeneous spin polarization and of plain spin helix.\nThis effect is in the focus of our investigation.\nLet us consider a two-dimensional electrons confined\nin a quantum well or heterostructure with Rashba-type\nspin-orbit interaction17. The standard Hamiltonian with\nthe Rashba term is given by\nˆH=ˆ p2\n2m+α(ˆσ׈ p)·z, (7)\nwhereˆ p= (ˆpx,ˆpy) is the 2D electron momentum oper-\nator,mis the effective electron’s mass, ˆσis the Pauli-\nmatrix vector, αis the spin-orbit coupling constant and\nzis a unit vectorperpendicular to the confinement plane.\nIt is not difficult to show that in the case of Hamil-\ntonian (7) the quantum mechanical evolution of a spin\nof an electron with a momentum pcan be reduced to\na spin rotation with the angular velocity Ω = 2 αp//planckover2pi1about the axis determined by the unit vector n=p×z/p\n(see Appendix A). In this way, the spin-orbit coupling\nconstant αenters into equations through the parameter\nη= 2αm/planckover2pi1−1, which gives the spin precession angle per\nunit length.\nBesides this evolution, 2D electrons experience dif-\nferent bulk scattering events such as, for example, due\nto phonons or impurities. These scatterings random-\nize the electron trajectories. Correspondingly, the di-\nrection of spin rotation becomes fluctuating what causes\naverage spin relaxation (dephasing). This is the famous\nD’yakonov-Perel’spinrelaxationmechanism.3,4Thetime\nscale of the bulk scattering events can then be character-\nized by a single rate parameter, the momentum relax-\nation time τ. It is connected to the mean free path by\nℓ=vτ, where v=p/mis the mean electron velocity.\nTo take into account these scatterings we use a model\nof diffusive spin transport, which in the limit of small\nkℓ≪1, yields the spin drift-diffusion equations (B3-B5)\n(Appendix B provides derivation details).\nLet us consider dynamics of a radial spin helix relax-\nation. We assume that such a structure is created at the\ninitial moment of time with the spin polarization com-\nponents given by Eqs. (1-3). The exact solution of the\nradial spin drift-diffusion equations (C1-C2) with initial\nconditions (1-3) and constants γandCfrom Eqs. (B6)\ncan be written as (see Appendix C for more details)\nSr(r,t)\nS0=−d\ndk/integraldisplayk\n0dsJ1(sr)√\nk2−s2/bracketleftbigg\nkcosh/parenleftbigg/radicalbig\nη2+16s2ηDt\n2/parenrightbigg\n+/parenleftbig\nkη+4s2/parenrightbigsinh/parenleftBig/radicalbig\nη2+16s2ηDt\n2/parenrightBig\n/radicalbig\nη2+16s2\ne−(s2+3η2/2)Dt,(8)\nSz(r,t)\nS0=d\ndk/integraldisplayk\n0dssJ0(sr)√\nk2−s2/bracketleftbigg\ncosh/parenleftbigg/radicalbig\nη2+16s2ηDt\n2/parenrightbigg\n+(4k��η)sinh/parenleftBig/radicalbig\nη2+16s2ηDt\n2/parenrightBig\n/radicalbig\nη2+16s2\ne−(s2+3η2/2)Dt,(9)\nwhereJ1(r) andJ0(r) are the Bessel functions ofthe first\nand zeroth order correspondingly and Dis the diffusion\nconstant.\nEqs. (8-9) define completely the radial spin helix at\nany point rand at any moment of time t. The time\ndependence of spin polarization at the center of helix is\nof particular interest because the spin relaxation at this\npoint is the slowest. The radial component of spin po-\nlarization Srin the vicinity of r= 0 is close to zero (it\nfollows from symmetry considerations or directly from\nEq. (8)). Therefore, below, we derive asymptotic ex-\npressions at short and long times for Szonly. At short\ntimesDη2t≪1, an expansion of the RHS of Eq. (9) in3\ntand its integration over satr= 0 results in\nSz(0,t)\nS0= 1−2(k−η)2Dt+2\n3(k−η)(2k3−6k2η\n+6kη2−3η3)D2t2−/parenleftbigg8\n15k6−16\n5k5η+8k4η2−104\n9k3η3\n+32\n3k2η4−14\n3kη5+4\n3η6/parenrightbigg\nD3t3+O(t4).(10)\nIt follows from Eq. (10) that when the radial spin helix\nperiod is equal to spin precession length (this happens\nwhenk=η) the decay of Szat the center of helix starts\nwith a cubic term in t\nSz(0,t)\nS0= 1−10\n9(Dη2t)3+O(t4),fork=η,Dη2t≪1.\n(11)\nThis means that the spin relaxation at the center of the\nradial spin helix at short times is significantly suppressed\n(for this special wave number, k=η) and characterized\nby a rather long initial interval of non-exponential be-\nhaviour.\nThe asymptotic behaviour of SrandSzat long times,\nDη2t≫1, can be determined by taking into account the\ndominant contribution to the integrals in Eqs. (8) and\n(9). This contribution comes from the vicinity of a point\ns∈[0,k]correspondingtothemaximumofthe −λ−(s)in\nthe integration interval (see Eq. (C10) and Fig. 7). We\nshould consider three cases. In the first case, when 0 <\nk < km=√\n15η/4, the main contribution to the integrals\nin Eqs. (8) and (9) comesfrom the vicinity ofpoint s=k\nat the right end of the integration interval. In the second\ncase, when k=km, we should keep in mind that k=\nkmis a stationary point of λ−(s) entering the exponent.\nTherefore, in this case, the asymptotic behaviour differs\nfrom the case k < kmby a pre-exponential factor. In the\nthird case, when k > k m, the main contribution to the\nintegrals in Eqs. (8) and (9) arises from the vicinity of\nthe inner stationary point s=kmofλ−(s).\nThe asymptotic behaviour of Laplace-type integrals is\nobtained using standard for this purpose technics. From\nEq. (9) we get\nSz(r,t)\nS0=/parenleftBigg\n1+4k−η/radicalbig\nη2+16k2/parenrightBigg\nJ0(kr)/radicalbigg\n−πk\n8dλ−(k)\ndkt\n×e−λ−(k)t,for 0< k <√\n15\n4η,−kdλ−(k)\ndkt≫1,(12)\nSz(r,t)\nS0=√\n15(√\n15+3)\n32√\n2Γ/parenleftbigg3\n4/parenrightbigg\nJ0(kmr)(Dη2t)1\n4\n×e−λ−(km)t,fork=km=√\n15\n4η, Dη2t≫1,(13)\nSz(r,t)\nS0=−3\n32(4k+5η)η\n(k2−k2m)3/2J0(kmr)/radicalbiggπ\nDte−λ−(km)t,\nfork >√\n15\n4η,(k−km)2Dt≫1, Dη2t≫1.(14)0 1 2 3 4 5 6 7 8 9 0.0 0.2 0.4 0.6 0.8 1.0 \n  k =0.5 η\n  k =km\n  k =1.6 η\n Sz/S 0\nDη2t12\n3\nFIG. 2. (Color online) Time-dependence of Sz(0,t) in the\nradial spin helix at several values of k.\nFrom Eq. (12) we see that if the wave vector ksatis-\nfies the inequality 0 < k < k m, then, at long times, the\nspin polarization decay is mainly exponential and char-\nacterized by a relaxation time τ(k) = (λ−(k))−1. This\nrelaxation time increases with k. At the same time (ac-\ncordingly to Eq. (10)), the spin polarization decay at\nshort times also decreases with k. Therefore, the spin\nlife time of the radial spin helix increases with increase\nofk∈(0,km).\nAtk=km, as it follows from Eqs. (13-14),\nthe relaxation time reaches its maximum value τm=\n(λ−(km))−1= (7Dη2/16)−1. Moreover, since km≈\n0.97ηis very close to η, the conditions for short time\nsuppression of spin relaxation are almost optimal at this\nvalue ofk. Therefore, when k=km, the spin relaxation\nis significantly suppressed at both short and long times.\nWhenk > km, the asymptotic relaxation time is the\nsame as when k=km(see Eqs. (13,14)). However, even\nwhenkis close to km, the ratio of the absolute value of\nRHSofEq. (14)tothoseofEq. (13)issmall(oftheorder\nof [(k−km)2Dt]−3/4). In addition, in the asymptotic\nformula (14), the pre-exponential factor changes its sign.\nTherefore, the spin polarization Szmust turn to zero at\nsome moment of time, before it reaches the asymptotic\nbehaviour given by Eq. (14). Eq. (10) at k > η≈km\nalso predicts a relaxation increase with kwhenk > km.\nThus we conclude that the longest spin relaxationtime\nfor the spin polarization at the center of the radial spin\nhelix occurrs at the wave vector\nk=km=√\n15η/4 (15)\nand is given by\nτm= (7Dη2/16)−1. (16)\nIn addition, at this value of k, the dynamics of spin po-\nlarization in the vicinity of r= 0 is non-exponential at\nshort times, when the spin polarization remains almost\nconstant. These are the main results of our calculations.4\n(a)0 2 4 6 8 10 12 -0.01 0.00 0.01 0.02 \n  k= 0.6 η\n   k=k m\n  k= 1.4 η\n  Sz/S 0\nηrDη2t= 10 \n(b)0 2 4 6 8 10 12 -0.01 0.00 0.01   k= 0.6 η\n   k=k m\n  k= 1.4 η\n Sr/S 0\nηrDη2t=10 \nFIG. 3. Spatial dependence of Sz(a) andSr(b) in the radial spin helix at the indicated moment of time.\nThe relaxation of initially homogeneous spin polariza-\ntion (when k= 0) can be obtained from Eq. (9) in the\nlimitk→0. In this limiting case the factor before the\nexponential function e−λ−(s)tin Eq. (9) turns to zero in\nk= 0 limit. Therefore, at k= 0, the spin relaxation is\ndetermined by the exponential function e−λ+(0)t, where,\naccordingly to Eq. (C10), λ+(0) = 2Dη2.\nIn fact, the exact time dependence of Szatk= 0\ncoincideswith its asymptoticbehaviour. It can be clearly\nseen from both Eqs. (9) and (B5) that\nSz(t) =S0e−λ+(0)t=S0e−2Dη2t,fork= 0.(17)\nAccordingly to Eq. (17), the initially homogeneous spin\npolarization, directed along z-axis, decays exponentially\nwith a time constant τ(h)= (2Dη2)−1. We also note that\nthe applicability limits of the long times asymptotic ex-\npressions listed in Eqs. (12-14) do not allow calculations\nofSzatk= 0 ork=kmas a limiting case of Eq. (12).\nIn Fig. 2 we show the time dependence of the spin\npolarization component Sz(r= 0,t) calculated at several\nwave vectors ( k= 0.5η(solid line 1), k=km=√\n15η/4\n(solid line 2) and k= 1.6η(solid line 3)) using Eq. (9).\nThe dashed lines 1, 2 and 3 represent asymptotic behav-\nior at short times (given by Eq. (10)), and dotted lines\n1, 2, 3 show asymptotic behavior at long times (given by\nEqs. (12-14)) for the same values of the wave vectors k.\nThese curves reveal main features discussed above.\nThe spatial dependences of z- andr-components of\nspin polarization, calculated from Eqs. (9) and (8), are\npresented in Fig. 3 for a particular moment of time\nDη2t= 10 and wave vectors k= 0.6η(dashed lines),\nk=km(solid lines), and k= 1.4η(dotted lines). These\nplots clearly demonstrate that the maxima of SzandSr\nare reached at the wave vector k=km.\nFig. 4 depicts time dependence of Sz(r= 0,t) for ho-\nmogeneous spin polarization (calculated from Eq. (17)),\nplain spin helix (calculated from Eq. (D4)) and radial0 2 4 6 8 10 0.0 0.2 0.4 0.6 0.8 1.0   homogeneous polarization \n  plane spin helix, k=k m\n  radial spin helix, k=k m\n Sz/S 0\nDη2tr = x =   0, \nkm=0.97 η\nFIG. 4. (Color online) Time dependence of Sz(r= 0,t) for\nthree different initial spin configurations: homogeneous po -\nlarization, plain spin helix and radial spin helix. The radi al\nspin helix is characterized by the largest magnitude of spin\npolarization at any moment of time.\nspin helix. This plot demonstrates that the spin polar-\nization at r= 0 in the radial spin helix lives longer that\nthose in the case of homogeneous spin polarization and\nplain spin helix. It is interesting and important that at\nshort times this curve stays almost flat as expected. At\nlonger times, when the initial non-exponential relaxation\nregion ends, the relaxation of spin polarization in the\nradial spin helix occurs with the same time constant as\nthose of the plain spin helix (it can be shown analytically\nfrom Eqs. (D3-D4) that the longest spin relaxation time\nof the plain spin helix is given by Eq. (16) and occurs\natk=kmgiven by Eq. (15)). Therefore, in both cases,\nthe increase of the exponential relaxation time relative\nto the homogeneous spin polarization is equal to 32 /7.5\n0 50 100 150 200 -1.0 -0.5 0.0 0.5 1.0 \nSz\nSrSpin polarization (arb. units) S\nr (in units of m.f.p.) (b) \nFIG. 5. (Color online) Distributions of zcomponent (a)\nand all components (b) of spin polarization ( Sϕ= 0,S=/parenleftbig\nS2\nr+S2\nz/parenrightbig1/2) att= 100τin the radial spin helix. The red\n(dark) area in the center of radial spin helix in (a) and the\nmaximum of SandSzin the vicinity of r= 0 in (b) demon-\nstrate a longer spin lifetime of electrons located in this re gion.\nThis plot was obtained at ηℓ= 0.1 and the radial spin helix\nperioda= 64.77ℓ(this value of acorresponds to k=km).\nM.f.p. (mean free path) stands for ℓ.\nIII. MONTE CARLO SIMULATIONS\nIn order to obtain an additional insight on spin relax-\nation of the radial spin helix, we perform Monte Carlo\nsimulations employing an approach described in Refs. 5\nand 20. This Monte Carlo simulation method uses a\nsemiclassical description of electron space motion and\nquantum-mechanical description of spin dynamics (the\nlaterisbasedontheHamiltonian (7)). All specific details\nof the Monte Carlo simulations program can be found in\nthe references cited above and will not be repeated here.\nTo some extent, Monte Carlo simulations program nu-\nmerically solves Eqs. (B3-B5) taking into account rela-\ntions (A4) and (A5).\nAll numerical results related to the radial spin helix\nwere obtained using an ensemble of 108electrons initially0 200 400 600 800 1000 0.01 0.1 1\n  Sz(0,t) /Sz(0,0) \nTime ( τ) Radial spin helix \n Plain spin helix \n Homogeneous polarization \nFIG. 6. (Color online) Spin polarization as a function of tim e\nfor different initial spin polarization configurations. Thi s plot\nwas obtained using the parameters values ηℓ= 0.1 anda=\n64.77ℓ(the period of helices). The straight lines are fitting\ncurves selected as exp( −(t−t0)/t1). The parameters of the\nfitting curves are t0= 0,t1= 103τfor homogeneous spin\npolarization, t0= 0τ,t1= 435τfor plain spin helix, and\nt0= 165τ,t1= 430τfor radial spin helix.\nhomogeneouslydistributedwithin acircleofasufficiently\nlarge radius R= 200ℓto insure that the influence of\nboundary effects on spin polarization in a region in the\nvicinity of r= 0 is negligible. The initial configuration\nofz-component of spin polarization of these electrons is\npresented in Fig. 1(b). Our Monte Carlo simulations\nresults are in an excellent agreement with the theory of\nradial spin helix relaxation presented above.\nFig. 5(a) demonstrates the spatial dependence of Szin\ntheradialspin helixat t= 100τandFig. 5(b)depicts the\nradialdependence of the spin polarizationcomponents at\nthe same moment of time. In particular, it can be clearly\nseen that the decay of Szin the vicinity of r= 0 is\nslower than in other regions. The total spin polarization\nSshows oscillations that are related to a well-known fea-\nture of D’yakonov-Perel’ relaxation: the spin relaxation\ntime of the perpendicular to plane spin polarization is\nshorter than the spin relaxation time of the in-plane spin\npolarization. Similar oscillation of spin polarization am-\nplitude were previously found in the relaxation dynamics\nof the plain spin helix.9\nIn addition to the radial spin helix relaxation, we sim-\nulated the relaxation of homogeneous spin polarization\nand relaxation of plain spin helix. Monte Carlo simu-\nlations reveal that at long times the time dependence of\nspin polarization in all spin configurations (homogeneous\nspin polarization, plain spin helix and radial spin helix)\nexhibits an exponential decay.\nIn Fig. 6 we compare the time dependencies of spin\npolarization ( S(0,t)/S0) for three different initial polar-\nization configurations: the homogeneous spin polariza-\ntion (initial spin polarization is selected as Sx=Sy= 0,\nSz=S0), plain spin helix and radial spin helix. The6\nselected period of radial and plane spin helices ( k=km)\ncorresponds to the longest spin lifetime of these struc-\ntures at a fixed ηℓ= 0.1. It follows from Fig. 6 that\nthe spin polarization in the radial spin helix is the most\nrobust against the relaxation. We emphasize that the\nlong-time behavior of all curves can be perfectly fitted\nby an exponential law given in the caption of Fig. 6.\nThe numerically obtained increase in the spin lifetime of\nthe radial spin helix 435 /103∼4.2 is very close to the\ntheoretically predicted value 32 /7∼4.6. We also note\nthat a slightly longer spin relaxation time increase ( ∼6)\nreported in Ref. 9 for the plain spin helix can be related\nto a large value of ηℓ= 0.3 that is beyond the linear spin\ndrift-diffusion theory.\nIV. CONCLUSION\nThe dynamics of spin relaxation of radial spin helix\nwas investigated using spin drift-diffusion equations and\nMonte Carlo simulations. Starting with a clear model\nof diffusive spin transport, we derived spin drift-diffusion\nequations for electron spin polarization in 2D semicon-\nductor structures with Rashba spin-orbit coupling. The\ngeneral solution of these equations for the axial symmet-\nric case was found. Based on this solution, we studied\nthe evolution of the suggested long-lived spin structure -\nthe radial spin helix. It was shown that the relaxation of\nspin polarization in the vicinity of r= 0 in this structure\ndemonstrates an unusual long initial non-exponential re-\nlaxation behavior followed by an exponential decay. The\noptimal value of the radial spin helix wave vector was\nfound andcorrespondingexponentialrelaxationtime was\ncalculated. Qualitatively, the initial non-exponential de-\ncayfeaturecanbeexplainedbyexistenceofaninfiniteset\nof dephasing-free trajectories propagating through the\npointr= 0.\nIn ordertoadditionallycheckouranalyticalresults, we\nalso performed Monte Carlo simulations of the dynamics\nofradialspinhelixrelaxationusingthesameMonteCarlo\nsimulation technique as those described in Refs. 5 and\n20. Fig. 5 shows a representative result of our simulation\nin which it is clearly demonstrated that the polarization\ndecay at r= 0 is slowest. Our analytical and Monte\nCarlo simulations results are in a perfect agreement.\nTo conclude, the radial spin helix is a new structure\nexhibiting an unusual spin relaxation dynamics and rel-\natively long lifetime. Its property of slow relaxation dy-\nnamicsatshorttimesisveryinteresting. Experimentally,\nthe radialspin helix canbe createdbyspin injection from\na point electrode located at the center of a second ring-\nshape electrode or possibly by a modified spin gratings\ntechnique21.Appendix A: Rotation of a single spin subjected to\nRashba SO interaction\nLet us consider an electron with a momentum p=\n(px,py). The Hamiltonian (7) can be rewritten as\nˆH=p2\n2m+αpn·ˆσ, (A1)\nwheren=p×z/pis the unit vector.\nSince eigenvalues spin projection operator\nˆ s·n=ˆσ·n/2 on any direction are equal to ±1/2,\nthe energy levels of (A1) are given by\nE±=p2\n2m±αp, (A2)\nand it is clear that corresponding spin eigenfunctions are\nthe states with spin directed along and opposite to n.\nThe evolution operator acting only on spin variables of\nthe electron with momentum pis equal to\nˆU(t) =e−iˆHt//planckover2pi1= exp/braceleftbigg−ip2t\n2m/planckover2pi1/bracerightbigg\nexp/braceleftbigg\n−iΩtn·ˆσ\n2/bracerightbigg\n,\n(A3)\nwhere Ω = 2 αp//planckover2pi1.\nWe note that the evolution operator (A3) coincides\nwith the operator of finite rotation by an angle Ω t=\n2αpt//planckover2pi1aboutn-axis up to the phase factor e−ip2t/(2m/planckover2pi1).\nSince electron spin stransforms under rotations as the\nordinary vector, we obtain\ns(t) = cos(Ω t)s(0)+sin(Ω t)n×s(0)+2sin2(Ωt/2)n·s(0)n,\n(A4)\nwhere\nΩ = 2αp//planckover2pi1,n=p×z/p. (A5)\nAppendix B: Derivation of spin drift-diffusion\nequations\nLet usconsidera two-dimensionalnon-degenerateelec-\ntron gas and use a semiclassical approach to model\nthe electron space motion and quantum-mechanical ap-\nproachbased on the Hamiltonian (7) to describe the elec-\ntron spin dynamics. Moreover, we assume the electri-\ncal neutrality and absence of an external electromagnetic\nfield. Within our approach, 2D electrons are character-\nized by the momentum relaxation time τand the mean\nfree path ℓ, so that the average velocity of electrons is\nv=ℓ/τ. From elementary gas-kinetic considerations22\nwe can write an equation for the change of electron spin\npolarization ∆ S(x,y,t) in a region of dimensions 2 ℓ×2ℓ\nwith the center at ( x,y) during the time interval τ:\n(2ℓ)2∆S(x,y,t) =1\n4vτ(2ℓ){S′(x−2ℓ,y,t)\n+S′(x+2ℓ,y,t)+S′(x,y−2ℓ,t)+S′(x,y+2ℓ,t)(B1)\n−4S(x,y,t)}.7\nIn the right hand side of Eq. (B1), the first four terms\nare the spin polarization fluxes into the region from four\nsides with length 2 ℓ, and the last term is the flux out\nof this region. The prime symbols in Eq. (B1) denote\na change of spin polarization because of SO interaction-\ninduced spin precession by the angle 2Ω τ= 4αmℓ//planckover2pi1=\n2ηℓaccordingly to Eqs. (A4,A5). For example,\nS′(x−2ℓ,y,t) = cos(2 ηℓ)S(x−2ℓ,y,t)−sin(2ηℓ)y\n×S(x−2ℓ,y,t)+2sin2(ηℓ)y·S(x−2ℓ,y,t)y,(B2)\nwhereyis the unit vector along y−axis.\nIn order to obtain drift-diffusion equations for spin\npolarization, we substitute expressions for S′into Eq.\n(B1), andexpandtrigonometricalfunctionsuptoquadric\nterms with respect to small 2 ηℓandS′terms up to\nquadric terms with respect to 2 ℓ. The resulting system\nof drift-diffusion equations for spin polarization have a\nform\n∂Sx\n∂t=D∆Sx+C∂Sz\n∂x−2γSx,(B3)\n∂Sy\n∂t=D∆Sy+C∂Sz\n∂y−2γSy,(B4)\n∂Sz\n∂t=D∆Sz−C/parenleftbigg∂Sx\n∂x+∂Sy\n∂y/parenrightbigg\n−4γSz,(B5)\nwhere\nC= 2ηD, γ=1\n2η2D, (B6)\nand\nD=ℓ2\n2τ. (B7)\nHereDis the coefficient of diffusion, Cdescribes spin\nrotations, and γis the coefficient describing spin relax-\nation.\nIt is interesting to note that the same drift-diffusion\nequations (B3-B6) can be obtained for the model of 2D\nlocalized electrons on a lattice23in the hopping regime.\nHowever, in this case, the diffusion coefficient is equal to\nD=ℓ2/(4τ), where τis the characteristic hopping time\nandℓis the distance between lattice sites.\nAppendix C: Analytical solution of drift-diffusion\nequations in the axially symmetric case\nIn the axially symmetric case (assuming that Sr=\nSr(r,t),Sz=Sz(r,t) andSϕ= 0) Eqs. (B3-B5) can be\nwritten as\n∂Sr\n∂t=D/bracketleftbigg∂\nr∂r/parenleftbigg\nr∂Sr\n∂r/parenrightbigg\n−Sr\nr2/bracketrightbigg\n+C∂Sz\n∂r−2γSr,(C1)\n∂Sz\n∂t=D∂\nr∂r/parenleftbigg\nr∂Sz\n∂r/parenrightbigg\n−C∂(rSr)\nr∂r−4γSz.(C2)Let us find a general solution of Eqs. (C1-C2) for the\ncase of an infinite plane. We search a specific solution of\nthe above Eqs. in the form\nSr(r,t) =A(s,t)J1(sr), (C3)\nSz(r,t) =B(s,t)J0(sr), (C4)\nwhereJ1(r) andJ0(r) are the Bessel functions of the\nfirst and zeroth order correspondingly. Substituting ex-\npressions (C3-C4) into Eqs. (C1-C2) we obtain a system\nof ordinary differential equations for unknown functions\nA(s,t) andB(s,t) of positive parameter sand time t\ndA(s,t)\ndt=−(Ds2+2γ)A(s,t)−CsB(s,t),(C5)\ndB(s,t)\ndt=−CsA(s,t)−(Ds2+4γ)B(s,t).(C6)\nThe general solution of this system can be presented as\nA(s,t) =C+(s)sCe−λ+(s)t+C−(s)sCe−λ−(s)t,(C7)\nB(s,t) =C+(s)/parenleftBig\nγ+/radicalbig\nγ2+C2s2/parenrightBig\ne−λ+(s)t(C8)\n+C−(s)/parenleftBig\nγ−/radicalbig\nγ2+C2s2/parenrightBig\ne−λ−(s)t,\nwhere we denote\nλ±(s) =Ds2+3γ±/radicalbig\nγ2+C2s2,(C9)\nandC±(s) are arbitrary functions of positive parameter\ns. Using Eqs. (B6), Eq. (C9) can be rewritten in a more\nsimple form\nλ±(s) =1\n2D(2s2+3η2±η/radicalbig\nη2+16s2).(C10)\nThe special solutions (C3-C4) (with A(s,t) andB(s,t)\ngiven by Eqs. (C7-C9)) of the radial drift-diffusion equa-\ntions (C1-C2) have the following simple meaning. Ac-\ncordingly to Eqs. (C3-C4), the spatial dependencies of\nthe radial and z-components of spin polarization are pro-\nportional to the first and zeroth order Bessel function\nat any moment of time. The parameter sis similar\nto the wave vector kfor the plane case. The ampli-\ntudesA(s,t) andB(s,t) determine the time dependence\nof the radial and z-components of spin polarization. If\nC+(s)/ne}ationslash= 0,C−(s) = 0 ( C+(s) = 0,C−(s)/ne}ationslash= 0), then\nthese amplitudes are exponential functions of time with\nthe inverse relaxation time λ+(s) (λ−(s)) as it can be\nseen from Eqs. (C7-C8).\nWhereas the inverse relaxation time λ+(s) takes its\nminimum value at s= 0 and monotonically increases\nwith parameter s, the inverse relaxation time λ−(s) has\na minimum at\nsm=/radicalbigg\nC2\n4D2−γ2\nC2. (C11)\nAt this value of s,λ−(s) is equal to\nλm= 3γ−C2\n4D−γ2D\nC2. (C12)8\nUsing relations (B6), we find that the minimum value\nofλ−(s) is equal to\nλm=7\n16Dη2, (C13)\nand it occurs at\nsm=√\n15\n4η. (C14)\nFig. 7 shows λ±(s) given by Eq. (C10) as a function of\ns.\nIn order to obtain the general solution of Eqs. (C1-\nC2), we should integrate the special solutions (C3-C4)\nover positive parameter staking into account relations\n(C7-C9). Two arbitrary functions C±(s) entering Eqs.\n(C7-C9) can be found from specified initial conditions\nfor the radial and z-components of polarization in the\nform of its Fourier-Bessel transforms\n˜Sr(s) =/integraldisplay+∞\n0drrSr(r,0)J1(sr),(C15)\n˜Sz(s) =/integraldisplay+∞\n0drrSz(r,0)J0(sr).(C16)\nAs a result of algebraic transformations, we obtain the\nsolution of the initial value problem (C1-C2) in the case\nof the infinite plane\nSr(r,t) =/integraldisplay+∞\n0dssGrr(r,t,s)˜Sr(s)\n+/integraldisplay+∞\n0dssGrz(r,t,s)˜Sz(s),(C17)\nSz(r,t) =/integraldisplay+∞\n0dssGzr(r,t,s)˜Sr(s)\n+/integraldisplay+∞\n0dssGzz(r,t,s)˜Sz(s),(C18)\n0.0 0.5 1.0 1.5 2.0 2.5 3.002468\nλ(s)/Dη2λ+(s)\nλ−(s)\ns/η\nFIG. 7. (Color online) s-dependence of inverse relaxation\ntimesλ±(s). The minimum of λ−(s) corresponds to the wave\nvector giving the longest relaxation time.wheretheGreenfunctions Grr,Gzr,Grz,Gzzaredefined\nas follows\nGrr(r,t,s) =J1(rs)/bracketleftBig\ncosh/parenleftBig\nt/radicalbig\nγ2+C2s2/parenrightBig\n+γsinh/parenleftBig\nt/radicalbig\nγ2+C2s2/parenrightBig\n/radicalbig\nγ2+C2s2\ne−(Ds2+3γ)t,(C19)\nGrz(r,t,s) =−CJ1(rs)sinh/parenleftBig\nt/radicalbig\nγ2+C2s2/parenrightBig\n/radicalbig\nγ2+C2s2\n×se−(Ds2+3γ)t,(C20)\nGzr(r,t,s) =−CJ0(rs)sinh/parenleftBig\nt/radicalbig\nγ2+C2s2/parenrightBig\n/radicalbig\nγ2+C2s2\n×se−(Ds2+3γ)t,(C21)\nGzz(r,t,s) =J0(rs)/bracketleftBig\ncosh/parenleftBig\nt/radicalbig\nγ2+C2s2/parenrightBig\n−γsinh/parenleftBig\nt/radicalbig\nγ2+C2s2/parenrightBig\n/radicalbig\nγ2+C2s2\ne−(Ds2+3γ)t.(C22)\nSubstitutingtheinitialconditionsfortheradialspinhelix\n(1-3) into Eqs. (C15,C16) and performing integration\nwe find that the generalized functions ˜Sr(s) and˜Sz(s),\nwhich correspond to the initial conditions (1) and (3),\nact as follows\n/integraldisplay+∞\n0dssF(s)˜Sr(s) =−S0d\ndk/integraldisplayk\n0dskF(s)√\nk2−s2,(C23)\n/integraldisplay+∞\n0dssF(s)˜Sz(s) =S0d\ndk/integraldisplayk\n0dssF(s)√\nk2−s2,(C24)\nwhereF(s) is a smooth enough function.\nSubstituting Eqs. (C23,C24) into the general solution\n(C17,C18) and taking into account the expressions for\nthe Green functions (C19-C22), we obtain the explicit\nformulae for the solution of the drift-diffusion equations\n(C1-C2) for the initial conditions (1-3):\nSr(r,t) =−S0d\ndk/integraldisplayk\n0dsJ1(sr)√\nk2−s2/bracketleftBig\nkcosh/parenleftBig\nt/radicalbig\nγ2+C2s2/parenrightBig\n+/parenleftbig\nkγ+Cs2/parenrightbigsinh/parenleftBig\nt/radicalbig\nγ2+C2s2/parenrightBig\n/radicalbig\nγ2+C2s2\ne−(Ds2+3γ)t,(C25)\nSz(r,t) =S0d\ndk/integraldisplayk\n0dssJ0(sr)√\nk2−s2/bracketleftBig\ncosh/parenleftBig\nt/radicalbig\nγ2+C2s2/parenrightBig\n+(kC−γ)sinh/parenleftBig\nt/radicalbig\nγ2+C2s2/parenrightBig\n/radicalbig\nγ2+C2s2\ne−(Ds2+3γ)t.(C26)\nAppendix D: Relaxation of plain spin helix\nIn the case of the plane spin helix defined by the ini-\ntial conditions (4-6), the solution of the drift-diffusion9\nequations (B3-B5) has the same harmonical spatial de-\npendence at any moment of time. Therefore, we can seek\nthe solution of this system of equations in the form\nSx(x,t) =A(k,t)sin(kx),Sy(x,t) = 0,(D1)\nSz(x,t) =B(k,t)cos(kx).(D2)\nSubstituting expressions (D1-D2) into the system (B3-\nB5) we obtain the same system of ordinary differential\nequations (C5-C6) for the functions A(k,t) andB(k,t)\nas in the axially symmetric case. Eqs. (D1-D2) to-\ngether with Eqs. (C7-C9) determine solutions of the\ndrift-diffusionequations(B3-B5)withsuchaspecific har-\nmonic spatial dependence. The arbitrary amplitudes C+andC−are calculated from the initial conditions (Eqs.\n(4-6)). Finally, the solution of drift-diffusion equations\n(B3-B5) for the plane spin helix is given by\nSx(x,t) =−S0sin(kx)/bracketleftBig\ncosh/parenleftBig\nt/radicalbig\nγ2+C2k2/parenrightBig\n+ (Ck+γ)sinh/parenleftBig\nt/radicalbig\nγ2+C2k2/parenrightBig\n/radicalbig\nγ2+C2k2\ne−(Dk2+3γ)t,(D3)\nSz(x,t) =S0cos(kx)/bracketleftBig\ncosh/parenleftBig\nt/radicalbig\nγ2+C2k2/parenrightBig\n+(Ck−γ)sinh/parenleftBig\nt/radicalbig\nγ2+C2k2/parenrightBig\n/radicalbig\nγ2+C2k2\ne−(Dk2+3γ)t.(D4)\n∗pershin@physics.sc.edu\n1D.D.Awschalom, N.Samarth, andD.Loss, eds., Semicon-\nductor Spintronics and Quantum Computation (Springer-\nVerlag, 2002).\n2I. Zutic, J. Fabian, and S. Das Sarma, Rev. Mod. Phys.\n76, 323 (2004).\n3M. I. Dyakonov and V. I. Perel’, Sov. Phys. Solid State 13,\n3023 (1972).\n4M. I. Dyakonov and V. Y. Kachorovskii, Sov. Phys. Semi-\ncond.20, 110 (1986).\n5A. A. Kiselev and K. W. Kim, Phys. Rev. B 61, 13115\n(2000).\n6E. Y. Sherman, Appl. Phys. lett 82, 209 (2003).\n7M. Q. Weng, M. W. Wu, and Q. W. Shi, Phys. Rev. B 69,\n125310 (2004).\n8Y. V. Pershin and V. Privman, Phys. Rev. B 69, 073310\n(2004).\n9Y. V. Pershin, Phys. Rev. B 71, 155317 (2005).\n10L. Jiang, M. Weng, M. Wu, and J. Cheng, J. Appl. Phys.\n98, 113702 (2005).\n11B. A. Bernevig, J. Orenstein, and S.-C. Zhang, Phys. Rev.\nLett.97, 236601 (2006).\n12P. Schwab, M. Dzierzawa, C. Gorini, and R. Raimondi,Phys. Rev. B 74, 155316 (2006).\n13J. D. Koralek, C. P. Weber, J. Orenstein, B. A. Bernevig,\nS.-C. Zhang, S. Mack, and D. D. Awschalom, Nature 458,\n610 (2009).\n14P. Kleinert and V. V. Bryksin, Phys. Rev. B 79, 045317\n(2009).\n15M. Duckheim, D. L. Maslov, and D. Loss, Phys. Rev. B\n80, 235327 (2009).\n16I. V. Tokatly and E. Y. Sherman, Ann. Phys. 325, 1104\n(2010).\n17Y. Bychkov and E. Rashba, JETP Lett. 39, 78 (1984).\n18G. Dresselhaus, Phys. Rev. 100, 580 (1955).\n19C. P. Weber, J. Orenstein, B. A. Bernevig, S.-C. Zhang,\nJ. Stephens, and D. D. Awschalom, Phys. Rev. Lett. 98,\n076604 (2007).\n20S. Saikin, Y. Pershin, and V. Privman, IEE-Proc. Circ.\nDev. Syst. 152, 366 (2005).\n21A. R. Cameron, P. Riblet, and A. Miller, Phys. Rev. Lett.\n76, 4793 (1996).\n22F. Reif, Fundamentals of Statistical and Thermal Physics\n(McGraw-Hill, 1965).\n23Y. Pershin, Phys. E 23, 226 (2004)."    },    {        "title": "1811.07681v2.Correlation_effects_in_spin_models_in_the_presence_of_a_spin_bath.pdf",        "content": "Correlation effects in spin models in the presence of a spin bath\nÁlvaro Gómez-León1, Tim Cox2and Philip Stamp2\n1Instituto de Ciencia de Materiales, CSIC, Cantoblanco, Madrid E-28049, Spain and\n2Department of Physics and Astronomy, University of British Columbia,\n6224 Agricultural Road, Vancouver, British Columbia, V6T 1Z1, Canada\n(Dated: May 24, 2022)\nWe analyze the effect of a bath of spins interacting with a spin system in terms of the equation of\nmotion technique. We show that this formalism can be used with general spin systems and baths,\nand discuss the concrete case of a Quantum Ising model longitudinally coupled to the bath. We\nshow how the uncorrelated solutions change when spin-spin correlations are included, the properties\nof the quasiparticle excitations and the effect of internal dynamics in the spin bath.\nI. INTRODUCTION\nSpins systems are among the most studied models in\nphysics, but their properties are also closely related with\nsystems in many other fields of science1–3. These mod-\nels are interesting because they display many emergent\nphenomena due to collective effects, and in addition, be-\ncause of their relevance in the development of quantum\ncomputers4. Most works on spin models mainly consider\nclosed systems and study the equilibrium phase diagram,\nthe criticalexponents, and morerecently, alsothe quench\ndynamics5,6. Less it is known about these models in the\npresence of environments, specially for those described\nby a collection of localized modes. These are known to\nproduce radically different effects than the usual oscil-\nlator bath in certain regimes7–9. As the coupling to lo-\ncalized spin environments is generally not weak30, the\ncalculations are generally non-perturbative and must be\ndone carefully, which implies that in the case of quan-\ntum models, the effect of correlations can become quite\nimportant.\nFor practical purposes, the study of spin models has\nrecently become more important due to the develop-\nment of the first quantum simulators and quantum\ncomputers10–12. In this case, environments are ubiqui-\ntous and its interaction with the quantum computer dur-\ning the computation time needs to be properly under-\nstood – e.g., the fate of the quantum critical point (QCP)\nduring the quantum annealing process, in the presence of\nan environment. In the case of quantum computers made\nof flux qubits, spin environments can be produced by\nparamagnetic impurities and nuclear isotopes in the sub-\nstrate, and even if they weakly couple, their effect in the\ndynamics can be non-trivial13. Also it has been shown\nthat for superconducting qubits dielectric loss can domi-\nnate, and it is produced by effective two level systems14.\nTo understand the physical effect of spin bath environ-\nments we study a spin model coupled to a set of localized\nspins. We use the equation of motion (EOM) technique\nfor the Green’s functions, and a decoupling scheme based\non a hierarchical expansion in terms of correlations. This\napproach transforms the EOM into a Dyson’s equation\nfor the Green’s function, with explicit non-perturbative\nexpressions for the self-energy, which include a full fre-quency and momentum dependence. Hence it can also\nbe used to study quantum dynamics of excited states,\nas already shown in15. The lowest order solution agrees\nwith mean-field (MF) treatments and can explain some\ngeneral features of the effect of a baths of spins, as the\nsolutions are solely characterized by the connectivity be-\ntween Ising spins and bath spins (i.e., the three differ-\nent coordination numbers that one can define for this\nmodel). Higher order corrections depend on the spe-\ncific details of the bath, however, we numerically solve\nthe self-consistent equations for some specific cases and\ndemonstrate that quantum correlations tend to suppress\nthe effect of the bath of spins, with respect to the MF\ncalculations.\nII. MODEL AND METHOD\nWe consider a model with a central system and a\nbath. The central system is made of a set of interact-\ning spins coupled to an external field ~B. Similarly, the\nbath is made of a set of spins as well, but we assume\nthat these are non-interacting, although they can be af-\nfected by an external field ~\u000131. The total Hamiltonian is\nH=HS+HB, whereHSandHBare given by (Greek\nindices specify the direction of the spin vector and latin\nindices label the different sites):\nHS=\u0000X\n\u0016;iB\u0016S\u0016\ni\u0000X\n\u0016X\ni;j>iV\u0016\ni;jS\u0016\niS\u0016\nj(1)\nHB=\u0000X\n\u0016;l\u0001\u0016I\u0016\nl+X\n\u0016X\ni;lA\u0016\ni;lS\u0016\niI\u0016\nl(2)\nHScorresponds to a general spin model coupled to an\nexternal field ~Band with arbitrary interaction V\u0016\ni;j, while\nHBcorresponds to the bath Hamiltonian which couples\nto a field~\u0001and interacts with the central system via\nA\u0016\ni;l. To study the properties of the system we define the\nnext double-time Green’s functions16:\nG\u000b;\f\nn;m(t;t0) =\u0000ihS\u000b\nn(t) ;O\f\nm(t0)i (3)\nY\u000b;\f\nn;m(t;t0) =\u0000ihI\u000b\nn(t) ;O\f\nm(t0)i (4)arXiv:1811.07681v2  [cond-mat.str-el]  20 Sep 20192\nwhere ;indicates that the Green’s functions may be a\ntime ordered, retarded or advanced. Also O\f\nmis some\ngeneral spin operator at site m(i.e., it can represent a\nsystem or a bath spin). The equation of motion for the\nGreen’s function is:\ni@tG\u000b;\f\nn;m =\u000e(t\u0000t0)h\u0002\nS\u000b\nn;O\f\nm\u0003\n\u0006i+i\u000f\u0016\u000b\u001aB\u0016G\u001a;\f\nn;m(5)\n+i\u000f\u0016\u000b\u001a0\n@X\ni6=nV\u0016\ni;nG\u0016\u001a;\f\nin;m\u0000X\nlA\u0016\nn;lY\u0016\u001a;\f\nln;m1\nA\nwhere [:::]\u0006refers to anti-commutator or commuta-\ntor, respectively17,18(i.e., to a fermionic or bosonic\nGreen’s function). We have defined the three-spin cor-\nrelatorsG\u0016\u001a;\f\nin;m(t;t0) =\u0000ihS\u0016\niS\u001a\nn;O\f\nmiandY\u0017\u001a;\f\nln;m(t;t0) =\n\u0000ihI\u0017\nlS\u001a\nn;O\f\nmi.\nAn important feature of many-body systems is the hi-\nerarchical structure of the EOM, where interaction terms\nproducehigher-orderGreen’sfunctions. Todealwiththis\nhierarchy, one needs to devise a method to decouple the\ninfinite set of equations into smaller blocks, which cor-\nrectly captures the regime of interest. For that purpose\nwe separate the Green’s functions into their uncorrelated\nand correlated parts: G\u0016\u001a;\f\nin;m =hS\u0016\niiG\u001a;\f\nn;m+hS\u001a\nniG\u0016;\f\ni;m+\nG\u0016\u001a;\f\nin;m(we will use calligraphic letters to denote the cor-\nrelated parts). This separation is completely general and\njust transforms the initial EOM for the two-point func-\ntion in two coupled equations, one for the two-point func-\ntions and one for the correlated parts. Physically, the\ntermshS\u0016\niiG\u001a;\f\nn;m+hS\u001a\nniG\u0016;\f\ni;mare similar to the Hartree\nand Fock contributions. They capture the average effect\nof all the other spins on the two-point function, while\nthe correlated part G\u0016\u001a;\f\nin;mcontains the corrections due tocorrelations with additional spins.\nOur approximation to close the system of equations\nwill assume that correlated parts scale in some partic-\nular way with the parameters of the system, such that\nthey can be neglected when they are small enough19,20.\nWe will organize terms in inverse powers of the coordina-\ntion number Z, whereZgenerally denotes any coordina-\ntion number required to describe the model (for the Ising\nmodel just one coordination number is required, while for\nthe examples below we show that three different coordi-\nnation numbers are needed. In this case the smallest\ncoordination number will dominate the scaling). Doing\nthis, it is possible to systematically include higher or-\nder corrections by simply adding higher order correlated\nparts.\nIt must be mentioned that the scaling of correlations\nconverges more slowly in 1D (it goes as \u0018Z\u00001, with\nZ= 2). However, it is still an interesting case, because in\nlow dimensional systems quantum corrections are usually\nenhanced. This facilitates the study of quantum correc-\ntionsduetothespinbath, andforthisreasonwewillcon-\nsider the 1D case for the numerical results. However one\nmustkeepinmindthattheequationsaregeneralforarbi-\ntrary dimension. Furthermore, we will show that includ-\ning the correlated parts in the equation of motion, allows\nto recover the exact quantum critical point in absence of\nthe spin bath, thus providing a good benchmark for our\nresults. For accurate resultsin low dimensional cases, the\npresent technique could also be combined with different\nfermionization or bosonization techniques6, and straight-\nforwardly applying the same Green’s function analysis.\nLet us apply this decoupling scheme to Eq.5 and\nFourier transform to frequency domain. One finds:\n!G\u000b;\f\nn;m =h\u0002\nS\u000b\nn;O\f\nm\u0003\n\u0006i+iX\n\u0016\u000f\u0016\u000b\u001a0\n@B\u0016+X\ni6=nV\u0016\ni;nhS\u0016\nii\u0000X\nlA\u0016\nn;lhI\u0016\nli1\nAG\u001a;\f\nn;m (6)\n+iX\n\u0016\u000f\u0016\u000b\u001a0\n@X\ni6=nV\u0016\ni;n\u0010\nhS\u001a\nniG\u0016;\f\ni;m+G\u0016\u001a;\f\nin;m\u0011\n\u0000X\nlA\u0016\nn;l\u0010\nhS\u001a\nniY\u0016;\f\nl;m+Y\u0016\u001a;\f\nln;m\u00111\nA\nThe first line corresponds to the uncorrelated contribution, while the second line corresponds to the contribution from\ncorrelations between spins. Notice the appearance of the Green’s function Y\u0016;\f\nl;m=\u0000ihI\u0016\nl;O\f\nmi, which couples the\nequation of motion for the Ising spins with the one for the bath spins:\n!Y\u000b;\f\nn;m =h\u0002\nI\u000b\nn;O\f\nm\u0003\n\u0006i+iX\n\u0016\u000f\u0016\u000b\u001a \n\u0001\u0016\u0000X\niA\u0016\ni;nhS\u0016\nii!\nY\u001a;\f\nn;m (7)\n\u0000iX\n\u0016;i\u000f\u0016\u000b\u001aA\u0016\ni;n\u0010\nhI\u001a\nniG\u0016;\f\ni;m+Y\u001a\u0016;\f\nni;m\u0011\nFinally, one can write the equations in a more compact form using matrix notation:\n\u0010\n!\u0000^H\u0011\n\u0001^G= ^\u001f\u0006+^V\u0001^G (8)3\nwhere ^Gis a vector containing all the system and bath\ntwo-point functions, ^\u001f\u0006contains the source terms com-\ning from the commutators/anti-commutators, ^Gcontains\nall extra contributions from correlations, ^Vis the general\ninteraction matrix for the correlated parts, and ^His the\neffectiveHamiltonianforthetwo-pointGreen’sfunctions.\nIf the contribution from the correlated parts ^Gis small\nenough, the solution can be obtained by direct matrix in-\nversion. This is a good approximation near fixed points\nof the RG flow, where solutions are approximately de-\nscribed by free spin Hamiltonians with renormalized pa-\nrameters. Furthermore, if one defines ^g=\u0010\n!\u0000^H\u0011\u00001\n,\nthe previous equation resembles a Dyson’s equation, and\nfrom the correlated parts it can be shown that ^V\u0001^Gcan\nbe written as ^\u0006 (!)\u0001^G, giving rise to the self-energy and\nthe familiar Dyson’s equation.\n^G= ^g\u0001^\u001f\u0006+ ^g\u0001^\u0006\u0001^G (9)\nIII. UNCORRELATED SOLUTION\nTo close the system of equations one can neglect the\ncorrelated parts in Eq.8. Then the solution for the\nGreen’s function is given by a matrix inversion. Eqs.6\nand 7 have two contributions from each interaction term,\nsimilar to the well known Hartree and Fock terms. If\none ignores the Fock contribution, the coupling between\nGreen’sfunctionsatdifferentsitesvanishesandtheequa-\ntions are diagonal in real space. This is equivalent to\nderive an effective-medium Hamiltonian for each spin,\nwhere all correlations between sites are neglected. In this\ncase one finds poles at !s(n) =q\n(P\n\u000bh\u000bs(n))2for the\ncentral system, and at !b(n) =q\n(P\n\u000bh\u000b\nb(n))2for the\nbath, where h\u0016\ns(n) =B\u0016+P\ni6=nV\u0016\nn;iM\u0016\ni\u0000P\nlA\u0016\nn;lm\u0016\nl\nandh\u0016\nb(n) = \u0001\u0016\u0000P\niA\u0016\ni;nM\u0016\ni. As the Green’s functions\ndependonthelocalmagnetization M\u000b\nn=hS\u000b\nniandm\u000b\nn=\nhI\u000b\nni, they need to be determined self-consistency. Phys-\nically, this happens due to the non-linearities introduced\nby the interaction term. Defining the spectral function\nJ\u000b;\f(!) =i\u0002\ng\u000b;\f\nn;n(!+i\u000f)\u0000g\u000b;\f\nn;n(!\u0000i\u000f)\u0003\u0000\ne!\nT\u00061\u0001\u00001,\nthe self-consistency equations are obtained from the re-\nlation between the statistical average and the spectral\nfunctioni\n2\u000f\u0016\u0017\u000bM\u000b\nn=\u0001\nJ\u0017;\u0016(!)d!=2\u001916(the\u0006sign cor-\nresponds to fermionic or bosonic Green’s function, re-\nspectively). The previous solution for the Green’s func-\ntions, in combination with the solutions of these non-\nlinear equations characterizing the local magnetization\nM\u000b\nnandm\u000b\nl, fully determine the properties of the sys-\ntem if correlations can be neglected. Furthermore, as\nin this uncorrelated case the Green’s functions display\nsimple poles only, one can rewrite the integral over fre-\nquency as a sum over poles. Then the final form of theself-consistency equations is:\nM\u000b\nn=h\u000b\ns(n)\n2!s(n)tanh\u0012!s(n)\n2T\u0013\n(10)\nm\u000b\nn=h\u000b\nb(n)\n2!b(n)tanh\u0012!b(n)\n2T\u0013\n(11)\nThis result has been particularized for spin 1/2, but\nlarger spins can be studied similarly, as we discuss be-\nlow for the specific case of the Quantum Ising model.\nThis simple result is a good first estimate to see if the\nsolutions are characterized by a magnetic texture. For\nexample, it can capture the anti-ferromagnetic ordering\nfor the case Vi;j<0. With this information one can build\nmore complete solutions, based on the symmetries of the\nground states.\nQuantum Ising model longitudinally coupled to the spin\nbath:As a more specific case, let us consider the fer-\nromagnetic Quantum Ising model, longitudinally cou-\npled to a bath of spins (we set Vx;y\ni;j= 0,By;z= 0,\nAx;y\ni;l= 0and initially ~\u0001 = 0). In this case the poles for\nthe central system are !s=q\nB2+ (V0Mz\u0000ZBAmz)2,\nwhile the ones for the bath are !b=ZBSAMz. The\nself-consistency equations can be directly obtained from\nEq.10 (to simplify notation we rename V0=ZSV,Vz\ni;j=\nVandA=Az\ni;l). We have assumed homogeneous mag-\nnetization to simplify the expressions, which holds if the\nsystem has a uniform magnetization solution. This not\nobvious in the presence of the spin bath, as it can me-\ndiate interactions between the system spins, modulating\nthe initially spatially homogeneous solution. However,\nwe assume bath configurations which do not destroy the\nhomogeneous magnetization.\nWhen correlations are neglected, the physics is dom-\ninated by three coordination numbers or connectivities:\nZScorresponds to the number of spins which directly in-\nteract in the central system, ZBto the number of bath\nspins connected to each spin, and ZBSto the number\nof spins directly interacting with each bath spin. The\ncoordination number ZSis well known in the analysis\nof the Ising model, but with the addition of the bath\nof spins, now two additional coordination numbers are\nneeded. Interestingly, when correlations are neglected,\ntheresultsonlydependonthetopologyofthelattice(i.e.,\nthe graph connectivity), while the geometrical effects will\nappear when correlations between sites are added.\nAn obvious question now is how the different phase\ntransitions are affected by the bath. To obtain the Curie\ntemperature, Tcwe fixB= 0and expand to third order\nin powers of Mz. The solution for Mz= 0is found to be:\nTc'V0\n8+r\n\u0000V0\n2\u00012+ZBZBSA21+8j~Pj+4j~Pj2\n6\n4(12)\nIt is easy to see that Tc(A!0) =V0=4, in agreement\nwith the Curie-Weiss law for the Ising model. The crit-\nical temperature seems to increase with all coordination4\nnumbers and with the total spin of the bath\f\f\f~P\f\f\f(see de-\ntailsofthecalculationintheAppendixA,wherearbitrary\nspin value for the bath is assumed). This result requires\nsome clarification, as it is known that the T-dependence\nfor the longitudinal magnetization should not be affected\nif the spin couples to a single bath spin21. The reason for\nthis discrepancy is the lack of correlations between the\nIsing spin and the local bath spin for the uncorrelated\nsolution. If one considers instead the electro-nuclear ba-\nsis at each site, or includes correlations between the bath\nandtheIsingspin, bothresultsagree. Toconfirmthis, we\nhave calculated in the Appendix the deviation from the\nexact solution for the case of a single Ising spin coupled\nto a bath spin. It shows that the limits of high and low\ntemperature are well captured by the uncorrelated solu-\ntion. However, as one goes to intermediate temperatures,\ncorrelations between the two spins become important,\nand the exact solution deviates from the uncorrelated\none (Fig.5 in the Appendix). This provides a practical\nexample of the importance of correlations, specially near\nthe phase transitions, and also confirms the validity of\nour results near the fixed points of the theory.\nImportantly, our solution can describe the case in\nwhich the spin bath is non-local ( ZBS>1), which has\nnot been previously discussed. Also, as previously indi-\ncated, theelectro-nuclearcorrelationscanbeeasilyincor-\nporated by exactly diagonalizing the local electro-nuclear\nHamiltonian22.\nTo find the critical field Bcwhich characterizes the\nquantumphasetransition(QPT)wetakethelimit T!0\nand expand to first order in powers of Mz:\nMz'ZBA\f\f\f~P\f\f\f\n2r\nB2+A2\f\f\f~P\f\f\f2\nZ2\nB+B2V0Mz\n2\u0014\nB2+A2\f\f\f~P\f\f\f2\nZ2\nB\u00153=2(13)\nThis shows that Mz(T!0) = 0is not a solution, and\nthe QPT between the ferromagnetic and the paramag-\nnetic phases is blocked due to a remnant magnetization\ninduced by the bath. Furthermore, this remnant mag-\nnetization scales as the first term in Eq.13. This means,\nstrictly speaking, that there is no QPT, although trans-\nverse terms in the interaction can recover the QCP. In-\nterestingly, all three coordination numbers appear in the\nresult for the critical temperature, while ZBSis absent\nin the expression for the quantum critical point. Fig.1\ncompares the characteristic behavior of the longitudinal\nmagnetization in the presence of the spin bath for differ-\nent cases. It shows that at low temperatures, the longi-\ntudinal coupling to the bath blocks the QPT (red), but it\ncan be recovered by adding a transverse field \u0001x6= 0to\nthe bath spins (green). Increasing the temperature also\nunblocks the transition (blue), because it disorders the\nbath, but the phase transition is not strictly at T= 0.\nThe phase diagram in absence of correlations captures\nqualitatively the different phases, but the critical point is\noverestimated. For the 1D Ising model it is well known\nCorr.(A≠0)\nCorr.(A=0)\nRPA(A≠0)\nUncorr.(A≠0)\n0.0 0.5 1.0 1.5 2.00.00.10.20.30.40.5\nB/VMzFigure 1: Phase diagram for the uncorrelated solution as a\nfunctionof B=V. (Black)IsolatedIsingmodelatlowT.(Red)\nIsing model longitudinally coupled to a static bath for A=V =\n0:05andlow T. (Blue)Isingmodellongitudinallycoupledtoa\nstatic bath at higher T. Temperature disorders the spin bath\nandunblocksthephasetransition. (Green)Isingmodelatlow\nTlongitudinally coupled to a dynamical spin bath ( \u00016= 0),\nwhich unblocks the QPT as well. We have considered ZBS=\nZB= 1for the plot.\nthat the exact critical field corresponds to B=V = 1=2\n, and this discrepancy happens due to corrections pro-\nduced by correlations. In the last section we will show\nthataverygoodagreementisobtainedifcorrelatedparts\nare added, but let us first consider the effect of the Fock\nterms in the T= 0regime.\nIV. CORRELATIONS BETWEEN SPINS\nIn general, quantum systems are made of interacting\nparticles, and although their interactions may be weak,\nthe correlations generated by these interactions may still\nhave important consequences. For example, it is well\nknownthatnearclassicalandquantumphasetransitions,\nsystemsdisplaymacroscopiccorrelations, atlengthscales\nmuch larger than the typical length scales present in the\nmicroscopic model. This is a particular case of emer-\ngence, where the system behaves in a different way than\ntheparticlesintheunderlyingmicroscopicmodel. There-\nfore, it is clear that in some cases correlations are impor-\ntant, and results where correlations are neglected will be\nsignificantly affected. Another important case, present in\nlow dimensional quantum systems, is when due to con-\nfinement, the role of quantum fluctuations is enhanced.\nAwellknownconsequenceofthisistheabsenceofcertain\nphase transitions in low dimensional models.\nTo include the effect of correlations one just needs to\nkeep the terms that were previously neglected in the\nequation of motion. The simplest contribution is the\nFock term, which couples two-point Green’s functions at\ndifferent sites. With this term added, two-point corre-\nlations are captured. If the system is translationally in-\nvariant, the Green’s function is diagonal in k-space and5\nthe matrix inversion can be performed analytically. The\nmain effect of the Fock term is to make quasiparticles\ndispersive, and to interpolate between the fixed points\nof the theory, where correlations between sites can be\nneglected (e.g., between the B= 0and theV= 0lim-\nits of the Quantum Ising model). For the present case\nwith a spin bath, this correction can also correlate the\nIsing and the bath Green’s functions (see terms hS\u001a\nniY\u0016;\f\nl;m\nandhI\u001a\nniG\u0016;\f\ni;min Eq.6 and Eq.7, respectively). This im-\nplies that in general, magnons in the Ising model become\na mixture of Ising magnons and excitations in the spin\nbath. Finally, the correction due to the correlated part of\nthe three-point function requires to calculate a new equa-\ntion of motion, which in the present case will also encode\nmagnon-magnon interactions. Formally, it can be shown\nthat the equation of motion for the correlated part of the\nthree-point function can be written in the next form:\n\u0010\n!\u0000^HG\u0011\n\u0001^G=^\u00030+^\u0003\u0001^G (14)\nonce it is truncated by neglecting four-point correlations,\nwhich are expected to scale as Z\u0000232. In Eq.14 we have\ndefined HGas the “Hamiltonian” for the correlated part,\nwhich is obtained from its equation of motion, and ^\u0003\nand ^\u00030are the two types of source terms which can be\npresent. The solution can be written as:\n^G=^\u00030+^\u0003\u0001^G\n!\u0000^HG(15)\nInserting this result in Eq.8, yields the solution for the\ntwo-point function including correlations from the three-\npoint function:\n^G=^\u001f\u0006+^\u00060\n!\u0000^H\u0000^\u0006(16)\nwhere the self-energies are defined as ^\u0006\u0011^V\u0001^\u0003\n!\u0000HGand\n^\u00060\u0011^V\u0001^\u00030\n!\u0000HG. Equation 16 shows that in general, ^\u0006modi-\nfies the pole structure of the Green’s function , while ^\u00060\nmodifies the spectral weight.\nAlthough the formal description of the previous solu-\ntions seems quite simple, a full solution can be a chal-\nlenging task due to the self-consistency equations, and\nin general, one needs to make use of numerical methods.\nNevertheless, it is also possible to obtain some analyti-\ncalresultsbymeansofperturbativeexpansionsandother\napproximationmethods. Inthisworkwewillfocusonthe\neffect of two-spin correlations, which capture the magnon\nquasiparticles. The specific effect of the correlated parts\nwill be analyzed only for the correction to the longitu-\ndinal magnetization, while more general effects will be\ndiscussed in future works.\nCorrelations for the Quantum Ising model longitudi-\nnally coupled to the spin bath: Now we include correla-\ntions in the transverse Ising model, as the significance of\nthis model for current quantum computing architecturesiscrucial. TheadditionoftheFocktermmodifiesthepre-\nvious equations of motion, and can couple the Ising and\nbath spins. For the case with ~\u0001 = 0the main difference\nwiththeprevioussolutionisthechangeinthequasiparti-\ncle excitations, from localized spin flips to magnons with\ndispersion relation:\n!k=q\n(MzV0\u0000ZBAmz)2+B2\u0000BMxVk(17)\nNotice that the last term vanishes at both fixed points\nB= 0andV= 0, agreeing with the uncorrelated so-\nlutions. However, as one interpolates between the two,\nthe quasiparticles turn mobile and the band structure\nbecomes dispersive. Results including the Hartree and\nFock terms in the equations of motion are similar to\nthe Random Phase Approximation (RPA)22. The self-\nconsistency equations are now:\nMz=1\n2MzV0\u0000ZBAmz\n1\nNP\nk!kcoth\u0000!k\n2T\u0001 (18)\nMx=1\n2B\n1\nNP\nk!kcoth\u0000!k\n2T\u0001 (19)\nIn absence of the spin bath ( A= 0), the phase diagram\ndoes not change much with respect to the uncorrelated\nresult. The critical field Bcshifts towards slightly larger\nvalues, stabilizing the ferromagnetic phase. On the other\nhand, this approximation captures that, at the quantum\nphase transition, the magnon gap closes, as it can be seen\nin the spin-spin correlation function (Fig.2, top). When\nthe spin bath is included ( A6= 0), the difference between\nthe uncorrelated and correlated phase diagrams gets re-\nduced (see Fig.3, green and blue lines). The reason is\nthat two-point correlations between Ising spins dominate\nnear the unperturbed Ising critical point, which is now\nshifted. This can be seen in Fig.3, where the main dif-\nferences happen near the Ising critical point Bc'1:1V,\nand the two solutions match again for larger B. There-\nfore, spin-spin correlations do not change quantitatively\nthe phase diagram, in the presence of the spin bath. On\nthe other hand, the effect of the bath in the correlation\nfunctions is more crucial, as it leads to a mode soften-\ning at the critical point. Without the spin bath, the\nmagnon gap closes, signaling a divergence of the corre-\nlation length between Ising spins; however, the presence\nof the remnant field produced by the spin bath makes\nthe gap finite for all B, and the correlation length be-\ntween spins remains finite. Furthermore, the effect of a\ndynamical bath due to a transverse field \u00016= 0modi-\nfies this picture non-trivially. The fact that Eq.17 corre-\nsponds to Ising magnons under an effective longitudinal\nfield\u0018ZBAmzis not a coincidence. When the bath is\nstatic ( \u0001 = 0), system and bath do not get entangled,\nbut making the bath dynamical ( \u00016= 0) correlates both\nsystems and modifies the quasiparticle picture. Fig.4\nshows the appearance of an electro-nuclear mode with\nnon-vanishing spectral weight. This mode emerges near\n!'0when\u0014\u0011ZBZBSA2B\u0001mxMx6= 0and corre-\nsponds to a mixture of magnon and bath excitation, as6\nFigure 2: Complex part of the bosonic Green’s function\nGz;z\nk;k(!)as a function of kand!. (Top) In absence of the spin\nbath, the magnon gap closes at the critical point Bc'1:1V.\n(Bottom) Gap does not close at the critical point if the longi-\ntudinal coupling to the bath is included. We have considered\nA= 0:05Vand a 1D lattice with ZB=ZBS= 1. The results\nare expressed in logarithmic scale to enhance the contrast.\nCorr(A 0)\n0.00.51.01.52.00.00.10.20.30.40.5\nB/VMz\nFigure 3: MzvsB=Vfor the Ising model longitudinally cou-\npled to a bath of spins at T= 0. (Green) Uncorrelated solu-\ntion for A= 0:05VandZB= 1. (Blue) RPA solution includ-\ning for A= 0:05VandZB= 1. (Black) Solution including\ncorrelated parts without the spin bath ( A= 0). (Red) So-\nlution including correlations for a large spin bath ( A= 0:2V\nandZB= 100).\nit can be seen from the poles of the Green’s function:\n~!k=\u0006vuut\n2+!2\nk\n2\u0006s\n\u0014+\u0012\n2\u0000!2\nk\n2\u00132\n(20)\nFigure4: Emergenceofanelectro-nuclearmodeatthecritical\npoint when system and bath get correlated due to a finite\n\u0001 = 10\u00003B.\nwhere \n =p\n\u00012+Z2\nBSA2(details in the Appendix).\nThe presence of this mode has been observed in7and\ndiscussed in detail in ref.22. There it is shown that this\nmode does not carry spectral weight unless the bath is\ndynamical, andthatitallowstorecovertheQCP.Finally,\nwe discuss the impact of adding the correlated parts in\nthe equation of motion. As previously mentioned, their\neffect can be incorporated in terms of a self-energy, but\nin the general case, the self-energy will be a complicated\nfunction of frequency and momentum. To estimate their\neffect we analyze the longitudinal magnetization, which\ncan be addressed more easily by using the Majorana rep-\nresentation for spins23:\nS\u000b=\u0000i\n2\u000f\u000b\u00121\u00122\u0011\u00121\u0011\u00122(21)\nThis representation simplifies the self-consistency equa-\ntions, andmakestheself-energyforthelongitudinalmag-\nnetization \u0006zz\u0010\n~k;!\u0011\na function of !only. Furthermore,\nunlike the Wigner-Jordan transformation, this represen-\ntation is local and the separation between correlated and\nuncorrelated parts is analogous to the spins case. The\nderivation of the equations of motion, as well as the de-\ncoupling scheme is analogous to the case of the spin rep-\nresentation, as described in the Appendix.\nThe phase diagram including the correlated parts is\nshown in Fig.3. In absence of the spin bath ( A= 0,\nblackline), theferromagneticphase shrinksandthe QCP\nshifts toB=V'1=2, which is the exact result for the 1D\nIsing model. When the spin bath is included ( A6= 0,\nred line), the phase boundary shifts towards larger val-\nues ofB=V, as it happened in the uncorrelated phase\ndiagram. However, in this case the remnant magnetic\nfield produced by the spin bath is highly renormalized\nby the correlated parts. This can be seen from the spe-\ncific values chosen for the bath in Fig.3 ( A= 0:2Vand\nZB= 100), which for the uncorrelated solution would\nproduce a phase boundary shifted towards much larger\nvalues ofB=V. Importantly, although the QPT seems to\nbe recovered when correlated parts are included, this is7\nnot the case. One can check that Mz(T= 0) = 0 is not a\nfixed point of the self-consistency equations, although its\nfinal value is very small due to the large renormalization.\nConclusions: We have shown that a large variety of\nspin models interacting with spin baths can be treated\nusing the equation of motion technique. This approach\nallows for non-perturbative solutions, even to lowest or-\nder, and using our decoupling scheme one can include\ncorrections in a systematic way. Neglecting correlations\nyields the standard MF solution, while adding the lowest\norder correlations between pairs of spins reproduces the\nRPA results.\nAs a concrete example, we have analyzed the Quantum\nIsing model coupled to a local bath of spins, where each\nIsing spin is coupled to a set of ZBindependent spins.\nThis case is interesting due to the changes produced by\nthe longitudinal interaction on the critical properties. In\nthe absence of correlations we have obtained solutions\nfor arbitrary spin value in the bath, and demonstrated\nthat a remnant magnetic field produced by the bath can\nblock the quantum phase transition. When correlations\nare included, system and bath quasiparticles can mix,\ntransforming the standard magnons into electro-nuclear\nmodes. We have shown that the effect of internal dynam-\nics in the spin bath entangles the system and bath spins,and that this leads to the emergence of electro-nuclear\nmodes near the phase transition.\nWhen higher correlations are added (i.e., the corre-\nlated parts of the Green’s functions), some quantities are\naccurately captured. For the Quantum Ising model we\nhave obtained the exact critical point in 1D and shown\nthat the effect of the spin bath gets highly renormalized\nby the virtual processes.\nFuture directions for this work include the study of\nmorecomplicatedspinmodels, asforexamplewithtrans-\nverse terms and dipolar interactions, or considering more\ncomplex connectivity between bath and system (e.g., if\na bath spin can interact with more than a single Ising\nspin, theresultswouldappreciablychange). Toconclude,\neffective models of spin systems interacting with local-\nized modes are ubiquitous in experiments at low tem-\nperature. Some examples are: dangling bonds24, nuclear\nspins7,25,26, paramagnetic impurities27,28, and localized\nvibrational modes29. Therefore a a theoretical approach\nthat allows to systematically study them is very relevant\nfor the field.\nWe would like to thank R. D. McKenzie for helpful\ndiscussions. This work was supported by the National\nScientific and Engineering Research Council of Canada\nand A.G-L. acknowledges the Juan de la Cierva program.\n1E. B.-N. P.L. Krapivsky, S. Redner, A Kinetic View of\nStatistical Physics , Cambridge University Press, 2010.\n2S. Lloyd, J. Phys.: Conf. Ser. 302, 012037 (2011).\n3H. G. Hiscock et al., Proceedings of the National Academy\nof Sciences (2016).\n4A. Das and B. K. Chakrabarti, Rev. Mod. Phys. 80, 1061\n(2008).\n5K. Sengupta, S. Powell, and S. Sachdev, Phys. Rev. A 69,\n053616 (2004).\n6S. Sachdev, Quantum Phase Transitions , Cambridge Uni-\nversity Press, 2 edition, 2011.\n7H. M. Rønnow et al., Science 308, 389 (2005).\n8M. Schechter and P. C. E. Stamp, Phys. Rev. Lett. 95,\n267208 (2005).\n9N. V. Prokof’ev and P. C. E. Stamp, Reports on Progress\nin Physics 63, 669 (2000).\n10S. Boixo et al., Nature Physics 10, 218 (2014).\n11R. Barends et al., Nature 534, 222 (2016).\n12E. F. Dumitrescu et al., Phys. Rev. Lett. 120, 210501\n(2018).\n13S. Tomsovic, Proceedings from the Institute for Nuclear\nTheory-Vol.5: Tunneling in complex systems , World Sci-\nentific Pub Co Inc, 1998.\n14J. M. Martinis et al., Phys. Rev. Lett. 95, 210503 (2005).\n15C. Deng and X. Hu, Phys. Rev. B 73, 241303 (2006).\n16D. Zubarev, Soviet Physics Uspekhi 3, 320 (1960).\n17H.-Y. WANG, K.-Q. CHEN, and E.-G. WANG, Interna-\ntional Journal of Modern Physics B 16, 3803 (2002).\n18J. Schreiber, Physica B+C 96, 27 (1979).\n19A. Gómez-León, Phys. Rev. B 96, 064426 (2017).\n20A. Gómez-León, Phys. Rev. B 94, 035144 (2016).\n21J. H. Page, S. R. P. Smith, and D. R. Taylor, Journal ofPhysics C: Solid State Physics 17, 73 (1984).\n22R. D. McKenzie and P. C. E. Stamp, Phys. Rev. B 97,\n214430 (2018).\n23V. R. Vieira, Phys. Rev. B 23, 6043 (1981).\n24A.M.Holder, K.D.Osborn, C.Lobb, andC.B.Musgrave,\nPhysical review letters 111, 065901 (2013).\n25A. Morello, O. N. Bakharev, H. B. Brom, R. Sessoli, and\nL. J. de Jongh, Phys. Rev. Lett. 93, 197202 (2004).\n26E. Chekhovich et al., Nature materials 12, 494 (2013).\n27L. Luan et al., Scientific reports 5, 8119 (2015).\n28C. M. Quintana et al., Phys. Rev. Lett. 118, 057702\n(2017).\n29J. Lisenfeld et al., Scientific reports 6, 23786 (2016).\n30Typically oscillator bath couplings scale as 1=p\nNdue to\ntheir delocalized nature, being Nthe number of modes.\nTherefore perturbative calculations can, in most cases,\ncapture the main properties.\n31Interactions between bath spins can be easily introduced,\nbut it is known that large interactions would make the\nenvironment behave as an effective oscillator bath, as the\nmodes would not be localized anymore.\n32In the present case the system is characterized by three\ndifferent coordination numbers, and the smallest one will\ndominate this scaling.8\nAppendix A: Comparison between the uncorrelated and the exact solution for two spins\nHere we discuss the limitations of the uncorrelated solution in terms of a two spin system. Consider the next\nHamiltonian:\nH=\u0000BxSx\u0000BzSz\u0000\u0001zIz+AIzSz(A1)\nwhere a single spin is coupled to a single bath spin. The interaction between the two spins is longitudinal and\nproportional to A. In addition, the Ising spin has a longitudinal field Bzwhich mimics the effect of the field produced\nby all the other Ising spins in the Ising model, and a transverse field Bx. Also the bath spin can be biased by a\nlongitudinal field \u0001z. The exact calculation is possible due to the small size of the Hilbert space, and the statistical\naverage for the magnetization of the Ising spin can be obtained analytically:\nMz=Z\u00001Trn\ne\u0000H\nTSzo\n=e\u0000\u0001\n2T(A+ 2Bz)sinh\u0012p\n(A+2Bz)2+4B2\n4T\u0013\nZq\n(A+ 2Bz)2+ 4B2\u0000e\u0001\n2T(A\u00002Bz)sinh\u0012p\n(A\u00002Bz)2+4B2\n4T\u0013\nZq\n(A\u00002Bz)2+ 4B2(A2)\nwhere\nZ= 2e\u0001\n2Tcosh0\n@q\n(A\u00002Bz)2+ 4B2x\n4T1\nA+ 2e\u0000\u0001\n2Tcosh0\n@q\n(A+ 2Bz)2+ 4B2x\n4T1\nA (A3)\nis the partition function. On the other hand, we make use of the self-consistency equations for the magnetization,\nobtained from the uncorrelated solutions of the Green’s functions in the main text:\nMz=Bz\u0000Amz\n2q\nB2x+ (Bz\u0000Amz)2tanh0\n@q\nB2x+ (Bz\u0000Amz)2\n2T1\nA (A4)\nMx=Bx\n2q\nB2x+ (Bz\u0000Amz)2tanh0\n@q\nB2x+ (Bz\u0000Amz)2\n2T1\nA (A5)\nmz=\u0001z\u0000AMz\n2q\n(\u0001z\u0000AMz)2tanh0\n@q\n(\u0001z\u0000AMz)2\n2T1\nA (A6)\nThe comparison between the two solutions is shown in Fig.5. It shows that in the absence of a transverse field ( Bx'0)\nthe longitudinal bath does not affect the Curie temperature, which is missed by the uncorrelated solution. On the\nother hand, for non-vanishing transverse field (i.e., when the Ising model becomes “quantum”), the bath modifies the\nbehavior with temperature, specially in the low temperature regime.\nNotice that the exact solution in terms of Green’s functions is also possible in this case, by just including the corre-\nlated part. This indicates that the intermediate temperature regime needs to be characterized including correlations\n(as one would expect, specially at the Curie temperature, where the correlation length diverges).9\nA=10-3\nA=10-3(exact)\nA=5\nA=5(exact)\n0 1 2 3 40.00.10.20.30.4\nTMz\nA=10-3\nA=10-3(exact)\nA=5\nA=5(exact)\n0 1 2 3 40.00.10.20.30.40.5\nTMz\nFigure 5: Comparison between the self-consistent solution and the exact calculation. (Left) Case with Bz= 0:1andBx= 1,\nand (right) case with Bz= 1andBx= 0:1. In general, the uncorrelated solution and the exact one, perfectly agree at low and\nhigh temperature, while for intermediate values they differ (for weak Athey agree very well in general). When the system is\nin the analog of a FM phase, with Bx\u001cBz(right), the hyperfine coupling (even for large A) would not make an important\nchange in the Curie temperature (which happens for intermediate values of T). This is in contrast with the case Bx\u001dBz\n(left), where specially at low Tthe cases with small and large Aare quite different.\nAppendix B: Quantum Ising model coupled to a spin bath\nThe case of the Quantum Ising model is obtained from the general equation of motion by setting By;z= 0,Vx;y\ni;j= 0,\n\u0001x;y= 0andAx;y\ni;l= 0. Furthermore, if the bath spins are static ( ~\u0001 = 0), it is useful to consider for its basis the\nset of eigenstates jP;Pzi, wherePlabels the spin value P2h\f\f\f~P\f\f\f;\f\f\f~P\f\f\f\u00001;:::; 1=2i\nandPz2[P;P\u00001;:::;\u0000P]its\nprojection. Their equation of motion is calculated analogously, from the Heisenberg equation:\n@t^O=ih\nH;^Oi\n(B1)\nThe advantage of using this basis is that it allows to calculate the solutions for arbitrary spin value. The Hamiltonian\nreads in this basis:\nHS=\u0000BX\niSx\ni\u0000X\ni;j>iVi;jSz\niSz\nj\u0000\u0001zX\nlX\nP;PzPzXPz;Pz\nP;l+X\ni;lAi;lSz\niX\nP;PzPzXPz;Pz\nP;l(B2)\nwhere\nXPz;Pz\nP;l=jP;PzilhP;Pzjl\nis the projector onto the eigenstate of the bath at site l. Neglecting correlations one finds that the average value of\neach projector, and the magnetization mz\nn=P\nPz;PPzhXPz;Pz\nP;niare given by:\nhXa;a\nP;ni=e(a\u00001)hb\nT\n2\u0010\nehb\nT\u00001\u00112\ncoshh\nhb\nT\u0010\f\f\f~P\f\f\f+ 1\u0011i\n\u0000cosh\u0000hb\n2T\u0001 (B3)\nmz\nn=\u00001\n4csch\u0012hb\n2T\u00133 + cosh\u0000hb\nT\u0001\n\u00002\u0010\n2 +\f\f\f~P\f\f\f\u0011\ncoshh\nhb\n2T\u0010\n1 + 2\f\f\f~P\f\f\f\u0011i\n+ 2\f\f\f~P\f\f\fcoshh\nhb\n2T\u0010\n3 + 2\f\f\f~P\f\f\f\u0011i\ncosh\u0000hb\n2T\u0001\n\u0000coshh\nhb\nT\u0010\f\f\f~P\f\f\f+ 1\u0011i (B4)\nwherehb(n) = \u0001z\u0000P\niAi;nMz\ni. This solution is specific for half-integer spins, but the case for integer spins can be\ncalculated analogously. Finally, we use the solution for the fermionic Green’s function to obtain the self-consistency\nequations for the magnetization:\nMz=hs\n2!stanh\u0010!s\n2T\u0011\n(B5)\nMx=B\n2!stanh\u0010!s\n2T\u0011\n(B6)10\nWith these results one can now estimate the Curie temperature and the critical field. For the Curie temperature we\nare looking for solutions with B= 0. We consider the case where the bath is unbiased by an external field to simplify\nthe expressions ( \u0001z= 0). Expanding for small Mzup to third order, and solving for the value of Tthat makes\nMz= 0, we find:\nTc=V0\n8+1\n4vuut\u0012V0\n2\u00132\n+ZBZBSA21 + 8\f\f\f~P\f\f\f+ 4\f\f\f~P\f\f\f2\n6(B7)\nIt shows that for this approximation, the Curie temperature would increase with all the coordination numbers (re-\nmemberV0=ZsV), as well as with the total spin of the bath\f\f\f~P\f\f\f. As discussed in the previous section of the\nAppendix, this result is not correct for a local spin bath, because the Curie temperature should not be affected by\nthe longitudinally coupled bath. The correction is provided by the correlations between the Ising spins and the bath\nspins. On the other hand, this model can also have several Ising spins coupled to each bath spin, and in this case the\nCurie temperature can be affected. Similarly for the critical field one finds:\nMz'ZBA\f\f\f~P\f\f\f\n2r\nB2+A2\f\f\f~P\f\f\f2\nZ2\nB+B2V0Mz\n2\u0014\nB2+A2\f\f\f~P\f\f\f2\nZ2\nB\u00153=2+O\u0000\nM2\nz\u0001\n(B8)\nThenMz= 0is not a solution due to a remnant magnetization, which saturates for large A\u001dB.\nAppendix C: Effect of two-spin correlations\nIn order to include two-point correlations, we include the Fock term in the equation of motion. This couples Green’s\nfunctions at different lattice sites:\n!G\u000b;\f\nn;m =h\u0002\nS\u000b\nn;O\f\nm\u0003\n\u0006i+iX\n\u0016\u000f\u0016\u000b\u001a0\n@B\u0016+X\ni6=nV\u0016\ni;nM\u0016\ni\u0000X\nlA\u0016\nn;lm\u0016\nl1\nAG\u001a;\f\nn;m\n+iX\n\u0016\u000f\u0016\u000b\u001aM\u001a\nn0\n@X\ni6=nV\u0016\ni;nG\u0016;\f\ni;m\u0000X\nlA\u0016\nn;lY\u0016;\f\nl;m1\nA\n!Y\u000b;\f\nn;m =h\u0002\nI\u000b\nn;O\f\nm\u0003\n\u0006i+iX\n\u0016\u000f\u0016\u000b\u001a \n\u0001\u0016\u0000X\niA\u0016\ni;nhS\u0016\nii!\nY\u001a;\f\nn;m\u0000iX\n\u0016;i\u000f\u0016\u000b\u001aA\u0016\ni;nhI\u001a\nniG\u0016;\f\ni;m\nUsing a transformation to momentum space, while assuming spatially homogeneous solutions, we find:\n!G\u000b;\f\nk=h\u0002\nS\u000b;O\f\u0003\n\u0006i+iX\n\u0016\u000f\u0016\u000b\u001a!s;\u0016G\u001a;\f\nk+iX\n\u0016\u000f\u0016\u000b\u001aM\u001a\u0010\nV\u0016\nkG\u0016;\f\nk\u0000ZBA\u0016Y\u0016;\f\nk\u0011\n(C1)\n!Y\u000b;\f\nk=h\u0002\nI\u000b;O\f\u0003\n\u0006i+iX\n\u0016\u000f\u0016\u000b\u001a!b;\u0016Y\u001a;\f\nk\u0000iX\n\u0016\u000f\u0016\u000b\u001am\u001aZBSA\u0016G\u0016;\f\nk(C2)\nwhere!s;\u0016=B\u0016+P\ni6=nV\u0016\n0M\u0016\u0000ZBA\u0016m\u0016and!b;\u0016= \u0001\u0016\u0000ZBSA\u0016M\u0016.\nFor the case of the Ising model longitudinally coupled to a bath of spins and \u0001x= 0, the poles of the Ising spins\nGreen’s functions are at:\n!k=q\nB2+ (MzV0\u0000ZBAmz)2\u0000BMxVk (C3)\nThe corresponding self-consistency equations are obtained from the anti-commutator Green’s functions:\nMz=MzV0\u0000ZBAmz\n21\nNP\nk!kcoth\u0000!k\n2T\u0001 (C4)\nMx=B\n21\nNP\nk!kcoth\u0000!k\n2T\u0001 (C5)11\nIf the bath is static one can see that the bath occupation probabilities are unchanged and that Ising spins and bath\nspins are not entangled.\nIf the bath has internal dynamics, now driven by \u0001x6= 0, one finds that the poles are not simple Ising magnons\nanymore, but combinations of the bath and system excitations:\n!\u0006=vuut\n2+!2\nk\n2\u0006s\n\u0014+\u0012\n2\u0000!2\nk\n2\u00132\n(C6)\nwhere\u0014 =ZBZBSA2Bx\u0001xmxMx,!k=q\nB2x+ (Bz+V0Mz\u0000ZBAmz)2\u0000BxVkMxand \n =p\n\u00012x+ (\u0001z\u0000ZBSAMz). Analogously one can calculate the self-consistency equations for the magnetization,\nand the correlation functions:\nMz=!z\n2\u0018; Mx=Bx\n2\u0018(C7)\n\u0018\u00111\nNX\nk!+!\u0000\u0000\n!2\n+\u0000!2\n\u0000\u0001\n(1\u0000\u0015k)\u0002\u0000\n\n2\u0000!2\n\u0000\u0001\n!+tanh\u0000!\u0000\n2T\u0001\n\u0000\u0000\n\n2\u0000!2\n+\u0001\n!\u0000tanh\u0000!+\n2T\u0001\u0003\nwhere\n\u0015k=ZBA\u0001xMx\u0002\n!\u0000tanh\u0000!+\n2T\u0001\n\u0000!+tanh\u0000!\u0000\n2T\u0001\u0003\n\u0000\n\n2\u0000!2\n+\u0001\n!\u0000tanh\u0000!+\n2T\u0001\n\u0000\u0000\n\n2\u0000!2\n\u0000\u0001\n!+tanh\u0000!\u0000\n2T\u0001\n\u0002ZBSABxmx\u0002\n!\u0000tanh\u0000!+\n2T\u0001\n\u0000!+tanh\u0000!\u0000\n2T\u0001\u0003\n\u0000\n!2\nk\u0000!2\n+\u0001\n!\u0000tanh\u0000!+\n2T\u0001\n\u0000\u0000\n!2\nk\u0000!2\n\u0000\u0001\n!+tanh\u0000!\u0000\n2T\u0001\nTherefore the self-consistency equations can be written in an analogous form to the case ~\u0001 = 0, by just redefining\nthe function \u0018, which contains a mixing term proportional to \u0015k/ZBSZBA2Bx\u0001xMxmx. Importantly, in this case\nthe bath-bath correlators are also non-vanishing, as they get entangled through the Ising spins.\nAppendix D: Correlated parts\nThe equations of motion for the correlated parts are obtained analogously, by calculating the Heisenberg equations\nof motion:\n@tS\u001a\npS\u000b\nn=X\n\u0016;iB\u0016\u0000\n\u000f\u0016\u001a\u0012S\u0012\npS\u000b\nn+\u000f\u0016\u000b\u0012S\u001a\npS\u0012\nn\u0001\n+X\n\u0016X\ni;j>iV\u0016\ni;j\u0002\n\u000f\u0016\u001a\u0012\u0000\n\u000ej;pS\u0016\niS\u0012\np+\u000ei;pS\u0012\npS\u0016\nj\u0001\nS\u000b\nn+\u000f\u0016\u000b\u0012S\u001a\np\u0000\n\u000ej;nS\u0016\niS\u0012\nn+\u000ei;nS\u0012\nnS\u0016\nj\u0001\u0003\n\u0000X\n\u0016X\ni;lA\u0016\ni;lI\u0016\nl\u0000\n\u000ep;i\u000f\u0016\u001a\u0012S\u0012\npS\u000b\nn+\u000en;i\u000f\u0016\u000b\u0012S\u001a\npS\u0012\nn\u0001\n(D1)\n@tI\u001a\npS\u000b\nn=X\n\u0016;\u0012\u000f\u0016\u000b\u0012B\u0016I\u001a\npS\u0012\nn+\u000f\u0016\u000b\u0012X\n\u0016;\u0012X\ni;j>iV\u0016\ni;jI\u001a\np\u0000\n\u000ej;nS\u0016\niS\u0012\nn+\u000ei;nS\u0012\nnS\u0016\nj\u0001\n+X\n\u0016;\u0012\u000f\u0016\u001a\u0012\u0001\u0016I\u0012\npS\u000b\nn\n\u0000X\n\u0016X\ni;lA\u0016\ni;l\u0000\n\u000ei;n\u000f\u0016\u000b\u0012I\u0016\nlI\u001a\npS\u0012\nn+\u000el;p\u000f\u0016\u001a\u0012I\u0012\npS\u000b\nnS\u0016\ni\u0001\n(D2)\n@tI\u001a\npI\u000b\nn=X\n\u0016 \n\u0001\u0016\u0000X\niS\u0016\ni!\n\u0000\n\u000f\u0016\u001a\u0012A\u0016\ni;pI\u0012\npI\u000b\nn+\u000f\u0016\u000b\u0012A\u0016\ni;nI\u001a\npI\u0012\nn\u0001\n(D3)\nThe correlated part of the Green’s functions are obtained subtracting the uncorrelated contributions. For example for\nthe case ofG\u001a\u000b;\f\npn;mone hasi@tG\u001a\u000b;\f\npn;m =i@tG\u001a\u000b;\f\npn;m\u0000ihS\u001a\npi@tG\u000b;\f\nn;m\u0000ihS\u000b\nni@tG\u001a;\f\np;m. Finally, separating into uncorrelated12\nand correlated parts, and neglecting higher order correlators, one finds:\n!G\u001a\u000b;\f\npn;m' h\u0002\nS\u001a\npS\u000b\nn;O\f\nm\u0003\n\u0006i\u0000hS\u001a\npih\u0002\nS\u000b\nn;O\f\nm\u0003\n\u0006i\u0000hS\u000b\nnih\u0002\nS\u001a\np;O\f\nm\u0003\n\u0006i\n+iX\n\u0016;\u0012\u000f\u0016\u001a\u0012B\u0016G\u0012\u000b;\f\npn;m +iX\n\u0016;\u0012\u000f\u0016\u000b\u0012B\u0016G\u001a\u0012;\f\npn;m\n+iX\n\u0016\u000f\u0016\u001a\u0012X\ni6=p;nV\u0016\ni;p\u0010\nhS\u0016\niS\u000b\nnicG\u0012;\f\np;m+hS\u000b\nnS\u0012\npicG\u0016;\f\ni;m+hS\u0016\niiG\u0012\u000b;\f\npn;m +hS\u0012\npiG\u0016\u000b;\f\nin;m\u0011\n+iX\n\u0016\u000f\u0016\u000b\u0012X\ni6=p;nV\u0016\ni;n\u0010\nhS\u0016\niS\u001a\npicG\u0012;\f\nn;m+hS\u001a\npS\u0012\nnicG\u0016;\f\ni;m+hS\u0016\niiG\u001a\u0012;\f\npn;m +hS\u0012\nniG\u0016\u001a;\f\nip;m\u0011\n\u0000iX\n\u0016\u000f\u0016\u001a\u0012X\nlA\u0016\np;l\u0010\nhS\u000b\nnI\u0016\nlicG\u0012;\f\np;m+hS\u000b\nnS\u0012\npicY\u0016;\f\nl;m+hI\u0016\nliG\u0012\u000b;\f\npn;m +hS\u0012\npiY\u0016\u000b;\f\nln;m\u0011\n\u0000iX\n\u0016\u000f\u0016\u000b\u0012X\nlA\u0016\nn;l\u0010\nhS\u001a\npI\u0016\nlicG\u0012;\f\nn;m+hS\u001a\npS\u0012\nnicY\u0016;\f\nl;m+hI\u0016\nliG\u001a\u0012;\f\npn;m +hS\u0012\nniY\u0016\u001a;\f\nlp;m\u0011\n+i\n4X\n\u0012\u000f\u000b\u001a\u0012\u0000\nV\u000b\nn;pG\u0012;\f\np;m\u0000V\u001a\np;nG\u0012;\f\nn;m\u0001\n\u0000iX\n\u0016;\u0012\u000f\u001a\u0016\u0012V\u0012\nn;p\u0000\nhS\u000b\nnihS\u0016\npiG\u0012;\f\nn;m+hS\u000b\nnihS\u0012\nniG\u0016;\f\np;m+hS\u0012\nnihS\u0016\npiG\u000b;\f\nn;m\u0001\n+iX\n\u0016;\u0012\u000f\u000b\u0016\u0012V\u0016\np;n\u0000\nhS\u001a\npihS\u0016\npiG\u0012;\f\nn;m+hS\u001a\npihS\u0012\nniG\u0016;\f\np;m+hS\u0012\nnihS\u0016\npiG\u001a;\f\np;m\u0001\nSimilarly for the other Green’s functions:\n!Y\u001a\u000b;\f\npn;m' h\u0002\nI\u001a\npS\u000b\nn;O\f\nm\u0003\n\u0006i\u0000hI\u001a\npih\u0002\nS\u000b\nn;O\f\nm\u0003\n\u0006i\u0000hS\u000b\nnih\u0002\nI\u001a\np;O\f\nm\u0003\n\u0006i\n+iX\n\u0016;\u0012\u0000\n\u000f\u0016\u000b\u0012B\u0016Y\u001a\u0012;\f\npn;m +\u000f\u0016\u001a\u0012\u0001\u0016Y\u0012\u000b;\f\npn;m\u0001\n+iX\n\u0016;\u0012\u000f\u0016\u000b\u0012X\ni6=nV\u0016\nn;i\u0010\nhI\u001a\npS\u0016\niicG\u0012;\f\nn;m+hI\u001a\npS\u0012\nnicG\u0016;\f\ni;m+hS\u0012\nniY\u001a\u0016;\f\npi;m+hS\u0016\niiY\u001a\u0012;\f\npn;m\u0011\n\u0000iX\n\u0016;\u0012\u000f\u0016\u000b\u0012X\nl6=pA\u0016\nn;l\u0010\nhI\u001a\npI\u0016\nlicG\u0012;\f\nn;m+hI\u001a\npS\u0012\nnicY\u0016;\f\nl;m+hI\u0016\nliY\u001a\u0012;\f\npn;m +hS\u0012\nniW\u0016\u001a;\f\nlp;m\u0011\n\u0000iX\n\u0016;\u0012\u000f\u0016\u001a\u0012X\ni6=nA\u0016\ni;p\u0010\nhI\u0012\npS\u0016\niiG\u000b;\f\nn;m+hI\u0012\npS\u000b\nnicG\u0016;\f\ni;m+hS\u000b\nnS\u0016\niicY\u0012;\f\np;m+hI\u0012\npiG\u0016\u000b;\f\nin;m+hS\u0016\niiY\u0012\u000b;\f\npn;m\u0011\n\u0000iX\n\u0016;\u0012\u000f\u0016\u000b\u0012A\u0016\nn;p\u0002\u0000\nhI\u0016\npI\u001a\npi\u0000hI\u001a\npihI\u0016\npi\u0001\nG\u0012;\f\nn;m+hS\u0012\nni\u0000\nW\u0016\u001a;\f\npp;m\u0000hI\u001a\npiY\u0016;\f\np;m\u0001\u0003\n\u0000iX\n\u0016;\u0012\u000f\u0016\u001a\u0012A\u0016\nn;p\u0000\nhI\u0012\npiG\u000b\u0016;\f\nnn;m\u0000hS\u000b\nnihI\u001a\npiG\u0016;\f\nn;m+hS\u000b\nnS\u0016\nniY\u0012;\f\np;m\u0000hS\u000b\nnihS\u0016\nniY\u001a;\f\np;m\u0001\nThe two-bath spins correlation function is also needed:\n!W\u001a\u000b;\f\npn;m =h\u0002\nI\u001a\npI\u000b\nn;O\f\nm\u0003\n\u0006i\u0000hI\u001a\npih\u0002\nI\u000b\nn;O\f\nm\u0003\n\u0006i\u0000hI\u000b\nnih\u0002\nI\u001a\np;O\f\nm\u0003\n\u0006i\n+iX\n\u0016\u000f\u0016\u001a\u0012\"\n\u0001\u0016W\u0012\u000b;\f\npn;m\u0000X\niA\u0016\ni;p\u0010\nhI\u000b\nnI\u0012\npicG\u0016;\f\ni;m+hI\u000b\nnS\u0016\niicY\u0012;\f\np;m+hI\u0012\npiY\u000b\u0016;\f\nni;m +hS\u0016\niiW\u0012\u000b;\f\npn;m\u0011#\n+iX\n\u0016\u000f\u0016\u000b\u0012\"\n\u0001\u0016W\u001a\u0012;\f\npn;m\u0000X\niA\u0016\ni;n\u0010\nhI\u001a\npI\u0012\nnicG\u0016;\f\ni;m+hI\u001a\npS\u0016\niicY\u0012;\f\nn;m+hI\u0012\nniY\u001a\u0016;\f\npi;m+hS\u0016\niiW\u001a\u0012;\f\npn;m\u0011#\nThese set of equations of motion provides very complex expressions for the self-energy, and their role will be analyzed\nin future publications, however to estimate their contribution, we consider the specific case of the correction to the\nlongitudinal magnetization. Furthermore, we rewrite these equations of motion in terms of Majorana fermions to13\nsimplify the self-consistency equations:\nS\u000b\nn=\u0000i\n2\u000f\u000b\u00121\u00122\u0011\u00121\nn\u0011\u00122\nn (D4)\nI\u000b\nn=\u0000i\n2\u000f\u000b\u00121\u00122\r\u00121\nn\r\u00122\nn (D5)\n\u0011\u000b\nnbeing a Majorana fermion at site nthat fulfills the usual anti-commutation relation for fermions, and in addition\n\u00112= 1=2. Similarly the \r\u000b\nnrefer to the Majorana fermions for the bath spins. The Green’s functions required for the\ncalculation of the magnetization M\u000b\nn=\u0000i\n2\u000f\u000b\u00121\u00122h\u0011\u00121n\u0011\u00122niare:\nG\u000b;\f\nn;m(t;t0) =\u0000ih\u0011\u000b\nn(t) ;\u0011\f\nm(t0)i (D6)\nP\u000b;\f\nn;m(t;t0) =\u0000ih\r\u000b\nn(t) ;\r\f\nm(t0)i (D7)\nThe longitudinal magnetization is obtained from the diagonal two-point function, with equation of motion:\n!G\u000b;\f\nn;n=\u000e\u000b;\f+i\u000f\u0016\u000b\u0012h\u0016\ns(n)G\u0012;\f\nn;n (D8)\n+1\n2\u000f\u0016\u000b\u0012\u000f\u0017\u00121\u00122X\ni6=nV\u0016;\u0017\nn;iG\u00121\u00122\u0012;\f\niin;n\n\u00001\n2\u000f\u0016\u000b\u0012\u000f\u0017\u00121\u00122X\nlA\u0016;\u0017\nn;lY\u00121\u00122\u0012;\f\nlln;n\nwhich also requires the calculation of the bath spins Green’s function:\n!P\u000b;\f\nl;l=\u000e\u000b;\f+i\u000f\u0016\u000b\u0012h\u0016\nb(l)P\u0012;\f\nl;l(D9)\n\u00001\n2\u000f\u0016\u00121\u00122\u000f\u0017\u000b\u0012X\niA\u0016;\u0017\ni;lR\u00121\u00122\u0012;\f\niil;l\nwhere we have defined h\u0016\ns(n) =B\u0016+P\ni6=nV\u0016;\u0017\nn;iM\u0017\ni\u0000P\nlA\u0016;\u0017\nn;lm\u0017\nl,h\u0016\nb(l) =B\u0016\u0000P\niA\u0017;\u0016\ni;lM\u0017\ni, andY\u00121\u00122\u0012;\f\nlln;nandR\u00121\u00122\u0012;\f\niil;l\nare the correlated parts of the mixed-Green’s functions Y\u00121\u00122\u0012;\f\nlln;n=\u0000ih\r\u00123\nl\r\u00124\nl\u0011\u00121n;\u0011\f\nniandR\u00121\u00122\u0012;\f\niil;l=\u0000ih\u0011\u00121\ni\u0011\u00122\ni\r\u0012\nl;\r\f\nli,\nrespectively.\nTo lowest order one recovers the MF expressions presented in the first sections of the main text. In this case, as\nwe are calculating the diagonal Green’s function G\u000b;\f\nn;n, there is no contribution from a Fock term to lowest order, and\nthe contribution from magnons comes in from the correlated parts. Their equations of motion are:\n!G\u001b1\u001b2\u000b;\f\nppn;n =\u0000iX\n\u0016B\u0016\u0000\n\u000f\u0016\u0012\u001b1G\u0012\u001b2\u000b;\f\nppn;n +\u000f\u0016\u0012\u001b2G\u001b1\u0012\u000b;\f\nppn;n +\u000f\u0016\u0012\u000bG\u001b1\u001b2\u0012;\f\nppn;n\u0001\n+1\n2X\n\u0016;\u0017\u000f\u0017\u00121\u00122X\nkA\u0016;\u0017\np;k\u000f\u0016\u0012\u001b1\u0010\nh\r\u00121\nk\r\u00122\nkiG\u0012\u001b2\u000b;\f\nppn;n +h\u0011\u0012\np\u0011\u001b2\npiY\u00121\u00122\u000b;\f\nkkn;n\u0011\n+1\n2X\n\u0016;\u0017\u000f\u0017\u00121\u00122X\nkA\u0016;\u0017\np;k\u000f\u0016\u0012\u001b2\u0010\nh\r\u00121\nk\r\u00122\nkiG\u001b1\u0012\u000b;\f\nppn;n +h\u0011\u001b1\np\u0011\u0012\npiY\u00121\u00122\u000b;\f\nkkn;n\u0011\n+1\n2X\n\u0016;\u0017\u000f\u0017\u00121\u00122X\nkA\u0016;\u0017\nn;k\u000f\u0016\u0012\u000b\u0010\nh\r\u00121\nk\r\u00122\nk\u0011\u001b1\np\u0011\u001b2\npiCG\u0012;\f\nn;n+h\r\u00121\nk\r\u00122\nkiG\u001b1\u001b2\u0012;\f\nppn;n\u0011\n\u00001\n2X\n\u0016;\u0017\u000f\u0016\u00121\u00122X\ni6=p;nV\u0016;\u0017\ni;p\u000f\u0017\u0012\u001b1\u0010\nh\u0011\u00121\ni\u0011\u00122\niiG\u0012\u001b2\u000b;\f\nppn;n +h\u0011\u0012\np\u0011\u001b2\npiG\u00121\u00122\u000b;\f\niin;n\u0011\n\u00001\n2X\n\u0016;\u0017\u000f\u0016\u00121\u00122X\ni6=p;nV\u0016;\u0017\ni;p\u000f\u0017\u0012\u001b2\u0010\nh\u0011\u00121\ni\u0011\u00122\niiG\u001b1\u0012\u000b;\f\nppn;n +h\u0011\u001b1\np\u0011\u0012\npiG\u00121\u00122\u000b;\f\niin;n\u0011\n\u00001\n2X\n\u0016;\u0017\u000f\u0016\u00121\u00122X\ni6=p;nV\u0016;\u0017\ni;n\u000f\u0017\u0012\u000b\u0010\nh\u0011\u001b1\np\u0011\u001b2\np\u0011\u00121\ni\u0011\u00122\niiCG\u0012;\f\nn;n+h\u0011\u00121\ni\u0011\u00122\niiG\u001b1\u001b2\u0012;\f\nppn;n\u0011\n\u00001\n2X\n\u0016;\u0017\u000f\u0016\u00121\u00122V\u0016;\u0017\nn;p\u0000\n\u000f\u0017\u0012\u001b1h\u0011\u0012\np\u0011\u001b2\npi+\u000f\u0017\u0012\u001b2h\u0011\u001b1\np\u0011\u0012\npi\u0001\u0000\nG\u00121\u00122\u000b;\f\nnnn;n\u0000h\u0011\u00121\nn\u0011\u00122\nniG\u000b;\f\nn;n\u0001\n\u00001\n2X\n\u0016;\u0017\u000f\u0016\u00121\u00122\u000f\u0017\u0012\u000bV\u0016;\u0017\np;n\u0000\nh\u0011\u001b1\np\u0011\u001b2\np\u0011\u00121\np\u0011\u00122\npi\u0000h\u0011\u001b1\np\u0011\u001b2\npih\u0011\u00121\np\u0011\u00122\npi\u0001\nG\u0012;\f\nn;n14\n!Y\u001b1\u001b2\u000b;\f\nlln;n=\u0000iX\n\u0016B\u0016\u000f\u0016\u0012\u000bY\u001b1\u001b2\u0012;\f\nlln;n\u0000iX\n\u0016\u0001\u0016\u0010\n\u000f\u0016\u0012\u001b1Y\u0012\u001b2\u000b;\f\nlln;n+\u000f\u0016\u0012\u001b2Y\u001b1\u0012\u000b;\f\nlln;n\u0011\n\u00001\n2X\n\u0016;\u0017\u000f\u0016\u00121\u00122\u000f\u0017\u0012\u000bX\ni6=nV\u0016;\u0017\ni;n\u0010\nh\r\u001b1\nl\r\u001b2\nl\u0011\u00121\ni\u0011\u00122\niiCG\u0012;\f\nn;n+h\u0011\u00121\ni\u0011\u00122\niiY\u001b1\u001b2\u0012;\f\nlln;n\u0011\n+1\n2X\n\u0016;\u0017\u000f\u0017\u00121\u00122\u000f\u0016\u0012\u000bX\nr6=lA\u0016;\u0017\nn;r\u0010\nh\r\u001b1\nl\r\u001b2\nl\r\u00121\nr\r\u00122\nriCG\u0012;\f\nn;n+h\r\u00121\nr\r\u00122\nriY\u001b1\u001b2\u0012;\f\nlln;n\u0011\n+1\n2X\n\u0016;\u0017\u000f\u0016\u00121\u00122X\ni6=nA\u0016;\u0017\ni;l\u000f\u0017\u0012\u001b1\u0010\nh\r\u0012\nl\r\u001b2\nliG\u00121\u00122\u000b;\f\niin;n +h\u0011\u00121\ni\u0011\u00122\niiY\u0012\u001b2\u000b;\f\nlln;n\u0011\n+1\n2X\n\u0016;\u0017\u000f\u0016\u00121\u00122X\ni6=nA\u0016;\u0017\ni;l\u000f\u0017\u0012\u001b2\u0010\nh\r\u001b1\nl\r\u0012\nliG\u00121\u00122\u000b;\f\niin;n +h\u0011\u00121\ni\u0011\u00122\niiY\u001b1\u0012\u000b;\f\nlln;n\u0011\n+1\n2X\n\u0016;\u0017\u000f\u0016\u00121\u00122A\u0016;\u0017\nn;l\u0000\n\u000f\u0017\u0012\u001b1h\r\u0012\nl\r\u001b2\nli+\u000f\u0017\u0012\u001b2h\r\u001b1\nl\r\u0012\nli\u0001\nG\u00121\u00122\u000b;\f\nnnn;n\n+1\n2X\n\u0016;\u0017\u000f\u0017\u00121\u00122\u000f\u0016\u0012\u000bA\u0016;\u0017\nn;l\u0010\nh\r\u001b1\nl\r\u001b2\nl\r\u00121\nl\r\u00122\nli\u0000h\r\u001b1\nl\r\u001b2\nlih\r\u00121\nl\r\u00122\nli\u0011\nG\u0012;\f\nn;n\n!R\u001b1\u001b2\u000b;\f\nppl;l=\u0000iX\n\u0016B\u0016\u0010\n\u000f\u0016\u0012\u001b1R\u0012\u001b2\u000b;\f\nppl;l+\u000f\u0016\u0012\u001b2R\u001b1\u0012\u000b;\f\nppl;l\u0011\n\u0000iX\n\u0016\u0001\u0016\u000f\u0016\u0012\u000bR\u001b1\u001b2\u0012;\f\nppl;l\n\u00001\n2X\n\u0016;\u0017\u000f\u0016\u00121\u00122X\ni6=pV\u0016;\u0017\ni;p\u000f\u0017\u0012\u001b1\u0010\nh\u0011\u00121\ni\u0011\u00122\niiR\u0012\u001b2\u000b;\f\nppl;l+h\u0011\u0012\np\u0011\u001b2\npiR\u00121\u00122\u000b;\f\niil;l\u0011\n\u00001\n2X\n\u0016;\u0017\u000f\u0016\u00121\u00122X\ni6=pV\u0016;\u0017\ni;p\u000f\u0017\u0012\u001b2\u0010\nh\u0011\u00121\ni\u0011\u00122\niiR\u001b1\u0012\u000b;\f\nppl;l+h\u0011\u001b1\np\u0011\u0012\npiR\u00121\u00122\u000b;\f\niil;l\u0011\n+1\n2X\n\u0016;\u0017\u000f\u0016\u00121\u00122X\ni6=pA\u0016;\u0017\ni;l\u000f\u0017\u0012\u000b\u0010\nh\u0011\u001b1\np\u0011\u001b2\np\u0011\u00121\ni\u0011\u00122\niiCP\u0012;\f\nl;l+h\u0011\u00121\ni\u0011\u00122\niiR\u001b1\u001b2\u0012;\f\nppl;l\u0011\n+1\n2X\n\u0016;\u0017\u000f\u0017\u00121\u00122X\nr6=lA\u0016;\u0017\np;r\u000f\u0016\u0012\u001b1\u0010\nh\u0011\u0012\np\u0011\u001b2\npiP\u00121\u00122\u000b;\f\nrrl;l+h\r\u00121\nr\r\u00122\nriR\u0012\u001b2\u000b;\f\nppl;l\u0011\n+1\n2X\n\u0016;\u0017\u000f\u0017\u00121\u00122X\nr6=lA\u0016;\u0017\np;r\u000f\u0016\u0012\u001b2\u0010\nh\u0011\u001b1\np\u0011\u0012\npiP\u00121\u00122\u000b;\f\nrrl;l+h\r\u00121\nr\r\u00122\nriR\u001b1\u0012\u000b;\f\nppl;l\u0011\n+1\n2X\n\u0016;\u0017\u000f\u0016\u00121\u00122A\u0016;\u0017\np;l\u000f\u0017\u0012\u000b\u0000\nh\u0011\u00121\np\u0011\u00122\np\u0011\u001b1\np\u0011\u001b2\npi\u0000h\u0011\u001b1\np\u0011\u001b2\npih\u0011\u00121\np\u0011\u00122\npi\u0001\nP\u0012;\f\nl;l\n+1\n2X\n\u0016;\u0017\u000f\u0017\u00121\u00122A\u0016;\u0017\np;l\u0000\n\u000f\u0016\u0012\u001b1h\u0011\u0012\np\u0011\u001b2\npi+\u000f\u0016\u0012\u001b2h\u0011\u001b1\np\u0011\u0012\npi\u0001\u0010\nP\u000b\u00121\u00122;\f\nlll;l\u0000h\r\u00121\nl\r\u00122\nliP\u000b;\f\nl;l\u0011\n!P\u001b1\u001b2\u000b;\f\nrrl;l=\u0000iX\n\u0016\u0001\u0016\u0010\n\u000f\u0016\u0012\u001b1P\u0012\u001b2\u000b;\f\nrrl;l+\u000f\u0016\u0012\u001b2P\u001b1\u0012\u000b;\f\nrrl;l+\u000f\u0016\u0012\u000bP\u001b1\u001b2\u0012;\f\nrrl;l\u0011\n+1\n2X\n\u0016;\u0017\u000f\u0016\u00121\u00122X\niA\u0016;\u0017\ni;r\u000f\u0017\u0012\u001b1\u0010\nh\u0011\u00121\ni\u0011\u00122\niiP\u0012\u001b2\u000b;\f\nrrl;l+h\r\u0012\nr\r\u001b2\nriR\u00121\u00122\u000b;\f\niil;l\u0011\n+1\n2X\n\u0016;\u0017\u000f\u0016\u00121\u00122X\niA\u0016;\u0017\ni;r\u000f\u0017\u0012\u001b2\u0010\nh\u0011\u00121\ni\u0011\u00122\niiP\u001b1\u0012\u000b;\f\nrrl;l+h\r\u001b1\nr\r\u0012\nriR\u00121\u00122\u000b;\f\niil;l\u0011\n+1\n2X\n\u0016;\u0017\u000f\u0016\u00121\u00122\u000f\u0017\u0012\u000bX\niA\u0016;\u0017\ni;l\u0010\nh\u0011\u00121\ni\u0011\u00122\ni\r\u001b1\nr\r\u001b2\nriCP\u0012;\f\nl;l+h\u0011\u00121\ni\u0011\u00122\niiP\u001b1\u001b2\u0012;\f\nrrl;l\u0011\nwith the addition of the exact equation:\nGxyz;\f\nnnn;n (!) =i\n!(\u000ex;\fMx+\u000ey;\fMy+\u000ez;\fMz) (D10)15\nThese equations are exactly solved for the case of the quantum Ising model in the main text. Finally, we solve the\nself-consistency equations numerically, until full convergence. This produces the phase diagram shown in Fig.3."    },    {        "title": "2204.14087v1.Dynamics_of_a_Pair_of_Overlapping_Polar_Bright_Solitons_in_Spin_1_Bose_Einstein_Condensates.pdf",        "content": "Dynamics of a Pair of Overlapping Polar Bright Solitons in Spin-1 Bose-Einstein Condensates\nGautam Hegde, Sandra M Jose, and Rejish Nath\nDepartment of Physics, Indian Institute of Science Education and Research, Dr. Homi Bhabha Road, Pune 411 008, India\nWe analyze the dynamics of both population and spin densities, emerging from the spatial overlap between\ntwo distinct polar bright solitons in Spin-1 Spinor Condensates. The dynamics of overlapping solitons in scalar\ncondensates exhibits soliton fusion, atomic switching from one soliton to another and repulsive dynamics de-\npending on the extent of overlap and the relative phase between the solitons. The scalar case also helps us\nunderstand the dynamics of the vector solitons. In the spinor case, non-trivial dynamics emerge in spatial and\nspin degrees of freedom. In the absence of spin changing collisions, we observe Josephson-like oscillations\nin the population dynamics of each spin component. In this case, the population dynamics is independent of\nthe relative phase, but the dynamics of the spin-density vector depends on it. The latter also witnesses the\nappearance of oscillating domain walls. The pair of overlapping polar solitons emerge as four ferromagnetic\nsolitons irrespective of the initial phase di \u000berence for identical spin-dependent and spin-independent interac-\ntion strengths. But the dynamics of final solitons depends explicitly on the relative phase. Depending on the\nratio of spin-dependent and spin-independent interaction strengths, a pair of oscillatons can also emerge as the\nfinal state. Then, increasing the extent of overlap may lead to the simultaneous formation of both a stationary\nferromagnetic soltion and a pair of oscillatons depending on the relative phase.\nI. INTRODUCTION\nBecause of the spin-dependent interactions, spinor conden-\nsates are ideal for exploring coherent spin-mixing dynamics\n[1–9]. The resulting oscillations in the populations of the Zee-\nman states can be engineered by varying the initial popula-\ntions and the relative phases [10]. In a spin-1 condensate, the\noscillatory dynamics arises because of the collisional inter-\nconversion of two atoms in the m=0 state and one atom each\ninm=1 and m=\u00001 states. Such a spin-changing process\npreserves the net magnetization. Further, high controllability\nover the spin-dynamics can be accessed via external magnetic\nor microwave fields utilizing linear and quadratic Zeeman ef-\nfects.\nSpinor condensates also provide an opportunity to study\nvector solitons [11–17] in quasi-one-dimensional (Q1D)\nBECs, including the bright solitons [18–21]. Vector soli-\ntons are self-trapped wave packets with multiple compo-\nnents. In spin-1 condensates, the bright solitons are classi-\nfied into polar and ferromagnetic ones based on the spin state\nand the expectation value of the time-reversal operator [21].\nCollisional properties of polar-polar, polar-ferromagnetic and\nferromagnetic-ferromagnetic solitons have been studied in the\npast [19, 22] and that later lead to the discovery of an exotic\nsoliton called the oscillaton [21, 23]. Oscillatons are solitons\nwhere the total density profile remains stationary while the\npopulations of each m-components oscillate in time. It is also\npossible to observe oscillatory spin-dynamics in a single soli-\nton [24], and in a pair of colliding solitons [23].\nIn this paper, we analyze the dynamics of a pair of overlap-\nping polar bright solitons in spin-1 spinor condensates. Note\nthat the scenario is di \u000berent from the setups used to study\nsoliton collisions. In the latter case, the solitons are initially\nplaced very far apart and collide against each other with an\ninitial velocity. The dynamics of overlapping optical solitons\nare previously studied both theoretically [25–27] and experi-\nmentally [28–31], but similar studies of matter-wave solitons\nare lacking. Because of that, first, we look at the dynamics\nof overlapping bright solitons in scalar condensates. The lat-ter helps us understand the unique features emerging from the\nvectorial nature of the spinor solitons. The dynamics critically\ndepends on the initial phase di \u000berence and the extent of over-\nlap between the solitons in the scalar case. It is expected that\nthe relative phase plays a vital role since it decides the nature\nof force between solitons [32]. Depending on the phase dif-\nference, the soliton fusion, atomic flow from one soliton to\nanother and repulsive dynamics can occur. Similar scenarios\nare found in the dynamics of overlapping optical solitons in\ndi\u000berent non-linear media [25, 28–30]. In particular, the flow\nof atoms from one soliton to another mimics the phenomenon\nof optical switching, and we term it atomic switching for the\nmatter-wave solitons.\nThe vectorial nature of spinor solitons leads to rich dynam-\nics, both in population and spin densities. Besides the rela-\ntive phase and extent of overlap between the solitons, the ratio\nbetween the spin-dependent and spin-independent interaction\nstrengths also a \u000bects the dynamics. A simplified picture of the\ndynamics is attained using a rotated frame. In the absence of\nspin changing collisions, we observe Josephson-like oscilla-\ntions in the population dynamics of each spin component. For\natoms in each Zeeman state, the Josephson-like oscillations\nare explained using an e \u000bective potential created by the den-\nsity of other components. In this case, the population dynam-\nics are independent of the relative phase, but that of the spin-\ndensity vector depends on it. The dynamics of spin-density\nvector reveals the appearance of oscillating domain walls, de-\npending on the relative phase. When spin-independent and\nspin-dependent interactions are identical, the collision of a\npair of polar solitons results in four ferromagnetic solitons ir-\nrespective of the value of the initial relative phase. Among the\nfour ferromagnetic solitons, each pair exhibits dynamics iden-\ntical to that of scalar solitons that become more apparent in\nthe new rotated frame. When the ratio of spin-dependent and\nspin-independent interactions is half, we see the formation of\na pair of oscillatons. An additional stationary ferromagnetic\nsoliton emerges if the extend of overlap is su \u000eciently large,\ndepending on the relative phase.\nThe paper is structured as follows. In Sec. II, we discussarXiv:2204.14087v1  [cond-mat.quant-gas]  29 Apr 20222\nFigure 1. Dynamics of two overlapping identical bright solitons\nin scalar condensates for di \u000berent relative phase between them for\ng=(2\u0019l2\n?)=\u00002. The separation \u0001 = 10l?for (a)-(c) and \u0001 = 5l?for\n(d)-(f). (a) and (d) for \u001e=0, and in (a) the overlap leads to perpetual\noscillating dynamics between the soliton fusion and the initial con-\nfiguration. In (d) the system did not reverse back completely to the\ninitial configuration. (b) and (e) for \u001e=\u0019=2, an initial atomic flow\nalong the phase gradient leads to asymmetric final solitons. (c) and\n(f) For\u001e=\u0019, the solitons repel each other, but identical in their size\nand shape.\nthe dynamics of overlapping bright solitons in scalar conden-\nsates. In Sec. III, we discuss the dynamics of overlapping po-\nlar solitons in spin-1 condensates. In particular, the Sec. III A\ndiscusses the physical setup of spinor solitons, especially the\ngoverning Hamiltonian, the initial state, time-dependent cou-\npled non-linear Schr ¨odinger equations in the original and the\nrotated frame. In Sec. III B, we classify the dynamics based on\nthe ratio between the spin-dependent and spin-independent in-\nteraction strengths. In particular, the dynamics in the absence\nof spin-changing collisions is discussed in Sec. III B 1. The\ndynamics for identical spin-independent and spin-dependent\ninteractions are discussed in Sec. III B 2 and when the spin-\ndependent interaction strength is half of the spin-independent\ninteraction is discussed in Sec. III B 3. Finally, we summarize\nin Sec. IV.\nII. SCALAR CONDENSATES\nIn the following, we analyze the dynamics of two identical,\noverlapping Q1D bright solitons in scalar condensates for dif-\nferent initial relative phases and overlaps. The analytic formof the initial two-soliton wavefunction is\n (z;t=0)=Ah\nsech (k(z\u0000\u0001=2))+ei\u001esech (k(z+ \u0001=2))i\n;\n(1)\nwhere\u001eis the relative phase, kis the wavenumber, \u0001is the ini-\ntial separation between the solitons, which controls the extent\nof overlap and the normalization constant is\nA=1\n2s\nk\n1+k\u0001cos\u001ecsch k\u0001:\nFor large values of \u0001, the solitons do not overlap, and they re-\nmain at rest, maintaining their size and shape over time. Inter-\nestingly, a tiny spatial overlap between the solitons can trigger\nnon-trivial dynamics, depending critically on the phase di \u000ber-\nence. The nature of soliton interactions, whether being attrac-\ntive or repulsive, also depends on their relative phase. The role\nof relative phase on the collisional dynamics of matter-wave\nsolitons has been a subject of study both experimentally [33]\nand theoretically [34]. Here, we analyze the dynamics mainly\nfor three di \u000berent relative phases: \u001e=0,\u001e=\u0019=2 rad and\n\u001e=\u0019rad, which captures the di \u000berent dynamical scenarios.\nFor the scalar condensates, the dynamics is governed by the\nQ1D non-linear Schr ¨odinger equation,\ni~@\n@t (z;t)=\"\n\u0000~2\n2M@2\n@z2+g\n2\u0019l?2j (z;t)j2#\n (z;t);(2)\nwhere g=4\u0019~2asN=Mwith the s-wave scattering length as<\n0,Nis the total number of particles and l?=p~=m!?is the\nwidth of the transverse harmonic confinement of frequency\n!?. The wavenumber kin Eq. (1) depends on the interaction\nstrength via k=jgj=(2\u0019~!?l4\n?).\nWhen\u001e=0, the overlap leads to the attractive interac-\ntion between the solitons, making them fuse into a single soli-\nton. Not being in its lowest energy state, the fused soliton\ndisentangles back into the initial two-soliton configuration if\nthe overlap is su \u000eciently small [see Fig. 1(a) for \u0001 = 10l?].\nThis process repeats periodically in time, leading to the per-\npetual oscillating dynamics between soliton fusion and the ini-\ntial configuration as shown in Fig. 1(a). As the initial overlap\nbetween the solitons increases (decreasing \u0001), the period of\noscillation gets shorter [see Fig. 2(a) for the time period Tvs\n\u0001]. Strikingly, oscillations in Fig. 1(a) arise in the absence of\nharmonic confinement, making it in high contrast with those\nexhibited by trapped solitons [33, 34]. Beyond a certain over-\nlap or smaller \u0001, the solitons become non-separable after the\ninitial fusion as shown in Fig. 1(d) for \u0001 = 5l?and the final\nsoliton exhibits breathing dynamics because of the extra en-\nergy it carries. These results are found to be similar to that of\noverlapping optical solitons in plasma [25].\nAs the initial relative phase between the solitons increases,\nthey repel each other. For instance, when \u001e=\u0019=2 [see\nFig. 1(b)], not only do the solitons repel, but also there is\na flow of atoms along the direction of phase gradient, i.e.,\nalong@\u001e=@ z, at the initial stage of the dynamics. Similar to\nthe atomic flow, an energy transfer among overlapping opti-\ncal solitons has been observed, leading to optical switching\n[28, 29]. In a similar spirit, the atom transfer from one soliton3\n7 8 910 11 12\n∆/l⊥306090120ω⊥T(a)\n7 8 910 11 12\n∆/l⊥0.00.10.20.3(b) ∆M/M (φ=π/2)\nvω⊥/l⊥(φ=π)\nFigure 2. The soliton properties as a function of \u0001for di \u000berent initial\nrelative phase for the scalar case. (a) The time period of oscillations\nbetween soliton fusion and the initial configuration for \u001e=0. In (b),\nthe solid line depicts the mass di \u000berence \u0001M=Mbetween the final\nsolitons for \u001e=\u0019=2 rad and the dashed line shows the speed of the\nfinal symmetric solitons for \u001e=\u0019rad. For both figures, g=(2\u0019l2\n?)=\n\u00002.\nto another can serve as a control knob to direct the motion of\nmatter waves. Hence we call it atomic switching. In Fig. 1(b),\nthe atomic flow takes place from the positive to negative z-\ndirection. Such a transient atomic current makes the final soli-\ntons asymmetric in density, size and speed. The denser soliton\nmoves faster. The speed of the lighter soliton arises from the\nrecoil of the atoms flown out from the original soliton. To\nquantify the mass asymmetry, we introduce the mass di \u000ber-\nence between the left and right solitons, i.e., \u0001M=Ml\u0000Mr,\nwhere Ml=MR0\n\u00001j (z;t)j2dzandMr=MR1\n0j (z;t)j2dz. In\nFig. 2(b), the solid line shows \u0001Mobtained when the solitons\nare well separated after a su \u000eciently long time. As seen, \u0001M\nincreases with decreasing \u0001because the larger the initial over-\nlap more significant the exchange of particles. The atomic\nswitching is evident in Figs. 1(b) and 1(e), respectively for\n\u0001 =10l?and\u0001 =5l?.\nThe above results imply that there is a net momentum in the\nsystem for\u001e=\u0019=2. To quantify that, we calculate the average\nmomentum of the initial wave function in Eq. (1) and is,\nhpi=4A2(1\u0000k\u0001cothk\u0001)csch k\u0001sin\u001e: (3)\nIn Fig. 3(a), we show hpias a function of \u001efor\u0001 = 10l?\nand\u0001 = 5l?, which oscillates between positive and negative\nvalues as a function of \u001e. The latter indicates that by tuning\nthe relative phase, we can control the direction of the transient\natomic current or the atomic switching. For \u001e=0,hpi=0\nas expected, and for \u001e=\u0019=2 rad, there is a non-vanishing\ninitial momentum leading to the asymmetric dynamics shown\nin Figs. 1(b) and 1(e). For \u001e=\u0019=2 rad,hpiexhibits a non-\nmonotonous behavior as a function of \u0001, exhibiting a local\nmaximum, as shown in Fig. 3(b). At \u0001 =0, the solitons com-\npletely overlap and remain at rest and thus hpi=0. Also, as\n\u0001!1 ,hpiapproaches zero as both solitons become com-\npletely independent.\nThe solitons also repel each other for \u001e=\u0019[see Figs. 1(c)\nand 1(f)], but there is no transient atomic current due to the\nnodal point at z=0. Hence, the final solitons are identical,\n−1 0 1\nφ/π−0.10.00.1/angbracketleftp/angbracketrightl⊥/¯h(a)\n∆ = 5l⊥\n∆ = 10l⊥\n0 10 20\n∆/l⊥0.000.040.080.120.16/angbracketleftp/angbracketrightl⊥/¯h(b)Figure 3. (a) The average momentum hpias a function of \u001efor\u0001 =\n5l?(solid line) and \u0001 = 10l?(dashed line) (b)hpivs\u0001for\u001e=\u0019=2\nrad.g=(2\u0019l3\n?~!?)=\u00002 for both (a) and (b).\nand\u0001M=0. The final solitons propagate with equal and\nopposite velocity, but the speed of each soliton depends on\nthe separation \u0001. The final speed vof the soliton vs \u0001for\n\u001e=\u0019rad is shown as a dashed line in Fig. 2(b). The more\noverlap the initial solitons have, the faster they move. Sum-\nmarizing this section, we see that the initial phase di \u000berence\nand the extent of overlap critically a \u000bect the dynamics of over-\nlapping bright solitons in scalar condensates. The most inter-\nesting feature is the local and transient atomic current in at-\ntractive overlapping condensates due to an e \u000bective repulsion\narising from the phase di \u000berence between the solitons. This\nscenario is identical to optical switching, o \u000bering the possi-\nbility of engineering matter-wave transport via controlling the\nrelative phase.\nIII. SPIN-1 SPINOR CONDENSATES\nA. Setup, model, initial state and rotated frame\nThe system we consider are partially overlapping spin-1\ncondensates, they are described by the Hamiltonian,\nˆH=Z\ndz2666664\u0000~2\n2MX\nmˆ y\nmd2\ndz2ˆ m+1\n2¯c0: ˆn2:+1\n2¯c1:ˆF2:3777775;\n(4)\nwhere Mis the mass of a boson, ¯ c0;1=c0;1=2\u0019l2\n?are the inter-\naction parameters, ˆ mis the field operator which annihilates\na boson of the mth Zeeman state, ˆ n(z)=Pf\nm=\u0000fˆ y\nm(z)ˆ m(z)\nis the total density operator. The components of spin density\noperator are\nˆF\u00172x;y;z(z)=X\nm;m0(f\u0017)mm0ˆ y\nm(z)ˆ m0(z); (5)\nwith f\u0017being the\u0017th component of the spin-1 matrices. The\nsymbol : : denotes the normal ordering that places annihi-\nlation operators to the right of the creation operators. The\nspin-independent and spin-dependent interaction parameters\narec0=(g0+2g2)=3 and c1=(g2\u0000g0)=3, respectively with\ngF=4\u0019~2aFN=mrelated to the scattering length aF=0;2of4\nz=0 zΔ m=1m=0m=-1\nFigure 4. Schematic setup of overlapping bright polar solitons in\nQ1D spin-1 condensates, along z-axis. A polar soliton with popu-\nlation shared among m=\u00061 states on the left (red-shaded) and a\npolar soliton with population solely at m=0 on the right side (blue-\nshaded). The separation \u0001determines the extent of overlap between\nthe two solitons.\nthe total spin-Fchannel. Nis the total number of atoms.\nSince we are interested in bright solitons, we always keep\nc0<0.\nWithin the mean-field theory, the dynamics of the sys-\ntem is described by the quasi-one-dimensional (Q1D) Gross-\nPitaevskii equations (GPEs) [18, 19, 23]\ni~@ 1\n@t=\"\n\u0000~2\n2M@2\n@z2+¯c0n+¯c1Fz#\n 1+c1p\n2F\u0000 0;(6)\ni~@ 0\n@t=\"\n\u0000~2\n2M@2\n@z2+¯c0n#\n 0+¯c1p\n2F+ 1+¯c1p\n2F\u0000 \u00001;(7)\ni~@ \u00001\n@t=\"\n\u0000~2\n2M@2\n@z2+¯c0n\u0000¯c1Fz#\n \u00001+¯c1p\n2F+ 0;(8)\nwhere n(z;t)=P\nmj m(z;t)j2is the total density, F\u0017(z;t)=P\nm;m0 \u0003\nm(f\u0017)mm0 m0, and F\u0006=Fx\u0006iFy. We introduce \r=\n\u0000c1=jc0jas the ratio of spin-dependent and spin-independent\ninteractions. The validity of Eqs. (6)-(8) requires that \u00161D\u001c\n~!?, where!?is the transverse confinement frequency and\n\u00161Dis the chemical potential of the Q1D condensates. We\nsolve Eqs. (6)-(8) numerically to analyze the dynamics [35].\nAt\r=1, Eqs. (6)-(8) represent a completely integrable\nsystem and support N-soliton solutions including two-soliton\nones [18].\nInitial state : The schematic diagram of the initial state is\nshown in Fig. 4, in which two distinct polar solitons overlap\naround z=0. The general solution of a static polar soliton is,\n (z)=r\nk\n2\u001fsech kz; (9)\nwhere k=j¯c0j=(4l?) is the inverse-width or the wavenumber\nof the soliton wavepacket, and \u001fis the spin state, which takes\nthe general form [21, 23],\n\u001f=ei\u001c0BBBBBBBBB@\u00001p\n2sin\u0012\ncos\u0012\n1p\n2ei\u001esin\u00121CCCCCCCCCA; (10)where\u001cis a global phase. The initial wave function of a pair\nof overlapping polar solitons we consider is,\n (z)=A\u0002sech (k(z+ \u0001=2))\u001fl+sech (k(z\u0000\u0001=2))\u001fr\u0003;(11)\nwhere A=p\nk=2 is the normalization constant, and \u001fr=\n(0;ei\u001e1;0)Tand\u001fl=(1;0;ei\u001e2)T=p\n2 are the spin states of the\nsolitons in right and left of z=0, respectively. Note that,\nindividually both polar solitons are degenerate ground state\nsolutions of the time-independent GPEs for ¯ c1>0. Compar-\ning to Eq. (10), \u0012=\u0019=2 for the left soliton and that of the\nright one is \u0012=0. The angles, \u001e1and\u001e2are the initial phases\nof the individual solitons and the extent of overlap between\nthe solitons is again controlled by \u0001. For Eq. (11),hpi=0,\nindependent of the value of \u0001and\u001e1.\nThe spin density vector is a null vector for polar soli-\ntons [18, 19, 21], but once they overlap, it may take a non-\nzero value in the overlapping region. For the initial state in\nEq. (11), the local spin density vector is,\nF(z;t=0)=4A2cos(\u001e1\u0000\u001e2=2)\ncosh 2 kz+cosh k\u0001\u0012\nˆxcos\u001e2\n2+ˆysin\u001e2\n2\u0013\n:\n(12)\nThe vector F(z;t=0) lies in the xy-plane and forms an an-\ngle\u001e2with the x-axis. The orientation is same at every point\nalong the z-axis, and depends only on \u001e2. The relative angle,\n\u001e1\u0000\u001e2=2 determines the magnitude of F(z;t=0) and hence,\nthe net magnetization. Integrating over z, we obtain the net\nspin density vector,\nFT=4A2\u0001cos(\u001e1\u0000\u001e2=2) csch k\u0001\u0012\nˆxcos\u001e2\n2+ˆysin\u001e2\n2\u0013\n;\n(13)\nwhich is a conserved quantity. Without loss of generality, we\nset\u001e2=0, which fixes the direction of FTalong the x-axis.\nIn the following we analyze the dynamics mainly for \u001e1=\n0; \u0019=2 and\u0019rad. For\u001e1=0 and\u001e1=\u0019rad, the initial\nspin-density vector is along the positive and negative x-axis,\nrespectively whereas for \u001e1=\u0019=2 rad, F(z;t=0) vanishes.\nIn the overlapping region, for \r,0, spin-mixing dynamics\ntakes place, in which two atoms in m=0 collisionally convert\ninto one atom each in m=1 and m=\u00001, and vice versa. As\nshown below, the nature of dynamics can be classified based\non the value of \rand\u001e1.\nRotated frame : Since FTis a conserved quantity, we can\nmove to a frame in which the quantization axis is parallel to\nthe direction of FT[24], which provides us with a simplified\npicture for the dynamics. Such a rotated frame is used in [21,\n23] to describe oscillatons, which emerged out of the collision\nbetween a ferromagnetic and a polar soliton. In our case, the\nnew frame is obtained by a rotation of \u0019=2 about the y-axis,\n(\u001f0=ei\u0019fy=2\u001f) i.e., making the quantization axis along the x-\naxis. The relation between the new and old spinor components5\nFigure 5. Dynamics of two overlapping polar bright solitons for \r=\n0,\u0001 = 10l?and ¯c0=~!?=\u00002. (a)-(c) show the dynamics of j 1j2+\nj \u00001j2,j 0j2and the total density n(z;t), respectively. The results are\nindependent of the value of \u001e1.\nare\n 0\n1(z)=1\n2 1(z)+1p\n2 0(z)+1\n2 \u00001(z) (14)\n 0\n0(z)=\u00001p\n2 1(z)+1p\n2 \u00001(z) (15)\n 0\n\u00001(z)= 1\n2 1(z)\u00001p\n2 0(z)+1\n2 \u00001(z)!\n: (16)\nNote that for the setup we consider  1and \u00001are identical\nat any instant, and hence  0\n0(z) completely vanishes. Hence,\nin the new frame, the system e \u000bectively reduces to a two-\ncomponent condensate [36]. The corresponding GPEs are,\ni~@ 0\n\u00061\n@t=\"\n\u0000~2\n2M@2\n@z2+¯c+j 0\n\u00061j2+¯c\u0000j 0\n\u00071j2#\n 0\n\u00061;(17)\nwhere ¯ c\u0006=¯c0\u0006¯c1. The exchange coupling between the two\ncomponents  0\n\u00061is provided by ¯ c0\u0000¯c1. When\r=1, ¯c0\u0000¯c1\nvanishes, indicating that in the rotated frame, the system ef-\nfectively reduces to two independent scalar condensates de-\nscribed by the wave functions  0\n1and 0\n\u00001. Note that  0\n1and\n 0\n\u00001, in general do not exhibit identical dynamics. We use the\nrotated frame to gain insights into the dynamics wherever pos-\nsible.\nB. Dynamics\nA comprehensive study of the dynamics of overlapping po-\nlar solitons as a function of \u0001,\rand\u001e1is a tedious task. We\nrestrict the analysis to \u001e1=0; \u0019=2; \u0019rad,\u0001 = 5l?;10l?and\n0\u0014\r\u00141 and in particular, (i) \r=0, (ii)\r=1 and (iii)\n\r=0:5, which capture the most exciting scenarios.\n−0.3−0.2−0.10.00.1Vm=1\neffω⊥t= 0 ω⊥t= 125 ω⊥t= 160\n−10−5 0 5 10\nz/l⊥−0.3−0.2−0.10.00.1Vm=0\neffFigure 6. E \u000bective potential Vm\ne f fgenerated for mth component by\nother components for \r=0,\u0001 = 10l?and ¯c0=~!?=\u00002. (a) Vm=1\ne f f\nexhibits a double well potential with its local minima and separation\nbetween them varying in time. (a) Vm=0\ne f foscillates in time between a\nsingle minimum and double minima. The results are independent of\nthe value of \u001e1\n1.\r=0\nFirst, we discuss the case for which there are no spin chang-\ning collisions, i.e., \r=0. In this case, the total population in\neach Zeeman component remains constant, and the population\ndynamics is found to be independent of \u001e1. We also found that\nchanging \u0001does not a \u000bect the dynamics qualitatively. The dy-\nnamics for\r=0 and \u0001 = 10l?is shown in Fig. 5. In partic-\nular, Figs. 5(a)-5(c) show the dynamics of j 1j2+j \u00001j2,j 0j2\nand the total density n(z;t), respectively. E \u000bectively, we ob-\nserve Josephson-like oscillations of populations in each com-\nponent [see Figs. 5(a) and 5(b)] between the two regions (left\nand right of z=0) where the solitons are initially placed.\nThe oscillations are non-sinusoidal. The density oscillations\nmay give a false impression that spin-changing collisions are\ntaking place. To demonstrate the Josephson-like oscillations,\nfor each Zeeman state, we introduce the population imbalance\nbetween the left and right sides of z=0, i.e.,\nZm(t)=1\nNm Z0\n\u00001dzj mj2\u0000Z1\n0dzj mj2!\n; (18)\nwhere Nmis the total number of atoms in mth state, which is a\nconstant.\nThe density oscillations of each component can be intu-\nitively understand via an e \u000bective potential created by other\ncomponents. First consider the population dynamics of m=1\nstate, which is shown in Fig. 5(a) (identical for m=\u00001 state\nalso). Since most of the population in m=1 is initially at\nthe left side of z=0, we have Z1(t=0)\u00181. For ¯ c0<0,6\n−101Zmm=±1 m= 0\n0 200 400 600 800 1000 1200\nω⊥t6810δ/l⊥\nFigure 7. Dynamics of (a) the population imbalance Zmand (b) the\nseparation\u000e(t) between the two peaks in the total density \r=0,\n\u0001 =10l?and ¯c0=~!?=\u00002. The results are independent of the value\nof\u001e1.\nthe terms ¯ c0j \u00001j2and ¯c0j 0j2in Eq. (6) form a double well\npotential for atoms occupying m=1 state, see Fig. 6(a) for\nVm=1\ne f f(z;t)=¯c0(j 0j2+j \u00001j2) at di \u000berent instants. A non-\nzero fraction of m=1 atoms in the right region triggers the\ntunnelling from left to right, leading eventually to oscillatory\ndynamics shown in Fig. 5(a). At t=0,Vm=1\ne f fis asymmetric\ninzsince m=0 population on the right side is twice that of\nm=\u00001 state on the left. As time evolves, Vm=1\ne f fgets mod-\nified, in particular, the two minima gets closer to each other\n[see dashed and dotted lines in Fig. 6(a)], which amplifies the\ntunnelling rate. Eventually the potential gets inverted but with\na shorter separation between the minima, as shown by dotted\nline in Fig. 6(a). The potential gets inverted because of the\nswapping of population in m=0 (m=\u00001) from left (right) to\nright (left). By this time, the majority of population in m=1\nstate has already tunnelled to the right. At this points, the re-\nverse dynamic happens, and the system recover to the initial\ndensity configuration. The whole processes then repeats peri-\nodically in time.\nAn identical picture also holds for the population dynamics\nof the m=\u00001 state. In contrast, the population in m=0, ini-\ntially placed in the right side, experiences a di \u000berent dynami-\ncal potential [ Vm=0\ne f f(z;t)=¯c0(j 1j2+j \u00001j2)] from the popula-\ntions of m=\u00061, as shown in Fig. 6(b). As time evolves, the\npotential minimum, initially on the left (solid line), moves to\nthe right via a transient double-well potential and then reverts.\nThem=0 atoms always move towards the potential mini-\nmum in the opposite region leading to the oscillatory dynam-\nics shown in Fig. 5(b). Also, note that m=\u00061 populations\nremain immiscible with the m=0 component for the entire\ntime, except in the overlapping region. The total density is\nFigure 8. Dynamics of spin-density Fx(z;t) for (a)\u001e1=0, (b)\u001e1=\n\u0019=2 rad, and (c) \u001e1=\u0019rad. Other parameters are \r=0,\u0001 = 10l?\nand ¯c0=~!?=\u00002.\ncharacterized by an out-of-phase oscillatory dynamics of the\ntwo peaks, see Fig. 5(c).\nIn Fig. 7(a), we show the dynamics of population imbalance\nZmfor the same dynamics shown in Fig. 5. Figure 7(b) shows\nthe separation ( \u000e) between the two peaks in the total density or\nequivalently the distance between the two minima in Vm=1\ne f f(z;t)\n[see Fig. 5(c)]. At t=0,\u000e= \u0001 and it varies periodically in\ntime. The population dynamics get faster as \u000edecreases and\nvice versa. It is seen in Fig. 7(a) that the population of each\ncomponent in the initial region never vanishes. Thus, there\nare always some atoms in m=\u00061 states on the left region and\nm=0 on the right region, which helps the system periodically\nrecover to initial densities.\nEven though the population dynamics shown in Fig. 5 is in-\ndependent of \u001e1, it a\u000bects the spin density vector F(z;t=0)=\nFx(z;t) ˆxvia Eq. (12) and its dynamics. In Figs. 8(a)-8(c), we\nshow the dynamics of Fx(z;t) for\u001e1=0,\u001e1=\u0019=2 rad, and\n\u001e1=\u0019rad, respectively. Since Fx(z;t)/( 1+ \u00001) \u0003\n0+\n( \u0003\n1+ \u0003\n\u00001) 0, the spin density vector is significant only in the\noverlapping regions. For \u001e1=0,FT=R1\n\u00001Fx(z;t)dzis pos-\nitive, i.e., the e \u000bective spin is pointing along the positive x-\ndirection. At t=0, the spin density vector is along the x-axis\nat every point in the overlapping region. As time progresses,\nwe observe the formation of domains with positive and nega-\ntiveFx. In particular, regions of negative values of Fxemerge\nfrom both the edges of the overlapping region, separated by a\nregion of positive Fx[see Fig. 8(a)]. The domain walls, which\nseparate the regions of positive and negative Fx, move towards\nthe centre, squeezing the region of positive Fxin the middle.\nThe latter results in an increase in spin density at the cen-\ntre. Because of the conservation of FTand energy, they can-\nnot reach beyond a separation. Eventually, the domain walls\nmove back to the edges, and their position oscillates in time.\nFor\u001e1=\u0019rad, the dynamics of F(z;t) is identical to that of\n\u001e1=0 except that positive and negative regions are switched\nas shown in Fig. 8(c).\nThe dynamics of F(z;t) for\u001e1=\u0019=2 rad is drastically dif-\nferent from that of \u001e1=0 and\u001e1=\u0019rad, see Fig. 8(b). For\n\u001e1=\u0019=2 rad, and F(z;0) vanishes and hence FT. As time7\nFigure 9. Dynamics of two overlapping polar bright solitons for \r=\n1,\u0001 = 10l?, ¯c0=~!?=\u00002, and\u001e1=0. (a)-(c) show the dynamics\nofj 1j2+j \u00001j2,j 0j2and the total density n(z;t), respectively. (d)-\n(f) show the dynamics of j 0\n1j2,j 0\n\u00001j2and the total density n0(z;t)=\nj 0\n1j2+j 0\n\u00001j2, respectively. Note that n(z;t)=n0(z;t).\nprogresses, we see simultaneous growth of equal regions of\npositive and negative Fx. The growth happens from the edges\nof the overlapping region, separated by a region of vanishing\nFx. As time evolves, the size of the central region of Fx=0\nshrinks and the regions of Fx,0 grow. After some time, sur-\nprisingly, Fx(z;t) vanishes quite rapidly and then reemerges\nwith opposite polarity. Then, the central region with Fx=0\ngrows until the regions of Fx,0 on either sides diminish.\nThe whole process repeats periodically in time and leads to\nthe butterfly pattern in space-time, as seen in Fig. 8(b). Thus,\nwe have a unique scenario of engineering the spin dynamics\nin spinor condensates leaving the population dynamics unaf-\nfected, by tuning the relative phase \u001e1, in the absence of spin-\nindependent interactions.\n2.\r=1\nThe presence of spin-changing collisions a \u000bects the dy-\nnamics drastically. For \r=1, spin-dependent and spin-\nindependent interactions are of equal strength. In Figs. 9(a)-\n9(c), we show the dynamics of j 1j2+j \u00001j2,j 0j2and the\ntotal density n(z;t), respectively for \r=1,\u0001 = 10l?and\n\u001e1=0. The first visible e \u000bect of spin-changing collision is\nthat all Zeeman components become spatially miscible, lead-\ning to identical density patterns for all Zeeman components at\nFigure 10. Dynamics of spin-density Fx(z;t) for (a)\u001e1=0, (b)\n\u001e1=\u0019=2 rad, and (c) \u001e1=\u0019rad. Other parameters are \r=1\nand\u0001 =10l?.\nlonger times. After a su \u000eciently long time, the initial, over-\nlapping polar solitons converted into four solitons via spin-\nchanging collisions, two each on either side of z=0. Each\nsoliton is characterized by a population ratio of 1 : 2 : 1 among\nthe Zeeman states ( m=\u00001,m=0,m=1) with a spin state\n\u001f=(1;\u0006p\n2;1)T=2, which is a ferromagnetic soliton [21, 23].\nThe spin density vector shown in Fig. 10(a) confirms that the\nfinal solitons are indeed ferromagnetic but it possesses oppo-\nsite directions for the inner and outer solitons. The inner soli-\ntons have Fx(z;t)>0, i.e., the polarization axis is along the\npositive x-axis, whereas the outer ones have the net spin vec-\ntor along the negative x-axis. Thus, we have a scenario where\ntwo overlapping polar solitons are dynamically converted into\nfour ferromagnetic solitons.\nIn the rotated frame [Eqs. (14)-(16)], the scenario becomes\nrelatively simple and also provide additional details on the dy-\nnamics. Recall that for \r=1, the wavefunctions  0\n1(z) and\n 0\n\u00001(z) are decoupled, and e \u000bectively we have two indepen-\ndent scalar condensates. At t=0, the initial states with \u001e1=0\nare,\n 0\n\u00061(z)=p\nk\n2\"sech (k(z\u0000\u0001=2))p\n2\u0006sech (k(z+ \u0001=2))p\n2#\n:(19)\nAs seen in Eq. (19),  0\n1and 0\n\u00001represent two-soliton states,\nidentical to Eq. (1), with relative phase \u001e=0 and\u001e=\u0019\nrad, respectively. Concurrently, comparing the results of\nthe original and the rotated frames in Fig. 9, we see that\nj 0\n1(z;t)j2provides the dynamics of the two inner solitons [see\nFig. 9(d)], andj 0\n\u00001(z;t)j2gives that of the two outer solitons\n[see Fig. 9(e)]. They both match with the dynamics of scalar\nsolitons shown in Fig. 1(a) and Fig. 1(c), respectively for the\ninner and outer solitons. In general, depending on the extend\nof the initial overlap, the inner and outer solitons have di \u000ber-\nent masses. The ratio of masses between the inner and outer\nsolitons can be easily obtained via the wavefunctions in the\nrotated frame as,\nR1\n\u00001j 0\n1j2dz\nR1\n\u00001j 0\n\u00001j2dz=1+k\u0001csch k\u0001\n1\u0000k\u0001csch k\u0001:8\nFigure 11. Dynamics of two overlapping polar bright solitons for\n\r=1,\u0001 = 10l?, ¯c0=~!?=\u00002, and\u001e1=\u0019=2 rad. (a)-(c) show the\ndynamics ofj 1j2+j \u00001j2,j 0j2and the total density n(z;t), respec-\ntively. (d)-(f) show the dynamics of j 0\n1j2,j 0\n\u00001j2and the total density\nn0(z;t)=j 0\n1j2+j 0\n\u00001j2, respectively. Note that n(z;t)=n0(z;t).\nHence, we can control the size of the inner and outer ferro-\nmagnetic solitons by varying \u0001.\nThe population dynamics for \u001e1=\u0019rad is identical to that\nof\u001e1=0 as discussed above, but the spin density of inner and\nouter solitons are now flipped, compare Figs. 10(a) and 10(c).\nFor\u001e1=\u0019=2 rad, population and spin density dynamics are\nqualitatively di \u000berent from the \u001e1=0 case. The population\ndynamics for \u001e1=\u0019=2 rad is shown in Fig. 11 and the corre-\nsponding spin density dynamics is shown in Fig. 10(b). After\nthe initial spin-mixing, the dynamics results in four ferromag-\nnetic solitons and they move away from the centre on either\nside. Due to the initial phase di \u000berence of\u0019=2, it takes a fi-\nnite time to reach a miscible state as seen in Figs. 11(a)-11(c),\nvia a transient oscillatory dynamics. The initial states in the\nrotated frame for \u001e1=\u0019=2 are\n 0\n\u00061(z)=p\nk\n2\"sech (k(z+ \u0001=2))p\n2\u0006isech (k(z\u0000\u0001=2))p\n2#\n(20)\nrepresenting two independent two soliton solutions, but with\nopposite phase gradients. As we know from the scalar case\nfor\u001e=\u0019=2 rad [see Figs. 1(b) and 1(e)], there is a transient\natomic current along the direction of phase gradient. There-\nfore, in 0\n1(z), there is a transient atomic current from left to\nright, and in  0\n\u00001(z), it is from right to left. That results in a\npair of asymmetric solitons in both  0\n1(z) and 0\n\u00001(z). Since\nFigure 12. Dynamics of two overlapping polar bright solitons for \r=\n0:5,\u0001 = 10l?, ¯c0=~!?=\u00002, and\u001e1=0. (a)-(c) show the dynamics\nofj 1j2+j \u00001j2,j 0j2and the total density n(z;t), respectively. (d)-\n(f) show the dynamics of j 0\n1j2,j 0\n\u00001j2and the total density n0(z;t)=\nj 0\n1j2+j 0\n\u00001j2, respectively. Note that n(z;t)=n0(z;t).\nthe denser solitons move faster, the inner solitons are lighter,\ncontrasting the \u001e1=0 case where all solitons are identical. As\nseen in Fig. 10(b), the spin vectors point in opposite directions\namong the inner solitons and the same among the outer ones.\nAgain, by tuning \u0001, we can control the mass ratio between the\ninner and outer solitons, also the frequency of transient oscil-\nlation seen in the initial stage of the dynamics in Figs. 11(a)\nand 11(b).\n3.\r=0:5\nIn the following, we analyze the dynamics for \r=0:5 and\nobserve the emergence of a pair of non-identical oscillatons\npropagating in opposite directions. Oscillatons are solitons\nin which the total density profile remains stationary while the\npopulations in di \u000berent spin components oscillate with a con-\nstant frequency [21, 23]. In Figs. 12(a)-12(c), we show the\ndynamics ofj 1j2+j \u00001j2,j 0j2and the total density n(z;t),\nrespectively for \r=0:5,\u0001 = 10l?and\u001e1=0. In the ini-\ntial stage, spin-mixing leads to oscillatory dynamics. Later,\nthe condensates transform into a pair of independent oscilla-\ntons moving in opposite directions. Unlike the case of \r=1\nwhere we see four final solitons, for \r=0:5, we see only\ntwo final solitons. It is explained later using the wavefunc-\ntions in the new frame. The initial oscillatory dynamics takes9\nFigure 13. (a) The snapshot of the initial state of two polar solitons.\n(b) The snapshot of densities at !?t=220 (c) The density profile\nof the left oscillaton at !?t=420. (d) The density profile of the\nright oscillaton at !?t=445. For all plots \r=0:5,\u0001 = 10l?,\n¯c0=~!?=\u00002, and\u001e1=0. The solid lines shows j 1j2+j \u00001j2, dashed\nlines showj 0j2and dotted-dashed lines show the total density n(z).\nplace between the two configurations shown in Figs. 13(a)\nand 13(b). Figure 13(b) is a completely miscible state with\npeaks ofj \u00061j2coincide with that of j 0j2, which implies that\nthe spin-mixing is taking place. On the oscillatory dynamics,\neach spin component leaves a small trail of atoms on the op-\nposite sides, i.e., m=\u00061 in the right region and m=0 in\nthe left region. Eventually, they cause the formation of a pair\nof oscillatons. Figures 13(c) and 13(d) are density snapshots\ntaken at two di \u000berent instants after the oscillatons are formed.\nThe total density n(z;t) of each oscillaton is identical in both\nfigures, whereas the densities of di \u000berent components vary in\ntime due to the spin-mixing. The total population oscillates\nbetween m=0 and m=\u00061 within each oscillaton. In general,\nthe frequency of internal oscillations of the two oscillatons are\ndi\u000berent and can be tuned via the extent of overlap, i.e. vary-\ning\u0001. Sincehpi=0, the denser oscillaton travels slower than\nthe other.\nBefore the oscillatons are formed, the whole system reaches\na miscible state. The density pattern of this miscible state is\nsusceptible to the initial noise in the system, if any present.\nThe latter can thereby a \u000bect the properties of the final oscil-\nlatons, such as the mass asymmetry, velocities and the fre-\nquency of the internal oscillations. But general qualitative\nfeatures of the dynamics remain the same. Such sensitivity\nto initial noise is not seen in any other cases discussed here.\nFor\u001e1=0, the initial states in the rotated frame are given\nby the two soliton solutions in Eq. (19). For \r=0:5, in the ro-\ntated frame, we have a binary condensate with attractive intra\nFigure 14. Density profiles of the oscillaton in the rotated frame\nat two di \u000berent instants. The solid ( j 0\n1(z)j2) and dashed (j 0\n\u00001(z)j2)\nlines show the the stationary solution obtained by solving Eqs. (21).\nThe dotted line shows the solutions from the dynamics. (a) for the\noscillaton in the left and (b) for the oscillaton in the right. Both snap\nshots are taken at !?t=1000.\nand inter-component interactions governed by the Eq. (17).\nTherefore both  0\n1(z) and 0\n\u00001(z) have the tendency to stick\ntogether in the same spatial regions, and leading to two fi-\nnal oscillatons or solitons. Note that, unlike in the lab frame,\nan oscillaton is characterized by stationary density profile for\neach component in the rotated frame, as seen in Figs. 12(d)-\n12(e). The wavefunctions for a stationary oscillation in the\nnew frame can be written as  0\n\u00061(z)=\u0011\u0006exp[i(\u0016\u0006t+\u001e\u0006)]\n[21, 23], where \u0011\u0006satisfy two coupled ordinary di \u000berential\nequations,\n \n\u0000~2\n2md2\ndz2+(¯c0+¯c1)\u00112\n\u0006+(¯c0\u0000¯c1)\u00112\n\u0007!\n\u0011\u0006=\u0016\u0006\u0011\u0006:(21)\nTaking the values of \u0016\u0006and\u001e\u0006from the numerical results\nand solving Eqs. (21) we obtain the stationary solutions \u0011\u0006.\nThe excellent agreement shown in Fig. 14 confirm us that\nthe final solitons formed in the dynamics are oscillatons. In\nFigs. 14(a) and 14(b), we show the density profiles of  0\n\u00061(z)\nfor the oscillaton from numerics (dotted lines) and the solution\nof Eqs. (21) (solid and dashed lines) at two di \u000berent instants.\nFigure 14(a) is for the left and Fig. 14(b) is for the right oscil-\nlaton.\nInterestingly, for \u001e1=0, increasing the extent of overlap\nor decreasing \u0001introduces a qualitatively new feature to the\ndynamics. Contrary to \u0001 =10l?case, an additional stationary\nferromagnetic soliton is formed, centerd at z=0, see Fig. 15.\nComparing Figs. 15 and 12, we see that decreasing \u0001reduces\nthe frequency of the internal oscillations and increases the ve-\nlocity of the final oscillatons. The ferromagnetic soliton ex-\nhibits a strong breathing character due to its dynamical forma-\ntion.\nFinally, we discuss the case of \u001e1=\u0019=2 rad. Unlike that of\n\u001e1=0, the two final oscillatons are symmetric [see Fig. 16].\nThere is no transient oscillatory dynamics at the initial stage of\nthe dynamics because of repulsion emerging from the initial\nphase di \u000berence of\u0019=2. Also, spin mixing happens indepen-\ndently in the left and right regions due to the initial repulsion.\nConsequently, we have independent and identical oscillatons\nmoving away from each other. Also, the noise does not a \u000bect10\nFigure 15. Dynamics of two overlapping polar bright solitons for\n\r=0:5,\u0001 = 5l?, ¯c0=~!?=\u00002, and\u001e1=0 rad. (a) show the\ndynamics ofj 1j2+j \u00001j2, (b) ofj 0j2and (c) of the total density\nn(z;t), respectively.\nFigure 16. Dynamics of two overlapping polar bright solitons for\n\r=0:5,\u0001 = 10l?, ¯c0=~!?=\u00002, and\u001e1=\u0019=2 rad. (a) show the\ndynamics ofj 1j2+j \u00001j2, (b) ofj 0j2and (c) of the total density\nn(z;t), respectively.\nthe dynamics in contrast to the case of \u001e1=0.IV . SUMMARY AND OUTLOOK\nIn summarizing, we have analyzed the dynamics of two\noverlapping polar bright solitons in spin-1 condensates, which\ndepends critically on the relative phase, the extent of overlap\nand the ratio between spin-independent and spin-dependent\ninteractions. The same dynamics of scalar solitons revealed\ninteresting scenarios, particularly the possibility of observing\natomic switching. Atomic switching can find applications in\nimplementing atom-based networks, identical to optical net-\nworks. Overlapping polar solitons resulted in non-trivial dy-\nnamics in spatial and spin degrees of freedom. For vanishing\nspin-dependent interactions, we observed Josephson like os-\ncillations of each component in an e \u000bective double-well po-\ntential created by the density of other components. For identi-\ncal spin-dependent and spin-independent interactions, we ob-\nserved the formation of four ferromagnetic solitons. In the\nlast case, when the spin-dependent interaction strength is half\nof the spin-independent one, the dynamics led to the forma-\ntion of two oscillatons. Strikingly, the properties of the final\nbright solitons can be easily tuned by the initial state and the\ninteraction parameters. Our studies o \u000ber a new possibility for\nengineering matter waves. Extending the above analysis to a\npair of overlapping ferromagnetic solitons and ferromagnetic-\npolar solitons may lead to exciting and completely new dy-\nnamics than those reported here. We also expect complex dy-\nnamics to emerge from overlapping more than two solitons.\nV . ACKNOWLEDGMENTS\nR.N. acknowledges support from DST-SERB for the Swar-\nnajayanti fellowship File No. SB /SJF/2020-21 /19. We ac-\nknowledge National Supercomputing Mission (NSM) for pro-\nviding computing resources of ’PARAM Brahma’ at IISER\nPune, which is implemented by C-DAC and supported by the\nMinistry of Electronics and Information Technology (MeitY)\nand Department of Science and Technology (DST), Govern-\nment of India. G. H. acknowledges the funding from DST\nIndia through an INSPIRE scholarship.\n[1] Y . Kawaguchi and M. Ueda, Phys. Rep. 520, 253 (2012).\n[2] M.-S. Chang, Q. Qin, W. Zhang, L. You, and M. S. Chapman,\nNature Physics 1, 111 (2005).\n[3] J. Kronj ¨ager, C. Becker, P. Navez, K. Bongs, and K. Sengstock,\nPhys. Rev. Lett. 97, 110404 (2006).\n[4] A. T. Black, E. Gomez, L. D. Turner, S. Jung, and P. D. Lett,\nPhys. Rev. Lett. 99, 070403 (2007).\n[5] Y . Liu, S. Jung, S. E. Maxwell, L. D. Turner, E. Tiesinga, and\nP. D. Lett, Phys. Rev. Lett. 102, 125301 (2009).\n[6] C. Klempt, O. Topic, G. Gebreyesus, M. Scherer, T. Henninger,\nP. Hyllus, W. Ertmer, L. Santos, and J. J. Arlt, Phys. Rev. Lett.\n103, 195302 (2009).\n[7] Y . Eto, H. Shibayama, H. Saito, and T. Hirano, Phys. Rev. A\n97, 021602 (2018).[8] Z. Chen, T. Tang, J. Austin, Z. Shaw, L. Zhao, and Y . Liu, Phys.\nRev. Lett. 123, 113002 (2019).\n[9] B. Evrard, A. Qu, J. Dalibard, and F. Gerbier, Phys. Rev. A\n103, L031302 (2021).\n[10] H. Pu, C. K. Law, S. Raghavan, J. H. Eberly, and N. P. Bigelow,\nPhys. Rev. A 60, 1463 (1999).\n[11] T. M. Bersano, V . Gokhroo, M. A. Khamehchi, J. D’Ambroise,\nD. J. Frantzeskakis, P. Engels, and P. G. Kevrekidis, Phys. Rev.\nLett. 120, 063202 (2018).\n[12] X. Chai, D. Lao, K. Fujimoto, R. Hamazaki, M. Ueda, and\nC. Raman, Phys. Rev. Lett. 125, 030402 (2020).\n[13] A. Farolfi, D. Trypogeorgos, C. Mordini, G. Lamporesi, and\nG. Ferrari, Phys. Rev. Lett. 125, 030401 (2020).\n[14] S. Lannig, C.-M. Schmied, M. Pr ¨ufer, P. Kunkel, R. Strohmaier,\nH. Strobel, T. Gasenzer, P. G. Kevrekidis, and M. K.11\nOberthaler, Phys. Rev. Lett. 125, 170401 (2020).\n[15] I.-K. Liu, S.-C. Gou, and H. Takeuchi, Phys. Rev. Research 2,\n033506 (2020).\n[16] X. Chai, D. Lao, K. Fujimoto, and C. Raman, Phys. Rev. Re-\nsearch 3, L012003 (2021).\n[17] L.-Z. Meng, Y .-H. Qin, and L.-C. Zhao, Communications\nin Nonlinear Science and Numerical Simulation 109, 106286\n(2022).\n[18] J. Ieda, T. Miyakawa, and M. Wadati, Phys. Rev. Lett. 93,\n194102 (2004).\n[19] J. Ieda, T. Miyakawa, and M. Wadati, J. Phys. Soc. Jpn. 73,\n2996 (2004).\n[20] L. Li, Z. Li, B. A. Malomed, D. Mihalache, and W. M. Liu,\nPhys. Rev. A 72, 033611 (2005).\n[21] P. Sza ´nkowski, M. Trippenbach, E. Infeld, and G. Rowlands,\nPhys. Rev. A 83, 013626 (2011).\n[22] M. Wang, B. Tian, W.-R. Shan, X. L ¨u, and Y .-S. Xue, Nonlin-\near Dyn. 69, 1137 (2012).\n[23] P. Szankowski, M. Trippenbach, E. Infeld, and G. Rowlands,\nPhys. Rev. Lett. 105, 125302 (2010).\n[24] W. Zhang, O. E. M ¨ustecaplıo ˘glu, and L. You, Phys. Rev. A 75,\n043601 (2007).[25] V . Saxena, I. Kourakis, G. Sanchez-Arriaga, and E. Siminos,\nPhys. Lett. A 377, 473 (2013).\n[26] V . V . Afanasjev and V . A. Vysloukh, J. Opt. Soc. Am. B 11,\n2385 (1994).\n[27] V . V . Afanasjev and N. Akhmediev, Phys. Rev. E 53, 6471\n(1996).\n[28] M. Shalaby and A. Barthelemy, Opt. Lett. 16, 1472 (1991).\n[29] M. Shalaby, F. Reynaud, and A. Barthelemy, Opt. Lett. 17, 778\n(1992).\n[30] J. S. Aitchison, A. M. Weiner, Y . Silberberg, D. E. Leaird, M. K.\nOliver, J. L. Jackel, and P. W. E. Smith, Opt. Lett. 16, 15 (1991).\n[31] S. G. I. and S. Mordechai, Science 286, 1518 (1999).\n[32] U. Al Khawaja, Phys. Rev. E 81, 056603 (2010).\n[33] J. H. V . Nguyen, P. Dyke, D. Luo, B. A. Malomed, and R. G.\nHulet, Nat. Phys. 10, 918 (2014).\n[34] T. P. Billam, S. L. Cornish, and S. A. Gardiner, Phys. Rev. A\n83, 041602 (2011).\n[35] L. M. Symes, R. I. McLachlan, and P. B. Blakie, Phys. Rev. E\n93, 053309 (2016).\n[36] K. L. Lee, N. B. Jørgensen, I.-K. Liu, L. Wacker, J. J. Arlt, and\nN. P. Proukakis, Phys. Rev. A 94, 013602 (2016)."    },    {        "title": "2109.06872v2.The_implications_of_high_BH_spins_on_the_origin_of_BH_BH_mergers.pdf",        "content": "Draft version October 12, 2021\nPreprint typeset using L ATEX style emulateapj v. 12/16/11\nTHE IMPLICATIONS OF HIGH BH SPINS ON THE ORIGIN OF BH-BH MERGERS\nA. Olejak1, K. Belczynski1\n1Nicolaus Copernicus Astronomical Center, Polish Academy of Sciences, ul. Bartycka 18, 00-716 Warsaw, Poland,\n(aleksandra.olejak@wp.pl,chrisbelczynski@gmail.com)\nDraft version October 12, 2021\nABSTRACT\nThe LIGO/Virgo collaboration has reported 50 black hole|black hole (BH-BH) mergers and 8\ncandidates recovered from digging deeper into the detectors noise. The majority of these mergers\nhave low e\u000bective spins pointing toward low BH spins and e\u000ecient angular momentum transport\n(AM) in massive stars as proposed by several models (e.g., the Tayler-Spruit dynamo). However,\nout of these 58 mergers, 7 are consistent with having high e\u000bective spin parameter ( \u001fe\u000b>0:3).\nAdditionally, 2 events seem to have high e\u000bective spins sourced from the spin of the primary (more\nmassive) BH. These particular observations could be used to discriminate between the isolated binary\nand dynamical formation channels. It might seem that high BH spins point to a dynamical origin if\nAM in stars is e\u000ecient and forms low-spinning BHs. In such a case dynamical formation is required\nto produce second and third generations of BH-BH mergers with typically high-spinning BHs. Here\nwe show, however, that isolated binary BH-BH formation naturally reproduces such highly spinning\nBHs. Our models start with e\u000ecient AM in massive stars that is needed to reproduce the majority\nof BH-BH mergers with low e\u000bective spins. Later, some of the binaries are subject to a tidal spin-up\nallowing the formation of a moderate fraction ( \u001810%) of BH-BH mergers with high e\u000bective spins\n(\u001fe\u000b&0:4\u00000:5). In addition, isolated{binary evolution can produce a small fraction of BH-BH\nmergers with almost maximally spinning primary BHs. Therefore, the formation scenario of these\natypical BH-BH mergers remains to be found.\nSubject headings: stars: black holes, compact objects, massive stars\n1.INTRODUCTION\nThe LIGO/Virgo collaboration has announced detec-\ntion of gravitational waves from \u001850 double black\nhole (BH-BH) mergers (Abbott et al. 2019a,b; Fish-\nbach & Holz 2020; Abbott et al. 2021a). Additional\n8 BH-BH merger candidates have been recently re-\nported (Abbott et al. 2021b). The majority of all\nthese events have low e\u000bective spins parameters: \u001fe\u000b=\nm1a1cos\u00121+m2a2cos\u00122\nm1+m2\u00190;wheremiare BH masses,\nai=cJi=Gm2\niare dimensionless BH spin magnitudes\n(Jibeing the BH angular momentum (AM), cthe speed\nof light,Gthe gravitational constant), and \u0012iare angles\nbetween the individual BH spins and the system orbital\nAM.\nHowever, among the dectections there are also several\nBH-BH mergers which are characterized by higher (non-\nzero) positive e\u000bective spins. In Table 1 we list the pa-\nrameters of the \fve BH-BH mergers with highest e\u000bective\nspins reported by Abbott et al. (2021a) with additional\ntwo high e\u000bective spin systems reported by Abbott et al.\n(2021b).\nThe formation of close BH-BH systems is an open issue\nwith several formation channels proposed and discussed\nin the context of the LIGO/Virgo mergers. The major\nformation scenarios include the isolated binary evolution\n(Bond & Carr 1984; Tutukov & Yungelson 1993; Lipunov\net al. 1997; Voss & Tauris 2003; Belczynski et al. 2010b;\nDominik et al. 2012; Kinugawa et al. 2014; Hartwig et al.\n2016; de Mink & Mandel 2016; Mandel & de Mink 2016;\nMarchant et al. 2016; Spera et al. 2016; Belczynski et al.\n2016a; Eldridge & Stanway 2016; Woosley 2016; van den\nHeuvel et al. 2017; Stevenson et al. 2017; Kruckow et al.TABLE 1\nBH-BH mergers with high effective spins\nNo. Namea\u001fe\u000bm1m2a1\n1 GW190517 0 :52+0:19\n\u00000:1937:4+11:7\n\u00007:625:3+7:0\n\u00007:3{\n2 GW170729 0 :37+0:21\n\u00000:2550:2+16:2\n\u000010:234:0+9:1\n\u000010:1{\n3 GW190620 0 :33+0:22\n\u00000:2557:1+16:0\n\u000012:735:5+12:2\n\u000012:3{\n4 GW190519 0 :31+0:20\n\u00000:2266:0+10:7\n\u000012:040:5+11:0\n\u000011:1{\n5 GW190706 0 :28+0:26\n\u00000:2967:0+14:6\n\u000013:338:2+14:6\n\u000013:3{\n6 GW190403 0 :70+0:15\n\u00000:2788:0+28:2\n\u000032:922:1+23:8\n\u00009:00:92+0:07\n\u00000:22\n7 GW190805 0 :35+0:30\n\u00000:3648:2+17:5\n\u000012:532:0+13:4\n\u000011:40:74+0:22\n\u00000:60\na: Names are abbreviated. We include candidate detections\nas full astrophysical events. Parameters of \frst 5 events are\nfrom original LIGO/Virgo analysis (Abbott et al. 2021a), while\nthe remaining 2 are from deeper search into the detectors noise\n(Abbott et al. 2021b).\n2018; Hainich et al. 2018; Marchant et al. 2018; Spera\net al. 2019; Neijssel et al. 2019; du Buisson et al. 2020;\nBavera et al. 2020, 2021; Qin et al. 2021), the dense stel-\nlar system dynamical channel (Portegies Zwart & McMil-\nlan 2000; Miller & Hamilton 2002b,a; Portegies Zwart\net al. 2004; G ultekin et al. 2004, 2006; O'Leary et al.\n2007; Sadowski et al. 2008; Downing et al. 2010; An-\ntonini & Perets 2012a; Benacquista & Downing 2013;\nMennekens & Vanbeveren 2014; Bae et al. 2014; Chat-\nterjee et al. 2016; Mapelli 2016; Hurley et al. 2016; Ro-\ndriguez et al. 2016; VanLandingham et al. 2016; Askar\net al. 2017; Arca-Sedda & Capuzzo-Dolcetta 2017; Sam-\nsing 2017; Morawski et al. 2018; Banerjee 2018; Di Carlo\net al. 2019; Zevin et al. 2019; Rodriguez et al. 2018;\nPerna et al. 2019; Kremer et al. 2020), isolated multi-arXiv:2109.06872v2  [astro-ph.HE]  11 Oct 20212\nple (triple, quadruple) systems (Antonini et al. 2017;\nSilsbee & Tremaine 2017; Arca-Sedda et al. 2018; Liu &\nLai 2018; Fragione & Kocsis 2019), mergers of binaries in\ngalactic nuclei (Antonini & Perets 2012b; Hamers et al.\n2018; Hoang et al. 2018; Fragione et al. 2019) and pri-\nmordial BH formation (Sasaki et al. 2016; Green 2017;\nClesse & Garc\u0013 \u0010a-Bellido 2017; Carr & Silk 2018; De Luca\net al. 2020).\nBH spins and their orientations can play an important\nrole in distinguishing between various BH-BH formation\nmodels. If the BH spins are not small, then their orienta-\ntion may possibly distinguish between a binary evolution\norigin (predominantly aligned spins) and dynamical for-\nmation channels (more or less isotropic distribution of\nspin orientations). If the BHs formed out of stars have\nsmall spins (Spruit 2002; Zaldarriaga et al. 2017; Ho-\ntokezaka & Piran 2017; Fuller et al. 2019; Qin et al. 2019;\nOlejak et al. 2020; Bavera et al. 2020; Belczynski et al.\n2020) then BH-BH mergers with high e\u000bective spins may\nchallenge their isolated evolution origin. In dense stellar\nclusters, BHs may merge several times easily producing\nBHs with high spins and making a dynamical channel a\nprime site for such events (Gerosa & Berti 2017; Fishbach\net al. 2017). However, the assumption about the BH na-\ntal spin (and the AM transport e\u000eciency) also plays a\nrole in the e\u000bective spin distribution for the dynamical\nchannel (Banerjee 2021).\nIn this study we show that the current understanding\nof stellar/binary astrophysics (Belczynski et al. 2021) and\nthe degeneracy between the di\u000berent formation channels\ndo not allow for such a simple test of the origin of the\nLIGO/Virgo BH-BH mergers. To demonstrate this we\nshow that although the isolated binary evolution channel\nproduces mostly BH-BH mergers with low e\u000bective spins,\na small but signi\fcant fraction of mergers is expected to\nhave moderate or even high e\u000bective spins. Despite the\nassumption that stars slow down their rotation due to\ne\u000ecient AM transport, we \fnd that tidal interactions\nare capable of spinning up some stars allowing formation\nof rapidly spinning BHs (Detmers et al. 2008; Kushnir\net al. 2017; Qin et al. 2018).\n2.METHOD\nWe use the population synthesis code StarTrack (Bel-\nczynski et al. 2002, 2008) with a model of star forma-\ntion rates and metallicity distribution based on Madau\n& Dickinson (2014) described in Belczynski et al. (2020).\nWe employ the delayed core-collapse supernova (SN) en-\ngine for neutron star/BH mass calculation (Fryer et al.\n2012), with weak mass loss from pulsation pair insta-\nbility supernovae (Belczynski et al. 2016b). BH na-\ntal kicks are calculated from a Maxwellian distribution\nwith\u001b= 265 km s\u00001and decreased by fallback during\ncore-collapse; this makes a signi\fcant fraction of BHs\nform without a natal kick (Mirabel & Rodrigues 2003).\nWe assume our standard wind losses for massive O/B\nstars (Vink et al. 2001) and LBV winds (speci\fc pre-\nscriptions for these winds are listed in Sec. 2.2 of Bel-\nczynski et al. 2010a). BH natal spins are calculated under\nthe assumption that AM in massive stars is transported\nby the Tayler-Spruit magnetic dynamo (Spruit 2002) as\nadopted in the MESA stellar evolutionary code (Paxton\net al. 2015). Such BH natal spins take values in the range\na20:05\u00000:15 (see Belczynski et al. 2020). Note thatthe modi\fed classic Tayler-Spruit dynamo with a di\u000ber-\nent non-linear saturation mechanism of the Tayler insta-\nbility (Fuller et al. 2019; Fuller & Ma 2019) causes larger\nmagnetic \feld amplitudes, more e\u000ecient AM transport\nand even lower \fnal natal spins ( a\u00180:01). BH spin\nmay be increased if the immediate BH progenitors (Wolf-\nRayet: WR) stars in close binaries are subject to tidal\nspin-up. In our calculations for BH-WR, WR-BH and\nWR-WR binary systems with orbital periods in the range\nPorb= 0:1\u00001:3 d the BH natal spin magnitude is \ft from\nWR star spun-up MESA models (see eq.15 of Belczyn-\nski et al. (2020)), while for systems with Porb<0:1d\nthe BH spin is taken to be equal to a= 1. BH spins\nmay also be increased by accretion in binary systems.\nWe treat accretion onto a compact object during Roche\nlobe over\row (RLOF) and from stellar winds using the\nanalytic approximations presented in King et al. (2001)\nand Mondal et al. (2020). In the adopted approach the\naccumulation of matter on a BH is very ine\u000ecient so\naccretion does not noticeably a\u000bect the \fnal BH spin.\nHowever note that, e.g. van Son et al. (2020) or Bavera\net al. (2021) tested di\u000berent super Eddington accretion\nprescriptions \fnding that some BHs may be signi\fcantly\nspun-up by accretion.\nFor common the envelope (CE) evolution we assume\na 100% (\u000bCE= 1) orbital energy transfer for CE ejec-\ntion and we adopt 5% Bondi accretion rate onto the BHs\nduring CE (Ricker & Taam 2008; MacLeod & Ramirez-\nRuiz 2015; MacLeod et al. 2017). During the stable\nRLOF (whether it is a thermal- or nuclear-timescale mass\ntransfer: TTMT/NTMT) we adopt the following input\nphysics. If an accretor is a compact object (neutron star\nor BH) we allow for super-Eddington accretion with ex-\ncess transferred mass lost with an AM speci\fc to the\naccretor (Mondal et al. 2020). In all other cases, we al-\nlow a fraction of the transferred mass of fa= 0:5 to be\nlost from the binary with a speci\fc AM of the binary\norbitjloss= 1:0 (expressed in units of 2 \u0019A2=Porb,Abe-\ning an orbital separation; see eq. 33 of Belczynski et al.\n(2008)).\nRLOF stability is an important issue in the context of\nBH-BH system formation in the framework of the iso-\nlated binary evolution (Neijssel et al. 2019; Olejak et al.\n2021; Gallegos-Garcia et al. 2021; Belczynski et al. 2021).\nIn the standard StarTrack evolution we impose rather\nliberal limits for CE (dynamical-timescale RLOF) to de-\nvelop (see Belczynski et al. (2008): binaries with a donor\nstar more massive than 2 \u00003 times the mass of the ac-\ncretor are subject to CE. In this model (for simplicity\ntagged here as CE model) the vast majority of BH-BH\nmergers form through CE evolution, although we \fnd\nsome cases ( .1%) of BH-BH merger formation without\nany CE event. In the alternative model (non-CE model,\ndetailed description in Olejak et al. 2021) we allow CE\nto be suppressed for some systems even with mass ra-\ntio as high as 6 \u00008 (Pavlovskii et al. 2017). In this\nmodel the majority of the BH-BH mergers form without\nany CE event (the orbital decrease is obtained through\nangular momentum loss during stable RLOF), although\nsome (<10%) BH-BH mergers form with the assistance\nof CE.\nFor each model we calculate the evolution of 64 mil-\nlion massive, Population I/II binary systems. We use\nthe star formation history and chemical evolution of the3\nUniverse to obtain the BH-BH merger properties within\nan approximate reach of LIGO/Virgo (redshift z < 1).\nWe use the same method as described in Belczynski et al.\n(2020).\n3.RESULTS\nFigure 1 shows a typical example of binary system evo-\nlution without a CE phase leading to the formation a\nBH-BH merger with a tidally spun-up primary BH (re-\nstricted RLOF stability criteria; Olejak et al. 2021). The\nrather unequal-mass massive stellar system (112 M\fand\n68M\f) with a metallicity of Z= 0:002 goes through two\nRLOF events. The RLOF I is initiated by the more mas-\nsive star; \frst by an NTMT when the donor is still on\nthe main-sequence and then through a TTMT when the\ndonor evolves o\u000b main-sequence. After the RLOF I, the\nsystem mass ratio is reversed: the initially more mas-\nsive star lost over 80% of its mass while the companion\ngained\u001840M\f. Next, the initially more massive star\nends its evolution directly collapsing to the less{massive\n(secondary) BH with a mass of m2= 15M\fand spin\na2= 0:14. When the companion star expands, it ini-\ntiates a second stable RLOF. At the onset of RLOF II\nthe system has highly unequal masses: the donor is al-\nmost 6:5 times more massive than the BH. The thermal\ntimescale for a donor with mass Mdon\u001997M\f, radius\nRdon\u0019300R\fand luminosity Ldon\u00193\u0002106L\f(pa-\nrameters at the RLOF II onset1) calculated with the\nformula by Kalogera & Webbink (1996), is \u001cth\u0019330\nyr:It corresponds to a very high mass transfer rate\n_M=Mdon=\u001cth\u00190:3 M\fyr\u00001which does not allow the\nBH to accrete much mass (despite the fact that we allow\nfor super-Eddington accretion). Half of the donors mass\nis lost from the binary with the speci\fc AM of the BH (as\nthe matter was transferred to the vicinity of the BH ac-\ncretor). This has a huge e\u000bect on the orbital separation\nwhich decreases from A= 467R\fto onlyA= 7:1R\f.\nAfter RLOF II the binary consists of a BH and a WR star\nthat are close enough to allow for the tidal spin-up of the\nWR star. Finally, the WR star directly collapses to the\nmore massive (primary) BH with a mass m1= 36M\f\nand spina1= 0:68. The BH-BH system merges after\n\u001867 Myr.\nFigure 2 shows a typical CE evolution scenario (stan-\ndard StarTrack RLOF stability criteria) leading to the\nformation a BH-BH merger with both BHs spun-up by\ntidal interactions. At the beginning, the binary system\nof two\u001836M\fstars with Z= 0:0025 is on a wide\n(A\u00191340R\f) and eccentric orbit ( e= 0:1). When\nthe initially more massive star expands the system goes\nthrough a stable RLOF, after which the donor looses\nits H-rich envelope and the orbit circularizes. Soon af-\nter RLOF I, the system goes through another (unstable)\nRLOF initiated by the initially less massive companion\nstar. The ensuing CE evolution leads to signi\fcant or-\nbital contraction from A= 3100R\ftoA= 4:5R\fand\nleaves two WR stars subject to strong tidal interactions.\nBoth stars end their evolution at a similar time form-\ning via supernovae explosions two \u00189M\fBHs. At\nthe formation, both BHs get signi\fcant natal kicks that\n1Such parameter values are inline with other predictions for\nmassive stars e.g. using Geneva stellar evolution code (Yusof et al.\n2013).\nFig. 1.| Typical example of non-CE evolutionary scenario lead-\ning to the formation of BH-BH merger with tidally spun-up pri-\nmary:a1= 0:68 and\u001fe\u000b= 0:52. Binary system goes through two\nphases of RLOF with episodes of nuclear and thermal timescale\nmass transfer. RLOF I ends with the system mass ratio reversal.\nAfter RLOF II the system orbital separation signi\fcantly decreases\nand WR star is a subject to tidal spin-up by a BH. Soon there-\nafter the close BH-BH system is formed with a short merger time\nof\u001867 Myr (see Sec. 3).\nmakes the system orbit larger A\u001919R\fand eccentric\ne= 0:44, leading to a merger time of \u00186:7 Gyr.\nIn Table 2 we present the statistical spin proper-\nties of BH-BH systems merging at redshifts z < 1\nfor the two tested RLOF stability criteria models. In\nthe rows 1\u00006 we list the percentage of the BH-BH\nmergers with e\u000bective{spin parameter values \u001fe\u000b>\n0:0;0:1;0:2;0:3;0:4;0:5. In the rows 7\u00009 we list the\npercentages of BH-BH mergers with a highly spinning\nprimary BH a1>0:5;0:7;0:9 while the rows 10 \u000012\ngive the percentages of mergers with a highly spinning\nsecondary BH a2>0:5;0:7;0:9. The full distribution of\nthe primary{spin, the secondary{spin and the e\u000bective{\nspin parameter for both the CE and non-CE evolution,\nis plotted in Figure 3 in APPENDIX A.\n4.DISCUSSION AND CONCLUSIONS\nThe rapidly increasing number of detected BH-BH\nmergers does allow for some general population state-\nments (Roulet et al. 2021; Galaudage et al. 2021; Abbott\net al. 2021b). It appears that (i)majority (\u001870\u000090%) of\nBH-BH mergers have low e\u000bective spins consistent with\n\u001fe\u000b\u00190 and that (ii)small fraction (\u001810\u000030%) of\nmergers have positive non-zero spins that can be as high\nas\u001fe\u000b&0:5. Additionally, the population is consistent\nwith (iii)no systems having negative e\u000bective spins and4\nFig. 2.| Typical example of evolutionary scenario with CE\nphase leading to the formation of BH-BH merger with a1= 0:79,\na2= 0:79 and\u001fe\u000b= 0:77. First, the binary system goes through\nstable RLOF phase with episodes of nuclear and thermal timescale\nmass transfer initiated by the initially more massive star. Then\ninitially less massive star expands and initiates CE, after which\nthe orbital separation is signi\fcantly decreased. After CE, binary\nhosts two compact WR stars that are subject to tidal spin-up.\nBoth stars explode as supernovae and form BHs on eccentric orbit\nwith merger time of \u00186:7 Gyr (see Sec. 3).\nTABLE 2\nPredictions for BH-BH mergers from binary evolution\nNo. conditionaCE model non-CE model\n1\u001fe\u000b>0:0 97% 93%\n2\u001fe\u000b>0:1 95% 85%\n3\u001fe\u000b>0:2 70% 60%\n4\u001fe\u000b>0:3 36% 39%\n5\u001fe\u000b>0:4 10% 21%\n6\u001fe\u000b>0:5 2% 7%\n7a1>0:5 3% 34%\n8a1>0:7 2% 15%\n9a1>0:9 1% 1%\n10a2>0:5 52% 11%\n11a2>0:7 33% 7%\n12a2>0:9 12% 2%\na: We list fractions of BH-BH mergers (redshift z<1) produced in\nour two population synthesis models satisfying a given condition.(iv)a not isotropic distribution of e\u000bective spins (which\ncould indicate dynamical origin). Finally, (v)there is\nat least one case of a primary BH (more massive) in a\nBH-BH merger with very high spin ( a1>0:7 at 90%\ncredibility). These properties are noted to be broadly\nconsistent with BH-BH mergers being formed in an iso-\nlated binary evolution.\nIn our study we have tested whether we can reproduce\nthe above spin characteristics with our binary evolution\nmodels that employ e\u000ecient AM transport in massive\nstars and that impose tidal spin-up of compact massive\nWolf-Rayet stars in close binaries. The two presented\nmodels employ our standard input physics but allow for\nthe formation of BH-BH mergers assisted either by a CE\nor by a stable RLOF. We \fnd that the observed popula-\ntion and its spin characteristics ( i{v) is consistent with\nour isolated{binary{evolution predictions (see Tab. 2).\nIn particular, we \fnd that the majority of BH-BH merg-\ners have small positive e\u000bective spins: \u001870% mergers\nhave 0< \u001f e\u000b<0:3 (e\u000ecient AM transport), while a\nsmall fraction have signi\fcant spins: 36 \u000039% mergers\nhave\u001fe\u000b>0:3 and 2\u00007% mergers have \u001fe\u000b>0:5 (tidal\nspin-up). The fraction of systems with negative e\u000bective\nspins is small (3\u00007%) as most BHs do not receive strong\nnatal kicks in our simulations. Individual BH spins can\nreach high values. A large fraction (11 \u000052%) of sec-\nondary BHs may have signi\fcant spin values ( a2>0:5)\nas it is the less massive stars that are most often sub-\nject to tidal spin-up. Nevertheless, primary BHs may\nalso form with high spins (3 \u000034% witha1>0:5) if\nboth stars have similar masses and both are subject to\ntidal spin-up (see Fig. 2) or due to mass ratio reversal\ncaused by the RLOF (see Fig. 1). We also note the for-\nmation of a small fraction of almost maximally spinning\nBHs: 2\u000012% fora2>0:9 (secondary BH) and 1% for\na1>0:9 (primary BH). These results on e\u000bective spins\nand individual BH spins are consistent with the current\nLIGO/Virgo population of BH-BH mergers. Note that\nQin et al. (2021) came to di\u000berent conclusions, \fnding\nthe high-spinning detections challenging for the Tayler-\nSpruit dynamo, especially for the unequal mass event\nwith a high spinning primary (GW190403). Our non-CE\nmodel reproduces this type of mergers due to the mass\nratio reversal (see Fig. 1). In this channel, at the onset\nof the second stable RLOF, the donor may be even 5-6\ntimes more massive than the accretor, ending as an un-\nequal mass ( q\u00140:4) BH-BH merger. Qin et al. (2021)\nhave not considered the case of a stable RLOF in such\nunequal mass systems.\nThe above fractions correspond to just two di\u000berent\nmodes of spinning-up during the classical isolated binary\nBH-BH formation. Had we varied several other factors\nthat in\ruence BH spins and their orientations in BH-\nBH mergers, the ranges of these fractions would have\nbroadened. Some obvious physical processes that can af-\nfect BH spins and their orientations include: initial star\nspin alignment (or lack thereof) with the binary AM, the\nalignment of stellar spins (or lack thereof) during RLOF\nphases, the treatment of accretion, the initial mass ra-\ntio distribution that can alter the ratio of systems going\nthrough stable and unstable (CE) RLOF, and the natal\nkicks that can misalign spin orientations. Above all, the\nthree major uncertainties include the initial stellar rota-\ntion of stars forming BHs, the e\u000eciency of AM transport5\nand the strength of tides in close binary systems. All of\nthe above are only weakly constrained. Note that this\nis a proof-of-principle study that is limited only to BH\nspins in BH-BH mergers. In particular, we did not try\nto match BH masses and BH-BH merger rates for the\nhighly spinning LIGO/Virgo BHs. In this study we have\nonly shown that it is possible to produce highly spinning\nBHs by tidal interactions of stars in close binaries in evo-\nlution that includes and does not include CE. Our two\nexamples of evolution (Fig. 1 and 2) have much smaller\nmasses than the LIGO/Virgo mergers with highly spin-\nning BHs (Tab. 1). Note, however, that we have not\nused here the input physics that allows for the formation\nof BHs with mass over 50 M\f. Such model is already\nincorporated and tested within our population synthe-\nsis code (Belczynski 2020). An attempt to match allobserved parameters simultaneously is projected to hap-\npen in the future when LIGO/Virgo will deliver a larger\nsample of highly spinning BHs.\nGiven the results presented in this study, alas limited\nonly to BH spins, we conclude that (i)the isolated binary\nevolution channel reproduces well the BH spins of the\nLIGO/Virgo mergers (ii)if, in fact, the binary channel\nis producing the majority of the LIGO/Virgo BH-BH\nmergers, then this indicates that the AM transport is\ne\u000ecient in massive stars and the tidal interactions in\nclose binaries are strong.\nWe thank the anonymous reviewer, Jean-Pierre La-\nsota, Ilya Mandel and Sambaran Banerjee for their useful\ncomments on the manuscript. KB and AO acknowledge\nsupport from the Polish National Science Center (NCN)\ngrant Maestro (2018/30/A/ST9/00050).\nREFERENCES\nAbbott, B. P., Abbott, R., Abbott, T. D., Abraham, S., LIGO\nScienti\fc Collaboration, & Virgo Collaboration. 2019a, ApJ,\n882, L24\n|. 2019b, Physical Review X, 9, 031040\nAbbott, R., et al. 2021a, Physical Review X, 11, 021053\n|. 2021b, arXiv e-prints, arXiv:2108.01045\nAntonini, F., & Perets, H. B. 2012a, ApJ, 757, 27\n|. 2012b, ApJ, 757, 27\nAntonini, F., Toonen, S., & Hamers, A. S. 2017, ApJ, 841, 77\nArca-Sedda, M., & Capuzzo-Dolcetta, R. 2017, ArXiv e-prints\nArca-Sedda, M., Li, G., & Kocsis, B. 2018, arXiv e-prints,\narXiv:1805.06458\nAskar, A., Szkudlarek, M., Gondek-Rosi\u0013 nska, D., Giersz, M., &\nBulik, T. 2017, MNRAS, 464, L36\nBae, Y.-B., Kim, C., & Lee, H. M. 2014, MNRAS, 440, 2714\nBanerjee, S. 2018, MNRAS, 473, 909\n|. 2021, MNRAS, 500, 3002\nBavera, S. S., et al. 2020, A&A, 635, A97\n|. 2021, A&A, 647, A153\nBelczynski, K. 2020, ApJ, 905, L15\nBelczynski, K., Bulik, T., Fryer, C. L., Ruiter, A., Valsecchi, F.,\nVink, J. S., & Hurley, J. R. 2010a, ApJ, 714, 1217\nBelczynski, K., Dominik, M., Bulik, T., O'Shaughnessy, R., Fryer,\nC. L., & Holz, D. E. 2010b, ApJ, 715, L138\nBelczynski, K., Holz, D. E., Bulik, T., & O'Shaughnessy, R.\n2016a, Nature, 534, 512\nBelczynski, K., Kalogera, V., & Bulik, T. 2002, ApJ, 572, 407\nBelczynski, K., Kalogera, V., Rasio, F. A., Taam, R. E., Zezas,\nA., Bulik, T., Maccarone, T. J., & Ivanova, N. 2008, ApJS, 174,\n223\nBelczynski, K., et al. 2016b, A&A, 594, A97\n|. 2020, A&A, 636, A104\n|. 2021, arXiv e-prints, arXiv:2108.10885\nBenacquista, M. J., & Downing, J. M. B. 2013, Living Reviews in\nRelativity, 16, 4\nBond, J. R., & Carr, B. J. 1984, MNRAS, 207, 585\nCarr, B., & Silk, J. 2018, MNRAS, 478, 3756\nChatterjee, S., Rodriguez, C. L., Kalogera, V., & Rasio, F. A.\n2016, ArXiv e-prints\nClesse, S., & Garc\u0013 \u0010a-Bellido, J. 2017, Physics of the Dark\nUniverse, 15, 142\nDe Luca, V., Desjacques, V., Franciolini, G., Pani, P., & Riotto,\nA. 2020, arXiv e-prints, arXiv:2009.01728\nde Mink, S. E., & Mandel, I. 2016, MNRAS, 460, 3545\nDetmers, R. G., Langer, N., Podsiadlowski, P., & Izzard, R. G.\n2008, A&A, 484, 831\nDi Carlo, U. N., Giacobbo, N., Mapelli, M., Pasquato, M., Spera,\nM., Wang, L., & Haardt, F. 2019, MNRAS, 487, 2947\nDominik, M., Belczynski, K., Fryer, C., Holz, D., Berti, B., Bulik,\nT., Mandel, I., & O'Shaughnessy, R. 2012, ApJ, 759, 52\nDowning, J. M. B., Benacquista, M. J., Giersz, M., & Spurzem,\nR. 2010, MNRAS, 407, 1946\ndu Buisson, L., et al. 2020, arXiv e-prints, arXiv:2002.11630\nEldridge, J. J., & Stanway, E. R. 2016, MNRAS, 462, 3302\nFishbach, M., & Holz, D. E. 2020, ApJ, 891, L27\nFishbach, M., Holz, D. E., & Farr, B. 2017, ApJ, 840, L24\nFragione, G., Grishin, E., Leigh, N. W. C., Perets, H. B., &\nPerna, R. 2019, MNRAS, 488, 47\nFragione, G., & Kocsis, B. 2019, MNRAS, 486, 4781Fryer, C. L., Belczynski, K., Wiktorowicz, G., Dominik, M.,\nKalogera, V., & Holz, D. E. 2012, ApJ, 749, 91\nFuller, J., & Ma, L. 2019, ApJ, 881, L1\nFuller, J., Piro, A. L., & Jermyn, A. S. 2019, MNRAS\nGalaudage, S., Talbot, C., Nagar, T., Jain, D., Thrane, E., &\nMandel, I. 2021, arXiv e-prints, arXiv:2109.02424\nGallegos-Garcia, M., Berry, C. P. L., Marchant, P., & Kalogera,\nV. 2021, arXiv e-prints, arXiv:2107.05702\nGerosa, D., & Berti, E. 2017, Phys. Rev. D, 95, 124046\nGreen, A. M. 2017, ArXiv e-prints\nG ultekin, K., Miller, M. C., & Hamilton, D. P. 2004, ApJ, 616,\n221\n|. 2006, ApJ, 640, 156\nHainich, R., et al. 2018, A&A, 609, A94\nHamers, A. S., Bar-Or, B., Petrovich, C., & Antonini, F. 2018,\nApJ, 865, 2\nHartwig, T., Volonteri, M., Bromm, V., Klessen, R. S., Barausse,\nE., Magg, M., & Stacy, A. 2016, MNRAS, 460, L74\nHoang, B.-M., Naoz, S., Kocsis, B., Rasio, F. A., & Dosopoulou,\nF. 2018, ApJ, 856, 140\nHotokezaka, K., & Piran, T. 2017, ArXiv e-prints\nHurley, J. R., Sippel, A. C., Tout, C. A., & Aarseth, S. J. 2016,\nMNRAS, 33, e036\nKalogera, V., & Webbink, R. F. 1996, Astrophys. J., 458, 301\nKing, A. R., Davies, M. B., Ward, M. J., Fabbiano, G., & Elvis,\nM. 2001, ApJ, 552, L109\nKinugawa, T., Inayoshi, K., Hotokezaka, K., Nakauchi, D., &\nNakamura, T. 2014, MNRAS, 442, 2963\nKremer, K., et al. 2020, ApJS, 247, 48\nKruckow, M. U., Tauris, T. M., Langer, N., Kramer, M., &\nIzzard, R. G. 2018, ArXiv e-prints\nKushnir, D., Zaldarriaga, M., Kollmeier, J. A., & Waldman, R.\n2017, MNRAS, 467, 2146\nLipunov, V. M., Postnov, K. A., & Prokhorov, M. E. 1997,\nAstronomy Letters, 23, 492\nLiu, B., & Lai, D. 2018, ApJ, 863, 68\nMacLeod, M., Antoni, A., Murguia-Berthier, A., Macias, P., &\nRamirez-Ruiz, E. 2017, ApJ, 838, 56\nMacLeod, M., & Ramirez-Ruiz, E. 2015, ApJ, 803, 41\nMadau, P., & Dickinson, M. 2014, ARA&A, 52, 415\nMandel, I., & de Mink, S. E. 2016, MNRAS, 458, 2634\nMapelli, M. 2016, MNRAS, 459, 3432\nMarchant, P., Langer, N., Podsiadlowski, P., Tauris, T. M., &\nMoriya, T. J. 2016, A&A, 588, A50\nMarchant, P., Renzo, M., Farmer, R., Pappas, K. M. W., Taam,\nR. E., de Mink, S., & Kalogera, V. 2018, arXiv e-prints\nMennekens, N., & Vanbeveren, D. 2014, A&A, 564, A134\nMiller, M. C., & Hamilton, D. P. 2002a, ApJ, 576, 894\n|. 2002b, MNRAS, 330, 232\nMirabel, I. F., & Rodrigues, I. 2003, Science, 300, 1119\nMondal, S., Belczy\u0013 nski, K., Wiktorowicz, G., Lasota, J.-P., &\nKing, A. R. 2020, MNRAS, 491, 2747\nMorawski, J., Giersz, M., Askar, A., & Belczynski, K. 2018,\nArXiv e-prints\nNeijssel, C. J., et al. 2019, MNRAS, 490, 3740\nO'Leary, R. M., O'Shaughnessy, R., & Rasio, F. A. 2007,\nPhys. Rev. D, 76, 061504\nOlejak, A., Belczynski, K., & Ivanova, N. 2021, A&A, 651, A100\nOlejak, A., Fishbach, M., Belczynski, K., Holz, D. E., Lasota,\nJ. P., Miller, M. C., & Bulik, T. 2020, ApJ, 901, L396\nPavlovskii, K., Ivanova, N., Belczynski, K., & Van, K. X. 2017,\nMNRAS, 465, 2092\nPaxton, B., et al. 2015, ApJS, 220, 15\nPerna, R., Wang, Y.-H., Farr, W. M., Leigh, N., & Cantiello, M.\n2019, ApJ, 878, L1\nPortegies Zwart, S. F., Baumgardt, H., Hut, P., Makino, J., &\nMcMillan, S. L. W. 2004, Nature, 428, 724\nPortegies Zwart, S. F., & McMillan, S. L. W. 2000, ApJ, 528, L17\nQin, Y., Fragos, T., Meynet, G., Andrews, J., S\u001crensen, M., &\nSong, H. F. 2018, A&A, 616, A28\nQin, Y., Marchant, P., Fragos, T., Meynet, G., & Kalogera, V.\n2019, ApJ, 870, L18\nQin, Y., Wang, Y.-Z., Dong-Hong, Wu, Meynet, G., & Song, H.\n2021, arXiv e-prints, arXiv:2108.04821\nRicker, P. M., & Taam, R. E. 2008, ApJ, 672, L41\nRodriguez, C. L., Amaro-Seoane, P., Chatterjee, S., Kremer, K.,\nRasio, F. A., Samsing, J., Ye, C. S., & Zevin, M. 2018,\nPhys. Rev. D, 98, 123005\nRodriguez, C. L., Haster, C.-J., Chatterjee, S., Kalogera, V., &\nRasio, F. A. 2016, ApJ, 824, L8\nRoulet, J., Chia, H. S., Olsen, S., Dai, L., Venumadhav, T.,\nZackay, B., & Zaldarriaga, M. 2021, arXiv e-prints,\narXiv:2105.10580\nSadowski, A., Belczynski, K., Bulik, T., Ivanova, N., Rasio, F. A.,\n& O'Shaughnessy, R. 2008, ApJ, 676, 1162\nSamsing, J. 2017, ArXiv e-printsSasaki, M., Suyama, T., Tanaka, T., & Yokoyama, S. 2016, ArXiv\ne-prints\nSilsbee, K., & Tremaine, S. 2017, ApJ, 836, 39\nSpera, M., Giacobbo, N., & Mapelli, M. 2016,\nMem. Soc. Astron. Italiana, 87, 575\nSpera, M., Mapelli, M., Giacobbo, N., Trani, A. A., Bressan, A.,\n& Costa, G. 2019, MNRAS, 485, 889\nSpruit, H. C. 2002, A&A, 381, 923\nStevenson, S., Vigna-G\u0013 omez, A., Mandel, I., Barrett, J. W.,\nNeijssel, C. J., Perkins, D., & de Mink, S. E. 2017, Nature\nCommunications, 8, 14906\nTutukov, A. V., & Yungelson, L. R. 1993, MNRAS, 260, 675\nvan den Heuvel, E. P. J., Portegies Zwart, S. F., & de Mink, S. E.\n2017, MNRAS, 471, 4256\nvan Son, L. A. C., et al. 2020, ApJ, 897, 100\nVanLandingham, J. H., Miller, M. C., Hamilton, D. P., &\nRichardson, D. C. 2016, ApJ, 828, 77\nVink, J. S., de Koter, A., & Lamers, H. J. G. L. M. 2001, A&A,\n369, 574\nVoss, R., & Tauris, T. M. 2003, MNRAS, 342, 1169\nWoosley, S. E. 2016, ApJ, 824, L10\nYusof, N., et al. 2013, MNRAS, 433, 1114\nZaldarriaga, M., Kushnir, D., & Kollmeier, J. A. 2017, ArXiv\ne-prints\nZevin, M., Samsing, J., Rodriguez, C., Haster, C.-J., &\nRamirez-Ruiz, E. 2019, ApJ, 871, 91\nAPPENDIX A\nFig. 3.| Distribution of primary BH spin ( a1) { top panel; secondary BH spin ( a2) { middle panel; e\u000bective spin parameter ( \u001fe\u000b) {\nbottom panel; of BH-BH mergers at redshifts z <1:0. The results are for two tested models: the non-CE model plotted with red line\nand the CE model plotted with blue line. The \fgure is a supplement to the statistical spin predictions shown in Table 2 and described in\nSection 3."    },    {        "title": "1705.05395v1.Spin_and_tunneling_dynamics_in_an_asymmetrical_double_quantum_dot_with_spin___orbit_coupling.pdf",        "content": "arXiv:1705.05395v1  [cond-mat.mes-hall]  15 May 2017Spin and tunneling dynamics in an asymmetrical double quant um dot with\nspin - orbit coupling\nMadhav Singh1, Pradeep K Jha2and Aranya B Bhattacherjee3\n1Department of Physics and Astrophysics, University of Delh i, Delhi-110007, India\n2Department of Physics, DDU College, University of Delhi, Ne w Delhi\n3School of Physical Sciences, Jawaharlal Nehru University, New Delhi-110067, India\nIn this articlewestudy the spin and tunneling dynamics asafunction o fmagneticfield in a\none-dimensionalGaAs double quantum dot with both the Dresselhau sand Rashbaspin-orbit\ncoupling. In particular we consider different spatial widths for the s pin-up and spin-down\nelectronic states. We find that the spin dynamics is a superposition o f slow as well as fast\nRabi oscillations. It is found that the Rashba interaction strength as well as the external\nmagnetic field strongly modifies the slow Rabi oscillations which is partic ularly useful for\nsingle qubit manipulation for possible quantum computer applications.\nI. INTRODUCTION\nSemiconductor quantum dots (QDs), also called “artificial a toms” are analogous to real atoms\n[1] and have been one of the most intensively studied nanostr uctures due to the richness in their\nproperties [2]. Numerous interesting transport propertie s of QD based systems have been revealed\nsuch as Coulomb blockade, spin blockade and Kondo effect [3–7] . One of the challenges in spintron-\nics and quantum transport is fast and efficient manipulation o f electron spins in QDs. Infact by\ncontrolling the electric bias, tunable spin diodes have bee n realized experimentally [8, 9]. Manipu-\nlated electron spins in QDs are expected as a possible realiz ation of qubits [10]. In the presence of\nspin-orbit (SO) coupling, single electron QDs exhibit spin -flip dynamics which can be understood\nin terms of the electric dipole spin resonance (EDSR) [11, 12 ]. Using EDSR, coherent spin manip-\nulation in GaAs QDs have been demonstrated, where electric fi eld induced spin Rabi oscillations\nhave been observed [13]. Other various SO effects present in se miconductor nanostructures provide\nan efficient and reliable way to manipulate electron spins in t wo-dimensional (2D) electron gases\n[14–21]. Manipulating spin degrees of freedom has opened ne w possibilities to design fast and low\npower quantum devices for applications in quantum computin g and memory storage [22, 23].\nHowever for quantum information applications, single isol ated QDs are not suitable since in-\nterdot interaction is necessary to produce and manipulate m any-body states. Double QDs where\ntunneling plays a crucial role are more promising for quantu m information technologies [24]. The2\n/MinuΣ4 /MinuΣ2 0 2 402468\nx/Slash1dV/LParen1x/RParen1\nFigure 1: Schematic plot of the double well potential described by Eq uation (1).\ninterplay between tunneling and spin flip process is an impor tant factor in double QDs. In the\npresence of SO interaction, the electron states and the inte rdot tunneling becomes spin-dependent.\nSingleelectrondynamicsinaone-dimensionaldoubleQDwit hSOcouplingdrivenbyanexternal\nelectric and magnetic fields has been a subject of earlier num erous study [25–28]. In these works,\nthe mutual effect of coordinate and spin motion on the Rabi osci llations were carried out under\nthe assumption that the spatial widths of the wacefunction f or spin-up and spin-down states are\nsame along the direction of the confining potential. However since the magnetic field lifts the\nspin degeneracy, the wavefunctions in the two spin-split le vels are not the same [29]. Keeping this\nimportant point in view, in this article we study the spin and tunneling dynamics as a function of\nmagnetic field in a one-dimensional double QD with SO couplin g and different spatial widths for\nthe spin-up and spin-down states. Both the Dresselhaus and R ashba SO coupling are taken into\naccount.\nII. THE MODEL AND HAMILTONIAN\nWe consider an electron confined in a one dimensional double Q D described by the quadratic\npotential as shown in figure1.\nV(x) =U0(−2x2\nd2+x4\nd4), (1)\nwhere the two minima located at d and -d are separated by a barr ier of height U0.\nThe unperturbed system is described by the Hamiltonian:3\nH0=ˆp2\nx\n2m+V(ˆx), (2)\nwhere ˆpx=−i/planckover2pi1∂\n∂xis the momentum operator in the x−direction. Here m is the mass of the\nelectron. In the absence of any external fields and spin-orbi t coupling (SOC), the ground state is\nsplit into the doublet of even (symmetric) and odd (anti symm etric) states. The tunneling energy\n∆Eg<< V0determines the tunneling time Ttun=2π\n∆Egand ∆Egalso is the gap between the\nground and first excited state of H0. Under the influence of an external magnetic field BzalongZ\ndirection and electric field along the x direction the system is described by Hamiltonian\nH=H0−eEx+HSO−∆zσz\n2. (3)\nThe second term −eExdescribe the perturbation caused by external electric field . The Zeeman\ncoupling to the magnetic field is∆zσz\n2, where ∆ z=|g|µBBzis the Zeeman splitting. The Pauli\nmatrices are described by σi=x,y,z. Heregis the Land´ e factor of electron and µBis the Bohr\nmagneton. The spin orbit (SO) interaction is described by Ha miltonian HSOwhich is the sum of\nthe bulk originated Dresselhaus( β) and structure related Rashba ( α) terms,\nHSO= (β\n/planckover2pi1pxσx−α\n/planckover2pi1pxσy). (4)\nIn order to study the spin and tunnelling dynamics as a functi on of the applied magnetic field ,\nwe use a perturbation approach and diagonalize the Hamilton ianHin the truncated spinor basis\nΨm(x)| ↑>with corresponding eigenvalues Em,σ. Here Ψ m(x) are the eigenfunctions of H0in the\ndouble well potential with m=s(symmetric), m=a(antisymmetric) and σ= +(−) corresponds\nto the spin parallel (antiparallel) to the zaxis. The wavefunctionΨ s(x)(Ψa(x)) is even (odd)\nwith respect to the inversion of x. Assuming weak tunneling, these symmetric and antisymmetric\nfunctions can be written in the form :\nΨs,a(x) =ΨL(x)±ΨR(x)√\n2, (5)\nwhere Ψ L(x) and Ψ R(x) are localized wavefunction in the left and right dot repect ively.\nΨL(x) =1√8wxsin(π\n2+nxπ\nwx), (6)4\n01./Multiply10/Minus112./Multiply10/Minus113./Multiply10/Minus114./Multiply10/Minus115./Multiply10/Minus116./Multiply10/Minus11/Minus0.5/Minus0.4/Minus0.3/Minus0.2/Minus0.10.0\nTime/LParen1Secs/RParen1/LeΣΣΣRz/Greatera\n01./Multiply10/Minus112./Multiply10/Minus113./Multiply10/Minus114./Multiply10/Minus115./Multiply10/Minus116./Multiply10/Minus110.00.20.40.60.8\nTime/LParen1Secs/RParen1PR/LParen1t/RParen1b\nFigure 2: Dynamics of σz\nR(t) (a) and PR(t) (b) for the identical width (ID) case for B= 0.88T. The various\nparameters used are: α= 1×10−9eV−cm,β= 0.3×eV−cm, g=-0.45, wx= 25√\n2nm,wx=w′\nx\n01./Multiply10/Minus112./Multiply10/Minus113./Multiply10/Minus114./Multiply10/Minus115./Multiply10/Minus116./Multiply10/Minus11/Minus0.6/Minus0.4/Minus0.20.00.2\nTime/LParen1Secs/RParen1/LeΣΣΣRz/Greatera\n01./Multiply10/Minus112./Multiply10/Minus113./Multiply10/Minus114./Multiply10/Minus115./Multiply10/Minus116./Multiply10/Minus110.400.450.500.550.600.65\nTime/LParen1Secs/RParen1PR/LParen1t/RParen1b\nFigure 3: Dynamics of σz\nR(t) (a) and PR(t) (b) for the identical width (ID) case for B= 6.4T. The various\nparameters used are: α= 1×10−9eV−cm,β= 0.3×eV−cm, g=-0.45, wx= 25√\n2nm,wx=w′\nx.\nΨR(x) =1√8wxsin(π\n2−nxπ\nwx). (7)\nHerewxis the spatial width of the wavefunction. We will be restrict ing our analysis to\ntwo lowest states n= 1,2. The first four lowest energy states are |Ψ1>= Ψs(x)| ↑>,|Ψ2>=\nΨs(x)| ↓>,|Ψ3>= Ψa(x)| ↑>and|Ψ4>= Ψa(x)| ↓>. The spatial parts of the wavefunctions\nΨs(a)(x) are different for spin-up and spin-down states. The spin dege neracy is lifted due to the\nmagnetic field and as a result of which the wavefunctions in th e two spin split levels are not the\nsame. Hence the spatial spread of the wavefunctions are differ ent . The wavefunction correspond-\ning to the state with higher energy spreads out more outside t he potential well (spatial width w′\nx\n) compared to the wavefunction corresponding to the state wi dth lower energy level (spatial width\nwx) andw′\nx> wx. The full dynamics of the system can then be studied with the f unction5\n01./Multiply10/Minus112./Multiply10/Minus113./Multiply10/Minus114./Multiply10/Minus115./Multiply10/Minus116./Multiply10/Minus11/Minus0.4/Minus0.20.00.20.4\nTime/LParen1Secs/RParen1/LeΣΣΣRz/Greatera\n01./Multiply10/Minus112./Multiply10/Minus113./Multiply10/Minus114./Multiply10/Minus115./Multiply10/Minus116./Multiply10/Minus110.40.50.60.7\nTime/LParen1Secs/RParen1PR/LParen1t/RParen1b\nFigure 4: Dynamics of σz\nR(t) (a) and PR(t) (b) for the non-identical width (NID) case for B= 0.88T.\nThe various parameters used are: α= 1×10−9eV−cm,β= 0.3×eV−cm, g=-0.45, wx= 25√\n2nm,\nw′\nx= 1.0375wx.\n01./Multiply10/Minus112./Multiply10/Minus113./Multiply10/Minus114./Multiply10/Minus115./Multiply10/Minus116./Multiply10/Minus11/Minus0.4/Minus0.20.00.20.4\nTime/LParen1Secs/RParen1/LeΣΣΣRz/Greatera\n01./Multiply10/Minus112./Multiply10/Minus113./Multiply10/Minus114./Multiply10/Minus115./Multiply10/Minus116./Multiply10/Minus110.350.400.450.500.550.600.65\nTime/LParen1Secs/RParen1PR/LParen1t/RParen1b\nFigure 5: Dynamics of σz\nR(t) (a) and PR(t) (b) for the non-identical width (NID) case for B= 6.4T.\nThe various parameters used are: α= 1×10−9eV−cm,β= 0.3×eV−cm, g=-0.45, wx= 25√\n2nm,\nw′\nx= 1.0375wx.\n|Ψ>=/summationdisplay\nnξn(t)e−iEnt\n/planckover2pi1|Ψn>, (8)\nwhereξn(t) are the expansion coefficients . Once the wavefunctions are k nown, we can calculate\nthe probability PR(t) to find the electron in the right dot and expectation value of theithspin\ncomponent σi\nRin the right dot since we are interested in the quantum transp ort from the left to\nthe right quantum dot.\nPR(t) =/integraldisplay∞\n0Ψ†Ψdx, (9)6\n01./Multiply10/Minus112./Multiply10/Minus113./Multiply10/Minus114./Multiply10/Minus115./Multiply10/Minus116./Multiply10/Minus11/Minus0.4/Minus0.20.00.20.40.6\nTime/LParen1Secs/RParen1/LeΣΣΣRx/Greatera\n01./Multiply10/Minus112./Multiply10/Minus113./Multiply10/Minus114./Multiply10/Minus115./Multiply10/Minus116./Multiply10/Minus11/Minus0.6/Minus0.4/Minus0.20.00.20.40.6\nTime/LParen1Secs/RParen1/LeΣΣΣRy/Greaterb\nFigure 6: Dynamics of σx\nR(t) (a) and σy\nR(t) (b) for the non-identical width (NID) case for B= 0.88T.\nThe various parameters used are: α= 1×10−9eV−cm,β= 0.3×eV−cm, g=-0.45, wx= 25√\n2nm,\nw′\nx= 1.0375wx.\n01./Multiply10/Minus112./Multiply10/Minus113./Multiply10/Minus114./Multiply10/Minus115./Multiply10/Minus116./Multiply10/Minus11/Minus0.6/Minus0.4/Minus0.20.00.20.40.6\nTime/LParen1Secs/RParen1/LeΣΣΣRx/Greatera\n01./Multiply10/Minus112./Multiply10/Minus113./Multiply10/Minus114./Multiply10/Minus115./Multiply10/Minus116./Multiply10/Minus11/Minus0.6/Minus0.4/Minus0.20.00.20.40.6\nTime/LParen1Secs/RParen1/LeΣΣΣRy/Greaterb\nFigure 7: Dynamics of σx\nR(t) (a) and σy\nR(t) (b) for the non-identical width (NID) case for B= 6.4T.\nThe various parameters used are: α= 1×10−9eV−cm,β= 0.3×eV−cm, g=-0.45, wx= 25√\n2nm,\nw′\nx= 1.0375wx.\nσi\nR(t) =/integraldisplay∞\n0Ψ†σiΨdx. (10)\nThe explicit expressions for PR(t) andσi\nR(t) are given in the appendix A.\nIII. RESULT AND DISCUSSION\nWe are interested in the inter dot transition. To demonstrat e the nontrivial dynamics of the\ntunnelling and spin flip process, we plot the dynamics of PR(t) andσi\nR(t)(i=x,y,z) for different\nmagnetic fields.\nFor the ID case as shown in Fig.2 and Fig.3, the spin component σz\nR(t) displays an irregular7\n01./Multiply10/Minus112./Multiply10/Minus113./Multiply10/Minus114./Multiply10/Minus115./Multiply10/Minus116./Multiply10/Minus11/Minus0.3/Minus0.2/Minus0.10.00.10.20.3\nTime/LParen1Secs/RParen1/LeΣΣΣRx/Greatera\n01./Multiply10/Minus112./Multiply10/Minus113./Multiply10/Minus114./Multiply10/Minus115./Multiply10/Minus116./Multiply10/Minus11/Minus0.2/Minus0.10.00.10.2\nTime/LParen1Secs/RParen1/LeΣΣΣRy/Greaterb\nFigure 8: Dynamics of σx\nR(t) (a) and σy\nR(t) (b) for the non-identical width (NID) case for B= 11T.\nThe various parameters used are: α= 1×10−9eV−cm,β= 0.3×eV−cm, g=-0.45, wx= 25√\n2nm,\nw′\nx= 1.0375wx.\n0 2 4 6 8 102.662.682.702.72\nB/LParen1Tesla/RParen1/VertBar1/CΑpDeltΑ1/Plus/CΑpDeltΑ2/VertBar1/LParen1meV/RParen1a\n0 2 4 6 8 100.080.100.120.140.16\nB/LParen1Tesla/RParen1/VertBar1/CΑpDeltΑ1/Minus/CΑpDeltΑ2/VertBar1/LParen1meV/RParen1b\nFigure 9: Plot of ∆(1)+∆(2)(a) and ∆(1)−∆(2)(b) as a function of the magnetic field .\npattern and the mean value of σz\nR(t) (denoted by ( σz\nR(t))M) increases from −0.10 atBz= 11Tto\n−0.25 atBz= 0.88T. Similarly the dynamics of PR(t) is also irregular and the mean value of\nPR(t)(denoted by ( PR(t))M) increases from 0 .35 atBz= 11Tto 0.52 at Bz= 0.88T. The strong\noscillations of PR(t) around a mean value of 0 .52 for small and moderate fields Bz= 0.88T(Fig.\n2b),Bz= 2.82T(not shown) and Bz= 6.4T(Fig. 3b), indicates that the external field almost\nequalizes the probabilities to find the electron in the right and left dot . For large magnetic field of\n11T, the mean value around which PR(t) oscillates is 0 .35 (not shown) which shows that a strong\nmagnetic field of 11 Tsuppresses the tunneling from the left to the right well. Thi s indicates that\nfor small magnetic fields, a small spin polarization can be ac hieved in the tunneling under ID case.\nThe dynamics for the NID case is more regular and rich as depic ted in figure 4 and figure 5.\nThe dynamics comprises of low frequency motion due to the ter m ∆(1)−∆(2)superimposed by\nhigh frequency oscillations due to the involvement of term ∆(1)+∆(2)in the expression for PR(t)\nandσi\nR(t) (see appendix). For the NID case, ( σz\nR(t))Mis zero for all magnetic fields while ( PR)M8\n01./Multiply10/Minus112./Multiply10/Minus113./Multiply10/Minus114./Multiply10/Minus115./Multiply10/Minus116./Multiply10/Minus11/Minus0.4/Minus0.20.00.20.4\nTime/LParen1Secs/RParen1/LeΣΣΣRz/Greatera\n0 1 2 3 4 50246810\nΑ/LParen1/Multiply10/Minus9eV/Minuscm/RParen1/VertBar1/CΑpDeltΑ1/Minus/CΑpDeltΑ2/VertBar1/LParen1meV/RParen1,BZ/EquΑl0.88 Tb\nFigure10: Plot(a): Dynamicsof σx\nR(t)(a)forthenon-identicalwidth(NID) casefor B= 0.88T. Thevarious\nparameters used are: α= 2×10−9eV−cm,β= 0.3×eV−cm, g=-0.45, wx= 25√\n2nm,w′\nx= 1.0375wx.\nPlot (b): ∆(1)−∆(2)versus Rashba interaction strength ( α) for the NID case.\nincreases from 0 .32 atBz= 11Tto 0.54 atBz= 0.88T, which indicates that no spin- polarization\ncan be achieved in the tunneling.\nThe SO interaction is the sum of bulk originated Dresselhaus (β) and structure related Rashba\n(α) terms. Inthetunneling, spinprecession aroundboththe xas well as yaxis isobserved as shown\nin figure 6 and figure 7 for the NID case. The spin precession aro und the y axis occurs due to the\nRashbaSOterm whilethespinprecession aroundthe xaxis occursduetotheDresselhausSOterm.\nThe magnetic field produces the zeeman spin splitting of the l evel ∆ z=En↓−En↑=|g|µBBzfor\nthe ID case. On the other hand for the NID case, the spin splitt ing energy is given by [29],\n∆z= 2/radicalBigg/bracketleftbigggµBB\n2/bracketrightbigg2\n+α264\nwxw′\nxf(wx,w′\nx), (11)\nwhere\nf(wx,w′\nx) = cos2(πwx\n2w′\nx)cos2(πw′\nx\n2wx)/bracketleftbigg1\n((wx/w′\nx)2−1)((w′\nx/wx)2−1)/bracketrightbigg2\n. (12)\nFor the NID case, we note that the spin split energy is larger a s compared to that for ID case.\nWith the increase in magnetic field , the effect of SOC decreases , leading to smaller amplitudes\nof spin precession as can be seen by comparing figures 6 and 7 wi th figure 8. Comparing the\ndynamics of < σz\nR(t)>from Figs. 4 and Fig. 5, there is a minor increase in the Rabi fr equency\n|∆(1)+∆(2)|with increasing magnetic fieldwhich is consistent with prev ious theoretical [11, 12] and\nexperimental result [13]. On the other hand the slow oscilla tions|∆(1)−∆(2)|becomes slower with9\nincreasingmagneticfield. Thesetwoobservationsareclear ly depictedinFig.9where∆(1)+∆(2)and\n∆(1)−∆(2)areplotted as a function of the magnetic field in Fig.9a and Fi g.9b respectively. Thelow\nfrequency dynamics corresponds to the spectrum of the low en ergy states while the high frequency\noscillations involve the higher-energy states. As evident from Figures 6,7 and 8, increasing the\nmagnetic field, the slow oscillations of < σx\nR(t)>and< σy\nR(t)>gradually disappear at Bz= 11T.\nThus at high magnetic fields, only the higher energy states ar e involved in the dynamics. The\nsame conclusion cannot be reached for the ID case due to its ir regular dynamics. In addition, we\nobserve that for the ID case, both the initial state and spin p recession axis change with magnetic\nfield leading to different phase shifts between the spin compon ents. On the other hand, for the\nNID case, the initial state and the spin precession axis rema ins intact for different magnetic fields.\nThe incomplete Rabi spin flips results from the fact that the t he electron tunneling between the\npotential minimas establishes a competing spin dynamics wh ich prevents the electric field to flip\nthe spin efficiently ( σz\nR(t) to reach ±1). The slow tunneling dynamics of the NID case (Fig.4b and\n5b) allows the electron to stay in the right well for a longer d uration. The spin precession in this\nslow interminima motion induces a corresponding spin dynam ics which does not allow the electric\nfield to flip the spin efficiently and thus spin polarization is a bsent for the NID case. This is in\ncontrast to the ID case where the tunneling dynamics is extre mely fast and the electron oscillates\nrapidly between the left and the right well. The fast inter-m inima motion washes out the induced\nspin dynamics and thus allows the electric field to flip the spi n.\nSpintronicsbasedimplementation ofscalablequantumcomp utershasgeneratedmuchinterest in\nexploringcoherentcontrolofsinlequbitrotationusingRa bioscillations[29]. Inordertosuccessfully\nimplement single qubit rotation selectively in a sample com prising of multiple quantum dots using\nRabi oscillations, thespin splitting energy in the target q uantum dot has to bechanged appreciably\nusing Rashba interaction [29]. It was shown in ref. [29] that the energies of the spin-split levels\nof the lowest subband in a QD decreases with increasing Rashb a interaction strength. Since Rabi\noscillations play an important role in the control of single qubit rotation, we have studied here\nthe dynamics of < σz\nR(t)>as a function of Rashba interaction strength α. Fig.10a shows the\nplot of< σz\nR(t)>as a function of time for α= 2×10−9eV−cmandB= 0.88T. Clearly the\nfrequency of the slow oscillations |∆(1)−∆(2)|increases as compared to that in Fig. 4a where\nα= 1×10−9eV−cm. This is also depicted in Fig.10b where |∆(1)−∆(2)|is seen to increase\nwithα. The influence of αon|∆(1)+∆(2)|is not appreciable. These observations indicates that\nincreasing the Rashba interaction strength induces a rapid energy transfer between the low energy\nstates while energy transfer between the higher energy stat es essentially remains uneffected.10\nIV. CONCLUSION\nIn conclusion, we have studied the quantum spin and charge dy namics of a single electron\nconfined in one-dimensional GaAs double quantum dot with ras hba and Dresselhaus spin-orbit\ninteraction. We have made a comparitive study of two specific cases: In first case, the spatial\nparts of the wavefunctions are same for spin-up and spin-dow n states (ID case) while in the second\ncase we consider the spatial parts of the wavefunctions to be different for spin-up and spin-down\nstates (NID case). We demonstrate that the dynamics of the NI D case is more regular. The\ndynamics comprises of a slow as well as fast Rabi oscillation s. The slow oscillations corresponds to\ndynamics involving low energy states while fast oscillatio ns involve the higher energy states. For\nthe ID case, we found a small amount of spin polarization in th e tunneling but for the NID case, a\ncomplete absence of spin polarization in the tunneling is no ticed. We found that a coherent control\nof the slow Rabi oscillations is possible using the Rashba in teraction strength and the external\nmagnetic field which makes this scheme useful for single qubi t manipuation for quantum computer\napplications.\nAcknowledgments\nA. B acknowledges financial support from the University Gran ts Commission, New Delhi under\nthe UGC-Faculty Recharge Programme. M. S acknowledges finan cial support from the University\nGrants Commission for the junior research fellowship.11\nV. APPENDIX A\nIdentical Width Case\nσy\nR(t) =N2(0.84(a1b2−a2b1)sin(2∆(1)t)+0.998(d2a1−a2d1+b2c1−b1c2)sin([∆(1)+∆(2)]t)\n+ 0.998(c2a1−a2c1+b2d1−b1d2)sin([∆(1)−∆(2)]t)+0.84(c1d2−c2d1)sin(2∆(2)t);\nσz\nR(t) =N2(1\n2(a2\n2−a2\n1+b2\n2−b2\n1+c2\n2−c2\n1+d2\n2−d2\n1)+(a2b2−a1b1)cos(2∆(1)t)\n+ (c2d2−c1d1)cos(2∆(2)t)+8\n3π(a2c2−a1c1)cos([∆(1)−∆(2)]t)\n+8\n3π(a2d2−a1d1+b2c2−b1c1)cos([∆(1)+∆(2)]t));\nσx\nR(t) =N2(0.858(a1a2+b1b2+c1c2+d1d2)+0.858(a2b1+a1b2)cos(2∆(1)t)\n+ 0.858(d1c2+c1d2)cos(2∆(2)t)+0.998(a2d1+a1d2+b2c1+b1c2)cos([∆(1)+∆(2))]t)\n+ 0.998(a2c1+a1c2+b2d1+b1d2)cos([∆(1)−∆(2)]t);\nPR(t) =N2(1\n2(a2\n2+a2\n1+b2\n2+b2\n1+c2\n2+c2\n1+d2\n2+d2\n1)+8\n3π(a2d2+a1d1+b2c2+b1c1)cos([∆(1)+∆(2)]t)\n+8\n3π(b2d2+b1d1+a1c1+a2c2)cos([∆(1)−∆(2)]t)+(a1b1+a2b2)cos(2∆(1)t)\n+ (c1d1+c1d2)cos(2∆(2)t), (13)12\nNon-identical width case\nσx\nR(t) =N2(8\n3π(a1a2+b1b2+c1c2+d1d2)+8\n3π(a2b1+a1b2)cos(2∆(1)t)\n+8\n3π(c2d1+c1d2)cos(2∆(2)t)+(a2c1+a1c2+b2d1+b1d2)cos(∆(1)t);\nσy\nR(t) =N2(8\n3π(a1b2−a2b1sin(2∆(1)t)+8\n3π(c1d2−c2d1sin(2∆(2)t)\n+ (a1c2−a2c1+b2d1−b1d2)sin([∆(1)−∆(2)]t)+(a1d2−a2d1+b2c1−b1c2)sin([∆(1)+∆(2)]t);\nσz\nR(t) =N2(1\n2(a2\n2−a2\n1+b2\n2−b2\n1+c2\n2−c2\n1+d2\n2−d2\n1)+(a2b2−a1b1)cos(2∆(1)t)\n+ (c2d2−c1d1)cos(2∆(2)t)+8\n3π(a2d2−a1d1+b2c2−b1c1)cos([∆(1)+∆(2)]t)\n+8\n3π(a2c2−a1c1)cos([∆(1)−∆(2)]t);\nPR(t) =N2(1\n2(a2\n1+a2\n2+b2\n1+b2\n2+c2\n1+c2\n2+d2\n1+d2\n2)+8\n3π(a2d2+a1d1+b2c2+b1c1)cos([∆(1)+∆(2)t)\n+8\n3π(b2d2+b1d1+a1c1+a2c2)cos([∆(1)−∆(2)]t)+(a1b1+a2b2)cos([2∆(1)]t)\n+ (c1d1+c2d2)cos(∆(2))t). (14)13\n|a1|2=(−eEη+(/radicalbig\nα2+β2)sin(πw′\nx\n2wx)2√\nwxw′\nx\n4w2x−(w′\nx)2)2\n(∆E1\n2+∆z\n2+∆(1))2+(−eEη+(/radicalbig\nα2+β2)sin(πw′\nx\n2wx)2√\nwxw′\nx\n4w2x−(w′\nx)2)2; (15)\n|a2|2= 1− |a1|2; (16)\n|b1|2=(−eEη+(/radicalbig\nα2+β2)sin(πw′\nx\n2wx)2√\nwxw′\nx\n4w2x−(w′\nx)2)2\n(−∆E1\n2−∆z\n2+∆(1))2+(−eEη+(/radicalbig\nα2+β2)sin(πw′\nx\n2wx)2√\nwxw′\nx\n4w2x−(w′\nx)2)2; (17)\n|b2|2= 1− |b1|2; (18)\n|c1|2=(−eEη1−(/radicalbig\nα2+β2)cos(πw′\nx\nwx)2√\nwxw′\nx\nw2x−4(w′\nx)2)2\n(∆E2\n2−∆z\n2−∆(2))2+(−eEη1−(/radicalbig\nα2+β2)cos(πw′\nx\nwx)2√\nwxw′\nx\nw2x−4(w′\nx)2)2; (19)\n|c2|2= 1− |c1|2; (20)\n|d1|2=(−eEη1−(/radicalbig\nα2+β2)cos(πw′\nx\nwx)2√\nwxw′\nx\nb2−4(w′\nx)2)2\n(∆E2\n2−∆z\n2+∆(2))2+(−eEη1−(/radicalbig\nα2+β2)cos(πw′\nx\nwx)2√\nwxw′\nx\nw2x−4(w′\nx)2)2; (21)\n|d2|2= 1− |d1|2; (22)\nη=−64wxw′\nxcos(πw′\nx\nwx)+8πsin(πw′\nx\n2wx)(4w2\nx−(w′\nx)2\nπ2((w′\nx)2−4w2x)21\n(wxw′\nx)3\n2; (23)\nη1=64wxw′\nxsin(πw′\nx\n2wx)+16πcos(πw′\nx\nwx)(w2\nx−4(w′\nx)2\nπ2(−4(w′\nx)2+w2x)21\n(wxw′\nx)3\n2; (24)\nN=/radicalBigg\n1\n4+2(a1b1+a2b2+c2d1+c2d2). (25)\n∆(1)=1\n2/radicalbig\n(∆E1+∆z)2+4c2, (26)\n∆(2)=1\n2/radicalBig\n(∆E2−∆z)2+4(c′)2, (27)\nc=−eEη+/radicalbig\nα2+β22√\nbb′\n4b2−b′2sinπb′\n2b, (28)\nc′=−eEη1−/radicalbig\nα2+β22√\nbb′\n−4b2+b′2cosπb′\nb, (29)14\nHere ∆E1=E′\n2−E1and ∆E2=E2−E′\n1, where E1andE2are the lowest energies of ↑\nstate while E′\n1andE′\n2are the lowest energies of ↓state. Here a1,a2,b1,b2,c1,c2,d1andd2are the\neigenvalues.\n[1] J. P. Bird, Electron transport in Quantum Dots, (Spinger, Berlin , 2003).\n[2] S. Kohler, J. Lehmann and P. Hanggi, Phys. Rep. 405, 379 (2005).\n[3] P. Recher, E. V. Sukhorukov and D. Loss, Phys. Rev. Lett. 85, 1962 (2000).\n[4] D. Weinmann, W. Hausler and B. Kramer, Phys. Rev. Lett., 74, 984 (1995).\n[5] S. Takahashi and S. Maekawa, Phys. Rev. Lett., 80, 1758 (1998).\n[6] W. Liang et al., Nature (London), 417, 725 (2002).\n[7] J. Park et al., Nature (London), 417, 722 (2002).\n[8] A. Merchant and N. Markovi ´ c, Phys. Rev. Lett., 100, 156601 (2008).\n[9] K. Hamaya et al., Phys. Rev. Lett., 102, 236806 (2009).\n[10] G. Burkard, D. Loss and D. P. DiVincenzo, Phys. Rev. B., 59, 2070 (1999).\n[11] E. L. Rashba and AI. L. Efros, Phys. Rev. Lett. 91, 126405 (2003).\n[12] E. L. Rashba and AI. L. Efros, Appl. Phys. Lett. 83, 5295 (2003).\n[13] K. C. Nowack, F. H. L. Koppens, Yu. V. Nazarovand L. M. K. Va ndersypen, Science 318, 1430(2007).\n[14] R. Winkler, Spin-Orbit coupling effects in two-dimensional electro n and hole systems, Springer tracts\nin modern physics, 191, (Springer, Berlin, 2003).\n[15] Y.V.Pershinnd C. Piermarocchi, Phys. Rev. B 72, 245331 (2005).\n[16] J. S. Sheng and K. Chang, Phys. Rev. B 74, 235315 (2006).\n[17] E. Rasanen, A. Castro, J. Werschnik, A. Rubio, and E. K. U. Ph ys.Rev. Lett. 98, 157404 (2007).\n[18] W.-H. Kuan, C.-S. Tang, and C.-H. Chang, Phys. Rev. B 75, 155326 (2007).\n[19] D.-Y. Liu and J.-B. Xia, J. Appl. Phys. 115, 044313 (2014).\n[20] F. K. Joibari, Ya. M. Blanter, and G. E. W. Bauer, Phys. Rev. B 90, 155301 (2014).\n[21] S. Gumber,M. Gambhir, P. K. Jha, and Man Mohan J. Appl. Phys. 119, 073101 (2016).\n[22] S. A. Wolf et al., Science, 294, 1488 (2001).\n[23] Semiconductor spintronics and quantum computation, editied b y D. D. Awschalom, N. Samarth and\nD. Loss (Springer, Berlin, 2002).\n[24] J. R. Petta et al., Science, 309, 2180 (2005).\n[25] D. V. Khomitsky, L. V. Gulyaev and E. Ya. Sherman Phys. Rev. B 85, 125312 (2012).\n[26] D. V. Khomitsky and E. Ya. Sherman, Phys. Rev. B 79, 245321 (2009).\n[27] D. Khomitsky and E. Sherman, Nanoscale Research Letters 6, 212,(2011).\n[28] D. V. Khomitsky and E. Ya. Sherman, EPL, 90, (2010) 27010.\n[29] D. Bhowmik and S. Bandyopadhyay, Physica E, 41, 587 (2009)."    },    {        "title": "2005.01391v1.Spin_orbit_torque_magnonics.pdf",        "content": "1\nSpin-orbit-torque magnonics \nV. E. Demidov,1* S. Urazhdin,2 A. Anane,3 V. Cros,3 S. O. Demokritov1 \n1Institute for Applied Physic s and Center for Nanotechnology , University of Muenster, \nCorrensstrasse 2-4, 48149 Muenster, Germany   \n2Department of Physics, Emory University, Atlanta, GA 30322, USA \n3Unité Mixte de Physique, CNRS, Thales, Uni versité Paris-Saclay, 91767, Palaiseau, France  \nThe field of magnonics, which ut ilizes propagating spin waves f or nano-scale transmission \nand processing of information, has  been significantly advanced by the advent of the spin-orbit \ntorque. The latter phenomenon ca n allow one to overcome two mai n drawbacks of magnonic \ndevices – low energy efficienc y of conversion of electrical sig nals into spin wave signals, and \nfast spatial decay of spin waves in thin-film waveguiding struc tures. At first glance, the \nexcitation and amplification of s pin waves by spin-orbit torque s can seem to be \nstraightforward.  Recent researc h indicates, however, that the lack of the mode-selectivity in \nthe interaction of spin current s with dynamic magnetic modes an d the onset of dynamic \nnonlinear phenomena represent si gnificant obstacles. Here, we d iscuss the possible route to \novercoming these limitations, based on the suppression of nonli near spin-wave interactions in \nmagnetic systems with perpendicular magnetic anisotropy. We sho w that this approach \nenables efficient excitation of  coherent magnetization dynamics  and propagating spin waves \nin extended spatial regions, a nd is expected to enable practica l implementation of complete \ncompensation of spin-wave propagation losses. \nKeywords: spin current, ma gnonics, magnetic nanostructures \n* Corresponding author, e-mail: demidov@uni-muenster.de  \n  2\nI. Introduction \nHigh-frequency spin waves propa gating in thin ferromagnetic fil ms have been utilized for \nthe implementation of advanced microwave devices for many decad es.1-5 Unique features of \nthese waves include electronic tunability by the magnetic field , very short (millimeter to \nsubmicrometer) wavelengths in the microwave frequency range, as  well as controllability of \npropagation characteristics by the direction of the static magn etic field relative to the direction \nof wave propagation.3-5 These features made spin waves very attractive for the impleme ntation \nof a variety of devices for comm unication technologies, such as  microwave filters, phase \nshifters, delay lines, multiplexers, etc. With the rapid increa se in recent years of the operational \nfrequencies of conventional elec tronic systems, in particular t hose for computation and \ninformation processing, high-fre quency spin waves have become i ncreasingly attractive for the \nimplementation of integrated logic circuits and nanoscale wave- based computing systems.6-8 \nThis has motivated the emergence of a new research field - magn onics,9-12 - which explores the \npossibility to utilize spin wav es for transmission and processi ng of information at nanoscale as \nan alternative to conventional CM OS-based electronic circuits. \nDespite numerous advantages provided by spin waves, magnonic de vices currently suffer \nfrom two drawbacks. First, the inductive mechanism traditionall y utilized to convert electrical \nsignals into spin waves and back, is characterized by relativel y low conversion efficiency.13-18 \nEven if low conversion losses can be achieved in traditional ma croscopic spin-wave devices \nwith millimeter-scale dimensions, miniaturization of magnonic d evices down to the sub-\nmicrometer scale inevitably results in large conversion losses unacceptable for real-world \napplications. The second drawback of microscopic magnonic devic es is associated with large \npropagation losses of spin waves in nanometer-thick magnetic fi lms. In macroscopic spin-wave \ndevices based on high-quality micro meter-thick films of Yttrium  Iron Garnet (YIG), the decay \nlength of spin waves reaches several millimeters. However, in m icroscopic waveguides 3\npatterned from ultrathin YIG f ilms, the decay length typically does not exceed a few tens on \nmicrometers19-21.   \nTo make the microscopic magnonic devices technologically compet itive, it is necessary to \novercome their two main drawback s described above. This may bec ome possible th anks to the \nadvent of spin-torque phenomena22,23, which allow reduction of magnetic damping, and even \nits complete compensation, enabling excitation of high-frequenc y magnetization oscillations by \ndc electrical currents24,25. In particular, it has been show n that spin torques created by  spin-\npolarized electric currents or by non-local spin injection can excite propagating spin waves26-\n32, and that this excit ation mechanism is intrinsically fast31 and highly power-efficient.30 These \nachievements clearly demonstrated a large potential of spin tor ques for magnonics. However, \nthe devices proposed in Refs. 26-32 utilized current injection through a magnetic nanocontact , \nresulting only in local spin torque. It could not provide compe nsation of damping over spatially \nextended regions, which is of vital importance for magnonic app lications that require enhanced \nspin wave propagation. \nIn contrast to the conventional spin torques produced by the lo cal current injection through \nconducting magnetic materials, the spin-orbit torque (SOT) asso ciated with charge to spin \nconversion in the bulk or at interfaces in material systems wit h strong spin-orbit interaction,33-\n36 provides the ability to control magnetic damping in spatially extended regions of both \nconducting and insulating magnetic materials37-39. This may enable decay-free propagation of \nspin waves, or even their true amplification in magnetic nanost ructures. In recent years, this \npossibility was intensively  explored for macroscopic40-43 and microscopic44-48 magnonic \ndevices. The largest effect reported so far was achieved in mic roscopic magnonic waveguides \nbased on ultrathin YIG films,46 where a nearly tenfold increase in the propagation length of \ncoherent spin waves was demonstr ated. The high efficiency of th e system studied in Ref. 46 \npermitted experimental access to the operational regime close a nd even above the point at which 4\ndamping is expected to be completely compensated by SOT.  Contr ary to naive expectations, it \nwas found that the onset of nonlinear damping in the overdriven  magnon system does not allow \none to achieve complete compensation of spin-wave propagation l osses, required for true \namplification of cohere nt spin waves by SOT. \nNote that the same fundamental mechanisms also prevent SOT-indu ced excitation of \ncoherent magnetization oscillati ons in spatially extended syste ms.39,49 In Ref. 50, it was shown \nthat the adverse effects of non linear damping can be suppressed  by localizing the injection of \nspin current to a nanoscale regi on of the magnetic system. This  a p p r o a c h  e n a b l e d  t h e  \nimplementation of SOT-driven ma gnetic nano-oscillators that exh ibit coherent high-frequency \nmagnetization oscillations50-61. However, it also imposed strict limitations on the layout of these \ndevices, hindering their utiliz ation as sources of propagating spin waves. As a result, it took a \nsignificant time to understand how confined-area SOT-driven osc illators can be integrated with \na medium supporting spin-wave propagation.62,63 We emphasize that, while spin-wave \nexcitation by SOT was demonstrat ed in Refs. 62,63, these works have not resolved the general \nproblem of the adverse eff ects of nonlinear damping.  \nIn this brief perspective article, we present our vision of the  main outstanding issues \nhindering further developments in the field of SOT-driven magno nics, and outline an approach \nthat can help to overcome them. We first discuss the key recent  experimental results revealing \nthe physical nature of these issues. In particular, we show tha t the functionality of SOT \nmagnonic devices is strongly limite d by the lack of spin-wave m ode selectivity of the \ninteraction of spin currents with the magnetization, which resu lts in the simultaneous \nenhancement by SOT of many incoherent spin-wave modes. This enh ancement causes an onset \nof the nonlinear scattering of coherent spin waves, which count eracts their enhancement by the \nanti-damping effect of SOT. We discuss a recently proposed appr oach that allows efficient \ncontrol of nonlinear damping, by utilizing reduction of ellipti city of magnetization precession 5\nin magnetic films where the dipolar anisotropy is compensated b y the interfacial perpendicular \nmagnetic anisotropy (PMA). This approach has been recently show n to enable SOT-driven \nexcitation of coherent magnetization auto-oscillations in spati ally extended systems based on \nconductive64 and insulating65 magnetic materials. It was also demonstrated to enable efficie nt \nexcitation of coherent propagating spin waves in magnetic insul ators.65 We project that this \napproach can also enable the implementation of decay-free propa gation of spin waves, \nresolving, in this way, the main issues limiting the progress i n nano-magnonics.  \nII. Amplification of spin waves by SOT \nFigure 1 presents the results of the experiment that demonstrat ed highly efficient \ncompensation of propagation losses of spin waves by SOT46. The test devices used in this study \n(Fig. 1(a)) are 1 m-wide spin-wave waveguides patterned from a 20 nm-thick YIG fi lm \ncovered by an 8 nm-thick layer of Pt. Propagating spin waves ar e excited in the waveguide by \nmicrowave current flowing through a 3 m wide and 250 nm thick Au inductive antenna. \nAdditionally, a dc electrical current I is applied through the Pt layer of the waveguide. The \nelectrical current is converted into an out-of-plane pure spin current IS (see the inset in Fig. \n1(a)), due to the spin-Hall effect (SHE)33-36 in Pt. In turn, the spin current exerts an anti-damping \nspin-transfer torque37 on the magnetization M in YIG, which is expected to compensate the \npropagation losses of the coherent spin wave emitted by the ant enna.  \nThe effects of SOT on the spin-w ave propagation were studied by  using micro-focus \nBrillouin light scattering (BLS) spectroscopy.66  The probing laser light was focused through \nthe sample substrate into a diffraction-limited spot on the YIG /Pt film (Fig. 1(a)), and the \nmodulation of light due to its interaction with  magnetic excita tions in YIG was analyzed. The \nresulting signal – the BLS inten sity – is proportional to the i ntensity of magnetic oscillations at \nthe position of the probing spot, which enables mapping of the intensity of propagating spin \nwaves with high spatial r esolution (Fig. 1(b)). 6\nFigure 1(c) shows several represen tative dependences of spin-wa ve intensity on the \npropagation coordinate x, obtained at different dc currents in the Pt layer. As seen fr om these \ndata, spin waves in the waveguide experience well-defined expon ential decay ~ exp(-2 x/), \nwhere  is the decay length, defined as the distance over which the wa ve amplitude decreases \nby a factor of e. The decay length i ncreases with the increase of the dc current, as expected for \nthe effects of SOT on the effective magnetic damping. Figure 1( d) shows the current \ndependences of the decay length  and of its inverse value – the decay constant, which is \nproportional to the effective Gilbert damping constant. The dec ay length monotonically \nincreases at small currents, reaches a maximum at a certain cur rent I = IC, and then abruptly \ndecreases at larger currents. Independent measurements were use d to identify IC as the critical \ncurrent, at which SOT is expected to completely compensate the natural damping in YIG. Thus, \none could expect that  diverges at I = IC (the decay constant vanishes), and that the propagating \nwave becomes spatially amplified at I > IC. These naive expectations are clearly inconsistent \nwith the experimental findings. \nThis experiment demonstrated tha t SOT is capable to increasing the propagation length of \nspin waves by nearly an order of magnitude. For the studied sys tem, this corresponded to the \nincrease of the spin-wave intensity at the output of a 10 m-long transmission line by three \norders of magnitude. Simultaneously, this experiment showed tha t the simple picture of SOT \neffects on the damping of coherent propagating spin waves becom es invalid in the vicinity of \nthe point of the complete damping compensation. As will be disc ussed below, in the regime of \nlarge currents, it is necessary t o take into account not only t hat SOT reduces the effective \ndamping, but also that it strongly enhances incoherent magnetic  fluctuations. The nonlinear \nscattering of coherent waves from these fluctuations represents  an additional damping channel \ncounteracting the anti- damping effect of SOT. \n 7\nIII. Excitation of spin waves by SOT \nWhile SOT-induced coherent magne tic auto-oscillations have been  achieved in several \nnanomagnetic device geometries50-53, and many other novel SOT oscillators have been \nproposed in recent years54-61, none of them provided the possibility to generate coherent \npropagating spin waves. This is mainly associated with the limi tations imposed by the device \nlayout necessary to achieve coherent  auto-oscillations. Indeed,  as was shown in Refs. 39,49, \ncoherent oscillations cannot be excited in spatially extended s ystems with arbitrary shape, due \nto the nonlinear interactions with  incoherent spin-wave modes s trongly enhanced by SOT. To \nsuppress these nonlinear effects, spin current must be injected  into a nanoscale region of the \nmagnetic system, which imposes strict limitations on the layout  of SOT-driven auto-oscillators. \nAfter an intensive search for a suitable geometry, in Ref. 62, we proposed a new concept \nof nano-notch SOT auto-oscillato rs directly incorporated into a  magnonic nano-waveguide. \nThese devices (Fig. 2(a)) were based on 180 nm-wide nano-wavegu ides patterned from a \nPermalloy(Py)(15 nm)/Pt(4 nm) bi layer. Ion milling was used to pattern a rectangular 200 nm-\nwide and 10 nm-deep notch in the top Py layer of the waveguide,  forming a nano-oscillator that \nserves as the spin-wave source. When electric current I flows through the device, SHE in Pt \ninjects pure spin current IS into the Py layer, producing anti-damping SOT acting on its \nmagnetization M. The thickness-averaged magnitude of the anti-damping torque i s inversely \nproportional to the thickness of the magnetic layer. Thus, the SOT effects on the 5 nm-thick Py \nlayer in the nano-notch area ar e significantly larger than on t he 15 nm-thick Py waveguide. As \nthe current I is increased, damping becomes completely compensated in the na no-notch region, \nresulting in the local excitat ion of magnetizati on auto-oscilla tions. \nThese devices exhibit two-mode a uto-oscillations (Fig. 2(b)). T he frequency of the low-\nfrequency mode, excited at small currents, is smaller than the frequencies of propagating spin \nwaves in the 15 nm-thick waveguide. Correspondingly, the spatia l mapping of the oscillations 8\nby micro-focus BLS showed that this mode is localized in the na no-notch and does not emit \nspin waves into the waveguide (Fig. 3(c)). In contrast, the hig h-frequency mode, excited at \nlarger currents, was found to efficiently emit propagating spin  waves (Fig. 3(d)). This emission \nwas found to be strongly unidirec tional, with the preferential direction controlled by the \ndirection of the static magne tic field. Additionally, it was sh own that the propagation length of \nemitted spin waves is enhanced by non-zero spin current injecte d over the entire length of the \nwaveguide, by up to a factor of three. \nThe system proposed in Ref. 62 combines all the advantages prov ided by SOT to locally \nexcite propagating spin waves, and to simultaneously enhance th eir propagation characteristics. \nThe achieved enhancement can be further increased by the materi al engineering and geometry \noptimization. Additionally, the proposed approach can be easily  scaled to chains of SOT nano-\noscillators coupled via propaga ting spin waves, facilitating th e development of novel nanoscale \nsignal processing circuits such as logic and neuromorphic compu ting networks. We would like \nto note, however, that in spite of all these advantages, the pr oposed system also suffers from \nthe limitations associated with the nonlinear spin-wave scatter ing. The latter adversely affects \nthe oscillation characteristics of nano-notch devices at large driving currents, resulting in a \nstrong reduction of the intensity of emitted spin waves (Fig. 2 (b)), and does not allow one to \nachieve complete compensation of  their propagation losses. \nIV. Control of nonlinear damp ing in SOT-driven devices \nThe effects of SOT on the magnetization are often approximated as a simple modification \nof the effective magnetic damping37. This simple picture neglects the effects of fluctuations \nalways present at finite temperatures. Analysis taking into acc ount this contribution reveals \nanother important effect of SOT – namely, it drives the spin sy stem out of thermal equilibrium, \nresulting in the enhancement of magnetic fluctuations, which ca n equivalently be described as \nexcitation of a large number of incoherent spin waves (magnons)  spread over a broad interval 9\nof frequencies and wavelengths.39,67 The importance of this cont ribution has been recognized \nstarting with the first experimen ts on the interaction of spin currents with magnetization.49 Later \non, excitation of incoherent magnons by SOT and their propagati on received a significant \nattention as a mechanism for transmission of spin information i n magnetic insulators.68-72 \nHowever, the spectral characteri stics of magnons excited by SOT  remained unknown for a long \ntime. This issue was addressed experimentally in Ref. 67. In th is work, by using the BLS \ntechnique, we were able to study the effects of SOT on the magn on distribution in a Py/Pt \nbilayer over a significant spectral range. \nFigure 3(a) shows the BLS spectra reflecting the spectral densi ty of magnons recorded with \nand without dc current in Pt.  The spectrum obtained at I = 0 characterizes the magnons present \nin Py due to thermal fluctuations at room temperature. When the  current is applied, the \npopulations of all magnon states s ignificantly increase due to the effects of SOT. Note that the \nenhancement of the magnon population is most significant at low  f r e q u e n c i e s ,  a n d  r a p i d l y  \ndecreases with the increase of the frequency of magnons (Fig. 3 (b)), which can be associated \nwith the larger relaxation rate s of magnons with higher frequen cies. With the increase in the \nSOT strength, the dominant mode at the lowest frequency fmin is expected to transition to the \nauto-oscillation regime at its damping compensation point. Howe ver, the increase in the \namplitudes of all the other mode s leads to their nonlinear coup ling, which results  in the energy \nflow from the dominant mode (see, e.g., results of micromagneti c simulations Fig. 5d in Ref. \n64). This process represents an onset of additional (nonlinear)  damping, which prevents \ncomplete damping compensation by SOT.  \nThe adverse effects of nonlinear damping can be reduced by supp ressing the amplitudes of \nparasitic incoherent modes. This approach was used to achieve S OT-driven coherent \nmagnetization oscillations by utilizing local injection of spin  current.39,50 In this geometry, \nparasitic incoherent spin waves quickly escape from the localiz ed active area, resulting in 10\nreduced nonlinear damping of the  dominant mode. This approach g enerally restricts the active \nregion to nanoscale dimensions, limiting the achievable dynamic al coherence and the \npossibilities for the magnonic device integration. Moreover, th is approach cannot be extended \nto the compensation of propagati on losses of spin waves by SOT over extended spatial areas.  \nWe have recently proposed a new, more efficient approach to sup pression of nonlinear \ndamping, based on the direct control of the mode coupling mecha nisms.64 In this work, we \nshowed that the nonlinear spin wave coupling is predominantly d etermined by the ellipticity of \nmagnetization precession, which is controlled in thin magnetic films by the demagnetizing \neffects. We demonstrated that, by using magnetic films with sui tably tailored perpendicular \nmagnetic anisotropy (PMA), one can compensate the dipolar aniso tropy and achieve almost \ncircular precession, resulting in suppression of nonlinear damp ing. As a result, we were able to \nachieve complete damping compensation and excitation of coheren t magnetization auto-\noscillations by SOT, in a simple system with uniform spatially- extended injection of spin \ncurrent. \nFigure 4(a) shows the layout of the test device studied in this  work. It consists of a 5 nm-\nthick magnetic disk fabricated on top of an 8 nm-thick Pt strip , which plays the role of a spin-\ncurrent injector. We emphasize, that it is well known from earl y studies39,49, that in this \ngeometry it is impossible to achieve complete damping compensat ion and excitation of \nmagnetization auto-oscillations in magnetic systems with in-pla ne anisotropy, due to the \nadverse effects of the nonlinea r damping. Indeed , no transition  to auto-oscillations is observed \nwhen Permalloy is used as the magnetic disk material (Fig. 4(b) ). When the current I is applied \nthrough the Pt strip, the recorded BLS spectra show an enhancem ent of magnetic fluctuations \nin Py. The intensity of fluctuations gradually increases with i ncreasing I at currents smaller than \nthe critical value IC. However, at I > IC, the intensity of fluctuations saturates, while their \nspectral width signi ficantly increases.  11\nThe observed behaviors change dramatically, if the magnetic dis k is made from a Co/Ni \nbilayer, with the relative thic knesses of Co and Ni adjusted so  that the effective PMA field Ha \ncompensates the saturation magnetization 4 M of the bilayer (Fig. 4(c )). The compensation of \nthe dipolar anisotropy by PMA res ults in nearly c ircular magnet ization precession trajectory. \nThis can be contrasted with the magnetization precession of the  Py disk, which exhibits a large \nellipticity (see the insets in Fi gs. 4(b) and 4(c)). As seen fr om Fig. 4(c), at I < IC, the effects of \nSOT in CoNi disk are very similar to those in the Py disk.  How ever, at I > IC, a narrow intense \nspectral peak emerges for CoNi, marking a transition to the aut o-oscillation regime. This result \nshows that the nonlinear damping that prevents the onset of aut o-oscillations in the Py disk is \nsuppressed in CoNi. \nThe experimental results describ ed above, together with the mic romagnetic simulations of \nthese systems, clearly demonstrated a route for overcoming the limitations imposed by the \nnonlinear damping, by utilizing PMA materials with tailored ani sotropy strength. This approach \nallows one to achieve complete compensation of the magnetic dam ping, and excitation of \ncoherent magnetization auto-oscillations by SOT, without confin ing the spin-current injection \nto a nanoscale area. This can enable the implementation of SOT- driven oscillators with spatially \nextended active area, capable of generating microwave signals w ith technologically relevant \npower levels and coherence, and t hus circumventing the challeng es of phase locking of a large \nnumber of oscillators with nano- scale dimensions. The proposed approach also provides a route \nfor the implementation of spatially extended amplification of c oherent propagating spin waves. \nV. Highly efficient SOT-driven s pin-wave emission in YIG films with PMA \nThe main challenges in the application of the approach describe d above to metallic \nmultilayers with PMA are associated with their large magnetic d amping, and the strong spatial \ninhomogeneity of magnetic properties typical for these systems.  Recent progress in the \ndeposition of high-quality nanometer-thick films of YIG can all ow one to overcome these 12\nlimitations. In particular, it was shown in Ref. 73 that Bi dop ing of ultrathin YIG films can \nfacilitate a large PMA controllable by the epitaxial strain and  growth-induced anisotropies, \nwhile preserving their low-dampi ng characteristics. These prope rties make Bi-doped YIG films \nuniquely suitable for SOT-based magnonic devices utilizing supp ression of nonlinear damping \nby the anisotropy compensation. \nFigure 5(a) shows the layout of the SOT devi ce based on an exte nded 20-nm thick film of \nBi-doped YIG (BiYIG) (Bi 1Y2Fe5O12) with PMA.65 To achieve suppression of the nonlinear \ninteractions, the strength of PMA is tuned to exactly compensat e the dipolar anisotropy. The \ndevice utilizes a simple large-scale injector of spin current, formed by a 6-nm thick Pt strip line \nwith the width of 1 m and the length of 4 m. Similar simple spin-current injectors were \nutilized in many prior experime nts on YIG films without PMA.68-72 We emphasize that it was \nwell established in those studies, that SOT-driven emission of coherent propagating SW cannot \nbe achieved in in-plane magnetized  YIG, even at large driving c urrents exceeding IC. Instead, \nthe dominant mode saturates in the vicinity of the point of com plete damping compensation, \nsimilar to the Py disks on Pt (Fig. 4(b)).72 \nIn contrast, the devices based on BiYIG were found to exhibit a  clear transition to coherent \nauto-oscillations at moderate current densities in the Pt strip . By using local detection of the \nauto-oscillation spectra with BLS (Fig. 5(b)), we confirmed the  high spatial coherence of the \nauto-oscillations, whose freque ncy remains unchanged over the e ntire active area above the 1×4 \nµm Pt injector. In contrast to most of the previously demonstra ted SOT oscillators, the intensity \nof the auto-oscillation peak m onotonically increases with the i ncrease of the current strength \nover a broad range of currents (Fig. 5(c)), consistent with sup pression of nonlinear interactions \nthat typically result in the deg radation of the oscillation cha racteristics at large current densities \n(see, e.g., Fig. 2(b)). 13\nAnother consequence of the precise compensation of the dipolar anisotropy by PMA is the \nabsence of the nonlinear frequenc y shift of the auto-oscillatio ns with the increase of their \nintensity (Fig. 5(c)). The spectral coherence of auto-oscillati ons is known to be significantly \nenhanced when the nonlinear shift is small, as it hence reduces  the coupling between the \noscillation amplitude and its phase.74 In addition, the nonlinear frequency shift, common for in-\nplane magnetized devices (see,  e.g., Fig. 2(b)), is well-known to result in self-localization of \nauto-oscillation into a nonlinear spin-wave bullet.75,76 In the absence of the nonlinear shift, the \nself-localization is not expected  to occur, enabling emission o f coherent spin waves into the \nsurrounding film. Indeed, spatiall y resolved BLS measurements ( Fig. 5(d)) confirm efficient \nspin-wave emission from the active device area. As seen from th ese data, the oscillations \nexcited due to SOT ar e not localized in the area above the Pt l ine, but significantly extend into \nthe surrounding BiYIG film.  \nAnalysis performed in Ref. 65 showed that the wavelength of the  emitted spin waves is \nabout 300 nm, as determined by the difference in the PMA streng th in the bare BiYIG film and \nin the BiYIG/Pt bilayer. The wavelength can be controlled by va rying the difference between \nthese anisotropies. Additionally, the frequency of auto-oscilla tions and the wavelength of the \nemitted spin waves can be controlled by the current, in devices  where the strength of PMA is \nincreased beyond the point of exac t compensation of the dipolar  anisotropy. These conditions \ncorrespond to the positive nonlinear frequency shift, which all ows one to increase the auto-\noscillation frequency and decrease the wavelength of emitted sp in waves by the increase of the \ncurrent in the Pt line. \nThe results described above clearly demonstrate that suppressio n of the adverse nonlinear \ndamping in materials with PMA allows one to overcome the most s ignificant limitations \nhindering the practical realizati on of advantages provided by S OT in magnonics. We believe \nthat these findings should spur f urther progress in this field.  14\nVI. Conclusions \nDownscaling of magnonic devices p resents a large number of new challenges, which must \nbe addressed before spin-wave t echnology becomes a competitive alternative to conventional \nCMOS-based microelectronics. Si multaneously, downscaling provid es novel opportunities \nunavailable in traditional macros copic-scale spin-wave devices.  Spin-orbit torque  is one of the \nstriking examples of physical mechanisms that become particular ly efficient at nanoscale. \nUtilization of such phenomena is necessary for the successful d evelopment of nano-magnonics, \nenabling the implementation of nanodevices that are more effici ent and more attractive for real-\nworld applications than their previously demonstrated macroscop ic counterparts. Naturally, \nnovel physical phenomena, which have not yet been fully underst ood, require intense research, \noften revealing new unexpected d ifficulties and challenges as w ell as new opportunities. This \nis clearly the case with SOT-driven excitation and amplificatio n of spin waves. The results \nobtained in early experiments in the “small-amplitude” regime c ould be easily interpreted by \nusing a simple interpretation of  the effects of SOT in terms of  the variation of the effective \ndamping. However, with the development of highly-efficient SOT systems, where complete \ndamping compensation can be achieved, new challenges have emerg ed, which can only be \novercome based on the deep insi ght into nonlinear dynamic proce sses in strongly -driven spin \nsystems. We believe that the recently proposed approach discuss ed in this article will finally \nenable efficient integration of s pin-orbitronics and magnonics,  and will accelerate new \ndevelopments in the latter field. \n ACKNOWLEDGMENTS \nThis work was supported in part by the Deutsche Forschungsgemei nschaft (Project No. \n423113162), the NSF Grant No. ECCS-1804198, by the French ANR G rants 15-CE08-0030-\n01 (ISOLYIG) and the ANR-18-CE24-0021 (MAESTRO). The authors acknowledge the \nessential contribution of the PhD  students that have contribute d to some of the studies presented 15\nin this article: Martin Collet, Boris Divinskiy, Michael Evelt,  Diane Gouéré, Hugo Merbouche, \nand Lucile Soumah. \n \nData Availability \nThe data that support the findings of this study are available from the corresponding author \nupon reasonable request. \n \nReferences  \n \n1 J.D. Adam, “Analog signal proc essing with microwave magnetics, ” Proc. IEEE 76, 159-170 \n(1988). \n2 W. S. Ishak, “Magnetostatic wav e technology: A review,” Proc. IEEE 76, 171-187 (1988). \n3 P. Kaboš and V. S. Stalmachov, “Magnetostatic Waves and Their Application,” Chapman & \nHall, New York, 1994. \n4 A.G. Gurevich and G.A. Melkov, “Ma gnetization Oscillations and  Waves,” CRC, New \nYork, 1996. \n5 D. D. Stancil and A. Prabhakar, “ Spin Waves: Theory and Applic ations,” Springer, New \nYork, 2009. \n6 A. Khitun, M. Bao, and K. L. Wang , “Magnonic logic circuits,” J. Phys. D 43, 264005 \n(2010). \n7 S. Dutta, S.-C. Chang, N. Kani , D. E. Nikonov, S. Manipatruni,  I. A. Young, and A.  \nNaeemi, “Non-volatile Clocked Spi n Wave Interconnect for Beyond -CMOS Nanomagnet \nPipelines,” Sci. Rep. 5, 9861 (2015). \n8 G. Csaba, A. Papp, and W. Porod, “Perspectives of using spin w aves for computing and \nsignal processing,” Phys. Lett. A 381, 1471 (2017). \n9 S. Neusser, D. Grundler, “Magnonics: Spin waves on the nanosca le,” Adv. Mater. 21, 2927–\n2932 (2009). \n10 V. V. Kruglyak, S. O. Demokritov,  and D. Grundler, “Magnonics, “ J. Phys. D 43, 264001 \n(2010). \n11 B. Lenk, H. Ulrichs, F. Garbs, M. Münzenberg, “The building bl ocks of magnonics,” Phys. \nRep. 507, 107–136 (2011). 16\n12 A. Chumak, V. Vasyuchka, A. Serga, B. Hillebrands, “Magnon spi ntronics,” Nat. Phys. 11, \n453–461 (2015). \n13 B. A. Kalinikos, “Excitation of pr opagating spin waves in ferr omagnetic films,” IEE Proc. \nH. 127, 4–10 (1980). \n14 V. Vlaminck and M. Bailleul, “Spin-wave transduction at the su bmicrometer scale: \nExperiment and modeling,” Phys. Rev. B 81, 014425 (2010).  \n15 V. E. Demidov, M. P. Kostylev, K. Rott, J. Münchenberger, G. R eiss, and S. O. \nDemokritov, “Excitation of short- wavelength spin waves in magno nic waveguides,” Appl. \nPhys. Lett. 99, 082507 (2011). \n16 H. Yu, O. d’ Allivy Kelly, V. Cro s, R. Bernard, P. Bortolotti,  A. Anane, F. Brandl, F. \nHeimbach, and D. Grundler, “Approa ching soft X-ray wavelengths in nanomagnet-based \nmicrowave technology,” Nat. Commun. 7, 11255 (2016). \n17 C. Liu, J. Chen, T. Liu, F. H eimbach, H. Yu, Y. Xiao, J. Hu, M . Liu, H. Chang, T. \nStueckler, S. Tu, Y. Zhang, Y. Zhang, P. Gao, Z. Liao, D. Yu, K . Xia, N. Lei, W. Zhao, \nand M. Wu, “Long-distance propaga tion of short-wavelength spin waves,” Nat. Commun. \n9, 738 (2018). \n18 A. Papp, G. Csaba, H. Dey, M. Madami, W. Porod, and G. Carlott i, “Waveguides as \nsources of short-wavelength spin w aves for low-energy ICT appli cations,” Eur. Phys. J. B \n91, 107 (2018). \n19 P. Pirro, T. Brächer, A. V. Chu mak, B. Lägel, C. Dubs, O. Surz henko, P. Görnert, B. \nLeven, and B. Hillebrands, “Spin-wave excitation and propagatio n in microstructured \nwaveguides of yttrium iron garnet /Pt bilayers,” Appl. Phys. Let t. 104, 012402 (2014). \n20 M. B. Jungfleisch, W. Zhang, W. Jiang, H. Chang, J. Sklenar, S . M. Wu, J. E. Pearson, A. \nBhattacharya, J. B. Ketterson, M . Wu, and A. Hoffmann, “Spin wa ves in micro-structured \nyttrium iron garnet nanometer-thick films,” J. Appl. Phys. 117, 17D128 (2015). \n21 M. Collet, O. Gladii, M. Eve lt, V. Bessonov, L. Soumah, P. Bor tolotti, S. O. Demokritov, \nY. Henry, V. Cros, M. Bailleul, V. E. Demidov, and A. Anane, “S pin-wave propagation in \nultra-thin YIG based wave guides,” Appl. Phys. Lett. 110, 092408 (2017). \n22 J. C. Slonczewski, “Current-drive n excitation of magnetic mult ilayers,” J. Magn. Magn. \nMater. 159, L1–L7 (1996). \n23 L. Berger, “Emission of spin waves by a magnetic multilayer tr aversed by a current,” Phys. \nRev. B 54, 9353–9358 (1996). 17\n24 J. A. Katine, F. J. Albert, R . A. Buhrman, E. B. Myers, and D.  C. Ralph, “Current-Driven \nMagnetization Reversal and Spin-Wa ve Excitations in Co/Cu/Co Pi llars,” Phys. Rev. Lett. \n84, 3149-3152 (2000). \n25 M. Tsoi, A. G. M. Jansen, J. Bass, W.-C. Chiang, V. Tsoi, and P. Wyder, “Generation and \ndetection of phase-coherent curr ent-driven magnons in magnetic multilayers,” Nature 406, \n46-48 (2000). \n26 V. E. Demidov, S. Urazhdin and S . O. Demokritov, “Direct obser vation and mapping of \nspin waves emitted by spin-torque  nano-oscillators,” Nature Mat er. 9, 984-988 (2010). \n27 M. Madami, S. Bonetti, G. Cons olo, S. Tacchi, G. Carlotti, G. Gubbiotti, F. B. Mancoff, M. \nA. Yar, J. Åkerman, “Direct obser vation of a propagating spin w ave induced by spin-\ntransfer torque,” Nat. Nanotech. 6, 635–638 (2011). \n28 H. Ulrichs, V. E. Demidov, S. O . Demokritov, and S. Urazhdin, “Spin-torque nano-emitters \nfor magnonic applications,” Appl. Phys. Lett. 100, 162406 (2012). \n29 S. Urazhdin, V. E. Demidov, H. Ulr ichs, T. Kendziorczyk, T. Ku hn, J. Leuthold, G. Wilde \nand S. O. Demokritov, “Nanomagnoni c devices based on the spin-t ransfer torque,” Nat. \nNanotech. 9, 509-513 (2014). \n30 V. E. Demidov, S. Urazhdin, R. Li u, B. Divinskiy, A. Telegin, and S. O. Demokritov, \n“Excitation of coherent propaga ting spin waves by pure spin cur rents,” Nat. Commun. 7, \n10446 (2016). \n31 B. Divinskiy, V. E. Demidov, S. O. Demokritov, A. B. Rinkevich , and S. Urazhdin, “Route \ntoward high-speed nano-magnonics  provided by pure spin currents ,” Appl. Phys. Lett. 109, \n252401 (2016). \n32 A. Houshang, R. Khymyn, H. Fulara , A. Gangwar, M. Haidar, S. R . Etesami, R. Ferreira, P. \nP. Freitas, M. Dvornik, R. K. Dum as, J. Åkerman, “Spin transfer  torque driven higher-\norder propagating spin waves in nano-contact magnetic tunnel ju nctions,” Nat. Commun. \n9, 4374 (2018). \n33 A. Hoffmann, “Spin Hall Effects in Metals,” IEEE Trans. Magn. 49, 5172–5193 (2013). \n34 J. Sinova, S. O. Valenzuela , J. Wunderlich, C. H. Back, and T.  Jungwirth, “Spin Hall \neffects,” Rev. Mod. Phys. 87, 1213 (2015). \n35 A. Soumyanarayanan, N. Reyre n, A. Fert, and C. Panagopoulos, “ Emergent phenomena \ninduced by spin–orbit coupling at surfaces and interfaces,” Nat ure 539, 509–517 (2016). \n36 A. Manchon, J. Železný, I.M. M iron, T. Jungwirth, J. Sinova, A. Thiaville, K. Garello, and \nP. Gambardella, “Curre nt-induced spin-orbit torques in ferromag netic and \nantiferromagnetic systems,” Rev. Mod. Phys. 91, 035004 (2019). 18\n37 K. Ando, S. Takahashi, K. Harii, K. Sasage, J. Ieda, S. Maekaw a, and E. Saitoh, “Electric \nManipulation of Spin Relaxation U sing the Spin Hall Effect,” Ph ys. Rev. Lett. 101, 036601 \n(2008). \n38 Y. Kajiwara, K. Harii, S. Takah ashi, J. Ohe, K. Uchida, M. Miz uguchi, H. Umezawa, H. \nKawai, K. Ando, K. Takanashi, S. M aekawa, and E. Saitoh, “Trans mission of electrical \nsignals by spin-wave interconvers ion in a magnetic insulator,” Nature 464, 262–266 \n(2010). \n39 V. E. Demidov, S. Urazhdin, G. de  Loubens, O. Klein, V. Cros, A. Anane, and S. O. \nDemokritov, “Magnetization osci llations and waves driven by pur e spin currents,” Phys. \nRep. 673, 1-31 (2017). \n40 Z. Wang, Y. Sun, M. Wu, V. Tiberke vich, and A. Slavin, “Contro l of SpinWaves in a Thin \nFilm Ferromagnetic Insulator th rough Interfacial  Spin Scatterin g,” Phys. Rev. Lett. 107, \n146602 (2011). \n41 E. Padron-Hernandez, A. Azevedo, and S. M. Rezende, “Amplifica tion of spin waves in \nyttrium iron garnet films through t he spin Hall effect,” Appl. Phys. Lett. 99, 192511 \n(2011). \n42 L.J. Cornelissen, J. Liu, B.J. van Wees, and R.A. Duine, “S pin-Current-Controlled \nModulation of the Magnon Spin Conduc tance in a Three-Terminal M agnon Transistor,” \nPhys. Rev. Lett. 120, 097702 (2018). \n43 T. Wimmer, M. Althammer, L. Lie nsberger, N. Vlietstra, S. Gepr ägs, M. Weiler, R. Gross, \nand H. Huebl, “Spin Transport in a  Magnetic Insulator with Zero  Effective Damping,” \nPhys. Rev. Lett. 123, 257201 (2019). \n44 V. E. Demidov, S. Urazhdin, A. B . Rinkevich, G. Reiss, and S. O. Demokritov, “Spin Hall \ncontrolled magnonic microwaveguides,” Appl. Phys. Lett. 104, 152402 (2014). \n45 K. An, D. R. Birt, C.-F. Pai , K. Olsson, D. C. Ralph, R. A. Bu hrman, and X. Li, “Control of \npropagating spin waves via spin transfer torque in a metallic b ilayer waveguide,” Phys. \nRev. B 89, 140405(R) (2014). \n46 M. Evelt, V. E. Demidov, V. B essonov, S. O. Demokritov, J. L. Prieto, M. Muñoz, J. Ben \nYoussef, V. V. Naletov, G. de L oubens, O. Klein, M. Collet, K. Garcia-Hernandez, P. \nBortolotti, V. Cros and A. Anane , “High-efficiency control of s pin-wave propagation in \nultra-thin yttrium iron garnet by the spin-orbit torque,” Appl.  Phys. Lett. 108, 172406 \n(2016). 19\n47 O. Gladii, M. Collet, K. Garcia- Hernandez, C. Cheng, S. Xavier , P. Bortolotti, V. Cros, Y. \nHenry, J.-V. Kim, A. Anane, and M. Bailleul, “Spin wave amplifi cation using the spin Hall \neffect in permalloy/platinum bilayers,” Appl. Phys. Lett. 108, 202407 (2016). \n48 A. Navabi, Y. Liu, P. Upadhyaya, K . Murata, F. Ebrahimi, G. Yu , Bo Ma, Y. Rao, M. \nYazdani, M. Montazeri, Lei Pan, I. N. Krivorotov, I. Barsukov, Q. Yang, P. K. Amiri, Y. \nTserkovnyak, and K. L. Wang, “Contro l of Spin-Wave Damping in Y IG Using Spin \nCurrents from Topological Insul ators,” Phys. Rev. Applied 11, 034046 (2019). \n49 V. E. Demidov, S. Urazhdin, E. R . J. Edwards, M. D. Stiles, R.  D. McMichael, and S. O. \nDemokritov, “Control of magnetic  fluctuations by spin current,”  Phys. Rev. Lett. 107, \n107204 (2011). \n50 V. E. Demidov, S. Urazhdin, H. Ulri chs, V. Tiberkevich, A. Sla vin, D. Baither, G. Schmitz, \nand S. O. Demokritov, “Magnetic na no-oscillator driven by pure spin current,” Nature \nMater. 11, 1028-1031 (2012). \n51 L. Liu, C.-F. Pai, D. C. Ralph, R . A. Buhrman, “Magnetic Oscil lations Driven by the Spin \nHall Effect in 3-Terminal Magne tic Tunnel Junction Devices,” Ph ys. Rev. Lett. 109, \n186602 (2012). \n52 V. E. Demidov, S. Urazhdin, A . Zholud, A. V. Sadovnikov, and S . O. Demokritov, \n“Nanoconstriction-based spin-Hal l nano-oscillator,” Appl. Phys.  Lett. 105, 172410 (2014). \n53 Z. Duan, A. Smith, L. Yang, B. Youngblood, J. Lindner, V. E. D emidov, S. O. Demokritov, \nand I. N. Krivorotov, “Nanowire sp in torque oscillator driven b y spin orbit torques,” Nat. \nCommun. 5, 5616 (2014). \n54 M. Collet, X. de Milly, O. d’Allivy Kelly, V.V. Naletov, R. Be rnard, P. Bortolotti, J. Ben \nYoussef, V. E. Demidov, S. O. Dem okritov, J. L. Prieto, M. Muno z, V. Cros, A. Anane, G. \nde Loubens, and O. Klein, “Genera tion of coherent spin-wave mod es in yttrium iron garnet \nmicrodiscs by spin–orbit torque,” Nat. Commun. 7, 10377 (2016). \n55 T. Chen, R. K. Dumas, A. Ekl und, P. K. Muduli, A. Houshang, A.  A. Awad, P. Dürrenfeld, \nB. G. Malm, A. Rusu, J. Åkerman, “ Spin-torque and spin-Hall nan o-oscillators,” Proc. \nIEEE 104, 1919–1945 (2016). \n56 P. Dürrenfeld, A. A. Awad, A. H oushang, R. K. Dumas, and J. Åk erman, “A 20 nm spin \nHall nano-oscillator,” Nanoscale 9, 1285–1291 (2017). \n57 B. Divinskiy, V. E. Demidov, A. Kozhanov, A. B. Rinkevich, S. O. Demokritov, and S. \nUrazhdin, “Nanoconstriction spin-Ha ll oscillator with perpendic ular magnetic anisotropy,” \nAppl. Phys. Lett. 111, 032405 (2017). 20\n58 A. A. Awad, P. Dürrenf eld, A. Houshang, M. Dvornik, E. Iacocca , R. K. Dumas, and J. \nÅkerman, “Long-range mutual sync hronization of spin Hall nano-o scillators,” Nat. Phys. \n13, 292 (2017). \n59 B. Divinskiy, V. E. Demidov, S. Urazhdin, R. Freeman, A. B. Ri nkevich, and S. O. \nDemokritov, “Controllable exc itation of quasi-l inear and bullet  modes in a spin-Hall nano-\noscillator,” Appl. Phys. Lett. 114, 042403 (2019). \n60 N. Sato, K. Schultheiss, L. Kör ber, N. Puwenberg, T. Mühl, A. A. Awad, S.S.P.K. \nArekapudi, O. Hellwig, J. Fassb ender, and H. Schultheiss, “Doma in Wall Based Spin-Hall \nNano-Oscillators,” Phys. Rev. Lett. 123, 057204 (2019). \n61M. Zahedinejad, A. A. Awad, S. Muralidhar, R. Khymyn, H. Fulara , H. Mazraati, M. \nDvornik, and J. Åkerman, “Two-dime nsional mutually synchronized  spin Hall nano-\noscillator arrays for neuromor phic computing,” Nat. Nanotech. 15, 47-52 (2020). \n62 B. Divinskiy, V. E. Demidov, S. Urazhdin, R. Freeman, A. B. Ri nkevich, and S. O. \nDemokritov, “Excitation and Amp lification of Spin Waves by Spin –Orbit Torque,” Adv. \nMater. 30, 1802837 (2018). \n63 H. Fulara, M. Zahedinejad, R . Khymyn, A. A. Awad, S. Muralidha r, M. Dvornik, and J. \nÅkerman, “Spin-orbit torque–driven  propagating spin waves,” Sci . Adv. 5, eaax8467 \n(2019). \n64 B. Divinskiy, S. Urazhdin, S. O . Demokritov, and V. E. Demidov , “Controlled nonlinear \nmagnetic damping in spin-Hall nano-devices,” Nat. Commun. 10, 5211 (2019). \n65 M. Evelt, L. Soumah, A. B. Ri nkevich, S. O. Demokritov, A. Ana ne, V. Cros, J. Ben \nYoussef, G. de Loubens, O. Klei n, P. Bortolotti, and V. E. Demi dov, “Emission of \ncoherent propagating magnons by ins ulator-based spin-orbit-torq ue oscillators,” Phys. Rev. \nAppl. 10, 041002 (2018). \n66 V. E. Demidov and S. O. Demokritov, “Magnonic waveguides studi ed by micro-focus \nBrillouin light scattering,” IEEE Trans. Mag. 51, 0800215 (2015). \n67 V. E. Demidov, S. Urazhdin, B. Divinskiy, V. D. Bessonov, A. B . Rinkevich, V.V. \nUstinov, and S. O. Demokritov, “C hemical potential of quasi-equ ilibrium magnon gas \ndriven by pure spin current,” Nat. Commun. 8, 1579 (2017). \n68 L. J. Cornelissen, J. Liu, R. A. Duine, J. Ben Youssef, and B.  J. van Wees, “Long-distance \ntransport of magnon spin informa tion in a magnetic insulator at  room temperature,” Nat. \nPhys. 11, 1022-1026 (2015). 21\n69 S. T. B. Goennenwein, R. Schlitz, M. Pernpeintner, K. Ganzhorn , M. Althammer, R. Gross, \nand H. Huebl, “Non-local magnetore sistance in YIG/Pt nanostruct ures,” Appl. Phys. Lett. \n107, 172405 (2015). \n70 L. J. Cornelissen, K. J. H. Peter s, G. E. W. Bauer, R. A. Duin e, and B. J. van Wees, \n“Magnon spin transport drive n by the magnon chemical potential in a magnetic insulator,” \nPhys. Rev. B 94, 014412 (2016). \n71 D. Wesenberg, T. Liu, D. Balzar, M. Wu, and B. L. Zink, “Long- distance spin transport in a \ndisordered magnetic ins ulator,” Nat. Phys. 13, 987 (2017). \n72 N. Thiery, A. Draveny, V. V. Nal etov, L. Vila, J. P. Attané, C . Beigné, G. de Loubens, M. \nViret, N. Beaulieu, J. Ben You ssef, V. E. Demidov, S. O. Demokr itov, A. N. Slavin, V. S. \nTiberkevich, A. Anane, P. Borto lotti, V. Cros, and O. Klein,  “ Nonlinear spin conductance \nof yttrium iron garnet thin films  driven by large spin-orbit to rque,” Phys. Rev. B 97, \n060409 (2018). \n73 L. Soumah, N. Beaulieu, L. Qassy m, C. Carrétéro, E. Jacquet, R . Lebourgeois, J. Ben \nYoussef, P. Bortolotti, V. Cros , and A. Anane, “Ultra-low dampi ng insulating magnetic \nthin films get perpendicular,” Nat. Commun. 9, 3355 (2018). \n74 A. Slavin and V. Tiberkevich, “No nlinear Auto-Osc illator Theor y of Microwave \nGeneration by Spin-Polarized C urrent,” IEEE Trans. Magn. 45, 1875-1918 (2009). \n75 A. N. Slavin and V. S. Tiberkevich, “Spin Wave Mode Excited by  Spin-Polarized Current \nin a Magnetic Nanocontact is a S tanding Self-Localized Wave Bul let,” Phys. Rev. Lett. 95, \n237201 (2005). \n76 M. Dvornik, A. A. Awad, and J. Åkerman, “Origin of Magnetizati on Auto-Oscillations in \nConstriction-Based Spin Hall Na no-Oscillators,” Phys. Rev. Appl . 9, 014017 (2018). \n  22\n \n \n \n \n \n \n \n \nFig. 1. (a) Schematic of the SOT device based on the Pt/YIG bil ayer waveguide, and the \nexperimental setup. (b) Measured spatial map of the BLS intensi ty, which is proportional \nto the local spin-wave intensity. (c) Dependences of the spin-w ave intensity on the \npropagation coordinate, recorded at different dc currents in th e Pt layer, as labeled. \nSymbols – experimental data, curves – exponential fits. (d) Cur rent dependences of the \ndecay length (point-up triangles) and of its inverse value (poi nt-down triangles). IC marks \nthe critical current, at which S OT is expected to completely co mpensate the natural \ndamping. Reproduced from M. Evelt, V. E. Demidov, V. Bessonov, S. O. Demokritov, J. \nL. Prieto, M. Muñoz, J. Ben Youssef, V. V. Naletov, G. de Loube ns, O. Klein, M. Collet, \nK. Garcia-Hernandez, P. Bortolotti, V. Cros and A. Anane, Appl.  Phys. Lett. 108, 172406 \n(2016), with the permission of AIP Publishing. \n  \n23\n \n \n \n \n \n \n \n \nFig. 2. (a) Schematic of the SOT device base d on the notched Pt/Py bilayer nano-waveguide, \nand the experimental setup. (b) Normalized color-coded map of the BLS intensity in the \nfrequency-current coordinates. (c) and (d) Color-coded spatial maps of the BLS intensity, \nmeasured at the frequencies of t he localized and of the radiati ng modes, as labeled. Dashed \nlines on the maps show the outlines of the waveguide and of the  nano-notch.  Reproduced \nwith permission from B. Divinskiy, V. E. Demidov, S. Urazhdin, R. Freeman, A. B. \nRinkevich, and S. O. Demokritov, Adv. Mater. 30, 1802837 (2018). Copyright 2018 John \nWiley and Sons. \n  \n24\n \n \n \n \n \n \n \n \nFig. 3. Effects of SOT on spectral magnon distribution for an S OT device based on crossed Pt \nand Py strips. (a) BLS spectra, reflecting the spectral density  of magnons, recorded at I=0 and \n20 mA, as labeled. Vertical dashe d line marks the frequency of the lowest-energy magnon \nstate. (b) Enhancement of the magnon population at I=20 mA. Reproduced from V. E. \nDemidov, S. Urazhdin, B. Divinskiy, V. D. Bessonov, A. B. Rinke vich, V.V. Ustinov, and S. \nO. Demokritov, Nat. Commun. 8, 1579 (2017); licensed under a Creative Commons \nAttribution (CC BY) license. \n  \n25\n \n \n \n \n \n \n \n \n \n \nFig. 4. (a) Layout of the test SOT devices based on Py and CoNi disks on th e Pt strip. (b) and \n(c) BLS spectra of magnetic oscillations vs current for Py and CoNi disks, respectively. IC \nmarks the critical current, at which SOT is expected to complet ely compensate the natural \nlinear magnetic damping.  Insets illustrate the ellipticities o f magnetization precession in Py \nand CoNi, respectively. Reproduced from B. Divinskiy, S. Urazhd in, S. O. Demokritov, and \nV. E. Demidov, Nat. Commun. 10, 5211 (2019); licensed under  a Creative Commons \nAttribution (CC BY) license. \n  \n26\n\n\n\n\n\n\n\n\nFig. 5. (a) Schematic of the SOT devices utilizing Pt strip on extended BiY IG film . (b) \nRepresentative BLS spectrum of m agnetization auto-oscillations recorded at I=2.5 mA. Note \nthat the width of the measured spectral peak is determined by t he limited frequency resolution \nof BLS. (c) Color-coded BLS inte nsity in the current-frequency coordinates. (d) Color-coded \nspatial map of the BLS intensity recorded by rastering the prob ing laser spot over 20 m by 7 \nm area. Dashed lines show the c ontours of the Pt line. Reproduc ed with permission from M. \nEvelt, L. Soumah, A. B. Rinkevich , S. O. Demokritov, A. Anane, V. Cros, J. Ben Youssef, G. \nde Loubens, O. Klein, P. Bortolo tti, and V. E. Demidov, Phys. R ev. Appl. 10, 041002 (2018). \nCopyright 2018 by the American Physical Society. \n\n\n \n"    },    {        "title": "2304.00528v2.Interedge_spin_resonance_in_the_Kitaev_quantum_spin_liquid.pdf",        "content": "Interedge spin resonance in the Kitaev quantum spin liquid\nTakahiro Misawa,1Joji Nasu,2and Yukitoshi Motome3\n1Beijing Academy of Quantum Information Sciences, Haidian District, Beijing 100193, China\n2Department of Physics, Tohoku University, Sendai 980-8578, Japan\n3Department of Applied Physics, University of Tokyo, Tokyo 113-8656, Japan\n(Dated: September 11, 2023)\nThe Kitaev model offers a platform for quantum spin liquids (QSLs) with fractional excitations,\nitinerant Majorana fermions and localized fluxes. Since these fractional excitations could be utilized\nfor quantum computing, how to create, observe, and control them through the spin degree of freedom\nis a central issue. Here, we study dynamical spin transport in a wide range of frequency for the\nKitaev-Heisenberg model, by applying an AC magnetic field to an edge of the system. We find that,\nin the Kitaev QSL phase, spin polarizations at the other edge are resonantly induced in a specific\nspin component, even though the static spin correlations are vanishingly small. This interedge spin\nresonance appears around the input frequency over the broad frequency range. Comparing with the\ndynamical spin correlations, we clarify that the resonance is governed by the itinerant Majorana\nfermions with a broad continuum excitation spectrum, which can propagate over long distances,\nalthough it vanishes for the pure Kitaev model because of accidental degeneracy and requires weak\nHeisenberg interactions. We also find that the spin polarizations in the other spin components are\nweakly induced at an almost constant frequency close to the excitation gap of the localized fluxes,\nirrespective of the input frequency. These results demonstrate that the dynamical spin transport is\na powerful probe of the fractional excitations in the Kitaev QSL. Possible experimental realization\nof the interedge spin resonance is discussed.\nI. INTRODUCTION\nExotic quasiparticles emerging in solids have attracted\nmuch interest from both fundamental physics and in-\ndustry applications. A prominent example is Majorana\nparticles — charge-neutral spin-1 /2 particles that are\ntheir own antiparticles1,2. While they usually behave\nas fermions, in some two-dimensional cases they can\nbe regarded as anyons that obey neither Fermi-Dirac\nnor Bose-Einstein statistics. Such Majorana particles\nhave been intensively studied for applications to quan-\ntum computing by using the anyonic nature3,4.\nThe Kitaev model on a honeycomb lattice offers an\nideal platform for realizing the Majorana particles5. It is\nan exactly solvable model whose ground state is a quan-\ntum spin liquid (QSL). The Kitaev QSL hosts two types\nof emergent quasiparticles from the fractionalization of\nspins: itinerant Majorana fermions and localized fluxes.\nThese quasiparticles turn into Abelian anyons when the\ninteractions between spins are largely anisotropic, or non-\nAbelian anyons when an external magnetic field is ap-\nplied in the nearly isotropic cases3,5. The Kitaev model\ncould be realized in Mott insulators with the strong spin-\norbit coupling6, such as Na 2IrO37andα-RuCl 38. De-\ntailed comparisons between experimental results and the-\noretical calculations have revealed fingerprints of the Ma-\njorana particles in such candidate materials; for a re-\nview, see Ref. 9. Among them, the discovery of the\nhalf-quantized thermal Hall effect in α-RuCl 3was ground\nbreaking, providing direct evidence for the Majorana par-\nticles10–12, while it is still under debate13–15.\nSince the Majorana fermions and the fluxes in the Ki-\ntaev QSL are generated by the fractionalization of spins,\nthey are quantum entangled and inherently nonlocal.Indeed, the spatial correlations between the Majorana\nfermions are long-range with power-law decay16,17, al-\nthough the spin correlations are short-ranged and van-\nish beyond nearest-neighbor sites18. Furthermore, in the\npresence of defects or edges, the spin correlations can\nbe long-range due to low-energy excitations around the\ndefects or edges16,17,19–21.\nBy exploiting such nonlocal nature, it was recently\nshown that the itinerant Majorana fermions can con-\ntribute to long-range spin transport from an edge of the\nsystem22,23. The previous study has focused only on\nlow-energy properties, such as the velocity of the spin\npropagation determined by the slope of the gapless Majo-\nrana dispersion. However, the spin dynamics in the wider\nrange of frequency is expected to offer more important in-\nsights into the two types of fractional quasiparticles with\ndistinct excitation spectra. Such comprehensive study\nof nonlocal spin dynamics would also be a crucial step\ntoward quantum computing, by elucidating how to con-\ntrol and probe the fractional quasiparticles via the spin\ndegree of freedom.\nIn this paper, in order to deepen the understanding\nof the relationship between the fractional quasiparticles\nand the spin degree of freedom, we study nonlocal spin\ndynamics in the Kitaev QSL in the wide frequency range.\nApplying a local AC magnetic field to one edge of the sys-\ntem, we investigate how the spin excitations are excited\nand propagate to the other edge. At the edges of the Ki-\ntaev model, it is known that local magnetic fields excite\nthe fluxes accompanied by gapless Majorana excitations,\ncalled the Majorana zero modes5,16. In the present study,\nwe introduce not static but time-dependent local mag-\nnetic fields at one edge and investigate how the excited\nspin polarizations propagate through the system. FromarXiv:2304.00528v2  [cond-mat.str-el]  8 Sep 20232\n(a) armchair (b) zigzag\nxy\nzhin(t)\nSiout(t)hin(t) Siout(t)\nFIG. 1. Schematic pictures of the Kitaev-Heisenberg model in\nEq. (1) with (a) the armchair edges and (b) the zigzag edges.\nIn (a) [(b)], we set the periodic and open (open and periodic)\nboundary conditions in the horizontal and vertical directions,\nrespectively. The blue, green, and red bonds represent the x,\ny, and zbonds for the Kitaev interaction, respectively. We\napply an AC magnetic field hin(t) in the [111] direction in\nspin space at one site on the edge shown by the yellow circles,\nand observe spin polarization at a site on the opposite edge\nshown by the orange circles; see Eqs. (3), (4), and (6).\nthe comprehensive analysis of the spin-component depen-\ndence and the comparison with the results for magnet-\nically ordered phases, we show that the dynamical spin\ntransport is a good probe for both itinerant Majorana\nfermions and localized fluxes. Our results give an insight\non the way of creating and controlling of the fractional\nexcitations via the spin degree of freedom.\nThe organization of this paper is as follows. In Sec. II,\nwe introduce the model and the setup used in this study,\nwith the details of real-time evolution and the definitions\nof static and dynamical spin correlations. In Sec. III A,\nwe show the phase diagram and the static spin correla-\ntions in our model with edges. In Sec. III B, we show\nhow an AC local magnetic field at the edge induces the\nspin polarization at the opposite edge of the system in\nthe ferromagnetic phase, the Kitaev QSL, and the stripy\nphase. In Sec. III C, we analyze the results in compari-\nson with the dynamical spin correlations between edges,\nand discuss the origin of the interedge dynamical spin\ntransport. Section IV is devoted to a summary.\nII. MODEL AND METHOD\nIn this paper, we employ the Kitaev-Heisenberg model,\nwhose Hamiltonian is given by\nˆHKH=KX\nνX\n⟨i,j⟩νˆSν\niˆSν\nj+JX\n⟨i,j⟩ˆSi·ˆSj, (1)\nwhere ˆSν\nidenotes the νth component of the spin-1/2\noperator at ith site: ˆSi= (ˆSx\ni,ˆSy\ni,ˆSz\ni). The first\nterm represents the bond-dependent Ising-type interac-\ntion, called the Kitaev interaction, where ⟨i, j⟩νrep-resents the nearest-neighbor ν(=x, y, z ) bonds on the\nhoneycomb lattice, and the second term represents the\nspin-isotropic Heisenberg interaction for all the nearest-\nneighbor bonds; see Fig. 1. Following the previous stud-\nies7,24, we parametrize the two coupling constants as\n(K, J) =\u0012\nsinα,1\n2cosα\u0013\n. (2)\nNote that the amplitudes of interactions are halved com-\npared to the previous ones so that |K|= 1 in the pure\nKitaev cases with α=π/2 and 3 π/2. In the follow-\ning, we focus on the range of π≤α≤7π/4 where\nthe Kitaev interaction is ferromagnetic. In this region,\nthe bulk system with the periodic boundary conditions\nshows three phases in the ground state24: the ferro-\nmagnetic phase for π≤α≲1.40π, the Kitaev QSL\nphase for 1 .40π≲α≲1.58π, and the stripy phase for\n1.58π≲α≲1.81π.\nTo study spin correlations and dynamics on the edges,\nwe consider the model in Eq. (1) on a strip with the\nopen boundary condition in one direction and the pe-\nriodic boundary condition in the other. There are two\ntypes of such strips. One has the so-called armchair type\nedges on the open boundaries, and the other has the so-\ncalled zigzag edges. Figure 1 displays these two types for\n24-site clusters used in the following calculations. For\nboth clusters, we examine how a time-dependent local\nmagnetic field on one edge induces spin polarization at\nthe other edge. Specifically, we apply an AC magnetic\nfield in the [111] direction in spin space at iinth site on\nthe edge (shown by the yellow circle in Fig. 1) as\nˆH(t) =ˆHKH+hin(t)·ˆSiin, (3)\nwith\nhin(t) =heccos (Ω t), (4)\nwhere his the amplitude of the AC field, ec=\n(1,1,1)/√\n3, and Ω = 2 π/T is the frequency of the AC\nfield ( Trepresents the oscillation period). We take the\nmagnetic field along the [111] direction since it coupled\nto all spin components. For this Hamiltonian, we solve\nthe time-dependent Schr¨ odinger equation given by\nid|Φ(t)⟩\ndt=ˆH(t)|Φ(t)⟩, (5)\nstarting from the initial condition of |Φ(t= 0)⟩=|ΦGS⟩,\nwhere |ΦGS⟩is the normalized ground state of ˆHKH. The\nspin polarization on the opposite edge is calculated as\nSν\niout(t) =⟨Φ(t)|ˆSν\niout|Φ(t)⟩, (6)\nwhere ioutdenotes the site on the other edge directly\nopposite to the iinth site (shown by the orange circle in\nFig. 1). In the following calculations, we take h= 0.05 in\nEq. (4) and solve Eq. (5) by using HΦ25; we discretize the\ntime with ∆ t= 0.05, which is small enough to preserve\nthe unitarity of real-time evolution.3\nIn addition to the real-time dynamics of the spin polar-\nization, we calculate the static and dynamical spin corre-\nlations between the two edges for the ground state |ΦGS⟩,\nwhich are defined by\nCνν\nedge=⟨ΦGS|ˆSν\niinˆSν\niout|ΦGS⟩, (7)\nand\nCνν\nedge(ω) =1\n2πZ∞\n−∞⟨ΦGS|δˆSν\niin(t)δˆSν\niout|ΦGS⟩eiωtdt,\n(8)\nrespectively, where δˆSν\ni=ˆSν\ni− ⟨ΦGS|ˆSν\ni|ΦGS⟩and\nδˆSν\niin(t) =eiˆHKHtδˆSν\niine−iˆHKHt. Note that in Eq. (8) we\nconsider correlations between the deviations from the ex-\npectation values for the ground state to subtract the elas-\ntic components in the presence of magnetic ordering. In\nthe actual calculations of Eq. (8), we employ the follow-\ning formula in the spectral representation:\nCνν\nedge(ω)\n=−1\n4πImh\n⟨Ψ+|(EGS−ˆHKH+ω+iη)−1|Ψ+⟩\n−⟨Ψ−|(EGS−ˆHKH+ω+iη)−1|Ψ−⟩i\n,\n(9)\nwhere |Ψ±⟩= (δˆSν\niin±δˆSν\niout)|ΦGS⟩,EGSis the ground-\nstate energy, and ηis an infinitesimal positive constant;\nwe take η= 0.05 in the following calculations. We calcu-\nlate Eq. (9) using the continued-fraction expansion based\non the Lanczos method.\nIn the calculations of Eqs. (7) and (9), we apply a\nweak static magnetic field to all the spins at the edge\non the iinth side with hs= 0.005ecto lift the ground-\nstate degeneracy in the ferromagnetic Heisenberg model\nwith α=πand the pure Kitaev model with α= 3π/2\n(see Appendix A). For the other cases, the ground state\nis not degenerate, but we apply the same weak field for\ncomparison.\nIII. RESULTS\nA. Phase diagram and static spin correlations\nBefore going into the spin dynamics, we discuss the\nground states of the clusters with edges shown in Fig. 1.\nFigures 2(a) and 2(b) show the αdependences of the\nsecond derivative of the ground-state energy for the sys-\ntems with the armchair and zigzag edges, respectively.\nThe two peaks at α=αc1andα=αc2indicate phase\ntransitions between the Kitaev QSL and the magnet-\nically ordered phases. Figures 2(c) and 2(d) display\nthe static interedge spin correlations defined by Eq. (7).\nFrom these data, we identify three different phases: the\nferromagnetic phase with a positive spin correlation for\nα < α c1, the Kitaev QSL with almost zero correlation\n(e) armchair (f) zigzag\niinioutiin\niout\nFIG. 2. αdependences of the second derivative of the ground-\nstate energy for the systems with (a) the armchair edges and\n(b) the zigzag edges. Corresponding static interedge spin cor-\nrelations are plotted in (c) and (d). (e) and (f) show the\nschematic pictures of the stripy order in the case of armchair\nand zigzag edges, respectively.\nforαc1< α < α c2, and the stripy phase with a nega-\ntive (positive) correlation for the system with the arm-\nchair (zigzag) edges for α > α c2. The antiferromagnetic\nand ferromagnetic spin correlations in the stripy phase\nare understood from the schematic pictures in Figs. 2(e)\nand 2(f), respectively. We note that in the ferromagnetic\nand stripy phases the spin correlations are dominant in a\nspecific spin component Szdue to the presence of edges,\nexcept for α=πwhere the ground state is degenerate in\nthe absence of the weak magnetic field hs.\nOur phase diagrams obtained for the clusters with\nedges are nearly identical to that for the same size clus-\nter under the periodic boundary conditions24. For the\nsystem with the armchair (zigzag) edges, we find that\nthe phase boundary between the ferromagnetic and Ki-\ntaev QSL phases is at αc1≃1.41π(1.49π) and that be-4\ntween the Kitaev QSL and stripy phases is at αc2≃1.57π\n(1.65π). These estimates are close to those for the clus-\nters with the periodic boundary conditions, αc1≃1.40π\nandαc2≃1.58π24. This indicates that the bulk proper-\nties are not much affected by the introduction of edges\neven for clusters of this size. In the following sections,\nwe will compute the spin dynamics in the three phases:\nFor the ferromagnetic, Kitaev QSL, and stripy phases,\nwe take α= 1.25π, 1.52π, and 1 .67π, respectively, for\nboth cases of the armchair and zigzag edges.\nLet us comment on the symmetry of the two clusters\nin Fig. 1. In the bulk system of the Kitaev-Heisenberg\nmodel, there is a four-sublattice transformation which\ndoes not change the form of the Hamiltonian with re-\nplacing KandJbyK+Jand−J, respectively24. This\ntransformation leads to the relation between the phase\nboundaries as tan αc2=−tanαc1−1. In the system\nwith the armchair edges, the relation holds for our es-\ntimates of αc1andαc2, since the cluster respects the\nfour-sublattice symmetry. In contrast, in the case of the\nzigzag edges, the symmetry is lost, and αc1andαc2do\nnot satisfy the relation.\nB. Real-time spin dynamics\nWe now turn to discuss how the spin at the edge is po-\nlarized when the AC magnetic field is applied to the spin\nat the other edge. Below, we present the results for the\nsystems with armchair and zigzag edges in Secs. III B 1\nand III B 2, respectively, and discuss the interedge spin\nresonance in the Kitaev QSL in Sec. III B 3.\n1. Armchair edge\nFigure 3 displays the time evolution of spin polariza-\ntion at the ioutth site in Eq. (6) for the system with\narmchair edges in Fig. 1(a). In the main panels, we show\nthe results for the period of the oscillating field, T= 10,\n20, 40, 60, and 80. Meanwhile, in the insets, we plot the\nFourier transformed spectra obtained by\nSν(ω) =\f\f\f\f2\ntmaxZtmax\n0Sν\niout(t)eiωtdt\f\f\f\f, (10)\nwhere we take tmax= 500 so that the lowest-energy scale\n2π/tmax∼0.013 is well below the excitation energy of\nthe localized fluxes ( ∼0.07) in the pure Kitaev model5.\nWe first discuss the results in the ferromagnetic phase\nshown in Figs. 3(a)–3(c). In this case, both Sx\niout(t) and\nSy\niout(t) show considerable oscillations, while Sz\niout(t) does\nnot. These behaviors are understood from the spin order-\ning in the ground state: As shown in Fig. 2(c), the spins\nare ferromagnetically ordered in the zdirection, for which\nfluctuations appear dominantly in the transverse compo-\nnents, Sx\nioutandSy\niout, rather than the longitudinal one\nSz\niout. In the Fourier transformed spectra shown in the in-\nsets, we find that the dominant Sx(ω) and Sy(ω) alwaysshow a peak around ω= Ω = 2 π/T. This result indicates\nthat the spin polarization is induced dominantly at the\nsame frequency of the input AC magnetic field.\nNext, we turn to the results in the Kitaev QSL phase\nshown in Figs. 3(d)–3(f). In contrast to the above ferro-\nmagnetic case, we find that only Sy\niout(t) shows consider-\nable oscillations, while the others do not. This behavior\ncan be understood from the fractional excitations in the\nKitaev QSL as follows. In the exact solution for the pure\nKitaev model, as mentioned in Sec. I, the spins are frac-\ntionalized into itinerant Majorana fermions and localized\nfluxes. The former has gapless excitations, while the lat-\nter is gapped5. The spin excitation is a composite of these\ntwo, and hence gapped. Indeed, the spin excitations by\nˆSxorˆSzat the iinorioutth site are gapped since these\nspin operators do not commute with the flux operators\ndefined by products of six spins on the hexagons includ-\ning the iinorioutth site5. This suppresses Sx\niout(t) and\nSz\niout(t) in Figs. 3(d) and 3(f), respectively. In contrast,\nˆSyat the iinorioutth site commutes with the flux op-\nerators, since the hexagons lack the ybond. In addition\nto the hexagonal fluxes, in the cluster with the armchair\nedges, there are additional flux operators defined only\nby the edge spins. While ˆSyat the iinorioutth site do\nnot commute with these fluxes, the spin excitations re-\nmain gapless because of the degeneracy in the ground\nstate (see Appendix A). These allow the excitation by\nˆSy\niinto yield long-range spin propagation via the gapless\nitinerant quasiparticles and induce Sy\niout(t) in Fig. 3(e).\nAlthough the above argument is valid only for the pure\nKitaev case, similar behavior is expected to appear in\nthe Kitaev QSL phase in the presence of weak Heisen-\nberg interactions. This is the reason why only Sy\niout(t)\nshows significantly large oscillations in Figs. 3(d)–3(f).\nThe resonant behaviors in the Kitaev QSL phase ex-\nhibit the following characteristics. First, while Sy(ω) al-\nways shows a peak around the input frequency Ω as in\nthe ferromagnetic case, the peak height does not decrease\nbut rather increases with ω, as shown in the inset of\nFig. 3(e). This characteristic behavior will be discussed\nin Sec. III B 3. Second, we note that a weak Heisenberg\ninteraction is crucial for the long-range spin propagation\nsince the ground state degeneracy in the pure Kitaev case\nwith α/π= 1.5 prohibits the propagation, as we will dis-\ncuss in detail in Sec. III C. Finally, the above argument\nalso allows static interedge spin correlation in the ydi-\nrection, Syy\nedge, also to develop, but it is almost zero as\nshown in Fig. 2(c). This indicates that the interedge spin\ncorrelations in the Kitaev QSL can only be dynamically\nenhanced.\nLastly, we discuss the results in the stripy phase shown\nin Figs. 3(g)–3(i). In this case, the results are similar to\nthe ferromagnetic case in Figs. 3(a)–3(c). The reason\nis common: As shown in Fig. 2(c), the spins are anti-\nferromagnetically ordered in the zdirection in this stripy\nphase, and hence, the transverse components Sx\niout(t) and\nSy\niout(t) are induced dominantly at the input frequency.5\n−0.2−0.10.00.10.20.30.4(a)α/π =1.25 (ferro)\nSx\niout(t)\nT=10\nT=20\nT=40\nT=60\nT=80(b)α/π =1.25 (ferro)\nSy\niout(t)(c)α/π =1.25 (ferro)\nSz\niout(t)\n−0.2−0.10.00.10.20.30.4(d)α/π =1.52 (QSL)\nSx\niout(t)\nT=10\nT=20\nT=40\nT=60\nT=80(e)α/π =1.52 (QSL)\nSy\niout(t)(f)α/π =1.52 (QSL)\nSz\niout(t)\n0 100 200 300 400 500\nt−0.2−0.10.00.10.20.30.4(g)α/π =1.67 (stripy)\nSx\niout(t)\nT=10\nT=20\nT=40\nT=60\nT=80\n0 100 200 300 400 500\nt(h)α/π =1.67 (stripy)\nSy\niout(t)\n0 100 200 300 400 500\nt(i)α/π =1.67 (stripy)\nSz\niout(t)0.00 0.25 0.50 0.75\nω0.000.050.10\nSx(ω)\n0.00 0.25 0.50 0.75\nω0.000.050.10\nSy(ω)\n0.00 0.25 0.50 0.75\nω0.000.050.10\nSz(ω)\n0.00 0.25 0.50 0.75\nω0.000.050.10\nSx(ω)\n0.00 0.25 0.50 0.75\nω0.000.050.10\nSy(ω)\n0.00 0.25 0.50 0.75\nω0.000.050.10\nSz(ω)\n0.00 0.25 0.50 0.75\nω0.000.050.10\nSx(ω)\n0.00 0.25 0.50 0.75\nω0.000.050.10\nSy(ω)\n0.00 0.25 0.50 0.75\nω0.000.050.10\nSz(ω)\nFIG. 3. Time evolution of the spin polarization Sν\niout(t) in Eq. (6) for the system with armchair edges in (a)–(c) the ferromagnetic\nphase at α/π = 1.25, (d)–(f) the Kitaev QSL phase at α/π = 1.52, and (g)–(i) the stripy phase at α/π = 1.67: (a),(d),(g)\nν=x, (b),(e),(h) ν=y, and (c),(f),(i) ν=z. The data for the input oscillation periods T= 10, 20, 40, 60, and 80 are shown.\nThe insets display the corresponding Fourier transformed spin polarizations in Eq. (10). The vertical dashed lines denote the\nfrequencies corresponding to the values of T,ω= Ω = 2 π/T.\n2. Zigzag edge\nFigure 4 displays the results for the system with the\nzigzag edges in Fig. 1(b), obtained by the same condi-\ntions for the armchair case. For the ferromagnetic andstripy phases shown in Figs. 4(a)–4(c) and 4(g)–4(i), re-\nspectively, we find similar tendency to the armchair case:\nThe spin polarizations in the xandydirections are in-\nduced significantly, while that in the zdirection is rather\nsuppressed. This is again understood from the fact that6\n−0.2−0.10.00.10.20.30.4(a)α/π =1.25 (ferro)\nSx\niout(t)\nT=10\nT=20\nT=40\nT=60\nT=80(b)α/π =1.25 (ferro)\nSy\niout(t)(c)α/π =1.25 (ferro)\nSz\niout(t)\n−0.2−0.10.00.10.20.30.4(d)α/π =1.52 (QSL)\nSx\niout(t)\nT=10\nT=20\nT=40\nT=60\nT=80(e)α/π =1.52 (QSL)\nSy\niout(t)(f)α/π =1.52 (QSL)\nSz\niout(t)\n0 100 200 300 400 500\nt−0.2−0.10.00.10.20.30.4(g)α/π =1.67 (stripy)\nSx\niout(t)\nT=10\nT=20\nT=40\nT=60\nT=80\n0 100 200 300 400 500\nt(h)α/π =1.67 (stripy)\nSy\niout(t)\n0 100 200 300 400 500\nt(i)α/π =1.67 (stripy)\nSz\niout(t)0.00 0.25 0.50 0.75\nω0.000.050.10\nSx(ω)\n0.00 0.25 0.50 0.75\nω0.000.050.10\nSy(ω)\n0.00 0.25 0.50 0.75\nω0.000.050.10\nSz(ω)\n0.00 0.25 0.50 0.75\nω0.000.050.10\nSx(ω)\n0.00 0.25 0.50 0.75\nω0.000.050.10\nSy(ω)\n0.00 0.25 0.50 0.75\nω0.000.050.10\nSz(ω)\n0.00 0.25 0.50 0.75\nω0.000.050.10\nSx(ω)\n0.00 0.25 0.50 0.75\nω0.000.050.10\nSy(ω)\n0.00 0.25 0.50 0.75\nω0.000.050.10\nSz(ω)\nFIG. 4. Corresponding plots to Fig. 3 for the system with zigzag edges.\nthe ordered moments in each phase appear in the zdi-\nrection as shown in Fig. 2(d).\nMeanwhile, in the Kitaev QSL phase, as shown in\nFigs. 4(d)–4(f), we find that Sz\niout(t) shows significant\nlarge oscillations, while Sx\niout(t) and Sy\niout(t) do not. This\ncan also be understood from the flux excitations dis-\ncussed above for the armchair case. In the current case,\nthe zigzag edges lack the zbonds, and hence, the Szcom-\nponents can be excited without the gapped excitationsowing to the ground state degeneracy; see Appendix A.\nInterestingly, the amplitude of Sz\niout(t) varies nonmono-\ntonically with Tand takes the maximum value around\nT= 40, as shown in Fig. 4(f). In this case also, we ob-\nserve the peaks in Sz(ω) at almost the input frequencies,\nas shown in the inset of Fig. 4(f), while the peak heights\nshow nonmonotonic ωdependence. These features will\nbe discussed in the next section.7\n3. Interedge resonance\nIn Fig. 5, we show Ω = 2 π/Tdependences of the max-\nimum values of Sν(ω) (ν=x, y, z ) within the range\nof 2/tmax < ω/ 2π≤0.5, which are represented as\nSν(ωpeak), in the systems with (a)-(c) the armchair edges\nand (d)-(f) the zigzag edges. We also plot the values of\nωpeakas functions of Ω in each inset. Here, we choose\nthe lower limit as twice 2 π/tmaxto avoid an artifact near\n2π/tmaxand the upper limit to be sufficiently larger than\nthe bandwidth of the dynamical spin correlation26.\nIn both armchair and zigzag cases, Sx(ωpeak) and\nSy(ωpeak) have large values in the ferromagnetic and\nstripy phases, while Sz(ωpeak) are suppressed, since these\nphases show the spin orderings along the zdirection as\nmentioned above. The values of Sx(ωpeak) and Sy(ωpeak)\ndecrease as Ω increases. In addition, we find that the val-\nues of ωpeakare close to Ω as shown in each inset. These\nare indications of conventional magnetic resonances; the\nspin polarization is induced dominantly at the input fre-\nquency, as long as the magnetic excitations are available\nin the frequency range.\nIn contrast, the interedge spin resonance behaves dif-\nferently in the Kitaev QSL phase. First of all, we find\nthat the dominant polarizations, Sy(ωpeak) for the arm-\nchair case and Sz(ωpeak) for the zigzag case, show broad\npeaks in the wide frequency range up to Ω ≃1.5, while\nthe latter shows a sharp peak at a smaller Ω ≃0.2 as well.\nFor both cases, the relation ωpeak∼Ω holds, except for\nsmall Ω (Ω ≲0.2 for the armchair case and Ω ≲0.1 for\nthe zigzag case); the deviation might be due to the finite-\nsize effect. The broad responses with ωpeak∼Ω, as well\nas the sharp peak in the zigzag case, can be ascribed to\nthe itinerant Majorana fermions, whose density of states\nshows a continuum up to ω= 1.55. This point will be\nfurther discussed in Sec. III C.\nIn addition, for the other spin components with sup-\npressed polarizations, we find that ωpeakis almost con-\nstant irrespective of Ω27. This behavior could be ex-\nplained by the flux gap that governs the low-energy ex-\ncitations in these spin components. We note that the\nconstant values of ωpeak∼0.2 in the armchair case is\nlarger than the energy of the low-energy coherent peak\natω∼0.1 in the dynamical spin structure factor of the\npure Kitaev model in the thermodynamic limit26, but\nthis might also be due to the finite-size effect.\nIn the Kitaev QSL, Sx(ωpeak) is larger than Sz(ωpeak)\nin the armchair case, while Sx(ωpeak) and Sy(ωpeak) are\nalmost the same in the zigzag case. This is understood\nfrom the geometry of the Kitaev bonds in each cluster. In\nthe armchair case, as shown in Fig. 1(a), the zbonds on\nwhich ˆSzcomponents interact via the Kitaev interaction\nare along the edges and perpendicular to the direction\nfrom iintoiout, which may suppress the interedge spin\ntransport of the zcomponent. Meanwhile, in the zigzag\ncase shown in Fig. 1(b), both xandybonds are along the\nedges and related with each other by symmetry, leading\nto almost the same interedge resonances assisted by thezbonds connecting them.\nC. Comparison with dynamical spin correlations\nLet us discuss the characteristic interedge spin reso-\nnances in comparison with the dynamical interedge spin\ncorrelations Cνν\nedge(ω) defined in Eq. (8). Figure 6 displays\nCνν\nedge(ω) for the armchair and zigzag cases while varying\nα. In the following, we show that Cνν\nedge(ω) explains well\nthe intensities and Ω dependences of the induced spin\npolarizations in Fig. 5.\nIn the system with armchair edges, Cxx\nedge(ω) and\nCyy\nedge(ω) show considerable intensities over the broad ω\nrange, whereas Czz\nedge(ω) is almost zero except for the\nlow-ωweights near the phase boundaries at α=αc1and\nαc2. In the ferromagnetic phase for α≤αc1and the\nstripy phase for α≥αc2, this is again consistent with the\nfact that the spin moments are ordered along the zdirec-\ntion. A striking difference between Cxx\nedge(ω) and Cyy\nedge(ω)\nappears in the Kitaev QSL phase for αc1≤α≤αc2;\nCyy\nedge(ω) has large spectral weights over the broad ω\nrange, while Cxx\nedge(ω) is almost zero. Notably, the inten-\nsity of Cyy\nedge(ω) is stronger than those in the ferromag-\nnetic and stripy phases, while it vanishes for the pure\nKitaev case at α/π= 1.5 because of the degeneracy in\nthe ground state (see Appendix B). This strong Cyy\nedge(ω)\nexplains well the broad response in Sy(ωpeak) found in\nFig. 5(b). In addition, we note that Cxx\nedge(ω) has weak\nintensities at low ω∼0.1, as shown in Fig. 6(a). This\nalso explains well the small peak in Sx(ωpeak) found in\nFig. 5(a).\nMeanwhile, in the system with zigzag edges, Cνν\nedge(ω)\nin the ferromagnetic and stripy phases behave qualita-\ntively similarly to those in the armchair case. In the\nKitaev QSL phase, however, strong intensity appears in\nCzz\nedge(ω) over the broad ωrange, while Cxx\nedge(ω) and\nCyy\nedge(ω) are almost absent. Again, this explains well the\nbroad response in Sz(ωpeak) found in Fig. 5(f). Further-\nmore, the sharp peak at ω∼0.2 in Fig. 5(f) is also con-\nsistent with the strong intensity of Czz\nedge(ω) in Fig. 6(f).\nThe interedge resonances in the broad ωrange in the\nKitaev QSL phase are mediated by the itinerant Majo-\nrana fermions whose excitation spectrum has a contin-\nuum in the broad energy range. This is explicitly shown\nby calculating the dynamical spin correlations for the\npure Kitaev model at α/π = 1.5 by using the Majo-\nrana representation, which we denote CMaj\nedge(ω); see Ap-\npendix B for the details of the calculations. Figure 7\nshows the results of CMaj\nedge(ω) in comparison with Cyy\nedge(ω)\nandCzz\nedge(ω) around α/π= 1.5. Note that here we com-\npare their absolute values since the sign of CMaj\nedge(ω) in\nEq. (B3) is not well defined. We find that the broad\nresponses of Cyy\nedge(ω) and Czz\nedge(ω) in the vicinity of\nα/π= 1.5 appear in the same energy range of CMaj\nedge(ω)\nwith showing similar ωdependences. This indicates that8\n0.000.020.040.060.080.10\narmchair, Sx(ωpeak) (a)\nα=1.25 (ferro)\nα=1.52 (QSL)\nα=1.67 (stripy)\narmchair, Sy(ωpeak) (b)\n armchair, Sz(ωpeak) (c)\n0.0 0.5 1.0 1.5 2.0\nΩ0.000.020.040.060.080.10\nzigzag, Sx(ωpeak) (d)\nα=1.25 (ferro)\nα=1.52 (QSL)\nα=1.67 (stripy)\n0.0 0.5 1.0 1.5 2.0\nΩ\nzigzag, Sy(ωpeak) (e)\n0.0 0.5 1.0 1.5 2.0\nΩ\nzigzag, Sz(ωpeak) (f)0.0 0.5 1.0 1.5\nΩ0.00.51.01.5ωpeak\n0.0 0.5 1.0 1.5\nΩ0.00.51.01.5ωpeak\n0.0 0.5 1.0 1.5\nΩ0.00.51.01.5ωpeak\n0.0 0.5 1.0 1.5\nΩ0.00.51.01.5ωpeak\n0.0 0.5 1.0 1.5\nΩ0.00.51.01.5ωpeak\n0.0 0.5 1.0 1.5\nΩ0.00.51.01.5ωpeak\nFIG. 5. Maximum values of the Fourier transformed spin polarizations Sν(ωpeak) for the input oscillating magnetic field with\nΩ = 2 π/T in the systems with (a)–(c) the armchair edges and (d)–(f) the zigzag edges: (a) and (d) ν=x, (b) and (e) ν=y,\nand (c) and (f) ν=z. The insets display Ω dependences of ωpeak. The gray dashed line shows the relation ωpeak= Ω.\nthe broad responses in the Kitaev QSL phase are domi-\nnated by the itinerant Majorana excitations.\nWhile our results are limited to the small clusters, we\nexpect that the interedge resonances appear also in larger\nsystems since the itinerant Majorana fermions propagate\nover long distances in the Kitaev QSL. This is demon-\nstrated by calculating |CMaj\nedge(ω)|while changing the sys-\ntem width. Figure 8 shows the maximum intensity of\n|CMaj\nedge(ω)|as a function of the number of the unit cells\nin the direction perpendicular to the edges, L⊥. We find\nthat the dynamical correlations decay slowly: the zigzag\ncase roughly obeys ∝1/L⊥, while the armchair case\nshows crossover from ∝1/L⊥to∝1/L3\n⊥. The results\nappear to be consistent with the Majorana-mediated spin\ncorrelations17,22. Thus, we believe that, when domi-\nnated by the itinerant Majorana fermions, the interedge\ndynamical spin correlations become long-range in realspace, even in the presence of weak Heisenberg interac-\ntions28–30.\nCombining these results with the almost constant be-\nhaviors of ωpeakirrespective of Ω for the other suppressed\ncomponents in Fig. 5, we conclude that the interedge\nspin resonances in the Kitaev QSL are good probes of\ntwo types of fractional excitations, itinerant Majorana\nfermions and localized fluxes. The resonance in the spin\ncomponent which does not excite the fluxes on hexagons\nleads to broad responses with ωpeak∼Ω, as found in\nFigs. 5(b) and 5(f). This is a clear indication of the itin-\nerant Majorana excitations. Meanwhile, the responses in\nthe other spin components appear around a small con-\nstant ωpeak. This is an indication of the gapped flux\nexcitations. We emphasize that weak Heisenberg interac-\ntions are essential for the interedge spin resonances since\nallCνν\nedge(ω) vanish for the pure Kitaev model because of9\n012ωCxx\nedge(ω)armchairαc1αc2\n(a)\nCxx\nedge(ω)zigzagαc1αc2\n(b)\n012ωCyy\nedge(ω)(c)\nCyy\nedge(ω)(d)\n1.00 1.25 1.50\nα/π012ωCzz\nedge(ω)(e)\n1.00 1.25 1.50\nα/πCzz\nedge(ω)(f)\n−0.50.00.5\n−0.50.00.5\n−0.50.00.5\nFIG. 6. Dynamical interedge spin correlations Cνν\nedge(ω)\n[Eq. (8)] for (a),(c),(e) the armchair edge and (b),(d),(f) the\nzigzag edge: (a) and (b) ν=x, (c) and (d) ν=y, and (e)\nand (f) ν=z.\nthe ground-state degeneracy (see Appendix B).\nIV. SUMMARY\nIn summary, we have studied how an AC local mag-\nnetic field at an edge of the system induces spin polar-\nizations at the opposite edge in the Kitaev-Heisenberg\nmodel with ferromagnetic Kitaev interactions by using\nthe exact diagonalization. We found that in the Kitaev\nQSL phase the spin polarizations are resonantly induced\nin a particular spin component, in stark contrast to the\nmagnetically ordered phases where conventional mag-\nnetic resonances appear in the transverse spin compo-\nnents. The spin resonance in the Kitaev QSL shows the\nfollowing peculiar features, stemming from the fraction-\nalization of spin degree of freedom into two types of frac-\ntional excitations, itinerant Majorana fermions and local-\nized fluxes: (i) It appears dominantly in the spin compo-\nnent which does not excite flux excitations, (ii) the dom-\ninant resonance appears in a broad range of frequency,\nreflecting the continuum of Majorana excitations, (iii) it\nis accompanied by subdominant resonances in the other\nspin components at a small constant frequency corre-\n0.00.20.40.60.81.01.21.41.6ω(a)armchair\n1.451.501.55\nα/π(b)\n (c)zigzag\n1.451.501.55\nα/π(d)αc1\n0.00.10.20.30.40.5FIG. 7. Comparison between (a) |CMaj\nedge(ω)|calculated by\nEq. (B3) for the pure Kitaev model at α/π = 1.5 and (b)\nan enlarged plot of |Cyy\nedge(ω)|around α/π= 1.5 for the sys-\ntem with the armchair edges. (c) and (d) The corresponding\nplots for the zigzag case, where |Czz\nedge(ω)|is plotted in (d).\n100101102\nL⊥10−210−1100|CMaj\nedge(ω)|max\narmchair L/bardbl=4\nzigzag L/bardbl=3\nFIG. 8. Maximum of |CMaj\nedge(ω)|as a function of the number\nof the unit cells in the direction perpendicular to the edges,\nL⊥. The data are calculated for the clusters with the numbers\nof the unit cell along the edge, L∥= 4 and 3, for the armchair\nand zigzag cases, respectively; the result for the smallest L⊥\nin each case corresponds to that in Figs. 7(a) and 7(c).\nsponding to the flux excitation gap, (iv) both resonances\nvanish in the exact Kitaev QSL because of the ground-\nstate degeneracy and require weak Heisenberg interac-\ntions, and (v) they are induced only dynamically, despite\nthe disappearance of the static spin correlations. These\nresults elucidate that the nonlocal spin dynamics in the\nwide frequency range contains information on both two10\ntypes of fractional excitations in the Kitaev QSL, which\ncannot be captured by the spin transport in the low-\nenergy limit in the previous studies22,23. While our cal-\nculations were done for small size clusters, the interedge\nresonance is expected to be observed in larger systems,\nsince it is mediated by itinerant Majorana excitations\nthat propagate over long distances. These results indi-\ncate that the interedge dynamical spin resonance is useful\nfor probing the two types of fractional excitations in the\nKitaev QSL, which are usually difficult to observe only\nfrom static physical quantities.\nA straightforward experiment would be implemented\nby using a scanning tunneling microscope (STM) tip with\nmagnetic atoms or the atomic force microscopy (AFM)\nto apply an AC magnetic field at the edge and measure\nthe spin polarization at the opposite edge. This could be\nperformed, for example, for a thin flake of a candidate\nmaterial α-RuCl 3. Similar experiments would be pos-\nsible in interface or heterostructure of a Kitaev magnet\nand a ferromagnetic material, where the AC magnetic\nfield can be applied to the edge spins by the ferromag-\nnetic resonance. Careful analysis of the dynamics in each\nspin component and its dependence on the edge structure\nwould pave the way for creating and controlling the frac-\ntional excitations in the Kitaev QSL through the spin\ndegree of freedom.\nACKNOWLEDGMENTS\nWe wish to thank Y. Kato, K. Fukui, and T. Okubo\nfor fruitful discussions. This work was also supported\nby the National Natural Science Foundation of China\n(Grant No. 12150610462). TM was supported by Build-\ning of Consortia for the Development of Human Re-\nsources in Science and Technology, MEXT, Japan. This\nwork was supported by Grant-in-Aid for Scientific Re-\nsearch Nos. 19H05825, JP19K03742, and 20H00122 from\nthe Ministry of Education, Culture, Sports, Science\nand Technology, Japan. It is also supported by JST\nCREST Grant No. JPMJCR18T2 and JST PRESTO\nGrant No. JPMJPR19L5.\nAppendix A: Degeneracy in the pure Kitaev model\nwith edges\nIn this Appendix, we show that the ground state of the\npure Kitaev model with α= 3π/2 has the degeneracy\nin both cases of the armchair and the zigzag edges. In\nthe pure Kitaev model, one can define the flux operator\nˆWpby a product of six spins for each hexagon p, which\ncommutes with the Hamiltonian5. We show the examples\nin Fig. 9; note that ˆWpin (a) [(b)] commutes with ˆSy\niin\n(ˆSz\niin), since the hexagon lacks the y(z) bond at the iinth\nsite, as discussed in Sec. III B 1 (III B 2). In addition to\nthe six-spin flux operators, at the edges of the system\nthere are additional flux operators defined only by the\n(a) armchair (b) zigzag\nxy\nz\n01 2\n345 6\n7402\n513\nWp\nWout4WpWout3a\nWout3b\n^^\n^^\n^uij = −1\nuiiniout\nWout2^uiiniouty\nzzFIG. 9. Schematic pictures of the flux operators for (a) the\narmchair edges and (b) the zigzag edges. The numbers denote\nthe sites used for the definitions of the fluxes in Eq. (A1) and\nEq. (A2). We show examples of the six-spin flux operator\nˆWp, four-spin ( ˆW4) and two-spin flux operators ( ˆW2) and\nthe three-spin flux operators ( ˆW3aand ˆW3b). We also show\nthe interedge correlations of the localized Majorana particles\n(uiiniout) by the purple lines. In (a), we represent the zbonds\nwhere uijtakes -1 with red dashed lines. See Appendix B for\nuiinioutanduij.\nedge spins. For instance, the flux operators including the\noutput site are given by\nˆW4\nout= 24ˆSz\n0ˆSy\n1ˆSx\n2ˆSz\n3,ˆW2\nout= 22ˆSx\n0ˆSy\n7, (A1)\nfor the armchair case, and\nˆW3a\nout= 23ˆSx\n0ˆSz\n1ˆSy\n2,ˆW3b\nout= 23ˆSy\n0ˆSz\n5ˆSx\n4, (A2)\nfor the zigzag case; see Fig. 9. Since these flux operators\nˆWq(q= 4, 2, 3 a, and 3 b) commute with the Hamiltonian\natα= 3π/2, the ground state |ΦGS⟩is the eigenstate of\nthe flux operators as\nˆWq\nout|ΦGS⟩=W|ΦGS⟩, (A3)\nwhere the eigenvalue Wtakes +1 or −1. Meanwhile, all\nthe eigenstates of the Hamiltonian can be taken to be\nreal since the Kitaev Hamiltonian does not include the\ncomplex elements in the conventional basis set composed\nof the eigenstates of ˆSz\ni. Therefore, if the ground state\n|ΦGS⟩is unique, we obtain ⟨ΦGS|ˆWq\nout|ΦGS⟩= 0 since\nˆWq\noutis the pure imaginary operator including a single\nˆSy\ni. This contradicts with Eq. (A3), meaning that the\nassumption of a unique ground state is incorrect. Hence,\nthe ground state of the pure Kitaev model with the arm-\nchair and the zigzag edges must be degenerate. For the\n24-site clusters shown in Fig. 1, we numerically confirm\nthat the ground state has eightfold (fourfold) degener-\nacy for the clusters with the armchair (zigzag) edges.\nWe note that the numbers of the degenerate states can\nbe accounted for by the numbers of independent flux-\ntype operators traversing the system from one edge to\nthe other and those consisting of edge spins.11\nAppendix B: Dynamical spin correlations in the\npure Kitaev model\nIn this Appendix, we describe the method to calculate\nthe dynamical spin correlations for the pure Kitaev model\nin Figs. 7(a) and 7(c). We adopt the Majorana represen-\ntation of the Hamiltonian in Eq. (1) at α= 3π/2, which\nis given by5\nˆHK=1\n4X\n⟨i,j⟩νuν\nijiˆciˆcj, (B1)\nwhere the spin operator is represented as ˆSν\ni=i\n2ˆbν\niˆciby\nintroducing four Majorana fermions {ˆci,ˆbx\ni,ˆby\ni,ˆbz\ni}. Here,\nuν\nij=⟨ΦGS|ˆuν\nij|ΦGS⟩=⟨ΦGS|iˆbν\niˆbν\nj|ΦGS⟩; ˆuν\nijcommutes\nwith the Hamiltonian and uν\nijtakes±1.\nIn this Majorana representation, the interedge dynam-ical spin correlation is given by\n⟨ΦGS|ˆSν\niin(t)ˆSν\niout|ΦGS⟩\n=−1\n4uν\niiniout⟨ΦGS|iˆciin(t)ˆciout|ΦGS⟩, (B2)\nwhere ν=yandzfor the armchair and zigzag case, re-\nspectively, and uν\niinioutis defined for the unpaired ˆbν\niinand\nˆbν\niouton the edges as uν\niiniout=⟨ΦGS|iˆbν\niinˆbν\niout|ΦGS⟩. The\ncorrelations for the other spin components vanish. By\nsubstituting Eq. (B2) to Eq. (8), we obtain the dynam-\nical spin correlation between the edges in the Majorana\nrepresentation as\nCMaj\nedge(ω) =−uν\niiniout\n8πZ∞\n−∞⟨ΦGS|iˆciin(t)ˆciout|ΦGS⟩eiωtdt,\n(B3)\nwhere the time-dependent operator is defined by ˆHKin\nwhich uν\nijare chosen to realize the flux-free ground state\n|Φ0⟩: We take all uν\nij= +1 for the zigzag case, while\nwe flip uν\nijto−1 on the zbonds in one column for the\narmchair case as shown in Fig. 9(a). In both cases, how-\never, the sign of CMaj\nedge(ω) is indefinite due to the fac-\ntor of uν\niiniout; we plot the absolute value |CMaj\nedge(ω)|in\nFigs. 7 and 8, which corresponds to setting uν\niiniout=−1\nin Eq. (B3). Note that Eq. (B3) besides this factor cor-\nresponds to the propagator of the Majorana fermions ˆ ci.\n1E. Majorana, “Teoria simmetrica dell’elettrone e del\npositrone,” Il Nuovo Cimento 14, 171 (1937).\n2F. Wilczek, “Majorana returns,” Nature Phys 5, 614\n(2009).\n3A. Kitaev, “Fault-tolerant quantum computation by\nanyons,” Annals of Physics 303, 2–30 (2003).\n4M. Freedman, A. Kitaev, M. Larsen, and Z. Wang, “Topo-\nlogical quantum computation,” Bulletin of the American\nMathematical Society 40, 31 (2003).\n5A. Kitaev, “Anyons in an exactly solved model and be-\nyond,” Annals of Physics 321, 2 (2006).\n6G. Jackeli and G. Khaliullin, “Mott Insulators in the\nStrong Spin-Orbit Coupling Limit: From Heisenberg to a\nQuantum Compass and Kitaev Models,” Phys. Rev. Lett.\n102, 017205 (2009).\n7J. Chaloupka, G. Jackeli, and G. Khaliullin, “Kitaev-\nHeisenberg Model on a Honeycomb Lattice: Possible Ex-\notic Phases in Iridium Oxides A2IrO3,” Phys. Rev. Lett.\n105, 027204 (2010).\n8R. Yadav, N. A. Bogdanov, Vamshi M. Katukuri, S. Nishi-\nmoto, J. Van Den Brink, and L. Hozoi, “Kitaev exchange\nand field-induced quantum spin-liquid states in honeycomb\nα-RuCl 3,” Scientific reports 6, 1 (2016).\n9Y. Motome and J. Nasu, “Hunting Majorana Fermions in\nKitaev Magnets,” Journal of the Physical Society of Japan\n89, 012002 (2020).\n10Y. Kasahara, T. Ohnishi, Y. Mizukami, O. Tanaka, S. Ma,\nK. Sugii, N. Kurita, H. Tanaka, J. Nasu, Y. Motome,T. Shibauchi, and Y. Matsuda, “Majorana quantization\nand half-integer thermal quantum Hall effect in a Kitaev\nspin liquid,” Nature 559, 227 (2018).\n11M. Yamashita, J. Gouchi, Y. Uwatoko, N. Kurita, and\nH. Tanaka, “Sample dependence of half-integer quantized\nthermal Hall effect in the Kitaev spin-liquid candidate α-\nRuCl 3,” Phys. Rev. B 102, 220404(R) (2020).\n12T. Yokoi, S. Ma, Y. Kasahara, S. Kasahara, T. Shibauchi,\nN. Kurita, H. Tanaka, J. Nasu, Y. Motome, C. Hickey,\nS. Trebst, and Y. Matsuda, “Half-integer quantized\nanomalous thermal Hall effect in the Kitaev material can-\ndidate α-RuCl 3,” Science 373, 568 (2021).\n13J. A. N. Bruin, R. R. Claus, Y. Matsumoto, N. Kurita,\nH. Tanaka, and H. Takagi, “Robustness of the thermal\nHall effect close to half-quantization in α-RuCl 3,” Nature\nPhysics 18, 401 (2022).\n14P. Czajka, T. Gao, M. Hirschberger, P. Lampen-Kelley,\nA. Banerjee, N. Quirk, D. G. Mandrus, S. E. Nagler,\nand N. P. Ong, “Planar thermal Hall effect of topological\nbosons in the Kitaev magnet α-RuCl 3,” Nature Materials\n, 1 (2022).\n15´E. Lefran¸ cois, G. Grissonnanche, J. Baglo, P. Lampen-\nKelley, J.-Q. Yan, C. Balz, D. Mandrus, S. E. Nagler,\nS. Kim, Y.-J. Kim, N. Doiron-Leyraud, and L. Taillefer,\n“Evidence of a Phonon Hall Effect in the Kitaev Spin Liq-\nuid Candidate α-RuCl 3,” Phys. Rev. X 12, 021025 (2022).\n16A. J. Willans, J. T. Chalker, and R. Moessner, “Site di-\nlution in the Kitaev honeycomb model,” Phys. Rev. B 84,12\n115146 (2011).\n17A. Koga, Y. Murakami, and J. Nasu, “Majorana corre-\nlations in the Kitaev model with ordered-flux structures,”\nPhys. Rev. B 103, 214421 (2021).\n18G. Baskaran, S. Mandal, and R. Shankar, “Exact Re-\nsults for Spin Dynamics and Fractionalization in the Ki-\ntaev Model,” Phys. Rev. Lett. 98, 247201 (2007).\n19A. J. Willans, J. T. Chalker, and R. Moessner, “Disorder\nin a Quantum Spin Liquid: Flux Binding and Local Mo-\nment Formation,” Phys. Rev. Lett. 104, 237203 (2010).\n20D. Takikawa, M. G. Yamada, and S. Fujimoto, “Dissi-\npationless spin current generation in a Kitaev chiral spin\nliquid,” Phys. Rev. B 105, 115137 (2022).\n21M. O. Takahashi, M. G. Yamada, M. Udagawa,\nT. Mizushima, and S. Fujimoto, “Non-local spin corre-\nlation as a signature of Ising anyons trapped in vacancies\nof the Kitaev spin liquid,” arXiv preprint arXiv:2211.13884\n(2022).\n22T. Minakawa, Y. Murakami, A. Koga, and J. Nasu,\n“Majorana-Mediated Spin Transport in Kitaev Quantum\nSpin Liquids,” Phys. Rev. Lett. 125, 047204 (2020).\n23H. Taguchi, A. Koga, Y. Murakami, J. Nasu, and\nH. Tsuchiura, “Field-direction Dependence of Majorana-\nmediated Spin Transport,” JPS Conf. Proc. 38, 011152\n(2023).\n24J. Chaloupka, G. Jackeli, and G. Khaliullin, “Zigzag Mag-netic Order in the Iridium Oxide Na 2IrO3,” Phys. Rev.\nLett. 110, 097204 (2013).\n25M. Kawamura, K. Yoshimi, T. Misawa, Y. Yamaji,\nS. Todo, and N. Kawashima, “Quantum lattice model\nsolver HΦ,” Computer Physics Communications 217, 180\n(2017).\n26J. Knolle, D. L. Kovrizhin, J. T. Chalker, and R. Moess-\nner, “Dynamics of a Two-Dimensional Quantum Spin\nLiquid: Signatures of Emergent Majorana Fermions and\nFluxes,” Phys. Rev. Lett. 112, 207203 (2014).\n27The data for Sz(ωpeak) at Ω = 2 π/10≃0.63 and Sz(ωpeak)\nat Ω = 2 π/5≃1.26 in the armchair case deviate from the\nconstant behavior, and rather close to Ω. In these cases,\nwe also observe peaks around the constant values, but the\npeak heights are slightly smaller than those around Ω.\n28K. S. Tikhonov, M. V. Feigel’man, and A. Yu. Kitaev,\n“Power-law spin correlations in a perturbed spin model on\na honeycomb lattice,” Phys. Rev. Lett. 106, 067203 (2011).\n29S. Mandal, Subhro Bhattacharjee, K. Sengupta,\nR. Shankar, and G. Baskaran, “Confinement-\ndeconfinement transition and spin correlations in a\ngeneralized Kitaev model,” Phys. Rev. B 84, 155121\n(2011).\n30X.-Y. Song, Y.-Z. You, and L. Balents, “Low-Energy Spin\nDynamics of the Honeycomb Spin Liquid Beyond the Ki-\ntaev Limit,” Phys. Rev. Lett. 117, 037209 (2016)."    },    {        "title": "1204.0239v2.Spin_dynamics_in_the_Kapitza_Dirac_effect.pdf",        "content": "Spin dynamics in the Kapitza-Dirac effect\nSven Ahrens,1Heiko Bauke,1,\u0003Christoph H. Keitel,1and Carsten Müller1, 2,y\n1Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany\n2Institut für Theoretische Physik I, Heinrich-Heine-Universität Düsseldorf, Universitätsstraße 1, 40225 Düsseldorf, Germany\n(Dated: November 15, 2018)\nElectron spin dynamics in Kapitza-Dirac scattering from a standing laser wave of high frequency and high\nintensity is studied. We develop a fully relativistic quantum theory of the electron motion based on the time-\ndependent Dirac equation. Distinct spin dynamics, with Rabi oscillations and complete spin-flip transitions, is\ndemonstrated for Kapitza-Dirac scattering involving three photons in a parameter regime accessible to future\nhigh-power X-ray laser sources. The Rabi frequency and, thus, the di \u000braction pattern is shown to depend crucially\non the spin degree of freedom.\nPACS numbers: 03.65.Pm, 42.50.Ct, 42.55.Vc, 03.75.-b\nIntroduction The di \u000braction of an electron beam from\na standing wave of light is referred to as the Kapitza-Dirac\ne\u000bect [1,2]. This process points out the quantum wave nature\nof the electron and may be considered as an analogue of the\noptical di \u000braction of light on a grating, but with the roles of\nlight and matter interchanged. Predicted already in 1933, a\nclear experimental confirmation of the Kapitza-Dirac e \u000bect\nas originally proposed has been achieved only recently [ 3].\nIt has stimulated renewed theoretical interest in the process\n[4], advancing earlier treatments [ 5,6]. A related success-\nful experiment observed the (classical) scattering of electrons\nin a standing laser wave [ 7]. Both experiments operated at\nmoderate light intensities between 109W/cm2and1014W/cm2\nand near optical wavelengths. The existing theoretical stud-\nies, accordingly, have treated the electron quantum dynamics\nnonrelativistically. The nonrelativistic Kapitza-Dirac e \u000bect has\nbeen observed experimentally also using atomic beams [8].\nThe current development of novel light sources, envisaged\nto provide field intensities in excess of 1020W/cm2and field\nfrequencies in the hard X-ray domain [ 9,10], raises the ques-\ntion as to how the Kapitza-Dirac e \u000bect is modified in this\nhitherto unexplored parameter regime. This calls for a fully\nrelativistic treatment of the process within the Dirac theory,\nvalid at high electron energies and, in particular, accounting\nfor the electron spin. Relativistic considerations of quantum\nmechanical electron scattering in two counterpropagating light\nwaves were provided, but assuming nonequal laser frequen-\ncies and disregarding the electron spin degree of freedom [ 11].\nSpin signatures in the Kapitza-Dirac e \u000bect have so far been\nexamined within the nonrelativistic framework of the Pauli\nequation [ 12] and by solving the classical equations of motion\n[13]. Both studies found negligibly small spin e \u000bects. We note\nthat the influence of the electron spin has also been investi-\ngated with respect to free-electron motion [ 14], bound-electron\ndynamics [ 15], atomic photoionization [ 16], and Compton and\nMott [17] scattering in strong plane-wave laser fields.\nIn this Letter, we present a fully relativistic consideration\nof the Kapitza-Dirac e \u000bect within the framework of Dirac the-\nory. We focus on the relevance of spin-flip transitions when\nthe process occurs in X-ray laser fields of high intensity. Wedemonstrate that under suitable conditions, the di \u000braction prob-\nability depends considerably on the electron spin. Thus, the\nspin degree of freedom significantly influences the quantum\nmechanical scattering dynamics.\nRelativistic theory We consider a quantum electron\nwave packet in a standing linearly polarized light wave (see\nFig. 1) with maximal electric field strength E, wave vector k,\nand wavelength \u0015=2\u0019=jkj=2\u0019=k, respectively. The laser is\nmodeled by the vector potential\nA(x;t)=\u0000E\nkcos(k\u0001x) sin( ckt)w(t) (1)\nintroducing the speed of light cand the temporal envelope\nfunction w(t). The relativistic quantum dynamics of an electron\nwith mass mand charge\u0000eis governed by the Dirac equation\n[18]\ni@\n@t (x;t)=\u0014\nc\u0012\n\u0000ir+e\ncA(x;t)\u0013\n\u0001\u000b+\fmc2\u0015\n (x;t):(2)\nIn (2), we have introduced the vector \u000b=(\u000b1;\u000b2;\u000b3), where\n\u000biand\fare the Dirac matrices in standard representation [ 19].\nBE\npkpE\n\u0015=2p=pk+pE\n#\nFIG. 1: (Color online.) Schematic setup. An electron with momentum\npis incident at an angle #on a linearly polarized standing laser wave\nwith electric and magnetic components EandB. The momentum p\nhas components pEandpkalong the laser’s electric field Eand wave\nvector k, respectively. The electron is initially spin-polarized along\nthe electric field component. After Kapitza-Dirac scattering, parts of\nthe electron wave packet may have flipped their spin orientation.arXiv:1204.0239v2  [quant-ph]  24 Jul 20122\nThe monochromatic light wave (1) allows us to decompose the\nwave function  (x;t) into a discrete set of plane waves, viz.,\n (x;t)=X\nn;\u0010c\u0010\nn(t) \u0010\nn;p(x);  \u0010\nn;p(x)=u\u0010\nn;pei(p+nk)\u0001x:(3)\nThe function  \u0010\nn;p(x)denotes a free particle Dirac eigenfunc-\ntion of momentum p+nk(n=0;\u00061;\u00062;:::). The index\n\u00102f+\";+#;\u0000\";\u0000#g labels the sign of the energy and the\nspin projection along the laser electric field vector. Taking\nadvantage of the basis functions’ orthonormality the ansatz (3)\nyields\ni˙c\r\nn(t)=ih \r\nn;pj˙ i=\u000f\rE(p+nk)c\r\nn(t)\n\u0000w(t)esin(ckt)\n2kX\n\u0010hu\r\nn;pjE\u0001\u000bju\u0010\nn\u00001;pic\u0010\nn\u00001(t)\n\u0000w(t)esin(ckt)\n2kX\n\u0010hu\r\nn;pjE\u0001\u000bju\u0010\nn+1;pic\u0010\nn+1(t);(4)\nwhere we have introduced the relativistic energy momentum\ndispersion relation E(p)=p\nm2c4+c2p2and the signum \u000f\r,\nwhich is 1 for \r2f+\";+#gand\u00001 for\r2f\u0000\";\u0000#g.\nGeneralized Bragg condition The elastic scattering of\nan electron on a standing light wave may be characterized by a\nBragg condition [ 1] provided that the ponderomotive energy\nof the electron is small (so-called Bragg regime) [5, 20]. This\nBragg condition may be generalized to an inelastic process\nof absorbing and emitting an arbitrary number of photons by\nutilizing momentum conservation p0=p+(nr\u0000nl)kand\nenergy conservation E(p0)=E(p)+(nr+nl)ck;where pand\np0are the initial and final electron momenta. The integers nr\nandnldenote the net numbers of photons exchanged with the\nright- and left-traveling laser waves, respectively, with positive\n(negative) values indicating photon absorption (emission). The\nmomentum and energy conservation laws yield the relativistic\ngeneralized Bragg condition\ncos#\n\u0015p=\n\u0000nr\u0000nl\n2\u0015+nr\u0000nl\njnr\u0000nljnr+nl\n2s\n1\n\u00152\u00001\nnrnl0BBBB@sin2#\n\u00152p+1\n\u00152\nC1CCCCA(5)\nby introducing the angle #(see Fig. 1), the de Broglie wave-\nlength\u0015p=2\u0019=jpj, and the Compton wavelength \u0015C=\n2\u0019=(mc). To be consistent with the nonrelativistic limit nr\nandnlmust have opposite signs. Equation (5) reduces to the\nBragg condition of the two-photon Kapitza-Dirac e \u000bect [1,20]\nby setting nr=\u0000nl=1.\nFrom Eq. (5), it follows that for inelastic processes\n(nr+nl,0)either the initial electron momentum por the\nlaser photon momentum kmust be of the order of mc, i. e., rela-\ntivistic, except we allow for a very large number of interacting\nphotons. Thus, an analysis of inelastic Kapitza-Dirac scattering\ndemands a relativistic treatment by the Dirac equation.\n0 500 1000 1500 2000 2500\nT(laser periods)0.00.20.40.60.81.0probability|c+↑\n0|2\n|c+↑\n3|2\n|c+↓\n3|2FIG. 2: Rabi oscillations of the spin resolved di \u000braction probabilities\nas a function of the interaction time Tfor a three-photon Kapitza-\nDirac e \u000bect. Starting from a pure spin-up state the electron is either\ndi\u000bracted with probability jc+\"\n3(T)j2+jc+#\n3(T)j2or passes the laser\nbeam without momentum transfer with probability jc+\"\n0(T)j2. Laser\nparameters of this simulation correspond to two counterpropagating\nX-ray laser beams with a peak intensity of 2:0\u00021023W=cm2each\nand a photon energy of 3:1keV. The electron impinges at an angle\nof inclination of #=0:4°and a momentum of 176keV=c, in order to\nfulfill the Bragg condition (5).\nA setup for spin-sensitive Kapitza-Dirac scattering\nEquation (4) couples momentum components having momenta\nthat di \u000ber by\u0006kand equal or opposite spin orientation. An\nexplicit calculation of hu\r\nn;pjE\u0001\u000bju\u0010\nn\u00061;pireveals that if pandE\nare orthogonal then c\r\nn(t)couples only to components c\u0010\nn\u00061(t)\nhaving opposite spin orientation. Therefore, a distinct spin\ndynamics may be expected for Kapitza-Dirac scattering with\nan odd number of photons provided that the initial electron\nmomentum pis almost orthogonal to the electric field E. Thus,\nwe focus on a three-photon Kapitza-Dirac e \u000bect, i. e., nr=2,\nnl=\u00001, in the subsequent sections.\nThe condition (5) allows us to determine the initial electron\nmomentum pand the laser wave number kfor a resonant\nthree-photon Kapitza-Dirac e \u000bect. The impinging electron is\nmodeled by a plane wave; thus, the initial condition c+\"\n0(0)=1\nandc\u0010\nn(0)=0else will be applied for the remainder of the\nLetter. We solve the Dirac equation (4) until time Tto compute\nthe transition amplitudes c\u0010\nn(T).\nSpin-flip probability and Rabi frequency Figure 2\nshows the final occupation probabilities jc\u0010\nn(T)j2after laser-\nelectron interaction, which have been calculated by numerical\npropagation of the Dirac equation (4) by a combination of the\nexplicit and the implicit Euler method [ 21]. The laser profile\nwas modeled by an envelope function w(t)that starts with a\nsin2-shaped turn-on ramp of 10 laser cycles and finishes with\nasin2-shaped turn-o \u000bramp of 10 laser cycles having a flat\nplateau in between. The experimental setup was chosen to\nmeet the Bragg condition (5) for a three-photon process. As\nillustrated in Fig. 2, only zeroth-order andthird-order modes\nare nonzero after interaction. The occupation probabilities\noscillate in Rabi cycles of frequency \nR\njc+\"\n0(T)j2=cos2(\nRT=2); (6a)\njc+\"\n3(T)j2+jc+#\n3(T)j2=sin2(\nRT=2); (6b)3\n0.0 0.5 1.0 1.5 2.0\n|pE|/k0.00.20.40.60.81.0Pflip\nFIG. 3: The spin-flip probability Pflipas a function of the electron\nmomentumjpEjin electric field direction. The solid black line is given\nby (8). White squares result from simulations with the Dirac equation\n(4). All simulation parameters except pEare the same as those for\nFig. 2.\nsimilarly to the two-photon Kapitza-Dirac e \u000bect [2] and the\nKapitza-Dirac e \u000bect in atomic beams [ 8]. In the present case,\nhowever, the scattered portion of the electron wave packet\nconsists of two parts which are distinguished by opposite spin\norientations. The period of the Rabi oscillations is 2\u0019=\nR=\n1:9 fs for parameters of Fig. 2.\nFor short times with \nRT\u001c1, the Dirac equation (4) can\nalso be solved analytically via time-dependent perturbation\ntheory. This yields for parameters compatible with the three-\nphoton Bragg condition (5) the probabilities\njc+\"\n3(T)j2= 1\n2\n0T!2 5p\n2jpEj\nk!2\n; (7a)\njc+#\n3(T)j2= 1\n2\n0T!2\n; (7b)\nwith\n0=e3jEj3=(24m3c5k2). ThejEj3dependence clearly in-\ndicates the three-photon nature of the transition. From Eq. (7),\nwe can derive the spin-flip probability Pflipwithin the scattered\nportion of the electron wave packet\nPflip\u0011jc+#\n3(T)j2\njc+\"\n3(T)j2+jc+#\n3(T)j2=1\n25\n2\u0010jpEj\nk\u00112+1: (8)\nThe Rabi frequency may be derived by identifying (7) with the\nTaylor expansion of (6b) for short times T, resulting in\n\nR= \n 0s\n25\n2 jpEj\nk!2\n+1: (9)\nFigure 3 compares the spin-flip probability as a function of\njpEj=k, as obtained from the numerical solution of the Dirac\nequation (4), with our analytical result (8). The initial elec-\ntron momentum in laser propagation direction pkis adjusted\naccording to equation (5) for each value of pE. The analytical\nformula (8) shows very good agreement with the numerical\ndata.Considerations on nonrelativistic theory The spin-\nflip probability (8) can be understood on a qualitative level by\nanalyzing the leading order of the Foldy-Wouthuysen expan-\nsion [ 22] of the Dirac equation (2) which equals the nonrel-\nativistic Pauli equation. This equation features two coupling\nterms which are linear in the fields, namely eA\u0001p=(mc)and\ne\u001b\u0001B=(2mc), where \u001bdenotes the vector of Pauli matrices.\nBoth terms give rise to couplings between adjacent electron\nmomentum components (di \u000bering by\u0006k), with the first term\npreserving and the second term flipping the electron spin. Note\nthat such nearest-neighbor couplings are necessarily involved\nin Kapitza-Dirac scattering with an odd number of photons.\nThe relative strength of the A\u0001pterm as compared with the \u001b\u0001B\nterm is just 2jpEj=k. This results in the spin-flip probability\nPflip, nonrel. =1\n4\u0010jpEj\nk\u00112+1; (10)\nwhich agrees with (8) up to a scale parameter 25=8. The\nprobability (10) can also be derived more rigorously via time-\ndependent perturbation theory for the Pauli equation, which\nyields the nonrelativistic Rabi frequency\n\nR, nonrel. =243\n128\n0s\n4 jpEj\nk!2\n+1: (11)\nNote that the relativistic and nonrelativistic spin-flip proba-\nbilities (8) and (10) and the relativistic and nonrelativistic\nRabi frequencies (9) and (11) agree qualitatively. However,\nthe nonrelativistic expressions cannot be recovered from the\nrelativistic ones by taking a nonrelativistic limit. This is a\nconsequence of the three-photon Bragg condition (5) that en-\nforces a relativistic photon momentum and /or a relativistic\nelectron momentum highlighting the relativistic nature of the\nthree-photon Kapitza-Dirac e \u000bect.\nThe role of the spin Equation (7) indicates that spin-\npreserving transitions in the three-photon Kapitza-Dirac e \u000bect\nare completely suppressed for setups with p?E. This means\nthat under such conditions the scattering is rendered possi-\nble only because the electron does carry spin, which clearly\ndemonstrates the pivotal role the electron spin can play in\nKapitza-Dirac scattering processes.\nThe above argument suggests that for a spinless particle with\np?Ethethree-photon channel of Kapitza-Dirac scattering is\nnot accessible at all. This expectation is confirmed by Fig. 4 (a).\nIt compares numerical results on the Rabi frequency as follow-\ning from the Dirac equation (4) with corresponding numbers\nthat we obtained by solving the time-dependent Klein-Gordon\nequation [ 23]. The predictions significantly di \u000ber in the limit\npE!0. While the Rabi frequency converges to a finite value\nfor spin-half particles [see also (9)], it approaches zero for\nspinless particles, indicating that the scattering channel closes\nindeed. In the spinless case, the spin-flipped channel (7b) is\nmissing, which consequently yields the Rabi frequency\n\nR;spinless = \n 05p\n2jpEj\nk: (12)4\n0.00.20.40.60.81.0\n|pE|/k01234ΩR/Ω0a)\n0 3Klein-Gordon\n0 30.00.20.40.60.81.0probabilityDirac\ndiffraction orderb)\nFIG. 4: Panel (a): The Rabi frequency \nRas a function of the elec-\ntron momentumjpEjin electric field direction for the Dirac equa-\ntion (squares) and the Klein-Gordon equation (triangles). The solid\nblack line is given by (9) and the dashed black line is given by (12).\nPanel (b): The di \u000braction probability after an interaction time of\n0.36 fs for particles with and without spin for parameters as in Fig. 2.\nThe light (dark) gray bars represent the spin-up (spin-down) probabil-\nities. In the case of the Klein-Gordon equation there is no spin degree\nof freedom and, therefore, no dark gray bars appear. Note that the\ndi\u000braction probability depends on the spin degree of freedom.\nFor nonzero values of pE, also Klein-Gordon particles may be\nscattered. But the scattering probability still may be consider-\nably di \u000berent from the Dirac case, as the example in Fig. 4 (b)\nshows.\nExperimental realization An experimental realization\nof the three-photon Kapitza-Dirac e \u000bect may utilize intense\nphoton beams at near-future X-ray laser facilities to form stand-\ning waves. In our numerical simulations, we assumed 3.1 keV\nphotons as envisaged, for example, at the European X-ray free\nelectron laser facility (XFEL) [ 9], which is currently under\nconstruction. The design value of the peak power at this pho-\nton energy is 80 GW. Assuming a focus diameter of 7 nm [ 25],\na field intensity of about 2\u00021023W=cm2results. Laser pulses\nwith duration of about half a Rabi period [which is about 1 fs\nfor the parameters in Fig. 2)] are required for experimental re-\nalization [ 26]. Since the Rabi frequency is much lower than the\nlaser frequency, the photon energy and the electron momentum\nmust be fine-tuned to achieve a resonant transition. Numeri-\ncal simulations indicate that only electrons whose momentum\nvaries by 0:1keV=caround the mean value of 176keV=care\ndi\u000bracted. The photon pulse of the European XFEL with a\nseeded beam of a primary undulator is expected to be coher-\nent, featuring a photon energy uncertainty far below 0.1 keV\n[27]. We note that the electron may lose energy due to sponta-\nneous photoemission with resulting quantitative modification\nof the presented results. Spontaneous emission (scaling with\njEj2), however, is substantially suppressed as compared with\nthe very fast momentum transfer through the three-photon\nKapitza-Dirac e \u000bect which takes place on a femtosecond time\nscale during a Rabi period ( 1=\nR\u00181=jEj3). A numerical so-\nlution of the Landau-Lifshitz equation [ 28] indicates that the\nmomentum transfer into laser propagation direction caused\nby spontaneous emission is su \u000eciently small in order not to\nviolate the resonance condition (5) for the current parameters.\nFinally, the electron beam is di \u000bracted almost in the electron\npropagation direction by 3\u00023:1keV=c. Therefore, a spec-trometer with a resolution below 10keV=cshould be able to\nseparate the di \u000bracted electron beam from the not di \u000bracted\none. In the di \u000bracted beam, about one out of three electrons are\nspin flipped in the case of the scenario in Fig. 2. This spin-flip\nfraction is independent of the interaction time Tof the electron\nwith the laser, in accordance with (8).\nConclusions Pronounced spin e \u000bects in Kapitza-Dirac\nscattering involving three X-ray laser photons interacting with\na weakly relativistic electron beam have been revealed. To this\nend, we deduced a generalized Bragg condition and developed\na theoretical description of the quantum dynamics based on\nthe Dirac equation. The process features characteristic Rabi\noscillations and a competition between spin-preserving and\nspin-flipping nearest-neighbor couplings. The spin-flipping\ntransition becomes dominant in the limit of small angles of\ninclination, where three-photon Kapitza-Dirac scattering cru-\ncially relies on the nonzero electron spin. Our predictions\nmay be tested with the aid of near-future high-intensity XFEL\nsources.\n\u0003heiko.bauke@mpi-hd.mpg.de\nymueller@tp1.uni-duesseldorf.de\n[1]P. L. Kapitza and P. A. M. Dirac, Math. Proc. Cambridge Philos.\nSoc. 29, 297 (1933).\n[2] H. Batelaan, Rev. Mod. Phys. 79, 929 (2007).\n[3]D. L. Freimund, K. Aflatooni, and H. Batelaan, Nature 413, 142\n(2001); D. L. Freimund and H. Batelaan, Phys. Rev. Lett. 89,\n283602 (2002).\n[4]O. Smirnova, D. L. Freimund, H. Batelaan, and M. Ivanov, Phys.\nRev. Lett. 92, 223601 (2004); X. Li, J. Zhang, Z. Xu, P. Fu, D.-S.\nGuo, and R. R. Freeman, Phys. Rev. Lett. 92, 233603 (2004);\nP. Sancho, Phys. Rev. A 82, 033814 (2010).\n[5] M. V . Fedorov, Sov. Phys.–JETP 25, 952 (1967).\n[6]R. Gush and H. P. Gush, Phys. Rev. D 3, 1712 (1971); M. V .\nFedorov, Opt. Commun. 12, 205 (1974).\n[7]P. H. Bucksbaum, D. W. Schumacher, and M. Bashkansky, Phys.\nRev. Lett. 61, 1182 (1988).\n[8]P. L. Gould, G. A. Ru \u000b, and D. E. Pritchard, Phys. Rev. Lett.\n56, 827 (1986); C. Adams, M. Sigel, and J. Mlynek, Phys. Rep.\n240, 143 (1994); M. Freyberger, A. Herkommer, D. Krähmer,\nE. Mayr, and W. Schleich, in Adv. At., Mol., Opt. Phys. , V ol. 41,\nedited by B. Bederson and H. Walther (Academic Press, New\nYork, 1999) pp. 143–180.\n[9]M. Altarelli, R. Brinkmann, M. Chergui, W. Decking, B. Dob-\nson, S. Düsterer, G. Grübel, W. Grae \u000b, H. Graafsma, J. Hajdu\net al. , eds., The European X-Ray Free-Electron Laser Techni-\ncal design report (DESY XFEL Project Group European XFEL\nProject Team Deutsches Elektronen-Synchrotron Member of the\nHelmholtz Association, Hamburg, 2007).\n[10] B. W. J. McNeil and N. R. Thompson, Nature Photon. 4, 814\n(2010); P. Emma, R. Akre, J. Arthur, R. Bionta, C. Bostedt,\nJ. Bozek, A. Brachmann, P. Bucksbaum, R. Co \u000bee, F.-J. Decker\net al. , Nature Photon. 4, 641 (2010); G. Mourou and T. Tajima,\nScience 331, 41 (2011).\n[11] V . M. Haroutunian and H. K. Avetissian, Phys. Lett. A 51, 320\n(1975); M. V . Federov and J. K. McIver, Opt. Commun. 32, 179\n(1980).\n[12] L. Rosenberg, Phys. Rev. A 70, 023401 (2004).5\n[13] D. L. Freimund and H. Batelaan, Laser Phys. 13, 892 (2003).\n[14] P. Krekora, Q. Su, and R. Grobe, J. Phys. B: At., Mol. Opt.\nPhys. 34, 2795 (2001); M. W. Walser, D. J. Urbach, K. Z.\nHatsagortsyan, S. X. Hu, and C. H. Keitel, Phys. Rev. A 65,\n043410 (2002).\n[15] S. X. Hu and C. H. Keitel, Phys. Rev. Lett. 83, 4709 (1999).\n[16] F. H. M. Faisal and S. Bhattacharyya, Phys. Rev. Lett. 93, 053002\n(2004).\n[17] C. Szymanowski, R. Taïeb, and A. Maquet, Laser Phys. 8, 102\n(1998); D. Ivanov, G. Kotkin, and V . Serbo, Eur. Phys. J. C 36,\n127 (2004); F. Ehlotzky, K. Krajewska, and J. Z. Kami ´nski, Rep.\nProg. Phys. 72, 046401 (2009).\n[18] In this article, we use Gauss units and define ~=1.\n[19] B. Thaller, The Dirac Equation , Texts and Monographs in\nPhysics (Springer, 1992).\n[20] H. Batelaan, Contemp. Phys. 41, 369 (2000).\n[21] Other numerical schemes for solving the time-dependent Dirac\nequation have been developed recently in J. W. Braun, Q. Su,\nand R. Grobe, Phys. Rev. A 59, 604 (1999); S. Selstø, E. Lin-\ndroth, and J. Bengtsson, Phys. Rev. A 79, 043418 (2009); G. R.\nMocken and C. H. Keitel, Comp. Phys. Comm. 178, 868 (2008);\nH. Bauke and C. H. Keitel, Comp. Phys. Comm. 182, 2454(2011); F. Fillion-Gourdeau, E. Lorin, and A. D. Bandrauk,\nComp. Phys. Comm. 183, 1403 (2012).\n[22] L. L. Foldy and S. A. Wouthuysen, Phys. Rev. 78, 29 (1950).\n[23] An expansion of the wave function into momentum eigenfunc-\ntions of the Klein-Gordon equation in Hamiltonian form [ 24]\nyields a system of ordinary di \u000berential equations similar to (4).\n[24] H. Feshbach and F. Villars, Rev. Mod. Phys. 30, 24 (1958);\nM. Ruf, H. Bauke, and C. H. Keitel, J. Comp. Phys. 228, 9092\n(2009).\n[25] H. Mimura, S. Handa, T. Kimura, H. Yumoto, D. Yamakawa,\nH. Yokoyama, S. Matsuyama, K. Inagaki, K. Yamamura, Y . Sano\net al. , Nat. Phys. 6, 122 (2010).\n[26] A. A. Zholents and G. Penn, Phys. Rev. ST Accel. Beams 8,\n050704 (2005).\n[27] G. Geloni, V . Kocharyan, and E. Saldin, J. Mod. Opt. 58, 1391\n(2011).\n[28] L. D. Landau and E. Lifshitz, The Classical Theory of Fields ,\n4th ed. (Butterworth-Heinemann, Oxford, 1975); C. H. Keitel,\nC. Szymanowski, P. L. Knight, and A. Maquet, J. Phys. B: At.,\nMol. Opt. Phys. 31, L75 (1998); A. Di Piazza, K. Z. Hatsagort-\nsyan, and C. H. Keitel, Phys. Rev. Lett. 102, 254802 (2009)."    },    {        "title": "1506.06891v1.Magnetic_Excitations_of_Spin_Nematic_State_in_Frustrated_Ferromagnetic_Chain.pdf",        "content": "arXiv:1506.06891v1  [cond-mat.str-el]  23 Jun 2015Journal of thePhysical Society of Japan LETTERS\nMagneticExcitations ofSpin NematicStateinFrustrated Fe rromagneticChain\nHiroakiOnishi\nAdvanced Science ResearchCenter, Japan AtomicEnergy Agen cy, Tokai, Ibaraki, 319-1195, Japan\nBy exploiting density-matrix renormalization group techn iques, we investigate the dynamical spin structure factor o f\naspin-1/2Heisenbergchainwithferromagneticnearest-neighboran dantiferromagneticnext-nearest-neighborexchange\ninteractionsinanappliedmagneticfield.Inafield-induced spinnematicregime,wefindgaplesslongitudinalandgapped\ntransverse spin excitation spectra, in accordance with qua si-long-ranged longitudinal and short-ranged transverse spin\ncorrelations, respectively. The gapless point coincides w ith the dominant longitudinal spin correlation, whereas th e gap\nposition exhibits a characteristic fielddependence contra dicting the dominant transverse spin correlation.\nFrustrated quantummagnets are a class of interacting spin\nsystemsinwhichtherearecompetinginteractionsthatcann ot\nbe satisfied simultaneously. In general, the combined e ffects\nof frustration and quantum fluctuations prevent convention al\nmagneticorder,andweenvisagetheemergenceofnovelspin\nstates. Even when the system is magnetically disordered, we\ntypicallyfind“hidden”orderdescribedbymultiple-spinor der\nparameters, such as dimer, chiral, and multipole ones. In th is\npaper,weparticularlydiscussaquadrupolestate,calleda spin\nnematic (SN) state.1,2)A schematic picture of the SN state is\nthatspinsaligninaspontaneouslychosenaxis,whiletheys till\nfluctuate within the axis and their directions remain unfixed ,\nresembling the directional order of rod-shaped molecules i n\nthenematicliquidcrystal.\nThe occurrence of the SN state has been demonstrated for\nseveral kindsof frustratedquantumspin models.3–13)Among\nthem, a spin-1 /2 chain with ferromagnetic nearest-neighbor\nJ1<0 and antiferromagnetic next-nearest-neighbor J2>0\nexchangeinteractionsina magneticfield h, describedby\nH=J1/summationdisplay\niSi·Si+1+J2/summationdisplay\niSi·Si+2−h/summationdisplay\niSz\ni,(1)\nhas been attracting much currentinterest because of not onl y\nthe novelty of a theoretically predicted SN state,3–9)but also\nthe relevance to a wide variety of edge-shared copper-oxide\nchaincompounds.14–23)\nOnthetheoreticalside,theground-statephasediagramhas\nbeenstudiedindetail,6,7)andit isnowwell establishedthata\nSNstateexistsathighfields,wherethequadrupolecorrelat ion\n/angbracketleftS+\niS+\ni+1S−\njS−\nj+1/angbracketrightexhibits a quasi-long-rangeorder due to the\nformationoftwo-magnonboundstates.Thelongitudinalspi n\ncorrelation/angbracketleftSz\niSz\nj/angbracketrightalsoshowsaquasi-long-rangeorder,while\nthe transverse spin correlation /angbracketleftS+\niS−\nj/angbracketrightdecays exponentially.\nAccordingly, the SN state should have gapless excitations i n\nthe quadrupoleandlongitudinalspin channels,whereasthe re\nshouldbeafiniteexcitationgapinthetransversespinchann el,\ncorresponding to the binding energy of a magnon pair. Note\nthatintheSNregime,thequadrupolecorrelationisdominan t\nbelow the saturation, and the longitudinal spin-density-w ave\n(SDW)correlationbecomesdominantasthemagneticfieldis\ndecreased, although both correlations are quasi-long-ran ged.\nThus,hereafter,we referto this phase as the SN /SDWphase.\nWhenthemagneticfieldisfurtherdecreased,theground-sta te\nphasechangestoa vectorchiral(VC)phaseatlow fields.Concerning the experimentalrealization of the SN state in\nthefruatratedferromagneticchain,LiCuVO 4hasbeenstudied\nfrequently as a prototypical material.14–18)In a field-induced\nphase,neutrondiffractionexperimentshavefounda collinear\nspin-modulatedstructure,whichisconsistentwiththeore tical\nresults for the longitudinal SDW correlation in the SN /SDW\nphase.15,17)Thespincorrelationsexhibitshort-rangebehavior\nin all directions.17)These observations are indeed suggestive\nof the development of the quadrupole correlation. However,\nsince the direct observation of the non-magnetic SN state is\ndifficult,the orderparameterisstill notidentifiedyet.\nTo collect evidence that the SN state occurs in reality, it is\nimportantto find any specific indicationsin what we observe\nby using various microscopic probes. In this context, recen t\ntheoretical studies have pointed out that the NMR relaxatio n\nrate shows characteristic temperature and field dependenci es\nin the SN/SDW phase.24,25)NMR experiments for LiCuVO 4\nhave reported consistent results with theoretical predict ions,\nsignaling the formation of the bound magnon pairs.18)The\npropertiesof thelow-energyspin excitationspectrahavea lso\nbeendiscussed.24,26–28)\nIn this paper, to clarify the propertyof the SN /SDW phase\nfrom the viewpoint of the spin dynamics,we investigate spin\nexcitation spectra in a wide range of momentum and energy\nby numerical methods. We clearly find that the longitudinal\nspin excitation spectrum is gapless, while the transverse o ne\nis gapped, as naively expected from the behavior of the spin\ncorrelations. We discuss the field dependence of the spectra l\nweighttransferinrelationwiththedominantspincorrelat ion.\nA striking feature is that the momentum position of the gap\ndeviatesfromthatofthedominanttransversespincorrelat ion\nasthesystem approachesthesaturation.\nLet us consider the model (1) in an N-site chain, and take\nJ2=1 as the energy unit. We investigate the spin excitation\ndynamics at zero temperature by exploiting density-matrix\nrenormalizationgroup (DMRG) techniques.29,30)We employ\nthe finite-system algorithm in open boundaryconditions. We\ncomputethedynamicalspinstructurefactor,definedby\nSα(q,ω)=−1\nπIm/angbracketleftψG|Sα†\nq1\nω+EG−H+iηSα\nq|ψG/angbracketright,(2)\nwhere|ψG/angbracketrightisthegroundstatewitheigenenergy EG.Notethat\n|ψG/angbracketrightis given by the lowest-energy state in the subspace of a\ngiven magnetization m=M/N, whereM=/summationtext\niSz\ni. For the\n1J.Phys. Soc. Jpn. LETTERS\nFig. 1. (Color online) Intensity plots ofthedynamical spinstruct ure factor Sα(q,ω)atJ1=−1andJ2=1,obtained bydynamical DMRGcalculations with\nN=128. (a)Sz(q,ω) atm=0andh=0. (b)Sz(q,ω) and (c)Sx(q,ω) atm=0.125(=16/128) and h=0.649.\ncalculation of Sα(q,ω) atm, we set the magnetic field to be\nthe midpoint of the magnetization plateau of Min theN-site\nsystem.ηis a small broadening factor, and we set η=0.1\nunlessotherwisespecified.Ourparticularinterestistocl arify\nthe anisotropy between longitudinal Sz(q,ω) and transverse\nSx(q,ω)duetotheformationofthetwo-magnonboundstates\ninthefield-inducedSN /SDWphase.\nFor the analysis of the spin excitation spectrum, we use a\ndynamicalDMRGmethod,30)targetingthegroundstate |ψG/angbracketright,\nan excitated state Sα\nq|ψG/angbracketright, and the so-called correction vector\n[ω+EG−H+iη]−1Sα\nq|ψG/angbracketright.Here,wenotethatthetruncation\nerrorrapidlyincreasesasthenumberoftargetstatesincre ases.\nTherefore,to obtain Sx(q,ω) with keepinghighaccuracy,we\ncalculate S+(q,ω) andS−(q,ω) separately, and then use the\nrelationSx(q,ω)=[S+(q,ω)+S−(q,ω)]/4,insteadofdirectly\ncalculating Sx(q,ω).Notealsothatwecalculatethespectrum\natqandωafter one DMRG run with fixed qandω, so that\nwe need to perform a great number of DMRG runs to obtain\na fullspectrum.\nLet us start with a brief discussion on the spectrum at zero\nfield. In Fig. 1(a), we present the intensity plot of Sz(q,ω) at\nJ1=−1,J2=1 (energy unit throughout the paper), m=0,\nandh=0. Note that Sz(q,ω)=Sx(q,ω) ath=0 due to the\nSU(2)spinrotationsymmetry.Wefindasinusoidaldispersio n\nthatgivesthelowerboundaryofacontinuum.Thesinusoidal\ndispersion represents the spinon excitation, described by the\ndes Cloizeaux-Pearson mode,31)since the system decouples\ninto two antiferromagnetic chains if J1→0. We see a large\namountof spectral weight at a lowest-energypeak ( q0,ω0)=\n(π/2,0.04). Note here that the spectrum is asymmetric with\nrespectto q0,aspointedoutbythepreviousstudies.14,32)That\nis, the spectralweight mainlylies nearthe lowerboundaryo f\nthe continuum for q<q0, whereas it is distributed to a high\nenergy region for q>q0. On the other hand, ω0coincides\nwitha spinexcitationenergy,\n∆(N,M)=[E0(N,M+1)+E0(N,M−1)−2E0(N,M)]/2,(3)\nwhereE0(N,M) is the lowest energy of the N-site system in\nthe subspace of Math=0. We find that∆(N,0) shifts to\nlower energy as Nincreases, and it is extrapolated to almost\nzero in the limit of N→∞. Note that an exponentiallysmall\ngap has been predicted by renormalization group theory, but\nit ishardto detectsuchasmall gapnumerically.33)\nNow, let us look into the longitudinal and transverse spinexcitation spectra in the SN /SDW phase. In Figs. 1(b) and\n1(c),weshow Sz(q,ω)andSx(q,ω),respectively,at J1=−1,\nm=0.125,andh=0.649.ForSz(q,ω),alowest-energypeak\nisat(q0,ω0)=(0.375π,0.00).Thatis, q0movestowardsmall\nmomentum from the position at zero field. ω0is nearly zero,\nindicatingagaplessmodeforthelongitudinalspinexcitat ion.\nWe see that a certainamountof spectralweight is transferre d\nto the origin due to a finite uniform magnetization, which is\nalso indicative of a gapless mode. These gapless points are\nclearly visible due to the large intensity. Moreover, Sz(q,ω)\nseems to be gapless at q=π−q0andq=π, although we\ndo not observe significant intensity near the possible gaple ss\npointsinq>π/2.\nIn contrast, for Sx(q,ω), we observe a lowest-energypeak\nat(q0,ω0)=(31π/64,0.15).Thatis, q0remainsnear π/2even\nin the SN/SDW phase. On the other hand, ω0appearsto be a\nfiniteenergy.Infact, ω0agreeswiththespinexcitationenergy\n∆(N,M), andit is extrapolatedto a finite valuein the limit of\nN→∞withm=M/Nfixed, indicating a gapped mode for\nthe transverse spin excitation. Note that Sx(q,ω) consists of\nS−(q,ω) andS+(q,ω). As shown in Fig. 2, both spectra have\nthelowest-energypeakatthesameposition,whiletheovera ll\nstructure is different between them. We find that S−(q,ω) is\nhighly dispersive in a wide range of momentum and energy,\nandS+(q,ω) is less dispersive and it is mainly concentrated\nin asmall regionnearthelowest-energypeak.\nHere, let us discuss how the lowest-energy peak position\ndepends on min the whole range of m. In Fig. 3(a), we plot\nq0ofSz(q,ω)asafunctionof mforseveralvaluesof J1.Note\nthat we also find a sharp peak at the origin for finite m, but\nFig. 2. (Coloronline)Intensityplotsofthedynamicslspinstruct urefactors\n(a)S−(q,ω) and (b)S+(q,ω) atJ1=−1,J2=1,m=0.125, andh=0.649.\nNote that Sx(q,ω)=[S+(q,ω)+S−(q,ω)]/4 is plotted in Fig. 1(c).\n2J.Phys. Soc. Jpn. LETTERS\nFig. 3. (Color online) The lowest-energy peak position q0of (a)Sz(q,ω)\nand (b)S−(q,ω) for several values of J1atJ2=1 as a function of m. Solid\nsymbols denote the VC phase at low fields, and open symbols den ote the\nSN/SDW phase at high fields. Note that q0shows a stepwise change simply\ndue to finite-size e ffects. The resolution of the momentum is 2 π/Nand we\nuseN=128 in the present calculations. Here, the broadning factor is set to\nη=0.02 to determine the position of thelowest-energy peak preci sely.\nwefocusonthefield-inducedshiftofthepeakpositionwhich\noriginally locates near π/2 at zero field. At small m, where\nthe system is in the VC phase, q0shows little dependenceon\nm. At large m, where the system is in the SN /SDW phase, q0\nfollows the relation q0=(1/2−m)πregardless of J1, which\nsupportsthebosonizationresult.24)Notethat q0isrepresented\nbythedensityofboundmagnons1 /2−m.\nIn Fig. 3(b), we show the mdependenceof q0ofS−(q,ω).\nInthe VC phase at small m, we noticethat q0agreeswith the\nincommensurabilityofthe transversespincorrelation,as will\nbeshowninFig.4(d).Noteherethatithasbeenrevealedthat\ntheincommensuratewavenumberintheVCphaseisstrongly\nquantum renormalized toward π/2 compared with the pitch\nanglecos−1(−J1/4J2) ofthehelicalorderin theclassical spin\ncase.6)Indeed,weobservethat q0=π/2atJ1=−0.5and−1,\nalthoughthe correspondingclassical pitch angle is 0 .46πand\n0.42π, respectively. For small |J1|, we see that q0remains at\nπ/2below a thresholdvalue of mevenin the SN/SDW phase\natlargem.However, q0goesawayfrom π/2asmapproaches\nthesaturation,andthedeviationfrom π/2ismorepronounced\nfor larger|J1|. On the other hand,the exact energydispersion\noftheone-magnonexcitedstate inthe fullypolarizedstate is\nǫ1(q)=J1(cosq−1)+J2(cos2q−1)+h,(4)\nand its minimum is at q=cos−1(−J1/4J2), which coincides\nwiththe classical pitchangle.4)Thepresentnumericalresults\nat the saturation totally agree with this exact description . We\nmentionthat the bosonizationanalysis showsthat the botto m\noftheone-magnonbandisfoundat π/2,24)butitisvalidonly\nfor the weak-coupling regime |J1|≪J2and inapplicable in\nthe limit of m→1/2. We thus confirm that the bottom of\nthe one-magnonbandmovesfromthe quantumrenormalized\nincommensurate wave number to the classical pitch angle as\nmincreasesfromzerotothesaturation.Thisfeaturewouldbe\nusefultodetermineexchangecouplingsofrealmaterials.\nTo gain an insight into the properties of the field-induced\nspectralweighttransferandthedominantspincorrelation ,we\nexaminethespinstructurefactor,\nSα(q)=/angbracketleftψG|Sα†\nqSα\nq|ψG/angbracketright, (5)\nwith attention to sum rules on the integrated intensity of th e\nFig. 4. (Color online) (a) Sz(q) atJ1=−1, (b)Sx(q) atJ1=−1, and (c)\nSx(q) atJ1=−2.5 for several values of m. (d) Pitch angle Q, determined\nfrom the maximum point of S−(q), as a function of m. Solid symbols denote\nthe VC phase at low fields, and open symbols denote the SN /SDW phase at\nhigh fields. J2=1 is fixed. Data are obtained by DMRG calculations with\nN=128.\ndynamical spin structure factor. To obtain Sα(q), we perform\nground-state DMRG calculations independent of dynamical\nDMRGrunsfor Sα(q,ω),sincewecanobtainaccurateresults\nwithrelativelysmallcomputationalcost.Thesumrulefort he\nenergy-integratedintensityreads\nIα(q)≡/integraldisplaydω\n2πSα(q,ω)=Sα(q), (6)\nleading to the spin structure factor. As for the total intens ity\nIα≡/summationtext\nqSα(q),we have\nIz=Ix=N/4,I±=N/2∓M. (7)\nInFig.4(a),weshow Sz(q)forseveralvaluesof matJ1=−1.\nThere is a clear peak at q=π/2 form=0, and it shifts\ntoward small momentum with increasing m, as indicated by\nvertical arrows. The peak position of Sz(q) representing the\nSDWcorrelationisinagreementwiththelowest-energypeak\npositionq0ofSz(q,ω) in Fig. 3(a). We also find that Sz(q)\ngrows at q=0 asmincreases. In other momentum parts,\nSz(q)is suppressedso as to keep the total intensity Iz=N/4.\nAccordingly, Sz(q,ω)exhibitsthespectralweighttransfer,as\nseenin Fig.1(b).\nWe show Sx(q)atJ1=−1andJ1=−2.5in Figs. 4(b)and\n4(c), respectively. We find a sharp peak structure in the VC\nphase at small m, while the peak structure becomes broad in\nthe SN/SDW phase at large m. This is because the transverse\nspin correlationexhibits an algebraic decay in the VC phase ,\nand it is short-rangedin the SN /SDW phase. As mincreases,\nthe borad peak is leveled o ff, while the baseline goes up, and\neventuallywe finda completelyflat profile,i.e., Sx(q)=1/4,\natthesaturation.Ontheotherhand,thepeakpositiondepen ds\n3J.Phys. Soc. Jpn. LETTERS\nFig. 5. (Color online) The extrapolated gap ∆(m) for several values of J1\natJ2=1 as a function of m. Here we plot data in the SN /SDW phase. Note\nthat the zero-field gap is tiny,33)and the VC phase is supposed to begapless.\nonJ1andm. Indeed, we find a peak near q=π/2 regardless\nofmforJ1=−1 [Fig. 4(b)], while a peak is at q=25π/64\natm=0 and it moves to q=π/2 near the saturation for\nJ1=−2.5[Fig.4(c)].\nTo clarify the mdependenceof the dominant qcomponent\nof the transverse spin correlation,we determinea pitch ang le\nQfrom the maximum point of S−(q), as shown in Fig. 4(d).\nNote that S−(q) andS+(q) have a peak at the same position.\nIn the VC phase at small m,Qagrees with the lowest-energy\npeak position q0ofS−(q,ω), as denoted by solid symbols in\nFigs. 3(b) and 4(d). However, in the SN /SDW phase at large\nm,Qbehaves quite differently from q0asmapproaches the\nsaturation. We find that Qchanges toward π/2 regardless of\nJ1, rather thanthe classical pitchangle cos−1(−J1/4J2) asq0,\nindicatingthatthedominant qcomponentisnotequivalentto\nthemomentumpositionoftheopeninggap.Thisdiscrepancy\nis naturally understood in terms of the short-range nature o f\nthetransversespincorrelation.Itisinadequatetotaketh eonly\nleadingasymptotictermthatrepresentstheexponentialde cay\nofthetransversespincorrelationfunctioninordertodesc ribe\nthe excitation dynamics correctly. In fact, we have seen tha t\nSx(q)turnsinto theflat profileat thesaturation,meaningthat\nnotonlythedominant qcomponentbutalsoothercomponents\nequallycontributetothespectralweight.\nFinally,wepresentthedependenceoftheenergygapofthe\ntransversespin excitationin Fig. 5. We extrapolatefinite- size\ndatatothethermodynamiclimit N→∞byassumingalinear\nrelation∆(N,M)=∆(m)+a/N. Form/greaterorsimilar0.1, we find that\n∆(m) increases as|J1|becomes large for J1=−0.5,−1, and\n−1.5, indicating that the ferromagnetic exchange interaction\nstabilizesthetwo-magnonboundstate. ∆(m)turnstodecrease\nwith further increasing |J1|, shown for J1=−2. For small m,\n∆(m)israpidlyreducedas mdecreasesdowntotheboundary\nbetweentheSN/SDWandVCphases.\nIn summary,we have studied the spin excitation dynamics\nof the frustratedferromagneticchain in the magneticfield b y\nnumerical methods. In the field-induced SN regime, the spin\nexcitationspectraexhibithighlyanisotropicbehaviorbe tween\ngaplesslongitudinalandgappedtransversecompoments.Th e\nfielddependenceofthegaplesspointof Sz(q,ω)isconsistent\nwith the dominant longitudinal spin correlation. In contra st,\nthe gappositionof Sx(q,ω)showsa uniquefield dependence\nthatcontradictswiththedominanttransversespincorrela tion.\nWe hope that these features could be examined by inelasticneutronscatteringexperimentswhensearchingforsignatu res\nof theSN/SDWphase.On theotherhand,we shouldobserve\ngaplessexcitationsin the quadrupolechannel.It wouldbe a n\ninteresting future problem to study the quadrupoleexcitat ion\ndynamicsanditspossiblerelevancetoobservables.34)\nAcknowledgment The author thanks T. Masuda, S. Maekawa, M. Mori,\nand T. Sugimoto for discussions and comments. Part of comput ations were\ncarried out on the supercomputers at the Japan Atomic Energy Agency and\nthe Institute for Solid State Physics,the University of Tok yo.\n1) A.F.Andreev and I. A.Grishchuk, Sov.Phys.JETP 60, 267 (1984).\n2) For a recent review, see K. Penc and A. M. L¨ auchli, in Introduction to\nFrustratedMagnetism ,ed.C.Lacroix,P.Mendels,andF.Mila(Springer-\nVerlag, Berlin, 2011), p. 331.\n3) A.V. Chubukov, Phys.Rev. B 44,4693 (1991).\n4) L. Kecke, T. Momoi, and A. Furusaki, Phys. Rev. B 76, 060407(R)\n(2007).\n5) T. Vekua, A. Honecker, H.-J. Mikeska, and F. Heidrich-Mei sner, Phys.\nRev. B76, 174420 (2007).\n6) T. Hikihara, L. Kecke, T. Momoi, and A. Furusaki, Phys. Rev . B78,\n144404 (2008).\n7) J. Sudan, A. L¨ uscher, and A. M. L¨ auchli, Phys. Rev. B 80, 140402(R)\n(2009).\n8) M.E.ZhitomirskyandH.Tsunetsugu,Europhys.Lett. 92,37001(2010).\n9) M.Sato,T.Hikihara,andT.Momoi,Phys.Rev.Lett. 110,077206(2013).\n10) N. Shannon, T. Momoi, and P. Sindzingre, Phys. Rev. Lett. 96, 027213\n(2006).\n11) H.Tsunetsugu and M.Arikawa, J.Phys.Soc. Jpn. 75,083701 (2006).\n12) A.L¨ auchli, F.Mila, and K. Penc, Phys.Rev. Lett. 97, 087205 (2006).\n13) T. Momoi, P. Sindzingre, and K. Kubo, Phys. Rev. Lett. 108, 057206\n(2012).\n14) M. Enderle, B. Fåk, H.-J. Mikeska, R. K. Kremer, A. Prokofi ev, and W.\nAssmus,Phys.Rev. Lett. 104, 237207 (2010).\n15) T.Masuda,M.Hagihala, Y.Kondoh,K.Kaneko,andN.Metok i, J.Phys.\nSoc. Jpn. 80, 113705 (2011).\n16) L. E. Svistov, T. Fujita, H. Yamaguchi, S. Kimura, K. Omur a, A.\nProkofiev, A. I. Smirnov, Z. Honda, and M. Hagiwara, JETP Lett .93,\n21 (2011).\n17) M. Mourigal, M. Enderle, B. Fåk, R. K. Kremer, J. M. Law, A. Schnei-\ndewind,A.Hiess,andA.Prokofiev,Phys.Rev.Lett. 109,027203(2012).\n18) K.Nawa,M.Takigawa,M.Yoshida,andK.Yoshimura,J.Phy s.Soc.Jpn.\n82, 094709 (2013).\n19) M.Hase, H.Kuroe, K.Ozawa, O.Suzuki, H.Kitazawa, G.Kid o, and T.\nSekine, Phys.Rev. B 70, 104426 (2004).\n20) A. A. Bush, V. N. Glazkov, M. Hagiwara, T. Kashiwagi, S. Ki mura, K.\nOmura,L.A.Prozorova, L.E.Svistov, A.M.Vasiliev, and A.Z heludev,\nPhys.Rev. B 85, 054421 (2012).\n21) B.Willenberg, M.Sch¨ apers, K.C.Rule, S.S¨ ullow,M.Re ehuis, H.Ryll,\nB. Klemke, K.Kiefer, W. Schottenhamel, B. B¨ uchner, B. Oula ddiaf, M.\nUhlarz, R.Beyer, J.Wosnitza, and A.U.B.Wolter, Phys.Rev. Lett.108,\n117202 (2012).\n22) S.E.Dutton, M.Kumar, M.Mourigal, Z.G.Soos,J.-J.Wen, C.L.Bro-\nholm, N. H. Andersen, Q. Huang, M. Zbiri, R. Toft-Petersen, a nd R. J.\nCava, Phys.Rev. Lett. 108, 187206 (2012).\n23) K.Nawa,Y.Okamoto,A.Matsuo,K.Kindo,Y.Kitahara, S.Y oshida, S.\nIkeda, S.Hara,T.Sakurai, S.Okubo,H.Ohta,and Z.Hiroi,J. Phys.Soc.\nJpn.83,103702 (2014).\n24) M.Sato,T.Momoi,andA.Furusaki,Phys.Rev.B 79,060406(R)(2009).\n25) M.Sato, T.Hikihara, and T.Momoi,Phys.Rev. B 83,064405 (2011).\n26) O.A. Starykh and L.Balents, Phys.Rev. B 89,104407 (2014).\n27) H.Onishi, J.Phys.:Conf. Ser. 592, 012109 (2015).\n28) A. Smerald, H. T. Ueda, and N. Shannon, Phys. Rev. B 91, 174402\n(2015).\n29) S.R. White, Phys.Rev. Lett. 69, 2863 (1992).\n30) E.Jeckelmann, Phys.Rev. B 66, 045114 (2002).\n31) J.des Cloizeaux and J.J.Pearson, Phys.Rev. 128, 2131 (1962).\n32) J.Ren and J.Sirker, Phys.Rev. B 85,140410(R) (2012).\n33) C. Itoi and S.Qin, Phys.Rev. B 63, 224423 (2001).\n34) A.V¨ oll and S.Wessel,Phys.Rev. B 91,165128 (2015).\n4"    },    {        "title": "1405.4164v1.Inferring_hidden_states_in_a_random_kinetic_Ising_model__replica_analysis.pdf",        "content": "arXiv:1405.4164v1  [cond-mat.dis-nn]  16 May 2014Inferring hidden states in a random kinetic Ising\nmodel: replica analysis\nLudovica Bachschmid Romano and Manfred Opper\nDepartment of Artificial Intelligence, Technische Universit¨ at Ber lin, Marchstraße 23,\nBerlin 10587, Germany\nE-mail:ludovica.bachschmidromano@tu-berlin.de and\nmanfred.opper@tu-berlin.de\nAbstract. We consider the problem of predicting the spin states in a kinetic Ising\nmodel when spin trajectories are observed for only a finite fractio n of sites. In a\nBayesian setting, where the probabilistic model of the spin dynamics is assumed to\nbe known, the optimal prediction can be computed from the conditio nal (posterior)\ndistribution ofunobserved spins given the observedones. Using th e replica method, we\ncompute the error of the Bayes optimal predictor for parallel disc rete time dynamics in\nafullyconnectedspinsystemwithnonsymmetricrandomcouplings. T heresults, exact\nin the thermodynamic limit, agree very well with simulations of finite spin systems.\n1. Introduction\nThe problem of statistical inference in kinetic Ising models has recen tly attracted\nconsiderable interest in the statistical physics community, see e.g. [1–5]. These systems\ncan be viewed as simple models of networks of spiking neurons and pro vide a prototype\nmodel for which a reconstruction of the network from dynamical d ata can be studied.\nBased on a temporal sequence of observed spin variables, a major goal is to estimate\nthe couplings between sites. This task gets more complicated when a t some sites the\nspin trajectories are not observed. Besides the problem of inferr ing the couplings\nit is then also interesting to predict the states of the non observed spins when the\ncouplings are known. In fact, an iterative solution to the maximum like lihood problem\nfor estimating the couplings is the Expectation Maximization (EM) algorithm [6] which\nwould iterate between estimating hidden spin states (given the last e stimate of the\ncouplings) and reestimating the couplings. Unfortunately, exact in ference of hidden\nstates is not tractable for large networks, but algorithms which ar e based on statistical\nphysics approximations have recently been discussed [7,8]. Hence, it will be interesting\nand important to study a scenario for which the theoretically optima l performance for\npredicting hidden spins can be computed exactly. In this paper, we w ill show that\nsuch a solution can be found in the thermodynamic limit of an infinitely lar ge network\nwhen the couplings are random. Our approach will be based on the re plica method of\ndisordered systems which enables us to compute quenched averag es over the randomInferring hidden states in a random kinetic Ising model: rep lica analysis 2\ncouplings for thermodynamic quantities of the model. These thermo dynamic quantities\nare themselves functions of posterior averages (e.g. local magne tizations) of the hidden\nspins. The replica approach has been successfully applied in the past to a large variety\nof statistical learning problems for staticnetwork models (for a summary see [9–11]).\nWe will restrict ourselves to a model where the couplings are mutually independent\nrandom variables, i.e. where no symmetry between in-and outgoing c onnections are\nassumed. For such type of models (without the observations) var ious exact solutions\nfor the non equilibrium dynamics have been computed, see e.g. [1,5] a nd [12,13] for\nsoft spin models. From the point of view of equilibrium statistical phys ics the case\nof symmetric couplings might be interesting. Such a spin model would o bey detailed\nbalance and allow for a stationary Gibbs distribution. Unfortunately , for the Ising case,\nthe exact computation of time dependent correlation functions wh ich are necessary for\nour analysis seems not possible. On the other hand, from a point of v iew of neural\nmodeling, the assumption of symmetric couplings is not realistic [1,14], as synaptic\nconnections in biological networks are known to be strongly asymme tric. Hence, we\nbelieve that our restriction to asymmetric couplings is justified both from a modeling\nand a computational perspective.\n2. The model and Bayes optimal inference\nWe will consider a model with NIsing spins which are divided into two groups: a group\nof spinssi(t) at sites i=1,..., Nobs=λNwhich are observed during a time interval of\nTtime steps, and a group of hidden, i.e. unobserved spins, denoted b yσa(t) at sites\na= 1,...,N hid= (1−λ)N. We assume parallel Markovian dynamics for the entire spin\nsystem, which is governed by the transition probability\nP[{s,σ}(t+1)|{s,σ}(t)] =/productdisplay\niesi(t+1)gi(t)\n2cosh[gi(t)]/productdisplay\naeσa(t+1)ga(t)\n2cosh[ga(t)], (1)\nwhere the fields are defined as\ngi(t) =/summationdisplay\njJijsj(t)+/summationdisplay\nbJibσb(t), g a(t) =/summationdisplay\njJajsj(t)+/summationdisplay\nbJabσb(t),(2)\nin terms of the couplings Jand{s,σ}denotes all the possible spin vector configurations;\nwhen the time index is not specified we are considering the whole time se ries,t= 0...T.\nThe total probability for a spin trajectory is given by\nP({s,σ}) =1\n2NT−1/productdisplay\nt=0P[{s,σ}(t+1)|{s,σ}(t)], (3)\nwhere we have considered completely random initial condition P0[{s,σ}(0)] = 1/2N.\nTo make predictions on the unobserved spins σa(t), we assume that the model given\nby the couplings Jis perfectly known and the posterior, i.e. conditional probability ofInferring hidden states in a random kinetic Ising model: rep lica analysis 3\nthe hidden spins defined by\nP({σ}|{s}) =P({s,σ})\nP({s}), (4)\ngives the complete information for an optimal inference of hidden sp ins. Based on this\nprobabilistic information, the best possible prediction σopt\na(t) for the hidden spin at site\naand at time tis computed by\nσopt\na(t) = sign[ma(t)], (5)\nwhere the local magnetization is defined as the posterior expectat ion\nma(t) =/summationdisplay\n{σ}σa(t)P({σ}|{s}). (6)\nNote that this does not correspond to the most likely spin configuration {σ}, because\nwe have averaged out the configurations of spins σb(t′) forb∝ne}ationslash=aandt′∝ne}ationslash=t.\nGiven a true ‘teacher’ sequence {σ∗}of unobserved spins, we are interested in the\ntotal quality of the Bayes optimal prediction, i.e. in the expected pr obability of wrongly\npredicting a spin at site aand timet, given by the Bayes error\nε=/summationdisplay\n{s,σ∗}P({s,σ∗})Θ(−σ∗\na(t)ma(t)) =/summationdisplay\n{s}P({s})/summationdisplay\n{σ∗}P({σ∗}|{s})Θ(−σ∗\na(t)ma(t)),\n(7)\nwhere the step function Θ( x) = 1 forx>0 and 0 else. In the next section we will use\nthe replica method to compute the error in the thermodynamic limit N→ ∞, when the\ncouplingsJare assumed to be mutually independent Gaussian random variables, with\nzero mean and variance of the order 1 /N.\n3. Replica analysis\nThe posterior statistics of the hidden spins can be obtained from th e following partition\nfunction\nP({s}) =1\n2N/summationdisplay\n{σ}/productdisplay\ntP[{s,σ}(t+1)|{s,σ}(t)], (8)\nwhich equals the total probability of the observed spin configuratio ns and is also the\nnormalizer of the posterior probability. Typical performance in the thermodynamic\nlimit for random couplings are then computed from the quenched ave rage of the free\nenergyF=−∝an}bracketle{tlnP({s})∝an}bracketri}htJ,s, where the average is taken over the the couplings Jand\nover the observed spin configurations with their weights P({s}). Hence, the averaged\nfree energy is given by\nF=−/summationdisplay\n{s}∝an}bracketle{tP({s})logP({s})∝an}bracketri}htJ. (9)Inferring hidden states in a random kinetic Ising model: rep lica analysis 4\nThis average can be computed by the replica trick [9–11] in the follow ing way:\nF=−lim\nn→1d\ndnlog/summationdisplay\n{s}∝an}bracketle{tPn({s})∝an}bracketri}ht. (10)\nFor integer n, we have\n/summationdisplay\n{s}∝an}bracketle{tPn({s})∝an}bracketri}htJ=1\n2nN/summationdisplay\n{s}/summationdisplay\n{σ(1)}.../summationdisplay\n{σ(n)}/angbracketleftBigg/bracketleftBiggn/productdisplay\nα=1exp/braceleftBigg/summationdisplay\nitsi(t+1)gα\ni(t)\n+/summationdisplay\natσα\na(t+1)gα\na(t)−/summationdisplay\nitlog2cosh[gα\ni(t)]−/summationdisplay\natlog2cosh[gα\na(t)]/bracerightBigg/bracketrightBigg/angbracketrightBigg\nJ,(11)\nwith\ngα\ni(t) =/summationdisplay\njJijsj(t)+/summationdisplay\nbJibσα\nb(t), gα\na(t) =/summationdisplay\njJajsj(t)+/summationdisplay\nbJabσα\nb(t).(12)\nTo perform the average over the couplings Jij,Jib,JajandJab, which are assumed to be\nmutually independent Gaussian random variables with zero mean and v ariancek2/N,\nwe note that the fields gα\ni(t) andgα\na(t) are also Gaussian, which are independent for\ndifferent sites ianda, but will be dependent for different replica index αandβand also\npossibly for different times. This yields\n/angbracketleftBig\ngα\ni(t)gβ\ni(t′)/angbracketrightBig\n=/angbracketleftbig\ngα\na(t)gβ\na(t′)/angbracketrightbig\n=k2/parenleftbig\nλS(t,t′)+(1−λ)Qαβ(t,t′)/parenrightbig\n,\n∝an}bracketle{tgα\ni(t)gα\ni(t′)∝an}bracketri}ht=∝an}bracketle{tgα\na(t)gα\na(t′)∝an}bracketri}ht=k2(λS(t,t′)+(1−λ)Cα(t,t′)),(13)\nwhere we have defined the following order parameters\nCα(t,t′) =1\nNhid/summationdisplay\naσα\na(t)σα\na(t′) fort<t′, (14)\nQαβ(t,t′) =1\nNhid/summationdisplay\naσα\na(t)σβ\na(t′) forα<β,\nS(t,t′) =1\nNobs/summationdisplay\nisi(t)si(t′) fort<t′. (15)Inferring hidden states in a random kinetic Ising model: rep lica analysis 5\nIntroducing these definitions within δfunctions and expressing the δfunctions using\nconjugate (hatted) integration parameters, we get the following expression:\n/summationdisplay\n{s}∝an}bracketle{tPn(s)∝an}bracketri}htJ=c\n2nN/integraldisplay/productdisplay\nt,t′/productdisplay\nα<β/parenleftBig\ndQαβ(t,t′)dˆQαβ(t,t′)/parenrightBig\n/productdisplay\nt<t′/productdisplay\nα/parenleftBig\ndCα(t,t′)dˆCα(t,t′)/parenrightBig/productdisplay\nt<t′/parenleftBig\ndS(t,t′)dˆS(t,t′)/parenrightBig\nexp/parenleftBigg\niNhid/summationdisplay\nα/summationdisplay\nt<t′Cα(t,t′)ˆCα(t,t′)+iNhid/summationdisplay\nα<β/summationdisplay\ntt′Qαβ(t,t′)ˆQαβ(t,t′)\n+iNobs/summationdisplay\nt<t′S(t,t′)ˆS(t,t′)+NobslogEobs(C,Q)\n+NhidlogEhid(C,ˆC,Q,ˆQ)/parenrightBig\n,\n(16)\nwherecis a trivial constant non depending on N,\nEobs(C,Q) =/summationdisplay\n{s}/angbracketleftBigg\nexp/parenleftBigg/summationdisplay\ntαs(t+1)gα(t)−/summationdisplay\ntαlog2cosh[gα(t)]\n−i/summationdisplay\nt<t′ˆS(t,t′)s(t)s(t′)/parenrightBigg/angbracketrightBigg\ng\nEhid(C,ˆC,Q,ˆQ) =/summationdisplay\n{σ(1)}.../summationdisplay\n{σ(n)}/angbracketleftBigg\nexp/parenleftBigg/summationdisplay\ntσα(t+1)gα(t)−/summationdisplay\ntαlog2cosh[gα(t)]\n−i/summationdisplay\nα/summationdisplay\nt<t′ˆCα(t,t′)σα(t)σα(t′)−i/summationdisplay\nα<β/summationdisplay\ntt′ˆQαβ(t,t′)σα(t)σβ(t′)/parenrightBigg/angbracketrightBigg\ng,(17)\nand the average is over the Gaussian fields with statistics given by (1 3). In the limit\nN→ ∞, keeping the ratio λ=Nobs/Nfixed, the integrals over the order parameters\ncan be performed using the saddle point method, where we assume r eplica symmetry,\ni.e.Cα(t,t′) =C(t,t′)∀α,t<t′and,Qαβ(t,t′) =Q(t,t′)∀α<β,t,t′. We get\nlim\nN→∞1\nNlog/summationdisplay\n{s}∝an}bracketle{tPn({s})∝an}bracketri}ht= Extrfn(C,S,...),Inferring hidden states in a random kinetic Ising model: rep lica analysis 6\nwhere we have to take the extremum with respect to the order par ameters in the\nexpression\nfn(C,S,...) =i(1−λ)n/summationdisplay\nt<t′C(t,t′)ˆC(t,t′)+i(1−λ)(n2−n)\n2/summationdisplay\ntt′Q(t,t′)ˆQ(t,t′)\n+iλ/summationdisplay\nt<t′S(t,t′)ˆS(t,t′)+λlog/summationdisplay\n{s}/angbracketleftBigg/angbracketleftBigg/productdisplay\ntV(t)/angbracketrightBiggn\nζe/summationtext\nts(t)ν(t)/angbracketrightBigg\nψ,ν\n+(1−λ)log/angbracketleftBigg/angbracketleftBigg\nΓ0/productdisplay\ntZ(t)/angbracketrightBiggn\nξ,ζ/angbracketrightBigg\nφ,ψ−nlog2,(18)\nwhere we have introduced\nV(t) =es(t+1)(ψ(t)+ζ(t))\n2cosh(ψ(t)+ζ(t)), Z(t) =cosh[ψ(t)+ζ(t)+φ(t+1)+ξ(t+1)]\ncosh(ψ(t)+ζ(t)),(19)\nin terms of Gaussian independent random fields ψ(t),ζ(t),ν(t),ξ(t) andφ(t), with zero\nmean and covariances given by the following set of equations:\n∝an}bracketle{tψ(t)ψ(t′)∝an}bracketri}ht=k2(λS(t,t′)+(1−λ)Q(t,t′)), (20)\n∝an}bracketle{tζ(t)ζ(t′)∝an}bracketri}ht=k2(1−λ)(C(t,t′)−Q(t,t′)), (21)\n∝an}bracketle{tν(t)ν(t′)∝an}bracketri}ht=−iˆS(t,t′), (22)\n∝an}bracketle{tξ(t)ξ(t′)∝an}bracketri}ht=−i(ˆC(t,t′)−ˆQ(t,t′)), (23)\n∝an}bracketle{tφ(t)φ(t′)∝an}bracketri}ht=−iˆQ(t,t′), (24)\nfort′∝ne}ationslash=tand\n∝an}bracketle{tψ(t)ψ(t)∝an}bracketri}ht=k2(λ+(1−λ)Q(t,t)),∝an}bracketle{tζ(t)ζ(t)∝an}bracketri}ht=k2(1−λ)(1−Q(t,t)),(25)\n∝an}bracketle{tν(t)ν(t)∝an}bracketri}ht= 0, ∝an}bracketle{tξ(t)ξ(t)∝an}bracketri}ht=iˆQ(t,t), (26)\n∝an}bracketle{tφ(t)φ(t)∝an}bracketri}ht=−iˆQ(t,t), (27)\nfort′=t. The term Γ 0contains the initial condition for the fields φ,ξ(Appendix B).\nThe 3 sets of Gaussian variables in (23,24,26,27) have been introduce d to linearize the\nquadratic forms in equation (17). We can now perform the continua tion to noninteger\nnand obtain the free energy per spin lim N→∞F/Nas the stationary value of\nf(C,S,...) =−i(1−λ)/summationdisplay\nt<t′C(t,t′)ˆC(t,t′)−i(1−λ)\n2/summationdisplay\ntt′Q(t,t′)ˆQ(t,t′)\n−λ/summationtext\n{s}/angbracketleftBig\n∝an}bracketle{t/producttext\ntV(t)∝an}bracketri}htζlog∝an}bracketle{t/producttext\ntV(t)∝an}bracketri}htζe−i/summationtext\nts(t)ν(t)/angbracketrightBig\nψ,ν\n/summationtext\n{s}/angbracketleftBig\n∝an}bracketle{t/producttext\ntV(t)∝an}bracketri}htζe/summationtext\nts(t)ν(t)/angbracketrightBig\nψ,ν\n−(1−λ)/angbracketleftBig\n∝an}bracketle{tΓ0/producttext\ntZ(t)∝an}bracketri}htξ,ζlog∝an}bracketle{tΓ0/producttext\ntZ(t)∝an}bracketri}htξ,ζ/angbracketrightBig\nφ,ψ\n∝an}bracketle{tΓ0/producttext\ntZ(t)∝an}bracketri}htξ,ζ,φ,ψ.(28)Inferring hidden states in a random kinetic Ising model: rep lica analysis 7\nFrom equation (28) we can compute the self-averageing values of t he order parameters\nandtheirconjugates. Previousstudies[1,5,15]ofspinmodelswith asymmetriccouplings\nhave shown that spin correlations S(t,t′) decay after one time step. Hence, we expect\nthat also for our model the other two time order parameters are z ero fort∝ne}ationslash=t′. Indeed,\nwe can show (for an example, see Appendix A) that the results\nC(t,t′) =Q(t,t′) =ˆC(t,t′) =ˆQ(t,t′) =ˆS(t,t′) = 0\nare self-consistent solutions of the order parameter equations f ort′∝ne}ationslash=tand this solution\nis also supported by simulations. In this case, only the terms with t′=tgive non-zero\ncontribution in equetion (16) and the free energy of the system sim plifies to\nf(Q,ˆQ) =−i\n2(1−λ)T/summationdisplay\nt=0Q(t)ˆQ(t)−i\n2(1−λ)T/summationdisplay\nt=0ˆQ(t)\n−λT−1/summationdisplay\nt=0/summationdisplay\n{s}(t+1)/angbracketleftBig\n∝an}bracketle{tV(t)∝an}bracketri}htζtlog∝an}bracketle{tV(t)∝an}bracketri}htζt/angbracketrightBig\nψt\n−(1−λ)T−1/summationdisplay\nt=0/angbracketleftbigg/angbracketleftBig\n˜Z(t)/angbracketrightBig\nζtlog/angbracketleftBig\n˜Z(t)/angbracketrightBig\nζt/angbracketrightbigg\nφt+1,ψt/angbracketleftBig\n˜Z(t)/angbracketrightBig\nζt,φt+1,ψt−(1−λ)˜Γ0,(29)\nwhere\n˜Z(t) =cosh[ψ(t)+ζ(t)+φ(t+1)]\ncosh(ψ(t)+ζ(t)),\nQ(t)≡Q(t,t) and the initial condition ˜Γ0is given in Appendix B. The order parameter\nQ(t) gives the typical overlap of two independent spin configurations a t timetdrawn\nat random from the posterior distribution. By symmetry, it also des cribes the expected\noverlap of the hidden spins drawn from the posterior with the true ‘t eacher’ spins of\nthe model from which the observation data were generated. Henc e the limit Q(t) = 0\ndescribes a situations where the posterior gives no information on t he hidden spins. On\nthe other hand, Q(t) = 1, means that we can predict the hidden spins perfectly. We\nobtain the following equations for the order parameters:\nQ(t) =1/angbracketleftBig\n˜Z(t−1)/angbracketrightBig\nζt−1,φt,ψt−1/angbracketleftBigg/angbracketleftBig\ntanh˜A(t−1)˜Z(t−1)/angbracketrightBig2\nζt−1/angbracketleftBig\n˜Z(t−1)/angbracketrightBig\nζt−1/angbracketrightBigg\nφt,ψt−1, t= 1...T\n(30)Inferring hidden states in a random kinetic Ising model: rep lica analysis 8\nˆQ(t) =ik2(1−λ)/angbracketleftBig\n˜Z(t)/angbracketrightBig\nζt,φt+1,ψt/angbracketleftBigg/angbracketleftBig\n[tanh˜A(t)−tanh˜B(t)]˜Z(t)/angbracketrightBig2\nζt/angbracketleftBig\n˜Z(t)/angbracketrightBig\nζt/angbracketrightBigg\nφt+1,ψt\n+ik2λ/summationdisplay\n{s}(t+1)/angbracketleftBigg/angbracketleftBig\n[s(t+1)−tanh˜B(t)]V(t)/angbracketrightBig2\nζt\n∝an}bracketle{tV(t)∝an}bracketri}htζt/angbracketrightBigg\nψt, t= 0...T−1(31)\nwhere\n˜A(t) =ψ(t)+ζ(t)+φ(t+1), ˜B(t) =ψ(t)+ζ(t).\nThe equations for the initial and final conditions, Q(0) and ˆQ(T), are given in in\nAppendix B.\n4. Distribution of local magnetization\nIt is easy to extend the replica approach to the computation of oth er thermodynamic\nquantities such as functions of the local magnetizations. We find th at, in the\nthermodynamic limit, hidden spins can be viewed as mutually independen t random\nvariables which are coupled to random fields. The spins have local mag netizations\nm(t|ψ,φ) =/angbracketleftBig\ntanhA(t−1)˜Z(t−1)/angbracketrightBig\nζt−1/angbracketleftBig\n˜Z(t−1)/angbracketrightBig\nζt−1, (32)\nwhere the ‘inner’ averages over ζreflect the averaging out of the other spins. The\nmagnetizations depend on the random fields ψ(t−1),φ(t). These Gaussian fields reflect\nthe disorder originating from the random couplings. In computing ex pectations they\nget an extra statistical weight given by\nw(ψ,φ) =/angbracketleftBig\n˜Z(t−1)/angbracketrightBig\nζt−1/angbracketleftBig\n˜Z(t−1)/angbracketrightBig\nζt−1,ψt−1,φt(33)\nin the ‘outer’ average. Hence, the distribution of local magnetizat ions at an arbitrary\nsite and at time tis given by\npt(m) =∝an}bracketle{tw(ψ,φ)δ(m−m(t|ψ,φ)∝an}bracketri}htψt−1,φt, (34)\nfrom which the overlap Qis recovered as Q(t) =/integraltext1\n−1pt(m)m2dm. Finally, to get the\nBayes error we note that the (posterior) probability of a spin σequals1\n2(mσ+1). Hence\neq (7) is translated into\nε=1\n2/summationdisplay\nσ=±1/integraldisplay1\n−1pt(m)(mσ+1)Θ(−σm)dm=1\n2/parenleftbigg\n1−/integraldisplay1\n−1pt(m)|m|dm/parenrightbigg\n,(35)Inferring hidden states in a random kinetic Ising model: rep lica analysis 9\nwhere the last equality follows easily from the fact that pt(m) =pt(−m).\n5. Results\nWe have solved the order parameter equations (30) and (31) by ite rating equations (25,\n27, 30, 31) for different values of the load parameter λand coupling strength k(for an\nexample see figure 1). We start the recursion from the prior initial c onditionQ(0) = 0\nandtheniteratetheequations forwardandbackward, updating t heboundaryconditions\nat each iteration according to equation (42). The overlap is smallest at the boundary\nt= 0 andt=T, because there the information flow is only from one direction and is\nalso expected to decay over the time T.\nWhen the length Tof the spin trajectories grows, the order parameters Q(t) and\nˆQ(t) for times taway from the boundaries, i.e. 0 ≪t≪T, converge to stationary\nvaluesQstatandˆQstat. These can be directly computed from eqs. (30-31) by setting\nQ(t) =Q(t−1) =QstatandˆQ(t) =ˆQ(t+ 1) = ˆQstat. For given stationary order\nparameters we have then computed the distribution of local magne tizations and the\nBayes error. The Bayes error εis shown in figure (2) as a function of the load factor λ.\nIn the limit of no observations, λ= 0, the prediction on the the state of hidden spins\nis completely random and the error has the trivial value ε= 0.5. The error rapidly\ndecreases as λgets larger, but remains nonzero for λ= 1, indicating the presence of a\nresidual error in almost fully observed systems due to the stochas ticity of the Markov\nprocess. Since the couplings are responsible for the propagation o f information between\nspin sites, the Bayes error decreases as the coupling strength inc reases; in particular\nwe find that ε→0 fork→ ∞. This behaviour is illustrated in figure (3), where\nthe distribution p(m) of the local magnetization (eq. 34) is shown. For small kthe\ndistribution is close to a Gaussian centered at zero, with vanishing va riance ask→0,\nmeaning (see eq. 35) that nontrivial prediction on the magnetizatio n can be made. As\nkgrows larger the distribution broadens and above a critical value th e curve becomes\nbimodal. For large k, the distribution p(m) concentrates at m=±1, allowing for a\nperfect prediction of hidden spins.\nOur analytical results agree very well with simulations of spin system s with relative\nsmall number of spins. For these systems we could compute local ma gnetizations ma(t)\nexactly by enumeration. The Markovian spin dynamics facilitated the se computations\nwith the use of a forward–backward algorithm [16] well known for hidden Markov models\n(Appendix C). We compute Q(t) using\nQ(t) =1\nNhidNhid/summationdisplay\na=1Es,Jm2\na(t),\nwhereEs,Jdenotes the expectation over all possible observed spins and over the set of\nrandom couplings.Inferring hidden states in a random kinetic Ising model: rep lica analysis 10\n2 4 6 8t0.340.350.360.370.38Q(t)Replica\nF-B\nQ  stat\nFigure 1: Order parameter Qas a function of time, for a system with λ= 0.4 andk= 1.\nRed line: solution of the order parameter equations. Black dashed lin e: stationary value\nQstatof the order parameter. Blue points: Qfrom numerical simulation of a system\nwithNhid= 10 hidden spins, averaged over 10000 samples; the error bars re present the\nstandard deviation.\n0 0.2 0.4 0.6 0.8 1\nλ00.10.20.30.40.5 εReplica, k=1\nReplica, k=2\nF-B, k=1\nF-B, k=2\nFigure 2: Bayes error as a function of the load factor for k= 1 (solid red line, blue\ntriangles) and k= 2 (dashed red line, blue circles). Red lines: replica result, computed\nwith the stationary values of the order parameters. Blue points: n umerical simulation\nof a system with Nhid= 8 hidden spins, averaged over 2500 samples; the error bars\nrepresent the standard deviation; the Bayes error is computed a t timet=T/2, with\nT= 20.Inferring hidden states in a random kinetic Ising model: rep lica analysis 11\n-1 -0.5 0 0.5 1m00.511.52p(m)F-B\nReplica(A)\n-1 -0.5 0 0.5 1m00.20.40.60.8p(m)F-B\nReplica(B)\n-1 -0.5 0 0.5 1m02468p(m)F-B\nReplica(C)\nFigure 3: Distribution of local magnetization for load factor λ= 0.8 and coupling\nstrengthsk= 0.2(A),k= 0.6(B) andk= 2(C). Red line: analytical result (eq.\n34) assuming stationary values of the order parameters. Blue hist ogram: numerical\nsimulations averaged over 80000 samples for a system with 8 hidden s pins. The\nmagnetization is computed at time t=T/2, withT= 20.\n6. A comment on symmetric networks\nFrom the point of view of equilibrium statistical physics a correspond ing analysis for\nsymmetric couplings Jij=Jjimight be of interest. In this case our approach would\nlead to additional order parameters (e.g. response functions). M ore important, order\nparameters would be usually non–zero for t∝ne}ationslash=t′. Take for example the order parameter\nC(t,t′) =EJ\n/summationdisplay\n{s}P({s})/summationdisplay\n{σ}P({σ}|{s})σa(t)σa(t′)\n\n=EJ\n/summationdisplay\n{s},{σ}P({σ},{s})σa(t)σa(t′)\n,(36)Inferring hidden states in a random kinetic Ising model: rep lica analysis 12\nwhere the last line follows from Bayes theorem. Hence, C(t,t′) equals the usual\nspin correlation in a system of Nhid+Nobs=Nspins, where there is no difference\nbetween hidden and observed spins (because there is no conditioning on the latter\nones). Unfortunately, even for this simpler, more standard type of spin–glass model\n(studied extensively in the 1990s), exact analytical results for tw o time correlations\n(except for the case of uncorrelated couplings and Gaussian or sp herical spin models)\nwere not possible. A Monte Carlo approach to the effective non–Mar kovian single spin\ndynamics [15,17]couldbeadaptedtoourmodelbut itwouldrequiree xtensive nontrivial\nnumerical simulations with an increasing complexity when the time windo wTgrows.\nMoreover, this method cannot be easily extended to the stationar y case.\nTo circumvent this problem, one might be tempted to resort to equilib rium\ntechniques instead. Infact,forthecaseofsymmetriccouplings, theMarkoviandynamics\nof thejoint system of sandσspins has a well known stationary equilibrium distribution.\nThis static distribution is usually known as the Little model [18–20] and was frequently\ndiscussed in in the framework of Hopfield type neural networks with parallel dynamics.\nOn might then calculate learning properties of the static model by us ing again the\nreplica approach. While this should indeed be feasible (when replica sym metry breaking\neffects are neglected), one should note that this approach would c onsider a quite\ndifferent statistical ensemble. The equilibrium case would deal with th e probability\nP(σ(t)|s(t)) of spins at fixed large time t, whereas our dynamic ensemble is concerned\nwithP({σ}|{s}) with a conditioning on information {s}from the time history of past\nand future observations.\nHence, the problem of solving the model with symmetric couplings is fa r different\nfrom the asymmetric case studied in this paper and will be postponed to future work.\n7. Outlook\nIn this paper we have presented a first step in analyzing optimal Bay esian inference\nfor kinetic Ising models with observed and unobserved spins valid for large random\nsystems. The replica analysis revealed a fairly simple statistical pictu re of the posterior\ntrajectories of hidden spins. Spins at different time steps (and site s) are statistically\nindependent, but their local magnetizations depend on the propag ation of information\nfrom past and future spins which is expressed through order para meters.\nOne can expect that this simple picture derived for the disorder ave raged system\ncan be translated into equations for the local magnetizations of hid den spins which\nare valid for a typical singlesystem with fixed couplings and observations. In fact,\nsuch mean field equations generalizing the results of [1] to the case o f observations can\nbe derived from cavity arguments and could be used as an efficient alg orithm for the\ncomputation of local magnetizations in large random networks. This could then be used\nas an approximation in the E-Step of an EM algorithm [6] which aims at c omputing the\nmaximum likelihood estimator of the network couplings Jij, averaging out unobserved\nspins. We will discuss such an approach in a forthcoming paper.Inferring hidden states in a random kinetic Ising model: rep lica analysis 13\nIt will be interesting to extend this replica approach to other dynam ical models. As\nlong as we restrict ourselves to asymmetric random couplings one ca n expect that the\ncase of continuous time (at least for the stationary limit) models cou ld be treated. This\nwouldinclude e.g. continuous timeGlauberdynamics andcoupledstoch astic differential\nequations (soft spin models).\nAcknowledgements\nThis work is supported by the Marie Curie Training Network NETADIS ( FP7, grant\n290038).\nA. Self consistent solution for the two time order parameter s\nLet us consider, as an example, the stationary value of the order p arameterQ. From\nthe saddle point equation∂f\n∂ˆQ= 0 we find:\nQ(t,t′) =1\n∝an}bracketle{tΓ0/producttext\nτZ(τ)∝an}bracketri}htξ,ζ,φ,ψ/angbracketleftBigg\n∝an}bracketle{tΓ0tanhA(t−1)/producttext\nτZ(τ)∝an}bracketri}htξ,ζ∝an}bracketle{tΓ0tanhA(t′−1)/producttext\nτZ(τ)∝an}bracketri}htξ,ζ\n∝an}bracketle{tΓ0/producttext\nτZ(τ)∝an}bracketri}htξ,ζ/angbracketrightBigg\nφ,ψ,(37)\nwhere\nA(t) =ψ(t)+ζ(t)+φ(t+1)+ξ(t+1). (38)\nWewanttoshowthat Q(t,t′) = 0fort∝ne}ationslash=t′isaselfconsistentsolution. Ifourassumption\nholds for the order parameters on the right hand side of equation ( 37 ), the averages\nover the gaussian fields factorize over time, yielding:\nQ(t,t′) =∝an}bracketle{ttanhA(t−1)Z(t−1)∝an}bracketri}htξt−1,ζt−1,φt,ψt\n∝an}bracketle{tZ(t−1)∝an}bracketri}htξt−1,ζt−1,φt,ψt∝an}bracketle{ttanhA(t′−1)Z(t′−1)∝an}bracketri}htξt′−1,ζt′−1,φt′,ψt′\n∝an}bracketle{tZ(t′−1)∝an}bracketri}htξt′−1,ζt′−1,φt′,ψt′.(39)\nThe first two terms in the numerator of the above equation can be w ritten in terms of\nthe independent random variables x=ψ(t−1)+ζ(t−1) andy=φ(t)+ξ(t) as\n/angbracketleftbiggsinh(x+y)\ncosh(x)/angbracketrightbigg\nx,y=/angbracketleftbiggsinh(x)cosh(y)+cosh(x)sinh(y)\ncosh(x)/angbracketrightbigg\nx,y\n=∝an}bracketle{ttanh(x)∝an}bracketri}htx∝an}bracketle{tcosh(y)∝an}bracketri}hty+∝an}bracketle{tsinh(y)∝an}bracketri}hty= 0.(40)\nUsing a similar procedure, this argument can be extended to all the o ther order\nparameters.Inferring hidden states in a random kinetic Ising model: rep lica analysis 14\nB. Boundary conditions\nTheparametersΓ 0and˜Γ0containingtheinitialconditionshavethefollowingexpression:\nΓ0= 2cosh[φ(0)+ξ(0)], ˜Γ0=∝an}bracketle{tcosh(φ0)log(2cosh( φ0))∝an}bracketri}htφ0\n∝an}bracketle{tcosh(φ0)∝an}bracketri}htφ0.(41)\nThe initial and final condition for the order parameter are:\nQ(0) =∝an}bracketle{ttanh(φ0)sinh(φ0)∝an}bracketri}htφ0\n∝an}bracketle{tcosh(φ0)∝an}bracketri}htφ0, ˆQ(T) = 0. (42)\nC. Forward-backward algorithm\nIn order to compute the local magnetizations of hidden spins at eac h timet, we need\nthe posterior distribution P[{σ}(t)|{s}]1≤t < Tof the hidden spins at time t, given\nthe obserserved spins at all times.\nIt is convenient to divide the computation of P[{σ}(t)|{s}] in two parts, one\ninvolving the spins up to time t+1, the other the spins from t+2 toT:\nP[{σ}(t)|{s}] =P[{σ}(t)|{s}1:t+1,{s}t+2:T]\n∝P[{σ}(t)|{s}1:t+1]P[{s}t+2:T|{σ}(t),{s}(t+1)],(43)\nwhere the last line follows from Bayes’ rule and the conditional indepe ndence of {s}t+2:T\nand{s}1:tgiven{s}(t+1) and {σ}(t). The two terms in the right hand side of eq. (43)\ncan be computed by recursion through time. In particular, it can be shown [16] that\nthe first term, referred to as the “forward message”, fm[ {σ}(t)] =P[{σ}(t)|{s}1:t+1], is\nobtained by a forward recursion form 1 to tgoverned by the following equation:\nfm[{σ}(t)]∝P[{s}(t+1)|{σ,s}(t)]/summationdisplay\n{σ}(t−1)P[{σ}(t)|σ(t−1),{s}(t−1)]fm[{σ}(t−1)].\n(44)\nThesecond term, or“backwardmessage” bm[ {σ}(t+1)] =P[{s}t+2:T|{σ}(t),{s}(t+1)],\nis obtained by a backward recursion, running from Ttot+1 and obeying:\nbm[{σ}(t+1)] =/summationdisplay\n{σ}(t+1)P[{s}(t+2)|{σ,s}(t+1)]bm[{σ}(t+2)]P[{σ}(t+1)|{σ,s}(t)].\n(45)\n[1] M´ ezard, M. and Sakellariou, J.: Exact mean-field inference in asy mmetric kinetic Ising systems.\nJ Stat Mech 2011, L07001 (2011)\n[2] Roudi, Y. and Hertz, J.: Mean Field Theory for Nonequilibrium Netwo rk Reconstruction. Phys\nRev Lett 106, 048702 (2011)\n[3] Zeng, H.-L., Aurell, E., Alava, M., and Mahmoudi, H.: Network inferen ce using asynchronously\nupdated kinetic Ising model. Phys Rev E 83, 041135 (2011)Inferring hidden states in a random kinetic Ising model: rep lica analysis 15\n[4] Sakellariou, J., Roudi, Y., M´ ezard, M., and Hertz, J.: Effect of cou pling asymmetry on mean-field\nsolutions of the direct and inverse SherringtonKirkpatrick model. Philosophical Magazine 92,\n272–279 (2012)\n[5] Huang, H. and Kabashima, Y.: Dynamics of asymmetric kinetic Ising systems revisited.\narXiv:13105003 (2013)\n[6] Dempster, A. P., Laird, N. M., and Rubin, D. B.: Maximum Likelihood fr om Incomplete Data via\nthe EM Algorithm. Journal of the Royal Statistical Society Series B 39, 1–38 (1977)\n[7] Dunn, B. and Roudi, Y.: Learning and inference in a nonequilibrium Is ing model with hidden\nnodes.Phys Rev E 87, 022127 (2013)\n[8] Tyrcha, J. and Hertz, J.: Network inference with hidden units. MBE11, 149 (2014)\n[9] Opper, M. and Kinzel, W.: Statistical Mechanics of Generalization. In Domany, E., van Hemmen,\nJ., and Schulten, K., eds., Models of Neural Networks III . Springer-Verlag (1996)\n[10] Nishimori, H.: Statistical Physics of Spin Glasses and Information Proces sing: An Introduction .\nOxford University Press (2001)\n[11] Engel, A. and Van den Broeck, C.: Statistical Mechanics of Learning . Cambridge University Press\n(2001)\n[12] Crisanti, A. and Sompolinsky, H.: Dynamics of spin systems with ra ndomly asymmetric bonds:\nLangevin dynamics and a spherical model. Phys Rev A 36, 4922–4939 (1987)\n[13] Sompolinsky, H., Crisanti, A., and Sommers, H. J.: Chaos in Random Neural Networks. Phys\nRev Lett 61, 259–262 (1988)\n[14] Parisi, G.: Asymmetric neural networks and the process of lear ning.Journal of Physics A:\nMathematical and General 19, L675 (1986)\n[15] Eissfeller, H. and Opper, M.: Mean-field Monte Carlo approach to the Sherrington-Kirkpatrick\nmodel with asymmetric couplings. Phys Rev E 50, 709–720 (1994)\n[16] Russel, S. and Norvig, P.: Artificial Intelligence: A Modern Approach . Pearson Education (1995)\n[17] Scharnagl,A., Opper, M., and Kinzel, W.: On the relaxationofinfinit e-rangespin glasses. Journal\nof Physics A: Mathematical and General 28, 5721 (1995)\n[18] Little, W.: The existence of persistent states in the brain. Mathematical Biosciences 19, 101–120\n(1974)\n[19] Little, W. and Shaw, G. L.: A statistical theory of short and long term memory. Behavioral\nBiology14, 115–133 (1975)\n[20] Little, W. and Shaw, G. L.: Analytic study of the memory storage capacity of a neural network.\nMathematical Biosciences 39, 281–290 (1978)"    },    {        "title": "1212.4353v2.Thermoelectric_spin_accumulation_and_long_time_spin_precession_in_a_non_collinear_quantum_dot_spin_valve.pdf",        "content": "Thermoelectric spin accumulation and long-time spin precession in a non-collinear\nquantum dot spin valve\nBhaskaran Muralidharan1, 2and Milena Grifoni2\n1Department of Electrical Engineering and Center of Excellence in Nanoelectronics,\nIndian Institute of Technology Bombay, Powai, Mumbai-400076, India\n2Institut f ur Theoretische Physik, Universit at Regensburg, Regensburg D-93040, Germany\n(Dated: February 28, 2022)\nWe explore thermoelectric spin transport and spin dependent phenomena in a non-collinear quan-\ntum dot spin valve set up. Using this set up, we demonstrate the possibility of a thermoelectric\nexcitation of single spin dynamics inside the quantum dot. Many-body exchange \felds generated on\nthe single spins in this set up manifest as e\u000bective magnetic \felds acting on the net spin accumula-\ntion in the quantum dot. We \frst identify generic conditions by which a zero bias spin accumulation\nin the dot may be created in the thermoelectric regime. The resulting spin accumulation is then\nshown to be subject to a \feld-like spin torque due to the e\u000bective magnetic \feld associated with\neither contact. This spin torque that is generated may yield long-time precession e\u000bects due to the\nprevailing blockade conditions. The implications of these phenomena in connection with single spin\nmanipulation and pure spin current generation are then discussed.\nPACS numbers: 73.23.Hk, 85.50.Fi, 85.75.-d\nI. INTRODUCTION\nAchieving the control of individual spins [1] or col-\nlective spin degrees of freedom [2] forms an important\nfrontier of spintronics. In the collective case, the ma-\nnipulation of magnetization dynamics [3, 4] in magnetic\nnanostructures via spin transfer torques [5] or magnetic\ndomain wall dynamics [2, 6] has been a topic of much\nattention. At the same time, quantum dots provide an\nideal platform for realizing individual spin manipulation\nand control [1]. Control of magnetization dynamics forms\nthe basis of a wide range of applications from microwave\noscillators to magnetic storage [3, 4], while that of indi-\nvidual spins is an important paradigm towards spin based\nquantum computation [7].\nThe control and manipulation of individual spins in\nquantum dots [8{10] has become possible owing to the\nability to lock the number of electrons, as well as their\nindividual spins. While the electron number can be con-\ntrolled by a gate voltage due to Coulomb blockade, their\nnet spin accumulation may be controlled via spin block-\nade [1, 11]. Spin blockade is a condition when an electron\ncurrent \row under non-equilibrium conditions is forbid-\nden due to the interplay between Pauli exclusion principle\nand Coulomb interaction. The net spin accumulation via\nthe spin blockade mechanism may also be \fne tuned via\nthe use of a gate electrode. In a typical control exper-\niment, a gate electrode pulses the system in and out of\nspin blockade, thus permitting spin manipulation when\nthe electron current \row is forbidden, and read out when\nthe current \row is permitted [8{10].\nSpin blockade and spin manipulation in the aforemen-\ntioned works were discussed under a voltage bias. In re-\ncent times, there has been a lot of activity in the area of\nspin based thermoelectrics or spin caloritronics [12] and\nhence it is timely to investigate spin transport under the\napplication of a temperature gradient [13]. Speci\fcally,spin dependent thermoelectric e\u000bects in quantum dots\nhave also been theoretically investigated in a few recent\nworks [14{19]. In this paper we explore the possibility\nof spin manipulation by creating a non-equilibrium spin\naccumulation in the thermoelectric regime. While in the\npioneering spin manipulation experiments [1, 8{10] spin\nblockade occurs due to a blocking triplet state [20] in\na detuned double quantum dot set up with unpolarized\ncontacts, we focus on creating the spin accumulation via\na di\u000berent spin blockade mechanism in a non-collinear\nquantum dot spin valve described extensively in some\nearlier works [21, 22]. In our set up, unlike in the double\nquantum dot case, the spin blockade results from the spin\nselection and \fltering between spins in the quantum dot\nand the ferromagnetic degrees of freedom of the contacts\nwhose magnetization directions in general, may be non-\ncollinear. Also, in our set up, many-body exchange \felds\nare generated from an interplay between the Coulomb\ninteraction in the dot or metallic island and the ferro-\nmagnetic degree of freedom in the contacts [21{23]. The\ne\u000bective magnetic \feld thereby creates a \feld-like term in\nthe description of the spin dynamics inside the dot. This\n\feld like term is reminiscent of spin torque in magnetic\nstructures [5] and is responsible for the precessional spin\ndynamics inside the dot. In addition to the precessional\nterm, one has terms arising from the spin polarized cur-\nrent injection, as well as relaxation due to single-electron\ntunneling processes between either contact and the dot.\nWe show here that the precessional term that arises\nout of the above mentioned \feld-like spin torque may be\ncreated under a pure thermal gradient in the absence of a\nbias. The crucial aspect is that the non-equilibrium spin\naccumulation is induced as a result of a spin blockade\nmechanism, to be discussed, in the regime where double\noccupancy is suppressed due to Coulomb interaction. As\na result of long dwell times in the dot due to the block-\nade, the charge and spin relaxation components of thearXiv:1212.4353v2  [cond-mat.mes-hall]  13 Jun 20132\nspin dynamics are suppressed, thus yielding a long time\nprecession.\nThis \feld like spin torque itself translates to a net spin\nangular momentum transfer rate or a spin current be-\ntween the contacts and the dot [24]. The traditional\nviewpoint of a spin current is that of a spin polarized cur-\nrent resulting from the transport of spin polarized elec-\ntrons. The precessional terms also imply a net angular\nmomentum transfer rate mediated by exchange interac-\ntion, and may, in general, also be a\u000eliated with spin cur-\nrents [24]. Such spin currents resulting from a spin pre-\ncession may possibly be detected via optical or electrical\nmeans as demonstrated in some recent pioneering exper-\niments [25, 26]. Earlier works on spin dependent ther-\nmoelectrics in quantum dots primarily focused either on\nthe linear response thermoelectric regime [14, 18], or on\nthe generation of pure spin currents using non-magnetic\nquantum dots in the presence of a magnetic \feld [15],\nor magnetic quantum dots [16] with collinearly polarized\ncontacts, or novel e\u000bects that arise due to the coupling\nwith magnons [19]. But these works, however, do not fea-\nture the e\u000bects related to spin precession to be discussed\nhere.\nThe paper is organized as follows. The following sec-\ntion will describe the necessary formulation brie\ry, and\nwill cover the important aspects of the physics of angu-\nlar momentum transfer in relation to its coverage in this\npaper. We then discuss the important results and their\nimplications in section III. Section IV concludes the pa-\nper.\nII. SET UP AND FORMULATION\nIn the schematic of the quantum dot spin valve set up\nshown in Fig. 1(a), the quantum dot is weakly coupled\nto two non-collinearly polarized ferromagnetic contacts\nlabeled\u000b=L;R, each with a degree of polarization p\u000b,\nan electrochemical potential \u0016\u000b, and a temperature T\u000b.\nThe contact L(R) acts as the collector (injector) in the\nforward (reverse) bias direction. Second order transport\ntheory across quantum dots weakly coupled to ferromag-\nnetic contacts predicts that the interplay between the\nstrong Coulomb repulsion in the dot and the spin po-\nlarization of the itinerant electrons to and from the fer-\nromagnetically pinned contacts results in a many body\nexchange \feld like term [21, 22, 27] that drives the pre-\ncessional dynamics inside the dot. The non-equilibrium\nspin dynamics of the quantum dot spin accumulation ~S\nis composed of spin injection, relaxation and precession\nterms [22], as shown in Fig. 1(b).\nA. Model\nThe theoretical description of transport in our set up\nbegins by de\fning the overall Hamiltonian ^Hwhich is\nusually written as ^H=^HD+^HC+^HT, where ^HD;^HC\nFIG. 1. (Color online) Non-collinear quantum dot spin valve\ntransport set up. a) The set up consists of a quantum dot\nweakly coupled to ferromagnetic contacts \u000b=L;R, each with\na pinned magnetization axis ^ m\u000boriented along the majority\nspin and a degree of polarization p\u000b. The contact L(R) acts\nas the collector (injector) in the forward (reverse) bias direc-\ntion. The common coordinate axis is chosen to be oriented\nwith respect to that of the quantum dot, with the ^ xaxis be-\ning pointing in the (longitudinal) transport dimension. The\nangle between the contact magnetizations is \u0012. The set up\nmay be spin blockaded for a certain range of bias and for\ncertain values of \u0012. b) Spin dynamics comprise of spin preces-\nsion around the net direction of the e\u000bective exchange \feld\n~BL+~BRand relaxation introduced via charge tunneling to\nand from the contacts. c) Transport through the single level\nquantum dot in the sequential tunneling regime is modeled\nvia density matrix rate equations which may be viewed as\ntransitions between many-electron states labeled 0 through 3.\nd) Example of a transport set up that displays spin blockade\nwhen the average electron number inside the quantum dot\nis unity. Spin blockade results in an accumulation of spins\nantiparallel to the collector contact spin polarization.\nand ^HTrepresent the dot, reservoir and reservoir-dot\ncoupling Hamiltonians, respectively. In this paper, the\nquantum dot is modeled as a single orbital Anderson im-\npurity described by the one-site Hubbard Hamiltonian:\n^HD=X\n\u001b\u000f\u001b^n\u001b+U^n\"^n#; (1)\nwhere\u000f\u001brepresents the orbital energy, ^ n\u001b=^dy\n\u001b^d\u001bis\nthe occupation number operator of an electron with spin\n\u001b=\", or\u001b=#, andUis the Coulomb interaction energy\nbetween electrons of opposite spins occupying the same\norbital. The exact diagonalization of the dot Hamilto-\nnian then results in four Fock-space energy levels labeled\nby their total energies 0, \u000f\",\u000f#, and\u000f\"+\u000f#+U. In\nthis paper, we consider only a spin-degenerate level such\nthat\u000f=\u000f\"=\u000f#. The contact Hamiltonian is given\nby^HC=P\n\u000b=L;RP\nk\u001b\u000b\u000f\u000bk\u001b\u000b^n\u000bk\u001b\u000b, where\u000blabels the3\nleft/right reservoir ( LorRin our case) and the sum-\nmation is taken over the single particle states labeled\nfk\u001b\u000bg, and\u001b\u000b=\u0006denotes the majority and minority\nspin orientation in the contacts. The tunneling Hamil-\ntonian that represents the dot-contact coupling may in\ngeneral be written as:\n^HT=X\n\u000bk\u001b\u000b\u0010\nt\u000b^cy\n\u000bk\u001b\u000b^d\u001b\u000b+t\u0003\n\u000b^dy\n\u001b\u000b^c\u000bk\u001b\u000b\u0011\n(2)\nwhere (^cy;^c) and ( ^dy;^d) are the creation/annihilation op-\nerators of the reservoir states labeled fk\u001b\u000bgand of the\nquantum dot one particle states respectively, and t\u000bde-\nnotes the tunneling matrix element. Note that, in gen-\neral, the direction of majority and minority spins \u001b\u000b=\u0006\nin either contact and of the spin orientation \u001b=\";#in\nthe dot may be non-collinear. If the ^ zaxis of the spin po-\nlarization in contact \u000bmakes an angle ( \u0012\u000b;\u001e\u000b) with the\n^zaxis of the dot, one can rewrite the tunneling Hamilto-\nnian in Eq. (2) as [21]:\n^HT=X\n\u000bk\u0010\nt\u000b^cy\n\u000bk+(C\u000b^d\"+S\u000b^d#)\u0011\n+X\n\u000bk\u0010\nt\u000b^cy\n\u000bk\u0000(\u0000S\u0003\n\u000b^d\"+C\u0003\n\u000b^d#)\u0011\n+h:c:; (3)\nwhereC\u000b= cos(\u0012\u000b=2)ei\u001e\u000b=2;S\u000b= sin(\u0012\u000b=2)e\u0000i\u001e\u000b=2, and\nh:c:stands for the hermitian conjugate. In this work,\nwithout loss of generality, we let \u001e= 0 so that the ori-\nentations of the magnetization directions of the two con-\ntacts are in the ^ x\u0000^zplane. We can then de\fne the tun-\nneling rate for each spin \u001b\u000bassociated with contact \u000bas\n\u0000\u000b\u001b\u000b=2\u0019\n~P\nkjt\u000bj2\u000e(E\u0000\u000f\u000bk\u001b\u000b) =2\u0019\n~jt\u000bj2D\u000b\u001b\u000b, where\nD\u000b\u001b\u000brepresent the density of states (assumed constant\nin our case) of the majority and minority spins of the\ncontact. We can then de\fne a degree of polarization as-\nsociated with either contact as p\u000b= (\u0000\u000b+\u0000\u0000\u000b\u0000)=(\u0000\u000b++\n\u0000\u000b\u0000) = (\u0000\u000b+\u0000\u0000\u000b\u0000)=\u0000\u000b.\nB. Spin accumulation and spin currents\nThe calculation of the non-equilibrium spin accumula-\ntion~Sand of its dynamics follows from the evaluation\nof the reduced density matrix of the dot using the den-\nsity matrix formulation discussed extensively in [22]. For\nthis, one starts with the time evolution of the compos-\nite (dot + contacts) density matrix ^ \u001a(t) which is given\nby the Liouville equation. The reduced density matrix\n^\u001ared(t) of the dot is then obtained by performing a trace\nexclusively over the reservoir space. An expansion of the\nLiouville equation up to the second order in the tunnel-\ning Hamiltonian in the limit of weak contact coupling\n(~\u0000\u001ckBT) leads to the density matrix master equation\nfor the reduced density matrix of the system [21, 27{29].\nIn this paper, we consider the regime of sequential tun-\nneling ~\u0000\u000b<<kBT,Usuch that this description based\non a second order perturbation in the tunnelling matrixelementt\u000bwill su\u000ece. In this regime, transport as de-\nscribed via density matrix rate equations for the reduced\ndensity matrix of the dot [27] may be viewed as transi-\ntions between Fock space states of the dot as depicted\nin Fig. 1(c). We consider steady state transport in all\nour calculations and hence consider the steady state so-\nlution\u001aijof the reduced density matrix of the dot. The\ndiagonal terms \u001aiiof this density matrix represent the\nprobability of occupation of each many electron state i\nlabeled 0 through 3. The average spin of the dot along\nits ^zdirection is given by Sz=~\n2\u0000\u001a11\u0000\u001a22\n2\u0001\n. The o\u000b-\ndiagonal terms \u001a12,\u001a21relate to the average spin in the\nquantum dot along the remaining two axes such that\nSx=~\n2\u0000\u001a12+\u001a21\n2\u0001\n,Sy=i~\n2\u0000\u001a12\u0000\u001a21\n2\u0001\n. The spin dynamics\nassociated with the non-equilibrium spin accumulation ~S\nare then described by [22, 29]:\n2q\n~d~S\ndt=X\n\u000b\"\nJq\n\u000bp\u000b^m\u000b\u00002q\n~ ~S\u0000p2\n\u000b( ^m\u000b\u0001~S) ^m\u000b\n\u001cr;\u000b!#\n\u00002q\n~X\n\u000b~S\u0002~B\u000b; (4)\nwithJq\n\u000bbeing the terminal charge current, \u0000qbeing\nthe magnitude of the electron charge, p\u000bbeing the\ndegree of polarization of each contact, and 1 =\u001cr;\u000b=\n\u0000\u000b(1\u0000f\u000b(\u000f) +f\u000b(\u000f+U)) representing the inverse tun-\nneling lifetime due to coupling to the contacts. Here,\nf\u000b(\u000f) =f\u0010\n\u000f\u0000\u0016\u000b\nkBT\u000b\u0011\nrefers to the Fermi-Dirac distribu-\ntion of either contact held at an electrochemical po-\ntential\u0016\u000band at a temperature T\u000b. The many body\nexchange \feld may be interpreted as a magnetic \feld\n~B\u000b=p\u000b\u0000\u000b^m\u000b\n\u0019~R0dE\u0010\nf(E)\nE\u0000\u000f\u0000U+1\u0000f(E)\nE\u0000\u000f\u0011\n, with the prime\nin the integral denoting the Cauchy principal value. The\nexpression for the terminal charge current Jq\n\u000bis given by:\nJq\n\u000b=2q\n~\u0000\u000b[f\u000b(\u000f)\u001a00\n+1\u0000f\u000b(\u000f) +f\u000b(\u000f+U)\n2(\u001a11+\u001a22)\n\u0000(1\u0000f\u000b(\u000f+U))\u001a33\n\u0000p\u000b[(1\u0000f\u000b(\u000f) +f\u000b(\u000f+U)] ^m\u000b\u0001~S]: (5)\nIn the above equation, the current depends on the dot\noccupation probabilities given in terms of the diagonal\nterms of the density matrix \u001aiiand also the dot spin vec-\ntor~S. In the absence of spin \rip processes, one may\ndeduce the expression for terminal spin currents via a\nsimple continuity equation based on Eq. (4) for the spin\naccumulation in the quantum dot as2q\n~d~S\ndt=~Js\nL+~Js\nR,\nwhere~Js\nL(R)is the terminal spin current with its three\ncomponents representing transport of ^ x, ^y, and ^zpo-\nlarized spins along the direction of electrical current [5].\nOne may then write an expression for the terminal spin4\ncurrents as [24]:\n~Js\n\u000b=Jq\n\u000bp\u000b^m\u000b\u00002q\n~ ~S\u0000p2\n\u000b( ^m\u000b\u0001~S) ^m\u000b\n\u001cr;\u000b!\n\u00002q\n~~S\u0002~B\u000b\n=~Js\n\u000b;^m\u000b+2q\n~ \nd~S\ndt!\n\u000b;rel+2q\n~ \nd~S\ndt!\n\u000b;prec; (6)\nwith~Js\n\u000b;^m\u000brepresenting the component due to injection,\nwhich is in the direction of magnetization of the contact,\n2q\n~\u0010\nd~S\ndt\u0011\n\u000b;reland2q\n~\u0010\nd~S\ndt\u0011\n\u000b;precrepresenting the angular\nmomentum transfer rate, in units of charge current, due\nto relaxation and precession, respectively. The \frst term\nhas a straightforward interpretation simply as being the\nspin current carried by a spin polarized charge current.\nThe other terms represent angular momentum transfer\nrates associated with either contact. Speci\fcally, the\nprecession term that arises from a \feld-like spin torque\n\u001c\u000b=\u0010\nd~S\ndt\u0011\n\u000b;prec=~S\u0002~B\u000brepresents an angular mo-\nmentum transfer transverse to the magnetization of the\ncontact and to the spin in the dot. This term, although\nit has a qualitatively di\u000berent \ravor in comparison to the\n\frst, it may still be viewed as a spin current [24]. There-\nfore, in this paper, when we talk of spin currents, it is the\nnet terminal spin current given in Eq. (6) that is being\nconsidered.\nThe relative contribution of spin injection, damping\nand precession terms that are described by the \frst, sec-\nond and the third term in Eq. (4) may be tuned relative\nto each other via the application of a gate and bias volt-\nage. We therefore focus on the spin blockade regime in\nwhich a sizeable spin accumulation may be achieved, and\nwhere the relaxation and injection terms are vanishingly\nsmall in comparison to the precession term. A sample\ntransport energy con\fguration of the considered set up\nis depicted in Fig. 1(d) where spin accumulation may be\ninduced via spin \fltering. The accumulation is usually\ndirected anti-parallel to the spin polarization of the col-\nlector contact.\nC. Transport set up\nWe consider transport across the set up shown in\nFig. 1(a). The relative angle between the two contacts is\ntaken as\u0012=\u0019=2, with the left contact being polarized\nin the ^xdirection and the right contact being polarized\nin the ^zdirection. Indeed such a con\fguration has been\nexperimentally realized in the context of spin torque os-\ncillators [4] using a magnetic free layer as the channel.\nWe consider two cases: I) Symmetric case: the polar-\nizations of the two contacts are identical, pL=pR; II)\nAsymmetric case: the polarizations of the two contacts\nare di\u000berent, pL6=pR, making one contact of larger po-\nlarization in comparison to the other. The asymmetry\nin the degree of polarization has a profound consequence\nwhen a pure temperature gradient is applied. As we willshow in the upcoming analysis, due to this asymmetry,\na minor imbalance in the tunneling rates between the\naddition and removal process in the set up created by\na pure temperature gradient is enough to induce a non-\nequilibrium spin accumulation due to spin blockade and\nhence trigger a spin precession. We take pL=pR= 1\nfor the symmetric case, and pL= 1;pR= 0:2 for the\nasymmetric case. For our transport set up, we take the\ncontact couplings to be ~\u0000 = 0:01meV ; the Coulomb\ninteraction parameter is U= 40kBTL. When no temper-\nature gradient is applied, we choose TR=TL= 0:7K. In\nthe case of thermoelectric transport we have TR= 0:7K\nandTL= 0:9K.\nThe important spin transport e\u000bects to be discussed in\nthis paper focus on the regime of blockade and speci\fcally\naround zero-bias where charge currents are vanishingly\nsmall. There is hence a possibility of higher order trans-\nport processes such as cotunneling and Kondo e\u000bect in\ru-\nencing the physics of transport in this regime. For exam-\nple, it has been shown in the case of a collinear quantum\ndot spin valve set up that spin-\rip cotunneling processes\n[30] may signi\fcantly in\ruence the spin accumulation as\nwell as the overall conductivity close to zero bias. This\nhappens speci\fcally when the tunnel coupling energy be-\ncomes of the order of the ambient temperature or higher\n(~\u0000\u0015kBT) although the ambient temperature may be\nwell above the Kondo temperature. Furthermore, in the\nnon-collinear set up that we consider here, the fourth or-\nder expansion will involve a larger class of two-electron\ntunneling mechanisms resulting from the coherence terms\nof the density matrix [31]. Therefore, the magnitude of\nthe tunnel coupling energy relative to the ambient tem-\nperature must satisfy ~\u0000< kBT(9~\u0000\u0019kBTLin our\ncase) so that the predictions made here out of the second\norder theory may remain valid atleast in the conducting\nregion. This also ensures that the ambient temperature\nis well above the Kondo temperature and hence the in-\n\ruence of Kondo physics on the zero bias transport is\nalso absent. The results presented here are certainly a\nreasonable approximation close to the boundary of the\nCoulomb blockade region, while deep inside the block-\nade, cotunneling might (or might not) alter the \fndings\nwhich could be the subject of a future study.\nIII. RESULTS\nA. Spin blockade e\u000bects\nThe spin blockade mechanism relevant to our set up\nis critical in understanding the occurrence of the non-\nequilibrium spin accumulation. We hence \frst set out\nto illustrate how spin blockade may be identi\fed in the\naforementioned cases using a stability plot, i.e., a plot of\nthe di\u000berential conductance G=dJq\ndVappin a bias voltage\nand gate voltage plane. Here, the e\u000bect of the applica-\ntion of a gate \feld has been encapsulated as an e\u000bective5\nFIG. 2. (Color online). Di\u000berential conductance G=dJq\ndVapp\nversus gate and bias voltages showing the N= 1 sector for\nthe unpolarized case. Here the diamond edges mark the entry\nof a conducting energy level of the dot within the transport\nwindow. We choose TR=TL= 0:7K,~\u0000 = 0:01meV and\nU= 40kBTL.\ndetuning\u000f\u0000\u00160\nkBTof the energy level \u000fwith respect to the\nequilibrium electrochemical potential \u00160. Here,\u00160has\nbeen arbitrarily set at the transition energy between the\n0 particle and the 1 particle con\fgurations. At a \fnite\napplied bias of Vapp, assuming equal capacitive couplings\nof the dot with the two contacts, the contact electro-\nchemical potentials are given by \u0016L=\u00160+qVapp=2, and\n\u0016R=\u00160\u0000qVapp=2. In general, there could be an asym-\nmetric voltage drop due to unequal capacitive couplings\nleading to a distortion in the Coulomb blockade region in\nthe stability plot. The stability plot for an unpolarized\nset up is shown in Fig. 2. Upon increasing the bias be-\nyond the Coulomb blockade regime, one reaches the dia-\nmond edges , signaling the fact that the conducting energy\nlevel of the dot has now entered the transport window,\nand hence transitions between the 0 particle and 1 par-\nticle con\fgurations are energetically allowed. However,\nin comparison with the unpolarized case shown in Fig. 2,\nthose diamond edges are clearly absent in Fig. 3(a), and\npresent along only one bias direction in Fig. 3(d), in-\ndicating spin-blockade. Along the black cut shown in\nFig. 3(a), which corresponds to \u000f\u0000\u00160=\u000012kBTL, for\nexample, the conducting transition is expected to occur\nat an applied bias of Vapp=VT=\u000624kBTL=q. We will\nconsider spin dependent transport along this black cut in\nthe analysis to follow, by \frst elucidating the mechanism\nof spin blockade in our considered set up.\nThe plots of relevant transport properties as a function\nof applied bias for the symmetric and asymmetric case are\nFIG. 3. (Color online) Spin accumulation and currents. a)\nStability plot depicting the N= 1 sector for the case of\nsymmetric polarization pL=pR= 1, and\u0012=\u0019=2. Non-\nequilibrium spin transport across the black cut is considered\nin Figs. b), c). b) Spin accumulation vs applied bias in-\ndicating an Sx=\u00001=2 blocking state in the forward bias\ndirection and an Sz=\u00001=2 blocking state in the reverse bias\ndirection (see text). c) Resulting charge and spin currents\ndepicting pronounced ^ y-polarized spin currents JS\nyin the re-\ngion of Coulomb blockade. d) Stability plot in the case of\nthe asymmetric polarization pL= 1,pR= 0:2 case. In this\ncase the spin blockade and hence spin accumulation only oc-\ncurs along the forward bias direction. e) and f) Resulting\nspin accumulation and currents according to the dashed line\nin Fig. d). Coulomb blockaded regions at \fnite bias voltages\nfeature sizeable transversely polarized spin currents with van-\nishingly small charge and in-plane spin currents. Remaining\nparameters are the same as in Fig. 2.\nshown in the left and right panels of Fig. 3 respectively.\nThe spin blockade regime crucial to this work is qual-\nitatively di\u000berent from the ones observed in the double\nquantum dot structure, and occurs based on the following\nmechanism. Consider transport along the black cut in the\nstability diagrams in Figs. 3(a) and 3(d). Along the for-\nward bias direction, in our convention, the right contact is\nthe injector and the left contact is the collector. In both\nthe symmetric and the asymmetric case, spins injected\nfrom the right contact are in varying degrees +^ zpolar-6\nized, while the left contact that acts as the collector is\nfully polarized along the +^ xdirection, and acts as a spin\n\flter accepting only Sx= +1=2~electrons. By noting\nthatjSz=\u00061=2i=1p\n2(jSx= +1=2i\u0006jSx=\u00001=2i),\nthe spin \fltering at the acceptor leaves behind an accu-\nmulation of Sx=\u00001=2~spins in the dot. This results in\na transport blockade as the energetics forbid the blocked\nelectrons to tunnel back to the right contact. In the re-\nverse bias situation, excess spins along the \u0000^zdirection\naccumulate to produce a similar blockading e\u000bect for the\nsymmetric case. The bias range of the blockade is af-\nfected by the polarization of the collector contact. The\ne\u000bectiveness of the reverse bias blockading e\u000bect for the\nasymmetric case, therefore, is considerably diminished\nsince the right (collector) contact is partially polarized.\nThe spin blockade regime can be observed in Figs. 3(b)\nand 3(e) in the excess accumulation of spins along the\n\u0000^xor\u0000^zdirections, and the suppression of charge and\nin-plane spin currents in the post-threshold regions of\nFigs. 3(c) and 3(f). In the symmetric case, this occurs\nalong both bias directions, and in the asymmetric case\nonly along the forward bias direction.\nThe polarization of the injecting contact determines\nthe amplitude of the exchange \feld ~B\u000bassociated with\nit. In turn, the amplitude of the exchange \feld a\u000bects the\ne\u000bectiveness of the torque like term \u00002q\n~~S\u0002~B\u000bin Eq.\n(4) which induces a precession of the accumulated spin in\nthe dot, and hence can partially remove the spin block-\nade. Because in the asymmetric case the polarization of\nthe injector is smaller than in the symmetric case, the\nonset of spin blockade and its persistence are more pro-\nnounced in this situation as seen by comparing Fig. 3(b)\nand Fig. 3(e).\nB. Spin precession and associated spin currents\nDue to the prevailing blockade conditions, spins in-\njected from either contact are subject to precessional dy-\nnamics on a time scale amounting to the tunneling life-\ntime. In the steady state, the precessing spin eventually\naligns with the net e\u000bective magnetic \feld. In the bias\nregion 0\u0014V\u0014VTit can be shown in steady state that,\nSy(V) = 0, and Sx(V)=Sz(V) =BL(V)=BR(V). The\ne\u000bective spin accumulation is directed along the e\u000bective\nexchange \feld direction given by ~Beff=BL^mL+BR^mR.\nWhile the steady state solution simply points to the\nspins being aligned with the e\u000bective \feld such that\n~S\u0002~Beff= 0, the \feld-like spin torques associated with\neach contact \u001c\u000b=~S\u0002~B\u000bdo not vanish. As a result,\nthe angular momentum transfer rate and hence spin cur-\nrents associated with either contact is \fnite and given as\n~Js= (~Js\nL\u0000~Js\nR)=2. The associated transversely polar-\nized terminal spin currents are depicted by the ^ ycompo-\nnent of the spin current tensor (shown dashed blue) in\nFig. 3(c) and (f), and the charge currents and in-plane\nspin currents in the whole blockade region are e\u000bectively\nFIG. 4. (Color online) E\u000bect of temperature gradient. a) In\nthe case of symmetric polarization, for a small temperature\ngradient, the zero bias spin accumulation is always absent.\nThus, spin precession is absent at zero bias, resulting in a\nb) zero ^ypolarized spin current ( Js\ny(Vapp= 0) = 0). c) In\nthe asymmetric case, however, zero bias spin accumulation\noccurs and the resulting spin precession causes a d) non-zero\ntransverse spin current ( Js\ny(Vapp= 0)6= 0) (blue dashes).\nThe in-plane spin currents and charge currents are vanishingly\nsmall in this region.\nnegligible.\nC. E\u000bect of a temperature gradient\nIn the asymmetric case, as remarked before, the most\nimportant consequence of the above discussed spin block-\nade mechanism is the zero bias non-equilibrium spin ac-\ncumulation emerging with the application of a tempera-\nture gradient. A small temperature gradient (we choose\n\u0001T= 0:2K, such that TL= 0:9K, andTR= 0:7K)\nin the absence of a bias opens the possibility of charge\nand spin transport via thermoelectric operation [32]. As\nshown in Fig. 4(c), the asymmetric situation induces a\nzero bias spin accumulation due to a small imbalance\nbetween the tunneling rates of the left and the right con-\ntacts. In contrast, no zero bias spin blockade occurs\nin the symmetric polarization case shown in Fig. 4(a).\nThe accumulation results in a zero bias spin torque\n\u001c\u000b=~S\u0002~B\u000bat either contact and hence in an associated\n^y-polarized pure spin current as shown in Fig. 4(d). The\nspin accumulation is in the plane of magnetization of the\ntwo contacts, and the spin precession dynamics due to\nthe third term in Eq. (4) results in a non-zero rate of\ntransfer of transverse spin angular momentum.\nThe result presented here involves the generation of a7\nFIG. 5. (Color online) Polarization and angular dependence.\na) Dependence of the zero bias thermal pure spin current mag-\nnitude on the degree of polarization of the right contact. For\nthe unpolarized case and the fully polarized cases, the trans-\nverse spin current is zero. The pure spin current magnitude\npeaks atpR= 0:5. b) Angular dependence of the magnitude\nof the pure spin current with pR= 0:2. The asymmetry is\ndue to the fact that the situation pR= 0:2 corresponds to the\nmajority up-spin case.\nspin current due to an applied thermal gradient. In or-\nder to make a connection with energy conversion, one has\nto typically quantify the e\u000eciency of this process. The\nprocess e\u000eciency depends on the detection method and\nutilization of the spin current. For example, if this spin\ncurrent was detected via electrical means, the e\u000eciency\nof this process would depend on the power drawn in the\ncircuit [32]. Typically, in the case of charge thermoelec-\ntric e\u000bects, the maximum e\u000eciency in the linear response\nregime is related to a dimensionless metric called the \fg-\nure of merit zT, which is de\fned as zT=S2\u001bT\n\u0014, whereS\nis the Seebeck coe\u000ecient, \u001bis the electrical conductivity,\n\u0014is the thermal conductivity and Tis the ambient tem-\nperature. In the collinear polarization case, a few recent\nworks [14, 15] have de\fned a similar spin \fgure of merit\nZsT=S2\ns\u001bsT\n\u0014, whereSsis the spin Seebeck coe\u000ecient\nand\u001bsis the spin dependent conductivity. The premise\nof de\fning a spin dependent \fgure of merit was motivated\nby the linear response expansion of the charge, in-plane\nspin currents and heat currents. In principle, one could\nextend this for our non-collienar case by a linear response\nexpansion that includes the charge current, the in-plane\nspin current and the transversely polarized spin currents\nvia a four-component voltage drop [33] with an Onsager\nmatrix [12, 33] that couples with the heat current. How-\never, it is left to a more rigorous analysis to assess thevalidity as well as the merit in de\fning performance met-\nrics such as ZsTfor this case.\nWe now analyze the e\u000bect of varying the lead polariza-\ntions and angles. Keeping the left contact fully polarized\n(pL= 1), we plot the polarization and angle dependence\nof the magnitude of this zero-bias transverse spin current\nin Fig. 5. It is seen from Fig. 5(a) that the spin cur-\nrent magnitude is zero for the unpolarized and the fully\nsymmetric case and maximizes at pR= 0:5. The angu-\nlar dependence of this spin current magnitude between\n\u0012= 0 and\u0012=\u0019is shown in Fig. 5(b) for pR= 0:2.\nThe noted asymmetry simply arises from the fact that\nthe majority spins are along the Sz= +1=2 direction.\nHaving either contact fully polarized is still an idealiza-\ntion and was used in this paper in order to elucidate\nthe non-trivial physics that was to be conveyed. Real\nferromangetic contact polarizations in the best case ap-\nproach 30\u000040 percent. We therefore study the e\u000bect\nof varying both contact polarizations in Fig. 6 keeping\n\u0012=\u0019=2 so that a realistic range of contact polarization\nmagnitudes may be assessed. Here, the three curves de-\npict the variation of the zero bias spin current magnitude\nwith the left contact polarization pLfor three represen-\ntative values of pR. As expected, we note that the spin\ncurrent magnitude vanishes when either contact is unpo-\nlarized and when pL=pRand varies quasi-quadratically\nin between. Furthermore, making pL> pRresults in a\nquasi-linear variation of the spin current magnitude. The\nnoticed trends here indicate the possibility of realizing a\nsizeable spin current for a wide range of realistic polar-\nization magnitudes for the two contacts.\nD. Discussion and perspectives\nThe results presented so far might have important im-\nplications. First, the spin accumulation result presented\nhere opens the interesting possibility of spin initializa-\ntion via a small temperature gradient, in the absence\nof a bias. Second implication is the occurence of trans-\nversely polarized terminal spin currents due to the zero\nbias \feld-like spin torque. The relaxation dynamics typ-\nically result from a transition from the one electron state\ninto the zero electron state or to the two electron state,\nboth of which are spin zero states. Typical relaxation\ntimes in this case are very long and of the order of 10 \u0016s.\nThese long-time coherent spin rotations may have impor-\ntant applications with respect to spin manipulation via a\ngate pulse, such that the spin rotation may be read out\nonce blockade is removed by gating the dot energy level.\nFurthermore, the precession may be used to probe relax-\nation times due to other relaxation mechanisms within\nthe quantum dot [9].\nFinally, the question of detecting the spin currents dis-\ncussed here has numerous subtleties. It has been theoret-\nically established [26, 33, 34] and experimentally demon-\nstrated [35] that precessing spins in a free magnetic thin\nlayer that is coupled to pinned ferromagnetic or normal8\nFIG. 6. (Color online) Dependence of the zero bias spin cur-\nrent magnitude on the relative polarization between the con-\ntacts. The spin current vanishes when pL=pRand also when\neither contact is unpolarized and varies quadratically in be-\ntween. For pL>pRwe notice a quasi-linear variation in the\nspin current magnitude. The noticed trends here indicate the\npossibility of realizing a sizeable spin current magnitude for\na wide range of realistic polarization magnitudes for the two\ncontacts.\nmetallic contacts can result in the volume generation of\nspin currents. The ferromagnetically pinned contact of-\nten acts as a spin sink that will absorb the transversely\npolarized angular momentum \row. However, due to the\nconservation of angular momentum, a back acting torque\nwill induce a perturbation in the precessing spins. Such\na perturbation may be indirectly detected via the broad-\nening of the ferromagnetic resonance lines described in\n[36, 37]. A more direct method would be to use a free\nmagnetic thin layer within a spin relaxation length inbetween the collector contact and the quantum dot, and\nhence detecting the angular momentum transfer via the\nprecession of this layer. Alternatively, the magneto-optic\nKerr e\u000bect [25] may be used to directly detect the exci-\ntation due to this pure spin current. While it is shown\nin the context of magnetization dynamics that a similar\nmagnetization precession may be related to pure spin cur-\nrents [26, 33, 34, 36, 37], progress on understanding the\nimplications of similar phenomena with respect to single\nspin precession noted here would form an interesting and\nimportant extension of this work.\nIV. CONCLUSIONS\nIn this paper, we explored spin dependent phenomena\nin the thermoelectric regime of a non-collinear quantum\ndot spin valve set up. This work opens the interesting\npossibility of thermoelectric manipulation of single spins\nin a quantum dot transport set up. The spin torque and\nthe related spin dynamics discussed here are reminiscent\nof what is observed in the collective case as a spin torque\nin the magnetization dynamics of magnetic layers. We\nshowed that when the set up is biased deep into blockade\nwhere double occupancy is forbidden, a resulting zero\nbias thermoelectric spin torque may yield a long time\nspin precession. The implications of this with respect\nto single spin manipulation as well as its connection with\npure spin currents were discussed. Unlike in the collective\ncase, the spin dynamics inside quantum dot arrays may\nbe thought of as an ensemble of weakly interacting spins.\nElectrical or thermoelectric control of spin dynamics of\nindividual spins interacting via a quantum dot array may\nin general open exciting paradigms and possibilities.\nAcknowledgements: This work was partly supported\nby the Deutsche Forschungsgemeinschaft (DFG) under\nprogramme SFB 689. BM would like to acknowledge\n\fnancial support from the IIT Bombay IRCC-SEED\ngrant.\n[1] R. Hanson, L. P. Kouwenhoven, J. R. Petta, S. Tarucha,\nand L. M. K. Vandersypen, Rev. Mod. Phys. 79, 1217\n(2007).\n[2] S. Bader and S. Parkin, Annual Review of Condensed\nMatter Physics 1(1), 71 (2010).\n[3] S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Em-\nley, R. J. Schoelkopf, R. A. Buhrman, and D. C. Ralph,\nNature 425(6956), 380 (2003).\n[4] D. Houssameddine, U. Ebels, B. Dela et, B. Rodmacq,\nI. Firastrau, F. Ponthenier, M. Brunet, C. Thirion, J.-\nP. Michel, L. Prejbeanu-Buda, M. C. Cyrille, O. Redon,\netal., Nature Materials 6(6), 441 (2007).\n[5] D. Ralph and M. Stiles, Journal of Magnetism and Mag-\nnetic Materials 320(7), 1190 (2007).\n[6] S. S. P. Parkin, M. Hayashi, and L. Thomas, Science\n320(5873), 190 (2008).[7] D. Loss and D. P. DiVincenzo, Phys. Rev. A 57, 120\n(1998).\n[8] J. R. Petta, A. C. Johnson, J. M. Taylor, E. A. Laird,\nA. Yacoby, M. D. Lukin, C. M. Marcus, M. P. Hanson,\nand A. C. Gossard, Science 309(5744), 2180 (2005).\n[9] F. H. L. Koppens, J. A. Folk, J. M. Elzerman, R. Han-\nson, L. H. W. van Beveren, I. T. Vink, H. P. Tranitz,\nW. Wegscheider, L. P. Kouwenhoven, and L. M. K. Van-\ndersypen, Science 309(5739), 1346 (2005).\n[10] K. C. Nowack, F. H. L. Koppens, Y. V. Nazarov, and\nL. M. K. Vandersypen, Science 318(5855), 1430 (2007).\n[11] K. Ono, D. G. Austing, Y. Tokura, and S. Tarucha, Sci-\nence297(5585), 1313 (2002).\n[12] G. E. W. Bauer, Solid State Communications 150(11-\n12), 459 (2011).9\n[13] S. P. Dash, S. Sharma, R. S. Patel, M. P. De Jong, and\nR. Jansen, Nature 462(7272), 491 (2009).\n[14] R. \u0013Swirkowicz, M. Wierzbicki, and J. Barna\u0013 s, Phys. Rev.\nB80, 195409 (2009).\n[15] Y. Dubi and M. Di Ventra, Phys. Rev. B 79, 081302\n(2009).\n[16] R.-Q. Wang, L. Sheng, R. Shen, B. Wang, and D. Y.\nXing, Phys. Rev. Lett. 105, 057202 (2010).\n[17] M. Wierzbicki and R. wirkowicz, Journal of Magnetism\nand Magnetic Materials 324(8), 1516 (2012).\n[18] T. c. v. Rejec, R. \u0014Zitko, J. Mravlje, and A. Ram\u0014 sak, Phys.\nRev. B 85, 085117 (2012).\n[19] B. Sothmann and M. B uttiker, EPL (Europhysics Let-\nters)99(2), 27001 (2012).\n[20] B. Muralidharan and S. Datta, Phys. Rev. B 76, 035432\n(2007).\n[21] J. K onig and J. Martinek, Phys. Rev. Lett. 90, 166602\n(2003).\n[22] M. Braun, J. K onig, and J. Martinek, Phys. Rev. B 70,\n195345 (2004).\n[23] W. Wetzels, G. E. W. Bauer, and M. Grifoni, Phys. Rev.\nB72, 020407 (2005).\n[24] J. K onig, M. Braun, and J. Martinek, in\nCFN Lectures on Functional Nanostructures - Volume 2\n(Springer Berlin / Heidelberg, 2011), vol. 820, pp.\n103{124.[25] G. Woltersdorf, O. Mosendz, B. Heinrich, and C. H.\nBack, Phys. Rev. Lett. 99, 246603 (2007).\n[26] J. Zhang, P. M. Levy, S. Zhang, and V. Antropov, Phys.\nRev. Lett. 93(25), 4 (2004).\n[27] S. Braig and P. W. Brouwer, Phys. Rev. B 71, 195324\n(2005).\n[28] C. Timm, Phys. Rev. B 77, 195416 (2008).\n[29] R. Hornberger, S. Koller, G. Begemann, A. Donarini, and\nM. Grifoni, Phys. Rev. B 77, 245313 (2008).\n[30] I. Weymann, J. Barna\u0013 s, J. K onig, J. Martinek, and\nG. Sch on, Phys. Rev. B 72, 113301 (2005).\n[31] S. Koller, M. Grifoni, M. Leijnse, and M. R. Wegewijs,\nPhys. Rev. B 82, 235307 (2010).\n[32] B. Muralidharan and M. Grifoni, Phys. Rev. B 85,\n155423 (2012).\n[33] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys.\nRev. Lett. 88, 117601 (2002).\n[34] A. Brataas, G. E. W. Bauer, and P. J. Kelly, Physics\nReports 427(4), 163 (2006).\n[35] K. Ando, S. Takahashi, J. Ieda, H. Kurebayashi, T. Tryp-\niniotis, C. H. W. Barnes, S. Maekawa, and E. Saitoh,\nNature Materials 10(9), 655 (2011).\n[36] X. Wang, G. E. W. Bauer, B. J. van Wees, A. Brataas,\nand Y. Tserkovnyak, Phys. Rev. Lett. 97, 216602 (2006).\n[37] M. V. Costache, M. Sladkov, S. M. Watts, C. H. van der\nWal, and B. J. van Wees, Phys. Rev. Lett. 97, 216603\n(2006)."    },    {        "title": "1609.03389v2.Spin_decoherence_of_magnetic_atoms_on_surfaces.pdf",        "content": "Spin decoherence of magnetic atoms on surfaces\nF. Delgadoa,b,c,, J. Fern ´andez-Rossierd,e\naCentro de F´ ısica de Materiales, Centro Mixto CSIC-UPV /EHU, Paseo Manuel de Lardizabal 5, E-20018 Donostia-San\nSebasti´ an, Spain\nbDonostia International Physics Center (DIPC), Paseo Manuel de Lardizabal 4, E-20018 Donostia-San Sebasti´ an,\nSpain\ncIKERBASQUE, Basque Foundation for Science, E-48013 Bilbao, Spain\ndQuantaLab, International Iberian Nanotechnology Laboratory (INL), Av. Mestre Jos´ e Veiga, 4715-330, Braga,\nPortugal\nePermanent address: Departamento de F´ ısica Aplicada, Universidad de Alicante.\nAbstract\nWe review the problem of spin decoherence of magnetic atoms deposited on a surface. Re-\ncent breakthroughs in scanning tunnelling microscopy (STM) make it possible to probe the spin\ndynamics of individual atoms, either isolated or integrated in nanoengineered spin structures.\nTransport pump and probe techniques with spin polarized tips permit measuring the spin relax-\nation time T1, while novel demonstration of electrically driven STM single spin resonance has\nprovided a direct measurement of the spin coherence time T2of an individual magnetic adatom.\nHere we address the problem of spin decoherence from the theoretical point of view. First we\nprovide a short general overview of decoherence in open quantum systems and we discuss with\nsome detail ambiguities that arise in the case of degenerate spectra, relevant for magnetic atoms.\nSecond, we address the physical mechanisms that allows probing the spin coherence of mag-\nnetic atoms on surfaces. Third, we discuss the main spin decoherence mechanisms at work on\na surface, most notably, Kondo interaction, but also spin-phonon coupling and dephasing by\nJohnson noise. Finally, we briefly discuss the implications in the broader context of quantum\ntechnologies.\nKeywords: decoherence, relaxation, Kondo, adatoms, spin-phonon\nPACS: 72.15.Qm, 75.10.Jm, 75.30.GW, 75.30.Hx, 75.78.-n, 76.20. +q\nContents\n1 Introduction 3\n1.1 The relevance of decoherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3\n1.2 Magnetic adatoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4\n2 Decoherence, a general overview 7\n2.1 Quantum dissipative dynamics in open quantum systems: decoherence and re-\nlaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7\nEmail address: fernando.delgadoa@ehu.eus (F. Delgado)\nPreprint submitted to Elsevier October 15, 2018arXiv:1609.03389v2  [cond-mat.mes-hall]  25 Nov 20162.1.1 Coherence as a basis dependent quantity . . . . . . . . . . . . . . . . . . 9\n2.1.2 Coherence in a two level system (TLS) . . . . . . . . . . . . . . . . . . 9\n2.1.3 Decoherence as entanglemenet with the bath . . . . . . . . . . . . . . . 11\n2.1.4 Decoherence as phase uncertainty induced by a stochastic field . . . . . . 12\n2.2 Bloch-Redfield perturbative approach to the dissipative dynamics . . . . . . . . 13\n2.3 Bloch equation for a 2-level system . . . . . . . . . . . . . . . . . . . . . . . . 17\n2.4 Decoherence as a limit for spectral resolution in magnetic resonance . . . . . . . 18\n2.5 The quantum to classical transition and the spin-boson model . . . . . . . . . . . 18\n2.6 Other approaches to the relaxation and decoherence of spins . . . . . . . . . . . 20\n3 Spin Hamiltonian for magnetic adatoms 21\n3.1 Single spin Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21\n3.1.1 Integer vs half integer spins . . . . . . . . . . . . . . . . . . . . . . . . 22\n3.2 Pseudo-spin 1 =2 approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 24\n3.3 Hamiltonian for multi-spin systems . . . . . . . . . . . . . . . . . . . . . . . . . 25\n4 Decoherence due to Kondo exchange 26\n4.1 Kondo exchange interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26\n4.1.1 Kondo Hamiltonian in the two-level approximation . . . . . . . . . . . . 27\n4.2 General expressions for T1andT2due to Kondo exchange . . . . . . . . . . . . 28\n4.3 Decoherence of a single degenerate spin . . . . . . . . . . . . . . . . . . . . . . 28\n4.3.1 Decoherence of a single spin with degenerate spectrum: perturbative re-\nsults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28\n4.3.2 Decoherence of a single spin with degenerate spectrum: non-perturbative\nresults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30\n4.3.3 Quasiparticle phase shift as the origin of the pure dephasing . . . . . . . 31\n4.3.4 Decoherence for the degenerate doublet: the choice of basis set . . . . . 32\n4.4 Non-degenerate two level systems . . . . . . . . . . . . . . . . . . . . . . . . . 33\n4.4.1 Renormalization of the QST splitting . . . . . . . . . . . . . . . . . . . 34\n4.4.2 Perturbative Dynamics of the non-degenerate TLS . . . . . . . . . . . . 35\n4.4.3 Density matrix in the classical basis for the split TLS . . . . . . . . . . . 36\n4.4.4 Non-perturbative derivation of the decoherence assisted switching . . . . 36\n4.5 Relaxation and decoherence in spin chains and ladders . . . . . . . . . . . . . . 37\n4.5.1 T2for broken symmetry states in spin chains . . . . . . . . . . . . . . . 38\n4.5.2 The quantum to classical transitions in spin chains . . . . . . . . . . . . 39\n5 Other spin relaxation mechanisms: photons, phonons and nuclear spins 40\n5.1 Spin-phonon coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40\n5.2 Hyperfine interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43\n5.3 Spin relaxation and decoherence due to Johnson noise . . . . . . . . . . . . . . . 43\n6 Experimental methods 44\n6.1 Single spin Inelastic Electron Tunnelling Spectroscopy . . . . . . . . . . . . . . 44\n6.2 Spin polarized STM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45\n6.3 Methods to determine T1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46\n6.4 Determination of T2via EPR-STM . . . . . . . . . . . . . . . . . . . . . . . . . 47\n6.5 Quantum technologies: quantum sensors with magnetic adatoms . . . . . . . . . 48\n27 Outlook and conclusions 48\nA Bloch-Redfield tensor Rmm0;nn0 50\nA.1 Population scattering 1 =T1\u0011\u0000n;m. . . . . . . . . . . . . . . . . . 50\nA.2 Decoherence rates 1 =T2. . . . . . . . . . . . . . . . . . . . . . . 50\nA.3 Energy shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51\nB Bloch-Redfield tensor for the Kondo coupling 51\nB.1 Bloch-Redfield tensor of a pseudo-spin 1 =2 . . . . . . . . . . . . . 53\nB.2 Relation between the M-matrix and the Bloch-Redfield tensor . . . 53\nC Bosonic representation of the excitations of the Fermi gas and the\nspin-boson model 54\nD Steady-state solution of the Bloch equation for the TLS 55\n1. Introduction\nMajor technological revolutions have occurred when the humankind has been able to harness\nnatural resources, such as fire, electricity or nuclear energy. We are now in the verge of the so\ncalled second quantum revolution, that aims to harness two of the weirdest natural resources,\ncoherence and entanglement. This is a tall order that calls for a great dose of ingenuity, because\nkeeping quantum states in coherent superpositions that could be used towards our advantage\nrequires to defeat a rather powerful enemy, the infamous decoherence. Here, we review the\nphenomenon of spin decoherence in the the context of magnetic atoms deposited on surfaces.\n1.1. The relevance of decoherence\nThe interaction of quantum spins with their environment introduces relaxation and decoher-\nence in the otherwise fully coherent evolution of ideal closed quantum systems [1]. Spin relax-\nation and decoherence play a central role in many branches of physics. In the case of nuclear\nspins, the time scales associated to energy relaxation and decoherence, T1andT2respectively,\nprovide a very meaningful information of the environment that forms the basis of magnetic res-\nonance imaging techniques [2]. The spin relaxation and decoherence time scales set the limit of\nsensitivity in several existing magnetometry techniques, such as optically detected magnetic res-\nonance (ODMR) [3–5] and spin-exchange relaxation-free (SERF) atomic magnetometry [6], and\nare also one of the major constraints in the implementation of spin-based quantum computers,\nsuch as donors in silicon [7], electrons in quantum dots [8] and even molecular magnets [9, 10].\nDecoherence plays a prominent role in the modern interpretation of the foundations of quan-\ntum mechanics, the quantum measurement problem [11–15] and the quantum to classical tran-\nsition [16]. In the more specific context of magnetism, decoherence accounts for the emergence\nof the classical behavior [17]. For instance, in the quantum realm, the ground state of an integer\nspin Snanomagnet with uniaxial anisotropy can display quantum spin tunnelling [18, 19], so that\nthe ground state is non-degenerate, and it is separated from the first excited state by an energy\ngap known as quantum spin tunnelling splitting \u0001QST. The wave functions of both the ground\nstate and the first excited states are then linear superposition states [17]\nj\u001ei/jC1i+ei\u0012jC2i (1)\n3wherejC1iandjC2idescribe states with a well defined and mutually orthogonal magnetic mo-\nments, and\u0012is a (real) phase. These states have null expectation value of the magnetic moment,\nwhich highlights how di \u000berent they are from our experience in the classical realm. By contrast, a\nhalf-integer spin uniaxial magnet has two equivalent ground states, whose quantum states would\nbejC1iandjC2irespectively. As we discuss below, the coupling to the environment favors clas-\nsical states and makes states like (1) fragile [17], unless \u0001QSTis larger than all the relevant energy\nscales in the problem, which only happens for small Sat cryogenic temperatures.\nThe emergence of states with a non-zero atomic magnetization is even more intriguing in\nthe case of insulating antiferromagnets [20–22], such as MnO, that display the so called N ´eel\nstates, with a finite staggered magnetization, demonstrated in a seminal neutron di \u000braction ex-\nperiment [23]. A good starting point to describe insulating antiferromagnets is the Heisenberg\nmodel, which commutes with the total spin operator ˆS2\nTOT. As a result, the ground state is an\neigenstate of ˆS2\nTOT. For antiferromagnets this would be a state with STOT=0, which has a null\nexpectation value for every atom in the lattice [20] and it is thus very di \u000berent from the bro-\nken symmetry N ´eel states [24] that are actually observed. Therefore, other minor interactions\nsuch as magnetic anisotropy, together with the coupling to the environment, must account for the\nemergence of symmetry breaking N ´eel states, as we discuss with some detail below.\n1.2. Magnetic adatoms\nThe extraordinary series of experimental breakthroughs [25–69] in the manipulation (see\nFig. 1) and probe of magnetic atoms, mostly using scanning tunnelling microscopes (STM) and\nin some instances X ray magnetic circular dichroism (XMCD), have allowed the study of the\ncrossover from quantum to classical regime [48] in nanoengineered spin structures as well the\nexploration of coherent dynamics of individual magnetic atoms of surfaces [59]. An early de-\nvelopment that permitted probing magnetism with STM was the so called spin polarized STM\n(SP-STM). Based on the same physical principles that tunnel magnetoresistance [26], SP-STM\nyields the average magnetization of individual magnetic atoms on surfaces [32, 36]. These exper-\niments could be analyzed in terms non-quantized magnetic moments, typical of itinerant mag-\nnetic systems, a picture in line with the results of density functional calculations for magnetic\nmetals [70].\nOne of the first experimental spectroscopic fingerprints of spin-related phenomena was the\nobservation of in-gap Yu-Shiba-Rusinov (YSR) states [71–73] for Mn and Gd magnetic atoms on\na superconducting Nb(110) surface [27]. Soon after this observation, a characteristic Kondo dip\nat the Fermi energy in the dI=dVfor Cobalt on Au(111) was observed [28], which implied the\nscreening of the atomic magnetic moment and the formation of a correlated singlet state. Both\nthe Kondo peak and the YSR states are a expected consequences of the Kondo exchange inter-\naction between the magnetic adatom and the conduction electrons of the substrate. These Kondo\ninteractions are known to produce a finite spin lifetime for local spins [74–76], and thereby a\nbroadening in their spin spectral functions.\nThe development of single spin inelastic electron tunnelling spectroscopy (IETS) gave a\ndirect access to the atomic spin excitations of both individual magnetic atoms [29, 31] and\natomic spin chains fabricated adding atoms one by one [30]. In most instances these obser-\nvations [29–31, 33, 52, 55, 56, 59, 63, 77] were reported for magnetic atoms deposited on\ntop of an atomically thin insulating decoupling layer, such as Cu 2N/Cu(100) [31, 33, 52, 78],\nMgO /Ag(001) [55, 59, 77], or h-BN /Rh(111) [63], or on top of doped semiconductors sub-\nstrate [42], although they have also been observed in Fe /Pt(111) [50, 79]. The decoupling layer\n41st layer  (a) \n(f) \n(b) \n (c) \n(d) \n (e) Figure 1: STM topography images of small arrays of magnetic adatoms or molecules deposited on substrates. (a) From\nS. Loth et al. , Science 335, 196 (2012). Reprinted with permission from AAAS. (2 \u00026) and (2\u00024) Fe arrays on a\nCu2N/Cu(100) substrate. (b) From A.A. Khajetoorians et al. , Science 332, 1062 (2011). Reprinted with permission from\nAAAS. Spin-resolved topography image of the dI=dVof 4, 5 and 6 Fe chains of on a Cu(111) surface. (c) From A.A.\nKhajetoorians et al. , Nature Physics 8, 497 (2012), reprinted by permission of Macmillan Publishers Ltd: Nature Physics\ncopyright (2012). Hexagonal array of seven antiferromagnetically coupled Fe atoms on Cu(111). (d) Reprinted with\npermission from Bryant et al. 111, 127203 (2013). Copyright 2013 by the American Physical Society. Weakly coupled\nantiferromagnetic Fe dimer on Cu 2N/Cu(100). (e) Reprinted with permission from B.W. Heinrich et al. , Nanoletters 13,\n4840 (2013). Copyright 2013 American Chemical Society. Fe porphyrin on Au(111) (e) From Xi Chen et al. , Phys. Rev.\nLett. 111, 197208 (2008). Copyright 2008 by the American Physical Society. Stacking of two Co-Phthalocyanine layers\non a Pb(111) surface.\ndecreases the strength of the Kondo coupling with the underlying surface, preserving the local-\nized atomic-like nature of the spin excitations in the magnetic adatoms. This localized nature is\nin line with the fact that, in most cases [31, 33, 38, 52, 55, 59, 63, 77], the experiments could be\naccurately modelled using quantized spin model Hamiltonians [80], such as single spin models,\nvery often used in the context of transition metal ions in insulators [81] and molecular mag-\nnets [19], and the Heisenberg model commonly used to study magnetic insulators [82], which\ndescribes spin exchange interactions between localized moments.\nBroadening of the inelastic spin transitions, beyond the thermal 5.4 kBTfactor [83], has been\nmeasured for Fe on top of Cu(111) [45], in line with the predictions of theory for spin relaxation\ndue to Kondo coupling [84] and also with more sophisticated theoretical treatments [85, 86]. This\nbroadening is the spectral counterpart of a finite spin relaxation time T1in the time domain. The\ndevelopment of electrical pump-probe technique with STM [43] has made it possible to measure\nthe rather fast (ns) spin relaxation time T1of individual atoms [43, 59] and other atomically\nengineered structures, such as antiferromagnetic chains and ladders [48, 66] or ferromagnetic\nchains [56] and clusters [49].\nCompared to single dopants systems [87, 88], such as nitrogen-vacancy (NV) centers in di-\namond or single donor in silicon [89], the study of spin coherence of magnetic adatoms is in its\n5infancy. At the time of writing this review, there is only one experimental paper [59] that reports\nthe measurement of the decoherence time T2of an individual Fe atom on MgO /Ag(001) inferred\nfrom a continuous wave resonance experiment, rather than the more sophisticated spin echo\ntechniques used to probe the T2of individual Pdopants [90, 91] or shallow NV centers [92–94].\nOn the theory side, only a few papers address the problem of decoherence in this specific con-\ntext [17, 76, 95–97]. Yet, the potential of magnetic adatoms to explore quantum spin dynamics\nis enormous for several reasons. First, STM makes feasible to fabricate atom-by-atom magnetic\nnanostructures [30, 46, 48, 56, 62, 78], controlling at will the number of atoms, the spin Sof\nthe atoms, and their interatomic distance. In turn, this permits researchers to determine both the\nstrength and the sign of the exchange interaction between magnetic adatoms [53, 64]. The vast\nspace for combinations of spin S, exchange interactions, magnetic anisotropies and number of\natoms results in very di \u000berent types of magnetic behaviors, including both systems with broken-\nsymmetry N ´eel states [48] or quantum disordered spin chains [30]. Strong spin-correlation can\nlead, in some cases, to the emergence of non-trivial S=1=2 objects at the edges [98] as in chains\nof antiferromagnetically coupled S=1 atoms, objects that might be more robust with respect to\ndecoherence than conventional S=1=2 spins.\nAnother important resource to explore quantum coherent phenomena in magnetic atoms on\nsurfaces is the very large potential to engineer their coupling to the substrate by means of the ad-\nequate choice of materials, including both substrate and coating layer. Most of the experiments\nmentioned in this review are done using 4 types of metal (Cu, Ag, Pt, Rh), in some instances with\na coating layer, [Cu 2N/Cu(100), MgO /Ag(001) or h-BN /Rh(111)]. Using di \u000berent materials will\nbring many opportunities. Take the example of superconductors. The standard BCS [99] theory\nof superconductor predicts that spin relaxation lifetimes of a localized spin in a superconduc-\ntor, such as nuclei, can be significantly enhanced [100]. This has been already experimentally\ndemonstrated by means of STM spectroscopy for electronic spins in magnetic molecules on top\nof superconducting lead [51]. The discovery of zero-energy edge states in ferromagnetic chains\ndeposited on superconductors [54], which might be a physical realization of the Majorana modes,\nprovides additional motivation to place magnetic nanostructures on top of electronically interest-\ning substrates. In this regard, the exploration of adatom spin dynamics in the case of substrates\nwith peculiar transport properties, such as topological insulators, graphene, or the edge states of\nQuantum Hall and Quantum Spin Hall systems has barely been explored.\nMolecules are another extremely powerful resource to build on surfaces nanostructures with\nnon-trivial magnetic properties. IETS has been used to probe, with atomic resolution, the spin ex-\ncitations of individual magnetic molecules, such as Fe phthalocyanine (FePC) [37], with S=1,\nCobalt PC, with S=1=2 [34], and molecular magnets, such as Mn 12[44] and Fe 4[60]. In the\ncase of CoPC, the self-assembly of molecular stacks also allowed exploring the spin properties of\nvertically spin chains, a physical realization of the Hubbard model [34, 101]. Transport experi-\nments both on open-shell and single-molecule magnets have also been reported using mechanical\nbreak junction techniques that, in some instances, permits one to add a gate and study di \u000berent\ncharge states of the molecule [102–106]. Molecular magnets with build-in spin chains [10] have\nbeen shown to posses very long T1andT2times in diluted phases. Therefore, it would be very\ninteresting to probe them individually, using STM, and see how their coupling to a conducting\nelectrode changes T1andT2.\nWhereas atomically engineered magnetic nanostructures are definitely interesting by their\nown, they can also be used to probe the spin dynamics of nearby structures [69]. This concept\ncan be downscaled to the atomic limit: probing the spin of a single magnetic atom can provide\ninformation of a chemically di \u000berent neighboring atom [38, 68, 107] as well as the dynamics\n6Table 1: Typical orders of magnitude of the relaxation times T1and decoherence times T2in a variety of spin systems at\nT\u00191 K in di \u000berent environments.\nSystem Shallow Donors in Si NV-Centers QD’s Magnetic adatom\nT1(s) 10\u00002\u0000103[109–111]11\u000010 [112] 10\u00004[113, 114] 10\u000013\u0000104[45, 107, 115]2\nT2(s) 10\u00004\u00000:6 [109–111] 10\u00003\u000010\u00004[112] 10\u00007\u000010\u000052:10\u00007[55]\nof magnetic nanostructure [67]. In addition, the combination of di \u000berent magnetic atoms in\nthe same structure, such a spin chain, may result in unexpected new properties that show the\nenormous potential of spin doping in correlated systems [62]. Limits for upscale remain to be\nexplored: non-magnetic atomically engineered structure where more than eight thousand atoms\nwere nanostructured has been presented recently [108].\nWith this background, the main goal of this review paper is to serve as a guide for future\nexploration of spin coherence at the atomic scale. To do so, we shall try to promote cross fertil-\nization between the traditional STM /atomic scale magnetism community on one side, with other\nfields where coherent single /few spin physics has been successfully explored using di \u000berent in-\nstrumental techniques, most notably, optically detected magnetometry using NV centers, and\nsilicon qubits. The rest of this review is divided in four main blocks. The first one is devoted\nto provide a general theory background that yields a proper definition of coherence /decoherence\nand how to compute it. Particular attention will be given to two level systems (TLS), given\nthat in many instances one will deal with systems with either a doubly degenerate ground state,\nsuch as half-integer spin magnetic atoms, or systems where the two lowest energy states are\nwell separated from the higher energy excited states, such as spin chains with strong uniaxial\nanisotropy [17, 48, 56]. Then, in the second block we illustrate the analysis of decoherence\ndue to the main source in magnetic adatoms, the Kondo exchange coupling with the substrate\nelectrons. In the third block we discuss other decoherence mechanism for magnetic adatoms,\nincluding spin-phonon coupling and Zeeman coupling to the random environmental magnetic\nfield created by Johnson-Nyquist noise and shot noise. In the fourth block, we briefly review the\nstate of the art experimental status to measure spin dynamics with STM and we comment on the\nmain challenges to observe Rabi oscillations at the single-atom spin level. Finally, we finish with\na discussion and main results of our work.\n2. Decoherence, a general overview\n2.1. Quantum dissipative dynamics in open quantum systems: decoherence and relaxation\nIn principle, the study of the dynamics of a quantum state can be tackled in deceptively\nsimple terms, solving the time dependent Schr ¨odinger equation\ni~@j (t)i\n@t=Hj (t)i (2)\n1TheT1grows exponentially with 1 =T.T2is bounded by T1at high temperatures but then saturates at low tempera-\ntures at a value dependent on the dopant concentration. The values in the table corresponds to T\u00191:8 K.\n2It has been claimed that Ho adatoms on a Pt(111) surface leads to relaxation times exceeding the second time\nscale [116]. However, this claim is in clear contrast with XMCD measurements revealing no evidence of magnetic\nstability and a di \u000berent ground state configuration for Ho /Pt(111) [117], which violates the theoretical criterion proposed\nby Miyamachi et al. [116]. Moreover, more recent SP-STM and IETS-STM measurements have found no evidences of\nmagnetic moment of Ho on this substrate [68].\n7that describes the unitary evolution of the state j (t)i. This gives us the most complete informa-\ntion about the quantum system. In particular, the expectation value of any observable ˆAcan be\nthen computed as hˆA(t)i=h (t)jˆAj (t)i. However, strictly speaking, Hshould be the Hamilto-\nnian of the entire universe, in which case the Schr ¨odinger equation can not be solved, and even\nif it could, it would give us a bunch of unusable information.\nInstead of giving up, one always restricts Hto the system of interest, which we label as\nHS, and we focus on the dynamics of the restricted set of degrees of freedom of the system S\ndescribed byHS, see Fig. 2. To do so, we split the the Hamiltonian in 3 terms:\nH=HS+HR+V; (3)\nwhereHRis the Hamiltonian of the environment, i.e., the degrees of freedom explicitly excluded\nfromHS, andVdescribes the coupling between environment and the system. The next step is\nto derive a dynamical equation to describe S, including both the action of HSand the influence\nof the environment, which will be described in an statistical coarse grained manner.\nNow we need to find a dynamical equation to describe the S, including both the action of\nHSandV. This is precisely the central theme in the study of the so called open quantum\nsystems [118]. For the dynamical equation, we adopt the density matrix language to account\nfor the quantum dynamics of a small subsystem while tracing out the degrees of freedom of\nthe rest, which are less interesting or irrelevant for the observer, leading in general to a non-\nunitary evolution. Importantly, as in most cases the environment consist on a system with a\nmacroscopically large number of degrees of freedom, it can be modelled as a reservoir orbath\nthat remains in thermal equilibrium, neglecting the back-action of HSon the density matrix of\nthe reservoir. In contrast, the bath influences the dynamics of S, and more specifically, it is the\nultimate responsible of the decay of the quantum coherence of the otherwise isolated quantum\nsystem.\nFor any global (system plus reservoir) quantum state j (t)iwe can define the total density\noperator ˆ\u001aTot(t)=j (t)ih (t)j. In the spirit of the open quantum system approach, we introduce\nthereduced density operator [118]\nˆ\u001a(t)=TrR\u0002ˆ\u001aTot(t)\u0003; (4)\nwhere Tr R[:::]corresponds to the trace over the bath degrees of freedom. When represented\nin the basis of eigenstates of HS, labeled byjni, the reduced density matrix (DM) has a clear\nstatistical interpretation. The diagonal entries of that matrix, \u001ann\u0011Pn, give us the probability of\nfinding the system Sin a given eigenstate jni. These occupations P nsatisfy the normalization\nconditionP\nnPn=1 together with Pn\u00150. The o \u000b-diagonal entries \u001ann0(n,n0) are known as\ncoherences . They quantify the capability of the quantum system to be in a superposition state\nthat combines two di \u000berent eigenstatesjniandjn0i.\nIn general, the dynamical equation for the reduced DM can be written down as:\n@tˆ\u001a(t)=\u0000i\n~\u0002HS;ˆ\u001a(t)\u0003+R\u0002ˆ\u001a(t);t\u0003: (5)\nThe first term on the right hand side would describe the coherent dynamics if we neglect the\ncoupling of the system to the reservoir. The DM of a pure quantum state ˆ \u001a=j ih jsatisfies\nthe identity ˆ \u001a2=ˆ\u001a, a condition preserved by the first term. The second term accounts for the\ninfluence of the reservoir and it induces the non-unitary dynamics of interest here. This term is\nresponsible of both, relaxation and decoherence. In fact, in the language of DM, relaxation is\n8Environment  \nQuantum \nspin \nsystem  \nFigure 2: Scheme of an open quantum system . The central idea is to split the whole system into three: a small (and\nanalytically or numerical treatable) quantum spin system with Hamiltonian HS, which can be studied when isolated\nfrom the rest of the universe, the environment HR, which comprises all other degrees of freedom not included in the\nsystem, and the interaction Vbetween them (yellow region in the scheme).\nassociated to the decay of any departure of the diagonal elements (populations) on a time scale\nT1, while the decoherence corresponds to the decay of the o \u000b-diagonal terms on a time scale T2.\n2.1.1. Coherence as a basis dependent quantity\nAt this point, we must point out that the notion of coherences is basis-set dependent. In the\ncontext of spins, there are two natural basis sets. First, the basis defined by the eigenstates of the\nquantum systemHS, where the thermal equilibrium density matrix acquires a simple diagonal\nform. Therefore, coherence between eigenstates is a transient phenomena, and is bound to fade\naway. This is the phenomenon of decoherence. However, there is a second natural choice of basis\nset to express the density matrix, the basis of eigenstates of a spin observable, say ˆSz, convenient\ndue to the existence of specific probes to measure this observable. Whereas in some cases HS\ncommutes with ˆSz, so that it is possible to choose a basis set that diagonalizes both, in other\ninstances this is not the case, and we can have a density matrix that is diagonal in one basis, but\nnot in the other. Thus, coherence and decoherence, as defined above, depend on the choice of\nbasis set, or at a more pragmatic level, on the type of experiment used to probe the system. For\nthis reason, whenever we have such an ambiguity, hereafter we will use a subindex in the density\nmatrix to label the basis set in which the matrix is represented.\nThe fact that coherence depends on the basis set becomes particularly tricky when there is a\ndegenerate spectra. In that case, there are infinite possible choices of basis sets. Interestingly, the\ncoupling to the environment is not neutral regarding this choice, which leads to the definition of\nthe so called pointer states . We shall discuss this below.\nIt must be pointed out that coherent superpositions can be found in rather trivial situa-\ntions, even at the macroscopic scale. For instance, the quantum description of the classical\nmotion of a harmonic oscillator, given by the Glauber quantum states, entails coherent super-\npositions of the so called Fock states j\u0017i, eigenstates of the harmonic oscillator Hamiltonian\nHHO=~!\u0010\nˆn+1\n2\u0011\n[119]. In contrast, having a coherent superposition of macroscopically di \u000ber-\nent classical states, is a much harder task.\n2.1.2. Coherence in a two level system (TLS)\nThe simplest case in which we can illustrate the concepts of decoherence is a two level\nsystem (TLS). In addition, the TLS approximation can be often used to describe magnetic atoms\n9and nanostructures. Let us consider a S=1=2 spin in the presence of a magnetic field along the\nzaxis, with eigenstates Sz=\u00061\n2. The representation of the DM in the basis set of eigenstates \"\nand#reads as:\nˆ\u001a(t)= P\"(t)C\"#(t)\nC#\"(t)P#(t)!\n(6)\nor, writing it in terms of the Pauli matrices ~\u001c=\u0010\nˆ\u001cx;ˆ\u001cy;ˆ\u001cz\u0011\nand the identity I[120]:\nˆ\u001a=1\n2\u0010\nI+~P\u0001~\u001c\u0011\n(7)\nwhere\n~P=\u0010\n2Re( C\"#);2Im( C\"#);P\"\u0000P#\u0011\n: (8)\nWe can now easily compute the expectation value of any operator hˆA(t)i=Tr[ˆAˆ\u001a(t)] using the\nwell known result for Tr [ˆ\u001caˆ\u001cb]=2\u000eab, where aandbdenote the cartesian components. We thus\nsee thathˆSai=Pa=2, i.e.,hˆSzi=(P\"\u0000P#)=2 andhˆSx(t)i=2Re\u0002C\"#(t)\u0003. Thus, the longitudinal\nmagnetization is governed by the occupations, whereas the transverse magnetization is governed\nby the coherences.\nIn a free induction decay experiment, typical of NMR experiments, spins are driven out of\ntheir equilibrium magnetization along the direction of a static field (the zdirection), by means\nof a pulse of actransverse magnetic field. The re-establishment of the equilibrium situation is\ndescribed by the equations:\nhSzi(t)\u0000hSzijeq/e\u0000t=T1(9)\nhSx\u0006iSyi(t)/e\u0000t=T2e\u0006i!0t(10)\nwhere ~!0is the Zeeman splitting. These equations serve also to define T1andT2, as the spin\nrelaxation and spin decoherence time, respectively. In addition to these two timescales, in the\ncase of ensembles it is often convenient to introduce the dephasing time T\u0003\n2associated to the\npurely coherent dynamics of an ensemble of spins (or two level systems in general), whose\nenergy splitting \u0001is not exactly the same for all of them. In an ensemble measurement, both T2\nandT\u0003\n2result in a decay of the transverse spin signal. However, using spin-echo techniques, the\ne\u000bect of T\u0003\n2can be reverted [13], which permits to measure T2.\nOf course, Eqs. (7-8) are only meaningful if the basis set used to represent the DM is spec-\nified, i.e., a change of basis will lead to a di \u000berent matrix, yet the physical state would be the\nsame. The DM can describe either a pure state j i, in which case ˆ \u001a=j ih j, or mixed states in\nwhich our knowledge about the state of the system is partial. For a pure state it is trivial to verify\nthat ˆ\u001a2=ˆ\u001a, which leads to P=j~Pj=1. In the opposite limit we have the DM with P=0, with a\n50 percent occupation for each state and no coherences. In order to quantify the purity of a given\nDM we can use the so called V on Neumann entropy\nS=\u0000Tr\u0002ˆ\u001aln ˆ\u001a\u0003=\u0000X\ns=\u000611+sP\n2log2 1+sP\n2!\n: (11)\nFor a pure state, P=1 and the entropy vanishes S=0, while for P=0 the entropy is maximized,\nwith S=1.\nAn important result is that the property P=0 is preserved under any unitary change of basis.\nIn other words, the expression of the fully decohered density matrix is basis independent. When\n10coupled to a reservoir for a su \u000eciently long time, the DM is expected to evolve towards the equi-\nlibrium DM regardless of the initial state, given by ˆ \u001a=e\u0000\fHS=Z, with Zthe canonical partition\nfunction and \f=1=kBT. The only exception for this rule arises if there is some symmetry that\nprevents the system to change the occupations of some energy levels, as discuss below.\n2.1.3. Decoherence as entanglemenet with the bath\nWe now provide a very simple example of how decoherence can arise in a TLS due to its\ncoupling to a dissipative bath in the simplest case of pure dephasing, i.e., when transitions be-\ntween the eigenstates of the TLS are strictly forbidden, so that the diagonal terms in the density\nmatrix stay constant. Here we reproduce an argument given by Stern, Aharonov and Imry [121]\nto show the relation between decoherence and the e \u000bect of the system on the wave function of\nthe environment,j\u0011(t)i.\nLet us assume that the system and the environment remains decoupled from each other for\nt\u00140. The wave function for the system plus environment can be then written as:\nj\t(t=0)i=\u0012\ncos\u001e\n2jC1i+sin\u001e\n2ei\u0018jC2i\u0013\n\nj\u0011(0)i; (12)\nwhere\u0018and\u001eare real numbers characterizing completely the state of the system, modulo a\nphase. At t=0 the interactions between them are turned on and we assume that the only e \u000bect\nof this interaction is to change the wave function of the reservoir, which acts as a witness of\nwhich path, either C1orC2, is being taken by the system. Let ~!21be the energy di \u000berence\nbetween statesjC1iandjC2i. Then, the wave function at t\u00150 satisfies:\nj\t(t)i=cos\u001e\n2jC1i\nj\u00111(t)i+\n+sin\u001e\n2ei\u0018ei!21tjC2i\nj\u00112(t)i: (13)\nWe see that as long as j\u00111i,j\u00112i, state (13) is no longer a product state, and it is thus said\nthat the reservoir and the system are now entangled. The DM operator associated to this state is\nˆ\u001a\t(t)\u0011j\t(t)ih\t(t)j. If we define the reduced DM \u001a(t)=TrR\u0002ˆ\u001a\t(t)\u0003, we get that\nˆ\u001a(t)= cos2\u001e\n2c0(t)S12(t)\nc\u0003\n0(t)S21(t) sin2\u001e\n2!\n; (14)\nwhere we have introduced the bare coherence c0(t)=cos\u001e\n2sin\u001e\n2ei\u0018ei!21tand the time dependent\noverlap\nSi j=TrRh\nj\u0011i(t)ih\u0011j(t)ji\n=h\u0011i(t)j\u0011j(t)i: (15)\nOf course, we have S11=S22=1 andS1;2\u00141. In terms of the polarization vector defined in\nEq. (8), we get that\nP=q\ncos2\u001e+sin2\u001ejS12(t)j2: (16)\nTherefore, the e \u000bect of the environment mimics a which-path detector and it leads to a depletion\nof the coherence between the jC1iandjC2istates that is given exactly by the overlap between\nthe alternative reservoir wave functions: the more sensitive the reservoir is to the state of the\nsystem, the smaller the overlap S12and the larger the decoherence. Complete decoherence occurs\nwhen the alternative reservoir wave functions become orthogonal. This can also be seen from\n11an information perspective. Given that the wave function of any given system encodes the most\ncomplete knowledge that we can a \u000bord, decoherence increases the V on Neumann entropy (11)\nat a rate controlled by the amount of which-path information is stored in the environment. This\ncomplies with the rule of thumb that quantum systems are able to stick to a linear superposition\nstate as long as nobody is watching.\n2.1.4. Decoherence as phase uncertainty induced by a stochastic field\nWe now discuss a second simple picture for decoherence, also analyzed by Stern and cowork-\ners [121]. Following the notation of Slichter [1], we consider a model Hamiltonian for a spin\nS=1=2 interacting with a static field Bzand a random stochastic field ~b(t):\nH=g\u0016BBzSz+g\u0016B~b(t)\u0001~S: (17)\nThe stochastic field is characterized by a null average ba(t)=0 (a=x;y;z) and a steady noise:\nba(t)bb(t+\u001c)=b2a\u000ea;be\u0000t=\u001c\u0002(t); (18)\nwhere \u0002(t) is the Heaviside function. Here the basymbol represents an average over realizations\nof the stochastic classical field ~b,b2represents the amplitude of the noise associated to the\nmagnetic field and \u001cis the so called correlation time, that characterizes the pace at which the\nrandom field fluctuates. In the IS system, this noise has units of T2=Hz. This model applies to\nthe spin relaxation caused by the stochastic magnetic field coming from the thermal fluctuations\nof the current in a conductor [122], as we discuss below.\nUsing the Bloch-Redfield theory discussed below, the relaxation time T1and the decoherence\ntime T2can be written down as [1] :\n1\nT1=\u0012g\u0016B\n~\u00132h\nkxx(!0)+kyy(!0)i\n(19)\n1\nT2=1\n2T1+\u0012g\u0016B\n~\u00132\nkzz(0); (20)\nwhere ~!0=g\u0016BBand\nkab(!)=1\n2Z1\n\u00001ba(t)bb(t+\u001c)e\u0000i!tdt (21)\nis the Fourier transform of the correlation function. Expression (20) helps us to introduce several\nrelated concepts frequently used in fields like quantum optics or electron /nuclear spin resonance\n(ESR /NMR). The single spin decoherence time T2, also called dephasing time ortransversal\nrelaxation time in the context of ESR /NMR, involves two types of processes, as revealed by\nEq. (20). First, scattering processes that implies a population transfer between the the two\nstates, which accounts for the 2 T1contribution in Eq. (20), its maximum value ( T2\u00152T1).\nIn the context of quantum systems coupled to a reservoir, this is the so called nonadiabatic\ncontribution [123]. Second, scattering processes that do not involve population transfer between\nthe system states, which for our stochastic field, are proportional to kzz(0). They constitute the\nadiabatic contribution to decoherence.\nEquations (19-20) imply that there is a generic linear relation between the dissipative rates\nand the spectral function of the stochastic field. This result still holds when we consider spins\ncoupled to quantum operators of a reservoir, in which case the spin relaxation times are propor-\ntional to the dynamic response functions of the reservoir. Thus, T1andT2provide a local probe\n12for the properties of the reservoir, a central notion in magnetic resonance imaging [2] and in the\nfield of quantum sensors [124–126].\nEquations (19-20) also show how the stochastic field can produce population transfer between\nthe two states only when the transverse components¯b2xor¯b2yare finite (otherwise Szis a constant\nof motion), whose e \u000eciency is proportional to the weight of the noise spectral function at the\ntransition energy, which relates to the conservation of the total energy. Decoherence occurs\nwhen there are population scattering events and, in addition, it can also occur even if¯b2x=\n¯b2y=0. This last case is the pure dephasing case, seen above, but from a di \u000berent perspective:\nstochastic fluctuations of the energy splitting are straightforwardly equivalent to a loss of the\nphase coherence. Finally, Eqs. (19-20) illustrate that population scattering invariably entails\ndecoherence, with a pure dephasing contribution proportional to kzz(0).\nUsing expressions (18) and (21), we get that kaa(!)=b2a\u001c0=(1+!2\u001c2\n0), which leads to\n1\nT2=\u0012g\u0016B\n~\u00132266664\u001c0\n1+!2\u001c2\n0h\nb2x+b2yi\n+b2z\u001c0377775: (22)\nIn the pure dephasing limit, where b2x=b2y=0, we have T2/\u001c\u00001\n0, which is the so called\nmotional narrowing. This result can be understood within the following picture. Let us consider\na stochastic field that can assume only 2 values b\u0006=\u0006(b2z)1=2. The phase acquired during the\ntime interval \u001c0in which the field stays active is \u000e\u001e=\u0006\u001c0g\u0016Bb+=~. After nsuch intervals, the\nspread of the phase, governed by a binomial distribution, equals \u0001\u001e2=n(\u000e\u001e)2=n(\u001c0g\u0016Bb+=~)2.\nNow, the number of such intervals in a time tisn=t=\u001c0. We estimate T2as the time it takes for\nthe phase spread to be equal to 1 radian:\n1\nT2=\u0012g\u0016B\n~\u00132\n\u001c0b2z: (23)\nThis result could of course be obtained from Eq. (22) in the pure dephasing limit ( b2x=b2y=0),\nwhere spin-flip transitions are forbidden.\n2.2. Bloch-Redfield perturbative approach to the dissipative dynamics\nIn this section we briefly review the Bloch-Redfield (BR) master equation theory to deal\nwith the dissipative dynamics of quantum systems weakly coupled to a reservoir. The biggest\nadvantage of this approach is that it can be applied to a great variety of systems, not restricted\nto individual magnetic atoms, and permits dealing with situations where both pure dephasing\nand population scattering are present. On the down side, the Bloch-Redfield treats the system-\nreservoir coupling to second order in perturbation theory, and it can only provide a description of\nthe dynamics on a coarse-grained time scale larger than the correlation time of the reservoir \u001cc,\nthe surface conduction electrons or lattice vibrations in our case. These limitations are not severe,\nand this technique have been successfully used to study the spin dynamics of a single or a few\nmagnetic atoms adsorbed on top of a monolayer of insulating material grown on a conducting\nsubstrate [17, 41, 52, 56, 127], and also for few-atom clusters on metals [49]. An excellent\nintroduction to the formalism can be found in Ref. [119], in the context of quantum optics, and\nalso in Ref. [118].\nThe starting point of the BR formalism is the Hamiltonian of Eq. (3), H=HS+HR+V.\nIn the case of magnetic adatoms, HSdescribes the atomic spins and usually accounts for the\n13local magnetic anisotropy, inter-spin interactions, and Zeeman interactions. If required, one can\nalso include the nuclear spins in the system degrees of freedom [128, 129], in which case HS\nshould include the hyperfine interactions as well. In the basis of eigenstates of HS, we can write\nHS=P\nnEnjnihnj.\nTheHRterm describes the environment Hamiltonian, which could corresponds to the elec-\ntronic bath of conduction electrons in the metallic substrate, a phonon bath, photons, or the bath\nof surrounding nuclear spins. Finally, the Vterm represents the interaction between the system\nof quantum spins and the degrees of freedom of the bath. For most systems, this interaction can\nbe written in the form\nV=X\n\u000bR\u000b\nS\u000b; (24)\nwhere R\u000bare reservoir operators and S\u000bspin operators, to be determined depending on the nature\nof the interaction. The composite index \u000bwill contain information about all the bath quantum\nnumbers and, if the system is coupled to more than one bath, also the bath label. We remark here\nthat the BR tensor contains only second order in Vcorrections to the dynamics of the quantum\nsystem due this coupling. In addition to condition (24), the BR also assumes a zero-average of\nthe system-bath coupling, i.e., Tr R\u0002ˆ\u001aRV\u0003=0, where ˆ\u001aRis the thermal equilibrium density matrix\nof the reservoir and the trace is over the reservoir degrees of freedom. Notice that in cases where\nthis last condition is not satisfied, one can always reinsert this trace into a renormalized system\nHamiltonian and interaction such that the new problem has a zero average interaction.3\nIf we introduce the basis of eigenstates of the system Hamiltonian, jni, the markovian evolu-\ntion of the reduced DM to second order in Vcan be written as:\n@t\u001anm(t)=\u0000i!nm\u001anm(t)+X0\nnn0Rnm;n0m0\u001an0m0(t); (25)\nwhere!nm=(En\u0000Em)=~andRnm;n0m0is the Redfield tensor. From the hermiticity of the density\nmatrix, one getsR\u0003\nmn;m0n0=Rnm;n0m0. The prime over the sum in Eq. (25) implies that only\nthe terms whose energies satisfy j!nm\u0000!n0m0j\u001c1=\u000etare included, where \u000et=~=(kBT) is the\ncoarse-grain time scale mentioned above. This is called the secular approximation [123].\nThus, the BR tensor depends both on the matrix elements of the system operators, Smn\n\u000b\u0011\nhmjS\u000bjni, as well as on the reservoir operator correlator, g\u000b\f(t)\u0011 hR\u000b(t)R\f(0)ijeqwhere the\nbrackets stand for statistical average over the reservoir equilibrium density matrix and R\u000b(t)=\neiHRt=~R\u000b(0)e\u0000iHRt=~. We provide here general expressions, decomposing the tensor entries in a\nsum of two terms, Rnn0mm0=R+\nnn0mm0+R\u0000\nnn0mm0, where [130] ,\nR+\nnn0mm0=X\n\u000b;\f1\n~20BBBBB@g\u000b\f(!nm)Sm0n0\n\u000bSnm\n\f\u0000\u000em0n0X\nn00g\u000b\f(!n00m)Snn00\n\u000bSn0m\n\f1CCCCCA (26)\nand\nR\u0000\nnn0mm0=X\n\u000b;\f1\n~20BBBBB@g\u0003\n\f\u000b(!n0m0)Sm0n0\n\u000bSnm\n\f\u0000\u000enmX\nn00g\u0003\n\f\u000b(!n00m0)Sm0n00\n\u000bSn00n0\n\f1CCCCCA (27)\nwhere g\u000b\f(!)\u0011R1\n0dte\u0000i!tg\u000b\f(t).\n3In the case of magnetic systems coupled to an electronic bath, this zero-average condition implies that, whenever the\nbath is spin polarized, one should proceed to remove the average Tr R\u0002ˆ\u001aRV\u0003[123].\n14The BR equation (25) describes the evolution of both the occupations \u001annand coherences\n\u001ann0(n,n0). Although in general their dynamics are coupled, in some instances it is possible\nto write down an equation for the occupations, Pn\u0011\u001annand the coherences separately. The\nresulting equation for the occupations is the so called Pauli master equation:\ndPn\ndt=X\nn0\u0000n0nPn0\u00000BBBBB@X\nn0\u0000nn01CCCCCAPn; (28)\nwhere \u0000nn0stands for the the scattering rates from state nton0. This equation has a transparent\nphysical interpretation: coupling to the reservoir results in scattering events that transfer weight\nfrom some states to others. When the reservoir is at thermal equilibrium, the steady occupations\nPnare given by the Boltzmann distribution.\nThe scattering rates \u0000nm\u00111=T1can be written down in terms of the Redfield coe \u000ecients [118]\n\u0000nm=2\n~2X\n\u000b;\fRe\u0010\ng\u000b;\f(!mn)\u0011\nSnm\n\u000bSmn\n\f: (29)\nThis general formula accounts for instance, for the T1obtained for a spin S=1=2 system\ncoupled to a stochastic magnetic field, Eq. (19), when we identify the indexes \u000b;\f with the\ncartesian coordinates and the reservoir correlator g\u000b;\fwith the magnetic field noise function k\u000b;\f.\nIn the case of the Kondo interactions discussed in this review, the quantum operator S\u000bis\nan atomic spin operator while the reservoir operator R\u000b\u0011~s(~rl) is the fermionic spin density,\nand their coupling is the isotropic spin interactionP\nl;aJlSa(l)sa(~rl). In this case, the reservoir\ncorrelator g\u000b;\fis related to the spin susceptibility of the bath, due to the fluctuation dissipation\ntheorem.\nEquation (25) also describes the evolution of coherences in the basis of eigenstates of the\nsystem Hamiltonian. In the simplest case when the transition energy between a given pair of\nstates is di \u000berent from that of every other pair of eigenstates, which automatically occurs for a\ntwo level system, the equation of motion for the coherence is independent from the rest:\n@t\u001anm(t)=\u0000i(!nm+\u000e\u0001nm)\u001anm(t)\u0000\rnm\u001anm(t); (30)\nwhere we have split the RB tensor into its real and imaginary parts Rnm;nm=\u0000\rnm\u0000i\u000e\u0001nm.\nEquation (30) has clear physical interpretation. The coupling to the reservoir produces two e \u000bects\non the dynamics of the coherence. First, it renormalizes the energy di \u000berence between the energy\nlevels. Second, and central for the purpose of this review, the coupling to the reservoir results in\na damping term, that results in a decay rate of the coherence given by\n1\nT2=\rnm: (31)\nImportantly, the decoherence rate has two types of contributions, the adiabatic andnonadia-\nbatic [123]. The later involve transitions between the levels nandmgoverned by the same type\nof rates and microscopic processes than T1:\n\rnonad:\nnm =1\n20BBBBB@X\nn0,n\u0000n;n0+X\nn0,m\u0000m;n01CCCCCA: (32)\nTheir interpretation is transparent: phase coherence is lost when population scattering occurs.\nThe adiabatic contribution to decoherence is more subtle, as it can happen even in the absence of\n15population scattering, and it connects directly with the ideas discussed in Secs. 2.1.3 and 2.1.4.\nThe adiabatic decoherence takes the form:\n\rad\nnm=1\n~2X\n\u000b;\fRe(g\u000b\f(0))\u0010\nSmm\n\f\u0000Snn\n\f\u0011\u0000Smm\n\u000b\u0000Snn\n\u000b\u0001: (33)\nThe physical interpretation of Eq. (33) is the following. If we have two states, nandm, such that\nthe expectation value of a given operator is di \u000berent, and the environment is sensitive to which\nof the two states the system stays at, even if it is not able to induce scattering between them,\ndecoherence will occur. To be more specific, let say n= +Sandm=\u0000Sfor a quantum spin. A\nmagnetic coupling to the environment will create decoherence between nandmeven if there is\nno population transfer. The adiabatic rate of such process would be: \rad\nSz;\u0000Sz'4S2Re(gzz(0))=~2,\nwhich again connects with the result (22).\nAs illustrated by Eq. (30), the coupling to the bath also produces a frequency shift \u000e\u0001nm\nof the original energy levels that can be decomposed as \u000e\u0001nm=(!n\u0000!m). Using the general\nexpressions ( 26) and (27), this frequency shift can be written as\n!m=1\n~2X\n\u000b;\fX\nn0Im\u0010\ng\u000b;\f(!n0m\u0011\nS\u000b\nmn0S\f\nn0m: (34)\nA well known example of this renormalization is the Lamb shift of the Hydrogen spectrum. In\nthe case of magnetic adatoms, the associated variation of the magnetic anisotropy due to the\nKondo exchange coupling with the substrate itinerant electrons has been recently reported [52].\nHere we should remark that result (34) is completely general, i.e., the only BR tensor elements\ncontributing to the energy shifts are those of the form Rnm;nm[118].\nIn summary, the BR theory accounts for the dissipative dynamics of the reduced density\nmatrix of a quantum system weakly coupled to a reservoir. The influence of the reservoir is\nincluded up to second order in the system-reservoir interaction, and the correlation functions of\nthe reservoir are assumed to have a very short memory, or in a more technical jargon, we adopt\nthe Markovian approximation. The BR theory accounts for 3 types of e \u000bects:\n1. The occupations of the quantum states (diagonal part of the density matrix) evolve in\ntime, experiencing transitions between the originally decoupled eigenstates of the isolated\nsystem.\n2. Decoherence. When the reservoir is in thermal equilibrium, the steady state solution of the\nBR is the equilibrium density matrix, which is diagonal in the basis of eigenstates of the\nisolated systemHS. Therefore, coherences in this basis are fragile and decay in time.4\n3. Renormalization of the energy levels. In the case of nanomagnets, the shift of the energy\nlevels translates into renormalized magnetic anisotropies and they can play an important\nrole. For instance, the quantum spin tunneling splitting of an isolated spin, which protects\nquantum coherence, can be quenched due to coupling to the environment.\n4The reduced density matrix can deviate from the thermal equilibrium distribution when there are some symmetries\nthat prevents transitions between eigenstates of HS. Moreover, in the case of a degenerate spectrum it is always possible\nto choose a basis in which coherences does not decay with time.\n162.3. Bloch equation for a 2-level system\nMost of the experiments measurements of the coherence time of a system involve driving\nthe system with an acsignal. Therefore, it is important to have a dissipative theory for the\ndensity matrix where the dynamical driving term is included. Here we review the Bloch approach\nfollowing Cohen-Tannoudji et al. [123]. The right hand side of the BR equation (25) contains\ntwo terms. The first is the commutator of the reduced density matrix with the Hamiltonian HS\nof the system, whereas the second accounts for the dissipative coupling to the environment. This\nderivation was done assuming that HSis time-independent.\nThe derivation of Bloch equation for the dissipative dynamics of a driven two-level system\nconsists, basically, in replacing HSbyHS+V1(t) in the first term of the right hand side of Eq.\n(25), whereV1(t) is the driving term. This approach is definitely justified in two limits. In the\nabsence of dissipation, the equation of motion of the density matrix is given by the commutator\nwith the time-dependent Hamiltonian. Second, in the absence of the driving term, we recover\nthe BR theory. Basically, this approximation amounts to assume that the coherent and incoherent\ndynamics are additive.\nFor an energy-split TLS, with energies EaandEb, the resulting equation of motion for the\ndensity matrix reads as:\n˙\u001aaa(t)=\u0000˙\u001abb(t)=1\n2T1\u0010\n\u001abb(t)\u0000\u001aaa(t)\u0000(\u001aeq\nbb\u0000\u001aeq\naa)\u0011\n+i\n~\u0010\n\u001aabVba\n1(t)\u0000\u001abaVab\n1(t)\u0011\n(35)\nand\n˙\u001aab(t)=˙\u001a\u0003\nba(t)=\u0000\u001aab(t)\nT2+i\n~\u0001\u001aab(t)\u0000i\n~(\u001aaa(t)\u0000\u001abb(t))Vab\n1(t); (36)\nwhereVba\n1(t)=hbjV1(t)jaiand\u0001 = Eb\u0000Ea\u00150. Equations (35) and (36) are the so called\nBloch equations. Here we assume that the time-dependent driving field has the periodic form\nVba\n1(t)=~\ncos!t, where \nis the Rabi frequency .\nThe Bloch equations in the context of electron /nuclear spin resonance are usually written in\na slightly di \u000berent way. In order to transform these equations into the most common form, we\nfirst introduce the e \u000bective average spin operators\nhS+(t)i=e\u0000i!t\u001aba(t)\nhS\u0000(t)i=ei!t\u001aab(t)\nhSz(t)i=1\n2(\u001aaa(t)\u0000\u001abb(t)); (37)\nand, as for the usual angular momentum operators, the xandy-componentshSxi=(hS+i+hS\u0000i)=2\nandhSyi=(hS+i\u0000hS\u0000i)=(2i). With the definition (37), one gets that the transversal terms hSx;yi\nwill contain two types of oscillations, provided j~!\u0000\u0001j=\u0001\u001c1: fast oscillating terms, of the form\ne\u0006i(!+\u0001=~)t, and slow oscillations of the form e\u0006i(!\u0000\u0001=~)t. Under the rotating-wave approximation\ndescribed in Ref. [123], the fast rotating terms are neglected. This is equivalent to describe\nthe evolution of this e \u000bective magnetic moment in a rotating frame such that the magnetization\nvector~M(t)\u0011(hSx(t)i;hSy(t)i;hSz(t)i) precesses at the e \u000bective Larmor frequency \u0001=~:\n8>>>><>>>>:h˙Sxi=\u000ehSyi\u00001\nT2hSxi;\nh˙Syi=\u0000\u000ehSxi+ \nhSzi\u00001\nT2hSyi;\nh˙Szi=\u0000\nhSyi\u00001\nT1(hSzi\u0000hS zi0)(38)\n17where\u000e= \u0001=~\u0000!is the detuning between the TLS splitting and the driving frequency and hSzi0\nis defined as in Eq. (37) with the initial occupations. Notice that to avoid unnecessary complica-\ntions in the notation, we have just dropped the argument ( t) to indicates that these quantities are\ngiven in the rotating frame .\nEquations (38) are formally identical to the phenomenological Bloch Equations describing\nthe dynamics of a macroscopic magnetic moment ~M(t). Thus, these equations can be used to\ndescribe the dynamics of a localized quantum spin (or an array of localized spins behaving as a\nTLS) driven by a classical radiofrequency field of frequency !and Rabi frequency \ncoupled to\na bath, which induces relaxation on a time scale T1and decoherence on T2.\nThe system of di \u000berential equations (38) admits a steady state solution of the form (see\nAppendix D for the details):\nhSzi\u0000hS zi0\nhSzi0=\u0000T1T2\n2\n1+T2\n2\u000e2+T1T2\n2(39)\nhSxi\nhSzi0=\u0000T2\u000ehSyi\nhSzi0=T2\n2\u000e\n1+T2\n2\u000e2+T1T2\n2: (40)\n2.4. Decoherence as a limit for spectral resolution in magnetic resonance\nEquation (39) provides the basis for magnetic resonance experiments. The deviation from the\nequilibrium of the longitudinal magnetization can be controlled changing the detuning \u000ebetween\nthe precession frequency \u0001=~and the frequency !of the driving field. When sweeping over \u000e,\nEq. (39) describes a resonance curve whose maximum is given by\n T1T2\n2\n1+T1T2\n2!\n: (41)\nNotice that the maximal deviation scales linearly with \n2when the dimensionless parameter\nx=T1T2\n2is small. The full width at half maximum is then given by:\n\u000e2\nFWHM =1\nT2\n2\u0010\n1+T1T2\n2\u0011\n: (42)\nIn the limit when the driving amplitude \nis small enough, or more precisely, when x\u001c1,\nthe width of the resonance is 1 =T2. In the opposite limit, the full width at half maximum is\ngiven by \npT1=T2. Thus, in both cases increasing T2results in a reduction of the FWHM of\nthe resonance curve. As we discuss in Sec. 6, experiments of spin resonance on an individual\nmagnetic atom using STM have been recently reported [59, 107], demonstrating the possibilities\nof single spin as a magnetometer. The accuracy of this quantum sensor is determined by the\nFWMH and thereby, is ultimately limited by the decoherence rate.\n2.5. The quantum to classical transition and the spin-boson model\nThe BR theory shows how a quantum state that is a linear superposition of two eigenstates\nwith di \u000berent energies decays with a characteristic time scale T2due to the interaction with the\nenvironment. We now discuss another important aspect of decoherence. Take a quantum system\nwhose ground state is a linear combination of two eigenstates of an operator ˆA. For instance, ˆA\ncould be a projection of the spin operator, ˆSz, or a pseudospin operator ˆ \u001czin a two level system,\n18such a diatomic molecule or a double quantum well. Quantum mechanically, the eigenstates of\nthe Hamiltonian can perfectly be linear superpositions of states with di \u000berent eigenstates of the\noperator ˆA. To be more specific, let us examine a relevant and simple example provided by an\nanisotropic integer spin system, governed by the Hamiltonian\nH=DS2\nz+E(S2\nx\u0000S2\ny); (43)\nwith a strong easy axis anisotropy ( \u0000D\u001dE>0). We discuss the simplest non-trivial case,\nS=1, which describes, for instance, iron phthalocyanine deposited on an oxidized copper\nsurface [37]. Thus, the ground state doublet (for E=0) is formed by the eigenstates of ˆSzwith\nSz=\u00061. As the in-plane anisotropy term is turned on, the degeneracy of the doublet is lifted,\nresulting in the so called quantum spin tunneling splitting [19] \u0001QST=E2\u0000E1. Importantly, the\nresulting eigenvectors are linear combinations of the states with Sz=\u00061:\njQ1i=1p\n2(j+1i+j\u00001i)\njQ2i=1p\n2(j+1i\u0000j\u0000 1i): (44)\nHence, the eigenstates of the Hamiltonian have a built-in coherence between the states with well\ndefined Sz. In the weak-coupling BR theory, when dissipation is included, the steady state density\nmatrix for this system would be given by ˆ \u001a=P\nnPEq:\nnjQnihQnj, where PEq:\nnare the Boltzmann\nfactors. When kBT\u001cjDj, the energy gap that separates the Sz=\u00061 doublet from the Sz=0\nstate, this equilibrium density matrix can be expressed in the fj+1i;j\u00001igbasis as\n\u001aSz=1\n21+1\n2Tanh \u0001QST\n2kBT! 0 1\n1 0!\n: (45)\nThus, in thermal equilibrium the DM conserves a finite coherence of order Tanh( \u0001QST=2kBT)\nwhen expressed in the basis set of eigenstates of ˆSz. Interestingly, the density matrix (45) also\ndescribes integer spins characterized by Hamiltonian (43) with arbitrarily large S. This im-\nplies having coherence between states with opposite (and arbitrarily large) magnetic moment,\ndefinitely at odds with classical systems. Of course, the QST splitting rapidly decreases as S\nincreases, reducing the temperature range for which quantum coherence is predicted. In partic-\nular, for the Hamiltonian (43), it is basically given by \u0001QST/E(E=D)S\u00001. Thus, if we take\nfor instance the Fe 8molecular magnet, where S=10,D=\u00000:295 K and E=jDj \u00190:19, a\n\u0001QS T\u00187 nK has been predicted [131], a temperature very di \u000ecult to reach experimentally.\nHence, in the weak coupling approach, the coupling to the reservoir seems to be ine \u000bective to\ndestroy coherence in the Szbasis, as long as \u0001QST=2kBTis not too small.\nGiven the perturbative nature of the result (45), it seems pertinent to ask ourself what happens\nwhen the strength of the coupling to the environment increases and, in particular, if there another\nmechanism quenching the coherences not accounted for in (45). For that matter, we consider\nthe coupling of a TLS, described by H=\u00010\n2ˆ\u001cxin the Sz-basis, to a bosonic environment that is\nsensitive to the value of Sz:\nHSB=\u00010\n2ˆ\u001cx+ˆ\u001czp\u000bX\n0<k<kcgk\u0010\nby\nk+bk\u0011\n+X\n0<k<kc~vFkby\nkbk; (46)\nwhere vFis the Fermi velocity and gk=~vF(\u0019k=L)1=2(Lis a quantization box length). This\nis the well known spin-boson model, proposed to study the competition between the quantum\n19tunnelling, driven by the\u00010\n2ˆ\u001cxterm, and the coupling to the reservoir, that is sensitive which state\nˆ\u001czthe system is at. For \u00010=0 the model can be solved exactly [11], and it predicts a vanishing\ncoherence even at T=0, in full agreement with the argument of Sec. 2.1.3. For a finite value of\n\u00010, the spin-boson model can be solved exactly in some specific limits. A central result of this\nmodel is that, due to the coupling to the environment, the QST splitting becomes renormalized,\naccording to the following equation [11]:\n\u0001\n\u00010\u0019(1\u0000\u0002(\u000b)) \u00010\n~!c!\u000b\n1\u0000\u000b\n; (47)\nwhere \u0002is the step function, and !cis a cut-o \u000bfrequency. For \u000b>1 the QST splitting vanishes\nstrictly, which removes the protection of coherence discussed above. For \u000b < 1 the QST is\nseverely renormalized except for very small \u000b. Interestingly, at weak coupling ( \u000b\u001c1) the\nrenormalization of the QST can also be captured by the BR theory [76].\nThe results of the spin-boson model acquires special relevance in the context of nanomagnets\ncoupled to the itinerant electrons through a Kondo exchange interaction, a system that in many\ninstances can be mapped to the spin-boson model described above [17, 132]. In any event, the\ncoupling to the environment destroys the spin coherence between states with opposite Szby\nrenormalizing the QST splitting.\n2.6. Other approaches to the relaxation and decoherence of spins\nIn this section we have presented a general overview of the decoherence problem from both\na formal point of view, where decoherence and relaxation is seen basically as the e \u000bect of the\ncoupling to the environment, an a practical point of view, with two methodologies to analyze it. In\naddition, in Sec. 4 we shall further particularized to the problem of Kondo induced decoherence\nin spin systems. However, there are many other possible approaches to treat the problem of spin\ndecoherence. Within the linear response theory, one can study the dynamical e \u000bects by looking\nat the behavior of the dynamical susceptibility. By using the Kubo formula [133], one obtains\nthe (complex) frequency dependent magnetic susceptibility, which can be related to the retarded\ncorrelation function. The first derivative of its imaginary part with respect to the frequency\nis related to dissipation (relaxation and decoherence), while a first derivative of the real part\ncorresponds to the shifts of the energy levels, the e \u000bects already derived from the Bloch-Redfield\ntheory 2.2. First frequency derivative of the dynamic susceptibility has a frequency-dependence\nsimilar to the susceptibility and can be written as the product of two Green functions, making a\nseparation of the shifts /dissipation possible for small couplings.\nIn addition to the particular treatment of the dissipative dynamics, there is a second implicit\nassumption in our whole treatment: the quantum system can be treated as a spin system, i.e,\nfluctuations of other degrees of freedom, such as charge, can be neglected. This does not need\nto be the case in the strong fluctuation regime where the local moments are fully suppressed\nor in the intermediate Hund’s impurity regime [134], where charge fluctuations occurs as a re-\nsult of the strong hybridization, but the local magnetic moment still survives. Furthermore,\nwhen the structural changes of the spin array modifies the local magnetic anisotropy of each\nspin due to the surface rearrangements [135], a more complete description including the orbital\ndegrees of freedom may be required, as it happens in the case of Co chains on Cu 2N/Cu(100)\nsurface [61]. Another examples where the spin description may fail are magnetic molecules on\nsurfaces where conformational changes may occur either induced by charging e \u000bects or by the\nfield of the tip [136, 137]. This is likely to occur when transport takes place close to a resonance,\n20i.e, the addition (or removal) energy of an electron of the molecule is close to the chemical\npotential of the electrodes [34, 39, 101].\nIn these cases, a more complete description of the electronic transport is required, taking into\naccount not only the spin degrees of freedom but also the charge. One possible treatment of this\nproblem is through a generalized multiorbital Anderson model. Under this description, the cor-\nrelation between electrons in the magnetic impurities are fully taken into account. For instance,\nin the case of transition metal adatoms, the electrons in the d-levels of the adatoms are treated\nas a many body system, with the e \u000bects of the surroundings already accounted for in the crystal\nand ligand fields [138, 139]. Itinerant electrons can thus hop in and out of these orbitals, and\ntheir dynamical e \u000bects may be treated within the Green function formalism [140]. Within this\nformalism, the e \u000bect of the electronic reservoirs is summarized in two central quantities, the on-\nsite energy levels of the d-electrons, and the hybridization function. In fact, these two quantities\ncan be estimated from ab-initio calculations [141, 142]. Of course, depending on the regime of\nparameters, one can analyze the properties of this multiorbital Anderson problem using di \u000berent\napproaches: from the perturbation method in Ref. [101] to more refined numerical techniques\nlike the non-crossing [141] and one-crossing approximations [52], or numerical renormalization\ngroup analysis [143] among others.\n3. Spin Hamiltonian for magnetic adatoms\nIn this section we review the Hamiltonian describing magnetic atoms on a surface, including\nboth the part that, within the open quantum system picture, we consider the “system”, as well as\nits coupling to other degrees of freedom, the “baths”. This includes single spin Hamiltonian and\nthe spin-spin interactions, relevant for engineered nanostructures such as spin chains.\n3.1. Single spin Hamiltonian\nThe low energy physics of individual quantum spins, such as magnetic atoms and molecules,\ncan be very often described with an e \u000bective single spin Hamiltonian that describes the magnetic\nanisotropy and the Zeeman coupling within the subspace of the ground state spin S. The ground\nstate multiplet of open-shell isolated atoms has, ignoring spin-orbit coupling (in the range of\n50 meV for 3 dtransition metals), (2 L+1)(2S+1) states. When the coupling to the surface\nis included, most often the orbital momentum is quenched, so that the low energy manifold\nhas only 2 S+1 levels. In that case, the single ion Hamiltonian can be written as an even\nfunction of the spin operators ˆSathat describes the magnetic anisotropy preserving time-reversal\nsymmetry, plus the Zeeman interaction with the external applied field. Quite often, this e \u000bective\nspin Hamiltonian is written in terms of a complete set of high-order spin operators\nHS=X\nk=2;4;6kX\nq=\u0000kBq\nkˆOq\nk(S); (48)\nwhere ˆOq\nk(S)are the (tesseral tensor) Stevens operators [81] and Bq\nkare real coe \u000ecients. The\nHamiltonian coe \u000ecients Bq\nkare determined by the symmetry of the the spin system, the crystal\nfield, hybridization and, importantly, the spin-orbit interaction. Several groups have tried to de-\ntermine the anisotropy parameters from first principles calculations, although in general this is a\nhard problem. This technique has been applied to transition metal atoms adatoms [139, 143–145]\n21or even rare earth like Ho on Pt(111) [116]. Multiplet calculations with rescaled Coulomb inte-\ngrals where the atomic spin-orbit interaction is used [146, 147] have also been used to compute\nBq\nk. This method requires to calculate spatial averages hrniover the atomic wavefunctions, but\nmost frequently they are taken as fitting parameters [59, 77, 115].\nA second approach to determine the single spin anisotropy coe \u000ecients consist on fitting the\nIETS spectra [31, 33, 52, 53, 65] For this matter, it is important to account for the adatom crystal\nfield symmetry. For instance, for magnetic adatoms, the dominant (quadratic) uniaxial term is\nusually written as DˆS2\nz(D=3B0\n2), while the quadratic in-plain anisotropy, typical of adsorption\nsites with symmetry C2v, is written as E\u0010ˆS2\nx\u0000ˆS2\ny\u0011\n, as in Eq. (43), where E=B2\n2. This is the\ncase of the transition metal atoms studied on Cu 2N/Cu(100) [78], MgO /Ag(100) [55, 59, 77], or\nh-BN /Rh(111) [63]. Whereas in some cases the z-axis is the o \u000b-plane direction, in others the z-\naxis may lie in the surface plane [31]. Another example of interest in the field of magnetic atoms\non surfaces is the adsorption on sites with C3vsymmetry, such as the Pt(111) surface [68, 116],\nwhere the lowest order transversal term takes the form B3\n4ˆO3\n4=B3\n4=2n\nSz;S3\n++S3\n\u0000o\n, i.e., it mixes\nstates with spin projection Szdi\u000bering in three units [116]. Higher order term may also appear\nin molecular magnets due to their high intrinsic spins [148].\nIn some high symmetry instances orbital momentum is not fully quenched. This is the case\nof Fe on top of MgO. In this unquenched case, the low energy levels are described by [59, 77]:\nHS=DˆL2\nz+F0(ˆL4\nx+ˆL4\ny)+\u0015~L\u0001~S+\u0016B~B\u0001\u0010~L+g~S\u0011\n(49)\nwhere L=2 and S=2. The most important consequence of this finite orbital momentum is\nthat the spin-orbit coupling induces a splitting of the energy levels linear in \u0015, resulting in an\nenhanced magnetic anisotropy. Thus, for the Fe /MgO, the low energy spectrum of this e \u000bective\nHamiltonian is a two level system with approximate quantum numbers Sz\u00192Lz\u0019\u00062. This\ndoublet lies 14 meV below the next excited state, so that for many practical purposes, the system\nbehaves like a two level system.\n3.1.1. Integer vs half integer spins\nThe behavior of integer and half-integer spins, described with the same Hamiltonian, can\nbe radically di \u000berent. According to Kramers’ theorem, half-integer spins have at least doubly\ndegenerate spectrum, on account of time reversal symmetry, whereas integer spins can be non-\ndegenerate at B=0. In the case of strong uniaxial anisotropy, where the dominant term in\nthe Hamiltonian is HS=DˆS2\nz, with D<0, the ground state doublet is made of the two states\nwith Sz=\u0006S, see Fig. 3. The e \u000bect of the the remaining terms in the Hamiltonian depends\ndramatically on the parity of 2 S+1. To be specific we discuss here the case of Hamiltonian (43),\nbut the discussion applies to general spin Hamiltonians as well [149].\nIn the case of half-integer spins, as well as some integer-spins where there are symmetry\nprotected degeneracies [115, 149, 150], there is a strict degeneracy of the ground state doublet at\nzero magnetic field [151], and we can always choose the eigenvectors of HSsuch as the uniaxial\nterm DˆS2\nzis diagonal in the E=0 limit:\njC1i / j\u0000 Si+O\u0010\n\"S\u00001\u0011\nj\u000e\u001e1i\njC2i / j +Si+O\u0010\n\"S\u00001\u0011\nj\u000e\u001e2i; (50)\nwhereO\u0010\n\"S\u00001\u0011\nj\u000e\u001eiis the contribution to the wave function coming from states with smaller jSzj\nand it is controlled by \", a small number that in the case of the Hamiltonian (43), is given by\n22E \na) \n….. ….. \nb) \nc) \n- \n+ \nFigure 3: E \u000bects of the magnetic anisotropy on quantum magnets. (a) Energy levels scheme of a nanomagnet with an\nuniaxial, easy axis ( D<0 ) anisotropy. The two degenerate ground states correspond to the classically oriented states\nalong the easy axis. (b) E \u000bect of the transversal Eterms (/(S2\nx\u0000S2\ny) ) on an exemplifying S=5=2 half-integer spin,\ninvolving mixing between states represented with the same color. (c) Idem to (b) but for an integer spin ( S=1 in the\nexample displayed). The new mixed states are bonding and antibonding linear combinations of the classical states, split\nin energy by \u0001QS T.\n\"=E=jDj.5Since this type of situation is compatible with the conventional classical picture\nof a magnet with two equivalent ground states with a finite and opposite average magnetization\nMz'hC1jSzjC1i=\u0000hC2jSzjC2i, we refer to these spins as type C orclassical type of spins . We\nstress, however, that since these two wave functions correspond to degenerate states, any linear\ncombination ofjC1iandjC2iwill also be a valid choice to describe the ground state doublet.\nHowever, the coupling to the environment will select the jC1iandjC2istates as the most stable,\na process discussed in Sec. 4.3.4.\nBy contrast, the zero-field spectrum of integer spins with finite transversal anisotropy Eis\nnon-degenerate due to quantum spin tunneling, so that E2=E1+\u0001, where \u0001/E\"S\u00001in the case\nof Hamiltonian (43). It is then apparent that, as Sincreases, \u0001decreases. As long as \u0001,0, the\nwave functions of the doublet are uniquely defined, modulus a global phase, and approximately\ngiven by [19]:\njQ1i / j Si+(\u00001)S\u00001j\u0000Si+O\u0010\n\"S\u00001\u0011\nj\u000e\u001ei\njQ2i / j Si+(\u00001)Sj\u0000Si+O\u0010\n\"S\u00001\u0011\nj\u000e\u001ei; (51)\nwhere the last term is a small contribution coming from states with jSzj<S. Thus, the wave\nfunctions are approximately given by the bonding and antibonding combination of the states with\nSz=\u0006S, while the bonding-antibonding nature of the ground state wave function alternates as S\nincreases. We refer to these states as type Q orquantum type of states because they are quantum\nsuperpositions of the two classical ground states, and have thereby very di \u000berent properties, such\nas the vanishing expectation values of the magnetic moment [17].\n5If 3E>Dwe relabel the Hamiltonian, so that the new easy axis is again z.\n233.2. Pseudo-spin 1=2approximation\nIn those instances where there is a gap between the ground state doublet and the rest of the\nenergy levels, one can truncate the Hilbert space and treat the anisotropic spins as two level\nsystems. This occurs for the Hamiltonian (43) when \u0000D\u001dE;kBT. This condition holds, for\ninstance, in the case of Fe or Co on Cu 2N [31, 33].6When projected over this subspace, the spin\nrotational invariance inherent to the Heisenberg coupling between magnetic adatoms, ~S(l)\u0001~S(l0),\nor the Kondo interaction with the surface electron spin density, ~S(l)\u0001~s(~rl), is broken on account\nof the single ion magnetic anisotropy. To see this, we represent the spin operators ˆSa(l) in the\nsubspace of dimension 2 of the ground state doublet. The resulting matrices of dimension 2 can\nbe written down as linear combinations of Pauli matrices ˆ \u001caacting on the space of the TLS:\nˆSa(l)=X\nb=x;y;z\u0014ab(l)ˆ\u001cb: (52)\nSpecifically, we write this up for the cases of a single spin described by Hamiltonian (43). The\nexpressions are then quite di \u000berent for the half-integer and integer spin. In the case of half-\ninteger, using the wave functions (50), we have \u0014ab/\u000eab, with\n\u0014zz\u0019S+O(\"2S\u00001)\n\u0014xx;\u0014yy\u0018S\"2S\u00001: (53)\nThus, in the limit of pure uniaxial anisotropy with S>1=2, the only operator with non-zero\nmatrix elements in the ground state manifold is the ˆSzoperator, and the Heisenberg coupling\n~S\u0001~s(0) takes the form of an Ising interaction ˆSzsz(0)'Sˆ\u001czsz(0). Another interesting case occurs\nfor half-integer spins where D>0, such that the ground state doublet has Sz=\u00061=2. This is\nrelevant for instance for Cobalt on Cu 2N (S=3=2) [152]. In this case the operators ˆSxand ˆSy\nhave non-vanishing matrix elements within the Sz=\u00061=2 doublet, but their strength is larger\nthan in the ˆSzterm. For S=3=2, the representation of the ˆSxmatrix in the subspace of the\nSz=\u00061=2 doublet give matrix elements twice as large as those obtained for a real S=1=2 spin.\nIn contrast, the representation of the ˆSzoperator gives the same matrix elements in both cases.\nThe resulting exchange interactions in the S=3=2 case are thereby anisotropic.\nWe now consider the representation of the spin operator in the 2 \u00022 space within the Q-basis\n(51), where [17]\n~S\u0011ˆzhQ1jˆSzjQ2i 0 1\n1 0!\n: (54)\nIn other words, the atomic spin-flip operators ˆSxand ˆSyhave no e \u000bect in the sub-space of states\njQ1iandjQ2i, and only the ˆSzoperator has a non-vanishing contribution whose e \u000bect is to induce\ntransitions among these states. It is important to note that for systems with a degenerate ground\nstate, it is a matter of choice whether we use the Qor the Cbasis.\nThe important take home message here is that the conventional spin-isotropic interactions,\nworking in the subspace of low energy selected by the large magnetic anisotropy, result in\nanisotropic e \u000bective spin Hamiltonians. This is of course a resource that might be used to do\nquantum simulations [152].\n6The 2-level approximation will be valid to describe the spin dynamics as long as excitation of higher energy states,\neither by thermal fluctuations or by scattering with transport electrons under a finite bias voltage, are negligible.\n243.3. Hamiltonian for multi-spin systems\nOne of the most appealing features of surface magnetism is the possibility to assemble mag-\nnetic structures, adding atoms one by one. In addition to the single spin anisotropy, these atoms\ninteract with each other through di \u000berent kind of sin-spin interactions. A broad class of magnetic\natoms on surfaces can be described with the spin Hamiltonian\nHS=X\nl\u0014NH0(l)+NX\nl,l0X\na;b=x;y;zJa;b(l;l0)Sa(l)Sb(l0) (55)\nwhere the first term describes the single ion magnetic anisotropy, given by Eq (48), while the\nsecond one represents a spin-spin exchange interaction. Several atomically engineered nanos-\ntructures, such as spin chains of Fe and Mn on Cu 2N [30, 38, 48, 53, 56, 67, 69, 152], can\nbe modelled assuming that spin-spin interaction is rotational invariant (Heisenberg type, Ja;b=\nJH\u000ea;b) and neighbor isotropic, either antiferromagnetic (AF) [30, 48, 152] or ferromagnetic\n(FM) [53, 56], while the second neighbor interaction is either negligible, or much smaller [80,\n153]. Dzyaloshinskii-Moriya (DM) interaction coupling, where the inter-spin interaction takes\nthe formP\nabJabSa(l)Sb(l0)\u0011~Dll0\u0001\u0010~S(l)\u0002~S(l0)\u0011\n, with~Dll0a vector whose orientation is given\nby some high-symmetry direction of the surface, is known to be relevant for magnetic atoms on\ntop of heavy metal surfaces, such as Pt [154] and Ir [155], resulting in non-collinear broken sym-\nmetry states, such as spin spirals for Fe chains on Ir [155] or Mn on W [40, 156], and skyrmions\nfor monolayers of Fe on Ir [157, 158]. Both indirect coupling mediated by the surface elec-\ntrons and super-exchange mediated by non-magnetic surface atoms are believed to contribute to\nthese interactions [159]. In the following we only consider first neighbor Heisenberg interactions\n~S(l)\u0001~S(l\u00061), the dominant coupling between magnetic atoms adsorbed on a decoupling layer\nsuch as Cu 2N, MgO or AlO.\nDepending on the competition between the single ion anisotropy and the Heisenberg interac-\ntion, the geometry of the structure (chain, ladder, cluster), and the spin parity, the ground state\nof the system can have very di \u000berent nature. From the experimental information, obtained using\nboth IETS and time resolved magnetization switching using spin polarized STM, together with\nthe information from the model Hamiltonian, one can infer the di \u000berent types of ground states.\nEven if we limit the discussion to chains and ladders formed with transition metals on Cu 2N, the\nfollowing ground states have been reported:\n1. Antiferromagnetically correlated spin ground state, without broken symmetries, for Mn\nchains ( S=5=2) with up to N=10 atoms [30]. In the case of even (odd) the spin of the\nground state is SG=0 (SG=5=2). The exchange interaction for the Mn dimer is JH'6\nmeV , much larger than the single ion anisotropy D\u0019\u00000:04 meV .\n2. Broken symmetry AF ground states (Ising type classical N ´eel states), for Fe chains along\ntheNrich direction, and N\u00153 [48]. For N=2 no signal of broken symmetry is observed\n[41].\n3. Broken symmetry FM ground states (Ising type), for Fe chains along the Cu rich direction,\nandN\u00153 [56].\n4. Distributed Kondo e \u000bect in hybrid spin Fe-Mn Nspin chain [62]. The Fe-Mn dimer ( N=1)\nhave a strong AF interaction that leads to a ground state spin SG=1=2, that results in a\nzero bias Kondo peak. As Nincreases, staying odd, the Kondo e \u000bect is preserved, but it is\nmore prominent in the side of the structure opposite to the Fe-Mn, reflecting the non-local\nand rather counterintuitive nature of the ground state of the chain.\n255. Correlated pseudo-spin 1 =2 XXZ model [152]. A chain of Cobalt atoms with S=3=2 and\nstrong uniaxial anisotropy that favors the Sz=\u00061=2 doublet realizes the XXZ model.\nThis list illustrates that the competition between uniaxial anisotropy, exchange interaction\nand Kondo coupling can result in a large variety of possible ground states. This can be un-\nderstood, in part, when the anisotropy is large enough to permit a truncation of the single spin\nHamiltonian, keeping only two levels. This approximation leads then to anisotropic pseudo-spin\n1=2 interactions. For instance, for an integer spin chain governed by the single ion anisotropy\nHamiltonian (43) and Heisenberg exchange JH, and provided D<0 withjDj\u001dE;JH, we can\nretain the doublet with maximal Szat each atomic spin. The resulting e \u000bective Hamiltonian\nreads [17]:\nHS\u0019X\nn\u0001\n2ˆ\u001cz(n)+jX\nnˆ\u001cz(n)ˆ\u001cz(n+1) (56)\nwhere \u0001is the quantum spin tunneling splitting of an individual adatom and j\u0011JHjhG1jˆSz(l)jG2ij2.\nOf course, Hamiltonian (56) is nothing but the quantum Ising model in a transverse field (QIMTF).\nThis is approximately the case of the FM [56] and AFM [48] Fe chains on Cu 2N/Cu(100). In\ncontrast, in the case of D\u001dE;kBT0, the two level truncation that keeps only the Sz=\u00061=2\ndoublet leads to the XXZ model [152]. We thus see that atomically engineered magnetic struc-\ntures with atomic spins with S>1=2 can be used to realize e \u000bective spin 1 =2 models and might\nbe used for quantum simulation [152].\n4. Decoherence due to Kondo exchange\nIn this section we treat the problem of decoherence of magnetic atoms on a surface due to the\nKondo exchange interaction with the itinerant electrons of the underlying conductor. We treat in\ndetail the case of an individual magnetic atom and, later on, we briefly address the problem of\nspin decoherence of finite size chains of exchanged-coupled magnetic atoms.\nWe assume that at t=0 some quantum demon has been able to prepare the system in a\nlinear superposition of two states that have very di \u000berent magnetic properties. In the case of an\nindividual quantum spin, the wave function would be:\nj (t=0)i=1p\n2(jSz= +Si+jSz=\u0000Si): (57)\nOur goal is to determine for how long this coherent superposition can be maintained, taking into\naccount that the spin is exchange coupled to the electron gas of the surface. This characteristic\ndecoherence time will be denoted as T2.\n4.1. Kondo exchange interaction\nThe Kondo exchange interaction with the surface electrons can be written down as:\nVK=NX\nlJl~Sl\u0001~s(~rl); (58)\nwhere~Slis the spin of the l-magnetic adatom and ~s(~rl) is the surface spin density evaluated at\nthe position ~rlof the lmagnetic atom,\n~s(~rl)=X\n~k~k0\u001b\u001b0ei\u0010~k\u0000~k0\u0011\n\u0001~rl~\u001c\u001b\u001b0\n2cy\n~k;\u001bc~k0\u001b0; (59)\n26where cy\n~k;\u001bindicates the creation operator of a conduction electron with momentum ~k, and spin\nprojection\u001balong the quantization axis z. In the case of a single magnetic atom, we always\nchoose its location at the origin, so that the phase factor in the Hamiltonian goes away. For many\natoms the phase factor can not be gauged away and it can play an important role. We assume that\nthe conduction electrons can be described within the independent particle picture:\nHsurface =X\n~k;\u001b\u000fkcy\n~k;\u001bc~k\u001b(60)\nFor simplicity, in this work we limit the discussion to the case of a non-polarized electrodes. A\nsimilar discussion could be carried out for the coupling to a spin-polarized bath [84]. Most often,\nthe exchange constant Jl\u0011Jis the same for all the spins, which is a quite reasonable assumption\nfor magnetic adatoms adsorbed on equivalent lattice sites on the surface.\n4.1.1. Kondo Hamiltonian in the two-level approximation\nAs we discussed in Sec. 3.2, in many instances we can restrict the (2 S+1) Hilbert space of\nmagnetic adatoms down to just two levels. Using equations (52) and (58), we can write down\nthe Kondo exchange Hamiltonian, projected into the subspace of the ground state doublet, as:\nVK=JX\nl\u0014NX\na;b\u0014ab(l)ˆ\u001cbsa(~rl): (61)\nwhereP\nb\u0014ab(l)ˆ\u001cbis the representation of the atomic spin operator ˆSa(l) in the basis set of the\ntwo levels. In the case of degenerate two levels, this representation is not unique. If we choose\nthe basis set as to diagonalize the ˆSzoperator, for a single spin that we place at the origin of\ncoordinates ( ~r1=0), the representation of the Kondo coupling in the truncated basis takes the\nform\nVC\nK=jzˆ\u001czsz(0)+jxˆ\u001cxsx(0)+jyˆ\u001cysy(0); (62)\nwhere jz'JS\u001djx;jy. Thus, the single ion anisotropy results in an e \u000bective anisotropic\nexchange. In the case where the eigenstates of the TLS are also eigenstates of ˆSz, withjSzj>1=2,\nwe have jx=jy=0 and the coupling anisotropy is maximal as the truncated Kondo Hamiltonian\ntakes the form of an Ising-Kondo model [160], VC\nK=jzˆ\u001czsz(0). As a consequence, population\nscattering is strictly forbidden between the states with di \u000berent atomic Sz, yet the coupling to the\nenvironment is able to induce decoherence, as we describe in Sec. 2.1.3.\nAnother interesting case is provided by a TLS whose wave functions are given by states as\nthose in Eq. (51). This situation arises naturally for integer spins described with the single ion\nHamiltonian (43), with \u0000D\u001djEj>0. In that case we have ~S=hQ1jSzjQ2i(0;0;ˆ\u001cx) [17], which\nleads to the single impurity Kondo Hamiltonian\nVQ\nK=jzˆ\u001cxsz(0) (63)\nwhere jz=JhQ1jSzjQ2i. In other words, the atomic spin-flip operators ˆSxand ˆSyhave no e \u000bect\nin the sub-space of states jQ1iandjQ2i, and only the ˆSzoperator has a non-vanishing contribution\nwhose e \u000bect is to induce transitions among these states [17].\n274.2. General expressions for T 1and T 2due to Kondo exchange\nThe general expression for a T1like transition rate between two eigenstates of a spin chain\ndue to Kondo coupling to the reservoir is given by [97]:\n\u0000MM0=\u0019J2\n2~2X\n~k;~k0f(\u000fk)(1\u0000f(\u000fk0))\u0002\u001fM;M0(~k\u0000~k0)\u000e(\u000fk+EM\u0000\u000fk0\u0000EM0); (64)\nwhere\n\u001fM;M0(~k\u0000~k0)\u00112NX\nl;l0=1ei(~k\u0000~k0)\u0001(~rl\u0000~rl0)X\naSa\nMM0(l)Sa\nM0M(l0)=X\na\f\f\f\fSa\nM;M0(~k\u0000~k0)\f\f\f\f2\n(65)\nwhereSa\nM;M0(~k\u0000~k0)=P\nlei(~k\u0000~k0)\u0001~rlSa\nMM0(l) and Sa\nMM0(l)\u0011hMjˆSa(l)jM0iwith a=x;y;z. The\ninterpretation of Eq. (65) is quite transparent. The rate contains the f(1\u0000f) factor that weights the\noccupation of the initial quasiparticle state and the availability of the final quasiparticle state. The\n\u000efunction ensures the overall conservation of energy, while energy exchange between system\nand reservoir is allowed. The spin structure function, \u001fM;M0(~k\u0000~k0) accounts both for the spin\nconservation and for the non-local couplings that naturally arise when the sum over all the atoms\nin the structure is squared in order to obtain a scattering rate.\nFor the decoherence rates, we obtain two contributions, very much like in the single spin\ncase. The first comes from T1-like population scattering processes [123]:\n\rnonad:\nM;M0=1\n20BBBBB@X\nN,M\u0000M;N+X\nN,M0\u0000M0;N1CCCCCA; (66)\nwhere \u0000M;M0are the scattering rates defined in Eq. (64). The adiabatic contribution corresponds\nto processes that occur even in the absence of changes in populations of the jMistates. It is\ndriven by elastic scattering processes with the reservoir and it is often known as pure dephasing.\nIn our case, the adiabatic decoherence rate is given by:\n\rad:\nM;M0=\u0019J2\n2~X\n~k;~k0f(\u000fk)(1\u0000f(\u000fk0))\u001fad:\nM;M0(~k\u0000~k0)\u000e(\u000fk\u0000\u000fk0): (67)\nThe matrix elements \u001fad:\nM;M0(~q) are given by (see Appendix A.2 for details):\n\u001fad:\nM;M0(~q)=X\na\f\f\f\f\f\f\fX\nl\u0010\nei~q\u0001~rlSa\nMM(l)\u0000e\u0000i~q\u0001~rlSa\nM0M0(l)\u0011\f\f\f\f\f\f\f2\n=X\na\f\f\fSa\nM;M(~q)\u0000Sa\nM0;M0(\u0000~q)\f\f\f2: (68)\n4.3. Decoherence of a single degenerate spin\n4.3.1. Decoherence of a single spin with degenerate spectrum: perturbative results\nWe now consider the simplest case of an individual magnetic atom with spin Sand uniaxial\nanisotropy,HS=\u0000jDjˆS2\nz, which leads to a degenerate ground state. We choose as a basis set the\neigenstates of ˆSz, dubbed as classical, see Eq. (50).\nWithin the perturbative BR theory, the decoherence rate has two types of contributions, adi-\nabatic and nonadiabatic. The latter are due to T1-like process that change the population of the\n28two statesj+Siandj\u0000Si, see Eq. (66). The Kondo interaction can only induce transitions with\n\u0001Sz=\u00061. Therefore, for S>1\n2, the nonadiabatic channel implies inelastic transitions from the\nstates of the ground state doublet to an excited state, separated in energy by \u0001 =jDj(2S\u00001).\nEnergy conservation implies that a thermally excited electron-hole pair across the Fermi energy\nhas to be annihilated . The scattering rate for these T1-like processes excitation of the spin is:\n1\nT1=\u0019\n2~(\u001aJ)2S\u0001\ne\f\u0001\u00001(69)\nwhere\u001ais the density of states at the Fermi level of the surface electrons. As the temperature goes\nto zero, the density of thermally excited electron-hole pairs vanish exponentially, and excitation\nrate 1=T1/e\u0000\f\u0001vanishes altogether.\nThe relaxation rate of the Sz=S\u00001 excited state (or S z=\u0000S+1) to the ground state by\nspontaneous emission of an electron-hole pair across the Fermi energy can be obtained from Eq.\n(69) by reversing the sign of \u0001. In the low temperature limit ( \f\u0001\u001d1), this relaxation rate scales\nlike1\nT1\f\f\f\frelax\u0011\rad:'\u0019\n2~(\u001aJ)2S\u0001 (70)\nThe linear relation between the transition energy \u0001and the relaxation of excited states has been\nobserved experimentally for Fe adatoms on Cu(111) surface [45] and also for an antiferromag-\nnetically coupled chain of N=3 Fe atoms on Cu 2N [69]\nFor the nonadiabatic decoherence of the ground states, the relevant rate is given by Eq. (69).\nThus, at su \u000eciently low temperatures this contribution is suppressed exponentially, and we are\nleft with the adiabatic contribution, whose rate is given by [95, 161]:\n1\nTad\n2\u0011\rad:=\u0019\n2~(\u001aJ)2S2kBT: (71)\nThis result shows that the bigger S, the faster the decay of the coherent superposition of the\nstates +Sand\u0000S. It also shows that even without inelastic scattering, the phase coherence of the\nsuperposition state is fragile. The temperature dependence in this case comes from a phase space\nargument. Elastic scattering requires the presence of an electron and a hole at the same energy.\nThe density of electron hole pairs scales linearly with kBT. Perturbative results work well when\n\u001aJ\u001c1. For instance, taking \u001aJ=0:1, and S=2, and kBT=100 mK, the decoherence time\nisT2'1:2 ns. Therefore, even in the most favorable case of a system where both spin flip\nand inelastic scattering are suppressed, elastic spin-preserving Kondo interactions are extremely\ndetrimental for atomic spin coherence. Baumann et al. have reported T2'120 ns for kBT=0:6\nK for Fe on MgO. If the decoherence rate in that system was governed exclusively by Eq. (71),\nand assuming S=2 for that system, we would infer \u001aJ'3\u000210\u00003. Thus, this is a good upper\nlimit for this quantity in that system.\nThe perturbative results seems to indicate that, in the limit of T=0, the coupling to the\nenvironment would not be able to decohere the linear superposition. Actually, we show below\nthat this conclusion is wrong, and elastic spin-conserving Kondo process would result in a finite\ndecoherence rate even at T=0. The failure of the BR theory relates, in this case, to the fact that\nthe the theory only describes processes that are slow compared to the correlation time of the bath\n\u001cc. In the case of the Fermi sea, \u001cc\u0019~\nkBT. Thus, as the temperature goes to zero, the theory can\nonly describe processes that are very slow, and eventually this prevents the proper description of\nT2.\n294.3.2. Decoherence of a single spin with degenerate spectrum: non-perturbative results\nWe now compute the adiabatic decoherence without using perturbation theory and without\ninvoking the Markovian limit, both essential ingredients of the BR theory. This is possible in this\nspecific case, because we can use two well established methods. First, we perform the partial\nwave expansion of the itinerant electron states of the surface. For a single Kondo impurity, only\ntheswave contributes [162]. This turns the original problem in a one-dimensional one, for which\nwe can apply the bosonization description of the spin density operator as described in Appendix\nC. Second, and as described in Sec 4.1.1, we restrict the spin Hilbert space to the ground state\ndoublet. We end up with the following Hamiltonian for the TLS coupled to the electron-hole\npairs of the Fermi sea, described with bosonic operators:\nHSBC=~vFX\n0\u0014k\u0014kckby\nkbk;+p\u000bˆ\u001czX\n0<k<kcgk(by\nk+bk): (72)\nNotice that this model is a particular version of the spin boson model when the tunneling term\nvanishes, see Eq. (C.6). Hamiltonian (72) is diagonalized by the following unitary transforma-\ntion [163]:\nUHSBCU\u00001=h0+\u0001\n2ˆ\u001cz (73)\nwhere U=exp(\u0000iˆ\u001cz\b=2) and\n\b\u0011ik=kcX\nk>0 k\n\u0019L!1=2\u0010\nby\nk\u0000bk\u0011\n: (74)\nThis permits us to calculate the evolution of the wave function of the bosonic operators that\ndescribe the electron-hole pair excitations of the bath. Using the argument of Sec. 2.1.3, we can\ncompute the decay of the coherence, given by Eq. (15). With the help of transformation (73),\none arrives toS12(t)=eK(t)[163], where\nK(t)=4\n\u0019~Z!c\n0d!J(!)\n!2F(!;T;t); (75)\nwith\nF(!;T;t)=coth ~!\n2kBT!\n(cos!t\u00001)\u0000isin(!t); (76)\nandJ(!) the spectral density. For a Ohmic bath, the case relevant for a Fermi gas, we have\nJ(!)/!, and we can obtain two very interesting limiting results. For finite temperature, and\nnot too short times, t>~\nkBT, we have:\nS12(t)\u0019e\u0000\u0000te\u0000iIm[K(t)](77)\nwith\n\u0000 =2\u0019\u000bkBT=~: (78)\nThis results is the same that can be obtained using the Bloch-Redfield approach, outlined in\nthe previous subsection. It basically means that, even in the absence of spin-flip interactions,\nthe electron gas is able to decohere a “Schr ¨odinger-cat ”-like state, Eq. (15), where the spin is\nprepared in a superposition of the two states with opposite magnetization along the easy axis. In\n30the opposite limit, 1 =!c<t<~=kBT, which becomes specially relevant as Tgoes to zero, we\nhave K(t)\u0019\u00002\u000bln(!ct), which yields\nS12(t)\u0019(!ct)\u00002\u000b: (79)\nThis result is interesting on two counts. First, it shows that even at T=0 the bath is able to\ndecohere the spin. Second, it permits covering a limit that can not be addressed by the Bloch-\nRedfield approach, as we discussed in Sec. 2.2.\nEquations (77) and( 79) can be reinterpreted as follows. Since they clearly show a transition\nfrom an exponential decay law in Eq. (77) to a power law in Eq. (79), one can define a transition\ntemperature T\u0003=~!c=kBexp(\u00001=2\u000b) below which the power-law decay dominates over the later\nexponential one [163]. Thus, we can distinguish between a low temperature regime ( T<T\u0003)\nwhere the dephasing rate is temperature independent and approximately given by kbT\u0003=~, and a\nhigh temperature one ( T>T\u0003) with the dephasing rate \u0000given by Eq. (78).\n4.3.3. Quasiparticle phase shift as the origin of the pure dephasing\nBoth the perturbative BR approach and the non-perturbative method based on the exact solu-\ntion of the bosonized Hamiltonian show that the Kondo interaction of an anisotropic spin with a\nFermi gas is able to decohere the linear superposition state of Eq. (57). This occurs with scatter-\ning events that preserve both the energy and the angular momentum of the two particles involved\nin the scattering, the quasiparticles and the magnetic atom, and although linear momentum is\ntransferred, this information is averaged out. Thus, and obvious question arises: how does the\nthe interaction modify the environment wave function, which is the ultimate responsible of the\ndecoherence in the absence of scattering?\nFor the spin-conserving interaction considered here, the environment actually is formed by\ntwo independent reservoirs, the Fermi gas for \"electrons and the Fermi gas for #electrons. If\nwe focus on the\"reservoir, it is apparent that the interaction with the atomic spin in the +Sstate\nresults in a phase shift di \u000berent from the one when the atomic spin is in the \u0000Sstate. This is\ntrivially seen in the case of one dimensional Fermions interacting with a delta function V0\u000e(x),\nfor which the phase shift transmission coe \u000ecient is given by t(\u000f)=1\n1+iV0\u001a(\u000f). For the Kondo-Ising\nproblem, we can write down V0=JS\u001b, where Sand\u001b=\u00061=2 are the atomic and quasiparticle\nspins respectively. In the weak coupling limit, the phase shift is:\n\u000e\u001b(S)'\u001aJS\u001b (80)\nwhere\u001a=m\n~2k(\u000f)is the density of states at the energy of the quasiparticle, that we omit from the\narguments for simplicity. Importantly, for a fixed quasiparticle spin, the phase shift depends on\nwhether the atomic spin is in the +Sor\u0000Sstate. Using this result, the pure dephasing (71) can\nbe written down as:1\nTad\n2=\u0019kBT\n2~X\n\u001b(\u000e\u001b(S)\u0000\u000e\u001b(\u0000S))2: (81)\nSo, it is natural to think that the environment reads the information about the system via the\nspin dependent phase shifts. This is in line with the standard results of the quantum impurity\nphenomena, such as the Fermi edge singularity [164] and the Kondo e \u000bect [165], where some\nimportant results can be expressed in terms of the quasiparticle scattering phase shift [164, 166].\nThe connection between spin decoherence and the quasiparticle phase shift opens up an in-\nteresting venue of research: in the case of spin structures, the scattering with multiple point\nscatterers can results in dramatic reductions of the phase shifts, that would lead to enhanced\ncoherent lifetimes for the atomic spins.\n314.3.4. Decoherence for the degenerate doublet: the choice of basis set\nThe discussion above has considered decoherence for spins with uniaxial anisotropy, such\nthatSzis a good quantum number. Thus, the Kondo interaction with the surface electrons was\nnotcapable of inducing transitions from Szto\u0000Sz, on account of the conservation of angular\nmomentum (provided that 2 S>1). Hence, in that situation, the choice of the eigenstates of\nˆSzas basis set is quite natural, although not unique. However this situation where scattering is\nforbidden is exceptional and, in general, for two level systems both decoherence and population\nscattering can occur leading to coupled dynamical equations. In this case, typical of half-integer\nspin at zero field, the distinction between T1andT2becomes ill defined, or at least it has to be\nreferred to a specific choice of the basis set.\nInterestingly, sometimes the coupling to the environment is fully responsible of selecting\nthe final states among the infinite possible choices of the isolated system. The states with this\nproperty receives the name of pointer states [12]. This is known as einselection process [12] and\nit bears obvious resemblance with the quantum measurement problem.\nIn the context of half-integer anisotropic spins, Kramers’ theorem ensures at least a dou-\nble degeneracy of the energy spectrum at zero magnetic field. Thus, the transverse anisotropy\nHamiltonian E(S2\nx\u0000S2\ny) can not lift the degeneracy of the Sz=\u0006Sdoublet when Sis half-\ninteger. However, it does modify the wave functions. The dimensionless parameter that controls\nthe mixing is \"=E=jDj, that in the following is assumed to be small. We illustrate the process of\neigenselection taking the case of a S=3=2 spin, but the discussion could be easily generalized.\nThe wave functions of the ground state doublet can be written as:\n(jC1i=cos\u0012\n2j+3=2i+sin\u0012\n2j\u00001=2i\njC2i=cos\u0012\n2j\u00003=2i\u0000sin\u0012\n2j+1=2i; (82)\nwhere\u0012\u0019p\n3\"+O(\"2). Thus, the mixing angle \u0012is imposed by the magnetic anisotropy.\nGiven that Kondo interactions can only connect states that di \u000ber in at most one unit of angular\nmomentum ( \u0001Sz=\u00061;0), having\"=0 makes the scattering rate between states jC1iandjC2i\nimpossible.\nStates (82), which hereafter we call classical states and thus the notation jC1iandjC2i, have\na well defined magnetization along the easy axis of the system [17]. However, any other pair of\nstates linear combination of jC1iandjC2i, such as\n jˆ\n1i\njˆ\n2i!\n= cos\u001e\n2sin\u001e\n2ei\u0018\n\u0000sin\u001e\n2e\u0000i\u0018cos\u001e\n2! jC1i\njC2i!\n; (83)\nwhere (\u001e;\u0018) are spherical coordinates in the unit sphere, is an equally valid choice of quantum\nstates for the doublet. Notice that the angles \u001eand\u0018are a matter of choice, in contrast with the\nangle\u0012, that is given by the Hamiltonian. We now study how the decoherence induced by the\nKondo coupling to a metallic substrate depends on \u001e. Since the only role of the angle \u0018is to\ninterchange the real and imaginary part of the coherences, in the following we just take \u0018=0.\nAs noticed in Sec. 2.1.2, the density matrix for a TLS has only 3 independent real quantities\nthat can be encoded in the vector ~P, see Eq. (8). The BR master equation of a degenerate TLS\ncan be then written as\n@~P(t)\n@t=M\u0001~P(t); (84)\n32where the matrix Mis a functional of the Redfield coe \u000ecients, as shown in the Appendix B.2.\nFor the matrix Mit is convenient to introduce the notation\nM=2666666664\u0000\r=2 Mr;iMr;p\nMi;r\u0000\r0=2Mi;p\nMp;r Mp;i\u0000\u00003777777775; (85)\nwhere p,randistand for populations, real and imaginary part of the coherences, respectively.7\nThus, in this notation \u0000stands for the transition rate between the eigenstates of HSwhile\r;\r0\ndenote decoherence rates. From Eqs. (84) and (85), it is clear that the evolution of the occupa-\ntions imbalance and real and imaginary parts of the coherences are fully decoupled from each\nother whenever Mk;j=0. In the case of the dissipative dynamics of the anisotropic S=3=2 spin\nthat we are considering here, we obtain the following closed expressions for the entries of (85):\n\u0000(\u001e)\u00199\u0019kBT\n4~(\u001aJ)2h\n(1\u0000cos(2\u001e))+\"2(1+3 cos(2\u001e))i\n(86)\n\r(\u001e)\u00199\u0019kBT\n2~(\u001aJ)2h\n(1+cos(2\u001e))+\"2(1\u00003 cos(2\u001e))i\n(87)\n\r0(\u001e)\u00199\u0019kBT\n~(\u001aJ)2\u0010\n1+\"2\u0011\n(88)\nMp;r=4Mr;p\u0019\u00009\u0019kBT\n2~(\u001aJ)2\u0010\n1\u00003\"2\u0011\nsin(2\u001e): (89)\nThe rest of the Mi jentries are zero for this system. From these equations, we see that the classical\nbasis (\u001e=0) is special on three counts:\n1. It minimizes the elastic population scattering (86).\n2. It decouples the evolution of coherences and occupations\n3. It maximizes the decoherence rate. In other words, the environment is particularly e \u000ecient\ndestroying coherent superpositions established between the states jC1iandjC2i.\nFor these reasons, the choice of states jC1iandjC2ias a basis set to study this class of\nsystems is very convenient. Of course, other choices of basis set are also legitimate, but make\nthe description of the dynamics more complicated.\n4.4. Non-degenerate two level systems\nNow we turn our attention to the type-Q systems, for which the spectrum is non-degenerate\nand the ground state is a linear combination of two states with opposite magnetic moment. The\nrepresentation of the density matrix in the basis of eigenstates of the spin Hamiltonian is thus\nunique. Within the BR picture, the coupling to the reservoir will lead to a steady state density\nmatrix that is diagonal in the eigenstate basis, and thereby it will have build-in coherences when\nrepresented in the Cbasis, as long as kBTis not much larger than the energy splitting \u0001QST.\nIt na¨ıvely looks like, within the BR theory, the coupling to the reservoir is not able to prevent\nthe coherence, i.e., the linear superposition of two states with opposite magnetization. In the\nfollowing we see how this is not really true, even within the BR picture.\n7As we are interested in the relaxation and decoherence rates, here we neglect the induced energy shift (imaginary\npart ofR12;12). This could be incorporated as a renormalized energy di \u000berence ˜\u0001.\n334.4.1. Renormalization of the QST splitting\nSo far we have considered in some detail how the coupling to the reservoir changes the\ndynamical evolution of the reduced density matrix. There is another aspect of this coupling that\nplays an important role in the case of type- Qground state, namely, the renormalization of the\nenergy di \u000berences in the spectrum. This type of e \u000bect was first observed in the context of atomic\nphysics: the coupling of the hydrogen atom to the vacuum photon field results in the so called\nLamb shift, a renormalization of the 1 s\u00002ptransition whose treatment requires to take care of\nthe singularities and played a major role in the development of quantum field theory.\nIn the context of magnetic adatoms, the notion of renormalization of the energy levels was\nused to explain the correlation between the changes in the transition energy between the Sz=\n\u00061=2 ground state doublet and the Sz=\u00063=2 excited states, on one hand, and the intensity of\nthe zero bias Kondo peak on the other. It was argued that [52, 76], given that the height of the\nKondo peak depends on \u001aJand it was correlated to the inelastic transition energy, this energy\nshift had also to depend on \u001aJ. The perturbative BR theory would naturally account for this\nenergy shifts. The same theory can also be applied to integer spins with quantum spin tunneling.\nFor instance, for a spin S=1 described with Hamiltonian (43) with D<0 and E,0, the\nsplitting is renormalized according to [76]:\n\u00010;1= \u0001 0 \n1\u00003\n2(\u001aJ)2ln 2W\n\u0019kBT!!\n(90)\nwhere Wis the bandwidth of the itinerant electrons that appears as an ultraviolet cut-o \u000bin the\ntheory [52].\nAn obvious consequence of Eq. (90) is that there is a critical value of \u001aJat which the\nsplitting vanishes. However, the BR perturbative approach used to get this results may break\ndown before this critical value, which together with the presence of the logarithmic term calls for\na critical questioning of the validity of the perturbative approach. Non-perturbative numerical\ncalculations based on the One Crossing approximation for a multi-orbital Anderson model that\nincludes magnetic anisotropy confirmed the validity of the perturbative calculation both for half-\ninteger [52] and integer [86]. In the last case, the QST splitting goes to zero as \u001aJincreases\nabove a threshold value.\nA more elegant nonperturbative description of this phenomenon can be obtained from the\nbosonization procedure described in Appendix C. This permits us to map the problem of an\nindividual quantum spin exchanged coupled to an electron gas into the spin-boson model [17].\nThe mapping between the two models then leads to the following identity between the coupling\nof the TLS and the baths [17]:\n\u000b\u0011jhQ1jSzjQ2ij2(\u001aJ)2: (91)\nMaking use of the well known results for the spin boson model, Eq. (47), one can see that the\nQST vanishes as \u000bgoes beyond one [17]. In the limit of small \u000b, we can expand Eq. (47) in a\nTaylor series to yield\n\u0001 = \u0001 0 \n1\u0000\u000bln~!c\n\u00010!\n: (92)\nThe result (92) basically reproduces the perturbative result (90), with the main di \u000berence that\nkBTwithin the Log function has now been replaced by \u00010, the bare zero-field-splitting. The\nmain and completely general result is that the coupling to the environment reduces the quantum\nspin tunneling splitting and, if su \u000eciently strong, it completely cancels it.\n344.4.2. Perturbative Dynamics of the non-degenerate TLS\nIn Sec. 4.3.2 we have already studied the dynamics of the reduced density matrix describing\nan e\u000bective two-level degenerate spin interacting with an electron gas via a Kondo interaction.\nNow we tackle this question in the case of a non-degenerate TLS, with an energy splitting \u0001,0.\nIn particular, by using the BR theory, we will analyze in detail the situation when the relaxation\nrate\u0000is comparable to the splitting \u0001.\nWe focus in the case when \u0001arises from quantum spin tunnelling. Without loss of generality,\nwe assume here that it corresponds to an anisotropic spin described by Hamiltonian (48). The\ndissipative dynamics is generated by the Kondo coupling, given by Eq. (63), where ˆ \u001cxinduces\ntransitions between the two eigenstates of the TLS. Importantly, in the jCibasis Kondo scattering\ndoes not produce scattering. In the limit when \u0001is very small, it is convenient to keep in mind\nthese two complementary points of view, as we discuss below.\nSince we are considering a non-degenerate system, coherences and occupations evolves in a\nfully independent way. For the specific decoherence mechanism considered here, the whole DM\ndynamics is determined uniquely by two parameters, the splitting \u0001and the relaxation rate \u0000.\nWe first derived the evolution of the occupation. Simple analytical expressions for the dy-\nnamics of ˆ\u001a(t) can be obtained for this particular case. If the initial occupation imbalance is \u000e\u001a0,\nthe population di \u000berence\u000e\u001a=P\"\u0000P#satisfy\n\u000e\u001a(t)=e\u0000t\u0000 \n\u000e\u001a0+\u000e\u0000\n\u0000!\n\u0000\u000e\u0000\n\u0000; (93)\nwhere\u000e\u0000 =R22;11\u0000R 11;22. This simple time evolution is depicted in Fig. 4a). As expected, the\nsteady state occupations is only determined by the ratio between Boltzmann factors, R22;11=R11;22.\nNotice that the evolution tends to the one of a degenerate TLS in the classical basis as \u0001!0.\nWe now consider the solution of Eq. (84) for the coherences, which can behave in a quite\ndi\u000berent way depending on the relation between \u0001and\u0000. This gives rise two di \u000berent regimes:\nanunderdamped oscillatory regime for~2\u00002\u00004\u00012<0 and an overdamped oscillatory regime\nfor~2\u00002\u00004\u00012>0.\nUnderdamped regime: ~2\u00002<4\u00012\nIn this regime the splitting \u0001is the largest energy scale. Coherences C\"#are given by\nC\"#(t)\nC0=e\u0000t\u0000=2\n#h\nsin#t\n2(\u0000e\u0000i\f\u00002i\u0001\n~ei\f)+#ei\fcos#t\n2i\n; (94)\nwhere we have defined C\"#(0)=C0ei\fwith C0>0 and we have introduced the rate #2=\nj\u00002\u00004\u00012=~2j. We thus see that the coherence oscillates with an amplitude that decays with a\ncharacteristic time1\nT2=\u0000\n2=1\n2T1: (95)\nThe oscillation period T=4\u0019=# is di\u000berent from~\n\u0001, a renormalization that also happens in\nan underdamped classical harmonic oscillator. This situation is illustrated o Fig. 4b) where, in\naddition to the real and imaginary part of C\"#(t), we plot the envelope jc(t)j.\nOverdamped regime: ~2\u00002>4\u00012>0\nWe now consider the case where \u0000is the dominant energy scale. For t\u001d2=#, we can write\nC\"#(t)\nC0\u0019#\u00001e\u0000t=T2h\n\u0000e\u0000i\f+ei\f(#\u00002\u0001=~)i\n; (96)\n35with\n1\nT2\u0011 \u0000\u0000#\n2!\n(97)\nThe coherence no longer oscillates and, importantly, the timescale for the decoherence depends\nboth on \u0000and on \u0001. In the limit of small \u0001,~\u0000\u001d\u0001we can write down\n1\nT2\u0019\u0001\n~ \u0001\n\u0000~!\n\u001c\u0000: (98)\nTherefore, a strong suppression of decoherence rate 1 =T2occurs in this regime, as depicted in\nFig. 4c). This rather counterintuitive results can be better understood if we reinterpret the results\nin the classical basis as explained in the next section.\n4.4.3. Density matrix in the classical basis for the split TLS\nEquations (93-94,96) describe the diagonal ( \u000e\u001a) and o \u000b-diagonal ( C\";#) parts of the density\nmatrix for the split TLS expressed in the basis of eigenstates (i.e., the basis (83) with \u001e=\u0019\n2). We\ncan write down the same density matrix in the basis set of the classical states:\nˆ\u001aC(t)=\"1\n2+Re[C\";#(t)]\u000e\u001a(t)\n2+iIm[C\";#(t)]\n\u000e\u001a(t)\n2\u0000iIm[C\";#(t)]1\n2\u0000Re[C\";#(t)]#\nC; (99)\nwhere the subindex “ C” denotes that ˆ \u001ais written in the classical basis of states jC1;2i. This equa-\ntion shows how population scattering in the Q-basis ( \u001e=\u0019\n2), which leads to the decay of \u000e\u001a,\nresults in a decay of the o \u000b-diagonal terms of the DM expressed in the classical basis. Analo-\ngously, decoherence in the Q-basis can be interpreted in terms population transfer in the Cbasis.\nThis result permits us to re-interpret the Eqs. (93-96) of the dynamics of the split TLS obtained.\nFor the system considered here, angular momentum conservation prohibits direct Kondo scatter-\ning between the classical states. On the other hand, the coherent evolution of the system permits,\nvia quantum spin tunneling \u0001, connecting the classical states, giving rise to Rabi oscillations in\nthe diagonal of the density matrix (99). In the overdamped regime these oscillations are sup-\npressed, and result in an incoherent population transfer governed by a exponential decay with T2\ngiven by Eq. (98). This mechanism for population transfer combines thus the coherent quantum\nspin tunneling and the dissipative coupling to the environment, and has been proposed by Gauy-\nacq and Lorente [96] to explain the switching between N ´eel states observed for AF chains of Fe\natoms on Cu 2N [48]. Following Ref. [96], we refer to this mechanism as the decoherence as-\nsisted switching , although perhaps it would be more accurate to refer to it as quantum tunneling\nassisted switching.\n4.4.4. Non-perturbative derivation of the decoherence assisted switching\nWe now provide an independent derivation for the decoherence assisted switching mecha-\nnism. For that matter, we use again the mapping of the Kondo model for the split TLS to the\nspin boson model of Eq. (C.6). Within this model, the time-dependent spin autocorrelation of\nthe system is given by P(t)=hˆ\u001bz(t)i, where\u001bz=\u00061 labels the two classical states. We assume\nthat at t=0 the system is prepared in the \u001bz= +1 state. Using the noninteracting-blip approx-\nimation [11], one gets P(t)=g(t)e\u0000t=\u001cwhere g(t) is an oscillating function (constant) of order\nunity for\u000b<1=2 (\u000b\u00151=2), and\n\u001c\u00001=(\u00012=~2!c)p\u0019\n2˜\u0000(\u000b)\n˜\u0000(\u000b+1=2) \u0019kBT\n~!c!2\u000b\u00001\n; (100)\n360 2 4 6 8\nt/T201c(t)/c(0)0 2 4 6 8\nΓτ-101c(t)/c(0)-101δρ(t)(a) (b) (c)Figure 4: Time evolution of the density matrix of an e \u000bective two level system corresponding to an integer spin. (a)\nOccupations di \u000berence for \u00001;2=0:1\u0000and\u00002;1=0:9\u0000. (b) and (c) Real (blue) and imaginary (red) parts of the\ncoherence c(t) in the underdamped ( \u0001 = 10~\u0000) and overdamped ( \u0001 = 0:1~\u0000) regimes respectively. Notice that in panel\n(c) the time axis has been scaled by T2\u0019100=\u0000. In all cases, \u000e\u001a0=1,c(0)=1=2.\nwhere ˜\u0000(x) is the Gamma function. Expression (100) is valid whenever \u000b\u00151, for which \u0001is\nrenormalized to zero [see Eq. (47)], or, for \u000b<1, if\n\u000bkBT\u001d\u0001 \u0001\n~!c!\u000b=(1\u0000\u000b)\n: (101)\nIn both cases, \u0000\u001d\u0001, so that we are in the overdamped limit discussed above. Equation (100)\nalready shows that the switching time \u001cis proportional to \u00012. This result is in line with the\nperturbative result in Sec. 4.5. In fact, when the coupling constant \u000bgoes to zero, we get that\nEq. (100) takes the simple form\n\u001c\u00001\u0019\u00012\n2~\u0019\u000bkbT; (102)\nwhich taking into account that \u0000 = 2\u0019\u000bkbT=~, reproduces the perturbative result (98) to lowest\norder in the coupling \u000b.\n4.5. Relaxation and decoherence in spin chains and ladders\nWe now address the problem of spin relaxation and decoherence of finite size spin chains due\nto their Kondo coupling to a nearby electron gas. Many of the results obtained in the previous\nsections will be useful. An important concept to keep in mind is that decoherence a \u000bects quantum\n37states, rather than the spins. Whereas in the single spin case this distinction if less important, in\nthe case of multi-spin chain it does play a key role. For instance, a single scattering event between\ntwo states might flip the spin of all the atoms in the chain. The problem of spin relaxation of a\nHeisenberg-coupled spin array due to Kondo exchange is formally connected with the so called\nKondo lattice model, where otherwise independent itinerant fermions are exchanged coupled\nto a lattice of quantum spins [167–169]. Even if direct exchange between the local spins is\nsometimes not explicitly written down in the the Kondo lattice Hamiltonian, indirect exchange\ninteractions emerge, which can compete with the Kondo correlations [169]. The Kondo lattice\nmodel is a rich many-body problem, with several di \u000berent types of electronic order [167]. Here\nwe treat the Kondo coupling as a perturbation within the BR theory. In addition, the systems of\ninterest have a rather small number of spins. These two simplifications make the problem easier\nthan the Kondo lattice model and tractable by means of numerical diagonalization. On the other\nhand, in order to model magnetic adatoms we have to include the e \u000bect of magnetic anisotropy,\nsometimes overlooked in the Kondo lattice model. As we discussed above, spin chains and\nladders do behave very di \u000berently depending on the interplay between magnetic anisotropy of\nindividual spins, exchange interactions, the number Nof spins in the system, and the parity of\nN, as discussed in Sec. 3.3. This a \u000bects both T1andT2.\nWe now discuss two di \u000berent problems. First, we discuss how the spin decoherence times T2\nof an Ising spin chain depends on the properties of the spin chain, such as the number of atoms N,\nand the atom-atom distance d. Second we discuss the role played by Kondo induced decoherence\nin the emergence of classical behavior in spin chains made of atoms that, when isolated, have\nfinite quantum spin tunneling splitting.\n4.5.1. T 2for broken symmetry states in spin chains\nIn the context of magnetic adatoms, the BR theory has been applied to study the spin re-\nlaxation T1, and to a lesser extend T2, of a variety of finite size spin chains and ladders due to\nKondo exchange [17, 84, 96, 170]. We discuss here the decoherence time of Ising chains without\nspin-flip terms in the Hamiltonian, so that Szis a good quantum number, and the ground state is\ndoubly degenerate. We consider the case of AF coupling, so that the lowest two states, which we\nlabel as 1 and 2, correspond to the two classical N ´eel state. Hence, we can write the matrix ele-\nments of the local spins operators as Sa\n11(l)=\u0000Sa\n22(l)=S(\u00001)l. Following our recent work [97],\nand using Eq. (67), the relevant form factor from Eq. (68) for a chain of atoms lying along the x\naxis is given by:\n\u001fad:\n1;2(~q)=S2\f\f\f\f\f\f\fX\nn(\u00001)n\u0010\neiqxna+e\u0000iqxna\u0011\f\f\f\f\f\f\f2\n: (103)\nThe resulting pure decoherence rate is given by [97]:\n1\nT\u0003\n2\u0019\u0019(\u001aJ)2\n8~kBT\u0003AFM(kFd;N): (104)\nwhereJ=JjSz\n1;2j2and\u0003AFM(kFd;N) is a dimensionless function that represents the average\nof the structure factor (103) over the Fermi surface, and it is therefore di \u000berent for fermions in\nD=1;2;3 dimensions. For D>1 and kFd>1, i.e., in the limit where the interatomic distance\nis larger than the Fermi wavelength, relevant for metals, the function \u0003AFM(kFd;N)'Nwith\nsome small oscillations as a function of kFd[97]. Thus, the decoherence rate of an Ising spin\nchain with N spins is N times quicker than the single spin, Eq (71) .\n38The origin of the oscillations of the decoherence time as a function of kFdis interesting\nin itself [97]. Within the BR theory, it is apparent that both decoherence and energy shifts in\nan open quantum system are two sides of the same coin: the reactive and dissipative response\nof the system to the coupling with a reservoir. The reactive coupling results in a shift of the\nenergy levels. In the case of a spin chain coupled to an electron gas, this shift of the energy\nlevels is nothing but the RKKY interactions [97], which is known to oscillate as a function of\nkFd. Therefore, the oscillations in T2are expected, being mathematically related to the RKKY\ninteraction. They can also be understood in terms of phase shifts of the quasiparticles. As we\ndiscussed in Sec. 4.3.3, the pure dephasing rate is proportional to the variation in the quasiparticle\nscattering phase shift for the two states of the spin(s). Elastic scattering between waves with\nwavenumber kFand a structure with period dis expected to depend in an oscillatory manner on\nkFd, on account of multiple scattering interference.\nThis type of e \u000bect depends crucially on the phase factor ei\u0010~k\u0000~k0\u0011\n\u0001~rlthat appears in the Kondo\nHamiltonian, overlooked by some previous work in the context of magnetic adatoms. It’s omis-\nsion is equivalent to ignore the local nature of Kondo coupling,P\nn~Sn\u0001~s(~rn) and replace it by\na coupling with the total spin ~s(0)\u0001P\nn~Sn. This coupling commutes with the total atomic spin\noperator,~ST\u0011P\nn~Sn, and it is thereby not capable of producing transitions between eigenstates\nwith di \u000berent S. As a result, the Kondo interaction without the phase factors can not induce spin\ntransitions between chain states with di \u000berent total spin ST.\n4.5.2. The quantum to classical transitions in spin chains\nIn the seminal work of Loth et al. [48], by using spin-polarized STM experiments, it was\nshown how chains of N>6 Fe atoms at T\u00181 K deposited on Cu 2N would acquire spontaneous\nlocal, forming the N ´eel state, expected on account of the AF interaction of Fe atoms along the\nnitrogen rich direction. This AF character was further confirmed by an accurate exploration of\nthe Fe dimer [53]. In addition, a time-resolved tracking of the magnetization made it possible\nto observe random telegraph noise with two states, revealing switching between the two N ´eel\nstates. As noted by Loth and coworkers, the system provided a unique opportunity to study the\ntransition between the classical behavior, observed for chains, and the quantum behavior of the\nsingle Fe atom in the same surface [17] inferred from the IETS spectroscopy [31].\nIn this system, both the local magnetic anisotropy [31] as well as the Fe-Fe exchange [53]\nare well determined from IETS. This provides an accurate description in terms of the Hamilto-\nnian [17]\nH=X\nlDˆSz(l)2+E\u0010ˆSx(l)2\u0000ˆSy(l)2\u0011\n+JX\nl=1;N1~S(l)\u0001~S(l+1): (105)\nIn the J=0 limit, this interaction describes an ensemble of independent anisotropic spins.\nForS=2, the case relevant for Fe on Cu 2N, the ground state is unique, and the expectation value\nof the atomic spin operators, ~M(l)\u0011h~S(l)iis strictly zero. In the opposite limit of very large J,\nwe can ignore the anisotropy, and we have again a unique ground state, with S=0, and null ~M(l)\nfor all atoms in the chain. In order to reproduce the phenomenology observed experimentally,\nwith two equally likely N ´eel states, with finite ~M(l), we would need a doubly degenerate ground\nstate. Mathematically, this situation appears for instance in the limit E=0 andjDj\u001d J. For\nthe values of D,EandJobtained from IETS, we can compute the energy di \u000berence \u00010between\nthe ground state and first excited state, and we can study their wave functions. Interestingly, we\nobtain that \u00010decreases exponentially as a function of N, as shown in the Fig. 5. Moreover, the\nwave functions of the ground and first excited states are bonding and anti-bonding combinations\n39of N ´eel states. Thus, the combination of single-ion anisotropy and exchange interactions shrink\n\u00010. From the numerical diagonalization with the best fitting parameters we obtain that \u00010(N)\u0019\n\u0001Fee\u0000N=0:74[17], where \u0001Fe\u00190:15 meV is the zero field splitting of the single Fe.\nThus, the energy scale that protects the quantum behavior of finite size spin chains decreases\nexponentially as the number of atoms in the chain increases, but it does not cancel. This can be\nseen in the case of the quantum Ising model in a transverse field. Even when the AF interaction\ndominates, leading to the doubly degenerate ground state in the thermodynamic limit, for finite\nsize chains there is a small splitting [17]. However, when the splitting is small enough, we can\nargue that the classical behavior for the spin chain takes over, on account of the dissipative Kondo\ncoupling with the surface electrons. This switching can be better understood in the N ´eel basis\nas we discussed in Sec. 4.4.4. There we showed that the decoherence (of the N ´eel states) in\nthis limit is Ntimes faster the one of an individual atom. For instance, T2(N=8) is in the 100\nps range, whereas the Rabi time ~=\u0001(8) is well above microseconds. Thus, we are in the limit\n~T\u00001\n2\u001d\u0001(N). This accounts for the observed random telegraph noise with switching times\nslower than milliseconds [96].\n5. Other spin relaxation mechanisms: photons, phonons and nuclear spins\nWe now consider additional sources of spin relaxation and decoherence a \u000becting spins in-\ntrinsic to magnetic atoms surfaces. In the previous section we have considered a mechanism\nthat is characteristic from metals, namely, Kondo interactions. Given that in many instances the\nunderlying metal and the magnetic atom are separated by an atomically thin insulating material,\nwe briefly discuss the dominant spin relaxation mechanisms for magnetic atoms in insulating\nhosts: spin-phonon coupling and hyperfine interactions with nearby nuclear spins. In addition\nto these, there is another mechanism that is known to result in spin decoherence: a conductor\ncreates a stochastic magnetic field due to the unavoidable thermal fluctuations of the current\naround its zero average (Johnson noise). These mechanisms could be particularly relevant when\nthe Kondo interaction is suppressed, something that might be achieved using superconducting\nsubstrates [51, 54], or decoupling layers with less transparent tunnel barrier, achieved either with\na thicker spacer or a wider band-gap material, such as MgO [55, 59, 107, 115] or h-BN [63].\n5.1. Spin-phonon coupling\nSpin-phonon interaction Vs\u0000phis an important source of spin relaxation for paramagnetic\ncenters in insulating materials where the density of itinerant carriers is negligible, blocking\nKondo exchange. It can be the dominant mechanism when the density of spin centers is small,\nso that dipolar coupling is negligible. The spin-phonon (SP) interaction also a \u000bects the parame-\nters in the static spin Hamiltonian, giving rise to shifts in the g-value, the fine structure splitting,\nand the hyperfine interaction [81]. In addition, phonons may also induce a spin-spin interaction\nbetween ions. Such e \u000bects arise even at zero temperature from the zero-point vibrations of the\nlattice, and though they increase at finite temperatures, they are often rather small.\nRelaxation processes mediated by phonons involve the emission or absorption of a quan-\ntum by the spin system, which should be absorbed or emitted by the lattice vibrations. Thus,\na transition between two spin levels will be driven e \u000eciently if the lattice can produce such an\noscillatory electromagnetic field. One possible mechanism is the one proposed by Waller [171]:\nthe local dipolar magnetic field created by neighboring ions, which depends on the ions dis-\ntance, fluctuates due to the lattice vibration. A second process, more relevant in diluted magnetic\n40b a Figure 5: Reprinted with permission from F. Delgado et al. , Europhysics Letters 109, 57001 (2015) (a) QST splitting\nof the Ising chain vs. g=2jjHj=\u00010for a N=20 chain (black line) and the infinite chain (red line). gc=1 marks\nthe quantum phase transition. Inset: QST splitting of the S=2 Heisenberg spin chain together with higher-energy\nexcitations (orange lines) vs. gfor the Fe chains with D=\u00001:5 meV and E=0:3 meV (the diamond marks the\nexperimental condition [48, 53] with g\u001927 ). (b) Chain size dependence of \u0001in the QIMTF for g=0:5<gc(weak\nsize dependence), and g=1;2 (exponential dependence that leads to a type-C ground state for large N). Inset: size\ndependence of \u0001for the experimental parameters, both for the AF [48, 53] and FM chains [56], showing an exponential\ndependence.\ncenters, consists of modulation of the crystal electric field or ligand field through motion of the\nelectrically charged ions under the action of the lattice vibrations [81], essentially a dynamic\norbit-lattice interaction, which influences the spin levels through the spin-orbit coupling.\nSpin relaxation due to one-phonon emission scales with the density of states of phonons at\nthe spin transition energy \u0001:8\n1=T1/\u001aph(\u0001)= \u0001nB(\u0001)\u0006(\u0001); (106)\nwhere \u0006(\u0001) is the number of phonon modes per unit volume in a frequency range \u0001=~;(\u0001+d\u000f)=~\nandnB(\u0001) the Bose-occupation factor. For the phonon radiation bath one has, taking into account\n8Being a small perturbation, one can apply the Bloch-Redfield approach of Sec. 2.2 to study the relaxation and\ndecoherence processes induced by Vs\u0000ph.\n41that there are two transverse polarizations an one longitudinal wave motion:\n\u0006\u0001=3\u00012\n~2\u001920BBBB@2\nv3\nt+1\nv3\nl1CCCCA; (107)\nwhere vtandvlare the transverse and longitudinal wave velocities.\nThe SP induced relaxation rates involves matrix elements of the crystal and ligand electric\nfield perturbations. Thus, the specific form of Vs\u0000phdepends in general on the particular phonon\nmode and the symmetry of the environment of the magnetic atoms [81]. It always contains an\neven power of atomic spin operators, like the zero field single ion Hamiltonian, on account of\ntime reversal symmetry, and it is linear in the atomic displacement operator. Chudnovsky and\ncoworkers [172] made a particularly elegant derivation of a universal spin-phonon coupling for\nthe transverse phonons. As very often the matrix elements also scale with the energy, a T\u00001\n1/\n\u00013=v5is commonly found [81, 172], with vthe wave velocity. As typically the ratio between\nthe transversal and longitudinal wave velocities satisfies vt=vl.0:7 [81], the contribution from\nlongitudinal modes can be usually neglected. A particularly simple an beautiful result found by\nChudnovsky and coworkers [172] is that the SP relaxation rate at zero applied field is given by\n1\nTph\n1=j\u0004j2\n12\u0019~4\u00015\n\u001amv5\nt; (108)\nwhere\u001amis the mass density and \u0004is a dimensionless matrix element of the spin operator.9\nThe application of the BR theory for spin relaxation of a spin center due to spin-phonon cou-\npling has been implemented by Leuenberger et al. [173], which considered the case of uniaxial\nmolecular magnet crystals. The same analysis could be applied for individual magnetic atoms\nwith the single ion Hamiltonian (43). The quadratic spin operators that enters into the general\nform (24) can be written in that case as:\nS\u000b\u0011q1(\u000b)S2\n++q2(\u000b)S2\n\u0000+q3(\u000b)S+Sz+q4(\u000b)S\u0000Sz+h:c:; (109)\nwhere qi(\u000b) are numerical coe \u000ecients that depends on the single ion parameters DandE. By\nusing a Bloch-Redfield master equation to study the spin dynamics, they obtained an excellent\nagreement with experimental data in a Mn 12-acetate crystal [173]. Ganzhorn et al. [174] also\nstudy the e \u000bects of the SP coupling of a TbPc 2molecular spin with a longitudinal phonon mode\nin a carbon nanotube and suggested that it could induce the suppression of quantum tunnelling of\nmagnetization, similar to the quenching of the QST by Kondo exchange explained in Sec. 4.4.1.\nIt is particularly interesting to consider the e \u000bect of the spin-phonon coupling on the low\nenergy doublet for the dominant uniaxial term DS2\nz. In that case, given the form (109), one\ngets that after the projection in the subspace spanned by jQ1iandjQ2i,Vs\u0000ph=0, i.e., the\nspin phonon coupling can not induce relaxation or decoherence between the low energy states\njQii. Similarly, in the case of a half-integer spins with Hamiltonian (43), the pure decoherence\nbetween the classical states jC1iandjC2ialso cancels. In other words, the spin-phonon coupling\ndoes not provide an e \u000ecient pure-decoherence mechanism for these spins. Of course, spin-\nphonon coupling can still produce decoherence via inelastic events, but these can be thermally\nsuppressed.\n9Here we have defined the matrix elements \u0004as dimensionless parameters, contrary to Ref. [172]\n425.2. Hyperfine interactions\nHyperfine interactions account for the spin coupling between electronic and nuclear degrees\nof freedom. It has two main components: the so called Fermi contact interaction term, that is\nonly non-zero whenever the electronic spin density is non-vanishing at the nuclear site, i.e., for\nelectrons with a finite s-wave component. In addition, there is the dipole-dipole coupling, that\ncan give a contribution for the hyperfine interaction of p,dandfelectrons with the nuclear spin\nin the same atom.\nSpin decoherence due to nuclear spins is known to play a major role in the case of electrons\nconfined in semiconductor quantum dots [175]. In that case, a single electron visits thousands\nof nuclei. For the atomic spins considered here, contact interaction is not particularly relevant\nbecause the magnetic moment lies mostly on delectrons, for which the contact term vanishes,\nand also because the local moment is localized in just one atom. Hyperfine interaction with the\nsame-atom nuclear spin will result in a small splitting of the energy levels, the so called hyperfine\nstructure. For instance, for the delectrons of51Mn, one has A\u00190:3\u00001\u0016eV [176], while for\ntheI=9=2 nuclear spin of the209Bi embedded in Si A\u00196:1\u0016eV [111]. These splittings might\nbe resolved with IETS at extremely low temperature [128] and are definitely within reach with\nthe spectral resolution achieved with STM-ESR [59]. In the case of Fe, only57Fe has a nuclear\nspin I=1=2, and a natural abundance of 2 percent. The hyperfine coupling for57Fe3+in bulk\nMgO [81] is A\u00190:1\u0016eV .\nThe dipolar coupling of a single electronic spin with an ensemble of surrounding spins could\nlead to decoherence. For an electronic spin ~Slocated at the origin, the dipolar interaction with\nthe nuclear spins ~Ii, located at~ri, takes the form reads:\nHdip=\u0000gI\u0016Ngs\u0016B\n4\u0019X\ni1\nr3\ni\u0010\n3\u0010~S\u0001ˆri\u0011\u0010~Ii\u0001ˆri\u0011\n\u0000~S\u0001~Ii\u0011\n: (110)\nIf we treat the nuclear spins as classical moments, we could write down the dipolar coupling\nHamiltonian for the electronic spin as Hdip=g\u0016BP\naSaHa, where Hawould be the acomponent\nof the nuclear spin field. If we assume this field takes random values, associated to the statistical\nthermal fluctuations of the nuclear spins, it is possible to estimate the electronic spin decoherence\ntime, using Eq. (20). The magnetic field created by a proton at 3Å is 40 \u0016T. Thus, we can\nestimate a lower limit forg\u0016B\n~'6 kHz. If we assume that the inverse nuclear spin correlation\ntime\u001c0is also in that range, this will give decoherence times in the milliseconds time scale. Of\ncourse, this is a very rough estimate of the order of magnitude.\n5.3. Spin relaxation and decoherence due to Johnson noise\nWe have just seen that nuclear spins, as sources of random magnetic fields, can induce deco-\nherence of remote spins. Conductors are known to be sources of random magnetic fields, so that\nis not surprising that they have been identified as possible sources of relaxation and decoherence\nfor cold atom spins [177], electronic spin qubits [178], spins in quantum dots [179], Phosphorus\ndonors in Si [91], and for S=1 NV centers on surfaces [180].\nThe random magnetic fields close to a conductor are created by the thermally induced statis-\ntical fluctuations of the current, the so called Johnson noise [181, 182]. In a quantum language,\nthe electronic currents in the conductor emit photons that interact with the remote spins. Mod-\nulations of the transverse (longitudinal) component of the magnetic field result in relaxation\n(decoherence), as inferred from the general equations (19) and (20). The spin relaxation and de-\ncoherence rates are thus directly proportional to the amplitude of the transverse and longitudinal\n43components of the magnetic noise, at the spin-transition frequency \u0001=~in the case of T1, and at\nzero frequency for T2. When the motion of the electrons in the conductor is di \u000busive ( kBT\u001d\u0001),\nand when the conductor is relatively far from the spin, so that the far field approximation is\nvalid,10an expression for the magnetic noise spectral density Sz\nBcan be easily computed [180]:\nSz\nB=\u0012\u00160\n4\u0019\u00132kBT\u001be\nd; (111)\nwhere\u001beis the electric conductivity. Applying Fermi’s golden rule one can arrive to a rather\nsimple analytical result for the T1relaxation due to Johnson magnetic noise [179, 180]:\n1\nT1=3g2\u00162\nB\n2~Sz\nB: (112)\nA controlled experiment with a wedge shape conductor has been carried out, where the T1\nof NV centers could be measured as a function of their distance to the conductor, finding spin\nrelaxation rates in the range of 1 =T1'1=3 ms\u00001ford=50 nm, results in good agreement\nwith this theory . A naive scaling of this result to distances of d\u00190:5 nm, would lead to\nT1\u00183\u0016sat the same temperature. However, at that point the far field expression no longer\napplies and a more sophisticated treatment of the evanescent components of the field becomes\nnecessary [177, 179]. Qualitatively, the correlation time of the magnetic noise in this regime\nis determined by the ballistic time of flight of electrons through the relevant interaction region,\nresulting in a saturation of the noise spectral density (and the spin relaxation rate) as either the\nNV approaches the surface or the mean free path becomes longer at lower temperatures, with its\nlimiting value given by [180]:\nSz\nB=2\u00162\n0kBT\n\u0019ne2\nmevF(113)\nwhere meis the electron e \u000bective mass and vFis their Fermi velocity. Again, good agreement\nwith the experiments was found in this limit when NVcenters are at d\u00194 nm, with T1in the\nrange of 0:5 ms at T=27 Kelvin. Whereas a more detailed analysis is probably necessary\nto assess how this mechanism a \u000bects magnetic adatoms, perhaps including the quantum e \u000bects\nin the Johnson noise, these experimental results provide a good starting point for the order of\nmagnitude. In addition, the shot noise associated to current flow across the STM-surface junction\nwill also contribute to the magnetic noise mechanism.\n6. Experimental methods\nIn this section we briefly overview the di \u000berent methods that are used to probe atomic spins\non surfaces with STM, with emphasis in spin dynamics.\n6.1. Single spin Inelastic Electron Tunnelling Spectroscopy\nThe technique of inelastic electron tunnelling spectroscopy was first applied in tunnel junc-\ntions back in the sixties [83, 183]. Three decades later it was implemented with an STM, first\n10This condition holds when the skin depth of the metal \u000eeis much larger than the distance dbetween the metal and\nthe magnetic impurity. For instance, for the Ag surface used in Ref. [180], this corresponds to d\u001c1\u0016m, an easily\nachievable condition.\n44for single molecule vibrational excitations [184], and a few years later, for the spin excitations\nof individual magnetic atoms [29–31]. In all cases the working principle is the following: when\nthe excess bias voltage energy, eV, of a transport electron is larger than the transition energy \u0001\nof some other localized degree of freedom in the barrier, an inelastic transport channel opens in\nwhich the electron tunnelling is accompanied by the inelastic excitation of the local degree of\nfreedom. The experimental fingerprint is a step \u0001Gin the dI=dVat the bias voltages V=\u0006\u0001=e.\nQuite frequently, the second derivative d2I=dV2is plotted, which o \u000bers an increased contrast.\nThe inelastic transitions are then identified as characteristic peaks (dips) at eV= \u0001(eV=\u0000\u0001).\nA shift of the excitation energies as a function of an applied magnetic field provides a clear\nfingerprint of the magnetic nature of a given excitation [29–31]. Interestingly, in many of these\nmagnetic excitations, the magnitude of \u0001Gis of the same order of magnitude than the zero bias\nconductance. This can be accounted for by the fact that both elastic and spin-flip inelastic events\noccur via cotunelling [101]. By contrast, the vibrational steps are frequently quite small, spe-\ncially for large molecules. Very often the evolution of the dI=dVspectra for various magnetic\nfields permits one to infer the e \u000bective spin Hamiltonian of a given structure [30, 31, 42] provid-\ning a starting point for subsequent modelling.\nThe spectral resolution provided by this method is limited by the unavoidable thermal smear-\ning of the Fermi distributions of the quasiparticles in the electrodes, given by 5 :4kBT[83, 183].\nIn addition, the amplitude of the lock-in potential provides an additional broadening, although is\nnormally smaller than the thermal smearing.\nIn a first seminal experimental work by S. Hirjibehedin et al. [31], it was shown that height\nof the inelastic step in the dI=dVwas proportional toP\na=x;y;zjhijSajfij2, where iand fare the\ninitial and final states of the spin Hamiltonian. This proportionality establishes a spin selection\nrule\u0001Sz=0;\u00061. Later it was demonstrated that this spin selection rule is a consequence of the\nKondo exchange interaction that dominates the inelastic scattering [80].\nIn the case of structures with several magnetic atoms, the dI=dVspectra at di \u000berent atoms\nhave the steps at the same bias voltage, reflecting the collective nature of the spin excitations, but\nthe height of the steps \u0001Gat atom nis expected to be proportional toP\na=x;y;zjhijSa(n)jfij2[80],\nwhich can be strongly spatially modulated [56]. This provides a tool to image the spin excitations\nin these structures, that has been used to image the spin waves in ferromagnetically coupled Fe\nspin chains [56], and in Cobalt spin chains on Cu 2N [61]. Dramatic variations of the intensity of\nthe Kondo peak have also been reported in Mn-Fe Nspin chains [62].\n6.2. Spin polarized STM\nThe technique of spin polarized STM (SP-STM), pioneered by R. Wiesendanger [26, 36],\nwas the first spin-sensitive STM based probe. A spin-polarized tip and a magnetic surface form\nan atomic-scale magnetic tunnel junction. When the relative orientation of the magnetic moments\nof tip and sample can be controlled independently, by application of an external magnetic field\nin most instances, the conductance of the system can present changes, providing thereby a spin-\ndependent signal. The technique has been used to explore a variety of surfaces with di \u000berent\ntypes of magnetic order [36] and, more relevant to this review, the magnetization of individual\nmagnetic atoms deposited on non-magnetic surfaces [32, 45, 47, 50, 79, 185, 186].\nThe early approach to achieve spin polarizaton in SP-STM was the use of tips of magnetic\nmaterials. More recently, the use of non-magnetic tips with just a few magnetic atoms in the apex\nof the tip, picked from the surface, has been demonstrated as a viable alternative [41, 43, 48, 56,\n59] to obtain magnetic contrast, although this approach requires the application of a magnetic\nfield to freeze paramagnetic fluctuations of the small magnetic cluster in the tip.\n456.3. Methods to determine T 1\nSeveral di \u000berent techniques have been applied to measure, or to infer, the spin relaxation\ntime of individual magnetic atoms using a STM. In the case when the intrinsic broadening ~T\u00001\n1\nassociated to the finite spin lifetime is larger than the thermal splitting, it can be extracted from\nthe full width at half maximum of the d2I=dV2spectra. Using this approach, the lifetime of\nFe atoms on Cu(111) was measured to be T1\u0019~=(2\u0001E)\u0019200 fs at 0:3 K [45]. In addition,\nKhajetoorians et al. , observed a linear scaling of the spin relaxation rate 1 =T1with the excitation\nenergy, tuned with the magnetic field, in agreement with Eq. (70).\nA second way to infer T1from the IETS spectra is based on the experimental results by\nLoth and coworkers [41], that observed how some inelastic steps appear when the current was\nincreased by reducing the tip-surface distance. The connection with T1is the following: when\nthe pace at which the current excites the spin, given by Iinelastic=e, is faster than the pace at\nwhich the atomic spin relaxes, T\u00001\n1, a non-equilibrium occupation of the excited atomic spin\nstatejX1ibuilds. This makes it possible to observe new inelastic steps corresponding to transi-\ntions fromjX1ito higher energy excited states jX2i. The inelastic current can be then estimated\nasIinelastic=e'\u0001GV, where \u0001Gis the height of the inelastic step associated to the primary inelas-\ntic transition, from ground state to jX1i. This method works better when the excitation energy\nEjX2i\u0000EjX1iis larger than the primary excitation EjX1i\u0000EjGi. In the case of a Mn dimer on a\nCu2N/Cu(100) surface, T1can be estimated around 30 and 5 ps for the first and second excited\nstates [41].\nThe pioneering implementation, in 2010, of electrical pump and probe measurements [43]\nwith a spin polarized STM provided a direct measurement of T1for an individual magnetic entity,\na dimer of Fe and Cu on top of Cu 2N. The working principle is the following: an electrical bias\npulse capable of driving the atomic spin out of equilibrium is applied on the tip-surface junction.\nA second smaller probe is send, after a time lapse \u001c, and the conductance is measured. As long as\n\u001c<T1, the atomic spin is still excited, and the magnetoresistive component of the conductance is\ndi\u000berent from the one prior to the pump pulse. This permitted a direct access to spin-transitions\noccurring on a time scale of a few tens of ns [43]. This technique has been applied for instance to\nstudy the relaxation time of a single Fe and Co atom on MgO /Ag(100) [55, 59]. or small arrays\nof magnetic adatoms on Cu 2N [48, 56, 69].\nYet another method to determine T1is possible for magnetic structures with two magnetic\nstates that are stable enough to resolve their random telegraph noise using a spin polarized tip [36,\n48, 56]. This requires that the switching time is slower than the time it takes to record the\nconductance, typically 1 ms. This is the case of long Fe spin chains on Cu 2N, coupled either\nAF [48] or ferromagnetically [56]. The histogram of the switching times \u001cscan be fit to an\nexponential function that allows one to fit the switching rate, T\u00001\n1. In the case of Fe chains\non Cu 2N, the switching rates of antiferromagnetic chains decreases dramatically for smaller\nchains, making it impossible to resolve the switching dynamics of individual Fe atoms with\nconventional measurements. Remarkably, the spin dynamics of individual Ho atoms on MgO\nhas been recently probed using this method [107], thanks to the extremely long switching time\nof Ho on this surface. Independent XMCD experiments for Ho adatoms on MgO demonstrate\nthe appearance of magnetic remanence for a single atom up to 30 Kelvin, with relaxation times\nexceeding 1500 seconds at 10 K [115]. A similar claim had been done for Ho on top of Pt [116].\nHowever, later XMCD measurements revealed no evidence of magnetic stability and a di \u000berent\nground state [117], while more recent SP-STM and IETS-STM measurements have not found\nany evidence of magnetic moment of Ho on Pt(111) [68].\n46Figure 6: Spin relaxation times of an Fe-Cu dimer on a Cu 2N/Cu(100) substrate measured by pump-probe scanning\ntunnelling microscopy. From S. Loth et al. , Science 329, 1628 (2010). Reprinted with permission from AAAS. (A)\nPump-probe measurements for di \u000berent magnetic fields on an Fe-Cu dimer. (B) T1as a function of magnetic field for\nthe two Fe-Cu dimers shown in the accompanying 5-nm by 5-nm STM topographs.\n6.4. Determination of T 2via EPR-STM\nThe implementation of electron paramagnetic resonance of an individual magnetic atom us-\ning an STM [59] has been one of the most recent dramatic developments in the field of scanning\nprobes. In a conventional EPR experiment, a static field B0splits the spin levels of the target\natoms, and the microwave ac magnetic field drives spin transitions. When the applied field B0\nis tuned to set a spin transition in resonance with the driving field frequency !0, the absorption\nis maximal. In c.w. experiments, the absorption of microwaves is probed as a function of B0,\nresulting in spectra with narrow resonance lines that provide valuable information about the lo-\ncal spins. Fitting this curve with the steady state solution of Bloch equation permits one to infer\nT2[81]. EPR is extensively used in a variety of fields, including biology, chemistry or physics.\nThe spectral resolution of EPR is limited by both instrumentation and by the intrinsic line width\nof the target spins. Spectral resolutions in the range of a few MHz permit probing the hyperfine\nstructure [187, 188], vastly superior to the spectral resolution of STM. The down side of conven-\ntional EPR is spatial resolution, or sensitivity: state-of-the-art EPR setups handle volumes larger\nthan 1\u0016m3, and the minimal number of spins that can be detected is in the range of 107[189].\nIn order to carry out a single spin resonance experiment with an STM it is necessary both to\ndrive the spins with an acsignal and to be able to probe their response. These are two challenging\nrequirements for a charge sensitive instrument. In order to drive the spin, Baumann et al. applied\nan ac. voltage in the 20-30 GHz range superimposed to the dcbias with an STM. They applied\nthis signal on top of an individual Fe atom on a MgO(100) layer grown on top of Ag [59]. The\ncoupling to the spin is believed to occur [59, 190, 191] via a combination of several e \u000bects: the\ntip electric field slightly distorts the position of the Fe atom, that in turn changes the crystal field\nof the atom; in combination with the spin-orbit coupling, transitions are induced between the two\nlowest energy states of the system.\nThe detection relies on the spin sensitivity of the tip that hosts one magnetic atom whose\norientation is fixed by the large external magnetic field. For a fixed value of the dc bias, the\ncurrent of the STM-tip-surface junction is scanned as a function of the RF frequency f, giving\naI(f) curve. A very narrow peak, with a full width at half maximum \u000ef'21 MHz, was found\nwhen fis in resonance with the lowest energy excitation f0= \u0001=h. Application of a small o \u000b-\nplane magnetic field tunes f0, resulting in a shift of the peak. Importantly, as the frequency of\n47the RF is changed, Baumann et al. [59] varied the power of the external RF source in order to\nmaintain a constant RF amplitude VRFat the STM tunnel junction.\nSpin contrast is verified by pump-probe experiments that are used to determine T1, in the\nrange of 100 \u0016s, much longer than the 0.2 ps of Fe on Cu(100) [45], and also Mn dimers on\nCu2N (5-30 ps). This highlights how the decoupling layer can increase dramatically the spin\nrelaxation time.\nThe determination of T2, together with the driving Rabi frequency \n, is done through fitting\nof the I(f) curves for various amplitudes VRFto the steady state solution of the Bloch equation\nfor a driven two level system [Eq. (39)]. Values of T2'210\u000650 ns were found, and \n'\n2:6\u00060:3rad=\u0016s[59]. The energy resolution of this experiment is 10 nanovolts, outperforming by\n3-4 orders of magnitude the one obtained with IETS. This setup can be used as magnetometer,\nas we discuss in the next subsection, thanks to its remarkable sensitivity to tiny variations of\nmagnetic field.\n6.5. Quantum technologies: quantum sensors with magnetic adatoms\nThe combination of all these single spin probing and manipulation techniques is opening now\nthe possibility to devise various sensing strategies. In the case of the STM-EPR experiment, the\nresonance curve of the Fe atom can be used as a probe for the magnetic moment of atoms nearby.\nTaking advantage of the atomic manipulation capabilities of STM, it is possible to change the\ndistance between the detection Fe atom under the tip, and the atoms nearby. When both the probe\nand target atoms are Fe on MgO, the shift in the resonance of the probe atom is expected to be\ngiven by\u00160m2\nFe=(4\u0019d3), where mFeis the magnetic moment of the Fe atom and dis their distance,\nthat can be determined with pm resolution [67]. This permits an absolute measurement of the\nmagnetic moment of Fe on this surface. Once the probe magnetic moment is calibrated, it can be\nused to measure the magnetic moments of di \u000berent atoms and structures. This has been used to\nprobe the magnetic moment of Ho atoms on MgO [107]. In this case, the spectral resolution is\nlimited by the intrinsic T2of Fe on MgO.\nA second example is the magnetometer designed by Yan and coworkers [69], a nanoengi-\nneered Fe spin chain on Cu 2N that acts as a probe, using the linear scaling between the spin\nrelaxation rate T\u00001\n1and its transition energy \u0001. The magnetic field created by a second nano-\nengineered structure, at a approximately 3 nm, results in a modification of the transition energy\nof the probe \u000e\u0001, that in turn, modifies the T1time of the probe, measured with the pump-probe\ntechnique. This setup can be used to detect the random telegraph switching of the source nanos-\ntructure, provided it is much longer than the T1of the probe structure. The working principle of\nthis setup benefits from a rapid spin relaxation of the probe.\n7. Outlook and conclusions\nA conclusion of this review is that the exploration of the spin coherence of surface spins by\nmeans of STM is just starting but has a promising future ahead. The development of the field will\ndepend crucially on the capability to increase the relaxation and coherence times of the surface\nspins. Given that Kondo interactions are the dominant source of spin relaxation, this translates\ninto a reduction of \u001aJ, the product of the density of electronic states at the Fermi level and the\nKondo exchange constant. The reduction of Jcan be achieved using thicker decoupling layers\nwith a wider gap. MgO and BN seem very promising materials in this regard.\nAnother route to reduce the e \u000bect of Kondo interactions is to use superconducting substrates,\nfor which the density of states at the Fermi energy vanishes. In fact, it has been shown that the\n48spin dynamics of individual magnetic adatoms slows down on superconducting substrates [51].\nThe modification of the the nuclear spin relaxation follows a non-monotonic curve, well under-\nstood within BCS theory. Future theory work should address this problem in the limit when the\nspin excitation energy is larger than the superconducting gap, in contrast with the nuclear spin\ncase, as well as the theory for T2due to the coupling with the superconductor.\nThe use of graphene as a conducting substrate might be helpful to reduce \u001aand increase T1\nthereby [192], on account of the vanishing density of states at the Dirac points, together with the\nsmall density of nuclear spins. In addition, graphene has is own very interesting spinful point\ndefects, such as chemisorbed hydrogen [193] and zigzag edges [194].\nIn addition to the reduction of \u001aJ, there are other more subtle ways to reduce the e \u000bect of\nKondo exchange. The spin relaxation rate of the Kondo exchange is controlled by the symmetry\nof the atomic spin wave functions, that in turn depends on the symmetry of the substrate. A\nproper choice of the adsorbate magnetic moment, the adsorption site and the surface can provide\nan almost full quench of the Kondo induced relaxation [115, 116, 195]. Multi-spin structures\nhave been predicted to provide additional opportunities to tune the spin relaxation [130].\nFinding other magnetic atoms and surfaces that permit STM-EPR experiments, beyond Fe on\nMgO, will be be another milestone for the development of this field. From the theory side it will\nbe important to understand in detail the mechanisms that make it possible to couple acelectric\nfields to the spin of the surface atoms [196, 197]. Another challenge for theory is to have a\nrealistic description of the system on two counts. First, to have a proper quantum spin model for\na given system, inferred from experiments or derived from first principles [139], or a combination\nof both [77]. Second, addressing the problem of the dissipative coupling of the surface spins to\ntheir environment. This problem can become particularly challenging in the case of strong Kondo\ncoupling that lead to the Kondo e \u000bect. On the other hand, in that limit we expect a very strong\ndecoherence, and from this perspective, it is a less interesting limit. Finally, a theory for the STM\ncurrent as a full functional of the atomic spin density matrix across, including coherences, might\nbe necessary to describe transport in the presence of the RF field.\nThe roadmap for the experimental development of the field of coherent manipulation of sur-\nface spins with STM will be inspired by the accomplishments with NV centers and P donors in\nsilicon. For that matter, it would be convenient to extend the pump-probe techniques [43] to the\ncoherent domain, to be able to perform coherent control experiments with pulsed RF perturba-\ntions. Time will tell how far can we make it in the development of quantum computers based\non atomic surface spins or if the remarkable feats a \u000borded by the exquisite quantum control of\nindividual spins can be combined with the amazing potential of STM to engineer spin structures.\nAcknowledgements\nWe acknowledge fruitful discussions and private communications , with A. Ardavan, S. Bau-\nmann, M. A. Cazalilla A. Ferr ´on, A. Heinrich, C. Hirjibehedin, N. Lorente, S. Loth, A. Morello,\nA. F. Otte, and J. Wrachtrup. JFR acknowledges financial supported by MEC-Spain (FIS2013-\n47328-C2-2-P) and Generalitat Valenciana (ACOMP /2010/070), Prometeo. This work is funded\nby ERDF funds through the Portuguese Operational Program for Competitiveness and Interna-\ntionalization COMPETE 2020, and National Funds through FCT- The Portuguese Foundation\nfor Science and Technology, under the project “PTDC /FIS-NAN /4662/2014” (016656). FD ac-\nknowledges funding by the Ministerio de Econom ´ıa y Competitividad (MINECO, Spain), with\ngrant MAT2015-66888-C3-2-R.. We thank J. Sinova for financial support in the SPICE work-\n49shop on “Magnetic adatoms as building blocks for quantum magnetism”. FD acknowledges\nhospitality of the Departamento de F ´ısica Aplicada, Universidad de Alicante.\nA. Bloch-Redfield tensor Rmm0;nn0\nIn this Appendix we provide the general expressions for the population scattering rates Rnn;mm\n(n,m), decoherence rates Re\u0002Rnn0;mm0\u0003(n,n0andm,m0), and energy shifts Im\u0002Rnn0;mm0\u0003. For\nthat matter, it is particularly convenient to introduce the reservoir correlation function g\u000b\f(t)=\nhR\u000b(t)R\f(0)ijeq, which can be recast as\ng\u000b\f(t)\u0011X\nrPrX\nr0Rrr0\n\u000bRr0r\n\fei(\u000fr\u0000\u000fr0)t=~; (A.1)\nwhere Rrr0\n\u000b=hrjR\u000bjr0i. Here we introduced the reservoir eigenvectors jriassociated to the eigen-\nvalues\u000fr, together with the thermal occupations Pr. The di \u000berent transition amplitudes and\nenergy shifts appearing in the Bloch-Redfield tensor can be written in terms of the Fourier trans-\nform\ng\u000b\f(!)=Z1\n0dt e\u0000i!tg\u000b\f(t) (A.2)\nThus, using Eq. (A.1) together with the Sokhotsky’s formula, one can write\ng\u000b;\f(!)=X\nrPrX\nr0Rrr0\n\u000bRr0r\n\f \n\u0019\u000e(!\u0000!rr0)\u0000iP1\n!\u0000!rr0!\n; (A.3)\nwherePstands for the Cauchy principal part.\nA.1. Population scattering 1=T1\u0011\u0000n;m\nForn,m, one gets after the substitution in Eqs. (26-27) that the scattering rate \u0000n;m=Rmm;nn\ncan be written in the form\n\u0000nm=2\u0019\n~2X\n\u000b;\fX\nrPrX\nr0Rrr0\n\u000bRr0r\n\fSnm\n\u000bSmn\n\f\u000e(!mn\u0000!rr0);\n(A.4)\nor, in terms of the correlation function g\u000b\f(!nm),\n\u0000nm=2\n~2X\n\u000b;\fReh\ng\u000b\f(!nm)i\nSnm\n\u000bSmn\n\f: (A.5)\nNotice that this tensor elements are real and positive, as corresponds to transition rates between\nthe eigenstates of the isolated system HS. They reproduce the result of the Fermi Golden Rule.\nA.2. Decoherence rates 1=T2\nThe evolution of the coherences \u001anmare dominated by the BR tensor components Rnm;nm. This\nsituation may change when there are other coherences \u001an0m0with degenerate Bohr frequencies,\ni.e.,j!n0m0\u0000!nmj\u001c 1=\u000et, in which case one can use the general expressions of Rnm;n0m0. For\nthe componentsRnm;nm, one can generally write Rnm;nm=\u0000\rnm\u0000i\u000e\u0001nm, with\rnm(\u000e\u0001) the real\n50(imaginary parts). We can split the decoherence rate \rnminto a non-adiabatic component \rnonad:\nnm ,\nwhich involves scattering between di \u000berent quantum system states, and a adiabatic one, where\nthe system does not actually changes its state. For the non-adiabatic one gets\n\rnonad:\nnm =1\n20BBBBB@X\nn0,n\u0000n;n0+X\nn0,m\u0000m;n01CCCCCA: (A.6)\nThe genuine decoherence mechanism, responsible of the pure decoherence, is given by the adia-\nbatic component\n\rad\nnm=\u0019\n~2X\nrPrX\nr0\f\f\f\f\f\f\fX\n\u000bRrr0\n\u000b\u0010\nSnn\n\u000b\u0000Smm\n\f\u0011\f\f\f\f\f\f\f2\n\u000e(!rr0); (A.7)\nwhich after introducing the correlator g\u000b\f(!) can be written in the form (33).\nSimilarly, one can also evaluate the decoherence rate due to the coupling with other co-\nherences. Using the general expressions (26) and (27), one gets for n,n0andm,m0the\ndecoherence rates\nRe\u0002Rnm;n0m0\u0003=X\n\u000b\f2\n~2Reh\nSm0m\n\u000bSnn0\n\f\u0010\ng\u000b\f(!nn0)+g\u0003\n\u000b\f(!mm0)\u0011i\n: (A.8)\nA.3. Energy shifts\nThe imaginary part of the BRF tensor leads to a modification of the bare Hamiltonian which\ncan be accounted for as a new renormalized Hamiltonian. The energy shifts \u000e\u0001nm\u0011\u000e!n\u0000\u000e!m\nassociated toRnm;nmcan be written in a similar way to the decoherence rates. Then, the general\nexpression is given by\n\u000e!m=1\n~2X\n\u000b;\fX\nn0Imh\ng\u000b;\f(!n0m)i\nSmn0\n\u000bSn0m\n\f: (A.9)\nIn addition to this energy shifts induced by the rates Rnm;nm, there can be additional terms coming\nfrom coupling with other coherences. By writing the Liouville operator as\nL(ˆ\u001a)=\u0000i\n~\u0002He\u000b;ˆ\u001a\u0003+RRe(ˆ\u001a); (A.10)\nwhereRRestands for the real part of the BR tensor, one can arrive to the following result [118]:\nhnjHe\u000bjmi=En\u000enm+X\n\u000b\fX\nn0Imh\ng\u000b\f(!n;n0)i\nSnn0\n\u000bSn0m\n\f: (A.11)\nImportantly, this e \u000bective Hamiltonian satisfies\u0002HS;He\u000b\u0003=0 [118].\nB. Bloch-Redfield tensor for the Kondo coupling\nIn this appendix we give explicit expressions for the Bloch-Redfield tensor in terms of the\nbath correlators.\n51In the case of the Kondo coupling with the conduction electrons of an electronic bath, which\ncould be the metallic substrate or an STM tip in the case of magnetic adatoms, the bath operators\nR\u000btakes the form\nR\u000b\u0011X\n\u001b;\u001b0X\n~k;~k0ei(~k\u0000~k0)\u0001~rl\u001ca\n\u001b;\u001b0\n2cy\n~k\u001bc~k0\u001b0; (B.1)\nwhere\u000b\u0011(l;a), with a=x;y;zandllabelling the atoms while S\u000b\u0011J(l)Sa(l).11\nConsidering fermionic bath states resulting from the creation of a single electron-hole pair,\none can obtain after some algebra that\ng\u000b;\f(!)\u0011\u000ea;b\n2X\n~k~k0f(\u000f~k)\u0010\n1\u0000f(\u000f~k0)\u0011\nei\u0010~k\u0000~k0\u0011\n:\u0001(~rl\u0000~rl0)ei!~k~k0t(B.2)\nwhere\f\u0011(l0;b) and f(x) is the Fermi-Dirac occupation distribution. Here we have assumed for\nsimplicity that the only reservoir quantum number are the spin \u001band the wavevector ~k. Notice\nthat the assumption of null expectation value of the interaction is equivalent in this case to assume\na spin-unpolarized electronic reservoir. Then, assuming a constant density of states \u001aat the Fermi\nlevel and using Eqs. (B.1-B.2), which leads to a analytical expression of the energy integrals,\ntogether with Eqs. (26-27), the real part of the BRF tensor can be then written as\nRnn0;mm0=\u0019\n~\u001a2h\n\u0000\u000en0;m0X\nn00\u0003n;n00;n00;mG(\u0001mn00)\n+\u0003 n;m;m0;n0G(\u0001mn)\u0000\u000en;mX\nn00\u0003m0;n00;n00;n0G(\u0000\u0001n00m0)\n+\u0003 n;m;m0;n0G(\u0000\u0001n0m0)i\n; (B.3)\nwithG(\u0001)= \u0001=(1\u0000e\u0000\f\u0001) and\f=1=kbT. The spin-dependent matrix elements are defined as\n\u0003nn0;mm0=1\n4X\nl;l0;aJ(l)J(l0)\u0015ll0(kF)Sa\nnn0(l)Sa\nmm0(l0) (B.4)\nwhere Sa\nnm(l)\u0011hEnjSa\nljEmi. In addition, we have introduced the factors \u0015ll0(kF) that depends\non the geometrical distribution of the spins together with the dimensionality of the electron gas\nand Fermi wavenumber. Assuming that in the small neighborhood of the Fermi level, where the\nproduct of the Fermi functions in Eq. (B.2) are non-zero, we can approximate j~kj\u0019kF, we have\nthat\n\u0015l;l0(kF)=1\n\n2Z\ndˆkdˆk0exph\n\u0006ikF\u0010ˆk\u0000ˆk0\u0011\n\u0001\u0000~rl\u0000~rl0\u0001i\n; (B.5)\nwith ˆk=~k=j~kjand\n =R\ndˆk. The phase integral \u0015l;l0(kF) is a function of the Fermi wavenumber\nkFand the dimensionality of the electron gas. For a linear chain of equidistant spins one can find\nexplicit analytical expressions of \u0015ll0(kF) [130].\n11When the system is coupled to more than one electronic bath at di \u000berent chemical potentials, as it would be the case\nfor adatoms subjected to a tunnel current as when studied by STM, the indexes \u000b;\fshould also include an electrode index\nthat must be summed up in V. This will give place to intra-electrodes and also inter-electrode scattering events [84].\n52B.1. Bloch-Redfield tensor of a pseudo-spin 1=2\nIn this section we provide the explicit expressions for all the Bloch-Redfield tensor elements\nRnn0;mm0in the case of a 2-level spin system. For convenience, we define \u0001 =E2\u0000E1\u00150. From\nthe hermiticity of the density matrix operator ˆ \u001aone has thatR\u0003\nnn0;mm0=Rn0n;m0m. Using Eq. (B.3),\nwe get the pure population rates\nR22;11=\u0000R 11;11=\u0019\u001a2\n~\u00031;2;2;1G(\u0000\u0001);\nR11;22=\u0000R 22;22=\u0019\u001a2\n~\u00031;2;2;1G(\u0001);\n(B.6)\nand the scattering rates between populations and coherences\nR11;21=R\u0003\n11;12=\u0019\u001a2\n~h\n\u0000\u00031;2;2;2G[0]\u0000\u00031;1;1;2G[\u0001]\n+\u0003 1;2;1;1(G[0]+G[\u0001])i\n;\nR22;21=R\u0003\n22;12=\u0019\u001a2\n~h\n\u0000\u00032;1;11G[0]\n\u0000\u00032;2;12G[\u0000\u0001]+ \u0003 2;1;2;2(G[0]+G[\u0000\u0001])i\n;\nR12;11=R\u0003\n21;11=\u0019\u001a2\n~h\n\u00031;1;1;2\u0000\u00031;2;2;2\u0003G[\u0000\u0001]\nR12;22=R\u0003\n21;22=\u0019\u001a2\n~h\n\u0000\u00031;1;1;2+ \u0003 1;2;2;2i\nG[\u0001]:\n(B.7)\nIn addition, there are two di \u000berent transition rates between the coherences\nR12;12=R\u0003\n21;21=\u0000\u0019\u001a2\n~\u00031;2;2;1(G[\u0000\u0001]+G[\u0001])\nR12;21=R\u0003\n21;12=\u0019\u001a2\n~\u00031;2;1;2(G[\u0000\u0001]+G[\u0001]): (B.8)\nB.2. Relation between the M-matrix and the Bloch-Redfield tensor\nWhen the time evolution of the linearly independent components of the density matrix of a\ndegenerate 2-level system, encoded in the vector ~Pof Eq. (84), are written in terms of the BR\ntensor components Rnm;n0m0, the di \u000berent matrix elements in Eq. (85) are given by\n\u0000 =R11;22+R22;11;\n\r=\u00002Re\u0002R12;12+R21;12\u0003;\n\r0=\u00002Im\u0002R12;12+R21;21\u0003;\nMp;r=\u00002Re\u0002R11;12\u0000R 22;12\u0003;\nMp;i=2Im\u0002R11;12\u0000R 22;12\u0003;\nMr;p=\u0000Re\u0002R12;11\u0000R 12;22\u0003=2;\nMr;i=\u0000Im\u0002R12;12\u0000R 12;21\u0003;\nMi;p=\u0000Im\u0002R12;11\u0000R 12;22\u0003=2;\nMi;r=Im\u0002R12;11+R12;21\u0003=2: (B.9)\n53C. Bosonic representation of the excitations of the Fermi gas and the spin-boson model\nThe mapping to the spin-boson model makes use of the so called bosonization technique [198]\nthat permits mapping the excitations of a one dimensional interacting Fermi system into a theory\nof bosons that, in some instances, reduces to a free boson theory. For 2D and 3D baths, it is still\npossible to use the bosonization technique when it comes to describe non-interacting fermions\ncoupled to a local impurity with rotational symmetry, such as the single impurity Kondo Hamilto-\nnian. Thus, one can use the partial the wave decomposition of the scattering states and keep only\nthes-wave states, which basically leads to a set of decoupled one dimensional channels [162].\nNotice that, in general, the rotation invariance condition is not satisfied by the spin-phonon cou-\npling.\nIntroducing the linear dispersion \u000fk=~vFjkj, valid in a small region around the Fermi energy,\nthe free electron Hamiltonian can be rewritten as\nh0=~vFX\nk\u001bjkjcy\nk;\u001bck\u001b: (C.1)\nNext, we define the following bosonic operators\nbk=\u0012\u0019\nkL\u00131=2X\nq\u001b\u001bcy\nq\u0000k;\u001bcq\u001b; (C.2)\nwhere Lis the length of a normalization box such that the wave vectors are quantized as kn=\n2\u0019n=Land the limit L!1 is finally taken. Hence, by\nkbasically creates a spin-flip electron hole\npair. Then,HRcan be written as\nh0=~vFX\nk\u0014kcjkjby\nkbk; (C.3)\nwhere kcis a momentum cut-o \u000bof the order of the bandwidth introduced to remove wavevectors\nbeyond the scale of the Fermi wavenumber kF.\nThe bosonization technique is particularly suitable for the Ising coupling. In fact, when the\nKondo exchange coupling projected into the low energy doublet takes the form\nVK\u0019jzˆ\u001czsz(0); (C.4)\nthe bosonized version of the Kondo Hamiltonian involves only the spin density operator [11]. As\nexplained in Sec. 3.2, this is the case of the quantum type of nanomagnets, such as integer spins\ndescribed by Hamiltonian (43). The Ising part of the fermion spin density can be then written\ndown in terms of the bosonic operators as:\nsz(0)=X\n0<k<kc k\n\u0019L!1=2\n(by\nk+bk): (C.5)\nIn the following we introduce two important quantities, the density of conduction electrons\nstates at the Fermi energy, given by \u001a=(2\u0019~vF)\u00001and the dimensionless coupling constant\n\u000b=(\u001ajz)2. Using Eqs. (C.3) and (C.5), we can can write the Hamiltonian of the truncated\nTLS interacting with the electrons gas in the bosonized form. This can be easily done in the two\nlimiting cases in which the Kondo interaction is reduced to the Ising form. First, for the integer\n54spin, described by the coupling Hamiltonian (63) in the basis (51). By making a \u0019=2 rotation\naround the yaxis, we can write the resulting Hamiltonian in the bosonized form as\nHSBQ=h0\u0000\u0001\n2ˆ\u001cx+ˆ\u001czp\u000bX\n0\u0014k\u0014kcgk\u0010\nby\nk+bk\u0011\n; (C.6)\nwith gk=~vf(\u0019k=L)1=2. This corresponds to the spin-boson model for an Ohmic spectral den-\nsity [11], a model introduced to study the dissipative dynamics of a TLS. Although exact an-\nalytical results are not available, it admits nonperturbative solutions for some phase space pa-\nrameters [11]. Furthermore, these approximate solutions are in good agreement with numerical\nanalysis based on numerical renormalization group [199].\nThe second case of interest corresponds to half-integer anisotropic spin with uniaxial anisotropy,\ndescribed by Hamiltonian (62) in the basis of classical states (50) for E=0:\nHSBC=h0+ˆ\u001czp\u000bX\n0\u0014k\u0014kcgk(by\nk+bk): (C.7)\nThe spin-boson model is mathematically equivalent to a set of displaced harmonic oscillators [11].\nThe sign of the displacement is provided by ˆ \u001cz, which in the case of half-integer spins is the ori-\nentation of the magnetic moment along the easy axis.\nIt is worth mentioning that, in general, even a single spin Kondo-coupled to the electron\ngas leads to a more complex result, since it may involve sx;y(0) operators that requires a more\ncomplex treatment [11]. In addition, in the case spin arrays, the coupling constant is no longer\nmomentum independent, which results in deviations from the Ohmic behavior [130, 132].\nD. Steady-state solution of the Bloch equation for the TLS\nIn this appendix derive the steady state solution of the Bloch equations equations (35) and\n(36). We assumeVba\n1(t)=Vbacos!tand we look for their steady state solution. For that matter,\nwe assume that the diagonal entries of ˆ \u001a(t) are time independent whereas the o \u000b-diagonal part\noscillates in phase with the perturbation, i. e.\n\u001aaa(t)\u0011Pa; \u001a bb(t)\u0011Pb; \u001a ab(t)=cei!t(D.1)\nwhere cis complex and PaandPbare real constants. The ansatz (D.1) is compatible with\nthe rotating wave approximation, where the fast rotating terms e\u0006i(!ab+!)tare removed from the\nequation of motion of ˆ \u001a(t). In terms of a classical magnetization field dynamics, this amounts to\ndescribe the evolution of the magnetization in a frame rotating at the Larmor frequency. Then,\nthe steady state version of Eqs. (35-36) reduces to\n0=1\n2T1\u0010\n(Pb\u0000Pa)\u0000(Peq\nb\u0000Peq\na)\u0011\n+i\n2\n(c\u0000c\u0003) (D.2)\ni!c=\u0000c\nT2+i\u0001\n~c\u0000i\n2\n(Pb\u0000Pa); (D.3)\nwhere for simplicity we have assumed a real Rabi frequency \n = Vba=~. We can write these\nequations down as:\n0BBBBBBBB@1=T2\u0000i\u000e 0\u0000i\n=2\n0 1=T2+i\u000e i\n=2\ni\n=2\u0000i\n=2 1=(2T1)1CCCCCCCCA0BBBBBBBB@c\nc\u0003\nPb\u0000Pa1CCCCCCCCA=0BBBBBBBB@0\n0\n(Peq\nb\u0000Peq\na)=(2T1)1CCCCCCCCA; (D.4)\n55where we have introduced the detuning \u000e= \u0001=~\u0000!. If we define \u000eP=Pb\u0000Pa(and\u000ePeq=\nPeq\nb\u0000Peq\na), the steady state occupation imbalance is given by:\n \u000eP\u0000\u000ePeq\n\u000ePeq!\n=\u0000\n2T1T2\n1+T2\n2\u000e2+T1T2\n2: (D.5)\nNotice that, by definition, \u000016\u000eP61, whilejcj61=2. We also notice that the micro-\nscopic Bloch equations (35) and (36) slightly di \u000ber from the macroscopic dynamical evolution\nof the magnetization components after making the identification (37). Basically, the role of the\nrelaxation time T1of the occupations is substituted by 2 \u001c1, with\u001c1the relaxation time of the\nz-component of the magnetization.\nReferences\n[1] C. P. Slichter, Principles of magnetic resonance, V ol. 1, Springer Science & Business Media, 2013.\n[2] I. L. Pykett, J. H. Newhouse, F. S. Buonanno, T. J. Brady, M. R. Goldman, J. P. Kistler, G. M. Pohost, Principles\nof nuclear magnetic resonance imaging, Radiology 143 (1) (1982) 157–168.\n[3] B. Cavenett, Optically detected magnetic resonance (ODMR) investigations of recombination processes in semi-\nconductors, Adv. Phys. 30 (4) (1981) 475–538.\n[4] F. Jelezko, J. Wrachtrup, Single defect centres in diamond: A review, Phys. Status Solidi (a) 203 (13) (2006)\n3207–3225.\n[5] G. Balasubramanian, I. Chan, R. Kolesov, M. Al-Hmoud, J. Tisler, C. Shin, C. Kim, A. Wojcik, P. R. Hem-\nmer, A. Krueger, et al., Nanoscale imaging magnetometry with diamond spins under ambient conditions, Nature\n455 (7213) (2008) 648–651.\n[6] I. Kominis, T. Kornack, J. Allred, M. Romalis, A subfemtotesla multichannel atomic magnetometer, Nature\n422 (6932) (2003) 596–599.\n[7] B. Kane, A silicon-based nuclear spin quantum computer, Nature 393 (1998) 133–137.\n[8] D. Loss, D. P. DiVincenzo, Quantum computation with quantum dots, Phys. Rev. A 57 (1) (1998) 120–126.\ndoi:10.1103 /PhysRevA.57.120.\n[9] M. N. Leuenberger, D. Loss, Quantum computing in molecular magnets, Nature 410 (2001) 789–793.\n[10] A. Ardavan, O. Rival, J. J. L. Morton, S. J. Blundell, A. M. Tyryshkin, G. A. Timco, R. E. P. Winpenny, Will\nspin-relaxation times in molecular magnets permit quantum information processing?, Phys. Rev. Lett. 98 (2007)\n057201.\n[11] A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg, W. Zwerger, Dynamics of the dissipative\ntwo-state system, Rev. Mod. Phys. 59 (1987) 1–85. doi:10.1103 /RevModPhys.59.1.\n[12] W. H. Zurek, Decoherence, einselection, and the quantum origins of the classical, Rev. Mod. Phys. 75 (2003)\n715–775. doi:10.1103 /RevModPhys.75.715.\n[13] D. Suter, G. A. ´Alvarez, Colloquium : Protecting quantum information against environmental noise, Rev. Mod.\nPhys. 88 (2016) 041001. doi:10.1103 /RevModPhys.88.041001.\n[14] S. L. Adler, Why decoherence has not solved the measurement problem: a response to PW Anderson, Studies in\nHistory and Philosophy of Science Part B 34 (1) (2003) 135–142.\n[15] G. Bacciagaluppi, The role of decoherence in quantum mechanics, in: E. N. Zalta (Ed.), The Stanford Encyclope-\ndia of Philosophy, fall 2016 Edition, 2016.\n[16] W. H. Zurek, Decoherence and the transition from quantum to classical, Physics Today 44 (1991) 36–44.\n[17] F. Delgado, S. Loth, M. Zielinski, J. Fern ´andez-Rossier, The emergence of classical behaviour in magnetic\nadatoms, EPL (Europhysics Letters) 109 (5) (2015) 57001.\n[18] A. Garg, Topologically quenched tunnel splitting in spin systems without Kramers’ degeneracy, EPL (Europhysics\nLetters) 22 (3) (1993) 205.\n[19] D. Gatteschi, R. Sessoli, J. Villain, Molecular nanomagnets, Oxford University Press, New York, 2006.\n[20] P. W. Anderson, Concepts in solids: lectures on the theory of solids, V ol. 58, World Scientific, 1997.\n[21] P. W. Anderson, Basic notions of condensed matter physics, Benjamin /Cummings Pub. Co., London, Advanced\nBook Program, 1984.\n[22] H. C. Donker, H. De Raedt, M. I. Katsnelson, Decoherence wave in magnetic systems and creation of N ´eel\nantiferromagnetic state by measurement, Phys. Rev. B 93 (2016) 184426. doi:10.1103 /PhysRevB.93.184426.\n[23] C. G. Shull, W. Strauser, E. Wollan, Neutron di \u000braction by paramagnetic and antiferromagnetic substances, Phys.\nRev. 83 (2) (1951) 333.\n56[24] M. Katsnelson, V . Dobrovitski, B. Harmon, N ´eel state of an antiferromagnet as a result of a local measurement in\nthe distributed quantum system, Phys. Rev. B 63 (21) (2001) 212404.\n[25] D. M. Eigler, E. K. Schweize, Positioning single atoms with a scanning tunnelling microscope, Nature 344 (1990)\n524–5.\n[26] R. Wiesendanger, H.-J. G ¨untherodt, G. G ¨untherodt, R. Gambino, R. Ruf, Observation of vacuum tunneling of\nspin-polarized electrons with the scanning tunneling microscope, Phys. Rev. Lett. 65 (2) (1990) 247.\n[27] A. Yazdani, B. Jones, C. Lutz, M. Crommie, D. Eigler, Probing the local e \u000bects of magnetic impurities on super-\nconductivity, Science 275 (5307) (1997) 1767–1770.\n[28] V . Madhavan, W. Chen, T. Jamneala, M. Crommie, N. Wingreen, Tunneling into a single magnetic atom: spectro-\nscopic evidence of the Kondo resonance, Science 280 (5363) (1998) 567–569.\n[29] A. J. Heinrich, J. A. Gupta, C. P. Lutz, D. M. Eigler, Single-atom spin-flip spectroscopy, Science 306 (2004)\n466–469.\n[30] C. F. Hirjibehedin, C. P. Lutz, A. J. Heinrich, Spin coupling in engineered atomic structures, Science 312 (2006)\n1021.\n[31] C. Hirjibehedin, C.-Y . Lin, A. Otte, M. Ternes, C. P. Lutz, B. A. Jones, A. J. Heinrich, Large magnetic anisotropy\nof a single atomic spin embedded in a surface molecular network, Science 317 (2007) 1199.\n[32] F. Meier, L. Zhou, J. Wiebe, R. Wiesendanger, Revealing magnetic interactions from single-atom magnetization\ncurves, Science 320 (2008) 82.\n[33] A. F. Otte, M. Ternes, K. von Bergmann, S. Loth, H. Brune, C. P. Lutz, C. F. Hirjibehedin, A. J. Heinrich, The\nrole of magnetic anisotropy in the Kondo e \u000bect, Nature Physics 4 (2008) 847.\n[34] X. Chen, Y .-S. Fu, S.-H. Ji, T. Zhang, P. Cheng, X.-C. Ma, X.-L. Zou, W.-H. Duan, J.-F. Jia, Q.-K. Xue, Probing\nsuperexchange interaction in molecular magnets by spin-flip spectroscopy and microsocpy, Phys. Rev. Lett. 101\n(2008) 197208.\n[35] S.-H. Ji, T. Zhang, Y .-S. Fu, X. Chen, X.-C. Ma, J. Li, W.-H. Duan, J.-F. Jia, Q.-K. Xue, High-resolution scanning\ntunneling spectroscopy of magnetic impurity induced bound states in the superconducting gap of Pb thin films,\nPhys. Rev. Lett. 100 (22) (2008) 226801. doi:10.1103 /PhysRevLett.100.226801.\n[36] R. Wiesendanger, Spin mapping at the nanoscale and atomic scale, Rev. Mod. Phys. 81 (2009) 1495–1550.\n[37] N. Tsukahara, K. Noto, M. Ohara, S. Shiraki, N. Takagi, Y . Takata, J. Miyawaki, M. Taguchi, A. Chainani, S. Shin,\nM. Kawai, Adsorption-induced switching of magnetic anisotropy in a single iron(ii) phthalocyanine molecule on\nan oxidized Cu(110) surface, Phys. Rev. Lett. 102 (2009) 167203.\n[38] A. F. Otte, M. Ternes, S. Loth, C. P. Lutz, C. F. Hirjibehedin, A. J. Heinrich, Spin excitations of\na Kondo-screened atom coupled to a second magnetic atom, Phys. Rev. Lett. 103 (10) (2009) 107203.\ndoi:10.1103 /PhysRevLett.103.107203.\n[39] Y . S. Fu, T. Zhang, S. H. Ji, X. Chen, X. C. Ma, J. F. Jia, Q. K. Xue, Identifying charge states of molecules with\nspin-flip spectroscopy, Phys. Rev. Lett. 103 (2009) 257202.\n[40] D. Serrate, P. Ferriani, Y . Yoshida, S.-W. Hla, M. Menzel, K. V on Bergmann, S. Heinze, A. Kubetzka, R. Wiesen-\ndanger, Imaging and manipulating the spin direction of individual atoms, Nature Nanotechnology 5 (5) (2010)\n350–353.\n[41] S. Loth, K. von Bergmann, M. Ternes, A. F. Otte, C. P. Lutz, A. J. Heinrich, Controlling the state of quantum\nspins with electric currents, Nature Physics 6 (2010) 340–344.\n[42] A. A. Khajetoorians, B. Chilian, J. Wiebe, S. Schuwalow, F. Lechermann, R. Wiesendanger, Detecting excitation\nand magnetization of individual dopants in a semiconductor, Nature 467 (2010) 1084–1087.\n[43] S. Loth, M. Etzkorn, C. P. Lutz, D. M. Eigler, A. J. Heinrich, Measurement of fast electron spin relaxation times\nwith atomic resolution, Science 329 (5999) (2010) 1628–1630. doi:10.1126 /science.1191688.\n[44] S. Kahle, Z. Deng, N. Malinowski, C. Tonnoir, A. Forment-Aliaga, N. Thontasen, G. Rinke, D. Le, V . Turkowski,\nT. S. Rahman, S. Rauschenbach, M. Ternes, K. Kern, The quantum magnetism of individual manganese-12-acetate\nmolecular magnets anchored at surfaces, Nano Letters 12 (1) (2011) 518–521.\n[45] A. A. Khajetoorians, S. Lounis, B. Chilian, A. T. Costa, L. Zhou, D. L. Mills, J. Wiebe, R. Wiesendanger,\nItinerant nature of atom-magnetization excitation by tunneling electrons, Phys. Rev. Lett. 106 (2011) 037205.\ndoi:10.1103 /PhysRevLett.106.037205.\n[46] A. Khajetoorians, J. Wiebe, B. Chilian, R. Wiesendanger, Realizing all-spin–based logic operations atom by atom,\nScience 332 (6033) (2011) 1062–1064.\n[47] A. Khajetoorians, J. Wiebe, B. Chilian, S. Lounis, S. Bl ¨ugel, R. Wiesendanger, Atom-by-atom engineering and\nmagnetometry of tailored nanomagnets, Nature Physics 8 (6) (2012) 497–503.\n[48] S. Loth, S. Baumann, C. P. Lutz, D. M. Eigler, A. J. Heinrich, Bistability in atomic-scale antiferromagnets, Science\n335 (6065) (2012) 196–199. doi:10.1126 /science.1214131.\n[49] A. A. Khajetoorians, B. Baxevanis, C. H ¨ubner, T. Schlenk, S. Krause, T. O. Wehling, S. Lounis, A. Lichtenstein,\nD. Pfannkuche, J. Wiebe, et al., Current-driven spin dynamics of artificially constructed quantum magnets, Science\n339 (6115) (2013) 55–59.\n57[50] A. A. Khajetoorians, T. Schlenk, B. Schweflinghaus, M. dos Santos Dias, M. Steinbrecher, M. Bouhas-\nsoune, S. Lounis, J. Wiebe, R. Wiesendanger, Spin excitations of individual Fe atoms on Pt(111):\nImpact of the site-dependent giant substrate polarization, Phys. Rev. Lett. 111 (2013) 157204.\ndoi:10.1103 /PhysRevLett.111.157204.\n[51] B. Heinrich, L. Braun, J. Pascual, K. Franke, Protection of excited spin states by a superconducting energy gap,\nNature Physics 9 (12) (2013) 765–768.\n[52] J. C. Oberg, M. R. Calvo, F. Delgado, M. Moro-Lagares, D. Serrate, D. Jacob, J. Fern ´andez-Rossier, C. F. Hir-\njibehedin, Control of single-spin magnetic anisotropy by exchange coupling, Nature Nanotechnology 9 (2014)\n64–68. doi:doi:10.1038 /nnano.2013.264.\n[53] B. Bryant, A. Spinelli, J. J. T. Wagenaar, M. Gerrits, A. F. Otte, Local control of single atom magnetocrystalline\nanisotropy, Phys. Rev. Lett. 111 (2013) 127203. doi:10.1103 /PhysRevLett.111.127203.\n[54] S. Nadj-Perge, I. K. Drozdov, J. Li, H. Chen, S. Jeon, J. Seo, A. H. MacDonald, B. A. Bernevig, A. Yazdani,\nObservation of Majorana fermions in ferromagnetic atomic chains on a superconductor, Science 346 (6209) (2014)\n602–607. doi:10.1126 /science.1259327.\n[55] I. G. Rau, S. Baumann, S. Rusponi, F. Donati, S. Stepanow, L. Gragnaniello, J. Dreiser, C. Piamonteze, F. Nolting,\nS. Gangopadhyay, et al., Reaching the magnetic anisotropy limit of a 3d metal atom, Science 344 (6187) (2014)\n988–992.\n[56] A. Spinelli, B. Bryant, F. Delgado, J. Fern ´andez-Rossier, A. Otte, Imaging of spin waves in atomically designed\nnanomagnets, Nature Materials 13 (8) (2014) 782–785.\n[57] A. Spinelli, M. Gerrits, R. Toskovic, B. Bryant, M. Ternes, A. F. Otte, Competition between Kondo screening and\nexchange interaction in pairs of atoms, arXiv:1411.4415 (2014).\n[58] S. Yoshida, Y . Aizawa, Z.-h. Wang, R. Oshima, Y . Mera, E. Matsuyama, H. Oigawa, O. Takeuchi, H. Shigekawa,\nProbing ultrafast spin dynamics with optical pump-probe scanning tunnelling microscopy, Nature Nanotechnology\n9 (8) (2014) 588–593.\n[59] S. Baumann, W. Paul, T. Choi, C. P. Lutz, A. Ardavan, A. J. Heinrich, Electron paramagnetic resonance of\nindividual atoms on a surface, Science 350 (6259) (2015) 417–420.\n[60] J. A. Burgess, L. Malavolti, V . Lanzilotto, M. Mannini, S. Yan, S. Ninova, F. Totti, S. Rolf-Pissarczyk, A. Cornia,\nR. Sessoli, et al., Magnetic fingerprint of individual Fe4 molecular magnets under compression by a scanning\ntunnelling microscope, Nature Communications 6.\n[61] B. Bryant, R. Toskovic, A. Ferr ´on, J. Lado, A. Spinelli, J. Fern ´andez-Rossier, A. Otte, Controlled complete\nsuppression of single-atom inelastic spin and orbital cotunnelling, Nano Letters (2015) 6542–6546.\n[62] D.-J. Choi, R. Robles, S. Yan, J. A. Burgess, S. Rolf-Pissarczyk, J.-P. Gauyacq, N. Lorente, M. Ternes, S. Loth,\nEntanglement-induced Kondo screening in atomic spin chains, arXiv:1507.04785 (2015).\n[63] P. Jacobson, T. Herden, M. Muenks, G. Laskin, O. Brovko, V . Stepanyuk, M. Ternes, K. Kern, Quantum engineer-\ning of spin and anisotropy in magnetic molecular junctions, Nature Communications 6 (2015) 8536.\n[64] A. Spinelli, M. Gerrits, R. Toskovic, B. Bryant, M. Ternes, A. Otte, Exploring the phase diagram of the two-\nimpurity Kondo problem, Nature Communications 6.\n[65] S. Yan, D.-J. Choi, J. A. Burgess, S. Rolf-Pissarczyk, S. Loth, Three-dimensional mapping of single-atom mag-\nnetic anisotropy, Nano Letters 15 (3) (2015) 1938–1942.\n[66] S. Yan, D.-J. Choi, J. A. Burgess, S. Rolf-Pissarczyk, S. Loth, Control of quantum magnets by atomic exchange\nbias, Nature Nanotechnology 10 (1) (2015) 40–45.\n[67] T. Choi, W. Paul, S. Rolf-Pissarczyk, A. Macdonald, K. Yang, F. Natterer, C. Lutz, A. Heinrich, Magnetic dipole-\ndipole sensing at atomic scale using electron spin resonance STM, Bulletin of the American Physical Society.\n[68] M. Steinbrecher, A. Sonntag, M. dos Santos Dias, M. Bouhassoune, S. Lounis, J. Wiebe, R. Wiesendanger,\nA. Khajetoorians, Absence of a spin-signature from a single Ho adatom as probed by spin-sensitive tunneling,\nNature Communications 7.\n[69] S. Yan, L. Malavolti, J. A. Burgess, S. Loth, Non-locally sensing the spin states of individual atomic-scale nano-\nmagnets, arXiv:1601.02723.\n[70] D. W. S. Heinze, P. Kurz, G. Bihlmayer, S. Bl ¨ugel, Resolving complex atomic-scale spin structures by spin-\npolarized scanning tunneling microscopy, Phys. Rev. Lett. 86 (2001) 4132.\n[71] L. Yu, Bound state in superconductors with paramagnetic impurities, Acta Phys. Sin. 21 (1965) 75.\n[72] H. Shiba, Classical spins in superconductors, Prog. Theo. Phys. 40 (3) (1968) 435–451. doi:10.1143 /PTP.40.435.\n[73] A. Rusinov, On the theory of gapless superconductivity in alloys containing paramagnetic impurities, Sov. Phys.\nJETP 29 (1969) 1101.\n[74] J. Korringa, Nuclear magnetic relaxation and resonnance line shift in metals, Physica 16 (7) (1950) 601–610.\n[75] D. C. Langreth, J. W. Wilkins, Theory of spin resonance in dilute magnetic alloys, Phys. Rev. B 6 (9) (1972) 3189.\n[76] F. Delgado, C. Hirjibehedin, J. Fern ´andez-Rossier, Consequences of Kondo exchange on quantum spins, Surface\nScience 630 (0) (2014) 337 – 342. doi:http: //dx.doi.org /10.1016 /j.susc.2014.07.009.\n[77] S. Baumann, F. Donati, S. Stepanow, S. Rusponi, W. Paul, S. Gangopadhyay, I. G. Rau, G. E. Pacchioni, L. Grag-\n58naniello, M. Pivetta, J. Dreiser, C. Piamonteze, C. P. Lutz, R. M. Macfarlane, B. A. Jones, P. Gambardella, A. J.\nHeinrich, H. Brune, Origin of perpendicular magnetic anisotropy and large orbital moment in Fe atoms on MgO,\nPhys. Rev. Lett. 115 (2015) 237202. doi:10.1103 /PhysRevLett.115.237202.\n[78] T. Choi, J. A. Gupta, Building blocks for studies of nanoscale magnetism: adsorbates on ultrathin insulating\nCu2N, J. Phys.: Condensed Matter 26 (39) (2014) 394009.\n[79] A. Khajetoorians, M. Steinbrecher, M. Ternes, M. Bouhassoune, M. dos Santos Dias, S. Lounis, J. Wiebe,\nR. Wiesendanger, Tailoring the chiral magnetic interaction between two individual atoms, Nature Communi-\ncations 7. doi:10.1038 /ncomms10620.\n[80] J. Fern ´andez-Rossier, Theory of single-spin inelastic tunneling spectroscopy, Phys. Rev. Lett. 102 (2009) 256802.\n[81] A. Abragam, B. Bleaney, Electron Paramagnetic Resonance of Transition Ions, Oxford University Press, Oxford,\n1970.\n[82] E. Manousakis, The spin-1 /2 Heisenberg antiferromagnet on a square lattice and its application to the cuprous\noxides, Rev. Mod. Phys. 63 (1991) 1–62. doi:10.1103 /RevModPhys.63.1.\n[83] R. C. Jaklevic, J. Lambe, Molecular vibration spectra by electron tunneling, Phys. Rev. Lett. 17 (1966) 1139.\n[84] F. Delgado, J. Fern ´andez-Rossier, Spin dynamics of current-driven single magnetic adatoms and molecules, Phys.\nRev. B 82 (13) (2010) 134414. doi:10.1103 /PhysRevB.82.134414.\n[85] M. dos Santos Dias, B. Schweflinghaus, S. Bl ¨ugel, S. Lounis, Relativistic dynamical spin excitations of magnetic\nadatoms, Phys. Rev. B 91 (7) (2015) 075405.\n[86] D. Jacob, J. Fern ´andez-Rossier, Competition between quantum spin tunneling and Kondo e \u000bect, Eur. Phys. J. B\n89 (10) (2016) 210. doi:10.1140 /epjb/e2016-70402-2.\n[87] P. M. Koenraad, M. E. Flatt ´e, Single dopants in semiconductors, Nature Materials 10 (2) (2011) 91–100.\n[88] J. Fern ´andez-Rossier, Single-atom devices: quantum engineering, Nature Materials 12 (6) (2013) 480–481.\n[89] F. A. Zwanenburg, A. S. Dzurak, A. Morello, M. Y . Simmons, L. C. Hollenberg, G. Klimeck, S. Rogge, S. N.\nCoppersmith, M. A. Eriksson, Silicon quantum electronics, Rev. Mod. Phys. 85 (3) (2013) 961.\n[90] J. J. Pla, K. Y . Tan, J. P. Dehollain, W. H. Lim, J. J. Morton, D. N. Jamieson, A. S. Dzurak, A. Morello, A\nsingle-atom electron spin qubit in silicon, Nature 489 (7417) (2012) 541–545.\n[91] J. T. Muhonen, J. P. Dehollain, A. Laucht, F. E. Hudson, R. Kalra, T. Sekiguchi, K. M. Itoh, D. N. Jamieson, J. C.\nMcCallum, A. S. Dzurak, et al., Storing quantum information for 30 seconds in a nanoelectronic device, Nature\nNanotechnology 9 (12) (2014) 986–991.\n[92] T. Rosskopf, A. Dussaux, K. Ohashi, M. Loretz, R. Schirhagl, H. Watanabe, S. Shikata, K. M. Itoh, C. L. De-\ngen, Investigation of surface magnetic noise by shallow spins in diamond, Phys. Rev. Lett. 112 (2014) 147602.\ndoi:10.1103 /PhysRevLett.112.147602.\n[93] B. A. Myers, A. Das, M. C. Dartiailh, K. Ohno, D. D. Awschalom, A. C. Bleszynski Jayich,\nProbing surface noise with depth-calibrated spins in diamond, Phys. Rev. Lett. 113 (2014) 027602.\ndoi:10.1103 /PhysRevLett.113.027602.\n[94] Y . Romach, C. M ¨uller, T. Unden, L. J. Rogers, T. Isoda, K. M. Itoh, M. Markham, A. Stacey, J. Meijer, S. Pez-\nzagna, B. Naydenov, L. P. McGuinness, N. Bar-Gill, F. Jelezko, Spectroscopy of surface-induced noise using\nshallow spins in diamond, Phys. Rev. Lett. 114 (2015) 017601. doi:10.1103 /PhysRevLett.114.017601.\n[95] F. Delgado, J. Fern ´andez-Rossier, Storage of classical information in quantum spins, Phys. Rev. Lett. 108 (2012)\n196602.\n[96] J.-P. Gauyacq, N. Lorente, Decoherence-governed magnetic-moment dynamics of supported atomic objects, J.\nPhys.: Condensed Matter 27 (45) (2015) 455301.\n[97] F. Delgado, J. Fern ´andez-Rossier, Spin decoherence of magnetic adatoms due to Kondo exchange,\narXiv:1608.07462 (2016).\n[98] F. Delgado, C. D. Batista, J. Fern ´andez-Rossier, Local probe of fractional edge states of S=1 Heisenberg spin\nchains, Phys. Rev. Lett. 111 (2013) 167201. doi:10.1103 /PhysRevLett.111.167201.\n[99] J. Bardeen, L. N. Cooper, J. R. Schrie \u000ber, Theory of superconductivity, Phys. Rev. 108 (5) (1957) 1175.\n[100] L. Hebel, C. P. Slichter, Nuclear spin relaxation in normal and superconducting aluminum, Phys. Rev. 113 (6)\n(1959) 1504.\n[101] F. Delgado, J. Fern ´andez-Rossier, Cotunneling theory of atomic spin inelastic electron tunneling spectroscopy,\nPhys. Rev. B 84 (2011) 045439. doi:10.1103 /PhysRevB.84.045439.\n[102] J. Park, A. N. Pasupathy, J. I. Goldsmith, C. Chang, Y . Yaish, J. R. Petta, M. Rinkoski, J. P. Sethna, H. D. Abru ˜na,\nP. L. McEuen, et al., Coulomb blockade and the Kondo e \u000bect in single-atom transistors, Nature 417 (6890) (2002)\n722–725.\n[103] R. Vincent, S. Klyatskaya, M. Ruben, W. Wernsdorfer, F. Balestro, Electronic read-out of a single nuclear spin\nusing a molecular spin transistor, Nature 488 (7411) (2012) 357–360.\n[104] S. Thiele, F. Balestro, R. Ballou, S. Klyatskaya, M. Ruben, W. Wernsdorfer, Electrically driven nuclear spin\nresonance in single-molecule magnets, Science 344 (6188) (2014) 1135–1138.\n[105] R. Gaudenzi, E. Burzur?, D. Reta, I. de P. R. Moreira, S. T. Bromley, C. Rovira, J. Veciana, H. S. J. van der Zant,\n59Exchange coupling inversion in a high-spin organic triradical molecule, Nano Letters 16 (3) (2016) 2066–2071.\ndoi:10.1021 /acs.nanolett.6b00102.\n[106] R. Frisenda, R. Gaudenzi, C. Franco, M. Mas-Torrent, C. Rovira, J. Veciana, I. Alcon, S. T. Bromley, E. Burzur?,\nH. S. J. van der Zant, Kondo e \u000bect in a neutral and stable all organic radical single molecule break junction, Nano\nLetters 15 (5) (2015) 3109–3114. doi:10.1021 /acs.nanolett.5b00155.\n[107] F. D. Natterer, K. Yang, W. Paul, P. Willke, T. Choi, T. Greber, A. J. Heinrich, C. P. Lutz, Reading and writing\nsingle-atom magnets, arXiv:1607.03977.\n[108] F. Kal \u000b, M. Rebergen, E. Fahrenfort, J. Girovsky, R. Toskovic, J. Lado, J. Fern ´andez-Rossier, A. Otte, A kilobyte\nrewritable atomic memory, Nature Nanotechnology (2016) 1748–3395doi:10.1038 /nnano.2016.131.\n[109] A. M. Tyryshkin, S. Tojo, J. J. Morton, H. Riemann, N. V . Abrosimov, P. Becker, H.-J. Pohl, T. Schenkel, M. L.\nThewalt, K. M. Itoh, S. A. Lyon, Electron spin coherence exceeding seconds in high-purity silicon, Nature Mate-\nrials 11 (2) (2012) 143–147.\n[110] G. W. Morley, M. Warner, A. M. Stoneham, P. T. Greenland, J. van Tol, C. W. M. Kay, G. Aeppli, The initialization\nand manipulation of quantum information stored in silicon by bismuth dopants, Nature Materials 9 (2010) 725–\n729.\n[111] R. E. George, W. Witzel, H. Riemann, N. V . Abrosimov, N. N ¨otzel, M. L. W. Thewalt, J. J. L. Morton, Electron\nspin coherence and electron nuclear double resonance of Bi donors in natural Si, Phys. Rev. Lett. 105 (6) (2010)\n067601. doi:10.1103 /PhysRevLett.105.067601.\n[112] S. Takahashi, R. Hanson, J. van Tol, M. S. Sherwin, D. D. Awschalom, Quenching spin decoherence in diamond\nthrough spin bath polarization, Phys. Rev. Lett. 101 (2008) 047601. doi:10.1103 /PhysRevLett.101.047601.\n[113] T. Fujisawa, D. G. Austing, Y . Tokura, Y . Hirayama, S. Tarucha, Allowed and forbidden transitions in artificial\nhydrogen and helium atoms, Nature 419 (6904) (2002) 278–281.\n[114] M. Kroutvar, Y . Ducommun, D. Heiss, M. Bichler, D. Schuh, G. Abstreiter, J. J. Finley, Optically programmable\nelectron spin memory using semiconductor quantum dots, Nature 432 (7013) (2004) 81–84.\n[115] F. Donati, S. Rusponi, S. Stepanow, C. W ¨ackerlin, A. Singha, L. Persichetti, R. Baltic, K. Diller, F. Patthey,\nE. Fernandes, J. Dreiser, ˇZ.ˇSljivan ˇcanin, K. Kummer, C. Nistor, P. Gambardella, H. Brune, Magnetic remanence\nin single atoms, Science 352 (6283) (2016) 318–321. doi:10.1126 /science.aad9898.\n[116] T. Miyamachi, T. Schuh, T. M ¨arkl, C. Bresch, T. Balashov, A. St ¨ohr, C. Karlewski, S. Andr ´e, M. Marthaler,\nM. Ho \u000bmann, et al., Stabilizing the magnetic moment of single holmium atoms by symmetry, Nature 503 (7475)\n(2013) 242–246.\n[117] F. Donati, A. Singha, S. Stepanow, C. W ¨ackerlin, J. Dreiser, P. Gambardella, S. Rusponi, H. Brune,\nMagnetism of Ho and Er atoms on close-packed metal surfaces, Phys. Rev. Lett. 113 (2014) 237201.\ndoi:10.1103 /PhysRevLett.113.237201.\n[118] H.-P. Breuer, F. Petruccione, The theory of open quantum systems, Oxford University Press on Demand, 2002.\n[119] C. Cohen-Tannoudji, J. Dupont-Roc, G. Grynberg, Photons and Atoms: Introduction to Quantum Electrodynam-\nics, Wiley and Sons, Inc., New York, 1989.\n[120] S. Haroche, J.-M. Raimond, Exploring the Quantum: Atoms, Cavities, and Photons, Oxford Graduate Texts,\nOxford, 2006.\n[121] A. Stern, Y . Aharonov, Y . Imry, Phase uncertainty and loss of interference: A general picture, Phys. Rev. A 41\n(1990) 3436–3448. doi:10.1103 /PhysRevA.41.3436.\n[122] M. P. A. Jones, C. J. Vale, D. Sahagun, B. V . Hall, E. A. Hinds, Spin coupling between cold atoms and the thermal\nfluctuations of a metal surface, Phys. Rev. Lett. 91 (2003) 080401. doi:10.1103 /PhysRevLett.91.080401.\n[123] C. Cohen-Tannoudji, G. Grynberg, J. Dupont-Roc, Atom-Photon Interactions, Wiley and Sons, INC., New York,\n1998.\n[124] V . Giovannetti, S. Lloyd, L. Maccone, Quantum-enhanced measurements: Beating the standard quantum limit,\nScience 306 (5700) (2004) 1330–1336. doi:10.1126 /science.1104149.\n[125] J. A. Jones, S. D. Karlen, J. Fitzsimons, A. Ardavan, S. C. Benjamin, G. A. D. Briggs, J. J. L. Morton, Magnetic\nfield sensing beyond the standard quantum limit using 10-spin noon states, Science 324 (5931) (2009) 1166–1168.\ndoi:10.1126 /science.1170730.\n[126] T. Tanaka, P. Knott, Y . Matsuzaki, S. Dooley, H. Yamaguchi, W. J. Munro, S. Saito, Proposed robust\nentanglement-based magnetic field sensor beyond the standard quantum limit, Phys. Rev. Lett. 115 (2015) 170801.\ndoi:10.1103 /PhysRevLett.115.170801.\n[127] F. Delgado, J. J. Palacios, J. Fern ´andez-Rossier, Spin-transfer torque on a single magnetic adatom, Phys. Rev.\nLett. 104 (2010) 026601.\n[128] F. Delgado, J. Fern ´andez-Rossier, Inelastic electron tunneling spectroscopy of a single nuclear spin, Phys. Rev.\nLett. 107 (2011) 076804. doi:10.1103 /PhysRevLett.107.076804.\n[129] F. Delgado, R. Aguado, J. Fern ´andez-Rossier, Probing a single nuclear spin in a silicon single electron transistor,\nApplied Physics Letters 101 (7) (2012) 072407. doi:http: //dx.doi.org /10.1063 /1.4746260.\n[130] F. Delgado, J. Fern ´andez-Rossier, RKKY oscillations in the spin relaxation and decoherence of magnetic nanos-\n60tructures, arXiv:1608.07462 (2016).\n[131] A. L. Barra, D. Gatteschi, R. Sessoli, High-frequency EPR spectra of [Fe 8O2(OH) 12(tacn) 6]Br8: A critical ap-\npraisal of the barrier for the reorientation of the magnetization in single-molecule magnets, Chemistry ? A Euro-\npean Journal 6 (9) (2000) 1608–1614.\n[132] N. V . Prokof’ev, P. C. E. Stamp, Theory of the spin bath, Rep. Prog. Phys. 63 (4) (2000) 669.\n[133] G. D. Mahan, Many-Particle Physics, Plenum Press, New York, 1990.\n[134] A. Khajetoorians, M. Valentyuk, M. Steinbrecher, T. Schlenk, A. Shick, J. Kolorenc, A. Lichtenstein, T. Wehling,\nR. Wiesendanger, J. Wiebe, Tuning emergent magnetism in a Hund’s impurity, Nature Nanotechnology 10 (2015)\n958–964. doi:10.1038 /nnano.2015.193.\n[135] P. Ruiz D ´ıaz, O. V . Stepanyuk, V . S. Stepanyuk, E \u000bects of interatomic coupling on magnetic anisotropy and order\nof spins on metallic surfaces, J. Phys. Chem. C 119 (46) (2015) 26237–26241. doi:10.1021 /acs.jpcc.5b10211.\n[136] B. W. Heinrich, G. Ahmadi, V . L. M ¨uller, L. Braun, J. I. Pascual, K. J. Franke, Change of the magnetic coupling of\na metal?organic complex with the substrate by a stepwise ligand reaction, Nano Letters 13 (10) (2013) 4840–4843.\ndoi:10.1021 /nl402575c.\n[137] B. W. Heinrich, L. Braun, J. I. Pascual, K. J. Franke, Tuning the magnetic anisotropy of single molecules, Nano\nLetters 15 (6) (2015) 4024–4028. doi:10.1021 /acs.nanolett.5b00987.\n[138] A. Ferr ´on, F. Delgado, J. Fern ´andez-Rossier, Derivation of the spin Hamiltonians for Fe in MgO, New J. Phys.\n17 (3) (2015) 033020.\n[139] A. Ferr ´on, J. L. Lado, J. Fern ´andez-Rossier, Electronic properties of transition metal atoms on Cu 2N(100), Phys.\nRev. B 92 (2015) 174407. doi:10.1103 /PhysRevB.92.174407.\n[140] H. Haug, A.-P. Jauho, Quantum kinetics in transport and optics of semi-conductors, Springer-Verlag, Berlin, 1996.\n[141] R. Koryt ´ar, N. Lorente, Multi-orbital non-crossing approximation from maximally localized Wannier functions:\nthe Kondo signature of copper phthalocyanine on Ag(100), J. Phys.: Condensed Matter 23 (35) (2011) 355009.\n[142] H. T. Dang, M. dos Santos Dias, A. Liebsch, S. Lounis, Strong correlation e \u000bects in theoretical stm studies of\nmagnetic adatoms, Phys. Rev. B 93 (2016) 115123. doi:10.1103 /PhysRevB.93.115123.\n[143] R. ˇZitko, J. Bon ˇca, Many-particle e \u000bects in adsorbed magnetic atoms with easy-axis anisotropy: the case of Fe on\nCuN/Cu(100) surface, New J. of Phys. 12 (6) (2010) 063040.\n[144] C.-Y . Lin, B. A. Jones, First-principles calculations of engineered surface spin structures, Phys. Rev. B 83 (2011)\n014413. doi:10.1103 /PhysRevB.83.014413.\n[145] M. Etzkorn, C. F. Hirjibehedin, A. Lehnert, S. Ouazi, S. Rusponi, S. Stepanow, P. Gambardella, C. Tieg,\nP. Thakur, A. I. Lichtenstein, A. B. Shick, S. Loth, A. J. Heinrich, H. Brune, Comparing XMCD and DFT with\nSTM spin excitation spectroscopy for Fe and Co adatoms on Cu 2N=Cu(100), Phys. Rev. B 92 (2015) 184406.\ndoi:10.1103 /PhysRevB.92.184406.\n[146] R. D. Cowan, The theory of atomic structure and spectra, Univ of California Press, Berkeley, 1981, 1981.\n[147] G. van der Laan, B. T. Thole, Strong magnetic X-ray dichroism in 2 pabsorption spectra of 3 dtransition-metal\nions, Phys. Rev. B 43 (1991) 13401–13411. doi:10.1103 /PhysRevB.43.13401.\n[148] D. Gatteschi, R. Sessoli, Quantum tunneling of magnetization and related phenomena in molecular materials,\nAngewandte Chemie International Edition 42 (3) (2003) 268–297.\n[149] M. Prada, The geometric phase of spin adsorbates on C ncrystal surfaces, arXiv:1605.03371 (2016).\n[150] C. Karlewski, M. Marthaler, T. M ¨arkl, T. Balashov, W. Wulfhekel, G. Sch ¨on, Magnetic adatoms as memory bits:\nA quantum master equation analysis, Phys. Rev. B 91 (2015) 245430. doi:10.1103 /PhysRevB.91.245430.\n[151] H. A. Kramers, General theory of the paramagnetic rotation in crystals, Proc. Amsterdam Acad. 33 (1930) 959.\n[152] R. Toskovic, R. van den Berg, A. Spinelli, I. S. Eliens, B. Bryant, J.-S. Caux, A. Otte, Atomic spin-chain realiza-\ntion of a model for quantum criticality, Nature Physics (2016) 1745–2481doi:10.1038 /nphys3722.\n[153] A. N. Rudenko, V . V . Mazurenko, V . I. Anisimov, A. I. Lichtenstein, Weak ferromagnetism in Mn nanochains on\nthe CuN surface, Phys. Rev. B 79 (2009) 144418. doi:10.1103 /PhysRevB.79.144418.\n[154] J. Honolka, T. Y . Lee, K. Kuhnke, A. Enders, R. Skomski, S. Bornemann, S. Mankovsky, J. Min ´ar, J. Staunton,\nH. Ebert, M. Hessler, K. Fauth, G. Sch ¨utz, A. Buchsbaum, M. Schmid, P. Varga, K. Kern, Magnetism of FePt\nsurface alloys, Phys. Rev. Lett. 102 (2009) 067207. doi:10.1103 /PhysRevLett.102.067207.\n[155] M. Menzel, Y . Mokrousov, R. Wieser, J. E. Bickel, E. Vedmedenko, S. Bl ¨ugel, S. Heinze, K. von Bergmann,\nA. Kubetzka, R. Wiesendanger, Information transfer by vector spin chirality in finite magnetic chains, Phys. Rev.\nLett. 108 (2012) 197204. doi:10.1103 /PhysRevLett.108.197204.\n[156] P. Ferriani, K. von Bergmann, E. Y . Vedmedenko, S. Heinze, M. Bode, M. Heide, G. Bihlmayer, S. Bl ¨ugel,\nR. Wiesendanger, Atomic-scale spin spiral with a unique rotational sense: Mn monolayer on W(001), Phys. Rev.\nLett. 101 (2008) 027201. doi:10.1103 /PhysRevLett.101.027201.\n[157] S. Heinze, K. V on Bergmann, M. Menzel, J. Brede, A. Kubetzka, R. Wiesendanger, G. Bihlmayer, S. Bl ¨ugel,\nSpontaneous atomic-scale magnetic skyrmion lattice in two dimensions, Nature Physics 7 (9) (2011) 713–718.\n[158] N. Romming, C. Hanneken, M. Menzel, J. E. Bickel, B. Wolter, K. von Bergmann, A. Kubetzka, R. Wiesendanger,\nWriting and deleting single magnetic skyrmions, Science 341 (6146) (2013) 636–639.\n61[159] B. Schweflinghaus, B. Zimmermann, M. Heide, G. Bihlmayer, S. Bl ¨ugel, Role of Dzyaloshinskii-Moriya\ninteraction for magnetism in transition-metal chains at Pt step edges, Phys. Rev. B 94 (2016) 024403.\ndoi:10.1103 /PhysRevB.94.024403.\n[160] A. E. Sikkema, W. J. L. Buyers, I. A \u000feck, J. Gan, Ising-Kondo lattice with transverse field: A possible f-moment\nHamiltonian for UR 2Si2, Phys. Rev. B 54 (1996) 9322–9327. doi:10.1103 /PhysRevB.54.9322.\n[161] A. M. Shakirov, Y . E. Shchadilova, A. N. Rubtsov, P. Ribeiro, Master equation approach to coherent dynamics\nand transport in engineered atomic spin devices, ArXiv:1609.01318.\n[162] K. D. Schotte, U. Schotte, Tomonaga’s model and the threshold singularity of X-ray spectra of metals, Phys. Rev.\n182 (1969) 479–482. doi:10.1103 /PhysRev.182.479.\n[163] A. Shnirman, G. Sch ¨on, Dephasing and renormalization in quantum two-level systems, in: Quantum Noise in\nMesoscopic Physics, Springer, 2003, pp. 357–370.\n[164] P. W. Anderson, Infrared catastrophe in Fermi gases with local scattering potentials, Phys. Rev. Lett. 18 (1967)\n1049–1051. doi:10.1103 /PhysRevLett.18.1049.\n[165] J. Kondo, Resistance minimum in dilute magnetic alloys, Prog. Theor. Phys. 32 (1964) 37–49.\n[166] P. Nozi `eres, C. T. DE Dominicis, Singularities in the X-ray absorption and emission of metals. III. One-body\ntheory exact solution, Phys. Rev. 178 (1969) 1097–1107. doi:10.1103 /PhysRev.178.1097.\n[167] O. Zachar, S. A. Kivelson, V . J. Emery, Exact results for a 1D Kondo lattice from bosonization, Phys. Rev. Lett.\n77 (1996) 1342–1345. doi:10.1103 /PhysRevLett.77.1342.\n[168] A. C. Hewson, The Kondo Problem to Heavy Fermions, Cambridge University Press, Cambridge, UK, 1997.\n[169] A. C. Neto, B. A. Jones, Non-Fermi-liquid behavior in U and Ce alloys: Criticality, disorder, dissipation, and\nGri\u000eths-McCoy singularities, Phys. Rev. B 62 (22) (2000) 14975.\n[170] J. P. Gauyacq, N. Lorente, Magnetic reversal of a quantum nanoferromagnet, Phys. Rev. B 87 (2013) 195402.\ndoi:10.1103 /PhysRevB.87.195402.\n[171] I. Waller, ¨Uber die magnetisierung von paramagnetischen kristallen in wechselfeldern, Zeitschrift f ¨ur Physik\n79 (5) (1932) 370–388. doi:10.1007 /BF01349398.\n[172] E. M. Chudnovsky, D. A. Garanin, R. Schilling, Universal mechanism of spin relaxation in solids, Phys. Rev. B\n72 (2005) 094426. doi:10.1103 /PhysRevB.72.094426.\n[173] M. N. Leuenberger, D. Loss, Spin tunneling and phonon-assisted relaxation in Mn 12-acetate, Phys. Rev. B 61\n(2000) 1286–1302. doi:10.1103 /PhysRevB.61.1286.\n[174] M. Ganzhorn, S. Klyatskaya, M. Ruben, W. Wernsdorfer, Strong spin-phonon coupling between a single-molecule\nmagnet and a carbon nanotube nanoelectromechanical system, Nature Nanotechnology 8 (3) (2013) 165–169.\n[175] D. D. Awschalom, D. Loss, N. Samarth, Semiconductor spintronics and quantum computation, Springer-Verlag\nBerlin Heidelberg, 2013.\n[176] W. M. Walsh, J. Jeener, N. Bloembergen, Temperature-dependent crystal field and hyperfine interactions, Phys.\nRev. 139 (1965) A1338–A1350. doi:10.1103 /PhysRev.139.A1338.\n[177] S. Scheel, P.-K. Rekdal, P. L. Knight, E. A. Hinds, Atomic spin decoherence near conducting and superconducting\nfilms, Phys. Rev. A 72 (4) (2005) 042901.\n[178] A. Morello, J. J. Pla, F. A. Zwanenburg, K. W. Chan, K. Y . Tan, H. Huebl, M. M ¨ott¨onen, C. D. Nugroho, C. Yang,\nJ. A. v. Donkelaar, A. D. C. Alves, D. N. Jamieson, C. C. Escott, L. C. L. Hollenberg, R. G. Clark, A. S. Dzurak,\nSingle-shot readout of an electron spin in silicon, Nature 467 (2010) 687–691.\n[179] A. Poudel, L. S. Langsjoen, M. G. Vavilov, R. Joynt, Relaxation in quantum dots due to evanescent-wave Johnson\nnoise, Phys. Rev. B 87 (2013) 045301. doi:10.1103 /PhysRevB.87.045301.\n[180] S. Kolkowitz, A. Safira, A. High, R. Devlin, S. Choi, Q. Unterreithmeier, D. Patterson, A. Zibrov, V . Manucharyan,\nH. Park, et al., Probing Johnson noise and ballistic transport in normal metals with a single-spin qubit, Science\n347 (6226) (2015) 1129–1132.\n[181] J. B. Johnson, Thermal agitation of electricity in conductors, Phys. Rev. 32 (1) (1928) 97.\n[182] H. Nyquist, Thermal agitation of electric charge in conductors, Phys. Rev. 32 (1) (1928) 110.\n[183] J. Lambe, R. C. Jaklevic, Molecular vibration spectra by inelastic electron tunneling, Phys. Rev. 165 (1968) 821–\n832. doi:10.1103 /PhysRev.165.821.\n[184] B. C. Stipe, M. A. Rezaei, W. Ho, Single-molecule vibrational spectroscopy and microscopy, Science 280 (5370)\n(1998) 1732–1735. doi:10.1126 /science.280.5370.1732.\n[185] L. Zhou, J. Wiebe, S. Lounis, E. Vedmedenko, F. Meier, S. Bl ¨ugel, P. H. Dederichs, R. Wiesendanger, Strength\nand directionality of surface Ruderman-Kittel-Kasuya-Yosida interaction mapped on the atomic scale, Nature\nPhys. 6 (2010) 187 – 191.\n[186] L. Zhou, F. Meier, J. Wiebe, R. Wiesendanger, Inversion of spin polarization above individual magnetic adatoms,\nPhys. Rev. B 82 (1) (2010) 012409. doi:10.1103 /PhysRevB.82.012409.\n[187] G. Morley, B. Herbert, S. Lee, K. Porfyrakis, T. Dennis, D. Nguyen-Manh, R. Scipioni, J. Van Tol, A. Horsfield,\nA. Ardavan, et al., Hyperfine structure of Sc@C 82from ESR and DFT, Nanotechnology 16 (11) (2005) 2469.\n[188] R. E. George, J. P. Edwards, A. Ardavan, Coherent spin control by electrical manipulation of the magnetic\n62anisotropy, Phys. Rev. Lett. 110 (2013) 027601. doi:10.1103 /PhysRevLett.110.027601.\n[189] A. Blank, C. R. Dunnam, P. P. Borbat, J. H. Freed, High resolution electron spin resonance microscopy, J. Magn.\nReson. 165 (1) (2003) 116–127.\n[190] J. Lado, A. Ferr ´on, J. Fern ´andez-Rossier, Universal mechanism for electron paramagnetic resonance of individual\nadatoms, ArXiv:1611.01110.\n[191] P. Berggren, J. Fransson, Electron paramagnetic resonance of single magnetic moment on a surface, Scientific\nreports 6.\n[192] C. Cervetti, A. Rettori, M. G. Pini, A. Cornia, A. Repoll ´es, F. Luis, M. Dressel, S. Rauschenbach, K. Kern,\nM. Burghard, et al., The classical and quantum dynamics of molecular spins on graphene, Nature Materials 15 (2)\n(2016) 164–168.\n[193] H. Gonz ´alez-Herrero, J. M. G ´omez-Rodr ´ıguez, P. Mallet, M. Moaied, J. J. Palacios, C. Salgado, M. M. Ugeda,\nJ.-Y . Veuillen, F. Yndurain, I. Brihuega, Atomic-scale control of graphene magnetism by using hydrogen atoms,\nScience 352 (6284) (2016) 437–441.\n[194] P. Ru \u000eeux, S. Wang, B. Yang, C. S ´anchez-S ´anchez, J. Liu, T. Dienel, L. Talirz, P. Shinde, C. A. Pignedoli,\nD. Passerone, et al., On-surface synthesis of graphene nanoribbons with zigzag edge topology, Nature 531 (7595)\n(2016) 489–492.\n[195] C. H ¨ubner, B. Baxevanis, A. A. Khajetoorians, D. Pfannkuche, Symmetry e \u000bects on the spin switching of adatoms,\nPhys. Rev. B 90 (2014) 155134. doi:10.1103 /PhysRevB.90.155134.\n[196] Z. Nussinov, M. F. Crommie, A. V . Balatsky, Noise spectroscopy of a single spin with spin-polarized STM, Phys.\nRev. B 68 (2003) 085402.\n[197] A. V . Balatsky, M. Nishijima, Y . Manassen, Electron spin resonance-scanning tunneling microscopy, Adv. Phys.\n61 (2) (2012) 117–152.\n[198] V . Emery, J. Devreese, Highly conducting one-dimensional solids, Plenum, New York, 1979.\n[199] K. Hur, Entanglement entropy, decoherence, and quantum phase transitions of a dissipative two-level system,\nAnn. Phys. 323 (9) (2008) 2208–2240.\n63"    },    {        "title": "0912.3607v2.Diffusive_versus_local_spin_currents_in_dynamic_spin_pumping_systems.pdf",        "content": "arXiv:0912.3607v2  [cond-mat.mes-hall]  7 Apr 2010Diffusive versus local spin currents in dynamic spin pumping\nsystems\nAkihito Takeuchi,∗Kazuhiro Hosono, and Gen Tatara\nDepartment of Physics, Tokyo Metropolitan University, Hac hioji, Tokyo 192-0397, Japan\n(Dated: September 17, 2018)\nAbstract\nUsing microscopic theory, we investigate the properties of a spin current driven by magnetization\ndynamics. In the limit of smooth magnetization texture, the dominant spin current induced by\nthe spin pumping effect is shown to be the diffusive spin current, i.e., the one arising from only a\ndiffusion associated with spin accumulation. That is to say, t here is no effective field that locally\ndrives the spin current. We also investigate the conversion mechanism of the pumped spin current\ninto a charge current by spin-orbit interactions, specifica lly the inverse spin Hall effect. We show\nthat the spin-charge conversion does not always occur and th at it depends strongly on the type\nof spin-orbit interaction. In a Rashba spin-orbit system, t he local part of the charge current is\nproportional to the spin relaxation torque, and the local sp in current, which does not arise from\nthe spin accumulation, does not play any role in the conversi on. In contrast, the diffusive spin\ncurrent contributes to the diffusive charge current. Alterna tively, for spin-orbit interactions arising\nfrom random impurities, the local charge current is proport ional to the local spin current that\nconstitutes only a small fraction of the total spin current. Clearly, the dominant spin current\n(diffusive spin current) is not converted into a charge curren t. Therefore, the nature of the spin\ncurrent is fundamentally different depending on its origin an d thus the spin transport and the\nspin-charge conversion behavior need to be discussed toget her along with spin current generation.\n1I. INTRODUCTION\nSpintronics1,2is a novel technology that enables control of both charge and spin of elec-\ntrons. To accomplish this aim, establishing methods for generation a nd observation of spin\ncurrents is an urgent issue. For generation in nonmagnetic conduc tors, several methods have\nbeen proposed. The standard way is to use the spin pumping effect in ferromagnetic-normal\nmetal junctions.3–7In those systems, a spin current can be induced by the precession of the\nmagnetization caused by an external alternating field and is then pu mped into the normal\nmetal. This spin pumping effect is widely used in experiments. A second t echnique is non-\nlocal spin injection in ferromagnetic-normal metal junctions.8,9In this case, a spin-polarized\ncurrent is induced in a ferromagnet by applying an electric field. The s pin-polarized current\nresults in a spin imbalance at the interface that gives rise to a diffusive flow of spin without\na charge current in the normal metal. A third technique is to use the spin Hall effect,10–15\nwhere the spin current flows in a transverse direction to an applied e lectric voltage in the\npresence of a spin-orbit interaction. Very recently, the spin Seeb eck effect was discovered\nenabling a generation of a spin current from thermal gradients in fe rromagnets.16Thus, spin\ncurrent generation can be realized by several mechanisms using va rious magnetic, electric,\nthermoelectric, and quantum relativistic (spin-orbit) effects.\nIn contrast, direct measurement of spin currents is still an open is sue. Spin current\ndetection has so far been performed by measuring subsidiary obse rvables that arise from\nspin currents. The first such observation was accomplished by Kat oet al.14by measuring\noptically the spin accumulation that appears as a result of spin curre nts at the edges of\nsamples (GaAs and InGaAs) in spin Hall systems. The critical issue, h owever, is that\nspin currents are not conserved. Therefore, spin accumulation a nd spin currents are not\nin direct correspondence, in sharp contrast to charge currents that are conserved. The\ninversespinHalleffectwasproposedasamethodformeasuringspin currentselectrically.17–19\nThe idea is based on the argument that spin currents flowing in the pr esence of spin-orbit\ninteractions induce an electric voltage through the inverse proces s associated with the spin\nHall effect.17,20,21(The electric detection of spin transport was first demonstrated at the\ninterface between ferromagnetic and paramagnetic metals by Joh nson and Silsbee.8) Being\nelectric, the inverse spin Hall effect has been widely utilized to detect spin currents.16,22–24\nTheoreticaljustificationfortheinverse spinHalleffecthasbeenc arriedoutonvarioussys-\n2tems.20,21,25–31Takahashi and Maekawa20,21investigated the inverse spin Hall effect because\nof nonlocal spin injection in ferromagnetic-nonmagnetic metal jun ctions, and demonstrated\nphenomenologically the relationship between charge current ( jc) and spin current ( js), i.e.,\njc∝σ×js, whereσis the spin polarization direction. Generation of the charge current\nby magnetization dynamics in magnetic junctions was studied phenom enologically by Wang\net al.25and Xiao et al.26By using microscopic theory, the direct connection between spin\ncurrents pumped by magnetization dynamics and induced charge cu rrents was revealed in\ndisordered metallic systems27,28and in disordered Rashba systems.29–31The result for metal-\nlic systems followed phenomenological predictions, jc∝σ×js, whereas pumped charge cur-\nrents in Rashba systems deviated from this simple relation. The naive picture that all spin\ncurrents are converted into charge currents is, therefore, inc orrect. What has been missing\nin this picture is the distinction between spin currents induced by som e effective field and\nthose arising from spin accumulation diffusion. The first one, local sp in current, is a contri-\nbution proportional to a local value of the magnetization texture. The other contribution is\na diffusive spin current which contains a long-range diffusion factor. As we will show below,\nthis difference is crucial in discussing the inverse spin Hall effect but c an not be addressed by\nphenomenological schemes. For applications, these two spin curre nts need to be considered\nseparately since the direction of the local (field-driven) spin curre nt is controlled by the field,\nwhile there is no way to control the direction of the diffusive spin curr ent.\nThe first aim of the present paper is to study these spin currents a rising from magnetiza-\ntion dynamics (spin pumping effect) in the absence of spin-orbit inter actions by providing a\nfully quantum-mechanical treatment of the conduction electrons . (We note that spin-orbit\ninteractions are not essential in discussing the spin pumping effect.) Generation of spin cur-\nrents because of a precessing magnetization was first studied by S ilsbeeet al.3They showed\nthat this precession generates an accumulation of spin at the inter face between ferromag-\nnetic and nonmagnetic domains and that the spin diffusion, i.e., spin cur rent, arises from\nthis spin accumulation. Tserkovnyak et al.4argued that the pumped spin current can be\nexpressed in the form js=a˙S+bS×˙S, introducing phenomenological parameters aandb\n(aandbare proportional to their mixing conductances and Sdenotes the magnetization di-\nrection). They assumed that the spin-flip relaxation rate is sufficien tly larger than their spin\ninjection rate and the spin accumulation does not build up at the inter face. Therefore, their\npumping mechanism is different from that of Silsbee et al.Later, Brataas et al.considered\n3the effect of a back flow arising from the spin accumulation at the inte rface.5There have\nbeen, therefore, predicted two types of spin currents: one aris ing from a nonlocal diffusion\nof the spin accumulation and the other a local contribution driven by an effective field. We\nshow in the present paper that the diffusive spin current originating from spin accumulation\ndominates in the case of the slow-varying magnetization (disordere d system). This result is\nin agreement with the prediction by Silsbee et al.3The scenario of the spin pumping effect\nwithout spin accumulation by Tserkovnyak et al.4does not hold in the present situation,\nbut it may apply to cases where rapid-varying magnetization occurs at the interface of a\nferromagnetic-normal metal junctions.\nThe second aim of the present paper is to clarify the spin-charge co nversion mechanism\nbased on microscopic theory and derive its conversion formula. By c alculating a spin current\nin the absence of spin-orbit interactions and a charge current with spin-orbit interactions\n(treated as linear-order perturbation), we will reveal the spin-t o-charge current conversion\nphenomena via the spin-orbit interaction. We will demonstrate that this conversion mecha-\nnismdiffersdepending onthetypeofspin-orbitinteraction, smooth nessofthemagnetization\nprofile, and the disorder (electron mean free path). In particular , the origin of spin currents,\nnamely, whether spin currents are generated by effective fields or arise from spin accumula-\ntion, turns out to be crucially important for spin-charge conversio n efficiency. Depending on\nthe spin-orbit interaction causing the conversion, we consider two cases: Rashba spin-orbit\ninteractions and spin-orbit interaction originating from random impu rity scattering in met-\nals. In the former, the dominant spin current is converted into a ch arge current. In contrast,\nin the latter, the correlation between the dominant spin current an d pumped charge currents\nis very weak. Instead, the local spin current (a small fraction of t he pumped spin current)\nis converted. Identifying the origin of spin currents is, therefore , crucial for realizing high\nconversion efficiency.\nII. SPIN PUMPING EFFECT\nA. Model\nWeconsider adisorderedelectronsystem coupledwithlocalizedspin S(x,t)(Fig.1). The\nlocalized spin can have spatial and temporal dependences, but we c onsider only the slowly\n4FIG. 1: (Color online) Schematic illustration of spin curre nt generation. The localized spin\nSis assumed to be slowly varying in space (compared to the elec tron mean-free path) and time\n(compared to the electron lifetime). Precession of the loca lized spin Sin the ferromagnet generates\na flow of the spin in the contiguous normal metal.\nvarying case [namely, that the spatial correlation length is larger th an the electron mean\nfree path (see below)]. The total Hamiltonian of the system is given a sH(t) =H0+Hex(t),\nwhere\nH0=/summationdisplay\nkεkc†\nkck+ui\nV/summationdisplay\nn/summationdisplay\nk,peip·rnc†\nk−pck,\nHex(t) =−J/summationdisplay\nk,q/parenleftbig\nc†\nk−qˆσck/parenrightbig\n·Sq(t).(1)\nThe first term H0describes free electrons in the presence of spin-independent impu rity\nscatterers and Hexrepresents the exchange interaction between the conducting ele ctron and\nthe local spin. We have introduced in momentum space annihilation (cr eation) operators\nc(†)\nkfor conduction electrons with kinetic energy εk≡/planckover2pi12k2/2m, whileuiis the strength\nof the impurity scatterers, Vis the system volume, rnrepresents the position of the nth\nimpurity,Jis the exchange coupling constant, ˆσcharacterizes a vector of Pauli matrices,\nthe caret signifies a matrix, and Sqdenotes the Fourier transform of the localized spin (or\nmagnetization). We note that impurity scattering gives rise to an ela stic electron lifetime\nτ. Let us stress that spin-orbit interactions are not considered in t his section, since it is not\nessential to spin current generation and to spin-charge convers ion.\nThe electron-spin density is defined as ρα\ns(x,t)≡(/planckover2pi1/2)tr/angbracketleftbig\nψ†(x,t)ˆσαψ(x,t)/angbracketrightbig\nH, where\n5FIG. 2: Diagrammatic representations of the spin current de nsityjs. Diagram (a) describes first-\norder contributions in Jand (b) second-order contributions. The wavy lines denote t he interaction\nwith the local spin Sand the gray shaded region represents the vertex correction from impurity\nscatterers.\nψ(†)(x) is the Fourier transform of c(†)\nk, tr denotes the trace over spin indices, and ∝angb∇acketleft···∝angb∇acket∇ightH\nrepresents the expectation value estimated for the total Hamilto nianH. The spin current\ndensity is defined (without spin-orbit interaction) as\njα\ns(x,t) =−i/planckover2pi13\n2mV/summationdisplay\nk,qe−iq·xtr/bracketleftBig\nkˆσαˆG<\nk−q\n2,k+q\n2(t,t)/bracketrightBig\n, (2)\nwhere the spin current jα\nsihas two direction components, one associated with the flow of\nspin in direction iand the other associated with a spin polarization in direction α. Here we\nused the lesser component of the path-ordered Green’s function defined asG<\nkα,k′α′(t,t′)≡\n(i//planckover2pi1)/angbracketleftbig\nc†\nk′α′(t′)ckα(t)/angbracketrightbig\nH.\nB. Pumped spin current\nWe carry out the calculation in a weak exchange coupling regime, J≪/planckover2pi1/τ. This\nassumption would be satisfied in a junction of normal metal and a fer romagnet, as shown in\nFig. 1, since the interface is usually disordered.32The Feynman diagrams contributing to the\nspin current at first and second orders in the exchange coupling Jare shown in Fig. 2. We\nassume a spatially smooth magnetization structure qℓ≪1, whereqis the momentum of the\nmagnetization and ℓdenotes the mean free path for conduction electrons, and assum e a slow\nprecession of magnetization Ω τ≪1, where Ω is a frequency of magnetization dynamics.\nContributions from Fig. 2(a) reads\nj(1)α\ns(x,t) =−2/planckover2pi13J\n3mV/summationdisplay\nk,k′,q/summationdisplay\nω,Ωe−iq·x+iΩtSα\nq,ΩqΩf′\nω/bracketleftBig\nImεkgr\nk,ω(ga\nk,ω)2/bracketrightBig/parenleftbigg\n1+niu2\ni\nVΠra\nq;ω,Ωgr\nk′,ωga\nk′,ω/parenrightbigg\n.\n(3)\n6Hereniis the impurity concentration, fω=θ(εF−/planckover2pi1ω) denotes the Fermi distribution\nfunction at zero temperature [ θ(ω) is the step function and εFis the Fermi energy], ga(gr)\ndenotes the advanced (retarded) free Green’s function given by ga\nk,ω= (gr\nk,ω)∗=/bracketleftbig\n/planckover2pi1ω−εk−\n(i/planckover2pi1/2τ)/bracketrightbig−1, and Πradescribes the diffusion ladder defining the vertex correction,\nΠra\nq;ω,Ω≡∞/summationdisplay\nn=0/parenleftbigg/summationdisplay\nkniu2\ni\nVgr\nk−q\n2,ω−Ω\n2ga\nk+q\n2,ω+Ω\n2/parenrightbiggn\n. (4)\nThe dominant first-order contribution to the spin current is calcula ted as\nj(1)α\ns(x,t) =/planckover2pi1νJτD\nV∇/angbracketleftbig˙Sα(x,t)/angbracketrightbig\n, (5)\nwhereνdenotes the density of states, D denotes a diffusion coefficient give n as 2εFτ/3m,\nand∝angb∇acketleft···∝angb∇acket∇ightdescribes long-range diffusion because of random impurity scatter ing\n/angbracketleftbig\nA(x,t)/angbracketrightbig\n≡1\nV/integraldisplay\nd3x′/integraldisplay\ndt′/summationdisplay\nq/summationdisplay\nΩe−iq·(x−x′)+iΩ(t−t′)A(x′,t′)\nDq2τ+iΩτ, (6)\nwhereA(x,t) is an arbitrary function of both xandt. (We show details of the calculations\nin Appendix A.) Similarly, we obtain second-order contributions in J[Fig. 2(b)] as j(2)\ns=\njp(2)\ns+jsc(2)\ns, where the dynamic component (pumped spin current) jp(2)\nsis given by\njp(2)α\ns(x,t) =−4/planckover2pi12J2τ\n3mV/summationdisplay\nk,k′,q,q′/summationdisplay\nω,Ω,Ω′e−iq·x+iΩt/parenleftbig\nSq′,Ω′×Sq−q′,Ω−Ω′/parenrightbigαΩ′f′\nω\n×/bracketleftBig\nImεkgr\nk,ω(ga\nk,ω)2/bracketrightBig/bracketleftbigg\n(q+q′)+niu2\niq\nVΠra\nq;ω,Ωgr\nk′,ωga\nk′,ω/bracketrightbigg\n≃−2νJ2τ2D\nV∇/angbracketleftbig/bracketleftbig\nS(x,t)×˙S(x,t)/bracketrightbigα/angbracketrightbig\n, (7)\nand the equilibrium component (spin super-current) jsc(2)\nsis given by\njsc(2)α\ns(x,t) =i/planckover2pi12J2\n3mV/summationdisplay\nk,q,q′/summationdisplay\nω,Ω,Ω′e−iq·x+iΩt/parenleftbig\nSq′,Ω′×Sq−q′,Ω−Ω′/parenrightbigαq′f′\nωIm(ga\nk,ω)2\n≃−/planckover2pi12νJ2\n12mεFV/bracketleftbig\nS(x,t)×∇S(x,t)/bracketrightbigα. (8)\nThis equilibrium flow is a spin super-current and arises from the angula r difference between\ntwo localized spins (or magnetizations) S1×S2, i.e., from the spin Josephson effect.33In\nEqs. (5) and (7), the dynamic component without vertex correct ion does not contribute\nbecause it is smaller than that with vertex correction by 1 order of m agnitude (qℓ)2≪1.\n7As a result, the pumped spin current for a smooth-varying magnet ization texture is a long-\nrange diffusive flow (Eqs. (5) and (7)). From Eqs. (5) and (7), the pumped spin current,\nj(p)\ns≡j(1)\ns+jp(2)\ns, can be written as a gradient of an effective potential µs,\nj(p)α\ns(x,t) =−∇µα\ns(x,t), (9)\nwhere\nµα\ns(x,t)≡ −/planckover2pi1νJτD\nV/angbracketleftbig˙Sα(x,t)/angbracketrightbig\n+2νJ2τ2D\nV/angbracketleftbig/bracketleftbig\nS(x,t)×˙S(x,t)/bracketrightbigα/angbracketrightbig\n(10)\nThis effective spin potential arises from dissipations of magnetizatio n energy S×˙Sand˙S.\nOur result, Eqs. (9) and (10), indicates that the sources of the s pin current are ˙SandS×˙S\nand this result appears to agree with phenomenological predictions for the spin pumping\neffect.4However, the spin current here has been diffusion-averaged and is not simply a local\nfunction of the source.\nSimilarly, spin density is given by\nρα\ns(x,t) =−/planckover2pi1νJτD\nV∇2/angbracketleftbig\nSα(x,t)/angbracketrightbig\n+2νJ2τ2\nV/angbracketleftbig/bracketleftbig\nS(x,t)×˙S(x,t)/bracketrightbigα/angbracketrightbig\n. (11)\nIt satisfies the spin diffusion equation:\n˙ρα\ns(x,t)−D∇2ρα\ns(x,t) =τα\nρ(x,t), (12)\nwhereτρrepresents the spin relaxation given as\nτα\nρ(x,t)≡ −/planckover2pi1νJD\nV∇2Sα(x,t)+2νJ2τ\nV/bracketleftbig\nS(x,t)×˙S(x,t)/bracketrightbigα. (13)\nThe effective spin potential is written as\nµα\ns(x,t) = Dρα\ns(x,t)−/planckover2pi1νJD\nVSα(x,t). (14)\nSpecifically, it is the spin density excluding the direct spin polarization b ecause of the ex-\nchange interaction, ρs−(/planckover2pi1νJ/V)S, multiplied by D. Therefore, the total spin current is\nreducible to\njα\ns(x,t) =jsc(2)α\ns(x,t)+j(p)α\ns(x,t)\n=−/planckover2pi12νJ2\n12mεFV/bracketleftbig\nS(x,t)×∇S(x,t)/bracketrightbigα−∇µα\ns(x,t). (15)\n8The last term, a diffusive contribution, can be written in terms of the spin density by using\nEq. (14) and we finally obtain\njα\ns(x,t) =j(sc)α\ns(x,t)−D∇ρα\ns(x,t), (16)\nwhere we have defined the total equilibrium spin current as\nj(sc)α\ns(x,t) =/planckover2pi1νJD\nV∇Sα(x,t)−/planckover2pi12νJ2\n12mεFV/bracketleftbig\nS(x,t)×∇S(x,t)/bracketrightbigα. (17)\nAs seen in Eq. (16), the dominant contribution of the dynamic spin cu rrent derives com-\npletely fromadiffusion ofthespin polarization. Equations (16)and(1 7)arethemainresults\ndescribing the spin pumping phenomena.\nFor a charge current jc, it is described generally as\njc(x,t) =σcE(x,t)+jsc(x,t)−D∇ρc(x,t), (18)\nwhereσcis an electrical conductivity, Erepresents an electric field, jscis a super-current\nwhich exists in an equilibrium situation (corresponding to j(sc)\nsof the spin current), and ρc\ndenotes a charge density. Comparing the two expressions, Eqs. ( 16) and (18), we imme-\ndiately see a striking difference between the charge and the spin, na mely, there is no field\nthat induces a spin current at least in the present perturbative re gime without spin-orbit\ninteractions. In other words, the spin pumping effect does not dire ctly generate the spin\ncurrent itself, but causes a spin imbalance or spin accumulation that then gives rise to a\ndiffusive spin current. This result is in agreement with the study by Sils beeet al.,3while\nthe prediction by Tserkovnyak et al.,4stating that spin pumping is not associated with spin\naccumulation, does not hold in the smooth spin case. The diffusive spin current represented\nby the vertex correction found here would correspond to the phe nomenologically discussed\nback-flow because of spin accumulation at the interface.5\nAbsence of aneffective field for the spin current is crucial in spintro nics. In fact, in charge\nelectronics, the charge and its current are independently contro llable and can be measured\nby different mechanisms, for instance, by capacitance means for c harges and Amp` ere’s law\nfor currents. This is not the case in spin transport phenomena. If we use the spin pumping\nmechanism in a disordered system, the spin current is always accomp anied by spin accumu-\nlation according to Eq. (16).\n9Fortunately, we know that there is an effective field for a spin curre nt if we use spin-orbit\ninteractions, specifically, as in the spin Hall system. By including spin- orbit interactions,\nthe spin current is generalizable to\njα\nsi(x,t) =σSHǫijαEj(x,t)+j(sc)α\nsi(x,t)−D∇iρα\ns(x,t), (19)\nwhereσSHrepresents the spin Hall conductivity proportional to the spin-or bit interaction\nandǫijαdenotes the Levi-Civita antisymmetric tensor. Therefore, charg e and spin currents,\nEqs. (18) and (19), now look symmetric. However, there appears a crucial difference when\none includes spin-orbit interactions, namely, the violation of the con servation law for spin.\nIn fact, spin and its current in the presence of spin-orbit interact ions satisfy the identity\n˙ρα\ns(x,t)+∇·jα\ns(x,t) =Tα\ns(x,t), (20)\nwhereTsrepresents the spin-relaxation torque arising from the spin-orbit interaction.34\n(This equation is equivalent to the diffusion equation for the spin dens ity.35–37) The non-\nconservation of spin causes definitional ambiguity of a spin current . Since a definition of spin\ncurrent is absolutely related to a definition of spin relaxation torque , spin currents cannot be\ndefined uniquely under the spin-orbit interaction. (As a possible solu tion, a gauge covariant\nderivative was proposed.38) Because of the spin relaxation torque Ts, the measurement of\nthe spin current can never be carried out by simply measuring the sp in density induced at\nthe edge since the induced spin current can disappear because of t he term Tswhile being\ntransported. (In this sense, the observation of the spin Hall effe ct in Ref. 14 cannot be con-\nsidered as a direct observation of the spin current.) At present, t here has been no indication\nof an emission of an observable field (either electric or magnetic) fro m the spin current and,\ntherefore, incontrast to charge currents which are observable by detecting anAmp` ere’s field,\nelectromagnetic detection of spin currents is not possible. [The abs ence of electromagnetic\nfields induced by spin currents is reasonable from Maxwell’s equations because the equations\nas determined by special relativity and U(1) gauge invariance canno t be modified by the\nspin current.] Here a serious dilemma for spintronics arises. Specific ally, spin currents and\nspin densities are independently controllable only if one switches on sp in-orbit interactions,\nbut such interactions make it impossible to detect the spin current b y measuring the spin\naccumulation.\n10FIG. 3: (Color online) Relationship between pumped spin cur rent and interface condition in\na ferromagnetic-normal metal junction. (a) The diffusive spi n current is dominant where the\nmagnetization structure Svaries slowly as compared to the electron mean free path ℓin normal\nmetal. (b) The local spin current is dominant where the magne tization structure varies rapidly at\nthe interface.\nC. Diffusive spin current vs. local spin current\nIn the spin pumping effect, we have demonstrated that the diffusive spin current is dom-\ninant in a slow-varying magnetization structure, subject to ℓ≪λ(ℓis the electron mean\nfree path and λrepresents the length scale of the magnetization structure) as d epicted in\nFig. 3(a). In an actual spin pumping system, this condition would be s atisfied since the\nmean-free path in such experimental systems is known to be very s hort, for example, less\nthan 1 nm size in Ni 81Fe19/Pt sample.32Strictly speaking, a local spin current can also exist\nbut this contribution is negligibly small compared with the diffusive spin c urrent. There-\nfore, in this situation, it is impossible that local and nonlocal spin curr ents coexist, as found\nin a study by Brataas et al.5The situation can be different with a clean sample or very\nsharp magnetization profile, subject to ℓ≫λ, as shown in Fig. 3(b). In this case, the\nlocal spin current without spin accumulation could become dominant a nd the diffusive spin\ncurrent arising from spin accumulation is small. We therefore expect that the mechanism of\nTserkovnayk et al.without spin accumulation may be valid. The spin current induced by\nthe spin pumping effect, therefore, depends much on the disorder or the electron mean-free\npath.\nIn this section, we did not take into account spin-orbit interactions . It is essential when\n11we need to discuss spin-charge conversions, which we do in the follow ing section.\nIII. SPIN-CHARGE CONVERSION\nWe now consider the conversion mechanism of the pumped spin curre nt into the charge\ncurrent through the inverse spin Hall effect. In this section, we co nsider two types of spin-\norbit interactions, the Rashba type and such interactions becaus e of random impurity scat-\ntering, and we finally obtain the exact spin-charge conversion form ula.\nA. Rashba spin-orbit interaction systems\nWe first consider charge currents driven by Rashba spin-orbit inte ractions. The Rashba\nspin-orbit coupling was first found as a peculiar effect in a two-dimens ional electron-gas sys-\ntem.39However recent studies have revealed that the Rashba system ap pears quite generally\nat surfaces of nonmagnetic materials without inversion symmetry a nd that this Rashba\ncoupling can be quite large.40–43Therefore, there is a possibility that these surface effects\ncontribute greatly to the inverse spin Hall effect in ferromagnetic- normal metal junction\nsystems.\nThe microscopic theory in the presence of Rashba spin-orbit intera ctions was demon-\nstrated by Ohe et al.29They considered a two-dimensional electron-gas system, where t he\nmaximum number of diffusion ladder needs to be taken into account. T he result was ex-\ntended to a three-dimensional and spatially dependent Rashba cou pling case.30However,\nthe account payed little attention to the charge conservation law b ecause the analyses were\nintended only to see whether the spin-charge current conversion indeed occurs or not. In the\nfollowing, we consider a three-dimensional system and evaluate the dominant contribution\nincluding one diffusion ladder as a vertex correction. By considering t he vertex correction,\nwe maintain charge conservation throughout the calculation.\nThe Rashba spin-orbit interaction is given by\nHso=−/summationdisplay\nkEso·/bracketleftbig\nk×/parenleftbig\nc†\nkˆσck/parenrightbig/bracketrightbig\n, (21)\nwhereEsodescribes thespin-orbit field (or strength ofthe Rashba coupling) . In thepresence\nof this interaction, the anomalous velocity resulting from the spin-o rbit interaction modifies\n12FIG. 4: Dominant contributions to the pumped charge current by magnetization dynamics in a\nRashba system: (a) is the normal charge current, jn\nc, and (b) describes corrections to the charge\ncurrentarisingfromtheRashbacoupling, jso\nc. DoublelinesrepresentRashbaspin-orbitinteractions\n(SOIs). The Rashba spin-orbit coupling gives the anomalous velocity to the conduction electrons,\nthereby modifying the definition of the charge current.\nthe charge current. By defining the charge density as ρc(x,t)≡ −etr/angbracketleftbig\nψ†(x,t)ψ(x,t)/angbracketrightbig\nH, the\ncharge current density is given by jc=jn\nc+jso\nc, where the normal charge current jn\ncis\njn\nc(x,t) =ie/planckover2pi12\nmV/summationdisplay\nk,qe−iq·xtr/bracketleftBig\nkˆG<\nk−q\n2,k+q\n2(t,t)/bracketrightBig\n, (22)\nand the correction from the Rashba coupling jso\ncis defined as\njso\nc(x,t) =ie\nV/summationdisplay\nk,qe−iq·xtr/bracketleftBig/parenleftbig\nEso׈σ/parenrightbigˆG<\nk−q\n2,k+q\n2(t,t)/bracketrightBig\n. (23)\nWe treat the Rashba spin-orbit interaction perturbatively by impos ing the constraint\nEsokF≪/planckover2pi1/τ, withkFbeing theFermiwavelength. Forslow-varying magnetizationtextu res,\nthe contribution from both the first-order Rashba and the first- order exchange interactions\nvanishes identically.29,30(Strictly speaking, a contribution proportional to ∇Eso∇˙Sarises if\nRashba spin-orbit interactions are inhomogeneous.30) We now consider the charge current\nto first order in the Rashba coupling and second order in the exchan ge coupling.\nIn Fig. 4, we present the Feynman diagrams for the dominant contr ibution which is\n13calculated (see details in Appendix B) as\njci(x,t) =ieJ2τ\nmV/summationdisplay\nk,k′,q,q′/summationdisplay\nω,Ω,Ω′e−iq·x+iΩt/bracketleftBig\nEso×/parenleftbig\nSq′,Ω′×Sq−q′,Ω−Ω′/parenrightbig/bracketrightBig\njΩ′f′\nωgr\nk,ωga\nk,ω\n×/braceleftbigg\n−m\n/planckover2pi1δij+/planckover2pi1qiqj\n3πν/bracketleftBig\nImεk′gr\nk′,ω(ga\nk′,ω)2/bracketrightBig\nΠra\nq;ω,Ω/bracerightbigg\n≃−4eνJ2τ2\n/planckover2pi12V/braceleftBig\nEso×/bracketleftbig\nS(x,t)×˙S(x,t)/bracketrightbig/bracerightBig\ni\n−4eνJ2τ3D\n/planckover2pi12V∇i/braceleftBig\n∇·/angbracketleftbig\nEso×/bracketleftbig\nS(x,t)×˙S(x,t)/bracketrightbig/angbracketrightbig/bracerightBig\n. (24)\nThis pumped charge current is expressible in terms of the dynamic co mponent of the spin\nrelaxation torque T(dy)\nsand the charge density31\njci(x,t) =ǫijαaR\njT(dy)α\ns(x,t)−D∇iρc(x,t), (25)\nwhereaR≡ −2eτEso//planckover2pi12and\nT(dy)α\ns(x,t) =2νJ2τ\nV/bracketleftbig\nS(x,t)×˙S(x,t)/bracketrightbigα,\nρc(x,t) =2νJ2τ2\nVaR·/angbracketleftbig\n∇×/bracketleftbig\nS(x,t)×˙S(x,t)/bracketrightbig/angbracketrightbig\n.(26)\nWe first note that Eqs. (25) and (26) indicate that only the damping of the local spin,\nS×˙S, is converted into a charge current by Rashba coupling. The equilibr ium spin current,\nEq. (17), does not contribute, as is reasonable from energy cons ervation considerations.\nSincejcis expressible in terms of T(dy)\nsandρc, the naive formula for the inverse spin Hall\neffect, i.e.jcproportional locally to js, does not hold in disordered Rashba systems. Instead,\nEq. (25) indicates that the generation mechanism of the local char ge current in a Rashba\nsystem is the inverse effect of the spin-relaxation torque.\nThe non-local part in Eq. (25) arises from a diffusion of the charge d ensity, which is\nwritten in terms of the dynamic component of the pumped spin curre nt [Eq. (7)],\nρc(x,t) =−1\nDǫijαaR\nijp(2)α\nsj(x,t). (27)\nTherefore, the (diffusive) spin current induces a charge polarizat ion via Rashba spin-orbit\ninteractions, but not a charge current itself. Since diffusive spin cu rrents dominate in slow-\nvarying magnetization structures, we expect that the ratio of th e charge current to the\npumped spin current is high [see Fig. 5(a)].\n14FIG. 5: (Color online) Conversion mechanism of the pumpedsp in current into a charge current via\nspin-orbit interactions (SOIs). (a) In the Rashbasystem, t he diffusivespin current is just converted\ninto a charge current. (b) With spin-orbit interactions cau sed by random impurity scattering, the\ndiffusive spin current is present but not converted. Thelocal spin current is converted into a charge\ncurrent.\nB. Random impurity-induced spin-orbit interaction systems\nWe now focus on spin-orbit interactions caused by random impurities . This interaction\nis defined as\nHso=iuiλso\nV/summationdisplay\nn/summationdisplay\nk,peip·rn(k×p)·/parenleftbig\nc†\nk−pˆσck/parenrightbig\n, (28)\nwhereλsois the spin-orbit strength. We have here assumed that the random impurity spin-\norbit interaction arises from the same impurities giving rise to the elec tron lifetime τ. In\nthis case, the correction current resulting from the spin-orbit int eraction is given by\njso\nc(x,t) =−euiλso\nV2/summationdisplay\nn/summationdisplay\nk,q,pe−iq·xeip·rntr/bracketleftBig/parenleftbig\np׈σ/parenrightbigˆG<\nk−q−p\n2,k+q−p\n2(t,t)/bracketrightBig\n.(29)\nAs shown in Ref. 28, the charge current is derived in the form\njci(x,t) =aimpǫijαj(local)α\nsj(x,t)−D∇iρc(x,t), (30)\nwhereaimp≡2eλsok2\nF/εFτand\nj(local)α\ns(x,t) =/planckover2pi1νJτD\nV/parenleftbigg\n∇˙Sα(x,t)−2Jτ\n/planckover2pi1/braceleftBig\n∇/bracketleftbig\nS(x,t)×˙S(x,t)/bracketrightbigα+/bracketleftbig\nS(x,t)×∇˙S(x,t)/bracketrightbigα/bracerightBig/parenrightbigg\n,\nρc(x,t) =2νJ2τ3Daimp\nVǫijα/angbracketleftbig/bracketleftbig\n∇iS(x,t)×∇j˙S(x,t)/bracketrightbigα/angbracketrightbig\n.\n(31)\n15We have here denoted the local spin current as j(local)\ns. Equation (30) seems consistent with\nthe naive formula of the inverse spin Hall effect, jc∝σ×js. However, one should note\nthat this local spin current is a very small correction to the dominan t spin current [given by\n−D∇ρsin Eq. (16)]. Therefore, most of the spin currents generated by t he spin pumping\neffect are not converted into a charge current; only a small fract ion [of order ( qℓ)2≪1]\ndevelops into a charge current [Fig 5(b)]. Thus, the spin-charge co nversion efficiency is\nsmall in the presence of random impurity-induced spin-orbit interac tions.\nC. Conversion mechanism\nBy comparing Eqs. (25) and (30) to the general formulation of the charge current\n[Eq. (18)], the spin relaxation torque T(dy)\nsand the local spin current j(local)\nsact as effective\nelectric fields for Rashba and random impurity-induced spin-orbit int eractions, respectively.\nTherefore, as we mentioned above, the spin current changes only the constitutive relations\nassociated with electromagnetic fields but does not change the Max well’s equations them-\nselves. Correctly, there is a spin current caused by spin-orbit inte ractions. This spin current,\nhowever, produces a second-order charge current with respec t to spin-orbit interactions at\nleast. It should be negligible compared to the above results.\nFrom the above results, we see that spin accumulation at the interf ace plays a crucial\nrole in determining spin pumping and spin-charge conversion mechanis ms. In fact, the\npumped charge current is proportional to the spin accumulation wh en Rashba interactions\nare present, but does not occur with the spin accumulation for ran dom impurity-induced\nspin-orbit interactions. Therefore, measuring spin accumulation a t the interface would pro-\nvide impetus to determine the dominant spin-orbit interaction and to clarify the spin-charge\nconversion mechanism.\nIV. CONCLUSIONS\nWe have studied aspects of spin pumping and the spin-charge conve rsion mechanism\nthrough spin-orbit interactions in the disordered electron system . We showed that the spin\ncurrent generated by the spin dynamics is a diffusive process arising from a dynamics-\ninduced spin accumulation. There is, therefore, no effective field th at drives the spin current\n16directly in the disordered case. We have confirmed that a charge cu rrent is induced by\nthese spin-orbit interactions. This process involves the conversio n of a pumped spin current\ninto a charge transport, but the mechanism has turned out not to be as simple as an\nearlier phenomenological proposal, jc∝(σ×js),17,20,21had anticipated. In fact, the spin-\ncharge conversion depends largely on the type of spin-orbit intera ction. For Rashba spin-\norbit interactions, the charge current is given by a local contribut ion proportional to the\nspin relaxation torque and a diffusive contribution arising from the diff usive spin current.\nTherefore, the naive formula for the inverse spin Hall effect does n ot hold in the Rashba\nsystems. In contrast, for random impurity-induced spin-orbit int eractions, the local part\nof the charge current is written as a very small fraction of the spin current [smaller by\nO(ℓ2/λ2), whereℓandλare the electron mean-free path and the coherence length scale o f\nthe magnetization, respectively]. However, the dominant spin curr ent is not converted into\na charge current in the presence of impurities. Thus, the naive inve rse spin Hall effect does\nnot occur either. Our result indicates that the spin-charge conve rsion formula as proposed\nearlier using phenomenological arguments is too simple and the whole p henomenon needs\ndiscussing together with the origin of spin currents.\nAppendix A: details of calculations for the spin pumping effect\nWe perform here calculations of the pumped spin current using stan dard perturbation\nexpansion techniques. We treat the exchange coupling up to the se cond order. The electron\nspin and its relaxation torque densities are defined in terms of Green ’s function as\nρα\ns(x,t) =−i/planckover2pi1\n2V/summationdisplay\nk,qe−iq·xtr/bracketleftBig\nˆσαˆG<\nk−q\n2,k+q\n2(t,t)/bracketrightBig\n,\nTα\ns(x,t) =−i/planckover2pi1J\nV/summationdisplay\nk,qe−iq·xtr/braceleftBig/bracketleftbig\nˆσ×S(x,t)/bracketrightbigαˆG<\nk−q\n2,k+q\n2(t,t)/bracerightBig\n,(A1)\nrespectively. Before carrying out the calculation, we introduce th e Dyson equation:\nGkα,k′α′(t,t′) =δkk′δαα′gkα(t,t′)\n+ui\nV/integraldisplay\nCKdT/summationdisplay\nn/summationdisplay\npeip·rngkα(t,T)Gk+pα,k′α′(T,t′)\n−J/integraldisplay\nCKdT/summationdisplay\nβ/summationdisplay\nqgkα(t,T)/bracketleftbig\nσαβ·Sq(T)/bracketrightbig\nGk+qβ,k′α′(T,t′),(A2)\n17FIG. 6: The lesser component of the Green’s function involvi ng the vertex correction can be\ndivided into the three terms. The diagram is partitioned off b y defining the diffusion ladder as\nthe boundary. We denote contributions from the left-hand si de as Γ, the middle representing the\ndiffusion ladder as Π, and the right-hand side as Λ. The spiral l ine represents all interactions and\ntherefore the contribution from Λ depends on the interactio ns.\nwhereGkα,k′α′(t,t′)≡ −(i//planckover2pi1)/angbracketleftbig\nTCK/bracketleftbig\nckα(t)c†\nk′α′(t′)/bracketrightbig/angbracketrightbig\nH(TCKbeing the path-ordering operator\ndefined on the Keldysh contour C K) andgis the free Green’s function which is obtained\nfrom the free Hamiltonian H0. The Dyson equation is very useful in carrying out the per-\nturbation expansion because this equation can be solved iteratively . Here we assume a weak\nexchange coupling regime, J≪/planckover2pi1/τ, and therefore we can treat the exchange interaction\nperturbatively. To evaluate the lesser component of G(t,t′) =/integraltext\nCKdTG1(t,T)G2(T,t′), we\nuse the following44\nG<(t,t′) =/integraldisplay∞\n−∞dT/bracketleftBig\nGr\n1(t,T)G<\n2(T,t′)+G<\n1(t,T)Ga\n2(T,t′)/bracketrightBig\n,\nGa(r)(t,t′) =/integraldisplay∞\n−∞dT/bracketleftBig\nGa(r)\n1(t,T)Ga(r)\n2(T,t′)+Ga(r)\n1(t,T)Ga(r)\n2(T,t′)/bracketrightBig\n.(A3)\nThe lesser component of the free Green’s function satisfies g<\nk,ω=fω/parenleftbig\nga\nk,ω−gr\nk,ω/parenrightbig\n.\n181. First-order calculations in J\nFirst, weshowthecalculationofthespincurrenttofirst-orderint heexchangeinteraction.\nThe diagram in Fig. 2(a) is written as\nj(1)α\ns(x,t) =i/planckover2pi13J\n2mV∞/summationdisplay\nn=0/parenleftbiggniu2\ni\nV/parenrightbiggn/summationdisplay\nqe−iq·x/summationdisplay\n{ki}n\ni=0tr/braceleftBigg\nk0ˆσα/bracketleftBiggn/productdisplay\ni=0/integraldisplay\nCi+1\nKdti+1ˆgki−q\n2(ti,ti+1)/bracketrightBigg\n×/bracketleftbig\nˆσ·Sq(tn+1)/bracketrightbig/bracketleftBiggn/productdisplay\ni=0/integraldisplay\nCn+i+1\nKdtn+i+1ˆgkn−i+q\n2(tn+i+1,tn+i+2)/bracketrightBigg/bracerightBigg<\nt0=t2n+2=t.(A4)\nBy using tr/parenleftbig\nˆσαˆσβ/parenrightbig\n= 2δαβand taking the lesser component, the equation reads\nj(1)α\ns(x,t) =i/planckover2pi13J\nmV/summationdisplay\nk,k′,q/summationdisplay\nω,Ωe−iq·x+iΩtSα\nq,Ωk/parenleftBig\nΛaa(1)\nk,q;ω,Ω−Λrr(1)\nk,q;ω,Ω+Λra(1)\nk,q;ω,Ω\n+Γaa\nk,q;ω,ΩΠaa\nq;ω,ΩΛaa(1)\nk′,q;ω,Ω−Γrr\nk,q;ω,ΩΠrr\nq;ω,ΩΛrr(1)\nk′,q;ω,Ω+Γra\nk,q;ω,ΩΠra\nq;ω,ΩΛra(1)\nk′,q;ω,Ω/parenrightBig\n.\n(A5)\nA diagram involving the vertex correction is divided into three parts: the left-hand side,\nthe diffusion ladder (middle), and the right-hand side of the diagram s hown in Fig. 6. The\ncontribution from the left hand side is given as\nΓaa(rr)\nk,q;ω,Ω≡niu2\ni\nVga(r)\nk−q\n2,ω−Ω\n2ga(r)\nk+q\n2,ω+Ω\n2,\nΓra\nk,q;ω,Ω≡niu2\ni\nVgr\nk−q\n2,ω−Ω\n2ga\nk+q\n2,ω+Ω\n2.(A6)\nThe diffusion ladder arising from the vertex correction is written as\nΠaa(rr)\nq;ω,Ω≡∞/summationdisplay\nn=0/parenleftbigg/summationdisplay\nkΓaa(rr)\nk,q;ω,Ω/parenrightbiggn\n,\nΠra\nq;ω,Ω≡∞/summationdisplay\nn=0/parenleftbigg/summationdisplay\nkΓra\nk,q;ω,Ω/parenrightbiggn\n.(A7)\nThe contribution from the right-hand side of a diagram depends on t he diagram and in\nFig. 2(a) is given by\nΛaa(rr)(1)\nk,q;ω,Ω≡fω−(+)Ω\n2ga(r)\nk−q\n2,ω−Ω\n2ga(r)\nk+q\n2,ω+Ω\n2,\nΛra(1)\nk,q;ω,Ω≡/parenleftbig\nfω+Ω\n2−fω−Ω\n2/parenrightbig\ngr\nk−q\n2,ω−Ω\n2ga\nk+q\n2,ω+Ω\n2.(A8)\n19Assuming slow dynamics Ω τ≪1 and a spatially smooth local spin structure qℓ≪1, we\nobtain the result\nj(1)α\ns(x,t)≃ −2/planckover2pi13J\n3mV/summationdisplay\nk,k′,q/summationdisplay\nω,Ωe−iq·x+iΩtSα\nq,ΩqΩf′\nω/bracketleftBig\nImεkgr\nk,ω(ga\nk,ω)2/bracketrightBig/parenleftbigg\n1+niu2\ni\nVΠra\nq;ω,Ωgr\nk′,ωga\nk′,ω/parenrightbigg\n≃/planckover2pi1νJτD\nV∇/angbracketleftbig˙Sα(x,t)/angbracketrightbig\n. (A9)\nHere we note simplifications in the ksummation\n/summationdisplay\nkgr\nkga\nk≃2πντ\n/planckover2pi1, (A10)\n/summationdisplay\nkεkgr\nk(ga\nk)2≃i2πνεFτ2\n/planckover2pi12, (A11)\nwhere we have put gk≡gk,/planckover2pi1ω=εF. The diffusion ladder arising from the vertex correction is\nalso given as\nΠaa(rr)\nq;0,Ω≃1, (A12)\nΠra\nq,0,Ω≃∞/summationdisplay\nn=0/braceleftBigg\nniu2\ni\nV/summationdisplay\nk/bracketleftBigg\n/parenleftbig\n1−iτΩ/parenrightbig\ngr\nkga\nk−2/planckover2pi1τq2\n3mImεkgr\nk(ga\nk)2/bracketrightBigg/bracerightBiggn\n≃/summationdisplay\nn=0/parenleftBig\n1−Dq2τ−iΩτ/parenrightBign\n=1\nDq2τ+iΩτ. (A13)\nSince the product of only ga(orgr) gives a very small contribution which is of order 1 /εF\ncompared to the coefficient of gaandgr, the diffusion ladder reduces approximately to\nunity. We cannot, however, ignore the product of only ga(orgr) completely because that\ncontribution corresponds to an equilibrium current.\nSimilarly, the spin density is also calculated in the form\nρ(1)α\ns(x,t) =i/planckover2pi12J\nV/summationdisplay\nk,q/summationdisplay\nω,Ωe−iq·x+iΩtSα\nq,Ω/parenleftBig\nΛaa(1)\nk,q;ω,Ω−Λrr(1)\nk,q;ω,Ω+Πra\nq;ω,ΩΛra(1)\nk,q;ω,Ω/parenrightBig\n≃i/planckover2pi12J\nV/summationdisplay\nk,q/summationdisplay\nω,Ωeiq·x+iΩtSα\nq,Ωf′\nω/parenleftbiggi\nτ+ΩΠra\nq;ω,Ω/parenrightbigg\ngr\nk,ωga\nk,ω\n≃ −/planckover2pi1νJτD\nV∇2/angbracketleftbig\nSα(x,t)/angbracketrightbig\n. (A14)\nTo first order in the exchange coupling, the spin-relaxation torque corresponding to diagram\nFig. 7(a) vanishes as a consequence of trˆ σα= 0. Therefore, the pumped spin current to first\n20FIG. 7: Diagrammatic representations of the spin-relaxati on torque of the conduction electrons.\n(a) This contribution comes from first-order terms in Jbut vanishes. (b) The leading contribution\narises from second-order exchange interaction terms. This contribution corresponds to local spin\ndamping S×˙S.\norder inJfollows\n˙ρ(1)α\ns(x,t)+∇·j(1)α\ns(x,t) = 0. (A15)\nHence, the spin of conduction electrons is conserved.\n2. Second-order calculations in J\nHere we derive the spin current to second order in the exchange co upling as shown in\nFig. 2(b). This is calculated in the same manner as the first-order ca lculations. The pumped\nspin current reads\nj(2)α\ns(x,t) =/planckover2pi13J2\nmV/summationdisplay\nk,k′,q,q′/summationdisplay\nω,Ω,Ω′e−iq·x+iΩt/parenleftbig\nSq′,Ω′×Sq−q′,Ω−Ω′/parenrightbigαk/parenleftBig\nΛaa(2)\nk,q,q′;ω,Ω,Ω′\n−Λrr(2)\nk,q,q′;ω,Ω,Ω′+Λra(2)\nk,q,q′;ω,Ω,Ω′+Γra\nk,q;ω,ΩΠra\nq;ω,ΩΛra(2)\nk′,q,q′;ω,Ω,Ω′/parenrightBig\n≃/planckover2pi12J2\n3mV/summationdisplay\nk,k′,q,q′/summationdisplay\nω,Ω,Ω′e−iq·x+iΩt/parenleftbig\nSq′,Ω′×Sq−q′,Ω−Ω′/parenrightbigαf′\nω/braceleftbigg\niq′Im(ga\nk,ω)2\n−4τΩ′/bracketleftBig\nImεkgr\nk,ω(ga\nk,ω)2/bracketrightBig/bracketleftbigg\n(q+q′)+niu2\niq\nVΠra\nq;ω,Ωgr\nk′,ωga\nk′,ω/bracketrightbigg/bracerightbigg\n≃−/planckover2pi12νJ2\n12mεFV/bracketleftbig\nS(x,t)×∇S(x,t)/bracketrightbigα−2νJ2τ2D\nV∇/angbracketleftbig/bracketleftbig\nS(x,t)×˙S(x,t)/bracketrightbigα/angbracketrightbig\n.(A16)\nIn this case, we should pay particular attention to tr/parenleftbig\nˆσαˆσβˆσγ/parenrightbig\n=i2ǫαβγand the contribution\nfrom the right-hand side of the diagram being given as\nΛaa(rr)(2)\nk,q,q′;ω,Ω,Ω′≡fω−(+)Ω\n2ga(r)\nk−q\n2,ω−Ω\n2ga(r)\nk−q−2q′\n2,ω−Ω−2Ω′\n2ga(r)\nk+q\n2,ω+Ω\n2,\nΛra(2)\nk,q.q′;ω,Ω,Ω′≡/parenleftBig\nfω−Ω−2Ω′\n2−fω−Ω\n2/parenrightBig\ngr\nk−q\n2,ω−Ω\n2ga\nk−q−2q′\n2,ω−Ω−2Ω′\n2ga\nk+q\n2,ω+Ω\n2\n+/parenleftBig\nfω+Ω\n2−fω−Ω−2Ω′\n2/parenrightBig\ngr\nk−q\n2,ω−Ω\n2gr\nk−q−2q′\n2,ω−Ω−2Ω′\n2ga\nk+q\n2,ω+Ω\n2.(A17)\n21The pumped spin current contains an equilibrium flow given by estimatin g (ga)2,\n/summationdisplay\nk(ga\nk)2≃ −iπν\n2εF. (A18)\nThe spin density is calculated in a similar manner\nρ(2)α\ns(x,t) =/planckover2pi12J2\nV/summationdisplay\nk,q,q′/summationdisplay\nω,Ω,Ω′e−iq·x+iΩt/parenleftbig\nSq′,Ω′×Sq−q′,Ω−Ω′/parenrightbigα\n×/parenleftBig\nΛaa(2)\nk,q,q′;ω,Ω,Ω′−Λrr(2)\nk,q,q′;ω,Ω,Ω′+Πra\nq;ω,ΩΛra(2)\nk,q,q′;ω,Ω,Ω′/parenrightBig\n≃i2/planckover2pi1J2τ\nV/summationdisplay\nk,q,q′/summationdisplay\nω,Ω,Ω′e−iq·x+iΩt/parenleftbig\nSq′,Ω′×Sq−q′,Ω−Ω′/parenrightbigαΩ′f′\nωΠra\nq;ω,Ωgr\nk,ωga\nk,ω\n≃2νJ2τ2\nV/angbracketleftbig/bracketleftbig\nS(x,t)×˙S(x,t)/bracketrightbigα/angbracketrightbig\n. (A19)\nTo second-order in the exchange coupling, the spin of conduction e lectrons is not conserved\nand follows the general spin continuity equation\n˙ρ(2)α\ns(x,t)+∇·j(2)α\ns(x,t) =T(2)α\ns(x,t). (A20)\nHere the spin relaxation torque is represented by the diagram in Fig. 7(b) and we obtain\nT(2)α\ns(x,t) =−i2/planckover2pi1J2\nV/summationdisplay\nk,q/summationdisplay\nω,Ωe−iq·x+iΩt/bracketleftbig\nS(x,t)×Sq,Ω/bracketrightbigα/parenleftBig\nΛaa(1)\nk,q;ω,Ω−Λrr(1)\nk,q;ω,Ω+Λra(1)\nk,q;ω,Ω/parenrightBig\n≃−i2/planckover2pi1J2\nV/summationdisplay\nk,q/summationdisplay\nω,Ωe−iq·x+iΩt/bracketleftbig\nS(x,t)×Sq,Ω/bracketrightbigαf′\nω/bracketleftbiggi/planckover2pi1q2\n6mIm(ga\nk,ω)2+Ωgr\nk,ωga\nk,ω/bracketrightbigg\n≃−/planckover2pi12νJ2\n12mεFV/bracketleftbig\nS(x,t)×∇2S(x,t)/bracketrightbigα+2νJ2τ\nV/bracketleftbig\nS(x,t)×˙S(x,t)/bracketrightbigα.(A21)\nAppendix B: Magnetization-pumped charge current in a Rashba system\nHere, we calculate the charge current stemming from magnetizatio n pumping in the\npresence of Rashba spin-orbit interactions. In the Rashba syste m, the Dyson equation is\nmodified as\nGkα,k′α′(t,t′) =δkk′δαα′gkα(t,t′)\n+ui\nV/integraldisplay\nCKdT/summationdisplay\nn/summationdisplay\npeip·rngkα(t,T)Gk+pα,k′α′(T,t′)\n−J/integraldisplay\nCKdT/summationdisplay\nβ/summationdisplay\nqgkα(t,T)/bracketleftbig\nσαβ·Sq(T)/bracketrightbig\nGk+qβ,k′α′(T,t′).\n−/integraldisplay\nCKdT/summationdisplay\nβ/summationdisplay\nqgkα(t,T)/bracketleftbig\nEso·(k×σαβ)/bracketrightbig\nGkβ,k′α′(T,t′).(B1)\n22Since the spin-orbit interactions are generally weak, subject to EsokF≪/planckover2pi1/τ, we perform\nthe perturbation expansion with respect to both the exchange int eraction and the Rashba\nspin-orbit interaction. By iteration, we obtain the leading contribut ion shown in Fig. 4,\njci(x,t) =−2eJ2\nV/summationdisplay\nk,k′,q,q′/summationdisplay\nω,Ω,Ω′e−iq·x+iΩt/bracketleftBig\nEso×/parenleftbig\nSq′,Ω′×Sq−q′,Ω−Ω′/parenrightbig/bracketrightBig\nj\n×/bracketleftbigg/planckover2pi12ki\nm/parenleftBig\n˜Λaa(2)\nj;k,q,q′;ω,Ω,Ω′−˜Λrr(2)\nj;k,q,q′;ω,Ω,Ω′+˜Λra(2)\nj;k,q,q′;ω,Ω,Ω′+Γra\nk,q;ω,ΩΠra\nq;ω,Ω˜Λra(2)\nj;k′,q,q′;ω,Ω,Ω′/parenrightBig\n+δij/parenleftBig\nΛaa(2)\nk,q,q′;ω,Ω,Ω′−Λrr(2)\nk,q,q′;ω,Ω,Ω′+Λra(2)\nk,q,q′;ω,Ω,Ω′/parenrightBig/bracketrightbigg\n≃ieJ2τ\nmV/summationdisplay\nk,k′,q,q′/summationdisplay\nω,Ω,Ω′e−iq·x+iΩt/bracketleftBig\nEso×/parenleftbig\nSq′,Ω′×Sq−q′,Ω−Ω′/parenrightbig/bracketrightBig\njΩ′f′\nωgr\nk,ωga\nk,ω\n×/braceleftbigg\n−m\n/planckover2pi1δij+/planckover2pi1qiqj\n3πν/bracketleftBig\nImεk′gr\nk′,ω(ga\nk′,ω)2/bracketrightBig\nΠra\nq;ω,Ω/bracerightbigg\n≃−4eνJ2τ2\n/planckover2pi12V/braceleftBig\nEso×/bracketleftbig\nS(x,t)×˙S(x,t)/bracketrightbig/bracerightBig\ni−4eνJ2τ3D\n/planckover2pi12V∇i/braceleftBig\n∇·/angbracketleftbig\nEso×/bracketleftbig\nS(x,t)×˙S(x,t)/bracketrightbig/angbracketrightbig/bracerightBig\n.\n(B2)\nHere we have defined contributions from the right-hand side of the diagrams as\n˜Λaa(rr)(2)\ni;k,q,q′;ω,Ω,Ω′≡fω−(+)Ω\n2/bracketleftbiggm\n/planckover2pi12∂\n∂ki/parenleftBig\nga(r)\nk−q\n2,ω−Ω\n2ga(r)\nk−q−2q′\n2,ω−Ω−2Ω′\n2ga(r)\nk+q\n2,ω+Ω\n2/parenrightBig\n−2/parenleftbigg\nk−q−2q′\n2/parenrightbigg\niga(r)\nk−q\n2,ω−Ω\n2/parenleftBig\nga(r)\nk−q−2q′\n2,ω−Ω−2Ω′\n2/parenrightBig2\nga(r)\nk+q\n2,ω+Ω\n2/bracketrightbigg\n,\n˜Λra(2)\ni;k,q,q′;ω,Ω,Ω′≡/parenleftBig\nfω−Ω−2Ω′\n2−fω−Ω\n2/parenrightBig/bracketleftbiggm\n/planckover2pi12∂\n∂ki/parenleftBig\ngr\nk−q\n2,ω−Ω\n2ga\nk−q−2q′\n2,ω−Ω−2Ω′\n2ga\nk+q\n2,ω+Ω\n2/parenrightBig\n−2/parenleftbigg\nk−q−2q′\n2/parenrightbigg\nigr\nk−q\n2,ω−Ω\n2/parenleftBig\nga\nk−q−2q′\n2,ω−Ω−2Ω′\n2/parenrightBig2\nga\nk+q\n2,ω+Ω\n2/bracketrightbigg\n+/parenleftBig\nfω+Ω\n2−fω−Ω−2Ω′\n2/parenrightBig/bracketleftbiggm\n/planckover2pi12∂\n∂ki/parenleftBig\ngr\nk−q\n2,ω−Ω\n2gr\nk−q−2q′\n2,ω−Ω−2Ω′\n2ga\nk+q\n2,ω+Ω\n2/parenrightBig\n−2/parenleftbigg\nk−q−2q′\n2/parenrightbigg\nigr\nk−q\n2,ω−Ω\n2/parenleftBig\ngr\nk−q−2q′\n2,ω−Ω−2Ω′\n2/parenrightBig2\nga\nk+q\n2,ω+Ω\n2/bracketrightbigg\n.(B3)\nThe charge density is written in terms of the lesser component of th e nonequilibrium Green’s\nfunction\nρc(x,t) =ie/planckover2pi1\nV/summationdisplay\nk,qe−iq·xtrˆG<\nk−q\n2,k+q\n2(t,t). (B4)\n23In a similar manner to the charge current, the charge density is calc ulated as\nρc(x,t) =−2e/planckover2pi1J2\nV/summationdisplay\nk,q,q′/summationdisplay\nω,Ω,Ω′e−iq·x+iΩt/bracketleftBig\nEso×/parenleftbig\nSq′,Ω′×Sq−q′,Ω−Ω′/parenrightbig/bracketrightBig\ni\n×/parenleftBig\n˜Λaa(2)\ni;k,q,q′;ω,Ω,Ω′−˜Λrr(2)\ni;k,q,q′;ω,Ω,Ω′+Πra\nq;ω,Ω˜Λra(2)\ni;k,q,q′;ω,Ω,Ω′/parenrightBig\n≃4eJ2τ2\n/planckover2pi1V/summationdisplay\nk,q,q′/summationdisplay\nω,Ω,Ω′e−iq·x+iΩt/bracketleftBig\nEso×/parenleftbig\nSq′,Ω′×Sq−q′,Ω−Ω′/parenrightbig/bracketrightBig\niqiΩ′f′\nωΠra\nq;ω,Ωgr\nk,ωga\nk,ω\n≃4eνJ2τ3\n/planckover2pi12V∇·/angbracketleftbig\nEso×/bracketleftbig\nS(x,t)×˙S(x,t)/bracketrightbig/angbracketrightbig\n. (B5)\nThe charge and its current densities that we have obtained satisfy the following charge\nconservation law:\n˙ρc(x,t)+∇·jc(x,t) = 0, (B6)\nindicating that our calculation has been performed correctly.\nAcknowledgments\nThe authors are grateful to S. Murakami and E. Saitoh for fruitf ul comments. This work\nwas supported by a Grant-in-Aid for Scientific Research in Priority A rea “Creation and\ncontrol of spin current” (Contract No. 1948027) from the Minist ry of Education, Culture,\nSports, Science and Technology, Japan, by the Kurata Memorial H itachi Science and Tech-\nnology Foundation, and by the Sumitomo Foundation. A.T. is financially supported by the\nJapan Society for the Promotion of Science for Young Scientists.\n∗atake@phys.metro-u.ac.jp\n1H. Ohno, Science 281, 951 (1998).\n2S.A. Wolf, D.D. Awschalom, R.A. Buhrman, J.M. Daughton, S. v on Moln´ ar, M.L. Roukes, A.Y.\nChtchelkanova, and D.M. Treger, Science 294, 1488 (2001).\n3R.H. Silsbee, A. Janossy, and P. Monod, Phys. Rev. B 19, 4382 (1979).\n4Y. Tserkovnyak, A. Brataas, and G.E.W. Bauer, Phys. Rev. Let t.88, 117601 (2002).\n5A. Brataas, Y. Tserkovnyak, G.E.W. Bauer, and B.I. Halperin , Phys. Rev. B 66, 060404(R)\n(2002).\n6S. Mizukami, Y. Ando, and T. Miyazaki, Phys. Rev. B 66, 104413 (2002).\n247M.V. Costache, M. Sladkov, S.M. Watts, C.H. van der Wal, and B .J. van Wees, Phys. Rev.\nLett.97, 216603 (2006).\n8M. Johnson and R.H. Silsbee, Phys. Rev. Lett. 55, 1790 (1985).\n9F.J. Jedema, A.T. Filip, and B.J. van Wees, Nature 410, 345 (2001).\n10J.E. Hirsch, Phys. Rev. Lett. 83, 1834 (1999).\n11S. Murakami, N. Nagaosa, and S.-C. Zhang, Science 301, 1348 (2003).\n12J. Sinova, D. Culcer, Q. Niu, N.A. Sinitsyn, T. Jungwirth, an d A.H. MacDonald, Phys. Rev.\nLett.92, 126603 (2004).\n13J.-I. Inoue, G.E.W. Bauer, and L.W. Molenkamp, Phys. Rev. B 70, 041303(R) (2004).\n14Y.K. Kato, R.C. Myers, A.C. Gossard, and D.D. Awschalom, Sci ence306, 1910 (2004).\n15J. Wunderlich, B. Kaestner, J. Sinova, and T. Jungwirth, Phy s. Rev. Lett. 94, 047204 (2005).\n16K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae, K. An do, S. Maekawa, and E. Saitoh,\nNature455, 778 (2008).\n17E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Appl. Phys. Le tt.88, 182509 (2006).\n18S.O. Valenzuela and M. Tinkham, Nature 442, 176 (2006).\n19T. Kimura, Y. Otani, T. Sato, S. Takahashi, and S. Maekawa, Ph ys. Rev. Lett. 98, 156601\n(2007).\n20S. Takahashi and S. Maekawa, Phys. Rev. Lett. 88, 116601 (2002).\n21S. Takahashi and S. Maekawa, J. Phys. Soc. Jpn. 77, 031009 (2008).\n22L. Vila, T. Kimura, and Y.C. Otani, Phys. Rev. Lett. 99, 226604 (2007).\n23T. Seki, Y. Hasegawa, S. Mitani, S. Takahashi, H. Imamura, S. Maekawa, J. Nitta, and K.\nTakanashi, Nature Mater. 7, 125 (2008).\n24K. Ando, S. Takahashi, K. Harii, K. Sasage, J. Ieda, S. Maekaw a, and E. Saitoh, Phys. Rev.\nLett.101, 036601 (2008).\n25X. Wang, G.E.W. Bauer, B.J. van Wees, A. Brataas, and Y. Tserk ovnyak, Phys. Rev. Let. 97,\n216602 (2006).\n26J. Xiao, G.E.W. Bauer, and A. Brataas, Phys. Rev. B 77, 180407(R) (2008).\n27J. Shibata and H. Kohno, Phys. Rev. Lett. 102, 086603 (2009).\n28K. Hosono, A. Takeuchi, and G. Tatata, J. Phys. Soc. Jpn. 79, 014708 (2010).\n29J.-I. Ohe, A. Takeuchi, and G. Tatara, Phys. Rev. Lett. 99, 266603 (2007).\n30A. Takeuchi and G. Tatara, J. Phys. Soc. Jpn. 77, 074701 (2008).\n2531A. Takeuchi, K. Hosono, and G. Tatara, J. Phys.: Conf. Ser. 200, 062031 (2010).\n32E. Saitoh (private communication).\n33H. Katsura, N. Nagaosa, and A.V. Balatsky, Phys. Rev. Lett. 95, 057205 (2005).\n34N. Nakabayashi, A. Takeuchi, K. Hosono, K. Taguchi, and G. Ta tara, arXiv:1002.0207.\n35E.G. Mishchenko, A.V. Shytov, and B.I. Halperin, Phys. Rev. Lett.93, 226602 (2004).\n36R. Raimondi, and P. Schwab, EPL 87, 37008 (2009).\n37M. Duckheim, D.L. Maslov, D. Loss, Phys. Rev. B 80, 235327 (2009).\n38I.V. Tokatly, Phys. Rev. Lett. 101, 106601 (2008).\n39E.I. Rashba, Sov. Phys. Solid State 2, 1109 (1960).\n40O. Krupin, G. Bihlmayer, K. Starke, S. Gorovikov, J.E. Priet o, K. D¨ obrich, S. Bl¨ ugel, and G.\nKaindl, Phys. Rev. B 71, 201403(R) (2005).\n41C.R. Ast, J. Henk, A. Ernst, L. Moreschini, M.C. Falub, D. Pac il´ e, P. Bruno, K. Kern, and M.\nGrioni, Phys. Rev. Lett. 98, 186807 (2007).\n42T. Nakagawa, O. Ohgami, Y. Saito, H. Okuyama, M. Nishijima, a nd T. Aruga, Phys. Rev. B\n75, 155409 (2007).\n43I.M. Miron, G. Gaudin, S. Auffret, B. Rodmacq, A. Schuhl, S. Piz zini, J. Vogel, and P. Gam-\nbardella, Nature Mater. 9, 230 (2010).\n44H. Haug and A.-P. Jauho, Quantum Kinetics in Transport and Optics of Semiconductors\n(Springer, 2007).\n26"    },    {        "title": "1603.03628v2.Time_dependent_spin_and_transport_properties_of_a_single_molecule_magnet_in_a_tunnel_junction.pdf",        "content": "Time-dependent spin and transport properties of a single-molecule magnet in a tunnel\njunction\nH. Hammar1and J. Fransson1,\u0003\n1Department of Physics and Astronomy, Uppsala University, Box 530, SE-751 21 Uppsala\n(Dated: March 28, 2022)\nIn single-molecule magnets, the exchange between a localized spin moment and the electronic\nbackground provides a suitable laboratory for studies of dynamical aspects of both local spin and\ntransport properties. Here we address the time-evolution of a localized spin moment coupled to\nan electronic level in a molecular quantum dot embedded in a tunnel junction between metallic\nleads. The interactions between the localized spin moment and the electronic level generate an\ne\u000bective interaction between the spin moment at di \u000berent instances in time. Therefore, we show\nthat, despite being a single spin system, there are e \u000bective contributions of isotropic Heisenberg, and\nanisotropic Ising and Dzyaloshinski-Moriya character acting on the spin moment. The interactions\ncan be controlled by gate voltage, voltage bias, the spin polarization in the leads, in addition to external\nmagnetic fields. Signatures of the spin dynamics are found in the transport properties of the tunneling\nsystem, and we demonstrate that measurements of the spin current may be used for read-out of the\nlocal spin moment orientation.\nPACS numbers: 73.63.Rt, 75.30.Et, 72.25.Hg, 75.78.-n\nI. INTRODUCTION\nSingle-molecule magnets provide interesting work-\nbench opportunities to study quantum phenomena re-\nlated to their individual properties as well as promis-\ning potential for quantum information technology and\nquantum computation based on spintronics devices.\nEasy control of single magnetic moments paved the way\nfor a deeper exploration of, e.g., magnetic anisotropies\nand exchange interaction, as well as new routes for sig-\nnificantly less energy consuming active electronics de-\nvices and information storage.\nMolecular magnets o \u000ber a platform for studies of mag-\nnetic properties on a fundamental level due to their\nintrinsic discreteness. Experimentally, this has paved\nthe way for electronic control and detection of the\nmagnetization of individual molecules [1–3], magnetic\nanisotropy, and exchange interaction of single atoms\nsuch as, e.g., Co and Mn on a surface [4–8], and tuning of\nthe magnetic anisotropy in molecular magnets [9]. Fur-\nthermore, spatial anisotropies have been observed for\nthe Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction\n[10] as well as signatures of superexchange interaction\nand the long-range Kondo e \u000bect between single mag-\nnetic molecules [11–13]. These advances in experimental\ntechniques have led to realizations of magnetically stable\natomic scale configurations [14–16] that are important\nsteps toward the creation of stable magnetic memory\ndevices at the atomic scale. Magnetic molecules contain-\ning transition metal atoms, e.g., M-phthalocyanine and\nM-porphyrins where M denotes a transition metal ele-\nment (Cr, Mn, Fe, Co, Ni, Cu) [17–21], as well as single\n\u0003Electronic address: Jonas.Fransson@physics.uu.semolecules comprising complexes of transition metal ele-\nments [22, 23] and antiferromagnetic rings [24–29] have\nbeen explored in many di \u000berent contexts.\nFor technological applications, on the other hand, the\npotential of molecular magnets and magnetic materials\nis unlimited. A range of di \u000berent spintronics devices\nhave been proposed, both using spin currents [30] or\nspin torque [31]. Such devices include molecular spin-\ntransistors, molecular spin-valves, molecular multidot\ndevices [32], etc. These can potentially be used both as\nbuilding blocks of quantum computers [33] and as quan-\ntum simulators [34]. There are already several experi-\nmental realizations of these kinds of devices, including\nmagnetic memories and spin qubits [35–38].\nOn the theoretical side, we have witnessed great\nprogress over the course of the past decade in devel-\nopments of the theory for, e.g., single molecular mag-\nnets and magnetization dynamics. There have been sev-\neral studies of magnetic exchange interaction and the\npossibilities for electrical control of the interaction and\nspin transport [39–44]. Under non-equilibrium condi-\ntions, magnetic molecules show signatures of intrinsic\nanisotropic exchange interactions that can be used to\ncontrol molecular spin [41, 45], something that may lead\nto read-and-write capabilities with currents in spintron-\nics devices [46–48]. Non-equilibrium studies of trans-\nport properties have, moreover, suggested that vibra-\ntions coupled to the spin degrees of freedom may induce\nelectrical currents that can provide interesting properties\nfor, e.g., mechanical control of single magnetic molecules\n[49, 50]. Superconducting spintronics also paves the way\ntoward enhancing central e \u000bects of spintronics devices\n[51–54].\nThe majority of the reported theoretical progress is,\nhowever, has been limited to stationary, or Markovian,\nprocesses. Although this is an important regime, both\nfor fundamental studies as well as for technological ap-arXiv:1603.03628v2  [cond-mat.mes-hall]  19 Aug 20162\nFIG. 1: The system studied in this work consisting of a lo-\ncal magnetic moment coupled to a QD in a tunnel junction\nbetween ferro- and non-magnetic leads.\nplications, it is nonetheless crucial to control also tran-\nsient properties induced by sudden on-sets and varia-\ntion of the external conditions applied to the system.\nRegarding spin dynamics, the Landau-Lifshitz-Gilbert\n(LLG) equation is often postulated as the platform for\ntheoretical studies, despite the fact that the (exchange\nand damping) parameters for this equation are typi-\ncally taken on phenomenological grounds or from ex-\nperiments. These parameters are, in addition, assumed\nto have a negligible time-dependence, something that\ncannot be taken for granted in nanoscale systems. Pre-\nvious derivations of the LLG equation [39, 55] clearly\nillustrate that the electronically mediated exchange inter-\nactions depend strongly on the magnetization dynamics\nand are, hence, intrinsically dynamical quantities as well.\nThe non-linearity of the dynamical equations indicates,\nmoreover, that it is non-trivial to decide whenever the\ntime-dependence of the interaction parameters can be\nneglected.\nTo begin to depart from the ad hoc treatments of the\ndynamics of spins coupled to electron currents, in this\npaper we perform time-dependent studies and analyses\nbeyond the Markovian and adiabatic approximations for\nboth the spin-dynamics and the tunneling current. In\naddition, we include the interdependence between the\ncurrent through the molecule and the localized magnetic\nmoment by considering both action and back-action in\nthe description. This can be regarded as the first loop in a\nself-consistent calculation, however, we do not perform\nour calculations to full self-consistency.\nThe model system, onto which we apply our devel-\noped method, is comprised of a magnetic molecule that\nis embedded in the tunnel junction between metallic\nleads. The leads themselves may support spin-polarized\ncurrents. Here, the magnetic molecule consists of two\ncomponents, namely, a quantum dot (QD) level and a\nlocalized magnetic moment, that interact via exchange.\nThe QD level is tunnel coupled to the leads. Hence, the\ncurrent flowing through the metal-QD-metal complexis expected to probe the presence of the localized mag-\nnetic moment, and, vice versa, the localized magnetic\nmoment is expected to depend on the current. Taking\nthis observation as an initial condition for our studies,\nwe construct a calculation scheme in which the dynam-\nics of the localized magnetic moment is described by a\ngeneralized version of the Landau-Lifshitz-Gilbert equa-\ntion [39, 42, 55, 56]. The e \u000bective spin-spin interactions\nare mediated by the tunneling current flowing across\nthe junction. The current, on the other hand, depends\ndirectly on the presence and dynamics of the localized\nmagnetic moment. We include this dependence by feed-\ning the time-evolution of the spin dynamics into the cur-\nrent, which causes the current-dependent temporal spin\nfluctuations to generate signatures back into the current.\nThe e \u000bective spin model derived in Sec. II, depends\nonly on the parameters included in our microscopical\nmodel – there are no ad hoc contributions in the descrip-\ntion in addition to the basic model. However, within the\nrealms of the model, there is a current mediated spin-\nspin interaction generated in the e \u000bective spin model,\nwhich describes interactions between the spin at time t\nand time t0. Hence, although there is only one spin in\nthe system, it is still justified to introduce the concept of\nspin-spin interaction since the spin at di \u000berent times can\nbe regarded as di \u000berent spins.\nSeparation of the magnetic molecule into a QD level\nand a localized magnetic moment is justified for, e.g.,\nM-phthalocyanines and M-porphyrins. In these com-\npounds, the transition metal d-levels, which are deeply\nlocalized, constitute the localized magnetic moment.\nThe s- and p-orbitals in the ligands, on the other hand,\ngenerate the spectral intensity at the highest occu-\npied molecular orbital (HOMO) and lowest unoccupied\nmolecular orbital (LUMO) levels which may be consid-\nered as the QD level(s) in our model.\nPreviously, Bode et al. [57] performed a similar theo-\nretical treatment of this problem using a non-equilibrium\nBorn-Oppenheimer approximation. Here, however, we\ngo beyond the adiabatic limit and extend the model to the\nnon-Markovian regime in order to treat memory e \u000bects\nand its impact on the exchange interaction. Hence, the\ninteraction fields in the spin equation of motion are not\nonly time-dependent but also dependent on their time-\nevolutions. A major di \u000berence with this formulation is\nthat all retardation e \u000bects are included in the time inte-\ngration of the interaction fields, and it is, therefore, not\nmeaningful to discuss quantities such as Gilbert damp-\ning since such parameters are defined in the adiabatic\nlimit.\nIn general, there also exist stochastic field acting on\nthe localized spin as a result of its interaction with the\nsurrounding electrons. Here, we have chosen to omit\nthe action of these fields, despite their importance for\na full description of the physics [58]. However, since\nwe consider the physics in the wide band limit, these\nelectronically induced stochastic fields are of Gaussian\nwhite noise character with no voltage bias dependence;3\nsee Appendix A. The stochastic field in this limit will,\ntherefore, merely play the role of a structureless thermal\nnoise field. As the main focus of our work is on the\ndynamics of the localized magnetic moment and the ex-\nchange interactions, we notice that our results are valid\nwhenever the energies of the interactions are larger than\nthe corresponding energies of these thermal noise fields.\nAdding a Langevin term, which arises from the quantum\nfluctuation in the spin action [55], into the spin equation\nof motion could be an interesting extension of the model\nused in this work, which would be the objective for a\nseparate study.\nThe paper is organized as follows. In Sec. II we dis-\ncuss the basic set-up of the formalism we employ in this\nstudy. After defining the model for the magnetic molec-\nular QD, we derive the equations for the spin moment\nand the tunneling current. Numerical results from these\nequations are presented in Sec. III and we summarize\nand conclude the paper in Sec. IV.\nII. METHOD\nTo be specific, we consider a magnetic molecule em-\nbedded in a tunnel junction between metallic leads that\nmay support spin-polarized currents, see Fig. 1 for refer-\nence. The magnetic molecule comprises a localized mag-\nnetic moment Scoupled via exchange to the highest oc-\ncupied molecular orbital (HOMO) or lowest unoccupied\nmolecular orbital (LUMO) level, henceforth referred to\nas the QD level. We define our system Hamiltonian as\nH=H\u001f+HT+HQD+HS: (1)\nHere,H\u001f=P\nk2\u001f;\u001b(\"k\u001f\u001b\u0000\u0016\u001f)cy\nk\u001f\u001bck\u001f\u001bis the Hamiltonian\nfor the lead \u001f=L=R, where cy\nk\u001f\u001b(ck\u001f\u001b) creates (annihi-\nlates) an electron in the lead with energy \"k\u001f\u001b, momen-\ntum kand spin\u001b=\";#. We have introduced the chemical\npotential\u0016\u001ffor the leads and the voltage bias across the\njunction defined as V=\u0016L\u0000\u0016R. Tunneling between the\nleads and the QD level is described by HT=HTL+HTR,\nwhereHT\u001f=T\u001fP\nk\u001b2\u001fcy\nk\u001f\u001bd\u001b+H:c:. The single-level QD\nis represented by HQD=P\n\u001b\"\u001bdy\n\u001bd\u001b, where dy\n\u001b(d\u001b) cre-\nates (annihilates) an electron in the QD with energy\n\"\u001b=\"0+g\u0016BB\u001bz\n\u001b\u001b=2 and spin\u001b. We include the Zeeman\nsplit due to the external magnetic field B=Bˆzwhere g is\nthe gyromagnetic ratio and \u0016Bthe Bohr magneton. The\nlocal spin is described by HS=\u0000g\u0016BS\u0001B\u0000vs\u0001S, where v\nis the interacting rate between the local spin and the elec-\ntron spin s=P\n\u001b\u001b0dy\n\u001b\u001b\u001b\u001b0d\u001b0=2, whereas\u001b\u001b\u001b0is the vector\nof Pauli matrices.\nA. Equation of motion of the local magnetic moment\nUsing the methods in, e.g., Refs. [39, 42, 44, 55, 56] and\nAppendix A, we derive an e \u000bective spin model for thelocalized magnetic moment S(t) from which we obtain\nthe equation of motion\n˙S(t)=\u0000g\u0016BS(t)\u0002Be\u000b(t)+1\neS(t)\u0002Z\nJ(t;t0)\u0001S(t0)dt0:(2)\nHere, in order to arrive at this result we have neglected\nlongitudinal spin fluctuations ( @tjSj=0) and rapid quan-\ntum fluctuations. The e \u000bective magnetic field is defined\nas\nBe\u000b(t)=B+1\neg\u0016BZ\n\u000fj(t;t0)dt0; (3)\nwhere Bis the external magnetic field while the second\nterm provides the internal magnetic field due to the elec-\ntron flow, where\n\u000fj(t;t0)=ie\u000fv\u0012(t\u0000t0)h[s(0)(t);s(t0)]i; (4)\nHere,\u000f=diagf\"\"\"#gand the charge s(0)=P\n\u001b\u001b0dy\n\u001b\u001b0\n\u001b\u001b0d\u001b0=2, where\u001b0is the identity matrix. This\ntwo-electron Green function (GF) is approximated by a\ndecoupling into single electron GFs according to\n\u000fj(t;t0)\u0019iev\u0012(t\u0000t0)sp\u000f\u0010\nG<(t0;t)\u001bG>(t;t0)\n\u0000G>(t0;t)\u001bG<(t;t0)\u0011\n; (5)\nwhere G<=>(t0;t) is the lesser /greater matrix GF of the\nQD defined by G<(t;t0)=fihcy\n\u001b0(t0)c\u001b(t)ig\u001b\u001b0and G>(t;t0)=\nf(\u0000i)hc\u001b(t)cy\n\u001b0(t0)ig\u001b\u001b0. In Eq. (5) sp denotes the trace over\nspin 1 /2 space.\nThe current J(t;t0)=i2ev2\u0012(t\u0000t0)h[s(t);s(t0)]iis the elec-\ntron spin-spin correlation function which mediates the\ninteractions between the localized magnetic moment at\ntimes tand t0. As with the internal magnetic field, we\ndecouple this two-electron GF according to\nJ(t;t0)\u0019ie\n2v2\u0012(t\u0000t0)sp\u001b\u0010\nG<(t0;t)\u001bG>(t;t0)\n\u0000G>(t0;t)\u001bG<(t;t0)\u0011\n: (6)\nThis current mediated interaction can be decomposed\ninto an isotropic Heisenberg, JH, interaction and the\nanisotropic Dzyaloshinski-Moriya (DM), D, and Ising,\nI, interactions. This can be seen from the product S\u0001J\u0001S,\nwhich is the corresponding contribution in the e \u000bec-\ntive spin model [44] to S(t)\u0002J(t;t0)\u0001S(t0) in the spin\nequation of motion. Using the general partitioning\nG=G0\u001b0+G1\u0001\u001b, where G0and G1describes the elec-\ntronic charge and spin, it is straight forward to see that\nspS\u0001\u001bG\u001bG\u0001S\n=spS\u0001\u001b(G0\u001b0+G1\u0001\u001b)\u001b(G0\u001b0+G1\u0001\u001b)\u0001S\n=sp\u0010\nS\u0001G1+[SG0+iS\u0002G1]\u0001\u001b\u0011\u0010\nG1\u0001S\n+[G0S\u0000iG1\u0002S]\u0001\u001b\u0011\n; (7)4\nwhere we have used the identity ( A\u0001\u001b)(B\u0001\u001b)=A\u0001B+\ni[A\u0002B]\u0001\u001b. As the Pauli matrices are traceless, the above\nexpression reduces to\n2\u0010\nS\u0001(G1G1)\u0001S+[SG0+iS\u0002G1]\u0001[G0S\u0000iG1\u0002S]\u0011\n:(8)\nAfter a little more algebra we obtain the Heisenberg ( JH),\nanisotropic Ising ( I) and anisotropic Dzyaloshinsky-\nMoriya ( D) interactions\nJH(t;t0)=iev2\u0012(t\u0000t0)\u0010\nG<\n0(t0;t)G>\n0(t;t0)\n\u0000G>\n0(t0;t)G<\n0(t;t0)\u0000G<\n1(t0;t)\u0001G>\n1(t;t0)\n+G>\n1(t0;t)\u0001G<\n1(t;t0)\u0011\n; (9a)\nI(t;t0)=iev2\u0012(t\u0000t0)\u0010\nG<\n1(t0;t)G>\n1(t;t0)\n\u0000G>\n1(t0;t)G<\n1(t;t0)+h\nG<\n1(t0;t)G>\n1(t;t0)\n\u0000G>\n1(t0;t)G<\n1(t;t0)it\u0013\n; (9b)\nD(t;t0)=\u0000ev2\u0012(t\u0000t0)\u0010\nG<\n0(t0;t)G>\n1(t;t0)\n\u0000G>\n0(t0;t)G<\n1(t;t0)\u0000G<\n1(t0;t)G>\n0(t;t0)\n+G>\n1(t0;t)G<\n0(t;t0)\u0011\n: (9c)\nThis leads to that we can partition the current mediated\nspin-spin interaction in the spin equation of motion into\nS(t)\u0002J(t;t0)\u0001S(t0)=JH(t;t0)S(t)\u0002S(t0)\n+S(t)\u0002I(t;t0)\u0001S(t0)\n\u0000S(t)\u0002D(t;t0)\u0002S(t0): (10)\nIn absence of spin-dependence in the QD GF, that is,\nforG1=0, it is clear that only the Heisenberg interaction\nJHremains, since both Iand Dexplicitly depend on G1.\nThere are di \u000berent sources that generates a finite G1, e.g.,\nspin injection from the leads, Zeeman split QD level, but\nalso the interaction with the localized magnetic moment\ngives an essential contribution. In this paper, we include\ne\u000bects from all three sources.\nThe spin equation of motion derived here goes far\nbeyond the LLG equation as it includes all retardation\ne\u000bects under the time-integration, something which is\nessentially missing in the LLG equation except for the\nstatic exchange interaction and Gilbert damping. Start-\ning from Eq. 2 and restricting to the adiabatic limit it\nis possible to derive the conventional LLG equation, see\nRef. [39]. For clarity this is also done in the Appendix\nB. This also implies that Eq. 2 includes the important\nGilbert damping and spin-transfer torque, as discussed\nin Ref. [57] and [59]. Higher order retardation e \u000bects\n(dissipation, moment of inertia, etc) are included in the\ntime-integralR\nJ(t;t0)\u0001S(t0)dt0.B. Quantum dot GF\n1. Bare quantum dot Green function\nNext, we derive the GF for the QD, which is defined\nasG(t;t0)=f(\u0000i)hTcy\n\u001b0(t0)c\u001b(t)ig\u001b\u001b0where T is the contour-\nordering operator. We introduce a bare GF g\u001b(t;t0) as the\nsolution to the equation\n(i@t\u0000\"\u001b)g\u001b(t;t0)=\u000e(t\u0000t0)+Z\n\u0006\u001b(t;\u001c)g\u001b(\u001c;t0)d\u001c: (11)\nThe bare GF then describes the electronic structure of the\nQD when coupled to the leads through the self-energy\n\u0006\u001b(t;t0)=P\n\u001fP\nk2\u001fjT\u001fj2gk\u001b(t;t0), however, without any\ncoupling to the local spin moment, as illustrated in Fig.\n2 (a). Here,\ngk\u001b(t;t0)=(\u0000i)Te\u0000iRt\nt0\"k\u001b(\u001c)d\u001c(12)\nis the GF for the lead \u001f, including the time-dependence\nimposed by the voltage bias.\nThe self-energy \u0006=diagf\u0006\"\u0006#gis treated in the wide-\nband limit (WBL), which for the retarded /advanced and\nlesser /greater forms are given by\n\u0006r=a\n\u001b(t;t0)=(\u0007i)\u000e(t\u0000t0)\u0000\u001b=2; (13a)\n\u0006<=>\n\u001b(t;t0)=(\u0006i)X\n\u001f\u0000\u001f\n\u001bK<=>\n\u001f(t;t0); (13b)\nwhere \u0000\u001b=P\n\u001f\u0000\u001f\n\u001band \u0000\u001f\n\u001b=2jT\u001fj2P\nk2\u001f\u000e(!\u0000\"k\u001b),\nwhereas\nK<=>\n\u001f(t;t0)=Z\nf\u001f(\u0006!)e\u0000i!(t\u0000t0)+iRt\nt0\u0016\u001f(\u001c)d\u001cd!\n2\u0019: (14)\nHere, f(\u0006!) is the Fermi function. The WBL allows to\nwrite the retarded /advanced zero GF as\ngr=a\n\u001b(t;t0)=(\u0006i)\u0012(\u0006t\u0007t0)e\u0000i(\"\u001b\u0007i\u0000\u001b=2)(t\u0000t0): (15)\nBy defining the coupling parameters \u0000\u001f\n0=P\n\u001b\u0000\u001f\n\u001band\n\u0000\u001f\n1=P\n\u001b\u001bz\n\u001b\u001b\u0000\u001f\n\u001bˆ zand introducing the spin-polarization in\nthe leads p\u001f2[\u00001;1], such that \u0000\u001f\n\u001b= \u0000\u001f\n0(1+\u001bz\n\u001b\u001bp\u001f)=2, we\ncan write \u0000\u001f\n1=p\u001f\u0000\u001f\n0ˆ z. With this notation we can in-\ntroduce the coupling matrix \u0000= \u0000 0\u001b0+\u00001\u0001\u001b, where\n\u00000=P\n\u001f\u0000\u001f\n0and\u00001=P\n\u001f\u0000\u001f\n1. Analogously, we write\nthe retarded /advanced and lesser /greater self-energies as\n\u0006r=a= \u0006r=a\n0\u001b0+\u0006r=a\n1\u0001\u001band\u0006<=>(t;t0)= \u0006<=>\n0\u001b0+\u0006<=>\n1\u0001\u001b,\nwhere\n\u0006r=a\n0(t;t0)=(\u0006i)\u000e(t\u0000t0)\u00000=2; (16a)\n\u0006r=a\n1(t;t0)=(\u0006i)\u000e(t\u0000t0)\u00001=2; (16b)\n\u0006<=>\n0(t;t0)=(\u0006i)X\n\u001f\u0000\u001f\n0K<=>\n\u001f(t;t0); (16c)\n\u0006<=>\n1(t;t0)=(\u0006i)X\n\u001f\u0000\u001f\n1K<=>\n\u001f(t;t0): (16d)5\nFIG. 2: Sketch of the system without (a) and with (b) a lo-\ncal magnetic moment and coupled to the leads. In the latter\ncase the interactions with the spin moment induce an e \u000bective\nZeeman split.\nUsing this notation we partion the bare GF in terms\nof its charge and magnetic components according to g=\ng0\u001b0+\u001b\u0001g1. The retarded /advanced form of gcan then\nbe written\ngr=a\n0(t;t0)=(\u0006i)\u0012(\u0006t\u0007t0)X\n\u001be\u0000i(\"\u001b\u0007i\u0000\u001b=2)(t\u0000t0)=2; (17a)\ngr=a\n1(t;t0)=(\u0006i)\u0012(\u0006t\u0007t0)X\n\u001b\u001bz\n\u001b\u001be\u0000i(\"\u001b\u0007i\u0000\u001b=2)(t\u0000t0)ˆ z=2:(17b)\nAnalogously, the lesser /greater forms of gare given by\ng<=>(t;t0)\u0011Z\ngr(t;\u001c)\u0006<=>(\u001c;\u001c0)ga(\u001c0;t0)d\u001cd\u001c0\n=g<=>\n0(t;t0)\u001b0+\u001b\u0001g<=>\n1(t;t0); (18)\nwhere (time-dependence of the propagators in the inte-\ngrands is suppressed)\ng<=>\n0(t;t0)=Z\u0010\ngr\n0\u0006<=>\n0ga\n0+gr\n1\u0006<=>\n0\u0001ga\n1\n+gr\n0\u0006<=>\n1\u0001ga\n1+gr\n1\u0001\u0006<=>\n1ga\n0\u0011\nd\u001cd\u001c0; (19a)\ng<=>\n1(t;t0)=Z\u0010\ngr\n0\u0006<=>\n1ga\n0+gr\n1\u0001\u0006<=>\n1ga\n1\n+gr\n0\u0006<=>\n0ga\n1+gr\n1\u0006<=>\n0ga\n0\u0011\nd\u001cd\u001c0: (19b)\n2. Dressed quantum dot Green function\nThe next step is to include the interactions with the\nlocal magnetic moment into the description. We achieve\nthis goal by defining the dressed QD GF as the first order\nexpansion in terms of the local moment, that is,\nG(t;t0)=g(t;t0)+\u000eG(t;t0)\n=g(t;t0)\u0000vI\nCg(t;\u001c)hS(\u001c)i\u0001\u001bg(\u001c;t0)d\u001c: (20)\nwhere gis the bare GF and \u000eGis the correction from\nthe interactions with the local magnetic moment. As\nabove, we write G=G0\u001b0+\u001b\u0001G1, where G0=g0+\u000eG0and G1=g1+\u000eG1, whereas the corrections are given by\n\u000eG0(t;t0)=\u0000vI\nC\u0010\ng0hSi\u0001g1\n+g1\u0001hSig0+i[g1\u0002hSi]\u0001g1\u0011\nd\u001c; (21a)\n\u000eG1(t;t0)=\u0000vI\nC\u0010\ng0hSig0+i[g1\u0002hSi]g0\n+ig0[hSi\u0002g1]+i[g1\u0002hSi]\u0002g1\u0011\nd\u001c: (21b)\nWe refer to Appendix C for details about the\nlesser /greater forms of the charge and magnetic com-\nponents of\u000eG.\nIt should be noticed that the presence of the local spin\nmoment gives rise to a spin-polarization of the QD level\ndue to the local exchange interaction, see Fig. 2 (b) for an\nillustration. The e \u000bect is particularly strong whenever\nthere is an intrinsic spin-polarization in either the leads\nand/or the QD, in which case g1,0. Then, the local\nspin moment a \u000bects the properties of both the charge\nand magnetic structure of the QD. Nevertheless, even\nfor spin-degenerate leads and QD, that is, for g1\u00110,\nthe QD level acquires a finite spin-dependence. This is\nlegible in the expression for \u000eG1, where the first term\nonly depends on the magnetic properties of the local\nspin moment and the charge density in the QD. Thus, by\ncalculating the electronic structure in the QD as function\nof the local spin moment opens for tracing signatures of\nthe local spin dynamics in the properties of the QD.\nC. Current\nThe properties of the QD are probed by means of the\nelectron currents flowing through the system. In this\nway, the goal is to pick up signatures of the spin dynam-\nics in the transport properties as these should influence\nthe electronic structure of the QD. The electron currents\ncan be decomposed into charge and spin currents, ICand\nIS, respectively. Here, we calculate the currents flowing\nthrough the left interface between the leads and the QD.\nAccordingly, we define\nIC\nL(t)=\u0000e@tX\nk\u001b2Lhnk\u001bi=iesp@tX\nkG<\nk(t;t); (22a)\nIS\nL(t)=\u0000e@tX\nk\u001b\u001b02Lhcy\nk\u001b\u001b\u001b\u001b0ck\u001b0i=iesp\u001b@tX\nkG<\nk(t;t):\n(22b)\nUsing standard methods we can write the charge current\nas\nIC\nL(t)=\u00002e\n¯hspIm\u0000LZt\n\u00001\u0010\nK>\nL(t;t0)G<(t0;t)\n+K<\nL(t;t0)G>(t0;t)\u0011\ndt0: (23)\nFollowing the same route as initiated above, we partition\nthe current into a spin-independent and spin-dependent6\npart according to IC\nL(t)=IC\n0(t)+IC\n1(t), where\nIC\n0(t)=4e\n¯h\u0000L\n0ImZt\n\u00001\u0010\nK>\nLG<\n0+K<\nLG>\n0\u0011\ndt0; (24a)\nIC\n1(t)=\u00004e\n¯h\u0000L\n1\u0001ImZt\n\u00001\u0010\nK>\nLG<\n1+K<\nLG>\n1\u0011\ndt0: (24b)\nAnalogously to the charge current, we write the spin\ncurrent as\nIS\nL(t)=\u00002e\n¯hspIm\u001b\u0000LZt\n\u00001\u0010\nK>\nL(t;t0)G<(t0;t)\n+K<\nL(t;t0)G>(t0;t)\u0011\ndt0; (25)\nwhere IS\nL(t)=IS\n0(t)+IS\n1(t) and\nIS\n0(t)=\u00004e\n¯h\u0000L\n0ImZt\n\u00001\u0010\nK>\nLG<\n1+K<\nLG>\n1\u0011\ndt0; (26a)\nIS\n1(t)=\u00004e\n¯hImZt\n\u00001\u0014\nK>\nL\u0010\n\u0000L\n1G<\n0+i\u0000L\n1\u0002G<\n1\u0011\n+K<\nL\u0010\n\u0000L\n1G>\n0+i\u0000L\n1\u0002G>\n1\u0011\u0013\ndt0: (26b)\nThese expressions for the charge and spin currents\nsuggest that any local dynamics that is picked up by the\nelectronic structure of the QD should provide signatures\nin its transport properties. Next, we analyze the impact\nof the local dynamics on the transport properties.\nIII. RESULTS\nA. Stationary limit\nBefore embarking into the full time-dependent prop-\nerties of the system, we review some of the expected\nresults for the stationary regime in order to provide a\nbenchmark for our calculations. In the stationary limit\nall the time-dependences induced from the on-set of the\napplied voltage bias have decayed which leads to the\nbare QD GF becomes time local and we can, therefore,\nstudy the energetic properties of the QD. Then, the local\nmagnetic moment, hSi, can be regarded as a constant\nspin-polarization and a source for coupling between the\nspin states, in agreement with Ref. [43]. The Fourier\ntransform of the bare QD GF is, therefore, written on the\nform\ngr=a\n0(!)=1\n2X\n\u001bgr=a\n\u001b(!); (27a)\ngr=a\n1(!)=1\n2X\n\u001b\u001bz\n\u001b\u001bgr=a\n\u001b(!);ˆz; (27b)\nwhere\ngr=a\n\u001b(!)=1\n!\u0000\"0\u0006i\u0000\u001b=2; (28)\nFIG. 3: Charge and spin current for a static local magnetic\nmoment in a tunnel junction. (a) Charge current ICas function\nof\"0and (b) spin current ISas function of \"0. Here, we used\n\u00000=v=1 meV , T=1 K,B=1 T,pL=pL=0 and V=2 mV such\nthat VL=V=2 and VR=\u0000V=2.\nand the self-energies become\n\u0006<=>\n0(!)=(\u0006i)X\n\u001f\u0000\u001f\n0f\u001f(\u0006!); (29a)\n\u0006<=>\n1(!)=(\u0006i)X\n\u001f\u0000\u001f\n1f\u001f(\u0006!); (29b)\nsince K<=>\n\u001f(!)=f(\u0006!) in the stationary limit. In the sta-\ntionary limit the interaction parameters, moreover, sim-\nplify in the limit \u000f!0 to\nJ(H)=\u0000v2Z1\n!+\u000f\u0000!0\u0010\nG<\n0(!)G>\n0(!0)\u0000G>\n0(!)G<\n0(!0)\n\u0000G<\n1(!)\u0001G>\n1(!0)+G>\n1(!)\u0001G<\n1(!0)\u0011d!\n2\u0019d!0\n2\u0019;(30a)\nI=\u0000v2Z1\n!+\u000f\u0000!0\u0010\nG<\n1(!)G>\n1(!0)\u0000G>\n1(!)G<\n1(!0)\n+h\nG<\n1(!)G>\n1(!0)\u0000G>\n1(!)G<\n1(!0)it\u0013d!\n2\u0019d!0\n2\u0019;(30b)\nD=v2\n2ReZ\u0010\nG<\n0(!+\u000f)G>\n1(!)\u0000G>\n0(!+\u000f)G<\n1(!)\n\u0000G<\n1(!+\u000f)G>\n0(!)+G>\n1(!+\u000f)G<\n0(!)\u0011d!\n2\u0019: (30c)\nConsidering a symmetric and spin-independent back-\nground, i.e. non-magnetic contacts p\u001f=0, and a constant\nlocal magnetic moment, S, the local spin-polarization\ngives rise to finite spin currents ISin the system, see\nEqs. (25) – (26) (note \u0000S=0). In Fig. 3 we plot the\ncalculated (a) charge ( IC) and (b) spin current ( IS) as a\nfunction of the gate voltage, V, for a QD with a bare level\nat\"0=0. While the charge current behaves as expected\nfor a single-level QD, given by\nIC\nL=e\n4\u0019¯h\u00002\n0ZfL(!)\u0000fR(!)\n(!\u0000\u000f0)2+(\u00000=4)2d!; (31)\nthe features in the spin current for gate voltages near zero\ngive a clear indication of the induced spin-polarization7\nfrom the local spin moment. Due to the local spin mo-\nment induced e \u000bective Zeeman split in the QD, as shown\nin Fig. 2 (b), the spin current is strongly peaked at \u0016\u001f=\"0.\nAs can be seen in Fig. 2, either one of the spin up- or\ndown channels will be more favorable for the tunneling\nelectrons, thus causing a net spin current in either di-\nrection depending on the configuration of the electron\nlevel of the leads. This is an important feature as it can\nbe used in order to read out the state of the local spin\nmoment from the spin current.\nRegarding the Heisenberg interaction, recalling that,\ne.g., Gr\n0(!)=gr\n0(!)=1=(!\u0000\"0+i\u00000=4) for non-magnetic\nleads, it can be readily seen that the charge contribution\nto the Heisenberg exchange is given by\n2v2\n\u0019X\n\u001f\u0000\u001f\n0Z\nf\u001f(!)!\u0000\"0\n[(!\u0000\"0)2+(\u00000=4)2]2d!: (32)\nThis suggests a spin-spin interaction which is strongly\npeaked around \u0016\u001f=\"0. Similarly, the contribution from\nthe local spin-polarization, G1=\u0000vg0hSig0, acquires the\nform\n\u00004v4\n\u0019jhSij2X\n\u001f\u0000\u001f\n0Z\nf\u001f(!)(!\u0000\"0)(!\u0000\"0)2\u0000(\u00000=4)2\n[(!\u0000\"0)2+(\u00000=4)2]4d!;\n(33)\nwhich is also strongly peaked at \u0016\u001f=\"0. However,\nas the integrand of this component changes sign at\n!=\"0;\"0\u0006\u00000=4, the contribution from the QD spin-\npolarization goes through local minima at \"0\u0006\u00000=4 and\na local maxima at \"0, as a function of the chemical poten-\ntial\u0016\u001f. We therefore expect a competition between the\ncharge and magnetic components which may lead to a\nchange of sign in the Heisenberg interaction, depending\nboth on the properties of the system as well as on the\nexternal conditions. This is illustrated by the computed\nHeisenberg exchange plotted in Fig. 4 (a) as a function of\nFIG. 4: Heisenberg interaction JHof a static local magnetic\nmoment in a tunnel junction in the z-direction, S=Szˆ z. Panel\n(a) shows the Heisenberg interaction for non-magnetic leads\np\u001f=0 as a function of bias voltage V . Here, the gate voltage is\nset to\"0=0 meV and\"0=1 meV and the plots shifted for clarity\n(scale is the same). Panel (b) shows the Heisenberg interaction\nfor antiferromagnetic leads, pL=\u0000pR=0:5, as a function of bias\nvoltage V . Here, we used \u00000=v=0:1 meV , T=1 K,B=0 T and\nV=2 mV .\nFIG. 5: Ising interaction Iof a static local magnetic moment in\na tunnel junction in the z-direction, S=Szˆ z. Panel (a) shows\nthe Ising interaction for non-magnetic leads p\u001f=0 as a func-\ntion of bias voltage V . Here, the gate voltage is set to \"0=0\nmeV and\"0=1 meV and the plots shifted for clarity (scale is\nthe same). Panel (b) shows the Ising interaction for antiferro-\nmagnetic leads, pL=\u0000pR=0:5, as a function of bias voltage V .\nOther parameters as in Fig. 4.\nthe voltage bias for di \u000berent gate voltages, showing the\nchanging character from negative to positive interaction\nas the chemical potential \u0016\u001fapproaches the QD level.\nFor ferromagnetic leads aligned anti-ferromagnetically\nin Fig. 4 (b), pL=\u0000pR=0:5, we notice an anisotropic be-\nhavior as the sign of the interaction switches with respect\nto the polarity of the voltage bias. This is in agreement\nwith previous studies of anti-ferromagnetically aligned\nleads coupled to molecular spins [45].\nThe Ising interaction Iessentially behaves in a simi-\nlar manner, however, this contribution requires a finite\nspin-polarization ( G1,0) to become non-vanishing. For\nnon-magnetic leads, p\u001f=0, this spin-polarization is pro-\nvided by the local spin moment and we find that the\nIsing interaction acquires the form given in Eq. (33),\nup to multiplying constants. This is also verified by\nthe numerically computed Ising interaction, shown in\nFig. 5 (a) as function of the voltage bias for di \u000berent\ngating conditions. Again, for ferromagnetic leads in\nanti-ferromagnetic alignment, pL=\u0000pR=0:5, there is a\nswitching behavior with respect to the polarity of the\nvoltage bias.\nFIG. 6: DM interaction Dof a static local magnetic moment in\na tunnel junction in the z-direction, S=Szˆ z, for antiferromag-\nnetic leads, pL=\u0000pR=0:5. Panel (a) shows the DM interaction\nas a function bias voltage V where the gate voltage is set to\n\"0=0 meV and\"0=1 meV . Panel (b) shows the DM interaction\ndi\u000berent gate voltage \"0. Other parameters as in Fig. 4.8\nFIG. 7: Time-dependent evolution of the exchange interaction parameters as a function of gate voltage \"0after an onset of a\nstep-like finite bias voltage of V =2 mV . Panel (a) shows the strength of the Heisenberg interaction, panel (b) shows the Izzpart of\nthe Ising interaction and panel (c) shows the z-component of the DM interaction. Here we used the parameters \u00000=v=0:1 meV ,\nT=1 K,B=1 T,pL=pL=0.\nA similar switching behavior appears in the DM in-\nteraction D, which is only considered for ferromagnetic\nleads aligned anti-ferromagnetically, see Fig. 6 (a), where\nthe DM interaction is plotted as a function of the volt-\nage bias and for di \u000berent gating conditions. Varying the\ngate voltage, it can be seen that there is a finite DM in-\nteraction only whenever the QD electron level, \"0, lies in\nthe window between the chemical potentials in the leads\nspanned by the voltage bias. This is understood since the\nDM interaction results from net current flow interacting\nwith the local spin moment, as it requires simultaneous\nbreaking of time-reversal and inversion symmetries to\nbe finite.\nWe comment finally on the relevance for calculating\nthe interaction parameters in the stationary limit. This\nquestion is justified since the e \u000bective spin Hamiltonian\nin the stationary limit would assume the form\nHe\u000b\nS=\u0000J(H)S\u0001S\u0000D\u0001S\u0002S\u0000S\u0001I\u0001S: (34)\nHere, one can notice that S\u0001S=jSj2, which is a constant\nof motion, whereas as S\u0002S\u00110. Both these identities\nrelies on the fact that the spin Sis time-independent in\nthe stationary limit. Actually, only the Ising interaction\nis physically motivated, providing an anisotropy field\non the spin. For collinear spin-polarization in the sur-\nrounding system, this contribution reduces to the form\nIzzS2\nz, which is the ordinary Ising Hamiltonian.\nWe justify the calculations and analysis of the station-\nary limit interaction parameters by that we can under-\nstand and interpret much of the time-dependent fea-\ntures, discussed in the remainder of this paper, from the\nresults obtained in the stationary limit. In addition, our\nresults also demonstrate that despite the dynamics may\nbe trivial, the fields that mediate the interactions between\nthe dynamical object need not be trivial.B. Time-dependent exchange interaction\nAs we are interested in the transient dynamics, we\nstudy the e \u000bect of an abrupt on-set of the voltage bias\napplied as a step-like function V\u001f\u0012(t\u0000t0) symmetrically\nover the junction such that VL=R=\u0006V=2. Before the on-set\nof the voltage bias, the local spin is subject to the static\nexternal magnetic field B=Bˆz, giving Sx=Sxysin!Lt,\nSy=Sxycos!LtandSz=Szwhere S2\nxy=S2\nx+S2\ny, whereas\njSj2=S2\nxy+S2\nzand!L=g\u0016BjBj, and we assume an initial\npolar angle of \u0019=4.\nThe time-dependence of the interaction parameters,\ncf. Eq. (9), has to be calculated as function of the gate\nvoltage and voltage bias at each time-step. In Fig. 7 we\nplot the time-evolution of the Heisenberg, Ising, and DM\ninteraction parameters as function of the gate voltage,\nwhere we integrated over all t0, hence, showing JH(t),\nIzz(t) and Dz(t). Considered in this fashion, the plots il-\nlustrate the time-evolution of the exchange interaction\nthat would be expected in the adiabatic approximation,\nthat is,R\nJ(t;t0)\u0001S(t0)dt0\u0019R\nJ(t;t0)dt0\u0001S(t)+\u0001\u0001\u0001. In the tran-\nsient regime, the interaction parameters changes contin-\nuously, both due to the changing characteristics of the\nsystem and the feed-back through the system from the\nchanging local magnetic moment. In the long time limit,\nit may be noticed that the interaction strength peaks for\nall three types of interactions when the QD electron level\n\"0is resonant with one of the chemical potentials of the\nleads\u0016\u001f=eV\u001f. We, hence, retain the properties of the\nsystem in the stationary regime.\nWhen going beyond the adiabatic approximation, one\ncannot strictly separate the interaction parameters from\nthe time-evolution of the spin, see for instance Eq. (2).\nIt is then more comprehensible to directly study and\nanalyze each component of the equation of motion. Ac-\ncordingly, in Fig. 8 we plot the rates of change in the\nrespective panels9\nFIG. 8: Contribution to the local magnetic moment equation of motion for di \u000berent exchange interaction parameters as a function\nof gate voltage \"0. In (a) the change ˙Sz(\u000fj) in z-direction depending on the induced internal magnetic field due to the charge\nflow. In (b)-(d) the change in the z-direction depending on the Heisenberg, Ising and DM interaction is shown respectively. Other\nparameters as in Fig. 7.\n(a)˙Sz(t;\"j)=\u0000S(t)\u0002R\n[\u000fj(t;t0)]zdt0=e,\n(b)˙Sz(t;J(H))=\u0000S(t)\u0002R\nJ(H)(t;t0)Sz(t0)dt0=e,\n(c)˙Sz(t)=\u0000S(t)\u0002R\nIzz(t;t0)Sz(t0)dt0=e, and\n(d) ˙Sz(t)=\u0000S(t)\u0002R\nDz(t;t0)Sz(t0)dt0=e.\nThe rate of change caused by the current induced mag-\nnetic field, ˙Sz(t;\"j), shown in Fig. 8 (a), initially provides\na large contribution to the spin dynamics while it tends\nto zero in the long time limit. This is to be expected since\nthe time variations of the charge current are largest im-\nmediately after the on-set of the voltage bias. Far beyond\nthe transient regime initiated by this on-set, the temporal\nvariations in the charge current are much smaller which,\ntherefore, also leads to a smaller induced magnetic field.\nSimilar behavior appears for the Heisenberg, Ising and\nDM interactions, shown in Fig. 8 (b) – (d), where they ini-\ntially provide a large contribution to the spin dynamics in\nthe transient regime. Resulting from the time-dependent\ninteraction parameters, there is a finite contribution to\nthe spin dynamics for large time scales in the stationary\nlimit, in agreement with the time-independent solution.\nFIG. 9: Time-dependent evolution of the field from the\nanisotropic DM interaction in x- and y-direction as a function\nof gate voltage \"0after an onset of a step-like finite bias voltage\nof V=2 mV . Other parameters as in Fig. 7.The di \u000berence, however, is that there is a finite con-\ntribution in the stationary limit of the DM-interaction,\nsomething not observed in the time-independent solu-\ntion for non-magnetic leads. The reason of this e \u000bect\nis the time-dependent feature of the DM-fields. In Fig.\n9, the e \u000bective field from the DM interaction is shown\nin the x- and y-components of the vector defined by\n(D\u0001S)(t)=R\nD(t;t0)\u0001S(t0)dt0. The fields for the Heisen-\nberg and Ising contribution, i.e.R\nJH(t;t0)S(t0)dt0andR\nI(t;t0)\u0002S(t0)dt0, are similar to the basic parameters\nshown in Fig. 7,R\nJH(t;t0)dt0andR\nI(t;t0)dt0.\nC. Non-magnetic leads\nNext, we study the time-dependent solution of the\ncharge and spin currents and the evolution of the local\nspin moment for non-magnetic leads, p\u001f=0. Here, we\nuse\u00000=v=0:1 meV , T =1 K, B =1 T,pL=pL=0,\"0=0\nmeV and V =2 mV . Our computed charge current ICis\nshown in Fig. 10 for step-like voltage biases Vwith dif-\nferent amplitudes, turned on at time t=0. The contour in\nFig. 10 (a) shows the time-evolution of IC(t) as function of\nV, while the plots in Fig. 10 (b) correspond to the traces\nindicated in panel (a). The current acts as a response\nfunction to the step-like voltage bias according to a well\nknown and expected scheme and eventually reaches the\nstationary regime. A direct influence of amplitude of the\nvoltage bias is the increasing the frequency of the current\noscillations as the voltage bias grows as well as increas-\ning decay time. For non-magnetic leads the influence of\nthe local magnetic moment S(t) on the current is negli-\ngible. Hence, the essential time-dependent properties of\nthe charge current are captured by results provided in\nRefs. [60, 61].\nIn the case of non-magnetic leads, both \u00001=0 and\ng1=0, thus the only contribution to the spin-dependent\ndressed GF comes from \u0000vg0hSig0. With no applied gate\nvoltage this will lead to zero spin currents in the station-\nary limit, cf. Fig. 3 (b). This can be seen in Fig. 1110\nFIG. 10: Charge current as a function of time, IC(t), for di \u000berent\nstep-like bias voltages V . In (a) a contour of the time evolution\nof the charge current as a function of bias voltage V is shown.\nIn (b) the cuts in (a) are shown for di \u000berent bias voltage V . Here\n\u00000=v=2=0:1 meV , T =1 K, B =1 T,pL=pL=0 and\"0=0 meV .\n(a) - (c) where we show the spin currents in the x-, y-,\nand z-direction for di \u000berent bias voltages. The local spin\nmoment, initially at a polar angle \u0019=4, will align in the\nz-direction due to the damping e \u000bect of the charge back-\nground, see Fig. 11 (d)-(e). The process will be slower\nfor a finite bias voltage than for zero voltage as there are\nanisotropic e \u000bects in the system. The back-action via the\ncurrent in the junction causes this slower dynamics as\nit counteracts the motion in Fig. 8 (notice that we have\nzero gate voltage). Here, the induced internal magnetic\nfield and DM interaction causes the spin to align in the\npositive z-direction, whereas the Heisenberg and Ising\ninteractions cause it to flip in the negative z-direction.\nSimilar to this, the transient e \u000bects in the spin currents\nin Fig. 11 (a) - (c) are due to both the presence of the\nlocal spin moment and the current through the system,\nas it depends on G1=\u0000vg0hSig0. As the dynamics of\nthe local spin moment has a small dependence of the\nvoltage bias, the time- and voltage-dependent changes\nof the spin current are thus mainly due to the electronflow in the system, given by g0.\nIf there is no exchange coupling between the QD and\nthe local spin moment, i.e. v=0, the spin moment would\njust continue to rotate in the magnetic field and there\nwould not be any induced spin currents in the system.\nThis can be seen in Fig. 12 (b)-(c) for a finite charge\ncurrent through the system in Fig. 12 (a). The interaction\nbetween the current through the QD and the local spin\nmoment increases as we turn on the exchange coupling,\nthus changing the direction of the spin. The scaling\nbehavior depends on the interaction between the QD\nand the local spin moment. The internal magnetic field\nscales linearly with the exchange coupling, \u000fj/v, and\nthe current mediated interaction scales quadratically, J/\nv2. In turn, these equations depends on the back-action\nfrom the spin moment through the GF, defined as G1=\n\u0000vg0hSig0. The scaling of the exchange coupling is thus\nv4and in Fig. 12 (c) we can observe a 16 times faster\nprocess for every doubling of the exchange coupling.\nThe spin current Fig. 12 (b) scales linearly with exchange\ncoupling, due to G1=\u0000vg0hSig0, and the charge current\nFig. 12 (a) is independent of the exchange coupling as it\nonly depends on g0for non-magnetic leads.\nThe QD electron level \u000f0is adjusted by the gate voltage\nV . In the stationary limit, the charge current peaks when\nthe QD electron level lies between the chemical potential\nof the leads, and it quickly diminishes for higher and\nlower gate voltage. Because of the e \u000bective Zeeman\nsplit in the QD there will be finite spin currents that are\nstrongly peaked at \u0016\u001f=\"0. This can also be observed\nin the long time limit for di \u000berent gate voltage, shown\nin Fig. 12 (d)-(e). Due to the asymmetry of the time-\ndependent solution of the bare GF, where g0(t;t0) is not\nthe same as g0(t0;t) around the onset of bias voltage,\nthe currents become asymmetric for small time scales\ndepending on the sign of the gate voltage. As this is a\nshort-term e \u000bect, the asymmetry vanishes as it reaches\nsteady state. The evolution of the local spin moment,\nshown in Fig. 8, depends on the gate voltage as the\nstrength of the interaction strongly depends on the gate\nvoltage, see Fig. 7 and Fig. 8. For zero gate voltage,\nthe local spin moment simply aligns with the leads, as in\nFig. 11 (f). When a finite gate voltage is applied, the QD\nlevel is not symmetrically in between the leads, which\ngives both a spin-polarized current due to the Zeeman\nsplit and in turn changes the action on the spin moment.\nThis causes the spin moment to reach di \u000berent solutions\nin the steady state depending on both the isotropic and\nanisotropic interactions shown in Fig. 8.\nRecalling that the charge current for non-magnetic\nleads is given in the stationary limit by Eq. 31 we no-\ntice that it scales with the tunneling coupling \u00000as an\nLorentzian. This can be seen in the steady state limit in\nFig. 12 (g) where the charge current is plotted against dif-\nferent tunneling coupling. The dependence of \u00000in both\nthe spin currents and the interaction strength is as in Eqs.\n32 – 33, where it is of the form \u00000=(1+ \u00002\n0)2,\u00000=(1+ \u00004\n0)2\nand\u00003\n0=(1+\u00004\n0)2. This will give a high contribution for a11\nFIG. 11: Spin currents IS(t) as a function of time for di \u000berent step-like bias voltage (a) IS\nLx, (b) IS\nLyand (c) IS\nLz. Normalized local\nmagnetic moment S(t) as a function of time for di \u000berent step-like bias voltage (d) Sx, (e)Syand (f) Sz. Here the same values as in\nFig. 10 is used.\nnarrow range and can be seen in Fig. 12 (h)-(i), where\nthe spin current and evolution of the local spin moment\nis plotted against di \u000berent tunneling coupling.\nD. Ferromagnetic leads\nWe now study the system for di \u000berent magnetization\nof the leads. In Fig. 13 the di \u000berent columns show dif-\nferent spin polarization of the leads. To enhance the\ne\u000bects, the simulation was run with T =1\u0016K and B =1\n\u0016T. In Fig. 13 column (a) the magnetization is changed\nfor both leads, pL=pR=p, in a ferromagnetic alignment.\nAs can be seen in the stationary limit, the spin current\nis then net positive or net negative due to the spin po-\nlarization of the leads, as expected. The charge current\ndecreases some for strongly polarized leads as one of the\nspin species diminishes, thus only tunneling through ei-\nther the spin-up or -down channel (see Fig. 2 (b) for\nillustration). If the drain lead is kept non-magnetic, i.e.\npL=p;pR=0, this behavior becomes clearer; see Fig. 13\ncolumn (b). Here, a majority of spin-up or spin-down\nelectrons enters the QD while both have equal prob-\nability to exit. The spin current through the junction\nbecomes less polarized because only one of the leads\nis ferromagnetic, while the other is non-magnetic. The\nchanges in charge current are large, because of the non-\ncollinear arrangement of the leads. Similar changes in\ncharge and spin currents appear when the source is keptnon-magnetic, i.e. pL=0;pR=p, see Fig. 13 column (d).\nWhen the source or drain lead is kept ferromag-\nnetic, i.e., p\u001f=0:5, the charge and spin current behavior\nchanges slightly; see Fig. 13 column (c) and (e). We\nget the highest charge current when the leads are in a\nferromagnetic configuration, while it is the lowest in an\nanti-ferromagnetic configuration. The same happens for\nthe spin current, where it is the highest in a ferromag-\nnetic configuration and it goes to zero /negative when the\nleads are in an anti-ferromagnetic configuration. This is\nwell expected and agrees with a simple model of a QD\nbetween magnetic leads.\nWe notice that there is a slight di \u000berence between the\nspin currents in Fig. 13 column (b) and (d) and the same\nfor column (c) and (e). This is partly due to the asymme-\ntry of the source and drain leads, which causes an e \u000bect\non small time scales when the bias voltage is turned on,\nwhere a spin-up injection causes a peak in the spin cur-\nrent, and a spin-down injection causes a bottom. It is also\ndue to the influence of the local spin moment in the junc-\ntion which reaches di \u000berent stationary solutions (see the\nbottom row of Fig. 13), as the dressed GF depends on the\nconnection with the spin moment, as is given in Eq. 20.\nThe di \u000berence is shown in Fig. 14 for the case in which\none lead is kept non-magnetic and the spin polarization\nof the other lead is shifted. The plot shows the di \u000berence\n[IS\nz(pL=0;pR=p)\u0000IS\nz(pL=p;pR=0)]=IC. Hence, because\nof the given polarity of the voltage bias, the system is\nnot invariant with respect to inversion symmetry, and12\nFIG. 12: Evolution of the currents and local magnetic moment when adjusting di \u000berent parameters. For di \u000berent exchange\ncoupling, v, the figures show (a) the charge current IC, (b) the spin current IS\nzand (c) the local magnetic moment in the z-direction\nSz. For di \u000berent gate voltage, \"0, the figures show (d) the charge current IC, (e) the spin current IS\nzand (f) the local magnetic\nmoment in the z-direction Sz. For di \u000berent tunneling coupling, \u00000, the figures show (g) the charge current IC, (h) the spin current\nIS\nzand (i) the local magnetic moment in the z-direction Sz. Here the same values as in Fig. 7 is used.\nthe configurations pL=0;pR=pandpL=p;pR=0 are not\nequivalent under finite voltage bias.\nFor ferromagnetic leads, the interaction between the\nQD and the spin moment changes. This causes the lo-\ncal spin moment to change direction for di \u000berent spin-\npolarized configurations of the leads. The evolution of\nthe spin moment for di \u000berent configurations is shown\nin the bottom row in Fig. 13. Due to the anisotropic\ninteractions Iand D, the local spin moment anti-aligns\nwith the source lead while it aligns with the drain lead;\nsee Figs. 13(b) and 13(c). The e \u000bect is stronger from the\nsource lead than the drain lead, as the source determines\nthe electron flow into the system. In the small-time scale,\nright after the gate voltage is turned on, we can observe adip in the evolution of the spin moment before its reaches\nits stationary solution. This we can describe energeti-\ncally as a double well potential, where the spin moment\nneeds enough energy in order to pass a barrier and reach\na spin-down solution. If it does not have enough energy\nto pass the barrier, it stays in a spin-up solution; see Fig.\n15. In the present case, the spin is considered classical\nand the transition will be continuous, while a quantum\nspin would tunnel through the potential barrier for high\nenough energies. As we see in Figs. 13(b) and 13(c), this\nconfiguration favours a spin up solution, while when\nthe current is driven in the opposite direction, shown\nin Figs. 13(d) and 13(e), it favors a spin-down solution.\nThis is illustrated by the di \u000berent depths of the potential13\nFIG. 13: Evolution of the currents and local magnetic moment for di \u000berent polarization p of the leads indicated by the arrows in\nthe second row, where the left arrow indicates the polarization of the left lead and the right arrow indicates the right lead. The first\nrow shows the charge current IC, the second row the spin currents IS\nzand the third row shows the local magnetic moment in the\nz-direction Sz. In column (a) the magnetization is changed for both leads, pL=pR=p. In column (b) the right lead is non-magnetic\nand the left lead is changed, pR=0 and pL=p. In column (c) the right lead is ferromagnetic and the left lead is changed, pR=0:5\nandpL=p. In column (d) the left lead is non-magnetic and the right lead is changed, pR=pandpL=0. In column (e) the left lead\nis ferromagnetic and the right lead is changed, pR=pandpL=0:5. Here \u00000=0:1 meV , v=0:25 meV , V =2 mV , T =1\u0016K, B=1\u0016T,\npL=pL=0 and\u000f0=0 meV .\nwells in Fig. 15. Due to the anisotropies in the sys-\ntem, the spin moment ends up di \u000berently depending on\nthe anti-ferromagnetic configurations of the leads. The\nanti-ferromagnetic configuration is the opposite in the\nbottom of the figures in (c) and (e) in the bottom row in\nFig. 13, which will cause di \u000berent stationary solutions\nof the local spin moment. Thus, depending on the di-\nrection of the current through the dot one can use the\ne\u000bective spin torque in order to control the spin, which\nis in agreement with previous studies [45].\nIV . SUMMARY AND CONCLUSION\nIn summary, we have studied the time evolution of a\nlocal magnetic moment in a tunnel junction. We have\nshown that one can control and read-out the local mag-\nnetic moment using gate voltage and using magnetic\nleads in an anti-ferromagnetic set-up. This is in agree-\nment with previous works[45–47] and is a promising\nfeature in order to perform electrical control and read-\nout of magnetic molecules.\nWe have shown that non-trivial exchange interaction\nappears in the time-dependent domain, especially forsmall time scales. Anisotropic e \u000bects occur due to time-\ndependency which will e \u000bect the direction of the mag-\nnetic moment. A large e \u000bective magnetic field is a signif-\nicant e \u000bect that occurs for small time scales and adjusts\nthe evolution of the local magnetic moment, an e \u000bect\nnot usually considered as it vanishes for the stationary\nsolution. Considering time-dependent exchange inter-\naction is thus important in small time-scale calculations\nand shows the potential for a deeper understanding of\nthe exchange interaction. This leads to further questions\non how important the time-dependency is for large scale\nspin-dynamics calculations, something suitable for fur-\nther investigation.\nThis work and previous studies have found that there\nis rich physics within this framework and that it is im-\nportant on the quantum scale to take time-dependent\nnon-equilibrium e \u000bects into consideration when analyz-\ning time-dependent phenomena. Anisotropic exchange\ninteractions play an important role when studying time-\ndependent phenomena. With recent experimental ad-\nvances, we believe it to be possible to verify our findings\nwith state-of-the-art experiments.14\nFIG. 14: Di \u000berence in spin current normalized by charge cur-\nrent in the stationary limit, [ IS\nz(pL=0;pR=p)\u0000IS\nz(pL=p;pR=\n0)]=IC, depending on direction of current with one lead non-\nmagnetic and the other lead with di \u000berent spin polarization.\nThis plot corresponds to the di \u000berence between columns (b)\nand (d) in the second row of Fig. 13. The di \u000berence is due to\nthe di \u000berent directions of the local spin moment.\nFIG. 15: Double well potential illustrating the possible solu-\ntions for the local spin moment depending on starting position.\nAs can be seen in Fig. 13 the spin moment starts up and then\ntries to reach a down solution. Depending on strength of the\nanisotropies in the system it either passes the energy barrier or\nnot.\nAcknowledgments\nWe thank J. -X. Zhu for inspiring discussions in the\ninitial stages of this work. We also thank K. Bj ¨ornson,\nT. L¨othman and J. D. Vasquez Jaramillo for fruitful dis-\ncussions and for comments on our results. Financial\nsupport from Vetenskapsrådet is acknowledged. This\nwork is part of the CINT User Proposal # U2015A0056.\n[1] J. R. Hauptmann, J. Paaske, and P . E. Lindelof, Nat Phys\n4, 373 (2008).\n[2] S. Loth, K. von Bergmann, M. Ternes, A. F. Otte, C. P . Lutz,\nand A. J. Heinrich, Nat Phys 6, 340 (2010).\n[3] S. Wagner, F. Kisslinger, S. Ballmann, F. Schramm, R. Chan-\ndrasekar, T. Bodenstein, O. Fuhr, D. Secker, K. Fink,\nM. Ruben, et al., Nat Nano 8, 575 (2013).\n[4] C. F. Hirjibehedin, C. P . Lutz, and A. J. Heinrich, Science\n312, 1021 (2006).\n[5] P . Wahl, P . Simon, L. Diekh ¨oner, V . S. Stepanyuk, P . Bruno,\nM. A. Schneider, and K. Kern, Phys. Rev. Lett. 98, 056601\n(2007).\n[6] F. Meier, L. Zhou, J. Wiebe, and R. Wiesendanger, Science\n320, 82 (2008).\n[7] T. Balashov, T. Schuh, A. F. Tak ´acs, A. Ernst, S. Ostanin,\nJ. Henk, I. Mertig, P . Bruno, T. Miyamachi, S. Suga, et al.,\nPhys. Rev. Lett. 102, 257203 (2009).\n[8] S. Voss, O. Zander, M. Fonin, U. R ¨udiger, M. Burgert, and\nU. Groth, Phys. Rev. B 78, 155403 (2008).\n[9] K. Bairagi, A. Bellec, V . Repain, C. Chacon, Y. Girard,\nY. Garreau, J. Lagoute, S. Rousset, R. Breitwieser, Y.-C.\nHu, et al., Phys. Rev. Lett. 114, 247203 (2015).\n[10] L. Zhou, J. Wiebe, S. Lounis, E. Vedmedenko, F. Meier,\nS. Bl ¨ugel, P . H. Dederichs, and R. Wiesendanger, Nat Phys\n6, 187 (2010).\n[11] X. Chen, Y.-S. Fu, S.-H. Ji, T. Zhang, P . Cheng, X.-C. Ma,\nX.-L. Zou, W.-H. Duan, J.-F. Jia, and Q.-K. Xue, Phys. Rev.\nLett. 101, 197208 (2008).\n[12] A. F. Otte, M. Ternes, S. Loth, C. P . Lutz, C. F. Hirjibehedin,\nand A. J. Heinrich, Phys. Rev. Lett. 103, 107203 (2009).\n[13] H. Pruser, M. Wenderoth, P . E. Dargel, A. Weismann, R. Pe-\nters, T. Pruschke, and R. G. Ulbrich, Nat Phys 7, 203 (2011).[14] A. A. Khajetoorians, J. Wiebe, B. Chilian, and R. Wiesen-\ndanger, Science 332, 1062 (2011).\n[15] S. Loth, S. Baumann, C. P . Lutz, D. M. Eigler, and A. J.\nHeinrich, Science 335, 196 (2012).\n[16] A. A. Khajetoorians, B. Baxevanis, C. H ¨ubner, T. Schlenk,\nS. Krause, T. O. Wehling, S. Lounis, A. Lichtenstein,\nD. Pfannkuche, J. Wiebe, et al., Science 339, 55 (2013).\n[17] H. Wende, M. Bernien, J. Luo, C. Sorg, N. Ponpandian,\nJ. Kurde, J. Miguel, M. Piantek, X. Xu, P . Eckhold, et al.,\nNat Mater 6, 516 (2007).\n[18] I. Fern ´andez-Torrente, K. J. Franke, and J. I. Pascual, Phys.\nRev. Lett. 101, 217203 (2008).\n[19] A. Chiesa, S. Carretta, P . Santini, G. Amoretti, and\nE. Pavarini, Phys. Rev. Lett. 110, 157204 (2013).\n[20] K. V . Raman, A. M. Kamerbeek, A. Mukherjee, N. Atodire-\nsei, T. K. Sen, P . Lazic, V . Caciuc, R. Michel, D. Stalke, S. K.\nMandal, et al., Nature 493, 509 (2013).\n[21] S. Fahrendorf, N. Atodiresei, C. Besson, V . Caciuc,\nF. Matthes, S. Bl ¨ugel, P . K ¨ogerler, D. E. B ¨urgler, and C. M.\nSchneider, Nat Commun 4(2013).\n[22] S. Carretta, P . Santini, G. Amoretti, T. Guidi, R. Caciu \u000bo,\nA. Candini, A. Cornia, D. Gatteschi, M. Plazanet, and J. A.\nStride, Phys. Rev. B 70, 214403 (2004).\n[23] A. Cornia, A. C. Fabretti, P . Garrisi, C. Mortal `o, D. Bonac-\nchi, D. Gatteschi, R. Sessoli, L. Sorace, W. Wernsdorfer,\nand A.-L. Barra, Angewandte Chemie International Edi-\ntion 43, 1136 (2004), ISSN 1521-3773.\n[24] J. van Slageren, R. Sessoli, D. Gatteschi, A. A. Smith,\nM. Helliwell, R. E. P . Winpenny, A. Cornia, A.-L. Barra,\nA. G. M. Jansen, E. Rentschler, et al., Chemistry – A Euro-\npean Journal 8, 277 (2002), ISSN 1521-3765.\n[25] S. Carretta, J. van Slageren, T. Guidi, E. Liviotti, C. Mon-15\ndelli, D. Rovai, A. Cornia, A. L. Dearden, F. Carsughi,\nM. A \u000bronte, et al., Phys. Rev. B 67, 094405 (2003).\n[26] F. Troiani, A. Ghirri, M. A \u000bronte, S. Carretta, P . Santini,\nG. Amoretti, S. Piligkos, G. Timco, and R. E. P . Winpenny,\nPhys. Rev. Lett. 94, 207208 (2005).\n[27] S. Carretta, P . Santini, G. Amoretti, T. Guidi, J. R. D. Copley,\nY. Qiu, R. Caciu \u000bo, G. Timco, and R. E. P . Winpenny, Phys.\nRev. Lett. 98, 167401 (2007).\n[28] C. J. Wedge, G. A. Timco, E. T. Spielberg, R. E. George,\nF. Tuna, S. Rigby, E. J. L. McInnes, R. E. P . Winpenny, S. J.\nBlundell, and A. Ardavan, Phys. Rev. Lett. 108, 107204\n(2012).\n[29] A. Candini, G. Lorusso, F. Troiani, A. Ghirri, S. Carretta,\nP . Santini, G. Amoretti, C. Muryn, F. Tuna, G. Timco, et al.,\nPhys. Rev. Lett. 104, 037203 (2010).\n[30] C. Chappert, A. Fert, and F. N. Van Dau, Nat Mater 6, 813\n(2007).\n[31] N. Locatelli, V . Cros, and J. Grollier, Nat Mater 13, 11\n(2014).\n[32] L. Bogani and W. Wernsdorfer, Nat Mater 7, 179 (2008).\n[33] M. N. Leuenberger and D. Loss, Nature 410, 789 (2001).\n[34] V . P . Antropov, M. I. Katsnelson, M. van Schilfgaarde, and\nB. N. Harmon, Phys. Rev. Lett. 75, 729 (1995).\n[35] M. Mannini, F. Pineider, P . Sainctavit, C. Danieli, E. Otero,\nC. Sciancalepore, A. M. Talarico, M.-A. Arrio, A. Cornia,\nD. Gatteschi, et al., Nat Mater 8, 194 (2009).\n[36] G. A. Timco, S. Carretta, F. Troiani, F. Tuna, R. J. Pritchard,\nC. A. Muryn, E. J. L. McInnes, A. Ghirri, A. Candini, P . San-\ntini, et al., Nat Nano 4, 173 (2009).\n[37] M. Mannini, F. Pineider, C. Danieli, F. Totti, L. Sorace,\nP . Sainctavit, M. A. Arrio, E. Otero, L. Joly, J. C. Cezar,\net al., Nature 468, 417 (2010).\n[38] S. Carretta, P . Santini, G. Amoretti, M. A \u000bronte, A. Can-\ndini, A. Ghirri, I. S. Tidmarsh, R. H. Laye, R. Shaw, and\nE. J. L. McInnes, Phys. Rev. Lett. 97, 207201 (2006).\n[39] S. Bhattacharjee, L. Nordstr ¨om, and J. Fransson, Phys. Rev.\nLett. 108, 057204 (2012).\n[40] S. Zhang and S. S.-L. Zhang, Phys. Rev. Lett. 102, 086601\n(2009).\n[41] A. Szilva, M. Costa, A. Bergman, L. Szunyogh, L. Nord-\nstr¨om, and O. Eriksson, Phys. Rev. Lett. 111, 127204 (2013).\n[42] J. Fransson, Phys. Rev. B 77, 205316 (2008).\n[43] M. Filipovi ´c, C. Holmqvist, F. Haupt, and W. Belzig, Phys.\nRev. B 87, 045426 (2013).\n[44] J. Fransson, J. Ren, and J.-X. Zhu, Phys. Rev. Lett. 113,\n257201 (2014).\n[45] M. Misiorny, M. Hell, and M. R. Wegewijs, Nat Phys 9, 801\n(2013).\n[46] C. Timm and F. Elste, Phys. Rev. B 73, 235304 (2006).\n[47] M. Misiorny and J. Barna ´s, Phys. Rev. B 75, 134425 (2007).\n[48] V . Moldoveanu, I. V . Dinu, B. Tanatar, and C. P . Moca,\nNew Journal of Physics 17, 083020 (2015).\n[49] R. H ¨artle, M. Butzin, O. Rubio-Pons, and M. Thoss, Phys.\nRev. Lett. 107, 046802 (2011).\n[50] P . Roura-Bas, L. Tosi, and A. A. Aligia, Phys. Rev. B 87,\n195136 (2013).\n[51] J. Linder and J. W. A. Robinson, Nat Phys 11, 307 (2015).\n[52] P . Stadler, C. Holmqvist, and W. Belzig, Phys. Rev. B 88,\n104512 (2013).\n[53] C. Holmqvist, S. Teber, and M. Fogelstr ¨om, Phys. Rev. B\n83, 104521 (2011).\n[54] C. Holmqvist, M. Fogelstr ¨om, and W. Belzig, Phys. Rev. B\n90, 014516 (2014).\n[55] J. Fransson and J.-X. Zhu, New J. Phys. 10, 013017 (2008).[56] J. Fransson, Nanotechnology 19, 285714 (2008).\n[57] N. Bode, L. Arrachea, G. S. Lozano, T. S. Nunner, and\nF. von Oppen, Phys. Rev. B 85, 115440 (2012).\n[58] L. Arrachea and F. von Oppen, Physica E: Low-\ndimensional Systems and Nanostructures 74, 596 (2015).\n[59] D. Ralph and M. Stiles, Journal of Magnetism and Mag-\nnetic Materials 320, 1190 (2008), ISSN 0304-8853.\n[60] N. S. Wingreen, A.-P . Jauho, and Y. Meir, Phys. Rev. B 48,\n8487 (1993).\n[61] A.-P . Jauho, N. S. Wingreen, and Y. Meir, Phys. Rev. B 50,\n5528 (1994).\n[62] K. chao Chou, Z. bin Su, B. lin Hao, and L. Yu, Physics\nReports 118, 1 (1985), ISSN 0370-1573.\nAppendix A: E \u000bective action and stochastic fields\nWe use the closed time-path Green function (CTPGF)\nformalism [62] and thus calculate the partition function\n(with closed path contour ordering operator TC),\nZ[Sn(t)]=trTCexp[ iS]; (A1)\nS=SWZWN+I\nCHdt; (A2)\nHere,SWZWN=R\nSq(t)\u0001[Sc(t)\u0002@tSc(t)]dt=jSj2is the Wess-\nZumino-Witten-Novikov (WZWN) term describing the\nBerry phase accumulated by the local spins. The trace\nis taken over the conduction electrons in the two leads\nin order to provide an e \u000bective spin action, which in\nthe present situation represents the interaction of the\nmagnetic spins with a non-equilibrium environment.\nAs the modell is defined on the Keldysh contour,\nwhich is necessary for general non-equilibrium condi-\ntions, we have to keep track of whether a spin operator\nis defined for times on the upper (lower) part of the\ncontour. We do this by assigning the superscripts u(l).\nThen, it is convenient to define the new spin operators\nSc=(Su+Sl)=2 and Sq=Su\u0000Sland following the proce-\ndure in Ref. [55], we find that the e \u000bective action can be\nwritten as\nS=SWZWN+g\u0016BZ\nB(t)\u0001Sq(t)dt+1\neZ\n\u000fj(t)\u0001Sq(t)dt\n+1\neZ\nSq(t)\u0001J(t;t0)\u0001Sc(t0)dtdt0\n+1\neZ\nSq(t)\u0001JK(t;t0)\u0001Sq(t0)dtdt0: (A3)\nHere, the fields \u000fj(t) and J(t;t0) are given in the main text,\nwhereas JK(t;t0)=iev2hfs(t);s(t0)gidefines the electroni-\ncally mediated interactions between the spin operators\nSq(t) and Sq(t0). This coupling between Sq(t) and Sq(t0)\nprovides a contribution to the model which is of di \u000berent\nnature than the one between Sq(t) and Sc(t0).\nThe operators Scand Sqcan be considered as slow\nand fast variables and the resulting equation of motion\npresented in this paper concerns the dynamics of the\nslow variable. The equation of motion is obtained from16\nthe saddle-point solution of the functional derivative of\nSwith respect to Sq, that is, by di \u000berentiating out the\nrapid dynamics from the discussion. Concerning the\nfirst four terms in the action, this leaves a model that\nis linear in the slow variable Sc, and by finally cross-\nmultiplying from the left by Scunder the assumption\nthat@tjScj2=0, we retain the equation of motion given\nin Eq. 2.\nRegarding the last contribution to the action, however,\nthe functional di \u000berentiation results in a term that is lin-\near in Sq. In order to handle this complication there are,\nat least, two routes to solving the problem. First, one\ncan also consider an equation of motion for the rapid\ndynamics, as well, which results in a coupled system\nof equations for the slow and fast variables. The sec-\nond route would be to perform a Hubbard-Stratonovich\ntransformation of the term in the action that is bilinear\ninSq. By this procedure, the problem is linearized at\nthe price of introducing a stochastic field, represented\nby the Gaussian random variables \u0018(t). These random\nfields provides a quantum fluctuation description of the\nspin correlations.\nThe second route leads to the fact that Eq. (2) also\ncontains a contribution of the form \rS(t)\u0002\u0018(t), which\ncan be interpreted as a random magnetic field acting\non the spin. It can be shown that the random vari-\nable\u0018is defined by the electronic correlations through\n(g\u0016B)2h\u0018(t)\u0018(t0)i=\u0000i2JK(t;t0)=e[55]. In the wide band\nlimit for the electronic states, it is easy to show that the\nspin-spin correlation function JK(t;t0)/\u000e(t\u0000t0), which\nshows that the stochastic field is of Gaussian white noise\ncharacter in our set-up.\nAppendix B: Derivation of the Landau-Lifshitz-Gilbert\nequation\nThe Landau-Lifshitz-Gilbert (LLG) equation is usually\ndefined as\n˙S=S\u0002(\u0000\rB+ˆG˙S) (B1)\nwhere Bis the e \u000bective magnetic field acting on the spin,\nand ˆGis the Gilbert damping. Starting from the spin\nequation of motion used in this work, defined in Eq.2, and restriction to the adiabatic limit, we can retain\nthe LLG equation [39]. In this limit one can assume\nthat the spin is slowly varying with time, which allows\nto Taylor expand according to t0, i.e. S(t0)=S(t)\u0000(t\u0000\nt0)˙S(t)+O(¨S(t)). From this we obtain\nZ\nJ(t;t0)\u0001S(t0)dt0\u0019Z\nJ(t;t0)dt0S(t)\n\u0000Z\nJ(t;t0)(t\u0000t0)dt0˙S(t): (B2)\nHere, the first term corresponds to the internal magnetic\nfield due to the spin background whereas the second\nterm is related to the Gilbert damping. The equation of\nmotion then simplifies to\n˙S(t)=S(t)\u0002\u0010\n\u0000g\u0016BBe\u000b(t)+ˆG˙S(t)\u0011\n(B3)\nwhere the first term represents the total e \u000bective mag-\nnetic field\nBe\u000b(t)=B+1\neg\u0016BZ\n\u000fj(t;t0)dt0+1\neg\u0016BZ\nJ(t;t0)dt0S(t);\n(B4)\nwhereas the Gilbert damping is given by\nˆG=\u00001\neZ\nJ(t;t0)(t\u0000t0)dt0: (B5)\nAppendix C: Lesser /greater dressed quantum Green\nfunction\nThe lesser /greater forms of the correction to the\ndressed GF becomes\n\u000eG<=>(t;t0)=\u0000vZ\u0010\ngr(t;\u001c)hS(\u001c)i\u0001\u001bg<=>(\u001c;t0)\n+g<=>(t;\u001c)hS(\u001c)i\u0001\u001bga(\u001c;t0)\u0011\nd\u001c; (C1)\nand decomposing into the charge and magnetic compo-\nnents we have\n\u000eG<=>\n0(t;t0)=\u0000vZ\u0010\ngr\n0(t;\u001c)hS(\u001c)i\u0001g<=>\n1(\u001c;t0)s+g<=>\n0(t;\u001c)hS(\u001c)i\u0001ga\n1(\u001c;t0)+gr\n1(t;\u001c)\u0001hS(\u001c)ig<=>\n0(\u001c;t0)\n+g<=>\n1(t;\u001c)\u0001hS(\u001c)iga\n0(\u001c;t0)+ih\ngr\n1(t;\u001c)\u0002hS(\u001c)ii\n\u0001g<=>\n1(\u001c;t0)+ih\ng<=>\n1(t;\u001c)\u0002hS(\u001c)ii\n\u0001ga\n1(\u001c;t0)\u0011\nd\u001c; (C2a)\nG<=>\n1(t;t0)=\u0000vZ\u0010\ngr\n0(t;\u001c)hS(\u001c)ig<=>\n0(\u001c;t0)+g<=>\n0(t;\u001c)hS(\u001c)iga\n0(\u001c;t0)+ih\ngr\n1(t;\u001c)\u0002hS(\u001c)ii\ng<=>\n0(\u001c;t0)\n+ih\ng<=>\n1(t;\u001c)\u0002hS(\u001c)ii\nga\n0(\u001c;t0)+igr\n0(t;\u001c)h\nhS(\u001c)i\u0002g<=>\n1(\u001c;t0)i\n+ig<=>\n0(t;\u001c)h\nhS(\u001c)i\u0002ga\n1(\u001c;t0)i\n+ih\ngr\n1(t;\u001c)\u0002hS(\u001c)ii\n\u0002g<=>\n1(\u001c;t0)+ih\ng<=>\n1(t;\u001c)\u0002hS(\u001c)ii\n\u0002ga\n1(\u001c;t0)\u0011\nd\u001c: (C2b)"    },    {        "title": "1902.06481v2.Spin_polarized_currents_driven_by_spin_dependent_surface_screening.pdf",        "content": "arXiv:1902.06481v2  [cond-mat.mes-hall]  21 Feb 2019Spin-polarized currents driven by spin-dependent surface screening\nPiotr Graczyk1,∗and Maciej Krawczyk2,†\n1Institute of Molecular Physics, Polish Academy of Sciences ,\nM. Smoluchowskiego 17, 60-179 Poznan, Poland\n2Faculty of Physics, Adam Mickiewicz University in Poznan, U multowska 85, 61-614 Poznan, Poland\nWe have examined spin polarization of the electron current i n a ferromagnetic metal induced by\nthe spin-dependent surface screening at the dielectric-fe rromagnetic metal (D-FM) interface. In an\napplied ac voltage, the dynamic band splitting driven by the changes in the screening charge at the\nD-FM interface develops spin accumulation. The resultant s pin accumulation gradient produces a\ntime-dependent spin current. We have derived the rate of the spin accumulation dependent on the\nrate of the screening charge density accumulation within th e Stoner band model. The spin-charge\ndynamics in the system is then modeled by a set of diffusive equ ations with the contributions from\nspin-dependent surface screening and spin-dependent cond uctivity. We show for MgO-Cu-Co-MgO\nsystem that the spin-dependent screening in thin Co film prod uces spin accumulation 7 times higher\nthan that obtained by the spin-dependent conductivity in th ick Co films. We propose an experimen-\ntal approach to validate our numerical predictions and to di stinguish between spin accumulation\ninduced by the spin-dependent conductivity and by the spin- dependent surface screening.\nThe phenomenon of spin-dependent surface screening\n(SS) leads to the charge-mediated magnetoelectric effect\n(ME) which is the key mechanism for the electric con-\ntrol of magnetic properties in ultrathin multiferroic het-\nerostructures [1, 2]. The effect has been described first\ntheoretically by Zhang in 1999 [3] and then confirmed by\nab initio calculations [4–6]. It has been shown experi-\nmentally that SS may be utilized to change the magne-\ntization, anisotropy or coercivity of ultrathin ferromag-\nnetic film [7–14], to induce magnetic ordering transition\n[15, 16] or even induce surface magnetism in nonmagnetic\nmaterials [17]. In multiferroic tunnel junctions SS pro-\nvides significant contribution to the tunneling resistance\n[18, 19] and opens possibilities to manipulate the spin\npolarization of tunneling electrons [20, 21]. Owing to the\nsurface magnetic anisotropy that is controlled by SS, spin\nwaves may be excited parametrically or resonantly with\nthe ac voltage [22–26].\nSpin-dependent surface screening changes the absolute\nvalue of magnetization at the dielectric-ferromagnetic\nmetal (D-FM) interface, in contrast to, studied more ex-\ntensively, strain-driven ME effects [2, 27–29], which ro-\ntate the magnetization direction. Although strain-driven\nME remains significant for relatively thick layers, it suf-\nfers from ageing effects, clamping and integration prob-\nlems with current technology.\nIn this letter we study the generation of spin-polarized\ncurrents by spin-dependent surface screening effect in D-\nFM-D system subjected to the ac voltage. The change of\nthe voltage results in the change of the screening charges\nat the insulator-ferromagnet interface. Since the screen-\ning is spin-dependent, this leads to the appearance of\nnonequilibrium spin density at the interface and then\nformed spin-density gradient generates diffusive spin cur-\nrent. We discuss this effect within the Stoner band\n∗graczyk@amu.edu.pl\n†krawczyk@amu.edu.pl\nFIG. 1. The schematic of considered multilayers. In system\nA there is no D-FM interface while in system B the single\nD-FM interface is present.\nmodel and derive the term that describes SS in drift-\ndiffusive formalism. With this model we perform numer-\nical simulations and find the spin current with the con-\ntributions from spin-dependent conductivity (SC) and\nspin-dependent surface screening (SS) for MgO-Cu-Co-\nMgO system. Finally, we propose the strategy to distin-\nguish experimentally between that two current polariza-\ntion mechanisms.\nConsidered systems are shown in Fig. 1. The mul-\ntilayer consists of two 2-nm-thick MgO films separated\nby Co film of different thicknesses. In system A the Co\nlayer is separated from MgO by 2-nm-thick nonmagnetic\nmetal (Cu) at both interfaces while in the system B it is\nseparated from MgO only from the top. Therefore, in the\nsystem A there is no D-FM interface, while in the sys-\ntem B it is present. Consequently, screening charges will\nappear in the ferromagnet only at the bottom interface\nin the system B when the voltage is applied to the mul-\ntilayer structure through bottom and top electrodes (e2\nand e1, respectively). It is assumed that charge and spin\ntransport occur only perpendicular to the layers, along x2\naxis.\nThe coupled charge-spin dynamics driven by the pre-\nscribed ac voltage is considered on the base of the dif-\nfusive model [30, 31]. The free charge density current is\ndescribed in the metal layers by the equation:\nJf=σE−D∂nf\n∂x+βDe\nµB∂s\n∂x, (1)\nwhich includes contributions from electron drift driven by\nelectric field E, diffusion driven by the gradient of free\ncharge density nfand diffusive spin polarisation (SC),\nrespectively. Here, σis conductivity, Dis diffusion con-\nstant,βis spin asymmetry coefficient, µBis Bohr magne-\nton andeis charge of electron. The displacement current\nis described by the equation:\nJb=ǫ0ǫr∂E\n∂t, (2)\nwhereǫ0ǫris permittivity of the material. The conserva-\ntion of free ( i=f) and bound ( i=b) charge density ni\nis described by continuity equation:\n∂ni\n∂t=−∂Ji\n∂x. (3)\nElectric potential Vis given by the Gauss Law:\nǫ0∂2V\n∂x2=nf+nb,\nE=−∂V\n∂x,(4)\nwith the boundary conditions at the electrodes:\nV|x=e1=U=U0sin(2πf0t),\nV|x=e2= 0,(5)\nwheref0is the frequency of the oscillating voltage U.\nThe spin current Jsin the ferromagnetic material is\nmodeled by the equation [31]:\nJs=−D∂s\n∂x−βµB\ne/parenleftbigg\nσE−D∂nf\n∂x/parenrightbigg\n, (6)\nwhich describes spin current driven by gradient of spin\naccumulation s(diffusion) and spin-conductivity term.\nThe continuity equation for the spin accumulation sis:\n∂s\n∂t=−∂Js\n∂x−s\nT1+fs, (7)\nwhich describes the rate of change of sdue to gradient of\nthe spin current, the spin-flip relaxation with the char-\nacteristic time T1.fsis an additional source from spin-\ndependent charge screening. The formula for fsis derived\nbelow. We use in the following paragraphs n≡nf.\nWe will consider the voltage-driven magnetization and\nspin density change in cobalt within the Stoner model\n[32]. Due to the exchange splitting of the band dfor\nFIG. 2. The schematic diagrams of sanddbands in Co (a)\nat electric field E= 0and in equilibrium; (b) at E/negationslash= 0with\nequal band shift −Vc; (c) atE/negationslash= 0with screening exchange\nsplitting Vs. (d) Dynamic nonequilibrium charges (with re-\nspect to equilibrium spin level s0)∆n↑and∆n↓developing as\na consequence of the dynamic exchange splitting. In (e) the\nequilibrium state with the exchange splitting of the screen ing\ncharges is shown.\nthe majority and minority spins and partially filled mi-\nnority subband (Fig. 2a) there exists equilibrium spin\ndensity (magnetization) in the bulk of the ferromagnet\nwhich is equal to the difference between charge den-\nsities of the spin-up ( ↑) and spin-down ( ↓) subbands,\nm=µB(n↑−n↓)/e. The voltage applied to the system\nshifts the potential of the bands at the D-FM interfaces\n(Fig. 2b) leading to the accumulation of the screening\ncharge density n=δn↑\n0+δn↓\n0whereδnσ\n0is accumulated\nscreening charge in the subband σ. Since the density\nof states are different between subbands at the Fermi\nlevel (∆ρ=ρ↑−ρ↓<0for Co), then screening charge\nchanges the value of the magnetization at the interface\nbyδm0=µB(δn↑\n0−δn↓\n0)/e[4].\nAnother important feature of the screening in the fer-\nromagnet is that the changes in the band occupancy in-\nfluence the exchange splitting [5], i.e. the potential is\nspin-dependent [3, 18]:\nVσ=Vc−I\ne(δnσ−δn−σ), (8)\nwhereIis exchange constant. Spin-dependent potential\n(8) means that the subbands shift relative to each other\nbyVs=V↑−V↓=−2Iδm. (Fig. 2c). From Eqs. (5)\nand (11) in Ref. [3] we get the relation between Vsand\nscreening charge n=δn↑+δn↓in the linearized Thomas-\nFermi model:\nVs=−2I∆ρ\nρ+4Iρ↑ρ↓n, (9)3\nwhereρ=ρ↑+ρ↓. This shift has important conse-\nquences. Since ρ↑∝negationslash=ρ↓, the shift changes the equi-\nlibrium charge level with respect to the Fermi level by\nEF−s0(Fig. 2d) where s0=−e(V↑ρ↑+V↓ρ↓)/ρ\nis spin equilibrium level. This leads to the changes in\nequilibrium charge densities δnσ∝negationslash=δnσ\n0and potential\nVc. Therefore, the surface magnetization value changes\nbyδm∝negationslash=δm0. Moreover, the screening length in fer-\nromagnet λFdeviates from the conventional Thomas-\nFermi value λ=/parenleftbig\nǫ0/e2ρ/parenrightbig1/2and thus the capacitance\nis reduced [3]. We can implement this effect to our nu-\nmerical model where the screening length is defined as\nλ= (ǫ0D/σ)1/2by dividing drift term and multiplying\ndiffusion term in Eq. (1) by γn=λF/λ≈1.1for Co.\nWe found that modification of the screening length does\nnot influence the effect of the spin-polarized current gen-\neration which is considered here so it will be neglected in\nfurther part.\nSince screening charge and thus Vs(Eq. (9)) is time-\ndependent under the ac volatage, the change of the shift\nbetween subbands ∆Vsis the source of nonequilibrium\nspin density ∆s∝∆Vs. From Fig. 2d we see that with\nthe respect to the s0:\n∆s=µB\ne(∆n↑−∆n↓)\n=µB\ne(−eV↑−s0)ρ↑−µB\ne(−eV↓−s0)ρ↓\n=−2µB∆Vsρ↑ρ↓/ρe.(10)\nTaking time derivatives we get from (9) and (10) the\nsource term fsto be:\nfs=4Iρ↑ρ↓\nρ∆ρ\nρ+4Iρ↑ρ↓µB\ne∂n\n∂t=γsµB\ne∂n\n∂t.(11)\nIn equilibrium the final change of magnetization δm=\nµB(δn↑−δn↓)/eis a consequence of uniform potential\nshift of the bands and their relative exchange splitting\n(Fig. 2e).\nEquations (1)-(7) with source term (11) are solved\nnumerically by finite element method in Comsol Mul-\ntiphysics with time-varying voltage applied to the elec-\ntrodes with amplitude U0= 8V and frequency f0= 200\nMHz. The relative permittivity is ǫr= 10for MgO and\nǫr= 1for metals. We assume the conductivity of metals\nσ= 1.2·107S/m and the diffusion constant D= 4·10−3\nm2/s. The spin asymmetry coefficient for Co is β= 0.5\n[31], the spin-flip relaxation time is T1= 0.9ps [31] and\nthe magnetoelectric coefficient calculated from Eq. (11)\nand parameters given in Ref. [3] is γs=−0.25. We as-\nsume also the magnetization of Co film lies in-plane of\nthe film. Below we analize the distribution of amplitudes\nof spin accumulation sand spin current Jsbetween MgO\nlayers with the contributions from SC and SS.\nFigure 3 shows the calculated spatial distribution of\nthe spin accumulation and the spin current (for the sys-\ntem B and thickness of Co film LF=51 nm) at the mo-\nment when they reach maximum values, i.e., when V\nFIG. 3. Spatial distribution of (a) spin accumulation and\n(b) spin current through the thickness of Cu-Co bilayer in\nsystem B with separated contributions from spin-dependent\nconductivity and spin-dependent surface screening. The da sh-\ndotted line indicates Cu/Co interface.\ncrosses zero value and charge current flows in the + xdi-\nrection. We separated the contributions which come from\nSC and SS. For the current driven by the spin-dependent\nconductivity, since β >0thenτ−> τ+, i.e., the relax-\nation time for spin down is bigger than for spin up and\nthere is more spin down electrons flowing than spin up\nelectrons. Consequently, Jshas negative sign (Fig. 3b,\nblue line) and its maximum is at the center of Cu/Co\nbilayer. Within the distance proportional to the spin-\ndiffusion length ( Lsf=60 nm for Co) from the ferromag-\nnet surfaces there builds up a negative spin accumulation\ngradient (Fig. 3a, blue line) that stands against SC term\n(compare Eq. (6)) [33]. In the Cu layer there is weak\nexponential decay of sfrom the Cu-Co interface.\nFor the current driven by the spin-dependent surface\nscreening, since γs<0, the outflow of negative charges\nfrom the F-D interface generates negative spin accumula-\ntion (Fig. 3a, red line). The resulting spin accumulation\ngradient induces spin current in the direction + x(Fig.\n3b, red line). The maximum value of Jsis close to the\nD-FM interface.\nWe have shown that the spin current driven by SS is a\nconsequence of the spin accumulation gradient build-up.\nThis is in contrast to the SC-polarized current, where\nspin accumulation gradient is a result of this current and\nstands against it.\nThe effects from SS and SC sum up to give resultant\nspin accumulation and spin current (black lines in Fig.\n3). In Co ( β >0,γs<0) they add up destructively\nand for Co thickness of 51 nm discussed above, that con-\ntributions balance out at the Cu layer as shown at the\nleft-hand side of dash-dotted line in Fig. 3b.\nIt is possible to distinguish between SC and SS gen-\nerated spin currents comparing samples where SS would\nbe absent (system A) and samples where SS would be\npresent (system B). Moreover, spin-dependent conduc-\ntivity is a phenomenon that acts in a bulk while spin-\ndependent surface screening is an interface effect. Thus,\nin thick ferromagnetic layer SC shall dominate while SS\nshould manifest itself in thin ferromagnetic layers.\nWe consider the values of spin accumulation and spin4\nFIG. 4. (a) Spin accumulation and (b) spin current in the Cu\nlayer in dependence on the thickness of Co layer for system A\nand system B.\ncurrent in the top Cu layer and we change the thickness\nof Co layer. The black lines in Fig. 4 show sandJs\ncalculated for the system A. Since there is no D-FM in-\nterface, the spin polarization of the current is a result of\nspin-dependent conductivity. The flow of spin-polarized\ncharges builds up a spin accumulation as discussed above.\nFor a thin Co layer ( LF<2Lsf) the spin accumulation\ncounteracts the spin current significantly. For the thick-\nness larger than 2Lsfthe influence of the interfacial ac-\ncumulation loses its significance and the value of spin-\npolarized current reaches maximum value of 105 A/s in\nCu (and the value of Js≈ −βJf= 7000 A/s in the bulk\nof Co layer). The values of sandJssaturate for thick\nCo films (see insets in Fig. 4).\nThe red lines in Fig. 4 show sandJsfor system\nB. Because of D-FM interface is present in the system,\nthere are contributions both from SC and SS to the spin\ntransport. Spin-dependent conductivity dominates for\nLF> Lsf. If the distance from the D-FM interface to\nCu layer is less than 2Lsf, the currents induced by spin-\ndependent surface screening come into picture and the\nvalues of sandJsdeviates from those for system A. At\nLF= 51nm SS balances out SC contribution in Cu and\nforLF<51nm spin current driven by SS dominates in\nthe Cu layer. The maximum values obtained for LF=\n2 nm are s=650 mA/m and Js=730 A/s which are 7times higher than those obtained for thick Co films by\nspin-dependent conductivity.\nThe dependences of sandJsonLFin the systems con-\nsidered above give us simple tool to distinguish between\nSS- and SC- driven currents. If SS contribution is absent\nin the system then s→0andJs→0asLF→0for\nLF< Lsf. If SS contribution is nonzero, sandJstend\nto nonzero values as LF→0forLF< Lsf. It is rather\ncrucial to choose ferromagnet with relatively long spin-\ndiffusion length like Co, since the effect is hard to resolve\nforLF> Lsf.\nIn summary, the mechanism of spin current genera-\ntion by the spin-dependent surface screening effect has\nbeen described within the Stoner model. The mecha-\nnism has been implemented to the drift-diffusion model.\nThen the specific dielectric-ferromagnetic heterostruc-\ntures have been considered numerically to show the prop-\nerties of SS-driven spin currents. We found the spin accu-\nmulation and the spin polarization of the current driven\nby spin-dependent surface screening is 7 times higher for\n2-nm-thick Co than those obtained by spin-dependent\nconductivity for thick Co layers. Since the strength of\nthe effect is dependent on the density of the screening\ncharge at the D-FM interface, it may be enhanced with\nthe use of dielectrics of high permittivity. Therefore, it\nis interesting to consider the influence of SS-driven spin\ncurrents onto magnetization dynamics, in particular, the\npossibility to decrease spin wave damping through cur-\nrents induced by that mechanism. The proposed systems\nand the method of extraction of SS-driven spin accumu-\nlation is suitable to the X-Ray magnetic circular dichro-\nism experiment [34, 35], where the measurement of spin\naccumulation dynamics is element-specific and thus may\nbe constrained to the signal from Cu layer and directly\ncompared to the dependences shown in Fig. 4.\nACKNOWLEDGMENTS\nThe study has received financial support from\nthe National Science Centre of Poland under grant\n2018/28/C/ST3/00052.\n[1]J. D. Burton and E. Y. Tsymbal, “Magnetoelectric in-\nterfaces and spin transport,” Philosophical Transactions\nof the Royal Society A: Mathematical, Physical and Engi-\nneering Sciences , vol. 370, no. 1977, pp. 4840–4855, 2012.\n[2]C. A. F. Vaz, “Electric field control of magnetism in\nmultiferroic heterostructures,” Journal of Physics: Con-\ndensed Matter , vol. 24, p. 333201, 2012.\n[3]S. Zhang, “Spin-Dependent Surface Screening in Ferro-\nmagnets and Magnetic Tunnel Junctions,” Physical Re-\nview Letters , vol. 83, no. 3, pp. 640–643, 1999.\n[4]C.-g. Duan, J. P. Velev, R. F. Sabirianov, Z. Zhu, J. Chu,\nS. S. Jaswal, and E. Y. Tsymbal, “Surface Magnetoelec-\ntric Effect in Ferromagnetic Metal Films,” Physical Re-view Letters , vol. 101, p. 137201, 2008.\n[5]M. K. Niranjan, J. D. Burton, J. P. Velev, S. S. Jaswal,\nand E. Y. Tsymbal, “Magnetoelectric effect at the Sr-\nRuO3/BaTiO3 (001) interface: An ab initio study,” Ap-\nplied Physics Letters , vol. 95, p. 052501, 2009.\n[6]J. M. Rondinelli, M. Stengel, and N. A. Spaldin,\n“Carrier-mediated magnetoelectricity in complex ox-\nide heterostructures,” Nature Nanotechnology , vol. 3,\nno. January, pp. 46–50, 2008.\n[7]H. J. Molegraaf, J. Hoffman, C. A. Vaz, S. Gariglio,\nD. Van Der Morel, C. H. Ahn, and J. M. Triscone,\n“Magnetoelectric effects in complex oxides with compet-\ning ground states,” Advanced Materials , vol. 21, no. 34,5\npp. 1–5, 2009.\n[8]M. Weisheit, S. Fähler, A. Marty, Y. Souche,\nC. Poinsignon, and D. Givord, “Electric field-induced\nmodification of magnetism in thin-film ferromagnets,”\nScience , vol. 315, no. 5810, pp. 349–351, 2007.\n[9]T. Maruyama, Y. Shiota, T. Nozaki, K. Ohta, N. Toda,\nM. Mizuguchi, A. A. Tulapurkar, T. Shinjo, M. Shiraishi,\nS. Mizukami, Y. Ando, and Y. Suzuki, “Large voltage-\ninduced magnetic anisotropy change in a few atomic lay-\ners of iron,” Nature Nanotechnology , vol. 4, p. 158, jan\n2009.\n[10]M. Endo, S. Kanai, S. Ikeda, F. Matsukura, and H. Ohno,\n“Electric-field effects on thickness dependent magnetic\nanisotropy of sputtered MgO/Co40Fe40B20/Ta struc-\ntures,” Applied Physics Letters , vol. 96, p. 212503, may\n2010.\n[11]Y. Shiota, T. Maruyama, T. Nozaki, T. Shinjo, M. Shi-\nraishi, and Y. Suzuki, “Voltage-assisted magnetization\nswitching in ultrathin Fe 80Co20 alloy layers,” Applied\nPhysics Express , vol. 2, no. 6, p. 063001, 2009.\n[12]Y. Shiota, T. Nozaki, F. Bonell, S. Murakami, T. Shinjo,\nand Y. Suzuki, “Induction of coherent magnetization\nswitching in a few atomic layers of FeCo using voltage\npulses,” Nature Materials , vol. 11, p. 39, nov 2011.\n[13]T. Seki, M. Kohda, J. Nitta, and K. Takanashi, “Co-\nercivity change in an FePt thin layer in a Hall device\nby voltage application,” Applied Physics Letters , vol. 98,\np. 212505, may 2011.\n[14]D. Chiba, S. Fukami, K. Shimamura, N. Ishiwata,\nK. Kobayashi, and T. Ono, “Electrical control of the fer-\nromagnetic phase transition in cobalt at room tempera-\nture,” Nature Materials , vol. 10, p. 853, oct 2011.\n[15]J. D. Burton and E. Y. Tsymbal, “Prediction of elec-\ntrically induced magnetic reconstruction at the mangan-\nite/ferroelectric interface,” Physical Review B , vol. 80,\np. 174406, 2009.\n[16]C. A. F. Vaz, J. Hoffman, Y. Segal, J. W. Reiner, R. D.\nGrober, Z. Zhang, C. H. Ahn, and F. J. Walker, “Ori-\ngin of the magnetoelectric coupling effect in PZT/LSMO\nmultiferroic heterostructures,” Physical Review Letters ,\nvol. 104, no. March, p. 127202, 2010.\n[17]Y. Sun, J. D. Burton, and E. Y. Tsymbal, “Electrically\ndriven magnetism on a Pd thin film,” Physical Review B ,\nvol. 81, p. 064413, 2010.\n[18]M. Y. Zhuravlev, S. Maekawa, and E. Y. Tsymbal, “Ef-\nfect of spin-dependent screening on tunneling electrore-\nsistance and tunneling magnetoresistance in multiferroic\ntunnel junctions,” Physical Review B , vol. 81, no. 10,\np. 104419, 2010.\n[19]Y. W. Yin, M. Raju, W. J. Hu, J. D. Burton, Y.-M. Kim,\na. Y. Borisevich, S. J. Pennycook, S. M. Yang, T. W.\nNoh, a. Gruverman, X. G. Li, Z. D. Zhang, E. Y. Tsym-\nbal, and Q. Li, “Multiferroic tunnel junctions and ferro-\nelectric control of magnetic state at interface (invited), ”\nJournal of Applied Physics , vol. 117, p. 172601, 2015.\n[20]D. Pantel, S. Goetze, D. Hesse, and M. Alexe, “Re-\nversible electrical switching of spin polarization in mult i-\nferroic tunnel junctions,” Nature Materials , vol. 11, no. 4,\npp. 289–293, 2012.\n[21]C. Sen, W. A. S. Aldulaimi, O. Mohammadmoradi, and\nI. B. Misirlioglu, “Loss of spin polarization in ferromagne t\n/ ferroelectric tunnel junctions due to screening effects,”\nJournal of Physics D: Applied Physics , vol. 52, p. 015305,2019.\n[22]B. Rana, Y. Fukuma, K. Miura, H. Takahashi, and\nY. Otani, “Excitation of coherent propagating spin waves\nin ultrathin CoFeB film by voltage-controlled magnetic\nanisotropy,” Applied Physics Letters , vol. 111, no. 5,\np. 052404, 2017.\n[23]R. Verba, V. Tiberkevich, I. Krivorotov, and A. Slavin,\n“Parametric Excitation of Spin Waves by Voltage-\nControlled Magnetic Anisotropy,” Physical Review Ap-\nplied, vol. 1, no. 4, p. 044006, 2014.\n[24]R. Verba, M. Carpentieri, G. Finocchio, V. Tiberke-\nvich, and A. Slavin, “Excitation of Spin Waves in\nan In-Plane-Magnetized Ferromagnetic Nanowire Using\nVoltage-Controlled Magnetic Anisotropy,” Physical Re-\nview Applied , vol. 7, p. 064023, 2017.\n[25]Y.-j. Chen, H. K. Lee, R. Verba, J. A. Katine, I. Bar-\nsukov, V. Tiberkevich, J. Q. Xiao, A. N. Slavin, and\nI. N. Krivorotov, “Parametric Resonance of Magnetiza-\ntion Excited by Electric Field,” Nano Letters , vol. 17,\npp. 572–577, 2017.\n[26]T. Nozaki, Y. Shiota, S. Miwa, S. Murakami, F. Bonell,\nS. Ishibashi, H. Kubota, K. Yakushiji, T. Saruya,\nA. Fukushima, S. Yuasa, T. Shinjo, and Y. Suzuki,\n“Electric-field-induced ferromagnetic resonance excita-\ntion in an ultrathin ferromagnetic metal layer,” Nature\nPhysics , vol. 8, no. 6, pp. 491–496, 2012.\n[27]P. Graczyk, R. Schäfer, and B. Mroz, “Magnetoelastic\ncoupling between NiFe thin film and LiCsSO4 studied by\nKerr microscopy,” Journal of Physics D: Applied Physics ,\nvol. 48, no. 42, p. 425002, 2015.\n[28]P. Graczyk, A. Trzaskowska, K. Zaleski, and B. Mroz,\n“Tunable properties of spin waves in magnetoelas-\ntic NiFe/GMO heterostructure,” Smart Materials and\nStructures , vol. 25, no. 7, p. 075017, 2016.\n[29]P. Graczyk, A. Trzaskowska, K. Adrjanowicz, B. Peplin-\nska, and B. Mroz, “Large magnetoelastic coupling\nin NiFe/LiCsSO4 and NiFe/KH2PO4 heterostructures,”\nJournal of Alloys and Compounds , vol. 656, pp. 825–829,\n2016.\n[30]T. Valet and A. Fert, “Theory of the perpendicular mag-\nnetoresistance in magnetic multilayers,” Physical Review\nB, vol. 48, no. 10, pp. 7099–7113, 1993.\n[31]Y. H. Zhu, B. Hillebrands, and H. C. Schneider, “Signal\npropagation in time-dependent spin transport,” Physical\nReview B , vol. 78, no. 5, p. 054429, 2008.\n[32]E. C. Stoner, “Collective electron ferronmagnetism,” Pro-\nceedings of the Royal Society of London. Series A. Mathe-\nmatical and Physical Sciences , vol. 165, no. 922, pp. 372–\n414, 1938.\n[33]V. Zayets, “Spin and charge transport in materials with\nspin-dependent conductivity,” Physical Review B , vol. 86,\nno. 17, p. 174415, 2012.\n[34]R. Kukreja, S. Bonetti, Z. Chen, D. Backes, Y. Acre-\nmann, J. A. Katine, A. D. Kent, H. A. Dürr, H. Ohldag,\nand J. Stöhr, “X-ray Detection of Transient Magnetic\nMoments Induced by a Spin Current in Cu,” Physical\nReview Letters , vol. 115, no. 9, p. 096601, 2015.\n[35]J. Li, L. R. Shelford, P. Shafer, A. Tan, J. X. Deng, P. S.\nKeatley, C. Hwang, E. Arenholz, G. Van Der Laan, R. J.\nHicken, and Z. Q. Qiu, “Direct Detection of Pure ac Spin\nCurrent by X-Ray Pump-Probe Measurements,” Physical\nReview Letters , vol. 117, no. 7, p. 076602, 2016."    },    {        "title": "1610.09689v1.Kinetic_analysis_of_spin_current_contribution_to_spectrum_of_electromagnetic_waves_in_spin_1_2_plasma__Part_I__Dielectric_permeability_tensor_for_magnetized_plasmas.pdf",        "content": "arXiv:1610.09689v1  [physics.plasm-ph]  30 Oct 2016Kinetic analysis of spin current contribution to spectrum o f electromagnetic waves in\nspin-1/2 plasma, Part I: Dielectric permeability tensor fo r magnetized plasmas\nPavel A. Andreev∗\nFaculty of physics, Lomonosov Moscow State University, Mos cow, Russian Federation.\n(Dated: September 24, 2018)\nThe dielectric permeability tensor for spin polarized plas mas is derived in terms of the spin-1/2\nquantum kinetic model in six-dimensional phase space. Expr essions for the distribution function\nand spin distribution function are derived in linear approx imations on the path of dielectric perme-\nability tensor derivation. The dielectric permeability te nsor is derived the spin-polarized degenerate\nelectron gas. It is also discussed at the finite temperature r egime, where the equilibrium distri-\nbution function is presented by the spin-polarized Fermi-D irac distribution. Consideration of the\nspin-polarized equilibrium states opens possibilities fo r the kinetic modeling of the thermal spin\ncurrent contribution in the plasma dynamics.\nPACS numbers: 52.25.Xz, 52.25.Dg, 52.35.Hr, 75.30.Ds\nKeywords: quantum kinetics, separate spin evolution, elec tromagnetic waves, spin waves, degenerate electron\ngas\nI. INTRODUCTION\nSpin effects in plasmas were introduced via hydrody-\nnamic formalism [1–5]. Later, a kinetic model was rein-\nstalled in [6] recapturing results of [7–9], where the phase\nspace was extended up to eight dimensions to include the\nspin dependence of the distribution function of particles\nwith fixed module of the spin. Both approaches attract\nattentionofalotofresearchers. Anotherkineticmodelof\nspin-1/2 particles was reinstalled in [10–15] recapturing\nresults of [16–21], where the distribution functions were\nconsideredin the traditionalsix dimensionalphasespace.\nExperimental conformation of results obtained by Dyson\n[17] is found in [22]. In this model, the spin evolution\nappears via presence of the vector distribution function–\nthe spin distribution function in addition to the tradi-\ntional distribution function. A quantum-relativistic ki-\nnetic equations were derived applying the Wigner distri-\nbution function [23]. It was derivedfrom a single-particle\nDirac equation. Hence, it may neglect relativistic inter-\nparticle effects arising between particles of comparable\nmasses [24].\nNext step in the development of models for the spin-\n1/2 quantum plasmas was the derivation of the sepa-\nrated spin evolution quantum hydrodynamics [25], [26]\n(see [27] for a generalized model with exchange interac-\ntion) and separated spin evolution quantum kinetics [28].\nIn this model the electrons are separated on two sub-\nspecies: electrons with spin-up and electrons with spin-\ndown [25, 29]. The separated spin evolution quantum\nhydrodynamics is derived from the single particle Pauli\nequation [25]. A many-particle derivation was suggested\nfor the separated spin evolution quantum kinetic deriva-\ntion [28]. The derivation based on the Pauli equation\ndiscover a specific structure of the spin-spin interaction\n∗Electronic address: andreevpa@physics.msu.ruforce field. Moreover, it demonstrates unconservation of\nthe particle number in each subspecies. Different behav-\nior of spin-up and spin-down electrons required different\npressure. Corresponding equation of state was presented\nin [25]. All these features of the separated spin evolu-\ntion are obtained in Refs. [25], [26] and [28], while a two\nfluid model of electrons was mention in literature earlier\n[29], [30], [31]. Incompleteness of earlier model bound to\nincorrect coefficients in spin depending terms [32].\nSpinevolutionleadstothethermalpartofthespinflux\nor the spin current. The thermal part of the spin current\nisananalogofthepressureexistingintheEulerequation.\nIt is expected that this spin current considerably affects\nspin properties ofplasmas. However, its analysis requires\nan equation of state. Corresponding equation of state\nis found recently with application of the separated spin\nevolution quantum hydrodynamics [33]:\nn(∂t+u·∇)µ\n−¯h\n2mµe∂β[nµ×∂βµ]+ℑ=2µe\n¯hn[µ×B],(1)\nwhereµ=M/n,M=M(r,t) is the magnetization of\nelectron gas, n=n(r,t) is the concentration of particles,\nu(r,t)isthevelocityfield, Bisthemagneticfield, ¯ histhe\nreduced Planck constant, mis the mass of particle, µeis\nthemagneticmomentofparticle, ∇and∂βarethevector\nand tensor notations for the spatial derivatives, and ℑis\nthe divergence of the thermal part of the spin current,\nits explicit form is found for the degenerate electron gas\n[33]:\nℑP=(3π2)2/3¯h\nm(n2/3\n↑−n2/3\n↓)[M,ez],(2)\nwithn↑=n−Mz/|µe|, n↓=n+ Mz/|µe|, where\nµeis the magnetic moment of electron, and subindex P\nshows that equation (2) is derived from the non-linear\nPauli equation, indexes ↑and↓refer to the spin-up and2\nspin-down states, correspondingly. It is an analog of the\nFermi pressure. Therefore, it is called the Fermi spin\ncurrent. It leads to the modification of spin-plasma wave\nspectrum [33]. The cut-off frequency of the spin waves is\nmodified by the Fermi spin current. If cut-off frequency\nof the spin-plasma wave is larger than the plasma fre-\nquencythelinearinteractionofthe spin-plasmawaveand\nthe ordinary wave appears. Polarization of spin-plasma\nwave propagating parallel to the external magnetic field\nchanges as well. Hence, a left-hand polarized wave ap-\npears instead of the right-hand polarized [33]. Modifica-\ntionsofspectrumandcut-offfrequencyappearattheper-\npendicular propagation either. Similar equation of state\nwas derived from the separated spin evolution quantum\nkinetics as a moment of the spin distribution function\n[28]:Jxx\nK=Jyy\nK=·3π¯hµe(6π2)1/3(n4/3\n↑−n4/3\n↓)/32mand\nJxy\nK=Jxz\nK=Jyx\nK=Jyz\nK=Jzx\nK=Jzy\nK=Jzz\nK= 0,\nwhere subindex Kdemonstrates that this result is ob-\ntained from kinetic model being defined as follows Jαβ\nK=\nµe/integraltext\nSα\n0(p)vβdp. Separate spin evolution affects extraor-\ndinarywavesdirectly via the difference ofthe Fermi pres-\nsuresforspin-upandspin-downelectrons[34]. While, the\nspin evolution and the Fermi spin current do not affect\nthese waves.\nExistence of the Fermi spin current is closely related to\ndifference of the Fermi pressure for the spin-up and spin-\ndown electrons [33]. Difference of the Fermi pressures it-\nself reveals in a modification (an increase) of the pressure\ncontribution in the properties of longitudinal waves. Dif-\nference of the Fermi pressures presented in the two fluid\nmodel ofelectron gasleads to extraphenomena: a pairof\nbulk spin-electron acoustic waves [25], [26], spin-electron\nacoustic soliton [27], surface spin-electron acoustic wave\n[35], a pair of bulk spin-electron-positron acoustic waves\nin addition to the pair of the bulk spin-electron acous-\ntic waves existing in electron-positron-ion plasmas [36],\nspin-electron acoustic soliton and spin-electron-positron\nacoustic soliton in electron-positron-ionplasmas [37]. All\nthese waves are longitudinal waves.\nThe Fermi spin current contributes to the transverse\nwaves [33]. Results of application of the Fermi spin cur-\nrent require the proper generalization by means of a ki-\nnetic model. This paper is devoted to the development\nof corresponding kinetic model. Presenting derivation of\nthe dielectric permeability tensor for spin polarized plas-\nmas is the first step in this direction.\nThis paper is organized as follows. In Sec. II, basic\nnonrelativistic kinetic equations for spin-1/2 plasmas are\npresented. In Sec. III, linearized kinetic equations and\ntheir solutions for the isotropic equilibrium distribution\nfunction are obtained. In Sec. IV, the dielectric perme-\nability tensor is found. It is presented in general form\nand for the spin-polarized Fermi step distribution func-\ntion. In Sec. V, a summary of the obtained results is\npresented.II. QUANTUM KINETIC MODEL FOR\nSPIN-1/2 PLASMAS\nDifferent quantum kinetic approacheshavebeen devel-\noped [6, 13–15], [28], [38–44]. Some of them do not in-\nclude the spin evolution [38, 39, 41–44], but consider the\nexchange part of the Coulomb interaction [39], [44], or\nconsider non-ideal plasmas with strong interaction [40],\n[41].\nThe kinetic equation for the distribution function\nf(r,p,t) in spin-1/2plasmasappearsas follows[13], [15]:\n∂tf+v·∇rf+qe/parenleftbigg\nEext+1\ncv×Bext/parenrightbigg\n·∇pf+µe(∇β\nrBα\next)∇β\npSα\n−q2\ne/integraldisplay\n∇rG(r,r′)·∇pf2(r,p,r′,p′,t)dr′dp′\n−µ2\ne/integraldisplay\n(∇α\nrGβγ(r,r′))∇α\npSβγ\n2(r,p,r′,p′,t)dr′dp′= 0,\n(3)\nwhereG(r,r′) = 1/|r−r′|is the Green function of\nCoulomb interaction, Gαβ(r,r′) =∂α∂β(1/|r−r′|) +\n4πδαβδ(r−r′) is the Green function of spin-spin in-\nteraction, ∇pis the derivative on the momentum p=\nmv. It contains the two-particle distribution function\nf2(r,p,r′,p′,t)inthetermdescribingtheCoulombinter-\naction. The spin distribution function S(r,p,t) arises in\ntheterm describinginteractionofthespins (the magnetic\nmoments) with the external magnetic field Bext. The\ntwo-particle spin distribution function Sαβ(r,p,r′,p′,t),\nwhich is a second rank tensor, appears in the term de-\nscribing the spin-spin interaction. The two-particle dis-\ntribution functions can be reduced to the one-particle\ndistribution functions in the self-consistent (mean-field)\napproximation. However, equation (3) requires an addi-\ntional equation. The equation of evolution of the spin-\ndistribution function S(r,p,t), which arrears as follows\n[13], [15]:\n∂tSα+v·∇rSα\n+qe/parenleftbigg\nEext+1\ncv×Bext/parenrightbigg\n·∇pSα+µe(∇β\nrBα\next)∇β\npf\n+q2\ne/integraldisplay\n∇rG(r,r′)·∇pMα\n2(r,p,r′,p′,t)dr′dp′\n−µ2\ne/integraldisplay\n(∇γ\nrGαβ(r,r′))∇γ\npNβ\n2(r,p,r′,p′,t)dr′dp′\n−2µe\n¯hεαβγ/parenleftigg\nSβ(r,p,t)Bγ\next(r,t)3\n+µe/integraldisplay\nGγδ(r,r′)Sβδ\n2(r,p,r′,p′,t)dr′dp′/parenrightigg\n= 0.(4)\nEquation (4) contains the mixed spin-number of parti-\ncles two-particle distribution function Mα\n2(r,p,r′,p′,t)\nand the number of particles-spin two-particle distribu-\ntion function Nβ\n2(r,p,r′,p′,t). The quantum terms con-\ntaining the Planck constant and higher derivatives of the\ndistribution functions are neglected in equations (3) and\n(4).\nThe two-particle distribution functions containing in\nthe kinetic equations have the following representation\nin the self-consistent field approximation\nf2(r,p,r′,p′,t) =f(r,p,t)f(r′,p′,t),(5)\nSαβ\n2(r,p,r′,p′,t) =Sα(r,p,t)Sβ(r′,p′,t),(6)\nMα\n2(r,p,r′,p′,t) =Sα(r,p,t)f(r′,p′,t),(7)\nand\nNα\n2(r,p,r′,p′,t) =f(r,p,t)Sα(r′,p′,t).(8)\nEquations (3) and (4) are obtained at the neglecting\nthe quantum terms explicitly containing the Planck con-\nstant in the kinetic equations (for details see [13], [15]).\nIt is assumed in equation (3) that the kinetic cur-\nrent is reduced to the distribution function J(r,p,t) =\npf(r,p,t) [13](seeequations12and13). Similarapprox-\nimationismadeforthekineticspincurrent Jαβ(r,p,t) =\npαSβ(r,p,t) in the second term in the kinetic equation\n(4) [13] (see equations 36 and 37).\nAs the result we have next set of equations describing\nthe evolution of spin-1/2 plasma in the self-consistent\nfield approximation [13], [14], [15]\n∂tf+v·∇rf+qe/parenleftbigg\nE+1\ncv×B/parenrightbigg\n·∇pf+∇α\nrBβ·∇α\npSβ= 0,\n(9)\nand\n∂tSα+v·∇rSα+qe/parenleftbigg\nE+1\ncv×B/parenrightbigg\n·∇pSα\n+∇β\nrBα·∇β\npf−2µe\n¯hεαβγSβBγ= 0.(10)\nInternal electromagneticfield in equations (9) and (10)\nappears in the electro-magneto-static limit, the electric\nfield\nEint=qe/integraldisplay\n∇G(r,r′)f(r′,p,t)dr′dp(11)\nsatisfying quasi-static equations ∇×E= 0 and\n∇·E= 4πqe/integraltextf(r,p,t)dp, (12)and the magnetic field\nBα\nint=µe/integraldisplay\nGαβ(r,r′)Sβ(r′,p,t)dr′dp(13)\nsatisfying the following quasi-static equations ∇·B= 0\nand\n∇×B= 4πµe∇×/integraltextS(r,p,t)dp. (14)\nTo study the plasma dynamics we need to generalize\nequations (11)-(14) arising at the non-relativistic deriva-\ntion up to the full set of the Maxwell equations\n∇·E= 4πρ,∇×E=−1\nc∂tB,∇·B= 0,(15)\nand\n∇×B=1\nc∂tE+4π\ncj+4π∇×M,(16)\nwhereρ=qe/integraltext\nf(r,p,t)dp+qin0i,j=qe/integraltext\nvf(r,p,t)dp,\nM=µe/integraltext\nS(r,p,t)dpis the magnetization.\nExisting kinetic research shows that spin effects leads\nto damping effects for the electrostatic wave modes [45],\nexisting of spin waves [6], and splitting of Bernstein\nmodes (each mode splits on three branches at the ac-\ncount of the anomalous part of magnetic moment of elec-\ntrons)[46]. Kineticgivesanadvanceddescriptionincom-\npare with hydrodynamics. However, some effects, which\ndo not enter result of traditional hydrodynamics, can be\nfound in extended hydrodynamic models [47], [48].\nIII. LINEARIZED KINETIC EQUATIONS AND\nSOLUTIONS FOR DISTRIBUTION FUNCTIONS\nFor the derivation of the dielectric permeability tensor\nwe need to consider linear on small perturbations kinetic\nequations. Assuming that in equilibrium we have non-\nzerof0(p),B0=B0ez,S0=S0(p)ez, withp=|p|,\nwe find from kinetic equations (3) and (4) the following\nlinearized Fourier transformed kinetic equations\n−ıωδf+ıv·kδf+qe\ncB0(v×ez)·∇pδf\n+qeδE·∇pf0+ıµeδBzk·∇pS0= 0,(17)\nand\n−ıωδS+ı(v·k)δS+qe\ncB0((v×ez)·∇p)δS+ıµe(k·∇p)f0δB\n+qe(δE·∇p)S0+2µe\n¯h(B0×δS−S0×δB) = 0,(18)\nwhere the wave vector has the following structure k=\n{kx,0,kz}that corresponds to the oblique propagation4\nof waves relatively to the external magnetic field. It cor-\nresponds to the isotropic equilibrium distribution func-\ntions. Moregeneralcaseofequilibrium distribution func-\ntion and solutions for linear distribution functions are\npresented in Appendix A.\nAs it follows from equation (18) projections of the spin\ndistribution function Ssatisfy the following linear equa-\ntions\nΩe∂ϕδSx+ı(ω−k·v)δSx+ΩµδSy\n=ıµe(k·∇p)f0δBx+2µe\n¯hS0zδBy,(19)\nand\n−ΩµδSx+Ωe∂ϕδSy+ı(ω−k·v)δSy\n=ıµe(k·∇p)f0δBy−2µe\n¯hS0δBx,(20)\nwhereδSxandδSyare bound to each other; and\nΩe∂ϕδSz+ı(ω−k·v)δSz\n=qe(δE·∇p)S0z+ıµe(k·∇p)f0δBz(21)\nwhich is independent from perturbations of other distri-\nbution functions. The following notations are used for\nthe charge cyclotron frequency Ω e=qeB0/mcand the\nmagnetic moment cyclotron frequency Ω µ= 2µeB0/¯h.\nThey are equal to each other if the anomalous part of\nmagnetic moment of electron is neglected. At the transi-\ntion from equations (17) and (18) to equations (61)-(63)\nwe used that ( v×ez)·∂p= (1/m)∂ϕ.\nEquation for δf(17) is independent from other equa-\ntions. Equationfor δSz(63) isindependent either. Equa-\ntions for δSx(61) and δSy(62) make a set of equations\nand should be solved together.\nComparing equation (17) with spinless case it can be\nseen that it differs by a single term on the right-hand\nside. Therefore, solution of equation (17) can be found\nin the traditional form\nδf=1\nΩe/integraldisplayϕ\nC0/parenleftbigg\nqe(v·δE)|ϕ′∂f0\n∂ε+ıµe(k·v)|ϕ′δBz∂S0z\n∂ε/parenrightbigg\n×\n×exp/parenleftbigg\nı/integraldisplayϕ′\nϕ1\nΩe(ω−k·v|ϕ′′)dϕ′′/parenrightbigg\ndϕ′,(22)\nbut containing two terms under integral instead of one.\nIn formula (22) and below symbol |ϕ′means that it\nis a function of ϕ′. Above, the differentiation on the\nmomentum is replaced by the differentiation on energy\nε=p2/2m,∇p=v∂ε.\nEquation (63) can be solved similarly. As the result,\nthe following solution is found\nδSz=1\nΩe/integraldisplayϕ\nC3/parenleftbigg\nqe(v·δE)|ϕ′∂S0z\n∂ε+ıµe(k·v)|ϕ′δBz∂f0\n∂ε/parenrightbigg\n××exp/parenleftbigg\nı/integraldisplayϕ′\nϕ1\nΩe(ω−k·v|ϕ′′)dϕ′′/parenrightbigg\ndϕ′.(23)\nSpin part of the distribution functions appears via\nthe magnetic field perturbations, which can be repre-\nsented via the electric field for the further derivation\nof the dielectric permeability tensor δBx=−kzcδEy/ω,\nδBy=c(kzδEx−kxδEz)/ω,δBz=kxcδEy/ω.\nNext, the followinganzaccanbeused forsimplification\nof set of equations (61) and (62)\nδSx=P(ϕ)exp/parenleftbigg\n−ı/integraldisplayϕ\nC1\nΩe(ω−k·v|ϕ′)dϕ′/parenrightbigg\n,(24)\nand\nδSy=R(ϕ)exp/parenleftbigg\n−ı/integraldisplayϕ\nC1\nΩe(ω−k·v|ϕ′)dϕ′/parenrightbigg\n.(25)\nIt gives the following equations for functions P(ϕ) and\nR(ϕ):\nΩe∂ϕP(ϕ)+ΩµR(ϕ) = exp/parenleftbigg\nı/integraldisplayϕ\nCω−k·v|ϕ′\nΩedϕ′/parenrightbigg\n×\n×/parenleftbigg\nıµe(k·∇p)f0δBx+2µe\n¯hS0zδBy/parenrightbigg\n,(26)\nand\n−ΩµP(ϕ)+Ωe∂ϕR(ϕ) = exp/parenleftbigg\nı/integraldisplayϕ\nCω−k·v|ϕ′\nΩedϕ′/parenrightbigg\n×\n×/parenleftbigg\nıµe(k·∇p)f0δBy−2µe\n¯hS0δBx/parenrightbigg\n.(27)\nThe structure of functions P(ϕ) andR(ϕ) can be found\nby solving homogeneous part of equations (26) and (27):\nP(ϕ) =ıp(ϕ)exp/parenleftbiggıΩµ\nΩeϕ/parenrightbigg\n−ır(ϕ)exp/parenleftbigg\n−ıΩµ\nΩeϕ/parenrightbigg\n,(28)\nand\nR(ϕ) =p(ϕ)exp/parenleftbiggıΩµ\nΩeϕ/parenrightbigg\n+r(ϕ)exp/parenleftbigg\n−ıΩµ\nΩeϕ/parenrightbigg\n.(29)\nEquations for p(ϕ) andr(ϕ) can be obtained at the sub-\nstituting of expressions (28) and (29) into equations (26)\nand (27):\nı∂ϕp(ϕ)·exp/parenleftbiggıΩµ\nΩeϕ/parenrightbigg\n−ı∂ϕr(ϕ)·exp/parenleftbigg\n−ıΩµ\nΩeϕ/parenrightbigg\n= Πx,\n(30)\nwhere\nΠx=1\nΩeexp/parenleftbigg\nı/integraldisplayϕ\nC1\nΩe(ω−k·v|ϕ′)dϕ′/parenrightbigg\n×5\n×/parenleftbigg\nıµe(k·∇p)f0δBx+2µe\n¯hS0zδBy/parenrightbigg\n,(31)\nand\n∂ϕp(ϕ)·exp/parenleftbiggıΩµ\nΩeϕ/parenrightbigg\n+∂ϕr(ϕ)·exp/parenleftbigg\n−ıΩµ\nΩeϕ/parenrightbigg\n= Πy,\n(32)\nwhere\nΠy=1\nΩeexp/parenleftbigg\nı/integraldisplayϕ\nC1\nΩe(ω−k·v|ϕ′)dϕ′/parenrightbigg\n×\n×/parenleftbigg\nıµe(k·∇p)f0δBy−2µe\n¯hS0δBx/parenrightbigg\n.(33)\nIndependent equations for functions p(ϕ) andr(ϕ) can\nbe found as combinations of equations (30) and (32):\n∂ϕp(ϕ) =1\n2ıexp/parenleftbigg\n−ıΩµ\nΩeϕ/parenrightbigg\n(Πx+ıΠy),(34)and\n∂ϕr(ϕ) =1\n2ıexp/parenleftbiggıΩµ\nΩeϕ/parenrightbigg\n(−Πx+ıΠy).(35)\nTheseequationscanbeeasilyintegrated. Theintegration\ngives the following solutions for functions p(ϕ) andr(ϕ):\np(ϕ) =1\n2ı/integraldisplayϕ\nC1exp/parenleftbigg\n−ıΩµ\nΩeϕ′/parenrightbigg\n(Πx(ϕ′)+ıΠy(ϕ′)),(36)\nand\nr(ϕ) =1\n2ı/integraldisplayϕ\nC2exp/parenleftbiggıΩµ\nΩeϕ′/parenrightbigg\n(−Πx(ϕ′)+ıΠy(ϕ′)).(37)\nThis calculation leads to the following expressionsfor the\nspin distribution functions:\nδSx=µe\n2Ωe/bracketleftigg/integraldisplayϕ\nC1exp/parenleftbiggıΩµ\nΩe(ϕ−ϕ′)/parenrightbigg\nexp/parenleftbigg\nı/integraldisplayϕ′\nϕω−k·v|ϕ′′\nΩedϕ′′/parenrightbigg/parenleftigg\n(δBx+ıδBy)(ık·v|ϕ′)∂f0\n∂ε+2S0z\n¯h(δBy−ıδBx)/parenrightigg\ndϕ′\n+/integraldisplayϕ\nC2exp/parenleftbiggıΩµ\nΩe(ϕ′−ϕ)/parenrightbigg\nexp/parenleftbigg\nı/integraldisplayϕ′\nϕω−k·v|ϕ′′\nΩedϕ′′/parenrightbigg/parenleftigg\n(δBx−ıδBy)(ık·v|ϕ′)∂f0\n∂ε+2S0z\n¯h(δBy+ıδBx)/parenrightigg\ndϕ′/bracketrightigg\n,(38)\nand\nδSy=µe\n2ıΩe/bracketleftigg/integraldisplayϕ\nC1exp/parenleftbiggıΩµ\nΩe(ϕ−ϕ′)/parenrightbigg\nexp/parenleftbigg\nı/integraldisplayϕ′\nϕω−k·v|ϕ′′\nΩedϕ′′/parenrightbigg/parenleftigg\n(δBx+ıδBy)(ık·v|ϕ′)∂f0\n∂ε+2S0z\n¯h(δBy−ıδBx)/parenrightigg\ndϕ′\n+/integraldisplayϕ\nC2exp/parenleftbiggıΩµ\nΩe(ϕ′−ϕ)/parenrightbigg\nexp/parenleftbigg\nı/integraldisplayϕ′\nϕω−k·v|ϕ′′\nΩedϕ′′/parenrightbigg/parenleftigg\n(ıδBy−δBx)(ık·v|ϕ′)∂f0\n∂ε−2S0z\n¯h(δBy+ıδBx)/parenrightigg\ndϕ′/bracketrightigg\n.(39)\nSolutions (38) and (39) together with solutions (22) and\n(64) can be used for derivation of dielectric permeability\ntensor. Constants C0,C1,C2andC3are chosen that\ndistribution functions δfandδSare periodic functions\nof angleϕ:δf(ϕ+2π) =δf(ϕ) andδS(ϕ+2π) =δS(ϕ),\nsoCi=∞, wherei= 0,1,2,3.\nIV. DIELECTRIC PERMEABILITY TENSOR\nFOR MAGNETIZED SPIN-1/2 PLASMAS\nAfterFouriertransformationoftheMaxwellequations,\nthe magnetic field taken from equation k×δE=ωδB/cis substituted into equation (16). It leads to\n/bracketleftbigg\nk2δαβ−kαkβ−ω2\nc2εαβ(ω)/bracketrightbigg\nδEβ= 0,(40)\nwhere the dielectric permeability tensor appears as\nεαβ(ω)δEβ=δαβδEβ+4πı\nωqe\nm/integraldisplay\npαδfdp\n−4πµec\nω/integraldisplay\nεαβγkβδSγdp, (41)\nandδαβis the Kronecker symbol. For further analysis\nit is useful to distinguish a part of conductivity tensor6\ncaused by the current\nσαβ\n1(ω)δEβ=qe\nm/integraldisplay\npαδfdp (42)\nand another part caused by the curl of magnetization\nσαβ\n2(ω)δEβ=ıµec/integraldisplay\nεαβγkβδSγdp.(43)\nThe standard resolution between εαβandσαβis used:\nεαβ=δαβ+(4πı/ω)σαβ.\nA. General form of dielectric permeability tensor\nfor isotropic distribution functions\nAn explicit form of the conductivity tensor (42), (43)\nis presented in Appendix B. After the integration on ϕ\nandϕ′in equations (67)-(73) (which are presented in\nAppendix B)we find the followingdielectric permeability\ntensor\nεαβ=δαβ+/integraldisplay\ndp∞/summationdisplay\nn=−∞Λαβ(n)\nω−kzvz−nΩe\n+/summationdisplay\nr=+,−/integraldisplay\ndp∞/summationdisplay\nn=−∞Λαβ\nS,r(n)\nω−kzvz−nΩe+rΩµ\n−4πµ2\nek2\nxc2\nω2/integraldisplay\ndp∞/summationdisplay\nn=−∞J2\nn∂f0\n∂εδαyδβy+∆αδβy,(44)\nwhere\nΛαβ(n) =4πq2\ne\nω∂f0\n∂εΠαβ\nCl(n)\n+4πµ2\nek2\nxc2\nωJ2\nn∂f0\n∂εδαyδβy+4πqeµec\nωΠαβ\nS(n)∂S0z\n∂ε,(45)\nwith\n/hatwideΠCl(n) =\nΩ2\ne\nk2xn2J2\nnıv⊥Ωe\nkxnJnJ′\nnvzΩe\nkxnJ2\nn\n−ıv⊥Ωe\nkxnJnJ′\nnv2\n⊥(J′\nn)2−ıv⊥vzJnJ′\nn\nvzΩe\nkxnJ2\nnıv⊥vzJnJ′\nnv2\nzJ2\nn\n\n(46)\ndescribing the classical part and presented in many text-\nbooks (see for instance [49], [50]), and the tensor\nΠαβ\nS(n) =\n0ıΩenJ2\nn 0\n−ıΩenJ2\nn2kxv⊥JnJ′\nn−ıkxvzJ2\nn\n0ıkxvzJ2\nn0\n(47)\ndescribes the spin evolution leading to the same reso-\nnances as the classic evolution ω=kzvz+nΩe. The\nBesselfunctions Jnandtheirderivatives J′\nnareusedhere.\nIn this section all Bessel functions Jnare functions of\nkxv⊥/Ωe.Next, present tensors describing spin evolution mani-\nfesting itself at shifted resonances ω=kzvz+nΩe±Ωµ:\n/hatwideΛS+(n) =4π\nωµ2\nec2\n2ωJ2\nn/parenleftbigg∂f0\n∂ε(kzvz+nΩe)+2S0z\n¯h/parenrightbigg\nˆK,\n(48)\nand\n/hatwideΛS−(n) =4π\nωµ2\nec2\n2ωJ2\nn/parenleftbigg∂f0\n∂ε(kzvz+nΩe)−2S0z\n¯h/parenrightbigg\n(ˆK)∗,\n(49)\nwhere\nˆK=\nk2\nz−ık2\nz−kxkz\nık2\nzk2\nz−ıkxkz\n−kxkzıkxkzk2\nx\n,(50)\nand symbol ∗means the complex conjugation.\nThe last term in the dielectric permeability tensor (44)\narises as follows\n∆=4πqeµec\nω2/integraldisplay∞/summationdisplay\nn=−∞∂S0z\n∂ε×\n×{−ıΩenJ2\nn,−kxv⊥JnJ′\nn,−ıkxvzJ2\nn}dp.(51)\nThe x and z projections of vector ∆are equal to zero.\nThe x projection is equal to zero due to explicit sum-\nmation on nand the z projection is equal to zero due\nto integration over angles (some details are discussed in\nthe Appendix C). Therefore, the last term in the dielec-\ntric permeability tensor (44) gives contribution in ele-\nmentεyyonly. This contribution has the following form:\n∆yδαyδβy, with ∆ y=−kx4πqeµec\nω2/integraltext∂S0z\n∂εv⊥J0J′\n0dp.\nB. Dielectric permeability tensor for the\nspin-polarized Fermi step distribution function\nThedielectricpermeabilitytensorisobtainedabovefor\nthegeneralformofisotropicdistributionfunction. Inthis\nsubsection, a special form of the dielectric permeability\ntensor is obtained for the spin-1/2 partially polarized 3D\nelectron gas. To this end the equilibrium distribution\nfunctions are chosen as follows f0(p) = [ϑ(pF↑−p) +\nϑ(pF↓−p)]/(2π¯h)3andS0z(p) = [ϑ(pF↑−p)−ϑ(pF↓−\np)]/(2π¯h)3, wherepFs= (6π2n0s)1\n3¯h.\nAs it follows from equation (2), the Fermi spin current\nis related to difference between concentrations of elec-\ntrons with different spin projections. In kinetic descrip-\ntion such difference comesfrom S0zwhich is proportional\nto difference of the Fermi steps of the spin-up and spin-\ndown electrons.\nThe derivatives of the distribution functions are pro-\nportional to the Dirac delta function δ(p−pFs), since\n∂ϑ(pFs−p)/∂p=−δ(p−pFs). Therefore, the integrals\nover the module of the momentum module can be easily\ncalculated. As the result we find:7\nεαβ=δαβ−/summationdisplay\ns=↑,↓/integraldisplay\nsinθdθ∞/summationdisplay\nn=−∞/bracketleftbigg/tildewideΛαβ(n,s)\nω−kzvFscosθ−nΩe+mpFs\nπ¯h3µ2\nec2\n2ω2/summationdisplay\nr=+,−J2\nn(kzvFscosθ+nΩe)καβ\nr\nω−kzvFscosθ−nΩe+rΩµ\n−1\nπ¯h3µ2\nec2\n¯hω2/summationdisplay\nr=+,−/integraldisplaypFs\n0p2dpr(−1)isJ2\nnκαβ\nr\nω−kzvz−nΩe+rΩµ−mpFs\nπ¯h3/parenleftbigg\nµ2\nek2\nxc2\nω2J2\nn+qeµekxc\nω2vFssinθJnJ′\nn/parenrightbigg\nδαyδβy/bracketrightbigg\n,(52)\nwhereκαβ\n+=Kαβ,καβ\n−= (Kαβ)∗,i↑= 0,i↓= 1, and\n/tildewideΛαβ(n,s) =3ω2\nLs\n2ωv2\nFsΠαβ\nCl(n,s)+m2vFs/parenleftbiggµ2\ne\nπ¯h3k2\nxc2\nωJ2\nnδαyδβy+qeµe\nπ¯h3(−1)isc\nωΠαβ\nS(n,s)/parenrightbigg\n, (53)\nwith\n/hatwideΠCl(n,s) =\nΩ2\ne\nk2xn2J2\nn ıvFssinθΩe\nkxnJnJ′\nnvFscosθΩe\nkxnJ2\nn\n−ıvFssinθΩe\nkxnJnJ′\nnv2\nFssin2θ(J′\nn)2−ıv2\nFssinθcosθJnJ′\nn\nvFscosθΩe\nkxnJ2\nnıv2\nFssinθcosθJnJ′\nnv2\nFscos2θJ2\nn\n (54)\nwhichhasstructuresimilartowell-knownfromtextbooks\n[50], but it separately describes electrons with spin-up\nand spin-down, and the tensor\nΠαβ\nS(n,s) =\n\n0 ıΩenJ2\nn 0\n−ıΩenJ2\nn2kxvFssinθJnJ′\nn−ıkxvFscosθJ2\nn\n0ıkxvFscosθJ2\nn 0\n\n(55)\ndescribes the spin evolution leading to the same res-\nonances as the classic evolution ω=kzvFscosθ+\nnΩe. Here all Bessel functions Jnare functions of\nkxvFssinθ/Ωe.\nV. CONCLUSION\nA single fluid spin-1/2 quantum kinetics of electrons\nhas been applied for a derivation of the dielectric perme-\nability tensor of spin polarized electron gas. This model\nconsists of two kinetic equations for each species of par-\nticles: a generalization of the Vlasov equation containing\nthe spin-spin interaction as an additional term with the\nspin distribution function and a vector equation for the\nspin distribution function. Necessity ofthe applicationof\nthis model with the spin polarized equilibrium distribu-\ntion functions follows from the existence of the thermal\npartofspincurrentortheFermispin currentfordegener-\nate electrons which affects properties of transverse waves\nincluding the spin-plasma waves. More detailed descrip-\ntion of spin current effects requires a kinetic model which\nhas been presented here. Necessary details of the solu-\ntion of linearized set of kinetic equation required for thederivation of the dielectric permeability tensor have been\npresented.\nAcknowledgments\nThe author thanks Professor L. S. Kuz’menkov for\nfruitful discussions. The work was supported by the\nRussian Foundation for Basic Research (grant no. 16-\n32-00886) and the Dynasty foundation.\nVI. APPENDIX A: LINEAR PART OF THE\nDISTRIBUTION FUNCTIONS FOR\nGENERALIZED EQUILIBRIUM SPIN\nDISTRIBUTION FUNCTIONS\nIn equilibrium the set of kinetic equations (9) and (10)\ncan be presented in the following form\n∂ϕf0e↑= 0, ∂ϕf0e↓= 0, ∂ϕf0i= 0,(56)\nand\n∂ϕS0e,x=S0e,y, ∂ϕS0e,y=−S0e,x,(57)\nwheretime andspacederivativesofthe distribution func-\ntions are equal to zero, the equilibrium electric field is\nequal to zero. The equilibrium magnetic field is equal to\nthe external field: B0=Bext=B0ez.\nEquations (57) give the general form of dependence\nof equilibrium spin distribution functions on momentum\nS0x=C(p/bardbl,p⊥)cos(ϕ+ϕ0),S0y=C(p/bardbl,p⊥)sin(ϕ+ϕ0).\nConstant Ccan be equal to zero. It gives the isotropic\nequilibrium considered in the paper. For isotropic f0and8\nS0zand nonzero constant C, functions S0xandS0ycan\nbe presented in the following form [28]:\nS0x=1\n(2π¯h)3/parenleftbigg\nΘ(pF↑−p)−Θ(pF↓−p)/parenrightbigg\ncos(ϕ+ϕ0),\n(58)\nS0y=1\n(2π¯h)3/parenleftbigg\nΘ(pF↑−p)−Θ(pF↓−p)/parenrightbigg\nsin(ϕ+ϕ0).\n(59)In equation (17), there is a change of the last term\nin the following way: δBz(k∇p)S0z→δBβ(k∇p)S0β=\nδBβ[(kp)∂pS0β/p+εβγzS0γεµνzkµpν]. So, solution (22)\nis modified to\nδf=1\nΩe/integraldisplayϕ\nC0/parenleftbigg\nqe(v·δE)∂f0\n∂ε+ıµe(k·v)/parenleftbigg\nδB·∂S0\n∂ε/parenrightbigg\n+ıµe([δB,S0]z)·([k,p]z)/parenrightbigg\nexp/parenleftbigg\nı/integraldisplayϕ′\nϕ(ω−k·v|ϕ′′)\nΩedϕ′′/parenrightbigg\ndϕ′.(60)\nEquation (18) contains full vector S0. So, changing the explicit form of S0we have the spin distribution function\nevolution equation in the generalized regime. Equations (61)-(63) a re modified to\nΩe∂ϕδSx+ı(ω−k·v)δSx+ΩµδSy\n=qeδE·∇pS0x+ıµe(k·∇p)f0δBx+2µe\n¯hS0zδBy−2µe\n¯hS0yδBz−S0y(mv2δBz−mvz(vδB)),(61)\n−ΩµδSx+Ωe∂ϕδSy+ı(ω−k·v)δSy\n=qeδE·∇pS0y+ıµe(k·∇p)f0δBy−2µe\n¯hS0δBx+2µe\n¯hS0xδBz+S0x(mv2δBz−mvz(vδB)),(62)\nand\nΩe∂ϕδSz+ı(ω−k·v)δSz=qe(δE·∇p)S0z+ıµe(k·∇p)f0δBz+2µe\n¯h(δBxS0y−δByS0x) (63)\n(17) and (18) to equations (61)-(63).\nTherefore, solution (64) is modified to\nδSz=1\nΩe/integraldisplayϕ\nC3/parenleftbigg\nqe(v·δE)∂S0z\n∂ε+ıµe(k·v)δBz∂f0\n∂ε+2µe\n¯h(δBxS0y−δByS0x)/parenrightbigg\nexp/parenleftbigg\nı/integraldisplayϕ′\nϕ(ω−k·v|ϕ′′)\nΩedϕ′′/parenrightbigg\ndϕ′,(64)\nIn both regimes solutions for δSxandδSyare presented by equations () and (), but in the generalized case f unctions\nΠxand Π yhave the following form found from equations (61) and (62)\nΠx=1\nΩeexp/parenleftbigg\nı/integraldisplayϕ\nC1\nΩe(ω−k·v|ϕ′)dϕ′/parenrightbigg/parenleftbigg\nıµe(k·∇p)f0δBx+2µe\n¯hS0zδBy−2µe\n¯hS0yδBz−S0y(mv2δBz−mvz(vδB))/parenrightbigg\n,\n(65)\nand\nΠy=1\nΩeexp/parenleftbigg\nı/integraldisplayϕ\nC1\nΩe(ω−k·v|ϕ′)dϕ′/parenrightbigg/parenleftbigg\nıµe(k·∇p)f0δBy−2µe\n¯hS0δBx+2µe\n¯hS0xδBz+S0x(mv2δBz−mvz(vδB))/parenrightbigg\n.\n(66)\nVII. APPENDIX B: AN INTERMEDIATE\nFORM OF THE CONDUCTIVITY TENSOR\nMoreexplicitformoftheconductivitytensorcausedby\nthe current is obtained at substituting of the distributionfunction δf\nσαβ\n1(ω) =qe\nm1\nΩe/integraldisplay\ndppα/integraldisplayϕ\nC0dϕ′/parenleftbigg\nqevβ|ϕ′∂f0\n∂ε9\n+ıµe(k·v)|ϕ′kxc\nωδyβ∂S0z\n∂ε/parenrightbigg\nexp/parenleftbigg\nı/integraldisplayϕ′\nϕω−k·v|ϕ′′\nΩedϕ′′/parenrightbigg\n.\n(67)\nThe conductivity tensor caused by the curl of magne-\ntization has a complicate structure. Hence, it is splitted\non three parts:\nσxβ\n2(ω)δEβ=−ıµec/integraldisplay\nkzδSydp, (68)\nσyβ\n2(ω)δEβ=ıµec/integraldisplay\n(kzδSx−kxδSz)dp,(69)and\nσzβ\n2(ω)δEβ=ıµec/integraldisplay\nkxδSydp. (70)\nExplicit forms of functions (68)-(70) appear at substi-\ntution of the distribution functions δSx,δSy, andδSz.\nThus, the following expression can be found for σxβ\n2(ω):\nσxβ\n2(ω)δEβ=−µ2\nekzc2\n2Ωeω/integraldisplay/braceleftigg/integraldisplayϕ\nC1exp/parenleftbiggıΩµ\nΩe(ϕ−ϕ′)/parenrightbigg\nexp/parenleftbigg\nı/integraldisplayϕ′\nϕω−k·v|ϕ′′\nΩedϕ′′/parenrightbigg/bracketleftigg/parenleftbigg\n−(ık·v|ϕ′)∂f0\n∂ε+ı2S0z\n¯h/parenrightbigg\nkzδEy\n+/parenleftbigg\nı(ık·v|ϕ′)∂f0\n∂ε+2S0z\n¯h/parenrightbigg\n(kzδEx−kxδEz)/bracketrightigg\ndϕ′+/integraldisplayϕ\nC2exp/parenleftbiggıΩµ\nΩe(ϕ′−ϕ)/parenrightbigg\nexp/parenleftbigg\nı/integraldisplayϕ′\nϕω−k·v|ϕ′′\nΩedϕ′′/parenrightbigg\n×\n×/bracketleftigg/parenleftbigg\n(ık·v|ϕ′)∂f0\n∂ε+ı2S0z\n¯h/parenrightbigg\nkzδEy+/parenleftbigg\nı(ık·v|ϕ′)∂f0\n∂ε−2S0z\n¯h/parenrightbigg\n(kzδEx−kxδEz)/bracketrightigg\ndϕ′/bracerightigg\ndp. (71)\nNeglecting the anomalous magnetic moment of electron we have Ω µ= Ωe. Hence, σxβ\n2(ω) consists of two parts\nproportional to eı(ϕ−ϕ′)ore−ı(ϕ−ϕ′). Elements σyβ\n2(ω) have the following explicit form:\nσyβ\n2(ω)δEβ=ıµ2\nekzc2\n2Ωeω/integraldisplay/braceleftigg/integraldisplayϕ\nC1exp/parenleftbiggıΩµ\nΩe(ϕ−ϕ′)/parenrightbigg\nexp/parenleftbigg\nı/integraldisplayϕ′\nϕω−k·v|ϕ′′\nΩed��′′/parenrightbigg/bracketleftigg/parenleftbigg\n−(ık·v|ϕ′)∂f0\n∂ε+ı2S0z\n¯h/parenrightbigg\nkzδEy\n+/parenleftbigg\nı(ık·v|ϕ′)∂f0\n∂ε+2S0z\n¯h/parenrightbigg\n(kzδEx−kxδEz)/bracketrightigg\ndϕ′+/integraldisplayϕ\nC2exp/parenleftbiggıΩµ\nΩe(ϕ′−ϕ)/parenrightbigg\nexp/parenleftbigg\nı/integraldisplayϕ′\nϕω−k·v|ϕ′′\nΩedϕ′′/parenrightbigg\n×\n×/bracketleftigg/parenleftbigg\n−(ık·v|ϕ′)∂f0\n∂ε−ı2S0z\n¯h/parenrightbigg\nkzδEy+/parenleftbigg\n−ı(ık·v|ϕ′)∂f0\n∂ε+2S0z\n¯h/parenrightbigg\n(kzδEx−kxδEz)/bracketrightigg\ndϕ′\n−kx/integraldisplayϕ\nC3/parenleftbiggqe\nµe(v·δE)|ϕ′∂S0z\n∂ε+ı(k·v)|ϕ′kxc\nωδEy∂f0\n∂ε/parenrightbigg\nexp/parenleftbigg\nı/integraldisplayϕ′\nϕω−k·v|ϕ′′\nΩedϕ′′/parenrightbigg\ndϕ′/bracerightigg\ndp.(72)\nSimilar to σxβ\n2(ω),σyβ\n2(ω) consists of two parts propor-\ntional to eı(ϕ−ϕ′)ore−ı(ϕ−ϕ′). Elements σzβ\n2can be sim-\nply presented via σxβ\n2\nσzβ\n2(ω) =−kx\nkzσxβ\n2(ω) (73)\nin accordance with equations (68) and (70).VIII. APPENDIX C: CALCULATION OF\nSEVERAL ELEMENTS IN DIELECTRIC\nPERMEABILITY TENSOR\nFor calculation of z-projection of ∆(51) we need\nto consider integral over θ. Dependence on θ\ncomes from vzJ2\nn(v⊥). So, we have integral In=/integraltext\nsinθdθcosθJ2\nn(asinθ), where a≡kxvFs/Ωe. Using10\nJ−n(z) = (−1)nJn(z) andJ2\n−n(z) =J2\nn(z) we reduce\nour calculations to nonnegative Bessel functions n≥0.\nExplicit integration can be performed at the application\nof series expansion of the Bessel function:\nJn(z) =∞/summationdisplay\nk=0(−1)kzn(z/2)2k\n2nk!Γ(n+k+1). (74)\nIntegrating we obtain that In∼sin2k+2l+2n+2θ|π\n0= 0,\nwherek,l,n≥0 andlis used in expansion of the sec-\nond Bessel function presented under the integral. x-\nprojection of ∆is proportional to nJ2\nnwe can explicitly\ncalculate sum of these terms and find that it is equal tozero:\n∞/summationdisplay\nn=−∞nJ2\nn=∞/summationdisplay\nn=1nJ2\nn+−∞/summationdisplay\nn=−1nJ2\nn\n=∞/summationdisplay\nn=1(nJ2\nn+(−n)J2\n−n) = 0, (75)\nwhere we have used J−n(z) = (−1)nJn(z). Similarly,\napplying equation J′\nn(z) =Jn−1(z)−n\nzJn(z), we find/summationtext∞\nn=−∞JnJ′\nn=J0J′\n0. It simplifies y-projection of ∆.\n[1] L. S. Kuz’menkov, S. G. Maksimov, and V. V. Fedoseev,\nMoscow Univ. Phys. Bull. No. 5, 1 (2000).\n[2] L. S. Kuz’menkov, S. G. Maksimov, V. V. Fedoseev,\nTheor. Math. Phys. 126, 110 (2001).\n[3] L. S. Kuz’menkov, S. G. Maksimov, and V. V. Fedoseev,\nTheor. Math. Phys. 126, 212 (2001).\n[4] P. A. Andreev, L. S. Kuz’menkov, Moscow Univ. Phys.\nBull.62, N.5, 271 (2007).\n[5] M. Marklundand G. Brodin, Phys.Rev.Lett. 98, 025001\n(2007).\n[6] G. Brodin, M. Marklund, J. Zamanian, A. Ericsson, and\nP. L. Mana, Phys. Rev. Lett. 101, 245002 (2008).\n[7] Yu. Kagan, Zh. Eksp. Teor. Fiz. 41, 844 (1961).\n[8] Yu. Kagan, Zh. Eksp. Teor. Fiz. 41, 1536 (1961).\n[9] Yu. Kagan, Zh. Eksp. Teor. Fiz. 51, 1893 (1966).\n[10] V. N. Oraevsky, V. B. Semikoz, Astroparticle Physics 18,\n261 (2002).\n[11] V. N. Oraevsky, V. B. Semikoz, A. S. Volokitin,\narXiv:hep-ph/0203020 (2002).\n[12] V. N. Oraevsky, V. B. Semikoz, Phys. At. Nucl. 66, 466\n(2003) [Yadernaya Fizika 66, 494 2003].\n[13] P. A. Andreev, arXiv:1212.0099.\n[14] J. Hurst, O. Morandi, G. Manfredi, and P.-A. Hervieux,\nEur. Phys. J. D 68, 176 (2014).\n[15] P. A. Andreev, Physica A 432, 108 (2015).\n[16] H. C. Torrey, Phys. Rev. 104, 563 (1957).\n[17] F. J. Dyson, Phys. Rev. 98, 349 (1955).\n[18] V. P. Silin, Sov. Phys. JETP 30, 421 (1956).\n[19] M. Ya. Azbel, V. I. Gerasimenko, I. M. Lishich, Sov.\nPhys. JETP 31, 357 (1956).\n[20] M. Ya. Azbel, V. I. Gerasimenko, I. M. Lishich, Sov.\nPhys. JETP 32, 1212 (1957).\n[21] M. Ya. Azbel, V. I. Gerasimenko, I. M. Lishich, Sov.\nPhys. JETP 35, 691 (1958).\n[22] G. Feher, A. F. Kip, Phys. Rev. 98, 337 (1955).\n[23] Jun Zhu, and Peiyong Ji, Plasma Phys. Control. Fusion\n54, 065004 (2012).\n[24] A. Yu. Ivanov, P. A. Andreev, L. S. Kuz’menkov, Prog.\nTheor. Exp. Phys. 2015, 063I02 (2015).\n[25] P. A. Andreev, Phys. Rev. E 91, 033111 (2015).\n[26] P. A. Andreev, L. S. Kuz’menkov, Annals of Physics 361,\n278 (2015).\n[27] P. A. Andreev, Phys. Plasmas 23, 012106 (2016).[28] P. A. Andreev, Phys. Plasmas 23, 062103 (2016).\n[29] L. S. Kuzmenkov, D. E. Harabadze, Russ. Phys. J. 47,\n437 (2004).\n[30] G. Brodin, M. Marklund, and G. Manfredi, Phys. Rev.\nLett.100, 175001 (2008).\n[31] G. Brodin, A. P. Misra, and M. Marklund, Phys. Rev.\nLett.105, 105004 (2010).\n[32] P. A. Andreev, arXiv:1410.7077.\n[33] P. A. Andreev, L. S. Kuz’menkov, arXiv:1510.03468.\n[34] P. A. Andreev, and M. Iv. Trukhanova,\narXiv:1603.07506.\n[35] P. A. Andreev and L. S. Kuz’menkov, Appl. Phys. Lett.\n108, 191605 (2016).\n[36] P. A. Andreev, Z. Iqbal, Phys. Rev. E 93, 033209 (2016).\n[37] Z. Iqbal, P. A. Andreev, Phys. Plasmas 23, 062320\n(2016).\n[38] M. Hillery, R. F. O’Connell, M. O. Scully, E. P. Wigner,\nPhysics Reports 106, 121 (1984).\n[39] Y. L. Klimontovich and W. D. Kraeft, Theor. Math.\nPhys.19, 574 (1974).\n[40] Yu. L. Klimontovich and W. Ebeling, Soviet Physics\nJETP36, 476 (1973).\n[41] Yu.L.Klimontovich, SovietPhysicsUspekhi 16, Number\n4, 512 (1974).\n[42] L. S. Kuz’menkov and S. G. Maksimov, Theor. Math.\nPhys.131641 (2002).\n[43] S. Maximov, Physica D: Nonlinear Phenomena, 238,\n1937 (2009).\n[44] J. Zamanian, M. Marklund, and G. Brodin, Eur. Phys.\nJ. D69, 25 (2015).\n[45] F. A. Asenjo, Phys. Lett. A 373, 4460 (2009).\n[46] A. Hussain, M. Stefan and G. Brodin, Phys. Plasmas 21,\n032104 (2014).\n[47] P. A. Andreev, L. S. Kuz’menkov, Int. J. Mod. Phys. B\n29, 1550077 (2015).\n[48] J. Zamanian, M. Stefan, M. Marklund, and G. Brodin,\nPhys. Plasmas 17, 102109 (2010).\n[49] L. Landau and E. M. Lifshitz, Statistical Physics, part II\n(Pergamon, New York, 1980).\n[50] A. F. Aleksandrov, L. S. Bogdankevich, A. A. Rukhadze,\nPrinciples of plasma electrodynamics , Berlin; New York:\nSpringer-Verlag, 1984."    },    {        "title": "1810.07438v1.Unified_theory_of_magnetization_dynamics_with_relativistic_and_nonrelativistic_spin_torques.pdf",        "content": "Unified theory of magnetization dynamics with relativistic and nonrelativistic spin\ntorques\nRitwik Mondal,\u0003Marco Berritta, and Peter M. Oppeneer\nDepartment of Physics and Astronomy, Uppsala University, P. O. Box 516, Uppsala, SE-75120, Sweden\n(Dated: August 18, 2021)\nSpin torques play a crucial role in operative properties of modern spintronic devices. To study\ncurrent-drivenmagnetizationdynamics, spin-torquetermsprovidingtheactionofspin-polarizedcur-\nrents have previously often been added in a phenomenological way to the Landau-Lifshitz-Gilbert\nequation describing the local spin dynamics, yet without derivation from fundamental principles.\nHere, starting from the Dirac-Kohn-Sham theory and incorporating nonlocal spin transport we rig-\norously derive the various spin-torque terms that appear in current-driven magnetization dynamics.\nIn particular we obtain an extended magnetization dynamics equation that precisely contains the\nnonrelativistic adiabatic and relativistic nonadiabatic spin-transfer torques (STTs) of the Berger and\nZhang-Li forms as well as relativistic spin-orbit torques (SOTs). We derive in addition a previously\nunnoticed relativistic spin-torque term and moreover show that the various obtained spin-torque\nterms do not appear in the same mathematical form in both the Landau-Lifshitz and Landau-\nLifshitz-Gilbert equations of spin dynamics.\nPACS numbers: 75.78.-n, 75.70.Tj, 72.25.-b\nI. INTRODUCTION\nAlong with the traditional Landau-Lifshitz-Gilbert\n(LLG)[1–3]equationofmotionofmagnetizationdynamics,\nthere exist additional spin torques that have emerged as a\nnew way to manipulate spins in spintronics [4, 5] and spin-\norbitronics [6]. These torques stem from the spin-polarized\ncurrent density flowing in a magnetic material. There are\ntwo kinds of torques that have been identified in experi-\nments: (1) spin-transfer torques (STTs) that arise due to\nthe one-directional current flow perpendicular to the sam-\nple’s (interface) plane. As a consequence angular momen-\ntum is exchanged between two layers that have different\nspin orientations. These torques have been instrumental to\ndescribe the domain wall motion of heterostructured mag-\nnetic layers [7–11] and ultrafast magnetization dynamics\n[12,13]. Thespin-transfertorquesareduetoanonrelativis-\ntic effect which is reflected in the fact that the current den-\nsity is given by the nonrelativistic velocity operator. There\nexists also (2) the spin-orbit torque (SOT), whose origin\nis rather different from that of the STT, and, in addition,\nit acts in the plane of the magnetic film [14]. It is a rel-\nativistic effect where the angular momentum is exchanged\nbetween the crystal lattice and the spin degrees of freedom\nthus enabling to control the magnetic state even in a sin-\ngle magnetic layer. The spin-orbit interaction produces a\ncurrent which strongly depends on the magnetization and\nthe electric field. As the magnetization precesses around an\neffective magnetic field, the direction of the magnetization\nchanges and thus the current generated from spin-orbit in-\nteraction is not unidirectional unlike the STT, but rather\nit remains in a plane depending on the magnetization. Re-\ncently, there have been successful attempts to manipulate\nmagnetization through SOTs in magnetic heterostructures\n[15–22], in ferromagnet-antiferromagnet bilayers [23], an-\ntiferromagnets [24], magnetic-doped topological insulators\n\u0003Present address: Fachbereich Physik, Universität Konstanz, DE-\n78457 Konstanz, Germany; ritwik.mondal@uni-konstanz.de[25], fast spin switching [26], and femtosecond control of\nmagnetization in ferromagnets [27].\nTosimulatetheeffectofspintorquesontheensuingmag-\nnetization dynamics in a magnetic system, the spin torques\nhave often simply been added to the LLG equation,\n@M\n@t=\u0000\rM\u0002He\u000b+\u000bM\u0002@M\n@t+T;(1)\nwhereM(r;t)is a local magnetization, He\u000bthe effective\nmagnetic field, \rthe gyromagnetic ratio, \u000bthe Gilbert\ndamping parameter, and Tthe spin torque. Despite the\nfact that the LLG equation is intensively used in spin-\ndynamicssimulations(see, e.g., [28–31]), itisaphenomeno-\nlogical equation proposed originally on the basis of physical\nintuition of the pioneers in the field [1, 2, 32]. A derivation\nof Eq. (1) from fundamental principles is thus still sollici-\ntated.\nOn the theoretical side, explicit forms of the spin torque\nThavebeeninvestigated(see, e.g., [10,33–37]). Itisknown\nthat spin torques arise from the nonequilibrium spin den-\nsity [10]. The common procedure to derive them is to con-\nsider an interaction Hamiltonian of conduction electrons\nand a local magnetization in the “ s-d” exchange Hamilto-\nnian forms\u0001M, wheresandMdenote the itinerant elec-\ntrons and local magnetization, respectively. The torque is\nthen proportional to m\u0002M, withm(r;t) =hsithe itiner-\nant spin density. The nonequilibrium spin density, denoted\nas\u000em, exerts the torque \u000em\u0002M, which is the origin of\nthespin-transfertorqueinferromagnets[10,34,35,37], fer-\nromagnetic semiconductors [38], and Weyl semimetals [39].\nAlthoughthesederivationshavebeenwellunderstood, they\nprovide an explicit form of the spin-transfer torque only.\nThe spin-orbit torque is fundamentally more difficult, be-\ncause it depends on a small relativistic term. To derive ex-\nplicit expressions for the SOT often a particular form, such\nas Rashba spin-orbit interaction [40], has been considered\n(see, e.g., [41–43]). A derivation of the spin equation of mo-\ntion with spin torques on the basis of relativistic electronic\nstructure theory has not yet been given.\nRecently, we have shown that the LLG equation can be\nfully derived in a relativistic Dirac-Kohn-Sham framework,arXiv:1810.07438v1  [cond-mat.mtrl-sci]  17 Oct 20182\nleading to an explicit expression for the Gilbert damping\n[44]. Apart from the Gilbert damping, previously other\ntypes of damping, such as the spin nutation or magnetic\ninertia have been included in an extended LLG equation\nof motion through a second order time-derivative of the\nmagnetization [45–47]. We have shown lately that this in-\nertialdampingtorquecanalsobefullyderivedintheDirac-\nKohn-Sham framework and arises from higher-order spin-\norbit coupling terms which act on ultrashort timescales\nonly [48]. Along the same direction, we have provided a\ngeneralized theory for spin dynamics, wherein we derived\nthe most general form of the intrinsic Gilbert damping and\nmagnetic inertia parameters from relativistic Dirac theory\n[49]. The derived forms suggest that both these parame-\nters can be expressed as a series of higher-order relativis-\ntic corrections and that they are tensors. It deserves to\nbe mentioned that there exist still other variants of the\nLLG equations. Bar’yakhtar et al.introduced an exchange\ndamping torque, as a second order spatial derivative of the\nexchange field [50, 51]. To describe longitudinal spin relax-\nation the Landau-Lifshitz-Bloch equation has been derived\n[52, 53]. The effect of transverse spin diffusion has further-\nmorebeenconsideredinthecaseofferromagnetswithinthe\nLLG framework [54]. Another recent work also proposed\nthe existence of a torque which involves both spatial and\ntemporal derivatives of the magnetization [55].\nIn this article we aim to derive the current-driven mag-\nnetization dynamics from fundamental principles. To this\nend we start from the Dirac-Kohn-Sham (DKS) Hamilto-\nnian and first derive, in the semirelativistic limit, all spin-\nrelated nonrelativistic and relativistic Hamiltonian terms\nusing an unitary transformation. Next, we proceed to the\nspin dynamics including spin-polarized currents, utilizing a\ncontinuity equation, which involves the spin density and\nthen rigorously calculate the spin torques. We observe\ntwo fundamentally different dynamics, the nonrelativistic\nmagnetization dynamics, derived solely from the nonrela-\ntivistic Hamiltonian, and the relativistic magnetization dy-\nnamics which mainly is related to the spin-orbit coupling.\nOur derivation leads to the current-induced spin-transfer\ntorque of Berger type [8], which is a nonrelativistic effect.\nThe Zhang-Li nonadiabatic spin-transfer torque [10] is ob-\ntained when the spin-dynamics equation is rewritten in the\nLandau-Lifshitz (LL) form. The derivation of the relativis-\ntic spin dynamics contrarily leads to various electric field-\ninduced torques, which are essentially spin-orbit torques.\nIn the following we first introduce the relativistic DKS\nHamiltonian with external fields. The magnetization dy-\nnamics with currents and external electric field is derivedin Sec. III and discussed in Sec. IV, and our conclusions are\ngiven in Sec. V.\nII. RELATIVISTIC HAMILTONIAN\nFORMULATION\nWe start from the Dirac-Kohn-Sham Hamiltonian which\ndescribes a relativistic spin-1\n2quantum particle efficiently\nin the effective mean-field formulation. For a magnetic ma-\nterial the DKS Hamiltonian is given as [44, 56–59]\nHDKS=c\u000b\u0001(p\u0000eA) +\u0000\n\f\u00001\u0001\nmc2+V1+e\b1\n\u0000\u0016B\f\u0006\u0001Bxc; (2)\nwhereVstands for the unpolarized Kohn-Sham selfcon-\nsistent potential, Bxcdefines the spin-polarized exchange-\ncorrelation potential in the material, A=A(r;t)is the\nmagnetic vector potential due to an applied electromag-\nnetic field (e.g., light), e\b(r;t)is the scalar potential of this\nfield,p=\u0000i~r, the momentum operator and \u0016B=e~\n2m,\nthe Bohr magneton. 1is the 4\u00024identity matrix and \u000b,\n\f, and\u0006are the well-known Dirac matrices defined e.g., in\nRef. [60]. Evidently there are two fundamentally different\nfields present in the DKS Hamiltonian, one is the Maxwell\nfields from the externally applied source and the other one\nis exchange field which couples to the spin degrees of free-\ndom and has a very different origin. Consequently, the\nexchange field does not fulfill the Maxwell equations and\ncan therefore not be treated as a minimal coupling to the\nmomentum, i.e., p\u0000eAxc[57, 61].\nNext, we want to investigate the relativistic spin dynam-\nics of spin-polarized electrons including the effect of spin\ncurrents. To achieve this we need the full Pauli Hamilto-\nnian for electrons in a magnetic system, including all the\nrelativistic corrections that stem from the DKS Hamilto-\nnian. ToreachthisweemploytheFoldy-Wouthuysen(FW)\ntransformation [60, 62] on the DKS Hamiltonian. The FW\ntransformation is a canonical and unitary transformation\nwhich is iteratively applied [63]. The main idea of the\niterative transformation is to make the off-diagonal (i.e.,\nparticle–antiparticle) elements smaller until the separation\nof particle and antiparticle is guaranteed for any given mo-\nmentum [62]. This leads to a semirelativistic, extended\nPauli (EP) Hamiltonian, HEP, which includes all spin-\ndependent interactions and relativistic corrections as well\nas the exchange field and external electromagnetic fields\n[44, 61]. For convenience sake, we explicitly separate the\nextended Pauli-like Hamiltonian in nonrelativistic and rel-\nativistic Hamiltonian terms, i.e., HEP=HNR+HR, which\nare defined as [64]\nHNR=(p\u0000eA)2\n2m+V\u0000\u0016B\u001b\u0001(B+Bxc) +e\b (3)\nHR=\u0000\u0016B\u001b\u0001Bxc\ncorr\u0000(p\u0000eA)4\n8m3c2\u0000e~2\n8m2c2r\u0001Etot\u0000e~\n8m2c2\u001b\u0001[Etot\u0002(p\u0000eA)\u0000(p\u0000eA)\u0002Etot]\n+i\u0016B\n4m2c2[(p\u0002Bxc)\u0001(p\u0000eA)]: (4)\nAll the appearing relativistic corrections involving the exchange interaction (with exception of the last term in Eq. (4))3\nhave here been added together to give the relativistic correction to the exchange field [65],\nBxc\ncorr=\u00001\n8m2c2n\u0002\np2Bxc\u0003\n+ 2(pBxc)\u0001(p\u0000eA) + 2(p\u0001Bxc)(p\u0000eA) + 4[Bxc\u0001(p\u0000eA)](p\u0000eA)o\n:(5)\nThus, the exchange fields appearing in the DKS Hamil-\ntonian (Bxc) and in the extended Pauli-like Hamiltonian\n(Bxc+Bxc\ncorr) are not identical. The other fields that are\npresent in the above-derived relativistic Hamiltonian HEP\nare: the external magnetic field, B(r;t) =r\u0002A(r;t)as\nwell as the total electric field that is the sum of internal,\nEintand external, Eextcontributions as Etot=Eint+Eext.\nThe internal electric field stems from the ions and the inter-\nactions of electrons, which even exists without any pertur-\nbation i.e.,Eint=\u00001\nerV. On the other hand the external\nfield is defined as Eext=\u0000@A\n@t\u0000r\b. The meaning of each\nof the terms in the Hamiltonians (3) and (4) are immedi-\nately explained, see Ref. [61] for details.\nIII. MAGNETIZATION DYNAMICS\nNowthatwehavealltherequiredtermsintheEPHamil-\ntonian we can proceed to derive the full magnetization dy-\nnamics with spin-current terms included.\nThe magnetization is, according to its definition, given\nby the expectation value of the spin density [66]\nM(r;t) =g\u0016B\n^S\u000e(r\u0000^r)\u000b\n; (6)\nwhere\u0016B=e~=2mis the usual Bohr magneton and the\nLandégfactor,g\u00192for spin degrees of freedom. Note\nthatwehavedefinedmagnetizationasspinangularmomen-\ntum per unit volume in our earlier investigations [44, 48],\nwhich is valid in the case of homogeneous spin distribu-\ntions. However, for inhomogeneous spin distributions the\nindividual spins have to be considered, as is done in Eq.\n(6). Note that we do not consider the magnetization con-\ntribution due to the orbital moment, because this quantity\nis quenched for the common transition metals (e.g., Fe, Ni,\nCo etc.). The equation of motion of the magnetization is\nobtained by taking the time derivative on both sides of Eq.\n(6), which gives\n@M\n@t=g\u0016BD@^S\n@t\u000e(r\u0000^r) +^S@\u000e(r\u0000^r)\n@tE\n:(7)\nNext we use the Heisenberg representation of operators\nand the corresponding dynamical equation of motion. This\nleads to the full magnetization dynamics\n@M\n@t=g\u0016B\ni~D\u0002^S;H\u0003\n\u000e(r\u0000^r) +^S[\u000e(r\u0000^r);H]E\n:(8)\nFor the first commutator of spin angular momentum with\nthe Hamiltonian [^S;H]to be nonzero, the Hamiltonian\nmust depend on the spin as otherwise the commutator van-\nishes. This commutator has been considered and worked\noutinanearlierstudy[44], wherewehavealreadydiscussed\ntheoriginofmagnetizationprecessionandofGilbertdamp-\ning that follow completely from the first term in Eq. (8).All these phenomena are directly related to the local mag-\nnetization dynamics. The second commutator, in contrast,\ninvolves the density operator and the Hamiltonian, which\nin turn defines the orbital current density. The latter will\ncontribute to the nonlocal processes in the magnetization\ndynamics, as e.g., current-induced magnetization dynam-\nics, spin-transfer torque, spin-orbit torques etc.\nIn the following we derive rigorously the effect of the\nsecond term on the right-hand side of Eq. (8) within the\nmagnetization dynamics. For an electron, the charge den-\nsity operator will be given as ^\u001a=e\u000e(r\u0000^r)[67]. From the\ncontinuity equation we get @^\u001a=@t =\u0000r\u0001^j, where ^jis the\ncurrent density operator which has to be obtained from the\nHamiltonians Eqs. (3) and (4). Using these considerations\ntogether with the fact that the spatial derivative of spin\nangular momentum is zero, we arrive at [35, 68, 69]\n@M\n@t+r\u0001JS=T; (9)\nwhereJSis the spin current and Tis the torque acting on\nthespinsthatoriginatesfromthecommutatorofspinangu-\nlar momentum with the Hamiltonian in Eq. (8). Note that\nthis continuity equation is different from the usual ones be-\ncause of the appearing torque term. We also note that spin\nrelaxation processes due to the electron-phonon scattering\nhave not been included in the present study. Incorporating\nscattering processes such as spin relaxation to the lattice\nor spin-defect scattering would give rise to nonadiabatic\neffects [10]. The DKS spin-current tensor is defined as\nJS=g\u0016B\neD\n^j\n^SE\n=1\nej\nM: (10)\nThis tensor depends on the charge current density jand\nthe magnetization.\nTo obtain expressions for the spin-current terms we need\nto calculate the terms in Eq. (8). At this point it is impor-\ntant to note that neither the spin nor the orbital degrees\nof freedom commute with the Hamiltonian due to the spin-\norbit coupling and the other 1=c2corrections. This in turn\nimplies that the equilibrium density operator ^\u001acannot be\nexpressed exactly as ^\u001a= ^\u001aS\n^\u001aOwhere ^\u001aSis the reduced\ndensity operator for the spin degrees of freedom and ^\u001aOthe\nreduced density operator of the orbital degrees of freedom.\nConsidering an observable Oacting on the orbital degrees\nof freedom in Hilbert space (for instance the momentum or\nthe orbital angular momentum or some function depending\non them) and ^Sthe spin, then, due to the impossibility to\nseparate orbital and spin parts of the density matrix, we\nare in principle not allowed to write\nTr[^\u001a^SO] = Tr[^\u001aS^S] Tr[^\u001aOO] =MhOi:(11)\nTo nonetheless be able to employ this approximation it\nis important to realize that the nonseparability (entangle-\nment) of the orbital and spin parts of the density matrix4\nis due to the spin-orbit coupling and its corrections (since\nit prevents each of these quantities to be conserved). How-\never, in ferromagnetic materials the energy separation of\nthe spin states is mostly due to the exchange field which\nis orders of magnitude larger than the spin-orbit coupling\nand its corrections. As a consequence, the separation of\nthe density operator as a direct product of spin and orbital\nparts is a good approximation, and therefore we can em-\nploy Eq. (11). Moreover, due to the large splitting of spin\nbands and the continuous smooth behavior of the energy\nlevels as function of momentum, the out-of-equilibrium dy-\nnamics on the latter degrees of freedom is much faster than\nthe dynamics of the spin degrees of freedom.\nThe tensor form of the spin current has been discussed\nearlier in the context of time-dependent spin-density func-\ntional theory [68, 70, 71]. In the nonrelativistic Kohn-Sham\nHamiltonian the current density stems from the commuta-\ntorofkineticenergywiththeparticledensity. However, the\npicture is different when we consider the relativistic DKS\nformalism.\nThe associated relativistic current density can be derived\nfrom the usual prescription of Landau and Lifshitz [72] in\nwhich it is given by the variation of interaction energy with\nrespect to the magnetic vector potential as [61, 73]\n\u000ehHinti=\u0000eZ\ndrj\u0001\u000eA: (12)\nAs the interaction Hamiltonians in Eqs. (3) and (4) contain\nnonrelativistic and relativistic parts, we can consequently\ndecompose the current density contributions into a nonrel-\nativistic current density jNRand a relativistic current den-\nsityjR. The total current density can then be expressed as\nj=jNR+jR. Similarly, the corresponding torques have\nnonrelativistic and relativistic origins, and we can formally\nwrite the magnetization dynamics as:\n@M\n@t\f\f\f\nNR+1\neh\nM(r\u0001jNR) + (jNR\u0001r)Mi\n=TNR;(13)\n@M\n@t\f\f\f\nR+1\neh\nM(r\u0001jR) + (jR\u0001r)Mi\n=TR:(14)\nNote that this formulation is done to shorten the notation;\nthe complete @M=@tdynamics is given below. In Eqs. (13)\nand (14) we have used that the divergence of a dyadic ten-\nsor productA\nBis simply given by the following identity,\nr\u0001(A\nB) = (r\u0001A)B+ (A\u0001r)Bfor any two vectors\nAandB. It is interesting to note that we can thus dis-\ntinguish the dynamical equations for magnetization due to\nnonrelativistic and relativistic origins, however the same is\nnot possible while defining magnetization only.\nIn the following we calculate the relativistic and nonrel-\nativistic dynamics.\nA. Nonrelativistic magnetization dynamics\nThe current density and torques that appear in the non-\nrelativistic part [Eq. (13)] can be calculated from the non-\nrelativistic Hamiltonian Eq. (3), where the nonrelativistic\npart of the effective exchange field is Bxchas to be consid-\nered. Let be the spinor solution to the EP Hamiltonian.The associated current density then becomes [73, 74]\njNR(r) =i~\n2m\u0000\n r y\u0000 yr \u0001\n\u0000e\nmA y \n+1\nmr\u0002\u0000\n yS \u0001\n: (15)\nThe first term derives from the commutator of the particle\ndensity with the kinetic energy p2=2m, the second term is\nthe current due to the external perturbation from the elec-\ntromagnetic field, i.e., the term involving A\u0001p. The last\ntermis derived fromthe“Zeeman-like” term inEq. (13)and\nis due to the fact that the external magnetic field can be ex-\npressed within the Coulomb gauge as B=r\u0002A[61, 73].\nA detailed derivation of this magnetic current term can be\nfound in Ref. [61]. As we mentioned before, the exchange\nfield solely stems from the Pauli exclusion principle and de-\npends on spin degrees of freedom only, hence, it cannot be\ntreated as Maxwellian field [57]. Thus, there is no contri-\nbution of current density coming from an exchange field.\nUsing the definition of magnetization, the last term could\nbe written in the form1\nmg\u0016Br\u0002M, which also defines the\nbound current density according to classical electrodynam-\nics [75].\nAlong the same reasoning the nonrelativistic torques on\nthe spins will stem from the commutator of spin angular\nmomentum with the Hamiltonian. Understandably, only\nthe “Zeeman-like” couplings with external and exchange\nfields will contribute to these torques. In the magnetiza-\ntion dynamics, these torques can be summed up and de-\nscribe the precession of the magnetization vector around\nan effective nonrelativistic field [44],\nTNR=\u0000\r0M\u0002HNR\ne\u000b; (16)\nwhere\r0=\u00160\ris the effective gyromagnetic ratio [76] and\nthegyromagneticratioisdefinedas \r=gjej=2m[44]. Here,\nthe effective nonrelativistic field, HNR\ne\u000b, is due to the Zee-\nman coupling of external and exchange fields to the spins.\nAdiabatic spin-dynamics simulations have previously been\ndeveloped for elemental ferromagnets on the basis of the\nnonrelativistic Kohn-Sham Hamiltonian [77].\nNow that we have all the required expressions we can\nproceed to analyze the magnetization dynamics given by\nEq. (13). Let us focus first on the the second term i.e.,\nM(r\u0001jNR). Due to the Coulomb gauge, r\u0001A= 0and\nalsothedivergenceofacurlisalwayszero, r\u0001(r\u0002M) = 0.\nThus, theexternalperturbationandthespin-polarizedZee-\nman current do not contribute to the dynamics and we are\nleft with the first term in Eq. (15). Over a volume that con-\ntains the sample, the divergence of current density is zero\nas there is no source nor sink. On the other hand, the third\nterm in Eq. (13), i.e. (jNR\u0001r), bears much importance for\nthe magnetization dynamics as it treats the magnetization\ninhomogeneity and its spatial variation. Writing down the\nnonrelativistic dynamics, we have\n@M\n@t\f\f\f\nNR=\u0000\r0M\u0002HNR\ne\u000b\u00001\ne\u0000\njNR\u0001r\u0001\nM:(17)\nInterestingly, wehavethusderivedtheBerger-typecurrent-\ninduced spin-transfer torque [8], which is expressed as\nTSTT =1\ne\u0000\njNR\u0001r\u0001\nMin the adiabatic limit (see, e.g.,5\nRefs. [11, 36, 37, 39, 78–81]). It is independent of the\nelectron relaxation lifetime. Note that we have deliber-\nately not taken into consideration the nonadiabatic current\ndensity; the latter would lead to the corresponding spin-\ntransfer torque that essentially includes the spin-flip and\nscatterings processes [10].\nB. Relativistic magnetization dynamics\nTo obtain the relativistic current density the spin-orbit\ncoupling and other relativistic effects need to be incorpo-\nrated explicitly in the continuity equation. Thereto we\nwritethecurrentdensityas jR=jsoc+jother, wherethelat-\ntertermrepresentstheother(non-SOC)relativisticcurrent\ndensity. The main relativistic contribution to the current\ndensity comes from the spin-orbit coupling and it has the\nfollowing form [61, 73]\njsoc(r) =\u0000e\n2m2c2\u0000\n yS \u0001\n\u0002Etot=\u0018M\u0002Etot;(18)\nwhere we defined \u0018=\u0000e\n2m2c2g\u0016B. The direction and mag-\nnitude of this current depends on the magnetization and\nelectric field. As the magnetization precesses, the direction\nof the current changes and thus generates a torque. As\nmentioned earlier, the total electric field has two contribu-\ntions, the internal and external field. In turn, the internal\nelectric field has also two contributing components: (1) the\ninternal electric field without any application of external\nfield which we denote E0\nint, and (2) the response of the\nmaterial with the applied external field, which in a linear\napproximation gives [82]\nEint=E0\nint+ \u0000\u0001Eext: (19)\nIn general \u0000is a material dependent tensor of rank 2 which\nreduces to a scalar for an isotropic material. For simplicity,\nin the following, and throughout this work we use \u0000as a\nscalar parameter. The zero-response internal electric field\nhas the form E0\nint=\u00001\nerV. Thus, the current density due\nto the spin-orbit coupling will be given as\njsoc(r) =\u0018M\u0002E0\nint+\u0018(1 + \u0000)M\u0002Eext:(20)\nNotethatthefirsttermherewouldexistevenintheabsence\nof any applied field. It accounts for the bound spin currents\ninducedbythespin-orbitcoupling. Thecurrentdensitydue\nto the spin-orbit coupling are at the heart of the anomalous\nHalleffect, wheretheoff-diagonalconductivityisafunction\nof the magnetization [83–86]. If we take a general E-field\ndirected along the z-axis, the spin-orbit current stays in the\nplane perpendicular to the electric field, i.e., the xyplane.\nThis phenomenon is pictorially shown in Fig. 1. Thus, this\ncurrent plays an important role in the case of interfaces in a\nlayered system of a ferromagnet and a heavy metal, where\nthe spin-orbit strength is large [16–18, 87, 88]. However,\nit is also interesting to note that this current can have a\nsignificant impact on a plane for a single-layered system,\nwhen the spin-orbit coupling is large.\nLet us look at the details of the spin-orbit current in the\ncase of ferromagnets. Consider a ferromagnet where the\nmagnetization is in the z-direction i.e., M=Mz^z. Thex\nandycomponents of the electric fields will contribute to\nFM HM\nE=E0ˆ zjNR\njsoc\nzx\nyFigure 1. Schematic illustration of the two different currents in\na ferromagnet (FM)–heavy metal (HM) bilayer. The nonrela-\ntivistic current jNRis unidirectional and provides a torque to\nthe ferromagnetic spin component that is perpendicular to the\nz-axis. In contrast, the relativistic spin-orbit current jsoclies in\nthexy-plane and provides a torque on the spin which is along\nthez-direction.\nthe current because of the cross product. The spin-orbit\ncurrent is then written as\njsoc=\u0000\u0018(1 + \u0000)Mz(Ey^x\u0000Ex^y):(21)\nUsing thatj=\u001b\u0001E, the off-diagonal conductivity matrix\nelements can be written as \u001bxy=\u0000\u001byx/\u0000\u0018(1 + \u0000)Mz,\nwhich represents the anomalous Hall effect. In this context,\nwe mention that \u0018(1+\u0000)has been argued in Ref. [86] to be\nthe anomalous Hall conductivity. These relativistic current\ndensities enter into the relativistic magnetization dynamics\nof Eq. (14) as M(r\u0001jsoc)and(jsoc\u0001r)M. Unlike the\nnonrelativistic current density, as jsocdepends on the mag-\nnetization texture, the divergence provides a dependence\non the inhomogeneity in the magnetization. Calculating\nnow the divergence of the current density due to spin-orbit\ncoupling, we obtain\nr\u0001jsoc=\u0018r\u0001\u0000\nM\u0002Etot\u0001\n=\u0018h\nEtot\u0001\u0000\nr\u0002M\u0001\n\u0000M\u0001\u0000\nr\u0002Etot\u0001i\n=\u0018h\nEtot\u0001\u0000\nr\u0002M\u0001\n+ (1 + \u0000)M\u0001@B\n@ti\n:(22)\nIn the last step we have used Faraday’s law of induction,\nr\u0002Eext=\u0000@B=@t, and that for the internal electric field\nthe curl of a gradient is always zero. The magnetic induc-\ntion follows a linear relationship with magnetization such\nthatB=\u00160(H+M)[44]. Using this relation and mak-\ning the fundamental assumption of transversal spin motion,\nnamely, that the magnitude of magnetization is conserved\nduring the dynamics [89], which leads to M\u0001@M=@t= 0,\nand we are left with only \u00160M\u0001@H=@tfor a general time-\ndependent field. However, for a harmonic external field, in-\ntroducing a differential susceptibility \u001f, the time-derivative\nleads to@B=@t=\u00160(1 +\u001f\u00001)@M=@tand the last term on\nthe right-hand side of Eq. (22) vanishes. For simplicity we\ncontinue our analysis with the harmonic field in the follow-\ning. Results for a nonharmonic driving field are given in\nAppendix A.\nThe other current density term in the relativis-\ntic magnetization dynamics will be simply given as6\n\u0018[(M\u0002Etot)\u0001r]M. On the same footing, the relativis-\ntic torques will be derived from the commutators of spin\nangularmomentumwiththeHamiltonianofspin-orbitcou-\npling and other interaction terms related to the exchange\nfields. In a previous study we have already discussed that\nthe other relativistic effects related to the exchange field\ncreate an effective field, HR\ne\u000bwhich is responsible for the\nrelativistic precession [44]. We have also shown that the\nspin-orbit coupling related to the internal field, contributes\nto the relativistic precession as well [44]. However, the ex-\nternal spin-orbit coupling in the form of \u001b\u0001(Eext\u0002p)con-\ntributes to the damping and relaxation processes, in partic-\nular, to the intrinsic Gilbert damping [44, 90]. The damp-\ning parameter Athat we have derived is a tensor, which\nincludes a scalar Gilbert damping, an anisotropic Ising-like\ntensorial damping and an antisymmetric chiral damping.\nFor a harmonic field, the Gilbert damping tensor has two\ncharacteristically different contributions: one is electronic\nin nature and is given by an expectation value of the prod-\nuct of position and momentum operator as hr\u000bp\fi, and the\nother is magnetic, and given through the imaginary part\nof the susceptibility tensor [44, 48]. We remark here that\nfor a general time-dependent magnetic field, the Gilbert\ndamping involves an additional torque – the field-derivative\ntorque. Its effect is to make the damping effectively fre-\nquency dependent. Also, if the applied field very sharply\nincreases, e.g., step-like, this term exerts a huge but instan-\ntaneous torque which might offer new ways to manipulate\nthe spins. As we have obtained the field-derivative torque\nin the LLG form of the spin dynamics, we derive in Ap-\npendix A the form of this torque in the Landau-Lifshitz\nspin dynamics where it leads to an additional term to the\neffective magnetic field.\nThe other relativistic interaction Hamiltonian part, \u001b\u0001\n(Etot\u0002A)is easily recognized as a gauge invariant part of\nthe spin-orbit coupling. Without this term the spin-orbit\ncoupling is not gauge invariant [82]. For the internal elec-\ntric field, this part of the Hamiltonian in Eq. (4) vanishes,\nleaving thus only the contribution from the external field.\nThe optical spin angular momentum density within parax-\nial approximation gives Js=\u000f0(Eext\u0002A)[91, 92]. The\nmagnetization dynamics due to the interaction with the\noptical angular momentum can be written as\n@M\n@t\f\f\f\nopt=\u0000\rM\u0002Bopt; (23)\nwhere we have introduced the optomagnetic field that can\nbe written as Bopt=e\n2mc2(Eext\u0002A)[44, 82, 93]. The\nensuing dynamics essentially defines the optical spin-orbit\ntorque acting on the local magnetization. Note that the\noptomagnetic field is helicity dependent, thus possibly en-\nabling helicity-dependent phenomena, as e.g., all-optical\nhelicity-dependent magnetization switching [94–96]. Sev-\neral experimental evidences for the possible manipulation\nof spins by means of an optical spin-orbit torque have been\nreported recently [97–99].\nTaking together all these above-derived terms, the rela-tivistic magnetization dynamics can be written as\n@M\n@t\f\f\f\nR+\u0018\neh\nM(Etot\u0001(r\u0002M)) + ((M\u0002Etot)\u0001r)Mi\n=\u0000\r0M\u0002HR\ne\u000b+M\u0002\u0012\nA\u0001@M\n@t\u0013\n\u0000\rM\u0002Bopt:\n(24)\nThis equation is one of the central results of this work\nbecause it derives the existence of field-induced SOTs\nthat are first order in the gradient of magnetization.\nWe denote the field-induced spin-orbit torques as TSOT,\nTSOT=\u0000\u0018\ne[M(E\u0001(r\u0002M)) + ((M\u0002E)\u0001r)M]. We\nhave dropped the subscript of the electric field and from\nnow onwards, for convenience, we use Etot\u0011E. It is\nimportant to remember that these electric field-induced\ntorques can occur with the intrinsic as well as the exter-\nnally applied field.\nInterestingly, theformofthesecondterminthe TSOThas\nexactly the form that has been predicted using solely sym-\nmetry considerations [100]. It is proportional to the electric\nfield; usingE=\u001a\u0001jit can be rewritten in terms of the\ncurrent density, where \u001ais the resistivity tensor. The first\nterm in the SOT, M(E\u0001(r\u0002M)), has to our knowledge\nnot be reported before. For the 2nd-term, let us briefly con-\nsider a simple case where the electric field is directed along\nthez-axis,E=jE0j^z(see Fig. 1). This term then produces\na torque which is proportional to Ez\u0010\nMy@\n@x\u0000Mx@\n@y\u0011\nM.\nWhen the electric field is along a particular direction, this\ntorque accounts for the inhomogeneity of the magnetiza-\ntionalongtheplaneperpendiculartothat E-fielddirection.\nThis torque resembles somewhat the Gilbert local damping\nterm, but with the difference that this term is determined\nby thespatialderivative of the magnetization.\nC. Complete magnetization dynamics\nThenonrelativisticmagnetizationdynamicsproducesthe\ncurrent-induced torque and the relativistic magnetization\ndynamics gives the E-field induced torques. Having all the\nterms we write the full magnetization dynamics as\n@M\n@t=\u0000\r0M\u0002He\u000b+M\u0002\u0010\nA\u0001@M\n@t\u0011\n\u0000\rM\u0002Bopt\u00001\ne\u0000\njNR\u0001r\u0001\nM\n\u0000\u0018\neh\nM\u0000\nE\u0001(r\u0002M)\u0001\n+\u0000\n(M\u0002E)\u0001r\u0001\nMi\n:\n(25)\nThis expression gives the field-driven magnetization dy-\nnamics of LLG form in a system with a nonuniform mag-\nnetization. The first two terms on the right-hand side have\nthe usual physical interpretations. The third term defines\nthe optical spin-orbit torque on the magnetization, related\nto a relativistic contribution to the inverse Faraday effect,\nwhere the optomagnetic field is proportional to E\u0002E?i.e.,\nintensity for a general electric field [93]. The fourth term is\nthe Berger [8] adiabatic STT as has already been pointed\nout before, and finally, the last two terms are the SOTs.7\nAs a side remark, we observe that Eq. (25) cannot be ob-\ntained from the usual LLG equation through substitution\nof convective derivates, i.e.@\n@t!@\n@t+v\u0001r, to account for\nthe spin-polarized particle flow with velocity v.\nFor practical purposes, it is often needed to write the\nspin dynamics in the form of the Landau-Lifshitz equation,\nwhere the@M=@tis eliminated from the Gilbert damp-\ning. For a scalar Gilbert damping parameter, the trans-\nformation is rather straightforward, and one obtains the\nLL equation that is mathematical equivalent to the LLGequation. This transformation is no longer straightforward\nfor the case of tensorial Gilbert damping [44]. The Gilbert\ndamping tensor can be written as Aij=\u000b\u000eij+Iij+Aij,\nwhere the first term is the isotropic, Heisenberg-like damp-\ning [\u000b=1\n3Tr(A)], the second term is the anisotropic, Ising-\nlike damping [Tr (I) = 0] and Aij=\"ijkDkis the chiral\ndamping. For sake of brevity we define A=\u000b1+I. The\ntransformation of the full magnetization dynamics given in\nEq. (25) to the LL form leads to (see Appendix B for de-\ntails)\n\u0000\n\t2+F\u0001\n\u0001@M\n@t=\u0000\r0\tM\u0002He\u000b\u0000\r0M\u0002[A\u0001(M\u0002He\u000b)]| {z }\nLandau\u0000Lifshitz equations\u0000\r\tM\u0002Bopt\u0000\rM\u0002[A\u0001(M\u0002Bopt)]| {z }\noptical spin\u0000orbit torques\n\u0000\t\ne\u0000\njNR\u0001r\u0001\nM\u00001\neM\u0002\u0002\nA\u0001\u0000\njNR\u0001r\u0001\nM\u0003\n| {z }\ncurrent\u0000induced STT\u0000\t\u0018\neh\nM\u0000\nE\u0001(r\u0002M)\u0001\n+\u0000\n(M\u0002E)\u0001r\u0001\nMi\n| {z }\n\feld\u0000induced even SOT\n\u0000\u0018\neM\u0002n\nA\u0001h\nM\u0000\nE\u0001(r\u0002M)\u0001\n+\u0000\n(M\u0002E)\u0001r\u0001\nMio\n| {z }\n\feld\u0000induced odd SOT; (26)\nwith the general form of the tensor F=\u000b2M21\u0000(\u000b1+\nI)(M\u0001I\u0001M) + (M\u0001I)2\u0000M2I2+M(M\u0001I2)and\t =\n1 +M\u0001Dwhich serves as a normalization parameter. One\ncan recognize that the first two terms on the right-hand\nside of Eq. (26) are exactly the LL equation of spin motion\nwith tensorial damping parameter. The additional terms\ngive the influences of the various derived torques.\nA distinction between the LL and LLG magnetization\ndynamics (Eqs. (26) and (25)) that has not be highlighted\nin earlier investigations can now be recognized. We find\nthat the STT and SOT terms do not appear in exactly\nthe same form in both the LL and LLG equations of spin\ndynamics. Although previously often the STT and SOT\nterms have been added simply in the same way to the LL\nor LLG equation of motion, our derivation suggests that\nthe torque terms should appear in distinct forms.\nIV. DISCUSSION\nIn the past several expressions have been proposed for\nspin-transfer torques caused by spin-polarized currents in\nferromagnetic systems with inhomogeneous magnetization\ndistributions [7–9, 11, 36, 37, 43, 78–81, 100–103]. The\nderivations have been based on various considerations, such\nas the “s-d” model, total electro-magnetic free energy, spin-\nwave theory, spin conservation, symmetry considerations,\nand micromagnetic simulations. These investigations have\nidentified the adiabatic Berger STT, TSTT/(jNR\u0001r)M\n[8], which can be seen as the continuous limit of the Slon-\nczewski spin-transfer torque [7] for infinitely thin successive\nlayers.\nThe nonadiabatic STT term, M\u0002(jNR\u0001r)M(second\nterm on 2ndline of Eq. (26)), has been obtained previ-\nously, by Zhang and Li [10] and others [37, 43, 80, 81].The need for this term has been stressed by micromag-\nnetic simulations [80, 104]. These simulations introduced\na so-called nonadiabaticity parameter \fthat defines the\nrelative strengths of the nonadiabatic and adiabatic STTs,\nwhose value was estimated to be comparable to the Gilbert\ndamping\u000b[80]. Our Dirac theory derivation gives that \f\nis equal toA(using that \t = 1for isotropic damping).\nWe note however that these relations are derived here for\ntheintrinsic Gilbert damping and nonadiabaticity param-\neter. Additional damping effects, as e.g., nonlocal damping\nand spin pumping contributions could change the effective\nGilbert damping [54, 105–107], thus also making it differ-\nent from\f. In our case, the nonadiabaticity parameter can\nbe accounted for through the Gilbert damping parameter\nbecause the electronic structure expressions for the damp-\ning parameter contain the broadening that is the relaxation\ntime (see e.g., Refs. [44] and [48] for expressions).\nThe optical spin-orbit torques in Eq. (26) have the same\nform as those of the precession and damping in the stan-\ndard LL equation. However, the first optical spin-orbit\ntorque (odd in magnetization) is already of relativistic ori-\ngin through the relativistic optomagnetic field, and the sec-\nond optical spin-orbit torque (even in magnetization) is of\nhigher order in the relativistic effects because it contains\nthe optomagnetic field (which is relativistic by our defini-\ntion) as well as the Gilbert damping parameter.\nThe remaining terms in Eq. (26) (and Eq. (25)) arise\nas SOTs due to the spin-orbit coupling. It deserves to be\nnoted that for these SOTs a relativistic derivation on the\nbasis of the Dirac theory is indispensible. While identical\nforms of the nonrelativistic STTs can be derived within\nvarious theories, for the SOTs the precise form of the spin-\norbit related terms in the Hamiltonian is essential. One\nterm, the second term in the field-induced even SOT in\nEq. (26), was proposed earlier on the basis of symmetry8\nconsiderations [100].\nWe observe that there exist two different SOTs in Eq.\n(26): (i) the even-in-magnetization - even SOT, and (ii) the\nodd-in-magnetization - odd SOT. Both of them account for\nthe inhomogeneity of the magnetization. Previous investi-\ngations [14, 16, 19, 108–110] have identified two contribu-\ntions to the SOT, one having fieldlike form, being odd-in-\nM,Todd\nSOT=toddM\u0002(^z\u0002E)and one having dampinglike\nform, being even-in- M,Teven\nSOT=tevenM\u0002[(^z\u0002E)\u0002M],\nwhere ^zis the unit vector perpendicular to the interface\nor surface and the E-field or current is typically in the xy-\nplane. These SOTs are often discussed phenomenologically\nin the literature in the case of surfaces or interfaces having\nRashba-like couplings which are derived from zeroth order\nin the gradient of magnetization [100, 111, 112] where the\nRashba-like field is expressed as ^z\u0002E. However, these\nexpressions do not account for the magnetization inhomo-\ngeneity, even though they are field-induced. Our formula-\ntion has the advantage that it provides an explicit deriva-\ntion that leads to general expressions for the SOTs that\naccount for inhomogeneous magnetization.\nSeveral recent studies have investigated SOTs without\nassuming a special form of spin-orbit interaction. In par-\nticular,ab initio calculations of SOTs have been performed\nfor several magnetic metal – nonmagnetic metal bilayers\nand trilayers [113–116]. These investigations provide mi-\ncroscopic information on the size and origin of the SOT at\nthe interfaces, whereas our formulation provides the rela-\ntivistic functional dependence of the SOTs on the electric\nfield and inhomogeneous magnetization. In addition, our\nequations are directly suitable for micromagnetic simula-\ntions.\nIn our derivation, the first and third and fourth terms of\nthe second line in Eq. (26) cannot be treated as a simple\ntorque as the torque has to be perpendicular to the mag-\nnetization. The third and fourth terms in the second line\nand the torques in the last line of Eq. (26) provides the\nSOT in the adiabatic and nonadiabatic limit, respectively.\nAs these SOTs derive from the spin-orbit coupling, they\nexplain the spin-torques due to the Hall effect [100]. This\nmeans that the current and the E-field direction have to\nbe perpendicular to each other. It should also be observed\nthat the even-SOT torques depend on the chiral damping\nDthrough the parameter \t, therefore they could play an\nimportantroletomanipulatemagnetictextures. Theterms\nin the last line of Eq. (26) provide the nonadiabatic SOT as\nthe relaxation times are accounted for through the Gilbert\ndamping tensor, A. At the same time, these torques are\nodd in the magnetization. For a scalar Gilbert damping\nparameter, the first term vanishes, however, for tensorial\ndamping it does not vanish. The second term, on the other\nhand, can be seen as a nonadiabatic SOT which is odd in\nmagnetization. The odd SOTs do not depend on the chiral\ndamping \t. A fundamental difference between the here-\nderived even and odd SOTs is that the even SOT is first\norder relativistic (scales with 1=c2), however, the odd SOT\nis a higher order relativistic torque (scales with 1=c4).\nFinally, we make a comparison between the here-derived\nspin torques and the possible forms of spin torques pre-\ndicted on the basis of symmetry considerations in the in-\nvestigation by van der Bijl and Duine [100]. They consid-ered all possible torques that are first-order linear in the\ngradient of the magnetization and applied field E. In this\nway they identified (E\u0001r)MandM\u0002(E\u0001r)M, the\nequivalents of the current-induced STT torques. They fur-\nther obtained [(M\u0002E)\u0001r]M, the field-induced even SOT,\nwhich they, consistent with our relativistic derivation, as-\ncribe to the Hall current. They also proposed several other\npossible torque forms, which however are not given by our\nderivation. Conversely, our first term of the field-induced\neven SOT in Eq. (25) could not be obtained by them be-\ncause it does not have pure torque form. Apart from these\nspin torques, we have derived the optical spin-orbit torques\nwhich are higher-order than linear in the applied E-field\nand have thus not been considered by van der Bijl and\nDuine [100].\nV. CONCLUSIONS\nStarting from the Dirac-Kohn-Sham theory we have de-\nrived the full magnetization dynamics with the effects of\nspin-polarized currents included. We have shown that the\nnonrelativistic and relativistic contributions to the dynam-\nics can be separated and we have derived their distinct\nmagnetization dynamics contributions. Our results show\nthat the nonrelativistic magnetization dynamics consist of\nLarmor precession and the adiabatic STT. On the other\nhand, the relativistic magnetization dynamics is more com-\nplicated. The latter consists of relativistic counterparts of\nprecession, Gilbertdampingwithatensorialdissipationpa-\nrameter, optical spin-orbit torque, the nonadiabatic STT,\nand adiabatic and nonadiabatic SOTs that derive from cur-\nrent density due to spin-orbit coupling. These SOTs take\ninto account the inhomogeneity in the magnetization and\nareE-field induced.\nThe current-induced nonrelativistic torque explains the\nBerger STT within the adiabatic limit. The corresponding\ncurrentisunidirectionalandperpendiculartothesurfaceor\ninterfaces in a multilayered system, thus it can explain sev-\neral STT-induced phenomena. On the other hand the spin-\norbit current depends on the electric field and the magneti-\nzation, limiting the current to lie within a plane that is per-\npendicular to the nonrelativistic current. This relativistic\ncurrent gives a spin-orbit torque on the magnetization. We\nhave shown that the here-derived SOTs are field-induced\nand, in contrast to previous expressions, take into account\nthe magnetization inhomogeneity in a preferred plane (not\nnecessarily the surface or interfaces). The derived expres-\nsions therefore describe a more general form of the SOTs.\nIn the later part, once we have derived the full mag-\nnetization dynamics including the nonrelativistic and rela-\ntivistic contributions, we have transformed the LLG form\nof the magnetization dynamics to the LL form. By doing\nso, we re-derive the nonadiabatic torques as well. We ob-\ntain the Zhang-Li nonadiabatic STT, and show that the\nnonadiabaticity parameter \fstrongly depends on the in-\ntrinsic Gilbert damping parameter and scales linearly. We\nhave further obtained an optical spin torque, which, if high\nenough, can lead to a spin switching. Consequently, we\nhave derived the full magnetization dynamics with the ef-\nfectsofspin-polarizedcurrentsincludedwithinafundamen-9\ntal framework, and within this unified framework we have\nderived all the dynamical processes i.e., magnetization pre-\ncession, damping, optical spin-orbit torque, adiabatic and\nnonadiabatic spin-transfer torque, adiabatic and nonadia-\nbatic spin-orbit torques.\nACKNOWLEDGMENTS\nWe thank U. Nowak, U. Ritzmann and A. Aperis for\nvaluable discussions. This work has been supported bythe Swedish Research Council (VR), the Knut and Al-\nice Wallenberg Foundation (Contract No. 2015.0060), the\nEuropean Union’s Horizon2020 Research and Innovation\nProgramme under grant agreement No. 737709 (FEMTO-\nTERABYTE, http://www.physics.gu.se/femtoterabyte).\nAppendix A: Landau-Lifshitz spin dynamics for a general time-dependent field\nFor a general time-dependent magnetic field, we have recently shown that the traditional Landau-Lifshitz-Gilbert\nequation of motion is not valid anymore, and we derived an improved LLG equation that includes the field-derivative\ntorque, as given by the expression [44]\n@M\n@t=\u0000\r0M\u0002He\u000b+M\u0002\u0014\n\u0016A\u0001\u0012@M\n@t+@H\n@t\u0013\u0015\n; (A1)\nwhereM\u0002@H\n@tis the field-derivative torque, and the following equation for the damping tensor has been obtained\n\u0016Aij=\u0000e\u00160\n8m2c2X\nnhripj+pjrii\u0000hrnpn+pnrni\u000eij: (A2)\nNote that, due to the additional term @H=@tin the damping term, \u0016A6=A. Fori6=j, the different components of ri\nandpjcommute with each other and thus the matrix \u0016Aijis symmetric. Now we split the damping tensor in two parts:\n\u0016Aij=\u000b\u000eij+ \u0003ij, where\u000b=1\n3Tr(\u0016Aij). Therefore the trace of the symmetric matrix \u0003ijwill be zero. Equation (A1) is\nthen rewritten as\n@M\n@t=\u0000\r0M\u0002He\u000b+\u000bM\u0002\u0012@M\n@t+@H\n@t\u0013\n+M\u0002\u0014\n\u0003\u0001\u0012@M\n@t+@H\n@t\u0013\u0015\n: (A3)\nTo evaluate the last two terms in Eq. (A3), we perform a suitable vector multiplication,\nM\u0002\u0012@M\n@t+@H\n@t\u0013\n=\u0000\r0M\u0002(M\u0002He\u000b) +\u000bM\u0012\nM\u0001@H\n@t\u0013\n\u0000\u000bM2\u0012@M\n@t+@H\n@t\u0013\n+M\u0014\nM\u0001\u001a\n\u0003\u0001\u0012@M\n@t+@H\n@t\u0013\u001b\u0015\n\u0000M2\u0014\n\u0003\u0001\u0012@M\n@t+@H\n@t\u0013\u0015\n+M\u0002@H\n@t;(A4)\nand similarly,\nM\u0002\u0014\n\u0003\u0001\u0012@M\n@t+@H\n@t\u0013\u0015\n=\u0000\r0M\u0002[\u0003\u0001(M\u0002He\u000b)] +\u000bM\u0002\u0014\n\u0003\u0001\u001a\nM\u0002\u0012@M\n@t+@H\n@t\u0013\u001b\u0015\n+M\u0002\u0003\u0001\u001a\nM\u0002\u0014\n\u0003\u0001\u0012@M\n@t+@H\n@t\u0013\u0015\u001b\n: (A5)\nNow, following the same procedure as described in Ref. [44] we arrive at\n( 1+ G)\u0001\u0012@M\n@t+@H\n@t\u0013\n=\u0000\r0(M\u0002He\u000b)\u0000\r0M\u0002[(\u000b 1+ \u0003)\u0001(M\u0002He\u000b)]\n+\u0002\n\u000b2M 1\u0000\u000b(M\u0001\u0003)\u0003\u0012\nM\u0001@H\n@t\u0013\n+\u000bM\u0002@H\n@t+@H\n@t; (A6)\nwhere the newly derived tensor Gdepends on the magnetization and is represented as (see e.g., Eq. (B11) of Ref. [44])\nG=\u000b2M21\u0000(\u000b 1+ \u0003)[M\u0001\u0003\u0001M]\u0000M2\u00032+ (M\u0001\u0003)2+M(M\u0001\u00032): (A7)\nEquation (A6) is the Landau-Lifshitz equation of spin motion for a general time-dependent field. Note that if H= 0, the\ndynamics collapses to the original LL equation. Although in general the damping is tensorial, it is of interest (e.g., for10\nnumerical simulations) to derive a compact form of the LL equation of motion for a general field yet for isotropic, scalar\ndamping. Thus, taking \u0003 = 0, the concrete form of the LL equation of spin motion with applied general field and scalar\ndamping will be\n\u0000\n1 +\u000b2M2\u0001@M\n@t=\u0000\r0M\u0002\u0012\nHe\u000b\u0000\u000b\n\r0@H\n@t\u0013\n\u0000\r0\u000bM\u0002\u0014\nM\u0002\u0012\nHe\u000b\u0000\u000b\n\r0@H\n@t\u0013\u0015\n; (A8)\nwhere the field-derivative torque appears in the spin precession and in the spin dissipation terms.\nAppendix B: Landau-Lifshitz equation with relativistic and nonrelativistic spin torques\nIn Sec. IIIC we derived the full LLG magnetization dynamics including the nonrelativistic and relativistic spin-current\ncontributions; the equation of motion was found to be given by\n@M\n@t=\u0000\r0M\u0002He\u000b+M\u0002\u0010\nA\u0001@M\n@t\u0011\n\u0000\rM\u0002Bopt\u00001\ne\u0000\njNR\u0001r\u0001\nM+\u0018\neh\nM\u0000\nE\u0001(r\u0002M)\u0001\n+\u0000\n(M\u0002E)\u0001r\u0001\nMi\n:\n(B1)\nIn its most general form the Gilbert damping Aijis a tensor. This tensor can be decomposed as Aij=\u000b\u000eij+Iij+Aij,\nwhere the first two terms are its symmetric part and the third term is its antisymmetric part. The symmetric part can be\ndecomposed in the isotropic, Heisenberg-like damping [ \u000b=1\n3Tr(Asym)], and an anisotropic, Ising-like damping [Tr (I) = 0].\nHowever, the antisymmetric part leads to a chiral, Dzyaloshinskii-Moriya-like damping represented by a corresponding\nvectorD, asAij=\"ijkDk, with\"the Levi-Civita tensor. The spin dissipation dynamics can then be expressed as\nM\u0002\u0010\nA\u0001@M\n@t\u0011\n=\u000bM\u0002@M\n@t+M\u0002\u0014\nI\u0001@M\n@t\u0015\n+M\u0002\u0014\nD\u0002@M\n@t\u0015\n: (B2)\nAs shown previously [44], the Dzyaloshinskii-Moriya-like damping contributes to the renormalization of the dynamics and\ntherefore, the spin dynamics in Eq. (B1) can be written as\n\t@M\n@t=\u0000\r0M\u0002He\u000b+\u000bM\u0002@M\n@t+M\u0002\u0014\nI\u0001@M\n@t\u0015\n\u0000\rM\u0002Bopt\n\u00001\ne\u0000\njNR\u0001r\u0001\nM\u0000\u0018\neh\nM\u0000\nE\u0001(r\u0002M)\u0001\n+\u0000\n(M\u0002E)\u0001r\u0001\nMi\n; (B3)\nwhere we define \t = 1 +M\u0001D.\nNext, our objective is to obtain the Landau-Lifshitz form of spin dynamics with spin-current torques. To this end we\nfollow the usual procedure to derive the LL spin dynamics, i.e., we have to calculate the second term of Eq. (B3). To do\nso, we take a cross product with the magnetization on the both sides of the last equation,\n\tM\u0002@M\n@t=\u0000\r0M\u0002(M\u0002He\u000b)\u0000\u000bM2@M\n@t\u0000M2\u0014\nI\u0001@M\n@t\u0015\n+M\u0012\nM\u0001\u0014\nI\u0001@M\n@t\u0015\u0013\n\u0000\rM\u0002(M\u0002Bopt)\u00001\neM\u0002\u0002\u0000\njNR\u0001r\u0001\nM\u0003\n\u0000\u0018\neM\u0002h\nM\u0000\nE\u0001(r\u0002M)\u0001\n+\u0000\n(M\u0002E)\u0001r\u0001\nMi\n:(B4)\nHowever, the calculation of the third term on the right-hand side of Eq. (B3) involves much more. We calculate this term\nas,\n\tM\u0002\u0014\nI\u0001@M\n@t\u0015\n=\u0000\r0M\u0002[I\u0001(M\u0002He\u000b)] +\u000bM2\u0014\nI\u0001@M\n@t\u0015\n+\u000b(M\u0001I\u0001M)@M\n@t\u0000\u000bM\u0014\nM\u0001\u0012\nI\u0001@M\n@t\u0013\u0015\n+ (M\u0001I\u0001M)\u0012\nI\u0001@M\n@t\u0013\n\u0000(M\u0001I)\u0014\nM\u0001\u0012\nI\u0001@M\n@t\u0013\u0015\n+M2\u0012\nI2\u0001@M\n@t\u0013\n\u0000M\u0014\nM\u0001\u0012\nI2\u0001@M\n@t\u0013\u0015\n\u0000\rM\u0002[I\u0001(M\u0002Bopt)]\u00001\neM\u0002\u0002\nI\u0001\u0000\njNR\u0001r\u0001\nM\u0003\n\u0000\u0018\neM\u0002n\nI\u0001h\nM\u0000\nE\u0001(r\u0002M)\u0001\n+\u0000\n(M\u0002E)\u0001r\u0001\nMio\n: (B5)\nNext, we substitute these last two equations back into Eq. (B3) and obtain the full magnetization dynamics with spin-\ncurrent torques in LL form that can be written as\n\u0000\n\t2+F\u0001\n\u0001@M\n@t=\u0000\r0\tM\u0002He\u000b\u0000\r0M\u0002[(\u000b1+I)\u0001(M\u0002He\u000b)]\u0000\r\tM\u0002Bopt\u0000\rM\u0002[(\u000b1+I)\u0001(M\u0002Bopt)]\n\u0000\t\ne\u0000\njNR\u0001r\u0001\nM\u00001\neM\u0002\u0002\n(\u000b1+I)\u0001\u0000\njNR\u0001r\u0001\nM\u0003\n\u0000\t\u0018\neh\nM\u0000\nE\u0001(r\u0002M)\u0001\n+\u0000\n(M\u0002E)\u0001r\u0001\nMi\n\u0000\u0018\neM\u0002n\n(\u000b1+I)\u0001h\nM\u0000\nE\u0001(r\u0002M)\u0001\n+\u0000\n(M\u0002E)\u0001r\u0001\nMio\n; (B6)11\nwhere the tensor Fis given by F=\u000b2M21\u0000(\u000b1+I)(M\u0001I\u0001M) + (M\u0001I)2\u0000M2I2+M(M\u0001I2), while considering the fact\nthat Tr(Iij) = 0here. Lastly, we note that if one for sake of simplicity considers only the scalar Gilbert damping parameter\n(i.e., I= 0,D= 0), the resulting spin dynamics resembles the LL equation of motion but including the additional current\nand field-induced torque terms,\n(1 +\u000b2M2)@M\n@t=\u0000\r0M\u0002He\u000b\u0000\u000b\r0M\u0002(M\u0002He\u000b)\u0000\rM\u0002Bopt\u0000\r\u000bM\u0002(M\u0002Bopt)\n\u00001\ne\u0000\njNR\u0001r\u0001\nM\u0000\u000b\neM\u0002\u0002\u0000\njNR\u0001r\u0001\nM\u0003\n\u0000\u0018\neh\nM\u0000\nE\u0001(r\u0002M)\u0001\n+\u0000\n(M\u0002E)\u0001r\u0001\nMi\n\u0000\u000b\u0018\neM\u0002\u0002\u0000\n(M\u0002E)\u0001r\u0001\nM\u0003\n:(B7)\n[1] L. D. Landau and E. M. Lifshitz, Phys. Z. Sowjetunion 8,\n101 (1935).\n[2] T.L.Gilbert, Ph.D.thesis, IllinoisInstituteofTechnology,\nChicago, 1956.\n[3] T. L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004).\n[4] S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M.\nDaughton, S. von Molnár, M. L. Roukes, A. Y. Chtchelka-\nnova, and D. M. Treger, Science 294, 1488 (2001).\n[5] A. D. Kent and D. C. Worledge, Nat. Nanotechn. 10, 187\n(2015).\n[6] T. Kuschel and G. Reiss, Nat. Nanotechn. 10, 22 (2015).\n[7] J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996).\n[8] L. Berger, Phys. Rev. B 54, 9353 (1996).\n[9] M. D. Stiles and A. Zangwill, Phys. Rev. B 66, 014407\n(2002).\n[10] S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 (2004).\n[11] D. C. Ralph and M. D. Stiles, J. Magn. Magn. Mater. 320,\n1190 (2008).\n[12] T. H. Kim, P. Grünberg, S. H. Han, and B. Cho, Sci. Rep.\n6, 35077 (2016).\n[13] R. Cheng, M. W. Daniels, J.-G. Zhu, and D. Xiao, Phys.\nRev. B91, 064423 (2015).\n[14] P. Gambardella and I. M. Miron, Phil. Trans. Roy. Soc.\nLondon A 369, 3175 (2011).\n[15] K. Ando, S. Takahashi, K. Harii, K. Sasage, J. Ieda,\nS. Maekawa, and E. Saitoh, Phys. Rev. Lett. 101, 036601\n(2008).\n[16] I. M. Miron, K. Garello, G. Gaudin, P.-J. Zermatten,\nM. V. Costache, S. Auffret, S. Bandiera, B. Rodmacq,\nA. Schuhl, and P. Gambardella, Nature 476, 189 (2011).\n[17] L. Liu, C.-F. Pai, Y. Li, H. W. Tseng, D. C. Ralph, and\nR. A. Buhrman, Science 336, 555 (2012).\n[18] L. Liu, O. J. Lee, T. J. Gudmundsen, D. C. Ralph, and\nR. A. Buhrman, Phys. Rev. Lett. 109, 096602 (2012).\n[19] K. Garello, I. M. Miron, C. O. Avci, F. Freimuth,\nY.Mokrousov, S.Blügel, S.Auffret, O.Boulle, G.Gaudin,\nand P. Gambardella, Nat. Nanotechn. 8, 587 (2013).\n[20] G. Yu, P. Upadhyaya, Y. Fan, J. G. Alzate, W. Jiang,\nK.L.Wong, S.Takei, S.A.Bender, L.-T.Chang, Y.Jiang,\nM. Lang, J. Tang, Y. Wang, Y. Tserkovnyak, P. K. Amiri,\nand K. L. Wang, Nat. Nanotechn. 9, 548 (2014).\n[21] C.K.Safeer, E.Jué, A.Lopez, L.Buda-Prejbeanu, S.Auf-\nfret, S. Pizzini, O. Boulle, I. M. Miron, and G. Gaudin,\nNat. Nanotechn. 11, 143 (2016).\n[22] M. Baumgartner, K. Garello, J. Mendil, C. O. Avci,\nE. Grimaldi, C. Murer, J. Feng, M. Gabureac, C. Stamm,\nY.Acremann, S.Finizio, S.Wintz, J.Raabe, andP.Gam-\nbardella, Nat. Nanotechn. 12, 980 (2017).[23] Y. Fan, P. Upadhyaya, X. Kou, M. Lang, S. Takei,\nZ. Wang, J. Tang, L. He, L.-t. Chang, M. Montazeri,\nG. Yu, W. Jiang, T. Nie, R. N. Schwartz, Y. Tserkovnyak,\nand K. L. Wang, Nat. Mater. 13, 699 (2014).\n[24] P. Wadley, B. Howells, J. Železný, C. Andrews, V. Hills,\nR. P. Campion, V. Novák, K. Olejník, F. Maccherozzi,\nS. S. Dhesi, S. Y. Martin, T. Wagner, J. Wunderlich,\nF. Freimuth, Y. Mokrousov, J. Kuneš, J. S. Chauhan,\nM. J. Grzybowski, A. W. Rushforth, K. W. Edmonds,\nB. L. Gallagher, and T. Jungwirth, Science 351, 587\n(2016).\n[25] S. Fukami, C. Zhang, S. Duttagupta, A. Kurenkov, and\nH. Ohno, Nat. Mater. 15, 535 (2016).\n[26] K. Garello, C. O. Avci, I. M. Miron, M. Baumgartner,\nA. Ghosh, S. Auffret, O. Boulle, G. Gaudin, and P. Gam-\nbardella, Appl. Phys. Lett. 105, 212402 (2014).\n[27] T. J. Huisman, R. V. Mikhaylovskiy, J. D. Costa,\nF. Freimuth, E. Paz, J. Ventura, P. P. Freitas, S. Blügel,\nY. Mokrousov, T. Rasing, and A. V. Kimel, Nat. Nan-\notechn.11, 455 (2016).\n[28] W. F. Brown, Micromagnetics (Wiley, New York, 1963).\n[29] U. Nowak, “Classical spin models,” in Handbook of Mag-\nnetism and Advanced Magnetic Materials , edited by\nH. Kronmüller and S. Parkin (J. Wiley, New York, 2007).\n[30] R. F. L. Evans, W. J. Fan, P. Chureemart, T. A. Ostler,\nM. O. A. Ellis, and R. W. Chantrell, J. Phys.: Condens.\nMatter26, 103202 (2014).\n[31] M. O. A. Ellis, R. F. L. Evans, T. A. Ostler, J. Barker,\nU. Atxitia, O. Chubykalo-Fesenko, and R. W. Chantrell,\nLow Temp. Phys. 41, 705 (2015).\n[32] V. G. Bar’yakhtar and B. A. Ivanov, Low Temp. Phys. 41,\n663 (2015).\n[33] S. Zhang, P. M. Levy, and A. Fert, Phys. Rev. Lett. 88,\n236601 (2002).\n[34] A. Shpiro, P. M. Levy, and S. Zhang, Phys. Rev. B 67,\n104430 (2003).\n[35] G. Tatara and P. Entel, Phys. Rev. B 78, 064429 (2008).\n[36] A. Brataas, A. D. Kent, and H. Ohno, Nat. Mater. 11,\n372 (2012).\n[37] K.-W. Kim, K.-J. Lee, H.-W. Lee, and M. D. Stiles, Phys.\nRev. B92, 224426 (2015).\n[38] D. Culcer, M. E. Lucassen, R. A. Duine, and R. Winkler,\nPhys. Rev. B 79, 155208 (2009).\n[39] K. Nomura and D. Kurebayashi, Phys. Rev. Lett. 115,\n127201 (2015).\n[40] A. Manchon, H. C. Koo, J. Nitta, S. M. Frolov, and R. A.\nDuine, Nat. Mater. 14, 871 (2015).\n[41] A. Manchon and S. Zhang, Phys. Rev. B 78, 21240512\n(2008).\n[42] A. Manchon and S. Zhang, Phys. Rev. B 79, 094422\n(2009).\n[43] K.-W. Kim, H.-W. Lee, K.-J. Lee, and M. D. Stiles, Phys.\nRev. Lett. 111, 216601 (2013).\n[44] R. Mondal, M. Berritta, and P. M. Oppeneer, Phys. Rev.\nB94, 144419 (2016).\n[45] M.-C.Ciornei, J.M.Rubí, andJ.-E.Wegrowe,Phys.Rev.\nB83, 020410 (2011).\n[46] M. Fähnle and C. Illg, J. Phys.: Condens. Matter 23,\n493201 (2011).\n[47] D. Böttcher and J. Henk, Phys. Rev. B 86, 020404 (2012).\n[48] R. Mondal, M. Berritta, A. K. Nandy, and P. M. Oppe-\nneer, Phys. Rev. B 96, 024425 (2017).\n[49] R. Mondal, M. Berritta, and P. M. Oppeneer, J. Phys.:\nCondens. Matter 30, 265801 (2018).\n[50] V. G. Bar’yakhtar, Sov. Phys. JETP 60, 863 (1984).\n[51] V. G. Bar’yakhtar, V. I. Butrim, and B. A. Ivanov, JETP\nLett.98, 289 (2013).\n[52] D. A. Garanin, Physica A 172, 470 (1991).\n[53] O. Chubykalo-Fesenko, U. Nowak, R. W. Chantrell, and\nD. Garanin, Phys. Rev. B 74, 094436 (2006).\n[54] Y. Tserkovnyak, E. M. Hankiewicz, and G. Vignale, Phys.\nRev. B79, 094415 (2009).\n[55] Y. Li and W. E. Bailey, Phys. Rev. Lett. 116, 117602\n(2016).\n[56] A. H. MacDonald and S. H. Vosko, J. Phys. C: Solid State\nPhys.12, 2977 (1979).\n[57] H. Eschrig and V. D. P. Servedio, J. Comput. Chem. 20,\n23 (1999).\n[58] W. Greiner, Relativistic quantum mechanics. Wave equa-\ntions(Springer, Berlin, 2000).\n[59] T.Kraft, P.M.Oppeneer, V.N.Antonov, andH.Eschrig,\nPhys. Rev. B 52, 3561 (1995).\n[60] P. Strange, Relativistic Quantum Mechanics: With Appli-\ncations in Condensed Matter and Atomic Physics (Cam-\nbridge University Press, 1998).\n[61] R. Mondal, M. Berritta, K. Carva, and P. M. Oppeneer,\nPhys. Rev. B 91, 174415 (2015).\n[62] L. L. Foldy and S. A. Wouthuysen, Phys. Rev. 78, 29\n(1950).\n[63] A. J. Silenko, Phys. Rev. A 93, 022108 (2016).\n[64] R. Mondal, “Relativisitic theory of laser-induced magneti-\nzation dynamics”, Ph.D. thesis, Uppsala University, Upp-\nsala, Sweden (Oct. 2017).\n[65] Note that when the momentum operator and a (vector)\nfunction are enclosed in round brackets the momentum\noperator acts only on this function.\n[66] R. M. White, Quantum Theory of Magnetism: Magnetic\nProperties of Materials (Springer, Heidelberg, 2007).\n[67] S. Y. Cho, Phys. Rev. B 77, 245326 (2008).\n[68] K. Capelle, G. Vignale, and B. L. Györffy, Phys. Rev.\nLett.87, 206403 (2001).\n[69] J. Shi, P. Zhang, D. Xiao, and Q. Niu, Phys. Rev. Lett.\n96, 076604 (2006).\n[70] Q.-f. Sun and X. C. Xie, Phys. Rev. B 72, 245305 (2005).\n[71] Z. An, F. Q. Liu, Y. Lin, and C. Liu, Sci. Rep. 2, 388\n(2012).\n[72] L. D. Landau and E. M. Lifshitz, Quantum Mechanics:\nNon-Relativistic Theory , 3rd ed., Vol. 3 (Butterworth-\nHeinemann, Oxford, 1981).\n[73] Y. Hinschberger and P.-A. Hervieux, Phys. Rev. B 88,\n134413 (2013).\n[74] M. Nowakowski, Am. J. Phys. 67, 916 (1999).\n[75] J. D. Jackson, Classical electrodynamics , 3rd ed. (Wiley,\nNew York, NY, 1999).\n[76] M. Lakshmanan, Phil. Trans. Roy. Soc. London A 369,1280 (2011).\n[77] S. V. Halilov, H. Eschrig, A. Y. Perlov, and P. M. Oppe-\nneer, Phys. Rev. B 58, 293 (1998).\n[78] J. P. Ansermet, IEEE Trans. Magn. 40, 358 (2004).\n[79] Z. Li and S. Zhang, Phys. Rev. Lett. 92, 207203 (2004).\n[80] A. Thiaville, Y. Nakatani, J. Miltat, and Y. Suzuki, EPL\n(Europhysics Letters) 69, 990 (2005).\n[81] R. Cheng and Q. Niu, Phys. Rev. B 88, 024422 (2013).\n[82] R. Mondal, M. Berritta, C. Paillard, S. Singh, B. Dkhil,\nP. M. Oppeneer, and L. Bellaiche, Phys. Rev. B 92,\n100402(R) (2015).\n[83] A. Husmann and L. J. Singh, Phys. Rev. B 73, 172417\n(2006).\n[84] E. Roman, Y. Mokrousov, and I. Souza, Phys. Rev. Lett.\n103, 097203 (2009).\n[85] N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, and\nN. P. Ong, Rev. Mod. Phys. 82, 1539 (2010).\n[86] L. Bellaiche, W. Ren, and S. Singh, Phys. Rev. B 88,\n161102 (2013).\n[87] V. P. Amin and M. D. Stiles, Phys. Rev. B 94, 104419\n(2016).\n[88] V. P. Amin and M. D. Stiles, Phys. Rev. B 94, 104420\n(2016).\n[89] Thebasicassumptionofnorm-conservingspinmotionthat\nunderlies the LLG dynamics implies that non-transversal\nspin motions, as expressed e.g. by the Landau-Lifshitz-\nBar’yakhtar equation [50] or Landau-Lifshitz-Bloch equa-\ntion [52, 53], cannot be derived here.\n[90] M. C. Hickey and J. S. Moodera, Phys. Rev. Lett. 102,\n137601 (2009).\n[91] L. Allen, S. M. Barnett, and M. J. Padgett, Optical An-\ngular Momentum (Institute of Physics, Bristol, 2003).\n[92] S. M. Barnett, J. Mod. Opt. 57, 1339 (2010).\n[93] R. Mondal, M. Berritta, and P. M. Oppeneer, J. Phys.:\nCondens. Matter 29, 194002 (2017).\n[94] C. D. Stanciu, F. Hansteen, A. V. Kimel, A. Kirilyuk,\nA. Tsukamoto, A. Itoh, and T. Rasing, Phys. Rev. Lett.\n99, 047601 (2007).\n[95] C.-H. Lambert, S. Mangin, B. S. D. C. S. Varaprasad,\nY. K. Takahashi, M. Hehn, M. Cinchetti, G. Malinowski,\nK. Hono, Y. Fainman, M. Aeschlimann, and E. E. Fuller-\nton, Science 345, 1403 (2014).\n[96] M. Berritta, R. Mondal, K. Carva, and P. M. Oppeneer,\nPhys. Rev. Lett. 117, 137203 (2016).\n[97] P. Němec, E. Rozkotová, N. Tesařová, F. Trojánek,\nE. De Ranieri, K. Olejník, J. Zemen, V. Novák, M. Cukr,\nP. Malý, and T. Jungwirth, Nat. Phys. 8, 411 (2012).\n[98] A. J. Ramsay, P. E. Roy, J. A. Haigh, R. M. Otxoa, A. C.\nIrvine, T. Janda, R. P. Campion, B. L. Gallagher, and\nJ. Wunderlich, Phys. Rev. Lett. 114, 067202 (2015).\n[99] G.-M. Choi, A. Schleife, and D. G. Cahill, Nat. Commun.\n8, 15085 (2017).\n[100] E. van der Bijl and R. A. Duine, Phys. Rev. B 86, 094406\n(2012).\n[101] Y. B. Bazaliy, B. A. Jones, and S.-C. Zhang, Phys. Rev.\nB57, R3213 (1998).\n[102] J. Fernández-Rossier, M. Braun, A. S. Núñez, and A. H.\nMacDonald, Phys. Rev. B 69, 174412 (2004).\n[103] G. Tatara and H. Kohno, Phys. Rev. Lett. 92, 086601\n(2004).\n[104] G. S. D. Beach, M. Tsoi, and J. L. Erskine, J. Magn.\nMagn. Mater. 320, 1272 (2008).\n[105] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys.\nRev. B66, 224403 (2002).\n[106] H. T. Nembach, J. M. Shaw, C. T. Boone, and T. J. Silva,\nPhys. Rev. Lett. 110, 117201 (2013).\n[107] W. Wang, M. Dvornik, M.-A. Bisotti, D. Chernyshenko,13\nM. Beg, M. Albert, A. Vansteenkiste, B. V. Waeyenberge,\nA. N. Kuchko, V. V. Kruglyak, and H. Fangohr, Phys.\nRev. B92, 054430 (2015).\n[108] P. M. Haney, H.-W. Lee, K.-J. Lee, A. Manchon, and\nM. D. Stiles, Phys. Rev. B 87, 174411 (2013).\n[109] H. Kurebayashi, J. Sinova, D. Fang, A. C. Irvine, T. D.\nSkinner, J. Wunderlich, V. Novák, R. P. Campion, B. L.\nGallagher, E. K. Vehstedt, L. P. Zarbo, K. Výborný, A. J.\nFerguson, and T. Jungwirth, Nat. Nanotechn. 9, 211\n(2014).\n[110] K.-S. Lee, D. Go, A. Manchon, P. M. Haney, M. D. Stiles,\nH.-W.Lee, andK.-J.Lee,Phys.Rev.B 91,144401(2015).[111] Y. A. Bychkov and E. I. Rashba, Sov. Phys. - JETP Lett.\n39, 78 (1984).\n[112] V. M. Edelstein, Solid State Commun. 73, 233 (1990).\n[113] F. Freimuth, S. Blügel, and Y. Mokrousov, Phys. Rev. B\n90, 174423 (2014).\n[114] F. Freimuth, S. Blügel, and Y. Mokrousov, Phys. Rev. B\n92, 064415 (2015).\n[115] S. Wimmer, K. Chadova, M. Seemann, D. Ködderitzsch,\nand H. Ebert, Phys. Rev. B 94, 054415 (2016).\n[116] F. Mahfouzi and N. Kioussis, Phys. Rev. B 97, 224426\n(2018)."    },    {        "title": "2201.01025v1.Quantum_dynamics_of_Gaudin_magnets.pdf",        "content": "Quantum dynamics of Gaudin magnets\nWen-Bin He,1, 2, 3Stefano Chesi,1, 4, 3,\u0003H.-Q. Lin,1, 4and Xi-Wen Guan2, 5, 6,y\n1Beijing Computational Science Research Center, Beijing 100193, China\n2State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics,\nWuhan Institute of Physics and Mathematics, APM, Chinese Academy of Sciences, Wuhan 430071, China\n3The Abdus Salam International Center for Theoretical Physics, Strada Costiera 11, 34151 Trieste, Italy.\n4Department of Physics, Beijing Normal University, Beijing 100875, China\n5NSFC-SPTP Peng Huanwu Center for Fundamental Theory, Xi’an 710127, China\n6Department of Theoretical Physics, Research School of Physics and Engineering,\nAustralian National University, Canberra ACT 0200, Australia\n(Dated: January 5, 2022)\nQuantum dynamics of many-body systems is a fascinating and significant subject for both theory and experi-\nment. The question of how an isolated many-body system evolves to its steady state after a sudden perturbation\nor quench still remains challenging. In this paper, using the Bethe ansatz wave function, we study the quan-\ntum dynamics of an inhomogeneous Gaudin magnet. We derive explicit analytical expressions for various local\ndynamic quantities with an arbitrary number of flipped bath spins, such as: the spin distribution function, the\nspin-spin correlation function, and the Loschmidt echo. We also numerically study the relaxation behavior of\nthese dynamic properties, gaining considerable insight into coherence and entanglement between the central\nspin and the bath. In particular, we find that the spin-spin correlations relax to their steady value via a nearly\nlogarithmic scaling, whereas the Loschmidt echo shows an exponential relaxation to its steady value. Our results\nadvance the understanding of relaxation dynamics and quantum correlations of long-range interacting models\nof Gaudin type.\nAlthough mature paradigms, like the celebrated Fermi liq-\nuid theory [1–3] and Luttinger liquid theory [4, 5], enable us\nto understand well a wide class of many-body systems in the\nstationary state, their dynamical evolution is much less un-\nderstood. In fact, quantum dynamics of many-body systems\nalways presents itself with formidable challenges, due to the\ndi\u000eculty of deriving eigenfunctions analytically and the ex-\nponentially growing complexity of numerics. Despite these\ntheoretical di \u000eculties, significant progresses have been made\nin the last decade, which greatly improves our understanding\nof dynamical properties of many-body systems. For exam-\nple, the Kibble-Zurek mechanism (KZM) [6] reveals power-\nlaw relations between the typical length scale \u0018of defect do-\nmains, the relaxation time \u001c, and the rate of change of the\ndriving parameter \u0015, namely\u0018\u0018\u0015\u0000\u0017and\u001c\u0018\u0015\u0000z\u0017, which has\nbeen observed in ion Coulomb crystals [7]. Thermalization\nof isolated many-body systems is also a di \u000ecult topic. The\neigenstate thermalization hypothesis (ETH) [8, 9] asserts that,\nfor a closed systems evolving to the steady state, the diago-\nnal and micro-canonical ensembles are equivalent. In recent\nexperiments with ultracold atoms [10, 11], T. Kinoshita et. al\nshowed that Bose gases in one-dimensional traps do not ther-\nmalize, even after thousands of collisions [12]. The ETH can\nbe violated by integrable models [13], due to the existence of\ninfinitely many conserved charges, or in the presence of many-\nbody localization [14]. The study of dynamics is of crucial im-\nportance for many-body quantum systems out of equilibrium,\nas it provides important clues on if and how their equilibrium\nstate can be reached.\n\u0003Electronic address: stefano.chesi@csrc.ac.cn\nyxwe105@wipm.ac.cnTherefore, while thermalization usually concerns the\nsteady-state properties of many-body systems after a long-\ntime evolution, an interesting question is on how the many-\nbody system evolves at intermediate times. For a system with\na short-range interaction, like the Heisenberg spin chain and\nthe Lieb-Lininger model, light-cone dynamics dominates the\nrelaxation of local observables to steady-state values, such as\nspin polarization and correlation functions [15]. However, the\nrelaxation process of local and global quantities to the steady\nstate is less explored in models with long-range interactions.\nIn this paper, we use Gaudin magnets, i.e. the inhomoge-\nneous central spin models, to study the relaxation dynamics\nof many-body systems. Such magnets with long-range inter-\nactions may describe the decoherence of a spin-qubit due to\nthe interaction with surrounding spins, for example an elec-\ntron interacting with nuclear spins in quantum dots or defect\ncenters [16, 17]. Quantum dynamics of the Gaudin model\nhas been extensively studied with a variety of methods as, for\nexample, exact diagonalization and quantum many-body ex-\npansions [17–22], the Bethe ansatz method for the polarized\nbath with one flipped bath spin [23], hybrid methods combin-\ning the Bethe ansatz and Monte Carlo sampling [24, 25], the\nDMRG method and semiclassical approaches [26–29].\nRecent work described in detail the collapse and revival\ndynamics of the homogeneous central spin model [30], and\nthe partition of eigenstates between entangled and separable\nstates [31, 32].\nWhile an early description of Gaudin magnets using the\nBethe ansatz technique considered one flipped bath spin, i.e.\nM=1 [23], here we study the time evolution of the cen-\ntral spin problem with an arbitrary number of flipped spins\nM, starting from an initial state j\b0i=s\u0000\na1\u0001\u0001\u0001s\u0000\naMj *i =\nja1;\u0001\u0001\u0001;aMi, wherej*iis the fully polarized state and ai=\n0;1;2;:::Nlabel the positions of spins in the system. ThearXiv:2201.01025v1  [cond-mat.str-el]  4 Jan 20222\nquantum dynamics of the model with arbitrary number of\nflipped spins has been already studied through Bethe ansatz\n[24, 25], as well as various other theoretical approaches, lead-\ning to important insights such as a description of quantum\ndecoherence by the Chebyshev expansion method [33], non-\nMarkovian dynamics [20], persistent spin correlations [34],\netc. Here we study the time evolutions of several interest-\ning quantities analytically and numerically: the spin distri-\nbution function, the spin-spin correlation function, and the\nLoschmidt echo. Evaluating these, we find that the time de-\npendence of the spin polarization function reveals a break-\ndown of thermalization, such that the steady-state of the\nGaudin magnets cannot be described by an equilibrium en-\nsemble. Instead, the relaxation to steady-state values of the\nspin-spin correlation function suggests a logarithmic decay,\nand we find that at intermediate times the Loschmidt echo re-\nlaxes exponentially to its steady-state value, using both the\nBethe ansatz solution and Matrix Product States (MPS) nu-\nmerical approach.\nI. MODEL AND BETHE ANSATZ EQUATIONS\nThe Gaudin magnet describes a central spin at position “0”\ncoupled to bath spins through long-range interactions. The\nHamiltonian reads\nH=Bsz\n0+2NX\nj=1Ajs0\u0001sj; (1)\nwhere the bath spins are labelled with 1 !N. For con-\nvenience, we parametrize the magnetic field as B=\u00002=g.\nFurthermore, we write Aj=1=(\u000f0\u0000\u000fj) for the inhomoge-\nneous interaction strength determined in experiments. Set-\nting\u000f0=0, the\u000fj(with j=1;:::; N) correspond to the\nenergy levels of a discrete BCS model associated with H,\nwhich can be constructed from the conserved quantities Hj=\nBsz\nj+P\nk,jsj\u0001sk=(\u000fj\u0000\u000fk) [35–37]. Following the algebraic\nBethe ansatz method, and considering a subspace where M\nspins are flipped-down with respect to the reference state j*i,\nthe eigenstates are given below [35, 36]:\nj\u00171;\u0001\u0001\u0001;\u0017Mi=MY\n\u000b=1B\u0017\u000bj*i=MY\n\u000b=1NX\nj=0s\u0000\nj\n\u0017\u000b\u0000\u000fjj*i;(2)\nwhere the Mparametersf\u0017\u000bgshould satisfy the following\nBethe ansatz equations\nNX\nj=01\n\u0017\u000b\u0000\u000fj=2\ng+MX\n\f,\u000b;\f=12\n\u0017\u000b\u0000\u0017\f; (3)\nwith\u000b=1;2;:::; M. The equations (3) are also called\nRichardson-Gaudin equations. There are CM\nN+1sets of solu-\ntions to Eq. (3), which correspond to the number of choices of\nflipping Mspins [38]. These CM\nN+1statesj\u00171;\u0001\u0001\u0001;\u0017Mispan the\nsubspace with Mspins flipped-down. In Appendix D we givedetails of the numerical method for solving the Bethe Ansatz\nEq. (3). Moreover, their eigenenergy is given by\nE=B\n2+1\n2NX\nj=11\n\u000f0\u0000\u000fj\u0000MX\n\u000b=11\n\u000f0\u0000\u0017\u000b: (4)\nWe exploited the above eigenfunctions to study the quan-\ntum dynamics of the central spin model. The initial state (see\nintroduction) is a simple product state j\b0i=ja1;\u0001\u0001\u0001;aMi,\nwherefa1;\u0001\u0001\u0001;aMgis a set of mutually unequal site indexes\nwhich can range from “0” to “ N”. After the interaction is\nswitched on, the wave function evolves as\nj (t)i=X\nkj\u001ekih\u001ekj\b0ie\u0000iEkt; (5)\nwhere we introduced the orthonormalized wave functions\nj\u001eki=N\u0017kj\u00171;k;\u0001\u0001\u0001;\u0017M;ki, with N\u0017kthe normalization\ncoe\u000ecient[39] (see Appendix B). The index “ k” labels all the\nstates corresponding to the roots of Eq. (3). Since the initial\nstate has Mspins which are flipped-down, we only need to\nconsider the Bethe ansatz basis in this particular subspace. A\ncritical quantity in Eq. (5) is the overlap between the initial\nstate and the eigenfunctions, given as follows:\nh\b0j\u001eki=N\u0017kX\nP2fa1;\u0001\u0001\u0001;aMgMY\n\u000b=11\n(\u0017\u000b;k\u0000\u000fP\u000b); (6)\nwhere “P” means summing over all permutation of indexes\nfa1;\u0001\u0001\u0001;aMg, see Appendices A and B. The overlap can be\nalso written ash\b0j\u001eki=N\u0017kperm(1=(\u0017\u000b;k\u0000\u000fP\u000b), where\n“perm” is the permanent of matrix 1 =(\u0017\u000b;k\u0000\u000fP\u000b), similar to\nthe determinant [35, 36]. It is worth noting that both perma-\nnent and determinant representation can give the exact quan-\ntum dynamics of the Gaudin magnets. However, the numer-\nical tasks from the two representations is quite di \u000berent as\ndeterminants can be evaluated in polynomial time. The in-\nterested readers can find more details about permanent and\ndeterminant representations in [40–42]. For either represen-\ntation, it still remains challenging to get dynamical quantities\nfor the Gaudin magnets with a large system size, and values\nup to N=48 were reached [24].\nFrom expressions (5) and (6), the evolution of observables\n(for example the spin distribution) can be calculated rather\nstraightforwardly. Based on this approach, we will present in\nthe rest of the paper our key analytical and numerical results\non the quantum dynamics of Gaudin magnets.\nII. SPIN DISTRIBUTION\nBy making use of Eq. (5), the time-dependent polarization\nof site jis immediately written as\nsz\nj(t)=X\nkX\nk0h\b0j\u001ekih\u001ekjsz\njj\u001ek0ih\u001ek0j\b0ie\u0000iwkk0t; (7)\nwhere wkk0=Ek0\u0000Eksimplifies toPM\n\u000b=11=(\u000f0\u0000\u0017\u000b;k)\u0000PM\n\u000b=11=(\u000f0\u0000\u0017\u000b;k0). While the overlaps are given in Eq. (6), the3\n−0.5−0.45−0.4−0.35−0.3sz0(b)−0.5−0.4−0.3sz0\n  \n(a)B= 0.5\nB= 1\nB= 10\n−0.500.5szj\n{7,8,9,10}{1,2,3,4,5,6}\n(c)\n0 100 200 300 400 500 600 700 800 900 1000−0.500.5szj\n{2,4,6,8}{1,3,5,7,9,10}\n(d)\nFIG. 1: Time evolution of the spin polarization of individual spins,\nobtained from Eq. (3) assuming an exponentially decaying coupling\nconstant Aj=A=Nexp(\u0000j=N). We take N=10 and use dimen-\nsionless units, setting A=1. In (a)(c) we choose an initial state\nj\bAi=j0;7;8;9;10i. In (b)(d) the initial state is j\bBi=j0;2;4;6;8i.\nFigures (a) and (b) present the dynamical polarization of central spins\nwith di \u000berent magnetic fields for the two initial states. Figures (c)\nand (d) show the dynamical evolution of bath spins at B=0:5 for the\ntwo initial states.\nremaining di \u000eculty is to obtain the matrix elements of the ob-\nservable in the eigenfunction basis. As detailed in Appendix\nB, one can insert the complete set of states ja1;\u0001\u0001\u0001;aMi,\nthus writing the matrix elements in terms of overlaps of type\nh\b0j\u001eki. We finally obtain the exact expression:\nsz\nj(t)=1\n2\u0000X\nkX\nk0cos(wkk0t)X\nj1<\u0001\u0001\u0001<jMMX\n\u000b=1\u000ej j\u000b\n\u0002X\nPX\nQjN\u0017kj2\nMQ\n\u000b=1(\u0017\u000b;k\u0000\u000fP\u000b)(\u0017\u000b;k\u0000\u000fQ\u000b)\n\u0002X\nPX\nQjN\u0017k0j2\nMQ\n\u000b=1(\u0017\u000b;k0\u0000\u000fP\u000b)(\u0017\u000b;k0\u0000\u000fQ\u000b); (8)\nwhere, here and throughout the article, “ P” and “Q” indicate\nsumming over all the permutations of indexes fa1;\u0001\u0001\u0001;aMg\nandfj1;\u0001\u0001\u0001;jMg, respectively.\nAlthough it is not immediately obvious from Eq. (8), sz\nj(t)\nrecovers the correct initial value when we take the limit t!0.\nDetailed proof can be found in Appendix B. There we also\nshow that, as expected, mz=P\njsz\nj(t)=(N+1)=2\u0000Mis con-\nserved by Eq. (8). The evolution of spin distribution function\nfor a more general initial state is just an extension of above\nresult Eq. (8).\nWe have evaluated Eq. (8) for two representative initial con-\nditions, i.e., the domain-wall state j\bAi=s\u0000\n0\u0010Q\na>N=2+1s\u0000\na\u0011\nj*\niand the anti-ferromagnetically ordered state j\bBi=QN=2\u00001\na=0s\u0000\n2aj *i, and present in Fig. 1 the resulting polariza-\n0 100 200 300 400 500 600 700 800 900 1000−0.5−0.45−0.4−0.35−0.3\nt¯Sz\n0\n  \n(a)\n|ΦA/angbracketright\n|ΦB/angbracketrightB=0.5\nB=1\nB=10\n8 10 12 14 16 18 200.020.030.040.050.06¯Sz\n0,A−¯Sz\n0,B\nN(b)FIG. 2: (a) Time average of the central spin polarization, computed\nfor di \u000berent values of BandN=10. The solid lines are for the\ninitial statej\bAiwhile the dashed lines are for the initial state j\bBi.\n(b) Dependence of ¯ sz\n0;A\u0000¯sz\n0;B(i.e., the di \u000berence in the time-averaged\ncentral spin polarization between the two initial conditions) on the\nbath spin number N. Here we use a fixed value of B=0:5. The red\ncircles are from the exact Bethe ansatz equations, whereas the blue\ndiamonds are obtained by the Matrix Product State (MPS) method\n(see Sec. IV).\ntion of individual spins. The dynamical evolution of the cen-\ntral spin is shown in panels (a) and (b), and is allowed by the\ninteraction with the bath spins, which leads to change of the\norientation of the central spin through flip-flop processes. It is\ncharacterized by a quick decay to the steady-state value, with\nthe persistence of a oscillatory behavior due to finite-size ef-\nfects [24]. There is a competition between magnetic field and\ncoupling to the bath spins, evident from the evolution curves\nat di\u000berent values of Bshown in Fig. 1(a) and (b). The phys-\nical mechanism behind the suppressed evolution at large Bis\nthe increase of the energy gap between states with sz\n0=\u00061=2,\nwhich inhibits changes of orientation of the central spin. In\nAppendix C, we also present the detailed derivation of the re-\nduced density matrix and von Neumann entropy for the central\nspin. The general behavior of the central spin polarization in\nFig. 1 is very similar to the oscillatory time dependence ob-\nserved for the von Neumann entropy, see Fig. 7.\nFor models with short-range interactions, it is well-known\nthat the time-evolution of non-equilibrium states is character-\nized by a light-cone dynamics of local observables [43–46].\nOn the other hand, the Gaudin magnets is a long-range model\nand, as shown in Fig. 1(c) and (d), the spin polarization of the\nbath spins evolve simultaneously into their steady-state value.\nBeing infinite-range, the interaction of the Gaudin magnet can\ntake e \u000bect without any propagation time.\nComparing the two initial states, we see that with j\bAithe\nbath spins need a much longer time than j\bBito evolve into a\nsteady state. In fact, for the initial state j\bAithe two groups of\nbath spins remain well distinct, and the oscillation in the spin\npolarization nearly approach periodically their initial values\n(1=2 or\u00001=2). On the other hand, the polarizations for the\ninitial statej\bBishow an initial quick decay, leading to os-4\n0246810t= 20 (e) t= 50 (f)\n  \n−0.25−0.2−0.15−0.1−0.0500.050.10.150.20.25\n0 5 100246810\nmt= 100 (g)\n0 5 10t= 1000 (h)\nm0246810 nt= 20 (a) t= 50 (b)\n0 5 100246810\nmnt= 100 (c)\n0 5 10t= 1000 (d)\nm\nFIG. 3: Left panels: the spin-spin correlation function Gz\nmmat di\u000berent times, with a bath size N=10, magnetic field B=0:5, and initial state\nj\bAi=j0;7;8;9;10i. Right panels: spin-spin correlation function Gz\nmmwith the initial state j\bBi=j0;2;4;6;8i.\ncillations around 0. The evolution for the initial state j\bAi\ndisplays a strong memory of the initial domain-wall order. In\norder to further understand such peculiar dynamics, in later\nsections we will study the time evolution of the spin-spin cor-\nrelations and the Loschmidt echo.\nIf the time-average ¯Oof an observable coincides its micro-\ncanonical ensemble average, i.e., ¯O=hOi, the system reaches\nthermalization. Here the time average of the spin distribution\n¯sz\nj(t)=1=tRt\n0sz\nj(\u001c)d\u001ccan be simply obtained from Eq. (8),\nby replacing cos( wkk0t) with sinc( wkk0t)=sin(wkk0t)=wkk0t. We\nsee in Fig. 2(a) that the time-average of spin polarization of\nthe central spin converges to clearly di \u000berent values for the\ntwo initial states. The di \u000berence in the steady-state spin po-\nlarizations does not decrease by increasing N, see Fig. 2(b), in\nwhich smaller values of N(red circles) are obtained by exact\nmethod, relatively larger values of N(blue diamond) are ob-\ntained by MPS method. The combined results of BA and MPS\nmethod verify the persistence of a finite di \u000berence ¯ sz\n0;A\u0000¯sz\n0;B\nfor the steady-state value. This observation confirms a break-\ndown of thermalization in the inhomogeneous central spin\nmodel, due to the restricted time evolution implied by inte-\ngrability. Since the existence of conserved quantities, like\nHj=Bsz\nj+P\nk,jsj\u0001sk=(\u000fj\u0000\u000fk), the time-evolution cannot be\nergodic, neither in the whole Hilbert space nor in a subspace\nwith fixed M.\nIII. SPIN-SPIN CORRELATIONS\nThe spin-spin correlation function is defined as Gz\nmn(t)=\nh (t)jsz\nmsz\nnj (t)iand, as discussed in Appendix B, which canbe computed in a way similar to the spin density. The explicit\nexpression reads:\nGz\nmn(t)=X\nk;k0X\nj1<\u0001\u0001\u0001<jM0BBBBB@1\n2\u0000MX\n\u000b=1\u000em j\u000b1CCCCCA0BBBBB@1\n2\u0000MX\n\u000b=1\u000en j\u000b1CCCCCA\n\u0002X\nP;QjN\u0017kj2\nMQ\n\u000b=1(\u0017\u000b;k\u0000\u000fP\u000b)(\u0017\u000b;k\u0000\u000fQ\u000b)\n\u0002X\nP;QjN\u0017k0j2\nMQ\n\u000b=1(\u0017\u000b;k0\u0000\u000fP\u000b)(\u0017\u000b;k0\u0000\u000fQ\u000b)cos(wkk0t);(9)\nwhich characterizes longitudinal correlation between spins.\nIn Fig. 3, we show the spin-spin correlation function at dif-\nferent times, both for the initial state j\bAi(left panels) and\nj\bBi(right panels). On a relatively short timescale ( t<50),\nthe correlation function is very similar to t=0 for both initial\nstates and when the evolution time approaches t\u0018102the cor-\nrelation function has decreased to smaller values. However,\nit can be noted that in the left panels of Fig. 3 (referring to\nj\bAi), the domain-wall order has not faded away completely\neven after a long evolution time t\u0018103. This is in contrast\nto panel (h) on the right side, where the anti-ferromagnetic\nalignment has nearly disappeared. This confirms our previ-\nous observation that the domain-wall configuration is more\nfavourable to retain memory of the initial state (see also Fig. 5\nfor the Loschmidt echo).\nFrom the contour plot of the correlations in Fig. 3, the corre-\nlations between the bath spins Gz\nmnwith m;n>0 tend to zero.\nMeanwhile the correlations between the central spin and bath5\n10010110210310410500.050.10.150.20.25Tm\nt  \n(c)m=0\nm=1\n100101102103104105\nt(d)00.050.10.150.20.25|Gz0n|\n  A\n(a)n=1\nn=NB\n(b)\nFIG. 4: Panels (a) and (b) plot the spin-spin correlation function\njGz\n0njfor spins n=1;N. Panles (c) and (d) average the spin-\nspin correlation function according to Eq. (10). The bath size is\nN=10 and the magnetic field B=0:5. Left panels: initial state\nj\bAi=j0;7;8;9;10i. Right panels: initial state j\bBi=j0;2;4;6;8i.\nspins Gz\nm0with m>0 oscillate near initial value. After long\nenough time evolution ( t\u0018103), the correlations between the\nbath spins almost evolve toward to zero for the state j\bBi.\nConsidering in more detail the right side of Fig. 3, we see\nin panel (g) that the correlations Gz\nmnwith m;n&5 are more\nrobust, and also the values of Gz\n0nwith larger nremain rather\nsimilar to the initial values. This behavior is a consequence of\nthe small coupling strength Ajwhen jis large, which implies\na longer timescale for those bath spins. However, after a su \u000e-\nciently long time evolution ( t\u0018103) the correlations between\nall spins are close to zero. The vanishing of correlations in\npanel (h) indicates that the steady state is like a paramagnetic\nstate.\nTo further analyze the evolution of Gz\nmn(t), we characterize\nthe typical correlation strength for spin mas\nTm=vtP\nn,m(Gz\nmn(t))2\nN; (10)\nwhere the self-correlation terms Gz\nmmare excluded. In Fig. 4,\nwe show the absolute value of correlation function and the\ntypical correlation strength for the two initial states, i.e., the\nleft two panels (a,c) are for the initial state j\bAiand the right\ntwo panels (b,d) for j\bBi. For both ferromagnetic and anti-\nferromagnetic alignment, the correlation strength decays from\ninitial maximum value of 1 =4 towards smaller values, with\nTm=0 corresponding to disordered spin alignment. We can\nsee that there are two stages in the relaxation of correlations.\nAfter the initial decay (up to t.10, see the first black dash\nline), there is an intermediate regime in which the correlation\nstrength has a logarithmic relaxation (up to t\u0018102, near the\nsecond black dash line), i.e., \u0001Tm/\u0000lnt. In this interval the\nlinear decay shown in Fig. 4 (on a semi-logarithmic scale) has\nits steepest slope. Finally, in the long-time limit the central\nspin is strongly entangled with bath spins and the value of\n0 100 200 300 400 500 600 700 800 900 100010−210−1100L(t)\nt(b)10−210−1100L(t)\n  \n(a)B= 0.5\nB= 1\nB= 2\nB= 10FIG. 5: Time dependence of the Loschmidt echo for two di \u000berent\ninitial state. Panel (a) with the initial state j\bAi, and panel (b) with\ninitial statej\bBi.\nTmreaches its steady-state value. Due to finite-size e \u000bects,\nthe decay is modified by oscillations, see especially panel (b),\nbut the qualitative form and timescales are quite robust with\nrespect to the value of mand the type of initial state.\nIV . LOSCHMIDT ECHO AND MPS APPROACH\nThe Loschmidt echo, defined as L(t)=jh\b0j (t)ij2, quan-\ntifies the memory of the initial state [47], thus can provide us\na clear understanding of the di \u000berence between the evolution\nfor the two initial states. By using Eq. (5), we express L(t) as:\nL(t)=X\nkX\nk0jh\b0j\u001ekij2jh\u001ek0j\b0ij2e\u0000iwkk0t; (11)\nwhere the overlaps are given in Eq. (6). Finally, we obtain:\nL(t)=X\nkX\nk02666666666666664X\nPjN\u0017kj\nMQ\n\u000b=1(\u0017\u000b;k\u0000\u000fP\u000b)37777777777777752\n\u00022666666666666664X\nPjN\u0017k0j\nMQ\n\u000b=1(\u0017\u000b;k0\u0000\u000fP\u000b)37777777777777752\ncos(wkk0t): (12)\nWe show representative examples of the Loschmidt echo in\nFig. 5. As seen in panel (a), for a weak magnetic field ( B\u0018\n0:5) the initial state j\bAileads to a long-time evolution with\nlarge oscillations around a mean value ¯L(t)\u00180:25. Instead,\npanel (b) shows that L(t)\u00180 in the long-time limit, thus the\nsystem nearly loses its memory of the initial state j\bBi.\nIn the first part of the time evolution, besides the presence of\nfast small amplitude oscillations, the Loschmidt echo displays6\n0 50 100 150 200 250 30010−1100\ntL(t)\n  \n|ΦA/angbracketright\n|ΦB/angbracketright0 200 40005001000150020002500\ntχ\nFIG. 6: Loschmidt echo as function of time tfor two di \u000berent initial\nstates by using the MPS method with N=50. The red line is for the\ninitial statesj\bAi, and the cyan line is for the initial state j\bBi.\na clear exponential dependence \u0001L(t)/exp(\u0000\rt), up to an\nintermediate time scale t\u0018102. According to our numerical\ncalculation, the decay coe \u000ecients is similar for both initial\nstates, i.e.,\rA'\rB\u00180:03 with B=0:5. Such exponential\nscaling behavior of the Loschmidt echo has been widely found\nin dynamical evolution of quantum many-body system [48–\n51].\nIn order to confirm the scaling relaxation behavior of\nLoschmidt echo for a larger system size, we further per-\nform an approximate calculation based on the MPS method\n[52, 53], as implemented in the Itensor library [54]. There are\ntwo important input parameters of the MPS simulations: the\nvariational tolerance \u000fof the MPS wavefunction, which rep-\nresents the error of the MPS state in approximating the true\nwavefunction (we use \u000f=10\u00009in our simulations), and the\nbond dimension \u001f, which indicates the maximum dimension\nof the matrices entering the variational ansatz. In the given im-\nplementation of the MPS method, the bond dimension is au-\ntomatically increased to meet the desired tolerance. In Fig. 6,\nwe show the Loschmidt echo for a bath size N=50 and the\ntwo types of initial states considered in this work, j\bAi(red\nlines) andj\bBi(cyan lines). Also in this case the Loschmidt\necho is characterized by an initial exponential decay, and this\nbehavior persists up to intermediate times ( t\u0019102).\nUnfortunately, we are not able to access the long-time dy-\nnamics at N=50 based on the MPS approach, as the en-\ntanglement in the wavefunction grows with time. This reflects\nitself on a rapid increase of the bond dimension \u001f, which even-\ntually reaches numerically intractable values. The time depen-\ndence of\u001fin the simulation of Fig. 6 is shown in the inset. As\nseen, the bond dimension \u001fof initial statej\bBigrows faster in\ntime thanj\bAi. Nevertheless, the range of times we can simu-\nlate is su \u000ecient to establish conclusively the initial exponen-\ntial form of decay of Loschmidt echo. In classical systems, the\nexponential decay of Loschmidt echo usually implies chaotic\ndynamics [50, 55, 56]. For the central spin model, the expo-\nnential decay of Loschmidt echo may indicate the dynamicsis chaotic here as well [57].\nV . CONCLUSION\nIn summary, based on Bethe Anstaz techniques, we have\nanalytically studied the time evolution of several dynamic\nquantities of Gaudin magnets with arbitrary number Mof\nflipped-down spins, such as spin distribution function, spin-\nspin correlation function, and Loschmidt echo. Furthermore,\nwe have numerically studied the scaling behavior of the re-\nlaxation dynamics, up to relatively large system sizes. Sig-\nnificantly, the correlation function relaxes to its steady value\nwith a logarithmic dependence, and the Loschmidt echo re-\nveals an exponential loss of the memory of the initial state.\nOur results highlight several interesting features of the dy-\nnamical evolution of a quantum many-body system with long\nrange interactions, thus advancing the understanding of deco-\nherence for single qubits coupled to a spin bath at the many-\nbody level. With respect to universal features of quantum\nmany-body systems, much e \u000bort is still necessary to better\nunderstand the non-equilibrium quantum dynamics [58]. In\nfuture studies, it would be desirable to extend our methods to\ninvestigate scrambling in many-body system [59, 60], and the\nout-of-time-of-correlation function (OTOC) [61] of central-\nspin models. The details of such dynamical evolution of spin\ncorrelations may be accessible experimentally, e.g., through\nNMR techniques [62].\nAcknowledgments\nWe would like to thank R. Fazio for useful discussions and\nF. Iemini and D. Ferreira their kind help with the MPS sim-\nulation package. W.B.H. acknowledges support from NSAF\n(Grant No. U1930402). X.W.G. is supported by the key NSFC\ngrant No. 11534014 and No. 11874393, and the National\nKey R&D Program of China No. 2017YFA0304500. S.C.\nacknowledges support from NSFC (Grants No. 11974040\nand No. 1171101295) and the National Key R&D Program\nof China No. 2016YFA0301200. H. Q. L. acknowledges\nfinancial support from National Science Association Funds\nU1930402 and NSFC 11734002, as well as computational\nresources from the Beijing Computational Science Research\nCenter.\nAppendix A: Bethe ansatz basis\nThe eigenstates of Eq. (1) can be obtained with Bethe\nansatz (BA) approach, as discussed in detail in Ref. 36. Here\nwe discuss several properties of these states which will find\nwide use in the following derivations. Since the Hilbert\nspace of the Gaudin model can be decomposed into the di-\nrect sum of subspaces with Mflipped-down spins, we intro-\nduce two basis sets for such subspaces. One is the natural ba-\nsisjj1;\u0001\u0001\u0001;jMi=S\u0000\nj1;\u0001\u0001\u0001;S\u0000\njMj*i, and another is the Bethe7\nansatz basisj\u00171;k;\u0001\u0001\u0001;\u0017M;ki=B\u0017M;k;\u0001\u0001\u0001;B\u00171;kj*i. The corre-\nsponding completeness relations are simply given by:\nX\nMX\nj1;\u0001\u0001\u0001;jMjj1;\u0001\u0001\u0001;jMihj1;\u0001\u0001\u0001;jMj=I; (A1)\nX\nM;kjN\u0017kj2j\u00171;k;\u0001\u0001\u0001;\u0017M;kih\u00171;k;\u0001\u0001\u0001;\u0017M;kj=I; (A2)\nwherejN\u0017kjare the normalization coe \u000ecients for the Bethe\nansatz basis. The two sets of states jj1;\u0001\u0001\u0001;jMiand\nj\u00171;k;\u0001\u0001\u0001;\u0017M;kiare both complete in the subspace with M\nflipped-down spins. Including all the M, the complete basis\nof the whole Hilbert space is obtained.\nThe explicit form of the Bethe ansatz eigenfunctions reads\nj\u00171;k;\u0001\u0001\u0001;\u0017M;ki=MY\n\u000b=1B\u0017\u000bj*i=MY\n\u000b=1X\njs\u0000\nj\n\u0017\u000b\u0000\u000fjj*i\n=X\nj1<\u0001\u0001\u0001<jM2666666666666664X\nQ1\nMQ\n\u000b=1(\u0017\u000b;k\u0000\u000fQ\u000b)3777777777777775jj1;\u0001\u0001\u0001;jMi;(A3)\nwhere, as in the main text, Qare the permutations of indexes\nfj1;\u0001\u0001\u0001;jMg. Note also that any of the two “ j” indexes are\nnecessarily unequal. The overlap of state ja1;\u0001\u0001\u0001;aMiwith a\nBA eigenstatej\u00171;k;\u0001\u0001\u0001;\u0017M;kican be simply obtained from the\nabove expression as:\nha1;\u0001\u0001\u0001;aMj\u00171;k;\u0001\u0001\u0001;\u0017M;ki=X\nP1\nMQ\n\u000b=1(\u0017\u000b;k\u0000\u000fP\u000b); (A4)\nwherePare the permutations of indexes fa1;\u0001\u0001\u0001;aMg. The\nabove formula can be also written in terms of the permanent\nof the matrix Mk j=1=(\u0017\u000b;k\u0000\u000faj) [41].\nThe scalar product between eigenstates can be obtained by\ninsertion ofP\nj1;\u0001\u0001\u0001;jMjj1;\u0001\u0001\u0001;jMihj1;\u0001\u0001\u0001;jMj:\nh\u0017M;k;\u0001\u0001\u0001;\u00171;kj\u00171;k0;\u0001\u0001\u0001;\u0017M;k0i=\u000ekk0\njN\u0017kj2\n=X\nj1<\u0001\u0001\u0001<jM2666666666666664X\nQ1\nMQ\n\u000b=1(\u0017\u000b;k\u0000\u000fQ\u000b)3777777777777775\n\u00022666666666666664X\nQ1\nMQ\n\u000b=1(\u0017\u000b;k0\u0000\u000fQ\u000b)3777777777777775; (A5)\nwhere we used the overlaps Eq. (A4) between the two basis\nsets. The above expression gives the normalization coe \u000ecient\njN\u0017kj2.We can also derive another useful formula:\nhj1;\u0001\u0001\u0001;jMja1;\u0001\u0001\u0001;aMi=X\nQ2fj1;\u0001\u0001\u0001;jMgMY\n\u000b=1\u000ea\u000b;jQ\u000b\n=X\nkX\nQX\nPjN\u0017kj2\nMQ\n\u000b=1(\u0017\u000b;k\u0000\u000fQ\u000b)MQ\n\u000b=1(\u0017\u000b;k\u0000\u000fP\u000b)(A6)\nwhich is obtained by inserting the resolution of the identity in\nterms of the BA eigenstates.\nAppendix B: Evaluation of physical observables\nWe now discuss the explicit evaluation of Eq. (7), where\nwe recall thatj\u001eki=N\u0017kj\u00171;k;\u0001\u0001\u0001;\u0017M;kiare the normalized\neigenfunctions. Therefore, for an initial state in the form\nj\b0i=s\u0000\na1\u0001\u0001\u0001s\u0000\naMj*i=ja1;\u0001\u0001\u0001;aMi, the overlap matrix el-\nementsh\b0j\u001ekiare immediately found from Eq. (A4), and are\ngiven in Eq. (6). Instead, to compute the matrix element of\nsz\nj, we use the fact that such operator is diagonal in the natural\nbasisjj1;\u0001\u0001\u0001;jMiand:\nhj1;\u0001\u0001\u0001;jMjsz\njjj1;\u0001\u0001\u0001;jMi=1\n2\u0000X\n\u000b=1\u000ej j\u000b: (B1)\nBy inserting the completeness relation in terms of the\njj1;\u0001\u0001\u0001;jMistates, and computing the overlaps with eigen-\nstatesj\u001ekias in Eq. (6), it is easy to find that:\nh\u001ekjsz\njj\u001ek0i=N\u0003\n\u0017kN\u0017k0X\nj1<\u0001\u0001\u0001<jM0BBBBB@1\n2\u0000MX\n\u000b=1\u000ej j\u000b1CCCCCA\n\u00022666666666666664X\nQ1\nMQ\n\u000b=1(\u0017\u000b;k\u0000\u000fQ\u000b)37777777777777752666666666666664X\nQ1\nMQ\n\u000b=1(\u0017\u000b;k0\u0000\u000fQ\u000b)3777777777777775: (B2)\nFinally, combining overlaps and matrix element gives:\nsz\nj(t)=X\nk;k0cos(wkk0t)X\nj1<\u0001\u0001\u0001<jM0BBBBB@1\n2\u0000MX\n\u000b=1\u000ej j\u000b1CCCCCA\n\u00022666666666666664X\nPX\nQjN\u0017kj2\nMQ\n\u000b=1(\u0017\u000b;k\u0000\u000fP\u000b)MQ\n\u000b=1(\u0017\u000b;k\u0000\u000fQ\u000b)3777777777777775\n\u00022666666666666664X\nPX\nQjN\u0017k0j2\nMQ\n\u000b=1(\u0017\u000b;k0\u0000\u000fP\u000b)MQ\n\u000b=1(\u0017\u000b;k0\u0000\u000fQ\u000b)3777777777777775:(B3)\nIn the above formulas, as usual, Pare permutations of the\nindexesfa1;\u0001\u0001\u0001;aMg(from the initial state) and Qare permu-\ntations of the indexes fj1;\u0001\u0001\u0001;jMg. The term proportional to\n1/2 (from the parenthesis in the first line) can be further sim-\nplified as follows. First we note that this term contains a factor8\nof type Eq. (A5), which is /\u000ekk0. Therefore, we can simply\nset cos wkk0t=1 in this particular contribution. After noticing\nthis, we can use Eq. (A6) to perform the summations over k\nandk0and finally recover the 1 /2 appearing in Eq. (8) of the\nmain text.\nUsing a similar method, we can derive the spin-spin corre-\nlation function, given by\nGz\nmn(t)=X\nkX\nk0h\b0j\u001ekih\u001ekjsz\nmsz\nnj\u001ek0ih\u001ek0j\b0ie\u0000iwkk0t:(B4)\nWhile the overlaps are found in Eq. (6), the matrix elements\nofsz\nmsz\nncan be obtained in a way completely analogous to\nEq. (B2):\nh\u001ekjsz\nmsz\nnj\u001ek0i\n=N\u0003\n\u0017kN\u0017k0X\nj1<\u0001\u0001\u0001<jM0BBBBB@1\n2\u0000MX\n\u000b=1\u000em j\u000b1CCCCCA0BBBBB@1\n2\u0000MX\n\u000b=1\u000en j\u000b1CCCCCA\n\u00022666666666666664X\nQ1\nMQ\n\u000b=1(\u0017\u000b;k\u0000\u000fQ\u000b)37777777777777752666666666666664X\nQ1\nMQ\n\u000b=1(\u0017\u000b;k0\u0000\u000fQ\u000b)3777777777777775; (B5)\nleading to Eq. (9) in the main text.\nFinally, we discuss how to recover the initial conditions,\nand prove the conservation of magnetization. When time t!\n0, the spin distribution becomes\nsz\nj(0)=1\n2\u0000X\nkX\nk0X\nj1<\u0001\u0001\u0001<jMMX\n\u000b=1\u000ej j\u000b\n\u0002X\nPX\nQjN\u0017kj2\nMQ\n\u000b=1(\u0017\u000b;k\u0000\u000fP\u000b)(\u0017\u000b;k\u0000\u000fQ\u000b)\n\u0002X\nPX\nQjN\u0017k0j2\nMQ\n\u000b=1(\u0017\u000b;k0\u0000\u000fP\u000b)(\u0017\u000b;k0\u0000\u000fQ\u000b); (B6)\nwhich is in a form suitable to Eq. (A6). Performing the sum-\nmations over kandk0we get:\nsz\nj(0)=1\n2\u0000X\nj1<\u0001\u0001\u0001<jM0BBBBB@MX\n\u000b\u000ej j\u000b1CCCCCA26666664X\nQMY\n\u000b\u000ea\u000b;jQ\u000b37777775\n=1\n2\u0000MX\n\u000b\u000ej;a\u000b; (B7)\nwhich recovers the initial spin distribution. For the total mag-\nnetization, computing mz=P\njsz\nj(t) leads toP\nj;\u000b\u000ej j\u000b=Min\nthe second term of Eq. (7). Finally, the summation over k;k0\ncan be computed as discussed after Eq. (B3), leading to the\nexpected result mz=(N+1)=2\u0000M.\n00.10.20.30.40.50.6 S\n  \nB= 0.5\nB= 1B= 2\nB= 10\n0 200 400 600 800 100000.10.20.30.40.5\ntSFIG. 7: Time dependence of the von Neumann entropy at di \u000berent\nmagnetic fields. upper subplot: initial state j\bAi=j0;7;8;9;10i.\nlower subplot: initial state j\bBi=j0;2;4;6;8i.\nAppendix C: Reduced density matrix of the central spin\nThe density matrix \u001as=j (t)ih (t)jis given by\n\u001as=X\nk;k0j\u00171;k;\u0001\u0001\u0001;\u0017M;kih\u00171;k0;\u0001\u0001\u0001;\u0017M;k0jeiwkk0t\n\u00022666666666666664X\nPjN\u0017kj2\nMQ\n\u000b=1(\u0017\u000b;k\u0000\u000fP\u000b)37777777777777752666666666666664X\nPjN\u0017k0j2\nMQ\n\u000b=1(\u0017\u000b;k0\u0000\u000fP\u000b)3777777777777775:(C1)\nwhere, by using Eq. (A3):\nj\u00171;k;\u0001\u0001\u0001;\u0017M;kih\u00171;k0;\u0001\u0001\u0001;\u0017M;k0j\n=X\nj1<\u0001\u0001\u0001<jMX\nl1<\u0001\u0001\u0001<lMjj1;\u0001\u0001\u0001;jMihl1;\u0001\u0001\u0001;lMj\n\u00022666666666666664X\nQ1\nMQ\n\u000b=1(\u0017\u000b;k\u0000\u000fQ\u000b)37777777777777752666666666666664X\nR1\nMQ\n\u000b=1(\u0017\u000b;k0\u0000\u000fR\u000b)3777777777777775: (C2)\nHereRindicates the permutations of fl1;\u0001\u0001\u0001;lMg.\nNow we consider the reduced density matrix of central spin,\ndefined by\u001acs=Tr1!N[\u001as]. With our choice of initial state,\nthere are no o \u000b-diagonal elements and the result is in the form:\n\u001acs=A(t)j\"ih\"j +D(t)j#ih#j: (C3)\nThe first contribution arises from the terms of Eq. (C2) where\nji=li>0 (i.e., site 0 is not flipped) and A(t) has the following9\nexpression:\nA(t)=X\nk;k0X\n0<j1<\u0001\u0001\u0001<jMcos(wkk0t)\n\u00022666666666666664X\nPjN\u0017kj2\nMQ\n\u000b=1(\u0017\u000b;k\u0000\u000fP\u000b)37777777777777752666666666666664X\nPjN\u0017k0j2\nMQ\n\u000b=1(\u0017\u000b;k0\u0000\u000fP\u000b)3777777777777775\n\u00022666666666666664X\nQ1\nMQ\n\u000b=1(\u0017\u000b;k\u0000\u000fQ\u000b)37777777777777752666666666666664X\nQ1\nMQ\n\u000b=1(\u0017\u000b;k0\u0000\u000fQ\u000b)3777777777777775:(C4)\nThe second contribution corresponds to j1=l1=0 (i.e., the\ncentral spin is flipped) and D(t) reads:\nD(t)=X\nk;k0X\nj1=0<j2\u0001\u0001\u0001<jMcos(wkk0t)\n\u00022666666666666664X\nPjN\u0017kj2\nMQ\n\u000b=1(\u0017\u000b;k\u0000\u000fP\u000b)37777777777777752666666666666664X\nPjN\u0017k0j2\nMQ\n\u000b=1(\u0017\u000b;k0\u0000\u000fP\u000b)3777777777777775\n\u00022666666666666664X\nQ1\nMQ\n\u000b=1(\u0017\u000b;k\u0000\u000fQ\u000b)37777777777777752666666666666664X\nQ1\nMQ\n\u000b=1(\u0017\u000b;k0\u0000\u000fQ\u000b)3777777777777775;(C5)\nWith the help of Eqs. (A5) and (A6), we can prove A+D=1.\nThe von Neumann entropy is simply given as S=\u0000AlnA\u0000\nDlnD, and a representative plot is shown in Fig. 7.\nAppendix D: Numerical method for solving BA roots\nIn our work we solve the BA equations (3) using the numer-\nical method presented in [38]. For convenience of the reader,\nin this Appendix we present the numerical method in detail.\nFor general M, the asymptotic solution to Eq. (3) with g!0\nreads:\n\u0017\u000b(0)\u0019\u000fj+g\n2+O(g2): (D1)\nWe then introduce the quantities\n\u0003j=MX\n\u000b=11\n\u0017\u000b\u0000\u000fj; (D2)\nand transform Eq.(3) to the following form:\n\u00032\nj=\u0000X\nl,j\u0003j\u0000\u0003l\n\u000fj\u0000\u000fl+2\ng\u0003j: (D3)\nThe advantage of using the \u0003jvariables is in avoiding the sin-\ngular dependence of \u0017\u000bon system parameters. With \u0017\u000b(0) as\ninitial value, we use Newtonian iteration to find the solution\nof the above equations at fixed g.\n−4 −3 −2 −1 0−14−12−10−8−6−4−20\ngRe(ν)\n−4 −3 −2 −1 0−3−2−10123\ngIm(ν)FIG. 8: Dependence of BA parameters on g. The top (bottom) panel\nshows thir real (imaginary) part. System parameters: N=15,M=8,\nandA=2.\nAfter obtaining \u0003j, the\u0017jcan be found as follows. We first\ntransform Eq. (D2) into\n\u0003jMY\n\u000b=1(\u0017\u000b\u0000\u000fj)=MX\n\u000b=1Y\n\f,\u000b(\u0017\f\u0000\u000fj): (D4)\nThe above Eq. (D4) gives a linear system in the pivariables:\np1=X\n\u000b\u0017\u000b;\np2=X\n\u000b<\f\u0017\u000b\u0017\f;\n:::\npM=X\n\u000b1<:::<\u000b M\u0017\u000b1:::\u0017\u000bM; (D5)\nwhich are elementary symmetric polynomials of the BA pa-\nrameters\u0017\u000bandp0can be defined as 1. Finally, after obtaining\nthe values of piwe consider the Mth-order polynomial:\nF(\u0016)=MY\n\u000b(\u0016\u0000\u0017\u000b)=p0\u0016M\u0000p1\u0016M\u00001+:::(\u00001)MpM;(D6)10\nwhose zeroes give the desired BA parameters \u0017\u000b. An example\nof the dependence of the BA parameters f\u0017gongis shown in\nFig. 8. Close to g=0 the BA parameters \u0017are all real, butcomplex conjugate pairs are formed by neighboring roots at\nlarger values ofjgj. The interested reader can also see [38] for\nthe description of this method.\n[1] L. D. Landau, Sov. Phys. JETP 3, 920 (1957).\n[2] L. D. Landau, Sov. Phys. JETP 5, 101 (1957).\n[3] L. D. Landau, Sov. Phys. JETP 8, 70 (1959).\n[4] A. Luther, and V . J. Emery, Phys. Rev. Lett. 33, 589(1974).\n[5] J. M. Luttinger, J. Math. Phys. (N.Y .) 4, 1154 (1963).\n[6] W. H. Zurek, Nature 317, 10(1985).\n[7] K. Pyka, et.al., Nat commun. 4, 2291 (2016).\n[8] J. M. Deutsch, Phys. Rev. A 43, 2046 (1991).\n[9] M. Srednicki, Phys. Rev. E 50, 888 (1994).\n[10] A. Polkovnikov, K. Sengupta, A. Silva, and M. Vengalattore,\nRev. Mod. Phys. 83, 863 (2011).\n[11] X.-W. Guan, M. T. Batchelor, and C.-H. Lee, Rev. Mod. Phys.\n85, 1633 (2013).\n[12] T. Kinoshita, T. Wenger, D. S. Weiss, Nature 440, 900 (2006).\n[13] M. Rigol, V . Dunjko, V . Yurovsky, and M. Olshanii, Phys. Rev.\nLett. 98, 050405 (2007).\n[14] D. Hetterich, N. Y . Yao, M. Serbyn, F. Pollmann, and B.\nTrauzettel, Phys. Rev. B 98, 161122(R) (2018).\n[15] L. Bonnes, F.H.L. Essler, and A.M.L ¨auchli, Phys. Rev. Lett.\n113, 187203 (2014).\n[16] W. A. Coish and L. Baugh, Phys. Status Solidi B 246, 2203\n(2009).\n[17] W. Yang, W.-L. Ma, and R.-B. Liu, Rep. Prog. Phys. 80, 016001\n(2017).\n[18] A. V . Khaetskii, D.Loss, and L.Glazman, Phys. Rev. Lett. 88,\n186802(2002).\n[19] A. V . Khaetskii, D. Loss, L. Glazman, Phys.Rev. B. 67,\n195329(2003).\n[20] W. A. Coish and D. Loss, Phys. Rev. B 70, 195340 (2004).\n[21] Ł. Cywi ´nski, W. M. Witzel, and S. Das Sarma, Phys. Rev. Lett.\n102, 057601 (2009).\n[22] W. A. Coish, J. Fischer, and D. Loss, Phys. Rev. B 81, 165315\n(2010).\n[23] M. Bortz and J. Stolze, Phys.Rev. B. 76, 014304 (2007).\n[24] A. Faribault and D. Schuricht, Phys. Rev. Lett. 110, 040405\n(2013).\n[25] A. Faribault and D. Schuricht, Phys. Rev. B 88, 085323 (2013).\n[26] D. Stanek, C. Raas, and G. S. Uhrig, Phys. Rev. B 88, 155305\n(2013).\n[27] D. Stanek, C. Raas, and G. S. Uhrig, Phys. Rev. B 90, 064301\n(2014).\n[28] P. Schering, G. S. Uhrig, and D. S. Smirnov, Phys. Rev. Re-\nsearch 1, 033189 (2019).\n[29] P. Schering, P. W. Scherer, and G. S. Uhrig, Phys. Rev. B 102,\n115301 (2020).\n[30] W.-B. He, S. Chesi, H.-Q. Lin, and X.-W. Guan, Phys. Rev. B\n99, 174308 (2019).\n[31] T. Villazon, A. Chandran, P. W. Claeys, Phys. Rev. Research 2,\n032052(R) (2020).\n[32] N. Wu, X.-W. Guan, J. Links, Phys. Rev. B 101, 155145 (2020).\n[33] V . V . Dobrovitski and H. A. De Raedt, Phys. Rev. E 67, 056702\n(2003).\n[34] G. S. Uhrig, J. Hackmann, D. Stanek, J. Stolze, and F.B. An-ders, Phys. Rev. B 90, 060301(R) (2014).\n[35] J. Dukelsky, S. Pittel, G. Sierra, Rev. Mod. Phys. 76, 643\n(2004).\n[36] H.-Q. Zhou, J. Links, R. H. McKenzie, and M. D. Gould,\nPhys.Rev.B 65, 060502(R) (2002).\n[37] X.-W. Guan, A. Foerster, J. Links and H.-Q. Zhou, Exact results\nfor BCS systems, JHEP proceedings, unesp2002.\n[38] O. El Araby, V . Gritsev, and A. Faribault, Phys. Rev. B 85,\n115130 (2012).\n[39] See the appendix for details. There we present brief deriva-\ntions for the time evolution of the spin distribution function,\nthe spin-spin correlation function, and the Loschmidt echo for\nthe Gaudin magnet with arbitrary number of flipped spins M.\n[40] A. Faribault, P. Calabrese, and J. S.Caux, Phys. Rev. B 77,\n064503 (2008).\n[41] A. Faribault and D. Schuricht, J. Phys.A: Math. Theor. 45,\n485202 (2012).\n[42] P. W. Claeys, D. V . Neck, S. D. Baerdemacker, SciPost Phys. 3,\n028 (2017).\n[43] W. Liu and N. Andrei, Phys. Rev. Lett. 112, 257204 (2014).\n[44] E. H. Lieb, D. W. Robinson, Commun. Math. Phys. 28, 251\n(1972).\n[45] M. Cheneau, et al., Nature 481, 484 (2012).\n[46] B. Wouters, J. De Nardis, M. Brockmann, D. Fioretto, M. Rigol,\nand J.-S. Caux, Phys. Rev. Lett. 113, 117202 (2014).\n[47] M. Heyl, A. Polkovnikov, and S. Kehrein, Phys. Rev. Lett. 110,\n135704 (2013).\n[48] C. Lupo and M. Schir ´o, Phys. Rev. B 94, 014310 (2016).\n[49] F. Tonielli, R. Fazio, S. Diehl, and J. Marino, Phys. Rev. Lett.\n122, 040604 (2019).\n[50] R. Dubertrand and A. Goussev, Phys. Rev. E 89, 022915 (2014).\n[51] B. Yan, L. Cincio, and W. H. Zurek, Phys. Rev. Lett. 124,\n160603 (2020).\n[52] G. Vidal, Phys. Rev. Lett. 91, 147902 (2003),\n[53] U. Schollwock, Annals of Physics 326, 96 (2011),\n[54] Itensor Libarary, http: //www.itensor.org /index.html\n[55] G. Veble and T. Prosen, Phys. Rev. Lett. 92, 034101 (2004).\n[56] Y . S. Weinstein and C. S. Hellberg, Phys. Rev. E 71, 016209\n(2005).\n[57] J. Emerson, Y . S. Weinstein, S. Lloyd, and D. G. Cory, Phys.\nRev. Lett. 89, 284102 (2002)\n[58] D. Suter and G.A. Alvarez, Rev. Mod. Phys. 88, 041001\n(2016).\n[59] H. Shen, P. Zhang, R. Fan, and H. Zhai, Phys. Rev. B 96,\n054503 (2017).\n[60] C.W. von Keyserlingk, T. Rakovszky, F. Pollmann, and S.L.\nSondhi, Phys. Rev. X 8, 021013 (2018).\n[61] A. Smith, J. Knolle, R. Moessner, and D. L. Kovrizhin, Phys.\nRev. Lett. 123, 086602 (2019).\n[62] M. Niknam, L. F. Santos, and D. G. Cory, Phys. Rev. Lett. 127,\n080401 (2021)."    },    {        "title": "1302.2121v1.Spin_dynamics_of_cold_fermions_with_synthetic_spin_orbit_coupling.pdf",        "content": "arXiv:1302.2121v1  [cond-mat.quant-gas]  8 Feb 2013Spin dynamics of cold fermions with synthetic spin-orbit co upling\nI. V. Tokatly\nNano-bio Spectroscopy group and ETSF Scientific Developmen t Centre, Departamento de F´ ısica de Materiales,\nUniversidad del Pa´ ıs Vasco UPV/EHU, E-20018 San Sebast´ ıa n, Spain and\nIKERBASQUE Basque Foundation for Science, 48011, Bilbao, S pain\nE. Ya. Sherman\nDepartment of Physical Chemistry, Universidad del Pa´ ıs Va sco UPV/EHU, 48080 Bilbao, Spain and\nIKERBASQUE Basque Foundation for Science, 48011, Bilbao, S pain\nWe consider spin relaxation dynamics in cold Fermi gases wit h a pure-gauge spin-orbit coupling\ncorresponding to recent experiments. We show that such expe riments can give a direct access to the\ncollisional spin drag rate, and establish conditions for th e observation of spin drag effects. In the\nrecent experiments the dynamics is found to be mainly ballis tic leading to new regimes of reversible\nspin relaxation-like processes.\nPACS numbers: 03.75.Ss, 05.30.Fk, 67.85.-d\nThe development of spintronics, the branch of physics\nstudying spin-determined dynamical and transport phe-\nnomena was mainly related to solid-state structures. In\nthese systems, spin-orbit coupling (SOC), one of the\nkey elements of spintronics, is well-understood [1–4] and\nmany interesting spin-related effects have been studied\nexperimentally and theoretically. Very recently a de-\ntailed study of spin dynamics in ultracold atomic gases\nhas become experimentally feasible [6–9]. In particular,\ntwosystemswhereSOCisproducedbyaspecialdesignof\noptical fields, attracted a great deal of attention. One of\nthem is the spin-orbit coupled Bose-Einsteincondensates\n[5, 6], with the pseudospin 1 /2 degree of freedom. The\nother class is represented by the cold fermion isotopes\n40K in Ref. [7] and much lighter6Li studied in Ref. [9].\nIn both cases, in addition to the SOC, an effective Zee-\nman magnetic field can be produced optically.\nTypicallyinsolids,e.g. indopedsemiconductors,adis-\norder, randomizing motion of electrons, plays the dom-\ninant role in the spin dynamics. The electron-electron\ncollisions become crucial only at high temperatures, or\nin intrinsic semiconductors with optically pumped elec-\ntrons and holes [10]. From this point of view, cold atomic\ngases offer a unique possibility of seeing basic effects of\ninteractions in the pure form since the disorder is absent\nthere. The interatomic collisions lead to the spin drag\ndetermining the spin diffusion and, as we will see below,\ncan be important for the spin dynamics in cold Fermi\ngases with SOC.\nIt is well-appreciated that in the presence of strong\nSOC the effects of interatomic interactions in the spin\ndynamics are difficult to analyze as this requires tracing\nessentially coupled orbital and spin subsystems. Fortu-\nnately, these dynamics become uncoupled not only for\nvanishing SOC, but also when it corresponds to an ef-\nfective non-Abelian vector potential [11–23] of a pure\ngauge form, which happens in a broad class of systems.\nRemarkably, the three-dimensional (3D) fermionic gaseswith SOC realized in recent experiments [7, 9, 24] belong\nto this interesting class. For a pure gauge SOC the be-\nhavior of the physical system maps to that of a system\nwithout SOC, which allows to consider effects of SOC\nof an arbitrary strength. In this case all qualitative fea-\ntures of the spin dynamics are the same as for a generic\nSO field, but the analysis is much easier. Here we study\nspin dynamics for systems with a pure gauge SO cou-\npling, where the entire pattern even if it is complicated\nby the interatomic interactions, can be explored using a\nsingle formula. As a byproduct of our analysis we show\nthat studying the spin relaxation in SO coupled systems\nmay offer an alternative route to experimentally access\nthe spin drag resistivity in SOC-free systems.\nWe represent the Hamiltonian of a 3D Fermi gas of\natoms with the mass mand a linear in momentum SOC\nas follows [22, 23]:\nˆH=/integraldisplay\nd3r/braceleftBigg\nψ†/bracketleftBigg/parenleftbig\ni∇j+Aa\njσa/parenrightbig2\n2m+Bbσb/bracketrightBigg\nψ/bracerightBigg\n+ˆHint,\n(1)\nwhereψis a spinor fermionic field operator, σawith\na={x,y,z}are the Pauli matrices, the space com-\nponents Aa\njof an effective SU(2) potential parametrize\nSOC, and Bbis the Zeeman field (the time component\nof the SU(2) potential) [25]. The term ˆHint[ψ†,ψ] in\nEq. (1) describes the interparticle interactions and the\nexternaltrapping potential. In the followingwe assume a\nshort-rangeinteractioncharacterizedbya s-wavescatter-\ning length as. For simplicity we consider a homogeneous\nsystem, assumingthe validity of the local approximation.\nThe spin relaxation is probed by applying the Zeeman\nfield of the form B(t) =Bθ(−t). That is, we first bring\nthe system to an equilibrium polarized state in the pres-\nence of the static Zeeman field which is then switched off\natt= 0. After the polarizing field is released, the spin\npolarization relaxes to zero because of the combined ac-\ntion ofthe SOC-induced spin precessionand the particle-2\nparticle collisions.\nIn conventional semiconductors the SOC is usually\nweak, that is/vextendsingle/vextendsingleAa\nj/vextendsingle/vextendsingle≪pF,wherepFis the Fermi momen-\ntum. As a result, only the vicinity of the Fermi surface\nis important for the spin dynamics. Moreover, the weak\ncoupling practically does not influence the initial spin-\npolarized state. With optical fields one can achieve the\nultrastrong coupling where the effective SU(2) field/vextendsingle/vextendsingleAa\nj/vextendsingle/vextendsingle\nisclosetoorevenlargerthantheFermi-orthemeanther-\nmal momentum. In this ultrastrong coupling regime the\ncollisionalrelaxationprocessescanbecomeirrelevantand\nthe dynamics is mainly due to the nonuniform spin pre-\ncession, that is its broadening in the momentum space.\nIn addition, the strong SOC significantly modifies the\ninitial state making it strongly non-semiclassical.\nWhile the description of the spin dynamics for a strong\ngeneric SOC is very complicated, significant simplifica-\ntions occur in the special case of a pure gauge SO field\nof the following “direct product” form\nAa\nj=1\n2qjna. (2)\nHerenis a unit vector pointing along the SO precession\naxis (momentum independent in this case), and the vec-\ntorqdetermines the strength of SOC for every direction\nof the particles’ momentum coupled to the spin degrees\nof freedom. Exactly this type of the SU(2) field has been\nrealized in the recent experiments with cold Bose and\nFermi gases [6, 7, 9].\nBecause different components of the SU(2) potential in\nEq. (2) commute, it can be gauged away by a local SU(2)\nrotation of the fermion fields:\nψ(r) =U(r)/tildewideψ(r),U(r) = exp/bracketleftBigi\n2qjrjnaσa/bracketrightBig\n.(3)\nAfter the rotation, the Hamiltonian of Eq. (1) reads\nˆH=/integraldisplay\nd3r/braceleftbigg\n/tildewideψ†/bracketleftbigg\n−∇2\n2m+σ/tildewideB(r)/bracketrightbigg\n/tildewideψ/bracerightbigg\n+ˆHint[/tildewideψ†,/tildewideψ],\n(4)\nwhere/tildewideB(r) = tr{σU−1(r)(σB)U(r)}/2 is the trans-\nformed Zeeman field. For definiteness we consider the\nconfiguration of fields generated in the experiments of\nRef. [7]. This corresponds to n= (0,0,1),q= (q,0,0),\nandB= (B,0,0), which translates to the following heli-\ncoidal structure [26–29] of the transformed Zeeman field\n/tildewideB(r) =B[/hatwidexcos(qx)+/hatwideysin(qx)]. (5)\nSince the interparticle interaction and the trap potential\narespin independent the formof ˆHintisuntouchedbythe\nSU(2) rotation of Eq. (3). Therefore the physical system\nwith the pure gauge SOC subjected to a uniform polar-\nizing field B=/hatwidexBis mapped to a dual system without\nSOC in the presence of an inhomogeneous Zeeman field\n– the magnetic helix of Eq. (5).The mapping of the spin relaxation for a pure gauge\nSOC to the diffusive spin dynamics in the SO-free sys-\ntem has been already noted in the context of conven-\ntional semiconductors [22, 23] where the spin relax-\nation/diffusion is mostly determined by a disorder. Ap-\nparently this is an absolutely general statement, which\nin particular implies that the spin diffusion coefficient\nDsand the spin drag rate Γ sin cold atomic gases can be\nmeasured by monitoring the dynamics of a global aver-\naged spin, provided the proper SOC in created.\nAfter the transformation of Eq. (3) the spin dynam-\nics can be easily found using the linear response theory\nfor homogeneous, SOC-free systems. Namely, we con-\nsider a response to a week Zeeman field Eq. (5) with\nB(t) =Bθ(−t). The spin density /tildewideS(r,t) in the trans-\nformed system follows the helicoidal pattern of Eq. (5)\nwith a time-dependent amplitude S(t):\n/tildewideS(r,t) =S(t)[/hatwidexcos(qx)+/hatwideysin(qx)].(6)\nThe amplitude S(t) ofthe spin helix can be related to the\nstandard spin response function χσσ(q,ω) at the wave\nvectorq(see, e.g., Ref. [30])\nS(t) =S(0)/integraldisplay∞\n−∞dω\n2πi/bracketleftbiggχσσ(q,ω)\nχσσ(q,0)−1/bracketrightbigge−iωt\nω.(7)\nTo recover the spin dynamics in the original system we\napply the inverse SU(2) transformation to the spin dis-\ntribution /tildewideS(r,t) in Eq. (6). This unwinds the helix and\nleads to the evolutions of the spatially uniform physical\nspin distribution S(t) =S(t)/hatwidexpresented below.\nEquation (7) relates spin dynamics in the physical sys-\ntem with SOC to the smearing out of an inhomogeneous\nspin texture in a system without it. We emphasize that\nEq. (7) is exact and valid for any phase without spon-\ntaneous ferromagnetic ordering, either normal or super-\nfluid one. The SOC enters the spin dynamics solely via\nthe wave vector dependence of the spin-spin correlation\nfunction χσσ(q,ω) of the SOC-free system. Hence by\nmonitoring the spin evolution for different values of q\none can access the entire momentum and frequency de-\npendence of χσσ(q,ω).\nWe begin with an illustration of the simplest universal\ntype of spin dynamics occurring for small SO couplings,\nvatq≪Γs, wherevatis the typical atomic velocity such\nas the Fermi velocity vFor the mean thermal velocity\nvTin the degenerate and nondegenerate system, respec-\ntively. In this case the expression in the square brackets\nin Eq. (7) takes the diffusive form iω/(iω+Dsq2), which\nis guaranteed by the spin conservation law. Hence the\nspin relaxes exponentially S(t) =S(0)exp/parenleftbig\n−Dsq2t/parenrightbig\nby\ntheDyakonov-Perel’mechanism[31]. Therelaxationrate\nfor a given SOC is directly determined by the spin diffu-\nsion coefficient of the dual SOC-free system.\nFor an arbitrary strong coupling the dynamics is much\nricher. To explore the behavior in the full range of SOC,3\ndensities, and temperatures, we employ a spin version\nof the “conserving” relaxation time scheme by Mermin\n[32]. In this scheme one approximates the collision inte-\ngral in the linearized kinetic equation for the spin part\nsp(q,t) of the Wigner function by the following form,\nIcol=−Γs[sp(q,t)−seq\np(q,t)], where seq\np(q,t) is the\nequilibrium function corresponding to the instantaneous\nvalue of the average spin S(q,t). For any inhomogene-\nity wave vector qthis approximation recovers the correct\nstaticandhigh-frequencyresponse, andrespectsthelocal\nspin conservation. The dynamical spin response function\nin this scheme takes the form\nχσσ(q,ω) =χ0(q,ω+iΓs)\n1−iΓs\nω+iΓs/bracketleftbigg\n1−χ0(q,ω+iΓs)\nχ0(q,0)/bracketrightbigg,(8)\nwhereχ0(q,/tildewideω) is the Lindhard function\nχ0(q,/tildewideω) =/summationdisplay\npfp+q/2−fp−q/2\n/tildewideω−(ǫp+q/2−ǫp−q/2)(9)\nwithfpbeing the Fermi function, and ǫp=p2/2m.\nThe spin drag relaxation rate Γ senters the Mer-\nmin’s scheme as a phenomenological parameter originat-\ning from interatomic collisions. Although collisions con-\nserve the total momentum, they cause a friction between\ndifferent spin species, and thus lead to relaxation and\ndamped response in the spin channel. Strictly speaking,\nthe spin drag rate Γ sdepends on q. It is however clear\nthat for large q/greaterorsimilarpFthe average spin relaxes mostly\nbecause of fast nonuniform precession (ballistic motion).\nThe spin drag contribution becomes qualitatively impor-\ntant in the opposite limit of q≪pF. Therefore below\nwe adopt the standard expression based on the Born ap-\nproximation at q= 0 [33, 34]\nΓs=/parenleftbigg4πas\nm/parenrightbigg2/summationdisplay\nkk2\n3mnT/integraldisplay∞\n0dω\nπ[Imχ0(k,ω)]2\nsinh2(ω/2T),(10)\nwhereTis the temperature, and nis the concentration\nof the atoms. The spin drag rate of Eq. (10) has the\nfollowing characteristic temperature dependence\nΓs∼EF(pFas)2×\n\n(T/EF)2, T≪EF\n(T/EF)1/2, T≫EF.(11)\nThe low- Tbehavior is typical for the Fermi liquid, while\nin the high- Tlimit it describes the frequency of collisions\nbetween particles with a scattering cross-section ∼a2\ns\nmoving at the mean thermal velocity vT∼/radicalbig\nT/m. For\nthe repulsive interaction the Stoner instability criterion\nlimits the product pFas< π/2 [35], thus preventing one\nfrom having a fast relaxation in the degenerate gas.\nEquations (7)-(10) provide us with a simple unified\ndescription of the spin dynamics in a wide range of tem-\nperatures, SO couplings, densities and interactions. Thisscheme captures all main physical effects which influence\nthe spin relaxation. The difference in the Fermi functions\nin Eq. (9) describes the modification of the initial state\nby the SO coupling. It selects a part of the momentum\nspace with the particles polarized by the initial Zeeman\nfield, as it is shown in Fig. 1 for the very strong coupling.\nThe difference in energies in the denominator in Eq. (9)\ndeterminesthefrequencyofthespinprecession, whilethe\nimaginary shift of ωbyiΓsdescribes the spin drag effect.\nNow we use Eqs. (7)-(10) to explore different regimes of\nspin dynamics and discuss those not achievable in the\nsolid-state experiments.\nFIG. 1: (Color online.) Schematic plot of precession of spin s\nof atoms with different momenta sp(t), as shown by short\narrows, in the case of very strong SOC with q >2pF. Filled\ncircles show spin-split thermally broadened Fermi spheres .\nWe consider first the low-temperature degenerate gas,\nT≪EF. If SOC is weak, so that q≪pF, only particles\nnear the Fermi surface participate in the dynamics. The\ncharacteristic time scale is set up by 1 /qvF, that is the\ntime for ballistically traversing the period of the spin he-\nlix. The relation of this time to the spin drag rate Γ sde-\ntermines the regimeofthe spin dynamics. If qvF/Γs≫1,\nthe dynamics is ballistic and the evolution of the total\nspin is dominated by the inhomogeneous p-dependent\nspin precession of the Fermi-surface particles. Appar-\nently this is always the case at sufficiently low tempera-\ntures,T→0. In the opposite limit of qvF/Γs≪1 the\ndynamics is diffusive, which corresponds to the conven-\ntional Dyakonov-Perel’ spin relaxation mechanism [31].\nFrom Eq. (11) we find that with the increase of the tem-\nperature the ballistic regime crosses over to the diffusive\none atTdb∼(vF/as)/radicalbig\nq/pF. Condition Tdb≪EFim-\npliesq≪pF(pFas)2for the SOC strength. Taking into\naccount that, e.g., in Ref. [7], pFas∼10−2, we conclude\nthat to reach the diffusive regime one has to increase the\ninteraction or to make the SOC considerably weaker.\nApart from determining the spin drag rate, the tem-\nperature effects modify the dynamics via broadening the\nFermi distribution in Eq. (9). Indeed, the broadening of\nthe velocity distribution by δvF∼(T/EF)vFcauses the\nbroadening in the precession rate by the typical value of4\n00.5 11.5 22.5\nt [1/EF]-0.500.51S(t)/S(0)q = pF\nq = 2pF\nq = 4pF (a)T=EF /10\n0 0.2 0.4 0.6\nt [1/EF]-0.200.20.40.60.81S(t)/S(0) q=pTpT=(mT)1/2T=4EF\n q=2pT\n q=4pT\n(b)\nFIG. 2: (Color online.) Time dependence of the total spin at\nstrong spin-orbit coupling. (a) low temperature, degenera te\ngas, (b) high temperature, classical gas. The values of qand\nTare shown in the plots.\nδΩ = (T/EF)qvF. In the ballistic regime the resulting\ntime dependence of the total spin is\nS(t) =S(0)sin(vFqt)\nsinh(πtδΩ)πT\nEF, (12)\nwhere the rapid increase in sinh( πtδΩ) att >1/δΩ leads\nto the exponential damping [36] in the spin oscillations\ninitially following the sin( vFqt)/vFqttime dependence.\nAtδΩ/Γs/lessorsimilar1,that is at T/greaterorsimilarT2\ndb/EFthe asymptotic\nbehavior is determined by Γ srather than by δΩ.\nFor strong SOC the dynamics is mostly ballistic, but\nstill quite nontrivial and diverse. As it can bee seen in\nFig.2(a), at q=pFthespinrelaxestozeromonotonically,\nwhile pronounced large amplitude oscillations appear for\nultrastrong SOC with q≫pF. The reason is that in the\nlatter case we have two well separated spin-split Fermi-\nspheres. The position ofthe sphere centerdetermines the\nmean precession rate, and the broadening is determined\nby the Fermi momentum pF. An interesting intermedi-\nate regime occurs at q= 2pFwhen the spin-split Fermi-\nsurfaces touch each other and the time dependence of the\nspin takes the form\nS(t) =S(0)sin2(4EFt)\n(4EFt)2. (13)\nNow we consider the high-temperature case, T≫EF.\nIn thislimit themean freepath ∼vT/Γsdoesnot dependon the temperature [see Eq. (11)]. Therefore the dynam-\nical regime is temperature-independent – the dynamics is\nballistic if ( q/pF)/(pFas)2≫1 and diffusive otherwise.\nThe condition of diffusive relaxation is equivalent to the\ncondition Tdb≪EF. As a result, if q≥pF(pFas)2, the\nspindynamicsisballisticatanytemperature. Inthiscase\nthe width of the thermal Maxwell distribution is much\nlargerthan the scale of SOC and the inhomogeneous pre-\ncessionisequivalenttotheFouriertransformoftheGaus-\nsian momentum distribution with the coordinate qt/m.\nFor a weak SOC in the ballistic regime the spin relax-\nation is Gaussian S(t) =S(0)exp/bracketleftBig\n−(vTqt)2/6/bracketrightBig\nwith\nthe time scale of 1 /qvT. At larger scattering length as,\nthat is at larger Γ s, we enter the diffusive regime where\nthe Gaussian damping slows down to the exponential\nDyakonov-Perel’ relaxation with the timescale Γ s/q2v2\nT.\nIn the nondegenerate gas the function S(t) is practically\nalwaysmonotonic. To detect signatures of the oscillatory\nspin precession one needs to go to the ultrastrong SOC\nand to satisfy experimentally the condition q≫pT.\nIt is worth noting that in general for the weak SOC\nthe spin dynamics in the ballistic limit is determined by\nthe parameter vatqsince this is the only characteristics\nof the low-energy excitation spectrum entering Eq. (9).\nFor the strong coupling with clearly separated spin-split\nmomentum distributions we have two main parameters\ndescribing spin dynamics: spin precession rate of the or-\nder ofq2/2mand spin relaxationrate ofthe orderof qvat,\nas can be seen from comparison of Fig.2(a) and (b) for\nthe low- and high-temperature gases, respectively. This\nlinks the spin relaxation dynamics to the spectrum of\nelectron-hole excitations in the Fermi gas.\nWe emphasize that the above limiting regimes of\nthe spin dynamics are described within a single uni-\nfied scheme, which is completely determined by four\nexperimentally controllable parameters, the density n,\nthe temperature T, the scattering length as, and the\nSOC strength q. Apparently all crossovers and transi-\ntion regimes are also covered by this approach.\nTo conclude, we have shown that the spin dynamics\nin cold Fermi gases can be mapped on the spectrum of\nparticle-hole excitations. The spin drag rate can be ob-\ntained from the relaxation of the uniform spin density\nwithout generating an inhomogeneous spin distribution\nin the real-space,providedthe conditionsofdiffusive spin\ndynamics in terms of the SOC strength, concentration,\nand the scattering length are satisfied.\nIVT acknowledges funding by the Spanish MEC\n(FIS2007-65702-C02-01) and ”Grupos Consolidados\nUPV/EHU del Gobierno Vasco” (IT-319-07). This work\nof EYS was supported by the University of Basque\nCountry UPV/EHU under program UFI 11/55, Spanish\nMEC (FIS2012-36673-C03-01), and ”Grupos Consolida-\ndos UPV/EHU del Gobierno Vasco” (IT-472-10).5\n[1] R. Winkler, Spin-orbit Coupling Effects in Two-\nDimensional Electron and Hole Systems , Springer Tracts\nin Modern Physics (2003).\n[2] J. Fabian, A. Matos-Abiague, C. Ertler, P. Stano, and I.\nZutic, Acta Physica Slovaca 57, 565 (2007).\n[3]Spin Physics in Semiconductors , Springer Series in Solid-\nState Sciences, Ed. by M.I. Dyakonov, Springer (2008)\n[4] M.W. Wu, J.H. Jiang, and M.Q. Weng, Physics Reports\n493, 61 (2010).\n[5] T. Stanescu, B. Anderson, and V. Galitski, Phys. Rev. A\n78, 023616 (2008)\n[6] Y.-J. Lin, K. Jimenez-Garcia, and I. B. Spielman, Nature\n471, 83 (2011)\n[7] P. Wang, Z.-Q. Yu, Z. Fu, J. Miao, L. Huang, S. Chai, H.\nZhai, and J. Zhang, Phys. Rev. Lett. 109, 095301 (2012)\n[8] A. Sommer, M. Ku, G. Roati, and M. W. Zwierlein, Na-\nture472, 201 (2011)\n[9] L. W. Cheuk, A. T. Sommer, Z. Hadzibabic, T. Yefsah,\nW. S. Bakr, and M. W. Zwierlein, Phys. Rev. Lett. 109,\n095302 (2012)\n[10] M. M. Glazov and E. L. Ivchenko, Pis’ma Zh. Eksp.\nTeor. Fiz. 75476 (2002) [JETP Lett. 75, 403 (2002)];\nM. M. Glazov and E. L. Ivchenko, JETP 991279 (2004);\nD. Stich, J. Zhou, T. Korn, R. Schulz, D. Schuh, W.\nWegscheider, M. W. Wu, and C. Sch¨ uller, Phys. Rev.\nLett.98, 176401 (2007); L. H. Teng, P. Zhang, T. S. Lai,\nand M. W. Wu, Europhys. Lett. 84, 27006 (2008); M. D.\nMower, G. Vignale, and I. V. Tokatly, Phys. Rev. B 83,\n155205 (2011); X. Liu and J. Sinova, Phys. Rev. B 86,\n174301 (2012).\n[11] V. P. Mineev and G. E. Volovik, Journal of Low Temper-\nature Physics 89, 823 (1992).\n[12] J. Fr¨ ohlich and U. M. Studer, Rev. Mod Phys. 65, 733\n(1993).\n[13] I. L. Aleiner and V. I. Fal’ko, Phys. Rev. Lett. 87, 256801\n(2001).\n[14] L. S. Levitov and E. I. Rashba, Phys. Rev. B 67, 115324\n(2003).\n[15] Y. Lyanda-Geller, Phys. Rev. Lett. 80, 4273 (1998), Y.\nLyanda-GellerandA.D.Mirlin, Phys.Rev.Lett. 72.1894\n(1994), J. B. Miller, D. M. Zumb¨ uhl, C. M. Marcus, Y.\nB. Lyanda-Geller, D. Goldhaber-Gordon, K. Campman,\nand A. C. Gossard, Phys. Rev. Lett. 90, 076807 (2003).[16] S.-R. Eric Yang and N. Y. Hwang, Phys. Rev. B 73,\n125330 (2006).\n[17] Q. Liu, T. Ma, and S.-C. Zhang, Phys. Rev. B 76, 233409\n(2007).\n[18] N. Hatano, R. Shirasaki, and H. Nakamura, Phys. Rev.\nA75, 032107 (2007).\n[19] J.-S. Yang, X.-G. He, S.-H. Chen, and C.-R. Chang,\nPhys. Rev. B 78, 085312 (2008).\n[20] B.W.A. Leurs, Z. Nazario, D.I. Santiago, and J. Zaanen,\nAnnals of Physics 323, 907 (2008).\n[21] R. Raimondi and P. Schwab, Europhys. Lett. 87, 37008\n(2009); M. Milletar, R. Raimondi, and P. Schwab, Euro-\nphys. Lett. 82, 67005 (2008).\n[22] I. V. Tokatly, Phys. Rev. Lett. 101, 106601 (2008).\n[23] I. V. Tokatly and E.Ya. Sherman, Annals of Physics 325,\n1104(2010), I.V.TokatlyandE.Ya.Sherman,Phys.Rev.\nB82, 161305 (2010).\n[24] Theoretical analysis of the optical fields leading to th is\ntype of spin-orbit coupling was done by X.-J. Liu, M. F.\nBorunda, X. Liu, and J. Sinova, Phys. Rev. Lett. 102,\n046402 (2009).\n[25] The units with /planckover2pi1≡1 are employed and the summation\nover repeated indices is assumed throughout the paper.\n[26] B. A. Bernevig, J. Orenstein, and S.-C. Zhang, Phys.\nRev. Lett. 97, 236601 (2006).\n[27] M.-H. Liu, K.-W. Chen, S.-H. Chen, C.-R. Chang, Phys.\nRev. B74235322 (2006).\n[28] J. D. Koralek, C. P. Weber, J. Orenstein, B. A. Bernevig,\nShou-Cheng Zhang, S. Mack, and D. D. Awschalom, Na-\nture458, 610 (2009).\n[29] V. A. Slipko and Y. V. Pershin, Phys. Rev. B 84, 155306\n(2011).\n[30] D. Forster, Hydrodynamic Fluctuations, Broken Symme-\ntry, and Correlation Functions (Addison-WesleyPublish-\ning, Reading, Masachusetts, 1994).\n[31] M. I. Dyakonov and V. I. Perel’, Sov. Phys. Solid State\n13, 3023 (1972).\n[32] N. D. Mermin, Phys. Rev. B 1, 2362 (1970).\n[33] I.D’AmicoandG.Vignale, Phys.Rev.B 62, 4853(2000).\n[34] M. Polini and G. Vignale, Phys. Rev. Lett. 98, 266403\n(2007).\n[35] R. A. Duine, M. Polini, H.T.C. Stoof, and G. Vignale,\nPhys. Rev. Lett. 104, 220403 (2010).\n[36] M. M. Glazov, Solid State Commun. 142, 531 (2007)."    },    {        "title": "1912.09054v2.Spin_rotation_by_resonant_electric_field_in_few_level_quantum_dots__Floquet_dynamics_and_tunneling.pdf",        "content": "arXiv:1912.09054v2  [cond-mat.mes-hall]  30 Jul 2020Spin rotation by resonant electric field in few-level quantu m dots: Floquet dynamics\nand tunneling\nD.V. Khomitsky,1,∗E.A. Lavrukhina,1and E.Ya. Sherman2,3\n1Department of Physics, National Research Lobachevsky Stat e University of Nizhni Novgorod,\n603950 Gagarin Avenue 23, Nizhny Novgorod, Russian Federat ion\n2Department of Physical Chemistry, The University of the Bas que Country, 48080 Bilbao, Spain\n3IKERBASQUE Basque Foundation for Science, Bilbao, Spain\n(Dated: July 31, 2020)\nWe study electric dipole spin resonance caused by sub-terah ertz (THz) radiation in a multilevel\nfinite-size quantum dot formed in a nanowire focusing on the r ange of driving electric fields ampli-\ntudes where a strong interplay between the Rabi spin oscilla tions and tunneling from the dot to\ncontinuum states can occur. A strong effect of the tunneling o n the spin evolution in this regime\noccurs due to formation of mixed spin states. As a result, the tunneling strongly limits possible\nspin manipulations time. We demonstrate a backaction of the spin dynamics on the tunneling and\nposition of the electron. The analysis of the efficiency of the spin manipulation in terms of the\nsystem energy shows that tunneling decreases this efficiency . Fourier spectra of the time-dependent\nexpectation value of the electron position show a strong effe ct of the spin-orbit coupling on their\nlow-frequency components. This results can be applied to op erational properties of spin-based\nnanodevices and extending the range of possible spin resona nce frequencies to the THz domain.\nI. INTRODUCTION\nImplementation ofelectronspinsin nanoscalesemicon-\nductors such as quantum dots and other heterostructures\nas scalable hardware elements for quantum information\nprocessing [1, 2] requires tools for fast and efficient spin\nmanipulation in these systems. A preferable way of such\nmanipulations is related to application by electric rather\nthan by magnetic fields since well-controllable electric\nfield can be produced on demand by modern nanoscale\nelectrical engineering tools. The venue for such a manip-\nulation is offered either by the natural or by engineered\npresence of spin-orbit coupling (SOC) in these semicon-\nductor structures, leading to the electric dipole spin res-\nonance (EDSR) [3–6], and making it an efficient tool of\nmanipulation of electron spins in solids. This ability to\nmanipulate spin by electric field appears, in general, due\nto the fact that spin precession is related to the electron\ndisplacement caused by application of external electric\nfield [7].\nThere are three basic systems where the electric dipole\nspin resonance can be observed [8]. First realization is\ngivenbyitinerantelectronsinbulkcrystals[3–6], surfaces\n[9] with spin-orbitcoupling[10], andin artificialsemicon-\nductor structures such as heterojunctions and quantum\nwells [11] and superlattices [12]. Second realization, the-\noretically analyzed in Refs. [13–23] and experimentally\nrealized for electrons [24] and holes [25], is related to car-\nriers in single and double semiconductor quantum dots\n(a similar approach can be applied for carriers in carbon\nnanotubes [26]). Due to astrongconfinement byexternal\npotential, here the spectrum of carriers is discrete, and\nthe carrier always remains in the dot, either two- [24] or\n∗khomitsky@phys.unn.ruone-dimensional [27], even when a strong electric field is\napplied. Third realization occurs for potentials vanish-\ning at large distances with the examples being: electrons\n(holes) on donors (acceptors) [28–31], in finite-range po-\ntential produced by various charged gates [32–36], and in\nsurface-based self-assembled quantum dots [37–39].\nAn important example of spin manipulation in system\nwith the confining potential vanishing at large distances\nis given by gate-defined quantum dots- [40] and P donors\nin Si [41], where the hyperfine coupling to donor nuclear\nspins results in experimentally achieved possibilities of\nspin qubit manipulations [42–46]. In a spin ensemble of\nthese donors, the coherent Rabi dynamics has been ob-\nserved [47]. However, the effect of SOC in these systems\nis weaker than that achievable in other nanostructures\nsuchas InSb quantum wiresand dots and othermaterials\nwith large tunable SOC. Since the large SOC generally\nleadsto fasterspin flip and shorteroperatingtimes, these\nsemiconductor structures are of interest for the applica-\ntions, and will be considered here in detail.\nThis third realization is of our interest here since low-\nfrequency electric fields can ionize the electron states by\ntunneling and, thus, cause additional nontrivial spin dy-\nnamics resulting from the spatial spread of the electron\nwavefunction. For example, even for the inter-minima\ntunneling in a double quantum dot at relatively short\ndistances the coupled spin and spatial dynamics may\ncause strong nonlinearities in the spin precession Rabi\nfrequencydependence on the driving field amplitude [16].\nThus, we are interested in Floquet dynamics in a tun-\nneling system, where a combination of two long time\nscales becomes important. The first long time scale is\nthe time of spin flip driven by electric field, inverselypro-\nportional to the field amplitude. The second one is the\ntunneling time due to driving by electric field produc-\ning coupling of localized stats to the continuum, being\nstrongly, approximately exponentially, dependent on the2\ndopant layergate-produced\npotential\ntop gateground state\nFIG. 1. Schematic presentation of a structure including a\ngated nanowire on a doped bulk substrate.\ninverse field amplitude. If both these times are of the\nsame order of magnitude, the coupled spin and charge\ndensity dynamics, studied in this paper, becomes highly\nnontrivial. In a multilevel quantum dot the tunneling is\na well-defined semiclassical process, in contrast to that in\nshallow quantum dots [48]. As a result of a stronger con-\nfinement, the amplitude of the electric field required to\noperate a spin flip is considerably larger and, therefore,\nthe involvement of the continuum states is qualitatively\ndifferent.\nA relevant issue here is the range of the frequencies\nrequired for the spin resonance. The major part of ex-\nperimental setups produces frequencies within the tens\nGHz-range. However, fast and decoherence-robust spin\nmanipulation may require higher, sub-THz frequencies.\nThereasonforthis requestis duetothe fact thatthe pos-\nsible spin Rabi frequencies decrease with the decrease in\nthe Zeeman splitting, and, therefore will require stronger\ndriving fields. Combination of progress in development\nof THz sources [49–51] and demand for high driving fre-\nquencies for the EDSR in two-dimensional systems [52]\nand for fast spin manipulation in conventional nanos-\ntructures will make these techniques applicable in spin-\nmanipulation technologies.\nAn interesting aspect of this dynamics is that usu-\nally the EDSR assumes a pure spin state, defined on the\nBlochsphere,wherethespinrotationoccurswithaminor\nchangeintheelectronorbitalstate. However,inthepres-\nence of spin-orbit coupling the spin states become mixed.\nThis circumstance strongly modifies the entire spin evo-\nlution. For applications, the efficiency of the spin-flip\nprocess, that is the ratio of the Zeeman energy to the to-\ntal energy acquired by electron as a result of the electric\nfield action, is of interest. We will demonstrate how the\nefficiency depends on the system parameters. Moreover,\nthe spin-orbit coupling causes a feedback of the spin dy-\nnamics on the tunneling efficiency and resulting electron\nposition. Even at a relatively weak spin-orbit coupling\nthis effect can be strong, as we will demonstrate below.\nThis paper is organized as follows. In Sec. II we in-\ntroduce the Hamiltonian of our system and discuss time-\nindependent and periodic contributions to the electron(a)\n(b)0\n0/c40 /c41 ) 4 / 1 (, /c43nTxV\n/c40 /c41 ) 4 / 3 (, /c43nTxV\nFIG. 2. Schematic illustration of combined tunneling and\nspin precession caused by a periodic external field. Arrows i n\nthe circles located at the points of tunneling-induced esca pe\nshow that dependent on the direction of the tunneling, spin\nprecesses in opposite directions.\nenergy. In Sec. III we describe the computational model\nof electron states and their dynamics and provide the\nspecific system parameters. In Sec. IV the main results\nare presented: spin dynamics, structure of the electron\nwavefunction, time dependencies of mean energy and ef-\nficiency of spin flip, localization probabilities, coordinate\nevolution, and its Fourier power spectra. Finally, in Sec.\nVwegiveourconclusionsandmakerelationtothe design\nof devices using the effects described in the paper.\nII. HAMILTONIAN AND OBSERVABLES\nWe consider a narrow nanowire, elongated along the\nx−axis located on the top of a doped substrate, as\nschematically presented in Fig. 1. Assuming that\nthe transversal modes are not excited, we characterize\nthe electron motion by a two-component wave function\nψ(x,t) = [ψ1(x,t),ψ2(x,t)]T,where T stands for the\ntransposition, and use the effective mass approximation\nHamiltonian with:\nH(t) =¯h2k2\n2m+V(x,t)+∆\n2σz+ασyk.(1)\nHeremis the electron effective mass, and k=−i∂/∂xis\nthe wave vector operator. The time-dependent potential\nis the sum of the static confinement potential U(x), as\ncan be produced by electrostatic gate shown schemati-\ncally by a rectangle above the nanowire in Fig. 1, and\nexternal driving giving\nV(x,t) =U(x)+Fxsinωdt, (2)\nwhereF≡eE0,eis the fundamental charge and E0\nis the field amplitude. The Rashba SOC HR=ασyk\nwhile, more complicated forms of spin-orbit coupling can3\nbe found in Refs. [53–56]. The constant magnetic field\nB= (0,0,B) produces the Zeeman term in the Hamil-\ntonian. Here the Zeeman splitting |∆|=µB|gB|,where\nµBis the Bohr magneton, and gis the Land´ e-factor.\nThe spin-split Zeeman partners participate in the spin\nresonance driven by the external periodic electric field in\n(2). The reason for this participation is that due to the\npresence of SOC the eigenstates of (1) at F= 0 always\ncontain both spin components. Therefore, spin-flip tran-\nsitions still be caused by the electric field described with\nthe position operator, producing the EDSR. In this case,\nthe well-defined spin Rabi oscillations occur if one ne-\nglects the tunneling effects, that is coupling to the other\norbital and continuum states. The dynamical interac-\ntion between discrete and continuum states during the\nevolution driven by periodic electric field causes nontriv-\nial effects to be discussed in the following Sections. One\nof the effects is illustrated schematically in Fig. 2: in the\npresence of SOC the direction of spin precession is differ-\nent, depending on the direction of the tunneling escape,\nmaking a controlled spin flip rather challenging task es-\npecially if the tunneling is triggered by the alternating\nfield responsible for the spin evolution.\nTo describe the evolution we solve the nonstationary\nSchr¨ odinger equation\ni¯h∂tψ(x,t) =H(t)ψ(x,t), (3)\nwhere∂t≡∂/∂t,and calculate expectation values for\nexperimentally measurable observables Oas:\n/angbracketleftO(t)/angbracketright=/integraldisplay∞\n−∞ψ†(x,t)Oψ(x,t)dx. (4)\nThe observables will be specified later in the text.\nIII. MODEL AND NUMERICAL APPROACH\nA. Basis states and model of the dynamics\nWe choose the static potential U(x) =\n−U0/cosh2(x/d),assuming that it is deep such\nthat the parameter ξ≡U0md2/¯h2≫1.Although the\nexact bound states for this potential are well-known\nanalytically [60] in terms of the Legendre functions, it\nis practical to consider simple approximations based on\nthe expansion at x≪dasU(x) =−U0/parenleftbig\n1−x2/d2/parenrightbig\n.\nAs a result, the lowest bound states can be described\nin terms of a harmonic oscillator with the energy\ninterval ¯hωho=U0//radicalbig\nξ/2≪U0and the Gaussian width\nd/(2ξ)1/4≪d.The numerically accurate eigenenergies\nE(0)\nnand basis states of the Hamiltonian ¯ h2k2/2m+U(x)\nare found by expansion in the basis of periodic functions\nsin(x/λn+δn) whereλn= 2L/(πn) andδn=πn/2,\nn= 1,2,...satisfying zero boundary conditions at\nx=±L,where 2Lis a large wire length.Then, we numerically produce basis of two-component\neigenstatesφn(x) of the spinful Hamiltonian\nH0=¯h2k2\n2m+U(x)+∆\n2σz+ασyk (5)\nwith the spin-split eigenenergies Ensuch asH0φn(x) =\nEnφn(x). As a result, the evolution of wave function is\npresented as\nψ(x,t) =/summationdisplay\nncn(t)e−iEnt/¯hφn(x). (6)\nThus, the problem is reduced to obtaining full set of co-\nefficientscn(t) by numerical solution of a large system\nof linear differential equations, as will be presented be-\nlow with some computations details given in Ref. [48].\nHere we choose the interval 2 L= 320dand take 4000 ba-\nsis statesφn(x).The initial state is chosen as ψ(x,0) =\nφ1(x),that is the ground state of H0,where for the re-\nalistic parameters one has /angbracketleftσz(0)/angbracketright ≈1.\nB. System parameters\nIn the following analysis we consider a gate-formed\nquantum dot based on InSb quantum wire, see Fig. 1.\nFor this material m= 0.0136 of a free electron mass [57]\nandg=−50.6.The magnitude of SOC αin InSb can be\ntuned by the gate voltage up to 100 meVnm [58, 59].\nFor the quantum dot parameters we accept U0=\n27 meV and d= 50 nm with five discrete quantum\ndot levelsE(0)\n1,...,E(0)\n5being formed [60] with ω0=\n(E(0)\n2−E(0)\n1)/¯h= 13.43 ps−1, and the associated period\nT0= 2π/ω0= 0.468 ps gives the natural time scale for\nthe evolution. We consider the magnetic field B= 0.447\nT, providing the Zeeman splitting of the ground state\ncorresponding to the resonant driving frequency (includ-\ning the contribution of spin-orbit coupling) ωd= 2 ps−1\n(0.32THz), givingtheratiotothefundamentalfrequency\nωd/ω0= 0.149. Such Zeeman splitting requires a sub-\nTHz-range driving hardware, as it has been discussed in\nthe Introduction. The advantages of this large splitting\ncome from the following circumstances: it is known that\nsmaller Zeeman splitting gives smaller values of spin-flip\nmatrix elements of the nonstationary part in (2) [48],\nwhich results in a higher driving field amplitude Fre-\nquired to operate the spin flip in a desired time. As\na result, the leakage to the continuum will be strongly\nenhanced, as we show below. Thus, the larger driving\nfrequencies may provide the desired spin flip time with\nloweroperatingfield which fostersthe applicabilityofthe\nproposed mechanism.\nWe consider the effect of a relatively weak SOC α= 5\nmeVnm and a relatively strong α= 25 meVnm for vari-\nousdriving amplitudes. Takingintoaccount the material\nparameters for spin-obit coupling produced by external\nelectricfieldinInSb[61], weobtainthattheeffectivetwo-\ndimensional donor dopant layer concentration, as shown4\nin Fig. 1, corresponding to this spin-orbit coupling, is in\ntherangeof1011−1012cm−2.Sincethenonuniformstatic\nelectric field forming the quantum dot has a z−compo-\nnent, it can contribute to the Rashba coupling. Tak-\ning into account that this field component cannot exceed\n∼U0/ed,and using the corresponding material param-\neter [61], we obtain that this contribution to the SOC\ncoupling is less than 2.5 meVnm, considerably smaller\nthan the accepted values for InSb nanowire.\nNow we discuss the effect of the Dresselhaus SOC, typ-\nical for III-V semiconductors. The corresponding Hamil-\ntonianhas the form HD=αD(κσ), whereαDis the cou-\npling constant and κx=kx/parenleftbig\nk2\ny−k2\nz/parenrightbig\n,with other com-\nponents obtained by cyclic permutation. This κ−depen-\ndence makes the Dresselhaus SOC in nanowires strongly\ndependent on their shape and orientation. For exam-\nple, for the wire grown along the crystallographic x-axis,\nwe obtainH[x]\nD=αDkx/parenleftbig\n/angbracketleftk2\ny/angbracketright−/angbracketleftk2\nz/angbracketright/parenrightbig\n, where/angbracketleftk2\ni/angbracketrightis the\nshape-dependent expectation value of the corresponding\noperator [62, 63]. As a result, this coupling can be made\nzero for the highly symmetric wires of the appropriate\ncross-section. For this growth direction, the Dresselhaus\ncoupling is similar to the Rashba coupling in Eq. (5).\nFor the wires grown along the yorz-axis, it will modify\nonly the direction of the SOC field without considerable\nchanges in the spin dynamics. Taking into account the\nestimateαD≈0.76 eVnm3[61] we obtain for typical val-\nues/parenleftbig\n/angbracketleftk2\ny/angbracketright−/angbracketleftk2\nz/angbracketright/parenrightbig\n∼10−2nm−2the coupling of the order\nof10meVnm, notconsiderablymodifyingtheparameters\nof the driving fields of interest for the present work.\nIV. SPIN DYNAMICS AND FEEDBACK ON\nTHE TUNNELING AND POSITION\nA. Time dependence of spin: the role of the\ntunneling\nIn Fig. 3 we show the time dependence of /angbracketleftσz(t)/angbracketrightcor-\nresponding to Eq. (4) with O=σzgiving\n/angbracketleftσz(t)/angbracketright=/summationdisplay\nn1,n2c∗\nn1(t)cn2(t)e−i(En2−En1)t/¯h/angbracketleftφn1|σz|φn2/angbracketright.\n(7)\nHere and below the time is measured in the units of T0,\nand we track the evolution on calculation time scales of\nabout 650T0for small SOC and 200 T0for large SOC.\nThese ranges are determined by the spin flip process\nwhich needs to be well-developed. As a result, one can\nobtain from the spin dynamics the characteristic spin\nevolution time. One can see that a sizable decrease in\nthistimecanbeobservedformoderateincreaseof Ffrom\n0.16 to 0.2 meV/nm which is in the framework of usual\nRabi frequency dependence on the driving field ampli-\ntude. When the field grows further, the electron escapes\nthe quantum dot quickly, before well-established Rabi\nspin oscillations are developed. This effect is stronger at\nlowervaluesofSOC α= 5meVnm. Onecanseealsothatat the fields F≥0.2 meV/nm the Rabi oscillations are\naccompanied by visible damping. We attribute this ef-\nfect to the interactions between localized and continuum\nstates which is enhanced at higher driving fields. The\ncontinuum states form a dense set of levels with alter-\nnating spin projections due to the Zeeman splitting [48].\nTransitionstothesestatesdrivenbyexternalelectricfield\nlead to the wavefunction consisting of spin-up and spin-\ndown states, and their relative contributions to the total\nnorm have a tendency to equalize when the electron is\npushed into the continuum.\nTo understand the interplay of tunneling and spin flip\nwefirst evaluate the upper limit for the semiclassicaltun-\nneling rate wtunin the model triangular potential Fx\nneglecting tunneling from the excited states as [64]:\nlnω0\nwtun<4√\n2\n3ξ1/2U0\nFd. (8)\nEquation (8) gives the upper estimate of the quantity\nsince the real tunneling barrier is lower and more narrow\nthan the triangular one. Although this equation shows\nthat the tunneling probability rapidly increases at Fd\nbeing a sizable fraction of U0,the actual tunneling rate,\nvery strongly F−dependent, can only be obtained nu-\nmerically. Since spin-flip Rabi frequency Ω Ris a linear\nfunction of F(being strictly linear when the two-level\napproximation is applicable) with Ω RT0≪1, the effect\nof tunneling on the spin-flip processes occurs in a rela-\ntively narrow range of the driving fields when two rates\nare comparable.\nFor a stronger Rashba coupling the spin dynamics\nbecomes faster and well-defined Rabi oscillations have\nenough time to develop before the tunneling occurs, as it\ncanbe seen in Fig. 3(b). However,when the field exceeds\n0.2meV/nm, the electrontunnels from thedot soquickly\nthat the spin flip has no time to be established, regard-\nless of the spin-orbit coupling strength, and the contin-\nuum states play the dominant role. In other words, high\nvalues ofFproduce the tunneling rate wtun(see Eq.(8))\nsatisfying the inequality Ω R<∼wtun,where the spin flip\ntime is longer than the tunneling escape time. As to\nthe driving field range desired for controlling spin flips,\na small amplitude produces slow Rabi oscillations, which\non long times may be hampered by various decoherence\nprocesses. So, it is oneofourgoalsto estimate accurately\nthe optimal range of electric fields where the driving is\nstrong enough to trigger fast spin flips while being not\nsufficiently strong to cause the quantum dot ionization.\nIt is important that the spin states we consider are\nmixed ratherthan pure. This can be seenwith the (2 ×2)\nspin density matrix\nρ(t) =/integraldisplayL\n−Lψ(x,t)ψ†(x,t)dx. (9)\nAtt= 0,we have |ψ2(x,0)| ≪ |ψ1(x,0)|with/angbracketleftσz(0)/angbracketright ≈\n1,such that the initial spin state has a high purity with\ntrρ2(0)≈1.Note that a state is pure if and only if5\n0100200 300 400 500600\nt/T0-101 〈σz(t)〉\n0 50 100 150\n t/T001〈σz(t)〉 α=5 meVnm\n0.16 meV/nm0.22 meV/nm\nα=25 meVnm\n0.2 meV/nm\n0.22 meV/nm\n0.18 meV/nm\n0.16 meV/nm0.18 meV/nm0.2 meV/nm0.5\n-0.5\n-0.50.5(a)\n(b)\nFIG. 3. Spin projection /an}bracketle{tσz(t)/an}bracketri}htforα= 5 meVnm (a) and\nα= 25 meVnm (b) for various driving field amplitudes\nas marked near the curves. At a relatively high F≥0.2\nmeV/nm the spin dynamics demonstrates visible damping of\nthe Rabi oscillations, as caused by the interaction with con -\ntinuum states increasing with the driving field amplitude.\nψ1(x,t) =cψ2(x,t),wherecis a complex constant. With\nthe course of time, two components of ψ(x,t) begin to\ndevelop different shapes, producing spin density matrix\nin Eq. (9) with tr ρ2(t)<1 characterizing a mixed spin\nstate [65]. An example of the structure of mixed spin\nstates in the driven spin dynamics is shown in Fig. 4\nat the end of the computational evolution t= 655T0for\nα= 5 meVnm and F= 0.2 meV/nm. Although at this\ntime and driving field, the electron density still has a\nwell-defined peak inside the confining potential, a broad\ncontribution of the continuum states with a relatively\nlow total probability density has already been formed at\nd<|x|<L,possessing more than half of the total norm.\nThis Figure shows that ψ1(x,t) andψ2(x,t) are\nstrongly different, both in the real and imaginary parts.\nDensities of spin components |ψ1(x,t)|2and|ψ2(x,t)|2\n(not shown in the Figure) are shifted with respect to\neach other, corresponding to Fig. 2, where direction of\nthe tunneling determines the direction of the spin preces-\nsion.\nBy evaluating the spin flip times in Fig. 3 one may\n ψ(x) components (nm-1/2)\n-200 -100 0 100 200\nx [nm] ψ(x) components (nm-1/2)Reψ1(x)\nReψ2(x)\nImψ2(x)\nImψ1(x)F=0.2 meV/nm \nα=5 meVnm(a)\n(b)00\n0.02-0.02\n-0.02-0.04\nFIG. 4. Components of the electron spinor ψ(x,t) function\natt= 655T0: (a) real parts, (b) imaginary parts.\nnotice that they are ofthe order of(30 ...40)Tdforα= 5\nmeVnm and (5 ...8)Tdforα= 25 meVnm where Td=\n6.71T0isthedrivingfieldperiod. AdifferencetoRef. [48]\nis that the electric field amplitude required to operate a\nspin flip in a multilevel dot is three orders of magnitude\nlarger than that for the shallow dot. We attribute these\ndifferences to the considerably stronger localization and\nenergy scales of a multilevel dot, where one needs larger\nelectric fields to induce the spin flip in a desired time.\nThese larger fields lead to a broader energy interval of\nthe continuum states contributing to the position and\nspin evolution.\nTo summarize this subsection, we mention that with\nthe expression for spin-flip matrix elements of coordinate\n[28, 29] for the Zeeman-split ground state in quantum\ndots of interest, one can estimate the tunneling-limited\nmaximalRabispinfrequencyΩ[max]\nRasΩ[max]\nR∼ωdd/Lso,\nwhereLso= ¯h2/mαis the spin precessionlength. In gen-\neral,thelength Lsoisdeterminedbythejointeffectofthe\nRashba and Dresselhaus couplings, and the existence of\nthis maximal Rabi frequency is model-independent. Tak-\ning into account that for our set of parameters at α= 5\nmeVnm,Lso≈10−4cm,we obtain a good agreement\nwith the numerical results presented in Fig. 3. Note\nthat such a limit has to exist for all finite-potential sys-\ntems, including P donors in Si. However, every realiza-\ntion needs a special analysis, which can be a topic of a\nfurther research.6\n0100200 300 400 500600\n t/T0050100150〈E(t)〉  (meV)\n100 150 200\n t/T02550〈E(t)〉 (meV)α=5 meVnm\nα=0F=0.2 meV/nm0.22 meV/nm\n0.20 meV/nm\n0.18 meV/nm\n0.16 meV/nm\n5 meVnm25 meVnm(a)\n(b)\nFIG. 5. Time dependence of mean energy /an}bracketle{tE(t)/an}bracketri}ht(a) for SOC\nparameter α= 5 meVnm and the same electric field ampli-\ntudes as in Fig. 3, and (b) for fixed F0= 0.2 meV/nm and\ndifferent SOC parameters α= 0,α= 5, and α= 25 meVnm.\nB. Efficiency of spin flip\nThe studies of coupled spin and coordinate dynamics\ncan be facilitated by tracking the time dependence of the\nenergy-related variables such as /angbracketleftE(t)/angbracketrightdefined as\n/angbracketleftE(t)/angbracketright=/summationdisplay\nn|cn(t)|2En (10)\nwhere the sum is taken over all basis states with coeffi-\ncientscn(t) in the wavefunction (6) for Endefined with\n(4) for the unperturbed Hamiltonian (5).\nWith the external driving, the energy is pumped into\nthe system, and the period-averaged expectation value\nof/angbracketleftE(t)/angbracketrightgrows with time. When /angbracketleftE(t)/angbracketrightpasses the\nthreshold /angbracketleftE(t)/angbracketright= 0 between the localized and contin-\nuum states, the electronhas effectively tunneled from the\nquantum dot into the continuum.\nIn Fig. 5(a) we plot /angbracketleftE(t)/angbracketrightforα= 5 meVnm and the\nsame electric fields as in Fig. 3, and in Fig. 5(b) we show\nexamples for fixed F= 0.2 meV/nm and different SOC\nparameters. As expected, the tunneling time quickly de-\ncreases with increasing field amplitude, and for strong\nfields greater than 0 .2 meV/nm the electron escapes into0100 200 300 400 500\n t/T0 η(t)\n0 50 100 150 200\n t/T0 η(t)α=5 meVnm\nα=25 meVnm0.16 meV/nm\n0.18 meV/nm\n0.2 meV/nm\n0.16 meV/nm\n0.18 meV/nm\n0.2 meV/nm(a)\n(b)0.1\n0.20.2\n0.4\nFIG. 6. Time dependent spin-flip efficiency (11): (a) α= 5\nmeVnm, (b) α= 25 meVnm, for some of the driving fields\n(marking the plots) from Fig. 3.\ncontinuum during ∼10 periods of driving. The influ-\nence of SOC on the energy evolution can be illustrated\nby Fig. 5(b): at the given time interval and at zero SOC\nthe energy grows faster than for finite SOC presented\nin the Figure. This effect can be attributed to stronger\ncoupling of all spin-resolved α−dependent states via the\nSOC, which slows to some extent the motion to higher\nenergy states.\nIn addition to the time dependence /angbracketleftE(t)/angbracketright, it is of in-\nterest to study the spin-flip efficiency η(t) defined as the\nratio of the energy pumped into the spin degree of free-\ndom to the total change of the mean energy:\nη(t) =∆\n2/angbracketleftσz(t)/angbracketright−/angbracketleftσz(0)/angbracketright\n/angbracketleftE(t)/angbracketright−/angbracketleftE(0)/angbracketright, (11)\nwhere/angbracketleftE(0)/angbracketright ≡E1.In Fig. 6 we show this η(t) for some\nof the electric fields from Fig. 3. One can see that its\ntime dependence is qualitatively independent of the SOC\nstrength. It reaches the maximum when the first full or\nalmost full spin flip is achieved. After this, the mag-\nnitude ofη(t) decreases with time. The explanation is\nstraightforward: the spin dynamics and in particular the\nZeemanenergy havean oscillatingcharacter(at low driv-\ning strength) or oscillating plus decaying character (at7\nhigh driving strength). So, the numerator in (11) does\nnot grow with time, while the denominator in general in-\ncreases, as it can be seen from the energy dependence in\nFig. 5. As to the magnitude of (11), it is greater for big-\nger SOC strength since the stronger SOC coupling pro-\nvides a quicker spin flip with lower corresponding change\nof the mean energy, giving the smaller denominator in\n(11) at the moment when the maximum numerator is\nreached.\nC. Feedback on the tunneling\nTo visualize the electron escape from the quantum dot,\nwe use the stay probability defining the weight of the lo-\ncalizedeigenfunctionsinthediscretepartofthespectrum\nas:\nP(t) =/summationdisplay\ni(loc)|ci(t)|2, (12)\nwhere summation is taken over the contribution of local-\nizedφi(x)−states.\nThe localization probability (12) is shown in Fig. 7 for\ntwo time intervals. One can see that for short evolution\ntimes shown in Fig. 7(a) both values of SOC strength\nproduce similar evolution P(t),and the values of (12) for\nsmallerαareingeneralslightlyhigherthanforthe bigger\none. The situation is, however, somewhat different on\nlong times to 600 T0shown in Fig. 7(b) for small value\nofα= 5 meVnm together with the α= 0 realization.\nHere twoP(t)−curves for a given driving strength and\ndifferent SOC swap more frequently than in Fig. 7(a).\nAn important element for understanding of the feed-\nbackeffect ofspinmotiononthe time-dependent position\nand tunneling is the anomalous [66] spin-dependent ve-\nlocity [60]:\nvso=i\n¯h[ασyk,x] =α\n¯hσy, (13)\nand the corresponding acceleration:\nd\ndtvso=i\n¯h∆\n2α\n¯h[σz,σy] =α∆\n¯h2σx.(14)\nThis term leads to variation in the velocity due to the\nspin precession, as was observed in experiments on high-\nfrequency conductivity of two-dimensional electron gas\n[67], and the effect of time-dependent /angbracketleftσy(t)/angbracketrighton/angbracketleftx(t)/angbracketright\nappears as a result. The corresponding local probability\nflux satisfying equation ∂t(ψ†(x,t)ψ(x,t)) +∂xj(x,t) =\n0,where∂x≡∂/∂x,is given by:\nj(x,t) =i¯h\n2m/bracketleftbig\nψ(x,t)∂xψ†(x,t)−ψ†(x,t)∂xψ(x,t)/bracketrightbig\n+α\n¯hψ†(x,t)σyψ(x,t). (15)\nNote that for a relatively small coupling constant α=\n5 meVnm,the maximum value of the vsovelocityα/¯h≈\n8 nm/ps can sufficiently modify the tunneling process.0 50 100 150 200\n t/T0P(t)\n0100200 300 400 500600\n t/T0P(t)α=5 meVnm\nα=0\n0.22 meV/nm0.18 meV/nm\n0.2 meV/nm0.18 meV/nm0.16 meV/nmα=25 meVnm\nα=5 meVnm0.2 meV/nm\n0.22 meV/nm(a)\n(b)0.20.40.60.810.20.40.60.81\n0\n0\nFIG. 7. Time dependence of localization probability (12) fo r\n(a) short evolution times for α= 5 (dashed lines) and α= 25\n(solid lines) meVnm and (b) long evolution times for α= 0\n(solid lines) and α= 5 (dashed lines) meVnm. The plots are\nmarked with the values of the driving field amplitudes.\nD. Evolution of the position\n1. Time-dependence of the expectation values\nAlong with the spin evolution, the expectation value\nof the coordinate /angbracketleftx(t)/angbracketright, is of interest for understanding\nof the dynamics. It provides information of the electron\nlocalization domain, and can help in studies of the tun-\nneling effect. For example, the sufficient criteria for tun-\nneling into the continuum may be formulated in terms of\nthe/angbracketleftx(t)/angbracketrightamplitude: if it steadily exceeds the quantum\ndot sized, the electron is out of the dot. Also, this ex-\npectation value produces a time-dependent electric field\noutside of the quantum dot, which under certain condi-\ntions, can be experimentally tracked.\nWe calculate the dynamics of /angbracketleftx(t)/angbracketrightin analogy with\n(7) where the spin operator is replaced by xwith some\nexamples of coordinate dynamics presented in Fig. 8.\nThe upper panel shows the dynamics for the low ampli-\ntudeF= 0.16 meV/nm and the bottom panel shows the\ndynamics for F= 0.22 meV/nm. By looking in Fig. 8(a)\nonecanconcludethat forlowdrivingamplitude F= 0.168\nmeV/nm the electron is still confined within the quan-\ntum dot, and its spin exhibits well-defined Rabi oscilla-\ntions (see Fig. 3). When the driving amplitude grows,\nthe delocalization trend is manifested, and for the high-\nest amplitude F= 0.22 meV/nm the amplitude of /angbracketleftx(t)/angbracketright\nwell exceeds the quantum dot size, as one can see in Fig.\n8(b).\nOne may compare the results presented in Fig. 8 with\nthe evolution of the localization probability (12) in Fig.\n7. It is clear that for the given time tthe localization\nprobabilityissignificantlylowerathigherdrivingfieldsas\nmanifested both in small P(t) and in large /angbracketleftx(t)/angbracketrightampli-\ntude. Thus, thedelocalizationtrendistrackedsimultane-\nously in the coordinate and Hilbert spaces. The strong\ndependence of the delocalization (ionization in atomic\nphysics) probability on the low-frequency electric field\namplitude has an exponential character[64] albeit with a\nrather undetermined model-dependent prefactor. Thus,\nour numerical findings shed light onto the amplitudes\nabove which an efficient tunneling to continuum takes\nplace. In addition, we note that despite two possible\ndirections of escape in a periodic field, the probability\ndensity becomes broad, but remains single-peaked, as it\nis seen in Fig.4, making /angbracketleftx(t)/angbracketrighta reliable characteristic of\nthe expectation value of the electron position.\nThe displacements presented in Fig. 8 can be com-\npared with the results for oscillations amplitude x[loc]\n0for\na particle localized in the potential U0x2/d2, and a free\nelectron driven by a periodic field, x[cl]\n0,where\nx[loc]\n0=d2\n2U0F, x[cl]\n0=1\nmω2\ndF. (16)\nForF= 0.16 meV/nm ,estimated amplitude x[loc]\n0≈\n8 nm,in a good agreement with the calculations at\nt <100T0(not shown in the Figure). Here the ve-\nlocity amplitude v[loc]\n0=ωdx[loc]\n0≈16 nm/ps,mean-\ning that the anomalous velocity α/¯hplays essential role\nin the electron dynamics. For a delocalized electron,\nEq. (16) yields for F= 0.22 meV/nm ,the amplitude\nx[cl]\n0≈720 nm,considerably larger than the calculated\nvalues. This difference is due to the fact that in the\ntunneling ionization a broadly distributed probability\ndensity rather than a well-defined wavepacket is being\nformed, resultinginarelativelysmallamplitudeof /angbracketleftx(t)/angbracketright.\nThis can be illustrated by the example for the wave-\nfunction component profile shown in Fig. 4 which can\nbe compared with the localization probability in Fig.\n7(b): forF= 0.2 meV/nm the low value of localiza-\ntion probability for the final moment of computation\n(P≈0.2) corresponds to Fig. 4 which describes a wave-\nfunction with substantial impact of continuum states\ndemonstrating a widely delocalized profile rather than\na well-localized wavepacket. As to the velocity, taking\ninto account that the numerically obtained amplitude,\nx[tun]\n0≈150 nm we find that the corresponding veloc-\nity amplitude v[tun]\n0≈300 nm/ps,being comparable to100 125 150 175 200\n t/T0020〈x(t)〉  (nm)\n100 125 150 175 200\nt/T0-200-1000100200〈x(t)〉 (nm)α= 0\nα= 0 F= 0.22 meV/nmF=0.16 meV/nm\n25 meVnm\n25 meVnm(a)\n(b)\nFIG. 8. Time dependence of mean value for the coordinate\n/an}bracketle{tx(t)/an}bracketri}htforα= 0 and for high α= 25 meVnm, shown for\nthe driving field amplitude (a) F= 0.16 meV/nm and (b)\nF= 0.22 meV/nm. For low driving amplitude F= 0.16\nmeV/nm |/an}bracketle{tx(t)/an}bracketri}ht| ≪d,while for F= 0.22 meV/nm the\namplitude of /an}bracketle{tx(t)/an}bracketri}htexceeds the quantum dot size. Here\nmax{|/an}bracketle{tx(t)/an}bracketri}ht|}<3d,meaning that the electron motion is still\nstrongly influenced by the U(x)−potential and approximately\nlimited by the tunneling escape points at ±E1/FwhereE1is\nthe ground state energy.\nthe maximum anomalous velocity at α= 25 meVnm ,\nconfirming the importance of spin-orbit coupling for the\nelectron displacement. At this point, we can define the\ncorresponding wavevector, k[tun]\n0≡v[tun]\n0m/¯hand the\nspin precession rate Ω[tun]\n0= 2αk[tun]\n0/¯h.For the given\nset of system parameters and α= 5 meVnm ,we obtain\natF= 0.22 meVnm,Ω[tun]\n0≈0.53 ps−1.In the panel\n(a) of Fig. 3 on can see that at F= 0.22 meV/nm ,the\ncharacteristic time of spin evolution (time, at which the\nexpectation value /angbracketleftσz(t)/angbracketright= 0), is close to 45 ps, consid-\nerably larger than π/Ω[tun]\n0≈6 ps.This difference means\nthat the main effect of the tunneling on the spin evolu-\ntion is due to formation of the mixed states rather than\ndue to the driven tunneling-induced spin precession.\nBy analyzing the results in Fig. 3 - Fig. 8, one can\nsee that the tunneling becomes efficient at few periods\nof driving field when the driving amplitude exceeds the\nvalues of about 0 .18...0.2 meV/nm. At lower fields the9\ntunneling is slow enough to allow for a well-defined spin\nflip. For the fields higher than 0 .22 meV/nm the po-\ntential well opens so effectively that the electron escapes\ninto continuum before the well-established Rabi oscilla-\ntions are developed. So, one needs to choose some opti-\nmal intermediate driving fields to achieve good spin flip\nin a proper time. Besides, the very strong dependence of\nthe tunneling probability on the driving field amplitude\nrequires a well-defined window of the field amplitude in\norder to achieve the desired spin flip if one is interested\nin still keeping the electron inside the hosting quantum\ndot.\nAs for the α−dependence of the tunneling, one can\nsee that the patterns of position /angbracketleftx(t)/angbracketright, the localization\nP(t),and the energy /angbracketleftE(t)/angbracketrightare similar but still quanti-\ntatively different depending on the SOC strength. For\nexample, for a stronger SOC the localization probability\nis slightly lower at all times and for most of the driving\nfields, i.e. the tunneling, in general, becomes faster with\nincreasing SOC. We attribute this effect to a more in-\ntense EDSR development at higher SOC. Stronger SOC\nenlarges the matrix elements for the states with opposite\nspins, leading not only to a faster spin flip but also to a\nlarger occupation of higher energy levels, speeding up, to\nsome extent, the tunneling.\n2. Fourier analysis of the dynamics\nIt is of interest to study the driven evolution in terms\nofits spectralproperties, namely, the Fourierpowerspec-\ntrum applied to a finite sequence of data for /angbracketleftx(t)/angbracketrightcol-\nlected on the evolution interval T=NT0withNpoints\nspaced byT0. Here it is appropriate to use the discrete\nversion of the Fourier transform written as:\nXp=N−1/summationdisplay\nn=0/angbracketleftx(nT0)/angbracketrighte−inωp/ω0. (17)\nIn (17)Xpis the output for the Fourier harmonic at\nfrequencyωp=ω0p/N,and (17) gives Fourier data for\nthe frequency interval 0 ...ω0/2 with frequency spacing\nω0/N, so the index ptakes the values 1 ...N/2,where it\nis convenientto expressthe frequenciesin units the of ωd.\nFor our numerical simulations we have the values N=\n200...600, where the discrete spectraobtained from (17)\nform a dense quasicontinous set of states.\nHere one may expect the local maxima at both driving\nfrequencyωdand in the low-frequency part associated\nwith slow spin evolution coupled to the coordinate via\nSOC. In Fig. 9 we show examples of Fourier power spec-\ntra to compare the α= 0 system with the system hav-\ning moderate SOC for driving with F= 0.16 meV/nm\nandF= 0.22 meV/nm. We may see that the presence\nof SOC modifies mainly the low frequency part of Xp\nin the∼10−2THz frequency domain. This is natu-\nral due to the SOC coupling of the coordinate with spin\nwhich affects mainly lower frequencies associated with ωp /ωd02468Fourier intensity | Xp|2 (nm2)\nωp /ωd020406080100Fourier intensity | Xp|2 (nm2)F=0.16 meV/nm\nα=5 meVnmα=5 meVnmα=0\nF=0.22 meV/nm(a)\n(b)\nα=0\n00.20.40.60.8 100.20.4 0.6 0.8 1\nFIG. 9. Fourier intensities |Xp|2forα= 0 and α= 5 meVnm\n(as marked near the plots): (a) F= 0.16 meV/nm, (b) F=\n0.22 meV/nm.\nthe spin evolution which is lower than the driving fre-\nquency of the electric field. The peak at ωp/ωd= 1\nis present for all SOC parameters and driving field am-\nplitudes and corresponds to the dominant frequency of\ndriven dynamics. Another interesting feature of Fourier\nspectra is the difference between the zero- and nonzero\nSOC cases which increases at high driving fields as it\ncan be seen by comparing panels (a) and (b) in Fig. 9.\nWith the increase in the driving field, at α= 0,the\ncontribution of the large-displacement dynamics at fre-\nquencyωddominates over the low-frequency part of the\ndisplacement spectrum. However, a finite SOC due to\nthe anomalous velocity produces a still relatively intense\nlow-frequency motion.\nIt is interestingto note that in the non-relativisticnon-\nlinearatomic optics, frequent recollisionsofa driven elec-\ntron with the singular Coulomb potential of the remain-\ning ion [68] lead to a strong enhancement in the high-\nfrequency radiation. In our calculations we take into ac-\ncount a relativistic effect in the form of the spin-orbit\ncouping, albeit for a nonsingular potential, where a re-\nsult of these frequent recollisions is seen as a relatively\nsmall amplitude of /angbracketleftx(t)/angbracketrightcompared to x[cl]\n0in Eq. (16),\nas shown in Fig. 8(b). It will be of a future interest to10\nQD\nspin manipulationfield sin F t/c119\nspin readouttunneling-\nachieved point\nd\nFIG. 10. Sketch of a device utilizing the joint effect of spin\nrotation and tunnelingto the continuum. At t=t0the EDSR\nspin manipulation starts, and at some t1> t0the readout be-\ngins for the electron outside of the dot. The readout process is\nbe performed outside of the QD, thus reducing the perturba-\ntion effects of the measurement on the electron spin rotation\nin the dot and, in turn, of the spin rotation on the readout.\nstudyhigh-frequencyharmonicsproducedinthepresence\nof these relativistic effects.\nV. CONCLUSIONS AND DISCUSSION\nWe studied the electric dipole spin resonance with si-\nmultaneous tunneling and position evolution in a multi-\nlevel quantum dot formed in a nanowire as caused by a\ndriving field in the sub-THz range. We demonstrated a\nstrongeffect oftunneling on the driven spin flip processes\nwith disappearance of well-defined Rabi spin oscillations\nwhen the tunneling rate is of the order of the correspond-\ning Rabi frequency. For multilevel quantum dots this\nmatching happens in the electric field amplitudes of the\norder of 0.1 meV/nm, being in our realization close to\n0.2 meV/nm. Thus, the tunneling induced by the driv-\ning electric field, strongly limits the maximal spin-flip\nRabi frequency and the reliability of the spin manipu-\nlation. Although this effect is common for all systems,\nwhere electrons are localized in finite-size potential such\nas donors, gate-based quantum dots, and lattice defects,\nevery realization requires a system-specific analysis. The\nstrong effect of the tunneling on the spin dynamics is\ndue to the formation of the mixed spin states, shifting\nthe spin vector inside the Bloch sphere. In addition,\nwe demonstrated backaction of the spin dynamics on the\ntunneling probability and position of the electron. This\nbackaction is qualitatively attributed to the anomalous\nspin-dependent velocity, proportional to the spin-orbit\ncoupling strength. These effects should be taken into\naccount in the development of techniques for fast oper-\nations in the electron spin-based qubit architecture insemiconductor nanostructures.\nFinally, we would like to briefly discuss possible device\nimplementation of the above predicted effects. A sketch\nof a device utilizing the tunneling to the continuum ac-\ncompanying the spin rotation is shown in Fig.10. The\nEDSR spin manipulation for electron in the QD starts at\nt=t0,and att=t1the readout begins at the readout\narea for the electron tunneled to continuum as shown in\nFig.10. At the arrival at the readout point, the electron\nhas the spin components determined both by the spin\nmanipulation in the dot and by the propagation to the\nreadout point. The latter can be accurately found from\nthe knowledge of the spin precession in the nanowire in-\ncluding the spin precession length discussed in Sec.IV.A.\nThus, the abilities ofspin manipulationscanbeenhanced\nby the tunneling process. The readout process shown in\nFig.10 is performed outside ofthe QD areadue to the de-\nlocalization of the electron wavefunction, thus reducing\nthe mutually perturbing effects of the EDSR spin ma-\nnipulation and spin measurement. This is illustrated by\nFig.4 where the sizable out-of-the-dot contribution of the\nwavefunction is visible and the corresponding spin com-\nponents can be detected with high fidelity. Such setup is\ndifferent from the setup using the second QDrequired for\nspin readout in the EDSR experiments [24, 27] where the\nPauli spin blockade has been utilized to enhance or sup-\npress the current through the whole structure indicating\nthe spin flip. We expect that our scheme with spatially\nseparatedspin rotationand readoutintervals canprovide\nalternative ways for experimental spin manipulation se-\ntups and their applications.\nACKNOWLEDGEMENTS\nD.V.K. and E.A.L. are supported by the Ministry of\nScience and Higher Education of the Russian Federa-\ntion through the State Assignment No 0729-2020-0058.\nE.A.L.wassupported bythe grantofthe Presidentofthe\nRussian Federation for young scientists MK-6679.2018.2.\nE.Y.S. acknowledges support by the Spanish Ministry of\nScience, Innovation and Universities, and the European\nRegional Development Fund FEDER through Grant No.\nPGC2018-101355-B-I00 (MCIU/AEI/FEDER, UE) and\nthe Basque Country Government through Grant No.\nIT986-16. E.Y.S. is grateful to S. Studenikin for valu-\nable discussion.\n[1] D. Loss and D. P. DiVincenzo, Quantum computation\nwith quantum dots , Phys. Rev. A 57, 120 (1998).\n[2] G. Burkard, D. Loss, and D. P. DiVincenzo, Coupled\nquantum dots as quantum gates , Phys. Rev. B 59, 2070\n(1999).[3] E. I. Rashba, Properties of semiconductors with an ex-\ntremum loop. 1. Cyclotron and combinational resonance\nin a magnetic field perpendicular to the plane of the loop ,\nSov. Phys. Solid State 2, 1109 (1960).\n[4] G. Bemski, Spin Resonance of Conduction Electrons in\nInSb, Phys. Rev. Lett. 4, 62 (1960).11\n[5] R.L.Bell, Electric Dipole Spin Transitions in InSb , Phys.\nRev. Lett. 4, 52 (1962).\n[6] Y.A. Bratashevskii, A.A. Galkin, and Y.M. Ivanchenko,\nResonance absorption by band carriers in InSb , Sov.\nPhys. Solid State 5, 260 (1963).\n[7] T.ˇCade˘ z, J.H. Jefferson, and A. Ramˇ sak, Exact Nonadi-\nabatic Holonomic Transformations of Spin-Orbit Qubits ,\nPhys. Rev. Lett. 112, 150402 (2014); A. Ramˇ sak, T.\nˇCade˘ z, A. Kregar, and L. Ulˇ cakar, Exact spin-orbit qubit\nmanipulation , The European Physical Journal Special\nTopics227, 353 (2018).\n[8] For a review, see: E. I. Rashba and V. I. Sheka, Electric-\nDipole Spin Resonances , arXiv:1812.01721.\n[9] J. Iba˜ nez-Azpiroz, A. Eiguren, E. Ya. Sherman, and\nA. Bergara, Spin-Flip Transitions Induced by Time-\nDependent Electric Fields in Surfaces with Strong Spin-\nOrbit Interaction , Phys. Rev. Lett. 109, 156401 (2012).\n[10] F.T. Vas’ko, Spin splitting in the spectrum of two-\ndimensional electrons due to the surface potential , JETP\nLetters30, 541 (1980).\n[11] E. I. Rashba and Al. L. Efros, Orbital Mechanisms of\nElectron-Spin Manipulation by an Electric Field , Phys.\nRev. Lett. 91, 126405 (2003).\n[12] D.V.Khomitsky, Manipulating the spin texture in a spin-\norbit superlattice by terahertz radiation , Phys. Rev. B 77,\n113313 (2008).\n[13] V.N.Golovach, M.Borhani, andD.Loss, Electric-dipole-\ninduced spin resonance in quantum dots , Phys. Rev. B\n74, 165319 (2006).\n[14] J. H. Jiang, M. Q. Weng, and M. W. Wu, Intense tera-\nhertz laser fields on a quantum dot with Rashba spin-orbit\ncoupling, J. Appl. Phys. 100, 063709 (2006).\n[15] X. Hu, Yu-Xi Liu, and F. Nori, Strong coupling of a spin\nqubit to a superconducting stripline cavity , Phys. Rev. B\n86, 035314 (2012).\n[16] D.V.Khomitsky, L.V.Gulyaev, andE.Ya.Sherman, Spin\ndynamics in a strongly driven system: Very slow Rabi\noscillations , Phys. Rev. B 85, 125312 (2012).\n[17] M. Borhani and X. Hu, Spin manipulation and relaxation\nin spin-orbit qubits , Phys. Rev. B 85, 125132 (2012).\n[18] M. P. Nowak, B. Szafran, and F. M. Peeters, Resonant\nharmonic generation and collective spin rotations in elec-\ntrically driven quantum dots , Phys. Rev. B 86. 125428\n(2012).\n[19] Y. Ban, Xi Chen, E. Ya. Sherman, and J. G. Muga,\nFast and Robust Spin Manipulation in a Quantum Dot\nby Electric Fields , Phys. Rev. Lett. 109, 206602 (2012).\n[20] Rui Li, J. Q. You, C. P. Sun, and F. Nori, Controlling a\nNanowire Spin-Orbit Qubit via Electric-Dipole Spin Res-\nonance, Phys. Rev. Lett. 111, 086805 (2013); Rui Li and\nJ. Q. You, Anisotropic exchange coupling in a nanowire\ndouble quantum dot with strong spin-orbit coupling , Phys.\nRev. B90, 035303 (2014).\n[21] J. Romh´ anyi, G. Burkard, and A. P´ alyi, Subharmonic\ntransitions and Bloch-Siegert shift in electrically drive n\nspin resonance , Phys. Rev. B 92, 054422 (2015).\n[22] M. T. Veszeli and A. P´ alyi, Fast electron spin flips\nvia strong subcycle electric excitation , Phys. Rev. B 97,\n235433 (2018).\n[23] P. Stipsi´ c and M. Milivojevi´ c, Control of a spin qubit in\na lateral GaAs quantum dot based on symmetry of gating\npotential , Phys. Rev. B 101, 165302 (2020); S. Miladi´ c,\nP. Stipsi´ c, E. Dobardˇ zi´ c, and M. Milivojevi´ c, Electrical\ncontrol of a spin qubit in InSb nanowire quantum dots:Strongly suppressed spin relaxation in high magnetic field ,\nPhys. Rev. B 101, 155307 (2020).\n[24] K. C. Nowack, F. H. L. Koppens, Yu. V. Nazarov, and L.\nM. K. Vandersypen, Coherent Control of a Single Elec-\ntron Spin with Electric Fields , Science 318, 1430 (2007).\n[25] S. Studenikin, M. Korkusinski, M. Takahashi, J. Duca-\ntel, A. Padawer-Blatt, A. Bogan, D. Guy Austing, L.\nGaudreau, P. Zawadzki, A. Sachrajda, Y. Hirayama, L.\nTracy, J. Reno, and T. Hargett, Electrically tunable ef-\nfective g-factor of a single hole in a lateral GaAs/AlGaAs\nquantum dot , Commun. Phys. 2, 159 (2019).\n[26] E. N. Osika and B. Szafran, Spinorbit interaction in bent\ncarbon nanotubes: resonant spin transitions , J. Phys.:\nCondens. Matter 27, 435301 (2015).\n[27] S. Nadj-Perge, S. M. Frolov, E. P. A. M. Bakkers, and\nL. P. Kouwenhoven, Spin-orbit Qubit in a Semiconductor\nNanowire , Nature (London) 468, 1084 (2010).\n[28] E. I. Rashba and V.I. Sheka, Combination resonance at\nlocal centers of large radius , Sov. Phys. Solid State 6, 114\n(1964).\n[29] E. I. Rashba and V.I. Sheka, Combination resonance at\nacceptor centers , Sov. Phys. Solid State 6, 451 (1964).\n[30] X. Linpeng, T. Karin, M. V. Durnev, R. Barbour, M.\nM. Glazov, E. Ya. Sherman, S. P. Watkins, S. Seto, and\nK.-M. C. Fu, Longitudinal spin relaxation of donor-bound\nelectrons in direct band-gap semiconductors , Phys. Rev.\nB94, 125401 (2016).\n[31] X. Linpeng, M. L.K. Viitaniemi, A. Vishnuradhan, Y.\nKozuka, C. Johnson, M. Kawasaki, and K.-M. C. Fu,\nCoherence Properties of Shallow Donor Qubits in Zn O ,\nPhys. Rev. Applied 10, 064061 (2018).\n[32] J. H. Davies, I. A. Larkin, and E. V. Sukhorukov, Model-\ning the patterned two-dimensional electron gas: Electro-\nstatics, J. Appl. Phys. 77, 4504 (1995).\n[33] A. G. Malshukov, C. S. Tang, C. S. Chu, and K. A. Chao,\nSpin-current generation and detection in the presence of\nan ac gate , Phys. Rev. B 68, 233307 (2003).\n[34] L. Serra, D.S´ anchez, andR.Lopez, Rashba interaction in\nquantum wires with in-plane magnetic fields , Phys. Rev.\nB72, 235309 (2005).\n[35] T. Zibold, P. Vogl, and A. Bertoni, Theory of semicon-\nductor quantum-wire-based single- and two-qubit gates ,\nPhys. Rev. B 76, 195301 (2007).\n[36] A. F. Sadreev and A. S. Pilipchuk, Bound states in the\ncontinuum in zigzag quantum wire enforced by a finger\ngate, JETP Letters 100, 585 (2015).\n[37] W. Sheng and P. Hawrylak, Spin polarization in self-\nassembled quantum dots , Phys.Rev.B 73, 125331 (2006).\n[38] S. Takahashi, R. S. Deacon, K. Yoshida, A. Oiwa, K.\nShibata, K. Hirakawa, Y. Tokura, and S. Tarucha, Large\nAnisotropy of the Spin-Orbit Interaction in a Single InAs\nSelf-Assembled Quantum Dot , Phys. Rev. Lett. 104,\n246801 (2010).\n[39] K. Gawarecki, Spin-orbit coupling and magnetic-field de-\npendence of carrier states in a self-assembled quantum\ndot, Phys. Rev. B 97, 235408 (2018).\n[40] P. Harvey-Collard, N. T. Jacobson, C. Bureau-Oxton,\nR.M. Jock, V. Srinivasa, A.M. Mounce, D.R. Ward, J.M.\nAnderson, R.P. Manginell, J.R. Wendt, T. Pluym, M.P.\nLilly, D.R. Luhman, M. Pioro-Ladri` ere, and M.S. Car-\noll,Spin-orbit Interactions for Singlet-Triplet Qubits in\nSilicon, Phys. Rev. Lett. 122, 217702 (2019).\n[41] W.M. Witzel, I. Monta˜ no, R.P. Muller, and M.S. Car-\nroll,Multiqubit gates protected by adiabaticity and dy-12\nnamical decoupling applicable to donor qubits in silicon ,\nPhys. Rev. B 92, 081407(R) (2015).\n[42] J.T. Muhonen, J.P. Dehollain, A. Laucht, S. Simmons,\nR. Kalra, F.E. Hudson, A.S. Dzurak, A. Morello, D.N.\nJamieson, J.C. McCallum, and K.M. Itoh, Coherent con-\ntrol via weak measurements in 31 P single-atom electron\nand nuclear spin qubits , Phys. Rev. B 98, 155201 (2018).\n[43] B.Weber, Y.-L.Hsueh, T. F.Watson, R.Li, A.R.Hamil-\nton, L. C. L. Hollenberg, R. Rahman, and M. Y. Sim-\nmons,Spinorbit coupling in silicon for electrons bound to\ndonors, npj Quantum Information 4, 61 (2018).\n[44] K.W. Chan, W. Huang, C.H. Yang, J.C.C. Hwang, B.\nHensen, T. Tanttu, F.E. Hudson, K.M. Itoh, A. Laucht,\nA. Morello, and A.S. Dzurak, Assessment of a Silicon\nQuantum Dot Spin Qubit Environment via Noise Spec-\ntroscopy, Phys. Rev. Appl. 10, 044017 (2018).\n[45] P. Ross, B.C. Rose, C.C. Lo, M.L.W. Thewalt, A.M.\nTyryshkin, S.A. Lyon, and J.L. Morton, Electron Spin\nResonance of P Donors in Isotopically Purified Si De-\ntected by Contactless Photoconductivity , Phys.Rev.Appl.\n11, 054014 (2019).\n[46] B. Hansen, W.W. Huang, C.-H. Yang, K.W. Chan, J.\nYonedal, T. Tanttu, F.E. Hudson, A. Laucht, K.M. Itoh,\nT.D. Ladd, A. Morello, and A.S. Dzurak, A silicon\nquantum-dot-coupled nuclear spin qubit , Nat. Nanotech.\n15, 13 (2020).\n[47] B.C. Rose, A.M. Tyryshkin, H. Riemann, N.V. Abrosi-\nmov, P. Becker, H.-J. Pohl, M.L.W. Thewalt, K.M. Itoh,\nand S.A. Lyon, Coherent Rabi Dynamics of a Superradi-\nant Spin Ensemble in a Microwave Cavity , Phys. Rev. X\n7, 031002 (2017).\n[48] D.V. Khomitsky, E.A. Lavrukhina, and E.Ya. Sherman,\nElectric dipole spin resonance at shallow donors in quan-\ntum wires , Phys. Rev. B 99, 014308 (2019).\n[49] S.D. Ganichev and W. Prettl, Intense Terahertz Exci-\ntation of Semiconductors , Series on Semiconductor Sci-\nence and Technology 14 (Oxford University Press, Ox-\nford, UK, 2006).\n[50] J. Takeda, K. Yoshioka, Y. Minami, and I. Katayama,\nNanoscale electron manipulation in metals with intense\nTHz electric fields , J. Phys. D: Appl. Phys. 51, 103001\n(2018).\n[51] A. Singh, A. Pashkin, S. Winnerl, M. Welsch, C. Beckh,\nP. Sulzer, A. Leitenstorfer, M. Helm, and H. Schneider,\nUp to 70 THz bandwidth from an implanted Ge photocon-\nductive antenna excited by a femtosecond Er:fibre laser ,\nLight: Science and Applications 9, 30 (2020).\n[52] M.Brooks andG.Burkard, Electric dipole spin resonance\nof two-dimensional semiconductor spin qubits , Phys.Rev.\nB101, 035204 (2020).\n[53] F. Mireles and G. Kirczenow, Ballistic spin-polarized\ntransport and Rashba spin precession in semiconductornanowires , Phys. Rev. B 64, 024426 (2001).\n[54] M. M. Gelabert, L. Serra, D. S´ anchez, and R. L´ opez,\nMultichannel effects in Rashba quantum wires , Phys.Rev.\nB81, 165317 (2010).\n[55] A. N. M. Zainuddin, S. Hong, L. Siddiqui, S. Srinivasan,\nand S. Datta, Voltage-controlled spin precession , Phys.\nRev. B84, 165306 (2011).\n[56] L. Xu, Xin-Qi Li, and Qing-Feng Sun, Revisit the spin-\nFET: Multiple reflection, inelastic scattering and lateral\nsize effects , Sci. Rep. 4, 7527 (2014).\n[57] I. Sa¨ ıdi, S. B. Radhia, and K. Boujdaria, Band param-\neters of GaAs, InAs, InP, and InSb in the 40-band kp\nmodel, J. Appl. Phys. 107, 043701 (2010).\n[58] M. A. Leontiadou, K. L. Litvinenko, A. L. Gilbertson, C.\nR. Pidgeon, W. R. Branford, L. F. Cohen, M. Fearn, T.\nAshley, M. T. Emeny, B. N. Murdin, and S. K. Clowes,\nExperimental determination of the Rashba coefficient in\nInSb/InAlSb quantum wells at zero magnetic field and\nelevated temperatures , J. Phys.: Condens. Matter 23,\n035801 (2011).\n[59] P. W´ ojcik, A. Bertoni, and G. Goldoni, Tuning\nRashba spin-orbit coupling in homogeneous semiconduc-\ntor nanowires , Phys. Rev. B 97, 165401 (2018).\n[60] L. D. Landau and E. M. Lifshitz, Quantum Mechanics\n- Nonrelativistic Theory , 3rd ed., Course of Theoretical\nPhysics Vol.3(Elsevier, Amsterdam, Netherlands, 1977).\n[61] R. Winkler, Spin-Orbit Coupling Effects in Two-\nDimensional Electron and Hole Systems (Springer,\nBerlin, 2003).\n[62] Forelectrons, see: M.I.DyakonovandV.Y.Kachorovski i,\nSpin relaxation of two-dimensional electrons on noncen-\ntrosymmetric semiconductors , Sov. Phys. Semicond. 20,\n110 (1986).\n[63] For holes: E.I. Rashba and E.Ya. Sherman, Spin-orbital\nband splitting in symmetric quantum wells , Phys. Lett. A\n129, 175 (1988).\n[64] N. B. Delone and V. P. Krainov, Atoms in Strong Light\nFields, (Springer Series in Chemical Physics), Heidelberg\n(1985).\n[65] K. Blum, Density Matrix Theory and Applications\n(Springer Series onAtomic, Optical, andPlasma Physics,\nSpringer, Berlin, 2012).\n[66] E. N. Adams and E. I. Blount, Energy bands in the pres-\nence of an external force field II: Anomalous velocities ,\nJ. Phys. Chem. Solids 10, 286 (1959).\n[67] S. A. Tarasenko, A. V. Poshakinskiy, E. L. Ivchenko, I.\nStepanov, M. Ersfeld, M. Lepsa, and B. Beschoten, Zit-\nterbewegung of Spin Split Electrons , JETP Letters 108,\n326 (2018).\n[68] M. Lewenstein, Ph. Balcou, M. Yu. Ivanov, A. L’Huillier ,\nand P. B. Corkum, Theory of high-harmonic genera-\ntion by low-frequency laser fields , Phys. Rev. A 49, 2117\n(1994)."    },    {        "title": "1410.5978v1.Spin_dynamics_of_Mn_impurities_and_their_bound_acceptors_in_GaAs.pdf",        "content": "arXiv:1410.5978v1  [cond-mat.mes-hall]  22 Oct 2014Spin dynamics of Mn impurities and their bound acceptors in G aAs\nM.R. Mahani, A. Pertsova and C.M. Canali\nDepartment of Physics and Electrical engineering, Linnaeu s University, 391 82 Kalmar, Sweden\n(Dated: January 27, 2021)\nWe present results of tight-binding spin-dynamics simulat ions of individual and pairs of substi-\ntutional Mn impurities in GaAs. Our approach is based on the m ixed quantum-classical scheme\nfor spin dynamics, with coupled equations of motions for the quantum subsystem, representing the\nhost, and the localized spins of magnetic dopants, which are treated classically. In the case of\na single Mn impurity, we calculate explicitly the time evolu tion of the Mn spin and the spins of\nnearest-neighbors As atoms, where the acceptor (hole) stat e introduced by the Mn dopant resides.\nWe relate the characteristic frequencies in the dynamical s pectra to the two dominant energy scales\nof the system, namely the spin-orbit interaction strength a nd the value of the p-dexchange coupling\nbetween the impurity spin and the host carriers. For a pair of Mn impurities, we find signatures\nof the indirect (carrier-mediated) exchange interaction i n the time evolution of the impurity spins.\nFinally, we examine temporal correlations between the two M n spins and their dependence on the\nexchange couplingand spin-orbit interaction strength, as well as on the initial spin-configuration and\nseparation between the impurities. Our results provide ins ight into the dynamic interaction between\nlocalized magnetic impurities in a nano-scaled magnetic-s emiconductor sample, in the extremely-\ndilute (solotronics) regime.\nPACS numbers: 75.50.Pp,71.55.Eq,75.78.-n,75.30.Hx\nKeywords: time-dependent Schr¨ odinger equation, spin dyn amics, single Mn-dopants in GaAs\nI. INTRODUCTION:\nRecently, remarkable progress has been achieved\nin describing the electronic and magnetic proper-\nties of individual dopants in semiconductors, both\nexperimentally[1–3] and theoretically [4–6], offering\nexciting prospects in future electronic devices. In\nview of novel potential applications, which involve\ncommunication between individual magnetic dopants,\nmediated by electronic degrees of freedom of the host,\nthe focus of this research field has been shifting towards\nfundamental understanding and control of spin dynamics\nof these atomic-scale magnetic centers. Importantly,\nthe development of advanced spectroscopic techniques\nhas opened up the possibility to probe the dynamics of\nsingle spin impurities experimentally [7]. Some specific\nexamples include inelastic tunneling spectroscopy of\natomic-scale magnetic structures [8], optical manipu-\nlation of spin-centers in semiconductors [9], magnetic\nresonance imaging [10–12] and time-resolved scanning\ntunneling spectroscopy [13] on single spins. These\nadvances pose new challenges for theory, calling for a\nfully microscopic time-dependent description of spin\ndynamics of individual spin impurities in the solid-state\nenvironment.\nThe most suitable approach to study the time evolution\nof spin systems, which is applicable at ultrashort time\n(/lessorsimilar100 ps) and length ( /lessorsimilar1 nm) scales and does not\nrely on phenomenological parameters, is ab initio spin\ndynamics (SD). A natural framework for this approach\nis provided by the extension of density functional\ntheory (DFT) to time domain and noncollinear spins\n(time-dependent spin DFT, or TD-SDFT), with several\npractical schemes developed to date [14–19]. How-ever, due to computational demands, the system sizes\nthat can be treated with this approach are limited to\nonly a handful of atoms [19]. In practice, numerical\nimplementations typically rely on approximations.\nA well-known example is the adiabatic ab initio SD\nmodel by Antropov et al.[14, 15], which is based on\na Born-Oppenheimer(BO)-type approximation for spin\ndegrees of freedom in materials with localized magnetic\nmoments. Because of the difference in the characteristic\nenergy scales for itinerant and localized spins, this model\ntreats the directions of the local magnetic moments as\nslow classical variables, while averaging over the fast\nelectronic degrees of freedom. Such separation leads\nto an equation of motion for the classical moments\ninteracting with an effective field.\nAntropov’s SD model provides the theoretical frame-\nwork for large-scale implementations of atomistic spin\ndynamics, such as the one by Skubic et al.[18]. In the\nlatter approach, the equation of motion for the localized\nmagnetic moments is augmented by a phenomenological\ndamping term (in analogy with Landau-Lifshitz-Gilbert\nequation) and by a stochastic (Langeven) term, which\naccounts for the effect of finite temperature. This\nscheme was used to investigate magnetic ordering and\ncorrelations in clusters of Mn-doped GaAs at finite\ntemperature [20]. We also mention that, apart from\nstandard atomistic spin-dynamics approaches, beyond\nmean-field dynamical models have been developed [21],\nwhich specifically address ultrafast photoenhanced mag-\nnetization dynamics in dilute magnetic semiconductors\n(DMSs), in particular (Ga, Mn)As [22].\nIn this work we employ the mixed quantum-classical\nSD model [23], which is similar in spirit albeit different\nin some important aspects from Antropov’s adiabatic\nSD, to study the dynamics of individual and pairs of2\nMn impurities in GaAs. Although our model assumes\na partition into a quantum subsystem, representing\nthe electrons of the host (GaAs), and a classical sub-\nsystem, representing the localized magnetic moments\nof the dopants (Mn), the interaction between the two\nsubsystems is treated at the level of the Ehrenfest ap-\nproximation [24], as opposed to the BO approximation.\nThis results in a system of coupled equations of motion,\nwith the two subsystems evolving simultaneously and\nexperiencing each other as time-varying classical fields.\nAs already known from its application to electron-ion\ndynamics [24], such scheme is able to capture non-\nadiabatic effects on the fast electronic time scale (fs),\nwhich are inaccessible by the BO approximation. Re-\ncently, Ehrenfest SD has been used to study the effects\nof electrostatic gating in atomistic spin conductors [25].\nIn order to describe the underlying electronic structure\nof the semiconductor host, we use a microscopic tight-\nbinding (TB) model for GaAs with parameters fitted to\nDFT calculations. This gives our approach a compu-\ntational edge compared to purely ab initio SD, which\nrelies on the Kohn-Sham Hamiltonian [14]. The spins\nof Mn dopants are introduced in the Hamiltonian as\nclassical vectors exchange coupled to the instantaneous\nspin-densities of the nearest-neighbors As atoms, in\nthe spirit of the p-dexchange interaction. In the static\nregime, this TB model has already proved successful in\ndescribing experimentally observed properties of single\nMn dopants and their associated acceptor states in\nGaAs [5, 6].\nHere we explore the solotronics limit of DMSs [26]\nin the time domain by probing explicitly the time\nevolution of individual substitutional Mn dopants in\na finite nanometer-size cluster of GaAs. From the\ntime-dependent spin-trajectories of single-impurity\nspins, we identify the characteristic energy scales of the\ndynamics, associated with intrinsic interactions present\nin the system, namely the p-dexchange interaction\nand the spin-orbit interaction (SOI). Furthermore, we\nstudy the time evolution of two Mn-impurity spins\nupon an arbitrary perturbation of one of the spins.\nThe SD simulations allow us to address explicitly the\ntime-dependent (dynamic) indirect exchange interaction\nbetween the spins, which is expected to differ from its\nstatic counterpart [27], and is relevant for nanostructures\nwith magnetic impurities [28]. We map out the temporal\ncorrelations between the two localized spins, expressed\nin terms of a classical spin-spin correlation function, as\na function of the SOI, p-dexchange interaction, spatial\nseparation, and the initial orientations of the spins (\ne.g. ferromagnetic, antiferromagnetic and noncollinear\nconfigurations). The variations in strength and time\nscale of the correlations, inferred from behavior of\nthe spin-spin correlation function for different system\nparameters, can be used as an indicative measure of\ndephasing of individual impurity spins due to their\ninteraction with the host and other impurities.\nThe paper is organized as follows. In Section II wedescribe the details of our theoretical approach. In\nSection III we discuss the results of numerical simula-\ntions. Section IIIA is concerned with the effect of SOI\nand exchange coupling strength on the time evolution\nof a single Mn spin. In Section IIIB we study the\ndynamical interplay between two spatially-separated\nMn impurities, coupled indirectly by carrier-mediated\nexchange interaction. Finally we draw some conclusions.\nII. THEORETICAL MODEL\nA. Tight-binding Hamiltonian\nWe begin to lay out the computational framework of\nourSDsimulationsbyfirst definingthe respectiveHamil-\ntonians of the two subsystems. The time-dependent\nHamiltonian of the GaAs host (quantum subsystem), in-\ncorporating a number of substitutional Mn atoms on Ga\nsites, takes the following form\nˆHel(t) =/summationdisplay\nii′,µµ′,σtii′\nµµ′ˆc†\niµσˆci′µ′σ+Jpd/summationdisplay\nm/summationdisplay\nn[m]/vectorSm(t)·ˆ/vector sn\n+/summationdisplay\ni,µµ′,σσ′λi/an}bracketle{tµ,σ|ˆ/vectorL·ˆ/vector s|µ′,σ′/an}bracketri}htˆc†\niµσˆciµ′σ′\n+e2\n4πε0εr/summationdisplay\nm/summationdisplay\niµσˆc†\niµσˆciµσ\n|/vector ri−/vectorRm|+VCorr, (1)\nwherei(i′) is the atomic index that runs over all atoms,\nmruns over the Mn atoms, and n[m] over the nearest-\nneighbors (NN) of m-the Mn atom; µ(µ′) labels atomic\norbitals (s,px,y,z) andσ(σ′) is the spin index; tii′\nµµ′are\nthe Slater-Koster parameters [29] and ˆ c���\niµσ(ˆciµσ) is the\ncreation(annihilation) operator. Below, we briefly dis-\ncuss the meaning of all the terms on the right-hand side\nof Eq. (1). For a more detailed description the reader is\nreferred to Ref. 5.\nThe first term in Eq. (1) is the nearest-neighbors\nsp3Slater-Koster Hamiltonian [30, 31] that reproduces\nthe band structure of bulk GaAs [29]. The second\nterm represents the antiferromagnetic ( p-d) exchange\ncoupling between the Mn spin /vectorSm(originating from\nthed-levels of the dopant and treated here as a classi-\ncal vector) and the nearest-neighbor As p-spins,ˆ/vector sn=\n1/2/summationtext\nπσσ′ˆc†\nnπσ/vector τσσ′ˆcnπσ′, where/vector τσσ′are elements of the\nPauli matrices /vector τ={τα}(α=x,y,z) and the orbital in-\ndexπrunsoverthreeAs p-orbitals. We chosethevalueof\nthe exchangecoupling Jpd= 1.5 eV, which has been been\nreported in the literature [32, 33]. Note that /vectorSmis a unit\nvector and the magnitude of the Mn magnetic moment\n(5/2) is absorbed by the exchange coupling parameter.\nThe third term represents the one-body intra-atomic\n(on-site) SOI, whereˆ/vectorLis the orbital moment operator, ˆ/vector s3\nis the spin operator, |µ,σ/an}bracketri}htare spin- and orbital-resolved\natomic orbitals, corresponding to atom i, andλiare the\nrenormalized spin-orbit-splitting parameters [29] ( λAs=\n0.14 eV,λGa= 0.058 eV and λMn= 0.058/2 eV).\nThe fourth term represents the long-range repulsive\nCoulomb potential, dielectrically screened by the host\nmaterial, with ǫr= 12 for bulk GaAs [34, 35]. /vector riand\n/vectorRmdenote the position of atom iand them-th classical\nspin, respectively. The last term, VCorr, is the central-cell\ncorrection to the impurity potential. This consists of the\non-site part Von, acting on the Mn ion, and the off-site\npartVoff, which affects the NN As atoms and is impor-\ntant for capturing the physics of the p-dhybridization,\nin addition to the exchange interaction ( Jpd). The value\nVon= 1.0 eV is inferred from the Mn ionization energy,\nand we set Voff= 2.4 eV to reproduce the experimen-\ntally observed position of the Mn-induced acceptor level\nin bulk GaAs [36–39].\nWe note that the time-dependence in the electronic\nHamiltonian is carried by the Mn classical spins, /vectorSm(t).\nAt timet, the classical Hamiltonian of the m-th Mn spin\nis written as\nHS(t) =Jpd/summationdisplay\nn[m]/an}bracketle{tˆ/vector sn/an}bracketri}ht(t)·/vectorSm(t). (2)\nThis describes the exchange coupling between /vectorSmand\nthe total instantaneous spin-density of the NN As p-\nspins, defined as a sum of expectation values /an}bracketle{tˆ/vector sn/an}bracketri}ht(t) =\nTr/bracketleftBig\nˆρ(t)ˆ/vector sn/bracketrightBig\n, where ˆρ(t) is the density matrix of the elec-\ntronic subsystem at time t.\nThe dynamical properties of substitutional Mn atoms\nin a GaAs matrix are obtained by performing time-\ndependent SDsimulations fora super-cell-typestructure,\nconsisting of a cubic cluster with 32 atoms and periodic\nboundary conditions applied in three dimensions. The\nequations of motion which, together with the definitions\nof the quantum and classical Hamiltonians [Eqs. (1) and\n(2)], makeupthe coreoftheSDsimulation, aredescribed\nin the next section.\nB. Equations of motion\nThe time evolution of the quantum subsystem is gov-\nerned by the Liouville-von Neumann equation for the\ndensitymatrix,whilethelocalizedspins, representingthe\nMn magnetic moments, evolve according to its classical\nanalogue. Thus the system of coupled equations of mo-\ntion reads\n\n\ndˆρ(t)\ndt=i\n/planckover2pi1/bracketleftBig\nˆρ(t),ˆHel(t)/bracketrightBig\nd/vectorSm(t)\ndt=/braceleftBig\n/vectorSm(t),HS(t)/bracerightBig\n,(3)where [,] denotes the commutator and {,}the Poisson\nbracket. Using the classicalanalogueofthe commutation\nrelations for /vectorSm[40], we can calculate explicitly the Pois-\nson bracket and the classical equation of motion becomes\nd/vectorSm(t)\ndt=Jpd\nS/summationdisplay\nn[m]/angbracketleftBig\nˆ/vector sn/angbracketrightBig\n(t)×/vectorSm(t) (4)\nwhereSis the magnitude of /vectorSm.\nAs one can see from Eq. (3), the quantum and the\nclassical subsystems evolve according to their respective\nequations of motion but are coupled through instanta-\nneous exchange terms, which enter the time-dependent\nHamiltonians ˆHel(t) [Eq. 1] and HS(t) [Eq. 2]. Such cou-\npled system represents the Ehrenfest approximation to\nspin dynamics [24]. This is in contrast to the BO approx-\nimation, in which the fast (quantum) degrees of freedom\nare integrated out. The equations of motion are inte-\ngrated numerically using the fourth-order Runge-Kutta\nalgorithm. The typical duration of the simulation is of\nthe order of 10 ps. We chose a time step dt= 0.001 fs,\nwhich insures that the total energy is conserved within\nan error of 10−9eV.\nIII. RESULTS AND DISCUSSION\nA. Single Mn impurity\nWe first consider a single Mn impurity replacing a Ga\nin the center of a 4-atom GaAs cluster. A smaller clus-\nter size allows us to increase the simulation time up to\n10 ps, in order to understand the evolution of the spins\nand all the energy scales involved in the dynamics. In\nSection IIIB the size of the cluster will be increased to\n32 atoms to study the SD in the presence of two Mn\nspins.\nBefore the start of the simulation (at t=0), the orien-\ntation of the classical Mn spin is fixed along the [001]\ndirection. This corresponds to the z-axis,/vectorSm(0)/bardblz,\norθ=ϕ= 0 in spherical coordinates, where φis the\nazimuthal angle and θis the polar angle. The calcu-\nlations of magnetic anisotropy landscape for a Mn im-\npurity in bulk GaAs, described with the classical-spin\nmodel used in this work, have shown that the [001] di-\nrection is the easy axis, while the plane perpendicular\nto it (x-y) is the hard plane [5]. For this equilibrium\norientation, the electronic density matrix is constructed\nas ˆρ(0) =/summationtext\nνfν|ψν/an}bracketri}ht/an}bracketle{tψν|, where{ψν}are the eigenfunc-\ntions of ˆHel(0) (see Eq. 1) and fνare Fermi-Dirac occu-\npation numbers. The initial NN As spin-density, enter-\ning the classical Hamiltonian in Eq. (2), is calculated as\n/an}bracketle{t/vector sAs/an}bracketri}ht(0) =/summationtext\nn[m]/an}bracketle{tˆ/vector sn(0)/an}bracketri}ht=/summationtext\nn[m]Tr/bracketleftBig\nˆρ(0)ˆ/vector sn/bracketrightBig\n.\nIn order to initiate the SD, the Mn spin is tilted from its\npreferentialaxisbyangles θandϕ. Thisprocedurerepre-\nsents an arbitrary external perturbation, applied locally4\n−0.4−0.200.20.40.6Stot\n0 2 4 6 8 10−0.6−0.4−0.200.20.40.6\nt (ps)Stot(a)\n(b)Sztot\nSxtotSytot\nSytot\nSztotSxtot\nFIG. 1. (Color online) Time evolution of the total spin of the\nsystem/vectorStotfor two choices of the perturbation, (a) θ=φ=\n5◦, and (b) θ=φ= 45◦. Different components of the total\nspin are marked by arrows.\n0 2 4 6 8 10−0.500.51\nt (ps)Sz\n2.72 2.75 2.78−0.100.1\n<szAs>SzMn\nFIG. 2. (Color online) Time evolution of the z-components\nof the spins of the Mn impurity ( SMn\nz) and its NN As atoms\n(/angbracketleftsAs\nz/angbracketright). The inset shows the zoom-in into smaller time scale.\nto the Mn magnetic moment, e.g. a laser pulse or an ex-\nternal magnetic field. As an output of the simulation we\nobtain the time-dependent Cartesian components of the\nMn spin, which will be denoted as /vectorSMnthroughout this\nsection, as well as the time-dependent expectation value\nof the spin at any given atom iof the cluster, /an}bracketle{t/vector si/an}bracketri}ht. We\nwill focus in particular on the total spin /an}bracketle{t/vector sAs/an}bracketri}htof the NN\nAs atoms, where the Mn-induced spin-polarized acceptor\nstate resides, and the total spin of the system defined as\n/vectorStot=/vectorSMn+/summationtext\ni/an}bracketle{t/vector si/an}bracketri}ht, whereiruns over all Ga and As\natoms.\nFigure 1 shows the time evolution of the three compo-\nnents of the total spin /vectorStotfor two different choices ofthe perturbation, namely θ=ϕ= 5◦andθ=ϕ= 45◦.\nThe first choice corresponds to a weak perturbation since\nat the start of the simulation the Mn spin deviates only\nslightly from its preferential axis. We will use this type\nof perturbation in this section. Throughout the next sec-\ntion, IIIB, we will use a slightly stronger perturbation,\nθ=ϕ= 10◦, unless specified otherwise.\nAs one can see from Fig. 1(a), all three components of\n/vectorStotexhibit long-period ( ≈5.5 ps) oscillations, which\nappear by turning on the SOI. Note that without SOI\nthe three components of the total spin are constants of\nmotion and do not change during the time evolution.\nWe conclude that in the case of a weak perturbation\nthe dynamics of the total magnetic moments is mainly\ndriven by the SOI. This is expected since the Mn spin re-\nmains in equilibrium with the spins of the NN As atoms.\nHowever, a strong perturbation ( θ=φ= 45◦) brings\nabout the interplay between /vectorSMnand/an}bracketle{t/vector sAs/an}bracketri}ht, governed by\nthe exchange coupling Jpd. This results in short-period\n(≈500 fs) oscillations, superimposed on the long-period\nandlarge-amplitudeprecessiondue toSOI[seeFig.1(b)].\nA similar effect, namely the appearance of pronounced\noscillations due to Jpd, will be observed if we artificially\nscaleupthe exchangeconstant. Note alsothatthe strong\nperturbation forces the z-component of the total spin to\noscillate between two easy axes (parallel or anti-parallel\nto thez-axis), while in the case of the weak perturbation\n/vectorStotremains above the x-yplane (Stot\nz>0).\nThe comparison between panels 1(a) and (b) also indi-\ncates that the resulting dynamics and the oscillation\nfrequency are sensitive to initial conditions, especially for\nstrong perturbations beyond the linear-response regime.\nFigure 2 shows the time evolution ofthe z-componentsof\nthe Mn spin and the spin of the NN As atoms. Similarly\ntoStot\nz[Fig. 1(a)], the dynamics is governed by the long-\nperiod precession. However, the zoom-in into smaller\ntimes reveals fast oscillations due to Jpd.SMn\nzand/an}bracketle{tsAs\nz/an}bracketri}ht\nare oscillating in anti-phase, in accordance with the an-\ntiferromagnetic nature of the p-dexchange coupling.\nIn order to understand how the characteristic energy\nscales of the system control its dynamics, we perform SD\nsimulations with modified values of exchange and SOI\nparameters. As a references set of parameters, we con-\nsider the values of the spin-orbit splittings λiand the\nexchange interaction Jpdtypically used in our TB model\nfor (Ga,Mn)As. Next, we consider two cases: ( i) the\nexchange interaction is unchanged and the SOI strength\nis 10λiand (ii) the SOI is unchanged and the exchange\ninteraction is 4 Jpd. The time evolution of the Mn and\nNN As spins, calculated with the reference set and with\nthe modified parameters, is shown in Figs. 3(a) and (b),\nrespectively.\nFrom the simulation with the reference set, we identify\ntwo main periods (frequencies), namely the long-period\n(low-frequency) oscillations and the short-period (high-\nfrequency) oscillations. As mentioned before, these two\ncharacteristicperiods aremost likelyassociatedwith SOI\nand with exchange interaction, respectively, since Jpdis5\n−0.400.40.81SzMn\n0 0.2 0.4 0.6 0.8−2.5−2−1.5−1−0.500.5\nt (ps)<szAs>(a)\n(b)Ref.\nRef.\n10λi10λi\n4Jpd4Jpd\nFIG. 3. (Color online) Time evolution of the z-components\nof the Mn spin (a) and the spin of the NN As atoms (b) for\nthree different choices of exchange and SOI parameters: ( i)\nthe reference case (see text for details), ( ii) the case in which\nλiis increased by a factor of 10 and ( iii) the case in which\nJpdis increased by a factor of 4. Only the first 800 fs of the\ntotal simulation time of 10 ps are shown.\n0 0.2 0.4 0.6 0.8−1.5−1−0.500.511.5\nt (ps)Sxtot\n0 0.05 0.1 0.15 0.200.511.52x 106\nω(1015 Hz)dFT[Sxtot]Ref.\nRef.(a)\n(b)4Jpd10λi\n10λi4Jpd\nFIG. 4. (Color online) (a) Time evolution of the x-component\nof the total spin of the system and (b) its discrete Fourier\ntransform, for three different choices of exchange and SOI\nparameters. Notations are the same as in Fig. 3.several orders of magnitude larger than λi. This is fur-\nther confirmed by the simulations with the modified pa-\nrameters. Increasing the SOI strength, while keeping Jpd\nunchanged, leads to a decrease of the long period. At the\nsame time the period of rapid oscillations, superimposed\non the dominant long-period precession, remains practi-\ncally unchanged.\nIncreasing the exchange parameter by a factor of 4 yields\na significant decrease of the short period oscillations.\nHowever, there is also a noticeable change (decrease)\nin the long period oscillations. This is due to the fact\nthat, in principle, one should not expect the two energy\nscalesto affect the dynamics in a completely independent\nway. The complex dynamics of our combined quantum-\nclassical system results from the interplay of the inter-\nactions (rather than simply from a superposition of har-\nmonic motions with different frequencies). The change\ninJpdhas the strongest effect on both short-period and\nlong-period dynamics since it is the largest energy scale\nin the system.\nThe effect of the two energy scales can be also ob-\nserved in the time evolution of the total spin. The time-\ndependentx-component of the total spin and its discrete\nFourier transform (dFT) are shown in Fig. 4(a) and (b),\nrespectively. For all three choices of parameters the long-\nperiod oscillations due to SOI are the most pronounced.\nThe period of these oscillations is approximately 5 .5 ps\nfor the reference set, which is equivalent to the frequency\nof 1012Hz. The latter can be associated with the po-\nsition of the major peak in the corresponding dFT [see\nFig. 1(b)]. Increasing the SOI strength leads to an in-\ncrease of the characteristic period and to a shift of the\ncharacteristic frequency to the higher range. An increase\nof the exchange coupling parameter has a similar effect\nalthough the shift is smaller.\nB. Two Mn impurities\nIn this section we focus on the dynamical interaction\nbetween two spatially separated Mn spins. We use a\nGaAs cluster illustrated in Fig. 5. The effects of the in-\nteraction parameters (exchange coupling and SOI), ini-\ntial configurationofthe spins and the separationbetween\nthe impurity atoms on the dynamics will be investigated.\nThroughout this section the two classical spins are la-\nbeled as/vectorS1and/vectorS2.\nWe first consider the situation when the two Mn spins\nare initially pointing along the z-direction. At the start\nof the simulation, one of the Mn spins is tilted by a small\nangle,θ=ϕ= 10◦, from its initial orientation. Figure 6\nshowsthe timeevolutionofthe x-componentsofbothMn\nspinsand the spinsofthe correspondingNNAs atomsfor\nthree different values of the exchange interaction (simi-\nlar curves are obtained for other spin-components). The\ndynamics is almost identical for the two Mn spins, which\nis a signature of the ferromagnetic coupling between the\ntwo. Each of the localized spins is coupled antiferromag-6\nFIG. 5. (Color online) A finite cluster of GaAs, consisting of\n32 atoms, with two Mn impurities. The separation between\nthe impurities is d≈0.8 nm.\nnetically to the spins of its NN As atoms, resulting in\ncharacteristic anti-phase oscillations.\nThe SD simulations reveal a short delay, or response\ntime (of the order of 1 fs for Jpd= 1.5 eV), between the\nperturbationandtheresponseofthesecondspin. There-\nsponse time decreases with increasing the strength of the\nexchange interaction Jpdbetween each of the individual\nMn spins and the spins of the NN As atoms. This means\nthatthespinperturbation, generatedbytheprecessionof\nthe first (tilted) spin is transferred more efficiently to the\nsecond spin, when the exchange interaction between the\nMn impurity spins and the host carriers is stronger. As\na result, the effective dynamical interaction between the\ntwo impurity spins is enhanced. We can identify a char-\nacteristic time-interval τd(time delay) such that when\nthe two curves, corresponding to the two Mn spins, are\nshifted byτd, they appear to oscillate in a similar way\n(τd= 10 fs forJpd= 1.5 eV). This is essentially the time\nneeded to establish the correlation between the two Mn\nspins. The value of τddecreaseswith Jpd. When the time\ndelay is taken into account, the short period oscillations\nof the spins can be simply superimposed on each other,\nas shown in Figs. 7(a) and (b).\nIn order to understand how the electronic subsystem\n(GaAs host) mediates the propagation of spin distur-\nbance between the localized spins, depending on the\nsystem parameters, we calculate the classical spin-spin\ncorrelation function [23]\nC(S1\nx,S2\nx,∆t) =1\nN/integraldisplay\nS1\nx(t)S2\nx(t+∆t)dt,(5)\nwhere ∆tis thetime lag andNis the normalization\nfactor\nN=/braceleftbigg/integraldisplay/bracketleftbig\nS1\nx(t)/bracketrightbig2dt/bracerightbigg1\n2\n·/braceleftbigg/integraldisplay/bracketleftbig\nS2\nx(t)/bracketrightbig2dt/bracerightbigg1\n2\n.(6)\nHere we focus on the temporal correlations between the\ntransverse ( x) components of the spins, corresponding−1−0.500.51Sx\n−1−0.500.51Sx\n0 0.5 1 1.5−2−1012\nt (ps)Sx\n0 0.5 1 1.5\nt (ps)(b)\n(c) (d)\n(e) (f)<sxAs>\nSxMn(a)\nJpd=1.5 Jpd=1.5\nJpd=0.75\nJpd=6 Jpd=6Jpd=0.75\nFIG. 6. (Color online) The time evolution of the x-\ncomponents of the two Mn spins (blue curves) and the spins\nof their respective NN As atoms (red curves) for (a,b) Jpd=\n1.5 eV, (c,d) Jpd= 0.75 eV and (e,f) Jpd= 6 eV. Left panels\nare for the first (tilted) Mn spin and right panels are for the\nsecond Mn.\n−1−0.500.5SxMn\n−1−0.500.5SxMn\n0 0.04 0.08 0.12 0.16 0.2−1−0.500.5\nt (ps)SxMnSxMn\n1\nSxMn\n1\nSxMn\n1SxMn\n2\nSxMn\n2(b)\n(c)(a)SxMn\n2\nJpd=0.75Jpd=1.5\nJpd=6τd\nτd\nFIG. 7. (Color online) Time evolution of the x-components\nof the two Mn spins during the first 200 fs from the start of\nthe simulation, for (a) Jpd= 1.5 eV, (b) Jpd= 0.75 eV and\n(c)Jpd= 6 eV. Notations are the same as in Fig. 6. Note\nthat the blue curves (tilted spin) in panels (a) and (b) are\nshifted by the time delay τd(see text for details) to show the\nsimilarity in the time evolution of the two spins at later tim es.7\nto deflections away from the easy axis. However, longi-\ntudinal components have also been considered and the\nresulting correlation functions show a similar behavior.\nA few remarks are in order about the definition and\nthe meaning of the classical cross-correlation function in\nEqs. (5) and (6). In principle, for continuous functions\nof time, the integrals in the cross-correlation function\nare taken over the total simulation time Tin the limit\nT→ ∞, i.e./integraltext\n→lim\nT→∞T/integraltext\n0. For functions sampled on a\nfinite time-interval T, the integrals are replaced by finite\nsums and the correlation function becomes [41]\nC(S1\nx,S2\nx,∆t) =1\nNNst−∆t/summationdisplay\ni=0S1\nx(ti)·S2\nx(ti+∆t),(7)\nwhereti=i·dt, ∆t=j·dtandjruns from 1 to Nst,\nwithNst=T/dtbeing the total number of time steps.\nThe normalization factor now reads\nN=/braceleftBiggNst/summationdisplay\ni=0S1\nx(ti)·S1\nx(ti)/bracerightBigg1\n2\n·/braceleftBiggNst/summationdisplay\ni=0S2\nx(ti)·S2\nx(ti)/bracerightBigg1\n2\n.\n(8)\nIn this work we adopt the latter definition.\nThe cross-correlation function C, defined above, mea-\nsures the similarity in the temporal profiles of two signals\n(in this case the two Mn spins) over the total simulation\ntime, as a function of the time lag ∆ tapplied to one of\nthem (this simply means that one of the signals is probed\nat a later time t+∆tcompared to the other one). The\nproperties of the correlation function are described be-\nlow.\nThe amplitude of Cat agiven∆ tisadirect measureof\nthe correlationbetween the two signals: if the correlation\nfunction has a node ( C= 0), the two signals are com-\npletely uncorrelated, while C= 1 means that the signals\nare essentially identical; note that due to the normal-\nization factor, max {|C|} ≤1. Let us first consider two\nsinusoidal signals, which have the same period but differ\nby a phase. The correlationfunction Cin this caseis also\na sinusoidal function with the same period. The value of\nCat zero time lag depends on the phase difference be-\ntween the two signals and varies in the interval [ −1,1].\nIn the ideal case of an infinite integration interval, such\ncorrelation function would oscillate forever without de-\ncay, with an amplitude changing between −1 to 1. If we\nnow add a random contribution to the second signal, the\namplitude of oscillations of the correlation function will\ndecrease, since the coherence between the two signals is\nhindered.\nHowever, for signals sampled on a finite time-interval,\nan additional issue affects the amplitude of the correla-\ntion function. Indeed, the integration interval, i.e. the\nnumber of terms in the finite sum in Eq. (7), becomes\nsmaller as the time lag increases. Therefore, the ampli-\ntude of the correlation function decreases with the time\nlag and vanishes for ∆ tequal to the total simulation time00.51\n  \n00.5\n  \n00.5\n  \n00.5\n  \n00.5\n  \n00.5\n  \n00.5\n  \n0 0.5 1 1.500.5\n  \n∆t (ps)−101\n−101\n−101\n−101\n−101\n−101\n−101\n01020304050−101\nt (fs)Jpd=0.75\nJpd=1.125\nJpd=1.5\nJpd=2\nJpd=6\nJpd=10\nJpd=20\nJpd=30ae−∆t/τSxMn\n2\nSxMn\n1|C(Sx1,Sx2,∆t)| Sx1,2(a) (b)\nFIG. 8. (Color online) (a) The absolute value of the corre-\nlation function/vextendsingle/vextendsingleC(S1\nx,S2\nx,∆t)/vextendsingle/vextendsingle, plotted as a function of the\ntime lag ∆ t, for different values of J. The fitting exponential\nfunctions a·e−∆t/τare shown in red. (b) Time evolution of\nthex-component of the first (blue curve) and second spin (red\ncurve).\n(even for two identical sinusoidal signals). This decay is\nlinear as a function of the time lag in the above example\nand it is negligible when the maximum time lag is much\nsmaller than the integration interval.\nHence, we conclude that for realistic signals, two dis-\ntinct factors affect the amplitude of the correlation func-\ntion: (i) the coherence, or the phase difference between\nthe two signals at a given time lag, which may result in\nan increase or a decrease of the amplitude and is physi-\ncallymeaningful, and ( ii) the decayofthe amplitude as a\nfunction of the time lag due to finite integration interval,\nwhich is inherent in the definition of the correlationfunc-8\nTABLE I. Parameters for the fitting functions in Fig. 8(a).\nThe correlation function for different Jpd.\nJpd(eV) a τ(fs)\n0.75 1.0 155\n1.125 0.66 389\n1.5 0.69 328\n2.0 0.6 508\n6.0 0.69 586\n10.0 0.34 718\n20.0 0.6 200\n30.0 0.61 15\ntion for sampled signals [see Eq. (7)]. In practice, it is\nproblematic to separate the artificial decay due to finite\nintegration intervals from the decay due to the gradual\nloss of coherence (dephasing) between the signals with\nincreasing time lag. However, any increase or non-linear\ndecay ofCcan be associated with the level of coherence.\nIt is also meaningful to compare the oscillating behav-\nior and the decay of the correlation function for different\nsystem parameters, with respect to a reference parame-\nter set. As shown below, the information extracted from\nthe correlation function will be used mainly as a compar-\native measure of the temporal correlations between the\ntwo Mn spins.\nFigure 8(a) shows the absolute value of C(S1\nx,S2\nx,∆t) as\na function of the time lag ∆ t, for different values of the\nexchange coupling Jpd. The curves are fitted to an expo-\nnential function\nf(∆t) =a·e−∆t/τ. (9)\nwhereais a constant and τis the characteristic decay\ntime, with parameters listed in Table I. The constant a\nindicates the degree of correlation between the two spins\nwhileτis the combined measure of the intrinsic decay\ndue to limited integration interval and the fluctuations\ndue to the phase difference between the two spins for in-\ncreasing time lag. While the period of the oscillations\nand the constant aare more robust, the value of τde-\npends on the integration interval T(hereT= 2 ps) and\nit gradually goes to zero as the time lag becomes com-\nparable to T. Although the value of τfor a given Jpd\ndoes not necessarily indicate the decay due to dephasing\nof the two spins, comparing this value for different panels\nin Fig. 8(a) provides insight into how quickly the corre-\nlation dies out for different Jpd.\nTo further clarify the issues related to the definition of\nthe correlationfunction, we presentin Fig. 9 the absolute\nvalue ofC(S1\nx,S2\nx,∆t) calculated using a different inte-\ngration scheme. Here the total simulation time is T=\n4 ps and the integration interval is [0+∆ t,T/2+∆t] ps,\ni.e. the sum in Eq. (7) runs from i=jtoi=Nst/2+j.\nTherefore, as the time lag increase, the integration inter-\nval remains constant ( T/2 = 2 ps). The results obtained\nwith this integration scheme clearly show that the decay\nof the correlation function seen in Fig. 8(a) is, indeed,00.51\n00.51\n00.51\n0 0.5 1 1.500.51\n∆t(ps)Jpd=2\nJpd=10\nJpd=30Jpd=1.125|C(Sx1,Sx2,∆t)|\nFIG. 9. (Color online) (a) The absolute value of the correla-\ntion function/vextendsingle/vextendsingleC(S1\nx,S2\nx,∆t)/vextendsingle/vextendsingle, plotted as afunction of the time\nlag ∆t, for different values of Jpd. Note that the integration\nscheme for the correlation function used here differs from th e\nrest of the paper (see text for details).\npartly caused by the dephasing between the two spins\nat larger time lags. Notably, the decay in the first two\npanels of Fig. 9, which is notcaused by the decrease in\nthe integration interval, is in agreement with the cor-\nresponding panels in Fig. 8(a). The slower decay for\nJpd= 10 eV and the very fast decay for Jpd= 30 eV\nare also in agreement with Fig. 8(a) and with the values\nof the fitting parameters in Table I. We should also men-\ntion that for all cases considered, the exponential fitting\ncurve has better fitting statistics than a linear fit. This\nfurther confirms that τcaptures part of the decay com-\ning from the dephasing of the two spins. However, in the\nrest of the paper, we use the value of τas a comparative\nrather than absolute measure of the dephasing.\nWe will now discuss in detail the behavior of the cor-\nrelation function for different values of Jpd, shown in\nFig. 8(a). The peak in the amplitude of the correlation\nfunctionoccursataveryearlytimecomparedtothetotal\nsimulation time. This confirms the observation that the\ntransfer of the spin perturbation between the two spins,\nmediatedbythehostcarriers,happensonaveryfasttime\nscale, e.g. ∼10 fs (see Fig. 7). The degree of correlation\nat ∆t= 0,a, is maximum for Jpd= 0.75 and nearly con-\nstant (a≈0.6−0.7) for other values (except a somewhat\nspecial intermediate case of Jpd= 10 eV). In contrast,\nthe decay time τvaries significantly with Jpd. It reaches\nits maximum for Jpd≈10 eV and decreases afterward.\nFor exchange interactions in this range, the system is in\na transient regime, before it undergoes a transition to a\nnewdynamicalstatein whichthe Mn spinsareeffectively\ndecoupled.\nThe long-period oscillations of the correlation function\nforJpd<10 eV indicate that the frequency of precession\nof the tilted spin of precession of the tilted spin is not\ncomparable to, i.e. it is much smaller than the character-9\nistic electronic frequency or “electron ticking time”. The\nlatter is the inverse of the time needed for the spin per-\nturbation to travel forth and back between the two spins.\nForJpd/greaterorsimilar10 eV these frequencies become comparable,\nand the correlation function oscillates very rapidly as a\nfunction of the time lag. It is also for a Jpdaround this\nvalue that the temporal correlations become long-ranged\n(the value τis maximum in Table I). This essentially\nmeans that the dynamical coupling between the local-\nized spins is the strongest in this regime. Going over to\nyet larger values of Jpd, the carriers are too localized to\ntransfer the spin perturbation efficiently, and therefore\nthe spins become decoupled. This is also consistent with\nthe short decay time of the correlation function τ≈15 fs\nforJpd= 30 eV.\nThis behavior is consistent with time evolution of\nthe transverse components of the spins [right panels of\nFig. 8]. Except for a short time delay for small values\nofJpd, the motion of the two spins is correlated for\nJpd/lessorsimilar10 eV. However, for very large values of Jpd\nthe second spin completely looses the high-frequency\ncomponent and even oscillates in anti-phase with the\nfirst spin, disrupting the ferromagnetic coupling between\nthe two.\nForabetter understandingofthe correlationbetweenthe\ntwo spins, we consider Mn atoms at different separations.\nThe time-dependent trajectories of the two spins, when\nthey are nearest neighbors (separation d= 0.4 nm),\nreveal that the time delay for the second spin, observed\nin Fig. 7, is now shorter due to closer distance. More\nimportantly the two spins are now behaving as they are\nstrongly coupled. Figure 10(a) shows the correlation\nfunction and the exponential fitting curves for the case\nwhen the Mn atoms are nearest neighbors, for different\nvalues of the exchange coupling (the parameters of the\nfitting are listed in Table II). As one can see, even for\nvery large Jpd= 30 eV, the spins are not completely\ndecoupled: the decay time τis roughly an order of\nmagnitude larger than in the case of larger separation.\nComparing Tables I and II gives us some information\nabout the dependence of the correlation between the\nspins on their separation. The characteristic decay time\nis consistently larger at smaller separation indicating\nthat spins stay correlated longer. Note that the strength\nof correlation, given by parameter a, does not change\ndramatically for the values of Jpdconsidered here and\nis close to the one found at larger separation ( ais\nmaximum for Jpdand is slightly smaller than average\nfor extremely large Jpd).\nIt is instructive to analyze the dependence of the\ncorrelation function on the initial orientation of the two\nMn spins, as it provides insight into which configuration\nis more favorable for long-range temporal correlations.\nSuch analysis is presented in Fig. 10(b). At t= 0 the\nfirst spin is pointing along [001] direction while the\nsecond spin is tilted by an angle θ(we setϕ= 0). We\nconsider a ferromagnetic (FM), θ= 0, antiferromagnetic\n(AFM),θ=π, and non-collinear (NC), 0 < θ < π ,00.51\n  \n00.51\n  \n00.51\n  \n00.51\n  \n0 0.5 1 1.500.51\n∆t (ps)  ae−∆t/τ|C(Sx1,Sx2,∆t)|\nJpd=0.75\nJpd=1.125\nJpd=1.5\nJpd=6\nJpd=30(a)\n00.51\n  \n00.51\n  \n00.51\n  \n00.51\n  \n0 0.5 1 1.500.51\n  \n∆t (ps)ae−∆t/τ θ=0\nθ=π/2θ=π/4θ=π/18|C(Sx1,Sx2,∆t)|\nθ=π(b)\nFIG. 10. (Color online) (a) The absolute value of the correla -\ntion function/vextendsingle/vextendsingleC(S1\nx,S2\nx,∆t)/vextendsingle/vextendsingle, plotted as afunction of the time\nlag (blue curves), and the corresponding exponential fittin g\nfunctions (red curves), for different values of Jpd. The Mn\nspins are nearest-neighbors. (b) The same correlation func -\ntion for the case when the second spin is initiated at differen t\nangles (θvaries from 0 to πandφ= 0,Jpdis fixed at the\nreference value).\nTABLE II. Parameters for the fitting curves in Fig. 10 (a).\nCorrelation function for different Jpdwhen the Mn atoms are\nnearest neighbors.\nJpd(eV) a τ(fs)\n0.75 0.63 740\n1.125 0.67 731\n1.5 0.68 719\n6.0 0.85 513\n30.0 0.56 169\nconfigurations. The parameters for the exponential\nfitting functions are listed in Table III. The spins are\nmost correlated for the FM configuration. In the NC\ncase, the spins are still correlated and the characteristic\ndecay time is even larger then in the FM case. However,\nthe level of correlation is smaller, in particular for\nθ=π/2. For the AFM configuration, the spins become\ncompletely decoupled, with the correlation function\ndecaying rapidly within the first ∼20 fs.10\n00.51\n  \n00.51\n  \n00.51\n  \n0 0.5 1 1.500.51\n  \n∆t (ps)ae−∆t/τ0.5λi\nλi\n2λi\n6λi(a) |C(Sx1,Sx2,∆t)|\n−1−0.500.51\n−1−0.500.51\n−1−0.500.51\n0 0.5 1 1.5−1−0.500.51\nt (ps)SzMn\n2SzMn\n1SzMn\n0.5λi\nλi\n2λi\n6λi(b)\nFIG. 11. (Color online) (a) The absolute value of the corre-\nlation function/vextendsingle/vextendsingleC(S1\nx,S2\nx,∆t)/vextendsingle/vextendsingle, plotted as a function of the\ntime lag (blue curves), and the corresponding exponential fi t-\nting functions (red curves) for different values of SOI. (b)\nTime evolution of of the z-components of the two spins for\nthe same values of SOI.\nTABLE III. Parameters for the fitting curves in Fig. 10(b).\nCorrelation function for different initial configurations o f Mn\nspins.\nthe direction of the second spin at t= 0\nφ= 0 a τ(fs)\nθ= 0 0.67 383\nθ=π/18 0.56 560\nθ=π/4 0.56 452\nθ=π/2 0.45 798\nθ=π 0.7 22.4\nTABLE IV. Parameters for the fitting curves in Fig. 11(a).\nCorrelation function for different SOI strength ( λi). Theλi\nare the reference values of the spin-orbit interaction stre ngth\nfor the three different atoms in the system.\nλi a τ(fs)\n0.5λi 1.11 141.6\nλi 0.69 328.2\n2λi 0.65 502.3\n6λi 0.64 682.5Finally, we probe the correlation function for different\nSOI [Fig. 11(a)]. The strength of SOI directly affects the\nmagnetic anisotropy of the Mn magnetic moment, i.e.\nthe magnetic anisotropy increases with increasing λi.\nBased on this, one might expect that for larger SOI the\ntwo Mn spins will not be able to cross the hard plane\nduring their time evolution (the anisotropy barrier in the\nx-yplane defined by θ=π/2 and arbitrary ϕ). However,\nthe analysis of the time-dependent longitudinal compo-\nnents of the two spins reveals the opposite. According to\nFig. 11(b), S1,2\nz(t) oscillate, switching between [001] to\n[00¯1] directions, for all value of λiconsidered. Therefore\nthe anisotropy energy is not high enough to prevent the\nMn spins from crossing the hard plane. However, as λi\nincreases, the Mn spins tend to spend less time around\nthe hard plane and oscillate more rapidly between the\ntwo easy-axis directions.\nInterestingly, the temporal correlations between the\nspins become stronger. As one can see from Table IV,\nwhich contains the parameters of the fitting functions,\nτincreases with λiwhileasaturates to a value of 0 .6\ntypical for our system. Similar to the case of increasing\nJpd(forJpd<10 eV), the frequency of the long-period\noscillations in the dynamics of the Mn spins increases\nand becomes comparable (although still smaller) to\nthe electron ticking rate. This results in more rapidly\noscillating correlation functions and larger decay times.\nIV. CONCLUSIONS\nIn this paper we studied the time evolution of substi-\ntutional Mn impurities in GaAs, using quantum-classical\nspin-dynamics simulations, based on the microscopic\ntight-binding model and the Ehrenfest approximation.\nWe described the effect of spin-orbit and exchange\ninteractions on the dynamical spectrum. A remarkable\nfeature emerging from our simulations are long-period\n(���5.5 ps) oscillations of the total magnetic moment due\nto the presence of the spin-orbit interaction.\nFurthermore, we studied the spin dynamics in a system\nconsistingoftwoseparatedMnimpurities. Wecalculated\nthe classical spin-spin correlation functions as a measure\nof the effective dynamical coupling between the impurity\nspins. Our results demonstrate the dependence of this\ncoupling on the properties of the electronic subsystem\n(GaAs), as well as on the spatial configuration of the\nimpurity atoms and the initial orientation of their spins.\nWe find that increasing the exchange coupling facili-\ntates the dynamical communication between the spins,\nmediated by the electronic degrees of freedom of the\nhost. However, very strong exchange interaction tends\nto decouple the two spins as the host carriers become\nlocalized. For smaller spacial separation between the\nimpurity atoms, the characteristic decay time of the\ncorrelation function increases significantly for all values\nofJpd. When the spins are initialized in the same11\ndirection (or slightly tilted) they remain ferromagneti-\ncally coupled, with a characteristic in-phase oscillation\npattern, over times of the order of few hundred fem-\ntoseconds. In contrast to this, when starting from an\nantiferromagnetic configuration, the decay time decrease\nto tens of femtoseconds. Increasing the spin-orbit\ninteraction strength seems to have a similar effect as the\nexchange interaction (at least for small values of Jpd),\ni.e. the temporal correlations between the two spins\nbecome more long-ranged. We note, however, that the\nenergy scale associated with the spin-orbit interaction is\norders of magnitude smaller than that of the exchange\ninteraction.\nThe role of the electronic subsystem of the host ma-\nterial, as well as the effects of inter- and intra-atomic\ninteractions (exchange and spin-orbit interactions), onthe dynamics of individual atomic-scale magnets are\nimportant questions that are becoming accessible exper-\nimentally. The present study contributes to fundamental\nunderstanding of spin dynamics in such devices, which\nis crucial for future applications in solotronics.\nACKNOWLEDGMENTS\nThis work was supported by the Faculty of Technology\nat Linnaeus University, by the Swedish Research Coun-\ncil under Grant Number: 621-2010-3761, and the Nord-\nForsk research network 080134 “Nanospintronics: theory\nand simulations”. Computational resources have been\nprovided by the Lunarc center for scientific and technical\ncomputing at Lund University.\n[1] A. M. Yakunin, A. Y. Silov, P. M. Koenraad, J. H.\nWolter, W. Van Roy, J. De Boeck, J.-M. Tang, and\nM. E. Flatt´ e, Phys. Rev. Lett. 92, 216806 (2004).\n[2] D. Kitchen, A. Richardella, J.-M. Tang, M. E. Flatt´ e,\nand A. Yazdani, Nature 442, 436 (2006).\n[3] J. Bocquel, V. R. Kortan, C. Sahin, R. P. Campion, B. L.\nGallagher, M. E.Flatt´ e, andP. M. Koenraad, Phys.Rev.\nB87, 075421 (2013).\n[4] J.-M. TangandM. E. Flatt´ e, Phys. Rev.Lett. 92, 047201\n(2004).\n[5] T. O. Strandberg, C. M. Canali, and A. H. MacDonald,\nPhys. Rev. B 80, 024425 (2009).\n[6] M. R. Mahani, M. F. Islam, A. Pertsova, and C. M.\nCanali, Phys. Rev. B 89, 165408 (2014).\n[7] A. A. Khajetoorians, B. Baxevanis, C. H¨ ubner,\nT.Schlenk,S.Krause, T.O.Wehling, S.Lounis, A.Licht-\nenstein, D. Pfannkuche, J. Wiebe, and R.Wiesendanger,\nScience339, 55 (2013).\n[8] C. F. Hirjibehedin, C. P. Lutz, and A. J. Heinrich,\nScience312, 1021 (2006).\n[9] G. D. Fuchs, V. V. Dobrovitski, D. M. Toyli, F. J. Here-\nmans, and D. D. Awschalom, Science 326, 1520 (2009).\n[10] H. J. M. D. Rugar, R. Budakian and B. W. Chui, Nature\n430, 329 (2004).\n[11] M. Grinolds, P. Maletinsky, S. Hong, M. Lukin,\nR. Walsworth, and A. Yacoby, Nature Physics 7, 687\n(2011).\n[12] M. Grinolds, M. Warner, K. De Greve, Y. Dovzhenko,\nL. Thiel, R. Walsworth, S. Hong, P. Maletinsky, and\nA. Yacoby, Nature nanotechnology 9, 279 (2014).\n[13] S. Loth, M. Etzkorn, C. P. Lutz, D. M. Eigler, and A. J.\nHeinrich, Science 329, 1628 (2010).\n[14] V. P. Antropov, M. I. Katsnelson, B. N. Har-\nmon, M. van Schilfgaarde, and D. Kusnezov,\nPhys. Rev. B 54, 1019 (1996).\n[15] V. P. Antropov, M. I. Katsnelson, M. van Schilfgaarde,\nand B. N. Harmon, Phys. Rev. Lett. 75, 729 (1995).\n[16] S. Kurth, G. Stefanucci, C.-O. Almbladh, A. Rubio, and\nE. K. U. Gross, Phys. Rev. B 72, 035308 (2005).\n[17] M. Fhnle, R. Drautz, R. Singer,\nD. Steiauf, and D. Berkov,Computational Materials Science 32, 118 (2005).\n[18] B. Skubic, J. Hellsvik, L. Nordstrm, and O. Eriksson,\nJournal ofPhysics: CondensedMatter 20,315203(2008).\n[19] M. Stamenova and S. Sanvito,\nPhys. Rev. B 88, 104423 (2013).\n[20] J. Hellsvik, B. Skubic, L. Nordstr¨ om, B. Sanyal,\nO. Eriksson, P. Nordblad, and P. Svedlindh,\nPhys. Rev. B 78, 144419 (2008).\n[21] O. Morandi, P.-A. Hervieux, and G. Manfredi, New\nJournal of Physics 11, 073010 (2009).\n[22] J. Wang, I. Cotoros, K. M. Dani, X. Liu, J. K. Furdyna,\nand D. S. Chemla, Phys. Rev. Lett. 98, 217401 (2007).\n[23] M. Stamenova and S. Sanvito, in The Oxford Handbook\non Nanoscience and Technology , Vol. 1 (Oxford Univer-\nsity Pres, Oxford, 2010).\n[24] A. P. Horsfield, D. R. Bowler, and A. J. Fisher, Journal\nof Physics: Condensed Matter 16, L65 (2004).\n[25] A. Pertsova, M. Stamenova, and S. Sanvito, Phys. Rev.\nB84, 155436 (2011).\n[26] P. M. Koenraad and M. E. Flatt´ e, Nat. Mater. 10, 91\n(2011).\n[27] J. Fransson, Phys. Rev. B 82, 180411 (2010).\n[28] A. T. Costa, R. B. Muniz, and M. S. Ferreira, New\nJournal of Physics 10, 063008 (2008).\n[29] D. J. Chadi, Phys. Rev. B 16, 790 (1977).\n[30] J. C. Slater andG. F.Koster, Phys.Rev. 94, 1498(1954).\n[31] D. A. Papaconstantopoulos and M. J. Mehl, J. Phys.:\nCond. Mat. 15, R413 (2003).\n[32] C. Timm andA. H. MacDonald, Phys. Rev.B 71, 155206\n(2005).\n[33] H. Ohno, Science 281, 951 (1998).\n[34] K. Teichmann, M. Wenderoth, S. Loth, R. G. Ulbrich,\nJ. K. Garleff, A. P. Wijnheijmer, and P. M. Koenraad,\nPhys. Rev. Lett. 101, 076103 (2008).\n[35] D.-H. Lee and J. A. Gupta, Nano Lett. 11, 2004 (2011).\n[36] W. Schairer and M. Schmidt, Phys. Rev. B 10, 2501\n(1974).\n[37] T. Lee and W. W. Anderson, Solid State Commun. 2,\n265 (1964).\n[38] R. A. Chapman and W. G. Hutchinson, Phys. Rev. Lett.\n18, 443 (1967).12\n[39] M. Linnarsson, E. Janzen, B. Monemar, M. Kleverman,\nand A. Thilderkvist, Phys. Rev. B 55, 6938 (1997).\n[40] K.-H. Yang and J. O. Hirschfelder,\nPhys. Rev. A 22, 1814 (1980).[41] J. R. Buck, M. M. Daniel, and A. C. Singer, Com-\nputer Explorations in Signals and Systems Using MAT-\nLAB, 2nd Edition (Prentice Hall, Upper Saddle River,\nNJ, 2002)."    },    {        "title": "2211.03706v1.Simulation_of_the_Einstein_de_Haas_effect_combining_molecular_and_spin_dynamics.pdf",        "content": "Simulation of the Einstein{de Haas e\u000bect combining\nmolecular and spin dynamics\nW. Dednama,b,\u0003, C. Sabaterb, A. E. Bothaa, E. B. Lombardia, J.\nFern\u0013 andez-Rossierc, M. J. Caturlab\naDepartment of Physics, Unisa Science Campus, University of South Africa, Johannesburg\n1710, South Africa\nbDepartamento de F\u0013 \u0010sica Aplicada and Unidad asociada CSIC, Universidad de Alicante,\nCampus de San Vicente del Raspeig, E-03690 Alicante, Spain.\ncInternational Iberian Nanotechnology Laboratory, 4715-310 Braga, Portugal\nAbstract\nThe spin and lattice dynamics of a ferromagnetic nanoparticle are studied via\nmolecular dynamics and with semi-classical spin dynamics simulations where\nspin and lattice degrees of freedom are coupled via a dynamic uniaxial anisotropy\nterm. We show that this model conserves total angular momentum, whereas\nspin and lattice angular momentum are not conserved. We carry out simulatios\nof the the Einstein{de Haas e\u000bect for a Fe nanocluster with more than 500\natoms that is free to rotate, using a modi\fed version of the open-source spin-\nlattice dynamics code ( SPILADY ). We show that the rate of angular momentum\ntransfer between spin and lattice is proportional to the strength of the magnetic\nanisotropy interaction. The addition of the anisotropy allows full spin-lattice\nrelaxation to be achieved on previously reported timescales of \u0018100 ps and\nfor tight-binding magnetic anisotropy energies comparable to those of small Fe\nnanoclusters.\nKeywords: Spin-lattice dynamics, Molecular dynamics simulations, Magnetic\nanisotropy, Conservation of total angular momentum, Angular momentum\nexchange, Einstein{de Haas e\u000bect\n\u0003Corresponding author\nEmail address: dednaw@unisa.ac.za (W. Dednam)\nPreprint submitted to Computational Materials Science November 8, 2022arXiv:2211.03706v1  [cond-mat.mtrl-sci]  7 Nov 20221. Introduction\nFirst observed by Einstein and De Haas [1] in 1915, the so-called Einstein{\nde Haas (EdH) e\u000bect involves the rotation of a magnetic object through change\nin its magnetization. The converse e\u000bect, i.e. magnetization through a rota-\ntional change, was discovered contemporaneously by Barnett [2]. Both e\u000bects\nrely on the conservation of the total angular momentum and are governed by\nthe same gyromagnetic tensor components, satisfying the Onsager reciprocal\nrelations [3{5]. Thus, for any particular physical system, experimental con\fr-\nmation of either e\u000bect is thought to be su\u000ecient to establish conservation of\ntotal angular momentum [6]. Arguably, the EdH e\u000bect is more widely known\nbecause of its routine use in measuring the gfactors of various ferromagnetic\nmaterials: Fe, Co, Ni, etc. [7]. Recently, the EdH e\u000bect has also been observed\nin a ferromagnetic thin \flm, deposited on a microcantilever [8].\nDespite several theoretical studies [10{13], a microscopic description of the\ntwo e\u000bects, either in terms of computer simulations or theoretically, remains a\nchallenging and open problem. On the theoretical front the most recent work by\nMentink et al. [13] shows that, to accurately determine the timescale of angu-\nlar momentum exchange between electronic and lattice degrees of freedom, and\nto establish whether the exchange is local or non-local, it is necessary to cast\nboth quantum subsystems in the angular momentum representation. In prac-\ntice, this involves \\dressing\" every atom with phonons that exchange angular\nmomentum locally with the multi-electron atom's spins and electronic orbital\nangular momenta. The quantum many-body Hamiltonian obtained in this way\nis fully rotationally invariant and a static non-perturbative variational calcula-\ntion shows that phonons quench the electronic orbital angular momentum of\nthe atoms as the coupling strength between the lattice and electronic degrees of\nfreedom increases. A dynamic non-perturbative variational calculation by the\nsame authors [13], on the other hand, reveals the emergence of high frequency\npeaks in the magnetic susceptibility as the electron-phonon coupling strength\nincreases. These peaks are consistent with the timescale of the ultra-fast EdH ef-\n2fect that was observed recently in experiments of laser-induced demagnetization\nin an isolated ferromagnetic sample [38].\nTransfer of angular momentum between spin and motional degrees of free-\ndom ultimately occurs due to spin-orbit interactions that couple these degrees of\nfreedom. In the spin-lattice dynamics (SLD) model, spin-orbit interactions lead\nto magnetic anisotropy terms that depend explicitly on the atomic positions,\nthereby coupling spin and lattice degrees of freedom [9, 15{22]. Importantly,\ntotal angular momentum must be conserved.\nIn this work we implement the SLD model of Ref. [17]. The model is based\non a truncated Taylor expansion of the magnetic anisotropy energy [23]. We\ndemonstrate that it is able to reproduce spin-lattice relaxation times \u0018100 ps,\nconsistent with previously reported values [15, 17, 20]. Our implementation of\nthe SLD model is applicable to combined molecular and spin dynamics of sys-\ntems of arbitrary symmetry (including bulk, clusters, defect systems, etc.) It\nallows the modelling of a wide range of potential new systems at \fnite tem-\nperatures, for example: non-collinear magnetic domain walls in ferromagnetic\natomic-sized contacts or wires [24{26], chiral spin textures in magnetic nanopar-\nticles [27, 28], skyrmions [29, 30], and an atomistic description of the Barnett\n[2] and Barkhausen [31] e\u000bects, amongst others.\nThe remaining material is organized as follows. In Section 2, we give a brief\noutline of the SLD model of ferromagnetic exchange previously implemented in\nSPILADY [32] and demonstrate analytically that it conserves angular momentum.\nHowever, we show that angular momentum exchange is not possible when only\nthe isotropic ferromagnetic exchange term is considered in the model. We there-\nfore show analytically in Section 2 that, by adding the second-order anisotropy\ncorrection of Ref. [17] to the SLD model, not only is total angular momentum\nconserved, but also exchange of angular momenta between spin and lattice is\nfacilitated. In Section 3, we discuss our choice of semi-classical potentials and\nparameters to perform SLD simulations of ferromagnetic iron nanoclusters and\nthen demonstrate numerically that our implementation replicates the Einstein{\nde Haas e\u000bect in a small iron nanocluster. Finally, we conclude the paper by\n3summarizing our \fndings in Section 4.\n2. Models and analytical considerations\n2.1. General description of the SLD model\nOur implementation builds on the well-established technique known as spin-\nlattice dynamics (SLD) [9, 15, 20, 32, 33]. This is a semi-classical molecular\ndynamics approach which allows the simultaneous evolving of atomic spin and\nposition degrees of freedom. The atom-atom interactions in this model are\ndescribed by the widely-used embedded atom method (EAM) potential [34].\nThe EAM potential has been implemented in SPILADY [32, 33] along with a\ngeneralized Heisenberg model of ferromagnetic exchange:\nH=NX\ni\u0000 !pi2\n2m+V+Hspin (1)\nIn Eq. (1),\u0000 !piis the momentum operator for the ith atom and Vis the in-\nteratomic EAM potential energy function that models the basic interactions\nbetween the atoms, i.e. excluding the spin interactions. Expressed in terms\nof the spin angular momentum vectors of particles iandj, the spin Hamilto-\nnianHspinis a sum of the Zeeman term ( HZeeman ), the generalized Heisenberg\nexchange term ( Hexch) and the anisotropy term ( Hanis):\nHspin=HZeeman +Hexch+Hanis (2)\nwith\nHZeeman =\u0000X\ni~Mi\u0001~Bext=X\nig\u0016B~Si\u0001~Bext; (3)\nand\nHexch=\u00001\n2X\niJij(rij)h\n~Si\u0001~Sj\u0000\f\f\f~Si\f\f\f\f\f\f~Sj\f\f\fi\n: (4)\nThe anisotropy term, Hanis, is discussed in the next section. The Zeeman term\nin Eq. (3) is due to an external magnetic \feld ~Bext, where~Mi=\u0000g\u0016B~Siis\nthe magnetic dipole moment, g= 2:002319 is the electron's gyromagnetic factor\n4and\u0016Bis the Bohr magneton. The exchange term, Eq. (4), is isotropic in\nform, with rij=j~ ri\u0000~ rjjbeing the magnitude of the displacement ~ rij=~ ri\u0000~ rj\nbetween the ith andjth atoms. ~Siis the (dimensionless) atomic spin or spin\nangular momentum vector at site iand the quantities Jij(rij) are obtained from\na shifted \ftting function of the form [17, 33]\nJij(rij) =J0(1\u0000rij=rc)3\u0002 (rc\u0000rij); (5)\nin whichJ0is the only \ftting parameter, \u0002 ( rc\u0000rij) is the Heavyside step\nfunction, and rcis the cut-o\u000b radius.\nNext, we discuss the conservation of total angular momentum in the Heisen-\nberg model of ferromagnetic exchange implemented in SPILADY .\n2.2. Conservation of total angular momentum in SLD simulations\nWe brie\ry revisit the textbook derivation of angular momentum conservation\nin a system of classical particles. We de\fne the total angular momentum of a\nsystem of classical particles:\n~L=X\ni~Li (6)\nwhere~Li=mi~ ri\u0002~ viis the angular momentum of an individual particle. The\nrate of change of the total angular momentum is governed by the following\nequation:\nd~L\ndt=X\nid~Li\ndt=X\ni~ ri\u0002~Fi (7)\nWe can break down the force acting on particle iinto two parts, the forces\ncoming from external sources outside of the system and the internal forces,\n~Fi=\u0010\n~Fi;ext+~Fi;int\u0011\n, where the internal forces are given by\n~Fi;int=X\nj6=i~fji (8)\nand where ~fjiis the force that particle jexerts oni. By Newton's third law,\n~fji=\u0000~fij. Therefore, for any pair of particles iandj, the contributions to the\n5rate of change of the angular momentum deriving from their mutual interaction\nonly, are:\nd~Li\ndt\f\f\f\f\f\nint;j+d~Lj\ndt\f\f\f\f\f\nint;i=~ ri\u0002~fji+~ rj\u0002~fij= (~ ri\u0000~ rj)\u0002~fji (9)\nNow, if the internal forces ~fijare central, then they do not contribute to\nthe rate of change of angular momentum of the system, and we are left with\nonly the external forces. Moreover, a force ~fij=\u0000~rU(j~ ri\u0000~ rjj) that can be\nobtained as a gradient of a potential U, such as the EAM potential [34] used\nin this work, is automatically central and, by virtue of being expressed as a\ngradient of a scalar function, conservative.\nThis shows that, for an isolated system of classical particles, in the absence of\nexternal forces, and with internal forces that are central, the rotational angular\nmomentum is conserved.\nEquation (3) shows that the atomic spins in our implementation are dimen-\nsionless. Consequently, in order to calculate the dimensionless classical angular\nmomentum, we de\fne ~\u0015=~L\n~with ~= 6:58211899\u000210\u000016eV.s in the follow-\ning expressions involving the classical angular momentum. So, substituting the\nHeisenberg exchange term, Eq. (4), into Eq. (9) yields for the rate of change of\nthe lattice angular momentum:\nd~\u0015i\ndt\f\f\f\f\f\nexch;j+d~\u0015j\ndt\f\f\f\f\f\nexch;i=1\n~h\n~ ri\u0002~fji+~ rj\u0002~fiji\n=1\n~h\n~ ri\u0002\u0000~riHexch;j+~ rj\u0002\u0000~rjHexch;ii\n=\u0000~ ri\u00021\n~rijdJij(rij)\ndrijh\n~Si\u0001~Sj\u0000\f\f\f~Si\f\f\f\f\f\f~Sj\f\f\fi\n~ rij+~ rj\u00021\n~rjidJji(rji)\ndrjih\n~Sj\u0001~Si\u0000\f\f\f~Sj\f\f\f\f\f\f~Si\f\f\fi\n~ rij\n= (\u0000~ ri+~ rj)\u00021\n~rijdJij(rij)\ndrijh\n~Si\u0001~Sj\u0000\f\f\f~Si\f\f\f\f\f\f~Sj\f\f\fi\n~ rij\n=1\n~rijdJij(rij)\ndrijh\n~Si\u0001~Sj\u0000\f\f\f~Si\f\f\f\f\f\f~Sj\f\f\fi\n~ rij\u0002~ rij= 0\n(10)\nWe therefore see that the classical or rotational angular momentum is con-\nserved in an isotropic model of ferromagnetic exchange.\nWith a view to establishing conservation of spin angular momentum in SLD\n6models, we need to derive an expression analogous to Eq. (9) involving the rates\nof change of the angular momentum of spins iandj. To that end, we start with\nthe equation of motion for the classical spin momentum of particle i:\nd~Si\ndt=\u0000~Si\u00021\n~r~SiHspin (11)\nwhere the energy of the spin system, Hspin, is in general a function of ~Si,~Sj\nand of the coordinates of the atoms and their velocities.\nWe \frst compute the e\u000bective \feld associated with the \frst two terms of Eq.\n(2), classically. For the \frst term, HZeeman , we obtain:\nr~SiHZeeman =g\u0016B~Bi (12)\nThis term produces a rate of change of the spin momentum when ~Siis not\nparallel to ~Bi, and, consequently, we do not consider total angular momentum\nconservation in the presence of an external magnetic \feld any further.\nThe second term, Hexch, produces central forces when we compute the asso-\nciated gradients. Since individual spin magnitudes are conserved in our model,\nwe have for the spin torques:\nr~SiHexch=\u00001\n2X\njJ(rij)~Sj (13)\nBy analogy with Eq. (9), we \fnd that for any pair of particles iandjthe\ncontribution to the rate of change of the spin angular momentum coming from\nthis term vanishes:\nd~Si\ndt\f\f\f\f\f\nexch;j+d~Sj\ndt\f\f\f\f\f\nexch;i=\nJ(rij)\n~\u0010\n~Si\u0002~Sj+~Sj\u0002~Si\u0011\n= 0(14)\nIn other words, isotropic interactions of the scalar form ~S1\u0001~S2preserve the spin\nangular momentum pairwise.\nTherefore, the Heisenberg model of ferromagnetic exchange in Eq. (4) not\nonly conserves total angular momentum, but does so separately within the spin\n7and lattice sub-reservoirs, and cannot account for angular momentum transfer\nbetween lattice and spin degrees of freedom.\nIn the next section, we consider the correction Hanis, which couples the spin\nand lattice degrees of freedom, allowing transfer of angular momentum between\nspin and lattice, while still conserving total angular momentum.\n2.3. Addition of magnetic anisotropy\nTo permit exchange of angular momentum between spins and lattice, we\nhave implemented in the SPILADY code [32] the model of anisotropy described\nin Ref. [17]. We exclude the \frst-order anisotropy correction of Ref. [17] since\nit does not preserve time-reversal symmetry in the absence of an external \feld.\nTherefore we only consider the second-order (uniaxial) anisotropy correction of\nRef. [17] in our implementation:\nHanis=\u0000C2X\ni;\u000b;\fS\u000b(i)\u0003\u000b;\f(i)S\f(i); (15)\nIn Eq. (15), \u0003 \u000b;\f(i) =@2\u001ai(rij)\n@xi\u000b@xi\f, with\u000b;\f = 1;2;3 Cartesian components, is\nthe Hessian matrix element with respect to lattice site i, of the semi-classical\nmany-body , EAM electron density-like, function:\n\u001ai(rij) =8\n<\n:P\nj6=i\u0010\n1\u0000rij\nr0\u00114\ne\u0010\n1\u0000rij\nr0\u0011\nrij\u0014r0\n0 rij>r0;(16)\nwherer0is the cut-o\u000b distance between second and third nearest neighbours\nandC2is a proportionality constant which determines the relative strength of\nthe anisotropy term [17].1\nThus, the contribution of the anisotropy to the e\u000bective local magnetic \feld\nat atom ifrom neighbouring spins within the cut-o\u000b r0becomes:\nr~SiHanis=\u0000C2X\nj6=i\u0014\ng(rij)~Si+1\nrijdg(rij)\ndrij\u0010\n~ rij\u0001~Si\u0011\n~ rij\u0015\n(17)\n1The interested reader is referred to Ref. [35] for full details of our implementation in\nSPILADY of the algorithms described in Refs. [16, 17, 36, 37].\n8whereg(rij)=1\nrijd\u001e(rij)\ndrijwithrij=j~ ri\u0000~ rjjand\u001e(rij) is the expression under\nthe summation in Eq. (16).\nEvaluatingd~Si\ndt=\u0000~Si\u00021\n~r~SiHanisfor the spin at site iyields\nd~Si\ndt=C2\n~X\nk6=i\u00141\nrikdg(rik)\ndrik\u0010\n~ rik\u0001~Si\u0011\n~Si\u0002~ rik\u0015\n(18)\nIn the same way, for the spin at site j,\nd~Sj\ndt=C2\n~X\nk6=j\u00141\nrjkdg(rjk)\ndrjk\u0010\n~ rjk\u0001~Sj\u0011\n~Sj\u0002~ rjk\u0015\n(19)\nFinally, for the jth term fromd~Si\ndtand theith term fromd~Sj\ndt, summing gives\nd~Si\ndt\f\f\f\f\f\nj+d~Sj\ndt\f\f\f\f\f\ni=C2\n~rijdg(rij)\ndrijh\u0010\n~ rij\u0001~Si\u0011\n~Si\u0002~ rij+\u0010\n~ rij\u0001~Sj\u0011\n~Sj\u0002~ riji\n(20)\nsince two instances of ~ rji= -~ rijoccur in the second term so their minus signs\ncancel when the indices of iandjare swapped.\nNext, we consider the lattice torque term, showing that it exactly compen-\nsates the spin torque term of Eq. (20).\nThe expression for the \\anisotropy\" force on the spin at site idue to all\nneighbours jwithin the cut-o\u000b of Eq. (16) is:\n~fanis;i=1\n2C2X\nj6=i1\nrij\u0014dg(rij)\ndrijn\n~Si\u0001~Si+~Sj\u0001~Sjo\n~ rij\n\u0000dg(rij)\ndrij8\n><\n>:\u0010\n~ rij\u0001~Si\u00112\nr2\nij\u0000\u0010\n~ rij\u0001~Sj\u00112\nr2\nij9\n>=\n>;~ rij\n+2dg(rij)\ndrijn\u0010\n~ rij\u0001~Si\u0011\n~Si+\u0010\n~ rij\u0001~Sj\u0011\n~Sjo\n+1\nrijd2g(rij)\nd2rij\u001a\u0010\n~ rij\u0001~Si\u00112\n+\u0010\n~ rij\u0001~Sj\u00112\u001b\n~ rij\u0015(21)\nThen, by substituting Eq. (21) into (9), and considering only the pair of\n9particlesiandj, we obtain after some algebra,\nd~\u0015i\ndt\f\f\f\f\f\nj+d~\u0015j\ndt\f\f\f\f\f\ni=1\n~~ rij\u0002~fji\n=C2\n~rijdg(rij)\ndrijh\u0010\n~ rij\u0001~Si\u0011\n~ rij\u0002~Si+\u0010\n~ rij\u0001~Sj\u0011\n~ rij\u0002~Sji\n;(22)\nwhere we have used the fact that the cross product of a vector with itself is zero.\nClearly, Eqs. (20) and (22) compensate each other when the cross products\nare commuted in Eq. (22) to appear in the same order as in Eq. (20). We have\ntherefore shown analytically that the model of Ref. [17] permits exchange of\nangular momentum, while at the same time guaranteeing conservation of total\nangular momentum.\n3. Numerical simulation results\nWe validate our implementation of the anisotropy correction by studying the\nphenomenon known as the Einstein{de Haas e\u000bect [1]. In its simplest version,\nthe EdH experiment involves using a strong magnetic \feld to change the magne-\ntization of a small freely suspended cylinder made of a ferromagnetic material.\nThe sudden change in magnetization in the cylinder brought on by the strong\nmagnetic \feld then manifests as a physical rotation about the axis of the sample\nas it evolves towards a new equilibrium magnetization due to the conservation\nof the total (spin + rotational) angular momentum. The e\u000bect is well-known,\nbut its microscopic origin has only been recently demonstrated in ultrafast (fem-\ntosecond scale) laser-induced demagnetization experiments [38], though in that\nparticular case the electronic degrees of freedom are also involved since the laser\nprimarily excites the electrons [39, 40].\n3.1. Simulation parameters for Fe\nIn our numerical simulations of the EdH e\u000bect, we use a small prolate\nspheroid made of iron. We choose this shape because the anisotropy is uni-\naxial and a positive value of C2will see the spins aligning preferentially along\nthe long axis of the spheroid. It consists of 505 Fe atoms cut out from a bulk\n10periodic simulation cell of body-centered cubic (BCC) Fe(001). The particle is\nperfectly symmetric about its central axis along the zdirection, i.e., the xandy\ncomponents of each of its constituent atoms sum to 0 with respect to this axis.\nTo describe the interatomic interactions, we use the Malerba et al. [41] EAM\npotential with a per-atom cohesive energy of \u00004:122 eV/atom. This contrasts\nwith the 0 eV cohesive energy of the Harmonic and Morse potentials used in\nRefs. [9, 22]. The EAM potential used in this work is more realistic in that it\nis able to more accurately reproduce material properties [41], including defects\nand the ab-initio energy of a free Fe(001) surface.\nThen, for the Heisenberg exchange term in Eq. (4), we employ the same\nparameter values as in Table II of Ref. [22]: J0= 0:904 eV and rc= 3:75\u0017A.\nFinally, for the anisotropy term in Eq. (15), we use C2= 0:025 eV. \u0017A2\nandr0= 3:5\u0017A.2This value of C2has been estimated by determining the\ndi\u000berence in magnetic anisotropy energy (MAE) of the relaxed spheroid, with\nspins and atoms frozen, compared to the MAE with spins perpendicular to the\n\frst con\fguration, resulting in \u0001MAE \u00180:2 meV/atom, in good agreement\nwith the value of \u00180:2 meV/atom reported in Fig. 7 of Ref. [42] for a similar-\nsized Fe nanocluster using a tight-binding formalism.3\n3.2. SLD simulation of the Einstein{de Haas e\u000bect\nWe simulate the EdH e\u000bect by mirroring an experiment where an initially\ndemagnetized sample (such as via heat shock) is allowed to relax to an equilib-\nrium magnetized state, while observing the rotational angular momentum which\n2We have discovered that the units of C1andC2are quoted incorrectly in Ref. [17], since\nthey should be eV. \u0017A and eV. \u0017A2, respectively.\n3We note that our implementation does not model the weak static background anisotropy\nof 3dtransition metals, which, for example, gives rise to their preferred bulk magnetic easy\naxes. Rather, it accounts for dynamic anisotropy due to local symmetry-breaking about a\ngiven lattice site that may dominate at su\u000eciently high temperatures approaching the Debye\ntemperature of the metal [23]. Nevertheless, dynamic many-body e\u000bects, such as those used\nin this model, play an important role [13] in fast demagnetization processes in ferromagnetic\nmaterials that cannot be modelled by static background anisotropy alone.\n11is induced by spins \ripping and precessing as the sample reaches its equilibrium\nmagnetization.\nWe therefore use an input con\fguration for our SLD simulation which con-\nsists of fully randomized spins, as shown in Fig. 1 (a). The sample is then\nallowed to relax for a period of 1 ns, reaching an equilibrium magnetized state\nafter\u0018100 ps, as illustrated in Fig. 1 (b), where we see the spins have mostly\naligned parallel to the easy axis.4.\nIn Fig. 1 (c) we plot the time evolution of the average per-atom spin and\nrotational angular momentum for C2= 0:025 eV. \u0017A2. Both the rotational lat-\ntice and spin angular momenta are initially near 0. As the simulation proceeds,\nangular momentum is transferred from the spins as they rotate (during the\nprocess of reaching a \fnal equilibrium magnetized state), to rotational lattice\nangular momentum, while conserving the total angular momentum. The mag-\nnitude of the total average angular momentum, or red line, in Fig. 1 (c) has\nbeen calculated as\n\f\f\fD\n~\u0015+~SE\f\f\f=\u0010\n[h\u0015xi+hSxi]2+ [h\u0015yi+hSyi]2+ [h\u0015zi+hSzi]2\u00111=2\n: (23)\nWe note that the average per atom spin angular momentum of \u00180:8\f\f\f~S\f\f\f\ninitial\nis higher than expected at the temperature5of\u0018500 K shown in Fig. 1 (d), due\nto the model of Refs. [17, 22, 33] which does not account for size dependence of\nmagnetization in nanoparticles [44, 45].\nIn Fig. 2 (a) we plot the di\u000berence in magnetic anisotropy energy \u0001MAE\nof the spin populations at 90 °to one another as a function of C2; it is seen that\n\u0001MAE is linear in C2, verifying that the MAE is directly proportional to C2.\n4See Supplemental Material at http://link.xxx.org/supplemental/xxxx/xxxx.xxx for\nmovies of the atoms and spins of the spheroid in Fig. 1 for C2= 0:025;0:05;0:075;0:1;0:15\nand 0:2 eV. \u0017A2showing how the nanocluster evolves to the \fnal states depicted in Fig. 1\n5SPILADY calculates the lattice temperature from the equipartition of kinetic energy in\nthree dimensions and we implemented the generalized expression for the spin temperature\ndeveloped by Nurdin et al. [43] in SPILADY .\n12Figure 1: (a) Initial ( t= 0) spin con\fguration of the Fe(001) prolate spheroid. The atoms\n(not shown) are placed near the sites they would occupy in a perfectly ordered BCC lattice of\nFe(001) and the spins are initially randomly oriented. (b) After t= 100 ps, the spins mostly\npoint upward. The color legend refers to the magnitude of the (dimensionless) atomic spins on\nthe positive z-axis, i.e., the direction of saturation magnetization of Fe, 2 :2\u0016B, divided by the\nelectron's gyromagnetic factor, g= 2:002319, and 1 :0\u0016B. (c) Conservation of total average\nper-atom angular momentum (red) as a sum of contributions from the atomic spins\f\f\fD\n~SE\f\f\f\n(green) and lattice \u0000\f\f\fD\n~\u0015E\f\f\f(purple) for C2= 0:025 eV. \u0017A2. For convenience, the negative\nof\f\f\fD\n~\u0015E\f\f\fhas been plotted so that\f\f\fD\n~SE\f\f\fcoincides with the general orientation ofD\n~SE\nin\n(b). Hyperbolic tangent \fts to\f\f\fD\n~SE\f\f\f(orange) and \u0000\f\f\fD\n~\u0015E\f\f\f(yellow) are also shown. (d) The\nlattice temperature (purple), initially 0 K, rises rapidly while the spin temperature (green),\ninitially \u001812000 K, converges equally rapidly to the equilibrium temperature of \u0018500 K for\nC2= 0:025 eV. \u0017A2.\nFurther, we verify that the strength of the anisotropy correction C2deter-\nmines the rate of exchange of angular momentum between spins and lattice\nrotational angular momentum. To this end, and for ease of comparison, we \ft\nthe absolute values of the average per-atom lattice and spin angular momenta\nto the hyperbolic tangent function atanh (bt) using the \frst 500 ps of simula-\n13Figure 2: (a) The magnetic anisotropy energy di\u000berence scales linearly with C2(R2= 0:993)\nand the range of \u0001MAE values are \u00181 meV/atom [42]. (b) The dependence on C2of the\nrate of change in angular momentum is clear, with stronger anisotropic interactions (larger\nC2) resulting in more rapid change in angular momenta. (Here, R2= 0:978 in the case of the\nspins and 0 :988 in the case of the lattice.)\ntion data in each case, as illustrated in Fig. 1 (c) for the particular case of\nC2= 0:025 eV. \u0017A2. The rate of change of angular momentum is the greatest\nnear the origin; therefore we use the derivatives of the \fts to this function at\nt= 0 as measure of the rate of exchange of angular momentum. In Fig. 2 (b)\nwe plot the rates of change of spin and lattice rotational angular momenta as\n14a function of C2. It is seen that the rate of exchange of angular momenta is\nlinear inC2, verifying that this rate is directly proportional to the strength of\nthe magnetic anisotropy.\nFigure 3: Conservation of total angular momentum for the full simulation duration and di\u000ber-\nent values of the anisotropy coe\u000ecient C2: (a) 0:025, (b) 0:075, (c) 0:1, (d) 0:15 eV. \u0017A2. Total\nangular momentum (black) does not vary during the simulation, while the individual compo-\nnents, the average per atom spin (green curve) and lattice rotational moment (red curve) do\nvary. As the value of C2increases, the coordinates of the average per-atom spin and lattice\nangular momenta take less time to approach the long ( z) axis of the spheroid via exchange\nof angular momentum. This is also evident from the values of the derivatives at t= 0 of the\nhyperbolic tangent \fts to the absolute values of the spin and lattice angular momenta in Fig.\n2.\nIn order to provide visual con\frmation of the trend in Fig. 2 (b), we plot\nin Fig. 3 the time evolution of the coordinates of the per-atom average angular\n15momenta of the spins (green), lattice (red) and their sum (black) for a range of\nvalues ofC2. Since the anisotropy is uniaxial in this work, the green and red\ntraces tend to cluster near the z-axis, and with opposite signs. The symmetry\nof the spin Hamiltonian implies that there is no preferred \fnal direction of\nthe average per-atom spin and lattice angular momenta as long as they are\nmutually opposite and generally point along the cluster's easy axis ( zin this\ncase). Therefore, randomly, either of the spin or the lattice rotational angular\nmomentum can be oriented along the positive zaxis (see e.g. the traces of Fig.\n3 (c) vs the other three traces); it is the relative orientation of rotational versus\nspin angular momenta which is important, and which are always opposite and\nequal, conserving total angular momentum.\nFinally, we note that, to our knowledge, the only previous SLD simulation\nwe are aware of that reproduces the EdH e\u000bect (Ref. [9]) applies an external\nmagnetic \feld of 50 Tesla throughout the entire demagnetization process of a\nfree face-centered cubic Co cluster. Also, only the zcomponents of the spin and\nlattice angular momenta are plotted in Ref. [9], casting doubt on whether total\nangular momentum is actually conserved in their EdH simulation. (An external\n\feld can induce a non-zero overall spin torque due to individual contributions\nnot cancelling out in a pairwise manner.)\n4. Conclusion\nWe have shown analytically that the dynamic many-body model of uniaxial\nanisotropy described in Ref. [17] conserves the total angular momentum and\nenergy of the system. We also implemented this model in the spin-lattice dy-\nnamics code SPILADY [32], and also make the source code available online [46].\nOur implementation reproduces the full Einstein{de Haas e\u000bect in an isolated\niron nanoparticle, consistent with the conservation of total angular momentum\nby the model over a range of values of the magnetic anisotropy coe\u000ecient C2.\nThe value of this coe\u000ecient not only determines the rate at which exchange\nbetween spin and rotational angular momenta occurs, but also how quickly the\n16angular momenta align parallel to the long axis of the nanoparticle for positive\nC2.\nThe advantage of the approach followed in this work, compared to other\nsimilar generally applicable semi-classical models, is that the realistic EAM po-\ntential is used to describe interactions between the atoms, allowing accurate\nreproduction of ab-initio free surface energies and defects. Another advantage\nis that the angular momentum exchange occurs in a very stable and consistent\nmanner, with full spin-lattice relaxation occurring in agreement with previously\nreported times\u0018100 ps. Moreover, neglecting shape anisotropy, the di\u000ber-\nence in per-atom magnetic anisotropy energy between spin populations oriented\nat 90 °to each other in the same frozen atomic con\fguration, yields order-of-\nmagnitude agreement, at the lower range of the magnetic anisotropy interaction\nstrength,C2, with previously reported tight-binding values for Fe nanoclusters\nof comparable size. The development of dynamic anisotropy models due to local\nsymmetry breaking at \fnite temperatures is therefore crucial to modeling fast\ndemagnetization processes.\nOur implementation of anisotropy can be employed to study any single-\nelement ferromagnetic system in which local coupling between spins and atoms\ncannot be ignored. It is also possible to extend this approach to multiple-\nelement systems once SPILADY , the underlying spin-lattice dynamics code used\nin this implementation, makes provision for this capability. For this reason it is\nlikely to \fnd applications in a signi\fcant number of situations where uniaxial\nmagnetization is important and coupling between lattice and spins plays a non-\nnegligible role.\nAcknowledgements\nWe acknowledge \fnancial support from the Ministry of Science and Inno-\nvation of Spain (grant No. PID2019-109539GB-4). This work was supported\nby the Generalitat Valenciana through PROMETEO2017/139 and PROME-\nTEO/2021/017. C. S. gratefully acknowledges \fnancial supports from General-\n17itat Valenciana with (CDEIGENT2018/028). JFR acknowledges funding from\nFCT grant PTDC/FIS-MAC/2045/2021. The SLD calculations in this paper\nwere performed on the high-performance computing facilities of the University\nof South Africa.\nReferences\n[1] A. Einstein, W. Haas, Experimental proof of the existence of amp\u0012 ere's\nmolecular currents, KNAW, Proceedings, 18 I, 1915, Amsterdam 18 I (1915)\n696{711.\nURL http://www.dwc.knaw.nl/DL/publications/PU00012546.pdf\n[2] S. J. Barnett, Magnetization by Rotation, Phys. Rev. 6 (1915) 239. doi:\n10.1103/PhysRev.6.239 .\n[3] L. Landau, E. Lifshitz, L. Pitaevskii, Electrodynamics of Continuous Me-\ndia, Course of Theoretical Physics, Butterworth-Heinemann, Oxford, 1995.\n[4] G. E. W. Bauer, S. Bretzel, A. Brataas, Y. Tserkovnyak, Nanoscale mag-\nnetic heat pumps and engines, Phys. Rev. B 81 (2010) 024427.\n[5] G. E. Bauer, E. Saitoh, B. J. V. Wees, Spin caloritronics, Nat. Mater. 11\n(2012) 391.\n[6] K. Fukushima, S. Pu, Z. Qiu, Eddy magnetization from the chiral Barnett\ne\u000bect, Phys. Rev. A 99 (2019) 032105.\n[7] G. G. Scott, Review of gyromagnetic ratio experiments, Rev. Mod. Phys.\n34 (1962) 102{109.\n[8] T. M. Wallis, J. Moreland, P. Kabos, Einsteinde Haas e\u000bect in a NiFe \flm\ndeposited on a microcantilever, Appl. Phys. Lett. 89 (2006) 122502.\n[9] M. A\u0019mann, U. Nowak, Spin-lattice relaxation beyond gilbert damping, J.\nMagn. Magn. 469 (2018) 217{223. doi:10.1016/j.jmmm.2018.08.034 .\n18[10] A. A. Kovalev, G. E. W. Bauer, A. Brataas, Current-driven ferromagnetic\nresonance, mechanical torques, and rotary motion in magnetic nanostruc-\ntures, Phys. Rev. B 75 (2007) 014430.\n[11] R. Jaafar, E. M. Chudnovsky, D. A. Garanin, Dynamics of the Einstein-de\nHaas e\u000bect: Application to a magnetic cantilever, Phys. Rev. B 79 (2009)\n104410.\n[12] E. M. Chudnovsky, D. A. Garanin, Rotational states of a nanomagnet,\nPhys. Rev. B 81 (2010) 214423.\n[13] J. H. Mentink, M. I. Katsnelson, M. Lemeshko, Quantum many-body dy-\nnamics of the Einstein-de Haas e\u000bect, Phys. Rev. B 99 (2019) 064428.\n[14] N. Bloembergen, S. Shapiro, P. Pershan, J. Artman, Cross-relaxation in\nspin systems, Physical Review 114 (2) (1959) 445.\n[15] D. Beaujouan, P. Thibaudeau, C. Barreteau, Anisotropic magnetic\nmolecular dynamics of cobalt nanowires, Phys. Rev. B 86 (2012) 174409.\ndoi:10.1103/PhysRevB.86.174409 .\nURL https://journals.aps.org/prb/abstract/10.1103/PhysRevB.\n86.174409\n[16] D. Perera, Atomistic simulations of magnetic models with coupled trans-\nlational and spin degrees of freedom, Ph.D. thesis, University of Georgia\n(2015).\n[17] D. Perera, M. Eisenbach, D. M. Nicholson, G. M. Stocks, D. P. Landau,\nReinventing atomistic magnetic simulations with spin-orbit coupling, Phys.\nRev. B 93 (2016) 060402. doi:10.1103/PhysRevB.93.060402 .\nURL https://doi.org/10.1103/PhysRevB.93.060402\n[18] M. A\u0019mann, Atomistic simulation of spin-lattice dynamics, Ph.D. thesis,\nUniversity of Konstanz (2018).\n19[19] J. Hellsvik, D. Thonig, K. Modin, D. Iusan, A. Bergman, O. Eriksson,\nL. Bergqvist, A. Delin, General method for atomistic spin-lattice dynamics\nwith \frst principles accuracy, Phys. Rev. B 99 (2018) 104302. doi:10.\n1103/PhysRevB.99.104302 .\n[20] J. Tranchida, S. Plimpton, P. Thibaudeau, A. Thompson, Massively paral-\nlel symplectic algorithm for coupled magnetic spin dynamics and molecular\ndynamics, J. Comp. Phys. 372 (2018) 406{425. doi:10.1016/j.jcp.2018.\n06.042 .\n[21] P. Nieves, J. Tranchida, S. Arapan, D. Legut, Spin-lattice model for cubic\ncrystals, Phys. Rev. B 103 (2021) 094437. doi:10.1103/PhysRevB.103.\n094437 .\nURL https://link.aps.org/doi/10.1103/PhysRevB.103.094437\n[22] M. Strungaru, M. O. A. Ellis, S. Ruta, O. Chubykalo-Fesenko, R. F. L.\nEvans, R. W. Chantrell, Spin-lattice dynamics model with angular mo-\nmentum transfer for canonical and microcanonical ensembles, Phys. Rev.\nB 103 (2) (2021) 1{13. doi:10.1103/physrevb.103.024429 .\n[23] M. Eisenbach, D. Perera, D. P. Landau, D. M. Nicholson, J. Yin, G. Brown,\nMagnetic materials at \fnite temperatures: Thermodynamics and combined\nspin and molecular dynamics derived from \frst principles calculations, J.\nPhys.: Conf. Ser. 640 (2015) 012019.\n[24] W. Dednam, C. Sabater, O. Tal, J. J. Palacios, A. E. Botha, M. J. Caturla,\nRe\fned electron-spin transport model for single-element ferromagnetic sys-\ntems: Application to nickel nanocontacts, Phys. Rev. B 102 (2020) 245415.\ndoi:10.1103/PhysRevB.102.245415 .\nURL https://link.aps.org/doi/10.1103/PhysRevB.102.245415\n[25] J. M. Caridad, D. Mccloskey, J. F. Donegan, V. Krsti\u0013 c, Controllable growth\nof metallic nano-helices at room temperature conditions, Appl. Phys. Lett.\n105 (2014) 233114. doi:10.1063/1.4904091 .\n20[26] S. Finizio, M. Foerster, M. Buzzi, B. Kr uger, M. Jourdan, C. A. F. Vaz,\nJ. Hockel, T. Miyawaki, A. Tkach, S. Valencia, F. Kronast, G. P. Car-\nman, F. Nolting, M. Kl aui, Magnetic anisotropy engineering in thin \flm\nni nanostructures by magnetoelastic coupling, Phys. Rev. Applied 1 (2014)\n021001. doi:10.1103/PhysRevApplied.1.021001 .\nURL https://link.aps.org/doi/10.1103/PhysRevApplied.1.021001\n[27] D. B. Reeves, J. B. Weaver, Approaches for modeling magnetic nanoparticle\ndynamics, Critical Reviews in Biomedical Engineering 42 (1) (2014) 85.\n[28] J.-Y. Chaulaeau, et al., Electric and antiferromagnetic chiral textures at\nmultiferroic domain walls, Nature Materials 4 (2020) 386{390. doi:10.\n1038/s41563-019-0516-z .\n[29] A. Fert, N. Reyren, V. Cros, Magnetic skyrmions: advances in physics and\npotential applications, Nature Reviews Materials 2 (2017) 17031. doi:\n10.1038/natrevmats.2017.31 .\n[30] J. Z\u0013 azvorka, F. Jakobs, D. Heinze, N. Keil, S. Kromin, S. Jaiswal,\nK. Litzius, G. Jakob, P. Virnau, D. Pinna, K. Everschor-Sitte, L. R\u0013 ozsa,\nA. Donges, U. Nowak, M. Kl aui, Thermal skyrmion di\u000busion used in\na reshu\u000fer device, Nature Nanotechnology 14 (2019) 658{661. doi:\n10.1038/s41565-019-0436-8 .\n[31] B. D. Cullity, C. D. Graham, Introduction to magnetic materials, 2nd Edi-\ntion, Wiley, New Jersey, 2009.\n[32] P.-W. Ma, S. L. Dudarev, C. H. Woo, Spilady: A parallel cpu and gpu\ncode for spinlattice magnetic molecular dynamics simulations, Comp. Phys.\nComm. 207 (2016) 350 { 361. doi:https://doi.org/10.1016/j.cpc.\n2016.05.017 .\n[33] P.-W. Ma, C. H. Woo, S. L. Dudarev, Large-scale simulation of the spin-\nlattice dynamics in ferromagnetic iron, Phys. Rev. B 78 (2) (2008) 024434.\n21doi:10.1103/PhysRevB.78.024434 .\nURL http://link.aps.org/doi/10.1103/PhysRevB.78.024434\n[34] M. S. Daw, M. I. Baskes, Semiempirical, quantum mechanical calculation\nof hydrogen embrittlement in metals, Phys. Rev. Lett. 50 (17) (1983) 1285.\n[35] W. Dednam, Atomistic simulations of competing in\ruences on electron\ntransport across metal nanocontacts, Ph.D. thesis, University of Ali-\ncante/University of South Africa (2019).\n[36] M. Krech, A. Bunker, D. Landau, Fast spin dynamics algorithms for\nclassical spin systems, Comput. Phys. Commun. 111 (1) (1998) 1{13.\ndoi:10.1016/S0010-4655(98)00009-5 .\n[37] D. P. Landau, S.-H. Tsai, M. Krech, A. Bunker, Improved spin dynamics\nsimulations of magnetic excitations, Int. J. Mod. Phys. C 10 (08) (1999)\n1541{1551.\n[38] C. Dornes, Y. Acremann, M. Savoini, M. Kubli, M. Neugebauer, E. Abreu,\nL. Huber, G. Lantz, C. Vaz, H. Lemke, E. Bothschafter, M. Porer, V. Es-\nposito, L. Rettig, M. Buzzi, A. Alberca, Y. Windsor, P. Beaud, U. Staub,\nS. Johnson, The ultrafast einstein-de haas e\u000bect, Nature 565 (01 2019).\ndoi:10.1038/s41586-018-0822-7 .\n[39] E. Beaurepaire, J.-C. Merle, A. Daunois, J.-Y. Bigot, Ultrafast spin dy-\nnamics in ferromagnetic nickel, Phys. Rev. Lett. 76 (1996) 4250{4253.\ndoi:10.1103/PhysRevLett.76.4250 .\nURL https://link.aps.org/doi/10.1103/PhysRevLett.76.4250\n[40] P.-W. Ma, S. L. Dudarev, C. H. Woo, Spin-lattice-electron dynamics sim-\nulations of magnetic materials, Phys. Rev. B 85 (2012) 184301. doi:\n10.1103/PhysRevB.85.184301 .\nURL https://link.aps.org/doi/10.1103/PhysRevB.85.184301\n[41] L. Malerba, M. C. Marinica, N. Anento, C. Bj orkas, H. Nguyen, C. Do-\nmain, F. Djurabekova, P. Olsson, K. Nordlund, A. Serra, D. Terentyev,\n22F. Willaime, C. S. Becquart, Comparison of empirical interatomic poten-\ntials for iron applied to radiation damage studies, J. Nucl. Mater. 406 (1)\n(2010) 19 { 38. doi:10.1016/j.jnucmat.2010.05.017 .\nURL http://www.sciencedirect.com/science/article/pii/\nS0022311510002291\n[42] D. Li, A. Smogunov, C. Barreteau, F. Ducastelle, D. Spanjaard, Magne-\ntocrystalline anisotropy energy of Fe (001) and Fe (110) slabs and nan-\noclusters: A detailed local analysis within a tight-binding model, Physical\nReview B 88 (21) (2013) 1{9. doi:10.1103/PhysRevB.88.214413 .\n[43] W. B. Nurdin, K.-D. Schotte, Dynamical temperature for spin systems,\nPhys. Rev. E 61 (2000) 3579{3582. doi:10.1103/PhysRevE.61.3579 .\nURL https://link.aps.org/doi/10.1103/PhysRevE.61.3579\n[44] G. Dos Santos, R. Aparicio, D. Linares, E. N. Miranda, J. Tranchida,\nG. M. Pastor, E. M. Bringa, Size- and temperature-dependent magne-\ntization of iron nanoclusters, Phys. Rev. B 102 (2020) 184426. doi:\n10.1103/PhysRevB.102.184426 .\nURL https://link.aps.org/doi/10.1103/PhysRevB.102.184426\n[45] G. Dos Santos, R. Meyer, R. Aparicio, J. Tranchida, E. M. Bringa,\nH. M. Urbassek, Spin-lattice dynamics of surface vs core magnetization\nin Fe nanoparticles, Applied Physics Letters 119 (16) (2021) 012404.\ndoi:10.1063/5.0055606 .\nURL https://aip.scitation.org/doi/10.1063/5.0055606\n[46] P.-W. Ma, et al., computer code SPILADY, with anisotropy corrections\nmade by W. Dednam (2018).\nURL https://github.com/wdednam/spilady/tree/anisotropy\n23"    },    {        "title": "0910.3593v1.Derivatives_of_spin_dynamics_simulations.pdf",        "content": "1 \n  \n \n \n  \n \nDerivatives of spin dynamics simulations \n \nIlya Kuprov†, Christopher T. Rodgers‡ \n†Oxford e-Research Centre, University of Oxford, \n7 Keble Road, Oxford, OX1 3QG, UK. \n‡Oxford Centre for Clinical Magnetic Resonance Research, \nLevel 0, John Radcliffe Hospital, Oxford, OX3 9DU, UK. \n \n \n \n  \n \n \n \n \nEmail: ilyaATkuprovDOTcom , pubs-cATrodgersDOTorgDOT uk 2 \n Abstract \nWe report analytical equations  for the derivatives of spin  dynamics simulations with \nrespect to pulse sequence and sp in system parameters. The met hods described are significantly \nfaster, more accurate and more reliable than the finite difference approximations typically \nemployed. The resulting derivatives  may be used in fitting, optim ization, performance evaluation \nand stability analysis of spin dynamics simulations and experiments. \nKeywords:  NMR, EPR, simulation, analytical deriva tives, optimal control,  spin chemistry, \nradical pair. 3 \n Introduction \nThe response of a magnetic resonance experi ment to a small perturbation in the spin \nsystem or pulse sequence parameters is us eful in two contexts: spectral fitting 1-5 and pulse \nsequence design using optimal control theories 6-9.  \nSpectral fitting algorithms require gradients for opt imization step control 10-12. In small-\nscale calculations, gr adient-free methods 13-14 are adequate, but as the system gets bigger, \nparameters proliferate and such gradient-free methods become sl ow and hence impractical. As \nour colleagues in electronic structure theory quickly discovered 15-16, for large systems analytical \ngradients are required for efficient optimization. \nIn pulse sequence design using optimal control, an important question is the robustness of \nthe resulting waveforms with resp ect to non-idealities in the control operators and external noise \n17-18 as well as the natural variability in the practi cally encountered spin systems. A derivative of \nthe coherence transfer efficiency with respect to e.g. a systematic phase distortion in the RF \nchannel can be interpreted as a measure of stabi lity of the pulse sequence with respect to that \nnotorious instrumental problem. \nImportantly, the derivatives in question of ten cannot be obtained numerically: modern \nlarge-scale simulation algorithms have multiple dynamic cut-offs and tolerances (in \northogonalization, matrix inversion, singular value decomposition and other essential \nmathematical procedures) meaning that a small pe rturbation in a parameter may trigger a step \nchange in the simulation result,  e.g. inclusion or rejecti on of a particular vect or in the basis. In \nother words, many algorithms are not numerically di fferentiable with respect to their parameters. \nThey may also display high levels of numerica l noise due to the fin ite precision of machine \narithmetic. Even when they are reasonably accurate, numerical derivatives have a high \ncomputational cost in such large-scale simula tions: typically between two and four separate \nsimulations per parameter. \nWe report in this communication the analytic al equations for the derivatives of spin \ndynamics simulations with respect  to both pulse sequence and spin system parameters. The 4 \n equations reported are significantly faster, much  more accurate, and, importantly, much more \nreliable than finite difference approximations. \nTheory \nSpin dynamics simulations may be formulated either in Hilbert space, where operators \nare represented as nn matrices, or in Liouville space, where operators are represented as 2n-\nelement vectors and superoperators are 2 2nn matrices 19. We describe the spin system in terms \nof its density operator ˆt . In Hilbert space, this evolves according to the Liouville–von \nNeumann equation 19: \n  ˆˆ ˆ i[ , ]tHt tt, (1) \nin which ˆHt  is the Hamiltonian operator. In Lio uville space, the density operator evolves \nunder the system Liouvillian superoperator ˆˆLt , which may include contributions from \nrelaxation and chemical kinetics: \n  ˆ ˆˆ ˆ i.tLt tt (2) \nBoth ˆHt  and ˆˆLt  are assumed to be piecewise continuous. Differentiation may be performed \nby several approaches in  both representations. \nI. Time domain derivative superoperator \nStarting in Liouville space, it is natural to seek a superoperator ˆˆAt  that acts on the \ndensity operator gi ving its derivative ˆt with respect to a simulation parameter α: \n    ˆˆˆˆ ˆˆ    i.e.      . Att At t  (3) \nIntegrating Equation (2) analytically for an infinitesimal time step t yields \n   ˆˆ ˆˆ exp i tt L t t   , (4) 5 \n which is free from time derivatives (which do not necessarily commute with  ) and may be \ndirectly differentiated with respect to the parameter: \n   ˆˆˆˆ ˆˆ ˆ e x pi e x pi tt L t t L t t               (5) \nCombining Equation (5) with the definition given in Equation (3) yields a time propagation law \nfor the differentiation superoperator: \n  ˆˆ ˆˆ ˆ ˆ ˆˆ ˆˆ ˆ ˆ exp i exp i exp i exp i At t L t L t L t At L t                    (6) \nTaking the limit 0t  and neglecting 2Ot  and higher terms, gives the equation of motion \nfor the differentiation superoperator: \n ˆˆˆˆˆˆˆˆi[ , ] iALA Lt   (7) \nwhere ˆˆL is a Liouvillian derivative with respect to  the simulation parameter in question. The \ninitial conditions are in most cases ˆˆ00A and  ˆ00, reflecting the fact that the \nsimulation starts from a user-speci fied state that does not depend on . Even though the \nevolution of the differentiation superoperator is governed by the same Liouvillian, it is \nindependent from the evolution of the density matrix. \nAnalogous reasoning yields the followi ng equations for the second derivative \nsuperoperator ˆˆW: \n    \n2ˆˆ ˆ ˆˆ ˆ ˆ ˆˆ ˆ ˆˆ ˆ ˆ exp i exp i exp i exp i\nˆ ˆˆ ˆ ˆ ˆ ˆˆ ˆ ˆ exp i exp i exp i exp iWt t LtWt Lt Lt B t Lt\nLt A t Lt Lt Lt\n                             \n                        (8) \n  ˆˆˆ ˆˆ ˆˆ ˆ ˆ ˆ ˆˆ ˆˆ ˆ ˆi[ , ] i i iWLW L B t L A t Lt        (9) \nwhere ˆˆBt is a first derivative superoperator with respect to th e second parameter . Higher \norder derivatives may be obt ained in a similar way. 6 \n II. Liouville-space derivative density matrix \nA second approach identifies an equation of motion for the derivative of the density \nmatrix in Liouville space.  We begin by taking the 0t limit of Equation (5), which yields the \nfollowing equation of motion for the derivative density matrix \n ˆ ˆˆˆˆˆˆiiLLt\n   (10) \nThis has the same general form as Equation (2) with the addition of a se cond driving term that \ndepends on the density matrix itself. \nRepeating these steps for the second derivative yields the following equations for the second \nderivative density matrix dynamics:  \n      ˆ ˆˆˆ ˆ ˆˆ ˆˆˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ;e x p i t t P t Pt Pt Pt P L t                        (11) \n ˆ ˆˆ ˆ ˆˆˆ ˆ ˆˆˆ ˆˆi iiiLL L Lt\n           (12) \nHigher order derivatives may be obtained in a similar manner. \nIII. Hilbert-space derivative density matrix \nIt is also possible to find an equivalent equation of motion for the derivative of the \ndensity matrix in Hilbert space. Integrating Equation (1) for an infinitesimal time step t \n  ˆˆ ˆˆ exp i exp i tt H t t H t     (13) \nand differentiating with respect to the parameter  gives \n  \n\nˆˆ ˆˆ exp i exp i\nˆˆ ˆ exp i exp i\nˆˆ ˆ exp i exp i .tt H t t H t\nHt t Ht\nHt t Ht\n\n\n        \n         \n        (14) \n 7 \n Taking the 0t  limit then yields the following Hilbert-space equation of motion for the \nderivative density matrix \n ˆˆˆˆˆ i[ , ] i[ , ]HHt\n    (15) \nLike Equation (10) above, this expression for the derivative density matrix is the same as \nEquation (1) with an additional driving term that depends on the density matrix itself. Once \nagain, higher derivatives may be obtained by analogous reasoning. For example, the second \nderivative density matrix satisfies \n     \n    ˆˆ ˆ ˆ ˆˆ ˆ ˆ ˆˆ ˆ ˆ ˆ\nˆˆ ˆ ˆ ˆ ˆˆ ˆ ˆˆ ˆˆ ˆ ˆ ˆ ;\nˆˆ ˆ ˆ exp i ;      exp itt t t t t\ntt t tt\nHt Ht       \n         \n              \n              \n         (16) \n ˆˆ ˆˆˆ ˆˆ ˆˆ i[ , ] i[ , ] i[ , ] i[ , ]H HHHt\n           (17) \nIV. Evaluation of key derivatives \nThe individual derivatives involved in the eq uations above are all ve ry straightforward. \nThe Hamiltonian and Liouvillian derivative with respect to a parameter returns an operator or \nsuperoperator corresponding to that parameter. For example, in Hilbert space the derivative of the Hamiltonian with respect to \nnka  is \n     \nZˆˆˆ ˆ ˆ ˆ ˆ ...          ii j n k\nii j\nii j nkHHS a S S S Sa    \n (18) \nAnd the derivative of the corresponding Liouvillian is \n \n  T\nTTˆˆˆˆ ˆˆ\nˆˆˆ ˆˆˆ ˆˆ ˆˆ ˆˆ nk nk\nnk nk nkLH II H\nLH HI I SS I ISSaa a  \n            (19) \nThe parameter derivative of the matrix exponen tial may be computed from its power series \ndefinition (or using more sophisticated techniques 20-22), taking into account  the fact that ˆˆL 8 \n might not commute with ˆˆL and using sufficiently small time increments so that ˆˆL may be \ntreated as being piecewise constant \n 1\n1\n10i ˆˆ ˆˆ ˆˆ ˆˆ exp i!nn\nkn k\nnktLt L L Ln\n\n    (20) \nand similarly for the Hamiltonian exponential. For very large sparse  matrices encountered in spin \ndynamics this Taylor series, ironically, works best because matrix sparsity is usually inherited \nafter multiplication, particularly if a clean-up pass ( 0  for all ik ik AA   ) is performed at \neach stage.  \nScaling and squaring may be used if necessa ry to accelerate convergence. It is worth \nnoting that the scaling and squaring procedur e is different for the derivative exponential 20: \n ˆˆ ˆ ˆˆˆ ˆ ˆii i i ˆˆ exp i exp exp exp exp22 2 2Lt Lt Lt LtLt                            (21) \nas opposed to the standard exponential procedure: \n 2ˆˆi ˆˆ exp i exp2LtLt  (22) \nAlthough the superoperator form ulation given by Equation (7) and the equation of motion \nformulation given by Equations (10) and (15) are fo rmally equivalent, the latter are likely to be \nmore computationally efficient because Equations (5) and (14) do not require Li ouville-space \nmatrix-matrix multiplications, which do occur in Equa tion (6).  It is important to note that all \nthree time-domain formalisms outlined above are numerically stable and compatible with time-\ndependent Hamiltonians and Liouvillians. \nV. Frequency domain spectrum derivatives \n In Liouville space simulations with a static Liouvillian superoperator, the positive-time \nFourier transform of the Liouvi lle–von Neumann equation is 9 \n  \n0\n0ˆ ˆˆˆi ˆˆˆˆ ˆˆ           i\nˆˆ0tLtIL t \n \n (23) \nwhere0ˆ is the initial density matrix,  is frequency, and ˆˆI is a unit matrix of the same \ndimensions as the Liouvillian s uperoperator. Differentiating with respect to the parameter α and \nrearranging yields \n 1ˆˆ ˆˆˆ ˆ ˆ IL L  \n   (24) \nIn most experiments and simulations, a specific observable – we shall call it Q – is monitored. \nHence, the quantity of interest is the derivative of Q: \n     1\n† †† ˆˆ ˆ ˆˆ ˆ ˆˆ ˆ ˆˆ ˆ QQ Q I L L Q               \n (25) \n 1\n† †ˆˆ ˆ ˆˆ ˆˆ ˆ QQ I L L  \n     (26) \nwhere the dagger sign denotes a Hermitian conjugate. Once the “derivative observable row \nvector” †ˆQ  has been computed, the derivative spect rum may be obtained quite simply with \na vector–vector product followed by an inverse Fourier transform. Note that the row vector \n†1ˆˆ ˆ ˆˆ()QI L  may be computed directly 23 without ever requiring th e time-consuming evaluation \nof the full 1 ˆˆˆˆ()IL  inverse. Large savings ar e also possible if derivativ es with respect to other \nparameters are sought because the †1ˆˆ ˆ ˆˆ()QI L  intermediate row vector does not need to be \nrecalculated. Also, specific points in the spectrum may be monitored without the need to \nsimulate the time trajectory in its entirety. \nSimilar to the first derivative treatment above, the frequency domain equations for the \nsecond derivative density matrix Fourier transform are: \n     \n   1\n11 1ˆˆ ˆ ˆ ˆˆˆ ˆ ˆ ˆ ˆˆ ˆ ˆ\nˆˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆˆˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆIL L L L\nIL L L L L L L L     \n         \n   \n          \n         \n (27) 10 \n where the intermediate results could also be re-u sed to calculate further derivatives with respect \nto different parameters. It is worth noting that  the extremely sparse st ructure of Liouvillian \nderivatives, which contain at most only a ha ndful of non-zeros, means that the Hessian ˆˆL in \nEquation (27) is very sparse. \nVI. Eigensystem differentiation \nA complementary strategy for calculating de rivatives with respect to a simulation \nparameter is available in systems that have a piecewise constant or time-independent \nHamiltonian ˆH that can be diagonalized reasonably quickly. We seek the derivative of an \nobservable \n ˆˆii\n0ˆˆ ˆ Tr .Ht HtQt e e Q       (28) \nin which the parameter derivatives of the initial density matrix 0ˆ and the observable operator \nˆQ are typically zero or simple to determine. Equation (28) is easily ev aluated using the product \nrule once the derivativ e of matrix exponen tial has been found. \nWe begin by diagonalizing the Hamiltonian 24: \n †† ˆˆ ˆ ˆ =           exp i exp iH VDV Ht V Dt V    , (29) \nwhere V is a matrix of eigenvectors and ˆD is a diagonal matrix of eigenvalues. The derivative \nof the matrix exponential is given by \n †\n††ˆˆˆ ˆ ˆ e x pi = e x pi i e x pi e x piVD VHt Dt V V t Dt V V Dt                (30) \nEvaluation of the eigenvalues and eigenvector derivatives has been studied extensively 25-33. The \nfollowing sections adapt the a pproach of Andrew and Tan 34 to the quantum mechanical context. \nNon-degenerate spectrum \nBy definition, the thi eigenvalue i of the Hamiltonian ˆH and its corresponding \neigenvector i satisfy 11 \n  ˆ\nii i H       (31) \n 1ii  (32) \nDifferentiating both equations with respect to the parameter  using the product rule and \ncancelling yields the well know n Hellmann-Feynman theorem 35 for the eigenvalue derivative \n ˆ .ii i H       (33) \nOnce all i  are known, the derivative of Equation (31) can be rearranged into \n      ˆˆ\nii ii HH           II  (34) \nin which I is the identity opera tor and all terms except i  are known. The difficulty lies in \nthe fact that the matrix ˆ\ni H I is singular, so the solutions are not unique. This reflects \nthe fact that the phase of eigenvectors is, in general, arbitrary. For example, let P be a diagonal \nmatrix of phase factors ie with unit magnitude. Then if VU P  \n †* † † ˆˆ ˆ ˆ=H VDV UPDP U UDU   (35) \nso both U and V are equally acceptable as eigenvector s. When Equation (29) is solved \nnumerically, P is chosen essentially at random. In order to obtain a well-defined solution to \nEquation (34), it is simplest to impose an additional constraint 34 on the eigenvectors \n  00 10ii ii        (36) \nfor  in a small region around the point 0 where Equation (34) is to be solved. Thus, we solve \ninstead \n ˆ ˆ\n0 0i ii i\ni iH H  \n         I (37) \nwhich is well behaved and provides the required eigenvector derivatives. \nDegenerate spectrum \nWhen there are degenerate eigenvalues at 0, one must consider each block of \ndegenerate eigenvalues together \n  ˆ\nii i H   Λ  (38) 12 \n where  i  is an nr matrix whose columns are the eigenvectors in this r-fold \ndegenerate set and  iiΛ I. Numerical routines for diagonalizing ˆH will return an \narbitrary linear combination of these degenerate  eigenvectors: †\ni P , where P is an as-\nyet-unknown rr unitary matrix. This linear combina tion may mix eigenvect ors corresponding \nto different eigenvalues deri vatives, which would make ˆD not a diagonal matrix. An \nappropriate choice of P is necessary to prevent this complication. \nNow, by the same logic as above \n      ††\n0ˆ\nii i PP H PP     Λ  (39) \nwhere the rn matrix  i  is the conjugate transpose of  i  and iΛ  is the \ndiagonal matrix of eigenvalues derivatives. Equation (39) is a small ( rr) eigenproblem, which \nis easily solved for iΛ  and the true eigenvectors  i  after having computed \n   †\n0ˆ\nii PH P   . \nThe equation analogous to E quation (34) is therefore \n      ˆˆ\nii i i i HH            I Λ  (40) \nTo stabilise this equation, we once again use the normalisation condition \n   00 ii ii     I0  (41) \nand form the block matrix \n \n ˆ ˆiii i\niiH H    \n          I\n0 Λ 0 (42) \nwhich is easily solved for  i . The value of iΛ obtained should be iden tical to that from \ndiagonalizing Equation (39), whic h provides a check of the numeri cal stability of the method at \nrun time. \nThis algorithm applies equally well to partia l derivatives, and can be extended to higher \nderivatives, beginning by differen tiating Equations (33) or (38)  several times and solving 13 \n eigenproblems analogous to Equation (39) for each  degenerate block at each stage. Similar \nprocedures are described in 28-29,34. \nIllustrative results \n Figure 1 shows the deriva tive of the theoretical 1H NMR spectrum of strychnine with \nrespect to one of the scalar couplings, calcu lated using Equation (5).  The procedure used \namounts to what may be loosely called co-propagation : the derivative density matrix is \npropagated forward in time alongside the main simulation and uses the ˆt  values that the \nmain simulation generates. Figure 2 gives a more  sophisticated example (2D COSY), making use \nof the fact that the J-coupling derivatives of a hard pulse are zero and therefore only the time \nevolution steps need to be differentiated. The un predictable behaviour of the error resulting from \nusing a finite-difference approximation is illustrate d in Figure 1C: as the finite-difference step  \ngets smaller, the error initially follows the expected 4Oh  scaling. However, numerical noise \nstarts creeping in around 0.01 Hzh  and thereafter the accuracy dete riorates rapidly for smaller \nsteps. \n As expected, these ca lculations take much longer if E quation (6) is used to find the \nparameter derivative, because matrix-matrix mu ltiplications in Liouville space are required. \nHowever, if multiple simulations must be carried  out from different initial conditions (e.g. the \ndirect dimension of a 2D experi ment), this superoperator formul ation could be faster, since it \nwould avoid repeated co mputation of the differe ntiation superoperators. \n The Fourier domain differentiat ion is significantly faster if only a few points are required \nin the resulting spectrum (a frequent situation in e.g. chemical kinetics fitting where a spectrum \nneeds to be differentiated with respect to the kinetic  rate constant). It also  has the benefit of not \nrequiring apodization. An example of such a derivative is given in Figure 3.  A less obvious differentiation pa rameter – waveform truncation level – is demonstrated in \nFigure 4. The resulting derivative gives a me asure of stability of the magnetic resonance \nexperiment with respect to the c lipping of the wings of a Gaussian pulse. As intuitively expected, 14 \n waveform clipping has no effect if the signal is positioned exactly on resonance, but does \nintroduce imperfections into the ex citation of off-resonance peaks. \nFigure 5 demonstrates eigens ystem differentiation using a simple model Hamiltonian \n  \n  \n 2cos sin 0 4 0 0 cos sin 0\nˆ sin cos 0 0 10 0 sin cos 0\n00 1 0 0 3 00 1H   \n   \n  \n    \n     (43) \nwhose eigenvectors and eigenvalues are known anal ytically. Notice that there are no difficulties \nfor degenerate eigenvalues at 10 / 3, 1 or 4 . Most importantly, all elements of the matrix \nexponential ˆiHe are accurate to within  an absolute error of 1310 throughout this range. Similar \ntests with 33 and 44 complex model Hamiltonian produced comparable accuracy in the all \nimportant matrix exponentials. \nFigure 6 is a more sophisticated example fr om low-field electron paramagnetic resonance \nof radical pairs 25,36. In such systems, the reaction yield is  governed by the strength of an applied \nmagnetic field. The simplest radical pa ir comprises two electrons, and a single 1H nucleus in one \nof the radicals and has Hamiltonian \n  0e 0 A z B z Aˆˆ ˆ ˆˆ . HB B S S a S I     (44) \nNeglecting the reaction kinetics for now, the proba bility of a radical pair created in a pure \nelectronic singlet state rema ining in that state is \n  ˆˆ ˆ ˆ ††\n00 S 0 Sˆˆ ˆ ˆ ˆˆ ˆˆ ;T r T riHt iHt iDt iDtStB e e P e V Ve VP V       (45) \nwhere ˆD is a diagonal matrix containing Hamiltonian eigenvalues, ˆV is the corresponding \neigenvector matrix, SˆP is the electron singlet projection operator and the initial density matrix \n0Sˆ ˆ /2P  because the initial nuclear spin stat e is random. Using the “exponential model” 36-39 \nfor reaction kinetics, the tota l yield of singlet product \n   00\n0;d .kt\nSB StB k e t\n (46) 15 \n Here, Equation (46) may be integrated analytically if ˆH has been diagonalised, which more than \ncompensates for the effort required for diagonalisa tion. In this case, the method of eigensystem \ndifferentiation is therefore the most attractive. \nFigure 6 compares 00 d/ dSB B   computed by eigensystem differentiation with an exact \ncalculation made analytically for speci fic values of the hyperfine constant a and rate constant k. \nAgreement is essentially exact thr oughout the range that was plotted. \nConclusion \nIn simulations of spin dynamics, the formalism outlined above provides a \nstraightforward, accurate and numerically stable avenue to determine derivatives with respect to \nspin system or pulse sequence parameters. The resulting derivatives may be used in fitting, optimization, performance evalua tion and stability analysis of spin dynamics experiments and \nexperimental results. \nAcknowledgements \n This work was supported by EPSRC (E P/F065205/1, EP/H003789/1) and NIS (PHY05-\n51164). The authors are grateful to Niels Chr. Nielsen  and Alan Andrew for helpful advice. IK \ngratefully acknowledges the Kavli Institute for Th eoretical Physics at UCSB. CTR is funded by \nNIHR Biomedical Research Centre Programme and Merton College, Oxford. 16 \n References \n1 A. D. Bain, D. M. Rex, and R. N. Smith, Magn. Reson. Chem. 39 (3), 122 (2001). \n2 L. C. Wang, A. V. Kurochkin, and E.  R. P. Zuiderweg, J. Magn. Reson. 144 (1), 175 (2000). \n3 S. Dusold, E. Klaus, A. Sebald, M. Bak, and N. C. Nielsen, J. Am. Chem. Soc. 119 (30), 7121 \n(1997). \n4 H. J. M. Degroot, S. O. Smith, A. C. Kolbert, J. M. L. Courtin, C. Winkel, J. Lugtenburg, J. \nHerzfeld, and R. G. Griffin, J. Magn. Reson. 91 (1), 30 (1991). \n5 S. Castellano and A. A. Bothner-By, J. Chem. Phys. 41 (12), 3863 (1964). \n6 Z. Tosner, T. Vosegaard, C. Kehlet, N. Khaneja,  S. J. Glaser, and N. C. Nielsen, J. Magn. \nReson. 197 (2), 120 (2009). \n7 I. I. Maximov, Z. Tosner, and N. C. Nielsen, J. Chem. Phys. 128 (18) (2008). \n8 N. Khaneja, T. Reiss, C. Kehlet, T. Schulte-H erbruggen, and S. J. Gl aser, J. Magn. Reson. 172 \n(2), 296 (2005). \n9 T. E. Skinner, T. O. Reiss, B. Luy, N. Khaneja, and S. J. Glaser, J. Magn. Reson. 163 (1), 8 \n(2003). \n10 J. Nocedal and S. J. Wright, Numerical optimization , 2nd ed. (Springer, New York ; London, \n2006). \n11 N. I. M. Gould, D. Orban, and P. L. Toint, Numerical methods for large-scale nonlinear \noptimization . (Council for the Central Laboratory of the Research Councils, Didcot, 2004). \n12 R. Dunkel, C. L. Mayne, J. Curtis, R. J. Pugmire, and D. M. Grant, J. Magn. Reson. 90 (2), \n290 (1990). \n13 J. A. Nelder and R. Mead, Comp. J. 7 (4), 308 (1965). \n14 D. A. Pearlman, J. Biomol. NMR 8 (1), 49 (1996). \n15 D. R. Yarkony, Modern electronic structure theory . (World Scientific, London, 1995). \n16 P. Pulay, Mol. Phys. 17 (2), 197 (1969). \n17 A. F. Bartelt, M. Roth, M. Mehendal e, and H. Rabitz, Phys. Rev. A 71 (6) (2005). \n18 F. Shuang and H. Rabitz, J. Chem. Phys. 121 (19), 9270 (2004). \n19 R. R. Ernst, G. Bodenhausen, and A. Wokaun, Principles of nuclear magnetic resonance in \none and two dimensions . (Clarendon, Oxford, 1987). \n20 T. C. Fung, Int. J. Numer. Meth. Eng. 59 (10), 1273 (2004). \n21 C. Moler and C. Van Loan, SIAM Rev. 45 (1), 3 (2003). \n22 I. Najfeld and T. F. Havel, Adv. Appl. Math. 16 (3), 321 (1995). \n23 G. H. Golub and C. F. Van Loan, Matrix computations , 2nd ed. (The Johns Hopkins \nUniversity Press, Baltimore, 1989). \n24 J. H. Wilkinson, The algebraic eigenvalue problem . (Clarendon Press, Oxford,, 1965). \n25 U. E. Steiner and T. Ulrich, Chem. Rev. 89 (1), 51 (1989). 17 \n 26 A. L. Andrew, K. W. E. Chu, and P. Lancaster, SIAM J. Matrix Anal. Appl. 14 (4), 903 \n(1993). \n27 A. L. Andrew, Comp. Tech. Appl., 51 (1998). \n28 A. L. Andrew and R. C. E. Tan, Comm. Num. Meth. Eng. 15 (9), 641 (1999). \n29 A. L. Andrew and R. C. E. Tan, Num. Lin. Alg. Appl. 7 (4), 151 (2000). \n30 A .  L .  A n d r e w  a n d  R .  C .  E .  T a n ,  i n  Linear operators and matrices : the Peter Lancaster \nanniversary volume , edited by I. Gohberg and H. Langer (Birkhauser Verlag, Boston, 2002), \nVol. 130, pp. 43. \n31 K.-W. E. Chu, SIAM J. Num. Anal. 27 (5), 1368 (1990). \n32 Z. H. Xu and B. S. Wu, Int. J. Num. Meth. Eng. 75 (8), 945 (2008). \n33 P. Lancaster, Numerische Mathematik 6, 377 (1964). \n34 A. L. Andrew and R. C. E. Tan, SIAM J. Matrix Anal. Appl. 20 (1), 78 (1998). \n35 R. P. Feynman, Phys. Rev. 56 (4), 340 (1939). \n36 C. T. Rodgers, Pure Appl. Chem. 81 (1), 19 (2009). \n37 R. Kaptein and J. L. Oosterhoff, Chem. Phys. Lett. 4 (4), 195 (1969). \n38 R. G. Lawler and G. T. Evans, Industrie Chimique Belge 36 (12), 1087 (1971). \n39 B. Brocklehurst and K. A. Mc Lauchlan, Int. J. Rad. Biol. 69 (1), 3 (1996). \n40 I. Kuprov, J. Magn. Reson. 195 (1), 45 (2008). \n41 I. Kuprov, N. Wagner-Rundell, and P. J. Hore, J. Magn. Reson. 189 (2), 241 (2007). \n \n  18 \n Figure captions \nFigure 1.  A: Fragments of the theoretical 90°–acquire 1H NMR spectrum of strychnine. \nSimulations were performed for a 22-spin syst em (with magnetic parameters obtained from a \nseparate GIAO B3LYP/cc-pVDZ calculation using Gaussian03), employing state space \nrestriction 40-41 to k-spin orders around every spin, where k is the number of J-coupled neighbors \n(or k=4 if that is larger), and with a diagonal DD-CSA relaxa tion superoperator. B: Analytical \nderivative of the same spectrum with respect to one of the scalar couplings (solid line) and \nnumerical derivative with respect to the same parameter (circles) using an 4Oh  four-point \ncentral finite difference approximation. Wall clock times for propaga tion with analytical \nderivatives: 7.6 s; and for the numerical derivatives: 29.5 s. C: Maximum absolute difference \nbetween the analytical derivative and the finite -difference approximation as a function of the \nfinite-difference step size. \nFigure 2. A: Fragment of the 1H–1H COSY spectrum of strychnine . Simulations were performed \nas described in Figure 1. B: Analytical derivative of the same  COSY spectrum with respect to \none of the scalar couplings. Wall clock times for propagation: original spectrum: 12 min; \nanalytical derivative: 15 min.  For comparison the equivalent 4Oh  four-point central finite \ndifference approximation took 49 min. \nFigure 3.  A: 1H NMR spectra of a strongly coupled two-spin system undergoing symmetric \nchemical exchange with rate constants varying between , kJ  (bottom trace) and \n, kJ   (top trace). B: Analytical derivative s of the same spectra with respect to the \nexchange rate constant k. (Each spectrum was scaled vertically  to make its features visible.)  \nFigure 4. Derivative of the excitation pr ofile of a 90° Gaussian puls e with respect to waveform \ntruncation. A: The function   tanh tanh fxk x k x    used to approximate the \nwaveform truncation. The three traces correspond to k = 10, 102, 103.  B: Gaussian waveform \nshowing the truncation parameter (i n units of standard deviation). C: Top: Magnitude-mode 1D \nNMR spectrum of a linear chain of  33 protons with regularly spa ced chemical shifts and equal \nnearest-neighbor J-couplings C: Bottom: Analytical derivative of the spectrum above with \nrespect to the truncation parameter  when 3.0 . 19 \n Figure 5.  Comparison of the derivatives of the eigenvalues and eigenvectors of ˆH defined \nin Equation (43). Exact, analytical solutions ar e shown as lines, numerical solutions from the \neigensystem differentiation algorithm are shown as crosses. Notice the good behaviour when \neigenvalue degeneracies occur (at 4, 1 and 10/3). Left:  First derivative with respect to  \nof the eigenvalues of ˆH. Right:  Absolute values of each component of the eigenvector \nderivatives with respect to . The absolute value is plotted to mitigate the arbitrary phase \nfactors in the numerical solutions. \nFigure 6.  Derivative of singlet yield with respect to field strengt h in a radical pair with hyperfine \ncoupling a to a single spin-1/2 nucleus, using th e exponential model w ith rate constant \n/10 ka . Numerical results (crosses) were obtained  from Equation (46) using the eigensystem \ndifferentiation algorithm, exact valu es (line) were obtained using a computer algebra system to \ngive Equations (S1) – (S3) in the Supporting Information.   \n   10 20 30 40 50 60 190 200 210 220\nfrequency / Hz\nMaximum absolute error, a.u.\n∆ / HzA\nBC  \n700 800 900 950 1000 1050 1100 115060070080090010001100\n  \n700 800 900 950 1000 1050 1100 115060070080090010001100\nFrequency, a.u. Frequency, a.u.Frequency, a.u.\nFrequency, a.u.A B0 -10 -20 10 20\nfrequency / Hzlog(k / s-1)\n0.01.02.03.04.0A B\n0 -10 -20 10 20\nfrequency / Hz-4 -2 2 0 4 -4 -2 2 0 4 -60.20.40.60.8\n0.10.2\nα\nx10A B C\n(std. devs.) (std. devs.) frequency / kHz-4 -2 0 2 4 6−5 0 5−10−50510Eigenvalue derivatives\nα  \nNumeric\nExact\n−5 0 500.51abs( Eigenvector derivatives )\nα0 1 2 3−0.8−0.40\n  \nNumeric\nExact\nω0/a(dΦS/dω0)/a"    },    {        "title": "1806.11100v2.Spin_Inertia_of_Resident_and_Photoexcited_Carriers_in_Singly_Charged_Quantum_Dots.pdf",        "content": "arXiv:1806.11100v2  [cond-mat.mes-hall]  27 Aug 2018Spin Inertia of Resident and Photoexcited Carriers in Singl y-Charged Quantum Dots\nE. A. Zhukov,1E. Kirstein,1D. S. Smirnov,2D. R. Yakovlev,1,2\nM. M. Glazov,2D. Reuter,3A. D. Wieck,4M. Bayer,1,2and A. Greilich1\n1Experimentelle Physik 2, Technische Universit¨ at Dortmun d, 44221 Dortmund, Germany\n2Ioffe Institute, Russian Academy of Sciences, 194021 St.Pete rsburg, Russia\n3Universit¨ at Paderborn, Department Physik, 33098 Paderbo rn, Germany\n4Angewandte Festk¨ orperphysik, Ruhr-Universit¨ at Bochum , 44780 Bochum, Germany\nThe spin dynamics in a broad range of systems can be studied us ing circularly polarized optical\nexcitation with alternating helicity. The dependence of sp in polarization on the frequency of helicity\nalternation, knownas thespininertiaeffect, isusedhereto studythespindynamicsinsingly-charged\n(In,Ga)As/GaAs quantum dots (QDs) providing insight into s pin generation and accumulation\nprocesses. We demonstrate that the dependence of spin polar ization in n- andp-type QDs on the\nexternal magnetic field has a characteristic V- and M-like sh ape, respectively. This difference is\nrelated to different microscopic mechanisms of resident car riers spin orientation. It allows us to\ndetermine the parameters of the spin dynamics both for the gr ound and excited states of singly-\ncharged QDs.\nLong carrier spin coherence and spin relaxation times\nare the main prerequisites for a system to be suited for\nquantum information technologies1,2. The main route to\nextend these times is to isolate the studied system from\nitsenvironment. At thesametime, suchisolationreduces\nthe possibility for a fast state readout and manipulation.\nCarrier spins confined in low-dimensional semiconductor\nstructures, in particular, singly-charged (In,Ga)As quan-\ntum dots (QDs), offer a unique balance of accessibility\nand robustness3. The direct optical band gap with a gi-\nantopticaldipolemomentofasemiconductorQD4allows\nexploiting the optical excitation, the exciton, as an aux-\niliary state for ultra-fast conversion between the optical\ncoherence of a laser pulse (picosecond duration) and the\nlong-living spin coherence of an isolated resident carrier\n(microsecond coherence). This opportunity has triggered\nconsiderable activity in optical operations with QDs, in-\ncluding spin initialization5, nondestructive readout6, and\nfast spin manipulation7.\nThe way the optical coherence is transferred to the\nspin-coherence in a QD involves the process of optical\norientation and excitation of a trion, the exciton which is\nbound to the resident carrier. After excitation, the trion\nrecombines stochastically and leaves behind a polarized\nsinglecarrierspinin the groundstate, which ismeasured.\nGenerally,thespinrelaxationdynamicswithinatrionde-\ntermines the final spin polarization. The standard opti-\ncal methods to accessthe relaxationdynamics of resident\nspinsarethe Hanleeffect8andtime-resolvedpump-probe\ntechniques9. However, these techniques provide limited\naccessto the spin generationprocessand dynamics ofthe\nphotoexcited states4.\nIn this paper, we access the parameters of the spin\ndynamics in the trion employing the recently developed\nSpin Inertia (SI) technique10. In SI, the spin dynam-\nics in the system is related to the helicity modulation of\nthe excitation polarization11–13. The maximal value of\nthe spin polarization, created by the circularly polarized\npump pulses, is then traced as a function of the helicity\nmodulation frequency fm. The spin polarization staysconstant for fm/lessorsimilarT−1\ns, whereTsis the spin relaxation\ntime of a carrier, and becomes reduced if the modula-\ntion frequency is larger than T−1\ns. The cut-off frequency\ndefines the spin relaxation time10,14.\nIn the presented experiment we study the spin relax-\nation in a model system, namely an ensemble of singly-\ncharged (In,Ga)As/GaAs QDs, doped either with resi-\ndent electrons or holes. The maximal spin polarization\nis determined by the interplay of the carrier spin gener-\nation and relaxation dynamics. Importantly, applying a\nlongitudinalmagneticfield, asubstantialdifferenceinthe\nmagnetic field behavior is observed for resident electron\nand hole spins. The origin of the observed difference and\nits relation to the SI represent the main topic of the pa-\nper. Using the developed theory, we are able to exploit\nthe full potential of the experiment. We describe the\nshape of the magnetic-field-dependent traces and extract\ninformation on the photoexcited carrier spin dynamics,\nwhich can be accessed from the signal accumulated long\nafter the radiative trion decay. Furthermore, we deter-\nmine spin relaxation times, longitudinal gfactor values\nand hyperfine interaction strengths in the ground and\nexcited states of the system.\nWe study n- andp-doped ensembles of singly-charged\nself-assembled QDs. Both samples contain 20 layers of\nmolecular beam epitaxy grown (In,Ga)As QDs separated\nby 60nm GaAs barriers. The average QD density is\nabout 1010cm−2per layer. The p-doped sample has a\nbackground level of p-type doping due to residual carbon\nimpurities. The n-doped sample was obtained by incor-\nporating δ-sheets of Si 20nm below each QD layer. The\nsamples are mounted in the variable temperature insert\nof a magneto-optical bath cryostat ( T= 1.5−300K),\nand are excited by the laser close to the maximum of the\nphotoluminescence at 1.412eV (878nm) for the n-type\nQDs, andat1.392eV(891nm)forthe p-typeQDs15. The\nlaser spot diameters on the sample are 300 µm. As the\nsamples are different in their doping types and concen-\ntrations, different excitation conditions are required for\nstrongest signal level. The magnetic field Bis applied2\nin Faraday geometry, along the optical z-axis. Details\non the sample characterization are given in Refs. [ 16and\n17].\nThe spin polarization is created by circularly polar-\nized pump pulses of 1.5ps duration emitted by a mode-\nlocked Ti:Sapphire laser operating at a repetition pe-\nriodTR= 13.2ns. The pump helicity is modulated\nat frequency fmbetween σ+andσ−polarization by\nan electro-optical modulator (EOM). Linearly polarized\nprobe pulses measure the induced spin polarization along\nthe optical axis via the ellipticity signal in transmission\ngeometry analyzed by a quarter wave plate and a wollas-\nton prism, using a balanced diode bridge10. This signal\nis called Faraday Ellipticity (FE).\nIn the presented experiments, we fix the pump-probe\ndelay at a negative value ( τpp=−50ps) and measure\nthe FE dependence on the longitudinal magnetic field,\napplied along the excitation direction and orthogonal to\nthe sample surface. Using the signal at negative time\ndelay greatly simplifies the interpretation of the results,\nas it can only arise from the spin polarization of the res-\nident long-living carriers. In the studied QDs the trion\nrecombination time of 400ps is much shorter than TR18.\nFigures1(a) and1(b) demonstrate the measured FE\nas a function of magnetic field for different modulation\nfrequencies for n- andp-doped samples. We call this de-\npendence a polarization recovery curve (PRC). For both\nsamples the FE increases with increase of magnetic field\naroundB= 0. In the case of strong carrier localization,\nthe dominant spin relaxation mechanism is the hyperfine\ninteraction with host lattice nuclear spins. Application\nof the external magnetic field leads to an effective decou-\npling of the resident carrier spin from the nuclear spin\nbath and stabilizes the optically oriented spins along the\nz-axis10,20, so FE increases. At higher magnetic fields\nthe behavior is different: The electron spin polarization\nsaturates (in the range around 300mT), while the hole\nspin polarization decreases after the initial increase and\nthen saturates within a similar range of fields. The ob-\nserved difference in the field dependencies of PRC is re-\nlated to the type of carriers (electrons vs. holes) and the\nspin relaxation dynamics in the trion states, which are\nschematically shown in the insets in Figs. 1(a) and1(b).\nThe SI method gives access to the relaxation dynamics\nof the resident carriers, as shown by the exemplary data\nsets in Figs. 1(c) and1(d). Figure 1(c), shows the fmde-\npendenceoftheFEatfixedmagneticfield( B= 300mT),\nextracted from the data shown in Fig. 1(a). Using the fit\nwith the form10:\nFE(fm)\nFE(0)=1/radicalbig\n1+(2πfmTs)2, (1)\nwe determine the characteristic spin lifetime Ts= 0.7µs\nof the resident electrons at the pump power of 4mW10.\nIn this case the spin polarization is far below the satu-\nration level. The case of strong pumping is discussed in\nRef. [14]. The value of Tsdepends on the pump power\ndue to the influence of the photoexcitation10,21, hence, to0 100 200 30000.20.40.60.811.2\nB (mT)Faraday ellipticity (arb. units)\n0 100 200 30000.20.40.60.8\nB (mT)fm = 40 kHz\n390\n970\n3800fm = 25 kHz 50\n100\n200n - doped p - doped\n(a) (b)\n1001011021031040\nfm (kHz)Faraday ellipticity\n1001011021031040\nfm (kHz)Ts (4mW) = 0.7 µsB = 300 mT\n(c) (d)Ts (1.5mW) = 5.9 µsB = 17 mT\n0 5 10012\nP (mW)1/Ts (µs-1)\n1.3 µs0 1 2 300.10.2\nP (mW)1/Ts(µs-1)\n7.3 µsFitGSES\nFIG. 1. Spin polarization of carriers as a function of magnet ic\nfield at different modulation frequencies of pump helicity: ( a)\nn-doped sample, laser pump/probe power is 4 /0.8mW,T=\n6K; (b) p-doped sample, pump/probe power is 1 .5/0.5mW,\nT= 1.7K. Gray curves are calculations after Eq. ( 2) with\nparameters giveninTab. I19. Insetsshowexamples ofpossible\nground (GS) and excited state (ES) spin configurations: thin\ngreen and thick black arrows represent the electron and hole\nspins, respectively. Panels (c) and (d) show the frequency\ndependencies of carrier spin polarizations at fixed magneti c\nfield for the n-doped and p-doped samples, respectively. Red\nline is a fit by Eq. ( 1). Insets are linear extrapolations of the\npower dependent spin lifetime Tsdown to zero pump power\ngiving the spin relaxation times in equilibrium, T(0)\ns.\ndetermine the longitudinal spin relaxation time in equi-\nlibrium, T(0)\ns, we apply different pump powers Pand\nextrapolate Tsdown to zero power using a linear fit to\nthe inverse of the data, see inset in Fig. 1(c). It gives\nT(0)\ns= 1.3µs forB= 300mT in n-doped QDs. Fig-\nure1(d) shows the corresponding frequency dependence\nfor thep-doped sample around the maximum of the FE\nsignal (at B= 17mT) and P= 1.5mW. The procedure\ndescribed above yields Ts= 5.9µs andT(0)\ns= 7.3µs.\nThis demonstrates the ability of SI to access very long\nspin relaxation times, exceeding TRby almost three or-\nders of magnitude. Spin relaxation times in the mi-\ncrosecond range are in good agreement with other mea-\nsurements13,15,21, while the difference from the results of\nRef. [22] can be explained by the difference in the exper-\nimental protocol23.\nImportantly, the PRCshavedifferent shapes for n-and\np-doped samples. This situation changes with increasing\ntemperature. Figure 2demonstrates the evolution of the\nPRCforthe p-dopedsampleatlowmodulationfrequency\nfm= 2kHz, in the temperature range T= 13−20K,3\n0 100 200 300\nB (mT)Faraday ellipticityT (K)\n13\n14\n15\n16\n20M-like\nV-like\nFIG. 2. Temperature dependence of PRC for the p-doped\nsample. Laser pump/probe power is 2 /0.5mW,fm= 2kHz.\nwhere it is changing from a M to a V shape. In com-\nparison, the temperature dependence of the V-type PRC\nfor then-type sample does not show a shape change (not\nshown here).\nThe full theory of SI is developed in the accompanying\ntheoretical paper14. Here we briefly summarize the main\nresults and apply the developed theory to the description\nof the experimental results. The FE signal, FE(fm), is\ndetermined by the absolute value of the spin polarization\nat modulation frequency fm. The spin dynamics forboth\ntypes of carriers at small magnetic fields ( <1T) is dom-\ninated by the hyperfine interaction with the host lattice\nnuclei. The resident carrier spin in each QD precesses\nin the random Overhauser field, while the nuclear spins\nare subject to the Knight field created by the carrier and\nexperience the quadrupolar interaction induced by the\nstrain in QDs24. As a result, the spin polarization of an\nensemble obeys a linear, but non-Markovian equation of\nmotion. In the model of Merkulov, Efros, and Rosen25\ntwo-thirds of the electron spin polarization is lost during\ntime scale of spin precession in the random Overhauser\nfield,ω−1\nN. The rest of spin polarization is destroyed at\nlonger times, such as the nuclear spin correlation time\nτcrelated, e.g., to the nuclear quadrupole interaction,\nor the spin relaxation time τsunrelated to the hyperfine\ninteraction.\nThe spin dynamics can be described by the Green’s\nfunction G(fm) in the frequency domain. Theoretical\nanalysisof the SI measurement protocol14shows that the\nSI signal is proportional to:\nFE(fm)∝QG(fm). (2)\nHere the factor Qis the probability of trion spin flip dur-\ning its lifetime. According to the optical selection rules,\ntrionexcitationandrecombinationpreservethetotalspin\nalong the excitation axis z. Therefore Qdetermines the\nefficiency of spin polarization in QDs26. Importantly, it\nis sensitive to the external magnetic field27, which ulti-\nmately gives rise to the different shape of PRC for differ-\nent carriers. One can show, that14:\nQ=(ωT\nNτ0/λT)2\n1+(ΩT\nLτ0)2+τ0\nτTs, (3)TABLE I. Parameters of resident and photoexcited carriers\nspin dynamics, extracted from the fits in Fig. 1.\ngzzωN, MHz λ τc,µsτs,µsτT\ns,µs\nElectron −0.61 70 1 0 .2 0.5<1\nHole −0.45 16 5 0 .26 5 .2 0.035\nwhereωT\nNis the characteristic trion spin precession fre-\nquency in the fluctuations of the Overhauser field along\nthezdirection, λTis the parameter of the hyperfine in-\nteraction anisotropy of the trion, ΩT\nL=gT\nzzµBB//planckover2pi1is the\ntrion spin precession frequency in the longitudinal ex-\nternal magnetic field with gT\nzzbeing the longitudinal g\nfactor and µBbeing the Bohr magneton, and τT\nsis the\ntrion spin relaxation time unrelated to the hyperfine in-\nteraction. The laser repetition period TRis assumed to\nbe much longer than the trion lifetime τ0.\nThe Green’s function of the spin dynamics accounting\nfor the finite nuclear spin correlation time is28:\nG(fm) =τfA\n1−Aτf/τc, (4)\nwhere 1/τf= 1/τs+1/τc−2πifmand\nA=/integraldisplay\ndΩNF(ΩN)1+Ω2\nzτ2\nf\n1+Ω2τ2\nf, (5)\nwithΩNbeing the frequency of spin precession in the\nrandom Overhauser field in a single QD and Ω=ΩN+\nΩLbeing the total spin precession frequency in a single\nQD. The distribution of the Overhauserfield is Gaussian:\nF(ΩN) =λ2\n(√πωN)3exp/parenleftBigg\n−Ω2\nN,x+Ω2\nN,y\nω2\nN/λ2−Ω2\nN,z\nω2\nN/parenrightBigg\n,\n(6)\nwhereωNis the typical spin precession frequency in the\nQD ground state and λdescribes the anisotropy of the\nhyperfine interaction. The latter is relevant ( λ >1)29for\nresident holes and for negatively charged trions, where\nthe hole spin is unpaired, while the electrons form a sin-\nglet state with total spin zero, see inset in Fig. 1(a). We\nstress that the parameters of spin dynamics are different\nin the ground and excited states, so we used the super-\nscript “T” in Eq. ( 3) for the parameters of the trion spin\ndynamics.\nEquations( 2)—(5) allowus to describe the experimen-\ntal data shown in Fig. 1and to determine the parame-\nters of spin dynamics, which are summarized in Tab. I19.\nBelow we describe qualitatively, how these parameters\ndetermine the PRC shape. As one can see from Eq. ( 2),\nthe frequency dependence of the FE is described by the\nGreen’s function G(fm). In general, this dependence is\ngoverned by the coupled spin dynamics of carrier and\nnuclei and differs from Eq. ( 1). However in sufficiently\nstrong magnetic fields, where Ω L> ωNEq. (1) is valid,\nprovided Ts=τs. In this case, the cut-off in the fre-\nquency dependence of the FE signalyields the spin relax-\nation time in the ground state, τs. All other parameters4\n00.20.40.60.81\nresident electron\n0 100 200 3000123456\nB (mT)Spin lifetime, T1 (µs)\nresident hole(a)\n00.0010.0020.0030.0040.005\nhole in X- trionn - doped\np - doped\n00.010.020.030.040.05\nGeneration rate, Q\nelectron in X+ trion\n(b)\nFIG. 3. Decomposition of the PRC (dashed blue lines, shaded\nforbettervisibility, arb. units)intomagneticfielddepen dence\nof spin polarization of resident and photoexcited carriers : (a)\nn-doped, (b) p-doped samples. Black curves give spin lifetime\nof resident carriers and red — generation rate Qdetermined\nby spin relaxation of the photoexcited carriers. Parameter s\nof the calculation are the same, as in Fig. 1.\nof the spin dynamics can be extracted from the PRC at\nsmall frequency.\nAt the smallest modulation frequencies fm≪τ−1\ns, the\nFE signal does not depend on fmand the Green’s func-\ntionatzeromodulationfrequencydeterminestheaverage\nspin relaxation time T1≡G(0). Therefore one simply\nhasFE(fm= 0)∝Q(B)T1(B), where both factors de-\npend on magnetic field. The decomposition of FE into\nthese two contributions is shown in Fig. 3. Here the PRC\ncurves calculated numerically (blue dashed curves with\ngray filling) are shown together with the dependencies\nT1(B) (black lines) and Q(B) (red lines) calculated for\nthe same parameters as in Tab. I.\nThe dip in the FE signals around zero magnetic field\nis determined by the dependence of T1on magnetic field.\nIt is similar for both samples, as one can see in Fig. 3:\nthe spin relaxation time increases with an increase of the\nlongitudinal magnetic field, and then saturates. Its full\nwidth at half maximum is related to the longitudinal g\nfactor of the carrier and the characteristic nuclear field\nfluctuation, ωN, as∼/planckover2pi1ωN/(gzzµB)14. Since the hyper-\nfine interaction for electrons is stronger30, the width of\ntheT1curve is larger for the n-type sample, than for the\np-type. We note that the saturation of the longitudinal\nspinrelaxationtimeatlargemagneticfieldsevidencesthe\npresence of spin relaxation mechanism unrelated to hy-\nperfineinteraction,whichisdescribedbythephenomeno-\nlogical time τs. The deviations of the modelled behavior\nfrom the experimental one for the n-type sample in the\nintermediate field range may be caused by specific corre-\nlations in the nuclear dynamics, like quadrupole-induced\nspin relaxations, which are not included at this stage.\nThe ratio of the average spin relaxation time at largeandzeromagneticfieldsisdeterminedbythenuclearspin\ncorrelation time, τc14:\nT1(B=∞)\nT1(B= 0)= 3+2τs\nτc. (7)\nThereforetheincreaseof T1inbothsamplesbymorethan\n3 times indirectly evidences the finite hyperfine field cor-\nrelationtime. We notethat the τcobtained fromthefit is\nsimilar for both samples. This indicates that the nuclear\nspin dynamics is related to the quadrupole interaction,\nwhich is similar for both samples31.\nFigure3shows that the main difference between the\nFE dependence on Bfor the two samples is the spin gen-\neration rate, or the trion spin-flip probability Q(B). In\npositively charged QDs the trion consists of two heavy\nholes in the singlet state and an electron, see inset in\nFig.1(b). In this case, the trion spin relaxation at zero\nmagnetic field is related to the hyperfine interaction of\nan unpaired electron spin with the host lattice nuclei in\nthe QD. This is described by the first term in Eq. ( 3).\nWith the increase of magnetic field, the electron spin\ngetseffectivelydecoupledfromthe nuclearspinbath, and\nthe trion spin-flip probability Qdecreases, as shown in\nFig.3(b). The width of the dependence Q(B) is given by\n2/planckover2pi1/(gT\nzzµBτ0). Using the τ0= 400ps18, we find the trion\ngfactorgT\nzz=−0.4, which is close to the bare electron\ngfactor. Moreover, assuming that the characteristic nu-\nclear field is the same as for n-type QDs ωT\nN= 70MHz,\nwe find the trion spin relaxation time unrelated to the\nhyperfine interaction τT\ns= 35ns.\nFor negatively charged QDs the trion consists of two\nelectrons in the singlet state and a hole with unpaired\nspin, as shown in the inset in Fig. 1(a). In this case, the\nspin dynamics of trion is determined by hole spin relax-\nation. The hyperfine interaction of negatively charged\ntrion is weaker than of positively charged trion. As a re-\nsult the trion spin relaxation in n-type QDs is unrelated\nto hyperfine interaction and Qdoes not depend on B,\nsee the red curve in Fig. 3(a). Mathematically the sec-\nond term in Eq. ( 3) dominates in this case, which allows\nus to estimate τT\ns<1µs in then-type sample.\nInterestingly, the dependence Q(B) can change with a\ntemperature increase, provided τT\nsdepends on T. This\ntakes place, for example, for the phonon-assisted spin re-\nlaxation mechanism, although the precise origin of τT\nsfor\nboth types of quantum dots is not fully clear so far. The\nspin relaxation speeds up with an increase of tempera-\nture, so the dependence Q(B) can become flat even for\np-type QDs. This explains the change of the shape of the\nFE signal from M-like to V-like, shown in Fig. 2.\nIt is instructive to compare the SI technique with the\nspin noise technique32,33. Indeed, both give access to the\nFourier transform of the spin dynamics Green’s function,\nwhich can be measured for different magnetic fields. In\nfact, one can show, that the spin noise spectrum is pro-\nportional to Re Gzz(fm)14. Thus, it is characterized by\nthe same parameters as given in Tab. I, but in a different\nway, which makes these two approaches complementary.5\nTo conclude, we show that the SI measurement gives\naccess to various parameters of the spin dynamics not\nonly of resident charge carriers but also of the photoex-\ncited electron-hole complexes.ACKNOWLEDGMENTS\nWe acknowledge the financial support by the Deutsche\nForschungsgemeinschaftin the frame of the International\nCollaborative Research Center TRR 160 (Project A5),\npartial support by the Russian Foundation for Basic Re-\nsearch (Grant No. 15-52-12012), and Basis Foundation.\n1D. Loss and D. P. DiVincenzo, Phys. Rev. A 57, 120\n(1998).\n2T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura,\nC. Monroe, and J. L. O’Brien, Nature 464, 45 (2010).\n3F. Henneberger and O. Benson, Semiconductor quantum\nbits(CRC Press, 2016).\n4M. I. Dyakonov, ed., Spin physics in semiconductors\n(Springer International Publishing AG, 2017).\n5M. Atat¨ ure, J. Dreiser, A. Badolato, A. H¨ ogele, K. Karrai,\nand A. Imamoglu, Science 312, 551 (2006).\n6D. Kim, S. E. Economou, S. C. B˘ adescu, M. Scheib-\nner, A. S. Bracker, M. Bashkansky, T. L. Reinecke, and\nD. Gammon, Phys. Rev. Lett. 101, 236804 (2008).\n7A. J. Ramsay, Semicon. Sci. Tech. 25, 103001 (2010).\n8W. Hanle, Z. Phys. 30, 93 (1924).\n9D.D.Awschalom, D. Loss, andN.Samarth, eds., Semicon-\nductor Spintronics and Quantum Computation (Springer-\nVerlag, Berlin, 2002).\n10F. Heisterkamp, E. A.Zhukov, A.Greilich, D.R.Yakovlev,\nV. L. Korenev, A. Pawlis, and M. Bayer, Phys. Rev. B 91,\n235432 (2015).\n11M. Ikezawa, B. Pal, Y. Masumoto, I. V. Ignatiev, S. Y.\nVerbin, and I. Y. Gerlovin, Phys. Rev. B 72, 153302\n(2005).\n12I. A. Akimov, D. H. Feng, and F. Henneberger, Phys. Rev.\nLett.97, 056602 (2006).\n13F. Fras, B. Eble, P. Desfonds, F. Bernardot, C. Testelin,\nM. Chamarro, A. Miard, and A. Lemaˆ ıtre, Phys. Rev. B\n84, 125431 (2011).\n14D. S. Smirnov, E. A. Zhukov, E. Kirstein, D. R. Yakovlev,\nD. Reuter, A. D. Wieck, M. Bayer, A. Greilich, and M. M.\nGlazov, Phys. Rev. B (2018).\n15P. Glasenapp, D. S. Smirnov, A. Greilich, J. Hackmann,\nM. M. Glazov, F. B. Anders, and M. Bayer, Phys. Rev. B\n93, 205429 (2016).\n16S. Varwig, A. Schwan, D. Barmscheid, C. M¨ uller,\nA. Greilich, I. A. Yugova, D. R. Yakovlev, D. Reuter, A. D.\nWieck, and M. Bayer, Phys. Rev. B 86, 075321 (2012).\n17A. Greilich, D. R. Yakovlev, and M. Bayer (Springer-\nVerlag, Berlin, 2010).\n18A. Greilich, M. Schwab, T. Berstermann, T. Auer, R. Oul-\nton, D. R. Yakovlev, V. Stavarache, D. Reuter, A. Wieck,\nand M. Bayer, Phys. Rev. B 73, 045323 (2006).\n19Forp-type QDs we also get gT\nzz=−0.4,ωT\nN= 70 MHz,λT= 1, see discussion in the main text. For n-type QDs\nwe getgT\nzz=−0.45,ωT\nN= 16 MHz, λT= 5,τT\ns= 0.1µs.\nHowever, these parameters weakly affect the results. The\nlongitudinal g-factors were also studied in theRefs. [ 34and\n35].\n20M. Y. Petrov, G. G. Kozlov, I. V. Ignatiev, R. V. Cher-\nbunin, D. R. Yakovlev, and M. Bayer, Phys. Rev. B 80,\n125318 (2009).\n21Y. Li, N. Sinitsyn, D. L. Smith, D. Reuter, A. D. Wieck,\nD. R. Yakovlev, M. Bayer, and S. A. Crooker, Phys. Rev.\nLett.108, 186603 (2012).\n22D. Heiss, S. Schaeck, H. Huebl, M. Bichler, G. Abstreiter,\nJ. J. Finley, D. V. Bulaev, and D. Loss, Phys. Rev. B 76,\n241306 (2007).\n23In the case of Ref. [ 22] the spin states are measured after\nthelong darkperiod, while in other techniquestheperiodic\nexcitation influences the system on a shorter timescales.\n24B. Urbaszek, X. Marie, T. Amand, O. Krebs, P. Voisin,\nP. Maletinsky, A. H¨ ogele, and A. Imamoglu, Rev. Mod.\nPhys.85, 79 (2013).\n25I. A. Merkulov, A. L. Efros, and M. Rosen, Phys. Rev. B\n65, 205309 (2002).\n26J. S. Colton, D. Meyer, K. Clark, D. Craft, J. Cutler,\nT. Park, and P. White, J. Appl. Phys. 112, 084307 (2012).\n27F. Fras, F. Bernardot, B. Eble, M. Bernard, B. Siarry,\nA. Miard, A. Lemaˆ ıtre, C. Testelin, and M. Chamarro, J.\nPhys. Condens. Matter 25, 202202 (2013).\n28M. M. Glazov, Phys. Rev. B 91, 195301 (2015).\n29λ= 1 describes fully isotropic case.\n30P. Desfonds, B. Eble, F. Fras, C. Testelin, F. Bernardot,\nM. Chamarro, B. Urbaszek, T. Amand, X. Marie, J. M.\nG´ erard, et al., Appl. Phys. Lett. 96, 172108 (2010).\n31J. Hackmann, P. Glasenapp, A. Greilich, M. Bayer, and\nF. B. Anders, Phys. Rev. Lett. 115, 207401 (2015).\n32V. S. Zapasskii, Adv. Opt. Photon. 5, 131 (2013).\n33J.H¨ ubner, F.Berski, R.Dahbashi, andM.Oestreich, Phys.\nStatus Solidi B 251, 1824 (2014).\n34I. A. Yugova, A. Greilich, E. A. Zhukov, D. R. Yakovlev,\nD. Reuter, A. D. Wieck, and M. Bayer, Phys. Rev. B 75,\n195325 (2007).\n35A. Schwan, B.-M. Meiners, A. Greilich, D. R. Yakovlev, A.\nD. B. Maia, A. A. Quivy, A. B. Henriques, and M. Bayer,\nApp. Phys. Lett. 99, 221914 (2011)."    },    {        "title": "1203.6684v1.Soliton_Magnetization_Dynamics_in_Spin_Orbit_Coupled_Bose_Einstein_Condensates.pdf",        "content": "Soliton Magnetization Dynamics in Spin-Orbit Coupled Bose-Einstein Condensates\nO. Fialko,1J. Brand,1and U. Z ulicke2\n1Centre for Theoretical Chemistry and Physics and New Zealand Institute for Advanced Study,\nMassey University, Private Bag 102904 NSMC, Auckland 0745, New Zealand\n2School of Chemical and Physical Sciences and MacDiarmid Institute for Advanced Materials and Nanotechnology,\nVictoria University of Wellington, PO Box 600, Wellington 6140, New Zealand\nRing-trapped Bose-Einstein condensates subject to spin-orbit coupling support localized dark\nsoliton excitations that show periodic density dynamics in real space. In addition to the density\nfeature, solitons also carry a localized pseudo-spin magnetization that exhibits a rich and tunable\ndynamics. Analytic results for Rashba-type spin-orbit coupling and spin-invariant interactions pre-\ndict a conserved magnitude and precessional motion for the soliton magnetization that allows for\nthe simulation of spin-related geometric phases recently seen in electronic transport measurements.\nPACS numbers: 03.75.Lm, 67.85.Fg, 03.65.Vf, 71.70.Ej\nThe recent realization of arti\fcial light-induced gauge\npotentials for neutral atoms [1] has added a power-\nful new instrument to the atomic-physics simulation\ntoolkit [2]. In particular, possibilities to induce Zeeman-\nlike and spin-orbit-type couplings in (pseudo-)spinor\natom gases [3] render them ideal laboratories to investi-\ngate the intriguing interplay of spin dynamics and quan-\ntum con\fnement that has been the hallmark of semicon-\nductor spintronics [4, 5]. At the same time, the unique\naspects of Bose-Einstein-condensed atom gases [6] asso-\nciated, e.g., with their intrinsically nonlinear dynamics,\npromise to give rise to novel behavior under the in\ruence\nof synthetic spin-orbit couplings [7{15].\nOne of the special properties resulting from nonlinear-\nity in Bose-Einstein condensates (BECs) is the existence\nof solitary-wave excitations [16]. Basic types of these\nare distinguished by the shape of their localized density\nfeature: dark (gray) solitons are associated with a full\n(partial) depletion of a uniform condensate density in a\n\fnite region of space, whereas bright solitons are local-\nized density waves on an empty background. A further\ncharacteristic associated with solitons is the phase gra-\ndient of the condensate order parameter centered at the\nposition of the density feature. In multi-component sys-\ntems, the dynamics of soliton excitations is found to be\nenriched by the additional degrees of freedom [17{20].\nWe have studied solitons in ring-trapped pseudo-spin-\n1=2 condensates with spin-invariant repulsive atom-atom\ninteractions subject to a Rashba-type [21, 22] spin-orbit\ncoupling and \fnd that they exhibit a third feature: a\npseudo-magnetization vector with conserved magnitude\nand rich dynamics that unfolds in tandem with the soli-\nton's periodic propagation in real space. Figure 1 shows\nan example and also illustrates the interesting fact that\nthe magnetization directions at the beginning and the\nend of a full cycle of the soliton's motion are generally not\nparallel. The appearance of such a geometric phase [23]\nand the precessional time evolution of the solitonic mag-\nnetization is reminiscent of the spin dynamics of electrons\ntraversing a mesoscopic semiconductor ring [24{28].In the following, we consider several soliton con\fgura-\ntions and obtain analytical results for their density and\nmagnetization pro\fles as well as the magnetization dy-\nnamics associated with their motion. We start by intro-\nducing the basic theoretical description of our system of\ninterest. Using the basis of a spatially varying local spin\nframe [26] for the condensate spinor, the nonlinear Gross-\nPitaevskii equation [6] for the spin-orbit-coupled ring\nBEC turns out to be of Manakov-type [17], making it pos-\nsible to apply standard methods [18, 20] to \fnd solitary-\nwave solutions. Accounting for the presence of spin-orbit\ncoupling adds an important twist: Spinors have to sat-\nisfy non-standard boundary conditions, which introduce\nbackground-density \rows in the local spin frame that\ncontribute to the nontrivial magnetization dynamics ex-\nhibited by the moving solitons in the lab frame.\nWe consider a two-component BEC trapped in the xy\nplane and con\fned to a ring of radius R. The atoms\nare assumed to be in the lowest quasi-onedimensional\nFIG. 1. Time evolution of a gray-bright soliton's magnetiza-\ntion in a ring-trapped BEC with Rashba spin-orbit coupling.\nDuring a full cycle of the soliton's motion on the ring, the\nmagnetization vector follows a trajectory on the surface of a\nsphere. The magnetization vectors at the beginning and the\nend of a cycle (indicated by arrows) di\u000ber by an angle #that is\nrelated to a spin-related geometric phase. Soliton parameters\n(see text):vs=c= 0:5, tan\u0011= 2,g= 100,\u0014=\u00000:01.arXiv:1203.6684v1  [cond-mat.quant-gas]  29 Mar 20122\nsubband [29] and subject to a spin-orbit coupling of the\nfamiliar Rashba form [21, 22] \u000bR[\u001bx(\u0000i@y)\u0000\u001by(\u0000i@x)]\nas well as a spin-rotationally invariant contact inter-\naction. (Here \u001bx;y;z are the spin-1/2 Pauli matrices.)\nThe energy functional of such a system [30] is given by\nE[\t] =R\nd'\ty(H\u0000\u0016) \t, where 'is the azimuthal\nangle,\u0016the chemical potential, \t = (  \"; #)Tthe two-\ncomponent (pseudo-spin-1 =2) spinor order parameter in\nthe representation where the ( z) direction perpendicular\nto the ring's plane is the spin-quantization axis, and\nH=E0\u0014\n\u0000@2\n'+g\n2\ty\t\n+ tan\u0011\u0000\n\u001b+e\u0000i'+\u001b\u0000ei'\u0001\u0010\n\u0000i@'+\u001bz\n2\u0011\u0015\n:(1)\nWe use\u001b\u0006\u0011(\u001bx\u0006i\u001by)=2 to denote raising and lower-\ning operators for spin-1/2 components, E0=~2=(2MR2)\nis the energy scale for quantum con\fnement of atoms\nwith massMin a ring of radius R,E0gis the two-body\ncontact-interaction strength, and tan \u0011= 2MR\u000b R=~2is\na dimensionless measure of the spin-orbit coupling.\nThe e\u000bect of Rashba spin-orbit coupling in a ring ge-\nometry can be elucidated by performing a suitable SU(2)\ntransformation. De\fning \t = U\u001fandHloc=U\u00001HU,\nwithU= e\u0000i'\u001bz=2ei\u0011\u001by=2ei'\u001bz=(2 cos\u0011), we \fnd\nHloc=E0\u0014\n\u0000@2\n'\u0000(tan\u0011)2\n4+g\n2\u001fy\u001f\u0015\n: (2)\nThe transformation U\u00001amounts to a '-dependent rota-\ntion of the pseudo-spin quantization axis [26], followed by\na spin-dependent gauge transformation. We will refer to\nthe original representation where the spin-quantization\naxis coincides with the axis of the ring as the lab frame ,\nwhereas the representation in which the Hamiltonian of\nthe system is diagonal in pseudo-spin space [i.e., given\nbyHlocof Eq. (2)] will be the local spin frame [26]. Note\nthat the spinors \t in the lab frame are periodic func-\ntions of', whereas the spinors \u001f= (\u001f+;\u001f\u0000)Tfrom the\nlocal spin frame have to satisfy the boundary conditions\n\u001f\u0006(') =\u001f\u0006('+ 2\u0019)e\u0006iAwith a spin dependent phase\ntwist originating from the spin-orbit coupling, where\nA=\u0019\u00121\ncos\u0011\u00001\u0013\n: (3)\nKnowledge of the local-spin-frame spinors enables the\ncalculation of expectation values for any observables ac-\ncessible to measurement in the lab frame. The total\ndensityn=j \"j2+j #j2\u0011j\u001f+j2+j\u001f\u0000j2is obviously\nthe same irrespective of which representation is chosen in\nspin space. The pseudo-spin-1/2 projections in the lab\nframe correspond to de\fnite atomic states, hence their\ndensity pro\fles n\"(#)= \ty([1 + (\u0000)\u001bz]=2)\t are of in-\nterest. In addition, we will consider the magnetization-\ndensity vector s= \ty\u001b\t in the lab frame, with \u001b=\n(\u001bx;\u001by;\u001bz) being the vector of Pauli matrices.We analyze the properties of localized excitations in\nspin-orbit-coupled ring-trapped BEC based on the time-\ndependent Gross-Pitaevskii equation [6] \u000eE[\u001f]=\u000e\u001f\u0003\n\u001b=\ni~@\u001f\u001b=@t. After rescaling to use the dimensionless time\nvariable\u001c=tE0=~, it has the form\ni@\u001f\u001b\n@\u001c=h\n\u0000@2\n'+g\u0010\nj\u001f+j2+j\u001f\u0000j2\u0000n0\u0011i\n\u001f\u001b(4)\nfor the two components of the spinor \u001f= (\u001f+;\u001f\u0000)T,\nwheren0= [\u0016+ (tan\u0011)2=4]=(gE0) is the uniform (back-\nground) density consistent with the chemical potential\n\u0016. While the spin-orbit coupling has formally disap-\npeared from the nonlinear equation (4), it is still im-\nplicitly present via the boundary conditions that the in-\ndividual components \u001f\u0006(';\u001c) must satisfy.\nWe have obtained several soliton solutions of Eqs. (4)\nusing established techniques [17, 18, 20] and implemented\nthe appropriate boundary conditions. Before giving fur-\nther details, we like to summarize a few general features.\nThe soliton spinors in the local-spin-frame representation\nturn out to be of the form\n\u001f(s)\n\u001b= \u0007(s)\n\u001b('\u0000vs\u001c) eivb\u001b'=2\u0000iv2\nb\u001b\u001c=4; (5)\nwhere \u0007(s)\n\u001b(\u0018) are complex amplitude functions encoding\nthe speci\fc soliton-like density features, vsis the propa-\ngation speed of the soliton, and vb\u001bare background \row\nvelocities of the individual spinor components that are\nnecessary to implement the boundary conditions arising\ndue to the presence of spin-orbit coupling. The density\nn(s)\n\"(#)and magnetization density s(s)exhibit spatially lo-\ncalized features. Subtracting s(s)from the magnetiza-\ntion density s(s)\nbof the condensate background yields the\nmagnetization density that is associated with the soli-\nton excitation only. Its integral S(s)=R\nd'[s(s)\nb\u0000s(s)]\nis the vector of total soliton magnetization, which is\nan additional property of localized excitations in multi-\ncomponent BECs. For soliton solutions of the form (5),\nS(s)has constant magnitude. Its temporal evolution is\nmost conveniently described by a set of four angles as\nde\fned in Fig. 2(c). While the tilt angles \fand\f0are\ntime-independent, the angles \u000band\u000b0vary linearly in\ntime, signifying the precession of S(s)around tilted z0\naxis with the universal result\n\f=\u0011; \u000b =vs\u001c+\u0019: (6)\nThez0axis is tilted by the angle \u0011characterizing the\nspin-orbit coupling and it rotates around the zaxis with\nthe same angular velocity vsthat characterizes the soli-\nton propagation. The second tilt angle \f0is found to de-\npend only on the soliton pro\fle \u0007 \u0006('), while the preces-\nsion frequency d\u000b0=d\u001chas complicated dependences on\nthe parameters of the soliton solutions. Figure 2 shows\nexemplary magnetization dynamics for gray-bright and3\nFIG. 2. Magnetization dynamics of a gray-bright soliton [panel (a)] and a gray-gray soliton with zero background magnetization\nin the local spin frame [panel (b)]. Parameters used are g= 100 (100), tan \u0011= 0:2 (0:5),vs=c= 0:5 (0.2), and \u0014=\u00000:5 for\nthe gray-bright (gray-gray) case. (c) Angles used to describe the two-step precessional motion of S. The angle \u000b0is measured\nwith respect to an x0axis that is perpendicular to both the zandz0axes.\ngray-gray solitons. Interestingly, we \fnd that the mag-\nnetization vector is usually not parallel to its initial di-\nrection after the soliton has completed a full cycle of its\nmotion around the ring as, e.g. seen in \fgure 1. The an-\ngle#between the magnetization directions at the start\nand the end of a cycle turns out to be \fnite only as a\nconsequence of spin-orbit coupling, as it depends promi-\nnently on the phase Agiven in Eq. (3) that also governs\nspin-dependent interference in mesoscopic ring conduc-\ntors [27].\nIn order to \fnd explicit soliton solutions, we introduce\n\u0018='\u0000u\u001c, whereuis a velocity parameter, and initially\nlook for solutions of the form \u001f\u001b(';\u001c) =p\nn\u001b(\u0018) ei\u0012\u001b(\u0018).\nThis allows us to rewrite Eq. (4) in the form\n\u0000u@n\u001b\n@\u0018+ 2@\n@\u0018\u0012\nn\u001b@\u0012\u001b\n@\u0018\u0013\n= 0;(7a)\nu@\u0012\u001b\n@\u0018+1pn\u001b@2pn\u001b\n@2\u0018\u0000\u0012@\u0012\u001b\n@\u0018\u00132\n\u0000g(n\u0000n0) = 0;(7b)\nwheren=n++n\u0000. Single component solutions for\n\u001b= ~\u001bare easily found by integration of Eqs. (7) to yield\n\u001f~\u001b(\u0018)/\u0007(\u0018) and\u001f\u0000~\u001b= 0, with the well-known dark\nsoliton solution on the in\fnite line [6]\n\u0007(\u0018) =pn0\u0014\niu\nc+\rutanh\u0012\n\ru\u0018\n\u0018D\u0013\u0015\n: (8)\nHere\r2\nu= 1\u0000u2=c2,c2= 2gn0, 1=\u00182\nD=gn0=2. The soli-\nton pro\fle (8) is appropriate for su\u000eciently strong non-\nlinearity, where \u0018D=\ru\u001c2\u0019[31]. However, \u0007( \u0018) does\nnot satisfy the proper boundary condition since it has a\nphase step \u0001 \u0012=\u00002 arccos(u=c). To compensate for the\nphase step and ensure the correct phase shift associated\nwith the gauge transformation U, we perform a Galilean\ntransformation on (8), which yields\n\u001fSC\n~\u001b(';\u001c) = \u0007(\u0018\u0000vb\u001c) eivb'=2\u0000iv2\nb\u001c=4: (9)\nHerevb=\u0000(\u0001\u0012+ ~\u001bA)=\u0019is the background velocity\nimposed by the boundary condition. Thus the single-\ncomponent soliton solution is of the form (5), with \u0007SC\n~\u001b=\n\u0007 and \u0007SC\n\u0000~\u001b= 0, and propagation speed vs=u+vb.A straightforward calculation yields sSC= ~\u001bj\u0007('\u0000\nvs\u001c)j2(\u0000sin\u0011cos';\u0000sin\u0011sin';cos\u0011)Tfor the magne-\ntization density of the single-component soliton solution.\nIn essence, the density depletion at the soliton's posi-\ntion gives rise to a reduction of the magnetization den-\nsitysSC\nb= ~\u001bn0(\u0000sin\u0011cos';\u0000sin\u0011sin';cos\u0011)Tasso-\nciated with the background. Thus sSC\nb\u0000sSCconsti-\ntutes the magnetization density associated with the soli-\nton itself, as it is the change in the background mag-\nnetization density due to the presence of the localized\nexcitation. For the single-component soliton, this corre-\nsponds to a peak in magnetization density at the soli-\nton's position. The total magnetization vector is ob-\ntained by integrating that peak in real space, which yields\nSSC= ~\u001b(\u0000sin\u0011cos(vs\u001c);\u0000sin\u0011sin(vs\u001c);cos\u0011)T. This\nmagnetization vector is precessing in a perfectly synchro-\nnized fashion with the soliton's motion around the ring\n[cf. Fig. 2(c) with Eq. (6) and \f0= 0], i.e.,#SC= 0.\nWe now consider a solution of Eqs. (7) that is a gray-\nbright (GB) soliton in the local spin frame. We assume\nthat the densities approach constant values n+!n0+\n(gray part) and n\u0000!0 (bright part) far away from\nthe soliton's position. To decouple Eq. (7b), we use the\nansatz [20] n\u0000=\u0014(n+\u0000n0+) with\u00001\u0014\u0014\u00140. We\napply a Galilean boost to both components to match the\nphase of the gray part only, hence they are of the form\n(5) with \u0007GB\n+(\u0018) given by \u0007( \u0018) from Eq. (8) but with\nrescaledc2= 2gn0+(1 +\u0014), 1=\u00182\nD=gn0+(1 +\u0014)=2, and\n\u0007GB\n\u0000(\u0018) =p\n\u0000\u0014n0+\rueiu\u0018=2\ncosh(\ru\u0018=\u0018D): (10)\nFurthermore, vb\u0000=vb+andvs=u+vb+. Figure 3\nshows the density pro\fles [panel (a)] and magnetization-\ndensity pro\fle [panel (c)] associated with a GB soliton.\nThe vector SGBof total magnetization for a GB soli-\nton precesses concomitantly with the soliton's motion; cf.\nFig. 2(c) with Eq. (6) and tan \f0= (p\u0000\u0014u\u0019)=[(1\u0000\u0014)c\ru],\n\u000b0=\u0000vs\u001c(1 +A=\u0019). Figures 1(a) and 2(a) show exam-\nples of possible time evolutions of the GB-soliton magne-\ntization. The magnitude of the magnetization vector is4\n/LParen1a/RParen1\n0.00.20.40.60.81.00.00.20.40.60.81.0\n/CurlyPhi/Slash12Π\n/LParen1b/RParen1\n0.00.20.40.60.81.00.00.20.40.60.81.0\n/CurlyPhi/Slash12Π\n/LParen1c/RParen1\n0.00.20.40.60.81.0/Minus0.50.00.51.0\n/CurlyPhi/Slash12Π\n/LParen1d/RParen1\n0.00.20.40.60.81.0/Minus0.50.00.51.0\n/CurlyPhi/Slash12Π\nFIG. 3. Lab-frame spin densities [(a),(b)] and zcompo-\nnent of the magnetization density [(c),(d)] for a stationary\ngray-bright soliton [(a),(c)] and a stationary gray-gray soliton\n[(b),(d)]. Panels (a) and (b) show the total density (solid yel-\nlow curve) and individual-spin (dashed blue = \", dotted red\n=#) densities normalized to the background-density value. In\n(c) and (d), the total density is plotted again for reference as\nthe solid yellow curve, together with the magnetization pro-\n\fless(s)\nz(dashed blue curve), and the magnetization density\ns(s)\nbz\u0000s(s)\nzassociated with the soliton excitations only (dotted\nred curve). Parameters are tan \u0011= 1:0,g= 100, and (for the\ngray-bright soliton) \u0014=\u00000:5.\nfound to be S= 2(1\u0000\u0014)\u0018D\ru=cos\f0. For a GB soliton,\nthe magnetization vector turns out to be notaligned with\nits initial direction after completion of a full cycle of its\nmotion around the ring. A straightforward calculation\nyields sin(#GB=2) = sin\f0sinA. As\f0is a known func-\ntion of the soliton parameters, a measurement of #GBwill\nyield the spin-related geometric phase A.\nThe solutions representing gray-gray (GG) solitons in\nthe local spin frame are obtained by Hirota's method [18].\nThe spinor components are of the form (5) with\n\u0007GG\n\u001b(\u0018) =pn0\u001b\u0014\niu\u001b\nc\u001b+\ru\u001btanh(a\u0018)\u0015\n: (11)\nHere,a2=\r2\nu+=\u00182\nD++\r2\nu\u0000=\u00182\nD\u0000,vs= 2au\u001b=c\u001b\ru\u001b+vb\u001b,\nc2\n\u001b= 2gn0\u001b, 1=\u00182\nD\u001b=gn0\u001b=2. The back-ground \rows\nare given by vb\u001b=\u0000(\u0001\u0012\u001b+\u001bA)=\u0019, where \u0001\u0012\u001b=\n\u00002 arccos(u\u001b=c\u001b). The independent parameters charac-\nterizing a GG soliton are the ratio n0+=n0\u0000(or, equiv-\nalently, the background magnetization in the local spin\nframe) and the speed vsof the soliton. All other param-\neters can be found by solving transcendental equations\ngiven just after Eq. (11). For simplicity, we consider the\ncase of a GG soliton with zero background magnetiza-\ntion in the local spin frame (i.e., n0+=n0\u0000\u0011n0=2).\nFigure 3 shows results for spinor-density [panel (b)] and\nmagnetization-density [panel (d)] pro\fles.\nThe time evolution of the magnetization vector asso-\nciated with a moving GG soliton is characterized by theangles de\fned in Fig. 2(c) with Eq. (6), \f0=\u0019=2, and\n\u000b0=!\u001c+\u0019=2, where!=\u0000(vb+\u0000vb\u0000)vs=2 + (v2\nb+\u0000\nv2\nb\u0000)=4\u0000(1+A=\u0019)vs. Figure 2(b) illustrates this dynam-\nics which, for small \u0011, corresponds to a slow rotation of\nthe magnetization vector in the ring's plane with super-\nimposed fast small-amplitude oscillations in the normal\ndirection. As in the case of the GB soliton, the magneti-\nzation vector does not evolve back to its initial direction\nafter a period of the soliton's ring revolution. The angle\nbetween magnetizations at the start and the end of the\ncycle is found to be #GG= 2\u0019j!j=vs!2AforA\u001c 1.\nAgain, the dependence of #GGonAenables determina-\ntion of the latter by measuring the former.\nIn conclusion, we have investigated the properties\nof soliton excitations in ring-trapped spin-orbit-coupled\nBECs. We \fnd that a magnetization degree of freedom is\ngenerally associated with a soliton, and that the magne-\ntization vector precesses around an axis that is rotating\nsynchronously with the soliton's orbital motion around\nthe ring. The magnetization direction at the end of a\ncycle of revolution does not coincide with the initial di-\nrection for the gray-bright and gray-gray cases, making it\npossible to measure a spin-orbit-related geometric phase.\nOur work opens up new avenues for the realization and\nmanipulation of magnetic soliton excitations in BECs.\nIt also creates the opportunity to study spin-dependent\ninterference and scattering e\u000bects that, until now, were\nonly accessible in semiconductor nanostructures.\nThis work was supported by the Marsden fund (con-\ntract no. MAU0910) administered by the Royal Society\nof New Zealand.\n[1] Y.-J. Lin, R. L. Compton, K. Jimenez-Garcia, J. V.\nPorto, and I. B. Spielman, Nature 462, 628 (2009).\n[2] J. Dalibard, F. Gerbier, Juzeli\u0016 unas, and P. Ohberg, Rev.\nMod. Phys. 83, 1523 (2011).\n[3] Y.-J. Lin, K. Jimenez-Garcia, and I. B. Spielman, Nature\n(London) 471, 83 (2011).\n[4] D. D. Awschalom, D. Loss, and N. Samarth, eds.,\nSemiconductor Spintronics and Quantum Computation\n(Springer, Berlin, 2002).\n[5] I. Zuti\u0013 c, J. Fabian, and S. D. Sarma, Rev. Mod. Phys.\n76, 323 (2004).\n[6] L. Pitaevskii and S. Stringari, Bose-Einstein Condensa-\ntion (Clarendon Press, Oxford, 2003).\n[7] T. D. Stanescu, B. Anderson, and V. Galitski, Phys.\nRev. A 78, 023616 (2008).\n[8] M. Merkl, A. Jacob, F. E. Zimmer, P. Ohberg, and\nL. Santos, Phys. Rev. Lett. 104, 073603 (2010).\n[9] C. Wang, C. Gao, C.-M. Jian, and H. Zhai, Phys. Rev.\nLett.105, 160403 (2010).\n[10] T.-L. Ho and S. Zhang, Phys. Rev. Lett. 107, 150403\n(2011).\n[11] C.-J. Wu, I. Mondragon-Shem, and X.-F. Zhou, Chin.\nPhys. Lett. 28, 097102 (2011).\n[12] S.-K. Yip, Phys. Rev. A 83, 043616 (2011).5\n[13] S. Sinha, R. Nath, and L. Santos, Phys. Rev. Lett. 107,\n270401 (2011).\n[14] H. Hu, B. Ramachandhran, H. Pu, and X.-J. Liu, Phys.\nRev. Lett. 108, 010402 (2012).\n[15] Y. Zhang, L. Mao, and C. Zhang, Phys. Rev. Lett. 108,\n035302 (2012).\n[16] R. Carretero-Gonz\u0013 aez, D. J. Frantzeskakis, and P. G.\nKevrekidis, Nonlinearity 21, R139 (2008).\n[17] S. V. Manakov, Zh. Eksp. Teor. Fiz. 65, 505 (1973), [Sov.\nPhys. JETP 38, 248 (1974)].\n[18] A. P. Sheppard and Y. S. Kivshar, Phys. Rev. E 55, 4773\n(1997).\n[19] P. Ohberg and L. Santos, Phys. Rev. Lett. 86, 2918\n(2001).\n[20] J. Smyrnakis, M. Magiropoulos, G. M. Kavoulakis, and\nA. D. Jackson, Phys. Rev. A 81, 063601 (2010).\n[21] E. I. Rashba, Fiz. Tverd. Tela (Leningrad) 2, 1224 (1960),\n[Sov. Phys. Solid State 2, 1109 (1960)].\n[22] Y. A. Bychkov and E. I. Rashba, J. Phys. C 17, 6039(1984).\n[23] A. Shapere and F. Wilczek, eds., Geometric Phases in\nPhysics (World Scienti\fc, Singapore, 1989).\n[24] D. Loss, P. Goldbart, and A. V. Balatsky, Phys. Rev.\nLett.65, 1655 (1990).\n[25] A. G. Aronov and Y. B. Lyanda-Geller, Phys. Rev. Lett.\n70, 343 (1993).\n[26] J. Splettstoesser, M. Governale, and U. Z ulicke, Phys.\nRev. B 68, 165341 (2003).\n[27] D. Frustaglia and K. Richter, Phys. Rev. B 69, 235310\n(2004).\n[28] F. Nagasawa, J. Takagi, Y. Kunihashi, M. Kohda, and\nJ. Nitta, Phys. Rev. Lett. 108, 086801 (2012).\n[29] This limitation is not crucial, as \fnite-width e\u000bects and\nhigher subbands could be treated straightforwardly.\n[30] M. Merkl, G. Juzeli\u0016 unas, and P. Ohberg, Eur. Phys. J.\nD59, 257 (2010).\n[31] Otherwise periodic solutions involving elliptic functions\nhave to be used. See, e.g., L. D. Carr, C. W. Clark, and\nW. P. Reinhardt, Phys. Rev. A 62, 063610 (2000)."    },    {        "title": "1308.5291v1.Spin_Dynamics_in_Graphene_and_Graphene_like_Nanocarbon_Doped_with_Nitrogen_the_ESR_Analysis.pdf",        "content": " \n \n1 \n Spin Dynamics in Graphene and Graphene –like Nanocarbon Doped with Nitrogen  : the \nESR Analysis  \nAshwini P. Alegaonkar1, Arvind Kumar2, Satish K. Pardeshi1 and Prashant S. Alegaonkar2 \n1Dept . of Chemistry, University of Pune, Pune, 411 007, MS, India  \n2Dept . of Applied Physics, Defence Institute of Advanced Technology, Girinagar, Pune 411 025, MS, India  \nABSTRACT  \nNano–engineer ed spin degree of freedom in carbon system may offer desired  \nexchange –coupling with optimum spin –orbit interaction which is essential, to construct solid \nstate qubit s, for fault–tolerant quantum computation . The purpose of this communication is to \nanalyze spin dynamics of, basically, four types of systems:  (i) Graphene (system with \ninversion symmetry), (ii) Graphene –like nanocarbons (GNCs , broken inversion symmetry \nand heterostructure , sp2+ sp3, environment), and (iii) their nitrogen doped derivatives. The \nspin transport data  was obtained using the electron spin resonance spectroscopy (ESR) \ntechnique, carried out  over 123–473K  temperature range . Analysis of shape, linewidths of \ndispersion derivatives,, and g –factor anisotropy has been carried out. Spin parameters such \nas, spin–spin, T sl, spin–lattice, T sl, relaxation time, spin–flip parameter, b, spin relaxation \nrate, spin, momentum relaxation rate, , pseudo chemical potential,      density of states, , \neffective magnetic moment, µ eff, spin–, defect–concentration, and Pauli susceptibility, spin \nhas been estimated , and examined for their temperature as well as interdepende nce. Details of \nthe analysis are presented. The quantitative study  underlined the following facts: (i) by and \nlarge, spin dynamics in Graphene and GNCs is significantly differen t, (ii) transport of spin \nbehaves in opposite fashion , after doping nitrogen , in both the systems, (iii) reduction in the \nmagnetization has been observed  for both GNCs and Graphene, after doping nitrogen , (iv) \nhyperfine interactions have been observed in all classes of s ystems  except in GNCs , (v) \nnitrogen doped GNCs seems to be appropriate for qubit designing.   \n   \n \n2 \n I. INTRODUCTION  \nHarnessing quantum laws  in nano carbon system seem s to be promising for  information \nprocessing  that outperforms  their classical counterparts  for unconditional, secured \ncommunication . 1 Such  quantum systems  interact with their environment via up or down spin \nof electron , in the form of strings of quantum bit s (qubit s).2-3 In general, t he spin degree of \nfreedom of electron endowed to its lattice is on the talking term  and could be exploited , \nprimarily, using  spin–orbit–(SO) and exchange –interactions.4 Nonetheless , the crucial \nrequirements on spin density wave propagation are quantum  entanglement,5 coupling the spin \nto lattice vibration, 6 coherence,7 and hyperfine interaction  of the electron spin with  the \nsurrounding n uclear spin .8 In Graphene,  contributions to SO interaction  is due to  intrinsic,9 \nBychkov–Rashba ,10 ripple ,4 and extrinsic11. These i nteractions  are supposed to be weak , in \neV regime ,12 due to low atomic number  of carbon .4 The exchange–interactions originating \ndue to  spin–flip and spin splitting are predicted to be identical for disordered – and ordered–\nsp2 Graphene  network . In fact, t hese interactions are supposed to independent of SO \ncoupling. 4  The spin–lattice relaxation time is estimated in 10–100 ns range13 with no or low \nhyper fine interaction s, due zero–spin 12C nuclei (abundance  99.0–98.9%) .14 In recent  \nexperiments , the spin–lattice  relaxation time is measured as short as 60 –150 ps .15-16 The \nKlein tunnelling paradox show ed leaky  spin density wave  propagation due to  difficulty in \ncreating gap –tuneable Graphene quantum dot. 17 The tunnel –couple dots has found hindrance \nin Heisenberg –spin exchange –coupling due to valley degeneracy that exist in the vicinity of \nDirac point in Graphene .18 The role of strength and contribution of SO component s is still a \nsubject of discussion , 4,9,11,12 whereas,  significant emphasis  has been given on the study of \nvirtual spin degree of freedom ,19-22 neglecting the fourfold spin –degenerated states of the \nactual  spin in the Graphene  band structure . 23 Realistic Graphene contain inversion, broken \ninversion symmetry, intrinsic, and extrinsic heterostructure lattice environment. And , present  \n \n3 \n scenario demands experimental validation for studying spin transport in realistic Graphene \nsuper lattice  which could address a few issues mentioned above . Principally, spin relaxation \ntime is inseparable from SO interactions and has direct dependence on transport parameters \nlike spin relaxation rate, spin flip probability, momentum relaxation rate, relaxation time, etc. \nAnalyzing them  may develop fundamental  insight in designing tuneable spin degrees of \nfreedom ; crucial for favourable SO – and exchange –interactions .  \nThe focus of this communication is  to study s pin dynamics of  Graphene , Graphene–\nlike nanocarbons  (GNCs) , and their nitrogen  doped derivatives . The spin transport data  was \nobtained using the electron spin resonance  spectroscopy (ESR) technique , carried out over a \ntemperature range  123–473K , on the bulk powder  specimen . Spin transport parameters has \nbeen estimated , examined , and analyzed for their temperature as well as interdependence. \nThe study  revealed that, there is significant difference in the spin transport properties of  \nGraphene and GNCs , moreover , transport of spin  behave s in opposite fashion after doping  \nnitrogen . The overall reduction in the magnitude of magnetization  has been observed, after \nnitrogen doping  in both the systems . The  hyperfine interactions ha ve been observed in all \nclasses of s ystems  except GNCs.  Understanding spin transport  may provide  clue for \nconstruction of solid state qubit s which is crucial for future quantum computing devices. \nDetails are presented.  \nII. EXPERIMENTL DETAILS      \nBasically, we have two types of carbon environments (i) Graphene, pure sp2 carbon (i.e. \nsystem with inversion symmetry), and (ii) GNCs, disordered sp2 network (broken inversion \nsymmetry and heterostructure). In the present work, Graphene –like nanocarbon sheets \n(GNCs) were synthesized according to protocol reported.24 The materia l obtained by our \nsynthesis methodology was graphene –like nano carbon  and not graphene. The basic structure \nof GNCs is of carbophane family in which carbon is in more than one hybridization state i.e.  \n \n4 \n mixture of sp2 and sp3 carbon atoms. In such structures carbocyclic molecule contain \ncovalently bridged aromatic rings called phanes.42 This system is typically 2 –5 layers and \neach layer contains heavy local disorder  in the form of configuration and missing carbon \natoms . While synthesizing, transformation of ¼ of the sp2 graphitic carbon to sp3 carbon \ntakes place so that remaining ¾ are all in the benzoid confirmation and within sp2 cluster no \nsp3 carbon present. It has presence only at the adjacent site to sp2 cluster in each layer. Every \nsp3 is covalently bonded within sp3 zone and at the interface to deviates from covalent nature \nwhile bonding with the sp2 carbon, in the form of three benzoid ring. The remaining covalent \nbond to each sp3 carbon provides covalent interlayer bond. Hence, we ha ve observed \nminimum 2 layers of GNCs. Details are provided in the reference mentioned above.                \nTreatment with tetrakis (dimethylamino) ethylene  (TDAE) for nitrogen doping   \nNitrogen doping was carried out using tetrakis (dimethylamino) ethylene  (TDAE) \ncompound. Initially, the suspension of GNCs was prepared in 25 ml of tetra hydro furan \n(THF) and 0.1 mg of TDAE was added in GNCs suspension. After adding TDAE, the \nsuspension was sonicated for 30 min followed by room temperature stirring for 8 –10 h. T he \nsuspension was allowed to settle for about 5 h. The vacuum filtration was carried out using \nPTFE filter (pore size ~1.2 µm) . The GNCs treated with TDAE was termed as N –GNCs. In \nsimilar fashion, Graphene and N–Graphene were prepared.  The powder obtained was used \nfor examined for structure –property relationship using preliminary characterization such as \nFourier transform infrared –, and UV –Visible–spectroscopy. Details are provided in \nSupporting information.    \nElectron spin resonance measurements on the sys tems          \nThe e lectron spin resonance (ESR) measurements were carried out using a standard \nESR set up equipped with electromagnet, microwave bridge , resonant cavity, waveguide \ncircuitry and spectrometer consol. Initially, a known amount of sample, under investigation,  \n \n5 \n was placed in the rectangular cavity consisting of cylindrical sample transfer ports. The mode \nof the cavity was TE 102. The sample was  positioned at cavity centre where the magnetic \ncomponent of the microwave standing wave attains the local maxima. This configuration \nprovides maximum sensitivity to the measurement. The microwave energy was injected and \ncoupled via an iris screw. The crit ical coupling, achieved using iris screw controlled the \namount of incident and reflected microwave radiation in the cavity. The measurements were \nperformed at microwave frequency of ~ 9.1 GHz. (X –band). The maximum microwave \noutput was varied from sample t o sample in the range 985 µW to 1000 µW. Further, for \nreceiver mode, phase shifter enables one to match the phase of microwave signal reflected \nfrom cavity with the phase of microwave injected in the reference arm of the circulator. The \nvalue of variable p hase shifter was kept fixed at zero. The quality factor of the cavity was \n12000. In general, the ESR signal is weak and submerge in the background level noise. By \nenhancing the sensitivity of spectrometer one can amplify the obtained ESR signal. In the \npresent measurements, the modulation frequency was kept constant at 100 kHz by adjusting \nthe band pass filter parameter of the lock –in–amplifier. The static magnetic field was swept \nslowly at a spectrum –point time constant 0.1 sec over the range 300 mT to 370  mT with the \namplitude of modulation frequency kept at 6 kHz. The field centre was 336 mT and ESR line \nwidth, ∆H = 0.05 mT. The signal –to–noise ratio was computed to be 20 at 300K with 1s per \nspectrum–point constant. The sensitivity of the system was 7.0  10 9 spins / 0.1 mT and \nresolution 2.35 µT. All measurements were carried out at a temperature  range 123 –473K . \nAnd the first derivative of the paramagnetic absorption signal was recorded for samples. T he \npristine Graphene and GNCs after doping nitrogen we re designated, respectively, as N –\nGraphene and N –GNCs . The highest concentration samples were taken from each batch of \nN–Graphene and N –GNCs.  \nIII. RESULTS, ANALYSIS , AND DISCUSSION   \n \n6 \n Basically, w e have two types of carbon environment s (i) Graphene, pure sp2 carbon (i.e. \nsystem with inversion symmetry ), and (ii) GNCs , disordered sp2 network (broken inversion \nsymmetry and heterostructure). Particularly; GNCs  carbon phanes contains mixture of sp2 \nand sp3 carbon. Such systems are sensitive to interband optical tr ansitions which occur  \nbetween 1 –3 eV. The characteristic  vibration band in Raman spectrum shows peak at 1670 \ncm–1 with full width half maximum 100 cm–1.43 These are typically 2 –5 layer ed system in \nwhich cluster s of sp2 are isolated by sp3 carbon zones having ra ndom patterning in them.  \nWhat nitrogen doping dose? It provides is the extrinsic environment in addition to intrinsic \nsymmetry that exists in two distinct systems.   \nA. Line width , and shape–the results  \nA great deal of information can be obtained from the careful analysis of the width and \nshape of a resonance absorption line. Two common line shapes are Gaussian and Lorentzian. \nFIG. 1 shows ESR dispersion derivatives,   \n   as a function of magnetic field. The upper panel \nis ESR for G raphene and N –Graphene. The spectrum is found to be asymmetrical for \nGraphene, over the measured temperature regime. For both, the fitted line shape is found to \nbe intermediate between Gaussian and Lorentzian.  \nFor Graphene, the line –width is found to be v aried from 1.1027 ± 0.0071 (123K) to \n1.2311 ± 0.0063 mT (473K). For N –Graphene, the degree of asymmetry of the ESR line is \nincreased and the width in the range 1.1495 ± 0.0066 (123K) to 1.3217 ± 0.0035 mT (473K). \nAfter nitrogen doing in Graphene, there is slight increase in the line –width with \ninhomogeneous broadening. The inhomogeneous broadening is indicative of three sources (i) \nmore than two spin –lattice relaxation time, T sl, are merged into one overall line, (ii) charge \ninhomogeneities (the so called puddles). The charge inhomogeneities over the volume of the \nsample exceeds the natural line –width,  \n    , where γ is gyromagnetic ratio and prevent \nrelaxing electron from reaching the Dirac point, and have the average minimal charge density   \n \n7 \n  \nFIG. 1. A comparison of ESR dispersion derivatives, dY/dH, as a function magnetic field for \n(i) Graphene, ( ii) N–Graphene, ( iii) GNCs and ( iv) N–GNCs over the measured temperature \nrange (a)–(f)123–473 K.  For all classes of samples narrowing of linewidth is observed . All \nsamples show unsymmetrical absorption derivative  line–shape  on low field side,  except \nsymmetrical (iii). Open ellipse connected by an arrow shows inset  (in each panel)  is \nindicative of hyperfine interactions . The explanation is given in text.       \nas 109 cm–2. The spins associated with these charge inhomogeneities, in various parts of the \nsample, find  themselves in different field strength, and the resonance is broaden ed in an \ninhomogeneous manner.25 The source (iii) is hyperfine interactions (HFI) , the weak magnetic \ninteraction s between the spin of unpaired electron s and nuclear spin . The interaction  could be  \nits own nucleus or ensemble of nearest neighboring nucle i spins . Lower panel in F IG. 1 \nshows ESR spectra for GNCs and N –GNCs. For GNCs, the fitted line shape is found to be \n \n \n8 \n perfect ly Lorentzian, whereas, for N –GNCs mix  feature,  Lorentzian –Gaussian. The line–\nwidths are somewhat narrowed down after doping nitrogen to GNCs. Their magnitudes are \n1.2964±0.0083 to 1.2669±0.0091 , for GN Cs, and 0.7091±0.0049 to 0.6710±0.0038 , for N–\nGNCs, over the measured temperature range. The signature of hyperfine structure is observed \nin Graphene a s well as N–Graphene  including N –GNCs . These weak lines are zoomed in \neach inset of FIG.1. These lines ar e not accounted for by considering forbidden transition.  In \ngeneral, the emergence of HFI  in gra phene comes from the nuclear spins of the 13C isotope, \nhaving natural abundance  1–1.1 %. The dominant 12C has zero spin. These pairs of lines are \nseparated from allowed lines by nuclear resonance frequency of the proton, at the field H r \nused for the ESR measurement. The interaction is present over entire region of magnetic field \nand at all temperatures. The presence of HFI indicate s that such interaction coul d originate \nfrom presence of adatom of odd nuclei like 1H. The weak lines arise from matrix of protons \nwhich undergo spin –flip when electron spins of surrounding trapped hydrogen atoms are \nreoriented.  The coupling is dipolar and the intensity of the weak lines varies approximately \nas :      and, there could be         hyperfine components for n nuclei with the nuclear \nspin I. The total number of hyperfine component ,     , could be:                    \nwhere the symbol       , denotes the formation of a product.26 Except GNCs , other system \nhas shown additional weak lines , present over the swept magnetic field.  \nB. Analysis of spin dynamics of the system   \nIn the subsequent section, a comprehensive exercise is presented to compute spin \ntransport parameters, using theory of spin relaxation. These parameters may shed light on \ndynamic of the spin to perform molecular architecting of sp2 network. As a first step towards \ncomputing the se parameters, peak –to–peak width , Hpp, and corresponding g –factor  are the \ncrucial parameters . The peak width ∆H pp has an azimuth angular dependence, θ, i.e. \norientation of applied field with respect to the orientation of carbon planes of the samples.  \n \n9 \n Since, measurements were conducted on the bu lk powder specimen then: ∆H pp(θ) = \n          +          . Thus, the  measured ∆H pp consists of contribution from    as well \nas   . Further, the value of g –factor, at which the resonance has occurred under applied \nmicrowave frequency, characterizes the magnetic moment and gyro –magnetic ratio \nassociated with unpaired electrons in the material. If the angular momentum of a system is \nsolely due to spin angular momentum, the tensor g –factor should be isotropic, with the value, \n2.00232. Any anisotropy or deviation from this value result involves (i) contributions of the \norbital angular momentum from excited state, and (ii) spin –spin contribu tion. This results \ninto the effective g –factor. Thus, magnitude of effective g–factor is estimated for all the \nsystems under investigation. The effective g–factor is observed to be less than the g –factor \nfor a typically intrinsic spin angular momentum of a  free electron  i.e. non–degenerated Pauli \ngas, 2.0023 2. The difference , g, between 2.0023 2 and obtained  effective g–factor for the \nsystem is computed. The observed variation is indicative of corresponding orbital angular \nmagnetic moment could be contributed to the spin magnetic moment. Moreover, t he values \nindicate that, the local magnetic environment in carbon system is distinctly different than that \nexist in a conventional magnetic material. The computed ∆H pp and corresponding g for our \nsystems  is enlisted in Table I. In subsequent discussions, temperature averaged magnitude of \nmeasured and estimated physical quantities have been quoted using the symbol <…> T. It \nindicates  average  of the quantity along with the standard deviation over the measured \ntemperature range. For Graphene, the value of <∆H pp >T is 0.68678 ± 0.05321  mT, whereas, \nfor N–Graphene, GNCs, and N –GNCs is , respectively, 0.64131 ± 0.06356, 0. 70428  ± \n0.02990, and 0.43470 ± 0.0268  (all are in mT) . The linewidh of GNCs is observe d to be large \ncompared to linewidth of Graphene. After doping nitrogen, there is a linewidth narrowing \ntrend in both the systems , as discussed above . The value of <∆g> T for Graphene is 0.00516 ± \n2.0510–4 and 0.00563 ± 4.015 10–4, for N–Graphene. For GNCs, magnitude of <∆g> T is  \n \n10 \n 0.00455 ± 7.86 10–5 and 0.00427 ± 9.48 10–5, for N–GNCs . The small values of linewidth, \nand small deviation in effective g–factor suggest that spin do not originate from transition \nmetal impurities but from  only carbon inherited spin species.26 For Graphene , <∆g> T is \nincreased after nitrogen doping, whereas, it is, comparably, smaller for GNCs and reduced \nafter nitrogen doping. This opposite behaviour underline s following facts: (i)  orbital angular \nmomentum,  and spin–spin contributions are increased in Graphene after doping nitrogen, (ii) \nthe overall strength of these contributions seems to be small  in GNCs system and reduced \nfurther for N –GNCs, (iii) response of spin to lattice environment , i.e. inversion (Graphene),  \n \nTable I. Estimated peak –to–peak linewidth, ∆H pp, and, g, for Graphene and GNCs and their  \nnitrogen doped derivates. The value of g is difference between g –factor for non –\ndegenerated Pauli gas to g –factor obtained for our sa mples, at resonance field. The row <A> T \nindicate temperature averaged value of ∆H pp and g.        \nbroken inversion symmetry (GNCs) , and with extrinsic adatom, seems to be different, (iv) \nelectron s of nitrogen adatom may also contribute to  orbital angular momentum  component .  \n \n \n11 \n To evaluate contribution of each component we have carried out  analysis o n spin–spin \nrelaxation and spin –orbit interactions in the discussions presented below. The magnitude of \n∆Hpp bares important information about spin dynamics of the system, specifically, T ss, which \ncorresponds to electron spin –spin relaxation time.  Due to external perturbation, the deformed \nspin system regains the state of equilibrium ‘up’ or ‘down’ over the char acteristic time scale \nand is termed as spin –spin relaxation time. The entity ∆H pp and T ss are correlated by equation,  \n       \n     , where,   , has magnitude 1.760859 X 1011 / sec–T, for electrons. The \ncomputed values of T ss are enlisted in Table III. To describe briefly, for Graphene, the values \nof T ss are varied from 8.82 to 6.94 ps ( <Tss>T = 8.32 ± 0.58 ps), whereas, they are in the range \n7.82 to 7.37 ps ( <Tss>T = 8.93 ± 0.80 ps), for N –Graphene. For GNCs, the values are 7 .71 to \n8.47 ps ( <Tss>T = 8.08 ± 0.265 ps), whereas, for N –GNCs , they are increased to a range 14.62 \nto 13.77 ps ( <Tss>T = 13.10 ± 0.8 1 ps). Tss is almost identical for Graphene and GNCs, and \nafter doping nitrogen, there is slight increase for N –Graphene. In contrast, after doping \nnitrogen in GNCs,  Tss is increased to a high value. This indicate s that, spin–spin momentum \nis retarded significantly for N –GNCs.  \nThe principal parameter governing spintronic usability is spin –lattice relaxation time, T sl, \nwhich quantifies variation of non –thermal spin state around the lattice. The magnitude of T sl \nis computed using relation:  \n             \n              . The variation in T sl as a function of \ntemperature for the measured carbon system is shown in Figure 2. For spintronic \napplications, theoretical estimate for T sl is 1–100 ns, whereas, experimental spin –transport \nmeasurements on Graphene showed that T sl is as short as 60 –150 ps ec.15 FIG. 2 shows, \nvariations in the magnitude of T sl as a function of temperature for the systems under \ninvestigation. For Graphene, T sl varies from 6.83 to 22.76 ns, whereas, for N –Graphene \nobserved variation is comparatively marginal from 6.06 to 24.23  ns. The magnitude of T sl is \nfound to varied from 5.97 to 23.14 ns, for GNCs and 11.31 to 41.00 ns, for N –GNCs. For  \n \n12 \n Graphene, N –Graphene and GNCs the values are closer and variations are almost identical.  \nIn fact, for these systems, after 300K the value of T sl is almost independent in increase of the \ntemperature . Basically, electron spin relax by transferring energy selectively to those lattice \nmodes with which they resonate. Thus, the resonant modes are on talking term with the spins. \nAnd one can modify their cross talk  by introducing break in symmetry inversion \n(i.e.disorder)  or adatom in the sp2 carbon network.  However, a t higher temperature, the two \nspins levels seems to be somewhat broadened  for the systems other than N –GNCs , resulting  a \nsaturation behaviour . In principle, the orbital and the spin angular moment have been \nconsidered separately; it is important to know the extent to which these are coupled. As a first \napproximation the two may be considered independently later introducin g a small correction \nto account for the so called spin –orbit (SO) interaction.  For pure, radical free carbon system \nhave, essentially, zero orbital angular momentum ; the SO interaction is usually very small for \nsuch systems hence for most purposes attentio n may be focused wholly upon the spin angular \nmomentum. However, SO interaction must necessarily be included in a discussion of the ESR \nbehaviour, as presented below.    \nSO interaction is one of the most prominent modes of spin relaxation.11 There are three \nprincipal sources of SO coupling, in Graphene: intrinsic, Bychkov –Rashba (BR, related to \nstructural symmetry break) and ripples (related to inevitable wrinkles/folding edges). \nHowever, it is not possible to estimate contribution of each componen t experimentally. \nTheoretical estimate for intrinsic SOCC ranges 1 µeV –0.2 meV and BR 10 –36 µeV/V/nm.  \nUntil recently, it has been reported that curved carbon surfaces could have zero –spin splitting \nSOCC upto 3.4 meV. [R31] The quantification of SO interact ion could be done using \ncorrelation: ∆g = α    \n   , where α, is band structure dependent constant ≈ 1, and L i is spin–  \n \n13 \n  \nFIG. 2. Variation in spin –lattice relaxation time, T sl, over the measured temperature range for \n(a) N–GNCs, (b) N –Graphene, (c) Graphene, and (d) GNCs. For N–GNCs, almost linear \nchange in T sl is observed from ~ 11 to 41 ns, whereas,  for other systems ~ 5 to 2 5 ns.  \norbit coupling constant (SOCC). From the estimated value of ∆g (in Table I.) and \nobtaining the magnitude of – bandwidth, ∆, one can estimate L i. A correlation of ESR \nline–width, ∆Hpp, and optical bandwidth , ∆, has already been noted. 27-28 The values of, ∆, for \nGraphene, N –Graphene, GNCs , and N–GNCs , is respectively, 4.18, 5.27, 4.06, and 3.79 eV. \nTable II shows variations in the magnitude of L i, over the measured temperature range, for \nour systems. The magnitude of L i is found to be three orders of magnitude larger than \nmeasured previously.29  \n \n \n14 \n  \nTABLE II. Indicating magnitudes of spin –orbit coupling constant, L i and spin relaxation rate, \nspin, computed over the measured temperature range.  The last row indicate, <A> T, i.e. \ntemperature averaged value of L i and spin.  \nThe existent knowledge about the SO interaction in graphene is not yet complete.4 \nFrom  the perspectives of solid state qubits construction it is important to understand the  role \nof electronic spin in ordered (Graphene) and disordered (GNCs) sp2 network. Table II show s \nvalue of <L i>T for Graphene is 21.271 ± 0.86 meV , whereas, it found to be increased to 29.63 \n± 0.21 meV for N –Graphene.  This indicates that, the orbital angular momentum contribution \nis increased in Graphene, after nitrogen doping. Comparatively,  for GNCs,  <Li>T is low \n18.49 ± 0.32 meV and reduced furth er to 16.17 ± 0.37 meV , for N–GNCs . The GNCs are \ndisordered sp2 network, in contrast to  ordered Graphene. The observed variations could also \npossibly be attributed to different spin and orbital angular momentum response of nitrogen \nadatom  in ordered and disordered sp2 carbon network leading to different different type of \nexchange interactions.4 Thus, analysis of spin –spin (T ss) and orbital contribution (L i) together \n \n \n15 \n with lattice relaxation (T sl) indicates that spin degrees of freedom seems  to be somewhat \nenhanced in N –GNCs.  For realistic applications, spin system with enhanced degrees of \nfreedom is advantageous to manipulate, flip and toggle the spin density wave. The SO \ninteraction couples electronic states with opposite spin projections in different bands, \ntypically identified as spin –up and spin –down state and could be quantified as spin –flip \nparameter, b.30 The perturbation theory set the limit as b      . In case of Graphene and \nallied systems, the SO interaction couples  and  bands with L i << . FIG. 3 shows \nvariations in spin–flip parameter b over the measured temperature range for our systems.  \nThe value of b, for Graphene, is 4.94 10–3–5.0410–3 and 5.17 10–3–6.0310–3, for N–\nGraphene. For GNCs, it is 4.60 10–3–4.5710–3 and for N –GNCs 4.35 10–3–4.2010–3. The \nvalues are over the measured temperature range.  \n \n \n \n16 \n FIG. 3. Spin –flip parameter, b, obtained experimentally using Elliot –Yafet relation  with \neigen Bloch states,      , for spin–Hamiltonian , where, k and r are wave vectors associated \nwith spin density wave. Profile (a) N–Graphene, (b) Graphene, (c) GNCs , and (d) N –GNCs. \nThe shaded region shows room temperature variations in flipping parameter , b, for our \nsystems  and arrows indicates their opposite behaviour.       \nThe important feature is , b has lower value for GNCs (<b>T = 4.5510–3) compared \nto Graphene (<b>T = 4.5510–3) and reduced further after nitrogen loading  in GNCs(< b>T = \n4.2710–3). In contrast,  <b>T is 5.6310–3 for N–Graphene and value is seems to be \nincreased with respect to <b>T estimat ed for Graphene. For realistic spin device s, b should \nbe conditioned to ~ 2510–3.32  \nFurther , the data obtained from electron spin resonance spectra could principally be \nused to set up spin –Hamiltonian for the system. The spin –Hamiltonian for such system  is \ncomposed of various terms such as electronic energy (H ele), crystal field (H cf,),spin–orbit \n(HSO), spin–spin (H SS), Zeeman energy (HZe), hyperfine structure (HHFI), quadrupole (HQ), and \nnuclear spin energy , (H N), but, the dominant mode s could  be H SO and H HFI for the system s \nunder considerations . For the pure carbon system s, Hele and H cf could be  neglected due to \nabsence of paramagnetic centres, and radical free environment , whereas, H SS and H Ze could \nbe of the same magnitudes and are orders of magnitudes smaller than H SO and hence \nneglected. The H Q and H N could also be neglected due to the select ed temperature region and \nare predominantly operative only in the sub–mK range. We could add contribut ion of HHFI \ndue to its presence obtained the ESR line–width , except in GNCs . Thus, the spin –\nHamiltonian read s as33: \n[                      +             ]           (1) \n[                          +              ]                  (2)  \n \n17 \n Where, a and b are complementary sublattice –period of the honeycomb super lattice, \nand k, r are density wave vectors, respectively, in real and momentum space. The s econd \nterm represents nuclear hyperfine interaction at ith site and could be neglected for GNCs , but, \nexist for other three systems .     \n \nFIG. 4. Estimated momentum relaxation rate,  (in eV) as a function of temperature (in K) \nfor (a) N –GNCs, (b) GNCs, (c) Graphene, and (d) N –Graphene. Arrows indicate entirely \nopposite behaviour for Graphene and GNCs. Open ellipses shows room temperature and low \ntemperature behaviour of  momentum relaxation rate, , for the systems.       \nThe spin relaxation rate, spin, is related to T sl via a relation:              , where,   \n= 6.59    10–16 eV–s and enlisted in Table II . The relation between , spin, and momentum \nrelaxation rate, , is  given by :             \n   \n . FIG. 4  shows variations in as a \n \n \n18 \n function  of temperature for our systems . Our observations for the obtained result  are as \nfollows.  The value of ,, decreases gradually as the temperature increases. F or Graphene,   is \nbetween 3.94 –1.25 meV range, with <   >T = 1.94±0.93 meV. For N –Graphene, GNCs and \nN–GNCs, the range of, , is, respectively, 4.06 –0.82 meV (<   >T = 1.59±1.09 meV), 5.22 –\n1.36 meV  (<  >T = 2.25±1.20 meV),  and 5.87–1.66 meV (<   >T = 2.83±1.39 meV).  This \nindicates that, magnitude of       GNCs is greater than that for Graphene. Afte r doping \nnitrogen,  is increased for GNCs,  whereas, it is reduced for N–Graphene. At room \ntemperature, the values of   are almost same for GNCs and N –GNCs  (indicated by open \nellipse) . This is indicat ive of nitrogen loading has no effect on the change in momentum \nrelaxation rate in GNCs system, however, scenario is opposite, for Graphene  (open ellipse) . \nAt the lowest measured temperature, exact opposite behaviour has been observed (open \nellipse). The presence of nitrogen adatom adsorbed and its position in  sp2 network ha s \nprofound effect on the spin configurations of these systems and consequently reflected in the  \nvariations in  momentum relaxation rate, . The spin has functional dependence on pseudo \nchemical potential,     i.e. local bonding environment and is given by relation ( 3) : \n                             \n              \n    +  \n       \n             \n                     (3) \nWhere, µ, is chemical  potential and D, continuum cut off parameter is ≈ 3.00 eV. \nSince µ =    , then the  magnitude of    is estimated for all samples . The pseudo chemical \npotential,     is also connected with  the expression of density of states (DOS)  (  , spin). The \nDOS  (  , spin) is measured in units of stats/eV –atom and given by:   \n (  , spin) =                  \n          (4)     \n           Where, A c =           \n   per atoms is unit cell for graphene and    is Fermi velocity of \ncarriers at Fermi level. Thus, t he value of  (  , spin) is computed using the estimated values  \n \n19 \n of    , spin, as above. It is noteworthy that, for the intrinsic component, when    » , \n(intrinsic) ≈    \n   , which is an Elliot –Yafet–like result.30 Where, (intrinsic) is intrinsic \nmomentum relaxation rate. Further, the ripple relaxation contribution becomes dominant only \nwhen   »    . The comparison of computed value s of  and    indicate that, intrinsic, and BR \ncould be the operative coupling channels  for our systems . The interdependence of spin \ndynamic parameters has been studied . FIG. 5 shows (i) variations in     with temperature, (ii) \nDOS variations vs , (iii)  Tsl dependence on   , and (iv)  Tss evolution with DOS. The spin \norbit  coupling allows transition between states of the –band near the Dirac point with the \nstates from –band at the same point and opens –up a gap in the energy dispersion . These \ntransition s imply a change of the spin degree of freedom, i.e. b, the  spin–flip process  \nresult ing in to the variations in  pseudo chemical potential,    of our systems .     \n \n \n \n20 \n FIG. 5. (i)  Pseudo chemical potential,      computed for the system, (ii) variations in density of \nstates,, as a function of momentum relaxation rate, , (iii) change in spin-lattice relaxation \ntime, Tsl with pseudo chemical potential,   , and (iv) change in spin-spin relaxation rate, Tss, vs \ndensity of states, . The data is plotted for temperature range 123 –473 K. And the arrow \nindicate s contrast behaviour of GNCs and Graphene, after nitrogen loading.  \nPanel (i) in FIG. 5 shows  that, value of     , for Graphene , ranges 1.9403 to 2.0383 eV \nwith <    >T = 2.0125±0.0337 eV. They are  at higher side, for GNCs , which ranges 1.8405 to \n1.9804 eV  with <    >T = 1.9336±0.04367 eV . This is indicative of midgap  opening , due to \nweaker <Li>T for GNCs , compared with the magnitude of <L i>T for Graphene . For GNCs, \nafter nitrogen doping,    is reduce d further and ranges 1.6823 to 1.8349 eV with <    >T = \n1.7927 ±0.0506  eV. This  shows th e subsequent broadening of  midgap , as reflected in further \nreduction in the obtained <Li>T for N–GNCs. In contrast, increase in    , for N–Graphene,  \n(range : 2.4845 to 2.6015 eV and <    >T = 2.5737 ±0.03958  eV) indicate s the lowering of the \nmidgap with higher  value  of <L i>T. The DOS at Fermi level plays crucial role, due the \ninterlink between    , and  via . From panel (ii) one can see that, the highest DOS are \navailable for N –Graphene compared to other  system s. The value of  ranges from 0.0958 to \n0.10037 stat/eV –atom with mean value < >T = 0.09929±0.00153 stat/eV –atom . The \nobserved increase in DOS could be  due to donation of loan pair of electrons from nitrogen  to \nGraphene. In case of GNCs, each standalone vacancy generates  three under coordinated \ncarbon atoms and four dangling electrons  keeping  sp2 state invariant . This gives rise to four \nenergy levels, three lie near Fermi energy level and fourth lie above  the Fermi level . The \nlowest  is fully occupied with two of electrons from dangling bonds keeping third semi filled \nand upper two fully empty. As a result, one can see the magnitude of DOS is smaller and \nranges 0.071 to 0.0764 stat/eV –atom with < >T = 0.0748  ± 0.00168 stat/eV –atom, for GNCs , \nin comparison with Graphene (range: 0.07486 to 0.07894 stat/eV –atom and < >T = 0.0748 ±  \n \n21 \n 0.0013 stat/eV –atom) and its nitrogen derivate. Nitrogen loading further reduces DOS in the \nrange 0.0649 to 0.07049 stat/eV –atom, wit h mean  <>T = 0.06913 ± 0.00193 stat/eV –atom \nfor N–GNCs. The observed decrease could be attributed to exchange interaction that \ndegenerate levels below the Fermi level.  Since the spin –polarized levels are derived from the \ndangling electrons, the spin densi ties could found to be localized on the undercoordinated \natoms. With the assumption of standalone vacancy per three undercoordinated carbon atoms \none can found lowest magnitude of effective magnetic moment, µ eff for GNCs ( displayed in \nTable III.).   \n \nFIG. 6. (i) Change in momentum relaxation rate, , with pseudo chemical potential ,     arrow s \nindicate how t wo systems behaves after nitrogen loading . The data is plotted for the measured \ntemperature range. , (ii) computed spin density (s/m3) for Graphene and GNCs and their \nnitrogen derivative s as a function of temperature , T, (iii) degenerated Pauli spin \n \n \n22 \n susceptibility, spin, for spin   \n  particles in the system as a function of  T–1, and (iv) variations \nin spin*T as a function of T , for the systems , arrow s indicate magnitude of change is large for \nGNCs system compared to Graphene .       \nFurther, lattice–electron, and electron –electron interactions play important role in \nmagnetization of honeycomb carbon.8 Panel (iii) and (iv) shows variations between T sl vs     \nand  Tss vs . For N–GNCs, small variations in     (1.60 –1.83 eV) caused larger changes in T sl \n(~ 11–41 ns). The variation seems to be higher compared to other systems. Two prominent \nfeatures has been observed for the plot of Tss vs . First, increase in  for Graphene after \nnitrogen loading with marginal increase in T ss, and second increase in T ss with marginal \ndecrease in DOS. However, the observed variations are consistent with discussions made \nabove. FIG. 6 (i) shows variations in  as a function of      The quantity  has     dependence. \nOver the observed variations , the computed curves are for total contribution of L i i.e intrinsic, \nBR–type, ripple, and extrinsic (in case of doped systems). However, the extrinsic contribution \nis added in opposite way for both the system. The ripple relaxation contribution depends on  \nonly when µ« , where it resembles an Eilliot –Yafet rel ation        ln (D / ), where D  \n3 eV, continuum cut-off parameter . However, in our case  ranges in meV, whereas, \ncomputed     is in the range of  eV. Thus, one can neglect ripple contribution. Further,  the \nperformance of ESR is given by limit of detection  of signal i.e.  the number concentration of \nnon–degenerated spin     particles measured over the temperature . The limit of detection \ngives a signal –to–noise ratio, S/N, and for our system S/N is 20 for 0.05 mT line –width  and \nthe value limit of detection for our system is 5  109 spins/mT. F urther, f or the ESR system, \nthe gain of the band amplifier and phase –sensitive detector was tuned much smaller than the \nmeasured line –width. As a result, the recorded line –shape transformed into  the first \nderivative,       , of the absorption line,Y. By measuring the a rea, A, under the obtained  \n \n23 \n resonance absorption derivative,      , one can  compute the number density of unpaired \nspins in the samples. The A , usually , is proportional to the spin concentrations (measured in \ns/m3) in the samples.  From FIG. 1 , we compute d s/m3 for the systems under investigation, \nusing relation[R34]: \n                    \n                                                              (5) \nWhere,   ,    ,and     is the jth component of absorption line –width, its first derivative, and \ncorresponding field, respectively. FIG. 6(ii) shows variations in spin density as a function of \ntemperature for our systems , moreover , panel (iv) is the complementary data, for defect \nconcentra tion, obtained from the susceptibility,       calculations of the samples. One can \nsee that, the concentration of spins for GNCs is  varied  from 8.771025 to 8. 791025 with \nmean <S c>T = 8.691025 ± 9.491025 s/m3.Whereas, for Graphene , it is in the range \n7.931025–8.571025 with <S c>T = 8.251025 ± 1.801024 s/m3. After nitrogen doping, \nmagnitude of spin concentration ,<S c>T, is decreased drastically to 5.361025 ± 1.151024 \ns/m3, for GNCs.  However, for N –Graphene the change , <S c>T, is marginal 7.6 6  1025 ± 1.98 \n 1024 s/m3. Usually, the localized spins are assigned either to unpaired spins of the free \nradical molecules or defects such as dangling bond, trapped carriers, and vacancies. The exact \nnumber could be estimated in the form of Pauli susceptibility of such spins and is given by  \nfollowing relation30:  \n      =          \n            (5) \nWhere,    is permeability of vacuum,    is Bhor magneton, N/ V is spin \nconcentration , kB, Boltzmann’s constant, and temperature, T. The variations in       as a \nfunction of T–1 is shown in panel (iii) of FIG.6 . The temperature dependence of total       \nhas two components one      i.e. orbital diamagnetic response and    due to  itinerant nature \nof electrons and the Pauli paramagnetism due to their spin. At low temperature,        \n \n24 \n originates from the DOS. The value of      , for N–GNCs,  varies from 3.05  10–6 A–m2/mT \nto 8.42  10–7 A–m2/mT, over the measured temperature range. For N–GNCs, a large \ndeviation has been observed from the linearity , which could be attributed to \nantiferromagnetic ordering in GNCs, after doping nitrogen. The graphene honeycomb \nsuperlattice consists of two complemen tary sublattices A and B and segregation of \nconduction electron spin density, nc, over the sublattices generates (i) ferromagnetic (     \n      ), (ii) ferrimagnetic       0,       ), and (iii) antiferromagnetic (            ) \nordering .[R35] The observed behaviour of GNCs indicate that, on each sublattice holes and \nelectrons could get segregates. For, Graphene,      , is 5.05–1.3910–6 A–m2/mT, where as \nfor N–Graphene, and GNCs its value is, respectively, 4.29–1.3610–6 A–m2/mT and 2.74–\n1.3810–6 A–m2/mT.  The vales are quoted over the measured temperature range. Frequently, \nthe Curie law is also plotted in modified form T as a function of T. Panel (iv) shows \nvariations in T as a function of T. The extrapolated intersection of each  profile in panel (iv) \nindicates its proportionality to the number of paramagnetic species. For GNCs the value of \nthe intersection is highest, whereas, for N –GNCs it has a lowest magnitude. The effective \nmagnetic moment, µ eff, is computed using values of obtained       for the samples. The Bohr \nMagneton,   , is used as a basic physical constant for magnetic moment of electrons per \nsingle atom. The    expresses an intrinsic electron magnetic dipole moment which is ~ 1 \nBohr Magneton for electron.36-37 The study of   of single carbon atom is critical because it \nverify how many    single carbon atom contains. The effective magnetic moment,      is \ncomputed by use of relation38:                          . The computed values are \nenlisted in Table III.  \nBy and large, our      calculation shows that, for a cluster of 1000 carbon atoms \nthere are ~ 72 interacting spins for Graphene and after nitrogen doping the value remains  \n \n25 \n almost same ~ 69. For GNCs, this  number is somewhat high ~ 74 and reduced substantially \nto ~ 58 spins for N –GNCs, per thousand carbon atoms. This shows number of electrons \nparticipating in spin interaction is reduced significantly, for GNCs, after nitrogen doping. \nThis provides enhanced spin degrees of freedom to spin bath.  \n \nTABLE III: Computed values of spin –spin relaxation time,T ss, and effective magnetic \nmoment,    , for Graphene, GNCs and their nitrogen doped derivatives. The last row \nindicate, <A> T, i.e. temperature averaged valu e of T ss and     .  \nThus, what follows in, from the analyses of spin dynamics? The Graphene and GNCs \nsystems are fundamentally working in opposite way. Another question; which system is well \nsuited for solid state qubit construction?;  seems to be N –GNCs.  The unit cell of pure \nGraphene is described by two inequivalent triangular sublattices (A and B)  with two \nindependent k –points K and K ′. They are  having two inequivalent corners of Brillouin zone \nnear the Fermi level which crosses the –band. This provides  an exotic fourfold degeneracy \n \n \n26 \n of the low energy spin degenerate states described by two sets of two dimensional chiral \nspinors which follows Dirac –Wyal equation . 39 It describe s the electronic state of the system \nwhere K and K ′ near t he Fermi level is  located. Hence,  pure Graphene has one electron per \ncarbon atom in the –band so band below the Fermi level is full (electron like state) and band \nabove the Fermi level is empty (hole like states).  Thus, from the perspective of spin degrees \nof freedom offered to electron, pure Graphene seems to be tight system. But, o ne could \nexpect more interesting and diversified lattice environment for spin degrees of freedom by \nGNCs  to be offered to electron . Although, GNCs form two dimensional ar ray of six member \ncarbon ring ; in contrast to Graphene, these arrays are in the benzenoid carbophane (sp2) \nconnected to sp3 zone via 1,4 hexadine type confirmation.  \n \nFIG. 7. Schematic depiction of spin splitting around Fermi level of Graphene , GNCs  \nand their nitrogen derivates . For GNCs coupled level split to three levels.  The values quoted \nare effective magnetic moment, µ eff, in terms of µ B for their temperature averaged \nmagnitudes.   \n \n \n27 \n Thus, GNCs consists of conformational and vacancy disorder. Vacan cies also plays \nimportant role in disordered system. Once the vacancy is created, the structure is relaxed \nleaving three under coordinated carbon atoms. These atoms get displaced away from their \nequilibrium position and move towards their immediate neighbo ur uniformly along the bonds \nand each of them will have dangling bonds which are still in the sp2 state. The four dangling \nbonds gives rise to four energy level three lie near the Fermi energy and fourth lie below the \nFermi energy level fully occupied with  two of electrons from dangling bond. The remaining \nthree orbital split with the polarized level for spin up and spin down  electrons. The remaining \ntwo electrons will now occupy the spin up energy levels giving rise to a net magnetic \nmoment. 40 Thus, magnetism, fundamentally, requires an unpaired electron with lattice atom \nthat carries a net, non –zero magnetic moment. These moments can be coupled via exchange \ninteraction with other electrons. The degrees  of freedom offered to electron by such s ystem \nwill be more along the interlayer sp2+sp3 bonded clusters. For N-GNCs,  the spin splitting \nobserved to be comparably high . However, the observed decrease in       in ca se of N–\nGNCs shows that the moments, coupled via exchange interactions, are reduced. Electron –\nelectron and lattice –electron interactions play important role in magnetization of honeycomb \ncarbon.31 [R41] 32 Nitrogen donates  one of the electrons from its loan pair. The donation is \nexchange based and exchange takes place with   \n  empt y p z orbital. The itinerant –electron gets \nbound and spin coordinated with localized–π–electron at carbon lattice. The coupling takes \nplace in the screen–columbic repulsive environment. During exchange –coupling, the \nperturbation offered by uncoordinated iti nerant–electron, to empty p z orbital,  could distort σ–\nbond sensitively. The schematic of spin splitting is shown in F IG. 7. Schematics shows  spin \nexchange coupling of nitrogen electron with hole of carbon system . As a result, the strength \nof spin–orbit coupling and carrier concentration at Fermi level could be affected. The present  \n \n28 \n analysis  indicates that  local bond configuration of GNCs after nitrogen d oping seems to be \nfavourable for setting up the qubits.   \nCON CLUSIONS  \nTo conclude, what we have studied is the spin dynamics, in Graphene, GNCs, and \ntheir nitrogen derivates, over 123 to 473 K, using electron spin resonance technique. \nBasically, we have four types of carbon lattice environments: (i) system with inversion \nsymmetry (pure sp2 carbon, Graphene), and (ii) broken inversion symmetry and \nheterostructure (disordered sp2 network, GNCs), (iii) extrinsic nitrogen adatom. The spin \ndynamics in both the systems are observed to be behaved, somewhat, oppositely a fter \nnitrogen doping. Analysis of line width and shape indicated that, for Graphene, line width is \nfound to be varied, asymmetrically, from 1.1027 ± 0.0071 (123K) to 1.2311 ± 0.0063 mT \n(473K), with increase in the degree of asymmetry for N –Graphene and ran ging between \n1.1495 ± 0.0066 (123K) to 1.3217 ± 0.0035 mT (473K). The line –widths are found smaller \nfor GNCs (1.2964±0.0083 to 1.2669±0.0091) and narrowed down further for N –GNCs \n(0.7091±0.0049 to 0.6710±0.0038). The observed variations are attributed to i nvolvement of \nmore than one relaxation time, anisotropic spin orientation of charge inhomogeneities, and \nhyperfine interactions. For Graphene, <∆H pp >T is 0.68678 ± 0.05321 mT, whereas, for N –\nGraphene, GNCs, and N –GNCs it is, respectively, 0.64131 ± 0.0635 6, 0.70428 ± 0.02990, \nand 0.43470 ± 0.0268 (in mT). The value of <∆g> T for Graphene is 0.00516 ± 2.05 10–4 and \n0.00563 ± 4.015 10–4, for N–Graphene. For GNCs, <∆g> T is 0.00455 ± 7.86 10–5 and \n0.00427 ± 9.48 10–5, for N–GNCs. The small values of linewidths,  and analysis of g –factor \nreveals anisotropic deviation for the systems which indicate variations in the contributions of \norbital angular momentum and spin -spin relaxation momentum. The spin -spin relaxation time \n(Tss), is <T ss>T is 8.32 ± 0.58 ps, for Grap hene, whereas, for N –Graphene, 8.93 ± 0.80 ps, for \nGNCs, 8.08 ± 0.265 ps, and increased, for N –GNCs, upto 13.10 ± 0.81 ps. The spin -lattice  \n \n29 \n relaxation time (T sl), for Graphene, is varied from 6.83 to 22.76 ns, whereas, for N –Graphene \nfrom 6.06 to 24.23 ns,  for GNCs, 5.97 to 23.14 ns and 11.31 to 41.00 ns, for N –GNCs. The \nextent of orbital and spin angular moment coupling have been studied by estimating spin –\norbit coupling constant (L i). The estimated <L i>T is 18.49 ± 0.32 meV  for GNCs and reduced \nfurther, to 16.17 ± 0.37 meV, after nitrogen doping; however, in contrast, for Graphene its \nvalue is high (21.271 ± 0.86 meV) and increased upto 29.63 ± 0.21 meV, for N –Graphene. \nThis is primary indication that, N -GNCs is a system wit h enhanced spin degrees of freedom. \nThe computed value of spin flip parameter ( b) is 4.94 10–3–5.0410–3, for Graphene, and \n5.1710–3–6.0310–3, for N–Graphene. For GNCs, it is in low range 4.60 10–3–4.5710–3 \nand from realistic view point, for N –GNCs, 4 .3510–3–4.2010–3. The spin relaxation rate \n(spin) is observed to be decreased gradually over 123 to 473 K. For Graphene, < spin>T is \n4.98±2.19×10–8 eV, for N -Graphene, 4.75±2.59×10–8 eV, for GNCs, 5.19±2.63×10–8 eV, for \nN–GNCs, 3.17±1.45×10–8 eV. This indicates that, spin rates are quiet retarded for N –GNCs. \nThe momentum relaxation rate, <   >T is 1.94±0.93 meV, for Graphene, whereas, for N –\nGraphene, GNCs and N –GNCs, respectively, 1.59±1.09 meV, 2.25±1.20 meV, and \n2.83±1.39 meV. The spin has functional  dependence on pseudo chemical potential (   ) with \nthe condition    » . The computed value of <    >T for Graphene is 2.0125±0.0337 eV, \n1.9336±0.04367 eV for GNCs, 1.7927±0.0506 eV, for N –GNCs and 2.5737±0.03958 eV, for \nN–Graphene. The observed changes are  attributed to variations in the midgap states. This has \neffect on density of states, the value of < >T is0.0748 ± 0.00168 states/eV –atom for GNCs, \nhowever, for N –GNCs, it is reduced to 0.06913 ± 0.00193 states/eV –atom. For Graphene, it \nis 0.07764±0.0013 states/eV–atom  and enhanced to 0.09929±0.00193 states/eV –atom for N –\nGraphene. The spin concentration <S c>T for Graphene is found to be lower 8.25 1025 ± \n1.801024 s/m3, with ~ 70 interacting spins per thousand carbon atoms and reduced \nmarginally to 7.66 1025±1.981024 s/m3, with almost same number of interacting spins. For  \n \n30 \n GNCs, the variations are drastic, it indicate that <S c>T is 8.691025 ± 9.491025 s/m3 with ~ \n74 interacting spins per thousand carbon atoms, and with nitrogen doping it reduced to \n5.361025 ± 1.151024 s/m3  with ~ 57 interacting spins per same number of carbon atoms. \nThe spin susceptibility, for N –GNCs, is deviated largely from linearity which leads us to a \nconclusion that antiferromagnetism could be the operative channel in N –GNCs.  . For, \nGraphene,      , is 5.05–1.3910–6 A–m2/mT, where as for N –Graphene, and GNCs its value \nis, respectively, 4.29–1.3610–6 A–m2/mT and 2.74–1.3810–6 A–m2/mT.  For realistic qubit  \ndesign, transport parameters such as spin –orbit coupling constant, spin –lattice relaxation \ntime, and spin –flip process should be optimum with optimum spin dynamics, at room \ntemperature. Analysis gives us the clue to design qubit and it seems that N –GNCs c ould be \nthe well suited candidate for this.  \n \nACKNOWLEDGMENT  \nAK and PSA are thankful to the Ministry of Defence, Govt. of India for giving grant through \nER–IPR, DIAT –DRDO nanoproject.     \n   \n \n31 \n REFERENCES  \n1 Neumann, P.; Kolesov, R.; Naydenov, B.; Beck, J.; Rempp, F.; Steiner, M.; Jacques, V.; \nBalasubramanian, G.; Markham, M. L.; Twitchen, D.J.; Pezzagna S.; Meijer, J.; Twamley, J.; \nJelezko, F.; Wrachtrup, J. Quantum Register Based on Coupled Electron Spins in a Room -\nTemperature Solid. Nat.Phys., 2010 , 6, 249 -253. \n2Merali, Z. The Power of Discord. Nature, 2011 , 474, 24 -26. \n3Carter, S.G.; Sweeney, T.M.; Kim, M.; Kim, C.S.; Solenov D.; Ecomomou, S. E.; Reinecke, \nT.L.; Yang, L.; Bracker, A.S.; Gammon, D.; Quantum Control of a Spin Qubit Coupled to a \nPhotonic Cyrsta l Cavity. Nature Photonics, 17 Mar. 2013 DOI 10.1038/NPHOTON.2013.41  \n4 Huerta -Hernando, D.; Guinea, F.; Brataas A. Spin -Orbit Coupling in Curved Graphene, \nFullerenes, Nanotubes, and Nanotube Caps. Arxiv.cond -mat/0606580v3 [cond -mat..mes -hall] \n25 Oct 2006.  \n5Monz, T. et al., Phys. Rev. Lett. 2011 , 106, 130506.  \n6Golovach, V.N.; Khaetskii, A. V.; Loss, D. Phonon -Induced Decay of the Electron Spin in \nQuantum Dots. Phys. Rev. Lett., 2004 , 93, 016601.  \n7Childress, L.; Dutt, M.V.G.; Taylor, J.M.; Zibrov, A.S.; Jrlez ko, F.; Wrachtrup, J.; Hemmer, \nP.R.; Lukin, M.D. Coherent Dynamics of Coupled Electron and Nucler Spin Qubits in \nDiamond. Science, 2006 , 314, 281 -285. \n8Yazyev, O. V. Hyperfine Interactions in Graphene and Related Carbon Nanostructures. arxiv \n: cond -mat/070 2424v3 [cond -mat-mtrl-sci] 18 Sept 2007.  \n9Kane, C.L.; Mele, E.J. Quantum Spin Hall Effect in Graphene. 2005 , 95, 226801.  \n10Bychkov Y.A.; Rashba, E.I. Theory of the Effect of Spin –Orbit Coupling on Magnetic \nResonance in Some Semiconductors   J. Phys. C 1984, 17, 6039.  \n11Simon, F.; Muranyi, F.; Dora, B. Electron Spin Dynamics and Electron Spin Resonance in \nGraphene.xiv; 1004.0210v1 [cond -mat.str -el] 1 Apr 2010    \n \n32 \n 12Gimitra, M.; Konschuh, S.; Ertler C.; Draxl, C.A.; Fabian, J. Band -Structure To pologies of \nGraphene; Spin -Orbit Coupling Effects from First Principles. Phys. Rev. B 2009, 80, 315431 -\n235434.  \n15Bolotin, K.I. et. al., Sol.Stat.Comm. 2008, 146, 351 -xxx.          \n14Karplus, M.; Fraenkel, G.K. name of paper J. Chem. Phys. 1961, 35, 1312 -xxxx. \n15Tombros, N.;  Jozsa .C.;  Popinciuc, T.; Jonkman H.; Bart M.; Wees J. W.  Electronic Spin \nTransport and Spin Precession in Single Graphene Layers at Room Temperature. Nature  \n2007 ,448,571–574. \n16Tombros, N. et al missing authors. Paper name, Phys. Rev. Lett. 2008, 101, 046601 -\n046605.  \n17Katsnelson, M.I.; Novoselov, K.S.; Geim, A.K. Klein Paradox in Graphene. Nature Phys., \n2, 2006, 620 -625.    \n18McClure, J.W. Diamagnetism of Graphite. Phys. Rev. 1956, 104, 666 -671. \n19Peres, N.M.R.; G uinea, F.; Castro Neto A.H. Electronic Properteis of Disordered Two –\nDimensional Carbon. Phys. Rev. B 2006, 73, 125411   \n20Khveshchenko, D.V. Long –Range–Correlated Disorder in Graphene. Phys. Rev. Lett. 2006, \n97, 036802.  \n21Bery, L.; Fertig, H.A. Edge States  and Quantized Hall Effect in Graphene. Phys. Rev. B, \n2006, 73, 195408.  \n22Mccann, E.; Kechedzhi, K.; Falko, V.I.; Suzuura, H.; Ando, T.; Altshuler, B.L. Weak \nLocalization Magnetoresistance in Graphene. Phys. Rev. Lett. 2006, 97 146805.  \n23Zhang, Y.; Talap atra, S.; Kar, S.; Vajtai, R.; Nayak, S. K.; Ajayan P. M. First –Principles \nStudy of Defect –Induced Magnetism in Carbon. Phys. Rev. Lett.  2007 ,99, 107201 –107201.   \n \n33 \n 24Kumar A.; Patil, S.; Joshi, A.; Bhoraskar, V.; Datar, S.; Alegaonkar, P. Mixed Phase, sp2–\nsp3 Bonded and Disordered Few Layer Graphene –like Nanocarbon: Synthesis and \nCharacterizations. Appl. Surf. Sci.  2013 ,271, 86–92. \n25Feldman, B.E.; Martin, J.; Yacoby, A. name of paper Nature Phys, 2009, 5, 889 -xxx   \n26 Electron Spin Resonance, A Comprehensive Treatise on Experimental Technique, Poole, \nC.P. Intersicence Publishers,a division of John Wiley and Sons, NY, 1967.  \n27Barklie, R.C.; Collins, M.; Silva S.R.P. EPR Linewidth Variation, Spin Relaxation Times, \nand Exchange in Amorphous Hydrogenated Carbon. P hys. Rev. B. 2000, 61, 3546 -3554.  \n28Sattel, S.; Robertson J.; Ehrhardt, H.; name of paper, J. Appl. Phys. 1997, 82, 4566 -xxxx       \n29Jozsa C. et. al. name of paper Phys. Rev. B 2009 , 80, 241403 (R).  \n30Ochoa, H.; Castro Neto A. H.; Guinea, F. Elliot –Yafet  Mechanism in Graphene. \narXiv:1107.3386v2 [cond –mat.mtrl–sci] 2011 . \n31Steele, G.A.; Pei, F.; Laird E.A.; Jol, J.M.; Meerwaldt, H.B.; Kouwenhoven, L. P. Large \nSpin-Orbit Coupling in Carbon Nanotubes. Nature Comms. DOI:10.1038/ncomms2584, \n2013.  \n32Weeks, C.; Hu, J.; Alicea, J.; Franz, M.; Wu, R.; arXiv:1104.328v1[cond -mat.mes -hall], \n2011.  \n33Elliott, R. Theory of the Effect of Spin –Orbit Coupling on Magnetic Resonance in Some \nSemiconductors. Phys. Rev.  1954 , 96, 266–272. \n34Electron Spin Resonance Elementary Theory and Practical Applications, J.E. Wertz and J. \nR. Bolton, McGraw Hill, 1972  \n35Yazyev O.V. Emergence of Magnetism in Graphene Materials and Nanostructures. \narXiv:1004.2034v1[cond -mat.mes -hall]12Apr 2010.  \n36Mahajan, A.; Rangwala, A. Electricity and Magnetism, Tata McGraw –Hill, 2007 .  \n \n34 \n 37Niyogi, S.; Bekyarova, E. ; Itkis, M. E.; McWilliams, J. L.;  Hamon, M. A.; Haddon, R. C. \nSolution Properties of Graphite and Graphene. J. Am. Chem. Soc. 2006 ,128,7720 –7721.  \n38Aubin, S.M.; Ruiz, D.; Rumberger, E.; Sun, Z,; Albela, B.; Wemple, M.W.; Dilley, N.R.; \nRibas, J.; Maple, M.B.; Christou, G.; Hendrickson, D.N. Resonant Magnetization Tunneling \nin Single -Molecule Magnets. Mo. Cryst. And Liq. Cryst. 1999 , 335, 371 -389. \n39 Talapatra, S.; Ganeshan, P.G.; Kim,T., Vajtai, R.; Huang, M.; Shima, M.; Ramanath, G.; \nShrivastva, D.; Deevi, S.C.; Ajayan, P. M. Temperature Chaos, Rejuvenation, and Memory in \nMigdal–Kadanoff Spin Glasses. Phys. Rev. Letts.  2003 ,91, 097201 –097204.  \n40Jozsa, C .; Maassen, T.; Popinciuc; M.; Zomer, P.J.; Vrligura, A.; Jonkman, H.T.; Van \nWess; B.J. Linear Scattering between Momentum - and Spin Scattering in Graphene, \nArxiv:0910.1054v2[cond -mat-mes-hall] 28 Nov 2009.  \n41Anderson S. H.; Chung D. D. L. Exfoliation of I ntercalated Graphite. Carbon  1984 , 22, \n253–258. \n42Baughman, R.H.; Hammond W.B. Carbophanes: a New Family of Crystalline Phases of \nCarbon . J. Chem. Phys. 1987 , 87,6687 -6693.  \n43Yin M.T. Cohen M.L. Phys. Rev. Lett. 1983 , 50, 2006 -2010.  "    },    {        "title": "1005.5330v2.Spinning_compact_binary_inspiral_II__Conservative_angular_dynamics.pdf",        "content": "arXiv:1005.5330v2  [gr-qc]  20 Nov 2010Spinning compact binary inspiral II: Conservative angular dynamics\nL´ aszl´ o´Arp´ ad Gergely1,2⋆\n1Department of Theoretical Physics, University of Szeged,\nTisza Lajos krt 84-86, Szeged 6720, Hungary\n2Department of Experimental Physics,\nUniversity of Szeged, D´ om t´ er 9, Szeged 6720, Hungary\n⋆E-mail: gergely@physx.u-szeged.hu\n(Dated: September 10, 2018)\nWe establish the evolution equations of the set of independe nt variables characterizing the 2PN\nrigorous conservative dynamics of a spinning compact binar y, with the inclusion of the leading\norder spin-orbit, spin-spin and mass quadrupole - mass mono pole effects, for generic (noncircular,\nnonspherical) orbits. More specifically, we give a closed system of first order ordinary differential\nequations for the orbital elements of the osculating ellipse and for th e angles characterizing the spin\norientations with respect to the osculating orbit.\nWe also prove that (i) the relative angle of the spins stays co nstant for equal mass black holes,\nirrespective of their orientation, and (ii) the special con figuration of equal mass black holes with\nequal, but antialigned spins, both laying in the plane of mot ion (leading to the largest recoil found in\nnumerical simulations) is preserved at 2PN level of accurac y, with leading order spin-orbit, spin-spin\nand mass quadrupolar contributions included.\nI. INTRODUCTION\nCompact binaries composed of neutron stars or stellar size black ho les are among the most likely sources to emit\ngravitational waves in the frequency range of the Earth-based g ravitational wave detectors LIGO and Virgo [1].\nSupermassive black holes in the mass range of 3 ×106÷3×109solar masses reside in the centers of galaxies and\nfollowing the merger of their host galaxies, they also merge. In the p rocess they create powerful gravitational waves,\ndetectable in the lower mass range by the space mission LISA [2].\nBy definition the inspiralis the regime of the orbital evolution, during which the post-Newton ian (PN) parameter\nε=Gm/c2r≈(v/c)2(wherem≡m1+m2is the total mass, randvthe orbital separation and relative velocity\nof the binary) is small and where the leading order dissipation is due to the gravitational waves. As two galaxies\nmerge, their supermassive black holes are subject to both gravita tional radiation and dynamical friction. The former\novercomesthe latter at about εin= 10−3(the actual number only weakly depends on both the stellar distribu tion and\nmass [3]). During the inspiral which follows, the parameter εincreases. When εapproaches its value at the innermost\nstable orbit, the PN description becomes increasinglynonaccurate , therefore the subsequent plungeis better described\nby numerical evolutions, or as an alternative, by expressions trac ed back to the PN approach, arising either from the\neffective one-body model, calibrated to numerical relativity simulatio ns [4] or from a phenomenological transition\nphase, with coefficients again calibrated by comparison with specific, numerically generated waveforms [5]. Finally,\ntheringdown follows, when the newly formed black hole radiates away its physical c haracteristics, with the exception\nof mass, spin and possibly electric charge (for a review of quasinorm al modes of black holes see Ref. [6]).\nThe spin and quadrupole moment of the supermassive black hole at th e Galactic center can be measured via\nastrometricmonitoringofstarsorbitingatmilliparsecdistances[7], andthiscanalsobeatest ofthegeneralrelativistic\nno-hair theorem.\nThe spin affects the horizon of the black holes, therefore those ob servations on black holes which indicate the size of\nthe horizon, will also lead to indirect spin magnitude estimates. (Estimating the quadrupole moment from observing\na two-dimensional projection of the horizon would be less straigtfo rward.) Both stellar size and supermassive black\nholes can have accretion disks and jets in their active periods. When ever these observations are connected to the\npresence of a jet and the direction of it can be identified (projectio n effects may again obstruct this), we also obtain\ninformation on the black hole spin axis, assuming it is aligned with with the symmetry axis of the magnetic field a nd\nhence the jet direction. Jets from rotating black holes have been s hown to be stable [8]. Spin direction can be also\ninferred from observations on the radiation of the accretion disk.\nSuch observational spin estimates can be made at least by four met hods:\ni) Reverberation mapping of the observed optical/X-ray lines (highly excited Mg, O, C) in active galaxies to\ndetermine the radius and velocity pattern of the Broad Line Region. This depends on the metric, decreasing with\nincreasing spin. From such considerations the mass, spin and spin or ientation of the black holes can be estimated [9].\nIn particular, information on the spin direction of the central black hole of the Seyfert galaxy Mrk 110 was obtained\nby estimating the central black hole mass in two independent ways. F irst, assuming that the broad emission lines are\ngenerated in gas clouds orbiting within an accretion disk, the mass co uld be determined as function of the inclination2\nangle of the accretion disk. Secondly, detecting the gravitational redshifted emission in the variable fraction of the\nbroad emission lines, a central black hole mass, which is independent o n the orientation of the accretion disk could\nbe deduced [10].\nii) There is a strong effort towards imaging with millimeter Very Long Bas eline Interferometry (VLBI) the event\nhorizons of Sagittarius A* (SgrA*) and Virgo A (M87), which again de pend on the spin. For SgrA*, the radio source\nat the center of our Milky Way, millimeter and infrared observations r equire the existence of a horizon [11]. Analysing\nthe peaks of the power density spectra in the light curves of X-ray flares from the Galactic Center black hole the\nmass and spin were inferred [12]. A compact emission region (bright sp ot) in a circular orbit and the lightcurves of\nits observed flux and polarisation depend on the mass and spin of the black hole. The emitted polarisation fraction\nis polarised orthogonally to the spin axis of the black hole [13]. Unlike Sgr A*, M87 exhibits a powerful radio jet,\nallowing future VLBI data to constrain the size of the jet footprint , the jet collimation rate, and the black hole spin\n[14].\niii) The size of the jet launching region in Active Galactic Nuclei (AGN) is d etermined by the Blandford-Znajek\neffect, which in turn depends on the spin [15]. Measurement of the dia meter of the jet base (e.g. in M87) gives\nevidence for small sizes, regarded as signature of a large spin [16].\niv) The low-energy cutoff in the energetic electron spectra of the j ets suggested by the radio spectra [17] is con-\nveniently explained by the pion decay resulting from proton-proton collisions [18]. The latter mechanism require\na relativistic temperature in the accretion disk near the foot of the jet, which translates to the central black hole\nspinning extremely fast [19].\nSome of these methods will certainly also work for stellar size black ho les due to scale invariance arguments in\naccretion phenomena. The jet/disk geometry has been constrain ed for the stellar black holes XTE J1118+480 and\nGX 339-4 [20].\nFrom all these observations we conclude that it is necessary to inclu de the spin and quadrupole moment of black\nholes when modeling their binary systems.\nInthispaperweinvestigatethe2PNrigorousconservativedynamic sduringtheinspiralofaspinningcompactbinary\nsystem, by including leading orderspin-orbit (SO), spin-spin (SS) an d mass quadrupole - massmonopole (QM) effects,\nfor generic (noncircular, nonspherical) orbits. Because of these interactions the spins undergo a precessional motion\n[21]-[22]. Various aspects related to the leading order contribution t o both the conservative and dissipative part of\nthe dynamics due to the SO interaction were discussed in Refs. [23]- [25], while the corrections represented by the\nSS coupling in Refs. [25]-[26], and by the QM coupling in Refs. [27]-[29]. Th e radial motion under the Newton-\nWigner-Pryce spin supplementary condition [30], with all these contr ibutions included is fully solved in Ref. [31].\nThe Hamiltonian approach including spins has been also widely discussed [32]. Based on numerical work, empirical\nformulasfor the final spin have been proposedin Refs. [33]. Zoom-w hirlorbits, which were known to exist for particles\norbiting Kerr black holes [34], also appear in the framework of the PN f ormalism [35], their likeliness increasing with\nthe spin [36]. Gravitational wave emission can lead to a spin-flip [37]-[38] in X-shaped radio galaxies [37], [39]. It is\na more recent result that for mass ratios 0 .3÷0.03 the combined effect of SO precession and gravitational radiation\nwill result in a substantial spin-flip already in the inspiral phase [3], [40 ].\nWith the spins and mass quadrupole moments included, the number of variables in the configuration space increases\ndrastically. In Ref. [41] a minimal and conveniently chosen set of independent variabl esfor such a system was\nestablished. Notational correspondence of some of these variab les with quantities employed in Refs. [42]-[44], where\nthe dynamics of spinning compact binaries (without mass quadrupola r contributions) has been also discussed, is\nestablished in Appendix A. Beside the masses m1, the number of independent variables characterizing the total an d\norbital angular momenta ( JandL, respectively) and spins Siwas shown to be 6, chosen either as\n(a) 3 angles (spanned by the Newtonian orbital angular momentum LNwith the total angular momentum J\nand with the spins Si, denoted as αandκi, respectively) and 3 scales (the normalized magnitudes of the spins\nχi≡(c/G)/parenleftbig\nSi/m2\ni/parenrightbig\nand the magnitude of the total angular momentum J), or equivalently as\n(b) 5 angles and a scale. In this case the dimensionless spin magnitude sχicould be replaced by the azimuthal\nanglesψiof the spins, measured in the plane of motion from a suitably defined n ode lineˆl(the intersection of the\nplanes perpendicular to the total orbital momentum J≡L+S1+S2and to the Newtonian orbital angularmomentum\nLN). The relation between the two sets of variables is given by Eqs. (46 )-(47) of [41].\nIn the present paper we discuss the conservative dynamics of these two sets of independent variables. All lengths\ninvolved(Jandχi)areconstantsofmotion,as Jisconservedto2PNaccuracy[23]andthespinsundergoaprecess ional\nmotion [21]-[22]. Therefore our goal reduces to the study of the dy namics of the angular variables . In the process we\nalso derive the evolutions of the parameters ( ar,er) of the osculating ellipse; of the spin relative angle γspan by ˆS1\nandˆS2; and of the periastron, given by the Laplace-Runge-Lenz vector AN.\nWe start with a discussion of the evolutions under a generic perturb ing force in Sec. II. First we monitor how the\nKeplerian dynamical constants evolve. This allows us to determine bo th the evolution of ( ar,er) and of the vectors\nˆLN,ˆAN, and ascending node vector ˆl. We also determine here how the evolution of the true anomaly param eter3\nχp(measured from ˆANto the actual location ˆ rof the reduced mass particle) is modified by the perturbing force.\nThe specific perturbing force components generated by PN, 2PN, SO, SS and QM effects are listed in Appendix B,\ntogether with the components of the spin precession angular veloc ity.\nEmploying these results, also the spin evolution equations discussed in detail in [41], we are able to derive in Sec.\nIII the evolution of α. Equation (14) and (15) of [41] show that once the evolution of χpandαare established, the\nevolution of the angle ψpmeasured from ˆltoˆAN, and of the angle −φnmeasured from ˆlto an arbitrary inertial axis\nˆ x⊥J(see Fig 1 of [41]) also follow, which complete the characterization of t he evolution of the Euler angles. Then,\nin Sec. IV we derive the evolutions of κi,γ, andψi. With this we fulfill the task of characterizing the evolution of\nthe variables composing the independent sets (a) and (b).\nWe discuss special spin configurations in Sec. V and present our Con cluding Remarks in Sec. VI.\nNotations and conventions. The gravitational constant Gand speed of light care kept in all expressions. For any\nvectorVwe denote its magnitude by Vand its direction by ˆV.\nThe reduced mass is µ≡m1m2/m. We assume that m1≥m2, thus the mass ratio ν≡m2/m1≤1 and the\nsymmetric mass ratio η≡µ/m=ν/(1+ν)2∈[0,0.25].\nThemassquadrupolemomentoriginatesentirelyfromrotation,bein gthereforecharacterizedbyasinglequadrupole-\nmoment scalar Qi=−/parenleftbig\nG2/c4/parenrightbig\nwχ2\nim3\ni, with the parameter w∈(4,8) for neutron stars, depending on their equation\nof state, stiffer equations of state giving larger values of w[27], [45] and w= 1 for rotating black holes [46]. The\nnegative sign arises because the rotating compact object is centr ifugally flattened, becoming an oblate spheroid.\nThe inertial system Kihas the arbitrary inertial xaxisˆ xandˆJas itszaxis. We also define the noninertial systems\nKLandKAwithˆLNas the common zaxis, thexaxes being ˆlandˆAN, respectively. Then the yaxes are ˆ m=ˆLN׈l\nforKLandˆQN=ˆLN׈ANforKA.\nII. EVOLUTIONS IN TERMS OF A GENERIC PERTURBING FORCE\nAlthough there is no notion of gravitational force within general re lativity, in the PN regime the motion of a\ncompact binary can be regarded as a perturbed Keplerian motion, w ith perturbations coming from the difference in\nthe predictions of general relativity with respect to Newtonian gra vity. Therefore one can adopt the terminology of\ncelestial mechanics, regarding the modifications induced by genera l relativity as perturbing forces.\nAny perturbed Keplerian motion is characterized by an acceleration\na=−Gm\nr2ˆ r+∆a. (1)\nWe find convenient to express ∆ ain the basis KAwith basis vectors/braceleftbig\nf(i)/bracerightbig\n= (ˆAN,ˆQN,ˆLN) as\n∆a=3/summationdisplay\ni=1aif(i). (2)\nA. Keplerian dynamical constants\nStarting from the definitions of the Keplerian constants of motion EN≡µv2/2−Gmµ/r,LN≡µr×v, and\nAN≡v×LN−Gmµˆ r, it is straightforward to show that\n˙EN=µv·∆a, (3)\n˙LN=µr×∆a, (4)\n˙AN= ∆a×LN+v×˙LN\n=µ[2(v·∆a)r−(r·∆a)v−(r·v)∆a]. (5)\nBy employing the decomposition of randvin the basis KA, given by Eqs. (B1)-(B2), also the decomposition of the\nperturbing force acting on the unit mass (2), and finally the generic formula for the time-derivative of any vector V,\n˙V=˙VˆV+Vd\ndtˆV, (6)4\nwe obtain for the magnitudes\n˙EN=−a1Gmµ2\nLNsinχp+a2µ(AN+Gmµcosχp)\nLN,\n˙LN= (a2cosχp−a1sinχp)µr ,\n˙AN=a2LN+(a2cosχp−a1sinχp)µr(AN+Gmµcosχp)\nLN. (7)\nand for the directions\nd\ndtˆLN=a3µ\nLNr/parenleftBig\nsinχpˆAN−cosχpˆQN/parenrightBig\n,\nd\ndtˆAN=/bracketleftbigg\n−a1LN\nAN+Gmµ2\nLNANrsinχp(a2cosχp−a1sinχp)/bracketrightbigg\nˆQN−a3µr\nLNsinχpˆLN, (8)\nwhereris given in terms of the true anomaly parameter χpby the standard formula (B20).\nB. Radial semimajor axis arand radial eccentricity er\nWe note that the constraint A2\nN= (Gmµ)2+2EL2\nN/µis preserved by the evolutions (7), therefore only two of these\nequations are independent. From them we can also derive evolution e quations for the parameter pr=L2\nN/Gmµ2\nand eccentricity er=AN/Gmµof the conic orbit. For bounded orbits we could introduce the semima jor axisar=\npr//parenleftbig\n1−e2\nr/parenrightbig\n=L2\nN/Gmµ2/parenleftbig\n1−e2\nr/parenrightbig\n=−Gmµ/2ENof the osculating ellipse instead, and derive evolution equations for\nthe pair (ar, er). In this way we obtain two Lagrange planetary equations :\n˙ar=2a3/2\nr\n[Gm(1−e2r)]1/2[−a1sinχp+a2(er+cosχp)], (9)\n˙er=/bracketleftBigg\nar/parenleftbig\n1−e2\nr/parenrightbig\nGm/bracketrightBigg1/2a2/parenleftbig\n1+2ercosχp+cos2χp/parenrightbig\n−a1(er+cosχp)sinχp\n(1+ercosχp). (10)\nHere we have employed the true anomaly parametrization (B20) writ ten in terms of osculating ellipse orbital elements\nr=ar/parenleftbig\n1−e2\nr/parenrightbig\n1+ercosχp. (11)\nC. The noninertial system KA\nRewriting Eqs. (8) in the form of precession equations by inserting ˆAN=ˆQN׈LN,ˆQN=−ˆAN׈LNin the first\nexpression and ˆQN=ˆLN׈AN,ˆLN=−ˆQN׈ANin the second; also computing the time derivative of ˆQNfrom its\ndefinition gives\n˙f(i)=ΩA×f(i), (12)\nwith the angular velocity vector\nΩA=a3µrcosχp\nLNˆAN+a3µrsinχp\nLNˆQN−/bracketleftbigg\na1LN\nAN+(a1sinχp−a2cosχp)Gmµ2rsinχp\nLNAN/bracketrightbigg\nˆLN.(13)\nWith this we have established the time evolution of the noninertial bas isKA.\nThe PN order of ΩAisO(ΩA) =ε−1/2O(ai/c). Employing the contributions to aifrom Appendix B and Eq. (58)5\nof [41] one finds\nO/parenleftbig\nΩPN\nA/parenrightbig\n=O(ε)O(1,η)O(T−1),\nO/parenleftbig\nΩ2PN\nA/parenrightbig\n=O/parenleftbig\nε2/parenrightbig\nO/parenleftbig\n1,η,η2/parenrightbig\nO(T−1),\nO/parenleftbig\nΩSO\nA/parenrightbig\n=O/parenleftBig\nε3/2/parenrightBig/bracketleftBigg2/summationdisplay\nk=1O/parenleftbig\n1,ν2k−3/parenrightbig\nχk/bracketrightBigg\nO(T−1),\nO/parenleftbig\nΩSS\nA/parenrightbig\n=O/parenleftbig\nε2/parenrightbig\nO(η)χ1χ2O(T−1),\nO/parenleftBig\nΩQM\nA/parenrightBig\n=O/parenleftbig\nε2/parenrightbig\nO(η)/bracketleftBigg2/summationdisplay\nk=1O/parenleftbig\nν2k−3/parenrightbig\nwkχ2\nk/bracketrightBigg\nO(T−1), (14)\nwithTbeing the radial period, defined as twice the time elapsed between co nsecutive ˙r= 0 configurations.\nA couple of immediate remarks are in order:\n(1) Ifa3= 0 (no perturbing force is pointing outside the plane of motion), ˆLN(the plane of motion) is conserved,\nwhile both ˆANandˆQNundergo a precessional motion about ˆLN(in the conserved plane of motion).\n(2) Ifa1=a2= 0 (the perturbing force is perpendicular to the plane of motion), t henˆANundergoes a precessional\nmotion about ˆQNand vice-versa, while ˆLNprecesses about r.\nD. True anomaly χp\nAs the basis/braceleftbig\nf(i)/bracerightbig\nis comoving with the plane of motion and the periastron, the position v ectorr=xif(i)[withxi\ngiven by Eq. (B1)] changes according to v= ˙xif(i)+xi˙f(i)= ˙xif(i)+xiΩA×f(i). A straightforward computation,\nemploying\n˙x1= ˙rcosχp−r˙χpsinχp,˙x2= ˙rsinχp+r˙χpcosχp,˙x3= 0, (15)\nthen leads to\nLN=µr2/bracketleftBig\n˙χp+/parenleftBig\nΩA·ˆLN/parenrightBig/bracketrightBig\nˆLN. (16)\nFrom here\n˙χp+/parenleftBig\nΩA·ˆLN/parenrightBig\n=LN\nµr2, (17)\nTherefore the deviation from the Newtonian expression is due to th e component of ΩAalongˆLN. The importance\nof Eq. (17) lies in allowing to pass from time derivatives to derivatives w ith respect to χpin the evolution equations\n(7), (9)-(10), which then become ordinary differential equations .\nIt is also immediate to derive v2and calculate ENas\nEN=µ/parenleftbig\n˙r2+r2˙χ2\np/parenrightbig\n2−Gmµ\nr+µr2˙χp/parenleftBig\nΩA·ˆLN/parenrightBig\n+µr2\n2/parenleftBig\nΩA·ˆLN/parenrightBig2\n. (18)\nBy inserting Eq. (17), we obtain the radial equation\n˙r2=2EN\nµ+2Gm\nr−L2\nN\nµ2r2. (19)\nRemarkably, all terms arising from the precession of the basis vect ors cancelled out and we formally recovered the\nradial equation for the Keplerian motion. This is not surprising, as th e dynamical quantities EN,LNrefer to the\nosculating ellipse.\nE. Ascending node ˆl\nThe basis vectors of KLare related to the basis vectors of KAby a rotation in the x-yplane with angle −ψp, thus\nˆl= cosψpˆAN−sinψpˆQN, (20)\nˆ m= sinψpˆAN+cosψpˆQN. (21)6\nThe time derivative of the direction of the ascending node is therefo re found as\nd\ndtˆl=−˙ψp/parenleftBig\nsinψpˆAN+cosψpˆQN/parenrightBig\n+cosψpd\ndtˆAN−sinψpd\ndtˆQN\n= cosψp/parenleftBig\nΩA−˙ψpˆLN/parenrightBig\n׈AN−sinψp/parenleftBig\nΩA−˙ψpˆLN/parenrightBig\n׈QN\n=/parenleftBig\nΩA−˙ψpˆLN/parenrightBig\n׈l. (22)\nSimilarly we can derive the evolution of ˆ mas\nd\ndtˆ m=˙ψp/parenleftBig\ncosψpˆAN−sinψpˆQN/parenrightBig\n+sinψpd\ndtˆAN+cosψpd\ndtˆQN\n= cosψp/parenleftBig\nΩA−˙ψpˆLN/parenrightBig\n׈QN+sinψp/parenleftBig\nΩA−˙ψpˆLN/parenrightBig\n׈AN\n=/parenleftBig\nΩA−˙ψpˆLN/parenrightBig\n׈ m. (23)\nAs it was to be expected, the unit vectors ˆlandˆ mundergo a precession characterized by the angular velocity vecto r\nΩL=ΩA−˙ψpˆLN. (24)\nIII. EULER ANGLE EVOLUTIONS\nNow we have all necessary elements for deriving the evolution of the angles which enter the set of indepen-\ndent variables. First we remark, that the time derivative of the defi nition of the argument of the periastron\nψp= arccos/parenleftBig\nˆl·ˆAN/parenrightBig\n, by employing Eqs. (12) and (22) gives an identity.\nA. Inclination α\nFrom the definition of the inclination α= arccos/parenleftBig\nˆJ·ˆLN/parenrightBig\n, employing the constancy of Jup to 2PN [23] and the\nderived precession equation for ˆLNwe find\n−sinα˙α=ˆJ·d\ndtˆLN=ˆJ·/parenleftBig\nΩA׈LN/parenrightBig\n=ΩA·/parenleftBig\nˆLN׈J/parenrightBig\n=−sinαΩA·ˆl, (25)\nthus\n˙α=a3µrcos(ψp+χp)\nLN. (26)\nB. Longitude of the ascending node −φn\nBy employing Eq. (14) of [41] we find the evolution of the azimuthal an gle−φnof the ascending node ˆlas\n˙φn=−a3µrsin(ψp+χp)\nLNsinα. (27)\nQuite naturally, both the orbital inclination and the ascending node c an be changed only by a force perpendicular to\nthe orbit.\nC. Argument of the periastron ψp\nFrom Eq. (15) of [41] and Eq. (27) the evolution of ψp+χpemerges as\n˙ψp+ ˙χp=LN\nµr2−a3µrsin(ψp+χp)\nLNtanα. (28)7\nAgain, only the perturbing force component along ˆLNcontributes. Combining Eqs. (28) and (17) leads to the\nevolution equation of the third Euler angle.\nΩA·ˆLN−˙ψp=a3µrsin(ψp+χp)\nLNtanα. (29)\nThe left-hand side is ΩL·ˆLN, such that the unit vectors ˆlandˆ mundergo a precession characterized by the angular\nvelocity vector\nΩL=a3µr\nLN/bracketleftbigg\ncosχpˆAN+sinχpˆQN+sin(ψp+χp)\ntanαˆLN/bracketrightbigg\n. (30)\nThe first two terms of the bracket combine to ˆ r. If there is no perturbing force perpendicular to the orbit, ˆlandˆ m\nstay unchanged.\nThe evolution of ψpin detail reads\n˙ψp=−a1LN\nAN−(a1sinχp−a2cosχp)Gmµ2rsinχp\nLNAN−a3µrsin(ψp+χp)\nLNtanα. (31)\nEquations (26), (27) and (31) are Lagrange planetary equations for the angular orbital elements. With the use\nof Eq. (17), by passing from time derivatives to derivatives with res pect toχp, these become ordinary differential\nequations. During the inspiral the perturbing force components aiarise as a combination of relativistic (PN and\n2PN), SO, SS and QM contributions, and are given in Appendix B.\nIV. SPIN ANGLE EVOLUTIONS\nA. Spin polar angles κi\nThe spin polar angles κi= arccos/parenleftBig\nˆSi·ˆLN/parenrightBig\nevolve due to the spin precessions (see Appendix B) and the evolutio n\nofˆLN, as\n−sinκi˙κi=/parenleftBig\nΩA׈LN/parenrightBig\n·ˆSi+ˆLN·/parenleftBig\nΩi׈Si/parenrightBig\n= (ΩA−Ωi)·/parenleftBig\nˆLN׈Si/parenrightBig\n. (32)\nIn order to proceed, we need the expression (B3) of the spin, suc h that\nˆLN׈Si= sinκi/bracketleftBig\nsin(ψp−ψi)ˆAN+cos(ψp−ψi)ˆQN/bracketrightBig\n, (33)\nand we find\n˙κi=/parenleftBig\nΩi·ˆAN/parenrightBig\nsin(ψp−ψi)+/parenleftBig\nΩi·ˆQN/parenrightBig\ncos(ψp−ψi)−a3µr\nLNsin(ψp+χp−ψi). (34)\nThe relative orientation of spins with respect to the orbital angular momentum is unchanged only if the perturbing\nforce lies in the plane of motion ( a3= 0) and if the spin precession axis is along ˆLN. The latter condition is obeyed\nby the SO precession, but not by its SS and QM corrections (except for perfect perpendicularity of the spins to the\norbital plane, when also a3= 0 holds, see Appendix B, thus ˙ κi= 0). Starting from this and the remark a3∝ O/parenleftbig\nε3/2/parenrightbig\n,\nalso the estimates (B35) we find\nO(˙κi) =O/parenleftBig\nε3/2/parenrightBig\nO(η)/bracketleftbig\nwiχi+O/parenleftbig\nν2i−3/parenrightbig\nχj/bracketrightbig\nO(T−1). (35)\nB. Relative spin angle γ\nFor this we take the derivative of its definition γ= arccos/parenleftBig\nˆS1·ˆS2/parenrightBig\nand obtain\n−sinγ˙γ= (Ω1−Ω2)·/parenleftBig\nˆS1׈S2/parenrightBig\n. (36)8\nIf the spins are either aligned or antialigned with each other, such th atˆS1׈S2= 0, then ˙γ= 0, irrespective of the\nmass ratio.\nOtherwise, by employing Eqs. (56) of [41] and also/parenleftBig\nˆS1׈S2/parenrightBig\n·ˆSi= 0, we rewrite the condition (36) as\n−c2r3\n3Gsinγ˙γ=/parenleftBigg/parenleftbig\nν−ν−1/parenrightbig\n2LN+/braceleftbig\nˆ r·/bracketleftbig/parenleftbig\n1−w2ν−1/parenrightbig\nS2−(1−w1ν)S1/bracketrightbig/bracerightbig\nˆ r/parenrightBigg\n·/parenleftBig\nˆS1׈S2/parenrightBig\n. (37)\nEqual mass ( ν= 1) black holes ( wi= 1) trivially imply ˙ γ= 0, irrespective of the orientations of the spins.\nFor the generic case from Eq. (B3) we have\nˆS1׈S2= [cosκ1sinκ2sin(ψp−ψ2)−sinκ1cosκ2sin(ψp−ψ1)]ˆAN\n+[cosκ1sinκ2cos(ψp−ψ2)−sinκ1cosκ2cos(ψp−ψ1)]ˆQN\n+sinκ1sinκ2sin(ψ2−ψ1)ˆLN, (38)\nthen\n/parenleftBig\nˆS1׈S2/parenrightBig\n·ˆ r= cosκ1sinκ2sin(ψp+χp−ψ2)−sinκ1cosκ2sin(ψp+χp−ψ1),\n/parenleftBig\nˆS1׈S2/parenrightBig\n·ˆLN= sinκ1sinκ2sin(ψ2−ψ1). (39)\nThus we can rewrite Eq. (37) in detail as\n−c2r3\n3GLNsinγ˙γ=/parenleftbig\nν−ν−1/parenrightbig\n2sinκ1sinκ2sin(ψ2−ψ1)\n+/bracketleftbigg/parenleftbig\n1−w2ν−1/parenrightbigS2\nLNsinκ2cos(ψ−ψ2)−(1−w1ν)S1\nLNsinκ1cos(ψ−ψ1)/bracketrightbigg\n×[cosκ1sinκ2sin(ψ−ψ2)−sinκ1cosκ2sin(ψ−ψ1)], (40)\nwhereψ=ψp+χp. Again, it is manifest, that the relative angle of the spins stays cons tant for equal mass black\nholes, irrespective of their orientation.\nStarting from the above remark, Eq. (36) and the estimates (B35 ) we find\nO(˙γ) =O(ε)O/parenleftbig\nν3−2i/parenrightbig\nO(T−1). (41)\nThus the angle γchanges faster than κi.\nC. Spin azimuthal angles ψi\nEq. (30) of [41]\nˆSi= sinκicosψiˆl+sinκisinψiˆ m+cosκiˆLN (42)\ngivesψi= arctan/bracketleftBig/parenleftBig\nˆ m·ˆSi/parenrightBig\n//parenleftBig\nˆl·ˆSi/parenrightBig/bracketrightBig\nfor the spin azimuthal angles, unless κi= 0,π(the spins are aligned or\nantialigned to the Newtonian orbital angular momentum) or ψi=π/2,3π/2 (the projections of the spins in the plane\nof motion are perpendicular to the node line).\nIn the generic case the spin azimuthal angles evolve according to\n/parenleftbig\n1+tan2ψi/parenrightbig˙ψi/parenleftBig\nˆl·ˆSi/parenrightBig\n=/parenleftBig\nˆ m−tanψiˆl/parenrightBig\n·d\ndtˆSi+/parenleftbiggd\ndtˆ m−tanψid\ndtˆl/parenrightbigg\n·ˆSi. (43)\nAs bothˆlandˆ mprecesses about ΩL, whileˆSiaboutΩi, we find\n˙ψisinκi=/parenleftBig\ncosψiˆ m−sinψiˆl/parenrightBig\n·/bracketleftBig\n(Ωi−ΩL)׈Si/bracketrightBig\n, (44)\nor, by employing Eqs. (20)-(21):\n˙ψisinκi=/bracketleftBig\nsin(ψp−ψi)ˆAN+cos(ψp−ψi)ˆQN/bracketrightBig\n·/bracketleftBig\n(Ωi−ΩL)׈Si/bracketrightBig\n, (45)9\nwith the vector products Ωi׈SiandΩL׈Si, given by Eqs. (B39) and (B6), respectively. We obtain\n˙ψi=/parenleftBig\nΩi·ˆLN/parenrightBig\n+/bracketleftBig/parenleftBig\nΩi·ˆQN/parenrightBig\nsin(ψp−ψi)−/parenleftBig\nΩi·ˆAN/parenrightBig\ncos(ψp−ψi)/bracketrightBig\ncotκi\n−a3µr\nLN[cotαsin(ψp+χp)−cotκicos(χp+ψp−ψi)]. (46)\nWith this we have completed the derivation of all required evolution eq uations.\nStarting from Eq. (46) and the estimates (B35) we find\nO/parenleftBig\n˙ψi/parenrightBig\n=O(ε)O(1,η)O(T−1). (47)\nThe change in the azimuthal angle of the spins is one PN order higher t han the Keplerian orbital evolution.\nV. SPECIAL CONFIGURATIONS\nAs a by-product of the calculations carried on in this paper we have r ecovered the known result that the plane\nof motion is changed only by perturbing forces pointing outside the p lane of motion, thus by the SO, SS and QM\nperturbations. Wehaveshownthattherelativeangleofthespinss taysconstantforequalmassblackholes,irrespective\nof their orientation. We have also proven that unless the spins are p erpendicular to the plane of motion ( κi= 0), the\npolar spin angles will change under these perturbations.\nThe nonprecessing ( κi= 0) and precessing (generic κi) cases have been discussed separately in the literature (see\nRefs [47] and [48], respectively) in connection with the recoil of the fi nal black hole [49]. From among the precessing\ncases the antialigned spin configuration with the spins laying in the orb ital plane has received special attention, as\nnumerical investigations have shown that it leads to the highest kick velocity.\nWe have now the means to investigate such a configuration analytica lly. First we specialize to spins laying in the\norbital plane, κi=π/2. After some algebra, Eq. (34) gives\n˙κi=G2m2η\n2c3r3/parenleftBig\nKSO\ni+KSS\ni+KQM\ni/parenrightBig\n, (48)\nKSO\ni=−sin(ψp+χp−ψi)\n1+AN\nGmµcosχp2/summationdisplay\nk=1/parenleftbig\n4ν2k−3+3/parenrightbig\nχk\n×/bracketleftbigg\n2cos(ψp+χp−ψk)+AN\nGmµ[2cos(ψp−ψk)−3sinχpsin(ψp+χp−ψk)]/bracketrightbigg\n,\nKSS\ni=ν2j−3χj[3sin(2ψp+2χp−ψi−ψj)+sin(ψj−ψi)],\nKQM\ni= 3wiχisin(2ψp+2χp−2ψi).\nAll contributions KSO\ni,KSS\ni,KQM\niare of the same order. In general the expression for ˙ κidoes not vanish, not even\nin the special case of equal mass ( ν= 1), maximally spinning ( χi= 1) black holes ( wi= 1) on circular orbit ( AN= 0),\nwhen\n˙κi=−G2m2η\nc3r3[2sin(2ψp+2χp−2ψi)+2sin(2ψp+2χp−ψi−ψj)+3sin(ψj−ψi)]. (49)\nTherefore in general a configuration with the spins in the plane of mo tion is not preserved.\nHowever in the special case ψj=ψi+πand equal mass ( ν= 1), equal spin ( χ2=χ1) black holes ( wi= 1) we find\na3= 0,\nΩi·ˆAN=G2m2η\nc3r3χ1cos(ψp−ψi),\nΩi·ˆQN=−G2m2η\nc3r3χ1sin(ψp−ψi),\nΩi·ˆLN=7G\n2c2r3Jcosα , (50)10\nsuch that according to Eq. (34) ˙ κi= 0.1\nThen one has to check, whether the condition imposed on ψiis consistent with their evolution. With a3= 0 Eq.\n(B6) gives ΩL׈Si= 0, while from Eq. (B39) we get\nΩi׈Si=7G\n2c2r3Jcosα/bracketleftBig\nsin(ψp−ψi)ˆAN+cos(ψp−ψi)ˆQN/bracketrightBig\n, (51)\nsuch that Eq. (46) simplifies to\n˙ψi=7G\n2c2r3Jcosα . (52)\nAs the right-hand side does not depend on the index i, the imposed antialignment of the spins can be maintained\nover time. This is also evident from Eq. (36). We have also checked th at the constraints (46)-(47) of [41] are trivially\nobeyed.\nTherefore the special configuration of equal mass black holes with equal, but antialigned spins, bo th laying in the\nplane of motion is preserved by the conservative PN dynamics , with leading order SO, SS and QM contributions\nincluded. This stands as the main result of this section.\nEquation (48) of [41] allows us to rewrite\n˙ψi=7G\n2c2r3LN(1+ǫPN+ǫ2PN), (53)\nwith the coefficients (given by Eqs. (39)-(40) of [41]) specified for e qual mass as\nǫPN=1\n8/parenleftBigv\nc/parenrightBig2\n+13\n4Gm\nc2r, (54)\nǫ2PN=3\n128/parenleftBigv\nc/parenrightBig4\n−13\n32Gm\nc2r/parenleftbigg˙r\nc/parenrightbigg2\n+63\n32Gm\nc2r/parenleftBigv\nc/parenrightBig2\n+/parenleftbiggGm\nc2r/parenrightbigg2\n. (55)\nVI. CONCLUDING REMARKS\nIn this paper we have established the conservative evolution equat ions of the two independent sets of variables\ncharacterizing a spinning compact binary during its inspiral, establish ed in [41], with leading order SO, SS and QM\ncontributions included. As the lengths Jandχiare constants, this reduces to angular evolutions. The evolutions of\nthe variables complementing the set ( J, χi), the inclination αand the spin polar angles κiwere given as Eqs. (26) and\n(34). The evolution equations for the spin azimuthal angles ψi(replacingχias independent variables) were given by\nEq. (46). These time derivatives (and all others computed throug hout the paper) can be transformed to derivatives\nwith respect to χpby employing Eq. (17) in the form\nd\ndt=/parenleftbiggLN\nµr2−ΩA·ˆLN/parenrightbiggd\ndχp. (56)\nThe true anomaly χpbecomes the only independent variable by employing the parametriza tionr(χp), Eqs. (B20)-\n(B21).\nThe system is closed by the evolution of the argument of the periast ronψpgiven as Eq. (31), the last two\nEqs. (7) giving ˙ANand˙LN, the analytical expression (B17)-(B19) of the perturbing accele ration components ai,\nthe expressions (B36)-(B37) of the of the spin precessional ang ular velocity components/parenleftBig\nΩi·ˆAN/parenrightBig\nand/parenleftBig\nΩi·ˆQN/parenrightBig\n,\nfinally the vector products Ωi׈SiandΩL׈Si, given by Eqs. (B39) and (B6), respectively.\nTherefore we have derived a closed system of first order ordinary differential equations for the variables ( α,κi,ψi,\nψp,AN,LN) evolving in terms of the true anomaly χp, ready for numerical evolution. From this set ( α,κi,ψi) are\nindependent variables characterizing the spinning binary configura tion, while ( ψp,AN,LN) characterize the orbit.\n1The SO contribution to ˙ κivanishes, while the SS and QM contributions cancel. A glance atKQM\nigiven by Eqs. (48) shows that\nwithout imposing the black hole condition wi= 1 the SS and QM contributions do notcancel, therefore the result does not hold for\nequal mass, identically spinning neutron stars.11\nIn another way of counting, replacing ( AN,LN) and their evolutions by the orbital elements ( ar,er) and Eqs.\n(9)-(10), respectively; also including the evolution Eq. (27) for th e longitude of the ascending node −φnwe have\nobtained evolutions for (i) the orbital elements ( ar,er,α,ψp,−φn) characterizing the perturbed Keplerian motion\nand for (ii) the spin angles ( κi,ψi) characterizing the spin orientations with respect to this perturb ed Keplerian orbit.\nAs a by-product, we have proven that the relative angle of the spin s stays constant for equal mass black holes,\nirrespective of their orientation.\nAlso, unlessthespinsareperpendiculartotheplaneofmotion, the p olarspinangleschangeundertheperturbations.\nThere is one notable exception under this rule: the special configur ation of equal mass black holes with equal,\nbut antialigned spins, both laying in the plane of motion is preserved by the conservative dynamics. This is the\nconfiguration which led to maximal recoil found in numerical simulation s [48], and our investigations show that it is\nconserved during the inspiral to a 2PN accuracy, with leading order spin-orbit, spin-spin and mass quadrupole effects\nincluded.\nAcknowledgements\nI am grateful to Thomas Krichbaum for references on observatio nal spin estimates. This work was partially sup-\nportedbyCOSTActionMP0905”BlackHolesinaViolentUniverse”andH ungarianScientificResearchFund(OTKA)\nGrant no. 69036.\nAppendix A: Comparison of notations with related literatur e\nIn this Appendix we compare the notations established in [41] and tho roughly employed in this paper with corre-\nsponding notations in the literature.\nFirst we establish the correspondence of the Euler angles ( −φn,α,ψp) employed in [41] and standard celestial\nmechanics angular orbital elements in Table I. The celestial mechanic s angular orbital elements (Ω ,ι,ω) are defined\nwith respect to a reference plane and a reference direction conta ined within it, both inertial. The node line is defined\nas the intersection of the reference plane with the plane of motion; the angle span by it with the reference direction\nis thelongitude of the ascending node Ω; the relative angle of the two planes is the inclination ιand the angle span\nby the ascending node with the direction of the periastron in the argument of the periastron ω. The Euler angles\n(−φn,α,ψp) employed in [41] are defined similarly, but with respect to the inertial systemKiwithˆ xandˆJstanding\nas thexandzaxes [any ˆ x⊥ˆJstanding as the reference direction and the reference plane given by/parenleftBig\nˆ x,ˆ y=ˆJ׈ x/parenrightBig\n].\nTABLE I: Comparison of the notations in Paper Iand standard celestial mechanics angular orbital elements .\nRef. [41] Celestial mechanics\nEuler angles ( −φn,α,ψp) angular orbital elements (Ω ,ι,ω)\nTrue anomaly χp true anomaly v\nEquation (14) of [41] ˙φn=−˙αtan(ψp+χp)\nsinα˙Ω = ˙ιtan(ω+v)\nsinι\nEquation (15) of [41] ˙ψp+ ˙χp=LN\nµr2+˙φncosα ˙ω+ ˙v=LN\nµr2−˙Ωcosι\nThe various systems of reference necessary for the description of the motion were also discussed in Refs. [42] and\n[43]. We establish the correspondencein Table II. While in these paper s a quasicircularorbit was assumed, the results\nof [41] hold for generic orbits. A correspondence can be establishe d as long as ˆJcan be viewed as an inertial axis.\nFinally we compare the notations of Ref. [44] with the notations of [4 1] in Table III.\nAppendix B: Decomposition of the acceleration and spin angu lar velocity vectors in the system KAduring\nthe inspiral\nIn this Appendix we give the decomposition of the accelerations and o f the precessional angular velocities of the\nspins in the system KA. The ingredients we need are Eqs. (19)-(20) of [41] for the deco mposition of the position and12\nTABLE II: Comparison of the notations in Refs. [42] and [43] a nd [41].\nReference [43] (based on [42]) Reference [41]\nOrbit circular elliptical,AN/negationslash= 0\nCorresponding quantities (ω, Mω)/parenleftBig\nLN\nµr2=c3\nGmε3/2,c3\nGε3/2/parenrightBig\nCorresponding equation numbers (16), (18) (25) and (26), (15)\nPlane of motion orthonormal base ( n,λ)nonorthogonal base/parenleftbigr\nr,v\nv/parenrightbig\nOrthonormal inertial source system/parenleftbig\neS\nx,eS\ny,eS\nz/parenrightbig\nKi=/parenleftBig\nˆx,ˆy,ˆJ/parenrightBig\n, ifˆJ≡eS\nz\nOrthonormal basis in the plane of motion/parenleftbig\neS\n1,eS\n2/parenrightbig/parenleftBig\nˆl,ˆm/parenrightBig\n, ifˆJ≡eS\nz\nEuler angles (ΦS,ι,α)/parenleftbig\nψ=ψp+χp,α,π\n2−φn/parenrightbig\n, ifˆJ≡eS\nz\nLine of sight in the ( x,z) plane Θ = arccos/parenleftBig\neS\nz·ˆN/parenrightBig\nΘ = arccos/parenleftBig\nˆJ·ˆN/parenrightBig\n,ˆy=ˆJ׈N\nsinΘ\nTABLE III: Comparison of the notations in Refs. [44] and [41] .\nReference [44] Reference [41]\nOrbit circular elliptical,AN/negationslash= 0\nCorresponding quantities/parenleftbig\nωorb, v3=Mωorb/parenrightbig/parenleftBig\nLN\nµr2=c3\nGmε3/2,c3\nGε3/2/parenrightBig\nInertial axis J0ˆJ\nInertial orthonormal basis ⊥J0(toˆJ) (ˆx,ˆy) (ˆx,ˆy)\nInertial orthonormal basis ⊥ˆLN(theKLbasis)(ˆxL,ˆyL)/parenleftBig\nˆl,ˆm/parenrightBig\nBasis comoving with µ (n,λ)/parenleftBig\nˆr,ˆLN׈r/parenrightBig\nSymmetric mass ratio ν∈[0,0.25] η∈[0,0.25]\nPolar and azimuthal angles of ˆLNinKi (ι,α)/parenleftbig\nα,3π\n2−φn/parenrightbig\nPhase Φ(t) ψ=ψp+χp\nvelocity vectors:\nˆ r= cosχpˆAN+sinχpˆQN, (B1)\nv=Gmµ\nLN/bracketleftbigg\n−sinχpˆAN+/parenleftbigg\ncosχp+AN\nGmµ/parenrightbigg\nˆQN/bracketrightbigg\n. (B2)\nIn the system KLthe spin is given by Eq. (30) of [41]. By employing Eqs. (20)-(21) we re write it in the system KA\nas\nˆSi= sinκi/bracketleftBig\ncos(ψp−ψi)ˆAN−sin(ψp−ψi)ˆQN/bracketrightBig\n+cosκiˆLN. (B3)\nWe also need\nˆ r׈Sk= cosκk/parenleftBig\nsinχpˆAN−cosχpˆQN/parenrightBig\n−sinκksin(ψp+χp−ψk)ˆLN, (B4)\nv׈Sk=Gmµ\nLNcosκk/bracketleftbigg/parenleftbigg\ncosχp+AN\nGmµ/parenrightbigg\nˆAN+sinχpˆQN/bracketrightbigg\n−Gmµ\nLNsinκk/bracketleftbigg\ncos(ψp+χp−ψk)+AN\nGmµcos(ψp−ψk)/bracketrightbigg\nˆLN, (B5)\nand\nΩL׈Si=a3µr\nLN/braceleftbigg/bracketleftbiggsinκi\ntanαsin(ψp−ψi)sin(ψp+χp)+cosκisinχp/bracketrightbigg\nˆAN\n+/bracketleftbiggsinκi\ntanαcos(ψp−ψi)sin(ψp+χp)−cosκicosχp/bracketrightbigg\nˆQN\n−sinκisin(ψp+χp−ψi)ˆLN/bracerightBig\n. (B6)13\nEqs. (3)-(5) of [41] give\nSi=G\ncm2ην2i−3χi, (B7)\nQi=−G2\nc4wim2ην2i−3χ2\nimi. (B8)\n1. Acceleration\nThe general relativistic, SO, SS and QM contributions to the acceler ation, with the SO part given in the Newton-\nWigner-Pryce spin supplementary condition (NWP SSC) [25],[28], by emp loying Eqs. (B7)-(B8) are:\n∆a=aPN+a2PN+aNWP\nSO+aSS+aQM, (B9)\nwith\naPN=Gm\nc2r2/braceleftbigg/bracketleftbigg\n2(2+η)Gm\nr−(1+3η)v2+3\n2η˙r2/bracketrightbigg\nˆ r+2(2−η)˙rv/bracerightbigg\n, (B10)\na2PN=−Gm\nc4r2/braceleftBigg/bracketleftBig3\n4(12+29η)/parenleftbiggGm\nr/parenrightbigg2\n+η(3−4η)v4+15\n8η(1−3η) ˙r4\n−3\n2η(3−4η) ˙r2v2−η\n2(13−4η)Gm\nrv2−/parenleftbig\n2+25η+2η2/parenrightbigGm\nr˙r2/bracketrightBig\nˆ r\n−1\n2/bracketleftbigg\nη(15+4η)v2−/parenleftbig\n4+41η+8η2/parenrightbigGm\nr−3η(3+2η) ˙r2/bracketrightbigg\n˙rv/bracerightBigg\n, (B11)\naNWP\nSO=G2m2η\nc3r32/summationdisplay\nk=1/parenleftbig\n4ν2k−3+3/parenrightbig\nχk/braceleftbigg3LN\n2µr/parenleftBig\nˆLN·ˆSk/parenrightBig\nˆ r−/parenleftBig\nv׈Sk/parenrightBig\n+3˙r\n2/parenleftBig\nˆ r׈Sk/parenrightBig/bracerightbigg\n(B12)\naSS=−3G3m3η\nc4r4χ1χ2/braceleftBig/bracketleftBig/parenleftBig\nˆS1·ˆS2/parenrightBig\n−5/parenleftBig\nˆ r·ˆS1/parenrightBig/parenleftBig\nˆ r·ˆS2/parenrightBig/bracketrightBig\nˆ r+/parenleftBig\nˆ r·ˆS2/parenrightBig\nˆS1+/parenleftBig\nˆ r·ˆS1/parenrightBig\nˆS2/bracerightBig\n,(B13)\naQM=−3G3m3η\n2c4r42/summationdisplay\nk=1wkν2k−3χ2\nk/braceleftbigg/bracketleftbigg\n1−5/parenleftBig\nˆ r·ˆSk/parenrightBig2/bracketrightbigg\nˆ r+2/parenleftBig\nˆ r·ˆSk/parenrightBig\nˆSk/bracerightbigg\n. (B14)\nAfter inserting Eqs. (B1)-(B5), the projections ai= ∆a·f(i)withf(i)= (ˆAN,ˆQN,ˆLN), they can be readily found.\nFor explicit expressions we also need\nˆ r·ˆSk= sinκkcos(ψp+χp−ψk), (B15)\nˆS1·ˆS2= cosκ1cosκ2+sinκ1sinκ2cos(ψ2−ψ1). (B16)\nThe acceleration components are14\na1=aPN\n1+a2PN\n1+aSO\n1+aSS\n1+aQM\n1, (B17)\naPN\n1=Gm\nc2r2/braceleftbigg/bracketleftbigg\n2(2+η)Gm\nr−(1+3η)v2+3\n2η˙r2/bracketrightbigg\ncosχp−2(2−η)˙rGmµ\nLNsinχp/bracerightbigg\n,\na2PN\n1=−Gm\nc4r2/braceleftBigg/bracketleftBig3\n4(12+29η)/parenleftbiggGm\nr/parenrightbigg2\n+η(3−4η)v4+15\n8η(1−3η) ˙r4\n−3\n2η(3−4η) ˙r2v2−η\n2(13−4η)Gm\nrv2−/parenleftbig\n2+25η+2η2/parenrightbigGm\nr˙r2/bracketrightBig\ncosχp\n+/bracketleftbigg\nη(15+4η)v2−/parenleftbig\n4+41η+8η2/parenrightbigGm\nr−3η(3+2η) ˙r2/bracketrightbiggGmµ˙r\n2LNsinχp/bracerightBigg\n,\naSO\n1=G2m2η\nc3r3/bracketleftbigg/parenleftbigg3LN\n2µr−Gmµ\nLN/parenrightbigg\ncosχp+3˙r\n2sinχp−AN\nLN/bracketrightbigg2/summationdisplay\nk=1/parenleftbig\n4ν2k−3+3/parenrightbig\nχkcosκk,\naSS\n1=−3G3m3η\nc4r4χ1χ2/braceleftBig\ncosκ1cosκ2cosχp+sinκ1sinκ2\n×/bracketleftbig\n[cos(ψ2−ψ1)−5cos(ψp+χp−ψ1)cos(ψp+χp−ψ2)]cosχp\n+cos(ψp+χp−ψ2)cos(ψp−ψ1)+cos(ψp+χp−ψ1)cos(ψp−ψ2)/bracketrightbig/bracerightBig\n,\naQM\n1=−3G3m3η\n2c4r42/summationdisplay\nk=1wkν2k−3χ2\nk/braceleftBig\ncosχp−sin2κkcos(ψp+χp−ψk)\n×[5cosχpcos(ψp+χp−ψk)−2cos(ψp−ψk)]/bracerightBig\n,\na2=aPN\n2+a2PN\n2+aSO\n2+aSS\n2+aQM\n2, (B18)\naPN\n2=Gm\nc2r2/braceleftbigg/bracketleftbigg\n2(2+η)Gm\nr−(1+3η)v2+3\n2η˙r2/bracketrightbigg\nsinχp+2(2−η)˙rGmµ\nLN/parenleftbigg\ncosχp+AN\nGmµ/parenrightbigg/bracerightbigg\na2PN\n2=−Gm\nc4r2/braceleftBigg/bracketleftBig3\n4(12+29η)/parenleftbiggGm\nr/parenrightbigg2\n+η(3−4η)v4+15\n8η(1−3η) ˙r4\n−3\n2η(3−4η) ˙r2v2−η\n2(13−4η)Gm\nrv2−/parenleftbig\n2+25η+2η2/parenrightbigGm\nr˙r2/bracketrightBig\nsinχp\n−/bracketleftbigg\nη(15+4η)v2−/parenleftbig\n4+41η+8η2/parenrightbigGm\nr−3η(3+2η) ˙r2/bracketrightbiggGmµ˙r\n2LN/parenleftbigg\ncosχp+AN\nGmµ/parenrightbigg/bracerightBigg\n,\naSO\n2=G2m2η\nc3r3/bracketleftbigg/parenleftbigg3LN\n2µr−Gmµ\nLN/parenrightbigg\nsinχp−3˙r\n2cosχp/bracketrightbigg2/summationdisplay\nk=1/parenleftbig\n4ν2k−3+3/parenrightbig\nχkcosκk,\naSS\n2=−3G3m3η\nc4r4χ1χ2/braceleftBig\ncosκ1cosκ2sinχp+sinκ1sinκ2\n×/bracketleftbig\n[cos(ψ2−ψ1)−5cos(ψp+χp−ψ1)cos(ψp+χp−ψ2)]sinχp\n−cos(ψp+χp−ψ2)sin(ψp−ψ1)−cos(ψp+χp−ψ1)sin(ψp−ψ2)/bracketrightbig/bracerightBig\n,\naQM\n2=−3G3m3η\n2c4r42/summationdisplay\nk=1wkν2k−3χ2\nk/braceleftBig\nsinχp−sin2κkcos(ψp+χp−ψk)\n×[5sinχpcos(ψp+χp−ψk)+2sin(ψp−ψk)]/bracerightBig\n,15\nand\na3=aSO\n3+aSS\n3+aQM\n3, (B19)\naSO\n3=G2m2η\nc3r32/summationdisplay\nk=1/parenleftbig\n4ν2k−3+3/parenrightbig\nχksinκk\n×/braceleftbiggGmµ\nLN/bracketleftbigg\ncos(ψp+χp−ψk)+AN\nGmµcos(ψp−ψk)/bracketrightbigg\n−3˙r\n2sin(ψp+χp−ψk)/bracerightbigg\n,\naSS\n3=−3G3m3η\nc4r4χ1χ2[cosκ1sinκ2cos(ψp+χp−ψ2)+cosκ2sinκ1cos(ψp+χp−ψ1)],\naQM\n3=−3G3m3η\n2c4r42/summationdisplay\nk=1wkν2k−3χ2\nksin2κkcos(ψp+χp−ψk).\nIn the above expressions we still need to employ Eqs. (20)-(21) an d (23) of [41]\nr=L2\nN\nµ(Gmµ+ANcosχp), (B20)\n˙r=AN\nLNsinχp, (B21)\nv2=(Gmµ)2+A2\nN+2GmµA Ncosχp\nL2\nN, (B22)\nin order to rewrite r, ˙randv2in terms of the chosen dynamical variables.\nAlso, asLNis not among the chosen independent variables, we need to express it in terms of them. For this, first\nwe give the SO part of the orbital angular momentum in the NWP SSC:\nLNWP\nSO=Gµ\n2c2r2/summationdisplay\nk=1/parenleftbig\n4+3ν3−2i/parenrightbig\nSi/bracketleftBig\nˆ r×/parenleftBig\nˆ r׈Si/parenrightBig/bracketrightBig\n=G2m3\n4c3rη22/summationdisplay\ni=1/parenleftbig\n4ν2i−3+3/parenrightbig\nχi/braceleftBig\nsinκi[cos(2χp+ψp−ψi)−cos(ψp−ψi)]ˆAN\n+sinκi[sin(2χp+ψp−ψi)+sin(ψp−ψi)]ˆQN−2cosκiˆLN/bracerightBig\n. (B23)\nAs expected its order is\nO/parenleftbiggLNWP\nSO\nLN/parenrightbigg\n=O/parenleftBig\nε3/2/parenrightBig\nO(η)O/parenleftbig\n1,ν2i−3/parenrightbig\nχi. (B24)\nThe total angular momentum J=L+S1+S2in the system KAthen becomes, using Eqs. (B3) and (B23):\nJˆJ=Gm2\ncη2/summationdisplay\ni=1χisinκi\n×/braceleftbigg/bracketleftbigg\nν2i−3cos(ψp−ψi)−Gm\n4c2rη/parenleftbig\n4ν2i−3+3/parenrightbig\n[cos(ψp−ψi)−cos(2χp+ψp−ψi)]/bracketrightbigg\nˆAN\n−/bracketleftbigg\nν2i−3sin(ψp−ψi)−Gm\n4c2rη/parenleftbig\n4ν2i−3+3/parenrightbig\n[sin(ψp−ψi)+sin(2χp+ψp−ψi)]/bracketrightbigg\nˆQN/bracerightbigg\n+/braceleftBigg\nLN(1+ǫPN+ǫ2PN)+Gm2\ncη2/summationdisplay\ni=1/bracketleftbigg\nν2i−3−Gm\n2c2rη/parenleftbig\n4ν2i−3+3/parenrightbig/bracketrightbigg\nχicosκi/bracerightBigg\nˆLN, (B25)\nHereǫPNandǫ2PNare given by Eqs. (39)-(40) of [41]. The projections along the basis vectorsˆl,ˆ m,ˆLNof theKL16\nsystem are2\n0 =2/summationdisplay\ni=1χisinκi/bracketleftbigg\nν2i−3cosψi+Gm\n4c2rη/parenleftbig\n4ν2i−3+3/parenrightbig\n[cos(2χp+2ψp−ψi)−cosψi]/bracketrightbigg\n, (B26)\ncJsinα\nGm2η=2/summationdisplay\ni=1χisinκi/bracketleftbigg\nν2i−3sinψi+Gm\n4c2rη/parenleftbig\n4ν2i−3+3/parenrightbig\n[sin(2χp+2ψp−ψi)−sinψi]/bracketrightbigg\n,(B27)\nJcosα=LN(1+ǫPN+ǫ2PN)+Gm2\ncη2/summationdisplay\ni=1/bracketleftbigg\nν2i−3−Gm\n2c2rη/parenleftbig\n4ν2i−3+3/parenrightbig/bracketrightbigg\nχicosκi. (B28)\nThe last equation enables us to express LNto 2PN accuracy in terms of the chosen independent variables:\nLN=J/parenleftbig\n1−ǫPN−ǫ2PN+ǫ2\nPN/parenrightbig\ncosα\n−Gm2\ncη2/summationdisplay\ni=1/bracketleftbigg\n(1−ǫPN)ν2i−3−Gm\n2c2rη/parenleftbig\n4ν2i−3+3/parenrightbig/bracketrightbigg\nχicosκi. (B29)\nWe also give the series expansion of its reciprocal:\n1\nLN=1+ǫPN+ǫ2PN\nJcosα+/parenleftbiggGm2\ncJcosα/parenrightbiggη\nJcosα/bracketleftbigg\n(1+ǫPN)χν−Gm\n2c2rη(4χν+3χ+)/bracketrightbigg\n+/parenleftbiggGm2\ncJcosα/parenrightbigg2η2\nJcosα/bracketleftbigg\n(1+ǫPN)χν−Gm\nc2rη(4χν+3χ+)/bracketrightbigg\nχν\n+/parenleftbiggGm2\ncJcosα/parenrightbigg3η3\nJcosαχ3\nν+/parenleftbiggGm2\ncJcosα/parenrightbigg4η4\nJcosαχ4\nν, (B30)\nwhere we employed the notations\nχ+=2/summationdisplay\ni=1χicosκi=χ1cosκ1+χ2cosκ2,\nχν=2/summationdisplay\ni=1ν2i−3χicosκi=ν−1χ1cosκ1+νχ2cosκ2. (B31)\nNote that the 2PN contribution of 1 /LNis rather messy (fourth rank in the spins), nevertheless for our p urposes we\nneed it only to 1PN accuracy (it enters only in PN terms or higher, and the desired accuracy is 2PN).\nWe also give here the detailed expression in terms of orbital elements ofǫPN, which is necessary at this accuracy:\nǫPN=1−3η\n2/parenleftBigv\nc/parenrightBig2\n+(3+η)Gm\nc2r\n=(7−η)(Gmµ)2+(1−3η)A2\nN+4(2−η)GmµA Ncosχp\n2c2L2\nN\n=(Gmµ)2\n2c2J2cos2α/bracketleftbig\n(1−3η)e2\nr+4(2−η)ercosχp+(7−η)/bracketrightbig\n. (B32)\n2. Spin angular velocity\nThe spin undergoes a pure precession, therefore its magnitude is u nchanged, while its direction changes as\nd\ndtˆSi=Ωi׈Si, (B33)\n2These are the equations in the NWP SSC corresponding to Eqs. ( 46)-(48) of [41], which were written in the covariant SSC.17\nwhere, after employing Eqs. (B7)-(B8), (B1), (B3), and (B15) in Eqs. (56) of [41] the angular velocity vector is found\nas\nΩi=ΩSO\ni+ΩSS\ni+ΩQM\ni, (B34)\nΩSO\ni=G/parenleftbig\n4+3ν3−2i/parenrightbig\n2c2r3L\nNˆLN,\nΩSS\ni=G2m2η\n2c3r3ν2j−3χj/bracketleftBig\nsinκj/braceleftBig\n[3cos(ψp+χp−ψj)cosχp−cos(ψp−ψj)]ˆAN\n+[3cos(ψp+χp−ψj)sinχp+sin(ψp−ψj)]ˆQN/bracerightBig\n−cosκjˆLN/bracketrightBig\n,\nΩQM\ni=G2m2η\n2c3r33wiχisinκicos(ψp+χp−ψi)/parenleftBig\ncosχpˆAN+sinχpˆQN/parenrightBig\n,\nwithj∝ne}ationslash=i. Their PN order is\nO/parenleftbig\nΩSO\ni/parenrightbig\n=O(ε)O/parenleftbig\n1,ν3−2i/parenrightbig\nO(T−1),\nO/parenleftbig\nΩSS\ni/parenrightbig\n=O/parenleftBig\nε3/2/parenrightBig\nO(η)O/parenleftbig\nν2i−3/parenrightbig\nχjO(T−1),\nO/parenleftBig\nΩQM\ni/parenrightBig\n=O/parenleftBig\nε3/2/parenrightBig\nO(η)wiχiO(T−1). (B35)\nThe projections employed in the main text are\nΩi·ˆAN=G2m2η\n2c3r3/braceleftbig\nν2j−3χjsinκj[3cos(ψp+2χp−ψj)+cos(ψp−ψj)]\n+3wiχisinκi[cos(ψp+2χp−ψi)+cos(ψp−ψi)]}, (B36)\nΩi·ˆQN=G2m2η\n2c3r3/braceleftbig\nν2j−3χjsinκj[3sin(ψp+2χp−ψj)−sin(ψp−ψj)]\n+3wiχisinκi[sin(ψp+2χp−ψi)−sin(ψp−ψi)]}, (B37)\nΩi·ˆLN=G/parenleftbig\n4+3ν3−2i/parenrightbig\n2c2r3Jcosα−G2m2η\n2c3r3/bracketleftbig/parenleftbig\n4ν2i−3+3/parenrightbig\nχicosκi+ν2j−3/parenleftbig\n5+3ν3−2i/parenrightbig\nχjcosκj/bracketrightbig\n.(B38)\nWe also need\nΩi׈Si=/bracketleftBig/parenleftBig\nΩi·ˆLN/parenrightBig\nsinκisin(ψp−ψi)+/parenleftBig\nΩi·ˆQN/parenrightBig\ncosκi/bracketrightBig\nˆAN\n+/bracketleftBig/parenleftBig\nΩi·ˆLN/parenrightBig\nsinκicos(ψp−ψi)−/parenleftBig\nΩi·ˆAN/parenrightBig\ncosκi/bracketrightBig\nˆQN\n−sinκi/bracketleftBig/parenleftBig\nΩi·ˆAN/parenrightBig\nsin(ψp−ψi)+/parenleftBig\nΩi·ˆQN/parenrightBig\ncos(ψp−ψi)/bracketrightBig\nˆLN. (B39)\n[1] The LIGO Collaboration, Phys. Rev. D80, 047101 (2009); Anand S. Sengupta for the LIGO Scientific Col laboration and\nthe Virgo Collaboration, J. Phys. Conf. Ser. 228, 012002 (2010).\n[2] K. G. Arun, S. Babak, E. Berti, N. Cornish, C. Cutler, J. Ga ir, S. A. Hughes, B. R. Iyer, R. N. Lang, I. Mandel, E. K.\nPorter, B. S. Sathyaprakash, S. Sinha, A. M. Sintes, M. Trias , C. Van Den Broeck, M. Volonteri, Class. Quantum Grav.\n26, 094027 (2009); R. N. Lang, S. A. Hughes, Class. Quantum Grav. 26, 094035 (2009).\n[3] L.´A. Gergely, P. L. Biermann, Astrophys. J. 697, 1621-1633 (2009).\n[4] Y. Pan, A. Buonanno, L. T. Buchman, T. Chu, L. E. Kidder, H. P. Pfeiffer, M. A. Scheel, Phys. Rev. D81, 084041 (2010).\n[5] R. Sturani, S. Fischetti, L. Cadonati, G. M. Guidi, J. Hea ly, D. Shoemaker, A. Vicer´ e, J. Phys. Conf. Ser. 243, 012007\n(2010).\n[6] E. Berti, V. Cardoso, A. O. Starinets, Class. Quantum Grav. 26, 163001 (2009).\n[7] C. M. Will, Astrophys. J. Letters ,674, L25 (2008); D. Merritt, T. Alexander, S. Mikkola, C. M. Will ,Phys. Rev. D81,\n062002 (2010).\n[8] J. C. McKinney, R. D. Blandford, Monthly Not. Royal Astron. Soc. 394, L126 (2009).\n[9] B. M. Peterson, New Astron. Rev. 52, 240 (2008); A. C. Fabian et al., Nature459, 540 (2009).\n[10] W. Kollatschny, Astron. Astrophys. 412, L61 (2003).18\n[11] S. S. Doeleman et al., Nature455, 78 (2008); A. E. Broderick, A. Loeb, R. Narayan, Astrophys. J. ,701, 1357 (2009).\n[12] B. Aschenbach, N. Grosso, D. Porquet, P. Predehl, Astron. Astrophys. 417, 71 (2004).\n[13] A. E. Broderick, A Loeb, Monthly Not. Royal Astron. Soc. 363, 353 (2005).\n[14] A. E. Broderick, A. Loeb, Astrophys. J. ,697, 1164 (2009).\n[15] D. Garofalo, Astrophys. J. 699, 400 (2009).\n[16] T. P. Krichbaum et al., J. Phys.: Conf. Ser. 54, 328 (2006); T. P. Krichbaum et al., ASP Conf. Ser. 386, 186 (2008).\n[17] J. P. Leahy, T. W. B. Muxlow, P. W. Stephens, Monthly Not. Royal Astron. Soc. 239, 401 (1989); C. L. Carilli, R. A.\nPerley, J. W. Dreher, J. P. Leahy, Astrophys. J. 383, 554 (1991); A. Celotti, A. C. Fabian, Monthly Not. Royal Astron.\nSoc.264, 228 (1993); W. J. Duschl, H. Lesch, Astron. Astrophys. 286, 431 (1994).\n[18] P. L. Biermann, R. G. Strom, H. Falcke, Astron. Astrophys. 302, 429 (1995); Gopal-Krishna, P. L. Biermann, P. J. Wiita,\nAstrophys. J. 603, L9 (2004).\n[19] H. Falcke, P. L. Biermann, Astron. Astrophys. 293, 665 (1995); H. Falcke, M. A. Malkan, P. L. Biermann, Astron.\nAstrophys. 298, 375 – 394 (1995).\n[20] D. Maitra, S. Markoff, C. Brocksopp, M. Noble, M. Nowak, J . Wilms, Monthly Not. Royal Astron. Soc. 398, 1638 (2009).\n[21] B. M. Barker, R. F. O’Connell, Phys. Rev. D 12, 329 (1975).\n[22] B. M. Barker, R. F. O’Connell, Gen. Relativ. Gravit. 11, 149 (1979).\n[23] L. E. Kidder, C. M. Will, A. G. Wiseman, Phys. Rev. D47, R4183 (1993).\n[24] T. A. Apostolatos, C. Cutler, G. J. Sussman, K. S. Thorne ,Phys. Rev. D 49, 6274 (1994); F. D. Ryan, Phys. Rev. D\n53, 3064 (1996); R. Rieth, G. Sch¨ afer, Class. Quantum Grav. 14, 2357 (1997); L. ´A. Gergely, Z. Perj´ es, M. Vas´ uth, Phys.\nRev.D57, 876 (1998); L. ´A. Gergely, Z. Perj´ es, M. Vas´ uth, Phys. Rev. D57, 3423 (1998); L. ´A. Gergely, Z. I. Perj´ es, M.\nVas´ uth, Phys. Rev. D58, 124001 (1998); R. F. O’Connell, Phys. Rev. Letters 93, 081103 (2004); C. M. Will, Phys. Rev.\nD71, 084027 (2005); J. Zeng, C. M. Will, Gen. Rel. Grav. 391661-1673 (2007); J. Maj´ ar, M. Vas´ uth, Phys. Rev. D77,\n104005 (2008); N. J. Cornish, J. Shapiro Key, Phys. Rev. D82044028 (2010).\n[25] L. E. Kidder, Phys. Rev. D52, 821 (1995).\n[26] T. A. Apostolatos, Phys. Rev. D52, 605 (1995); T. A. Apostolatos, Phys. Rev. D54, 2438 (1996); L. ´A. Gergely, Phys.\nRev.D61, 024035 (1999); L. ´A. Gergely, Phys. Rev. D62, 024007 (2000); B. Mik´ oczi, M. Vas´ uth, L. ´A. Gergely, Phys.\nRev.D71, 124043 (2005); H. Wang, C. M. Will, Phys. Rev. D75, 064017 (2007); J. Maj´ ar, Phys. Rev. D80, 104028\n(2009); A. Klein, Ph. Jetzer, Phys.Rev. D81124001 (2010).\n[27] E. Poisson, Phys. Rev. D57, 5287 (1998),\n[28] L.´A. Gergely, Z. Keresztes, Phys. Rev. D67, 024020 (2003).\n[29] E. E. Flanagan, T. Hinderer, Phys. Rev. D75, 124007 (2007); ´E. Racine, Phys. Rev. D78, 044021 (2008).\n[30] T. D. Newton, E. P. Wigner, Rev. Mod. Phys. 21, 400 (1949); M. H. L. Pryce, Proc. Roy. Soc. Ser. A195, 62 (1948).\n[31] Z. Keresztes, B. Mik´ oczi, L. ´A. Gergely, Phys. Rev. D72, 104022 (2005). The coefficients G2m2L2/c2µ2in the third line\nof Table I. and GµL/2c2rin the first line of Table II. should read as GmL2/c2µ2andGµ/2c2r, respectively; in Eq. (18)\nthe term (dt/dχ)PNshould have a factor of µ/Land the sign of ( dt/dχ)SOshould be switched.\n[32] Th. Damour, P. Jaranowski, G. Sch¨ afer, Phys. Rev. D77, 064032 (2008); J. Steinhoff, G. Sch¨ afer, S. Hergt, Phys. Rev. D\n77, 104018 (2008); Th. Damour, P. Jaranowski, G. Sch¨ afer, Phys. Rev. D78, 024009 (2008); E. Barausse, ´E. Racine, A.\nBuonanno, Phys. Rev. D80, 104025 (2009); J. Steinhoff, H. Wang, Phys. Rev. D81, 024022 (2010); X. Wu, Y. Xie, Phys.\nRev.D81, 084045 (2010).\n[33] L. Rezzolla, E. Barausse, E. Nils Dorband, D. Pollney, C . Reisswig, J. Seiler, S. Husa, Phys. Rev. D78044002 (2008);\nM. C. Washik , J. Healy, F. Herrmann, I. Hinder, D. M. Shoemake r, P. Laguna, R. A. Matzner, Phys. Rev. Lett. 101\n061102 (2008); A. Buonanno, L. E. Kidder, L. Lehner, Phys. Rev. D77026004 (2008); W. Tichy, P. Marronetti, Phys.\nRev.D78, 081501(R) (2008); E. Barausse, L. Rezzolla, Astrophys. J. Lett. 704L40-L44 (2009); U. Sperhake, V. Cardoso,\nF. Pretorius, E. Berti, T. Hinderer, N. Yunes, Phys. Rev. Lett. 103, 131102 (2009); J. Healy, P. Laguna, R. A. Matzner,\nD. M. Shoemaker, Phys. Rev. D81, 081501 (2010); E. Barausse, The importance of precession in modelling the direction\nof the final spin from a black-hole merger, E-print: arXiv:0911.1274 (2009); M. Kesden, U. Sperhake, E . Berti,Phys. Rev.\nD81, 084054 (2010); C. O. Lousto, M. Campanelli, Y. Zlochower, H . Nakano, Class. Quantum Grav. 27, 114006 (2010).\n[34] K. Glampedakis, D. Kennefick, Phys. Rev. D66, 044002 (2002); J. Levin, G. Perez-Giz, Phys. Rev. D79, 124013 (2009);\nG. Perez-Giz, J. Levin, Phys. Rev. D79, 124014 (2009).\n[35] J. Levin, R. Grossman, Phys. Rev. D79, 043016 (2009); R. Grossman, J. Levin, Phys. Rev. D79, 043017 (2009).\n[36] J. Healy, J. Levin, D. Shoemaker, Phys. Rev. Lett. 103, 131101 (2009).\n[37] J. P. Leahy, A. G. Williams, Mon. Not. Royal. Astron. Soc. 210, 929 (1984).\n[38] Ch. Zier, P. L. Biermann, Astron. Astroph. 377, 23 (2001); D. Merritt, R. Ekers, Science297, 1310 (2002); Gopal-Krishna,\nP. L. Biermann, P. J. Wiita, Astrophys. J. Letters 594, L103 (2003); F. K. Liu, Mon. Not. Royal. Astron. Soc. 347, 1357\n(2004); Gopal-Krishna, P. L. Biermann, L. ´A. Gergely, P. J. Wiita, On the origin of X-shaped radio galaxies , in preparation\n(2010).\n[39] J. P. Leahy, R. A. Perley, Astron. J 102, 537 (1991); A. R. S. Black, S. A. Baum, J. P. Leahy, R. A. Perle y, J. M. Riley,\nP. A. G. Scheuer, Mon. Not. Royal. Astron. Soc. 256, 186 (1992).\n[40] L.´A. Gergely, P. L. Biermann, L. I. Caramete, Class. Quantum Grav. 27, 194009 (2010).\n[41] L.´A. Gergely, Phys. Rev. D81, 084025 (2010). The power of νin Eq. (57) should read 3 −2i.\n[42] L. S. Finn, D. F. Chernoff, Phys. Rev. D47, 2198 (1993).\n[43] A. Buonanno, Y. Chen, M. Vallisneri, Phys. Rev. D67, 104025 (2003); Erratum-ibid. D 74, 029904 (2006).19\n[44] K. G. Arun, A. Buonanno, G. Faye, E. Ochsner, Phys. Rev. D79, 104023 (2009).\n[45] W. G. Laarakkers, E. Poisson, Astrophys. J. 512, 282 (1999).\n[46] K. S. Thorne, Rev. Mod. Phys. 52, 299 (1980).\n[47] F. Herrmann, I. Hinder, D. Shoemaker, P. Laguna, R. A. Ma tzner,Astrophys. J. 661, 430 (2007); J. D. Schnittman, A.\nBuonanno, J. R. van Meter, J. G. Baker, W. D. Boggs, J. Centrel la, B. J. Kelly, S. T. McWilliams, Phys. Rev. D77,\n044031 (2008).\n[48] J. A. Gonzalez, M. D. Hannam, U. Sperhake, B. Brugmann, S . Husa,Phys. Rev. Lett. 98, 231101 (2007); F. Herrmann,\nI. Hinder, D. Shoemaker, P. Laguna, R. A. Matzner, Phys. Rev. D76, 084032 (2007); M. Campanelli, C. O. Lousto,\nY. Zlochower, D. Merritt, Phys. Rev. Lett. 98, 231102 (2007); M. Campanelli, C. O. Lousto, Y. Zlochower, D . Merritt,\nAstrophys. J. 659, L5, (2007); S. Dain, C. O. Lousto, Y. Zlochower, Phys. Rev. D78, 024039 (2008); C. O. Lousto, Y.\nZlochower, Phys. Rev. D79, 064018 (2009).\n[49] J. D. Schnittman, A. Buonanno, Astrophys J. 662, L63 (2007); ´E. Racine, A. Buonanno, L. Kidder, Phys. Rev. D80,\n044010 (2009); Z. Keresztes, B. Mik´ oczi, L. ´A. Gergely, M. Vas´ uth, J. Phys.: Conf. Series 228, 012053 (2010)."    },    {        "title": "2205.09367v1.Spin_spin_coupling_based_quantum_and_classical_phase_transitions_in_two_impurity_spin_boson_models.pdf",        "content": "Spin-spin coupling-based quantum and classical phase transitions in two-impurity spin-boson\nmodels\nR. Grimaudo,1A. Messina,2H. Nakazato,3, 4A. Sergi,5, 6, 7and D. Valenti1\n1Dipartimento di Fisica e Chimica \\Emilio Segr\u0012 e\", Universit\u0012 a degli Studi di Palermo,\nviale delle Scienze, Ed. 18, I-90128, Palermo, Italy\n2Dipartimento di Matematica ed Informatica, Universit\u0012 a degli Studi di Palermo, Via Archira\f 34, I-90123 Palermo, Italy\n3Department of Physics, Waseda University, Tokyo 169-8555, Japan\n4Institute for Advanced Theoretical and Experimental Physics, Waseda University, Tokyo 169-8555, Japan\n5Dipartimento di Scienze Matematiche e Informatiche,\nScienze Fisiche e Scienze della Terra, Universit\u0012 a degli Studi di Messina,\nViale F. Stagno d'Alcontres 31, 98166 Messina, Italy\n6Istituto Nazionale di Fisica Nucleare, Sez. di Catania, 95123 Catania, Italy\n7Institute of Systems Science, Durban University of Technology, P.O. Box 1334, Durban 4000, South Africa\n(Dated: May 20, 2022)\nThe class of two-interacting-impurity spin-boson models with vanishing transverse \felds on the\nspin-pair is studied. The model can be exactly mapped into two independent standard single-\nimpurity spin-boson models where the role of the tunnelling parameter is played by the spin-spin\ncoupling. The dynamics of the magnetization is analysed for di\u000berent levels of (an)isotropy. Further,\nthe existence of a decoherence-free subspace as well as of both classical and quantum (\frst-order\nand Kosterlitz-Thouless type) phase transitions, in the Omhic regime, is brought to light.\nThe dynamics of any open quantum system is pro-\nfoundly in\ruenced by its surrounding environment which\nis at the origin of decoherence and/or dissipation man-\nifestation [1]. The former e\u000bect plays a leading role in\ndetermining the transition from quantum to classical be-\nhavior. In the last decades, it has attracted much atten-\ntion mostly in the \feld of quantum state manipulation\nand quantum computation [2].\nAn important model exhibiting quantum dissipation is\nthe so called single-impurity spin-boson model (SISBM)\nwhich describes a single spin-1/2 coupled to a bosonic\nquantum bath [3]. The SISBM has been thoroughly stud-\nied in wide regions of the parameter space with diverse\nmethods and techniques since the 1980s [3{6]. It encap-\nsulates e\u000bects stemming from quantum decoherence, dis-\nsipation, and relaxation on the otherwise coherent spin\nevolution [3]. Furthermore, the model exhibits a nontriv-\nial ground-state behavior, since it displays a quantum\nphase transition as a function of system-bath coupling\nstrength [7, 8], attributed to zero-point rather than ther-\nmal \ructuations within the bath [9{11]. Applications are\nnumerous, ranging from quantum optics to quantum in-\nformation and computation [12{30].\nThe interest towards decoherence and dissipation as\nwell as quantum phase transitions (QPTs) in two-\nimpurity spin-boson models (TISBMs) with competing\ninteractions has remarkably grown in the last two decades\n[31{44]. The TISBM is currently under attention to\ndetermine the existence of critical points and then the\npresence of quantum and/or classical phase transitions\n[32, 33]. On the basis of numerical approaches, di\u000berent\nresults have been proposed; however, until now, a uni-\nvocal response is missing [32, 33]. Moreover, a remark-\nable question to be addressed is whether, in presence of\nthe impurity-impurity coupling, the transition is of the\nKosterlitz-Thouless (K-T) type [32, 33].In this work we study the class of TISBMs useful for\ndescribing bi-nuclear units [45, 46], where the transverse\n(x) \feld on the spins is absent and a non-isotropic spin-\nspin Heisenberg interaction is considered. We show that\nthe dynamical problem can be exactly and analytically\nreduced to that of two independent SISBMs, wherein the\nrole of the transverse \feld is e\u000bectively played by the two-\nspin coupling(s). First, we bring to light the existence of\na decoherence-free subspace, charcterized then by a dissi-\npationless spin-dynamics. Further, basing on the results\npreviously obtained for the SISBM in the Ohmic case\n[3, 7, 8], we derive the behaviour of the magnetization of\nthe system as well as the presence of both classical and\nQPTs. In particular, two types of QPT are present: a\n\frst-order QPT (due to a level crossing) and a K-T QPT.\nModel. Consider the following model (in units of ¯h):\nH=W1\n2ˆsz\n1+W2\n2ˆsz\n2+N\nå\nj=1wjˆa†\njˆaj\u0000\ngx\n2ˆsx\n1ˆsx\n2\u0000gy\n2ˆsy\n1ˆsy\n2\u0000gzˆsz\n1ˆsz\n2+2\nå\nk=1N\nå\nj=1ck j\n2\u0010\nˆa†\nj+ˆaj\u0011\nˆsz\nk;\n(1)\nwhich describes two interacting spin-1/2's subject to lo-\ncal longitudinal ( z) \felds and coupled to a common bath\nof quantum harmonic oscillators. Wiandwjare the char-\nacteristic frequencies of the i-th spin and the j-th mode,\nrespectively. ˆsl\nk(k=1;2,l=x;y;z) are the Pauli opera-\ntors of the spins, while ajanda†\njare the annihilation and\ncreation boson operators of each \feld mode.\nThanks to the existence of the constant of motion ˆsz\n1ˆsz\n2,\nthe model can be unitarily transformed into eH=eHa\barXiv:2205.09367v1  [quant-ph]  19 May 20222\neHb,with\neHa=b=Wa=b\n2ˆsz\na=b\u0000ga=b\n2ˆsx\na=b\u0007gzˆ1a=b+\nN\nå\nj=1wjˆa†\njˆaj+N\nå\nj=1ca=b\nj\n2\u0010\nˆa†\nj+ˆaj\u0011\nˆsz\na=b;(2)\nwhere Wa=b=W1\u0006W2,gb=a=gx\u0006gy, and ca=b\nj=c1j\u0006c2j.\neHaandeHbare e\u000bective Hamiltonians governing the dy-\nnamics of the two-spin-boson system (TSBS) within each\ndynamically invariant subspace.\nThe two independent subdynamics are equivalent to\ntwo e\u000bective SISBMs: I) the coupling between the two\ntrue spins provides the e\u000bective transverse magnetic \feld\n(gaandgb); II) the longitudinal \feld results from precise\ncombinations ( WaandWb) of the two \felds applied to the\nactual spin-1/2's; III) the coupling with the quantum os-\ncillator bath is mediated by appropriate combinations ( ca\nj\nandcb\nj) of the coupling parameters of the two spin-1/2's\nwith each boson mode. Therefore, all the results obtained\nfor the SISBM can be applied to each subdynamics and\nexploited to get information about the TSBS dynamics.\nDecoherence-Free Subspace. Ifcb\nj=0the subspace b\nis a decoherence-free subspace. It means that, although\nthe actual spins interact with the bath, they experience a\ndissipationless dynamics within such a subspace as if the\nbath were absent. Therefore, the initial or produced en-\ntanglement between the spins would not degrade, despite\nthe presence of the spin-bath coupling term.\nThis circumstance is of remarkable importance in\nquantum computation, where controlling the dissipative\nspin-spin-boson dynamics in nonequilibrium conditions,\ne.g., in the presence of time-dependent external \felds, is\ncrucial [39, 43]. We emphasize that the analytical treat-\nment employed to unitary transform the TISBM is not\na\u000bected by any time-dependence of the Hamiltonian pa-\nrameters.In this way, appropriate time variations of the\nlocal \felds and/or the coupling parameters can be engi-\nneered in order to generate unperturbed quantum gates\nacting on the two spins.\nConditions. Consider the bath in a thermal state\nand the two-spin system prepared in the state r(0) =\nj++ih++j, which is mapped to the single-spin state\nra(0) =j+iah+j. In this instance, the dynamics is entirely\nrestricted to the subspace a(spanned byfj++i;j\u0000\u0000ig )\nand the mean value of the total magnetization hˆSzi\u0011\nhˆsz\n1i+hˆsz\n2i, as well as the mean value of the single mag-\nnetizations of the two spins, can be easily obtained from\nhˆsz\nai:\nTrfˆra(t)ˆsz\nag=hˆsz\nai=hˆsz\n1i=hˆsz\n2i=hˆSzi\n2: (3)\nIn the asymptotic low-temperature limit the bath spec-\ntral density function J(w) =påj(ca\nj)2d(w\u0000wj) is deter-\nmined by the low-energy part of the spectrum and its\nstandard parametrization is\nJ(w) =2paw1\u0000s\ncws; 0<w<wc;s>\u00001;(4)where wcis a cut-o\u000b frequency and ais the dimension-\nless parameter accounting for the dissipation strength [3].\nThe spectral exponent sde\fnes three regimes: Ohmic\n(s=1), sub-Ohmic ( s<1) and super-Ohmic ( s>1).\nOhmic Regime. The ohmic case ( s=1) presents a large\nvariety of di\u000berent behaviours depending on the region of\nthe parameter space taken into account [3]. First, con-\nsider the case W1=W2=0. We underline that all the fol-\nlowing results are valid under two conditions [3]: 1) Wa=b,\ngb=aandkBTare small compared to the bath cut-o\u000b fre-\nquency wc; 2) the `interesting' times are large compared\ntow\u00001\nc[3].\nFora=1=2the two-spin magnetization reads [3]\nhˆSz(t)i=2exp\u001a\n\u0000p\n2g2\na\nwct\u001b\n: (5)\nThis result is valid at all temperatures ( T=0andT6=0)\ncompatible with the condition kBT\u001cwc[3]. The expo-\nnential decaying rate of the two-spin probability depends\non the ratio g2\na=wc, meaning that the characteristic time-\nscale of the system is determined by the spin-spin cou-\npling parameter. Precisely, it depends on the di\u000berence\ngx\u0000gy, so that: i) in case of isotropy ( gx=gy) the system\ntends to remain in its initial state; ii) a slight di\u000berence\nbetween the two coupling parameters, instead, causes an\nexponential decay of the magnetization(s) towards the\nequilibrium value.\nFora<1(6=1=2) two cases can be considered: kBT&\negaandkBT.ega, with [3]\nega=ga\u0012ga\nwc\u0013a=1\u0000a\n: (6)\nIn the \frst case the magnetization results to be [3]\nhˆSz(t)i=2expf\u0000t=tg; (7a)\nt\u00001=pp\n2G(a)\nG(a+1=2)g2\na\nwc\u0012pkBT\nwc\u00132a\u00001\n; (7b)\nwhere Gis the gamma function. The previous expression\ndescribes an exponential relaxation with a rate µT2a\u00001.\nIn the second case: i) if 1=2<a<1the time behav-\nior is most likely an incoherent relaxation with an a-\ndependent rate of order eg\u00001\na; ii) if 0<a<1=2the system\nexhibits damped incoherent oscillations [3]. These re-\nsults show that, depending on the ratio of the spin-spin\nenergy coupling to the thermal energy, di\u000berent dynam-\nics arise. Therefore, the spin-spin interaction, besides\nthe decaying rate, sets the limit temperature dividing\nthe two dynamical regions. Physical systems character-\nized by di\u000berent couplings exhibit thus a di\u000berent criti-\ncal temperature and/or di\u000berent behaviours at the same\ntemperature. Three scenarios can be considered. In nu-\nclear magnetic resonance, the spin-spin coupling typically\nranges from 10 Hz to 300 Hz, depending on the molecule\n[47]. For microwave-driven trapped ions the interaction\nstrength can reach the kHz range [48]. Rydberg atoms3\nand ions, due to the huge electric-dipole moments of the\nRydberg states, are characterized by an e\u000bective spin-\nspin coupling which can reach a few MHz [49, 50]. In\nthe three cases the critical temperature Tc=ega=kBsepa-\nrating the two dynamical regimes results Tc\u00190:1\u00001nK,\nTc\u001910nK,Tc\u001910mK, respectively.\nWhen a>1: i) if T6=0, the two-spin dynamics consists\nin the same exponential relaxation written in Eq. (7),\ncharacterized by a rate µT2a\u00001[3]; ii) for T=0, instead,\nthe two-spin system experiences the localization regime,\nthat is, it is frozen in its initial condition [3].\nIt is worth noticing that all the previous dynamical\nbehaviours rely only on internal parameter characterizing\nthe physical system: the spin-spin coupling, the spin(s)-\nbath coupling and the bath cut-o\u000b frequency. This aspect\nsuggests a sort of self-organization of the system and an\nauto-determination of the system dynamics.\nAn appropriate non-vanishing bias ( Wa\u001cwc), large\ncompared to the renormalized tunneling frequency ( W\u001d\nega), makes the system to relax from the upper to the lower\nstate, even at zero temperature [3]. Thus, the physical\ne\u000bect of a su\u000ecient bias is that to suppress the coherent\noscillations shown, in some cases, by the unbiased system.\nMixing Subspaces . All the previous results can be even\napplied on the subdynamics b, withj+\u0000ias initial state\nof the two spins and Waandgareplaced with Wbandgb,\nrespectively. Of course, the Hamiltonian parameters are\nconstrained to ful\fll gb\u001cwcandWa\u001cwc. In this case\nnet magnetization vanishes, hˆSzi=0, sincehˆsz\n1i=\u0000hˆsz\n2i.\nIf we consider the initial condition ( j++i+j+\u0000i)=p\n2,\nboth subspaces are involved. In this circumstance, we\nhave to indipendently solve the dynamics in each sub-\nspace and `merge' the results. It must be taken into\n01(a) x=0.01c;y=0\n<z\n1>\n<z\n2>\n<z>\n01(b) x=2y=0.01c\n0 500 1000 1500 2000 2500 3000\nct\n01(c) x=y=0.01c\nAverage Magnetizations\nFigure 1: Time behaviour of both the single-spin and\nthe net magnetizations for s=1,aa=1=2, and three\nlevels of isotropy: (a) minimum, (b) intermediate, (c)\nmaximum. The two spins start from ( j++i+j+\u0000i)=p\n2,\nwhile the bath from the thermal state.account that the two subdynamics are characterized by\ndi\u000berent (e\u000bective) couplings with the bath and then by\ndi\u000berent as, say aaandab. Consider the following case:\ns=1,aa=1=2andab=0. The latter condition stems\nfrom c1j=c2j;8j, so that cb\nj=0.\nThe time behaviour of the net magnetization hˆSziis\ndetermined by the time evolution in the subspace asince\nno contribution stems from the subspace b. Therefore,\nthis time, the value of the magnetization is half times\nthat obtained when the system is initially prepared in\nj++i[Eq. (5)].\nRather, a relevant di\u000berence is found for hˆsz\n1iand\nhˆsz\n2i. Previously indeed we had hˆsz\n1i=hˆsz\n2i=hˆsz\nai(Eq.\n(5)). Now, the time-behaviour of each spin magnetiza-\ntion reads\nhˆsz\n1;2i=hˆsz\nai\u0006h ˆsz\nbi\n2=e\u0000p\n2g2a\nwct\u0006cos(gbt)\n2: (8)\nThe cosin -term stems from the exact solution of the de-\nterministic dynamics in the subspace b.\nFrom Figs. 1(a) and 1(b) we see that the level of\nanisotropy in\ruences both the frequency of the single-\nspin oscillations and the decaying rate of the net mag-\nnetization. In case of isotropy ( gx=gy), instead, the\ntwo spins exhibit dissipationless oscillations and the net\nmagnetization is constant. This circumstance is related\nto the vanishing value of gawhich rules the exponen-\ntial relaxation (playing the role of the e\u000bective transverse\n\feld in the subdynamics a). Therefore, by studying both\nthe oscillation frequency of each spin magnetization and\nthe exponential decaying rate of the net magnetization,\nthe coupling parameters gxandgyand then the level of\n(an)isotropy of the two-spin system can be estimated.\nQPT. Depending on the parameter-space region, the\nground state (GS) of the two-spin system belongs to ei-\nther the aorbspace and coincides with the GS of the\n\fctitious spin-qubit aorb. In this way, we can write the\ntwo possible GSs and the related ground energies of the\nTISBM on the basis of the expressions obtained for the\nSISBM. The ansatz proposed in Ref. [8] provides, in the\nOmhic case, a good approximation of both the GS and\nthe ground energy of the SISBM. Basing on these results,\nthe GS of the TISBM, in the Omhic regime, results to\nhave one of the following forms\njGSia=Aa\"\nj++iÕ\nkj0+\nki#\n+Ba\"\nj\u0000\u0000iÕ\nkj0\u0000\nki#\n;(9a)\njGSib=Ab\"\nj+\u0000iÕ\nkj0+\nki#\n+Bb\"\nj\u0000+iÕ\nkj0\u0000\nki#\n;(9b)\nwherej0\u0006\nki=D(b\u0006\nk)j0i[8];j0istands for the GS of the\nquantum oscillator bath, D(bk) =expfbk(ˆa†\nk\u0000ˆak)g(bk\nreal) are the bosonic displacement operators, and with4\n0.0 0.5 1.0\n4\n2\n0/c\n(a)×101\nx=2y=102c\n1=22=107c\nk=0.0\nk=0.25\nk=0.5\nk=0.750.00 0.05\n0.0 0.5 1.0\n4\n2\n0/c\n(b)×101\nx=2y=103c\n1=22=109c\nk=0.0\nk=0.25\nk=0.5\nk=0.750.000 0.005\n0.0 0.5 1.0\n4\n2\n0/c\n(c)×101\nx=2y=103c\n1=2=102c\nk=0.0\nk=0.25\nk=0.5\nk=0.75\nFigure 2: Dependence of L=la\n0\u0000lb\n0ona=aa=ab=k. The black horizontal line represents L=0.\nAa=b=\u0000(1+Ra=b)Wa=b\u0000ha=b\nNa=b; Ba=b=g0\na=b\nNa=b;(10a)\nRa=b=2aa=bwc\nca=b+wc; ca=b=q\n(g0\na=b)2+W2\na=b; (10b)\nha=b=q\n(g0\na=b)2+W2\na=b(1+Ra=b)2; (10c)\ng0\na=b=ga=b\u0012ca=b\nca=b+wc\u0013aa=b\nexpfaa=bwc=(ca=b+wc)g;\n(10d)\nNa=bbeing the normalization factors. The self-consistent\nequation for g0\nb=a, in the scaling limit ca=b=wc!0, reads\ng0\na=b= \nga=beaa=b\nwaa=b\nc!1=1\u0000aa=b\n;Wa=b\u001cTa=b\nK;(11a)\ng0\na=b=ga=b\u0012Wa=b\nwc\u0013aa=b\n; Wa=b\u001dTa=b\nK;(11b)\nwhere Ta=b\nK=ga=b(ga=b=D)aa=b=1\u0000aa=bis the Kondo energy\nwhich scales the energies of the system, with Dbeing\na cuto\u000b introduced to regularize the integral for the\nground-state energy [7, 8]. We underline that, in our\ncase, the scaling Kondo energy strictly depends on the\nspin-spin interaction strength.\nThe related energies can be cast as follows\nla=b\n0=1\n2 aa=bwc(W2\na=b\u0000ca=bwc)\nca=b(ca=b+wc)\u0000ha=b!\n: (12)\nTherefore the spectrum of the TISBM is constituted by\ntwo sets of eigenvalues, each of which is the spectrum of\nthe related e\u000bective SISBM.\nSince the latter does not present level crossing, by\nstudying the di\u000berence L\u0011la\n0\u0000lb\n0the subspace where\nthe GS of the TSBS is placed can be deduced. In Fig.\n2(a) the dependence of Lona=aa=ab=kfor di\u000ber-\nent values of the parameter k\u0011ab=aais shown. Thea-dependent straight lines emerge by considering the ex-\npression of Lin the limits Wa=b!0andga=b\u001cwc, which\nbecomes\nL\u0019(k\u00001)wca+[g0\nb(a)\u0000g0\na(a)]\n2: (13)\nThe presence of a QPT at ac2[0:01;0:05] is clearly vis-\nible, as well as the dependence of the critical point ( ac) on\nk. The GS of the TSBS is then placed in the b(a) space\nfora<ac(a>ac). The critical value ac, for which such\na QPT occurs, is sensitive to the Hamiltonian param-\neters, as shown in Fig. 2(b), where ac2[0:001;0:005].\nThis circumstance can be traced back to the di\u000berent\nvalue taken on in the two cases by the intercept of the\nstraight lines in Eq. (13), which is ( gb\u0000ga)=2=gy, with\nthe slope remaining unchanged. In the isotropic case\ngx=gy(ga=0), the QPT is still present since the in-\ntercept is gb=2.\nIn the QPT the spin-pair moves from the vanishing ( b)\nto the non-vanishing ( a) magnetization subspace. In this\nway, the total magnetization plays the role of the order\nparameter. It jumps from zero to a constant value, which\ncan be derived on the basis of the result obtained for the\nSISBM, namely [7, 8]:\nhˆSzi=\u0000Cz(a)Wa\nTa\nK;Wa\u001cTa\nK; (14a)\nCz(a) =4eb\n2(1\u0000a)\nppG[1+1=(2\u00002a)]\nG[1+a=(2\u00002a)]; (14b)\nb=aln(a)+(1\u0000a)ln(1\u0000a): (14c)\nThe transition can be then classi\fed as a \frst-order QPT\n[9].\nWhen W1;W2\u001dgx;gyas in Fig. 2(d), no QPT occurs:\nthe GS of the TSBS is jGSiaand the net magnetiza-\ntion expression is equal to Eq. (14). This is due to the\nfact that, in this case, Lcan be approximated as in Eq.\n(13), but this time the intercept takes on the negative\nvalue\u0000W2. For homogeneous magnetic \felds, W1=W2\n(Wb=0), the QPT is still absent since the intercept reads\n\u0000Wa=2.5\nIn the limit W1;W2!0,a=1is another critical point.\nFor the SISBM in the Ohmic case, by mapping the\nmodel onto the anisotropic Kondo model with bosoniza-\ntion techniques, it has been proved that the K-T tran-\nsition is present at a=1(in the scaling limit and for\nvanishing z-magnetic \feld) [3]. This critical value of a\nseparates a localized phase at a>1(the spin is inj+i\norj\u0000i), characterized by a renormalized vanishing tun-\nnel splitting, from a delocalized phase at a<1with an\ne\u000bective non-vanishing tunnel energy [3]. Therefore, bas-\ning on our approach, we can claim that at a=1the\nTSBS undergoes a quantum phase transition of the K-T\ntype, with a consequent localization of the two spins in\nthe statej++iorj\u0000\u0000i (the \fctitious spin-1/2 alocalizes\ninj+iaorj\u0000ia). In our case, the tunneling parameter\nwhich renormalizes to 0 for a>1, causing the localiza-\ntion phenomenon, consists in the spin-spin coupling. We\npoint out that, with respect to the previous works [31],we obtain a di\u000berent critical value of afor which a K-\nT transition occurs in the TISBM. However, it must be\nreminded that our TISBM is di\u000berent from those until\nnow analysed: the absence of an external transverse \feld\napplied to the spin-pair causes such a remarkable di\u000ber-\nence. The interesting aspects of the present model are\nboth the possibility of rigorously deriving the existence\nof a K-T transition at a=1and the fact that such a tran-\nsition relies on the presence of a non-vanishing transverse\nspin-spin interaction.\nIn conclusion, the present work, besides reporting non-\ntrivial dynamical e\u000bects emerging in the TISBM, has\nshown the potentiality of the exact approach used. The\nlatter, exploited in other parameter-space regions, as well\nas in the sub-Omhic and super-Omhic regimes, can lead\nto a plethora of new results based on those obtained for\nthe SISBM.\n[1] H.-P. Breuer and F. Petruccione,\nThe theory of open quantum systems (Oxford Uni-\nversity Press on Demand, 2002).\n[2] M. A. Nielsen and I. L. Chuang,\nQuantum Computation and Quantum Information\n(Cambridge University Press, 2010).\n[3] A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A.\nFisher, A. Garg, and W. Zwerger, Rev. Mod. Phys. 59,\n1 (1987).\n[4] K. Le Hur, P. Doucet-Beaupr\u0013 e, and W. Hofstetter, Phys.\nRev. Lett. 99, 126801 (2007).\n[5] M. Vojta, N.-H. Tong, and R. Bulla, Phys. Rev. Lett.\n94, 070604 (2005).\n[6] R. Bulla, N.-H. Tong, and M. Vojta, Phys. Rev. Lett.\n91, 170601 (2003).\n[7] K. L. Hur, Ann. Phys. (NY) 323, 2208 (2008).\n[8] A. Nazir, D. P. S. McCutcheon, and A. W. Chin, Phys.\nRev. B 85, 224301 (2012).\n[9] M. Vojta, Rep. Prog. Phys. 66, 2069 (2003).\n[10] D. Rossini and E. Vicari, Phys. Rep. 936, 1 (2021), co-\nherent and dissipative dynamics at quantum phase tran-\nsitions.\n[11] A. Carollo, D. Valenti, and B. Spagnolo, Phys. Rep. 838,\n1 (2020).\n[12] S. Kundu and N. Makri, J. Phys. Chem. B 125, 8137\n(2021).\n[13] A. J. Dunnett and A. W. Chin, Entropy 23(2021),\n10.3390/e23010077.\n[14] A. Lemmer, C. Cormick, D. Tamascelli, T. Schaetz, S. F.\nHuelga, and M. B. Plenio, New J. Phys. 20, 073002\n(2018).\n[15] S. Lerma-Hern\u0013 andez, D. Villase~ nor, M. A. Bastarrachea-\nMagnani, E. J. Torres-Herrera, L. F. Santos, and J. G.\nHirsch, Phys. Rev. E 100, 012218 (2019).\n[16] J. Lepp akangas, J. Braum uller, M. Hauck, J.-M. Reiner,\nI. Schwenk, S. Zanker, L. Fritz, A. V. Ustinov, M. Weides,\nand M. Marthaler, Phys. Rev. A 97, 052321 (2018).\n[17] R. Puebla, J. Casanova, O. Houhou, E. Solano, and\nM. Paternostro, Phys. Rev. A 99, 032303 (2019).\n[18] S. Wenderoth, H.-P. Breuer, and M. Thoss, Phys. Rev.A104, 012213 (2021).\n[19] L. Magazz\u0012 u, S. Denisov, and P. H anggi, Phys. Rev. E\n98, 022111 (2018).\n[20] G. De Filippis, A. de Candia, L. M. Cangemi, M. Sassetti,\nR. Fazio, and V. Cataudella, Phys. Rev. B 101, 180408\n(2020).\n[21] Y.-Z. Wang, S. He, L. Duan, and Q.-H. Chen, Phys. Rev.\nB100, 115106 (2019).\n[22] Y.-Z. Wang, S. He, L. Duan, and Q.-H. Chen, Phys. Rev.\nB101, 155147 (2020).\n[23] L.-T. Shen, J.-W. Yang, Z.-R. Zhong, Z.-B. Yang, and\nS.-B. Zheng, Phys. Rev. A 104, 063703 (2021).\n[24] E. Aurell, B. Donvil, and K. Mallick, Phys. Rev. E 101,\n052116 (2020).\n[25] A. Miessen, P. J. Ollitrault, and I. Tavernelli, Phys. Rev.\nResearch 3, 043212 (2021).\n[26] D. Villase~ nor, S. Pilatowsky-Cameo, M. A. Bastarrachea-\nMagnani, S. Lerma-Hern\u0013 andez, L. F. Santos, and J. G.\nHirsch, New J. Phys. 22, 063036 (2020).\n[27] M. Pino and J. J. Garc\u0013 \u0010a-Ripoll, New J. Phys. 20, 113027\n(2018).\n[28] L. Magazz\u0012 u, P. Forn-D\u0013 \u0010az, R. Belyansky, and et al. , Nat.\nCommun. 9(2018), 10.1038/s41467-018-03626-w.\n[29] N. Lambert, S. Ahmed, M. Cirio, and et al. , Nat. Com-\nmun. 10(2019), 10.1038/s41467-019-11656-1.\n[30] J. Casanova, R. Puebla, H. Moya-Cessa, and et al. , npj\nQuantum. Inf. 4(2018), 10.1038/s41534-018-0096-9.\n[31] D. Dolgitzer, D. Zeng, and Y. Chen, Opt. Express 29,\n23988 (2021).\n[32] Y.-Z. Wang, S. He, L. Duan, and Q.-H. Chen, Phys. Rev.\nB103, 205106 (2021).\n[33] N. Zhou, Y. Zhang, Z. L u, and Y. Zhao, Ann. Phys.\n(Berlin) 530, 1800120 (2018).\n[34] P. N agele and U. Weiss, Physica E: Low-dim. Syst.\nNanostruct. 42, 622 (2010), proceedings of the interna-\ntional conference Frontiers of Quantum and Mesoscopic\nThermodynamics FQMT '08.\n[35] M. J. Storcz, F. Hellmann, C. Hrelescu, and F. K. Wil-\nhelm, Phys. Rev. A 72, 052314 (2005).\n[36] M. Garst, S. Kehrein, T. Pruschke, A. Rosch, and M. Vo-6\njta, Phys. Rev. B 69, 214413 (2004).\n[37] D. P. S. McCutcheon, A. Nazir, S. Bose, and A. J. Fisher,\nPhys. Rev. B 81, 235321 (2010).\n[38] J. Bonart, Phys. Rev. B 88, 125139 (2013).\n[39] P. P. Orth, D. Roosen, W. Hofstetter, and K. Le Hur,\nPhys. Rev. B 82, 144423 (2010).\n[40] H. Zheng, Z. L u, and Y. Zhao, Phys. Rev. E 91, 062115\n(2015).\n[41] A. Winter and H. Rieger, Phys. Rev. B 90, 224401 (2014).\n[42] P. N agele, G. Campagnano, and U. Weiss, New J. Phys.\n10, 115010 (2008).\n[43] M. Thorwart and P. H anggi, Phys. Rev. A 65, 012309\n(2001).\n[44] M. J. Storcz and F. K. Wilhelm, Phys. Rev. A 67, 042319\n(2003).\n[45] R. Calvo, J. E. Abud, R. P. Sartoris, and R. C. Santana,Phys. Rev. B 84, 104433 (2011).\n[46] L. M. B. Napolitano, O. R. Nascimento, S. Cabaleiro,\nJ. Castro, and R. Calvo, Phys. Rev. B 77, 214423 (2008).\n[47] L. M. K. Vandersypen and I. L. Chuang, Rev. Mod. Phys.\n76, 1037 (2005).\n[48] S. Weidt, J. Randall, S. C. Webster, K. Lake, A. E. Webb,\nI. Cohen, T. Navickas, B. Lekitsch, A. Retzker, and\nW. K. Hensinger, Phys. Rev. Lett. 117, 220501 (2016).\n[49] A. Gaetan, Y. Miroshnychenko, T. Wilk, A. Chotia,\nM. Viteau, D. Comparat, P. Pillet, A. Browaeys, and\nP. Grangier, Nat. Phys. 5, 115 (2009).\n[50] E. Urban, T. A. Johnson, T. Henage, L. Isenhower,\nD. Yavuz, T. Walker, and M. Sa\u000bman, Nat. Phys. 5,\n110 (2009)."    },    {        "title": "1112.0667v2.Spectroscopy_of_composite_solid_state_spin_environments_for_improved_metrology_with_spin_ensembles.pdf",        "content": "arXiv:1112.0667v2  [cond-mat.mes-hall]  7 Dec 2011Spectroscopy of composite solid-state spin environments f or\nimproved metrology with spin ensembles\nN. Bar-Gill,1,2L. M. Pham,3C. Belthangady,1D. Le Sage,1P.\nCappellaro,4J. R. Maze,5M. D. Lukin,2A. Yacoby,2and R. Walsworth1,2\n1Harvard-Smithsonian Center for Astrophysics, Cambridge, MA 02138, USA\n2Department of Physics, Harvard University, Cambridge MA, U SA\n3School of Engineering and Applied Sciences, Harvard Univer sity, Cambridge, MA 02138, USA\n4Department of Nuclear Science and Engineering, MIT, Cambri dge MA, USA\n5Department of Physics, Pontificia Universidad Catolica de C hile, Santiago 7820436, Chile\nFor precision coherent measurements with ensembles of quan tum spins the relevant Figure-of-\nMerit (FOM) is the product of polarized spin density and cohe rence lifetime, which is generally\nlimited by the dynamics of the spin environment. Here, we app ly a coherent spectroscopic technique\nto characterize the dynamics of the composite solid-state s pin environment of Nitrogen-Vacancy\n(NV) centers in room temperature diamond. For samples of ver y different NV densities and impurity\nspin concentrations, we show that NV FOM values can be almost an order of magnitude larger than\npreviously achieved in other room-temperature solid-stat e spin systems, and within an order of\nmagnitude of the state-of-the-art atomic system. We also id entify a new mechanism for suppression\nof electronic spin bath dynamics in the presence of a nuclear spin bath of sufficient concentration.\nThis suppression could inform efforts to further increase th e FOM for solid-state spin ensemble\nmetrology and collective quantum information processing.\nUnderstanding and controlling the coherence of quan-\ntum spins in solid-state systems is crucial for preci-\nsion metrology [1–3], quantum information science [4–\n6] and basic research on quantum many-body dynam-\nics [7, 8]. Examples of such systems include Nitrogen-\nVacancy (NV) color centers in diamond [9–11], phospho-\nrous donors in silicon [12, 13] and quantum dots [14]. A\nFigure-of-Merit (FOM) for precision coherent measure-\nments with quantum spins is the product ofthe polarized\nspin density ( nNV) and the coherence lifetime ( T2), with\nthe phase-shift sensitivity δφscaling as [1, 15, 16]\nδφ∝1√nNVT2≡1√\nFOM. (1)\nIncreasing this FOM requires an understanding of the\nsources of decoherence in the system, and their interplay\nwith spin density. For example, in the solid state T2is\ntypicallylimited byinteractionwith anenvironment(i.e.,\nbath) of paramagnetic spin impurities; whereas nNVis\nlimited by fabrication issues, the associated creation of\nadditional spin impurities in the environment, and the\nability to polarize (state prepare) the quantum spins.\nThe paradigm of a central spin coupled to a spin envi-\nronment has been studied intensively for many years (see\ne.g., [17, 18]); and quantum control methods have been\ndeveloped to extend the spin coherence lifetime by re-\nducing the effective interaction with the environment. In\nparticular, dynamical decoupling techniques pioneered in\nthe field of NMR [19–22] have recently been applied suc-\ncessfully to extend the effective T2of single NV-diamond\nelectronic spins by more than an order of magnitude [23–\n25].\nHere we study experimentally the FOM of ensembles\nof NV color centers in diamond (Fig. 1(a,b)) as a systemfor precision spin metrology such as magnetometry. To\ncharacterize the dynamics of the composite solid-state\nspin bath that limits the NV electronic spin coherence\ntime, consisting of both electronic spin (N) and nuclear\nspin (13C) impurities, we apply the technique of spec-\ntral decomposition [26, 27]. We describe this technique\nand its experimental application to three different dia-\nmond samples with a wide range of NV densities and\nimpurity spin concentrations (see Table 1). With the\nuse of dynamical decoupling we realize very large values\nfor the spin coherence FOM ∼2×1014[ms/cm3] for two\nof these diamond samples, which is almost an order of\nmagnitude larger than in other room-temperature solid-\nstatesystems[28]; threeordersofmagnitudegreaterthan\nhas been realized in NV-diamond systems for which T2\nhas been measured [15, 29]; and comparable to high-NV-\ndensity samples for which only the inhomogeneouscoher-\nence lifetime ( T∗\n2) was measured [30]. In addition, the\nlarge NV FOM values realized here are only an order of\nmagnitude smaller than the state-of-the-art alkali-vapor\nSERF (Spin Exchange Relaxation Free) magnetometer\n[16], which suffers from additional practical limitations\ncompared to room-temperature solid-state devices. We\nalso find unexpectedly long correlationtimes for the elec-\ntronic spin baths in two diamond samples with natural\nabundance of13C nuclear spin impurities. We identify a\nnew mechanism involving an interplay between the elec-\ntronic and nuclear spin baths that can explain the ob-\nserved suppression of electronic spin bath dynamics. In\nfuture work this effect could be used to engineer dia-\nmond samples for even larger NV spin FOM exceeding\nthecurrentstate-of-the-art,e.g., forprecisionmagnetom-\netry and collective quantum information applications.\nThe coherence of a two-level quantum system can be2\n(a)\n (b)\n(c)\n (d)\nFIG. 1. Quantum spin in a solid-state environment (NV-\ncenter in diamond) and the applied spin-control pulse-\nsequences. (a) Lattice structure of diamond with a Nitrogen -\nVacancy (NV) color center. (b) Magnetic environment of the\nNV center electronic spin:13C nuclear spin impurities and N\nelectronic spin impurities. (c) Energy-level schematic of the\nNV center. (d) Hahn-echo and multi-pulse (CPMG) spin-\ncontrol sequences.\nrelated to the magnitude of the off-diagonal elements of\nthe system’s density matrix. Specifically, we deal here\nwith an ensemble of NV electronic spins in a finite ex-\nternal magnetic field, which can be treated as an effec-\ntive two-level ensemble spin system with quantization\n(z) axis aligned with the nitrogen-vacancy axis. When\nthe NV spins are placed into a coherent superposition\nof spin eigenstates, e.g., aligned with the x-axis of the\nBloch sphere, the measurable spin coherence is given by\nC(t) =Tr[ρ(t)Sx]. Due to coupling of the NV spins to\ntheir magnetic environment, the coherence is lost over\ntime, with the general form\nC(t) =e−χ(t), (2)\nwhere the functional χ(t) describes the time dependence\nof the decoherence process. In the presence of a modu-\nlation acting on the NV spins (e.g., a resonant RF pulse\nsequence), described by a filter function in the frequency\ndomainFt(ω) (see below), it can be shown that the de-\ncoherence functional is given by [31, 32]\nχ(t) =1\nπ/integraldisplay∞\n0dωS(ω)Ft(ω)\nω2, (3)\nwhereS(ω) is the spectral function describing the cou-\npling of the system to the environment. Eq. (3) holds\nin the approximation of weak coupling of the NV spinsto the environment, which is appropriate for the (dom-\ninantly) electronic spin baths of the diamond samples\ndiscussed here [18] (see below).\nThe spectral decomposition technique allows us to de-\ntermineS(ω) from straightforwarddecoherence measure-\nments of an ensemble of NV spins, as outlined here. If\nwe describe the magnitude of the coupling between the\nNV spins and the spin bath by a fluctuating energy term\nǫ(t), thenS(ω) is the Fourier transform of the second-\norder correlator of ǫ(t):\nS(ω) =1\n¯h2/integraldisplay∞\n0dteiωt∝angbracketleftǫ(0)ǫ(t)∝angbracketright. (4)\nIt can be seen from Eq. (3) that if an appropri-\nate modulation is applied to the NV spins such that\nFt(ω)/(ω2t) =δ(ω−ω0), i.e., if a Dirac δ-function (or\nclose approximation) is localized at a desired frequency\nω0, thenχ(t) =tS(ω0)/π. Therefore, by measuring the\ntime-dependence of the spin ensemble’s coherence C(t)\nwhen subjected to such a “spectral δ-function” modula-\ntion, we can extract the spin bath’s spectral component\nat frequency ω0:\nS(ω0) =−πln(C(t))/t. (5)\nThis procedure can then be repeated for different values\nofω0to provide complete spectral decomposition of the\nspin environment.\nA close approximation to the ideal spectral filter func-\ntionFt(ω)describedabovecanbeprovidedbyavariation\non the well-known CPMG pulse sequence for dynamical\ndecoupling of a spin system from its environment [21]\n(Fig. 1(d)). The CPMG pulse sequence is an exten-\nsion of the Hahn-echo sequence [19] (Fig. 1(d)), with n\nequally-spaced π-pulses applied to the system after ini-\ntially rotatingit into the x-axis with a π/2-pulse. The re-\nsulting filter function for this sequence FCPMG(ω) covers\na narrow frequency region (given by π/t, wheretis total\nlength of the sequence) which is centered at ω0=πn/t,\nand is given by [33]:\nFCPMG\nn(ωt) = 8sin2/parenleftbiggωt\n2/parenrightbiggsin4/parenleftbigωt\n4n/parenrightbig\ncos2/parenleftbigωt\n2n/parenrightbig.(6)\nWe note that the narrow-band feature of the CPMG fil-\nteressentiallydefinestheeffectivenessofthissequencefor\ndynamical decoupling. We apply a deconvolution proce-\ndure to correct for deviations of this filter function from\nthe ideal Dirac δ-function (see Supplement).\nThe composite solid-state spin environment of interest\nhere, which is dominated by a bath of ���uctuating N elec-\ntronic spin impurities, causes decoherence of the probed\nNV spins through magnetic dipolar interactions. In the\nregime of low external magnetic fields and room tem-\nperature (relevant to the present experiments), the bath\nspins are randomly oriented, and their flip-flops (spin3\nTABLE I. Comparison of key characteristics and extracted\n“average-fit” Lorentzian spin bath parameters for the NV-\ndiamond samples studied in this work.\n12C Apollo HPHT\nMeas. technique ensemble ensemble confocal\nN concentration ∼1 ppm ∼100 ppm ∼50 ppm\nNV density ∼1014[cm−3]∼1016[cm−3]∼1012[cm−3]\n13C concentration 0.01% 1.1% 1.1%\nEcho (1-pulse) T2240(6)µs 2(1)µs 5(1)µs\nT2scaling n0.43(6)n0.65(5)n0.7(1)\nMax. achieved T22.2 ms 20µs 35µs\n∆ (expected) ≈60 kHz ≈6 MHz ≈3 MHz\n∆ (measured) 30(10) kHz 7(3) MHz 1(1) MHz\nτc(expected) ≈15µs≈0.17µs≈0.34µs\nτc(measured) 10(5)µs 3(2)µs 10(5)µs\nFOM [ms /cm3] 2×10142×10141010\nstate exchanges) can be considered as random uncorre-\nlated events [18]. Therefore, the resulting spectrum of\nthe bath’s coupling to the NV spins can be reasonably\nassumed to be Lorentzian [31, 34]:\nS(ω) =∆2τc\nπ1\n1+(ωτc)2. (7)\nThis spin bath spectrum is characterized by two param-\neters: ∆ is the average coupling strength of the bath\nto the probed NV spins; and τcis the correlation time\nof the bath spins with each other, which is related to\ntheircharacteristicflip-floptime. Ingeneral,thecoupling\nstrength ∆ is given by the averagedipolar interaction en-\nergy between the bath spins and the NV spins, and the\ncorrelation time τcis given by the inverse of the dipo-\nlar interaction energy between neighboring bath-spins.\nSince such spin-spin interactions scale as 1 /r3, whereris\nthe distance between spins, it is expected that the cou-\npling strength scales as the bath spin density nspin(i.e.,\n∆∝nspin), and the correlation time scales as the inverse\nof this density (i.e., τc∝1/nspin).\nNote also that the multi-pulse CPMG sequence used\nin this spectral decomposition technique extends the NV\nspin coherence lifetime by suppressing the time-averaged\ncoupling to the fluctuating spin environment. In gen-\neral, the coherence lifetime T2increases with the num-\nber of pulses nemployed in the CPMG sequence. For\na Lorentzian bath, in the limit of very short correlation\ntimesτc≪T2, the sequence is inefficient and T2∝n0\n(no improvementwith number ofpulses). In the opposite\nlimit of very long correlation times τc≫T2, the scaling\nisT2∝n2/3[1, 22, 35] (see also recent work on quantum\ndots [36]).\nExperimentally, we manipulate the |0∝angbracketright-|1∝angbracketrightspin man-\nifold of the NV triplet electronic ground-state using astatic magnetic field and microwave pulses; and employ\na 532 nm laser to initialize and provideoptical readout of\nthe NV spin states (Fig. 1(c); for details see Supplement\nand [10]).\nWe applied the spectral decomposition technique de-\nscribed above and extracted the spin bath parameters ∆\nandτcas well as the NV spin coherence FOM for three\ndiamond samples with differing NV densities and con-\ncentrations of electronic and nuclear spin impurities (see\nTable I): (1) An isotopically pure12C diamond sample\ngrown with chemical vapor deposition (CVD) techniques\n(Element 6), was studied using an NV ensemble micro-\nscope [15]. This sample has a very low concentration of\n13C nuclear spin impurities (0 .01%), a moderate concen-\ntration of N electronic spin impurities ( ∼1 ppm), and a\nmoderate NV density ( ∼1014[cm−3]). (2) A thin-layer\nCVD diamond sample (Apollo) with natural13C con-\ncentration (1 .1%), high N concentration ( ∼100 ppm),\nand large NV density ( ∼1016[cm−3]) was also studied\nusing the NV ensemble microscope. (3) A type 1b high-\npressurehigh-temperature(HPHT) diamond sample (El-\nement 6) with natural13C concentration, high N concen-\ntration (∼50 ppm), and low NV density ( ∼1012[cm−3])\nwas studied using a confocal apparatus able to measure\nsingle NV centers.\nIn Sample 1 (12C) the overall spin environment for the\nprobed NV spins is expected to be simply described by\na bath of N electronic spin impurities. We studied this\nsample becauseof its combinationof insignificantnuclear\nspin impurities and moderate electronic spin impurity\nconcentration, suggesting a favorable (large) FOM value.\nWe selected Sample 2 (Apollo) for characterization in or-\nder to compare the NV spin coherence FOM at high N\nconcentration and NV density with the FOM at moder-\nate N concentration and NV density in the12C sample.\nCharacterization of Sample 3 (HPHT) was important for\nconfirmingthat similar electronic spin bath dynamics are\ndetermined by the spectral decomposition technique for\nsingle NV spins and NV ensembles in the high N concen-\ntration limit (even though we expect a small FOM value\ndue to the low NV density of the HPHT sample).\nBoth the HPHT and Apollo samples have a natural\n1.1% abundance of13C nuclear spin impurities: two or-\nders of magnitude higher density than in the12C dia-\nmond sample. Nonetheless, for the high N concentration\nin the HPHT and Apollo samples, the simple physical\npicture outlined above, of NV electronic spins coupled\nto independent N electronic and13C nuclear spin baths,\nimplies that the spin environment and NV spin decoher-\nence should still be dominated by the bath of N electron\nspins.\nThe results for the12C sample are summarized in Fig.\n2. As displayed in Fig. 2(a), we find that the best-fit\nLorentzian spin bath spectrum (fitted to the average of\nall data points, see Supplement) has coupling strength\nof ∆ = 30 ±10 kHz and correlation time τc= 10±54\n0 0.2 0.4 0.6 0.8 10510\nω [106 rad/s]Sn (ω) [a.u.]∆ ≈ (30−50) kHz, τc ≈ (5−15) µs\n∆ = 30(8) kHz, τc = 10(5) µs\n(a)\n020406080100120012\nn (number of CPMG pulses)T2 [ms]\n  \ndata\nfit to data\ncalculated\n(b)\nFIG. 2. Application of the spectral decomposition techniqu e\nto an isotopically pure12C NV-diamond sample. (a) De-\nrived values for the spin bath spectral functions Sn(ω) for\nall CPMG pulse sequences (blue dots) and average values\nat each frequency (red crosses); best fit Lorentzian for the\nmean spectral function /angbracketleftSn(ω)/angbracketrightn(solid black line); and range\nof best-fit Lorentzians for the individual spectral functio ns\nSn(ω) for each CPMG pulse sequence (grey band). (b) Scal-\ning ofT2with the number nof CPMG pulses: derived from\nNV spin coherence decay data Cn(t) (dots); fit of dots to a\npower law T2= 245µs×n0.43(6)(solid line); and synthesized\nfrom the average-fit Lorentzian spin bath spectrum ∆ = 30\nkHz,τc= 10µs (open squares).\nµs, which agrees well with the range of values we find for\nthe Lorentzian spin bath spectra Sn(ω) fit to each pulse\nsequence separately (see Supplement), ∆ ≃30−50 kHz,\nτc≃5−15µs. These values are also in reasonable agree-\nment with the expected “N dominated bath” values for\n∆ andτc, giventhis12C NV-diamond sample’sestimated\nconcentrations of13C and N spins (see Table I).\nIn Fig. 2(b) we plot the NV spin coherence lifetime T2,\ndetermined from the measured coherence decay Cn(t) for\nannpulse CPMG control sequence, as a function of the\nnumber of CPMG pulses out to n= 128 (dots); and a\npower-law fit to this relationship, from which the scaling\nT2∝n0.43(6)is found (solid line). This power law ex-\nponent<2/3 is consistent with the spin bath τcbeing\nabout an order of magnitude smaller than the Hahn-echo\nT2for the12C diamond sample (see Table I). The max-\nimal coherence time achieved (using a 512-pulse CPMG\nsequence) was 2 .2 ms (see Table I). To check the consis-\ntency of our data analysis procedure and assumptions,\nwe used the best-fit Lorentzian spin bath spectrum dis-cussed above as an input to calculate the T2(n) scaling\nby synthesizing an NV spin coherence decay Cn(t) for\neach CPMG pulse sequence; and then fit to find T2for\neach synthesized Cn(t) (empty squares in Fig. 2(b)).\nFrom this process we extract a scaling T2∼n0.4(1), in\ngood agreement with the scaling derived directly from\ntheCn(t) data.\n104105106107103105107\nω [rad/s]Sn (ω) [a.u.]\n12C sampleApollo sample\nHPHT sample\n(a)\n10203040506070809010−610−510−410−3\nn (number of CPMG pulses)T2 [s]\n  \nHPHT Sample: T2 ∝ n0.7(1)\nApollo Sample: T2 ∝ n0.65(5)12C Sample: T2 ∝ n0.43(6)\n(b)\nFIG. 3. Comparative application of the spectral decomposi-\ntion technique to the12C, HPHT and Apollo NV-diamond\nsamples. (a) Spin bath spectral functions and associated\nLorentzian fits using the average-fit method (solid lines) an d\nthe individual-fit method (color bands). (b) Scaling of T2\nwith the number of CPMG pulses: dots - derived from NV\nspin coherence decay data Cn(t); solid lines - fit of dots to\na power law; open squares - synthesized from the average-fit\nLorentzian spin bath spectrum.\nThe results of the spectral decomposition procedure\nfor all three samples are displayed in Fig. 3 and Table I.\nFig. 3(a) plots the derived spin bath spectral functions\nSn(ω)andassociatedLorentzianfits; andFig. 3(b)shows\nthe measured scaling of T2with the number of CPMG\npulses. From the full set of data for each sample we\nextract the coupling strength and correlation time of the\nbest average-fit Lorentzian, which we verified to recreate\ncorrectlythedecoherencecurvesandthe T2scaling, using\nthe synthesized data analysis procedure outlined above.\nImportantly, as listed in the bottom row of Table I,\nwe found very large values for the spin coherence FOM\n∼2×1014[ms/cm3] for the12C and Apollo samples.\nThese FOM values are 3 orders of magnitude larger than\npreviously obtained in other room temperature NV di-\namond systems for which T2has been measured, with\neither single spins [29] or ensembles [15]; and comparable5\nto the FOM for a high-NV-density sample for which only\nT∗\n2was measured [30]. The present results should in-\nform improved precision spin metrology, e.g., magnetom-\netry using NV diamond. The similar large FOM values\nfor the12C and Apollo samples are qualitatively consis-\ntent with the trade-off of increased NV density causing\ncorrespondingly decreased T2due to increased N impu-\nrity concentration, for a similar N-to-NV conversion effi-\nciency∼0.1% for these CVD samples. The HPHT sam-\nple has very low N-to-NV conversion efficiency ∼10−7,\nand hence has a much smaller FOM.\nAlso as seen in Table I, there is reasonable agreement\nbetween the measured and expected values for the cou-\npling strength ∆ in all three NV-diamond samples, with\n∆ scaling approximately linearly with the N concentra-\ntion. Similarly, the measured and expected values for the\ncorrelationtime τcagreewell for the12C sample with low\nN concentration. However, we find a striking discrep-\nancy between the measured and expected values of τcfor\nthe two samples with 1 .1%13C concentration and high\nN concentration (HPHT and Apollo): both these sam-\nples have measured spin bath correlation times that are\nmore than an order of magnitude longer than given by\nthe above simple electronic spin bath model, though the\nrelative values of τcfor the HPHT and Apollo samples\nscale inversely with N concentration, as expected.\nThe qualitative agreement between the results of the\nApollo and HPHT samples indicates that the measured\nbehavior of the electronic spin bath at high N concentra-\ntion is not an artifact of the few NVs measured in the\nHPHT sample, and is not related to ensemble averaging\nin the Apollo sample.\nThe surprisingly long spin bath correlation times for\nthe13C diamond samples, revealed by the spectral de-\ncomposition technique, imply a suppression of the elec-\ntronic (N) spin bath dynamics at higher nuclear spin\n(13C) concentration. We explain this suppression as a\nresult of the random, relative detuning of electronic spin\nenergy levels due to interactions between proximal elec-\ntronic and nuclearspin impurities. The13C nuclear spins\ncreate a local Overhauser field for the N electronic spins,\nwhich detunes the energy levels of neighboring N spins\nfrom each other, depending on their (random) relative\nproximity to nearby13C spins [37]. This detuning then\nsuppresses the flip-flops of N spins due to energy con-\nservation, effectively increasing the correlation time of\nthe electronic spin bath in the presence of a finite con-\ncentration of nuclear spin impurities (compared to the\ncorrelation time in the12C sample).\nThe ensemble average effect of the composite\nelectronic-nuclear spin bath interaction is to induce an\ninhomogeneous broadening ∆ Eof the N resonant elec-\ntronic spin transition. The electronic spin flip-flop rate\nRis modified by this inhomogeneous broadening accord-ing to the expression [38, 39]\nR≃π\n9∆2\nN\n∆E, (8)\nwhere ∆ Nis the dipolar interaction between N electronic\nspins. In this physical picture, ∆ Eis proportional to\nthe concentration of13C impurities and to the hyper-\nfine interaction energy between the N electronic and13C\nnuclear spins; whereas ∆ Nis proportional to the N con-\ncentration. Taking into account the magnetic moments\nand concentrations of the N electronic and13C nuclear\nspin impurities in our samples, and the large contact hy-\nperfine interaction for N impurities in diamond [40, 41],\nwe estimate ∆ E∼10 MHz and ∆ N∼1 MHz for the\nHPHT and Apollo samples. This estimate implies ap-\nproximatelyanorderofmagnitudesuppressionof Rcom-\npared to the bare flip-flop rate ( Rbare∼∆N), which is\nconsistent with the long spin bath correlation times for\nthe13C diamond samples, as revealed by the spectral\ndecomposition technique (see Table I).\nIn summary, we applied a spectral decomposition\ntechnique to three NV-diamond samples with different\ncomposite-spinenvironmentstocharacterizethebathdy-\nnamics and determine the NV spin coherence Figure-\nof-Merit (FOM). Using this technique we realized for\ntwo samples a FOM ∼2×1014[ms/cm3], which is al-\nmost an order of magnitude larger than in other room-\ntemperature solid-state spin system; and within one or-\nder of magnitude of the largest spin coherence FOM\nachieved in atomic systems – an alkali vapor SERF mag-\nnetometer [16] (the highest sensitivity magnetometer to\ndate). The effective NV spin coherence time achieved\nwith the12C sample (2 .2 ms for 512 CPMG pulses) was\nnearlyT1limited and comparable to the best coherence\ntimes achieved with single NV spins, suggesting possible\napplications in collective quantum information process-\ning. For samples with a finite concentration of13C nu-\nclearspin impurities, this technique revealeda significant\nsuppressionof the N electronic spin bath dynamics by in-\nteractions between the nuclear and electronic spin baths.\nFurtheroptimizationoftheNVspincoherenceFOMmay\nbepossiblebyengineeringthissuppression,e.g., with13C\nconcentration higher than the natural value. This possi-\nbility isin contrastto the alkalivaporSERF FOM,which\nisfundamentallylimitedasatomicdensityisincreasedby\nspin destruction collisionsbetween atoms [16], and there-\nfore holds the promise of NV-diamond devices exceeding\nthe current state-of-the-art.\nFinally, the spectral decomposition technique pre-\nsented here, based on well-known pulse sequences and a\nsimple reconstruction algorithm, can be applied to other\ncompositesolid-statespin systems, such asquantum dots\nand phosphorous donors in silicon. Such measurements\ncould providea powerfulapproachfor the study ofmany-\nbodydynamicsofcomplexspinenvironments,andenable\nidentification of optimal parameter regimes for a wide6\nrangeofapplicationsin quantuminformationscience and\nmetrology.\nWe gratefully acknowledge fruitful discussions with\nPatrick Maletinsky and Shimon Kolkowitz; and assis-\ntance with samples by Daniel Twitchen and Matthew\nMarkham (Element 6) and Patrick Doering and Robert\nLinares (Apollo). This work was supported by NIST,\nNSF and DARPA (QuEST and QuASAR programs).\n[1] Taylor, J. M. et al. High-sensitivity diamond magne-\ntometer with nanoscale resolution. Nat. Phys. 4, 810–816\n(2008).\n[2] Maze, J. R. et al. Nanoscale magnetic sensing with an\nindividual electronic spin in diamond. Nature455, 644–\n647 (2008).\n[3] Balasubramanian, G. et al. Nanoscale imaging magne-\ntometry with diamond spins under ambient conditions.\nNature455, 648–651 (2008).\n[4] Jiang, L. et al. Repetitive readout of a single electronic\nspin via quantum logic with nuclear spin ancillae. Science\n326, 267–272 (2009).\n[5] Neumann, P. et al. Single-shot readout of a single nuclear\nspin.Science329, 542–544 (2010).\n[6] McCamey, D. R., Van Tol, J., Morley, G. W. & Boehme,\nC. Electronic spin storage in an electrically readable nu-\nclear spin memory with a lifetime >100 seconds. Science\n330, 1652–1656 (2010).\n[7] Nielsen, M. A. & Chuang, I. L. Quantum Computa-\ntion and Quantum Information ((Cambridge University\nPress, 2000).\n[8] Zurek, W. H. Decoherence, einselection, and the quan-\ntum origins of the classical. Rev. Mod. Phys. 75, 715–775\n(2003).\n[9] Wrachtrup, J. & Jelezko, F. Processing quantum infor-\nmation in diamond. Journal of Physics: Condensed Mat-\nter18, S807 (2006).\n[10] Childress, L. et al. Coherent dynamics of coupled electron\nand nuclear spin qubits in diamond. Science 314, 281–\n285 (2006).\n[11] Dutt, M. V. G. et al. Quantum register based on in-\ndividual electronic and nuclear spin qubits in diamond.\nScience316, 1312–1316 (2007).\n[12] Feher, G. & Gere, E. A. Electron spin resonance exper-\niments on donors in silicon. ii. electron spin relaxation\neffects.Phys. Rev. 114, 1245–1256 (1959).\n[13] Tyryshkin, A. M., Lyon, S. A., Astashkin, A. V. & Rait-\nsimring, A. M. Electron spin relaxation times of phos-\nphorus donors in silicon. Phys. Rev. B 68, 193207 (2003).\n[14] Hanson, R., Kouwenhoven, L. P., Petta, J. R., Tarucha,\nS. & Vandersypen, L. M. K. Spins in few-electron quan-\ntum dots. Rev. Mod. Phys. 79, 1217–1265 (2007).\n[15] Pham, L. M. et al. Magnetic field imaging with nitrogen-\nvacancy ensembles. New Journal of Physics 13, 045021\n(2011).\n[16] Allred, J. C., Lyman, R. N., Kornack, T. W. & Romalis,\nM. V. High-sensitivity atomic magnetometer unaffected\nby spin-exchange relaxation. Phys. Rev. Lett. 89, 130801\n(2002).\n[17] Dobrovitski, V. V., Feiguin, A. E., Awschalom, D. D. &Hanson, R. Decoherence dynamics of a single spin versus\nspin ensemble. Phys. Rev. B 77, 245212 (2008).\n[18] Barnes, E., Cywi´ nski, /suppress L. & Das Sarma, S. Master equa-\ntion approach to the central spin decoherence problem:\nthe uniform coupling model and the role of projection\noperators. ArXiv e-prints (2011). 1108.1199.\n[19] Hahn, E. L. Spin echoes. Phys. Rev. 80, 580–594 (1950).\n[20] Carr, H. Y. & Purcell, E. M. Effects of diffusion on free\nprecession in nuclear magnetic resonance experiments.\nPhys. Rev. 94, 630–638 (1954).\n[21] Meiboom, S. & Gill, D. Modified spin-echo method for\nmeasuring nuclear relaxation times. Review of Scientific\nInstruments 29, 688–691 (1958).\n[22] Slichter, C. P. Principles of magnetic resonance\n(Springer-Verlag, 1990).\n[23] de Lange, G., Wang, Z. H., Rist, D., Dobrovitski, V. V. &\nHanson, R. Universal dynamical decoupling of a single\nsolid-state spin from a spin bath. Science 330, 60–63\n(2010).\n[24] Ryan, C. A., Hodges, J. S. & Cory, D. G. Robust de-\ncoupling techniques to extend quantum coherence in di-\namond. Phys. Rev. Lett. 105, 200402 (2010).\n[25] de Lange, G. et al. Controlling the quantum dynamics of\na mesoscopic spin bath in diamond. Arxiv: 1104.4648v1\n(2011).\n[26] Almog, I. et al. Direct measurement of the system-\nenvironment coupling as a tool for understanding deco-\nherence and dynamical decoupling. Arxiv: 1103.1104v1\n(2011).\n[27] Bylander, J. et al. Noise spectroscopy through dynamical\ndecoupling with a superconducting flux qubit. Nature\nPhysics 7, 565–570 (2011).\n[28] Du, J. et al. Preserving electron spin coherence in solids\nby optimal dynamical decoupling. Nature461, 1265–8\n(2009).\n[29] Balasubramanian, G. et al. Ultralong spin coherence time\nin isotopically engineered diamond. Nat Mater 8, 383–\n387 (2009).\n[30] Acosta, V. M. et al. Diamonds with a high density of\nnitrogen-vacancy centers for magnetometry applications.\nPhys. Rev. B 80, 115202 (2009).\n[31] Klauder, J. R. & Anderson, P. W. Spectral diffusion\ndecay in spin resonance experiments. Phys. Rev. 125,\n912–932 (1962).\n[32] Gordon, G., Erez, N. & Kurizki, G. Universal dynami-\ncal decoherence control of noisy single- and multi-qubit\nsystems. J. Phys. B 40, S75 (2007).\n[33] Cywi´ nski, L., Lutchyn, R. M., Nave, C. P. & Das Sarma,\nS. How to enhance dephasing time in superconducting\nqubits.Phys. Rev. B 77, 174509 (2008).\n[34] Kittel, C. & Abrahams, E. Dipolar broadening of mag-\nnetic resonance lines in magnetically diluted crystals.\nPhys. Rev. 90, 238–239 (1953).\n[35] de Lange, G., Rist` e, D., Dobrovitski, V. V. & Hanson,\nR. Single-spin magnetometry with multipulse sensing se-\nquences. Phys. Rev. Lett. 106, 080802 (2011).\n[36] Medford, J. et al. Scaling of Dynamical Decoupling for\nSpin Qubits. ArXiv e-prints (2011). 1108.3682.\n[37] Portis, A. M. Spectral diffusion in magnetic resonance.\nPhys. Rev. 104, 584–588 (1956).\n[38] de Sousa, R. & Das Sarma, S. Electron spin coherence\nin semiconductors: Considerations for a spin-based solid-\nstate quantum computer architecture. Phys. Rev. B 67,\n033301 (2003).7\n[39] Witzel, W. M., Carroll, M. S., Morello, A., Cywi´ nski, L .\n& Das Sarma, S. Electron spin decoherence in isotope-\nenriched silicon. Phys. Rev. Lett. 105, 187602 (2010).\n[40] Cox, A., Newton, M. E. & Baker, J. M.13C,14N and15N\nendor measurements on the single substitutional nitrogen\ncentre (p1) in diamond. Journal of Physics: CondensedMatter6, 551 (1994).\n[41] Barklie, R. C. & Guven, J.13C hyperfine structure and\nrelaxation times of the p1 center in diamond. J. Phys. C\n14, 3621 (1981)."    },    {        "title": "1712.03009v1.Spin_Hall_and_spin_swapping_torques_in_diffusive_ferromagnets.pdf",        "content": "Spin Hall and spin swapping torques in diffusive ferromagnets\nChristian Ortiz Pauyac,1, 2Mairbek Chshiev,2Aurelien Manchon,1, 3and Sergey A. Nikolaev4, 2,∗\n1King Abdullah University of Science and Technology (KAUST),\nPhysical Science and Engineering Division, Thuwal 23955-6900, Saudi Arabia\n2Univ. Grenoble Alpes, CNRS, CEA, Grenoble-INP, INAC-Spintec, 38000 Grenoble, France\n3King Abdullah University of Science and Technology (KAUST), Computer,\nElectrical and Mathematical Science and Engineering (CEMSE), Thuwal 23955-6900, Saudi Arabia\n4Ural Federal University, 10 Mira str., 620002 Ekaterinburg, Russia\n(Dated: December 11, 2017)\nA complete set of the generalized drift-diffusion equations for a coupled charge and spin dynamics\nin ferromagnets in the presence of extrinsic spin-orbit coupling is derived from the quantum ki-\nnetic approach, covering major transport phenomena, such as the spin and anomalous Hall effects,\nspin swapping, spin precession and relaxation processes. We argue that the spin swapping effect\nin ferromagnets is enhanced due to spin polarization, while the overall spin texture induced by the\ninterplay of spin-orbital and spin precessional effects displays a complex spatial dependence that can\nbe exploited to generate torques and nucleate/propagate domain walls in centrosymmetric geome-\ntries without use of external polarizers, as opposed to the conventional understanding of spin-orbit\nmediated torques.\nPACS numbers: 85.75.-d, 73.40.Gk, 72.25.-b, 72.10.-d\nIntroduction . The exploitation of spin-orbit coupling\n(SOC) effects to probe and control the magnetization\nin nanodevices has been extensively studied, uncover-\ning many physical phenomena, such as the anomalous\nHall effect [1], spin Hall effect [2], tunneling anisotropic\nmagnetoresistance [3], electrically controlled perpendicu-\nlar magnetic anisotropy [4], and relativistic spin torques\n[5–7]. The latter observed in multilayers comprising fer-\nromagnets and normal metals display both spin-orbit\ntorques induced by the interfacial inverse spin-galvanic\neffect [6] and spin-transfer torques associated with the\nspin Hall effect in an adjacent non-magnetic layer [7].\nSpin-orbit torques generated in a single ferromagnetic\nlayer are of great importance to enable electrical control\nof the magnetization without use of an external polarizer,\nand they offer many promising advantages compared to\nspin-transfer torques, such as high scalability and stabil-\nity. Thus, finding novel routes to excite magnetization\ndynamics by means of spin-orbit torques is essential for\nrealizing high-performance spintronic devices.\nMeanwhile, new ways to generate spin accumulation\nare also of strong interest. More recently, a new mecha-\nnism referred to as spin swapping, which converts a pri-\nmary spin current into a secondary spin current with in-\nterchanged spin and flow directions, was proposed to ex-\nist in normal metals and semiconductors in the presence\nof spin-orbit coupled impurities [8]. However, whether\nspin swapping in ferromagnets can produce a measur-\nable effect remains an open question that has not yet\nbeen addressed. On the one hand, the exchange mag-\nnetic field present in ferromagnets tends to destroy the\ninduced spin accumulation. On the other hand, not only\ndoes SOC act constructively in generating spin accumu-\nlation, it also leads to the spin-memory loss [9]. Overall,\nthe possibility to employ these effects in ferromagnetsstrongly depends on the transport regime as a function\nof many parameters describing a given system.\nIn this letter, we explore the nature of the extrinsic\nspin Hall and spin swapping effects in diffusive ferromag-\nnets and demonstrate that these effects can offer poten-\ntial advantages in contrast to non-centrosymmetric mag-\nnetic multilayers involving heavy metals. To this end, we\ndevelop a set of coupled spin-charge diffusive equations\nby using the non-equilibrium Green’s function formal-\nism and taking into account scattering off the impurity\ninduced SOC potential. Based on these equations, we\nproceed to study the interplay between spin-orbital and\nspin precessional effects that can be used to demonstrate\ncurrent-driven manipulation of the magnetization in cen-\ntrosymmetric magnets.\nDerivation of the coupled spin-charge drift-diffusion\nequation . We consider a single ferromagnetic layer in\nthe standard s-dmodel [10] defined as ˆH=ˆp2\n2mˆσ0+Jˆσ·\nm+ˆHimp, wheremis the effective electron’s mass, ˆp\nis the momentum operator, Jis the exchange coupling,\nˆσ0is the identity matrix, ˆσis the Pauli matrix vector,\nandmis the unit vector of the spatial magnetization\nprofile. Here, the third term stands for the impurity po-\ntential given by randomly distributed NimpuritiesRj,\nˆHimp=/summationtextN\nj[V(r−Rj)ˆσ0+ξSO\n¯hk2\nFˆσ·(∇V(r−Rj)׈p)],\nwhereV(r−Rj) =viδ(r−Rj) is the on-site impu-\nrity potential, ξSOis the SOC parameter (defined as a\ndimensionless quantity), and kFis the Fermi wave vec-\ntor. In the Keldysh formalism for an interacting system\ndriven out of equilibrium, the Dyson equation for the\nnon-equilibrium Green’s function ˆGKis written as:\n[ˆGR]−1∗ˆGK−ˆGK∗[ˆGA]−1=ˆΣK∗ˆGA−ˆGR∗ˆΣK,(1)arXiv:1712.03009v1  [cond-mat.mes-hall]  8 Dec 20172\nwhere ˆGi≡ˆGi(r,t;r/prime,t/prime) and ˆΣi≡ˆΣi(r,t;r/prime,t/prime) with\ni=K,R,A are the real space real time Keldysh,\nretarded and advanced Green’s functions and self-\nenergies, respectively, and ˆGR(A)=ˆG−1\n0−ˆΣR(A), where\nˆG−1\n0=i¯h∂t−ˆHis a non-interacting Green’s func-\ntion for the system without impurities [11]. Having\napplied the Wigner transformation ˆGK(r,t;r/prime,t/prime) =/integraltextdE\n2πdk\n(2π)3eik·(r−r/prime)−iE\n¯h(t−t/prime)ˆgK\nk,E(R,T) withR= (r+\nr/prime)/2 andT= (t+t/prime)/2, we employ the so-called gra-\ndient approximation to linearize convolutions ( ∗) in the\nDyson equation (1) and obtain the following quantum ki-\nnetic equation for the non-equilibrium distribution func-\ntion ˆgk=i/integraltextdE\n2πˆgK\nk,E(R,T):\n¯h∂Tˆgk−i[ˆgk,Jˆσ·m] +¯h2\nm(k·∇R) ˆgk=/integraldisplaydE\n2πC,(2)\nwhere C= (ˆΣKˆGA−ˆGRˆΣK) + ( ˆΣRˆgK−ˆgKˆΣA) is the\ncollision integral that accounts for the scattering and re-\nlaxation events, respectively. Scattering off the impuritypotential in the right-hand side of Eq. (2) is considered\nup to third order by impurity averaging over disorder\nwith concentration ni, so that the spin-dependent mo-\nmentum and spin-flip relaxations, side-jump [12], spin\nswapping and skew-scattering [13] processes are prop-\nerly taken into account. In the diffusive limit, where\nthe mean-free path is much less compared to the size of\nthe system, one can partition the distribution function\nˆgk= ˆσ0−2ˆhkinto the isotropic charge µc, spinµand\nanisotropic ˆjcomponents, ˆhk=µcˆσ0+µ·ˆσ+ˆj·ˇk, where\nˇk=k/|k|[14]. First, integrating Eq. (2) multiplied by\nˇkover the Brillouin zone gives us the corresponding ex-\npression for ˆj≡ˆj(µc,µ). Secondly, integrating Eq. (2)\nitself leads to the generalized continuity equation for the\nchargeµcand spinµdensities, so that their time depen-\ndence is given as a divergence of the charge jCand spin\nJS\nj(itsjth spin component) currents, respectively. We\nrefer the reader to Ref. [15] for more detail concerning\nthe derivation. Finally, in the weak exchange coupling\nlimit (J/lessmuchεF, whereεFis the Fermi energy), the result-\ning drift-diffusion equations up to leading orders in the\nexchange interaction and SOC have the following form:\n∂Tµc=D∇2[µc+βµ·m] =−∇·jC, (3)\n∂Tµ=−∇·JS+1\nτLm×µ+1\nτφm×(m×µ)−1\nτsfµ, (4)\njC/D=˜jC/D+∇×/bracketleftBig\nαsj(2µ−βµcm) +αsk(µ−βµcm) + (αsjτ0\nτL+αskτ0\nτL−αswβ)m×µ/bracketrightBig\n, (5)\nJS\nj/D=˜JS\nj/D+∇×/bracketleftBig\nαsjej(2µc−βµ·m) +αskej(µc−βµ·m)−αswej×(µ−βµcm) (6)\n+τ0\nτL(αsj+αsk)(ej×m)µc+ (αsjβ+αskβ−αswτ0\nτL)(ej×m)×µ−(αswτ0\nτL+αsjβ+αskβ)ej×(m×µ)/bracketrightBig\n,\nwhere ˜jC=−D∇(µc+βm·µ) and ˜JS\nj/D=−∇(µj+\nβµcmj) +τ0\nτL∇(m×µ)j+τ0\nτφ∇(m×(m×µ))jare the\ncharge and spin currents in the absence of SOC, respec-\ntively;β=J\n2εFis the polarization factor; αsw=2ξSO\n3,\nαsj=ξSO\nlFkF, andαsk=vimkFξSO\n3π¯h2are the dimension-\nless spin swapping, side-jump, and skew scattering co-\nefficients, respectively; D=τ0v2\nF\n3is the diffusion coef-\nficient,vFis the Fermi velocity, and lF=τ0vFis the\nmean-free path. Here,1\nτ0=2πv2\niniD0\n¯his the spin inde-\npendent relaxation time, where D0=mkF\n2π2¯h2is the spin\nindependent density of states,1\nτsf=8\n9ξ2\nSO\nτ0is the spin-\nflip relaxation time,1\nτL=2J\n¯his the spin precession time\naround the magnetization, known as the Larmor preces-\nsion time, and1\nτφ=4J2τ0\n¯h2refers to the spin dephasing\nterm. The set of the drift-diffusion equations (3)–(6) isthe central result of this paper. On the one hand, in the\nabsence of SOC, our approach is in-line with the gen-\neralized drift-diffusion theory [16], which captures main\nfeatures of the transverse spin transport in ferromag-\nnets, such as the Larmor precession and spin dephasing\nterms. On the other hand, in the case of normal met-\nals (β→0,τL→∞ ,τφ→∞ ) our equations are in\nagreement with Shen et al. [17], while some other works\nfail to include the correct symmetry of the spin swapping\nterm [15, 18–20]. In the presence of both, the exchange\ninteraction and extrinsic SOC, the anomalous Hall ef-\nfect is present in Eq. (5) (second and third terms) from\nboth the side-jump and skew scattering processes [21],\nwhile spin polarization and spin precession give rise to\nadditional terms to the anomalous charge and spin cur-\nrents in Eqs. (5) and (6). Overall, the resulting charge\nand spin accumulation profiles appear to be much more3\nFIG. 1. Spin accumulation profiles µx,µy, andµz, as calculated for the rectangular geometry of 100 ×50 nm2: a)µyin the\ncase of Larmor precession only, b) µxandµzwhen only the spin swapping and Larmor precession terms are considered, c)\nµxandµzwhen only the spin Hall effect and Larmor precession are considered, d) refers to the full drift-diffusion equations.\nHere, the spin current polarized along the yaxis is flowing along the xaxis,εF= 0.7 eV,J= 0.02 eV, and ξSO= 0.3. The\nspin diffusion length is lsf=/radicalbig\nDτsf= 5 nm, the Larmor precession length is lL=√DτL= 2.6 nm, the spin dephasing length\nislφ=/radicalbig\nDτφ= 4.8 nm, the mean free path is lF=τ0vF= 2.5 nm, the Fermi velocity is vF= 5×105m/s, and the Fermi\nwave-vector is kF= 4.3 nm−1. The grey and yellow spheres denote the non- and spin-polarized charge currents, while the red\nand blue spheres denote the spin-up and spin-down components, respectively.\ncomplex, as opposed to normal metals. In magnetic sys-\ntems with SOC, the competition between these effects is\ngoverned by the ratio of the corresponding characteristic\nlengths: spin precession, spin dephasing, and spin dif-\nfusion lengths. In ferromagnets with a strong exchange\ncoupling, where the spin dephasing length is shorter than\nthe spin-flip relaxation, the spin Hall and spin swapping\neffects are expected to vanish far from the interface, as\nthe strong exchange field tends to destroy induced spin\ncurrents. Consequently, any transverse spin component\nwill eventually align or anti-align with the magnetization\n[22]. In contrast, in the weak exchange coupling limit\n(J/lessmuchεF) the spin dephasing length is larger or com-\nparable with the spin-flip relaxation, and these coupled\neffects can become prominent.\nSpin accumulation profile . To study the interplay of\nthe effects in question, the drift-diffusion equations (3)-\n(6) can be solved numerically by putting malong the\nyaxis and imposing adequate boundary conditions [23].\nThe results calculated for the rectangular geometry are\nshown in Fig. 1. In the absence of SOC, the spin current\ndensityJS\nxy∼∇xµyis induced in the magnetic layer and\ndue to Larmor precession the spin accumulation µyis lo-\ncalized at the normal metal/ferromagnet interfaces along\nthe transport direction (Fig. 1a), in agreement with the\nValet-Fert theory [24]. When the impurity induced SOC\nis present, the transverse spin accumulation is expected\nto build up at the lateral edges. First off, let us analyze\nadditional contributions to the spin swapping term. As\nseen from Eq. (6), the “spin swapping” spin current hasthe following form:\nJsw\nij/D=αsw(∇jµi−δij∇kµk)\n+αswβ(δijmk∇k−mi∇j)µc\n+ (αsjτ0\nτL+αskτ0\nτL)(δijmk∇k−mi∇j)µc,(7)\nwhere summation over repeated indexes is implied. In\nnormal metals, due the spin swapping mechanism the\nprimary spin current ˜JS\nji∼∇jµigives rise to the sec-\nondary spin current Jsw\nij∼αsw(∇jµi−δij∇kµk) and\nthe generated spin accumulation µidecays over a length\nscale given by the spin-flip relaxation length and survives\nonly close to the interface [19]. However, in ferromagnets\nan additional non-vanishing spin accumulation ∼αswβ\nbuilds up due to spin polarization and develops smoothly\nat the lateral edges (Fig. 1b, x-component). Interest-\ningly, there is an extra term coming from the spin Hall\neffect and precessional motion (third term in Eq. (7)),\nwhich also exhibits the symmetry of the spin-swapping\ncurrent, even though it is not actually attributed to the\nspin swapping effect itself. The rest of the effects are\nrelated to Larmor precession. For example, let us con-\nsider a toy model, where mpoints along the yaxis and\nthe spin Hall effect, spin polarization, and higher order\nterms to the precessional motion are discarded, so the4\nFIG. 2. Spin accumulation profiles µx(left) andµz(right), as\ncalculated for the diamond-shaped geometry of 100 ×50 nm2.\nThe top panel describes the idea of nucleation and propaga-\ntion of reversed magnetic domains, where the red and blue\ncircles correspond to the spin-up and spin-down components\nof the spin accumulation, respectively. Here, the black arrows\nshow the magnetization direction in the xyplane, and its ori-\nentation with respect to the zaxis is given by θ= 60◦. The\nrest of the calculation parameters are given in Fig. 1.\nspin current density in Eq. (6) is reduced to:\nJS\nyx\nD∼−∇yµx−τ0\nτL∇yµz+αsw∇xµy, (8)\nJS\nyz\nD∼−∇yµz+τ0\nτL∇yµx. (9)\nAs one can see, there is an additional term in Eq. (9) that\ncouplesJS\nyxtoJS\nyzand builds up µz, which eventually\nmanifests itself due to the spin swapping effect (Fig. 1b,\nz-component). Similar results are obtained if we neglect\nspin swapping and focus on the spin Hall effect instead,\nwhere the spin accumulations do not vanish far from the\ninterfaces (Fig. 1c). If we solve our toy models separately\nfor the side-jump and spin swapping terms, the following\nrelation holds for the maxima of the corresponding spin\naccumulations:\nµx\nµz∝αswβ\nαsj=1\n3lFkFβ, (10)\nso that spin swapping is about one order of magnitude\nsmaller than the spin Hall effect (as also seen in Fig. 1)\nin diffusive ferromagnets ( J/lessmuchεF) as a result of µxde-\npending strongly on the polarization factor β.\nCurrent driven magnetization switching . We propose\na possible way to exploit the spin Hall and spin swap-\nping effects to reversibly control the magnetization in\ncentrosymmetric ferromagnets that can be realized even\nin the absence of adjacent non-magnetic layers. Nor-\nmally, spin-orbit torques are observed in ferromagneticfilms lacking inversion symmetry through the Rashba ef-\nfect, which is essentially inherent to non-centrosymmetric\nstructures. However, geometry itself can play an impor-\ntant role building up distorted spin accumulation pro-\nfiles and giving rise to non-zero local spin-orbit torques.\nFor example, let us consider a centrosymmetric diamond-\nshaped geometry. The resulting spin accumulation pre-\nsented in Fig. 2 turns out to be highly asymmetric (while\nthe net spin accumulation is zero) and peaks at the oppo-\nsite edges that can be used to nucleate reversed magnetic\ndomains. Once nucleated at the corners, the flowing cur-\nrent can either expand or shrink the reversed magnetic\ndomain by current-driven domain wall motion, as shown\nin the top panel of Fig. 2. A similar scenario in control-\nling the magnetization (albeit without considering the\nspin Hall or spin swapping mechanisms) has been stud-\nied in Ref. [25].\nConclusion . We derived a complete set of the drift-\ndiffusion equations for the coupled charge and spin trans-\nport in diffusive ferromagnets in the presence of extrinsic\nSOC. While combining major effects, such as the spin\nand inverse spin Hall effects, anomalous Hall effect and\nspin swapping, these equations reveal some new intrigu-\ning features. In particular, we showed that in ferromag-\nnets the resulting spin accumulation exhibits a complex\nspatial profile, where the spin swapping effect is enhanced\ndue to spin polarization, while spin precession gives rise\nto additional contributions to the anomalous charge and\nspin currents. These effects can be employed to generate\nspin-orbit mediated torques and reversibly control the\nmagnetization in centrosymmetric structures. Our re-\nsults call for experimental approbation in current-driven\nmagnetization dynamics, where suitable materials may\ninclude magnetic alloys with heavy impurities, such as\nCo-Pt, Fe-Au [26] or CuMn-Pt [27].\nAcknowledgements. A.M and C.O.P were supported by\nthe King Abdullah University and Technology (KAUST).\nM.C. and S.A.N. were supported by the programme\n“Emergence et partenariat strat´ egique” at Univ. Greno-\nble Alpes and the PRESTIGE Postdoctoral Fellowship\nProgramme co-funded by EU. S.A.N. acknowledges sup-\nport from the Act 211 Government of the Russian Fed-\neration, contract No 02.A03.21.0006\n∗saishi@inbox.ru\nC.O.P. and S.A.N. contributed equally in this work.\n[1] N. Nagaosa, J. Sinova, S. Onoda, A. H. McDonald, and\nN. P. Ong, Rev. Mod. Phys. 82, 1539 (2010)\n[2] M. I. Dyakonov and V. I. Perel, Sov. Phys. JETP Lett.\n13, 467 (1971); ibis, Physics Letters A 35, 459 (1971);\nJ. E. Hirsch, Phys. Rev. Lett. 83, 1834 (1999).\n[3] C. Gould, C. Ruster, T. Jungwirth, E. Girgis,\nG. M. Schott, R. Giraud, K. Brunner, G. Schmidt, and\nL. W. Molenkamp, Phys. Rev. Lett. 93, 117203 (2004);5\nH. Saito, S. Yuasa, and K. Ando, Phys. Rev. Lett. 95,\n086604 (2005); L. Gao, X. Jiang, S. Yang, J. D. Burton,\nE. Y. Tsymbal, and S. S. P. Parkin, Phys. Rev. Lett. 99,\n226602 (2007).\n[4] M. K. Niranjan, C. G. Duan, S. S. Jaswal, E. Y. Tsymbal,\nAppl. Phys. Lett. 96, 222504 (2010).\n[5] I. M. Miron, K. Garello, G. Gaudin, P.-J. Zermatten,\nM. V. Costache, S. Auffret, S. Bandiera, B. Rodmacq,\nA. Schuhl, P.Gambardella, Nature 476, 189-193 (2011).\n[6] I. M. Miron, G. Gaudin, S. Auffret, B. Rodmacq,\nA. Schuhl, S. Pizzini, J. Vogel, and P. Gambardella. Nat,\nMat. 9, 230 (2010); A. Manchon and S. Zhang, Phys.\nRev. B 78, 212405 (2008).\n[7] L. Liu, C-F. Pai, Y. Li, H. W. Tseng, D. C. Ralph, and\nR. A. Buhrman. Science 336, 555 (2012).\n[8] M. B. Lifshits and M. I. Dyakonov, Phys. Rev. Lett. 103,\n186601 (2009)\n[9] J. Fabian, A. Matos-Abiague, C. Ertler, P. Stano,\nI.ˇZuti´ c, Acta Physica Slovaca 57, 565 (2007).\n[10] S. P. Shubin and S. V. Vonsovski, Proc. Roy. Soc. A145,\n159 (1934).\n[11] J. Rammer and H. Smith, Rev. Mod. Phys. 58, 323\n(1986).\n[12] L. Berger, Phys. Rev. B 2, 4559 (1970); S. K. Lyo and\nT. Holstein, Phys. Rev. Lett. 29, 423 (1972).\n[13] S. Takahashi and S. Maekawa, J. Phys. Soc. Jpn. 77,\n031009 (2008).\n[14] R. V. Shchelushkin and A. Brataas, Phys. Rev. B 71,\n045123 (2006).[15] Supplemental Materials.\n[16] C. Petitjean, D. Luc, and X. Waintal, Phys. Rev. Lett.\n109, 117204 (2012).\n[17] K. Shen, R. Raimondi, and G. Vignale, Phys. Rev. B 90,\n245302 (2014); ibis, Phys. Rev. B 92, 035301 (2015).\n[18] R. V. Shchelushkin and A. Brataas, Phys. Rev. B 72,\n073110 (2005).\n[19] H. Saidaoui and A. Manchon, Phys. Rev. B 92, 024417\n(2015).\n[20] H. Saidaoui and A. Manchon, Phys. Rev. Lett. 117,\n036601 (2016).\n[21] W.-K. Tse and S. Das Sarma, Phys. Rev. Lett 96, 056601\n(2006); A. Crepieux and P. Bruno, Phys. Rev. B 64,\n014416 (2001); V. K. Dugaev, A. Crepieux, and P. Bruno,\nPhys. Rev. B 64, 104411 (2001).\n[22] T. Taniguchi, J. Grollier, and M. D. Stiles, Phys. Rev.\nAppl. 3, 044001 (2015).\n[23] In our calculations the following boundary conditions are\nimposed: JS\nj= 0 at the boundary, while the charge\ndensities at the edges in the propagation direction are\nµcL=k2\nF/2πandµcR= 0, respectively.\n[24] T. Valet and A. Fert, Phys. Rev. B 48, 7099 (1993).\n[25] C. K. Safeer, E. Ju´ e, A. Lopez, L. Buda-Prejbeanu,\nS. Auffret, S. Pizzini, O. Boulle, I. M. Miron, and\nG. Gaudin, Nat. Nanotech. 11, 143 (2016).\n[26] A. Fert, A. Friederich, and A. Hamzic, J. Magn. Magn.\nMater. 24, 231 (1981).\n[27] J. J. Prejean, M. J. Joliclerc, and P. Monod, J. Physique\n41, 427 (1980).Supplementary Materials:\nSpin Hall and spin swapping torques in diffusive ferromagnets\nChristian Ortiz Pauyac,1, 2Mairbek Chshiev,2Aurelien Manchon,1, 3and Sergey A. Nikolaev4, 2,∗\n1King Abdullah University of Science and Technology (KAUST),\nPhysical Science and Engineering Division, Thuwal 23955-6900, Saudi Arabia\n2Univ. Grenoble Alpes, CNRS, CEA, Grenoble-INP, INAC-Spintec, 38000 Grenoble, France\n3King Abdullah University of Science and Technology (KAUST), Computer,\nElectrical and Mathematical Science and Engineering (CEMSE), Thuwal 23955-6900, Saudi Arabia\n4Department of Theoretical Physics and Applied Mathematics,\nUral Federal University, 19 Mira str., 620002 Ekaterinburg, Russia\n(Dated: December 11, 2017)arXiv:1712.03009v1  [cond-mat.mes-hall]  8 Dec 20172\nCONTENTS\nList of model and transport parameters 3\nI. General formalism 4\nII. Self-energy 6\nA. Self-consistent Born approximation 6\nB. Skew-scattering 8\nIII. Relaxation time 10\nIV. Averaged velocity operator 12\nV. Quantum transport equations 13\nVI. Ferromagnetic solution without spin-orbit coupling 17\nVII. Ferromagnetic solution with spin-orbit coupling 18\nVIII. Spin swapping symmetry 21\nReferences 213\nLIST OF MODEL AND TRANSPORT PARAMETERS\nkF Fermi wavevector\nm Electron’s effective mass\nJ Exchange constant in the s-dmodel\nvi Impurity potential\nni Impurity concentration\nξSO Dimensionless spin-orbit coupling constant\nεF=¯h2k2\nF\n2mFermi energy\nvF=¯hkF\nmFermi velocity\nβ=J\n2εFSpin polarization factor\nD0=mkF\n2π2¯h2Spin independent density of states per spin at the Fermi level\n1\nτ0=2πv2\niniD0\n¯hSpin independent relaxation time\n1\nτL=2J\n¯hLarmor precession time\n1\nτφ=4J2τ0\n¯h2Spin dephasing relaxation time\n1\nτsf=8\n9ξ2\nSO\nτ0Spin-flip relaxation time\nlF=τ0vF Mean-free path\nD=τ0v2\nF\n3Diffusion coefficient\nαsw=2\n3ξSO Dimensionless spin swapping constant\nαsj=ξSO\nlFkFDimensionless side-jump constant\nαsk=vimkF\n3π¯h2ξSO Dimensionless skew scattering constant4\nI. GENERAL FORMALISM\nWe start with a free-electron Hamiltonian ˆH0and its Fourier transform ˆHk:\nˆH0=−¯h2\n2m∇2ˆσ0+Jˆσ·mF−−→ ˆHk=¯h2k2\n2mˆσ0+Jˆσ·m. (1)\nHere, the first term stands for the kinetic energy, where mandkare the electron’s effective mass and wave\nvector, respectively; the second term refers to the exchange interaction in the so-called s-dmodel, where m=\n(cosφsinθ,sinφsinθ,cosθ) is the magnetization unit vector parametrized in spherical coordinates, Jis the exchange\ncoupling parameter, ˆσis the Pauli matrix vector, and ˆ σ0is the identity matrix. The unperturbed Green’s function is\ndefined by ˆHk:\nˆGR(A)\n0,kE=/bracketleftBig\nE−ˆHk±iη/bracketrightBig−1\n=/summationdisplay\ns=±|s/angbracketright/angbracketlefts|\nE−Eks±iη=1\n2/summationdisplay\ns=±ˆσ0+sˆσ·m\nE−Eks±iη, (2)\nwheresrefers to the spin index, |s/angbracketrightis the corresponding eigenstate:\n|s/angbracketright=\nse−iφ/radicalBig\n1+scosθ\n2/radicalBig\n1−scosθ\n2\n, (3)\nEks=Ek+sJ,Ek= ¯h2k2/2m, andηis a positive infinitesimal.\nNext, we consider the impurity Hamiltonian with spin-orbit coupling:\nˆHimp=/summationdisplay\nRiV(r−Ri)ˆσ0+ξSO\n¯hk2\nF/summationdisplay\nRi(∇V(r−Ri)׈p)·ˆσ, (4)\nwhere ˆp=−i¯h∂ris the momentum operator, V(r−Ri) =viδ(r−Ri) is the on-site potential at the impurity site Ri,\nξSOis the spin-orbit coupling parameter (defined as a dimensionless quantity), and kFis the Fermi wavevector. Here,\nwe neglect the localization effects and electron-electron correlations, and assume a short-range impurity potential. In\nthe reciprocal space, it can be written as:1\nˆHkk/prime= Ω/angbracketleftk|ˆHimp|k/prime/angbracketright, (5)\nwhere the momentum eigenstates are defined as /angbracketleftr|k/angbracketright= Ω−1/2eik·r, and Ω is the volume of the system. Then, by\nusing the following identities:\n/integraldisplay\nΩdrf(r)δ(r−ri) =f(ri),/integraldisplay\nΩdrf(r)∇δ(r−ri) =−/integraldisplay\nΩdr∇f(r)δ(r−ri), (6)\nwe obtain:\nˆHkk/prime=/summationdisplay\nRi/integraldisplay\nΩdr/bracketleftbigg\nviδ(r−Ri)ˆσ0e−i(k−k/prime)·r−iviξSO\nk2\nFe−ik·r/parenleftBig\n∇δ(r−Ri)×∂r/parenrightBig\n·ˆσeik/prime·r/bracketrightbigg\n=V(k−k/prime)/bracketleftbigg\nˆσ0+iξSO\nk2\nFˆσ·(k×k/prime)/bracketrightbigg\n,(7)\nwhereV(k−k/prime) is the Fourier transform of the impurity on-site potential:\nV(k−k/prime) =vi/summationdisplay\nRie−i(k−k/prime)·Ri. (8)\nWe proceed to write a kinetic equation by means of the Keldysh formalism:\nˆG−1=ˆG−1\n0−Σ, ˆG=/parenleftbiggˆGRˆGK\n0ˆGA/parenrightbigg\n, ˆΣ=/parenleftbiggˆΣRˆΣK\n0ˆΣA/parenrightbigg\n, (9)5\nwhere ˆGandˆΣare the Green’s function and self-energy in the Keldysh space; the indexes R,AandKstand for the\nretarded, advanced and Keldysh components, respectively, and ˆG−1\n0=i¯h∂t−ˆH0. In the semiclassical approximation,\na set of diffusive equations for the non-equilibrium charge and spin densities can be derived through the distribution\nfunction ˆgk≡ˆgk(R,T) defined as the Wigner representation of the Keldysh Green’s function ˆGK:2\nˆGK(r1,t1;r2,t2)W−→ˆGK(R+r\n2,T+t\n2;R−r\n2,T−t\n2)≡ˆGK(r,t;R,T)\nF−→ˆGK(r,t;R,T) =/integraldisplaydE\n2πdk\n(2π)3ˆgK\nkE(R,T)e−i\n¯hEteir·k\n− →ˆgk=i/integraldisplaydE\n2πˆgK\nkE(R,T),(10)\nwhere the relative r=r1−r2,t=t1−t2and center-of-mass R= (r1+r2)/2,T= (t1+t2)/2 coordinates are\nintroduced. In the dilute limit, we can employ the Kadanoff-Baym anzats:\nˆGkE(R,T) =/parenleftbiggˆGR\nkEˆgK\nkE(R,T)\n0 ˆGA\nkE/parenrightbigg\n(11)\nand\nˆgK\nkE(R,T) =ˆGR\nkEˆgk(R,T)−ˆgk(R,T)ˆGA\nkE. (12)\nThe Keldysh Green’s function ˆGKsatisfies the Kadanoff-Baym equation:2\n[ˆGR]−1∗ˆGK−ˆGK∗[ˆGA]−1=ˆΣK∗ˆGA−ˆGR∗ˆΣK, (13)\nHaving applied the Wigner transformation, we use the so-called gradient approximation, where the convolution A∗B\nof two functions is expressed as:\n(A∗B )kE(R,T)/similarequalAB−i¯h\n2(∂TA∂EB−∂EA∂TB)\n−i\n2(∇kA·∇ RB−∇ RA·∇ kB).(14)\nTaking into account that ˆGR(A)andˆΣR(A)do not depend on the center-of-mass coordinates, we obtain:\ni¯h∂TˆgK+ [ˆgK,Jˆσ·m] +i\n2/braceleftBig\n∇kˆHk,∇RˆgK/bracerightBig\n=ˆΣKˆGA−ˆGRˆΣK+ˆΣRˆgK−ˆgKˆΣA\n−i¯h\n2/parenleftBig\n∂TˆΣK∂EˆGA+∂EˆGR∂TˆΣK/parenrightBig\n+i¯h\n2/parenleftBig\n∂EˆΣR∂TˆgK+∂TˆgK∂EˆΣA/parenrightBig\n−i\n2/parenleftBig\n∇kˆΣR·∇RˆgK+∇RˆgK·∇kˆΣA/parenrightBig\n+i\n2/parenleftBig\n∇RˆΣK·∇kˆGA+∇kˆGR·∇RˆΣK/parenrightBig\n,(15)\nwhere [·,·] and{·,·}stand for a commutator and anticommutator, respectively. In steady state, we have:\n[ˆgK,Jˆσ·m] +i\n2/braceleftBig\n∇kˆHk,∇RˆgK/bracerightBig\n=ˆΣKˆGA−ˆGRˆΣK+ˆΣRˆgK−ˆgKˆΣA\n−i\n2/parenleftBig\n∇kˆΣR·∇RˆgK+∇RˆgK·∇kˆΣA/parenrightBig\n+i\n2/parenleftBig\n∇RˆΣK·∇kˆGA+∇kˆGR·∇RˆΣK/parenrightBig\n.(16)\nFinally, in the dilute limit, we can assume that the self-energy is almost constant and neglect its derivatives on the\nright-hand side:\n[ˆgK,Jˆσ·m] +i\n2/braceleftBig\n∇kˆHk,∇RˆgK/bracerightBig\n=ˆΣKˆGA−ˆGRˆΣK+ˆΣRˆgK−ˆgKˆΣA, (17)\nor\n[ˆgK,Jˆσ·m] +i¯h2\nm(k·∇R) ˆgK=ˆΣKˆGA−ˆGRˆΣK+ˆΣRˆgK−ˆgKˆΣA. (18)6\nFIG. 1.a) Diagrammatic expansion for scatterings off the static impurity potential. b) Self-consistent Born approximation. c)\nSkew-scattering diagrams.\nII. SELF-ENERGY\nA. Self-consistent Born approximation\nLet us consider first and second orders of the diagrammatic expansion for the scattering off the static impurity\npotential (Fig. 1 a). The Green’s function ˆGand the corresponding self-energy ˆΣare defined in the wave vector\nrepresentation as follows:\nˆG(k,t1;k/prime,t2) =ˆG0(k,t1;k/prime,t2) +/summationdisplay\n{ki}ˆG0(k,t1;k1,t2)/angbracketleftk1|ˆHimp|k2/angbracketrightˆG0(k2,t1;k/prime,t2)\n+/summationdisplay\n{ki}ˆG0(k,t1;k1,t2)/angbracketleftk1|ˆHimp|k2/angbracketrightˆG0(k2,t1;k3,t2)/angbracketleftk3|ˆHimp|k4/angbracketrightˆG0(k4,t1;k/prime,t2) +...\n=ˆG0(k,t1;k/prime,t2) +/summationdisplay\n{ki}ˆG0(k,t1;k1,t2)ˆΣ(k1,t1;k2,t2)ˆG(k2,t1;k/prime,t2),(19)\nwhere ˆG0is a free propagator. To consider a particle moving in a random potential we take the average over different\nspatial configurations of the ensemble of Nimpurities. Upon impurity-averaging the first order term in Eq. (19) is\na constant and can be renormalized away. For the second order term, we take into account all two-line irreducible\ndiagrams corresponding to the double scattering off the same impurity and neglect the so-called crossing diagrams,\nwhere impurity lines cross and give a small contribution to the region of interest, E/similarequalEFandk/similarequalkF.1This is nothing\nelse but the self-consistent Born approximation (Fig. 1 b). Using Eq. (7) for the impurity potential, the self-energy is\ngiven by three terms:\nˆΣ1a(k1,t1;k4,t2) =1\nΩ2/summationdisplay\nk2k3ˆG(k2,t1;k3,t2)/angbracketleftV(k1−k2)V(k3−k4)/angbracketright, (20)\nˆΣ1b(k1,t1;k4,t2) =i\nΩ2ξSO\nk2\nF/summationdisplay\nk2k3/bracketleftBig\n[(k1×k2)·ˆσ]ˆG(k2,t1;k3,t2)\n+ˆG(k2,t1;k3,t2)[(k3×k4)·ˆσ]/bracketrightBig\n/angbracketleftV(k1−k2)V(k3−k4)/angbracketright,(21)7\nˆΣ1c(k1,t1;k4,t2) =−1\nΩ2ξ2\nSO\nk4\nF/summationdisplay\nk2k3[(k1×k2)·ˆσ]ˆG(k2,t1;k3,t2)[(k3×k4)·ˆσ]/angbracketleftV(k1−k2)V(k3−k4)/angbracketright, (22)\nwhere impurity averaging leads to:\n/angbracketleftV(k1−k2)V(k3−k4)/angbracketright=v2\ni/angbracketleftBigg/summationdisplay\nRie−i(k1−k2)·Rie−i(k3−k4)·Ri/angbracketrightBigg\n=v2\niNδk1+k3,k2+k4. (23)\nTo proceed with the Wigner transformation, we change the variables:\nk1=k+q\n2k2=k/prime+q/prime\n2k3=k/prime−q/prime\n2k4=k−q\n2\nand\nT=t1+t2\n2t=t1−t2,\nthat gives:\nˆΣ1a(k,t;q,T) =1\nΩ2/summationdisplay\nk2k3ˆG(k/prime,t;q/prime,T)δq,q/prime, (24)\nˆΣ1b(k,t;q,T) =i\nΩ2ξSO\nk2\nF/summationdisplay\nk2k3/bracketleftbigg/bracketleftbig\n((k+q\n2)×(k/prime+q/prime\n2))·ˆσ/bracketrightbigˆG(k/prime,t;q/prime,T)\n+ˆG(k/prime,t;q/prime,T)/bracketleftbig\n((k/prime−q/prime\n2)×(k−q/prime\n2)))·ˆσ/bracketrightbig/bracketrightbigg\nδq,q/prime,(25)\nˆΣ1c(k,t;q,T) =−1\nΩ2ξ2\nSO\nk4\nF/summationdisplay\nk2k3/bracketleftbig\n((k+q\n2)×(k/prime+q/prime\n2))·ˆσ/bracketrightbigˆG(k/prime,t;q/prime,T)/bracketleftbig\n((k/prime−q/prime\n2)×(k−q\n2))·ˆσ/bracketrightbig\nδq,q/prime. (26)\nThe Kronecker function reflects that translation invariance is recovered, and we have in the continuum limit:\nˆΣ1a(k,t;q,T) =v2\nini/integraldisplaydk/prime\n(2π)3ˆG(k/prime,t;q,T), (27)\nˆΣ1b(k,t;q,T) =ˆΣsw(k,t;q,T) +ˆΣsj(k,t;q,T)\n=iv2\niniξSO\nk2\nF/integraldisplaydk/prime\n(2π)3/bracketleftBig\n(k×k/prime)·ˆσ,ˆG(k/prime,t;q,T)/bracketrightBig\n+iv2\niniξSO\n2k2\nF/integraldisplaydk/prime\n(2π)3/braceleftBig\n[q×(k/prime−k)]·ˆσ,ˆG(k/prime,t;q,T)/bracerightBig\n,(28)\nˆΣ1c(k,t;q,T) =−v2\niniξ2\nSO\nk4\nF/integraldisplaydk/prime\n(2π)3[(k×k/prime+1\n2q×k/prime+1\n2k×q)·ˆσ]\n·ˆG(k/prime,t;q,T)[(k/prime×k−1\n2k/prime×q−1\n2q×k)·ˆσ].(29)\nHaving Fourier transformed with respect to q, that gives the Wigner coordinate R:\nˆG(r1;r4) =/integraldisplaydk1\n(2π)3/integraldisplaydk4\n(2π)3ˆG(k1;k4)ei(k1r1−k4r4)\n=/integraldisplaydk\n(2π)3/integraldisplaydq\n(2π)3ˆG(k+1\n2q;k−1\n2q)ei(k+1\n2q)·(R+1\n2r)e−i(k−1\n2q)·(R−1\n2r)\n=/integraldisplaydk\n(2π)3/integraldisplaydq\n(2π)3ˆG(k;q)eiq·Reik·r=ˆG(r;R),(30)8\nwe get the final form for the self-energy in the mixed representation:\nˆΣ1a\nkE(R,T) =v2\nini/integraldisplaydk/prime\n(2π)3ˆGk/primeE(R,T),\nˆΣ1b\nkE(R,T) =ˆΣsw\nkE(R,T) +ˆΣsj\nkE(R,T)\n=iv2\niniξSO\nk2\nF/integraldisplaydk/prime\n(2π)3/bracketleftBig\n(k×k/prime)·ˆσ,ˆGk/primeE(R,T)/bracketrightBig\n+v2\niniξSO\n2k2\nF/integraldisplaydk/prime\n(2π)3∇R/braceleftBig\n(k/prime−k)׈σ,ˆGk/primeE(R;T)/bracerightBig\n,\nˆΣ1c\nkE(R,T) =−v2\niniξ2\nSO\nk4\nF/integraldisplaydk/prime\n(2π)3[(k×k/prime)·ˆσ]ˆGk/primeE(R,T)[(k/prime×k)·ˆσ].(31)\nAt the level of the self-consistent Born approximation, the self-energy is given by the following contributions. The first\nterm ˆΣ1astands for the standard elastic scattering off the on-site impurity potential. To first order of ξSO, there are\ntwo terms, ˆΣswandˆΣsj, related to the side-jump and spin swapping contributions, respectively. Finally, the second\norder ofξSOyields the Elliot-Yafet spin relaxation mechanism (all gradient terms ∼ξ2\nSO∇Rare neglected on account\nof their smallness).\nLet us rewrite these contributions for the retarded, advanced and Keldysh Green’s functions (taking into account\nthat ˆGR(A)does not depend on the center-of-mass coordinates):\nˆΣkE(R,T) =ˆΣ1a\nkE(R,T) +ˆΣ1b\nkE(R,T) +ˆΣ1c\nkE(R,T), (32)\nwhich is equivalent to:\nˆΣR\nkE=v2\nini/integraldisplaydk/prime\n(2π)3/bracketleftbigg\nˆσ0+iξSO\nk2\nF(k×k/prime)·ˆσ/bracketrightbigg\nˆGR\nk/primeE/bracketleftbigg\nˆσ0−iξSO\nk2\nF(k×k/prime)·ˆσ/bracketrightbigg\n, (33)\nˆΣA\nkE=v2\nini/integraldisplaydk/prime\n(2π)3/bracketleftbigg\nˆσ0+iξSO\nk2\nF(k×k/prime)·ˆσ/bracketrightbigg\nˆGA\nk/primeE/bracketleftbigg\nˆσ0−iξSO\nk2\nF(k×k/prime)·ˆσ/bracketrightbigg\n, (34)\nand\nˆΣK\nkE(R,T) =v2\nini/integraldisplaydk/prime\n(2π)3/bracketleftbigg\nˆσ0+iξSO\nk2\nF(k×k/prime)·ˆσ/bracketrightbigg\nˆgK\nk/primeE(R,T)/bracketleftbigg\nˆσ0−iξSO\nk2\nF(k×k/prime)·ˆσ/bracketrightbigg\n+v2\niniξSO\n2k2\nF/integraldisplaydk/prime\n(2π)3∇R/braceleftbig\n(k/prime−k)׈σ,ˆgK\nk/primeE(R;T)/bracerightbig\n.(35)\nB. Skew-scattering\nTo take into account skew-scattering, one has to go beyond the Born approximation. Starting from third order\ndiagrams (Fig. 1 c), we obtain the following expressions for the self-energy to first order in ξSO:\nˆΣ2a(k1,t1;k6,t2) =i\nΩ3ξSO\nk2\nF/summationdisplay\nk2,k3,k4,k5[(k1×k2)·ˆσ]ˆG0(k2,t1;k3,t2)ˆG0(k4,t1;k5,t2)\n·/angbracketleftV(k1−k2)V(k3−k4)V(k5−k6)/angbracketright,(36)\nˆΣ2b(k1,t1;k6,t2) =i\nΩ3ξSO\nk2\nF/summationdisplay\nk2,k3,k4,k5ˆG0(k2,t1;k3,t2)[(k3×k4)·ˆσ]ˆG0(k4,t1;k5,t2)\n·/angbracketleftV(k1−k2)V(k3−k4)V(k5−k6)/angbracketright,(37)\nˆΣ2c(k1,t1;k6,t2) =i\nΩ3ξSO\nk2\nF/summationdisplay\nk2,k3,k4,k5ˆG0(k2,t1;k3,t2)ˆG0(k4,t1;k5,t2)[(k5×k6)·ˆσ]\n·/angbracketleftV(k1−k2)V(k3−k4)V(k5−k6)/angbracketright,(38)9\nwhere impurity averaging in the triple scattering off the same impurity potential is implied:\n/angbracketleftV(k1−k2)V(k3−k4)V(k5−k6)/angbracketright=v3\ni/angbracketleftBigg/summationdisplay\nRie−i(k1−k2)·Rie−i(k3−k4)·Rie−i(k5−k6)·Ri/angbracketrightBigg\n=v3\niNδk1+k3+k5,k2+k4+k6.(39)\nHere, we do not consider triple scatterings off the on-site impurity potential without spin-orbit coupling, which gives\na negligible correction to the elastic relaxation time ( ∼βvi\nτ0). Changing the variables:\nk1=k+q\n2k2=k/prime+q/prime\n2k3=k/prime−q/prime\n2,\nk4=k/prime/prime+q/prime/prime\n2k5=k/prime/prime−q/prime/prime\n2k6=k−q\n2(40)\nand\nT=t1+t2\n2t=t1−t2 (41)\nleads in the continuum limit to:\nˆΣ2a(k,t;q,T) =iv3\niniξSO\nk2\nF/integraldisplaydk/prime\n(2π)3/integraldisplaydk/prime/prime\n(2π)3/integraldisplaydq/prime\n(2π)3\n/bracketleftbigg/parenleftBig\nk+q\n2/parenrightBig\n×/parenleftbigg\nk/prime+q/prime\n2/parenrightbigg\n·ˆσ/bracketrightbigg\nˆG(k/prime,t;q/prime,T)ˆG(k/prime/prime,t;q−q/prime,T),(42)\nˆΣ2b(k,t;q,T) =iv3\niniξSO\nk2\nF/integraldisplaydk/prime\n(2π)3/integraldisplaydk/prime/prime\n(2π)3/integraldisplaydq/prime\n(2π)3\nˆG(k/prime,t;q/prime,T)/bracketleftbigg/parenleftbigg\nk/prime−q/prime\n2/parenrightbigg\n×/parenleftbigg\nk/prime/prime+q\n2−q/prime\n2/parenrightbigg\n·ˆσ/bracketrightbigg\nˆG(k/prime/prime,t;q−q/prime,T),(43)\nˆΣ2c(k,t;q,T) =iv3\niniξSO\nk2\nF/integraldisplaydk/prime\n(2π)3/integraldisplaydk/prime/prime\n(2π)3/integraldisplaydq/prime\n(2π)3\nˆG(k/prime,t;q/prime,T)ˆG(k/prime/prime,t;q−q/prime,T)/bracketleftbigg/parenleftbigg\nk/prime/prime−q\n2+q/prime\n2/parenrightbigg\n×/parenleftBig\nk−q\n2/parenrightBig\n·ˆσ/bracketrightbigg\n.(44)\nLet us assume that any inhomogeneity in a system is smooth, so we can neglect all gradient terms ∼q. Having\nFourier transformed with respect to qandt, we obtain:\nˆΣ/primekE(R,T) =ˆΣ2a\nkE(R,T) +ˆΣ2b\nkE(R,T) +ˆΣ2c\nkE(R,T), (45)\nwhere\nˆΣ2a\nkE(R,T) =iv3\niniξSO\nk2\nF/integraldisplaydk/prime\n(2π)3/integraldisplaydk/prime/prime\n(2π)3[(k×k/prime)·ˆσ]ˆGk/primeE(R,T)ˆGk/prime/primeE(R,T), (46)\nˆΣ2b\nkE(R,T) =iv3\niniξSO\nk2\nF/integraldisplaydk/prime\n(2π)3/integraldisplaydk/prime/prime\n(2π)3ˆGk/primeE(R,T)[(k/prime×k/prime/prime)·ˆσ]ˆGk/prime/primeE(R,T), (47)\nˆΣ2c\nkE(R,T) =iv3\niniξSO\nk2\nF/integraldisplaydk/prime\n(2π)3/integraldisplaydk/prime/prime\n(2π)3ˆGk/primeE(R,T)ˆGk/prime/primeE(R,T)[(k/prime/prime×k)·ˆσ]. (48)\nTo first order in ξSO, we can express the retarded and advanced Green’s functions by using the Sokhotski formula:\nˆGR(A)\nkE=/parenleftBig\nˆE−ˆHk−ˆΣR(A)\nkE/parenrightBig−1\n≈1\n2/summationdisplay\ns=±ˆσ0+sˆσ·m\nE−Eks±i¯h\n2τs\n=1\n2/summationdisplay\ns=±(ˆσ0+sˆσ·m) [∓iπδ(E−Eks)],(49)10\nwhereτsis the spin dependent relaxation time. As a result, the retarded and advanced components of the skew-\nscattering self-energy vanish, and we deal with its Keldysh part, which survives for ˆΣ2aandˆΣ2conly:\nˆΣ/primeK\nkE(R,T) =iξSO\nk2\nFv3\nini/integraldisplaydk/prime\n(2π)3/integraldisplaydk/prime/prime\n(2π)3(k×k/prime)·ˆσ/parenleftBig\nˆGR\nk/primeEˆgK\nk/prime/primeE(R,T) + ˆgK\nk/primeE(R,T)ˆGA\nk/prime/primeE/parenrightBig\n+iξSO\nk2\nFv3\nini/integraldisplaydk/prime\n(2π)3/integraldisplaydk/prime/prime\n(2π)3/parenleftBig\nˆGR\nk/primeEˆgK\nk/prime/primeE(R,T) + ˆgK\nk/primeE(R,T)ˆGA\nk/prime/primeE/parenrightBig\n(k/prime/prime×k)·ˆσ(50)\nor\nˆΣ/primeK\nkE(R,T) =iξSO\nk2\nFv3\nini/integraldisplaydk/prime\n(2π)3/integraldisplaydk/prime/prime\n(2π)3(k×k/prime)·ˆσˆgK\nk/primeE(R,T)ˆGA\nk/prime/primeE\n+iξSO\nk2\nFv3\nini/integraldisplaydk/prime\n(2π)3/integraldisplaydk/prime/prime\n(2π)3ˆGR\nk/primeEˆgK\nk/prime/primeE(R,T)(k/prime/prime×k)·ˆσ.(51)\nThis expression can be further simplified, as we integrate over k:\n∓iπ\n2/integraldisplaydk/prime\n(2π)3/summationdisplay\ns=±(ˆσ0+sˆσ·m)δ(E−Eks) =∓iπ\n2(D↑+D↓) (ˆσ0+δˆσ·m), (52)\nwhereD↑(↓)is the spin-dependent density of states and δ= (D↑−D↓)/(D↑+D↓), so the final form of ˆΣ/primeK\nkE(R,T)\nis given by:\nˆΣ/primeK\nkE(R,T) =−π\n2ξSO\nk2\nFv3\nini(D↑+D↓)/integraldisplaydk/prime\n(2π)3(k×k/prime)·ˆσˆgK\nk/primeE(R,T) [ˆσ0+δˆσ·m]\n−π\n2ξSO\nk2\nFv3\nini(D↑+D↓)/integraldisplaydk/prime\n(2π)3[ˆσ0+δˆσ·m] ˆgK\nk/primeE(R,T)(k×k/prime)·ˆσ.(53)\nIn the weak exchange coupling limit ( J/lessmuchεF), we can express D↑(↓)≈D0(1∓β), whereβ=J\n2εFis the spin\npolarization factor and D0=mkF\n2π2¯h2is the spin independent density of states per spin at the Fermi level. Then, we\nobtain:\nˆΣ/primeK\nkE(R,T) =−πv3\niniD0ξSO\nk2\nF/integraldisplaydk/prime\n(2π)3(k×k/prime)·ˆσˆgK\nk/primeE(R,T) [ˆσ0−βˆσ·m]\n−πv3\niniD0ξSO\nk2\nF/integraldisplaydk/prime\n(2π)3[ˆσ0−βˆσ·m] ˆgK\nk/primeE(R,T)(k×k/prime)·ˆσ(54)\nor\nˆΣ/primeK\nkE(R,T) =−¯h\n2vi\nτ0ξSO\nk2\nF/integraldisplaydk/prime\n(2π)3(k×k/prime)·ˆσˆgK\nk/primeE(R,T) [ˆσ0−βˆσ·m]\n−¯h\n2vi\nτ0ξSO\nk2\nF/integraldisplaydk/prime\n(2π)3[ˆσ0−βˆσ·m] ˆgK\nk/primeE(R,T)(k×k/prime)·ˆσ,(55)\nwhere1\nτ0= 2πv2\niniD0/¯his the spin independent relaxation time.\nIII. RELAXATION TIME\nThe imaginary part of the retarded and advanced self-energies is related to the momentum relaxation time, which\nis given by the elastic scattering off the on-site impurity potential and Elliot-Yafet mechanism:\nˆΣR(A)\nkE=∓i¯h\n2ˆτk. (56)\nTaking into account Eqs. (33) and (2), we get:\n1\nˆτk=πv2\nini\n¯h/summationdisplay\ns=±/integraldisplaydk/prime\n(2π)3/bracketleftbigg\nˆσ0+iξSO\nk2\nFn·ˆσ/bracketrightbigg\n(ˆσ0+sˆσ·m)/bracketleftbigg\nˆσ0−iξSO\nk2\nFn·ˆσ/bracketrightbigg\nδ(E−Ek/primes), (57)11\nwheren=k×k/prime. In terms of spherical coordinates, dk=k2dksinθdθdφ withk∈[0,kF],θ∈[0,π] andφ∈[0,2π],\nit is easy to show:\n/integraldisplay\nkidk= 0, (58)\nwherekiis theith cartesian coordinate of k, and all terms linear in nvanish. Thus, we get:\n1\nˆτk=πv2\nini\n¯h/summationdisplay\ns=±/integraldisplaydk/prime\n(2π)3/bracketleftbigg\nˆσ0+sˆσ·m+ξ2\nSO\nk4\nF(n·ˆσ)(n·ˆσ) +sξ2\nSO\nk4\nF(n·ˆσ)(ˆσ·m)(n·ˆσ)/bracketrightbigg\nδ(E−Ek/primes), (59)\nor having used ( a·ˆσ)(b·ˆσ) = (a·b)ˆσ0+i(a×b)·ˆσ:\n1\nˆτk=πv2\nini\n¯h/integraldisplaydk/prime\n(2π)3/bracketleftbigg\n(1 +ξ2\nSO\nk4\nFn2)ˆσ0(δ(E−Ek/prime+) +δ(E−Ek/prime−))\n+/parenleftbigg\nˆσ·m+ξ2\nSO\nk4\nFˆσ·(2n(n·m)−mn2)/parenrightbigg\n(δ(E−Ek/prime+)−δ(E−Ek/prime−))/bracketrightbigg\n.(60)\nLet us also consider the following expressions:\n/integraldisplaydk/prime\n(2π)3k/prime\nik/prime\nj(δ(E−Ek/prime+)±δ(E−Ek/prime−)) = 0 for i/negationslash=j,\n/integraldisplaydk/prime\n(2π)3k/prime2\ni(δ(E−Ek/prime+)±δ(E−Ek/prime−)) =1\n6π2/integraldisplay\ndk/primek/prime4(δ(E−Ek/prime+)±δ(E−Ek/prime−)),\n/integraldisplaydk/prime\n(2π)3n2(δ(E−Ek/prime+)±δ(E−Ek/prime−)) =1\n3π2k2/integraldisplay\ndk/primek/prime4(δ(E−Ek/prime+)±δ(E−Ek/prime−)).(61)\nWe can rewrite Eq. (60) as:\n1\nˆτk=v2\nini\n2π¯h/integraldisplay\ndk/prime/bracketleftbigg\n(k/prime2+2\n3ξ2\nSO\nk4\nFk2k/prime4)ˆσ0(δ(E−Ek/prime+) +δ(E−Ek/prime−))\n+/parenleftbigg\nˆσ·mk/prime2−2\n3ξ2\nSO\nk4\nFk/prime4(ˆσ·k)(k·m)/parenrightbigg\n(δ(E−Ek/prime+)−δ(E−Ek/prime−))/bracketrightbigg\n.(62)\nNext, we can employ the following relation for the delta-function:\nδ(E−Eks) =δ(k−ks)\n¯h2ks\nm, (63)\nwherek±=/radicalbig\n2m(εF∓J)/¯h, andεF=¯h2k2\nF\n2mis the Fermi energy. Then, integrating over k/primegives:\n1\nˆτk=v2\nini\n2π¯h/parenleftBigm\n¯h2(k++k−)ˆσ0+2m\n3¯h2ξ2\nSO\nk4\nFk2(k3\n++k3\n−)ˆσ0+m\n¯h2(k+−k−)ˆσ·m\n−2m\n3¯h2ξ2\nSO\nk4\nF(k3\n+−k3\n−)(ˆσ·k)(k·m)/parenrightBig\n.(64)\nFinally, in the weak exchange coupling limit ( J/lessmuchεF), we can perform a Taylor expansion, k±≈kF(1∓β). Neglecting\nhigher order terms ∼βξ2\nSO, we obtain:\n1\nˆτk=1\nτ0/parenleftbigg\nˆσ0−βˆσ·m+2\n3ξ2\nSO\nk2\nFk2ˆσ0/parenrightbigg\n, (65)\nwhere1\nτ0= 2πv2\niniD0/¯his the spin-independent relaxation time due to scattering off the impurity potential.12\nIV. AVERAGED VELOCITY OPERATOR\nFor educational purposes, let us derive the averaged velocity operator in diffusive ferromagnets with extrinsic spin-\norbit coupling.4Within the Lippmann-Schwinger equation, the scattered state /bardblk,s/angbracketrightcan be written in the first order\nofˆHimp:\n/bardblk,s/angbracketright=|k,s/angbracketright+/summationdisplay\nk/primeˆGR\n0,k/prime/angbracketleftk/prime|ˆHimp|k/angbracketright|k/prime,s/angbracketright\n=|k,s/angbracketright−1\nΩiπ\n2/summationdisplay\nk/primes/prime(ˆσ0+s/primeˆσ·m)( ˆσ0−iξSO\nk2\nFˆσ·(k×k/prime))V(k/prime−k)δ(E−Ek/primes/prime)|k/prime,s/angbracketright,(66)\nand\n/angbracketleftk,s/bardbl=/angbracketleftk,s|+/summationdisplay\nk/prime/angbracketleftk/prime,s|/angbracketleftk|ˆHimp|k/prime/angbracketrightˆGR\n0,k/prime\n=/angbracketleftk,s|+1\nΩiπ\n2/summationdisplay\nk/primes/prime/angbracketleftk/prime,s|( ˆσ0+iξSO\nk2\nFˆσ·(k×k/prime))(ˆσ0+s/primeˆσ·m)V(k−k/prime)δ(E−Ek/primes/prime).(67)\nThe corresponding matrix elements of the velocity operator can be found as:\nvss/prime\nkk/prime=−i\n¯h/angbracketleftk,s/bardbl[ˆr,ˆH]/bardblk/prime,s/prime/angbracketright=−i\n¯h/angbracketleftk,s/bardbl[ˆr,ˆH0+ˆHimp]/bardblk/prime,s/prime/angbracketright, (68)\nwhere\n−i\n¯h[ˆr,ˆH] =−i\n¯h[ˆr,ˆH0+ˆHimp] =−i¯h\nm∇ˆσ0+ξSO\n¯hk2\nF/summationdisplay\nRiˆσ×∇V(r−Ri). (69)\nLet us express these terms separately neglecting higher order terms ∼ξ2\nSO:\n−i\n¯h/angbracketleftk,s/bardbl[ˆr,ˆH0]/bardblk/prime,s/prime/angbracketright=−i¯h\nm/angbracketleftk,s/bardbl∇/bardblk/prime,s/prime/angbracketright\n=−i¯h\nm/angbracketleftk|∇|k/prime/angbracketrightδss/prime−iπ\n2/summationdisplay\ns/prime/prime/integraldisplaydk/prime/prime\n(2π)3V(k/prime/prime−k/prime)δ(E−Ek/prime/primes/prime/prime)/angbracketleftk|∇|k/prime/prime/angbracketright/angbracketlefts|(ˆσ0+s/prime/primeˆσ·m)( ˆσ0−iξSO\nk2\nFˆσ·(k/prime×k/prime/prime))|s/prime/angbracketright\n+iπ\n2/summationdisplay\ns/prime/prime/integraldisplaydk/prime/prime\n(2π)3V(k−k/prime/prime)δ(E−Ek/prime/primes/prime/prime)/angbracketlefts|( ˆσ0+iξSO\nk2\nFˆσ·(k×k/prime/prime))(ˆσ0+s/prime/primeˆσ·m)|s/prime/angbracketright/angbracketleftk/prime/prime|∇|k/prime/angbracketright,\n(70)\nand\n−i\n¯h/angbracketleftk,s/bardbl[ˆr,ˆHimp]/bardblk/prime,s/prime/angbracketright=ξSO\n¯hk2\nF/summationdisplay\nRi/angbracketleftk,s/bardblˆσ×∇V(r−Ri)/bardblk/prime,s/prime/angbracketright\n=i\nΩξSO\n¯hk2\nF/angbracketlefts|ˆσ|s/prime/angbracketright×(k−k/prime)V(k−k/prime)\n+1\nΩπ\n2ξSO\n¯hk2\nF/summationdisplay\ns/prime/prime/integraldisplaydk/prime/prime\n(2π)3V(k/prime/prime−k/prime)V(k−k/prime/prime)δ(E−Ek/prime/primes/prime/prime)/angbracketlefts|ˆσ×(k−k/prime/prime)(ˆσ0+s/prime/primeˆσ·m)|s/prime/angbracketright\n−1\nΩπ\n2ξSO\n¯hk2\nF/summationdisplay\ns/prime/prime/integraldisplaydk/prime/prime\n(2π)3V(k−k/prime/prime)V(k/prime/prime−k/prime)δ(E−Ek/prime/primes/prime/prime)/angbracketlefts|(ˆσ0+s/prime/primeˆσ·m)ˆσ×(k/prime/prime−k/prime)|s/prime/angbracketright.(71)\nUpon impurity averaging we obtain:\nvss/prime\nk=¯h\nmkδss/prime+v2\niniπ\n2ξSO\n¯hk2\nF/summationdisplay\ns/prime/prime/integraldisplaydk/prime/prime\n(2π)3δ(E−Ek/prime/primes/prime/prime)/angbracketlefts|{ˆσ×(k−k/prime/prime),(ˆσ0+s/prime/primeˆσ·m)}|s/prime/angbracketright\n=¯h\nmkδss/prime+v2\niniπξSO\n¯hk2\nF/summationdisplay\ns/prime/prime/integraldisplaydk/prime/prime\n(2π)3δ(E−Ek/prime/primes/prime/prime)/angbracketlefts|ˆσ×k+s/prime/primem×k|s/prime/angbracketright,(72)\nor in the limit J/lessmuchεF:\nˆvk=¯h\nmkˆσ0+1\nτ0ξSO\nk2\nFˆσ×k−β\nτ0ξSO\nk2\nFm×kˆσ0. (73)13\nV. QUANTUM TRANSPORT EQUATIONS\nHaving integrated Eq. (18) over energy, we arrive at the kinetic equation written for the distribution function ˆ gk:\n−i[ˆgk,Jˆσ·m] +¯h2\nm(k·∇R) ˆgk=coll, (74)\nwhere the collision integral is defined as:\ncoll=Jk+Ik, (75)\nand\nJk=/integraldisplaydE\n2π/parenleftBig\nˆΣK\nkEˆGA\nkE−ˆGR\nkEˆΣK\nkE/parenrightBig\n, (76)\nIk=/integraldisplaydE\n2π/parenleftBig\nˆΣR\nkEˆgK\nkE−ˆgK\nkEˆΣA\nkE/parenrightBig\n. (77)\nLet us proceed with its detailed derivation. Taking into account the Kadanoff-Baym anzats (12) for ˆ gK\nkE, the integration\nover energy (up to a given Fermi level εF) can be performed by using the residue theorem:\n/integraldisplaydE\n2πˆGR\nkEˆgk/primeˆGA\nk/primeE=/integraldisplaydE\n8π/summationdisplay\ns,s/primeˆσ0+sˆσ·m\nE−Eks+i¯h\n2τsˆgk/primeˆσ0+s/primeˆσ·m\nE−Ek/primes/prime−i¯h\n2τs/prime\n=−i\n4/summationdisplay\ns,s/prime(ˆσ0+sˆσ·m)ˆgk/prime(ˆσ0+s/primeˆσ·m)\nεF−Ek/primes/prime−i¯h\n2/parenleftBig\n1\nτs/prime+1\nτs/parenrightBig,(78)\nand/integraldisplaydE\n2πˆGR\nk/primeEˆgk/primeˆGA\nkE=/integraldisplaydE\n8π/summationdisplay\ns,s/primeˆσ0+s/primeˆσ·m\nE−Ek/primes/prime+i¯h\n2τs/primeˆgk/primeˆσ0+sˆσ·m\nE−Eks−i¯h\n2τs\n=i\n4/summationdisplay\ns,s/prime(ˆσ0+s/primeˆσ·m)ˆgk/prime(ˆσ0+sˆσ·m)\nεF−Ek/primes/prime+i¯h\n2/parenleftBig\n1\nτs+1\nτs/prime/parenrightBig,(79)\nwhile/integraldisplaydE\n2πˆGR(A)\nkEˆgk/primeˆGR(A)\nk/primeE= 0, (80)\nwhere the retarded and advanced Green’s functions are defined as:\nˆGR(A)\nkE=1\n2/summationdisplay\ns=±ˆσ0+sˆσ·m\nE−Eks±i¯h\n2τs. (81)\nAssuming the scattering term in the denominator to be small and transport properties to be described solely by the\nelectrons close to the Fermi level, we can rewrite these expressions with the Sokhotski formula:\n/integraldisplaydE\n2πˆGR\nkEˆgk/primeˆGA\nk/primeE=π\n2/summationdisplay\ns/primeˆgk/prime(ˆσ0+s/primeˆσ·m)δ(εF−Ek/primes/prime), (82)\nand/integraldisplaydE\n2πˆGR\nk/primeEˆgk/primeˆGA\nkE=π\n2/summationdisplay\ns/prime(ˆσ0+s/primeˆσ·m)ˆgk/primeδ(εF−Ek/primes/prime). (83)\nStarting from the Born approximation (35), we have:\nˆΣK\nkEˆGA\nkE=v2\nini/integraldisplaydk/prime\n(2π)3/bracketleftbigg\nˆGR\nk/primeEˆgk/primeˆGA\nkE+iξSO\nk2\nFn·ˆσˆGR\nk/primeEˆgk/primeˆGA\nkE\n−iξSO\nk2\nFˆGR\nk/primeEˆgk/primen·ˆσˆGA\nkE+ξ2\nSO\nk4\nFn·ˆσˆGR\nk/primeEˆgk/primen·ˆσˆGA\nkE/bracketrightbigg\n+v2\nini\n2ξSO\nk2\nF/integraldisplaydk/prime\n(2π)3/braceleftBig\n(k/prime−k}׈σ,ˆGR\nk/primeE∇Rˆgk/prime/bracerightBig\nˆGA\nkE,(84)14\nand\nˆGR\nkEˆΣK\nkE=−v2\nini/integraldisplaydk/prime\n(2π)3/bracketleftbigg\nˆGR\nkEˆgk/primeˆGA\nk/primeE+iξSO\nk2\nFˆGR\nkEn·ˆσˆgk/primeˆGA\nk/primeE\n−iξSO\nk2\nFˆGR\nkEˆgk/primeˆGA\nk/primeEn·ˆσ+ξ2\nSO\nk4\nFˆGR\nkEn·ˆσˆgk/primeˆGA\nk/primeEn·ˆσ/bracketrightbigg\n−v2\nini\n2ξSO\nk2\nF/integraldisplaydk/prime\n(2π)3ˆGR\nkE/braceleftBig\n(k/prime−k)׈σ,∇Rˆgk/primeˆGA\nk/primeE/bracerightBig\n.(85)\nBy using the following relations:\nˆσaˆσb=iεabcˆσc+δabˆσ0,\n(a·ˆσ)(b·ˆσ) = (a·b)ˆσ0+i(a×b)·ˆσ,(86)\nthese terms give:\n/integraldisplaydE\n2π/bracketleftBig\nˆGR\nk/primeEˆgk/primeˆGA\nkE+ˆGR\nkEˆgk/primeˆGA\nk/primeE/bracketrightBig\n=πˆgk/primeδT+π{ˆgk/prime,ˆσ·m}δJ, (87)\n/integraldisplaydE\n2πiξSO\nk2\nF/bracketleftBig\n[n·ˆσ,ˆGR\nk/primeEˆgk/prime]ˆGA\nkE+ˆGR\nkE[n·ˆσ,ˆgk/primeˆGA\nk/primeE]/bracketrightBig\n=\n=iπξSO\nk2\nF[n·ˆσ,ˆgk/prime]δT+iπξSO\nk2\nF(n·ˆσˆgk/primem·ˆσ−m·ˆσˆgk/primen·ˆσ)δJ\n−πξSO\nk2\nF{(n×m)·ˆσ,ˆgk/prime}δJ,(88)\n/integraldisplaydE\n2πξ2\nSO\nk4\nF/bracketleftBig\nn·ˆσˆGR\nk/primeEˆgk/primen·ˆσˆGA\nkE+ˆGR\nkEn·ˆσˆgk/primeˆGA\nk/primeEn·ˆσ/bracketrightBig\n=\n=πξ2\nSO\nk4\nFn·ˆσˆgk/primen·ˆσδT+πξ2\nSO\nk4\nFn·m{ˆgk/prime,n·ˆσ}δJ\n+iπξ2\nSO\nk4\nF(n×m)·ˆσˆgk/primen·ˆσδJ−iπξ2\nSO\nk4\nFn·ˆσˆgk/prime(n×m)·ˆσδJ,(89)\n/integraldisplaydE\n2π/bracketleftBig/braceleftBig\n(k/prime−k}׈σ,ˆGR\nk/primeE∇Rˆgk/prime/bracerightBig\nˆGA\nkE+ˆGR\nkE/braceleftBig\n(k/prime−k)׈σ,∇Rˆgk/primeˆGA\nk/primeE/bracerightBig/bracketrightBig\n=\n=π{(k/prime−k)׈σ,∇Rˆgk/prime}δT+π{(k/prime−k)׈σ,{∇Rˆgk/prime,m·ˆσ}}δJ,(90)\nwhere the following notations are used:\nδT=δ(εF−Ek/prime+) +δ(εF−Ek/prime−),\nδJ=1\n2[δ(εF−Ek/prime+)−δ(εF−Ek/prime−)].(91)\nFor the skew-scattering self-energy Eq. (55), we have:\nˆΣ/primeK\nkEˆGA\nkE=−¯h\n2vi\nτ0ξSO\nk2\nF/integraldisplaydk/prime\n(2π)3n·ˆσˆGR\nk/primeEˆgk/prime[ˆσ0−βˆσ·m]ˆGA\nkE\n−¯h\n2vi\nτ0ξSO\nk2\nF/integraldisplaydk/prime\n(2π)3[ˆσ0−βˆσ·m]ˆGR\nk/primeEˆgk/primen·ˆσˆGA\nkE,(92)\nand\nˆGR\nkEˆΣ/primeK\nkE=¯h\n2vi\nτ0ξSO\nk2\nF/integraldisplaydk/prime\n(2π)3ˆGR\nkEn·ˆσˆgk/primeˆGA\nk/primeE[ˆσ0−βˆσ·m]\n+¯h\n2vi\nτ0ξSO\nk2\nF/integraldisplaydk/prime\n(2π)3ˆGR\nkE[ˆσ0−βˆσ·m] ˆgk/primeˆGA\nk/primeEn·ˆσ,(93)15\nthat gives:\n/integraldisplaydE\n2π/bracketleftBig/braceleftBig\nn·ˆσ,ˆGR\nk/primeEˆgk/prime/bracerightBig\nˆGA\nkE+ˆGR\nkE/braceleftBig\nn·ˆσ,ˆgk/primeˆGA\nk/primeE/bracerightBig/bracketrightBig\n=\n=π{ˆgk/prime,n·ˆσ}δT+π{n·ˆσ,{ˆgk/prime,ˆσ·m}}δJ,(94)\n/integraldisplaydE\n2π/bracketleftBig\nn·ˆσˆGR\nk/primeEˆgk/primeˆσ·mˆGA\nkE+ˆσ·mˆGR\nk/primeEˆgk/primen·ˆσˆGA\nkE\n+ˆGR\nkEn·ˆσˆgk/primeˆGA\nk/primeEˆσ·m+ˆGR\nkEˆσ·mˆgk/primeˆGA\nk/primeEn·ˆσ/bracketrightBig\n=\n=π(ˆσ·mˆgk/primen·ˆσ+n·ˆσˆgk/primeˆσ·m)δT+πn·m{ˆgk/prime,ˆσ·m}δJ+πm2{ˆgk/prime,n·ˆσ}δJ\n+iπ((n×m)·ˆσˆgk/primeˆσ·m−ˆσ·mˆgk/prime(n×m)·ˆσ)δJ.(95)\nFinally, we obtain:\nJk=πa1/integraldisplaydk/prime\n(2π)3/bracketleftBig\nˆgk/primeδT+{ˆgk/prime,ˆσ·m}δJ\n+iξSO\nk2\nF[n·ˆσ,ˆgk/prime]δT+iξSO\nk2\nF(n·ˆσˆgk/primem·ˆσ−m·ˆσˆgk/primen·ˆσ)δJ−ξSO\nk2\nF{(n×m)·ˆσ,ˆgk/prime}δJ\n+ξ2\nSO\nk4\nFn·ˆσˆgk/primen·ˆσδT+ξ2\nSO\nk4\nFn·m{ˆgk/prime,n·ˆσ}δJ\n+iξ2\nSO\nk4\nF(n×m)·ˆσˆgk/primen·ˆσδJ−iξ2\nSO\nk4\nFn·ˆσˆgk/prime(n×m)·ˆσδJ\n+1\n2ξSO\nk2\nF{(k/prime−k)׈σ,∇Rˆgk/prime}δT+1\n2ξSO\nk2\nF{(k/prime−k)׈σ,{∇Rˆgk/prime,m·ˆσ}}δJ\n−πa2/integraldisplaydk/prime\n(2π)3/bracketleftBig\n{ˆgk/prime,n·ˆσ}δT+{n·ˆσ,{ˆgk/prime,ˆσ·m}}δJ\n−β(ˆσ·mˆgk/primen·ˆσ+n·ˆσˆgk/primeˆσ·m)δT−βn·m{ˆgk/prime,ˆσ·m}δJ−βm2{ˆgk/prime,n·ˆσ}δJ\n−iβ((n×m)·ˆσˆgk/primeˆσ·m−ˆσ·mˆgk/prime(n×m)·ˆσ)δJ/bracketrightBig\n,(96)\nwherea1=niv2\nianda2=¯h\n2vi\nτ0ξSO\nk2\nF.\nIn a similar manner, by using Eqs. (33) and (34) we proceed with the second part of the collision integral Ik:\nIk=−a1/integraldisplaydE\n2π/integraldisplaydk/prime\n(2π)3/bracketleftBig/parenleftbig\nˆσ0+iξSO\nk2\nFn·ˆσ/parenrightbigˆGR\nk/primeE(ˆσ0−iξSO\nk2\nFn·ˆσ)ˆgkˆGA\nkE\n+ˆGR\nkEˆgk/parenleftbig\nˆσ0+iξSO\nk2\nFn·ˆσ/parenrightbigˆGA\nk/primeE(ˆσ0−iξSO\nk2\nFn·ˆσ)/bracketrightBig\n,(97)\nwhere\n/integraldisplaydE\n2π/bracketleftBig\nˆGR\nk/primeEˆgkˆGA\nkE+ˆGR\nkEˆgkˆGA\nk/primeE/bracketrightBig\n=πˆgkδT+π{ˆgk,ˆσ·m}δJ, (98)\n/integraldisplaydE\n2πiξSO\nk2\nF/bracketleftBig\n[n·ˆσ,ˆGR\nk/primeE]ˆgkˆGA\nkE+ˆGR\nkEˆgk[n·ˆσ,ˆGA\nk/primeE]/bracketrightBig\n=\n=−πξSO\nk2\nF(n×m)·{ˆgk,ˆσ}δJ,(99)\n/integraldisplaydE\n2πξ2\nSO\nk4\nF/bracketleftBig\nn·ˆσˆGR\nk/primeEn·ˆσˆgkˆGA\nkE+ˆGR\nkEˆgkn·ˆσˆGA\nk/primeEn·ˆσ/bracketrightBig\n=\n=πξ2\nSO\nk4\nFn2ˆgkδT+πξ2\nSO\nk4\nF(2(n·m)n−n2m)·{ˆgk,ˆσ}δJ,(100)16\nso we obtain:\nIk=−πa1/integraldisplaydk/prime\n(2π)3/bracketleftBig\nˆgkδT+{ˆgk,ˆσ·m}δJ−ξSO\nk2\nF(n×m)·{ˆgk,ˆσ}δJ\n+ξ2\nSO\nk4\nFn2ˆgkδT+ξ2\nSO\nk4\nF(2(n·m)n−n2m)·{ˆgk,ˆσ}δJ/bracketrightBig\n.(101)\nFinally, the collision integral is written as:\ncoll=πa1/integraldisplaydk/prime\n(2π)3/bracketleftBig\n(ˆgk/prime−ˆgk)δT+{ˆgk/prime−ˆgk,ˆσ·m}δJ\n+iξSO\nk2\nF[n·ˆσ,ˆgk/prime]δT+iξSO\nk2\nF(n·ˆσˆgk/primem·ˆσ−m·ˆσˆgk/primen·ˆσ)δJ−ξSO\nk2\nF{(n×m)·ˆσ,ˆgk/prime}δJ\n+ξ2\nSO\nk4\nFn·ˆσˆgk/primen·ˆσδT+ξ2\nSO\nk4\nFn·m{ˆgk/prime,n·ˆσ}δJ\n+iξ2\nSO\nk4\nF(n×m)·ˆσˆgk/primen·ˆσδJ−iξ2\nSO\nk4\nFn·ˆσˆgk/prime(n×m)·ˆσδJ\n−ξ2\nSO\nk4\nFn2ˆgkδT−ξ2\nSO\nk4\nF(2(n·m)n−n2m)·{ˆgk,ˆσ}δJ\n+1\n2ξSO\nk2\nF{(k/prime−k)׈σ,∇Rˆgk/prime}δT+1\n2ξSO\nk2\nF{(k/prime−k)׈σ,{∇Rˆgk/prime,m·ˆσ}}δJ/bracketrightBig\n−πa2/integraldisplaydk/prime\n(2π)3/bracketleftBig\n{ˆgk/prime,n·ˆσ}δT+{n·ˆσ,{ˆgk/prime,ˆσ·m}}δJ\n−β(ˆσ·mˆgk/primen·ˆσ+n·ˆσˆgk/primeˆσ·m)δT−βn·m{ˆgk/prime,ˆσ·m}δJ−βm2{ˆgk/prime,n·ˆσ}δJ\n−iβ((n×m)·ˆσˆgk/primeˆσ·m−ˆσ·mˆgk/prime(n×m)·ˆσ)δJ/bracketrightBig\n.(102)\nBy neglecting higher order terms βδJ∼β2in skew-scattering and introducing a more familiar distribution function\nˆgk= ˆσ0−2ˆhk, we get:\ncoll=−2πa1/integraldisplaydk/prime\n(2π)3/bracketleftBig\n(ˆhk/prime−ˆhk)δT+{ˆhk/prime−ˆhk,ˆσ·m}δJ\n+iξSO\nk2\nF[n·ˆσ,ˆhk/prime]δT+iξSO\nk2\nF(n·ˆσˆhk/primem·ˆσ−m·ˆσˆhk/primen·ˆσ)δJ−ξSO\nk2\nF{(n×m)·ˆσ,ˆhk/prime}δJ\n+1\n2ξSO\nk2\nF{(k/prime−k)׈σ,∇Rˆhk/prime}δT+1\n2ξSO\nk2\nF{(k/prime−k)׈σ,{∇Rˆhk/prime,m·ˆσ}}δJ\n+ξ2\nSO\nk4\nFn·ˆσˆhk/primen·ˆσδT+ξ2\nSO\nk4\nFn·m{ˆhk/prime,n·ˆσ}δJ\n+iξ2\nSO\nk4\nF(n×m)·ˆσˆhk/primen·ˆσδJ−iξ2\nSO\nk4\nFn·ˆσˆhk/prime(n×m)·ˆσδJ\n−ξ2\nSO\nk4\nFn2ˆhkδT−ξ2\nSO\nk4\nF(2(n·m)n−n2m)·{ˆhk,ˆσ}δJ/bracketrightBig\n+ 2πa2/integraldisplaydk/prime\n(2π)3/bracketleftBig\n{ˆhk/prime,n·ˆσ}δT+{n·ˆσ,{ˆhk/prime,ˆσ·m}}δJ−β(ˆσ·mˆhk/primen·ˆσ+n·ˆσˆhk/primeˆσ·m)δT/bracketrightBig\n,(103)\nwhile the Keldysh equation (74) is rewritten as:\n−2/parenleftbigg\n−i[ˆhk,Jˆσ·m] +¯h2\nm(k·∇R)ˆhk/parenrightbigg\n=coll. (104)17\nVI. FERROMAGNETIC SOLUTION WITHOUT SPIN-ORBIT COUPLING\nLet us consider Eq. (104) without extrinsic spin-orbit coupling:\n−i[ˆhk,Jˆσ·m] +¯h2\nm(k·∇R)ˆhk=πa1/integraldisplaydk/prime\n(2π)3/bracketleftbig\n(ˆhk/prime−ˆhk)δT+{ˆhk/prime−ˆhk,ˆσ·m}δJ/bracketrightbig\n. (105)\nBy introducing Ω = i¯h/τ0,ˆU=ˆσ·m, and:\nˆK=−¯h2\nm(k·∇R)ˆhk+πa1/integraldisplaydk/prime\n(2π)3/bracketleftbigˆhk/primeδT+{ˆhk/prime,ˆσ·m}��J/bracketrightbig\n, (106)\nin the weak exchange coupling limit ( J/lessmuchεF) we have:\nˆhk=i\nΩˆK+β\n2{ˆhk,ˆU}+J\nΩ[ˆU,ˆhk]. (107)\nThis equation is solved iteratively:\nˆhk=i\nΩ/parenleftbigg\n1 + 2J2\nΩ2+ 8J4\nΩ4+ 32J6\nΩ6+.../parenrightbigg\nˆK+i\nΩβ2\n2/parenleftbig\n1 +β2+β4+.../parenrightbigˆK\n+i\nΩβ\n2/parenleftbig\n1 +β2+β4+.../parenrightbig\n{ˆU,ˆK}+i\nΩJ\nΩ/parenleftbigg\n1 + 4J2\nΩ2+ 16J4\nΩ4+.../parenrightbigg\n[ˆU,ˆK]\n+i\nΩβ2\n2/parenleftbig\n1 +β2+β4+.../parenrightbigˆUˆKˆU−2i\nΩJ2\nΩ2/parenleftbigg\n1 + 4J2\nΩ2+ 16J4\nΩ4+.../parenrightbigg\nˆUˆKˆU,(108)\nor by using 1 + x+x2+x3...=1\n1−xforx/lessmuch1:\nˆhk=i\nΩ/parenleftbiggΩ2−2J2\nΩ2−4J2+β2\n2(1−β2)/parenrightbigg\nˆK+i\nΩβ\n21\n1−β2{ˆU,ˆK}\n+iJ\nΩ2−4J2[ˆU,ˆK] +i\nΩ/parenleftbiggβ2\n2(1−β2)−2J2\nΩ2−4J2/parenrightbigg\nˆUˆKˆU.(109)\nSinceJ2/Ω2/lessmuch1 andβ2/lessmuch1, this solution is well justified. By substituting Ω, ˆKand ˆUand removing the delta-\nfunctions, we have:\n¯h\nτ0ˆhk=−¯h2\nm(k·∇R)ˆhk+τ2\n0\nm2J2\n1 +4J2τ2\n0\n¯h2(k·∇R)/parenleftBig\nˆhk−ˆσ·mˆhkˆσ·m/parenrightBig\n−β2\n2(1−β2)¯h2\nm(k·∇R)/parenleftBig\nˆhk+ˆσ·mˆhkˆσ·m/parenrightBig\n−β\n2(1−β2)¯h2\nm(k·∇R){ˆhk,ˆσ·m}\n+¯h\nτ0/integraldisplaydˇk/prime\n4πˆhk/prime+τ0\n¯h2J2\n1 +4J2τ2\n0\n¯h2/parenleftbigg/integraldisplaydˇk/prime\n4πˆσ·mˆhk/primeˆσ·m−/integraldisplaydˇk/prime\n4πˆhk/prime/parenrightbigg\n−iJ\n1 +4J2τ2\n0\n¯h2/integraldisplaydˇk/prime\n4π[ˆσ·m,ˆhk/prime] +τ0¯h\nmiJ\n1 +4J2τ2\n0\n¯h2(k·∇R)[ˆσ·m,ˆhk],(110)\nwhere ˇk=k/|k|. In the diffusive limit vFτ/lessmuchL, whereLis the system size, we can partition the distribution function\nˆhkinto the isotropic charge µcand spinµand anisotropic ˆj·ˇkcomponents, ˆhk=µcˆσ0+µ·ˆσ+ˆj·ˇk. This form\nis nothing else but the generalized p-wave approximation for the distribution function. Upon integrating Eq. (110)\nmultiplied by ˇkoverdˇk/4πand neglecting higher order terms ∼β2, we obtain the following expression for ˆj:\n¯h\nτ0ˆj=−¯h2\nmk∇(µcˆσ0+µ·ˆσ+βµcˆσ·m+βµ·mˆσ0)\n−τ0¯h\nm2J\n1 +4J2τ2\n0\n¯h2k∇ˆσ·(m×µ)−τ2\n0\nm4J2\n1 +4J2τ2\n0\n¯h2k∇ˆσ·(m×(m×µ)).(111)18\nThe charge and spin currents (its jth component in the spin space) can be defined as:\n˜jC=1\n4/integraldisplaydˇk\n4πTr{ˆvk,ˆhk}=vF\n6Trˆj, (112)\nand\n˜JS\nj=1\n4/integraldisplaydˇk\n4πTr/bracketleftbig\nˆσj{ˆvk,ˆhk}/bracketrightbig\n=vF\n6Tr/bracketleftbig\nˆσjˆj/bracketrightbig\n, (113)\nwhere the velocity operator is defined as ˆvk=¯h\nmkˆσ0, andvF=¯h\nmkFis the Fermi velocity. Thus, neglecting higher\norder terms∼J3gives:\n˜jC=−D∇(µc+βµ·m), (114)\nand\n˜JS\nj\nD=−∇(µj+βµcmj)−τ0\nτL∇(m×µ)j−τ0\nτφ∇(m×(m×µ))j, (115)\nwhereD=τ0v2\nF/3 is the diffusion coefficient, 1 /τL= 2J/¯his the Larmor precession time, and 1 /τφ= 4J2τ0/¯h2is\nthe spin dephasing time.\nThe corresponding equations for the charge and spin densities are obtained by integrating Eq. (110) over ˇkand\nneglecting terms ∼J∇2and∼β∇2:\n−¯h2\nmk\n3∇·ˆj+τ0\n¯h4J2\n1 +4J2τ2\n0\n¯h2ˆσ·(m×(m×µ)) +2J\n1 +4J2τ2\n0\n¯h2ˆσ·(m×µ) = 0. (116)\nTaking Tr [...] and Tr [ ˆσ...] and neglecting terms ∼J3leads to:\n0 =D∇2(µc+βµ·m) =−∇· ˜jC(117)\nand\n0 =−∇· ˜JS+1\nτL(m×µ) +1\nτφ(m×(m×µ)) (118)\nfor the charge and spin components, respectively. Finally, by recovering time-dependence from Eq. (16) we obtain:\n∂Tµc=−∇· ˜jC(119)\nand\n∂Tµ=−∇· ˜JS+1\nτL(m×µ) +1\nτφ(m×(m×µ)). (120)\nThus, Eqs. (114), (115), (119) and (120) define a set of the drift-diffusion equations for ferromagnets in the absence\nof extrinsic spin-orbit coupling.\nVII. FERROMAGNETIC SOLUTION WITH SPIN-ORBIT COUPLING\nTo derive drift-diffusion equations including extrinsic spin-orbit coupling, we employ the same p-wave approximation\nforˆhk. Then, we have:\n−i[ˆhk,Jˆσ·m] = 2Jˆσ·(µ×m)−i[ˆj·ˇk,Jˆσ·m], (121)\n(k·∇R)ˆhk= (k·∇R)µcˆσ0+ (k·∇R)µ·ˆσ+ (k·∇R)ˆj·ˇk (122)19\nfor the left-hand side of the Keldysh equation (104), and:\n{∇Rˆhk/prime,(k/prime−k)׈σ}= 2 (∇R×(k/prime−k))·ˆσµc−2(k/prime−k)·(∇R×µ) ˆσ0\n+{∇Rˆj·ˇk/prime,(k/prime−k)׈σ},(123)\n{(k/prime−k)׈σ,{∇Rˆhk/prime,m·ˆσ}}= 4(k/prime−k)·(m×∇Rµc)ˆσ0+ 4ˆσ·(∇R×(k/prime−k))µ·m\n+{(k/prime−k)׈σ,{∇Rˆj·ˇk/prime,m·ˆσ}}(124)\nn·ˆσˆhk/primen·ˆσ−n2ˆhk= 2(n×(n×µ))·ˆσ+n·ˆσˆj·ˇk/primen·ˆσ−n2ˆj·ˇk (125)\nfor the collision integral (103). Upon integrating Eq. (104) over dˇk/4πand neglecting terms ∼ξ2\nSOβ, we obtain in\nthe limitJ/lessmuchεF:\n2Jˆσ·(µ×m) +1\n3¯h2k\nm∇R·ˆj=−8\n9¯h\nτ0k2\nk2\nFξ2\nSOµ·ˆσ\n+1\n6¯h\nτ0ξSO\nkF/parenleftbig\n∇R·(ˆj׈σ)−∇R·(ˆσ׈j)/parenrightbig\n+1\n6¯h\nτ0ξSO\nkFβ/bracketleftbig\n∇R·(ˆj׈σ) +∇R·(ˆσ׈j),m·ˆσ/bracketrightbig\n+2\n3¯h\nτ0ξSO\nkFβ∇R·(m׈j).(126)\nOne more equation is derived by averaging Eq. (104) over ˇkmultiplied by ˇkand neglecting terms ∼ξ2\nSOβ:\n−iJ/bracketleftbigˆj,ˆσ·m/bracketrightbig\n+¯h2k\nm∇R(µcˆσ0+µ·ˆσ) =−¯h\nτ0/parenleftBig\n1 +2\n3k2\nk2\nFξ2\nSO/parenrightBig\nˆj+1\n2¯h\nτ0β/braceleftbigˆj,ˆσ·m/bracerightbig\n−i\n3¯h\nτ0k\nkFξSO/parenleftbigˆj׈σ+ˆσ׈j/parenrightbig\n+i\n3¯h\nτ0k\nkFξSOβ/parenleftbig\nm·ˆσˆj׈σ+ˆσ׈jm·ˆσ/parenrightbig\n+1\n3¯h\nτ0k\nkFξSOβ/parenleftbigˆσ·ˆj+ˆj·ˆσ)m−1\n3¯h\nτ0k\nkFξSOβ/braceleftbigˆσ,ˆj·m/bracerightbig\n+¯h\nτ0k\nkFξSO\nkF/parenleftbig\n∇R׈σ(µc−βµ·m) +∇R×(µ−βµcm) ˆσ0/parenrightbig\n+1\n6m\nπ¯hvi\nτ0ξSOk/parenleftbigˆσ׈j−ˆj׈σ/parenrightbig\n+1\n3m\nπ¯hvi\nτ0ξSOβk/parenleftbig\nm·ˆσˆj׈σ−ˆσ׈jm·ˆσ/parenrightbig\n+1\n6m\nπ¯hvi\nτ0ξSOβk/parenleftbig\nm·ˆσˆσ׈j−ˆj׈σm·ˆσ/parenrightbig\n−2\n3m\nπ¯hvi\nτ0ξSOβkm׈j.(127)\nThe equations above define a set of the generalized drift-diffusion equations, which can now be solved approximately\nwhile keeping leading orders in ξSOandβ. Then, starting from a ferromagnetic solution given by Eq. (111) the\nanisotropic component of the density matrix is obtained by solving Eq. (127):\nˆj=−τ0vF∇ˆµ0+/parenleftbiggξSO\nkF+τ0vik2\nF\n3π¯hξSO/parenrightbigg\n∇×µˆσ0−τ0vik2\nF\n3π¯hξSOβ(∇×m)µcˆσ0\n−/parenleftbigg2\n3ξSOβτ0vF−τ0vik2\nF\n3π¯hξSOτ0\nτL/parenrightbigg\n∇×(m×µ)ˆσ0+/parenleftbiggξSO\nkF+τ0vik2\nF\n3π¯hξSO/parenrightbigg\n∇× ˆσµc\n−ξSO\nkFβ∇×(m×(ˆσ×µ))−τ0vik2\nF\n3π¯hξSOβ∇×((m׈σ)×µ)−τ0vik2\nF\n3π¯hξSOβ(∇× ˆσ)µ·m\n+/parenleftbiggξSO\nkF+τ0vik2\nF\n3π¯h/parenrightbiggτ0\nτL∇×(ˆσ×m)µc−2\n3τ0vFξSO∇×(ˆσ×(µ−βµcm))\n−2\n3τ0vFξSOτ0\nτL∇×((ˆσ×m)×µ+ˆσ×(m×µ)),(128)20\nwhere\nˆµ0= (µc+βµ·m)ˆσ0+ (µ+βµcm)·ˆσ+τ0\nτL(m×µ)·ˆσ+τ0\nτφ(m×(m×µ))·ˆσ. (129)\nHere, the first term of ˆjcomes from the ferromagnetic solution by moving the right-hand side of Eq. (127) into\nEq. (106). Plugging this solution in Eq. (126) leads to:\n2J\n¯h(µ×m)·ˆσ+8\n9ξ2\nSO\nτ0µ·ˆσ=\n=D∇·/bracketleftBig\n∇ˆµ0+αsj/bracketleftbigˆσ×∇(2µc−βµ·m)−∇× (2µ−βµcm)/bracketrightbig\n+αsk/bracketleftbigˆσ×∇(µc−βµ·m)−∇× (µ−βµcm)/bracketrightbig\n+αsw∇×(ˆσ×µ)\n−∇×/bracketleftbig\n(αsjτ0\nτL+αskτ0\nτL+αswβ)(ˆσ×m)µc+ (αsjτ0\nτL+αskτ0\nτL−αswβ)(m×µ)/bracketrightbig\n+∇×/bracketleftbig\n(αsjβ+αskβ+αswτ0\nτL)ˆσ×(m×µ)−(αsjβ+αskβ−αswτ0\nτL)(ˆσ×m)×µ/bracketrightbig/bracketrightBig\n,(130)\nwhereαsw=2ξSO\n3,αsj=ξSO\nlFkFandαsk=vimkF\n3π¯h2ξSOare the spin swapping, side-jump and skew-scattering coefficients,\nrespectively, and lF=τ0vFis the mean-free path. As seen, Eq. (130) can be regarded as a generalized continuity\nequation for the density matrix µcˆσ0+µ·ˆσ, and its right-hand side is nothing else but the divergence of the full\ncurrentjCˆσ0+JS·ˆσ, where the dot product is over spin components. Thus, the corresponding expressions for the\ncharge and spin currents (its jth spin component) can be readily written as:\njC/D=˜jC/D+αsj∇×(2µ−βµcm) +αsk∇×(µ−βµcm) + (αsjτ0\nτL+αskτ0\nτL−αswβ)∇×(m×µ) (131)\nand\nJS\nj/D=˜JS\nj/D+αsj∇×ej(2µc−βµ·m) +αsk∇×ej(µc−βµ·m)−αsw∇×(ej×µ)\n+∇×(αsjτ0\nτL+αskτ0\nτL+αswβ)(ej×m)µc−∇× (αsjβ+αskβ+αswτ0\nτL)(ej×(m×µ))\n+∇×(αsjβ+αskβ−αswτ0\nτL)((ej×m)×µ),(132)\nor\nJS\nij/D=˜JS\nij/D−αsj/epsilon1ijk∇k(2µc−βµnmn)−αsk/epsilon1ijk∇k(µc−βµnmn)−αsw(δij∇kµk−∇jµi)\n+ (αsjτ0\nτL+αskτ0\nτL+αswβ)(δij∇kmk−mi∇j)µc\n−(αsjβ+αskβ+αswτ0\nτL)/epsilon1ikn∇k(mnµj−µnmj)\n+ (αsjβ+αskβ−αswτ0\nτL)(/epsilon1ikn∇kmnµj+/epsilon1ijk∇kmnµn),(133)\nwhere/epsilon1ijkis the Levi-Civita symbol, and summation over repeated indexes is implied. Here, the first and second\nsubscripts correspond to the spatial and spin components, respectively. Finally, by recovering time dependence in\nEq. (130) we obtain the remaining equations for the charge and spin densities:\n∂Tµc=D∇2(µc+βµ·m) =−∇·jC(134)\nand\n∂Tµ=−∇·JS+1\nτL(m×µ) +1\nτφ(m×(m×µ))−1\nτsfµ, (135)\nwhere 1/τsf= 8ξ2\nSO/9τ0is the spin-flip relaxation time. The set of Eqs. (131), (132), (134) and (135) is the central\nresult of this work.21\nVIII. SPIN SWAPPING SYMMETRY\nIn this section we verify the symmetry of the spin swapping term and compare our results with the previous ones\nderived for normal metals.\nIn their original work Dyakonov and Lifshits give the following definition of the spin current qijdue to scattering\noff the spin-orbit coupling potential:3\nqij=q(0)\nij−αsh/epsilon1ijkq(0)\nk+αsw(q(0)\nji−δijq(0)\nkk), (136)\nwhereq(0)\nkandq(0)\nijstand for the primary charge and spin currents in the absence of spin-orbit coupling, respectively,\nandαshrepresents the overall spin Hall effect. Thus, it is argued that the spin swapping effect always appears in the\nform given above.\nLet us consider our solution in the case of normal metals ( β= 0 andJ= 0):\nJS\nj=−D∇µj+Dαsh∇×ejµc−Dαsw∇×(ej×µ) (137)\nor\nJS\nij=−D∇iµj−Dαsh/epsilon1ijk∇kµc+Dαsw(∇jµi−δij∇kµk). (138)\nTaking into account that q(0)\nij≈−D∇iµjandq(0)\nk≈−D∇kµc, it is seen that Eq. (132) displays the correct symmetry\nup to a sign coming from the definition of the spin-orbit coupling potential, Eq. (4). This form is also in agreement\nwith some previously published results.5,6\nFinally, it is worth comparing our equations with those that fail to include spin swapping in the form given by\nEq. (136). For example, in Ref. [7] the spin swapping term appeared with the following symmetry:\ne2JS\nj/σN=−∇µj/2 +αsjej×∇µc−αswej×(∇×µ)/2\n=−∇µj/2 +αsjej×∇µc−αsw(∇µj−∇jµ)/2,(139)\nwhereσNis the bulk conductivity. It is clear that the symmetry of spin swapping is wrong: e.g. qxxshould contain a\nterm∼αsw(−q(0)\nyy−q(0)\nzz), which is absent in the expression above. There is the same symmetry problem in Eq. (2) of\nRef. [8], where the spin swapping term reads as −αswˆσ×∇×µ(or−αswej×∇×µfor the spin current component).\n∗saishi@inbox.ru\n1J. Rammer, ”Quantum Transport Theory” , Westview Press (2004).\n2J. Rammer and H. Smith, Rev. Mod. Phys. 58, 323 (1986).\n3M. B. Lifshits and M. I. Dyakonov, Phys. Rev. Lett. 103, 186601 (2009).\n4S. K. Lyo and T. Holstein, Phys. Rev. Lett. 29, 423 (1972).\n5R. V. Shchelushkin and Arne Brataas, Phys. Rev. B 73, 169907 (2005).\n6K. Shen, R. Raimondi, and G. Vignale, Phys. Rev. B 90, 245302 (2014). ibis, Phys. Rev. B 92, 035301 (2015).\n7H. Saidaoui and A. Manchon, Phys. Rev. Lett. 117, 036601 (2016).\n8R. V. Shchelushkin and Arne Brataas. Phys. Rev. B 72, 073110 (2005)."    },    {        "title": "1704.08837v1.Quantum_Spin_Lenses_in_Atomic_Arrays.pdf",        "content": "`Quantum Spin Lenses' in Atomic Arrays\nA. W. Glaetzle,1, 2, 3, 4K. Ender,1, 2D. S. Wild,5S. Choi,5H. Pichler,6, 5M. D. Lukin,5and P. Zoller1, 2\n1Institute for Theoretical Physics, University of Innsbruck, A-6020 Innsbruck, Austria\n2Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, A-6020 Innsbruck, Austria\n3Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543\n4Clarendon Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU, United Kingdom\n5Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA\n6ITAMP, Harvard-Smithsonian Center for Astrophysics, Cambridge, Massachusetts 02138, USA\n(Dated: May 1, 2017)\nWe propose and discuss `quantum spin lenses', where quantum states of delocalized spin excita-\ntions in an atomic medium are `focused' in space in a coherent quantum process down to (essentially)\nsingle atoms. These can be employed to create controlled interactions in a quantum light-matter\ninterface, where photonic qubits stored in an atomic ensemble are mapped to a quantum register\nrepresented by single atoms. We propose Hamiltonians for quantum spin lenses as inhomogeneous\nspin models on lattices, which can be realized with Rydberg atoms in 1D, 2D and 3D, and with\nstrings of trapped ions. We discuss both linear and non-linear quantum spin lenses: in a non-linear\nlens, repulsive spin-spin interactions lead to focusing dynamics conditional to the number of spin\nexcitations. This allows the mapping of quantum superpositions of delocalized spin excitations to\nsuperpositions of spatial spin patterns, which can be addressed by light \felds and manipulated. Fi-\nnally, we propose multifocal quantum spin lenses as a way to generate and distribute entanglement\nbetween distant atoms in an atomic lattice array.\nI. INTRODUCTION\nIn quantum information processing [1] with atoms,\nqubits are typically represented by internal atomic states,\ne.g. as long-lived spin excitations within the atomic\nground state manifold [2]. Ideally, qubits are stored in\nsingle atoms, and for these qubits to be identi\fable and\naddressable, we typically require localization of the atoms\nin well-de\fned spatial regions. Spatial control, and lo-\ncalization of single atoms is a pre-requisite to implement\nsingle and two-qubit operations, allowing addressing of\nindividual qubits with laser light, and providing entan-\ngling operations between adjacent qubits by \fnite range\ninteractions. Recent atomic physics experiments have\ndemonstrated in a remarkable way the basic ingredients\nof single atom manipulation and addressing for trapped\natoms and ions, and controlled interaction and entangle-\nment between atomic spin qubits with Rydberg atoms\n[3, 4], trapped ions [5{8], cavity QED setups [9], and\nquantum interfaces [10{12].\nIn contrast to localized qubits stored in single trapped\natoms, atomic ensembles provide us with qubits in the\nform of delocalized spin excitations [13, 14]. Delocalized\nspin qubits arise naturally in light-atomic ensemble in-\nterfaces in both free space and cavity assisted setups.\nHere incident photons representing a `\rying qubit' are\nabsorbed in an atomic ensemble with enhanced interac-\ntions bene\fting from a large atom number N, as in an\noptically thick medium, and converted into a spin exci-\ntation, which may be delocalized over the whole atomic\ncloud [15{19]. In order to create controlled interactions\nbetween such delocalized qubits it is desirable to con-\nvert delocalized spin qubits into localized qubits in the\natomic array representing quantum memory. Thus ide-\nally we want operations | a lens for spin excitations |on the atomic array, which allow in a coherent process\n`focusing' of qubits to a well-de\fned and localized region,\nand ultimately to a single atom.\nIn this paper we propose and discuss linear and non-\nlinear `quantum spin lenses' and their physical realiza-\ntion in quantum optical setups. We will \frst identify\nHamiltonians to realize linear spin lenses , which map in\na coherent process a delocalized to localized spin exci-\ntation, and vice versa . This has immediate application\nas a quantum atom-light interface, where incident pho-\ntonic qubits are sequentially stored in an atomic array,\nand focused to a quantum register of spatially localized\nspin qubits represented by single atoms [see Fig. 1(a)].\nMoreover, we can generalize the concept of the `quantum\nspin lens' to a multifocal lens . In particular, this allows a\nsingle delocalized spin excitation to be mapped to a spa-\ntial EPR-like superposition state, thus providing a way\nto distribute or generate entanglement between (distant)\natoms [c.f. Fig. 1(c)]. Finally, we will discuss the design\nofnon-linear spin lenses , adding \fnite range (repulsive)\nspin-spin interactions to the spin-lens Hamiltonian. Thus\nfocusing dynamics will be conditional on the number of\ninitial spin excitations, and an initial quantum superposi-\ntion state of delocalized spins will be mapped to a super-\nposition of spatial spin patterns [c.f. Fig. 1(d)]. Remark-\nably, this provides a tool to manipulating the individ-\nual terms (corresponding to a speci\fc excitation number)\nin the superposition state by spatially addressing in the\natomic medium. As noted above, the relevant spin mod-\nels are naturally implemented in existing atomic and solid\nstate quantum optical setups, and we will illustrate this\nbelow with the examples of neutral atoms with Rydberg-\nmediated spin-spin interactions in 1D, 2D and 3D atomic\nlattices [20{23] using laser-dressing techniques [24{27],\nas well as with strings of trapped ions [7, 8].arXiv:1704.08837v1  [quant-ph]  28 Apr 20172\nFigure 1. (a { d) Various scenarios of a quantum light-atom interface illustrating a quantum spin lens . Incident photonic\nqubits are initially stored as delocalized spin excitations in an atomic array, and focused to single atoms (see Sec. II). (a) Basic\nprocess of write and focusing operations (see Secs. IIA,B). Qubits stored in single atoms allow quantum gate operations to be\nimplemented between adjacent atoms, e.g. via Rydberg gates. (b) 1D setup with atomic ensemble stored inside a cavity (see\nSec. IID). (c) Generation of EPR type states by constructing lenses with multiple focal points (see Sec. II C). (d) Focusing\ndynamics with a non-linear spin lens with repulsive spin-spin interactions with range rB, illustrated for two spin excitations\n(see Sec. III). (e) Density plots illustrating focusing dynamics of spin excitations in 2D arrays as a function of time taccording\nto spin-lens Hamiltonians (1) or (10). (i) Focusing of an initially delocalized `qubit 2' in presence of a hole (representing, e.g. a\npreviously stored `qubit 1') (see Sec. II). (ii) Two-focus lens illustrating generation of EPR pairs (see Sec. IIC). Parameters:\n50\u000250 lattice, initial width of the Gaussian wave packet \u001b0= 10alattice spacings. (iii) Focusing of two spin excitations\nwith a non-linear quantum lens with repulsive spin-spin interactions of range rB. The two excitations are focused to a ring,\nreminiscent of a quantum crystal. Parameters: 31 \u000231 lattice. (see Sec. III).\nII. LINEAR QUANTUM SPIN LENSES:\nFOCUSING DYNAMICS OF SINGLE SPIN\nEXCITATIONS\nWe are interested here in a scenario illustrated in\nFig. 1(a), where an incident wave packet E, represent-\ning a qubit \u000bj0i+\fj1ias a superposition of vacuum and\na one-photon wave packet, is stored as a delocalized spin\nexcitation in a medium of Ntwo-level atoms. These two-\nlevel systems can be physically represented by long-lived\natomic hyper\fne ground states two-level atoms jgi;jei,\nwith all atoms initially prepared in the ground state, and\nwe assume atoms trapped in an array. Storage of a pho-\ntonic qubit in the atomic medium is achieved, for exam-\nple, in a Raman process [28, 29] , where the incident pho-\nton is absorbed and atoms, initially prepared in jgitrans-\nferred tojei. Writing to atomic quantum memory thus\ncorresponds to a mapping of the photonic qubit to the\natomic state \u000bjGi+\f^S+jGi. Here ^S+=PN\nn=1 n^\u001b(n)\n+,\na sum of Pauli raising operators for atoms n, creates a\ndelocalized excitation distributed over the atoms accord-\ning to an amplitude  n, acting on the `vacuum state'\njGi\u0011jg1;:::;gNiwith all atoms in the ground state.\nThis mapping of photons to spin excitations should be\nunderstood in the spirit of the Holstein-Primako\u000b ap-\nproximation, where excitations in the atomic medium are\nessentially bosonic for small excitation fractions.\nA quantum spin lens aims to achieve a mapping of the\ndelocalized atomic spin excitation to (ideally) a singleatom, ^S+jGi! ^\u001b(nf)\n+jGiin a coherent quantum pro-\ncess, and preserving the superposition character of the\nqubit. Below we will \frst discuss spin-lens Hamiltonians\nthat focus initially delocalized single spin excitations dur-\ning the associated unitary time evolution. We will call\nthis focusing dynamics of single excitations linear spin\nlenses, with nonlinear spin lenses as focusing of multiple\ninteracting spin excitations to be discussed in the follow-\ning section.\nThe focusing of single spin excitations discussed be-\nlow can be generalized immediately to kphotonic qubits,\nprovided we store and focus them sequentially , i.e. inci-\ndent photonic qubits are absorbed and focused in the\natomic medium one by one in spatially separated atoms\nn1;:::;n\u0017representing a quantum register [30]. Due to\nthe spatial localization, these atomic qubits can now be\nindividually addressed, and we can operate on them with\nsingle and two-qubit gate operations, implemented, for\nexample, as Rydberg gates [see Fig. 1(a) and panel ( i) of\nFig. 1(e)].\nA. Spin lens Hamiltonian with nearest-neighbor\n`\rip-\rop' interactions\nFocusing of a delocalized excitation in a spin chain is\nachieved with the Hamiltonian\n^H=\u0000JX\nnh\n^\u001b(n)\n+^\u001b(n+1)\n\u0000 + H.c.i\n+X\nnVn^\u001b(n)\nz;(1)3\n0 500 1000 1500−300−100100300\n0 1000 2000 3000\n0.0010.0020.0070.0180.0500.135 (b) (a)\nJtx xp1 2 3 4 1 2 3 4\nJt−300−100100300pxn/a(c)\n0 100 200−300−100100300\nJt(d)xn/a\nxn/a\n−10 −5 0 5 1000.6\nxn/a\nFigure 2. Time evolution of an initially delocalized spin excitation on a lattice according to the spin-lens Hamiltonian (1) for\nthe (a) thick and (b) thin lens for a 1D string of L= 800 spins as a function of time t. The red line, \u001b(t), indicates the\ncontinuum result. Insets 1-4 above illustrate the dynamics of the Wigner function at four speci\fc times. (c) Zoom to illustrate\nthe \fnal optimized spatial excitation probability per atom illustrating that focusing on the scale of (essentially) single atoms\ncan be obtained. The green bars correspond to a focusing dynamics of the thick lens with initial width \u001b0= 100aand \fnal\nwidth\u001bf= 2:7a. The yellow bars correspond to a second focusing stage with a thin lens starting with \u001b0= 2:7ato \fnal width\n\u001bf= 1:2a. (d) Bloch oscillations in focusing dynamics at the edge \u001bBOof the lens (see text).\nwhere ^\u001b(n)\n\u0016are Pauli spin 1 =2 operators at lattice site n.\nThe \frst term describes hopping of the spin excitation\n(kinetic energy), which for the moment we take as near-\nest neighbor hopping, while the second term is a spa-\ntially dependent energy shift Vn=v0(n\u0000nf)2. While in\nEq. (1) we write a 1D model, the present discussion can\nin a straightforward way be generalized to higher dimen-\nsions.\nThe Hamiltonian (1) is motivated by analogy to an op-\ntical lens with Vnimprinting a phase on the n-th spin cen-\ntered around lattice site nf, reminiscent of the refractive\nmaterial of a lens [31, 32]. The analogy to an optical lens\nis best illustrated by visualizing the focusing dynamics of\nsingle spin excitations with a Wigner phase space distri-\nbution as a function of time [see Fig. 1(e) and Fig. 2(a,b),\nupper panel]. We write the wave function of the single\nspin excitation as j (t)i=P\nn n(t)^\u001b(n)\n+jGiwith ampli-\ntude n(t) initially delocalized as a wave packet of spatial\nwidth\u001b0over the lattice, and we de\fne a Wigner func-\ntion on the lattice as [33, 34] ( ~= 1)\nWlat(xn;k) =a\n\u0019Z\u0019=2a\n\u0000\u0019=2adqhk\u0000qj ih jk+qie\u00002iqxn:\nHerexn=an(n2Z) are discrete lattice po-\nsitions with athe lattice spacing, and momentum\nka2(\u0000\u0019;\u0019) is 2\u0019-periodic, and we denote by\njki= (a=2\u0019)1=2P\nneikxn^\u001b(n)\n+jGispin waves with mo-\nmentumkon an in\fnite lattice. A momentum space rep-\nresentation of the time-dependent Schr odinger equation\nwith Hamiltonian (1) shows that the dynamics is the one\nof a quantum pendulum. The \frst term in (1) gives rise to\na Bloch band dispersion relation \u000f(k) = 2J[1\u0000cos(ka)],\nand the quadratic potential term maps to a Laplacian in\nk, i.e. a kinetic energy term.\nFocusing dynamics is best illustrated in the continuum\nlimit, i.e. we assume that the spin dynamics is smooth\non the scale of the lattice spacing, and the wave func-\ntion in momentum space remains localized to a region atthe bottom of the Bloch band. Thus the dispersion re-\nlation is well approximated by \u000f(k)\u0019J(ka)2+O(ka)4\nfor small momenta ka\u001c1, and the Wigner func-\ntionWlat(xn;k) maps to the standard Wigner function\nW(x;p) for the continuous variables xn!x2Rand\nk!p2R. The Hamiltonian (1) becomes an e\u000bec-\ntive harmonic oscillator (HO), H=p2=(2m) +m!2x2=2\nwith momentum pand position x. Here we have de-\n\fned a frequency != 2pv0J, massm= 1=(2Ja2), and\nwe denote by `= (~=m!)1=2the HO length. Under\nthis Hamiltonian an initial Wigner function, W(x;p; 0),\nsimply performs a (classical) rigid rotation in phase\nspace,W(x;p;t) =W(\u0016x(x;p;t );\u0016p(x;p;t ); 0), where po-\nsition, \u0016x(x;p;t ) =m!x cos!t\u0000psin!t, and momentum,\n\u0016p(x;p;t ) =pcos!t+m!x sin!t, describe elliptical tra-\njectories in phase space as a function of time. Thus, an\ninitial wave packet with width \u001b0in position space is\ntransformed after a quarter of period, tf=\u0019=(2!), to a\nspatially localized state with width \u001bf=`2=\u001b0\u001c`, as\nfamiliar from squeezed states [2] [35].\nIn this continuum approximation the single particle\nSchr odinger equation from (1) is formally equivalent to\nthe paraxial Helmholtz equation [32]. The role of time in\nthe Schr odinger equation is replaced by the axial dimen-\nsion in the paraxial Helmholtz equation, and the poten-\ntial translates to a spatially dependent refractive index.\nThis allows to interpret most of the focussing dynamics in\nthe language of classical optics. So far we considered fo-\ncusing of a delocalized excitation in a potential Vn, which\nis `on' during the whole dynamics. In analogy to optics\nthis corresponds to light propagating in a graded index\nmultimode \fber. In the following we will refer to this\ndynamics as a `thick lens'. This is to distinguish from\na second scenario, discussed below, where focussing is\nachieved by a `thin lens'.\nIn Fig. 2(a) we illustrate focusing dynamics of a `thick\nlens' for spin excitations with the lattice model (1) in a\nparameter regime where the continuum approximation is4\nwell justi\fed (see below for details). For an initial Gaus-\nsian wave packet with spatial width \u001b0, corresponding\nto a (in the continuum limit) cigar shaped Wigner func-\ntionW(x;p;0) = (1=\u0019) exp[\u0000(x=\u001b0)2\u0000(\u001b0p)2], the spa-\ntial width \u001b(t)2=\u001b2\n0[cos2!t+ (`=\u001b0)4sin2!t] starts\noscillating as a function of time [see red line in panel (a)]\nand has periodic minima at every quarter of a period\n!tf=\u0019=2 where\u001bf\u0011\u001b(tf) =`2=\u001b0. The \fnal\nwidth in real space (after a quarter of a period) corre-\nsponds to the Fourier transform of the initial wavefunc-\ntion, i.e. (x;tfoc) =Fx0f (x0;0)g(x=`2), as for an op-\ntical lens. The focusing in real space is illustrated in\nFig. 2, where panels 1-4 show the corresponding phase\nspace dynamics of the Wigner function.\nInstead of the `always-on' Hamiltonian (1) of the `thick\nlens', focusing can also be obtained in a pulsed scheme,\nwhere the quadratic potential term is switched on for\na short time only. This imprints a position dependent\nmomentum kick \u0001 ka=\u00002\u001e0(n\u0000nf) (with\u001e0>0)\nonto the initial wavefunction via the quadratic phase\nshift ^U\u001e0= exp[\u0000i\u001e0P\nn(n\u0000nf)2^\u001b(n)\nz], followed by a\nfree evolution of the spin system via (1) with v0= 0,\nas illustrated in Fig. 2(b). This is in analogy with a `thin\nlens' in classical optics, where a thin refractive material\nimprints a position dependent phase onto the incoming\nplane wave. The Wigner function in phase space \frst\nacquires a momentum kick p!p\u00002\u001e0x=a2, followed\nby free evolution corresponding to a shear motion of the\nWigner function, i.e. W(x;p;t ) =W(x\u0000pt=m;p; 0),\nas illustrated in panels 1-4 of Fig. 2(b). In contrast\nto the `thick lens' there is a single focal time Jtf=\n2(\u001b0=a)4\u001e0=[4\u001e2\n0(\u001b0=a)4+ 1] (\u00191=(2\u001e0) for\u001b0\u001da)\nwhere a Gaussian wavefunction has its minimum width\n\u001bf=\u001b0=p\n4\u001e2\n0(\u001b0=a)4+ 1.\nFigure 2(c) shows the spatial excitation probability for\nlattice sites around the focus. The green bars correspond\nto the dynamics illustrated in panel (a). An excitation\ninitially delocalized over \u001b0=a= 100 lattice sites gets\nlocalized on \u001bf=a= 2:7 lattice sites using a `thick' lens\nincluding corrections up to sixth order (described in IIB).\nOne can improve the focusing by using multiple pulses or\neven combining the two di\u000berent schemes. For example,\nwe can further focus the spin excitation from \u001b0=a= 2:7\nlattice sites to \u001bf=a= 1:2 lattice sites, shown as the\nyellow bars in panel (c), by adjusting the lens strength\nto the new initial condition.\nB. Lattice Corrections { Dephasing and Bloch\nOscillations\nCorrections to the continuum limit become important\nwhen the delocalized excitation is focused to a spatial re-\ngion on the order of the lattice spacing, and the Wigner\nfunction extends close to the border of the \frst Brillouin\nzone. This happens for `su\u000eciently strong potential' in\nEq. (1), which leads to aberration and Bloch oscillations\ndue to deviation from a quadratic dispersion relation.\nσ0/a)\n(v0\n/J)1 2 3−8−6−4\n−2\nσf/a)\n1234\n5\n(\nσ\n0/a)2 3 4−6−5−4−3−2\n5123\nσf/a)\n(φ0\n)\nσf/a\nσ\n0/a)σf/aσ0/a)\n(a) (b)1 2 32468\n2\n4\n6\nφcorr246\n1 2 3\n2\n4\n6\n(ii)(i)\n(ii)(i)10\nlog10\nlog\n10\nloglog(v\n(10\nloglog(v\n(10\nloglog(v\n(10\nloglog(v\n(10\nloglog(v\n(10\nloglog(vFigure 3. (a) Final width \u001bfas a function of initial width \u001b0\nand (i) potential strength v0or (ii) phase \u001e0for the (i) `thick'\nand (ii) `thin' lens setup. Dephasing due to the non-quadratic\ndispersion relation starts to dominate at v0=vopt(\u001b0) and\n\u001e0=\u001eopt(\u001b0) (black dashed line) while Bloch oscillations\nstart atv0=vBO(\u001b0) and\u001e0=\u001eBO(\u001b0) (red dashed line)\nfor `thick' and `thin' lenses, respectively. (b) Minimum \fnal\nwidth\u001bffor the optimized lens strength as a function of ini-\ntial width for (i) `thick' and (ii) `thin' lens setup. Dark blue\ndots, light blue squares and green triangles correspond to the\nnumerically obtained \fnal width for a quadratic, quartic, and\nsixth order potential. The black lines are a guide to the eye,\nshowing the respective scalings of Eq. (6).\nIn the following we discuss the limitations this imposes\non the achievable \fnal width and show how the e\u000bects\nof the non-quadratic dispersion relation can be compen-\nsated using non-parabolic lens potentials.\nThe main results are summarized in Fig. 3(a) where\nthe numerically obtained \fnal width \u001bfis plotted as a\nfunction of the initial width \u001b0and lens strength v0or\u001e0\nfor the (i) `thick' and (ii) `thin' lens on a lattice, respec-\ntively. Simulations have been performed on a 1D chain\nwithL= 106(L= 105) spins according to Eq. (1) for\nthe thin (thick) lens setup. In contrast to the continuum\npicture, where a stronger lens leads to a tighter spatial\nfocus and a faster focusing time, the numerical results\nshow that there exist optimal `lens potentials' (see be-\nlow) scaling as\nvopt\u0018\u0012a\n\u001b0\u00138=3\nand\u001eopt\u0018\u0012a\n\u001b0\u00134=3\n; (2)\nfor the `thick' and `thin' lens setups, illustrated as black\ndashed lines in Figs. 3(a). At this optimal lens strength\nthe \fnal achievable width scales as\n\u001bf=a\u0014\u0010\u001b0\na\u00111=3\n: (3)\nfor both the `thick' and `thin' lens with \u0014obtained nu-\nmerically. Focusing works well below this optimal lens\nstrength, in excellent agreement with Figs. 2(a,b). The\nscaling of Eq. (3) (black line) is in perfect agreement with5\nthe numerically evaluated \fnal width (blue dots) shown\nin Fig. 3(b) for (i) the `thick' and (ii) the `thin' lens with\n\u0014= 0:68 and\u0014= 0:80, respectively.\nIn the following we discuss the two main e\u000bects of the\nlattice: abberation, giving rise to the optimal lens poten-\ntials of Eq. (2), and Bloch oscillations at the edge of the\nlens shown in Fig. 2(d). We show that the corresponding\naberration can be addressed using potentials and pulse\nshapes that include quartic and higher order terms.\nAbberation: Deviations from the continuum model can\nbe understood as arising from the non-linear group ve-\nlocityvg(k) = 2Jasin(ka) on the lattice, which implies\nthat wave packets with large momenta propagate more\nslowly in a lattice than in the continuum limit, where\nv(c)\ng(k) = 2Ja2k. The di\u000berence between the two veloci-\nties is only negligible provided the path di\u000berence during\nthe focusing time is small compared to the \fnal size of\nthe wave packet, i.e. [ v(c)\ng(k)\u0000vg(k)]tf\u001c\u001bf. Expand-\ning the sine up to third order and evaluating the equa-\ntion at the maximum momentum k\u00181=\u001bfwe obtain\nEq. (2) (see appendix A). The di\u000berence in the group ve-\nlocities further explains the ` s'-shaped distortion of the\nWigner function observed panels 1-4 of Fig. 2(a,b) as the\nnon-linear group velocity induces a non-rigid rotation in\nphase space.\nBloch oscillations: At an even larger potential\nstrength, the wings of the wave packet with an exten-\nsion larger than\n\u001bBO= 2ar\nJ\nv0(4)\nwill undergo Bloch oscillations (see appendix A), as illus-\ntrated in Fig. 2(d). This limits the lens strength to val-\nues well-below vBO= 4J(a=\u001b0)2, indicated as red dashed\nline in Figs. 3(a). At this critical lens potential the (lo-\ncal) potential gradient V0\nn= 2v0(n\u0000nf) gives rise to\nBloch oscillations with an amplitude \u0001 n= 2J=V0\nnand\nfrequency!BO=V0\nn=2 [36]. If this frequency becomes\nof the order of the focusing time focusing becomes inef-\nfective. A similar argument for the thin lens yields the\nmaximum pulse strength \u001eBO\u0018a=\u001b0. In contrast to\nBloch oscillations in the thick lens, for the thin lens fo-\ncusing is limited by phase wraps as the momentum kicks\nimparted by the pulse extend beyond the \frst Brillouin\nzone.\nCorrection of aberration: Quartic deviations from the\ndispersion relation limit the \fnal width of the spin wave\nto\u001bf\u0018(\u001b0=a)1=3. Similar to aspherical lenses, the e\u000bect\nof dephasing due to the non-quadratic dispersion rela-\ntion, can be compensated using more general potentials\n(`thick' lens) and imprinted phase pro\fles (`thin' lens),\nof the form\nVn=Q=2X\nq=1v2q(n\u0000nf)2q; (5)\nincluding additionally quartic ( Q= 4), sixth ( Q= 6) or\neven higher order terms. Such higher order terms will\n10\n17\n30\n83\n100\n500\n1000(b) (a)Figure 4. (a) Dispersion relation \u000f\u000b(k) for long-range `\rip-\n\rop' interactions (see text) for di\u000berent exponents \u000band\nnearest-neighbor (NN) interactions. (b) Rescaled \fnal width\n\u001bf=\u001b1=3\n0for di\u000berent exponents \u000band for di\u000berent initial con-\nditions\u001b0. The data points on the very right correspond to\nnearest-neighbor interactions.\naccelerate the wings of the wave packet stronger com-\npensating for their smaller group velocities on the lat-\ntice. Using a similar argument as in appendix A, one can\nshow that for an appropriate choice of v2qthis leads to\nan improved scaling\n\u001bf\u0018\u0010\u001b0\na\u00111=(Q+1)\n: (6)\nFig. 3(b) plots the \fnal width as a function of the ini-\ntial width for ( i) the `thin' and ( ii) the `thick' lens setup\nusing a lens strength up to Q= 2;4 and 6. The numeri-\ncally obtained \fnal width agrees well with the scaling of\nEq. (6).\nFor the `thin' lens an optimized form of Eq. (5) can\nbe derived analytically using a semi-classical model with\ncontinuous spatial variable an!xand a Bloch band\ndispersion\u000f(k) giving rise to a non-linear group velocity\nvg(k). Given the imprinted phase pro\fle \u001e(x), the initial\nwave packet receives a position dependent momentum\nkick\u001e0(x)=a. In order to focus all parts of the wave\npacket to the focal point xf= 0 at the same time tf, we\nrequirex=vg(k(\u001e0))tfwhich yields\n\u001e(x) =\u0000(x=a) arcsin[\u001e0(x=a)]\u0000q\n\u001e2\n0\u0000(x=a)2;(7)\nwithtf= 1=(2J\u001e0). Panel (ii) of Fig. 3(b) shows the \fnal\nwidth obtained using (7) with x=naon a lattice (yellow\ndiamonds), which shows a clear improvement over the\nparabolic phase pro\fle. Note that \u001e(x) is only real-valued\nup tojx=aj=\u001e0due to the maximum group velocity\nvg(\u0019=2) = 2Jon a lattice, since only parts of the wave\npackets with distance x < 2Jtfcan be focussed within\nthe focusing time.\nC. Multifocal Lenses and Generation of EPR states\nInstead of the single focus lens, as in (1), we can employ\ndouble well, or multi-well potentials. The corresponding\npotentials can be generated as spin dependent optical\npotentials, and an array of spin lenses can be realized6\nwith large spacing optical lattices. A multifocal lens op-\nerating on a single initial delocalized spin excitation will\ngenerate a superposition state of excitations at the focal\npoints. For two foci, for example, we can generate an\nEPR type state\n^S+jGi\u0000!\u0010\n^\u001b(n1)\n++ ^\u001b(n2)\n+\u0011\njGi: (8)\nThus we generate a superposition (EPR state) between\nspins at lattice site n1andn2, as schematically illustrated\nin panel (ii) of Fig. 1(c).\nThe time evolution of the corresponding multi-\nfocal lens is visualized in panel ( ii) of Fig. 1(e).\nIn the upper half plane ( yn>0) we used the\n2D potential V(xn;ym) =v0[(xn\u0000xf)2+ (ym\u0000yf)2]\nwith focal points ( xf;yf) = (0;p\n2\u000210)awhile in the\nlower half plane ( yn<0) we used V(xn;ym) with\n(xf;yf) = (0;\u0000p\n2\u000210)a. Note that in Fig. 1(e) we ro-\ntated the potential by 45 degrees. The potential strength\nis optimized to v0= 4:5\u000210\u00003Jfor an initially sym-\nmetric Gaussian wave function with radial spatial width\n\u001b0= 10aon a 2D lattice with Lx=Ly= 50 lattice sites.\nD. Long-range `\rip-\rop' interactions\nImplementations of Hamiltonian (1) with Rydberg\natom in optical lattices or strings of trapped ions mo-\ntivate a model\n^H=\u0000X\nn;mJnh\n^\u001b(m)\n+^\u001b(m+n)\n\u0000 + H.c.i\n+X\nnVn^\u001b(n)\nz;(9)\nwith long-range `\rip-\rop' interactions Jn=J0=n\u000b. In\nparticular, dipolar and van der Waals interactions be-\ntween Rydberg (dressed) atoms allow to realize \u000b= 3 and\n\u000b= 6 [3], respectively, while 0 < \u000b < 3 can be realized\nwith strings of ions [5{8]. The \frst term of (9) gives rise\nto a dispersion relation \u000f\u000b(k) = 2P\nn[1\u0000cos(nka)]=n\u000b,\nas shown in Fig. 4(a). While for \u000b= 6 the disper-\nsion relation \u000f00\n6(0) =\u00194=90\u00191:08 closely resembles the\none of the nearest-neighbor `\rip-\rop' interactions of (1),\nfor\u000b < 3 the dispersion relation exhibits a kink, e.g.\n\u000f2(k) = (\u0019=2)jkaj\u0000(ka)2=4, resulting in a linear group\nvelocity at small momenta with a discontinuity at k= 0.\nThis leads to strong aberration and ine\u000ecient focusing.\nWe note that this can be corrected using an adiabatic\nlens schemes [35].\nFig. 4(b) shows \u0014\u0011\u001bf=(a2\u001b0)1=3[see Eq. (3)] for dif-\nferent realizations of \u000bfor the `thick' lens setups. While\nfor large values of \u000bthe scaling of Eq. (3) agrees with\nthe numerically obtained \fnal width, for smaller values\nthe linear dispersion relation leads to strong deviations.\nThe smallest \fnal width (smallest \u0014) is obtained for large\nvalues of\u000band ultimately with nearest-neigbor interac-\ntions, however, \u000b= 6 almost perfectly resembles nearest-\nneighbor interactions.\n−20020\nxn\n0.2\n0.6\n0.10.3\n0 5 10 15 20 25\nJt\n0.10.3−20020\n−20020\n(i)(ii)\n(iii)\n(a)\nxn\nxn\n(b)−10 0 100.5\nxnexcitation density\n−20 20\n0.2\n0.10.3\n0.6\n0.40Figure 5. Time evolution of the excitation probability p(\u0017)\nn(t)\nfor (i)\u0017= 1, (ii)\u0017= 2 and (iii) \u0017= 3 initially delocalized spin\nexcitation according to the non-linear spin-lens Hamiltonian\n(10) with interaction range rB= 4:1aandJz= 5\u0002103J\n(b) Excitation probability p(\u0017)\nn(tf) at the focal time tffor\n\u0017= 1 (yellow), \u0017= 2 (green) and \u0017= 3 (blue) excitations,\ndemonstrating spatial spin pattern formation depending on\nthe number of excitations \u0017.\nIII. A NON-LINEAR QUANTUM SPIN LENS:\nFOCUSING AND SPATIAL SORTING OF\nMULTI-PHOTON STATES\nWhile the lenses discussed so far are linear lenses op-\nerating on single spin excitations, we can also design\nnon-linear lenses , where the focusing dynamics depend\non the number of spin excitations in the medium via\nspin-spin interactions. Returning to the light-matter in-\nterface discussed at the beginning of Sec. II, we now\ngeneralize to an incident multiphoton superposition state\njEi=P1\n\u0017=0c\u0017j\u0017i. For a write process to atomic quantum\nmemory using a Raman scheme involving a pair of atomic\nground state levels (as described in Ref. [37]), this multi-\nphoton state will be mapped to a superposition of (dilute)\ndelocalized spin excitations, jEi!P\n\u0017c\u0017(^S+)\u0017jGi=p\n\u0017!\n(representing hardcore bosons). Repulsive spin-spin in-\nteractions, which become relevant during the focusing\ndynamics when the excitation density increases, will map\nthis superposition state to a superposition of spatial spin\npatterns in an atomic quantum memory. We note that\nthis provides a means of manipulating the individual\nterms in the superposition state by spatially addressing\nthe atomic spins with a laser. These transformed su-\nperposition states of spins can then be mapped back to\nphotons in a defocusing and read operation, providing ef-\nfective nonlinearities and manipulation of quantum states\non the single photon level.\nNon-linear quantum lenses can be implemented by gen-\neralizing the Hamiltonian (1) to\n^H=\u0000JX\nnh\n^\u001b(n)\n+^\u001b(n+1)\n\u0000 + H.c.i\n+X\nnVn^\u001b(n)\nz\n+X\nnX\nmJ(m)\nz^\u001b(n)\nz^\u001b(n+m)\nz;(10)\nwith the last term a long-range J(m)\nz=Jz=m6spin-spin\ninteraction and blockade distance rB=a(Jz=J)1=6. We7\nemphasize that the spin-spin interactions in (10) arise\nnaturally in Rydberg (dressed) gases and in trapped ion\nspin models. Time evolution according to the above\nHamiltonian will propagate the initial quantum state to\na strongly correlated many-body quantum state,\n1X\n\u0017=0c\u00171p\n\u0017!\u0010\n^S+\u0011\u0017\njGi\u0000!1X\n\u0017=0c\u0017j \u0017i (11)\nwith\nj \u0017i=X\nn1;:::;n\u0017 (\u0017)\nn1;:::;n\u0017^\u001b(n1)\n+:::^\u001b(n\u0017)\n+jGi;(12)\nand (\u0017)\nn1;:::;n\u0017the spatial wave functions for \u0017spin exci-\ntations.\nFigure 5 illustrates these focusing dynamics of inter-\nacting spins according to (10) for an initial superposi-\ntion state consisting of exactly one, two or three delo-\ncalized spin excitations as a function of time. We plot\nthe excitation probability as a function of position in the\nlattice,p(\u0017)\nn\u0011trf^\u001b(n)\n+^\u001b(n)\n\u0000j \u0017ih \u0017jgat lattice site nfor\n\u0017= 1;2;3, which clearly exhibits the spatial mapping\nand resolution of spin patterns associated with j \u0017iof\nEq. (12). This allows to perform gate operations on spa-\ntially localized atoms, e.g. atoms n=\u00064 orn=\u00067,\nin order to manipulate the \u0017= 2 or\u0017= 3 contribution\nof the superposition state individually. We note that the\nsmall excitation fraction between the peaks, e.g. popu-\nlation of atoms with n=\u00001;0;1 for\u0017= 2 (green bars),\ncan be traced back to states in the initial wave function\nwhere two excitations were closer than rB. This frac-\ntion of states becomes smaller by decreasing the initial\nexcitation density, i.e. increasing the atom number.\nThe above can be immediately generalized to higher\ndimensions. In particular, Fig. 1(e), bottom panel (iii),\nillustrates focusing of two spin excitations ( \u0017= 2) in 2D.\nIn this case the repulsive spin-spin interactions give rise\nto a superposition of states with two excitations sepa-\nrated by a characteristic distance rBaround the single-\nexcitation focus forming a ring, reminiscent of a quantum\ncrystal.\nIV. IMPLEMENTATION WITH RYDBERG\nATOMS IN 2D AND 3D ARRAYS\nThe quantum spin lenses proposed in the previous sec-\ntions can be implemented with atoms stored in optical\ntrap arrays, including large spacing optical lattices and\nand optical tweezers [38{40] in 1D, 2D and 3D, or alter-\nnatively with trapped ions in 1D [5{8].\nBelow we describe \frst a realization of a linear spin-\nlens Hamiltonians of the type (1) in 1D, but in par-\nticular also in 2D and 3D with alkali Rydberg atoms,\nwhere the spin degree of freedom of Sec. II is repre-\nsented by a pair of levels involving a long-lived atomic\nhyper\fne ground state, and a highly-excited Rydberg\nstate. As an example, we consider87Rb atoms and\na(n,n|n,n)\na(n,n+1|n,n+1)\na(n,n+1|n+1,n)\n20 30 4050607010-610-410-21102\nnC6(GHzm6)(a) (b)\n0.5\n0.7\n0.9\n0.00.51.01.52.02.50.00.20.40.60.81.0\nrV,W\n(n,n+1) (n,n+2)\n20 30 405060700.10.413\nn(c)(d)~ ~\n~~~Figure 6. (a) Atomic level scheme (see text). (b) Van der\nWaals coe\u000ecients between Rydberg nS1=2andn0S1=2states\nas a function of principal quantum number n. Negative values\nare plotted as outlined markers, positive values as \flled. (c)\nExchange interaction strength \u0018(n;n+\u000en)\u0011a(n+\u000enjn;n+\n\u000en)=a(n;n+\u000enjn;n+\u000en) as a function of principal quantum\nnumbern. (d) E\u000bective interaction potentials of Eq. (15) for\n\u0018= 0:5, 0.7 and 0.9, e.g., corresponding to n= 29;90 and 27.\nchoosejgi=j5S1=2;F= 2;mF= 2ias the spin down\nandjsi=jnS1=2;mj= 1=2ias the spin up state [see\nFig. 6(a)]. Note that in this section we denote by nthe\nprincipal quantum number.\nLong-range spin exchange interactions J(rij) =J=r6\nij\nbetween spins iandjin 3D can be achieved by weakly\ndressing the atomic ground state jgiby admixing with\na laser a second Rydberg state jri=jn0S1=2;mj= 1=2i\nwith (e\u000bective) Rabi frequency \n and detuning \u0001 \u001d\n.\nThis particular choice of Rydberg states leads to spin\ncouplingsJ(rij), which are isotropic in 3D, i.e. a purely\nradial dependence as a function of the distance rij=\njri\u0000rjjfor a large range of principal quantum num-\nbers [41, 42]. We note that, e.g. dipolar exchange inter-\nactions can be engineered by dressing the ground state\nwith Rydbergjn0Pj;mjistates resulting in anisotropic\n\rip-\rop interactions of the form Jij\u0018J=r3\nij[43].\nTo obtain the desired \rip-\rop term in Eq. (1) we\n\frst consider two atoms and derive an e\u000bective Hamil-\ntonian for the dynamics between the dressed ground\nstate and the Rydberg state. We start with a micro-\nscopic Hamiltonian, ^Hmic=^H(1)\nA+^H(2)\nA+^Hint, where\nthe \frst two terms account for the two driven atoms\nwith ^H(i)\nA=\u0000\u0001jriihrj+ (\n=2)jgiihrj+ h:c:written in a\nrotating frame . A small magnetic \feld and a circularly\npolarized laser beam allows dressing of the ground state\nwith a speci\fc Zeeman sublevel of the Rydberg state.\nThe key element of the implementation is the van\nder Waals interaction, ^HvdW, between the nS1=2and8\nn0S1=2Rydberg states. Choosing two s-states en-\nsures that the resulting vdW interactions are isotropic\nin 3D over a large range of principal quantum num-\nbersn(see appendix B). The exchange interaction be-\ntween the degenerate states jnS1=2;1=2i\njn0S1=2;1=2i\nandjn0S1=2;1=2i\njnS1=2;1=2idominantly arises via vir-\ntual population of j(n\u00001)Pj;mji\nj(n0+1)Pj0;m0\njiRyd-\nberg states [see Fig. 6(a)] and strongly depends on nand\nn0. As a particular example to demonstrate the tunabil-\nity of the resulting spin interactions we discuss the case\nn0=n\u00001 for which the exchange process is maximized\n[see Fig 6(c)]. The interaction has the structure\n^HvdW=1\nr60\n@c110 0\n0c12w12\n0w12c121\nA; (13)\nwritten in the basis of states jr;ri,jr;siandjs;riwhere\nwe neglect thejs;siinteractions, since we start initially\nwith only one excitation and considering a linear lens.\nThe generalized vdW coe\u000ecients cij=a(ni;njjni;nj)\n(diagonal) and wij=a(ni;njjnj;ni) (exchange) are\nshown in Figure 6(a) and derived in appendix B. The\nlinear behavior (on the log-scale) of c11shows the typi-\ncaln11scaling of vdW interactions. The resonances in\nc12andw12forn= 25 and n= 40 stem from van-\nishingly small energy di\u000berences between to the states\njnP3=2;nP1=2iandjnP3=2;nP3=2i, respectively. Close to\none of the F orster resonances diagonal and o\u000b-diagonal\ninteractions become approximately equal, c12\u0019w12,\nwhich are both dominated by a single channel.\nAdiabatic elimination of jriin the limit \n\u001c\u0001 leads\nto an e\u000bective long-range spin model between the dressed\nground statejgiand the Rydberg state jsiof the form\n^He\u000b=X\nij\u0014\nV(ij)\nsg^\u001b(i)\ngg^\u001b(j)\nss+1\n2W(ij)\nsg^\u001b(i)\ngs^\u001b(j)\nsg\u0015\n(14)\nwith Pauli operators ^ \u001bij=jiihjjand e\u000bective laser ad-\nmixed interactions V(ij)\nsg= \n2=(4\u0001) ~V(rij) andW(ij)\nsg=\n\n2=(2\u0001) ~W(rij) given by\n~V=~r12+ ~r6\n(~r6+ 1)2\u0000\u00182and ~W=\u0018~r6\n(~r6+ 1)2\u0000\u00182:(15)\nHere, ~r= (j\u0001j=c12)1=6r, (\u0001<0) is a dimensionless dis-\ntance and\u0018=w12=c12is the relative exchange strength\n(see Fig. 6). Note that we have dropped the AC Stark\nshift, which a\u000bects all sstates equally. The potentials\nare shown in Fig. 6(a) as a function of interatomic dis-\ntance. As a particular example we consider dressing to\nthen= 60 Rydberg state with (two-photon) Rabi fre-\nquency \n=2\u0019= 10MHz and detuning \u0001 =2\u0019=\u000020MHz\nwith a lattice constant aadjusted to the maximum of\n~Wwhich results in J=2\u0019= 0:36 MHz resulting in a\ntypical focusing times for an initial width \u001b0= 100a\naroundtf= 5\u0016s which increases linearly with the ini-\ntial width This compares well with the lifetime of the\n1000 10 1 100\n2D5thin\nnumber5of5holes\n2D5thick\n1D5thin\n1D5thick\n01\n0.8\n0.6\n0.4\n0.2\n0Pfoc\nσ0/a(b) (a)\n/a10-2\nδ\nlog(σfδ/σf(0))\n10-410-310-1102103\n10112345Figure 7. (a) Integrated probability Pfocto \fnd the focused\nspin wave inside a circle of radius 3 around the set focal point\nas a function of the number of holes in the lattice. (b) Final\nwidth\u001b\u000e\nfof a spin wave in a disordered system compared\nto the \fnal width \u001b(0)\nfwithout disorder as a function of the\ninitial width \u001b0and the disorder strength \u000e. The results were\nobtained for a thick lens including corrections up to sixth\norder (described in IIB), where the potential strength and the\nfocusing time were chosen to minimize \u001b(0)\nffor each value of\n\u001b0. The line represents \u000e\u00181=tfoc, indicating the breakdown\nof focusing due to disorder (details in text).\nstatejsi=j60S1=2;mj= 1=2iwith\u001c60S= 252\u0016s [44]\nresulting in tf=\u001c\u00190:02.\nInstead of the spin models with atomic ground and Ry-\ndberg states representing a spin 1 =2 system, one can also\nemploy dressing schemes, where spin is represented by a\npair of long-lived and trapped atomic ground states, and\nspin hopping and interaction terms are obtained by ad-\nmixing with a laser Rydberg interactions [26, 27]. Such\nschemes may be convenient for the non-linear lenses de-\nscribed in Sec. III.\nV. EFFECTS OF DISORDER ON FOCUSING\nDYNAMICS\nIn this section we analyze the robustness of the fo-\ncusing dynamics against two types of static disorder in\nthe spin lattice: ( i) holes in the lattice and ( ii) static\n\ructuations of the atomic position resulting in a random\ndistribution of long-range spin couplings V(ij)\nsgandW(ij)\nsg\nin Eq. (14).\nNon-unity \flling: Missing atoms in the lattice may\narise due to imperfect loading or when a previously fo-\ncused spin wave is stored in a di\u000berent hyper\fne ground\nstate. In addition to being excluded from the hopping\nmatrixWsg, each hole is surrounded by an e\u000bective po-\ntential due to the modi\fcation of Vsgin Eq. (14). We nu-\nmerically investigated the e\u000bect of randomly distributed\nholes in both one and two dimensions. Fig. 7(a) shows\nthe spin excitation probability within a radius 3 aaround\nthe focus of the lens, Pfoc, for a 1D and 2D spin lattice\nofN= 70 andN= 70\u000270 sites, respectively. Each\ndata point is obtained by averaging over 1000 random\nhole realizations for the 1D example (400 realizations for\n2D), starting with a Gaussian wavefunction with initial\nwidth\u001b0= 14a. The width of the statistical distribution9\nof the \fnal probability is indicated by the error bars.\nFor the 1D case, a single hole already has a signi\f-\ncant detrimental e\u000bect on the \fnal wave packet, which\nwe attribute to the fact that Wsg\u00181=r6closely resem-\nbles nearest-neighbor hopping. By contrast, in 2D the\nfocusing-scheme is almost una\u000bected by a small number\nof holes ( .10), as the spin wave can `\row' around the\nholes. This is further illustrated in panel (i) of Fig. 1(e)\nas a sequence of snapshots showing the 2D lattice of spins\nas a function of time. We expect the focusing scheme to\nbe even more robust in three dimensions as there are\nmore paths to avoid the holes.\nStatic disorder in atomic positions: As a second form\nof disorder we analyze the e\u000bect of \ructuations of the\nlong-range spin couplings VsgandWsg. Such models\nhave previously been discussed in the context of Ryd-\nberg atoms trapped in optical tweezers [45]. We as-\nsume that the position of the n-th atom is given by\nrn=r(0)\nn+dn, with r(0)\nnthe position on a regular lat-\ntice and the displacement dn\u0018 N (0;\u000e2) drawn from\na normal distribution with zero mean and standard de-\nviation\u000e. This results in a change of the interatomic\nseparation rn\u0000rm=r(0)\nnm+dnm, in turn modify-\ning the diagonal and hopping potentials of Eq. (14)\ntoWsg(rij)\u0019Wsg(r(0)\nij) +dijW0\nsg(r(0)\nij) andVsg(rij)\u0019\nVsg(r(0)\nij)+dijV0\nsg(r(0)\nij). We note that the \frst order term\nin the expansion of Wsgmay vanish for certain separa-\ntions, e.g. nearest neighbors, if the maximum of the in-\nteraction potential ~Wis commensurate with the lattice.\nHowever, the \frst order term will be present for all other\nseparations, as well as in the expansion of Vsgwhich ex-\nhibits no maximum or minimum as a function of distance\n(see Fig. 6).\nDisorder tends to localize the eigenstates of the system\nand thus prevents focusing when the localization length\nis smaller than the initial width of the wave packet [46].\nHowever, the focusing \fdelity may be signi\fcantly re-\nduced even if the localization length is large. To quan-\ntify the role of disorder we estimate the energy broad-\nening of plane wave states due to position disorder. In\na regular lattice, plane wave states with momentum k\nare energy eigenstates following the dispersion relation\n\u000f(k). In the presence of weak disorder, states with sim-\nilar momenta are coupled together such that the energy\nof a plane wave acquires an uncertainty on the order of\n\u0001\u000f(k) =q\nhkj(^H\u0000^H0)2jki, where ^Hand ^H0denote\nthe Hamiltonian of the disordered and the clean system,\nrespectively. Since our scheme sensitively relies on theinterference between di\u000berent momentum states, we ex-\npect that focusing ceases to be e\u000bective when the focus-\ning time exceeds tfoc&1=\u0001\u000f. Given that \u0001 \u000f\u0018\u000eto\nlowest order in \u000e, this suggests that there exists a criti-\ncal disorder strength \u000ec\u00181=tfocabove which the disor-\nder strongly a\u000bects the focusing dynamics. Indeed, this\nsimple argument correctly predicts the breakdown of fo-\ncusing as demonstrated in Fig. 7(b). We note that we\nnumerically veri\fed that the argument applies equally\nwell to the `thin lens'.\nVI. CONCLUSION AND OUTLOOK\nIn this work we have shown that lenses for spin qubits\ncan be designed for atomic lattice gases, allowing focus-\ning of delocalized spin excitations in quench dynamics to\nessentially single atoms. In addition, we have provided an\nimplementation of a spin lens based on Rydberg-dressed\nspin-spin interactions. The present work de\fnes a novel\nlight-matter interface, where incoming photons are stored\nin delocalized atomic excitations in an atomic medium,\nwith spin focusing providing the link and mapping to\nstorage of qubits in single atoms. We note that exist-\ning experimental setups with Rydberg atoms [20{23] en-\nabling the physical realization of 1D and 2D XY-spin\nmodels can provide \frst proof-of-principle experiments:\nhere a single delocalized spin excitation as initial con-\ndition could be generated using the Rydberg-blockade\nmechanism in an atomic lattice [38, 43, 47], and with fo-\ncusing dynamics implemented as described in the present\nwork. This scenario could also be demonstrated with the\nspin models realized with trapped ions [7, 8]. Finally, we\nexpect that optimal coherent control techniques both for\nspatial and temporal model parameters should allow for\nsigni\fcant improvement of `spin lenses' [48].\nAcknowledgments: We acknowledge discussions with\nM. Heyl and UQUAM partners C. Gross and I. Bloch.\nWork at Innsbruck is supported by the Austrian Sci-\nence Fund SFB FoQuS (FWF Project No. F4016-N23),\nthe European Research Council (ERC) Synergy Grant\nUQUAM, EU H2020 FET Proactive project RySQ and\nScalable Ion-Trap Quantum Network (SciNet). Work at\nHarvard is supported through NSF, CUA, AFOSR Muri\nand the Vannevar Bush Faculty Fellowship. AWG ac-\nknowledges funding from the National Research Founda-\ntion and the Ministry of Education of Singapore. SC ac-\nknowledges support from Kwanjeong Educational Foun-\ndation. HP is supported by the NSF through a grant for\nthe ITAMP at Harvard University and the Smithsonian\nAstrophysical Observatory.\n[1] T. D. Ladd, F. Jelezko, R. La\ramme, Y. Nakamura,\nC. Monroe, and J. L. O'Brien, Nature 464, 45 (2010).\n[2] C. Gardiner and P. Zoller, in The Quantum World of\nUltra-Cold Atoms and Light Book II: The Physics ofQuantum-Optical Devices (World Scienti\fc, 2015) pp. 1{\n524.\n[3] M. Sa\u000bman, T. G. Walker, and K. M\u001clmer, Rev. Mod.\nPhys. 82, 2313 (2010).10\n[4] I. Bloch, J. Dalibard, and S. Nascimbene, Nat Phys 8,\n267 (2012).\n[5] P. Schindler, D. Nigg, T. Monz, J. T. Barreiro, E. Mar-\ntinez, S. X. Wang, S. Quint, M. F. Brandl, V. Nebendahl,\nC. F. Roos, et al. , New Journal of Physics 15, 123012\n(2013).\n[6] S. Debnath, N. Linke, C. Figgatt, K. Landsman,\nK. Wright, and C. Monroe, Nature 536, 63 (2016).\n[7] P. Jurcevic, B. P. Lanyon, P. Hauke, C. Hempel, P. Zoller,\nR. Blatt, and C. F. Roos, Nature 511, 202 (2014).\n[8] P. Richerme, Z.-X. Gong, A. Lee, C. Senko, J. Smith,\nM. Foss-Feig, S. Michalakis, A. V. Gorshkov, and\nC. Monroe, Nature 511, 198 (2014).\n[9] A. Reiserer and G. Rempe, Rev. Mod. Phys. 87, 1379\n(2015).\n[10] H. J. Kimble, Nature 453, 1023 (2008).\n[11] T. Northup and R. Blatt, Nature photonics 8, 356 (2014).\n[12] D. Hucul, I. Inlek, G. Vittorini, C. Crocker, S. Debnath,\nS. Clark, and C. Monroe, Nature Physics 11, 37 (2015).\n[13] M. D. Lukin, Rev. Mod. Phys. 75, 457 (2003).\n[14] K. Hammerer, A. S. S\u001crensen, and E. S. Polzik, Rev.\nMod. Phys. 82, 1041 (2010).\n[15] M. D. Eisaman, A. Andr\u0013 e, F. Massou, M. Fleischhauer,\nA. S. Zibrov, and M. D. Lukin, Nature (London) 438,\n837 (2005).\n[16] L. Lamata, D. R. Leibrandt, I. L. Chuang, J. I. Cirac,\nM. D. Lukin, V. Vuleti\u0013 c, and S. F. Yelin, Phys. Rev.\nLett. 107, 030501 (2011).\n[17] T. Peyronel, O. Firstenberg, Q.-Y. Liang, S. Ho\u000berberth,\nA. V. Gorshkov, T. Pohl, M. D. Lukin, and V. Vuleti\u0013 c,\nNature (London) 488, 57 (2012).\n[18] G. Heinze, C. Hubrich, and T. Halfmann, Phys. Rev.\nLett. 111, 033601 (2013).\n[19] D. Maxwell, D. J. Szwer, D. Paredes-Barato, H. Busche,\nJ. D. Pritchard, A. Gauguet, K. J. Weatherill, M. P. A.\nJones, and C. S. Adams, Phys. Rev. Lett. 110, 103001\n(2013).\n[20] K. M. Maller, M. T. Lichtman, T. Xia, Y. Sun, M. J.\nPiotrowicz, A. W. Carr, L. Isenhower, and M. Sa\u000bman,\nPhys. Rev. A 92, 022336 (2015).\n[21] H. Labuhn, D. Barredo, S. Ravets, S. de L\u0013 es\u0013 eleuc,\nT. Macr\u0012 \u0010, T. Lahaye, and A. Browaeys, Nature (Lon-\ndon)534, 667 (2016).\n[22] J. Zeiher, R. van Bijnen, P. Schausz, S. Hild, J.-y. Choi,\nT. Pohl, I. Bloch, and C. Gross, Nature Physics advance\nonline publication (2016).\n[23] Y.-Y. Jau, A. M. Hankin, T. Keating, I. H. Deutsch, and\nG. W. Biedermann, Nat Phys 12, 71 (2016), article.\n[24] N. Henkel, R. Nath, and T. Pohl, Phys. Rev. Lett. 104,\n195302 (2010).\n[25] G. Pupillo, A. Micheli, M. Boninsegni, I. Lesanovsky,\nand P. Zoller, Phys. Rev. Lett. 104, 223002 (2010).\n[26] A. W. Glaetzle, M. Dalmonte, R. Nath, C. Gross,\nI. Bloch, and P. Zoller, Phys. Rev. Lett. 114, 173002\n(2015).\n[27] R. M. W. van Bijnen and T. Pohl, Phys. Rev. Lett. 114,\n243002 (2015).\n[28] I. Novikova, A. V. Gorshkov, D. F. Phillips, A. S.\nS\u001crensen, M. D. Lukin, and R. L. Walsworth, Phys.\nRev. Lett. 98, 243602 (2007).\n[29] A. V. Gorshkov, A. Andr\u0013 e, M. Fleischhauer, A. S.\nS\u001crensen, and M. D. Lukin, Phys. Rev. Lett. 98, 123601\n(2007).\n[30] Sequential mapping of qubits requires transfer of qubitsstored in the excited states jeito another excited state\nje0itohide these qubits from the focusing dynamics of\nthe following qubits. Note that these previously stored\nqubits appear as holes (defects) in the focusing dynamics\nof the consecutive qubits, as discussed in Sec. V.\n[31] A. W. Lohmann, Journal of the Optical Society of Amer-\nica A 10, 2181 (1993).\n[32] W. Schleich, Quantum Optics in Phase Space (Wiley,\n2015).\n[33] M. V. Berry, Philosophical Transactions of the Royal So-\nciety of London A: Mathematical, Physical and Engineer-\ning Sciences 287, 237 (1977).\n[34] J. a. P. Bizarro, Phys. Rev. A 49, 3255 (1994).\n[35] In contrast to focusing with quench dynamics in a HO,\nas described above, one could also localize the wave\nfunction in an adiabatic ramp of the harmonic oscilla-\ntor, i.e. by sweeping !!!f. An initial Gaussian wave\npacket, which is matched to represent the HO ground\nstate with!with width `would then be mapped to the\n\fnal ground state with width `f\u0011p\n~=m!f, with time\nrequired\u001d1=!.Quench dynamics discussed in this pa-\nper results in fast focusing tf\u00181=!, and does not require\ngood knowledge of the initial wave function or control\nover the applied trapping potential to match the initial\nwave packet to the HO ground state. In contrast, an adi-\nabatic scheme can be expected to be more robust against\nimperfect parameters in Hamiltonian (1).\n[36] M. Ben Dahan, E. Peik, J. Reichel, Y. Castin, and C. Sa-\nlomon, Phys. Rev. Lett. 76, 4508 (1996).\n[37] M. Fleischhauer and M. D. Lukin, Phys. Rev. A 65,\n022314 (2002).\n[38] M. Endres, H. Bernien, A. Keesling, H. Levine, E. R.\nAnschuetz, A. Krajenbrink, C. Senko, V. Vuletic,\nM. Greiner, and M. D. Lukin, ArXiv e-prints (2016),\narXiv:1607.03044 [quant-ph].\n[39] D. Barredo, S. de L\u0013 es\u0013 eleuc, V. Lienhard, T. Lahaye, and\nA. Browaeys, ArXiv e-prints (2016), arXiv:1607.03042\n[quant-ph].\n[40] K. M. Maller, M. T. Lichtman, T. Xia, Y. Sun, M. J.\nPiotrowicz, A. W. Carr, L. Isenhower, and M. Sa\u000bman,\nPhys. Rev. A 92, 022336 (2015).\n[41] R. M. W. van Bijnen, Quantum Engineering with Ultra-\ncold Atoms , Ph.D. thesis, Technische Universiteit Eind-\nhoven (2011).\n[42] S. Whitlock, A. W. Glaetzle, and P. Hannaford, Journal\nof Physics B: Atomic, Molecular and Optical Physics 50,\n074001 (2017).\n[43] D. Barredo, H. Labuhn, S. Ravets, T. Lahaye,\nA. Browaeys, and C. S. Adams, Phys. Rev. Lett. 114,\n113002 (2015).\n[44] I. I. Beterov, I. I. Ryabtsev, D. B. Tretyakov, and V. M.\nEntin, Phys. Rev. A 79, 052504 (2009).\n[45] M. Marcuzzi, J. c. v. Min\u0013 a\u0014 r, D. Barredo, S. de L\u0013 es\u0013 eleuc,\nH. Labuhn, T. Lahaye, A. Browaeys, E. Levi, and\nI. Lesanovsky, Phys. Rev. Lett. 118, 063606 (2017).\n[46] P. W. Anderson, Phys. Rev. 109, 1492 (1958).\n[47] P. Schau\u0019, M. Cheneau, M. Endres, T. Fukuhara, S. Hild,\nA. Omran, T. Pohl, C. Gross, S. Kuhr, and I. Bloch,\nNature (London) 491, 87 (2012), arXiv:1209.0944\n[physics.atom-ph].\n[48] T. Caneva, T. Calarco, and S. Montangero, Phys. Rev.\nA84, 022326 (2011).11\nAppendix A: Lattice Corrections\nIn this Appendix we discuss dephasing and Bloch os-\ncillations on the lattice and derive Eq. (2) for the optimal\nlens strength and Eq. (3) for the scaling of the \fnal width.\nDephasing: The optimal potential strengths vcand\noptimal pulse strength \u001ecof Eq. (2) for the `thick'\nand `thin' lens, respectively, can be derived from the\nBloch-band dispersion relation \u000f(k) = 2J[1\u0000cos(ka)]\nand its deviations from the quadratic dispersion rela-\ntion\u000f(k)\u0000J(ka)2\u0019J(ka)4=12 +O(ka)6. If these devi-\nations become of the order of the inverse focusing time,\ni.e.J(ka)4=12\u0018t\u00001\nfoc(v0), then plane wave eigenstates\nwill dephase during the focusing dynamics. This hap-\npens for parts of the Wigner function exceeding a critical\nmomentum kca= [2304v0=(\u00192J)]1=8for the `thick' lens\nandkca= (24\u001e0)1=4for the `thin' lens setup. During\nfocusing the distribution of momentum states populated\nwill become broader with the largest width in momen-\ntum space, i.e. kf\u00181=\u001bf, at the focusing time. This\nlimits the minimum \fnal width and restricts the the `lens\npotential' to values below vcfor the `thick' lens, as well\nas the critical pulse strength \u001ecfor the `thin' lens.\nBloch oscillations: The onset of Bloch oscillations at\n\u001bBOof Eq. (4) and the corresponding critical potential\nstrengthvBOcan be understood in a semi-classical model\nfor a particle in a quadratic potential with a Bloch-band\ndispersion relation, following the (semi-classical) equa-\ntions of motion _ x= 2Jasin(ka) and _k=\u00002v0xfor\nposition and momentum, respectively. These equations\nare equivalent to a motion of a classical particle with\na quadratic dispersion relation in a modi\fed potential\nVBO(x) = [(2v2\n0x2\n0\u00004v0J)a2x2\u0000(v2\n0a2)x4]=2. Depend-\ning on its initial position x0being smaller or larger than\n\u001bBOthe particle will either experience a single well or a\ndouble well potential. Fig. 2(d) shows the corresponding\neigenfunctions of the lattice Hamiltonian (1). Eigenfunc-tions which have an extension less than \u001bBOare well de-\nscribed by discretized harmonic oscillator eigenfunctions\ncentered around the origin, while eigenfunctions with an\nextension larger than \u001bBOstart to get localized at the\nminima of the double wells of VBO(x). Thus, Bloch os-\n−100−50 05010050100150200eigenvector\nxn/a\nFigure 8. Bloch oscillations: eigenfunctions of Hamiltonian\n(1) illustrating Wannier-Stark localized eigenfunctions at the\nedge of the lens (see text).\ncillations start to dominate the focusing dynamics at a\ncritical potential strength, vBO= 4J(a=\u001b0)2, which is\nindicated as the red dashed line in Figs. 3(a).\nAppendix B: Rydberg interaction between87Rb\natoms inn1S1=2andn2S1=2Rydberg states\nFor distances large enough, such that the dipole in-\nteraction matrix element between two S-states and two\nP-states is larger than the energy di\u000berence \u0001 Fbetween\nthese pair states, i.e., Vdip>\u0001F, we can treat vdW\ninteractions perturbatively. The vdW interaction Hamil-\ntonian between n1S1=2andn2S1=2Rydberg states can\nbe described by a 16 \u000216 matrix of the form\nHvdW=0\nB@M(n;njn;n) 0 0 0\n0M(n;n0jn;n0)M(n;n0jn0;n) 0\n0M(n0;njn0;n)M(n0;njn0;n) 0\n0 0 0 M(n0;n0jn0;n0)1\nCA (B1)\nThe vdW coe\u000ecients are given by\nM(n1;n2jn3;n4) =hn1S1=2m1;n2S1=2m2jHvdWjn3S1=2m3;n4S1=2m4i (B2)\nis a 4\u00024 matrix in the subspace of Zeeman levels m=\u00061=2 and the vdW interaction operator\nHvdW=X\nn\u000b;j\u000b;m\u000bX\nn\f;j\f;m\fVddjn\u000bPj\u000bm\u000b;n\fPj\fm\fihn\u000bPj\u000bm\u000b;n\fPj\fm\fjVdd\nEn1+En2\u0000En\u000b\u0000En\f; (B3)\ncouplingS-states with energy En1andEn2to intermediate P-states with energies En\u000bandEn\fvia dipole-dipole\ninteractions\nVdd(r) =\u0000r\n24\u0019\n51\nr3X\n\u0016;\u0017C1;1;2\n\u0016;\u0017;\u0016+\u0017Y\u0016+\u0017\n2(#;')\u0003d\u0016d\u0017:12\n◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆\n◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◆a(n,n|n,n)\n◇b(n,n|n,n)\n20 30 40 50 60 70 8005101520253035\nnC6/n11(A.U.)\n■■■\n■■\n■\n■■■■■■■\n■\n■■\n■\n■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■\n□□□□\n□□□□□□□□□□□□□□\n□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□\n■a(n,n+1|n,n+1)\n□b(n,n+1|n,n+1)\n20 30 40 50 60 70 80-100-50050100150\nnC6/n11(A.U.)\n●●\n●\n●●\n●\n●\n●●●●\n●\n●\n●\n●●●\n●\n●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●\n○○○○\n○○\n○○○○○○○○○○○○\n○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○\n●a(n,n+1|n+1,n)\n○b(n,n+1|n+1,n)\n20 30 40 50 60 70 80-100-50050100150\nnC6/n11(A.U.)\nFigure 9. Generalized vdW coe\u000ecients of Eq. (B4) as a function of principal quantum number nfor thenS1=2and (n+ 1)S1=2\nRydberg states.\nHere, dis the atomic dipole operator and r= (r;#;' ) is the vector between the two atoms in spherical coordinates.\nWithd\u0016we denote the \u0016-th spherical components ( \u0016;\u00172f\u0000 1;0;1g) of the atomic dipole operator, Cj1;j2;J\nm1;m2;Mare\nClebsch-Gordan coe\u000ecients and Ym\nlare spherical harmonics. Using Wigner-Eckart's theorem the vdW interactions\ncan be split up in an angular and radial part\nM(n1;n2jn3;n4) =1\nr6[a(n1;n2jn3;n4)14+b(n1;n2jn3;n4)D0]; (B4)\nwith generalized isotropic and anisotropic vdW coe\u000ecients\na=1\n81h\n7C(1)\n6+ 25C(2)\n6+ 11\u0010\nC(3)\n6+C(4)\n6\u0011i\n;\nb=1\n27h\nC(3)\n6+C(4)\n6\u0000C(1)\n6\u0000C(2)\n6i\n;(B5)\nand14the 4\u00024 identity matrix and\nD0=0\nBB@cos(2#)e\u0000i'sin(2#)e\u0000i'sin(2#) 2e\u00002i'sin2(#)\nei'sin(2#)2\n3\u0000cos(2#)\u0000cos(2#)\u00005\n3\u0000e\u0000i'sin(2#)\nei'sin(2#)\u0000cos(2#)\u00005\n32\n3\u0000cos(2#)\u0000e\u0000i'sin(2#)\n2e2i'sin2(#)\u0000ei'sin(2#)\u0000ei'sin(2#) cos(2 #)1\nCCA(B6)\nwritten in the basis fj1\n21\n2i;j1\n21\n2i;j1\n21\n2i;j1\n21\n2igof Zee-\nman states in the j= 1=2 Rydberg manifold and account-\ning for the anisotropy and mixing between the Zeeman\nsublevels. With C(\u0017)\n6we denote the radial part of the\nmatrix elements\nC(\u0017)\n6(n1;n2jn3;n4) =X\nn\u000b;n\fR\u000b\n1R\f\n2R\u000b\n3R\f\n4\n\u000e\u000b\f(B7)\nwhich accounts for the overall strength of the interaction\nand is independent of the magnetic quantum numbers.\nHere,Rk\ni=R\ndrr2 ni;`i;ji(r)\u0003r nk;`k;jk(r) is the radial\nintegral and \u0017accounts for the four channels to interme-\ndiatePjstates.\nFigure 9 shows the numerically calculated aandbcoef-\fcients corresponding to the di\u000berent blocks in Eq. (B1)\nas a function of the principal quantum number n. Both\naandbshow two F orster resonances around n\u001924\nandn\u001938 where the channels to fnP3=2;nP1=2gand\nfnP3=2;nP3=2gstates become close in energy, respec-\ntively. Apart from these resonances the anisotropy co-\ne\u000ecientbis several orders smaller then the diagonal co-\ne\u000ecientawhich allows to safely neglect mixing of Zee-\nman sublevels and results in an almost perfect isotropic\ninteraction.\nFor two atoms initially in the jn1S1=2;1=2iand\njn2S1=2;1=2istates this allows to reduce the dynamics to\nthe four states S1=2states with principal quantum num-\nbersjn1;n1i,jn1;n2i,jn2;n1iandjn2;n2iand magnetic\nquantum number 1 =2. The corresponding Hamiltonian\nrestricted to this basis has the form\nHvdW=1\nr60\nB@a(n1;n1jn1;n1) 0 0 0\n0a(n1;n2jn1;n2)a(n1;n2jn2;n1) 0\n0a(n2;n1jn2;n1)a(n2;n1jn2;n1) 0\n0 0 0 a(n0;n0jn0;n0)1\nCA (B8)13\nwithadenoted as C6in the main text and plotted in\nFig. 6(a) as a function of the principal quantum number\nn.\nFor the long-range Hamiltonian of Eq. (14) (cf.\nFig. 6(c)) next-nearest neighbor hopping is around 10%\nof the nearest neighbor hopping element. In Fig. 4 wecompare the performance of a spin lens implemented\nwith the potentials arising from the Rydberg interac-\ntions, compared to the case with ideal nearest neighbor\nhopping. Our numerical results indicate that long-range\nhopping terms slightly increase the speed of the scheme\nand decrease the \fnal width."    },    {        "title": "0908.0273v1.Signatures_of_dynamically_polarized_nuclear_spins_in_all_electrical_lateral_spin_transport_devices.pdf",        "content": "APS/123-QED\nSignatures of dynamically polarized nuclear spins in all-electrical lateral spin\ntransport devices\nG. Salis, A. Fuhrer, and S. F. Alvarado\nIBM Research, Zurich Research Laboratory, S aumerstrasse 4, 8803 R uschlikon, Switzerland\n(Dated: July 20, 2006)\nThe e\u000bect of nuclear spins in Fe/GaAs all-electrical spin-injection devices is investigated. At\ntemperatures below 50 K, strong modi\fcations of the non-local spin signal are found that are char-\nacteristic for hyper\fne coupling between conduction electrons and dynamically polarized nuclear\nspins. The perpendicular component of the nuclear Overhauser \feld depolarizes electron spins near\nzero in-plane external magnetic \feld, and can suppress such dephasing when antialigned with the\nexternal \feld, leading to satellite peaks in a Hanle measurement. The features observed agree well\nwith a Monte Carlo simulation of the spin di\u000busion equation including hyper\fne interaction, and\nare used to study the nuclear spin dynamics and to quantify the Overhauser \feld that is related to\nthe spin polarization of the injected electrons.\nPACS numbers:\nThe interdependence of nuclear and electron spin dy-\nnamics in semiconductors caused by the contact hyper-\n\fne interaction leads to a rich variety of phenomena that\nsigni\fcantly alter the behavior of independent electron\nand nuclear systems. For instance, fully polarized nu-\nclear spins in GaAs create an e\u000bective magnetic \feld\nof 5.3 T acting on the spin of conductance-band elec-\ntrons. This interaction has implications for applications\nin quantum information processing and spintronics. On\nthe one hand it can be employed as a means to e\u000eciently\ncontrol the electron spin state,1but on the other hand\nit also leads to spin dephasing. Conduction-band elec-\ntron spins in a semiconductor can be e\u000eciently polar-\nized either by means of optical orientation2or spin in-\njection from ferromagnetic contacts.3,4,5,6The hyper\fne\ninteraction leads to a \rip-\rop spin scattering between\nthe electron and nuclear spins that dynamically trans-\nfers the spin polarization to the nuclear system.7,8The\nstatic part of the hyper\fne interaction can be described\nby an e\u000bective (Overhauser) magnetic \feld Bnthat acts\non the electron spins and has been detected optically2,9\nand in transport experiments.10,11The restriction of elec-\ntron spin pumping to small quantum-con\fned regions\nallows one to study nuclear polarization in semiconduc-\ntor heterostructures12,13,14and quantum dots.10,11,15All-\nelectrical injection and detection of electron spins were\nrecently demonstrated in bulk GaAs16,17and Si,18,26and\nit was suggested that the linewidth of Hanle peaks is in-\n\ruenced by dynamic nuclear polarization (DNP) at lower\ntemperatures.19\nHere, we investigate the consequences of DNP in an\nall-electrical non-local spin device consisting of Fe injec-\ntion and detection contacts and a n-doped GaAs spin\ntransport channel. From measurements of the non-local\nvoltageUnlat the detection contact, we obtain quantita-\ntive information on the Overhauser \feld Bnand on the\nnuclear spin dynamics in the GaAs channel. By apply-\ning an external magnetic \feld B=Bx^ x+Bz^ z[see de\fni-\ntion of coordinate system and sample layout in Fig. 1(a)],\nwe investigate the interdependence of nuclear and elec-tron spin dynamics by (i) in-plane sweeps of Bxat con-\nstantBz, and by (ii) perpendicular (Hanle) sweeps of\nBzat constant Bx. For (i), we probe the depolarization\npeak inUnlatBx= 0, previously reported in Ref. 16,\nwhich we here explain in terms of a Hanle-type electron\nspin dephasing arising from a perpendicular Bn. In (ii),\nwe observe that apart from the Hanle peak at Bz= 0,\ntwo satellite peaks occur at \fnite and opposite Bzval-\nues provided a \fnite Bxis oriented parallel to the spin-\npolarization vector of the injected electrons. We show\nthat these satellite peaks occur when Bn+B= 0, lead-\ning to a reduction of spin dephasing. A comparison of\nthe measurements with a numerical model allows us to\nextract quantitative values for Bn, the sign of injected\nelectron spins, and a lower limit for injected spin polariza-\ntion. We \fnd that majority spins are injected into GaAs,\nand that minority spins get accumulated in GaAs when\nelectrons are extracted from the semiconductor. A lower\nlimit of 1% for the spin-polarization in the GaAs channel\nis estimated at 25 K and a current of 30 \u0016A through a\ncontact area of 360 \u0016m2.\nThe spin-devices were prepared by epitaxially growing\na 1\u0016m thick n-doped GaAs epilayer with Si doping con-\ncentration of 5\u00021016cm\u00003onto an undoped GaAs(001)\nwafer. The doping concentration within 15 nm below the\nsurface is 6\u00021018cm\u00003, followed by a gradual reduction\nto 5\u00021016cm\u00003within 15 nm. The highly-doped surface\nregion allows one to obtain a thin Schottky barrier for\ne\u000ecient charge carrier injection.5,6The substrates, pro-\ntected by an As capping layer, are then transferred into\nan ultra-high vacuum chamber for Fe growth by thermal\nsublimation. Prior to deposition of a 4{6 nm thick Fe\n\flm, the As capping was removed by heating the wafer\nto 400\u000eC for one hour. The GaAs surface was inspected\nby scanning tunneling microscopy to have a c4 \u00024 recon-\nstruction. A \fnal 2{4 nm thick Au layer protects the Fe\n\flm from oxidation. Samples were annealed in situ at\n220\u000eC for 10 min before further processing. By means\nof optical or e-beam lithography and ion milling, the Fe\nlayer was patterned into stripes that are 60 \u0016m long andarXiv:0908.0273v1  [cond-mat.mes-hall]  3 Aug 20092\n6 and 2\u0016m wide (parallel to the [110] orientation of the\nGaAs substrate) serving as injection (2) and detection\n(3) contacts, as indicated in Fig. 1(a), which shows a\nscheme of a sample. Unless stated otherwise, the separa-\ntion between injection and detection contacts was 3 \u0016m.\nA 100 nm thick layer of Al 2O3isolates large Au/Ti bond\npads for contacting the Fe bars from the substrate. In-\njection and detection of electron spins are achieved in the\nnon-local geometry.16,20A currentIinjis drawn from con-\ntact (1) to contact (2) such that spin is injected at con-\ntact (2) for Iinj>0 and spin \fltering occurs for Iinj<0.\nThe nonlocal voltage Unlis measured between contact\n(3) and contact (4) using both dc and ac lock-in tech-\nniques. Both approaches yield equivalent results, and in\nthe following we use a superscript to di\u000berentiate ac ( Iac\ninj)\nfrom dc (Iinj) excitation of the injection current. Mea-\nsurements were performed in two di\u000berent cryostats with\nvariable-temperature inserts and a superconducting mag-\nnet system. One of them allows application of magnetic\n\feldsBxandBzalong two independent axes.\n(a) (b)\n(c)Bx (mT)\n-185-184-183\n-10100Unl (mV)\nt (min)\nDU0 (mV)\n0.00.51.01.5\n0510Unl\n \n(d)\nIIIIII\nIVV\n(mV)\n-404051015\nBx (mT)t (min)-10IinjUnl\n1234IIIIIIVIV\nDU0UnlS0BnB\nBxBz\nxyz\n<S>\nFIG. 1: (color online) (a) Coordinate system with external\nmagnetic \feld and spin vectors and a scheme of the non-local\nsample geometry and measurement setup with two central\nferromagnetic injection (2) and detection(3) bars and two\nouter reference bars (1) and (4). (b) The nonlocal voltage\nUnlexhibits jumps for both upward (black) and downward\n(blue) sweeps of Bx. The jumps are related to magnetization\nswitching of the injection and detection bars into parallel and\nantiparallel con\fgurations. The central peak in Unlis a mea-\nsure of nuclear polarization as is indicated in a sweep of Bx\n(c) after waiting for 10 min at Bx=\u000050 mT and a time tat\nBset\nx= 6:5 mT [position IV in (b)]. The \ftted height \u0001 U0of\nthe peak vs. tis shown in (d) for di\u000berent waiting \felds Bset\nx\nlabeled I-V in (b) and (d).\nFigure 1(b) shows the nonlocal voltage Unlmeasured\nas a function of Bxat temperature T= 5 K, obtained at\nIac\ninj=1.6\u0016A. In an upward sweep of Bx,Unljumps to a\nhigher value at Bx\u00195 mT when the \frst bar reverses its\nmagnetization, and drops back down at the reversal of\nthe second bar, i.e., when the magnetizations are parallelagain. In the following, we subtract an o\u000bset \u0016Unlfrom\nUnlso thatUnl= 0 lies exactly in the middle between the\ntwo jumps, marking the nonlocal voltage level with zero\nelectron spin polarization. In addition to the jumps, Unl\nforms a peak at Bx= 0 mT, indicative of a loss of average\nelectron spin polarization at contact (3). Its height \u0001 U0\ndepends on the history before performing the Bxsweep\nand appears within a time scale of several minutes, which\nis characteristic of nuclear spin-lattice relaxation times\n\u001c1.21\nTo demonstrate that the depolarization peak is related\nto nuclear spin polarization, we performed a series of\nmeasurements in which DNP was built up and then re-\nversed while monitoring \u0001 U0. The system is initialized at\nBx=\u000050 mT for 10 min with Iinj=\u00001:6\u0016A such that\nnuclear spins get dynamically polarized until they reach a\nsaturation value. Then Bxis swept to a value Bset\nx. The\nnuclear spins adiabatically follow the external \feld, and\nifBxcrosses zero reverse their direction in space.22After\nwaiting a time tatBx=Bset\nx, the depolarization peak is\nimmediately recorded by sweeping BxacrossBx= 0 mT,\nwithIac\ninj= 1:6\u0016A. This was repeated for several values of\ntto obtain data as shown in Fig. 1(c) for Bset\nx= 6:5 mT\n[corresponding to arrow IV in Fig. 1(b)], where \u0001 U0\frst\ndecreases, passes through a minimum at t\u00196 min and\nsaturates again at t >10 min. This time-dependence of\n\u0001U0is a strong indication that it is a measure of the nu-\nclear polarization. Substantial nuclear spin polarization\nhIican be built up by hyper\fne-induced \rip-\rop spin\nscattering if an average electron spin polarization hSiis\nsustained, such as in our case by injection or \fltering\nof spin-polarized electrons from the Fe contact (2). The\ne\u000bective Overhauser magnetic \feld Bn/hIi, which in\nsteady state can be described by2\nBn=fbnB\u0001hSi\nB2B: (1)\nHere,f\u00141 is a leakage factor that takes into ac-\ncount the possibility of nuclear spin relaxation by other\nchannels than through a hyper\fne-induced \rip-\rop pro-\ncess, andbn=\u00005:3 T in GaAs9is the maximum \feld for\n100% nuclear spin polarization. Equation (1) neglects\nthe Knight shift and the e\u000bect of dipole-dipole interac-\ntion between nuclear spins that is only important for typ-\nicallyB < 1 mT, where it leads to a drop of hIito zero at\nB= 0. In Eq. 1,hSican be replaced by S0=S0^ x, which\ndenotes the average spin polarization of the GaAs elec-\ntron density without precession, see Fig. 1(a). Depending\non the sign of BxS0,Bnis aligned parallel or antiparallel\ntoB. WhenBxchanges sign, the spatial direction of\nDNP does not change, which means that after adiabatic\nreversal of the nuclear polarization at Bx= 0 mT, the\nnuclear polarization will \frst decrease and then repolar-\nize into the opposite direction. This is exactly what is\nobserved in \u0001 U0. A similar decrease and subsequent in-\ncrease of \u0001U0are measured if the sign of Iinjis reversed,\nwhereby electron spins with the opposite sign will accu-3\nmulate below the injection contact (data not shown). In\nFig. 1(d), \u0001 U0is plotted as a function of tfor di\u000berent\nBset\nx, labeled I to V in Fig. 1(b). For position (I) and (V),\n\u0001U0does not fall to zero but slightly increases before sat-\nuration. Because Bset\nxremains negative (I) or is positive\nand large enough such that the magnetization of both\ninjection and detection contact reverses (V), no cross-\ning ofhIi= 0 is necessary to attain steady state. Only\nfor the two intermediate \felds (III and IV), where Bx\nhas crossed zero but the magnetization of the injection\ncontact has not yet reversed, will \u0001 U0drop to zero and\nreappear afterwards. For Bset\nx= 0, build-up of nuclear\nspin polarization is prevented, see trace II in Fig. 1(d),\nbecause of ine\u000ecient DNP for B\u0001hSi= 0.\n-5050246DB (mT)\nDB0\n2DB(a) (b)\n20406000.4\nT(K)DU0/DUnl  (c)(d)(e)\n  -10010-4-2024\n  \nBz = -3mT\nBx (mT)Unl (mV)D\nUnlDU0\nBz (mT)\n24DU0 (mV)\n-101Bz (mT)D\nB (mT)1.0\n0.5\n-101Bz (mT)\nFIG. 2: (color online) (a) Unlas a function of Bxat a \fxed\nBz=\u00003 mT at 20 K. The red curve is a Lorentzian with a\nhalf width \u0001 B=\u00003 mT and a height \u0001 U0= 4:2\u0016V that\nhas been adjusted for best \ft to the depolarization peak at\nBx= 0. For antiparallel magnetization of the central contacts\nthe Lorentzian is inverted and o\u000bset by the maximum spin\nsignal \u0001Unl. (b) Peak width \u0001 Bas a function of Bz. At\nvanishingBz, \u0001Bsaturates at a \fnite value \u0001 B0= 0:8 mT.\n(c) \u0001U0(squares) dips at Bz= 0 mT because of the nuclear\nlocal dipolar \feld. The solid red line is calculated using a\nmodel as described in the text. (d) Measured values of \u0001 B\naroundBz= 0 mT (squares), showing a local peak, and model\n(solid red line) as described in the text. (e) Temperature\ndependence of \u0001 U0=\u0001Unl, which is slightly larger than 0 :5\nfor low temperatures and vanishes for T\u001550 K.\nTo understand the reason for the occurrence of the\npeak inUnl, we have carried out measurements of Unl\nversusBxfor di\u000berent Bzshown in Fig. 2. The nu-\nclear spin polarization is initialized at Iinj= 20\u0016A and\nBx=\u000050 mT for 15 min. Bxis then swept up and\ndown with a sweep rate of 25 mT/min, for di\u000berent Bz\nfrom\u00006 to 6 mT in steps of 0.5 mT. Figure 2(a) shows\na trace measured at Bz=\u00003 mT andT= 20 K. \u0001U0\nextends slightly beyond \u0001 Unl=2, de\fned as half the sep-\naration between Unlfor parallel and antiparallel mag-\nnetization. The peak can be \ftted by a Lorentz curve\n/(1 +B2\nx=\u0001B2)\u00001, where the half width at half maxi-\nmum \u0001Bof the peak follows ( B2\nz+\u0001B2\n0)1=2with \u0001B0=0:8 mT, see Fig. 2(b). The Lorentz shape with a width\n\u0001B=Bzsuggests that the depolarization peak is due\nto the rotation of the total magnetic \feld Btot=B+Bn\nin thexzplane asBxis swept through zero. For su\u000e-\nciently large Btot, the electron spins precess fast enough\nthathSipoints along (or against) Btot.Unlis given by\nthe projection of hSionto^ x, and thus becomes propor-\ntional toB2\nx=(B2\nx+B2\nz), i.e., follows a Lorentz curve with\na half width equal to Bz, as observed in the experiment\nfor largerjBzj[the red curve in Fig. 2(a) is a Lorentzian\n\ft with \u0001B=jBzj= 3 mT and \u0001 U0as the only free \ft\nparameter].\nNext we discuss why \u0001 Band \u0001U0do not disappear\natBz= 0. We have repeated measurements as the ones\nshown in Fig. 2(b), but with higher resolution around\nBz= 0 and at T= 25 K. As shown in Fig. 2(c), a\ndip in \u0001U0appears at Bz= 0 mT with a full width\nat half maximum of about 0.5 mT and a decrease from\n4.5 to 2\u0016V. This is evidence of the presence of a small\n\feld component Byalong ^ ythat orients Bninto the\nydirection, accompanied by a partial depolarization of\nthe nuclear spins because of dipole-dipole interaction be-\ntween the nuclear spins. For a local dipole \feld BL,\nBn=B0\nnB2=(B2+B2\nL), whereB0\nnis the nuclear \feld\nforB\u001dBL. In GaAs, BL\u00191 mT.9An expression\nforUnlis obtained for arbitrary Btotby separatinghSi\ninto its components along and perpendicular to Btot. We\n\fndUnl=\u0000\u0001Unl\n2(cos2\u000b+H(Btot)=H(0) sin2\u000b), where\nH(Btot) is the Hanle lineshape as de\fned in Eq. (1) of\nRef. 16 and \u000b= arctanq\nB2y+B2z=Bx. The term pro-\nportional to cos2\u000b(sin2\u000b) corresponds to the component\nofhSiparallel (perpendicular) to Btot. \u0001U0is given by\nthe value of Unlat\u000b= 0,\u0001Unl\n2(1\u0000H(Btot)=H(0)), and\ntherefore follows a typical Hanle curve: For B < B L,\n\u0001U0decreases because Bndepolarizes and thus Btot\ndecreases, making Hanle-type spin dephasing less e\u000e-\ncient. For intermediate Btot,H(Btot) becomes nega-\ntive and therefore \u0001 U0>\u0001Unl=2. The solid line in\nFig. 2(c) shows the calculated \u0001 U0using the model de-\nscribed above and reproducing the observed dip. As pa-\nrameters, we used a di\u000busion constant D= 0:002 m2/s,\nspin lifetime \u001cs= 10 ns,B0\nn= 47 mT,BL= 1 mT and\nBy= 0:2 mT. Interestingly, in this model, \u0001 Bdoes not\ndrop to zero, but even increases around Bz= 0 mT. Fig-\nure 2(d) shows the measured \u0001 Bhaving a local peak at\nBz= 0, as well as the results of a \ft to the model using\nthe same parameters as above (solid red line). In the\nmodel, the local increase in \u0001 Bis because of a change\nof the form of the depolarization peak whose height \u0001 U0\ndecreases while its tails remain unchanged because they\nare determined by the term proportional to cos2\u000b. Com-\npared to the model, the measured \u0001 Bexhibits a more\npronounced local peak at Bz= 0. We note that the sizes\nof \u0001U0and \u0001BatBz= 0 are determined by the inter-\nplay of many di\u000berent mechanisms, from which our model\ntakes into account the electron spin dynamics along the\ndi\u000busive path between injection and detection contacts, a4\nhomogeneous Overhauser \feld aligned with the external\n\feld, a decrease of nuclear polarization at B= 0 because\nof the local dipole \feld, and a \fnite magnetic \feld com-\nponent along ^ y. We have neglected the Knight shift that\nreorients the direction of the Overhauser \feld, as well\nas the possibility of electron spin dephasing in a locally\n\ructuating Overhauser \feld, e\u000bects that both will a\u000bect\nthe details of UnlaroundB= 0.\nThe temperature dependence of \u0001 U0=\u0001UnlatBz\u0019\n1 mT is shown in Fig. 2(e). For T < 20 K, \u0001U0=\u0001Unl\nremains slightly above 0.5, indicative of the intermediate\n\feld regime. At T\u001525 K, \u0001U0=\u0001Unldecreases. Within\nthe explanation given above, Btotmust there be of the\nsame order as the Hanle peak width, i.e. about 3 mT.\nFrom measurements of satellite peaks in Hanle con\fgura-\ntions (explained later), we can determine Bnand extrap-\nolate values well above 20 mT for 25 K and initialization\natBx=\u000050 mT, which is much larger than 3 mT, thus\nsupporting the model that involves the nuclear dipolar\nmechanism that reduces Bn. AboveT= 50 K, where\nDNP becomes ine\u000ecient, \u0001 U0is no longer observable.\n-2002040-80-4004080-404\nBz (mT)\nIinj(mA)\n-10100DUnl (mV)\nU\nnl(mV)Unl(mV)\n(b)(c)\n-40Bx = -1.6 mT\n-404(a) Iinj = 30mA\nBx = 2.5 mT\nBx = -2.5 mT\nT = 25K\nFIG. 3: (a) Hanle measurements of UnlversusBzfor \fxed\nBx=\u00062:5 mT and parallel (antiparallel) magnetization of\nthe central contact bars indicated by black (red) arrows. (b)\nMaximum spin signal \u0001 Unlas a function of Iinj. For spin in-\njection, i.e. Iinj>0, the spin signal goes through a maximum\nwith increasing Iinjthat correlates with the satelite peak po-\nsition in the Hanle measurement in (c) as a function of Bz\nandIinj. For spin \fltering, i.e. Iinj<0, no satellite peaks ap-\npear because the nuclear \feld Bnis parallel to the externally\napplied \feld. Data in (a) and (c) were averaged over an up\nand down sweep of Bz.\nAs mentioned in Ref. 19, nuclear spin polarization also\nmodi\fes the lineshape of the Hanle curve UnlversusBz.\nAn even more profound e\u000bect occurs when Bnpoints\nagainstBand is so large that the two \felds cancel. In\nsuch a situation, Hanle-type spin dephasing is strongly\nreduced, leading to two satellite peaks in Unlat \fnite\nand opposite Bzvalues. Figure 3(a) shows such measure-ments at \fxed Bx= 2:5 mT (solid lines) and \u00002:5 mT\n(dashed lines), and for the magnetization of the detection\ncontact oriented parallel (black) and antiparallel (red) to\nthat of the injection contact. The magnetization of the\nlatter is oriented along positive Bx. According to Eq. (1),\nBnpoints against or along B, depending on the sign of\nBxS0. The appearance of the satellite peaks requires that\nBxS0>0. ForIinj>0, we observe the satellite peaks\natBx<0 and forIinj<0 atBx>0 [see Fig. 4(a)].\nTherefore,S0is negative (and antialigned with the mag-\nnetization Mof the injection contact) in the case of spin\ninjection and positive (aligned with M) for spin \fltering.\nIn agreement with previous observations,16,23this means\nthat majority spins are injected from Fe into GaAs.\nThe position of the satellite peaks provides a direct\nmeasure of the nuclear \feld because there, B=\u0000Bn.\nThe sign and magnitude of S0can be controlled with the\ninjection current Iinj. As shown in Fig. 3(b), \u0001 Unlre-\nverses its sign at Iinj= 0. ForIinj>0, \u0001Unlreaches a\npeak and decreases again, whereas for negative Iinjit sat-\nurates. The separation of the satellite Hanle peaks mea-\nsured atBx=\u00001:6 mT and shown in Fig. 3(c) follows the\nsame behavior as \u0001 Unl, indicating that the nuclear \feld\nmonotonically depends on S0. This is also evidence that\nthe peak in \u0001 UnlforIinj>0 directly re\rects a maximum\nspin polarization in the GaAs channel and is not due to\na dependence of the detector sensitivity on the injection\ncurrent that could indirectly occur through a spreading\nresistance. Because of the opposite direction of S0for\nIinj<0, no satellites are observed in Fig. 3(c). Similarly,\nthe appearance of the satellites can be controlled by the\norientation of the magnetization of the injection contact\n(data not shown).\nFigure 4(a) shows a colorscale plot of measured Unl\nas a function of BxandBz. For small Bx, we observe\na linear dependence of the satellite peak separation on\nBx. From the condition B+Bn= 0 and Eq. 1, i.e.\nwhen assuming that Bnis at its saturation value for\nall measured B, the satellite peak positions are given\nbyBz=\u0006p\nBx(\u0000fbnS0\u0000Bx). As\u0000fbnS0\u001dBxfor\nour measurements, we expect a quadratic dependence,\nBx/B2\nz, at the satellite peaks. In an optical orientation\nmeasurement with oblique magnetic \feld, similar satel-\nlite peaks were observed in the circular polarization of\nphotoluminescence as a function of Bz.24Also there, a\nlinear increase of the satellite peak separation was ob-\nserved, which was interpreted as a B-dependent leakage\nfactorf. As we will demonstrate with a numerical sim-\nulation, our data can be explained without assuming a\n\feld-dependent f, but taking into account the long \u001c1\nof nuclear spins whose polarization does not reach sat-\nuration at individual \feld values within a sweep of Bz.\nWe performed the numerical simulation of the di\u000busing\nelectron spins using a Monte Carlo approach by assigning\none-dimensional spatial coordinates yi, velocities viand\nthree-dimensional spin directions sito electrons labeled\ni= 1 ton. At constant time intervals \u000et,yiare updated\ntoyi+vi\u000et, and to a fraction of the nelectrons a new5\n \"0 . 1 \n0 \n0 . 1 \n0 . 2 \n  \n00.2<Sx>\nBZ (mT)\n060246\nUnl (mV)(a)\n−20020−20246\nBZ (mT)(b)\n048y (mm)\n00.2(c)Bx=2.5mT\n<Sx>\n  \n020B-Bn-Bn, B (mT)(d)  \n  \n   Bx (mT) B x (mT)<\nSx>\n-2002000.2\n  \n(e)\nFIG. 4: (a) UnlversusBzfor di\u000berent Bx, measured at\nIinj=\u000030\u0016A andT= 25 K. Magnetizations were set parallel\nalong the positive xdirection. (b) Calculated spin polariza-\ntionhSxiat detection contact using Monte Carlo simulation.\nD= 0:002 m2s\u00001,\u001cs= 10 ns ,bnfS0= 45 mT in (b)-(e). The\ndashed black lines indicate expected peak positions for a sat-\nuratedBnaccording to Eq. 1. (c) Map of hSxiversusyand\nBzforBx= 2:5 mT. Spin is injected at y<0, and data in (b)\nis averaged at the detection contact located at 3 <y< 5\u0016m.\n(d)B(blue) and\u0000BnversusBzforBx= 2:5 mT.Bnwas cal-\nculated for saturated condition (black) and for re\u001c1= 250 mT\n(red) averaged at the detection contact (dashed line) and in\nbetween the contacts (solid lines). Dots indicate positions\nwhereB=\u0000Bnand satellite peaks are expected. (e) Hys-\nteresis and broadening of satellite peaks in calculated hSxiat\ndetection contact for locally varying Bn[red, same parameters\nas in (d)] and uniform/saturated Bn.\nrandom velocity is assigned, thus simulating the di\u000busive\nscattering process characterized by the di\u000busion constant\nD. The new velocity viis distributed between \u0000vFand\nvFaccording to the projection of a two-dimensional vec-\ntor of length vFonto theyaxis. Spin coordinates siare\nregularly updated by calculating the rotation about the\nlocally varying B+Bnand by accounting for a spin decay\nat rate 1=\u001cs. At a constant rate, spin-polarized electrons\nare injected by assigning coordinates yiwithin the injec-\ntion contact area ( \u00006 to 0\u0016m) to new electrons i. We\nletBnlocally evolve with a time constant \u001c1towards the\nsaturation nuclear \feld as calculated by Eq. 1, thus ac-\ncounting for the fact that typical sweep rates rin the\nexperiment are faster than 1 =\u001c1. We neglect nuclear spin\ndi\u000busion because of the small di\u000busion constant (103\u0017A/s\nhas been measured in Ref25). The simulation is run for a\ntime 5\u001cs, ensuring a converging self-consistent solution.\nThe nonlocal voltage Unlis proportional to the electron\nspin componenthSxiaveraged over the detector contact\nat 3\u0016m< y < 5\u0016m, and is plotted in Figure 4(b) as a\nfunction of BxandBz. In Fig. 4(c), a map of hSxiversus\nyandBzis shown for Bx= 2:5 mT. In the simulation,\n\u001cs= 10 ns,D= 2\u000210\u00003m2/s andr\u001c1= 250 mT areused. We obtain an excellent match with the experimen-\ntal data in Fig. 4(a) with fbnS0= 45 mT, where S0is\nthe averaged x-component ofhSifor 0< y < 3\u0016m and\natBz= 0. The solid line in Fig 4(b) indicates the in-\ncrease withpBxof the satellite peak separation that is\nexpected when Bnreaches its saturated value for all \feld\npositions. In contrast to this, the simulation reproduces\nthe linear increase for small Bx. In Fig. 4(d), calculated\n\u0000Bn(red) is shown versus Bzaveraged in between the\ntwo contacts (solid line) and below the detection con-\ntact (dashed line). Because r\u001c1is much larger than the\nsweep range of\u000640 mT,Bndoes not follow the satu-\nrated value as Bzis swept (shown as black line), but is\nrather uniform at Bn\u0019\u0000fbnS0Bxh1\nBi, whereh1\nBiis the\ntime-average of1\nBfor a sweep of Bz, andS0is averaged\nin between the contacts. From this, a splitting that is\nlinear inBxdirectly follows. In addition, Bnexhibits a\nsmall asymmetry with \u0006Bz, leading to an asymmetry of\nthe two satellite peak positions, as shown in Fig. 4(e).\nFrom the data in Fig. 4(d) one sees that Bndepends on\ny. Therefore B=\u0000Bnis not ful\flled over the entire dis-\ntance between injection and detection contacts, leading\nto a reduced height of the satellite peaks. In Fig. 4(e),\nthe red line is a linecut through the data in Fig. 4(b) at\nBx= 2:5 mT, whereas for the data of the black line, Bn\nis uniformly \fxed to the saturation value predicted by\nEq. 1 with fbnS0= 45 mT. In the latter case, the satel-\nlite peaks reach the full height because at B=\u0000Bn,\nthe total \feld disappears everywhere in the sample. The\ndecrease of the satellite peak height is signi\fcantly un-\nderestimated in the simulation, compare with Fig. 3(a).\nThis indicates that Bnmight even be more inhomoge-\nneous in the sample than estimated with the Monte Carlo\nsimulation.\nThe measured size of fbnS0=45 mT allows a lower es-\ntimate of the injected spin polarization. The value bn=\n\u00005:3 T known from literature (Ref. 9) limits S0to about\n1% forf= 1. To obtain a rough estimate of the polariza-\ntion of the injected current from this, we have to account\nfor the ratio of injected electrons to the 5 \u00021016cm\u00003\nelectrons that are already in the sample. Within a spin\nlifetime\u001cs,Iinj\u001cs=e= 1:3\u0002106electrons are injected and\ndi\u000buse into a volume 60 \u0016m\u0002(6\u0016m+p\n(D\u001cs))\u00021\u0016m,\ncorresponding to a density of 2.1 \u00021015cm\u00003, i.e. the in-\njected spins make up about 1 =24 of the electron density.\nAccordingly, the spin polarization of injected electrons is\nat least 20% for f= 1,Iinj=\u000030\u0016A andT= 25 K.\nIn conclusion, we have found that the non-local volt-\nageUnlin an all-electrical spin injection and detection\ndevice exhibits distinct signatures of dynamically polar-\nized nuclear spins that can be used to measure the Over-\nhauser e\u000bective magnetic \feld Bnand to study nuclear\nspin dynamics. We obtained a quantitative understand-\ning of the depolarization peak in an in-plane magnetic\n\feld sweep. Because the peak height sensitively depends\non small stray \felds on the order of 0.1 mT and because of\nnuclear dipole-dipole interaction, a quantitative relation\nbetween the shape/size of the peak and Bnis di\u000ecult6\nto obtain. However, a quantitative measurement of Bn\nis achieved by observing the satellite peaks that occur\nin a Hanle measurement when B+Bn= 0. By com-\nparison with a self-consistent simulation of spin di\u000busion\nand hyper\fne interaction, we obtain a value for bnfhSxi\nof 45 mT at 25 K and Iinj=\u000030\u0016A, from which the sign\nof injected spin polarization can be determined and its\nmagnitude estimated. We can explain our data using\na leakage factor fthat does not depend on the exter-\nnal magnetic \feld. The observed nuclear spin signatures\nenable the study of nuclear spin dynamics including nu-clear spin resonance in small semiconductor/ferromagnet\nstructures by a transport measurement. Of speci\fc in-\nterest is to extend this method to investigate hyper\fne\ninteraction in other semiconductor materials like silicon26\nor graphene27where the spins can not easily be accessed\noptically.\nWe acknowledge fruitful discussions with Rolf Al-\nlenspach, Reto Schlittler and Leo Gross, and techni-\ncal support from Meinrad Tschudy, Daniele Caimi, Ute\nDrechsler and Martin Witzig.\n1D. J. Reilly, J. M. Taylor, J. R. Petta, C. M. Marcus, M. P.\nHanson, and A. C. Gossard, Science 321, 817 (2008).\n2F. Meier and B. P. Zakharchenya, Optical Orientation (El-\nsevier, Amsterdam, 1984).\n3Y. Ohno, D. K. Young, B. Beschoten, F. Matsukura,\nH. Ohno, and D. D. Awschalom, Nature 402, 790 (1999).\n4R. Fiederling, M. Keim, G. Reuscher, W. Ossau,\nG. Schmidt, A. Waag, and L. W. Molenkamp, Nature 402,\n787 (1999).\n5H. J. Zhu, M. Ramsteiner, H. Kostial, M. Wassermeier,\nH.-P. Sch onherr, and K. H. Ploog, Phys. Rev. Lett. 87,\n016601 (2001).\n6A. T. Hanbicki, B. T. Jonker, G. Itskos, G. Kioseoglou,\nand A. Petrou, Appl. Phys. Lett. 80, 1240 (2002).\n7G. Lampel, Phys. Rev. Lett. 20, 491 (1968).\n8J. Strand, B. D. Schultz, A. F. Isakovic, C. J. Palmstr\u001cm,\nand P. A. Crowell, Phys. Rev. Lett. 91, 036602 (2003).\n9D. Paget, G. Lampel, B. Sapoval, and V. I. Safarov, Phys.\nRev. B 15, 5780 (1977).\n10K. Ono and S. Tarucha, Phys. Rev. Lett. 92, 256803\n(2004).\n11J. R. Petta, J. M. Taylor, A. C. Johnson, A. Yacoby, M. D.\nLukin, C. M. Marcus, M. P. Hanson, and A. C. Gossard,\nPhys. Rev. Lett. 100, 067601 (2008).\n12S. E. Barrett, R. Tycko, L. N. Pfei\u000ber, and K. W. West,\nPhys. Rev. Lett. 72, 1368 (1994).\n13J. A. Marohn, P. J. Carson, J. Y. Hwang, M. A. Miller,\nD. N. Shykind, and D. P. Weitekamp, Phys. Rev. Lett. 75,\n1364 (1995).\n14G. Salis, D. D. Awschalom, Y. Ohno, and H. Ohno, Phys.\nRev. B 64, 195304 (2001).\n15D. Gammon, A. L. Efros, T. A. Kennedy, M. Rosen, D. S.Katzer, D. Park, S. W. Brown, V. L. Korenev, and I. A.\nMerkulov, Phys. Rev. Lett. 86, 5176 (2001).\n16X. Lou, C. Adelmann, S. A. Crooker, E. S. Garlid,\nJ. Zhang, K. S. M. Reddy, S. D. Flexner, C. J. Palmstr\u001cm,\nand P. A. Crowell, Nat Phys 3, 197 (2007).\n17M. Ciorga, A. Einwanger, U. Wurstbauer, D. Schuh,\nW. Wegscheider, and D. Weiss, Phys. Rev. B 79, 165321\n(2009).\n18O. M. J. van 't Erve, A. T. Hanbicki, M. Holub, C. H. Li,\nC. Awo-A\u000bouda, P. E. Thompson, and B. T. Jonker, Appl.\nPhys. Lett. 91, 212109(2007).\n19C. Awo-A\u000bouda, O. M. J. van 't Erve, G. Kioseoglou, A. T.\nHanbicki, M. Holub, C. H. Li, and B. T. Jonker, Appl.\nPhys. Lett. 94, 102511 (2009).\n20F. J. Jedema, A. T. Filip, and B. J. van Wees, Nature 410,\n345 (2001).\n21J. Lu, M. J. R. Hoch, P. L. Kuhns, W. G. Moulton, Z. Gan,\nand A. P. Reyes, Phys. Rev. B 74, 125208 (2006).\n22P. Van Dorpe, W. Van Roy, J. De Boeck, and G. Borghs,\nPhys. Rev. B 72, 035315 (2005).\n23S. A. Crooker, M. Furis, X. Lou, C. Adelmann, D. L.\nSmith, C. J. Palmstr\u001cm, and P. A. Crowell, Science 309,\n2191 (2005).\n24W. Farah, M. Dyakonov, D. Scalbert, and W. Knap, Phys.\nRev. B 57, 4713 (1998).\n25D. Paget, Phys. Rev. B 25, 4444 (1982).\n26I. Appelbaum, B. Huang, and D. J. Monsma, Nature 447,\n295 (2007).\n27N. Tombros, C. Jozsa, M. Popinciuc, H. T. Jonkman, and\nB. J. van Wees, Nature 448, 571 (2007)."    },    {        "title": "1806.00766v2.Quantum_Spin_Dynamics_in_a_Normal_Bose_Gas_with_Spin_orbit_Coupling.pdf",        "content": "arXiv:1806.00766v2  [cond-mat.quant-gas]  30 Jul 2018Quantum Spin Dynamics in a Normal Bose Gas with Spin-orbit Co upling\nWai Ho Tang1and Shizhong Zhang1,∗\n1Department of Physics and Center of Theoretical and Computa tional Physics,\nThe University of Hong Kong, Hong Kong, China\n(Dated: July 1, 2021)\nIn this Letter, we investigate spin dynamics of a two-compon ent Bose gas with spin-orbit coupling\nrealised in cold atom experiments. We derive coupled hydrod ynamic equations for number and spin\ndensities as well as their associated currents. Specialisi ng to quasi-one-dimensional situation, we\nobtain analytic solutions of the spin helix structure and it s dynamics in both adiabatic and diabatic\nregimes. In the adiabatic regime, the transverse spin decay s parabolically in the short-time limit and\nexponentially in the long time limit, depending on initial p olarisation. In contrast, in the diabatic\nregime, transverse spin density and current oscillate in a w ay similar to the charge-current oscillation\nin an undamped LC circuit. The effects of Rabi coupling on the s hort-time spin dynamics is also\ndiscussed. Finally, using realistic experimental paramet ers for87Rb, we show that the time scales\nfor spin dynamics is of order of milliseconds to a few seconds and can be observed experimentally.\nIntroduction. It has long been recognised that collec-\ntive spin dynamics of quantum mechanical origin can ex-\nist in a dilute gas at temperature T/greaterorsimilarTd, whereTdis\nthe degeneracy temperature. It arises due to indistin-\nguishability of identical atoms in binary scattering and\nis known as the identical spin rotation effect (ISRE) [ 1–\n3]. This effect is operative for both bosons [ 4–6] and\nfermions [ 7,8] and has led to the observations of spin\nwaves and anomalous spin segregation for weakly inter-\nacting bosons [ 9] and fermions [ 10]. Similar effects also\noccurs in a degenerate Fermi liquid like3He where it\nleads to anomalous spin diffusion known as the Leggett-\nRice effect [ 11,12]. Recently, Leggett-Rice effect has also\nbeen observed in unitary Fermi gas in both two [ 13] and\nthree dimensions [ 14,15].\nThe ISRE effects explored so far are limited to sys-\ntems with spin-SU(2) symmetry where the total spin\nis a good quantum number and its dynamics decouples\nfrom that of the density [ 4–8]. In this Letter, we in-\nvestigate the spin dynamics of a normal Bose gas with\nspin-orbit coupling (SOC) that was recently realized in\ncold atom experiments [ 16–29]. The coupling between\nspin and orbit degrees of freedom breaks the SU(2) sym-\nmetry and leads to more intricate dynamics that has no\nanalog in usual dilute gases discussed above. In particu-\nlar, we show how the long-wavelength and low-frequency\nhydrodynamic equations are modified in the presence of\nSOC, and how it leads to the appearance of persistent\nspin helical (PSH) structure. The decay of the spin heli-\ncal structure is discussed in both adiabatic and diabatic\nlimits. The general equations we obtain should serve as\nthe starting point for investigating spin dynamics in a\nspin-orbit coupled Bose gas such as spin waves and their\nattenuations. Spin dynamics for Fermi gas with spin-\norbital coupling has been discussed in Refs.[ 30–37].\nGeneral Setup. For definiteness, let us consider a gas\nof bosonic atoms87Rb of mass mwith two hyperfine-\nZeeman sub-levels |F,mF/angb∇acket∇ight ≡ |1,0/angb∇acket∇ight ≡ |↑/angb∇acket∇ightand|1,−1/angb∇acket∇ight ≡\n|↓/angb∇acket∇ightthat are coupled by a pair of Raman lasers with mo-mentum transfer q=qˆxalong the ˆx-direction. We set\nthe two-photon detuning to be zero for simplicity in the\nfollowing discussion. The harmonic trapping potential\nV(r), independent of spin, is assumed to be strong in the\nˆy- andˆz-directions but weak in the ˆx-direction and the\nsystem can be considered quasi-one-dimensional. The s-\nwave interaction is almost SU(2) invariant for87Rb and\nisgivenbyasinglecouplingconstant g. TheHamiltonian\ncan be written as ˆH=/integraltextd3r/summationtext\nµ,ν=↑,↓ψ†\nµ(r)Hµνψν(r)+\n1\n2g/integraltext\nd3r: ˆn(r)ˆn(r): whereHµνis given by\nHµν=/bracketleftbigg−/planckover2pi12∇2\n2m+V(r)/bracketrightbigg\nδµν−i/planckover2pi1q\nmσz\nµν∂x+/planckover2pi1ΩR\n2σx\nµν.\n(1)\nˆψµ(r) (ˆψ†\nµ(r)) is the annihilation (creation) operator\nfor boson with spin µat position r. ΩRis the two-\nphoton Rabi coupling. The number and spin densities\nare then given by ˆ n(r) =/summationtext\nµˆψ†\nµ(r)ˆψµ(r) and ˆsi(r) =\n1\n2/summationtext\nµ,νˆψ†\nµ(r)σi\nµνˆψν(r), respectively. σiare the Pauli ma-\ntrices. In what follows, we use arrow on top of an oper-\nator to indicate that it is a vector in spin space while\nboldface ˆx,ˆy,ˆzdescribes the spatial direction. Proper-\nties of condensation described by ˆHhave been discussed\nextensively in the literature, including its phase diagram\nand collective excitations [ 38–50] as well as spin dynam-\nics [51].\nTransport Equations. We first derive the continuity\nequations for number and spin densities and also identify\nthe modifications to the associated number and spin cur-\nrents due to spin-orbit coupling. We restrict ourselves\nto transport along the ˆx-direction. Using Heisenberg’s\nequation of motion i/planckover2pi1/angb∇acketleft∂tˆA/angb∇acket∇ight=/angb∇acketleft[ˆA,ˆH]/angb∇acket∇ightwithˆAbeing the\nnumber ˆn(r) and spin ˆ/vector s(r) densities, we find immediately\n∂tn+∂xj0= 0 (2)\nwhere the number current along ˆx-directionj0=/angb∇acketleftˆj0/angb∇acket∇ight\nwithˆj0=−i/planckover2pi1/(2m)/summationtext\nµ(ˆψ†\nµ∂xˆψµ−∂xˆψ†\nµˆψµ)+(2q/m)ˆsz.\nWe note that due to SOC, the number current is coupled2\nto the ˆz-component of the spin density. This redefinition\nis recently found to cause the violation of irrotationality\nof velocity field in spin-orbit couple condensate and the\nreduction of the quantum of circulation [ 49].\nIn the presence of spin-orbit coupling, the total spin\nis no longer conserved and the definition of spin current\noperatorˆ/vectorjisnotentirelyobvious. Inourcase,weidentify\nthe spin current by grouping all the gradient term in the\ncontinuity equation for spin density\n∂t/vector s+∂x/vectorj= ΩRˆx×/vector s+(2q//planckover2pi1)ˆz×/vectorj (3)\nwhere the spin current operator is given byˆ/vectorj=\n−i/planckover2pi1/(4m)/summationtext\nµ,νˆψ†\nµ/vector σµν∂xˆψν+H.c.+ˆnq/(2m)ˆz. We note\ntwo important modifications due to SOC. Firstly, the\nspin-current is now coupled to the total density of the\nsystem. Secondly, apart from the usual spin precessing\nterm due to Rabi coupling, there is an additional precess-\ning term, proportional to the strength of SOC, of spin\ncurrent along ˆ z-direction in Eq.( 3). We note that the\nmodified definition of spin current operatorˆ/vectorjcan also be\nmotivated from semiclassical considerations. Let the dis-\ntribution function (a matrix in spin space) be given by\nˆf(r,p,t), then one can define the spin current as\n/vectorj(r,t) =1\n2Tr/integraldisplayd3p\n(2π/planckover2pi1)3ˆf(r,p,t)1\n2/parenleftbigg\n/vector σ∂H\n∂px+∂H\n∂px/vector σ/parenrightbigg\n.\n(4)\nThe symmetrization is necessary because of non-\ncommutivity of /vector σand∂H/∂p x. Since∂H/∂p x=\npx/m+ (q/m)ˆσz. The first term px/mcorresponds to\nthe standard spin-current operator, while the second\nterm (q/m)ˆσzonly modifies the ˆ z-component ofthe spin-\ncurrent by an additional term ˆ nq/(2m).\nUsing the operator forms of the number and spin cur-\nrents, it is now straightforward to obtain their equations\nof motions, which are much more complicated because\nof the involvement of the momentum flux tensors. How-\never, in the normal state above the degeneracy tempera-\nture, the momentum flux tensors can be simplified using\nBoltzmann distribution (recall T/greaterorsimilarTd) and gradient ex-\npansion (for detailed derivation, see Supplemental Mate-\nrial [52]). As a result, we obtain\n∂tj0+kBT\nm∂xn=2q\nmΩRsy−g\n2m∂x/parenleftbigg3\n4n2+/vector s2/parenrightbigg\n(5)\n∂t/vectorj+α∂x/vector s=/parenleftBig\nΩRˆx+g\n/planckover2pi1/vector s/parenrightBig\n×/vectorj+2qα\n/planckover2pi1ˆz×/vector s\n+qnΩR\n2mˆy−3g\n4m(∂xn)/vector s−γ/vectorj ,(6)\nwhereα=kBT/m+ng/(4m). A phenomenological spin\ncurrent relaxation term −γ/vectorjis added to Eq.( 6). In the\nabsence of the spin-orbit coupling (Ω R= 0 andq= 0),\nEqs.(3,6) reduce to the standard Leggett-Rice form for a\ndegenerateFermiliquid[ 11,12]. Itisnoteworthythatthespin gradient term ng/(4m)∂x/vector sin Eq.(6) is usually omit-\nted in comparison to the Leggett-Rice term ( g//planckover2pi1)/vector s×/vectorj\nwhen the spatial variation of /vector sis small. In the presence\nof SOC, however, it has to be retained because the nat-\nural scale of variation for /vector sis set by the spin-orbit scale\nqwhich can be quite large. In addition, due to the fast\ntemporal variation of spin density, it is necessary to go\nbeyond the adiabatic approximation |∂t/vector s//vector s|/lessorsimilarγusually\nassumed in literature and discuss the dynamics in the\ndiabatic regime as well.\nEquations ( 2,3,5,6) form the basic equations for the\nspindynamicsofaSOCbosonabovethedegeneracytem-\nperature. In following, we first discuss the limit when the\neffects of Rabi coupling Ω Ris small, or what is equiva-\nlent, for time t≪1/ΩR, and discuss the existence of\npersistent spin helix (PSH) at wave vector k= 2q(here-\nafter/planckover2pi1= 1) and its decay when kdeviates from 2 q. The\neffects of Rabi term on PSH will be discussed in the end\nof the Letter.\nPersistent spin helical structure. The full set of equa-\ntions allow an exact solution corresponding to persis-\ntent spin helix with uniform density n=n0, spin den-\nsitysz=sz,0and/vector s2≡/vector s·/vector sthat are independent of\ntime. If we write the transverse spin /vector s⊥=sxˆx+syˆyin\nterms ofs±=sx±isy, and likewise for the spin cur-\nrents/vectorj(x,t) =/vectorj⊥(x,t) +jz(t)ˆz. Then for the spin heli-\ncal structure with definite wave number k, we can write\ns±(x,t) =e±ikx˜s±(t) and similarly j±(x,t) =e±ikx˜j±(t)\nand obtain the following set of equations\n∂t˜s+=−i(k−2q)˜j+, (7)\n∂t˜j+= (iλsz,0−γ)˜j+−i[α(k−2q)+λjz]˜s+,(8)\n∂tjz=λIm[˜s−˜j+]−γjz, (9)\nwhereλ=g//planckover2pi1andImdenotestheimaginarypart. When\nk= 2q, the transversespin ˜ s+is independent of time and\ncorresponds to a static spin helical structure in which\nspin density rotates about ˆ z-axis with wave vector 2 qin\ntheˆxdirection,\n/vector spsh=s⊥,0cos(2qx)ˆx+s⊥,0sin(2qx)ˆy+sz,0ˆz .(10)\nIn semiconductor heterostructure, it is understood that\nthe persistent spin helix is due to an emergent SU(2)\nsymmetry in the presence of spin-orbit coupling [ 53–55].\nIn the long time limit t≫1/γ, it is easy to see that both\njzand˜j+decays to zero, according to Eqs.( 8,9).\nVicinity of PSH. In the following, we investigate the\ndynamics of spin helical structure when its wave vec-\ntor deviates away from 2 q, described by the parameter\nε≡k/(2q)−1. Here it is important to distinguish two\nregimes. In the adiabatic regime where the spin currents\ncan relax much faster than the spin densities and can\nthus follow adiabatically the time evolution of spin den-\nsity,|∂t/vector s//vector s|/lessorsimilarγ, we can set ∂t˜j+=∂tjz= 0 in Eqs.( 8,9)\nin the steady state. Writing ˜ s+(t)≡s⊥(t)exp[iθ(t)], we3\nobtain the following set of equations\n(γ2+λ2s2\nz,0)lns⊥(t)\ns⊥,0+λ2\n2/bracketleftbig\ns2\n⊥(t)−s2\n⊥,0/bracketrightbig\n=−αγ(k−2q)2t ,(11)\nθ(t) =λsz,0\nγln/bracketleftbiggs⊥(t)\ns⊥,0/bracketrightbigg\n, (12)\njz=−λs2\n⊥α(k−2q)\nγ2+λ2(s2\n⊥+s2\nz,0), (13)\n˜j+= ˜s+α(k−2q)(λsz,0−iγ)\nγ2+λ2(s2\n⊥+s2\nz,0), (14)\nwheres⊥,0=s⊥(t= 0). Substitution of k= 2qre-\ncovers the previous solution of PSH. When k/negationslash= 2q, the\ntransverse spin magnitude decays according to Eq.( 11).\nDepending on the relative magnitude of s⊥andsz,0, one\ncan distinguish two qualitatively different behaviours.\n(i) When |s⊥,0| ≥ |s⊥(t)| ≫ |sz,0|, namely, when spins\nare polarized close to the xy-plane, the first term on the\nleft of Eq.( 11) is negligible, hence the transverse spin\nmagnitude decays parabolically in the short time limit\nt≪τpara,\ns⊥(t)≈s⊥,0/radicalBigg\n1−t\nτpara, τpara=λ2s2\n⊥,0\n2αγ(k−2q)2,(15)\nwhere the time constant τparadepends quadratically on\nthe interaction parameter λand inversely on the spin\ncurrent relaxation rate γ. As expected, it diverges when\nk= 2q.\n(ii) In the long time limit t≫τparawhen|s⊥(t)| ≪ |sz,0|,\nthe decay becomes exponential in the adiabatic regime\nwith a different time constant τexp\ns⊥(t)≈s⊥,0e−t/τexp, τexp=γ2+λ2s2\nz,0\nαγ(k−2q)2.(16)\nFig.1shows the excellent agreement between above ana-\nlytic formulae and numerical results. We note that the\ndynamicalequation( 11)is similarin formtotheLeggett-\nRice equation derived for a degenerate Fermi liquid [ 12],\nexcept for the explicit appearance of spin-orbit coupling\nterm on the right hand side of Eq.( 11). We emphasize\nthat Eqs.( 15,16) apply so long as 1 /ΩR≫τpara,τexp\neven in the presence of a small Rabi coupling.\nDiabatic Regime. When the wave vector kof the\nspin helix deviates significantly away from the PSH wave\nvector 2q, the adiabatic condition fails. In the dia-\nbatic regime when |∂t/vector s//vector s| ≫γ, we can neglect λ|jz|\nin Eq.(8), and as a result ˜ s+(t),˜j+(t) form a closed dy-\nnamical system in Eqs.( 7,8). With the initial conditionFIG. 1. (Colour online) Short- and long-time behaviours of\nthe transverse spin in the adiabatic regime. Short-time dec ay\nfor initial spin density polarised close to the xy-plane. The\ndecay is parabolic (left panel). Long-time decay for initia l\nspin density polarised close to the z-axis. The decay is expo-\nnential (right panel). Numerical simulations of the full se t of\nEqs. (2,3,5,6) (black solid) agree very well with the analyti-\ncal results (red dashed) given by Eq.( 11). Blue dashed lines\nshows the asymptotic results, Eqs.( 15,16). The deviation\nat the tail of left panel between the simulation and analytic\nequation( 11)indicates thefailure ofadiabatic approximation.\n(˜s+,˜j+,jz)t=0= (s⊥,0,0,jz,0), one obtains [ 52]\n˜s+(t) =s⊥,0e−γt\n2+iλsz,0\n2t/bracketleftbigg\ncosΓt+γ−iλsz,0\n2ΓsinΓt/bracketrightbigg\n,\n(17)\n˜j+(t) =i√αs⊥,0e−γt\n2+iλsz,0\n2tsin(Γt)sign(2q−k),(18)\njz(t) =jz,0e−γt−e−γtλs2\n⊥,0\n4(k−2q)\n×/bracketleftbigg\n1+γt−cos(2Γt)−γ\n2sin(2Γt)\nΓ/bracketrightbigg\n,(19)\nwhere Γ =√α|k−2q|. In obtaining the above simplified\nexpressionswehaveassumedthat thepolarizationofspin\nare close to the xy-plane and as a result Γ ≫γ,λsz,0.\nFIG. 2. (Colour online) Left panel shows time dependence of\ny-component of spin density for spin density polarized along\nˆx-direction at t= 0. Numerical result (black dashed) agrees\nvery well with the analytical result (red solid), Eq.( 17). Right\npanel is thetrajectory of transverse spin componentin the xy-\nplane. Also indicated in the graph are the fast oscillations of\nthe magnitude of transverse spin (Γ) and its slow rotations\nwith rate λsz,0/2.\nThe dynamics of the transversecomponents consists of4\nthree parts: fast oscillation in magnitude with frequency\nΓ, slowprecessingoftheaxisofoscillationwith frequency\nλsz,0/2 and the damping of oscillation amplitude with\nthe rate of γ/2, as shown in Fig. 2. If one neglects the\nsmall correction of the sine function in Eq.( 17), then\nthere is an exact π/2 phase difference between the os-\ncillations of ˜ s+(t) and˜j+(t), similar to the un-dampened\nLC circuit, in contrast to the over-damped case where ˜j+\nfollows adiabatically the dynamics of ˜ s+.\nThe region of adiabaticity for various initial polariza-\ntionssz,0ands⊥,0are determined (approximately) by\nthe condition |∂t/vector s//vector s| ∼γ. It is shown in Fig. 3that\nclose to PSH, the adiabatic region prevails for most of\nthe parameter regime except when the spin polarisation\nis small. On the other hand, as one moves away from\nPSH, the region of non-adiabatic evolution grows much\nlarger. Starting from an arbitrary initial conditions, the\nspindynamicsmighttraversebothadiabaticanddiabatic\nregimes and becomes much richer. In particular, close to\nthe boundaries, it is necessary to deal with the full set of\nhydrodynamicequations( 2,3,5,6) that wederived before.\nFIG. 3. (Colour online) The approximate demarcation of adi-\nabatic from dabaticregions based onthecondition |∂t/vector s//vector s| ∼γ\n(redshadedlines). The region ofadiabaticity becomes smal ler\nwhen the wave number of spin helix deviates away from 2 q.\nQuenching of Rabi coupling on PSH. Intheaboveanal-\nysis, wehaveassumedthat the Rabi couplingis weakand\ncan be neglected. Inclusion of Rabi term results in com-\nplex dynamics, for example an extra precession of spin\ndensityandspincurrentdensityinEqs.( 3,6). In theequi-\nlibrium state, due to the breaking of the emergent SU(2)\nsymmetry, PSHis nolongerstableand decayswith arate\nthat is determined by Ω R.\nConsidering the situation that the Rabi coupling is\nturned on suddenly at t= 0 and remains fixed, all spin\nhelices except the PSH with k= 2qvanish long before\nt= 0. In the following the short-time effect of the Rabi\ncouplingonthePSHwillbestudied. Wecanseparatethe\ndensities and current densities into two parts, one from\nthe PSH while the other from the leading correction dueto Rabi coupling which vanishes for t≤0,\nn(x,t) =n0+δn(x,t), (20)\n/vector s(x,t) =/vector spsh+δ/vector s(x,t), (21)\nj0(x,t) = 0+δj0(x,t), (22)\n/vectorj(x,t) = 0+δ/vectorj(x,t). (23)\nSubstituting the above expressions into the transport\nequations Eqs.( 2,3,5,6) and only keeping terms linear in\nsmall derivations and the Rabi coupling, we are led to an\ninhomogeneous diffusion equation of the form\n∂t/vectorδV(x,t) =ˆH(x,∂x)/vectorδV(x,t)+/vector g(x),(24)\nwith/vectorδV= (δn,δj0,δsz,δjz,δsx,δjx,δsy,δjy)Tand\n/vector g(x) = ΩR(0,2qspsh,y/m,spsh,y,0,0,0,−sz,0,qn0/(2m))T.\nThe explicit form of ˆH(x,∂x) is given in Supplemental\nMaterial [ 52] and the solution is given by\n/vectorδV(x,t) =ˆH−1/bracketleftBig\nexp(ˆHt)−ˆI/bracketrightBig\n/vector g(x).(25)\nTo characterize the decay of transverse component of\nspin density due to Rabi coupling, we define a quantity\nthat measures the amplitude of the spin helical structure\nR(t) =1\n|/vector s(x,0)|(π/q)/integraldisplayπ\nq\n0Re[s+(x,t)e−i2qx]dx ,(26)\nwhere|/vector s(x,0)|=/radicalBig\ns2\n⊥,0+s2\nz,0is the initial spin mag-\nnitude. For t >0, Rabi coupling destroys the helical\nstructure and results in decay of R(t). The short-time\nbehaviour is described by the leading terms of the series\nEq.(25). When Ω Rt≪1,\nR(t)≈s⊥,0/radicalBig\ns2\n⊥,0+s2\nz,0/bracketleftbigg\n1−(ΩRt)2\n4/bracketrightbigg\n+O(t4).(27)\nExperimental Considerations. For87Rb used in spin-\norbit coupling experiment [ 16], the typical density is\naboutn0= 2×1013cm−3. For our calculation, we\nassumeT= 700nK, well above the typical condensa-\ntion temperature. The Raman laser defines a scale of\nwave number kL= (√\n2π/804.1nm)/10, chosen to be 10\ntimes smaller than that in Ref.[ 16] to make the spin he-\nlical structure more visible. In the numerical calcula-\ntions presented, we chose q= 0.5/planckover2pi1kLand Rabi coupling\n/planckover2pi1ΩR= 0.5/planckover2pi12k2\nL/(2m), appropriate to experimental situ-\nations. We assume the system is initially polarized with\n|/vector s0|=smax=n0/2. The intrinsic spin current relaxation\nrateγis chosen to be approximately 20Hz, appropriate\nto87Rb [4,9]. Numerical simulations show that the time\nscale for spin dynamics in the adiabatic regime is of or-\nder of seconds, while it is of order of milliseconds in the\ndiabatic regime, and can be observed experimentally. To\ninitialize the system in a particular spin helical state, one5\ncan start with Rb atoms in the |↓/angb∇acket∇ightstate with no Raman\nlasers and apply a radio-frequency pulse to achieve a de-\nsiredszpolarization. Afterwards, a small magnetic field\nwith linear gradient ∆ Bcan be applied to create the\nspin helical structure with wave vector k. To investigate\nthe stability of spin helical structure, one can now turn\non the Raman fields which create the spin-orbit coupling\nwith strength q, and measure the evolution of transverse\nspin component.\nThisworkissupportedbyHongKongResearchGrants\nCouncil, GRF HKU 17305217, CRF C6026-16W, and\nthe Croucher Foundation under the Croucher Innovation\nAward.\n∗shizhong@hku.hk\n[1] C. Lhuillier and F. Lalo¨ e, J. Phys. France 43, 197 (1982);\nJ. Phys. France 43, 225 (1982).\n[2] B. R. Johnson, J. S. Denker, N. Bigelow, L. P. L´ evy, J. H.\nFreed, and D. M. Lee, Phys. Rev. Lett. 52, 1508 (1984).\n[3] L. P. L´ evy and A. E. Ruckenstein, Phys. Rev. Lett. 52,\n1512 (1984).\n[4] M.¨O. Oktel and L. S. Levitov, Phys. Rev. Lett. 88,\n230403 (2002).\n[5] J. N. Fuchs, D. M. Gangardt, and F. Lalo¨ e, Phys. Rev.\nLett.88, 230404 (2002).\n[6] J. E. Williams, T. Nikuni, and C. W. Clark, Phys. Rev.\nLett.88, 230405 (2002).\n[7] S. S. Natu and E. J. Mueller, Phys. Rev. A 79, 051601\n(2009).\n[8] F. Pi´ echon, J. N. Fuchs, and F. Lalo¨ e, Phys. Rev. Lett.\n102, 215301 (2009).\n[9] H. J. Lewandowski, D. M. Harber, D. L. Whitaker, and\nE. A. Cornell, Phys. Rev. Lett. 88, 070403 (2002).\n[10] X. Du, L. Luo, B. Clancy, and J. E. Thomas, Phys. Rev.\nLett.101, 150401 (2008).\n[11] A. J. Leggett and M. J. Rice, Phys. Rev. Lett. 20, 586\n(1968).\n[12] A. J. Leggett, J. Phys. C: Solid State Phys. 3, 448 (1970).\n[13] M. Koschorreck, D. Pertot, E. Vogt, and M. Kohl, Nat.\nPhys.9, 405 (2013).\n[14] A. B. Bardon, S. Beattie, C. Luciuk, W. Cairncross, D.\nFine, N. S. Cheng, G. J. A. Edge, E. Taylor, S. Zhang, S.\nTrotzky, and J. H. Thywissen, Science 344, 722 (2014).\n[15] S.Trotzky,S.Beattie, C. Luciuk, S.Smale, A.B.Bardon ,\nT. Enss, E. Taylor, S. Zhang, and J. H. Thywissen, Phys.\nRev. Lett. 114, 015301 (2015).\n[16] Y.-J. Lin, K. Jim´ enez-Garc´ ıa, and I. B. Spielman, Nat ure\n471, 83 (2011).\n[17] P. Wang, Z.-Q. Yu, Z. Fu, J. Miao, L. Huang, S. Chai, H.\nZhai, and J. Zhang, Phys. Rev. Lett. 109, 095301 (2012).\n[18] L. W. Cheuk, A. T. Sommer, Z. Hadzibabic, T. Yefsah,\nW. S. Bakr, and M. W. Zwierlein, Phys. Rev. Lett. 109,\n095302 (2012).\n[19] J.-Y. Zhang, S.-C. Ji, Z. Chen, L. Zhang, Z.-D. Du, B.\nYan, G.-S. Pan, B. Zhao, Y.-J. Deng, H. Zhai, S. Chen,\nand J.-W. Pan, Phys. Rev. Lett. 109, 115301 (2012).\n[20] R. A. Williams, M. C. Beeler, L. J. LeBlanc, K. Jim´ enez-\nGarc´ ıa, and I. B. Spielman, Phys. Rev. Lett. 111, 095301(2013).\n[21] C. Qu, C. Hamner, M. Gong, C. Zhang, and P. Engels,\nPhys. Rev. A 88, 021604 (2013).\n[22] A. J. Olson, S.-J. Wang, R. J. Niffenegger, C.-H. Li, C.\nH. Greene, and Y. P. Chen, Phys. Rev. A 90, 013616\n(2014).\n[23] C. Hamner, C. Qu, Y. Zhang, J. Chang, M. Gong, C.\nZhang, and P. Engels, Nat. Commun. 5, 4023 (2014).\n[24] N. Q. Burdick, Y. Tang, and B. L. Lev, Phys. Rev. X 6,\n031022 (2016).\n[25] B. Song, C. He, S. Zhang, E. Hajiyev, W. Huang, X.-J.\nLiu, and G.-B. Jo, Phys. Rev. A 94, 061604 (2016).\n[26] L. Huang, Z. Meng, P. Wang, P. Peng, S.-L. Zhang, L.\nChen, D. Li, Q. Zhou, and J. Zhang, Nat. Phys. 12, 540\n(2016).\n[27] J. Li, W. Huang, B. Shteynas, S. Burchesky, F. C ¸. Top,\nE. Su, J. Lee, A. O. Jamison, and W. Ketterle, Phys.\nRev. Lett. 117, 185301 (2016).\n[28] Z. Meng, L. Huang, P. Peng, D. Li, L. Chen, Y. Xu, C.\nZhang, P. Wang, and J. Zhang, Phys. Rev. Lett. 117,\n235304 (2016).\n[29] Z. Wu, L. Zhang, W. Sun, X.-T. Xu, B.-Z. Wang, S.-C.\nJi, Y. Deng, S. Chen, X.-J. Liu, and J.-W. Pan, Science\n354, 83 (2016).\n[30] T. D. Stanescu, C. Zhang, and V. Galitski, Phys. Rev.\nLett.99, 110403 (2007).\n[31] I. V. Tokatly and E. Y. Sherman, Phys. Rev. A 87,\n041602 (2013).\n[32] M. C. L¨ uffe, J. Danon, and T. S. Nunner, Phys. Rev. B\n87, 125416 (2013).\n[33] S. S. Natu and S. Das Sarma, Phys. Rev. A 88, 033613\n(2013).\n[34] T. Yu and M. W. Wu, Phys. Rev. A 88, 043634 (2013).\n[35] J. Radi´ c, S. S. Natu, and V. Galitski, Phys. Rev. Lett.\n112, 095302 (2014).\n[36] T. Yu and M. W. Wu, Phys. Rev. A 92, 013607 (2015).\n[37] I. V. Tokatly and E. Y. Sherman, Phys. Rev. A 93,\n063635 (2016).\n[38] C. Wang, C. Gao, C.-M. Jian, and H. Zhai, Phys. Rev.\nLett.105, 160403 (2010).\n[39] T.-L. Ho and S. Zhang, Phys. Rev. Lett. 107, 150403\n(2011).\n[40] H. Zhai, Int. J. Mod. Phys. B 26, 1230001 (2012).\n[41] Y. Li, L. P. Pitaevskii, and S. Stringari, Phys. Rev. Let t.\n108, 225301 (2012).\n[42] Y. Li, G. I. Martone, L. P. Pitaevskii, and S. Stringari,\nPhys. Rev. Lett. 110, 235302 (2013).\n[43] S.-C. Ji, J.-Y. Zhang, L. Zhang, Z.-D. Du, W. Zheng,\nY.-J. Deng, H. Zhai, S. Chen, and J.-W. Pan, Nat. Phys.\n10, 314 (2014).\n[44] S.-C. Ji, L. Zhang, X.-T. Xu, Z. Wu, Y. Deng, S. Chen,\nand J.-W. Pan, Phys. Rev. Lett. 114, 105301 (2015).\n[45] H. Zhai, Rep. Prog. Phys. 78, 026001 (2015).\n[46] Y. Li, G. I. Martone, and S. Stringari, “Spin-orbit-\ncoupled Bose-Einstein condensates,” in Annual Review\nof Cold Atoms and Molecules (World Scientific, 2015)\nChap. 5, pp. 201-250.\n[47] Y.-H. Hou and Z. Yu, Sci. Rep. 5, 15307 (2015).\n[48] Y. Zhang, M. E. Mossman, T. Busch, P. Engels, and C.\nZhang, Front. Phys. 11, 118103 (2016).\n[49] S. Stringari, Phys. Rev. Lett. 118, 145302 (2017).\n[50] J. Hou, X.-W. Luo, K. Sun, T. Bersano, V. Gokhroo,\nS. Mossman, P. Engels, and C. Zhang, Phys. Rev. Lett.\n120, 120401 (2018).6\n[51] Ting Fung Jeffrey Poon and X.-J. Liu, Phys. Rev. A 93,\n063420 (2016).\n[52] See Supplemental Material for derivation of equations of\nmotion of densities and currents, time evolution of spin\nhelix in both adabatic and diabatic regimes, evolution\nmatrixˆHand solution of /vectorδV(x,t) under quenching of\nRabi coupling on PSH.[53] B. A. Bernevig, J. Orenstein, and S.-C. Zhang, Phys.\nRev. Lett. 97, 236601 (2006).\n[54] J. D. Koralek, C. P. Weber, J. Orenstein, B. A. Bernevig,\nS.-C. Zhang, S. Mack, and D. D. Awschalom, Nature\n458, 601 (2009).\n[55] J. Schliemann, Rev. Mod. Phys. 89, 011001 (2017).arXiv:1806.00766v2  [cond-mat.quant-gas]  30 Jul 2018Supplemental Material for\n“Persistent Spin Helix and its Instability in Spin-orbit Co upled Normal Bose Gas”\nWai Ho Tang and Shizhong Zhang\nDepartment of Physics and Center of Theoretical and Computa tional Physics,\nThe University of Hong Kong, Hong Kong, China\n(Dated: July 1, 2021)\nDERIVATION OF EQUATIONS OF MOTION\nDynamics of Densities\nGiven the Hamiltonian of system ˆH=/integraltext\nd3r/summationtext\nµ,νψ†\nµ(r)Hµνψν(r)+1\n2g/integraltext\nd3r: ˆn(r)ˆn(r): withµ,ν=↑,↓and\nHµν=/bracketleftbigg−/planckover2pi12∇2\n2m+V(r)/bracketrightbigg\nδµν−i/planckover2pi1q\nmσz\nµν∂x+/planckover2pi1ΩR\n2σx\nµν. (1)\nˆψµ(r) (ˆψ†\nµ(r)) is the annihilation (creation) operator of a boson with spin compon entµat position r.σiare the Pauli\nmatrices. The dynamics of number density ˆ n(r) =/summationtext\nµˆψ†\nµ(r)ˆψµ(r) and spin densities ˆ si(r) =1\n2/summationtext\nµ,νˆψ†\nµ(r)σi\nµνˆψν(r)\nis governed by the equations of motion derived from Heisenberg’s eq uationi/planckover2pi1/an}b∇acketle{t∂tˆA/an}b∇acket∇i}ht=/an}b∇acketle{t[ˆA,ˆH]/an}b∇acket∇i}ht, explicitly\n∂tn+∇·j0=−2\nmq·∇sz, (2)\n∂tsz+∇·jz= ΩRsy−1\n2mq·∇n . (3)\n∂tsx+∇·jx=−2\n/planckover2pi1q·jy, (4)\n∂tsy+∇·jy=−ΩRsz+2\n/planckover2pi1q·jx, (5)\nwith spin-orbit coupling q=qˆxand the help of Wick’s theorem:\n/an}b∇acketle{tˆψ†\nαˆψ†\nβˆψmˆψn/an}b∇acket∇i}ht=/an}b∇acketle{tˆψ†\nαˆψm/an}b∇acket∇i}ht/an}b∇acketle{tˆψ†\nβˆψn/an}b∇acket∇i}ht+/an}b∇acketle{tˆψ†\nαˆψn/an}b∇acket∇i}ht/an}b∇acketle{tˆψ†\nβˆψm/an}b∇acket∇i}ht+/an}b∇acketle{tˆψ†\nαˆψ†\nβ/an}b∇acket∇i}ht/an}b∇acketle{tˆψmˆψn/an}b∇acket∇i}ht, (6)\nthe last term called anomalous term is omitted in the present case of n ormal phase. The number current j0=/an}b∇acketle{tˆj0/an}b∇acket∇i}ht\nwithˆj0=−i/planckover2pi1/(2m)/summationtext\nµ(ˆψ†\nµ∇ˆψµ−∇ˆψ†\nµˆψµ) and spin currents ji=/an}b∇acketle{tˆji/an}b∇acket∇i}htwithˆji=−i/planckover2pi1/(4m)/summationtext\nµ,νˆψ†\nµσi\nµν∇ˆψν+H.c.are\ndefined in conventional manner, i.e. only taking kinetic term of Hamiltonian into account. The e quations of motion\nof these conventional currents are necessary to complete the s ystem’s dynamics.\nDynamics of Current Densities\nAfter straightforwardbut tedious calculation, equationsofmotio n ofconventionalnumberand spin currentdensities\nare obtained,\n∂tj0,α+∂β/an}b∇acketle{tˆΠ0\nαβ/an}b∇acket∇i}ht=−n\nm∂αV(r)−3g\n4m∂α(n2)−g\nm∂α(s2\nx+s2\ny+s2\nz)−2(q·∇)\nmjz,α, (7)\n∂tjz,α+∂β/an}b∇acketle{tˆΠz\nαβ/an}b∇acket∇i}ht=−sz\nm∂αV(r)+2g\n/planckover2pi1(sxjy,α−syjx,α)−3g\n2msz∂αn−g\n2mn∂αsz−(q·∇)\n2mj0,α+ΩRjy,α,(8)\n∂tjx,α+∂β/an}b∇acketle{tˆΠx\nαβ/an}b∇acket∇i}ht=−sx\nm∂��V(r)+2g\n/planckover2pi1(syjz,α−szjy,α)−3g\n2msx∂αn−g\n2mn∂αsx−2qβ/an}b∇acketle{tˆΠy\nαβ/an}b∇acket∇i}ht\n/planckover2pi1, (9)\n∂tjy,α+∂β/an}b∇acketle{tˆΠy\nαβ/an}b∇acket∇i}ht=−sy\nm∂αV(r)+2g\n/planckover2pi1(szjx,α−sxjz,α)−3g\n2msy∂αn−g\n2mn∂αsy+2qβ/an}b∇acketle{tˆΠx\nαβ/an}b∇acket∇i}ht\n/planckover2pi1−ΩRjz,α.(10)2\nRepeated index βmeans summation of β=x,y,z. The generalized momentum flux tensors ˆΠi\nαβare defined as\nˆΠ0\nαβ=/parenleftbigg/planckover2pi1\n2m/parenrightbigg2\n(∂αˆψ†\n↑∂βˆψ↑−ˆψ†\n↑∂α∂βˆψ↑+∂αˆψ†\n↓∂βˆψ↓−ˆψ†\n↓∂α∂βˆψ↓)+H.c. , (11)\nˆΠz\nαβ=1\n2/parenleftbigg/planckover2pi1\n2m/parenrightbigg2\n(∂αˆψ†\n↑∂βˆψ↑−ˆψ†\n↑∂α∂βˆψ↑−∂αˆψ†\n↓∂βˆψ↓+ˆψ†\n↓∂α∂βˆψ↓)+H.c. , (12)\nˆΠx\nαβ=1\n2/parenleftbigg/planckover2pi1\n2m/parenrightbigg2\n(∂αˆψ†\n↑∂βˆψ↓−ˆψ†\n↑∂α∂βˆψ↓+∂βˆψ†\n↑∂αˆψ↓−∂α∂βˆψ†\n↑ˆψ↓)+H.c. , (13)\nˆΠy\nαβ=1\n2i/parenleftbigg/planckover2pi1\n2m/parenrightbigg2\n(∂αˆψ†\n↑∂βˆψ↓−ˆψ†\n↑∂α∂βˆψ↓+∂βˆψ†\n↑∂αˆψ↓−∂α∂βˆψ†\n↑ˆψ↓)+H.c. . (14)\nIn contrast to Oktel’s work where the densities is nearly uniform ove r the time duration studied, spin helix having\nlarge wave number k∼2q//planckover2pi1implies that the gradient terms associated with exchange interactio n are comparable\nwith the Leggett-Rice terms, hence these gradient terms should b e retained in present studies.\nQuasi-one-dimensional limit\nThe calculation of ensemble average of momentum flux tensors are n ot trivial in presence of spin-orbit coupling.\nOnly the component ( αβ) = (xx) of the tensors are concerned as quasi-one-dimensional limit is tak en below. In\nthe case without Rabi coupling, the number and longitudinal spin den sities are conserved so their corresponding\ncurrent densities are well defined. These currents are modified by spin-orbit coupling as shown in main text. Their\nmodifications can also be obtained by performing gauge transforma tion, which eliminates the spin-orbit coupling at\nthe expense of introducing space-dependent spin quantisation ax is,\n˜ψ= exp(iqxσz)ψ . (15)\nTherefore the gradient expansion of momentum flux tensors /an}b∇acketle{tΠ0\nxx/an}b∇acket∇i}htand/an}b∇acketle{tΠz\nxx/an}b∇acket∇i}htcould be obtained by first carrying\nout the standard gradient expansion on energy momentum flux with out SOC ( ˜Π0\nxx,˜Πz\nxx) and then using the gauge\ntransformation to obtain that for /an}b∇acketle{tΠ0\nxx/an}b∇acket∇i}htand/an}b∇acketle{tΠz\nxx/an}b∇acket∇i}ht. As a result, we obtain,\n/an}b∇acketle{tΠ0\nxx/an}b∇acket∇i}ht=/parenleftbiggkBT\nm−q2\nm2/parenrightbigg\nn−4q\nmjz, (16)\n/an}b∇acketle{tΠz\nxx/an}b∇acket∇i}ht=/parenleftbiggkBT\nm−q2\nm2/parenrightbigg\nsz−q\nmj0. (17)\nHowever, the transverse spin components in general does not sa tisfy the equations of continuity owing to spin-orbit\ncoupling, as a result, the definition of the correspondingenergy-m omentum tensors is not uniquely fixed. In our study,\nwe recognise that the “streaming” term (group velocity) is not mod ified in the ˆ xand ˆy-directions and as a result, we\nshall apply the standard gradient expansion to their correspondin g energy momentum flux tensor,\n/an}b∇acketle{tΠx\nxx/an}b∇acket∇i}ht=kBT\nmsx, (18)\n/an}b∇acketle{tΠy\nxx/an}b∇acket∇i}ht=kBT\nmsy. (19)\nSince the external potential V(r) is uniform along ˆx-direction but has strong confinement along ˆy-ˆzplane, the\nsystem is in quasi-one-dimensional limit, in which n(x,y,z) is replaced by the central peak value n(x,0,0) (similarly\nfor other quantities), the Gaussian distribution profile along ˆy-ˆzplane is averaged out and results in replacement of\ninteraction strength g→g/2. After adopting redefinitions of currents defined in main text, on e have the equations of\nmotion in quasi-one-dimensional limit,\n∂tn+∂xj0= 0, (20)\n∂t/vector s+∂x/vectorj= ΩRˆx×/vector s+(2q//planckover2pi1)ˆz×/vectorj , (21)\n∂tj0+kBT\nm∂xn=2q\nmΩRsy−g\n2m∂x/parenleftbigg3\n4n2+/vector s2/parenrightbigg\n, (22)\n∂t/vectorj+α∂x/vector s=/parenleftBig\nΩRˆx+g\n/planckover2pi1/vector s/parenrightBig\n×/vectorj+2qα\n/planckover2pi1ˆz×/vector s+qnΩR\n2mˆy−3g\n4m(∂xn)/vector s−γ/vectorj , (23)3\nwhereα=kBT/m+ng/(4m), the relaxation of spin current −γ/vectorjis phenomenologically added. Vectorial quantities\nare in spin space.\nADIABATIC EVOLUTION OF SPIN HELIX\nHereafter /planckover2pi1= 1. The spin dynamics of spin helix simplifies and given by Eqs.(7-9) in main text, explicitly\n∂t˜s+=−i(k−2q)˜j+, (24)\n∂t˜j+= (iλsz,0−γ)˜j+−i[α(k−2q)+λjz]˜s+, (25)\n∂tjz=λIm[˜s−˜j+]−γjz. (26)\nFork/ne}ationslash= 2q, there is no persistent spin helix and the spin dynamics could be classi�� ed into two regimes. In this section\nthe adiabatic regime where the spin density varies slowly is discussed. The diabatic regime where the spin density is\nchanging fast will be the subject of the next section.\nThe dynamical equation of spin current is inhomogeneous so the spin current consists of a homogeneous solution\nand an inhomogeneous solution. The former dies out in time scale abou tγ−1and is unimportant, while the latter is\nwhat we desire. The spin current will acquire the inhomogeneous valu e in the duration ∆ t∼γ−1, which is obtained\nby setting∂t/vectorj= 0. The solution is valid only when every terms in the equation are slowly varying in the time interval\n∆t∼γ−1, i.e.|∂t/vector s//vector s|/lessorsimilarγ, the validity will be justified at the end.\nAdiabatic assumption ∂t˜j+=∂tjz= 0 implies\njz=λ\nγIm[˜s−˜j+], (27)\n˜j+=−i˜s+\nγ−iλsz,0[α(k−2q)+λjz]. (28)\nBy writing ˜ s±(t)≡s⊥(t)e±iθ(t),jzcould be solved by substituting Eq.( 28) into Eq.( 27), and hence ˜j+:\njz=−λs2\n⊥α(k−2q)\nγ2+λ2(s2\n⊥+s2\nz,0), (29)\n˜j+= ˜s+α(k−2q)(λsz,0−iγ)\nγ2+λ2(s2\n⊥+s2\nz,0). (30)\nAs∂t˜s+=eiθ[∂ts⊥+i(s⊥∂tθ)], separation of real and imaginary parts results in the rates of ch ange of the transverse\nspin magnitude s⊥(t) and phase θ(t),\n∂ts⊥=−s⊥γα(k−2q)2\nγ2+λ2(s2\n⊥+s2\nz,0), (31)\n∂tθ=λsz,0\nγ∂ts⊥\ns⊥. (32)\nDirect integration implies algebraic equations of s⊥(t) andθ(t),\n(γ2+λ2s2\nz,0)lns⊥(t)\ns⊥,0+λ2\n2/bracketleftbig\ns2\n⊥(t)−s2\n⊥,0/bracketrightbig\n=−α(k−2q)2γt , (33)\nθ(t) =λsz,0\nγlns⊥(t)\ns⊥,0. (34)\nAbove solution is valid only when it is consistent with the adiabatic assum ption,\n|∂t/vector s//vector s|=/radicalBigg\n(∂ts⊥)2+(s⊥∂tθ)2\ns2\n⊥+s2\nz,0/lessorsimilarγ . (35)4\nFor PSHk= 2qthis condition is always satisfied as the left hand side vanishes. For k= (1+ε)2q/ne}ationslash= 2qthe condition\nbecomes\n(k−2q)2=ε2(2q)2/lessorsimilarγ/radicalBig\nγ2+λ2s2\nz,0/radicalBig\ns2\n⊥+s2\nz,0\ns⊥γ2+λ2(s2\n⊥+s2\nz,0)\nα. (36)\nIn conclusion, adiabatic regimes is valid only in the vicinity of PSH, i.e. sma llε.\nDIABATIC EVOLUTION OF SPIN HELIX\nWe have seen that the adiabatic condition is only valid in the vicinity of PS H (say|ε|/lessorsimilar0.05). This precision is hard\nto control in experiments, it means we have to study the general r egime as well. As previous, we only have interest\nin the dynamics of spin helix, which is governed by Eqs.( 24-26). Attention should be paid to Eq.( 25), whenkis far\naway from 2 q, the time-dependent term jz(t) is dominated by the term with α, this neglect justified below benefits us\nthat the coefficients are time-independent and that the dynamics o f ˜s+and˜j+are closed. Once the transverse spin\nand current are known, the longitudinal current jz(t) can be solved by direct integration on Eq.( 26).\nThe equations of motion of spin helix can be rewritten as below,\n∂t/bracketleftbigg˜s+\n˜j+/bracketrightbigg\n=/bracketleftbigg0 −i(k−2q)\n−i[α(k−2q)+λjz]iλsz,0−γ/bracketrightbigg/bracketleftbigg˜s+\n˜j+/bracketrightbigg\n= Λ/bracketleftbigg˜s+\n˜j+/bracketrightbigg\n. (37)\nThe time dependence of jzresults in that of eigenvalues and eigenstates. We approximate jzby its equilibrium\nvalue,jz(t→ ∞) = 0 thus the eigenvalues and eigenvectors become time independen t, the general solution can be\ndecomposed into the form\n/bracketleftbigg˜s+(t)\n˜j+(t)/bracketrightbigg\n=C1eΛ1t/bracketleftbigg˜s+\n˜j+/bracketrightbigg\n1+C2eΛ2t/bracketleftbigg˜s+\n˜j+/bracketrightbigg\n2, (38)\nwhere constants C1,2are fixed by the initial condition (˜ s+,˜j+)t=0= (s⊥,0,0). Vanishing initial current is assumed\nfor convenience but it has captured enough features. Therefor e the transverse components of spin density and spin\ncurrent are known,\ns+(x,t) =s⊥,0\n2(Γ1+iΓ2)exp/parenleftbigg\n−γt\n2+iλsz,0\n2t+ikx/parenrightbigg\n×{(γ−iλsz,0)sin[(Γ 1+iΓ2)t]+2(Γ 1+iΓ2)cos[(Γ 1+iΓ2)t]},\n(39)\nj+(x,t) =s⊥,0\n2(Γ1+iΓ2)exp/parenleftbigg\n−γt\n2+iλsz,0\n2t+ikx/parenrightbigg\n×{−i2α(k−2q)sin[(Γ 1+iΓ2)t]}, (40)\nwhere\nΓ1+iΓ2=/radicalBigg\nα(k−2q)2−/parenleftbiggγ\n2−iλsz,0\n2/parenrightbigg2\n. (41)\nThe longitudinal component of spin current obeying Eq.( 26) with initial condition jz(0) =jz,0is given by\njz(t) =jz,0e−γt−e−γtλs2\n⊥,0α(k−2q)\n4(Γ2\n1+Γ2\n2)/bracketleftbigg\ncosh(2Γ 2t)+γ\n2sinh(2Γ 2t)\nΓ2−cos(2Γ 1t)−γ\n2sin(2Γ 1t)\nΓ1/bracketrightbigg\n.(42)\nIf the time dependence of jz(t) is retained, the eigenvalues of Eq.( 37) become\nΛ1,2=−γ\n2+iλsz,0\n2±/radicalBigg\nα(k−2q)2−/parenleftbiggγ\n2−iλsz,0\n2/parenrightbigg2\n+λjz(k−2q). (43)\nThe first two terms remain unchanged but the square root term Γ 1+iΓ2becomes time dependent. Estimate of\nmagnitude shows that\nα(k−2q)2≫λ|jz||k−2q| ∼ |γ\n2−iλsz,0\n2|2, (44)5\nhence the approximation Γ 1+iΓ2≈√α|k−2q|= Γ simplifies the expressions,\ns+(x,t) =s⊥,0exp/parenleftbigg\n−γt\n2+iλsz,0\n2t+ikx/parenrightbigg/parenleftbigg\ncosΓt+γ−iλsz,0\n2ΓsinΓt/parenrightbigg\n, (45)\nj+(x,t) =i√αs⊥,0exp/parenleftbigg\n−γt\n2+iλsz,0\n2t+ikx/parenrightbigg\nsin(Γt)sign(2q−k), (46)\njz(t) =jz,0e−γt−e−γtλs2\n⊥,0\n4(k−2q)/bracketleftbigg\n1+γt−cos(2Γt)−γ\n2sin(2Γt)\nΓ/bracketrightbigg\n. (47)\nRecalling that λ|jz| ≪Γ is assumed when de-coupling the equations of motion, for self-con sistency we substitute\nthe amplitude of jz\nλ|jz|/Γ∼(λs⊥,0/Γ)2≪1, (48)\nthe last inequality is identical to the limit taken in Eq.( 44).\nQUENCHING OF RABI COUPLING ON PSH\nAfter substituting Eqs.(20-23) in main text into Eqs.( 20-23) and dropping quadratic terms of fluctuations, one can\nobtain an inhomogeneousdiffusion equation for the dynamics of fluct uation/vectorδV= (δn,δj0,δsz,δjz,δsx,δjx,δsy,δjy)T,\n∂t/vectorδV(x,t) =ˆH(x,∂x)/vectorδV(x,t)+/vector g(x) (49)\nwith source term /vector g(x) = ΩR(0,2qspsh,y/m,spsh,y,0,0,0,−sz,0,qn0/(2m))T. The evolution matrices are given by\nˆH=/bracketleftbigg\nH11H12\nH21H22/bracketrightbigg\n, (50)\nH11=\n0 −∂x 0 0\n−/parenleftbigkBT\nm+3gn0\n4m/parenrightbig\n∂x0−g\nm(∂xsz,0+sz,0∂x) 0\n0 0 0 −∂x\n−3gsz,0\n4m∂x0 −α∂x −γ\n, (51)\nH12=\n0 0 0 0\n−g\nm(∂xspsh,x+spsh,x∂x) 0 −g\nm(∂xspsh,y+spsh,y∂x)+2qΩR\nm0\n0 0 +Ω R 0\n0 −λspsh,y 0 + λspsh,x+ΩR\n,(52)\nH21=\n0 0 0 0\n−3gspsh,x\n4m∂x0 0 + λspsh,y\n0 0 −ΩR 0\n−3gspsh,y\n4m∂x+qΩR\n2m0 0 −λspsh,x−ΩR\n, (53)\nH22=\n0−∂x0−2q\n−α∂x−γ−2qα−λsz,0\n0 2q0−∂x\n2qα λs z,0−α∂x−γ\n. (54)\nThe inhomogeneous diffusion equation can be solved by repeating inte grations,\n/vectorδV(x,t) =/integraldisplayt\n0/vector g(x)dt′+/integraldisplayt\n0ˆH(x,∂x)/vectorδV(x,t′)dt′\n=/integraldisplayt\n0/vector g(x)dt′+/integraldisplayt\n0/integraldisplayt′\n0ˆH(x,∂x)/vector g(x)dt′′dt′+/integraldisplayt\n0/integraldisplayt′\n0ˆH2/vectorδV(x,t′′)dt′′dt′\n=···=/parenleftBigg∞/summationdisplay\nn=1tn\nn!ˆHn−1/parenrightBigg\n/vector g(x) =ˆH−1/bracketleftBig\nexp(ˆHt)−ˆI/bracketrightBig\n/vector g(x), (55)6\nand imposing initial condition /vectorδV(x,0) = 0, i.e. only the inhomogeneous solution is concerned. Identical s olution\ncould be obtained alternatively by letting the form of /vectorδV(x,t) as\n/vectorδV(x,t) =ˆS(t)/vector g(x) withˆS(0) = 0, (56)\ntherefore the diffusion equation becomes\n∂t[ˆS(t)/vector g(x)] =ˆH(x,∂x)[ˆS(t)/vector g(x)]+/vector g(x) (57)\n∂tˆS(t) =ˆH(x,∂x)ˆS(t)+ˆI, (58)\nand implies\nˆS(t) =∞/summationdisplay\nn=1tn\nn!ˆHn−1=ˆH−1/bracketleftBig\nexp(ˆHt)−ˆI/bracketrightBig\n. (59)\nAbove solution is valid only for /vectorδV≈/vector g(x)t≪/vectorV0, i.e. Ω Rt≪1."    },    {        "title": "2005.03905v2._s___d__model_for_local_and_nonlocal_spin_dynamics_in_laser_excited_magnetic_heterostructures.pdf",        "content": "s-dmodel for local and nonlocal spin dynamics in laser-excited\nmagnetic heterostructures\nM. Beens,1,\u0003R.A. Duine,1, 2and B. Koopmans1\n1Department of Applied Physics, Eindhoven University of Technology\nP.O. Box 513, 5600 MB Eindhoven, The Netherlands\n2Institute for Theoretical Physics, Utrecht University\nLeuvenlaan 4, 3584 CE Utrecht, The Netherlands\n(Dated: September 24, 2020)\nAbstract\nWe discuss a joint microscopic theory for the laser-induced magnetization dynamics and spin\ntransport in magnetic heterostructures based on the s-dinteraction. Angular momentum transfer\nis mediated by scattering of itinerant selectrons with the localised ( delectron) spins. We use the\ncorresponding rate equations and focus on a spin one-half delectron system, leading to a simpli\fed\nanalytical expression for the dynamics of the local magnetization that is coupled to an equation for\nthe non-equilibrium spin accumulation of the selectrons. We show that this description converges\nto the microscopic three-temperature model in the limit of a strong s-dcoupling. The equation\nfor the spin accumulation is used to introduce di\u000busive spin transport. The presented numerical\nsolutions show that during the laser-induced demagnetization in a ferromagnetic metal a short-\nlived spin accumulation is created that counteracts the demagnetization process. Moreover, the\nspin accumulation leads to the generation of a spin current at the interface of a ferromagnetic and\nnon-magnetic metal. Depending on the speci\fc magnetic system, both local spin dissipation and\ninterfacial spin transport are able to enhance the demagnetization rate by providing relaxation\nchannels for the spin accumulation that is built up during demagnetization in the ferromagnetic\nmaterial.\n1arXiv:2005.03905v2  [cond-mat.mes-hall]  23 Sep 2020I. INTRODUCTION\nExciting magnetic systems with ultrashort laser pulses gives rise to fascinating physics.\nFirst, it was shown that a femtosecond laser pulse can quench the magnetization of a ferro-\nmagnetic thin-\flm on a subpicosecond timescale [1]. Later, all-optical magnetization switch-\ning was discovered in GdFeCo alloys [2], which proved the high potential of using ultrashort\nlaser pulses for future data writing technologies. Moreover, it was demonstrated that the\nlaser pulse generates a spin current [3, 4]. In non-collinear magnetic heterostructures the\nultrafast generated spin current exerts a spin-transfer torque [5, 6], leading to the excitation\nof Terahertz standing spin waves [7, 8]. Understanding all these ultrafast phenomena paves\nthe way towards faster magnetic data technologies, and bridges the boundaries between\nphotonics, spintronics and magnonics.\nDespite the vast experimental developments within the \feld, the microscopic origin of\nthe observed demagnetization rates is still heavily debated. Various microscopic processes\nhave been proposed as being the dominant mechanism, such as (i) the coherent interaction\nbetween the photons and the spins [9, 10], (ii) spin-dependent transport of hot electrons [11],\nand (iii) local spin dynamics as triggered by laser heating or excitation [1, 12{21]. In the\nlatter case, the models often rely on the assumption that heating of the electrons increases\nthe amount of spin-\rip scattering events, resulting in the transfer of angular momentum. An\nexample of this type of models is the microscopic three-temperature model (M3TM) [15],\nwhere it is assumed that the magnetization dynamics is dominated by Elliott-Yafet electron-\nphonon scattering. Arguably, other types of scattering mechanisms can also account for the\nobserved demagnetizations rates, such as Elliott-Yafet electron-electron scattering [14] and\nelectron-magnon scattering [16, 19]. The latter stems from the s-dinteraction in ferro-\nmagnetic transition metals, that couples the local magnetic moments ( delectrons) and free\ncarriers (selectrons). Similar models were derived to describe the ultrafast magnetization\ndynamics in semiconductors [22] and ferrimagnetic alloys [23].\nAnother important question is what mechanism drives the optically induced spin currents\nin magnetic heterostructures. First, it could be directly related to the proposed superdif-\nfusive spin currents created in the magnetic material [11, 24]. Secondly, the laser-induced\nthermal gradients can generate a spin current resulting from the spin-dependent Seebeck\ne\u000bect [25, 26]. Recently, it was proposed that the spin-polarized electrons are generated at\n2a rate given by the temporal derivative of the magnetization [5]. Interestingly, this implies\nthat the demagnetization and the generated spin current are driven by the same physical\nmechanism. The s-dinteraction, which mediates angular momentum transfer between the\nlocal magnetic moments and itinerant electrons, is a principal candidate [5, 19, 27].\nIn this work, we discuss an extended s-dmodel for laser-induced magnetization dynamics\nthat includes spin transport. The model describes that during demagnetization an out-\nof-equilibrium spin accumulation is created in the selectron system [19, 22], which leads\nto the generation of a spin current in magnetic heterostructures [5, 27]. We apply the s-\ndmodel to investigate the interplay between the local magnetization dynamics and spin\ntransport in laser-excited magnetic heterostructures. The numerical solutions of the rate\nequations show a qualitative agreement with the experiments and support the view that\nthes-dinteraction could be the main driving force of the observed ultrafast phenomena\n[5, 19, 27]. Furthermore, the crucial role of the spin accumulation is emphasized, namely\n(i) the generated spin accumulation has a negative feedback on the demagnetization process\n[19, 22] and (ii) this bottleneck can be removed by either local spin-\rip processes or by\nelectron spin transport. Hence, both local and nonlocal processes play a crucial role in\nthe magnetization dynamics. Finally, we discuss the limit in which the s-dmodel becomes\nequivalent to the M3TM, and we conclude with an outlook.\nWe start with an overview of the derivation of the s-dmodel in Section II and we highlight\nthe simpli\fcations we use compared to the derivations reported in Ref. [19] and Ref. [22].\nImportantly, we show that the s-dmodel can be written in a mathematical form analog to\nthe M3TM. In Section III, we model the demagnetization experiments and discuss the role\nof the spin accumulation. We describe the laser-induced dynamics in a collinear magnetic\nheterostructure in Section IV. We explain how the di\u000berent demagnetization rates of the\nparallel and antiparallel con\fguration can be understood from the s-dmodel. In Section V,\nwe describe a bilayer consisting of a ferromagnetic and non-magnetic metallic layer. Here,\nwe introduce di\u000busive spin transport, similar to the modeling as presented in [5, 27, 28].\nWe speci\fcally address the interplay between the local magnetization dynamics and spin\ntransport. We investigate the role of the layer thickness on the magnetization dynamics and\nwe analyze the temporal pro\fle of the injected spin current in the non-magnetic layer.\n3II. MODEL\nIn this section, we give an overview of the derivation of the s-dmodel for ultrafast\nmagnetization dynamics in transition metal ferromagnets. Although our approach is closely\nrelated to the derivation as presented in Ref. [19], it mathematically resembles the results\nfor magnetic semiconductors [22]. We keep our notation consistent with these references\nand we highlight the modi\fcations that are needed to reach the simpli\fed s-dmodel that is\nused in the remainder of this paper.\nAnalogous to Ref. [19], we de\fne the ferromagnetic transition metal in terms of two\nseparate electronic systems, corresponding to the 3 dand 4selectrons. A schematic overview\nof the model is presented in Fig. 1(a). The delectrons are the main contributor to the\nmagnetic properties of the system and are relatively localised. Therefore, we approximate\nthedelectron system as a lattice of localised spins. At each lattice site there is only one spin\nand the atomic magnetic moment is given by \u0016at= 2S\u0016B, where\u0016Bis the Bohr magneton\nandSis the spin quantum number. We neglect the orbital angular momentum.\nIn this work, we describe the localised spin system within a Weiss mean \feld approach,\nsimilar to the description used in the M3TM [15]. The Hamiltonian of the delectrons is\nexpressed as\n^Hd= \u0001X\nj^Sd;z\nj; (1)\nwhere ^Sd;z\njis thezcomponent of the spin at lattice site jand \u0001 is the exchange splitting.\nHence, each spin corresponds to a system of 2 S+ 1 energy levels splitted by energy \u0001. Note\nthat this description of the delectron system does not consider spin-wave excitations, which\nmakes it di\u000berent from the approach in Ref. [19].\nTheselectrons are described as a free electron gas. They are coupled to the localised\nspins through the on-site s-dinteraction, given by [19]\n^Hsd=JsdVatX\nj^Sd\nj\u0001^s(rj): (2)\nHere,Jsdis thes-dexchange coupling constant, Vatis the atomic volume, ^Sd\njis the spin\n43d electrons\nPhonons4s electrons\nf fE4s\nEF3d\nf fE4s3d\nf fE4s\nEF3d(a)\n(b) (c) (d)τsd\nμs-Figure 1. Schematic overview of the s-dmodel for ultrafast demagnetization [19, 22]. (a) The\nsystem is divided into a subsystem of localised 3 delectrons and itinerant 4 selectrons. The laser\npulse heats up the selectrons. Angular momentum is transferred between the sanddsubsystems\nby thes-dinteraction. Secondly, angular momentum can dissipate out of the combined system by\nadditional spin-\rip processes in the ssystem, e.g., Elliott-Yafet electron-phonon scattering. (b)-(d)\nSchematically show the occupation of the energy levels in the dandssubsystems during the laser\nheating (for S= 1=2). (b) Indicates the ground state ( Te= 0 K). (c) Shows that the broadening\nof the Fermi-Dirac distribution allows spin-\rip transitions around the Fermi level. This process is\naccompanied by a spin-\rip of a local dspin. Theselectrons thermalize rapidly and a non-zero\nspin accumulation \u0016sis created, as is indicated in Fig. (d).\noperator of the spin at lattice site j, and ^s(rj) is the spin density operator of the selectrons\nat position rjof lattice site j.\nWe express ^s(rj) in terms of the electron creation and annihilation operators in momen-\ntum space. This yields [22]\n^Hsd=X\njX\nkk0h\nJ\u0003\njkk0cy\nk0#ck\"^Sd+\nj+ H.c.i\n; (3)\nwhere the coupling strength is parametrized by the matrix element Jjkk0.^Sd\u0006\njcorresponds\nto the spin ladder operator for the spin at lattice site j. The operator cy\nk\u001b(ck\u001b) creates\n(annihilates) an selectron with momentum kand spin\u001b. In the transition from Eq. (2) to\n5Eq. (3) the terms proportional to the zcomponents are omitted and rewritten in terms of a\nmean-\feld energy shift in the Hamiltonian for the selectrons [19]. The similar energy shift\nin thedelectron system (a shift of \u0001) plays a minor role and is neglected.\nEquation (3) describes the spin-\rip scattering of selectrons with the localised spins, which\nmediates angular momentum transfer between the sanddelectron systems, but conserves\nthe total angular momentum. Hence, these scattering events change the total spin in the\nzdirection of the delectron system. To calculate the resulting magnetization dynamics,\nwe apply perturbation theory using the density matrix formalism. We only show the most\nimportant steps, for more details we refer to Ref. [22], where an equivalent calculation is\npresented for semiconductors. In contrast to Ref. [22], our system does include a direct ( d-d)\nexchange interaction between the localised spins, as represented by Eq. (1).\nFirst, we assume that the density matrix of the complete system can be factorized in\nterms of a density matrix ^ \u001aCfor the carriers ( selectrons) and ^ \u001aSfor the localised spins\n(delectrons). Secondly, we assume that after excitation there is no coherence between the\nspins. In other words, the time scale at which the spins dephase is the shortest time scale\nwithin the system, such that the density matrix ^ \u001aSis diagonal. The diagonal elements of\n^\u001aSare given by the occupation numbers \u001aS\nmsms=fmsfor each energy level msof a single\nspin, where mscorresponds to the zcomponent of the spin. In this Boltzmann approach,\nthe ensemble average of the spin in the zdirection is given by h^Sd;zi=PS\nms=\u0000Smsfms.\nIn order to \fnd the magnetization dynamics, we calculate the time derivative of all occu-\npation numbers fms. The mathematical description follows from the Liouville-von Neumann\nequation, and a coarse-grained description of the time evolution of the density operator [29].\nThe coarse-graining step size, interval \u000et, determines the time resolution of the model and\nshould be su\u000eciently small compared to the observed demagnetization time \u001cM. Moreover,\nwe assume that the time interval \u000etsatis\fes the conditions for the Markov approximation,\ni.e.,\u000etshould be much larger than the correlation time of the electrons and the density\nmatrix changes relatively slowly [18, 22]. Secondly, it is assumed that the time interval is\nmuch larger than the time scale associated with the energy transfer, in this case that yields\n\u000et\u001d~=\u0001 [18]. This is the standard limit underlying Fermi's golden rule, i.e., the condition\nleads to the transitions having a well-de\fned energy conservation represented by the Dirac\ndelta function. Hence, we should have that ~=\u0001\u001c\u000et\u001c\u001cM. Since ~=\u0001\u001810 fs (having\n\u0001\u0018kBTCand Curie temperature TC\u00181000 K) and \u001cMis of the order of\u0018100 fs, the valid-\n6ity of this limit is not trivial. However, it is expected that the role of all the approximations\nis relatively weak and only a\u000bects the results quantitatively.\nFinally, using the diagonality of the density matrix ^ \u001aSand the explicit form of the inter-\naction Hamiltonian ^Hsd, the rate equation can be written as [22]\ndfms\ndt=\u0000(Wms\u00001;ms+Wms+1;ms)fms (4)\n+Wms;ms\u00001fms\u00001+Wms;ms+1fms+1;\nwhereWms\u00061;msare the transition rates from level mstoms\u00061. The transition rates\nare calculated using Fermi's golden rule, analogous to the derivation of the M3TM [15].\nWe assume that the selectrons thermalize rapidly due to Coulomb scattering and can be\ndescribed by Fermi-Dirac statistics. Here, the distributions for the spin up and spin down\nselectrons have a common temperature Te, but are allowed to have a distinct chemical\npotential for which the di\u000berence is de\fned as the spin accumulation \u0016s=\u0016\"\u0000\u0016#[19, 22].\nIn the limit that the Fermi energy is much larger than all other energy scales, the transition\nrate is given by [22, 23]\nWms\u00061;ms=\u0019\n2~J2\nsdS\u0006\nmsD\"D#(\u0001\u0000\u0016s)exp\u0012\n\u0007\u0001\u0000\u0016s\n2kBTe\u0013\n2 sinh\u0012\u0001\u0000\u0016s\n2kBTe\u0013: (5)\nHere,S\u0006\nms=S(S+ 1)\u0000ms(ms\u00061) andD\";#(in units eV\u00001atom\u00001) is the density of states\nat the Fermi level for the spin up and spin down selectrons respectively. Equation (5)\nmathematically quanti\fes the amount of available phase space for transitions induced by\nthes-dinteraction. Figures 1(b)-1(d) schematically show the changes to the occupation of\nthedandselectron states as a result of laser heating the system. Figure 1(c) shows that\nthe thermal broadening of the Fermi-Dirac functions allows for transitions between the two\nspin directions of the selectrons, which is accompanied by a \rip of a localised delectron\nspin. Theselectrons thermalize rapidly and the new distributions have a shifted chemical\npotential, i.e., a non-zero spin accumulation is created, as is depicted in Fig. 1(d).\nThe dynamics of the spin accumulation \u0016scan be derived analogously and directly follows\nfrom spin angular momentum conservation. Now we de\fne the normalized magnetization\n7md=\u0000h^Sd;zi=Sof the localised magnetic moments. In equilibrium, the experimentally\ndetectable magnetization is dominated by md. This is not straightforward after excitation\nbecause of the induced exchange of angular momentum between the sanddelectrons.\nIn general, the magneto-optical signal in typical pump-probe experiments is a linear su-\nperposition of the contribution of the sanddelectrons. For a S= 1=2 system we have\n\u0001 = 2kBTCmd, and the dynamics is described by the two equations\nd\u0016s\ndt=\u001asddmd\ndt\u0000\u0016s\n\u001cs; (6)\ndmd\ndt=1\n\u001csd\u0010\nmd\u0000\u0016s\n2kBTC\u0011\u0014\n1\u0000mdcoth\u00102mdkBTC\u0000\u0016s\n2kBTe\u0011\u0015\n; (7)\nwhere we used the de\fnition \u001asd=\u0016D\u00001\u0000Jsd=2, with \u0016D= 2D\"D#=(D\"+D#) from Ref.\n[19]. Note that the term proportional to Jsdresults from the energy gap between the spin\nup and spin down selectrons arising from the s-dinteraction (as was introduced in the\ntransition from Eq. (2) to Eq. (3)), which can be both positive and negative depending on\nthe sign of Jsd. Moreover, we de\fned \u001c\u00001\nsd= (\u0019=~)J2\nsdD\"D#kBTC, which is closely related to\nthe demagnetization rate. We introduced the phenomenological term proportional to \u001c\u00001\ns,\nwhich describes all spin-\rip scattering processes that dissipate angular momentum out of\nthe combined electronic system [19], e.g., this term includes Elliott-Yafet electron-phonon\nscattering.\nEquation (7) clearly shows the similarities with the standard form of the equation for the\nlongitudinal magnetization relaxation of a spin S= 1=2 system within a mean-\feld approach.\nFor instance, in the limit \u001cs!0 the spin accumulation directly vanishes and the equilibrium\ncondition is given by md= tanh(mdTC=Te). In this limit there is no net spin polarization, i.e.,\ntheselectrons can be considered as spinless, which is exactly the assumption underlying the\nM3TM [15]. We note that although this expression closely resembles the expression presented\nin Ref. [15], the prefactor corresponds to a completely di\u000berent physical mechanism. More\ndetails about the relation with the M3TM will be discussed below.\nAlthough we have a simple de\fnition of the parameters \u001asdand\u001csd, the estimation of\nthese parameters is far from straightforward. We approximated the dandselectrons as\ntwo distinct systems, localised and itinerant electrons. In the real system there is no such\nclear separation because of s-dhybridization. E\u000bectively, we separated the `band-like' and\n8`local magnetic' properties of the combined electronic system ( dands), which makes it\ncomplex to estimate the relevant value of \u0016D. Hence, it is convenient to treat both \u001csdand\n\u001asdas e\u000bective parameters. In the upcoming sections, we interpret \u001csdas the experimentally\nretrieved demagnetization time and we choose the constant \u001asd= 1 eV. The exact values\nshould be retrieved from carefully \ftting the model to the experiments, which is beyond the\nscope of this theoretical paper.\nFinally, \u0016D\u00001(D\u00001\n\";#) scales with the width of the conduction band and is typically much\nlarger than Jsd=2, i.e., we have \u001asd=\u0016D\u00001\u0000Jsd=2\u0019\u0016D\u00001. Then, we can de\fne the\nmagnetization of the total spin system ( sanddelectrons) as mtot=md\u0000\u001a\u00001\nsd\u0016s, which is\nconserved by the s-dinteraction and will be used in the following analyses.\nIn the next section we discuss the important role of the spin accumulation by describing\nthe laser-induced demagnetization experiments using the numerical solutions of Eqs. (6)\nand (7).\nIII. ULTRAFAST DEMAGNETIZATION\nIn order to investigate the typical laser-induced dynamics of the local magnetization and\nspin accumulation speci\fcally, we consider a system with magnetic parameters \u001csd= 0:2 ps\nandTC= 1000 K. To model the laser heating we de\fne the temporal pro\fle of the laser\npulse asP(t) = (P0=(\u001bp\u0019)) exp[\u0000(t\u0000t0)2=\u001b2], whereP0is the absorbed laser pulse energy\ndensity and \u001bdetermines the pulse duration, which is set to 50 fs. We use the standard\ntwo-temperature model to \fnd the dynamics of the selectron temperature Teand phonon\ntemperature Tp[30]. The two-temperature model describes the equilibration of TeandTpby\nelectron-phonon scattering. We include a heat dissipation term that transfers heat out of the\nphonon system on a time scale \u001cD= 20 ps. For the heat capacities and the electron-phonon\ncoupling constant we use the values for cobalt given in Ref. [15]. We calculate the dynamics\nof the magnetization and spin accumulation by solving Eqs. (6) and (7) numerically. We do\nthis for multiple values of \u001cs. The results are presented in Figs. 2(a)-(d).\nFigure 2(a) shows the laser heating of the selectrons and the equilibration of the electron\ntemperature with the phonon temperature. Figures 2(b)-(d) display the laser-induced dy-\nnamics of the spin systems for di\u000berent values of the spin-\rip scattering time \u001cs, as indicated\nby the di\u000berent line types. Figure 2(b) shows the magnetization of the delectronsmdas\n9T(K) md μs(meV) mtot(a)\n(b)\n(c)\n(d)\nt(ps)\n20040060080010001200\n0.800.850.900.951.00\n-80-60-40-20020\n0 1 2 3 40.800.850.900.951.00Te\nTp\nτs=0.01 ps\nτs=0.1 ps\nτs=1.0 ps\nτs=0.01 ps\nτs=0.1 ps\nτs=1.0 ps\nτs=0.01 ps\nτs=0.1 ps\nτs=1.0 psFigure 2. Ultrafast demagnetization described by the s-dmodel. Figure (a) shows the temporal\npro\fle of the selectron temperature Teand phonon temperature Tp, after laser-pulse excitation at\nt= 0 withP0= 12\u0001108Jm\u00003. (b)-(d) Present the laser-induced dynamics of the spin systems,\nusingTC= 1000 K and \u001csd= 0:2 ps. Here, the line types indicate the calculations for di\u000berent\nvalues of\u001cs, which are given in the \fgure. (b) Shows the resulting magnetization dynamics in the\ndelectron system. (c) Shows the temporal pro\fle of the spin accumulation \u0016sand (d) shows the\ndynamics of the total magnetization mtot.\na function of time. The temporal pro\fle of the spin accumulation \u0016sis presented in Fig.\n2(c). Finally, Fig. 2(d) displays the total magnetization mtotas a function of time. Figures\n2(b) and 2(d) clearly show that the demagnetization of mdandmtotis maximized for the\nsmallest\u001cs.\nThe calculations show that the creation of a spin accumulation has a negative feedback\ne\u000bect on the demagnetization (of both mdandmtot), i.e., the short-lived spin accumulation\nacts as a bottleneck [19, 22]. The bottleneck can be removed by the additional spin-\rip\nrelaxation processes in the selectron system, which happen at a rate given by \u001c\u00001\ns. This\n100.0 0.5 1.0 1.5 2.00.99700.99720.99740.99760.9978\nt(ps)m\nτsd=0.0 ps\nτs=0.2 psmtot\nmdmM3TMFigure 3. The s-dmodel in the limit of a strong s-dcoupling (\u001csd!0), compared to the micro-\nscopic three-temperature model (M3TM) [15]. The plot shows the magnetizations md(red) and\nmtot(green) as a function of time after laser-pulse excitation at t= 0 withP0= 0:1\u0001108Jm\u00003\nand\u001b= 50 fs. The remaining magnetic parameters are given by TC= 1000 K and \u001cs= 0:2 ps.\nThe dotted black line indicates the magnetization calculated with the basic M3TM (using demag-\nnetization time scale \u001cM=\u001cs= 0:2 ps [31]).\nmeans that in the limit \u001csd\u001c\u001csthe demagnetization rate strongly depends on \u001cs. In the\nextreme case \u001csd!0, which corresponds to an in\fnitely strong s-dinteraction, mdand\u0016s\nare equilibrated instantaneously and their relation can be found by setting Eq. (7) equal\nto zero. Now the dandselectrons can be treated as a single spin system with magnetiza-\ntionmtotof which the subsequent dynamics is governed by Teand the additional spin-\rip\nscattering processes of the selectrons. These additional scatterings include Elliott-Yafet\nelectron-phonon scattering. Hence, in analogy with the M3TM [15], the system behaves\nas a single spin system with a characteristic demagnetization rate that is associated with\nElliott-Yafet electron-phonon scattering. More speci\fcally, in the low-\ruence limit and hav-\ning temperatures well below the Curie temperature, mtot(t) converges to the magnetization\ndynamics from the M3TM, which is visualized in Fig. 3. Here, md(t) andmtot(t) follow\nfrom thes-dmodel using P0= 0:1\u0001108Jm\u00003and\u001cs= 0:2 ps in the limit of a strong s-d\ncoupling (\u001csd!0). All other system parameters are kept equal to the calculations of Fig.\n2. The dotted black line is the magnetization described by the M3TM for the same system,\nusing the demagnetization time scale \u001cM=\u001cs= 0:2 ps [31], which shows a clear overlap\nwith the total magnetization mtot.\nOn the other hand, in the limit \u001csd\u001d\u001csthe spin accumulation relaxes e\u000eciently and the\nbottleneck e\u000bect is negligible. In this limit, the magnetizations mdandmtotconverge and\n11-0.2 0.0 0.2 0.4 0.6 0.8 1.00.880.900.920.940.960.98\nt(ps)md\nF1AP\nN\nF2F1PA\nN\nF2Figure 4. The laser-induced magnetization dynamics of the F/N/F structure in the antiparallel\nand parallel con\fgurations, described by the s-dmodel. We used a low-energetic laser pulse with\nP0= 1\u0001108Jm\u00003and\u001b= 70 fs. The diagram shows the magnetization mdof F layer 1 as a\nfunction of time. The color scheme indicates the speci\fc con\fguration, as is indicated in the inset.\nThe F layers have magnetic parameters TC= 600 K,\u001csd= 0:1 ps and\u001cs= 0:02 ps.\ntheir dynamics can be well described by Eq. (7) without the terms involving \u0016s(similar to\nthe limit\u001cs!0). Up to a prefactor, the magnetizations mdandmtotare now described\nby the same mathematical expression as in the M3TM. However, the physical origin of the\nultrafast demagnetization is di\u000berent.\nIn conclusion, in both regimes there is a clear relation with the M3TM. Nevertheless,\nin a real system it is expected that \u001csdand\u001cscan be of the same order and a short-lived\nspin accumulation in\ruences the magnetization dynamics. Finally, Fig. 2(c) shows that\nfor a decreasing \u001csthe spin accumulation becomes directly proportional to the temporal\nderivative of the magnetization md, as can be mathematically derived from Eq. (6) in the\nlimit\u001cs!0. These typical curves for \u0016sresemble the measurements in the experimental\ninvestigations of the optically generated spin currents at the interface of a ferromagnetic and\nnon-magnetic metal [5].\nIn the following sections we investigate the role of spin transport on the demagnetization\nprocess.\n12IV. F/N/F STRUCTURES: PARALLEL VERSUS ANTIPARALLEL\nIn the previous section we showed that during laser-pulse excitation a spin accumulation\nis generated that counteracts the demagnetization process. In this section, we show that\nspin transport can act as an additional mechanism for removing this bottleneck e\u000bect. We\nmodel the experiments with collinear magnetic heterostructures [3, 32]. More speci\fcally, we\naddress the results presented in Ref. [3], in which a magnetic heterostructure is investigated\nthat consists of two identical Co/Pt multilayers separated by a Ru spacer layer. The authors\npresent a comparison of the demagnetization of the parallel and antiparallel aligned states\nof the heterostructure. The measurements showed that the antiparallel con\fguration has a\nlarger demagnetization rate and amplitude, which can be explained by the generation of a\nspin current that enhances the demagnetization process. In the following, we will show that\nthese results can be understood and reproduced by the presented s-dmodel.\nHence, we consider a system containing two identical ferromagnetic (F) layers with a non-\nmagnetic (N) layer in between. We further refer to this system as the F/N/F structure. We\ninvestigate the di\u000berent laser-induced demagnetization rates for the parallel and antiparallel\ncon\fguration of the F/N/F structure. The systems are schematically depicted in the inset\nof Fig. 4. By de\fnition, F layer 1 is pointing up in both con\fgurations, whereas F layer\n2 is pointing in the up and down directions for the parallel and antiparallel con\fgurations\nrespectively.\nWe assume all the layers to be very thin, such that we can take the temperature, magne-\ntization and spin accumulation homogeneous within each layer. We de\fne a magnetization\nmd;iand\u0016s;ifor each F layer i. Because of the very small thickness of the N layer we assume\nthat the electron transport is in the ballistic regime. In that case, we can approximate that\nthe spin transport in the non-magnetic layer is purely driven by the di\u000berence in the spin\naccumulation of both F layers. Within these limits the spin accumulations satisfy\nd\u0016s;i\ndt=\u001asddmd;i\ndt\u0000\u0016s;i\n\u001cs;i\u0000\u0016s;i\u0000\u0016s;j\n\u001cB; (8)\nwherei6=jandi;j2 f1;2g. The last term, which is introduced phenomenologically,\nrepresents the spin transfer between the F layers driven by ballistic electron transport and\nenforces the spin accumulations to equilibrate. The prefactor is de\fned in terms of the\n13time scale\u001cB. We use that \u001cB\u00181 fs based on the assumptions that the Fermi velocity is\nvF\u0018106ms\u00001and the thickness of the N layer is dN\u00181 nm. Note that the transport\nterm depends on the spin accumulation at the same time coordinate, i.e., the distinct F\nlayers feel changes in the opposing layer instantaneously. In the real experiment there might\nbe a small delay. However, we expect that this e\u000bect can be neglected in our calculations.\nFinally, we stress that this particular form of the transport term can only be used for two\nstrictly identical F layers, as was the case in Ref. [3].\nFor the F layers we use the magnetic parameters \u001csd= 0:1 ps,\u001cs= 0:02 ps andTC=\n600 K, which are approximated values corresponding to the Co/Pt multilayers used in the\nexperiments [3]. Furthermore, we apply a low-energetic laser pulse with P0= 1\u0001108Jm\u00003\nand assume that the system is heated homogeneously. In this speci\fc case, we set the pulse\nduration to \u001b= 70 fs [3]. For convenience, we still use the heat capacities and electron-\nphonon coupling constant of pure cobalt [15].\nThe results are displayed in Fig. 4. The red and blue curves show the magnetization md\nof F layer 1 for the parallel and antiparallel con\fguration respectively. It is veri\fed that\nmtot(not shown) behaves very similar. In agreement with the experiments, we observe a\nlarger demagnetization rate and amplitude for the antiparallel con\fguration. This can be\neasily understood from the transport term in Eq. (8). In the parallel con\fguration we have\n\u0016s;1=\u0016s;2at any time and the transport term vanishes. In contrast, for the antiparallel\ncon\fguration we have \u0016s;1=\u0000\u0016s;2, the transport does not vanish and behaves as an extra\nchannel for angular momentum transfer. This extra channel assists the reduction of the spin\naccumulation, thereby leading to a larger demagnetization. Equivalently, in the antiparallel\ncon\fguration the spin current in the non-magnetic layer is non-zero and has exactly the\ncorrect polarization to enhance the demagnetization rates in both F layers. Finally, Fig. 4\nalso shows that the demagnetization curves of the two con\fgurations converge at t\u0018400 fs,\nwhich is in agreement with the experiments [3].\nIn the next section, we analyze the temporal pro\fle of the spin current generated in an\nF/N structure in the di\u000busive regime. Furthermore, we investigate the role of the thickness\nof the layers.\n14(a) (b)\n51015202530350.180.190.200.210.220.230.240.25\ndF(nm)Δmd\n0 10 20 30 40 500.00.10.20.30.40.5\nP0(108Jm-3)Δmd\ndF = 5 nm\ndF = 20 nmFN\n0 xdF dNFigure 5. The laser-induced magnetization dynamics in an F/N structure. The magnetic pa-\nrameters of the ferromagnetic layer are given by \u001csd= 0:3 ps,\u001cs= 0:2 ps andTC= 1388 K. (a)\nThe maximum demagnetization \u0001 md(averaged over the F layer) as a function of the ferromag-\nnetic layer thickness dF. The non-magnetic layer thickness is set to dN= 200 nm and we used\nP0= 30\u0001108Jm\u00003. The inset shows the system schematically. (b) The maximum demagnetization\n\u0001mdas a function of P0fordF= 5 nm (blue) and dF= 20 nm (red). For both systems we have\ndN= 200 nm. The black dashed line indicates the demagnetization of the bulk ferromagnet in the\nabsence of an F/N interface ( dF=1anddN= 0).\nV. F/N STRUCTURES: DIFFUSIVE SPIN TRANSPORT\nFinally, we introduce di\u000busive spin transport in the simpli\fed s-dmodel and we show\nthat in magnetic heterostructures spin di\u000busion within the selectron system can signi\fcantly\nenhance the demagnetization rate. Here, we model a system consisting of a ferromagnetic\n(F) layer and a non-magnetic (N) layer. In contrast to the similar approach reported in Refs.\n[5, 27, 28], we calculate the local magnetization dynamics dmd=dtdirectly from the s-dmodel\nusing Eq. (7), that serves as a source for spin-polarized selectrons via Eq. 6. Thereby, we\ncan speci\fcally address the mutual in\ruence of the dynamics of the local magnetization md\nand spin accumulation \u0016sboth as a function of position and time.\nAs indicated in the inset of Fig. 5(a), we de\fne the thickness of the F layer and N layer as\ndFanddNrespectively. Spin transport is described in the di\u000busive regime, where both layers\nare treated on an equal footing [33]. For convenience, we assume that the system is heated\nhomogeneously, i.e., there are no thermal gradients present. Hence, the demagnetization of\nthe F layer is the only source of the spin current and there is no spin-dependent Seebeck\ne\u000bect included in this calculation.\n15It is assumed that that the interface is transparent for spins, such that the spin accumu-\nlation is continuous at the interface. Imposing that there is no charge transport, the spin\ncurrent density can be expressed as js=\u0000(\u0016\u001b=e2)@\u0016s=@x[28, 33], with \u0016 \u001b= 2\u001b\"\u001b#=(\u001b\"+\u001b#)\nand\u001b\";#the spin-dependent electrical conductivity. Combining this with the continuity\nequations for spin-up and -down electrons [28, 34], gives that introducing di\u000busive spin\ntransport in Eq. (6) leads to [28]\n@\u0016s\n@t\u00001\n\u0016\u0017@\n@x\u0014\u0016\u001b\ne2@\u0016s\n@x\u0015\n=\u001asd@md\n@t\u0000\u0016s\n\u001cs; (9)\nwhere all variables explicitly depend on the spatial coordinate xand we assumed that the\nsystem is homogeneous in the lateral directions. The interface is at x= 0. Here, \u0016 \u0017=\n2\u0017\"\u0017#=(\u0017\"+\u0017#) with\u0017\";#the spin-dependent density of states in units per volume per energy\n(\u0016\u0017=\u0016D=V at). The N layer ( x>0) is characterised by \u001b\"=\u001b#and\u0017\"=\u0017#, and the absence\nof the (s-dinteraction) source term. Equation (9) is further simpli\fed by imposing that \u0016 \u0017\nis independent of x, where we have assumed that in the F layer \u0017\"\u0018\u0017#and\u001b\"\u0018\u001b#. These\nchoices are made for convenience and are consistent with the underlying assumptions of the\ns-dmodel, that the delectrons in the F layer do not contribute to the conductive properties\nof the material. In that case, Eq. (9) reduces to\n@\u0016s\n@t=\u001asd@md\n@t\u0000\u0016s\n\u001cs+@\n@x\u0014\nDdi\u000b@\u0016s\n@x\u0015\n: (10)\nwith the di\u000busion coe\u000ecient Ddi\u000b= \u0016\u001b=(\u0016\u0017e2) [27, 28]. Finally, we set the spin currents at\nthe edges equal to zero js(\u0000dF) =js(dN) = 0.\nEquation (10) is solved numerically, where we discretized the system using a \fnite di\u000ber-\nence method. Note that the spatial derivative of the di\u000busion coe\u000ecient Ddi\u000bis only non-zero\nat the interface. In these calculations, the F layer corresponds to pure cobalt for which we\nuse the di\u000busion coe\u000ecient Ddi\u000b= 250 nm2ps\u00001[27]. Furthermore, we use \u001csd= 0:3 ps and\n\u001cs= 0:2 ps. For the N layer we take Ddi\u000b= 9500 nm2ps\u00001and\u001cs= 25 ps, which correspond\nto the di\u000busion coe\u000ecient and spin-\rip relaxation time for Copper [27].\nFigure 5(a) shows a calculation of the F/N structure excited with a laser pulse with energy\ndensityP0= 30\u0001108Jm\u00003and pulse duration \u001b= 50 fs, where we used the heat capacities\nand electon-phonon coupling constant of cobalt [15]. The diagram shows the maximum\n16demagnetization \u0001 md=md;0\u0000md;min(averaged over the F layer) as a function of the\nF layer thickness dF, wheremd;0is the initial (equilibrium) value of md, andmd;minis the\nminimum of mdafter excitation. The thickness of the N layer is kept constant and set to dN=\n200 nm. It clearly shows that the demagnetization becomes larger when the F layer thickness\ndecreases. Intuitively, the injection of spins into the non-magnetic layers can enhance the\ndemagnetization signi\fcantly as long as the F layer is relatively thin. This conclusion is\ncorroborated by the results presented in Fig. 5(b), which shows the demagnetization \u0001 md\nas a function of P0. The results are plotted for dF= 5 nm and dF= 20 nm. The dashed line\nindicates the demagnetization of a bulk ferromagnet in the absence of an N layer ( dF=1\nanddN= 0). The calculations show that for a relatively thin F layer spin injection into the\nN layer can lead up to \u001830% more demagnetization.\nNow we discuss the dynamics of the injected spin current itself. We do this by calculating\nthe spin accumulation at the outer edge of the N layer \u0016s(dN). The results are shown in\nFig. 6, which displays the spin accumulation as a function of time for three di\u000berent values\nofdNthat are given in the \fgure. The F layer thickness is kept constant at dF= 10 nm.\nIn agreement with the experimental investigations [5, 27], the diagram clearly shows that\nfor an increasing dNthe minimum of \u0016s(dN) shifts in time and is reduced. This behavior\ncan be understood from the di\u000busive character of the spin transport. Here, the temporal\npro\fle of\u0016s(dN) is highly sensitive to the speci\fc material that composes the F layer and\nthe corresponding e\u000bective parameters, as is expected from the experimental and numerical\ninvestigation using various materials for the F layer [27].\nA more quantitative comparison with the experiments would require addressing spin\ntransport beyond the di\u000busive regime and implementing a \fnite penetration depth of the\nlaser pulse in the modeling. However, we focused our discussion on the dynamics that stems\nfrom thes-dinteraction and we speci\fcally investigate the role of \u0016sindependent of the\nthermal properties of the system. In that case, the model shows that in the presence of\nonly thes-dinteraction, the typical experimental observations can be explained and show a\nqualitative agreement [5, 27].\n170 1 2 3 4 5 6-6-5-4-3-2-101\nt(ps)dN=100 nm\ndN=150 nm\ndN=200 nm μs (meV) (dN)\nFN\n0 xdF dN Figure 6. The di\u000busive spin current injected in the non-magnetic (N) layer with thickness dN,\ncoupled to a ferromagnetic (F) layer with thickness dF. The diagram shows the spin accumulation\nat the outer edge of the N layer (position x=dN, indicated by the red dotted line in the inset) as a\nfunction of time. Furthermore, we set P0= 20\u0001108Jm\u00003and\u001b= 50 fs. The magnetic parameters\nare identical to the values used in Fig. 5. The line types indicate the results for three di\u000berent\nvalues ofdN. The thickness of the F layer is kept constant and set to dF= 10 nm.\nVI. CONCLUSION AND DISCUSSION\nIn conclusion, we discussed a simpli\fed s-dmodel that we used to describe laser-induced\nmagnetization dynamics in magnetic heterostructures and to study the interplay between lo-\ncal and nonlocal spin dynamics. The presented numerical calculations emphasize the critical\nrole of the spin accumulation. During demagnetization a spin accumulation is created, which\ncounteracts the demagnetization process. Both local spin-\rip scatterings and spin transfer\nto a non-magnetic layer can reduce this spin accumulation e\u000bectively and, depending on the\nsystem, can both play a dominant role in the characterization of the demagnetization rate.\nImportantly, the modeling shows that even in the absence of any other interaction, the s-d\ninteraction could account for the typically observed ultrafast phenomena, such as being a\ndriving force for laser-induced spin transport in magnetic heterostructures.\nThe presented analyses show that the simpli\fed s-dmodel provides a versatile descrip-\ntion of ultrafast magnetization dynamics, which converges to the M3TM for a strong s-d\ncoupling and possesses the additional feature that spin transport can be included straight-\nforwardly. However, one needs to keep in mind that the model is a simpli\fed description\nof the underlying physics. As was earlier discussed, the delectrons are not perfectly lo-\n18calised. Moreover, the delectron spins are described using a Weiss model, i.e., spin wave\nexcitations are neglected. In a more complete description, the delectrons are described\nas a magnonic system and the s-dinteraction corresponds to electron-magnon scattering\n[19]. That has the advantage that spin transport driven by magnon transport can be in-\ncluded, which is expected to give a non-negligible contribution to the spin transport at the\ninterface between a ferromagnetic and non-magnetic metal [34]. In that case, the electronic\nand magnonic contribution to the spin transport can be treated on an equal footing by\nintroducing a magnon chemical potential [34, 35]. This description should allow for both\nchemical potential gradients and thermal gradients. For instance, thermal gradients can be\ninduced by a \fnite penetration depth of the laser pulse and can drive a spin current via\nthe electronic spin-dependent Seebeck e\u000bect [25, 26] and the magnonic spin Seebeck e\u000bect\n[36, 37]. Nevertheless, we expect that the dominant contributions to the dynamics can be\nwell described by the simpli\fed s-dmodel including spin transport and it provides a useful\npathway to investigate the underlying physics.\nThis work is part of the research programme of the Foundation for Fundamental Research\non Matter (FOM), which is part of the Netherlands Organisation for Scienti\fc Research\n(NWO). R.D. is member of the D-ITP consortium, a program of the NWO that is funded\nby the Dutch Ministry of Education, Culture and Science (OCW). This work is funded by\nthe European Research Council (ERC).\n\u0003Corresponding author: m.beens@tue.nl\n[1] E. Beaurepaire, J.-C. Merle, A. Daunois, and J.-Y. Bigot, Ultrafast spin dynamics in ferro-\nmagnetic Nickel, Phys. Rev. Lett. 76, 4250 (1996).\n[2] C.D. Stanciu, F. Hansteen, A.V. Kimel, A. Kirilyuk, A. Tsukamoto, A. Itoh, and Th. Rasing,\nAll-optical magnetic recording with circularly polarized light, Phys. Rev. Lett. 99, 047601\n(2007).\n[3] G. Malinowski, F. Dalla Longa, J.H.H. Rietjens, P.V. Paluskar, R. Huijink, H.J.M. Swagten,\nand B. Koopmans, Control of speed and e\u000eciency of ultrafast demagnetization by direct\ntransfer of spin angular momentum, Nat. Phys. 4, 855 (2008).\n[4] A. Melnikov, I. Razdolski, T.O. Wehling, E.Th. Papaioannou, V. Roddatis, P. Fumagalli,\n19O. Aktsipetrov, A.I. Lichtenstein, and U. Bovensiepen, Ultrafast transport of laser-excited\nspin-polarized carriers in Au/Fe/MgO (001), Phys. Rev. Lett. 107, 076601 (2011).\n[5] G.-M. Choi, B.-C. Min, K.-J. Lee, and D.G. Cahill, Spin current generated by thermally driven\nultrafast demagnetization, Nat. Commun. 5, 4334 (2014).\n[6] A.J. Schellekens, K.C. Kuiper, R.R.J.C. De Wit, and B. Koopmans, Ultrafast spin-transfer\ntorque driven by femtosecond pulsed-laser excitation, Nat. Commun. 5, 4333 (2014).\n[7] I. Razdolski, A. Alekhin, N. Ilin, J.P Meyburg, V. Roddatis, D. Diesing, U. Bovensiepen,\nand A. Melnikov, Nanoscale interface con\fnement of ultrafast spin transfer torque driving\nnon-uniform spin dynamics, Nat. Commun. 8, 15007 (2017).\n[8] M.L.M. Lalieu, P.L.J. Helgers, and B. Koopmans, Absorption and generation of femtosecond\nlaser-pulse excited spin currents in noncollinear magnetic bilayers, Phys. Rev. B 96, 014417\n(2017).\n[9] G.P. Zhang and W. H ubner, Laser-induced ultrafast demagnetization in ferromagnetic metals,\nPhys. Rev. Lett. 85, 3025 (2000).\n[10] J.-Y. Bigot, M. Vomir, and E. Beaurepaire, Coherent ultrafast magnetism induced by fem-\ntosecond laser pulses, Nat. Phys. 5, 515 (2009).\n[11] M. Battiato, K. Carva, and P. M. Oppeneer, Superdi\u000busive spin transport as a mechanism of\nultrafast demagnetization, Phys. Rev. Lett. 105, 027203 (2010).\n[12] B. Koopmans, J.J.M. Ruigrok, F. Dalla Longa, and W.J.M. de Jonge, Unifying ultrafast\nmagnetization dynamics, Phys. Rev. Lett. 95, 267207 (2005).\n[13] N. Kazantseva, U. Nowak, R.W. Chantrell, J. Hohlfeld, and A. Rebei, Slow recovery of the\nmagnetisation after a sub-picosecond heat pulse, Europhys. Lett. 81, 27004 (2007).\n[14] M. Krau\u0019, T. Roth, S. Alebrand, D. Steil, M. Cinchetti, M. Aeschlimann, and H.C. Schnei-\nder, Ultrafast demagnetization of ferromagnetic transition metals: The role of the Coulomb\ninteraction, Phys. Rev. B 80, 180407(R) (2009).\n[15] B. Koopmans, G. Malinowski, F. Dalla Longa, D. Steiauf, M. F ahnle, T. Roth, M. Cinchetti,\nand M. Aeschlimann, Explaining the paradoxical diversity of ultrafast laser-induced demag-\nnetization, Nat. Mat. 9, 259 (2010).\n[16] A. Manchon, Q. Li, L. Xu, and S. Zhang, Theory of laser-induced demagnetization at high\ntemperatures, Phys. Rev. B 85, 064408 (2012).\n[17] B.Y. Mueller and B. Rethfeld, Thermodynamic \u0016T model of ultrafast magnetization dynamics,\n20Phys. Rev. B 90, 144420 (2014).\n[18] P. Nieves, D. Serantes, U. Atxitia, and O. Chubykalo-Fesenko, Quantum Landau-Lifshitz-\nBloch equation and its comparison with the classical case, Phys. Rev. B 90, 104428 (2014).\n[19] E. G. Tveten, A. Brataas, and Y. Tserkovnyak, Electron-magnon scattering in magnetic het-\nerostructures far out of equilibrium, Phys. Rev. B 92, 180412(R) (2015).\n[20] K. Krieger, J.K. Dewhurst, P. Elliott, S. Sharma, and E.K.U. Gross, Laser-induced demag-\nnetization at ultrashort time scales: Predictions of TDDFT, Journal of chemical theory and\ncomputation 11, 4870 (2015).\n[21] W. T ows and G.M. Pastor, Many-body theory of ultrafast demagnetization and angular mo-\nmentum transfer in ferromagnetic transition metals, Phys. Rev. Lett. 115, 217204 (2015).\n[22]  L. Cywi\u0013 nski and L.J. Sham, Ultrafast demagnetization in the sp-d model: A theoretical study,\nPhys. Rev. B 76, 045205 (2007).\n[23] V.N. Gridnev, Ultrafast heating-induced magnetization switching in ferrimagnets, Journal of\nPhysics: Condensed Matter 28, 476007 (2016).\n[24] M. Battiato, K. Carva, and P.M. Oppeneer, Theory of laser-induced ultrafast superdi\u000busive\nspin transport in layered heterostructures, Phys. Rev. B 86, 024404 (2012).\n[25] G.-M. Choi, C.-H. Moon, B.-C. Min, K.-J. Lee, and D.G. Cahill, Thermal spin-transfer torque\ndriven by the spin-dependent Seebeck e\u000bect in metallic spin-valves, Nat. Phys. 11, 576 (2015).\n[26] A. Alekhin, I. Razdolski, N. Ilin, J.P. Meyburg, D. Diesing, V. Roddatis, I. Rungger, M. Sta-\nmenova, S. Sanvito, U. Bovensiepen, and A. Melnikov, Femtosecond spin current pulses gen-\nerated by the nonthermal spin-dependent Seebeck e\u000bect and interacting with ferromagnets in\nspin valves, Phys. Rev. Lett. 119, 017202 (2017).\n[27] I.-H. Shin, B.-C. Min, B.-K. Ju, and G.-M. Choi, Ultrafast spin current generated by electron{\nmagnon scattering in bulk of ferromagnets, Japanese Journal of Applied Physics 57, 090307\n(2018).\n[28] J. Kimling and D.G. Cahill, Spin di\u000busion induced by pulsed-laser heating and the role of\nspin heat accumulation, Phys. Rev. B 95, 014402 (2017).\n[29] R.C. Iotti, E. Ciancio, and F. Rossi, Quantum transport theory for semiconductor nanostruc-\ntures: A density-matrix formulation, Phys. Rev. B 72, 125347 (2005).\n[30] S.I. Anisimov, B.L. Kapeliovich, and T.L. Perelman, Electron emission from metal surfaces\nexposed to ultrashort laser pulses, Zh. Eksp. Teor. Fiz 66, 375 (1974).\n21[31] The time scale \u001cMcorresponds to the prefactor \u001c\u00001\nM=RTp=TCfrom the standard M3TM [15],\nand is taken as a constant.\n[32] D. Rudolf, C. La-O-Vorakiat, M. Battiato, R. Adam, J.M. Shaw, E. Turgut, P. Maldon-\nado, S. Mathias, P. Grychtol, H.T. Nembach, T.J. Silva, M. Aeschlimann, H.C. Kapteyn,\nM.M. Murnane, C.M. Schneider, and P.M. Oppeneer, Ultrafast magnetization enhancement\nin metallic multilayers driven by superdi\u000busive spin current, Nat. Commun. 3, 1 (2012).\n[33] A. Slachter, F.L. Bakker, J.-P. Adam, and B.J. van Wees, Thermally driven spin injection\nfrom a ferromagnet into a non-magnetic metal, Nat. Phys. 6, 879 (2010).\n[34] M. Beens, J.P. Heremans, Y. Tserkovnyak, and R.A. Duine, Magnons versus electrons in\nthermal spin transport through metallic interfaces, Journal of Physics D: Applied Physics 51,\n394002 (2018).\n[35] L.J. Cornelissen, K.J.H. Peters, G.E.W. Bauer, R.A. Duine, and B.J. van Wees, Magnon spin\ntransport driven by the magnon chemical potential in a magnetic insulator, Phys. Rev. B 94,\n014412 (2016).\n[36] K. Uchida, J. Xiao, H. Adachi, J. Ohe, S. Takahashi, J. Ieda, T. Ota, Y. Kajiwara,\nH. Umezawa, H. Kawai, G.E.W. Bauer, S. Maekawa, and E. Saitoh, Spin Seebeck insula-\ntor, Nat. Mat. 9, 894 (2010).\n[37] J. Xiao, G.E.W. Bauer, K.C. Uchida, E. Saitoh, and S. Maekawa, Theory of magnon-driven\nspin Seebeck e\u000bect, Phys. Rev. B 81, 214418 (2010).\n22"    },    {        "title": "0801.1775v1.Giant_dynamical_Zeeman_split_in_inverse_spin_valves.pdf",        "content": "arXiv:0801.1775v1  [cond-mat.mes-hall]  11 Jan 2008Giant dynamical Zeeman split in inverse spin valves\nX. R. Wang\nDepartment of Physics, The Hong Kong University of Science a nd Technology, Clear Water Bay, Hong Kong, China\n(Dated: November 4, 2018)\nThe inversion of a spin valve device is proposed. Opposite to a conventional spin valve of a non-\nmagneticspacersandwichedbetweentwoferromagnetic meta ls, aninversespinvalveisaferromagnet\nsandwiched between two non-magnetic metals. It is predicte d that, under a bias, the chemical\npotentials of spin-up and spin-down electrons in the metals split at metal-ferromagnet interfaces,\na dynamical Zeeman effect. This split is of the order of an appl ied bias. Thus, there should be\nno problem of generating an eVsplit that is not possible to be realized on the earth by the us ual\nZeeman effect.\nPACS numbers:\nSpintronicsbecomesanemergentsubfieldincondensed\nmatter physics since the discovery of giant magnetoresis-\ntance in 1988 by Fert[1] and Grunberg[2]. Spintronics\nis about the control and manipulation of both electron\ncharge and electron spin. Many interesting phenomena\nare related to the interplay between electron spin and its\ncharge degrees of freedom. For example, electron trans-\nport can be manipulated by the magnetizationconfigura-\ntions. This is the basisofgiant magnetoresistance[1, 2, 3]\nand tunneling magnetoresistance[4, 5] phenomena and\ndevices. Its inverse effect, known as spin-transfer torque\n(STT)[6, 7, 8], wasalsodiscovered. TheSTTopensanew\nway to manipulate magnetization other than a magnetic\nfield[9], which has been much of recent focus in the field\ndue to its potential applications in information storage\nindustry.\nIn this letter, an inverse spin valve structure of a fer-\nromagnet sandwiched between two non-magnetic metals\nis studied when a bias is applied to the device. Due to\nthe spin-dependent electron transport of the structure,\nthe chemical potentials of spin-up (SU) and spin-down\n(SD) electrons at the metal-ferromagnet interfaces split\nby the magnitude of an applied bias. We term this split\ndynamical Zeeman effect. This split can be of the order\nofeVif propermaterials are used. To introduce a similar\nsplit by the usual Zeeman effect, a magnetic field unreal-\nizable on the earth is required. Thus, a giant dynamical\nZeeman effect is predicted.\nA conventional spin valve is a layered structure of a\nnon-magnetic spacer sandwiched between two ferromag-\nnetic metals. The spin valve is called a giant magnetore-\nsistance device if the spacer is a normal metal while it\nis a tunneling magnetoresistance device for an insulator\nspacer. The electrontransportofa spin valvedepends on\ntherelativepolaritiesofthetwomagnets. Aninversespin\nvalve is also a layered structure with a ferromagnet sand-\nwiched between two non-magnetic metals as illustrated\nin Fig. 1a. M1 andM2 are two normal metals with\nidentical SU and SD electron density of states (DOS) as\ndepicted schematically in the left and right diagrams of\nFig. 1b. To simplify our analysis, a half-metal (HM)\nspacer is considered first so that only electrons of one\ntype of spins (spin-up) can pass through it. The DOSs ofSU and SD electrons for a half-metal is illustrated by the\nmiddle diagram of Fig. 1b. Under a bias V, SU electrons\ninM1 flow into the empty SU electron states in M2 via\nthe empty SU electron states in the half-metal, shown\npictorially by the curved arrows in Fig. 1b. The flow of\nthe electrons creates chemical potential differences (∆ µi\nin Fig. 1b) between SU and SD electrons in both M1\nandM2 near the metal-ferromagnet interfaces.\nV \nM1 M2\nE\nEE\nM1\nM2∆µ1\n∆µ2∆V\nDOS\nDOS(a)\n(b)HM\nHM\nFIG.1: Schematicillustration oftheinversespinvalve(a) and\nrelative chemical potentials of spin-up and spin-down elec -\ntrons in nonmagnetic metals and half-metal (b). The curved\narrows indicate the electron flow. ∆ µ1and the ∆ µ2are the\nchemical potential splits between spin-up and spin-down el ec-\ntrons at the left and right metal-ferromagnet interfaces. ∆ V\nis the effective bias on the half-metal.\nIn order to understand why the chemical potentials of\nSU and SD electrons at the M1−HMandM2−HM\ninterfaces split under a bias, let us consider an extreme\ncase in which spin relaxation time in both M1 andM2\nis infinite long (no spin relaxation so that SU and SD\nelectrons are isolated from each other). Since electron\ntransport in a nanostructure depends on how a bias is\napplied[10], we assume, to be precise and without losing\ngenerality, that the electron chemical potential in M1\nis initially moved up by eVwhile that of M2 is kept\nunchanged. Initially, SU electrons flow into the empty\nSU electron states in M2 because the half-metal prevent\nSD electrons from flowing. The same amount electrons\nwill be pumped back from M2 toM1 by a battery to\nkeep the electron neutrality in M1 andM2. However, a2\nbattery does not distinguish electronspin, and as a result\nit will pump equal amount of SU and SD electrons. In\nother words, SU electrons flow out of M1 and into M2\nvia the half-metallic spacer. In the meanwhile, an equal\namount of electrons with half of them being in the SU\nstate and the other half in the SD state are drawn out of\nM2and aresupplied into M1by the battery. As a result,\nM1accumulatesmoreSDelectrons,and M2accumulates\nmore SU electrons. Thus, the chemical potential of the\nSD electrons will be higher than that of SU electrons\ninM1. Vice versa, the chemical potential of the SU\nelectrons will be higher than that of SD electrons in M2.\nThe chemical potential splits keep increasing until the\nchemical potential of SU electrons in M1 equals that in\nM2. At this point, the steady state with no current in\nthe circuit is reached and chemical potential splits in M1\nandM2 are established with ∆ µ1+∆µ2=eV.\nThe spin relaxation[11] alwaysexistsin amaterial, and\nthe magnitude of the chemical potential split for SU and\nSD electrons depends sensitively on the spin relaxation.\nThis sensitivity can be seen from another extreme case\nthat the spin flipping is so fast that SU and SD electrons\ncan be converted into each other any time at no cost\n(meaning SU and SD electrons are at equilibrium with\nrespect to each othersat all times in M1andM2). Thus,\nitisimpossibletocreateanychemicalpotentialdifference\nfor SU and SD electrons in M1 andM2. The reality, of\ncourse, is somewhere in between (the two extreme cases).\nThe spin relaxation time of a real material is finite, and\nit depends on the strength of spin-orbital coupling, hy-\nperfine interaction and other interactions that cause spin\nflipping. In order to include the spin relaxation time\nquantitatively, consider an ideal model at zero tempera-\nture. Assume the spin flip occurs only near the interfaces\nwithin a width of spin diffusion length ξ1inM1 andξ2in\nM2. This is justified because both SU and SD electrons\nin the rest parts of the circuit (other than the half-metal)\nshould have same chemical potential. Thus number of\nelectrons flipped from up-spin state to down-spin state\nis the same as that from down-spin to up-spin. Further-\nmore, let us assume the resistance of the half-metal is R\n(for SU electrons, the resistance for SD electrons is infin-\nity due to the half-metallic nature of the middle spacer).\nThe equations of the motion of the device can be ob-\ntained by considering electron flow diagram of Fig. 2.\nThere are two reservoirs in M1 andM2 each. One is for\nSU electrons, and the other is for SD electrons (denoted\nby rectangular boxes). SU electrons in M1 can flow into\nSU electron reservoir of M2. The current Idepends on\nresistance Rand chemical potential difference ∆ V(Fig.\n1b) of SU electrons in M1 andM2,\nI=∆V\nR. (1)\nIf one neglects the direct tunneling of SD electron from\nM1 toM2 through the half-metal ( I′= 0 in Fig. 2), SD\nelectrons can only go to M2 by first flipping their spinsBatteryI\nI I1 2\nI'\nM1 SD electrons SD electronsSU electronsSU electrons\nM2\nI+I' I+I'(I+I')/2(I+I')/2\n(I+I')/2(I+I')/2Reservoir of\nReservoir of Reservoir ofReservoir of\nFIG. 2: Schematic illustration of current flow from and into\nspin-up and spin-down states in M1 andM2. At the steady\nstate, current flowing into any reservoir should be equal to\nthose flowing out.\nand converting themselves into SU electrons. Let τ1and\nτ2be the spin flipping time (spin-relaxation time T1[11])\ninM1 andM2, corresponding to the flipping rate of\n1/τ1and 1/τ2, respectively. Due to the conversion of SD\nelectronsto SU electronsthat is the product of the excess\nSD electrons n1∆µ1ξ1Aand the single electron flipping\nrate, the current from SD reservoirto SU reservoirin M1\nis\nI1=n1∆µ1eξ1A\nτ1, (2)\nwheren1is the density of states of SD electrons in M1 at\nthe Fermi level. Ais the cross section of M1. Similarly,\nthe current due to the conversion of SU electrons to SD\nelectrons in M2 is\nI2=n2∆µ2eξ2A\nτ2. (3)\nAt the steady state, there is no net electron build up\nanywhere in the circuit. Since the current through the\nbattery is unpolarized, spin-up electrons will be mixed\nup with spin-down electrons and be pumped by the bat-\ntery from M2 intoM1. Thus, the SU electrons make up\nhalf of the current while other half is made up from the\nSD electrons, and the balancing conditions and external\nconstraint require\nI1=I/2,\nI2=I/2, (4)\n∆µ1/e+∆µ2/e+∆V=V.\nSolving Eqs. (1), (2),(3), and (4), the dynamical Zeeman\nsplit ∆µ1and ∆µ2are\n∆µ1=(eV)τ1/(n1e2ξ1A)\n2R+τ1/(n1e2ξ1A)+τ2/(n2e2ξ2A),\n∆µ2=(eV)τ2/(n2e2ξ2A)\n2R+τ1/(n1e2ξ1A)+τ2/(n2e2ξ2A).(5)\nIt is interesting to see that the largest dynamical Zeeman\nsplit occur at R= 0, a short circuit for spin-up (SU)3\nelectrons! In this particular case, one, of course, needs\nto use metals with proper material parameters such that\ncurrent density is not too large to cause the metal break-\ning down by heating. Further discussions on this issue\nare given soon.\nThe half-metal may be replaced by a usual ferromag-\nnetic metal. In this case, SD electrons in M1 can also\nflow directly into M2, contributing an extra current I′\nto the circuit\nI′=V\nR′, (6)\nwhereR′is the resistance of ferromagnet for SD elec-\ntrons. Without losing generality, the minority carriers of\nthe ferromagnet are assumed to be the SD electrons, and\nR′is also assumed to be larger than R. The chemical po-\ntential difference of SD electrons in M1 andM2 equals\nVas it is shown in Fig. 1b. Eq. (4) should be modified\naccordingly as\nI1= (I−I′)/2,\nI2= (I−I′)/2, (7)\n∆µ1/e+∆µ2/e+∆V=V.\nThis set of equations with non-zero I′can be solved, and\nthe dynamicalZeemansplit ∆ µ1and ∆µ2, in comparison\nwith that of half-metal case, are reduced by a factor of\n(1−R/R′)\n∆µ1=eV(1−R/R′)τ1/(n1e2ξ1A)\n2R+τ1/(n1e2ξ1A)+τ2/(n2e2ξ2A),\n∆µ2=eV(1−R/R′)τ2/(n2e2ξ2A)\n2R+τ1/(n1e2ξ1A)+τ2/(n2e2ξ2A).(8)\nOne should not be surprised about this reduction from\nour early explanations of the origin of this split. Ob-\nviously, previous results Eq. (5) are recovered when\nR′=∞. Also, there are no chemical potential splits in\nM1 andM2 when the spacer is non-magnetic ( R=R′).\nTherefore, agoodhalf-metal(goodconductorforthe ma-\njority carriers and good insulator for the minority carri-\ners) should be used if one wants to maximize the split. In\nthe following discussion, R′=∞case is considered only.\nTheconventionalwayofintroducinganenergysplitfor\nSU and SD electrons is through the Zeeman effect. Due\ntothesmallvalueofBohrmagneton,anorderof10 Tfield\ncan only induce about 1 meVenergy split. However, the\ndynamical Zeeman split predicted here could easily be of\nthe order of 1 eV, a truly giant Zeeman effect. Compare\nwith the static Zeeman effect, this split is equivalent to\na field in the order of 104T, an impossible magnetic field\non the earth!It is interesting to notice that a large spin-dependent\nchemical potential difference means a large dynamical\nmagnet. For ∆ µi= 1eVand a typical electron den-\nsity of states of 1022eV−1cm−3for a metal, the dynami-\ncal magnetization is about 105A/mwhich is comparable\nwith manymagnets. Thus onecan useMOKE(magneto-\noptical Kerr effect) to ‘see’ the dynamical magnetization.\nUpon verification of this dynamical magnetism, it should\nbe interesting to explore the potential applications of\nthis electric-field controlled magnet. Another possible\napplication of the predicted phenomena is polarized elec-\ntron/light source[12]. Since electrons of one type of spins\noccupy higher energy levels than those of the opposite\nspin, the predicted effect can be used as a polarized elec-\ntron source, or light source when the electrons flip their\nspins and emitted well-defined polarized photons. Thus,\nthe phenomenon can be used to make tunable light emit-\nting diode or laser in very wide frequency range. The\nlarge chemical potential difference may also be used to\nenhance electron magnetic resonance signal. The elec-\ntron magnetic resonance is useful in probing material\nproperties in various technologieswith wide applications,\nincluding imaging in information processing.\nIt should be pointed out that we have not considered\nthe spatial distribution of the chemical potential split.\nOurresultsmaybe modified quantitatively, but not qual-\nitatively, when the detailed distribution is taken into ac-\ncount. This is because a factor characterized the dis-\ntribution should be added to equations (3) and (4). It\nshould also be noted that the giant dynamical Zeeman\nsplit reported here is different from the split in the DOS\nof a ferromagnet. In the usual ferromagnets, electron\nDOS is spin dependent, but the chemical potential of\nboth SU and SD electrons are the same at the equilib-\nrium. However, the dynamical Zeeman split predicted\nhere occurs inside a non-magnetic metal where the elec-\ntron DOS is spin-independent, but the spin-up and spin-\ndown electrons fill their DOSs to different levels, leading\nto a chemical potential difference.\nIn conclusion, we propose an inverse spin valve struc-\nture consisting of a ferromagnet sandwiched between\ntwo normal metals. Under a bias, we predicted a gi-\nant dynamical Zeeman split of the chemical potential\nfor spin-up electrons and spin-down electrons at metal-\nferromagnet interfaces. This prediction is yet to be con-\nfirmed by experiments.\nThe author would like to thank Prof. Yongli Gao and\nProf. John Xiao for the useful discussions. This work\nis supported by UGC, Hong Kong, through CERG grant\n(# 603007 and SBI07/08.SC09).\n[1] M.N. Baibich, J.M. Broto, A. Fert, F.N. Van Dau, F.\nPetroff, P. Etienne, G. Creuzet, A. Friederich, and J.Chazelas, Phys. Rev. Lett. 61, 2427 (1988).\n[2] G. Binach, P. Grunberg, F. Saurenbach, and W. Zinn,4\nPhys. Rev. B 39, 4828 (1989).\n[3] R.F.C. Farrow, C.H. Lee, and S.S.P. Parkin, IBM J. Res.\nDev. bf 34, 903 (1990).\n[4] T. Miyazaki and N. Tezuka, J. Magn. Magn. Mater. 139,\nL231 (1995).\n[5] J.S. Moodera et al., Phys. Rev. Lett. 74, 3273 (1995).\n[6] J.C. Slonczewski, J. Magn. Magn. Mater. 159, L1-L7\n(1996).\n[7] L. Berger, Phys. Rev. B 54, 9353 (1996).\n[8] X.R. Wang, and Z.Z. Sun, Phys. Rev. Lett. 98, 077201\n(2007).[9] Z.Z. Sun, and X.R. Wang, Phys. Rev. Lett. 97, 077205\n(2006); Phys. Rev. B 71, 174430 (2005); ibid73, 092416\n(2006);ibid73, (2007); T. Moriyama, R. Cao, J.Q. Xiao,\nJ. Lu, X.R.Wang, Q. Wen, andH.W. Zhang, Appl.Phys.\nLett.90, 152503 (2007).\n[10] S.D. Wang, Z.Z. Sun, N. Cue, H.Q. Xu, and X.R. Wang,\nPhys. Rev. B 65, 125307 (2002).\n[11] X.R. Wang, Y.S. Zheng, and S. Yin, Phys. Rev. B 72,\nR121303 (2005).\n[12] X.R. Wang, unpublished."    },    {        "title": "1012.4115v1.Ultrafast_Hole_Spin_Dynamics_in_Optically_Excited_Bulk_GaA.pdf",        "content": "arXiv:1012.4115v1  [cond-mat.mtrl-sci]  18 Dec 2010Ultrafast Hole-Spin Dynamics in Optically Excited Bulk GaA s\nMichael Krauß and Hans Christian Schneider∗\nDepartment of Physics and Research Center OPTIMAS,\nUniversity of Kaiserslautern, 67663 Kaiserslautern, Germ any\nDavid J. Hilton†\nDepartment of Physics, University of Alabama at Birmingham , Birmingham, AL, 35294-1170\n(Dated: July 17, 2018)\nWe present experimental and theoretical results on hole-sp in dynamics in bulk GaAs after ultra-\nfast optical excitation. The experimental differential tra nsmission are compared with a dynamical\ncalculation of the momentum-resolved hole distributions, which includes the carrier-carrier, carrier-\nphonon and carrier-impurity interaction at the level of Bol tzmann scattering integrals. We obtain\ngood agreement with the experimentally determined hole-sp in relaxation times, but point out that\ndepending on how the spin-polarization dynamics is extract ed, deviations from an exponential de-\ncay at short times occur. We also study theoretically the beh avior of the spin-relaxation for heavily\np-doped GaAs at low temperatures.\nResearch on spin dynamics and spintronics in semiconductors and me tals covers an immense variety of applications\nin information storage and manipulation in carrier spins in solid state sy stems.1–3Both storage and manipulation of\nspins are limited by spin relaxation, sothat an understanding ofspin r elaxationprocessesis important.4,5For ultrafast\nspin dependent dynamics on picosecond timescales, simplified relaxat ion-time approximations are no longer justified\nand a fully microscopic understanding of the complex spin dynamics is needed.6–8While electron spin-dynamics\ncontinue to be extensively investigated in semiconductors due to th eir extremely long spin-relaxation times,9hole-\nspin dynamics are intriguing for different reasons. In III-V semicon ductors, hole dynamics is inherently different\nfrom that of electrons because of the strong spin-orbit coupling o f the hole states. Thus, hole-spin relaxation and\nmomentum (energy) relaxation occur on a comparable ultrashort t imescale.\nIn this paper, we connect differential transmission measurements of the hole-spin dynamics10with a microscopic\ncalculation that includes the anisotropic band-structure as well as carrier-carrier and carrier-phonon/impurity scat-\ntering mechanisms at the level of Boltzmann scattering integrals. A lthough our approach does not take into account\nall the intricacies of the dynamical multi-band screening (including ph onon-plasmon coupling)11,12and interband\npolarizations, it captures essential aspects of the spin dynamics o f holes.13We show that this calculation compares\nwell with measured hole-spin relaxation times, but that deviations fr om an exponential spin decay occur for short\ntimes depending on how the relaxation time is extracted. Further, w e present results for the hole-spin relaxation at\nhigh p-doping, which underlies a recent treatment of GaAs-based f erromagnetic semiconductors.14\nRecent theoretical treatments of hole spin (and charge) dynamic s in bulk GaAs have focused on the calculation of\nenergy-dependent spin relaxation-rates due to phonon/impurity scattering,15the investigation of coherent hole (spin)\ndynamics in GaAs,16,17and the influence of Dyakonov-Perel type precession effects due to spin-dependent splittings\nbetween hole bands in GaAs quantum wells.18Very recently, additional contributions due to “small” band struct ure\neffects andthe spin-orbitcontributionto the hole-phononinterac tionforholesin bulk GaAs havealsobeen analyzed.19\nFrom an experimental point of view, hole-spin polarizations can be cr eated efficiently using optical orientation\ntechniques.20Because of the ultrashort lifetimes21involved, it is difficult to unambiguously study hole dynamics\nin degenerate pump-probe experiments (i.e., experiments with the s ame pump and probe wavelengths) due to the\ncompetingpresenceofthecoherentartifactonthesametimesca le22and,toalesserdegree,theelectronspindynamics.9\nHole spin study, therefore, requires measurement of the spin-de pendent carrier dynamics on ultrashort timescales\nusing a time-delayed, mid-infrared probe beam (i.e., a nondegenerat e pump-probe experiment), which was realized\nonly much later10than similar experiments on electrons.9\nWe focus here on the dynamics of the spin-polarized heavy hole subb and. We have performed nondegenerate,\npolarization-resolved pump-probe spectroscopy at 300K to meas ure the hole spin relaxation lifetime in intrinsic\nGaAs as a function of input pump fluence, F. In this setup, 1W of average power from an 80fs titanium:sapphire\nlaser (Spectra Physics Tsunami) operating at an 80MHz repetition r ate is used to pump a custom-designed optical\nparametric oscillator based on periodically-poled lithium niobate, as de scribed in further detail in ref. 23. A portion of\nthe titanium:sapphire beam ( ≤300mW) is circularly polarized using a quarter wave plate and is used to photoexcite\nspin-polarized electron-hole pairs in the GaAs sample. The pump beam induces a circular birefringence and results\nin a time-dependent rotation of the transmitted probe polarization . To isolate the dynamics of the hole spin state\nfrom the electron spin, we probe the occupancy of the heavy hole a nd/or light hole subbands using a synchronized\nmid-infrared idler pulse ( λ= 3200 nm), which probes the induced transitions from the split off su bband into the\nheavy hole subband. This wavelength directly probes states from t he lower split-off hole subband to the heavy hole2\n−400−200 020040060080000.050.10.150.20.250.30.35\nDelay(fs)∆T/T(a.u.)\n  \n  \n13.8 µJ cm−2, τs = 0.130 ps\n11.4 µJ cm−2, τs = 0.135 ps\n8.9 µJ cm−2, τs = 0.145 ps\n6.0 µJ cm−2, τs = 0.120 ps\n3.8 µJ cm−2, τs = 0.125 ps\n2.6 µJ cm−2, τs = 0.130 ps\nFIG. 1. Fluence dependence of the time-resolved differentia l transmission, ∆ T/T(t), at 300 K.\nsubband at the same quasimomentum ( k≈0.3 nm−1) as the initial photoexcitation at 800nm.\nFigure 1 shows the measured field rotation, ∆ T/T/parenleftbig\nt/parenrightbig\n, of the mid-IR probe from F= 2.6 to 13.8 µJ cm−2. The\ninput probe beam is linearly polarized (ˆ x), which is an equal admixture of both the co-circularly polarized (ˆ σ+) and\ncounter circularly polarized(ˆ σ−). A MgF 2linear polarizer is rotated 90◦with respect to the input polarization (i.e.,\nalong ˆy) and is used to measure the field rotation of the transmitted probe , which depends on the time-dynamics of\nthe the heavy hole occupancy state near the quasimomentum, k, of photogeneration. Using the measured absorption\ncoefficient ofGaAs,24the resultingcarrierconcentrationsrangefrom 5 ×1016cm−3(F= 2.6µJcm−2) to2×1017cm−3\n(F= 13.8µJcm−2). We fit the measured field rotation to a single exponential model an d account for the finite pump\nand probe pulse widths by convolving the model function with a gauss ian with a width of 120fs (see ref. 10 for more\ndetails of the fitting model). We find no significant trend in the extrac ted lifetimes, which range from 120fs to 145fs,\nwithin the experimental error ( ±20%) of this experiment.\nThe calculation of the hole-spin dynamics follow the procedure outline d in ref. 13. The electron and hole states\naround the fundamental bandgap needed as input for the dynamic al calculation are determined from an 8-band Kane\nmodelH(/vectork) with six hole and two electron bands containing terms up to second o rder ink.25By diagonalizing H(/vectork)\nwe obtain the “intelligent basis” (in the sense of ref. 26) for the hole states:|ν,/vectork/angbracketrightand their energy eigenvalues εν,/vectork.\nThe label ν= (b,p) includes the band index b= E, HH, LH, SOH, (for electrons, heavy holes, light holes, and split- off\nholes) and the pseudospin p= 1,2. The pseudospin can be introduced because the quasiparticle disp ersions of all\nfour types of carriers are (nearly) doubly degenerate.\nThe dynamical equations governing the time evolution of the incoher ent carrier distributions, nν,/vectork, under the\ninfluence of optical fields and scattering are:13,27\n∂\n∂tnν,/vectork= Γin\nν,/vectork(1−nν,/vectork)−Γout\nν,/vectorknν,/vectork(1)\n(plus a contribution from the optical excitation, which neglects the hole band coherences between hole states of the\n“intelligent basis,” or, equivalently, only describes contributions to t he hole spin relaxation of the Elliott type26that\narise from the /vectorkdependent mixing of different spins in the states |ν,/vectork/angbracketright. Contributions to the spin relaxation via\ncoherent spin precession,i.e., Dyakonov-Perel-type spin dynamics,19are neglected.\nThe dynamical in-scattering rate consists of the carrier-carrier interaction contribution (i.e., eq. (2) in Ref. 13) and\nthe carrier-phonon interaction\nΓin\nν,/vectork|c−p=2π\n/planckover2pi1/summationdisplay\nν1,/vector q,λ|Mq,λ|2|/angbracketleftν1,/vectork+/vector q|ν,/vectork/angbracketright|2nν1,/vectork+/vector q[(1+Nq,λ)δ(∆E−)+Nq,λδ(∆E+)]. (2)\nwhere ∆E±=εν,k−εν1,|/vectork+/vector q|±/planckover2pi1ωq,λ. There is also a contribution similar to eq. (2) due to carrier-impurity scattering.7\nHere,Nq,λ,Mq,λ, andωq,λare the phonon occupation numbers, carrier-phononcoupling mat rix elements, and phonon\ndispersions for LO and LA phonons, respectively.28The out-scattering rates Γoutare obtained from Γinby exchanging\n(1−n) withnand (1+ N) withN. Equation (2) and its counterpart for Coulomb scattering describ es two-particle\nscattering processes connecting states |ν,/vectork/angbracketright → |ν1,/vectork+/vector q/angbracketright(and|ν2,/vectork1+/vector q/angbracketright → |ν3,/vectork1/angbracketright) with different average spin.\nThe pronounced anisotropy of the single-particle states, which is d ue to the spin-orbit interaction, is included in\nour calculation in the overlaps and the carrier energies, ǫν,/vectork, which enter the energy-conserving delta functions in the3\n0 0.5 1 1.5 2\nx 101750100150200\nExcited Carrier Density (cm−3)Spin−Relaxation Time (fs)\nFIG. 2. Theoretical (curve) and experimental values (symbo ls) for fluence dependence of hole-spin relaxation in intrin sic GaAs\nat room temperature.\nscattering rates. The anisotropies and high-dimensional scatter ing integrals result in a challenging numerical problem.\nWe incorporate the effects of the anisotropy by expanding nν,/vectork(t) into spherical harmonics Yℓ,m(ˆk) up to order ℓ= 4\nand retain only the expansion coefficients with radial or cubic symmet ry, as these are the dominant symmetries of\nthe Hamiltonian H.25This procedure, together with the use of static screening, vs(q) =e2/[ǫ0ǫbg(q2+κ2)], reduces\nthe numerical complexity. Additional simplifications are achieved by in cluding only the heavy-hole bands in the\ndynamical calculation as the number of optically excited light holes is mu ch smaller,13and by replacing the non-\nequilibrium electron distributions by equilibrium distributions with the lat tice temperature. We then calculate the\nspin polarization of the HH band. In ref. 13, it was shown that the sp in polarization is not identical to the DT signal\nthat is measured, but the deviations are small enough for the accu racy of the present comparison.\nWe include the carrier excitation by an ultrashort optical pulse using a ˆσ+polarized plane wave traveling in the\n(001) direction (ˆ z) to generate an initial condition for the carrier distributions\nnν,/vectork(t= 0) =/summationdisplay\nµ|/vectordµν(/vectork)·/vectorE|2g(/planckover2pi1ω−εµ,/vectork−εν,/vectork). (3)\nHere,/vectordµν(/vectork) =e/angbracketleftµ,/vectork|/vector r|ν,/vectork/angbracketrightare the dipole-matrix elements, /vectorEis the laser field, and /planckover2pi1ωis the photon energy of the\nexciting field. A Gaussian broadening function g/parenleftbig\nε/parenrightbig\npeaked at ε=/parenleftbig\n/planckover2pi1ω−εµ,/vectork−εν,/vectork/parenrightbig\naccounts for the spectral width\nof the pulse (15meV).\nTo compare with experimental results in Fig. 2, we convert the expe rimental laser fluence into excited carrier\ndensities, andextracttherelaxationtimefromanexponentialfit t othespinpolarizationattheprobe-laserwavelength.\nThe experimental results are in good agreement with our calculation s and a recent theoretical study.19. Fig. 2 shows\nthat hole spin-relaxation is rather insensitive to excited carrier den sities on the order of 1017cm−3. The calculation\nalso predicts a very weak temperature dependence with a hole spin r elaxation time of 250fs for an excited density of\n1017cm−3at 4K, which is somewhat shorter than the result predicted by the E lliott-Yafet spin relaxation-time .15\nIn Fig. 3, we simulate the momentum dependence of the spin polarizat ion dynamics. We model the excitation by\nan 800nm pump pulse that excites a carrier density of 1017cm−3. This leads to anisotropic initial hole distributions\nthat are peaked at momentum k= 0.3nm−1(after integration over the angular variables). Fig. 3 shows differe nt spin\npolarizations calculated using the angle-averaged distributions n(k,t) for different modulus of the hole momentum k.\nThe decay of the spin polarization near the peak position of the initial distribution is nearly exponential, as can be\nseen from the linear curve shape on the logarithmic scale. The evolut ion of the spin polarization for larger and smaller\nmomenta (i.e, away from the momenta of the initially photoexcited hole s) is significantly faster and non-exponential\nfor the initial ∼100fs. After the initial non-exponential dynamics the spin-relaxa tion becomes exponential again with\nthe same time constant. The spin relaxation-time therefore depen ds on the way it is extracted from the calculation,\nas opposed to the case of electron spin-relaxation in p-doped GaAs where the spin relaxation time can be quite\nrigorously defined.29The dashed lines in Fig. 3 correspond to experimental situations whe re pump and probe pulses\nare detuned, as was investigated in ref. 10 (see Fig. 5 therein), wh ere no change in spin relaxation times for different\ndetuning between pump and probe pulses was found. The apparent discrepancy likely results from the analysis of the\nexperimental data via a convolution of Gaussians, which are used to describe both the pulse shapes and the dynamics\nof the material response. The analysis of the pump-probe experim ent therefore presupposed an exponential decay of\nthe polarization, which turns out to be only somewhat longer than th e experimental time resolution of about 100fs10.4\n0 50 100 150 20010−1100\nTime (fs)Normalized Polarization\n  \nk=0.1nm−1\nk=0.3nm−1 (peak)\nk=0.5nm−1\nFIG. 3. Computed hole-spin polarization dynamics at differe nt hole momenta for fixed excitation at k= 0.3 nm−1in intrinsic\nbulk GaAs. The excited carrier density is n= 1017cm−3.\n1018101910200100200300400500\nDoping Density (cm−3)Spin−Relaxation Time (fs)\nFIG. 4. Computed p-doping dependence of hole-spin relaxati on in GaAs at low temperatures (solid line). The dashed line\nresults if only scattering by from ionized donors is include d.\nFor degenerate pump and probe, this analysis yields the exponentia l decay of the spin polarization in agreement with\nthe calculated result in Fig. 3. For non-degenerate situations the n on-exponential features in Fig. 3 on timescales\nbelow 100fs are not resolved. Their resolution would require a time re solution well below 100fs.\nFinally, we examine the hole-spin relaxation dynamics in heavily p-doped GaAs at low temperatures (5K) theo-\nretically. By choosing these parameters, we estimate the hole-spin relaxation time for GaMnAs in the ferromagnetic\nphase at low temperatures, which has similar carrier doping levels. We model the optical excitation of nonequilibrium\ncarriers in p-doped GaAs by a pump laser with a wavelength of 400nm. Choosing this wavelength ensures that the\noptical transitions are not blocked by holes even for very high dopin g densities. We assume initial carrier distributions\nof the form n0\nk=ndop\nk+nexc\nk, which contain both the influence of the doping and the optical excit ation. All itinerant\nholes introduced by the p-doping are assumed to be thermalized at t he lattice temperature; they are modeled by\nunpolarized Fermi-Dirac electron distributions ndop=f(ǫe\nk) at lattice temperature with carrier density equal to the\ndensity of dopants. For the optical excitation we assume again an u ltrashort pulse with a width of 15meV. Fig. 4\nshows that the hole-spin relaxation drops to approximately 10fs fo r doping densities typical for GaAs based magnetic\nsemiconductors. At these high doping densities, the rapid momentu m relaxation due to scattering from the ionized\ndonors provides the main relaxation mechanism. The assumption of a hole spin relaxation time of ∼10fs in ref. 14\nmakes it possible to neglect a spin-bottleneck effect for ferromagn etic GaMnAs. Our results, based on a dynamical\ncalculation, support this assumption.5\nIn conclusion we examined ultrafast hole-spin relaxation using differe ntial transmission measurements and a mi-\ncroscopic theoretical approach for different excitation condition s. Theory and experiment show that the hole-spin\nrelaxation time is approximately 100 fs and independent of moderate changes of the pump fluence at room temper-\nature. The concept of spin-relaxation time is not, in general, applica ble for non-degenerate pump-probe schemes,\nbut the deviations from the exponential decay of the spin-polariza tion are confined to time scales of less than 100fs,\nand can therefore only be resolved by a time resolution well below 100 fs. These results also demonstrate the limited\nvalidity of a single spin relaxation-time for holes. Strong p-type doping shortens relaxation times significa ntly due to\nthe scattering from ionized donors. The calculated spin relaxation t imes support the assumption of very fast hole-spin\nrelaxation in ferromagnetic GaMnAs at low temperatures.\nWe acknowledge support by the DFG through GRK 792 and a grant fo r CPU time from the the NIC J¨ ulich. We\nare grateful to C. L. Tang and M. W. Wu for helpful discussions.\n∗hcsch@physik.uni-kl.de\n†dhilton@uab.edu\n1D. D. Awschalom, D. Loss, and N. Samarth, eds., Semiconductor Spintronics and Quantum Computation (Springer, Berlin,\n2002).\n2S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton, S. v on Molnar, M. L. Roukes, A. Y. Chtchelkanova, and\nD. M. Treger, Science 294, 1488 (2001).\n3I.ˇZuti´ c, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004).\n4L. J. Sham, J. Phys.: Cond. Matt. 5, A51 (1993).\n5N. S. Averkiev, L. E. Golub, and M. Willander, J. Phys.: Cond. Matt.14, R271 (2002).\n6M. M. Glazov and E. L. Ivchenko, JETP Lett. 75, 403 (2002).\n7M. Q. Weng and M. W. Wu, Phys. Rev. B 68, 075312 (2003).\n8H. C. Schneider, J. P. W¨ ustenberg, O. Andreyev, K. Hiebbner , L. Guo, J. Lange, L. Schreiber, B. Beschoten, M. Bauer,\nand M. Aeschlimann, Phys. Rev. B 73, 081302(R) (2006).\n9J. M. Kikkawa and D. D. Awschalom, Phys. Rev. Lett. 80, 4313 (1998).\n10D. J. Hilton and C. L. Tang, Phys. Rev. Lett. 89, 146601 (2002).\n11J. H. Collet, Phys. Rev. B 47, 10279 (1993).\n12M. Woerner and T. Elsaesser, Phys. Rev. B 51, 17490 (1995).\n13M. Krauss, M. Aeschlimann, and H. C. Schneider, Phys. Rev. Le tt.100, 256601 (2008).\n14/suppress L. Cywi´ nski and L. J. Sham, Phys. Rev. B 76, 045205 (2007).\n15Z. G. Yu, S. Krishnamurthy, M. van Schilfgaarde, and N. Newma n, Phys. Rev. B 71, 245312 (2005).\n16A. Dargys, Phys. Rev. B 70, 125207 (2004).\n17D. Culcer, C. Lechner, and R. Winkler, Phys. Rev. Lett. 97, 106601 (2006).\n18C. Lu, J. L. Cheng, and M. W. Wu, Phys. Rev. B 73, 125314 (2006).\n19K. Shen and M. W. Wu, Phys. Rev. B 82, 115205 (2010).\n20F. Meier and B. P. Zakharchenya, eds., Optical Orientation (North-Holland, Amsterdam, 1984).\n21F. Ganikhanov, K. C. Burr, D. J. Hilton, and C. L. Tang, Phys. R ev. B60, 8890 (1999).\n22C. W. Luo, Y. T. Wang, F. W. Chen, H. C. Shih, and T. Kobayashi, O ptics Express, 17, 11321 (2009).\n23K. Burr, C. Tang, M. Arbore, and M. Fejer, Opt. Lett. 22, 1458 (1997).\n24J. Blakemore, J. Appl. Phys. 53, 520 (1982).\n25R. Winkler, Spin-Orbit Coupling Effects in Two-Dimensional Electron and Hole Systems , Springer Tracts in Modern Physics,\nVol. 191 (Springer, Berlin, 2003).\n26J. Fabian, A. Matos-Abiague, C. Ertler, P. Stano, and I. ˇZuti´ c, Acta Physica Slovaca 57, 565 (2007).\n27J. H. Collet, Phys. Rev. B 47, 10279 (1993).\n28P. Y. Yu and M. Cardona, Fundamentals of Semiconductors , 3rd ed. (Springer, Berlin, 2001).\n29H. C. Schneider and M. Krauß, in Ultrafast Phenomena in Semiconductors and Nanostructure M aterials XIV , Proceedings\nof SPIE, Vol. 7600 (SPIE, Bellingham, WA, 2010) p. 1D."    },    {        "title": "2108.11243v2.Probing_Kondo_spin_fluctuations_with_scanning_tunneling_microscopy_and_electron_spin_resonance.pdf",        "content": "Probing Kondo spin \ructuations with scanning tunneling microscopy and electron\nspin resonance\nYinan Fang,1, 2Stefano Chesi,1, 3, 4,\u0003and Mahn-Soo Choi5,y\n1Beijing Computational Science Research Center, Beijing 100193, People's Republic of China\n2School of Physics and Astronomy, Yunnan University, Kunming 650091, China\n3ICTP, Strada Costiera 11, I-34151 Trieste, Italy\n4Department of Physics, Beijing Normal University, Beijing 100875, People's Republic of China\n5Department of Physics, Korea University, Seoul 02841, South Korea\n(Dated: November 3, 2021)\nWe theoretically analyze a state-of-the-art experimental method based on a combination of elec-\ntron spin resonance and scanning tunneling microscopy (ESR-STM), to directly probe the spin\n\ructuations in the Kondo e\u000bect. The Kondo impurity is exchange coupled to the probe spin, and\nthe ESR-STM setup detects the small level shifts in the probe spin induced by the spin \ructuations\nof the Kondo impurity. We use the open quantum system approach by regarding the probe spin\nas the \\system\" and the Kondo impurity spin as the \ructuating \\bath\" to evaluate the resonance\nline shifts in terms of the dynamic spin susceptibility of the Kondo impurity. We consider various\ncommon adatoms on surfaces as possible probe spins and estimate the corresponding level shifts.\nIt is found that the sensitivity is most pronounced for the probe spins with transverse magnetic\nanisotropy.\nI. INTRODUCTION\nEver since the \frst theoretical explanation, dating back\nmore than 50 years [1], the Kondo e\u000bect has attracted\ngreat interest not only in condensed matter physics but\nalso in other areas of physics [2{7]. It serves as a phys-\nical paradigm to understand other strong correlation ef-\nfects and helped the developments of powerful theoretical\nmethods [8{12]. Recent advances in nanotechnology, al-\nlowing the control of electronic states down to a single\nmagnetic impurity, have brought renewed interest in this\ntopic [13{15], and more exotic Kondo physics has been\ndesigned and explored [16{18].\nDespite this progress, the answer to some of the most\nfundamental questions, such as the size of the Kondo\nscreening cloud [19{22] (which is an essential feature of\nthe Kondo e\u000bect), still remain elusive due to the lack of\ndirect experimental probes. One of the measurements\nclosest to a direct assessment of the Kondo screening\ncloud considers the compressibility (i.e., the charge sus-\nceptibility) of the Kondo impurity [23]. By measuring\nthe suppression of the compressibility in the presence\nof a Kondo-enhanced transport current, the authors of\nRef. 23 have demonstrated that the Kondo e\u000bect is due\nto spin \ructuations without charge \ructuations. How-\never, the experiment probed only the charge sector but\nnot directly the spin \ructuations themselves.\nHere, we propose a method to directly probe the spin\n\ructuations of the Kondo e\u000bect by taking advantage of\nthe excellent spatial ( \u0017A) and energy ( \u0016eV) resolution [24{\n27] of scanning tunneling microscopy (STM) and spec-\ntroscopy (STS). Before discussing our scheme, we note\n\u0003stefano.chesi@csrc.ac.cn\nychoims@korea.ac.krhow STM and STS have allowed many remarkable exper-\niments, such as the observation of charge density oscilla-\ntions surrounding a magnetic adatom [28, 29], and later\nof the Kondo resonance in Ce or Co adatoms [25, 30].\nVarying the lateral tip-adatom distance has revealed links\nbetween the asymmetry of the Kondo resonance and dif-\nferent types of tunneling matrix elements [25, 31]. The\ncapability of manipulating adatoms, vividly illustrated\nby the quantum mirage [32, 33], has allowed control-\nlable studies on the e\u000bect of interaction between two\nnearby Kondo impurities [34, 35], such as in the two-\nimpurity Kondo model [36, 37]. Equipped further with\nspin-polarized tips, STM can also investigate magnetic\nproperties such as magnetic anisotropy [38, 39] and mag-\nnetic exchange coupling [35] at the atomic level.\nThe setup we consider here is based on a combination\nof electron spin resonance and scanning tunneling mi-\ncroscopy (ESR-STM, see Fig. 1) [40{42]. The key idea is\nto detect the small level shifts in the probe spin due to the\nspin \ructuations of the Kondo impurity. In essence, the\nlevel shifts are similar to the Lamb shift of atomic levels\ndue to the quantum \ructuations of the electromagnetic\n\feld [43, 44]. In fact, the charge compressibility mea-\nsurement [23] was along the same principle, by detecting\nthe shift of the cavity resonance frequency due to the\ncharge \ructuations on the quantum dot. In that case,\nthe bosonic nature (i.e. linear system) of the cavity pho-\ntons allows one to estimate the shift by means of the\nrandom phase approximation or similar methods.\nIn our case, the probe spin is a highly non-linear system\nand the theoretical analysis is more complicated. We ex-\npress and estimate the level shifts in terms of the dynamic\nspin susceptibility based on the open quantum system\napproach, by regarding the probe spin as the \\system\"\nand the Kondo impurity spin as the \ructuating \\bath\".\nTesting various common adatoms (as the probe spin) on\nsurfaces, we illustrate that the level shifts are within thearXiv:2108.11243v2  [cond-mat.mes-hall]  2 Nov 20212\nSP-tip\nprobe spin\nmetal substrateKondo \nimpurity\nFIG. 1. Schematics of a Kondo impurity (right dashed box)\ncoupled to a probe spin measured by an ESR-STM device\n(left dashed box). The Kondo impurity is tunnel-coupled\nwith electrons in the metal substrate, and can be described\nby the Anderson model [45]. The probe spin is adsorbed on\nthe top of an insulating layer (yellow), and modulates the\ntunneling current through the spin-polarized (SP) tip under\nmicro-wave voltage drive Vrf. Upper-right inset: Schematic\nplot of the low-energy spectrum of a typical probe spin with\nS= 2 (e.g., Fe on MgO) from the e\u000bective spin Hamiltonian\nEq. (2), see also Fig. 6 in Appendix C. The transition fre-\nquencyf0between the low-energy states j0iandj1i, modi\fed\nby the dynamical spin \ructuations of the Kondo system, is of\nprimary interest in this study.\nresolution of ESR-STM. We \fnd that transverse mag-\nnetic anisotropy (in addition to the common axial one)\nof the probe spin enhances the level shifts.\nThis article is organized as follows: In Sec. II we\npresent the model of the Kondo impurity and discuss\nhow it is coupled to, and detected through, the ESR-\nSTM-based probe spin. In Sec. III we develop the gen-\neral theory underlying this work. We derive the shifts in\nthe energy levels of the probe spin, and connect them to\nthe dynamical spin susceptibility of the Kondo impurity.\nThen, those general discussions are applied in Sec. IV\nto speci\fc examples from various common impurities ap-\npeared in the literature. Section V concludes the article\nwith some additional remarks. To maintain the major\nscope of the work clearer, we defer some technical details\nto Appendices A, B, and C.\nII. SETUP AND MODEL\nWe propose to utilize the ESR-STM techniques to per-\nform an accurate spectroscopic analysis of spin \ructua-\ntions in the Kondo e\u000bect. Our ESR-STM-based setup is\nschematically shown in Fig. 1 and consists of a Kondo\nimpurity magnetically coupled to a probe spin. It is gov-\nerned by the Hamiltonian\n^H=^HKondo +^Hprobe +^Hexchange; (1)where the three di\u000berent terms refer to the system, probe,\nand system-probe interaction Hamiltonians, respectively.\n^Hexchange is typically small, and the Kondo impurity can\nbe treated as providing an e\u000bective environment to the\nprobe spin. The \ructuating environment, i.e., the Kondo\nimpurity, induces small shifts to the energy levels of the\nprobe spin. Such shifts will then be re\rected in the po-\nsition of ESR lines detected by the STM tip. The aim of\nthe setup is therefore to indirectly probe the spin \ructu-\nations of the Kondo system through the displaced ESR\nresonance line of the probe spin. We will provide below\nthe theoretical framework to compute such e\u000bects.\nA. Probe impurity and ESR-STM detection\nThe spin of the probe impurity (hereafter simply called\ntheprobe spin ) is coupled to the metal substrate through\nan insulating layer as shown in Fig. 1 (while the metallic\nsubstrate is necessary for the STM current, the insulating\nlayer prevents the probe spin from being directly coupled\nto the metallic substrate, which would screen the probe\nspin). It is addressed with an STM device. To a good\napproximation, several relevant magnetic impurities can\nbe described by the e\u000bective spin Hamiltonian [35, 46{\n50]:\n^Hprobe =Ds^Sz\np^Sz\np+Es(^Sx\np^Sx\np\u0000^Sy\np^Sy\np)\n+g\u0016BB\u0001^Sp;(2)\nwhere ^S\u0016\npis the\u0016-th component of the probe-spin angular\nmomentum ( \u0016=x;y;z ),DsandEsare the axial and\ntransverse anisotropy parameters, resepectively, Bis the\nmagnetic \feld, and gis theg-factor of the probe spin.\nTypical values of the parameters from the literature are\nlisted in Table I. Physically, the DsandEsterms arise as\nsecond-order corrections from the spin-orbit interaction,\nin the presence of crystal-\feld anisotropy induced by the\ninsulating substrate [40, 51].\nThe spin-dependent level structure of the probe spin is\ndetected accurately by the ESR-STM measurement with\na spin-polarized tip. Note that the same setup can ac-\ntively manipulate/detect the spin state of the probe spin\nas well. All these are achieved by feeding a micro-wave\ndriveVrfof frequency fon the bias voltage of the spin-\npolarized tip, which induces a time-dependent change to\nthe local electrostatic environment of the probe spin. The\ndrive induces spin transitions via the spin-orbit coupling.\nWe will be interested below in the transition between the\ntwo lowest levels of the probe spin, which we denote as\nj0i(the ground state) and j1i(the \frst excited state).\nSince typically Es\u001cDs(see Table I), at small magnetic\n\feld these two states form a doublet. When Ds<0:\nj0i;j1i'jS;\u0006Si; (3)\nwhile forDs>0 andSis half-integer (like in the case\nof Co) the low-energy states are j0i;j1i'jS;\u00061=2i. Be-\ncause the probe spin states involved by the ESR drive3\nTABLE I. The parameters in the e\u000bective spin Hamiltonian\nfor some reported probe spins. DsandEs(in unit of meV)\nare the axial and transverse magnetic anisotropy parameters\nin Eq. (2). gis the electron g-factor,Sis the collective\nspin angular momentum. Base A/B indicates the material\nfor the insulating layer (A) and the metallic STM substrate\n(B), while the binding site and local symmetry group (Sym.)\nof the probe spin are also speci\fed.\nType S Base Site Sym. DsEsg\nMn [50] 5/2 Cu 2N/Cu Cu D2-0.039 0.007 1.90\nFe [50] 2 Cu 2N/Cu Cu D2-1.55 0.31 2.11\nCe [52] 3/2 Cu 2N/Cu Cu D2-1.3 0.18\nFe [40, 51, 53] 2 MgO/Ag O C4v-4.7 0 2.6\nCo [35, 38, 54] 3/2 Cu 2N/Cu Cu 2.75 0 2.19\nCo [55] 3/2 MgO/Ag O C4v27.5\nTi [56, 57] 1/2aMgO/Ag - - 1.98\naThe Ti probe spin in Refs. 56 and 57 has spin 1 =2 because of a\nhydrogen atom attached to the Ti adatom. It has been pointed\nout that for clean Ti on MgO one would have S= 1 [56].\ndi\u000ber in their magnetic moments, the spin transition\nwill cause a change in tunneling current I(f) through\nthe probe spin from the spin-polarized tip to the sub-\nstrate. Technical details of the tunneling current I(f)\nare explained in Appendix A, following previous litera-\nture [40, 58, 59]. The signal is approximately given by:\nI(f)'I0+Ip\u0014\n2T1=T2\n4\u00192(f\u0000f0)2+ \n2T1=T2+ 1=T2\n2\u0015\n;(4)\nwhere \n is the Rabi frequency of the drive and f0:=\n(E1\u0000E0)=2\u0019~is the ESR transition frequency. 1 =T1\nand 1=T2are, respectively, the relaxation and dephas-\ning rates of the probe spin. The explicit expressions for\nthe background ( I0) and saturation ( Ip) currents are de-\nscribed in the Appendix A [see Eqs. (A13)], and include\nboth contributions from elastic and inelastic tunneling\n[58, 59].\nAs we explain in detail in Appendix A, the tunnel-\ning current Eq. (4) is derived under the assumption of a\nsmall bias voltage between the spin-polarized tip and the\nsubstrate, i.e., the bias is much smaller then Ds. Thus,\nonly the lowest states j0i;j1iof the probe spin need to\nbe considered. Under this two-level approximation, it is\nknown that the change in tunneling current I(f) induced\nby the drive is simply proportional to the steady-state\npopulation P1(f) of the upper level [40]. When the inco-\nherent excitation rate \u0000 0!1from the ground state j0ito\nthe excited state j1iis negligible, a simpli\fed expression\nforP1(f) can be obtained by solving the Bloch equations\n[40], and is given by the square parenthesis of Eq. (4). As\na result, the ESR signal I(f) has a Lorentzian lineshape\ncentered around the 0-1 transition frequency, which al-\nlows to map out the ESR spectrum by sweeping the drive\nfrequencyf.B. Kondo impurity\nThe Kondo impurity is shown in the right dashed box\nof Fig. 1. Unlike the probe spin (coupled to an insu-\nlating host), it is strongly coupled to the metallic sub-\nstrate, which justi\fes neglecting anisotropy e\u000bects (an\nanisotropy in the Kondo coupling is irrelevant in the\nrenormalization group sense as the strong-coupling \fxed\npoint is isotropic). One could describe it with a standard\nversion of the Anderson model [45]:\n^HKondo = X\n\u001b\u000f\u001b^nd;\u001b+U^nd;\"^nd;#!\n+X\nk;\u001b\"k^cy\nk;\u001b^ck;\u001b\n+X\nk;\u001bVk\u0010\n^dy\n\u001b^ck;\u001b+ ^cy\nk;\u001b^d\u001b\u0011\n(5)\nwhere the \frst term (in the brackets) refers to the Kondo\nimpurity, the second to the substrate, and the third (sec-\nond line) to the electron tunneling between the Kondo\nimpurity and the substrate. In Eq. (5), ^d\u001band ^ck;\u001b\nare annihilation operators for the impurity and sub-\nstrate, respectively, and ^ nd;\u001b=^dy\n\u001b^d\u001bgives the occupa-\ntion of the Kondo impurity level by an electron with spin\n\u001b=\";#. Here we assume that the Kondo impurity has\nspinSK= 1=2. For larger spins, the Kondo e\u000bects are\neven richer. However, the dynamic spin \ructuations due\nto such exotic Kondo e\u000bects can also be detected in the\nsame method, and the resulting frequency shift can be\nestimated in an essentially similar method described be-\nlow.\nA crucial quantity characterizing magnetic \ructuations\nof the Kondo impurity and hence a\u000becting the ESR-STM\nsignal, is the dynamical spin susceptibility:[60]\ni~\u001f\u0016(!) =\u0000(g\u0016B)2Z1\n0dtei!tD\n[^S\u0016\nK(t);^S\u0016\nK(0)]E\n;(6)\nwhere the spin angular momentum operator of the\nKondo impurity is given by ^S\u0016\nK:=1\n2P\ns;s0^dy\ns\u001b\u0016\ns;s0^ds0;\nwith\u001b\u0016the\u0016-th Pauli matrix, and ^S\u0016\nK(t) =\nei^HKondot=~^S\u0016\nKe\u0000i^HKondot=~. While\u001f\u0016(!) is highly non-\ntrivial to compute theoretically and di\u000ecult to access\nexperimentally, the Bethe Ansatz method provides one\nrelevant result [61]\n\u001f\u0016(0) =(g\u0016B)2\n4\u0019kBTK; (7)\nwheregis theg-factor,\u0016Bis the Bohr magneton, and\nTKis the Kondo temperature [62]. The full dynamic\nspin susceptibility could be obtained, e.g., through the\nnumerical renormalization group (NRG) method [63]. A\nqualitative feature of \u001f\u0016(!) is the presence of a peak at\n!= 0 of width kBTK=~, which develops when the tem-\nperature of the Kondo system is lowered to T\u0014TK.\nTherefore, for qualitative estimations at low tempera-\ntures andj!j.kBTK=~, one may resort to the following4\napproximate form (setting \u0016Bto unity):\n\u001f\u0016(!)\u0019\u001f\u0016(0)ikBTK\n~!+ikBTK; (8)\nwhere Re\u001f\u0016(!) gives a Lorentzian peak and Im \u001f\u0016(!) fol-\nlows from the the Kramers-Kronig relations. As we will\nsee, it is Im \u001f\u0016(!) which determines the shift in ESR fre-\nquencies of the probe spin.\nImportantly, the sole characteristic energy scale of the\nKondo impurity, i.e., the Kondo temperature TK, is large\nin our problem. For example, it is known that the\nKondo temperatures of a Co impurity can reach nearly\n100 K\u00189 meV when deposited without an insulating\nlayer on Ag(111) [64]. Among Kondo systems with an in-\nsulating layer, despite TKmay signi\fcantly degrade as a\nresult of partial decoupling to the metallic surface, mod-\nerate values of\u001815 K have been reported for Ce clusters\ndeposited on a monolayer of Cu 2N on top of Cu(100) [52].\nOne needs to apply an external magnetic \feld for the\nESR-STM technique, but we work here in a regime where\nthe Zeeman splitting of the Kondo spin is much smaller\nthankBTK. As a reference, typical values of the magnetic\n\feldBapplied in ESR-STM experiments are around 5 T\n[40, 41], giving g\u0016BB=kB\u00187 K. If for a given system the\nKondo temperature is not su\u000eciently large, and working\nat smaller values of Bis not an option, another interest-\ning possibility is to modify the local magnetic environ-\nment by introducing additional magnetic impurities. It\nwas demonstrated that the e\u000bect of nearby magnetic im-\npurities can o\u000bset the external magnetic \feld acting on\nthe Kondo spin, and a single Kondo peak can be observed\nat a \fnite value of B[35].\nC. Interaction between impurity spins\nTo probe the spin \ructuations of the Kondo impurity,\na sizable coupling is necessary between the probe im-\npurity (spin) and the Kondo impurity, which could be\nmediated through the substrate or come from the direct\ndipole-dipole coupling. The corresponding Hamiltonian\nis written as\n^Hexchange =X\n\u00162fx;y;zgJ\u0016^S\u0016\np^S\u0016\nK; (9)\nwhich describes either isotropic ( J\u0016=J) or XXZ (Jz6=\nJx=Jy) interactions. Depending on the system\nand underlying mechanism, previously reported coupling\nstrengths vary in a wide range. Closely spaced impuri-\nties can exhibit strong magnetic interactions, dominated\nby direct exchange, which are able to destroy or strongly\nmodify the Kondo e\u000bect [35, 65]. For our purpose we\nwill avoid this regime, by requiring J\u0016\u001ckBTK. Dipo-\nlar couplings between two adatoms detected by ESR-\nSTM spectroscopy lie in a suitable range, with reported\nESR frequency shifts of 0 :02\u00003 GHz [41] (translating\ntoJ\u0016\u00180:1\u000010\u0016eV). The ESR shifts induced by suchdipolar interaction are thus negligibly smaller than the\nbare transition frequency, f0'23 GHz in Ref. [41].\nThis parameter regime allows us to treat the Kondo\nimpurity as an additional bath for the probe spin. Since\na typical Kondo temperature corresponds to spin-\rips\non a time-scale of THz, while typical ESR transition fre-\nquencies are a few tens of GHz [40, 41, 56], the sepa-\nration of time-scales implies that the probe spin cannot\nresolve individual spin \rips of the Kondo impurity, but is\nonly in\ruenced by the average e\u000bect of spin-\ructuations.\nThe scenario is similar to the open-system dynamics in\nthe presence of a Markovian bath, where only the fast\nvarying bath correlation functions enter the equation of\nmotion [66]. An explicit application of this approach to\nour system is pursued in the next section.\nIt is optimal to put the Kondo impurity directly on\ntop of the metallic substrate for stronger coupling [67]\nwhereas the probe spin is better absorbed on a thin insu-\nlating layer for sharper ESR resonance lines (to prevent\nit from being screened by the Kondo e\u000bect). On the\nother hand, the Kondo and probe spins should be close\nenough to have a substantial exchange coupling J\u0016. One\nway to achieve these goals is to employ insulating nano-\nislands, which have been realized with Al 2O3, Cu 2N or\nMgO [50, 51, 68]. Positioning an adatom at the edge of\nan insulating nano-island, where a signi\fcant Kondo ef-\nfect appears, is another way and has also been realized\nin experiments [39].\nIII. EFFECTIVE ENVIRONMENT OF THE\nPROBE SPIN\nThe time-scale separation and weak coupling between\nthe probe spin and Kondo impurity justify treating the\nKondo impurity as an e\u000bective bath. Following standard\nprocedures [66], the master equation for the reduced den-\nsity operator ^ \u001ap(t)\u0011TrKondof^\u001a(t)gof the probe spin can\nbe derived as\nd\ndt^\u001a(I)\np(t) =\u00001\n~2Z1\n0d\u001cTr\nKondonh\n^H(I)\nexchange(t);\nh\n^H(I)\nexchange(t\u0000\u001c);^\u001a(I)\np(t)\n^\u001aKondoiio\n;(10)\nwhere the superscript I indicates that the operators are\nde\fned under the interaction picture with respect to\n^Hprobe +^HKondo , and the thermal equilibrium ^ \u001aKondo/\nexp[\u0000\f^HKondo ] is assumed for the Kondo impurity. The\nexplicit evaluation of Eq. (10) is discussed in Appendix\nB, with the \fnal form including two e\u000bects:\nd\ndt^\u001a(I)\np(t) =1\ni~[^HLamb;^\u001a(I)\np(t)] +R[^\u001a(I)\np(t)]: (11)\nThe dissipative term R[^\u001a(I)\np] is given by Eq. (B2) and\ndescribes the relaxation and dephasing induced by the\nKondo system. It is expressed and estimated more ex-\nplicilty in Appendix B. In principle, the dissipative term5\n-5-4-3-2-1 012345-0.100.10.20.30.40.50.6\nE/(kBTK)WȰ(E)\nT/TK=1T/TK=0.1T/TK=0.02\nT/TK=0.2\nFIG. 2. Energy weighting function W\u0016(E), de\fned in\nEq. (14), and evaluated under the Lorentzian approximation.\nThe black curves are from a numerical evaluation of Eq. (15)\nwhile the red dashed curve is the zero temperature analytical\nresult Eq. (16).\nleads to a \fnite resonance line width in the ESR-STM\nexperiment. However, in reality the ESR line width is de-\ntermined by the instrumental factors such as the strong\ndriving RF \feld and the tip-sample vibrations [40]. For\nthis reason, we do not further discuss the dissipative term\nin this work and focus on the coherent (Lamb shift) term.\nThe shift in the probe spin energy levels Emis:\n^HLamb =X\nm\u000eEmjmihmj; (12)\nwherejmiis the eigenstate of ^Hprobe with energy Em.\nBy explicit evaluation (see Appendix B), we obtain:\n\u000eEm=\u0000X\n\u0016J2\n\u0016\n\u0019kBTKX\nnjS\u0016\nnmj2W\u0016(Enm); (13)\nwhereEnm:=En\u0000EmandS\u0016\nmn=hmj^S\u0016\npjniis the spin\ntransition matrix element of the probe spin. The energy\nweighting factor W\u0016(E) is given by:\nW\u0016(E) = PrZ1\n\u00001d!\n\u0019fB(~!=kBT)Im [\u001f\u0016(!)=\u001f\u0016(0)]\nE=~\u0000!;\n(14)\nwherefB(z) = (ez\u00001)\u00001and the symbol Pr means the\nCauchy principle value. Note that, in the regime of our\ninterest, the ratio \u001f\u0016(!)=\u001f\u0016(0) is a universal function of\n~!=kBTK. From these expressions we see that the energy\nshifts\u000eEmdepend crucially on two factors: (i) the spin\ntransition matrix elements S\u0016\nmnand (ii) the dynamical\nsusceptibility \u001f\u0016(!), contained in the weighting factor\nW\u0016.For concreteness, we consider the Lorentzian approxi-\nmation of\u001f\u0016(!), Eq. (8), giving:\nW\u0016(E) = PrZ1\n\u00001dx\n\u0019\u0014xfB(x=\u001c)\n1 +x2\u00151\n\"\u0000x; (15)\nwhere\u001c=T\nTKand\"=E\nkBTK. Figure 2 shows the gen-\neral behaviors of the energy weighting factor W\u0016(E) as a\nfunction of energy for di\u000berent temperatures. In the zero\ntemperature limit we obtain:\nW\u0016(E) =\u0019=2 +\"lnj\"j\n\u0019(1 +\"2);(\u001c!0) (16)\nwhich works well if T=TK.10\u00002(cf. Fig. 2). Another\ninteresting limit is for \"!0, givingW\u0016(E= 0) = 1=2.\nTheE= 0 result is independent of \u001c, also seen in Fig. 2.\nFinally, the shift in spin transition frequency f0:=\n(E1\u0000E0)=2\u0019~, which can be detected using an ESR-\nSTM device, is obtained as:\n\u000ef0=J2\n\u0016=(2\u0019~)\n\u0019kBTKX\n\u0016;n\u0002\njS\u0016\nn0j2W\u0016(En0)\u0000jS\u0016\nn1j2W\u0016(En1)\u0003\n:\n(17)\nTo analyze the behavior of \u000ef0, we will carry out in the\nnext section the numerical evaluations of these expres-\nsions for typical probe spins.\nIV. ESR TRANSITION FREQUENCIES\nWe show in Fig. 3 the frequency shifts obtained for the\nprobe spins listed in Table I. The shifts have a sensitive\ndependence on the type of impurity, insulating layer, and\nsubstrate, as well as the anisotropy of the exchange inter-\naction. We generally \fnd that Mn on Cu 2N is the most\nsensitive probe spin, while Fe on MgO always yields a\nvery small shift and will be considered separately below.\nWith this exception, the range of shift agrees reasonably\nwell with the general scale predicted by Eq. (17), where\nthe prefactor evaluates to '80 kHz with J\u0016'1\u0016eV and\nkBTK= 1 meV (the parameters assumed in Fig. 3). In\nparticular, in the isotropic case ( Jz=Jx;y) all the shifts\nare in the range of 10 \u000025 kHz, with the variation between\ndi\u000berent probe spins due to the spin operator matrix el-\nements and weighting factors appearing in Eq. (17).\nWhile it is immediate to understand how the over-\nall frequency scale is in\ruenced by a change of J\u0016and\nTK, the e\u000bects related to jS\u0016\nnmj2W\u0016(Enm) are much less\ntransparent. For example, sharp dips are induced at cer-\ntain values of the Jz=Jx;yratio, due to a sign change\nof\u000ef0. The precise condition for these vanishing shifts,\ni.e.,P\nnjS\u0016\nn1j2W\u0016(En1) equals toP\nnjS\u0016\nn0j2W\u0016(En0), de-\npends in a nontrivial way on the magnetic \feld, exchange\ncoupling, as well as temperature. It is also interesting to\nnotice that some probe spins (in particular, Mn and Co)\nprefer an in-plane exchange coupling /^Sx\np^Sx\nK+^Sy\np^Sy\nK,6\n00.2 0.4 0.6 0.8 110-510-410-310-210-1100\n0 0.2 0.4 0.6 0.81Mn/CuN\nFe/CuN\nCe/CuN\nFe/MgO\nCo/MgO\nCo/CuN\nFIG. 3. Frequency shifts \u000ef0as a function of exchange cou-\npling anisotropy for probe spins listed in Table I (we use g= 2\nif the g-factor is not speci\fed). Other parameters: Bx= 6 T,\nBz= 0:2 T,T= 0:5 K, andkBTK= 1 meV. In the left panel\nJx;y= 1\u0016eV and in the right panel Jz= 1\u0016eV.\nwhich facilitates a larger \u000ef0, while in other cases (Fe\nand Ce on Cu 2N) the limit of an Ising interaction with\nJx;y= 0 is most favorable.\nWe now return on the special case of Fe on MgO which,\ndespite being one of the commonly employed probe spins\nin ESR-STM devices, shows the smallest energy shifts\namong those listed in Fig. 3. From the point of view\nof the spin Hamiltonian Eq. (2), the smallness of \u000ef0\nis attributed to the large spin of the impurity, S= 2,\nand the high symmetry of the binding site, which does\nnot allow the transverse anisotropy term ( Es= 0). To\ncon\frm it, in Appendix C we considering a more micro-\nscopic Hamiltonian, commonly used in modeling ESR-\nSTM studies [40]. Under the conditions of large Sand\nuniaxial anisotropy ( Es= 0), the unperturbed Hamilto-\nnianDs^Sz\np^Sz\npleads to a low-energy doublet jS;\u00062i, for\nwhich the direct matrix elements of the spin operators\nvanish, i.e.,jS\u0016\n01j2= 0 in Eq. (17). While the pertur-\nbation\u000e^V=\u0000g\u0016BBx^Sx\npinduced by a transverse \feld\nallows tunneling between the two levels, this can only\noccur through higher order processes. The o\u000b-diagonal\nmatrix elementsj\u000eVS;\u0000SjbetweenjS;\u0006Sihad been cal-\nculated in the literature, which in the large Slimit reads\n[69, 70]\nj\u000eVS;\u0000Sj\u00192DsS3=2\n\u00191=2\u0012eg\u0016BBx\n4SDs\u00132S\n; (18)\nand decreases rapidly with S, thus leading to small val-\nues ofjS\u0016\n01j2. While this matrix element is small, jS\u0016\n0nj2\nandjS\u0016\n1nj2can become sizable when considering the con-\ntributions to Eq. (17) from excited states, i.e., n6= 0;1.\nHowever, these contributions are suppressed by the unfa-\nvorable dependence of the weighting factor by the large\n0 1 2 3 4 5 6 70.05\n0.04\n0.03\n0.02\n0.01\n0\n-0.01\n (T)xB0 (MHz) f\nδ  (meV)mE\n-1.5-1-0.500.511.52FIG. 4. Lower panel: Frequency shift \u000ef0as a function of\nmagnetic \feld Bxin thex-direction, with Bz= 0:2 T. Param-\netersDs=\u00000:039meV,Es= 0:007meV and g= 1:90 corre-\nspond to a Mn probe spin. We assumed here an isotropic mag-\nnetic interaction, Jx=Jy=Jz= 1\u0016eV, andkBTK= 1 meV.\nThe solid line is computed for T= 0:5 K, with the red shaded\nregion showing the variation on \u000ef0caused by \u0001 T= 0:1 K\ntemperature change. The red dash-dotted curve is for T= 0.\nUpper panel: Energy spectrum of the probe spin.\ntransition frequencies E0n;E1n\u0018Ds.\nIn passing, we note how the other impurity spins listed\nin Table I avoid the aforementioned problems of Fe on\nMgO. For probe spins at binding sites with low symme-\ntry (described by the point group D2), the \fnite value\nofEsbreaks the spin selection rule and allows relatively\nlarge values ofjS\u0016\n01j2. It mostly results in large shifts for\nprobe spins such as Mn, Fe and Ce (see the \frst three\nrows in Table I). On the other hand, although Co are sit-\nting on the binding sites with a high symmetry (the point\ngroupC4v), the positive value of Dsleads to a low-energy\ndoublet of the type j0i;j1i'j 3=2;\u00061=2i. In this case,\nthe spin transition matrix elements are sizable and leads\nto relatively large level shifts. Finally, the larger levels\nshift predicted in Mn on Cu 2N are related to the small\nvalues ofDs;Es\u001ckBTK(see Table I), allowing for con-\nsiderable contributions from virtual transitions through\nsome excited states.7\n/\n/\n0 0.5 1 1.5 20\n0.1\n0.2\n0.3\n0.4\n0.5\n0.6\n0.7\n0.8\n0.9\n1\n0.0350.040.0450.050.055\nFIG. 5. Orientation dependence of the frequency shift j\u000ef0j\nfor a magnetic \feld with a constant magnitude B= 6 T.\n\u0012and\u001eare the polar and azimuth angles that specify the\ndirection of B. The color bar is in units of MHz. Except\nBx;y;z, other parameters are the same as in Fig. 4.\nBefore concluding this section, we examine in more\ndetail the frequency shift in the favorable case of a Mn\nprobe spin on Cu 2N. The lower panel of Fig. 4 shows the\nmagnetic \feld dependence of \u000ef0, which has a compli-\ncated behavior and sign change for B\u00143 T. A compari-\nson to the low energy spectrum (upper panel) suggests a\nconnection between \u000ef0= 0 and level anti-crossing in the\nlower-energy doublet. On the other hand, the behavior\nof\u000ef0atB\u00153 T is dominated by the Zeeman term,\nand shows a nearly quadratic increase of j\u000ef0jversusBx.\nFurthermore, since the energy splitting of the probe spin\ndepends on the interplay between the Zeeman term and\nthe magnetic anisotropies, a signi\fcant dependence of \u000ef0\nonB=Bmay be expected for probe spins with biaxial\nanisotropy ( Es6= 0), such as Mn on Cu 2N. This is evi-\ndent from Fig. 5, showing that when the magnetic \feld\nis pointing in the x-yplane (\u0012=\u0019=2),j\u000ef0jcan increase\nby\u001820% by tuning the magnetic \feld direction from\nthe hard axis ( \u001e= 0;\u0019) to\u001e=\u0019=2;3\u0019=2. It is also clear\nfrom Fig. 5 that the largest shift \u000ef0is induced by a \feld\nBpointing along the easy axis ( \u0012= 0;\u0019).\nV. CONCLUSION\nIn this work, we have explored a possibility for access-\ning the spin \ructuations from a Kondo impurity, based\non the ESR-STM technique. The spin \ructuations in\nthe Kondo impurity e\u000bectively shift the resonance tran-\nsition line in the probe spin, which can be detected with\nthe ESR-STM setup [40, 50]. We estimate theoretically\nthe shifts in the resonance line for various types of probe\nspins on top of surfaces and express then in terms of the\ndynamical spin susceptibility.\nWe \fnd that the expected amount of shifts on thetransition frequency f0are most favorable under the ex-\nistence of a suitable transverse magnetic anisotropy, or\nwithj\u0001Szj= 1 selection rules for the ESR-active states.\nThese \fndings are similar to the previous studies on\nKondo e\u000bect with the STM setups where, due to mag-\nnetic anisotropy, the conditions for the Kondo resonance\nfrom single impurity on metallic surface were shown to\nbe di\u000berent for impurities with S >1 [38, 54]. Through\ncomparison among various existing impurities employed\nin the ESR-STM studies, it turns out that Mn on Cu 2N\nmight be suitable for detecting the spin \ructuations.\nDespite of the small frequency shift \u000ef0for Fe on\nMgO, this type of ESR probe is very promising because\nof its large perpendicular magnetic anisotropy and the\nlong life-time of spin excited state [51, 53, 55]. We ex-\npect that signi\fcant improvement may be achieved, e.g.,\nthrough multiple-step transitions involving the excited\nstates, similar methods had already been discussed in\nthe context of single molecular magnets [71{75].\nAppendix A: Calculation of I(f)\nWe summarize in this Appendix the main features of\nthe tunneling current between the STM tip and the sub-\nstrate, and discuss how it allows to detect the energy\nlevels of the probe spin. We start by the Hamiltonian of\nthe STM measurement device without an ESR drive:\n^HSTM=^Hprobe +^Hs+Ht+^Vtun; (A1)\nwhere ^Hprobe is given in Eq. (2), with eivenvalues and\neigenstates Enandjni, respectively. The other terms of\n^HSTM include the free Hamiltonians for the SP-tip and\nsubstrate:\n^Ht=X\nk;\u001b\"T\nk;\u001b^ty\nk;\u001b^tk;\u001b; ^Hs=X\nq;\u001b\"S\nq;\u001b^sy\nq;\u001b^sq;\u001b;(A2)\nas well as their tunnel coupling in the presence of the\nprobe spin [58, 59]:\n^Vtun=X\nk;qX\n\u001bn\u0010\nTk;q^ty\nk;\u001b^sq;\u001b+ h:c:\u0011\n+X\n\u0016X\n\u001b0 \nT0\nk;q\u001b\u0016\n\u001b;\u001b0\n2^S\u0016\np^ty\nk;\u001b^sq;\u001b0+ h:c:!\n(A3)\n'X\n\u000bX\nk;\u001bX\nq;\u001b0\u0012\nT(\u000b)\u001b\u000b\n\u001b;\u001b0\n2^S\u000b\np^ty\nk;\u001b^sq;\u001b0+ h:c:\u0013\n:\n(A4)\nThe \frst (second) line of Eq. (A4) describes elastic (in-\nelastic) tunneling, with tunneling strength given by Tk;q\n(T0\nk;q). In the third line of Eq. (A4), the index \u000bsums\nover 0, and \u0016=x;y;z , with ^S\u000b=0\np=^Ipthe identity oper-\nator and\u001b0the two-by-two identity matrix. In the spirit\nof the commonly employed wide band limit approxima-\ntion, the tunnel couplings Tk;\u001bandT0\nq;\u001b0are assumed to8\nbe constant [76], i.e., T(0)\u0011T0andT(\u0016)=T0\n0while\n\u0010=T0\n0=T0quanti\fes the relative strength of inelastic to\nelastic tunneling.\nIn the weak-tunneling limit, the relaxation rate from\nstatejmito statejniof the probe spin could be calculated\nusing Fermi golden rule expressions [58, 59]:\n~\u0000tip!sub\nm!n=\u0019\n2X\nk;k0X\n\u001b;\u001b0jX\n\u000bT(\u000b)\u001b\u000b\n\u001b0;\u001bS\u000b\nnmj2[1\u0000fS(\"S\nk0;\u001b0)]\n\u0002fT(\"T\nk;\u001b)\u000e(Em+\"T\nk;\u001b\u0000En\u0000\"S\nk0;\u001b0);\n(A5)\nand\n~\u0000sub!tip\nm!n=\u0019\n2X\nk;k0X\n\u001b;\u001b0jX\n\u000bT(\u000b)\u001b\u000b\n\u001b;\u001b0S\u000b\nnmj2[1\u0000fT(\"T\nk;\u001b)]\n\u0002fS(\"S\nk0;\u001b0)\u000e(Em+\"S\nk0;\u001b0\u0000En\u0000\"T\nk;\u001b);\n(A6)\nwhereS\u000b\nnm=hnj^S\u000b\npjmiare matrix elements of the probe\nspin,fS;T(\") = [exp[\f(\"\u0000\u0016S;T)] + 1]\u00001are the Fermi-Dirac distribution function of the substrate and SP-tip,\nand we take real tunneling amplitudes. \u0000tip!sub\nm!nrefers\nto the transition rate of the probe spin from jmitojni\ndue to an electron tunneling from tip to substrate while\n\u0000sub!tip\nm!ndescribes the same transition due to a reverse\ntunneling. The total transition rate is\n\u0000m!n= \u0000tip!sub\nm!n+ \u0000sub!tip\nm!n: (A7)\nFollowing the standard approach of replacing the sum-\nmation over k;k0by integrals over energy, and assuming\na constant (spin-resolved) density of states \u001aT\u001band\u001aS\u001b0\nfor the SP-tip and the substrate respectively, we can re-\ncast Eqs. (A5) and (A6) to the following form:\n~\u0000tip!sub\nm!n=\u0019\n2X\n\u000b;\u000b0T(\u000b)S\u000b\nnmM\u000b\u000b0G(Emn\u0000eV)T(\u000b0)S\u000b0\nmn;\n~\u0000sub!tip\nm!n=\u0019\n2X\n\u000b;\u000b0T(\u000b0)S\u000b0\nnmM\u000b0\u000bG(Emn+eV)T(\u000b)S\u000b\nmn;\n(A8)\nwhere the matrix Mcontains products of type \u001aT\u001b\u001aS\u001b0\nand is de\fned as follows (basis ordering \u000b= 0;x;y;z ):\nM=2\n6664\u001aS\"\u001aT\"+\u001aS#\u001aT# 0 0 \u001aS\"\u001aT\"\u0000\u001aS#\u001aT#\n0 \u001aS\"\u001aT#+\u001aS#\u001aT\"i\u001aS\"\u001aT#\u0000i\u001aS#\u001aT\" 0\n0\u0000i\u001aS\"\u001aT#+i\u001aS#\u001aT\"\u001aS\"\u001aT#+\u001aS#\u001aT\" 0\n\u001aS\"\u001aT\"\u0000\u001aS#\u001aT# 0 0 \u001aS\"\u001aT\"+\u001aS#\u001aT#3\n7775: (A9)\nIn Eq. (A8), the function G(E) =E=(1\u0000e\u0000\fE) is due to\nthe overlapping of the two Fermi-Dirac distribution func-\ntions andeV=\u0016S\u0000\u0016Tis the applied bias, de\fned such\nthat electrons tunnel from the SP-tip to the substrate if\nV > 0 (eincludes sign).\nIn the absence of ESR drive, the populations Pm=\nhmj^\u001apjmiof the probe spin in state jmican be found by\nsolving the rate equations [58, 59]:\n_Pm(t) =X\nn(\u0000n!mPn(t)\u0000\u0000m!nPm(t)); (A10)\nwhich yield the stationary values Pm(1). The d.c. cur-\nrentIdccan be evaluated as:\nIdc=\u0000eX\nm;n\u0000\n\u0000tip!sub\nm!n\u0000\u0000sub!tip\nm!n\u0001\nPm(1):(A11)\nWhen the probe spin is driven by micro-wave \feld\nat frequency f, one has to generalize the rate equation\nEq. (A10) to a master equation, which leads to a modi-\n\fcation of the steady state populations Pm(1) (thus, of\nthe tunneling current). To discuss the e\u000bect in a simple\nand relevant limit, we suppose that the temperature and\napplied bias are su\u000eciently small to neglect all the exci-\ntation rates, i.e., \u0000 m!n\u00190 ifn>m . Then,P0(1) = 1in the absence of the drive and a \fnite P1(1) is induced\nby the ESR excitation. Under these conditions, the sys-\ntem can be restricted to the lowest two eigenstates j0i\nandj1iof^Hprobe, with the tunneling current taking the\nform:\nI(f) =I0+IpP1(1); (A12)\nwhere we used Eq. (A11) together with P0(1) = 1\u0000\nP1(1). The background ( I0) and saturation ( Ip) cur-\nrents are the same of Eq. (4) of the main text, and are\ngiven by:\nI0=\u0000e\u0010\n\u0000tip!sub\n0!0\u0000\u0000sub!tip\n0!0\u0011\n;\nIp=\u0000I0\u0000eX\nn=0;1\u0010\n\u0000tip!sub\n1!n\u0000\u0000sub!tip\n1!n\u0011\n; (A13)\nin terms of the transition rates de\fned in Eq. (A8).\nTo compute P1(1) under the two-level approximation,\nwe describe the dynamical evolution through the Bloch\nequation [40]:\nd\ndt^\u001ap=1\ni~[^HTLS(t);^\u001ap]\n+ \u00001!0D[^L0;1]^\u001ap+ 2\rdpD[^L1;1]^\u001ap;(A14)9\nwhere ^Lm;n:=jmihnj,D[^L] is a superoperator de\fned\nbyD[^L]^A:=^L^A^Ly\u00001\n2f^Ly^L;^Agfor a linear operator ^A:\nThe unitary dynamics in Eq. (A14) is determined by\n^HTLS(t) =X\nn=0;1Enjnihnj+~\n2\u0000\nj0ih1jei2\u0019ft+ h:c:\u0001\n;\n(A15)\nwith Rabi frequency \n. The second line of Eq. (A14)\ndescribes the dissipative dynamics with phenomenologi-\ncal pure dephasing rate \rdp. The stationary solution of\nEq. (A14) is well-known:\nP1(1) =1\n2\n2T1=T2\n4\u00192(f\u0000f0)2+ \n2T1=T2+ 1=T2\n2;(A16)\nwhere the longitudinal and transverse relaxation rates are\nrespectively given by 1 =T1= \u0000 1!0and 1=T2= \u0000 1!0=2+\n\rdp:Finally, substituting Eq. (A16) in Eq. (A12) leads\nto Eq. (4) of the main text.\nAppendix B: Master equation and ^HLamb\nHere, we give the detailed derivation for the energy\nshifts Eq. (13). We start from the master equation Eq.\n(10), which is the standard form derived under the Born-\nMarkov approximation, valid in the weak-coupling limit\n[66, 77]. It is worth mentioning that, in writing such\nmaster equation, we also assume:\nh^S\u0016\nKiKondo\u0011TrKondof^S\u0016\nK^\u001aKondog= 0: (B1)\nThis condition of a negligible spin polarization is consis-\ntent with the general discussion in the main text, where\nwe required that kBTKis much larger than other energy\nscales (and, in particular, than the Zeeman energy).\nTo obtain an explicit form of the master equation in\nEq. (10), we further invoke the rotating wave approxima-\ntion (RWA) under the eigenstate representation of ^Hprobe\n[66], which gives\nd\ndt^\u001a(I)\np(t) =1\ni~X\n\u0016;m;nh\njS\u0016\nmnj2W\u0016(Enm)jmihmj;^\u001a(I)\np(t)i\n+2\u0019\n~X\n\u0016J2\n\u00160\n@X\nm6=nA>\n\u0016(Emn=~)jS\u0016\nmnj2D[jnihmj]\n+A>\n\u0016(0)D\"X\nmS\u0016\nmmjmihmj#!\n^\u001a(I)\np(t);\n(B2)\nwhereA>\n\u0016(!) is the diagonal part of the greater spectral\nfunction:\n2\u0019~A>\n\u0016\u0017(!) =Z1\n\u00001dtei!tD\n^S\u0016\nK(t)^S\u0017\nKE\n; (B3)\nand the time dependence in the Kondo impurity opera-\ntors ^S\u0016\nK(t) =ei^HKondot=~^S\u0016\nKe\u0000i^HKondot=~is due to the in-\nteraction picture. Note that the greater spectral functionis diagonal, A>\n\u0016\u0017(!) =A>\n\u0016(!)\u000e\u0016\u0017, due the spin isotropy\nof the Anderson Hamiltonian Eq. (5).\nThe \frst line of Eq. (B2) is responsible for the Lamb\nshift term ^HLamb in Eq. (11). Weighting factor W\u0016(!) is\nrelated toA>\n\u0016(!) through the Hilbert transform\nW\u0016(\u0000E) :=J2\n\u0016PrZ1\n\u00001d!0A>\n\u0016(!0)\nE=~\u0000!0; (B4)\nW\u0016(E) can be written in terms of the susceptibility by\nmaking use of the \ructuation-dissipation (FD) relations\nforA>\n\u0016\u0017(!):[78]\nA>\n\u0016\u0017(!) =\u000e\u0016\u0017Im\u001f\u0016(!)\n\u0019(g\u0016B)2[1 +fB(~!=kBT)]: (B5)\nBy substituting Eq. (B5) in Eq. (B4), noting the prop-\nerty Im\u001f\u0016(\u0000!) =\u0000Im\u001f\u0016(!);and de\fning the normal-\nized weighting factor as\nW\u0016(E) :=\u0000W\u0016(E)(g\u0016B)2\n\u001f\u0016(0)J2\u0016;\nwe obtain Eq. (14) of the main text.\nThe second and third line of Eq. (B2) describe the\ndissipative e\u000bects of the Kondo impurity on the probe\nspin and constitute dissipation term R[^\u001a(I)\np] in Eq. (11).\nA tedious but straightforward inspection of the diagonal\npart ofR[^\u001a(I)\np] is given by\nhmjR[^\u001a(I)\np]jmi=X\nn(6=m)\u0012\n\u0000n!mhnj^\u001a(I)\np(t)jni\n\u0000\u0000m!nhmj^\u001a(I)\np(t)jmi\u0013\n;(B6)\nwhere the relaxation rate \u0000 n!mfrom statejnito state\njmiis given by\n~\u0000n!m= 2X\n\u0016J2\n\u0016\n(g\u0016B)2jS\u0016\nnmj2Im\u001f\u0016\u0012Enm\n~\u0013\n\u0002\u0014\n1 +fB\u0012Enm\nkBT\u0013\u0015\n;(B7)\nwhere we have used (B5). Similarly, the o\u000b-diagonal part\nofR[^\u001a(I)\np] is given by\nhmjR[^\u001a(I)\np(t)]jni=\u00001\n2\rmnhmj^\u001a(I)\np(t)jni\n\u00001\n2X\nl(6=m)\u0000m!lhmj^\u001a(I)\np(t)jni\n\u00001\n2X\nl(6=n)\u0000n!lhmj^\u001a(I)\np(t)jni(B8)\nwhere\n~\rmn=T=TK\n\u0019kBTKX\n\u00162J2\n\u0016(S\u0016\nmm\u0000S\u0016\nnn)2: (B9)10\nThese expressions allow us to estimate the relaxation\ntimeT1and the dephasing time T2within the two-level\napproximation [see also the Bloch equation in Eq. (A14)]\nand in the low-temperature limit ( kBT\u001chf0\u001ckBTK)\nby\n2\u0019=T 1=j\u00001!0j\u00192\u0012hf0\nkBTK\u0013X\n\u0016J2\n\u0016=~\n\u0019kBTKjS\u0016\n10j2(B10)\nand\n2\u0019=T 2=j(\u00000!1+ \u00001!0)=2 +\r01=2j\n\u0019j\u00001!0=2 +\r01=2j\n\u0019X\n\u0016J2\n\u0016=~\n\u0019kBTK\u001a\njS\u0016\n10j2hf0\nkBTK(B11)\n+ (S\u0016\n11\u0000S\u0016\n00)2kBT\nkBTK\u001b\n: (B12)\nBased on the ESR line shape in Eq. (4) [or, equivalently,\nEq. (A16)], one can further estimate the line width \u0001 f\n[we de\fne it as the full width at half maximum (FWHM)]\nby\nh\u0001f\u00192vuut X\n\u0016J2\u0016jS\u0016\n10j2\n\u0019~kBTK\u0002hf0\nkBTK!2\n+1\n2(~\n)2:\n(B13)\nIt has values in the sub-MHz range or less. Note that the\nESR line width is mainly determined by the instrumental\nfactors such as the strong driving RF \feld and tip-sample\nvibrations [40], which is typically in the MHz range.\nAppendix C: Alternative e\u000bective spin Hamiltonian\nIn the main text, the probe spin is described through\nthe spin Hamiltonian (2). By restricting ourselves to im-\npurities with C4vsymmetry (see Table I), we consider\nhere a more fundamental model which takes into accounts\nexplicitly the orbital degrees of freedom.\nA standard derivation of the e\u000bective spin Hamilto-\nnian (2) treats the spin-orbit coupling (SOC) as a pertur-\nbation, compared to the orbital level splittings. This as-\nsumption is usually well justi\fed for 4 felectrons [46, 48],\nbut might be violated for adatoms with 3 delectrons\n[48, 49]. In this case, the crystal \feld Hamiltonian\n^HCF=P\ni^H(i)\nCFis decomposed into various terms with\ndecreasing strengths j^H(1)\nCFj>j^H(2)\nCFj> :: >j^H(k)\nCFj, and\nthe spin-orbit term ^HSOCa\u000bects the system at some in-\ntermediate levelj^H(1)\nCFj>j^HSOCj>j^H(k)\nCFj, thus a treat-\nment taking into account explicitly the crystal-\feld is\nmore appropriate. Among interesting impurities falling\ninto this regime is Fe on MgO/Ag(100), which was suc-\ncessfully exploited as probe spin in ESR-STM experi-\nments [40, 41]. Its energy level structure is schematically\nshown in Fig. 6.\nFIG. 6. Energy level structure of an Fe probe spin with C4v\nlocal symmetry. Left panel: Structure of the \fve L= 2 or-\nbital levels. The two lowest orbitals are further mixed by the\nF0crystal \feld term appearing in Eq. (C1). Right panel: De-\ntailed structure of the low-energy subspace, corresponding to\n\fveS= 2 sub-levels.\nIn case the total orbital Land total spin Sangular mo-\nmentums of the probe atom's electrons are still approxi-\nmately good quantum numbers, the crystal \feld Hamil-\ntonian ^HCFcan be constructed by the method of opera-\ntor equivalence [49, 79]. For impurities with C4vorC1v\nlocal symmetry, the general form of ^HCFreads [40]:\n^HCF=D^Lz\np^Lz\np+E0\u0010\n^Lz\np\u00114\n+F0\u0010\n^L4\n++^L4\n\u0000\u0011\n:(C1)\nThe other two terms entering the total Hamiltonian,\n^Hatom =^HCF+^HSOC+^HZ, are the spin-orbit coupling:\n^HSOC=\u0015z^Lz\np^Sz\np+\u0015?\n2\u0010\n^L+^S\u0000+^L\u0000^S+\u0011\n; (C2)\nand the Zeeman splitting:\n^HZ=g\u0016BB\u0001^Sp+gL\u0016BB\u0001^Lp: (C3)\nConsidering Fe on MgO/Ag(100), theoretical and exper-\nimental studies give the values D=\u0000433meV,E0= 0,\nF0= 2:19meV,\u0015z=\u0015?=\u000012:6meV,g= 2 andgL= 1\n[40, 51]. As shown in Fig. 6, the F0term splitsjL;\u00062i\nstates by 48 F0, which is of the same order as the spin-\norbit coupling term 2 \u0015z;?Sz\np. Thus, it is not a good ap-\nproximation to treat the SOC as a perturbation for the\ntwo lowest eigenstates of ^HCF.\nDespite this apparent di\u000eculty, in Fig. 7 we compare\nthe ESR frequency shift \u000ef0computed from ^Hatom or\nthe e\u000bective spin Hamiltonian Eq. (2). Besides a rela-\ntively small di\u000berence (which, in principle, can be com-\npensated by small adjustments of the e\u000bective parame-\nters in Table I) the two models are in agreement in pre-\ndicting a small value \u000ef0.30 Hz, weakly sensitive to\nthe anisotropy of the exchange interaction.\nWe can also derive an alternative e\u000bective model, by\nrestricting ourselves to the subspace spanned by jL;\u00062i\nand eliminating direct transitions to the higher orbital11\n00.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.0250.0260.0270.0280.0290.030.0310.0320.033\nFull model\nTwo-orbitals\nEffective spin\nFIG. 7. ESR frequency shift as a function of the ratio Jz=Jx;y,\nfor an Fe probe spin on MgO [cf. the black dashed curve\nin Fig. 3]. The green and black solid curves are calcu-\nlated from the full model Eq. (C4) and the two-orbitals model\nEq. (C9), respectively. The red circles are obtained from the\ne\u000bective spin Hamiltonian Eq. (2), with parameters listed in\nTable I. Other parameters: L=S= 2,D=\u0000433 meV,\nF0= 2:19 meV,\u0015z=\u0015?=\u000012:6 meV,Bz= 0:2 T,\nBx= 6 T,T= 0:5 K,kBTK= 1 meV, and Jx=Jy= 1\u0016eV.\nlevels. To this end, we make the following partition of\nthe total Hamiltonian:\n^Hatom =^H0+^H1+^H2; (C4)\nwhere the unperturbed Hamiltonian is:\n^H0=D(^Lz\np)2+F0(^L4\n++^L4\n\u0000) +\u0015z^Lz\np^Sz\np; (C5)\nwhich includes here both crystal-\red and spin-orbit cou-\npling terms. The two perturbations are given by:\n^H1=\u0016B^Lx\npBx+1\n2\u0015?(^L+^S\u0000+^L\u0000^S+); (C6)\nas well as the remaining of the Zeeman term (assuming\nBin thex-zplane)\n^H2=\u0016BBz(^Lz\np+ 2^Sz\np) + 2\u0016BBx^Sx\np: (C7)\nWe obtain the e\u000bective Hamiltonian by applying a\nSchrie\u000ber-Wol\u000b transformation [80]:\n^H0\nprobe\u0019^H0+1\n2[^H1;^S] +e\u0000^S^H2e^S; (C8)where ^Ssatis\fes ^S=\u0000^Syand [ ^S;^H0] = ^H1, i.e., it is\nchosen to eliminate ^H1to the lowest order. Since ^H2is\nalready small, we neglect in the last term of Eq. (C8)\nthe corrections induced by the unitary transformation.\nFurthermore, ^Sinvolves transitions between the jL;\u00062i\nandjL;\u00061isubspaces, and we can set the energy de-\nnominators equal to 3 D. As illustrated in Fig. 7, this\napproximation neglects corrections to the eigenenergies\ninduces by F0and and\u0015zwhich, however, are much\nsmaller than D. Finally, after dropping a small shift\nequal to (\u0016BBx)2=(3D), we obtain:\n^H0\nprobe =4D\u00006D0\ns+ 24F0^\u001cx+D0\ns(^Sz\np)2+\u00150\nz^Sz\np^\u001cz\n+ 2\u0016BBz(^Sz\np+ ^\u001cz) +\u0016BgxBx^Sx\np; (C9)\nwhere ^\u001c\u0016are Pauli matrices of a pseudo spin-1/2 particle\nspanned by the two lowest orbital states:\n^\u001cx=jL;2ihL;\u00002j+jL;\u00002ihL;2j;\n^\u001cz=jL;2ihL;2j\u0000jL;\u00002ihL;\u00002j; (C10)\nand the coe\u000ecients in Eq. (C9) are given by:\nD0\ns=\u00150\nz\u00002\u0015z=\u0000\u00152\n?\n3D; gx= 2\u0012\n1 +\u0015?\n3D\u0013\n:(C11)\nThe e\u000bective spin Hamiltonian Eq. (C9) is valid provided\nthatF0;j\u0015z;?j\u001cDare satis\fed, and is applicable to\nadatoms with C4vorC1vlocal symmetry. In Fig. 7,\nwe see that the ESR frequency shift \u000ef0obtained from\n^H0\nprobe is in excellent agreement with the full model.\nIn closing, we note that the parameters of the two-\norbitals e\u000bective Hamiltonian also allow for a transparent\ndescription of ESR-STM setups. For example, according\nto Ref. 40, the ESR drive modi\fes the local electrostatic\nenvironment in a time-dependent way, yielding a cosine\nmodulation of F0. Such coupling appears explicitly in\nEq. (C9) and, for a given strength of the F0modulation,\nit would be possible to compute in a rather straightfor-\nward way the Rabi frequency of the ESR transition. In-\nstead, the ESR drive enters the e\u000bective parameter of\nEq. (2) in a more complicated manner.\nACKNOWLEDGMENTS\nY.F. acknowledges support from NSFC (Grant No.\n12005011). S.C. acknowledges support from NSFC\n(Grant No. 11974040) and NSAF (Grant No.\nU1930402). M.-S.C. has been supported by the Na-\ntional Research Function (NRF) of Korea (Grant Nos.\n2017R1E1A1A03070681 and 2018R1A4A1024157) and\nby the Ministry of Education through the BK21 Four\nprogram.\n[1] J. Kondo, Prog. Theor. Phys. 32, 37 (1964). [2] M. Tachiki and S. Maekawa, Phys. Rev. B 29, 2497\n(1984).12\n[3] G. R. Stewart, Rev. Mod. Phys. 73, 797 (2001).\n[4] H. Alloul, J. Bobro\u000b, M. Gabay, and P. J. Hirschfeld,\nRev. Mod. Phys. 81, 45 (2009).\n[5] L. I. Glazman and M. E. Raikh, JETP Lett. 47, 452\n(1988).\n[6] E. J. H. Lee, X. Jiang, R. Aguado, G. Katsaros, C. M.\nLieber, and S. De Franceschi, Phys. Rev. Lett. 109,\n186802 (2012).\n[7] J.-D. Pillet, P. Joyez, R. Zitko, and M. F. Go\u000bman, Phys.\nRev. B 88, 045101 (2013).\n[8] K. G. Wilson, Rev. Mod. Phys. 47, 773 (1975).\n[9] T. A. Costi, A. C. Hewson, and V. Zlatic, J. Phys.: Con-\ndens. Matter 6(1994).\n[10] R. Bulla, T. A. Costi, and T. Pruschke, Rev. Mod. Phys.\n80, 395 (2008).\n[11] N. Andrei, Phys. Rev. Lett. 45, 379 (1980).\n[12] N. Andrei and C. Destri, Phys. Rev. Lett. 52, 364 (1984).\n[13] D. Goldhaber-Gordon, H. Shtrikman, D. Mahalu,\nD. Abusch-Magder, U. Meirav, and M. A. Kastner, Na-\nture391(1998).\n[14] S. M. Cronenwett, T. H. Oosterkamp, and L. P. Kouwen-\nhoven, Science 281(1998).\n[15] W. G. van der Wiel, S. D. Franceschi, T. Fujisawa, J. M.\nElzerman, S. Tarucha, and L. P. Kouwenhoven, Science\n289(2000).\n[16] M.-S. Choi, R. L\u0013 opez, and R. Aguado, Phys. Rev. Lett.\n95, 067204 (2005).\n[17] J. S. Lim, M.-S. Choi, M. Y. Choi, R. L\u0013 opez, and\nR. Aguado, Phys. Rev. B 74, 205119 (2006).\n[18] T. Delattre, C. Feuillet-Palma, L. G. Herrmann,\nP. Mor\fn, J.-M. Berroir, G. F\u0012 eve, B. Pla\u0018 cais, D. C. Glat-\ntli, M.-S. Choi, and C. Mora, Nat. Phys. 5(2009).\n[19] I. A\u000feck and P. Simon, Phys. Rev. Lett. 86, 2854 (2001).\n[20] I. A\u000feck, L. Borda, and H. Saleur, Phys. Rev. B 77,\n180404 (2008).\n[21] J. Park, S.-S. B. Lee, Y. Oreg, and H.-S. Sim, Phys. Rev.\nLett. 110, 246603 (2013).\n[22] I. V. Borzenets, J. Shim, J. C. H. Chen, A. Ludwig, A. D.\nWieck, S. Tarucha, H.-S. Sim, and M. Yamamoto, Nature\n579, 210 (2020).\n[23] M. M. Desjardins, J. J. Viennot, M. C. Dartiailh, L. E.\nBruhat, M. R. Delbecq, M. Lee, M.-S. Choi, A. Cottet,\nand T. Kontos, Nature 545, 71 (2017).\n[24] G. Binnig and H. Rohrer, Rev. Mod. Phys. 59, 615\n(1987).\n[25] V. Madhavan, W. Chen, T. Jamneala, M. F. Crommie,\nand N. S. Wingreen, Science 280(1998).\n[26] R. Wiesendanger, Rev. Mod. Phys. 81, 1495 (2009).\n[27] H. Pruser, M. Wenderoth, P. E. Dargel, A. Weismann,\nR. Peters, T. Pruschke, and R. G. Ulbrich, Nat. Phys. 7\n(2011).\n[28] M. F. Crommie, C. P. Lutz, and D. M. Eigler, Nature\n363, 524 (1993).\n[29] T. Kawasaka, H. Kasai, W. A. Dino, and A. Okiji, J.\nAppl. Phys. 86, 6970 (1999).\n[30] J. Li, W.-D. Schneider, R. Berndt, and B. Delley, Phys.\nRev. Lett. 80, 2893 (1998).\n[31] O. Ujsaghy, J. Kroha, L. Szunyogh, and A. Zawadowski,\nPhys. Rev. Lett. 85, 2557 (2000).\n[32] H. C. Manoharan, C. P. Lutz, and D. M. Eigler, Nature\n403(2000).\n[33] G. A. Fiete and E. J. Heller, Rev. Mod. Phys. 75, 933\n(2003).[34] W. Chen, T. Jamneala, V. Madhavan, and M. F. Crom-\nmie, Phys. Rev. B 60(1999).\n[35] A. F. Otte, M. Ternes, S. Loth, C. P. Lutz, C. F. Hir-\njibehedin, and A. J. Heinrich, Phys. Rev. Lett. 103,\n107203 (2009).\n[36] E. Sela and I. A\u000feck, Phys. Rev. Lett. 102, 047201\n(2009).\n[37] J. Bork, Y.-h. Zhang, L. Diekhoner, L. Borda, P. Simon,\nJ. Kroha, P. Wahl, and K. Kern, Nat. Phys. 7, 901 (2011).\n[38] A. F. Otte, M. Ternes, K. von Bergmann, S. Loth,\nH. Brune, C. P. Lutz, C. F. Hirjibehedin, and A. J. Hein-\nrich, Nat. Phys. 4(2008).\n[39] J. C. Oberg, M. R. Calvo, F. Delgado, M. Moro-Lagares,\nD. Serrate, D. Jacob, J. Fernandez-Rossier, and C. F.\nHirjibehedin, Nat. Nano. 9(2013).\n[40] S. Baumann, W. Paul, T. Choi, A. Lutz, Christopher\nP. Ardavan, and A. J. Heinrich, Science 350, 417 (2015).\n[41] T. Choi, W. Paul, S. Rolf-Pissarczyk, A. J. Macdonald,\nF. D. Natterer, K. Yang, P. Willke, C. P. Lutz, and A. J.\nHeinrich, Nat. Nano. 12, 420 (2017).\n[42] P. Willke, W. Paul, F. D. Natterer, K. Yang, Y. Bae,\nT. Choi, J. Fern\u0013 andez-Rossier, A. J. Heinrich, and C. P.\nLutz, Sci. Adv. 4, eaaq1543 (2018).\n[43] W. E. Lamb and R. C. Retherford, Phys. Rev. 72, 241\n(1947).\n[44] H. A. Bethe, Phys. Rev. 72, 339 (1947).\n[45] A. C. Hewson, The Kondo Problem to Heavy Fermions\n(Cambridge University Press, New York, 1993).\n[46] M. H. L. Pryce, Proc. Phys. Soc. A 63, 25 (1950).\n[47] B. Bleaney and K. W. H. Stevens, Rep. Prog. Phys. 16,\n108 (1953).\n[48] R. Boca, Theoretical Foundations of Molecular Mag-\nnetism (Elsevier, Amsterdam, 1999).\n[49] A. Abragam and B. Bleaney, Electron Paramagnetic Res-\nonance of Transition Ions (Clarendon Press, Oxford,\n1970).\n[50] C. F. Hirjibehedin, C.-Y. Lin, A. F. Otte, M. Ternes,\nC. P. Lutz, B. A. Jones, and A. J. Heinrich, Science 317,\n1199 (2007).\n[51] S. Baumann, F. Donati, S. Stepanow, S. Rusponi,\nW. Paul, S. Gangopadhyay, I. G. Rau, G. E. Pacchioni,\nL. Gragnaniello, M. Pivetta, et al., Phys. Rev. Lett. 115,\n237202 (2015).\n[52] M. Ternes, C. P. Lutz, A. J. Heinrich, and W.-D. Schnei-\nder, Phys. Rev. Lett. 124, 167202 (2020).\n[53] W. Paul, K. Yang, S. Baumann, N. Romming, T. Choi,\nC. Lutz, and A. Heinrich, Nat. Phys. 13, 403 (2017).\n[54] M. Ternes, A. J. Heinrich, and W.-D. Schneider, J. Phys.:\nCondens. Matter 21(2009).\n[55] I. G. Rau, S. Baumann, S. Rusponi, F. Donati,\nS. Stepanow, L. Gragnaniello, J. Dreiser, C. Piamonteze,\nF. Nolting, S. Gangopadhyay, et al., Science 344(2014).\n[56] K. Yang, Y. Bae, W. Paul, F. D. Natterer, P. Willke, J. L.\nLado, A. Ferr\u0013 on, T. Choi, J. Fern\u0013 andez-Rossier, A. J.\nHeinrich, et al., Phys. Rev. Lett. 119, 227206 (2017).\n[57] K. Yang, W. Paul, F. D. Natterer, J. L. Lado, Y. Bae,\nP. Willke, T. Choi, A. Ferron, J. Fernandez-Rossier, A. J.\nHeinrich, et al., Phys. Rev. Lett. 122, 227203 (2019).\n[58] J. Fern\u0013 andez-Rossier, Phys. Rev. Lett. 102, 256802\n(2009).\n[59] F. Delgado and J. Fernandez-Rossier, Phys. Rev. B 82,\n134414 (2010).13\n[60] The additional `-' sign in Eq. (6) comes from the same\nsign in the implied perturbation term, \u0000P\n\u0016g\u0016BB\u0016^S\u0016\nK,\ndue to an external magnetic \feld that is consistent with\nEq. (7).\n[61] N. Andrei, in Low-Dimensional Quantum Field Theories\nfor Condensed Matter Physicists , edited by S. Lundqvist\n(World Scienti\fc, 1995), vol. 6 of Low-Dimensional\nQuantum Field Theories for Condensed Matter Physi-\ncists, chap. Integrable Models in Condensed Matter\nPhysics, pp. 457{551.\n[62] Denoted by Toin Eq. (50) of Ref. [61], this temperature\nscale is di\u000berent by a numerical prefactor from the con-\nventional de\fnition of the Kondo temperature in terms\nof the logarithmic divergence in the perturbation argu-\nments.\n[63] T.-F. Fang, N.-H. Tong, Z. Cao, Q.-F. Sun, and H.-G.\nLuo, Phys. Rev. B 92, 155129 (2015).\n[64] P. Wahl, L. Diekh oner, M. A. Schneider, L. Vitali,\nG. Wittich, and K. Kern, Phys. Rev. Lett. 93, 176603\n(2004).\n[65] P. Wahl, P. Simon, L. Diekh oner, V. S. Stepanyuk,\nP. Bruno, M. A. Schneider, and K. Kern, Phys. Rev.\nLett. 98, 056601 (2007).\n[66] H. P. Breuer and F. Petruccione, The Theory of Open\nQuantum Systems (Oxford University Press, New York,\n2002).\n[67] While magnetic adatoms such as Co coupled to a metal\nsubstrate via a thin insulating layer can certaintly ex-hibit Kondo e\u000bects [35, 38], the relatively low Kondo\ntemperature complicates the analysis of the signal; see\nSection III.\n[68] A. J. Heinrich, J. A. Gupta, C. P. Lutz, and D. M. Eigler,\nScience 306, 466 (2004).\n[69] D. A. Garanin, J. Phys. A 24, L61 (1991).\n[70] D. A. Garanin and E. M. Chudnovsky, Phys. Rev. B 56,\n11102 (1997).\n[71] M. N. Leuenberger and D. Loss, Nature 410(2001).\n[72] K. Park, M. A. Novotny, N. S. Dalal, S. Hill, and P. A.\nRikvold, Phys. Rev. B 65, 014426 (2001).\n[73] I. D. Tokman, V. I. Pozdnjakova, G. A. Vugalter, and\nA. V. Shvetsov, Phys. Rev. B 77, 094414 (2008).\n[74] A. V. Shvetsov, G. A. Vugalter, and A. I. Grebeneva,\nPhys. Rev. B 74, 054416 (2006).\n[75] Y. Wu and X. Yang, Phys. Rev. B 76, 054425 (2007).\n[76] A.-P. Jauho, N. S. Wingreen, and Y. Meir, Phys. Rev. B\n50, 5528 (1994).\n[77] E. Fick and G. Sauermann, The Quantum Statistics of\nDynamic Processes (Springer-Verlag, Berlin, 1990).\n[78] A \ructuation-dissipation relation usually appears with a\n`-' sign. It is missing here because of the implied per-\nturbation term,\u0000P\n\u0016g\u0016BB\u0016^S\u0016\nK; see also Eqs. (6) and\n(7).\n[79] K. W. H. Stevens, Proc. Phys. Soc. A 65, 209 (1952).\n[80] J. R. Schrie\u000ber and P. A. Wol\u000b, Phys. Rev. 149, 491\n(1966)."    },    {        "title": "1903.09729v2.Exceptional_points_in_classical_spin_dynamics.pdf",        "content": "Exceptional points in classical spin dynamics\nAlexey Galda1, 2and V. M. Vinokur2\n1James Franck Institute, University of Chicago, Chicago, Illinois 60637, USA\n2Materials Science Division, Argonne National Laboratory,\n9700 S. Cass Avenue, Argonne, Illinois 60439, USA\n(Dated: October 21, 2019)\nNon-conservative physical systems admit a special kind of spectral degeneracy, known as ex-\nceptional point (EP), at which eigenvalues and eigenvectors of the corresponding non-Hermitian\nHamiltonian coalesce. Dynamical parametric encircling of the EP can lead to non-adiabatic evolu-\ntion associated with a state \rip, a sharp transition between the resonant modes. Physical conse-\nquences of the dynamical encircling of EPs in open dissipative systems have been explored in optics\nand photonics. Building on the recent progress in understanding the parity-time ( PT)-symmetric\ndynamics in spin systems, we use topological properties of EPs to implement chiral non-reciprocal\ntransmission of a spin through the material with non-uniform magnetization, like helical magnet.\nWe consider an exemplary system, spin-torque-driven single spin described by the time-dependent\nnon-Hermitian Hamiltonian. We show that encircling individual EPs in a parameter space results in\nnon-reciprocal spin dynamics and \fnd the range of optimal protocol parameters for high-e\u000eciency\nasymmetric spin \flter based on this e\u000bect. Our \fndings o\u000ber a platform for non-reciprocal spin\ndevices for spintronics and magnonics.\nINTRODUCTION\nIn magnetic structures, non-reciprocal spin transport\nis usually mediated by either surface modes [1] or asym-\nmetric exchange interaction [2, 3], giving rise to asym-\nmetric spin-wave dispersion with respect to the propaga-\ntion direction [4{6]. Here we present a general method for\nsetting non-reciprocal spin transport based on the non-\nHermitian dynamics around exceptional points (EPs),\nspectral degeneracies arising in non-conservative sys-\ntems endowed with simultaneous parity-time ( PT) in-\nvariance [7, 8]. The emerging e\u000bect of non-reciprocal\ntransport is topological, hence fundamentally universal,\nand stems from the adiabaticity breaking unique non-\nHermitian systems. The resultant non-reciprocity is\nhighly tunable and robust with respect to noise.\nNon-Hermitian extension of Hamiltonian formalism\nprovides a quantitative description of non-conservative\ndynamics in out-of-equilibrium systems [9]. Non-\nHermitian Hamiltonians admit a special class of degen-\neracy points, called exceptional points, where some of\ntheir eigenvalues and the corresponding eigenvectors co-\nalesce. This property is unique to non-Hermitian opera-\ntors and must be distinguished from Hermitian spectrum\ndegeneracies, like diabolical points, where only eigen-\nvalues coalesce. Non-Hermitian degeneracies are stable\nagainst both Hermitian or non-Hermitian perturbations,\nand have far reaching physical implications due to their\ntopological structure that have been explored both the-\noretically [10{13] and experimentally [14{16].\nAppearance of EPs in the eigenspectrum of non-\nHermitian Hamiltonians is a direct consequence of the\nPTsymmetry-breaking [7, 8]. Hamiltonians invariant\nunder the simultaneous operations of parity ( P) and\ntime-reversal (T) form a special class of non-Hermitian\nHamiltonians that admit a fully real eigenspectrum,\nprovided that characteristic amplitudes of their anti-Hermitian parts are below certain threshold values. The\nPTsymmetry-breaking marks the transition from the\nreal to complex eigenspectrum of PT-symmetric Hamil-\ntonians. The physical interpretation of PTsymmetry\nas \\balanced gain and loss\" [17] triggered an increasing\nnumber of experimental realizations of PT-symmetric\nsystems and established PT symmetry-breaking as a\nnon-equilibrium phase transitions between stationary\nand nonstationary dynamics in driven dissipative sys-\ntems.\nWe demonstrate that dynamical encircling a second-\norder EP in the parameter space of a classical magnetic\nsystem can lead to non-trivial spin transport properties\nnever seen before. We consider a linear non-Hermitian\nHamiltonian describing dynamics of a single classical spin\nin the external time-dependent magnetic \feld. This is a\ngeneric model for ballistic spin transport experiments, in\nwhich spin current is generated by passing non-polarized\nelectrons through a material with non-uniform magneti-\nzation, e.g. helical magnet. We show that the combi-\nnation of applied spin-transfer torque (STT) and time-\ndependent magnetic \feld, coordinated in the form of\na protocol that takes the system around an EP of the\ncorresponding non-Hermitian Hamiltonian, enables non-\nreciprocal spin dynamics and transport with a high de-\ngree of spin-polarization e\u000eciency.\nTo derive the propagation of individual spins through\na medium with spatially non-uniform magnetic \feld, we\nemploy the model of a stationary spin in time-dependent\nmagnetic \feld. We introduce an external imaginary mag-\nnetic \feld to account for the e\u000bects of non-conservative\nforces on spin dynamics, e.g. spin-torque. Indeed, it was\nshown that the action of Slonczewski STT [18] is equiv-\nalent to that of applied imaginary magnetic \feld in the\nnon-Hermitian Hamiltonian formalism [20]. We propose\na design of a classical single-spin system with the inher-\nentPTsymmetry, such that the energy spectrum of thearXiv:1903.09729v2  [cond-mat.mes-hall]  18 Oct 20192\nFIG. 1. Asymmetric spin-\flter e\u000bect achieved by sending nonpolarized electrons through a medium with spatially non-uniform\nmagnetization and biased by spin-torque. The combination time-dependent e\u000bective magnetic \feld and spin-torque experi-\nenced by individual spins creates a protocol of parametrically encircling an EP of the corresponding e\u000bective non-Hermitian\nHamiltonian's complex eigenspectrum, resulting in non-reciprocal spin transport properties. Spins sent left-to-right (a) and\nright-to-left (b) come out polarized in di\u000berent directions, controlled by the magnitudes of spin-torque and e\u000bective magnetic\n\feld experienced by the spins. Blue and red \rat arrows (anti-)parallel to the y-axis represent spin accumulations generated, e.g.,\nby spin Hall e\u000bect. As a result, spin-torque is exerted on incident spins propagating through a medium (gray) with spatially\nvarying magnetic \feld (orange arrows).\ncorresponding non-Hermitian Hamiltonian admits a pair\nof second-order EPs. We then consider a cyclic variation\nof the applied magnetic \feld, such that a single EP is en-\ncircled in the two-dimensional parameter space ( hx;hy)\nof the model, where hxandhyare projections of applied\nmagnetic \feld. Encircling protocols can be constructed\nin such a way that only a clockwise or anti-clockwise di-\nrection, depending on the chirality of the EP, yields the\n\fnal spin state that is di\u000berent from the initial one at\nthe beginning of the protocol, despite identical initial and\n\fnal system parameters. The e\u000bect relies on the break-\ndown of the adiabatic theorem in non-Hermitian systems,\nand results in non-reciprocal time evolution of spin.\nWe perform numerical simulations to verify the robust-\nness of the non-reciprocal e\u000bect and to identify optimal\nprotocol parameters for achieving maximum asymmetric\nspin-polarization e\u000eciency in the proposed scheme. In\nparticular, for encircling protocols centered around an\nisolated EP, we calculate deviations from perfect non-\nreciprocity as a function of protocol frequency, radius,\nand initial conditions. Our results show that the high-\nest degree of spin-\flter asymmetry between the clock-\nwise and anti-clockwise circular protocols is achieved for\nslow-frequency parametric trajectories starting in a wide\nregion near the line of PT-symmetry and with a large\nenough radii to encircle both EPs, as opposed to encir-\ncling an individual EP.\nExperimental realization of the proposed spin evolu-\ntion protocols enables a novel type of high-e\u000eciency\nasymmetric spin-\flter devices possessing a number of\nunique properties. First, high e\u000eciency of the spin-\flter\ne\u000bect is guaranteed by the topology of trajectories en-\ncircling the EP. Second, the proposed spin-\flter device,\nFig. 1, is asymmetric, i.e. breaks time-reversal symmetry.\nAs a result, the polarization directions of spins entering\nthe device from the left and from the right are di\u000berent.\nThird, both spin-polarization directions are controlled by\nthe amplitudes of the bias STT and external magnetic\n\feld, rendering a highly tunable and e\u000ecient system.RESULTS\nThe most general form of a linear spin Hamiltonian\ncontains three independent parameters [19], which, with-\nout any loss of generality, can be chosen as two compo-\nnents of the real magnetic \feld along the xandyaxes,\nand a single component of the imaginary \feld along the\nyaxis:\nH0=hxSx+ (hy+i\f)Sy; (1)\nwhere\fis the dimensionless amplitude of applied spin-\ntorque [20].\nTo describe single-spin dynamics governed by the\nnon-Hermitian Hamiltonian (1), we employ SU(2)\nspin-coherent states [21{23]: jzi= ezS+jS;\u0000Si, where\nS\u0006\u0011Sx\u0006iSy, andz2Cis the standard stereographic\nprojection of the spin direction on a unit sphere,\nz= (sx+isy)=(1\u0000sz), withsi\u0011Si=S. For the purpose\nof this work, we focus on the classical limit of large spin,\nS!1 , in which case one can write the generalized spin\nequation of motion as follows [20]:\n_z=i(1 + \u0016zz)2\n2S@H0(z;\u0016z)\n@\u0016z; (2)\nwhereH0(z;\u0016z) is the expectation value in spin-coherent\nstates:\nH0(z;\u0016z) =hzjH0jzi\nhzjzi: (3)\nFor the spin Hamiltonian (1), this yields the following\nevolution equation:\n_z(t) =\u0000ihx\u0000i(hy+i\f)\n2\u0014\nz2\u0000hx+i(hy+i\f)\nhx\u0000i(hy+i\f)\u0015\n:(4)\nThe eigenvalue problem for the spin Hamiltonian (1) in\nthe classical limit reduces to evaluating the expectation\nvalues, Eqs. (3), at the \fxed points of the correspond-\ning classical spin dynamics on a unit sphere, Eq. (2){(4).3\nFIG. 2. (a) Diabolical point (DP) of real energy spectrum of the Hermitian Hamiltonian (5) with \f= 0. DP bifurcates into\ntwo EPs, shown in (b), (c) for \f= 1 in the real and imaginary parts of eigenspectrum, correspondingly. Na \u0010ve consideration\npredicts that a 4 \u0019-rotation (red trajectory) around an EP would bring the system back to its initial state (up to a phase\nfactor). (d){(f) Numerical simulations of state evolution during clockwise (purple) and anti-clockwise (green) encircling of the\nEP, illustrating non-adiabatic state-\rip events as sharp jumps between two spectral surfaces. The system's \fnal state after\na complete 2 \u0019-rotation depends on the encircling direction, illustrating non-reciprocity of non-Hermitian parametric driving.\nThe non-reciprocal time evolution is shown for the cases of (d) encircling of a single EP, (e) circular trajectory in the vicinity\nof the EP, and (f) complete encircling of both EPs.\nFor the linear non-Hermitian Hamiltonian (1), this is also\nequivalent to the eigenvalue problem for a two-level sys-\ntem described by the Hamiltonian:\nH=hx\u001bx+ (hy+i\f)\u001by; (5)\nwhere\u001biare Pauli matrices, which yields the following\ncomplex eigenspectrum:\nE1;2=\u0006q\nh2x+ (hy+i\f)2: (6)\nThe two \fxed points of classical spin dynamics governed\nby the Hamiltonian (1), referred to as AandBin the\ntext, are de\fned by the following expression in stereo-\ngraphic projection coordinates:\nzA;B=\u0006s\nhx+i(hy+i\f)\nhx\u0000i(hy+i\f): (7)\nTo derive non-reciprocal time evolution of classical\nspin, we consider the eigenspectrum of the Hamilto-\nnian (5) in two-dimensional parameter space of mu-\ntually orthogonal magnetic \felds, ( hx;hy), at \fxed \f.\nFigs. 2(a){2(c) show the real and imaginary parts of the\neigenspectrum (6) at (a) \f= 0 and (b), (c) \f= 1. In\nthe \frst (Hermitian) case, the spectral surfaces meet at\na single diabolical point (DP) hx=hy=\f= 0. It is a\nspectral degeneracy that occurs due to simple symmetries\nand disappears under generic perturbations. In the Her-\nmitian case, states on both spectral surfaces are stable,with two \fxed points of classical spin dynamics classi-\n\fed as centers. Distinct surface colors in Figs. 2(b){2(f)\ncorrespond to opposite signs of imaginary parts of the\neigenvalues, with the states on the blue (orange) energy\nsurface being dynamically (un)stable, i.e. corresponding\nto spin directions of (un)stable equilibria. At \f6= 0, i.e.\nin the non-Hermitian case, the diabolical point splits into\ntwo chiral EPs located at ( hx=\u0006\f;hy= 0). The black\nsolid linehy= 0 in Figs. 2(b){2(f) is the line of PTsym-\nmetry, where the Hamiltonian (1) is PT-symmetric [20,\n19]. The interval of this line between the EPs corre-\nsponds to the regime of broken PTsymmetry, with the\neigenvalues forming complex conjugate pairs, while on\nthe part of the line outside of this interval the eigen-\nvalues are purely real. Note that in the general case,\nhy6= 0, the Hamiltonian (1) is not PT-symmetric and\nhas a complex eignespectrum. It is the reducibility of the\nnon-Hermitian Hamiltonian to the PT-symmetric form\nthat produces EPs in the eigenspectrum.\nFigs. 2(b), 2(c) show the na \u0010ve picture of encircling\na single EP in 2-dimensional parameter space, with the\nstate evolution following a 4 \u0019-periodic closed trajectory\n(red line) by visiting both, top and bottom, spectral sur-\nfaces before returning to the starting point. This picture\nis generally incorrect in the case of dynamic encircling of\nan EP with control parameters varied cyclically at \fnite\nspeed. Instead of adiabatic evolution, the system expe-\nriences sudden jumps between the two spectral surfaces,\nin the direction from the unstable to the stable one, see\nFigs. 2(d){2(f). The jumps occur not immediately upon4\ncrossing the line of PTsymmetry into the unstable re-\ngion, but after a certain time interval, illustrating the\ne\u000bect of stability loss delay [15].\nLet us consider the physics of encircling an isolated\nEP in parameter space ( hx; hy) of the model (1) by ap-\nplying time-dependent magnetic \feld. Encircling an EP\nreveals its chiral nature and provides access to topological\nnon-reciprocal physical e\u000bects unique to non-Hermitian\nsystems. In Fig. 2(d), we present parametric trajectories\nstarting on di\u000berent energy surfaces and going around\nthe EP in opposite directions. For \fxed model param-\neters (hx; hy;\f), thez-component of the starting point\nof each trajectory, i.e. the value of complex energy at\nt= 0, is determined by the initial spin orientations s(0),\nwhich can take any direction on the unit sphere. For the\npurpose of Figs. 2(b){2(f), we only consider the initial\nspin orientations corresponding to the two \fxed points\nof spin dynamics A and B, which con\fnes the start and\nend points of the trajectories in Figs. 2(b){2(f) to the\nspectral surfaces and best illustrates non-reciprocal dy-\nnamics. Numerical simulations of spin evolution show\nthat the end states for parametric trajectories starting\non the top and bottom spectral surfaces in Fig. 2(d) and\nencircling the EP in closed loops of the same orienta-\ntion always coincide. On the contrary, protocols with\nopposite encircling directions lead to mutually di\u000berent\nend states, either on the top or bottom energy surfaces,\nand, therefore, di\u000berent \fnal spin orientations after any\nnumber of complete revolutions around the EP. It has\nbeen pointed out in the literature that in order to ob-\nserve such non-reciprocal behaviour, the start/end points\nof closed loop trajectories much lie on the PT-symmetry\nline, where both eigenstates are neither stable nor unsta-\nble [15, 16, 24{28]. In fact, the e\u000bect of non-reciprocal dy-\nnamics stems from the process of crossing the line of PT-\nsymmetry in opposite directions and not strictly from the\ntopology of parametric trajectories. It turns out that to\nachieve non-reciprocity it is su\u000ecient to realize a closed\nloop protocol in the vicinity of a single EP without ac-\ntually encircling it [29], see Fig. 2(e), or to encircle both\nEPs, as shown in Fig. 2(f).\nConsider spin dynamics governed by the Hamilto-\nnian (5) under the following two protocols of encircling\nthe isolated EP ( \f= 1; hx=h0; hy= 0):\nP1;2(t)=[hx(t); hy(t)]=[h0+R0cos(\u001e(t)); R 0sin(\u001e(t))];\n\u001e(t) =\u001e0\u0007!t: (8)\nFor the purpose of this work, we focus on circular tra-\njectories centered at the EP, with h0= 1, see Fig. 2(d).\nSuppose the system is initialized at t= 0 with phase\n\u001e0= 0 and subjected to the protocol P1, which takes\nthe system clockwise around the EP. Regardless of the\ninitial spin orientation, i.e. location of the starting point\nwith respect to the two spectral surfaces, the spin will\neventually saturate in the direction of the stable \fxed\npoint and quickly converge to the state on the top en-\nergy surface, even if it was initialized in the unstable\n\fxed point (in which case the transition occurs rapidlyvia a non-adiabatic state-\rip process after a \fnite time\ndelay). The clockwise protocol P1ensures that the sys-\ntem's \fnal state after a full rotation around the EP is\non the bottom spectral surface, as shown in Fig. 2(d).\nFollowing a completely analogous argument, one can see\nthat the encircling protocol P2, which drives the system\nanti-clockwise around the EP, results in a di\u000berent \fnal\nstate - on the top spectral surface, regardless of the ini-\ntial spin direction at t= 0. This illustrates the chiral\nnature and non-reciprocity of parametrically driven dy-\nnamics near the EP.\nAs we have pointed out, in order to exploit the non-\nreciprocal properties discussed above, the encircling pro-\ntocols must start and end near the line of PTsymmetry.\nOnly such trajectories in parameter space can take the\nsystem to di\u000berent \fnal states upon crossing the line of\nPTsymmetry in opposite directions, e.g. by completing\na single clockwise or anti-clockwise closed loop around\nthe EP. The question of tolerance to deviations from this\ncondition, as well as optimization of other protocol pa-\nrameters to achieve consistent non-reciprocal behaviour,\nhave not been thoroughly studied in the literature. Here\nwe investigate the range of initial conditions and param-\neters of the encircling protocols P1;2optimal for observ-\ning the non-reciprocal evolution. We performed numer-\nical simulations of spin dynamics for a range of initial\nphases\u001e0, radiiR0, and driving frequencies !of the en-\ncircling protocols, Eq. (8). The \fnal state of the system\nafter executing the protocols P1;2correspond to the \fxed\npointszA;B, see Eq. (7). To quantify deviation of the end\nspin state after each encircling protocol from the corre-\nsponding \fxed point, we de\fne deviation angles \u000fi(\u001e0;!)\nbetween the spin direction at the end of a protocol Pi,\ni= 1;2, and the direction corresponding to \fxed point\nzA;B. As an absolute measure of deviation from perfect\nnon-reciprocity, we take the maximum deviation between\nthe two protocols, \u000f(\u001e0;!)\u0011max[\u000f1(\u001e0;!);\u000f2(\u001e0;!)].\nIn Fig. 3(a) we present the results of numerical sim-\nulations of spin dynamics following the protocols P1;2,\nEq. (8), with h0= 1 and!= 0:1, centered around the\nEP, for a range of parameters R0and\u001e0, which determine\nthe starting point of both protocols in the ( hx; hy)-plane.\nThe color map shows the maximum deviation parameter\n\u000f, measured in radians, in the logarithmic scale. A wide\nrange of initial protocol parameters ( hx;hy) that favor\nhigh-e\u000eciency non-reciprocal spin-\flter e\u000bect lie in the\ncentral blue region around the EP. The interval directly\nbetween the EPs exhibits poor non-reciprocity because it\nis adjacent to the region of broken PT-symmetry of the\nHamiltonian (1) with hy= 0, where the eigenspectrum\nis purely imaginary, see Fig. 2(b){2(c), corresponding to\none stable and one unstable \fxed points of spin dynam-\nics. As a result, the \fnal states of the system in the\nvicinity of this region in the ( hx;hy)-plane are almost\nidentical regardless of the protocol encircling direction.\nThe boundary of the low deviation region is set by spe-\nci\fc properties of the stability loss delay e\u000bect, discussing\nwhich is beyond the scope of the present work and will5\nFIG. 3. (a) Color map of maximum deviation parameter, \u000f, of the \fnal spin direction from the \fxed points zA;Bafter encircling\nprotocolsP1;2, as a function of initial protocol phase \u001e0(deviation from the line of PT-symmetry, hy= 0) and protocol\nfrequency!, plotted in logarithmic scale. The deviation \u000fis a measure of e\u000bectiveness of the non-reciprocal spin-\flter e\u000bect.\n(b) Dependence of the deviation parameter on the initial phase for a range of protocol frequencies, showing that slower encircling\nprotocols yield higher non-reciprocal e\u000eciency. The plot corresponds to encircling protocols starting and ending on the black\ndashed line in panel (a). (c) The deviation parameter as a function of protocol radius R0. A sharp jump in deviation from\nperfect non-reciprocity is observed for large radii within the considered range, corresponding to the boundary of the high-\ne\u000eciency region in panel (a), shown in blue. The size of the region with small deviation parameter \u000fshrinks with increasing\nfrequency. The plot corresponds to encircling protocols starting and ending on the white dashed line in panel (a).\nbe addressed elsewhere. Note that the highest-e\u000eciency\nnon-reciprocal protocols in Fig. 2(a) have a relatively\nlarge radius, such that they encircle both EPs. This is\nan important result that extends the idea of encircling a\nsingle EP to achieve non-reciprocity to parametric tra-\njectories that go around both isolated EPs to achieve an\neven more robust e\u000bect.\nIn Fig. 3(b), the deviation quantity \u000fis plotted for pro-\ntocols starting on the circle of radius R0= 0:5 around\nthe EP, shown by the black dashed line in Fig. 3(a),\nfor a range of initial phases \u001e0and protocol frequencies\n!. This result illustrates that lower encircling frequen-\ncies result in smaller deviation parameter \u000f, i.e. higher\ne\u000eciency of the non-reciprocal e\u000bect. Fig. 3(c) shows\ndependence of the deviation parameter on protocol ra-\ndiusR0for the same range of frequencies !, calculated\nat \fxed initial phase \u001e0= 0. The initial conditions\nfor the protocols studied in Fig 3(c) are chosen on the\nline ofPT-symmetry, shown as a white dashed line in\nFig. 3(a). In the limit of slow encircling speeds, the proto-\ncols with larger radius result in smaller deviation \u000f, even\nwhen both EPs are encircled by the protocols ( R0>2).\nThe example of non-reciprocal spin dynamics that resultsfrom parametric encircling of both EPs is presented in\nFig. 2(f). While Fig. 3(c) shows relatively low degree of\ndeviation from non-reciprocity in the case of parametric\ntrajectories with small radius R0, the practical impor-\ntance of protocols that lie in the immediate vicinity of\nthe EP is weakened by the fact that eigenstates coalesce\nin the EP. As a result, the angle between spin orienta-\ntion given by the \fxed points zAandzBdisappears in\nthe limitR0!0.\nNote that it is not necessary to encircle an EP to ob-\nserve non-reciprocity, see Fig. 2(e). Periodic trajectories\nin the vicinity of an EP can achieve the same e\u000bect [29].\nWhat is crucial for realizing non-reciprocal dynamics is\nthe direction in which the protocols cross the line of PT-\nsymmetry, where \fxed points of the corresponding dy-\nnamics change their stability properties. In fact, para-\nmetric trajectories may not enclose any area at all and\nstill achieve a high-e\u000eciency non-reciprocal state con-\nversion. The study of various trajectory types and the\nrobustness of resultant non-reciprocal dynamics will be\nconsidered elsewhere.6\nMETHODS\nThe numerical simulations were carried out in Wolfram\nMathematica [30]. Each data point in Figs. 3(a){3(d) was\nobtained by averaging results of 10 separate simulation\nruns with arbitrary initial spin orientations uniformly dis-\ntributed on a unit sphere at protocol initiation.\nDISCUSSION\nTo summarize, there has been a remarkable advance in\nexploiting non-Hermitian systems and their distinct fea-\nture, spectral exceptional points. The stage for this non-\nHermitian physics is however restricted mostly to optical\nand nanophotonic systems. Here we extended the ex-\nploration of EP singularities onto another broadest class\nof physical systems, spin systems. We demonstrated that\nspin systems can host EP singularities and thus become a\ncrucial element for realizing devices exhibiting non-trivial\ntopological transport seen so far only in optical systems.\nThis establishes that non-reciprocal e\u000bects are ubiquitous\nin open dissipative systems with gain and loss. Imple-\nmentation of topological spin transfer enabled us to pro-pose a high-e\u000eciency asymmetric spin \flter device based\non non-reciprocal spin dynamics around an EP of the cor-\nresponding non-Hermitian Hamiltonian. As a prototypi-\ncal model, we studied time evolution of a single classical\nspin in presence of time-dependent magnetic \feld and\nspin-transfer torque. Appearance of EPs in the Hamilto-\nnian's eigenspectrum allows for encircling protocols that\nresult in di\u000berent \fnal spin states despite identical initial\nconditions. We considered two protocols with opposite\nencircling orientations and showed that such a scheme\ncan be used as a high-e\u000eciency spin \flter. Optimal pro-\ntocol parameters are suggested, indicating that encircling\ntrajectories in parameter space of applied magnetic \felds\nmust start and end near the line of PTsymmetry, with\nslower encircling frequencies resulting in higher e\u000eciency\nof the asymmetric spin-\flter e\u000bect.\nACKNOWLEDGEMENTS\nA.G. and V.M.V. were supported by the U.S. Depart-\nment of Energy, O\u000ece of Science, Basic Energy Sciences,\nMaterials Sciences and Engineering Division.\n1Damon, R. W. & Eshbach, J. R. Magnetostatic modes of\na ferromagnet slab. J. Phys. Chem. Solids 19, 308-320\n(1961).\n2Dzyaloshinsky, I. E. A thermodynamic theory of \\weak\"\nferromagnetism of antiferromagnetics. J. Phys. Chem.\nSolids 4, 241{255 (1958).\n3Moriya, T. Anisotropic superexchange interaction and\nweak ferromagnetism. Phys. Rev. 120, 91 (1960).\n4Zakeri, K. et al. Asymmetric spin-wave dispersion on\nFe(110): direct evidence of the Dzyaloshinskii-Moriya in-\nteraction. Phys. Rev. Lett. 104, 137203 (2010).\n5Iguchi, Y., Uemura, S., Ueno, K & Onose, Y., Nonrecipro-\ncal magnon propagation in a noncentrosymmetric ferro-\nmagnet LiFe 5O8.Phys. Rev. B. 92, 184419 (2015).\n6Sato, T. J. et al. Magnon dispersion shift in the induced\nferromagnetic phase of noncentrosymmetric MnSi, Phys.\nRev. B 94, 144420 (2016).\n7Bender, C. M. & Boettcher, S. Real spectra in non-\nHermitian Hamiltonians having PTsymmetry. Phys. Rev.\nLett.80, 5243 (1998).\n8Bender, C. M. Boettcher, S. & Meisinger, P. N. PT-\nsymmetric quantum mechanics. J. Math. Phys. 40, 2201{\n2229 (1999).\n9Moiseyev, N. Non-Hermitian quantum mechanics. (Cam-\nbridge Univ. Press, 2011).\n10Heiss, W. D. Repulsion of resonance states and exceptional\npoints. Phys. Rev. E 61, 929 (2000).\n11Heiss, W. D. & Harney, H. L. The chirality of exceptional\npoints. Eur. Phys. J. D 17, 149{151 (2001).\n12Heiss, W. D. Exceptional points - their universal occurrence\nand their physical signi\fcance. Czech. J. Phys. ,54, 1091{\n1099 (2004).13Berry, M. V. Physics of nonhermitian degeneracies. Czech.\nJ. Phys. 54, 1039{1047 (2004).\n14Dembowski, C. et al. Experimental observation of the topo-\nlogical structure of exceptional points. Phys. Rev. Lett. 86,\n787 (2001).\n15Doppler, J. et al. Dynamically encircling an exceptional\npoint for asymmetric mode switching. Nature 537, 7618\n(2016).\n16Choi, Y. et al. Extremely broadband, on-chip optical non-\nreciprocity enabled by mimicking nonlinear anti-adiabatic\nquantum jumps near exceptional points. Nat. Commun. 8,\n14154 (2017).\n17Ruschhaupt, A., Delgado, F. & Muga, J. G. Physical real-\nization of PT-symmetric potential scattering in a planar\nslab waveguide. J. Phys. A: Math. Gen. 38, L171 (2005).\n18Slonczewski, J. C. Current-driven excitation of magnetic\nmultilayers. J. Magn. Magn. Mater. 159, L1L7 (1996).\n19Galda, A. & Vinokur, V. M. Linear dynamics of classical\nspin as M obius transformation. Sci. reports 7, 1168 (2017).\n20Galda, A. & Vinokur, V. M. Parity-time symmetry break-\ning in magnetic systems. Phys. Rev. B 94, 020408 (2016).\n21Lieb, E. H. The classical limit of quantum spin systems.\nComm. Math. Phys. 34, 327{340 (1973).\n22Stone, M., Park, K.-S. & Garg, A. The semiclassical prop-\nagator for spin coherent states. Journ. Math. Phys. 41,\n8025{8049 (2000).\n23Radcli\u000be, J. M. Some properties of coherent spin states. J.\nPhys. A: Gen. Phys. 4, 313 (1971).\n24Uzdin, R., Mailybaev, A. & Moiseyev, N. On the observ-\nability and asymmetry of adiabatic state \rips generated by\nexceptional points. J. Phys. A: Math. Theor. 44, 435302\n(2011).7\n25Gilary, I., Mailybaev, A. A. & Moiseyev, N. Time-\nasymmetric quantum-state-exchange mechanism. Phys.\nRev. A 88, 010102 (2013).\n26Milburn, T. J. et al. General description of quasiadiabatic\ndynamical phenomena near exceptional points. Phys. Rev.\nA92, 052124 (2005).\n27Hassan, A. U. et al. Dynamically encircling exceptional\npoints: exact evolution and polarization state conversion.Phys. Rev. Lett. 118, 093002 (2017).\n28Zhang, X. L., Wang, S., Hou, B. & Chan, C. T. Dynamically\nencircling exceptional points: in situ control of encircling\nloops and the role of the starting point. Phys. Rev. X 8,\n021066 (2018).\n29Hassan, A. U. et al. Chiral state conversion without encir-\ncling an exceptional point. Phys. Rev. A 96, 052129 (2017).\n30Inc., W. R. Mathematica, Version 11.0. Champaign, IL,\n2016."    },    {        "title": "0709.4561v1.Semi_Classical_Dynamics_in_Quantum_Spin_Systems.pdf",        "content": "arXiv:0709.4561v1  [math-ph]  28 Sep 2007Semi-Classical Dynamics in Quantum Spin Systems\nJ¨ urg Fr¨ ohlich1∗Antti Knowles1†Enno Lenzmann2‡\n1Institute of Theoretical Physics, ETH H¨ onggerberg, CH-80 93 Z¨ urich, Switzerland.\n2Department of Mathematics, Massachusetts Institute of Tec hnology, Cambridge, MA 02139-4307.\nNovember 1, 2021\nAbstract\nWe consider two limiting regimes, the large-spin and the mea n-field limit, for the dynamical\nevolution of quantum spin systems. We prove that, in these li mits, the time evolution of a class of\nquantum spin systems is determined by a corresponding Hamil tonian dynamics of classical spins.\nThis result can be viewed as a Egorov-type theorem. We extend our results to the thermodynamic\nlimit of lattice spin systems and continuum domains of infini te size, and we study the time\nevolution of coherent spin states in these limiting regimes .\n1 Introduction\nThe purpose of this letter is to study classical limits of quantum spin s ystems. Work in this direction\nwas undertaken already at the beginning of the seventies. Yet, mo st of the mathematical results\nestablished so far only concern time-independent aspects, such a s the classical limit of quantum\npartition functions for spin systems [4, 5]. Here we consider the dyn amical evolution of quantum spin\nsystems in limiting regimes; see also [7]. In particular, we discuss i) the large-spin limit andii) the\nmean-field/continuum limit . As our main results, we prove that the time evolution of a large class of\nquantum spin systems approaches the time evolution of classical sp ins. Our results can be regarded\nasEgorov-type theorems, asserting that quantization commutes with time evolution in the class ical\nlimit; see [1] for a similar result on classical and quantum Bose gases. A long the way, we discuss\nthermodynamic limits and the time evolution of coherent spin states, in the two limits mentioned\nabove.\nAn example of an evolution equation for classical spins is the Landau-Lifshitz equation\n∂tM=M∧Hex(M), (1)\nwhich is widely used in the study of ferromagnetism. Here M=M(t,x)∈S2denotes a classical spin\nfield with values on the unit sphere, and ∧stands for the vector product in R3. A standard choice for\nthe exchange field is Hex(M) =J∆M, where ∆ denotes the Laplacian and Jis the exchange coupling\nconstant. Equation (1) then becomes\n∂tM=J M∧∆M . (2)\nThis form of the Landau-Lifshitz equation has been studied in the ma thematical literature; see for\ninstance [2, 3] and references given there. In physical terms, (2 ) describes the dynamics of spin waves\nin a ferromagnet with nearest neighbor exchange interactions in a c lassical regime; see [6].\n∗juerg@itp.phys.ethz.ch\n†aknowles@itp.phys.ethz.ch\n‡lenzmann@math.mit.edu\n1In this paper we consider the Landau-Lifshitz equation with an exch ange field Hex(M) given by\nan integral operator applied to M, and generalizations thereof. Equation (1) then takes the form\n∂tM(t,x) =M(t,x)∧/integraldisplay\nJ(x,y)M(t,y)dy. (3)\nThe integral kernel J(x,y) describes the exchange interactions between classical spins bey ond the\nnearest-neighborapproximation in the continuum limit. A formal arg ument on how to derive (2) from\n(3) is given in a remark in Sect. 3.4.\nOur paper is organized as follows. In Sect. 2, we study the dynamics of finite lattice systems of\nquantum spins in the limit where their spin sapproaches ∞. The main result of Sect. 2 is formulated\nin Theorem 1 below. In order to prepare the ground for this theore m and its proof, we first introduce\na class of Hamilton functions for classical spins and define their quan tization by means of a normal-\nordering prescription. At the end of Sect. 2, we pass to the therm odynamic limit, and we discuss the\ntime evolution of coherent spin states.\nIn Sect. 3, we present a similar analysis for the mean-field limit of quan tum spin systems defined\non a lattice with spacing h >0 in the continuum limit, h→0. The main result of Sect. 3 is stated in\nTheorem 4 below.\n2 Large-Spin Limit\n2.1 A system of classical spins\nLet Λ be a finite subset of the lattice Zd(or any other lattice). A classical spin system on Λ is\ndescribed in terms of the finite-dimensional phase space\nΓΛ:=/productdisplay\nx∈ΛS2,\ni.e. we associate an element M(x) of the unit two-sphere S2⊂R3with each site x∈Λ. The phase\nspace Γ Λis conveniently coordinatized as follows. For each site x∈Λ, let (M1(x),M2(x),M3(x))\ndenote the three cartesian components of a unit vector M(x)∈S2, and define the complex coordinate\nfunctions ( M+(x),Mz(x),M−(x)) onS2by\nM±(x):=M1(x)±iM2(x)√\n2, M z(x):=M3(x).\nWe define a Poisson structure1on ΓΛby setting\n{Mi(x),Mj(y)}=i˜εijkδ(x,y)Mk(x). (4)\nHereδ(x,y) stands for the Kronecker delta, and the indices i,j,krun through the index set I:=\n{+,z,−}, where the symbol ˜ εijkis defined as ˜ ε±∓z=±1, ˜ε±z±=∓1, ˜εz±±=±1, and ˜εijk= 0\notherwise.\nFor our purposes it is convenient (but not necessary) to replace S2with the closed unit ball\nB1(0)⊂R3. To this end, we introduce a larger “phase space” (a Poisson manifo ld)\nΞΛ:=/productdisplay\nx∈ΛB1(0),\nequipped with the l∞-norm. The algebra\nPΛ:=C/bracketleftbig\n{Mi(x) :i∈I,x∈Λ}/bracketrightbig\nof complex polynomials is a Poisson algebra with Poisson bracket deter mined by (4). We equip PΛ\nwith the norm /bardblA/bardbl∞:= supM∈ΞΛ|A(M)|, and we denote its norm closure by AΛ. Note that, by the\nStone-Weierstrass theorem, AΛis the algebra of continuous complex-valued functions on Ξ Λ.\n1Actually Γ Λis symplectic, with symplectic structure determined by the usual one on S2.\n2A fairly general class of Hamilton functions HΛon ΞΛmay be described as follows: We asso-\nciate with each multi-index α∈NI×Zdsatisfying |α|:=/summationtext\nx∈Zd/summationtext\ni∈Iαi(x)<∞a complex number\nV(α). Using the trivial embedding NI×Λ⊂NI×Zddefined by adjoining zeroes, we consider Hamilton\nfunctions of the form\nHΛ:=/summationdisplay\nα∈NI×ΛV(α)Mα. (5)\nIn order to obtain a real-valued HΛ, we require that V(α) =V(α), where the “conjugation” αof a\nmulti-index αis defined as αi(x):=αi(x), with·: (+,z,−)/ma√sto→(−,z,+). Furthermore, we impose\nthe following bound on the interaction potential:2\n/bardblV/bardbl:=/summationdisplay\nn∈Nsup\nx∈Zd/summationdisplay\nα∈NI×Zd\n|α|=n|α(x)||V(α)|en<∞. (6)\nIt is then easy to see that the series (5) converges in norm and tha t the set of allowed interaction\npotentials Vis a Banach space. The Hamiltonian equation of motion reads ˙A={HΛ,A}, for any\nobservable A∈AΛ. In particular, a straightforward calculation yields\nd\ndtMi(t,x) =/summationdisplay\nα∈NI×ΛV(α)/summationdisplay\nj,ki˜εjikαj(x)Mα−δj(x)+δk(x)(t), (7)\nwhere the multi-index δi(x) is defined by [ δi(x)]j(y):=δijδ(x,y).\nWe record the following well-posedness result for the dynamics gene rated by the class of Hamilto-\nnians introduced above.\nLemma 1. LetΛbe a (possibly infinite) subset of Zd. LetM0∈ΞΛ. Then the Hamiltonian equation\n(7)has a unique global-in-time solution M∈C1(R,ΞΛ)that satisfies M(0) =M0. Moreover, the\nsolution Mdepends continuously on the initial condition M0, and we have the pointwise conservation\nlaw|M(t,x)|=|M(0,x)|for allt.\nProof.Local-in-time existence and uniqueness follows from a simple contrac tion mapping argument\nfor the integral equation associated with (7). We omit the details. A lso, continuous dependence on\nM0follows from standard arguments. Finally, the claim that |M(0,x)|=|M(t,x)|for alltcan be\neasily verified by using (7), which implies thatd\ndtM(t,x) is perpendicular to M(t,x).\nRemarks. 1. In what follows, we denote the flow map M0/ma√sto→M(t) byφt\nΛ. Note that, under our\nassumptions, (7) also makes sense for infinite Λ ⊂Zd, whereas the Hamiltonian HΛdoes not have a\nlimit when |Λ| → ∞.\n2. The last statement implies that the magnitude of each spin remains constant in time, i.e. the\nspins precess. In particular, if M0∈ΓΛ, it follows that M(t)∈ΓΛfor allt. Mathematically, this is\nsimply the statement that the symplectic leaves of the Poisson manif old ΞΛremain invariant under\nthe Hamiltonian flow.\n3. Time-dependent potentials V(t,α) may be treated without additional complications, provided that\nthe map t/ma√sto→V(t) is continuous (in the above norm) and supxin (6) is replaced by supx,t. The\nweaker assumption that t/ma√sto→V(t,α) is continuous for all αimplies Lemma 1 with the slightly weaker\nstatement that M∈C(R,ΞΛ) is a classical solution of (7).\nExample. Consider the Hamiltonian\nHΛ(t) =−/summationdisplay\nx∈Λh(t,x)·M(x)−1\n2/summationdisplay\nx,y∈ΛJ(x,y)M(x)·M(y), (8)\nwhereM(x) = (M1(x),M2(x),M3(x)). Here h(t,x)∈R3is an “external magnetic field” satisfying\nsupt∈R,x∈Zd|h(t,x)|<∞. We also require the map t/ma√sto→h(t,x) to be continuous for all x∈Zd.\n2Note that this condition may be weakened by replacing enwithern, for any r >0. It may be checked that this\ndoes not affect the following results.\n3The exchange coupling J:Zd×Zd→Ris assumed to be symmetric and to satisfy J(x,x) = 0 for\nallx. Finally we assume, in accordance with condition (6), that supx∈Zd/summationtext\ny∈Zd|J(x,y)|<∞. The\ncorresponding equation of motion for M(t,x) is given by\nd\ndtM(t,x) =M(t,x)∧/bracketleftbigg\nh(t,x)+/summationdisplay\ny∈ΛJ(x,y)M(t,y)/bracketrightbigg\n, (9)\ni.e. theLandau-Lifschitz equation for a classical lattice spin system.\n2.2 A system of quantum spins\nIn this section we formulate the quantum analogue of the system of classical spins from the previous\nsection. We associate with each point x∈Zda finite-dimensional Hilbert space Hs\nx≡Hx:=C2s+1\ndescribing a quantum-mechanical spin of magnitude s. (Here, and in the following, we refrain from\ndisplaying the explicit s-dependence whenever it is not needed.) Furthermore, we associa te with each\nfinite set Λ ⊂Zdthe product space HΛ:=/circlemultiplytext\nx∈ΛHx, and we define the algebra /hatwideAΛas the algebra of\n(bounded) operators on HΛ, equipped with the operator norm /bardbl·/bardbl.\nThe spins are represented on HΛby a family {/hatwideSi(x) :i= 1,2,3, x∈Λ}of operators, where\n/hatwideSi(x) is thei’th generator of the spin- s-representation of su(2) on Hx, rescaled by s−1. In analogy to\nthe complex coordinatization of the classical phase space Γ Λin the previous section, we replace the\noperators ( /hatwideS1(x),/hatwideS2(x),/hatwideS3(x)) with ( /hatwideS+(x),/hatwideSz(x),/hatwideS−(x)) as follows:\n/hatwideS±(x):=/hatwideS1(x)±i/hatwideS2(x)√\n2,/hatwideSz(x):=/hatwideS3(x),for allx∈Zd.\nAn easy calculation yields /bardbl/hatwideS±(x)/bardbl ≤1 and/bardbl/hatwideSz(x)/bardbl= 1 ifs≥1, and/bardbl/hatwideS±(x)/bardbl=√\n2 and/bardbl/hatwideSz(x)/bardbl= 1\nifs= 1/2. Furthermore one finds the fundamental commutation relations\n/bracketleftbig/hatwideSi(x),/hatwideSj(y)/bracketrightbig\n=1\ns˜εijkδ(x,y)/hatwideSk(x) (10)\nwithi,j,k∈I.\n2.3 Quantization\nIn orderto quantize polynomialsin PΛweneed a concept ofnormalordering. We saythat a monomial\n/hatwideSi1(x1)···/hatwideSip(xp) isnormal-ordered ifik< il⇒k < l, where<is defined on Ithrough + < z <−.\nWe then define normal-ordering by\n:/hatwideSi1(x1)···/hatwideSip(xp):=/hatwideSiσ(1)(xσ(1))···/hatwideSiσ(p)(xσ(p)),\nwhereσ∈Spis a permutation such that the monomial on the right side is normal-or dered. Next, we\ndefinequantization /hatwide·:PΛ→/hatwideAΛby setting\n(Mi1(x1)···Mip(xp))c=:/hatwideSi1(x1)···/hatwideSip(xp):\nand by linearity of /hatwide·. We set /hatwide1 = /BD. Note that, by definition, /hatwide·is a linear map (but, of course, not\nan algebra homomorphism) and satisfies ( /hatwideA)∗=/hatwideA.\n2.4 Dynamics in the large-spin limit\nFor each finite Λ ⊂Zdwe define the Hamiltonian /hatwideHΛas the quantization of HΛ. More precisely, we\nquantize (5) term by term and note that the resulting series conve rges in operator norm. Because HΛ\nis real, the operator /hatwideHΛis self-adjoint on the finite-dimensional Hilbert space HΛand generates a\none-parameter group of unitary propagators Us(t;/hatwideHΛ) (equal to e−isbHΛtif/hatwideHΛis time-independent).\n4We introduce the short-hand notation\nαt\nΛA:=A◦φt\nΛ, A∈AΛ,\n/hatwideαt\nΛA:=Us(t;/hatwideHΛ)∗AUs(t;/hatwideHΛ),A ∈/hatwideAΛ,\nwhereφt\nΛis the Hamiltonian flow on Ξ Λ. Note that both αt\nΛand/hatwideαt\nΛare norm-preserving.\nWe are now able to state and prove our main result for the case of a fi nite lattice Λ. Roughly it\nstates that time evolution and quantization commute in the s→ ∞limit. This is a Egorov-type result .\nTheorem 1. LetA∈PΛandε >0. Then there exists a function A(t)∈PΛsuch that\nsup\nt∈R/bardblαt\nΛA−A(t)/bardbl∞≤ε, (11)\nand, for any t∈R,\n/vextenddouble/vextenddouble/hatwideαt\nΛ/hatwideA−/hatwidestA(t)/vextenddouble/vextenddouble≤ε+C(ε,t,A)\ns, (12)\nwhereC(ε,t,A)is independent of Λ.\nProof.Without loss of generality we assume that A=Mβfor some β∈NI×Λ. For simplicity of\nnotation we also assume, here and in the following proofs, that HΛis time-independent. Consider the\nLie-Schwinger series for the time evolution of the classical spin syst em,\n∞/summationdisplay\nl=0tl\nl!/braceleftbig\nHΛ,A/bracerightbig(l), (13)\nwhere/braceleftbig\nHΛ,A/bracerightbig(l)=/braceleftBig\nHΛ,/braceleftbig\nHΛ,A/bracerightbig(l−1)/bracerightBig\nand/braceleftbig\nHΛ,A/bracerightbig(0)=A. In order to compute the nested\nPoisson brackets we observe that\n{Mα,Mβ}=/summationdisplay\nx∈Λ/summationdisplay\ni,j,k∈Ii˜εijkαi(x)βj(x)Mα+β−δi(x)−δj(x)+δk(x), (14)\nas can be seen after a short calculation. Iterating this identity yield s\n/braceleftbig\nHΛ,A/bracerightbig(l)=il/summationdisplay\nα1,...,αl/summationdisplay\nx1,...,xl/summationdisplay\ni1,...,il/summationdisplay\nj1,...,jl/summationdisplay\nk1,...,kl/bracketleftBiggl/productdisplay\nq=1˜εiqjqkqV(αq)αq\niq(xq)/bracketleftbigg\nβ+q−1/summationdisplay\nr=1/parenleftbig\nαr−δir(xr)−δjr(xr)+δkr(xr)/parenrightbig/bracketrightbigg\njq(xq)/bracketrightBigg\nMβ+Pl\nr=1/parenleftbig\nαr−δir(xr)−δjr(xr)+δkr(xr)/parenrightbig\n. (15)\nIn order to estimate this series, we recall that /bardblMγ/bardbl∞≤1 and rewrite it by using that\n/summationdisplay\nα1,...,αl=∞/summationdisplay\nn1,...nl=1/summationdisplay\n|α1|=n1···/summationdisplay\n|αl|=nl.\nWe then proceed recursively, starting with the sum over αl,xl,il,jl,kland, at each step, using that\n/summationdisplay\n|α|=n/summationdisplay\nx/summationdisplay\ni,j,k|˜εijk|αi(x)γj(x)|V(α)| ≤ |γ|/bardblV/bardbl(n),\nwhere\n/bardblV/bardbl(n):= sup\nx∈Zd/summationdisplay\n|α|=n|V(α)||α(x)|.\n5In this manner we find that\n/vextenddouble/vextenddouble/vextenddouble/braceleftbig\nHΛ,A/bracerightbig(l)/vextenddouble/vextenddouble/vextenddouble\n∞\n≤∞/summationdisplay\nn1,...,nl=1|β|(|β|+n1)···(|β|+n1+···+nl−1)/bardblV/bardbl(n1)···/bardblV/bardbl(nl)\n≤∞/summationdisplay\nn1,...,nl=1(|β|+n1+···+nl)l/bardblV/bardbl(n1)···/bardblV/bardbl(nl)\n≤l!∞/summationdisplay\nn1,...,nl=1e|β|+n1+···+nl/bardblV/bardbl(n1)···/bardblV/bardbl(nl)\n=l!e|β|/bardblV/bardbll.\nThus, for |t|</bardblV/bardbl−1, the series (13) converges in norm, and an analogous estimate of t he remainder\nof the Lie-Schwinger expansion of αt\nΛAshows that (13) equals αt\nΛA. As all estimates are independent\nof Λ, the convergence is uniform in Λ.\nThe quantum-mechanical case is similar. Consider the Lie-Schwinger series for the time evolution\nof the quantum spin system:\n∞/summationdisplay\nl=0tl\nl!(is)l/bracketleftbig/hatwideHΛ,/hatwideA/bracketrightbig(l), (16)\nwhere/bracketleftbig/hatwideHΛ,/hatwideA/bracketrightbig(l)=/bracketleftBig\n/hatwideHΛ,/bracketleftbig/hatwideHΛ,/hatwideA/bracketrightbig(l−1)/bracketrightBig\nand/bracketleftbig/hatwideHΛ,/hatwideA/bracketrightbig(0)=/hatwideA. In order to estimate the multiple com-\nmutators, we remark that, from (4) and (10) and since both {·,·}andis[·,·] are derivations in\nboth arguments, we see that ( is)l/bracketleftbig/hatwideHΛ,/hatwideA/bracketrightbig(l)is equal to the expression obtained from/braceleftbig\nHΛ,A/bracerightbig(l)by\nreordering the terms appropriately and by replacing Mi(x) with/hatwideSi(x). In particular (assuming s≥1)\n/vextenddouble/vextenddouble/vextenddouble(is)l/bracketleftbig/hatwideHΛ,/hatwideA/bracketrightbig(l)/vextenddouble/vextenddouble/vextenddouble≤l!eβ/bardblV/bardbll,\nand we deduce exactly as above that (16) equals /hatwideαt\nΛ/hatwideAfort </bardblV/bardbl−1.\nTo show the claim of the theorem for |t|</bardblV/bardbl−1we first remark that/hatwiderαt\nΛ(A) is well-defined\nthrough its convergent power series expansion. Now as shown abo ve, each term of ( is)l/bracketleftbig/hatwideHΛ,/hatwideA/bracketrightbig(l), as a\npolynomial in /hatwideAΛ, is equal to a reorderingof the corresponding term of/braceleftbig\nHΛ,A/bracerightbig(l). IfPis a monomial\n(with coefficient 1) of degree pin the generating variables {/hatwideSi(x)}and˜Pa monomial obtained from\nPby any reordering of terms, the commutation relations (10) imply th at\n/bardblP−˜P/bardbl ≤p2\ns.\nThus/parenleftBig/braceleftbig\nHΛ,A/bracerightbig(l)/parenrightBigc\n= (is)l/bracketleftbig/hatwideHΛ,/hatwideA/bracketrightbig(l)+Rl,\nwhere, recalling the expression (15) and the estimates following it, w e see that the “loop terms” Rl\nare bounded by\n/bardblRl/bardbl ≤1\ns∞/summationdisplay\nn1,...,nl=1(|β|+n1+···+nl)l+2/bardblV/bardbl(n1)···/bardblV/bardbl(nl)\n≤(l+2)!\nse|β|/bardblV/bardbll.\nTherefore, if |t|</bardblV/bardbl−1,\n/bardbl/hatwideαt\nΛ/hatwideA−/hatwidestαt\nΛA/bardbl ≤e|β|\ns∞/summationdisplay\nl=0(l+2)(l+1)(t/bardblV/bardbl)l≤C(t,A)\ns,\n6whereC(t,A) is independent of Λ.\nIn order to extend the result to arbitrary times we proceed by iter ation. The crucial observations\nthat enable this process are that the convergence radius /bardblV/bardbl−1is independent of |β|andαt\nΛ,/hatwideαt\nΛare\nnorm-preserving. Let t∈Rand choose ν∈Nsuch that τ:=t/νsatisfies|τ|</bardblV/bardbl−1. In order to\niterate we need to introduce a cutoff in the series (13) and (15). Th e series (13) consists of an infinite\nsum of terms in PΛwhich are be indexed by ( l,α1,...,αl,x1,...,x l). Now let ε >0 be given. Since\nthe series converges in norm there is a finite subset\nB1=B1(ε)⊂ {(l,α1,...,αl,x1,...,x l)}=∞/uniondisplay\nl=0/parenleftbig\nNI×Λ/parenrightbigl×Λl\nsuch that the norm of the series restricted to the complement of B1is smaller than ε/ν. This\ninduces a splitting ατ\nΛA=αB1A+αBc\n1A(in self-explanatory notation), such that αB1A∈PΛand\n/bardblαBc\n1A/bardbl∞≤ε/ν. Similarly, one splits /hatwideατ\nΛ/hatwideA=/hatwideαB1/hatwideA+/hatwideαBc\n1/hatwideAwhere, after an eventual increase of B1,\n/bardbl/hatwideαBc\n1/hatwideA/bardbl ≤ε/ν.\nNow we use the above result for |τ|</bardblV/bardbl−1:\n/hatwideατ\nΛ/hatwideA=/hatwiderαB1A+R1\ns+/hatwideαBc\n1/hatwideA\nwhereR1is some bounded operator. Since αB1A∈PΛwe may repeat the process on the time interval\n[τ,2τ]:\n/hatwideατ\nΛ/hatwideατ\nΛ/hatwideA=/hatwideατ\nΛ/hatwiderαB1A+/hatwideατ\nΛR1\ns+/hatwideατ\nΛ/hatwideαBc\n1/hatwideA\n=/parenleftbig\nαB2αB1A/parenrightbigc+/hatwideαBc\n2/hatwiderαB1A+/hatwideατ\nΛ/hatwideαBc\n1/hatwideA+R2+/hatwideατ\nΛR1\ns\nContinuinginthismanneroneseesthat, since αt\nΛand/hatwideαt\nΛarenorm-preserving, A(t):=αBν···αB1A∈\nPΛsatisfies\n/bardblαt\nΛA−A(t)/bardbl∞≤ε,\nas well as\n/bardbl/hatwideαt\nΛ/hatwideA−/hatwidestA(t)/bardbl ≤ε+C(ε,t,A)\ns.\n2.5 The thermodynamic limit\nThe above analysis was done for a finite subset Λ, but the observed uniformity in Λ allows for a\nstatement of the result directly in limit Λ = Zd. We pause to describe how this works.\nConcentrate first on the quantum case. If Λ 1⊂Λ2, an operator A1∈/hatwideAΛ1may be identified in\nthe usual fashion with an operator A2∈/hatwideAΛ2by setting A2=A1⊗ /BDΛ2\\Λ1. We shall tacitly make\nuse of this identification in the following. It induces the norm-preser ving mapping /hatwideAΛ1→/hatwideAΛ2of the\nabstract C∗-algebras and the isotony relation /hatwideAΛ1⊂/hatwideAΛ2. Observables of the quantum spin system\nin the thermodynamic limit are then elements of the quasi-local algebra\n/hatwideA:=/logicalordisplay\nΛ⊂Zdfinite/hatwideAΛ,\nwhich is the C∗-algebra defined as the closure of the normed algebra generated b y the union of all\n/hatwideAΛ’s, where Λ is finite. The spins are represented on /hatwideAby a family {/hatwideSi(x) :i∈I, x∈Zν}of\noperators.\nThe dynamics of the system is determined by a one-parameter grou p (/hatwideαt)t∈Rof automorphisms of\n/hatwideA. Its existence is a corollary of the proof of Theorem 1.\n7Lemma 2. LetA ∈/hatwideAΛ0for some finite Λ0⊂Zdandt∈R. Then the following limit exists in the\nnorm sense:\nlim\nΛ→∞/hatwideαt\nΛA=:/hatwideαtA,\nwhereΛ→ ∞means that Λeventually contains every finite subset. By continuity this extends to a\nstrongly continuous one-parameter group (/hatwideαt)t∈Rof automorphisms of /hatwideA.\nProof.For|t|</bardblV/bardbl−1the series (16) is bounded in norm, uniformly in Λ, so to show converg ence of\nthe series it suffices to show the convergence of/bracketleftbig/hatwideHΛ,/hatwideA/bracketrightbig(l)for each l∈N, which is an easy exercise.\nThus/hatwideαtAis well-defined for any polynomial A. By continuity, /hatwideαtextends to an automorphism\nof/hatwideA. Since/hatwideαtA ∈/hatwideAand/hatwideαtis a one-parameter group, we may extend it to all times by iteration.\nStrong continuity follows since /hatwideαtA, for small tand polynomial A, is defined through a convergent\npower series:\nlim\nt→0/bardbl/hatwideαtA−A/bardbl = 0.\nBy continuity, this remains true for all A ∈/hatwideA.\nFor classical spin systems we recall that, for finite Λ, we have AΛ=C/parenleftbig/producttext\nx∈ΛB1(0);C/parenrightbig\n, aC∗-\nalgebra under /bardbl·/bardbl∞. As above, for Λ 1⊂Λ2, we identify A1∈AΛ1with a function A2∈AΛ2by\nsettingA2=A1⊗1Λ2\\Λ1. We thus get a norm-preserving mapping AΛ1→AΛ2of the abstract\nC∗-algebras and the relation AΛ1⊂AΛ2. Define the classical quasi-local algebra as\nA:=/logicalordisplay\nΛ⊂ZdfiniteAΛ.\nNote that Ais equal to the space of continuous complex functions on/producttext\nx∈ZdB1(0), equipped with the\nproducttopology(thisisanimmediateconsequenceoftheTychono ffandStone-Weierstrasstheorems).\nThe spins are represented on Aby a family {Mi(x) :i∈I, x∈Zν}of functions. Existence of\nthe dynamics follows exactly as above.\nLemma 3. LetA∈AΛ0for some finite Λ0⊂Zdandt∈R. Then the following limit exists in /bardbl·/bardbl∞:\nlim\nΛ→∞αt\nΛA=:αtA.\nBy continuity this extends to a strongly continuous one-par ameter group (αt)t∈Rof automorphisms of\nA. Furthermore, αtA=A◦φt, whereφt=φt\nZdis the Landau-Lifschitz flow defined in the remark\nafter Lemma 1.\nNow set P:=C[{Mi(x) :i∈I, x∈Zd}]. Then the proof of Theorem 1 yields the following\nTheorem 2. LetA∈Pandε >0. Then there exists a function A(t)∈Psuch that\nsup\nt∈R/bardblαtA−A(t)/bardbl∞≤ε, (17)\nand, for any t∈R,\n/vextenddouble/vextenddouble/hatwideαt/hatwideA−/hatwidestA(t)/vextenddouble/vextenddouble≤ε+C(ε,t,A)\ns. (18)\nRemark. In particular, the result applies to classical equations of motion of t he form (9) where the\nsum over yranges over Zd.\n2.6 Evolution of coherent states\nDenote by Si:=s/hatwideSithe unscaled spin operator in the spin- s-representation of su(2). For the polar\nangles (θ,ϕ)∈[0,π]×[0,2π) corresponding to the unit vector M∈S2we define the coherent state in\nC2s+1as\n|M/an}bracketri}ht:= expθ√\n2/bracketleftBig\neiϕS−−e−iϕS+/bracketrightBig\n|s/an}bracketri}ht,\n8where|s/an}bracketri}htis the highest-weight state, i.e. Sz|s/an}bracketri}ht=s|s/an}bracketri}ht. Note that |M/an}bracketri}ht=eiα·S|s/an}bracketri}ht, whereα=θnand\nnis the unit vector (sin ϕ,−cosϕ,0).\nSetA:=θ√\n2/bracketleftBig\neiϕS−−e−iϕS+/bracketrightBig\nandU:=eAso that|M/an}bracketri}ht=U|s/an}bracketri}ht. Then using\nU∗SiU=∞/summationdisplay\nk=01\nk![...[Si,A],...,A],\nwe find\nU∗S1U= sinθcosϕSz+1√\n2cos2θ\n2(S++S−)−1√\n2sin2θ\n2/parenleftbig\ne−2iϕS++e2iϕS−/parenrightbig\n,\nU∗S2U= sinθsinϕSz+1√\n2icos2θ\n2(S+−S−)−1√\n2isin2θ\n2/parenleftbig\ne2iϕS+−e−2iϕS−/parenrightbig\n,\nU∗S3U= cosθSz−1√\n2sinθ/parenleftbig\ne−iϕS++eiϕS−/parenrightbig\n. (19)\nAs a consequence note that\n/an}bracketle{tM,/hatwideSM/an}bracketri}ht=\nsinθcosϕ\nsinθsinϕ\ncosθ\n=M . (20)\nIn order to derive our main result for coherent spins states we nee d the following lemma, which\nfollows from direct calculations.\nLemma 4. For any unit vector M∈S2, we have that\n/vextendsingle/vextendsingle/vextendsingle/an}bracketle{tM,/hatwideSi1···/hatwideSipM/an}bracketri}ht−Mi1···Mip/vextendsingle/vextendsingle/vextendsingle≤p/radicalbigg\n2\ns.\nNow let M:Zd→S2be a configuration of classical spins on the lattice. Then Mdefines a state\nρMon/hatwideAas follows. For finite Λ, consider the product state\n|MΛ/an}bracketri}ht:=/circlemultiplydisplay\nx∈Λ|M(x)/an}bracketri}ht ∈HΛ.\nThen, for A ∈/hatwideAΛ, we set\nρM(A):=/an}bracketle{tMΛ,AMΛ/an}bracketri}ht,\nand extend the definition of ρMto arbitrary A ∈/hatwideAby continuity.\nLetM:R×Zd→S2be the solution of the Hamiltonian equation of motion (7) with initial\nconditions M(0,x) =M(x). The following result links the quantum time evolution for coherent s pin\nstates with the corresponding classical configuration in the large- spin limit.\nTheorem 3. Lett∈R,A∈PandM:Zd→S2. Then\nlim\ns→∞ρM(/hatwideαt/hatwideA) =A(M(t)),\nuniformly in ton compact time intervals.\nProof.The proof is essentially a corollary of Lemma 4 and the proof of Theor em 1. First, let |t|<\n/bardblV/bardbl−1. We know from (13) and (16) that\nA(M(t)) =∞/summationdisplay\nl=0tl\nl!lim\nΛ→∞/braceleftbig\nHΛ,A/bracerightbig(l)(M),\nas well as\nρM(/hatwideαt\nΛ/hatwideA) =∞/summationdisplay\nl=0tl\nl!(is)lρM/parenleftBig\nlim\nΛ→∞/bracketleftbig/hatwideHΛ,/hatwideA/bracketrightbig(l)/parenrightBig\n.\n9Now Lemma 4 implies that\n/vextendsingle/vextendsingleρM/parenleftbig/hatwidestMα/parenrightbig\n−Mα/vextendsingle/vextendsingle≤ |α|/radicalbigg\n2\ns.\nArguing exactly as in the proof of Theorem 1, we get the bound\n/vextendsingle/vextendsingleρM(/hatwideαt/hatwideA)−A(M(t))/vextendsingle/vextendsingle≤C(t,A)√s. (21)\nArbitrary times are reached by iteration as in the proof of Theorem 1.\n3 Mean-Field Limit\nThis section is devoted to the dynamics of a quantum spin system in th e mean-field/continuum limit.\nMoreprecisely,weconsiderasystemofquantumspinsonalatticewit hspacing h >0. Thelimit h→0\nyields again a Egorov-type result: The quantum dynamics approach es the dynamics of a classical spin\nsystem defined on a continuum set. As in the previous section, we als o discuss the thermodynamic\nlimit and the time evolution of coherent states.\n3.1 A system of quantum spins on a lattice\nLet Λ⊂Rdbe bounded and open. We associate with each spacing h >0 the finite lattice\nΛ(h):=hZd∩Λ.\nAt each lattice site x∈Λ(h)there is a spin of (fixed) magnitude s. The Hilbert space of this quantum\nsystem is\nH(h)\nΛ:=/circlemultiplydisplay\nx∈Λ(h)C2s+1.\nThe algebra of bounded operators on H(h)\nΛis denoted by /hatwideA(h)\nΛ.\nThe spins are represented on H(h)\nΛby a family {/hatwideSi(x) :i= 1,2,3, x∈Λ(h)}of operators, where\n/hatwideSi(x) is thei’th generator of the spin- s-representation of su(2), rescaled by hd/s. As usual, we replace\nthe operators ( /hatwideS1,/hatwideS2,/hatwideS3) with (/hatwideS+,/hatwideSz,/hatwideS−). They satisfy the bounds /bardbl/hatwideS±/bardbl ≤hdand/bardbl/hatwideSz/bardbl=hdif\ns≥1, as well as /bardbl/hatwideS±/bardbl=√\n2hdand/bardbl/hatwideSz/bardbl=hdifs= 1/2. The commutation relations now read\n/bracketleftbig/hatwideSi(x),/hatwideSj(y)/bracketrightbig\n=hd\ns˜εijkδ(x,y)/hatwideSk(x), (22)\nwithi,j,k∈I.\n3.2 A continuum theory of spins\nLet Λ⊂Rdbe a bounded, open set. A system of classical spins on Λ is represen ted in terms of the\nPoisson “phase space”3\nΞΛ:=/braceleftbig\nM∈L∞(Λ;R3) :/bardblM/bardbl∞≤1/bracerightbig\n,\nwhich we equip with the L∞-norm. In analogy to Section 2, we use the complex coordinates\n(M+,Mz,M−) instead of ( M1,M2,M3), so that the Poisson bracket on Ξ Λsatisfies\n/braceleftbig\nMi(x),Mj(y)/bracerightbig\n=i˜εijkδ(x−y)Mk(x), (23)\nfori,j,k∈I.\n3As in the previous section, one may introduce a symplectic ph ase space Γ Λconsisting of all M∈ΞΛsuch that\n|M(x)|= 1 a.e.\n10In order to describe a useful class of observables on Ξ Λ, we introduce the space B(p),p∈N, which\nconsists of all functions finC(Rpd;C3p) that are symmetric in their arguments, in the sense that\nPf=f, where\n(Pf)i1...ip(x1,...,x p):=1\np!/summationdisplay\nσ∈Spfiσ(1)...iσ(p)(xσ(1),...,x σ(p)).\nOn the space B(p)we introduce the norms\n/bardblf/bardbl(h)\n1:=hpd/summationdisplay\ni1,...,ip∈I/summationdisplay\nx1,...,xp∈hZd|fi1...ip(x1,...,x p)|,\n/bardblf/bardbl(h)\n∞,1:= sup\nx/summationdisplay\ni1,...,ip∈Ih(p−1)d/summationdisplay\nx2,...,xp∈hZd|fi1...ip(x,x2,...,x p)|.\nWe shall be interested in observables arising from f∈B(p)satisfying\nlimsup\nh→0/bardblf/bardbl(h)\n1<∞. (24)\nNote that Fatou’s lemma implies that /bardblf/bardbl1≤limsuph→0/bardblf/bardbl(h)\n1.\nWe define PΛas the “polynomial” algebra of functions on Ξ Λgenerated by functions of the form\nMΛ(f):=/summationdisplay\ni1,...,ip/integraldisplay\nΛpdx1···dxpfi1...ip(x1,...,x p)Mi1(x1)···Mip(xp),\nwheref∈B(p)satisfies (24). PΛis clearly a Poisson algebra. We equip it with the norm /bardblA/bardbl∞:=\nsupM∈ΞΛ|A(M)|so that\n/bardblMΛ(f)/bardbl∞≤ /bardblf/bardbl1. (25)\n3.3 Quantization\nForf∈B(p)let us define\n/hatwideSΛ(f):=/summationdisplay\ni1,...,ip/summationdisplay\nx1,...,xp∈Λ(h)fi1...ip(x1,...,x p)/hatwideSi1(x1)···/hatwideSip(xp). (26)\nIffsatisfies (24), we find that\n/bardbl/hatwideSΛ(f)/bardbl ≤ /bardblf/bardbl(h)\n1. (27)\nAs above, quantization /hatwide·:PΛ→/hatwideA(h)\nΛis defined by /hatwiderMΛ(f) =:/hatwideSΛ(f):and linearity. Here :·:\ndenotes the normal-ordering of the spin operators introduced ab ove. Also, we set /hatwide1 = /BD. Again,\n(/hatwideA)∗=/hatwideA.\n3.4 Dynamics in the mean-field limit\nWe consider a family V=/parenleftbig\nV(n)/parenrightbig∞\nn=1of functions, where V(n)∈B(p)satisfies\nV(n)\ni1...in(x1,...,x n) =V(n)\ni1...in(x1,...,x n),\nwhere, we recall, ·onImaps (+ ,z,−) to (−,z,+). We define the Hamilton function on Ξ Λthrough\nHΛ:=∞/summationdisplay\nn=1MΛ(V(n)). (28)\nSet\n/bardblV/bardbl(h):=∞/summationdisplay\nn=1nen/bardblV(n)/bardbl(h)\n∞,1. (29)\n11We impose the condition /bardblV/bardbl:= limsuph→0/bardblV/bardbl(h)<∞.In the continuum limit, we observe that\n∞/summationdisplay\nn=1nensup\nx/summationdisplay\ni1,...,in/integraldisplay\ndx2···dxn/vextendsingle/vextendsingleV(n)\ni1...in(x,x2,...,x n)/vextendsingle/vextendsingle≤ /bardblV/bardbl, (30)\nas can be seen using Fatou’s lemma. It now follows easily that, for eac h bounded set Λ, the sum (28)\nconverges in /bardbl·/bardbl∞on ΞΛand yields a well-defined real Hamilton function HΛ.\nThe Hamiltonian equation of motion reads\nd\ndtMi(t,x) =i∞/summationdisplay\nn=1n/summationdisplay\ni1,...,in,j/integraldisplay\ndx2···dxnV(n)\ni1...in(x,x2,...,x n)\n˜εi1ijMj(t,x)Mi2(t,x2)···Min(t,xn).(31)\nBy standard methods, we find the following global well-posedness re sult for (31).\nLemma 5. LetΛ⊂Rdby any open subset of RdandM0∈ΞΛ. Then(31)has a unique solution\nM∈C1(R,ΞΛ)that satisfies M(0) =M0. Moreover, the solution Mdepends continuously on the\ninitial condition M0. Finally, we have the pointwise conservation law |M(t,x)|=|M(0,x)|for allt.\nRemarks. 1. As in Section 2, we denote the norm-preserving Hamiltonian flow by φt\nΛ.\n2. Time-dependent potentials V(t) may be treated exactly as in the previous section.\nExample. Consider\nHΛ=−/integraldisplay\nΛdx h(t,x)·M(x)−1\n2/integraldisplay\nΛ×Λdxdy J(x,y)M(x)·M(y),\nwhich yields the Landau-Lifshitz equation of motion\nd\ndtM(t,x) =M(t,x)∧/bracketleftbigg\nh(t,x)+/integraldisplay\nΛdy J(x,y)M(t,y)/bracketrightbigg\n.\nRemark. In a formal way, the Landau-Lifshitz equation (2) mentioned in the introduction can be\nobtained from (3) with J=J(|x−y|) by Taylor expanding M(t,y) up to second order in y−x. This\nleads to (2) after rescaling time by t/ma√sto→αtwhereα=1\n2d/integraltext\nJ(|x|)|x|2dx.\nThe quantum dynamics is generated by the Hamiltonian /hatwideHΛ∈/hatwideA(h)\nΛdefined as the quantization of\nHΛ. More precisely, each term of HΛis quantized and it may be easily verified that the resulting series\nconverges in operator norm. The fact that HΛis real immediately implies that /hatwideHΛis self-adjoint. As\nabove we introduce the short-hand notation\nαt\nΛA:=A◦φt\nΛ, A∈AΛ,\n/hatwideαt\nΛA:=Uh(t;/hatwideHΛ)∗AUh(t;/hatwideHΛ),A ∈/hatwideA(h)\nΛ.\nHere,Uh(t;/hatwideHΛ) is the quantum mechanical propagator, equal to eish−dbHΛtif/hatwideHΛis time-independent.\nWe are now in a position to state our main result on the mean-field dyna mics of the quantum\nsystem on the finite lattice Λ(h)in the continuum limit, as h→0.\nTheorem 4. LetΛ⊂Rdbe open and bounded, A∈PΛandε >0. Then there exists a function\nA(t)∈PΛsuch that\nsup\nt∈R/bardblαt\nΛA−A(t)/bardbl∞≤ε, (32)\nand, for any t∈R,/vextenddouble/vextenddouble/hatwideαt\nΛ/hatwideA−/hatwidestA(t)/vextenddouble/vextenddouble≤ε+C(ε,t,A)hd, (33)\nwhereC(ε,t,A)is independent of Λ.\n12Proof.One finds, for f∈B(p)andg∈B(q),\n/braceleftbig\nMΛ(f),MΛ(g)/bracerightbig\n=pqMΛ(f ⇀ g) (34)\nwheref ⇀ g∈B(p+q−1)is defined by\n(f ⇀ g)i1...ip+q−1(x1,...,x p+q−1)\n:=iP/summationdisplay\ni,j˜εiji1fii2...ip(x1,...xp)gjip+1...ip+q−1(x1,xp+1,...,x p+q−1).(35)\nWe have the estimate\n/bardblf ⇀ g/bardbl1≤ /bardblf/bardbl∞,1/bardblg/bardbl1, (36)\nwhere\n/bardblf/bardbl∞,1:= sup\nx/summationdisplay\ni1,...,ip/integraldisplay\ndx2...dxp|fi1...ip(x,x2,...xp)|.\nWithout loss of generality, we assume that A=MΛ(f) for some f∈B(p)satisfying the bound\n(24). Iterating\n/braceleftbig\nHΛ,MΛ(f)/bracerightbig\n=∞/summationdisplay\nn=1npMΛ(V(n)⇀ f)\nwe obtain that\n/braceleftbig\nHΛ,MΛ(f)/bracerightbig(l)=∞/summationdisplay\nn1,...,nl=1/bracketleftbig\npn1/bracketrightbig/bracketleftbig\n(p+n1−1)n2/bracketrightbig\n···/bracketleftbig\n(p+n1+···+nl−1−l+1)nl/bracketrightbig\nMΛ/parenleftBig\nV(nl)⇀/parenleftbig\nV(nl−1)⇀ ...(V(n1)⇀ f)/parenrightbig/parenrightBig\n,\nwith norm\n/vextenddouble/vextenddouble/braceleftbig\nHΛ,MΛ(f)/bracerightbig(l)/vextenddouble/vextenddouble\n∞≤/summationdisplay\nn1,...,nl/bracketleftbig\npn1/bracketrightbig/bracketleftbig\n(p+n1−1)n2/bracketrightbig\n···/bracketleftbig\n(p+n1+···+nl−1−l+1)nl/bracketrightbig\n/bardblV(nl)/bardbl∞,1···/bardblV(n1)/bardbl∞,1/bardblf/bardbl1\n≤l!/summationdisplay\nn1,...,nl(p+n1+···+nl)l\nl!n1···nl/bardblV(nl)/bardbl∞,1···/bardblV(n1)/bardbl∞,1/bardblf/bardbl1\n≤ep/bardblf/bardbl1l!/bracketleftbigg/summationdisplay\nnnen/bardblV(n)/bardbl∞,1/bracketrightbiggl\n≤ep/bardblf/bardbl1l!/bardblV/bardbll, (37)\nby (30). Therefore, for |t|</bardblV/bardbl−1, the series\n∞/summationdisplay\nl=0tl\nl!/braceleftbig\nHΛ,A/bracerightbig(l)(38)\nconverges in /bardbl·/bardbl∞toαt\nΛA.\nThe quantum case is dealt with in a similar fashion, with the additional co mplication caused by\nthe ordering of the generators {/hatwideSi(x)}. This does not trouble us, however, as an exact knowledge of\nthe ordering is not required. It is easy to see that, for fandgas above,\nish−d/bracketleftBig\n/hatwideSΛ(f),/hatwideSΛ(g)/bracketrightBig\nis equal, up to a reordering of the spin operators, to pq/hatwideSΛ(f ⇀ g). Iterating this shows that\n(ish−d)l/bracketleftbig/hatwideHΛ,/hatwideA/bracketrightbig(l)\n13is equal, up to a reordering of the spin operators, to\n∞/summationdisplay\nn1,...,nl=1/bracketleftbig\npn1/bracketrightbig/bracketleftbig\n(p+n1−1)n2/bracketrightbig\n···/bracketleftbig\n(p+n1+···+nl−1−l+1)nl/bracketrightbig\n/hatwideSΛ/parenleftBig\nV(nl)⇀/parenleftbig\nV(nl−1)⇀ ...(V(n1)⇀ f)/parenrightbig/parenrightBig\n,\nConsequently an estimate analogous to (37) yields, for s≥1,\n/vextenddouble/vextenddouble(ish−d)l/bracketleftbig/hatwideHΛ,/hatwideA/bracketrightbig(l)/vextenddouble/vextenddouble≤ep/bardblf/bardbl(h)\n1l!(/bardblV/bardbl(h))l,\nwhich readily implies the bound\n/vextenddouble/vextenddouble/vextenddouble∞/summationdisplay\nl=0tl\nl!(ish−d)l/bracketleftbig/hatwideHΛ,/hatwideA/bracketrightbig(l)/vextenddouble/vextenddouble/vextenddouble≤ep/bardblf/bardbl(h)\n1∞/summationdisplay\nl=0(|t|/bardblV/bardbl(h))l. (39)\nIfs= 1/2, the first line of (37) gets the additional factor√\n2n1+···+nl+p. This may be dealt with by\nreplacing the factor ( p+n1+···+nl)lin the second line of (37) with ( rp+rn1+···+rnl)l/rl. The\ndesired bound then follows for 0 < r≤1−1\n2log2. Note that in this case the convergence radius for t\nis reduced to r/bardblV/bardbl−1. For ease of notation, we restrict the following analysis to the case s≥1, while\nbearing in mind that the extension to s= 1/2 follows by using the above rescaling trick.\nNow, by definition of /bardblV/bardbl, for any |t|</bardblV/bardbl−1there is an h0such that (39) converges in norm to\n/hatwideαt\nΛ/hatwideAfor allh≤h0, uniformly in hand Λ.\nIn order to establish the statement of the theorem for short time s|t|</bardblV/bardbl−1, we remark that\nthe commutation relations (22) imply the bound\n/vextenddouble/vextenddoubleA−B/vextenddouble/vextenddouble≤hd\nsp2/bardblf/bardbl(h)\n1,\nfor arbitrary reorderings, AandB, of the same operator /hatwideSΛ(f), withf∈B(p)for some p <∞.\nIf we define/hatwidestαt\nΛAthrough its norm-convergent power series, we therefore get\n/vextenddouble/vextenddouble/hatwideαt\nΛ/hatwideA−/hatwidestαt\nΛA/vextenddouble/vextenddouble\n≤hd\ns∞/summationdisplay\nl=0|t|l\nl!/summationdisplay\nn1,...,nl/bracketleftbig\npn1/bracketrightbig/bracketleftbig\n(p+n1−1)n2/bracketrightbig\n···/bracketleftbig\n(p+n1+···+nl−1−l+1)nl/bracketrightbig\n(n1+···+nl−l+1)2/bardblV(nl)/bardbl(h)\n∞,1···/bardblV(n1)/bardbl(h)\n∞,1/bardblf/bardbl(h)\n1\n≤hd\ns∞/summationdisplay\nl=0|t|l/summationdisplay\nn1,...,nl(p+n1+···+nl)l+2\nl!n1···nl/bardblV(nl)/bardbl(h)\n∞,1···/bardblV(n1)/bardbl(h)\n∞,1/bardblf/bardbl(h)\n1\n≤hd\ns∞/summationdisplay\nl=0|t|lep/bardblf/bardbl(h)\n1(l+2)(l+1)/bracketleftbigg/summationdisplay\nnnen/bardblV(n1)/bardbl(h)\n∞,1/bracketrightbiggl\n≤hd\nsep/bardblf/bardbl(h)\n1∞/summationdisplay\nl=0(l+2)(l+1)(|t|/bardblV/bardbl(h))l\n=O(hd),\nwhere in the last step we have used the fact that the sum converge nces uniformly in h, forhsmall\nenough, as seen above.\nArbitrary times are reached by iteration of the above result.\n143.5 The thermodynamic limit\nThe above result may again be formulated in the thermodynamic limit as Λ→hZd. We only sketch\nthe arguments, which are almost identical to those of Section 2.5.\nThe quantum quasi-local algebra is\n/hatwideA(h):=/logicalordisplay\nΛ⊂⊂Rd/hatwideA(h)\nΛ,\nThe existence of dynamics is guaranteed by the following statement .\nLemma 6. Leth >0and suppose A ∈/hatwideA(h)\nΛ0for some bounded and open Λ0⊂Rd. Then, for any\nt∈R, the following limit exists in the norm sense:\nlim\nΛ→∞/hatwideαt\nΛA=:/hatwideαtA,\nBy continuity this extends to a strongly continuous one-par ameter group (/hatwideαt)t∈Rof automorphisms of\n/hatwideA(h).\nThe classical quasi-local algebra is\nA:=/logicalordisplay\nΛ⊂⊂RdPΛ.\nLemma 7. LetA∈PΛ0for some open and bounded Λ0⊂⊂Rd. Then, for any t∈R, the following\nlimit exists in /bardbl·/bardbl∞:\nlim\nΛ→∞αt\nΛA=:αtA,\nBy continuity this extends to a strongly continuous one-par ameter group (αt)t∈Rof automorphisms of\nA. Furthermore, αtA=A◦φt, whereφt=φt\nRdis the Landau-Lifschitz flow defined in Lemma 5.\nNow, for f∈B(p),M(f) and/hatwideS(f) are well-defined in the obvious way. Define Pas the algebra\ngenerated by functions of the form M(f), where fsatisfies (24).\nTheorem 5. LetA∈Pandε >0. Then there exists a function A(t)∈Psuch that\nsup\nt∈R/bardblαtA−A(t)/bardbl∞≤ε, (40)\nand, for any t∈R,/vextenddouble/vextenddouble/hatwideαt/hatwideA−/hatwidestA(t)/vextenddouble/vextenddouble≤ε+C(ε,t,A)hd. (41)\n3.6 Evolution of coherent states\nIn this section, our “smearing functions” fare assumed to have compact support, i.e. to belong to\nthe space\nB(p)\nc:=B(p)∩Cc(Rpd;C3p).\nIn addition, we require the interaction potential Vto be of finite range in the sense that there exists\na sequence Rn>0 such that if |xi−xj|> Rnfor some pair ( i,j) thenV(n)\ni1...in(x1,...,x n) = 0.\nNext, wetakesomeinitialclassicalspinconfiguration M∈C(Rd;S2), or,moregenerally,afunction\nM:Rd→S2whose points of discontinuity form a null set. We shall study the time evolution of\nproduct states ρMon/hatwideA(h)that reproduce the given classical state M. For open and bounded Λ ⊂Rd,\nwe define the product state\n|MΛ/an}bracketri}ht:=/circlemultiplydisplay\nx∈Λ(h)|M(x)/an}bracketri}ht,\nwhere|M(x)/an}bracketri}htis the coherent spin state corresponding to the unit vector M(x). ForA ∈/hatwideA(h)\nΛ, define\nρM/parenleftbig\nA/parenrightbig:=/an}bracketle{tMΛ,AMΛ/an}bracketri}ht,\n15which we extend to arbitrary A ∈/hatwideA(h)by continuity.\nFor our main result on the time evolution of coherent states, we firs t record the following auxiliary\nresult whose elementary proof we omit.\nLemma 8. Letf∈B(p)\ncsatisfy(24). Then\nlim\nh→0ρM/parenleftbig/hatwideS(f)/parenrightbig\n=M(f). (42)\nThe last result in this paper links the quantum time evolution of cohere nt spin states with the\nclassical evolution in the mean-field/continuum limit when the lattice sp acinghtends to 0.\nTheorem 6. Lett∈R,A∈PandMbe as described above. Let M(t)be the solution of (31)onRd\nwith initial configuration M. Then\nlim\nh→0ρM/parenleftbig\n/hatwideαt/hatwideA/parenrightbig\n=A(M(t)),\nuniformly in ton compact time intervals.\nProof.The proof is a corollary of the proof of Theorem 4. First, let |t|</bardblV/bardbl−1and pick an ε >0.\nChoosea cutoff such that the tails of the thermodynamic limits of the series (38) and (39) are bounded\nbyε. We therefore have to estimate a finite sum of terms of the form\n/vextendsingle/vextendsingleρM/parenleftbig/hatwideS(g)/parenrightbig\n−M(g)/vextendsingle/vextendsingle,\nwhereg∈B(p(g))\ncbecause of our assumptions on V. By Lemma 8, for hsmall enough, these are all\nbounded by ε, and the claim for small times follows. Finally, by iteration, we extend t he result to\narbitrary times.\nAcknowledgement. We thank Simon Schwarz for very useful discussions.\nReferences\n[1] J. Fr¨ ohlich, A. Knowles, and A. Pizzo, Atomism and Quantization , J. Phys. A 40(2007), no. 12,\n3033–3045.\n[2] S. Gustafson, K. Kang, and T.-P. Tsai, Schr¨ odinger Flow near Harmonic Maps , Comm. Pure\nAppl. Math. 60(2007), no. 4, 463–499.\n[3] M. Kruˇ z´ ık and A. Prohl, Recent Developments in the Modeling, Analysis, and Numeric s of Fer-\nromagnetism , SIAM Rev. 48(2006), no. 3, 439–483.\n[4] E. H. Lieb, The Classical Limit of Quantum Spin Systems , Comm. Math. Phys. 31(1973), 327–\n340.\n[5] B. Simon, The Classical Limit of Quantum Partition Functions , Comm. Math. Phys. 71(1980),\n247–276.\n[6] P.-L. Sulem, C. Sulem, and C. Bardos, On the Continuous Limit for a System of Classical Spins ,\nComm. Math. Phys. 107(1986), no. 3, 431–454.\n[7] R. F. Werner, The Classical Limit of Quantum Theory , quant-ph/9504016v1.\n16"    },    {        "title": "1409.2178v1.Spin_transport_in_non_degenerate_Si_with_a_spin_MOSFET_structure_at_room_temperature.pdf",        "content": "Spin transport in non-degenerate Si  with a spin MOSFET structure at room temperature  Tomoyuki Sasaki 1,#, Yuichiro Ando 2,3,#, Makoto Kameno 2,#, Takayuki Tahara 3, Hayato Koike 1,Tohru Oikawa 1, Toshio Suzuki 4 and Masashi Shiraishi 2,3, $ 1. Advanced Technology Development Center, TDK Corporation, Japan. 2. Graduate School of Engineering Science, Osaka University, Japan. 3. Department of Electronic Science and Engineering, Kyoto University, Japan. 4. AIT, Akita Prefectural Industrial Center, Akita, Japan.  # These three authors contributed equally to this work. $ Corresponding author (mshiraishi@kuee.kyoto-u.ac.jp)  Abstract  Spin transport in non-degenerate semiconductors is expected to pave a way to the creation of spin transistors, spin logic devices and reconfigurable logic circuits, because room temperature (RT) spin transport in Si has already been achieved. However, RT spin transport has been limited to degenerate Si, which makes it difficult to produce spin-based signals because a gate electric field cannot be used to manipulate such signals. Here, we report the experimental demonstration of spin transport in non-degenerate Si with a spin metal-oxide-semiconductor field-effect transistor (MOSFET) structure. We successfully observed the modulation of the Hanle-type spin precession signals, which is a characteristic spin dynamics in non-degenerate semiconductor. We obtained long spin transport of more than 20 µm and spin rotation, greater than 4π at RT. We also observed gate-induced modulation of spin transport signals at RT. The modulation of spin diffusion length as a function of a gate voltage was successfully observed, which we attributed to the Elliott-Yafet spin relaxation mechanism. These achievements are expected to make avenues to create of practical Si-based spin MOSFETs. I. INTRODUCTION      The ability to control the flow of electrons based on their spin angular momentum by using two ferromagnetic electrodes, separated by a spin channel, i.e., a spin valve, has led to significant breakthroughs in the study of spin transport phenomena and their applications, resulting in tremendous advances in the field of spintronics [1-5]. Various techniques, such as electrical, dynamical and thermal methods, for generating pure spin currents have opened up the possibility of spin-based information devices [6,7]. Metals and semiconductors are important materials in spintronics, both have been subjects of rigorous studies. Since the first demonstration of spin transport in a hot-electron transistor in 2007 [8], various approaches have been used to achieve spin accumulation in Si, including devices based on electrical non-local 4-terminal [9-14], non-local 3-terminal [15-18] and dynamical [19] methods. Consequently, successful demonstration of spin transport in degenerate n-type [11] and p-type Si [19] at room temperature (RT) has been experimentally achieved.      Silicon is an attractive material for spintronics for the following reasons: (1) it has lattice inversion symmetry, resulting in good spin coherence. This is very advantageous when creating spin transistors (so-called “Sugahara-Tanaka” type spin transistors [3]); (2) Si is ubiquitous and non-toxic, which makes it an environment-friendly material that is important for the realization of a greener society; (3) Si-based electronics is already well established, and the enormous range of existing technologies and infrastructures can be fully utilized for mass-producing Si-based spin devices. Hence, Si is now regarded as one of the most promising materials for pushing technology beyond the complementary metal-oxide-semiconductor (CMOS). To accelerate the development of Si spintronics, it is essential to achieve long-range spin transport in non-degenerate Si at RT. Another important goal is the manipulation and modulation of spin signals using an external magnetic field and a gate voltage. This is because Si spin transistors, which would allow spin logic circuits to be fabricated [2,3], are based on the concept of conventional CMOS devices, in which source, drain and gate electrodes are used with a non-degenerate charge carrier channel. By using a hot-electron transistor structure, spin transport has been demonstrated at 260 K in non-generate n-type Si (resistivity 1-10 Ω cm) [20], and a spin transport length of 2 mm was achieved in intrinsic Si at 85 K [21]. However, this device structure is not considered appropriate for practical applications, because the output signal is on the order of several picoamperes, and the device itself is not easily integrated. It would therefore be difficult to fabricate spin-based logic circuits using this approach.       The purpose of this study was to investigate the possibility of controlling spin transport in non-degenerate n-type Si at RT by using spin metal-oxide-semiconductor field-effect-transistor (MOSFET) structure. Modulation of spin signals, in addition to spin diffusion length, is achieved by application of a gate electric field. A spin rotation of greater than 4π is realized by using an external magnetic field, and the results are well reproduced by a spin drift diffusion model. A spin transport length of more than 20 µm is demonstrated. The achievement of long-range spin transport at RT and the ability to easy manipulate spin signals using external fields in non-degenerate Si are expected to pave the way for the development of practical Si-based spin MOSFETs.   II. EXPERIMENTAL      The spin transport experiments in the present study were carried out using an electrical method. We have previously developed a variety of experimental techniques for observing spin transport signals, including non-local 4-terminal [10,11], modified non-local 3-terminal [22,23] and local 2-terminal [22] methods. Successful transport of a pure spin current and a spin-polarized current was achieved in degenerate n-type Si, and the results were supported by the observation of Hanle spin precession and magnetoresistance (MR). The same experimental platforms were also used in the present study. Spin-dependent transport and gate-induced modulation of the spin signals in the non-degenerate Si were measured using a 4-probe system (Janis ST-500) and a Physical Property Measurement System (PPMS, Quantum Design) at a temperature of 300 K. An external magnetic field was applied parallel and perpendicular to the Si channel plane in order to induce a MR effect and Hanle spin precession, respectively. For all measurements, the direction of spin flow was from contact 2 (pinned layer) to contact 3 (free layer) as shown in Fig. 1. The Si spin device was fabricated on a silicon-on-insulator substrate with the structure Si(100 nm)/SiO2(200 nm)/bulk Si (see Fig. 1). The upper Si layer was phosphorous (P) doped by ion implantation. Using a 4-terminal method, the resistivity of the Si channel was determined to be 160 Ω µm, indicating that the dopant concentration was about 2!1018 cm-3 [24-31]. Before depositing the ferromagnetic electrodes, the Si in these regions was highly doped to a concentration of about 5!1019 cm-3, as determined by secondary ion mass spectroscopy [25]. Because of the insertion of the highly-doped region, the contact resistance decreases, resulting in the decrease of the whole resistance of the device. The decrease of the sample resistance allows efficient detection of spin signals. After the natural oxide layer on the Si channel was removed using a HF solution, Ti(3 nm)/Fe(13 nm)/MgO(0.8 nm) was grown on the etched surface by molecular beam epitaxy. Then, we etched out the Ti (3 nm)/Fe (3 nm) layers and Ta (3nm) was grown on the remaining Fe. In order to form FM contacts, the Si channel was etched to a depth of 25 nm by ion milling. The contacts had dimensions of 0.5×21 µm2 and 2×21 µm2, respectively. The Si channel surface and sidewalls at the FM contacts were buried by SiO2. The nonmagnetic electrodes, with dimensions of 21×21 µm2, were made from Al and were produced by ion milling. The gap between the FM electrodes was varied from 1.4 to 21.3 µm. The gate electric field was applied from the backside of the device.   III. RESULTS AND DISCUSSION      Figure 2 shows the results of MR measurements carried out at 300 K. Local MR appeared to be present, with resistance hysteresis occurring between about ±200 and ±350 mT, as shown in Fig. 2(a). The sample resistance at 10 mA and the detected spin signal were 376 Ω and 170 µV, respectively, i.e., the MR ratio was 0.005%. In order to verify that the observed MR is related to the magnetization of the ferromagnetic (FM) electrodes, a modified 3-terminal method was used, as shown in Figs. 2(b) and 2(c). Clear MR was observed when the free FM layer was set as the base electrode (“MRF” configuration), but none was found when the pinned FM layer was set as the base electrode (“MRP” configuration). This is consistent with previous studies by our group [22,23], in which spin transport was achieved at RT in degenerate n-type Si. Here, we briefly summarize a principle of our modified non-local 3-terminal scheme (see also ref.22). The notable is that spin accumulation voltage is obtained from the ferromagnetic contact only under the spin extraction condition. The spin accumulation voltage obtained by the modified non-local 3-terminal measurement under the spin extraction condition is almost the same magnitude with that of local measurements in our previous study, indicating that the magnetoresistance in local scheme is caused by the spin accumulation voltage in one contact. This is the reason why the resistance hysteresis can be seen when the magnetization alignment of the detection FM electrode was reversed. Furthermore, as shown in Fig. 2(d), an apparent minor loop was observed, which was related to magnetization alignment of the FM electrodes. All of the above results provide evidence for successful spin transport in the non-degenerate n-type Si layer.      To obtain further confirmation of successful spin transport in the non-degenerate Si, Hanle-effect experiments were carried out. Hanle-type spin precession is not only an indicator of spin transport, but also of how easily the spins can be manipulated by an external field, in this case, a magnetic field. Figure 3 shows typical Hanle spin precession signals from a sample with a gap of 21.3 µm between the FM layers. The MRF configuration was used for the Hanle-effect experiment and the applied electric current was varied from 2 to 4 mA. A sample with a large gap was chosen in order to determine how many times the spins can rotate during transport. As shown in Fig. 3(a), symmetric oscillation of the spin signals as a function of the magnetic field was observed for both the parallel and anti-parallel magnetization configurations. Crossing of the parallel and anti-parallel signals occurred at about ±30 mT, indicating that the average spin rotation angle was π/2. It should be noted that spin rotation by up to 4π was realized, and such frequent rotation has previously only been found for intrinsic Si at low temperature [8]. In addition, a spin transport length of more than 20 µm was realized. In degenerated Si at RT, the spin transport length is limited to within 3 µm [10,11]. The spin drift diffusion equation is: !S!t=D!2S!x2\"v!S!x\"S!,     (1) where S(x,t) is spin density, v is the spin drift velocity, x is position, t is time, and τ is the spin lifetime. This allows a quantitative estimation of the spin transport length. The analytic solution to Eq. (1) is [32]: V(B)I=±P2DT2!Aexp(!L\"N+L2\"N2v#)(1+$2T2)!14\"exp[!L\"N(1+$2T2+12!1)]\"cos[arctan($T)2+L\"N1+$2T2!12], (2) where P is the spin polarization, A is the cross-sectional area of the channel, L is the length of the gap between the two FM electrodes,  is the Larmor frequency, g is the g-factor for the electrons (g = 2 in this study),  is the Bohr magneton, is the Dirac constant, and the spin diffusion length is given by . Under spin drift, . The solid lines in Fig. 3(a) represent fits to the experimental data using Eq. (2). It can be seen that for both the parallel and anti-parallel configurations, the Hanle signals up to 4π are well reproduced using the spin drift diffusion model. Since the spin drift velocity was calculated to be 3720 m/s based on the resistivity, the spin lifetime and the spin diffusion length in the non-degenerate Si at RT were estimated to be 0.84 ns and 1.4 µm, respectively. Although the spin diffusion length is much shorter than the gap size in this sample, spin signals were successfully observed, which indicates the strong contribution of spin drift to spin transport.      Figure 3(b) shows the bias current dependence of the Hanle signals. It is clear that all of the peaks exhibit a monotonic shift to higher field with increasing bias current. No such shift was observed in the absence of a bias electric field in the spin channel in the non-local 4-terminal method. As shown in Fig. 3(b), the oscillations of the Hanle signals at 2, 3 and 4 mA are again well reproduced by the theory. The spin drift velocities at 3 and 4 mA are estimated to !=gµBB!µB!!N=D\"T!1=v24D+1!be 5570 and 7430 m/s, respectively. The good agreement between the experimental and theoretical results provides confirmation that the peak shift is due to the spin drift effect, and not to experimental errors.       Since successful spin transport was verified, an attempt was made to modulate the spin signal using a gate electric field. Figure 4 shows the gate voltage dependence of spin accumulation voltages that was detected in the MRF configuration as spin voltages, the channel resistivity, and MR ratio, respectively of the sample with the gap length of 1.7 µm ((a)-(c)) and 6.25 µm ((d)-(f)). The MR signal was almost constant for a negative gate voltage, but decreased monotonically with increasing positive gate voltage. Importantly, this behavior is qualitatively the same as that for the sample resistance as a function of the gate voltage, i.e., n-type FET characteristics are indicated. It is noteworthy that this is also consistent with the results of a theoretical analysis by Takahashi and Maekawa [33]. The MgO tunneling barrier is used in our spin device, resulting in the tunneling spin injection from the Fe to the non-degenerate Si. The spin voltage is, in theory, linearly proportional to the channel resistance in the tunneling spin injection, which is contrary to the Ohmic spin injection [33]. As shown in the figure, the spin voltage decrease as the channel resistance decreases by the gate voltage application, which is consistent with the theory. Thus, our experimental results indicate that the modulation of the spin voltages was successfully achieved by using the gate voltage application, which is due to both the non-degenerate Si spin channel and the existence of the tunneling barrier. Although the spin signal modulation is not large, probably due to the comparatively high doping level in the Si, this is the first demonstration of a Si-based spin MOSFET operating at RT.       The successful gate-voltage-induced modulation of the spin signals allows expecting the modulation of spin transport parameters, i.e., modulation of spin diffusion length and spin drift velocity as a function of the gate voltages. Figure 5 shows the gate voltage dependence of the Hanle signals of the sample with 6.25 µm gap, where the gate voltage was set to be -25, 0 and +50 V. As can be seen, the Hanle signals exhibit different behavior when the gate voltage was changed from -25 to +50 V. The theoretical fitting enables to estimate the spin diffusion length and the spin drift velocity, and it is clarified that they are modified as a function of the gate voltages as shown in Table I. The spin drift velocity decreases as the gate voltage increase because the electron accumulation, resulting in a decrease of resistivity, takes place under the positive gate voltage application. More importantly, the spin diffusion length also decreases under the positive gate voltage application. The electron accumulation induces an increase of a carrier density in the Si channel. Since the Elliott-Yafet type spin relaxation holds in Si [34], the increase of the carrier density enhances the spin relaxation, which attributes to the suppression of the spin diffusion length under the positive gate voltage application.   IV. SUMMARY      Spin transport at RT was successfully achieved in non-degenerate n-type Si using a spin MOSFET structure. A spin rotation of more than 4π was realized using an external magnetic field, and the results were well reproduced by a spin drift diffusion model. Long-range spin transport over a distance of more than 20 µm was also demonstrated. Modulation of the spin signal was shown to be possible using the gate voltage. The spin diffusion length was suppressed by the positive gate voltage application, which was attributed to the carrier accumulation, resulting in enhancement of the Elliot-Yafet type spin relaxation. These results are expected to open the door to the development of novel spin-based information devices using Si.  ACKNOLEDGMENTS      One of the authors (M.S.) thanks Prof. Y. Suzuki of Osaka University for his valuable suggestions and comments on the theoretical analyses. The part of this study performed by M.K. was supported by the Japan Society for the Promotion of Science (JSPS).  References 1. S. Datta and B. Das, Electronic analog of the electro‐optic modulator, Appl. Phys. Lett. 56, 665 (1990). 2. H. Dery, P. Dalal, L. Cywinski, and L.J. Sham, Spin-based logic in semiconductors for reconfigurable large-scale circuits, Nature 447, 573 (2007).  3. T. Matsuno, S. Sugahara and M. Tanaka, Novel Reconfigurable Logic Gates Using Spin Metal–Oxide–Semiconductor Field-Effect Transistors, Jpn. J. Appl. Phys. 43, 6032 (2003). 4. G. Binasch, P. Grünberg, F. Saurenbach, and W. Zinn, Enhanced magnetoresistance in layered magnetic structures with antiferromagnetic interlayer exchange, Phys. Rev. B39, 4828(R) (1989).  5. M. N. Baibich, J. M. Broto, A. Fert, A. F. Nguyen Van Dau, F. Petroff, P. Eitenne, G. Creuzet, A. Friederich, and J. Chazelas, Giant Magnetoresistance of (001)Fe/(001)Cr Magnetic Superlattices, Phys. Rev. Lett. 61, 2472 (1989). 6. M. Johnson and R.H. Silsbee, Interfacial charge-spin coupling: Injection and detection of spin magnetization in metals, Phys. Rev. Lett. 55, 1790 (1985). 7. F. J. Jedema ,H. B. Heersche, A. T. Filip, J. J. A. Baselmans, and B. J. van Wees, Electrical detection of spin precession in a metallic mesoscopic spin valve, Nature 416, 713 (2002). 8. I. Appelbaum, B.Q. Huang and D.J. Monsma, Electronic measurement and control of spin transport in silicon, Nature 447, 295 (2007).  9. O.M.J. van’t Erve, A.T. Hanbicki, M. Holub, C.H. Li, C. Awo-Affouda, P. Thompson and B.T. Jonker, Electrical injection and detection of spin-polarized carriers in silicon in a lateral transport geometry, Appl. Phys. Lett. 91, 212109 (2007). 10. T. Sasaki, T. Oikawa, T. Suzuki, M. Shiraishi, Y. Suzuki and K. Tagami, Appl. Phys. Express. 2, 053003 (2009). 11. T. Suzuki, T. Sasaki, T. Oikawa, M. Shiraishi, Y. Suzuki, and K. Noguchi, Electrical Spin Injection into Silicon Using MgO Tunnel Barrier, Appl. Phys. Express 4, 023003 (2011). 12. M. Shiraishi, Y. Honda, E. Shikoh, Y. Suzuki, T. Shinjo, T. Sasaki ,T. Oikawa and T. Suzuki, Spin transport properties in silicon in a nonlocal geometry, Phys. Rev. B83, 241204(R) (2011). 13. Y. Aoki, M. Kameno, Y. Ando, E. Shikoh, Y. Suzuki, T. Shinjo, M. Shiraishi, T. Sasaki, T. Oikawa and T. Suzuki, Investigation of the inverted Hanle effect in highly doped Si, Phys. Rev. B86, 081201(R) (2012). 14. M. Kameno, Y. Ando, E. Shikoh, T. Shinjo, T. Sasaki, T. Oikawa, Y. Suzuki, T. Suzuki and M. Shiraishi, Effect of spin drift on spin accumulation voltages in highly doped silicon, Appl. Phys. Lett. 101, 122413 (2012).  15. S.P. Dash, S. Sharma, R.S. Patel, M.P. de Jong and R. Jansen, Electrical creation of spin polarization in silicon at room temperature, Nature 462, 491 (2009). 16. C.H. Li, O.M.J. van’t Erve and B.T. Jonker, Electrical injection and detection of spin accumulation in silicon at 500 K with magnetic metal/silicon dioxide contacts, Nature Comm. 2, 245 (2011). 17. K.R. Jeon, B-C. Min, I-J. Shin, C-Y. Park, H-S. Lee, Y-H. Jo and S-C. Shin, Electrical spin accumulation with improved bias voltage dependence in a crystalline CoFe/MgO/Si system, Appl. Phys Lett. 98, 262102 (2011). 18. N.W. Gray and A. Tiwari, Tunable all electric spin polarizer, Appl. Phys. Lett. 98, 102112 (2011). 19. E. Shikoh, K. Ando, K, Kubo, E. Saitoh, T. Shinjo and M. Shiraishi, Spin-Pump-Induced Spin Transport in p-Type Si at Room Temperature, Phys. Rev. Lett. 110, 127201 (2013). 20. B.Q. Huang, H.-J. Jang and I. Appelbaum, Geometric dephasing-limited Hanle effect in long-distance lateral silicon spin transport devices, Appl. Phys. Lett. 93, 162508 (2008). 21. Y. Lu and I. Appelbaum, Reverse Schottky-asymmetry spin current detectors, Appl. Phys. Lett. 97, 162501 (2010). 22. T. Sasaki, T. Suzuki, Y. Ando, H. Koike, T. Oikawa, Y. Suzuki and M. Shiraishi, Local magnetoresistance in Fe/MgO/Si lateral spin valve at room temperature, Appl. Phys. Lett. 104, 052404 (2014). 23. T. Sasaki, T. Oikawa, T. Suzuki, M. Shiraishi, Y. Suzuki and K. Noguchi, Local and non-local magnetoresistance with spin precession in highly doped Si, Appl. Phys. Lett. 98, 262503 (2011). 24. S. M. Sze, Semiconductor Devices : Physics and Technology, 2nd Ed. (Wiley & Sons, 2002). 25. See Supplemental Material at [URL will be inserted by publisher] for the detail of the sample preparation and the temperature dependence of the sample resistivity.  26. N. F. Mott, Conduction in non-crystalline systems IX. the minimum metallic conductivity, Philos. Mag. 26, 1015 (1972). 27. N. F. Mott, The minimum metallic conductivity in three dimensions, Philos. Mag. B 44, 265 (1981). 28. N. F. Mottand and M. Kaveh, Diffusion and logarithmic corrections to the conductivity of a disordered non-interacting 2D electron gas: power law localization, J. Phys. C14, L177 (1981). 29. N. F. Mottand and M. Kaveh, Electron correlations and logarithmic singularities in density of states and conductivity of disordered two-dimensional systems, J. Phys. C14, L183 (1981) 30. N. F. Mott, Review Lecture : metal-insulator transitrion, Proc. R. Soc. London Ser. A 382, 1 (1982). 31. T. F. Rosenbaum, R. F. Milligan, M. A. Paalanen, G. A. Thomas, R. N. Bhatt and W. Lin, Metal-insulator transition in a doped semiconductor, Phys. Rev. B27, 7509 (1983). 32. M. Kameno, Y. Ando, T. Shinjo, H. Koike, T. Sasaki, T. Oikawa, T. Suzuki and M. Shiraishi, Spin drift in highly doped n-type Si, Appl. Phys. Lett. 104, 092409 (2014).  33. S. Takahashi and S. Maekawa, Spin injection and detection in magnetic nanostructures, Phys. Rev. B67, 052409 (2003). 34. T. Sasaki, T. Oikawa, T. Suzuki, M. Shiraishi, Y. Suzuki and K. Noguchi, Temperature dependence of spin diffusion length in silicon by Hanle-type spin precession, Appl. Phys. Lett. 96, 122101 (2010).    Figure Captions  Figure 1. Schematic structure of non-degenerate Si spin device. Electrodes 1 and 4 are nonmagnetic electrode made from Al, and electrodes 2 and 3 are ferromagnetic electrodes with MgO tunneling barriers. A highly-doped region was formed beneath the ferromagnetic electrodes, enabling efficient spin injection into the non-degenerate Si (see also ref. 26). The Si spin channel was formed on SiO2. The gap length between the ferromagnetic electrodes, L, was varied from 1.4 to 21.3 µm, and the channel width was set to 21 µm.  Figure 2. Magnetoresistance (MR) effects in Si spin device. The schematics to the left of the experimental results indicate the circuit used for each measurement. Here, the gate voltage was not applied, and the external magnetic field was applied along y-axis. The red and blue solid lines in each panel represent the MR signals obtained under a forward and backward sweep of the external magnetic field, respectively. The bias electric current was 10 mA for each measurement, and the measurement temperature was 300 K. (a) MR signals observed using the electrical local 2-terminal method. Resistance hysteresis can be seen. (b) MR signals observed using the modified non-local 3-terminal method in the MRP configuration. No signal is seen in this configuration. (c) MR signals observed using the modified non-local 3-terminal method with the MRF configuration. Resistance hysteresis again appears. (d) Minor MR loop using the modified non-local 3-terminal method with the MRF configuration. The loop appears in the magnetic field region between two regions of resistance hysteresis. See the main text for details.   Figure 3. Hanle spin precession signals from non-degenerate Si without an application of a gate voltage. (a) Hanle signals for anti-parallel (blue open squares) and parallel (red open circles) magnetization configurations. Parabolic and linear background signals are subtracted from the raw data by taking the average of the signals and by linear fitting, respectively. The bias electric current was 2 mA and the measurement temperature was 300 K. The gap between the FM electrodes was 21.3 µm. A spin rotation of more than 4π could be observed. The blue and red solid lines are theoretical fits using Eq. (2), which reproduce the experimental results well. (b) Bias electric current dependence of the Hanle signals. The bias current was 2, 3 or 4 mA, corresponding to spin drift velocities of 3720, 5570 and 7430 m/s, respectively. Red, black and blue arrows show peak positions of π and 2π spin rotation at 2, 3 and 4 mA, respectively. Pale red, black and blue solid lines are theoretical fits using Eq. (2).   Figure 4. Si spin MOSFET operation at RT of the sample with 1.7 µm gap ((a)-(c)) and with 6.25 µm gap ((d)-(f)). (a) and (d) are channel resistivity (FET characteristics), (b) and (e) spin voltages, and (c) and (f) MR ratio. A similar dependence on the gate voltage is seen in both cases, which can be explained using the theoretical model of Takahashi and Maekawa [27].  Figure 5. Gate voltage dependence of the Hanle signals from the sample with 6.25 µm gap. The gate voltage was set to be -25, 0, +50 V. The open circles are experimental data and the solid lines are theoretical fitting lines.    Table and Table caption   Table I. Estimated spin drift velocity and spin diffusion length as a function of the gate voltages.           Vg (V)            -25        0        50 --------------------------------------------------------------------------------- Spin drift velocity (m/s)     4480       2620     1570 Spin diffusion length (µm)   4.53       1.66      0.85     Figures \n Figure 1 T. Sasaki et al.   \n \n Fig. 2(a)-(d) T. Sasaki et al. \n Fig. 3(a) T. Sasaki et al.  \n Fig. 3(b) T. Sasaki et al.\n Fig. 4 T. Sasaki et al.    \n0\u0003100\u0003-100\u0003Gate Voltage (V)\u0003MR ratio (%)\u00030\u00030.02\u0003Resistivity (! µm)\u0003300\u0003200\u00030\u0003Spin voltage (µV)\u0003150\u0003100\u000350\u00030\u0003(a)\u0003(b)\u0003(c)\u0003\n(e)\u0003(d)\u0003\n0\u0003100\u0003200\u00030.04\u0003(f)\u0003\n-100\u0003MR ratio (%)\u00030\u00030.01\u0003Resistivity (! µm)\u0003800\u0003400\u00030\u0003Spin voltage (µV)\u0003150\u0003100\u000350\u00030\u00030.02\u0003\n0.03\u0003Gate Voltage (V)\u0003\n100\u0003\n Fig. 5 T. Sasaki et al.  \n"    },    {        "title": "2109.05775v1.Revisiting_the_quantum_open_system_dynamics_of_central_spin_model.pdf",        "content": "Revisiting the quantum open system dynamics of central spin model\nSamyadeb Bhattacharya\u0003\nCenter for Security Theory and Algorithmic Research,\nInternational Institute of Information Technology, Gachibowli, Hyderabad, India\nSubhashish Banerjeey\nIndian Institute of Technology Rajasthan, Jodhpur, India\nIn this article we revisit the theory of open quantum systems from the perspective of fermionic\nbaths. Speci\fcally, we concentrate on the dynamics of a central spin half particle interacting with a\nspin bath. We have calculated the exact reduced dynamics of the central spin and constructed the\nKraus operators in relation to that. Further, the exact Lindblad type cannonical master equation\ncorresponding to the reduced dynamics is constructed. We have also brie\ry touch upon the aspect\nof non-Markovianity from the backdrop of the reduced dynamics of the central spin.\nI. INTRODUCTION\nA fundamental problem in quantum physics, and one\nwhich has transcended itself to accommodate an unprece-\ndented number of interdisciplinary research domains, is\nrelated to the subject of open quantum systems (OQSs)\n[1, 2]. In the most general sense, these systems can be un-\nderstood as consisting of localized quantum systems un-\nder the in\ruence of much bigger quantum systems, which\ncan be considered as the environment. Incidentally, in\nthe quantum world, physical systems are unlikely to be\nisolated from such environmental in\ruences. Systems\nwith potential for implementation of quantum informa-\ntion theoretic and computational protocols like ion traps\n[3], quantum dots [4], NMR qubits [5], polarized photons\n[6], Josephson junction qubits [7], Quantum Walks [8{10]\nand many others [11{15] are all exposed to one extent or\nother, to their corresponding environments. It is thus im-\nperative to understand the characteristic traits of open\nsystem dynamics for such quantum systems submerged in\ndi\u000berent types of baths. For quantum systems interact-\ning with Markovian environments, their non-classicality\neventually fades over time, thus nullifying any quantum\nadvantage which can be exploited in some information\ntheoretic protocol. Even in the \feld of quantum ther-\nmodynamics, the characteristically quantum traits like\nentanglement [16] or coherence [17, 18] signi\fcantly en-\nhances the performance of quantum thermal devices [19].\nThus, it is imperative to engineer baths in such a way so\nas to retain non-classical features of the system for large\ndurations.\nDetermination of global dynamics of the quantum sys-\ntem and its environment [10] is essentially a many body\nproblem. For exact determination of the the reduced dy-\nnamics of the system, the total global dynamics of the\nsystem plus environment must be unravelled, which is\noften not possible in reality. The reason behind that is\nthe fact that often in many body problems the evolu-\n\u0003samyadeb.b@iiit.ac.in\nysubhashish@iitj.ac.intion of microscopic quantum systems in consideration,\ngenerally gets extremely involved due to the interaction\nwith the complex environment. To deduce the local evo-\nlution of the quantum system of interest, it is a gen-\neral procedure to consider the bath as a huge collec-\ntion of harmonic oscillators or fermionic spin half enti-\nties [1, 2] usually identi\fed as bath. These baths are\ncategorized into two separate universal classes of quan-\ntum environment [20]. In the harmonic oscillator en-\nvironmental model, the environment is conceived as a\ncollection of non-interacting harmonic oscillators. Most\nprominent of such models are spin-boson [21] and the\nCaldeira-Leggett model [11, 12, 22, 23] originating from\na schematic put forward by Feynman and Vernon [24].\nThese particular paradigms of open system dynamics\nhave been widely investigated in the backdrop of many\ndi\u000berent physical phenomena under Markovian approx-\nimation [1, 2, 25, 26]. On the contrary, the class of\nfermionic bath models are still relatively less investigated,\nin spite of the fact that such fermionic baths are of piv-\notal interest in the quantum theory of magnetism [27],\nquantum spin glasses [28], theory of conductors and su-\nperconductors [29]. Deducing the reduced exact dynam-\nics of a quantum system interacting with a spin bath\nmodel is simultaneously of paramount importance yet a\ndi\u000ecult task. Indeed, in usual cases the reduced dynam-\nics cannot be derived exactly without applying several\napproximation methods, both local and nonlocal in time\n[1, 30{35].\nIn this work, we are going to review a method to derive\nthe exact reduced dynamics [36{39] of a spin half system\ninteracting with a special type of spin environment. One\nof the most signi\fcant aspect of this particular formalism\nis that, it is amongst the very few cases where the exact\nreduced dynamics can be derived without applying any\nmajor approximation technique. We primarily focus on\nthe `central spin' system, where the homogeneous interac-\ntions couple a central two-level system to a background\nof spin environment. It is the fermionic bath counter-\npart of the famous `spin-boson' oscillator model. This\nmodel adequately describes, e.g., the tunnelling dynamics\nof nanoscopic and mesoscopic magnets and superconduc-arXiv:2109.05775v1  [quant-ph]  13 Sep 20212\ntors. Here we demonstrate how to average over (or `in-\ntegrate out') spin bath modes, using Holstein-Primako\u000b\ntransformation [40, 41], to \fnd the central spin dynam-\nics. The formal technique involves transformation of the\nnon-interacting bath spins into a bosonic representation,\nwhich allows us to average out the bath modes. After\n\fnding the reduced dynamics of the central spin by this\nmethod, we will derive the Kraus operators and \fnally\nthe exact Lindblad [25] master equation for this model.\nII. THE REDUCED DYNAMICS OF A SPIN\nHALF PARTICLE FOR A CENTRAL SPIN\nMODEL\nIn this section, we present the central spin model,\nwhere a single spin half particle described as the system,\ninteracts centrally with a collection of non-interacting\nspin half particles conceived as the fermionic environ-\nment.\nWe consider a spin half particle interacting uniformly\nwith a collection of non-interacting spin half particles.\nThe total Hamiltonian of this spin system and the spin\nenvironment is given by\nH=Hs+HB+HSB; (1)\nwhere the system, environment and the interaction\nHamiltonians are respectively given by\nHs=~!0\n2\u001bz0;\nHB=~!\n2NPN\ni=1\u001bzi;\nHSB=~\u0001\n2p\nNPN\ni=1(\u001bx0\u001bxi+\u001by0\u001byi+\u001bz0\u001bzi):(2)\nHere\u001bi0(i=x;y;z ) are the Pauli matrices for the sys-\ntem and\u001bij(i=x;y;z andj= 1;2;::::N ) are the\nsame for the Nnumber of environment spins. Both the\ncharecteristic bath and interaction frequencies has been\nrescaled as !=N and \u0001=p\nN, respectively. Our goal is\nto represent this total Hamiltonian in a simple enough\nform, so that we can work with it to achieve an exact\nsolution of the corresponding dynamical equation for the\nsystem and the environment. For that purpose, we are\ngoing to utilize a method called the Holstein-Primako\u000b\ntransformation, which will allow us to write the total\nHamiltonian in a form similar to a distorted Rabi oscil-\nlation. In the following we exercise this method on our\nsystem.\nLet us use the total angular momentum operator for\nthe bath spins Jl=PN\ni=1\u001bli(l=x;y;z; +;\u0000). With the\nhelp of this, without altering the physics behind it, let us\ntransform the environment and interaction Hamiltonian\nasHB=~!\n2NJz; HSB=~\u0001\n2p\nN(\u001bx0Jx+\u001by0Jy+\u001bz0Jz):\n(3)\nFurther using the ladder operators 2 J+=Jx+iJy,\n2J\u0000=Jx\u0000iJyand similarly \u001b+; \u001b\u0000for the system\nspin, the interaction Hamiltonian can be rewritten as\nHSB=~\u0001p\nN\u0012\n\u001b+0J\u0000+\u001b\u00000J++\u001bz0Jz\n2\u0013\n:\nWe now use the Holstein-Primako\u000b transformation for\nthe total angular momentum of the environmental spin\noperators\nJ+=p\nNby\u0012\n1\u0000byb\n2N\u00131=2\n;J\u0000=p\nN\u0012\n1\u0000byb\n2N\u00131=2\nb;\n(4)\nwhereb; byare the bosonic annihilation and creation op-\nerators respectively, with the property [ b; by] = 1. Using\nthese transformations, the Hamiltonians of equation (2)\ncan be rewritten as\nHB=\u0000~!\n2\u0010\n1\u0000byb\nN\u0011\n;\nHSB=~\u0001\u0014\n\u001b+0\u0010\n1\u0000byb\n2N\u00111=2\nb+\u001b\u00000by\u0010\n1\u0000byb\n2N\u00111=2\u0015\n+~\u0001\n2\u001bz0\u0010\n1\u0000byb\nN\u0011\n:\n(5)\nEquipped with this transformed Hamiltonian, we are now\ntechnically dealing with a single spin interacting with\na single oscillator mode, though the underlining physics\nremains unchanged.\nIn the following, with the help of the previously dis-\ncussed transformation we now deduce the exact reduced\ndynamics of the system spin half particle after per-\nforming the total dynamical evolution for the system\nand environment and then discarding the bath degrees\nof freedom. In order to do that, we assume the ini-\ntial completely decoupled system-bath joint state to be\n\u001aSB(0) =\u001aS\n\u001aB, which basically makes sure the com-\nplete positivity of the reduced dynamics. Furthermore,\nwe consider the initial bath state to be a thermal state\n\u001aB= exp(\u0000HB=KT )=Z, whereK; T; Z are respectively\nthe Boltzmann constant, temperature of the bath and the\npartition function. Let us further consider the evolution\nof the joint system-bath state j\u001e(0)i=j1i\njxi, under\nthe previously discussed Hamiltonian, where j1iis the\nexcited state of the system and jxiis an arbitrary bath\nstate. After the total evolution described by the uni-\ntaryU= exp(\u0000iHt=~), let the initial state evolved into\nj\u001e(t)i=\u00111(t)j1ijy1i+\u00112(t)j0ijy2i. For the purpose of\nsolving the dynamics, let us further consider two opera-\ntors ^M1(t) and ^M2(t) in the environment space such that,\n^M1(t)jxi=\u00111(t)jy1iand ^M2(t)jxi=\u00112(t)jy2i. Now us-\ning the Schr odinger equation corresponding to the to-\ntal evolutiond\ndtj\u001e(t)i=\u0000i\n~Hj\u001e(t)i, we get the following\nequations3\nd\ndt^M1(t) =\u0000i\u0010\n!0\n2\u0000!\u0000\u0001\n2\u0010\n1\u0000byb\nN\u0011\u0011\n^M1(t)\u0000i\u0001\u0010\n1\u0000byb\n2N\u00111=2\nb^M2(t);\nd\ndt^M2(t) =i\u0010\n!0\n2+!+\u0001\n2\u0010\n1\u0000byb\nN\u0011\u0011\n^M2(t)\u0000i\u0001by\u0010\n1\u0000byb\n2N\u00111=2^M1(t):(6)\nIf we now further substitute ^M0\n1(t) = ^M1(t) and ^M0\n2(t) =by^M2(t), then we have\nd\ndt^M0\n1(t) =\u0000i\u0000!0\n2\u0000!\u0000\u0001\n2\u0000\n1\u0000^n\nN\u0001\u0001^M0\n1(t)\u0000i\u0001\u0000\n1\u0000^n\n2N\u00011=2(^n+ 1) ^M0\n2(t);\nd\ndt^M0\n2(t) =i\u0000!0\n2+!+\u0001\n2\u0000\n1\u0000^n+1\nN\u0001\u0001^M0\n2(t)\u0000i\u0001\u0000\n1\u0000^n\n2N\u00011=2^M0\n1(t):(7)\nHere ^n=bybis the number operator. This equation (7)\ncan now be solved and the solution will be a function of\nboth the number operator ^ nand timet. We can further\nconsider the eigenstate jniof the number operator, sothat we have ^M0\n1(t)jni=M0\n1(n;t)jniand ^M0\n2(t)jni=\nM0\n2(n;t)jni. Using this we can determine the evolution\nof the reduced state of the qubit ( j1ih1j),by tracing out\nthe environment basis ( jni). Therefore the qubit excited\nstate evolves under the given dynamics as\n\b(j1ih1j) =1\nZNX\nn=0\u0000\njM0\n1(n;t)j2j1ih1j+ (n+ 1)jM0\n2(n;t)j2j0ih0j\u0001\nexp\u0012\n\u0000~!\n2KT\u0010n\nN\u00001\u0011\u0013\n; (8)\nwith\njM02\n1(n;t)j2= 1\u00004\u00012(1\u0000n=2N)(n+ 1)jM02\n2(n;t)j2;\njM02\n2(n;t)j2=sin2(\ft=2)\n\f2;\n\f2=\u0000\n!0\u0000!\n2N+ \u0001\u0000\n1\u00002n+1\n2N\u0001\u00012+ 4\u00012(n+ 1)\u0000\n1\u0000n\n2N\u0001\n:\n(9)\nSimilarly, let us consider j\u001f(0)i=j0i\njxiandj\u001f(t)i=^M3(t)j0ijy3i+^M4(t)j1ijy4i. Let us consider the transfor-\nmation ^M3(t) = ^M0\n3(t) and ^M4(t) =b^M0\n4(t). Now if we\nfollow similar procedure as demonstrated above, we come\nto the following characteristic equations\nd\ndt^M0\n3(t) =i\u0000!0\n2+!+\u0001\n2\u0000\n1\u0000^n\nN\u0001\u0001^M0\n3(t)\u0000i\u0001^n\u0000\n1\u0000^n\u00001\n2N\u0001^M0\n4(t);\nd\ndt^M0\n4(t) =i\u0000!0\n2\u0000!\u0000\u0001\n2\u0000\n1\u0000^n\nN\u0001\u0001^M0\n4(t)\u0000i\u0001^n\u0000\n1\u0000^n\u00001\n2N\u0001^M0\n3(t);\n(10)\nSolving equation (10) we get that\n\b(j0ih0j) =1\nZNX\nn=0\u0000\nnjM0\n4(n;t)j2j1ih1j+jM0\n3(n;t)j2j0ih0j\u0001\nexp\u0012\n\u0000~!\n2KT\u0010n\nN\u00001\u0011\u0013\n: (11)\nwith\njM3(n;t)j2=sin2(\f0t=2)\n\f02;\njM4(n;t)j2= 1\u00004n\u00012\u0000\n1\u0000n\u00001\n2N\u0001\njM1(n;t)j2;\n\f02=\u0000\n!0\u0000!\n2N+ \u0001\u0000\n1\u0000n\nN\u0001\u00012+ 4\u00012n\u0000\n1\u0000n\u00001\n2N\u0001\n:\n(12)The o\u000b-diagonal components of the system density ma-\ntrix can be calculated as\n\b(j1ih0j) =\u0010(t)j1ih0j; (13)4\nwith\n\u0010(t) =1\nZPN\nn=0e\u0000!t=2N\u0010\ncos(\ft=2)\u0000i\u000fsin(\ft=2)\n\f\u0011\u0010\ncos(\f0t=2) +i\u000f0sin(\f0t=2)\n\f0\u0011\nexp\u0000\n\u0000~!\n2KT\u0000n\nN\u00001\u0001\u0001\n;\n\u000f=1\n\f\u0000\n!0\u0000!\n2N+ \u0001\u0000\n1\u00002n+1\n2N\u0001\u0001\n;\n\u000f0=1\n\f0\u0000\n!0\u0000!\n2N+ \u0001\u0000\n1\u0000n\nN\u0001\u0001\n:(14)\nTherefore the reduced density matrix of the system\nqubit\n\u001a(t) =\u0012\n\u001a11(t)\u001a12(t)\n\u001a\u0003\n12(t)\u001a22(t)\u0013\n(15)\nis given as\n\u001a11(t) = (1\u0000\u000b1(t))\u001a11(0) +\u000b2(t)\u001a22(0);\n\u001a22(t) = 1\u0000\u001a11(t);\n\u001a12(t) =\u0010(t)\u001a12(0);(16)\nwith\n\u000b1(t) =PN\nn=04\u00012\u0000\n1\u0000n\n2N\u0001\n(n+ 1)sin2(\ft=2)\n\f2;\n\u000b2(t) =PN\nn=04n\u00012\u0000\n1\u0000n\u00001\n2N\u0001sin2(\f0t=2)\n\f02:(17)\nA. Operator sum representation\nAnother important aspect of open system dynam-\nics is to express a completely positive trace preserv-\ning evolution in terms of Kraus operators or in another\nwords, operator sum representation, given as \u001a(t) =P\niKi(t)\u001a(0)Ky\ni(t). The Kraus operators can be con-\nstructed from the eigen spectrum of the Choi state cor-\nresponding to the dynamical map. The Choi state of\nthe dynamical map \b( \u0001) can be derived by applying the\nmap on one side of the maximally entangled state j i\nasI\nj\bih\bj, where Iis the identity matrix. For our\nparticular case, this matrix is given by\nC(t) =0\nBBB@1\u0000\u000b1(t)\n20 0\u0010(t)\n2\n0\u000b1(t)\n20 0\n0 0\u000b2(t)\n20\n\u0010\u0003(t)\n20 01\u0000\u000b2(t)\n21\nCCCA: (18)\nFrom the eigen spectrum of this Choi state, we can\ncalculate the Kraus operators asK1(t) =p\n\u000b2(t)\u0012\n0 1\n0 0\u0013\n;\nK2(t) =p\n\u000b1(t)\u0012\n0 0\n1 0\u0013\n;\nK3(t) =q\n\u00001(t)\n1+\u00032\n1(t)\u0012\n\u00031(t)ei\u0012(t)0\n0 1\u0013\n;\nK4(t) =q\n\u00002(t)\n1+\u00032\n2(t)\u0012\n\u00032(t)ei\u0012(t)0\n0 1\u0013\n;(19)\nwhere\n\u00001;2=\u0010\n1\u0000\u000b1(t)+\u000b2(t)\n2\u0011\n\u00061\n2p\n(\u000b1(t)\u0000\u000b2(t))2+ 4j\u0010(t)j2;\n\u00031;2=p\n(\u000b1(t)\u0000\u000b2(t))2+4j\u0010(t)j2\u0007(\u000b1(t)\u0000\u000b2(t))\n2j\u0010(t)j;\n\u0012(t) = arctan [ \u0010I(t)=\u0010R(t)];\n(20)\nwhere\u0010I(t); \u0010R(t) are the imaginary and real part of \u0010(t)\nrespectively.\nB. Cannonical master equation\nNow, our goal is to \fnd the generator corresponding\nto the completely positive trace preserving evolution. In\nother words, here we are going to construct the cannon-\nical master equation for the evolution we demonstrated\npreviously. Derivation of the exact master equation cor-\nresponding to a given quantum dynamical map is consid-\nered to be one of the most fundamental issues in the the-\nory of open quantum systems. This is because, the can-\nnonical or Lindblad type master equation of a quantum\nevolution, paves the path for understanding various phys-\nical processes like dissipation, absorption, dephasing and\nthe decohering process in general. Moreover, theoretical\nand also practical studies of quantum scale heat engines,\nrefrigerators, diodes, transistors and other such devices\nhas gained paramount importance in recent times,since\nthey are paving the way for realization of quantum com-\nputers in the near future. In this context, construction of5\nLindblad master equations for practically implementable\nreservoir engineering models are of considerable interest\nfrom the perspective of quantum thermodynamics,where\na very few number of quantum systems are coupled to\ntheir respective heat baths in general. In those situa-\ntions, the canonical Lindblad type master equation in\nthe spin bath models can provide a novel path to explore\nthe thermodynamics in hithertho unexplored strong cou-\npling and non-Markovian regions which presumably have\nfar reaching impacts to enhance the performance of many\nquantum thermal devices.\nIn the following, we construct the exact Lindblad type\ncanonical master equation [36, 37] for the central spin\nhalf particle interacting centrally with a collection of spin\nhalf particles, starting from a completely positive trace\npreserving map given in equation (16). The dynamical\nmap expressed in equation (16) is also notationally ex-\npressed as\u001a(t) = \b(\u001a(0). Let us consider that the master\nequation corresponding to this map is given by\nd\ndt\u001a(t) =L(\u001a(t)) (21)\nThe above equation is characterized by the time de-\npendent generator L(\u0001). We now consider the following\nmethod to construct this Lindblad type master equation\nfor the evolution of the central spin given by equation\n(16).\nLet us consider an orthonormal basis set of Hermitian\noperatorsfGkg. By de\fnition, they have the following\nproperties\nTr[GkGl] =\u000ekl; Gy\nk=Gk:A dynamical map of the form \u001a(t) = \b(\u001a(0)), can be\nrepresented as\n\b(\u001a(0)) =X\nk;lTr[Gk\b(Gl)]Tr[Gl\u001a(0)]Gk= [F(t)r(0)]GT;\n(22)\nwhereF(t) is a matrix with elements Fkl(t) =\nTr[Gk\b(Gl)] andr(0) is a column vector with elements\nrl=Tr[Gl\u001a(0)]. By di\u000berentiating equation (22), we get\n_\u001a(t) = [ _F(t)r(0)]GT: (23)\nSimilarly, let us construct a matrix L(t) with elements\nLkl(t) =Tr[GkL(Gl)] and a column vector r(t) with el-\nementsrl(t) =Tr[Gl\u001a(t)]. Therefore, we can represent\nthe master equation (21) as\nL(\u001a(t)) =X\nk;lTr[GkL(Gl)]Tr[Gl\u001a(t)]Gk= [L(t)r(t)]GT:\n(24)\nNow comparing equation (23) and (24) and using the\nidentityF(t)r(0) =r(t), we get\nL(t)F(t) =_F(t))L(t) =_F(t)F(t)\u00001: (25)\nIf theF(t) matrix for some given quantum evolution\nis invertible, then we can always \fnd the corresponding\nL(t) matrix and hence the exact master equation. For\nour speci\fc situation, fortunately the corresponding F(t)\nmatrix is invertible and we \fnd the exact expression of\nL(t), which is given in the following.\nL(t) =0\nBBBBBB@0 0 0 0\n0d\ndtlnj\u0010(t)j \u0000d\ndtln\u0012\n1 +\f\f\f\u0010R(t)\n\u0010I(t)\f\f\f2\u0013\n0\n0d\ndtln\u0012\n1 +\f\f\f\u0010R(t)\n\u0010I(t)\f\f\f2\u0013\nd\ndtlnj\u0010(t)j 0\n_\u000b2(t)\u0000_\u000b1(t)\u0000_\u000b2(t)+ _\u000b2(t)\n1\u0000(\u000b1(t)+\u000b2(t))(\u000b1(t) +\u000b2(t)) 0 0 \u0000_\u000b2(t)+ _\u000b2(t)\n1\u0000(\u000b1(t)+\u000b2(t))1\nCCCCCCA:\n(26)\nHereafter using this matrix (26), we get the following\nset of di\u000berential equations for the elements of the density\nmatrix\u001a(t).\n_\u001a11(t) =\u0000_\u001a22(t) =Lz0+Lzz\n2\u001a11(t) +Lz0\u0000Lzz\n2\u001a22(t);\n_\u001a12(t) = (Lxx+iLxy)\u001a12(t);\n(27)\nwhereLklare the matrix elements of L(t) withk;l=\nf0;x;y;zg. This set of equations is essentially the dy-\nnamical master equation for the density matrix corre-\nsponding to the evolution we considered. But, as we can\nclearly see that it is not in the cannonical Lindblad form.To understand the process of dissipation, absorption, de-\nphasing and other phenomena in an orderly fashion, one\nneeds to construct the Lindblad form of the master equa-\ntion. Therefore to obtain the desired form of the master\nequation, let us consider the following form\n_\u001a(t) =L(\u001a(t)) =X\nkAk(t)\u001a(t)Bk(t)y; (28)\nwhereAk(t) andBk(t) are matrices represented as\nAk(t) =X\niGiaik(t); Bk(t) =X\niGibik(t):6\nBy virtue of this speci\fc decomposition, the master\nequation (27) can be rewritten as\n_\u001a(t) =X\nijCij(t)Gi\u001a(t)Gj;\nwithCij(t) =P\nkaik(t)bjk(t)\u0003. Doing some algebraic\nmanipulation, we arrive at the following master equation\nof the Lindblad form.\n_\u001a(t) =i\n~[\u001a(t);H(t)] +P\nij=fx:y;zgCij(t)\u0002\nGi\u001a(t)Gj\u00001\n2fGjGi;\u001a(t)g\u0003\n;\n(29)\nwith\nH(t) =i~\n2(D(t)\u0000D(t)y); D(t) =C00(t)\n8I+X\niCi0(t)\n2Gi:\nHere the curly braces stand for anti-commutator. There-\nfore the cannonical Lindblad form of the master equation\nlooks like\n_\u001a(t) =i\n~\n(t)[\u001a(t);\u001bz] +\rd(t)[\u001bz\u001a(t)\u001bz\u0000\u001a(t)]\n+\r\u0000(t)[\u001b\u0000\u001a(t)\u001b+\u00001\n2f\u001b+\u001b\u0000;\u001a(t)g]\n+\r+(t)[\u001b+\u001a(t)\u001b\u0000\u00001\n2f\u001b\u0000\u001b+;\u001a(t)g];(30)\nwith\n\n(t) =\u00001\n2d\ndtln\u0012\n1 +\f\f\f\u0010R(t)\n\u0010I(t)\f\f\f2\u0013\n;\n\r\u0000(t) =h\nd\ndt\u000b1(t)\u0000\u000b2(t)\n2\u0000\u000b1(t)\u0000\u000b2(t)+1d\ndtln(1\u0000\u000b1(t)\u0000\u000b2(t))\n2i\n;\n\r+(t) =\u0000h\nd\ndt\u000b1(t)\u0000\u000b2(t)\n2\u0000\u000b1(t)\u0000\u000b2(t)\u00001d\ndtln(1\u0000\u000b1(t)\u0000\u000b2(t))\n2i\n;\n\rd(t) =1\n4d\ndth\nln\u0010\n1\u0000\u000b1(t)\u0000\u000b2(t)\nj\u0010(t)j2\u0011i\n:\nThe \frst term in the right hand side of equation (30) in\nthe commutator corresponds to the unitary part, having\nfrequency \n( t). The second, third and fourth terms are\nchronologically the dephasing, dissipation and absorption\nterms with rates \rd(t); \r\u0000(t); \r+(t), respectively.\nIII. DYNAMICS OF NON-MARKOVIANITY\nBoth the qualitative and quantitative analysis of quan-\ntum non-Markovianity is of fundamental importance in\nthe theory of open quantum dynamics. Over the past\ndecade, there has been numerous proposals for quanti\fng\nnon-Markovianity based on CP divisibility [35, 42] and\nnon-Markovianity witness [10, 35, 38, 39, 42{54]. One\nof the prominent non-Markovianity measures based on\nthe composition of the dynamical map was introduced\nin [35], the so called RHP measure. In this method of\ncharacterization, non-Markovianity is quanti\fed as the\namount of deviation from divisibility of a dynamics.A divisible quantum completely positive trace pre-\nserving dynamics \b D(t2;t1) is considered as such a\ndynamical map which can be divided into in\fnitely\nmany completely positive trace preserving maps like the\nfollowing.\n\bD(t2;t1) = \bD(t2;t0)\u000e\bD(t0;t00)\u000e\n:::\u000e\bD(t(n\u00001)0;tn0)\u000e\bD(tn0;t1);(31)\nfor allt1; t2andt1\u0014tn0\u0014:::\u0014t0\u0014t2. For a dy-\nnamical map \b(\u0001), which does not follow this property,\nthe amount of \"indivisibility\" can be quanti\fed as the\nshift from complete positivity of some intermediate map\n\b(t+\u001c;t). This amount can be calculated from the Choi\nstate as\nN(t) = lim\n\u001c!0+jjI\n\b(t+\u001c;t)j ih jjj1\u00001\n\u001c;\nwherejj(\u0001)jj1=p\nTr(\u0001)y(\u0001) stands for the trace norm\nof a matrix. Note that though we are dealing here with\nqubit systems, the procedure is viable for any dimen-\nsional quantum systems.\nFor quantum evolutions having Lindblad type genera-\ntors,N(t) can only be a \fnite non-zerp quantity when\none or more of the Lindblad coe\u000ecients are negative\nat certain time t, i.e., the divisibility of a Lindblad\ndynamics breaks down, only if Lindblad coe\u000ecients\nare negative. Therefore for simplicity, we can consider\nthe negativity of \r\u0000(t); \r+(t) or\rd(t) as a proper\nindicator of non-Markovianity. In the following plots, we\ndemonstrate the time evolution of some of the Lindblad\ncoe\u000ecients for our speci\fc dynamics, to understand\nhow its non-Markovian features change with variation\nof parameters like interaction strength, number of bath\nspins and temperature of the bath.\nNote that we have only considered the temporal\ndynamics of \r\u0000(t) for the sake of brevity. The other\nLindblad parameters will also show similar type of\nnon-Markovian behaviour. From the plots it is clear\nthat the dynamics in question is non-Markovian and this\nnon-Markovianity increases with increasing interaction\nstrength, bath temperature and also the number of bath\nspins. We can see from the plots that, as we increase\nthe numner of bath spins, interaction strengths, and\ntemperature of the bath, the non-Markovian \ructuation\nof information \row from the system and the back\row\nof information from environment into the system also\nincreases. This clearly indicates that the bath param-\neters have major roles to play in the non-Markovian\nbehaiviour of the system dynamics. Nevertheless,\nwe conclude that all non-Markovian environmental\ninteractions will follow the same sort of behaviour as\nthe case considered in this article. The phenomenon of\nquantum non-Markovianity is still not fully resolved and7\n0 5 10 15 20-0.006-0.004-0.0020.0000.0020.0040.006\ntγdis(t)\nFIG. 1. (Colour online)\nHere we have plotted \r\u0000(t) with time tfor di\u000berent values\nof interaction strength \u0001. We have considered !0=!= 1,\nnumber of bath spins N= 100 and temperature T= 1. The\nred thick, green dashed and blue dotted plots are for \u0001 =\n0:005;0:003;0:01, respectively.\n0 5 10 15 20-0.006-0.004-0.0020.0000.0020.0040.006\ntγdis(t)\nFIG. 2. (Colour online)\nHere we depict \r\u0000(t) with respect to time tfor di\u000berent values\nof temperature T. We have considered !0=!= 1, number\nof bath spins N= 100 and interaction strength \u0001 = 0 :01.\nThe red thick, green dashed and blue dotted plots are for\nT= 0:1;1;10, respectively.is a heavily researched area of study in quantum science.\nThe spin bath paradigm introduced in this work, has\nthe potential to deeply impact this \feld of study.\n0 5 10 15 20-0.03-0.02-0.010.000.010.020.03\ntγdis(t)\nFIG. 3. (Colour online)\nThe behavior of \r\u0000(t) as a function of time tfor di\u000berent\nvalues ofN. We have considered !0=!= 1, interaction\nstrength \u0001 = 0 :01 and temperature T= 1. The red thick,\ngreen dashed and blue dotted plots are for N= 100;200;500,\nrespectively.\nIV. CONCLUSION\nIn this article, we have revisited the open quantum dy-\nnamical aspects of central spin system interacting with\na spin bath. The tools needed have been discussed.\nFor the model chosen, the exact reduced dynamics of\nthe spin is derived and the Kraus operators constructed\nfrom it. Furthermore, we have also reviewed a speci\fc\ntechnique to construct Lindblad type cannonical master\nequations in detail. Using this method, we have con-\nstructed the exact Lindblad type master equation for the\ncentral spin. Moreover, we have also discussed some as-\npects of non-Markovianity of the central spin. This re-\nview o\u000bers a substantial material for both general tech-\nniques of the theory of open quantum systems and the\ntheory of fermionic baths.\n[1] H. P. Breuer and F. Petruccione, The theory of\nopen quantum systems (Oxford University Press, Great\nClarendon Street, 2002).\n[2] S. Banerjee, Open Quantum Systems (Springer Singa-\npore, 152 Beach Rd, 21-01 Gateway East, Singapore\n189721, 2018).\n[3] J. I. Cirac and P. Zoller, Phys. Rev. Lett. 74, 4091 (1995).\n[4] D. Loss and D. P. DiVincenzo, Phys. Rev. A 57, 120\n(1998).[5] D. G. Cory, A. F. Fahmy, and T. F. Havel, Proceedings\nof the National Academy of Sciences 94, 1634 (1997),\nhttps://www.pnas.org/content/94/5/1634.full.pdf.\n[6] R. Okamoto, J. L. O'Brien, H. F. Hofmann,\nand S. Takeuchi, Proceedings of the Na-\ntional Academy of Sciences 108, 10067 (2011),\nhttps://www.pnas.org/content/108/25/10067.full.pdf.\n[7] V. Bouchiat, D. Vion, P. Joyez, D. Esteve, and M. H.\nDevoret, Physica Scripta T76, 165 (1998).8\n[8] C. M. Chandrashekar, R. Srikanth, and S. Banerjee,\nPhys. Rev. A 76, 022316 (2007).\n[9] S. Banerjee, R. Srikanth, C. M. Chandrashekar, and\nP. Rungta, Phys. Rev. A 78, 052316 (2008).\n[10] S. Bhattacharya, S. Banerjee, and A. K. Pati, Quantum\nInformation Process 17, 236 (2018).\n[11] S. Banerjee and R. Ghosh, Phys. Rev. A 62, 042105\n(2000).\n[12] S. Banerjee and R. Ghosh, Phys. Rev. E 67, 056120\n(2003).\n[13] F. Galve, G. L. Giorgi, and R. Zambrini, Phys. Rev. A\n81, 062117 (2010).\n[14] J. Naikoo, A. K. Alok, S. Banerjee, and S. U. Sankar,\nPhys. Rev. D 99, 095001 (2019).\n[15] A. K. Alok, S. Banerjee, and S. Uma Sankar, Nuclear\nPhysics B 909, 65 (2016).\n[16] N. Brunner, M. Huber, N. Linden, S. Popescu, R. Silva,\nand P. Skrzypczyk, Phys. Rev. E 89, 032115 (2014).\n[17] J. B. Brask and N. Brunner, Phys. Rev. E 92, 062101\n(2015).\n[18] M. T. Mitchison, M. P. Woods, J. Prior, and M. Huber,\nNew Journal of Physics 17, 115013 (2015).\n[19] G. Thomas, N. Siddharth, S. Banerjee, and S. Ghosh,\nPhys. Rev. E 97, 062108 (2018).\n[20] N. V. Prokof 'ev and P. C. E. Stamp, Reports on Progress\nin Physics 63, 669 (2000).\n[21] A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A.\nFisher, A. Garg, and W. Zwerger, Rev. Mod. Phys. 59,\n1 (1987).\n[22] U. Weiss, Quantum Dissipative Systems , 2nd ed., Series\nin Modern Condensed Matter Physics, Vol. 10 (World\nScienti\fc, 1999).\n[23] A. O. Caldeira and A. J. Leggett, Annals of Physics 149,\n374 (1983).\n[24] R. Feynman and F. Vernon, Annals of Physics 24, 118\n(1963).\n[25] G. Lindblad, Communications in Mathematical Physics\n48, 119 (1976).\n[26] V. Gorini, A. Kossakowski, and E. C. G. Sudarshan,\nJournal of Mathematical Physics 17, 821 (1976).\n[27] J. B. Parkinson and D. J. J. Farnell, An Introduction\nto Quantum Spin Systems , Lect. Notes Phys., Vol. 816\n(Springer, Berlin, 2010).\n[28] T. F. Rosenbaum, Journal of Physics: Condensed Matter\n8, 9759 (1996).\n[29] A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A.\nFisher, A. Garg, and W. Zwerger, Rev. Mod. Phys. 59,\n1 (1987).\n[30] S. Nakajima, Progress of Theoretical Physics 20, 948\n(1958).[31] R. Zwanzig, The Journal of Chemical Physics 33, 1338\n(1960).\n[32] S. Chaturvedi and F. Shibata, Zeitschrift f ur Physik B\nCondensed Matter 35, 297 (1979).\n[33] H.-P. Breuer, D. Burgarth, and F. Petruccione, Phys.\nRev. B 70, 045323 (2004).\n[34] E.-M. Laine, J. Piilo, and H.-P. Breuer, Phys. Rev. A\n81, 062115 (2010).\n[35] A. Rivas, S. F. Huelga, and M. B. Plenio, Phys. Rev.\nLett. 105, 050403 (2010).\n[36] M. J. W. Hall, J. D. Cresser, L. Li, and E. Andersson,\nPhys. Rev. A 89, 042120 (2014).\n[37] E. Andersson, J. D. Cresser, and M. J. W.\nHall, Journal of Modern Optics 54, 1695 (2007),\nhttps://doi.org/10.1080/09500340701352581.\n[38] S. Bhattacharya, A. Misra, C. Mukhopadhyay, and A. K.\nPati, Phys. Rev. A 95, 012122 (2017).\n[39] C. Mukhopadhyay, S. Bhattacharya, A. Misra, and A. K.\nPati, Phys. Rev. A 96, 052125 (2017).\n[40] T. Holstein and H. Primako\u000b, Phys. Rev. 58, 1098\n(1940).\n[41] X.-Z. Yuan, H.-S. Goan, and K.-D. Zhu, Phys. Rev. B\n75, 045331 (2007).\n[42] D. Chru\u0013 sci\u0013 nski, A. Kossakowski, and A. Rivas, Phys.\nRev. A 83, 052128 (2011).\n[43] S. e. Haseli, Phys. Rev. A 90, 052118 (2014).\n[44] T. Chanda and S. Bhattacharya, Annals of Physics 366,\n1 (2016).\n[45] F. F. e. Fanchini, Phys. Rev. Lett. 112, 210402 (2014).\n[46] S. Luo, S. Fu, and H. Song, Phys. Rev. A 86, 044101\n(2012).\n[47] X.-M. Lu, X. Wang, and C. P. Sun, Phys. Rev. A 82,\n042103 (2010).\n[48] R. Vasile, S. Maniscalco, M. G. A. Paris, H.-P. Breuer,\nand J. Piilo, Phys. Rev. A 84, 052118 (2011).\n[49] N. Awasthi, S. Bhattacharya, A. Sen(De), and U. Sen,\nPhys. Rev. A 97, 032103 (2018).\n[50] S. Bhattacharya, B. Bhattacharya, and A. S. Majumdar,\nJournal of Physics A: Mathematical and Theoretical 54,\n035302 (2020).\n[51] S. Bhattacharya, B. Bhattacharya, and A. S. Majumdar,\nJournal of Physics A: Mathematical and Theoretical 53,\n335301 (2020).\n[52] A. G. Maity, S. Bhattacharya, and A. S. Majumdar,\nJournal of Physics A: Mathematical and Theoretical 53,\n175301 (2020).\n[53] N. P. Kumar, S. Banerjee, R. Srikanth, V. Ja-\ngadish, and F. Petruccione, Open Systems\n& Information Dynamics 25, 1850014 (2018),\nhttps://doi.org/10.1142/S1230161218500142.\n[54] R. Shrikant, U Srikanth and S. Banerjee, Scienti\fc Re-\nports 10, 15049 (2020)."    },    {        "title": "0907.3094v2.Current_induced_dynamics_in_non_collinear_dual_spin_valves.pdf",        "content": "arXiv:0907.3094v2  [cond-mat.mes-hall]  25 Sep 2009APS/123-QED\nCurrent-induced dynamics in non-collinear dual spin-valv es\nP. Bal´ aˇ z,1,∗M. Gmitra,2and J. Barna´ s1,3\n1Department of Physics, Adam Mickiewicz University, Umulto wska 85, 61-614 Pozna´ n, Poland\n2Institute of Phys., P. J. ˇSaf´ arik University, Park Angelinum 9, 040 01 Koˇ sice, Slov ak Republic\n3Institute of Molecular Physics, Polish Academy of Sciences Smoluchowskiego 17, 60-179 Pozna´ n, Poland\n(Dated: October 22, 2018)\nSpin-transfer torque and current induced spin dynamics in s pin-valve nanopillars with the free\nmagnetic layer located between two magnetic films of fixed mag netic moments is considered theo-\nretically. The spin-transfer torque in the limit of diffusiv e spin transport is calculated as a function\nof magnetic configuration. It is shown that non-collinear ma gnetic configuration of the outermost\nmagnetic layers has a strong influence on the spin torque and s pin dynamics of the central free\nlayer. Employing macrospin simulations we make some predic tions on the free layer spin dynamics\nin spin valves composed of various magnetic layers. We also p resent a formula for critical current\nin non-collinear magnetic configurations, which shows that the magnitude of critical current can be\nseveral times smaller than that in typical single spin valve s.\nPACS numbers:\nI. INTRODUCTION\nMagnetic multilayers are ones of the most relevant ele-\nments in the developmentofcutting-edgetechnologies. If\ncurrent flows along the axis of a metallic hybrid nanopil-\nlar structure, a spin accumulation builds up in the vicin-\nity of the normal-metal/ferromagnet interface. More-\nover, the current produces spin-transfer torque (STT)\nwhich acts on magnetic moments of the ferromagnetic\nlayers1,2. As a consequence, magnetization of a particu-\nlar layer may be driven to an oscillation mode3, or can\nbe switched between possible stable states4. However,\ncurrent-induced control of magnetic moments requires a\nrather high current density. Therefore, reduction of the\ncritical current density remains the most challenging re-\nquirementfrom the pointofviewofpossibleapplications.\nSince the critical currents are related to STT, which in\nthe diffusive transport regime is proportional to spin ac-\ncumulation at the normal-metal/ferromagnet interface,\none may alternatively rise a question of possible ways to\nenhance the spin accumulation.\nOne of the possibilities to decrease the critical current\ndensity in metallic structures is the geometry proposed\nby L. Berger5, in which the free magnetic layer is located\nbetween two pinned magnetic layers of opposite magne-\ntizations. Indeed, in such a dual spin valve (DSV) geom-\netry both interfaces of the free layer can generate STT,\nand this may decrease the critical current several times5.\nTo examine the influence of an additional magnetic\nlayer on the free layer’s spin dynamics one needs to find\nfirst the STT acting on the central layer for arbitrary di-\nrectionofitsmagnetizationvector. Todothis, weemploy\nthe spin-dependent diffusive transport approach6, based\non Valet-Fert description7. In this paper we present a\ncomprehensive survey of STT-induced effects in DSVs\nfor generally non-collinear magnetic configuration of the\noutermost magnetic films. We also study asymmetric\nexchange-biased DSV, in which magnetic moment of one\nof the outer magnetic layers is fixed to an antiferromag-netic layer due to exchange anisotropy. We show that a\nnon-standard (wavy-like) angular dependence of STT –\noriginally predicted for asymmetric spin valves6,8– can\nalso occur in DSV geometry. Such a non-standard an-\ngular dependence of the torque is of some importance\nfrom the application point of view, as it allows to induce\nsteady-state precessional modes without external mag-\nnetic field8,10. Furthermore, we examine current-induced\nspin dynamics within the macrospin model, and derive a\nformulaforcriticalcurrentswhich destabilizetrivialfixed\npointsofthecentralspin’sdynamicsinnon-collinearcon-\nfigurations of the outermost magnetizations.\nIn section II we describe the model and introduce ba-\nsic formula for STT and spin dynamics in DSVs. Then,\nin section III we study dynamics in a symmetric DSV\ngeometry, where we compare STT and spin dynamics in\nDSVs and single spin valves (SSVs). Finally, in section\nIV we analyze an exchange-biased DSV structure with\nnoncollinear magnetic moments of the outermost mag-\nnetic layers. Finally, we conclude in section V.\nII. MODEL\nWe consider a multilayer nanopillar structure,\nFL/NL/FC/NR/FR,consistingofthreeferromagnetic(F)\nlayers separated by normal-metal (N) layers; see Fig. 1.\nSpin moment of the central layer, F C, is free to rotate,\nwhile the right F Rand left F Lferromagnetic layers are\nmuch thicker and their net spin moments are assumed to\nbe fixed for current densities of interest. Fixing of these\nmoments can be achieved either by sufficiently strong\ncoercieve fields, or by exchange anisotropy at interfaces\nwith antiferromagnetic materials.\nIn the Landau-Lifshitz-Gilbert phenomenological de-\nscription, the dynamics of a unit vector along the net\nspin moment ˆsof the central (free) magnetic layer is de-2\nFIG. 1: Schematic of a dual spin valve with fixed magnetic\nmoments of the outer magnetic layers, F Land F R, and free\ncentral magnetic layer, F C, separated by normal-metal layers\nNLand N R. Here, ˆSL,ˆSR, andˆsare unit vectors along the\nspin moments of the F L, FR, and F Clayers, respectively.\nscribed by the equation\ndˆs\ndt+αˆs×dˆs\ndt=Γ, (1)\nwhereαis the Gilbert damping parameter. The right-\nhandsiderepresentsthetorquesduetoeffectivemagnetic\nfield and spin-transfer,\nΓ=−|γg|µ0ˆs×Heff+|γg|\nMsdτ, (2)\nwhereγgis the gyromagneticratio, µ0is the vacuum per-\nmeability, Msstands for the saturation magnetization of\nthe free layer, and ddenotes thickness of the free layer.\nConsidering the thin ferromagnetic free layer of ellipti-\ncal cross-section, the effective magnetic field Heffcan be\nwritten as\nHeff=−Hextˆez−Hani(ˆs·ˆez)ˆez+Hdem,(3)\nand includes applied external magnetic field ( Hext), uni-\naxial anisotropy field ( Hani), and the demagnetization\nfield (Hdem), where ˆezis the unit vector along the z-axis\n(easy axis), compare scheme in Fig. 1. The demagnetiza-\ntion field can be written in the form Hdem= (ˆs·¯N)Ms=\n(Hdxsx,Hdysy,Hdzsz), where ¯Nis a diagonal demagne-\ntization tensor.\nSpin-transfer torque\nGenerally, STT acting on a magnetic layer is deter-\nmined by the electron spin angular momentum absorbedfrom conduction electrons within a few interfacial atomic\nlayersof the ferromagnet11. Thus, the STT acting on the\ncentral layer F Ccan be calculated as τ= (/planckover2pi1/2)(jL\n⊥−jR\n⊥),\nwherejL\n⊥andjR\n⊥are the spin current components per-\npendicular to magnetic moment of the free layer and\ncalculated at the corresponding left and right normal-\nmetal/ferromagnet interfaces. The torque consists of in-\nplaneτ/bardbl(in the plane formed by magnetic moments of\nthe two interacting films) and out-of-plane τ⊥(normal to\nthis plane) parts, τ=τ/bardbl+τ⊥, which have the following\nforms\nτ/bardbl=Iˆs×/bracketleftBig\nˆs×/parenleftBig\naLˆSL−aRˆSR/parenrightBig/bracketrightBig\n,(4a)\nτ⊥=Iˆs×/parenleftBig\nbLˆSL−bRˆSR/parenrightBig\n, (4b)\nwhereIis the current density, and ˆSLandˆSRare the\nunit vectors pointing along the fixed net-spins of the F L\nand FRlayers, respectively. The parameters aL,aR,bL,\nandbRdepend, generally, on the magnetic configuration\nand material composition of the system, and have been\ncalculated in the diffusive transport limit6. According to\nfirst-principles calculations of the mixing conductance12,\nthe out-of-plane torque in metallic structures is about\ntwoordersofmagnitudesmallerthanthein-planetorque.\nAlthough this component has a minor influence on crit-\nical currents, it may influence dynamical regimes, so we\ninclude it for completeness in our numerical calculations.\nLet us consider now how the STT affects spin\ndynamics of the free magnetic layer. We rewrite\nEq.(1) in spherical coordinates ( φ,θ), which obey ˆs=\n(cosφsinθ,sinφsinθ,cosθ); see Fig. 1. Defining unit\nbase vectors, ˆeφ= (ˆez׈s)/sinθandˆeθ=ˆeφ׈s, Eq.(1)\ncan be rewritten as\nd\ndt/parenleftbigg\nφ\nθ/parenrightbigg\n=1\n1+α2/parenleftbigg\nsin−1θ−αsin−1θ\nα 1/parenrightbigg/parenleftbigg\nvφ\nvθ/parenrightbigg\n,(5)\nwhere the overall torques vθandvφ, changing the angles\nθandφ, respectively, read\nvθ=Γ·ˆeθ=−|γg|µ0(Hdx−Hdy)cosφsinφsinθ+|γg|\nMsdτθ, (6a)\nvφ=Γ·ˆeφ=−|γg|µ0/bracketleftbig\nHext+(Hani+Hdxcos2φ+Hdysin2φ−Hdz)cosθ/bracketrightbig\nsinθ+|γg|\nMsdτφ. (6b)\nThe first terms in Eqs. (6) describe the torques due to demagnetiz ation and anisotropy fields of the free layer,3\nwhileτθ=τ/bardbl\nθ+τ⊥\nθandτφ=τ/bardbl\nφ+τ⊥\nφarethecorresponding\ncomponents of the current-induced torque, which origi-\nnate from τ/bardblandτ⊥.\nAs we have already mentioned above, the main con-\ntribution to STT comes from τ/bardbl. Assuming now that\nmagnetic moment of the left magnetic layer is fixed\nalong the z-axis,ˆSL= (0,0,1), and magnetic moment\nof the right magnetic layer is rotated by an angle Ω from\nthez-axis and fixed in the layer’s plane (see Fig. 1),\nˆSR= (0,sinΩ,cosΩ), one finds\nτ/bardbl\nθ= (aL−aRcosΩ)Isinθ+aRIsinΩsinφcosθ,(7a)\nτ/bardbl\nφ=aRIcosφsinΩ. (7b)\nThe component τ/bardbl\nθconsists of two terms. The first one is\nanalogous to the term which describes STT in a SSV.\nHowever, its amplitude is now modulated due to the\npresence of F R. In turn, the second term in Eq. (7a)\nis nonzero only in noncollinear magnetic configurations.\nFrom Eq. (7b) follows that τ/bardbl\nφis also nonzero in non-\ncollinear configurations, and only if the magnetization\npoints out of the layer’s plane ( φ/negationslash=π/2). When mag-\nnetic moments of the outer magnetic layers are parallel\n(Ω = 0), τ/bardbl\nθ= (aL−aR)Isinθ. For a symmetric DSV,\naL(θ) =aR(θ), and hence STT acting on ˆsvanishes. On\nthe other hand, in the antiparallel configuration of ˆSL\nandˆSR(Ω =π), the maximal spin torque enhancement\ncan be achieved, τ/bardbl\nθ= (aL+aR)Isinθ.\nSimilar analysis of τ⊥leads to the following formulas\nforτ⊥\nθandτ⊥\nφ:\nτ⊥\nθ=bRIcosφsinΩ, (8a)\nτ⊥\nφ=−(bL−bRcosΩ)Isinθ+bRcosθsinφsinΩ.(8b)\nThus, if the outer magnetic moments are collinear, τ⊥\nθ=\n0, whileτ⊥\nφreduces to τ⊥\nφ=−(bL−bR)Isinθfor Ω = 0,\nandτ⊥\nφ=−(bL+bR)Isinθfor Ω =π. Hence, in symmet-\nric spin valves, where bL=bR,τ⊥\nφvanishes in the parallel\nconfiguration of the outermost magnetic moments and is\nenhanced in the antiparallel configuration.\nIn the following sections we investigate STT and its\neffects on critical current and spin dynamics. We start\nfrom symmetric spin valves in antiparallel magnetic con-\nfiguration (Ω = π). Then, we proceed with asymmetric\nexchange-biased dual spin valves.\nIII. SYMMETRIC DSVS\nLet us consider first symmetric DSVs with antiparallel\norientation of magnetic moments of the outermost ferro-\nmagneticfilms: ˆSL= (0,0,1)andˆSR= (0,0,−1). Asin-\ndicatedbyEqs.(7)and(8), suchaconfigurationmaylead\nto enhancement of STT in comparison to that in SSVs.\nThus, let us analyze first STT in two types of structures:\nthe double spin valve F(20)/Cu(10)/F(8)/Cu(10)/F(20)in the antiparallel configuration, and the corresponding\nsingle spin valve F(20)/Cu(10)/F(8). The numbers in\nbrackets correspond to layer thicknesses in nanometers.\nIn Fig. 2 we show the angular dependence of STT for\nDSVs and SSVs, when the vector ˆschanges its orien-\ntation, described by angle θ, in the layer plane ( φ=\nπ/2). The magnetic layers made of Permalloy, Ni 80Fe20\n(Fig. 2a), and of Cobalt (Fig. 2b) are considered. Due\nto the additional fixed layer (F R), STT in DSVs is\nabout twice as large as in SSVs, which is consistent\nwith Berger’s predictions2. Additionally, the angular de-\npendence of STT acting on the free layer in Co/Cu/Co\nspin valves is more asymmetric than in Py/Cu/Py. This\nasymmetry, however, disappears in Co/Cu/Co/Cu/Co\nDSVsduetosuperpositionofthecontributionsfromboth\nfixed magnetic layers to the STT.\n 0 0.1 0.2 0.3 0.4\n 0 0.25 0.5 0.75 1 \nθ / πτθ(a)\n 0 0.25 0.5 0.75 100.050.10.150.2 \nθ / π(b)\nFIG. 2: Spin transfer torque τθin symmetric DSVs,\nF(20)/Cu(10)/F(8)/Cu(10)/F(20), in the antiparallel con -\nfiguration, Ω = π, (solid lines), and STT in SSVs,\nF(20)/Cu(10)/F(8) (dashed lines), where (a) F = Permal-\nloy, (b) F = Cobalt. STT is shown in the units of /planckover2pi1|I|/|e|,\nand calculated for φ=π/2. The material parameters as in\nRef.13.\nThe enhancement of STT in dual spin valves may lead\nto two important improvements of the spin dynamics:\nreductionofthecriticalcurrentneededtotriggerthespin\ndynamics, and decrease of the switching time. The latter\nis defined as the time needed to switch the magnetization\nfrom one stable position to the opposite one. The fixed\npoints of the dynamics of ˆsare given by the equations\nvθ= 0 and vφ= 0. If Ω = π, they are satisfied for θ= 0,\nandθ=π. Employing the ’zero-trace’stability condition\nof the linearized Landau-Lifshitz-Gilbert equation14, we\nfind the critical current destabilizing the initial ( θ= 0)\nstate in the form\nI0\nc,DSV=αµ0Msd\na0\nL+a0\nR/parenleftbigg\nHext+Hani+Hdx+Hdy\n2−Hdz/parenrightbigg\n,\n(9)\nwherea0\nLanda0\nRare calculated for θ→0. Addition-\nally, we have omitted here the terms resulting from τ⊥\nbecause of their small contribution to the critical cur-\nrent. Equation (9) is analogous to the expression for\ncritical current in SSV13. According to our calculations,\nthe critical current in DSVs with F = Cobalt is 6-times\nsmaller than in SSVs, as reported by Berger5. However,\nif F = Permalloy, the introduction of a second fixed mag-\nnetic layer reduces the critical current only by a factor\nof 2. This difference arises from the dependence of spin4\naccumulation on spin-flip length which is about 10-times\nlonger in Cobalt than in Permalloy6.\nTo estimate the switching time in DSVs as well as in\nSSVs we employ the single-domain macrospin approx-\nimation to the central layer. Equations (1), including\nEq. (3) and Eq. (4), completely describe the dynamics\nof central layer’s spin, ˆs. In our simulations we assumed\na constant current of density I. The positive current\n(I >0) is defined for electrons flowing from F Rtowards\nFL(current then flows from F Ltowards F R); opposite\ncurrent is negative. Apart from this, the demagnetiza-\ntion field of the free layer of elliptical cross-section with\nthe axes’ lengths 130nm and 60nm has been assumed,\nwhile external magnetic field was excluded, Hext= 0.\nFor each value of the current density, from Eq. (1) we\nfound evolution of ˆsstarting from an initial state until ˆs\nswitched to the opposite state (provided such a switch-\ning was admitted). In all simulation, the spin was ini-\ntiallyslightlytiltedinthelayerplanefromtheorientation\nˆs= (0,0,1), assuming θ0= 1◦andφ0=π/2. A success-\nful switching, with the switching time ts, is counted when\nsz(ts)<−0.99, where sz(t) is the exponentially weighted\nmoving average15,sz(t) =η sz(t) + (1−η)sz(t−∆t),\n∆tis the integration step, and the weighting parame-\nterη= 0.1. The moving average szis calculated only\nin time intervals, where sz(t) remains continuously be-\nlow the value of −0.9; otherwise sz(t) =sz(t). Fig. 3\ncompares the switching times in DSVs and in the corre-\nsponding SSVs. In both cases shown in Fig. 3, a con-\nsiderable reduction of the switching time is observed in\nDSVs. Similarly as for the critical current, the reduction\nof current required for switching in DSVs with Cobalt\nlayers is larger from that in DSVs with Permalloy layers.\n02468\n01234tsw [ns]\nI / I0(a)\n     \n0.51.52.53.5 \nI / I0(b)\nFIG. 3: Switching time in DSVs\nF(20)/Cu(10)/F(8)/Cu(10)/F(20) in the antiparallel mag-\nnetic configuration, Ω = π, (solid lines), and in SSVs\nF(20)/Cu(10)/F(8) (dashed lines), where (a) F = Permalloy,\n(b) F = Cobalt. The switching time is shown as a function\nof the normalized current density I/I0, withI0= 108Acm−2.\nThe other parameters as in Fig.2.\nIV. EXCHANGE-BIASED DSV\nLet us consider now an asymmetric exchange-biased\nDSV structure with an antiferromagnetic layer IrMn ad-\njacent to the F Rlayer in order to pin its magneticmoment in a required orientation, i.e. the structure\nCo(20)/Cu(10)/Py(4)/Cu(4)/Co(10)/IrMn(8). The left\nmagnetic layer, F L= Co(20), is assumed to be thick\nenough so its magnetic moment is fixed, ˆSL=ˆez. In\nturn, magnetic moment of the right ferromagnetic layer,\nFR= Co(10), is fixed in the layer plane at a certain angle\nΩ with respect to ˆSLdue to the exchange-bias coupling\nto IrMn.\nFrom Eqs. (6), (7), and (8) follows, that in a general\ncase (Ω/negationslash= 0), the points θ= 0 and θ=πare no more\nsolutions of the conditions for fixed points, vθ= 0 and\nvφ= 0,becauseoftheadditionaltermsinSTT,whichap-\npear in non-collinear situations (discussed in section II).\nThese additional terms lead to a nontrivial θ-dependence\nof STT, and to a shift of the fixed points out of the\ncollinear positions.\nTo analyze this effect in more details, let us consider\nfirst STT assuming ˆsin the layer plane ( φ=π/2). Ac-\ncording to Eq. (7b) and Eq. (8b), and to the fact that the\nparameters bare much smaller than a, the component τφ\nis verysmall. In Fig. 4(a) we show the secondcomponent\nof the torque, namely τθ, as a function of the angle θand\nfor different values of the angle Ω. The configurations\nwhereτθ= 0 are presented by the contour in the base\nplane of Fig. 4(a).\nIn the whole range of the angle Ω, two ’trivial’ zero\npoints are present. Additionally, for small angles close\nto Ω = 0, two additional zero points occur. The ap-\npearance of these additional zero points closely resem-\nbles non-standard wavy-like STT angular dependence,\nwhich has been already reported in single spin valves\nwith magnetic layers made of different materials6,8,16.\nSuch a θ-dependence dictates that both zero-current\nfixed points ( θ= 0,π) become simultaneously stabi-\nlized (destabilized) for positive (negative) current. This\nbehavior is of special importance for stabilization of\nthe collinear configurations, and for microwave genera-\ntion driven by current only (without external magnetic\nfield)6,8,17. We note, that in contrast to SSVs, the wavy-\nlikeθ-dependence in exchange-biased DSVs is related\nrather to asymmetric geometry of the multilayer than\nto bulk and interface spin asymmetries. This trend is\ndepicted in Fig. 4(b), where we show variation of STT\nfor different thicknesses of F Rat Ω = 0. The wavy-like\ntorque angular dependence appears for the thickness of\nFRmarkedly different from that of F L.\nMaking use of the above described STT calculations,\nextended to arbitrary orientation of ˆs, we performed nu-\nmerical simulations of spin dynamics induced by a con-\nstant current in zero external magnetic field ( Hext= 0).\nThe sample cross-section was assumed in the form of an\nellipse with the axes’ lengths 130nm and 60nm. As be-\nfore, the numerical analysis has been performed within\nthe macrospin framework, integrating equation of mo-\ntion, Eq.(1). Inthe simulationweanalyzedthelong-term\ncurrent-induced spin dynamics started from the initial\nstate corresponding to θ0= 1◦,φ0=π/2.\nAs one might expect, the current-induced spin dynam-5\nθ / π\nΩ / π 0\n 0.5\n 1\n 1.5\n 2 0 0.25 0.5 0.75 1(a)\nθ / π\nΩ / π-0.2-0.1 0 0.1 0.2\nτθ\n 0\n 0.5\n 1\n 1.5\n 2 0 0.25 0.5 0.75 1τθ(a)-0.04-0.02 0 0.02\n 0 0.25 0.5 0.75 1 \nθ / πτθ4 nm\n6 nm\n8 nm\n20 nm(b)\nFIG. 4: (a) STT acting on the central magnetic layer\nin Co(20)/Cu(10)/Py(4)/Cu(4)/Co(10)/IrMn(8) exchange-\nbiased DSV as a function of the angle θ, calculated for\nΩ =kπ/4, k= 0,1,2,3,4. The contour plot in the base\nplane corresponds to zeros of τθ. (b) Wavy-like STT angular\ndependence in exchange biased DSV for Ω = 0, calculated for\ndifferent thicknesses of the F Rlayer. STT is shown in the\nunits of/planckover2pi1|I|/|e|. The other parameters as in Fig.2.\nics of the free layer depends on the angle Ω, current den-\nsityI, and on current direction. To designate different\nregimes of STT-induced spin dynamics, we constructed\na dynamical phase diagram as a function of current and\nthe angle Ω. The diagram shows the average value /angbracketleftsz/angbracketright\nof thezcomponent of the free layer net-spin in a sta-\nble dynamical regime (see Fig. 5a), as well as its disper-\nsionD(sz) =/radicalBig\n/angbracketlefts2z/angbracketright−/angbracketleftsz/angbracketright2(see Fig. 5b). The average\nvalue/angbracketleftsz/angbracketrightprovides an information on the spin orienta-\ntion, whereas the dispersion distinguishes between static\nstates (for which D= 0) and steady precessional regimes\n(whereD >0), in which the zcomponent is involved.\nFor each point in the diagram, a separate run from the\ninitial biased state φ0=π/2 andθ0= 1◦was performed.\nIn the/angbracketleftsz/angbracketrightdiagram, Fig. 5(a), one can distinguish three\nspecific regions. Region (i) covers parameters for which\na weak dynamics is induced only: ˆsfinishes in the equi-\nlibrium stable point which is very close to ˆs= (0,0,1).\nAs the angle Ω increases, STT becomes strong enough to\ncauseswitching,asobservedintheregion(ii). Thehigher\nthe angle Ω, the smaller is the critical current needed for\ndestabilization of the initial state. For smaller Ω, the\ncurrent-induced dynamics occurs for currents flowing in\nthe opposite direction, see the region (iii). This behav-\nior is caused by different sign of STT in the initial state.\nFrom/angbracketleftsz/angbracketrightwe conclude, that neither of the previously\nmentioned stable states is reached. To elucidate the dy-\nnamics in region (iii), we constructed the corresponding\nmap ofD(sz), Fig. 5(b). This map reveals three different\nmodes of current-induced dynamics. For small current\namplitudes, in-plane precession (IPP) around the initial\nstable position is observed. The precessional angle rises 0 0.25 0.5 0.75  1\n -3-2-1 0 1 2 3I / I0\n-1-0.5 0 0.5 1\nsz average(a)\n(i) (ii)\n(iii)\n00.10.20.30.40.5\nΩ / π-3-2-10I / I0\n 0 0.2 0.4 0.6 0.8\nsz dispersion(b)\n(c)\n(d)OPP\nSSIPP\n-1-0.500.51-1-0.500.510.50.751sz\nsxsyszIPP\n-0.55-0.5-0.45-1-0.500.51 -1-0.500.51sz\nsxsysz OPP\n29.0529.0629.0729.0829.09\n0510R [fΩ m2]\ntime [ns]29.129.229.329.4\n0 5 10 \ntime [ns]\nFIG. 5: (Color online) Dynamical phase diagram for\nCo(20)/Cu(10)/Py(4)/Cu(4)/Co(10)/IrMn(8) exchange-\nbiased double spin valve as a function of the angle Ω and\nnormalized current density I/I0(withI0= 108Acm−2): (a)\naverage value of the szspin component; (b) dispersion of\ntheszspin component; (c) typical precessional orbits; (d)\nresistance oscillations associated with the IPP for Ω = 0 .4π\nandI=−1.3I0(left part), and with the OPP for Ω = 0 .1π\nandI=−1.3I0(right part). The other parameters as in\nFig.3.\nwith increasing current amplitude. Above a certain criti-\ncalvalue of I, the precessionsturn toout-of-planepreces-\nsions (OPPs). In a certain range of Ω, the OPPs collapse\nto a static state (SS), where the spin ˆsremains in an\nout-of-plane position close to ±ˆex.\nAsˆSRdeparts from the collinear orientation, the crit-\nical current needed for destabilization of the initial state\nincreases. This growth is mostly pronounced close to\nΩ =π/2. To describe the critical current, we analyzed\nEq. (5) with respect to the stability of ˆsin the upper po-6\nsition. Assuming that even in noncollinear configuration\nthe stable position of ˆsis close to θ= 0, we have lin-\nearized Eq. (5) around this point for arbitrary Ω. Then,\nfor the critical current needed for destabilization of the\nconsidered stable state we find\nI0\nc,EBDSV≃αµ0Msd/parenleftBig\nHani+Hdx+Hdy\n2−Hdz/parenrightBig\na0\nL(Ω)−a0\nR(Ω)cosΩ,(10)\nwherea0\nL(Ω) and a0\nR(Ω) are calculated (for each config-\nuration) assuming θ→0. Comparison of Eq. (10) with\nthe results of numerical simulations is shown in Fig. 5.\nWhen considering the opposite stable point as the initial\nstate, we need to take aLandaRforθ→π. Clearly, to\ndestabilize the θ=πstate one needs current of opposite\ndirection.\nCurrent-induced oscillations are usually observed ex-\nperimentally viathe magnetoresistance effect3. When\nelectric current is constant, then magnetic oscillations\ncause the corresponding resistance oscillations, which in\nturn lead to voltage oscillations. The later are measured\ndirectly in experiments. In Fig. 5(d) we show oscilla-\ntions in the system resistance associated with the IPP\n(left) and OPP (right). As the amplitude of the oscilla-\ntions correspondingto the OPP mode is sufficiently large\nto be measured experimentally, the amplitude associated\nwith the IPP mode is relatively small. This is the reason,\nwhy IPP mode is usually not seen in experiments.\nV. DISCUSSION AND CONCLUSIONS\nWehavecalculatedSTTinmetallicdualspinvalvesfor\narbitrary magnetic configuration of the system, but with\nmagnetic moments of the outer magnetic films fixed in\ntheirplaneseitherbylargecoercievefieldsorbyexchange\nanisotropy. In the case of symmetric DSV structures, we\nfound a considerableenhancement ofSTT in the antipar-\nallel magnetic configuration. This torque enhancement\nleads to reduction of the critical current for switching as\nwell as to reduction of the switching time. The switch-ing improvement has been found to be dependent on the\nspin-flip lengths in the magnetic and nonmagnetic lay-\ners. According to our numerical simulations, an ultrafast\nsubnanosecond current-induced switching processes can\noccurin DSVs with antiparallelmagnetic momentsofthe\noutermost magnetic films.\nIn exchange-biased spin valves we have identified con-\nditions which can lead to various types of spin dynam-\nics. For Ω /lessorsimilarπ/2, the negative current excites the central\nmagneticlayer,whileforΩ /greaterorsimilarπ/2,oppositecurrentdirec-\ntion is needed. We have evaluated parameters for which\nswitchingtoanewstablestateortoaprecessionalregime\nappears. This is especially interesting from the applica-\ntion point of view. However, the static state (SS) should\nbe treated carefully. Such a state is often connected with\nthewavy-like angular dependence of STT13,16,17. The\nSS becomes stable in the framework of macrospin model,\nwhen STT disappearsin acertain noncollinearconfigura-\ntion. However, experimentally such a state has not been\nobserved so far16. It has been shown in more realistic\nmicromagnetic approach, that this static state may cor-\nrespond to out-of-plane precessions18. The reason of this\nis the finite strength of exchange coupling in magnetic\nfilm, which does not fully comply with the macrospin\napproximation in some cases.\nFinally, we have derived an approximate formula for\nthe critical current in exchange biased DSVs, valid for\nnon-collinear magnetic configurations. This formula,\nEq.(10), is in good agreement with the numerical sim-\nulations.\nAcknowledgment\nThe work has been supported by the the EU\nthrough the Marie Curie Training network SPIN-\nSWITCH (MRTN-CT-2006-035327). M.G. alsoacknowl-\nedges support VEGA under Grant No. 1/0128/08, and\nJ.B. acknowledges support from the Ministry of Science\nandHigherEducationasaresearchprojectinyears2006-\n2009.\n∗Electronic address: balaz@amu.edu.pl\n1J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996).\n2L. Berger, Phys. Rev. B 54, 9353 (1996).\n3S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Emley,\nR. J. Schoelkopf, R. A. Buhrman, and D. C. Ralph, Nature\n425, 380 (2003).\n4J. A. Katine, F. J. Albert, R. A. Buhrman, E. B. Myers,\nand D. C. Ralph, Phys. Rev. Lett. 84, 3149 (2000).\n5L. Berger, J. Appl. Phys. 93, 7693 (2003).\n6J. Barna´ s, A. Fert, M. Gmitra, I. Weymann, and\nV. K. Dugaev, Phys. Rev. B 72, 024426 (2005).\n7T. Valet and A. Fert, Phys. Rev. B 48, 7099 (1993).\n8M. Gmitra and J. Barna´ s, Phys. Rev. Lett. 96, 207205\n(2006).9A. A. Kovalev, A. Brataas, and G. E. W. Bauer, Phys.\nRev. B66, 224424 (2002).\n10O. Boulle, V. Cros, J. Grollier, L. G. Pereira, C. Deranlot,\nF. Petroff, G. Faini, J. Barna´ s, and A. Fert, Nature Phys.\n3, 492 (2007).\n11M. D. Stiles and A. Zangwill, Phys. Rev. B 66, 014407\n(2002).\n12K. Xia, P. J. Kelly, G. E. W. Bauer, A. Brataas, and\nI. Turek, Phys. Rev. B 65, 220401(R) (2002).\n13M. Gmitra and J. Barna´ s, Appl. Phys. Lett. 89, 223121\n(2006).\n14S. Wiggins, Introduction to Applied Nonlinear dynamical\nSysystems and Chaos (Springer-Verlag, 1990).\n15S. W. Roberts, Technometrics 42, 97 (2000).7\n16O. Boulle, V. Cros, J. Grollier, L. G. Pereira, C. Deranlot,\nF. Petroff, G. Faini, J. Barna´ s, and A. Fert, Phys. Rev. B\n77, 174403 (2008).\n17P. Balaz, M. Gmitra, and J. Barna´ s, Phys. Rev. B 79,144301 (2009).\n18E. Jaromirska, P. Bal´ aˇ z, L. L´ opez D´ ıaz, J. Barna´ s, to be\npublished."    },    {        "title": "1309.0306v1.Temperature_Dependence_of_Spin_Hall_Angle_of_Palladium.pdf",        "content": "1  Temperature Dependence of Spin Hall Angle of\t\r Palladium  Zhenyao Tang1, Yuta Kitamura1, Eiji Shikoh2, Yuichiro Ando1, Teruya Shinjo1, and Masashi Shiraishi1,* 1. Graduate School of Engineering Science, Osaka University, Toyonaka 560-8531, Japan 2. Graduate School of Engineering, Osaka City University, Osaka 558-8585, Japan *E-mail address: shiraishi@ee.es.osaka-u.ac.jp  Abstract   In this study, the temperature dependence of the spin Hall angle of palladium (Pd) was experimentally investigated by spin pumping. A Ni80Fe20/Pd bilayer thin film was prepared, and a pure spin current was dynamically injected into the Pd layer. This caused the conversion of the spin current to a charge current owing to the inverse spin Hall effect. It was found that the spin Hall angle varies as a function of temperature, whereby the value of the spin Hall angle increases to ca. 0.02 at 123 K.             2       In recent years, spincurrentronics has been attracting significant attention for potential future applications because of the low energy consumption of spin current. 1) Spin current can be generated by inducing spins into spin sink materials via spin pumping. 2–5) The spin current can then be detected by utilizing the inverse spin Hall effect (ISHE), 2,3,6) which can convert a pure spin current to a charge current by spin–orbit interaction. Many studies have applied this method in order to evaluate the spin Hall angle (θSHE ), 7) to inject spins into condensed matter, 8,9) and estimate the spin diffusion length and spin relaxation time, 10,11) but most of these studies have been conducted at room temperature (RT). The recent success of dynamical spin injection and transport in Si at room temperature10) allows the establishment of spin-based logic by using a dynamical method since Si is one of the most suitable materials for beyond CMOS spin devices. In the spin logic using the dynamical method, readout of a spin current is realized by the ISHE. In the case of Si, Pt is not suitable for the “spin-charge converter”, because a Pt-silicide alloy can be easily formed, which impedes the conversion. Hence, Pd is a better material for the converter. For future investigations of Si-based spin logic using the dynamical method, the temperature dependence of the logic device performance is also quite important and the temperature dependence of θSHE, as a good index of spin-charge conversion, should be investigated. Herein, we report the temperature dependence of the spin Hall angle of palladium (Pd), which is a commonly used spin sink material exhibiting the ISHE. The spin Hall angle of Pd was qualitatively evaluated by changing the temperature from 130 K to RT. This evaluation yields an important spin-related physical parameter that can be investigated in future studies by applying the spin pumping method.      Figure 1(a) shows the schematic illustration of a Ni80Fe20(Py)/Pd bilayer sample. A 25-nm-thick permalloy (Py) film and a 5-nm-thick Pd film were prepared on an oxidized silicon substrate by electron beam evaporation. Both Py and Pd layers were rectangular with an area of 2 × 1 mm2. The Pd layer was connected to the positive and negative ends of a nano-voltmeter as shown in Fig. 1. Spins from the ferromagnetic Py were injected into the Pd 3  layer by the dynamical spin injection method. The sample system was placed at the center of a TE011 microwave cavity in an electron spin resonance (ESR) instrument with a frequency (f) of 9.12 GHz. An external magnetic field H was applied to the Py/Pd bilayer at an angle of θH, as shown in Fig. 1(a). The ferromagnetic resonance (FMR) condition was determined using the following equation: 12,13) ,     (1) where ω = 2πf, γ, HFMR, and Ms are the gyromagnetic ratio of the ferromagnet, FMR field, and saturation magnetization of Py, respectively. Figure 1(b) shows the FMR spectra of Py with and without Pd. The linewidth of the spectrum corresponding to Py/Pd is larger than that corresponding to Py alone, which is attributed to the modulation of the Gilbert damping constant (α) owing to successful spin pumping into the Pd layer. The FMR spectra and output dc voltages of the Py/Pd bilayer at RT resulting from the ISHE in the Pd are shown in Figs. 1(c) and 1(d), respectively. The FMR intensities and resonance fields were nearly identical for all values of θH, and an electromotive force was induced when θH  was set to 0 and 180°, while the signal was flat at θH = 90°. Because this is consistent with the symmetry of the ISHE (σ×∝scJJ), the observed electromotive force was ascribed to the ISHE of Pd, which was due to a pure spin current generated dynamically by spin pumping at the Py/Pd interface. A theoretical fitting was then performed in order to separate the ISHE and anomalous Hall effect (AHE) signals by using the following equation. 14)  baHHHHHVHHVVFMRFMRAHEFMRISHE++Γ+−−Γ−+Γ+−Γ=22222)()(2)(,     (2) where VISHE is the electromotive force owing to the ISHE, VAHE is the output voltage resulting from the AHE, and H is the external static magnetic field. The value of HFMR was experimentally determined to be 89.51 mT at 0 and 180° at RT. The variables Γ, a, and b are the fitting parameters. As shown in Fig. 1(e), a theoretical fitting using eq. (2) )4()(2sFMRFMRMHHπγω+=4  effectively reproduced the experimental results, and VISHE was estimated to be 7.66 µV at RT.       Figure 2 shows the microwave power dependence of the electromotive force from the Pd of the Py/Pd sample. The electromotive force from the Pd layer VISHE was proportional to the microwave power. As shown in the inset of Fig. 2, the power dependence of VISHE was in good agreement with the theoretical prediction, 2) indicating that the generation of the observed electromotive force is attributed to the ISHE in Pd. More importantly, the unsaturated FMR spectra enabled the estimation of the spin Hall angle of Pd. A pure spin current was injected at the Py/Pd interface via spin pumping under the resonance condition, and the generated spins diffused into the Pd layer and were converted to a charge current (jc), as shown in Fig. 1(a). The spin current density js was theoretically calculated as 15),]4)4[(8]4)4(4[222222222ωγππαωγπγπγ+++=↑↓sssrsMMMhgj where h is the microwave magnetic field, and it was set to 0.16 mT at a microwave power of 200 mW. The variable gr!\"and the constant ! are the real part of the mixing conductance 14) and the Dirac constant, respectively. Note that ↑↓rgis given by ),(32/PyPdPyBPysrWWgdMg−=↑↓ωµγπ7, 15) where g,Bµ, dPy, WPy/Pd, and WPy are the g-factor, Bohr magneton, thickness of the Py layer, FMR spectral width of the Py/Pd film, and FMR spectral width of the Py film, respectively. In this study, dPy, WPy/Pd, and WPy were 25 nm, 3.03 mT, and 2.49 mT, respectively. The electromotive force due to the ISHE can then be expressed in the simplest form as follows: 7,15)  sPdPdPyPyPdPdPdSHEISHEjedddwV)2()2/tanh(σσλλθ+=,     (3) where w is the length of the Py layer defined as in Fig. 1(a), dPy and σPy are the thickness (25 nm) and electric conductivity (4.48\t\r × 106 Ω−1 m−1) of the Py layer at RT, and dPd and σPd are the thickness (5 nm) and electric conductivity (3.50 × 106 Ω−1 m−1) of the Pd layer, respectively. From the above calculations, the spin Hall angle θSHE was estimated to be 0.011 at RT, which is reasonably consistent with that obtained in a previous study. 7) 5       To estimate the spin Hall angle at various temperatures, the temperature evolution of the linewidth (W) of a simple Py film, resonance field (HFMR), and saturation magnetization (4πMs) were measured and evaluated as shown in Fig. 3(a), and the results were physically reasonable. Here, note that σPy, σPd, VISHE, and λPd changed with the temperature, and thus, the temperature dependences of σPy and σPd were also evaluated [Fig. 3(b)], and the estimated values of VISHE obtained from eq. (2) at different temperatures are shown in the inset of Fig. 3(c). The spin diffusion length of Pd, denoted as λPd, has been reported to be 9 nm at RT and 25 nm at 4.2 K, 16,17) so λPd can be estimated at each temperature by interpolating the values at RT and at 4.2 K (In fact, λPd is estimated to be 9.12 nm at 133 K by our calculation), since the spin lifetime is inversely proportional to temperature in this temperature region. The change in λPd yields little contribution to the change in θSHE. Thus, θSHE of Pd can be estimated by solving eq. (3) and using the conductivities of Pd and Py and the estimated spin diffusion length of Pd at each temperature. The results of this evaluation are shown in Fig. 3(d). It was found that θSHE increased to 0.020 monotonically as the temperature decreased to 130 K. It is surprising that the temperature evolution of θSHE of Pd exhibits opposite behavior compared with that of Pt18,19). Whereas Pt and Pd belong to the same group in the periodic table, the electron configurations are different, which may induce such difference 20). In fact, their electron configuration of the d-orbital allows a different sign of θSHE 21). The origin of this characteristic temperature dependence of θSHE of Pd is beyond the scope of this study, but will be investigated in the near future.      In conclusion, in this study, the temperature dependence of the spin Hall angle of Pd was estimated by spin pumping. This approach enables the quantitative estimation of the temperature dependence of the spin transport properties of Si, graphene, and other materials by combining spin pumping and ISHE.     6  References 1) F. J. Jedema, A. T. Filip, and B. J. van Wees: Nature (London) 410 (2001) 345. 2) Y. Tserkovnyak and A. Brataas: Phys. Rev. Lett. 88 (2002) 117601. 3) S. Mizukami, Y. Ando, and T. Miyazaki: Phys. Rev. B 66 (2002) 104413. 4) P. E. Tannenwaldet and M. H. Seavey: J. Phys. Radium 20 (1959) 323.  5) Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I. Halperin: Rev. Mod. Phys. 77 (2005) 1375. 6) A. Brataas, Y. Tserkovnyak, G. E. W. Bauer, and B. I. Halperin: Phys. Rev. B 66 (2002) 060404(R). 7) K. Ando and E. Saitoh: J. Appl. Phys. 108 (2010) 113925. 8) Y. Kitamura, E. Shikoh, Y. Ando, T. Shinjo, and M. Shiraishi: Sci. Rep. 3 (2013) 1739. 9) M. Koike, E. Shikoh, Y. Ando, T. Shinjo, S. Yamada, K. Hamaya, and M. Shiraishi: Appl. Phys. Express 6 (2013) 023001. 10) E. Shikoh, K. Ando, K. Kubo, E. Saitoh, T. Shinjo, and M. Shiraishi: Phys. Rev. Lett. 110 (2013) 127201. 11) Z. Tang, E. Shikoh, H. Ago, K. Kawahara, Y. Ando, T. Shinjo, and M. Shiraishi: Phys. Rev. B 87 (2013) 140401(R). 12) K. Ando, Y. Kajiwara, S. Takahashi, S. Maekawa, K. Takemoto, M. Takatsu, and E. Saitoh: Phys. Rev. B 78 (2008) 014413. 13) C. Kittel: Introduction to Solid State Physics, (Wiley, New York, 2005) 8th ed. 14) E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara: Appl. Phys. Lett. 88 (2006) 182509. 15) K. Ando, S. Takahashi, J. Ieda, Y. Kajiwara, H. Nakayama, T. Yoshino, K. Harii, Y. Fujikawa, M. Matsuo, S. Maekawa, and E. Saitoh: J. Appl. Phys. 109 (2011) 103913. 7  16) J. Foros, G. Woltersdorf, B. Heinrich and A. Brataas: J. Appl. Phys. 97 (2005) 10A714. 17) H. Kurt, R. Loloee, K. Eid, W. P. Pratt, and J. Bass: Appl. Phys. Lett. 81 (2002) 4787. 18) L. Liu, R. A. Buhrman, and D. C. Ralph: arXiv:1111.3702v3. 19) L. Vila, T. Kimura, and Y. Otani: Phys. Rev. Lett. 99 (2007) 226604. 20) G. Y. Guo, S. Murakami, T.-W. Chen, and N. Nagaosa: Phys. Rev. Lett. 100 (2008) 096401. 21) M. Morota, Y. Niimi, K. Ohnishi, D. H. Wei, T. Tanaka, H. Kontani, T. Kimura, and Y. Otani: Phys. Rev. B83 (2011) 174405.   8  Figure captions  Figure 1(a) Schematic illustration of the Py/Pd bilayer sample used in this study. H represents an external magnetic field and θH the angle of the external magnetic field. The dimensions of the sample are shown in the figure. (b) FMR spectra of the Py layer (red line) and the Py/Pd layer on the SiO2 substrate (black line). An increase in the line width, W, can be observed, which is attributed to the modulation of the Gilbert damping constant α, indicating successful spin injection into the Pd layer by spin pumping. (c) Magnetic field angle (θH) dependence of the FMR signal dI(H)/dH for the Py/Pd bilayer sample at room temperature (RT). (d) Magnetic field angle (θH) dependence of the electromotive force measured from the Py/Pd bilayer sample at RT. (e) External magnetic field dependence of the electromotive force VISHE for the Py/Pd bilayer film at 200 mW and at RT. Open circles represent the experimental data. The solid curve shows the fitting result, which was calculated using eq. (2). Orange and blue dashed lines show the contribution from ISHE and AHE, respectively.  Figure 2. Microwave power dependence of the electromotive force in the Pd film. A magnetic field was applied parallel to the film plane (θΗ = 0°). The output voltage increased with increasing microwave power. The inset shows the power dependence of the contribution of ISHE.  Figure 3(a). (upper panel) Temperature dependence of the FMR linewidth of the single Py film and the Pd/Py film. The inset shows the difference in the linewidth between the Py and Pd/Py films as a function of temperature. (lower panel) Ferromagnetic resonance field (HFMR) and saturation magnetization (4πMs). Saturation magnetization decreased with increasing temperature. An increase in the resonance field with increasing temperature can also be observed owing to the relation shown in eq. (1). (b) Temperature dependence of the resistivity of the Pd and Py 9  layers. Red and blue circles indicate the experimental data for Pd and Py, respectively. Red and blue lines show the linear fittings. (c) Temperature dependence of the output electromotive force measured for the Py/Pd sample. The insets show the temperature dependence of the contribution from ISHE (VISHE) and an expanded graph of the electromotive force. (d) Temperature dependence of the spin Hall angle θSHE of Pd. The spin Hall angle decreased with increasing temperature.                      10   Figure 1(a).                  \n(a) 11  Figure 1(b).              \n(b) \n-10010\nPyH-HFMR (mT)\n  dI(H)/dH (arb. unit)Py/Pd12    \n Figures 1(c) and (d)            \n(c) \n(d) \n-30030  θH=0°θH=90°θH=180°H-HFMR (mT)V (µV)5µV\n-40040θH=180°θH=90°  θH=0°dI(H)/dH (arb. unit)H-HFMR (mT)13     \nFigure 1(e)  \n(e) \n-40040   Exp. ISHE AHE Calc.V (µV)H-HFMR (mT)4 µV@RT14  Figure 2.               \u000e\u0015\u0011\u0011\u0015\u0011\u0011\u0012\u0011\u0011\u0013\u0011\u0011\u0011\u0015\u0019\n\u0001\u0001\u0001\u0013\u0011\u0011N8\u0001\u0012\u0016\u0011N8\u0001\u0012\u0011\u0011N8\u0001\u0016\u0011N8\u0001\u0012\u0011N8\u0001\u0011N87\u0001\tµ7\n)\u000e)'.3\u0001\tN5\n\u0016µ77*4)&\u0001\tµ7\n\u0001\u0001\u00017*4)&\u0001\u0001MJOFBS\u0001GJU1\u0001\tN8\n15   \n     Figure 3(a).   15020025030084868890  HFMR 4!Ms\nT (K)HFMR (mT)0.981.001.024!Ms (T)2.53.03.54.0\n   Py only Pd/PyW (mT)1502002503000.00.51.0!W (mT)\n  T (K)16              Figures 3(b) and (c)                 \n(b) \n(c) \u0012\u0017\u0011\u0013\u0015\u0011\u0014\u0013\u0011\u0011\u000f\u0012\u0017\u0011\u000f\u0013\u0011\u0011\u000f\u0013\u0015\u0011\u000f\u0013\u0019\n\u0001\u0001\u00011E\u00011Zρ\u0001\tµΩN\n5\u0001\t,\n17   \n   Figure 3(d)   1502002503000.0120.0160.020!SHE  !SHE\nT (K)\n(d) "    },    {        "title": "1710.10833v2.Probe_of_Spin_Dynamics_in_Superconducting_NbN_Thin_Films_via_Spin_Pumping.pdf",        "content": "1\n \n \nProbe of Spin Dynamics in Superconducting NbN Thin\n \nFilms via \n \nSpin Pumping\n \n \nYunyan\n \nYao\n1,2\n,\n \nQi\n \nSong\n1,2\n,\n \nYota\n \nTakamura\n3,4\n,\n \nJuan\n \nPedro\n \nCascales\n3\n,\n \nWei\n \nYuan\n1,2\n,\n \nYang\n \nMa\n1,2\n,\n \nYu\n \nYun\n1,2\n,\n \nX.\n \nC.\n \nXie\n1,2\n,\n \nJagadeesh\n \nS.\n \nMoodera\n3,5\n,\n \nand\n \nWei\n \nHan\n1,2*\n \n1\nInternational Center for Quantum Materials, School of Physics, Peking University, Beijing \n100871, China\n.\n \n2\nCollaborative Innovation Center of Quantum Matter, Beijing 100871, China\n.\n \n3\nPlasma Science and Fusion\n \nCenter and Francis Bitter Magnet Laboratory, Massachusetts \nInstitute of Technology, Cambridge, MA 02139, USA\n.\n \n4\nSchool of Engineering, Tokyo Institute of Technology, Tokyo 152\n-\n8550, Japan.\n \n5\nDepartment of Physics, Massachusetts Institute of Technology, Camb\nridge, MA 02139, USA\n \n*Correspondence to: \nweihan@pku.edu.cn (W.H.)\n \n \n \nAbstract\n:\n \nThe emerging field of superconductor (SC) spintronics has attracted \nintensive attentions recently\n. \nMany fantastic spin dependent properties in SC\ns\n \nhave been discovered, including large \nmagnetoresistance, long spin lifetimes and the giant spin Hall effect\n, \netc\n. Regarding the spin \ndynamic\ns\n \nin \ns\nuperconducting \nthin \nfilms, \nfew studies has been reported yet. \nHere, we report the \ninvestig\nation of the spin d\nynamics in a\n \ns\n-\nwave superconducting NbN film via spin pumping from \nan adjacent insulating ferromagnet GdN\n \nfilm\n. \nA\n \nprofound coherence peak of the Gilbert damping \nof GdN \nis observed slightly below the superconducting critical temperature of the NbN\n, \nwhich \nag\nrees well\n \nwith recent\n \ntheoretical \nprediction\n \nfor \ns\n-\nwave SCs \nin the presence of impurity spin\n-\norbit \nscattering\n. \nThis observation is also \na manifestation of the dynamic\n \nspin injection\n \ninto \nsuperconducting NbN thin film.\n \nOur results \ndemonstrate that spin pumping could be used \nt\no probe \nthe dynamic spin susceptibility\n \nof \nsuperconducting thin films, thus pave the \nway for future \ninvestigation of spin dynamics \nof\n \ninterfacial and \ntwo \ndimensional\n \ncrystalline \nSC\ns\n.\n \n 2\n \n \nI.\n \nINTRODUCTION\n \nThe \ninterplay between s\nuperconductivity and \nspintronics \nhas\n \nbeen intensively investigated \nin \nthe last decade\n \n[\n1\n-\n3\n]\n.\n \nIn the content of superconductivity, the \nferromagnetic \nmagnetization \nhas\n \nbeen found to \nplay\n \nan important role in the superconducting critical temperature (T\nC\n) in the \nferromagnet\n \n(FM)\n/\nsuperconductor (\nSC\n)\n \njunctions\n \n[\n4\n-\n9\n]\n.\n \nBesides, \nunusual\n \nspin\n-\npolarized\n \nsupercurrent \nhas been observed \nin \nferromagnetic \nJosephson junctions\n \n[\n2\n,\n3\n,\n10\n-\n13\n]\n. \nFurthermore, \nnon\n-\nabelian Majorana fermions have been proposed for \np\nx\n \n+ ip\ny\n \nsuperconductor surfaces driven \nby magnetic proximity effect, whic\nh has the potential for topological quantum computation \n[\n14\n]\n. \nIn the content of spintronics\n \n[\n15\n,\n16\n]\n, \nthe co\noper pairs \nin \nSCs\n \nhave exhibited many fa\nntas\ntic spin\n-\ndependent properties\n \nthat \nare promising \nfor future\n \ninformation technologies.\n \nFor instance, l\narge \nmagnetoresistance has been \nachi\ne\nved\n \nin the spin valve with \nSCs \nbetween two ferromagnetic films\n \n[\n7\n,\n17\n]\n, \nextremely \nl\nong spin lifetimes \nand large spin Hall \neffect have \nbeen discovered for the \nquasiparticles\n \nof\n \nSCs\n, which exceeds several orders compared to the normal states\n \nabove T\nC\n \n[\n18\n,\n19\n]\n. \nDespite these inte\nnsive studies of the spin\n-\ndependent \npr\noperties \nin the FM/SC junctions\n, \nthe spin dynamics of \nthe \nsupercond\nu\nc\nting thin films\n \nhas not been \nreported\n \nyet.\n \n \nVery \nintriguingly\n, \nit has been\n \nproposed \nrecently \nt\no investigate\n \nthe\n \nspin dynamics in \nsuperconducting \nfilms via spin pumping\n \n[\n20\n]\n, a well\n-\nestablished technique to \nperform d\nynamic\n \nspin injection and to probe the dynamic spin susceptibility in various materials, including metal, \nse\nmiconductors, Rashba 2DEGs, and topological insulators, etc\n \n[\n21\n-\n30\n]\n.\n \n \nIn this letter\n, we report the \nexperimental investigation \nof \nthe spin dynamics in \ns\n-\nwav\ne \nsuperconducting NbN thin film\ns\n \nvia spin pumping. \nA profou\nnd coherence peak of the Gilbert \ndamping\n \nof GdN\n \nis observed slightly below the superconducting critical temperature\n \n(T\nC\n)\n \nof the \nNbN \nin the \nNbN/GdN/NbN\n \ntrilayer samples\n, which indicates \ndynamic spin injection into the \nN\nbN \nthin film\n. Besides, \nthe \ninterface\n-\nenhanced Gilbert damping\n \nprobes \ndynamic spin susceptibility\n \nin \nthe \nNbN thin films \nin the presence of impurity spin\n-\norbit\n \nscattering\n, \nwhich is consistent w\nith \nthe\n \nrecent theoretical study\n \n[\n20\n]\n. \nOur experimental results further \ndemonstrat\ne that\n \nspin pumping \ncould \nbe\n \na \npowerful tool to study the spin dynamics\n \nin \nthe emerging two dimensional SCs\n \n[\n31\n-\n33\n]\n. \n \nII. EXPERIMENTAL DETAILS\n \nThe NbN (t)/GdN (d)/NbN (t) trilayer samples \nare\n \ngrown on Al\n2\nO\n3\n \n(~5 nm)\n-\nbuffered thermally \noxidized Si substrates by d.c. reactive magnetron sputtering at 300 \n°\nC in an ultrahigh vacuum 3\n \n \nchamber. The \nNbN layers \nare\n \ndeposited from a pure Nb target (99.95%) in Ar and N\n2\n \ngas mixture \nat a pressure of 2.3 mTorr (20% N\n2\n), and \nGdN films \nare\n \ndeposited from a pure Gd target (99.9%) \nin Ar and N\n2\n \ngas mixture at a pressure of 2.8 mTorr (6% N\n2\n).\n \nThe NbN and GdN layers \nare \nof \ntextured crystalline quality with a preferred direction along (111)\n-\norientation, as evidenced by X\n-\nray diffraction results (\nF\nig. S1)\n \n[\n34\n]\n. After the growth, a thin Al\n2\nO\n3\n \nlayer (~ 10 nm) was deposited \nin situ\n \nas a capping layer to avoid sample degradation with air exposure. The Curie temperature of \nthe GdN film was determined via the offset\n-\nmagnetization as a function of the temperature us\ning \na Magnetic Properties Measurement System (MPMS; Quantum Design).\n \nThe T\nC\n \nof the SC NbN \nthin films was measured by four\n-\nprobe resistance technique as a function of the temperature in a \nPhysical Properties Measurement System (PPMS; Quantum Design) using s\ntandard ac lock\n-\nin \ntechnique at low frequency of 7 Hz.\n \nThe FMR spectra of the multilayer samples were measured \nusing the coplanar wave guide technique with a vector network analyzer (VNA, Agilent E5071C) \nin the variable temperature insert of PPMS. The samp\nles were attached to the coplanar wave guide \nusing insulating silicon paste. During the measurement, the amplitudes of forward complex \ntransmission coefficients (S\n21\n) were recorded as a functio\nn of the\n \nin\n-\nplane\n \nmagnetic field from ~ \n4\n000 to 0 Oe at various\n \ntemperatures under different microwave frequencies\n \nand microwave power \nof 5 dBm\n.  \n \nIII. RESULTS and DISCUSSION\n \nFig\n.\n \n1\n(a)\n \nillustrates \nspin pumping and \nthe interfacial \ns\n-\nd\n \nexchange coupling between \nspins\n \nin \nthe NbN layer and \nm\nagnetic moment\ns\n \nin the GdN laye\nr\n. \nDue to the interfacial \ns\n-\nd\n \nexchange \ninteraction (\nJ\nsd\n), the time\n-\ndependent magnetization in the GdN layer pumps a quasiparticles\n-\nmediated spin current into the NbN layer, and the spin\n-\nflip scattering of quasiparticles \naccompanies a quantum process of mag\nnon annihilation in the GdN layer, giving rise to enhanced \nGilbert damping\n \n[\n20\n]\n. Hence, the interfacial s\n-\nd exchange coupling provides the route to detect the \nspin dynami\ncs of the NbN layer by measuring the magnetization dynamics of GdN\n.\n \nThe spin \npumping is performed by measuring Gilbert damping of the GdN \nfilm \nin the NbN (t)/GdN (d)/NbN \n(t) trilayer heterostructures\n \nvia ferromagnetic resonance (FMR) technique \n[\n35\n]\n. \nNbN is a\n \ns\n-\nwave \nSC\n \nwith short coherence length of ~ 5 nm\n \nand \nspin diffusion length of ~ 7 nm\n \n[\n19\n,\n36\n]\n. \nGdN is an \ninsulating FM\n. \nA 10 nm \nNbN film is used f\nor the NbN (t)/GdN (d)/NbN (t) \nsamples to justify the \nassumption that the spin backflow from SC is small in the theoretical study \n[\n20\n]\n.\n \nIn a typical sample \nof N\nbN (10)/GdN (5)/NbN (10)\n \n(with thickness in nm)\n, the Curie temperature (T\nCurie\n) of GdN is 4\n \n \ndetermined to be ~ 38 K \nfrom \nthe temperature dependence of \nmagnetic moments\n \n(Fig. 1\n(b)\n), and \nT\nC\n \nof NbN is \nobtained to be \n~ 10.8 K\n \nvia\n \nresistivity vs. temperature \nmeasurement \n(Fig. 1\n(c)\n)\n. \nThe \nT\nC\n \nis slightly affected by the \nexternal in\n-\nplane magnetic\n \nfield of 4000 Oe\n \ndue to the large critical \nfield.\n \n \nFig. 1\n(d)\n \nshows a typical FMR signal (S\n21\n) vs.\n \nthe magnetic field measured \nat \nT\n \n= \n10 K, with \na microwave excitation \nfrequency (\nf\n) of 15 GHz. \nThe half linewidth (\nH\n\n) could be obtained by \nthe Lorentz fitting of the \nFMR \nsignal\n \nfollowing the relationship\n \n[\n37\n]\n:\n \n2\n21 0\n2 2\n( )\n( ) ( )\nres\nH\nS S\nH H H\n\n\n  \n \n \n \n \n \n(1)\n \nwhere S\n0\n \nis the coefficient for the transm\nitted microwave power, \nH\n \nis the external magnetic field, \nand \nres\nH\nis the resonance magnetic field. Gilbert damping (\n\n) is determined using numerical \nfitting of\n \nH\n\nvs. \nf\n \n(Fig. 1\n(e)\n) \nbased \non the spin\n-\nrelaxation\n \nmechanism \n(\nFig. S2\n)\n \n[\n34\n,\n38\n]\n:\n \n0\n2\n4 2\n1 (2 )\nf f\nH H A\nf\n  \n  \n   \n\n \n \n \n \n(2)\n \nwhere \n0\nH\n\n \nis related to the inhomogeneous properties,  \n\n \nis the gyromagnetic ratio, \nA\n \nis t\nhe \nspin\n-\nrelaxation coefficient, and \n\n \nis the \nspin\n-\nrelaxation time constant\n. \n \nThe Gilbert damping \nof GdN \nis studied\n \nas a function of the temperature for two trilayer samples\n \nwith the same interface\n:\n \nNbN (2)/GdN (5)/NbN (2) and NbN (10)/GdN (5)/NbN (10)\n. \nThe 2 nm \nnm Nb\nN layer\n \nis not superconducting\n \ndown to 2 K \nwhile the \nNbN (10)/GdN (5)/NbN (10)\n \nexhibits \na T\nC\n \nof ~ 10.8 K \n(Fig. 2(a)\n). \nThe non\n-\nsuperconducting \nfeature of 2 nm NbN\n \nsample \nand lower \nT\nC\n \nof t\nhe 10 nm NbN \nsample \ncompared to over 16 K\n \nfor \nsingly crystalline \nNbN\n \ncould be \nattributed\n \nto \nreduced thickness, \npolycrystalline property, and magnetic proximity effect\n \n[\n1\n,\n36\n,\n39\n]\n. \nInterestingly, \na profound\n \ncoherence peak of the Gilbert damping \nis observed\n \nin the NbN (10)/GdN (5)/NbN (10)\n, \nbut not on \nNbN (2)/GdN (5)/NbN (2), \nas shown in Fig. 2\n(b)\n. \nThis feature is also evident \nfrom the \ntemperature dependence of the half\n \nlinewidth\n \n(Fig. S\n3\n)\n \n[\n34\n]\n.\n \nThe peak of the Gilbert damping in \nthe NbN (10)/GdN (5)/NbN (10) is observed at ~ 8.5 K, which is slightly below the T\nC\n \n(~ 10.8\n \nK) \nof the 10 nm NbN \nlayer\ns\n. These results indicate the successful dynamic spin injection into the 10 \nnm NbN \nsuperconducting \nlayer, which authenticates a charge\n-\nfree method to injec\nt spin\n-\npolarized \ncarriers into\n \nSC\ns\n \nbeyond previous repor\nts of electrical spin injection\n \n[\n18\n,\n40\n-\n42\n]\n. Furthermore, the 5\n \n \nobservation of the profound coherence peak \nat \nT\n \n= \n~\n \n0.8\n \nT\nC\n \nis expe\ncted \nbased on \nrecent theoretical \nstudies \nof the spin dynamic\ns for \ns\n-\nwave \ns\nupercond\nu\nc\nting thin films\n \nvia spin pumping\n \n[\n20\n]\n. \nSince \nthe thickness of NbN layer (d = 10 nm) i\ns longer than \nits\n \nspin diffusion length of ~ 7 nm, the spin \nbackflow effect, which \nis expected to\n \nreduce\n \nthe damping peak,\n \nis negligible\n \nhere\n \n[\n20\n,\n21\n]\n. \nAccording to the theory, \nGilbert damping is related to the interfacial \ns\n-\nd\n \nexchange interaction and \nthe imaginary part of the dynamic spin susceptibility of the SC\n \n[\n20\n]\n.\n \n \n \n \n \n \n2\nIm ( )\nR\nsd q\nq\nJ x\n \n\n\n \n \n \n \n \n(3)\n \nFor the \ns\n-\nwave superconducting NbN thin films, the superconducting gap \n∆\n \nforms\n \nbelow T\nC\n. At \nthe temperature slightly below T\nC\n, two coherence peaks of the density states exist around the edge \nof the superconducting gap following the BCS theory\n \n[\n43\n]\n, and these peaks in turn give rise to the \nenhancement of t\nhe dynamic spin susceptibility\n \nin the pre\nse\nnce of \nimpurity spin\n-\norbit scattering\n.\n \nQuantitatively\n,\n \nthe ratio of the peak value of Gil\nb\nert damping over \nthat \nat \nT\n \n~ \nT\nC\n \nis\n \n~ 1.8\n, which \nalso\n \nagree\ns\n \nwell wit\nh the theoretical calculation\n \nusing \nprevious\n \nexperimental values of spin \ndiffusion length (~ 7 nm), phase\n \ncoherence length (~ 5 nm), and mean free path (~ 0.3 nm)\n \nfor \nNbN\n \nthin films \n[\n19\n,\n20\n,\n36\n]\n. \nAs the temperature further decreases, the number of quasiparticles \ndecreases \nrapidly \nas the \n∆\n \ngrows, \ngiving rise to the fast decrease of Gilbert damping\n \nbelow \nT\n \n=\n \n~ \n7\n \nK\n. \n \nNext, the thickness of th\ne GdN laye\nr is varied to further study spin pumping\n \nand the spin \ndynamics in NbN layer. For\n \nboth \ns\namples of NbN (10)/GdN (d)/NbN (10) with d \n= \n10\n \nnm and d \n=\n \n30 nm, a profound coherence peak of Gilbert damping is observed slightly below the \nsuperconducting \ntemperat\nure of NbN, as shown in Figs. 3\n(a)\n \nand 3\n(b)\n \n(red circles). While for all \nthe samples of NbN (2)/GdN (d)/NbN (2), no such coherence peak of the Gilbert damping is \nnoticeable (green circles\n \nin Figs. 3(a) and 3(b)\n). The role of the effective \nmagnetization\n \nin the \nobserved \ncoherence peak\n \nhas been ruled out since it exhibits similar temperature dependence for \nNbN (2\n)/GdN (d\n)/NbN (2) and NbN (10)/G\ndN (d\n)/NbN (10) sampl\nes (\nF\nig. S\n4\n)\n \n[\n34\n]\n. \nClearly, \na \nmore profound damping peak for \nthe d = 10 nm sample\n \nis \nobserved\n \ncompared to \nd = 5 nm and \nd = \n30 nm \nsamples. \nFig. 3\n(c)\n \nshows t\nhe ratio of the peak Gilbert damping over the value at \nT\n \n= \n~ \nT\nC\n \nas a function of the GdN thickness\n, which\n \nis in the range from \n1.8 to 2.8\n. \nThe T\nC\n \nexhibit little \nvariation as a function of the GdN thickness\n \n(Fig. S5)\n \n[\n34\n]\n. \nFor a deeper understanding of the \nunderlying mechanism\n \nto account for the thickness dependence of the \nratio,\n \nfurther\n \ntheoretical and 6\n \n \nexperim\ne\nn\nt\nal\n \nstudies \nw\nould be essential\n. One possible cause might be related to the interface \nproximity exchange effect and/or \nthe \npresence of magnetic loose spins leading to scattering at the \ninterface, which could a\nffect the spin diffusion length, \ncoherence length\n, and mean free path\n \nof the \nN\nbN layer. \n \nTo further confirm \nthat \nthe observed coherence peak of the Gilbert damping aris\nes\n \nfrom the\n \ns\n-\nd\n \nexchange interaction\n \nat the SC/FM interface\n, \nthe interface\n-\ninduced Gilbert damping\n \n(\nS\n\n)\n \nis \nstudied as a function of temperatur\ne. \nS\n\n \nis obtained from the thickness dependence of \ntotal \nGilbert \ndamping \nat each temperature\n \n[\n34\n]\n, as shown in \nFig. 4\n(a)\n. \nT\nhe \ncontribution \nfrom GdN itself is \nsignificantly small\n \ncompared to\n \nS\n\n. \nThe unpe\nrturbed peak at ~ 8.5 K (Fig. 4\n(b)\n) unambiguously \ndemonstrates that the origin of the coherence peak in the Gilbert damping is indeed due to the \ninterfacial \ns\n-\nd\n \nexchange interaction between the magnetization of GdN and the spins \nof the \nquasiparticles\n \nin superconducting NbN thin films\n. \nThe \nslightly upturn of the interface\n-\ninduced \nGilbert damping starting at \nT\n \n= 11 K might be associated with the fluctuation superconductivity \nand/or the higher superconducting transition temperature for some regions of the NbN films than \nthe zero resistance tem\nperature. \nFor comparison, the interface\n-\ninduced Gilbert damping in the \nsamples of NbN (2)/GdN (d)/NbN (2) does not \nexhibit\n \nany \nsignature\n \nof coherence peak \nbetween \n4 and 15 K\n \n(\nF\nig. S6\n)\n \n[\n34\n]\n.\n \n \nNoteworthy is that our results are essentially different from previous reports of spin pumping \ninto \nsuperconducting\n \nNb films using the ferromagnetic metal permalloy\n \n(Py)\n, where a monotonic \ndecrease of\n \nthe Gilbert damping is reported when the temperature decreases\n \n[\n44\n,\n45\n]\n. \nThis feature \nis also \nconfirmed\n \nin our studies \nby measuring the temperature dependence of\n \nGilbert damping of \nPy in \nNb\n(100)/Py(20\n)/Nb(100)\n \n(F\nig. S7)\n \n[\n34\n]\n. \nSince \nP\ny\n \nis a \nFM\n \nmetal, the interface \nexchange \ninteraction\n \nwould be \nsignificant\n \nto \nstrongly\n \nor even \ncompletely\n \nsuppress\n \nthe\n \nsuperconducting gap\n \nof Nb\n \nat the \nNb/Py interface \n[\n17\n,\n20\n,\n43\n]\n. \nHence, no coherence peak of Gilbert damping is expected\n \nbased on the \ntheoretical\n \nstudy \nthat assumes the completely suppression of the gap at the interface\n \n[\n45\n]\n. \nBesid\nes, the strong suppression of the superconducting gap at the Nb/Py interface is in good \nagreement of previous tunnel spectroscopy measurements of vanishing superconducting gap at the \ninterface between Nb and Ni\n \n[\n4\n6\n]\n. \nHowever, \nfor \nFM insulating GdN, charge carriers from NbN do \nnot penetrate into GdN, thus not weakening the \nsuperconductivity\n \nat the interface\n, resulting the \nsurvival of the\n \nsuperconducting gap \n[\n47\n,\n48\n]\n. \n 7\n \n \nThe experimental \ndemonstration of the dynamic spin \nsusceptibility\n \nin \nsuperconducting\n \nthin \nfilms\n \nvia\n \nspin pumping \ncould be \nessential\n \nfor the field of superconductors.\n \nFor bulk SCs,\n \ndynamic \nspin susceptibility has been studied from the \ntemperature \ndependent \nspin \nrel\na\nxation rate via\n \nthe\n \nnuclear magnetic resonance\n \n[\n43\n,\n49\n,\n50\n]\n. \nIt also provides an avenue for  identifying unconventional \nsuperconducting paring mechanisms\n \n[\n51\n]\n, while limited to mostly bulk SCs due to low signal\n-\nto\n-\nnoise ratio\n \n[\n43\n,\n49\n-\n51\n]\n. S\npin pumping \nmethod \nhas much better \nsignal\n-\nto\n-\nnoise ratio\n \nso it could be \nvery \nsensitive\n \nto \nprobe the dynamic spin susceptibility\n \nof \nsuperconducting \nthin films\n.\n \nFurthermore, \nthe \nspin pumping offers a special \ntechnique \nto probe the \npair breaking strength, \nthe impurity s\npin\n-\norbit scattering, and the \nmagnetic proximity effect in the SC/FM junctions, which is \nalso \nof \nconsiderable interest\n \nfor the field of SC \nspintronics\n \n[\n1\n-\n3\n]\n. \n \nIV. CON\nCLUSION\n \nIn conclusion, \nthe \nspin \ndynamic\ns\n \nof \nsuperconducting \nNbN \nfilm\ns\n \nare\n \ninvestigated\n \nvia spin \npumping from \nan adjacent FM insulating layer\n. \nA profound coherence peak of the Gilbert damping \nis observed below T\nC\n,\n \nwhich indicates the dynamic spin injection \ninto the superconducting NbN \nfilms.\n \nOur results \npaves the way for\n \nfuture investigation of interface impurity spin\n-\norbit scattering, \nand pair breaking strength in FM/SC junctions, as w\nell as the spin dynamics in the \ninterfacial and \ntwo dimensional \ncrystalline SCs.\n \nIt may also be useful for the search for Majorana fermions in the \nSC/ferromagnetic insulator heterostructures.\n \n \n \nAcknowledgments\n \nWe acknowledge the fruitful discussion with Yuan Li and Tao Wu. Y.Y., Q.S., W.Y., Y.M., \nY.Y., X.C.X., and W.H.\n \nacknowledge the financial support from National Basic Research \nPrograms of China (973 program Grant Nos. 2015CB921104 and 2014CB920902) and National \nNatural Science Foundation of China (NSFC Grant No. 11574006). Y.T., J.P.C, and J.S.M. \nacknowledge the gra\nnts NSF DMR\n-\n1700137 and ONR N00014\n-\n16\n-\n1\n-\n2657. Y.T. also \nacknowledges the JSPS Overseas Research Fellowships and the Fundacion Seneca (Region de \nMurcia) posdoctoral fellowship (19791/PD/15). W.H. also acknowledges the support by the 1000 \nTalents Program for\n \nYoung Scientists of China.\n \n 8\n \n \nReferences\n:\n \n[1]\n \nA. I. Buzdin, Proximity effects in superconductor\n-\nferromagnet heterostructures.\n \nRev. Mod. \nPhys.\n \n77\n, 935 (2005).\n \n[2]\n \nJ. Linder and J. W. A. Robinson, Superconducting spintronics.\n \nNat. Phys.\n \n11\n, \n307 (2015).\n \n[3]\n \nM. Eschrig, Spin\n-\npolarized supercurrents for spintronics.\n \nPhys. Today\n \n64\n, 43 (2011).\n \n[4]\n \nJ. Y. Gu, C. Y. You, J. S. Jiang, J. Pearson, Y. B. Bazaliy, and S. D. Bader, Magnetization\n-\nOrientation Dependence of the Superconducting Transition Te\nmperature in the Ferromagnet\n-\nSuperconductor\n-\nFerromagnet System: CuNi/Nb/CuNi.\n \nPhys. Rev. Lett.\n \n89\n, 267001 (2002).\n \n[5]\n \nZ. Yang, M. Lange, A. Volodin, R. Szymczak, and V. V. Moshchalkov, Domain\n-\nwall \nsuperconductivity in superconductor\n-\nferromagnet hybrids.\n \nNa\nt. Mater.\n \n3\n, 793 (2004).\n \n[6]\n \nI. C. Moraru, W. P. Pratt, and N. O. Birge, Magnetization\n-\nDependent T\nC\n \nShift in \nFerromagnet/Superconductor/Ferromagnet Trilayers with a Strong Ferromagnet.\n \nPhys. Rev. Lett.\n \n96\n, 037004 (2006).\n \n[7]\n \nG.\n-\nX. Miao, K. Yoon, T. S. Sant\nos, and J. S. Moodera, Influence of Spin\n-\nPolarized Current \non Superconductivity and the Realization of Large Magnetoresistance.\n \nPhys. Rev. Lett.\n \n98\n, 267001 \n(2007).\n \n[8]\n \nH. Jeffrey Gardner, A. Kumar, L. Yu et al., Enhancement of superconductivity by a \nparallel \nmagnetic field in two\n-\ndimensional superconductors.\n \nNat. Phys.\n \n7\n, 895 (2011).\n \n[9]\n \nJ. Q. Xiao and C. L. Chien, Proximity Effects in Superconductor/Insulating\n-\nFerromagnet \nNbN/GdN Multilayers.\n \nPhys. Rev. Lett.\n \n76\n, 1727 (1996).\n \n[10]\n \nT. Kontos, M. April\ni, J. Lesueur, F. Genêt, B. Stephanidis, and R. Boursier, Josephson \nJunction through a Thin Ferromagnetic Layer: Negative Coupling.\n \nPhys. Rev. Lett.\n \n89\n, 137007 \n(2002).\n \n[11]\n \nR. S. Keizer, S. T. B. Goennenwein, T. M. Klapwijk, G. Miao, G. Xiao, and A. Gupta,\n \nA \nspin triplet supercurrent through the half\n-\nmetallic ferromagnet CrO\n2\n.\n \nNature\n \n439\n, 825 (2006).\n \n[12]\n \nJ. W. A. Robinson, J. D. S. Witt, and M. G. Blamire, Controlled Injection of Spin\n-\nTriplet \nSupercurrents into a Strong Ferromagnet.\n \nScience\n \n329\n, 59 (2010).\n \n[13]\n \nK. Senapati, M. G. Blamire, and Z. H. Barber, Spin\n-\nfilter Josephson junctions.\n \nNat. Mater.\n \n10\n, 849 (2011).\n \n[14]\n \nJ. D. Sau, R. M. Lutchyn, S. Tewari, and S. Das Sarma, Generic New Platform for \nTopological Quantum Computation Using Semiconductor Hetero\nstructures.\n \nPhys. Rev. Lett.\n \n104\n, \n040502 (2010).\n \n[15]\n \nS. A. Wolf, D. D. Awschalom, R. A. Buhrman et al., Spintronics: A Spin\n-\nBased Electronics \nVision for the Future.\n \nScience\n \n294\n, 1488 (2001).\n \n[16]\n \nI. Zutic, J. Fabian, and S. Das Sarma, Spintronics: Fundame\nntals and applications.\n \nRev. \nMod. Phys.\n \n76\n, 323 (2004).\n \n[17]\n \nG.\n-\nX. Miao, A. V. Ramos, and J. S. Moodera, Infinite Magnetoresistance from the Spin \nDependent Proximity Effect in Symmetry Driven bcc\n-\nFe/V/Fe Heteroepitaxial Superconducting \nSpin Valves.\n \nPhys. Rev. Lett.\n \n101\n, 137001 (2008).\n \n[18]\n \nH. Yang, S.\n-\nH. Yang, S. Takahashi, S. Maekawa, and S. S. P. Parkin, Extremely long \nquasiparticle spin lifetimes in superconducting aluminium using MgO tunnel spin injectors.\n \nNat. \nMater.\n \n9\n, 586 (2010).\n \n[19]\n \nT. Waka\nmura, H. Akaike, Y. Omori et al., Quasiparticle\n-\nmediated spin Hall effect in a \nsuperconductor.\n \nNat. Mater.\n \n14\n, 675 (2015).\n \n[20]\n \nM. Inoue, M. Ichioka, and H. Adachi, Spin pumping into superconductors: A new probe \nof spin dynamics in a superconducting thin f\nilm.\n \nPhys. Rev. B\n \n96\n, 024414 (2017).\n 9\n \n \n[21]\n \nY. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I. Halperin, Nonlocal magnetization \ndynamics in ferromagnetic heterostructures.\n \nRev. Mod. Phys.\n \n77\n, 1375 (2005).\n \n[22]\n \nY. Ohnuma, H. Adachi, E. Saitoh, and S. Maeka\nwa, Enhanced dc spin pumping into a \nfluctuating ferromagnet near T\nC\n.\n \nPhys. Rev. B\n \n89\n, 174417 (2014).\n \n[23]\n \nK. Ando, S. Takahashi, J. Ieda et al., Inverse spin\n-\nHall effect induced by spin pumping in \nmetallic system.\n \nJ. Appl. Phys.\n \n109\n, 103913 (2011).\n \n[24]\n \nK.\n \nAndo, S. Takahashi, J. Ieda et al., Electrically tunable spin injector free from the \nimpedance mismatch problem.\n \nNat. Mater.\n \n10\n, 655 (2011).\n \n[25]\n \nY. Shiomi, K. Nomura, Y. Kajiwara et al., Spin\n-\nElectricity Conversion Induced by Spin \nInjection into Topologi\ncal Insulators.\n \nPhys. Rev. Lett.\n \n113\n, 196601 (2014).\n \n[26]\n \nJ. C. R. Sánchez, L. Vila, G. Desfonds et al., Spin\n-\nto\n-\ncharge conversion using Rashba \ncoupling at the interface between non\n-\nmagnetic materials.\n \nNat. Commun.\n \n4\n, 3944 (2013).\n \n[27]\n \nQ. Song, J. Mi, D. \nZhao et al., Spin injection and inverse Edelstein effect in the surface \nstates of topological Kondo insulator SmB\n6\n.\n \nNat. Commun.\n \n7\n, 13485 (2016).\n \n[28]\n \nE. Lesne, Y. Fu, S. Oyarzun et al., Highly efficient and tunable spin\n-\nto\n-\ncharge conversion \nthrough Rashba\n \ncoupling at oxide interfaces.\n \nNat. Mater.\n \n15\n, 1261 (2016).\n \n[29]\n \nQ. Song, H. Zhang, T. Su et al., Observation of inverse Edelstein effect in Rashba\n-\nsplit \n2DEG between SrTiO\n3\n \nand LaAlO\n3\n \nat room temperature.\n \nSci. Adv.\n \n3\n, e1602312 (2017).\n \n[30]\n \nR. Ohshima, Y. \nAndo, K. Matsuzaki et al., Realization of d\n-\nelectron spin transport at room \ntemperature at a LaAlO\n3\n/SrTiO\n3\n \ninterface.\n \nNat. Mater.\n \n16\n, 609 (2017).\n \n[31]\n \nN. Reyren, S. Thiel, A. D. Caviglia et al., Superconducting Interfaces Between Insulating \nOxides.\n \nScience\n \n317\n, 1196 (2007).\n \n[32]\n \nS. He, J. He, W. Zhang et al., Phase diagram and electronic indication of high\n-\ntemperature \nsuperconductivity at 65K in single\n-\nlayer FeSe films.\n \nNat. Mater.\n \n12\n, 605 (2013).\n \n[33]\n \nY. Saito, T. Nojima, and Y. Iwasa, Highly crystalline \n2D superconductors.\n \nNat. Rev. Mater.\n \n2\n, 16094 (2016).\n \n[34]\n \nSee Supplemental Material at [URL will be inserted by publisher] for the crystalline \nstructure characterization, the analysis of Gilbert damping, the measurement of effective \nmagnetization, the det\nermination of interface\n-\ninduced Gilbert damping, Temperature dependence \nof Gilbert damping of Nb/Py/Nb trilayer samples, and the impact of spin pumping exepriments on \nsuperconducting temperature of the NbN layers.\n \n[35]\n \nY. Zhao, Q. Song, S.\n-\nH. Yang et al., \nExperimental Investigation of Temperature\n-\nDependent Gilbert Damping in Permalloy Thin Films.\n \nScientific Reports\n \n6\n, 22890 (2016).\n \n[36]\n \nS. P. Chockalingam, M. Chand, J. Jesudasan, V. Tripathi, and P. Raychaudhuri, \nSuperconducting properties and Hall effect o\nf epitaxial NbN thin films.\n \nPhys. Rev. B\n \n77\n, 214503 \n(2008).\n \n[37]\n \nZ. Celinski, K. B. Urquhart, and B. Heinrich, Using ferromagnetic resonance to measure \nthe magnetic moments of ultrathin films.\n \nJ. Magn. Magn. Mater.\n \n166\n, 6 (1997).\n \n[38]\n \nC. L. Jermain, S. V. \nAradhya, N. D. Reynolds et al., Increased low\n-\ntemperature damping \nin yttrium iron garnet thin films.\n \nPhys. Rev. B\n \n95\n, 174411 (2017).\n \n[39]\n \nA. M. Goldman, Superconductor\n-\nInsulator Transitions.\n \nInt. J. Mod. Phys. B\n \n24\n, 4081 \n(2010).\n \n[40]\n \nC. H. L. Quay, D. Chev\nallier, C. Bena, and M. Aprili, Spin imbalance and spin\n-\ncharge \nseparation in a mesoscopic superconductor.\n \nNat. Phys.\n \n9\n, 84 (2013).\n 10\n \n \n[41]\n \nF. Hübler, M. J. Wolf, D. Beckmann, and H. v. Löhneysen, Long\n-\nRange Spin\n-\nPolarized \nQuasiparticle Transport in Mesoscopic\n \nAl Superconductors with a Zeeman Splitting.\n \nPhys. Rev. \nLett.\n \n109\n, 207001 (2012).\n \n[42]\n \nT. Wakamura, N. Hasegawa, K. Ohnishi, Y. Niimi, and Y. Otani, Spin Injection into a \nSuperconductor with Strong Spin\n-\nOrbit Coupling.\n \nPhys. Rev. Lett.\n \n112\n, 036602 (2014).\n \n[43]\n \nM. Tinkham, \nIntroduction to superconductivity \n(Dover Publications, 2004).\n \n[44]\n \nC. Bell, S. Milikisyants, M. Huber, and J. Aarts, Spin Dynamics in a Superconductor\n-\nFerromagnet Proximity System.\n \nPhys. Rev. Lett.\n \n100\n, 047002 (2008).\n \n[45]\n \nJ. P. Morten, A.\n \nBrataas, G. E. W. Bauer, W. Belzig, and Y. Tserkovnyak, Proximity\n-\neffect\n–\nassisted decay of spin currents in superconductors.\n \nEurophys. Lett.\n \n84\n, 57008 (2008).\n \n[46]\n \nM. A. Sillanpää, T. T. Heikkilä, R. K. Lindell, and P. J. Hakonen, Inverse proximity effect\n \nin superconductors near ferromagnetic material.\n \nEur. Phys. Lett.\n \n56\n, 590 (2001).\n \n[47]\n \nP. G. De Gennes, Coupling between ferromagnets through a superconducting layer.\n \nPhys. \nLett.\n \n23\n, 10 (1966).\n \n[48]\n \nB. Li, N. Roschewsky, B. A. Assaf et al., Superconducting\n \nSpin Switch with Infinite \nMagnetoresistance Induced by an Internal Exchange Field.\n \nPhys. Rev. Lett.\n \n110\n, 097001 (2013).\n \n[49]\n \nY. Masuda and A. G. Redfield, Nuclear Spin\n-\nLattice Relaxation in Superconducting \nAluminum.\n \nPhys. Rev.\n \n125\n, 159 (1962).\n \n[50]\n \nL. C. \nHebel and C. P. Slichter, Nuclear Spin Relaxation in Normal and Superconducting \nAluminum.\n \nPhys. Rev.\n \n113\n, 1504 (1959).\n \n[51]\n \nN. J. Curro, T. Caldwell, E. D. Bauer et al., Unconventional superconductivity in PuCoGa\n5\n.\n \nNature\n \n434\n, 622 (2005).\n \n[52]\n \nP. E. Seiden\n, Ferrimagnetic Resonance Relaxation in Rare\n-\nEarth Iron Garnets.\n \nPhys. Rev.\n \n133\n, A728 (1964).\n \n[53]\n \nM. Sparks, \nFerromagnetic Relaxation Theory \n(McGraw\n-\nHill, New York, 1964).\n \n[54]\n \nR. Arias and D. L. Mills, Extrinsic contributions to the ferromagnetic resonan\nce response \nof ultrathin films.\n \nPhys. Rev. B\n \n60\n, 7395 (1999).\n \n[55]\n \nK. Lenz, H. Wende, W. Kuch, K. Baberschke, K. Nagy, and A. Jánossy, Two\n-\nmagnon \nscattering and viscous Gilbert damping in ultrathin ferromagnets.\n \nPhys. Rev. B\n \n73\n, 144424 (2006).\n \n[56]\n \nO. d'Al\nlivy Kelly, A. Anane, R. Bernard et al., Inverse spin Hall effect in nanometer\n-\nthick \nyttrium iron garnet/Pt system.\n \nAppl. Phys. Lett.\n \n103\n, 082408 (2013).\n \n[57]\n \nX. Liu, W. Zhang, M. J. Carter, and G. Xiao, Ferromagnetic resonance and damping \nproperties of Co\nFeB thin films as free layers in MgO\n-\nbased magnetic tunnel junctions.\n \nJ. Appl. \nPhys.\n \n110\n, 033910 (2011).\n \n[58]\n \nJ. F. Sierra, V. V. Pryadun, S. E. Russek et al., Interface and Temperature Dependent \nMagnetic Properties in Permalloy Thin Films and Tunnel Junct\nion Structures.\n \nJ. Nanosci. \nNanotechnol.\n \n11\n, 7653 (2011).\n \n[59]\n \nC. Kittel, On the Theory of Ferromagnetic Resonance Absorption.\n \nPhys. Rev.\n \n73\n, 155 \n(1948).\n \n[60]\n \nG. Counil, J.\n-\nV. Kim, T. Devolder, C. Chappert, K. Shigeto, and Y. Otani, Spin wave \ncontributions\n \nto the high\n-\nfrequency magnetic response of thin films obtained with inductive \nmethods.\n \nJ. Appl. Phys.\n \n95\n, 5646 (2004).\n \n[61]\n \nM. L. Schneider, T. Gerrits, A. B. Kos, and T. J. Silva, Gyromagnetic damping and the role \nof spin\n-\nwave generation in pulsed induct\nive microwave magnetometry.\n \nAppl. Phys. Lett.\n \n87\n, \n072509 (2005).\n 11\n \n \n[62]\n \nP. Krivosik, N. Mo, S. Kalarickal, and C. E. Patton, Hamiltonian formalism for two \nmagnon scattering microwave relaxation: Theory and applications.\n \nJ. Appl. Phys.\n \n101\n, 083901 \n(2007).\n \n \n \n \n \n \n 12\n \n \nFigure 1\n \n \nFig. 1. \nSpin pumping into the \nsuperconducting\n \nNbN thin films. \n(a)\n \nSchematic of the\n \ninterfacial\n \ns\n-\nd\n \nexchange interaction (\nJ\nsd\n) between the spins in NbN layer and the rotating magnetization of GdN \n13\n \n \nlayer under the ferromagnetic resonance\n \n(FMR)\n \nconditions. \n(b) \nThe magnetic moment as a \nfunction of the temperature. Inset: The magnetic hysteresis loop at \nT\n \n= \n5 K. \n(c) \nThe four\n-\nprobe \nresistance as a function of the temperature\n \nwith \nin\n-\nplane \nmagnetic field at 0 and 4000 Oe\n. \n(d) \nThe \ntypical \nFMR \nsignal\n \nmeasured at \nT\n \n= \n10 K and \nf\n \n= \n15 GHz. The red line indicates the Lorentz fitting \ncurve to obtain the half linewidth based on equation (1). \n(e)\n \nThe half linewidth\n \nas a function of the \nmicrowave\n \nfrequency at \nT\n \n= \n10K. The red solid line is \nthe fitting c\nurve based on spin\n-\nrelaxation \nmodel. The results in Fig. 1\n(b\n-\nd)\n \nare obtained on the NbN (10)/GdN (5)/NbN (10) sample.\n \n \n \n 14\n \n \nFigure 2\n \n \n \n \n \nFig. 2. \nSpin dynamics \nof \nthe \nsuperconducting\n \nNbN thin films probed via \nspin pumping\n. The \nnormalized four\n-\nprobe resistance\n \n(\na\n)\n \nand Gilbert damping (\nb\n) \nas a function of the temperature for \nthe samples of NbN (2)/GdN (5)/NbN (2) and NbN (10)/GdN (5)/NbN (10), respectively. \n \n \n15\n \n \n \n \nFigure 3\n \n \nFig. 3. \nThe GdN thickness effect on spin dynamics \nof \nthe superconducting \nNbN thin films\n. \n(a\n-\nb)\n \nThe Gilbert damping as a function of the temperature for the samples of NbN (\nt\n)/GdN (10)/NbN \n(\nt), \nand NbN (\nt\n)/GdN (30)/NbN (\nt\n). Insets\n: The normalized four\n-\nprobe resistance as a function of \nthe temperature for the samples of NbN (10)/GdN (10)/NbN (10) and NbN (10)/GdN (30)/NbN \n(10). \n(c)\n \nThe \nratio of \nthe \npeak Gilbert damping \nover \nthe value\n \nat \nT\n \n= \n~ T\nC\n \n(\npeak C\n/ (~ T )\n \n) \nas a \nfun\nction of the GdN layer thickness.\n \n16\n \n \nFigure 4\n \n \n \nFig. 4. \nThe interface\n-\ninduced Gilbert damping\n \nat the \nNbN/GdN interface\n. \n(a)\n \nThe Gilbert damping \nas a function of the GdN thickness for the samples of NbN (10)/GdN (d)/NbN (10) \nat \nT\n \n= \n11\n,\n \n8.5\n, \nand 6 K\n, respect\nively\n. \n(b)\n \nThe interface\n-\ninduced Gilbert damping as a function of temperature\n \nfor \nNbN (10)/GdN (d)/NbN (10)\n. \n \n \n \n1\n \nSupplementary Materials for:\n \n \nProbe of Spin Dynamics in Superconducting NbN Thin Films via \n \nSpin Pumping\n \n \nYunyan\n \nYao\n1,2\n,\n \nQi\n \nSong\n1,2\n,\n \nYota\n \nTakamura\n3,4\n,\n \nJuan\n \nPedro\n \nCascales\n3\n,\n \nWei\n \nYuan\n1,2\n,\n \nYang\n \nMa\n1,2\n,\n \nYu\n \nYun\n1,2\n,\n \nX.\n \nC.\n \nXie\n1,2\n,\n \nJagadeesh\n \nS.\n \nMoodera\n3,5\n,\n \nand\n \nWei\n \nHan\n1,2*\n \n1\nInternational Center for Quantum Materials, School of Physics, Peking University, Beijing \n100871, China.\n \n2\nCollaborative Innovation Center of Quantum Matter, Beijing 100871, Chi\nna.\n \n3\nPlasma Science and Fusion Center and Francis Bitter Magnet Laboratory, Massachusetts \nInstitute of Technology, Cambridge, MA 02139, USA.\n \n4\nSchool of Engineering, Tokyo Institute of Technology, Tokyo 152\n-\n8550, Japan.\n \n5\nDepartment of Physics, Massachusetts\n \nInstitute of Technology, Cambridge, MA 02139, USA\n \n*Correspondence to: \nweihan@pku.edu.cn (W.H.)\n \n 2\n \nS1.\n \nDetermination\n \nof\n \nGilbert\n \ndamping\n \nA\n \nnonlinear\n \nbehavior\n \nof\n \nhalf\n \nlinewidth\n \n(\nH\n\n)\n \nvs.\n \nmicrowave\n \nfrequency\n \n(\nf\n)\n \nis\n \nobserved\n \non\n \nthe\n \nNbN\n \n(t)/GdN\n \n(d)/NbN\n \n(t)\n \nsamples,\n \nas\n \nshown\n \nin\n \nFig.\n \nS2(a\n-\nc).\n \nTo\n \nour\n \nbest\n \nknowledge,\n \nt\nhis\n \nnonlinear\n \nbehavior\n \ncould\n \nbe\n \nattributed\n \nto\n \ntwo\n \nmechanisms,\n \nnamely\n \nspin\n-\nrelaxation\n \n[\n38\n,\n52\n]\n,\n \nand\n \ntwo\n-\nmagnon\n \nscattering\n \n[\n53\n,\n54\n]\n.\n \nFor\n \nthe\n \nspin\n-\nrelaxation\n \nmechanism,\n \nH\n\n \nis\n \nrelated\n \nto\n \na\n \ntemperature\n-\ndependent\n \nspin\n-\nrelaxation\n \ntime\n \nconstant\n \n(\n\n),\n \nand\n \ncan\n \nbe\n \nexpressed\n \nby\n \n[\n38\n]\n:\n \n \n \n0\n2\n4 2\n1 (2 )\nf f\nH H A\nf\n  \n  \n    \n\n \n \n(S1)\n \nwhere\n \n0\nH\n\n \nis\n \nassociated\n \nwith\n \nthe\n \ninhomogeneous\n \nproperties,\n \n\n \nis\n \nthe\n \ngyromagnetic\n \nratio,\n \nand\n \nA\n \nis\n \nthe\n \nspin\n-\nrelaxation\n \ncoefficient.\n \nWhile,\n \nthe\n \nrelationship\n \nof\n \nH\n\n \nand\n \nf\n \nbased\n \non\n \ntwo\n-\nmagnon\n \nscattering\n \nmechanism\n \ncan\n \nbe\n \nexpressed\n \nby\n \n[\n54\n,\n55\n]\n:\n \n \n\n\n\n\n2\n2\n1\n2\n2\n(2 ) 2 2\nsin\n(2 ) 2 2\neff eff\neff eff\nf M M\nH\nf M M\n    \n    \n\n \n \n \n \n(S2)\n \nwhere\n \n\n \nis\n \nthe\n \ntwo\n-\nmagnon\n \nscattering\n \ncoefficient,\n \nand\n \neff\nM\nis\n \nthe\n \neffective\n \nmagnetization.\n \nClearly,\n \nas\n \nshown\n \nin\n \nFig.\n \nS2(a\n-\nc),\n \ntwo\n-\nmagnon\n \nscattering\n \nmechanism\n \nfails\n \nto\n \ngive\n \nreasonable\n \nfittings\n \n(blue\n \nlines)\n \nto\n \nour\n \nexperimental\n \nresults\n \n(black\n \nsquares).\n \nWhileas,\n \nour\n \nexperimental\n \nresults\n \nagree\n \nwell\n \nwith\n \nthe\n \nfitting\n \ncurves\n \nbased\n \non\n \nthe\n \nspin\n-\nrelaxation\n \nmechanism\n \n(red\n \nlines).\n \nBesides,\n \nit\n \nis\n \nobserved\n \nthat\n \nthe\n \nnonlinearity\n \nof\n \nH\n\n \nvs.\n \nf\n \nincreases\n \nas\n \nthe\n \nGdN\n \nlayer\n \nthickness\n \nincreases,\n \nwhich\n \nis\n \nalso\n \nopposite\n \nto\n \nthe\n \ntwo\n-\nmagnon\n \nscat\ntering\n \nmechanism\n.\n \nSince\n \ntwo\n-\nmagnon\n \nscattering\n \narises\n \nfrom\n \nthe\n \ndefects\n \nat\n \nthe\n \ninterface,\n \na\n \nmore\n \nconspicuously\n \nnonlinear\n \nbehavior\n \nis\n \nexpected\n \nas\n \nthe\n \nFM\n \nfilms\n \nbecome\n \nthinner\n \n[\n55\n-\n57\n]\n.\n \nAnother\n \nfeature\n \nthat\nH\n\n \ndecreases\n \nas\n \nf\n \nincreases\n \n(Fig.\n \nS2(c))\n \ncannot\n \nbe\n \nexplained\n \nby\n \nthe\n \ntwo\n-\nmagnon\n \nscattering\n \nmechanism\n \neither.\n \nHence,\n \nthe\n \nspin\n-\nrelaxation\n \nmechanism\n \nis\n \nused\n \nto\n \nanalyze\n \nthe\n \nexperimental\n \nresults\n \nof\n \nH\n\n \nvs.\n \nf\n,\n \nand\n \nGilbert\n \ndamping\n \ncould\n \nbe\n \nobtained\n \nsubsequently.\n \n \nS2.\n \nRole\n \nof\n \nthe\n \neffective\n \nmagnetization\n \n 3\n \nIn\n \nprevious\n \nstudies,\n \nit\n \nha\ns\n \nbeen\n \nsuggested\n \nthat\n \nthe\n \nenhancement\n \nof\n \nthe\n \nGilbert\n \ndamping\n \ncould\n \nbe\n \nalso\n \nassociated\n \nwith\n \nthe\n \nchange\n \nof\n \neff\nM\n \n[\n35\n,\n58\n]\n.\n \nTo\n \nrule\n \nout\n \nthe\n \neffect\n \nof\n \neff\nM\n,\n \nwe\n \nsystemically\n \nstudy\n \nthe\n \ntemperature\n \ndependence\n \nof\n \neff\nM\nbetween\n \nthe\n \nNbN\n \n(2)/GdN\n \n(d)/NbN\n \n(2)\n \nand\n \nNbN\n \n(10)/GdN\n \n(d)/NbN\n \n(10)\n \nsamples.\n \nThe\n \n4\neff\nM\n\n \nis\n \nbased\n \non\n \nthe\n \nKittel\n \nformula\n \n[\n59\n]\n:\n \n \n \n \n \n \n \n1\n2\n( )[ ( 4 )]\n2\nres res eff\nf H H M\n\n\n\n \n \n \n \n               \n(S3)\n \nwhere\n \nf\n \nis\n \nthe\n \nmicrowave\n \nfrequency,\n \nand\n \nres\nH\nis\n \nthe\n \nresonance\n \nmagnetic\n \nfield.\n \nAs\n \nshown\n \nin\n \nfig.\n \nS4(a),\n \nthe\n \nfitting\n \ncurve\n \n(red\n \nline)\n \nagrees\n \nwell\n \nwith\n \nthe\n \nexperimental\n \nresults.\n \nSimilar\n \ntemperature\n \ndependences\n \nof\n \n4\neff\nM\n\nfor\n \nthe\n \nNbN\n \n(2)/GdN\n \n(d)/NbN\n \n(2)\n \nand\n \nNbN\n \n(10)/GdN\n \n(d)/NbN\n \n(10)\n \nsamples\n \nare\n \nobserved\n \n(Figs.\n \nS4(c\n-\nd)).\n \nThis\n \nobservation\n \nconfirms\n \nthat\n \nthe\n \nobserved\n \ncoherence\n \npeak\n \nof\n \nthe\n \nGilbert\n \ndamping\n \nat\n \nT\n \n=\n \n~\n \n8.5\n \nK\n \nis\n \nnot\n \nrelated\n \nto\n \nthe\n \neffective\n \nmagnetization.\n \n \nS3.\n \nDetermination\n \nof\n \ninterface\n-\ninduced\n \nGilbert\n \ndamping\n \nPrevious\n \nstudies\n \nhave\n \nidentified\n \nthe\n \nmajor\n \nsources\n \nthat\n \ncontribute\n \nto\n \nthe\n \nGilbert\n \ndamping\n \nof\n \nthe\n \nFM\n \nfilms,\n \nincluding\n \nbulk\n \ndamping\n \n(\nB\n\n)\n \nand\n \nthe\n \ninterface\n-\ninduced\n \nGilbert\n \ndamping\n \n(\nS\n\n)\n \nthat\n \nis\n \ndue\n \nto\n \nspin\n \npumping\n \nand\n \nother\n \ninterface\n \neffects\n \n[\n20\n-\n22\n]\n.\n \nAs\n \nshown\n \nin\n \nFig.\n \n4(a)\n \nand\n \nFig.\n \nS6(a),\n \na\n \nslightly\n \nincrease\n \nof\n \nthe\n \nGilbert\n \ndamping\n \nis\n \nobserved\n \nas\n \nthe\n \nthickness\n \nof\n \nthe\n \nGdN\n \nlayer\n \nincreases\n \nfrom\n \n20\n \nto\n \n30\n \nnm.\n \nThis\n \nfeature\n \ncould\n \nbe\n \nassociated\n \nwith\n \nnon\n-\nhomogeneous\n \nfield\n \nexcitation\n \nwhen\n \nthe\n \nw\nidth\n \nof\n \ncoplanar\n \nwave\n \nguide\n \nis\n \nvery\n \nsmall\n \ncompared\n \nto\n \nthe\n \nsamples\n \n[\n60\n,\n61\n]\n,\n \nand\n \nthe\n \neddy\n \ncurrent\n \nloss\n \neffect\n \n[\n62\n]\n.\n \nBoth\n \nof\n \nthese\n \neffects\n \ncontribute\n \nto\n \nthe\n \nGilbert\n \ndamping\n \nthat\n \nis\n \nproportional\n \nto\n \nd\n2\n.\n \nHence,\n \nthe\n \ntotal\n \nGilbert\n \ndamping\n \ncan\n \nbe\n \nexpressed\n \nby:\n \n2\n1\n( ) '\nB S\nd\nd\n   \n  \n \n \n \n \n(S4)\n \nAnd\n \nthe\n \ninterface\n-\ninduced\n \nGilbert\n \nda\nmping\n \nand\n \nbulk\n \ndamping\n \ncould\n \nbe\n \nobtained\n \nsubsequently.\n \nBased\n \non\n \nthe\n \nfitted\n \nresults,\n \nB\n\n \nis\n \nsignificantly\n \nsmall\n \ncompared\n \nto\n \nS\n\n.\n \nThe\n \n~10%\n \nerror\n \nbar\n \nof\n \nS\n\n \nat\n \neach\n \ntemperature\n \nmakes\n \nit\n \nhard\n \nto\n \nobtain\n \nthe\n \naccurate\n \nvalues\n \nof\n \nB\n\n,\n \nwhich\n \nrequire\n \nfuture\n \nstudies.\n \nComparing\n \nthe\n \ninterface\n-\ninduced\n \nGilbert\n \ndamping\n \nfor\n \nNbN\n \n(10)/GdN\n \n(d)/NbN\n \n(10)\n \nand\n \nNbN\n 4\n \n(2)/GdN\n \n(d)/NbN\n \n(2)\n \nsamples\n \n(Fig.\n \n4(b)\n \nand\n \nFi\ng.\n \nS6(b)),\n \nthe\n \nprofound\n \ncoherence\n \npeak\n \nis\n \nonly\n \nobserved\n \non\n \nNbN\n \n(10)/GdN\n \n(d)/NbN\n \n(10)\n \nsamples\n \nat\n \n~\n \n8.5\n \nK.\n \nThis\n \nobservation\n \nunambiguously\n \ndemonstrates\n \nthat\n \nthe\n \norigin\n \nof\n \nthe\n \ncoherence\n \npeak\n \nin\n \nthe\n \nGilbert\n \ndamping\n \narises\n \nfrom\n \nthe\n \ninterfacial\n \ns\n-\nd\n \nexchange\n \ninter\naction\n \nbetween\n \nthe\n \nmagnetic\n \nmoments\n \nof\n \nGdN\n \nand\n \nthe\n \nspins\n \nof\n \nthe\n \nquasiparticles\n \nof\n \nNbN\n \nin\n \nits\n \nsuperconducting\n \nstate.\n \nS4.\n \nThe\n \nimpact\n \nof\n \nspin\n \npumping\n \nexperiments\n \non\n \nthe\n \nT\nC\n \nBeyond\n \nthe\n \nmagnetic\n \nfield\n \neffect\n \non\n \nthe\n \nsuperconducting\n \nfilms\n \n(Fig.\n \n1(c)),\n \nthe\n \nspin\n \npumping\n \nexperiments\n \nwith\n \nvarious\n \nmicrowave\n \nexcitations\n \ndo\n \nnot\n \naffect\n \nthe\n \nsuperconducting\n \nfilms\n \neither.\n \nAs\n \nshown\n \nin\n \nFigs.\n \nS8(a)\n \nand\n \nS8(b),\n \nthe\n \nT\nC\n \nexhibit\n \na\n \nsmall\n \nvariation\n \np\nrobed\n \non\n \nthe\n \ntypical\n \nsample\n \nNbN\n \n(10)/GdN\n \n(5)/NbN\n \n(10).\n \n \n \n \n 5\n \nFigure\n \nS\n1\n \n \n \n \n \nFigure\n \nS\n1.\n \nC\nrystalline\n \nstructure\n \nof\n \nthe\n \n(111)\n-\ntextured\n \nGdN\n \nand\n \nNbN\n \nfilms.\n \n(a\n-\nb)\n,\n \nXRD\n \nresults\n \nmeasured\n \non\n \nthe\n \nNbN\n \n(10)/GdN\n \n(50)/NbN\n \n(10)\n \nsample.\n \nTwo\n \nmain\n \nobservable\n \npeaks\n \ncorrespond\n \nto\n \nGdN\n \n(111)\n \nand\n \nNbN\n \n(111).\n \n \n \n \n6\n \nFigure\n \nS\n2\n \n \n \n \nFig\nure\n \nS\n2\n.\n \nComparison\n \nof\n \nthe\n \nspin\n-\nrelaxation\n \nand\n \ntwo\n-\nmagnon\n \nscattering\n \nmechanisms\n.\n \n(a\n-\nc)\n \nThe\n \nexperimental\n \nresults\n \nof\n \n\nH\n \nvs.\n \nf\n \nat\n \nT\n \n=\n \n10\n \nK\n \nand\n \nthe\n \nfitting\n \ncurves\n \n(red\n/green\n \nlines\n \nfor\n \nthe\n \nspin\n \nrelaxation/\ntwo\n-\nmagnon\n \nscattering)\n \non\n \nNbN\n \n(10)/GdN\n \n(d)/NbN\n \n(10)\n \nsamples\n \nwith\n \nd\n \n=\n \n5,\n \n10,\n \nand\n \n30\n \nnm\n \nrespectively.\n \n \n7\n \n \nFigure\n \nS\n3\n \n \n \n \nFigure\n \nS\n3.\n \nHalf\n \nlinewidth\n \nas\n \na\n \nfunction\n \nof\n \ntemperature\n \nfor\n \nNbN\n \n(10\n)/GdN\n \n(\n5)/NbN\n \n(10\n)\n \n(a)\n \nand\n \nNbN\n \n(10)/GdN\n \n(10)/NbN\n \n(10)\n \n(b)\n \nsamples.\n \n \n \n \n8\n \nFigure\n \nS\n4\n \n \n \n \nFigure\n \nS\n4.\n \nC\nharacterization\n \nof\n \neffective\n \nmagnetization.\n \n(a)\n \nThe\n \ndetermination\n \nof\n \neff\nM\n \nvia\n \nKittel\n \nf\normula\n \n(red\n \nline)\n \nfrom\n \nthe\n \nexperimental\n \nresults\n \nof\n \nf\n \nvs.\n \nres\nH\n.\n \n(b\n-\nd)\n,\n \n4\neff\nM\n\n \nas\n \na\n \nfunction\n \nof\n \ntemperature\n \nfor\n \nthe\n \nNbN\n \n(\n2\n)/GdN\n \n(d)/N\nbN\n \n(2\n)\n \n(\ngreen\n \ncircles)\n \nand\n \nNbN\n \n(\n10\n)/GdN\n \n(d)/N\nbN\n \n(10\n)\n \nsamples\n \n(red\n \ncircles)\n \nwith\n \nd\n \n=\n \n5\n \nnm\n \n(\nb\n)\n,\n \n10\n \nnm\n \n(\nc\n),\n \nand\n \n30\n \nnm\n \n(\nd\n),\n \nrespectively.\n \n \n \n9\n \n \nFigure\n \nS5\n \n \n \n \n \n \n \n \nFigure\n \nS\n5.\n \nT\nC\n \nof\n \nNbN\n \nas\n \na\n \nfunction\n \nof\n \nthe\n \nGdN\n \nthickness\n.\n \nThese\n \nresults\n \nare\n \nobtained\n \non\n \nthe\n \nNbN\n \n(10)/GdN\n \n(d)/NbN\n \n(10)\n \nsamples\n.\n \n \n \n \n10\n \nFigure\n \nS\n6\n \n \n \nFigure\n \nS6.\n \nI\nnterface\n-\ninduced\n \nGilbert\n \ndamping\n \nin\n \nNbN\n \n(2)/GdN\n \n(d)/NbN\n \n(2)\n.\n \n(a)\n \nThe\n \nGilbert\n \ndamping\n \nconstant\n \nas\n \na\n \nfunction\n \nof\n \nthe\n \nGdN\n \nthickness\n \nat\n \nT\n \n=\n \n11,\n \n8,\n \nand\n \n6\n \nK\n,\n \nrespectively\n.\n \nSolid\n \nlines\n \nare\n \nthe\n \nfitting\n \ncurves\n \nbased\n \non\n \nthe\n \nequation\n \n(S4)\n.\n \n(b)\n \nThe\n \ninterface\n-\ninduced\n \nGilbert\n \ndamping\n \nas\n \na\n \nfunction\n \nof\n \ntemperature.\n \n \n \n11\n \nFigure\n \nS7\n \n \n \n \n \n \nFigure\n \nS\n7.\n \nTemperature\n \ndependence\n \nof\n \nGilbert\n \ndamping\n \nof\n \nPy\n \nin\n \nthe\n \nNb\n \n(100)/Py\n \n(20)/Nb\n \n(100)\n \nsample.\n \n \n \n12\n \nFigure\n \nS8\n \n \n \n \n \n \nFigure\n \nS\n8\n.\n \nThe\n \nimpact\n \nof\n \nspin\n \npumping\n \nexperiments\n \non\n \nthe\n \nT\nC\n \non\n \nthe\n \ntypical\n \nNbN\n \n(10\n)/GdN\n \n(5\n)/NbN\n \n(10)\n \nsample\n.\n \n(a)\n \nThe\n \nfour\n-\nprobe\n \nresistance\n \nvs.\n \ntemperature\n \nunder\n \n4000\n \nOe\n \nwith\n \nvarious\n \nmicrowave\n \nexcitation\n \nfrequencies.\n \n(b)\n \nThe\n \nfour\n-\nprobe\n \nresistance\n \nvs.\n \ntemperature\n \naround\n \nthe\n \nFMR\n \nresonance\n \nconditions\n \nof\n \nGdN\n.\n \n \n \n \n"    },    {        "title": "1301.5851v1.Soliton_dynamics_of_an_atomic_spinor_condensate_on_a_Ring_Lattice.pdf",        "content": "Soliton dynamics of an atomic spinor condensate on a Ring Lattice\nIndubala I. Satija1, Carlos L. Pando L.2and Eite Tiesinga3\n1School of Physics, Astronomy and Computational Sciences, George Mason University, Fairfax, VA 22030, USA\n2IFUAP , Universidad Autonoma de Puebla, Apdo, Postal J-48, Puebla 72570, Mexico and\n3Joint Quantum Institute, National Institute of Standards and Technology and the University of Maryland,\n100 Bureau Drive, Stop 8423, Gaithersburg, MD 20899-8423, USA\n(Dated: May 28, 2022)\nWe study the dynamics of macroscopically-coherent matter waves of an ultra-cold atomic spin- 1or spinor\ncondensate on a ring lattice of six sites and demonstrate a novel type of spatio-temporal internal Josephson\neffect. Using a discrete solitary mode of uncoupled spin components as an initial condition, the time evolution\nof this many-body system is found to be characterized by two dominant frequencies leading to quasiperiodic\ndynamics at various sites. The dynamics of spatially-averaged andspin-averaged degrees of freedom, however,\nis periodic enabling an unique identification of the two frequencies. By increasing the spin-dependent atom-\natom interaction strength we observe a resonance state, where the ratio of the two frequencies is a characteristic\ninteger multiple and the spin-and-spatial degrees of freedom oscillate in “unison”. Crucially, this resonant state\nis found to signal the onset to chaotic dynamics characterized by a broad band spectrum. In a ferromagnetic\nspinor condensate with attractive spin-dependent interactions, the resonance is accompanied by a transition\nfrom oscillatory- to rotational-type dynamics as the time evolution of the relative phase of the matter wave of\nthe individual spin projections changes from bounded to unbounded.\nPACS numbers: 03.75.Ss,03.75.Mn,42.50.Lc,73.43.Nq\nI. INTRODUCTION\nSpinor condensates are atomic Bose-Einstein condensates\n(BEC) with an internal spin degree of freedom that com-\nbine magnetism with condensation. Examples are optically-\ntrapped spinor condensates of atomic rubidium-87[1–4],\nsodium[5, 6], or chromium [7] with either spin or angular mo-\nmentumf= 1, 2, or 3 condensates with a three-, five-, or\nseven-component vector order parameter. In contrast to mix-\ntures of two or more atomic states[8] or mixtures of several\natomic species, spin-changing collisions in spinor conden-\nsates permit coherent dynamics among the hyperfine states.\nIn a typical process for two f= 1atoms, one in spin compo-\nnentm=\u00001and one inm= +1 , can reversibly scatter into\ntwo atoms with spin component m= 0, which conserves the\nglobal magnetization of the condensate. This coherent spin\nmixing leads, nevertheless, to oscillations of the spin popu-\nlations, and is an analogue of Josephson oscillations in ultra-\ncold atoms[9–13]. Hence, just as collisional interactions allow\nfor a single-component condensate to be spatially coherent,\nspin-changing collisions, driven by internal interactions, allow\ncoherence among internal degrees of freedom. A positive or\nnegative sign of the strength of the spin-changing interaction\ndetermines whether the systems behave anti-ferromagnetic or\nferromagnetic, respectively [10, 14]. Spinor physics has also\nbeen studied in optical lattices with exactly two atoms per lat-\ntice site [15].\nIn view of the nonlinear nature due to interparticle inter-\nactions and high degree of control in the experiments, BEC\nsystems are ideal systems for visualizing a wide variety of\nnonlinear phenomena. This includes solitary waves, the lo-\ncalized nonlinear traveling waves that retain their shape, size\nand speed during propagation [16]. Experimental realization\nof solitons in a homogeneous single component BEC systems\nis a hallmark of the quantum coherence associated with manybody systems.[17–19] There has also been theoretical studies\nof solitons in a homogeneous spin-1 condensate [20, 21]. Fur-\nthermore, theoretical and experimental studies of double-well\nbosonic Josephson junctions have unveiled novel phenomena\nsuch as broken symmetry macroscopic quantum self-trapping\n[22]and\u0019-modes[23–25] as well as symmetry restored swap-\nping modes[22]. In addition, numerical investigation of BEC\nsystems, spatially separated into a ring lattice have demon-\nstrated chaotic dynamics, deterministic dynamics with sensi-\ntive dependence on initial conditions.[26–28]\nIn this paper, we explore the quantum coherent time evo-\nlution of a spin-1 BEC that is spatially separated into six\nweakly-coupled sites arranged to form a ring geometry, see\nFig. 1. For a large number of atoms such a system is well\nrepresented by a three-component wavefunction or order pa-\nrameter,~\b(~ x;t), that satisfies a nonlinear Gross-Pitaevskii\nequation[9, 10]. Here, we restrict our calculations to the case\nwith no external magnetic field. More importantly, we assume\nthat all sites and spin components have at all times the same\nidentical localized spatial-mode function, \u001e(~ x). That is, for\nthemth-component of ~\b(~ x;t)we have\n\bm(~ x;t) =LX\nn=1 m\nn(t)\u001e(~ x\u0000~ xn); (1)\nwhereLis the number of sites, ~ xnis the center of site n, and\nthe dimensionless  m\nn(t)are complex time-dependent ampli-\ntudes. The overlap between spatial-mode functions at differ-\nent sites leads to tunneling between neighboring sites.\nMore advanced numerical modeling uses time varying\nmode functions [29, 30] or directly solves for the three-\ndimensional Gross-Pitaevskii equation[18]. As a first study\nin the soliton dynamics of a spinor condensate we believe that\nour simplified approach is justified. It shows the generaliza-\ntion of the spinor Josephson effect as well as the route to chaosarXiv:1301.5851v1  [cond-mat.quant-gas]  24 Jan 20132\n123456lattice index, n0.30.40.50.60.70.8ρnn=1n=2n=3n=4n=5n=6|Φ(x)|2\nFIG. 1: (color online) Panel a) shows a schematic of a ring lattice of\nsix sites (red dashed circle with labeled markers) containing a f=\n1spinor condensate. The solid black line represents the superfluid\npopulation along the ring as a polar graph. Panel b) shows the initial\nfractional population, \u001an, as a function of site index nused in our\ndynamical simulations. The distribution corresponds to an excited\n(period-6) solution of the DNLSE. For each site the population of\nthe three magnetic sublevels of the spinor is the same.\nin its clearest form. It should be noted that the ring lattice with\nless than three sites is an integrable system. BEC on a lattice\nwith three or more than three sites exhibits amazing degree of\ncomplexity including coexistence of regular and irregular be-\nhavior that is known to accompany chaotic dynamics. In our\ndetailed numerical exploration of ring lattices with different\nnumbers of sites, the key characteristics of the time evolution\nas discussed in this paper are independent of the number of\nsites. Here, we will present our results for a ring lattice of six\nsites only.\nOur initial state is a discrete solitary wave that is an ex-\ncited eigenstate of the discrete nonlinear Schr ¨odinger equa-\ntion (DNLSE) satisfied by the  m\nn(t)in the absence of spin-\nchanging interactions. The spin-changing interaction then\ninduces oscillatory dynamics, an internal Josephson effect\nwhere different spin components evolve with different ampli-\ntudes and phases. In reality the spin-changing Hamiltonian\ncan not be turned off and this state needs to be engineered\nby a combination of resonant electro-magnetic radiation that\ncontrollably creates superpositions of m-states [6] and off-\nresonant spatially-dependent light forces that induce solitons\n[17, 18].\nIn this article we show that for weak spin-changing interac-\ntions the dynamics on the ring lattice is dominated by two\nfrequencies. Interestingly, one of these frequencies, which\nwe denote as !, describes the periodic behavior of spatially-\naveraged spin populations. The second frequency, denoted by\n\n, describes the dynamics of the site-dependent population\nsummed over the three spin components. In other words, the\ndynamics of collective or averaged coordinates is pendulum-\nlike. As we discuss below, this is somewhat surprising as the\nspatial profiles of the three spin components evolve differ-\nently and the dynamics on the ring should deviate from that\nfor a “simple” trapped spinor condensate. The single-mode\napproximation for the latter system has been shown to leadto pendulum-type physics [10, 12]. Our study suggests that\nthe origin of this “regular” global dynamics has its roots in\nthe strong correlations among different lattice sites and spin\nprojections and is not rooted in thermal averaging over many\ndegrees of freedom. We note that !is solely determined by\nthe spin-changing interaction, whereas \nshows a rather weak\ndependence. We find that \nis mainly controlled by the size\nof the ring.\nBy increasing the absolute value of the spin-dependent in-\nteraction strength, the two frequencies can be mode locked.\nThis resonance condition is found to describe the onset to\nchaotic dynamics. In other words, the spinor condensate ex-\nhibits a transition from quasi-periodic dynamics to chaotic\ndynamics where at the onset to transition, all local and\nglobal degrees of freedom oscillate in unison . Consequently,\nthe initial soliton profile reappears periodically, providing a\nunique demonstration of a novel-type of spatio-temporal in-\nternal Josephson effect. Furthermore, in contrast to the anti-\nferromagnetic case, in the ferromagnetic condensate, the res-\nonance is accompanied by a transition from bounded to un-\nbounded dynamics. In the anti-ferromagnetic case, the dy-\nnamics remains bounded and oscillatory. Nevertheless, be-\nyond the resonance additional frequencies appear.\nFor sake of simplicity, we restrict our simulations to zero\nmagnetization. This is enforced by choosing initial condi-\ntions where the m= +1 andm=\u00001components have\nsame initial wave function. In fact, we choose an initial state\nwhere all three components have the same initial wave func-\ntion. An alternate initial state with zero magnetization and\ncontains mostly m=\u00061atoms lead to similar results. It will\nnot be described here.\nIn section II we review the mean-field equations that de-\nscribe the evolution of the spinor condensate on a lattice as\nwell as the underlying single-mode approximation approxi-\nmation on each lattice site. The latter has been shown to pro-\nvide a reasonable description for the continuum system. For\nour ring lattice, the SMA suggests collective coordinates to\ndescribe the global dynamics. The initial state is described\nin Sec. III. In section IV, we show numerical simulations\nfor weak positive, anti-ferromagnetic spin-dependent integra-\ntions and show that the dynamics, although complex, is dom-\ninated by two frequencies. In section V, we discuss the reso-\nnance condition that occurs when the spin-dependent interac-\ntion strength is increased. At resonance we observe an onset\nto chaos. Section VI briefly illustrates the ferromagnetic case.\nWe conclude in Sec. VII.\nII. MEAN-FIELD EQUATIONS FOR A SPINOR\nCONDENSATE AND COLLECTIVE COORDINATES\nAn atomic spinor condensate with large atom number is de-\nscribed within mean-field theory by a complex vector order\nparameter or condensate wave function \t(~ x;t), whose evolu-\ntion is governed by a multi-component Gross-Pitaevskii equa-\ntion [9, 10]. We will assume that the order parameter can\nbe approximated by Eq. 1, where a single time-independent\nmode function, \u001e(~ x), determines the spatial dependence in3\neach well. We denote this by the L-site single-mode approx-\nimation (L-SMA) in analogy to the single-mode approxima-\ntion (SMA) for a spinor condensate in a dipole trap.\nThe interactions between two spin-1 atoms have\na spin-independent and a spin-dependent or (spin-\nchanging) contribution. Within a mean-field the-\nory and the L-SMA their strength is given by\ncq= 4\u0019~2gq=(2\u0016)R\nd~ xj\u001e(~ x)j4=R\nd~ xj\u001e(~ x)j2withq= 0and\n2 for the spin-independent and spin-dependent contribution,\nrespectively. Here \u0016is the reduced mass for two atoms\nand~is the reduced Planck constant. The lengths gqare\ng0= (a0+2a2)=3andg2= (a2\u0000a0)=3, wherea0anda2are\nscattering lengths for s-wave collisions of two f= 1 bosons\nwith total angular momentum F= 0 and2, respectively.\nFor stable condensed gases we require that c0>0. We\nnote thatc2>0for anti-ferromagnetic Na and c2<0for\nferromagnetic87Rb.\nThe dynamics of the  m\nn(t)are governed by the DNLSE\ni~_ \u00001\nn=\u0000J( \u00001\nn\u00001+ \u00001\nn+1) (2)\n+(c0+c2)(j \u00001\nnj2+j 0\nnj2) \u00001\nn\n+(c0\u0000c2)j 1\nnj2 \u00001\nn+c2( 0\nn)2( 1\nn)\u0003\ni~_ 0\nn=\u0000J( 0\nn\u00001+ 0\nn+1) +c0j 0\nnj2 0\nn (3)\n+(c0+c2)(j 1\nnj2+j \u00001\nnj2) 0\nn+ 2c2 1\nn \u00001\nn( 0\nn)\u0003\ni~_ 1\nn=\u0000J( 1\nn\u00001+ 1\nn+1) + (c0+c2)(j 1\nnj2+j 0\nnj2) 1\nn\n+(c0\u0000c2)j \u00001\nnj2 1\nn+c2( 0\nn)2( \u00001\nn)\u0003; (4)\nwhereJis the positive site-to-site tunneling energy and we\nuse periodic boundary conditions. In the absence of the last\nterm on the right hand side of Eqs. 2-4, the set of equations de-\nscribes a three-species condensate. Spin-changing terms make\na spinor condensate unique as they induce population oscilla-\ntions between mlevels. Equations 2-4 conserve total atom\nnumber and magnetization. In other words,P\nnmj m\nn(t)j2\nandP\nnmmj m\nn(t)j2are conserved.\nWe will monitor the local population of each spin state as\nwell as global or collective coordinates, such as spatially- and\nspin-averaged population and phases. It is therefore conve-\nnient to define  m\nn(t) =p\n\u001amn(t) exp[i\u001em\nn(t)]with popula-\ntions\u001am\nn(t)and phases\u001em\nn(t). Following Refs. [9, 10] for a\nsingle-mode spinor condensate natural local canonical coordi-\nnates are\nZn=\u001a0\nn\u0000(\u001a+1\nn+\u001a\u00001\nn) and\rn=\u001e\u00001\nn+\u001e1\nn\u00002\u001e0\nn:(5)\nas well as globally-averaged coordinates Z=P\nnZn=Land\n\r=P\nn\rn=L. Throughout this article we call \rnand\r\nspinor phases. Other useful population averages are\n\u001am=1\nLLX\nn=1\u001am\nnand\u001bn=1\n31X\nm=\u00001\u001am\nn: (6)\nIn a dipole trap, or equivalently for L= 1, a simple spinor\nmodel is given by the SMA. The canonical variables Zand\r\nthen satisfy the pair of equations\n_Z=c2\n~(1\u0000Z2) sin\rand _\r=\u00002c2\n~Z(1 + cos\r);(7)which are independent of the spin-independent interaction\nwith strength c0. For small Zand\rthe dynamics are har-\nmonic. In general, however, these coupled equations describe\na nonlinear pendulum whose length depends upon the mo-\nmentum.\nFor aL > 2ring of lattice sites the collective variables Z\nand\rsatisfy\n_Z=c2\n~X\nn(\u001b2\nn\u0000Z2\nn) sin\rn (8)\n_\r=\u00002c2\n~X\nnZn(1 + cos\rn) +J\n~W (9)\nwhereW= Re[P\nn(2 0\nn+1= 0\nn\u0000 +1\nn+1= +1\nn\u0000 \u00001\nn+1= \u00001\nn)].\nThe spin-independent interaction strength c0does not explic-\nitly appear in these equations. As already mentioned in the\nintroduction and further discussed in next section, this sug-\ngests that the spatially-averaged Zand\rwill exhibit periodic\noscillations, with frequency !, that are solely determined by\nthe spin-changing interaction. Finally, the spin-averaged pop-\nulations at each site satisfy\n_\u001bn=\u0000J\n~Im 1X\nm=\u00001( m\nn)\u0003[ m\nn+1+ m\nn\u00001]!\n; (10)\nwhich does not explicitly depend on c2. Hence, we expect\nthat the\u001bnoscillate periodically, characterized by frequency\n\nand its harmonics.\nIII. INITIAL DISCRETE SOLITON\nFor a ring lattice of six sites, we use the initial condition\nshown in Fig. 1. Commonly referred to as a discrete soliton,\nit is a stationary solution of the DNLSE in the absence of the\nspin-changing interaction [26, 31] (i.e. with c2= 0) andc0=\nJ. Along with phase \u001em\nn= 0 or\u0019for even or odd site index\nn, respectively, the initial state populations are the same for\nthe three components.\nThis stationary solution, corresponding to a solution\n m\nn(t) = m\nn(t= 0) exp(\u0000i\u0016t=~)where\u0016is the chemi-\ncal potential, is obtained by numerically solving the result-\ning nonlinear map, as Eqs. 2-4 reduce to a set of Ltwo-\ndimensional cubic maps. Following Ref. [28], we find that for\nL= 6the solutions are 6-fold degenerate and for a range of \u0016\nthe localized soliton mode in Fig. 1 is the only stable solution.\nIn fact, we have used \u0016= 2:5J. Intriguingly, for these values\nof the chemical potential the homogeneous solution with the\nsame density at all sites is unstable.\nIt should also be noted that known localized soliton-type\nsolutions with zero phase for all sites correspond to attractive\nspin-independent interactions with c0<0. To obtain local-\nized solutions for repulsive interactions, one needs to consider\nsolutions with phases different from zero. In general, it can be\nshown that for a lattice with even number of sites, the mapping\n m\nn!(\u00001)n( m\nn)\u0003relates a soliton solution for attractive in-\nteractions with those with repulsive interactions. For a single-\ncomponent condensate discrete solitons have been studied ex-\ntensively for ring lattices of various sizes [26–28].4\nFIG. 2: (color online) Time dynamics of an anti-ferromagnetic spinor\nsoliton on a six-site ring for several spin- and/or spatially-averaged\ndegrees of freedom assuming a small positive spin-changing inter-\naction energy. Time is in units of ~=J, whereJis the tunneling\nenergy between the sites. Calculations are performed for c2= 0:1J\nandc0=J. Panel a) shows the dynamics of the spatially-averaged\nspinor population Z(black line) and the spinor population Znfor\nthe individual sites n(colored lines). The symmetry of the initial\nsoliton around n= 3 implies that only four of the six sites have a\ndistinct time evolution. Panel b) shows the spinor phases \r(black\nline) and\rn(colored lines). Finally, panel c) shows the site-specific\npopulation\u001bn, averaged over the three spin components.\nExperiments with single-component Bose condensates in\ndouble-well potentials [24] have observed Josephson oscilla-\ntions and quantum self-trapping in the limit J\u001cc0. The\nopposite limit can also be reached leading to tunneling of\n(nearly-)independent atoms. Here, we chose a compromise\nwithc0=J. As an aside we note that with our initial state\nandc0=Jwe have implicitly specified the chemical poten-\ntial and thus atom number in each site.\nIV . QUASIPERIODIC DYNAMICS OF AN\nANTI-FERROMAGNETIC SPINOR\nFigure 2 shows the time evolution of the spinor soliton on a\nsix-site ring, described in terms of the spinor coordinates Zn\nandZ, spinor phases \rnand\r, as well as populations \u001bn. We\nuse a small positive spin-changing interaction strength c2for\nan anti-ferromagnetic spinor. We observe that the time depen-\ndence ofZand\rare nearly sinusoidal. The site-dependent\nZn,\rnand\u001bn, however, oscillate at a higher frequency. They\ndo so in a non-sinusoidal manner with sharper minima than\nmaxima or vice versa. The phases \rnonly show small excur-\nFIG. 3: (color online) Power spectrum or Fourier transform of\nthe time evolution of the spatially-averaged spin population \u001am=+1\n(red), spatially-averaged spinor phase \r(green), and spin-averaged\npopulation\u001bn=1at siten= 1 (blue). The parameters and initial\nstate are as in Fig. 2 and the frequency is in units of J=~. The top\nand bottom panel show the power spectra on a linear and logarith-\nmic scale, respectively. Also indicated are the dominant frequencies\n!and\n, which have been assigned as due to the spin-dependent\nand spin-independent interactions, respectively. In the bottom panel\nhigher harmonics of \ncan be observed.\nsions around the average \r. Finally, we note that the spinor\nphases are bounded for oscillatory motion.\nFigure 3 shows the power spectrum of three of the time\ntraces shown in Fig. 2. It highlights the existence of two\ndominant frequencies, !and\n, as well as weaker higher\nharmonics in \nindicating non-sinusoidal periodic behavior.\nThe spatially-averaged degrees of freedom predominantly os-\ncillate with a frequency !, which from simulations with other\nsmallc2is found to be proportional to the absolute value of c2.\nIn contrast, spin-averaged local populations oscillate nearly-\nsinusoidal with frequency \n, which from other simulations is\nfound to weakly depend on the spin-changing interaction but\nis inversely proportional to the number of lattice sites. Fi-\nnally, the local dynamics for individual spin projections is\nquasiperiodic with a slow frequency !and a faster beat fre-\nquency at multiples of \n. This illustrates the correlations that\nexist among the spatial and spin components. Detailed stud-\nies with various ring sizes indicate that the spinor dynamics\nis characterized by two frequencies, irrespective of the size of\nthe ring.5\nFIG. 4: (color online) Time series for an anti-ferromagnetic\nspinor showing the resonance condition for increasing spin-changing\nstrengthc2. Time is in units of ~=Jandc0=J. In all panels Z(red\ncurve) and\u001bn=1(blue curve) are shown. Panels a), b), and c) show\nresults forc2= 0:5J,0:65J, and 1:0J, respectively. The resonance\ncondition occurs for c2= 0:65J, where the spatially-averaged Z\nand spin-averaged \u001bn=1oscillate at the same rate. For c2>0:65J\nthe traces are chaotic.\nV . RESONANCE CONDITION FOR THE\nANTI-FERROMAGNETIC CONDENSATE\nFigures 1 and 2 showed that for small c2=c0two frequencies\n!and\nare very distinct. For increasing positive c2at fixed\nc0, as shown by time traces in Fig. 4 and power spectra in\nFig. 5, both frequencies increase although at a different rate.\nFor a critical spin-changing strength c2when!= \n the anti-\nferromagnetic spinor reaches a “resonance” state. All three\nspin-components at all the sites of the ring then oscillate as a\nsingle entity. For c0=Jthis resonance occurs at c2= 0:65J.\nFor large values of c2the behavior becomes chaotic, which is\napparent as broad-band features in the corresponding power\nspectrum shown in Fig. 5.\nFourier analysis of collective as well as local coordinates\nshows that the resonance state corresponds to matching of not\nonly the dominant frequencies !and\n, but also some of the\nother less prominent frequencies (not visible in the linear plot)\nas illustrated in the log-linear plot on the middle row of Fig. 5.\nIn fact, at resonance, the dominant or the primary peak is ac-\ncompanied by secondary satellite peaks, equally spaced on the\neither side of the primary peak.\nFigure 6 further illustrates that at resonance all sites oscil-\nlate in phase with same frequency. The spatial profile of the\nsoliton reemerges periodically without any significant change\nfrom the initial profile. Thus, the resonance state in an ordered\nFIG. 5: (color online) Power spectra showing the resonance condi-\ntion for increasing spin-changing strength c2for the same parame-\nters as in Fig. 4. The frequency is in units of J=~. In all panels the\nspatially-averaged \u001am=+1 (red curve) and spin-averaged \u001bn=1(blue\ncurve) are shown. Panels on the left and right show spectra on a\nlinear and logarithmic scale, respectively, while from top to bottom\nc2= 0:5J,0:65Jand1:0J.\nFIG. 6: (color online) Spin-averaged soliton profiles, \u001bn, for the anti-\nferromagnetic spinor before and at resonance for five times spaced\nby time interval 2\u0019=\n, starting with the initial state (red curve). We\nusec0=Jand panels a) and b) correspond to c2= 0:5Jand\nc2= 0:65J, respectively. The soliton reforms close to its initial\nstate at resonance when c2= 0:65J.\nstate where wave functions of all the components of the spinor\ncondensate at all L-sites of the ring lattice oscillate in unison\nand the dynamics is well characterized one frequency. The\nfact that the soliton profile reemerges periodically provides a\nunique demonstration of quantum coherence and a novel-type\nof internal spatio-temporal Josephson effect.6\nFIG. 7: (color online) Time evolution of spatially-averaged popula-\ntionZ(red) and spin-averaged population \u001bn=1(blue) for a ferro-\nmagnetic spinor condensate with negative c2. Panel a), b), and c)\nshow traces for c2=\u00000:075J,\u00000:09J, and\u00000:095J, respec-\ntively, corresponding to cases before, at, and beyond resonance. We\nusec0=Jand time is in units of ~=J.\nFIG. 8: Phase portraits or parametric plots of the time-evolution of\nthe spinor population Zand phase\rfor a ferromagnetic spinor at\nc2=\u00000:075J, panel a), and c2=\u00000:095J, panel b). The system\nis at resonance in panel b). The dynamics in panel b) is accompanied\nby unbounded motion of the spinor phase. We use c0=J.VI. FERROMAGNETIC CONDENSATE\nWe now briefly describe the dynamics of a ferromag-\nnetic spinor condensate with negative c2. For a small spin-\nchanging interaction, the dynamics is similar to that of an\nanti-ferromagnetic spinor. Namely, the spatially-averaged be-\nhavior is pendulum-like with dominant frequency !, while the\nlocal population oscillates with frequency \n. An example is\nshown in Fig. 7. The local density for each spin component\noscillates with both frequencies. A resonance state can again\nbe achieved by increasing jc2jbut, forc0=J, now occurs\nwhen!=\n = 3 .\nIn contrast to the anti-ferromagnetic case, however, the res-\nonant transition is accompanied by unbounded dynamics or\nphase winding where the phase of the condensate becomes un-\nbounded as shown in Fig. 8. This behavior manifests itself as\na zero-frequency mode in a power spectrum. In other words,\nchaotic dynamics with its broad-band spectrum is accompa-\nnied by a transition from an oscillatory to rotational mode for\nthe collective degrees of freedom.\nVII. DISCUSSION\nIn summary, numerical explorations of mean-field equa-\ntions of spinor condensate on a ring lattice with a small\nnumber of sites reveals strikingly correlated dynamics of the\nmany-body system. Even though the condensate is separated\nintoL-sites, soliton dynamics at all sites can be characterized\nby just two frequencies. With quasiperiodic dynamics at lo-\ncal sites for individual spin components, spatially averaged\nbehavior for each spin component as well as spin-averaged\ndynamics at each sites is found to be periodic. The fact that\nthe time series describing local dynamics is characterized by\ntwo frequencies and these two frequencies untangle in the col-\nlective degrees of freedom is rooted in correlations among dif-\nferent spin degrees of freedom at various sites of the lattice.\nHowever, proper understanding of these correlations remains\nan open challenge. Our study with ring lattices of various\nsizes show that two-frequency characterizations of the spin- 1\ncondensate on a ring lattice is valid irrespective of the number\nof sites on the lattice.\nOur study provides a new illustration of a quasiperiodic\nroute to chaotic dynamics in a many-body system where the\ncritical point is known to be characterized by a resonant\nstate. Simple models of dynamical systems such as the one-\ndimensional circle map[32] are paradigms of quasiperiodic\nroute to chaos, where the critical point corresponds to pa-\nrameters where the two frequencies are mode-locked. The\nquasiperiodic route to chaos is a well-established scenario\nin dynamical systems exhibiting a transition from regular to\nchaotic dynamics [32]. However, the fact that the critical point\ndescribes periodic dynamics is a novelty in many-body sys-\ntems rooted in the coherence associated with spinor conden-\nsate. In this case, the critical point is a highly-ordered many-\nbody state exhibiting a new type of spatial-temporal coher-\nence. At the critical point where the two dominant frequen-\ncies are in resonance, the ring lattice with L-sites oscillates7\nin unison with a single characteristic frequency. This internal\nJosephson effect where an unscathed soliton profile reemerges\nperiodically provides a novel illustration of both spatial andtemporal coherence.\nThis research is supported by Office of Naval Research, the\nCONACYT-M ´exico and the US Army Research Office.\n[1] T. Kuwamoto, K. Araki, T. Eno, and T. Hirano, Phys. Rev. A\n69, 063604 (2004).\n[2] M.-S. Chang, C. D. Hamley, J. A. Sauer, K. M. Fortier,\nW. Zhang, L. You, and M. S. Chapman, Phys. Rev. Lett. 92,\n140403 (2004).\n[3] H. Schmaljohann, M. Erhard, J. Kronj ¨ager, M. Kottke, S. van\nStaa, L. Cacciapuoti, J. J. Arlt, K. Bongs, and K. Sengstock,\nPhys. Rev. Lett. 92, 040402 (2004).\n[4] J. Guzman, G.-B. Jo, A. N. Wenz, K. W. Murch, C. K. Thomas,\nand D. M. Stamper-Kurn, Phys. Rev. A 84, 063625 (2011).\n[5] A. T. Black, E. Gomez, L. D. Turner, S. Jung, and P. D. Lett,\nPhys. Rev. Lett. 99, 070403 (2007).\n[6] Y . Liu, S. Jung, S. E. Maxwell, L. D. Turner, E. Tiesinga, and\nP. D. Lett, Phys. Rev. Lett. 102, 125301 (2009).\n[7] B. Pasquiou, E. Mar ´echal, G. Bismut, P. Pedri, L. Vernac,\nO. Gorceix, and B. Laburthe-Tolra, Phys. Rev. Lett. 106,\n255303 (2011).\n[8] C. J. Myatt, E. A. Burt, R. W. Ghrist, E. A. Cornell, and C. E.\nWieman, Phys. Rev. Lett. 78, 586 (1997).\n[9] H. Pu, C. K. Law, S. Raghavan, J. H. Eberly, and N. P. Bigelow,\nPhys. Rev. A 60, 14631470 (1999).\n[10] W. Zhang, D. L. Zhou, M.-S. Chang, M. S. Chapman, and\nL. You, Phys. Rev. A 72, 013602 (2005).\n[11] L. Santos and T. Pfau, Phys. Rev. Lett. 96, 190404 (2006).\n[12] R. Barnett, J. D. Sau, and S. Das Sarma, Phys. Rev. A 82,\n031602 (2010).\n[13] E. Yukawa and M. Ueda, Phys. Rev. A 86, 063614 (2012).\n[14] R. Barnett, A. Turner, and E. Demler, Phys. Rev. Lett. 97,\n180412 (2006).\n[15] A. Widera, F. Gerbier, S. F ¨olling, T. Gericke, O. Mandel, and\nI. Bloch, Phys. Rev. Lett. 95, 190405 (2005).\n[16] A soliton is a solitary wave with a special collision property,\nviz., it retains its identity even after collision with another soli-\ntary wave. Following a common practice in physics lierature,\nwe will refer to nonlinear localized solitary waves as solitionshere.\n[17] S. Burger, K. Bongs, S. Dettmer, W. Ertmer, K. Sengstock,\nA. Sanpera, G. V . Shlyapnikov, and M. Lewenstein, Phys. Rev.\nLett. 83, 5198 (1999).\n[18] J. Denschlag, J. E. Simsarian, D. L. Feder, C. W. Clark, L. A.\nCollins, J. Cubizolles, L. Deng, E. W. Hagley, K. Helmerson,\nW. P. Reinhardt, et al., Science 287, 97 (2000).\n[19] R. V . Mishmash, I. Danshita, C. W. Clark, and L. D. Carr, Phys.\nRev. A 80, 053612 (2009).\n[20] M. Wadati and N. Tsuchida, J. Phys. Soc. Jpn. 75, 014301\n(2006).\n[21] J. Ieda, T. Miyakawa, and M. Wadati, Laser Physics 16, 678\n(2006).\n[22] I. I. Satija, R. Balakrishnan, P. Naudus, J. Heward, M. Edwards,\nand C. W. Clark, Phys. Rev. A 79, 033616 (2009).\n[23] A. Smerzi, S. Fantoni, S. Giovanazzi, and S. R. Shenoy, Phys.\nRev. Lett. 79, 4950 (1997).\n[24] M. Albiez, R. Gati, J. Folling, S. Hunsmann, M. Cristiani, and\nM. K. Oberthaler, Phys. Rev. Lett. 95, 010402 (2005).\n[25] K. W. Mahmud, H. Perry, and W. P. Reinhardt, Phys. Rev. A 71,\n023615 (2005).\n[26] C. L. Pando L. and E. J. Doedel, Phys. Rev. E 69, 036603\n(2004).\n[27] P. Buonsante, P. G. Kevrekidis, V . Penna, and A. Vezzani, Phys.\nRev. E 75, 016212 (2007).\n[28] C. L. Pando L. and E. Doedel, Physica D 238, 687 (2009).\n[29] T. Anker, M. Albiez, R. Gati, S. Hunsmann, B. Eiermann,\nA. Trombettoni, and M. K. Oberthaler, Phys. Rev. Lett. 94,\n020403 (2005).\n[30] O. E. Alon, A. I. Streltsov, and L. S. Cederbaum, Phys. Lett. A\n362, 453 (2007).\n[31] C. Pando L. and E. J. Doedel, Phys. Rev. E 71, 056201 (2005).\n[32] E. Ott, Chaos in Dynamical Systems (Cambridge University\nPress, 1993), 1st ed."    },    {        "title": "2012.02674v1.Robust_Spin_Interconnect_with_Isotropic_Spin_Dynamics_in_Chemical_Vapour_Deposited_Graphene_Layers_and_Boundaries.pdf",        "content": "1 Robust Spin Interconnect  with Isotropic Spin Dynamics  in \nChemical Vapour Deposited Graphene Layers and Boundaries \nDmitrii Khokhriakov1, Bogdan Karpiak1, Anamul Md. Hoque1, Bing Zhao1, \nSubir Parui2, Saroj P. Dash1,3* \n1Department of Microtechnology and Nanoscience, Chalmers University of Technology, \nSE-41296, Göteborg, Sweden,  \n2K.U. Leuven, 3001 Leuven, Belgium,  \n3Graphene center, Chalmers University of Technology, SE -41296, Göteborg, Sweden.  \nA\nbstract \nThe u\ntilization of large- area graphene grown by chemical vapour  deposition (CVD) is crucial \nfor the development of scalable spin interconnects  in all- spin-based memory and logic circuits . \nHowever, the fundamental  influence of the presence of multilayer graphene patches  and their \nboundaries on spin dynamics  has not been addressed yet, which is necessary for basic \nunderstanding and application of robust spin interconnects . Here, we report universal spin \ntransport and dynamic  properties  in specially devised single layer, bi -layer , and tri -layer \ngraphene channels and their layer boundaries  and folds  that are usually  present in CVD  \ngraphene samples. We observe uniform spin lifetime with  isotropic spin relaxation for spins \nwith different orientations in  graphene layers and their  boundaries  at room temperature . In all \nthe inhomogeneous graphene channels, t he spin lifetime anisotropy ratios for spins polarized \nout-of-plane and in- plane are measured to be close to uni ty. Our analysis shows the \nimportance of both Elliott -Yafet and D’yakonov -Perel’ mechanisms, with an increasing role of \nthe latter mechanism in multilayer channels. These results of universal and isotropic spin \ntransport  on large- area inhomogeneous CVD graphene with multilayer patches and their \nboundaries and folds at room temperature prove its  outstanding spin interconnect \nfunctionality , beneficial  for the development of scalable spintronic circuits . \nK\neywords:  graphene, multilayer , spin transport, spin precession, spin lifetime anisotropy.  \nC\norresponding author:  dmikho@chalmers.se; saroj.dash@chalmers.se  2 Spintronics aims at utilizing the electron spin degree of freedom for developing future data \nstorage and processing devices, promising faster operation and lower power consumption.1,2 \nSo far, substantia l progress has been achieved in non-volatile spintronic memory technology, \nwhere the spin-transfer torque and spin- orbit torque have enabled electrical control and \nswitching of the magnetization.3–5 For the realization of  spin transistors and all- spin-logic \ndevices,  such non- volatile nanomagnetic elements should be connected by coherent spin \nchannels with the possibility to manipulate the propagating spins .2,6 Significant research efforts \nwere devoted to  finding a suitable spin interconnect exhibiting long spin lifetime and spin \ndiffusion length, with spintronic studies of metals ,7 semiconductors ,8–11 and two -dimensional \nmater ials.12–14 \nGraphene has emerged as a promising interconnect material for spintronic applications due \nto its long spin diffusion length and low spin- orbit coupling.14 Whereas  the research on a single -\ndevice level is mainly done with the exfoliated graphene, large- scale fabrication prompts the \nuse of a scalable production technique such as chemical vapor deposition (CVD). However, \nas opposed to a single- crystalline nature of exfoliated flakes, CVD growth can naturally lead \nto the formation of grain boundaries15,16 and multilayer graphene patches .17,18 For the \nsuccessful integration of CVD graphene as a material of choice for the realization of long-\ndistance spin interconnects in emerging spintronic devices, it is essential to investigate and \nunderstand the effect of these naturally -occur ring inhomogeneities on spin relaxation \nproperties.  Particularly, o ne of the critical requirements for its application in spin interconnects \nis an experimental demonstration of universal and isotropic spin transport in CVD graphene, \nmultilayer patches, and their boundaries  at room temperature . \nTheoretical studies of multilayer graphene and layer boundaries predicted  their modified  band \nstructure , interlayer carrier hopping effects ,19,20 and the emergence of  valley polarization,21 \nedge states ,22 and interface Landau levels .23 Experimentally, electr ical transport properties of \nmultilayer boundaries have been studied in epitaxial graphene24,25 and, recently, in CVD-\ngrown channels .26 Owning to the transition from linear to parabolic energy dispersion at the \nlayer boundary, a localized conductance minimum was observed due to poor wave function \nmatching at the interface,25 whereas  the presence of edge -channel transport along the \nboundary  was confirmed by quantum transport measurements .26 In spintronics,  theoretical \ninvestigations predicted that the spin relaxation by  CVD- specific defects such as grain \nboundaries and ripples should not limit the device performance,27 and experimental studies \nwith continuous CVD graphene channels have confirmed their spin transport capabil ities.28–32 \nHowever, the spin scattering, dynamics , and spin lifetime anisotropy in multilayer CVD \nsamples and their boundaries have not yet been determined experimentally . \nHere, we demonstrate robust spin transport and isotropic relaxation in monolayer, bilayer , and \ntrilayer graphene patches and across their layer boundaries that  are naturally  present  in CVD \ngrow n graphene . The spin dynamics in continuous multilayer patches  and through their \nboundaries and folds are probed by using spin valve and Hanle spin precession experiments  \nat room temperature . The detailed measurements of spin precession in oblique magnetic fields  \nand at various doping levels provide information about the spin dynamics , spin lifetime \nanisotropy  and spin relaxation mechanisms . Our study suggests that inhomogeneities  in large-\narea CVD graphene do not deter its  spintronic device performance, showing universal ly \nisotropic spin transport propert ies at room temperature, which makes CVD graphene suitable \nfor the realization of spin interconnects.  3 Results and Discussion \nFigure 1. A spintronic device with multilayer CVD graphene and layer boundaries . a, A schematic \nof a multilayer graphene patch with diffusion and precession of spin -polarized electrons . b, An optical \nmicroscope image of a multilayer CVD graphene patch on SiO 2/Si substrate with an outlined area used \nfor device fabrication  spanning single- , bi-, and tri -layer graphene (SLG, BLG, TLG) areas, as well as \nthe boundaries between them.  c, A micrograph of a nanofabricated CVD graphene spintronic device \nused to investigate spin dynamics in SLG, BLG, TLG , and their layer boundaries with ferromagnetic \n(FM) tunnel contacts and nonmagnetic (NM) electrodes.  \nThe graphene spintronic devices were specifically designed to investigate spin relaxation and \ndynamics in single -layer ( SLG), bilayer (BLG), and trilayer (TLG) patches and across their \nlayer boundaries. Figure s 1a and 1b show  a schematic and an optical picture of a multilayer \nCVD graphene patch transferred on a Si/SiO 2 substrate. The optical contrast allows to identify \nthe S LG, BLG , and TLG regions. Depending on the CVD process parameters, adlayers of \ngraphene have been shown to grow either underneath or on top of the first-formed layer, which \nis referred to as the Volmer –Weber (VW) and Stranski– Krastanov  regimes, respectively .33 In \naddition, the different growth regimes were utilized to produce Bernal -stacked34 (twist angle \n0°), turbostratic35 (twist angle 30°) , and arbitrary -twist-angle36,37 graphene. Knowledge of the \nrelative rotation angle is essential because if the graphene layers are not perfectly aligned,  \nthey form a moiré pattern, which  alters its electronic properties  and, for specific angles ( e.g., \nat a magic angle ~1.1°), gives rise to exotic electronic states including Mott insulator s38 and \nunconventional superconductor s.39 According to the information provided by the manufacturer \n(Grolltex Inc.), the graphene adlayers in our samples were formed beneath the initial layer, \nindicating the VW growth. As the corners of the BLG and TLG appear aligned,40,41 and the VW \nregime predominantly yields Bernal -stacked graphene,33 we consider that our patches have \nno twist angle and are not subjected to a moiré potential.  \nA CVD graphene device was patterned in a stripe shape spanning multilayer areas and their \nboundaries by electron beam lithography and oxygen plasma etching (Fig.1c) . For the \nspintronic characterization of the devices, ferromagnetic tunnel contacts  (Co/TiO 2) (FM) and \nnonmagnetic reference contact s (Au/Ti) were fabricated by subsequent EBL, metal \nevaporation and lift -off processes  (see details in Methods) .  \n10 μmTLGBLG\nSLGa b\nc\nNM\nFMSLG TLG\nBLG\n5 μm\n4 Spin transport and precession in multilayer graphene patches  \nF\nigure 2. Spin transport and precession in multilayer CVD graphene.  a, A schematic of the device \nand the nonlocal measurement geometry  for spin transport and precession studies . b, The spin valve \nand Hanle spin precession signal s measured in CVD SLG, BL G, and TLG channels. The channel \nlengths L are noted in respective plots.  The measurements were performed with I  = –50 μA and Vg = 0 \nV.c, Spin lifetime, spin diffusion constant and spin diffusion length in SLG, BLG , and TLG extracted as\na function of the carrier concentration ( n) in graphene  channels .\nSpin transport in graphene was characterized using nonlocal  (NL) spin valve (SV) and Hanle \nspin precession experiments  at room temperature.  The spin accumulation in graphene is \ncreated by passing a current I between the injector ferromagnetic contact and graphene, while  \nthe NL spin -dependent voltage is detected by another ferromagnetic contact placed at a length \nL away from the injector  (Fig. 2a) . To perform the spin  valve measurement, we sweep the in-\nplane m agnetic field By along the easy axis of the ferromagnetic contacts while  recording the \nnonlocal resistance. Sharp changes in R NL=V/I are measured when the magnetization of the \ninjector or detector switches giving either parallel or antiparallel configuration, as shown in \nFigure 2b. \nHanle spin precession measurements were performed by measuring the nonlocal resistance  \nwhile  sweeping an out -of-plane magnetic field Bz, which induces spin precession and \ndephasing in the graphene channel. The data, shown in Fig. 2b , are fitted with eq. 1,  \na\nb cFM NM\nGr\nx\nyz5 \n 𝑅𝑅NL(𝐵𝐵)=𝑅𝑅𝑅𝑅�𝑃𝑃2𝑅𝑅□𝜆𝜆s�\n𝑊𝑊gr𝑅𝑅−𝐿𝐿\n𝜆𝜆s��                                                          (1) \nwith \n𝜆𝜆s�=𝜆𝜆s\n�1+𝑖𝑖𝜔𝜔L𝜏𝜏s \nwhere 𝑃𝑃 is the average spin polarization of ferromagnetic contacts , 𝑅𝑅□ is the sheet resistivity \nof graphene, 𝑊𝑊gr is the graphene channel width, 𝜔𝜔L=𝑔𝑔𝜇𝜇B \nℏ𝐵𝐵 is the Larmor spin precession \nfrequency, 𝜇𝜇B is Bohr magneton, 𝐿𝐿 is the channel length, 𝜆𝜆s=�𝐷𝐷s𝜏𝜏s is the spin diffusion length \nwith 𝐷𝐷s and 𝜏𝜏s being the spin diffus ion coefficient and spin lifetime, respectively.  A small \ncharge- based background signal was subtracted from the data by performing measurements \nwith parallel and antiparallel configurations of the ferromagnetic electrodes (see Figure S 1). \nThe corresponding SV and Hanle measurements in channels of SLG, BLG , and TLG are \nshown in Fig. 2 b, establishing the presence of the spin transport and allowing to characterize \nspintronic parameters of the system. Performing such measurements at different gate \nvoltages allows us to trace the parameters with changing carrier density in graphene, as \nshown in Fig. 2 c. Strong p-type doping of our CVD samples  (see Figure S2)  restricts the \naccessible range of carrier concentration to mostly negative n values  correspond ing to hole \nconduction.  We observe three main features in our data: (i) the spin parameters of SLG are \nhigher than that of BLG and TLG, (ii) all parameters tend to reduce as the carrier density is \nreduced, (iii) the degree of tunability of the parameters reduces with the increasing number of layers.   \nThe observed larger parameter values of SLG may indicate its superior capability for spin \ntransport, but the differ ence can also appear due to  the larger channel length of SLG \ncompared to BLG and TLG .\n29 Although the spin signal amplitude decreases exponentially with \nthe channel length, long channels c an be beneficial to reduce the adverse effects of additional \ndefects and doping induced by the contacts into the underlying graphene ,42 which can diminish \nthe spin parameters in shorter channels. In addition, whereas  the presence of multiple \ngraphene layers can be expected to protect the inner conducting paths from the scattering on \nmagnetic surface impurities and increase the spin performance of multilayer graphene ,43,44 the \ninterlayer hopping of carriers may introduce spin- dependent scattering effects .45  \nThe reduction of spin transport parameters as the graphene Fermi level is tuned to the charge \nneutrality point  (CNP)  reflects the intrinsic relation between the carrier density and the diffusion \nconstant, which also influences the spin lifetime due to the proportional ity between 𝜏𝜏s and 𝐷𝐷s \ncharacteristic of the Elliot -Yafet spin relaxation mechanism .46 In addition, the gate dependence \nstudies can be affected  by the conductivity mismatch between the contacts and graphene ,42 \nwhich in our devices leads to the reduction of spin signal  magnitude  near the CNP . However, \nas the resistance of our FM contacts ( Rc = 2 – 12 kΩ) is always larger than the channel spin \nresistance given by 𝑅𝑅sgr=𝑅𝑅□𝜆𝜆s\n𝑊𝑊𝑔𝑔𝑔𝑔≲ 1 kΩ, the back –scattering of the spins into the FM electrodes \nshould be suppressed and not have a strong influence on the obtained parameters.  Finally, \nthe smaller degree of tunability of the parameters with the carrier density in thicker channels \nis consistent with the notion of gate field screening, where the bottom layers of multilayer graphene are strongl y coupled to the gate whereas  the upper layers experience exponentially \nsmaller field, and therefore are less tunable.\n43 6 Spin dynamics  in multilayer CVD graphene patches  \nFigure 3. S pin dynamics in multilayer  CVD graphene.  a, Schematic of the spin precession in \ngraphene with an oblique magnetic field. The field is applied at an angle 𝛽𝛽 to the FM contact easy axis \n(along 𝑦𝑦) in the 𝑦𝑦𝑦𝑦-plane. b, Spin precession signals measured at different angles  𝛽𝛽 in SLG, BLG , and \nTLG at  Vg = 0 V . c, Graphs visualizing the spin anisotropy, where the black dots represent the \nnormalized data from respective panels in (b), the red line is a fit to eq.  2, and the dashed gr ay line is a  \nguide to an eye for 𝜉𝜉 = 1. d, Anisotropy values obtained in SLG, BLG, and TLG  channels as a function \nof the carrier density. The measurements were performed with I  = –50 μA  at room temperature. \nTo get an insight into the nature of the spin dynamics in the CVD graphene multilayers , we \nstudy the spin lifetime anisotropy , which is defined as the ratio of relaxation times for spins \npolarized out -of-plane and in- plane 𝜉𝜉=𝜏𝜏⊥\n𝜏𝜏ǁ. To quantify the spin anisotropy ratio, we employ \nspin precession in oblique magnetic fields .47 In this method, the spin precession is studied with \nthe external magnetic field applied at an angle 𝛽𝛽 to the contact magnetization in the 𝑦𝑦𝑦𝑦-plane  \n(Fig. 3a). Such measurement configuration leads to dephasing of spin components that are \nnot parallel to the magnetic field. Thus, the resulting nonlocal resistance after complete \ndephasing is purely defined by the spin projection onto the field axis  fixed by the angle 𝛽𝛽. \nFigure 3b shows the obtained spin precession signals for a set of 𝛽𝛽 in SLG, BLG , and TLG  \nchannels. To visualize 𝜉𝜉 in each channel, Fig. 3c shows the normalized value of the nonlocal \nresistance in the fully dephased regime RNLβ as a function of cos2(𝛽𝛽∗), where 𝛽𝛽∗ is the angle \nof the magnetic field corrected by a small angle 𝛾𝛾 that describes an out-of-plane tilting of the \ninjector and detector magnetization by the magnetic field. The values of RNLβ are obtained by \na\nb\ncd\nz\nx\nyβs\nB\nn = −5.9 ×1012 cm-2n = −6 ×1012 cm-2n = −3.3 ×1012 cm-2\nanisotropy ξ\nξ ξ ξ7 averaging data points in a fully dephased regime marked by the shaded areas centered at B sat \n= 0.2 T t o mitigate the measurement noise. Figure 3c allows to visualize the anisotropy, since \nfor a completely isotropic system , the data would follow the 𝜉𝜉 = 1 line (dotted gray) whereas 𝜉𝜉 \n>1(< 1) should render points above (below) that line. To obtain a quantitative estimation, the\ndata are fitted with eq. 2 (red curves in Fig. 3c )47\n𝑅𝑅𝑁𝑁L𝛽𝛽\n𝑅𝑅NL0=�cos2(𝛽𝛽)+1\n𝜉𝜉sin2(𝛽𝛽)−1\nexp �−𝐿𝐿\n𝜆𝜆∥��cos2(𝛽𝛽)+1𝜉𝜉sin2(𝛽𝛽)−1��cos2(𝛽𝛽∗)         (2)\n, where L  is the channel length, 𝜆𝜆∥=�𝐷𝐷s𝜏𝜏ǁ is the in -plane spin diffusion length, 𝛽𝛽∗=𝛽𝛽 −𝛾𝛾 , \nand 𝛾𝛾 is the small angle of contact magnetization tilting out -of-plane between B  = 0 and B sat. \nThe extracted anisotropy values 𝜉𝜉 ~ 0.9 – 1 are close to unity, indicating mainly isotropic spin \nrelaxation within the error margins of the experiment.  Further, oblique spin precession \nmeasurements were repeated in three distinct carrier density regimes (low, medium and high \ndensity) , giving a general overview of the system behavior  as shown in Fig.  3d. The absence \nof a visible trend indicates that the CVD graphene does not develop noticeable directional \nspin-orbit fields  in the studied carrier density regimes . This observ ation is similar to  previous \nreports on exfoliated SLG, where only small and gate- independent anisotropy (𝜉𝜉  ~ 0.8 –  1) \nwas reported at both cryogenic  and room temperatures.47–49 Similarly, isotropic spin relaxation \nwas previously observed in exfoliated BLG at room temperature , developing anisotropic and \ngate- dependent features at low temperatures .50,51 \nWhereas isotropic spin relaxation in multilayers of CVD graphene emphasizes its excellent \nperformance as a spin in terconnect material, a realization of active control over spin anisotropy \nin graphene can allow developing unique  spintronic functionalities . To this end, gate -\ndependent anisotropy behavior may be possible to achieve in proximity -modified graphene by \nincorporating it in heterostructures  with high -SOC TMDCs .52,53 In addition, intriguing proximity -\ninduced SOC effects have been reported in heterostructures of TMDCs and topological \ninsulators with both exfoliated54–58 and CVD graphene.59–63 8 Spin transport and dynamics  across layer boundaries in CVD graphene \nFigure 4. Spin dynamics across the layer boundar ies in CVD graphene.  a, A schematic of the \ndevice and measurement geometry  to probe spin dynamics across the layer boundaries . b, The spin \nvalve and Hanle signal s in the channels with SLG -BLG and BLG -TLG boundaries . The data is obtained \nwith I = –50 μA and Vg = 0 V. c,  Spin precession signals across the SLG -BLG and BLG -TLG boundaries \nmeasured at different angles between the external magnetic field and contact magnetization in the 𝑦𝑦𝑦𝑦-\nplane.  d, Visualization of the spin anisotropy, where the black dots represent the normalized data from \nrespective panels in (c), the red line is a fit to eq. 2, and the dashed gray line is a guide to an eye for 𝜉𝜉 \n= 1. e, Spin lifetime, spin diffusion constant, spin diffusion length, and spin lifetime anisotropy ratio 𝜉𝜉 \nacross SLG -BLG (blue) and BLG -TLG (pink) boundaries as a function of the carrier concentration. \nValues for SLG, BLG, and TLG are shown in the background for comparison.  \nIn addition to the presence of patches with different number s of layers in CVD graphene, large-\nscale fabrication may result in channels incorporating layer boundaries.  To study the effect of \nsuch boundaries on spin transport and spin lifetime anisotropy, we performed SV, Hanle, and \noblique spin precession experiments in  the channels having an SLG-BLG and BLG -TLG \nboundary  (Fig. 4a ). Figure 4b shows the obtained SV and Hanle signals across SLG -BLG and \nBLG-TLG boundaries , demonstrating their robust spintronic performance . Figure 4c shows \nthe spin precession curves obtained with the magnetic field applied at different angles in  SLG-\nBLG and BLG-TLG channel s. The extracted anisotropy values  (Fig. 4d)  are close to unity, \nsuggesting similar spin lifetime for spins polarized in and out of the graphene plane, which \nindicates that the interlayer hopping does not introduce anisotropic SOC -like scattering fields. \na\nb ce\n-10 -5 00.51.01.5anisotropy ξξξ\n-0.1 0.0 0.1-101\nB (T)RNL (Ω)SLG-BLG\nL = 4  μm\n-0.1 0.0 0.1-0.40.00.4\nB (T)RNL (Ω)BLG-TLG\nL = 3.3  μmd\nn = −5.5 ×1012 cm-2n = −2.8 ×1012 cm-2\nSLG-BLG BLG-TLGIs\nI V VIs9 \n With an application of  the gate voltage,  we assess the dependence of spin transport \nparameters on the carrier concentration, as shown in Fig. 4 e. The parameter values and their \ntunability across the BLG -TLG boundary  are similar to  those obtained in continuous BLG and \nTLG channels , whereas t he SLG -BLG channel displays a larger range of values in its carrier \ndensity  similarly  to the SLG.  A general reduction of  𝐷𝐷s and 𝜏𝜏s in thicker channels can be seen ; \nhowever, the small er values of 𝜏𝜏s in the SLG -BLG channel  seem to deviate from the trend \nobserved in other channels. It should be noted that 𝐷𝐷s and 𝜏𝜏s are independent spin transport \nparameters , affected differently by various aspects of device structure and fabrication \nprocesses.  Previous studies of spin transport in graphene on various substrates showed that \n𝐷𝐷s depends strongly on the substrate properties ,64,65 whereas  𝜏𝜏s is expected to be strongly \naffected by t he channel contaminations and the presence of magnetic impurities. Thus, such \nbehavior of 𝜏𝜏s in the SLG -BLG channel may be caused by the local variation of the density of \nspin scatterers. Spin lifetime anisotropy  in channels with layer boundaries remains close to \nunity with an apparent upturn near the charge neutrality point, which, however, may be a \nconsequence of a larger uncertainty  in this regime due to the increase in the noise level of the \nmeasurements.  In ad dition to the spin precession experiments with oblique magnetic fields, \nwe used the xHanle measurement geometry, in which the spin precession in the  𝑦𝑦𝑦𝑦-plane is \ninduced by the magnetic field aligned in the 𝑥𝑥 -direction ( Bx). The measured curves could be \nfitted with the anisotropy parameter in the range of 𝜉𝜉 ~ 0.9  – 1.1, showing a good agreement \nbetween the values obtained with the two techniques (see S I Note 1).  \nSpin transport in channels having a boundary between graphene areas of different \nthicknes ses can help to identify  the effects of interlayer diffusion of spin- polarized carriers and \nthe related spin scattering timescale.  As the graphene adlayers in our samples are grown \nbeneath the continuous top layer , and all contacts are coupled to t hat upper layer, the  detected \nspin transport could be weakly affected by the interlayer hopping . In experiments, the spin \nprecession characteristics through layer boundaries do not show significant differences \ncompared to the results obtained in continuous  BLG and TLG channels . Comparing all studied \nchannels, we observe predominantly isotropic spin scattering in all devices  with similar \nspintronic performance, except for notic eably higher spin transport parameters in the SLG \nchannel . Whereas this difference can stem from local variations in graphene properties and/or \nthe more extended channel s, it can also indicate that the interlayer spin scattering present in \nother channels leads to the reduction of spin transport characteristics.  \n  10 Spin dynamics  across a fold i n monolayer CVD graphene  \nFigure 5. Spin dynamics across a fold in monolayer CVD graphene.  a, An optical image of the \nmonolayer CVD graphene with a fold. The device was patterned, as shown by the outline. b , The \nchannel resistivity as a function of the back gate voltage, showing a Dirac point at V g = 39 V. c,d,  The \ngate voltage dependence of the spin lifetime and spin diffusion length in the graphene channel with a \nfold. e, Hanle spin precession curves obtained for different angles of the applied magnetic field in the \n𝑦𝑦𝑦𝑦-plane. f, Normalized amplitude of the spin signal in a dephased regime plotted vs squared cosine of \nthe field angle, allowing to visualize anisotropy. These  measurements were performed in the channel \nwith L = 7.5 μm at Vg = –50 V and I = –300 μA. All the experiments  were conducted at room temperature.  \nTo investigate how the folds in a CVD graphene sheet can affect spin transport and spin \nlifetime anisotropy, we fabricated a device having such a fold in the channel, as outlined in \nFig. 5a (see the final device picture in Fig . S4). The sheet resistivity of the channel is shown \nin Fig. 5b, indicating the Dirac point at V g = 39 V. A r obust spin transport is observed in this \nchannel at various gate voltages, with spin lifetime 𝜏𝜏s ~ 400 ps  and spin diffusion length 𝜆𝜆 ~ 4 \nμm. The obtained spin transport parameters are comparable with those extracted in the \nchannels without irregularities, indicating that the presence of folds does not compromise the \nspin transport performance of wafer -scale CVD graphene.  \nTo investigate the effects of the fold on spin lifetime anisotropy, we performed spin precession experiments in oblique magnetic fields , as shown in Figs. 5e,f. The obtained anisotropy values \nare close to unity and do not show any trend with  the gate voltage (Fig. 5g), which indicates \nthat the presence of a fold in graphene does not induce a noticeable anisotropic SOC at room temperature. The reduction of the signal -to-noise ratio due to the conductivity mismatch effect \nnear the graphene Dirac point did not allow us to quantify the anisotropy for V\ng > 10 V reliably . \n10 μm\nanisotropy ξa b\nc\nde\nf\ng11 Figure 6. Spin relaxation mechanisms in inhomogeneous CVD graphene channels . a, Estimation \nof the contributions of spin scattering mechanisms in multilayer CVD graphene channels  and their \nboundaries . The experimental data are fitted  with eq. (3) . b, The extracted SOC energy scale for the \nElliott -Yafet (EY) and D’yakonov -Perel’ (DP) mechanisms in different channels.  Open circles \ncorrespond to the SLG channel with a fold.  \nSpin scattering mechanisms \nTo further investigate spin scattering in our inhomogeneous CVD graphene channels with \nmultilayer patches, boundaries and folds, we consider the two main spin relaxation \nmechanisms. The Elliott -Yafet (EY) mechanism  assumes spin -flip events during electron \nscattering, leading to the direct proportionality between spin lifetime 𝜏𝜏 𝑠𝑠  and momentum \nscattering time 𝜏𝜏𝑝𝑝 , wh ereas the D’yakonov -Perel’ (DP) mechanism describes spin- flip \nscattering due to spin precession in effective magnetic fields between the scattering events, resulting in the spin lifetime being inversely proportional to the momentum scattering time. These mechanisms are associated with spin- orbit coupling (SOC)  that induces band splitting, \ncharacterized b y the energy scale parameters  Δ\n𝐷𝐷𝐷𝐷 and Δ𝐸𝐸𝐸𝐸 with 𝜏𝜏𝑠𝑠,𝐷𝐷𝐷𝐷−1= 4�Δ𝐷𝐷𝐷𝐷\nℏ�2\n𝜏𝜏𝑝𝑝 and 𝜏𝜏𝑠𝑠,𝐸𝐸𝐸𝐸=\n�𝐸𝐸𝐹𝐹\nΔ𝐸𝐸𝐸𝐸�2\n𝜏𝜏𝑝𝑝, where 𝐸𝐸𝐹𝐹 is the Fermi energy relative to the graphene Dirac point .61,66 Assumi ng\nbot\nh mechanisms being relevant, we can define the total spin lifetime of the system 𝜏𝜏𝑠𝑠 =\n�𝜏𝜏𝑠𝑠,𝐷𝐷𝐷𝐷−1+𝜏𝜏𝑠𝑠,𝐸𝐸𝐸𝐸−1�−1, giving\n𝐸𝐸F2𝜏𝜏𝑝𝑝\n𝜏𝜏𝑠𝑠=∆𝐸𝐸𝐸𝐸2+ 4�Δ𝐷𝐷𝐷𝐷𝐸𝐸F𝜏𝜏𝑝𝑝 \nℏ�2\n. (3) \nFigure 6a shows the experimental data for the studied channels fitted with eq. ( 3). The \nextracted Δ𝐷𝐷𝐷𝐷 and Δ𝐸𝐸𝐸𝐸 are shown in Fig.  6b, yielding  comparable values with previous \nreports67. We o bserve similar Δ𝐸𝐸𝐸𝐸 values in different channels , whereas Δ𝐷𝐷𝐷𝐷 increases non-\nmonotonically with increasing number of graphene layers.   \nOur results indicate an increased role of the DP mechanism in multilayer graphene, in \nagreement with  previous studies .28,44,68 However, other reports have found that the spin \nrelaxation in both SLG and multilayer exfoliated graphene is limited by the EY mechanism .43 \nAnother theory has been proposed, assigning spin relaxation in both SLG and BLG to the \nresonant scattering on magnetic impuritie s, such as hydrogen adatoms and polymer \nresidues .69–71 This theory predicts opposite trends of the spin relaxation time in SLG and BLG  \nnear the CNP due to their different density of states and the scales of the energy fluctuations.  \na b\n SLG w. fold\n SLG-BLG\n BLG\n BLG-TLG\n TLG SLG0.61.11.6\n6011016012 \n In SLG, the  𝜏𝜏𝑠𝑠  is predicted to i ncrease with increasing carrier density , whereas in BLG it is \npredicted to have a nonmonotonic trend with a minimum at the intermediate doping and \nincreasing 𝜏𝜏𝑠𝑠  towards both the CNP and strongly -doped regime.  Our results in SLG CVD are \nconsistent with this pictur e, however , all other inhomogeneous CVD channels with multilayer \ngraphene patches , boundaries and folds display a n apparent reduction of 𝜏𝜏𝑠𝑠 at large carrier \nconcentrations, where the spin lifetime is limited by the D’yakonov -Perel’ mechanism . \nConclusion  \nIn conclusion, we demonstrate the robust spin interconnect performance of CVD graphene \nand spin dynamics in its characteristic defects such as multilayer patches , boundaries  and \nfolds.  Our measurements show that these naturally occurring inhomogeneities in CVD \ngraphene do not have a significant detrimental impact on its spin transport properties. The \nvalues of spin lifetime and spin diffusion length  and the degree of their gate- tunability  are \nsimilar in multilayer patches  of different thickness . Using spin precession in oblique magnetic \nfields, we confirm mostly isotropic spin relaxation in all studied channels, where the anisotropy \nratio is in the range  of 𝜉𝜉 ~ 0.9 – 1.1. The i nvestigations of spin relaxation indicate the \nimportance of both Elliott -Yafet and D’yakonov -Perel’ mechanisms, with an increasing Δ𝐷𝐷𝐷𝐷 in \nmultilayer patches . These experimental observations of robust spin transport in graphene \nmultilayer  patches  and across their layer boundaries  and folds  at room temperature  confirm \nthe ex cellent performance of large -area CVD graphene as a spin interconnect material, with \npromising applications in emerging spin- based technologies and scalable spintronic circuits.  \n \nMethods  \nThe CVD graphene spintronic devices on highly doped Si substrate with a thermally grown \n285 nm SiO 2 layer were patterned by electron beam lithography (EBL) and oxygen plasma \netching. The large area CVD graphene (obtained from Grolltex Inc) was grown on Cu foil and \ntransferred onto a 4- inch SiO 2/Si wafer with p re-fabricated alignment markers. Both the \nnonmagnetic (Ti/Au) and ferromagnetic (TiO 2/Co) tunnel contacts were defined using EBL, \nfollowed by electron beam evaporation and lift -off processes. The ferromagnetic contacts were \nproduced by electron beam evapor ation of 6 Å of Ti in two steps, each followed by in situ  \noxidation in a pure oxygen atmosphere to form a TiO 2 tunnel barrier layer. In the same \nchamber, we deposited 40 nm of Co, after which the devices were finalized by lift -off in acetone \nat 65°C. The c ontact resistance was in the range of Rc = 2 – 12 kΩ for FM electrodes, and R c \n= 2 – 5 kΩ for NM electrodes. In the measurements, a bias current was applied using a Keithley \n6221 current source, and the nonlocal voltage was detected by a Keithley 2182A \nnanovoltmeter; the gate voltage was applied using a Keithley 2400 source meter.  \n \nAcknowledgments  \n \nAuthors acknowledge the financial support from EU Graphene Flagship (Core 2 No. 785219 \nand Core 3 No. 881603), Swedish Research Council VR project grants (No. 2016- 03658),  2D \nTech Center (Vinnova),  Graphene center, and the EI Nano and AOA Materials program at \nChalmers University of Technology.  \n \nSupporting Information  \n \nThe Supporting Information is available free of charge at https://pubs.acs.org/  13 \n Hanle spin precession with parallel and antiparallel FM configurations, conductivity and \nmobility data, spin precession with the in- plane field Bx, picture of a finished device with a \ngraphene fold.  \n \n \nReferences  \n \n(1)  Chappert, C.; Fert, A.; Van Dau, F. N. The Emergence of Spin Electronics in Data Storage. Nat. Mater. \n2007, 6 (11), 813 –823. https://doi.org/10.1038/nmat2024.  \n(2)  Behin-Aein, B.; Datta, D.; Salahuddin, S.; Datta, S. Proposal for an All -Spin Logic Device with Built -In \nMemory. Nat. Nanotechnol.  2010 , 5, 266.  \n(3)  Kawahara, T.; Ito, K.; Takemura, R.; Ohno, H. Spin-Transfer Torque RAM Technology: Review and \nProspect. Microelectron. Reliab.  2012, 52 (4), 613 –627. \nhttps://d oi.org/https://doi.org/10.1016/j.microrel.2011.09.028.  \n(4)  Sinova, J.; Valenzuela, S. O.; Wunderlich, J.; Back, C. H.; Jungwirth, T. Spin Hall Effects. Rev. Mod. \nPhys.  2015, 87 (4), 1213– 1260. https://doi.org/10.1103/RevModPhys.87.1213.  \n(5)  Yu, G.; Upadh yaya, P.; Fan, Y.; Alzate, J. G.; Jiang, W.; Wong, K. L.; Takei, S.; Bender, S. A.; Chang, L.-\nT.; Jiang, Y.; Lang, M.; Tang, J.; Wang, Y.; Tserkovnyak, Y.; Amiri, P. K.; Wang, K. L. Switching of Perpendicular Magnetization by Spin– Orbit Torques in the Absence of External Magnetic Fields. Nat. \nNanotechnol.  2014, 9 (7), 548 –554. https://doi.org/10.1038/nnano.2014.94.  \n(6)  Datta, S.; Das, B. Electronic Analog of the Electro-Optic Modulator. Appl. Phys. Lett. 1990, 56 (7), 665 –\n667. https://doi.org/10.1063/1.102730.  \n(7)  Fukuma, Y.; Wang, L.; Idzuchi, H.; Takahashi, S.; Maekawa, S.; Otani, Y. Giant Enhancement of Spin \nAccumulation and Long-Distance Spin Precession in Metallic Lateral Spin Valves. Nat. Mater. 2011 , 10 \n(7), 527.  \n(8)  Lou, X.; Adelmann, C.; Crooker, S. A.; Garlid, E. S.; Zhang, J.; Reddy, K. S. M.; Flexner, S. D.; \nPalmstrøm, C. J.; Crowell, P. A. Electrical Detection of Spin Transport in Lateral Ferromagnet –\nSemiconductor Devices. Nat. Phys.  2007, 3 (3), 197 . \n(9)  Dash, S. P.; Sharma, S.; Patel, R. S.; de Jong, M. P.; Jansen, R. Electrical Creation of Spin Polarization in Silicon at Room Temperature. Nature  2009, 462 (7272), 491.  \n(10)  Oltscher, M.; Eberle, F.; Kuczmik, T.; Bayer, A.; Schuh, D.; Bougeard, D.; Ciorga, M.; Weiss, D. Gate -\nTunable Large Magnetoresistance in an All -Semiconductor Spin Valve Device. Nat. Commun.  2017, 8 \n(1), 1807. https://doi.org/10.1038/s41467-017 -01933-2.  \n(11)  Sharma, S.; Dash, S. P.; Saito, H.; Yuasa, S.; Van Wees, B. J.; Jansen, R. Anisotropy of Spin \nPolarization and Spin Accumulation in Si/Al\n2O3/Ferromagnet Tunnel Devices. Phys. Rev. B - Condens. \nMatter Mater. Phys. 2012, 86 (16), 165308. https://doi.org/10.1103/PhysRevB.86.165308.  \n(12)  Han, W.; Kawakami, R. K.; Gmitra, M.; Fab ian, J. Graphene Spintronics. Nat. Nanotechnol.  2014 , 9, 794.  \n(13)  Tombros, N.; Jozsa, C.; Popinciuc, M.; Jonkman, H. T.; van Wees, B. J. Electronic Spin Transport and Spin Precession in Single Graphene Layers at Room Temperature. Nature 2007, 448 (7153), 571 –574. \nhttps://doi.org/10.1038/nature06037.  \n(14)  Roche, S.; Åker man, J.; Beschoten, B.; Charlier, J. -C.; Chshiev, M.; Prasad Dash, S.; Dlubak, B.; Fabian, \nJ.; Fert, A.; Guimarães, M.; Guinea, F.; Grigorieva, I.; Schönenberger, C.; Seneor, P.; Stampfer, C.; \nValenzuela, S. O.; Waintal, X.; van Wees, B. Graphene Spintroni cs: The European Flagship Perspective. \n2D Mater.  2015, 2 (3), 30202. https://doi.org/10.1088/2053-1583/2/3/030202.  \n(15)  Cummings, A. W.; Duong, D. L.; Nguyen, V. L.; Van Tuan, D.; Kotakoski, J.; Barrios Vargas, J. E.; Lee, Y. H.; Roche, S. Charge Transpor t in Polycrystalline Graphene: Challenges and Opportunities. Adv. \nMater.  2014, 26 (30), 5079– 5094. https://doi.org/10.1002/adma.201401389.  \n(16)  Tsen, A. W.; Brown, L.; Levendorf, M. P.; Ghahari, F.; Huang, P. Y.; Havener, R. W.; Ruiz -Vargas, C. S.; \nMuller, D. A.; Kim, P.; Park, J. Tailoring Electrical Transport Across Grain Boundaries in Polycrystalline \nGraphene. Science  2012, 336 (6085), 1143– 1146. https://doi.org/10.1126/science.1218948.  \n(17)  Batzill, M. The Surface Science of Graphene: Metal Interfaces, CVD Synthesis, Nanoribbons, Chemical Modifications, and Defects. Surf. Sci. Rep.  2012 , 67 (3), 83 –115. \nhttps://doi.org/https://doi.org/10.1016/ j.surfrep.2011.12.001.  \n(18)  Chen, X.; Zhang, L.; Chen, S. Large Area CVD Growth of Graphene. Synth. Met.  2015, 210, 95–108. \nhttps://doi.org/https://doi.org/10.1016/j.synthmet.2015.07.005.  \n(19)  Nilsson, J.; Castro Neto, A. H.; Guinea, F.; Peres, N. M. R. Electronic Properties of Bilayer and Multilayer \nGraphene. Phys. Rev. B  2008 , 78 (4), 45405. https://doi.org/10.1103/PhysRevB.78.045405.  \n(20)  McCann, E.; Koshino, M. The Electronic Properties of Bilayer Graphene. Reports Prog. Phys.  2013 , 76 \n(5), 56503. https://doi.org/10.1088/0034-4885/76/5/056503.  \n(21)  Nakanishi, T.; Koshino, M.; Ando, T. Transmission t hrough a Boundary between Monolayer and Bilayer \nGraphene. Phys. Rev. B  2010 , 82 (12), 125428. https://doi.org/10.1103/PhysRevB.82.125428.  \n(22)  Hasegawa, Y.; Kohmoto, M. Electric -Field -Induced Penetration of Edge States at the Interface b etween \nMonolayer and Bilayer Graphene. Phys. Rev. B  2012, 85 (12), 125430. \nhttps://doi.org/10.1103/PhysRevB.85.125430.  \n(23)  Koshino, M.; Nakanishi, T.; Ando, T. Interface Landau Levels in Graphene Monolayer -Bilayer Junctions. 14 \n Phys. Rev. B  2010, 82 (20), 205436. https://doi.org/10.1103/PhysRevB.82.205436.  \n(24)  Willke, P.; Schneider, M. A.; Wenderoth, M. Electronic Transport Properties of 1D -Defects in Graphene \nand Other 2D -Systems. Ann. Phys.  2017, 529 (11), 1700003. https://doi.org/10.1002/andp.201700003.  \n(25)  Giannazzo, F.; Deretzis, I.; La Magna, A.; Roccafor te, F.; Yakimova, R. Electronic Transport at \nMonolayer -Bilayer Junctions in Epitaxial Graphene on SiC. Phys. Rev. B  2012, 86 (23), 235422. \nhttps://doi.org/10.1103/PhysRevB.86.235422.  \n(26)  Rui, D.; Sun, L.; Kang, N.; Li, J.; Lin, L.; Peng, H.; Liu, Z.; Xu,  H. Q. Realization and Transport \nInvestigation of a Single Layer -Twisted Bilayer Graphene Junction. Carbon 2020 , 163, 105– 112. \nhttps://doi.org/https://doi.org/10.1016/j.carbon.2020.03.006.  \n(27)  Cummings, A. W.; Dubois, S. M.-M.; Charlier, J. -C.; Roche, S. Universal Spin Diffusion Length in \nPolycrystalline Graphene. Nano Lett.  2019, 19 (10), 7418– 7426. \nhttps://doi.org/10.1021/acs.nanolett.9b03112.  \n(28)  Avsar, A.; Yang, T. -Y.; Bae, S.; Balakrishna n, J.; Volmer, F.; Jaiswal, M.; Yi, Z.; Ali, S. R.; Güntherodt, \nG.; Hong, B. H.; Beschoten, B.; Özyilmaz, B. Toward Wafer Scale Fabrication of Graphene Based Spin \nValve Devices. Nano Lett.  2011 , 11 (6), 2363– 2368. https://doi.org/10.1021/nl200714q.  \n(29)  Kamalakar, M. V.; Groenveld, C.; Dankert, A.; Dash, S. P. Long Distance Spin Communication in Chemical Vapour Deposited Graphene. Nat. Commun.  2015, 6, 6766. \nhttps://doi.org/10.1038/ncomms7766.  \n(30)  Drögeler, M.; Banszerus, L.; Volmer, F.; Taniguchi, T.; W atanabe, K.; Beschoten, B.; Stampfer, C. Dry -\nTransferred CVD Graphene for Inverted Spin Valve Devices. Appl. Phys. Lett. 2017, 111 (15), 152402. \nhttps://doi.org/10.1063/1.5000545.  \n(31)  Khokhriakov, D.; Karpiak, B.; Hoque, A. M.; Dash, S. P. Two -Dimensiona l Spintronic Circuit Architectures \non Large Scale Graphene. Carbon 2020, 161, 892 –899. https://doi.org/10.1016/j.carbon.2020.01.103.  \n(32)  Gebeyehu, Z. M.; Parui, S.; Sierra, J. F.; Timmermans, M.; Esplandiu, M. J.; Brems, S.; Huyghebaert, C.; Garello, K.;  Costache, M. V; Valenzuela, S. O. Spin Communication O ver 30 μm Long Channels of \nChemical Vapor Deposited Graphene on SiO\n2. 2D Mater.  2019 , 6 (3), 34003. \nhttps://doi.org/10.1088/2053-1583/ab1874.  \n(33)  Ta, H. Q.; Perello, D. J.; Duong, D. L.; Han, G. H.; Gorantla, S.; Nguyen, V. L.; Bachmatiuk, A.; Rotkin, S. V; Lee, Y. H.; Rümmeli, M. H. Stranski –Krastanov and Volmer –Weber CVD Growth Regimes t o Control \nthe Stacking Order in Bilayer Graphene. Nano Lett.  2016, 16 (10), 6403– 6410. \nhttps://doi.org/10.1021/acs.nanolett.6b02826.  \n(34)  Hwang, J. -S.; Lin, Y.-H.; Hwang, J. -Y.; Chang, R.; Chattopadhyay, S.; Chen, C.- J.; Chen, P.; Chiang, H.-\nP.; Tsai, T.-R.; Chen, L. -C.; Chen, K. -H. Imaging Layer Number and Stacking Order t hrough Formulating \nRaman Fingerprints Obtained f rom Hexagonal Single Crystals of Few Layer Graphene. Nanotechnology  \n2012, 24 (1), 15702. https://doi.org/10.1088/0957-4484/24/1/015702.  \n(35)  Pezzini, S.; Mišeikis, V.; Piccinini, G.; Forti, S.; Pace , S.; Engelke, R.; Rossella, F.; Watanabe, K.; \nTaniguchi, T.; Kim, P.; Coletti, C. 30° -Twisted Bilayer Graphene Quasicrystals f rom Chemical Vapor \nDeposition. Nano Lett.  2020, 20 (5), 3313 –3319. https://doi.org/10.1021/acs.nanolett.0c00172.  \n(36)  Lu, C. -C.; Lin, Y. -C.; Liu, Z.; Yeh, C.-H.; Suenaga, K.; Chiu, P.-W. Twisting Bilayer Graphene \nSuperlattices. ACS Nano  2013 , 7 (3), 2587– 2594. https://doi.org/10.1021/nn3059828.  \n(37)  Wu, X.; Chuang, Y.; Contino, A.; Sorée, B.; Brems, S.; Tokei, Z.; Heyns, M.; Huygh ebaert, C.; \nAsselberghs, I. Boosting Carrier Mobility of Synthetic Few Layer Graphene on SiO\n2 by Interlayer Rotation \nand Decoupling. Adv. Mater. Interfaces  2018, 5 (14), 1800454. https://doi.org/10.1002/admi.201800454.  \n(38)  Cao, Y.; Fatemi, V.; Demir, A.; Fang, S.; Tomarken, S. L.; Luo, J. Y.; Sanchez -Yamagishi, J. D.; \nWatanabe, K.; Taniguchi, T.; Kaxiras, E.; Ashoori, R. C.; Jarillo -Herrero, P. Correlated Insulator \nBehaviour at Half -Filling in Magic -Angle Graphene Superlattices. Nature  2018, 556 (7699), 80– 84. \nhttps://doi.org/10.1038/nature26154.  \n(39)  Cao, Y.; Fatemi, V.; Fang, S.; Watanabe, K.; Taniguchi, T.; Kaxiras, E.; Jarillo -Herrero, P. \nUnconventional Superc onductivity in Magic -Angle Graphene Superlattices. Nature 2018, 556 (7699), 43–\n50. https://doi.org/10.1038/nature26160.  \n(40)  Yu, Q.; Jauregui, L. A.; Wu, W.; Colby, R.; Tian, J.; Su, Z.; Cao, H.; Liu, Z.; Pandey, D.; Wei, D.; Chung, T. F.; Peng, P.; Guisi nger, N. P.; Stach, E. A.; Bao, J.; Pei, S.-S.; Chen, Y. P. Control and \nCharacterization of Individual Grains and Grain Boundaries in Graphene Grown by Chemical Vapour \nDeposition. Nat. Mater.  2011, 10 (6), 443 –449. https://doi.org/10.1038/nmat3010.  \n(41)  Wang, Z. -J.; Dong, J.; Cui, Y.; Eres, G.; Timpe, O.; Fu, Q.; Ding, F.; Schloegl, R.; Willinger, M.-G. \nStacking Sequence and Interlayer Coupling in Few -Layer Graphene Revealed by in Situ  Imaging. Nat. \nCommun.  2016, 7 (1), 13256. https://doi.org/10.1038/ncomm s13256.  \n(42)  Han, W.; Pi, K.; McCreary, K. M.; Li, Y.; Wong, J. J. I.; Swartz, A. G.; Kawakami, R. K. Tunneling Spin \nInjection into Single Layer Graphene. Phys. Rev. Lett. 2010, 105 (16), 167202. \nhttps://doi.org/10.1103/PhysRevLett.105.167202.  \n(43)  Maassen, T.; Dejene, F. K.; Guimarães, M. H. D.; Józsa, C.; van Wees, B. J. Comparison b etween \nCharge and Spin Transport in Few -Layer Graphene. Phys. Rev. B  2011 , 83 (11), 115410. \nhttps://doi.org/10.1103/PhysRevB.83.115410.  \n(44)  Han, W.; Kawakami, R. K. Spin Relaxation in Single -Layer and Bilayer Graphene. Phys. Rev. Lett. 2011 , \n107 (4). https://doi.org/10.1103/PhysRevLett.107.047207.  \n(45)  Guinea, F. Spin-Orbit Coup ling in a Graphene Bilayer and in Graphite. New J. Phys.  2010 , 12 (8), \n083063. https://doi.org/10.1088/1367-2630/12/8/083063.  15 \n (46)  Popinciuc, M.; Józsa, C.; Zomer, P. J.; Tombros, N.; Veligura, A.; Jonkman, H. T.; van Wees, B. J. \nElectronic Spin Transport  in Graphene Field-Effect Transistors. Phys. Rev. B  2009, 80 (21), 214427. \nhttps://doi.org/10.1103/PhysRevB.80.214427.  \n(47)  Raes, B.; Scheerder, J. E.; Costache, M. V; Bonell, F.; Sierra, J. F.; Cuppens, J.; Van de Vondel, J.; Valenzuela, S. O. Determinat ion of the Spin-Lifetime Anisotropy in Graphene Using Oblique Spin \nPrecession. Nat. Commun.  2016 , 7, 11444.  \n(48)  Ringer, S.; Hartl, S.; Rosenauer, M.; Völkl, T.; Kadur, M.; Hopperdietzel, F.; Weiss, D.; Eroms, J. \nMeasuring Anisotropic Spin Relaxation in G raphene. Phys. Rev. B  2018, 97 (20), 205439. \nhttps://doi.org/10.1103/PhysRevB.97.205439.  \n(49)  Tombros, N.; Tanabe, S.; Veligura, A.; Jozsa, C.; Popinciuc, M.; Jonkman, H. T.; van Wees, B. J. \nAnisotropic Spin Relaxation in Graphene. Phys. Rev. Lett. 2008, 101 (4), 46601. \nhttps://doi.org/10.1103/PhysRevLett.101.046601.  \n(50)  Leutenantsmeyer, J. C.; Ingla- Aynés, J.; Fabian, J.; van Wees, B. J. Observation of Spin-Valley -\nCoupling-Induced Large Spin- Lifetime Anisotropy in Bilayer Graphene. Phys. Rev. Lett. 2018, 121 (12), \n127702. https://doi.org/10.1103/PhysRevLett.121.127702.  \n(51)  Xu, J.; Zhu, T.; Luo, Y. K.; Lu, Y.-M.; Kawakami, R. K. Strong and Tunable Spin-Lifetime Anisotropy in Dual-Gated Bilayer Graphene. Phys. Rev. Lett.  2018, 121 (12), 127703. \nhttps://d oi.org/10.1103/PhysRevLett.121.127703.  \n(52)  Ghiasi, T. S.; Ingla -Aynés, J.; Kaverzin, A. A.; van Wees, B. J. Large Proximity -Induced Spin Lifetime \nAnisotropy in Transition -Metal Dichalcogenide/Graphene Heterostructures. Nano Lett.  2017 , 17 (12), \n7528– 7532. https://doi.org/10.1021/acs.nanolett.7b03460.  \n(53)  Benítez, L. A.; Sierra, J. F.; Savero Torres, W.; Arrighi, A.; Bonell, F.; Costache, M. V; Valenzuela, S. O. \nStrongly Anisotropic Spin Relaxation in Graphene– Transition Metal Dichalcogenide Heterostructures at \nRoom Temperature. Nat. Phys.  2018 , 14 (3), 303 –308. https://doi.org/10.1038/s41567-017 -0019-2.  \n(54)  Yan, W.; Txoperena, O.; Llopis, R.; Dery, H.; Hueso, L. E.; Casanova, F. A Two-Dimensional Spin Field-Effect Switch. Nat. Commun.  2016, 7. https://doi.org/10.1038/ncomms13372.  \n(55)  Safeer, C.  K.; Ingla -Aynés, J.; Herling, F.; Garcia, J. H.; Vila, M.; Ontoso, N.; Calvo, M. R.; Roche, S.; \nHueso, L. E.; Casanova, F. Room -Temperature Spin Hall Effect in Graphene/MoS\n2 van der Waals \nHeterostructures. Nano Lett.  2019, 19 (2), 1074– 1082. https://doi.org/10.1021/acs.nanolett.8b04368.  \n(56)  Benítez, L. A.; Savero Torres, W.; Sierra, J. F.; Timmermans, M.; Garcia, J. H.; Roche, S.; Costache, M. \nV; Valenzuela, S. O. Tunable Room -Temperature Spin Galvanic and Spin Hall Effects in van d er Waals \nHeterostructures. Nat. Mater.  2020, 19 (2), 170– 175. https://doi.org/10.1038/s41563-019 -0575-1.  \n(57)  Ghiasi, T. S.; Kaverzin, A. A.; Blah, P. J.; van Wees, B. J. Charge -To-Spin Conversion by the Rashba–\nEdelstein Effect in Two -Dimensional van d er Waals Heterostructures up to Room Temperature. Nano \nLett. 2019 , 19 (9), 5959– 5966. https://doi.org/10.1021/acs.nanolett.9b01611.  \n(58)  Li, L.; Zhang, J.; Myeong, G.; Shin, W.; Lim, H.; Kim, B.; Kim, S.; Jin, T.; Cavill, S.; Kim, B. S.; Kim, C.; \nLischner , J.; Ferreira, A.; Cho, S. Gate -Tunable Reversible Rashba– Edelstein Effect in a Few -Layer \nGraphene/2H -TaS 2 Heterostructure at Room Temperature. ACS Nano  2020, 14 (5), 5251– 5259. \nhttps://doi.org/10.1021/acsnano.0c01037.  \n(59)  Khokhriakov, D.; Hoque, A. M.;  Karpiak, B.; Dash, S. P. Gate -Tunable Spin-Galvanic Effect in Graphene-\nTopological Insulator van d er Waals Heterostructures at Room Temperature. Nat. Commun.  2020, 11 (1), \n3657. https://doi.org/10.1038/s41467- 020-17481-1.  \n(60)  Dankert, A.; Dash, S. P. El ectrical Gate Control of Spin Current in van d er Waals Heterostructures at \nRoom Temperature. Nat. Commun.  2017, 8 (May), 16093. https://doi.org/10.1038/ncomms16093.  \n(61)  Khokhriakov, D.; Cummings, A. W.; Song, K.; Vila, M.; Karpiak, B.; Dankert, A.; Roche , S.; Dash, S. P. \nTailoring Emergent Spin Phenomena in Dirac Material Heterostructures. Sci. Adv.  2018, 4 (9). \nhttps://doi.org/10.1126/sciadv.aat9349.  \n(62)  Hoque, A. M.; Khokhriakov, D.; Karpiak, B.; Dash, S. P. All -Electrical Creation and Control of Giant Spin-\nGalvanic Effect in 1T -MoTe 2/Graphene Heterostructures at Room Temperature. arXiv Prepr. \narXiv1908.09367 2019.  \n(63)  Karpiak, B.; Cummings, A. W.; Zollner, K.; Vila, M.; Khokhriakov, D.; Hoque, A. M.; Dankert, A.; \nSvedlindh, P.; Fabian, J.; Roche, S.; Dash, S. P. Magnetic Proximity in a van der Waals Heterostructure \nof Magnetic Insulator and Graphene. 2D Mater.  2019, 7 (1), 15026. https://doi.org/10.1088/2053-\n1583/ab5915.  \n(64)  Guimarães, M. H. D.; Veligura, A.; Zomer, P. J.; Maassen, T.; Vera-Mar un, I. J.; Tombros, N.; Van Wees, \nB. J. Spin Transport in High-Quality Suspended Graphene Devices. Nano Lett.  2012, 12 (7), 3512 –3517. \nhttps://doi.org/10.1021/nl301050a.  \n(65)  Serrano, I. G.; Panda, J.; Denoel, F.; Vallin, Ö.; Phuyal, D.; Karis, O.; Kamala kar, M. V. Two -Dimensional \nFlexible High Diffusive Spin Circuits. Nano Lett.  2019, 19 (2), 666– 673. \nhttps://doi.org/10.1021/acs.nanolett.8b03520.  \n(66)  Gurram, M.; Omar, S.; Van Wees, B. J. Electrical Spin Injection, Transport, and Detection in Graphene-Hexagonal Boron Nitride van d er Waals Heterostructures: Progress and Perspectives. 2D Materials . IOP \nPublishing Ltd June 1, 2018, p 032004. https://doi.org/10.1088/2053-1583/aac34d.  \n(67)  Zomer, P. J.; Guimarães, M. H. D.; Tombros, N.; Van Wees, B. J. Long -Distance Spin Transport in High-\nMobility Graphene on Hexagonal Boron Nitride. Phys. Rev. B - Condens. Matter Mater. Phys.  2012 , 86 \n(16). https://doi.org/10.1103/PhysRevB.86.161416.  \n(68)  Yang, T. Y.; Balakrishnan, J.; Volmer, F.; Avsar, A.; Jaiswal, M.; Samm, J.; Ali, S. R.; Pachoud, A.; Zeng, 16 \n M.; Popinciuc, M.; Güntherodt, G.; Beschoten, B.; Özyilmaz, B. Observation of Long Spin-Relaxation \nTimes in Bilayer Graphene at Room Temperature. Ph ys. Rev. Lett. 2011. \nhttps://doi.org/10.1103/PhysRevLett.107.047206.  \n(69)  Kochan, D.; Gmitra, M.; Fabian, J. Spin Relaxation Mechanism in Graphene: Resonant Scattering by Magnetic Impurities. Phys. Rev. Lett.  2014, 112 (11), 116602. \nhttps://doi.org/10.1103/PhysRevLett.112.116602.  \n(70)  Kochan, D.; Irmer, S.; Gmitra, M.; Fabian, J. Resonant Scattering by Magnetic Impurities as a Model for \nSpin Relaxation in Bilayer Graphene. Phys. Rev. Lett. 2015, 115 (19), 196601. \nhttps://doi.org/10.1103/PhysRevLett.115.196601.  \n(71)  Avsar, A.; Vera -Marun, I. J.; Tan, J. Y.; Koon, G. K. W.; Watan abe, K.; Taniguchi, T.; Adam, S.; \nÖzyilmaz, B. Electronic Spin Transport in Dual -Gated Bilayer Graphene. NPG Asia Mater.  2016, 8 (6), \ne274– e274. https://doi.org/10.1038/am.2016.65.  \n \n \n \n Supporting information  \nRobust Spin Interconnect with Isotropic Spin Dynamics in \nChemical Vapour Deposited Graphene Layers and Boundaries \nDmitrii Khokhriakov1, Bogdan Karpiak1, Anamul Md. Hoque1, Bing Zhao1, \nSubir Parui2, Saroj P. Dash1,3*\n1Department of Microtechnology and Nanoscience, Chalmers University of Technology, \nSE-41296, Göteborg, Sweden,  \n2K.U. Leuven, 3001 Leuven, Belgium,  \n3Graphene center, Chalmers University of Technology, SE -41296, Göteborg, Sweden.  \nFi\ngure S1. Hanle spin precession in CVD graphene devices with multilayer patches and their \nlayer boundaries with parallel and antiparallel FM configurations.  Hanle spin precession data for \nperpendicular magnetic field (B z) in SLG, BLG, TLG channels and across SLG -BLG and BLG -TLG \nboundaries, obtained with parallel (pink curve) and antiparallel (blue curve) configuration of the \nferromagnetic electrodes. By calculating the difference between the two curves, the parabolic charge-based background signal was removed in Figs. 2b and 4b. For SLG channel described in the main text, \nthe background was subtracted manually; the first panel here shows representative data from another \nSLG device.  All the measurements were performed at room temperature.  \n-0.4 0.0 0.4-101\nBz (T)RNL (Ω)\nBLG\n-0.4 0.0 0.4-0.30.00.30.6\nBz (T)RNL (Ω)\nBLG-TLG\n-0.4 0.0 0.4-101\nBz (T)RNL (Ω)\nSLG-BLG-0.4 0.0 0.4-101\nBz (T)RNL (Ω)\nTLG\n-0.4 0.0 0.4-0.30.00.3\nBz (T)RNL (Ω)\nSLG w. fold-0.4 0.0 0.4-0.80.00.8\nBz (T)RNL (Ω)\nSLGFigure S2. Conductivity and mobility in multilayer graphene channels and across their \nboundaries.  a, Conductivity of the studied channels as a function of the carrier density. b,c , \nConductivity and mobility extracted at n = –5×1012 cm–2 in different channels. The graphs indicate a \ngeneral trend of reducing mobility and conductivity with increasing number of graphene layers. This \ntrend in electrical performance of multilayer channels compared to SLG is consistent with sizeable \ninterlayer coupling expected for Bernal graphene .1 The mobility of our samples has typical values for \nwet-transfer CVD graphene, smaller than mobilties commonly observed in exfoliated devices. \nHowever, previous studies of spin transport in samples with different mobility,2,3 as well as in high-\nmobility exfoliated graphene on hBN,4 did not show a significant correlation between 𝜏𝜏s and 𝜇𝜇. Thus, \nwe expect our findings to be valid also for the exfoliated graphene samples of various mobilities.  \nSLG SLG-BLG BLG BLG-TLG TLG010002000n = −5× 1012 cm-2μ (cm2 V−1 s−1)\n−10 −5 0012 SLG\n SLG-BLG\n BLG\n BLG-TLG\n TLGσ (mS)\nn (×1012 cm-2)SLG SLG-BLG BLG BLG-TLG TLG012σ (mS)n = −5× 1012 cm-2a b cNote 1 \nFigure S3. Spin precession in CVD graphene devices with multilayer patches and their layer \nboundaries with B x field sweep . a, A schematic of the device with measurement geometry and spin \nprecession in B x. b, The measured xHanle signal in a TLG channel (data points) with fitting  (red line) . \nc, xHanle data obtained in SLG and BLG channels, as well as in the channels having SLG -BLG and \nBLG-TLG boundary.  All the measurements were performed at room temperature.  \nThe \nspin precession was also measured in all channels while sweeping the magnetic field Bx \nin the graphene plane along the channel  (Fig. S3a), yielding the xHanle curves presented in \nFig. S3b,c. These curves show a combination of spin precession in the 𝑦𝑦𝑦𝑦 plane and rotation \nof injector and detector polarization in the 𝑦𝑦𝑦𝑦 plane, which leads to the contacts aligning with \nthe filed in 𝑦𝑦-di\nrection above the saturation field Bsat ≈ 0.4 T.  Due to the spin precession \nhappening in the 𝑦𝑦𝑦𝑦 plane, both in- plane and out -of-plane spin lifetimes influence the \nmeasurement, and therefore this geometry allows to quantify the spin lifetime anisotropy ratio \n𝜉𝜉. Since the anisotropy information is fully contained in the spin precession contribution,  it is \nnecessary to  isolate it by  remov ing from the signal  the additional contribution originating from \nmagnetization rotation.  Normally, this contribution can be completely subtracted by performing \nxHanle measurements with parallel and antiparallel initial configurations of the ferromagnetic \ncontacts, since in such measurements the spin precession signal changes sign whereas the \nmagnetization rotation does not. Unfortunately, our device has failed before the \nmeasurements in the AP configuration could be performed. In addition, the two contributions \ncan be easily separated in long channels, in which complete spin dephasing can be achieved \nalready in small fields ; however, our channel lengths were limited by the device geometry and \nsize of the multilayer patch. Therefore, we could simulate  the magnetization rotation \ncontribution  in our device using the Wohlfarth- Stoner model, but the saturation level of our \nsignal at high fields differs from the signal value at B = 0, which hinders data interpretation \nsince at both B = 0 and | B| > Bsat the same value of spin signal is expected due to ferromagnetic \ncontacts being in the  parallel configuration  (𝑴𝑴𝒊𝒊∥ 𝑴𝑴𝒅𝒅∥ 𝑦𝑦 and  𝑴𝑴𝒊𝒊∥ 𝑴𝑴𝒅𝒅∥𝑦𝑦 r espectively) . In our \ndevice, t he application of the magnetic field gives rise to a n additional small parabolic \nbackground signal due to the graphene magnetoresistance; however, this magnetoresistance \nis positive and , therefore , cannot be the reason for the lower  signal at high fields.  Alternatively, \nthis difference in the signal  magnitudes may indicate varying spin lifetimes for spins aligned in \n𝑦𝑦 and 𝑦𝑦 di\nrections, but such anisotropy between the in- plane directions is not expected due to \nthe symmetry of the gr aphene lattice.  Thus, the origin of such missing signal is still \nunidentified, although this feature has also been observed in exfoliated single- layer graphene \ndevices .5  \n0.00.2RNL (Ω)\nSLG-BLG\n−0.8 0.0 0.801RNL (Ω)\nBx (T)BLG0.00.4RNL (Ω)\nBLG-TLG\n−0.8 0.0 0.80.00.2RNL (Ω)\nBx (T)SLGa\nbc\nzx\ny\n−0.8 −0.4 0.0 0.4 0.80.00.40.8RNL (Ω)\nBx (T)TLG\nUsing a spin precession model that includes both anisotropic spin relaxation and contact \nmagnetization rotation,6,7 we fit the data , as shown in Fig. S3 b. To account for the missing part \nof the signal in our fitting, we introduce an additional scaling factor, which, however, limits the accuracy of parameter extraction. Within the noise level of our experiment, all data can be \nadequately fitted with 𝜉𝜉 being in the range of 0.9 \n– 1.1, indicating isotropic spin relaxation, in \nagreement with the oblique Hanle spin precession experiments described in the main \nmanuscript.  \nF\nigure S4. Optical microscopy photograph of the CVD graphene device with a fold.  A micrograph \nof a nanofabricated CVD graphene spintronic device used to investigate spin dynamics across the fold \nin monolayer graphene. The arrow points at the fold in the graphene channel . The atomic force \nmicroscopy was used to extract the dimensions of the fold, which are shown in the schematic.  \nS\nupplementary References:  \n(1) W\nu, X.; Chuang, Y.; Contino, A.; Sorée, B.; Brems, S.; Tokei, Z.; Heyns, M.; Huyghebaert, C.;\nAsselberghs, I. Boosting Carrier Mobility of Synthetic Few Layer Graphene on SiO 2 by\nInterlayer Rotation and Decoupling. Adv. Mater. Interfaces  2018, 5  (14), 1800454.\n(2) V\nolmer, F.; Drögeler, M.; Maynicke, E.; Von Den Driesch, N.; Boschen, M. L.; Güntherodt, G.;\nBeschoten, B. Role of MgO Barriers for Spin and Charge Transport in Co/MgO/Graphe ne\nN\nonlocal Spin- Valve Devices. Phys. Rev. B - Condens. Matter Mater. Phys.  2013, 88  (16),\n161405.\n(3) D\nrögeler, M.; Volmer, F.; Wolter, M.; Terrés, B.; Watanabe, K.; Taniguchi, T.; Güntherodt, G.;\nStampfer, C.; Beschoten, B. Nanosecond Spin Lifetimes in Single- and Few -Layer Graphene–\nHBN Heterostructures at Room Temperature. Nano Lett.  2014 , 14 (11), 6050 –6055.\n(4) Z\nomer, P. J.; Guimarães, M. H. D.; Tombros, N.; Van Wees, B. J. Long- Distance Spi n\nT\nransport in High- Mobility Graphene on Hexagonal Boron Nitride. Phys. Rev. B - Condens.\nMatter Mater. Phys.  2012, 86 (16).\n(5) R\ninger, S.; Hartl, S.; Rosenauer, M.; Völkl, T.; Kadur, M.; Hopperdietzel, F.; Weiss, D.; Eroms ,\nJ\n.Meas\nuring Anisotropic Spin Relaxation in Graphene. Phys. Rev. B  2018 , 97 (20), 205439.\n(6) X\nu, J.; Zhu, T.; Luo, Y. K.; Lu, Y.- M.; Kawakami, R. K. Strong and Tunable Spin- Lifetime\nAnisotropy in Dual -Gated Bilayer Graphene. Phys. Rev. Lett.  2018, 121 (12), 127703.\n(7) G\nhiasi, T. S.; Ingla- Aynés, J.; Kaverzin, A. A.; van Wees, B. J. Large Proximity -Induced Spi n\nLi\nfetime Anisotropy in Transition- Metal Dichalcogenide/Graphene Heterostructures. Nano Lett.\n2017 , 17 (12), 7528 –7532.\n400 nm\n0.6 nm\n5 μm"    },    {        "title": "1111.5466v1.Rashba_spin_torque_in_an_ultrathin_ferromagnetic_metal_layer.pdf",        "content": "arXiv:1111.5466v1  [cond-mat.mtrl-sci]  23 Nov 2011Rashba spin torque in an ultrathin ferromagnetic metal laye r\nXuhui Wang∗and Aurelien Manchon†\nPhysical Science & Engineering Division, KAUST, Thuwal 239 55-6900, Kingdom of Saudi Arabia\n(Dated: August 24, 2018)\nIn a two-dimensional ferromagnetic metal layer lacking inv ersion symmetry, the itinerant electrons\nmediate the interaction between the Rashba spin-orbit inte raction and the ferromagnetic order\nparameter, leading to a Rashba spin torque exerted on the mag netization. Using Keldysh technique,\nin the presence of both magnetism and a spin-orbit coupling, we derive a spin diffusion equation\nthat provides a coherent description to the diffusive spin dy namics. The characteristics of the spin\ntorque and its implication on magnetization dynamics are di scussed in the limits of large and weak\nspin-orbit coupling.\nPACS numbers: 75.60.Jk,75.70.Tj,72.25.-b,72.10.-d\nI. INTRODUCTION\nBy transferring angular momentum between the elec-\ntronic spin and the orbital, spin-orbit coupling fills the\nneedforelectricalmanipulationofspindegreeoffreedom.\nOutstanding examples are the electrically generated bulk\nspin polarization1,2and the well-known spin Hall effect\n(SHE)3–5in a two dimensional electron gas where the\nspin-orbit interaction, particularly of the Rashba-type,6\nplays the leading role. Rashba spin-orbit interaction not\nonly introduces an effective field perpendicular to the lin-\near momentum but also provides the backbone to the\nspin-relaxation through the so-called D’yakonov-Perel\nmechanism,7which is dominant in a two-dimensional\nsystem. Besides its prominent role in semiconductors,\nRashba spin-orbit coupling is believed to exist at ferro-\nmagnetic/heavy metal as well as ferromagnetic/metal-\noxide interfaces, in which the inversion symmetry break-\ning offers a potential gradient empowering the spin-orbit\ncoupling.\nMeanwhile, magnetism continuously stimulates the in-\ndustrial and academic appetite. In the pursuit of fast\nmagnetization switching, Slonczewski-Berger spin trans-\nfer torque8employs a polarized spin current instead of\na cumbersome magnetic field. This celebrated scheme\ndemands non-collinear magnetic textures in forms of, for\nexample, spin valves or domain wall structures.9\nIn the presence of inversion symmetry breaking (such\nasasymmetricinterfaces),aferromagneticmetallayeras-\nsembles both magnetism and spin-orbit coupling, hence\noffering an alternative switching mechanism:10,11Spin-\norbit coupling transfers the orbital angular momentum\ncarried by an electric current to the electronic spin, thus\ncreating an effective magnetic field (Rashba field). As\nlong as the effective field is mis-aligned with the magne-\ntization direction, the so-called Rashba torque emerges,\nthus exciting the magnetization.\nCurrent-driven magnetization dynamics by spin-orbit\ntorque has been demonstrated by several experiments\non metal-oxide based systems.12–14In fact, the Rashba\ntorque can be categorized into to a broader family\nof spin-orbit interaction induced torque that has been\nobserved in diluted magnetic semiconductors.16–18Re-cently, Miron et al.,15has demonstrated the current-\ninducedmagnetizationswitchingusinga singleferromag-\nnet in Pt/Co/AlO xtrilayers, which further consolidates\nthe feasibility of the Rashba torque. The same type of\nspin-orbit coupling induced torque is predicted to im-\nprove current-driven domain wall motion,11,19which is\nsupported by experimental observations.20At this stage,\nwe are aware of an alternative explanation, as pointed\nout by Liu et al.,21in terms of the spin Hall effect (SHE)\noccurring in the underlying heavy metal layer, such as\nPt or Ta. The distinction between the spin Hall induced\neffect and the Rashba one is discussed in the last section\nof this article.\nIn searching for a general form of the Rashba torque\nin ferromagnetic metal layers,10we found an expression\nthat consists of two components:22An in-plane torque\n(∝m×(ˆy×m)) and an out-of-plane one ( ∝ˆy×m),\ngivenˆyisthein-planedirectiontransversetotheinjected\ncurrent and mis the magnetization direction. Numerical\nsolution on a two-dimensional nano-wire with one open\ntransport direction has been carried out to appreciate\nthe significance of diffusive motion on the spin torque.\nWe found that the in-plane component of the torque in-\ncreases when narrowing the magnetic wire22.\nIn the present article, we give a full theoretical deriva-\ntion of the coupled diffusive equation for spin dynam-\nics in a ferromagnetic metal layer and describe the form\nof the Rashba torque in both weak and strong Rashba\nlimits. In Sec. II, we combine the Keldysh formalism\nand the gradient expansion technique to derive a cou-\npled diffusion equation for charge and non-equilibrium\nspin densities. To demonstrate that the diffusion equa-\ntion provides a coherent framework to describe the spin\ndynamics, we dedicate Sec. III to the spin diffusion in\na ferromagnetic metal, which shows an excellent agree-\nment to early result on the same system. In Sec.IV, we\nillustrate that the absence of magnetism (in our diffusion\nequation) describes the well-know phenomenon of elec-\ntrically induced spin polarization. The cases of a weak\nand a strong spin-orbit coupling are discussed in Sec.V\nand Sec. VI, respectively, where we provide an analytical\nform of the Rashba torque in an infinite medium. In Sec.\nVII, we discuss the implication of the Rashba torque on2\nmagnetization dynamics as well as its distinction from\nspin Hall effect induced torque.\nII. DIFFUSION EQUATIONS\nThe system of interest is defined as a quasi-two-\ndimensional ferromagnetic metal layer rolled out in the\nxy-plane. Two asymmetric interfaces provide a confine-\nment in z-direction, along which the potential gradi-\nent generates a Rashba spin-orbit coupling. Therefore\na single-particle Hamiltonian for an electron of momen-\ntumˆkis (/planckover2pi1= 1 is assumed throughout)\nˆH=ˆk2\n2m+αˆσ·(ˆk׈z)+1\n2∆xcˆσ·m+Hi(1)\nwhereˆσis the Pauli matrix, mthe effective mass,\nandmthe magnetization direction. The ferromag-\nnetic exchange splitting is given by ∆ xcandαrepre-\nsents the Rashba constant (parameter). Hamiltonian\nˆHi=/summationtextN\nj=1V(r−rj) sums the contribution of the non-\nmagnetic impurity scattering potential V(r) localized at\nrj.\nTo derive a diffusion equation for the non-\nequilibriumchargeandspindensities, weemployKeldysh\nformalism.23Using Dyson equation, in a 2 ×2 spin space,\nwe obtain a kinetic equation that assembles the retarded\n(advanced) Green’s function ˆGR(ˆGA), the Keldysh com-\nponent of the Green function ˆGK, and the self-energy\nˆΣK, i.e.,\n[ˆGR]−1ˆGK−ˆGK[ˆGA]−1=ˆΣKˆGA−ˆGRˆΣK,(2)\nwhere all Green’s functions are full functions with inter-\nactions taken care of by the self-energies ˆΣR,A,K. The\nretarded (advanced) Green’s function in momentum and\nenergy space is\nˆGR(A)(k,ǫ) =1\nǫ−ǫk−ˆσ·b(k)−ˆΣR(A)(k,ǫ),(3)\nwhereǫk=k2/(2m) is the single-particle energy. We\nhave introduced a k-dependent effective field b(k) =\n∆xcm/2+α(k×z) of the magnitude bk=|∆xcm/2+\nα(k×z)|and the direction ˆb=b(k)/bk.\nNeglecting localization effect and electron-electron in-\nteractions, we assume a short-range δ-function type im-\npurity scattering potential. At a low concentration and\na weak coupling to electrons, the second-order Born ap-\nproximation is justified,23i.e., the self-energy is24\nˆΣR,A,K(r,r′) =δ(r,r′)\nmτˆGR,A,K(r,r) (4)\nwhere the momentum relaxation time reads\n1\nτ≈2π/integraldisplayd2k′\n(2π)2|V(k−k′)|2δ(ǫk−ǫk′),(5)whereV(k) is the Fourier transform of the scattering\npotential and the magnitude of momentum kandk′is\nevaluated at Fermi vector kF.\nThe quasi-classical distribution function ˆ g≡\nˆgk,ǫ(T,R), defined as the Wigner transform of the\nKeldysh function ˆGK(r,t;r′,t′), is obtained by inte-\ngrating out the relative spatial-temporal coordinates\nwhile retaining the center-of-mass ones R= (r+r′)/2\nandT= (t+t′)/2. As long as the spatial profile\nof the quasi-classical distribution function is smooth\nat the scale of Fermi wave length, we may apply the\ngradient expansion technique on Eq.(2),25which gives\nus a transport equation associated with macroscopic\nquantities. The left-hand side of the kinetic equation in\ngradient expansion becomes\n[ˆGR]−1ˆGK−ˆGK[ˆGA]−1\n≈[ˆg,ˆσ·b(k)]+i\nτˆg+i∂ˆg\n∂T\n+i\n2/braceleftbiggk\nm+α(ˆz׈σ),∇Rˆg/bracerightbigg\n,(6)\nwhere{·,·}denotes the anti-commutator. The relax-\nation time approximation indulges the right-hand side\nof Eq.(2) as\nˆΣKˆGA−ˆGRˆΣK\n≈1\nτ/bracketleftBig\nˆρ(ǫ,T,R)ˆGA(k,ǫ)−ˆGR(k,ǫ)ˆρ(ǫ,T,R)/bracketrightBig\n(7)\nwherewe haveintroducedthe densitymatrix by integrat-\ning out the momentum kin ˆg, i.e.,\nˆρ(E,T,R) =1\n2πN0/integraldisplayd2k′\n(2π)2ˆgk′,ǫ(T,R).(8)\nFor the convenience of discussion, time variable is\nchanged from Ttot. At this stage, we have a kinetic\nequation depending on ˆ ρas well as on ˆ g\ni[ˆσ·b(k),ˆg]+1\nτˆg+∂ˆg\n∂t+1\n2/braceleftbiggk\nm+α(ˆz׈σ),∇Rˆg/bracerightbigg\n=i\nτ/bracketleftBig\nˆGR(k,ǫ)ˆρ(ǫ)−ˆρ(ǫ)ˆGA(k,ǫ)/bracketrightBig\n. (9)\nA Fouriertransformationon temporal variableto the fre-\nquency domain ωleads to\nΩˆg−bk[ˆUk,ˆg] =iˆK, (10)\nwhere Ω = ω+i/τand the operator ˆUk≡ˆσ·ˆbsatisfies\nˆUkˆUk= 1. The right hand side of Eq.(10) is partitioned\naccording to\nˆK=−1\n2/braceleftbiggk\nm+α(ˆz׈σ),∇Rˆg/bracerightbigg\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\nˆK(1)\n+i\nτ/bracketleftBig\nˆGR(k,ǫ)ˆρ(ǫ)−ˆρ(ǫ)ˆGA(k,ǫ)/bracketrightBig\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\nˆK(0).(11)3\nThe equilibrium part is denoted by ˆK(0)while the gradi-\nent term ˆK(1)is regarded as perturbation. Functions ˆ g\nand ˆρare both in frequency domain. We solve Eq. (10)\nformally to find a solution to ˆ g\nˆg=i(2b2\nk−Ω2)ˆK+2b2\nkˆUkˆKˆUk−Ωbk[ˆUk,ˆK]\nΩ(4b2\nk−Ω2)≡L[ˆK].\n(12)\nAn iterationprocedureto solveEq.(12) hasbeenoutlined\nby Mishchenko et al.,in Ref.[24]. We follow this proce-\ndure here: According to the partition scheme on ˆK, we\nuseˆK(0)to obtain the zero-th order approximation as\nˆg(0)≡L[ˆK(0)(ˆρ)], which replaces ˆ ginˆK(1)to generate\na correction due to the gradient term, i.e., ˆK(1)(ˆg(0)).\nWe further insert ˆK(1)(ˆg(0)) back to Eq.(12) to obtain\na correction given by L[ˆK(1)(ˆg(0))], then we obtain the\nfirst order approximation to the quasi-classical distribu-tion function,\nˆg(1)= ˆg(0)+L[ˆK(1)(ˆg(0))]. (13)\nThe above procedure is repeated to any desired order,\ni.e.,\nˆg(n)= ˆg(n−1)+L[ˆK(1)(ˆg(n−1))].(14)\nIn this paper, the second order approximation is suffi-\ncient. The full expression of the second orderapproxima-\ntion istedious thusnot included in the following. Adiffu-\nsion equation is derived by an angle averagingin momen-\ntum space, which allowsalltermsthat areoddorderin ki\n(i=x,y) to vanish while the combinations such as kikj\ncontribute to the averaging by a factor k2\nFδij, givenkF\nthe Fermi wave vector.25Further more, a Fourier trans-\nform from frequency domain back to the real time brings\na diffusion type equation for the density matrix,\n∂\n∂tˆρ(t) =D∇2ˆρ−1\nτxcˆρ+1\n2τxc(ˆz׈σ)·ˆρ(ˆz׈σ)+iC[ˆz׈σ,∇ˆρ]−B{ˆz׈σ,∇ˆρ}\n+Γ[(m×∇)zˆρ−ˆσ·m∇ˆρ·(ˆz׈σ)−(ˆz׈σ)·∇ˆρˆσ·m]\n+1\n2Txc(ˆσ·mˆρˆσ·m−ˆρ)−i˜∆xc[ˆσ·m,ˆρ]−2R{ˆσ·m,(m×∇)zˆρ}, (15)\nwhere all quantities are evaluated at Fermi energy ǫF.\nIn a two-dimensional system, the diffusion constant D=\nτv2\nF/2 is given in terms of Fermi velocity vFand mo-\nmentum relaxation time τ. The renormalized exchange\nsplitting reads ˜∆xc= (∆xc/2)/(4ξ2+ 1) where ξ2=\n(∆2\nxc/4+α2k2\nF)τ2. The other parameters are\nC=αkFvFτ\n(4ξ2+1)2,Γ =α∆xcvFkFτ2\n2(4ξ2+1)2, R=α∆2\nxcτ2\n2(4ξ2+1),\n1\nτxc=2α2k2\nFτ\n4ξ2+1, B=2α3k2\nFτ2\n4ξ2+1,1\nTxc=∆2\nxcτ\n4ξ2+1.\nτxcis the relaxation time due to the so-called D’yakonov-\nPerel mechanism.1Equation (15) is valid in the dirty\nlimitξ≪1, whichenablestheapproximation1+4 ξ2≈1.\nCharge density nand the non-equilibrium spin density S\nare introduced by the vector decomposition on the den-\nsity matrix ˆ ρ=n/2 +S·ˆσ. In a real experimental\nsetup,12,15,20spin transport in ferromagnetic layers suf-\nfersfromrandommagneticscatterers, forwhichweintro-\nduce an isotropic spin-flip relaxation S/τsfphenomeno-\nlogically.\nEventually, we obtain a set of diffusion equations for\nthe charge and spin densities, i.e.,\n∂n\n∂t=D∇2n+B∇z·S\n+Γ∇z·mn+R∇z·m(S·m),(16)and\n∂S\n∂t=D∇2S−1\nτ/bardblS/bardbl−1\nτ⊥S⊥\n−∆xcS×m−1\nTxcm×(S×m)\n+B∇zn+2C∇z×S+2R(m·∇zn)m\n+Γ[m×(∇z×S)+∇z×(m×S)],(17)\nwhere∇z≡ˆz×∇. The spin density S/bardbl≡Sxˆx+Syˆyis\nrelaxed at a rate 1 /τ/bardbl≡1/τxc+ 1/τsfwhileS⊥≡Szˆz\nhas a rate 1 /τ⊥≡2/τxc+1/τsf.\nForabroadrangeoftherelativestrengthbetweenspin-\norbit coupling and the exchange splitting, i.e., αkF/∆xc,\nEq.(16) and Eq.(17) describe the spin dynamics in a fer-\nromagnetic layer. When the magnetism vanishes (∆ xc=\n0), theB-term provides a source that generates spin den-\nsity electrically.2,24On the other hand, when the spin-\norbit coupling is absent ( α= 0), the first two lines in\nEq.(17) describe a diffusive motion of spin density in\na ferromagnetic metal, which, to be shown in the next\nsection, agrees excellently with early results in the cor-\nresponding limit.26C-term describes the coherent pre-\ncession of the spin density around the effective Rashba\nfield. The precession of the spin density (induced by the\nRashba field) around the exchange field is described by\nthe Γ-term, thus a higher order (compared to C) in the4\ndirty limit for Γ = ∆ xcτC/2. TheR-term contributes to\nthe magnetization renormalization.\nIII. SPIN DIFFUSION IN A FERROMAGNET\nSpin diffusion in a ferromagnet has been discussed ac-\ntively in the field of spintronics.26–29In this section we\nshow explicitly that, by suppressing the spin-orbit cou-\npling, Eq.(17) describes the spin diffusion equation in the\ncorresponding limits.\nIn the present model, vanishing Rashba spin-orbit cou-\npling means α= 0, then Eq.(17) reduces to\n∂\n∂tS=D∇2S+1\nτ∆m×S\n−1\nτsfS−1\nTxcm×(S×m),(18)\nwhereτ∆≡1/∆xcis the time scale of the coherent pre-\ncession of the spin density around the magnetization.\nThis equation differs from the result of Zhang et al.,27\nonly by a dephasing of the transverse component of the\nspin density that is set by the time scale Txc. In a fer-\nromagnetic metal, we may divide the spin density into\nalongitudinal component that follows the magnetization\ndirection adiabatically, and a deviation that is perpendic-\nularto the magnetization, i.e., S=s0m+δSwheres0\nis the local equilibrium spin density. Such a partition,\nafter restoring the electric field by ∇→∇+eE∂ǫ, gives\nrise to\n∂\n∂tδS+∂\n∂ts0m\n=s0D∇2m+D∇2δS+DePFNFE·∇m\n−δS\nτsf−s0m\nτsf−δS\nTxc+∆xcm×δS,(19)\nwhere the magnetic order parameteris allowedto be spa-\ntial dependent, i.e., m=m(r,t). The energy derivative\nis treated as ∂ǫS≈PFNFmgivenPFthe spin polariza-\ntion and NFthe density of state, both at Fermi energy.\nFor a smooth magnetic texture in which the character-\nistic length scale of the magnetic profile is much larger\nthan the length scale for electron transport, we discard\nthe contribution D∇2δS.26The diffusion of the equilib-\nrium spin density follows s0D∇2m≈s0m/τsf. In this\npaper, we retain only terms that are first order in tem-\nporal derivative, which simplifies Eq.(19) to\n−1\nτ∆m×δS+/parenleftbigg1\nτsf+1\nTxc/parenrightbigg\nδS=\n−s0∂\n∂tm+DePFNFE·∇m.(20)The last equation can be solved exactly\nδS=τ∆\n1+ς2/bracketleftbiggPF\nem×(je·∇)m+ςPF\ne(je·∇)m\n−s0m×∂m\n∂t−ςs0∂m\n∂t/bracketrightbigg\n(21)\nwhereς=τ∆(1/τsf+ 1/Txc) and the electric current\nje=e2nτE/mis given in terms of electron density\nn. Apart from the inclusion of the dephasing of trans-\nversecomponentasimplementedinparameter ς, thenon-\nequilibrium spin density Eq.(21) agrees excellently with\nEq.(8) in Ref.[26].\nGiven the knowledge of the spin density, the spin\ntorque, defined as\nT=−1\nτ∆m×δS+1\nTxcδS, (22)\nis given by\nT=1\n1+ς2/bracketleftbigg\n−ηs0∂m\n∂t+βs0m×∂m\n∂t\n+ηPF\ne(je·∇)m−βPF\nem×(je·∇)m/bracketrightbigg\n(23)\nwhereη= 1 +ςτ∆/Txcandβ=τ∆/τsf. Assum-\ning a long dephasing time of the transverse component\n(i.e.,Txc→ ∞), thenη≈1 and Eq. (23) reproduces the\nEq.(9) in Ref.[26]. On the other hand, a short dephasing\ntime (of the transverse component) enhances parameter\nηtherefore increases the temporal spin torque (i.e., the\nfirst term in Eq.(23)).\nIV. ELECTRICALLY GENERATED SPIN\nDENSITY\nThe effect of an electrically generated non-equilibrium\nspin density due to spin-orbit coupling2can be ex-\ntractedfrom Eq.(17) bysetting exchangeinteractionzero\n(i.e., ∆ xc= 0). Retaining D’yakonov-Perel as the only\nspin relaxation mechanism and letting τsf=∞, Eq.(17)\nends up in\nD∇2S−1\nτxc(S+Szˆz)\n+2C(ˆz×∇)×S+B(ˆz×∇)n= 0 (24)\nwhich reduces to the results in the well-known spin Hall\neffect.24,30,31In the case of an infinite medium along\ntransport direction, i.e., ˆx-direction, Eq.(24) gives rise\nto a solution to the spin density\nS=τxcBeE1\nǫFnˆy=eEζ\nπvFˆy,\nwhere only the linear term in electric field has been re-\ntained. On the right hand side, we have used the charge5\ndensity in a 2D system n=k2\nF/(2π) and introduced the\nparameter ζ=αkFτas used in Ref. [24].\nIn the following sections, we explore the spin torque in\nthe presence of both exchange and Rashba field in an in-\nfinite medium. The primary focus is on two cases: Weak\nand a strong spin-orbit coupling, when comparing to the\nmagnitude of exchange splitting. In general, Eq.(17) is\napplicable through a broad range of relative strength be-\ntween spin-orbit coupling and exchange splitting. A full\nscale numerical simulation on the diffusion equation is\nbeyond the scope of this paper, we refer the readers to\nRef.[22] for further interests.\nV. WEAK SPIN-ORBIT COUPLING\nA weak Rashba spin-orbit coupling implies a small\nD’yakonov-Perel relaxation rate 1 /τxc∝α2, such that\nτxc≫τsf,τ∆, which allows spin relaxation to be dom-\ninated by random magnetic impurities. In this regime,\nwhen comparing to the magnitudes of Cand Γ, the con-\ntribution from BandRare at a higher order in α, thus\nto be disregarded. We consider a stationary state where\n∂S/∂t= 0. An electric ���eld applied along ˆx-direction,\ni.e.,E=Eˆx. In an infinite medium,10all the spatial\nderivatives vanishes ( ∇→0) and the dynamic equation\nreads\n−1\nτ∆m×S+1\nTxcm×(S×m)+1\nτsfS\n=2eECˆy×∂ǫS\n+eEΓ[ˆy×(m×∂ǫS)+m×(ˆy×∂ǫS)].(25)\nIn addition to the spin density induced by exchange\nsplitting, a weak spin-orbit interaction leads to a devi-\nation in spin density that can be considered as a per-\nturbation. Therefore, we may well apply the partition\nS=S⊥+S/bardblmto separate the longitudinal and the\ntransverse components. Eq.(25) is thus reduced to\n1\nτ∆m×S⊥−1\nT⊥S⊥−1\nτsfS/bardblm\n=−2eECP FNFˆy×m\n−eEΓPFNFm×(ˆy×m) (26)\nwhere 1/T⊥≡1/Txc+ 1/τsfand we have again em-\nployedthe approximationon the energy derivative ∂ǫS≈\nPFNFmand replaced the energy derivative of the\ncharge density by the density of states at Fermi energy\n(i.e.,∂ǫn≈n/ǫF=NF). We solve Eq.(26) to obtain a\nsolution to the non-equilibrium spin density\nS⊥=τ∆\n1+ς2eEPFNF[(2C+ςΓ)m×(ˆy×m)\n−(Γ−2ςC)(ˆy×m)]. (27)\nandS/bardbl= 0. In Eq.(27), the second component, oriented\nalong the direction ˆy×m, is actually perpendicular totheplanespannedbythemagnetizationdirectionandthe\neffective Rashba field (along ˆy), which, as to be shown\nbelow, contributes to a Rashba torque that fulfils the\nsymmetry described in a recent experiment.15The defi-\nnition Eq.(22) leads to a general expression for the spin\ntorque\nT=T⊥ˆy×m+T/bardblm×(ˆy×m),(28)\nwhich consists of an out-of-plane and anin-plane com-\nponents with magnitudes determined by\nT⊥=eEPFNF\n1+ς2(2ηC+βΓ), (29)\nT/bardbl=eEPFNF\n1+ς2(ηΓ−2βC). (30)\nTheplaneis defined by the magnetization direction m\nand the direction of the effective Rashba field that in the\npresent setting is aligned along ˆy-direction. Note that\nthe sign of the in-plane torque, Eq. (30), can change\ndepending on the interplay between spin relaxation and\nprecession.\nTo compare directly with the results in Ref.[10], we al-\nlow the spin relaxation time τsf→ ∞, therefore β≈0.\nWe also consider the transverse dephasing time to be\ninfinite.26,27Under these assumptions, η≈1 andς≈0\nand we have T⊥≈2eEPFNFCandT/bardbl≈eEPFNFΓ. In\nthe dirty limit, Γ ≪Cdue to ∆ xcτ≪1, therefore mak-\ning use of the relation for the polarization PF= ∆xc/ǫF\nand the Drude relation je=e2nτE/m, we obtain an\nout-of-plane torque\nT= 2αm∆xc\neǫFjeˆy×m, (31)\nwhich agrees excellently with the spin torque in an in-\nfinite system in the corresponding limit as derived in\nRef.[10].\nVI. STRONG SPIN-ORBIT COUPLING\nThe opposite limit to Sec.V is a strong spin-orbit\ncoupling. In this case, we consider the scenario that\nαkF≫∆xcand the D’yakonov-Perel relaxation mech-\nanism is dominating, i.e., 1 /τxc≫1/τsf, due to the\nfact 1/τxc∝α2. Therefore, it is not physical to sim-\nply assume that the direction of spin density is domi-\nnantlyalignedalongthemagnetizationdirection, aswhat\nis treated in the case of a weak spin-orbit coupling. A\nself-consistent solution from Eq.(17) to the spin density\nis more justified.\nAgain, as in Sec.V, we consider an infinite system\nwhere an electric field Eis applied at ˆx-direction. The\nmagnetization direction is left arbitrary. We approxi-\nmate the energy derivative by ∂ǫ≈1/ǫF. The above6\nassumptions simplify Eq.(17) to\n1\nτ∆S×m+1\nTxcm×(S×m)−2eEC\nǫFˆy×S\n+1\nτxc(S+Szˆz) =eE\nǫFnBˆy,(32)\nwherea strongspin-orbitcouplingrendersΓ and Rterms\nnegligible. By considering Txc≫τ∆,τxc, Eq. (32) re-\nduces to\n1\nτ∆S׈m+1\nτxc(S+Szˆz)\n−2eEC\nǫFˆy×S=eE\nǫFnBˆy,(33)\nwhich is a set of linear equations for the non-equilibrium\nspin density. We are interested in the linear response\nregime, which implies that at the distance as defined by\nthe Fermi wave length 1 /kF, we have eE/kF≪αkF.\nTherefore up to the first order in exchange splitting, we\nextract the spin density from the above equation to be\nS=eE\nǫFnτxcB/parenleftBig\nˆy−χˆy×m−χ\n2mxˆz/parenrightBig\n(34)\nwhereχ≡τxc/τ∆we have used the identity ˆy×m=\nmzˆx−mxˆz. This yields a spin torque\nT=αm∆xc\neǫFje(ˆy×m\n+χm×(ˆy×m)−χ\n2mxˆz×m/parenrightBig\n.(35)\nThis torque is slightly different from the weak Rashba\nlimit and has a strong implication in terms of magneti-\nzation dynamics. The torque is dominated by a field-like\ntorque along ˆy, similarly to the weak Rashba case. First,\nin contrast to the weak Rashba case [see Eq. (30)], the\nsign of the in-plane torque remains positive. Secondly,\nthe anisotropic spin relaxation coming from D’yakonov-\nPerel mechanism yields an additional component of spin\naccumulation that is oriented along ˆz. The implication\nof this torque on the current-driven magnetization dy-\nnamics is discussed in the next section.\nVII. DISCUSSION\nCurrent-induced magnetization dynamics in a sin-\ngle ferromagnetic layer has been observed in vari-\nous structures that involve interfaces between transi-\ntion metal ferromagnets, heavy metals and/or metal-\noxide insulators. Existing experimental systems are\nPt/Co/AlO x,12,13,15,20Ta/CoFeB/MgO,14Pt/NiFe and\nPt/Co bilayers.21Besides the structural complexity in\nsuch systems, an unclear form of spin-orbit coupling in\nthe bulk and interfaces places a challenge to understand\nthe nature of the torque.A. Validity of Rashba model\nThe celebrated Rashba-type effective interfacial spin-\norbit Hamiltonian was pioneered by E. I. Rashba to\nmodel the influence of asymmetric interfaces in semicon-\nducting 2DEG:6A sharp potential drop, emerging at the\ninterface (say, in the xy-plane) between two materials,\ngivesrisetoapotentialgradient ∇Vthatisnormaltothe\ninterface, i.e., ∇V≈ξ(r)ˆz. In case a rotational symme-\ntry exists in the interface plane, a spherical Fermi surface\nassumptionallowsthe spin-orbitinteractionHamiltonian\ntohavetheform ˆHR=αˆσ·(p׈z), whereα≈ ∝an}b∇acketle{tξ∝an}b∇acket∇i}ht/4m2c2.\nAs a matter of fact, in semiconducting interfaces where\nthe transport is described by a limited number of bands\naround a high symmetry point, the Rashba form can be\nrecovered through k·ptheory.32\nAs far as metallic interfaces areconcerned, a spin-orbit\nsplitting of the Rashba-type in the conduction band has\nbeen observedat Au surfaces,33Gd/GdO interfaces,34Bi\nsurfacesandcompounds,35andmetallicquantumwells.36\nThe presence of a Rashba interaction in graphene37and\nat oxide hetero-interfaces38has also been reported re-\ncently. It is quite interesting to notice that the sym-\nmetry breaking-induced spin splitting of the conduction\nband seems rather general and might not be restricted to\nheavy metal interfaces36.\nIn the case of transition metals, however, the free elec-\ntron approximation fails to characterize the band struc-\nture accuratelydue to both alargenumberofband cross-\ningattheFermienergyandastronghybridizationamong\ns,panddorbitals. Density functional theory (DFT) is a\nsuccessful tool to investigate the nature of spin-orbit in-\nteraction at metallic surfaces. For example, in Refs.[39],\nthe authors observe a band splitting that possesses simi-\nlarpropertiesasRashbaspin-orbitinteractionanddecays\nexponentially away from the surface.39Alternatively, the\nspin-orbit interaction at metallic surfaces has been ad-\ndressed using tight-binding models for the porbitals.40,41\nAt such sharp interfaces, the magnitude of the orbital\nangular momentum (OAM) is considered to play a dom-\ninant role at the onset of a Rashba-type spin splitting.\nThis finding is consistent with the long stand-\ning work on interfacial magnetic anisotropy at\na ferromagnet/heavy metal,42and more recently,\nferromagnetic/metal-oxide interface.43In such systems,\na perpendicular magnetic anisotropy arises from the\norbital overlap between the 3 dstates of the ferromagnets\nand the spin-orbit coupled states of the normal metal.\nThe observation of perpendicular magnetic anisotropy\nat Co/metal-oxide interfaces tends to support the major\nrole of large interfacial OAM in the onset of interfacial\nspin-orbit effects.41,43The presence of interfacial Rashba\nspin-orbit coupling has also been shown to produce\ninterfacial perpendicular magnetic anisotropy.44\nAll these previoustheoreticaland experimental studies\nstrongly suggest that the interfacial spin splitting exists\nin the presence of a large OAM and potential gradient.\nHowever, a microscopic description of realistic interfaces7\nis still missing. Although the Rashba spin-orbit interac-\ntion is a convenient Hamiltonian to extract qualitative\nbehaviors, its applicability to realistic metallic interfaces\nwith complex band structures remains to be tested.\nB. Spin Hall effect versus Rashba torque\nRecently, Liu et al.,21proposedtomanipulatethemag-\nnetization of a Pt/Co or Pt/NiFe bilayer using the spin\ncurrent generatedby spin Hall effect in the underlying Pt\nlayer. When injecting a charge current jeinto a normal\nmetal accommodating a strong spin-orbit coupling, the\nasymmetricspinscatteringinducesatransversepurespin\ncurrent that has the form J= (αH/e)je׈µ⊗ˆµ, where\nαHis the spin Hall angle and ˆµis the spin direction.45\nWhen impinging on the ferromagnetic layer deposited on\ntop of the Pt layer, the spin current transverse to the\nlocal magnetization is absorbed and generates a torque\nTSHE= (bH/e)(1−βm×)m×(ˆµ×m) (to be called\nSHE torque thereafter). Here, bH=αHjeµB/eis the\nspintorqueamplitudewheretheregularspinpolarization\nPis replaced by the spin Hall angle αH.βis the non-\nadiabaticity parameter proposed by Zhang and Li26and\nit stems from the presence of spin-flip scattering in the\nsystem. In the configuration adopted by Liu et al.,the\ncharge current is injected along ˆxand the torque is given\nby\nTSHE=αHµBje\ne(m×(ˆy×m)+βˆy×m).(36)\nNote that a more realistic model should account for spin\ndiffusion in Co and Pt, as discussed in Ref. [46]. An\nimportant conclusion is that, besides the correction in\nthe case of a strong Rashba coupling, both Rashba and\nSHE produce the same type of torque, see Eq.(28) and\nEq.(35) in this article.\nNevertheless, distinctions can be made. First, in the\nabsence of the corrections due to spin-flip and spin pre-\ncession, the Rashba torque reduces to the field-like term,\nˆy×m, whereas the SHE torque reduces to the (anti-\n)dampingterm m×(ˆy×m). Thisassertionmustbescru-\ntinizede carefully since the actual relative magnitude be-\ntween the field-like and the damping torques depends on\nthe width of the magnetic wire as well as on the detailed\nspin dynamics in presence of spin-flip and precession.22\nFurthermore, for such an ultra-small system the spin-flip\nscattering giving rise to the non-adiabaticity parameter\n(β) might be significantly different from the one mea-\nsured in a more conventional thin film.\nA second important difference arises from the fact that\nthe Rashba torque arises from spin-orbit fields generated\nbyinterfacial currents, whereas the SHE torque is due to\nthe current flowing in the bulkof the Pt layer. Therefore,\nforaconstantexternalelectricfield, varyingthethickness\nof Pt layer shall enhance the SHE torque, while keeping\nthe Rashba torque unchanged.The torques as a function of the Co layer thickness is\nmore difficult to foresee. Although one could claim that\nRashba spin-orbit interaction is expected to be localized\nat the interface, where the potential gradient is large,\nnumerical simulations show that the Rashba-type inter-\nactionsurvivesafew monolayers39(which istypically the\nthickness of the Co layer under consideration). In addi-\ntion, the presenceofquantumwellstatesmightalsomod-\nifythenatureofthespin-orbitinteractionintheultrathin\nmagnetic layer in a system such as Pt/Co/AlO x.36\nThe same is true for the SHE torque. The injection\nof spin current into a Co layer is accompanied by spin\nprecession that takes place over a very short decoherence\nlength. This decoherence length has been studied experi-\nmentally and theoretically in spin valves and found to be\nof the order of a few monolayers.47In the typical case of\n3 or 4 monolayer-thickferromagnets, the SHE torque can\nnot be considered as a purely interfacial phenomenon.\nC. Magnetization Dynamics\nIn Pt/Co/AlO xtrilayers, Miron et alhave observed\na current-driven domain wall nucleation,12an enhanced\ncurrent-driven domain wall velocity20and a current-\ndriven magnetization switching.15The symmetry of the\nspin torque required to explain the experimental findings\nagree well with Rashba torque proposed in Ref. 10. On\na similar structure, Pi et al13and Suzuki et al14also ob-\nserved an effective field torque that could be interpreted\nintermsoftheRashbatorque. Recently,Liu et al21inter-\npreted their experiments on Pt/NiFe and Pt/Co bilayers\nusing SHE in the underlying Pt layer.\n1. Magnetization switching\nAccording to our previous discussions, both Rashba\ntorque and SHE torque have a general form T=T⊥ˆy×\nm+T/bardblm×(ˆy×m). The first term acts like a field\noriented along the direction transverse current direction\nwhereas the second term acts like an (anti-)damping\nterm, mimicking a conventional spin transfer torque that\nwould arise from a polarizer pointing to ˆy.\nAsaconsequence,bothRashbatorqueandSHEtorque\npossess the appropriate symmetry to excite the magne-\ntization of a single ferromagnet and induce switching, as\nobservedby Miron et al15and Liuet al.21In the case of a\nlarge Rashba spin-orbit coupling, the torque acquires an\nadditional component that acts like an effective magnetic\nfield along ˆz, vanishing as the magnetization component\nmxis zero (see Section VI), which provides an additional\ntorque that helps destabilize the magnetization.8\n2. Current-driven domain wall motion\nThe influence of Rashba/SHE torque on a domain wall\ncan be illustrated within the rigid Bloch wall approxima-\ntion. The perpendicularly magnetized Bloch wall is pa-\nrameterized by m= (cosφsinθ,sinφsinθ,cosθ) where\nφ=φ(t) andθ(x,t) = 2tan−1[e(x−xc(t))/∆], where xc\nrefers to the center of the domain wall and ∆ is defined\nas the domain wall width. To describe the dynamics of\na Bloch wall, Landau-Lifshitz-Gilbert (LLG) equation\n∂tm=−γm×Heff+αG∂tm×m+τ(37)\nhas to be augmented by the current induced torque τ\nτ=bJ∇m−βbJm×∇m\n+bJ(τ⊥ˆy×m+τ/bardblm×(ˆy×m)\n+τzmxˆz×m).(38)\nThe torque τis written in the most general form, where\nthe first two terms are the regular adiabatic and the so-\ncalled non-adiabatic torques; the next two terms ( τ/bardbland\nτ⊥) emerge from the presenceof Rashbaand/orspin Hall\neffect and the last term τzappears only in large Rashba\nlimit (see Sec. VI). The magnitude of the adiabatic\ntorque is bJ=µBPje/e. The effective field is given by\nHeff=2A\nMs∇2m+HKmxˆx+H⊥mzˆz.(39)\nParameter γin LLG is the gyromagnetic ratio, αGis the\nGilbert damping, Ais the exchange constant, Msis the\nsaturation magnetization, HKis the in-plane magnetic\nanisotropy and H⊥is the combination of an out-of-plane\nanisotropyand ademagnetizingfield. Themagnetization\ndynamics can be obtained readily from Eqs. (37)-(39) by\nintegrating over the magnetic volume\n∂tφ+αG∂txc\n∆=/bracketleftbigg∆π\n2(τ/bardbl−τz\n2)cosφ−β/bracketrightbiggbJ\n∆(40)\nαG∂tφ−∂txc\n∆=−γHK\n2sin2φ+/parenleftbigg\n1+∆π\n2τ⊥cosφ/parenrightbiggbJ\n∆.\n(41)\nWe observe that the in-plane torque τ/bardbldistorts the do-\nmain wall texture, while the perpendicular torque τ⊥\ndrives the domain wall motion. The additional torque\nτz, arising in the large Rashba limit, only contributes to\nthe in-plane torque. Therefore, in the following, we will\nrefer to the in-plane torque as τ∗\n/bardbl=τ/bardbl−τz/2. Below the\nWalker breakdown ( ∂tφ= 0), the velocity is given by\n∂txc=−/parenleftbigg\nβ−∆π\n2τ∗\n/bardblcosφ/parenrightbiggbJ\nαG(42)\nγHK\n2sin2φ=/bracketleftbigg\nαG−β+∆π\n2(αGτ⊥+τ∗\n/bardbl)cosφ/bracketrightbiggbJ\nαG∆,\n(43)where the tilting angle φis given by the competition be-\ntween the magnetic anisotropy, the non-adiabatictorque,\nand the Rashba/SHEtorque. In the case ofweak Rashba\n(τz= 0), assuming τ/bardbl=βτ⊥and omitting the correction\nto the spin precession, we recover the results of Ref. [48].\nWhen neglecting the in-plane torque and accounting for\nthe perpendicular Rashba torque ( τ∗\n/bardbl= 0), the Rashba\ntorque only acts like an effective transverse field and en-\nhances the Walker breakdown limit20[see Eq. (43)].\nAccounting for the in-plane component τ/bardblarising ei-\nther from corrections to Rashba torque or from the SHE,\nthis torque appears to modify the domain wall veloc-\nity. Therefore, depending on the strength and the sign\nof Rashba/SHE torque as well as on the resulting tilt-\ning angle φ, it is possible to obtain a vanishing or even\na reversed domain wall velocity, as has been shown nu-\nmerically in Ref. [48] and illustrated in Eq. (42). A full\nscale numerical investigation is beyond the scope of this\narticle, but it will help understand the profound effect of\nRashba and SHE torque on the domain wall structures.\nVIII. CONCLUSION\nUsing Keldysh technique, in the presence of both mag-\nnetism and a Rashba spin-orbit coupling, we derive a\nspin diffusion equation that provides a coherent descrip-\ntion to the diffusive spin dynamics. In particular, we\nhave derived a general expression for the Rashba torque\nin the bulk of a ferromagnetic metal layer, at both weak\nand strong Rashba limits. We find that the torque is in\ngeneral composed of two components, a field-like torque\nand the other (anti-)damping one. Being aware of the\nrecent alternative interpretation on the current-induced\nmagnetization switching in a single ferromagnet, we have\ndiscussedthe differencebetweentheRashbaandtheSHE\ntorques. While exploring the common features, we found\nthat the magnetization dynamics driven by the Rashba\ntorque presents several interesting similarities to that in-\nduced by SHE torque. Nevertheless, further investiga-\ntion involving structural modification of the system is\nexpected to provide a deeper knowledge on the nature\nof the interfacial spin-orbit interaction as well as the\ncurrent-induced magnetization switching in a single fer-\nromagnet.\nAcknowledgments\nWe thank G. E. W. Bauer, J. Sinova, M. D. Stiles, X.\nWaintal, and S. Zhang for stimulating discussions. We\nare specially grateful to K. -J. Lee and H. -W. Lee for\ninspiring discussion about the magnetization dynamics.9\n∗Electronic address: xuhui.wang@kaust.edu.sa\n†Electronic address: aurelien.manchon@kaust.edu.sa\n1M. I. D’yakonov and V. I. Perel, Sov. Phys. JETP Lett.\n13, 467 (1971); Phys. Lett. A 35, 459 (1971).\n2V. M. Edelstein, Solid. Stat. Comm. 73, 233 (1989).\n3S. Murakami et al.,Science301, 1348 (2004).\n4J. Sinova et al., Phys. Rev. Lett. 92, 126603 (2004).\n5Y. K. Kato et al.,Science306, 1910 (2004).\n6Yu. A. Bychkov and E. I. Rashba, J. Phys. C: Solid. Stat.\nPhys.17, 6039 (1984).\n7M. I. D’yakonov and V. I. Perel, Sov. Phys. Solid State 13,\n3023 (1971).\n8J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996);\nL. Berger, Phys. Rev. B. 54, 9353 (1996).\n9M. D. Stiles and J. Miltat, Top. Appl. Phys. 101, 225\n(2006); D. C. Ralph and M. D. Stiles, J. Magn. Magn.\nMater.320, 1190 (2008); J. Z. Sun and D. C. Ralph, ibid.\n320, 1227 (2008).\n10A. Manchon and S. Zhang, Phys. Rev. B. 78, 212405\n(2008);ibid.79, 094422 (2009).\n11A. Matos-Abiague and R. L. Rodriguez-Suarez, Phys. Rev.\nB80, 094424 (2009); I. Garate and A. H. MacDonald, ibid.\n80, 134403 (2009); P. M. Haney, and M. D. Stiles, Phys.\nRev. Lett. 105, 126602 (2010).\n12I. M. Miron et al.,Nature Materials 9, 230 (2010).\n13U. H. Pi et al.,Appl. Phys. Lett. 97, 162507 (2010).\n14T. Suzuki et al.,Appl. Phys. Lett. 98, 142505 (2011).\n15I. M. Miron et al.,Nature (London) 476, 189 (2011).\n16A. Chernyshov et al.,Nature Physics 5, 656 (2010).\n17D. Fang et al.,Nature Nanotechnology 6, 413 (2011).\n18M. Endo et al.,Appl. Phys. Lett. 97, 222501 (2010).\n19K. Obata, and G. Tatara, Phys Rev. B 77, 214429 (2008).\n20I. M. Miron et al.,Nature Materials 10, 419 (2011).\n21L. Liu et al.,Phys. Rev. Lett. 106, 036601 (2011);\narXiv:1110.6846 (2011).\n22X. Wang and A. Manchon, arXiv:1111.1216 (2011).\n23J. Rammer and H. Smith, Rev. Mod. Phys. 58, 323 (1986).\n24E. G. Mishchenko et al.,Phys. Rev. Lett. 93, 226602\n(2004).\n25J. Rammer, Quantum Field Theory of Non-equilibrium\nStates(Cambridge University Press, Cambridge, 2007).\n26S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 (2004).\n27S. Zhang et al.,Phys. Rev. Lett. 88, 236601 (2002).28Y. Tserkovnyak et al.,Rev. Mod. Phys. 77, 1375 (2005).\n29Y. Tserkovnyak et al.,Phys. Rev. B 79, 094415 (2009).\n30A. A. Burkov et al.,Phys. Rev. B 70, 155308 (2004).\n31˙I. Adagideli and G. E. W. Bauer, Phys. Rev. Lett. 95,\n256602 (2005).\n32R. Winkler, Spin-Orbit Coupling Effects in Two-\nDimensional Electron and Hole Systems , Springer, Berlin,\n(2003).\n33S. LaShell et al.,Phys. Rev. Lett. 77, 3419 (1996); F.\nReinertet al.,Phys. Rev. B 63, 115415 (2001); G. Nicolay\net al., ibid .65, 033407 (2001); M. Hoesch et al., ibid .69,\n241401(R) (2004).\n34O. Krupin et al.,Phys. Rev. B 71, 201403(R) (2005); O\nKrupinet al.,New. J. Phys. 11, 013035 (2009).\n35Ph. Hofmann, Prog. Surf. Sci. 81, 191 (2006); T. Hirahara\net al.,Phys. Rev. B 76, 153305 (2007); H. Mirhosseini\net al., ibid .79, 245428 (2009); A. Takayama et al.,Phys.\nRev. Lett. 106, 166401 (2011); K. Ishizaka et al.,Nature\nMaterials 10, 521 (2011).\n36A. Varykhalov et al.,Phys. Rev. Lett. 101, 256601 (2008);\nJ. Hugo Dil et al., ibid .101, 266802 (2008); A. G. Rybkin\net al.,Phys. Rev. B 82, 233403 (2010).\n37Yu. S. Dedkov et al.,Phys. Rev. Lett. 100, 107602 (2008);\nO. Rader et al., ibid .102, 057602 (2009).\n38A. D. Caviglia et al.,Phys. Rev. Lett. 104, 126803 (2010).\n39G. Bihlmayer et al.,Phys. Rev. B 75, 195414 (2007); G.\nBihlmayer et al.,Surf. Sci. 600, 3888 (2006); M. Nagano\net al.,J. Phys.: Cond. Mater. 21, 064239 (2009).\n40L. Petersen and P. Hedeg˚ ard, Surf. Sci. 459, 49 (2000).\n41S. R. Park et al.,Phys. Rev. Lett. 107, 156803 (2011).\n42K. Kyuno et al.,J. Phys. Soc. Jpn. 61, 2099 (1992); D.\nWelleret al.,Phys. Rev. B 49, 12888 (1994); G. H. O.\nDaalderop et al., ibid .50, 9989 (1994).\n43H. X. Yang et al.,Phys. Rev. B 84, 054401 (2011); C.\nNistoret al., ibid .84, 054464 (2011).\n44A. Manchon, Phys. Rev. B 83, 172403 (2011).\n45J. E. Hirsch, Phys. Rev. Lett. 83, 1834 (1999); S. Zhang,\nibid.85, 393 (2000).\n46A. Vedyayev et al.,arXiv:1108.2589 (2011).\n47S. Wang et al.,Phys. Rev. B 77, 184430 (2008).\n48K.-W. Kim et al.,arXiv:1111.3422 (2011)."    },    {        "title": "1704.03133v1.Dynamics_of_Pure_Spin_Current_in_High_frequency_Quantum_Regime.pdf",        "content": "1 \n Dynamics  of Pure Spin Current in High -frequency Quantum Regime  \n \nShuichi Iwakiri ,1 Yasuhiro Niimi ,1,2 and Kensuke Kobayashi1,2 \n1Department of Physics, Graduate School of Science , Osaka  University,  Osaka  560-\n0043, Japan  \n2Center for Spintronics Research Network (CSRN),  \nGraduate School of Engineering Science, Osaka University, Osaka 560 -8531, Japan.  \nAbstract  \nPure spin current is a powerful tool for manipulating spintronic devices, \nand its dynamical behavior is an important issue. By using mesoscopic transport \ntheory for electron tunneling induced by spin accumulation, we investigate the \ndynamics of the spin current in the high -frequency quantum regime, where the \nfrequency is much larger than temperature and bias voltag e. Besides the thermal \nnoise, frequency -dependent finite noise emerges, signaling the spin current across \nthe tunneling barrier. We also find that the autocorrelation of the spin current \nexhibits sinusoidal oscillation in ti me as a consequence of the Pauli exclusion \nprinciple even without any net charge current.  2 \n Spin current , a flow of spin angular momentum , is one of the central concepts \nin spintronics [ 1-3]. It plays an important role  not only as a tool for controlling various \nspintronic device s but also as a probe for spin dynamics  in solids  [1-9]. Especially, p ure \nspin current , a flow  of spin without net charge current , is of great interest  because it is \nsupposed  to be dissipationless  [10-12]. Beside s various studies to utilize it, there are \nseveral attempts  to reveal  its fundamental nature  [13,14,15]. Nevertheless,  there remains \nmuch to be done to  understand  the spin current  itself and new measurement probes  are \nrequired . As spin is a quantum object, it is of significance to address quantum nature of \nthe spin current by investigating  its dynamics [15]. \nShot noise, or non -equilibrium current fluctuation, can be a unique tool to \ntackle this problem.  Conventionally, it  appears  when  electrons tunnel through  a potential \nbarrier  such that  the current noise power spectral density  𝑆I is expressed as  the Schottky \nformula 𝑆𝐼=2𝑒|𝐼| in the zero -frequency and zero-temperature limit  (𝐼: the current and \ne: electric charge) [1 6]. Interestingly, shot noise  is not trivial for spin current  [17,18]. \nConsider a pure spin current, where the current with spin -up electrons (𝐼↑) and that with \nspin-down ones (𝐼↓) move in the opposite direction  (say, 𝐼↑=−𝐼↓). While the re is no  net \ncharge current, the fluctuation of each current adds up according to the Schottky formula , \nyielding  a finite shot noise  of 𝑆𝐼=2𝑒(|𝐼↑|+|𝐼↓|). Recently , the relevance of this concept  3 \n of “spin shot noise ” was experimentally demonstrated in the (Ga(Mn)As -based all -\nsemiconductor lateral spin -valve device [18]. On the other hand , the above treatment  is \nmade for the zero -frequency limit  and t he shot noise at the finite frequency would enable \nus to obtain new information on transport dynamics .  \nRecently, the shot noise in this regime was experimentally add ressed [ 19,20,21]. \nThibault et al. [19] clarified an important  aspect of the current fluctuation  in the charge \ntransport  in a tunnel barrier . They measured noise spectr a in the high-frequency quantum \nregime, namely, the frequency region much higher than both temperature and bias voltage, \nand derived the autocorrelation of the charge current. They observed the oscillation of \nautocorrelation in time as a direct consequence of the Pauli exclusion principle in the \ncurrent . This is a clear manifestation  of the quantum nature of the charge current. However, \nsuch an attempt has been lacking  for the spin current . \nIn this Letter , expanding the concept of  the spin shot noise to the finite  \nfrequency  region , we show that the Pauli exclusion principle  is also relevant for  the spin \ncurrent  even when there is no net charge current . We evaluate the noise spectrum and \nautocorrelation of the spin current induced by the spin accumulation at the tunnel barrier . \nWe also discuss the experimental feasibility  of our results .  \nFigure 1(a) shows the setup we consider here , which is similar to the 4 \n conventional lateral -spin valve system  [23,24]. The device consists of one ferro magnetic \nlead (FM) that is magnetized along the lead , and two nonmagnetic lead s attached to it; \nmiddle lead  (NM1)  and right lead  (NM2) . There is a tunnel  barrier between NM1 and \nNM2. Note that we assume NM2  to be nonmagnetic just for simplicity , while  the \nfollowing result can be easily generalized for the ferromagnetic case by taking the spin \npolarization into account  phenomenologically . \nBy injecting a spin-polarized  current from FM to NM1 , spin accumulation is \ncreated inside NM1  along the x-axis (Fig.1 (a)) ; The chemical potentials of the spin -up \nand spin -down electrons become x-dependent , generating a spin current as shown in Fig. \n1(b). A part of this spin current  flows  down  along the z-axis into NM2  through the tunnel \nbarrier. The energy diagram in the vicinity of the barrier  is presented  in Fig.  1(c), where  \nthe chemical potentials of the spin -up and spin -down electrons steeply change at the \nbarrier . The resulting  potential difference at the barrier corresponds to the spin \naccumu lation  (𝛿𝜇). Here, we neglect  the effect of the spin diffusion a long the z-axis across \nthe barrier , as the signature of such an effect  was never detected in the  experiment  of \nRef.[1 8]. The spin diffusion does not  play an important role as long as we consider \ndevices similar to those in Ref.[1 8]. We apply 𝑉 as the voltage difference between NM1 \nand NM2  (Figure 1(c) shows the case of 𝑉=0). We consider  the noise generated at this  5 \n barrier . Note that, a s it occurs locally, the noise is irrelevant to the spin diffusion process  \nin NM2 . The spin flip  during  the tunneling can be neglected , which was validated  by the  \nrecent experiment [ 18]. \nThe calculation is performed using  the mesoscopic transport theory based on \nthe Landauer -Büttiker formalism  [24]. The tunnel barrier is  treated  as a one-dimensional  \nsingle channel scatterer with an energy -independent  transmission probability,  τ . Note \nthat i t is straightforward  to extend our analysis to the multi -channel case. We define a \ncurrent operato r 𝐼̂𝛾,𝜎  using the second quantization . Here , γ  and σ  denote leads \n(L:NM1 or R:NM2 ) and spin ( ↑ or ↓), respectively. By taking the quantum statistical \naverage , the mean  current is given in the well-known formula as follows:  \n 〈𝐼̂𝛾,𝜎〉 =𝑒\nℎ𝜏∫ 𝑑𝐸[𝑓𝛾,𝜎(𝐸,𝜇𝛾,𝜎,𝑇)−𝑓𝛾′,𝜎(𝐸,𝜇𝛾′,𝜎,𝑇)]∞\n−∞,   (1) \nwhere  ℎ  is the Planck constant , 𝐸  is the electron energy, and  𝑓𝛾,𝜎(𝐸)  is the Fermi \ndistribution function for electrons with chemical potential 𝜇𝛾,𝜎 and temperature 𝑇.  \nThe spin current operator is  defined  as 𝐼̂𝑆=𝐼̂↑−𝐼̂↓ [17]. Here, the spin-up and \nspin-down  channels are independen t of each other  as we can neglect the spin flip . By \nintegrating the Fermi distribution function , we obtain   〈𝐼̂𝛾,𝜎〉=𝑒\nℎ𝜏(𝜇𝛾,𝜎−𝜇𝛾′,𝜎)(1−\n𝛿𝛾,𝛾��) . Substituting  the chemical potential of each lead  𝜇𝐿↑/↓=𝜇0+𝑒𝑉\n2±𝛿𝜇\n2  and \n𝜇𝑅↑/↓=𝜇0−𝑒𝑉\n2, we obtain 〈𝐼̂𝐶〉=2𝑒2\nℎ𝜏𝑉 for the charge current , and 〈𝐼̂𝑆〉=𝑒\nℎ𝜏(𝑒𝑉+6 \n 𝛿𝜇) for the spin current . This is  consistent with the previous results [ 18].  \nNow we discuss  the noise spectrum.  By defin ing the current noise operator \n𝛿𝐼̂𝛾,𝜎=𝐼̂𝛾,𝜎−〈𝐼̂𝛾,𝜎〉, the noise spectrum is expressed  as follows : \n𝑆𝛾,𝜎(𝜈)=∫〈𝛿𝐼̂𝛾,𝜎(𝑡)𝛿𝐼̂𝛾,𝜎(𝑡+𝜏)〉𝑒2𝜋𝑖𝜈𝜏𝑑𝜏,∞\n−∞ \nwhere 𝑡 is time and  𝜈 is frequency  defined from −∞ to +∞. Following the work by \nMeair et al.  [17],  in the framework of the Landauer -Büttiker formalism , the spin -\ndependent transmission channels can be treated as if they form a parallel circuit. Thus the \ntotal noise spectru m of the spin current 𝑆𝑡𝑜𝑡𝑎𝑙 =∑ 𝑆𝛾,𝜎(𝜈)𝛾,𝜎   is analytically given as \nfollows :  \n𝑆𝑡𝑜𝑡𝑎𝑙 (𝜈,𝑇,𝑉,𝛿𝜇)=𝑒2\nℎ4𝜏2ℎ𝜈\n1−𝑒−ℎ𝜈\n𝑘𝐵𝑇+𝑒2\nℎ𝜏(1−𝜏)(𝑒𝑉+𝛿𝜇\n2+ℎ𝜈)\n1−𝑒−1\n𝑘𝐵𝑇(ℎ𝜈+𝑒𝑉+𝛿𝜇\n2)+𝑒2\nℎ𝜏(1−𝜏)(−𝑒𝑉−𝛿𝜇\n2+ℎ𝜈)\n1−𝑒−1\n𝑘𝐵𝑇(ℎ𝜈−𝑒𝑉−𝛿𝜇\n2) \n+𝑒2\nℎ𝜏(1−𝜏)(𝑒𝑉−𝛿𝜇\n2+ℎ𝜈)\n1−𝑒−1\n𝑘𝐵𝑇(ℎ𝜈+𝑒𝑉−𝛿𝜇\n2)+𝑒2\nℎ𝜏(1−𝜏)(−𝑒𝑉+𝛿𝜇\n2+ℎ𝜈)\n1−𝑒−1\n𝑘𝐵𝑇(ℎ𝜈−𝑒𝑉+𝛿𝜇\n2) .      (2) \nBy taking the zero frequency and zero temperature limit, we obtain  the expression  \nconsistent with that was given in Ref.[1 8]. The quantum nature of the current appears in \nthe finite -frequency component of the calculated noise [24].  As it is generated via the \nprocess where there is a finite energy difference between the initial and the final states, \nthe system absorbs/emits a photon to conserve the energy.  Actually , Eq.(2)  is understood \nin terms of one dimensional emission ( 𝜈<0) and absorption ( 𝜈>0) spectrum of \nphoton with an energy of ℎ𝜈±𝑒𝑉+𝛿𝜇\n2 for the spin -up channel and ℎ𝜈±𝑒𝑉−𝛿𝜇\n2 for 7 \n the spin -down  channel.  Thus  the quantum nature is naturally introduced  when we \nconsider the finite frequency noise.  When  the emission and absorption processes  occur \nwith the same probability, the noise spectr um can be symmetrized with regard to the \npositive and negative frequency:  𝑆sym(𝜈,𝑇,𝑉,𝛿𝜇)=𝑆(𝜈,𝑇,𝑉,𝛿𝜇)+𝑆(−𝜈,𝑇,𝑉,𝛿𝜇) \nwith 𝜈=[0,∞).  \nIn the rest of this paper , we focus only on the zero bias regime ( 𝑉=0, see Fig.  \n1(c)) , where there is no net charge current  across the barrier . For simplicity, w e redefine \n𝑆sym(𝜈,𝑇,𝛿𝜇)≡𝑆sym(𝜈,𝑇,𝑉=0,𝛿𝜇). Two remarks are made for the zero -frequency \nlimit . First, 𝑆sym(𝜈=0,𝑇,𝛿𝜇=0)≡𝑆0 gives the classical thermal ( Johnson -Nyquist ) \nnoise  [25]. Second, 𝑆sym(𝜈=0,𝑇,𝛿𝜇) reproduces the previous result [ 18]. Now, for the \nfinite frequency, in Fig.2, we show  𝑆sym(𝜈,𝑇,𝛿𝜇)/𝑆0 as a function of the normalized \nfrequency  (ℎ𝜈/𝑘𝐵𝑇) for the case s of 𝛿𝜇=0 (no spin accumulation ) and 𝛿𝜇/𝑘𝐵𝑇=1,\n3, and 5 (finite spin accumulat ion). While 𝑆sym(𝜈,𝑇,𝛿𝜇=0) again equals to  the well-\nknown thermal noise spectra , we found the increase of the noise  when there is finite spin \naccumulation . This indicates that the shot noise  is generated by the spin current even \nwithout any net charge current .  \nWhat does the increase of the noise mean? To understand this, we investigate the \ndynamics  of the system  in real time . Applying W iener-Khinchin theorem to Eq. (2), we 8 \n derive the autocorrelation of the spin current.  Before this treatment,  following Ref. [19], \nwe need to redefine  the current noise spectral density  as 𝑆sym′(𝜈,𝑇,𝛿𝜇)≡\n 𝑆sym(𝜈,𝑇,𝛿𝜇)−𝑆sym(𝜈,𝑇=0,𝛿𝜇), because 𝑆sym diverges such that 𝑆sym →4𝑒2𝜏𝜈 \nfor 𝜈→∞. This subtraction means that we ignore the contribution of the vacuum \nfluctuation in the noise spectrum  (see the dotted line in Fig.(2)) , and focus  ourselves  only \non the noise generated by the electron tunneling .  \nWe found that t he autocorrelation of the spin current is given as follows:  \n𝐶(𝑡,𝛿𝜇,𝑇)=4𝜏((1−𝜏)cos(𝛿𝜇𝑡\nℏ)+𝜏)(1\n2(ℏ\n𝑘𝐵𝑇)2\n−2(𝜋\n𝑒𝑘𝐵𝑇\n2ℏ𝑡−𝑒−𝑘𝐵𝑇\n2ℏ𝑡)2\n). (3) \nWe plot Eq. (3) as a function of time in units of  ℎ/𝛿𝜇 in Fig.  3. In the absence of the \nspin accumulation ( 𝛿𝜇=0), the thermal noise is the only origin  of the noise , making  the \nautocorrelation 𝐶(𝑡,𝛿𝜇=0,𝑇) monotonously decrease according to time . This means \nthat the quantum coherence of electron decays due to the thermal agitation with a \ncharacteristic time scale  of ℎ/𝑘𝐵𝑇. In the presence of the spin accumulation (𝛿𝜇≠0), \nthe autocorrelation oscillat es with the envelope 𝐶(𝑡,𝛿𝜇=0,𝑇). Thibault et al. observed \na similar oscillation when a charge current flows across the voltage -biased barrier , which \ndirectly indicates  that electron can tunnel only one by one due to the Pauli exclusion \nprinciple  [19]. In the same way, the present oscillation suggests  that th is principle also \naffects transport dynamics of the spin current  even when there is no net charge current .  9 \n Moreover , we found  an interesting  temperature -independent relation : \n𝐶(𝑡,𝛿𝜇,𝑇)\n𝐶(𝑡,0,𝑇)=𝜏+(1−𝜏)cos(𝛿𝜇\nℏ𝑡).    (4)  \nThis shows that the oscillation with a frequency 𝛿𝜇/ℎ is always present  regardless of  \ntemperature . \nThe time dependence  of the autocorrelation clearly shows the quantum nature of \nthe spin current. With the same analogy as discussed in Ref. [19], the oscillation of the \nautocorrelation can be understood as follows. The tunneling of the spin current occurs as \nspin-up and spin -down electrons sequentially come into the barrier. Due to the Pauli ’s \nexclusion principle, only one spin -up and one spin -down electron can tunnel in a certain \ntime. Here, spin -up and spin -down electrons have the same chemical potential difference \nwith the same absolute value but with the opposite signs. In the quantum regime, because \nthe energy before the tunneling and that after the tunneling are different, it takes a finite \ntime to resolve the electron ’s wave function of the two states, which is at least ℎ/𝛿𝜇 \nreflecting Heisenberg ’s uncertainty principl e.  \nThus the autocorrelation oscillates with the period of ℎ/𝛿𝜇, indicating that the \nPauli ’s exclusion principle acts on the spin current. Such quantum nature can be destroyed \ndue to the decoherence processes. In this case, because of the thermal agitation, the \ncoherence vanishes a s a function of time, and thus  the oscillation of the autocorrelation 10 \n decay as shown in Fig. 3. Here, ℎ/𝑘𝐵𝑇 has a dimension of time, which corresponds to \nthe coherence time.   \nThen we discuss the feasibility of the experiment. The measurement frequency \nmust be higher than 𝛿𝜇/ℎ to observe more than one period of the oscillation. At the same \ntime, it must be higher than 𝑘𝐵𝑇/ℎ to inhibit the thermal decay of electron coherence. \nFor example, when 𝑇=30 mK  and  𝛿𝜇=5 𝜇eV,  the noise spectra need to be \nmeasured up to the frequency higher than 1 GHz.  This can be realized using dilution \nrefrigerator  (its temperature range is typically from a few ten mK to room temperature)  \nwith high frequency noise measurement setup . For examp le, measurement sensitivity can \nreach up to 10−27  A2/Hz while the shot noise of the sample with 50 ohm and 5 μeV \nbias is of order of 10−26 A2/Hz according to the Schottky formula .  \nThen we discuss the feasibility of the experiment. The measurement frequency \nmust be higher than δμ/h to observe more than one period of the oscillation. At the same \ntime, it must be higher than 𝑘𝐵𝑇/ℎ to inhibit the thermal decay of electron coherence. \nFor example, when T=30 mK and δμ=5 μ eV , the noise spectra need to be measured up to \nthe frequency higher than 1 GHz. This can be realized using dilution refrigerator (its \ntemperature range is typically from a few ten mK to room temperature) with high \nfrequ ency noise measurement setup. For example, measurement sensitivity can reach up 11 \n to 10−27  A2/Hz while the shot noise of the sample with 50 ohm and 5 μeV bias is of \norder of 10−26 A2/Hz according to the Schottky formula.   \nFinally, we mention several points which we have ignored but may affect the \nbehavior of the noise and the autocorrelation. First, if there exist decoherence mechanism \nother than thermal agitation, it affects the decaying behavior. To investigate it \nexperimentally is an important issue as it  gives critical information on decoherence \nsources for the spin current. Second, the transmission may be energy -dependent in high \ntemperature regime, while we assumed the energy -independent transmission here. We \nbelieve th e assumption is valid in the energ y scale of a few ten mK and a few μeV. For \nexample, in the case of Fe/MgO, according to Fig.3 of Ref.[26] , the typical energy scale \ngoverning the spin dependent transport is at least of the order of a few meV . So we believe \nthat the energy dependence of the tunneling can be ignored if we consider such a very \nlow energy scale. However, this  assumption is certainly broken at high temperature (like \nroom temperature) and the energy dependence becomes very important.  Third, the \ntransmission may be channel -dependent with some materials whose unique band structure \ncan yield the channel dependence. For example, the conduction band of Fe/MgO has a \nstrong spin dependence. In our calculation, this can be included as spin ch annel dependent \ntransmission which is equivalent as including spin polarization of the electrodes as shown 12 \n before. We may experimentally detect the energy and channel dependence by check ing \nthe temperature dependence of Eq. (4) or changing materials, which  is surely very \ninteresting as it gives us new information. We may experimentally detect it through Eq.(4), \nwhich is surely very interesting as it conveys us new information.  \nIn summary, according to the Landauer -Büttiker  formalism, we have analytically \nderived the noise spectrum and the autocorrelation  of the spin current at the  tunnel \njunction. We show t he temperature -independent behavior of the autocorrelation  due to the \nPauli exclusion principle  for the spin current , which  can be detectable experiment ally. \nSuch an experiment would  enable us to directly address  several unexplored aspects of the \nspin current, for example, its quantum coherence and dissipationless property.  \n \nAcknowledgments  \nThis work was partially supported by JSPS KAKENHI Grant Numbers JP26220711, \nJP25103003, JP15H05854, JP16H05964, and JP26103002, Yazaki Memorial Foundation \nfor Science and Technology, RIEC, Tohoku University.  \n  13 \n References  \n1. S. Maekawa, H . Adachi, K . Uchida: J.  Phys.  Soc. Jpn. 82 (2013) 102002 . \n2. A. Hirohata and K. Takanashi: J. Phys. D: Appl. Phys. 47 (2014) 193001  \n3. I. Zutic, J . Fabian, and S. D. Sarma: Rev.Mod.Phys. 76 (2004) 323 . \n4. Y . Niimi, D . Wei, and Y . Otani: J.  Phys.  Soc.Jpn. 86 (2017) 011004 . \n5. D.H.  Wei, Y .  Niimi, B.  Gu, T.  Ziman, S.  Maekawa, and Y .  Otani : Nat.Comm. 3 (2012) \n1058 . \n6. J. Grollier, V . Cros, A. Hamzic, J. M. George, H. Jaf frès, A. Fert G. Faini J. Ben Youssef \nand H. Legall: Appl. Phys. Lett. 78 (2016)  3663 . \n7. E.Saitoh, M. Ueda, and H. Miyajima : Appl. Phys. Lett. 88 (2016) 182509 . \n8. T. Yang, T . Kimura, and Y .Otani : Nat. Phys.  4 (2008) 851 . \n9. A.  M. Deac, A. Fukushima, H. Kubota, H. Maehara, Y . Suzuki, S. Yuasa, Y . Nagamine , \nK. Tsunekawa, D.  D. Djayaprawira, and N. Watanabe : Nat.Phys.  4 (2008) 803 . \n10. S. Murakami, N.  Nagaosa, S.  C. Zhang: Science 301 (2003) 5638 . \n11. J. Sinova, D. Culcer, Q. Niu, N. A. Sinitsyn, T. Jungwirth, and A. H. MacDonald: Phys. \nRev. Lett. 92 (2004) 126603 . \n12. E.I.Rashba: Phys. Rev. B 68 (2003) 241315 . \n13. Y. Tserkovnyak, A . Brataas, and G . E. W. Bauer : Phys.Rev.B 66 (2002) 224403  14 \n 14. T. Taniguchi and W. M. Saslow: Phys.Rev.B.  90 (2014) 214407 . \n15. A. Kamra and W. Belzig: Phys.  Rev. Lett. 116 (2016) 146601 . \n16. W. Schottky, Ann. Phys. 362(23) (1918) 541.  \n17. J. Meair , P. Stano and P. Jacquod: Phys. Rev. B. 84 (2011) 073302.  \n18. T. Arakawa, J. Shiogai, M. Ciorga, M. Utz, D. Schuh, M. Kohda, J. Nitta, D. Bougeard,  \nD. Weiss, T. Ono, and K. Kobayashi: Phys. Rev. Lett. 114 (2015) 016601 . \n19. K. Thibault, J. Gabelli, C. Lupien, and B. Reulet: Phys. Rev. Lett. 114 (2015) 236604 . \n20. J. Basset, A. Yu. Kasumov, C. P. Moca, G. Zaránd, P. Simon, H. Bouchiat, and R. \nDeblock: Phys. Rev. Lett. 108 (2012) 046802 . \n21. A. L. Grimsmo, F. Qassemi, B. Reulet, and A. Blais: Phys. Rev. Lett.  116 (2016) \n043602 . \n22. S. Takahashi and S. Maekawa: Phys. Rev. B. 67 (2003) 052409 . \n23. F. J. Jedema, A. T. Filip, and B. J. Wees : Nature 410 (2001 ) 345-348. \n24. Y. M. Blanter and M. B üttiker: Phys. Rep. 336 (2000) 1 . \n25. H. Nyquist : Phys. Rev. 32 (1928)110.  \n26. K.Liu, K.Xia, and G. E. W. Bauer Phys. Rev. B 86, (2012) 020408 .   \n  15 \n Captions  \nFigure 1 (a) Schematic of the lateral spin valve setup. Left electrode is ferromagnetic  \n(FM) , middle and right ones are nonmagnetic  (NM1 and NM2) . There is a tunnel barrier \nbetween NM1 and NM2. (b) Schematic of spin accumulation in the x-axis inside NM1 . \n(c) Schematic of spin accumulation in the z-axis in the vicinity of the barrier between \nNM1 and NM2 . We consider the noise generated at the  barrier.    \n \nFigure 2  Normalized noise  spectrum 𝑆sym(𝜈,𝑇,𝛿𝜇)/𝑆0 as a function of the normalized \nfrequency  (ℎ𝜈/𝑘𝐵𝑇). Each curve show s noise spectr um with different  spin accumulation s \n𝛿𝜇/𝑘𝐵𝑇=0,1,3, and 5. We fix 𝜏=0.01. All the curve converge to the zero temperature \nnoise 𝑆𝑠𝑦𝑚(𝜈,𝑇=0,𝛿𝜇)/𝑆0  for 𝜈→∞ . The dotted line shows the zero temperature \nspectrum of 𝑆𝑠𝑦𝑚(𝜈,𝑇= 0,𝛿𝜇=0)/𝑆0   \n \nFigure 3 Autocorrelation of the spin current 𝐶(𝑡,𝛿𝜇,𝑇) with various ℎ/𝑘𝐵𝑇 ranging \nfrom 100 to 1000. 𝛿𝜇/𝑘𝐵𝑇=5 for solid curves , and 𝛿𝜇=0 for dotted line s. We fix \n𝜏=0.01. Clear oscillation with frequency ℎ/𝛿𝜇 is observed.  \n \n 16 \n  \nFigure 1 Iwakiri et al.   \n \n17 \n  \nFigure 2 Iwakiri et al.   \n \n \n \n \n \n \n18 \n Figure 3 Iwakiri et al.  \n \n"    },    {        "title": "1503.03171v2.Spin_wave_induced_spin_torque_in_Rashba_spin_orbit_coupling_system.pdf",        "content": "arXiv:1503.03171v2  [cond-mat.mtrl-sci]  24 Mar 2015Spin-wave-induced spin torque in Rashba spin-orbit coupli ng system\nNobuyuki Umetsu, Daisuke Miura, and Akimasa Sakuma\nDepartment of Applied Physics, Tohoku University, Aoba 6-6 -05, Aoba-ku,\nSendai 980-8579, Japan\n(Dated: 16 June 2021)\nWe study the effects of Rashba spin-orbit coupling on the spin torqu e induced by\nspin waves, which are the plane wave dynamics of magnetization. The spin torque\nis derived from linear response theory, and we calculate the dynamic spin torque by\nconsidering the impurity-ladder-sum vertex corrections. This dyn amic spin torque is\ndivided into three terms: a damping term, a distortion term, and a correction term\nfor the equation of motion. The distorting torque describes a phenomenon unique\nto the Rashba spin-orbit coupling system, where the distorted mot ion of magnetiza-\ntion precession is subjected to the anisotropic force from the Ras hba coupling. The\noscillation mode of the precession exhibits an elliptical trajectory, a nd the ellipticity\ndepends on the strength of the nesting effects, which could be red uced by decreasing\nthe electron lifetime.\n1I. INTRODUCTION\nThespace-dependent spin dynamics inferromagneticmetalshaveb eenthesubject ofcon-\nsiderable investigation over the past several years. This subject is of great significance for\ntechnological applications such as spin-transfer torque random- access memory (STT-RAM)\nand microwave generators or detectors. In the spintronics devic es, which are nanoscale\nmagnetic multilayer systems, inhomogeneous magnetization dynamic s are caused by an ex-\nternal applied field or thermal excitations due to the finite size effec ts1,2and the interface\neffects3. The spin torques, which arise in non-equilibrium conditions include som e terms\nwith different forms4–8, and the space-dependent spin torques caused by inhomogeneou s\nmagnetization dynamics are different from the current-induced sp in torques, namely, the\nspin transfer torque9–13and the spin-orbit torque14–19. The former torque is due to the\nindirect interaction between magnetization through conduction ele ctrons, while the latter\ntorque is due to the direct interaction between magnetization and c onduction electrons.\nGenerally, the space-dependent torque acting on magnetization a trcan be represented by\n/integraltext\ndr′dtM(r,t)טχ(r,r′,t,t′)M(r′,t′), where ˜χ(r,r′,t,t′) is the spin susceptibility tensor\n(in detail, see (9)). We are interested in the dynamic part of this tor que, which describes\nthe interaction between magnetization M(r) and the time-derivative of other magnetiza-\ntion˙M(r′), and it corresponds to Gilbert damping torque αM×˙Mwhen magnetization\nprecesses uniformly, where αis the Gilbert damping constant20. Although a large number\nof studies have considered magnetization damping in uniform preces sion systems5,21–26(i.e.,\nlocal damping), little is known about the damping mechanism in inhomoge neous magneti-\nzation dynamics27–30(i.e., nonlocal damping). Owing to the development of recent experi-\nmental techniques, detailed and unified theories of nonlocal dampin g are more desirable for\nspintronics applications.\nPrevious theoretical studies show that magnetic damping arising fr om spin diffusion is\nenhanced by the spin wave that describes the plane wave dynamics o f magnetization28. The\ndamping coefficient is represented by α=α0+ηq2, especially in the long wavelength limit,\nwhere the first term represents damping for uniform precession s ystems and the second\nterm is the contribution from spin wave motion with wave vector q. The coefficient ηis\nthe diffusion constant, which depends strongly on the lifetime of the conduction electrons.\nMoreover, experimental evidence for nonlocal damping has been r eported by the recent\n2work2, where the Gilbert damping constants for two modes (center mode and edge mode)\ndepend on the size of the nanoscale magnets, implying that additiona l damping torque was\ncaused by finite size effects.\nThe purpose of our work is to calculate the dynamic part of the spin t orque induced by\nthe spin wave motion of magnetization in a Rashba spin-orbit coupling ( RSOC) system. No\npreviousstudieshavediscussed theinhomogeneousmagnetization dynamics inSOCsystems.\nIn our previous work31, we discuss qualitatively the effects of SOC in the RSOC system\nwithout electrons scattering by impurities. In this paper, we show t he results including\neffects of the impurity scattering beyond the previous results in th e clean limit and discuss\nthe effects of impurity-ladder-sum vertex corrections.\nThe reminder of this paper is organized as follows. In Sec. II, we exp lain our model\nand formulation. In Sec. III, we show our calculation results for th e dynamic part of the\nspin-wave-induced spin torque. Finally, a summary of our work is give n in Sec. IV.\nII. MODEL AND FORMULATION\nThe exchange coupling between electrons and magnetization is repr esented by\nHex(t) =−2∆/integraldisplay\ndrn(r,t)·s(r), (1)\nwheren(r,t) =M(r,t)/Mis the unit vector of the time-dependent magnetization at r\nand ∆ is the exchange energy. Here, s(r) =ψ†(r)ˆσψ(r)/2 is the electron spin density\nwith the electron field operator ψ(†)(r) = [ψ(†)\n↑(r),ψ(†)\n↓(r)] and Pauli matrix vector ˆσ(the\nhat symbol indicates a 2 ×2 matrix). The direction of magnetization is parallel to z-axis\nat the equilibrium state, and the contributions from the transvers e component (denoted by\n⊥) are treated perturbatively. Based on linear response theory, t he time-dependent energy\nis written as\nEex(t) =1\ni¯h/integraldisplayt\n−∞dt′/angbracketleft[Hex(t),Hex(t′)]/angbracketright\n=−4∆2N/integraldisplay\ndt′/integraldisplay\ndr′n⊥(r,t) ˜χ(r,t,r′,t′)n⊥(r′,t′), (2)\nwhereNis the number of unit cells. The spin susceptibility tensor ˜ χcontains elements\n(l,m), which are given by\nχlm(r,t,r′,t′) =−N−11\ni¯h/angbracketleftbig/bracketleftbig\nsl(r,t),sm(r′,t′)/bracketrightbig/angbracketrightbig\nΘ(t−t′). (3)\n3The effective field is derived from Hex(t) =M−1Eex(t)/∂n(r,t), and the spin torque is\nobtained by Tst(r,t) =−γn(r,t)×Hex(t), whereγis the gyromagnetic ratio. In the first\napproximation, neglecting the products of perturbed quantities, the spin torque is written\nas\nTst(r,t) =γ2∆\nMnz×/angbracketleftbig\ns⊥(r,t)/angbracketrightbig\n, (4)\nwhere/angbracketleftbig\ns⊥(r,t)/angbracketrightbig\n= 2∆/integraltext\ndt′/integraltext\ndr′˜χ(r,t,r′,t′)n⊥(r′,t′) is the transverse component of the\nelectron spin density expectation value. In slow modulations of the m agnetization motion,\nn⊥(r′,t′)≈n⊥(r′,t)+(t′−t)˙n⊥(r′,t) is satisfied, and the spin torque is rewritten as\nTst(r,t)≈/integraldisplay\ndr′nz×/bracketleftig\n˜β(r−r′)n⊥(r′,t)+ ˜α(r−r′)˙n⊥(r′,t)/bracketrightig\n, (5)\nin an electron system with translation symmetry. Here,\n˜β(r) =4γ∆2N\nM˜χ(r,ω= 0), (6)\nis the coefficient tensor of the static part, and\n˜α(r) = i4γ∆2N\nMlim\nω→0∂\n∂ω˜χ(r,ω), (7)\nisthecoefficienttensorofthedynamicpart. TheFouriertransfor mationofspinsusceptibility\nis defined as ˜ χ(ω) =/integraltext\ndteiωt˜χ(t). The first term on the right-hand side of Eq.(5) is the\nRKKY-type interaction, which is regarded as the time-dependent c orrection for precession\ntorque caused by the static magnetic field; however, this term is irr elevant for our work. In\nthis study, we focus only on the dynamic spin torque, which is describ ed in second term on\nthe right-hand side of Eq.(5). Expanding the magnetization motion u sing a plane wave with\nwave vector q, the dynamic part of Tstinq-space is written as nzטαq˙n⊥\nq. Here,\n˜αq=4∆2\n¯hSlim\nω→0∂\n∂ωIm˜χq(ω), (8)\nwhere ¯hSis the spin angular momentum of magnetization in a unit volume v=V/N.\nUsing a simple calculation, it can be shown that the ω-derivative of the real part of\n˜χq=/integraltext\ndr˜χ(r)e−iq·rvanishes in ω→0. Hereafter, the matrix elements related to the\nz-component of ˜ αqare neglected because we are not concerned with the longitudinal m otion\nin this study.\nWecalculate ˜ χq(ω)using aGreenfunctiontechnique thatconsiders theelectron-imp urity\n(spin-independent) interactionwithinthefirstBornapproximation26. Thespinsusceptibility\n4is obtained using analytic continuation of the Matsubara Green’s fun ction, which is written\nas\nχlm\nq(iνn) =/integraldisplay1/T\n0dηeiνnη/angbracketleftbig\nTsl\nq(η)sm\n−q/angbracketrightbig\n=−T\n4/summationdisplay\nk/summationdisplay\nn′Tr/bracketleftig\nˆg(iωn′,k)ˆΓl(iωn′,iωn′+iνn,q)ˆg(iωn′+iνn,k+q)ˆσm/bracketrightig\n,(9)\nwhere ˆg(iωn,k) =−/integraltext1/T\n0dηeiωnη/angbracketleftig\nTψk(η)ψ†\nk/angbracketrightig\nis the one-particle Green’s function and\nˆΓl(iωn′,iωn′+iνn,q) is the vertex function. This vertex function satisfies the following\nWard identity:\nΓl\nσ1σ2(iωn′,iωn′+iνn,q) =σµ\nσ1σ2\n+¯h/τ\nǫF/summationdisplay\nρ1ρ2Πσ1ρ1ρ2σ2(iωn′,iωn′+iνn,q)Γl\nρ1ρ2(iωn′,iωn′+iνn,q),(10)\nwhere\nΠσ1ρ1ρ2σ2(iωn′,iωn′+iνn,q) =ǫF\nπνFN/summationdisplay\nkgσ1ρ1(iωn′+iνn,k+q)gρ2σ2(iωn′,k).(11)\nHere, ¯h/τ=πνFn0u2is the inverse electron lifetime τ[νFis the density of states per volume\nat the Fermi level, n0is the density of impurities, and uis the scattering constant of the\nshort range potential uvδ(r)]. We define the following 4 ×4 matrices:\nˇK=/parenleftbigg\n1−¯h/τ\nǫFˇΠ/parenrightbigg−1\n,ˇΠ =\nΠ↑↑↑↑Π↑↑↓↑Π↑↓↑↑Π↑↓↓↑\nΠ↑↑↑↓Π↑↑↓↓Π↑↓↑↓Π↑↓↓↓\nΠ↓↑↑↑Π↓↑↓↑Π↓↓↑↑Π↓↓↓↑\nΠ↓↑↑↓Π↓↑↓↓Π↓↓↑↓Π↓↓↓↓\n. (12)\nThen, the vertex function is given by\nΓl\nσ1σ2(iωn′,iωn′+iνn,q) =/summationdisplay\nρ1ρ2Kσ1ρ1ρ2σ2(iωn′,iωn′+iνn,q)σl\nρ1ρ2. (13)\nSubstituting Eq.(13) into Eq.(9) and performing the conventional k-sum andn′-sum in the\n5low-temperature limit, we obtain the following form of Eq.(8):\n˜αq=αqˆσ0+αx\nqˆσx+αy\nq(−i)ˆσy+αz\nqˆσz,\nαq=νF∆2\n2SǫFRe/summationdisplay\nσ/parenleftbigˇKAR\nqˇΠAR\nq−ˇKAA\nqˇΠAA\nq/parenrightbig\nσσ¯σ¯σ,\nαx\nq=νF∆2\n2SǫFIm/summationdisplay\nσσ/parenleftbigˇKAR\nqˇΠAR\nq−ˇKAA\nqˇΠAA\nq/parenrightbig\n¯σσ¯σσ,\nαy\nq=νF∆2\n2SǫFIm/summationdisplay\nσσ/parenleftbigˇKAR\nqˇΠAR\nq−ˇKAA\nqˇΠAA\nq/parenrightbig\nσσ¯σ¯σ,\nαz\nq=νF∆2\n2SǫFRe/summationdisplay\nσ/parenleftbigˇKAR\nqˇΠAR\nq−ˇKAA\nqˇΠAA\nq/parenrightbig\n¯σσ¯σσ, (14)\nwhere (ˇΠAR,AA\nq)σ1ρ1ρ2σ2= (ǫF/πνFN)/summationtext\nkgA\nσ1ρ1(ω= 0,k+q)gR,A\nρ2σ2(ω= 0,k). The super-\nscripts R and A denote the retarded Green’s function and the adva nced Green’s function,\nrespectively.\nIn Eq.(14), αqis the conventional Gilbert damping coefficient, and αy\nq, which is the\ncoefficient of nz×(−i)ˆσy˙n⊥\nq=−˙n⊥\nq, is the correction for the equation of motion. Neither\nαx\nqnorαy\nqhave been examined in previous studies because these terms disapp ear in non-\nSOC systems and in uniform precession systems. The physical mean ing of these terms is\ndiscussed in the next section.\nThe electron system of our model is a 2D electron gas incorporating RSOC26, which is\ndescribed by\nH=ǫF/summationdisplay\nkψ†\nk/parenleftbig\n|k0|2+Λk·ˆσ/parenrightbig\nψk, (15)\nwhereǫF= ¯h2k2\nF/2mis the Fermi energy and k0= (kx/kF,ky/kF) is the normalized wave\nvector. Here, Λkis defined as\nΛk=λk0(sinθkcosφk,sinθksinφk,cosθk), (16)\nwhereφk=−tan−1(kx/ky),θk= cos−1(−∆0/λk0), andλk0=/radicalbig\nλ2k2\n0+∆2\n0. Moreover,\n∆0= ∆/ǫF, andλisthestrengthofRSOC(theconventional Rashbaparameter αRiswritten\nasαR=λǫF/kF). Theeigenstatecorresponding totheeigenenergy of H,ǫk±=ǫF(k2\n0±λk0),\nis given by\n|k,+(−)/angbracketright= e−(+)iφk\n2cosθk\n2|↑(↓)/angbracketright+(−)e+(−)iφk\n2sinθk\n2|↓(↑)/angbracketright. (17)\n6Then, the matrix elements of the retarded and advanced Green’s f unction are written as\ngR,A\nσ1σ2(iωn,k) =/summationdisplay\nα=±/angbracketleftσ1|α/angbracketrightkgR,A\nα(iωn,k)/angbracketleftα|σ2/angbracketrightk,\ngR,A\nα(iωn,k) = (iωn+ǫF−ǫkα±i¯h/τ)−1,(+forR,−forA). (18)\nWe can calculate ˜ αqfrom Eq.(12), Eq.(13), Eq.(14), Eq.(17), and Eq.(18).\nIII. RESULTS AND DISCUSSION\nIn the numerical calculations, we set the following parameters: ∆ 0= 0.1,νF= 1/ǫF,\nλ= 0.3, andS= 1. We show the numerical results of two cases: ¯ h/τ= 0.01ǫFand\n¯h/τ= 0.05ǫF. However, we do not show the detail results of αy\nqin this section because we\nhave determined that this term is sufficiently small and can be neglect ed. In our results, the\nmaximum absolute value of αy\nqfor ¯h/τ= 0.01ǫFis about 0.07 and that for ¯ h/τ= 0.05ǫFis\nabout 0.06, which satisfies αy\nq≪1 (i.e.,αy\nq˙n⊥≪˙n⊥).\nA. Damping torque\nIn this subsection, we discuss the results of αq, which is the coefficient of conventional\ndamping torque, nz×˙n⊥\nq. Theq-dependence of αqis shown in Fig.1. It can be strictly\nshown that αqdoes not depend on the direction of q.\nIn the clean limit,\nαq=νF∆2\nSǫF/summationdisplay\nαβ/parenleftbig\nk2\nα−k2\nβ/parenrightbig2+αβλ2/parenleftbig\nk2\nα+k2\nβ/parenrightbig\n(k2\n−−k2\n+)2Iαβ(q0), (19)\nwhere\nIαβ(q0) =Θ(q0−|kα−kβ|)Θ(kα+kβ−q0)/radicalig/bracketleftbig\n(kα+kβ)2−q2\n0/bracketrightbig/bracketleftbig\nq2\n0−(kα−kβ)2/bracketrightbig,/parenleftbigg\nq0=q\nkF/parenrightbigg\n(20)\nrepresents the strength of the nesting effects for spin excitatio ns between the α-band and\ntheβ-band. Here, k±=/radicalig\n1+λ2/2∓/radicalbig\nλ2+λ4/4+∆2\n0is the radius of the Fermi sphere of\nthe±-band. Atq0= 0,k−±k+,2k±,αqdiverges because of the strong nesting caused by the\nadjoining Fermi surfaces. However, in a non-SOC system, αqdiverges only at q0=k−±k+29\nbecause intra-band transitions are forbidden (i.e., the contributio n fromα=βis zero). The\ndivergence at q= 0 in SOC systems is well understood from previous results24, but the\n7divergence at q0= 2k±arising from the intra-band transitions is a new result related to the\ninhomogeneous dynamics of magnetization.\nFromFig.1, itisconfirmedthatthe αqdivergence issuppressed bytheimpurity scattering\nof electrons, and the value at each peak decreases with decreasin g electron lifetimes. These\nresults imply that the nesting effects are reduced by broadening th e Fermi level due to the\nincrease in self-energy. There are cases that αqincreases with decreasing τbecause the\nline width of each peak increases with τ. These behaviors are also confirmed in the non-\nSOC system in our previous study29. This implies that αqincreases due to spin diffusion\noriginating in inter-band transitions.\nThebehaviors of αqnearq= 0areshown inFig.2. Thevalueof αqatq= 0corresponding\nto the Gilbert damping constant in a uniform precession system diver ges in the clean limit,\nbut it converges in the presence of impurities24. On the other hand, in a non-SOC system,\nαq=0is always zero regardless of the concentrations of impurities; this b ehavior is required\nfromtheangularmomentaconservationlaw, whichissatisfiedbytak ingthevertexcorrection\n(VC). However, in the presence of magnetic (spin-dependent) impurities, αq=0has a finite\nvalue due to the violation of the conservation law5,29. In a non-SOC system, the spin-\ndependency of impurities causes dramatic differences in the αqbehavior; however, the spin-\ndependence on impurities rarely appears in SOC systems26. Therefore, we do not need to\nprovide a detailed analysis of the effects of magnetic impurities in our R SOC system.\nFrom Fig.3, it is clear that results including the VC term are larger than those without\nthe VC term, especially at q≃0. However, in a non-SOC system, αqwith VC is zero at\nq= 0, andαqis smaller than the results without VC. The VC term reduces the over value\nof the contributions from spin diffusion originating in inter-band tran sitions. In an SOC\nsystem atq≃0, the contributions from intra-band transitions are much larger t han those\nfrom inter-band transitions. We conclude that the VC term of an SO C system enhances the\ncontributions from nesting effects, which originate in intra-band tr ansitions.\nB. Distorting torque\nIn this subsection, we discuss the results of both αx\nqˆσxandαz\nqˆσz. In the RSOC system,\nthese terms vanish at q= 0, but not at q >0. From our calculations, the sum of these terms\n8/HBar/Slash1Τ/EquΑl0/HBar/Slash1Τ/EquΑl0.01ΕF/HBar/Slash1Τ/EquΑl0.05ΕF\n0.0 0.5 1.0 1.5 2.0 2.501234\nq0Α\nFIG. 1. The q-dependence of αq(divided by νF∆2/SǫF= 10−2). The results in the clean limit are\nindicated by the dashed line.\n/HBar/Slash1Τ/EquΑl0\n/HBar/Slash1Τ/EquΑl0.05ΕF/HBar/Slash1Τ/EquΑl0.01ΕF\n0.000.020.040.060.080.10020406080100\nq0Α\nFIG. 2. Behavior of αqnearq= 0.\nis given by\nαx\nqˆσ+αz\nqˆσz=αxz\nqˆRq, (21)\nwhereˆRqis the reflection matrix in terms of the direction θq= tan−1(qy/qx):\nˆRq=\ncos2θqsin2θq\nsin2θq−cos2θq\n. (22)\n9VC/LParen10.01ΕF/RParen1\nNVC/LParen10.01ΕF/RParen1\nVC/LParen10.05ΕF/RParen1\nNVC/LParen10.05ΕF/RParen1\n0.000.010.020.030.040.05020406080\nq0Α\nFIG. 3. Comparison of αqinq0<0.05 including the vertex corrections (VC) and not including\nthe vertex corrections (NVC). The solid lines show the resul ts for VC, and the dashed lines show\nthe results for NVC. The blue lines are results for ¯ h/τ= 0.01ǫF, and the red lines are results for\n¯h/τ= 0.05ǫF.\nMoreover,αxz\nqis theq-dependent coefficient of new torque, nz׈Rq˙n⊥\nq. The direction of\nthis torque is shown in Fig.4. We call this the distorting torque because it applies q-\ndependent force to the precession motion. For example, the disto rting torque is parallel to\nthe damping torque when n⊥\nqis perpendicular to q; on the other hand, the distorting torque\nis anti-parallel to the damping torque when n⊥\nqis parallel to q.\nTheq-dependence of αxz\nqis shown in Fig.5. We confirm that the inequality equation,\nαxz\nq≤αq, is satisfied in the entire q-range. Therefore, the precession orbits necessarily\ndecrease with time.\nIn the clean limit,\nαxz\nq=νF∆2\nSǫFλ2/summationdisplay\nαβ/parenleftbig\nk2\nα−k2\nβ/parenrightbig2/q2\n0+αβ/parenleftbig\nk2\nα+k2\nβ/parenrightbig\n(k2\n−−k2\n+)2Iαβ(q0). (23)\nThepeaks of αxz\nqoccur atsamepointswiththoseof αqbecause theright handsideofEq.(23)\nincludesIαβ(q0). This equation implies that the distorting torque is unique to the RSO C\nsystem because αxz\nqis proportional to the square of λ. We find that the contribution from\nthe intra-band transitions of αxz\nq, which are given by 2 λ2νF∆2k2\nαIαα(q0)/SǫF(k2\n−−k2\n+)2, is\nequal to that of αq, but the contribution from the inter-band transitions is not equal. These\n10results reflect the differences in the radius of the Fermi surfaces and in the spin directions\nat the nesting points between the +-band and the −-band. These differences are attributed\nto thek-dependent spin states originating from RSOC. Moreover, the neg ative values of αq\nat 1.7<∼q0<∼2 imply that the larger contributions come from inter-band transitio ns than\nfrom intra-band transitions.\nInthepresence of impurities, the divergence of αxz\nqissuppressed similarly to theresults of\nαq. The behavior of αxz\nqnearq= 0 is shown in Fig.6. At q= 0,αxz\nqis always zero regardless\nof the presence or absence of impurities. However, in q→0,αxz\nqdiverges (i.e., discontinuity\natq= 0) in the absence of impurities, but αxz\nqconverges to zero (i.e., continuity at q= 0)\nin the presence of impurities. In the absence of impurities, αxz\nq=αqatq<k−−k+because\nonly intra-band transitions occur, but this equality is no longer satis fied in the absence of\nimpurities due to spin diffusion originating in the inter-band transitions . Consequently, αxz\nq\ndecreases atthepeaknear q= 0, andthispeakshiftsright withdecreasing electronlifetimes.\nFig.7 shows the differences between the results including VC and thos e without VC. The\nvalues with VC are larger than those without VC, and VC for αxz\nqclearly has effects similar\nto those for αq(see Fig.3).\nFIG. 4. Direction of nz׈Rq˙n⊥\nq.\n11/HBar/Slash1Τ/EquΑl0.05ΕF\n/HBar/Slash1Τ/EquΑl0.01ΕF/HBar/Slash1Τ/EquΑl0\n0.00.51.01.52.02.5/Minus101234\nq0Αxz\nFIG. 5. The q-dependence of αxz\nq(divided by νF∆2/SǫF= 10−2). The results in the clean limit\nare indicated by the dashed line.\n/HBar/Slash1Τ/EquΑl0\n/HBar/Slash1Τ/EquΑl0.01ΕF\n/HBar/Slash1Τ/EquΑl0.05ΕF\n0.000.020.040.060.080.1001020304050\nq0Αxz\nFIG. 6. Behavior of αxz\nqnearq= 0.\nC. Oscillation modes\nIn the small plane wave field h⊥\nq, if we can neglect ˜βq, then the linearized equation of\nmotion is given by\n˙n⊥\nq=−γn⊥\nq×Hz−γnz×h⊥\nq+nzטαq˙n⊥\nq. (24)\n12VC/LParen10.01ΕF/RParen1\nNVC/LParen10.01ΕF/RParen1\nVC/LParen10.05ΕF/RParen1\nNVC/LParen10.05ΕF/RParen1\n0.000.010.020.030.040.0502468101214\nq0Αxz\nFIG. 7. Comparison of αxz\nqinq0<0.05 including the vertex corrections (VC) and not including\nthe vertex corrections (NVC). The solid lines show the resul ts for VC, and the dashed lines show\nthe results for NVC. The blue lines are results for ¯ h/τ= 0.01ǫF, and the red lines are results for\n¯h/τ= 0.05ǫF.\nWe solve Eq.(24) by assuming harmonic-time-dependence of h⊥\nq(∼eiωt) without the cor-\nrection term αy\nq˙n⊥\nq. We obtain the eigenequation n±\nq=χ±\nqh±\nq, wheren±\nqis the oscillation\nmode corresponding to each component of the magnetic field, h±\nq(h+\nqis the component of\nright-hand rotation relative to the direction nzandh−\nqis the component of the opposing\ndirection). Assuming αq,αxz\nq≪1, we find\nn±\nq= cos(ωt)±isin(ωt)∓iαxz\nqsin(ωt+sin2θq), (25)\nand\nχ±\nq=γ\nγHz∓iαqω. (26)\nEq. (25) indicates that each magnetization precession describes a n elliptical trajectory.\nStrongerSOC,whichgeneratesalargemagnitudeof αxz\nq,producesamoreelongatedelliptical\norbit. When αxz\nq>0, the major axis of the elliptical orbit is parallel to q, while the major\naxis is perpendicular to qwhenαxz\nq<0.\nFrom Eq.(26), it is confirmed that χ±\nqis not affected by αxz\nqwhenαq,αxz\nq≪1. Therefore,\nαxz\nqcannot be determined experimentally in the same way as the estimatio n ofαq, which\ncan be measured from the line width of magnetic susceptibility. To exp erimentally estimate\nthe value of αxz\nq, direct observation of the elliptical orbit motion is required.\n13We suggest that the time-resolved magneto-optical Ker effect (T RMOKE)32is an appro-\npriatemethodfordetecting theelliptical trajectoryinRSOCsyste m. Fortheseobservations,\na large value of αxz\nqis required: both strong RSOC and nesting effects are required. In our\nresults for ¯h/τ= 0.01ǫF, the maximum value of αxz\nqis about 0.13, which is sufficient for\ndetection at q≃0.12kF≃O(10−1)˚A−1. However, an artificial excitation of spin wave which\nwave number is larger than 0.01 ˚A−1is technically difficult at this time. Moreover, for\nclear detection of the q-dependent precession motion, the spot size of TRMOKE must be\nreduced from 10 µm to a few nanometers, which is currently a difficult target. Thus, ne w\nexperimental techniques are necessary to resolve these issues.\nIV. SUMMARY\nWe calculate the dynamic part of the spin torque induced by the plane wave dynamics\nof magnetization in a RSOC system. In addition to the conventional d amping torque,\nour results show that distorting torque, which is a phenomenon uniq ue to RSOC systems,\noriginates from the inhomogeneous dynamics of magnetization. The magnitudes of these\ntorques depend on the strength of the nesting effects, and they are reduced by decreasing\nthe electron lifetime. The vertex corrections for these terms are sufficiently large, especially\nin the long wavelength limit, correcting the deficiency of contribution s from nesting effects,\nwhich originateintheintra-bandtransitions. Intheresonant plane wave field, theoscillation\nmode of magnetization precession exhibits an elliptical trajectory w hose major axis depends\non the direction of the wave vector. However, to observe this ellipt ical precession motion,\nimprovements in measurement sensitivity and experimental techniq ues are required.\nACKNOWLEDGMENTS\nThisworkwassupportedbyaGrant-in-AidforScientificResearchf romtheJapanSociety\nfor the Promotion of Science (24-5058) and by JSPS KAKENHI, Gra nt Number 25420686.\nREFERENCES\n1J. M. Shaw, T. J. Silva, M. L. Schneider, and R. D. McMichael, Phys. R ev. B79, 184404\n(2009).\n142H. T. Nembach, J. M. Shaw, C. T. Boone, and T. J. Silva, Phys. Rev. Lett.110, 117201\n(2013).\n3R. Urban, G. Woltersdorf, and B. Heinrich, Phys. Rev. Lett. 87, 217204 (2001).\n4S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 (2004).\n5H. Kohno, G. Tatara, and J. Shibata, J. Phys. Soc. Jpn. 75, 113706 (2006).\n6H. Kohno and J. Shibata, Journal of the Physical Society of Japan 76, 063710 (2007).\n7G. Tatara et al., Journal of the Physical Society of Japan 76, 054707 (2007).\n8A. Sakai and H. Kohno, Phys. Rev. B 89, 165307 (2014).\n9J. Slonczewski, J. Magn. Magn. Mat. 159, L1 (1996).\n10Y. B. Bazaliy, B. A. Jones, and S.-C. Zhang, Phys. Rev. B 57, R3213 (1998).\n11S. E. Barnes and S. Maekawa, Phys. Rev. Lett. 95, 107204 (2005).\n12J. Xiao, A. Zangwill, and M. D. Stiles, Phys. Rev. B 73, 054428 (2006).\n13Y. Tserkovnyak, H. J. Skadsem, A. Brataas, and G. E. W. Bauer, Phys. Rev. B 74, 144405\n(2006).\n14K. Obata and G. Tatara, Phys. Rev. B 77, 214429 (2008).\n15U. H. Pi et al., Applied Physics Letters 97, (2010).\n16I. Mihai Miron et al., Nat Mater 9, 230 (2010).\n17L. Liuet al., Science 336, 555 (2012).\n18A. V. Khvalkovskiy et al., Phys. Rev. B 87, 020402 (2013).\n19J. Linder and M. Alidoust, Phys. Rev. B 88, 064420 (2013).\n20T. Gilbert, IEEE Trans. Magn. 40, 3443 (2004).\n21E.ˇSim´ anek and B. Heinrich, Phys. Rev. B 67, 144418 (2003).\n22Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I. Halperin, R ev. Mod. Phys. 77,\n1375 (2005).\n23R. A. Duine, A. S. N´ u˜ nez, J. Sinova, and A. H. MacDonald, Phys. R ev. B75, 214420\n(2007).\n24V. Kambersk´ y, Phys. Rev. B 76, 134416 (2007).\n25K. Gilmore, Y. U. Idzerda, and M. D. Stiles, Phys. Rev. Lett. 99, 027204 (2007).\n26I. Garate and A. MacDonald, Phys. Rev. B 79, 064404 (2009).\n27V. Korenman and R. E. Prange, Phys. Rev. B 6, 2769 (1972).\n28Y. Tserkovnyak, E. M. Hankiewicz, and G. Vignale, Phys. Rev. B 79, 094415 (2009).\n29N. Umetsu, D. Miura, and A. Sakuma, J. Phys. Soc. Jpn. 81, 114716 (2012).\n1530A. Sakuma, J. Appl. Phys. 117, (2015).\n31N. Umetsu, D. Miura, and A. Sakuma, J. Appl. Phys. (2015, in press ).\n32J. Walowski et al., J. Phys. D: Appl. Phys. 41, 164016 (2008).\n16"    },    {        "title": "0906.2986v1.Entanglement_enhancement_for_two_spins_assisted_by_two_phase_kicks.pdf",        "content": "arXiv:0906.2986v1  [quant-ph]  16 Jun 2009Entanglement enhancement for two spins assisted by two phas e\nkicks\nGennadiy Burlak1, Isabel Sainz2, and Andrei B. Klimov2∗\n1Centro de Investigaci´ on en Ingenier´ ıa y Ciencias Aplicad as,\nUniversidad Aut´ onoma del Estado de Morelos, Cuernavaca, M orelos, M´ exico\n2Departamento de F´ ısica, Universidad de Guadalajara,\nRevoluci´ on 1500, Guadalajara, Jalisco, 44420, M´ exico.\nAbstract\nWe study the entanglement dynamics in a two-spin system gove rned by a bilinear Hamiltonian\nand assisted by phase kicks. It is found that the application of instant kicks to both spins at\nsome specific moments leads to enhancement of entanglement. This procedure also improves the\ntransient character of entanglement leading, for large spi ns, to a formation of a plateau for the\nI-concurrence. We have numerically investigated the spin-s pin dynamics for several values of spins\nand observed a substantial enhancement of entanglement in c omparison to the evolution without\nkicks.\n∗klimov@cencar.udg.mx\n1I. INTRODUCTION\nRecently, a significant amount of theoretical and experimental re search has been carried\nout for generating, manipulating and preserving entanglement, wh ich is the key ingredient\nfor developing quantum technologies. The main attention is usually pa id to studying en-\ntanglement properties of qubits (two-level systems) (see e.g. [1 , 2]), considered as a basic\nphysical resource. Although, higher dimensional systems (qudits ) could bemore appropriate\nfor certain physical applications, entanglement dynamics, genera tion and manipulation in\nhigher dimensions is not as well studied as in two-dimensions.\nIn particular, it was suggested in [3] that it is possible to increase the security, the bit\ntransmission rate, or both in quantum key distribution protocols by increasing the dimen-\nsionality of the involved systems. The possibility to perform quantum computation based\non qudits cluster states [4](has also been noted) . Nevertheless, j ust few years ago some\nstudies on entanglement dynamics in higher dimensions have been don e. For example, the\ndynamics of a initially entangled two-qudit state was studied in [5], while in [6] it is shown\nthat two non-interacting, initially separable spins can get entangled via a common purely\ndephasing environment. More general results about environment mediated generation of\nentanglement in higher dimensions can be found in [7]. Other aspects o f entanglement in\nhigher dimensional systems, like entanglement concentration for t wo qudits in [8], and its\nexperimental implementation for two qutrits [9] were reported. I t was also shown in [10] that\nthe quality of entanglement transfer in spin chains actually increase s with the dimension of\nthe spin.\nOn the other hand, higher dimensional systems (real, as high excite d nuclear spins or\neffective, as Dicke-like states) are of interest by themselves, esp ecially in the context of\nstudying the quantum-classical transition, when the system’s dime nsion becomes sufficiently\nlarge. However, it iswell-know thatthere areno reasons toexpect that theclassical behavior\ncan be approached in the large dimension limit for every state [11]. In particular, the\ncoherent spin states (CSS), |ζ,S/an}bracketri}ht, [12] are the most classical ones, and some specific linear\ncombination of them may reveal non-classical features, like entan glement, even in the limit\nof large values of S(see [13]).\nOne of the regular ways for generating and measuring entanglemen t in many spin 1/2\nsystem is through spin squeezing (see [14] and references therein ), where quantum correla-\n2tions naturally appear in the basic concept of squeezed spin states (SSS) [15]. The simplest\nHamiltonian which produces squeezing starting with coherent state s located on the equator\nof the Bloch sphere is H=gS2\nz,whereSz=/summationtext\njσ(j)\nz/2 is a collective spin operator [15].\nUnfortunately, there are two problems with such dynamical gener ation of squeezing and\nentanglement: a) the squeezing (and entanglement) is always tran sient, i.e. it appears\nperiodically ; b) it does not reach its maximal value, except the simples t two-qubit case.\nInthecase oflargetwo spins thesituation israther similar. It isposs ible togenerateinan\neasy way the entanglement between these spins, but it would resem ble the above mentioned\ndisadvantages as in the symmetric combination of many qubits.\nIn the present article we address the following question: is it possible to improve the\nentanglement properties of a two qudit system in a relatively simple wa y?\nWe will show that by applying instant kicks to both spins at some partic ular moments\nwe can not only enhance the entanglement, but, more importantly, essentially improve its\ntransient nature, i.e. reduce the distance between the maximum an d the minimum values, so\nthat the system evolves into an approximate “steady” state of th e interaction Hamiltonian\nwith a high value of entanglement for sufficiently large spins. We also dis cuss its possible\napplication to the purely dephasing environment [6, 7].\nII. THE MODEL\nLet us consider two spins S1andS2, of dimensions 2 S1+1 and 2 S2+1 respectively, which\ninteract according to the Ising-like Hamiltonian,\nH=gˆSz1ˆSz2. (1)\nThis kind of interaction was proposed to implement one way quantum c omputation with\nmany-level cluster states (see [4] and references therein). Sup pose that each spin is initially\nin the eigenstate of ˆSxiwith zero phase, ˆSxi|ζi= 1,Si/an}bracketri}ht=Si|ζi= 1,Si/an}bracketri}ht, which is a CSS\nplaced on the equator of the Bloch sphere,\n|ζi= 1,Si/an}bracketri}ht=1\n2SiSi/summationdisplay\nki=−Si/radicalBigg\n(2Si)!\n(Si+ki)!(Si−ki)!|ki,Si/an}bracketri}ht,\nwherei= 1,2 andˆSzi|ki,Si/an}bracketri}ht=ki|ki,Si/an}bracketri}ht, so that the joined state for the system composed\nby spins S1andS2is|Ψ0/an}bracketri}ht=|ζ1= 1,S1/an}bracketri}ht|ζ2= 1,S2/an}bracketri}ht.\n3Since|Ψ0/an}bracketri}htis not an eigenstate of the Hamiltonian, in the course of evolution the joined\nstate\n|Ψ(t)/an}bracketri}ht=ˆU(t)|Ψ0/an}bracketri}ht,ˆU(t) =e−itgˆSz1ˆSz2, (2)\nbecomes entangled at some instants. As an entanglement measure we will use the (normal-\nized)I-concurrence [16], that for pure states takes the form,\nCI=2S+1\n2S/parenleftbig\n1−Trˆρ2\n1(2)/parenrightbig\n,\nwhere ˆρ1(2)= Tr2(1)(ˆρ(t)) is the density matrix of the composed system ˆ ρ(t) =|Ψ(t)/an}bracketri}ht/an}bracketle{tΨ(t)|\ntraced by the spin S2(S1), andS= min(S1,S2), such that 0 ≤ CI≤1, being zero for\nseparable states and one for maximally entangled states.\nFor the state (2), CI= (2S+1)(1−P)/2S, where the purity P= Trˆρ2\n1(t) is\nP=1\n16S1S1/summationdisplay\nk,l=−S1[(2S1)!]2cos4S2(gt(k−l)/2)\n(S1+k)!(S1−k)!(S1+l)!(S1−l)!.\nFrom now on we will focus only on the symmetric case: S1=S2=S. Let us consider\nthe two simplest cases: S= 1/2,1 . For two qubits, S= 1/2,CI= sin2gt/2, i.e. it oscillates\nbetween zero and one, so that the spins are maximally entangled at t imesgt= (2n+1)πand\nthey are separable at times gt= 2nπ,n= 0,1,.... Nevertheless, for S= 1, the situation is\nalreadydifferent, since CI= 3(4−2cosgt−cos2gt−cos4gt)/8, reaches itsmaximum value\nCmax\nI≈0.88 at times gt≈2.2,4.1+2nπ,and thus the spins will never become maximally\nentangled, although they are disentangled at gt= 2nπ, n= 0,1,.... The entanglement\ndynamics have a quasi-periodical behavior, as is shown by the (gree n) dash-dotted line in\nFig. 1 (a).\nIn the large spin limit, S≫1, we can approximate the time average of CIas follows,\n/an}bracketle{tCI/an}bracketri}ht ≈2S+1\n2S/parenleftbigg2√\n2πS−1\n2πS/parenrightbigg\n, (3)\ngiving/an}bracketle{tCI/an}bracketri}ht ≈0.54 forS= 1, which is quite close to the exact numerical value /an}bracketle{tCI/an}bracketri}ht= 0.58,\neven if the approximation is done for S≫1.\nIII. ENHANCING ENTANGLEMENT\nIn order to enhance the maximum achievable entanglement let us app ly instant pulses\n(kicks) toeach spin atcertain times tk. Such kicks correspond torotationsaroundthe y-axis,\n40 2 4 6 8 1000.10.20.30.40.50.60.70.80.91C\nt  \nS=1, with phase kick\nS=1, no phase kick\n(a)0 2 4 6 8 1000.10.20.30.40.50.60.70.80.91C\ntS=10, with phase kick\nS=10, no phase kick\n23450.40.60.81\n(b)\nFIG. 1: (Color online) Time evolution of CI, without kicks (green, dash-dotted line) and with\nphase-kicks (blue, continuous line) for (a) S= 1, with the numerical optimization times t1= 1.6\nandt2= 3.9, and (b) S= 10, in this case t1= 1.9 andt2= 4.2 where the inset shows the details\nfor the second phase kick.\nand for the j-th spin are represented by the operators ˆRj=e−iπˆSyj/2. The main idea is as\nfollows: initially both spins are in eigenstates of ˆSxi, which can be represented by localized\ndistributions on the equator of the corresponding Bloch spheres. In the course of evolution\ngoverned by Eq.(1) the state become partially entangled. At some a ppropriate moment t1\nwe apply a π/2-rotation to both spins so that each component of the angular mo mentum\nbasis|ki,S/an}bracketri}htinstantly transforms into an eigenstate of ˆSxi. Afterwards, the evolution under\nthe Hamiltonian (1) is continued for a while, which continues coupling bo th spins. Then, at\nsome time t2we apply the inverse π/2 -rotation that transforms again each element |ki,S/an}bracketri}ht\ninto an eigenstate of ˆSxi, after that we allow the system to evolve further:\n|Ψ(t)/an}bracketri}ht=ˆU(t−t2)ˆR−1ˆU(t2−t1)ˆRˆU(t1−t0)|Ψ0/an}bracketri}ht,\nwhereˆR=ˆR1⊗ˆR2. The instants t1andt2are chosen in order to optimize the maximum\nvalue of achievable entanglement after the second kick. A similar pro cedure was proposed\nto optimally create squeezing in Dicke states [17] and entanglement f or continuous-variable\nsystems in [18, 19].\nThe reason for applying π/2-rotations can be explained as follows: we want to correlate\nas many components of the spin states as possible, which cannot be done only with the\nHamiltonian evolution, since the number of components of the angula r momentum basis,\ninitially involved in the CSS, is of order√\nS. By applying the kicks at some specific moments\n5010020030040000.010.020.03(a)\n010020030040000.010.020.03(b)\n010020030040000.010.020.030.040.05\nk(c)\n010020030040000.0050.01\nk(d)\nFIG. 2: (Color online). Distribution of the two-spin probab ility/an}bracketle{tΨ(t)|Ψ(t)/an}bracketri}htas function of the\nnumber of state kin the long basis {0,...,(2S+1)(2S+1)}forS= 10. (a) for the initial state,\nt= 0, (b) before the first phase kick at t < t1, (c) after the first kick, but before the second kick\natt1< t < t 2, and (d) after the second kick t > t2.\nweareabletospreadthedistributionmaking itmoreuniform[18, 19], w hiletheHamiltonian\nevolution produces correlation between spin components in the ang ular momentum basis,\nleading to entanglement.\nIn Fig. 2, we plot the distribution of the two-spin probability /an}bracketle{tΨ(t)|Ψ(t)/an}bracketri}htas a function\nof the state number k, corresponding to the basis |k1,10/an}bracketri}ht|k2,10/an}bracketri}ht, in the following way,\n|−10,10/an}bracketri}ht|−10,10/an}bracketri}htcorresponds to k= 0, thenext state |−10,10/an}bracketri}ht|−9,10/an}bracketri}htcorresponds to k=\n1, and so on. For the initial state (Fig. 2 (a)), the distribution is cen tered on |0,10/an}bracketri}ht|0,10/an}bracketri}ht,\nthat is,k= 2S(S+1)+1 = 221. The Hamiltonian evolution does not affect the distributio n\nof the amplitudes of the components of the basis k(|k1,S/an}bracketri}ht|k2,S/an}bracketri}ht) as shown by Fig. 2 (b).\nIt can be seen in Fig. 2 (c) that after the first instantaneous rota tion of each spin, the\ndistribution of /an}bracketle{tΨ(t)|Ψ(t)/an}bracketri}htspreads over the states k, this effect is increased after the second\ninstantaneous rotation of each spin (Fig. 2 (d)).\nFor semi-integer spins ( S= (2n+ 1)/2,n= 0,1,...), the situation is slightly different:\nthe initial distribution is similar to Fig. 2 (a), but instead of having a sing le maximum at\nthe center, it has two maxima, which leads to non-essential (but ob servable) differences in\nthe dynamics.\nThe analytical expressions are very cumbersome for higher spins, so that we have to use\nnumerical optimization for S≥3/2.\n60 2 4 6 8 10 120.40.50.60.70.80.91C\nSwith phase kick\nno phase kick\nFIG. 3: (Color online) The behavior of /an}bracketle{tCI/an}bracketri}htagainst the spin number S≤12. The (green) squares\njoined by the dashed line correspond to the two-spin system u nder the Hamiltonian evolution given\nin Eq. 1. The (blue) circles joined by a continuous line corre spond to the enhanced behavior, for\ntimest≥t2.\nAs noted before, for the simplest case, when S= 1/2, the system oscillates between the\nseparable and maximally entangled states and the kicks produce jus t a phase shift, that\ndoes not affect the dynamics.\nForS= 1, wehavebeenabletoanalyticallyoptimize CI, andfoundthattheentanglement\nacquires its maximum value when the kicks are applied at times t1≈1.6 andt2≈3.9 (which\nis also in accordance with the numerical results). For these times, CIreaches the maximum\nCmax\nI≈0.98 at times t≈7.1+2πn, which should be compared with that obtained by the\npure Hamiltonian evolution, where Cmax\nI≈0.88. However, it is more important that after\nthe second kick the quasi-periodical behavior of CIhas the minimal value Cmin\nI≈0.57, which\nmeans that the spins are entangled at any time t > t2> t1, as shown in Fig. 1 (a), and /an}bracketle{tCI/an}bracketri}ht\nincreases from /an}bracketle{tCI/an}bracketri}ht ≈0.58 to/an}bracketle{tCI/an}bracketri}ht ≈0.71.\nIn Fig. 1 (b), we plot the entanglement dynamics for S= 10. In this case, the effect of\nenhancing the maximum of entanglement for t > t2, is not that pronounced as for S= 1, it\nincreases from Cmax\nI≈0.90 toCmax\nI≈0.96. However, another important effect emerges: the\ntransient nature of entanglement becomes significantly improved, i.e. after the second kick\nCIrapidly oscillates with a very small amplitude around its average value /an}bracketle{tCI/an}bracketri}ht ≈0.95. So\nthat the spins are entangled for all times t > t2, with a high value. On the other hand, we\nobserve, that in the case of pure Hamiltonian evolution CIperiodically oscillates between its\n7maximum Cmax\nI≈0.90 and zero.\nThe average CIgrows slowly as Sincreases, and even if the maximum and minimum\nare getting closer, the rate of increase of the average is not as fa st as could be expected.\nThis is a consequence of the growing character of /an}bracketle{tCI/an}bracketri}htfor the Hamiltonian dynamics for\nlargeSas seen from Eq. (3). To represent such behavior clearly we plot in F ig. (3)/an}bracketle{tCI/an}bracketri}ht\nfor the Hamiltonian evolution (squares) and after applying the sequ ence of kicks (circles),\nnumerically obtained.\nIV. DISCUSSION\nWe have studied the dynamics of a two-spin entanglement assisted b y two phase kicks.\nThe instantaneous kicks were applied at times such that the maximum ofCIhas been\n(numerically) optimized. We found that such kicks not only lead to an e nhancement of the\ntwo-spins maximum entanglement, but also to a substantial increas ing of the minimum and\naverage values of CIfor times after the second kick. Such an effect becomes much more\npronounced for large spins S≫1, whenCItends to a constant value very close to unity.\nIt is worth noting that the interaction Hamiltonian similar to (1) appea rs in sev-\neral effective processes. In particular, when two non-interactin g spins are embedded in\na dephasing environment [6]. The Hamiltonian describing such evolution has the form\nH= Ω1Sz1+ Ω2Sz2+/summationtext\nkωka†\nkak+ (Sz1+Sz2)/summationtext\nkgk(ak+a†\nk), such that ωk≫Ω1,2,\ngk. The fast field is effectively decoupled from the spin subsystem and t hus can be\nadiabatically eliminated in a standard way [20] by applying a small unitary transforma-\ntionU= exp[(Sz1+Sz2)/summationtext\nkǫk(a†\nk−ak)],ǫk=gk/ωk,so that the effective Hamiltonian,\nHeff=UHU†(diagonal on the field variables) takes the form,\nHeff= (Ω1−ǫSz1)Sz1+(Ω2−ǫSz2)Sz2\n+/summationdisplay\nkωka†\nkak−2ǫSz1Sz2,\nwhereǫ=/summationtext\nkgkǫk. It can be observed that the effect of bosonic bath in this case is\nreduced to an effective interaction between spins, which may give ris e to a creation of spin\nentanglement and a quadratic phase shift that is not important for entanglement generation.\n8V. ACKNOWLEDGEMENTS\nThe work of G.B is partially supported by CONACyT grant 47220. The w ork of I.S. was\nsupported by CONACyT grant 74897.\n[1] S. Campbell and M. Paternostro, Phys. Rev. A 79, 032314 (2009); P. J. dosReis and\nS. S. Sharma, Phys. Rev. A 79, 012326 (2009); J. G. Oliveira, R. Rossi, and M. C. Nemes,\nPhys. Rev. A 78, 044301 (2008).\n[2] L Amico, R. Fazio, A. Osterloh, and V. Vedral, Rev. Mod. Ph ys.80, 517 (2008).\n[3] H. Bechmann-Pasquinucci and W. Tittel, Phys. Rev. A 61, 062308 (2000); M. Bouren-\nnane, A. Karlsson, and G. Bj¨ ork, ibid.64, 012306 (2001); S. P. Walborn, D. S. Lemelle,\nM. P. Almeida, and P. H. SoutoRibeiro, Phys. Rev. Lett. 96, 090501 (2006).\n[4] D. L.Zhou, B. Zeng, Z.Xu, andC.P. Sun,Phys.Rev. A 68, 062303 (2003); W. Hall, Qauntum\nInf. Comp. 7, 174 (2007).\n[5] A. R. R. Carvalho, F. Mintert, S. Palzer, and A. Buchleitn er, Europ. Phys. J. D 41, 425\n(2007).\n[6] J. Dajka and J. Luczka, Phys. Rev. A 77, 062303 (2008).\n[7] F. Benatti, A. M. Liguori, and A. Nagy, J. Math. Phys. 49, 042103 (2008).\n[8] M. Yang, A. Delgado, L. Roa, and C. Saavedra, Opt. Comm. 282, 1482 (2009).\n[9] A. Vaziri, J.-W. Pan, T. Jennewein, G. Weihs, and A. Zeili nger, Phys. Rev. Lett. 91, 227902\n(2003).\n[10] A. Bayat and V. Karimipour, Phys. Rev. A 75, 022321 (2007).\n[11] N. D. Mermin and G. M. Schwarz, Found. Phys. 12, 101 (1982).\n[12] J. M. Radcliffe, J. Phys. A 4, 313 (1971); F. T. Arecchi, E. Courtens, R. Gilmore, and\nH. Thomas, Phys. Rev. A 6, 2211 (1972).\n[13] C. C. Gerry, A. Benmoussa, E. E. Hach III, and J. Albert, P hys. Rev. A 79, 022111 (2009).\n[14] G. T´ oth, C. Knapp, O. G¨ uhne, and H. J. Briegel, Phys. Re v. A79, 042334 (2009).\n[15] M. Kitagawa and M. Ueda, Phys. Rev. A 47, 5138 (1993).\n[16] P. Rungta, V. Buˇ zek, C. M. Caves, M. Hillery, and G. J. Mi lburn, Phys. Rev. A 64, 042315\n(2001); P. Rungta and C. M. Caves, ibid.67, 012307 (2003).\n9[17] D. Shindo, A.Chavez, S.M. Chumakov, and A. B. Klimov, J. Opt. B.: Quant.Sem.Opt., 634\n(2004).\n[18] B. Kraus, K. Hammerer, G. Giedke, and J. I. Cirac, Phys. R ev. A67, 042314 (2003).\n[19] K. Hammerer, K. Mølmer, E. S. Polzik, and J. I. Cirac, Phy s. Rev. A 70, 044304 (2004).\n[20] A. B. Klimov and L. L. S´ anchez-Soto, Phys. Rev. A 61, 063802 (2000); A. B. Klimov,\nL. L. S´ anchez-Soto, A. Navarro, and E. C. Yustas, J. Mod. Opt 49, 2211 (2002).\n10"    },    {        "title": "1106.2989v1.Dynamics_of_Spin_Relaxation_in_Finite_Size_2D_Systems__an_Exact_Solution.pdf",        "content": "Dynamics of Spin Relaxation in Finite-Size 2D Systems: an Exact Solution\nValeriy A. Slipko1, 2and Yuriy V. Pershin1,\u0003\n1Department of Physics and Astronomy and USC Nanocenter,\nUniversity of South Carolina, Columbia, SC 29208, USA\n2Department of Physics and Technology, V. N. Karazin Kharkov National University, Kharkov 61077, Ukraine\nWe find an exact solution for the problem of electron spin relaxation in a 2D circle with Rashba\nspin-orbit interaction. Our analysis shows that the spin relaxation in finite-size regions involves\nthree stages and is described by multiple spin relaxation times. It is important that the longest\nspin relaxation time increases with decrease in system radius but always remains finite. Therefore,\nat long times, the spin polarization in small 2D systems decays exponentially with a size-dependent\nrate. This prediction is supported by results of Monte Carlo simulations.\nPACS numbers: 72.15.Lh, 72.25.Dc, 85.75.2d\nThe problem of D’yakonov-Perel’ [1, 2] spin relaxation\nin two-dimensional (2D) systems have attracted wide at-\ntention [1–14] because of its fundamental importance for\nthe field of spintronics [15, 16]. However, the spin re-\nlaxation in systems with boundaries is even more im-\nportant because boundaries are naturally present in all\nelectronic devices. There are only several examples in\nthe literature where the influence of boundary conditions\non D’yakonov-Perel’ spin relaxation have been explored\ntheoretically and/or experimentally. These examples in-\ncludeinvestigationsofspinrelaxationin2Dchannels[17–\n22], 2D half-space [23], 2D systems with antidots [24],\nlarge quantum dots [25, 26], and one-dimensional (1D)\nfinite-length wires [27]. Both available experimental and\ntheoretical results indicate that typically in the diffusive\nspin transport regime the electron spin life time is longer\nin systems with boundaries.\nIn this paper we find an exact solution for the prob-\nlem of electron spin relaxation in finite-size 2D systems.\nSpecifically, we consider the dynamics of electron spin\nrelaxation in a 2D circle made of a semiconductor struc-\nture with Rashba-type spin-orbit interaction. Naively,\none may think that in small systems the spin relaxation\nis incomplete as the spin precession angle across the sys-\ntem is small. However, in such a situation, the different\neffect plays role: the non-commutativity of spin rota-\ntions. Because of this effect, the electron spin precession\nangle can largely exceed the maximum rotation angle al-\nlowed by naive geometrical considerations. To the best\nof our knowledge, the spin relaxation in small systems\nwas investigated previously only in Ref. 25. This previ-\nous study provides only an asymptotic value of the spin\nrelaxation time without giving details on how the whole\nprocess of spin relaxation occurs. In the present paper,\nweshowthatthespinrelaxationprocessinfinite-sizesys-\ntems is intrinsically complex. The exact solution of this\nfundamental problem involves an infinite number of spin\nrelaxation constants. At long times, however, only the\nslowest decaying component survives and the spin po-\nlarization exhibit a slow size-dependent exponential de-\ncay. Our exact analytical solution is obtained using theLaplace transform and is confirmed by Monte Carlo sim-\nulations of spin dynamics. This work thus provides an\nimportant missing part of spin relaxation theory.\nLet us consider the dynamics of electron spin polariza-\ntion in a finite-size 2D electron system such as the circle\nshown in Fig. 1. The set of diffusion equations [4, 28, 29]\nfor spin polarization is given by\n1\nD@Sin\n@t= \u0001Sin+ 2\u0011rSz\u0000\u00112Sin; (1)\n1\nD@Sz\n@t= \u0001Sz\u00002\u0011r\u0001Sin\u00002\u00112Sz; (2)\nwhere SinandSzare the in-plane and zcomponents of\nspin polarization, respectively, D=l2=(2\u001c)is the diffu-\nsion coefficient, lis the mean free path, \u001cis the momen-\ntum relaxation time, and \u0011is the spin rotation angle per\nunit length (spin rotations are induced by the Rashba\nspin-orbit interaction [30]). The applicability limits of\nEqs. (1) and (2) are \u0015\u001cl\u001caandl\u001c\u0011\u00001, where\n\u0015is the electron’s de Broglie wavelength, and ais the\ncharacteristic system size (in the context of this work, a\nis the radius of the circle). We also note that Eqs. (1)\nand (2) are valid for any value of a\u0011. Eqs. (1) and (2) are\nsupplemented by the standard boundary conditions [28]\n\u0012@Sn\n@n+\u0011Sz\u0013\n\u0000= 0;\u0012@Sz\n@n\u0000\u0011Sn\u0013\n\u0000= 0;\u0012@S\u0007\n@n\u0013\n\u0000= 0;(3)\nz\naa\nFIG. 1: (Color online) Schematic of a circular 2D region of a\nradiusawith electron spin polarization initially pointing in z\ndirection perpendicular to the circle plane.arXiv:1106.2989v1  [cond-mat.mes-hall]  15 Jun 20112\nwhere nand\u0007are the normal in-plane and tangential\nvectors to the boundary \u0000, respectively.\nNext, we would like to reduce the set of Eqs. (1) and\n(2) to a single equation for Sz. Introducing u=r\u0001Sin\nandv= (r\u0002Sin)z, Eqs. (1) and (2) can be rewritten as\nut= \u0001u+ 2\u0001Sz\u0000u; (4)\n(Sz)t= \u0001Sz\u00002Sz\u00002u; (5)\nvt= \u0001v\u0000v: (6)\nHere, \u0001is the 2D Laplace operator, the time is measured\nin the units of ts= (D\u00112)\u00001, and the coordinates are\nmeasured in the units of \u0011\u00001. Such a convention is used\nbelow if not stated otherwise. Combining Eqs. (4) and\n(5) we readily get\n(Sz)tt\u00002\u0001(Sz)t+ 3(Sz)t+ \u00012Sz+ \u0001Sz+ 2Sz= 0:(7)\nThe Laplace transform of Eq. (7) is given by\n\u00012~Sz+ (1\u00002p)\u0001~Sz+ (p2+ 3p+ 2) ~Sz\n= (p+ 1)Szjt=0\u0000\u0001Szjt=0\u00002ujt=0;(8)\nwhere ~Szis the Laplace transform of Szto the complex\np-domain. In the above equation, the time derivative of\nSzatt= 0was substituted from Eq. (5).\nIn what follows we consider the relaxation of homoge-\nneous spin polarization pointing in zdirection perpen-\ndicular to the plane of 2D circle (see Fig. 1). Taking into\naccount the axial symmetry of the problem, the initial\nand boundary conditions read\nu(r;t= 0) =@(rSr(r;t= 0))\nr@r= 0;Sz(r;t= 0) =S0;(9)\n\u0012@Sr\n@r+Sz\u0013\f\f\f\f\nr=a= 0;\u0012@Sz\n@r\u0000Sr\u0013\f\f\f\f\nr=a= 0:(10)\nWe note that the function v=@(rS\u001e)=(r@r)is safely\ntaken out of the consideration since the solution v(r;t) =\n0satisfies Eq. (6) with the boundary condition\n@S\u001e=@rjr=a= 0and the initial condition v(r;0) = 0.\nApplying the initial conditions (9), Eq. (8) simplifies to\n\u00012~Sz(r)+(1\u00002p)\u0001~Sz(r)+(p2+3p+2)~Sz(r) = (p+1)S0:\n(11)\nThe general solution of Eq. (11) can be found by the\nfactorization of its left-hand side and be presented as\n~Sz(r) =A1J0(k1r) +A2J0(k2r) +B1N0(k1r)\n+B2N0(k2r) +S0\np+ 2;(12)\nwhereA1;2;B1;2are arbitrary constants, J0(z)andN0(z)\nare the zeroth order Bessel and Neumann functions, re-\nspectively, and\nk2\n1;2=\u0000p+1\n2\u00062ir\np+7\n16: (13)In the case of the circle, B1;2= 0sinceN0(x)diverges\nasx!0. Moreover, the actual choice of two branches\ncorresponding to \u0006in Eq. (13) is not essential. We may\nmake a branch cut along the line p<\u00007=16, Im(p) = 0\nin the plane of complex pand define two branches by the\nconditionsk2\n1;2(p= 0) = (1\u0006ip\n7)=2.\nAlthough the radial component of spin polarization\nequals zero at t= 0, it becomes different than zero at\nt>0similarly to the case of spin relaxation in rings [31].\nWith a help of Laplace transform of Eq. (5), we express\n~Sr(r)through ~Sz(r)as\n~Sr(r) =1\nrZr\n0d\u0018\u0018~u(\u0018) =1\n2@~Sz(r)\n@r\n\u0000p+ 2\n2rZr\n0d\u0018\u0018~Sz(\u0018) +1\n2rZr\n0d\u0018\u0018Sz(\u0018;t= 0):(14)\nThe boundary conditions (10) are used to find the val-\nues ofA1;2in Eq. (12). For this purpose, we Laplace\ntransform Eqs. (10) and employ Eq. (14) to find\n@2~Sz(a)\n@r2+p+ 2\na2Za\n0drr~Sz(r)\u0000p~Sz(a)\n=1\na2Za\n0drrSz(r;t= 0)\u0000Sz(a;t= 0);(15)\n@~Sz(a)\n@r+p+ 2\naZa\n0drr~Sz(r) =1\naZa\n0drrSz(r;t= 0):(16)\nA1;2are obtained from Eqs. (15) and (16) complemented\nby Eq. (12) and the initial conditions (9). Finally, the\nLaplace transform of z-component of spin polarization is\nwritten as\n~Sz(r) =2S0\n(p+ 2)D(p)\u0014\n(p+ 2\u0000k2\n2)J1(k2a)\nk2J0(k1r)\n\u0000(p+ 2\u0000k2\n1)J1(k1a)\nk1J0(k2r)\u0015\n+S0\np+ 2;(17)\nwhere the following notation is used:\nD(p) = 2(p+ 2)(k2\n2\u0000k2\n1)J1(k1a)J1(k2a)\nak1k2\n+[(p+ 2)(k2\n1\u00001)\u0000pk2\n2]J0(k1a)J1(k2a)\nk2\n\u0000[(p+ 2)(k2\n2\u00001)\u0000pk2\n1]J0(k2a)J1(k1a)\nk1:(18)\nThe Laplace transform of r-component of spin polariza-\ntion is found combining Eqs. (14), (17) and (9):\n~Sr(r) =S0\n(p+ 2)D(p)\u0014\n(k2\n1+p+ 2)(k2\n2\u0000p\u00002)J1(k2a)\nk2\n\u0002J1(k1r)\nk1\u0000(k2\n2+p+ 2)(k2\n1\u0000p\u00002)J1(k1a)\nk1J1(k2r)\nk2\u0015\n:(19)\nThe inverse Laplace transform of Eqs. (17) and (19)\nprovides the time-domain components of spin polariza-\ntion. It is important to note that the right-hand sides of3\n0 2 4 6 8 10 12 141E-30.010.1110100\n  -pn\naη  -p0    -p3\n  -p1    -p4\n  -p2    -p5\nFIG.2: (Coloronline)Firstsixpolesof ~Sz(r)and~Sr(r)versus\nthe ring radius. These poles are proportional to the slowest\nsix spin relaxation rates.\nEqs. (17) and (19) do not change under a permutation\nofk1andk2. This means that these functions are one-\nvalued functions in the whole complex plane of pdespite\nthe square root in Eq. (13). As a result, ~Sz(r)and~Sr(r)\nare meromorphic functions. The poles of these functions\nare defined by the equation D(p) = 0, which have an in-\nfinite number of roots pn,jpnj!1asn!1. All poles\nare characterized by Im (pn) = 0and Re (pn)<0. Note\nthat, generally, p=\u00002is not a pole of both ~Sz(r)and\n~Sr(r). Referring to Fig. 2, the positions of poles depend\non the circle radius. At small values of a\u0011, the values of\nalljpnj(n= 1;2;:::)increase with decrease of a\u0011except\nofjp0jwhose value decreases. Basically, this is the most\ninterestingpoledescribingtheasymptoticspinrelaxation\nat long times. p0is always located between \u00007=16and\n0and tends to zero when a\u0011!0, and to\u00007=16when\na\u0011!1. In the limit of small a, an analytical expression\nforp0can be found. Expanding Eq. (18) over small p\nanda, we get (using dimensional units for a)\n\u0000p0=1\n48(a\u0011)4\u00007\n768(a\u0011)6+O\u0010\n(a\u0011)8\u0011\n:(20)\nThis expression is basically valid when a\u0011 < 1. We also\nnote that the first term in the right-hand side of Eq. (20)\nwas previously reported in Ref. 25. It can also be shown\nthat forn= 1;2;:::,jpnj\u0018Cn(a\u0011)\u00002whena\u0011\u001c1.\nThe time-domain components of spin polarization are\nwritten using the normal dimensional units of time and\ncoordinates as\nSz(r;t) =+1X\nn=02S0epnD\u00112t\n(pn+ 2)D0(pn)\u0014\n(pn+ 2\u0000k2\n2n)J1(k2na\u0011)\nk2n\n\u0002J0(k1nr\u0011)\u0000(pn+ 2\u0000k2\n1n)J1(k1na\u0011)\nk1nJ0(k2nr\u0011)\u0015\n(21)\n0.0 0.2 0.4 0.6 0.8 1.0-0.50.00.51.0  t = 0           t = 1 ts \n  t = 0.1 ts      t = 100 ts \nSr/S0Sz/S0  \nr/aFIG. 3: (Color online) Radial distributions of SrandSzcom-\nponents of spin polarization at different moments of time.\nThis plot was obtained at a\u0011= 1.\nand\nSr(r;t) =+1X\nn=0S0epnD\u00112t\n(pn+ 2)D0(pn)\u0002\n(k2\n1n+pn+ 2)\n\u0002(k2\n2n\u0000pn\u00002)J1(k2na\u0011)\nk2nJ1(k1nr\u0011)\nk1n\n\u0000(k2\n2n+pn+ 2)(k2\n1n\u0000pn\u00002)J1(k1na\u0011)\nk1nJ1(k2nr\u0011)\nk2n\u0015\n:(22)\nEqs. (21) and (22) represent the main result of this work\ndescribing the time-dependence of spin relaxation in the\ncircle. Basically, three main stages of spin relaxation in\nsmall systems can be identified (this separation is appro-\npriate ata\u0011.3, when, as it follows from Fig. 2, p0is\nwell separated from all other poles). The first (initial)\nstage of spin relaxation takes place at t.a2=(16D),\nwherea2=(4D)is the time it takes for an electron to dif-\nfusively propagate over a distance a. During this stage,\nthe most of electrons in the circle’s center still do not\n’know about’ the presence of the boundary and, there-\nfore, the spin relaxation occurs essentially as in the bulk\n(accordingly to the standard 2D D’yakonov-Perel’ spin\nrelaxationtheory). Inthesecondstageofspinrelaxation,\nwhena2=(16D).t.4=(jp1jD\u00112), several exponentially\ndecaying terms play the main role in Eqs. (21) and (22).\nDuring this stage, a slow-decaying spin polarization pro-\nfile establishes. In such a profile, jSrjincreases with ras\nwe demonstrate in Fig. 3. The last third stage of spin\nrelaxation is a slow single-exponent decay of the slow-\ndecaying spin polarization profile established during the\nsecond stage. This process occurs at long times, namely,\nwhen 4=(jp1jD\u00112).t. The analytical expressions de-\nscribing the shape of the slow-decaying spin polarization\nprofile can be easily inferred from Eqs. (21) and (22) in\nthe long-time limit. All three stages of spin relaxation4\n0 2000 4000 60000.010.11\nexp(-(lη)2t/τ)   Theory\n  Monte Carlo\naη =2\n  Sz/S0\nt/τaη =1\nFIG. 4: (Color online) Time dependence of Szcomponent of\nspin polarization at r= 0obtained using Eq. (21) and Monte\nCarlo calculations. The dashed line corresponds to the usual\nD’yakonov-Perel’ relaxation in infinite 2D systems. This plot\nwas obtained using the parameter value \u0011l= 0:1.\ncan be easily distinguished in Fig. 4.\nIn order to obtain an additional insight on spin relax-\nation in 2D circle, we have performed extensive Monte\nCarlo simulations. All specific details of the Monte Carlo\nsimulations approach can be found in Refs. [17] and [32]\nand will not be repeated here. We just mention that\nthe Monte Carlo simulation program uses a semiclassi-\ncal description of electron space motion and quantum-\nmechanical description of spin dynamics. A spin conser-\nvation condition was used for electrons elastically scat-\ntered from system boundaries. Generally, all obtained\nMonteCarlosimulationresultsareinperfectquantitative\nagreement with our analytical predictions thus confirm-\ning our analytical theory of spin relaxation in finite-size\nsystems. A comparison of selected analytical and numer-\nical curves is given in Fig. 4.\nIn summary, we have found an analytical solution for\nthe problem of electron spin relaxation in the circle. It is\nshown that a small but non-vanishing spin relaxation ex-\nists even in small systems at long times. Consequently,\nit is not possible to completely eliminate the electron\nspin relaxation by reducing the system size, although the\nrelaxation rate is dramatically suppressed in small-size\nsystems. Basically, the spin relaxation process can be\nseparated into three stages including an initial region of\nfast bulk-type relaxation, a transition region where the\nrelaxation is described by a combination of several ex-\nponentially decaying functions and a region of slow ex-\nponential decay at long times. Our results can be easilyverified experimentally.\n\u0003Electronic address: pershin@physics.sc.edu\n1M. I. Dyakonov and V. I. Perel’, Sov. Phys. Solid State 13,\n3023 (1972).\n2M. I. Dyakonov and V. Y. Kachorovskii, Sov. Phys. Semi-\ncond. 20, 110 (1986).\n3E. Y. Sherman, Appl. Phys. lett 82, 209 (2003).\n4A. A. Burkov, A. S. Núñez, and A. H. MacDonald, Phys.\nRev. B 70, 155308 (2004).\n5S. Saikin, J. Phys.: Condens. Matter 16, 5071 (2004).\n6Y. V. Pershin, Phys. E 23, 226 (2004).\n7Y. V. Pershin, Phys. Rev. B 71, 155317 (2005).\n8I. S. Lyubinskiy and V. Y. Kachorovskii, Phys. Rev. B 73,\n041301 (2006).\n9Y. V. Pershin, Phys. Rev. B 75, 165320 (2007).\n10M. Q. Weng, M. W. Wu, and H. L. Cui, J. Appl. Phys.\n103, 063714 (2008).\n11P. Kleinert and V. V. Bryksin, Phys. Rev. B 79, 045317\n(2009).\n12Y. V. Pershin and V. A. Slipko, Phys. Rev. B 82, 125325\n(2010).\n13I. V. Tokatly and E. Y. Sherman, Ann. Phys. 325, 1104\n(2010).\n14I. V. Tokatly and E. Y. Sherman, Phys. Rev. B 82, 161305\n(2010).\n15I. Zutic, J. Fabian, and S. Das Sarma, Rev. Mod. Phys.\n76, 323 (2004).\n16M. Wu, J. Jiang, and M. Weng, Phys. Reports 439, 61\n(2010).\n17A. A. Kiselev and K. W. Kim, Phys. Rev. B 61, 13115\n(2000).\n18P. Schwab, M. Dzierzawa, C. Gorini, and R. Raimondi,\nPhys. Rev. B 74, 155316 (2006).\n19A. W. Holleitner, V. Sih, R. C. Myers, A. C. Gossard, and\nD. D. Awschalom, New J. Phys. 9, 342 (2007).\n20C.-H. Chang, J. Tsai, H.-F. Lo, and A. G. Mal’shukov,\nPhys. Rev. B 79, 125310 (2009).\n21S. Lüscher, S. M. Frolov, and J. A. Folk, Phys. Rev. B 82,\n115304 (2010).\n22J. Liu, T. Last, E. Koop, S. Denega, B. van Wees, and\nC. van der Wal, J. Supercond. Nov. Magn. 23, 11 (2010).\n23Y. V. Pershin, Phys. E 27, 77 (2005).\n24Y. V. Pershin and V. Privman, Phys. Rev. B 69, 073310\n(2004).\n25I. S. Lyubinskiy, JETP Lett. 83, 336 (2006).\n26E. J. Koop, B. J. van Wees, D. Reuter, A. D. Wieck, and\nC. H. van der Wal, Phys. Rev. Lett. 101, 056602 (2008).\n27V. A. Slipko and Y. V. Pershin, Phys. Rev. B 83, 193302\n(2011).\n28V. M. Galitski, A. A. Burkov, and S. Das Sarma, Phys.\nRev. B 74, 115331 (2006).\n29Y. V. Pershin and V. A. Slipko, arXiv:1007.0853v1 (2010).\n30Y. Bychkov and E. Rashba, JETP Lett. 39, 78 (1984).\n31V. A. Slipko and Y. V. Pershin, arXiv:1103.3231 (2011).\n32S. Saikin, Y. V. Pershin, and V. Privman, IEE-Proc. Circ.\nDev. Syst. 152, 366 (2005)."    },    {        "title": "2401.01334v2.Hyperfine_enhanced_gyroscope_based_on_solid_state_spins.pdf",        "content": "Hyperfine-enhanced gyroscope based on solid-state spins\nGuoqing Wang ( 王国庆),1, 2, 3, ∗Minh-Thi Nguyen,2, 3,†and Paola Cappellaro1, 2, 3, ‡\n1Department of Nuclear Science and Engineering,\nMassachusetts Institute of Technology, Cambridge, MA 02139, USA\n2Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA\n3Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA\nSolid-state platforms based on electro-nuclear spin systems are attractive candidates for rotation\nsensing due to their excellent sensitivity, stability, and compact size, compatible with industrial\napplications. Conventional spin-based gyroscopes measure the accumulated phase of a nuclear spin\nsuperposition state to extract the rotation rate and thus suffer from spin dephasing. Here, we propose\na gyroscope protocol based on a two-spin system that includes a spin intrinsically tied to the host\nmaterial, while the other spin is isolated. The rotation rate is then extracted by measuring the\nrelative rotation angle between the two spins starting from their population states, robust against\nspin dephasing. In particular, the relative rotation rate between the two spins can be enhanced\nby their hyperfine coupling by more than an order of magnitude, further boosting the achievable\nsensitivity. The ultimate sensitivity of the gyroscope is limited by the lifetime of the spin system\nand compatible with a broad dynamic range, even in the presence of magnetic noises or control\nerrors due to initialization and qubit manipulations. Our result enables precise measurement of\nslow rotations and exploration of fundamental physics.\nIntroduction.— Inertial sensing finds broad applica-\ntions, from tests of fundamental physics such as ge-\nometric phases and general relativity [1–6], to indus-\ntrial applications such as navigation in the absence of\nglobal positioning system [7–10]. In recent years, sens-\ning technologies based on quantum systems have shown\npromising performances in sensitivity, resolution, stabil-\nity, etc. compared with their classical counterparts [11].\nAmong these systems, nuclear spins in solid-state plat-\nforms, in particular nitrogen-vacancy (NV) centers in di-\namonds [12], have become an attractive candidate due to\ntheir long coherence times, ambient operation conditions,\nsmall size, and fabrication capabilities [13–18].\nAn inertial measurement requires the system to evolve\n(e.g. rotate) relative to an inertial reference frame. Ex-\nisting NV gyroscopes are mostly based on the nuclear\nspin of the14N (I= 1) atom [13–16] whose nuclear\nquadrupole term quantizes the nuclear spin along the NV\norientation ˆ zNVand constrains the nuclear spin to de-\ntect longitudinal rotations about ˆ zNV, that is, the crys-\ntal [111] direction. The rotation rate can be extracted\nfrom the accumulated dynamic phase using a Ramsey se-\nquence. In contrast, the nuclear spin of the15N atom is\na spin-1\n2without a quadrupole term and thus is released\nfrom such a constraint. Although the hyperfine interac-\ntion with the NV electronic spin still effectively applies\na large field that constrains the quantization along ˆ zNV,\nsuch a field vanishes for the NV spin state mS= 0. Thus,\nthe nuclear spin can be isolated from the crystal orien-\ntation and makes the system an ideal platform for iner-\ntial sensing. Moreover,15N nuclear spin has only two\nenergy levels, making it easier to polarize, control, and\nread out. Despite these benefits, the use of15N in NV\ngyroscopes remains less explored and the current sensing\nprotocols, limited to the conventional Ramsey methods,are still limited by the nuclear spin dephasing time T∗\n2n\n[19], which is degraded by the presence of transverse field\ninhomogeneities [20].\nIn this work, we develop a gyroscope protocol based\non the15N nuclear spin in diamond NV centers. Instead\nof using a superposition state to measure longitudinal ro-\ntations about ˆ zNV, we now use the population state to\nmeasure rotations along a transverse direction, thus be-\ning robust against spin dephasing. While the NV spin\nquantization axis co-rotates with the diamond, the nu-\nclear spin either adiabatically follows its eigenstate un-\nder a slow rotation (Fig 1(b)), or remains in its initial\nstate under a fast rotation (Fig 1(c)), both yielding rel-\native rotations between the two spins that can be used\nto extract the rotation rate, covering a broad dynamic\nrange. In particular, the NV-nuclear hyperfine interac-\ntion enhances the transverse Zeeman coupling of the nu-\nclear spin by a (magnetic field dependent) factor of 15 to\na few thousand, which can be used to boost the nuclear\nspin rotation rate and thus the gyroscope sensitivity by\na same factor. Numerical simulations demonstrate that\nour gyroscope protocol is robust against magnetic noise\nand only limited by the NV lifetime T1e. Besides rotation\nsensing, our gyroscope can provide insights into experi-\nmental tests of fundamental physics such as Lorentz in-\nvariance, relativistic geometric phases, and the Einstein\nde Haas effect [1, 21, 22].\nSystem.— The physical system of our gyroscope is\nbased on an electro-nuclear spin system hosted by point\ndefects in solid-state platforms. Specifically, we use NV\ncenters in diamond with enriched15N isotope to illus-\ntrate the protocol. The NV center consists of an elec-\ntronic spin-1 and a nitrogen nuclear spin-1\n2. Due to its\nisolation from the environment, the nuclear spin is used\nas the inertial sensor, while the electronic spin is used forarXiv:2401.01334v2  [quant-ph]  3 Jan 20242\nenha ncement\nlaser\nMW/RF\nN\nS\n/uni03A9\nyz\n(a)\nB\nBxBz\nαBx\nBθφ’\nNV (z)15N\nz\nx<111>\ny\nHyperfine-enhanced regime(b)\n(c)Inertial regimeB\nNV (z)15Nz\nx<111>\nyB\nBxBzθ\nφ’= 0B\n|α|Ω\nNV z axis\nSimula ted ev olution\nNuclear Eigenstateenha nced rotation\n|    |n(t)|2+zNV\nTheory S(t) \ncos(Ωt) \nInertial \nSensing Enhanced \nSensing(d)\nB=50G\n=0.5kH zEigenstate DeviationΔS\n0.2\n0.00.40.60.81.01.21.4\n10-3γnB\nB=50G\n=0.1kH z\nB=50G\n=1 MH zAdiabatic Range\nFIG. 1. Gyroscope principle. (a) Gyroscope based on an electro-nuclear spin system composed of NV electronic and nuclear\nspins in diamond. (b) Hyperfine-enhanced rotation regime (Ω ≪γnB). The relative rotation between the NV and the nuclear\nspin is enhanced due to the hyperfine interaction. For a diamond rotation rate Ω = (2 π)0.1kHz and magnetic field B= 50G,\nthe measured overlap of the final electronic and nuclear spin states (referenced with the diamond rotation) is simulated, with\nan accompanying schematic showing the NV ˆ zNVaxis rotation, nuclear spin evolution, and nuclear spin eigenstate. (c) Inertial\nregime (Ω ≫γnB). The nuclear spin remains in its initial state and the relative rotation corresponds directly to the rotation\nof the diamond. The simulated signal and system evolution is shown for a fast rotation Ω = (2 π)1 MHz. (d) Signal deviation\nfrom the ideal regimes presented in (b) and (c) and the corresponding eigenstate deviation of the nuclear spin Hamiltonian. In\nthe intermediate regime, it is hard to attribute the population signal to either hyperfine-enhanced or inertial regimes.\ninitializing and reading out the nuclear spin state [12].\nThe ground state Hamiltonian of the NV center can be\nwritten as\nH=⃗S·D·⃗S+⃗S·A·⃗I+γe⃗B·⃗S+γn⃗B·⃗I(1)\nwhere D,Aare zero-field splitting (ZFS) and hy-\nperfine tensors and γe= (2 π)2.802 MHz/G, γn=\n(2π)0.432 kHz/G are the gyromagnetic ratios of the elec-\ntronic and nuclear spin. When the ˆ zaxis is chosen\nto be along the N-to-V orientation intrinsic to the di-\namond crystal (henceforth referred to as ˆ zNVin the NV\nframe), both tensors are diagonal: the ZFS tensor has\na longitudinal term D= (2π)2.87 GHz; the longitudinal\nand transverse components of the hyperfine tensor are\nAzz= (2π)3.03 MHz and A⊥= (2π)3.65 MHz [23, 24].\nUnder an external magnetic field ⃗Bsatisfying γeB≪\nD, the electronic spin is quantized along the NV orien-\ntation by the large ZFS and is thus tied to the diamond\ncrystal. In contrast, the nuclear spin energy is only de-\nfined by the external magnetic field and hyperfine interac-\ntion with the electron spin, with an effective Hamiltonian\nin the NV frame [25–27]\nHI=γn[BzIz+αmsBxIx] +AzzmsIz, (2)\nwhere msis the electronic spin Zeeman state and αmsis\nan enhancement factor of the transverse Zeeman coupling\ninduced by mixing with the electronic spin states due to\nthe transverse hyperfine interaction [25, 26]. When the\nexternal magnetic field is small ( γeB≪D), such a factor\ncan be approximated to a constant αms= (1−2κ+3κm2\ns)\nwith κ≈γeA⊥/(γnD)≈8.26 [27, 28].\nIdeally, when there is no external magnetic field and\nmS= 0, the nuclear spin is effectively decoupled fromthe NV electronic spin and a physical rotation of the\ndiamond along a transverse direction ˆ yrotates the elec-\ntronic spin eigenstates while leaving the nuclear spin un-\nchanged. When the rotation rate satisfies Ω ≪D−γeB,\nthe NV electronic spin state adiabatically follows the NV\norientation axis ˆ zNV. In the NV frame, the nuclear spin\ninitialized to |mI= +1 /2⟩rotates in the z−xplane with\na rate −Ω due to its inertia. The nuclear spin population\nprojected onto the NV axis can be measured by mapping\nit to the electronic spin state with a CNOT gate (achieved\nby an electron spin π-pulse conditionally applied on a nu-\nclear spin state [28]) and then reading out through the\nfluorescence of the NV center. The readout signal is then\nS(t) = (1 + cos(Ω t))/2, from which the rotation rate can\nbe extracted.\nWhile this protocol is attractive as it does not require\ngimbals, working at a zero magnetic field is challenging\nbecause of the need for magnetic shielding. In addition,\nwe expect that working at zero field will make the nu-\nclear spin even more susceptible to magnetic noise, which\nis usually larger at lower frequencies and can lead to de-\nphasing and depolarization, harmful to the sensitivity. It\nis thus more practical to develop a general protocol for\nnonzero magnetic field conditions.\nHyperfine-enhanced regime.— Here we show how\nadding a magnetic field improves the rotation sensing\nperformance by amplifying the nuclear spin rotation. We\nassume to apply an external magnetic field Bin a static\nframe, and first consider sensing slow rotations of the\ndiamond, in the regime where Ω ≪γnB. With the\ndiamond initially aligned with ˆ zNValong the magnetic\nfield, both electronic and nuclear spins are initialized in\nthe z-eigenstates, |mS= 0, mI= 1/2⟩. When the dia-\nmond starts to rotate along ˆ yby an angle θ= Ωt,3\nthe ZFS and hyperfine tensors start to rotate. Con-\nversely, in the NV frame, the magnetic field starts to\nhave a transverse component. While the electronic spin\nis still mainly quantized along ˆ zNV, the nuclear spin adi-\nabatically follows the effective magnetic field direction\n⃗Beff=−α0BsinθˆxNV+BcosθˆzNV(Eq. (2)) (Fig. 1(b)).\nThe effective magnetic field for the nuclear spin rotates\nwith respect to the NV axis by an angle φ′satisfying\ntanφ′=−α0tanθ. For θ≪1, the nuclear spin rotation\nin the NV frame is amplified by a factor of |φ′/θ| ≈ |α0|,\ngiving an effective rate in the lab frame of Ω(1 + |α0|).\nThe rotation enhancement factor is magnetic-field depen-\ndent [28], increasing dramatically to its maximum value\nα0≈γe√\n2γn≈4.6×103near the GSLAC condition when\nγeB≈D[25, 26].\nTo extract the rotation rate Ω, we can measure the\nnuclear spin population along the ˆ zNVaxis by mapping\nthe nuclear spin state onto the NV, yielding a signal\nS(t) =1 + cos φ′(t)\n2=1\n2(1 +cos Ω tq\ncos2Ωt+α2\n0sin2Ωt)\n(3)\nIn Fig. 1(b), we numerically simulate the evolution of the\ngyroscope system under a rotation rate Ω = (2 π)100 Hz\nwith an external magnetic field B= 50 G. The evolu-\ntion of the NV electronic and nuclear spins is separated\nout by tracing out the other spin. The simulated nu-\nclear spin population along zNVis shown by the pink\ncurve in Fig. 1(b), which matches the theoretical predic-\ntion from Eq. (3) shown by the black dashed curve. We\nnote that the fast oscillations around the predicted evo-\nlution observed in simulations are a result of the nuclear\nspin precession about the effective magnetic field. As a\nreference, the diamond rotation is shown by the purple\ncurve.\nIn Fig. 2(a), we simulate the rotation enhancement fac-\ntor by explicitly calculating the rotation rate of the nu-\nclear spin. The maximum enhancement is achieved when\nthe magnetic field is aligned with the NV axis, where\nhowever the signal in Eq (3) doesn’t depend on α0. We\nthus instead calculate the rate of change in the nuclear\nspin population along ˆ xwhich is proportional to the ro-\ntation angle when it is small (see inset of Fig. 2(a)). The\nsimulation results match the theoretically predicted en-\nhancement factor to first order |α0| ≈γe\nγnA⊥/(D−γeB).\nWhen the nuclear Larmor precession frequency is com-\nparable to the rotation rate, Ω ∼γnB, the adiabaticity of\nthe nuclear spin evolution begins to break down (as seen\non the Bloch sphere in Fig. 1(d) for Ω = (2 π)0.5kHz), and\nit no longer follows the effective magnetic field. To fur-\nther study such effects, we compare simulations over dif-\nferent rotation rates for a fixed magnetic field in Fig. 1(d).\nFor each rotation rate Ω ≤γnB, we compute the average\ndeviation from the predicted signal in Eq. (3) on the left\ny-axis and compute the non-adiabaticity from the eigen-\n(b)\nB (G)e- spin Ramsey Inertial gyro / n spin Ramsey \nHyperfine-enhanced gyro(a)\nt (0 100 200\ns)0.50.550.6\nB=50G\n0.50.550.6\n010 20 30B=950GRotation enhancement\n0 200 400 600 800 100010-210-1100101\n0 200 400 600 800 1000\nB (G)02004006008001000\nSimulation\neA/(n(D-eB))\nSensitivity (mdeg/s/ Hz)FIG. 2. Simulation of the gyroscope protocol. (a)\nSimulated rotation enhancement factor for small θ≪1. The\ninsets show the time evolution of the nuclear spin population\nalong ˆ xNVused to extract the rotation rate under two ex-\nemplary conditions with small and large magnetic fields. (b)\nSensitivity comparison of different gyroscope protocols. For\na fair comparison, here we use the same readout efficiency\nC= 2% across protocols. For all three nuclear spin-based\nprotocols, we set the signal decay times to 1.5 T1e= 7.5 ms\nand dead time to td= 0.5 ms. The “e spin” protocol utilizes\nthe NV electronic spin resonance frequency shifts measured\nby a Ramsey sequence to sense the rotation rate (see supple-\nment [28] for details), where we set a typical spin dephasing\ntime T∗\n2e= 0.7µs and td= 0.05 ms [20].\nstate deviation |⟨+1\n2|dHI(t)\ndt| −1\n2⟩|2(where |mI=±1/2⟩\nare the nuclear spin eigenstates) on the right y-axis [28].\nAs the rotation rate increases, the eigenstate deviation\nincreases and after a threshold value, it becomes difficult\nto directly extract the rotation rate from the measured\nsignal. To maintain the adiabaticity and sense faster\nrotations in the hyperfine-enhanced regime, one can in-\ncrease the bias magnetic field to increase the nuclear spin\nenergy gap. Thus, quantifying the nuclear eigenstate de-\nviation allows us to define a bespoke maximum rotation\nrate that one can detect for a given magnetic field, de-\npendent on the desired precision of the signal.\nInertial regime.— While the enhanced rotation factor\nis limited to sensing a rotation rate that is much less than\nthe nuclear spin energy gap, the proposed gyroscope pro-\ntocol can still sense rotation rates in the regime where\nΩ≫γnB. Here, the electron spin is still quantized in\nthe crystal’s ˆ zNVdirection, thus following the diamond\nrotation adiabatically. The effective external magnetic\nfield on the nuclear spin rapidly oscillates and averages\nto zero similar to the B= 0 case, so the nuclear spin\nremains in its initial state, effectively decoupled from the\nelectron spin. In this scenario, the relative rotation ob-\nserved between the two spins corresponds directly to the\ndiamond’s rotation, and thus in the gyroscope protocol\nthe nuclear spin population signal is given by\nS(t) =1\n2(1 + cos θ) =1\n2(1 + cos(Ω t)). (4)\nWe simulate the evolution of our system under a fast\nrotation Ω = (2 π)1 MHz in Fig. 1(c) yielding a signal\nconsistent with that predicted by Eq. (4). Similar to the4\nhyperfine-enhanced regime, calculating the eigenstate de-\nviation (in Fig. 1(d)) identifies the minimum rotation rate\nrequired for high-fidelity measurements in this regime.\nThus, combining both regimes, our gyroscope mea-\nsures a broad dynamic range Ω ≲D−γeB, except for a\nsmall window near the nuclear spin energy gap Ω ∼γnB.\nTo maximize the enhanced rotation rate in the hyperfine-\nenhanced regime, one can choose to set the bias magnetic\nfield strength as close as possible to GSLAC. Neverthe-\nless, at high fields, misalignment of the electron spin from\nthe magnetic field, as well as its small energy gap, can\nmake initialization and readout of the protocol difficult\n[28]. In the following, we discuss and analyze such limits\nto the sensitivity of the gyroscope, including the system’s\nultimate lifetime and optimal performance.\nGyroscope Performance.— Quantum sensors’ perfor-\nmance is bound by their coherence or relaxation times,\nwhich limits the sensing time t. In the enhanced ro-\ntation regime, the nuclear spin adiabatically follows its\neigenstate and thus its decay induced by an external\nbath follows a spin relaxation process, which typically\nyields a relaxation time T1nmuch longer than the de-\nphasing time T∗\n2n. However, the NV electronic spin re-\nlaxation T1ehas been shown to be a source of deco-\nherence, typically limiting the nuclear spin dephasing,\nT∗\n2n= 1.5T1e[29]). Under the gyroscope rotation, an NV\nspin flip due to relaxation changes the nuclear spin quan-\ntization axis (Eq. (2)), leading to fast decay of what was\npreviously an eigenstate. We numerically simulate this\nprocess using the Lindblad equation with Lindblad op-\nerators Lk=√\nΓ|ms⟩⟨m′\ns|, where ms, m′\ns∈ {− 1,0,+1}\nand the jump rate is Γ = 1 /(3T1e). Fig. 3(a)i shows that\nin the slow rotation regime the gyroscope signal decay\ntime is T1n≈1.5T1e. In the inertial regime under fast\nrotations shown in Fig. 3(b)ii, the simulation yields a de-\ncay time T1n>1.5T1e. This increased decay time might\noriginate from effective dynamical decoupling effects due\nto the electron spin’s rapid rotation [28, 29].\nThe decay due to the electronic spin relaxation process\nsets the upper limit of the system coherence, but other\nnoise sources such as magnetic noise induced by the spin\nbath could dominate, especially in dense spin ensembles.\nWe further explore these effects by simulating the gy-\nroscope signal under magnetic field inhomogeneities in\nFig. 3(b). In the frame of the nuclear spin, the longi-\ntudinal magnetic noise causes a T∗\n2dephasing process\nthat results in a decay in its transverse spin components.\nIntuitively, this might lead to a decay of the system co-\nherence as well; however, even though the fast oscillation\ndecays with time, the main gyroscope signal remains rel-\natively robust against the magnetic noise. Such an ef-\nfect may even improve the performance of our gyroscope\nby suppressing the undesired fast oscillation signal that\noriginates from the spin precession around the effective\nmagnetic field.\nThus, the gyroscope signal decay is ultimately limited\n(b)\n(c)i. ii.\nB=50G\nΩ=10 MH z(a)\nii. i.B=50G\nΩ=0.1kH z\n/uni0394B=0.002G\nB (G) B (G)\nΩ<<ω\nAdiabatic Region>\n>\nInertial Region\nΩ>>ωFIG. 3. Performance analysis. (a) Gyroscope signal un-\nder electronic spin T1 relaxation at B= 50 G in (i) hyperfine-\nenhanced regime with Ω = (2 π)0.1 kHz ≪γnBand (ii) iner-\ntial regime with Ω = (2 π)10 MHz. The decay of the nuclear\nspin signal envelope is fit to S(t) =c0+c1e−t/τfor regime\n(i) and S(t) =c0+c1e−(t/τ)c2cos(Ω t) for regime (ii) to ob-\ntain the decay time T1n. (b) Gyroscope signal under static\nmagnetic noise at B= 50 G and Ω = (2 π)0.1 kHz. The\nvariance of the magnetic noise is set to be 0.002 G. The in-\nset shows the dephasing time T∗\n2nas a function of the noise\nstrength. (c) Sensitivity bound η(Ω, B) for a spin density\nnNV/4≈2.5×1014/mm3. Solid lines indicate the maximum\nand minimum rotation rates such that the theoretical signal\npredictions remain valid.\nby the lifetime T1n, giving a lower bound of the achievable\nsensitivity of the gyroscope [28],\nη≥√\n2γn\nγeet/T1n\nC√\nN√t+td\nt, (5)\nfor the hyperfine-enhanced regime, where tis the total\nsensing time and tdis the dead time for initialization\nand readout. With typical ensemble parameters and\nconditions (readout efficiency C∼2% [30]), for a vol-\nume V= 1mm3andN=nNVV/4≈2.5×1014sen-\nsor spins and T1n∼7.5 ms, the sensitivity limit reaches\nη∼1×10−3(mdeg/s)/√\nHz near GSLAC. In the inertial\nregime, the sensitivity follows Eq. (5) but without the\nenhancement factor ∼γn/(√\n2γe).\nNevertheless, implementation of the gyroscope requires\nthe consideration of many practical limitations; here, we\nbriefly discuss these factors and their influence on the sen-5\nsitivity. To take advantage of the hyperfine-enhanced ro-\ntation rate, the time for signal readout should be chosen\nsuch that the two spins are nearly aligned upon measure-\nment. In particular, it is a compromise between choos-\ning a longer sensing time for optimizing sensitivity and\nmaking the measurement as quickly as possible to main-\ntain the enhancement which decreases with the initial ro-\ntation angle. Furthermore, under large magnetic fields,\noptical polarization of the NV spin is difficult with mis-\nalignment from the magnetic field. Thus, to operate the\ngyroscope continuously, both the dead time and sensing\ntime are limited to integer multiples of the rotation pe-\nriodt+td≈kπ/Ω, which also guarantees that the NV is\nnearly aligned with the magnetic field during initializa-\ntion and readout. For these calibrations, an estimate of\nthe rotation rate needs to be initially acquired (e.g., by\nusing the NV spin transition frequency shift during the\nrotation). Different from typical Ramsey magnetometry,\nthe sensitivity of our gyroscope depends on the relative\nangle between two spins in the final state, thus the ac-\ntual sensitivity needs to be calculated taking these into\naccount (see supplement [28] for details).\nMoreover, during the rotation, initialization and read-\nout of the nuclear spin require selective control of the\nelectronic spin such as π-pulses only for a specific nu-\nclear spin state. Thus, for large magnetic fields and fast\nrotation rates, it is important to consider the resonance\nfrequency shift during the rotation, which affects the con-\ntrol fidelity. We conceptually characterize these effects by\nadding a pulse fidelity factor in the readout efficiency as-\nsuming a pulse duration ∼0.5µs in Fig. 3(c), where we\ncompute the lower bounded sensitivity of the gyroscope\nvalues of (Ω , B). Despite these restrictions, these effects\ncan be improved by optimal control techniques or adap-\ntive methods. Thus, our gyroscope remains a versatile\nplatform capable of reaching extremely good sensitivity\nfor bespoke rotation measurements.\nWe note that with the assistance of an external mag-\nnetic field, the NV electronic spin transition frequency\ncan also be used to extract the rotation rate for a trans-\nverse rotation, which can be estimated by Ramsey exper-\niments limited by a dephasing time T∗\n2e. In a spin ensem-\nble, the electron spin dephasing is short T∗\n2e≪T1e[20],\nwhich leads to a worse sensitivity than the nuclear spin\ngyroscopes as shown in Fig. 2(b).\nConclusion and Discussion.— We propose a novel gy-\nroscope protocol based on15NV centers in diamond.\nIn comparison to conventional Ramsey-type gyroscopes\nlimited to sensing a ˆ zrotation and suffering from spin\ndephasing, our protocol uses the more robust popula-\ntion state to sense a transverse rotation. In particular,\nwhen an external magnetic field is fixed in the lab frame,\nthe nuclear spin rotation rate can be significantly en-\nhanced by its hyperfine interaction with the electronic\nspin, achieving a sensitivity improvement of up to three\norders of magnitude. The protocol is robust against mag-netic noises and is only limited by T1relaxation times.\nWhile our analysis shows that nuclear spin lifetime T1n\nis still limited by the electron spin T1erelaxation time,\nwhich is also the ultimate limit to the coherence limit\nof the conventional Ramsey-type gyroscope, our protocol\nis intrinsically more robust to other sources of dephas-\ning noise that dominate Ramsey gyroscopes and an ad-\nditional, significant sensitivity improvement derives from\nthe rotation enhancement. The sensitivity could be fur-\nther improved by designing dynamical decoupling tech-\nniques to extend the nuclear spin lifetime beyond T1eby\ncanceling the deleterious effects of the electronic spin re-\nlaxation process [29].\nIn addition to building a highly sensitive and compact\ngyroscope under ambient conditions competitive with\natomic gyroscopes, our gyroscope can also exploit the\nelectro-nuclear spin system to provide insights into test-\ning fundamental physics. Thus, our gyroscope proves to\nbe a robust and versatile device, with broad opportuni-\nties for integration and applications.\nThis work was supported in part by DARPA DRINQS\nprogram (Cooperative Agreement No. D18AC00024).\nG.W. thanks MathWorks for their support in the form of\na Graduate Student Fellowship. The opinions and views\nexpressed in this publication are from the authors and\nnot necessarily from MathWorks.\n∗gqwang@mit.edu; These authors contributed equally.\n†minhthin@mit.edu; These authors contributed equally.\n‡pcappell@mit.edu\n[1] D. Maclaurin, M. W. Doherty, L. C. L. Hollenberg, and\nA. M. Martin, Measurable Quantum Geometric Phase\nfrom a Rotating Single Spin, Phys. Rev. Lett. 108,\n240403 (2012).\n[2] A. A. Wood, L. C. L. Hollenberg, R. E. Scholten, and\nA. M. Martin, Observation of a Quantum Phase from\nClassical Rotation of a Single Spin, Phys. Rev. Lett. 124,\n020401 (2020).\n[3] A. A. Wood, E. Lilette, Y. Y. Fein, V. S. Perunicic,\nL. C. L. Hollenberg, R. E. Scholten, and A. M. Mar-\ntin, Magnetic pseudo-fields in a rotating electron–nuclear\nspin system, Nat. Phys. 13, 1070 (2017).\n[4] M. Smiciklas, J. M. Brown, L. W. Cheuk, S. J. Smullin,\nand M. V. Romalis, New Test of Local Lorentz Invariance\nUsing a Ne 21 - Rb - K Comagnetometer, Phys. Rev. Lett.\n107, 171604 (2011).\n[5] W. A. Terrano and M. V. Romalis, Comagnetometer\nprobes of dark matter and new physics, Quantum Sci.\nTechnol. 7, 014001 (2021).\n[6] H. J. Arditty and H. C. Lef` evre, Sagnac effect in fiber\ngyroscopes, Opt. Lett. 6, 401 (1981).\n[7] V. M. N. Passaro, A. Cuccovillo, L. Vaiani, M. De Carlo,\nand C. E. Campanella, Gyroscope Technology and Appli-\ncations: A Review in the Industrial Perspective, Sensors\n17, 2284 (2017).\n[8] H. D. Escobar-Alvarez, N. Johnson, T. Hebble,\nK. Klingebiel, S. A. P. Quintero, J. Regenstein, and N. A.6\nBrowning, R-ADVANCE: Rapid Adaptive Prediction for\nVision-based Autonomous Navigation, Control, and Eva-\nsion, J. Field Robot. 35, 91 (2018).\n[9] B. Fang, I. Dutta, P. Gillot, D. Savoie, J. Lautier,\nB. Cheng, C. L. G. Alzar, R. Geiger, S. Merlet, F. P. D.\nSantos, and A. Landragin, Metrology with Atom Inter-\nferometry: Inertial Sensors from Laboratory to Field Ap-\nplications, J. Phys. Conf. Ser. 723, 012049 (2016).\n[10] L. Zhao, X. Shen, L. Ji, and P. Huang, Inertial measure-\nment with solid-state spins of nitrogen-vacancy center in\ndiamond, Adv. Phys.: X 7, 2004921 (2022).\n[11] C. L. Degen, F. Reinhard, and P. Cappellaro, Quantum\nsensing, Rev. Mod. Phys. 89, 035002 (2017).\n[12] M. W. Doherty, N. B. Manson, P. Delaney, F. Jelezko,\nJ. Wrachtrup, and L. C. Hollenberg, The nitrogen-\nvacancy colour centre in diamond, Phys. Rep. 528, 1\n(2013).\n[13] A. Ajoy and P. Cappellaro, Stable three-axis nuclear-spin\ngyroscope in diamond, Phy. Rev. A 86, 062104 (2012).\n[14] J.-C. Jaskula, K. Saha, A. Ajoy, D. Twitchen,\nM. Markham, and P. Cappellaro, Cross-Sensor Feedback\nStabilization of an Emulated Quantum Spin Gyroscope,\nPhy. Rev. Applied 11, 054010 (2019).\n[15] A. Jarmola, S. Lourette, V. M. Acosta, A. G. Birdwell,\nP. Bl¨ umler, D. Budker, T. Ivanov, and V. S. Malinovsky,\nDemonstration of diamond nuclear spin gyroscope, Sci.\nAdv. 7, eabl3840 (2021).\n[16] V. V. Soshenko, S. V. Bolshedvorskii, O. Rubinas, V. N.\nSorokin, A. N. Smolyaninov, V. V. Vorobyov, and A. V.\nAkimov, Nuclear Spin Gyroscope based on the Nitrogen\nVacancy Center in Diamond, Phys. Rev. Lett. 126, 6\n(2021).\n[17] A. A. Wood, E. Lilette, Y. Y. Fein, N. Tomek, L. P.\nMcGuinness, L. C. L. Hollenberg, R. E. Scholten, and\nA. M. Martin, Quantum measurement of a rapidly rotat-\ning spin qubit in diamond, Sci. Adv. 4, eaar7691 (2018).\n[18] G. Wang, A. R. Barr, H. Tang, M. Chen, C. Li, H. Xu,\nA. Stasiuk, J. Li, and P. Cappellaro, Characterizing tem-\nperature and strain variations with qubit ensembles for\ntheir robust coherence protection, Phys. Rev. Lett. 131,\n043602 (2023).\n[19] B. B¨ urgler, T. F. Sjolander, O. Brinza, A. Tallaire,\nJ. Achard, and P. Maletinsky, All-optical nuclear quan-\ntum sensing using nitrogen-vacancy centers in diamond,\nnpj Quantum Inf. 9, 1 (2023).\n[20] G. Wang, M.-T. Nguyen, D. W. de Quilettes, E. Price,\nZ. Hu, D. A. Braje, and P. Cappellaro, Emulated nu-\nclear spin gyroscope with15nv centers in diamond (2024),\narXiv:2401.01333 [quant-ph].\n[21] G. Stedman, Ring-laser tests of fundamental physics\nand geophysics, Reports on progress in physics 60, 615\n(1997).\n[22] R. Jaafar, E. Chudnovsky, and D. Garanin, Dynamics\nof the einstein–de haas effect: application to a magnetic\ncantilever, Physical Review B 79, 104410 (2009).\n[23] S. Felton, A. M. Edmonds, M. E. Newton, P. M. Mar-\ntineau, D. Fisher, D. J. Twitchen, and J. M. Baker, Hy-\nperfine interaction in the ground state of the negatively\ncharged nitrogen vacancy center in diamond, Phys. Rev.\nB79, 075203 (2009).\n[24] S. Lourette, A. Jarmola, V. M. Acosta, A. G.\nBirdwell, D. Budker, M. W. Doherty, T. Ivanov,\nand V. S. Malinovsky, Temperature Sensitivity of$ˆ{14}\\mathrm {NV}$ and $ˆ{15}\\mathrm {NV}$\nGround State Manifolds (2022), 2212.12169 [cond-mat,\nphysics:physics, physics:quant-ph].\n[25] M. Chen, M. Hirose, and P. Cappellaro, Measurement of\ntransverse hyperfine interaction by forbidden transitions,\nPhys. Rev. B 92, 020101 (2015).\n[26] S. Sangtawesin, C. A. McLellan, B. A. Myers, A. C. B.\nJayich, D. D. Awschalom, and J. R. Petta, Hyperfine-\nenhanced gyromagnetic ratio of a nuclear spin in dia-\nmond, New J. Phys. 18, 083016 (2016).\n[27] J. T. Oon, J. Tang, C. A. Hart, K. S. Olsson, M. J.\nTurner, J. M. Schloss, and R. L. Walsworth, Ramsey en-\nvelope modulation in NV diamond magnetometry, Phys.\nRev. B 106, 054110 (2022).\n[28] See Supplemental Material for details.\n[29] M. Chen, W. K. C. Sun, K. Saha, J.-C. Jaskula,\nand P. Cappellaro, Protecting solid-state spins from a\nstrongly coupled environment, New J. Phys. 20, 063011\n(2018).\n[30] J. F. Barry, J. M. Schloss, E. Bauch, M. J. Turner, C. A.\nHart, L. M. Pham, and R. L. Walsworth, Sensitivity op-\ntimization for NV-diamond magnetometry, Rev. Mod.\nPhys. 92, 015004 (2020), arXiv:1903.08176.\n[31] V. Jacques, P. Neumann, J. Beck, M. Markham,\nD. Twitchen, J. Meijer, F. Kaiser, G. Balasubramanian,\nF. Jelezko, and J. Wrachtrup, Dynamic Polarization of\nSingle Nuclear Spins by Optical Pumping of Nitrogen-\nVacancy Color Centers in Diamond at Room Tempera-\nture, Phys. Rev. Lett. 102, 057403 (2009).\n[32] T. W. Kornack, R. K. Ghosh, and M. V. Romalis, Nu-\nclear Spin Gyroscope Based on an Atomic Comagnetome-\nter, Phys. Rev. Lett. 95, 230801 (2005).Supplementary Materials\nDERIVATION OF ENHANCEMENT FACTOR\nHere, we derive the exact expression for the transverse Zeeman coupling enhancement factor of the nuclear spin for\nthe15NV center system for mS= 0 [25, 26]. The ground state Hamiltonian is given by Eq. (1). Under an applied\nmagnetic field Bz, the Hamiltonian can be decomposed into secular H||and non-secular terms H⊥:\nH||=DS2\nz+γeBzSz+γnBzIz+AzzSzIz (S1)\nH⊥=A⊥\n2(S+I−+S−I+). (S2)\nwhere S±=Sx±iSyandI±=Ix±iIy. The total Hamiltonian can then be diagonalized by rotating the zero\nquantum (ZQ) subspaces with UZQ=e−i(θ−σ−\ny+θ+σ+\ny)where σ+\ny=i(|+ 1,−1\n2⟩⟨0,+1\n2| − |0,+1\n2⟩⟨+1,−1\n2|) and σ−\ny=\ni(|0,−1\n2⟩⟨−1,+1\n2| − | − 1,+1\n2⟩⟨0,−1\n2|) and\ntan\u0000\n2θ+\u0001\n=2A⊥\nD+γeBz−γNBz−Azz/2(S3)\ntan\u0000\n2θ−\u0001\n=−2A⊥\nD−γeBz+γNBz−Azz/2(S4)\nApplying an additional transverse magnetic field in the NV frame (for simplicity and in context with our paper,\nconsider only Bx, but a magnetic field component Bywould see an equivalent enhancement) introduces an interaction\nHamiltonian Hx=Bx(γeSx+γNIx). Under the unitary transformation, ˆHx=UZQHxU†\nZQand keeping only the\nnuclear spin terms, we obtain the effective Hamiltonian for the nuclear spin (where αmsis the enhancement factor for\nthe electronic spin state ms)\nˆHxI=γN(α+1|+ 1⟩⟨+1|+α0|0⟩⟨0|+α−1| −1⟩⟨−1|) (S5)\nwith\nα0= cos\u0000\nθ+\u0001\ncos\u0000\nθ−\u0001\n−γe\nγnsin\u0000\nθ+−θ−\u0001\nThus, for an applied magnetic field ⃗B= (Bx,0, Bz) on the NV center, we obtain the effective nuclear spin Hamiltonian\nas presented in Eq. (2). For small fields α0≈15.5 and reaches a finite maximum value near GSLAC with α0≈γe√\n2γn\nshown in Fig. S1.\nPROTOCOL DETAILS\nNuclear Polarization and Readout\nThere are several methods to polarize the nuclear spin intrinsic to the NV center: we can optically pump the\nnuclear spin by setting the static magnetic field close to the ground or excited level crossing, or we can use a sequence\nof selective microwave and rf pulses to transfer polarization from the NV (Fig. S2(a).) Under misalignment of the\nNV spin, however, optical polarization of the nuclear spin is significantly suppressed [31]. Thus to achieve efficient\npolarization of the nuclear spin for our gyroscope sensing protocol, especially in the case of initialization during a\nrotation (which causes NV misalignment), it is preferable to polarize the nuclear spin to |mI= +1\n2⟩using a polarization\nsequence that coherently transfers the electron polarization to the nuclear spin.2\nFIG. S1. Predicted enhancement factor |α0|as a function of Bz.\nWorking in the mS= 0,−1 states, we first transfer the population from the |mS= 0, mI=−1\n2⟩spin state to\n|mS=−1, mI=−1\n2⟩using a selective microwave (mw) π-pulse with a Rabi frequency smaller than the hyperfine\nsplitting strength. We then apply a selective rf π-pulse that transfers the |mS=−1, mI=−1\n2⟩population to\n|mS=−1, mI= +1\n2⟩. Finally, a green laser pulse is applied to transfer the population back to the |mS= 0⟩manifold\nwhile preserving the nuclear spin state. The system is ultimately polarized to |mS= 0, mI= +1\n2⟩as an initial state\nof the rotation sensing protocol.\nOptical readout of the nuclear spin state is accomplished with mw mapping pulses at the end of the pulse sequence,\nas shown in Fig. S2(a).\nTo achieve efficient initialization and readout, we prefer the NV to be closely aligned with the magnetic field when\nperforming the initialization and readout operations. Indeed, the transition frequencies of the NV electronic and\nnuclear spins change during the rotation due to their misalignment with the magnetic field in the NV frame. Thus,\nwhen applying a pulse, these energy shifts can affect the pulse fidelity for the polarization and readout sequences.\nBecause this change is larger at higher magnetic fields (Fig. S2(b)), even though the enhancement factor is larger,\nthe sensitivity may be practically limited at large magnetic fields by errors in initialization and qubit manipulation.\nWe note that this challenge could be addressed by setting optimal pulse timing settings for the polarization/readout\nsequence in an adaptive way, as control optimization requires an initial estimation of the rotation rate or an indepen-\ndent measure of the resonance frequency by interleaving the gyroscope sequence with nuclear Ramsey or NV ODMR\nexperiments. For simplicity, here we discuss the worst-case scenario, where we assume to just use a single frequency\nfor the microwave and rf pulses. As relevant to quantum sensing, We consider the state fidelity for an initial state\n|ψi⟩and a target state ψtarget,F=|⟨ψtarget|U|ψi⟩|2. We can then evaluate the effect of the detuning induced by a\nθ-misalignment of the field on a π-pulse Uπ(t). For |ψi⟩=|0⟩and|ψtarget⟩=|1⟩, the π-pulse fidelity with detuning\nδand amplitude Ω dis given by F(Ωd, δ) =Ω2\nd\nΩ2\nRsin2(ΩRt\n2), with Ω R=p\nΩ2\nd+δ2the effective Rabi frequency and\nt=π/Ωdthe pulse duration. As shown in Fig. S2(b), when the magnetic field is not close to the GSLAC condition,\nthe nuclear spin transition frequencies are relatively stable against a small field misalignment, and the change of the\nelectron spin transition frequency is smaller than the typical Rabi frequency Ω ∼1 MHz. As a result, high-fidelity\ncontrol can be achieved even under the rotation as long as the Rabi frequency is larger than the transition frequency\nchange during the rotation and smaller than the difference ωmS=−1\nn −ωmS=−0\nn ≈3MHz, as shown in Fig. S2(c). In\nthe main text we accounted for these potential errors in qubit manipulation when estimating the sensitivity of the\nprotocol (Fig 3). In the sensitivity formula, we introduced a degraded contrast of the signal: C≥ FC0, where F(B) is\nthe B-field dependent pulse fidelity and C0∼2% the contrast for a typical NV ensemble. We note that the imperfect\nselectivity of the readout pulse can also lead to additional contrast degradation. These imperfections can be improved\nwith various optimal control strategies such as narrow-band and broad-band pulse designs.\nPerformance near the GSLAC\nWhile the enhancement factor α0reaches its maximum value ≈γe√\n2γn∼4×103near the GSLAC ( B≈D/γ e≈\n1024G), in that scenario the behavior of the gyroscope becomes more complex. Due to its small energy gap, the3\nlaser\nmw (NV)\nrf (N)\n πmS= 0   -1\nml = -1/2\n π\nmS= -1\nreadoutt mS= 0\nπ\n2( (mS= 0   -1\nml = -1/2\n π(a)\n(b)(c)evolutioninitialization readout\nFIG. S2. Example initialization and readout protocol (a) Basic sensing pulse sequence. The nuclear spin is dynamically\npolarized using selective microwave and rf pulses. After a sensing time t, the nuclear spin is mapped onto the NV and overlap is\nread out to extract the rotation rate. (b) NV electronic and nuclear spin transition frequency shifts (from alignment at θ=nπ) as\na function of magnetic field misalignment θat different magnetic field strengths. The top three panels are transition frequencies\nbetween the two nuclear spin levels under different magnetic fields. The bottom three panels are transition frequencies between\nthe two electronic spin states. (c) These frequency shifts are used to calculate the fidelity of the nuclear and electronic spin\n(selective on the nuclear spin state mI=±1/2)πpulses. The Rabi frequency for the nuclear spin used here 50 kHz and the\nelectronic spin is 1 MHz. This allows the assume pulse duration of tπ∼0.5µs used to compute the pulse fidelity factor in\nFig.3(c) in the main text.\nelectron spin not only is strongly mixed with the nuclear spin but it also no longer follows a completely adiabatic\nevolution. Nevertheless, simulation results indicate that even at large magnetic fields, the enhancement of the nuclear\nspin rotation compared to the electron spin is maintained. The hyperfine interaction still yields a relative rotation\nbetween the two spins, and the rotation rate can still be recovered (Fig. S3i). While the observed enhancement\nfactor near the GSLAC is lower than the analytical prediction by nearly a factor of 2, it could still provide gain a\nthree-order of magnitude sensitivity enhancement. As mentioned above, this enhancement might be reduced due to\ncontrol limitations, although adaptive methods will mitigate those issues.\nGyroscope Sensitivity\nHyperfine-Enhanced Gyroscope\nThe sensitivity of the gyroscope for an ensemble of Nspins is given by\nη=σet/τ√t+td\n2C√\nN|∂ΩS(t)|(S6)\nwhere tis the total sensing time, tdis the deadtime, τis the coherence time, Cis the readout efficiency parameter\nthat is typically set by the signal contrast and photon collection efficiency, Nis the total number of spins, and S(t)\nandσits standard deviation from the spin projection noise.4\n(a) B = 1000G\n(b) B = 1024G\ni. ii. \ni. ii. (c) \nFIG. S3. Hyperfine-enhanced gyroscope at large magnetic fields (a) i. Electro-nuclear spin system evolution during\na (2π)0.1kHz rotation at B = 1000 G. The left plot shows the measured nuclear spin population along the ˆ zNVaxis and the\nevolution of the electronic and nuclear spin states, compared to the reference diamond rotation and the predicted signal from\n3. An accompanying schematic of the NV and nuclear spin evolution on the Bloch sphere is shown on the right. ii. Simulated\nmeasurement signal of the nuclear spin population along ˆ xNVaxis, used to extract the rotation rate. (b) System evolution\nat GSLAC (B = 1024 G). (c) Simulated rotation angle enhancement factor for B≥1000G, compared to analytically-derived\nenhancement factor |α0|.\nIn practice, to optimize the sensitivity, we can maximize the signal slope by measuring the nuclear spin population\nalong the ˆ xNV(which can be accomplished by applying a π/2-pulse to the nuclear spin before readout) at the time of\nnear-alignment condition, yielding the signal:\nSx(t) =1\n2\n1 +α0sin(Ω t)q\nα2\n0sin2(Ωt) + cos2(Ωt)\n. (S7)\nTo ensure continuous operation of the protocol as well as efficient polarization and readout, we require t+td≈ ⌈Ωt\nπ⌉(π\nΩ)\nand assume the initialization and readout times ( td∼3µs under large laser power and fast control pulses) are small\nin comparison to period of the oscillation 2 π/Ω. The sensitivity then gives\nη(Ω≪γnB, B, t ) =1\n(α0\n2)\f\f\fcos(Ω t)\n(α2\n0sin2(Ωt)+cos(Ω t)2)3/2\f\f\f\net/τq\n⌈Ωt\nπ⌉(π\nΩ)\n2C√\nNt\n≥et/τ\nC√\nNα0√\nt(S8)\nHence the optimal sensitivity is achieved when the sensing time t≈kπ/Ω and the sensitivity has an enhancement\nofα0in comparison to the conventional scheme based on nuclear Ramsey (where |∂ΩS(t)|max=t/2). The ultimate\nbound on the sensitivity occurs near GSLAC, where α0≈γe√\n2γn, and with sensing time t≈τ/2.\nAssuming a typical NV ensemble diamond sample chip of volume V= 1 mm3with N= 2.5×1014spin sensors,\ndetection efficiency C∼2% and coherence time τ= 1.5T1e≈7.5ms, the sensitivity bound (Eq. (S8)) can achieve\nη(Ω≪γnB, B≈1024G , t≈τ/2)≈1×10−3(mdeg/s)/√\nHz, comparable with atomic gyroscopes [32].\nNevertheless, for slow rotations such that the signal decay time is on the same order as the rotation period when\nτ∼π/Ω, there is a tradeoff: the measurement needs to be made as quickly as possible after initialization of the\nelectron spin to maintain the high sensitivity enhancement factor in the signal, which has a conflict with maximizing\nthe sensing time. In this case, the sensitivity enhancement is dictated by the factor ∂tS(t). We note that for even\nlower rates where Ω τ≪1, the sensing time is again limited by the coherence time and the deadtime, thus reaching\nthe sensitivity bounds of the case where τ≫π/Ω.5\nInertial Gyroscope\nWhen sensing fast rotations Ω ≫γnB, the electron spin and nuclear spin still need to be nearly aligned with the\nmagnetic field when reading out, so we still require t+td≈ ⌈Ωt\nπ⌉(π\nΩ) and the sensitivity (measured along ˆ xNV) is\ngiven by\nη(Ω≫γnB, B, t ) =1\n(1\n2)|cos(Ω t)|\net/τq\n⌈Ωt\nπ⌉(π\nΩ)\n2C√\nNt\n≥et/τ\nC√\nN√\nt(S9)\nThe lower bound for the optimal sensitivity of the sensor can thus be approximated ηopt≈√\n2e\nC√\nNτ≈4.91\n(mdeg/s) /√\nHz for τ≈7.5ms.\nComparison with other protocols\nThe most conventional method in sensing a rotation is using the nuclear Ramsey. The Ramsey protocol using the\nnuclear spin has a signal of the form S(t) =1\n2(1 + sin(Ω t)).Thus, the sensitivity is similar to that given in Eq. S9:\nηN-Ramsey (t) =1\n(1\n2)|cos(Ω t)|\u0012et/τ√t+td\n2C√\nNt\u0013\n(S10)\nand yields a lower bound to the sensitivity ηopt≈√\n2e\nC√\nNτ. The coherence time for the Ramsey sequence is limited by the\nspin dephasing time T∗\n2n≤1.5T1e[29]. Thus, even in the fast rotation sensing regime (without hyperfine-enhancement)\nof the gyroscope developed in this work, the sensitivity is slightly improved compared to the conventional nuclear spin\nRamsey scheme, as we found τ >1.5T1ein numerical simulations.\nAlternatively, with the assistance of the external magnetic field along ˆ z(like in the setup discussed in our work),\nthe change in the electron spin transition frequency can also be used to extract the rotation rate (for a transverse\nrotation along ˆ y). In this case, similar to conventional magnetometry, this is tested with a Ramsey experiment with\nthe electron spin, ultimately limited by the spin dephasing time τ=T∗\n2e≪T1e[20]. The signal of the Ramsey is\ngiven by S(t) =1\n2(1 + sin( ωet)), where ωe=ωe(θ) is the rotation-angle dependent electron spin transition frequency.\nThus, the sensitivity is given by\nηNV-Ramsey (t) =1\n(1\n2)\f\fcos(ωet)∂ωe\n∂θ\f\f\u0012et/τ√t+td\n2C√\nNt2\u0013\n. (S11)\nIn an electronic spin ensemble, the spin dephasing time is typically 2 to 3 order-of-magnitude shorter than in a nuclear\nspin ensemble, while the derivative∂ωe\n∂θincreases with an increase in the magnetic field and can improve the sensitivity.\nThus it is not straightforward to analytically compare this scheme to other ones and here we discuss a few special\nconditions. Near GSLAC at B≈1000G,\f\f∂ωe\n∂θ\f\f\nmax≈(2π)4×103MHz/rad, and although T∗\n2e≪T1e, the sensitivity\nis improved compared to the nuclear spin Ramsey, with ηopt≈0.2 (mdeg/s) /√\nHz for T∗\n2e= 0.7µs, still worse than\nthe hyperfine-enhanced gyroscope.\nA sensitivity comparison of all conventional NV-based gyroscopes is shown in the main text Fig. 2e.\nAdiabatic Range for Hyperfine-Enhanced Sensing\nThe hyperfine-enhanced gyroscope protocol requires that the nuclear spin quantization axis follow the effective\nmagnetic field axis adiabatically. Because we assume that the gyroscope operates in regimes where the rotation rate is\nexpected to be significantly less than the ZFS D= 2π×2.87 GHz, the electron spin is expected to follow the eigenstate\nset by the crystalline axis of the diamond. Nevertheless, because the15N does not have a large quadrupole term, the\nenergy splitting of the nuclear spin is dependent on the magnetic field strength, and the rotation has to be sufficiently\nslow with respect to γnBsuch that the evolution of the nuclear spin remains adiabatic and the hyperfine-enhanced\nsignal can be measured. Thus, to provide a quantifiable range for the gyroscope, we derive the adiabatic criteria for\nthe nuclear spin under the rotation.6\nThe effective Hamiltonian for the nuclear spin under the rotation of rate Ω (in the NV-frame) is given by:\nH(t) =γn(Bz(t)Iz+α0Bx(t)Ix)−ΩIy (S12)\nwhere Bz(t) =Bcos(Ω t) and Bx(t) =−Bsin(Ω t). The time-dependent Hamiltonian has two orthogonal eigenstates\n|ψ±(t)⟩such that such that H(t)|ψ±(t)⟩=E±|ψ±(t)⟩. The time evolution operator is given by:\nU(t) =Texp\u0012\n−iZt\n0H(t′)δt′\u0013\n= exp\n−iX\nϕ=ψ±Zt\n0Eϕ(t′)|ϕ⟩⟨ϕ|dt′\n=X\nϕexp\u0012\n−iZt\n0Eϕ(t′)dt′\u0013\n|ϕ⟩⟨ϕ|,(S13)\nwhere Tis the time-ordering operator. Thus, we can write the time-dependent wavefunction:\n|ψ(t)⟩=X\nϕ=ψ±cϕ(t) exp\u0012\n−iZt\n0Eϕ(t′)dt′\u0013\n|ϕ⟩ (S14)\nfor coefficients cϕ. In our protocol, we initiate the nuclear spin in the |+1\n2⟩state, so let c+(0) = 1 , c−(0) = 0. From\nthe time-dependent Schrodinger equation, we obtain:\ni∂|ψ(t)⟩\n∂t=H(t)|ψ(t)⟩ (S15)\niX\nϕexp\u0012\n−iZt\n0Eϕ(t′)dt′\u0013\u0014dcϕ\ndt|ϕ⟩+cϕ∂|ϕ⟩\n∂t−iEϕcϕ|ϕ⟩\u0015\n=X\nϕcϕEϕexp\u0012\n−iZt\n0Eϕ(t′)dt′\u0013\n|ϕ⟩ (S16)\nWe multiply both sides with the state ⟨ϕ′̸=ϕ|and divide by iexp\u0010\n−iRt\n0Eϕ(t′)dt′\u0011\n:\ncϕ⟨ϕ′|∂ϕ\n∂t⟩+ exp\u0012\n−iZt\n0(Eϕ′−Eϕ)(t′)dt′\u0013\u0014dcϕ′\ndt+cϕ′⟨ϕ′|∂ϕ′\n∂t⟩\u0015\n= 0 (S17)\nthus giving an expression for the coefficient cϕ′:\ndcϕ′\ndt=−cϕ′⟨ϕ′|∂ϕ′\n∂t⟩+ exp\u0012\niZt\n0(Eϕ′−Eϕ)(t′)dt′\u0013\ncϕ⟨ϕ′|∂ϕ\n∂t⟩ (S18)\nThe adiabatic approximation requires that the state remains in its eigenstate and evolves independently of other\nstates and thus ⟨ϕ′|∂ϕ′\n∂t⟩ ≫ ⟨ ϕ′|∂ϕ\n∂t⟩. For the adiabatic part, we have the evolution\ndcϕ′\ndt=−cϕ′⟨ϕ′|∂ϕ′\n∂t⟩ (S19)\n∂ϕ′\n∂t=−iH(t)|ϕ′⟩=−iEϕ′(t)|ϕ′⟩ (S20)\ncϕ′(t) =cϕ′(0) exp\u0012\n−Zt\n0⟨ϕ′(t′)|∂ϕ′(t′)\n∂t⟩dt′\u0013\n=cϕ′(0) exp\u0012\niZt\n0Eϕ′(t′)dt′\u0013\n(S21)\nThus, in the adiabatic approximation of the evolution, the population in the states remain the same, while they\nacquire a (Berry) phase. The second term in Eq. ( S18) describes the nonadiabatic effects of the evolution and the\nnonadiabatic coupling ⟨ϕ′|∂ϕ\n∂t⟩determines the magnitude of effects of non-adiabaticity. We see that\n⟨ϕ′|\u0014∂\n∂t(H|ϕ⟩) =∂\n∂tH|ϕ⟩+H|∂ϕ\n∂t⟩=∂Eϕ\n∂t|ϕ⟩+Eϕ∂|ϕ⟩\n∂t\u0015\n⇒ ⟨ϕ′|∂H\n∂t|ϕ⟩+Eϕ′⟨ϕ′|∂ϕ\n∂t⟩=Eϕ⟨ϕ′|∂ϕ\n∂t⟩ (S22)\nand establish an adiabatic criteria for eigenstates |ϕ⟩,|ϕ′⟩=|ψ±⟩:\n\f\f\f\f⟨ϕ′|∂ϕ\n∂t⟩\f\f\f\f2\n=\f\f⟨ϕ′|∂H\n∂t|ϕ⟩\f\f2\n(Eϕ−Eϕ′)2≪E2\nϕ′ (S23)7\nExplicitly,\nH(t) =1\n2Bγn\u0014cos (Ω t) −α0sin (Ω t) +iΩ\n−α0sin (Ω t)−iΩ −cos (Ω t)\u0015\n(S24)\n∂H(t)\ndt=1\n2Bγn\u0014−Ω sin (Ω t)−α0Ω cos (Ω t)\n−α0Ω cos (Ω t) Ω sin (Ω t)\u0015\n(S25)\nSolving for the instantaneous eigenstates and eigenenergies of H(t), we obtain:\n|ψ∓(t)⟩=Bγn\"\n−−α0sin (Ω t)+iΩ\ncos (Ω t)±q\nα2\n0sin2(Ωt)+cos2(Ωt)+(Ω\nBγn)2\n1#\n(S26)\nE∓=∓Bγn\n2s\nα2\n0sin2(Ωt) + cos2(Ωt) + (Ω\nBγn)2 (S27)\nFrom Eq. (S23), the nuclear spin remains adiabatic when\nM(t) =\f\f\f⟨ψ+|∂ψ−\n∂t⟩\f\f\f2\nE2\n+≪1 (S28)\nThus, to maintain the adiabaticity of the nuclear spin during the evolution, we require M(t)≪1 during the\nrotation. By numerically computing M(t) for different values of Bover the course of θ= Ωt∈[0, π] for a chosen\nΩ = 0 .1kHz shown in Fig. S4, we see that M(t) reaches its minimum value at θ=π/2, where the effective magnetic\nfield is the largest. Calculating M(t) at a time t=t0allows us to define a cutoff ϵsuch that the system is sufficiently\nadiabatic to achieve a measurable signal of the rotation. ϵis defined as the maximal acceptable signal uncertainty\n⟨∆S⟩(as shown in main text Fig. 2c), which then sets a standard to quantify the projected range of frequencies\ndetectable by the hyperfine-enhanced gyroscope for a desired precision.\nFIG. S4. Nuclear Spin Eigenstate Deviation The eigenstate deviation M(t) as a function of the rotation angle (at a fixed\nrotation rate (2 π)0.1kHz), for various values of magnetic field strength. As Bincreases, M(t) is suppressed and the system\nbecomes adiabatic and we can perform sensing in the hyperfine-enhanced regime. As Bdecreases, M(t) increases, and the\nsystem is in the intermediate regime where a rotation signal is difficult to extract due to the non-adiabatic evolution of the\nnuclear spin. Nevertheless, for a sufficiently small B,M(t) begins to decrease again and we can sense in the inertial regime\nwhere the nuclear spin is fixed in its initial eigenstate during the rotation.\nAtθ=π/2, we can simplify\nMmin=4(Ω\nBγn)2\n(α2\n0+ (Ω\nBγn)2)2(S29)\nIf we require that the adiabatic condition for the nuclear spin satisfy Mmin< ξ2for some ξ, and assume thatΩ\nBγn≪1,\nto first order, we can approximate:\nΩ< ξα2\n0Bγn (S30)8\nThus, the maximum detectable rotation rate by the hyperfine-enhanced gyroscope is given by Ω max∝α2\n0B. Thus,\nthe enhancement factor also suppresses the non-adiabatic contribution of the rotation on the nuclear spin, so for\nlarge magnetic fields, the range for detectable rotation rates is significantly enhanced. In Fig. 2(c), we quantify the\nadiabatic range by ξ2∼1×10−5from choosing the regime where ⟨∆S⟩<0.01, thus allowing us to obtain the Ω max\nin main text Fig. 3(a)i.\nFor fast rotations, we require the nuclear spin to remain in its initial eigenstate and decoupled from the electron\nspin rotation. In this regime, as Ω ≫γeB, the effective nuclear Hamiltonian becomes time-independent H(t)≈ −ΩIy,\nandM(t)≪1, we can use similar analysis to characterize a minimum rotation rate for fast-rotation sensing Ω min\nshown in main text Fig. 3(a)ii. In the main text, we refer to M(t) as the eigenstate deviation.\nSIMULATION DETAILS\nTo simulate the dynamics of the system we can either work in the lab frame where the diamond, and thus the\nzero-field splitting and hyperfine interaction tensors, rotate; or work in the diamond frame, where the NV sees a\nrotating external magnetic field, as well as an effective field along the rotation axis due to the non-intertial frame.\nIn the following, we discuss the basic principles of both methods and use them to simulate the system dynamics\nand study the spin relaxation effect.\nLab Frame\nThe dynamics of a quantum system is described by the Hamiltonian of the system. The ground state Hamiltonian\nof the15NV center can be written as\nH=⃗S·D·⃗S+γe⃗B·⃗S+γn⃗B·⃗I+⃗S·A·⃗I (S31)\nwhere γe= 2π×2.8024 MHz/G and γn= 2π×0.4316 kHz/G are the gyromagnetic ratios of the electronic and nuclear\nspin. When the frame is chosen such that the zaxis is along the N-V orientation, the effective zero-field splitting (ZFS)\ntensorDis diagonal with the only nonzero diagonal term D= 2π×2.87 GHz along the zdirection, and the hyperfine\ntensorAincludes only diagonal terms as well with Azz= 2π×3.03 MHz and Axx=Ayy=A⊥= 2π×3.65 MHz.\nThe rotation of the diamond crystal can be described by the Euler angle ( α, β, γ ), represented by a rotation matrix\nR(α, β, γ ) =Rz(γ)Ry(β)Rz(α) with\nRz(α) =\ncos(α)−sin(α) 0\nsin(α) cos( α) 0\n0 0 1\n, R y(β) =\ncos(β) 0 sin( β)\n0 1 0\n−sin(β) 0 cos( β)\n (S32)\nsuch that\nR(α, β, γ ) =\ncos(α) cos( β) cos( γ)−sin(α) sin( γ)−sin(α) cos( β) cos( γ)−cos(α) sin( γ) sin( β) cos( γ)\ncos(α) cos( β) sin( γ) + sin( α) cos( γ) cos( α) cos( γ)−sin(α) cos( β) sin( γ) sin( β) sin( γ)\n−cos(α) sin( β) sin( α) sin( β) cos( β)\n.(S33)\nBoth of the ZFS and hyperfine interaction in the NV Hamiltonian are represented by 3-by-3 tensors, and as we\nintroduced above both of them are diagonal in the reference frame choosing N-V as the zaxis. To clarify the\ndifference between different frames, here we denote terms in the N-V frame with “ ′” such that\nD′=\n0 0 0\n0 0 0\n0 0 D\n,A′=\nAxx0 0\n0Ayy0\n0 0 Azz\n. (S34)\nThe ZFS and hyperfine terms of the Hamiltonian in the NV frame can be written as\n(S′\nx, S′\ny, S′\nz)·D′·\nS′\nx\nS′\ny\nS′\nz\n′\n,(S′\nx, S′\ny, S′\nz)·A′·\nI′\nx\nI′\ny\nI′\nz\n. (S35)\nUsing ( S′\nx, S′\ny, S′\nz)T=RT(Sx, Sy, Sz), we can get the ZFS and hyperfine tensors in the lab frame\nD=R·D′·RT,A=R·A′·RT. (S36)\nThe full Hamiltonian is then obtained by plugging these tensors into Eq. (S31).9\nNV Frame\nInstead of working in the lab frame and calculating the co-rotating ZFS and hyperfine tensors, we can work in the\nNV frame to avoid complicated tensor rotations. Instead, in this case, the external static magnetic field has a rotation\ndescribed by RTwhich is opposite to the diamond rotation, and the rotating frame transformation leads to additional\nterms in the Hamiltonian.\nTo simplify the analysis, here we consider a one-axis rotation ωalong y. In the NV frame, the system Hamiltonian\ncan be written as\nH=DS2\nz+AzzSzIz+A⊥(SxIx+SyIy)+γeB[cos(ωt)Sz−sin(ωt)Sx]+γnB[cos(ωt)Iz−sin(ωt)Ix]−ωSy−ωIy.(S37)\nThe simulation in the NV frame should give the same results as the simulation in the lab frame. In Fig. S5 we show\na simulation of the nuclear spin evolution with the same parameters as in the main text, which shows consistent\nbehavior.\n02468 10 12\nt (ms)00.20.40.60.81Signal\n| zB|n(t)|2\n|+zNV|n(t)|2\nTheory S(t)\na b\nenhanced rotation\nαΩ\nc\nNV (z)\n15N\nMagnetic /f_ield B\nNuclear spin eigenstate\nSimulated evolution15N\ntB\nFIG. S5. Gyroscope with hyperfine-enhanced sensitivity simulated in the NV frame. a. Effective dynamics of\nthe electro-nuclear system for the hyperfine-enhanced gyroscope in the NV frame. In the frame of the NV, the magnetic field\ncounter rotates to the rotation, while the nuclear spin’s rotation is enhanced from the magnetic field rotation. b. Simulation\nof the nuclear spin state evolution on the Bloch sphere. The diamond is rotated about yaxis with a rate ω= 2π×0.1 kHz,\nand the magnetic field is set to B= 50 G. c. The overlap of the final nuclear spin state with either its initial state or the state\nalong the Bfield direction.\nMaster Equation for Relaxation Time\nWhile the coherence evolution of a closed system can be calculated using the Schr¨ odinger’s equation, the evolution\nof an open system (e.g., depolarization and/or dephasing processes that exist due to the system-bath coupling) can\nbe derived using the Lindblad equation under the assumption of Markov bath,\n∂\n∂tρ(t) =−i[H(t), ρ(t)] +MX\nk=1\u0012\nLkρ(t)L†\nk−1\n2L†\nkLkρ(t)−1\n2ρ(t)L†\nkLk\u0013\n, (S38)10\nwhere Lkare Lindblad operators. To describe the spin relaxation dynamics in our system, the Lindblad operators\nareLk=√\nΓ|ms⟩⟨m′\ns|where ms, m′\ns={−1,0,+1}and the jump rate Γ = 1 /(3T1e) [29].\nIn a practical simulation, the differential equation in Eq. (S38) is converted to an integral equation and is solved\nby calculating ρ(t+δt) from ρ(t) stepwise. The simulation precision depends on the step size δt.\nDue to the large order of magnitude difference between the system energy scale ( ∼GHz) and the rotation rate ( <kHz)\nwhich is of interest in our study, numerically calculating the evolution using the Lindblad equation in Eq. (S38) is chal-\nlenging. To improve the simulation efficiency, we modify the Lindblad equation to eliminate the second approximation.\nThe detailed derivation is as follows.\nDerivation\nThe system evolution of an open quantum system can be described by a superoperator map. According to the\nKraus Representation Theorem, a concrete expression of such a map can be written in the form of\nS[ρ] =KX\nk=1MkρM†\nk, withKX\nk=1M†\nkMk=I (S39)\nwhere Mkare Kraus operators.\nWe can further write the evolution of ρfrom ttot+δtas\nρ(t+δt) =X\nkMk(δt)ρ(t)M†\nk(δt). (S40)\nTo separately account for contributions of the coherent evolution under the system Hamiltonian and the dephas-\ning/depolarization process due to system-bath coupling, we separate the set of MktoM0andMk(k > 1). The\ncoherent part is included in M0=e−iH(t)δt+δtK+O(δt2) with Khermitian, and the system-bath coupling effects\nare included in Mk=√\nδtLk+O(δt). To satisfy the Kraus sum normalization conditionPM†\nkMk=I, yielding\nM†\n0M0+X\nk>0M†\nkMk=I+δt(eiH(t)δtK+K†eiH(t)δt) +δtL†\nkLk+O(δt2) =I+δt(K+K†) +δtL†\nkLk+O(δt2),(S41)\nwe obtain K=−1\n2P\nk>0L†\nkLk. The evolution of ρis then calculated by\nρ(t+δt) =M0ρ(t)M†\n0+X\nk>0Mkρ(t)M†\nk=e−iH(t)δtρ(t)eiH(t)δt+δtX\nk>0\u0012\nLkρ(t)L†\nk−1\n2L†\nkLkρ(t)−1\n2ρ(t)L†\nkLk\u0013\n.(S42)\nWhen the jump rate is set to zero, the evolution in Eq. (S42) comes back to the form when considering only the\ncoherent evolution under the system Hamiltonian, only requiring the step size δtsmaller than the characteristic time\nscale Tof the change in H(t). And the second term linear in δtrequires that δtshould be much smaller than the\ncharacteristic jump time 1 /Γ.\nSimulation Results\nSimulations were performed for varying values of T1eto compute the system coherence time. Simulations using a\nmodified Lindblad equation in Eq. (S42) demonstrated a ∼85% improvement in time efficiency per repetition. In the\nadiabatic regime (at B= 50G and Ω = 0 .1kHz), we find the decay time of the signal yields consistently τ≈1.5T1e\nFig. S6(b), consistent with predictions from the spin fluctuator model [29]. Nevertheless, in the inertial regime under\nfast rotation in Fig. S6(a), the ratio τ/T1eis a bit more unclear due to dynamical decoupling effects from the rapid\nrotation of the electron spin; our results yield that decreases with T1e, as well as the rotation rate Ω.\nNuclear Spin Dephasing\nWe also consider the effect of inhomogenous dephasing due to static thermal noise in the adiabatic regime. We\ncompute the system coherence time under a non-Markovian process where the we impose static local field fluctuations11\n2468 10 12 14\nSetting T1 (ms)00.511.52Ratiosimulated /T1e\nratio=1.5\nba\nFIG. S6. Hyperfine-enhanced regime coherence time simulation (a) Decay of the gyroscope signal and fitted coherence\ntimes (following envelop fit S(t) =c0+c1e−t/τ) atB= 50G and Ω = (2 π)0.1kHz for various T1evalues. (b) Simulated ratios\nτ/T1efrom (a) as a function T1e.\nacross the ensemble. In particular, we simulate the evolution of the Hamiltonian averaged over Nspins where for\neach spin i:\nHi=H0+δBi·⃗S+δBi·⃗I (S43)\nwhere Biis sampled from a Gaussian distribution with mean ⟨δB⟩= 0 and we can describe the ensemble density\nmatrix\nρ(t) =Z\ne−iHitρ(0)eiHitP(δB)dδB (S44)\nwhere P(δB) =e−(δB)2/2(∆B)2\n√\n2π∆Bwhere ∆ B=p\n⟨δB2⟩is the strength (variance) of the magnetic field fluctuations. For\na typical NV ensemble sample, ∆ B∼0.1−0.5G.\nDuring the system evolution under a bias magnetic field B, the nuclear spin precesses with an effective Larmor\nfrequency ωeff=γnBeffwhere Beff=q\nB2z+α2(B2x+B2y) due to the transverse hyperfine enhancement. In the\nNV frame, the external magnetic field rotates counter-clockwise, and the nuclear spin precesses about the axis defined\nbyBx=−Bsin(Ω t)ˆx,Bz=Bcos(Ω t)ˆz. In cases where Beff≫∆B, because we assume that the static noise does not\nrotate with the diamond (and thus rotates with the bias magnetic field in the NV frame), the large energy gap of the\nnuclear spin suppresses the influence of the transverse noise δBxandδBy. Nevertheless, the δBznoise causes a pure\ndephasing process, and the ensemble of spins precess at different frequencies, resulting in a decay in the transverse\nspin components ( T∗\n2process). Under a Gaussian distribution for δB, we can characterize the decay T∗\n2=√\n2/∆B\nshown in Fig. 3b.\nThus during the gyroscope protocol, longitudinal noise results in a pure dephasing process for the nuclear spin,\nwhich reduces the effect of the ensemble nuclear precession in the measured signal. In particular, as seen in Fig. S5,\nthe simulated dynamics of the system results in a signal with oscillations largest at θ=nπ/2. In the presence of\nnoise, these oscillations are suppressed by the T∗\n2dephasing process. Thus, even though the transverse Bloch vector\ncomponents of the nuclear spin decays during the sensing protocol, a pure dephasing process does not result in a\ndecay of the overall measured signal that is used to extract the rotation rate and our protocol is robust against static\nmagnetic field fluctuations, in contrast to previous NV-gyro protocols.\nIn our simulations in main text Fig. 3(b), we assume that δBx, δBy= 0 for an ensemble of N= 1000 spins,\nand we compute T∗\n2by measuring the decay of the oscillations in the signal at θ=nπ/2, where the nuclear spin\nexperiences the same effective magnetic field strength. Varying ∆ Bz, we confirm that fluctuations in Bzresults in a T∗\n2\ndecay process. Despite the enhancement factor’s amplification of the transverse noise, because the effective magnetic\nfield is also amplified by the same magnitude, as long as Beff≫δBx,y, the dephasing process will be dominated12\nby the longitudinal noise. Considering only δBx,ynoise, we see that the dephasing of the signal is much weaker.\nFurthermore, in cases where the static fluctuations rotate with the diamond (thus remaining static in the NV frame),\nthe components of δBchange in strength with the rotation, but ultimately, a pure dephasing process dominated by\nthe longitudinal noise persists."    },    {        "title": "1612.09405v3.Derivation_of_Spin_Torques_of_the_Magnetic_System_in_Broken_Inversion_Symmetry.pdf",        "content": " \nDerivation of Spin Torques  of the Magnetic System in B roken Inversion Symmetry   \n \nH. Y . Kwon , S. P. Kang, N. J. Kim , C. K. Lee and C. Won  \nDepartment of Physics , Kyung Hee University , Seoul 02447 , Korea  \n \nWe propose  a new approach  to derive  spin torque  in system s of broken inversion \nsymmetry . It uses the concepts of asymmetric and directional  spin-spin interaction s \nto obtain  their effective field s. We applied the effective fields into the Landau -\nLifshitz  equation and obtained  spin torques . The model offer s a new and  general \napproach for spin dynamics, one that  effectively merges the Dzyaloshinskii -Moriya \ninteraction, spin transfer torque s, and spin-orbit torque  into the spin dynamics  \nequation. We discussed how our model is imposed on the spin dynamics and \ncompared our approach with the traditional discussions on spin dynamics.   I. INTRODUCTION  \nIn recent years, the magnetic properties and dynamics in the system of broken inversion symmetry have \nbecome one of the most intensely  investigated topics in magnetism studies. When the magnetic moments \nare located in the environment of broken inversion symmetry or when the directional preference is \napplied to the system, the energy  and the torque on the magnetic moments should include various \nadditional contribution  such as  the Dzyaloshinskii -Moriya interaction  (DMI), spin transfer torque (STT), \nspin-orbit torque  (SOT)  from  Rashba effect , spin hall (SH) effect , and etc.[1-12]. The broken inversion \nsymmetry may arise from the spin current as well as the crystalline structure. I n the case of DMI, the \ninversion symmetry is broken by the structure such as non -centrosymmetric crystalline structure, surface, \nor interface  with heavy metal . In the case of STT, the inversion symmetry is broken by the direction of \nspin current.  Among them, the DMI can be effectively applied in spin dynamics using its effective field , \nsimilarly as other interactions such as exchange interaction and anisotropy  are. However, the other \ncontributions STT and SOT  have been discussed with explicit torque terms not derived from effective \nfields.   \nIn this paper, we wi ll discuss a new approach to model the dynamics of spin -spin interaction s in broken \ninversion symmetry environment by introducing asymmetric and directional interaction into the spin \ndynamics equation.  We obtained the effective fields causing spin torques directly from the asymmetric \nand directional interaction and found the spin torques have the same form of spin transfer torque and spin \norbit torque.  We discuss how the above consideration affects the spin dynamics of  the system  and \ncompare how our model fits into other traditional approa ches.   \n  \nII. THEORY  \nThe spin-spin interaction  energy  in magnetic system is written as −∑ 𝑆⃗𝑖∙𝐽⃡∙𝑆⃗𝑗 <𝑖,𝑗>  ,  where 𝐽⃡  is a \ngeneral exchange interaction tensor , 𝑆⃗𝑖 and 𝑆⃗𝑗 are normalized spin vector s. When inversion symmetry exists in the magnetic system, the energy term can be  simplified as −𝐽∑ 𝑆⃗𝑖∙𝑆⃗𝑗 <𝑖,𝑗> , which is  usually  \ncalled the exchange interaction  energy, ℋex , where t he 𝐽  is a parameter related with exchange \ninteraction strength. For the exchange interaction case, the effective field, ℎ⃗⃗𝑖,𝐽(=−1\n𝜇0𝑚𝜕ℋex\n𝜕𝑆⃗𝑖), which is \ncommonly used scheme to study magnetic system, can be written by  𝐽\n𝜇0𝑚∑ 𝑆⃗𝑗𝑛𝑛\n𝑗  , where 𝑚  is the \nconstant magnitude  of the magnetic moment and 𝜇0 is the magnetic permeability of vacuum.  If a spin \nis in an environment with broken inversion symmetry,  the DMI is involved , adding the  energy term  \nℋDM=−∑ 𝐷⃗⃗⃗𝑖𝑗∙(𝑆⃗𝑖×𝑆⃗𝑗) <𝑖,𝑗>[13,14]. DMI corresponds to the off -diagonal term in the general exchange \ninteraction  tensor  𝐽⃡, and its effective field can be written by  ℎ⃗⃗𝑖,𝐷⃗⃗⃗=1\n𝜇0𝑚∑(𝑆⃗𝑗×𝐷⃗⃗⃗𝑖𝑗)𝑛𝑛\n𝑗 .  \nAdditionally, we propose that one should consider  the asymmetric  or directional preference of the spin-\nspin interaction  in the case of broken inversion symmetry;  a spin may interact more  on one side than the \nother  side, as presented in Fig. 1 . Actually, t he exchange interaction is related to 𝑆⃗𝑖∙𝑆⃗𝑗=𝑆𝑖,𝑧𝑆𝑗,𝑧+\n1\n2(𝑆𝑖+𝑆𝑗−+𝑆𝑖−𝑆𝑗+), and the later terms involving spin ladder operators are related to spin transfer from j \nto i, or from i to j. The asymmetry in the se terms will cause a directional spin flow,  and it is proportional \nto the current density and the strength of spin polarization of the current  from the broken inversion \nsymmetry. For example, assuming  there is a spin current flowing  in one direction , the interaction \nstrengths will have  a specific  directional preference , and the interaction s from left to right and from right \nto left may no longer be identical . Another example would be spins  on the edge of the system,  which  do \nnot have neighbor ing spins in the same direction. The magnetoelastic  effects , which are related to the \nlattice vibration , are another example of the directional difference  of an exchange interaction[15 ]. As \nrepresented in Fig. 1 (a), we consider ed that the exchange constants 𝐽 differ  by 𝛿𝐽, which can be positive  \nor negative  according to  the broken symmetry characteristic s.  \nThe effective field caused by the exchange interaction  with x-directional broken symmetry is given by  ℎ⃗⃗𝑖,ex=1\n𝜇0𝑚( (𝐽+𝛿𝐽\n2)𝑆⃗𝑖+𝑥+(𝐽−𝛿𝐽\n2)𝑆⃗𝑖−𝑥) \n          ≅1\n𝜇0𝑚( (𝐽+𝛿𝐽\n2)(𝑆⃗𝑖+𝜕𝑆⃗𝑖\n𝜕𝑥𝑎+1\n2𝜕2𝑆⃗𝑖\n𝜕𝑥2𝑎2)+(𝐽−𝛿𝐽\n2)(𝑆⃗𝑖−𝜕𝑆⃗𝑖\n𝜕𝑥𝑎+1\n2𝜕2𝑆⃗𝑖\n𝜕𝑥2𝑎2)) \n          =ℎ⃗⃗𝑖,𝐽+𝑎 𝛿𝐽\n𝜇0𝑚𝜕𝑆⃗𝑖\n𝜕𝑥, (1) \nwhere  𝑎 is the unit length of the system,  and the additional effective field term 𝛿ℎ⃗⃗𝑖,𝛿𝐽(=𝑎 𝛿𝐽\n𝜇0𝑚𝜕𝑆⃗𝑖\n𝜕𝑥) is \nthe leading additional effective field  of the asymmetric interaction.  Since we used normalized spin vector, \nthe magnitude of 𝑆⃗𝑖 is constant,  and 𝜕𝑆⃗𝑖\n𝜕𝑥 is perpendicular to the direction  of 𝑆⃗𝑖. Therefore, t he direction \nof the additional effective field is perpendicular to the spin direction , and the spin cannot be at rest  unless \nthe strength of 𝛿ℎ⃗⃗𝑖,𝛿𝐽 is zero.  Thus, this effective field result s in a dynamic feature of spins, such as  the \ndomain wall (DW)  motion or spin wave.  \nNaturally, we can consider the asymmetric DM I as shown in Fig. 1(b). For example, the effect would \nbe considered if a spin current additionally breaks a directional symmetry of the system with broken \ninversion symmetry due to the structural reason. The effective field due to  asymmetric  DMI is  \nℎ⃗⃗𝑖,𝐷𝑀=1\n𝜇0𝑚( 𝑆⃗𝑖+𝑥×(𝐷⃗⃗⃗+𝛿𝐷⃗⃗⃗\n2)−𝑆⃗𝑖−𝑥×(𝐷⃗⃗⃗−𝛿𝐷⃗⃗⃗\n2)) \n          ≅1\n𝜇0𝑚( (𝑆⃗𝑖+𝜕𝑆⃗𝑖\n𝜕𝑥𝑎)×(𝐷⃗⃗⃗+𝛿𝐷⃗⃗⃗\n2)−(𝑆⃗𝑖−𝜕𝑆⃗𝑖\n𝜕𝑥𝑎)×(𝐷⃗⃗⃗−𝛿𝐷⃗⃗⃗\n2)) \n          =ℎ⃗⃗𝑖,𝐷⃗⃗⃗+1\n𝜇0𝑚𝑆⃗𝑖×𝛿𝐷⃗⃗⃗, (2) \nwhere 𝛿𝐷⃗⃗⃗ is the additional vector caused by the asymmetry of the system.  The additional effective field \nfrom 𝛿𝐷⃗⃗⃗ is, therefore,  𝛿ℎ⃗⃗𝑖,𝛿𝐷⃗⃗⃗=1\n𝜇0𝑚𝑆⃗𝑖×𝛿𝐷⃗⃗⃗. The 𝛿ℎ⃗⃗𝑖,𝛿𝐷⃗⃗⃗ is perpendicular to 𝑆⃗𝑖 as 𝛿ℎ⃗⃗𝑖,𝛿𝐽 is, and it \nmakes the  spin to be dynamic.   This additional effective field  terms  will attribute two additional  torques  in the Landau -Lifshitz  (LL) \nequation . One term is related to a precession -like effect (spin rotation perpendicular to both the effective \nfield and the spin direction) and another  term is related to a damping effect (spin rotation within the plane \nof the effective  field and spin direction) . For general application s, we introduce another dimensionless \nparameter  𝑟Δ, allowing for the difference between the effects on precession  and the damping motion \ngenerated by  𝛿ℎ⃗⃗𝑖,Δ, where Δ=𝛿𝐽 or 𝛿𝐷⃗⃗⃗. The LL equation , including the additional effective field s, \nbecomes  \n𝜕𝑚⃗⃗⃗𝑖\n𝜕𝑡=−𝛾𝐿𝑚⃗⃗⃗𝑖×ℎ⃗⃗𝑖,𝛿=0−𝜆𝑚⃗⃗⃗𝑖×𝑚⃗⃗⃗𝑖×ℎ⃗⃗𝑖,𝛿=0−∑(𝑟Δ𝛾𝐿𝑚⃗⃗⃗𝑖×𝛿ℎ⃗⃗𝑖,Δ+𝜆𝑚⃗⃗⃗𝑖×𝑚⃗⃗⃗𝑖×𝛿ℎ⃗⃗𝑖,Δ)\nΔ, (3) \nwhere 𝑚⃗⃗⃗𝑖=𝑚𝑆⃗𝑖, ℎ⃗⃗𝑖,𝛿=0=ℎ⃗⃗𝑖,𝐽+ℎ⃗⃗𝑖,𝐷⃗⃗⃗, 𝛾𝐿=𝜇0𝛾, and 𝜆=𝛾𝐿𝛼\n𝑚. The 𝛾 and 𝛼 are the electron \ngyromagnetic ratio and the Gilbert damping constant respectively . The third and fourth terms of Eq. (3) \nare reduced to   \n𝑟𝛿𝐽𝛾𝐿𝑚⃗⃗⃗𝑖×𝛿ℎ⃗⃗𝑖,𝛿𝐽=𝑟𝛿𝐽𝛾𝐿𝑎 𝛿𝐽\n𝜇0𝑚2(𝑚⃗⃗⃗𝑖×𝜕𝑚⃗⃗⃗𝑖\n𝜕𝑥) \n𝜆𝑚⃗⃗⃗𝑖×𝑚⃗⃗⃗𝑖×𝛿ℎ⃗⃗𝑖,𝛿𝐽=−𝜆𝑎 𝛿𝐽\n𝜇0(𝜕𝑚⃗⃗⃗𝑖\n𝜕𝑥) \n𝑟𝛿𝐷⃗⃗⃗𝛾𝐿𝑚⃗⃗⃗𝑖×𝛿ℎ⃗⃗𝑖,𝛿𝐷⃗⃗⃗=𝑟𝛿𝐷⃗⃗⃗𝛾𝐿1\n𝜇0𝑚2(𝑚⃗⃗⃗𝑖×𝑚⃗⃗⃗𝑖×𝛿𝐷⃗⃗⃗) \n𝜆𝑚⃗⃗⃗𝑖×𝑚⃗⃗⃗𝑖×𝛿ℎ⃗⃗𝑖,𝛿𝐷⃗⃗⃗=−𝜆\n𝜇0(𝑚⃗⃗⃗𝑖×𝛿𝐷⃗⃗⃗). (4) \nIn Eq. (4), we used the properties that 𝛿ℎ⃗⃗𝑖,𝛿𝐽 is perpendicular to  𝑆⃗𝑖, because 𝑆⃗𝑖 varies while keeping \nits magn itude  and 𝑆⃗𝑖⊥𝜕𝑆⃗𝑖. \nHence, t he LL equation , including all the effective fields , becomes  𝜕𝑚⃗⃗⃗𝑖\n𝜕𝑡=−𝛾𝐿𝑚⃗⃗⃗𝑖×ℎ⃗⃗𝑖,𝛿=0−𝜆𝑚⃗⃗⃗𝑖×𝑚⃗⃗⃗𝑖×ℎ⃗⃗𝑖,𝛿=0−𝑟𝛿𝐽𝛾𝐿𝑎 𝛿𝐽\n𝜇0𝑚2(𝑚⃗⃗⃗𝑖×𝜕𝑚⃗⃗⃗𝑖\n𝜕𝑥)+𝜆𝑎 𝛿𝐽\n𝜇0(𝜕𝑚⃗⃗⃗𝑖\n𝜕𝑥)\n−𝑟𝛿𝐷⃗⃗⃗𝛾𝐿1\n𝜇0𝑚2(𝑚⃗⃗⃗𝑖×𝑚⃗⃗⃗𝑖×𝛿𝐷⃗⃗⃗)+𝜆\n𝜇0(𝑚⃗⃗⃗𝑖×𝛿𝐷⃗⃗⃗). (5) \nThe Eq. (5) can be  transformed into a form of the Landau -Lifshitz -Gilbert  (LLG)  equation  as shown in \nEq. (6),  \n𝜕𝑚⃗⃗⃗𝑖\n𝜕𝑡=−𝛾𝐺𝑚⃗⃗⃗𝑖×ℎ⃗⃗𝑖,𝛿=0+𝛼\n𝑚𝑚⃗⃗⃗𝑖×𝜕𝑚⃗⃗⃗𝑖\n𝜕𝑡−𝑢𝛿𝐽(𝜕𝑚⃗⃗⃗𝑖\n𝜕𝑥)+𝛽𝛿𝐽𝑢𝛿𝐽(𝑚⃗⃗⃗𝑖×𝜕𝑚⃗⃗⃗𝑖\n𝜕𝑥)−𝛾𝐺𝑚⃗⃗⃗𝑖×ℎ⃗⃗𝐹𝐿\n+𝛾𝐺𝑚⃗⃗⃗𝑖×𝑚⃗⃗⃗𝑖×(𝛽𝛿𝐷⃗⃗⃗ℎ⃗⃗𝐹𝐿). (6) \nThe relations among the parameters in Eq. (5) and Eq. (6) are  \n𝑢𝛿𝐽=𝛾𝐺𝑎 𝛿𝐽\n𝜇0𝑚𝛼\n1+𝛼2(𝑟𝛿𝐽−1), (6.a) \n𝛾𝐺ℎ⃗⃗𝐹𝐿=𝛾𝐺𝛿𝐷⃗⃗⃗\n𝜇0𝑚𝛼\n1+𝛼2(𝑟𝛿𝐷⃗⃗⃗−1), (6.b) \n𝛽∆=1\n𝑚𝛼(𝑟∆+𝛼2\n1−𝑟∆), (6.c) \nwhere ∆ =𝛿𝐽 or 𝛿𝐷⃗⃗⃗, 𝛾𝐺=𝛾𝐿(1+𝛼2).  \nIn Eq. ( 6), the first two terms are the precession  and damping terms in the original LLG equation . The \nthird term in Eq. ( 6), −𝑢𝛿𝐽(𝜕𝑚⃗⃗⃗⃗𝑖\n𝜕𝑥), corresponds to the adiabatic STT, where 𝑢𝛿𝐽 is the adiabatic STT \ncoefficient derived from our parameters in Eq. ( 6.a). It is the amplitude of the velocity vector 𝑢⃗⃗ used in \nusual STT studies, and it is related to spin-polarized  currents and given by  𝑢⃗⃗=𝒿𝑃𝑔 𝜇𝐵\n2𝑒𝑚𝒿̂, where 𝒿 is the \ncurrent density , 𝒿̂ is the direction of current flow,  and \nP  its spin polarization rate. The parameters 𝛿𝐽 \nand 𝑟𝛿𝐽 have the relation  𝛼(𝑟𝛿𝐽−1)𝛿𝐽=ℏ𝒿𝑃\n2𝑎𝑒 with 𝒿 and \nP . The fourth  term in Eq. ( 6), 𝛽𝛿𝐽𝑢𝛿𝐽(𝑚⃗⃗⃗𝑖×\n𝜕𝑚⃗⃗⃗⃗𝑖\n𝜕𝑥), corresponds to the non -adiabatic STT.  The dimensionless parameter 𝛽𝛿𝐽 corresponds to the non -\nadiabatic spin torque  component  𝛽 in usual  STT study[2]. The origin and characterizing parameters of 𝛽 have  been studied intensively since it has an important role in transverse and vortex domain wall \ndynamics[2,16-18]. It is known that the 𝛽 value is affected by various factors such as momentum \ntransfer[16,17 ], spin-flip scatterin g[18], and the magnetization gradient[14]. The experimentally measured 𝛽 \nhas a wide range from 0 to 18𝛼, with the  possibility of be ing negative[19-24]. \nOur model also includes the cases where  the coefficient of adiabatic torque term, 𝑢𝛿𝐽, is zero  and the \ncoefficient of non -adiabatic torque term, 𝛽𝛿𝐽𝑢𝛿𝐽, is non -zero under  the condition 𝑟𝛿𝐽=1 and 𝛿𝐽≠0. \nIn case of DW motion studies, this condition  means  that the pure non -adiabatic behavior of DW motion \nwill show when the directional preference of exchange interaction occur s for any reason without the spin \npolarized current . We guess that this phenomenon can observed in the two-dimensional magnetic \ninsulator  system , such as TmIG  film system,  with external electric field  applied in planar direction , which \ncan generate electrical polarization in the system.  \nThe last two terms of Eq.  (6), −𝛾𝐺𝑚⃗⃗⃗𝑖×ℎ⃗⃗𝐹𝐿  and 𝛾𝐺𝑚⃗⃗⃗𝑖×𝑚⃗⃗⃗𝑖×(𝛽𝛿𝐷⃗⃗⃗ℎ⃗⃗𝐹𝐿), are called field -like and \nanti-damping torque terms in usual SOT study. These are related to the torques induced by the intrinsic \nor extrinsic spin -orbit torque[10]. In our approach, if no specific situation is considered, 𝛿𝐷⃗⃗⃗ can have any \nthree -dimensional direction, so these terms can produce spin dynamics behavior such as the Rashba effect \nor spin Hall effect , depending  on the proper  choice of 𝛿𝐷⃗⃗⃗ direction. For example, t he Rashba \nHamiltonian  resulting inversion symmetry breaking in the direction perpendicular to the two -\ndimensional plane is known as ℋ𝑅=𝛼𝑅(𝜎⃗×𝑘⃗⃗)∙𝑧̂, where 𝛼𝑅 is the Rashba coupling  constant , ℏ𝑘⃗⃗ is \nthe electron’s momentum and 𝜎⃗ is the Pauli matrix vector[5]. The effective field of Rashba effect \nbecomes ℎ⃗⃗𝑅=𝛼𝑅\n𝜇0𝑚(𝑘⃗⃗×𝑧̂), and it is known that the field generates field -like torques dominantly. In this \ncase,  we can understand the Rashba effect using the 𝑟𝛿𝐷⃗⃗⃗=−𝛼2 and −𝛼 𝛿𝐷⃗⃗⃗=𝛼𝑅(𝑘⃗⃗×𝑧̂) conditions , \nwhich make  ℎ⃗⃗𝐹𝐿 become the effective field of Rashba effect.  As same way, the case that anti-damping \ntorque term is mainly considered, such as spin hall effect case, also can be interpreted using 𝑟𝛿𝐷⃗⃗⃗=1 with proper 𝛿𝐷⃗⃗⃗ choice.  If the real system should consider both field -like and anti-damping  terms, \nchoosing  a suitable value  of 𝑟𝛿𝐷⃗⃗⃗ provides the ratio between two torque terms.  \nIn addition, there is no reason that  𝛿𝐽 and 𝛿𝐷⃗⃗⃗ are restricted to be constants; they may be functions of \nexternal conditions breaking inversion symmetry in the system. For example,  our model can be expanded \nto use for the magneto -elastic effect, which is generated by lattice vibrations through  𝛿𝐽, with 𝛿𝐷⃗⃗⃗ having \nthe form of ~𝑒𝑖𝜔𝑡. In a similar way , the approach using our model can be extended to more general \ncauses of breaking i nversion symmetry in systems such as an external electric field, surface polarization, \na surficial or interfacial spin environment, spontaneous symmetry breaking, etc.  \nIn our approach , we obtained those additional terms, not from the mechanism of  physical  origins, but \nfrom the symmetry discussion. We do not confine our model to a special case, like STT or SOT, so  we \ncould find the unified form of additional torque term induced by inversion symmetry breaking as shown \nin Eq. (7).  \n𝜏⃗=−𝛾𝐺(𝛼\n1+𝛼2)∑(𝑟Δ−1)(𝛿ℎ⃗⃗𝑖,Δ−𝛽Δ𝑚⃗⃗⃗𝑖×𝛿ℎ⃗⃗𝑖,Δ)\nΔ. (7) \nThis unified form can be applied on various subjects by choosing a proper 𝛿ℎ⃗⃗𝑖,Δ. \n \nIII. SUMMARY  \n We discussed the effect of broken inversion symmetry on spin dynamics. We extend ed the LLG  \nequation with asymmetric and directional  spin-spin interaction s, including both  the exchange interaction \nand the DMI . The asymmetric and directional interaction results  in additional torque terms related to the \nSTT and SOT . In developing this theory, w e do not require an understanding of  the physical origins of \nthe parameters ; we only consider the effect s of the broken inversion symmetry. Therefore , STT and SOT \nare the natural consequence s of the broken inversion symmetry by a spin current  or electric field. Our \nmodel off ers an insightful understanding of the spin dynamics generated by broken  inversion  symmetry , and opens up a new perspective for us that STT and SOT can be interpreted as the directional preference \nof spin -spin interaction .  \n \nACKNOWLEDGMENT S \n This research was supported by a Grant from the National Research Foundation of Korea, funded by \nthe Korean Government ( 2015R1D1A1A01056971 ). \nREFERENCES  \n[1] N. Vernier, D. A. Allwood, D. Atkinson, M. D. Cooke, and R. P. Cowburn, Europhys. Lett.  65, 526 \n(2004).  \n[2] A. Thiaville, Y. Nakatani, J. Miltat,  and Y. Suzuki, Europhys. Lett. 69 (6), 990 (2005)  \n[3] O. A. Tretiakov and Ar. Abanov , Phys. Rev. Lett. 105, 157201 (2010).  \n[4] I. M. Miron, T . Moore, H . Szambolics , L. D. Buda -Prejbeanu,  S. Auffret , B. Rodmacq, S. Pizzini, J. \nVogel, M. Bonfim, A. Schuhl and G. Gaudin , Nat . Mat., 10, 419 (2011)  \n[5] E. Martinez,  S. Emori, and S. D. Beach, Appl. Phys. Lett. 103, 072406 (2013)  \n[6] V. P. Kravchuk, J. Magn. Magn. Mater. 367, 9 (2014).  \n[7] L. Liu, O. J. Lee, T. J. Gudmundsen, D. C. Ralph, and R. A. Buhrman, Phys. Rev. Lett. 109, 096602 \n(2012).  \n[8] A. Manchon and S. Zhang, Phys. Rev. B  79, 094422 (2009)  \n[9] L. Liu, C. F. Pai, Y. Li, H. W. Tseng, D. C. Ralph, and R. A. Buhrman, Science 336(6081), 555 (2012)  \n[10] A. V. Khvalkovskiy, V. Cros, D. Apalkov, V. Nikitin, M. Krounbi, K. A. Zvezdin, A. Anane, J . \nGrollier, and A. Fert, Phys. Rev.  B 87, 020402(R) (2013)  \n[11] A. V. Khvalkovskiy, K. A. Zvezdin, Ya. V. Gorbunov, V. Cros, J. Grollier, A. Fert, and A. K. \nZvezdin  Phys. Rev. Lett. 102, 067206 (2009)  \n[12] A. Manchon, arXiv:1204.4869 (2012)  \n[13] I. Dzyaloshinsky, J. Phys. Chem. Solids 4, 241 (1958).  \n  \n  \n \n[14] T. Moriya, Phys. Rev. 120, 91 (1960).  \n[15] X. X. Zhang and N. Nagaosa, arXiv:cond -mat/1608.06362.(2016)  \n[16] C. Burrowes, A. P. Mihai , D. Ravelosona, J. -V. Kim, C. Chappert, L. Vila, A. Marty,  Y. Sam son, \nF. Garcia -Sanchez, L. D. Buda -Prejbeanu, I. Tudosa, E. E. Fullerton and J. -P. Attané , Nat . Phys. 6, 17 \n(2010).  \n[17] M. Eltschka, M. Wotzel , J. Rhensius, S. Krzyk, U. Nowak, and M. Klaui , Phys. Rev. Lett. 105, \n056601 (2010).  \n[18] S.D. Pollard , L. Huang , K.S. Buchanan, D.A. Arena  and Y. Zhu , Nat. Comm. 3, 1028 (2012).  \n[19] G. Tatara and H. Kohno, Phys. Rev. Lett. 92, 086601 (2004).  \n[20] G. Tatara, H. Kohno, and J. Shibata, J. Phys. Soc. Jpn. 77, 031003 (2008).  \n[21] S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 (2004).  \n[22] I. Garate, K. Gilmore, M. D. Stiles, and A. H. MacDonald, Phys. Rev. B 79, 104416 (2009)  \n[23] S. Bohlens and D. Pfannkushe , Phys. Rev. Lett. 105, 177201 (2010)  \n[24] S. G. Je, S. C. Yoo, J. S. Kim, Y . K. Park, M. H. Park, J. Moon, B. C. Min, and S. B. Choe, Phys. \nRev. Lett. 118, 167205 (2017)  FIGURE CAPTIONS  \n \nFIG. 1. Schematic illustration of the asymmetric interaction between nearest n eighb or spins  for (a) \nexchange interaction and (b) DM I. (c)-(e) shows the examples in which the asymmetric and directional \ninteraction may occur in grid models. (c) The existence of fields or current, (b) edge or boundary of the \nsystem, or (e) non -symmetri c structure or distortion can be the reasons of asymmetric or directional \nspin-spin interaction as in (a) and (b).  \n"    },    {        "title": "0801.3391v1.W_like_states_of_N_uncoupled_spins_1_2.pdf",        "content": "arXiv:0801.3391v1  [quant-ph]  22 Jan 2008EPJ manuscript No.\n(will be inserted by the editor)\nW-like states of Nuncoupled spins1\n2\nE. Ferraro1, A. Napoli1, M. A. Jivulescu1,2, A. Messina1\n1CNISM and Dipartimento di Scienze Fisiche ed Astronomiche, Universit` a di Palermo, via Archirafi\n36, 90123 Palermo, Italy\n2Department of Mathematics, ”Politehnica” University of Ti mi¸ soara, P-ta Victoriei Nr. 2, 300006\nTimi¸ soara, Romania\nAbstract. The exact dynamics of a disordered spin star system, describ ing a cen-\ntral spin coupled to Ndistinguishable and non interacting spins1\n2, is reported.\nExploitingtheirinteractionwiththecentralsingle spins ystem,wepresentpossible\nconditional schemes for the generation of W-like states, as well as of well-defined\nangular momentum states, of the Nuncoupled spins. We provide in addition a\nway to estimate the coupling intensity between each of the Nspins and the central\none. Finally the feasibility of our procedure is briefly disc ussed.\n1 Introduction\nInteracting spin models play a central role in many physical context s providing a paradigm to\ndescribe a wide range of different systems. In condensed matter p hysics, for example, they can\nbe explored to analyze many properties of magnetic compounds. It is indeed well known that\na suitable general model of a magnet consists of Nspins coupled by exchange interaction with\narbitrary range and strength.\nThe interest toward spin systems, and more in particular toward sp in dynamics in semicon-\nductor structures, has remarkably increased in the last few year s also in connection with the\nnew emerging areas of quantum computation and information [1]-[3]. I n these contexts, spin\nmodels like Heisenberg spin chains or spin star systems, describing fo r example a single electron\nspin in a semiconductor quantum dot interacting with surrounding nu clear spins via hyperfine\ncoupling mechanisms, have been extensively studied [4]-[14].\nGenerally speaking spin models have proved to be promising candidate s for the generation\nand the control of assigned quantum correlations,as witnessed b y the numerous papers recently\nappeared in literature[15]-[18].\nIn this paper we concentrate on the possibility of manipulating at dem and the state of a\nsampleofNuncoupledspinsexploitingtheircommoninteractionwithanothersing lespincalled\ncentral system. The analysis we have developed, on the one hand p rovides possible procedures\nto guide the system toward pure states characterized by fixed co rrelation conditions, on the\nother hand suggests a way to to estimate the coupling strenght be tween each of the Nspins\nand the central one.\n2 Disordered spin star system\nHaving in mind as objective the possibility of generating fixed entangle ment conditions in a\nsystem ofNuncoupled spins1\n2, we exploit the interaction of each element of the system with\nanother spin1\n2hereafter called central one. The system we are talking about is illus trated for2 Will be inserted by the editor\nA\nFig. 1.An illustration of a disordered spin star system\nconvenience in Figure 1 and it can be described adopting the following h amiltonian model:\nH=H0+HI (1)\nwith\nH0=ωN/summationdisplay\nj=1σj\nz+ω0σA\nz, (2)\nHI=N/summationdisplay\nj=1αj(σA\n+σj\n−+σA\n−σj\n+), (3)\nThe Pauli operators labelled by the index Arefer to the central spin, while the others char-\nacterized by the index j(j= 1..N) refer to the Nspins. The coupling strength between the\nspinjand the central one is measured by the constant αjand, generally speaking, αjmay be\ndifferent from αiforj∝ne}ationslash=i. In realistic physical situations the coupling constants can change\nfor example proportionally to the distance from the central syste m [19]. We refer to the system\ndescribed by eq. (1) as disordered spin star system .\nStudying the dynamical properties of this model is for example of int erest in contexts like\nquantum dot coupled by hyperfine interaction with nuclear spins, or electronic spins bound to\nphosphorus atoms in a matrix of silica or germanio in presence of defe cts [20].\nThe symmetry properties of the hamiltonian model can be successf ully exploited in order to\nanalyze the dynamics of the system. It is easy to convince oneself t hat the component along the\nzaxes of the total angular momentum operator Sz=σA\nz\n2+1\n2/summationtextN\nj=1σj\nz=σA\nz\n2+Jzis a constant\nof motion. Then, starting from an eigenstate of Szthe system evolves in the correspondent\ninvariant Hilbert subspace. Let’s suppose in particular to prepare t he system in the following\nstate\n|ψ{i1,...ip}(0)∝an}bracketri}ht=|↑A∝an}bracketri}ht|↓...↑i1...↑ip...↓∝an}bracketri}ht, (4)\nwhere the central spin, as well as pof theNspins, namely the i1-th,i2-th,...ipth, are in their\nrespective up state |↑∝an}bracketri}htdefined asσi\nz|↑∝an}bracketri}ht=|↑∝an}bracketri}htwithi=A,1,..,N, whereas the others are in their\ndown state |↓∝an}bracketri}htwithσi\nz|↓∝an}bracketri}ht=−|↓∝an}bracketri}ht.\nWe in addition denote the state |↑A∝an}bracketri}ht|↓,↓,...,↓∝an}bracketri}ht, where all the uncoupled spins are down,\nby|ψ0(0)∝an}bracketri}ht.Will be inserted by the editor 3\nTaking into account the previous considerations we may claim that, a t time instant tthe\nstate of the system prepared in the state (4) can be written as\n|ψ{i1,...ip}(t)∝an}bracketri}ht=/summationtext\nj1<j2<...<j paj1,j2,...,jp(t)|↑A∝an}bracketri}ht|↓...↑j1...↑jp...↓∝an}bracketri}ht+\n+/summationtext\nj1<j2<...<j p+1bj1,j2,...,jp+1(t)|↓A∝an}bracketri}ht|↓...↑j1...↓...↑jp+1...↓∝an}bracketri}ht,(5)\nIn equation (5) each index ji(i= 1,..,p+1) runs from 1 to N. Thus the first term of the\nright hand side is a superposition of all the states in which the centra l spin, as well as pamong\ntheNuncoupled spins, are in their up state. The second term of equation (5) is similarly a\nlinear combination of all the states in which the central system is in its down state whereas\np+1 of theNspins are in their up state. We notice that the structure of |ψ0(t)∝an}bracketri}htcan be deduced\nfrom that of |ψ{i1,...ip}(t)∝an}bracketri}htsimply substituting to the first sum of eq. (5) the term a(t)|↑A∝an}bracketri}ht\n|↓...↓...↓∝an}bracketri}ht.Insertingeq.(5)inthetime-dependentSchr¨ odingerequationle adstothefollowing\nsystem of coupled equations for the probability amplitudes aj1,j2,...,jp(t) andbj1,j2,...,jp+1(t)\ni˙aj1,j2,...,jp(t) =∆aj1,j2,...,jp(t)+N/summationdisplay\nr=1(r/ne}ationslash=j1,..jp)αrbO({j1,...,jp}∪{r})(t) (6)\ni˙bj1,j2,...,jp+1(t) =−∆bj1,j2,...,jp+1(t)+N/summationdisplay\nr=1(r∈{j1,..jp})αraδr(j1,...,jp+1)(t) (7)\nwhere∆=ω−ω0. In eq.(6) we have introduced the operator Owhich adds the index rto the\nset of indices {j1,...,j p}arranging them in increasing order. We point out that this operator\nis well defined if rdoes not belong to the set {j1,...,j p}, otherwise the probability amplitude\nbO({j1,...,jp}∪{r})(t) would have pindices instead of ( p+1) becoming senseless. The operator δr\nappearing in turn in eq.(7) acts on the family of p+1 indices, recovering a set of pindices from\n{j1,j2,...,j p+1}by eliminating the index r. We have to mention that the above operator is\nwell defined if rbelongs to the set {j1,j2,...,j p+1}in order to assure the correct definition of\na probability amplitudes of the type aj1,j2,...,jp(t).\nThis system of differential equations can be easily decoupled when th e system is prepared\nin the state |ψ0(0)∝an}bracketri}ht. In this case eqs. (6) and (7) become\ni˙a(t) =∆a(t)+N/summationdisplay\nj=1αjbj(t) (8)\ni˙bj(t) =−∆bj(t)+αja(t), (9)\nand it is easy to show that a(t) fulfills the following Cauchy problem\na(0) = 1 (10)\n¨a(t) =−/parenleftigg\n∆2+N/summationdisplay\nr=1α2\nr/parenrightigg\na(t), (11)\nwhose solution is\na(t) = cos\n/radicaltp/radicalvertex/radicalvertex/radicalbtN/summationdisplay\nj=1α2\nj+∆2t\n−i∆/radicalig/summationtextN\nj=1α2\nj+∆2sin\n/radicaltp/radicalvertex/radicalvertex/radicalbtN/summationdisplay\nj=1α2\nj+∆2t\n(12)\nInserting eq. (12) in eq. (9) leads to a first order non homogeneou s linear differential equation\nforbj(t) which can be easily solved from the initial conditions bj(0) = 0 getting\nbj(t) =−iαj/radicalig/summationtextN\nj=1α2\nj+∆2sin\n/radicaltp/radicalvertex/radicalvertex/radicalbtN/summationdisplay\nj=1α2\nj+∆2t\n. (13)4 Will be inserted by the editor\nWhenthesystemispreparedinthestate(4),eqs.(6)and(7)may bestillmanagedinsuchaway\nto decouple the probability amplitudes of the type aj1,j2,...,jpfrom those of type bj1,j2,...,jp+1.\nWe do not present here such a procedure since in what follows we con centrate on the rich\ndynamical properties of the system evolving in accordance with |ψ0(t)∝an}bracketri}ht.\n3 Generation of W-like states\nThe results obtained in the previous Section suggest that measurin g the central spin we have\nthe possibility of guiding the system of interest, namely the Nuncoupled spins, toward a linear\ncoherent superposition of states characterized by the fact tha t only one spin is in the state |↑∝an}bracketri}ht\nwhereas the others are in the state |↓∝an}bracketri}ht.\nStarting from eq.(5) we may indeed claim that a measure of the obser vableσA\nzgives, with\nprobabilities/summationtextN\nj=1|bj(t)|2, the eigenvalue −1. In this case the Nspins are left in the normalized\nstate\n|Wgen∝an}bracketri}ht=1/radicalig/summationtextN\nj=1α2\nj+∆2(α1|↑,↓,...,↓∝an}bracketri}ht+...+αj|↓,...,↑,...,↓∝an}bracketri}ht+...+αN|↓,↓,...,↑∝an}bracketri}ht).\n(14)\nThe state given by eq.(14) looks like the well known W-state [21]\n|W∝an}bracketri}ht=1√\nN(|↑,↓,...,↓∝an}bracketri}ht+...+|↓,...,↑,...,↓∝an}bracketri}ht+...+|↓,↓,...,↑∝an}bracketri}ht). (15)\nthe only difference between the two states (14) and (15) being the weight of each component\nin the superposition. In the W-state all the states of the superpo sition appear indeed with\nthe same probability. On the other hand the state (14) we have obt ained with our procedure,\nreduces to the W-state when the central spin does not distinguish theNspins around it, that\nis whenαj≡α∀j= 1,...,N. For these reasons we call the state |Wgen∝an}bracketri}hta W-like state.\nIt is important to underline that the procedure we have discussed is a conditional one. In\nother words we may claim to generate the state |Wgen∝an}bracketri}htonly if the measurement of σA\nzgives the\neigenvalue −1. Starting from eq. (13) we can write the probability of success of our procedure\nas follows\nP=N/summationdisplay\nj=1α2\nj/summationtextN\nj=1α2\nj+∆2sin2\n/radicaltp/radicalvertex/radicalvertex/radicalbtN/summationdisplay\nj=1α2\nj+∆2t\n. (16)\nThus, appropriately choosing the time instant tat which the measure of σA\nzis performed,\nwe may generate the desired state with the highest probability that coincides with one in\ncorrespondence to ∆= 0. If indeed we measure the observable σA\nzat time instant\ntn=π(2n+1)\n2/radicalig/summationtextN\nj=1α2\nj+∆2, n= 0,1,2,... (17)\nthe probability of success becomes P=/summationtextN\nj=1α2\nj/summationtextN\nj=1α2\nj+∆2and thusP= 1 if∆= 0.\nIt is on the other hand of relevance to analyze the behaviour of suc h a probability with\nrespect to imprecisions in setting the time instant at which the measu rement of the observable\nσA\nzis performed. Let’s first of all observe that the probability of succ essP, as given by equation\n(16), is a periodic function of t, the period being T=π(/summationtextN\nj=1α2\nj+∆2)−1\n2. This circumstance\nsuggests to choice the time instant at which to perform the measur ement optimizing, as far as\npossible, our proposal on the experimental side too. To this end let ’s consider for simplicity the\ncase∆= 0 and indicate by t∗\nn=tn+δthe time instant at which the measurement if performed.\nIfδis small enough the probability of success does not appreciably get r educed. Let’s supposeWill be inserted by the editor 5\nin particular that |δ| ≤T10−1. Moreover it is also reasonable to assume that the relative error\nδ\ntnis of the order of 10−3thus implying that\ntn∼103δ≤102π/radicalig/summationtextN\nj=1α2\nj(18)\nIf our model describes, for example, hyperfine interaction of a loc alized electron with nuclei,\nthe order of magnitude of the coupling constant αjcan be estimated as 10−5eV[22], [23] in\ncorrespondenceto which tnmust belessthan orequalto π10−9sec.Thustn∼1nsiscompatible\nwith eq. (18) and it may be realized fixing n= 100. In figure 2 we plot the probability of success\nFig. 2. Probability of success to generate |Wgen/angbracketrightas function of x=δ\ntnin correspondence to ∆= 0\nand n=100.\nof our scheme given by eq. (16) versus x=δ\nt100putting∆= 0 andt=t100+δ. As foreseeable,\nthis probability of success remains of experimental interest for xup to 3·10−3. Thus we may\nconclude that the procedure is stable enough against unavoidable u ncertainties in the time\ninstant at which the measurement of σA\nzis done.\nBeforeconcluding this section it is interesting to emphasize that the coupling between the N\nspins and the central system generates entanglement between a ny two spins around the central\none. In order to estimate the amount of such an entanglement and analyze its time evolution\nwe evaluate the relative concurrence function. It is easy to prove that, starting from the initial\nstate|ψ0(0)∝an}bracketri}ht, the reduced density ρij(t) describing the system of the two spins iandjamong\ntheNuncoupled ones, can be written in the form\nρij(t) =\n0 0 0 0\n0|bi(t)|2bi(t)bj(t)∗0\n0bi(t)∗bj(t)|bj(t)|20\n0 0 0 |a(t)|2+/summationtext\nk/ne}ationslash=i,j|bk(t)|2\n(19)\nwhen expressed in the standard two-spin basis {|↑↑∝an}bracketri}ht,|↑↓∝an}bracketri}ht,|↓↑∝an}bracketri}ht,|↓↓∝an}bracketri}ht}. In equation (19) bj(t) and\na(t) are given by eqs. (13) and (12) respectively. Thus the concurre nce function Cij(t) turns\nout to be\nCij(t) =2|αi||αj|\n/summationtextN\nj=1α2\nj+∆2sin2\n/radicaltp/radicalvertex/radicalvertex/radicalbtN/summationdisplay\nj=1α2\nj+∆2t\n (20)\nAs expected, the degree of entanglement get established betwee n the two uncoupled spins iand\nj, oscillates with time t. In addition the maximum value of Cij(t) is proportional to |αi||αj|and6 Will be inserted by the editor\nwhenαk=α∀kand∆= 0 reaches the value2\nN. This means that, increasing the number of\nspins around the central one, leads to a weak and weak pair quantu m correlation in the system\nof interest.\n4 Generation of well-defined angular momentum states of the N uncoupled\nspins\nThe W-state in eq.(15) is a multipartite entangled state of the Nuncoupled spins around the\ncentral one. On the other hand it coincides with a particular superp osition of states in which\nN−1 spins have projection down while only one has projection up. This st ate can be thus\nobtained applying the collective ladder operator σ+=/summationtext\niσ(i)\n+on the state |↓,↓,...,↓∝an}bracketri}htthat is\n|W∝an}bracketri}ht ∝/summationtextN\ni=1σ(i)\n+|↓,↓,...,↓∝an}bracketri}ht. In other words the W-state is a common eigenstate of J2andJz,\nJ=1\n2/summationtextN\ni=1σibeing the collective angular momentum operator of the Nuncoupled spins, with\nJ2|W∝an}bracketri}ht=N\n2(N\n2+1)|W∝an}bracketri}ht (21)\nJz|W∝an}bracketri}ht= (−N\n2+1)|W∝an}bracketri}ht (22)\nThus|W∝an}bracketri}ht ≡ |J,M∝an}bracketri}htwhitJ=N\n2andM=−N\n2+1.\nThis observation suggests us the possibility to iterate our procedu re in order to generate all\nthe well defined angular momentum states |J=N\n2,M∝an}bracketri}htwithM=−N\n2...N\n2.\nLet’s indeed consider the spin star system in which the central spin in teracts in the same\nway with all the others uncoupled Nspins, that is αj≡α∀j. Under this condition the\nhamiltonian model (1) is invariant by permutation of an arbitrary cou ple of spins among the N.\nMoreover[σA2,H] = [J2,H] = [σA\nz+Jz,H] = [J2\nint,H] = 0,Jintbeing an intermediate angular\nmomentum resulting from the coupling of selected at will individual ang ular momentum of the\nNspins. These symmetry properties suggest to develop the dynamic s of our system exploiting\nthe coupled angular momentum basis {|J,M,ν∝an}bracketri}ht}for theNspins instead of the factorized one\npreviously used. The index νruns from 1 to νMAX(J) and allows us to distinguish between\ndifferent states of the basis characterized by the same JandM.\nIn the initial condition |ψ0(0)∝an}bracketri}httheNspins around the centralone arein the coupled angular\nmomentum state |J=N\n2,M=−N\n2,ν= 1∝an}bracketri}ht. In what follows we do not indicate anymore the\nthe indexνremaining it equal to one. Thanks to the symmetry properties of ou r system at a\ngeneric time instant twe can rewrite the state of the total system in the form\n|ψ(t)∝an}bracketri}ht=A1(t)| ↑A∝an}bracketri}ht|N\n2,−N\n2∝an}bracketri}ht+B1(t)| ↓A∝an}bracketri}ht|N\n2,−N\n2+1∝an}bracketri}ht (23)\nwhere\nA1= cos(p1αt), (24)\nB1=−isin(p1αt), (25)\nwith\np1=/radicaligg\nN\n2/parenleftbiggN\n2+1/parenrightbigg\n−/parenleftbigg\n−N\n2/parenrightbigg/parenleftbigg\n−N\n2+1/parenrightbigg\n≡√\nN. (26)\nThus as before, if we assume the central spin Ameasured in the state | ↓A∝an}bracketri}httheNuncoupled\nspins are projected onto the W-state |W∝an}bracketri}ht ≡ |N\n2,−N\n2+ 1∝an}bracketri}ht. As demonstrated in the previous\nSection the probability of success to generate the W-state coincid es with sin2(p1αt) and it\nis equal to 1 if the measurement is performed at time instants tn=π(2n+1)\n2αp1obtained from\nequation (17) putting αj≡α∀jand∆= 0. Suppose now to iterate the procedure preparing\nonce again the central spin in the up state. The new initial condition is then\n|ψ(0)∝an}bracketri}ht=| ↑A∝an}bracketri}ht|N\n2,−N\n2+1∝an}bracketri}ht (27)Will be inserted by the editor 7\nthat as easily demonstrable evolves as\n|ψ(t)∝an}bracketri}ht=A2(t)| ↑A∝an}bracketri}ht|N\n2,−N\n2+1∝an}bracketri}ht+B2(t)| ↓A∝an}bracketri}ht|N\n2,−N\n2+2∝an}bracketri}ht (28)\nwith\nA2(t) = cos(p2αt) (29)\nB2(t) =−isin(p2αt) (30)\nwherep2=/radicalig\nN\n2(N\n2+1)−(−N\n2+1)(−N\n2+2). Eq. (28) immediately implies that, measuring\nthe spinAin its down state, makes the Nspins to collapse onto the angular momentum state\n|N\n2,−N\n2+2∝an}bracketri}ht. It is possibleatthis point toconvinceoneselfthat, iterating k-times ourprocedure,\nwe generate the coupled angular momentum state |N\n2,−N\n2+k∝an}bracketri}ht. As far as the probability of\nsuccessPkthat afterkmeasurements the Nuncoupled spins are left in the state |N\n2,−N\n2+k∝an}bracketri}ht,\nit is easy to prove that it is given by\nPk=k/productdisplay\ni=1sin2(piαt) (31)\nwherepi=/radicalig\nN\n2(N\n2+1)−(−N\n2+i−1)(−N\n2+i).\nThus, if at the i−th step we have the possibility of choosing the time instant at which\nperforming the measurement act on the central spin Ain such a way that t(i)\nn=(2n+1)π\n2αpi, at\nleast in principle we may claim that the desired state |N\n2,−N\n2+k∝an}bracketri}htof theNspins are generated\nwith certainty.\n5 Estimating the order of magnitude of the coupling constant αj\nAs we are going to prove the dynamics of our system can be also succ essfully exploited in order\nto estimate the coupling intensity between each of the Ndistinguishable and non interacting\nNspins and the central one.\nLet’s indeed consider for simplicity the case ∆= 0 and, under this condition, concentrate\non the behaviour of the probability P0(t) to find the system, at a generic time instant t, in the\ninitial condition |ψ0(0)∝an}bracketri}ht. The results obtained in the previous Section immediately implies that\nsuch a probability is given by\nP0(t) =|a(t)|2= cos2\n/radicaltp/radicalvertex/radicalvertex/radicalbtN/summationdisplay\nj=1α2\njt\n. (32)\nThus, the probability of recovering the Nuncoupled spins as well as the central one in their\ninitial state is a periodic function of t, the period being in inverse relation to the quantity/summationtextN\nj=1α2\nj. If, as reasonable, we assume that all the coupling constants are of the same order,\nanalyzing the temporal behaviour of P0(t), we have the possibility of estimating the order of\nmagnitude of each αj. It is important to stress that the possibility of knowing at least the order\nof magnitude of the coupling constants αjplay a central role for example in all the cases in\nwhich the spin of an electron localized in a quantum dot is used as realiza tion of a quantum\nbit [22]. In these cases indeed the spin relaxation mechanism is mainly co nnected with its\ninteraction with bulk nuclear spins.\nLet’smoreoverobservethatthe knowledgeofthefrequencyoft hefunction P0(t) givenbyeq.\n(32) can be also exploited in order to estimate how much disordered t he spin star system model\n(1) is. Let’s suppose indeed that the Nspins of interest have been prepared in a W-like state8 Will be inserted by the editor\nfollowing the procedure previously discussed. At this point, if we mea sure the observable σj\nz\nfinding the eigenvalue +1, then the total system is projected onto the state |↓,↓,...,↑j,...,↓∝an}bracketri}ht\nin which only the j-th spin is in its upper state whereas the others are in their respect ive\ndown state. The probability of such an event, directly obtainable st arting from eq.(14), exactly\ncoincides with the quantityα2\nj/summationtextN\nj=1α2\nj.\nWe may thus conclude that knowing the probability of success to find thej-th spin in the\nup state |↑∝an}bracketri}htwhen the system of Nspins is prepared in a W-like state, allows us to give an\nestimation of the interaction strength between the j-th spin and the central one.\n6 Conclusion\nGenerally speaking the possibility of establishing on demand fixed enta nglement conditions\nin a multipartite system is an interesting objective both in its own and a lso in view of its\napplicative potentialities. In this paper in particular we have concent rated on a multipartite\nsystem composed by Nnot interacting spins1\n2. In order to guide this system toward assigned\nentangled states, we have exploited the interaction between each of theNsubsystems with\na single spin1\n2. The disordered spin star system thus obtained has been success fully used to\ngenerate W-like states as well as well-defined angular momentum sta tes of theNuncoupled\nspins.\nThe study of the exact dynamics of the disordered spin star syste m reported in this paper,\nhas provided the possibility to envisage a way to estimate at least the order of magnitude of\nthe coupling strength between each of the Nuncoupled spins and the central one. To gain\nthis information is, for example, of particular relevance when electr on spin relaxation plays\nan important role. It is indeed appropriate to remark that our syst em can be adopted to\ndescribe hyperfine interaction of a single electron spin with nuclei in q uantum dots and that\nthis interaction mechanism may be the dominant source of electron s pin relaxation.\nReferences\n1. A. Bayat, S. Bose, quant-ph/07064176 (2007)\n2. V. Kostak, G. M. Nikolopoulos, I. Jex, quant-ph/0702016 ( 2007)\n3. M. B. Plenio, S. Virmani, quant-ph/0702059 (2007)\n4. J. S. Pratt, Phys. Rev. A 69, 042312 (2004)\n5. G.L. Kamta, A. Starace, Phys. Rev. Lett. 88, 10 (2002)\n6. S. Hamieh, M.I. Katsnelson, Phys. Rev. A 72, 032316 (2005)\n7. P. Karbach, J. Stolze, Phys. Rev. A 72, 030301 (2005)\n8. D. Bruß, N. Datta, A. Ekert, L. C. Kwek, C. Machiavello, Phy s. Rev. A 72, 014301 (2005)\n9. V. Subrahmanyam, Phys. Rev. A 69, 034304 (2004)\n10. F. Pan, X. Guan, N. Ma, W.-J. Han, J. P. Draayerb, quant-ph /0702030 (2007)\n11. A. Hutton, S. Bose, Phys. Rev. Lett. 93, 237205 (2004)\n12. X.-Z. Yuan,H.-S. Goan, K.-D. Zhu, Phys. Rev. B 75, 045331 (2007)\n13. Y. Hamdouni, M. Fannes, F. Petruccione, Phys. Rev. B 73, 245323 (2006)\n14. Jan Fischer, H-P Breuer quant-ph 0708.0410v1 (2007)\n15. F. Verstraete, M. Popp, J.I. Cirac, Phys. Rev. Lett. 92, 027901 (2004)\n16. M.C. Arnesen, S. Bose, V. Vedral, Phys. Rev. Lett. 87, 017901 (2001)\n17. J.S. Pratt, Phys. Rev. Lett. 93, 237205 (2004)\n18. D. Binosi, G. De Chiara, S. Montangero, A. Recati, cond-m ath 0707.0266v2 (2007)\n19. D.D. Braktavatsala Rao, V. Ravishankar, V. Subrahmanya m, Phys. Rev. A 74, 022301 (2006)\n20. J. Schliemann, A. V. Khaetskii, D. Loss, Phys. Rev. B 66, 245303 (2002)\n21. R. Horodecki, P. Horodecki, M. Horodecki, K. Horodecki, quant-ph/0702225 (2007)\n22. I.A. Merkulov, A.L. Efros, M. Rosen, Phys. Rev. B 65, 205309 (2002)\n23. D.C. Mattis, The theory of magnetism made simple , World Scientific (2006)"    },    {        "title": "2203.06401v2.Predicting_Phonon_Induced_Spin_Decoherence_from_First_Principles__Colossal_Spin_Renormalization_in_Condensed_Matter.pdf",        "content": "Predicting Phonon-Induced Spin Decoherence from First-Principles:\nColossal Spin Renormalization in Condensed Matter\nJinsoo Park,1Jin-Jian Zhou,2Yao Luo,1and Marco Bernardi1,\u0003\n1Department of Applied Physics and Materials Science,\nCalifornia Institute of Technology, Pasadena, CA 91125, USA.\n2School of Physics, Beijing Institute of Technology, Beijing 100081, China.\nDeveloping a microscopic understanding of spin decoherence is essential to advancing quantum\ntechnologies. Electron spin decoherence due to atomic vibrations (phonons) plays a special role as\nit sets an intrinsic limit to the performance of spin-based quantum devices. Two main sources of\nphonon-induced spin decoherence { the Elliott-Yafet (EY) and Dyakonov-Perel (DP) mechanisms {\nhave distinct physical origins and theoretical treatments. Here we show calculations that unify their\nmodeling and enable accurate predictions of spin relaxation and precession in semiconductors. We\ncompute the phonon-dressed vertex of the spin-spin correlation function with a treatment analogous\nto the calculation of the anomalous electron magnetic moment in QED. We \fnd that the vertex\ncorrection provides a giant renormalization of the electron spin dynamics in solids, greater by many\norders of magnitude than the corresponding correction from photons in vacuum. Our work demon-\nstrates a general approach for quantitative analysis of spin decoherence in materials, advancing the\nquest for spin-based quantum technologies.\nSpin decoherence from phonons is a pressing question\nin quantum technology \u0000it governs spin transport [1{6]\nand limits the manipulation of quantum information [7{\n13] and the realization of reliable quantum devices [14{\n16]. Previous work has identi\fed two key sources of\nphonon-induced spin decoherence in the presence of spin-\norbit coupling (SOC) \u0000the Elliott-Yafet (EY) mecha-\nnism [17, 18], whereby electron-phonon ( e-ph) collisions\nchange the spin direction, and the Dyakonov-Perel (DP)\nmechanism [19] originating from spin precession between\ne-ph collisions. Historically, these two mechanisms have\nbeen described with distinct theoretical models [17{21],\nbut signi\fcant e\u000borts have been made to unify them, for\nexample using real-time evolution of spin ensembles [22{\n24] or analyzing quasiparticle broadening in model sys-\ntems [25{27].\nHowever, formulating a theory that encompasses both\nthe EY and DP mechanisms, and developing corre-\nsponding quantitative calculations of spin decoherence\nin real materials, are still outstanding challenges. Many-\nbody approaches combined with density functional the-\nory (DFT) and related \frst-principles calculations are\nparticularly promising to tackle this problem. These ab\ninitio methods have become a gold standard for calcu-\nlations ofe-ph interactions and transport phenomena in\nsolids [28{36]. Recent work has extended this framework\nto compute spin-\rip processes due to e-ph interactions,\nleading to predictions of EY spin decoherence within the\nspin relaxation time approximation (sRTA) [37]. It is\nwidely accepted that the sRTA neglects spin precession,\nand thus a di\u000berent formalism is needed to capture the\nDP mechanism [20, 24].\nInspired by the work of Kim et al. [38], which rigor-\nously proved that the Boltzmann equation is equivalent\nto the ladder vertex correction to the conductivity, we\nask if a similar many-body approach can be used to studyspin dynamics. The development of this framework, and\nof corresponding \frst-principles calculations, would pro-\nvide a viable tool to study phonon-induced spin decoher-\nence, mimicking the progress of \frst-principles studies of\ncharge transport [28{36]. In turn, accurate predictions of\nspin decoherence would advance both condensed matter\ntheory and spin-based quantum technology.\nHere we present a many-body theory of spin relaxation\nand develop precise ab initio calculations of phonon-\ninduced spin decoherence in semiconductors. Our ap-\nproach calculates the e-ph vertex corrections to the spin\nsusceptibility, with an accurate account of electronic and\nvibrational states, SOC, and e-ph interactions. We com-\npute the spin relaxation times (SRTs) of electron and hole\ncarriers in Si and GaAs \u0000two key candidates for spin-\nbased quantum computing \u0000and in monolayer WSe 2, a\n2D semiconductor with strong SOC. Our predicted SRTs\nare in excellent agreement with experiments over a wide\ntemperature range. We demonstrate that our formalism\ncan calculate both spin relaxation and spin precession,\nand capture EY and DP decoherence on equal footing;\nwe contrast these results with the sRTA, which lacks DP\ndecoherence and gives unphysical SRTs near the band\ngap. Our analysis shows that the e-ph interactions lead\nto a colossal renormalization of the electron spin dynam-\nics in solids, signi\fcantly modifying the SRTs and spin\nprecession rates (SPRs). The theory and computational\nmethod developed in this work pave the way for a deeper\nunderstanding of electron spin decoherence, with broad\nimplications for quantum materials and devices.\nTo describe phonon-induced spin decoherence, we con-\nsider the Kubo formula for the spin-spin correlation func-\ntion [39], and include the ladder vertex correction [38]\nfrome-ph interactions (see Fig. 1(a)). We derive a Bethe-\nSalpeter equation for the phonon-dressed spin vertex (in\nshort, spin-phonon BSE), as discussed in the Supplemen-arXiv:2203.06401v2  [cond-mat.mtrl-sci]  24 Aug 20222\nFIG. 1. Feynman diagrams for spin decoherence. (a) Bub-\nble diagram for the spin-spin correlation function including\nthe vertex correction. (b) Bethe-Salpeter equation for the\nphonon-dressed spin vertex in the ladder approximation.\ntal Material [40] and in the companion paper [41]. Our\nspin-phonon BSE is shown diagrammatically in Fig. 1(b),\nand can be written as:\ns\u0003k(\")=sk+1\nVX\n\u0017q\u0006gy\n\u0017kqh\nGAs\u0003GRi\nk+q;\n\"\u0006!\u0017qg\u0017kqF\u0006(T)\n(1)\nwhere all bolded quantities are matrices in Bloch basis.\nAbove, s\u0003k(\") =snn0k\u0003nn0k(\") is the phonon-dressed\nspin vertex, \u0003\u000b\nnn0k(\") is the vertex correction at energy \"\nfor the Cartesian direction \u000b, and snn0k=hn0kj~\n2^\u001bjnkiis\nthe bare spin vertex; GR=Aare the retarded/advanced in-\nteracting Green's functions [39], Vis the system volume,\nF\u0006(T) is a thermal occupation factor at temperature T,\nand [g\u0017kq]nm=gnm\u0017(k;q) aree-ph matrix elements [29].\nThe vertex correction \u0003 governs the spin dynamics by\nrenormalizing the microscopic SRTs and SPRs [40, 41].\nThe macroscopic SRTs are obtained as the thermal aver-\nage\n\u001c(s)\n\u000b\f=P\nnks\u000b\nnnks\f\nnnk\u001ce-ph\nnk\u0003\f\nnnk(\"nk)(\u0000dfnk\nd\")\nP\nnks\u000b\nnnks\f\nnnk\u0000\n\u0000dfnk\nd\"\u0001; (2)\nwhere\u001ce-ph\nnkaree-ph collision times [28, 39]. For \u000b=\f\nalong the external magnetic \feld, Eq. (2) gives the lon-\ngitudinal SRT, usually called T1, along the direction \u000b,\nwhile for a perpendicular magnetic \feld one obtains the\ntransverse SRT, T2(not computed here) [42]. The renor-\nmalized microscopic SRTs ( \u001c\u000b\nnn0k) and SPRs ( !\u000b\nnn0k),\nwhich are matrices in Bloch basis, are computed from\nthe vertex correction using\n1\n1\n\u001c\u000b\nnn 0k(\")+i!\u000b\nnn0k(\")\u0011\u0003\u000b\nnn0k(\")\ni(\u0006R\nnk\u0000\u0006A\nn0k) +i(\"nk\u0000\"n0k);\n(3)\nwith \u0006A=Rthe advanced/retarded e-ph self-energy [28].\nThe diagonal components with n=n0give the renormal-\nized microscopic SRTs, \u001c\f\nnnk=\u001ce-ph\nnk\u0003\f\nnnk(\"nk) entering\nEq. (2). We implement and solve Eqs. (1)-(3) in our\nperturbo code [29] (see Supplemental Material [40]).\nThe ground state, band structures, and phonon disper-\nsions are obtained using Quantum ESPRESSO [43].\nWe employ perturbo [29] to compute and interpolate\nthee-ph matrix elements and spin matrices, using amethod described in Ref. [37], starting from spinor\nWannier functions from the wannier90 code [44]. We\nmodel all materials in the intrinsic (i.e., undoped) limit,\nand accordingly compare our results with experiments\ncarried out on undoped samples. Additional details are\nprovided in Supplemental Material [40].\nUsing this formalism, in Fig. 2 we compute the\nmacroscopic SRTs in Eq. (2) as a function of tem-\nperature for Si, GaAs, and monolayer WSe 2. In Si,\na centrosymmetric material where spin decoherence\nis governed by the EY mechanism, the results are in\nexcellent agreement with experiments [5, 45, 46] in the\n100\u0000300 K temperature range. For example, the SRT\ncomputed at 300 K is 6.1 ns, in remarkable agreement\nwith the 6.0 ns value measured in Ref. [46]. Due to the\ndominant EY mechanism, in this case the sRTA, which\nneglects spin precession, also gives accurate SRTs.\nIn GaAs, the SOC induces a small ( \u00181 meV) split-\nting in the conduction band, so spin relaxation is\ndominated by the DP mechanism [21]. Figure 2(b)\nshows our calculated SRTs for electrons in GaAs as\nfunction of temperature; the excellent agreement with\nexperiments [47{50] is a strong evidence that the spin-\nphonon BSE describes correctly the DP mechanism.\nBy contrast, the sRTA, which captures only the EY\nmechanism, clearly fails in GaAs, predicting SRTs an\norder of magnitude greater than experiments.\nOur spin-phonon BSE achieves a similar accuracy for\ncalculations on hole carriers. In Fig. 2(c), we compute\nthe SRTs for hole spins in monolayer WSe 2, obtaining\nexcellent agreement with all available experimental\nresults between 20 \u000090 K [51{54]. Note that the valence\nband of WSe 2has a large (\u00180.4 eV) splitting due to\nSOC, leading to a precession rate far greater than the\nholee-ph collision rates; in this strong precession regime,\nthe spin dynamics is controlled by the diagonal part\nof the spin vertex and the DP mechanism becomes\nirrelevant, so EY spin decoherence dominates the SRTs.\nConversely, for heavy holes in GaAs (see Fig. 2(d))\nboth EY and DP spin decoherence are important. The\nagreement with experiment is noteworthy in this regime\nwhere both mechanisms are relevant: our computed\nSRT for holes in GaAs at 300 K is 200 fs, versus a 110 fs\nvalue measured by Hilton et al. [55].\nA key distinction between the EY and DP mechanisms\nis their dependence on the e-ph collision time, \u001ce-ph\nnkin\nEq. (2): the SRT is proportional to \u001ce-ph\nnkfor EY, and\ninversely proportional to \u001ce-ph\nnkfor DP. Our spin-phonon\nBSE can capture both of these trends, as we show in\nFig. 2(e) by arti\fcially increasing \u001ce-ph\nnk(by multiplying\nthee-ph matrix elements through a constant) and\nrecomputing the SRTs at 300 K for all four cases. In Si\nand WSe 2, where EY spin decoherence is dominant, we\n\fnd that the recomputed SRTs are directly proportional\nto thee-ph collision time, consistent with the EY3\n1002 003 004 00101001\n002 003 004 001010010001\n002 02 003 000.11101001,0001\n002 003 004 005 00.1110Spin relaxation time (ns) Spin-phonon BSE \nsRTA \nAppelbaum et al. \nLepine \nLancaster et al.S\ni (el)(a)S\npin relaxation time (ps) \nOertel et al. \nKimel et al. \nDzhioev et al. \nLai et al.G\naAs (el)(b)T\nemperature (K)Spin relaxation time (ns) Ersfeld et al. \nLi et al. \nYan et al. \nGoryca et al.W\nSe2 (h)(c)S\npin relaxation time (ps) Hilton et al.T\nemperature (K)GaAs (h)(d)0\n.80 .91 .01 .11 .20.80.91.01.11.2\n \nGaAs (el) \nGaAs (h) \nSilicon \nWSe2Spin relaxation time (τs)(e)e\n-ph collision time S OC band splitting0.80 .91 .01 .11 .20.70.80.91.01.11.21.3(f)\nFIG. 2. Spin relaxation times. (a)-(d) Computed spin relaxation times as a function of temperature, for (a) electrons in Si, (b)\nelectrons in GaAs, (c) holes in monolayer WSe 2, and (d) holes in GaAs. Results obtained from the spin-phonon BSE (black\nsolid line) are compared with sRTA calculations (red dashed line). Experimental results from Refs. [5, 45{54] are shown for\ncomparison. (e)-(f) The SRTs at room temperature for these four cases are recomputed by arti\fcially varying (e) the e-ph\ncollision time and (f) the SOC band splitting entering the spin-phonon BSE. In all cases, the axes are referenced to the real\nsystem values. The conventional DP spin relaxation trend (black dotted line) is also shown for comparison.\nmechanism [17, 18]. Conversely, for electron spins in\nGaAs, the SRTs are nearly inversely proportional to\nthee-ph collision time (see Fig. 2(e)), in agreement\nwith the DP mechanism [19]. (Note that the computed\ntrend slightly deviates from the conventional DP inverse\nproportionality because EY decoherence, although\nweak, is still present.) For hole spins in GaAs, the\nrecomputed SRTs exhibit a trend intermediate between\npure EY and DP, further supporting our conclusion\nthat both mechanisms are important for hole spins in\nGaAs [24, 56].\nSpin precession in the DP mechanism is induced\nby the SOC \feld, which is proportional to the band\nsplitting for each electronic state. To examine the role of\nDP spin decoherence, we arti\fcially vary the SOC band\nsplitting \u0001Eand for each new value we recompute the\nSRTs (see Fig. 2(f)). For WSe 2, varying the SOC band\nsplitting has no e\u000bect on the SRTs, showing that spin\ndecoherence is controlled by the EY mechanism. For\nelectrons in GaAs, the SRTs are highly sensitive to the\nSOC splitting, a clear evidence that our formalism can\ncapture the dominant DP mechanism. This dependence\nis weaker than in the conventional trend for pure DP,\n\u001c(s)/1=(\u0001E)2, due to the coexistence of EY deco-\nherence. For hole carriers in GaAs, the SRTs are less\nsensitive to the SOC splitting than for electrons, as the\ndecoherence originates from a balanced combination of\nboth EY and DP mechanisms. This analysis also shows\nthat band structure calculations accurately describing\nthe SOC band splitting are essential to predict spinprecession and DP decoherence.\nThe phonon-induced renormalization greatly modi\fes\nthe microscopic spin dynamics. Figure 3(a) compares\nsRTA and spin-phonon BSE calculations of the micro-\nscopic electron SRTs, \u001c(s)\nnk=\u001ce-ph\nnk\u0003z\nnnk(\"nk) de\fned\nbelow Eq. (3), in Si at 300 K for energies near the con-\nduction band minimum. The sRTA results are strongly\nenergy dependent, with an unphysical divergence at low\nenergy. By contrast, the results from the spin-phonon\nBSE are nearly energy independent. The vertex correc-\ntion makes spins with similar energy relax on the same\ntime scale\u0000a constant value of 6.1 ns nearly equal to\nthe macroscopic SRT \u0000and overcomes the limitations\nof the sRTA. A closer examination of the SRTs from the\nspin-phonon BSE (see Fig. 3(b)) reveals an oscillatory\npattern with a period of !O\u001960 meV, the energy\nof an optical phonon with strong e-ph coupling; this\npattern disappears when optical phonons are neglected.\nThis oscillation is a manifestation of the self-consistency\nof the spin-phonon BSE and its ability to capture\nstrong coupling e\u000bects beyond lowest-order perturbation\ntheory. We observe the same energy dependence and\nSRT oscillations due to optical phonons for hole spins in\nWSe 2.\nFigure 3(c) shows the computed vertex correction\n\u0003z\nnnk(\"nk) as a function of energy in Si. The vertex cor-\nrection from the e-ph interactions is colossal \u0000relative\nto the bare spin, it is of order \u0003 \u00001\u0019105, and thus eight\norders of magnitude greater than the corresponding\nvertex correction due to photons in vacuum [57, 58]4\nFIG. 3. Microscopic spin decoherence. (a) Microscopic electron SRTs in Si as a function of conduction band energy, computed\nwith the spin-phonon BSE (black) and sRTA (red). (b) Zoom-in of the spin-phonon BSE results in (a). (c) Vertex corrections\n\u0003nnkin Si (black dots) compared with the inverse e-ph collision times (green crosses). (d) Microscopic electron SRTs in GaAs\nfrom the spin-phonon BSE, shown as a function of conduction band energy and overlaid with a color map of the expectation\nvalue ofSzfor each electronic state; the sRTA results (red) are given for comparison. (e) Microscopic o\u000b-diagonal SRTs, \u001cnn0k\nin Eq. (3), overlaid with a color map of the SOC band splitting \u0001 E. (f) Renormalized electron SPRs in GaAs, !nn0kin Eq. (3),\nplotted as a function of SOC band splitting and overlaid with a color map of the conduction band energy; the bare electron\nSPRs (black dashed line) are given for comparison. All results are computed at 300 K, and the zero of the energy axis is the\nconduction band minimum.\n(with value \u0003\u00001\u00191:16\u000110\u00003, which corrects the\nelectron magnetic moment). The energy dependence of\nthe vertex correction is nearly identical to that of the\ninversee-ph collision times, thus explaining the origin\nof the constant trend with energy of the microscopic\nSRTs. We \fnd large vertex correction values (102\u0000105)\nalso in GaAs and WSe 2. These giant values account\nfor the large di\u000berences between e-ph collision times\n(femtoseconds) and SRTs (nanoseconds) in condensed\nmatter, and are key to accurately predicting long spin\ncoherence times of interest in quantum technologies.\nIn GaAs, due to the Dresselhaus SOC band splitting,\nthe bare spin vertex snn0kacquires large o\u000b-diagonal\n(n6=n0) components that precess in the e\u000bective SOC\nmagnetic \feld with a bare SPR of\"nk\u0000\"n0k. While the\nmacroscopic SRTs in Eq. (2) are determined only by\nthe band diagonal components snnk, the spin-phonon\nBSE couples the diagonal and o\u000b-diagonal components\nvia Eq. (1), so spin precession modi\fes the SRTs. The\nmicroscopic SRTs for electrons in GaAs (see Fig. 3(d))\nexhibit trends similar to Si \u0000the renormalized SRTs\nare nearly energy independent near the band edge, in\ncontrast with the rapidly varying SRTs predicted by the\nsRTA; an oscillating pattern is evident with period equalto the 30 meV longitudinal optical (LO) phonon energy,\na signature of strong coupling with LO phonons [32].\nYet, due to the spin precession, we also observe unique\ntrends not found in Si. The SRTs decrease at higher\nenergies due to the increasing spin precession (the SOC\nband splitting increases with energy), a manifestation of\nDP spin decoherence. In addition, the SRTs are strongly\nstate dependent as states with a smaller spin component\nalong the quantization axis, shown with lighter colors in\nFig. 3(d), are subject to stronger precession.\nThe relaxation of the o\u000b-diagonal spin components,\nquanti\fed by the o\u000b-diagonal SRTs \u001cnn0kin Eq. (3),\nreveals additional signatures of the DP mechanism.\nFigure 3(e) shows these o\u000b-diagonal electron SRTs for\nGaAs and highlights their correlation with the SOC\nband splitting. When the band splitting is small (black),\nprecession is negligible and the SRTs are identical to\nthe diagonal SRTs in Fig. 3(d). However, for increas-\ning values of the band splitting (lighter colors), spin\nprecession signi\fcantly enhances the SRTs. These\nintriguing microscopic phenomena are encoded in the\nvertex correction \u0003 in Eq. (3), which suppresses the real\npart 1=\u001cnn0kin the denominator, thus slowing down spin\nrelaxation. Similarly, the vertex correction signi\fcantly5\nslows down spin precession, as shown in Fig. 3(f) for\nGaAs. Electrons with a bare SPR of 1 meV drop to a\n\u001810\u00002meV precession rate after renormalization due\nto phonons. These renormalized SPRs are strongly\nenergy dependent, with higher electron energies leading\nto faster precession for spins with the same bare SPRs.\nThis microscopic dynamics reveals the rich interplay\nbetween spin relaxation and precession in materials.\nIn conclusion, our \fndings highlight the dramatic\ne\u000bects of phonon-induced renormalization on electron\nspins in solids. Our spin-phonon BSE can capture\nrenormalized spin dynamics beyond relaxation, shedding\nlight on the interplay between the EY and DP spin\ndecoherence mechanisms, and describing their diverse\nphysics on the same footing. This formalism reveals\nthat the long intrinsic spin coherence times in condensed\nmatter are due to the colossal vertex correction from e-\nph interactions. Our computationally a\u000bordable method\nenables precise predictions of spin decoherence, with\nbroad implications for spin-based quantum technologies\nand for advancing microscopic understanding of spin\ndynamics in solids.\nThis work was supported by the National Science Foun-\ndation under Grants No. DMR-1750613 and QII-TAQS\n1936350, which provided for method development, and\nGrant No. OAC-2209262, which provided for code devel-\nopment. J.P. acknowledges support by the Korea Foun-\ndation for Advanced Studies. This research used re-\nsources of the National Energy Research Scienti\fc Com-\nputing Center (NERSC), a U.S. Department of Energy\nO\u000ece of Science User Facility located at Lawrence Berke-\nley National Laboratory, operated under Contract No.\nDE-AC02-05CH11231.\n\u0003Corresponding author: bmarco@caltech.edu\n[1] G. Balasubramanian, I. Y. Chan, R. Kolesov, M. Al-\nHmoud, J. Tisler, C. Shin, C. Kim, A. Wojcik, P. R. Hem-\nmer, A. Krueger, T. Hanke, A. Leitenstorfer, R. Brats-\nchitsch, F. Jelezko, and J. Wrachtrup, Nature 455, 648\n(2008).\n[2] R. Jansen, Nat. Mater. 11, 400 (2012).\n[3] I. \u0014Zuti\u0013 c, J. Fabian, and S. Das Sarma, Rev. Mod. Phys.\n76, 323 (2004).\n[4] W. Han, R. K. Kawakami, M. Gmitra, and J. Fabian,\nNat. Nanotechnol. 9, 794 (2014).\n[5] I. Appelbaum, B. Huang, and D. J. Monsma, Nature 447,\n295 (2007).\n[6] J. \u0014Zelezn\u0013 y, P. Wadley, K. Olejn\u0013 \u0010k, A. Ho\u000bmann, and\nH. Ohno, Nat. Phys. 14, 220 (2018).\n[7] M. Veldhorst, C. H. Yang, J. C. C. Hwang, W. Huang,\nJ. P. Dehollain, J. T. Muhonen, S. Simmons, A. Laucht,\nF. E. Hudson, K. M. Itoh, A. Morello, and A. S. Dzurak,\nNature 526, 410 (2015).\n[8] X. Mi, M. Benito, S. Putz, D. M. Zajac, J. M. Taylor,\nG. Burkard, and J. R. Petta, Nature 555, 599 (2018).[9] S. J. Whiteley, G. Wolfowicz, C. P. Anderson,\nA. Bourassa, H. Ma, M. Ye, G. Koolstra, K. J. Satzinger,\nM. V. Holt, F. J. Heremans, A. N. Cleland, D. I. Schus-\nter, G. Galli, and D. D. Awschalom, Nat. Phys. 15, 490\n(2019).\n[10] G. Wolfowicz, F. J. Heremans, C. P. Anderson, S. Kanai,\nH. Seo, A. Gali, G. Galli, and D. D. Awschalom, Nat.\nRev. Mater. 6, 906 (2021).\n[11] A. Noiri, K. Takeda, T. Nakajima, T. Kobayashi, A. Sam-\nmak, G. Scappucci, and S. Tarucha, Nature 601, 338\n(2022).\n[12] A. Bourassa, C. P. Anderson, K. C. Miao, M. Onizhuk,\nH. Ma, A. L. Crook, H. Abe, J. Ul-Hassan, T. Ohshima,\nN. T. Son, G. Galli, and D. D. Awschalom, Nat. Mater.\n19, 1319 (2020).\n[13] L. Petit, H. G. J. Eenink, M. Russ, W. I. L. Lawrie,\nN. W. Hendrickx, S. G. J. Philips, J. S. Clarke, L. M. K.\nVandersypen, and M. Veldhorst, Nature 580, 355 (2020).\n[14] C. L. Degen, F. Reinhard, and P. Cappellaro, Rev. Mod.\nPhys. 89, 035002 (2017).\n[15] D. Vasyukov, Y. Anahory, L. Embon, D. Halbertal,\nJ. Cuppens, L. Neeman, A. Finkler, Y. Segev, Y. Mya-\nsoedov, M. L. Rappaport, M. E. Huber, and E. Zeldov,\nNat. Nanotechnol. 8, 639 (2013).\n[16] J. F. Barry, J. M. Schloss, E. Bauch, M. J. Turner, C. A.\nHart, L. M. Pham, and R. L. Walsworth, Rev. Mod.\nPhys. 92, 015004 (2020).\n[17] R. J. Elliott, Phys. Rev. 96, 266 (1954).\n[18] Y. Yafet, in Solid State Physics , Vol. 14, edited by\nF. Seitz and D. Turnbull (Academic Press, New York,\n1963) pp. 1{98.\n[19] M. Dyakonov and V. Perel, Sov. Phys. Solid State USSR\n13, 3023 (1972).\n[20] A. Baral, S. Vollmar, S. Kaltenborn, and H. C. Schneider,\nNew J. Phys. 18, 023012 (2016).\n[21] M. D. Mower, G. Vignale, and I. V. Tokatly, Phys. Rev.\nB83, 155205 (2011).\n[22] M. W. Wu, J. H. Jiang, and M. Q. Weng, Phys. Rep.\n493, 61 (2010).\n[23] J. Xu, A. Habib, R. Sundararaman, and Y. Ping, Phys.\nRev. B 104, 184418 (2021).\n[24] K. Shen and M. W. Wu, Phys. Rev. B 82, 115205 (2010).\n[25] F. Simon, B. D\u0013 ora, F. Mur\u0013 anyi, A. J\u0013 anossy, S. Garaj,\nL. Forr\u0013 o, S. Bud'ko, C. Petrovic, and P. C. Can\feld,\nPhys. Rev. Lett. 101, 177003 (2008).\n[26] P. Boross, B. D\u0013 ora, A. Kiss, and F. Simon, Sci. Rep. 3,\n3233 (2013).\n[27] L. Szolnoki, B. D\u0013 ora, A. Kiss, J. Fabian, and F. Simon,\nPhys. Rev. B 96, 245123 (2017).\n[28] M. Bernardi, Eur. Phys. J. B 89, 239 (2016).\n[29] J.-J. Zhou, J. Park, I.-T. Lu, I. Maliyov, X. Tong,\nand M. Bernardi, Comput. Phys. Commun. 264, 107970\n(2021).\n[30] T. Sohier, M. Calandra, C.-H. Park, N. Bonini,\nN. Marzari, and F. Mauri, Phys. Rev. B 90, 125414\n(2014).\n[31] W. Li, Phys. Rev. B 92, 075405 (2015).\n[32] J.-J. Zhou and M. Bernardi, Phys. Rev. B 94, 201201(R)\n(2016).\n[33] S. Ponc\u0013 e, E. R. Margine, and F. Giustino, Phys. Rev. B\n97, 121201 (2018).\n[34] J.-J. Zhou, O. Hellman, and M. Bernardi, Phys. Rev.\nLett. 121, 226603 (2018).\n[35] V. A. Jhalani, J.-J. Zhou, J. Park, C. E. Dreyer, and6\nM. Bernardi, Phys. Rev. Lett. 125, 136602 (2020).\n[36] J.-J. Zhou, J. Park, I. Timrov, A. Floris, M. Cococ-\ncioni, N. Marzari, and M. Bernardi, Phys. Rev. Lett. 127,\n126404 (2021).\n[37] J. Park, J.-J. Zhou, and M. Bernardi, Phys. Rev. B 101,\n045202 (2020).\n[38] S. Kim, S. Woo, and H. Min, Phys. Rev. B 99, 165107\n(2019).\n[39] G. D. Mahan, Many-Particle Physics , 3rd ed., Physics of\nSolids and Liquids (Springer, Berlin, 2000).\n[40] See the Supplemental Material for the derivation of\nthe Bethe-Salpeter equation for the phonon-dressed spin\nvertex, additional computational details, and the aug-\nmented iterative approach for solving the spin-phonon\nBSE, which include Refs. [20, 28, 29, 31, 32, 35, 37{\n39, 43, 44, 59{76].\n[41] J. Park, Y. Luo, J.-J. Zhou, and M. Bernardi,\narXiv:2208.09575 (2022).\n[42] A. A. Burkov and L. Balents, Phys. Rev. B 69, 245312\n(2004).\n[43] P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car,\nC. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococ-\ncioni, I. Dabo, A. D. Corso, S. de Gironcoli, S. Fabris,\nG. Fratesi, R. Gebauer, U. Gerstmann, C. Gougoussis,\nA. Kokalj, M. Lazzeri, L. Martin-Samos, N. Marzari,\nF. Mauri, R. Mazzarello, S. Paolini, A. Pasquarello,\nL. Paulatto, C. Sbraccia, S. Scandolo, G. Sclauzero, A. P.\nSeitsonen, A. Smogunov, P. Umari, and R. M. Wentzcov-\nitch, J. Phys.: Condens. Matter 21, 395502 (2009).\n[44] G. Pizzi, V. Vitale, R. Arita, S. Bl ugel, F. Freimuth,\nG. G\u0013 eranton, M. Gibertini, D. Gresch, C. Johnson,\nT. Koretsune, J. Iba~ nez-Azpiroz, H. Lee, J.-M. Lihm,\nD. Marchand, A. Marrazzo, Y. Mokrousov, J. I. Mustafa,\nY. Nohara, Y. Nomura, L. Paulatto, S. Ponc\u0013 e, T. Pon-\nweiser, J. Qiao, F. Th ole, S. S. Tsirkin, M. Wierzbowska,\nN. Marzari, D. Vanderbilt, I. Souza, A. A. Mosto\f, and\nJ. R. Yates, J. Phys. Condens. Matter 32, 165902 (2020).\n[45] D. J. L\u0013 epine, Phys. Rev. B 2, 2429 (1970).\n[46] G. Lancaster, J. A. van Wyk, and E. E. Schneider, Proc.\nPhys. Soc. 84, 19 (1964).\n[47] S. Oertel, J. H ubner, and M. Oestreich, Appl. Phys. Lett.\n93, 132112 (2008).\n[48] A. V. Kimel, F. Bentivegna, V. N. Gridnev, V. V. Pavlov,\nR. V. Pisarev, and T. Rasing, Phys. Rev. B 63, 235201\n(2001).\n[49] R. I. Dzhioev, K. V. Kavokin, V. L. Korenev, M. V.\nLazarev, N. K. Poletaev, B. P. Zakharchenya, E. A. Sti-\nna\u000b, D. Gammon, A. S. Bracker, and M. E. Ware, Phys.\nRev. Lett. 93, 216402 (2004).\n[50] T. Lai, X. Liu, H. Xu, Z. Jiao, J. Wen, and W. Lin, Appl.\nPhys. Lett. 88, 192106 (2006).\n[51] M. Ersfeld, F. Volmer, L. Rathmann, L. Kotewitz,\nM. Heitho\u000b, M. Lohmann, B. Yang, K. Watanabe,T. Taniguchi, L. Bartels, J. Shi, C. Stampfer, and\nB. Beschoten, Nano Lett. 20, 3147 (2020).\n[52] J. Li, M. Goryca, K. Yumigeta, H. Li, S. Tongay, and\nS. A. Crooker, Phys. Rev. Mater. 5, 044001 (2021).\n[53] T. Yan, S. Yang, D. Li, and X. Cui, Phys. Rev. B 95,\n241406 (2017).\n[54] M. Goryca, N. P. Wilson, P. Dey, X. Xu, and S. A.\nCrooker, Sci. Adv. 5, eaau4899 (2019).\n[55] D. J. Hilton and C. L. Tang, Phys. Rev. Lett. 89, 146601\n(2002).\n[56] M. Krau\u0019, M. Aeschlimann, and H. C. Schneider, Phys.\nRev. Lett. 100, 256601 (2008).\n[57] J. Schwinger, Phys. Rev. 73, 416 (1948).\n[58] D. Hanneke, S. Fogwell, and G. Gabrielse, Phys. Rev.\nLett. 100, 120801 (2008).\n[59] M. Cardona, Phys. Status Solidi A 188, 1209 (2001).\n[60] G. Stefanucci and R. van Leeuwen, Nonequilibrium\nMany-Body Theory of Quantum Systems: A Modern In-\ntroduction (Cambridge University Press, 2013).\n[61] J.-J. Zhou and M. Bernardi, Phys. Rev. Res. 1, 033138\n(2019).\n[62] J.-M. Lihm and C.-H. Park, Phys. Rev. B 101, 121102\n(2020).\n[63] P. B. Allen and V. Heine, J. Phys. C: Solid State Phys.\n9, 2305 (1976).\n[64] D. Vollhardt and P. W ol\re, Phys. Rev. B 22, 4666 (1980).\n[65] M. Rohl\fng and S. G. Louie, Phys. Rev. B 62, 4927\n(2000).\n[66] K. Shen, G. Vignale, and R. Raimondi, Phys. Rev. Lett.\n112, 096601 (2014).\n[67] J. P. Perdew, A. Ruzsinszky, G. I. Csonka, O. A. Vydrov,\nG. E. Scuseria, L. A. Constantin, X. Zhou, and K. Burke,\nPhys. Rev. Lett. 100, 136406 (2008).\n[68] J. Heyd, G. E. Scuseria, and M. Ernzerhof, J. Chem.\nPhys. 118, 8207 (2003).\n[69] M. J. van Setten, M. Giantomassi, E. Bousquet, M. J.\nVerstraete, D. R. Hamann, X. Gonze, and G. M. Rig-\nnanese, Comput. Phys. Commun. 226, 39 (2018).\n[70] T. Sohier, M. Calandra, and F. Mauri, Phys. Rev. B 94,\n085415 (2016).\n[71] S. Baroni, S. de Gironcoli, A. Dal Corso, and P. Gian-\nnozzi, Rev. Mod. Phys. 73, 515 (2001).\n[72] J. Park, J.-J. Zhou, V. A. Jhalani, C. E. Dreyer, and\nM. Bernardi, Phys. Rev. B 102, 125203 (2020).\n[73] G. Brunin, H. P. C. Miranda, M. Giantomassi, M. Royo,\nM. Stengel, M. J. Verstraete, X. Gonze, G.-M. Rignanese,\nand G. Hautier, Phys. Rev. Lett. 125, 136601 (2020).\n[74] G. Brunin, H. P. C. Miranda, M. Giantomassi, M. Royo,\nM. Stengel, M. J. Verstraete, X. Gonze, G.-M. Rignanese,\nand G. Hautier, Phys. Rev. B 102, 094308 (2020).\n[75] P. E. Bl ochl, O. Jepsen, and O. K. Andersen, Phys. Rev.\nB49, 16223 (1994).\n[76] S. Vollmar, D. J. Hilton, and H. C. Schneider, Phys. Rev.\nB96, 075203 (2017)."    },    {        "title": "1706.01013v2.Simulating_spin_dynamics_with_spin_dependent_cross_sections_in_heavy_ion_collisions.pdf",        "content": "arXiv:1706.01013v2  [nucl-th]  11 Oct 2017Simulating spin dynamics with spin-dependent cross sectio ns in heavy-ion collisions\nYin Xia,1,2Jun Xu∗,1Bao-An Li,3,4and Wen-Qing Shen1\n1Shanghai Institute of Applied Physics, Chinese Academy of S ciences, Shanghai 201800, China\n2University of Chinese Academy of Sciences, Beijing 100049, China\n3Department of Physics and Astronomy, Texas A &M University-Commerce, Commerce, TX 75429-3011, USA\n4Department of Applied Physics, Xi’an Jiao Tong University, Xi’an 710049, China\n(Dated: June 16, 2021)\nWe have incorporated the spin-dependent nucleon-nucleon c ross sections into a Boltzmann-\nUehling-Uhlenbecktransport model for the first time, using the spin-singlet and spin-triplet nucleon-\nnucleon elastic scattering cross sections extracted from t he phase-shift analyses of nucleon-nucleon\nscatterings in free space. We found that the spin splitting o f the collective flows is not affected by\nthe spin-dependent cross sections, justifying it as a good p robe of the in-medium nuclear spin-orbit\ninteraction. With the in-medium nuclear spin-orbit mean-fi eld potential that leads to local spin\npolarization, we found that the spin-averaged observables , such as elliptic flows of free nucleons and\nlight clusters, becomes smaller with the spin-dependent di fferential nucleon-nucleon scattering cross\nsections.\nPACS numbers: 25.70.-z, 13.88.+e, 21.10.Hw, 24.10.Lx, 21. 30.Fe\nI. INTRODUCTION\nThe spin-orbit interaction, which was previously in-\ntroduced to explain the magic number of finite nuclei,\nis critical in understanding the structures of rare iso-\ntopes and their impacts on astrophysics [1–5]. Heavy-\nion collisions provide the only way of studying prop-\nerties of nuclear matter as well as nuclear interactions\nat both finite densities and temperatures in terrestrial\nlaboratories, and a useful means of extracting proper-\nties of the in-medium nuclear spin-orbit interaction with\noptimal reaction conditions. Recently, we have devel-\noped a spin- and isospin-dependent Boltzmann-Uehling-\nUhlenbeck (SIBUU) transport model, by incorporating\nthe nucleon spin degree of freedom and the nuclear spin-\norbit interaction into the IBUU transport model [6, 7].\nWe found this model is useful in studying the spin dy-\nnamics in intermediate-energy heavy-ion collisions [8].\nIn particular, it was observed that the spin splittings\nof collective flows of free nucleons and light clusters\ncan be good probes of the in-medium spin-orbit interac-\ntion [9, 10]. However, in our previous studies, we applied\nthe spin-dependent mean-field potential for nucleons but\nemployed the spin-averaged nucleon-nucleon scattering\ncross sections. In order to have a complete framework\nof the spin-dependent transport approach and a better\ndescription of the spin dynamics in intermediate-energy\nheavy-ioncollisions, in the presentstudy weincorporated\nthe spin-singlet and spin-triplet cross sections for elas-\ntic nucleon-nucleon scatterings into the model. The lat-\nter are extracted based on the phase-shift analyses of\nnucleon-nucleon scatterings in free space. We found that\nthe spin splitting of the collective flow as a probe of the\nin-medium nuclearspin-orbitinteractionisalmostnotaf-\n∗corresponding author: xujun@sinap.ac.cnfected by the spin-dependent nucleon-nucleon scattering\ncross sections. However, the overall elliptic flows of free\nnucleons and light clusters are slightly smaller with the\nspin-dependent nucleon-nucleon scattering cross sections\ncompared with the spin-averaged ones, if there is local\nspin polarization induced by the spin-dependent mean-\nfield potential. A more complete BUU framework in-\ncluding both the spin-dependent potential and the spin-\ndependent cross sections has been established, providing\npossibilities of further exploring the interesting physics\nof spin dynamics in intermediate-energy heavy-ion colli-\nsions.\nII. SPIN-DEPENDENT CROSS SECTIONS\nFROM PHASE-SHIFT ANALYSES\nThe phase-shift analysis has been an effective way\nof decoupling nucleon-nucleon interactions into various\nchannels by fitting experimental nucleon-nucleonscatter-\ning data in terms of the scattering matrix [11–14]. There\nare series of studies on the energy-dependent phase-shift\nanalyses of nucleon-nucleon elastic scattering data in a\nwide energy range [15–17]. Using the phase-shift data\nin Ref. [15], we evaluate the spin-singlet and spin-triplet\nnucleon-nucleon elastic cross sections within the incident\nnucleon energy range between 1 and 500 MeV, where the\ninelastic scatterings are less important. For complete-\nness, we first recall the most relevant formulaes in the\nfollowing.\nWe begin with the general formula for the differential\ncross section of two-body collisions expressed directly in\ntermsofthe eigenphasesofthescatteringmatrixbyBlatt\nand Biedenharn [18]:\ndσα′s′;αs=g\n(2s+1)k2∞/summationdisplay\nL=0BL(α′s′;αs)PL(cosθ)dΩ,\n(1)2\nwheregis 1 for neutron-proton scatterings and 4 for\nproton-proton(neutron-neutron)scatterings, PL(cosθ)is\nthe Legendre polynomial, kis the center-of-mass (C.M.)\nmomentum in the two-body system, and the coefficients\nBL(α′s′;αs) =(−)s′−s\n4/summationdisplay\nJ1/summationdisplay\nJ2/summationdisplay\nl1/summationdisplay\nl2/summationdisplay\nl1′/summationdisplay\nl2′\n×Z(l1J1l2J2,sL)Z(l1′J1l2′J2,s′L)\n×R.P.[(δα′αδs′sδl1′l1−SJ1\nα′s′l1′;αsl1)∗\n×(δα′αδs′sδl2′l2−SJ2\nα′s′l2′;αsl2)] (2)\ncan be determined directly from the phase-shift data. In\ntheaboveexpression, δabrepresentstheKronecker δfunc-\ntion;α,s,l, andJrepresent the scattering channel, the\nspin of the scattering channel, the orbital angular mo-\nmentum, and the total angular momentum, respectively;\nSJ\nα′s′l′;αslis the scattering amplitude of a collision from\na channel αslto a channel α′s′l′; theZcoefficient rep-\nresents the selection rules introduced by Biedenharn et\nal.[19]; theR.P.[...]representstherealpartoftheexpres-\nsion in the bracket. In the limit of pure elastic nucleon-\nnucleon scatterings without spin flipping, α′=αand\ns′=sare always satisfied, so we omit the superscript α\nand use only sas the superscript in the following. s= 0\nands= 1 stand for the spin-singlet and spin-triplet scat-\ntering, respectively.\nLet’s first consider the spin-singlet and spin-triplet\nchannel for neutron-proton scatterings. For the spin-\nsinglet case with s= 0, there is only one channel l=J.\nUsingS= exp(2iδ0\nJ), Eq. (2) becomes\nBL(0;0) =/summationdisplay\nJ1/summationdisplay\nJ2/summationdisplay\nl1=J1/summationdisplay\nl2=J2Z(l1J1l2J2,0L)2\n×sinδ0\nJ1sinδ0\nJ2cos(δ0\nJ1−δ0\nJ2), (3)\nwhereδ0\nJis the phase-shift of the spin-singlet channel\nwith orbital angular momentum l=J. For the spin-\ntriplet case with s= 1, given l=J, there is still only\none channel with S= exp(2iδ1\nJ). When the neutron-\nproton scattering is affected by the tensor force in their\nspin-triplet state, the angular momentum lcan have two\nvalues, i.e., l=J±1. In the latter case, the general\nexpression of the Smatrix of a two-channel reaction is\nS= (4)\n/parenleftBigg\ncos2(ǫJ)e2iδ1\nJ−1+ sin2(ǫJ)e2iδ1\nJ+1 1\n2sin(2ǫJ)(e2iδ1\nJ−1−e2iδ1\nJ+1)\n1\n2sin(2ǫJ)(e2iδ1\nJ−1−e2iδ1\nJ+1) sin2(ǫJ)e2iδ1\nJ−1+ cos2(ǫJ)e2iδ1\nJ+1/parenrightBigg\n.\nIn the above, δ1\nJ±1is usually called the Biedenharn-Blatt\n(BB)phaseshiftofthespin-tripletchannelwith l=J±1,\nandǫJis the parameter describing the mixing proba-\nbility of the two coupling states. By using the energy-\ndependent neutron-proton phase-shift data as well as the\nmixing parameters for various channels in Tables III and\nIV of Ref. [15], we calculate the coefficient BLand thedifferential cross section. For the unpolarized neutron-\nproton cross section, we also take the summation of the\nisovector contribution T= 1, the isoscalar contribution\nT= 0, and their interference contribution to the coeffi-\ncientBL[20]. We note there is a simplified method for\ncalculating the spin-triplet case developed by Blatt and\nBiedenharn [21], and it leads to identical results.\nFor proton-proton scatterings, we only incorporate the\nnuclear contribution to the cross section into transport\nmodel simulations, but subtract the contribution of the\nlong-range Coulomb potential to the scatterings. For\nthe spin-singlet and spin-triplet proton-proton scatter-\nings with l=J, the scattering matrix Scan be expressed\nas\nS=e2iδ0(1)\nJ−e2iφJ+1, (5)\nwhereφJis the pure Coulomb phase shift for orbital\nangular momentum l=J, and it can be written as [22]\nφl=l/summationdisplay\nm=1arctan(η/m), (6)\nwithη=e2//planckover2pi1v≈(137β)−1whereβ=v/cis the reduced\nvelocity of the incident proton in the lab frame. In order\nto subtract the Coulomb contribution from the Smatrix\nfor the two channels of spin-triplet scatterings with l=\nJ±1, we express it as [23]\nS= 1 + (7)\n/parenleftBigg\ncos(2ǫJ)e2iδ1\nJ−1−e2iφJ−1isin(2ǫJ)ei(δ1\nJ+1+δ1\nJ−1)\nisin(2ǫJ)ei(δ1\nJ+1+δ1\nJ−1)cos(2ǫJ)e2iδ1\nJ+1−e2iφJ+1/parenrightBigg\n,\nwith\nδ0(1)\nl=δ0(1)\nl(N)+φl (8)\nwhereδ0(1)\nl(N) is called the nuclear bar phase shifts,\nwhich are taken from Table II of Ref. [15] for various\nproton incident energies. The way of subtracting the\nCoulomb contribution assumes that the Coulomb force\nactsonlyoutsidethe regionofthe nuclearforcewherethe\nWKB approximation is valid [23]. For the spin-singlet\nand spin-triplet proton-proton scatterings with l=J,\nthe expressions for the scattering matrix Sare the same\nfor BB phase shifts and bar phase shifts, as can be seen\nfrom Eq. (5). In this way the spin-dependent differential\nproton-protonelasticscatteringcrosssectionscanalsobe\nobtained.\nIII. PARAMETRIZATION OF THE\nSPIN-DEPENDENT CROSS SECTIONS\nStarting from the energy-dependent phase-shift results\nof nucleon-nucleon scatterings by Arndt et al.[15], and\nusing the method described above, we are now able to3\nobtain the differential cross sections for elastic nucleon-\nnucleon scatterings at various collision energies. Since\nthe higher-order terms of the Legendre polynomials van-\nish after intergration, the total cross section is deter-\nmined by the terms with L= 0 in Eq. (1). The total\ncross sections for both spin-singlet and spin-triplet elas-\ntic neutron-proton scatterings between 1 and 500 MeV\ncan be parameterized respectively as\nσ0\nnp= 9302.64/E3−982.187/E2−1.32×102+1.03E\n−3.06×10−3E2+4.85×10−6E3−3.19×10−9E4,(9)\nσ1\nnp=−27888.84×104/E3+17565.39/E2+13382.81/E\n−8.10×101+0.37E−2.75×10−4E2−9.60×10−7E3\n+ 1.37×10−9E4. (10)\nSimilarly, the total cross sections for both spin-singlet\nand spin-triplet elastic proton-proton scatterings be-\ntween 1 and 500 MeV can also be parameterized respec-\ntively as\nσ0\npp=−11877.31/E3+733.31/E2+17397.66/E−2.38×102\n+ 1.51E−4.9×10−3E2+8.37×10−6E3−5.58×10−9E4,\nσ1\npp=−1.20+0.79E−8.40×10−3E2+3.24×10−5E3,\n(1 MeV < E <100 MeV)\nσ1\npp= 1.72×101+0.16E−8.13×10−4E2+2.14×10−6E3\n−2.86×10−9E4+1.52×10−12E5.\n(100 MeV < E <500 MeV) (11)\nIn the above, σ0andσ1in mb are the cross sections for\nthe elatistic spin-singlet and spin-triplet scatterings, re-\nspectively, and Ein MeV is the kinetic energy of the\nincident nucleon in the lab frame. The spin-averaged\ncross section can be obtained from σ=σ0/4+3σ1/4. In\nFig. 1 we compared the total elastic scattering cross sec-\ntions obtained in the present study with those previously\nused in the IBUU transport model, with the latter taken\nfrom Ref. [24] parameterized as\nσpp(nn)= 8.76/β2−15.04/β+13.73+68.76β4,(12)\nσnp= 25.26/β2−18.18/β−70.67+113.85β,(13)\nwhere the cross section σis in mb, and β= /radicalbig\n1−M2\nNc4/(MNc2+E)2is the reduced velocity of the\nincident nucleon with MNbeing the nucleon mass. We\nfind that the previously used parameterized cross sec-\ntions are similar to the spin-averaged ones obtained in\nthe present study using the phase-shift results in the en-\nergyrangeconsidered. Notethat acutatverylowenergy\nregion, where the cross section may diverge, is usually\nused in transport model simulations.\nWehavealsoparameterizedthedifferentialspin-singlet\nand spin-triplet cross sections for elastic neutron-proton\nand proton-proton scatterings between 1 and 500 MeV/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54/s49/s48/s48/s49/s48/s49/s49/s48/s50/s49/s48/s51\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56/s49/s48/s48/s49/s48/s49/s49/s48/s50/s49/s48/s51/s32/s32/s32/s40/s109 /s98/s41/s110/s112\n/s32/s115/s112/s105/s110/s45/s97/s118/s101/s114/s97/s103/s101/s100\n/s32/s115/s112/s105/s110/s45/s115/s105/s110/s103/s108/s101/s116\n/s32/s115/s112/s105/s110/s45/s116/s114/s105/s112/s108/s101/s116\n/s32/s112/s114/s101/s118/s105/s111/s117/s115/s40/s97/s41 /s40/s98/s41\n/s32/s32\n/s112/s112/s40/s110/s110/s41\nFIG. 1: (color online) Total elastic spin-averaged, spin-\nsinglet, and spin-triplet cross sections obtained in the pr esent\nstudy for neutron-proton (left) and proton-proton (neutro n-\nneutron) (right) scatterings as functions of the reduced in ci-\ndent nucleon velocity βin the lab frame, compared with the\nprevious ones used in the IBUU transport model.\nin the following form:\ndσs\nnp(θ) =11/summationdisplay\nn=0as\nncosnθdΩ, (14)\ndσs\npp(θ) =5/summationdisplay\nm=0bs\n2mcos2mθdΩ. (15)\nIn the above equations, the cross sections are in mb, n\nandmare related to the angular momentum quantum\nnumbers of the orbital wave function, i.e., from s-waveto\nh-wave,asusedintheenergy-dependentphase-shiftanal-\nyses. With the differential crosssections from phase-shift\nresults at discrete energies, we are able to parameterize\nthe coefficients as\nnandbs\n2mas functions of the energy E\nto get continuous energy-dependent differential cross sec-\ntions between 1 and 500MeV. The s-wavecoefficients a0\n0,\na1\n0,b0\n0, andb1\n0, which lead to the total cross section, are\nparameterized respectively as\na0\n0=−47.99/E3−28.93/E2+740.42/E−12.11+7.53×10−2E\n−2.32×10−4E2+3.83×10−7E3−2.62×10−10E4,(16)\na1\n0=−2435.74/E3+1565.07/E2+1115.4/E−9.27+0.02E\n+ 1.43×10−4E2−6.04×10−7E3+6.08×10−10E4,(17)\nb0\n0=−987.99/E3+80.497/E2+1409.47/E−23.51+0.148E\n−4.488×10−4E2+7.203×10−7E3−4.72×10−10E4,(18)\nb1\n0=−0.142 +0.079E−8.78×10−4E2+3.36×10−6E3,\n(1 MeV < E <100 MeV)\nb1\n0= 1.89 +9.85×103E−6.90×10−5E2+1.84×10−7E3(19)\n−2.33×10−10E4+1.16×10−13E5.\n(100 MeV < E <500 MeV)\nFor other coefficients as\nnandbs\n2mcorresponding to larger\norbital angular momentum quantum numbers, polyno-4\nmial functions are used to fit their energy dependence,\nand the fitting results are showed in Table I.\n/s48 /s52/s53 /s57/s48 /s49/s51/s53/s48/s52/s56/s49/s50/s49/s54/s50/s48\n/s48 /s52/s53 /s57/s48 /s49/s51/s53 /s49/s56/s48/s48/s52/s56/s49/s50/s49/s54/s50/s48/s32/s32/s100 /s100 /s109/s98/s47/s115/s114\n/s32/s40/s100/s101/s103/s41/s32/s32/s32/s32/s32/s32/s32/s110/s112/s32\n/s69/s32/s61/s32/s49/s48/s48/s32/s77/s101/s86/s40/s97/s41 /s40/s98/s41/s32 /s32/s32\n/s32 /s32/s111/s114/s32\n/s32 /s32/s111/s114/s32\n/s32/s32 /s115/s112/s105/s110/s45/s97/s118/s101/s114/s97/s103/s101/s100/s32/s32/s32/s32/s32/s112/s112/s40/s110/s110/s41/s32\n/s69/s32/s61/s32/s49/s48/s48/s32/s77/s101/s86\nFIG. 2: (color online) Differential cross sections for neutr on-\nproton (left) and proton-proton (right) pairs with the same\nspin, different spins, and the spin-averaged differential cr oss\nsections as functions of the scattering polar angle θin the\nC.M. frame, at an incident nucleon energy of 100 MeV.\nIn transport model simulations of heavy-ion collisions,\nit is more convenient to use the cross sections determined\nbythe spinsofthe collidingnucleons, andthey can be ex-\npressed in terms of the spin-singlet and spin-triplet scat-\ntering cross section as\nσ↑↑(↓↓)\nNN=σ1\nNN, (20)\nσ↑↓(↑↓)\nNN= (σ1\nNN+σ0\nNN)/2, (21)\nwhereσ↑↑(↓↓)\nNN(σ↑↓(↑↓)\nNN) is the cross section for nucleon\npairs with the same (different) spin with respect to the\nangular momentum of the pair. The angular dependence\nof the differential cross sections to be used in SIBUU\ntransport model simulations is plotted in Fig. 2. These\nangular distributions reveal the nucleon interaction in\nvacuum. For example, in neutron-proton scatterings, the\nforward peak is due to the Wigner force while the back-\nward peak is due to the Majorana force [25]. On the\nother hand, scatterings between identical particles with\nthe same spin and isospin are not likely to have forward\nand backward peaks due to the strong Pauli repulsive\neffect.\nIV. EFFECTS IN HEAVY-ION SIMULATIONS\nIn the SIBUU transport model, the density of the ini-\ntial two nuclei is sampled according to the prediction of\nSkyrme-Hartree-Fock calculations, while the momentum\nof each nucleon is sampled according to its local den-\nsity and further boosted by the beam energy. The spinexpectation value of each nucleon is chosen as a unit vec-\ntor in the 4 πsolid angle, and it is randomly sampled in\nthe initial stage. As the system begins to evolve, the\ncoordinate /vector r, momentum /vector p, and spin /vector sof each nucleon\nfollow the equations of motion consistently derived from\nthespin-dependent Boltzmann-Vlasovequation[7]asfol-\nlows:\nd/vector r/dt=/vector p//radicalBig\np2c2+M2\nNc4+∇pUs,(22)\nd/vector p/dt=−∇U−∇Us, (23)\nd/vector s/dt=−i\n/planckover2pi1[/vector s,Us], (24)\nwhereUis the momentum- and spin-independent mean-\nfield potential, and the right-hand side of the third equa-\ntion denotes the commutator of each component of spin\nwith the spin-dependent mean-field potential Us. Par-\nticularly, the strength, the isospin dependence, and the\ndensity dependence of Usare still under debate and are\nhot topics in nuclear structure studies [8]. In our pre-\nvious studies, we have shown that nucleons with differ-\nent spins may have different dynamics with Us, and this\nleads to local spin polarization (see Fig. 1 of Ref. [6] and\nFig. 1 of Ref. [10]) as well as the spin splitting of col-\nlective flows of free nucleons and light clusters [6, 9, 10].\nHere we investigate the effects of nucleon-nucleon scat-\nterings with spin-dependent differential elastic cross sec-\ntions in heavy-ion collisions. Since the spin expectation\ndirection is known for each nucleon, the spin state of\na single nucleon and that of the colliding nucleon pair\ncan be obtained by projecting the spin expectation direc-\ntion onto the total angular momentum of the incoming\nnucleon pair. The differential scattering cross sections\nare then determined from the spin state as well as the\ncollision energy through the parameterizations given in\nSec. III, and they are technically incorporated according\nto the scattering treatment in Appendix B of Ref. [26].\nWefirstexaminetheeffectsofthespin-dependentcross\nsections on the spin up-down differential transverse flow\ndefined as [6, 9]\nFud(yr) =1\nN(yr)N(yr)/summationdisplay\ni=1si(px)i, (25)\nwhereN(yr) is the number of nucleons with rapidity yr,\n(px)iis the momentum of the ith nucleon in xdirection,\nandsiis 1(−1) for spin-up (spin-down) nucleons with\nrespect to the total angular momentum of the heavy-ion\ncollision system. As discussed in Refs. [6, 9], the spin-\ndependent potential Usgives an additional attractive\n(repulsive) potential to spin-up (spin-down) nucleons, re-\nsultingintheirdifferenttransverseflows. Asshowninthe\nleft panel of Fig. 3, Fudvanishes without Us, with the\nlatter the source of different potentials for spin-up and\nspin-down nucleons and thus their different transverse\nflows. With Us,Fudremain almost the same using the\nspin-averaged and spin-dependent nucleon-nucleon cross\nsections, as shown in the right panel of Fig. 3. We have5\nTABLE I: Coefficients for the polynomial fit of the energy depen dence of as\nn(n >0) andbs\n2m(m >0), i.e.,as\nn(bs\n2m) =C0+\nC1E+C2E2+C3E3(1MeV< E <50MeV) and as\nn(bs\n2m) =C4+C5E+C6E2+C7E3+C8E4+C9E5(50MeV < E <500 MeV).\nC0 C1C2(10−3)C3(10−5)C4 C5C6(10−3)C7(10−6)C8(10−9)C9(10−12)\na0\n1−6.23 −0.46 32 .24−42.98−2.20−2.84×10−30.14−0.26−0.064 0 .29\na0\n2−0.48 0 .54 −21.99 24 .794.71−7.55×10−20.74−2.41 3 .54 −1.98\na0\n30.47 −0.32 7 .92−4.77−8.82 0 .23 −1.98 6 .80−10.61 6 .39\na0\n4−4.07×10−211.08 5 .23−9.120.23 0 .12 −1.96 7 .48−12.09 7 .36\na0\n5−0.13 9 .77×10−2−11.76 15 .66−1.80−0.12 1 .37−5.28 8 .92 −5.75\na0\n64.80×10−2−3.77×10−24.57−5.473.41 −0.14 3 .03−12.81 21 .96−13.90\na0\n7−2.63×10−21.90×10−2−1.99 2 .071.43 −0.11 1 .13−4.02 6 .57 −4.01\na0\n82.12×10−2−1.55×10−21.68−2.230.10 0 .12 −2.67 11 .89−21.24 13 .77\na0\n9−2.52×10−31.49×10−3−0.052−0.902.43−6.06×10−2−0.30 1 .97−3.80 2 .55\na0\n10−4.21×10−54.64×10−4−0.19 1 .45−4.37 9.84×10−20.43−3.68 7 .86 −5.56\na1\n10.36 −8.13×10−27.28−10.152.72−1.39×10−2−0.11 0 .92−2.11 1 .59\na1\n2−0.18 −0.22 16 .87−21.241.10 1.07×10−1−1.00 4 .27−8.11 5 .63\na1\n31.17×10−21.15×10−2−1.34 2 .27−1.69 5.93×10−2−0.57 2 .10−3.42 2 .09\na1\n45.37×10−2−3.46×10−23.80−6.464.16−6.24×10−2−0.65 4 .21−8.40 5 .75\na1\n5−1.81×10−33.65×10−3−1.07 1 .97−2.72 6.95×10−2−0.36 0 .39 0 .41 −0.72\na1\n62.03×10−2−1.74×10−22.24−0.78−5.71 0 .25 −1.42 3 .63−4.75 2 .52\na1\n7−1.05×10−28.01×10−3−0.91 1 .061.76 −0.11 1 .41−5.04 7 .87 −4.64\na1\n81.32×10−3−9.45×10−40.09 0 .08−0.70 2.77×10−2−0.19 0 .53−0.66 0 .29\na1\n9−3.39×10−45.50×10−50.05−0.610.94−1.36×10−2−0.43 2 .03−3.58 2 .27\na1\n10−5.08×10−56.50×10−5−0.01 0 .09−0.08−1.16×10−30.11−0.48 0 .81 −0.48\nb0\n2−1.24−9.83×10−1−36.74 38 .868.51−9.91×10−20.47−1.27 1 .84 −1.04\nb0\n40.16 −0.12 14 .36−20.554.01 4.33×10−2−0.87 3 .07−4.73 2 .91\nb0\n61.44×10−2−1.02×10−20.99−0.35−1.62 0 .064 0 .0061 −0.27 0 .57 −0.61\nb0\n81.38×10−4−8.66×10−50.0048 0 .034−0.078 0 .0022 0 .0097 −0.042 0 .068 −0.041\nb1\n20.039 −0.019 −0.82 2 .06−1.44 0 .03 −0.15 0 .52−0.98 0 .67\nb1\n4−0.027 0 .023 −1.79 1 .63−0.488 0 .397 −1.61 3 .08−2.10 5 .63\nb1\n66.53×10−5−3.52×10−30.52−0.842.13−5.18×10−20.34−0.84 0 .65−0.073\nb1\n81.73×10−3−1.95×10−30.204 −0.18−0.63 4.94×10−2−0.68 2 .55−3.93 2 .25\nb1\n10−2.30×10−36.16×10−3−0.021 0 .15−0.29−3.78×10−30.23−1.04 1 .75 −1.07\nalso found that the spin up-down differential transverse\nflowremainsthe sameforneutrons andprotonsaswell as\nfor energetic nucleons, justifying the validity of Fudas a\ngood probe of the strength, the isospin dependence, and\nthe density dependence of the in-medium nuclear spin-\norbit potential [6, 9, 10].\nFigure 4 compares the resulting spin-averaged elliptic\nflows (v2=/angb∇acketleft(p2\nx−p2\ny)/(p2\nx+p2\ny)/angb∇acket∇ight) of free nucleons and\ndeuterons from the spin-averaged and spin-dependent\nnucleon-nucleon scattering cross sections with and with-\nout the spin-dependent mean-field potential. It is seen\nthatv2of free nucleons is the same from the spin-\naveraged and spin-dependent cross sections without Us,\nwhile the difference is observed with Us. The latter is\ndue to the local spin polarization induced by Us. As is\nknow,v2is sensitive to the shear viscosity of the sys-\ntem [27, 28], with the later related to the transport cross\nsection defined as\nσtr=/integraldisplaydσ\ndΩ(1−cos2θ). (26)\nFor a given total σ, a more forward- and backward-\npeaked differential cross section generally leads to asmaller transport cross section and a larger shear vis-\ncosity. Since locally there can be different numbers of\nspin-up and spin-down nucleons induced by the spin-\ndependent mean-field potential, the transport cross sec-\ntion can be different from the spin-averaged and spin-\ndependent cross sections. This is the reason why the\nslightly different v2is observed in Panel (c) of Fig. 4. We\nhavealsostudied the formationoflightclustersformedin\ntransport simulations, through a dynamical coalescence\nalgorithm from nucleons that are close in coordinate and\nmomentum space [10, 29]. The spin-dependent differen-\ntial cross sections lead to correlations between the scat-\ntering angles of final-state nucleons, resulting in the dif-\nferent final distribution of these light clusters. It is seen\nfrom Fig. 4 that slight v2difference is observed at midra-\npidities between results from the spin-averagedcross sec-\ntions and the spin-dependent ones, and the effect is fur-\ntherenhanced with the spin-dependent mean-fieldpoten-\ntial.6\n/s45/s49 /s48/s45/s53/s46/s48/s45/s50/s46/s53/s48/s46/s48/s50/s46/s53/s53/s46/s48\n/s45/s49 /s48 /s49/s45/s53/s46/s48/s45/s50/s46/s53/s48/s46/s48/s50/s46/s53/s53/s46/s48/s32/s32/s70\n/s117 /s100 /s32/s40/s77 /s101/s86/s47/s99/s41/s119/s47/s111/s32/s83/s79/s40/s98/s41\n/s32/s32\n/s121\n/s114/s47/s121/s98/s101/s97/s109\n/s114/s32/s115/s112/s105/s110/s45/s97/s118/s101/s114/s97/s103/s101/s100/s32\n/s32/s115/s112/s105/s110/s45/s100/s101/s112/s101/s110/s100/s101/s110/s116/s32/s119/s105/s116/s104/s32/s83/s79/s40/s97/s41\nFIG. 3: (color online) Rapidity dependence of the spin\ndifferential transverse flow from the spin-averaged and the\nspin-dependent cross sections with (right) and without (le ft)\nthe spin-orbit (SO) potential in Au+Au collisions at 100\nMeV/nucleon and an impact parameter b = 12 fm.\n/s48/s46/s56/s49/s46/s48/s49/s46/s50\n/s119 /s47/s111/s32/s83/s79\n/s32/s115/s112/s105/s110/s45/s97/s118/s101/s114/s97/s103/s101/s100/s32\n/s32/s115/s112/s105/s110/s45/s100/s101/s112/s101/s110/s100/s101/s110/s116/s32\n/s32/s32\n/s55/s56/s57/s49/s48\n/s119 /s47/s111/s32/s83/s79\n/s32/s32\n/s45/s48/s46/s54 /s45/s48/s46/s51 /s48/s46/s48 /s48/s46/s51 /s48/s46/s54/s48/s46/s56/s49/s46/s48/s49/s46/s50/s40/s100/s41\n/s40/s99/s41/s40/s98/s41\n/s119 /s105/s116/s104/s32/s83/s79/s118/s100 /s50/s32/s40/s37 /s41\n/s32/s118/s78 /s50/s32/s40/s37 /s41\n/s121\n/s114/s47/s121/s98/s101/s97/s109\n/s114\n/s32/s32/s40/s97/s41\n/s45/s48/s46/s54 /s45/s48/s46/s51 /s48/s46/s48 /s48/s46/s51 /s48/s46/s54/s55/s56/s57\n/s119 /s105/s116/s104/s32/s83/s79\n/s32/s32\nFIG. 4: (color online) Rapidity dependence of the elliptic fl ow\nof free nucleons (left) and deuterons (right) from the spin-\naveraged and the spin-dependent cross sections with (lower )\nand without (upper) the spin-orbit (SO) potential in Au+Au\ncollisions at 100 MeV/nucleon and an impact parameter b =\n12 fm.\nV. CONCLUSION AND OUTLOOK\nUsing the phase-shift results from nucleon-nucleon\nscattering data in free space, we have obtained the spin-dependent neutron-proton and proton-protondifferential\nelasticscatteringcrosssections. Wehavefurtherincorpo-\nrated them into the spin-dependent Boltzmann-Uehling-\nUhlenbeck transport model for the first time. The spin\nsplittings of collective flows, which were previously found\nasprobesofthein-mediumnuclearspin-orbitinteraction,\nare not affected by these spin-dependent cross sections.\nHowever, spin-averaged quantities, such as the elliptic\nflows of free nucleons and deuterons, can be slightly af-\nfected with both the spin-dependent mean-field potential\nand cross sections.\nWe note that it is still a big challenge to obtain the\nspin-dependent inelastic nucleon-nucleon scattering cross\nsections in the suitable energy range for intermediate-\nenergy heavy-ion collisions, due to the lack of the experi-\nmental data. On the other hand, the in-medium nucleon-\nnucleon scattering cross sections remain uncertain, even\nfor the spin-independent part. So far, the information of\nthe in-mediumcrosssectionsreliesonvariousmany-body\ntheories[30–32], whilein transportmodelsimulationsthe\nin-medium effective mass scaling [33, 34] or empirical pa-\nrameterizations [35] are generally used. We are still on\nthe way of looking for reliable probes of the in-medium\nnuclear spin-orbit interaction and studying interesting\nand relevant physics of spin dynamics in heavy-ion col-\nlisions, by using a dynamical framework as complete as\npossible.\nAcknowledgments\nWe thank Xiao-Ming Xu for helpful communications,\nand Chen Zhong for maintaining the high-quality per-\nformance of the computer facility. This work was sup-\nported by the Major State Basic Research Develop-\nment Program (973 Program) of China under Contract\nNos. 2015CB856904 and 2014CB845401, the National\nNatural Science Foundation of China under Grant Nos.\n11320101004, 11475243, and 11421505, the ”100-talent\nplan” of Shanghai Institute of Applied Physics under\nGrant Nos. Y290061011 and Y526011011 from the Chi-\nnese Academy of Sciences, the Shanghai Key Labora-\ntory of Particle Physics and Cosmology under Grant\nNo. 15DZ2272100, the U.S. Department of Energy, Of-\nfice of Science, under Award Number de-sc0013702, and\ntheCUSTIPEN(China-U.S.TheoryInstitute forPhysics\nwith Exotic Nuclei) under the US Department of Energy\nGrant No. DEFG02- 13ER42025.\n[1] P. G. Reinhard and H. Flocard, Nucl. Phys. A 584, 467\n(1995).\n[2] R. J. Furnstahl, John J. Rusnak, and Brian D. Serot,\nNucl. Phys. A 632, 607 (1998).[3] M. Bender et al., Phys. Rev. C 60, 034304 (1999).\n[4] J. P. Schiffer et al., Phys. Rev. Lett. 110, 169901 (2013).\n[5] L. Gaudefroy et al., Phys. Rev. Lett. 97, 092501 (2006).\n[6] J. Xu and B. A. Li, Phys. Lett. B 724, 346 (2013).7\n[7] Y. Xia, J. Xu, B. A. Li, and W. Q. Shen, Phys. Lett. B\n759, 596 (2016).\n[8] J. Xu, B. A. Li, W. Q. Shen, and Y. Xia, Front. Phys.\n10, 102501 (2015).\n[9] Y. Xia, J. Xu, B. A. Li, and W. Q. Shen, Phys. Rev. C\n89, 064606 (2014).\n[10] Y. Xia, J. Xu, B. A. Li, and W. Q. Shen, Nucl. Phys. A\n955, 41 (2016).\n[11] E. P. Winger and L. Eisenbud, Phys. Rev. 72, 29 (1947).\n[12] L. Wolfenstein, Phys. Rev. 82, 690 (1951).\n[13] W. Hauser and H. Feshbach, Phys. Rev. 87, 366 (1952).\n[14] M. H. Macgregor, Phys. Rev. 113, 1559 (1959).\n[15] R. A. Arndt et al., Phys. Rev. C 15, 1002 (1977).\n[16] R. A. Arndt et al., Phys. Rev. D 28, 97 (1983).\n[17] R. A. Arndt et al., Phys. Rev. D 35, 128 (1987).\n[18] J. M. Blatt and L. C. Biedenharn, Rev. Mod. Phys. 24,\n258 (1952).\n[19] L. C. Biedenharn, J. M. Blatt, and M. E. Rose, Rev.\nMod. Phys. 24, 249 (1952).\n[20] Y. Xia, Ph.D thesis, Shanghai Institute of Applied\nPhysics, 2017.\n[21] J. M. Blatt and L. C. Biedenharn, Phys. Rev. 86, 399\n(1952).\n[22] N. F. Mott and H. S. W. Massey, Theory of Atomic Col-\nlisions(Clarendon Press, Oxford, 1949), second edition,\np. 101.[23] H. P. Stapp, T. J. Ypsilantis, and N. Metropolis, Phys.\nRev.105, 302 (1957).\n[24] S. K. Charagi and S. K. Gupta, Phys. Rev. C 41, 1610\n(1990).\n[25] J. M. Blatt and V. F. Weisskopf, Theoretical Nuclear\nPhysics(Springer-Verlag, New York Inc., 1979), p. 178.\n[26] G. F. Bertsch and S. Das Gupta, Phys. Rep. 160, 189\n(1988).\n[27] H. C. Song, S. A.Bass, U. Heinz, T. Hirano, and C. Shen,\nPhys. Rev. Lett. 106, 192301 (2011).\n[28] C. L. Zhou, Y. G. Ma, D. Q. Fang, G. Q. Zhang, J. Xu,\nX. G. Cao, and W. Q. Shen, Phys. Rev. C 90, 057601\n(2014).\n[29] L. W. Chen, C. M. Ko, and B. A. Li, Nucl. Phys. A 729,\n809 (2003).\n[30] G. Q. Li and R. Machleidt, Phys. Rev. C 49, 566 (1994).\n[31] Q. F. Li, Z. X. Li, and G. J. Mao, Phys. Rev. C 62,\n014606 (2000).\n[32] H. F. Zhang, Z. H. Li, U. Lombardo, P. Y. Luo, F. Sam-\nmarruca, and W. Zuo, Phys. Rev. C 76, 054001 (2007).\n[33] V. R. Pandharipande and S. C. Pieper, Phys. Rev. C 45,\n791 (1991).\n[34] B.A.LiandL.W.Chen, Phys.Rev.C 72, 064611(2005).\n[35] D. Klakow, G. Welke, and W. Bauer, Phys. Rev. C 48,\n1982 (1993)."    },    {        "title": "1904.00564v1.Mechanism_of_Electron_Spin_Relaxation_in_Spiral_Magnetic_Structures.pdf",        "content": "arXiv:1904.00564v1  [cond-mat.mes-hall]  1 Apr 2019Mechanism of Electron Spin Relaxation in Spiral Magnetic St ructures\nI.I. Lyapilin∗\nInstitute of Metal Physics of Ural Branch of Russian Academy of Sciences, 620990 Yekaterinburg, Russia\nDepartment of Theoretical Physics and Applied Mathematics ,\nUral Federal University, Mira St. 19, 620002 Yekaterinburg , Russia\n(Dated: April 2, 2019)\nSpin dynamics in spiral magnetic structures has been invest igated. It has been shown that the\ninternal spatially dependent magnetic field in such structu res produces a new mechanism of spin\nrelaxation.\nI. INTRODUCTION\nOne of the central problems facing spintronics, such as\nthe development of methods of injection, generation, and\ndetection of spin-polarized charge carriers, is to analyze\nand study the mechanisms of spin relaxation. Spin-orbit\ninteraction(SOI)iswell-knowntohaveasignificantinflu-\nence on the mechanisms of electron spin relaxation. SOI\ncan impact on spin degrees of freedom through transla-\ntionalones1. SOIitselfdoesnotcausespinrelaxationbut\nin combination with the scattering of the electron mo-\nmentum leadstoits relaxation. Elliottexamined the pro-\ncess of electron spin relaxation through the momentum\nscattering at impurity centers, conditional upon induc-\ning the spin-orbit interaction by lattice ions2. Phonons\ncan also participate in spin relaxation. The nature of\nspin relaxation involving phonons is to modulate time-\ndependent spin-orbit relaxation by lattice oscillations3.\nAs a result, spin relaxation occurs. The Elliott-Jafet\nmechanism offers for the spin relaxation frequency ωs\nto be proportional to the electron momentum relaxation\nfrequency ωp. In systems without a center of inversion,\na fundamentally different mechanism of spin relaxation\ncan be realized. In such systems, the spin states ↑and\n↓are not degenerate Ek↑/ne}ationslash=Ek↓, but the condition is\nsatisfied Ek↑=E−k↓. The lack of a center of symmetry\ncontributes to splitting the states subjected to SOI. The\nsplitting can be described by entering an intrinsic mag-\nnetic field Bi(k) around which the electron spins precess\nat a Larmor frequency. The precession of the electron\nspin in the effective magnetic field together with the elec-\ntron momentum scattering leads to spin relaxation, and\nωs∼ω−1\np4. Below we consider the spin relaxation mech-\nanisminmagneticallyorderedstructures, whicharechar-\nacterized by a spiral arrangement of magnetic moments\nrelative to some crystal axes5. The simplest case of such\nstructures is an antiferromagnetic spiral or a helicoid;\nthey can be found in rare earth metals ( Eu, Tb, Dy ) and\nsomeoxidecompounds. The structureofthis type canbe\nrepresented as a sequence of atomic planes perpendicular\nto the axis of a helicoid. In this case, atoms in each of\nthe planes have ferromagnetically ordered magnetic mo-\nments. However, the magnetic moments in neighboring\nplanes turn at some angle θdepending on the ratio of ex-\nchange interactions. This is because of the coexistence of\npositive exchange interaction between the nearest atomicneighbors and negative exchange interaction between the\nneighbors following the closest ones. The components of\nthe magnetic moments of atoms oscillate in the plane\nof the magnetic layer Sx=S0xsinkz, S y=S0ycoskz.\nIf, in this case, Sz/ne}ationslash= 0, we have a ferromagnetic spiral\nwith a resulting moment. If also oscillates Szby a har-\nmonic law, a complex magnetic structure emerges. In\nsuch structures, the exchange interaction between zone\nchargecarriersand localizedmoments is described by the\nwell-known expression Hex=−J/summationtext\nis·Swherejis the\nexchangeconstant. Inthe mean-fieldapproximation,this\ninteraction can be represented as Hex=−m(r)Hef(r),\nwherem(r) =gµBS(r),g– is the spectroscopic split-\nting factor, µBis the Bohr magneton, and Hef(r) is the\ninternal effective magnetic field.\nIt is obvious that the spatial variation of the internal\nfield in spiral magnetic structures should affect the spin\ndynamics of conduction electrons. Indeed, the spin of an\nelectron in a state with quasi-momentum ( k) precesses in\nthe effective magnetic field Hef(r) only for a time of the\norder of the elastic scattering time τp. After scattering,\nthe electron goes into a state ( k′) where the effective\nmagnetic field has a different direction. Consequently,\nthe evolution of the spin dynamics under such conditions\nturns out to be associated with the electron momentum\nrelaxation.\nLet us look into a simple spiral when the effec-\ntive magnetic field rotates in a plane ( xy), where\nHef(H0cos(Qz), H0sin(Qz),0) ,Q= 2π/Λ, Λ is the\nperiod of the spiral structure. To gain insight into the\nspin dynamics, we estimate the spin relaxation mecha-\nnism realized under above conditions. The system at\nhand is assumed to expose to an electric field directed\nalong the axis z ( E= (0,0E0) . The Hamiltonian of the\nsystem can be written in the form H=Hk+Hs+HeE+\nHv+Hev,, whereHk, Hsare the operators of the kinetic\nand spin energy of electrons interacting with the lattice\nHevand the external electric field HeE=−e/summationtext\niEriand\nHvis the Hamiltonian of the lattice.\nHk=/summationdisplay\njp2\ni/2m,\nHs=−gµB/summationdisplay\njsiHef=−¯hωsf/summationdisplay\nj(s+eiQzj+s−\nje−iQzj)\n,(1)\nwhereωsf=geH0/2mocis the electron precession fre-2\nquency in the effective field, s±=sx±isy. The spin\ndynamics is determined by macroscopic equations of mo-\ntionforthespinsubsystemofelectrons. Themacroscopic\nequations of motion can be deduced by averaging micro-\nscopic equations of motion over the non-equilibrium sta-\ntistical operator ρ(t) . To begin with, we write down\nthe microscopic equations of motion for the longitudinal\ncomponents of the spin and momentum of electrons. We\nhave\n˙sz=iωsf{s+eiQz−s−e−iQz}+ ˙sz\nsv,(2)\n˙pz=−eEN+iQ¯hωsf{s+eiQz−s−e−iQz}+ ˙pz\nev(r),.\n(3)\n˙A= (i¯h)−1[A, H],˙Aij= (i¯h)−1[A, Hij].Averaging\nthe microscopic equations of motion over the operator\nρ(t), we arrive at < Ai>t=Sp{Aiρ(t)}. The explicit\nform of the operator ρ(t) can be found within the NSO\nmethod6. Suppose that we have carried out this proce-\ndure. As a result, we come to the following system of\nmacroscopic equations.\n<˙sz>t=iωsf{< s+eiQz>t−< s−e−iQz>t}+<˙sz\nev>t,\n(4)\nand\n<˙pz>=iQ¯hωsf{< s+eiQz>t−< s−e−iQz>t}−\n−eEn0+<˙pz\nev>t. (5)\nwhere< N >t=nis the electron concentration. The\ncollisional summands in the balance equations (4) and\n(5) can be represented as follows6,7:\n<˙sz\nev>t∼(sz, sz)0ωs, <˙pz\nev>t∼(pz, pz)0ωp,(6)\nwhere\n(A, B)0=1/integraldisplay\n0dτ Sp{Aρτ\n0∆Bρ1−τ\n0},(7)\nwhere ∆ A=A−< A > 0, <···>0=Sp{···ρ0},ρ0is\nthe equilibrium Gibbs distribution. ωsis the relaxationfrequency of the longitudinal spin components, and ωp\nis the momentum relaxation frequency; both latter are\ncalculated in the Born approximation over the scatter-\nelectron interaction.\nωs=gµB\nχ0/integraldisplay\n−∞dteǫt(˙sz\nev,˙sz\nev(t))0,\nωp=1\nmnT0/integraldisplay\n−∞dteǫt(˙pz\nev,˙pz\nev(t))0, ǫ→+0.(8)\nHereχis the paramagnetic susceptibility of an electron\ngas\nχ=(gµB)2\n2T·(s+, s−)0=gµB\nH0·< sz>0,(9)\nT≡kBTis the temperature in energy units. Taking\na stationary case into account, we finally obtain the ex-\npression for the spin relaxation frequency in a helicoidal\nmagnet:\nωs∼mnT\n¯hQ(sz, sz)0·ωp (10)\nThus, in spiral magnets as well as in the Elliott-Jafet\nmechanism, thespiralrelaxationisrelatedtothe electron\nmomentum relaxation and ωs∼ωp. However, against\nthe Elliott-Jafet mechanism, the spin relaxation mech-\nanism in spiral magnetic structures is due to the pres-\nence of the internal spatially-dependent magnetic field in\nthem. The evolution of the spin dynamics is determined\nby both the period of the magnetic structure and the\nmomentum relaxation frequency.\nACKNOWLEDGMENTS The research was carried\nout within the state assignment of Minobrnauki of Rus-\nsia (theme ”Spin No AAAA-A18-118020290104-2), sup-\nported in part by RFBR (project No. 19-02-00038/19).\n∗ligor47@mail.ru\n1E.I. Rashba, Electron spin operation by electric elds: spin\ndynamics and spin injection, Physica E 20, 189 (2004).\n2R. J. Elliott, Theory of the effect of spin-orbit coupling on\nmagnetic rresonance in some semiconductors, Phys. Rev.,\n96266 (1954).\n3Y. Y a f e t, Sol. g factors and spin-lattice relaxation of\nconduction electrons, in Solid State Physics, F. Seitz and\nD. Turnbull, Eds. New York: Academic, 141 (1963).\n4M. I. Dyakonov, V. I. Perel, Spin relaxation of conductionelectrons in noncentrosymetric semiconductors, Sov. Phys .\nSolid State, 13, 3023 (1972).\n5Yu. A. Izyumov, Modulated, or long-periodic, magnetic\nstructures of crystals” Sov. Phys. Usp 27, 845 (1984).\n6I. I. Lyapilin, M. S. Okorokov, V.V. Ustinov, Spin effects in-\nduced by thermal perturbation in a normal metal/magnetic\ninsulator system, Phys.Rev B. 91, 195309-7 (2015).\n7H.M. Bikkin, I. I. Lyapilin, Non equlibrium thermodynam-\nics and physical kinetics, Walter de Gruyter GmbH, Berlin\n2014, p. 359."    },    {        "title": "1606.03403v1.Nanoscale_confinement_of_ultrafast_spin_transfer_torque_exciting_non_uniform_spin_dynamics_by_femtosecond_spin_current_pulses.pdf",        "content": "Nanoscale con\fnement of ultrafast spin transfer torque exciting non-uniform spin\ndynamics by femtosecond spin current pulses\nIlya Razdolski,1,\u0003Alexandr Alekhin,1Nikita Ilin,1Jan P. Meyburg,2\nVladimir Roddatis,3Detlef Diesing,2Uwe Bovensiepen,4and Alexey Melnikov1,y\n1Physical Chemistry Dept., Fritz Haber Institute of Max Planck Society, Faradayweg 4-6, 14195 Berlin, Germany\n2Faculty of Chemistry, University of Duisburg-Essen, Universit atsstr. 5, 45141 Essen, Germany\n3Universit at G ottingen, Institut f ur Materialphysik,\nFriedrich-Hund-Platz 1, 37077 G ottingen, Germany\n4Faculty of Physics and Center for Nanointegration (CENIDE),\nUniversity of Duisburg-Essen, Lotharstr. 1, 47057 Duisburg, Germany\n(Dated: October 16, 2018)\nSpintronics had a widespread impact over the past decades due to transferring information by spin\nrather than electric currents [1{6]. Its further development requires miniaturization and reduction\nof characteristic timescales of spin dynamics combining the sub-nanometer spatial and femtosecond\ntemporal ranges [7]. These demands shift the focus of interest towards the fundamental open question\nof the interaction of femtosecond spin current (SC) pulses with a ferromagnet (FM). The spatio-\ntemporal properties of the impulsive spin transfer torque (STT) exerted by ultrashort SC pulses on\nthe FM open the time domain for probing non-uniform magnetization dynamics. Here we employ\nlaser-generated ultrashort SC pulses [8{10] for driving ultrafast spin dynamics in FM and analyzing\nits transient local source. Transverse spins injected into FM excite inhomogeneous high-frequency\nspin dynamics up to 0.6 THz, indicating that the perturbation of the FM magnetization is con\fned\nto 2 nm.\nApproaching the timescales of the underlying elementary processes, SCs with femtosecond pulse duration can pro-\nvide valuable fundamental insights into the ultrafast spin dynamics. In addition to manipulating the magnetization\nin multilayer structures [11{13], ultrashort SC pulses were shown to exert STT and thus drive the coherent magneti-\nzation dynamics in semiconductor \flms [14] or perpendicularly coupled magnetic bilayers [13, 15]. However, a nearly\nuniform in-depth STT limited the coherent magnetization dynamics to the homogeneous precession (with k= 0)\non the picosecond timescale, similarly to other known ultrafast excitation mechanisms [16{20]. An inhomogeneous\nFIG. 1: Laser-induced excitation of spin dynamics via spin current pulses. Laser pump pulse impinging on the\n\frst Fe layer (emitter) excites hot electrons at elevated energies E\";#. Because of the unequal energies and therefore di\u000berent\ntransmittance of the Fe/Au interface for the majority (green) and minority (red) hot electrons [10], the emission of hot electrons\ninto Au is largely spin-polarised. Having crossed the Au layer in a nearly ballistic regime, the electrons reach the FM2 layer and\ntransfer their spin to it. Owing to that, STT is exerted on the magnetization MCwhich is driven out of the equilibrium and\nstarts precessional dynamics. Due to the spatial con\fnement of the STT perturbation [22] (blue shaded area), spin waves with\na broad spectrum of non-zero wavevectors kare excited and can be probed by the magneto-optical Kerr e\u000bect in the collector.\n\u0003Electronic address: razdolski@fhi-berlin.mpg.de\nyElectronic address: melnikov@fhi-berlin.mpg.dearXiv:1606.03403v1  [cond-mat.mes-hall]  10 Jun 20162\nperturbation of magnetization mediated by the interfacial STT contribution [10, 22] can excite spin waves in a FM\n\flm, thus extending the spin dynamics into higher frequency range. Moreover, the spatial properties of the SC-driven\nSTT excitation can be inferred from the spectral analysis of these high-frequency spin waves. In this Letter, we realize\nthis approach in epitaxial Fe/Au/Fe/MgO(001) multilayers by means of all-optical excitation and detection of the\nstanding spin waves in a 15-nm thick Fe \flm via the STT mechanism (see Fig. 1). Further, we demonstrate the ex-\ncited complex mode structure of the non-uniform magnetization dynamics. We show that the ultrashort laser-induced\nSC pulses constitute a convenient tool to excite spin waves and study the interaction of spins with a non-collinear\nmagnetization.\nThe concept of our experiment is illustrated in Fig. 1. A 14 fs long laser pump pulse absorbed in the FM1\nlayer (emitter, thickness 16 nm) results in the emission of the subpicosecond SC pulse into Au via the non-thermal\nspin-dependent Seebeck e\u000bect [10]. Owing to their large lifetimes in Au [21], the transport of hot electrons in a quasi-\nballistic regime delivers spin angular momentum to the FM2 layer (collector, 14 nm-thick). With this spin polarization\northogonal to the collector magnetization MC, both re\rected and transmitted electrons in the SC pulse transfer the\ntransient angular momentum density \u0016(t) to FM2 and thus exert a STT on MC[22]. Locally, the interaction of \u0016(t)\nwith the magnetization MCis given by: [1]\n1\n\r@MC\n@t\f\f\f\f\nSTT=\u0015MC\u0002[~ \u0016(t)\u0002MC]: (1)\nThe ultrafast impulsive STT excitation triggers spin dynamics in the collector. Because of the strong localization\nof the delivered perturbation in the vicinity of the Au/Fe interface [10, 22], spin waves with non-zero k-vectors are\nexcited along with the homogeneous precession of magnetization MC(Fig. 2,a). The spin-wave dispersion for a thin\nmagnetic \flm (Fig. 2,b) is given by [26]:\nf(k) =\rp\n(Han+Dk2)\u0001(Han+Hdem+Dk2); (2)\nwhereHdem\u00192:1 T (Fig. 2,b), \r\u001928 GHz/T is the gyromagnetic ratio, and DFe= 280 A2\u0001meV [27]. In a \flm\nof a \fnite thickness, only the standing waves with kn=\u0019n=d are supported (Fig. 2,a-b), where the zero derivative\nof the magnetic moment at the interfaces [26] is ensured by the low Fe/Au interface anisotropy. Rich dynamics of\nthe time-resolved MOKE signals observed in our experiments (Fig. 2,c) is associated with the superposition of these\nlong-lived standing spin-wave modes.\nIn lieu of \ftting the raw data, we performed measurements in various combinations of the magnetization directions\nof FM1 and FM2, while keeping them perpendicular to each other. Based on the parity rules with respect to both ~ \u0016\nandMC(see Supplementary Note 1), we disentangle the polar and longitudinal MOKE contributions to magnetization\ndynamics in the FM2 layer (Fig. 2d). This procedure results in background-free data for MOKE rotation and ellipticity\n(Fig. 3).\nIt is clearly seen that the transient MOKE signals have a complex structure demonstrating oscillations at multiple\nfrequencies. To unravel their nature, we performed the Fourier analysis of the time-resolved MOKE data. The analysis\nshows the presence of four frequencies (Fig. 4,a) besides the fundamental excitation which corresponds to the uniform\nmagnetization precession ( k= 0). We argue that these frequencies indicate the excitation of the long-lived (up to\n500 ps) standing spin waves in the FM2 \flm. The striking match between the frequencies obtained from the Fourier\nanalysis and those calculated from the standing spin-wave dispersion illustrated in Fig. 4,a veri\fes our explanation.\nWe \ftted a set of the exponentially decaying oscillations with the \fve frequencies given by Eq. (2) to the experimental\ndata. The excellent quality of the \ftting results corroborate our understanding of the standing spin-wave excitation\n(Fig. 3, solid lines). The di\u000berence between the MOKE rotation and ellipticity data, as well as between the polar and\nlongitudinal MOKE e\u000bects is attributed to their unequal sensitivity to the in-depth magnetization pro\fle m(z) [5, 24]\nand thus to various standing spin-wave eigenmodes (see Supplementary Note 2).\nThese data unambiguously prove that the STT mechanism is capable of exciting high-frequency modes of spin\nprecession in FM \flms. Note that because the data shown in Figs. 2,3 were obtained in the absence of an external\nmagnetic \feld, the heating mechanism of the excitation of the magnetization precession relying on the ultrafast\nquenching of the magnetic anisotropy is inactive (see Supplementary Note 3).\nIn order to elucidate the STT-induced excitation, we now turn to the ultrafast timescale at t <1 ps. Figure 4,b\nshows the initial stage of the STT-induced magnetization dynamics, when the emitter and collector \flms are mag-\nnetized longitudinally and transversely, respectively. Due to angular momentum conservation, the longitudinal spin\npolarization of the SC pulse drives the rapid initial increase of the corresponding magnetization projection starting at\napproximately 50 fs delay, indicating the ballistic SC propagation through the 55 nm-thick Au spacer [9]. We found\nthe duration of the SC pulse \u001cSC.250 fs, in agreement with the results of the direct SC measurements [10]. On3\nFIG. 2: Laser-induced magnetization dynamics in the Fe/Au/Fe/MgO(001) multilayer. a, b , Spin excitations\nin a thin Fe(001) \flm magnetized in-plane. The demagnetizing \feld Hdemand the two in-plane easy axes with the e\u000bective\ncrystalline anisotropy \feld Hanresult in the uniform magnetization precession with a frequency f0\u001910 GHz [6]. A \flm with\nthe thickness dand the exchange sti\u000bness DFecan sustain standing spin waves with a discrete set of eigenwavevectors kn=\u0019n=d\nand frequencies fngiven by Eq. (2). c, Time-resolved transient polar MOKE rotation obtained in four di\u000berent geometries for\northogonal magnetizations of the emitter and collector. The periodic signals with multiple frequencies indicate the excitation\nof the standing spin-wave eigenmodes. d, Separation of the longitudinal and polar MOKE contributions. Electrons in the\nSC pulse propagate with the velocity vand magnetic moment ~ \u0016. For the longitudinal SC polarization ~ \u0016and transverse\nmagnetization of the collector MC, the solid (dashed) blue arrows depict the sign of the longitudinal, L (polar, P) transient\nmagnetization component \u0001 M. The dotted lines illustrate the appearance of the polar component of MCdue to the precession\nof magnetization.\nthe picosecond timescale only resonant spin waves from the initially excited wavepacket with a broad distribution of\nk-vectors endure. Thus, standing spin waves in the collector are formed, giving rise to the oscillatory dynamics seen\nin Fig. 4,b.\nThe transverse angular momentum is transferred from the injected electrons to the magnetization MCin the\nvicinity of the interface, locally driving magnetization dynamics according to Eq. 1. Thus, besides the SC pulse\nduration\u001cSC.250 fs allowing the excitation of the modes with frequencies up to 1 =2\u001cSC\u00192 THz, the spectrum\nof the excited magnons is limited by the k-vector spectrum of the delivered excitation, or, in other words, the STT\ncharacteristic depth \u0015STT. In Fig. 4,b it is seen that the \frst detected oscillation is the n= 4 mode with a frequency\noff4= 0:56 THz. As such, the n= 5 and the higher modes are supposedly not excited, due to either spatial or\ntemporal limitations. The estimated \u001cSCcomplies with the requirement for the e\u000ecient impulsive excitation of the\nn= 5 eigenmode, \u001cSC< T 5=2\u00190:6 ps. Thus, the temporal constraint can be excluded and we need to invoke the4\nFIG. 3: STT-induced magnetization dynamics. Transient MOKE signals in both rotation (top) and ellipticity (bottom).\nPolar and longitudinal MOKE e\u000bects are separated. Solid lines are the results of the \ftting procedure with a set of exponentially\ndecaying oscillations with their frequencies given by Eq. (2). The pronounced similarity between the two polar MOKE dynamics\n(red and blue dots) demonstrates the robustness of the measurements. The indicated \u0019=2 phase shift between the polar and\nlongitudinal MOKE components con\frms the precessional nature of the magnetization dynamics.\nspatial inhomogeneity-driven limitation on the excited eigenmodes. Figure 4,c illustrates that for the standing spin\nwaves with open ends, the critical STT excitation depth is about a quarter of the wavelength. This means that the\neigenmode with k5is not e\u000eciently excited if \u0015STT>1=4\u00022\u0019=k5\u00191:5 nm.\nWe note that there are other possible limitation mechanisms related to the lifetime of the eigenmodes. From the\n\ftting procedure, the eigenmode with k4= 4\u0019=dwas found to live for about 6 ps. The eigenmodes with even larger\nkmost likely have even shorter lifetimes (comparable to their periods or shorter) and thus render invisible in our\nexperiments. However, the very fact that the eigenmode with k4is unambiguously observed in the experiments can\nbe used for a rather conservative estimate \u0015STT.1=4\u00022\u0019=k4\u00192 nm. Together with the L-MOKE data shown in\nFig. 4b, this value indicates that the angular momentum transfer to the FM2 layer results in a 1.3\u000etilt of MCfrom\nits equilibrium direction within \u0015STT.\nWith the ultimately short (2 nm) characteristic depth, the hot electron-induced STT remains one the most e\u000ecient\nmechanisms for the excitation of the non-uniform magnetization dynamics with large k-vectors. For comparison, the\noptical penetration depth in transition metals is of the order of the skin depth \u000e\u001910\u000015 nm. In this case, the\nexcitation of the standing spin waves with non-zero k-vectors would become possible in relatively thick \flms only,\nwhich e\u000bectively reduces the eigenmodes frequencies down to the tens of GHz [6, 28]. In this regard, the ultrashort\nlaser-induced pulses of SC are a unique tool capable of exciting large- k, sub-THz spin waves in FM \flms. Our results\ndemonstrate the extreme abilities of the SC pulses at exciting non-uniform spin dynamics and elucidate the interaction\nof the SC with a non-collinear magnetization. We found that on average, a magnetic moment of about 1 \u0016Bper Fe\natom at the Au/Fe interface is transferred to the collector, showing high promise for the magnetization switching in\nthin FM layers. With these results in hand, further steps towards ultrafast spintronics can be expected in the near\nfuture.\nMethods\nSample fabrication. Epitaxial Fe/Au/Fe structures with 15 and 16 nm Fe layers and 55 nm Au spacer capped with 3 nm of\nAu were grown on optically transparent MgO(001) substrates with a thickness of 0.5 mm. The substrates were cleaned in the\nultrasonics baths of ethanol, isopropanol, acetone (15 min each at elevated temperature of \u0019310 K). After being mounted in\na UHV chamber, the substrates were exposed to oxygen with a partial pressure of 2 \u000210\u00003mbar at a temperature of 540 K\nfor 30 min in order to remove spurious amounts of diamond polish left by the last fabrication step. With UHV conditions\nreached, Fe and the \frst nm of the interstitial Au layer were evaporated at 460 K. The samples were then cooled down and\nthe additional amount of Au was evaporated at room temperature. Transparent substrate and thin capping Au layer provided\noptical access from both sides. The scanning transmission electron microscopy revealed excellent epitaxial quality and \ratness\nof interfaces [10] (see Supplementary Note 4).5\nFIG. 4: Excitation of the standing spin waves. a , Fourier spectrum of the experimental data, averaged over several\ndatasets (left panel) and the frequencies of the standing spin waves in a 15 nm-thick Fe \flm (right panel, red symbols). Red\nsolid line is the calculated spin waves dispersion curve from Eq. (2) with the indicated magnon sti\u000bness DFe. Along with the\nfundamental uniform precession ( n= 0), four higher modes are shown which were detected in the experiment (full dots). b,\nTransient MOKE rotation data on the ultrashort timescale. The rapid onset of the longitudinal component (green circles) is\nfollowed by a slower increase of the polar one (blue circles) due to the magnetization precession around its equilibrium. The\nred shaded area reproduces the SC pulse as measured in Ref. [10]. The purple line illustrates the 4 \u0000thspin wave eigenmode\nwith the frequency f4observed in the transient polar MOKE component. c, Excitation of the higher eigenmodes is limited by\nthe spatial in-depth pro\fle of the STT perturbation MC(red curve). The excitation region with a characteristic depth \u0015STT\nis sketched by the red shaded area. The excitation e\u000eciency given by the convolution of the STT excitation pro\fle and the\nin-depth pro\fle of a particular standing spinwave eigenmode (blue curves) is represented with red and blue shaded areas. In the\ncase of a homogeneous precession (the top eigenmode with n= 0), this overlap is always non-zero. The eigenmodes with small\nnon-zeronare still excited albeit with smaller e\u000eciency. However, for the eigenmode with a large n(bottom,n= 20), the\nconvolution yields positive and negative regions (red and blue shaded areas) which cancel each other out, thus greatly reducing\nthe excitation e\u000eciency.\nMagnetic characterization. Prior to the time-resolved experiments, magnetic properties of the samples were characterized\nby means of the static MOKE measurements. The hysteresis loops of the MOKE rotation and ellipticity were obtained from\nboth sides of the samples (see Supplementary Note 5). The sample was placed in such a way that its easy anisotropy axes were\noriented along the longitudinal and transverse magnetic \feld directions. During the measurements, a main magnetic \feld was\nscanned (from -20 to 20 Oe) across the longitudinal direction, while a weak auxiliary magnetic \feld (up to 5 Oe) was applied\nin the transverse direction. Both \felds were produced by electromagnets. This procedure allowed us to realize the two-step\nswitching of the magnetization from one longitudinal direction to another via the intermediate transversal direction (along the\nauxiliary magnetic \feld). In the MOKE measurements on the collector and emitter \flms, we obtained di\u000berent width of the\nhysteresis loops. This allowed us to attain an orthogonal magnetic state, where the magnetizations of the collector and the\nemitter were perpendicular to each other.\nTime-resolved measurements. In our back pump-front probe experiments we used p-polarized 800 nm, 14 fs output of a\ncommercial cavity-dumped Ti:sapphire oscillator (Mantis, Coherent) operated at 1 MHz, which was split at a power ratio 4:1\ninto pump and probe pulses. Both beams were chopped with di\u000berent frequencies and focused with o\u000b-axis parabolic mirrors\nto about 10 \u0016m spot size, resulting in a pump \ruence of the order of 10 mJ/cm2. The signal re\rected from the sample was split\ninto two identical shoulders in order to allow for simultaneous measurements of the MOKE rotation and ellipticity. Balanced\ndetection scheme in each shoulder was realized with the help of a Glan-laser prism and two photodiodes. In the ellipticity\nshoulder a quarter-wave plate was installed before the prism. The measurements were performed at room temperature. The\nzero time delay was determined before each measurement with the help of the cross-correlation second harmonic generation\nsignal.\n[1] Slonczewski, J. C. Current-driven excitation of magnetic multilayers. J. Magn. Magn. Mater. 159, L1{L7 (1996).\n[2] Berger, L. Emission of spin waves by a magnetic multilayer traversed by a current. Phys. Rev. B 54, 9353{9358 (1996).\n[3] Kiselev, S. I. et al. Microwave oscillations of a nanomagnet driven by a spin-polarized current. Nature 425, 380{383 (2003).\n[4] Krivorotov, I. N. et al. Time-Domain Measurements of Nanomagnet Dynamics Driven by Spin-Transfer Torques. Science\n307, 228{231 (2005).6\n[5] Woltersdorf, G. et al. Magnetization Dynamics due to Pure Spin Currents in Magnetic Double Layers. Phys. Rev. Lett. 99,\n246603 (2007).\n[6] Brataas, A., Kent, A. D. & Ohmo H. Current-induced torques in magnetic materials. Nat. Mater. 11, 372{381 (2012).\n[7] Bader, S. D. & Parkin, S. S. P. Spintronics. Annu. Rev. Condens. Matter Phys. 1, 71{88 (2010).\n[8] Malinowski, G. et al. Control of speed and e\u000eciency of ultrafast demagnetization by direct transfer of spin angular momen-\ntum. Nature Phys. 4, 855{858 (2008).\n[9] Melnikov, A. et al. Ultrafast Transport of Laser-Excited Spin-Polarized Carriers in Au/Fe/MgO(001). Phys. Rev. Lett. 107,\n076601 (2011).\n[10] Alekhin, A. et al. Generation of femtosecond spin current pulses via non-thermal spin-dependent Seebeck e\u000bect and their\ninteraction with ferromagnets in spin valves (ArXiv.org, 2016).\n[11] Rudolf, D. et al. Ultrafast magnetization enhancement in metallic multilayers driven by superdi\u000busive spin current. Nature\nCommun. 3, 1037 (2012).\n[12] Choi, G.-M. et al. Spin current generated by thermally driven ultrafast demagnetization. Nature Commun. 5, 4334 (2014).\n[13] Choi, G.-M. et al. Thermal spin-transfer torque driven by the spin-dependent Seebeck e\u000bect in metallic spin-valves. Nature\nPhys. 11, 576{582 (2015).\n[14] Nemec, P. et al. Experimental observation of the optical spin transfer torque. Nat. Phys. 8, 411{415 (2012).\n[15] Schellekens, A. J. et al. Ultrafast spin-transfer torque driven by femtosecond pulsed-laser excitation. Nat. Commun. 5,\n4333 (2014).\n[16] Ju, G. et al. Ultrafast Time Resolved Photoinduced Magnetization Rotation in a Ferromagnetic/Antiferromagnetic Ex-\nchange Coupled System. Phys. Rev. Lett. 82, 3705{3708 (1999).\n[17] Zhang, Q. et al. Coherent Magnetization Rotation and Phase Control by Ultrashort Optical Pulses in CrO 2Thin Films.\nPhys. Rev. Lett. 89, 177402 (2002).\n[18] Kimel, A. et al. Ultrafast non-thermal control of magnetization by instantaneous photomagnetic pulses. Nature 435,\n655-657 (2005).\n[19] Hansteen, F. et al. Femtosecond Photomagnetic Switching of Spins in Ferrimagnetic Garnet Films. Phys. Rev. Lett. 95,\n047402 (2005).\n[20] Afanasiev, D. et al. Laser Excitation of Lattice-Driven Anharmonic Magnetization Dynamics in Dielectric FeBO 3. Phys.\nRev. Lett. 112, 147403 (2014).\n[21] Zhukov, V. P. et al. Lifetimes and inelastic mean free path of low-energy excited electrons in Fe, Ni, Pt, and Au: Ab initio\nGW+T calculations. Phys. Rev. B 73, 125105 (2006).\n[22] Stiles, M. D. & Zangwill, A. Anatomy of spin-transfer torque. Phys. Rev. B 66, 014407 (2002).\n[23] Carpene, E. et al. Ultrafast three-dimensional magnetization precession and magnetic anisotropy of a photoexcited thin\n\flm of iron. Phys. Rev. B 81, 060415(R) (2010).\n[24] Hamrle, J. et al. Analytical expression of the magneto-optical Kerr e\u000bect and Brillouin light scattering intensity arising\nfrom dynamic magnetization. J. Phys. D: Appl. Phys. 43, 325004 (2010).\n[25] Wieczorek, J. et al. Separation of ultrafast spin currents and spin-\rip scattering in Co/Cu(001) driven by femtosecond\nlaser excitation employing the complex magneto-optical Kerr e\u000bect. Phys. Rev. B 92, 174410 (2015).\n[26] Gurevich, A. G. & Melkov, G. A. Magnetization Oscillations and Waves (CRC Press, 1996).\n[27] Mook, H. A. & Nicklow, R. M. Phys. Rev. B 7, 336{342 (1973).\n[28] van Kampen, M. et al. All-optical Probe of Coherent Spin Waves. Phys. Rev. Lett. 88, 227201 (2002).\nAcknowledgements The authors are indebted to A. Kirilyuk, S. Sanvito, S. O. Demokritov, M. Weinelt, P.\nOppeneer and T. Kampfrath for fruitful discussions, as well as M. Wolf for the \fnancial support. This work was\npartially funded by the Deutsche Forschungsgemeinschaft (ME 3570/1, Sfb 616) and by the EU 7-th framework\nprogram (CRONOS).\nCorrespondence\nCorrespondence and request for materials should be addressed to I.R. or A.M.7\nSupplementary Information\nSupplementary Note 1: Separation of the polar and longitudinal MOKE contributions\nThe spin transfer torque (STT)-induced contribution to the magnetization dynamics is given by [1]:\n1\n\r@M\n@t\f\f\f\f\nSTT=\u0015M\u0002[~ \u0016(t)\u0002M]: (3)\nHere\u0015is the scaling factor, ~ \u0016is the magnetic moment carried by the spin current and Mis the magnetization\nof a ferromagnet. It is seen that the STT-induced magnetization dynamics has di\u000berent parity with respect to both\n~ \u0016andM. Whereas variations of Mare odd with respect to ~ \u0016, at the same time they are even with respect to M\nitself. In other words, consider the collector magnetized transversely along the y-axis (My6= 0), and the emitter\nmagnetization (proportional to ~ \u0016) along the x-axis (longitudinal). Here, upon the impulsive STT excitation the\ncollector magnetization Macquires ax-projection Mx. The sign of Mxreplicates the sign of \u0016xand is independent\nof the sign of My.\nFurther, the magnetization precession around its equilibrium (parallel to the y-axis) leads to the emergence of the dy-\nnamic polar (out-of-plane) component Mz. Moreover, at the initial stage of the precession \u0001 Mz/dMz=dt/HanMx,\nwhereHanis the anisotropy \feld parallel to the y-axis and thus the non-perturbed direction of the magnetization My.\nIt is seen that the sign of the polar component Mzis determined by both signs of the projections \u0016x,My, whereas\nthe sign of the longitudinal component Mxdepends on the sign of the \u0016xonly. Owing to that, it becomes possible\nto separate the polar MOKE contribution from the longitudinal one when dealing with four datasets measured in\nvarious con\fgurations of the equilibrium magnetizations, \u0006\u0016x,\u0006My(Fig. 2c-d in the main Manuscript). Indeed,\nconsider the notation where U, D (R, L) as in up, down (right, left) denote the magnetization and magnetic moment\ndirections parallel and antiparallel to the y(x) axis. The experimental con\fgurations with two orthogonal magne-\ntizations discussed above can be written as RU, RD, LU, LD, where the \frst and the second symbols represent the\nmagnetization of the emitter (proportional to \u0016) and the collector M, respectively. As such, odd with respect to \u0016,\nthe pure longitudinal dynamics of Kerr rotation is given by\n'L=1\n4('RU+'RD\u0000'LU\u0000'LD);\nFurther, odd (even) with respect to the magnetization of the emitter (collector), the polar Kerr dynamics is given by\n'P=1\n4('RU\u0000'RD+'LU\u0000'LD):\nSupplementary Note 2: MOKE in-depth sensitivity\nDespite having no total magnetic moment, standing spin waves can be visualized in MOKE experiments because of\nthe in-depth selective sensitivity of MOKE. Figure 5 illustrates the calculated speci\fc MOKE rotation and ellipticity\nfor the 15 nm-thick Fe \flms. The data were calculated using the layer-by-layer approach based on the medium\nboundary and propagation matrices [2{5] for the p-polarized incident beam.\nThe total MOKE response \u0012;\"is given by a convolution of the sensitivity s(z) and the certain in-depth pro\fle of\nmagnetization m(z):\n\u0012;\"=Zd\n0s\u0012;\"(z)m(z)dz (4)\nTwo main considerations can be inferred from the data shown in Fig. 5. Firstly, it is seen that longitudinal MOKE\nrotation and ellipticity are predominantly sensitive to the opposite interfaces of the \flm. On the contrary, polar\nMOKE rotation and ellipticity exhibit very similar sensitivities to local magnetization m(z). As such, the di\u000berences\nin the transient rotation and ellipticity in the case of the longitudinal MOKE could be attributed to a di\u000berent\ndynamics at the two interfaces of the Fe \flm.8\nFIG. 5: Sensitivity s(z) of (a) longitudinal and (b) polar MOKE rotation (full blue circles) and ellipticity (open red circles)\nin a 15 nm-thick Fe \flm for the p-polarized light.\nSupplementary Note 3: Separation of the heating and STT contributions in the spin waves excitation\nThe experiments described in the main Manuscript were performed without external magnetic \feld. In order\nto study the heating e\u000bect on the transport-induced magnetization dynamics in Fe/Au/Fe trilayers, we performed\nadditional measurements with an external magnetic \feld ~Hon. This magnetic \feld was applied in the plane of the\nsample at an angle with respect to the easy axes of the two Fe \flms, so that the magnetizations of the emitter and\nthe collector were aligned orthogonally to each other (see Supplementary Note 5).\nFIG. 6: (a) Schematic of the heating and STT-induced magnetization dynamics for two opposite directions of H?\next, a projection\nof the external magnetic \feld perpendicular to the magnetization Mof the collector. Red and blue short arrows depict the\ninitial dynamics of the magnetization induced by the heating and STT mechanisms, respectively. (b-c) Heating- (red) and STT\n(blue)-induced contributions to the polar time-resolved MOKE signal (b) and their spectra (c).\nIn this con\fguration (Fig. 6,a), the equilibrium direction of the collector magnetization Mis determined by an\ninterplay of the external magnetic \feld ~Hext=~Hk\next+~H?\nextand the anisotropy \feld ~Han. The thermal action of the\nultrashort spin current pulse results in heating of the collector and thus quenching its magnetic anisotropy Han. As\nsuch, the equilibrium direction of the magnetization Mshifts giving rise to the precessional dynamics around He\u000b\nwith the frequency f0\u001910 GHz [6].\nFigure 6,a illustrates this mechanism along with the STT excitation discussed above. Note that according to the\nEq. (3), the STT (heating) mechanism is even (odd) with respect to the external \feld Hext. Thus, the direction of the\nheating-induced precession reverses together with the perpendicular projection applied external \feld H?\next. We note,\nhowever, that the STT mechanism does not depend on the external magnetic \feld as long as the directions of the\nmagnetizations of both emitter and collector are set. Thus, obtaining the MOKE data for the two opposite projections\nof the external \feld H?\nextperpendicular to the equilibrium collector magnetization (but keeping the magnetization M\nin place), we can separate the heating and STT contributions (Fig. 6,b). Whereas the STT-driven dynamics (blue\nsymbols) has a much larger amplitude and can boast a rich spectrum of frequencies, the heating-induced magnetization\ndynamics (red symbols) is small and consists of a single FMR mode only. The latter can be explained invoking the\ngood heat conductivity of Fe along with the large duration of heat pulses as compared to spin current ones. Both these\nfactors facilitate the excitation of the homogeneous precession with k= 0 (FMR) and inhibit any inhomogeneous9\n(withk>0) dynamics of thermal origin.\nSupplementary Note 4: Samples transmission electron microscopy\nThe Fe/Au/Fe/MgO(001) samples were capped with a 3 nm-thick Au protective layer. The structure of the\nsamples was characterized using a transmission electron microscope (TEM) TITAN 80-300 (FEI, USA) equipped\nwith a corrector of spherical aberration at image side. The microscope was operated at 300 kV. An example of\ncross-section specimen prepared by means of the Focused Ion Beam lift-out technique is shown in Fig. 7. The cross-\nsection was prepared at the spot used for the optical measurements. The micro-structural study reveals that both\nFe and Au \flms grow epitaxially and the roughness of Fe/MgO and Fe/Au interface varies from 0.7 to 1.5 nm. The\nprecise measurements of the layers thicknesses yield the values of 12 :7 and 14:6 nm for the Fe collector and emitter,\nrespectively, and 49 :4 nm for the Au spacer. We note that the collector thickness obtained from the microscopy is\nsomewhat smaller than the one used in the main manuscript for the data \ftting. However, because the independent\nparameter utilized in the \ftting routine is the ratio DFe=d2, this deviation can be mitigated by adjusting DFe. It is\nknown, for example, that thin Fe \flms of up to 24 monolayers demonstrate signi\fcantly lower sti\u000bness (160 meV A2\nas compared to 280 meV A2for the bulk Fe) [7]. In our case, the collector thickness dobtained using the TEM\nmicroscopy results in D\u0019225 meVA2, in between of these two values. Moreover, the lattice mismatch at the\nMgO/Fe interface leads to the appearance of dislocations and subsequent strain in the Fe \flm, which can also modify\nthe e\u000bective magnon sti\u000bness D. This strain could also induce the sti\u000bness anisotropy resulting in the inequality of\nDfor the magnons with their k-vectors parallel (as in Ref. [7]) and perpendicular (as in our work) to the \flm surface.\nFIG. 7: (a) Au/Fe/Au/Fe/MgO(001) cross-section TEM image. (b-c) High-resolution TEM images of \rat Fe/Au and Au/Fe\ninterfaces. The horizontal bar is 1 nm long. (d) Electron di\u000braction image.\nSupplementary Note 5: Realizing orthogonal magnetic con\fgurations\nStatic magnetic properties of the emitter and collector Fe layers were characterized by employing the magneto-\noptical Kerr e\u000bect (MOKE) measured independently. A thick (55 nm) Au spacer prevented magnetic coupling of the\ntwo Fe layers, as well as optical access to the collector from the emitter side and vice versa. As such, we performed\nsimultaneous characterization of the magnetic states of the collector and emitter using MOKE with two laser beams.\nWe note that thin Fe \flms have two in-plane orthogonal easy axes corresponding to the h100iandh010idirections\n(see also Ref. [8]). In our experiments, the sample was placed in such a way that one of these axes was in the incidence10\nplane. We found that both magnetizations were switched to the opposite direction along one of the easy axes while\nthe magnetic \feld in the longitudinal MOKE geometry HLwas swept from -20 to 20 Oe.\nIn order to force the magnetization along the other easy axis we have applied a small auxiliary transversal magnetic\n\feldHTperpendicular to the main one. We found that for the values of HTas small as 4 Oe the switching of\nmagnetization acquired a step in the middle (Fig. 8,b-c) corresponding to the zero MOKE rotation and ellipticity.\nAt the same time, when sweeping the transversal \feld while keeping the small HLon, we observed a sharp feature\nwith a non-zero the MOKE rotation and ellipticity response (Fig. 8,a,d). This indicates that the magnetization\nswitching proceeds via an intermediate state which corresponds to the orthogonal orientation of the magnetization,\nalong the second easy axis in the \flm. As such, when sweeping the main magnetic \feld HLacross the hysteresis loop\nin the presence of the auxiliary \feld HTwhich could be positive or negative, we were able to attain any of the four\norientations of the magnetization in the Fe \flm (Fig. 8,e-f).\nIt is seen in Fig. 8,a-d that the hysteresis loops of the emitter and the collector have di\u000berent widths, i.e. the\ncoercive \felds of the two \flms are unequal. This allows for the realization of a orthogonal con\fguration, where the\nmagnetizations of the emitter and the collector are perpendicular to each other. For instance, if in the magnetic\nset-up used for the measurements shown in Fig. 8,a-b longitudinal magnetic \feld of \u001912 Oe was applied, the emitter\nand the collector were magnetized along the longitudinal and transversal axes, respectively. Varying the roles of the\nlongitudinal and transversal magnetic \felds (main or auxiliary) as well as their signs, it was possible to realize all 8\northogonal magnetic con\fgurations of the multilayer sample.\nFIG. 8: (a-b) Hysteresis loops measured with MOKE from the emitter (a) and the collector (b) Fe \flms. (c-d) Schematic of\nthe magnetization switching using main HLand auxiliary HTmagnetic \felds applied perpendicular to each other along the\ntwo easy axes of magnetic anisotropy in Fe(100). The four coloured points at the sketch of the MOKE hysteresis loop (d)\ncorrespond to the four possible directions of magnetization (c).\n[1] Slonczewski, J. C. Current-driven excitation of magnetic multilayers. J. Magn. Magn. Mater. 159, L1{L7 (1996).\n[2] Zak, J., Moog, E. R., Liu, C. & Bader, S. D. Universal approach to magnetooptics. J. Mag. Mag. Mater. 89, 107 (1990).\n[3] Traeger, G., Wenzel, L. & Hubert, A. Computer Experiments on the Information Depth and the Figure of Merit in\nMagnetooptics. Phys. stat. sol. (a) 131, 201{227 (1992).\n[4] Hamrle, J. et al. Analytical expression of the magneto-optical Kerr e\u000bect and Brillouin light scattering intensity arising\nfrom dynamic magnetization. J. Phys. D: Appl. Phys. 43, 325004 (2010).\n[5] Wieczorek, J. et al. Separation of ultrafast spin currents and spin-\rip scattering in Co/Cu(001) driven by femtosecond laser\nexcitation employing the complex magneto-optical Kerr e\u000bect. Phys. Rev. B 92, 174410 (2015).\n[6] Carpene, E. et al. Ultrafast three-dimensional magnetization precession and magnetic anisotropy of a photoexcited thin \flm\nof iron. Phys. Rev. B 81, 060415(R) (2010).\n[7] Prokop, J. et al. Magnons in a Ferromagnetic Monolayer Phys. Rev. Lett. 102, 177206 (2009).11\n[8] Zhan, Q.-F. et al. Magnetic anisotropies of epitaxial Fe/MgO(001) \flms with varying thickness and grown under di\u000berent\nconditions. New. J. Phys. 11, 063003 (2009)."    },    {        "title": "1011.2788v1.Spin_Torque_Ferromagnetic_Resonance_Induced_by_the_Spin_Hall_Effect.pdf",        "content": " 1Spin Torque Ferromagnetic Resonance Induced by the Spin Hall Effect \nLuqiao Liu, Takahiro Moriyama, D. C. Ralph, and R. A. Buhrman \nCornell University, Ithaca, New York, 14853 \n \nABSTRACT \nWe demonstrate that the spin Hall effect in a thin film with str ong spin-orbit scattering \ncan excite magnetic precession in an adjacent ferromagnetic film. Th e flow of alternating current \nthrough a Pt/NiFe bilayer generates an oscillatin g transverse spin current in the Pt, and the \nresultant transfer of spin angular momentum to  the NiFe induces ferromagnetic resonance (FMR) \ndynamics. The Oersted field from the current also  generates an FMR signa l but with a different \nsymmetry. The ratio of these two signals allows a quantitative determinati on of the spin current \nand the spin Hall angle.   \n  2The spin Hall effect (SHE), the conversion of a longitudinal charge current density JC \ninto a transverse spin current density / 2SJe= , originates from spin-orbit scattering [1-4], \nwhereby conduction electrons with  opposite spin orientations in  a nonmagnetic metal [5] or \nsemiconductor [6] are deflected in opposite di rections. The SHE has attracted widespread \ninterest because it can generate pure spin currents from  a nonmagnetic source, a phenomenon \nthat could find important applications in future  spintronic devices. Seve ral techniques [5, 7, 8] \nhave been developed to determine the magnitude of the SHE, which is generally characterized by \nthe spin Hall angle, θSH = JS/JC. For thin-film Pt, estimates of θSH obtained using different \napproaches differ by more than an order of magnit ude [8-10], but already there have been efforts \nto utilize the spin current that arises from the SHE, first to tune the damping coefficient in a \nferromagnetic metal [8], and, most recently, to indu ce a spin wave oscillation in a ferrimagnetic \ninsulator having small damping [11]. Here we show  that the SHE can also be used to excite \ndynamics in an ordinary metallic ferromagnet.  Our experiment also allows a quantitative \ndetermination of the SHE strength that is self-calibrated, as explained below, enabling \nmeasurements of the spin currents generated by th e SHE with small experimental uncertainties.  \nWe study Pt/Permalloy bilayer films with a microwave-frequency (RF) charge current \napplied in the film plane (Permalloy = Py = Ni 81Fe19). An oscillating transv erse spin current is \ngenerated in the Pt by the SHE and injected into the adjacent Py (Fig. 1(a)), thereby exerting an \noscillating spin torque (ST) on the Py th at induces magnetization precession. When the \nfrequency and field bias satisfy  the FMR condition for the Py, st rong resonant precession results \nin a significant oscillation of  the bilayer resistance due to the anisotropic magnetoresistance \n(AMR) of the Py. This generates a DC voltage signal across the sample from the mixing of the \nRF current and the oscillating re sistance, similar to the signal th at arises from ST induced FMR  3in spin valves and magnetic tunne l junctions [12-15]. The resonan ce properties enable a direct \nquantitative measure of the spin  current absorbed by the Py. \nOur measurement setup is shown in Fig. 1(c). Pt/Py bilayers were grown by DC \nmagnetron sputter deposition. The in dividual layer thicknesses we re 4-15 nm, with specific \nvalues stated below. The star ting material for the Pt was  99.95% pure. Highly resistive Ta (1 nm) \nwas employed as the capping layer to preven t oxidation of the Py. The bilayers were \nsubsequently patterned into microstrips usi ng photolithography and ion milling. The samples’ \nwidths ranged from 1 to 20 μm and the lengths from 3 to 250 μm. By using a bias tee, we were \nable to apply a microwave current to our sample and at the same time me asure the DC voltage. A \nsweeping magnetic field Hext was applied in the film plane, with the angle θ between Hext and \nmicrostrip kept at 45° unless otherwise indi cated. The output power of  the microwave signal \ngenerator was varied from 0 to 20 dBm and th e measured DC voltage was proportional to the \napplied power, indicating that the induced precession was in the small angle regime. All the \nmeasurements we present were performed at room temperature with a power of 10 dBm.  \nWe model the motion of the Py magnetic moment ˆm by the Landau-Lifshitz-Gilbert \nequation containing the ST term [16]:  \n,\n0ˆˆˆˆ ˆ ˆ ˆ ˆ ()2eff S RF RF\nSdm dmmH m J m m mHdt dt e M tγα γ σ γμ=− × + × + × × − ×GG =.  (1) \nHere γ is the gyromagnetic ratio, α is the Gilbert damping coefficient, μ0 is the permeability in \nvacuum, Ms is the saturation magnetization of Py,  t is the thickness of the Py layer, ,/2SR FJe=  \nrepresents the oscillating spin curr ent density injected into Py,  HRF is the Oersted field \ngenerated by the RF current, Heff is the sum of Hext and the out-of-plane demagnetization field \n  4πMeff, and ˆσ is the direction of the injected spin moment. The third and fourth terms on the  4right hand side of Eq. (1) are the result of in-plane spin torque and the out-of-plane torque due \nto the Oersted field, respectiv ely (Fig. 1(a)).  The magnetic-re sonance mixing signal in response \nto a combination of in-plane a nd out-of-plane torques has been calculated in the context of ST-\ndriven FMR [14, 15], which we can  translate to our notation as: \nVmix=−1\n4dR\ndθγIRFcosθ\nΔ2πdf/dH()Hext=H0SFS(Hext)+AFA(Hext) [] ,    (2) \nwhere FSHext() =Δ2/Δ2+Hext−H0 ()2⎡⎣⎤⎦ is a symmetric Lorentzian  function centered at the \nfield H0 with linewidth Δ, FAHext() =FSHext() Hext−H0 () /Δ is an antisymmetric Lorentzian, \n(),0/2SR F s SJ e M t μ == , A=HRF1+4πMeff/Hext ()⎡⎣⎤⎦1/2, R is the resistance of the stripline, IRF \nis the microwave current through the stripline, and f is the resonance frequency. We therefore \nexpect the resonance signal to consist of two pa rts, a symmetric Lorentzian peak proportional to \nthe spin current density and an an tisymmetric peak proportional to HRF.   \nThe Oersted field HRF can be calculated from the geometry of the sample. Since the \nmicrowave skin depth is much greater than the Py thickness the current de nsity in the Py should \nbe spatially uniform, and in this case the Oersted field from the charge current density in the Py \nshould produce no net torque on the Py [see Fig.  1(b)]. The Oersted field can therefore be \ncalculated entirely from the current density JC,RF in the Pt layer. The microstrip width is much \nlarger than the Pt thickness, so the sample can  be approximated as an infinitely wide conducting \nplate and the Oersted field determined by Ampère's law, HRF=JC,RFd/2, where d is the Pt \nthickness. We checked HRF by numerical integration and the di fference is less than 0.1% from \nthe infinite plate approximation. Using this result , the ratio of the spin current density entering \nthe Py to the charge current density in the Pt can  then be determined quantitatively in a simple  5way from the ratio of the symmetric and anti symmetric components of the resonance curve \n()1/2, 0\neff ext\n,14 /SR F S\nCR FJ eM t dSMHJAμπ =+ ⎡ ⎤⎣ ⎦=.     ( 3 )  \nAll of the parameters entering Eq . (3) are either fundamental cons tants or quantities that can be \nmeasured directly, so this expression allows a measurement of JS,RF/JC,RF with small \nexperimental uncertainties. The measurement is se lf-calibrated in the sense that the strength of \nthe torque from the spin current is m easured relative to the torque from HRF, which can be \ncalculated easily from the geometry of the sample. \nAn additional contribution to the DC voltage can arise from spin pumping by the \nprecessing moment in combination w ith the inverse SHE in the Pt la yer, as observed in Ref. 10.  \nHowever, this effect is second order in θSH in our geometry and we calculate that it should \ncontribute a negligible voltage, about two orders of magnitude smaller than the signals we \nmeasure. \nFigure 2(a) shows the ST-FMR signals m easured on a Pt(6)/Py(4) (thicknesses in \nnanometers) sample for f = 5-10 GHz. As expected from Eq. (2), the resonance peak shapes can \nbe very well fit by the sum of symmetric and antisymmetric Lorentzian curves with the same linewidth for a given f (fits are shown as lines in Fig. 2(a)). The fact that the symmetric peak \nchanges its sign when H\next  is reversed (inset of Fig. 2(a)) ag rees with the form of spin torque \nˆˆˆST mm τσ∝× ×G given in Eq. (1), and excludes the possi bility that the signal is due to an \nunbalanced perpendicular Oerste d field torque, in direction ˆˆRF mH⊥× , which would yield \nsymmetric peaks with the same sign for opposite Hext. The resonant peak positions are \nsummarized in Fig. 2(b), and agree well with the Kittel formula \n() ( )1/2\n00 e f f /2 4 fH H Mγπ π=+ ⎡⎤⎣⎦. From a one-parameter fit to the resonance frequencies we  6determine that the demagnetization field 4πMeff= 0.805 ± 0.005 T for the Pt(6)/Py(4) bilayers.  \nWe have also measured th e saturation magnetization MS = 6.4 × 105 A/m in test samples [17].   \nTo verify the SHE origin of field-symmetr ic components of the FMR signals, we have \nstudied several different types of  control samples. In Fig. 2(c) we compare the FMR signals \nmeasured at 8 GHz for a Pt(15)/Py(15) and a Pt (6)/Py(4) sample. The signal for the Pt(6)/Py(4) \nsample contains a sizable field-symmetric component, with S/A = 0.63. Due to the increased \nthicknesses of the two layers, we expect from Eq. (3) that S/A for the Pt (15)/Py (15) should be \ngreatly reduced, approximately ∝1/td if the spin Hall currents in the two samples are similar. \nS/A for the Pt(15)/Py(15) is very small, S/A = 0.08 ± 0.05, near the noise floor for the fits of the \nsymmetric component (the uncertainty reflects  the standard deviation over five samples \nmeasured). The difference between the change in the S/A ratio expected from Eq. (3) (a factor of \n11.2, taking into account a small change in 4πMeff) and the measured reduction by a factor 8.0 \nmay be associated with a change in the magnit ude of the spin Hall current generated by the \ndifferent thicknesses of the Pt films when this thickness is comparable to the spin diffusion length (see below). \nWe also studied control samples with the laye rs Cu(6)/Py(4) and 4 nm of Py alone, with \nresults as shown in Fig. 2(d). The Cu/Py bila yer sample gives a purely antisymmetric signal, \nindicating that only the Oersted-fi eld contribution is present, as expected because of the very \nsmall SHE in Cu in comparison to that in Pt . For the Py(4)  sample, we would expect no \nresonance signal at all, since ther e is no SHE and as noted above if  the current density in the Py \nis uniform there should also be no net effect of  the Oersted field on the Py dynamics. However, \nwe do observe a very small, purely antisymmetric signal in the 4 nm Py sa mple. We suspect that \nthis may arise from an Oersted field due to non-unif orm current flow at the ends of the Py due to  7the electrode contacts. The lack of field-symmet ric components in the re sonance curves for the \ncontrol samples provides strong su pport that the symmetric component  we observe in Pt(6)/Py(4)  \ndoes indeed arise from the SHE in the Pt.   \nWith   4πMeff and MS determined , we can use Eq. (3) and the measured values of S/A to \ncalculate JS,RF/JC,RF. The results are shown in Fig. 2(e)  for the resonance curves spanning 5-\n10 GHz shown in Fig. 2(a).  We find JS,RF/JC,RF = 0.056 ± 0.005 for Pt(6)/Py(4). We \nmeasured more than ten Pt(6)/Py(4) samples w ith different lateral dimensions and the total \nvariation of JS,RF/JC,RF was < 15%. The dominant experimental uncertainty [and the small \nvariation with Hext visible in Fig. 2(e)] may be associated  with Oersted fields from non-uniform \ncurrents at the sample ends, as noted above for the single-layer Py sample.  Note that according to Eq. (2) S/A should not depend upon the angle of the a pplied DC field, as confirmed by the \nresults shown in Fig. 2(f).  \nAs an independent check we also employed an alternative method for determining the \nspin current density absorbed by the Py  layer, by measuring the FMR linewidth \nΔ as a function \nof DC current, similar to the technique introduced in Ref. 8. According to the theory of ST, a \nDC spin current IS,DC will increase or decrease the effective magnetic damping and hence Δ, \ndepending upon its relative or ientation with respect to the magnetic moment: [18] \neff 02s i n\n(2 ) 2S\next SJ f\nHM M t eπθαγπ μ⎛⎞Δ= +⎜⎟+ ⎝⎠=         (4) \nOur results obtained with a Pt(6)/Py(4) sample  ~1 μm wide are shown in Fig. 3. The measured \ndamping coefficient at zero current ( α ≈ 0.028) is significantly higher than that measured in a \nspin valve nanopillar sample ha ving a 4 nm Py free layer ( α≈ 0.01 ) [19]. This can be \nexplained by the spin pumping e ffect previously observed in the Py/Pt system [20, 21]. For a  8negative applied field ( Hext applied -135° from the current di rection in the microstrip), the \nlinewidth is broadened when I DC ramps from -0.7 mA to 0.7 mA; while for a positive field \n(Hext applied 45° from the current direction), th e trend is the opposite. By fitting the data \nshown in Fig. 3, and calculating the charge cu rrent density in the Pt using the measured \nresistivities 20Pt cm ρμ=Ω and 45Py cm ρμ=Ω , we find 10/ (0.9 0.012) 10cJα−Δ= ± ×  (A/cm2)-1. \nWith Eq. (4), this yields / 0.048 0.007scJJ =±  for Pt(6)/Py(4), which agrees well with the \nvalue 0.056 ± 0.005 determined from the FMR lineshape.  \nOur experiments yield values for JS/JC, the ratio of the spin current density (in units of \ncharge) absorbed by the Py to the charge current  density in the Pt film. For many applications, \nthis is the figure of merit of direct interest. Ho wever, for comparing to other experiments, it is \nalso of interest to determine the spin Hall angle θSH, the ratio of the spin current density inside \nbulk Pt to the charge current density. For a perf ectly transparent Pt/Py interface and for a Pt \nlayer much thicker than the spin diffusion length λsf, the quantities JS/JC and θSH should be \nequal. However, because our Pt/Py interface is li kely not perfectly transparent, and because our \nPt layers likely do not have thicknesses \u0015λsf, our results may underestimate the transverse \nspin current density appropriate  to bulk Pt. Therefore, our measurements imply a lower bound, \nθSH >  0.056 ± 0.005 for our Pt material. In the lim it of a transparent Pt/P y interface, for which \nthere should be no spin accumulation transverse to  the Py moment at the interface, we calculate \nusing drift-diffusion theory [22] that the spin Hall current density in a Pt film of thickness d \nshould be reduced from the bulk value by JSd()/JS∞()=1−sech d/λsf(). Using this \nexpression, our best estimate, based on compar ison between the Pt(15)/Py(15) and Pt(6)/Py(4) \nsamples is that 3sf nm λ≈ , and we can set an upper bound of 6sf nm λ< , lower than the low  9temperature value measured previously [23]. This gives a best estimate of 0.076SHθ= , and \nbounds 0.056 0.005 0.16SHθ ±< < , again in the limit of a tr ansparent Pt/Py interface.  \nWe mentioned above that pr evious measurements of θSH in Pt have differed by over an \norder of magnitude. Kimura et al. [9], using a Pt/Cu/Py lateral nonlocal geometry reported \nθSH= 0.0037. However, their 4-nm-thick Pt wires are in contact to 80-nm -thick Cu wires. We \nbelieve that the Cu likely shunted the charge current flowing in the Pt, resulting in a large \nunderestimation of θSH. Ando et al.  [8] by measuring magnetic damp ing in Pt/Py versus current, \nreported JS/JC = 0.03 and estimated 0.08SHθ= . We have shown that a technique closely \nrelated to the method of Ref. 8 gives result s that agree with our FMR method, although we \ndiffer with Ref. 8 regarding the form of our Eq. (4) and the drift-di ffusion analysis. Mosendz et \nal. [10], using a technique based on spin pump ing together with the inverse SHE, reported θSH= \n0.0067, later refined to θSH = 0.013 [24]. This result relied on an assumption that λsf= 10 nm \nfor Pt. Their value for θSHwould be 3 times larger, and in much better accord with our value, \nusing our estimate that λsf= 3 nm.  \nIn summary, we demonstrate th at spin current generated by the SHE in a Pt film can be \nused to excite spin-torque FMR in an adjacen t metallic ferromagnet (Py) thin film. This \ntechnique allows a straightforward determination of the efficiency of spin current generation, \nJs/Jc (the spin current density absorbed by the Py di vided by the charge curre nt density in the Pt), \nthat is self-calibrated, in that the torque due to  the spin current can be measured relative to the \ntorque from the Oersted field generated by the sa me charge current density in the Pt layer. We \nfind Js/Jc = 0.056 ± 0.005 for Pt(6)/Py(4), implying θSH >  0.056 for bulk Pt . This simple \ntechnique is an excellent soluti on for the quantitative measuremen t of the SHE efficiency in any  10metallic film that can be produced as part of  a ferromagnetic/non-magnetic metal bilayer. The \nrelatively large efficiency of spin current generation that we observe for Py/Pt is promising for applications which might utilize the SH E to manipulate ferromagnet dynamics. \nThis research was supported in part by th e Army Research Office, and by the NSF-NSEC \nprogram through the Cornell Center for Nanoscale Sy stems.  This work was performed in part at \nthe Cornell NanoScale Facility, which is  supported by the NSF through the National \nNanofabrication Infrastructure Network and bene fitted from the use of the facilities of the \nCornell Center for Materials Research , supported by the NSF-MRSEC program. 11REFERENCES \n1. J. E. Hirsch, Phys. Rev. Lett. 83, 1834 (1999). \n2. S. F. Zhang, Phys. Rev. Lett. 85, 393 (2000). \n3. J. Sinova  et al. , Phys. Rev. Lett. 92, 126603 (2004). \n4. S. Murakami, N. Nagaosa and S. C. Zhang, Science 301, 1348 (2003). \n5. S. O. Valenzuela and M. Tinkham, Nature 442, 176 (2006). \n6. Y. K. Kato  et al. , Science 306, 1910 (2004). \n7. E. Saitoh  et al. , Appl. Phys. Lett. 88, 182509 (2006). \n8. K. Ando  et al. , Phys. Rev. Lett. 101, 036601 (2008). \n9. T. Kimura  et al. , Phys. Rev. Lett. 98, 156601 (2007). \n10. O. Mosendz  et al. , Phys. Rev. Lett. 104, 046601 (2010). \n11. Y. Kajiwara  et al. , Nature 464, 262 (2010). \n12. A. A. Tulapurkar  et al. , Nature 438, 339 (2005). \n13. J. C. Sankey  et al. , Phys. Rev. Lett. 96, 227601 (2006). \n14. J. C. Sankey  et al. , Nat. Phys. 4, 67 (2008). \n15. H. Kubota  et al. , Nat. Phys. 4, 37 (2008). \n16. J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996). \n17. I. N. Krivorotov  et al. , Science 307, 228 (2005). \n18. S. Petit  et al. , Phys. Rev. Lett. 98, 077203 (2007). \n19. G. D. Fuchs  et al. , Appl. Phys. Lett. 91, 062507 (2007). \n20. S. Mizukami, Y. Ando and T. Miyazaki, J. Magn. Magn. Mater. 239, 42 (2002). \n21. Y. Tserkovnyak, A. Brataas and G. E. W. Bauer, Phys. Rev. Lett. 88, 117601 (2002). \n22. P. C. Vanson, H. Vankempen and P. Wyder, Phys. Rev. Lett. 58, 2271 (1987). \n23. H. Kurt  et al. , Appl. Phys. Lett. 81, 4787 (2002). \n24. A. Hoffman, private communication. \n \n  12FIGURE CAPTIONS \nFig. 1 (color online): (a) Schematic of a Pt/Py bi layer thin film illustrating the spin transfer \ntorque τSTT, the torque τH induced by the Oersted field HRF , and the direction of the damping \ntorque τα. θ denotes the angle betw een the magnetization M and the microstrip. Hext is the applied \nexternal field. (b) Left side view of the Pt/P y system, with the solid line showing the Oersted \nfield generated by the current flowing just in the Py layer, which should produce no net effect on \nthe Py AMR. (c) Schematic circuit for the ST-FMR measurement.   Fig. 2 (color online): (a) Spectra of ST-F MR on a Pt(6)/Py(4) sample measured under \nfrequencies of 5-10 GHz. The sample dimension is 20 μm wide × 110 μm long. Inset: ST-FMR \nspectrum of 8 GHz for both positive and negative H\next. (b) Resonance frequency f as a function \nof the resonant field H0. The solid curve represents a fit to  the Kittel formula. (c) FMR spectra \nmeasured for two Pt/Py bilayer samples, with fits  to Eq. (2). The data were taken at 8 GHz. (d) \nFMR spectra ( f = 8 GHz) on the Pt(6)/Py(4) sample (blue triangles) as well as control samples \nconsisting of Cu(6)/Py(4) (red circles) and Py(4) (black squares). (e) JS,RF/JC,RF values \ndetermined from the FMR analysis [Eq. (3)] at different f. (f) FMR signals measured for different \nexternal field angles θ (f = 8 GHz). The mixing voltages Vmix are normalized and offset to enable \ncomparison of the lineshapes.   Fig.3 (color online): The change of the FMR linewidth (left y axis) and Gilbert damping coefficient (right y axis) as a function of I\nDC for two orientations of the Py magnetization relative \nto the current direction. The data are taken at f = 8 GHz.  13\n \n  14 \n0 1000 2000-2002040\n  Vmix (μV)\nHext (Oe)5 GHz\n6 GHz\n7 GHz\n8 GHz\n9 GHz\n10 GHz\n  \n0 500 1000 15000246810\n  f (GHz)\nH0 (Oe)\n-600 -300 0 300 6000.00.81.62.43.2 \n \n75o60o45oNormalized V mix\nHext - H0 (Oe)θ=15o\n30o\n400 800 12000.000.020.040.06\n  Js / Jc\nH0 (Oe)0 1000 2000-20-1001020\n  Vmix (μV)\nHext (Oe) Pt (6) / Py (4)\n Cu (6) /Py (4)\n Py (4)\n0 1000 2000-40-2002040\n  Vmix (μV)\nHext (Oe)Pt (15) / Py (15)\nPt (6)/Py (4)(b)\n(f)(c) (d)\n(e)(a) 15    \n-0.8 -0.4 0.0 0.4 0.87578818487\n2.72.82.93.0 \n positive field\n negative fieldlinewidth (Oe)\nIDC (mA)X 10-2\n α   \n "    },    {        "title": "2205.00292v1.Dynamic_quantum_enhanced_sensing_without_entanglement_in_central_spin_systems.pdf",        "content": "Dynamic quantum-enhanced sensing without entanglement in central spin systems\nWenkui Ding,1,\u0003Yanxia Liu,1,2Zhenyu Zheng,1and Shu Chen1,3,4,y\n1Beijing National Laboratory for Condensed Matter Physics,\nInstitute of Physics, Chinese Academy of Sciences, Beijing 100190, China\n2Department of Physics, Yunnan University, Kunming 650500, China\n3School of Physical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China\n4Yangtze River Delta Physics Research Center, Liyang, Jiangsu 213300, China\n(Dated: May 3, 2022)\nWe propose a dynamic quantum sensing scheme by using a quantum many-spin system composed\nof a central spin interacting with many surrounding spins. Starting from a generalized Ising ring\nmodel, we investigate the error propagation formula of the central spin and it indicates that Heisen-\nberg scaling can be reached while the probe state only needs to be a product state. Particularly,\nwe derive an analytical form of the dynamic quantum Fisher information in a limit case, which\nexplicitly exhibits the Heisenberg scaling. By comparing with numerical results, we demonstrate\nthat the general case can be well approximated by the analytical result when the coupling strength\namong the surrounding spins is much weaker than the coupling strength between the central and\nsurrounding spins. This analytic result guides us to find the appropriate probe state and the proper\nmeasurement time, to achieve the Heisenberg scaling in realistic situations. Furthermore, we inves-\ntigate various effects which are important in practical quantum systems, including the central spin\nZeeman term, the anisotropy of the hyperfine interaction and the inhomogeneity of the hyperfine\ncoupling strength. Our result indicates that the dynamic quantum-enhanced sensing scheme seems\nfeasible in realistic quantum central spin systems, like semiconductor quantum dots.\nI. INTRODUCTION\nQuantum sensing [1, 2] and quantum metrology [3–\n5] are becoming the frontiers of quantum technologies\nnowadays. Specifically, theoretical research on quantum\nparameter estimation utilizing quantum properties has\nattracted great attentions recent years. On one hand,\nmost of the research on quantum metrology has been\nthe employment of entangled probe state to achieve the\nsub-shot-noise limit [6–10]. On the other hand, since the\nentangled probe state is extremely difficult to generate\nand very prone to decoherence [11–14], many techniques\nto realize the quantum-enhanced metrology without en-\ntanglement have also been proposed [15–17]. Particu-\nlarly, the quantum effects of quantum many-body sys-\ntems are employed to realize metrology schemes without\nentanglement. For example, the quantum phase tran-\nsition has been proposed to realize quantum-enhanced\nparameter estimation [18–21]. Besides, the non-linear\nmany-body Hamiltonian has also been proposed to en-\nhance the measurement precision, even reaching beyond\nthe conventional Heisenberg limit [22–27].\nThe central spin model, which consists of a central spin\ninteracting with many surrounding spins, can be used to\ndescribe realistic quantum many-spin systems, such as\nthe semiconductor quantum dots [28, 29], the Nitrogen-\nVacancy center in the diamond [30, 31], etc. Recently,\nthese solid-state central spin systems have been widely\ninvestigated in quantum information and quantum com-\nputation [29, 32], as well as quantum metrology and\n\u0003wenkuiding@iphy.ac.cn\nyschen@iphy.ac.cnquantum sensing [33–35]. Particularly, in Ref. [33, 34],\nthe central spin model has been proposed to achieve\nthe quantum-enhanced sensing. However, this sensing\nscheme is still based on the generation of the entangled\nprobe state. The non-linear interaction between the cen-\ntral spin and the surrounding spins implies that it is pos-\nsibletorealizethequantum-enhancedsensingofthemag-\nnetic field without entanglement. Recently a dynamic\nframework for criticality-enhanced quantum sensing in\nthe quantum Rabi model has been proposed [20]. This\nwork inspires us to look for a dynamic scheme for the\nrealization quantum-enhanced sensing without resorting\nto the entanglement. Particularly, we shall explore a dy-\nnamic routine to sense the magnetic field in a central spin\nmodel. We discover that the Heisenberg scaling can be\nachieved in our sensing scheme, while neither the entan-\ngled probe state nor the quantum criticality is required.\nA similar model known as ZZXX-model was studied in\nRef. [36] via the numerical calculation of the quantum\nFisher information, in which the authors concluded that\nthe Heisenberg scaling cannot be reached when only the\ncentral spin was measured [36]. However, in this work\nwe demonstrate that the Heisenberg scaling can still be\nreached dynamically by only measuring the central spin,\nwhen appropriate probe state and measurement time is\napplied. Especially, we can obtain an explicit analytic\nform of the dynamic quantum Fisher information for a\nlimit case of our studied central spin model, which cor-\nrectly predicts the dynamics of the ZZXX-model when\nthe number of surrounding spins becomes large enough.\nBesides, weinvestigatemorerealisticcentralspinsystems\nand our result indicates that quantum-enhanced magne-\ntometry using our dynamic sensing framework seems fea-\nsible. In particular, our scheme has the great advantagearXiv:2205.00292v1  [quant-ph]  30 Apr 20222\nthat neither entangled probe state nor quantum critical-\nity is required.\nThe rest of paper is organized as follows. In Sec.II,\nwe first describe the formalism for the calculation of\nquantum metrology and introduce our model system.\nIn Sec.III, we study both the local and global quantum\nFisher information for the central spin systems and de-\nrive analytical expression of the dynamic quantum Fisher\ninformation in a limit case. In Sec.IV, we discuss effects\nrelated to more realistic central spin system. A conclu-\nsion summary is given in the last section.\nII. FORMALISM AND MODEL\nBefore the study of concrete models, firstly we give\na brief review of the basics of quantum metrology for\nthe convenience of following calculations. The quantum\nfidelity between two quantum states ^\u001a1and^\u001a2is defined\nas,\nF(^\u001a1;^\u001a2) =Tr(qp\n^\u001a1^\u001a2p\n^\u001a1): (1)\nIn particular, when the quantum state is continuously\ndependent on parameter \u0015, we can define the fidelity sus-\nceptibility[37], whichisequivalenttothequantumFisher\ninformation [38],\nF\u0015=\u00004@2F(\u001a(\u0015);\u001a(\u0015+\u000e\u0015))\n@\u000e2\n\u0015j\u000e\u0015=0:(2)\nSpecifically, when the quantum state is a pure state,\nnamely\u001a(\u0015) =j\t(\u0015)ih\t(\u0015)j, we can calculate the quan-\ntum Fisher information as follows,\nF\u0015= 4(h\t(\u0015)j \u0000@\n@\u0015\u0000!@\n@\u0015j\t(\u0015)i\u0000jh \t(\u0015)j\u0000!@\n@\u0015j\t(\u0015)ij2):(3)\nFurthermore, for a unitary parameter imprint process [3,\n39],\nj\t(\u0015)i=e\u0000iH\u0015tj\t0i; (4)\nwhereH\u0015=H0+\u0015H1is a general parameter dependent\nHamiltonian, the quantum Fisher information is given by\nF\u0015= 4(h\t0jG2\n\u0015j\t0i\u0000jh \t0jG\u0015j\t0ij2):(5)\nHere, the transformed local generator\nG\u0015\u0011ieiH\u0015t@\n@\u0015e\u0000iH\u0015t\ncan be calculated as follows [40],\nG\u0015=Zt\n0eiH\u0015tH1e\u0000iH\u0015tds=\u0000i1X\nn=0(it)n+1\n(n+ 1)![H\u0015;H1]n;\n(6)\nwhere the commutation relation is defined as\n[H\u0015;H1]n+1= [H\u0015;[H\u0015;H1]n], with [H\u0015;H1]0=H1.The quantum Fisher information sets a bound to the\nestimation sensitivity, which is called the Cramér-Rao\nbound [38],\n\u0001\u000e\u0015(^A;\u0015)\u0015F\u00001\n2\n\u0015: (7)\nHere, the standard derivation of the measurement value\nis calculated via the error propagation formula,\n\u0001\u000e\u0015(^A;\u0015) =q\nh^A2i\u001a(\u0015)\u0000h^Ai2\n\u001a(\u0015)\nj@h^Ai\u001a(\u0015+\u000e\u0015)\n@\u000e\u0015j\u000e\u0015=0j; (8)\nwhere ^Ais the observable to be measured in the experi-\nment.\nWe begin with a generalized cental spin model [41]\nwhich is described by the Hamiltonian of the Ising ring\nin a transverse magnetic field interacting with a central\nspin [42],\nH=\u0000JNX\ni=1\u001bx\ni\u001bx\ni+1\u0000hNX\ni=1\u001by\ni+A\u001bz\n0NX\ni=1\u001bz\ni:(9)\nHere,\u001b\u000b=x;y;z\n0are the operators of the central spin,\n\u001b\u000b=x;y;z\ni=1;:::;Nare the operators of spins in the Ising ring, Jis\nthe strength of the Ising coupling, and Ais the coupling\nstrength between the central spin and surrounding spins\nin the Ising ring. Different from the usual central spin\nmodel studied in Ref.[41], here we apply the transverse\nmagnetic field along the y-axis, instead of the z-axis cou-\npled with the central spin. In this work, the strength of\ntransverse magnetic field his the parameter to be esti-\nmated.\nIII. LOCAL AND GLOBAL QUANTUM FISHER\nINFORMATION\nA. Local quantum Fisher information\nWe shall first consider the case with only the central\nspin being measured. In order to evaluate the perfor-\nmanceofutilizingthisparameter-dependentHamiltonian\nas the parameterization generator, we will first calculate\ntheerrorpropagationformulawhentheexpectationvalue\nof central spin is measured. Specifically, the initial state\n(or the probe state) is chosen to be a product state\nj\t0i=1p\n2(j\"i+j#i)\nj\bni; (10)\nwherej\"(#)iis the central spin state and j\bniis the\ncollective state of the spins in the Ising ring. The time\nevolutionofthesystemisgovernedby j\t(t)i=e\u0000iHtj\t0i\nand the central spin expectation value can be expressed\nas\nh\u001bx\n0(t)i=1\n2Re(h\bnjeiH+te\u0000iH\u0000tj\bni);(11)3\nwhere\nH\u0006=\u0000JNX\ni=1\u001bx\ni\u001bx\ni+1\u0000hNX\ni=1\u001by\ni\u0006A\n2NX\ni=1\u001bz\ni:(12)\nFurthermore, we have the relation\ne\u0000iH\u0006tj\bni=e\u0000i\u0012PN\ni=1\u001bx\nie\u0000iH(\u0006)\nefftei\u0012PN\ni=1\u001bx\nij\bni;(13)\nwhere\u0012= arctan(2h=A)and\nH(\u0006)\neff=\u0000JNX\ni=1\u001bx\ni\u001bx\ni+1\u0006r\nh2+A2\n4NX\ni=1\u001bz\ni:(14)\nWe now consider a specific product probe state given\nbyj\bni=j+;+;:::+i\u0011jN=2;Mx=N=2iwithj+i=\n1=p\n2(j\"in+j#in), namely all spins in the Ising ring are\npolarized along the x-axis. Now, we have\nh\bnjeiH+te\u0000iH\u0000tj\bni=e\u0000i\u0012N\u0002\nhN\n2;Mx=N\n2jeiH(+)\neffte2i\u0012PN\ni=1\u001bx\nie\u0000iH(\u0000)\nefftjN\n2;Mx=N\n2i:\n(15)\nNext, we discuss two important limiting situa-\ntions. IfJ\u001dp\nh2+A2=4, thenH(+)\neff=H(\u0000)\neff\u0019\n\u0000JPN\ni=1\u001bx\ni\u001bx\ni+1, andapproximately, j\bni=jN=2;Mx=\nN=2iis the eigenstate of H(+)\neffandH(\u0000)\neffwith eigenen-\nergy\u000f0. This leads to h\bnjeiH+te\u0000iH\u0000tj\bni \u0019 1,\nwhich indicates that no information on the parame-\nter can be retrieved by monitoring the central spin\nexpectation value. Meanwhile, for the opposite lim-\niting situation with J\u001cp\nh2+A2=4, approxi-\nmately,H(+)\neff\u0019p\nh2+A2=4PN\ni=1\u001bz\niandH(\u0000)\neff\u0019\n\u0000p\nh2+A2=4PN\ni=1\u001bz\ni. If we choose the evolution time\nt=t0=\u0019=p\nh2+A2=4, then we have\nh\bnjeiH+te\u0000iH\u0000tj\bni\n=e\u0000i\u0012Nh\bnjei\u0019PN\ni=1\u001bz\nie2i\u0012PN\ni=1\u001bx\niei\u0019PN\ni=1\u001bz\nij\bni\n=e\u0000i\u0012NhN\n2;Mx=\u0000N\n2je2i\u0012PN\ni=1\u001bx\nijN\n2;Mx=\u0000N\n2i\n=e\u00002i\u0012N:\n(16)\nUsing Eq.(11), the expectation value of the central spin\nis given by\nh\u001bx\n0(t0)i=1\n2cos(2\u0012N)\n=1\n2cos[2 arctan(2h\nA)N]:(17)\nBy substituting this result into the error propagation for-\nmula in Eq.(8), we obtain\nE\u00001\nh=h[\u001bx\n0(t0)]2i\u0000h\u001bx\n0(t0)i2\nj@h\u001bx\n0(t0)i\n@hj2\n=(A2+ 4h2)2\n16A2N2;(18)which indicates the Heisenberg scaling with respect to N,\ni.e. the number of spins in the Ising ring.\nSince here the measurement is only done on the cen-\ntral spin, now in order to estimate the precision bound\nof this local measurement, we need to calculate the lo-\ncalquantum Fisher information corresponding to the re-\nduced density matrix of the central spin. We can calcu-\nlate this quantity by using the formula for the quantum\nFisher information of one qubit [43]:\nF0\nh=(\nj@hVj2+(V\u0001@hV)2\n1\u0000jVj2;ifjVj<1;\nj@hVj2; ifjVj= 1;(19)\nwhere V= (2h\u001bx\n0(t)i;2h\u001by\n0(t)i;2h\u001bz\n0(t)i)isthespinvector\non the Bloch sphere. Following the same procedure for\nderiving Eq.(17), we have\nh\u001by\n0(t0)i=1\n2sin[2 arctan(2h\nA)N];(20)\nandh\u001bz\n0(t0)i= 0. Then, by substituting these expecta-\ntion values and their derivatives into Eq. 19, we get\nF0\nh=Eh=16A2N2\n(A2+ 4h2)2; (21)\nwhich happens to be the same form as the reciprocal of\nthe error propagation formula in Eq. 8. This indicates\nthat\u001bx\n0is indeed the optimal observable to saturate the\nsensitivity bound for this specific probe state and local\nmeasurement.\nInfact, employingthestandardIsingHamiltonianwith\ntransverse field (with A= 0in Eq. 9) to estimate pa-\nrameters has been investigated in Ref. [44]. Particularly,\nthe result in that paper demonstrated that quantum-\nenhanced metrology cannot be realized if the probe state\nis restricted to be the product state. Therefore, the inter-\naction of the central spin with the Ising ring introduced\nin our model is crucial to realize the quantum-enhanced\nsensing without entanglement.\nIn order to investigate the effect of finite Ising cou-\npling strength, we numerically calculate the local quan-\ntum Fisher information as function of Nfor different\nvalues ofJ. The result is plotted in Fig. 1(a) and it\nindicates that the analytical result for the case of J= 0\ncan give good approximation for the general case when\nthe coupling strength among the surrounding spins ( J)\nis much weaker than the coupling strength between the\ncentral and surrounding spins ( A). As the Ising coupling\nstrengthJincreases, the Heisenberg scaling deteriorates.\nThis implies that it is better to measure a magnetic field\nwith large amplitude (which can be realized by adding\na large reference magnetic field) to satisfy the condition,\nJ\u001cp\nh2+A2=4, for the optimal sensitivity.\nB. Global quantum Fisher information\nAs discussed above, the calculation of local quantum\nFisher information reveals that the case of J= 0can4\n0 5 10 15 20\nN0100200300local QFI\n0 5 10 15 20\nN05001000150020002500global QFI(a) (b)\nFigure 1. (a) The scaling of the local quantum Fisher infor-\nmation with respect to Nfor different Ising coupling strength\nJ. The dashed line corresponds to the analytic result for\nJ= 0in Eq. 21, while the squares correspond to J= 0:1and\nthe stars correspond to J= 0:2, respectively. (b) The scaling\nof the global quantum Fisher information with respect to N\nfor different Ising coupling strength J. The dashed line cor-\nresponds to the analytic result for J= 0in Eq. 35, while the\nsquares correspond to J= 0:1and the stars correspond to\nJ= 0:2, respectively. The other parameters in both figures\nare set to be A= 1,h= 1.\nrealize the quantum-enhanced sensing without entangle-\nment. Now we will focus on the case of J= 0and use\nEq. 5 and Eq. 6 to deduce the analytic form of the global\nquantum Fisher information. In comparison with the lo-\ncal quantum Fisher information, the global one gives the\nultimate sensitivity bound for all possible measurements,\nnamely, not restricted to measurements only on the cen-\ntral spin.\nBefore we calculate the dynamic quantum Fisher in-\nformation for the specific case with J= 0, we have to\ncalculate the transformed local generator using Eq. 6.\nThe Hamiltonian of the system now becomes,\nH=\u0000hNX\ni=1\u001by\ni+A\u001bz\n0NX\ni=1\u001bz\ni\n\u0011\u0000hIy+ASzIz:(22)\nHere, since the Hamiltonian does not contain the Ising\nterms any longer, we have used the collective spin oper-\natorI\u000b=x;y;z\u0011PN\ni=1\u001b\u000b=x;y;z\ni, and redefined the central\nspin operator Sz\u0011\u001bz\n0. For convenience, we may call the\ncentral spin the electron spin and the surrounding spin\nthenuclear spin in the following sections.\nWe now use Eq. 6 to calculate the transformed local\ngenerator. For the even commutation terms, we have\n[H;H 1]2n=\u0000A(A2S2\nz+h2)n\u00001(hSzIz+AS2\nzIy);(23)\nwhile for the odd terms, we have the commutation rela-\ntion as follows,\n[H;H 1]2n+1=iA(A2S2\nz+h2)nSzIy:(24)\nThen, the transformed local generator to estimate hiscalculated as follows,\nGh=ieiHt@\n@he\u0000iHt\n=\u0000tIy\u0000Asin(\nt)\u0000\nt\n\n3(hSzIz+AS2\nzIy)\n+Acos(\nt)\u00001\n\n2SzIx;(25)\nwhere since the central spin S= 1=2, we have the oscil-\nlation frequency \n =p\nA2S2z+h2=p\nA2=4 +h2.\nUsing Eq. 5, it is easy to verify that the maximized\nquantum Fisher information is [40]\nFmax= (Emax\u0000Emin)2; (26)\nwhereEmaxandEminare the maximal and minimal\neigenvalues of Gh, respectively. Here for convenience,\nwe denote the transformed local generator in Eq. 25 as\nGh\u0011\u000bIy+\fSzIz+\rSzIx: (27)\nSince [Gh;Sz] = 0, we can separately find the eigenvalues\nin theSz= 1=2subspace and the Sz=\u00001=2subspace.\nThe calculated result of the maximized quantum Fisher\ninformation is\nFmax= (4\u000b2+\f2+\r2)I2; (28)\nwhereI=N=2and it clearly shows the Heisenberg scal-\ning. Generally, we have the relation\nEh\u0014F0\nh\u0014Fmax: (29)\nThefirstrelationissimplyduetotheCramér-Raobound.\nThe second relation results from the fact that the local\nquantum Fisher information is obtained by tracing out\nthedegreeoffreedomofthenuclearspins, whichunavoid-\nably leading to the loss of information on the parameter.\nTo achieve this maximized quantum Fisher informa-\ntion, the probe state in Eq. 5 should be [45],\nj\t0i=1p\n2(jEmaxi+jEmini): (30)\nThe eigenstate corresponding to the maximal eigenvalue\nis\njEmaxi=j\"i\n (e\u0000iIz\u001ee\u0000iIy\u0012jI;Mz=Ii);(31)\nand the eigenstate corresponding to the minimal eigen-\nvalue is\njEmini=j\"i\n (e\u0000iIz\u001ee\u0000iIy\u0012jI;Mz=\u0000Ii);(32)\nwith\u0012= arctan(p\n4\u000b2+\r2\n\f)and\u001e= arctan(2\u000b\n\r), where\njI;Mziis the eigenstate of Iz. This probe state j\t0i\nis generally an entangled state, however, what interests\nus most is whether the Heisenberg scaling still main-\ntains when the probe state is restricted to be a product\nstate [46].5\nSimilar to the dynamical sensing scheme in Ref. [20],\nwe emphasize on the discussion of the dynamic quantum\nFisher information at a specific sensing time t=t0=\n2\u0019=\n. Now the transformed local generator in Eq. 25\nbecomes\nGh= (\u0019\n2A2\n\n3\u00002\u0019\n\n)Iy+2\u0019Ah\n\n3SzIz\n\u0011\u000b0Iy+\f0SzIz:(33)\nHere, we consider a specific product probe state,\nj\t0i=1p\n2(j\"i+j#i)\njI;Mz=Ii;(34)\nwherejI;Mz=Ii=j\"n;\"n;\u0001\u0001\u0001;\"ni. It is readily to\nverifyh\t0jGhj\t0i= 0, and using Eq. 5 we obtain\nFh= 4(h\t0jG2\nhj\t0i) = 2\u000b2\n0I+\f2\n0I2:(35)\nThis again obviously manifests the Heisenberg scaling\nwith respect to the number of nuclear spins ( I=N=2).\nIt needs to be mentioned that the probe state (Eq. 34)\nused to achieve the Heisenberg scaling in Fhis different\nfrom the probe state (Eq. 10) used to calculate the er-\nror propagation formula in the previous subsection. This\nis due to the fact that the global quantum Fisher infor-\nmation discussed in this subsection corresponds to the\nultimate bound that the measurement on the observable\ncan be done globally, not limited to the central spin only.\nSimilar to the discussions for the local quantum Fisher\ninformation, here we also numerically investigate the ef-\nfect of finite Ising coupling strength ( J6= 0) to the global\nquantum Fisher information. In Fig. 1(b), we illustrate\nthe scaling of the global quantum Fisher information\nwith respect to Nfor different Ising coupling strength\nJ. Compared to the local quantum Fisher information\nin Fig. 1(a), the scaling of the global quantum Fisher\ninformation is less sensitive to the strength of Ising cou-\npling, and the analytic result for J= 0in Eq. 35 approx-\nimates very well with the numerical result when the cou-\npling strength between the nuclear spins is much weaker\nthan the coupling strength among the surrounding spins.\nThis is reasonable, since the local quantum Fisher infor-\nmation corresponds to the reduced state of central spin,\nwhile the finite Ising coupling will accelerate the decoher-\nence of the central spin, the decoherence will unavoidably\nlead to loss of information on the parameter. Meanwhile,\nsince the global quantum Fisher information corresponds\nto the quantum state of the composite system (central\nspin and surrounding spins), the Ising coupling between\nthe nuclear spins may not lead to loss of the parameter\ninformation encoded in the total quantum state.IV. REALIZATION OF THE PROTOCOL IN\nREALISTIC QUANTUM SYSTEMS\nA. Effect of the electronic Zeeman term\nIn the above section, the coupling between the central\nspin and the surrounding spins may be implemented by\nthe interaction of the light with two-level atoms, where\nthe interaction of the central spin with the field is ne-\nglected. However, it becomes necessary to take into ac-\ncount the electronic Zeeman term if we want to imple-\nment our dynamic sensing scheme using solid-state spin\nsystems, like semiconductor quantum dots. By taking\ninto account the electronic Zeeman term, the Hamilto-\nnian of the system now becomes\nH=\u0000h(\u001by\n0+NX\ni=1\u001by\ni) +A\u001bz\n0NX\ni=1\u001bz\ni\n\u0011\u0000h(Sy+Iy) +ASzIz;(36)\nwhichcorrespondstotheZZXX-modelnumericallyinves-\ntigated in Ref. [36]. In this subsection we will show that,\nfor the dynamic sensing scheme proposed in this paper,\nthe dynamics and the dynamic quantum Fisher informa-\ntion corresponding to this Hamiltonian, can actually be\nwell approximated by the Hamiltonian in Eq. 22, where\nthe electronic Zeeman term is neglected.\nAs shown in Fig. 2, the performance of this approx-\nimation (by neglecting the electronic Zeeman term) is\nactually dependent on N, the number of nuclear spins.\nThis is reasonable, since for our dynamic sensing scheme,\nthe central spin is initialized along the x-axis and then it\nwill precess about the effective magnetic field, from the\nview of semi-classical picture. This effective magnetic\nfield consists of the magnetic field halong the y-axis and\nthe nuclear field, Bnuc/AhIzi, along the z-axis. As the\nnuclear spin number Nincreases, the nuclear field Bnuc\nbecomessignificantlylargerthanthemagneticfield hand\nthe approximated analytic result becomes even closer to\nthe exact numerical result. Thus, in realistic solid state\ncentral spin system, which contains a large number of\nnuclear spins, the analytic result obtained by neglecting\nthe electronic Zeeman term can be safely utilized in our\ndynamic sensing scheme. In other words, when the dy-\nnamic sensing scheme by employing the Hamiltonian in\nEq. 36 is applied, our analytic result can be used to de-\ntermine the appropriate probe state, to devise the proper\nmeasurement strategy or to estimate the overall sensitiv-\nity.\nB. Effect of the anisotropy of the hyperfine\ninteraction\nIn many realistic quantum central spin systems, the\nXX-term of the hyperfine interaction may not be ne-\nglected, which is described by the so-called XXZ central6\n0 5\nTime-0.200.20.4Sx\n0 5\nTime-0.200.20.4Sx\n0 5\nTime0246QFI\n0 5\nTime050100QFI\n0.5 1 1.5 2\nh0200400local QFI\n0.5 1 1.5 2\nh0500010000local QFI(a) (b)\n(c) (d)\n(e) (f)\nFigure 2. The effect of the electronic Zeeman term on the\nperformance of the dynamic sensing scheme. In all figures,\nthe solid lines correspond to the Hamiltonian that taking into\naccount the electronic Zeeman term, while the dashed lines\ncorresponds to the analytic result that neglecting this term.\nThe left column [(a),(c),(e)] corresponds to N= 8and the\nright column [(b),(d),(f)] corresponds to N= 40. The time\nevolution of the central spin expectation value hSx(t)iis com-\npared in (a) and (b), with A= 1andh= 1. The time evolu-\ntion of the global quantum Fisher information is compared in\n(c) and (d), with A= 1andh= 1, where the vertical dashed\nline in these figures corresponds to the measurement time t0\ndeduced from the analytic result. In (e) and (f), we plot the\nlocal quantum Fisher information for the central spin as a\nfunction of the external field h, while the coupling strength\nA= 1and the measurement time t=t0.\nspin model [47]. The Hamiltonian for such a general cen-\ntral spin system with homogeneous coupling is as follows,\nH=h\u0001(\u001b0+NX\nk=1\u001bk) +NX\nk=1[\u0001\n2(\u001b+\n0\u001b\u0000\nk+\u001b\u0000\n0\u001b+\nk) +A\u001bz\n0\u001bz\nk]\n=h\u0001(S+I) +\u0001\n2(S+I\u0000+S\u0000I+) +ASzIz;\n(37)\nwhere \u0001is the hyperfine coupling strength of the XX-\nterm andI\u0006=Ix\u0006iIy, etc. When \u0001 =A, namely no\nanisotropy in the hyperfine coupling, we can check that\n[h\u0001(S+I);H] = 0. It is easy to verify that Gh=n\u0001(S+I)\nwithn=h=h, whichindicatesthatnoquantumenhance-\nment can be obtained when the probe state is restricted\nto be only a product state. Therefore, the anisotropy\nin the hyperfine interaction is necessary to achieve the\n0 50 100 150 200\nN100105local QFI\n0 50 100 150 200\nN101102103104105global QFI\n(a) (b)Figure 3. Investigate the effect of the anisotropy of the hy-\nperfine interaction on the performance of the dynamic sensing\nscheme. The local quantum Fisher information for the cen-\ntral spin is plotted in (a), while the global quantum Fisher\ninformation is plotted in (b). In both figures, the solid line\ncorresponds to the analytic result when the electronic Zee-\nman term is neglected and the hyperfine anisotropy \u0001 = 0.\nThe dashed lines in both figures correspond to the standard\nquantum limit, which is the ultimate limit when \u0001 =A= 0\nin Eq. 37 and the probe state is restricted to be the product\nstate. In both figures, the circles correspond to \u0001 = 0,A= 1\nin Eq. 37, while the squares correspond to \u0001 = 0:1,A= 1\nand the stars correspond to \u0001 = 0:2,A= 1, respectively.\nquantum-enhanced sensing without entanglement for our\ndynamic sensing scheme. Furthermore, when \u00016=Aand\nthe magnetic field is applied along the z-axis, it is readily\nto verify that Gh=Sz+Izand again no quantum en-\nhancement is possible when the initial state is restricted\nto be a product state. Thus, in order to obtain the\nquantum-enhanced sensing without entanglement, first,\nthere should be anisotropy in the hyperfine coupling; sec-\nond, the magnetic field to be estimated cannot be applied\nalong the z-axis.\nWe numerically calculate the scaling of the dynamic\nquantum Fisher information with respect to the nuclear\nspin number Nfor various values of \u0001in Fig. 3. The\nresult indicates that, increasing the anisotropy in the hy-\nperfine coupling will lead to a better performance of our\ndynamic sensing scheme.\nC. Effect of the inhomogeneity of the hyperfine\ncoupling strength\nFor realistic solid-state quantum central spin systems,\nlike semiconductor quantum dots, the hyperfine coupling\nstrength between the central electron spin and surround-\ning nuclear spins is usually inhomogeneous. The Hamil-\ntonian describing such a system is\nH=\u0000h(\u001by\n0+NX\nk=1\u001by\nk) +NX\nk=1Ak\u001bz\n0\u001bz\nk;(38)\nwhereAkis the coupling strength between the central\nelectron spin and the k-th nuclear spin. Due to the inho-\nmogeneity of the coupling strength ( Ak6=A), the collec-\ntive nuclear spin operator cannot be used any longer and7\n0 5 10\nTime050010001500QFI\n0 5 10 15 20\nN0200400600QFI(a) (b)\nFigure 4. Investigate the performance of the inhomogeneity\nof the hyperfine coupling strength on the performance of the\ndynamic quantum sensing. (a). The solid line corresponds to\nthe time evolution of the global quantum Fisher information\nforN= 16nuclear spins with inhomogeneous hyperfine cou-\npling, while the dashed line corresponds to the homogeneous\ncase. The vertical dashed line corresponds to the measure-\nment time. (b). The global quantum Fisher information as\na function of the nuclear spin number N. The dashed line\ncorresponds to the analytic result of the homogeneous case,\nwhile the squares correspond to the numerical result of the\ninhomogeneous case.\nwehavetoresorttonumericstoresearchtheperformance\nof the dynamic sensing scheme.\nHere, we employ the Chebyshev method (by expand-\ning the time evolution operator in terms of Chebyshev\npolynomials [48]) to calculate the dynamics of this inho-\nmogeneous central spin system and retrieve the scaling\nof the dynamic quantum Fisher information with respect\ntoN. The result is shown in Fig. 4 and it indicates that\nthe performance of such an inhomogeneous sensor can\nbe well approximated by the homogeneous one, as long\nas the coupling strength in the homogeneous case is set\nto beA=P\nkAk=N. Crucially, the Heisenberg scaling\nstill exists for the inhomogeneously coupled central spin\nsystem.\nD. The realistic semiconductor quantum dot\nsystem\nWe now elaborately consider a practical solid-state\nquantum central spin system, namely the semiconduc-\ntor quantum dots, which has been widely researched in\nthe area of quantum computation [29, 32] and quantum\nsensing [33–35] recent years. The detailed Hamiltonian\ndescribing such a quantum many-spin system is as fol-\nlows [29, 49],\nH=HD+HZ+HHF; (39)with,\nHD=\r2\nn\n2NX\nj=1NX\nk=1[Ij\u0001Ik\nr3\njk\u00003(Ij\u0001rjk)(Ik\u0001rjk)\nr5\njk];\nHZ=h\u0001(\reS+\rnNX\nk=1Ik);\nHHF=NX\nk=1Ak[\u000e\n2(S+Ik\u0000+S\u0000Ik+) +SzIkz];(40)\nwhere Sis the electronic spin operator and Ikis the spin\noperator of the k\u0000th nuclei in the quantum dot. HD\ndescribes the dipole-dipole interaction between nuclear\nspins, where rjkis the position vector from the j-th nu-\nclei to thek\u0000th nuclei and \rnis the gyromagnetic ratio\nof the nuclear spin. HZcorresponds to the electronic\nZeeman term and nuclear Zeeman terms, where \reis the\ngyromagnetic ratio of the electron spin. HHFdescribes\nthe hyperfine interaction between the electron spin and\nnuclear spins, where \u000estands for the anisotropy in the\nhyperfine interaction. The hyperfine coupling strength\nAk/j\u001e(xk)j2, wherej\u001e(xk)j2is the electron density at\nthe site xkof thek-th nuclear spin.\nThe dipole-dipole interaction between nuclear spins in\nthe quantum dot resembles the role of the Ising cou-\npling in Eq. 9. Typically, in semiconductor quantum\ndots, the hyperfine coupling strength between the elec-\ntron spin and nuclear spins is several orders of magnitude\nstronger than the strength of the nuclear dipole-dipole\ncoupling [49, 50]. Similar to the discussion on the Ising\ncoupling in the previous section, this indicates that the\neffect from the nuclear dipolar coupling to the overall\nsensitivity of the dynamic sensing scheme can be negli-\ngible. Another point that needs to be mentioned is the\ndifference in the gyromagnetic ratio between the electron\nspin and the nuclear spins, namely \re6=\rn. However,\nthe discussions in Sec. IVA still applies despite the dif-\nference in the gyromagnetic ratio. This is because for\ntypical semiconductor quantum dots, the number of nu-\nclear spins N\u0018104\u0000106in a single quantum dot, and\ntheeffectivenuclearfiledexperiencedbytheelectronspin\nsuppresses the magnetic field. Thus, in realistic semicon-\nductor quantum dots, the electronic Zeeman term can\nbe safely neglected to analyze the performance of our\ndynamic sensing scheme.\nV. CONCLUSION\nIn summary, by studying the Ising ring model inter-\nacting with a central spin, we show that the Heisenberg\nscaling in the error propagation formula can be reached\nby only measuring the central spin dynamics. While the\nthe dynamic quantum Fisher information for the general\ncase can be numerically calculated, we can obtain an an-\nalytical form of the dynamic quantum Fisher informa-\ntion to estimate the magnetic field in a limit case, which8\ngives good approximation for the general case as long as\nJ\u001cp\nh2+A2=4is fulfilled. This analytic result explic-\nitly manifests that the Heisenberg scaling can be reached\nwith an appropriate product probe state and the proper\nmeasurement time. Furthermore, we gradually investi-\ngate more realistic central spin models analytically and\nnumerically and our results indicate that the Heisenberg\nscaling can still be achieved in practical quantum central\nspin systems, like semiconductor quantum dots. Our dy-\nnamic sensing scheme processes the great advantage that\nneither entangled probe state nor quantum phase transi-\ntionisrequiredtoobtainthequantumenhancement. Ourtheoretical result paves the way to the implementation of\nquantum-enhanced magnetometry without entanglement\nin practical quantum many-spin systems.\nACKNOWLEDGMENTS\nThe work is supported by National Key National Key\nResearchandDevelopmentProgramofChina(GrantNo.\n2021YFA1402104), the NSFC under Grants No.12174436\nand No.T2121001 and the Strategic Priority Research\nProgram of Chinese Academy of Sciences under Grant\nNo. XDB33000000.\n[1] C. L. Degen, F. Reinhard, and P. Cappellaro, Rev. Mod.\nPhys.89, 035002 (2017).\n[2] D. Budker and M. Romalis, Nat. Phys. 3, 227 (2007).\n[3] V. Giovannetti, S. Lloyd, and L. Maccone, Phys. Rev.\nLett.96, 010401 (2006).\n[4] V.Giovannetti,S.Lloyd,andL.Maccone,Nat.Photonics\n5, 222 (2011).\n[5] L. Pezzè, A. Smerzi, M. K. Oberthaler, R. Schmied, and\nP. Treutlein, Rev. Mod. Phys. 90, 035005 (2018).\n[6] D.J.Wineland,J.J.Bollinger,W.M.Itano,F.L.Moore,\nand D. J. Heinzen, Phys. Rev. A 46, R6797 (1992).\n[7] S. L. Braunstein, Phys. Rev. Lett. 69, 3598 (1992).\n[8] M. J. Holland and K. Burnett, Phys. Rev. Lett. 71, 1355\n(1993).\n[9] V. Giovannetti, S. Lloyd, and L. Maccone, Science 306,\n1330 (2004).\n[10] T. Nagata, R. Okamoto, J. L. O’brien, K. Sasaki, and\nS. Takeuchi, Science 316, 726 (2007).\n[11] S. F. Huelga, C. Macchiavello, T. Pellizzari, A. K. Ekert,\nM. B. Plenio, and J. I. Cirac, Phys. Rev. Lett. 79, 3865\n(1997).\n[12] D. Leibfried, E. Knill, S. Seidelin, J. Britton, R. B.\nBlakestad, J. Chiaverini, D. B. Hume, W. M. Itano, J. D.\nJost, C. Langer,R. Ozeri, R. Reichle,D. J. Wineland, Na-\nture438, 639 (2005).\n[13] J.KołodyńskiandR.Demkowicz-Dobrzański,Phys.Rev.\nA82, 053804 (2010).\n[14] B. Escher, R. de Matos Filho, and L. Davidovich, Nat.\nPhys.7, 406 (2011).\n[15] T.Tilma, S.Hamaji,W.J.Munro,andK.Nemoto,Phys.\nRev. A81, 022108 (2010).\n[16] F. Benatti and D. Braun, Phys. Rev. A 87, 012340\n(2013).\n[17] D. Braun, G. Adesso, F. Benatti, R. Floreanini, U. Mar-\nzolino, M. W. Mitchell, and S. Pirandola, Rev. Mod.\nPhys.90, 035006 (2018).\n[18] P. Zanardi, M. G. A. Paris, and L. Campos Venuti, Phys.\nRev. A78, 042105 (2008).\n[19] M. M. Rams, P. Sierant, O. Dutta, P. Horodecki, and\nJ. Zakrzewski, Phys. Rev. X 8, 021022 (2018).\n[20] Y. Chu, S. Zhang, B. Yu, and J. Cai, Phys. Rev. Lett.\n126, 010502 (2021).\n[21] U. Mishra and A. Bayat, Phys. Rev. Lett. 127, 080504\n(2021).\n[22] A. Luis, Phys. Rev. A 76, 035801 (2007).[23] S. Boixo, S. T. Flammia, C. M. Caves, and J. Geremia,\nPhys. Rev. Lett. 98, 090401 (2007).\n[24] S. Boixo, A. Datta, M. J. Davis, S. T. Flammia, A. Shaji,\nand C. M. Caves, Phys. Rev. Lett. 101, 040403 (2008).\n[25] S. M. Roy and S. L. Braunstein, Phys. Rev. Lett. 100,\n220501 (2008).\n[26] S. Choi and B. Sundaram, Phys. Rev. A 77, 053613\n(2008).\n[27] M. Napolitano, M. Koschorreck, B. Dubost, N. Behbood,\nR. Sewell, and M. W. Mitchell, Nature 471, 486 (2011).\n[28] I. A. Merkulov, A. L. Efros, and M. Rosen, Phys. Rev. B\n65, 205309 (2002).\n[29] R. Hanson, L. P. Kouwenhoven, J. R. Petta, S. Tarucha,\nand L. M. K. Vandersypen, Rev. Mod. Phys. 79, 1217\n(2007).\n[30] L.Childress, M.G.Dutt, J.Taylor, A.Zibrov, F.Jelezko,\nJ. Wrachtrup, P. Hemmer, and M. Lukin, Science 314,\n281 (2006).\n[31] R. Hanson, V. Dobrovitski, A. Feiguin, O. Gywat, and\nD. Awschalom, Science (2008).\n[32] J. M. Taylor, C. M. Marcus, and M. D. Lukin, Phys. Rev.\nLett.90, 206803 (2003).\n[33] G. Goldstein, P. Cappellaro, J. R. Maze, J. S. Hodges,\nL. Jiang, A. S. Sørensen, and M. D. Lukin, Phys. Rev.\nLett.106, 140502 (2011).\n[34] P. Cappellaro, G. Goldstein, J. S. Hodges, L. Jiang, J. R.\nMaze, A. S. Sørensen, and M. D. Lukin, Phys. Rev. A 85,\n032336 (2012).\n[35] W. Ding, W. Zhang, and X. Wang, Phys. Rev. A 102,\n032612 (2020).\n[36] J. M. E. Fraïsse and D. Braun, Ann. Phys. (Berlin) 527,\n701 (2015).\n[37] S.-J. Gu, Int. J. Mod. Phys. B 24, 4371 (2010).\n[38] S. L. Braunstein and C. M. Caves, Phys. Rev. Lett. 72,\n3439 (1994).\n[39] J. Liu, X.-X. Jing, and X. Wang, Sci. Rep. 5, 1 (2015).\n[40] S. Pang and T. A. Brun, Phys. Rev. A 90, 022117 (2014).\n[41] H. T. Quan, Z. Song, X. F. Liu, P. Zanardi, and C. P.\nSun, Phys. Rev. Lett. 96, 140604 (2006).\n[42] The spins considered in this article is always spin-1\n2, so\nhere the Pauli matrix is defined as \u001bz=1\n2\u0014\n1 0\n0\u00001\u0015\n, and\nlikewise for \u001bxand\u001by.\n[43] W. Zhong, Z. Sun, J. Ma, X. Wang, and F. Nori, Phys.\nRev. A87, 022337 (2013).9\n[44] M. Skotiniotis, P. Sekatski, and W. Dür, New J. Phys.\n17, 073032 (2015).\n[45] Due to the degeneracy of the spectrum of Gh, the choice\nof the optimal probe state is actually not unique.\n[46] S. Boixo, A. Datta, S. T. Flammia, A. Shaji, E. Bagan,\nand C. M. Caves, Phys. Rev. A 77, 012317 (2008).\n[47] W.-B. He, S. Chesi, H.-Q. Lin, and X.-W. Guan, Phys.\nRev. B99, 174308 (2019).[48] V. V. Dobrovitski and H. A. De Raedt, Phys. Rev. E 67,\n056702 (2003).\n[49] C. P. Slichter, Principles of Magnetic Resonance, 3rd ed.\n(Springer-Verlag, Berlin, 1996).\n[50] B. Urbaszek, X. Marie, T. Amand, O. Krebs, P. Voisin,\nP. Maletinsky, A. Högele, and A. Imamoglu, Rev. Mod.\nPhys.85, 79 (2013)."    },    {        "title": "2401.16738v1.A_novel_non_adiabatic_spin_relaxation_mechanism_in_molecular_qubits.pdf",        "content": "A novel non-adiabatic spin relaxation mechanism in molecular qubits\nPhilip Shushkova)\nDepartment of Chemistry, Indiana University, Bloomington, Indiana, 47405\n(Dated: 31 January 2024)\nThe interaction of electronic spin and molecular vibrations mediated by spin-orbit coupling governs spin relaxation in\nmolecular qubits. I derive an extended molecular spin Hamiltonian that includes both adiabatic and non-adiabatic spin-\ndependent interactions, and I implement the computation of its matrix elements using state-of-the-art density functional\ntheory. The new molecular spin Hamiltonian contains a novel spin-vibrational orbit interaction with non-adiabatic\norigin together with the traditional molecular Zeeman and zero-field splitting interactions with adiabatic origin. The\nspin-vibrational orbit interaction represents a non-Abelian Berry curvature on the ground-state electronic manifold and\ncorresponds to an effective magnetic field in the electronic spin dynamics. I further develop a spin relaxation rate\nmodel that estimates the spin relaxation time via the two-phonon Raman process. An application of the extended\nmolecular spin Hamiltonian together with the spin relaxation rate model to Cu(II) porphyrin, a prototypical S=1/2\nmolecular qubit, demonstrates that the spin relaxation time at elevated temperatures is dominated by the non-adiabatic\nspin-vibrational orbit interaction. The computed spin relaxation rate and its magnetic field orientation dependence are\nin excellent agreement with experimental measurements.\nI. INTRODUCTION\nMolecular qubits, paramagnetic systems that exhibit long\nspin-coherence times, are emerging as a promising platform\nfor the implementation of quantum information processing.1–4\nThe molecular electronic spin is an excellent candidate to\nencode quantum information because of protection by time-\nreversal symmetry. Taken together with the unprecedented\naccuracy of chemical methods to synthesize and assem-\nble molecular systems, there has been a recent surge in\nefforts to engineer molecular qubits that are suitable to\nadvance the much sought-after room-temperature quantum\ntechnologies.5–18\nA major limiting factor to the room-temperature quantum\ninformation storage in molecular qubits is the spin relax-\nation time T1, called also spin-lattice or longitudinal relax-\nation time,19that characterizes the timescale of thermal equi-\nlibration of the electronic spin by the molecular vibrational\nmotion.5–7,9The spin relaxation theory dates back to the pio-\nneering work of Van Vleck,20Mattuck and Strandberg,21and\nOrbach,22who focused on the relaxation dynamics of para-\nmagnetic impurities in crystals. The physical picture that\nemerged from their seminal contributions is that both adia-\nbatic and non-adiabatic processes can dominate the spin re-\nlaxation time depending on the specifics of the electronic\nstructure of the paramagnetic impurities. Furthermore, they\nshowed that one-phonon processes contribute to the the spin\nrelaxation dynamics only at very low temperatures and at el-\nevated temperatures spin relaxation is driven by two-phonon\nabsorption-emission processes.\nDensity functional theory and multi-reference wavefunc-\ntion approaches have provided a firm theoretical basis to pre-\ndict molecular spin interactions.23–28Fundamental to this suc-\ncess is the concept of the static spin Hamiltonian, an effective\nHamiltonian defined at the equilibrium nuclear geometry that\na)phgshush@iu.eduincorporates the influence of the molecular electronic struc-\nture in spin-dependent interactions.29,30The dynamical exten-\nsion of the static spin Hamiltonian by accounting for the nu-\nclear geometry-dependence of the traditional spin-dependent\ninteractions has underscored recent efforts to simulate the spin\nrelaxation dynamics of molecular qubits.31–38This approach\nto spin relaxation, however, does not account for the contribu-\ntion of the non-adiabatic interactions.\nThe goal of the paper is to derive a molecular spin Hamil-\ntonian that includes both adiabatic and non-adiabatic spin-\ndependent interactions and to implement the computation of\nthe matrix elements of this Hamiltonian using density func-\ntional theory. I achieve this goal by applying the Born-\nOppenheimer approximation in the adiabatic representation\nfor the electronic wavefunctions,39,40circumventing the need\nfor diabatization, and allowing seamless integration with\nlinear response density functional theory. Similar to the\nstatic spin Hamiltonian, I use unitary degenerate perturbation\ntheory41,42to derive the molecular spin Hamiltonian, which\nlimits its applicability to molecular systems with weak spin-\norbit interaction and orbitally non-degenerate ground elec-\ntronic states.30The derived molecular spin Hamiltonian con-\ntains a novel, non-adiabatic spin-vibrational orbit interaction\ntogether with the traditional molecular Zeeman and zero-field\nsplitting interactions29that have an adiabatic origin. I fur-\nther develop a rate model to estimate the contribution to the\nspin relaxation time of the interactions in the molecular spin\nHamiltonian. The rate model is specialized to elevated tem-\nperatures and evaluates the spin relaxation time via the two-\nphonon Raman process20using density functional calcula-\ntions on the isolated paramagnetic molecule. A pilot appli-\ncation of the molecular spin Hamiltonian together with the\nspin relaxation rate model to Cu(II) porphyrin,43a prototyp-\nicalS=1/2 molecular qubit,44,45demonstrates that the two-\nphonon spin relaxation time is dominated by the non-adiabatic\nspin-vibrational orbit interaction with doubly degenerate nor-\nmal modes being the major vibrational relaxation channel.\nThe paper is organized as follows: I present the derivation\nof the molecular spin Hamiltonian and of the spin relaxationarXiv:2401.16738v1  [physics.chem-ph]  30 Jan 20242\nrate model in Sec. II. I outline the numerical evaluation of\nthe molecular spin Hamiltonian matrix elements using density\nfunctional theory in Sec. III. I present the results of the appli-\ncation of the new approach to Cu(II) porphyrin in Sec. IV,\nand I conclude the paper with an outline of future directions\nin Sec. V.\nII. THEORY\nSpin-vibrational interactions involve the coupling of the\nelectronic spin with the molecular vibrations that is mediated\nby the electronic orbital motion. I start Sec. II by specifying\nthe molecular Hamiltonian and establishing the notation used\nin the paper. I then present an overview of unitary degenerate\nperturbation theory42that I apply to derive the molecular spin\nHamiltonian in Sec. II A and proceed with the application of\nthe Born-Oppenheimer approximation. In Sec. II A, I derive\nthe expression for the Berry connection46–48on the ground-\nstate spin manifold and arrive at a molecular spin Hamilto-\nnian that contains a novel spin-vibronic vector potential. In\nSec. II B, I apply a unitary gauge transformation40to a sym-\nmetrized gauge for the spin-vibronic vector potential and de-\nrive the spin-vibrational orbit interaction. Harmonic expan-\nsion of the resulting molecular spin Hamiltonian in Sec. II C\ngives one- and two-phonon spin-vibrational interactions with\nnon-adiabatic and adiabatic origins. In Sec. II D, I present a\nrate model for the computation of the spin relaxation time of\nS=1/2 molecular qubits at elevated temperature and at weak\nto moderate magnetic field intensities using the two-phonon\nspin-vibrational interactions of Sec. II C.\nA. Molecular Spin Hamiltonian\nI write the molecular Hamiltonian,29including relativistic\ninteractions and the interaction of the molecule with an exter-\nnal magnetic field as:\nˆHmolec=ˆTR+ˆHNR\nR+ˆHSOZ\nR. (1)\nˆTRin Eq. (1) is the nuclear kinetic energy, ˆHNR\nR- the non-\nrelativistic electronic Hamiltonian together with the spin-\nindependent, scalar relativistic corrections, and ˆHSOZ\nR- the\nspin-dependent relativistic interactions that include the spin-\norbit, the electronic Zeeman, and the electronic spin-spin in-\nteractions. Rstands for the collection of Cartesian nuclear co-\nordinates, and the subscript denotes the explicit dependence\nof the Hamiltonian terms on the nuclear configuration. In this\npaper, I do not include the interactions of the electronic spin\nwith the nuclear magnetic moments, which give rise to hyper-\nfine splitting of the electronic energy levels.\nI define the zero-order adiabatic electronic wavefunctions,\f\f\fK(0)\nRE\n, and the zero-order adiabatic electronic potential en-\nergy surfaces, E(0)\nK,R, as the eigenvectors and the eigenvalues\nof the non-relativistic electronic Hamiltonian:\nˆHNR\nR\f\f\fK(0)\nRE\n=E(0)\nK,R\f\f\fK(0)\nRE\n. (2)The eigenfunctions and eigenvalues are zero order with re-\nspect to the spin-dependent interactions, in which case the\nzero-order electronic states form multiplets of degenerate\nstates that correspond to different projection of the electronic\nspin angular momentum. To account for the degeneracy of the\nzero-order electronic spectrum, the multi-index Kstands for a\ncollection of quantum numbers K={kSM S}that specify the\norbital quantum number k, the spin quantum number S, and\nthe associated spin projection MS. I assume that the state de-\ngeneracy in the multiplets is only of spin origin and exclude\ndegeneracy in the multiplets that results from spatial symme-\ntry. The adiabatic eigenfunctions in Eq. (2) are parametrically\ndependent on the nuclear configuration, R, and form an or-\nthonormal set at each nuclear geometry.\nThe spin-dependent interactions in Eq. (1) couple the elec-\ntronic states of the non-relativistic Hamiltonian and remove\npartially or fully the degeneracy of the electronic spectrum.\nI apply unitary degenerate perturbation theory42to diagonal-\nize the inter-multiplet interactions due to the spin-dependent\nHamiltonian ˆHSOZ\nR. The perturbative treatment is justified for\nmolecules that consist of atoms of light elements, including\nearly transition metal and main group elements, because for\nthem the spin-dependent multiplet splittings are significantly\nsmaller than the energy differences between the zero-order\nelectronic multiplets. The perturbed wavefunctions, |KR⟩, in\nunitary perturbation theory are obtained by a unitary transfor-\nmation of the zero-order wavefunctions:\n|KR⟩=eˆGR\f\f\fK(0)\nRE\n, (3)\nwhere ˆGis the generator of the transformation and satisfies\nˆG†=−ˆGto ensure the unitarity of the transformation. The\ngenerator is obtained perturbatively from the requirement of\nvanishing inter-multiplet interactions in the unitarily trans-\nformed Hamiltonian, and the perturbed eigenvalues derive\nfrom the intra-multiplet blocks of the transformed Hamilto-\nnian:\nD\nJ(0)\nR\f\f\fe−ˆGR\u0000ˆHNR\nR+ˆHSOZ\nR\u0001\neˆGR\f\f\fK(0)\nRE\n=δJKEK′K,R.(4)\nMulti-indeces J,K,..refer to general multiplets, and I reserve\nthe multi-index Ifor the ground-state multiplet. HIJstands\nfor the matrix block between two different multiplets IandJ,\nwhereas HI′Istands for the matrix block of the multiplet I.δJK\nstands for a multi-index Kronecker delta symbol. Use of the\nBaker-Campbell-Hausdorff formula in Eq. (4) and collection\nof terms according to the order of perturbation provides the\nequation for the first-order generator:\nh\nˆHNR\nR,ˆG(1)\nRi\nJK+HSOZ\nJK,R=0, (5)\nwhich is solved for the inter-multiplet blocks of the genera-\ntorG(1)\nJKand is supplemented with the additional condition of\nvanishing intra-multiplet blocks G(1)\nK′K=0. The perturbation\nexpansion in Eq. (4) gives also the perturbed eigenvalues up\nto second order in the perturbation:\nEK′K,R=E(0)\nK,RδK′K+HSOZ\nK′K,R+1\n2h\nˆHSOZ\nR,ˆG(1)\nRi\nK′K.(6)3\nEq. (5) gives the ground-state electronic wavefunction to first\norder in the spin-orbit perturbation as:\n|IR⟩=\f\f\fI(0)\nRE\n+ˆG(1)\nR\f\f\fI(0)\nRE\n=\f\f\fI(0)\nRE\n−∑\nJ̸=I\f\f\fJ(0)\nRE HSOI\nJI,R\nE(0)\nJ,R−E(0)\nI,R.(7)\nComponents of multiplets with different projections of spin\nangular momentum are non-interacting at zero order; coupling\nbetween them arises in the first-order contribution to the elec-\ntronic wavefunction, the second term in Eq. (7), from the spin-\norbit interaction (SOI) in ˆHSOZ\nR. The perturbative expansion of\nthe ground-state multiplet energies from Eq. (6) is given by:\nEI′I,R=E(0)\nI,RδI′I+E(1)\nI′I,R+E(2)\nI′I,R\n=E(0)\nI,RδI′I+HSOZ\nI′I,R+∑\nJ̸=IHSOZ\nI′J,RHSOZ\nJI,R\nE(0)\nI,R−E(0)\nJ,R,(8)\nwhere EI′I,Ris a matrix within the ground-state manifold,\nEM′M,R. The first-order and second-order adiabatic energy\ncontributions, the second and third term in Eq. (8), give rise\nto the traditional molecular spin Hamiltonian with the molec-\nular Zeeman interaction, µB⃗BgR⃗S, characterized by the elec-\ntronic g-tensor and µBthe Bohr magneton, and the zero-field\nsplitting interaction, ⃗SDR⃗S, characterized by the electronic D-\ntensor. Both the g-tensor and D-tensor depend parametrically\non the nuclear positions, and the existence of the molecular\nspin Hamiltonian requires an orbitally-non-degenerate ground\nstate, which is the case for all systems that I consider in the\npaper. I establish the connection with the traditional spin\nHamiltonian and the spin-component structure of the adia-\nbatic wavefunctions in Eq. (7) using the Wigner-Eckart the-\norem in Appendix A.\nI apply the Born-Oppenheimer approximation39,40toun-\ncouple the electron-nuclear dynamics on different electronic\nmultiplets and preserve the coupled electron-nuclear dynam-\nicswithin an electronic multiplet. This treatment of the\nelectron-nuclear dynamics is justified when the energy dif-\nferences between electronic multiplets are significantly larger\nthan the vibrational quanta involved in the spin relaxation dy-\nnamics, whereas the energy differences within electronic mul-\ntiplets are of similar magnitude or smaller than the vibrational\nquanta of the nuclear dynamics. This regime holds for the\nS=1/2 systems considered in this work. The case where\ntwo or more electronic multiplets come close in energy and\ninteract non-adiabatically as it occurs in conical intersections\nis outside the scope of the present treatment. In the Born-\nOppenheimer approximation, the vibronic wavefunction for\nthe ground-state multiplet is expressed as the sum of prod-\nucts of the electronic wavefunction of a multiplet component,\n|iSM,R⟩, and the associated nuclear wavefunction, χiSM(R):\n\f\fΨBO\nI\u000b\n=∑\nM,M′\f\fiSM′,R\u000b\ncM′M,R0χiSM(R) =|IR⟩χI(R).(9)\nThe electronic wavefunctions in Eq. (9) are the perturbed\nwavefunctions of Eqs. (3) and (7), and the last equality impliesa sum over the multiplet components. I choose a diabatic rep-\nresentation within the multiplet manifold that conserves the\ncharacter of the multiplet components, the dominant projec-\ntion of spin angular momentum M, and the matrix cM′M,R0\nin Eq. (9) allows for a specific choice of the spin quantiza-\ntion axis at a fixed nuclear configuration. I use this matrix to\nspecify the reference spin quantization axis for the spin re-\nlaxation dynamics at the equilibrium nuclear configuration.\nIf the matrix cM′M,Rwere allowed to vary with nuclear po-\nsition and at each nuclear configuration were chosen as the\neigenvectors of the matrix, EI′I,R, in Eq. (8), then the result-\ning wavefunctions, ∑M,M′|iSM′,R⟩cM′M,R, would be the adi-\nabatic electronic wavefunctions of the multiplet components\nand the associated eigenvalues would be the adiabatic poten-\ntial energy surfaces of the multiplet components.\nTo derive the Born-Oppenheimer Hamiltonian for the\nground-state multiplet, I apply the nuclear kinetic energy op-\nerator to the Born-Oppenheimer wavefunction in Eq. (9) and\nobtain:\nˆTR\f\fΨBO\nI\u000b\n=|IR⟩ˆTRχI(R)\n−i∂µ|IR⟩ˆpµχI(R)+χI(R)ˆTR|IR⟩.(10)\nThe notation in Eq. (10) implies a sum over the components\nof the electronic multiplet. I use the Einstein convention for\nan unrestricted summation over repeated indeces, and I ex-\nplicitly include the sum symbol when there are restrictions on\nthe sum indeces and when there is a summation without in-\ndex repetition. µis the index of the mass-weighted nuclear\nCartesian coordinates, and ˆ pµis the nuclear momentum oper-\nator of coordinate µ.iis the imaginary unit, and I use atomic\nunits, such that ¯h=1. The action of the nuclear kinetic en-\nergy operator gives terms with nuclear derivatives acting only\non the nuclear, first term, and the electronic wavefunctions,\nthird term, as well as a mixed term with nuclear derivatives\nacting on both the electronic and nuclear wavefunctions.\nI project the molecular Hamiltonian, Eq. (1), in the ground-\nstate multiplet, Eq. (9), and use Eqs. (4) and (10) to derive the\nBorn-Oppenheimer molecular Hamiltonian:\nˆHBO\nI′I=δI′IˆTR−iAI′Iµ,Rˆpµ+DI′I,R+EI′I,R, (11)\nThe Hamiltonian in Eq. (11) is a matrix within the ground-\nstate electronic multiplet space and an operator in nuclear\nCartesian coordinate space. The first term is the nuclear ki-\nnetic energy, and the last term is the adiabatic potential en-\nergy matrix of the ground-state multiplet, Eq. (8). The second\nand third terms are non-adiabatic interactions and contain the\nfirst-derivative non-adiabatic coupling matrix,\nAI′Iµ,R=\nI′\nR\f\f∂µ|IR⟩, (12)\nand the second-derivative non-adiabatic coupling matrix,\nDI′I,R=\nI′\nR\f\fˆTR|IR⟩. (13)\nAI′Iµ,Ris the Berry connection,40,46,48,49which in this case\nhas a non-Abelian algebraic structure and arises as a result of\nnon-adiabatic interactions with excited electronic multiplets.4\nAll inter-multiplet non-adiabatic coupling matrices vanish be-\ncause of the application of the Born-Oppenheimer approxima-\ntion.\nI expand the Berry connection in the spin-orbit interaction\nto derive explicit expressions for the non-adiabatic coupling\nmatrix elements between multiplet components. I use the per-\nturbative expansion of the electronic wavefunctions, Eq. (3),\nand the Baker-Campbell-Hausdorff formula in Eq. (12) and\nobtain to first order in the SOI:\nAI′Iµ,R=D\nI′(0)\nR\f\f\fe−ˆGR∂µeˆGR\f\f\fI(0)\nRE\n=D\nI′(0)\nR\f\f\f∂µI(0)\nRE\n+D\nI′(0)\nR\f\f\fh\n∂µ,ˆG(1)\nRi\f\f\fI(0)\nRE\n.(14)\nThe zero-order term of the Berry connection A(0)\nI′Iµ,R, the first\nterm in the last equality in Eq. (14), is diagonal in the ground-\nstate multiplet space and is the generalization of the traditional\nBerry connection that arises from conical intersections.46,47\nBecause I consider systems away from conical intersections in\nthis paper, A(0)\nI′Iµ,Rvanishes. The first-order term in the spin-\norbit interaction, A(1)\nI′Iµ,R, the second term in Eq. (14), is the\nleading contribution to the Berry connection on the ground-\nstate manifold and it couples different multiplet components\nas a result of the interplay of spin-orbit and non-adiabatic\ninteractions with excited electronic states. Expansion of the\ncommutator in Eq. (14) gives an explicit formula for the first-\norder Berry connection in terms of zero-order adiabatic elec-\ntronic wavefunctions:\nA(1)\nI′Iµ,R=∑\nJ̸=IHSOI\nI′J,R\nE(0)\nI′,R−E(0)\nJ,RD\nJ(0)\nR\f\f\f∂µI(0)\nRE\n−∑\nJ̸=ID\nI′(0)\nR\f\f\f∂µJ(0)\nRE HSOI\nJI,R\nE(0)\nJ,R−E(0)\nI,R.(15)\nThe non-Abelian algebraic structure of A(1)\nI′Iµ,Rfollows from\nan application of the Wigner-Eckart theorem50to Eq. (15),\nwhich gives an expression for A(1)\nI′Iµ,Rin terms of the set of\nnon-commuting spin matrices, SM′M,α, with dimension equal\nto the dimensionality of the ground-state multiplet space:\nA(1)\nI′Iµ,R=iSM′M,αa(1)\nαµ,R,\na(1)\nαµ,R=2∑\nj̸=ihSOI\ni jα,R\nE(0)\ni,R−E(0)\nj,RD\nj(0)\nR\f\f\f∂µi(0)\nRE\n.(16)\nThe sum over the Cartesian index α=x,y,zis implicit\nin Eq. (16), and hSOI\ni jα,Ris the imaginary part of the one-\nelectron, mean-field spin-orbit coupling operator as given in\nAppendix A. The functions a(1)\nαµ,Rare the three Cartesian com-\nponents of a vector potential associated with each nuclear de-\ngree of freedom µ.a(1)\nαµ,Rare real functions of the nuclear\npositions, which ensures that A(1)\nI′Iµ,Ris an anti-Hermitian ma-\ntrix. Note that with the definition of the Berry connection in\nEq. (12), the product −i¯hA(1)\nI′Iµ,Ris a Hermitian matrix. Thelast equality of Eq. (16) expresses the first-order vector po-\ntential, a(1)\nαµ,R, in terms of matrix elements of zero-order adia-\nbatic electronic wavefunctions,\f\f\f{jSS}(0)\nRE\n=\f\f\fj(0)\nRE\n, with spin\nquantum number Sequal to the spin quantum number of the\nground-state multiplet and maximum projection of the spin\nangular momentum MS=S.\f\f\fi(0)\nRE\nis the maximum-spin-\nprojection ground-state electronic wavefunction and the sum\nin Eq. (16) runs over the maximum-spin-projection excited-\nstate electronic wavefunctions. The benefit of the Wigner-\nEckart theorem is to separate the exact spin-component struc-\nture from the electronic matrix elements and to express all rel-\nevant quantities in terms of electronic wavefunctions of fixed\nprojection of spin angular momentum. I implement Eq. (16)\ntogether with state-of-the-art density functional theory ap-\nproximations to evaluate the first-order Berry connection in\nSec. III.\nSimilar expansion of the second-derivative non-adiabatic\ncoupling matrix, Eq. (13), in the spin-orbit interaction gives:\nDI′I,R=D\nI′(0)\nR\f\f\fˆTR\f\f\fI(0)\nRE\n+D\nI′(0)\nR\f\f\fh\nˆTR,ˆG(1)\nRi\f\f\fI(0)\nRE\n=D(0)\nI′I,R+D(1)\nI′I,R.(17)\nThe zero-order term of the second-derivative non-adiabatic\ncoupling matrix, D(0)\nI′I,R, is similarly diagonal in the ground-\nstate multiplet space and vanishes away from conical inter-\nsections. The first-order contribution, D(1)\nI′I,R, can be entirely\nexpressed in terms of the first-order Berry connection:\nD(1)\nI′I,R=−∑\nµ1\n2D\nI′(0)\nR\f\f\f∂2\nµˆG(1)\nR\f\f\fI(0)\nRE\n−∑\nµD\nI′(0)\nR\f\f\f∂µˆG(1)\nR\f\f\f∂µI(0)\nRE\n=−1\n2∑\nµ∂µA(1)\nI′Iµ,R.(18)\nThe first line of Eq. (18) follows from the expansion of the\ncommutator with the nuclear kinetic energy in Eq. (17), and\nin the second line I apply an identity proved in Appendix B\ntogether with the definition of the first-order Berry connection\nin Eq. (14).\nCollecting the results from the perturbative treatment of the\nnon-adiabatic coupling matrices, Eqs. (14) and (18), and the\nadiabatic potential energy matrix, Eq. (8), gives an expansion\nof the Born-Oppenheimer molecular Hamiltonian, Eq.(11), in\nthe spin-orbit interaction:\nˆHBO\nI′I=δI′IˆTR−iA(1)\nI′Iµ,Rˆpµ−1\n2∑\nµ∂µA(1)\nI′Iµ,R\n+δI′IE(0)\nI,R+E(1)\nI′I,R+E(2)\nI′I,R.(19)\nUniting the Berry connection terms with the nuclear kinetic\nenergy gives a covariant form of this Hamiltonian:\nˆHBO\nI′I=−1\n2∑\nµ\u0010\n∂µ+A(1)\nI′Iµ,R\u00112\n+δI′IE(0)\nI,R+E(1)\nI′I,R+E(2)\nI′I,R.(20)5\nEq. (20) and Eq. (19) agree to first-order in the Berry connec-\ntion upon expansion of the kinetic energy term. I use Eq. (16)\nand Eq. (8) in Eq. (20) to derive the final form of the new\nmolecular spin Hamiltonian in terms of spin operators, ˆSα,\nacting in the ground-state multiplet space:\nˆHspin=−1\n2∑\nµ\u0010\n∂µ+iˆSαa(1)\nαµ,R\u00112\n+E(0)\nR+µBBαg(2)\nαβ,RˆSβ+ˆSαD(2)\nαβ,RˆSβ.(21)\nEq. (21), the extended molecular spin Hamiltonian, is a main\nresult of this paper. It contains a novel, spin-vibronic vec-\ntor potential that derives from the first-order Berry connec-\ntion, as well as the traditional spin Hamiltonian terms that\nderive from the adiabatic potential energy matrix. The spin-\nvibronic vector potential gives rise to an effective magnetic\nfield on the ground-state multiplet, and I explore its effects on\nthe spin-vibrational dynamics in the rest of this paper. The\nspin-orbit interaction makes a leading first-order contribution\nto the vector potential, unlike the adiabatic molecular Zeeman\nand zero-field splitting terms, which are of second order in the\nspin-orbit and orbital Zeeman interactions. I expect that the\nspin-vibronic vector potential dominates the spin-vibrational\ndynamics in weak external magnetic fields, and I demonstrate\nits fundamental role for the spin relaxation dynamics of a pro-\ntotypical molecular qubit. The external magnetic field also\nmakes a contribution to the Berry connection: at first order\nthe contribution to the Berry connection is diagonal in the\nground-state multiplet space and does not couple to the spin-\nvibrational dynamics.\nB. Gauge transformation\nI showed in Sec. II A that the spin-vibrational dynamics in\nthe ground-state multiplet is governed by the molecular spin\nHamiltonian in Eq. (21), which contains a novel, spin-vibronic\nvector potential together with the traditional spin Hamiltonian\ninteractions. In Sec. II B, I transform the representation of\nthe Hamiltonian to bring the molecular spin Hamiltonian to a\nsymmetrized form that is convenient for application in time-\ndependent perturbation theory.\nGauge transformations40,48are nuclear configuration-\ndependent single-valued unitary transformations of the elec-\ntronic multiplet wavefunctions:\n\f\f˜IR\u000b\n=|IR⟩UI′I,R=∑\nM′\f\fiSM′,R\u000b\nUM′M,R, (22)\nwhich preserve the dynamics generated by the Born-\nOppenheimer molecular Hamiltonian, Eq. (11). As a result of\nthe gauge transformation in Eq. (22), the spin-vibronic vec-\ntor potential, Eq. (12), transforms as a non-Abelian gauge\npotential,40,48\n˜AI′Iµ,R=U†\nI′I′′,RAI′′I′′′µ,RUI′′′I,R+U†\nI′I′′,R∂µUI′′I,R,(23)\nand the Born-Oppenheimer Hamiltonian undergoes a unitary\ntransformation,\n˜HBO\nI′I,R=U†\nI′I′′,RHBO\nI′′I′′′,RUI′′′I,R. (24)Physical observables are independent of the specific choice of\nelectronic representation; they are gauge-covariant quantities,\nwhich transform upon gauge transformations like the Hamil-\ntonian in Eq. (24). For instance, the gauge-covariant nuclear\nmomentum, called kinematic momentum ˆΠI′Iµ, differs from\nthe canonical momentum ˆ pµby the vector potential:\nˆΠI′Iµ=δI′Iˆpµ−iAI′Iµ,R, (25)\nand neither ˆ pµnorAI′Iµ,Rare separately gauge-covariant. The\nvector potential in Eq. (25) results in the non-commutativity\nof the kinematic momenta:\n\u0002ˆΠµ,ˆΠν\u0003\nI′I=−FI′Iµν,R, (26)\nunlike the well-known commutation relation of the canonical\nmomenta, [ˆpµ,ˆpν] =δµν. In Eq. (26), bold symbols denote\nmatrices in the ground-state multiplet space, and FI′Iµν,Ris\nthe field tensor (also called Berry curvature46):\nFI′Iµν,R=∂µAI′Iν,R−∂νAI′Iµ,R+\u0002\nAµ,R,Aν,R\u0003\nI′I,(27)\nwhich is a gauge-covariant measure of the strength of the in-\nduced effective magnetic field on the ground-state manifold.\nThe gauge-covariance of the field tensor requires that UI′IR\nis both unitary and single-valued. Expansion of the spin-\nvibronic vector potential to first order in the spin-orbit interac-\ntion as in Sec. II A gives the first-order spin-vibronic magnetic\nfield tensor:\nF(1)\nI′Iµν,R=∂µA(1)\nI′Iν,R−∂νA(1)\nI′Iµ,R=iSM′M,αf(1)\nαµν,R,(28)\nwith\nf(1)\nαµν,R=∂µa(1)\nαν,R−∂νa(1)\nαµ,R=∂µ∧a(1)\nαν,R. (29)\nIn Eq. (29), ∧denotes the antisymmetric product, the gener-\nalization of the cross-product ×to multiple dimensions. With\nthe first-order spin-vibronic vector potential, the kinematic\nmomentum becomes:\nˆΠM′Mµ=δM′Mˆpµ+SM′M,αa(1)\nαµν,R. (30)\nThe nuclear dynamics depends only on gauge-covariant quan-\ntities, and I derive the Heisenberg equation of motion for the\nkinematic momentum Eq. (30) with the Hamiltonian Eq. (21)\nto leading order in the spin-orbit interaction:\nd2ˆRµ\ndt2=ˆSα1\n2\u001adˆRν\ndtf(1)\nανµ,R−f(1)\nαµν,RdˆRν\ndt\u001b\n−∂µE(0)\nR−µBBα∂µg(2)\nαβ,RˆSβ−ˆSα∂µD(2)\nαβ,RˆSβ.(31)\nThe derivation of Eq. (31) uses the equation of motion for\nthe nuclear position δM′MdˆRµ/dt=ˆΠM′Mµwith the veloc-\nity operator dˆRµ/dt. The first term on the right-hand side of\nEq. (31) is a quantum Lorentz force, the quantum equivalent to\nthe classical Lorentz force51FL=v×Bfor a unit-charge par-\nticle moving in three dimensions in a magnetic field B, where\nBrelates to the electromagnetic tensor Fi jasεi jkBk=Fjiwith6\nthe completely antisymmetric tensor εi jk. Because of the non-\nAbelian algebraic structure of the spin-vibronic magnetic field\ntensor, there is a separate magnetic field for each generator of\nthe associated non-Abelian group SO(3) or SU(2). The rest of\nthe terms on the right-hand side are the forces that originate\nfrom the adiabatic potential energy matrix.\nI use the freedom of gauge transformation to explicitly de-\nrive the spin-vibronic magnetic field tensor contribution to the\nspin Hamiltonian in Eq. (21). With the hindsight of the har-\nmonic approximation, I first expand the spin-vibronic vector\npotential to linear order in the deviations uµfrom a fixed nu-\nclear configuration R0, which is equivalent to the multipole\nexpansion of the electromagnetic vector potential50:\nA(1)\nI′Iµ,R=A(1)\nI′Iµ,R0+∂νA(1)\nI′Iµ,R0uν. (32)\nI construct a single-valued unitary gauge transformation using\nthe generator approach UI′I,R=eΛI′I,Rwith the anti-Hermitian\ngenerator function:\nΛI′I,R=A(1)\nI′Iµ,R0uν+1\n2∂νA(1)\nI′Iµ,R0uνuµ, (33)\nthat transforms the molecular spin Hamiltonian to a symmet-\nric gauge. I use this gauge transformation in Eq. (23) to obtain\nthe first-order, symmetrized spin-vibronic vector potential:\n˜A(1)\nI′Iµ,R=1\n2\u0010\n∂νA(1)\nI′Iµ,R0uν−∂µA(1)\nI′Iν,R0uν\u0011\n. (34)\nGauge transforming Eq. (21) together with the vector potential\nin Eq. (34) gives to leading order in the spin-orbit interaction\nthe symmetrized molecular spin Hamiltonian:\nˆHspin=ˆH(0)+1\n4f(1)\nανµ,R0ˆSα\u0000\nuν∧ˆpµ\u0001\n+µBBαg(2)\nαβ,RˆSβ+ˆSαD(2)\nαβ,RˆSβ,(35)\nwith\nˆH(0)=−1\n2∑\nµ∂2\nµ+E(0)\nR\nand summation over repeated indeces. The significance of\nthe spin-vibronic vector potential is particularly revealing in\nthe symmetric gauge where it gives rise to a spin-vibrational\norbit interaction, second term in Eq. (35), which involves\nthe effective magnetic field f(1)\nανµ,R0ˆlνµinduced by the vi-\nbrational angular motion with vibrational angular momen-\ntum ˆlνµ=uν∧ˆpµ. The spin-vibrational orbit interaction is\nanalogous to the spin-electronic orbit interaction (SOI) and\nit is similarly symmetric upon both time-reversal and par-\nity transformations. Note that the time-reversal invariance\nrequires that the first-order magnetic field tensor f(1)\nανµ,Ris\nreal. This implies that in the absence of an external magnetic\nfield, the spin-vibrational orbit interaction preserves Kramers’\ntheorem50and guarantees that half-integer spin systems have\ndoubly degenerate electronic states.C. Harmonic approximation\nI carried out a multipole expansion of the spin-vibronic vec-\ntor potential followed by a gauge transformation to a sym-\nmetric gauge in Sec. II B to derive the spin-vibrational orbit\ninteraction in the molecular spin Hamiltonian Eq. (35). In\nSec. II C, I derive a harmonic approximation to the molec-\nular spin Hamiltonian in Eq. (35), which I use to obtain an\nexpression for the spin relaxation time in Sec. II D using time-\ndependent perturbation theory. I start with the case of har-\nmonic vibrational dynamics of molecules in Sec. II C 1, which\nI extend to molecular crystals in Sec. II C 2.\n1. Vibrations of molecules\nThe harmonic approximation to the vibrational dynamics\nof molecules52is based on the second-order expansion of the\nground-state potential energy surface E(0)\nRinˆH(0)around the\nequilibrium nuclear geometry R0:\nˆHvib=−1\n2∑\nµ∂2\nµ+1\n2∂µ∂νE(0)\nR0uµuν\n=1\n4∑\niωi\u0000\nP2\ni+Q2\ni\u0001(36)\nI introduce in Eq. (36) the dimensionless normal mode coor-\ndinates Qiand momenta Piof vibrations with frequencies ωi.\nThe mass-weighted Cartesian displacements and momenta re-\nlate to the normal mode coordinates and momenta by the lin-\near transformation:\nuµ=∑\niCµ,ir\n1\n2ωiQi,Qi=b†\ni+bi\nˆpµ=∑\niCµ,ir\nωi\n2Pi,Pi=i\u0010\nb†\ni−bi\u0011\nwith Cµ,ithe eigenvectors of the mass-weighted Hessian ma-\ntrix. The equations also give the expressions for the quanti-\nzation of QiandPiin terms of the ladder operators biandb†\ni.\nThe second-order expansion of the spin-dependent terms in\nthe Hamiltonian Eq. (35) gives: (i) the traditional static spin\nHamiltonian at the equilibrium nuclear configuration, which I\ncall the reference spin Hamiltonian:\nˆHspin-ref=µBBαg(2)\nαβ,R0ˆSβ+ˆSαD(2)\nαβ,R0ˆSβ; (37)\n(ii) one-phonon (1P) spin-vibrational coupling Hamiltonian\nwith adiabatic (AD) origin, deriving from the adiabatic po-\ntential energy matrix in Eq. (35):\nˆH1P-AD=µB∂νg(2)\nαβ,R0BαˆSβuν+∂νD(2)\nαβ,R0ˆSαˆSβuν;(38)\n(iii) two-phonon (2P) spin-vibrational coupling Hamiltonian\nwith non-adiabatic (NA) origin, deriving from the spin-\nvibrational orbit interaction in Eq. (35), as well as the adia-7\nbatic two-phonon spin-vibrational coupling Hamiltonian:\nˆH2P-NA=1\n4f(1)\nανµ,R0ˆSα\u0000\nuν∧ˆpµ\u0001\n,\nˆH2P-AD=µB\n2∂ν∂µg(2)\nαβ,R0BαˆSβuνuµ+\n1\n2∂ν∂µD(2)\nαβ,R0ˆSαˆSβuνuµ.(39)\nTransforming the spin-vibrational coupling Hamiltonians in\nEqs. (38) and (39) to normal mode representation gives: (i)\nthe one-phonon spin-vibrational interactions:\nˆH1P-AD=µB∂ig(2)\nαβBαˆSref\nβQi+∂iD(2)\nαβˆSref\nαˆSref\nβQi; (40)\n(ii) the two-phonon spin-vibrational interactions:\nˆH2P-NA=1\n4f(1)\nαjiˆSref\nα(Qj∧Pi),\nˆH2P-AD=µB\n2∂i∂jg(2)\nαβBαˆSref\nβQiQj+\n1\n2∂i∂jD(2)\nαβˆSref\nαˆSref\nβQiQj.(41)\nIn Eqs. (40) and (41), indeces iandjrefer to the dimension-\nless normal modes, and fαji=ωi∂jaαi−ωj∂iaαj.\n2. Vibrations of molecular crystals\nThe generalization of the harmonic approximation in\nSec. II C 1 to molecular crystals requires a straightforward\nchange of notation to account for the wavevector qdepen-\ndence of the normal mode coordinates Qiq, momenta Piq, and\nfrequencies ωiq:53\nˆHvib=1\n4∑\ni,qωiq\u0000\nP2\niq+Q2\niq\u0001\n. (42)\nThe sum in Eq. (42) goes over all normal mode branches\niand all values of the wavevector qin the first Brillouin\nzone. The mass-weighted Cartesian displacements uµLand\nmomenta ˆ pµLof atomic coordinate µin the cell centered at\nLin terms of the normal mode coordinates and momenta be-\ncome:\nuµL=∑\ni,qCµ,iqs\n1\n2Nωiqeiq·LQiq,Qiq=b†\niq+bi−q\nˆpµL=∑\ni,qCµ,iqr\nωiq\n2Neiq·LPiq,Piq=i\u0010\nb†\niq−bi−q\u0011\nwith Cµ,iqthe eigenvectors of the mass-weighted dynamical\nmatrix. The equations give also the quantization of the crys-\ntal vibrations in terms of the ladder operators biqandb†\niq. The\nspin relaxation rate model in Sec. II D 3 assumes that the para-\nmagnetic molecular center, a paramagnetic molecule in a crys-\ntal environment of diamagnetic molecular counterparts, is lo-\ncalized in the unit cell at the origin L=0 and relies only on thedynamics of the optical vibrational modes, for which I adopt\nan approximation that neglects the dependence of the opti-\ncal mode eigenvectors Cµ,iqon the wavevector q. Within this\napproximation, the paramagnetic molecular center displace-\nments and momenta in terms of the optical normal modes are\ngiven by:\nuµ=∑\ni,qr\n1\n2NωiCµ,iQiq,\nˆpµ=∑\ni,qr\nωi\n2NCµ,iPiq.(43)\nSubstituting Eq. (43) in Eq. (39) gives the expression for\nthe two-phonon spin-vibrational coupling Hamiltonian of the\nparamagnetic molecular center,\nˆH2P-NA=1\nN∑\niq,jk1\n4f(1)\nαjiˆSref\nα\u0000\nQjk∧Piq\u0001\nˆH2P-AD=1\nN∑\niq,jkµB\n2∂i∂jg(2)\nαβBαˆSref\nβQjkQiq+\n1\nN∑\niq,jk1\n2∂i∂jD(2)\nαβˆSref\nαˆSref\nβQjkQiq.(44)\nEq. (44) is the extension of the isolated molecule two-phonon\nspin-vibrational interactions Eq. (41) to a single paramagnetic\nmolecular spin center in a diamagnetic crystal environment. I\napply Eq. (44) in Sec. II D to compute the two-phonon Raman\nprocess contribution to the spin relaxation time.\nD. Spin relaxation rate model\n1. Rate model assumptions\nThe rate model in this paper focuses on the simulation of\nthe spin relaxation dynamics of S=1/2 molecular qubits near\nroom temperature and at weak to moderate magnetic field in-\ntensities. The rate model is motivated by experimental mea-\nsurements of the temperature and magnetic field dependence\nof the spin-lattice relaxation time ( T1time) of molecular qubit\ncrystals under diamagnetic dilution.5–9,11,14,43,44The assump-\ntions of the rate model are:\n• the model applies to single paramagnetic molecular\ncenters which are embedded in the crystal environment\nof diamagnetic molecular homologs.\n• the model targets early-transition-metal-based molec-\nular qubits, allowing the perturbative treatment of the\nspin-orbit interaction of Sec. II A. It further focuses\nonS=1/2 molecular qubits, for which the zero-field\ninteraction is unnecessary, leaving the spin-vibronic\nmagnetic-field and molecular Zeeman interactions as\nthe only spin-dependent contributions to the Hamilto-\nnian in Eq. (35).8\n• the model assumes that the spin relaxation dynam-\nics near room temperature is determined by the two-\nphonon spin-vibrational interactions in Eq.(44), which\ninvolve a broad range of vibrational modes in the re-\nlaxation dynamics by the virtual absorption-emission of\nphonons, the two-phonon Raman process.\n• the model includes only the intra-molecular, optical\nvibrational modes in the approximation of Eq. (43)\nas they are more efficient in modulating the intra-\nmolecular spin-dependent interactions.\n• the model approximates the density-of-states of an op-\ntical vibrational mode by a Gaussian function centered\nat the isolated molecule normal mode frequency. The\nwidth of the Gaussian function is treated as an empiri-\ncal parameter.\nSubsections II D 2 and II D 3 develop the rate model based on\ntime-dependent perturbation theory,20–22the molecular spin\nHamiltonian of Sec. II C, and the model assumptions.\n2. Reference state\nThe application of time-dependent perturbation theory to\nevaluate the two-phonon Raman process rate of the model of\nSec. II D 1 requires a definition of the initial and final states of\nthe relaxation dynamics.21I assume that as a result of thermal\nequilibration the initial and final states are the eigenvectors\nof the reference spin Hamiltonian Eq. (37) at the equilibrium\nnuclear configuration:\nˆU†ˆHspin-ref ˆU=EM\nˆSref\nα=ˆU†ˆSαˆU.(45)\nThis definition of reference states can be straightforwardly im-\nplemented in the molecular spin Hamiltonian by rotating the\nquantization axis of the reference spin operators ˆSref\nαas written\nin the second line of Eq. (45), where ˆUis the unitary matrix\ndiagonalizing the reference spin Hamiltonian. With this defi-\nnition, the quantization axis of ˆSref\nαclosely follows the direc-\ntion of the external magnetic field for S=1/2 paramagnetic\nmolecular centers with a near isotropic g-tensor.\n3. Two-phonon Raman process rate\nThe rate kb→afor the spin-flip transition from the initial\nstate bto the final state avia the two-phonon Raman process,\nwhere a phonon in mode jkis absorbed and a phonon in mode\niqis emitted, is given by time-dependent perturbation theory\nas:\nkb→a=2π\n¯h∑\niq,niq∑\njk,njkρniqρnjkδ\u0000\n¯hωiq−¯hωjk−∆ba\u0001\n\f\f\na,niq+1,njk−1\f\fˆH2P\f\fb,niq,njk\u000b\f\f2.(46)In Eq. (46), the sum ranges over all optical vibrational modes\naccording to the rate model assumptions in Sec. II D 1, their\nassociated wavevectors and occupation numbers niq.ρniqis\nthe thermal weight of mode iq,ωiq- the frequency of the\nmode, and ∆ba- the energy gap between the initial and final\nstates. δis the Dirac delta function that imposes energy con-\nservation. The virtual absorption-emission process in Eq. (46)\nallows optical modes with close frequencies to match ener-\ngetically the small energy gap in the spin-flip transition, jus-\ntifying the leading role of the two-phonon Raman process in\nthe near-room-temperature spin relaxation dynamics. Evalu-\nation of the two-phonon spin-vibrational Hamiltonian matrix\nelements using Eq. (44) gives: (i) for the non-adiabatic two-\nphonon coupling matrix elements\n\na,niq+1,njk−1\f\fˆH2P-NA\f\fb,niq,njk\u000b\n=\n=i f(1)\nαjiSref\nα,ab1\nN√njkp\nniq+1\n=H2P-NA\nji,ab1\nN√njkp\nniq+1,(47)\nwith the definition of H2P-NA\nji,abin the last line of Eq. (47); (ii)\nfor the adiabatic two-phonon coupling matrix elements\n\na,niq+1,njk−1\f\fˆH2P-AD\f\fb,niq,njk\u000b\n=\n=µB∂i∂jg(2)\nαβBαSref\nβ,ab1\nN√njkp\nniq+1\n=H2P-AD\nji,ab1\nN√njkp\nniq+1(48)\nwith the definition of H2P-AD\nji,abin the last line of Eq. (48). I\ninclude only the g-tensor contribution in Eq. (48) because I\nconsider only S=1/2 systems, for which the D-tensor term\nvanishes. With the model for the optical vibrational modes\nin Eq. (43), the Hamiltonian matrix elements are independent\nof the mode wavevector, resulting in the simplified expression\nfor the Raman process rate:\nkb→a=2π\n¯h∑\ni,j\f\f\fH2P-NA\nji,ab+H2P-AD\nji,ab\f\f\f2\n1\nN2∑\nq,kδ\u0000\n¯hωiq−¯hωjk−∆ba\u0001\n(¯n(ωiq)+1)¯n(ωjk).(49)\nI carry out the sum over the thermal weight in Eq. (49) to ob-\ntain the average thermal occupation of the vibrational mode\n¯n(ωiq). Furthermore, I carry out the sum over the mode\nwavevectors in Eq. (49) using the vibrational mode density of\nstates gi(ω) =1\nN∑qδ(ω−ωiq)to obtain the final expression\nfor the Raman process rate:\nkb→a=2π\n¯h2∑\ni,j\f\f\fH2P-NA\nji,ab+H2P-AD\nji,ab\f\f\f2\nZ\ndωgi(ω−ωba)gj(ω)(¯n(ω−ωba)+1)¯n(ω).(50)\nThe spin-lattice ( T1) relaxation time characterizes the\ntimescale of thermal equilibration of the spin system as a re-\nsult of the spin-vibrational interactions with the phonon bath.9\nFor the case of an S=1/2 spin system that interacts with a\nphonon bath of reciprocal temperature βwith rate constants\nkb→aandka→bsatisfying detailed balance, the T1time is given\nby:\n1\nT1=kb→a+ka→b≈2k. (51)\nThe last equality in Eq. (51) follows from the smallness of\nωbacompared to the vibrational mode frequencies, and kis\ndefined as the expression obtained from Eq. (50) for vanishing\nωba. Substitution of Eq. (50) in Eq. (51) gives the expression\nfor the contribution of the two-phonon Raman process to the\nT1time:\n1\nT1=4π\n¯h2∑\ni,j\f\f\fH2P-NA\nji,ab+H2P-AD\nji,ab\f\f\f2\nZ\ndωgi(ω)gj(ω)(¯n(ω)+1)¯n(ω).(52)\nThe implementation of Eq. (52) requires the vibrational mode\ndensity-of-states, for which I adopt the Gaussian approxima-\ntion from Sec. II D 1 with mode frequency ωiand width σi.\nWith the Gaussian approximation, I arrive at the final ex-\npression for the T1spin relaxation time of the rate model of\nSec. II D 1:\n1\nT1=4π\n¯h2∑\ni,j\f\f\fH2P-NA\nji,ab+H2P-AD\nji,ab\f\f\f2\n1q\n2π(σ2\ni+σ2\nj)e−1\n2(ωi−ωj)2\nσ2\ni+σ2\nje−β¯hωji\n\u0010\n1−e−β¯hωji\u00112,(53)\nwhere ωji=σ2\njωi+σ2\niωj\nσ2\ni+σ2\njis the center of the product of the mode\ndensity-of-states Gaussians. To obtain Eq. (53), I assumed\nthat the thermal probabilities change slowly at the scale of the\nmode density-of-states, allowing their evaluation at the center\nof the Gaussian product. Section III presents the computa-\ntion of the spin-vibrational Hamiltonian matrix elements in\nEq. (53) based on density functional theory calculations on\nthe isolated paramagnetic molecule. The only empirical pa-\nrameters that enter in the implementation of Eq. (53) are the\nwidths of the mode density-of-states, for which I use a simple\nmodel with mode widths: σi=10cm−1forωi<100cm−1,\nσi=5cm−1forωi<200cm−1, and σi=1cm−1forωi>\n200cm−1. The ab initio estimation of the mode density-of-\nstates requires a computational model of the molecular crystal,\nwhich is outside of the scope of the present work. I present re-\nsults in Sec. IV for the two-phonon Raman contribution to the\nT1time as a function of temperature and magnetic field orien-\ntation for the prototypical molecular qubit Cu(II) porphyrin.\nIII. IMPLEMENTATION\nI derived in Sec. II an extended molecular spin Hamilto-\nnian that contains a novel, non-adiabatic contribution fromthe spin-vibronic vector potential. In Sec. III, I present the\nnumerical evaluation of the spin-vibronic vector potential via\nstate-of-the-art density functional theory.\nThe starting point of the derivation is Eq. (16), which I eval-\nuate using linear response unrestricted density functional the-\nory. As derived in Refs.23,54, the first-order perturbed Kohn-\nSham electronic wavefunctions\f\f\fψ(1)\n0,αE\nwith respect to the\nspin-orbit interaction are given by:\n\f\f\fψ(1)\n0,αE\n=∑\na,i,σUSOI\naiσ,α|ψaσ\niσ⟩,(54)\nwhere I use the convention that i,j,..denote occupied molec-\nular orbitals (MOs) and a,b,..- unoccupied (virtual) MOs in\nthe ground-state determinant, and σis the MO spin projec-\ntion. The sum in Eq. (54) ranges over all single excitations\nfrom the occupied to the unoccupied MOs of both spin pro-\njections. The coefficients USOI\naiσsatisfy the coupled-perturbed\nself-consistent-field Kohn-Sham equations:\n∑\na,iMσσ\nb j,aiUSOI\naiσ,α=−hSOI\nbσjσ,α,(55)\nwith the magnetic electronic Hessian Mσ′σ\nb j,aigiven by:\nMσ′σ\nb j,ai=(εaσ−εiσ)δb j,aiδσσ′+\ncHF(\naσj′\nσ\f\fiσb′\nσ\u000b\n−\naσb′\nσ\f\fiσj′\nσ\u000b\n).(56)\nIn Eq. (56), εiσare the Kohn-Sham orbital energies, cHFis\nthe percentage of exact Hartree-Fock exchange in the density\nfunctional approximation, and ⟨aσj′\nσ|iσb′\nσ⟩are the electron\nrepulsion integrals. hSOI\nbσjσ,αin Eq. (55) are the matrix ele-\nments of the spin-orbit coupling operator as defined in Ap-\npendix A.\nWith Eq. (54) the spin-vibronic vector potential in Eq. (16)\nbecomes:\na(1)\nαµ,R=−2∑\na,i,σUSOC\niaσ,αR\naσR\f\f∂µiσR\u000b\n.(57)\nWith generalized-gradient density functional approximations,\nthe coupled-perturbed Kohn-Sham equations have a non-\niterative solution, and Eq. (57) gives:\na(1)\nαµ,R=2∑\na,i,σhSOI\niσaσ,αR\nεiσ,R−εaσ,R\naσR\f\f∂µiσR\u000b\n, (58)\nwhich is the result of Eq. (57) for electronic states repre-\nsented as single Kohn-Sham Slater determinants. To com-\nplete the evaluation of the spin-vibronic vector potential, I ob-\ntain the last term in the sum in Eq. (57) from the solution\nof the coupled-perturbed Kohn-Sham equations for the nu-\nclear displacements55,56. These equations are conventionally\nsolved when computing analytical second derivatives of the\nground-state energy, and using the results of analytical gra-\ndient theory,55,56I write the nuclear derivatives of the Kohn-\nSham MOs as:\n\f\f∂µiσR\u000b\n=∑\np,iUG\npiσ,µ|pσR⟩+\f\f∂µ˜iσR\u000b\n,(59)10\ni j ωi ωj|fi j|\nPBE/SV|fi j|\nPBE/TZ|fi j|\nPBE0/SV|∂i jg|\nPBE/SV|∂i jg|\nPBE/TZ|∂i jg|\nPBE0/SV\n10 10 126.8 126.8 0.000 0.000 0.000 0.042 0.043 0.061\n10 11 126.8 126.8 13.500 16.189 7.995 0.018 0.018 0.027\n11 11 126.8 126.8 0.000 0.000 0.000 0.042 0.043 0.061\n13 13 204.3 204.3 0.000 0.000 0.000 0.029 0.030 0.039\n13 14 204.3 204.4 3.356 2.885 4.824 0.007 0.007 0.009\n14 14 204.4 204.4 0.000 0.000 0.000 0.029 0.030 0.039\n12 16 196.5 233.6 0.000 0.000 0.000 0.071 0.076 0.117\n15 17 212.9 240.6 4.211 4.727 5.345 0.002 0.002 0.002\n18 19 288.9 288.9 13.961 14.213 15.205 0.003 0.003 0.012\n22 23 387.1 389.1 3.532 3.306 3.092 0.001 0.001 0.002\n23 25 389.1 433.4 8.129 8.826 9.726 0.001 0.001 0.002\n25 26 433.4 437.8 4.292 4.033 4.351 0.001 0.001 0.002\n28 29 454.3 454.3 7.686 8.606 2.158 0.000 0.000 0.000\nTABLE I. Two-phonon spin-vibrational Hamiltonian matrix elements for the leading pairs of normal modes. iandjare the indeces of the\nmodes, and ωiandωjare the mode frequencies in cm−1.|fi j|2=∑αf2\nαi jdenotes the magnitude of the spin-vibronic magnetic field tensor for\nmodes iandjin 10−3cm−1.|∂i jg|2=µB∑α̸=β∂i∂jg2\nαβi jdenotes the magnitude of the off-diagonal portion of the second derivative g-tensor\nfor modes iandjin 10−3cm−1/T. The matrix elements are computed by: PBE density functional and def2-SVP basis set (PBE/SV), PBE\ndensity functional and def2-TZVP basis set (PBE/TZ), and PBE0 density functional and def2-SVP basis set (PBE0/SV).\nFIG. 1. Major normal modes contributing to the two-phonon spin\nrelaxation time T1for the prototypical Cu(II) porphyrin S=1/2\nmolecular qubit. The four normal modes are doubly degenerate with\nnormal mode frequencies 126 .8cm−1(upper pair) and 288 .9cm−1\n(lower pair). The mode at 126 .8cm−1brings the molecule out of the\nsymmetry plane, whereas the mode at 288 .9cm−1develops within\nthe plane of Cu(II) porphyrin. Arrows portray the relative magnitude\nand direction of the atomic displacements. Color code: Carbon is\ndark gray, Nitrogen - blue, Copper - orange, and Hydrogen - light\ngray.\nwhere UG\npiσ,µare the solutions of the coupled-perturbed Kohn-\nSham equations for the nuclear displacement µ, and\f\f˜iσR\u000b\nde-\nnotes MOs with frozen orbital coefficients, such that the nu-\nclear derivative operates only on the atom-centered basis func-\ntions. Using Eq. (59) in Eq. (57), I arrive at the final formulafor the evaluation of the spin-vibronic vector potential:\na(1)\nαµ,R=−2∑\na,i,σUSOC\niaσ,αR\u0010\nUG\naiσ,µR+Saiσ,µR\u0011\n,(60)\nwhere Saiσ,µRis the right-hand derivative of the basis overlap\nmatrix in the MO basis. The numerical evaluation of Eq. (60)\nrequires two coupled-perturbed Kohn-Sham response calcu-\nlations: one response calculation for the spin-orbit interac-\ntion perturbation USOI\niaσ,αR, and a second response calculation\nfor the nuclear displacement perturbations UG\naiσ,µR. I imple-\nment Eq. (60) in an in-house version of the quantum chem-\nistry software package ORCA57, for which I adapted already\nexisting highly optimized algorithms for molecular g-tensor\ncalculations23and analytical second derivatives55. I applied\nin all calculations a widely-used mean-field approximation for\nthe spin-orbit coupling operator26that is already available in\nthe ORCA suite.\nThe implementation of the spin-vibronic vector potential\nEq. (60) is consistent with the numerical evaluation of the\ng-tensor23and the D-tensor54in linear response unrestricted\ndensity functional theory. The molecular g-tensor, for in-\nstance, is expressed as:\ng(2)\nαβ,R=−∑\na,i,σUSOI\niaσ,αRIm(Laiσ,βR),(61)\nwhere Im (Laiσ,βR)is the imaginary part of the electronic or-\nbital angular momentum operator in the MO basis. Both the\nspin-vibronic vector potential Eq. (60) and the molecular g-\ntensor Eq. (61) expressions contain the self-consistent pertur-\nbation of the Kohn-Sham orbitals with respect to the spin-orbit\ninteraction; the crucial difference is the second perturbing in-\nteraction: for the spin-vibronic vector potential, these are the\nnuclear displacement perturbations, whereas for the molecu-\nlar g-tensor this is the orbital Zeeman interaction. This dif-\nference determines the different orders of the two effective11\n103105107\nPBE/SVP PBE/TZVP PBE0/SVP1031051071/T1, s-1Non-adiabatic rate\n10-2100102\n 50  150  250\nT, KTotal rate\nAdiabatic rateCuPc \nCuTTP\nFIG. 2. Two-phonon spin relaxation rate 1 /T1as a function of tem-\nperature Tin K for Cu(II) porphyrin. Rate is in s−1and the y-axis is a\nlogarithmic scale (base 10). The rate is uniformly averaged over the\nmagnetic field orientation. Magnetic field intensity is B=330mT.\nUpper panel is the total rate, middle panel - the non-adiabatic con-\ntribution to the rate, and lower panel - the adiabatic contribution to\nthe rate. Panels share the same temperature range. Black circles rep-\nresent experimentally measured T1times for Cu(II)-phthalocyanine\n(CuPc) from Ref.14. Black diamond is an experimetally measured\nT1time for Cu(II) tetratolylporphyrin (CuTPP) from Ref.43. Color\ncode: orange - PBE/def2-SVP, red - PBE/def2-TZVP, and blue -\nPBE0/def2-SVP. Note the difference in scale of the adiabatic and\nthe non-adiabatic rate.i j ωi ωj DoS Th. Pr.1\nT1\n10 11 126.8 126.8 0.056 2.588 1.758\n13 14 204.4 204.4 0.282 0.949 1.172\n18 19 288.9 289.0 0.282 0.438 5.385\n22 23 387.1 389.2 0.102 0.215 0.039\n28 29 454.3 454.3 0.282 0.141 0.035\n33 34 662.9 662.9 0.282 0.044 0.036\nTABLE II. Two-phonon spin relaxation rate contributions for room\ntemperature T=298K, magnetic field intensity B=330mT, and\nmagnetic field orientation θ=90orelative to the molecular axis.\niand jare the indeces of the modes, and ωiandωjare the mode\nfrequencies in cm−1. DoS denotes the mode density-of-states contri-\nbution to the rate using the Gaussian model in Sec. II D. Th. Pr. is the\nthermal probability contribution to the rate. 1 /T1is the two-phonon\nspin relaxation rate in MHz. The spin-vibronic matrix elements are\ncalculated at PBE0/def2-SVP level of theory.\ninteractions: the spin-vibronic vector potential is first order in\nthe spin-orbit interaction, whereas the molecular g-tensor is\nfirst order in the spin-orbit interaction but also first order in\nthe small orbital Zeeman interaction, resulting in an overall\nsecond order expression.\nThe matrix elements that enter in the calculation of the\nspin relaxation time Eq. (44) require geometric derivatives\nof both the spin-vibronic vector potential and the molecular\ng-tensor. I calculate these derivatives by numerical differen-\ntiation using a central difference approximation and normal\nmode displacements. I apply a dimensionless mode displace-\nment (displacement in units of the zero-point amplitude) of\n0.5. The calculation protocol starts with a gas-phase opti-\nmization of the molecular geometry using the selected ba-\nsis set and density functional together with: resolution-of-\nidentity approximation for the Coulomb portion of the Fock\nmatrix,58fine grid for the exchange-correlation functional in-\ntegration, very tight criteria for the self-consistent-field con-\nvergence, and tight criteria for the geometry convergence, all\nper ORCA 3.0.0 definitions.57A calculation of the analyti-\ncal second derivatives follows using the same convergence\ncriteria as applicable. In this work, I computed the molec-\nular Hessian using the Perdew-Burke-Ernzerhof (PBE) den-\nsity functional59and the Ahlrichs def2-TZVP basis set.60–62\nI calculated the spin-vibronic magnetic field tensor and the\nmolecular g-tensor second geometrical derivatives using three\ndensity functional and basis set combinations: PBE functional\nand def2-SVP basis set, PBE functional and def2-TZVP basis\nset, and the hybrid PBE0 functional63,64and def2-SVP ba-\nsis set. The results of these calculations and the associated\nspin-lattice relaxation times for the Cu(II) porphyrin molecu-\nlar qubit are presented in Sec. IV.\nIV. RESULTS\nI apply the new molecular spin Hamiltonian to the spin re-\nlaxation dynamics of Cu(II) porphyrin, a prototypical S=1/2\nmolecular qubit that represents the common core of a homol-12\ni j f jix fjiy fjiz1\nT1\n10 11 0.000 0.000 1.999 1.758\n13 14 0.000 -0.003 -1.206 1.172\n18 19 0.000 0.000 3.801 5.385\n22 23 0.000 0.000 -0.773 0.039\n28 29 0.004 0.000 0.539 0.035\n33 34 0.000 0.000 0.971 0.036\nTABLE III. Spin-vibronic magnetic-field tensor components for the\nleading contributions to the two-phonon spin relaxation rate for room\ntemperature T=298K, magnetic field intensity B=330mT, and\nmagnetic field orientation θ=90orelative to the molecular symme-\ntry axis. iandjare the indeces of the modes. fjixdenotes the value\nof the spin-vibronic tensor x-component in 10−3cm−1. 1/T1is the\ntwo-phonon spin relaxation rate in MHz. The spin-vibronic matrix\nelements are calculated at PBE0/def2-SVP level of theory.\nogous series of Cu(II)-based molecular qubits. I present the\nresults for the two-phonon spin relaxation time using the spin\nrate model of Sec. II D, and I calculate the two-phonon spin-\nvibrational Hamiltonian matrix elements according to the im-\nplementation in Sec. III using three levels of density func-\ntional theory to investigate the density functional and basis\nset dependence of the computed matrix elements.\nTable I presents the magnitudes of the spin-vibronic mag-\nnetic field tensor and the off-diagonal second derivative g-\ntensor for the leading pairs of normal modes. The striking\nresult is that the spin-vibronic magnetic field matrix elements\nare two to three orders of magnitude larger than the deriva-\ntive g-tensor matrix elements, demonstrating that the domi-\nnating spin-vibrational coupling mechanism in this class of\nmolecular qubits is of non-adiabatic character. The results in\nTable I affirm that this conclusion is independent of both the\nbasis set size and the density functional approximation. The\nbasis set dependence of the matrix elements is mild, show-\ning that computationally affordable basis sets of double zeta\nsize can be employed to compute spin-vibronic coupling ma-\ntrix elements. The amount of Hartree-Fock exchange in the\ndensity functional (the PBE0 density functional has 25% ex-\nact exchange compared to 0% for the PBE functional) leads\nto larger deviations in the magnitudes of the matrix elements,\nbut the values are in quantitative agreement with one another,\nallowing the use of generalized gradient density functionals\nfor initial investigations.\nThe data in Table I further demonstrate that degenerate\nnormal modes have a disproportionately large contribution to\nthe spin-vibrational coupling mechanism in Cu(II) porphyrin.\nFigure 1 depicts the two normal modes that are the major con-\ntributors to the spin-vibrational coupling in the qubit. The\nmode at 126 .8cm−1brings the molecule out of the molec-\nular plane, whereas the mode at 288 .9cm−1develops in the\nmolecular plane. Table I shows that the spin-vibronic matrix\nelements of the 288 .9cm−1in-plane mode increase with the\namount of Hartree-Fock exchange in the density functional\nand this mode becomes the major spin-vibrational coupling\nchannel at PBE0/def2-SVP level of theory. The emerging\nphysical picture from the results in Table I and Fig. 1 is that\n fitNon-adiabatic rate\n 0  45  90\nθ, deg1/T1, s-1\nAdiabatic rate\nPBE/SVP PBE/TZVP PBE0/SVP10-410-2100102104106FIG. 3. Two-phonon spin relaxation rate 1 /T1as a function of the\nangle θin degrees relative to the C4axis of Cu(II) porphyrin. Rate is\nin s−1and the y-axis is a logarithmic scale (base 10). The rate is uni-\nformly averaged over the polar angle φin the plane of the molecule.\nTemperature is 100K and magnetic field intensity is 330 mT. Upper\npanel is the the non-adiabatic contribution to the rate and lower panel\n- the adiabatic contribution to the rate. Panels share the same angle\nrange. The non-adiabatic rate is practically equal to the total rate -\nnote the difference in the scales of the two panels. Color codes the\nlevel of theory: orange - PBE/def2-SVP, red - PBE/def2-TZVP, and\nblue - PBE0/def2-SVP. The dashed black line is a Asin2(θ)fit to the\nPBE0/def2-SVP data. The fit to the non-adiabatic contribution coin-\ncides with the computed results.\nnearly degenerate normal modes linearly superpose to give vi-\nbrational rotational modes, which strongly interact with the\nmolecular spin via the spin-vibrational orbit interaction (see\nend of Sec. II B). The spin-vibrational orbit interaction is of\nnon-adiabatic origin and results from the non-Abelian Berry\ncurvature on the ground-state electronic spin multiplet.\nFigure 2 presents the magnetic-field-orientation averaged\ntwo-phonon spin relaxation rate 1 /T1, resulting from the cou-\npling to the optical normal modes of the molecular qubit com-\nputed by the spin relaxation rate model of Sec. II D. The data\nin Figure 2 confirm that the total two-phonon relaxation rate is\ndetermined by the non-adiabatic spin-vibrational orbit interac-\ntion and the adiabatic contribution to the relaxation rate is five13\norders of magnitude smaller. Similar to the spin-vibronic cou-\npling, this conclusion holds true independently of the basis set\nsize and the density functional approximation. The calculated\nrates are in excellent agreement with experimentally measured\nT1times for both a powder of Cu(II) phthalocyanine14and sin-\ngle crystals of Cu(II) tetratolylporphirin43. The theory con-\nfirms that the leading spin relaxation mechanism in Cu(II)-\nbased molecular qubits is the spin-vibrational orbit interac-\ntion.\nThe rates in Fig. 2 display a characteristic activated dynam-\nics with increasing temperature, which results from the suc-\ncessive thermal population of the doubly degenerate normal\nmodes. Table II presents the major normal mode contribu-\ntions to the two-phonon spin relaxation rate for T=298K and\nB=330mT. The data demonstrate that the doubly degener-\nate normal modes at 288 .9cm−1, 126.8cm−1, and 204 .4cm−1\nare the major channels for two-phonon spin relaxation in the\nsystem, and the three normal mode pairs fully determine the\norder of magnitude of the rate. This conclusion suggests an\nappealing route to control the spin relaxation rate of S=1/2\nmolecular qubits by decreasing the molecular symmetry via\nremoval of symmetry axes greater than second order.\nFigure 3 plots the two-phonon spin relaxation rate as a\nfunction of the angle θbetween the molecular C4axis and\nthe external magnetic field for T=100K and B=330mT.\nThe relaxation rates are averaged over the angle φthat de-\nscribes the rotation of the magnetic field vector around the C4\naxis. The results in Figure 3 demonstrate the strong magnetic-\nfield orientation dependence of the two-phonon relaxation rate\nwith distinct functional dependence of the rate contributions\nonθ: the non-adiabatic contribution, which entirely domi-\nnates the total rate, shows a clear sin2(θ)dependence (dashed\nline), whereas the adiabatic contribution goes through a max-\nimum at 45oand is described poorly by the sin2(θ)func-\ntion. This orientational dependence of the two-phonon rate\nallows to distinguish theoretically and experimentally the non-\nadiabatic and the adiabatic mechanisms of spin relaxation.\nThe computed sin2(θ)dependence of the total rate is in com-\nplete agreement with the observed sin2(θ)trend of the exper-\nimentally measured T1times43, which presents an indepen-\ndent confirmation of the non-adiabatic spin relaxation mech-\nanism. Furthermore, as a consequence of the orientational\ndependence, the two-phonon spin relaxation rate for in-plane\norientation θ=90oof the magnetic field is several orders of\nmagnitude larger than the rate for perpendicular orientation\nθ=0o. This result leads to a greater than one T1anisotropy,\ndefined as the ratio of the in-plane to the perpendicular relax-\nation rate, that similarly parallels experimental observations.\nThe dependence of the two-phonon rate on the orienta-\ntion of the external magnetic field can be rationalized based\non the data in Table III, which presents the components of\nthe spin-vibronic magnetic field tensor for the leading normal\nmode pairs in the spin relaxation dynamics at T=298Kand\nB=330mT. Table III shows that the largest component of\nthe spin-vibronic magnetic field tensor is along the C4axis of\nCu(II) porphyrin and the in-plane components are smaller by\nseveral orders of magnitude. As a result, the induced effective\nmagnetic field fαi jˆli jby the spin-vibrational orbit interactionis entirely directed along the C4axis of the molecule, and the\nmost efficient relaxation occurs when the molecular spin is\noriented perpendicularly to the C4axis in the molecular plane.\nProvided that the molecular g-tensor at the equilibrium geom-\netry is very nearly axial, the molecular spin closely follows\nthe orientation of the external magnetic field, such that the in-\nplane molecular spin component varies as sin (θ)withθthe\nangle between the magnetic field vector and the C4axis. The\nrelaxation rate depends on the squared modulus of the spin-\nvibrational orbit interaction, resulting in the observed sin2(θ)\ndependence in Fig. 3.\nV. CONCLUSIONS\nI derive in the present paper an extended molecular spin\nHamiltonian that takes into account both the traditional adia-\nbatic spin-dependent interactions and includes a novel, non-\nadiabatic spin-vibrational interaction. The derivation employs\nthe Born-Oppenheimer approximation to decouple the dy-\nnamics on different electronic spin manifolds, which induces a\nnon-Abelian Berry connection on the ground-state electronic\nmultiplet. The resulting non-Abelian Berry curvature is the\nspin-vibrational orbit interaction, which couples the vibra-\ntional angular momentum to the electronic spin in complete\nanalogy to the spin-electronic orbit interaction. Thus, the vi-\nbrational angular motion induces an effective magnetic field\nin the spin dynamics on the electronic ground state and vice\nversa the electronic spin exerts a quantum Lorentz force on the\nnuclear motion. The spin-vibronic magnetic field tensor that\nquantifies the strength of coupling between the vibrational an-\ngular motion and the electronic spin is first order in the spin-\norbit interaction and first order in the non-adiabatic interac-\ntions between the electronic multiplets. As such, this previ-\nously unappreciated interaction is expected to dominate the\nspin relaxation dynamics at weak magnetic field intensities\nand elevated temperatures.\nI implement the computation of the matrix elements of\nthe molecular spin Hamiltonian using linear response density\nfunctional theory, which allows the ab initio prediction of the\nstrength of the different interactions and the unbiased compar-\nison between them. The density-functional implementation\nrequires two different kinds of coupled-perturbed calculations\nfor computing the response of the electronic wavefunctions to\nthe spin-orbit interaction and to the nuclear displacements. I\nfurther develop a rate model to estimate the spin relaxation\ntime using the molecular spin Hamiltonian computed for the\nisolated paramagnetic molecule. The rate model is special-\nized to isolated paramagnetic molecular centers in molecular\ncrystals and to elevated temperatures and estimates the spin\nrelaxation time via the two-phonon Raman process with the\nsole participation of the optical crystal vibrations. The only\nempirical parameters of the model are the widths of the mode\ndensity-of-states, for which I employ a Gaussian approxima-\ntion.\nI apply the molecular spin Hamiltonian together with the\nspin relaxation rate model to the prototypical S=1/2 molec-\nular qubit Cu(II) porphyrin. The striking result is that the14\nspin relaxation rate near room temperature is determined by\nthe new spin-vibrational orbit interaction, and that the com-\nputed spin relaxation rate is in remarkable agreement with\nexperimental measurements. This result is independent of\nboth basis set size and density functional approximation. The\nphysical picture that emerges for spin relaxation in Cu(II)-\nbased molecular qubits is that the vibrational angular motion\nof nearly degenerate normal modes strongly couples to the\nelectronic spin dynamics and is responsible for spin relaxation\nin the system. Furthermore, the spin relaxation time shows a\nstrong dependence on the orientation of the magnetic field to\ntheC4symmetry axis of the molecule and is distinct for the\nnon-adiabatic and the adiabatic relaxation mechanisms, which\nallows to distinguish them both theoretically and experimen-\ntally.\nThe current work lays the foundation for a further investiga-\ntion of a much broader set of S=1/2 qubits. Furthermore, the\napproach can be successfully applied to S>1/2 systems that\ninclude the emerging class of optically addressable molecular\nqubits. Taken together, the new molecular spin Hamiltonian\nformalism together with density functional theory is expected\nto provide a much-needed theoretical approach to simulate\nand understand spin relaxation dynamics in molecular qubits.\nACKNOWLEDGMENTS\nI would like to acknowledge start-up funds from Indiana\nUniversity, Bloomington and Tufts University.\nDATA AVAILABILITY\nThe data that support the findings of this study are available\nfrom the corresponding author upon reasonable request.\nAppendix A SPIN STRUCTURE VIA THE\nWIGNER-ECKART THEOREM\nIn Appendix A, I apply the Wigner-Eckart theorem to estab-\nlish the spin-operator equivalents of the effective interactions\nin the molecular spin Hamiltonian. The spin-dependent rel-\nativistic interactions ˆHSOZ, including the interaction with the\nweak external magnetic field, consists of: (i) the spin-orbit\ninteraction:\nˆHSOI=∑\ni,m(−1)mˆF(1)\n−m(i)ˆS(1)\nm(i), (62)\nwhich is a one-electron operator in the mean-field approxi-\nmation and contains an orbital-dependent part ˆF(i), a func-\ntion of the position and momentum of electron i, and a spin-\ndependent part ˆS(i), the spin operator of electron i; (ii) the\nZeeman interaction:\nˆHZ=µB∑\ni,m(−1)m\u0010\ngeB(1)\n−mˆS(1)\nm(i)+B(1)\n−mˆL(1)\nm(i)\u0011\n,(63)which includes the interaction of the magnetic field Bwith the\nelectron spin, first term, and the interaction with the orbital\nmotion of the electrons, second term. The sum over iruns\nover all electrons, and the sum over mruns over all spher-\nical components of the spherical tensors of rank (1).geis\nthe anomalous g-factor of the electron. Application of the\nWigner-Eckart theorem to the matrix elements of the spin-\norbit interaction gives:\nHSOI\nJI=\njSM′\f\fˆHSOI|jSM⟩\n=∑\nm(−1)m\nS1;Mm\f\fS1;SM′\u000b\n⟨jS||∑iˆF(1)\n−m(i)ˆS(1)(i)||iS⟩√\n2S+1\n=∑\nm(−1)mS(1)\nM′Mm⟨jSS|∑iˆF(1)\n−m(i)ˆS(1)\n0(i)|iSS⟩\nS\n=i∑\nαSM′MαIm⟨jSS|∑iˆFα(i)ˆSz(i)|iSS⟩\nS\n=i∑\nαSM′MαhSOI\njiα.(64)\nThe second line of Eq. (64) results directly from the Wigner-\nEckart theorem, where ⟨S1;Mm|S1;SM′⟩is a Clebsch-Gordan\ncoefficient in the nomenclature of Sakurai50and⟨jS||·||iS⟩\ndenotes a reduced matrix element. In the third line, I express\nthe Clebsch-Gordan coefficient using the matrix elements of\nthe electron spin operator, and in the fourth line, I account for\nthe purely imaginary nature of the resulting orbital-dependent\noperator. The last line of Eq. (64) provides the definition of\nhSOI\njiα. The sum over mis a sum over the spherical compo-\nnents of the rank-1 spherical tensors as before, and the sum\noverαis the equivalent sum over the Cartesian components of\nthe vector operators. Similar application of the Wigner-Eckart\ntheorem to the Zeeman interaction reveals the following spin-\ncomponent structure:\nˆHZ\nJI=\njSM′\f\fˆHZ|jSM⟩\n=µBge∑\nm(−1)mB(1)\n−mS(1)\nM′Mmδji\n+µB∑\nm(−1)mB(1)\n−m⟨jSS|ˆL(1)\nm|iSS⟩δM′M.(65)\nUse of Eqs. (64) and (65) in the second order term of Eq. (8),\nand keeping only the combination terms between the orbital\nZeeman and spin-orbit interactions gives the second order\ncontribution to the molecular g-tensor:\n∆g(2)\nαβ=1\nS2∑\nj̸=0Im⟨0SS|ˆLα|jSS⟩Im⟨jSS|ˆFβˆSz|0SS⟩\nE(0)\nj−E(0)\ni\n+1\nS2∑\nj̸=0Im⟨0SS|ˆFαˆSz|jSS⟩Im⟨jSS|ˆLβ|0SS⟩\nE(0)\nj−E(0)\ni.(66)\nEquation (66) is the sum-over-states expression of the sec-\nond order molecular g-tensor and demonstrates the relation\nbetween adiabatic terms in the molecular spin Hamiltonian15\nand the traditional static spin Hamiltonian interactions. Simi-\nlar account of the pure spin-orbit interaction terms in Eq. (8)\ngives the second-order sum-over-states expression for the\nmolecular D-tensor, which also involves matrix elements of\nthe spin-orbit interaction between spin manifolds that differ\nby one unit of angular momentum.\nFinally, the same-spin-manifold matrix elements of the\nfirst-order spin-orbit perturbing operator can be written using\nthe Wigner-Eckart theorem as:\nG(1)\nJI=\njSM′\f\fˆG(1)|iSM⟩=−∑\nJ̸=IHSOI\nJI\nE(0)\nJ−E(0)\nI\n=−iSM′Mα∑\nj̸=ihSOI\njiα\nE(0)\nj−E(0)\ni=iSM′Mαg(1)\njiα,(67)\nwhere in the last line I define the Cartesian components of the\nfirst-order perturbing operator gjiα.gjiαare purely real func-\ntions and are symmetric matrices with respect to the orbital\nindeces.\nAppendix B PROOF OF IDENTITY IN EQ. 18\nI prove in Appendix B an identity that is needed to trans-\nform the second derivative non-adiabatic coupling matrix in\nEq. (18):\nD\nI′(0)\nR\f\f\f∂µˆG(1)\nR\f\f\f∂µI(0)\nRE\n=\n=∑\nJ̸=I∂µGI′J\nJ\f\f∂µI\u000b\n+∑\nJ̸=IGI′J\n∂µJ\f\f∂µI\u000b\n+\n∑\nK,J,J̸=K\nI′\f\f∂µK\u000b\nGKJ\nJ\f\f∂µI\u000b\n=iSM′Mα(∑\nj̸=i∂µgi jα\nj\f\f∂µi\u000b\n+∑\nj̸=igi jα\n∂µj\f\f∂µi\u000b\n+\n∑\nk,j,j̸=k\ni\f\f∂µk\u000b\ngk jα\nj\f\f∂µi\u000b\n)\n=iSM′Mα(∑\nj̸=i\n∂µi\f\fj\u000b\n∂µgjiα+∑\nj̸=i\n∂µi\f\f∂µj\u000b\ngjiα+\n∑\nk,j,j̸=k\n∂µi\f\fj\u000b\ngjkα\n∂µk\f\fi\u000b\n)\n=D\n∂µI′(0)\nR\f\f\f∂µˆG(1)\nR\f\f\fI(0)\nRE(68)\nThe first line follows from the expansion of the first-order per-\nturbing operator, the second line employs Eq. (67), and the\nthird line uses the symmetry of gjiαand of the derivative ma-\ntrix elements in the case of real zero-order wavefunctions.\n1A. Gaita-Ariño, F. Luis, S. Hill, and E. Coronado, “Molecular spins for\nquantum computation,” Nature chemistry 11, 301–309 (2019).\n2M. S. Fataftah and D. E. Freedman, “Progress towards creating optically ad-\ndressable molecular qubits,” Chemical Communications 54, 13773–13781\n(2018).\n3M. Atzori and R. Sessoli, “The second quantum revolution: role and chal-\nlenges of molecular chemistry,” Journal of the American Chemical Society\n141, 11339–11352 (2019).\n4M. J. Graham, J. M. Zadrozny, M. S. Fataftah, and D. E. Freedman, “Forg-\ning solid-state qubit design principles in a molecular furnace,” Chemistry\nof Materials 29, 1885–1897 (2017).5N. P. Kazmierczak, K. M. Luedecke, E. T. Gallmeier, and R. G. Hadt, “T 1\nanisotropy elucidates spin relaxation mechanisms in an s= 1 cr (iv) optically\naddressable molecular qubit,” The Journal of Physical Chemistry Letters\n14, 7658–7664 (2023).\n6N. P. Kazmierczak and R. G. Hadt, “Illuminating ligand field contributions\nto molecular qubit spin relaxation via t 1 anisotropy,” Journal of the Amer-\nican Chemical Society 144, 20804–20814 (2022).\n7N. P. Kazmierczak, R. Mirzoyan, and R. G. Hadt, “The impact of ligand\nfield symmetry on molecular qubit coherence,” Journal of the American\nChemical Society 143, 17305–17315 (2021).\n8J. M. Zadrozny, J. Niklas, O. G. Poluektov, and D. E. Freedman, “Mil-\nlisecond coherence time in a tunable molecular electronic spin qubit,” ACS\ncentral science 1, 488–492 (2015).\n9M. J. Amdur, K. R. Mullin, M. J. Waters, D. Puggioni, M. K. Wojnar,\nM. Gu, L. Sun, P. H. Oyala, J. M. Rondinelli, and D. E. Freedman, “Chemi-\ncal control of spin–lattice relaxation to discover a room temperature molec-\nular qubit,” Chemical Science 13, 7034–7045 (2022).\n10C.-J. Yu, S. V on Kugelgen, M. D. Krzyaniak, W. Ji, W. R. Dichtel, M. R.\nWasielewski, and D. E. Freedman, “Spin and phonon design in modular ar-\nrays of molecular qubits,” Chemistry of Materials 32, 10200–10206 (2020).\n11M. S. Fataftah, M. D. Krzyaniak, B. Vlaisavljevich, M. R. Wasielewski,\nJ. M. Zadrozny, and D. E. Freedman, “Metal–ligand covalency enables\nroom temperature molecular qubit candidates,” Chemical Science 10,\n6707–6714 (2019).\n12M. S. Fataftah, S. L. Bayliss, D. W. Laorenza, X. Wang, B. T. Phelan, C. B.\nWilson, P. J. Mintun, B. D. Kovos, M. R. Wasielewski, S. Han, et al. , “Trig-\nonal bipyramidal v3+ complex as an optically addressable molecular qubit\ncandidate,” Journal of the American Chemical Society 142, 20400–20408\n(2020).\n13M. J. Graham, J. M. Zadrozny, M. Shiddiq, J. S. Anderson, M. S. Fataftah,\nS. Hill, and D. E. Freedman, “Influence of electronic spin and spin–orbit\ncoupling on decoherence in mononuclear transition metal complexes,” Jour-\nnal of the American Chemical Society 136, 7623–7626 (2014).\n14A. H. Follmer, R. D. Ribson, P. H. Oyala, G. Y . Chen, and R. G.\nHadt, “Understanding covalent versus spin–orbit coupling contributions\nto temperature-dependent electron spin relaxation in cupric and vanadyl\nphthalocyanines,” The Journal of Physical Chemistry A 124, 9252–9260\n(2020).\n15L. Tesi, E. Lucaccini, I. Cimatti, M. Perfetti, M. Mannini, M. Atzori,\nE. Morra, M. Chiesa, A. Caneschi, L. Sorace, et al. , “Quantum coherence\nin a processable vanadyl complex: new tools for the search of molecular\nspin qubits,” Chemical Science 7, 2074–2083 (2016).\n16M. Atzori, L. Tesi, E. Morra, M. Chiesa, L. Sorace, and R. Sessoli, “Room-\ntemperature quantum coherence and rabi oscillations in vanadyl phthalo-\ncyanine: toward multifunctional molecular spin qubits,” Journal of the\nAmerican Chemical Society 138, 2154–2157 (2016).\n17M. Atzori, S. Benci, E. Morra, L. Tesi, M. Chiesa, R. Torre, L. Sorace, and\nR. Sessoli, “Structural effects on the spin dynamics of potential molecular\nqubits,” Inorganic chemistry 57, 731–740 (2018).\n18M. Atzori, L. Tesi, S. Benci, A. Lunghi, R. Righini, A. Taschin, R. Torre,\nL. Sorace, and R. Sessoli, “Spin dynamics and low energy vibrations: in-\nsights from vanadyl-based potential molecular qubits,” Journal of the Amer-\nican Chemical Society 139, 4338–4341 (2017).\n19M. Goldman, “Formal theory of spin–lattice relaxation,” Journal of Mag-\nnetic Resonance 149, 160–187 (2001).\n20J. Van Vleck, “Paramagnetic relaxation times for titanium and chrome\nalum,” Physical Review 57, 426 (1940).\n21R. D. Mattuck and M. Strandberg, “Spin-phonon interaction in paramag-\nnetic crystals,” Physical Review 119, 1204 (1960).\n22R. Orbach, “Spin-lattice relaxation in rare-earth salts,” Proceedings of the\nRoyal Society of London. Series A. Mathematical and Physical Sciences\n264, 458–484 (1961).\n23F. Neese, “Prediction of electron paramagnetic resonance g values using\ncoupled perturbed hartree–fock and kohn–sham theory,” The Journal of\nChemical Physics 115, 11080–11096 (2001).\n24F. Neese and E. I. Solomon, “Calculation of zero-field splittings, g-values,\nand the relativistic nephelauxetic effect in transition metal complexes. ap-\nplication to high-spin ferric complexes,” Inorganic chemistry 37, 6568–\n6582 (1998).16\n25F. Neese, “Importance of direct spin- spin coupling and spin-flip excitations\nfor the zero-field splittings of transition metal complexes: A case study,”\nJournal of the American Chemical Society 128, 10213–10222 (2006).\n26F. Neese, “Efficient and accurate approximations to the molecular spin-orbit\ncoupling operator and their use in molecular g-tensor calculations,” The\nJournal of chemical physics 122(2005).\n27S. K. Singh, M. Atanasov, and F. Neese, “Challenges in multireference per-\nturbation theory for the calculations of the g-tensor of first-row transition-\nmetal complexes,” Journal of Chemical Theory and Computation 14, 4662–\n4677 (2018).\n28J. Gauss, M. Kallay, and F. Neese, “Calculation of electronic g-tensors\nusing coupled cluster theory,” The Journal of Physical Chemistry A 113,\n11541–11549 (2009).\n29R. McWeeny, “On the origin of spin-hamiltonian parameters,” The Journal\nof Chemical Physics 42, 1717–1725 (1965).\n30M. Pryce, “A modified perturbation procedure for a problem in paramag-\nnetism,” Proceedings of the Physical Society. Section A 63, 25 (1950).\n31A. Lunghi and S. Sanvito, “The limit of spin lifetime in solid-state elec-\ntronic spins,” The Journal of Physical Chemistry Letters 11, 6273–6278\n(2020).\n32A. Lunghi and S. Sanvito, “Multiple spin–phonon relaxation pathways in a\nkramer single-ion magnet,” The Journal of Chemical Physics 153(2020).\n33A. Lunghi, “Toward exact predictions of spin-phonon relaxation times: An\nab initio implementation of open quantum systems theory,” Science Ad-\nvances 8, eabn7880 (2022).\n34M. Briganti, F. Santanni, L. Tesi, F. Totti, R. Sessoli, and A. Lunghi, “A\ncomplete ab initio view of orbach and raman spin–lattice relaxation in a\ndysprosium coordination compound,” Journal of the American Chemical\nSociety 143, 13633–13645 (2021).\n35E. Garlatti, A. Albino, S. Chicco, V . Nguyen, F. Santanni, L. Paolasini,\nC. Mazzoli, R. Caciuffo, F. Totti, P. Santini, et al. , “The critical role of\nultra-low-energy vibrations in the relaxation dynamics of molecular qubits,”\nNature Communications 14, 1653 (2023).\n36A. Albino, S. Benci, L. Tesi, M. Atzori, R. Torre, S. Sanvito, R. Ses-\nsoli, and A. Lunghi, “First-principles investigation of spin–phonon coupling\nin vanadium-based molecular spin quantum bits,” Inorganic chemistry 58,\n10260–10268 (2019).\n37D. Reta, J. G. Kragskow, and N. F. Chilton, “Ab initio prediction of high-\ntemperature magnetic relaxation rates in single-molecule magnets,” Journal\nof the American Chemical Society 143, 5943–5950 (2021).\n38J. G. Kragskow, A. Mattioni, J. K. Staab, D. Reta, J. M. Skelton, and\nN. F. Chilton, “Spin–phonon coupling and magnetic relaxation in single-\nmolecule magnets,” Chemical Society Reviews (2023).\n39D. R. Yarkony, “Diabolical conical intersections,” Reviews of Modern\nPhysics 68, 985 (1996).\n40C. A. Mead, “The geometric phase in molecular systems,” Reviews of mod-\nern physics 64, 51 (1992).\n41J. Van Vleck, “On σ-type doubling and electron spin in the spectra of di-\natomic molecules,” Physical Review 33, 467 (1929).\n42H. Primas, “Generalized perturbation theory in operator form,” Reviews of\nModern Physics 35, 710 (1963).\n43J.-L. Du, G. R. Eaton, and S. S. Eaton, “Electron spin relaxation in vanadyl,\ncopper (ii), and silver (ii) porphyrins in glassy solvents and doped solids,”\nJournal of Magnetic Resonance, Series A 119, 240–246 (1996).\n44C.-J. Yu, M. D. Krzyaniak, M. S. Fataftah, M. R. Wasielewski, and D. E.\nFreedman, “A concentrated array of copper porphyrin candidate qubits,”Chemical science 10, 1702–1708 (2019).\n45C. M. Moisanu, R. M. Jacobberger, L. P. Skala, C. L. Stern, M. R.\nWasielewski, and W. R. Dichtel, “Crystalline arrays of copper porphyrin\nqubits based on ion-paired frameworks,” Journal of the American Chemi-\ncal Society 145, 18447–18454 (2023).\n46M. V . Berry, “Quantal phase factors accompanying adiabatic changes,” Pro-\nceedings of the Royal Society of London. A. Mathematical and Physical\nSciences 392, 45–57 (1984).\n47C. A. Mead and D. G. Truhlar, “On the determination of born–oppenheimer\nnuclear motion wave functions including complications due to conical inter-\nsections and identical nuclei,” The Journal of Chemical Physics 70, 2284–\n2296 (1979).\n48A. Shapere and F. Wilczek, Geometric phases in physics , V ol. 5 (World\nscientific, 1989).\n49C. A. Mead, “Molecular kramers degeneracy and non-abelian adiabatic\nphase factors,” Physical review letters 59, 161 (1987).\n50J. Sakurai and J. Napolitano, “Modern quantum mechanics. 2-nd edition,”\nPerson New International edition , 39 (2014).\n51J. D. Jackson, “Classical electrodynamics,” (1999).\n52E. B. Wilson, J. C. Decius, and P. C. Cross, Molecular vibrations: the theory\nof infrared and Raman vibrational spectra (Courier Corporation, 1980).\n53N. W. Ashcroft and N. D. Mermin, Solid state physics (Cengage Learning,\n2022).\n54F. Neese, “Calculation of the zero-field splitting tensor on the basis of hy-\nbrid density functional and hartree-fock theory,” The Journal of chemical\nphysics 127(2007).\n55D. Bykov, T. Petrenko, R. Izsák, S. Kossmann, U. Becker, E. Valeev, and\nF. Neese, “Efficient implementation of the analytic second derivatives of\nhartree–fock and hybrid dft energies: a detailed analysis of different ap-\nproximations,” Molecular Physics 113, 1961–1977 (2015).\n56J. Pople, R. Krishnan, H. Schlegel, and J. S. Binkley, “Derivative studies in\nhartree-fock and møller-plesset theories,” International Journal of Quantum\nChemistry 16, 225–241 (1979).\n57F. Neese, “The orca program system,” Wiley Interdisciplinary Reviews:\nComputational Molecular Science 2, 73–78 (2012).\n58K. Eichkorn, F. Weigend, O. Treutler, and R. Ahlrichs, “Auxiliary basis\nsets for main row atoms and transition metals and their use to approximate\ncoulomb potentials,” Theoretical Chemistry Accounts 97, 119–124 (1997).\n59J. P. Perdew, K. Burke, and M. Ernzerhof, “Generalized gradient approxi-\nmation made simple,” Physical review letters 77, 3865 (1996).\n60A. Schäfer, H. Horn, and R. Ahlrichs, “Fully optimized contracted gaussian\nbasis sets for atoms li to kr,” The Journal of chemical physics 97, 2571–\n2577 (1992).\n61A. Schäfer, C. Huber, and R. Ahlrichs, “Fully optimized contracted gaus-\nsian basis sets of triple zeta valence quality for atoms li to kr,” The Journal\nof chemical physics 100, 5829–5835 (1994).\n62F. Weigend and R. Ahlrichs, “Balanced basis sets of split valence, triple\nzeta valence and quadruple zeta valence quality for h to rn: Design and\nassessment of accuracy,” Physical Chemistry Chemical Physics 7, 3297–\n3305 (2005).\n63C. Adamo and V . Barone, “Toward reliable density functional methods\nwithout adjustable parameters: The pbe0 model,” The Journal of chemi-\ncal physics 110, 6158–6170 (1999).\n64M. Ernzerhof and G. E. Scuseria, “Assessment of the perdew–burke–\nernzerhof exchange-correlation functional,” The Journal of chemical\nphysics 110, 5029–5036 (1999)."    },    {        "title": "1206.2755v2.Observation_of_elastic_anomalies_driven_by_coexisting_dynamical_spin_Jahn_Teller_effect_and_dynamical_molecular_spin_state_in_paramagnetic_phase_of_the_frustrated_MgCr__2_O__4__.pdf",        "content": "arXiv:1206.2755v2  [cond-mat.str-el]  1 Oct 2012Observation of elastic anomalies driven by coexisting dyna mical spin Jahn-Teller effect\nand dynamical molecular spin state in paramagnetic phase of the frustrated MgCr 2O4\nTadataka Watanabe1, Shin-Ichi Ishikawa1, Haruki Suzuki1, Yusuke Kousaka2, and Keisuke Tomiyasu3\n1Department of Physics, College of Science and Technology (C ST),\nNihon University, Chiyoda, Tokyo 101-8308, Japan\n2Department of Physics and Mathematics, Aoyama-Gakuin Univ ersity, Sagamihara, Kanagawa 229-8558, Japan and\n3Department of Physics, Tohoku University, Aoba, Sendai 980 -8578, Japan\n(Dated: August 29, 2018)\nUltrasound velocity measurements of magnesium chromite sp inel MgCr 2O4reveal elastic anoma-\nlies in the paramagnetic phase that are characterized as due to geometrical frustration. The temper-\nature dependence of the tetragonal shear modulus ( C11−C12)/2 exhibits huge Curie-type softening,\nwhich should be the precursor to spin Jahn-Teller distortio n in the antiferromagnetic phase. The\ntrigonal shear modulus C44exhibits nonmonotonic temperature dependence with a chara cteristic\nminimum at ∼50 K, indicating a coupling of the lattice to dynamical molec ular spin state. These re-\nsults strongly suggest the coexistence of dynamical spin Ja hn-Teller effect and dynamical molecular\nspin state in the paramagnetic phase, which is compatible wi th the coexistence of magnetostructural\norder and dynamical molecular spin state in the antiferroma gnetic phase.\nPACS numbers: 72.55.+s, 75.30.-m, 75.40.Gb, 75.50.Xx\nI. INTRODUCTION\nGeometrically frustrated magnets have attracted con-\nsiderable interest because the vast degeneracy of their\nground states gives rise to a rich variety of novel\nphenomena1. One of the most highly frustrated spin sys-\ntems is the nearest-neighbor Heisenberg antiferromagnet\non the pyrochlore lattice, where the magnetic sites form\na network of corner-sharingtetrahedra. The prototypical\nexamples for this highly frustrated system are chromite\nspinelsACr2O4with non-magneticions A= Mg, Zn, and\nCd. The magnetic properties of these compounds are\nfully dominated by the Jahn-Teller (JT)-inactive Cr3+\nwith spin S= 3/2 residing on the pyrochlore network.\nThe high values of frustration parameter f=|ΘW/TN|,\nthe ratio of Weiss and N´ eel temperatures, indicate the\npresence of strong geometrical frustration: f≃30 with\nΘW= -370 K and TN= 12.5 K for MgCr 2O4,f≃33\nwith Θ W= -390 K and TN= 12.0 K for ZnCr 2O4, and\nf≃9 with Θ W= -70 K and TN= 7.8 K for CdCr 2O42.\nInACr2O4, the antiferromagnetic (AF) transition\natTNcoincides with a tetragonal lattice distortion\nwith a compression along the c-axis for MgCr 2O4and\nZnCr2O43,4, and an elongation for CdCr 2O45. This\nmagnetostructural ordering is explained by the spin JT\nmechanism via the magnetoelastic coupling, where local\ndistortions of the tetrahedra release the frustration in\nthe nearest-neighbor AF interactions6,7. The very small\nmagnitudes of the tetragonal lattice strain of the order\n10−3imply the presence of a nonuniform component in\nthe lattice distortion, where the neighboring tetrahedra\nare alternately stretched and compressed along the same\ncrystalaxis3–5,8. Indeed, arecentstudyofZnCr 2O4using\nsynchrotron X-ray and neutron scattering experiments\nrevealed complex spin-lattice order with the nonuniform\nlattice distortion9.\nIn the paramagnetic (PM) phase of ZnCr 2O4andMgCr2O4, inelastic neutron scattering (INS) experi-\nments observed quasielastic magnetic scattering, indi-\ncating the presence of strong spin fluctuations due to\nfrustration3,10–12. Interestingly, this quasigapless mode\nwas characterized as fluctuations of AF hexagonal spin\nclusters (AF hexamers) in the pyrochlore lattice11,12. It\nis natural to expect that this dynamical molecular spin\nstate vanishes in the AF phase as a consequence of the\nrelease of frustration. Actually, however, the INS studies\nrevealedthe appearanceof gapped molecular spin excita-\ntions in the AF phase, which indicates the coexistence of\nthe magnetostructural order and the zero point motion-\nlike spin molecules11.\nThe appearance of the magnetostructural order in\nACr2O4means that the spin-lattice coupling plays a sig-\nnificantrolein releasingthe frustration; thus, the phonon\nspectra are expected to be affected by the frustration.\nIndeed, infrared and Raman spectroscopies in ZnCr 2O4\nand CdCr 2O4revealed that the optical phonon spec-\ntra exhibit anomalies of softening on cooling in the PM\nphase, and splitting in the AF phase13–15. In this paper,\nwe present acoustic phonon information provided by ul-\ntrasound velocity measurements in magnesium chromite\nspinel MgCr 2O4. The modified sound dispersions by\nmagnetoelastic coupling allow one to extract detailed in-\nformation about the interplay between spin and lattice\ndegrees of freedom in a lower energy region near to the\nground states. Thus ultrasound velocity measurements\ncan be a useful tool in studying geometrically frustrated\nmagnets16–19. We find elastic-mode-dependent sound-\nvelocity anomalies in MgCr 2O4in the PM phase, similar\nto those observed in the past in ZnCr 2O416. The ori-\ngins of these elastic anomalies are discussed in terms of\nfrustration inherent in ACr2O4.2\nII. EXPERIMENTAL\nUltrasound velocities were measured in single crystals\nofMgCr 2O4preparedbythefloatingzonemethod, where\nthe phase comparison technique was used with longitu-\ndinal and transverse sound waves at a frequency of 30\nMHz. The ultrasounds were generated and detected by\nLiNbO 3transducers glued on the parallel mirror sur-\nfaces of the crystal. We measured sound velocities in all\nthe symmetrically independent elastic moduli in the cu-\nbic crystal: compression modulus C11, tetragonal shear\nmodulusC11−C12\n2≡Ct, and trigonal shear modulus C44.\nThe respective measurements of C11,Ct, andC44were\nperformed by using longitudinal wave with propagation\nk/bardbl[100] and polarization u/bardbl[100], transverse wave with\nk/bardbl[110] and u/bardbl[1¯10], and transverse wave with k/bardbl[110]\nandu/bardbl[001].\nIII. RESULTS\nFigures 1(a)-(c) depict the longitudinal sound velocity\nvLinC11and the transverse sound velocity vTinCtand\nC44asfunctions oftemperature ( T), respectively. All the\nelastic moduli exhibit a jump at TN= 13 K. And, more\nimportantly, all the elastic moduli exhibit elastic-mode-\ndependent anomalous T-dependence in the PM phase,\nfrom room temperature (300 K) down to TN, that is dif-\nferent from the T-dependence usually observed in solids,\nnamely monotonic hardeningwith decreasing T20. These\nelastic anomalies should have magnetic origins where the\nspin degrees of freedom play significant roles but also\ntaking into account the absence of orbital degeneracy in\ntheB-site Cr3+in MgCr 2O4. Such elastic anomalies are\nattributed to magnetoelastic coupling acting on the ex-\nchange interactions. In this mechanism, the exchange\nstriction arises from a modulation of the exchange inter-\nactions by ultrasound as follows:21\nHexs=/summationdisplay\nij[J(δ+ui−uj)−J(δ)]Si·Sj.(1)\nHereδ=Ri−Rjis the distance between two magnetic\nions, and uiis the displacement vector for the ion Ri.\nWhen a sound wave with polarization uand propagation\nkis given by u=u0exp[i(k·r−ωt)], where u0andω\nare the respective amplitude and frequency, the exchange\nstriction of Eq. (1) is rewritten as:22\nHexs=/summationdisplay\ni(∂J\n∂δ·u0)(k·δ)(Si·Si+δ)ei(k·Ri−ωt).(2)\nHere the exponential is expanded to first order because\nwith a 30-MHz ultrasound frequency kδ≪1. Eq. (2)\nimplies that both the longitudinal and transverse sound\nwaves can couple to the spin system via the exchange\nstriction mechanism, which depends on the directions ofTN6800\n6600\n6400\n6200\n3400\n3200\n3000\n2800\n2600vL (m/s)\n4450\n4400\n4350300250200150100500\n T (K)vT (m/s) vT (m/s)Transverse ultrasound \n k||[110], u||[110] \n f = 30 MHz (a) C11 \n(b) Ct\n(c) C44 Transverse ultrasound \n k||[110], u||[001] \n f = 30 MHz Longitudinal ultrasound \n k||u||[100] \n f =30 MHz \nFIG. 1: (Color online). Temperature dependence of the lon-\ngitudinal ( vL) and transverse ( vT) sound velocities. (a) C11,\n(b)(C11−C12)\n2≡Ct, and (c) C44.\npolarization uand propagation krelativeto the exchange\npathδ. ConcerningtheelasticanomaliesinMgCr 2O4, we\nemphasize that the anomalous Tvariations of the elas-\ntic moduli in the PM phase are divided into two types:\nCurie-type-1/ Tsofteningfor C11(T)andCt(T)and non-\nmonotonic T-dependence with a characteristic minimum\nat∼50KforC44(T). Togetherthesetypesimplythatthe\nelastic anomalies have different origins, which we shall\ndiscuss below.\nIV. DISCUSSION\nA. Curie-type softening in PM phase\nFirst we discuss the origin of the Curie-type softening\ninC11(T) andCt(T) in the PM phase. Taking into ac-\ncount that the relative change in softening is much larger\ninCt(T) (∆Ct\nCt∼50%) than in C11(T) (∆C11\nC11∼10%),\nand that the compression modulus C11is written as\nC11=CB+4\n3CtwithCBthe bulk modulus, the Curie-\ntype softening should originate from a softening in the\ntetragonal shear modulus Ct(T). Thus the Curie-type\nsoftening should be a precursor to the tetragonal struc-\ntural transition at TN. Such a precursor softening to\nthe structural transition is well known to occur as a re-\nsult of the Jahn-Teller (JT) effect in orbital-degenerate\nsystems21. However, the absence of orbital degeneracy\nfor theB-site Cr3+in MgCr 2O4rules out such orbital\neffects. Thus the Curie-type softening is most probably\ndue to a spin JT effect6,7, which is compatible with the\ncoincidence of AF ordering and tetragonal lattice com-\npression at TN. The spin JT distortion is analogous to\norbitaldistortioninthesensethatadegenerateelectronic3\nsystem lowers its energy by lifting the degeneracy under\nstructural distortion. In ACr2O4, the spin JT distor-\ntion lifts the spin degeneracy (frustration) inherent in\nthe pyrochlore lattice by modulating the balance of ex-\nchangeinteractionsviatheexchangestrictionmechanism\nexpressed as Eq. (1).\nFor the orbital JT effect, a JT-active ion and its set\nof ligands are considered to be a structural unit, where\nthe crystal-field striction mechanism leads to the orbital\nJT distortion. Tdependence of elastic modulus CΓ(T)\nin the orbital JT systems is explained assuming the cou-\npling of ultrasound to JT-active ions via the crystal-field\nstriction mechanism, and the presence of inter-JT-active-\nion interactions. A mean-field expression of CΓ(T) in the\norbital JT systems is given as:23–25\nCΓ(T) =C0\nΓ−g2\nΓNχΓ(T)\n(1−g′\nΓχΓ(T)),(3)\nwithC0\nΓthe elastic constant without the orbital JT ef-\nfect,gΓthe coupling constant of crystal field strain, N\nthe number of JT-active ions in the unit volume, g′\nΓthe\ninter-JT-active-ioninteraction, and χΓ(T) the strain sus-\nceptibility of a single JT-active ion. Eq. (3) expresses\nTdependence of the strain susceptibility which is the\nresponse function to an applied strain, and its form is\nanalogous to the mean-field expression of the magnetic\nsusceptibility which is the response function of the mag-\nnetization to an applied magnetic field. In the orbital\nJT systems, the degenerate ground state is considered\nto couple strongly and selectively to the elastic modu-\nlusCΓwhich has the same symmetry as the orbital JT\ndistortion. For such a JT-active elastic mode, the strain\nsusceptibility is dominated by Curie term χΓ(T)∼1/T\nat low temperatures. Then Eq. (3) is rewritten as:23–25\nCΓ(T) =C0\nΓT−Tc\nT−θ, (4)\nwithθthe inter-JT-active-ion interaction, and Tcthe\nsecond-order structural transition temperature. Here\nθis positive/negative when the interaction is ferrodis-\ntortive/antiferrodistortive. Although the background C0\nΓ\nin Eq. (4) generally exhibits a hardening with decreasing\nT20, we here assume C0\nΓis constant because its hard-\nening is negligibly small compared with the huge Curie-\ntype softening in CΓ(T). In contrast to the JT-active\nsoft mode CΓexpressed as Eq. (4), the elastic modulus\nCΓ′with the different symmetry from the orbital JT dis-\ntortion is not affected by the degenerate ground state;\nCΓ′(T) exhibits the absence of anomaly, CΓ′(T) =C0\nΓ′,\noriginating from χΓ′= 0 in Eq. (3).\nIn analogyto the orbital JT effect, the Curie-type soft-\nening inCt(T) in MgCr 2O4can be explained in the spin-\nJT-effect-scenarioassuming the coupling of a Cr3+struc-\ntural unit to the external strain of ultrasound via the ex-\nchangestrictionmechanismexpressedasEqs.(1)and(2), T (K)60\n50\n40\n30\n300250200150100500MgCr 2O4ZnCr 2O4Ct (GPa)\n Ct (GPa)   Tc (K)   θ (K) \n   54.4         0.7        -14 \n   57.4         0            -6 MgCr 2O4 \nZnCr 2O40\nFIG. 2: (Color online). The tetragonal shear modulus Ctin\nMgCr 2O4(open circles, from Fig. 1(b)) and ZnCr 2O4(open\nsquares, from Ref. [16]) as functions of temperature. The\nsolid and dotted curves are fits of the experimental data for\nMgCr 2O4and ZnCr 2O4to Eq. (4), respectively. Values for\nthe fitting parameters are listed in the table beneath the fig-\nure. The inset shows a schematic of the tetragonal lattice\ndeformation in Ct.\nandthepresenceofinter-structural-unitinteractions. Al-\nthough the structural unit of the spin JT effect has not\nbeen identified so far, the mean-field expression of the\nsoft mode CΓ(T) should have the same form as Eq. (4)\nwithC0\nΓthe elastic constant without the spin JT effect, θ\nthe inter-structural-unit interaction, and Tcthe second-\norderstructuraltransitiontemperature. ForMgCr 2O4, it\nisexpectedthatultrasoundin Ctgeneratesthetetragonal\nexchange striction which promotes the tetragonal spin-\nlattice correlation, while the trigonal exchange striction\nis generated in C44. Thus, since the spin JT distortion in\nMgCr2O4has the tetragonal symmetry4, the tetragonal\nshear modulus Ct(T) should exhibit Curie-type soften-\ning as expressed in Eq. (4), whereas its absence in the\ntrigonal shear modulus is expressed by C44(T) =C0\n44in\nanalogy to the orbital JT effect.\nIt is noted that the Curie-type softening in the\ntetragonal shear modulus Ct(T) is observed in not\nonly MgCr 2O4but also another highly frustrated spinel\nZnCr2O416. Figure2comparesthetetragonalshearmod-\nulusCtin MgCr 2O4(open circles, from Fig. 1(b)) with\nthat in ZnCr 2O4(open squares, from Ref. [16]) as func-\ntions of T; fits of the experimental data for MgCr 2O4\nand ZnCr 2O4to Eq. (4) are also plotted as solid and\ndotted curves, respectively. The plots reproduce the ex-\nperimental data given the fitted values of C0\nt,Tc, andθ\nlisted in the table beneath the figure. For MgCr 2O4and\nZnCr2O4, the respective fitted values of Tc= 0.7 K and 0\nK are lower than the experimentally-observed N´ eel tem-\nperatures of TN= 13 K and 12.5 K, Tc< TN, indicating\nthat the phase transition at TNis of first order3,4. The\nnegative fitted values of θmean the dominance of an-\ntiferrodistortive inter-structural-unit interactions which\nare stronger in MgCr 2O4(θ= -14 K) than in ZnCr 2O44\n(θ= -6 K). The dominance of such antiferrotype interac-\ntions in the PM phase seems to be compatible with the\nnonuniform lattice distortion in the AF phase revealed\nexperimentally in ZnCr 2O4, where neighboring tetrahe-\ndra are alternately stretched and compressed along the\nsame crystal axis9. In the antiferrotype nonuniform lat-\ntice order, the tetragonallattice strainperspinel unit cell\nshould be negligibly small, as is consistent with the very\nsmallmagnitudesofthetetragonallatticestrainobserved\ninACr2O4in the AF phase3–5.\nB. Softening with minimum in PM phase\nNext, we discuss the origin of the other salient elastic\nanomaly in the PM phase, the characteristic minimum\nat∼50 K inC44(T). As shown in Fig. 3, this anomaly\nis observed in not only MgCr 2O4(open circles, from Fig.\n1(c)) but also ZnCr 2O4(open squares, from Ref. [16]).\nTaking into account the absence of orbital degeneracy\nfor theB-site Cr3+in these compounds, the spin degrees\nof freedom should play a significant role for the pres-\nence of the anomaly in C44(T). One possible origin for\nthe anomaly in C44(T) is the spin JT effect. However,\nthe symmetry of the spin-lattice order in MgCr 2O4and\nZnCr2O4is tetragonal, which is different from the trig-\nonal symmetry of C443,4,9. Thus, in the spin-JT-effect\nscenario, as described above, while the tetragonal shear\nmodulus Ct(T)shouldexhibitCurie-typesofteninginthe\ncubic phase (PM phase) as expressed in Eq. (4), the trig-\nonal shear modulus C44(T) should be anomaly-free as\nC44(T) =C0\n44, which rules out the spin JT effect as a\npossible origin for the anomaly in C44(T). The other\npossible origin for the anomaly in C44(T) is the cou-\npling between the dynamical molecular spin state and\nthe acoustic phonons. We here note that the INS studies\nin MgCr 2O4aswellasZnCr 2O4observedthe quasielastic\nmagnetic scattering in the PM phase3,10–12, and charac-\nterized this quasigapless mode as the fluctuations of the\nAF hexagonal spin clusters (the AF hexamers) in the py-\nrochlore lattice11,12. As illustrated in the inset to Fig. 3,\nthe AF hexamers are aligned in the trigonal (111) planes\nof the spinel structure, and thus should be sensitive to\nthe trigonal lattice deformation which generates the trig-\nonal exchange striction. Therefore, the anomaly of the\ntrigonal shear modulus C44(T)in the PM phase should\nstem from the dynamical molecular spin state.\nThe ultrasound velocity measurements in the present\nstudy extract information about the acoustic phonon dis-\npersions at very small wave vectors of q∼10−5˚A−1. In\ncontrast, the molecular spin fluctuations observed in the\nINS spectra have large wave vectors of q∼1.5˚A−13,11.\nThus the observation of the anomaly in C44(T) suggests\nthat the strong coupling of the dynamical molecular spin\nstate to the lattice modifies the acoustic phonon disper-\nsions over a wide range in the Brillouin zone, even at\naroundzerowavevector. Themodification ofthe phonon\ndispersions would be strongest at around q∼1.5˚A−1.\nC44  (GPa)\n T (K)88 \n86\n84\n82\n80\n300250200150100500\nHexamer ZnCr 2O4MgCr 2O4\n C44  (GPa)   Δ1 (K)   G1, Γ5 (K)   Δ2 (K)   G2, Γ5 (K)   KΓ5 (K) \n     90.1          39         3600         136        10200       -19 \n     86.7          34         3160         111          9290        -19 MgCr 2O4 \nZnCr 2O40\nFIG. 3: (Color online). The trigonal shear modulus C44in\nMgCr 2O4(open circles, from Fig. 1(c)) and ZnCr 2O4(open\nsquares, from Ref. [16]) as functions of temperature. The\nsolid and dotted curves are fits of the experimental data for\nMgCr 2O4and ZnCr 2O4to Eq. (5), respectively. Values for\nthe fitting parameters are listed in the table beneath the fig-\nure. The inset depicts an AF hexamer in the Cr3+pyrochlore\nlattice.\nRegarding the elastic anomaly from the short-range\nspin correlations, the characteristic minimum in C44(T)\nin MgCr 2O4and ZnCr 2O4is similar to that in low-\ndimensional spin-dimer systems such as SrCu 2(BO3)227.\nNotably, the minimum in CΓ(T) in the spin dimer sys-\ntems is explained as a result of the presence of a finite\ngap for the local magnetic excitations which is sensitive\nto the strain27. In the spin-dimer systems, the quantita-\ntive analysis of the experimental data of CΓ(T) is given\nby applying a theoretical model which has the same form\nas Eq. (3). Assuming a singlet-triplet excitation gap ∆ 1\nfor a single spin dimer and a multitriplet excitation gap\n∆2, the contribution of the spin dimers to the elastic\nconstant takes the form,27\nCΓ(T) =C0\nΓ−G2\n1,ΓNχΓ(T)\n(1−KΓχΓ(T)),(5)\nwithC0\nΓthe background elastic constant, Nthe density\nof spin dimers, G1,Γ=|∂∆1/∂ǫΓ|the coupling constant\nfor a single spin dimer measuring the strain ( ǫΓ) depen-\ndence of the excitation gap ∆ 1,KΓthe (q= 0) dimer-\ndimer interaction, and χΓ(T) the strain susceptibility of\na single spin dimer. Eq. (5) is based on the coupling\nbetween the gapped spin triplet states and the acoustic\nphonons, which, in turn, means that this equation can be\napplied to all systems with the coupling between gapped\nstates and acoustic phonons regardless of whether it is\na singlet system or not. According to Eq. (5), the mini-\nmum in CΓ(T) appears when this elastic mode strongly\ncouples to the excited state at ∆ i; on cooling, CΓ(T) ex-\nhibits softening roughly down to T∼∆i, but recovery of\nthe elasticity (hardening) roughly below T∼∆i.\nIn analogy to the spin dimer systems in the presence\nof local excitations, therefore, the minimum in C44(T) in5\nMgCr2O4and ZnCr 2O4should arise from a gap in the\nmolecularspinexcitationswhichissensitivetothe strain.\nThis interpretation helps to understand the INS results\nfor MgCr 2O4and ZnCr 2O4. The broadquasielasticmag-\nnetic scattering spectrum in the PM phase comprises\ngapless spin fluctuations which probably form embryo of\nthe tetragonal spin-lattice order below TN, and gapped\nspin excitations which are considerably smeared3,10–12.\nAnd the observation of distinct gapped excitations in the\nAF phase is attributed to the suppression of spin fluctu-\nations by spin JT distortion3,10,11.\nSimilar to the spin dimer systems27, we now give a\nquantitative analysis of the experimental data of C44(T)\nin MgCr 2O4and ZnCr 2O4assuming an excitation gap\n∆1for a single AF hexamer and a multi-AF-hexamer ex-\ncitation gap ∆ 2. The contributionof the AF hexamers to\nthe elastic constant should take the same form as Eq. (5)\nwithC0\n44the background elastic constant, Nthe den-\nsity of AF hexamers, G1,Γ5=|∂∆1/∂ǫΓ5|the coupling\nconstant for a single AF hexamer measuring the strain\n(ǫΓ5) dependence of the excitation gap ∆ 1,KΓ5the (q=\n0) inter-AF-hexamer interaction, and χΓ5(T) the strain\nsusceptibility of a single AF hexamer. Here, we ignore\ntheT-dependence in the background C0\n44(the hardening\nwith decreasing T)20, because the nonmonotonic varia-\ntion inC44(T) is very large compared with the harden-\ning on cooling in C0\n44. Fits of the experimental data for\nMgCr2O4and ZnCr 2O4to Eq. (5) are depicted in Fig. 3\nas solid and dotted curves, respectively. Here, the value\nofN= 3.45×1027m−3is fixed in afirst approximation12.\nWith the fitted parameterslistedin thetable beneaththe\nfigure, fits of Eq. (5) are in excellent agreement with the\nexperimental data, reproducing the characteristic min-\nimum in C44(T). For MgCr 2O4(ZnCr 2O4), the fitted\ngap value of ∆ 1= 39 K (34 K) in the PM phase is rather\nsmaller than the gap of ∆ ≃50 K measured in the AF\nphase3,11, which implies that the suppressionof spin fluc-\ntuations by the spin JT distortion makes the gap in the\nAF phase, ∆, rather larger than that in the PM phase,\n∆1. The negative value of KΓ5= -19 K means that the\ninter-AF-hexamer interaction is antiferrodistortive. The\ncoupling constant G2,Γ5=|∂∆2/∂ǫΓ5|= 10200 K (9290\nK) is about three times largerthan G1,Γ5= 3600K (3160\nK), indicating that the higher excitations ∆ 2= 136 K\n(111 K) couple to the trigonal lattice deformation more\nstrongly than the lowest excitations ∆ 1.\nAccording to Eq. (2), not only C44but alsoC11should\ncouple to the AF hexamers via the exchange striction\nmechanism, while Ctshould be inactive. However, the\nnonmonotonic Tdependence is invisible in C11(T). In-\nstead,C11(T) exhibit Curie-type softening due to the\ndynamical spin JT effect. We notice here that the mag-\nnitude of the nonomonotonic Tdependence in C44(T),especially the relative change in hardening on cooling be-\nlow∼50 K down to TNinC44(T) (∆C44/C44∼1%),\nis rather smaller than the magnitude of Curie-type soft-\nening in C11(T) (∆C11/C11∼10%). Thus, in C11(T),\nthe weak nonmonotonic Tdependence would be hidden\nbehind the huge Curie-type softening.\nC. Hardening in AF phase\nAs described above, the elastic anomalies in MgCr 2O4\nand ZnCr 2O4in the PM phase strongly suggest the coex-\nistence of dynamical spin JT effect and dynamical molec-\nular spin state, both of which strongly couple to the acous-\ntic phonons. Conversely, in the AF phase, Ct(T) and\nC44(T) exhibit the usual hardening on cooling without\nanomaly, as shown in Figs. 2 and 3, respectively20. Ac-\ncordingto the INS experiments, while the spin JT distor-\ntion below TNpartially releases frustration, the dynami-\ncal molecular spin state persists even in the AF phase in-\ndicating the survival of some frustration3,11. Thus, com-\nbined with the above discussion on the elastic anomalies\nin the PM phase, we attribute the absence of anomalies\ninCt(T) andC44(T) in the AF phase to different origins:\nrespectively, the spin JT distortion (generating the par-\ntial release of frustration) for Ct(T), and the decoupling\nof the acoustic phonons from the dynamical molecular\nspin state (producing the decoupling from the remaining\nfrustration) for C44(T).\nV. SUMMARY\nTo summarize, ultrasound velocity measurements\nof MgCr 2O4revealed elastic-mode-dependent sound-\nvelocity anomalies in the PM phase. These elastic\nanomalies can be attributed to a coexistence of the dy-\nnamical spin JT effect and the dynamical molecular spin\nstate in the PM phase, the presence of which is compat-\nible with the coexistence of the magnetostructural order\nand the dynamical molecular spin state in the AF phase.\nFurther experimental and theoretical work is necessary\nto clarify the coexistence mechanism for the two different\ntypesoffrustrationeffectsinthe geometrically-frustrated\nchromite spinels.\nVI. ACKNOWLEDGMENTS\nThis work was partly supported by Grants-in-Aid\nfor Young Scientists (B) (21740266) and Priority Areas\n(22014001) from MEXT of Japan.\n1R. Moessner and A. Ramirez, Phys. Today 59, 24 (2006).\n2H. Ueda, H. Mitamura, T. Goto, and Y. Ueda, Prog.Theor. Phys. 159, 256 (2005).\n3S. -H. Lee, C. Broholm, T. H. Kim, W. Ratcliff II, and S.6\nW. Cheong, Phys. Rev. Lett. 84, 3718 (2000).\n4L. Ortega-San-Mart´ ın, A. J. Williams, C. D. Gordon, S.\nKlemme, and J. P. Attfield, J. Phys.: Condens. Matter 20,\n104238 (2008).\n5J. -H. Chung, M. Matsuda, S. -H. Lee, K. Kakurai, H.\nUeda, T. J. Sato, H. Takagi, K. -P. Hong, and S. Park,\nPhys. Rev. Lett. 95, 247204 (2005).\n6Y. Yamashita and K. Ueda, Phys. Rev. Lett. 85, 4960\n(2000).\n7O. Tchernyshyov, R. Moessner, and S. L. Sondhi, Phys.\nRev. Lett. 88, 067203 (2002).\n8O. Tchernyshyov, R. Moessner, and S. L. Sondhi, Phys.\nRev. B66, 064403 (2002).\n9S. Ji, S. -H. Lee, C. Broholm, T. Y. Koo, W. Ratcliff, S.\n-W. Cheong, and P. Zschack, Phys. Rev. Lett. 103, 037201\n(2009).\n10H. Suzuki and Y. Tsunoda, J. Phys. Chem. Solids 68, 2060\n(2007).\n11K. Tomiyasu, H. Suzuki, M. Toki, S. Itoh, M. Matsuura,\nN. Aso, and K. Yamada, Phys. Rev. Lett. 101, 177401\n(2008).\n12S. -H. Lee, C. Broholm, W. Ratcli, G. Gasparovic, Q.\nHuang, T. H. Kim, and S. -W. Cheong, Nature (London)\n418, 856 (2002).\n13A. B. Sushkov, O. Tchernyshyov, W. Ratcliff II, S. W.\nCheong, and H. D. Drew, Phys. Rev. Lett. 94, 137202\n(2005).\n14R. Vald´es Aguilar, A. B. Sushkov, Y. J. Choi, S. -W.\nCheong, and H. D. Drew, Phys. Rev. B 77, 092412 (2008).\n15Ch. Kant, J. Deisenhofer, T. Rudolf, F. Mayr, F. Schrettle,\nA. Loidl, V. Gnezdilov, D. Wulferding, P. Lemmens, andV. Tsurkan, Phys. Rev. B 80, 214417 (2009).\n16Y. Kino and B. L¨ uthi, Solid State Commun. 9, 805 (1971).\n17S. Bhattacharjee, S. Zherlitsyn, O. Chiatti, A. Sytcheva, J .\nWosnitza, R. Moessner, M. E. Zhitomirsky, P. Lemmens,\nV. Tsurkan, and A. Loidl, Phys. Rev. B 83, 184421 (2011).\n18T. Watanabe, S. Hara, and S. I. Ikeda, Phys. Rev. B 78,\n094420 (2008).\n19T. Watanabe, S. Hara, S. I. Ikeda, and K. Tomiyasu, Phys.\nRev. B84, 020409(R) (2011).\n20V. P. Varshni, Phys. Rev. B 2, 3952 (1970).\n21B. L¨uthi,Physical Acoustics in the Solid State (Springer,\n2005).\n22H. Stern, J. Phys. Chem. Solids 26, 153 (1965).\n23Y. Kino, L¨ uthi, and M. E. Mullen, J. Phys. Soc. Jpn. 33,\n687 (1972); Solid State Commun. 12, 275 (1973).\n24M. Kataoka and J. Kanamori, J. Phys. Soc. Jpn. 32, 113\n(1972).\n25H. Hazama, T. Goto, Y. Nemoto, Y. Tomioka, A.\nAsamitsu, and Y. Tokura, Phys. Rev. B 62, 15012 (2000).\n26Ch. Kant, M. Schmidt, Zhe Wang, F. Mayr, V. Tsurkan,\nJ. Deisenhofer, and A. Loidl, Phys. Rev. Lett. 108, 177203\n(2012).\n27S. Zherlitsyn, S. Schmidt, B. Wolf, H. Schwenk, B. L¨ uthi,\nH. Kageyama, K. Onizuka, Y. Ueda, and K. Ueda, Phys.\nRev. B 62, R6097 (2000); B. Wolf, S. Zherlitsyn, S.\nSchmidt, B. L¨ uthi, H. Kageyama, and Y. Ueda, Phys. Rev.\nLett.86, 4847 (2001).\n28H. Kageyama, M. Nishi, N. Aso, K. Onizuka, T. Yosihama,\nK. Nukui, K. Kodama, K. Kakurai, and Y. Ueda, Phys.\nRev. Lett. 84, 5876 (2000)."    },    {        "title": "0710.0075v1.Elliptic_functions_and_efficient_control_of_Ising_spin_chains_with_unequal_couplings.pdf",        "content": "arXiv:0710.0075v1  [quant-ph]  29 Sep 2007Elliptic functions and efficient control of Ising spin chains with unequal couplings\nHaidong Yuan,1,∗Robert Zeier,2,†and Navin Khaneja2,‡\n1Department of Mechanical Engineering, Massachusetts Inst itute of Technology,\n77 Massachusetts Avenue, Cambridge, Massachusetts 02139, USA\n2Harvard School of Engineering and Applied Sciences,\n33 Oxford Street, Cambridge, Massachusetts 02138, USA\n(Dated: October 31, 2018)\nIn this article, we study optimal control of dynamics in a lin ear chain of three spin 1 /2, weakly\ncoupled with unequal Ising couplings. We address the proble m of time-optimal synthesis of multiple\nspin quantum coherences. We derive time-optimal pulse sequ ence for creating a desired spin order\nby computing geodesics on a sphere under a special metric. Th e solution to the geodesic equation\nis related to the nonlinear oscillator equation and the mini mum time to create multiple spin order\ncan be expressed in terms of an elliptic integral. These tech niques are used for efficient creation of\nmultiple spin coherences in Ising spin-chains with unequal couplings.\nPACS numbers: 03.67.–a, 82.56.–b\nI. INTRODUCTION\nIn the absence of relaxation, experiments in coherent\nspectroscopy and quantum information processing con-\nsistofasequenceofunitarytransformationsonthe quan-\ntum system of interest. Pulse sequences that create a\ndesired unitary transformation in the minimum possible\ntime are of particular interest in experimental realiza-\ntions as they can reduce losses due to relaxation. This\nposes the problem of time optimal control of quantum\nsystems, which is of both theoretical and practical in-\nterest in the broad area of coherent control of quantum\nsystems.\nBesides application in spectroscopy, synthesizing uni-\ntary transformations in a time-optimal way using avail-\nable physical resources is also a problem in quantum\ninformation processing. It has received significant at-\ntention, and time-optimal control of two coupled qubits\n[1, 2, 3, 4, 5, 6, 7, 8, 9] is now well understood. Recently,\nthis problem has also been studied in the context of lin-\nearly coupled three-qubit topologies [10, 11, 12], where\nsignificant savings in the implementation time of trilin-\near Hamiltonians and logic gates between indirectly cou-\npled qubits were demonstrated over conventional meth-\nods. Many of these ideas have found applications in\nefficient ways to propagate coherences along Ising spin\nchains [13, 14]. However, the complexity of the general\nproblem oftime-optimal controlofmultiple qubit topolo-\ngies is only beginning to be appreciated.\nIn this article, we study the problem of finding the\nshortest pulse sequences for creating desired coherences\nin a linear chain of spins (of spin 1 /2) weakly coupled\nwith unequal Ising couplings. In particular, we start by\nconsidering the case of three linearly coupled spins and\n∗Electronic address: haidong@mit.edu\n†Electronic address: zeier@eecs.harvard.edu\n‡Electronic address: navin@hrl.harvard.edugeneralize our methods to linear spin chains. This study\nhas immediate applications to multi-dimensional nuclear\nmagnetic resonance (NMR) spectroscopy [15]. In multi-\ndimensional NMR experiments [15], starting from the\nthermal state of the spins, a multiple quantum coherence\nbetween spins is synthesized, which helps to correlate the\nfrequencies of various spins. Efficient pulse sequences for\ncreating such coherences help to improve the sensitivity\nof the experiments.\nOne approach for solving these problems reduces the\nefficient synthesis of multiple spin order to the geometri-\ncal question of finding shortest paths on the sphere under\na special metric [1, 8, 12, 16]. The metric enforces the\nconstraints on the quantum dynamics that arise because\nonly limited Hamiltonians can be realized [17, 18, 19].\nSuch analogies between optimization problems related to\nsteering dynamical systems with constraints and geom-\netry have been well explored in areas of control theory\n[20, 21] and sub-Riemannian geometry [22]. In this pa-\nper, we study in detail the metric and the geodesics that\narise from the problem of efficient synthesis of multiple\nspin coherences between qubits (spin 1 /2) that are indi-\nrectly coupled via unequal couplings to a third qubit.\nII. OPTIMAL CONTROL OF A LINEAR\nTHREE-SPIN SYSTEM WITH UNEQUAL\nCOUPLINGS\nWe consider a linear chain of three spins placed in a\nstatic externalmagnetic field in the zdirection with Ising\ntype couplings between next neighbors [23, 24]. In a\nsuitably chosen (multiple) rotating frame which rotates\nwith each spin at its resonantfrequency, the Hamiltonian\nthat governsthe free evolutionof the spin system is given\nby the coupling Hamiltonian\nHc= 2J12I1zI2z+2J23I2zI3z.\nWe use the notation Iℓν=/circlemultiplytext\njIaj, where aj=νfor\nj=ℓandaj= 0 otherwise (see [15]). The matrices2\nIx:= (0 1\n1 0)/2,Iy:=/parenleftbig0−i\ni0/parenrightbig\n/2, andIz:=/parenleftbig1 0\n0−1/parenrightbig\n/2, are\nthe Pauli spin matrices and I0:= (1 0\n0 1) is the 2 ×2-\ndimensional identity matrix.\nIf the Larmor frequencies of the spins are well sepa-\nrated, each spin can be selectively excited by an appro-\npriate choice of the amplitude and phase of the radio-\nfrequency (RF) field at its resonance frequency. The\ngoal of the pulse designer is to derive explicit controls\nfor the variables comprising of the frequency, amplitude\nand phase of the external RF field to effect a net unitary\nevolution U(t) most efficiently.\nWebeginwiththeproblemoffindingtheshortestpulse\nsequencethattransformtheinitialpolarization I1zonthe\nfirst spin to a multiple quantum coherence, i.e.,\nI1x→4I1zI2zI3z.\nExample 1. The conventional strategy for achieving\nthis transfer is given by the following stages\nI1x(Hc)τ1→2I1yI2zI2y→2I1yI2x(Hc)τ2→4I1yI2yI3z\nIn the first stage of the transfer, operator I1xevolves to\n2I1yI2zunder the natural coupling Hamiltonian Hcin\nτ1=π/(2J12) units of time. This operator is then ro-\ntated to 2 I1yI2xby applying a hard ( π/2)ypulse on the\nsecond spin, which evolves to 4 I1yI2yI3zunder the nat-\nural coupling, in τ2=π/(2J13) units of time. Finally,\nhard (π/2)xpulses on first and second spin prepare the\ndesired final state. The total evolution time is then sim-\nplyτ1+τ2=π/(2J12)+π/(2J13).\nWe now study time-optimal designs for achieving this\n(and more general) transfers. To simplify notation, we\nintroduce the following symbols /an}bracketle{tO/an}bracketri}ht:= Tr(Oρ), where\nTr denotes the trace, for the expectation values of oper-\natorsO. Letx1=/an}bracketle{tI1x/an}bracketri}ht,x2=/an}bracketle{t2I1yI2z/an}bracketri}ht,x3=/an}bracketle{t2I1yI2x/an}bracketri}ht,\nandx4=/an}bracketle{t4I1yI2yI3z/an}bracketri}htandX= (x1,x2,x3,x4)T. By ex-\npressing the time t, in units of 1 /J12, the evolution of the\nvectorXis given by\ndX\ndt=\n0−1 0 0\n1 0−u0\n0u0−k\n0 0k0\nX=B(u,k)X,(1)\nwherek=J23/J12andu=u(t) is the control param-\neter representing the amplitude of the y-pulse on the\nsecond spin. Now, the problem of optimal transfer is\nto find the optimal u(t) for steering the system from\n(1,0,0,0)Tto (0,0,0,1)Tin minimal time. We consider\nalso the more general case of transfering the system from\n(cosα,sinα,0,0)Tto (0,0,cosβ,sinβ)T, where we con-\nsider different α,β∈[0,π/2].\nExample 1 (continued). In this picture, the conven-\ntional method to transfer (1 ,0,0,0)Tto (0,0,0,1)Tis\ndescribed by setting u(t) = 0 for τ1units of time, trans-\nfering (1,0,0,0)Tto (0,1,0,0)T. Using the control u, we\nrotate (0 ,1,0,0)Tto (0,0,1,0)Tin arbitrary small time,\nas the control can be performed much faster as comparedr2\nx2x3\nθ\nFIG. 1: Auxiliary variables r2andθ.\nto the evolution of couplings. Then, we set u(t) = 0 and\nevolveτ2units of time under the coupling Hamiltonian,\ntransfering (0 ,0,1,0)Tto (0,0,0,1)T. The total time for\nthe transfer is τ1+τ2.\nWe now show how significantly shorter transfer times\nare achievable, if we relax the constraint that the selec-\ntive rotations on second spin are only hard pulses. We\nshow that if we let the selective operations be carried out\nby soft shaped pulses, along with evolution of the cou-\npling Hamiltonian, then we can achieve shorter transfer\ntimes. The pulse shapes can be numerically computed\nby formulating the problem as a derivation of geodesics\non a sphere under a special metric as detailed below.\nWe first make a change of variables (see Fig. 1). Let\nr1=x1,r2=/radicalig\nx2\n2+x2\n3,r3=x4,and tanθ=x3\nx2.\nUsingu(t), we can control the angle θ, so we can think\nofθas a control variable. Expressing the time in units of\n1/J12, the evolution of the system w.r.t. the coordinates\nriis given by\nd\ndt\nr1\nr2\nr3\n=\n0−cosθ(t) 0\ncosθ(t) 0 −ksinθ(t)\n0ksinθ(t) 0\n\nr1\nr2\nr3\n.\n(2)\nThe problem of transferring the system in Eq. (1) from\n(cosα,sinα,0,0)Tto (0,0,cosβ,sinβ)T, reduces to find-\ningθ(t), for steering the system from (cos α,sinα,0)T\nto (0,cosβ,sinβ)Tin minimal time in Eq. (2). We\nshow that this is equivalent to finding the correspond-\ning geodesic on the sphere, under the metric\ng=k2dr2\n1+dr2\n3\nk2r2\n2. (3)\nBy substituting for sin θ(t) and cos θ(t) from Eq. (2),\nthe transfer time τ=/integraltextτ\n0/radicalbig\n[sinθ(t)]2+[cosθ(t)]2dtre-\nduces to\nτ=1\nk/integraldisplayτ\n0/radicalig\n[k2(˙r1)2+(˙r3)2]/r2\n2dt=1\nk/integraldisplayτ\n0L dt.\nThus, minimizing τ, amounts to computing the geodesic\nunder the metric g. The Euler-Lagrange equations for\nthe geodesic take the form\nd\ndt/parenleftbigg∂L\n∂˙r1/parenrightbigg\n=/parenleftbigg∂L\n∂r1/parenrightbigg\nandd\ndt/parenleftbigg∂L\n∂˙r3/parenrightbigg\n=/parenleftbigg∂L\n∂r3/parenrightbigg\n.3\nNote,r2\n2= 1−r2\n1−r2\n3and along the geodesic, L=kis\nconstant. We get,\nd\ndt/parenleftbiggk2˙r1\nr2\n2/parenrightbigg\n=L2r1\nr2\n2andd\ndt/parenleftbigg˙r3\nr2\n2/parenrightbigg\n=L2r3\nr2\n2,(4)\nwhich implies that\nd\ndt/parenleftbiggk2˙r1r3−˙r3r1\nr2\n2/parenrightbigg\n= (k2−1)˙r1˙r3\nr2\n2.(5)\nLetf= (k2˙r1r3−˙r3r1)/r2\n2. From Eq. (4), we get\nd\ndt/parenleftbiggk2˙r1\nr2/parenrightbigg\n=−f˙r3\nr2.\nSubstitute with ˙ r1/r2=−cosθ(t) and ˙r3/r2=ksinθ(t),\nand get\nd\ndt/bracketleftbig\n−k2cosθ(t)/bracketrightbig\n=−fksinθ(t),\nso˙θ=−f/k. Differentiating again, we obtain from\nEq. (5), that\n¨θ=−1\nkd\ndtf=−k2−1\nk˙r1˙r3\nr2\n2= (k2−1)cosθsinθ.\nUsinga=k2−1, we rewrite this as,\n¨θ=a\n2sin2θ(t). (6)\nThe solution to Eq. (6), can be given in terms of an\nelliptic integral. Note that by multiplying both sides of\nEq. (6), with ˙θ, we get the equation of an ellipse in terms\nof the coordinates ( ˙θ,cos[θ])Tas\nc=˙θ2(t)+acos2θ(t), (7)\nwherecis a constant. Thus, we obtain that, ˙θ=\n±/radicalbig\nc−acos2θ(t) and\n/integraldisplayθ(t)\nθ(0)dσ√\nc−acos2σ=±t. (8)\nTheleft hand sideofEq.(8)isanelliptic integral[25, 26].\nEquation (6), can be integrated if θ(0) and˙θ(0) are both\nknown explicitly. In transfers considered subsequently,\nonlyθ(0) is known and therefore one has to numerically\nsearch over the possible values of ˙θ(0), such that the\nresulting trajectory X(t) achieves the desired transfer.\nNote, guessing an initial value of ˙θ(0) is same as search-\ning for the correct value of cin Eq. (7).\nIII. COMPUTATION OF OPTIMAL CONTROL\nLAWS\nNow, consider the problem of steering the system of\nEq. (2), from the initial state ( r1,r2,r3)T= (1,0,0)Tto≈2.72\n1.61.822.22.42.6\n1 2 3 4 5 6 7 8 9 10T\nk (a)≈0.8660.97\n0.880.90.920.940.96\n1 2 3 4 5 6 7 8 9 10T\nπ/2+π/(2k)\nk (b)\nFIG.2: Fordifferent k, weplot(a)theminimaltime Tinunits\nof 1/J12and (b) the corresponding ratio T/[π/2+π/(2k)].\nthe target state (0 ,cosβ,sinβ)T. Forβ=π/2, we get\nthe target state (0 ,0,1)T. Recall that,\n−k˙θ=f=−k2cosθr3\nr2−ksinθr1\nr2.(9)\nAtt= 0,r1(0) = 1 and r2(0) =r3(0) = 0. For fto be\nfinite, at t= 0, we should have sin θ(0) = 0. Therefore,\nin Eq. (6), the initial condition is θ(0) = 0. To solve\nEq. (6), we only need to know the initial value of ˙θ(0),\nwhich is the same as knowing the constant cin Eq. (7).\nGivenθ(0) = 0 and the value of ˙θ(0), we can numeri-\ncally solve first Eq. (6) and then Eq. (2). Consequently,\nwe can determine for each value of ˙θ(0), the smallest\ntimessuch that r1(s) = 0 by using a one-dimensional\nsearch, e.g., using the bisection method (see pp. 46–51 of\nRef. [27]). Thus, we can determine the value of ˙θ(0) such\nthatr2(s) = cos( β), again by using a one-dimensional\nsearch. In summary, we have reduced the original opti-\nmzation problem to (combined) one-dimensional search\nproblems.\nFor different values of k, we plot the minimal time\nT=T(k) to transfer ( r1,r2,r3)T= (1,0,0)Tto (0,0,1)T\nin Fig. 2(a) in units of 1 /J12. For example, when k= 1,\ni.e.,J12=J23=J, it takes 2 .72/Jamount of time,\nwhich is about 86 .6% ofπ/J, the time needed using the\nconventional method. Figure 2(b), shows the ratio of the\nthe minimal time T(k) to the time π/2+π/(2k)obtained\nusingtheconventionalstrategy. Wedefinethisratio η(k).\nIn both the plots k≥1 is considered, as the case for\nk <1 can be derived from this. Figure 3, shows the\noptimal control u(t) in Eq. (1) for different values of k.\nObserve that T(1/k) =kT(k). Letu(t,k) be the\ntime optimal control for steering the system of Eq. (1)\nfrom (1,0,0,0)Tto (0,0,0,1)T. Then the control v(t) =\nu(T−t,k), will steer the same system from (0 ,0,0,1)T\nto (1,0,0,0)Tin the same time, which is also minimal for\nthis transfer. Let Y= (x4,x3,x2,x1)Tand consider the\ncontrolv(t) =u(T−t,k). Then, we have\ndY\ndτ=kB(1/k,v)Y.\nBut, we have just remarked that the minimal time to\nsteerYfrom (1,0,0,0)Tto (0,0,0,1)TisT(k), and it4\n024681012\n0 0.5 1 1.5 2 2.5 3u\ntK=1\nK=2\nK=10\nFIG. 3: The figure shows the shape of the optimal control\nu(t) in Eq. (1) for k= 1,2,10. The time is in units of 1 /J12.\nThe control amplitude is in units of J12. Rightmost datapoint\nof each control u(t) is not displayed for reasons of numerical\naccuracy.\nfollows that T(1/k)/k=T(k). It also follows that the\noptimal control u(t,1/k) =u[T(k)−t/k,k], where t∈\n[0,kT(k)]. Note η(k) =η(1/k).\nRemark 1. In Fig. 4, we present the geodesics of the\nmetric of Eq. (3) for β=π/2 andk∈ {1/10,1,10},\nwhereβ= tan−1[r3(T)/r2(T)], i.e.,r1(T) =r2(T) = 0\nandr3(T) = 1. Observe that the geodesics for the case\nk >1,bendmoretowardsthepoint( r1,r2,r3) = (0,1,0),\nbeforeapproachingthefinalpoint(0 ,0,1). Inanintuitive\nway to understand this, consider first the limit k≫1.\nIn this limit, the minimum time to steer Eq. (2) from\n(1,0,0) to (0,0,1)is essentially the time required to steer\nthe system (of Eq. (2)) to the equator. This is achieved\nfastest by moving from (1 ,0,0), directly to (0 ,1,0) and\nkeepingθ(0) = 0. As kis decreased to 1, the geodesics\ngradually move away from (0 ,1,0), implying ˙θ(0) fork >\n1 is smaller than for the case k= 1. In the special case\nofk= 1, Eq. (6) reduces to ˙θ=C, a constant. This case\nwas been studied in detail in [12, 14]. Therefore, in the\nnumerical computation of optimal control, the search for\ntrue˙θ(0) can be restricted to the interval [0 ,C].\nIn Fig. 5, we plot the minimal time T=T(k,β)\nto transfer ( r1,r2,r3)T= (1,0,0)Tto (0,cosβ,sinβ)T,\nfor different values of kandβ= tan−1[r3(T)/r2(T)].\nFigure 6, shows the geodesics of the metric of Eq. (3)\nfork= 2, for two different values β∈ {π/4,π/2}of the\nterminal point (0 ,cosβ,sinβ)T.\nIf we transfer the system from ( x1,x2,x3,x4)T=\n(cosα,sinα,0,0)Tto (0,0,cosβ,sinβ)T, where α >0,\nwe do not know the value of θ(0). But from Eq. (9),\nwe have the relationship ˙θ(0) = sin[ θ(0)]cot(α) between\n˙θ(0) andθ(0). Thus, we can apply similar search meth-\nods for the right initial conditions as in the case of\nFIG. 4: Geodesics to transfer from ( r1,r2,r3)T= (1,0,0)Tto\n(0,0,1)T, wherek= 1/10,k= 1, and k= 10.\n123456789100π\n8π\n43π\n8π\n2\nπ\n2≈2.72\n1.71.92.12.32.5\nT\nkβ(rad)T\nFIG. 5: For different kandβ= tan−1[r3(T)/r2(T)], we plot\nthe minimal time Tin units of 1 /J12\nα= 0. For k= 2 and different values of αandβ, we\nplot in Fig. 7 the minimal time T=T(α,β) to transfer\n(r1,r2,r3)T= (cosα,sinα,0)Tto (0,cosβ,sinβ)T.\nIV. EFFICIENT CREATION OF COHERENCES\nIN ISING SPIN CHAINS WITH UNEQUAL\nCOUPLINGS\nWe now consider a generalization of the problem\ntreated above. We consider a linear chain of nspins,\nplaced in a static external magnetic field in the z-\ndirection,withunequalIsingtypecouplingsbetweennext\nneighbors [23, 24]. In a suitably chosen (multiple) rotat-5\nFIG. 6: Geodesics to transfer from ( r1,r2,r3)T= (1,0,0)Tto\n(0,cosβ,sinβ)T, wherek= 2 and β∈ {π/4,π/2}.\n0π\n8π\n43π\n8π\n20π\n8π\n43π\n8π\n20.51π\n2≈2.09\n00.51T\nα(rad)β(rad)T\nFIG. 7: For k= 2 and different values of αandβ, we plot\nthe minimal time Tin units of 1 /J12\ning frame, which rotates with each spin at its resonant\nfrequency, the free evolution of the spin system is given\nby the coupling Hamiltonian,\nHc= 2n−1/summationdisplay\nl=1Jl,l+1IlzI(l+1)z.\nAs before, we assume that the Larmor frequency of the\nspins are well separated compared to the coupling con-\nstantJl,l+1, so that we can selectively rotate each spin at\na rate much faster than the evolution of the couplings.\nWe first consider the problem of synthesizing a uni-\ntary transformation U, which efficiently transfers a sin-\ngle spin coherence represented by the initial operatorI1xto a multiple spin order represented by the opera-\ntor 2n−1/producttextn\nl=1Ilz.\nThe conventional strategy for achieving this transfer is\nachieved through the following stages\nI1xHc→2I1yI2zI2y→2I1yI2xHc→4I1yI2yI3z\nI3y→4I1yI2yI3x→ ··· → 2n−1(n−1/productdisplay\nl=1Ily)Inz\nwhere each evolution represents an appropriate evolution\nbyrotationangle π/2. Thefinalstate2n−1(/producttextn−1\nl=1Ily)Inz,\nis locally equivalent to 2n−1/producttextn\nl=1Ilz. The whole transfer\ninvolvesn−1 evolution steps under the natural Hamilto-\nnian, where the first step takes only π/(2Jl,l+1) units of\ntime, resulting in a total time of/summationtextn−1\nl=1π/(2Jl,l+1). We\nnow formulate the problem of this transfer as a problem\nof optimal control and derive time efficient strategies for\nachieving this transfer.\nTo simplify notation, we introduce the following sym-\nbols for the expectation values of operators that play a\nkey part in the transfer:\nx1=/an}bracketle{tI1x/an}bracketri}ht,x2=/an}bracketle{t2I1yI2z/an}bracketri}ht,\nx3=/an}bracketle{t2I1yI2x/an}bracketri}ht,x4=/an}bracketle{t4I1yI2yI3z/an}bracketri}ht,...,\nx2n−3=/an}bracketle{t2n−1I1yI2yI3y···I(n−1)x/an}bracketri}ht,\nandx2n−2=/an}bracketle{t2nI1yI2yI3y···I(n−1)yInz/an}bracketri}ht.\nLetX= (x1,x2,x3,...,x 2n−2)T. Expressing the time\nin units of 1 /J12, the evolution of the system is given by\ndX\ndt=\n0−k10 0 0 0 ···\nk10−u10 0 0 ···\n0u10−k20 0 ···\n0 0 k20−u20···\n0 0 0 u20−k3···\n0 0 0 0 k30···\n.....................\nX,(10)\nwhereulare the control parameters representing the am-\nplitude of the ypulse on spin l+1 and kl=Jl,l+1/J12.\nThe problem now is to find the optimal ul(t), steering\nthesystemfrom(1 ,0,0,···,0)Tto(0,0,···,0,1)Tin the\nminimal time.\nWe divide the optimal transfer problem into mul-\ntiple steps. Let l= 1,...,n−1. Consider\nthe step that steers ( x2l−1,x2l,x2l+1,x2l+2)Tfrom the\ninitial state (cos βl,sinβl,0,0)Tto the target state\n(0,0,cosβl+1,sinβl+1)T, by optimal choice of ul.\nd\ndt\nx2l−1\nx2l\nx2l+1\nx2l+2\n=\n0−kl0 0\nkl0−ul0\n0ul0−kl+1\n0 0kl+10\n\nx2l−1\nx2l\nx2l+1\nx2l+2\n,(11)\nWe denote the minimal time for this transfer as\nT(βl,βl+1,kl,kl+1), as it depends on the parameters,\n(βl,βl+1,kl,kl+1). Note that\nklT(βl,βl+1,kl,kl+1) =T(βl,βl+1,1,kl+1/kl).6\n2.002.022.042.062.082.102.122.142.162.18\n0π\n8γoptπ\n43π\n8π\n2J(γ)\nγ\nFIG. 8: The figure shows the plot of function J(γ) in Eq. (13),\nas a function of γ.\nWe can then break the original transfer in Eq. (10),\ninto sub-transfers denoted by\nβ1→β2··· →βl··· →βn−1,\nwhere both β1= 0 and βn−1=π/2. The control ulin\nEq. (11) should be so chosen that the time for transfer\nβltoβl+1is minimized. Furthermore, the choice of in-\ntermediate βlshould be made to minimize the total time\nof transfer\nJ1(0) =n−2/summationdisplay\nl=1T(βl,βl+1,kl,kl+1).\nThe notation for J1(0) denotes that the index lin the\nabove summation begins with l= 1 with initial angle\nβ1= 0. Now suppose, we have computed Jl+1(ξ) for\nallξ∈[0,π/2]. Then we can compute Jl(βl) from the\nequation below as\nJl(βl) = min\nη{Jl+1(η)+T(βl,η,kl,kl+1)}.(12)\nWecansolveEq.(12)iterativelystartingfrom l=n−2\nand proceeding backwards and remembering that\nJn−2(βn−2) =T(βn−2,π/2,kn−2,kn−1).\nNoteT(βl,η,kl,kl+1) in Eq. (12) can be explicitly com-\nputed, as we showed in the previous section. ThereforeEq. (12) helps us to compute the optimal ηand the cor-\nresponding optimal control ul.\nNow, the problem is reduced to a simple dynamic pro-\ngramming problem, which can be solved efficiently.\nExample 2. We consider as an example, an Ising spin\nchain with four spins with J12/(2π) = 91 Hz, J23/(2π) =\n15 Hz and J34/(2π) = 55 Hz. This system represents\nthe popular [28] HNCACO experiment in multidimen-\nsional NMR, where first spin is the proton, the second\none represent15N, the third and fourth one are13C. For\nthe transfer in Eq. (10), the dynamic programmingequa-\ntions for the minimum time is in units of 1 /J23, can be\nwritten as min γJ(γ), where\nJ(γ) ={T(0,γ,k1,1)+T(γ,π\n2,1,k2)}(13)\nwherek1=J12/J23andk2=J34/J23. Figure 8 shows\nthe plot of J(γ) and the minimum is achieved at γopt≈\n.193πand the corresponding minimum value of J(γopt)\nis≈2.01. The conventional transfer strategy will take\nπ(1 + 1/k1+ 1/k2)/2≈2.26, units of time, which is\n≈12.2% longer than the proposed efficient methodology.\nV. CONCLUSION\nIn this article, we studied the problem of efficient cre-\nation of multiple spin order in an Ising spin chain with\nunequal couplings. We first analyzed in detail the system\nof three linearly coupled spins. We showed that the time\noptimal pulse sequences for creating multiple spin order\ninthissystemcanbeobtainedbycomputinggeodesicson\na sphere under a special metric and that the solution to\nthe resulting Euler Lagrange equation is related to the\nsolution of a nonlinear oscillator equation ¨θ=Asinθ.\nWe showed that the minimum times and optimal control\nlaws for the studied problem can be explicitly computed\nand provide significant gains over conventional methods.\nThe methods developed in the three spin case were then\nexploited to find efficient strategies for manipulating the\ndynamics of Ising spin chains with unequal couplings. It\nis expected that these methods will find immediate ap-\nplications in coherent spectroscopy and quantum infor-\nmation processing.\n[1] N. Khaneja, R. Brockett, and S. J. Glaser, Phys. Rev. A\n63, 032308 (2001).\n[2] C. H. Bennett, J. I. Cirac, M. S. Leifer, D. W. Leung,\nN. Linden, S. Popescu, and G. Vidal, Phys. Rev. A 66,\n012305 (2002).\n[3] G. Vidal, K. Hammerer, and J. I. Cirac, Phys. Rev. Lett.\n88, 237902 (2002).\n[4] K. Hammerer, G. Vidal, and J. I. Cirac, Phys. Rev. A\n66, 062321 (2002).[5] T. O. Reiss, N. Khaneja, and S. J. Glaser, J. Magn. Re-\nson.154, 192 (2002).\n[6] R. Zeier, M. Grassl, and T. Beth, Phys. Rev. A 70,\n032319 (2004).\n[7] H. Yuan and N. Khaneja, Phys. Rev. A 72, 040301(R)\n(2005).\n[8] H. Yuan, Ph.D. thesis, Harvard University (2006).\n[9] R. M. Zeier, Lie-theoretischer Zugang zur Erzeugung\nunit¨ arer Transformationen auf Quantenrechnern (Uni-7\nversit¨ atsverlagKarlsruhe, Karlsruhe, 2006), Ph.D.thes is,\nUniversit¨ at Karlsruhe, 2006.\n[10] N. Khaneja, S. J. Glaser, and R. Brockett, Phys. Rev. A\n65, 032301 (2002).\n[11] T. O. Reiss, N. Khaneja, and S. J. Glaser, J. Magn. Re-\nson.165, 95 (2003).\n[12] N. Khaneja, B. Heitmann, A. Sp¨ orl, H. Yuan, T. Schulte-\nHerbr¨ uggen, and S. J. Glaser, Phys. Rev. A 75, 012322\n(2007).\n[13] N. Khaneja and S. J. Glaser, Phys. Rev. A 66, 060301(R)\n(2002).\n[14] H. Yuan, S. J. Glaser, and N. Khaneja, Phys. Rev. A 76,\n012316 (2007).\n[15] R. R. Ernst, G. Bodenhausen, and A. Wokaun, Prin-\nciples of Nuclear Magnetic Resonance in One and Two\nDimensions (Clarendon Press, Oxford, 1997), reprinted\nwith corrections.\n[16] H.YuanandN. Khaneja, in Proceedings of the 45th IEEE\nConference on Decision and Control, 2006 (San Diego,\nUSA, 2006), pp. 3117–3120.\n[17] M. A.Nielsen, M. R.Dowling, M. Gu, andA. C. Doherty,\nScience311, 1133 (2006).\n[18] M. A. Nielsen, Quantum Inf. Comput. 6, 213 (2006).\n[19] M. A.Nielsen, M. R.Dowling, M. Gu, andA. C. Doherty,Phys. Rev. A 73, 062323 (2006).\n[20] R. W. Brockett, in New Directions in Applied Mathemat-\nics, edited by P. J. Hilton and G. S. Young (Springer,\nNew York, 1982), pp. 11–27.\n[21] J. B. Baillieul, Ph.D. thesis, Harvard University (197 5).\n[22] R. Montgomery, A Tour of Subriemannian Geometries,\nTheir Geodesics and Applications , no. 91 in Mathemat-\nical surveys and monographs (American Mathematical\nSociety, Providence, 2002).\n[23] E. Ising, Z. Physik 31, 253 (1925).\n[24] W. J. Caspers, Spin systems (World Scientific, Singa-\npore, 1989).\n[25] R. W. Brockett and L. Dai, in Nonholonomic Motion\nPlanning , edited by Z. Li and J. F. Canny (Kluwer Aca-\ndemic Publishers, 1993), pp. 1–21.\n[26] E. T. Whittaker and G. N. Watson, A course of modern\nanalysis (CambridgeUniversityPress, Cambridge, 1963),\n4th ed.\n[27] R. L. Burden and J. D. Faires, Numerical Analysis\n(Thomson Brooks/Cole, Belmont, 2005), 8th ed.\n[28] J. Cavanagh, W. J. Fairbrother, A. G. Palmer, and\nN. J. Skelton, Protein NMR Spectroscopy: Principles and\nPractice (Academic Press, San Diego, 1996)."    },    {        "title": "1801.08375v2.Theory_of_high_resolution_tunneling_spin_transport_on_a_magnetic_skyrmion.pdf",        "content": "Theory of high-resolution tunneling spin transport on a magnetic skyrmion\nKriszti\u0013 an Palot\u0013 as1;2,\u0003Levente R\u0013 ozsa3, and L\u0013 aszl\u0013 o Szunyogh4;5\n1 Slovak Academy of Sciences, Institute of Physics, SK-84511 Bratislava, Slovakia\n2 University of Szeged, MTA-SZTE Reaction Kinetics and Surface Chemistry Research Group, H-6720 Szeged, Hungary\n3 University of Hamburg, Department of Physics, D-20355 Hamburg, Germany\n4 Budapest University of Technology and Economics,\nDepartment of Theoretical Physics, H-1111 Budapest, Hungary\n5 Budapest University of Technology and Economics,\nMTA-BME Condensed Matter Research Group, H-1111 Budapest, Hungary\n(Dated: May 24, 2021)\nTunneling spin transport characteristics of a magnetic skyrmion are described theoretically in\nmagnetic scanning tunneling microscopy (STM). The spin-polarized charge current in STM (SP-\nSTM) and tunneling spin transport vector quantities, the longitudinal spin current and the spin\ntransfer torque, are calculated in high spatial resolution within the same theoretical framework. A\nconnection between the conventional charge current SP-STM image contrasts and the magnitudes of\nthe spin transport vectors is demonstrated that enables the estimation of tunneling spin transport\nproperties based on experimentally measured SP-STM images. A considerable tunability of the\nspin transport vectors by the involved spin polarizations is also highlighted. These possibilities\nand the combined theory of tunneling charge and vector spin transport pave the way for gaining\ndeep insight into electric-current-induced tunneling spin transport properties in SP-STM and to the\nrelated dynamics of complex magnetic textures at surfaces.\nI. INTRODUCTION\nDue to the electron's intrinsic properties, electronic\ntransport includes both charge and spin transfer. Un-\nderstanding the mechanisms and increasing the e\u000eciency\nof current-induced magnetization switching (CIMS) have\na large impact on the ongoing development of magnetic\nspintronic devices. Consequently, CIMS attracted con-\nsiderable research interest in metallic spin valves and\nmagnetic tunnel junctions (MTJs) in the recent years\n[1]. So far, theoretical research of spin transport partly\nfocused on the calculation of the tunneling spin trans-\nfer torque (STT) in symmetric or asymmetric planar\nMTJs. The theoretical approaches cover free-electron\nmodels [2{4], a transfer Hamiltonian method [5], tight-\nbinding-based transport models [6{10], and scattering\nmethods [11], partly combined with \frst-principles tech-\nniques [12, 13]. The \frst direct measurement of the out-\nof-plane and in-plane components of the STT vector in\na planar MTJ was provided by Sankey et al. [14]. The\ntunability of the magnetotransport properties in material\ncombinations of current interest has also been predicted\nrecently [15].\nMagnetic skyrmions in thin \flms are real-space spin\ntextures possessing a topological charge [16], and in most\nof the studied cases they show a preferred chirality due\nto the emergence of the antisymmetric Dzyaloshinsky-\nMoriya exchange interaction [17, 18] in the magnetic layer\nwith broken inversion symmetry[19{23]. On the other\nhand, it has been demonstrated that frustrated Heisen-\nberg exchange interactions may also lead to the stabi-\nlization of localized skyrmionic spin con\fgurations with\n\u0003palotas@phy.bme.hudi\u000berent topologies [24{27]. A range of technological ap-\nplications of isolated skyrmions has been proposed for\nmagnetic data storage and transfer [28{32]. The \fnite\ntemperature stability of skyrmions has been investigated\nboth experimentally [22, 23, 33] and theoretically [34{\n36], and the annihilation of skyrmions has recently been\nextensively studied by using minimum energy path cal-\nculations [36{40].\nMagnetic skyrmions in ultrathin \flms can conveniently\nbe imaged by using spin-polarized scanning tunneling mi-\ncroscopy (SP-STM) [41{43], and \frst-principles calcula-\ntions have proven crucial in understanding the formation\nof these spin structures through determining microscopic\nmagnetic interactions [44{47]. Over the last two decades,\nthe powerful capabilities of SP-STM have been demon-\nstrated for obtaining local information on spin-polarized\ntunneling charge transport properties of a great variety\nof magnetic surfaces [48]. High-resolution SP-STM is\nexpected to further contribute, e.g., to the proper un-\nderstanding of the interaction of skyrmions with defects\nthat is currently an important challenge [49, 50]. Re-\ncently, there has been a growing interest in getting insight\ninto local STT properties in high spatial resolution in\nasymmetric MTJs. Naturally, the tunneling STT, which\nis concomitantly present with the charge current, con-\ntributes to local CIMS e\u000bects [51]. Two recent examples\non the atomic scale include Refs. 19 and 52, where lo-\ncal current pulses with opposite voltage polarities have\nbeen used with an STM tip to create and annihilate indi-\nvidual magnetic skyrmions in di\u000berent thin \flm systems.\nHowever, the detailed microscopic mechanism of the local\ntunneling STT and another spin transport component,\nthe longitudinal spin current (LSC) [53], is not clari\fed\nin these processes yet.\nIn this paper the generalization of high-resolution STMarXiv:1801.08375v2  [cond-mat.mtrl-sci]  23 Apr 20182\ncharge transport theories [54] is proposed to include vec-\ntor spin transport in STM junctions going beyond the\nassumption of collinear magnetic structure for both the\nsurface and the tip used in Ref. 55. A combined electron\ntunneling theory for the charge and vector spin trans-\nport in magnetic STM junctions is presented considering\ncomplex noncollinear surface magnetic structures within\nthe three-dimensional (3D) Wentzel-Kramers-Brillouin\n(WKB) electron tunneling framework. The theory is\nimplemented in the 3D-WKB-STM code [56], which is\nan established and e\u000ecient method for the simulation\nof spin-polarized scanning tunneling microscopy [57{61]\nand spectroscopy [62, 63] with an enhanced parameter\nspace for modeling tip geometries [64{66]. Besides the\nspin-polarized charge current above complex magnetic\ntextures in SP-STM [43, 57, 59, 67, 68], the proposed\nmethod allows for the high-resolution calculation of tun-\nneling spin transport vector quantities, the STT and\nthe LSC. The connections between the SP-STM image\ncontrasts of the charge current and the magnitudes of\nthe STT as well as the LSC are highlighted, and it is\nconcluded that estimations on the tunneling spin trans-\nport properties can be made based on experimental SP-\nSTM images. Moreover, a direct relationship between\nthe scalar spin polarization and the ratio of the out-of-\nplane and the in-plane STT components is given, and\nit is proposed that the measurement of the STT vec-\ntor components would enable the determination of the\nspin polarizations separately for the sample surface and\nthe tip. The in\ruence of these spin polarizations on the\ntunneling spin transport is also analyzed, and a consider-\nable tunability of the spin transport properties is pointed\nout. Finally, theoretical considerations are reported on\nthe measurement of the STT and the LSC vectors.\nThe paper is organized as follows. The 3D-WKB com-\nbined tunneling electron charge and vector spin transport\ntheory in magnetic STM considering noncollinear mag-\nnetic surfaces is presented in detail in Sec. II A, where the\ndominating atomic contributions for the interpretation of\nour results are described in Sec. II B. Relationships be-\ntween the charge current and the magnitudes of the LSC\nand the STT are reported in Sec. II C, and tunneling pa-\nrameters and visualization remarks are given in Sec. II D.\nThe vector spin transport (LSC and STT) properties of\na skyrmion in relation to the spin-polarized charge cur-\nrent are analyzed in Sec. III A, and the e\u000bect of the spin\npolarizations is investigated in Sec. III B. Summary and\nconclusions are found in Sec. IV, and spin transport vec-\ntor measurement considerations are given in Appendix.\nII. THEORY AND METHODS\nA. 3D-WKB theory of combined electron charge\nand spin tunneling\nThe 3D-WKB theoretical model for the combined tun-\nneling charge and vector spin transport in magnetic STMabove complex noncollinear magnetic surfaces is based on\nthe recently introduced electron tunneling model through\nSTM junctions built up from collinear magnetic surface\nand tip [55]. Note that in the present paper we do not re-\nstrict the formalism to \fxed spin quantization axes of the\nsample and the tip, as this would describe collinear mag-\nnets. Instead, the spatial dependence of the local atomic\nspin quantization axes of the noncollinear magnetic sur-\nface is allowed. For simplicity, we omit the orbital de-\npendence [58, 60] of the electronic structure taken into\naccount in Ref. 55, and the low bias limit is employed.\nConsider the following normalized (dimensionless) den-\nsity matrices in spin space for the sample surface ( S)\natom labeled by \" a\" and the tip ( T) apex atom at the\ncorresponding Fermi levels ES\nFandET\nF[55, 57, 69]:\n\u001aa\nS(ES\nF) =I+Pa\nS(ES\nF)\u0001\u001b; (1a)\n\u001a\nT(ET\nF) =I+PT(ET\nF)\u0001\u001b; (1b)\nwhereIis the 2\u00022 unit matrix and \u001b=\u0010\n\u001bx;\u001by;\u001bz\u0011\nis a vector composed of the Pauli matrices. Pa\nS(ES\nF) =\nma\nS(ES\nF)=na\nS(ES\nF) is the spin polarization vector of the\natom-projected density of states (PDOS) of the sample\nsurface atom \" a\", where ma\nS(ES\nF) andna\nS(ES\nF) denote the\nmagnetization (vector) and charge (scalar) character of\nthe corresponding PDOS at ES\nF, respectively. Similarly,\nPT(ET\nF) =mT(ET\nF)=nT(ET\nF), where PT(ET\nF),mT(ET\nF)\nandnT(ET\nF) denote the spin polarization PDOS vector,\nmagnetization PDOS vector, and charge PDOS of the tip\napex atom at ET\nF. These quantities can be calculated by\nusing ab initio electronic structure methods [57], or can\nbe treated as parameters as in the present paper.\nThe coupled dimensionless transport quantities, the\nscalar charge conductance ( ~I) and the generalized vec-\ntor spin torkance ( ~T), of the electron tunneling between\nthe sample surface atom \" a\" and the tip apex atom at the\ncommon Fermi levels ES\nF=ET\nF=EF(zero bias limit)\ncan be represented by traces in spin space, and they de-\npend on the tunneling direction tip !sample (T!S) or\nsample!tip (S!T) as\n~Ia;T!S(EF) =1\n2Tr\u0010\n\u001a\nT(EF)I\u001aa\nS(EF)\u0011\n=~Ia;S!T(EF) =1\n2Tr\u0010\n\u001aa\nS(EF)I\u001a\nT(EF)\u0011\n=~Ia(EF) = 1 + Pa\nS(EF)\u0001PT(EF); (2a)\n~Ta;T!S(EF) =1\n2Tr\u0010\n\u001a\nT(EF)\u001b\u001aa\nS(EF)\u0011\n=Pa\nS(EF) +PT(EF)\n+iPa\nS(EF)\u0002PT(EF); (2b)\n~Ta;S!T(EF) =1\n2Tr\u0010\n\u001aa\nS(EF)\u001b\u001a\nT(EF)\u0011\n=Pa\nS(EF) +PT(EF)\n\u0000iPa\nS(EF)\u0002PT(EF): (2c)\nSince Pa\nS(EF)\u0001PT(EF) = [Pa\nS(EF)sa\nS]\u0001[PT(EF)sT] =\nPa\nS(EF)PT(EF) cos\u001eawith\u001eathe angle between the3\nclassical spin unit vectors of the sample surface atom\n\"a\" (sa\nS=Sa\nS=jSa\nSj) and the tip apex ( sT=ST=jSTj),\nwhere Sa\nSandSTare the corresponding spin moments,\nthe charge conductance formula is formally equivalent to\nthe spin-polarized Terso\u000b-Hamann model [67, 68, 70, 71]:\n~Ia(EF) = 1 +Pa\nS(EF)PT(EF) cos\u001ea: (3)\nHere, the \frst and second term is the non-magnetic\nand magnetic (spin-polarized) part of the dimensionless\ncharge conductance at zero bias voltage, respectively.\nThe in-plane and out-of-plane components of the gen-\neralized vector spin torkance formula are obtained as the\nreal and imaginary parts of Eqs. (2b) and (2c), respec-\ntively:\n~Ta;in\u0000pl:(EF) =Pa\nS(EF) +PT(EF)\n= Ren\n~Ta;T!S(EF)o\n= Ren\n~Ta;S!T(EF)o\n; (4a)\n~Ta;out\u0000pl:(EF) =Pa\nS(EF)\u0002PT(EF)\n= Imn\n~Ta;T!S(EF)o\n=\u0000Imn\n~Ta;S!T(EF)o\n:(4b)\nHere, the in-plane component ~Ta;in\u0000pl:lies in the plane\nspanned by the Pa\nS(EF) and PT(EF) (orsa\nSandsT) vec-\ntors, depending on the considered lattice site a. However,\nthe direction of ~Ta;in\u0000pl:is not necessarily perpendicu-\nlar to the spin moment ( STorSa\nS) on which the gen-\neralized torkance is acting. Therefore, ~Tin\u0000pl:can be\ndecomposed into a dimensionless longitudinal spin con-\nductance part denoted by L, parallel to the spin mo-\nment directions: sTor local sa\nSunit vectors; and into a\ndimensionless spin torkance part denoted by k, perpen-\ndicular to the spin moment directions [11]. The decom-\nposition reads ~Ta;in\u0000pl:(EF) =~Ta;j;L(EF) +~Ta;j;k(EF)\nwithj2fT;Sg, wherekdenotes that this in-plane (Slon-\nczewski or damping-like) torkance vector is in the Sa\nS{ST\n(orsa\nS{sT) plane. The corresponding components are\n~Ta;T;L(EF) =h\n~Ta;in\u0000pl:(EF)\u0001sTi\nsT\n= (Pa\nS(EF) cos\u001ea+PT(EF))sT;(5a)\n~Ta;T;k(EF) =~Ta;in\u0000pl:(EF)\u0000~Ta;T;L(EF)\n=Pa\nS(EF) (sa\nS\u0000sTcos\u001ea)\n=Pa\nS(EF)sT\u0002(sa\nS\u0002sT); (5b)\n~Ta;S;L(EF) =h\n~Ta;in\u0000pl:(EF)\u0001sa\nSi\nsa\nS\n= (Pa\nS(EF) +PT(EF) cos\u001ea)sa\nS;(5c)\n~Ta;S;k(EF) =~Ta;in\u0000pl:(EF)\u0000~Ta;S;L(EF)\n=PT(EF) (sT\u0000sa\nScos\u001ea)\n=PT(EF)sa\nS\u0002(sT\u0002sa\nS): (5d)\nHere, theTorSindices denote the tip or sample side\nof the tunnel junction, on which the longitudinal spin\nconductance and the in-plane torkance are acting. As it\nis clear from Eq. (4a), these quantities are independent\nof the actual current \row direction T!SorS!T;for experimental evidence for the in-plane torkance, see,\ne.g., Fig. 3a in Ref. 14.\nThe out-of-plane component in Eq. (4b), ~Ta;out\u0000pl:, is\nperpendicular to the plane spanned by the Pa\nS(EF) and\nPT(EF) (or sa\nSandsT) vectors, and the dimensionless\nout-of-plane (\feld-like) torkance vector can be identi\fed\nas [11]\n~Ta;?(EF) =~Ta;out\u0000pl:(EF) =Pa\nS(EF)PT(EF)sa\nS\u0002sT\n=~Ta;T!S;?(EF) =\u0000~Ta;S!T;?(EF): (6)\nAccording to Eq. (4b), the out-of-plane torkance vector\nchanges sign by reversing the current \row direction; for\nexperimental evidence see, e.g., Fig. 3a in Ref. 14. Note\nthat throughout the paper the out-of-plane torkance and\ntorque is purely current-induced, resulting from electron\ntunneling through the vacuum barrier, and the equilib-\nrium torque [6] is not taken into account. Combining\nEqs. (4a), (4b), (5b), (5d) and (6), the following dimen-\nsionless torkance vectors are obtained:\n~Ta;T!S;T(EF) =~Ta;T;k(EF) +~Ta;?(EF); (7a)\n~Ta;S!T;T(EF) =~Ta;T;k(EF)\u0000~Ta;?(EF); (7b)\n~Ta;T!S;S(EF) =~Ta;S;k(EF) +~Ta;?(EF); (7c)\n~Ta;S!T;S(EF) =~Ta;S;k(EF)\u0000~Ta;?(EF); (7d)\nwhere ~Ta;T!S;Tand ~Ta;S!T;Tact on the spin moment\nof the tip apex atom at T!SandS!Ttunneling,\nrespectively. Similarly, ~Ta;T!S;Sand ~Ta;S!T;Sact on\nthe spin moment of the sample surface atom \" a\" at the\nindicated tunneling directions.\nAssuming elastic electron tunneling, the signed charge\ncurrent (Is) can be obtained from the charge conduc-\ntance within the 3D-WKB theory by the superposition\nof atomic contributions from the sample surface (sum\nover \"a\") [57] at the tip apex position RTand at bias\nvoltageVin the low bias limit as\nIs(RT;V) =e2\n2\u0019~VX\nah(RT\u0000Ra)~Ia(EF):(8)\nThe transmission function,\nh(r) = exph\n\u0000p\n8m\b=~2jrji\n(9)\n(with the electron's mass m, the reduced Planck constant\n~, and the e\u000bective work function \b), depends on the rel-\native position of the tip apex atom and the sample surface\natom \"a\" (RT\u0000Ra), and in the transmission all electron\nstates are considered as exponentially decaying spherical\nstates [68, 70, 71], neglecting orbital dependence [58].\nThe signed longitudinal spin current (LSC) vector\n(TL\ns) and the signed spin transfer torque (STT) vector\ncomponents ( Tk\nsandT?\ns) acting on the spin moment\nof the tip apex atom can be calculated from the cor-\nresponding longitudinal spin conductance and torkance4\ncomponents, respectively, as\nTTL\ns(RT;V) =eVX\nah(RT\u0000Ra)~Ta;T;L(EF);\n(10a)\nTTk\ns(RT;V) =eVX\nah(RT\u0000Ra)~Ta;T;k(EF);\n(10b)\nTT!S;T?\ns (RT;V) =eVX\nah(RT\u0000Ra)~Ta;T!S;?(EF);\n(10c)\nTS!T;T?\ns (RT;V) =eVX\nah(RT\u0000Ra)~Ta;S!T;?(EF):\n(10d)\nSimilarly, the signed LSC vector and the signed STT vec-\ntor components acting on the spin moment of surface\natom \"a\" at position Raare obtained as\nTaSL\ns(Ra;V) =eVh(RT\u0000Ra)~Ta;S;L(EF);\n(11a)\nTaSk\ns(Ra;V) =eVh(RT\u0000Ra)~Ta;S;k(EF);\n(11b)\nTa;T!S;S?\ns (Ra;V) =eVh(RT\u0000Ra)~Ta;T!S;?(EF);\n(11c)\nTa;S!T;S?\ns (Ra;V) =eVh(RT\u0000Ra)~Ta;S!T;?(EF):\n(11d)\nIn Eqs. (8), (10a)-(10d) and (11a)-(11d) the signed charge\ncurrent and the signed LSC and STT vectors have\nthe correct dimensions by multiplying the dimension-\nless atomic charge conductance, spin conductance and\ntorkance contributions with proper factors, i.e., Is=\ne2\n2\u0019~V~IandTL=k=?\ns =eV~TL=k=?, whereeis the elemen-\ntary charge and Vis the bias voltage.\nTaking the sign convention of the bias voltage into ac-\ncount, i.e., V > 0 atT!SandV < 0 atS!T\ntunneling, we \fnd that the LSC and the in-plane STT\nvectors actually change sign and the out-of-plane STT\nvectors do not change sign by reversing the bias polar-\nity, thus the direction of the current \row. These are\nclearly seen in Eqs. (10a)-(10d) and (11a)-(11d): in TL=k\ns\nVchanges sign and according to Eq. (4a) ~TL=kdo not,\nand in T?\nsbothVand ~T?change sign, the latter ac-\ncording to Eq. (6). These \fndings are in agreement with\nprevious spin transport interpretations [1]. The absolute\ncharge current is independent of the tunneling direction,\nand can be calculated as\nI(RT;V)\n=Is(RT;V > 0) =\u0000Is(RT;V < 0)\n=e2\n2\u0019~jVjX\nah(RT\u0000Ra)(1 +Pa\nS(EF)PT(EF) cos\u001ea):\n(12)Similarly, the spin transport vectors acting on the\nspin moment of the tip apex atom are obtained from\nEqs. (10a)-(10d) as\nTTL(RT;V)\n=TTL\ns(RT;V > 0) =\u0000TTL\ns(RT;V < 0)\n=ejVjX\nah(RT\u0000Ra) (Pa\nS(EF) cos\u001ea+PT(EF))sT;\n(13a)\nTTk(RT;V)\n=TTk\ns(RT;V > 0) =\u0000TTk\ns(RT;V < 0)\n=ejVjX\nah(RT\u0000Ra)Pa\nS(EF)sT\u0002(sa\nS\u0002sT);(13b)\nT?(RT;V)\n=TT!S;T?\ns (RT;V > 0) =TS!T;T?\ns (RT;V < 0)\n=ejVjX\nah(RT\u0000Ra)Pa\nS(EF)PT(EF)sa\nS\u0002sT;(13c)\nTT!S;T(RT;V > 0) =T?(RT;V) +TTk(RT;V);\n(13d)\nTS!T;T(RT;V < 0) =T?(RT;V)\u0000TTk(RT;V);\n(13e)\nand the spin transport vectors acting on the spin moment\nof surface atom \" a\" are obtained from Eqs. (11a)-(11d)\nas\nTaSL(Ra;V)\n=TaSL\ns(Ra;V > 0) =\u0000TaSL\ns(Ra;V < 0)\n=ejVjh(RT\u0000Ra) (Pa\nS(EF) +PT(EF) cos\u001ea)sa\nS;\n(14a)\nTaSk(Ra;V)\n=TaSk\ns(Ra;V > 0) =\u0000TaSk\ns(Ra;V < 0)\n=ejVjh(RT\u0000Ra)PT(EF)sa\nS\u0002(sT\u0002sa\nS); (14b)\nTa?(Ra;V)\n=Ta;T!S;S?\ns (Ra;V > 0) =Ta;S!T;S?\ns (Ra;V < 0)\n=ejVjh(RT\u0000Ra)Pa\nS(EF)PT(EF)sa\nS\u0002sT; (14c)\nTa;T!S;S(Ra;V > 0) =Ta?(Ra;V) +TaSk(Ra;V);\n(14d)\nTa;S!T;S(Ra;V < 0) =Ta?(Ra;V)\u0000TaSk(Ra;V);\n(14e)\nwhere the notations TTLandTaSLfor the LSC vec-\ntors, TTkandTaSkfor the in-plane STT vectors, and\nT?=P\naTa?for the out-of-plane STT vectors have\nbeen introduced.\nThe LSC and the STT acting on the sample for the\nscanning tip at position RTare de\fned as the sum of the\nvector spin transport quantities acting on the di\u000berent5\nsurface atoms \" a\",\nTSL(RT;V)\n=ejVjX\nah(RT\u0000Ra) (Pa\nS(EF) +PT(EF) cos\u001ea)sa\nS;\n(15a)\nTSk(RT;V)\n=ejVjX\nah(RT\u0000Ra)PT(EF)sa\nS\u0002(sT\u0002sa\nS);(15b)\nT?(RT;V)\n=ejVjX\nah(RT\u0000Ra)Pa\nS(EF)PT(EF)sa\nS\u0002sT;(15c)\nTT!S;S(RT;V > 0) =T?(RT;V) +TSk(RT;V);\n(15d)\nTS!T;S(RT;V < 0) =T?(RT;V)\u0000TSk(RT;V):\n(15e)\nAs discussed in Sec. II B, these values are dominated by\nthe contributions coming from the closest surface atom\nAbelow the STM tip. Note also the equivalence of\nEqs. (13c) and (15c).\nThroughout the paper the reported electronic charge\nand vector spin transport quantities correspond to a scan-\nning tip at position RTand to Eqs. (12), (13a)-(13e)\nand (15a)-(15e). These are key results of the present pa-\nper. The transport components in a simpli\fed fashion\nare summarized below for a better overview:\nI(RT)/X\nah(RT\u0000Ra)(1 +PSPTcos\u001ea);(16a)\nTTL(RT)/X\nah(RT\u0000Ra)(PT+PScos\u001ea)sT;(16b)\nTSL(RT)/X\nah(RT\u0000Ra)(PS+PTcos\u001ea)sa\nS;(16c)\nTTk(RT)/X\nah(RT\u0000Ra)PSsT\u0002(sa\nS\u0002sT);(16d)\nTSk(RT)/X\nah(RT\u0000Ra)PTsa\nS\u0002(sT\u0002sa\nS);(16e)\nT?(RT)/X\nah(RT\u0000Ra)PSPTsa\nS\u0002sT; (16f)\nand the total STT vectors are\nTT!S;T(RT) =T?(RT) +TTk(RT); (17a)\nTS!T;T(RT) =T?(RT)\u0000TTk(RT); (17b)\nTT!S;S(RT) =T?(RT) +TSk(RT); (17c)\nTS!T;S(RT) =T?(RT)\u0000TSk(RT): (17d)\nHere, it was assumed that Pa\nS(EF) =PSfor all sur-\nface atoms, and the notation PT(EF) =PTwas used\nfor simplicity. Note that the e\u000bective spin polarization\n(Pe\u000b=PSPT) only enters the charge current (Eq. (16a))\nand the out-of-plane torque (Eq. (16f)) expressions, and\nnot the longitudinal spin current or the in-plane torque.This means that Pe\u000bis not su\u000ecient to characterize the\nspin polarization of the magnetic tunnel junction con-\ncerning spin transport quantities, and PSandPTare in-\ndependent parameters in our model. The e\u000bect of these\nspin polarizations is investigated on the tunneling spin\ntransport properties of a magnetic skyrmion in Sec. III B.\nB. Dominating atomic contributions\nDue to the exponential decay of the tunneling trans-\nmission in Eq. (9), the atomic sums in the tunneling\ncharge and vector spin transport quantities in Eqs. (8),\n(10a)-(10d), (12), (13a)-(13c), (15a)-(15c) and (16a)-\n(16f) are convergent. They are dominated, and thus can\nbe approximated by the sum of the contributions from\nthe closest surface atoms below the tip position RT. Such\na set of surface atoms can be denoted by A(RT). The\nselection of surface atoms \" a\" in the set ofA(RT) de-\npends on a properly chosen convergence criterion [62].\nConsequently, Eqs. (8), (10a)-(10d), (12), (13a)-(13c),\n(15a)-(15c) and (16a)-(16f) can be interpreted as RT-\ndependent weighted averages over the set of surface\natomsA(RT), e.g.,Is(RT;V)\u0019e2\n2\u0019~V~IA(RT)(EF) with\n~IA(RT)(EF) =P\na2A(RT)h(RT\u0000Ra)~Ia(EF). Although\nin the paper the sum over \" a\" is performed for all sam-\nple atoms in the simulated area (for more details see Sec.\nII D), for the interpretation of the results the dominat-\ning contribution is considered to come from the closest\nsurface atom below the STM tip, which is denoted by A\nand characterized by the spin unit vector sA\nS. Clearly,\nall quantities denoted by Adepend on the lateral posi-\ntion of the tip, just as above: A(RT). Following this,\nthe tunneling electron charge and vector spin transport\ncomponents can be written as\nI(RT;V)/1 +PSPTcos\u001eA;\n(18a)\nTTL(RT;V)/(PScos\u001eA+PT)sT;\n(18b)\nTSL(RT;V)\u0019TASL(RA;V)/(PTcos\u001eA+PS)sA\nS;\n(18c)\nTTk(RT;V)/PSsT\u0002(sA\nS\u0002sT);\n(18d)\nTSk(RT;V)\u0019TASk(RA;V)/PTsA\nS\u0002(sT\u0002sA\nS);\n(18e)\nT?(RT;V)\u0019TA?(RA;V)/PSPTsA\nS\u0002sT;(18f)6\nwith cos\u001eA=sA\nS\u0001sT, and the magnitudes of the vector\nspin transport quantities are\njTTL(RT;V)j/jPScos\u001eA+PTj;\n(19a)\njTSL(RT;V)j\u0019jTASL(RA;V)j/jPTcos\u001eA+PSj;\n(19b)\njTTk(RT;V)j/jPSsin\u001eAj; (19c)\njTSk(RT;V)j\u0019jTASk(RA;V)j/jPTsin\u001eAj; (19d)\njT?(RT;V)j\u0019jTA?(RA;V)j/jPSPTsin\u001eAj;\n(19e)\njTT(RT;V)j/jPSsin\u001eAjq\n1 +P2\nT;\n(19f)\njTS(RT;V)j\u0019jTAS(RA;V)j/jPTsin\u001eAjq\n1 +P2\nS;\n(19g)\nwherejTTj=jTT!S;Tj=jTS!T;Tj,jTSj=jTT!S;Sj=\njTS!T;Sj, andjTASj=jTA;T!S;Sj=jTA;S!T;Sj. Note\nthat both STT components ( kand?), and thus the STT\nobey the expected sin \u001eA-dependence.\nC. Connections between the charge current and\nthe magnitudes of the LSC and the STT\nFollowing the previous section for the dominating\natomic contributions to the tunneling electron transport\nproperties, simple relationships between the charge cur-\nrent and the spin transport magnitudes can be derived.\nLet us assume that the charge current can be measured at\nopposite tip magnetization directions sTand\u0000sT. This\nresults in the charge currents I(sT)/1 +PSPTcos\u001eA\nandI(\u0000sT)/1\u0000PSPTcos\u001eA, from which the spin-\npolarized contribution [59] to the current (also known as\nmagnetic asymmetry [63], AI) can be expressed as\nAI=PSPTcos\u001eA=I(sT)\u0000I(\u0000sT)\nI(sT) +I(\u0000sT): (20)\nThis quantity takes values between \u00001 and +1, and\ncan directly be obtained in experiments in the di\u000beren-\ntial magnetic mode [72] of SP-STM, or in simulations\nemploying, e.g., the above-described 3D-WKB model.\nNote that similar magnetic asymmetry quantities can\nbe de\fned for the longitudinal spin currents, see Ap-\npendix. Following the above, cos \u001eA=AI=(PSPT),\njsin\u001eAj=p\nP2\nSP2\nT\u0000A2\nI=jPSPTj, and the charge and\nspin transport magnitudes assuming an sTtip magne-tization direction can be written as\nI/1 +AI; (21a)\njTTLj/jPT+AI=PTj; (21b)\njTSLj/jPS+AI=PSj; (21c)\njTTkj/q\nP2\nSP2\nT\u0000A2\nI=jPTj; (21d)\njTSkj/q\nP2\nSP2\nT\u0000A2\nI=jPSj; (21e)\njT?j/q\nP2\nSP2\nT\u0000A2\nI; (21f)\njTTj/q\nP2\nSP2\nT\u0000A2\nIq\n1 + 1=P2\nT; (21g)\njTSj/q\nP2\nSP2\nT\u0000A2\nIq\n1 + 1=P2\nS: (21h)\nThese equations establish a direct connection between\nthe magnitudes of the spin transport components and the\ncharge current through the spin-polarized contribution of\nthe latter,AI.\nConsidering the lateral ( x;y)-dependence of a scan-\nning tip, the approximate \u001eAcan be replaced by an\ne\u000bective\u001e(x;y) in a continuum description, which ex-\nactly reproduces the electron transport components in\nEqs. (16a)-(16f), and the above derivation also ap-\nplies in a point by point fashion for high-resolution\nimages using the AI(x;y) expression based on Eq.\n(20):AI(x;y) =PSPTcos\u001e(x;y) = [I(x;y;sT)\u0000\nI(x;y;\u0000sT)]=[I(x;y;sT) +I(x;y;\u0000sT)].\nD. Model parameters and visualization remarks\nIn the following we report on computational parame-\nters used in the electron charge and spin tunneling model.\nThe spin structure of the noncollinear magnetic surface\nis an input parameter of the 3D-WKB-STM code. The\nconsidered spin structure of a skyrmion was taken from\nRef. 27, where it was relaxed on a single-layer triangu-\nlar lattice with C3vcrystallographic symmetry containing\n128\u0002128 = 16384 lattice sites using spin dynamics simu-\nlations. The underlying magnetic interaction parameters\nof Fe in the (Pt 0:95Ir0:05)/Fe/Pd(111) ultrathin \flm sys-\ntem were obtained from ab initio calculations [47]. We\nnote that the antisymmetric Dzyaloshinsky-Moriya and\nthe frustrated Heisenberg magnetic exchange interactions\nare concomitantly present in this ultrathin magnetic \flm\n[27].\nThe absolute bias voltage is set to jVj= 1:5 meV and\nthe e\u000bective work function to \b = 5 eV. Motivated by the\nreported electronic structure of a recent work [73], PS=\n\u00000:5 together with PT=\u00000:8 are chosen in Sec. III A,\nthus resulting in Pe\u000b= +0:4, which value was also con-\nsidered in previous works discussing SP-STM character-\nistics of skyrmionic spin textures with di\u000berent topologies\n[43, 74]. For investigating the e\u000bect of the spin polariza-\ntions on the spin transport, the combinations of the sets\nPS2 f\u0000 0:5;+0:5gandPT2 f\u0000 0:8;\u00000:4;+0:4;+0:8g\nare considered in Sec. III B.7\nFIG. 1. (a) Spin structure of a skyrmion obtained from Ref.\n27, and its constant-current SP-STM images [43] using (b)\nan out-of-plane and (c) an in-plane magnetized tip (bright:\nhigher, dark: lower apparent height) according to Eq. (16a).\nTunneling charge and spin transport quantities are cal-\nculated in a scan area of 7.5 nm \u00026 nm. SP-STM images\nof the charge current are shown in constant-current mode\nusing a white-brown-black color palette corresponding\nto maximum-medium-minimum apparent heights. Em-\nploying the reported parameters, the current value of\nI= 10\u00004nA of the constant-current contours corre-\nsponds to about 6 \u0017A minimal tip-sample distance and\ncorrugation values between 30 and 40 pm [43]. The\nspin transport (STT and LSC) quantities (vectors and\nscalar magnitudes) are given in constant-height mode at\n6\u0017A tip-sample distance. The magnitudes of the STT\nand LSC are shown using a red-green-blue color palette\ncorresponding to maximum-medium-minimum values of\nthe individual images. While the STT and LSC vectors\nare calculated in the same high lateral resolution as the\ncharge current and the magnitudes of the STT and the\nLSC (1 \u0017A resolution for all), for visualization reasons the\nlateral resolution of the vector spin transport quantities\nis set to 5 \u0017A.\nIII. RESULTS AND DISCUSSION\nA. Tunneling charge and spin transport properties\nof a skyrmion\nTo utilize the above electron charge and spin tunnel-\ning theory of noncollinear magnetic surfaces, we consider\na magnetic skyrmion. Figure 1 reports the spin struc-\nture and SP-STM images of the charge current above\nthe skyrmion showing characteristic circular and two-\nlobe contrasts for out-of-plane and in-plane magnetized\ntips, respectively [42, 43]. It is known [67, 68] that the\ncharge current, I/1 +PSPTcos\u001eA, is sensitive to the\ne\u000bective spin polarization, Pe\u000b=PSPT, see Eq. (16a).\nTherefore, the charge current has maxima at cos \u001eA=\u00061\n(AI=\u0006Pe\u000b) and minima at cos \u001eA=\u00071 (AI=\u0007Pe\u000b)\nfor sgn(Pe\u000b) =\u00061.\nFigure 2 shows calculated LSC magnitudes and vec-\ntors above the skyrmion in Fig. 1 with the same di\u000ber-\nently magnetized tips and the chosen spin polarization\nparameters. The maxima of the magnitudes jTTLjand\njTSLj(red regions in Fig. 2) are found at cos \u001eA=\u00061\n(AI=\u0006Pe\u000b) for sgn(Pe\u000b) =\u00061, exactly as for the\ncharge current. However, the positions of the LSC min-ima (blue regions in Fig. 2) depend on the relation of\nPStoPT: ifjPSj>jPTjthenjTTLjhave minima at\ncos\u001eA=\u0000PT=PS(AI=\u0000P2\nT), and ifjPSj<jPTjthen\nthe minima are found at cos \u001eA=\u00071 (AI=\u0007Pe\u000b) for\nsgn(Pe\u000b) =\u00061. Similarly, ifjPTj>jPSjthenjTSLj\nhave minima at cos \u001eA=\u0000PS=PT(AI=\u0000P2\nS), and if\njPTj<jPSjthen the minima are found at cos \u001eA=\u00071\n(AI=\u0007Pe\u000b) for sgn(Pe\u000b) =\u00061. These result in an iden-\ntical contrast ofjTTLjin Fig. 2 to that of the correspond-\ning charge current in Fig. 1. The contrast of jTSLjis also\nqualitatively similar, except that its minima are found at\ncos\u001eA=\u00005=8,\u001eA= 0:715\u0019(AI=\u00000:25), which are\nshown as blue belts in Fig. 2. According to Eq. (16b),\ntheTTLvectors generally point to the sgn( PT)sTdirec-\ntion, except for the regions with small LSC magnitudes\nifjPSj>jPTj. Similarly, according to Eq. (16c), the TSL\nvectors generally point to the sgn( PS)sA\nSdirection, except\nfor the regions with small LSC magnitudes if jPTj>jPSj,\nthat is inside the mentioned blue belts in Fig. 2, where\ncos\u001eA<\u00005=8. A detailed overview of the e\u000bect of var-\nious combinations of PSandPTon the LSC is given in\nSec. III B.\nThe similarity of the LSC and the charge current im-\nage contrasts enables the estimation of the LSC based\non experimentally measured SP-STM images. The LSC\nmagnitudes can directly be related to the SP-STM im-\nages as discussed above, and the theoretical basis for this\nis outlined in Sec. II C. The orientation of the LSC vec-\ntors can be based on the knowledge of sTand the non-\ncollinear spin structure sa\nS. The latter can, in princi-\nple, be extracted from a series of SP-STM images with\ndi\u000berent tip magnetization directions [43, 75], and such\na procedure has been proven experimentally [52]. For\nthe estimation of the LSC magnitudes and vectors, the\nknowledge of Eqs. (16b)-(16c) and the spin polarizations\nPSandPTare essential.\nFigure 3 shows calculated STT magnitudes, out-of-\nplane and in-plane STT vector components, and to-\ntal STT vectors above the skyrmion in Fig. 1. We\n\fnd that the magnitudes of all STT components show\nthe same type of contrast, which is denoted by jTjin\nFig. 3. Such a behavior results from their dominating\njsin\u001eAj-dependence due to the vector product sA\nS\u0002sTin\nEqs. (18d)-(18f), with di\u000berent spin-polarization-related\nprefactors (Eqs. (19c)-(19g)). Thus, the STT minima\nand maxima are obtained where the spins of the skyrmion\nare in line (parallel or antiparallel) with and perpen-\ndicular to the tip magnetization direction, respectively.\nMoreover, the STT minima at sin \u001eA= 0 are found ex-\nactly at the maxima and minima of the charge current,\nwhere cos\u001eA=\u00061 (AI=\u0006Pe\u000b), and the STT max-\nima are obtained at sin \u001eA=\u00061 (cos\u001eA= 0,AI= 0).\nThis means that the STT is small (large) where the ab-\nsolute magnetic contrast of the charge current jAIjis\nlarge (small) [55], see also Eqs. (21d)-(21h). This enables\nthe estimation of the STT based on experimentally mea-\nsured SP-STM images, similarly to the LSC, and again\nthe knowledge of sa\nS,sT,PS, andPTis required, see the8\nFIG. 2. Longitudinal spin current (LSC) magnitudes (red: maximum, blue: minimum) and vectors acting on the scanning tip\n(jTTLjandTTL) and on the skyrmion ( jTSLjandTSL) for both T!SandS!Ttunneling directions using an out-of-plane\nand an in-plane magnetized tip according to Eqs. (16b) and (16c). Red and blue colors of the LSC vectors correspond to\npositive and negative out-of-plane ( z) vector components, respectively.\nFIG. 3. Spin transfer torque (STT) magnitudes jTj(red: maximum, blue: minimum) and vectors (out-of-plane component\n(T?, Eq. (16f)), in-plane component ( Tjk, Eqs. (16d) and (16e)), total ( T?\u0006Tjk), depending on the tunneling direction\nT!SorS!T, see Eqs. (17a)-(17d)) acting on the spin moments of the scanning tip ( j=T) and of the skyrmion ( j=S)\nusing an out-of-plane and an in-plane magnetized tip. Red and blue colors of the STT vectors correspond to positive and\nnegative out-of-plane ( z) vector components, respectively.\ntorque expressions in Eqs. (16d)-(16f) and (17a)-(17d).\nThe dependence of the STT on the combinations of PS\nandPTis investigated in Sec. III B.\nThe calculated STT vector components and vectors in\nFig. 3 show a wide variety depending on the tip mag-\nnetization orientation ( sT), the spin moment they are\nacting on (TorS), and the tunneling direction ( T!S\norS!T). For the out-of-plane magnetized tip ( sTkz,\n\frst row of Fig. 3) the T?andTTkvectors are perpen-\ndicular toz, i.e., they lie in the xysurface plane. The\nTTkvectors point to the direction of PSsA\nSprojected on\nthe surface plane. Thus, the T?\u0006TTkvectors are also\nin the surface plane. On the other hand, the TSkand\ntheT?\u0006TSkvectors have z-components proportional\ntoPT(Eqs. (16e)-(16f)), these are negative for TSkand\nT?+TSkand positive for T?\u0000TSkin Fig. 3. This\ndi\u000berence in the zcomponent of the total STT vectors\nacting on the sample for the two tunneling directionshas an important consequence for the possible rotation\nof the spins of the skyrmion due to the tunneling STT,\nclearly preferring one direction. For the skyrmion de-\npicted in Fig. 1(a) and the selected spin polarization pa-\nrameters we conclude that S!Ttunneling tends to\nannihilate the skyrmion since here the torque would ro-\ntate the spins outwards from the surface, while a spin\nrotation towards the surface would stabilize skyrmions\nin the case of T!Stunneling.\nFor the in-plane magnetized tip ( sTkx, second row\nof Fig. 3) the T?,TSkandT?\u0006TSkvectors lie in the\nxysurface plane outside the skyrmion above the ferro-\nmagnetic (FM) background, where sA\nSkz. Here, the T?\nvectors are obtained as z\u0002x=yand their direction ( \u0006y)\nis determined by the sign of PSPT. The TSkvectors are\nin line with sTabove the FM background, and their di-\nrection (\u0006x) is determined by the sign of PT. Here, the\nTTkvectors are in line with sA\nS, and their direction ( \u0006z)9\nis determined by the sign of PS. By summing up the com-\nponents, the total STT vectors T?\u0006TTkandT?\u0006TSk\nabove the FM background can be characterized by two\nangles\u000bTand\u000bS, which describe the inclination from\nthesA\nSkzandsTkxdirections, respectively. These an-\ngles are directly related to the spin polarizations of the\ntip and the sample,\njtan\u000bTj=jT?j\njTTkj=jPTj; (22a)\njtan\u000bSj=jT?j\njTSkj=jPSj: (22b)\nGiven the used spin polarization parameters in Fig. 3,\nthe two angles are \u000bT=\u00060:215\u0019and\u000bS=\u00060:148\u0019.\nThe corresponding inclinations of the total STT vectors\nabove the FM background are clearly visible in the sec-\nond row of Fig. 3. Note that \u0000\u0019=4\u0014\u000bT;\u000bS\u0014\u0019=4\nsince\u00001\u0014PT;PS\u00141, and according to Eqs. (22a)\nand (22b) this means that the magnitude of T?cannot\nexceed the magnitude of Tkconsidering purely current-\ninduced torques. Similarly, it was found in experiments\nperformed for planar MTJs [14] that the magnitude of\nthe out-of-plane torque is smaller than that of the in-\nplane torque. Equations (22a) and (22b) also imply that\nthe direct measurement of the STT vector components\nin magnetic tunnel junctions would allow an accurate de-\ntermination of the spin polarizations of the tip and the\nsample separately. Presently, Pe\u000b=PSPTcan be ob-\ntained from the measured charge current contrasts in the\ndi\u000berential magnetic mode [72] of SP-STM (see also Sec.\nII C), and the knowledge of the spin polarization of one\nside is needed to determine the spin polarization of the\nother side in the tunnel junction. Further implications\nfor STT measurements are given in Appendix.\nB. E\u000bect of the spin polarizations on the tunneling\ncharge and spin transport of a skyrmion\nIn the following, the tunneling charge and spin trans-\nport properties of the skyrmion in Fig. 1 are calcu-\nlated and discussed taking the following combinations\nof the spin polarizations: PS2f\u0000 0:5;+0:5gandPT2\nf\u00000:8;\u00000:4;+0:4;+0:8g.\nFigure 4 displays SP-STM images of the charge current\nand the magnitudes of the spin transport quantities LSC\nand STT obtained with out-of-plane and in-plane mag-\nnetized tips, depending on PSandPT. The SP-STM\ncontrast is reversed by changing the sign of Pe\u000b, see the\ncontrasts of Iin the middle four images in the \frst and\n\ffth columns of Fig. 4. The LSC magnitudes in the sec-\nond, third, sixth, and seventh columns of Fig. 4 show\nalmost identical contrasts with those of the correspond-\ning charge currents in the same row, and the contrast\nchange depending on the sign of Pe\u000bis also reproduced.\nSuch a behavior results from the expressions jTTLjand\njTSLjin Eqs. (19a) and (19b), respectively, and the di-\nrect connections between the LSC magnitudes and thecharge current are introduced in Sec. II C. The appear-\ning blue belts for the jTTLjandjTSLjcontrasts in Fig. 4\ncorrespond to the real minima depending on the relation\nofPStoPT, as explained at the discussion of Fig. 2 in\nSec. III A.\nFor the STT magnitudes we \fnd the same type of con-\ntrast for bothjTTjandjTSjand for all their compo-\nnents (?,k, total), which are commonly denoted as jTj\nin the fourth and eighth columns of Fig. 4. This is due to\ntheir dominating jsin\u001eAj-dependence (Eqs. (19c)-(19g))\nas discussed at Fig. 3 in Sec. III A. Figure 4 clearly shows\nthat the contrasts of the STT magnitudes are sensitive to\nthe magnetic structure only, and not to the involved spin\npolarizations. The STT minima (blue regions of STT\nin Fig. 4) and maxima (red regions of STT in Fig. 4)\nare obtained where the spins of the skyrmion are in line\n(parallel or antiparallel) with and perpendicular to the\ntip magnetization direction, respectively. Moreover, the\nSTT minima are found exactly at the maxima and min-\nima of the charge current, compare the corresponding I\nandjTjcontrasts in Fig. 4.\nFigure 5 shows calculated LSC vectors for the\nskyrmion in Fig. 1 for both T!SandS!\nTtunneling directions, depending on the considered\ncombinations of PSandPT, employing out-of-plane\nand in-plane magnetized tips. We \fnd the gen-\neral rule that TjL(PS;PT) =\u0000TjL(\u0000PS;\u0000PT), i.e.,\nTT!S;jL(PS;PT) =TS!T;jL(\u0000PS;\u0000PT) forj2fT;Sg.\nThe magnitudes of the TTLvectors correspond to the\nsecond and sixth columns of Fig. 4, and the magnitudes\nof the TSLvectors to the results shown in the third and\nseventh columns of Fig. 4.\nFigures 6 and 7 show calculated STT vectors and vec-\ntor components for the skyrmion in Fig. 1 for both T!S\nandS!Ttunneling directions, depending on the con-\nsidered combinations of PSandPT, employing an out-of-\nplane and an in-plane magnetized tip, respectively. We\n\fnd that the TTkvectors scale with PS, following its sign\nchange, and are independent of PT(Eq. (16d)). Simi-\nlarly, the TSkvectors scale with PT, also following its\nsign change, and are independent of PS(Eq. (16e)). On\nthe other hand, the T?vectors scale with PSPT, and\nthey follow the sign change of the e\u000bective spin polariza-\ntion (Eq. (16f)). Since the total STT vectors are the sum\nof the corresponding two components, Tj=T?\u0006Tjk\nwithj2fT;SgforT!SandS!Ttunneling di-\nrections, respectively, the STT results in Fig. 6 (with an\nout-of-plane magnetized tip) and in Fig. 7 (with an in-\nplane magnetized tip) give good indications on how the\nSTT vectors can be tuned by changing the spin polariza-\ntions of the sample and the tip in the tunnel junction.\nThis feature can turn out to be very useful if aiming\nat engineering the STT vectors at the atomic scale for\ntechnical applications in the future. We \fnd the gen-\neral rule that TT!S;j(PS;PT) =TS!T;j(\u0000PS;\u0000PT) for\nj2fT;Sg. Note that the magnitudes of all STT compo-\nnents and vectors show the same type of contrast with a\ngiven tip magnetization orientation, and these contrasts10\nFIG. 4. Dependence of the charge and spin transport magnitudes on the spin polarizations of the surface ( PS) and the tip\n(PT) in various combinations, for the skyrmion displayed in Fig. 1: Constant-current SP-STM images ( I; bright: higher, dark:\nlower apparent height), and magnitudes of the longitudinal spin current (\f\fTTL\f\fand\f\fTSL\f\f; red: maximum, blue: minimum)\nand the spin transfer torque ( jTj; red: maximum, blue: minimum) at 6 \u0017A tip-sample distance using an out-of-plane (+ z, in\n[111] crystallographic direction) and an in-plane (+ x, in [1 \u001610] crystallographic direction) magnetized tip. The tip magnetization\ndirections are explicitly shown. The color scales correspond to the data range of the individual images. Note that qualitatively\nvery similar contrasts are observed for the magnitudes of the STT vectors and their components, i.e., for jTj,jTkj, andjT?j.\nare reported in the fourth and eighth columns of Fig.\n4, respectively. Taking an out-of-plane magnetized tip\nand the skyrmion depicted in Fig. 1(a), we conclude that\nS!Ttunneling tends to annihilate the skyrmion if\nPT<0, and the opposite T!Stunneling direction\ntends to annihilate the skyrmion if PT>0 because in\nboth cases the TStorque would rotate the spins out-\nwards from the surface due to its positive zcomponent,\nsee the last two columns of Fig. 6.\nConsidering the di\u000berent scalings of the STT vector\ncomponents with PS(forTTk),PT(forTSk) orPSPT(forT?), we can state that Eqs. (22a) and (22b) generally\nhold true, and do not depend on the tip-sample geometry\nwhile in the tunneling regime. Deviations from this can\nbe expected close to contact, where the importance of the\nSTT contributions stemming from farther surface atoms\nbelow the STM tip apex is enhanced.\nWe \fnd opposite inclinations of the STT vectors in the\nmiddle of the skyrmion compared to those above the fer-\nromagnetic (FM) background in Fig. 7. This is due to the\nopposite directions of the spins in that region compared\nto the FM background. Note that the determination of11\nFIG. 5. Dependence of the longitudinal spin current (LSC) vectors ( TTLandTSL) on the spin polarizations of the surface\n(PS) and the tip ( PT) in various combinations for the skyrmion in Fig. 1. The LSC vectors are reported at 6 \u0017A tip-sample\ndistance for both T!SandS!Ttunneling directions using an out-of-plane (+ z, in [111] crystallographic direction) and\nan in-plane (+ x, in [1 \u001610] crystallographic direction) magnetized tip. The tip magnetization directions are explicitly shown in\nparentheses. Red and blue colors of the reported vectors correspond to positive and negative out-of-plane ( z) components,\nrespectively. The absolute maximal LSC magnitudes are 5.7 neV.\nPSandPTfrom the inclinations of the total STT vectors,\nor from the in-plane and out-of-plane STT components\nrefers to a certain bias voltage. In the presented model\nin Sec. II A we are restricted to very small bias and thus\npractically to the Fermi levels of both sides of the mag-\nnetic tunnel junction. Note, however, that the tunneling\nmodel can be extended to include energy dependence of\nthe contributing electronic states, and bias voltage e\u000bects\ncan be studied [55]. It is known that the bias voltage de-pendence of the charge current and the conductance com-\nplicates the determination of the energy-dependent spin\npolarizations signi\fcantly [63]. This is also expected in\ncase of the determination of PSandPTfrom the STT\nvector components at non-zero bias voltage in possible\nfuture experiments.12\nFIG. 6. Dependence of the spin transfer torque (STT) vector components and total STT vectors on the spin polarizations of\nthe surface ( PS) and the tip ( PT) in various combinations for the skyrmion in Fig. 1 at 6 \u0017A tip-sample distance using an out-\nof-plane magnetized tip (spin moment pointing along the + zor [111] crystallographic direction). The STT vector components\n(T?,Tjk) and vectors ( T?\u0006Tjk) are acting on the spin moments of the scanning tip ( j=T) and of the skyrmion ( j=S).\nRed and blue colors of the reported vectors correspond to positive and negative out-of-plane ( z) components, respectively. The\nabsolute maximal STT magnitudes are 4 neV.\nIV. SUMMARY AND CONCLUSIONS\nIn summary, a theoretical method for the combined\ncalculation of charge and vector spin transport of elasti-\ncally tunneling electrons in high spatial resolution above\ncomplex noncollinear magnetic surfaces in SP-STM was\ndeveloped. Connections between the SP-STM image con-\ntrasts of the charge current and the magnitudes of the\nlongitudinal spin current (LSC) and the spin transfer\ntorque (STT) were identi\fed and explained. It was pro-\nposed that this enables the estimation of tunneling spin\ntransport properties based on experimentally measured\nSP-STM images. A qualitative explanation was providedfor a preferred bias voltage polarity for the STT con-\ntribution of skyrmion deletion in SP-STM. It was also\nproposed that the direct measurement of the STT vec-\ntor components would enable the separate determination\nof the spin polarizations of the sample ( PS) and the tip\n(PT), even above a ferromagnetic surface. A considerable\ntunability of the spin transport vectors by the involved\nspin polarizations was also demonstrated that could in-\nspire the engineering of desired spin transport properties.\nThe high-resolution determination of the tunneling STT\nand LSC vectors paves the way for the future investiga-\ntion of current-induced magnetization switching in com-\nplex spin textures on surfaces due to local spin-polarized13\nFIG. 7. Dependence of the spin transfer torque (STT) vector components and total STT vectors on the spin polarizations\nof the surface ( PS) and the tip ( PT) in various combinations for the skyrmion in Fig. 1 at 6 \u0017A tip-sample distance using an\nin-plane magnetized tip (spin moment pointing along the + xor [1\u001610] crystallographic direction). The STT vector components\n(T?,Tjk) and vectors ( T?\u0006Tjk) are acting on the spin moments of the scanning tip ( j=T) and of the skyrmion ( j=S).\nRed and blue colors of the reported vectors correspond to positive and negative out-of-plane ( z) components, respectively. The\nabsolute maximal STT magnitudes are 4 neV.\ncurrents in SP-STM. As an example, the knowledge of\nthe local STT and LSC vectors is expected to deliver a\ndetailed microscopic insight into the creation, annihila-\ntion, and lateral manipulation of skyrmions and other\ncomplex surface magnetic objects in the future.\nACKNOWLEDGMENTS\nFinancial supports of the SASPRO Fellowship of the\nSlovak Academy of Sciences (project No. 1239/02/01),\nthe Hungarian State E otv os Fellowship, the National\nResearch, Development, and Innovation O\u000ece of Hun-gary under Projects No. K115575 and No. FK124100, the\nAlexander von Humboldt Foundation, and the Deutsche\nForschungsgemeinschaft via SFB668 are gratefully ac-\nknowledged.\nAppendix A: Spin transport vector measurement\nconsiderations\nLet us assume that in the magnetic STM junction the\nT?andTTkvector components can be measured at low\nbias voltage for at least one tunneling direction T!S\norS!T, and denote TTk=TT!S;Tk(=TTk\ns(V >14\n0)) =\u0000TS!T;Tk(=\u0000TTk\ns(V < 0)), see Eq. (13b). With\nthese the total STT vectors can be written as TT!S;T=\nT?+TTkandTS!T;T=T?\u0000TTk. Here, we show\nthat by knowing the two STT vector components T?\nandTTk, or alternatively TT!S;TandTS!T;T, the spin\npolarization of the tip ( PT) and the sample ( PS) and the\nmagnitudes of the STT components and the STT acting\non the sample surface can be determined. Taking the\nscalar product and using Eqs. (19c) and (19e) result in\nTT!S;T\u0001TS!T;T= (T?+TTk)\u0001(T?\u0000TTk)\n=jT?j2\u0000jTTkj2\n=P2\nSsin2\u001eA(P2\nT\u00001)\u00140:(A1)\nThe absolute value square of the total STT vector is\njTTj2=jTT!S;Tj2=jTS!T;Tj2=jT?\u0006TTkj2\n=jT?j2+jTTkj2=P2\nSsin2\u001eA(P2\nT+ 1):(A2)\nTaking the ratio of the above quantities results in a\ncorrelation-like formula,\n\u001aTT!S;T;TS!T;T=TT!S;T\u0001TS!T;T\np\nTT!S;T\u0001TT!S;Tp\nTS!T;T\u0001TS!T;T\n=\u00001\u0000P2\nT\n1 +P2\nT\u00140: (A3)\nUsing Eq. (22a), PT= tan\u000bT,\n\u001aTT!S;T;TS!T;T=\u00001\u0000tan2\u000bT\n1 + tan2\u000bT=\u0000cos(2\u000bT):(A4)\nThus, the angle \u000bTcan be determined,\n\u000bT=1\n2arccos(\u0000\u001aTT!S;T;TS!T;T); (A5)\nand the following relationships can be observed,\njT?j\njTTj=jPTjp\n1 +P2\nT=jsin\u000bTj;PTp\n1 +P2\nT= sin\u000bT;\njTTkj\njTTj=1p\n1 +P2\nT= cos\u000bT: (A6)\nKnowingPT= tan\u000bT, the magnitude of TSkisjTSkj=\njPTsin\u001eAj. Since we also know the out-of-plane STT\nvector T?, the same procedure can be repeated as above\nto determine PS= tan\u000bS,\n\u001aTT!S;S;TS!T;S=TT!S;S\u0001TS!T;S\np\nTT!S;S\u0001TT!S;Sp\nTS!T;S\u0001TS!T;S\n=jT?j2\u0000jTSkj2\njT?j2+jTSkj2\n=\u00001\u0000P2\nS\n1 +P2\nS=\u00001\u0000tan2\u000bS\n1 + tan2\u000bS=\u0000cos(2\u000bS):\n(A7)\nThus, the angle \u000bScan be obtained as\n\u000bS=1\n2arccos(\u0000\u001aTT!S;S;TS!T;S); (A8)and the following formulas relate the STT components\nto the STT magnitude,\njT?j\njTSj=jPSjp\n1 +P2\nS=jsin\u000bSj;PSp\n1 +P2\nS= sin\u000bS;\njTSkj\njTSj=1p\n1 +P2\nS= cos\u000bS: (A9)\nInspired by the magnetic asymmetry of the charge cur-\nrentAIin Eq. (20), similar quantities can be de\fned\nfor the LSC. For that reason, let us assume that the\nLSC vectors can be measured at opposite tip magneti-\nzation directions sTand\u0000sT. This results in the LSC\nvectors TTL(sT)/(PT+PScos\u001eA)sT,TTL(\u0000sT)/\n(PT\u0000PScos\u001eA)(\u0000sT),TSL(sT)/(PS+PTcos\u001eA)sA\nS,\nandTSL(\u0000sT)/(PS\u0000PTcos\u001eA)sA\nS, from which the fol-\nlowing magnetic asymmetry expressions can be obtained:\nATL=jTTL(sT)\u0000TTL(\u0000sT)j\njTTL(sT) +TTL(\u0000sT)j\n=jPTj\njPScos\u001eAj=P2\nT\njAIj=ASL\ncos2\u001eA;\nASL=jTSL(sT)\u0000TSL(\u0000sT)j\njTSL(sT) +TSL(\u0000sT)j\n=jPTcos\u001eAj\njPSj=jAIj\nP2\nS=ATLcos2\u001eA;(A10)\nandATL\u0015ASL\u00150. With these quantities the spin\ntransport magnitudes assuming an sTtip magnetization\ndirection can be written as\njTTLj/jPTj(1 + 1=ATL);\njTSLj/jPSj(1 +ASL);\njTTkj/jPSjp\n1\u0000ASL=ATL;\njTSkj/jPTjp\n1\u0000ASL=ATL;\njT?j/jPSPTjp\n1\u0000ASL=ATL;\njTTj/jPSjq\n1 +P2\nTp\n1\u0000ASL=ATL;\njTSj/jPTjq\n1 +P2\nSp\n1\u0000ASL=ATL:(A11)\nThese equations establish a direct connection between\nthe magnitudes of the spin transport components and\nthe LSC asymmetries. Using Eq. (20), the spin polar-\nizations can be expressed by solely using the magnetic\nasymmetries AI,ATLandASLas\nP2\nT=jAIj\u0001ATL;\nP2\nS=jAIj=ASL;\nP2\nTP2\nS=A2\nIATL=ASL;\nP2\nT=P2\nS=ATLASL: (A12)\nThis way, the spin transport magnitudes assuming an\nsTtip magnetization direction can be written using the15\nmagnetic asymmetries:\njTTLj/p\njAIj=ATL(1 +ATL); (A13)\njTSLj/p\njAIj=ASL(1 +ASL);\njTTkj/p\njAIj=ASL\u0000jAIj=ATL;\njTSkj/p\njAIjATL\u0000jAIjASL;\njT?j/jAIjp\nATL=ASL\u00001;\njTTj/q\njAIj=ASL\u0000jAIj=ATL+A2\nIATL=ASL\u0000A2\nI;\njTSj/q\njAIjATL\u0000jAIjASL+A2\nIATL=ASL\u0000A2\nI:\nThe connections between the ratios of the magnitudes of\nthe STT vector components (or the angles \u000bTand\u000bS)and the magnetic asymmetries read as follows:\njT?j\njTTj=jsin\u000bTj=s\njAIjATL\n1 +jAIjATL;\njTTkj\njTTj= cos\u000bT=1p\n1 +jAIjATL;\njT?j\njTSj=jsin\u000bSj=s\njAIj=ASL\n1 +jAIj=ASL;\njTSkj\njTSj= cos\u000bS=1p\n1 +jAIj=ASL: (A14)\nOther important relations can be written that might\nprove to be useful in the evaluation of future STT ex-\nperiments in magnetic STM junctions:\njcos\u001eAj=jAIj\njTTLj\u0001jTSLj(1 + 1=ATL)(1 +ASL)\n=r\nASL\nATL=jPTj\njPSj1\nATL=jPSj\njPTjASL=jAIj\njPSj\u0001jPTj;\njsin\u001eAj=jTTkj\u0001jTSkj\njT?j=r\n1\u0000ASL\nATL=s\n1\u0000A2\nI\nP2\nSP2\nT;\njTTj\njsin\u000bSj=jTSj\njsin\u000bTj=jsin\u001eAj\njcos\u000bSj\u0001jcos\u000bTj: (A15)\n[1] D.C. Ralph, M.D. Stiles, J. Magn. Magn. Mater. 320,\n1190-1216 (2008).\n[2] J.C. Slonczewski, J. Magn. Magn. Mater. 159, L1-L7\n(1996).\n[3] M. Wilczy\u0013 nski, J. Barna\u0013 s, R. \u0013Swirkowicz, Phys. Rev. B\n77, 054434 (2008).\n[4] A. Manchon, N. Ryzhanova, A. Vedyayev, M. Chshiev,\nB. Dieny, J. Phys. Condens. Matter 20, 145208 (2008).\n[5] J.C. Slonczewski, Phys. Rev. B 71, 024411 (2005).\n[6] I. Theodonis, N. Kioussis, A. Kalitsov, M. Chshiev,\nW.H. Butler, Phys. Rev. Lett. 97, 237205 (2006).\n[7] A. Kalitsov, M. Chshiev, I. Theodonis, N. Kioussis,\nW.H. Butler, Phys. Rev. B 79, 174416 (2009).\n[8] A. Kalitsov, W. Silvestre, M. Chshiev, J.P. Velev, Phys.\nRev. B 88, 104430 (2013).\n[9] C. Ortiz Pauyac, A. Kalitsov, A. Manchon, M. Chshiev,\nPhys. Rev. B 90, 235417 (2014).\n[10] A. Useinov, M. Chshiev, A. Manchon, Phys. Rev. B 91,\n064412 (2015).\n[11] J. Xiao, G.E.W. Bauer, A. Brataas, Phys. Rev. B 77,\n224419 (2008).\n[12] C. Heiliger, M.D. Stiles, Phys. Rev. Lett. 100, 186805\n(2008).\n[13] X. Jia, K. Xia, Y. Ke, H. Guo, Phys. Rev. B 84, 014401\n(2011).\n[14] J.C. Sankey, Y.-T. Cui, J.Z. Sun, J.C. Slonczewski,\nR.A. Burman, D.C. Ralph, Nat. Phys. 4, 67-71 (2008).\n[15] A.M. Ukpong, J. Phys. D: Appl. Phys. 51, 095302 (2018).\n[16] A.O. Leonov, T.L. Monchesky, N. Romming, A. Kubet-zka, A.N. Bogdanov, R. Wiesendanger, New J. Phys. 18,\n065003 (2016).\n[17] I. Dzyaloshinsky, J. Phys. Chem. Solids 4, 241-255\n(1958).\n[18] T. Moriya, Phys. Rev. Lett. 4, 228-230 (1960).\n[19] N. Romming, C. Hanneken, M. Menzel, J.E. Bickel,\nB. Wolter, K. von Bergmann, A. Kubetzka, R. Wiesen-\ndanger, Science 341, 636-639 (2013).\n[20] X. Z. Yu, Y. Onose, N. Kanazawa, J. H. Park, J. H. Han,\nY. Matsui, N. Nagaosa, Y. Tokura, Nature 465, 901-904\n(2010).\n[21] Y. Tokunaga, X. Z. Yu, J. S. White, H. M. R\u001cnnow,\nD. Morikawa, Y. Taguchi, Y. Tokura, Nat. Commun. 6,\n7638 (2015).\n[22] C. Moreau-Luchaire, C. Mouta\fs, N. Reyren, J. Sampaio,\nC.A.F. Vaz, N. Van Horne, K. Bouzehouane, K. Garcia,\nC. Deranlot, P. Warnicke, P. Wohlh uter, J.-M. George,\nM. Weigand, J. Raabe, V. Cros, A. Fert, Nat. Nanotech-\nnol.11, 444-448 (2016).\n[23] S. Woo, K. Litzius, B. Kr uger, M.-Y. Im, L. Caretta,\nK. Richter, M. Mann, A. Krone, R.M. Reeve,\nM. Weigand, P. Agrawal, I. Lemesh, M.-A. Mawass,\nP. Fischer, M. Kl aui, G.S.D. Beach, Nat. Mater. 15, 501-\n506 (2016).\n[24] T. Okubo, S. Chung, H. Kawamura, Phys. Rev. Lett.\n108, 017206 (2012).\n[25] A.O. Leonov, M. Mostovoy, Nat. Commun. 6, 8275\n(2015).\n[26] S.-Z. Lin, S. Hayami, Phys. Rev. B 93, 064430 (2016).16\n[27] L. R\u0013 ozsa, K. Palot\u0013 as, A. De\u0013 ak, E. Simon, R. Yanes,\nL. Udvardi, L. Szunyogh, U. Nowak, Phys. Rev. B 95,\n094423 (2017).\n[28] A. Fert, V. Cros, J. Sampaio, Nat. Nanotechnol. 8, 152-\n156 (2013).\n[29] N. Nagaosa, Y. Tokura, Nat. Nanotechnol. 8, 899-911\n(2013).\n[30] X. Zhang, M. Ezawa, Y. Zhou, Sci. Rep. 5, 9400 (2015).\n[31] R. Wiesendanger, Nat. Rev. Mater. 1, 16044 (2016).\n[32] A. Fert, N. Reyren, V. Cros, Nat. Rev. Mater. 2, 17031\n(2017).\n[33] O. Boulle, J. Vogel, H. Yang, S. Pizzini, D. de Souza\nChaves, A. Locatelli, T.O. Mente\u0018 s, A. Sala, L.D. Buda-\nPrejbeanu, O. Klein, M. Belmegueani, Y. Roussign\u0013 e,\nA. Stashkevich, S.M. Ch\u0013 erif, L. Aballe, M. Foerster,\nM. Chshiev, S. Au\u000bret, I.M. Miron, G. Gaudin, Nat.\nNanotechnol. 11, 449-454 (2016).\n[34] J. Hagemeister, N. Romming, K. von Bergmann,\nE.Y. Vedmedenko, R. Wiesendanger, Nat. Commun. 6,\n8455 (2015).\n[35] L. R\u0013 ozsa, E. Simon, K. Palot\u0013 as, L. Udvardi, L. Szunyogh,\nPhys. Rev. B 93, 024417 (2016).\n[36] I.S. Lobanov, H. J\u0013 onsson, V.M. Uzdin, Phys. Rev. B 94,\n174418 (2016).\n[37] P.F. Bessarab, V.M. Uzdin, H. J\u0013 onsson, Comput. Phys.\nCommun. 196, 335-347 (2015).\n[38] P.F. Bessarab, Phys. Rev. B 95, 136401 (2017).\n[39] S. Rohart, J. Miltat, A. Thiaville, Phys. Rev. B 95,\n136402 (2017).\n[40] D. Stosic, J. Mulkers, B. Van Waeyenberge, T.B. Luder-\nmir, M.V. Milo\u0014 sevi\u0013 c, Phys. Rev. B 95, 214418 (2017).\n[41] K. von Bergmann, A. Kubetzka, O. Pietzsch, R. Wiesen-\ndanger, J. Phys. Condens. Matter 26, 394002 (2014).\n[42] N. Romming, A. Kubetzka, C. Hanneken, K. von\nBergmann, R. Wiesendanger, Phys. Rev. Lett. 114,\n177203 (2015).\n[43] K. Palot\u0013 as, L. R\u0013 ozsa, E. Simon, L. Udvardi, L. Szunyogh,\nPhys. Rev. B 96, 024410 (2017).\n[44] S. Heinze, K. von Bergmann, M. Menzel, J. Brede, A. Ku-\nbetzka, R. Wiesendanger, G. Bihlmayer, S. Bl ugel, Nat.\nPhys. 7, 713-718 (2011).\n[45] B. Dup\u0013 e, M. Ho\u000bmann, C. Paillard, S. Heinze, Nat. Com-\nmun. 5, 4030 (2014).\n[46] E. Simon, K. Palot\u0013 as, L. R\u0013 ozsa, L. Udvardi, L. Szunyogh,\nPhys. Rev. B 90, 094410 (2014).\n[47] L. R\u0013 ozsa, A. De\u0013 ak, E. Simon, R. Yanes, L. Udvardi,\nL. Szunyogh, U. Nowak, Phys. Rev. Lett. 117, 157205\n(2016).\n[48] R. Wiesendanger, Rev. Mod. Phys. 81, 1495-1550 (2009).\n[49] C. Hanneken, A. Kubetzka, K. von Bergmann,\nR. Wiesendanger, New J. Phys. 18, 055009 (2016).[50] D. Stosic, T.B. Ludermir, M.V. Milo\u0014 sevi\u0013 c, Phys. Rev. B\n96, 214403 (2017).\n[51] S. Krause, G. Herzog, A. Schlenho\u000b, A. Sonntag,\nR. Wiesendanger, Phys. Rev. Lett. 107, 186601 (2011).\n[52] P.-J. Hsu, A. Kubetzka, A. Finco, N. Romming, K. von\nBergmann, R. Wiesendanger, Nat. Nanotechnol. 12, 123-\n126 (2017).\n[53] A. Manchon, S. Zhang, Spin torque e\u000bects in magnetic\nsystems: Theory, Chapter 8 in Handbook of Spin Trans-\nport and Magnetism, pp. 157-176, CRC Press (2016).\n[54] W.A. Hofer, A.S. Foster, A.L. Shluger, Rev. Mod. Phys.\n75, 1287-1331 (2003).\n[55] K. Palot\u0013 as, G. M\u0013 andi, L. Szunyogh, Phys. Rev. B 94,\n064434 (2016).\n[56] K. Palot\u0013 as, G. M\u0013 andi, W.A. Hofer, Front. Phys. 9, 711-\n747 (2014).\n[57] K. Palot\u0013 as, W.A. Hofer, L. Szunyogh, Phys. Rev. B 84,\n174428 (2011).\n[58] K. Palot\u0013 as, G. M\u0013 andi, L. Szunyogh, Phys. Rev. B 86,\n235415 (2012).\n[59] K. Palot\u0013 as, Phys. Rev. B 87, 024417 (2013).\n[60] G. M\u0013 andi, K. Palot\u0013 as, Appl. Surf. Sci. 304, 65-72 (2014).\n[61] P. Nita, K. Palot\u0013 as, M. Ja lochowski, M. Krawiec, Phys.\nRev. B 89, 165426 (2014).\n[62] K. Palot\u0013 as, W.A. Hofer, L. Szunyogh, Phys. Rev. B 83,\n214410 (2011).\n[63] K. Palot\u0013 as, W.A. Hofer, L. Szunyogh, Phys. Rev. B 85,\n205427 (2012).\n[64] G. M\u0013 andi, N. Nagy, K. Palot\u0013 as, J. Phys.: Condens. Mat-\nter25, 445009 (2013).\n[65] G. M\u0013 andi, G. Teobaldi, K. Palot\u0013 as, J. Phys.: Condens.\nMatter 26, 485007 (2014).\n[66] G. M\u0013 andi, G. Teobaldi, K. Palot\u0013 as, Prog. Surf. Sci. 90,\n223-238 (2015).\n[67] D. Wortmann, S. Heinze, P. Kurz, G. Bihlmayer,\nS. Bl ugel, Phys. Rev. Lett. 86, 4132-4135 (2001).\n[68] S. Heinze, Appl. Phys. A 85, 407-414 (2006).\n[69] W.A. Hofer, A.J. Fisher, J. Magn. Magn. Mater. 267,\n139-151 (2003).\n[70] J. Terso\u000b, D.R. Hamann, Phys. Rev. Lett. 50, 1998-2001\n(1983).\n[71] J. Terso\u000b, D.R. Hamann, Phys. Rev. B 31, 805-813\n(1985).\n[72] W. Wulfhekel, C.L. Gao, J. Phys.: Condens. Matter 22,\n084021 (2010).\n[73] D.M. Crum, M. Bouhassoune, J. Bouaziz, B. Schwe\ring-\nhaus, S. Bl ugel, S. Lounis, Nat. Commun. 6, 8541 (2015).\n[74] B. Dup\u0013 e, C.N. Kruse, T. Dornheim, S. Heinze, New J.\nPhys. 18, 055015 (2016).\n[75] J. Hagemeister, E.Y. Vedmedenko, R. Wiesendanger,\nPhys. Rev. B 94, 104434 (2016)."    },    {        "title": "1912.02337v2.Spin_pumping_into_a_spin_glass_material.pdf",        "content": "Spin pumping into a spin glass material\nYusei Fujimoto,1Masanori Ichioka,2, 1and Hiroto Adachi2, 1\n1Department of Physics, Okayama University, Okayama 700-8530, Japan\n2Research Institute for Interdisciplinary Science, Okayama University, Okayama 700-8530, Japan\n(Dated: March 24, 2022)\nSpin pumping is a recently established means for generating a pure spin current, whereby spins\nare pumped from a magnet into the adjacent target material under the ferromagnetic resonance\ncondition. We theoretically investigate the spin pumping from an insulating ferromagnet into spin\nglass materials. Combining a dynamic theory of spin glasses with the linear-response formulation\nof the spin pumping, we calculate temperature dependence of the spin pumping near the spin glass\ntransition. The analysis predicts that a characteristic peak appears in the spin pumping signal,\nre\recting that the spin \ructuations slow down upon the onset of spin freezing.\nI. INTRODUCTION\nSpin current is a \row of spin angular momentum [1].\nOver the last two decades, great progress has been made\nin generating, manipulating, and detecting the spin cur-\nrent [2, 3]. With regard to the spin current generation,\nas nicely reviewed in Ref. [4] the spin pumping is now\nestablished as a charge-free and versatile means [5{8]. In\nthis method a pure spin current, which is unaccompa-\nnied by a charge current, is pumped from a ferromagnet\ninto the adjacent spin sink material by a stimulus of mi-\ncrowaves satisfying the ferromagnetic resonance (FMR)\ncondition. Thanks to the advent of the spin pumping\ntechnique, the spin current physics has so far been inves-\ntigated in a variety of spin sink materials, ranging from\nnonmagnetic metals [9{13], semiconductors [14{16], mag-\nnetic metals [17{19], insulators [20], to more exotic sys-\ntems such as graphene [21{23], transition metal dichalco-\ngenides [24], organic materials [25, 26], and strongly spin-\norbit coupled materials [27, 28].\nRecently, the playground of the spin current physics\nhas been extended to disordered magnets or the so-called\nspin glass (SG) materials [29, 30]. The SGs are charac-\nterized by a freezing of random spins [31], and its nature\nhas long been studied both experimentally and theoreti-\ncally [32]. However, despite its long history of research,\nthe interplay of spin current and the SG ordering has\nnot yet been well examined. Thus, it is quite natural to\nask what happens if we inject a pure spin current into a\nSG material by the spin pumping. Experimentally, the\nspin pumping into a SG material was reported in 2011\nusing a Ag 90Mn10/Ni81Fe19bilayer [33]. To the best of\nour knowledge, however, no theoretical work on the spin\npumping into SG materials can be found in the literature.\nTherefore, developing a theory of spin pumping into the\nSG material is highly desirable.\nIn this paper, we theoretically investigate the spin\npumping into SG materials. Although a metallic magnet\nNi81Fe19was used as the spin injecting magnet in the\nprevious experiment [33], we consider here an insulat-\ning ferromagnet such as yttrium iron garnet for the spin\ninjector, since it makes the spin pumping signal more vis-\nible. Our strategy to calculate the spin pumping into SGmaterials is as follows. First, we use a linear-response\napproach to the spin pumping [34, 35]. The notion de-\nrived from the linear response approach, that the spin\npumping is intimately related to the dynamic spin sus-\nceptibility of the spin sink layer, has successfully been\napplied to the spin pumping into a ferromagnet [36], an-\ntiferromagnets [37, 38], and recently it was also applied\nto the spin pumping into superconductors [39{41]. Thus,\nwe relate the spin pumping signal to the dynamic spin\nsusceptibility of the SG layer. Next, we calculate the sus-\nceptibility of the SG layer by employing a dynamic theory\nof SGs [42, 43]. Not only that this theory is known to\nbe an alternative formulation of the static replica the-\nory [44{47], but also that the dynamic theory is more\nsuitable to discuss the dynamic quantity such as the dy-\nnamic spin susceptibility [48].\nIn the literature, the dynamic spin susceptibility near\nthe SG transition was calculated [48], but the result\nwas limited to the Ising case and to an extremely low-\nfrequency regime less than 10 KHz, which is out of the\nFMR condition. In the present paper, we extend the sus-\nceptibility calculation of Ref. [48] to the Heisenberg case\nand GHz frequency regime that is relevant to spin pump-\ning experiments, and calculate temperature dependence\nof the spin pumping into SG materials. With this, we\nshow that a characteristic peak structure appears near\nthe SG transition, which is a consequence of the slowing\ndown of spin \ructuations that is concomitant with the\nspin freezing of the system.\nThe plan of this paper is as follows. In the next section,\nwe introduce our microscopic model, and relate the spin\npumping with the dynamic spin susceptibility of impu-\nrity spins. In Sec. III, on the basis of the dynamic theory\nof SGs, we explain how to calculate the dynamic spin sus-\nceptibility of impurity spins. In Sec. IV, the spin pump-\ning signal into a SG material is calculated as a function\nof temperature. Finally, in Sec. V we discuss and sum-\nmarize our results.arXiv:1912.02337v2  [cond-mat.mes-hall]  13 May 20202\nFIG. 1. Schematics of the system considered in this paper,\nwhere the bilayer is composed of a SG material and a fer-\nromagnetic insulator (FI). Here, \u001bandSare, respectively,\nthe conduction-electron spin and the impurity spin in the SG\nlayer, and \nis the localized spin in the FI layer. A spin cur-\nrent with a helicity opposite to \n\rows from the FI layer to\nthe SG layer.\nII. MODEL\nWe consider a bilayer composed of a ferromagnetic in-\nsulator (FI) and a SG material, as shown in Fig. 1. More\nconcretely, we may think of yttrium iron garnet (YIG)\nfor the FI layer and Mn-doped Cu (Cu:Mn) for the SG\nlayer. We assume that a static magnetic \feld H0=H0^ z\nis applied to the FI/SG bilayer in the lateral direction,\nand that the anisotropy \feld is much smaller than H0\nsuch that it can be discarded.\nWe start from the following Hamiltonian:\nH=HSG+HFI\u0000SG; (1)\nwhere the \frst term,\nHSG=X\np\u0018pcy\npcp+JeSX\nra\u001b(ra)\u0001S(ra); (2)\ndescribes the SG layer [49, 50]. Here, the \frst term on the\nright-hand side describes the conduction electron kinetic\nenergy, and the second term the coupling between the\nconduction electron spin and magnetic impurity at an\nimpurity position ra, whereJeSis thesd-type exchange\ncoupling. Here, cy\np= (cy\np;\";cy\np;#) is the electron creation\noperator for spin projection \"and#,Sis an impurity\nspin,\u001b(r) =cy(r)^\u001bc(r) is the spin density operator with\n^\u001bbeing the Pauli matrices, and c(r) =N\u00001=2\nSGP\npcpeip\u0001r\nwithNSGbeing the number of lattice sites at the SG\nlayer.\nThe second term of Eq. (1),\nHFI\u0000SG=JintX\nrint\u001b(rint)\u0001\n(rint); (3)\ndescribes the interaction between the FI and SG layers.\nHere,Jintis the interfacial sdcoupling between the con-\nduction electron spins in the SG and the localized spins\nin the FI, where rintis a position at the FI/SG interface.\nSG\nFIFIG. 2. Diagrammatic representation of the magnon self-\nenergy giving the spin pumping signal. Here, \u001f(\u001b)is\nthe dynamic spin susceptibility of conduction-electron spins,\nwhereas\u001f(S)is that of impurity spins.\nIn order to investigate the spin pumping in the present\nsystem, we use the linear-response formulation of the spin\npumping [34, 35]. We consider the situation where an\nexternal microwave with the angular frequency !acis ap-\nplied to the FI/SG bilayer that drives the FMR of the\nFI side. The linear-response formulation uses the follow-\ning magnon language. In the absence of the adjacent\nSG layer, the uniform-mode (Kittel mode) magnon has\nan intrinsic damping rate \u000b0!ac, where\u000b0is the intrin-\nsic Gilbert damping constant. In the presence of the SG\nlayer, since the spin-relaxation rate due to the SG layer is\nadditive and hence an additional spin dissipation channel\nopens, there arises an additional magnon damping rate.\nTherefore, the total Gilbert damping constant \u000bfor the\nbilayer is given by\n\u000b=\u000b0+\u000e\u000b; (4)\nwhere\u000e\u000bis the additional Gilbert damping constant.\nThe relationship between this additional Gilbert damp-\ning constant and the spin current Ispumped into the SG\nlayer withz-axis polarization is given by [34]\nIs=\u000e\u000b\r~\nMsV!ac(\rhac)2\n(\rH0\u0000!ac)2+ (\u000b0!ac)2; (5)\nwhere\ris the gyromagnetic ratio, Msthe saturation\nmagnetization, Vthe volume of the magnet, and hacis\nthe amplitude of the external microwave.\nAccording to the linear-response formulation [34, 35],\nthe additional magnon damping rate can be calculated\nfrom the corresponding magnon self-energy \u0006( !) whose\nprocess involves the spin transfer across the interface\n(Fig. 2). In the present situation, up to the lowest or-\nder with respect to Jint, the self-energy is given by\n\u0006(!) =\u0000J2\nintNint\n~2NSGNFIX\nq\u001f(\u001b)\nq(!)JeS\u001f(S)\nq(!)JeS\u001f(\u001b)\nq(!);\n(6)\nwhereNintis the number of the localized spins \nat the\nFI/SG interface, and NFIis the number of lattice sites3\nin the FI layer. In the above equation, \u001f(\u001b)\nq(!) is the\nFourier transform of the retarded susceptibility of the\nconduction-electron spin \u001b, i.e.,\u001f(\u001b)\nq(t\u0000t0) = i\u0002(t\u0000\nt0)h[\u001b\u0000\nq(t);\u001b+\n\u0000q(t0)]i, where \u0002(t) is the step function and\nwe de\fned O\u0006=Ox\u0006iOyfor a vector operator O. By\ncontrast,\u001f(S)\nq(!) is the Fourier transform of the retarded\nsusceptibility of the impurity spin S, i.e.,\u001f(S)\nq(t\u0000t0) =\ni\u0002(t\u0000t0)h[S\u0000\nq(t);S+\n\u0000q(t0)]i,\nUsing the relation \u000e\u000b=\u0000!\u00001\nacIm\u0006(!ac) [34], the addi-\ntional Gilbert damping constant \u000e\u000bis expressed as\n\u000e\u000b\u0019J2\nintNint\n~2NFI\u0010\n\u001f(\u001b)\n0JeS\u001121\n!acIm\u001f(S)\nloc(!ac); (7)\nwhere we introduced a shorthand notation \u001f(\u001b)\n0=\n\u001f(\u001b)\nq=0(!= 0), and\u001f(S)\nloc(!) =N\u00001\nSGP\nq\u001f(S)\nq(!) is the local\nsusceptibility of impurity spins. In obtaining the above\nresult, we made use of the fact that \u001f(S)\nloc(!) is de\fned in\na smallqregionq.2\u0019=bwherebis the average distance\nof two magnetic impurities. In this small qregion, the\nconduction-electron spin susceptibility is approximated\nby the uniform and static component \u001f(\u001b)\n0, where this\nquantity is pure real.\nEquation (7) means that the additional Gilbert damp-\ning constant \u000e\u000bdue to the spin pumping is proportional\nto the imaginary part of the dynamic spin susceptibil-\nity\u001f(S)\nloc(!). In this expression, the strongest temper-\nature dependence upon the SG transition results from\nthe imaginary part of the dynamic spin susceptibility,\nIm[\u001f(S)\nloc(!)]. This means that, as long as the temperature\ndependence is concerned, the part other than Im[ \u001f(S)\nloc(!)]\ncan be regarded as being temperature independent, and\nthe temperature dependence is dominated by that of\nIm[\u001f(S)\nloc(!)]. We adopt this approximation in the nu-\nmerical calculation in Sec. IV.\nThe quantity \u001f(S)\nloc(!) is a correlation function between\ntwo impurity spins, and hence it can be evaluated using\nour knowledge on SGs. In the next section, we evaluate\n\u001f(S)\nloc(!) using a dynamic theory of SGs [42, 43].\nIII. DYNAMIC SPIN SUSCEPTIBILITY OF\nIMPURITY SPINS\nIn this section, by employing a dynamic theory of\nSGs [42, 43], we sketch our procedure for calculating\nthe dynamic spin susceptibility \u001f(S)\nloc(!) appearing in the\nspin pumping signal [Eq. (7)]. We emphasize that we\nuse the dynamic model of the Heisenberg spin glasses\ndeveloped by Sompolinsky and Zippelius [43], which is\nconstructed on top of a similar dynamic theory of Ising\nspin glasses [42].\nWe \frst integrate out the conduction-electron degrees\nof freedom in the Hamiltonian for the SG layer HSG.Then,HSGis transformed into the following form [49, 50]:\nHSG=1\n2X\ni6=jJijS(ri)\u0001S(rj); (8)\nwhereJijis nominally the RKKY interaction of the form\nJij=JeScos(2kFrij)=r3\nijwithrij=jri\u0000rjj. However,\nfollowing the standard approach to the SG problem [32],\nwe regards Jijas Gaussian random variables with zero\nmean and variance [ J2\nij]av=J2=NS, where [\u0001\u0001\u0001]avmeans\nthe random average over the distribution of Jij, andNS\nis the number of impurity spins.\nHamiltonianHSGin Eq. (8) is the same as the vec-\ntor spin version [51] of the Sherrington-Kirkpatrick (SK)\nmodel [45], so that we employ the established dynamical\napproach. Following Sompolinsky and Zippelius [43], we\n\frst replaceHSGwith its soft-spin version:\n\feHSG=1\n2X\n\u000bX\nij(r0\u000eij\u0000\fJij)S\u000b\niS\u000b\nj\u0000\fX\nih\u000b\ni\u0001S\u000b\ni;(9)\nwhere\f= 1=kBTis the inverse temperature, and each\ncomponent of the soft spin varies \u00001<S\u000b\ni<1, where\nthe Greek superscript \u000b=x;y;z speci\fes the direction\nin spin space. Here, we consider a paramagnet/SG tran-\nsition by ignoring any tendency to ferromagnetic order,\nsuch thatr0in Eq. (9) is chosen to be a positive constant.\nNext, we introduce the Langevin dynamics\n\u0000\u00001\n0@tS\u000b\ni=\u0000@(\feHSG)\n@S\u000b\ni+\u0018\u000b\ni(t); (10)\nwhere \u0000 0is the bare relaxation rate. In the above equa-\ntion,\u0018\u000b\ni(t) is a thermal noise represented by a Gaussian\nrandom variable with zero mean and variance\nh\u0018\u000b\ni(t)\u0018\u000b0\nj(t0)i\u0018= 2\u0000\u00001\n0\u000eij\u000e\u000b;\u000b0\u000e(t\u0000t0); (11)\nwhereh\u0001\u0001\u0001i\u0018means the average over \u0018\u000b\ni. In the following\ncalculation, the response function\nG\u000b\u000b0\nij(t\u0000t0) =\"\n@hS\u000b\ni(t)i\u0018\n@h\u000b0\nj(t0)#\nav; (12)\nplays an important role. This is because the spin suscep-\ntibility can be calculated by the relation\n\u001f\u000b\u000b0\nij(!) =\fG\u000b\u000b0\nij(!); (13)\nwhereG\u000b\u000b0\nij(!) is the Fourier transform of G\u000b\u000b0\nij(t\u0000t0).\nBesides, we need to de\fne the correlation function\nC\u000b\u000b0\nij(t\u0000t0) =h\nhS\u000b\ni(t)S\u000b0\nj(t0)i\u0018i\nav: (14)\nTo proceed further, we precisely follow the procedure\nof Refs. [42, 43, 52], which involves a lots of technical\nalgebra. Since reviewing the details of Refs. [42, 43, 52]\nis beyond our scope, we leave it to the original paper4\nand a famous textbook [53], and we brie\ry sketch the\nderivation of the self-consistent dynamical equation in\nthe mean-\feld limit. First, we rewrite Eq. (10) in terms of\na generating functional, and introduce an auxiliary \feld\nbS\u000b\nias was done by Martin, Siggia, and Rose [54]. Next,\nthis generating functional is averaged over Jijwithout\nusing replicas [55], which generates temporally nonlocal\nquartic interactions among S\u000b\niandbS\u000b0\ni. Then, we in-\ntroduce new auxiliary \felds Q\u000b\u000b0\n1;Q\u000b\u000b0\n2;Q\u000b\u000b0\n3andQ\u000b\u000b0\n4\nto decouple the quartic terms, and we evaluate the func-\ntional integral by using the saddle-point approximation.\nFollowing the above procedure, we obtain the new\nequation of motion for S\u000b\nicontaining the local self-\ninteraction, which in the frequency space is written as\n\u0000i!\n\u00000S\u000b(!) =X\n\u000b0h\u0000\n\u0000r0+\fh\u000b(!)\u0001\n\u000e\u000b\u000b0\n+(\fJ)2G\u000b\u000b0(!)i\nS\u000b0(!) +\u001e\u000b(!);(15)\nwhereG\u000b\u000b0(!) is the local response function G\u000b\u000b0\nii(!).\nHere and hereafter, the site index iis discarded. In the\nabove equation, \u001e\u000bis a new noise \feld satisfying\nh\u001e\u000b(!)\u001e\u000b0(!0)i\u001e= 2\u0019\u000e(!+!0)\u00142\u000e\u000b\u000b0\n\u00000+ (\fJ)2C\u000b\u000b0(!)\u0015\n;\n(16)\nwhereC\u000b\u000b0(!) is the Fourier transform of the local cor-\nrelation function C\u000b\u000b0\nii(t\u0000t0), andh\u0001\u0001\u0001i\u001eis the average\nover\u001e\u000b(!).\nIn the approach by Sompolinsky and Zippelius [43] the\nweak external-\feld limit was considered, such that after\nthe random average , physical quantities are diagonal in\nspin space. Therefore, the local correlation function is\nseparated as C(t) =q+\u0001C(t), whereC\u000b\u000b0(t) =C(t)\u000e\u000b\u000b0\nand q=C(t)jt!1. In parallel with this separation, the\nnoise \feld\u001e\u000bis divided into two parts,\n\u001e\u000b(!) =f\u000b(!) +u\u000b(!); (17)\nwhere the \frst term, f\u000b(!), satis\fes\nhf\u000b(!)f\u000b0(!0)i= 2\u0019\u000e(!+!0)\u000e\u000b\u000b0\u00142\n\u00000+ (\fJ)2\u0001C(!)\u0015\n(18)\nwithh\u0001\u0001\u0001i being the usual thermal average over f\u000b(!),\nwhereas the second term, u\u000b(!), satis\fes\n[u\u000b(!)u\u000b0(!0)]u= (2\u0019)2\u000e(!+!0)\u000e(!)\u000e\u000b\u000b0(\fJ)2q;(19)\nwhere [\u0001\u0001\u0001]uis the average over u\u000b(!).\nFrom Eqs. (18) and (19), we \fnd that f\u000b(!) is a usual\nthermal \ructuation, whereas u\u000b(!) represents a frozen\nrandom \feld that breaks the ergodicity. Since u\u000bacts as\na static random \feld, Sompolinsky and Zippelius intro-\nduced the \\unaveraged\" response function over u:\ng\u000b\u000b0(!;u) =@hS\u000b(!)i\n@h\u000b0(!); (20)whereg\u000b\u000b0(!;u) is related to G\u000b\u000b0(!) through\nG\u000b\u000b0(!) = [g\u000b\u000b0(!;u)]u: (21)\nNote that the average over uis separated as\n[\u0001\u0001\u0001]u= [[\u0001\u0001\u0001]^u]u; (22)\nwhere [\u0001\u0001\u0001]^umeans the angular average over a unit vector\n^u=u=u, and from Eq. (19) the average of a quantity\nQ(u) overuis given by\n[Q(u)]u=Z1\n04\u0019u2du\n[2\u0019(\fJ)2q]3=2e\u0000u2=[2(\fJ)2q]Q(u):(23)\nNow, we discuss the Dyson equation for g\u000b\u000b0(!;u). In\ndoing so, we introduce the matrix notation \u0014 g(!;u), where\nthe matrix element of \u0014 g(!;u) equalsg\u000b\u000b0(!;u), i.e.,\n\u0014g(!;u) =\u0010\ng\u000b\u000b0(!;u)\u0011\n: (24)\nWith this matrix notation the Dyson equation for\ng(!;u), which results from Eq. (15) with static random\n\feldu, is given by\n\u0014g(!;u)\u00001=\u0014G(0)(!)\u00001\u0000\u0014\u0006(!;u); (25)\nwhere the bare propagator is given by \u0014G(0)(!)\u00001= [r0\u0000\ni!=\u00000\u0000(\fJ)2G(!)]\u00141, and \u0014\u0006(!;u) is the self-energy com-\ning from the frozen u-\feld. Since we are interested in the\ndynamic behavior of \u0014G(!), we solve the Dyson equation\nby perturbation with respect to \u0001 \u0014G(!) =\u0014G(!)\u0000\u0014G(0).\nWe de\fne \u0014\u0011(!;u) = i!=\u00000+ (\fJ)2\u0001\u0014G(!)\u0000\u0001\u0014\u0006(!;u),\nwhere \u0001 \u0014\u0006(!;u) = \u0014\u0006(!;u)\u0000\u0014\u0006(0;u), and rewrite the\nDyson equation as \u0014 g(!;u)\u00001= \u0014g(0;u)\u00001\u0000\u0014\u0011(!;u). To\nproceed further, we disregard \u0001 \u0014\u0006(!;u) as it brings only\na small change [48]. This allows us to regard \u0014 \u0011(!;u)\nasu-independent, and the matrix \u0014 \u0011becomes diagonal in\nspin space, i.e., \u0014 \u0011(!) =\u0011(!)\u00141. Then, after expanding the\nDyson equation up to \u0014 \u00112and averaging the result over u,\nwe obtain a quadratic equation for \u0001 \u0014G(!) = \u0001G(!)\u00141:\n\u0001G(!)\u0000\n1\u0000(\fJ)2[\u0014g2\n0]u\u0001\n=\u0012\ni!\n\u00000+ (\fJ)2\u0001G(!)\u00132\n[\u0014g3\n0]u\n+i!\n\u00000[\u0014g2\n0]u; (26)\nwhere we introduced the shorthand notation [\u0014 gn\n0]u=\n[\u0014g(0;u)n]u. Note that [\u0014 gn\n0]uis diagnoal in spin space\nafter the average over u, such that it appears in Eq. (26)\nas a c-number. Equation (26) can be solved for \u0001 G(!).\nAfter using the relation in Eq. (13), we obtain\n\u001f(S)\nloc(!) =\u001f(S)\nloc(0)+eT=3\n2[\u0014g3\n0]u\u0012\nD\u0000r\nD2\u0000i4!\n\u00000[\u0014g2\n0]u[\u0014g3\n0]u\u0013\n;\n(27)\nwhereD= (eT=3)2\u0000[\u0014g2\n0]u\u00002i(!=\u00000)[\u0014g3\n0]u,eT=T=Tg.\nNote that, in contrast to the Ising spin glass [45], the5\n0.0 0.5 1.0 1.5 2.0\nT/Tg0.00.10.20.30.40.5(S)\nloc(0)\nFIG. 3. Static spin susceptibility \u001f(S)\nloc(0) as a function of\ntemperature calculated using Eq. (31).\nprenset Heisenberg spin glass has a transition tempera-\ntureTg=J=3kB[51].\nIn the following calculation, we take the mean-\feld\napproximation of the classical Heisenberg spins un-\nder a frozen u\feld. We use hS\u000bi=hSi^u\u000band\nhS\u000bS\u000b0i=\u000e\u000b\u000b0=3, wherehSi= coth(u)\u00001=uis the\nLangevin function. Then, recalling that [ hS\u000bS\u000b0i]u=\n[hSi2]u[^u\u000b^u\u000b0]^u= [hSi2]u\u000e\u000b\u000b0=3, the spin glass order pa-\nrameter qcan be calculated from\nq=1\n3[hSi2]u; (28)\nwhere the quantity [ hSi2s]uis given by\n[hSi2s]u=Z1\n04\u0019u2du\n[2\u0019(\fJ)2q]3=2e\u0000u2=[2(\fJ)2q]\n\u0002\u0012\ncoth(u)\u00001\nu\u00132s\n: (29)\nIn a similar manner, we use [\u0014 g0]u= (1\u00003q)=3, [\u0014g2\n0]u=\n(1\u00006q+ 3[hSi4]u)=9, and [\u0014g3\n0]u= (1\u00009q+ 9[hSi4]u\u0000\n3[hSi6]u)=27. At the SG transition, the Almeida-\nThouless condition [56] holds, which in the present nota-\ntion takes the form\n[\u0014g2\n0]u=\u0010\neT=3\u00112\n: (30)\nThen, following Sompolinsky and Zippelius [42] we as-\nsume that Eq. (30) holds not only at Tgbut also below\nTg, meaning that the SG phase is characterized by the\nmarginal instability condition.\nIV. RESULTS FOR SPIN PUMPING INTO SPIN\nGLASS MATERIALS\nIn this section, using the formalism developed in the\nprevious two sections, we calculate temperature depen-\ndence of the spin pumping signal. The key equation in\n0.0 0.5 1.0 1.5 2.0\nT/Tg024681012141618(T)/(T=2Tg)\n=0.01\n=0.02\n=0.05\n=0.10\n=1.00\nFIG. 4. Temperature dependence of the additional Gilbert\ndamping constant [Eq. (7)] in a SG/FI bilayer (Fig. 1), cal-\nculated for several values of e!ac=!ac=\u00000.\nthe present argument is Eq. (7), which relates the spin\npumping with the dynamic spin susceptibility of SG ma-\nterials.\nBefore presenting our results for the spin pumping that\nis intimately related to the dynamic spin susceptibility\n\u001f(S)\nloc(!), it is instructive to examine the static spin sus-\nceptibility\u001f(S)\nloc(0). This quantity can be calculated us-\ning [57]:\n\u001f(S)\nloc(0) =1\n3eT(1\u00003q); (31)\nwhere the additional factor 3 follows from the relation\n[g\u000b\u000b0]u=\u000e\u000b\u000b0(1\u00003q)=3. Figure 3 shows the static spin\nsusceptibility \u001f(S)\nloc(0) as a function of temperature, cal-\nculated from Eq. (31). The result reproduces the well-\nknown cusp structure at Tg[45, 51].\nNow we discuss the spin pumping into a SG material.\nIn discussing the spin pumping signal in the present case,\nthe important parameter is the ratio of the microwave\nangular frequency !acto the relaxation rate of localized\nspins \u0000 0, i.e.,e!ac\u0011!ac=\u00000. Since the magnitude of\n!acunder the FMR condition is of the order of 60 GHz,\nthe parameter e!acis determined by a material parameter\n\u00000. The spin relaxation time in a prototypical SG mate-\nrial Cu:Mn is reported to be of the order of picosecond\n(corresponding to \u0000 0\u0018103GHz) [58], and hence the\nparametere!acunder the FMR condition is estimated to\nbee!ac\u00180:1. Note that the previous calculation of the\ndynamic spin susceptibility [48] was done for very low\nfrequenciese!ac.10\u00004, which is far out of the FMR\ncondition.\nFigure 4 shows temperature dependence of the spin\npumping signal, calculated from Eqs. (7) and (27). First,\nwe see that a clear peak structure appears at Tg. Second,\nupon the increase of the parameter e!ac, the height of\nthe peak is reduced. This is because the peak structure\noriginates from the critical slowing down of spins that6\ndevelops on the verge of the spin freezing [53], so that\nthe e\u000bects of the slowing down are more prominent when\nthe paramagnetic state has a more rapid dynamics (i.e.,\nlarger \u0000 0) in comparison to the spin frozen state.\nAs shown in Fig. 4, the peak structure is visible for\na parameter region e!ac.0:1. Since the parameter e!ac\nfor a prototypical SG material Cu:Mn is estimated about\ne!ac\u00180:1 [58] as mentioned above, we expect that we can\nobserve a peak structure in the spin pumping signal near\nTg. Therefore, we propose a spin pumping experiment\nfor a Cu:Mn/YIG system in order to test our theoreti-\ncal prediction. Frequency ( !ac) dependence of the peak\nmentioned above, namely, the lower the frequency !ac,\nthe higher the peak, is a key to identify the predicted\nsignal.\nV. DISCUSSION AND CONCLUSION\nThe main result of this paper is the theoretical pre-\ndiction that the spin pumping into a SG material, whose\nsignal is proportional to the additional Gilbert damping\nconstant\u000e\u000b, can be enhanced near the SG transition.\nThe physics behind this enhancement is explained in the\nfollowing way. First, a SG material exhibits the critical\nslowing down upon the spin freezing [53], meaning that\nthe spin relaxation rate of the SG material is reduced.\nThen, recalling that the spin pumping represents an ad-\nditional damping of magnons in the spin injecting mag-\nnet, this type of enhancement in the spin pumping signal\ncan be interpreted as a kind of the inverse of the motional\nnarrowing [59, 60] as discussed in Ref. [61] (see Sec. VI\ntherein). It means that a reduction of the spin relaxation\nrate in the spin sink material results in a broadening of\nthe magnon damping in the adjacent magnet, leading to\nthe enhancement of the spin pumping.\nThe spin pumping has an advantage that it can mea-\nsure the dynamic spin susceptibility of a thin\flm sample.\nSo far, this fact has been applied to the spin pumping into\nferromagnets [34, 36], antiferromagnets [37, 38], and su-\nperconductors [39{41]. Extending the same idea to SGs\nwithin the linear-response approach, we formulated the\nspin pumping into a SG material, and shown that the\nsignal is expressed by using the local spin susceptibility\nof the SG material [Eq. (7)]. Moreover, the spin pumping\ninto a SG material is predicted to exhibit a characteristicpeak around Tg(Fig. 4).\nThe height of the predicted peak in the spin pumping\nsignal is controlled by the ratio of the microwave angu-\nlar frequency to the relaxation rate of impurity spins,\ni.e.,e!ac=!ac=\u00000. That is, a smaller e!acis better for\nan experimental detection of the peak. Conversely, it\nmeans that if the parameter e!acis too large, the peak is\nnot visible. To test our theoretical prediction, we hope\na future spin pumping experiment using an insulating\nmagnet with, e.g., Cu:Mn/YIG structure, since use of an\ninsulating magnet instead of a metallic magnet makes the\nspin pumping signal more visible owing to the smallness\nof the intrinsic Gilbert damping term \u000b0.\nBefore conclusion, we brie\ry comment on the previous\nexperiment of the spin pumping into a SG material us-\ning Ag 90Mn10/Ni81Fe19[33]. In that experiment, the SG\ntransition temperature is estimated to be Tg= 25 K from\nthe cusp in the susceptibility data of a thick Ag90Mn10\n\flm. Note that this thick \flm is di\u000berent from the thin\n\flm used for the spin pumping experiment. In the spin\npumping experiment, a weak temperature dependence of\nthe signal was measured around Tg, but no pronounced\npeak expected from the theoretical calculation (Fig. 4)\ncan be seen. The reason could be either i) the important\nparametere!ac=!ac=\u00000is too large in Ag 90Mn10for the\nenhanced spin pumping to be detected (see Fig. 4), or ii)\nthe SG transition temperature of the thin\flm sample is\nmuch lower than 25 K, since the thin\flm may have lower\nTgthan the thick \flm used to determine Tg= 25 K.\nTo conclude, we have theoretically examined the spin\npumping into a SG material. We have shown that the\ntemperature dependence exhibits a characteristic peak\nnear the SG transition, whose height is controlled by the\ndimensionless angular frequency e!ac=!ac=\u00000. This is a\nconsequence of the critical slowing down of spin \ructua-\ntions upon the spin freezing. Since the spin pumping has\nan advantage of being able to measure the spin dynamics\nof a thin\flm sample [39], we hope that the present the-\nory stimulates further experiments of the spin pumping\ninto SG materials.\nACKNOWLEDGMENTS\nThis work was \fnancially supported by JSPS KAK-\nENHI Grant Number 19K05253.\n[1] S. Maekawa, H. Adachi, K. Uchida, J. Ieda, and E.\nSaitoh, Spin Current: Experimental and Theoretical As-\npects, J. Phys. Soc. Jpn 82, 102002 (2013).\n[2] I. \u0015Zuti\u0013 c, J. Fabian, and S. D. Sarma, Spintronics: Fun-\ndamentals and applications Rev. Mod. Phys. 76, 323\n(2004).\n[3]Spin Current , edited by S. Maekawa, S. O. Valenzuela, E.\nSaitoh, and T. Kimura (Oxford University Press, Oxford,\n2012).[4] I. \u0015Zuti\u0013 c and H. Dery, Taming spin currents, Nat. Mater.\n10, 647 (2011).\n[5] R. Urban, G. Woltersdorf, and B. Heinrich, Gilbert\nDamping in Single and Multilayer Ultrathin Films: Role\nof Interfaces in Nonlocal Spin Dynamics, Phys. Rev. Lett.\n87, 217204 (2001).\n[6] S. Mizukami, Y. Ando, and T. Miyazaki, E\u000bect of spin\ndi\u000busion on Gilbert damping for a very thin permalloy\nlayer in Cu/permalloy/Cu/Pt \flms, Phys. Rev. B 66,7\n104413 (2002).\n[7] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Spin\npumping and magnetization dynamics in metallic multi-\nlayers, Phys. Rev. B 66, 224403 (2002).\n[8] E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Con-\nversion of spin current into charge current at room tem-\nperature: Inverse spin-Hall e\u000bect, Appl. Phys. Lett. 88,\n182509 (2006).\n[9] O. Mosendz, J. E. Pearson, F. Y. Fradin, G. E. W. Bauer,\nS. D. Bader, and A. Ho\u000bmann, Quantifying Spin Hall\nAngles from Spin Pumping: Experiments and Theory,\nPhys. Rev. Lett. 104, 046601 (2010).\n[10] C. W. Sandweg, Y. Kajiwara, A. V. Chumak, A. A.\nSerga, V. I. Vasyuchka, M. B. Jung\reisch, E. Saitoh, and\nB. Hillebrands, Spin Pumping by Parametrically Excited\nExchange Magnons, Phys. Rev. Lett. 106, 216601 (2011).\n[11] F. D. Czeschka, L. Dreher, M. S. Brandt, M. Weiler,\nM. Althammer, I.-M. Imort, G. Reiss, A. Thomas, W.\nSchoch, W. Limmer, H. Huebl, R. Gross, and S. T. B.\nGoennenwein, Scaling Behavior of the Spin Pumping Ef-\nfect in Ferromagnet-Platinum Bilayers, Phys. Rev. Lett.\n107, 046601 (2011).\n[12] V. Castel, N. Vlietstra, J. Ben Youssef, and B. J. van\nWees, Platinum thickness dependence of the inverse spin-\nHall voltage from spin pumping in a hybrid yttrium iron\ngarnet/platinum system, Appl. Phys. Lett. 101, 132414\n(2012).\n[13] C. Hahn, G. de Loubens, O. Klein, M. Viret, V. V.\nNaletov, and J. Ben Youssef, Comparative measurements\nof inverse spin Hall e\u000bects and magnetoresistance in\nYIG/Pt and YIG/Ta, Phys. Rev. B 87, 174417 (2013).\n[14] K. Ando, S. Takahashi, J. Ieda, H. Kurebayashi, T. Tryp-\niniotis, C. H. Barnes, S. Maekawa, and E. Saitoh, Electri-\ncally tunable spin injector free from the impedance mis-\nmatch problem, Nat. Mater. 10, 655 (2011).\n[15] J.-C. Rojas-Sanchez, M. Cubukcu, A. Jain, C. Vergnaud,\nC. Portemont, C. Ducruet, A. Barski, A. Marty, L. Vila,\nJ.-P. Attane, E. Augendre, G. Desfonds, S. Gambarelli,\nH. Ja\u000bres, J.-M. George, and M. Jamet, Spin pumping\nand inverse spin Hall e\u000bect in germanium, Phys. Rev. B\n88, 064403 (2013).\n[16] E. Shikoh, K. Ando, K. Kubo, E. Saitoh, T. Shinjo,\nand M. Shiraishi, Spin-pump-induced spin transport in\np-type Si at room temperature, Phys. Rev. Lett. 110,\n127201 (2013).\n[17] P. Hyde, L. Bai, D. M. J. Kumar, B. W. Southern, C.-M.\nHu, S. Y. Huang, B. F. Miao, and C. L. Chien, Electri-\ncal detection of direct and alternating spin current in-\njected from a ferromagnetic insulator into a ferromag-\nnetic metal, Phys. Rev. B 89, 180404(R) (2014).\n[18] J. B. S. Mendes, R. O. Cunha, O. Alves Santos, P. R. T.\nRibeiro, F. L. A. Machado, R. L. Rodriguez-Su\u0013 arez, A.\nAzevedo, and S. M. Rezende, Large inverse spin Hall ef-\nfect in the antiferromagnetic metal Ir 20Mn80, Phys. Rev.\nB89, 140406(R) (2014).\n[19] H. Saglam, W. Zhang, M. B. Jung\reisch, J. Sklenar, J. E.\nPearson, J. B. Ketterson, and A. Ho\u000bmann, Spin trans-\nport through the metallic antiferromagnet FeMn, Phys.\nRev. B 94, 140412(R) (2016).\n[20] H. Wang, C. Du, P. C. Hammel, and F. Y. Yang, Anti-\nferromagnonic Spin Transport from Y 3Fe5O12into NiO,\nPhys. Rev. Lett. 113, 097202 (2014).[21] A. K. Patra, S. Singh, B. Barin, Y. Lee, J.-H. Ahn, E. del\nBarco, E. R. Mucciolo, and B. Ozyilmaz, Dynamic spin\ninjection into chemical vapor deposited graphene, Appl.\nPhys. Lett. 101, 162407 (2012).\n[22] Z. Tang, E. Shikoh, H. Ago, K. Kawahara, Y. Ando, T.\nShinjo, and M. Shiraishi, Dynamically generated pure\nspin current in single-layer graphene, Phys. Rev. B 87,\n140401(R) (2013).\n[23] S. Dushenko, H. Ago, K. Kawahara, T. Tsuda, S. Kuwa-\nbata, T. Takenobu, T. Shinjo, Y. Ando, and M. Shiraishi,\nGate-Tunable Spin-Charge Conversion and the Role of\nSpin-Orbit Interaction in Graphene, Phys. Rev. Lett.\n116, 166102 (2016).\n[24] J. B. S. Mendes, A. Aparecido-Ferreira, J. Holanda, A.\nAzevedo, and S. M. Rezende, E\u000ecient spin to charge cur-\nrent conversion in the 2D semiconductor MoS 2by spin\npumping from yttrium iron garnet, Appl. Phys. Lett.\n112, 242407 (2018).\n[25] K. Ando, S. Watanabe, S. Mooser, E. Saitoh, and H.\nSirringhaus, Solution-processed organic spin-charge con-\nverter, Nat. Mater. 12, 622 (2013).\n[26] Z. Qiu, M. Uruichi, D. Hou, K. Uchida, H. M. Yamamoto,\nand E. Saitoh, Spin-current injection and detection in \u0014-\n(BEDT-TTF) 2Cu[N(CN) 2]Br, AIP Advances 5, 057167\n(2015).\n[27] Y. Shiomi, K. Nomura, Y. Kajiwara, K. Eto, M. Novak,\nK. Segawa, Y. Ando, and E. Saitoh, Spin-Electricity Con-\nversion Induced by Spin Injection into Topological Insu-\nlators, Phys. Rev. Lett. 113, 196601 (2014).\n[28] J. C. Rojas S\u0013 anchez, L. Vila, G. Desfonds, S. Gam-\nbarelli, J. P. Attan\u0013 e, J. M. De Teresa, C. Mag\u0013 en, and A.\nFert, Spin-to-charge conversion using Rashba coupling at\nthe interface between non-magnetic materials, Nat. Com-\nmun. 4, 2944 (2013).\n[29] Y. Niimi, M. Kimata, Y. Omori, B. Gu, T. Ziman, S.\nMaekawa, A. Fert, and Y. Otani, Strong Suppression of\nthe Spin Hall E\u000bect in the Spin Glass State, Phys. Rev.\nLett. 115, 196602 (2015).\n[30] D. Wesenberg, T. Liu, D. Balzar, M. Wu, B. L. Zink,\nLong-distance spin transport in a disordered magnetic\ninsulator, Nat. Phys. 13, 987 (2017).\n[31] P. W. Anderson, in Ill-Condensed Matter , edited by R.\nBalian, R. Maynard, and G. Toulouse (North-Holland,\nAmsterdam, 1979).\n[32] K. Binder and A. P. Young, Spin glasses: Experimen-\ntal facts, theoretical concepts, and open questions, Rev.\nMod. Phys. 58, 801 (1986).\n[33] R. Iguchi, K. Ando, E. Saitoh, and T. Sato, Spin current\nstudy of spin glass AgMn using spin pumping e\u000bect, J.\nPhys.: Conf. Ser. 266, 012089 (2011).\n[34] Y. Ohnuma, H. Adachi, E. Saitoh, and S. Maekawa, En-\nhanced dc spin pumping into a \ructuating ferromagnet\nnearTC, Phys. Rev. B 89, 174417 (2014).\n[35] Y. Ohnuma, H. Adachi, E. Saitoh, and S. Maekawa,\nMagnon instability driven by heat current in magnetic\nbilayers, Phys. Rev. B 92, 224404 (2015).\n[36] B. Khodadadi, J. B. Mohammadi, C. Mewes, T. Mewes,\nM. Manno, C. Leighton, and C. W. Miller, Enhanced\nspin pumping near a magnetic ordering transition, Phys.\nRev. B 96, 054436 (2017).\n[37] L. Frangou, S. Oyarz\u0013 un, S. Au\u000bret, L. Vila, S. Gam-\nbarelli, and V. Baltz, Enhanced Spin Pumping E\u000eciency\nin Antiferromagnetic IrMn Thin Films around the Mag-8\nnetic Phase Transition, Phys. Rev. Lett. 116, 077203\n(2016).\n[38] Z. Qiu, J. Li, D. Hou, E. Arenholz, A. T. N'Diaye, A. Tan,\nK. Uchida, K. Sato, S. Okamoto, Y. Tserkovnyak, Z. Q.\nQiu, and E. Saitoh, Spin-current probe for phase transi-\ntion in an insulator, Nat. Commun. 7, 12670 (2016).\n[39] M. Inoue, M. Ichioka, and H. Adachi, Spin pumping into\nsuperconductors: A new probe of spin dynamics in a su-\nperconducting thin \flm, Phys. Rev. B 96, 024414 (2017).\n[40] Y. Yao, Q. Song, Y. Takamura, J. P. Cascales, W. Yuan,\nY. Ma, Y. Yun, X. C. Xie, J. S. Moodera, and W. Han,\nProbe of spin dynamics in superconducting NbN thin\n\flms via spin pumping, Phys. Rev. B 97, 224414 (2018).\n[41] M. Umeda, Y. Shiomi, T. Kikkawa, T. Niizeki, J.\nLustikova, S. Takahashi, and E. Saitoh, Spin-current co-\nherence peak in superconductor/magnet junctions, Appl.\nPhys. Lett. 112, 232601 (2018).\n[42] H. Sompolinsky and A. Zippelius, Relaxational dynam-\nics of the Edwards-Anderson model and the mean-\feld\ntheory of spin-glasses, Phys. Rev. B 25, 6860 (1982).\n[43] H. Sompolinsky and A. Zippelius, Relaxational dynamics\nof the in\fnite-ranged spin glass with n-component spins,\nJ. Phys. C: Solid State Phys. 15, L1059 (1982).\n[44] S. F. Edwards and P. W. Anderson, Theory of spin\nglasses, J. Phys. F: Metal Phys. 5, 965 (1975).\n[45] D. Sherrington and S. Kirkpatrick, Solvable Model of a\nSpin-Glass, Phys. Rev. Lett. 35, 1792 (1975).\n[46] G. Parisi, In\fnite Number of Order Parameters for Spin-\nGlasses, Phys. Rev. Lett. 43, 1754 (1979).\n[47] G. Parisi, A sequence of approximated solutions to the\nS-K model for spin glasses, J. Phys. A 13, L115 (1980).\n[48] K. H. Fischer and W. Kinzel, The \feld and tempera-\nture dependences of the dynamic susceptibility of in\fnite-\nrange Ising spin glasses, J. Phys. C: Solid State Phys. 17,4479 (1984).\n[49] A. I. Larkin, V. I. Mel'nikov, and D. E. Khmelnitskii,\nVirial expantion for magnetic impurities in metals, Sov.\nPhys. JETP 33, 458 (1971).\n[50] K. H. Fischer, On the electrical resistivity of spin glasses,\nZ. Phys. B 34, 45 (1979).\n[51] S. Kirkpatrick and D. Sherrington, In\fnite-ranged mod-\nels of spin-glasses, Phys. Rev. B 17, 4384 (1978).\n[52] K. H. Fischer, Dynamics of vector spin glasses: The\nGabay-Toulouse line, Z. Phys. B 55, 317 (1984).\n[53]Spin Glasses , K. H. Fischer and J. A. Hertz (Cambridge\nUniversity Press, Cambridge, 1991).\n[54] P. C. Martin, E. D. Siggia, and H. A. Rose, Statisti-\ncal Dynamics of Classical Systems, Phys. Rev. A 8, 423\n(1973).\n[55] C. De Dominicis, Dynamics as a substitute for replicas\nin systems with quenched random impurities, Phys. Rev.\nB18, 4913 (1978).\n[56] J. R. L. de Almeida and D. J. Thouless, Stability of the\nSherrington-Kirkpatrick solution of a spin glass model,\nJ. Phys. A 11, 983 (1978).\n[57] K. H. Fischer, Theory of spin glasses: Susceptibility and\nspeci\fc heat in a magnetic \feld, Solid State Commun.\n18, 1515 (1976).\n[58] F. Mezei, Neutron spin echo study of spin glass dynamics\n(invited), J. Appl. Phys. 53, 7654 (1982).\n[59] R. Kubo, M. Toda, and N. Hashitsume, Statisti-\ncal Physics II: Nonequilibrium Statistical Mechanics\n(Springer, 1991).\n[60] C. P. Slichter, Principles of magnetic resonance\n(Springer, Berlin, 2010).\n[61] T. Taira, M. Ichioka, S. Takei, and H. Adachi, Spin dif-\nfusion equation in superconductors in the vicinity of TC,\nPhys. Rev. B 98, 214437 (2018)."    },    {        "title": "2203.10345v1.Spin_wave_excitation_and_directional_propagation_in_presence_of_magnetic_charges_in_square_artificial_spin_ice.pdf",        "content": "Spin wave excitation and directional propagation in presence of magnetic charges in\nsquare arti\fcial spin ice\nNimisha Arora\u0003and Pintu Dasy\nDepartment of Physics, Indian Institute of Technology Delhi, New Delhi, India\nArti\fcial spin ice is a special class of engineered lattice of highly shape anisotropic single domain\nmagnetic nanostructures which is used as one of the model systems to study the spin ice behavior\nobserved in pyrochlore oxides. The nanomagnets interact via dipolar interaction which results in\ncorrelated magnetization dynamics exhibiting macroscopic spin con\fguration states. Here, we ex-\nploit the interplay of underlying magnetic state and external bias \feld orientation to study controlled\nspin wave propagation in square Arti\fcial Spin Ice (sASI) by performing detailed micromagnetic\nsimulations. We report that careful selection of vertices with local magnetic charges can e\u000bectively\ndirect the anisotropic spin wave in presence of an external \feld. Further, we explore the in\ruence\nof local charges due to the excited state in even-coordinated vertices as well as uncompensated\ncharges due to odd-coordinated vertices on spin wave behavior. Our studies suggest that there is no\nperceptible di\u000berence on spin wave dynamical behavior due to the origin of local magnetic charge in\nsASI. Our results of controlled and directional spin wave propagation in sASI system may be useful\nfor low-power consumption based all magnonic on-chip devices.\nI. INTRODUCTION\nAs the conventional electronic charge transport based\ntechnological devices approaching the quantum limit,\nthere is a growing thrust on the energy e\u000ecient and faster\ndevices based on electron's spin. Recent development\nhas shown that collective precession of localized magnetic\nmoments (excitation of spin waves) can be used to carry\ninformation to a much larger distance in comparison to\nthe conduction electron's spin current. The quantum of\nspin waves (SWs) is called magnon and the \feld of study-\ning SWs in magnetic nanostructures has emerged under\nthe name of magnonics [1]. From a classical perspective,\nSWs can be treated as perturbations in phase-coherent\nprecession of magnetization vectors, which propagate in\ncoupled magnetic media. Due to the very short wave-\nlengths of SWs (1 \u0016m to 150 nm depending on the lattice\nconstant of the magnetic material), these are emerging\nas potential information carriers in energy e\u000ecient minia-\nturized spin based devices in the nanoscale regime. Thus,\nexcitation of SWs of tuned wavelength is one of the ma-\njor topics of current research in this area [1{6]. The other\nmajor considerations for the SWs based devices are the\npropagation distance, group velocity, phase velocity of\nthe SWs, etc. [1{6]. Several of these parameters depend\non the competition between the exchange and dipolar\ninteractions of the spins. Recent studies have reported\nseveral SW based functional device applications in phase\nshifters [7], microwave antenna [8], directional coupler [9],\nSW-based multiplexer [10], interferometer [11{13], grat-\ning [14], neuromorphic computing [15{17, 19], and mem-\nory register [18] etc. Furthermore, it has been reported\nthat phase incoherent electromagnetic wave can give rise\nto multiple phase-coherent magnons, which shows the\n\u0003narora1.physics@gmail.com\nypdas@physics.iitd.ac.inapplication in energy harnessing and information trans-\nfer from alternative sources [20]. Thus, it is clear that\nsuccessful implementation of magnonics in modern de-\nvices requires the SWs to be controllably guided through\nnanostructures. In order to acquire this ability, funda-\nmental and in-depth understanding of the behavior of\nSWs in magnetic materials of di\u000berent nanoscale geome-\ntries is very important. Recently a few experimental as\nwell as theoretical results on the investigation of this as-\npect of SWs have been reported in literature [21{26]. The\nmagnetic nanostructures, which are involved in the prop-\nagation of SWs, may in general interact via long range\ndipolar interactions. Therefore, such magnetostatic in-\nteraction may play a role on the propagation of SWs in\ndipolarly coupled magnetic nanostructures which is far\nfrom clearly understood. This clearly underlines the im-\nportance of investigation of general behavior of SWs un-\nder dipolarly coupled environment.\nIn this work, we have studied the behavior of excited\nSWs in highly shape anisotropic nanomagnets of Ni 80Fe20\narranged in the sASI geometry [27]. Typically, these\ndesigner nanostructured materials o\u000ber alluring possi-\nbilities to study physics of geometric frustration [27],\nemergent magnetic monopoles [28, 29], collective mag-\nnetization dynamics of interacting spins [30], and phase\ntransition [31], etc. at room temperature while o\u000bering\nthe possibility of visualizing individual macrospins. Ow-\ning to the possibility of controlling magnetic microstates\nby designing di\u000berent structures such as kagome [32],\nsASI [27], etc. where dipolar interactions play a sig-\nni\fcant role in de\fning the exact magnetic microstates,\nthese nanoscale magnetic structures are interesting can-\ndidates to study the interplay of dipolar interaction and\nSW propagation. Although, recently some attention\nhas been directed towards the SW behavior of ASI sys-\ntems [33{39], the role of dipolar interactions and associ-\nated magnetic charges on the SW behavior has not been\ninvestigated in details.arXiv:2203.10345v1  [cond-mat.mes-hall]  19 Mar 20222\nFIG. 1. (a) Schematic diagram of open-edged sASI system\nwith local transient pulse excitation applied at the center ver-\ntex along the z-axis (out of the plane) and external bias \feld\napplied in the xy-plane, making an angle \u0012with respect to\nthe x-axis. Inset below shows the 3D spatial pro\fle of the lo-\ncal excitation, (b) Schematic diagram of the closed-edge (CE)\nsASI, and (c) transient square pulse with rise time ( trise), fall\ntime (tfall), and duration ( tdur) of 20 ps each. nanomagnets\ndimensions for (a) and (b) here are: 300 nm \u0002100 nm \u000230\nnm with edge-to-edge inter-island gap of 150 nm.\nThe role of dipolar interaction in the behavior of SW\npropagation in sASI geometry (Fig. 1) are investigated by\nperforming a detailed micromagnetic analysis of propa-\ngating SWs in sASI systems with two types of vertices,\nviz., the ones with vertices of even coordination num-\nber while the other with vertices of mixed (even and\nodd) coordination number. Open-edged (OE) sASI cor-\nresponds to even coordination whereas closed-edged (CE)\nsASI (shown in Fig. 1) corresponds to mixed coordination\nvertices. In general, the vertices with odd coordination\nnos. are always magnetically charged whereas the ones\nwith even coordination may be uncharged or charged (ex-\ncited state) depending on the minimum energy state cor-\nresponding to the macrospin states of the involved nano-\nmagnets.\nII. METHOD\nFor the investigations, we carried out micromagnetic sim-\nulations for the sASI structures (see Fig. 1) using \f-\nnite di\u000berence discretization based (open source) GPU-\naccelerated software M UMAX3 [40]. Here, the evolu-\ntion of space and time dependent magnetization vector,\n~ m(~ r;t), is calculated at each cell of the discretized ge-\nometry by employing Landau-Lifshitz-Gilbert equation\n(Eq. 1).\n~ \u001cLL=@~ m\n@t=\rLL1\n1 +\u000b2[~ m\u0002~Beff+\u000b(~ m\u0002(~ m\u0002~Beff))]\n(1)with\u001cLLis the Landau-Lifshitz torque, ~ m(~ r;t) is the re-\nduced magnetization vector of unit length, \rLLis the\ngyro-magnetic ratio (rad/Ts), \u000bis the dimensionless\ndamping parameter, and ~Beffis the e\u000bective magnetic\n\feld.\nHere,~Beff=~Bext+~Bdemag +~Bexch+~Banis+~Bexc(t)+:::\nwith~Bext,~Bdemag ,~Banisand~Bexc(t) are the external bias\n\feld, demagnetization \feld, anisotropy \feld, and time\nvarying excitation (magnetic) \feld, respectively. For the\nmicromagnetic simulations, the nanomagnetic structures\nof Ni 80Fe20as shown in the schematic diagram Fig. 1(a,\nb) are discretized in cuboidal cells of 5 nm length, which\nis less than the exchange length (5.3 nm) of Ni 80Fe20.\nExperimentally reported value of saturation magnetiza-\ntionMsat= 8:6\u0002105A/m, exchange sti\u000bness constant\nAex= 1:3\u000210\u000011J/m, and damping coe\u000ecient \u000b= 0:5\nfor Ni 80Fe20are used throughout the study [41]. In or-\nder to determine the SW behavior, a reduced damping\ncoe\u000ecient of 0.008 is used. This enables a prolonged pre-\ncession of weak modes thereby allowing for a detailed\nanalysis of such modes [42].\nSWs are excited by perturbing the equilibrium macrospin\nstate corresponding to a static in-plane bias \feld ( Bbias)\nvia an out-of-plane temporal \feld pulse, see Fig. 1. The\nequilibrium (macro) spins con\fguration is \frst evalu-\nated by quasi-statically down-sweeping the external in-\nplane bias \feld from 250 mT to Bbiasand performing en-\nergy minimization using 4th order Runge-Kutta method.\nAt this state, a transient square pulse \feld of ampli-\ntudeBmax = 3 mT with rise-time trise, fall-time tfall,\nand pulse-duration tdurof 20 ps each is applied along z-\ndirection (out of the plane). The transient excitation\n\feld pulse is convoluted with Gaussian function in space\nto locally excite central vertex region of the OE geometry\n(see Fig. 1a). After the incidence of the pulse, time evolu-\ntion of reduced magnetization vector ( ~ m(t)) is recorded\nfor 4 ns at time-step (\u0001 t) of 10 ps. To extract the fre-\nquencies of the excited SW modes, fast Fourier transfor-\nmation (FFT) is performed on the recorded ~ m(t) data.\nThus, power spectra for SW modes are determined for a\ngiven magnetic structure. The nature and the origin of\nthe SW modes are analyzed by investigating the power\nas well as phase pro\fles of the excited SW modes at par-\nticular frequencies using a MATLAB code (see supple-\nmentary material). For the propagational characterstics\nof SW, fast Fourier transform of magnetization in unper-\nturbed neighbouring region is compared with the central\nperturbed vertex in two orthogonal direction. This study\nwas performed at two external \feld con\fgurations, viz.,\nsaturation and remanence for both even and mixed coor-\ndinated vertices. Additionally, to obtain a clear insight\nin to the role of magnetic charges on SW propagation,\na transient square pulse \feld is uniformly applied along\nthe z-direction on all nanomagnets in CE as well as OE\nvertex geometries. SW analysis is then performed in the\nrange of\u0006150 mT ofBbias, whereBbiasis applied along\none of the easy axes of the magnetic sublattice.3\nFIG. 2. SW spectra ( black solid line ) for open-edge vertex at saturation, i.e., Bbias= 200 mT applied along x-axis (a) and\nat remanence (d). Spectra for individual nanomagnets for bias \feld of 200 mT applied along the easy axes (E-islands, red solid\nline) and the hard axes (H-islands, red dotted line, shown in (a)) and at remanence shown using red solid line in (d). Inset in\n(a) shows the schematic diagram for applied \feld direction and that in (d) shows the schematics of macro-spin con\fgurations at\nremanence, (c) micromagnetic simulation results for internal spin con\fguration in the perpendicularly placed islands for Bbias\n= 200 mT as highlighted in the inset of (a). Power pro\fle of the excited SW modes for open edge vertex at saturation (b) and\nremanence (e).\nIII. RESULTS AND DISCUSSION\nFig. 2 shows the SW spectra for OE vertex geometry (ver-\ntices with even coordination) for saturated state corre-\nsponding to the bias \feld of Bbias;x= 200 mT as well\nas remanence ( Bbias;x= 0 mT) where the bias \feld is\napplied along x-direction as clari\fed in Fig. 2c. At sat-\nuration, the spectrum exhibits \fve peaks corresponding\nto the \fve di\u000berent modes of excitations of SWs in the\nfrequency range of 2 GHz - 17 GHz (see Fig. 2a). In the\nstudied system involving sixteen nanomagnets, the ap-\nplied \feld direction is along easy-axes and hard-axes of 8\nnanomagnets each (see Fig. 2a inset), henceforth identi-\n\fed as E-islands and H-islands, respectively (see Fig. 2c).\nAs the orientation of the external bias \feld applied in\nthis case corresponds to easy as well as hard directions\nof di\u000berent nanomagnets, we have analyzed SW behav-\nior of individual islands with external \feld along both\neasy and hard directions . This allows us to carry out an\nin-depth analysis of the role of individual nanomagnets\nin the overall SW spectra of the dipolar coupled system.The results are shown in Fig. 2(a). A comparison of the\nSW modes excited in the individual nanomagnets with\nthat of the coupled sASI system (Fig. 2(a)) shows that\nalthough the modes associated with coupled system dis-\nplay a nominal frequency shift of up to \u00180.3 GHz which\naccounts for the stray \feld of \u00183.6 mT, yet it consists of\nall the modes excited in individual E-as well as H-islands.\nThe shift may be a result of the in\ruence of additional\ndipolar \feld emanating from the interacting nanomag-\nnets. Thus, these results suggest that as the spins are\nstrongly pinned along the direction of the Bbiasat sat-\nurated state, the SW modes in individual nanomagnets\nremain largely una\u000bected in the dipolar coupled sASI sys-\ntem.\nIn order to determine the origin of these peaks in the\nobserved power spectra, we analyze the power pro\fles of\nthe excited SW modes corresponding to each of the \fve\nfrequencies where a peak is observed. The power pro-\n\fles are shown in Fig. 2b. We make the following four\ninteresting observations: i) signi\fcant mode-power for\nthe three lower-frequency modes viz., 2.3 GHz, 6.3 GHz,4\nand 10.9 GHz, appears to be concentrated primarily in\nthe H-islands. This indicates that the peaks observed at\nthose frequencies are primarily due to modes excited in\nthe H-islands. ii) with increasing frequencies, inhomo-\ngeneously distributed power is systematically reduced in\nthe H-islands. For the SW mode at 2.3 GHz, excited\nmode exhibits signi\fcant power in the four H-islands,\nwhereas for the mode at 10.9 GHz, the power is found\nto be mainly concentrated in the two H-islands coupled\nto the central vertex. iii) two higher frequency modes ap-\npearing at 11.9 GHz and 17 GHz are primarily excited at\nthe E-islands. Large power corresponding to these modes\nis primarily located at the E-islands of the central vertex.\niv) the excited SW modes in both E- and H-islands are in-\nhomogeneously distributed. In general, demagnetization\n\feld in con\fned magnetic nanostructures may be inho-\nmogeneous which results in multiple low energy potential\nwells where spins are in local equilibrium corresponding\nto the internal local \feld. Curling of magnetization ob-\nserved near the edges of H-islands is an example of one\nsuch inhomogeneity of demagnetization \feld (see Fig. 2c).\nSuch inhomogeneous demagnetizing \feld within the in-\ndividual sASI nanostrctures may in turn lead to inhomo-\ngeneously distributed local excitation of SWs in the sASI\nstructures. According to the spatial distribution of the\nmodes, we identify the modes observed at edges of nano-\nmagnets as edge-modes and the others excited within the\n\\bulk\" of the nano-magnets as bulk-modes (hereafter B-\nmode). Thus, we identify the 2.3 GHz mode as long-edge\n(LEH) mode and 11.9 GHz as short-edge (SE E) mode.\nThe su\u000eces denote the H- and E-islands, respectively.\nFrom the power distribution pro\fle at saturation we also\nobserve that the lower-frequency LE Hmodes appear to\nextend to the nearest neighbors (H-islands) restricted\nalong the direction perpendicular to that of Bbias. The\ndemagnetization \feld for H-islands is larger than that\nfor the E-islands thereby reducing the e\u000bective in-plane\n\feld (Be\u000b) in H-islands compared to that for E-islands.\nThe total \feld ~Btot(=~Be\u000b+~Bexc(t)) during temporal\npulse excitation along z-direction thus make a larger an-\ngle with respect to the in-plane direction of Be\u000bresulting\nin larger \ructuations in z-component of magnetization in\nH-islands. Since f/Btot, therefore, the perturbation\nin phase coherent precession which manifests itself as a\nSW has higher probability of transmittance/propagation\nto the nearest neighbour of H-islands which in this case\nalong the y-axis for lower frequency mode \u00182:3 GHz and\n6.4 GHz. Along the x-axis, adjacent H-islands are sepa-\nrated by much larger distance \u0018450 nm. At this sepa-\nration, resulting stray \feld variation due to \ructuations\nin z-magnetization is negligible. Thus, we don't observe\ntransmittance of SW mode along \feld direction or x-axis.\nAt remanence ( Bext= 0), the magnetization in each\nnanomagnet orients along their easy directions. Due\nto the magnetostatic interactions, low-energy 2-in/2-out\nspin ice states are observed for three vertices whereas the\nremaining two vertices exhibit magnetically charged ex-\ncited states, see inset of Fig. 2(d). The magnetic chargesare reconciled considering the nanomagnets as a dumb-\nell of two equal and opposite magnetic charges (dumb-\nell model) [43]. For this magnetic con\fguration of sASI\nsystem at remanence, number of SW modes are now re-\nduced to three from \fve as compared to the saturated\nstate. In this case, both E- and H-islands are now iden-\ntical with respect to an external transient \feld perturba-\ntion which accounts for the observed less nos. of peaks\nin remanence. The peaks are observed at the frequencies\nof 6.4 GHz, 9.9 GHz, and 10.4 GHz, respectively in the\nSW-spectra as shown in Fig. 2d. In order to clearly ana-\nlyze the role of dipolar interactions in the SW spectra in\nthis case, we have calculated the SW modes for individual\n(i.e., non interacting) nanomagnets with magnetization\nalong the easy directions. The simulation results show\nthat only two SW modes, viz., 6.4 GHz and 10.4 GHz,\nare excited in an individual nanomagnet of same dimen-\nsions thereby indicating a possible role of dipolar inter-\nactions on the excitation of 9.9 GHz-mode in such dipo-\nlar coupled nanostructures. Therefore to elucidate the\nFIG. 3. SW spectra of L-shaped geometry at remanence\nin two magnetic state con\fguration viz., tail-to-tail (mag-\nnetically charged vertex) and head-to-tail (magnetically un-\ncharged vertex). Inset shows the xy orientation for the opted\ngeometry and magnetic state with tail-to-tail as well as head-\nto-tail con\fguration.\norigin of this 9.9 GHz mode, we analyze a fundamental\ncell consisting of two dipolar-coupled nearest neighbour\nnanomagnets: one E- and one H-island in L-shaped ge-\nometry (see Fig. 3). The SW spectra of the L-shaped\ngeometry at remanence, where the magnetization in the\nislands orient as head-to-tail con\fguration resulting in\nzero magnetic charge at the meeting point (junction) of\nthe nanomagnets, exhibit two peaks at the frequencies\nof 6.3 GHz and 10.3 GHz, respectively. Since the individ-\nual nanomagnets show similar SW behavior, these ob-\nservations suggest that such head-to-tail (i.e., zero mag-\nnetic charge) con\fguration in L-shape geometry retains\nthe SW behavior of individual nanomagnets. However, as5\nthe magnetization of the L-shaped geometry is prede\fned\nas head-to-head or tail-to-tail con\fgurations leading to a\nmagnetic charge at the junction, the SW spectra show\na red-shift of\u00180:3 GHz with respect to that observed\nfor individual nanomagnets thereby exhibting a strong\nmode at 10 GHz. This demonstrates a clear role of mag-\nnetic charges due to dipolarly interacting nanomagnets\non the excitation of SWs. This 10 GHz-mode appears\nto be excited at 9.9 GHz in the sASI system with mixed\ntypes of charged and chargeless vertices. This suggests\nthat SW spectra for vertices with mixed charge state may\nexhibit SW modes as observed in Fig. 2d. In order to elu-\ncidate this observation further, simulations were carried\nout for the \fve-vertex sASI system containing prede\fned\nchargeless vertices. We observe an absence of the addi-\ntional mode at 9.9 GHz in the SW spectra (see supple-\nmentary materials, Fig. S1) which further demonstrates\nthat the origin of the 9.9 GHz peak in the SW spectra\nlies in the magnetically charged excited vertices. These\nresults also underlines the fact that the magnetostatic in-\nteraction between nanomagnets leading to 3-in/1-out or\n3-out/1-in type local charged vertices modi\fes the local\nmagnetic structure of the involved nanomagnets in such\na way that an additional SW mode is excited in these\nnanostructures. In the following, we identify this mode\nas charged vertex (CV) mode.\nFurther con\frmation on the role of strong dipolar inter-\naction in the excitation of the additional SW mode is\nobtained from the investigation of the SW modes of the\nOE vertex system with varying inter-island separations.\nThe spectral behavior was investigated by varying the\nedge-edge separation of the co-linear nanomagnets from\n100 nm to 1 \u0016m. We observe that the static magnetic\nbehavior (i.e., macrospin con\fguration) of the sASI sys-\ntem remains unaltered in the entire range of separation\n(i.e., 100 nm-1 \u0016m). However, the excitation of the addi-\ntional mode at 9.9 GHz is observed till the separation of\n\u0018225 nm (see supplementary information for details) be-\nyond which only the two modes, as excited for individual\nnanomagnets, are observed. Based on these observations,\nwe identify the separation up to 225 nm as the strongly\ninteracting regime. Next, in order to investigate the na-\nture of the excited SW modes at remanence in details,\nthe power and phase pro\fles of the modes are calculated.\nAs shown in Fig. 2(e), excitations for the three frequen-\ncies are observed in both type of nanomagnets. However,\nthe modes can be distinctly identi\fed according to the\ndistribution of the power pro\fles within the individual\nnanomagnets. The excitation of 6.4 GHz mode appears\nto be localized near the short edges (SE mode) whereas\nthe 9.9 GHz and 10.4 GHz modes are excited in the bulk\nof the nanomagnets. The power pro\fles clearly show that\nthe modes which are excited in the nanomagnets at the\ncentral vertex, propagate beyond the central vertex al-\nbeit with reduced intensity and di\u000berent propagational\ncharacteristics (see below).\nThe SE 6.4 GHz mode exhibits stronger power along the\ndirection perpendicular to the bias \feld than along the\nFIG. 4. Line power pro\fle for the excited SW mode at (a)\n6.4 GHz and (b) 10.4 GHz in remanent state, along the two or-\nthogonal directions as shown in the inset of (a) (Blue: along x,\nRed: along y). Highlighted regions correspond to nanomag-\nnets at central vertex which are under pulse \feld excitation.\n\feld direction. This is exempli\fed by plotting the line\npro\fle of the power map (Fig. 2e) along the x- and y-\ndirections through the central vertex as shown in Fig. 4.\nThe line pro\fle clearly shows that the power of the\n6.4 GHz mode excited at the central vertex nanomag-\nnets along the x-direction is reduced by almost 50% as\ncompared to that along y-direction. A nominal power\nof\u00182% at the other end of these nanomagnets is ob-\nserved only along y-direction. No power for excitation\nis observed in the nanomagnets of outer vertices. Thus,\nthe excited SE mode is strongly localized at the central\nvertex where the pulse \feld is applied and doesn't ex-\nhibit propagational character in dipolar-coupled nanos-\ntructures. For the bulk SW mode of \u001810:4 GHz, the\nobserved power of excitation at the two nanomagnets in-\nvolved at the central vertex is nearly the same for both E-\nand H-islands. However, for nanomagnets involved at the\nouter vertices, the observed power along y-axis is nearly\n60% of the maximum power observed at central vertex\n(see Fig. 4b). whereas a nominal power of only \u00182% of\nthe maximum power is observed at the nanomagnets of\nthe outer vertices along the \feld direction.. Thus, we ob-\nserve a signi\fcant SW propagation of the bulk 10.4 GHz\nmode along the direction perpendicular to the applied\nbias \feld direction (i.e., x-direction). For the CV mode\nobserved at 9.9 GHz, the power pro\fle is signi\fcantly dif-\nferent than the other two modes. Here, the power is\nfound to be concentrated mainly at the nanomagnets in-\nvolved at the central vertex and the nanomagnets with6\nreversed magnetization (-y direction) at the outer ver-\ntices with non-zero local charges (see Fig. 2e and inset\nof Fig. 2d). Here, the power pro\fle of this mode in the\nH-islands of the central vertex exhibit nodal lines per-\npendicular to the magnetization orientation. This pro\fle\nis the characteristic of backward volume like mode (BV).\nHowever, power of the mode is distributed in the bulk of\nthe rest of the nanomagnets. Thus, 9.9 GHz mode can\nbe identi\fed as hybridized BV and B mode.\nThe above analyses suggest that the SW modes (SE- and\nB-modes) in such unconnected highly shape anisotropic\nnanomagnets in sASI geometry show preferential propa-\ngation along the direction perpendicular to the external\nbias \feld direction. We note here that at ground state,\nthis is also the direction along which the two charged\nvertices are present in the system. Thus, so far it is\nnot clear if the propagation direction is a\u000bected by the\nexternal \feld or the presence of magnetic charges. In\norder to elucidate this further, we investigated the SW\nbehavior of the sASI system as a function of applied \feld\ndirection. By applying external bias \feld along di\u000berent\ndirections, the remanent magnetic states of the individ-\nual nanomagnets are tuned. The SW modes for these\ninvestigations are calculated for \u0012= 45\u000e;90\u000e;and 135\u000e,\nrespectively where, \u0012is the angle between the applied\n\feld direction and x-axis (see Fig. 1a). For \u0012= 45\u000e,\nthe magnetic con\fguration of the sASI system at rema-\nnence exhibits chargeless 2-in/2-out vertices in the sASI\nsystem as shown in Fig. 5a. The SW spectra for this\nchargeless sASI state exhibit only two modes which are\nexcited at 6.4 GHz (SE mode) and 10.4 GHz (B-mode),\nrespectively. Interestingly, excitation of only these two\nmodes were earlier observed for chargeless L-shaped ge-\nometry as discussed above. Here we observe a remark-\nable di\u000berence in the power pro\fle for 6.4 GHz mode cal-\nculated for this magnetic con\fguration corresponding to\n\u0012= 45\u000ein comparison to that observed for \u0012= 0\u000e. As\nshown in Fig. 5b,d, the SE mode for this magnetic con-\n\fguration is found to be excited in all the nanomagnets\nwhereas for \u0012= 0\u000ethe mode-power at the two farthest\nnanomagnets along x-axis is nearly zero suggesting no\nexcitation of the SW mode in those islands. For this\nchargeless state at \u0012= 45\u000e, the sASI system is magnet-\nically symmetric with respect to the applied bias \feld.\nTherefore, the SE mode is excited symmetrically in all\nthe nanomagnets. However, for \u0012= 0\u000e, the sASI sys-\ntem is magnetically asymmetric with respect to the bias\n\feld. Under this condition, magnetically charged sASI\nvertices are generated as discussed above. Our analy-\nsis suggests that these charged vertices for \u0012= 0\u000eare\nresponsible for the observed asymmetric propagation of\nthe SE mode. For 10.4 GHz B mode, it is apparent (see\nFig.5c) that the nanomagnets of the outer vertices posi-\ntioned along the \feld direction (enclosed in black dash-\ndot rectangular box) display nominal power as compare\nto the nanomagnets positioned perpendicular to the \feld\norientation. This observation is similar to the case of\n\u0012= 0\u000ewhere 10.4 GHz mode propagates along the or-\nFIG. 5. (a) Equilibrium magnetization state for OE vertex\nat remanence with \u0012= 45\u000e. Power pro\fle of the SE mode\nexcited at 6.4 GHz (b) and B mode excited at 10.4 GHz (c) in\nremanence with \u0012= 45\u000e. Black and red arrow in (c) denote\nthe bias \feld orientation and its orthogonal direction. Black\ndash dotrectangular box encloses the nanomagnets positioned\nalong the bias \feld orientation. (d) Zoomed power pro\fle of\nthe vertex region highlighted with red dotted circle in (b).\nthogonal to the bias \feld orientation. Thus, the 10.4 GHz\nB-mode exhibits an asymmetric propagation with respect\nto bias \feld. Absence of the CV mode in this case of\n\u0012= 45\u000eclearly validates our observations for a signi\f-\ncant role of dipolar interactions on the SW excitation in\nthis system. Our additional calculations performed for\n\u0012= 135\u000eshows a similar behavior of the SW modes as\nfor\u0012= 45\u000e. For\u0012= 90\u000ei.e., when the bias \feld is ap-\nplied along y-direction (see supplementary Fig.S3a), the\nremanent state exhibits two charged (outer) vertices po-\nsitioned along the \feld direction. Thus, these charged\nvertices for \u0012= 90\u000eappear in orthogonal direction to the\ncharged vertices observed for \u0012= 0\u000ecase. Interestingly,\nthe SW spectra for this case again show the CV mode\nin addition to the SE and B-modes (see supplementary\ninformation). Thus, for this charged state also, three\nmodes are excited as for \u0012= 0\u000ewith similar SW prop-\nagational characteristics. These results clearly demon-\nstrate the following three facts: one, the excited SWs\nin such unconnected nanomagnets propagate to other\nnanomagnets which are within the strongly interacting\n(dipole-dipole interaction) regime which is \u0018225 nm of\ninterisland separation. Remarkably, this suggests that\nthe dipolar coupling of the nanomagnets facilitates car-\nrying the speci\fc information of SWs from one nanomag-\nnet to another. Secondly, the presence of \\sources\" and\n\\sinks\" of magnetic \rux favors excitation of a speci\fc\nmode - 9.9 GHz for the given structure. Third, the prop-\nagation of B-mode (10.4 GHz) is tuned by the orientation\nof bias \feld. Also, we observe that the propagation of SE7\nFIG. 6. Equilibrium macrospins con\fguration in sASI array at remanence when initial bias \feld applied along one of the easy\naxes , i.e. 0\u000e(a) and at 45\u000e(b) with respect to the easy axis of one of the sublattices, (c) SW spectra for 5 \u00025 vertex array\nfor equilibrium state shown in (a, b). Line power pro\fle of the bulk SW mode excited in sASI array at remanence, across the\ntwo orthogonal direction (highlighted region shown in (a, b)) when initial bias \feld is oriented along one of the easy axes, i.e.\n0\u000e(d) and at 45\u000e(e) from the easy axis of one of the sublattice. Highlighted regions in (d, e) correspond to central vertex\nnanomagnets which are under \feld pulse excitation.\nmode is impeded by the \\source\" or \\sink\" of magnetic\n\rux (see Fig. 2(e)). Thus, our results show that by tun-\ning charged magnetic state in such structures along with\nbias \feld orientation, guided or directional SW propaga-\ntion can be realized in such unconnected nanostructures.\nAlthough the above analysis shows SW propagation\nin the unconnected nanostructures, the distance up to\nwhich SW propagates is not clear. In order to investi-\ngate this, we have calculated the behavior of SWs for\na larger array of vertices. An array of 5 \u00025 vertices at\nremanent state with two di\u000berent initial bias \feld con\fg-\nurations is considered for the analysis. For ~Bbiasapplied\nalong one of the easy axes of the nanomagnets, the mag-\nnetic con\fguration at remanent state exhibits two pairs\nof charged vertices (i.e., two \\sources\" and two \\sinks\"\nof \rux) separated by an uncharged 2-in/2-out type ver-\ntex along the external \feld direction as schematically de-\npicted in Fig. 6a. Thus, we observe similar micromag-\nnetic behavior as for the 5-vertex case discussed above\nexcept that in this case two pairs of charged vertices in-\nstead of one pair are observed for this array of 25 vertices.\nTo observe the SW propagation, again we apply a square\npulse \feld with spatial Gaussian distribution of similar\nspeci\fcation as before at the central uncharged vertex\n(see Fig. 6a) and calculate the SW spectra for the sys-\ntem. Here again, the calculated SW spectra exhibit threeexcited modes, however, at frequencies of 6.4 GHz (SE-\nmode), 10.4 GHz (CV-mode), and 10.8 GHz (B-mode) as\nshown in Fig. 6c. The power pro\fles are shown in the sup-\nplementary material in Fig. S4. In this case, 9.9 GHz and\n10.4 GHz mode as discussed above for 5-vertex state shift\nto 10.4 GHz and 10.8 GHz, respectively most likely due\nto the di\u000berent net stray \feld emanating from the consti-\ntuting islands. For ~Bbiasapplied along 45\u000e, the magnetic\ncon\fguration at remanence exhibit all 2-in/2-out charge-\nless vertices (see Fig. 6b) as in the case of 5 vertices dis-\ncussed earlier. SW spectra for this state again exhibit\ntwo modes at 6.4 GHz and 10.4 GHz similar to 5 vertices\ncase which corresponds to SE, and B-mode (see supple-\nmentary info for power pro\fles). Thus, the calculated\nSW spectra for an array of 25 vertices corresponding to\ntwo di\u000berent magnetic con\fgurations stabilized for Bbias\napplied at 0\u000eand 45\u000efurther con\frm the important role\nof charged vertices in the origin of the additional peak\ncorresponding to the CV mode.\nIn order to obtain more quantitative information about\nthe propagational characteristics of SW modes in an ar-\nray of 25 vertices, we investigate the power pro\fles (see\nsupplementary material) of the modes across the two or-\nthogonal directions highlighted in Figs. 6a,b. Line pro-\n\fle of power for magnetic con\fguration for \u0012= 0\u000e(ar-\nray with charged vertices) suggests that a signi\fcant8\nSW power propagate along the y-direction whereas for\n\u0012= 45\u000e, SWs propagate symmetrically in both the x-\nand y-direction (see Fig. 6d,e) owing to the geometri-\ncal symmetry of the system. To quantify the amount\nof propagation to the nearest vertex, we calculate the\ntransmission coe\u000ecient (P neighbor\u0000island=Pcentre\u0000island ) in\nboth the aforementioned cases where Pcentre\u0000island and\nPneighbhor\u0000island are corresponding powers at central and\na neighboring island, respectively. For \u0012= 0\u000e, we ob-\nserve that the B-mode shows e\u000ecient transmission to\nthe nearest neighbour vertices with transmission coe\u000e-\ncient\u00180.75 towards the direction perpendicular to the\ninitial bias \feld which reduces to 0.07 to the next nearest-\nneighbour vertex. However, along the \feld direction or\ndirection of charged vertices, transmission coe\u000ecient of\nonly\u00180:02 is observed indicating insigni\fcant transmis-\nsion along the direction of charged vertices. For \u0012= 45\u000e\nwith no charged vertices, symmetric transmission along\nboth the orthogonal directions with transmission coef-\n\fcient of\u00180.4 at nearest neighbor (see Fig. 6e) is ob-\nserved owing to the geometrical symmetry of the system.\nHowever, farthest nanomagnets of the nearest vertices\npositioned perpendicular to the \feld direction, display\nno power as compared to the nanomagnets positioned\nalong the \feld direction (see Fig.S4d of the supplemen-\ntary info). These results con\frm that although the bias\n\feld orientation plays a role in directional propagation\nof SWs, presence of charged vertices along with it sig-\nni\fcantly enhance the propagated power to the nearest\nvertex. Also a profound SW transmission of \u001875% can\nbe observed upto only \frst nearest neighbour thus lim-\niting the transmission upto \u0018450 nm (center-to-center\nvertex distance) in our case. The SE mode, however,\nbehave similarly as for OE vertices discussed before (see\nFig. S4c,d of supplementary info). For \u0012= 0\u000e, propaga-\ntion of SE mode is impeded along the charged vertices\nwhereas a symmetric propagation of SE mode is present\nfor chargeless vertices at \u0012= 45\u000e.\nIn large arrays of such sASI systems, often the structures\nare fabricated such that additional islands are patterned\nat the edges, thus forming CE systems, see Fig. 1(b) for\na schematic diagram. To complete our understanding\nof SWs in sASI systems, we next investigate the SW\nexcitation in CE-vertex. We note here that unlike for\nOE-vertices as discussed above, here the vertices are of\nmixed coordination. The vertices at edges are of coordi-\nnation no. 3 and that at the centre is of 4 (see Fig.1(b)).\nThis results in an uncompensated magnetic charge at\nthe odd coordinated 1-in/2-out or 2-in/1-out type ver-\ntices. Our earlier work [44] shows that for such mixed\ncoordinated vertices, the net distribution of charges at\nouter vertices screen and compensate any charge that is\nformed at inner/central even coordinated vertex. The\ncloud of charges at the border then form magnetic po-\nlarons [44{46] which is otherwise absent in case of OE\nvertices. Therefore, the behavior of SWs in magnetopo-\nlaronic environment is an open question which we address\nhere.Accordingly, we investigate the feasibility of forming\nmagnetic-polaron and its in\ruence on SW dynamics dur-\ning an external bias \feld sweep within \u0006150 mT. The\n\feld range of\u0006150 mT is chosen such that underlying\nspins are in non-saturated state and therefore the role of\ndipolar interactions is signi\fcant in stabilizing the mag-\nnetic state at each bias \feld. Our simulation results\nfor CE vertex show that within the aforementioned bias\n\feld regime, CE vertex does not exhibit non-zero mag-\nnetic charge at the center vertex whereas there is still\na magnetic charge cloud comprising of uncompensated\n\u0006Qmcharges (see inset of Fig. 7a) such that net mag-\nnetic charge of the system remains zero. The SW spec-\ntrum for this magnetic state at remanence shows two\nSW modes at 10.1 GHz and 10.6 GHz. However, spec-\ntrum for OE vertices under uniform excitation at rema-\nnence shows three SW modes at 6.4 GHz, 10.3 GHz, and\n10.6 GHz. Interestingly, we observe that weak mode at\n6.4 GHz for CE vertex with charge cloud is absent and\n10.3 GHz mode is red-shifted to 10.1 GHz. The red-shift\nmay be accounted due to the variation in the dipolar \feld\nenvironment for charge cloud in CE-vertex and localized\ncharge in OE-vertices. The absence of 6.4 GHz mode fur-\nther uphold our earlier statement that presence of local\nmagnetic charges may impede the propagation and exci-\ntation of SE mode (6.4 GHz). Details of the mode-pro\fles\nare given in supplementary information. We next deter-\nmined the evolution of the SW spectra in varying \feld.\nFor comparison, the resulting SW spectra for the two\ntypes of vertices viz., the CE and the OE vertices (dis-\ncussed above) at varying \felds are shown in Fig. 7 as 2D\nsurface plot. The Figs. 7(c) and (d) represent the evo-\nlution of SW modes, i.e., their excitation frequencies as\na function of static bias \feld applied along one of the\neasy axes of sASI. The simulations were carried out at\nevery 2 mT \feld in the entire \feld range of \u0006150 mT.\nInitially at 150 mT, both CE and OE vertices display\ntwo strong modes at \u00187.0 GHz and 15.4 GHz and three\nweak modes at\u00183 GHz, 4.2 GHz, and 10.6 GHz, respec-\ntively. Weak modes are not distinctively visible in 2D\nsurface plot due to very low relative intensity. The po-\nsition of these modes are shown by a rectangular dash\ndot box. For clarity, SW spectrum of CE and OE ver-\ntices at 150 mT is shown in Fig. 7(b). From the indi-\nvidual spectrum at 150 mT, weak modes can be easily\nidenti\fed. Furthermore, we observe that there is a slight\nshift in the frequencies ( \u00180.1 GHz) of the observed sw\nmodes which become signi\fcant( \u00181 GHz) for the lower\nfrequency mode at 3.7 GHz. Thus, mixed and even co-\nordinated vertices (CE and OE vertices) display nominal\ndiscrepancy in the frequencies of the SW mode which may\nbecome signi\fcant for the lower frequency mode. De-\ntails of the power pro\fles are provided in supplementary\ninformation. Further, we focus on the behavior of two\nprominent SW modes appearing at 15.5 GHz and 7 GHz\nwhich exist during complete \feld sweep regime and ex-\nhibit di\u000berent \feld dependence. Here, 15.5 GHz mode\nis the fundamental bulk mode associated to E-islands9\nFIG. 7. SW spectra for CE and OE vertices at remanence, i.e., Bbias=0 mT (a) and at Bbias=150 mT (b) applied along one of\nthe easy axes. Inset in (a) shows the schematic of macrospins arrangement and uncompensated charges at odd coordination\nvertices for CE vertex at remanence. (c) Dependence of SW mode frequency on bias \feld strength applied along one of the easy\naxes (left ordinate) and normalized magnetization vs bias \feld strength (right ordinate) plot during down-sweep for OE-vertices\nand (d) CE-vertex. white dash dot rectangular box show the position of weaker modes which are not apparent in the surface\nplot.\nwhereas 7 GHz mode is the fundamental bulk mode as-\nsociated to H-islands. Bulk mode associated to E-islands\nshows a linear response from 150 mT to a negative bias\n\feld strength of -122 mT and -128 mT for CE and OE\nvertices, respectively, with a positive slope during down-\nsweep. The observed SW response on external \feld sweep\ncan be understood by examining the variation in inter-\nnal spin con\fguration in E-islands. During the investi-\ngated \feld sweep regime, magnetic spins remain aligned\nalong the easy-axis and display a \frst sharp switching\nat -128 mT (OE vertex) and -122 mT (CE vertex) when\nZeeman \feld prevails over anisotropy \feld (see magneti-\nzation vs \feld plot superimposed on 2D SW frequencies\nvs \feld surface plot in Fig. 7). Thus, E-islands are sub-\njected to decreasing e\u000bective \feld during down-sweep of\nexternal bias \feld with underlying spin con\fguration re-\nmains invariant. Hence, frequency of fundamental SW\nmode arisen due to uniform spin alignment follow lin-\near trend in accord with larmour precession f/He\u000b\nup-to switching \feld. As the switching appears, funda-\nmental SW mode corresponding to E-island appearing at\n\u00185 GHz shifted to \u001814 GHz due to sudden variation in\nunderlying macrostate for both the cases. B H, on the\nother hand, behave di\u000berently as internal spin con\fgura-\ntion changes signi\fcantly during downsweep. As the bias\feld decreases from 150 mT to 0 mT, spins starts rotating\nto align itself along it's easy-axis viz., perpendicular to\nbias \feld orientation. As Field is further increased from\n0 mT to 150 mT in (-)ve x direction, spin reorients itself\nalong \feld direction by rotation. This results, in sym-\nmetric SW dispersion across 0 mT \feld. Furthermore,\nas the structure with mixed coordinated CE-vertex carry\na charge cloud around center vertex due to uncompen-\nsated charges at remanence, we observe a bifurcation at\n0 mT. However, for the even coordinated OE-vertex local\ncharges appear at two corner vertices along the bias \feld\norientation. This results in clear merging of SW modes\nand a shoulder peak.\nThus Detailed investigation performed for both geome-\ntries (see Fig.1) suggest that there is no perceptible dif-\nference on \feld dependence of spin wave dynamical be-\nhavior due to origin of local magnetic charge. However,\nSW modes excited in this sort of structures show sig-\nni\fcant mode tuning even by varying a single parame-\nter viz., Bias \feld strength. This can further be tune\nby giving angle between easy axes and external mag-\nnetic \feld as an additional degree of freedom. Hence,\nthe study suggest that these structures manifest itself as\na potential candidate for magnonic crystals. Our simu-\nlation work on OE vertex with localized excitation sug-10\ngests that appreciable directional propagation in prefer-\nential direction can be achieved in disconnected dipolar\ncoupled nanomagnets by carefully tuning the location of\nlocal charges or excited type-III state and bias \feld orien-\ntation. This suggest that information embedded in form\nof SW phase and amplitude can be processed without\nthe integration of current path which can be employ to\nresolve the persisting joule heating issue in cMOS based\ndevices. Hence, all magnonic directional propagation be-\nhavior in studied sASI shows prospect in information pro-\ncessing and de-multiplexing devices. Further, response of\nSW in these structures can be utilized for fundamental\nstudies such as the identi\fcation of higher order exci-\ntation e.g., monopole and anti-monopole from the CVmode.\nACKNOWLEDGMENTS\nWe acknowledge High Performance Computation (HPC)\nfacility of IIT Delhi. N.A. is thankful to the Council\nof Scienti\fc Industrial Research (CSIR), Government of\nIndia for research fellowship.\nREFERENCES\n[1] V. Kruglyak, S. Demokritov, and D. Grundler, Magnon-\nics. J. Phys. D 43, 260301 (2010).\n[2] S. Neusser and D. Grundler, Magnonics: spin waves on\nthe nanoscale, Adv. Mater. 21, 2927 (2009).\n[3] B. Lenk, H. Ulrichs, F. Garbs, and M. M /dieresis.ts1unzenberg,\nThe building blocks of magnonics, Phys. Rep. 507, 107\n(2011).\n[4] S. O. Demokritov and A. N. Slavin, Magnonics: From\nfundamentals to applications, Vol. 125 (Springer Science\n& Business Media, 2012).\n[5] S. A. Nikitov, D. V. Kalyabin, I. V. Lisenkov, A.\nN. Slavin, Y. N. Barabanenkov, S. A. Osokin, A. V.\nSadovnikov, E. N. Beginin, M. A. Morozova, Y. P.\nSharaevsky, et al., Magnonics: a new research area in\nspintronics and spin wave electronics, Physics-Uspekhi\n58, 1002 (2015).\n[6] H. Liu, C. Zhang, H. Malissa, M. Groesbeck, M. Kavand,\nR. McLaughlin, S. Jamali, J. Hao, D. Sun, R. A. David-\nson,et al. , Organic-based magnon spintronics, Nat. mat.\n17, 308 (2018).\n[7] Y. Au, M. Dvornik, O. Dmytriiev, and V. Kruglyak,\nNanoscale spin wave valve and phase shifter, Appl. Phys.\nLett. 100, 172408 (2012).\n[8] M. A. Abeed and S. Bandyopadhyay, Experimental\ndemonstration of an extreme subwavelength nanomag-\nnetic acoustic antenna actuated by spin{orbit torque\nfrom a heavy metal nanostrip, Adv. Mater. Technol. 5,\n1901076 (2020).\n[9] Q. Wang, P. Pirro, R. Verba, A. Slavin, B. Hillebrands,\nand A. V. Chumak, Recon\fgurable nanoscale spin-wave\ndirectional coupler, Sci. Adv. 4, e1701517 (2018).\n[10] K. Vogt, F. Y. Fradin, J. E. Pearson, T. Sebastian,\nS. D.Bader, B. Hillebrands, A. Ho\u000bmann, and H.\nSchultheiss, Realization of a spin-wave multiplexer, Nat.\nCommun. 5, 1 (2014).\n[11] M. Kostylev, A. Serga, T. Schneider, B. Leven, and, B.\nHillebrands, Spin-wave logical gates, Appl. Phys. Lett.\n87, 153501 (2005).\n[12] A. Papp, W. Porod, /acute.ts1A. I. Csurgay, and G. Csaba, /acute.ts1\nNanoscale spectrum analyzer based on spin-wave inter-\nference, Sci. Rep. 7, 1 (2017).\n[13] J. Chen, H. Wang, T. Hula, C. Liu, S. Liu, T. Liu, H.\nJia, Q. Song, C. Guo, Y. Zhang, et al., Recon\fgurable\nspinwave interferometer at the nanoscale,Nano Lett. 21,6237 (2021).\n[14] H. Yu, G. Duerr, R. Huber, M. Bahr, T. Schwarze,\nF. Brandl, and D. Grundler, Omnidirectional spin-wave\nnanograting coupler, Nat. Commun. 4, 1(2013).\n[15] J. Grollier, D. Querlioz, and M. D. Stiles, Spintronic\nnanodevices for bioinspired computing, Proc. IEEE 104,\n2024 (2016).\n[16] H. Arava, P. M. Derlet, J. Vijayakumar, J. Cui, N. S.\nBingham, A. Kleibert, and L. J. Heyderman, Computa-\ntional logic with square rings of nanomagnets, Nanotech-\nnology 29, 265205 (2018).\n[17] H. Arava, N. Leo, D. Schildknecht, J. Cui, J. Vijayaku-\nmar, P. M. Derlet, A. Kleibert, and L. J. Heyderman,\nEngineering relaxation pathways in building blocks of ar-\nti\fcial spin ice for computation, Phys. Rev. Appl. 11,\n054086 (2019).\n[18] F. Caravelli, G.-W. Chern, and C. Nisoli, Arti\fcial spin\nice phase-change memory resistors, New J. Phys. (2022).\n[19] J. Torrejon, M. Riou, F. A. Araujo, S. Tsunegi, G.\nKhalsa, D. Querlioz, P. Bortolotti, V. Cros, K. Yakushiji,\nA. Fukushima, et al., Neuromorphic computing with\nnanoscale spintronic oscillators, Nature 547, 428 (2017).\n[20] Y. Li, T. Polakovic, Y.L. Wang, J. Xu, S. Lendinez, Z.\nZhang, J. Ding, T. Khaire, H. Saglam, R. Divan, et al.,\nStrong coupling between magnons and microwave pho-\ntons in on-chip ferromagnet-superconductor thin-\flm de-\nvices, Phys. Rev. Lett. 123, 107701 (2019).\n[21] M. T. Ka\u000bash, S. Lendinez, and M. B. Jung\reisch, Nano-\nmagnonics with arti\fcial spin ice, Phys. Lett. A 402,\n127364 (2021).\n[22] S. Lendinez, M. Taghipour Ka\u000bash, and M. B.\nJung\reisch, Observation of mode splitting in arti\fcial\nspin ice: A comparative ferromagnetic resonance and\nbrillouin light scattering study, Appl. Phys. Lett. 118,\n162407 (2021).\n[23] J. C. Gartside, A. Vanstone, T. Dion, K. D. Stenning,\nD. M. Arroo, H. Kurebayashi, and W. R. Branford, Re-\ncon\fgurable magnonic mode-hybridisation and spectral\ncontrol in a bicomponent arti\fcial spin ice, Nat. Com-\nmun. 12, 1 (2021).\n[24] F. Caravelli and C. Nisoli, Logical gates embedding in\narti\fcial spin ice, New J. Phys. 22, 103052 (2020).\n[25] E. Iacocca, S. Gliga, and O. G. Heinonen, Tailoring spin\nwave channels in a recon\fgurable arti\fcial spin ice, Phys.11\nRev. Appl. 13, 044047 (2020).\n[26] S. Gliga, E. Iacocca, and O. G. Heinonen, Dynamics of\nrecon\fgurable arti\fcial spin ice: Toward magnonic func-\ntional materials, APL Mater. 8, 040911 (2020).\n[27] R. Wang, C. Nisoli, R. S. d. Freitas, J. Li, W. McConville,\nB. Cooley, M. Lund, N. Samarth, C. Leighton, V. H.\nCrespi, et al., Arti\fcial `spin ice' in a geometrically frus-\ntrated lattice of nanoscale ferromagnetic islands, Nature\n439, 303 (2006).\n[28] J. P. Morgan, A. Stein, S. Langridge, and C. H. Mar-\nrows, Thermal ground-state ordering and elementary ex-\ncitations in arti\fcial magnetic square ice, Nat. Phys. 7,\n75 (2011).\n[29] N. Keswani, R. J. Lopes, Y. Nakajima, R. Singh, N.\nChauhan, T. Som, D. S. Kumar, A. R. Pereira, and P.\nDas, Controlled creation and annihilation of isolated ro-\nbust emergent magnetic monopole like charged vertices\nin square arti\fcial spin ice, Sci. Rep. 11, 1 (2021).\n[30] M. B. Jung\reisch, J. Sklenar, J. Ding, J. Park, J. E. Pear-\nson, V. Novosad, P. Schi\u000ber, and A. Ho\u000bmann, High fre-\nquency dynamics modulated by collective magnetization\nreversal in arti\fcial spin ice, Phys. Rev. Appl. 8, 064026\n(2017).\n[31] V. Kapaklis, U. B. Arnalds, A. Harman-Clarke, E. T. Pa-\npaioannou, M. Karimipour, P. Korelis, A. Taroni, P. C.\nHoldsworth, S. T. Bramwell, and B. Hj /dieresis.ts1orvarsson, Melt-\ning arti\fcial spin ice, New J. Phys. 14, 035009 (2012).\n[32] Y. Qi, T. Brintlinger, and J. Cumings, Direct observation\nof the ice rule in an arti\fcial kagome spin ice, Phys. Rev.\nB77, 094418 (2008)\n[33] A. Talapatra and A. Adeyeye, Coupled magnetic nanos-\ntructures: Engineering lattice con\fgurations, Appl.\nPhys. Lett. 118, 172404 (2021).\n[34] A. K. Chaurasiya, A. K. Mondal, J. C. Gartside, K. D.\nStenning, A. Vanstone, S. Barman, W. R. Branford, and\nA. Barman, Comparison of spin-wave modes in connected\nand disconnected arti\fcial spin ice nanostructures using\nbrillouin light scattering spectroscopy, ACS nano (2021).[35] S. Saha, J. Zhou, K. Hofhuis, A. K /acute.ts1akay, V. Scagnoli,\nL. J. Heyderman, and S. Gliga, Spin-wave dynamics and\nsymmetry breaking in an arti\fcial spin ice, Nano Lett.\n21, 2382 (2021).\n[36] N. Arora and P. Das, Spin wave spectral probing of de-\ngenerate microstates in building-block of square arti\fcial\nspin ice, AIP Adv. 11, 035030 (2021).\n[37] D. M. Arroo, J. C. Gartside, and W. R. Branford, Sculpt-\ning the spin-wave response of arti\fcial spin ice via mi-\ncrostate selection, Phys. Rev. B 100, 214425 (2019).\n[38] E. Iacocca, S. Gliga, R. L. Stamps, and O. Heinonen,\nRecon\fgurable wave band structure of an arti\fcial square\nice, Phys. Rev. B 93, 134420 (2016).\n[39] F. Montoncello, L. Giovannini, W. Bang, J. Ketterson,\nM. Jung\reisch, A. Ho\u000bmann, B. W. Farmer, and L. E.\nDe Long, Mutual in\ruence between macrospin reversal\norder and spin-wave dynamics in isolated arti\fcial spinice\nvertices, Phys. Rev. B 97, 014421 (2018).\n[40] A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen, F.\nGarcia-Sanchez, and B. Van Waeyenberge, The design\nand veri\fcation of mumax3, AIP Adv. 4, 107133 (2014).\n[41] K. H. J. Buschow, Handbook of magnetic materials (El-\nsevier, 2003).\n[42] A. Barman and S. Barman, Dynamic dephasing of mag-\nnetization precession in arrays of thin magnetic elements,\nPhys. Rev. B 79, 144415 (2009).\n[43] C. Castelnovo, R. Moessner, and S. L. Sondhi, Magnetic\nmonopoles in spin ice, Nature 451, 42 (2008).\n[44] N. Keswani, R. Singh, Y. Nakajima, T. Som, and P. Das,\nAccessing low-energy magnetic microstates in square ar-\nti\fcial spin ice vertices of broken symmetry in static mag-\nnetic \feld, Phys. Rev. B . 102, 224436 (2020).\n[45] G.-W. Chern and P. Mellado, Magnetic monopole po-\nlarons in arti\fcial spin ices, EPL 114, 37004 (2016).\n[46] I. Gilbert, G.-W. Chern, S. Zhang, L. O'Brien, B. Fore,\nC. Nisoli, and P. Schi\u000ber, Emergent ice rule and magnetic\ncharge screening from vertex frustration in arti\fcial spin\nice, Nat. Phys. 10, 670 (2014)"    },    {        "title": "0811.2456v1.Dynamics_in_two_leg_spin_ladder_with_a_four_spin_cyclic_interaction.pdf",        "content": "arXiv:0811.2456v1  [cond-mat.str-el]  17 Nov 2008Dynamics in two-leg spin ladder with a four-spin cyclic inte raction\nS. Nishimoto\nLeibniz-Institut f¨ ur Festk¨ orper- und Werkstoffforschung Dresden, D-01171 Dresden, Germany\nM. Arikawa\nInstitute of Physics, University of Tsukuba 1-1-1 Tennodai , Tsukuba Ibaraki 305-8571, Japan\n(Dated: November 3, 2018)\nWe study two-leg Heisenberg ladder with four-spin cyclic in teraction using the (dynamical)\ndensity-matrix renormalization group method. We demonstr ate the dependence of the low-lying\nexcitations in the spin wave, staggered dimer order, and sca lar-chirality order structure factors on\nthe four-spin cyclic interaction. We find that the cyclic int eraction enhances spin-spin correlations\nwith wave vector around momentum ( qx,qy) = (π\n2,0). Also, the presence of long-range order in the\nstaggered dimer and scalar-chirality phases is confirmed by aδ-function peak contribution of the\nstructure factors at energy ω= 0.\nPACS numbers: 75.10.Jm, 75.30.Kz, 75.40.Gb, 75.40.Mg\nFormanyyears, it hadbeen generallybelieved that the\nmagnetic properties of undoped high- Tcmaterials can\nbe well described by the two-dimensional (2D) Heisen-\nberg model with nearest-neighbor exchange interaction\nJ. However, four-spin cyclic interaction Khas been in-\ncreasinglyrecognizedasanon-negligiblecorrectiontothe\nHeisenberg model. The cyclic interaction comes from the\nforth-order processes in the strong-coupling limit of the\nsingle-band Hubbard model at half filling.1The impor-\ntance of this interaction was initially proposed in the 2D\nsolid3He, which has the hard-core correlations between\nspin-1\n2fermions.2\nIn fact, a substantial value K= 0.24Jwas proposed\nfor 2D copper oxide La 2CuO4by an accurate fit of the\nmagnon dispersion.3A close value was also suggested by\nan analysis of the Raman-scattering data.4Such mag-\nnitude of the four-spin cyclic interaction must have a\nconsiderable influence at least quantitatively on the low-\nenergy spin physics. Similar situations have been re-\nported for several two-leg spin-ladder systems: the ex-\nchange interactions were estimated as J/bardbl=J⊥= 110\nmeV andK= 16.5 meV for La 6Ca8Cu24O41(Ref. 5);\nJ/bardbl= 186 meV, J⊥= 124 meV, and K= 31 meV for\nLa4Sr10Cu24O41(Ref. 6);J/bardbl= 165 meV, J⊥= 150 meV,\nandK= 15 meV for SrCu 2O3(Ref. 7), where J/bardblandJ⊥\nare exchange interactions in the leg and rung directions,\nrespectively.\nMotivated by those observations, the ground-state\nproperties of two-leg spin-1\n2Heisenberg ladder with the\nfour-spin cyclic interaction have been intensively stud-\nied.6,8,9,10,11Also, the effect of magnetic field on the\ngroundstatehasbeeninvestigated.12,13Furthermore,the\nspectral features of the spin structure factor have been\nexamined by the exact diagonalization,perturbation the-\nory, and density-matrix renormalization group (DMRG)\nmethod.14,15,16,17The spin dynamics for small cyclic in-\nteractions is thus well understood, but the dynamical\nproperties for other correlations and/or relatively large\ncyclic interactions are still open. In this paper, we study\nthe dynamical structure factors of staggereddimer order,\n/C2/CZ/C2/BR/C3/DD/DC\nFIG. 1: Lattice structure of the two-leg Heisenberg ladder.\nJ/bardbl(J⊥) is the exchange interaction in the leg (rung) direction\nandKis the four-spin cyclic interaction. The x- (y-) axis is\ndefined as the leg (rung) direction.\nscalar-chirality order, and spin waves for a wide range of\nthe four-spin cyclic interaction to give a deeper insight\ninto our knowledge of the low-lying excitations, using the\ndynamical DMRG (DDMRG) method.18\nThe Hamiltonian of the two-leg spin-1\n2Heisenberg lad-\nder with the four-spin cyclic interaction is given by\nH=J/bardbl/summationdisplay\nx,y/vectorSx,y·/vectorSx+1,y+J⊥/summationdisplay\nx/vectorSx,1·/vectorSx,2\n+K/summationdisplay\nx(Px+P−1\nx), (1)\nwith the cyclic permutation operator\nPx+P−1\nx=/vectorSx,1·/vectorSx,2+/vectorSx+1,1·/vectorSx+1,2+/vectorSx,1·/vectorSx+1,1\n+/vectorSx,2·/vectorSx+1,2+/vectorSx,1·/vectorSx+1,2+/vectorSx,2·/vectorSx+1,1\n+4(/vectorSx,1·/vectorSx,2)(/vectorSx+1,1·/vectorSx+1,2)\n+4(/vectorSx,1·/vectorSx+1,1)(/vectorSx,2·/vectorSx+1,2)\n−4(/vectorSx,1·/vectorSx+1,2)(/vectorSx,2·/vectorSx+1,1) (2)\nwhere/vectorSx,yis a spin-1\n2operatorat a site ( x,y) [see Fig. 1].\nFor simplicity, we focus on the case of J/bardbl=J⊥=Jand\ntakeJ= 1 as the unit of energy hereafter. The ground-\nstate phase diagram was obtained in Ref. 8 as follows:\nthe system has a rung singlet phase for −3.33<∼K<∼\n0.23, a staggered dimer long-range order (LRO) phase\nfor 0.23<∼K<∼0.5, a scalar-chirality LRO phase for2\n0.5<∼K<∼2.8, a dominant vector chirality phase for\n2.8<∼K, and a ferromagnetic phase for K<∼−3.33.\nLet us define the dynamical structure factor as\nA(/vector q,ω) =/summationdisplay\nν/angbracketleftψ0/vextendsingle/vextendsingleˆO−/vector q/vextendsingle/vextendsingleψν/angbracketright/angbracketleftψν/vextendsingle/vextendsingleˆO/vector q/vextendsingle/vextendsingleψ0/angbracketright\n×δ(ω−Eν+E0),(3)\nwhere|ψν/angbracketrightis theν-th eingenstate with the eigenenergy\nEνandˆO/vector qis the Fourier transformation of the quantity-\ndependent operator ˆO/vector r. Theδ-function is replaced by\na Lorenzian with width ηin our numerical calculations.\nWe now study the following three kinds of the dynamical\nstructure factor corresponding to three phases at −3.3<∼\nK<∼2.8. The first is spin structure factor S(/vector q,ω) with\nthe operator\nˆO/vector r=Sz\nx,y, (4)\nwhereSz\nx,yisz-component of the total spin, the second\nis dimer order structure factor D(/vector q,ω) with\nˆO/vector r=/vectorSx−1,y·/vectorSx,y−/vectorSx,y·/vectorSx+1,y, (5)\nand the third is scalar-chirality structure factor C(/vector q,ω)\nwith\nˆO/vector r=/vectorSx,1·(/vectorSx+1,1×/vectorSx+1,2). (6)\nBy integrating Eq. (3), we can easily obtain the static\nstructure factor\nA(/vector q) =/angbracketleftψ0/vextendsingle/vextendsingleˆO−/vector qˆO/vector q/vextendsingle/vextendsingleψ0/angbracketright. (7)\nWe employ the DDMRG method18which is an ex-\ntension of the standard DMRG method.19It has been\ndeveloped for calculating dynamical correlation func-\ntions at zero temperature in quantum lattice models.\nThis method has been successfully applied to the one-\ndimensional Heisenberg model.20We now calculate the\ndynamical structure factor (3) with applying the peri-\nodic boundary conditions in the leg ( x) direction. We\nfix the system length L= 32 andη= 0.1 if not other-\nwise stated. In the DDMRG calculation, a required CPU\ntime increases rapidly with the number of the density-\nmatrix eigenstates so that we would like to keep it as few\nas possible; meanwhile, the DDMRG approach is based\non a variational principle so that we have to prepare a\n‘good trial function’ ofthe ground state with the density-\nmatrix eigenstates as much as possible. Therefore, we\nkeepm= 600 to obtain true ground state in the first\nten DDMRG sweeps and keep m= 300 to calculate the\nspectral functions. In this way, the maximum truncation\nerror, i.e., the discarded weight, is about 3 ×10−4, while\nthe maximum errorin the ground-state and low-lying ex-\ncited states energies is about 10−2.\nTo begin with, we consider the spin structure fac-\ntor. In Fig. 2, we show the DMRG results of the static\nand dynamical spin structure factors for the rung sin-\nglet (K= 0.1), staggered dimer LRO ( K= 0.4), and0\r5\r10\r15\r\n0\r π\r0\r π\r\n0\r1\r2\r3\r4\r0\r1\r2\r3\r4\r0\r1\r2\r\n0\r1\r2\r\n0\r1\r2\r3\r\n0\r1\r2\r3\r\n/D5/DD\n/BP /BC /D5/DD\n/BP /AP/CB /B4 /DI /D5 /B5/D5/DC\n/D5/DC\n/C3 /BP /BC /BM /BD /C3 /BP /BC /BM /BD/C3 /BP /BC /BM /BG /C3 /BP /BC /BM /BG/C3 /BP /BC /BM /BJ /C3 /BP /BC /BM /BJ/CB/D4/CX/D2 /D7/D8/D6/D9\r/D8/D9/D6/CT /CU/CP\r/D8/D3/D6 /CB /B4 /DI /D5 /BN /AX /B5\n/D5/DC\n/BP/AP/BC/AP/BC/AP/BC/AX /AX\n/B4/CP/B5/B4/CQ/B5\n/AX /B4 /DI /D5 /B5/BC /AP /BC /AP/D5/DC\n/D5/DC/AX /B4 /DI /D5 /B5/BC /AP /BC /AP/D5/DC\n/D5/DC\nFIG. 2: (a) Static spin structure factor. (b) Dynamical spin\nstructure factor for K= 0.1 (Top), K= 0.4 (Middle), and\nK= 0.7 (Bottom). Left and right panels correspond to the\nresults for qy= 0 and qy=π, respectively. Insets: lower-\nedge of the two-spinon continuum ( qy= 0) and one-spinon\ndispersion ( qy=π). The dashed line denotes the perturbative\nresultω(qx,qy=π) = 1.186+0.558cos(qx)−0.271cos(2 qx)+\n0.071cos(3 qx).\nscalar-chirality LRO ( K= 0.7) phases. The dispersion\nrelationsω(/vector q) are also plotted in the insets. The spectra\nforqy= 0 andπexhibit the two-triplon and one-triplon\ncontributions, respectively. In the absence of the cyclic\ninteraction, i.e., K= 0,21,22it is known that the spin dis-\npersion has two minima at qx= 0,πand a maximum at\nqx∼2π/3 forqy= 0; whereas, two minima at qx= 0,π\nand a maximum at qx∼π/3 forqy=π. Those features\nhave been confirmed to remain qualitatively unchanged\natK<∼0.075.16ForK= 0.1, however, the minima at\n(qx,qy) = (π,0),(0,π) are no longer visible [see the in-\nsets of the top panels in Fig. 2(b)], i.e., the dispersions\nare nearly flat around ( qx,qy)∼(π,0),(0,π). The one-3\ntriplon excitation ( qy=π) is in good agreement with the\nperturbative result.14This is also consistent with other\nnumerical study.15\nWhen the cyclic interaction is further increased to\nK= 0.4, we can see a drastic change of the spectra\nfor bothqy= 0 andπ: especially, an enhancement\nof peaks around ( qx,qy) = (π\n2,0) and a reduction of\npeaks around ( qx,qy) = (π,π) are derived. It is be-\ncause the cyclic interaction leads to a repulsive inter-\naction between neighboring rung triplets. In addition, a\nnode emerges at qx=π\n2for bothqyvalues and a re-\nlationω(qx,qy= 0) =ω(π−qx,qy=π) appears to\nbe satisfied. They would indicate a two-fold degener-\nate ground state with a broken translational symmetry,\nwhich is consistent with the staggered dimer order state.\nForK= 0.7, the peaks around ( qx,qy) = (π\n2,0) are still\nmoreenhancedand, whereas,thelow-energyspectralfea-\ntures forqy=πseem to be much reduced. Actually, the\none-triplon contribution is shunted off to the high-energy\nexcitations since the static structure factor for qy=πis\nnot much suppressed. In short, from the standpoint of\nspin-spin correlation the four-spin cyclic interaction may\nwork for enhancing a spin-density wave with wave vector\n(qx,qy) = (π\n2,0) and for reducing the antiferromagnetic\ncorrelation S(π,π) [see Fig. 2(a)].\nNext, we turn to the dimer-orderstructure factor. Fig-\nure 3 shows the DMRG results of the static and dynam-\nical dimer order structure factors for K= 0.1, 0.4, and\n0.7. For comparison, the results of D(/vector q,ω) forK= 0\nare shown in the insets of the top panels of Fig. 3(b).\nIn the rung-singlet phase, the ground state is approxi-\nmately expressed as the product of local rung singlets\nwith gap ∆ ∼ O(J⊥). The lowest excitation comes\nfrom the formation of a leg singlet with coupling en-\nergy∼J/bardbl\n2as well as the collapse of two rung singlets.\nForK= 0, therefore, undispersive sharp peaks appear\naroundω=O(2∆−J/bardbl\n2)∼1.5 forqy= 0; whereas,\nthe spectra for qy=πconsist of broad continua at\nω>O(2∆−J/bardbl).\nWhen small cyclic interaction ( K= 0.1) is introduced,\nwe can see a strong influence on the continua around\n(qx,qy) = (π,π), i.e., they are significantly shifted to-\nwards lower energies. It implies that the gap ∆ is re-\nduced rapidly as Kincreases. For K= 0.4, the continua\nare further drastically changed: a pronounced peak ap-\npears at (qx,qy) = (π,π) andω∼0; also, most of the\nspectral weight concentrates around the peak. On the\nother hand, the spectral weights for qy= 0 are totally\nsuppressed. The pronounced peak is well fitted by a\nLorenzian with η= 0.1, as shown in the inset of the mid-\ndle panel of Fig. 3(b). In other words, the spectrum for\n(qx,qy) = (π,π)mayconsistofa δ-functionpeakat ω= 0\nand a gapfull continuum. They would be a signature of\nlong-rangestaggereddimer order. For K= 0.7, the spec-\ntral weights around ( qx,qy) = (π,π) are much reduced\nand a gap opens. It means that the staggered dimer\norder is no longer dominant in the ground state. Never-\ntheless, the spectral weights around ( qx,qy) = (π,π) are0\r1\r2\r3\r4\r5\r0\r1\r2\r3\r4\r5\r0\r1\r2\r3\r4\r5\r 0\r1\r2\r3\r4\r5\r\n0\r-0.5\r 0.5\r0\r5\r10\r\n0\r π\r0\r π\r\n/D5/DD\n/BP /BC /D5/DD\n/BP /AP/BW /B4 /DI /D5 /B5/D5/DC\n/D5/DC\n/C3 /BP /BC /BM /BD /C3 /BP /BC /BM /BD/C3 /BP /BC /BM /BG /C3 /BP /BC /BM /BG/C3 /BP /BC /BM /BJ /C3 /BP /BC /BM /BJ/BW/CX/D1/CT/D6/B9/D3/D6/CS/CT/D6 /D7/D8/D6/D9\r/D8/D9/D6/CT /CU/CP\r/D8/D3/D6 /BW /B4 /DI /D5 /BN /AX /B5\n/D5/DC\n/BP/AP/BC/AP/BC/AP/BC/AX /AX\n/B4/CP/B5/B4/CQ/B5\nFIG. 3: (a) Static dimer-order structure factor. (b) Dynam-\nical dimer-order structure factors for K= 0.1 (Top), K= 0.4\n(Middle), and K= 0.7 (Bottom). Left and right panels cor-\nrespond to the results for qy= 0 and qy=π, respectively.\nInsets of the top panels: D(/vector q,ω) forK= 0 with L= 16 and\nη= 0.2. Inset of the muddle panel: A Lorenzian fit of the\npeak at ( qx,qy) = (π,π) andω∼0 withη= 0.1.\nstill significant, as seen in Fig. 3(a), so that the dimer-\norder correlation could just be changed from long-range\norder to short-range order. It is consistent with the fact\nthat the staggered dimer order parameter is finite even\nin the scalar-chirality LRO phase.8\nFinally, we look at the scalar-chirality structure fac-\ntor. The DMRG results of the static and dynamical\nscalar-chirality structure factors for K= 0.1,K= 0.4,\nK= 0.7 are shown in Fig. 4. For K= 0.1, the lowest ex-\ncitations are described by (almost) undispersive peaks\naroundω∼ O(2∆−J/bardbl\n2) in analogy with the dimer-\norder structure factor. For K= 0.4, the spectra form\na continuum bounded by the branches ω(qx)∼Asin(qx)\nandω(qx)∼2Asin(qx/2), except that a gap opens at4\n0\r0.1\r0.2\r0.3\r0.4\r\n0\r π\r\n0\r1\r2\r3\r4\r5\r0\r1\r2\r3\r4\r5\r0\r1\r2\r3\r4\r5\r0\r-0.5\r 0.5\r/BV /B4 /DI /D5 /B5/D5/DC\n/C3 /BP /BC /BM /BD/C3 /BP /BC /BM /BG/C3 /BP /BC /BM /BJ/BV /B4 /DI /D5 /BN /AX /B5\n/D5/DC\n/BP/AP/BC/AX /AX /AX\n/B4/CP/B5/B4/CQ/B5\nFIG. 4: (a) Static scalar-chirality structure factor. (b) D y-\nnamical scalar-chirality structure factors for K= 0.1 (left),\nK= 0.4 (center), and K= 0.7 (right). Inset of the bottom\npanel: A Lorenzian fit of the peak at qx=πandω∼0 with\nη= 0.1. The dashed lines denote the lower and upper edges\nof the continuum.\nqx= 0 andπ. The existence of the gap implies that the\nscalar-chirality order still belongs to an excited state. If\nwe assume a complete staggered dimer order, i.e., theground state is the product of local dimer singlets, the\nscalar-chiralityoperator(6) maybe effectivelyreducedas\n/vectorSx,1·(/vectorSx+1,1×/vectorSx+1,2)/vextendsingle/vextendsingleψ0/angbracketright ≈(/vectorSx,1/2)/vextendsingle/vextendsingleψ0/angbracketright. Thus, the dis-\npersionsaresimilartothoseofthespinstructurefactorin\nthe one-dimensional spin-Peierls Heisenberg model.23,24\nForK= 0.7, we can see the closing ofthe gap and, more-\nover, the appearance of a dominant peak at qx=πand\nω∼0. This peak is well fitted by a Lorenzian with\nη= 0.1, as shown in the inset of the right panel of\nFig. 4(b). Hence, the spectrum for qx=πis composed\nof aδ-function peak at ω= 0 and a gapfull continuum,\nas isD(/vector q,ω) in the staggered dimer LRO phase. It must\nindicate the presence of the scalar-chirality LRO.\nIn summary, we study the two-leg Heisenberg ladder\nwith the cyclic four-spin interaction. The static and dy-\nnamical structure factors for the spin waves, staggered\ndimerorder,andthescalar-chiralityorderparametersare\ncalculated with the DDMRG method. We find that the\nspin-spin correlation with wave vector ( qx,qy) = (π\n2,0) is\nenhanced by the cyclic interaction. We also confirm the\npresence of long-range order in the staggered dimer and\nscalar-chirality phases by a δ-function peak contribution\nof the structure factor at ω= 0.\nWe thank M. Nakamura, S. Suga, T. Tohyama, T. Hik-\nihara, T. Momoi, I. Maruyama, S. Tanaya and Y. Hat-\nsugai for useful discussions. This work has been sup-\nported in part by the University of Tsukuba Research\nInitiative and Grants-in-Aid for Scientific Research,\nNo.20654034 from JSPS and No.220029004 (Physics of\nNew Quantum Phases in Super-clean Materials) and\nNo.20046002 (Novel States of Matter Induced by Frus-\ntration) on Priority Areas from MEXT for M.A.\n1M. Takahashi, J. Phys. C: Solid State Phys. 10, 1289\n(1977).\n2M. Roger, J.H. Hetherington, and J.M. Delrieu, Rev. Mod.\nPhys.55, 1 (1983).\n3A.A. Katanin and A.P. Kampf, Phys. Rev. B 66,\n100403(R) (2002).\n4Y. Honda, Y. Kuramoto, and T. Watanabe, Phys. Rev. B\n47, 11329 (1993).\n5M. Matsuda, K. Katsumata, R.S. Eccleston, S. Brehmer\nand H.-J. Mikeska, Phys. Rev. B 62, 8903 (2000).\n6S. Notbohm, P. Ribeiro, B. Lake, D.A. Tennant,\nK.P. Schmidt, G.S. Uhrig, C. Hess, R. Klingeler, G. Behr,\nB. Buchner, M. Reehuis, R.I. Bewley, C.D. Frost,\nP.Manuel, andR.S.Eccleston, Phys.Rev.Lett. 98, 027403\n(2007).\n7Y. Mizuno, T. Tohyama, and S. Maekawa, J. Low. Temp.\nPhys.117, 389 (1999).\n8A. L¨ auchli, G. Schmid, and M. Troyer, Phys. Rev. B 67,\n100409(R) (2003).\n9T. Hikihara, T. Momoi, and X. Hu, Phys. Rev. Lett. 90,\n087204 (2003).\n10K.P. Schmidt, H. Monien, and G.S. Uhrig, Phys. Rev. B\n67, 184413 (2003).\n11K. Hijii, S. Qin, and K. Nomura, Phys. Rev. B 68, 134403(2003).\n12A. Nakasu, K. Totsuka, Y. Hasegawa, K. Okamoto, and\nT. Sakai, J. Phys.: Condens. Matter 13, 7421 (2001).\n13T. Hikihara and S. Yamamoto, J. Phys. Soc. Jpn. 77,\n014709 (2008).\n14S. Brehmer, H.-J. Mikeska, M. M¨ uller, N. Nagaosa, and\nS. Uchida, Phys. Rev. B 60, 329 (1999).\n15N. Haga and S. Suga, Phys. Rev. B 66, 132415 (2002).\n16T.S. Nunner, P. Brune, T. Kopp, M. Windt, and\nM. Gr¨ uninger, Phys. Rev. B 66, 180404(R) (2002).\n17K. Inaba and S. Suga, Prog. Theor. Phys. Suppl. 159, 128\n(2005).\n18E. Jeckelmann, Phys. Rev. B 66, 045114 (2002).\n19S.R. White, Phys. Rev. Lett. 69, 2863 (1992); Phys. Rev.\nB48, 10345 (1993).\n20S. Nishimoto and M. Arikawa, Int. J. Mod. Phys. B 21,\n2262 (2007).\n21T. Barnes, E. Dagotto, J. Riera, E.S. Swanson, Phys. Rev.\nB47, 3196 (1993).\n22R. Eder, Phys. Rev. B 57, 12832 (1998).\n23A.M. Tsvelik, Phys. Rev. B 45, 486 (1992).\n24S. Haas and E. Dagotto, Phys. Rev. B 52, R14396 (1995)."    },    {        "title": "2307.16311v1.Ab_initio_predictions_of_spin_relaxation__dephasing_and_diffusion_in_solids.pdf",        "content": "Ab-initio predictions of spin relaxation, dephasing and diffusion in solids\nJunqing Xu1, 2and Yuan Ping3, 4,∗\n1Department of Physics, Hefei University of Technology, Hefei, Anhui, China\n2Department of Chemistry and Biochemistry, University of California, Santa Cruz, CA 95064, USA\n3Department of Materials Science and Engineering,\nUniversity of Wisconsin-Madison, WI, 53706, USA\n4Department of Physics, University of California, Santa Cruz, CA, 95064, USA\n(Dated: August 1, 2023)\nSpin relaxation, dephasing and diffusion are at the heart of spin-based information technology.\nAccurate theoretical approaches to simulate spin lifetimes ( τs), determining how fast the spin po-\nlarization and phase information will be lost, are important to the understandings of underlying\nmechanism of these spin processes, and invaluable to search for promising candidates of spintronic\nmaterials. Recently, we develop a first-principles real-time density-matrix (FPDM) approach to\nsimulate spin dynamics for general solid-state systems. Through the complete first-principles’ de-\nscriptions of light-matter interaction and scattering processes including electron-phonon, electron-\nimpurity and electron-electron scatterings with self-consistent spin-orbit coupling, as well as ab\ninitio Land´ e g-factor, our method can predict τsat various conditions as a function of carrier den-\nsity and temperature, under electric and magnetic fields. By employing this method, we successfully\nreproduce experimental results of disparate materials and identify the key factors affecting spin re-\nlaxation, dephasing, and diffusion in different materials. Specifically, we predict that germanene\nhas long τs(∼100 ns at 50 K), a giant spin lifetime anisotropy and spin-valley locking effect under\nelectric fields, making it advantageous for spin-valleytronic applications. Based on our theoretical\nderivations and ab initio simulations, we propose a new useful electronic quantity, named spin-flip\nangle θ↑↓, for the understanding of spin relaxation through intervalley spin-flip scattering processes.\nOur method can be further applied to other emerging materials and extended to simulate exciton\nspin dynamics and steady-state photocurrents due to photogalvanic effect.\nI. INTRODUCTION\nIn last two decades, spintronics in unconventional\nsemiconductors and metals, including two-dimensional\nmaterials and their heterostructures[1, 2], topological and\nmagnetic materials[3, 4], hybrid perovskites[5, 6] etc.,\nhave drawn significant interests owing to its unprece-\ndented potentials in microelectronics and next generation\nlow-power electronics. Spin, as a pure quantum mechani-\ncal object, is the fundamental information carrier instead\nof charge, with much less energy dissipation. Ideally one\nwants such information preserved as long as possible for\nstable manipulation. Therefore, understanding how spins\nrelax and transport is of central importance in spintron-\nics.\nSpin is an unconserved quantity in solids due to its\ncoupling with other quantities, such as electron orbital.\nTherefore, after excess spins being generated, spin can\nloose its polarization (relaxation) and phase (dephasing)\ndue to coupling with environment. One critical param-\neter describing the timescale of such processes is spin\nlifetime τsincluding T 1(relaxation) and T 2(dephasing),\nwhich is often required to be sufficiently long for stable\ndetection and manipulation of spin. Accurate and reli-\nable theoretical approaches to simulate τsare demanded\nfor the detailed understandings of spin dynamics and\ntransport phenomena, and designing new spintronics ma-\n∗yping3@wisc.eduterials and devices.\nPreviously, methods based on model Hamiltonian with\nempirical parameters[1, 7, 8] were extensively employed\nfor simulation of spin relaxation, dephasing, and diffusion\nin solids. While these methods provide some mechanis-\ntic insight, they do not serve as predictive tools for the\ndesign of new materials and may sometimes lead to qual-\nitatively incorrect predictions due to the use of simplified\nelectronic structures and interactions.\nOn the other hand, the existing first-principles\nmethodology for τshas been mostly based on Fermi’s\nGolden rule[9, 10] considering spin-flip transitions, which\nis only applicable to systems with inversion symmetry\nor high spin polarization, not suitable to lots of materi-\nals with broken inversion symmetry, promising for quan-\ntum computing and spintronics applications [1, 7]. Other\nfirst-principles techniques like Time-Dependent Density\nFunctional Theory (TDDFT)[11] are challenging for crys-\ntalline systems due to high computational cost for de-\nscribing phonon relaxations that require large supercells.\nMore importantly, long simulation time over nanoseconds\noften required by spin relaxation is a major difficulty for\nTDDFT, which is only practical for tens to a few hun-\ndred fs. While spin dynamics based on TDDFT has been\nrecently performed for ultrafast demagnetization of mag-\nnetic systems within tens of fs[12–14], the intrinsic time\nscale and supercell limitations mentioned above remain.\nA recent development of spin phonon relaxation based on\nspin-spin correlation function may be a promising path-\nway, where its applicability for various spin relaxation\nand dephasing pathways remains to be explored [15].arXiv:2307.16311v1  [cond-mat.mtrl-sci]  30 Jul 20232\nTo overcome these challenges, we recently devel-\noped a first-principles real-time density-matrix (FPDM)\napproach[16–18] to simulate spin dynamics and pump-\nprobe Kerr-rotation for general solid-state systems. The\napproach is free from empirical parameters and is thus\nof great predictive power. Through the complete first-\nprinciples descriptions of light-matter interaction and\nscattering processes including electron-phonon, electron-\nimpurity and electron-electron scatterings with self-\nconsistent spin-orbit coupling (SOC) and Land´ e g-factor,\nour method can predict τsas a function of temperature\nand carrier density, under electric and magnetic fields.\nThe method was applied to disparate materials, includ-\ning semiconductors and metals, with and without in-\nversion symmetry, in good agreement with experimental\nresults[16–21].\nIn this article, we briefly introduce the theory and im-\nplementation of our method, present its applications and\nshow its predictive power using a two-dimensional Dirac\nmaterial - Germanene as a showcase. Through detailed\ntheoretical analysis, we show how ab-initio simulations\nimprove our understandings of spin relaxation mecha-\nnisms and are used to identify the key quantities and/or\nfactors to spin relaxation, dephasing, and diffusion. At\nthe end, we discuss how our FPDM approach can be gen-\neralized/extended to simulate spin dynamics of excitons,\ninstead of free carriers, and transport properties such as\nphotocurrent and spin currents in broken inversion sys-\ntems (photogalvanic effect).\nII. THEORY\nA. Density-matrix (DM) master equation\n1. Quantum master equation\nTo provide a general formulation of quantum dynamics\nin solid-state materials, we start from the Liouville-von\nNeumann equation in the Schr¨ odinger picture,\ndρ(t)\ndt=−i\nℏ[H, ρ(t)], (1)\nH=H0+H′, (2)\nwhere H,H0, and H′are total, unperturbed and per-\nturbation Hamiltonians respectively. In this work,\nH0=He+Hph+Hphoton , (3)\nH′=He−light+He−ph+He−i+He−e, (4)\nwhere He,HphandHphoton are unperturbed single-\nparticle electronic, phonon, and photon Hamiltonian re-\nspectively. He−lightis the light-matter interaction term.\nHe−ph,He−iandHe−edescribe the electron-phonon (e-\nph), electron-impurity (e-i), and electron-electron (e-e)\ninteraction, respectively.\nIn practice, the many-body density matrix in Eq. 1\nis reduced to one-particle density matrix for electrons,where the environmental degree of freedom is traced\nout[22], with a proper truncated BBGKY hierarchy [23,\n24]. The total rate of change of the electronic DM ρis\nseparated into terms related to different parts of Hamil-\ntonian,\ndρ\ndt=dρ\ndt|coh+dρ\ndt|scatt, (5)\nwhere [ H, ρ] =Hρ−ρH.dρ\ndt|cohdescribes the coherent\ndynamics of electrons including the free-particle dynam-\nics, the field-induced dynamics, etc.dρ\ndt|scattcaptures the\nscattering between electrons and other particles. Their\ndetailed forms are given in the subsections below.\nTo obtain Eq. 5 which involves only the dynamics of\nelectrons or the electronic subsystem, we have assumed\nthe environmental subsystem is characterized by a huge\nnumber of degrees of freedom and is not perturbed by the\nelectronic subsystem; in this work it means there is no\ndynamics of phonons and photons. Formally this is the\nprerequisite for the Markovian approximation. This as-\nsumption is valid when the system is not far from initial\nequilibrium, e.g., the excited carrier density is not very\nhigh. The inclusion of dynamics of the environment, e.g.\nphonon degrees of freedom in the dynamics, or formally\nnon-Markovian approximation by taking into account the\nmemory effect of phonon bath, has been discussed in de-\ntails in Refs. 22, 25.\n2. Coherent terms and external fields\nIn general, a coherent term corresponding to a single-\nparticle electronic Hamiltonian Hspreads\ndρ\ndt|sp=−i\nℏ[Hsp, ρ]. (6)\nIn absence of external fields, the coherent dynamics\ncontains only one term\ndρ\ndt|free=−i\nℏ[He, ρ]. (7)\nHeis the unperturbed electronic Hamiltonian, practi-\ncally computed at the mean-field level, i.e. from density\nfunctional theory (DFT). With the electronic eigen-basis,\nHe,kmn =ϵknδmn, (8)\nwhere kis k-point index, mandnare band indices, ϵ\nis the single-particle energy, and δmnis Kronecker delta\nfunction.\nTherefore, we have\n\u0012dρ\ndt|free\u0013\nkmn=−i\nℏ(ϵkm−ϵkn)ρkmn. (9)\nWe note that the exchange-correlation potential in\nelectronic Hamiltonian could be time-dependent, which3\nwill introduce an additional term in coherent dynamics[8,\n26]. We will not discuss in detail here.\nThe inclusion of external fields in the DM ap-\nproach for solid-state materials is not trivial and was\nstudied by many theorists since 1950s[27–33]. Here,\nwe consider three spatially homogeneous fields as follows.\n2.1 A laser field. In this work, we approximate that\nthe light-mater interaction He−lightconsists of two parts\n- a semiclassical part Hlaserdescribing electrons moving\nin a laser field (i.e., an electromagnetic field caused by\nthe incident laser) and a quantum part describing the\nelectron-photon interaction in the vacuum without con-\nsidering the laser. The semiclassical part corresponds\nto light absorption and stimulated emission under a laser\nfield, while the quantum part corresponds to spontaneous\nemission. We discuss the coherent term due to the semi-\nclassical part first but discuss the quantum part in the\nnext subsection.\nThe Hamiltonian of a laser with frequency ωlaser is\napproximately[16]\nHlaser,kmn(ω, t) =e\nmeAlaser(t)·pkmnexp (iωlasert) +H.C.,\n(10)\nwhere H.C. is Hermitian conjugate. Alaser(t) is the\namplitude. Alaser(t) is real (complex) for linearly (circu-\nlarly) polarized light. pkmnis the momentum operator\nmatrix element. Note that to include the contribution\nfrom the nonlocal part of the pseudopotentials, momen-\ntum operator is computed with commutator [ r, He] =∇.\nFor a Gaussian pump pulse centered at time tcenter\nwith width τpump,\nAlaser(t) =Alaser1p√πτpumpexp\"\n−(t−tcenter )2\n2τ2pump#\n.\n(11)\nNote that the corresponding pump power is Ilaser =\nω2\nlaser|Alaser|2/(8πα), where αis fine structure constant.\nThe corresponding dynamics is\ndρ\ndt|laser=−i\nℏ[Hlaser, ρ]. (12)\n2.2 A static electric field along a non-periodic direc-\ntion. Such electric field along non-periodic direction\nEnpis directly included in the DFT calculation and is\nmodeled by a ramp or saw-like potential. Under such\ntreatment, HeisEnp-dependent. As all terms of the\nmaster equation (Eq. 5) are built on eigensystems from\nHe, they are all Enp-dependent. However, if electric field\nis applied along the periodic direction, one needs to prop-\nerly treat the periodic boundary condition (e.g. using the\nmodern theory of polarization [34, 35]).2.3 A magnetic field. We describe the effects of an\nexternal magnetic field Bextusing Zeeman Hamiltonian\nHZ,k\u0000\nBext\u0001\n=µBBext·(Lk+g0Sk), (13)\nwhere µBis Bohr magneton; g0is the free-electron g-\nfactor; SandLare the spin and orbital angular mo-\nmentum respectively. The simulation of Lis nontrivial\nfor periodic systems. With the Bl¨ och basis, the orbital\nangular momentum reads\nLk,mn =i\u001c∂ukm\n∂k\f\f\f\f×\u0010\nbHe\u0000\nBext= 0\u0001\n−ϵkmn\u0011\f\f\f\f∂ukn\n∂k\u001d\n,\n(14)\nϵkmn=ϵkm+ϵkn\n2, (15)\nwhere uis the periodic part of the single-particle wave-\nfunction; bHe(Bext= 0) is the zero-field Hamiltonian\noperator. Eqs. 14 - 15 can be proven equivalent to\nL=0.5∗(r×p−p×r) with rthe position operator.\nThe detailed implementation of Eq. 14 is described in\nRef. 36.\nThere are two ways to consider magnetic-field effects.\nThe first way is including HZ,k(Bext) inHeperturba-\ntively (instead of self-consistently), then the new eigen-\nsystems can be obtained by diagonalizing He(Bext̸= 0).\nThe second way is including HZ,k(Bext) inH′and the\ncorresponding coherent dynamics is −i\nℏ[HZ,k(Bext), ρ].\nIn practice, the two approaches lead to nearly the same\ndynamical quantities such as lifetimes, since HZ,k(Bext)\nis rather weak - e.g. Zeeman splitting under 1 Tesla is of\norder 0.1 meV for many solid-state systems. In addition,\nwe note that we do not consider the effect of very strong\nmagnetic field such as the appearance of Landau level in\nthis work.\n3. Scattering terms\nThe scattering part of quantum master equation can\nbe separated into contributions from several scattering\nchannels,\ndρ\ndt|scatt=X\ncdρ\ndt|c, (16)\nwhere clabels a scattering channel corresponding to an\ninteraction Hamiltonian Hc. Under Born-Markov ap-\nproximation and neglecting renormalization of single par-\nticle energies, in general we have[37]\ndρ12\ndt|c=1\n2X\n345\n(I−ρ)13Pc\n32,45ρ45\n−(I−ρ)45Pc,∗\n45,13ρ32\n+H.C., (17)\nwhere Pcis the generalized scattering matrix and H.C.\ndenotes Hermitian conjugate. The subindex, e.g., “1”, is\nthe combined index of k-point and band. The weights of\nk points must be considered when summing over k points.4\n3.1 Electron-phonon. The electron-phonon (e-ph)\nscattering matrix is given by[37]\nPe−ph\n1234 =X\nqλ±Aqλ±\n13Aqλ±,∗\n24 , (18)\nAqλ±\n13=r\n2π\nℏgqλ±\n13q\nδGσ(ω13±ωqλ)q\nn±\nqλ, (19)\nω13=ϵ1−ϵ3, (20)\nwhere qandλare phonon wavevector and mode, gqλ±is\nthe e-ph matrix element, resulting from the absorption\n(−) or emission (+) of a phonon, n±\nqλ=nqλ+ 0.5±0.5\nin terms of phonon Bose factors nqλ, and δG\nσrepresents\nan energy conserving δ-function broadened to a Gaus-\nsian of width σ. The explicit derivation of Lindblan-\ndian dynamics and scattering matrix can be found in\nRef. [37]. This form guarantees the positive definition\nof density matrix. gqλ±is computed fully from first-\nprinciples with self-consistent spin-orbit coupling based\non density-functional perturbation theory and Wannier\ninterpolation methods[38].\n3.2 Electron-impurity. Similar to the e-ph scattering,\nwe write electron-impurity scattering as\nPe−i\n1234=Ai\n13Ai,∗\n24, (21)\nAi\n13=r\n2π\nℏgi\n13q\nδGσ(ω13)p\nniVcell, (22)\ngi\n13=⟨1|∆Vi|3⟩, (23)\n∆Vi=Vi−V0, (24)\nwhere niandVcellare impurity density and unit cell vol-\nume, respectively, Viis the potential of the impurity sys-\ntem and V0is the potential of the pristine system. Here\nwe assume impurities are randomly distributed and im-\npurity density is sufficiently low so that the average dis-\ntance between neighboring impurities is sufficiently long\nwith nearly no interactions among impurities.\nIn this work, giof neutral and ionized impurities are\ncomputed differently as follows.\nFor neutral impurities, Viis computed with SOC using\na large supercell including an impurity. To speed up the\nsupercell convergence, we used the potential alignment\nmethod developed in Ref. 39.\nFor ionized impurities, we approximate ∆ Vias the\npotential of point charge and is simply the product\nof the impurity charge Zand the screened Coulomb\npotential[40]. Such approximate ∆ Viis accurate in the\nlong-range limit,[40] i.e., its Fourier transform ∆ Vi(q) is\naccurate when q→0, so that it is most suitable for\nintravalley relaxation process due to electron-impurity\nscatterings. The justification of this approximation is\nthat the long-range part is the most dominant in the\nelectrostatic interaction of ionzied impurities[40].3.3 Electron-electron interaction. The e-e scattering\nmatrix is given by[37]\nPe−e\n1234=2X\n56,78(I−ρ)65A15,37A∗\n26,48ρ78, (25)\nA1234=1\n2(A1234−A1243), (26)\nA1234=1\n2r\n2π\nℏh\nge−e\n1234(δG\nσ,1234)1/2+ge−e\n2143(δG\nσ,2143)1/2i\n,\n(27)\nge−e\n1234=⟨1 (r)|⟨2 (r′)|V(r−r′)|3 (r)⟩|4 (r′)⟩, (28)\nwhere V(r−r′) is the screened Coulomb potential and\nδG\nσ,1234=δG\nσ(ϵ1+ϵ2−ϵ3−ϵ4) is a Gaussian-broadened\nenergy conservation function. The screening is described\nby Random-Phase-Approximation (RPA) dielectric func-\ntion. We note that unlike the e-ph and e-i channels, Pe−e\nis a function of ρand needs to be updated during time\nevolution of ρ. This is a clear consequence of the two-\nparticle nature of e-e scattering. Pe−ecan be written\nas the difference between a direct term ( Pe−e,d) and an\nexchange term ( Pe−e,x),\nPe−e=Pe−e,d−Pe−e,x, (29)\nPe−e,d=X\n56,78(I−ρ)65A15,37A∗\n26,48ρ78, (30)\nPe−e,x=X\n56,78(I−ρ)65A15,37A∗\n26,84ρ78. (31)\nAccording to Ref. 22, the direct term is expected to dom-\ninate the dynamical scattering processes among conduc-\ntion electrons or valence electrons, allowing us to neglect\nthe exchange term here.\nCurrently, we use the static Random Phase Approxi-\nmation (RPA) dielectric function for the screening with-\nout local-field effects. Given the intraband transition is\nmore important, we take the approximate dielectric func-\ntion as:\nε(q) =εsεintra(q), (32)\nwhere ϵsis the static dielectric constant well-defined\nfor insulators, calculated by Density Functional Pertur-\nbation Theory (DFPT)[41]. εseffectively takes into ac-\ncount the interband contribution in the dielectric screen-\ning. εintra(q) is the intraband contribution which in-\nvolves only states with free carriers and is critical for\ndoped semiconductors (with free carriers). We computed\nit at RPA without local field effect,\nεintra(q) =1−Vbare(q)X\nkmn\nfk−q,m−fkn\nϵk−q,m−ϵkn×\n|⟨uk−q,m|ukn⟩|2\n,\n(33)\nwhere the sum runs over only states having free car-\nriers, e.g., for a n-doped semiconductor, mandnare\nconduction band indices. In the formula above, fis5\ntime-dependent non-equilibrium occupation instead of\nthe equilibrium one feq. Therefore, if the optical field\nAlaser(t) is activated, εintra(q) will be updated in ev-\nery time step, as fwill differ from feqand the dif-\nference depends on the excitation density. Vbare(q) =\ne2/\u0000\nVcellε0|q|2\u0001\nis the bare Coulomb potential with ε0\nvacuum permittivity.\nWe then have the matrix elements in reciprocal space,\nge−e\n1234=Vscr(q13)δk1+k2,k3+k4⟨u1|u3⟩⟨u2|u4⟩,\n(34)\nVscr(q13) =Vbare(q13)/ε(q13), (35)\nwhere Vscr(q) is the screened Coulomb potential\nandq13=k1−k3.⟨u1|u3⟩is the overlap matrix ele-\nment between two periodic parts of Bloch wave functions.\n(iv) Spontaneous emission. As discussed above, in\nthis work, the light-matter interaction consists of a semi-\nclassical part (absorption and simulated emission by a\nlaser field) and a quantum part. The semiclassical part\nhas been discussed above (Sec. II A 2). Here we discuss\nthe quantum part, which describes the electron-photon\ninteraction in the vacuum. Similar to the electron-\nphonon scattering, we write a similar form for electron-\nphoton interaction (the underlying consideration on pos-\nitive definition of density matrix is similar) [37].\nPsp−em\n1234 =X\nqλ±Aphoton ,qλ±\n13 Aphoton ,qλ±,∗\n24 , (36)\nAphoton ,qλ±\n13 =r\n2π\nℏgphoton ,qλ±\n13r\nδGσ\u0010\nω13±ωphoton\nqλ\u0011\n(37)\n×q\nnphoton ,±\nqλ, (38)\nwhere “ +” and “ -” correspond to photon emission and\nabsorption respectively, λis the photon mode, gphoton ,qλ±\n13\nis the electron-photon matrix element proportional to p\nmatrix element. The differences from Pe−phare: (i) As\nωphotonis comparable to the band gap which is much\ngreater than kBT,nphoton ,−\nqλ≈0 and nphoton ,+\nqλ≈1. This\nrepresents that photon absorption is not allowed in vac-\nuum, and every photon emission process emits one pho-\nton. (ii) As photon momentum qis tiny (at long wave-\nlength limit), for Aphoton\n13 , we have k1≈k3. Therefore,\nAphoton ,qλ+\n13 =r\n2π\nℏgphoton ,qλ+\n13r\nδGσ\u0010\nω13+ωphoton\nqλ\u0011\nδk1k3.\n(39)\nThe detailed form of Psp−em\n1234 will be given in our future\nwork.B. Spin lifetime and spin diffusion length\n1. Spin lifetime: relaxation and dephasing\nIn most spin lifetime τsexperiments, τsof ensemble\nspins are measured. The spin observable of an ensemble\nis\nStot\ni(t) =Tr [ siρ(t)] =X\nkX\nmnsi,kmn ρknm(t),(40)\nwhere siis spin Pauli matrix in Bl¨ och basis along di-\nrection i. For spin dynamics, the total excited or excess\nspin observable is often more relevant and is defined as,\nSex\ni(t) =Stot\ni(t)−Seq\ni(t), (41)\nwhere “eq” corresponds to the final equilibrium state.\nThe time evolution must start at an initial state (at\nt=t0) with a net spin i.e. δρ(t0) =ρ(t0)−ρeq̸= 0 such\nthat Sex\ni(t)̸= 0. We evolve ρ(t) through the quantum\nmaster equation Eq. 5 for a long enough simulation time,\ntypically from a few ps to a few hundred ns, until the\nevolution of Sex\ni(t) can be reliably fitted by\nSex\ni(t) =Sex\ni(t0) exp\u0014\n−t−t0\nτs,i\u0015\n(42)\n×cos [ω(t−t0) +ϕ]\nto extract the relaxation time, τs,i. Above, ωis oscillation\nfrequency due to Larmor precession.\nIn Ref. 16, we have shown that one can generate the\ninitial spin imbalance by applying a test magnetic field\natt=−∞, allowing the system to equilibrate with a net\nspin and then turning it off suddenly at t0, in order to\nmeasure spin relaxation.\nHistorically, spin relaxation time (or longitudinal time)\nT1and ensemble spin dephasing time (or transverse time)\nT∗\n2were used to characterize the decay of spin ensemble[8,\n42]. Suppose the spins of the system are polarized along\ndirection r0, possibly due to applying a constant external\nfieldB0along r0, and suppose the total excess/excited\nspin is along direction r1, ifr1||r0(orr1⊥r0),τsis called\nT1(orT∗\n2). For T∗\n2, the excess spin Sex(t) precesses with\na frequency proportional to |B0|.\nThe ensemble spin dephasing rate 1 /T∗\n2consists of re-\nversible part and irreversible part. The reversible part\nmay be removed by the spin echo technique. The irre-\nversible part is called spin dephasing rate 1 /T2, which\nmust be smaller than 1 /T∗\n2. According to Ref. 42, T2\nmay be also defined using Eq. 43 without the need of\nspin echo but instead of Stot\ni(t), we need another quan-\ntity - the sum of individual spin amplitudes\nSindiv\ni =X\nkX\nmn|si,kmn ρknm(t)|. (43)6\n2. Optical measurements\nExperimentally, τsis often studied through ultra-\nfast magneto-optical pump-probe measurements[43, 44]\nwhere excess spin is generated by a circularly-polarized\npump pulse and its evolution is detected by probe pulses.\nThe probe pulse interacts with the material similarly\nto the pump pulse, and could be described in exactly\nthe same way in principle. However, this would require\nrepeating the simulation for several values of the pump-\nprobe delay. Instead, since the probe is typically chosen\nto be of sufficiently low intensity, we use second-order\ntime-dependent perturbation theory to capture its inter-\naction with the system [16],\n∆ρprobe=1\n2X\n345\n\n[I−ρ(t)]13Pprobe\n32,45ρ(t)45\n−[I−ρ(t)]45Pprobe\n45,13ρ(t)32\n\n+H.C.,\n(44)\nPprobe\n1234 =X\n±Aprobe ,±\n13 Aprobe ,±,∗\n24 , (45)\nAprobe ,±\n13 =r\n2π\nℏe\nme\u0000\nAprobe·p\u0001q\nδGσ(ω13±ωprobe)\n(46)\nThe change of dielectric function ∆ ϵbetween the excited\nstate and ground state absorption detected by the probe\nis then\nIm∆ϵ=2π\n(ωprobe)3|Aprobe|2Tr\u0000\nH0∆ρprobe\u0001\n. (47)\nNote that ∆ ρprobecontains |Aprobe|2so that Im∆ ϵis\nindependent of Aprobe. The Im∆ ϵabove is a functional\nofρaccording to Eq. 44 and is an extension of the usual\nindependent-particle Im ϵdepending on just occupation\nfunction f[45]. After obtaining Im∆ ϵ, the real part Re∆ ϵ\ncan be obtained from the Krames-Kronig relation.\nBy summing up ∆ ϵwith the dielectric function for\nground state absorption, we can obtain the excited-state\nϵas inputs for Kerr and Faraday rotation calculations[46].\nThese correspond to the rotations of the polarization\nplane of a linearly-polarized light, reflected by (Kerr\nrotation) and transmitted (Faraday rotation) through\nthe material, after a pump excitation with a circularly-\npolarized light. Time-Resolved Kerr/Faraday Rotation\n(TRKR/TRFR) has been widely used to study spin dy-\nnamics of solids[47, 48]. In a TRKR experiment, a\ncircularly-polarized pump pulse is used to excite valence\nelectrons to conduction bands. The transitions approx-\nimately satisfy the selection rule of ∆ mj=±1 for left\nand right circularly-polarized pulses, respectively, where\nmjis secondary total angular momentum. TRKR works\nby measuring the polarization changes of reflected light,\nwhich qualitatively is proportional to the small popula-\ntion imbalance of electronic states with different mj.Specifically, the Kerr rotation angle θKis computed\nwith dielectric functions by\nθK=Im√ϵ+−√ϵ−\n1−√ϵ+√ϵ−, (48)\nwhere ±denotes the left and right circular polarization,\nrespectively.\n3. The work flow of a spin dynamics simulation\nAs shown in Fig. 1, a spin dynamics simulation has\nthree main steps:\n(i)DFT step. The ground-state electronic struc-\nture, phonons, as well as the e-ph and e-i matrix ele-\nments are firstly calculated using density functional the-\nory (DFT) with relatively coarse kandqmeshes in the\nDFT plane-wave code JDFTx[49]. The phonon calcula-\ntion uses the finite difference method with supercells. Al-\nternatively, we can compute the same quantities in Quan-\ntumEspresso, with phonon and e-ph couplings calculated\nby DFPT at the coarse mesh as well.\n(ii)Wannier step. We then transform all quanti-\nties from plane wave basis to maximally localized Wan-\nnier function basis[50], and interpolate them[38, 51–55]\nto substantially finer k and q meshes. The Wannier in-\nterpolation approach fully accounts for long-range polar\nterms (originally from LO-TO splitting) in the e-ph ma-\ntrix elements and phonon dispersion relations using the\napproaches of Ref. 56 and 57 for the 3D and 2D systems.\nThis part is performed in JDFTx code.\n(iii)Dynamics step. Starting from an initial state\nwith a spin imbalance, we evolve ρ(t) through the quan-\ntum master equation of Eq. 5. After obtaining spin ob-\nservable S(t) from ρ(t) (Eq. 40) and fitting S(t) to an\nexponentially oscillating decay curve (Eq. 43), the decay\nconstant τsis obtained. This part is performed in the\nDMD code interfacing with the JDFTx code.\n4. Carrier lifetime and spin diffusion length\n4.1 Semiclassical limit and carrier lifetime. At the\nsemiclassical limit, ρis replaced by (non-equilibrium) oc-\ncupation f, then the scattering term originally with a full\nquantum description in Eq. 17 required by DM dynamics\nbecomes [16]:\nd f1\ndt|c=X\n2̸=1\u0002\n(1−f1)Pc\n11,22f2−(1−f2)Pc\n22,11f1\u0003\n,(49)\nusing the facts that P11,22is real and “2=1” term is\nzero. “c” represent a scattering channel. Note that the\nweights of k points must be considered when summing\nover k points.7\nFIG. 1. The work flow of a spin dynamics simulation. The codes implemented for different steps are listed and explained in\nthe main text.\nSuppose fis perturbed from its equilibrium value by\nδf, i.e., f=feq+δf, then insert fafter perturbation\ninto Eq. 49 and linearize it,\nd f1\ndt|c=−X\n2̸=1\u0002\nPc\n11,22feq\n2+ (1−feq\n2)Pc\n22,11\u0003\nδf1,(50)\nusing the fact that δP11,22is always zero, even for the\ne-e scattering.\nDefine carrier/particle lifetime of state “1” τc\np,1by\nd f1\ndt|c=−δf1\nτc\np,1, we have\n1\nτc\np,1=X\n2̸=1\u0002\nPc\n11,22feq\n2+ (1−feq\n2)Pc\n22,11\u0003\n. (51)\nThe linewidth or the imaginary part of self-energy\nfor the scattering channel cis related to τc\npby\nImΣc\n1=ℏ/\u0000\n2τc\np,1\u0001\n.\n4.2 Spin diffusion length. The DM approach has been\nwidely employed to simulate transport properties[27, 58,\n59] including spin diffusion length ls.[8] The implementa-\ntion of simulating lsbased on FPDM approach for general\nsolid-state materials is however still under development.\nIn this work, lsare obtained from first-principles us-\ning the commonly-used relation[7, 8] based on the drift-\ndiffusion model,\nls=p\nDτs, (52)\nD=−µcnc/dnc\ndεµF, (53)\nwhere Dis the diffusion coefficient of carriers calculated\nusing the generalized Einstein relation[60]. The under-\nlying assumption here is the diffusion coefficient of car-\nrier being the same as spin, which was shown to be true\nin graphene-derived systems experimentally[61]. εµFis\nthe electron chemical potential. ncis the carrier den-\nsity. µcis the carrier mobility calculated by solving thelinearized Boltzmann equation in momentum-relaxation-\ntime approximation[62–64],\nµc,i=e\nncVuNkX\n1[feq]′\n1v2\n1,iτm,1, (54)\nwhere i=x, y, z .Nkis the number of k points. Vuis\nthe unit cell volume. ncis electron density. [ feq]′is the\nderivative of the Fermi-Dirac distribution function. vis\nthe band velocity. τmis the momentum relaxation time\nand is approximated as[62–64]\nτ−1\nm,1=X\n2̸=1{[P11,22feq\n2+ (1−feq\n2)P22,11] cosθ12},\n(55)\ncosθ12=v1·v2\nv1v2, (56)\nwhere vis the velocity vector. We find that τmis\nsimilar to the carrier lifetime τpexcept the angle factor\ncosθ12.\nIII. SPIN RELAXATION MECHANISM AND\nANALYSIS METHODS FOR SPIN RELAXATION\nThrough studying the relations between τsand impor-\ntant electronic quantities, switching on/off certain dy-\nnamic processes, or tuning the key factors affecting spin\ndynamics, our FPDM method is a powerful technique to\ndetermine spin relaxation mechanism and quantitatively\npredict spin lifetime in different materials at various con-\nditions.\nThere are several mechanisms causing spin relax-\nation and dephasing of electron carriers in non-magnetic\nsystems[7, 8]. Among them, the most important mech-\nanisms are Elliot-Yafet (EY) and Dyakonov-Perel (DP)\nmechanisms. EY represents spin relaxation due to spin-\nflip scattering. DP is activated when inversion symme-\ntry is broken, which results in random spin precession\nbetween adjacent scattering events. Similar to DP mech-\nanism, free induction decay (FID) mechanism is caused8\nby random spin precession but happens when scatter-\ning is weak enough. Below we show how to determine\nspin relaxation mechanism and analyse spin relaxation\nthrough ab-initio simulations. We note that our FPDM\napproach provides unified treatment for different mech-\nanisms. We also acknowledge that some approximate\nmodels were proposed in the past specific for each mech-\nanism, which was helpful for providing mechanistic in-\nsights and qualitative understanding. We will introduce\nthem in the following sections. By comparing FPDM\nand these approximate models, we can further grasp the\ndetailed physical picture of spin relaxation.\nA. Spin expectation value, spin texture and\ninternal magnetic field\nWe first define some important spin-related properties\nin spin dynamics.\nThe spin expectation value along idirection is defined\nas the diagonal element of spin matrix si, i.e.,\nSexp\ni,1=si,11. (57)\nNote that for degenerate bands, the matrix elements of\nsiare arbitrary, thus, we need to diagonalize simatrix\nin degenerate subspaces.\nThe spin texture presents when the spin up and down\ndegeneracy (Kramers pair) is lifted (e.g., due to broken\ninversion symmetry), and is the distribution of the spin\nexpectation value vector Sexp≡\u0000\nSexp\nx, Sexp\ny, Sexp\nz\u0001\n.\nSuppose originally a system has time-reversal and in-\nversion symmetries, so that every two bands form a\nKramers degenerate pair. Suppose the k-dependent spin\nmatrix vectors in Bloch basis of the Kramers degenerate\npairs are s0\nkwiths≡(sx, sy, sz). The inversion symme-\ntry broken induces k-dependent Hamiltonian terms[7]\nHISB\nk=µBg0Bin\nk·s0\nk, (58)\nwhere Bin\nkis the so-called internal magnetic fields at\npresence of SOC. Binsplits the degenerate pair and po-\nlarizes the spin along its direction. The definition of Bin\nk\nis\nBin\nk≡2∆kSexp\nk/(µBg0), (59)\nwhere ∆ is the band splitting energy. From Eq. 59,\nSexp\nk||Bin\nk(internal magnetic field Bin\nkis along the spin\ntexture direction Sexp\nk).\nB. Elliot-Yafet (EY) mechanism\nEY mechanism dominates spin relaxation when spin-\nup and spin-down are well defined (in absence of spin\nprecession), so that it dominates in two types of systems:(i) Materials with both time-reversal and spatial inver-\nsion symmetries, e.g., silicon, free-standing graphene in\nabsence of external fields. With such symmetries, every\ntwo bands form a Kramers degenerate pair[7]. There-\nfore, spin-up/down is well defined along an axis brby di-\nagonalizing the corresponding spin matrix sr=s·brin\ndegenerate subspaces. (ii) Systems with high spin polar-\nization e.g. due to intrinsic magnetization or spin-valley\nlockng, e.g., ferromagnets, transition metal dichalco-\ngenides (TMDs).\n1. Fermi’s golden rule (FGR)\n1.1 FGR with spin-flip transitions According to\nRef. [17], if a solid-state system is close to equilibrium\n(but not at equilibrium) and its spin relaxation is domi-\nnated by EY mechanism, its free carriers’ τsdue to spin-\norbit coupling and e-ph scattering approximately satisfies\n(for simplicity the band indices are dropped)\nτ−1\ns∝N−2\nk\nχX\nkqλ\n\n|g↑↓,qλ\nk,k−q|2nqλfeq\nk−q(1−feq\nk)\nδ(ϵk−ϵk−q−ωqλ)\n\n,(60)\nχ=N−1\nkX\nkfeq\nk(1−feq\nk), (61)\nwhere g↑↓is the spin-flip e-ph matrix element between\ntwo electronic states of opposite spins, nq,λis phonon oc-\ncupation at momentum qand mode λ, and feqis Fermi-\nDirac distribution function. We will further discuss g↑↓\nin the next subsection.\nAccording to Eq. 60 and 61, τ−1\nsis proportional to\n|g↑↓\nq|2and also the density of the spin-flip transitions.\nTherefore we propose a temperature ( T) and chemical\npotential ( εµF) dependent effective modulus square of\nthe spin-flip e-ph matrix element |eg↑↓|2and a scattering\ndensity of states DSas\n|eg↑↓|2=P\nkqwk,k−qP\nλ|g↑↓,qλ\nk,k−q|2nqλP\nkqwk,k−q, (62)\nDS=N−2\nkP\nkqwk,k−q\nN−1\nkP\nkfeq\nk(1−feq\nk), (63)\nwk,k−q=feq\nk−q(1−feq\nk)δ(ϵk−ϵk−q−ωc), (64)\nwhere ωcis the characteristic phonon energy chosen\nas the averaged energy of the phonons contributing to\nspin relaxation, and w k,k−qis the weight function. The\nmatrix element modulus square is weighted by nqλac-\ncording to Eq. 60 and 61. This rules out high-frequency\nphonons at low Twhich are not excited. w k,k−qselects\ntransitions between states separated by ωcand around\nthe band edge or εµF, which are “more relevant” transi-\ntions to spin relaxation.\nDScan be regarded as an effective density of spin-\nflip e-ph transitions satisfying energy conservation be-\ntween one state and its pairs. When ωc= 0, we9\nhave DS=R\ndϵ\u0010\n−d feq\ndϵ\u0011\nD2(ϵ)/R\ndϵ\u0010\n−d feq\ndϵ\u0011\nD(ϵ) with\nD(ϵ) density of electronic states (DOS). So DScan\nbe roughly regarded as a weighted-averaged DOS with\nweight\u0010\n−d feq\ndϵ\u0011\nD(ϵ).\nWith|eg↑↓|2andDS, we have the approximate relation\nfor spin relaxation rate,\nτ−1\ns∝|eg↑↓|2DS. (65)\nWe can similarly define an effective spin conserving\nmatrix element |eg↑↑|2by replacing g↑↓,qλ\nk,k−qtog↑↑,qλ\nk,k−qin Eq.\n62. Then we have the approximate relation for carrier\nrelaxation rate due to e-ph scattering,\n\nτ−1\np\u000b\n∝|eg↑↑|2DS. (66)\nAlthough such formula are not as general and accurate\nas our FPDM formulation, Eq. 65 and 66 can be used\nto understand the spin flip and spin conserving e-ph\nmatrix element’s relation to spin or carrier relaxation,\nand one can develop intuition on what type of phonons\ncontribute more to |eg↑↑|2or|eg↑↓|2. As our recent studies\nshow Fr¨ olich LO phonon strongly contributes to carrier\nrelaxation in CsPbBr 3, but much less important in its\nspin relaxation due to the spin-conserving nature of\nlong-ranged Fr¨ olich electron-phonon coupling [18].\n1.2 Generalized FGR In centrosymmetric systems\nwith strong spin-mixing and band degeneracy, Sexp\nkmay\nhave multiple values. For example, valence bands of sili-\ncon at Γ are four-fold degenerate and their Sexp\nkare (more\nprecisely, very close to) ±1/2 and ±1/6. In such cases,\nspin relaxation may be still driven by EY mechanism\nbut needs a generalized FGR formula[17] beyond spin-\nflip transition,\nτ−1\ns,i∝N−2\nk\nχiX\n12λ\n\n\f\f\f∆Sexp\ni,12gq,λ\n12\f\f\f2\nnqλ\nfeq\n2(1−feq\n1)δ(ϵ1−ϵ2−ωqλ)\n\n,\n(67)\nχi=N−1\nkX\n1feq\n1(1−feq\n1)Sexp,2\ni,1, (68)\n∆Sexp\ni,12=Sexp\ni,1−Sexp\ni,2, (69)\nwhere ∆ Sexpis the change of spin expectation value\nfor a pair of electronic states. Therefore, the transitions\nwith non-negligible ∆ Sexpcan contribute to spin relax-\nation. Again, unlike our FPDM method, this is approxi-\nmate formula, which can be used to understand how e-ph\nscattering between a pair of states changes spin expecta-\ntion value as we showcase bulk Si hole spin relaxation in\nRef. 17.\n2. Spin-mixing parameter b2\n2.1 State-resolved b2.Suppose the spin of a state “1”\nis highly polarized along idirection. Then in general,the wavefunction of state “1” can be written as Ψ 1(r) =\nai,1(r)α+bi,1(r)β, where aandbare the coefficients of\nthe large and small components of the wavefunction, and\nαandβare spinors (one up and one down for direction\ni). Define a2\ni,1=R\n|ai,1(r)|2drandb2\ni,1=R\n|bi,1(r)|2dr,\nthen a2\ni,1> b2\ni,1andb2\ni,1is just spin-mixing parameter of\nstate “1” along direction i[19].\na2\ni,1+b2\ni,1=1, (70)\n0.5\u0000\na2\ni,1−b2\ni,1\u0001\n=Sexp\ni,1, (71)\nTherefore,\nb2\ni,1=0.5−Sexp\ni,1. (72)\n2.2 EY relation. According to Eq. 60 and 66, we have\nτ−1\ns/τ−1\np∝|eg↑↓|2/|eg↑↑|2, (73)\nwhere τp= 1/\nτ−1\np\u000b\n.\nAs thermal averaging frequently appears in spin relax-\nation analysis, we define ⟨A⟩as the thermal average of\nelectronic quantity A,\n⟨A⟩=P\nkn\u0000\n−[feq]′\nkn\u0001\nAknP\nkn\u0000\n−[feq]′\nkn\u0001, (74)\nwhere [ feq]′is the derivative of the Fermi-Dirac distribu-\ntion function.\nAccording to Refs. 9, 65, 66, |eg↑↓|2/|eg↑↑|2∼\nb2\u000b\n, so\nthat\nτ−1\ns/τ−1\np∼\nb2\u000b\n. (75)\nThe above is a rough relation and cannot be used to\npredict τs(the error may be several or even ten times).\nPractically, we use the following approximate relation\nτ−1\ns/τ−1\np=4\nb2\u000b\n. (76)\nThis is called EY relation in this work.\n3. Spin-flip angle θ↑↓\nIn our previous paper[21], we proposed an important\nelectronic quantity for intervalley spin-flip scattering -\nthe spin-flip angle θ↑↓between two electronic states. For\ntwo states ( k, n) and ( k′, n′) with opposite spin direc-\ntions, θ↑↓is the angle between −Sexp\nknandSexp\nk′n′. The\nmotivation of examining θ↑↓is that: According to Ref.\n[67], due to time-reversal symmetry, the spin-flip matrix\nelement of the same band between kand−kis exactly\nzero, so that g↑↓is zero at lowest order for intervalley\ntransitions between two opposite valleys (e.g., Kand\n−K). In the first-order perturbation level,\f\fg↑↓\f\fbetween10\ntwo states is determined by θ↑↓between these two states\nand proportional to\f\fsin\u0000\nθ↑↓/2\u0001\f\f.\nSuppose (i) the inversion symmetry broken induces Bin\nk\n(Eq. 59) for a Kramers degenerate pair; (ii) there are two\nvalleys centered at wavevectors Qand−Qand (iii) there\nare two wavevectors k1andk2nearQand−Qrespec-\ntively. Due to time-reversal symmetry, the directions of\nBin\nk1andBin\nk2are almost opposite.\nWe can prove that for a general operator bA,\n\f\f\fA↑↓\nk1k2\f\f\f2\n≈sin2\u0010\nθ↑↓\nk1k2/2\u0011\f\f\fA↓↓\nk1k2\f\f\f2\n, (77)\nwhere A↑↓\nk1k2and A↓↓\nk1k2are the spin-flip and spin-\nconserving matrix elements between k1andk2respec-\ntively. We present the detailed derivation in SI Sec. SIII\nof Ref. 21.\nFrom Eq. 77, for the intervalley e-ph matrix elements,\nwe have\n\f\f\fg↑↓\nk1k2\f\f\f2\n≈sin2\u0010\nθ↑↓\nk1k2/2\u0011\f\f\fg↓↓\nk1k2\f\f\f2\n. (78)\nFinally, similar to Eq. 62, we propose an effective mod-\nulus square of sin2\u0010\nθ↑↓\nk1k2/2\u0011\n,\nsin2(θ↑↓/2) =P\nkqwk,k−qsin2\u0010\nθ↑↓\nk,k−q/2\u0011\nP\nkqwk,k−q. (79)\nWe found spin relaxation time (obtained from our\nFPDM method) linearly proportional to this quantity\n(sin2(θ↑↓/2)) when intervalley spin relaxation process\ndominates, as an example of substrate effects on spin re-\nlaxation of strong SOC Dirac materials in Ref. 21. Such\nangle relates to the spin expectation value direction be-\ntween initial ↑and final ↓states.\nC. Dyakonov-Perel (DP) and Free induction decay\n(FID) mechanisms\n1. Model relations\nFor non-magnetic materials, with nonzero k-dependent\ninternal magnetic field Bin\nkinduced by inversion symme-\ntry broken and spin-orbit coupling, the spins at kprecess\nabout Bin\nk. The Larmor precession frequency vector can\nbe defined as:\nΩk=∆kbSexp\nk, (80)\nwherebSexp\nkis the normalized Sexp\nk.\nWe define Ω⊥bras the component of Ωperpendicular\nto direction br. Suppose the fluctuation amplitude amongdifferent k-points of Ω⊥bris ∆Ω⊥brand numerically we\ndefine it as (using Eq. 74)\n∆Ω⊥br=rD\n|Ω⊥br− ⟨Ω⊥br⟩|2E\n. (81)\nAccording to Refs. 7 and 8, a nonzero ∆ Ωleads to\nfinite spin lifetime τ∆Ω\nsalongbrand the spin relaxation\nmechanism depends on the magnitude of τp∆Ω[7, 8] (the\nsubindex “ ⊥br” is dropped for simplicity):\n(i) DP mechanism if τp∆Ω≪1 (strong scattering\nlimit). We have the DP relation\n\u0000\nτ∆Ω\ns\u0001−1∼\u0000\nτDP\ns\u0001−1∼τp(∆Ω)2. (82)\n(ii) FID mechanism if τp∆Ω≳1 (weak scattering\nlimit). We have\n\u0000\nτ∆Ω\ns\u0001−1∼\u0000\nτFID\ns\u0001−1∼∆Ω. (83)\n(iii) Between (i) and (ii) regimes, there isn’t a good\napproximate relation for\u0000\nτ∆Ω\ns\u0001−1, but we may expect\nthat[8]\n\u0000\nτDP\ns\u0001−1<\u0000\nτ∆Ω\ns\u0001−1<\u0000\nτFID\ns\u0001−1. (84)\n2. Land´ e g-factor of free carriers\nBesides intrinsic spin-orbit coupling, one channel to\ninduce nonzero ∆ Ωis through gfactor fluctuations under\nmagnetic field. In Sec. II A 2, we have described how\nBextis considered in FPDM approach, therefore, with\nHz(Bext), the magnetic-field effects on spin dynamics\nare straightforwardly included in FPDM simulations.\nFor the description of the magnetic-field effects, Land´ e\ng-factor has been widely used[18]. For a single band and\na pair of bands, g-factor is well defined and relates to\nBext-induced change of energy, energy splitting and/or\nLarmor precession frequency. In this work, we limit our\ndiscussions to two Kramers degenerate bands under a\ntransverse Bext(perpendicular to the direction of the ex-\ncess/excited spin S≡(Sx, Sy, Sz)). In general, in the\ntwo-band case, g-factor is a tensor and may be defined\nby the relation Ωk(Bext) =µBBextgS\nk. Here, for simplic-\nity, we assume Ωk(Bext)||Bext, which is valid in many\nmaterials including CsPbBr 3. With this assumption and\nfrom Eq. 80,\ngS\nk=Ωk(Bext)·bBext\nµBBext. (85)\nΩk(Bext) is computed using Eq. 80.\nHowever, in many previous theoretical studies[68, 69],\ng-factors were defined based on pseudo-spins related to\nthe total magnetic momenta Jat, which are determined\nfrom the atomic-orbital models. The pseudo-spins can\nhave opposite directions to the actual spins ( S). Most11\nprevious experimental studies adopted the same conven-\ntion for the signs of carrier g-factors. Therefore, to com-\npare with g-factors obtained in those previous studies,\nwe introduce a correction factor CS→Jand define a new\ng-factor:\negk\u0010\nbBext\u0011\n=CS→JgS\nk. (86)\nCS→J=mat\nS/mat\nJwith mat\nJandmat\nSthe total and\nspin magnetic momenta respectively, obtained from the\natomic-orbital model[69]. CS→Jis independent from k-\npoint, and is ∓1 for electrons and holes respectively for\nCsPbBr 3[18].\nAsegkis different at different k, we can define its fluc-\ntuation amplitude as\n∆eg=D\n(eg− ⟨eg⟩)2E\n. (87)\nFrom the above equation, we have the fluctuation am-\nplitude of Ωk(Bext)\n∆Ω\u0000\nBext\u0001\n=µBBext∆eg. (88)\nA nonzero ∆Ω ( Bext) leads to spin dephasing under\nexternal magnetic field Bextand the mechanism may be\nDP or FID depending on the magnitude of τp∆Ωas dis-\ncussed above.\nD. Determination of spin relaxation mechanism\nIn general, the applicability of our FPDM approach\ndoes not depend on specific spin relaxation mecha-\nnism. To determine the dominant relaxation or dephas-\ning mechanism, we have the following two approaches by\nutilizing our FPDM calculations.\n1. Method 1: Comparing FPDM calculations with FGR or\nmodel relations\nThe first approach is to compare FPDM results di-\nrectly with simple models designed for various mecha-\nnisms as introduced in previous two sections. If FPDM\nagrees with one of the model relations, it is a good indi-\ncator of dominant mechanism.\n1.1 EY mechanism. Since spin precession is sup-\npressed when EY mechanism dominates, the DM mas-\nter equation in semiclassical limit from Eq. 49 and FGR\nformula with spin-flip scattering Eq. 60 should describe\nwell spin relaxation (suppose spin matrix is diagonalized\nin degenerate subspaces along the direction of the excess\nspin). Therefore, if the values and trends of τsby Eq. 49\nand 60 are similar to those obtained from FPDM calcu-\nlations, the dominating mechanism is likely the EY.1.2 DP or FID mechanism. With nonzero ∆ Ω,\nwhether spin relaxation is dominated by DP or FID\nmechanism may be determined by comparing the values\nand trends of τsby Eq. 82 or 83 with the FPDM results.\nWe have shown the success of such analysis in our previ-\nous work on spin relaxation and transport in germanene\nand silicene under electric field [19] as well as CsPbBr 3\nunder magnetic field [18].\n2. Method 2: Tuning the scattering strength or precession\nfrequency in FPDM calculations\nConventionally, spin relaxation mechanism is deter-\nmined from the relation between τsandτp- EY mech-\nanism leads to τs∝τpwhile DP mechanism leads to\nτs∝τ−1\np. Another way to look at this is through tuning\nthe strength of scatterings (e.g. physically, increasing im-\npurity concentration increases scattering). If τsdecreases\nwith increasing scattering, it’s likely EY; otherwise, it’s\nlikely DP mechanism, because τpalways decreases with\nincreasing scatterings.\nPractically in our FPDM calculations, this approach\ncan be implemented by introducing a scaling factor Fsc\nto tune the scattering strength, i.e. multiplying Fscto\nthe scattering termdρ\ndt|scattof the DM master equation\n(Eq. 5) but keep the coherent term (the first term of Eq.\n5) unchanged. This is equivalent to multiply Fscto all\nelements of the generalized scattering-rate matrix P(Eq.\n18, 21 and 25).\nTo avoid confusions , we name carrier and spin lifetimes\nafter introducing Fscasτ′\nsandτ′\nprespectively. Accord-\ning to Eq. 51, we always have τ′\np/τp= (Fsc)−1. On\nthe other hand, τ′\nsdepends on the spin relaxation mech-\nanism. For EY mechanism, τ′\ns/τ′\np=τs/τp=CEY, where\nCEYis a constant unrelated to Fsc. Therefore, it can be\nproven that τ′\ns/τs= (Fsc)−1=τ′\np/τp. Similarly, for DP\nmechanism, we can prove that τ′\ns/τs=Fsc=\u0000\nτ′\np/τp\u0001−1.\nFor FID mechanism, as τ′\nsis irrelevant to τ′\np, we have\nτ′\ns/τs≡1. Therefore, the relation between τ′\ns/τsand\nτ′\np/τpby tuning Fscis useful to understand spin relax-\nation mechanism.\nMoreover, the relation between τsand ∆Ω is also useful\nto understand spin relaxation mechanism. ∆Ω can be\ntuned easily by tuning the energy splitting.\nIV. APPLICATIONS\nA. Non-magnetic materials with inversion\nsymmetry\nWe first present results for systems with inversion sym-\nmetry traditionally described by EY mechanism. Fig-\nure 2(a) shows that our theoretical electron τsof Si as\na function of Tare in excellent agreement with experi-\nmental measurements[70, 71]. Note that previous first-12\nFIG. 2. Spin relaxation in Silicon (Si). Spin ( τs) and carrier ( τp) lifetimes of (a) electrons and (b) holes of Si. Exp. A and B\nare from Ref. 70 and 71. (c) Cumulative contributions to spin relaxation by change in spin, ∆s, per scattering event defined\nbased on Eqs. 67-69: electrons in Si exhibit spin flips with all contributions at ∆s = 1, whereas holes in Si and electrons in iron\nexhibit a broad distribution in ∆s [17].\nprinciples calculations[9] approximated spin-flip electron-\nphonon matrix elements from pseudospin wavefunction\noverlap and spin-conserving electron-phonon matrix ele-\nment, effectively assuming that the scattering potential\nvaries slowly on the scale of a unit cell; we make no such\napproximation in our FPDM approach. In contrast, holes\nin Si exhibit strong spin mixing with spin-2/3 character\nand spin expectation values no longer close to ℏ/2. Fig-\nure 2(b) shows our predictions for the hole spin lifetime\nτs,hwhich is much shorter than the electron spin as a\nresult of the strong mixing (450 fs for holes compared to\n7 ns for electrons at 300 K) and is much closer to carrier\nrelaxation time τp. Additionally, Fig. 2(c) shows that the\nchange in spin expectation values (∆ Sexp) per scattering\nevent (evaluated using Eqs. 67 -69) has a broad distribu-\ntion for holes in Si, indicating that they cannot be de-\nscribed purely by spin-flip transitions, while conduction\nelectrons in Si predominantly exhibit spin-flip transitions\nwith ∆ Sexp= 1.\nB. Materials with high spin polarization\nIn ferromagnets and antiferromagnets, spins are highly\npolarized, so that spin relaxation is likely dominated by\nEY mechanism. In Ref. 17, we simulated τsof bcc iron,\nwhere we found good agreement with experiments, with\na dominant EY mechanism (magnon was not considered\nhere). In nonmagnetic materials, when inversion sym-\nmetry is broken and SOC strength is moderate or large,\nit has been found that some of them such as transition\nmetal dichalcogenides also have highly polarized carrier\nspins and exhibit EY spin relaxation.1. Transition metal dichalcogenides (TMDs)\nMonolayer TMDs exhibit exciting features includ-\ning spin-polarized bands, valley-specific optical selection\nrules and spin-valley locking. In Ref. 72, 73, it has been\nshown that by introducing doping in monolayer TMDs,\nultraslow decays of Kerr rotations, which correspond to\nultralong spin/valley lifetimes of resident carriers espe-\ncially resident holes, can be observed at low tempera-\ntures. Those features establish TMDs’ advantages for\nspin-valleytronics and (quantum) information processing.\nAlthough spin/valley relaxation time of resident carriers\nin monolayer TMDs, most relevant to spinvalleytronics’\napplication, has been extensively examined, the underly-\ning relaxation mechanisms especially the effects of differ-\nent types of impurities have not been addressed through\nab-initio simulations.\nIn Ref. 16, we clarified the above problem by conduct-\ningab initio real-time dynamics simulations with relevant\nscattering mechanisms. We focused on WSe 2due to its\nlarger valence band SOC splitting and weaker interlayer\ncoupling compared with other TMDs and focused on dy-\nnamics of resident holes as τsof holes seem longer than\nelectrons.\nFor holes of monolayer WSe 2, spin/valley relaxation is\ncompletely determined by intervalley spin-flip scattering\nbetween KandK′valleys because of spin-valley lock-\ning. Previously, we reported spin/valley lifetimes of res-\nident holes of monolayer TMDs at T≥50 K with e-ph\nscattering[17]. At very low temperatures, e.g., 10 K, in-\ntervalley e-ph scattering is however not activated as the\ncorresponding phonon occupation is negligible, so that\nother scattering mechanisms need to be included. Note\nthat e-e scattering should not play an important role in\nspin relaxation of holes of TMDs. The reason is: The\ne-e scattering is a two-particle process where a transition\nis accompanied by another transition with energy and\nmomentum being conserved. Considering the fact that13\nFIG. 3. (a) The schematics of four types of impurities in\nWSe 2. (b) Spin lifetimes of holes of monolayer WSe 2with\na relatively low hole density 1011cm−2with impurities com-\npared with experimental data[73–76]. The choices of impurity\nconcentration niof different impurities are given in the main\ntext [16].\nonly the highest occupied band is involved in dynamics of\nTMD holes, for a e-e process, a K→K′(K′→K) spin-flip\ntransition must be accompanied by an opposite K′→K\n(K→K′) spin-flip transition. Overall, an e-e scattering\nprocess cannot change the spin of the system. Therefore,\nwe will include only e-ph and e-i scatterings.\nExperimentally there exists many types of impuri-\nties/defects in TMD samples. Here we pick four types of\nimpurities with different symmetries and chemical bonds\n(see Fig. 3(a)) - Se vacancy (Se vac.), two neighbor-\ning Se vacancies (2N-Se vac.), W vacancy (W vac.) and\ntwo Se vacancies with the same in-plane position (2S-Se\nvac.). According to Refs. 77–79, niranges from 8 ×1010\nto 1014cm−2depending on samples. Considering that\nSe vac. is often regarded as the most abundant impu-\nrity, we choose a reasonable niof Se vac. - 7 ×1011cm−2,\nwhich is taken relatively low for better comparison with\nexperimental τsshown in Fig. 3(b). Since the formation\nenergy of Se vac. seems lowest and niof larger impu-\nrities are found lower than smaller impurities[79], niof\n2N-Se vac. is chosen as 8 ×109cm−2lower than Se vac.\nand also for better comparison with experimental τs.ni\nof W vac. and 2S-Se vac. are chosen arbitrarily as we\nfound they have rather weak effects on spin relaxation\nand are 7 ×1011and 3.5 ×1011cm−2, respectively.From Fig. 3, we first find that assuming ninot so high,\natT>20 K, spin relaxation is almost driven by e-ph scat-\ntering and impurities can only affect spin relaxation at\nT≤20 K. For the effects on spin relaxation of different\nimpurities, we have 2N-Se vac. ≫Se vac. ≫W vac. ∼\n2S-Se vac.. Moreover, the temperature dependence of τs\nwith 2N-Se vac. is much weaker and in better agreement\nwith experiments than that with Se vac.. Therefore, the\nobserved weak temperature dependence in some experi-\nments is probably related the existence of larger impu-\nrities with lower symmetries. Our observations suggest\nthat the local symmetry and chemical bonds surround a\nimpurity have large impact on spin relaxation. To un-\nderstand why the effects of different impurities on spin\nrelaxation significantly differ, further theoretical investi-\ngations on how impurities affect impurity potentials and\ntheir SOC corrections are required.\n2. Germanene under an electric field or on a substrate\nIn Ref. 19, through FPDM simulations, we predicted\nthat monolayer germanene (ML-Ge) has giant spin life-\ntime anisotropy, spin-valley-locking (SVL) effect under\nnonzero perpendicular electric field Ezand long τs(∼100\nns at 50 K), which makes it advantageous for spin-\nvalleytronic applications. In 2D-material-based spin-\ntronic devices, the materials are usually supported on\na substrate. Therefore, for the design of those devices,\nit is crucial to understand substrate effects on spin re-\nlaxation. We further examined τsof ML-Ge on different\nsubstrates in Ref. 21.\nWe show band structures and spin textures of free-\nstanding and supported ML-Ge in Fig. 4, which are es-\nsential for understanding spin relaxation mechanisms. At\nEz= 0, ML-Ge has time-reversal and inversion sym-\nmetries, so that its bands are Kramers degenerate[7].\nA finite Ezor a substrate breaks the inversion sym-\nmetry and induces a strong out-of-plane Bin(and also\nSexp, Eq. 59), which splits the Kramers pairs into spin-\nup and spin-down bands[19]. Interestingly, we find that\nband structures of ML-Ge-SiH (Fig. 4c) and ML-Ge-GeH\n(not shown) are quite similar to free-standing ML-Ge\nunder Ez=-7 V/nm (ML-Ge@-7V/nm, Fig. 4b), which\nindicates that the impact of the SiH/GeH substrate on\nband structure (and also Bin) may be similar to a fi-\nniteEz. This similarity is frequently assumed in model\nHamiltonian studies[80, 81]. On the other hand, the\nband structures of ML-Ge-InSe (Fig. 4d) and ML-Ge-\nGaTe (not shown) have more differences from the free-\nstanding one under Ez, with larger band gaps, smaller\nband curvatures at Dirac Cones, and larger electron-hole\nasymmetry of band splittings. This implies that the im-\npact of the InSe/GaTe substrates can not be approxi-\nmated by applying an Ezto the free-standing ML-Ge,\nunlike SiH/GeH substrates. We further examine Sexpof\nsubstrate-supported ML-Ge. Importantly, from Fig. 4e\nand 4f, although Sexpof ML-Ge on substrates are highly14\nFIG. 4. Band structures and spin textures around the Dirac cones of ML-Ge systems with and without substrates. (a)-(d) show\nband structures of ML-Ge under Ez= 0 and under -7 V/nm and ML-Ge on silicane (SiH) and on InSe substrates respectively.\nThe red and blue bands correspond to spin-up and spin-down states. Due to time-reversal symmetry, band structures around\nanother Dirac cone at K′=−Kare the same except that the spin-up and spin-down bands are reversed. The grey, white,\nblue, pink and green balls correspond to Ge, H, Si, In and Se atoms, respectively. Band structures of ML-Ge on germanane\n(GeH) and GaTe are shown in Supplementary Figure 4 in the Supporting Information, and are similar to those of ML-Ge on\nSiH and InSe substrates, respectively. (e) and (f) show spin textures in the kx-kyplane and 3D plots of the spin vectors Sexp\nk1\non the circle |− →k|= 0.005 bohr−1of the band at the band edge around Kof ML-Ge on SiH and InSe substrates respectively.\nThe color scales Sexp\nzand the arrow length scales the vector length of in-plane spin expectation value [19].\npolarized along z(out-of-plane) direction, the in-plane\ncomponents of Sexpof ML-Ge-InSe (and ML-Ge-GaTe)\nare much more pronounced than ML-Ge-SiH (and ML-\nGe-GeH). Such differences are crucial to the out-of-plane\nspin relaxation as discussed later.\nWe compare out-of-plane τsdue to e-ph scattering be-\ntween the free-standing ML-Ge (with/without an electric\nfield) and ML-Ge on different substrates in Fig. 5. From\nFig. 5, we find that τsof ML-Ge under Ez= 0 and -\n7 V/nm are at the same order of magnitude for a wide\nrange of temperatures. The differences are only consid-\nerable at low T, e.g, by 3-4 times at 20 K. On the other\nhand, τsof supported ML-Ge are very sensitive to the\nspecific substrates. While τsof ML-Ge-GeH and ML-\nGe-SiH have the same order of magnitude as the free-\nstanding ML-Ge, in particular very close between ML-\nGe-GeH and ML-Ge@-7 V/nm, τsof ML-Ge-GaTe and\nML-Ge-InSe are shorter by at least 1-2 orders of magni-\ntude in the whole temperature range.\nSince spins of ML-Ge at Ez̸= 0 and on a substrate\nare highly polarized, spin relaxation in ML-Ge systems\nis dominated by EY mechanism caused by spin-flip scat-\ntering. According to Eq. 65 and 75, τ−1\nsis roughly pro-portional to density of states (DOS) and spin-mixing pa-\nrameter\nb2\nz\u000b\n. Indeed, in Ref. 21, we find that at 300\nK, the differences of τsof ML-Ge on different substrates\nare well explained by the differences of the products of\nDOS and\nb2\nz\u000b\n. However, at T≤50 K, the differences of\nthe products of DOS and\nb2\nz\u000b\nfor different substrates are\nabout 3-7 times, while the differences of τsfor different\nsubstrates can be as large as two orders of magnitude.\nTherefore, the substrate effects on τscan not be fully ex-\nplained by the changes of DOS and\nb2\nz\u000b\n, in particular at\nrelatively low T.\nSince τ−1\ns∝|eg↑↓|2DS(Eq. 65), to understand sub-\nstrate effects on τsat low T, we first compare τ−1\ns\nand|eg↑↓|2DSof different ML-Ge systems. In Ref. 21,\nwe found that τ−1\nsis almost linearly proportional to\n|eg↑↓|2DSat 20 K. As the variation of DSamong ML-\nGe on different substrates is at most three times, which\nis much weaker than the large variation of τ−1\ns, this in-\ndicates that the substrate-induced change of τsis mostly\ndue to the substrate-induced change of spin-flip matrix\nelements.\nTo have deeper intuitive understanding, we then pro-15\nFIG. 5. The out-of-plane τsof ML-Ge under Ez= 0, -7 V/nm\nand substrate-supported ML-Ge as a function of temperature\nwithout impurities. Here we show electron τsfor intrinsic\nML-Ge systems except that hole τsis shown for ML-Ge-InSe,\nsince electron τsare longer than hole τsat low Texcept ML-\nGe-InSe [21].\nFIG. 6. The relation between τ−1\nsandsin2(θ↑↓/2) multiplied\nby the scattering density of states DSat 20 K. DSis defined\nin 63. θ↑↓is the spin-flip angle between two electronic states.\nFor two states ( k, n) and ( k′, n′) with opposite spin direc-\ntions, θ↑↓is the angle between −Sexp\nknandSexp\nk′n′.sin2(θ↑↓/2)\nis defined in Eq. 79. The variation of DSamong different\nsubstrates is at most three times, much weaker than the vari-\nations of τ−1\nsand other quantities shown here [21].\npose an important electronic quantity for intervalley\nspin-flip scattering - the spin-flip angle θ↑↓between two\nelectronic states. For two states ( k1, n1) and ( k2, n2) with\nopposite spin directions, θ↑↓is the angle between −Sexp\nk1n1\nandSexp\nk2n2or equivalently the angle between −Bin\nk1and\nBin\nk2. At low T, due to large SOC splittings of conduction\nand valence bands of supported ML-Ge, intravalley spin-flip e-ph scattering processes are forbidden, because the\ncorresponding phonons have too high energies and are\nnot occupied. So spin relaxation in supported ML-Ge is\ndominated by intervalley spin-flip e-ph scattering. There-\nfore, θ↑↓is helpful for understanding spin relaxation in\nsupported ML-Ge.\nThe motivation of examining θ↑↓is that: Suppose two\nwavevectors k1andk2=−k1are in two opposite valleys\nQand - Qrespectively and there is a pair of bands, which\nare originally Kramers degenerate but splitted by Bin.\nDue to time-reversal symmetry, we have Bin\nk1=−Bin\nk2,\nwhich means the two states at the same band natk1\nandk2have opposite spins and θ↑↓between them is\nzero. Therefore, the matrix element of operator bAbe-\ntween states ( k1, n) and ( k2, n) -Ak1n,k2nis a spin-flip\none and we name it as A↑↓\nk1k2. According to Ref. 67,\nwith time-reversal symmetry, A↑↓\nk1k2is exactly zero. In\ngeneral, for another wavevector k3within valley - Qbut\nnot−k1,A↑↓\nk1k3is usually non-zero. One critical quan-\ntity that determines the intervalley spin-flip matrix ele-\nment A↑↓\nk1k3for a band within the pair introduced above\nisθ↑↓\nk1k3. Based on time-independent perturbation theory,\nwe can prove that\f\fA↑↓\f\fbetween two states is approxi-\nmately proportional to\f\fsin\u0000\nθ↑↓/2\u0001\f\f. The derivation is\ngiven in Sec. III B 3.\nAs shown in Figure 6c, τ−1\nsof ML-Ge on different\nsubstrates at 20 K is almost linearly proportional to\nsin2(θ↑↓/2)DS, where sin2(θ↑↓/2) is the statistically-\naveraged modulus square of sin\u0000\nθ↑↓/2\u0001\n. This indicates\nthat the relation |eg↑↓|2∝sin2(θ↑↓/2) is nearly perfectly\nsatisfied at low T, where intervalley processes dominate\nspin relaxation. We additionally examined the relation\nbetween τ−1\nsandsin2(θ↑↓/2)DSat 300 K and found that\nthe trend of τ−1\nsis still approximately captured by the\ntrend of sin2(θ↑↓/2)DS.\nSince θ↑↓is defined by Sexpat different states, τsis\nhighly correlated with Sexpand more specifically with\nthe anisotropy of Sexp(equivalent to the anisotropy of\nBin). Qualitatively, the larger anisotropy of Sexpleads\nto smaller θ↑↓(consistent with Fig. 4e and f) and longer\nτsalong the high-spin-polarization direction. This find-\ning may be applicable to spin relaxation in other materi-\nals whenever intervalley spin-flip scattering dominates or\nspin-valley locking exists, e.g., in TMDs[72], Stanene[82],\n2D hybrid perovskites with persistent spin helix[6], etc.\nC. DP (Dyakonov-Perel) systems\nIn many non-magnetic materials with broken inversion\nsymmetry, spin relaxation is dominated by DP mecha-\nnism.16\n1. GaAs\nSpin dynamics in GaAs has broad interest in spintron-\nics over past decades[7, 47, 83–85] and more recently[86–\n88], partly due to its long spin lifetime in n-type GaAs\nat low temperatures[47]. Despite various experimen-\ntal [47, 48, 83, 89, 90] and theoretical [7, 84, 85, 91–93]\n(mostly using parameterized model Hamiltonian) stud-\nies previously, the dominant spin relaxation mechanism\nin bulk GaAs at various Tand doping concentrations ni\nremains unclear.\nThrough FPDM simulations of n-GaAs at various T\nandniwith different scattering mechanisms in Ref. 16,\nwe pointed out that although at low temperatures and\nmoderate doping concentrations e-i scattering dominates\ncarrier relaxation, e-e scattering is the most dominant\nprocess in spin relaxation. Moreover, we found that the\nrelative contributions of phonon modes vary considerably\nbetween spin and carrier relaxation.\nFigure 7 shows τsandτpwith different niat 30 K\nwith individual and total scattering pathways, respec-\ntively. It is found that the roles of different scattering\nmechanism differ considerably between spin and carrier\nrelaxation processes. Specifically, for the carrier relax-\nation in Fig. 7b, except when niis very low (e.g. at\n1014cm−3), the e-i scattering dominates. On the other\nhand, for the spin relaxation in Fig. 7a, the e-e scattering\ndominates except at very high concentration (above 1017\ncm−3), while e-i scattering is only important in the very\nhigh doping region (close to or above 1017cm−3).\nFigure 7 shows the calculated τshas a maximum at\nni= 1-2 ×1016cm−3, and τsdecreases fast with ni\ngoing away from its peak position. This is in good agree-\nment with the experimental finding in Ref. 47, which also\nreported τsatni= 1016cm−3is longer than τsat other\nlower and higher niat a low temperature (a few Kelvin).\nThenidependence of τsmay be qualitatively interpreted\nfrom the DP relation[7] τs,z∼τp∆Ω2\n⊥z(Eq. 82), where\n∆Ω⊥zis the fluctuation amplitude of Larmor frequency\ndefined in Eq. 81. From Fig. 7, we find that with nifrom\n1014cm−3to 5×1015cm−3,τpdecreases rapidly (black\ncurve in Fig. 7b) and ∆Ω2\n⊥zremains flat in Fig. 7c, which\nmay explain why τsincreases in Fig. 7a based on the DP\nrelation; however, when ni>1016cm−3,τpdecreases\nwith a similar speed but ∆Ω2\n⊥zexperiences a sharp in-\ncrease, which may explain why τsdecreases in Fig. 7b\nand owns a maximum at 1016cm−3.\nNote that although the DP relation is intuitive to un-\nderstand the cause of doping-level dependence of spin\nlifetime, it may break down when we evaluate individ-\nual scattering processes. For example, when niincreases\nfrom 1014cm−3to 1015cm−3, both carrier lifetime τs\nandτs,zdue to e-i scattering decrease while the internal\nmagnetic field remains unchanged. Moreover, the sim-\nple empirical relation cannot possibly explain our first-\nprinciples results that the e-e and e-i scatterings have\nlargely different contributions in carrier and spin relax-\nation. First-principles calculations are critical to provideunbiased mechanistic insights to spin and carrier relax-\nation of general systems.\n2. Graphene on hBN\nGraphene samples exhibit exciting spintronic prop-\nerties such as long τsand lsat room temperature\n[1, 94]. In practice, graphene is usually supported by\na substrate[94–98]. Actually τsof the free-standing\ngraphene sample was found relatively low[99] ∼150 ps\nat 300 K (compared with the longest reported value ∼12\nns), probably due to free-standing graphene samples of-\nten have more imperfections. Therefore, it is important\nto understand spin relaxation in supported graphene.\nFrom Fig. 8, we find that the EY model for graphene,\nand the DP model for Gr+ E⊥and Gr-hBN, agree qual-\nitatively with FPDM predictions, but with some im-\nportant quantitative differences discussed next. First,\nthe EY model for graphene is more accurate for elec-\ntrons than for holes, for both τs∥andτs⊥(Fig. 8(a-b)).\nThe conventional DP model matches FPDM predictions\nquantitatively for both τs∥andτs⊥of Gr+ E⊥, but only\nforτs⊥of Gr-hBN.\nThe discrepancy of the conventional DP model for\nτs∥of Gr-hBN can be rectified by modifying the model.\nBriefly, the DP model assumes that Bineffectively\nchanges randomly each time the electron scatters. The\nin-plane magnetic field Bin\n||rotates over the Fermi circle\nand covers all in-plane directions, satisfying this condi-\ntion, in both Gr+ E⊥and Gr-hBN. However, the out-of-\nplane magnetic field Bin\n⊥, which matters only for τs∥and\nis present only for Gr-hBN, has the same direction within\neach valley. Consequently, only intervalley scattering will\nchange the Bin\n⊥for a given electron spin. As proposed in\nRef. 100, this can be captured by changing the DP model\nfrom ( τDP\ns,x)−1∼τp(∆Ω⊥x)2=τp\nΩ2\ny+ Ω2\nz\u000b\nas given by\nEq. 82 for in-plane xspins, to\n\u0000\nτmDP\ns,x\u0001−1≈τp\nΩ2\ny\u000b\n+τInter\np\nΩ2\nz\u000b\n, (89)\nwhere τInter\np is the intervalley scattering time (dotted\nline in Fig. 8(d)). This modified DP model agrees with\nFPDM predictions for τs∥of Gr-hBN (Fig. 8(a)).\nThe ratio τs⊥/τs∥(Fig. 8(c)) is nearly 1/2 for graphene,\nconsistent with the EY relation Eq. 75 and the fact thatD\nb2\n||E\n/\nb2\n⊥\u000b\nis also 1/2 (Fig. 8(e)). This ratio remains\nunchanged for Gr+ E⊥, but now because (∆ Ω⊥z)2=\n⟨Ω2\nx+ Ω2\ny⟩= 2⟨Ω2\ny⟩, while (∆ Ω⊥x)2=⟨Ω2\ny⟩since Ω z= 0\n(Fig. 8(f)), leading to τDP\ns,x= 2τDP\ns,zusing Eq. 82. This\nratio deviates substantially from 1/2 only for Gr-hBN\n(Fig. 8(c)) due to the substrate-induced Ω z̸= 0. The\nconventional DP model (Eq. 82) only captures part of\nthis dramatic effect seen in the FPDM calculations, while\nthe modifications in Eq. 89 account for Ω zcorrectly and\nagree with the FPDM results in Fig. 8(c).17\nFIG. 7. (a) τsand (b) τpofn-GaAs with different doping concentrations niat 30 K with different scattering mechanisms.\n“All” represents all the e-ph, e-i and e-e scattering mechanisms being considered. (c) ∆Ω2\n⊥zas a function of carrier density n,\nwhere ∆Ω⊥zis the fluctuation amplitude of Larmor frequency defined in Eq. 81 [16].\n4\n 2\n0 2 4\nn [x1012 cm2]\n0.00.51.01.52.0s/s\n(c)102104106s [ps]\n(a)\nGraphene\nGr.+E\nGr-hBN\n102104106s [ps]\n(b)100101\np [ps]\n(d)\nInter\np\n01020\nb2\ni [x108]\n(e)\nGraphene, i=z\nGraphene, i=x\n4\n 2\n0 2 4\nn [x1012 cm2]\n104\n102\n100\n2\ni [x108 fs2]\n(f)\nGr-hBN, i=x\nGr.+E, i=x\nGr-hBN, i=z\nFIG. 8. Theoretical spin and carrier relaxation in graphene\n(red lines), graphene at E⊥=0.4 V/nm (Gr+ E⊥, green lines)\nand graphene on hBN (Gr-hBN, blue lines) as a function of\nn(positive or negative for electron and hole doping respec-\ntively) at room temperature. (a) In-plane spin lifetime τs∥,\n(b) out-of-plane spin lifetime τs⊥and (c) their ratios τs⊥/τs∥\ncalculated using FPDM approach (solid lines) compared with\nthose estimated from EY/DP models with first-principles in-\nputs (dashed for conventional DP and dotted for modified\nDP with intervalley scattering contribution lines). (d) Total\nτp(three dashed lines) and intervalley only contribution for\nGr-hBN (dotted blue line), (e)\nb2\ni\u000b\nand (f)\nΩ2\ni\u000b\n, all predicted\nfrom first-principles for the DP and EY models estimates of\nτsin (a-c) [20].\nτsdecrease with increasing carrier density magni-\ntude in graphene (Fig. 8(a,b)), but this trend reverses\nfor both inversion-symmetry-broken cases, in agreement\nwith some experiments.[94, 101] The overall τsare re-\nduced by one-two orders of magnitude in free-standinggraphene by E⊥of 0.4 V/nm, down from µsto tens\nof ns.\nΩ2\ni\u000b\nof Gr-hBN in Fig. 8(f) is about 100\ntimes larger than that of Gr+ E⊥, further reducing τs\nof Gr-hBN to the ns scale, comparable to experimental\nmeasurements.[94, 96, 97, 102]\nFinally, τsis mostly symmetric between electrons and\nholes for Gr+ E⊥(Fig. 8(a, b)). However, we find hole τs\nto be typically 2-3 times smaller than electrons for both\nfree-standing graphene and Gr-hBN. On hBN, this asym-\nmetry is captured by the (modified) DP model and stems\nprimarily from the larger spin-splitting and hence\nΩ2\ni\u000b\nin the valence band compared to the conduction band\n(not shown), consistent with previous calculations.[103]\nImportantly, this effect depends sensitively on the sub-\nstrate, and even on hBN, could reverse for a different\nlayer stacking.[103] Consequently, experiments may find\nelectron-hole asymmetries of either sign depending on the\nsubstrate and precise structure,[94, 96–98, 101–103] and\nwe focus here on the comparison between FPDM predic-\ntions and DP model for the specific lowest-energy stack-\ning of Gr-hBN.\nD. Spin dephasing under magnetic field\nAs introduced in Sec. II B 1, the dephasing time (or\ntransverse time) of a spin ensemble is called T∗\n2and de-\nscribes the decay of the total excess spin S−Seqat an\nnonzero Bextperpendicular to S−Seq. As discussed\nin Sec. II A 2, T∗\n2can be simulated straightforwardly\nthrough FPDM simulations with Zeeman Hamiltonian\nconsidering both spin and orbital angular momenta (Eq.\n13 and 14). For the purpose of analysing and under-\nstanding spin dephasing, one key parameter is the Land´ e\ng-factor eg(Eq. 86). It value relates to Bext-induced\nenergy splitting (Zeeman effect) ∆ E(Bext) and Larmor\nprecession frequency Ω, satisfying Ω ≈∆E=µBBexteg.\nMore importantly, the g-factor fluctuation (near Fermi\nsurface or µF,c) ∆eg(Eq. 87) leads to a nonzero ∆Ω (Eq.\n88) and then T∗\n2due to DP or FID mechanism.\nSpintronics in halide perovskites has drawn significant18\nattention in recent years, due to highly tunable spin-orbit\nfields and intriguing interplay with lattice symmetry. Re-\ncently, we simulate[18] eg, ∆egandT∗\n2of a typical halide\nperovskite - CsPbBr 3. We find that T∗\n2is sensitive to\nBextatT <20 K but not at T≥20 K. At 4 K, we pre-\ndict that ( T∗\n2)−1is linear to BextatBext≥0.4 Tesla\nand the predicted slope of ( T∗\n2)−1is in good agreement\nwith experimental data. Moreover, we predict strong n-\ndependence of eg, ∆egandT∗\n2. Together with FPDM sim-\nulations of spin relaxation time T1of CsPbBr 3at various\nconditions, our work provides fundamental insights on\nhow to control and manipulate spin relaxation/dephasing\nin halide perovskites, which are vital for their spintronics\nand quantum information applications.\nV. OUTLOOKS\nThe first-principles density-matrix dynamics (FPDM)\napproach is an important technique for studying spin and\nelectron dynamics and transport, with outstanding ad-\nvantages: i) it can accurately describe various interac-\ntions, scattering processes, and spin precession simulta-\nneously; ii) it can describe processes far from equilib-\nrium; iii) it can simulate dynamical processes on various\ntime scales from femtoseconds to milliseconds. We have\nshown its success in simulating ultrafast spin dynamics,\nspin and charge transport, relaxation and dephasing. In\nthe future study, the FPDM approach still has a broad\nopen area for new theory development.\n1. Exciton dynamics\nExciton is a bound electron-hole pair generated by op-\ntical excitation. Excitons play an important role in opti-\ncal properties of semiconductors, in particular in low-\ndimensional systems. Exciton dynamics has been ex-\ntensively studied using different methods, in particular\nnon-equilibrium Greens’ function theory (NEGF) within\nKadanoff-Baym equations[45, 104, 105]. We focus on the-\noretical simulations of exciton spin relaxation based on\nLindbladian DM master equation with quantum descrip-\ntion of the exciton-phonon scattering. Understanding the\nexciton spin relaxation is also important for understand-\ning optical measurements of spin dynamics. By replacing\nelectrons to excitons, the existing FPDM approach can\nbe generalized to simulation spin dynamics of excitons.\nThe exciton dynamics can involve many types of pro-\ncesses including exciton-light interaction, exciton-phonon\nscattering, phonon-mediated exciton recombination and\ndissociation, exciton–exciton annihilation, etc. Initially,\nwe will focus on exciton-light interaction and exciton-\nphonon scattering, which are two most important pro-\ncesses leading to exciton spin relaxation.\nThe exciton-light interaction is responsible to exciton\nspin generation by light absorption with a circularly-\npolarized pump pulse and exciton spin relaxationthrough exciton-phonon scattering and exciton radiative\nrecombination. The light absorption for excitons is simi-\nlar to that for electrons (see Sec. II A 2) except that the\nmomentum matrix elements are now between two exci-\ntonic states [106–108].\nThe exciton-phonon scattering is responsible to exci-\nton spin relaxation. According to Refs. 109, 110, the\nexciton-phonon scattering matrix elements can be ap-\nproximately obtained from the excition wavefunction at\nfinite momentum and electron-phonon scattering ma-\ntrix elements. Recently we implemented the exciton-\nphonon scattering matrix elements and used them to sim-\nulate phonon-assisted indirect exciton radiative recombi-\nnation. In the density-matrix master equation, the term\nof the exciton-phonon scattering is rather similar to the\nelectron-phonon scattering, except that the excitons are\nbosons while the electrons are fermions. Therefore, the\nimplementation of the exciton-phonon scattering in the\nframe of FPDM approach can be straightforward.\n2. Circular photogalvanic effect (CPGE)\nCPGE is the effect that under the circularly-polarized\nlight a DC current may be induced in a solid-state mate-\nrial, and widely presents in materials without inversion\nsymmetry, in absence of p-n junction and applied elec-\ntric field. In recent years, CPGE has attracted grow-\ning interests in the fields of topological physics and spin-\noptotronics [111, 112]. In order to deeply understand\nCPGE, theoretical methods have been developed based\non perturbation theory. However, several critical issues\nremain: (i) the scattering mechanisms are highly sim-\nplified with a single relaxation time approximation; (ii)\nspontaneous recombination was rarely considered; (iii)\navailable theory requires different formulation for each\ncontribution i.e. interband or intraband contributions;\n(iv) many-body effects were not considered. These issues\nmay be resolved based on our FPDM approach.\nIf a constant laser field is applied to an inversion sym-\nmetry broken system, the quantum master equation for\ndescribing density matrix of the electronic system reads:\ndρ\ndt=dρ\ndt|laser+dρ\ndt|sp−em+dρ\ndt|e−ph, (90)\nwheredρ\ndt|laseris coherent dynamics due to a laser field\n(Eq. 12),dρ\ndt|sp−emanddρ\ndt|e−phare the spontaneous emis-\nsion (Eq. 36) and e-ph scattering terms (Eq. 17 and 18),\nrespectively.\nCPGE is measuring the steady state current when\na nonmagnetic system is under a constant circularly-\npolarized laser field for a long enough time. In the veloc-\nity gauge, the current density is computed using[32]19\nJ=Tr ( ρj), (91)\njkmn(t) =−e\u0012\nvkmn+δmne\nmeA(t)\u0013\n, (92)\nwhere vis the velocity operator matrix and v=p/me.\nThe average value of the second term of jis usually zero\nso that this term does not contribute to CPGE.\n3. More on scattering terms\nAlthough in our FPDM approach, the ab-initio\ntreatment of quantum scattering is rather general, the\nfollowing theoretical development can further improve\nthe applicability of our methods to different systems:\nAnharmonic phonons. In current implementation,\nphonons are harmonic and it is required that the ma-\nterial must be dynamically stable at zero temperature.\nHowever, for some soft materials (e.g., hybrid halide\nperovskites), significant anharmonic effects appear\nat high- Tphase, which requires extracting phonon\nproperties from finite-temperature simulations, e.g.\nusing ab-initio molecular dynamics. Such approach has\nbeen implemented [113] for studying phonon, electron-\nphonon, and carrier transport properties, which can\nbe also applied to study the anharmonic effect on spin\nrelaxation under our FPDM framework.\nFr¨ ohlich interaction and LO-TO splitting in doped\nsemiconductor. The intraband dielectric screening is\nnot considered for Fr¨ ohlich interaction and LO-TO\nsplitting in current implementation. This may be\nproblematic in doped semiconductor and may lead to\nsignificant errors at moderate or high carrier density.Since we have already implemented the intraband dielec-\ntric function (Eq. 33), its effect can be straightforwardly\nincluded in FPDM approach similar to Ref. 114.\nShort-range contribution to the electron-ionized-\nimpurity scattering. For the electron-ionized-impurity\nscattering, we currently only consider the long-range\ncontribution. This is appropriate for certain systems\nlike GaAs, but may be problematic when intravalley\nprocesses dominate spin relaxation or the short-range\nspin-flip electron-ionized-impurity scattering is unim-\nportant to spin relaxation. Therefore, it is important to\ninclude both long-range and short-range contributions\nto the electron-ionized-impurity scattering. Similar to\nRef. 56, the long-range and short-range parts are treated\nseparately - the short-range part is treated from first\nprinciples similar to neutral impurity, and the long-range\npart is simulated using the screened Coulomb potential.\nACKNOWLEDGEMENTS\nThis work is supported by National Science Founda-\ntion under grant No. DMR-1956015. This research used\nresources of the Center for Functional Nanomaterials,\nwhich is a US DOE Office of Science Facility, and the\nScientific Data and Computing center, a component of\nthe Computational Science Initiative, at Brookhaven Na-\ntional Laboratory under Contract No. DE-SC0012704,\nthe lux supercomputer at UC Santa Cruz, funded by\nNSF MRI grant AST 1828315, the National Energy Re-\nsearch Scientific Computing Center (NERSC) a U.S. De-\npartment of Energy Office of Science User Facility oper-\nated under Contract No. DE-AC02-05CH11231, the Ex-\ntreme Science and Engineering Discovery Environment\n(XSEDE) which is supported by National Science Foun-\ndation Grant No. ACI-1548562 [115].\n[1] A. Avsar, H. Ochoa, F. Guinea, B. ¨Ozyilmaz, B. J.\nvan Wees, and I. J. Vera-Marun, Rev. Mod. Phys. 92,\n021003 (2020).\n[2] J. F. Sierra, J. Fabian, R. K. Kawakami, S. Roche, and\nS. O. Valenzuela, Nat. Nanotechnol 16, 856 (2021).\n[3] J. H. Garcia, M. Vila, C.-H. Hsu, X. Waintal, V. M.\nPereira, and S. Roche, Phys. Rev. Lett. 125, 256603\n(2020).\n[4] L. ˇSmejkal, Y. Mokrousov, B. Yan, and A. H. MacDon-\nald, Nat. Phys. 14, 242 (2018).\n[5] Y.-H. Kim, Y. Zhai, H. Lu, X. Pan, C. Xiao, E. A.\nGaulding, S. P. Harvey, J. J. Berry, Z. V. Vardeny, J. M.\nLuther, et al. , Science 371, 1129 (2021).\n[6] L. Zhang, J. Jiang, C. Multunas, C. Ming, Z. Chen,\nY. Hu, Z. Lu, S. Pendse, R. Jia, M. Chandra, et al. ,\nNat. Photon. , 1 (2022).\n[7] I. ˇZuti´ c, J. Fabian, and S. D. Sarma, Rev. Mod. Phys.\n76, 323 (2004).[8] M. Wu, J. Jiang, and M. Weng, Phys. Rep. 493, 61\n(2010).\n[9] O. D. Restrepo and W. Windl, Phys. Rev. Lett. 109,\n166604 (2012).\n[10] D. V. Fedorov, M. Gradhand, S. Ostanin, I. V.\nMaznichenko, A. Ernst, J. Fabian, and I. Mertig, Phys.\nRev. Lett. 110, 156602 (2013).\n[11] M. A. L. Marques, N. T. Maitra, F. M. S. Nogueira,\nE. K. U. Gross, and A. Rubio, Fundamentals of\nTime-Dependent Density Functional Theory , Vol. 837\n(Springer Science & Business Media, 2012).\n[12] Z. Chen and L. Wang, Sci. Adv. 5, eaau8000 (2019).\n[13] S. R. Acharya, V. Turkowski, G. P. Zhang, and T. S.\nRahman, Phys. Rev. Lett. 125, 017202 (2020).\n[14] K. Krieger, J. K. Dewhurst, P. Elliott, S. Sharma, and\nE. K. U. Gross, J. Chem. Theory Comput. 11, 4870\n(2015).\n[15] J. Park, J.-J. Zhou, Y. Luo, and M. Bernardi, Phys.\nRev. Lett. 129, 197201 (2022).20\n[16] J. Xu, A. Habib, R. Sundararaman, and Y. Ping, Phys.\nRev. B 104, 184418 (2021).\n[17] J. Xu, A. Habib, S. Kumar, F. Wu, R. Sundararaman,\nand Y. Ping, Nat. Commun. 11, 2780 (2020).\n[18] J. Xu, K. Li, U. N. Huynh, J. Huang, R. Sun-\ndararaman, V. Vardeny, and Y. Ping, Preprint at\nhttps://arxiv.org/abs/2210.17074 (2023).\n[19] J. Xu, H. Takenaka, A. Habib, R. Sundararaman, and\nY. Ping, Nano Lett. 21, 9594 (2021).\n[20] A. Habib, J. Xu, Y. Ping, and R. Sundararaman, Phys.\nRev. B 105, 115122 (2022).\n[21] J. Xu and Y. Ping, npj Comput. Mater. 9, 47 (2023).\n[22] F. Rossi and T. Kuhn, Rev. Mod. Phys. 74, 895 (2002).\n[23] M. Bonitz, Quantum Kinetic Theory (Springer, Cham,\n2016).\n[24] V. M. Axt and A. Stahl, Z. Physik B - Condensed Mat-\nter93, 195 (1994).\n[25] R. C. Iotti and F. Rossi, Eur. Phys. J. B 90, 1 (2017).\n[26] A. Marini, in Journal of Physics: Conference Series ,\nVol. 427 (IOP Publishing, 2013) p. 012003.\n[27] W. Kohn and J. Luttinger, Phys. Rev. 108, 590 (1957).\n[28] J. Krieger and G. Iafrate, Phys. Rev. B 35, 9644 (1987).\n[29] E. Ciancio, R. C. Iotti, and F. Rossi, Phys. Rev. B 69,\n165319 (2004).\n[30] G. Kan´ e, M. Lazzeri, and F. Mauri, Phys. Rev. B 86,\n155433 (2012).\n[31] A. Sekine, D. Culcer, and A. H. MacDonald, Phys. Rev.\nB96, 235134 (2017).\n[32] G. Ventura, D. Passos, J. L. Dos Santos, J. V. P. Lopes,\nand N. Peres, Phys. Rev. B 96, 035431 (2017).\n[33] G. Iafrate, V. Sokolov, and J. Krieger, Phys. Rev. B\n96, 144303 (2017).\n[34] R. D. King-Smith and D. Vanderbilt, Phys. Rev. B 47,\n1651 (1993).\n[35] R. Resta, Rev. Mod. Phys. 66, 899 (1994).\n[36] C. Multunas, A. Grieder, J. Xu, Y. Ping, and S. R.,\nUnder Preparation (2022).\n[37] R. Rosati, R. C. Iotti, F. Dolcini, and F. Rossi, Phys,\nRev. B 90, 125140 (2014).\n[38] F. Giustino, Rev. Mod. Phys. 89, 015003 (2017).\n[39] R. Sundararaman and Y. Ping, J. Chem. Phys. 146,\n104109 (2017).\n[40] C. Jacoboni, Theory of Electron Transport in Semi-\nconductors: A Pathway from Elementary Physics to\nNonequilibrium Green Functions , Vol. 165 (Springer Sci-\nence & Business Media, 2010).\n[41] X. Wu, D. Vanderbilt, and D. R. Hamann, Phys, Rev.\nB72, 035105 (2005).\n[42] C. L¨ u, J. Cheng, M. Wu, and I. da Cunha Lima, Phys.\nLett. A 365, 501 (2007).\n[43] F. Meier and B. P. Zakharchenya, Optical Orientation\n(Elsevier, 2012).\n[44] M. P. M. Dean, Y. Cao, X. Liu, S. Wall, D. Zhu,\nR. Mankowsky, V. Thampy, X. M. Chen, J. G. Vale,\nD. Casa, et al. , Nat. Mater. 15, 601 (2016).\n[45] A. Molina-S´ anchez, D. Sangalli, L. Wirtz, and\nA. Marini, Nano Lett. 17, 4549 (2017).\n[46] N. Mainkar, D. A. Browne, and J. Callaway, Phys, Rev.\nB53, 3692 (1996).\n[47] J. M. Kikkawa and D. D. Awschalom, Phys. Rev. Lett.\n80, 4313 (1998).\n[48] A. V. Kimel, F. Bentivegna, V. N. Gridnev, V. V.\nPavlov, R. V. Pisarev, and T. h. Rasing, Phys, Rev.\nB63, 235201 (2001).[49] R. Sundararaman, K. Letchworth-Weaver, K. A.\nSchwarz, D. Gunceler, Y. Ozhabes, and T. A. Arias,\nSoftwareX 6, 278 (2017).\n[50] N. Marzari and D. Vanderbilt, Phys. Rev. B 56, 12847\n(1997).\n[51] A. M. Brown, R. Sundararaman, P. Narang, W. A. God-\ndard, and H. A. Atwater, ACS Nano 10, 957 (2016).\n[52] P. Narang, L. Zhao, S. Claybrook, and R. Sundarara-\nman, Adv. Opt. Mater. 5, 1600914 (2017).\n[53] A. M. Brown, R. Sundararaman, P. Narang, A. M.\nSchwartzberg, W. A. Goddard III, and H. A. Atwa-\nter, Phys. Rev. Lett. 118, 087401 (2017).\n[54] A. Habib, R. Florio, and R. Sundararaman, J. Opt. 20,\n064001 (2018).\n[55] A. M. Brown, R. Sundararaman, P. Narang, W. A. God-\ndard III, and H. A. Atwater, Phys. Rev. B 94, 075120\n(2016).\n[56] C. Verdi and F. Giustino, Phys. Rev. Lett. 115, 176401\n(2015).\n[57] T. Sohier, M. Gibertini, M. Calandra, F. Mauri, and\nN. Marzari, Nano Lett. 17, 3758 (2017).\n[58] R. C. Iotti and F. Rossi, Phys. Rev. Lett. 87, 146603\n(2001).\n[59] H. Watanabe and Y. Yanase, Phys. Rev. X 11, 011001\n(2021).\n[60] R. Kubo, Rep. Prog. Phys. 29, 255 (1966).\n[61] T. Maassen, F. Dejene, M. Guimar˜ aes, C. J´ ozsa, and\nB. Van Wees, Physical Review B 83, 115410 (2011).\n[62] C. J. Ciccarino, T. Christensen, R. Sundararaman, and\nP. Narang, Nano Lett. 18, 5709 (2018).\n[63] T. Gunst, T. Markussen, K. Stokbro, and M. Brand-\nbyge, Phys. Rev. B 93, 035414 (2016).\n[64] G. D. Mahan, Many-Particle Physics (Springer US,\n2000).\n[65] J. Fabian and S. D. Sarma, Phys. Rev. Lett. 81, 5624\n(1998).\n[66] W. Leyland, R. Harley, M. Henini, A. Shields, I. Farrer,\nand D. Ritchie, Phys. Rev. B 76, 195305 (2007).\n[67] Y. Yafet, in Solid state physics , Vol. 14 (Elsevier, 1963)\npp. 1–98.\n[68] Z. Yu, Sci. Rep. 6, 1 (2016).\n[69] E. Kirstein, D. Yakovlev, M. Glazov, E. Zhukov,\nD. Kudlacik, I. Kalitukha, V. Sapega, G. Dimitriev,\nM. Semina, M. Nestoklon, et al. , Nat. Commun. 13,\n1 (2022).\n[70] D. J. Lepine, Phys. Rev. B 6, 436 (1972).\n[71] B. Huang, D. J. Monsma, and I. Appelbaum, Phys.\nRev. Lett. 99, 177209 (2007).\n[72] P. Dey, L. Yang, C. Robert, G. Wang, B. Urbaszek,\nX. Marie, and S. A. Crooker, Phys. Rev. Lett. 119,\n137401 (2017).\n[73] J. Li, M. Goryca, K. Yumigeta, H. Li, S. Tongay, and\nS. A. Crooker, Phys. Rev. Mater. 5, 044001 (2021).\n[74] M. Goryca, N. P. Wilson, P. Dey, X. Xu, and S. A.\nCrooker, Sci. Adv. 5, eaau4899 (2019).\n[75] X. Song, S. Xie, K. Kang, J. Park, and V. Sih, Nano\nLett.16, 5010 (2016).\n[76] T. Yan, S. Yang, D. Li, and X. Cui, Phys. Rev. B 95,\n241406 (2017).\n[77] D. Edelberg, D. Rhodes, A. Kerelsky, B. Kim, J. Wang,\nA. Zangiabadi, C. Kim, A. Abhinandan, J. Ardelean,\nM. Scully, et al. , Nano Lett. 19, 4371 (2019).\n[78] D. Rhodes, S. H. Chae, R. Ribeiro P., and J. Hone,\nNat. Mater. 18, 541 (2019).21\n[79] M. Yankowitz, D. McKenzie, and B. J. LeRoy, Phys.\nRev. Lett. 115, 136803 (2015).\n[80] D. Van Tuan, F. Ortmann, A. W. Cummings, D. Sori-\nano, and S. Roche, Sci. Rep. 6, 1 (2016).\n[81] C. Ertler, S. Konschuh, M. Gmitra, and J. Fabian,\nPhys. Rev. B 80, 041405 (2009).\n[82] L. Tao and E. Y. Tsymbal, Phys. Rev. B 100, 161110\n(2019).\n[83] D. J. Hilton and C. L. Tang, Phys. Rev. Lett. 89, 146601\n(2002).\n[84] J. H. Jiang and M. W. Wu, Phys, Rev. B 79, 125206\n(2009).\n[85] A. Kamra and B. Ghosh, J. Appl. Phys 109, 024501\n(2011).\n[86] F. Dettwiler, J. Fu, S. Mack, P. J. Weigele, J. C. Egues,\nD. D. Awschalom, and D. M. Zumb¨ uhl, Phys. Rev. X\n7, 031010 (2017).\n[87] D. Huber, M. Reindl, S. F. C. Da Silva, C. Schimpf,\nJ. Mart´ ın-S´ anchez, H. Huang, G. Piredda, J. Edlinger,\nA. Rastelli, and R. Trotta, Phys. Rev. Lett. 121, 033902\n(2018).\n[88] V. V. Belykh, A. Y. Kuntsevich, M. M. Glazov, K. V.\nKavokin, D. R. Yakovlev, and M. Bayer, Phys. Rev. X\n8, 031021 (2018).\n[89] Y. Ohno, R. Terauchi, T. Adachi, F. Matsukura, and\nH. Ohno, Phys. Rev. Lett. 83, 4196 (1999).\n[90] P. E. Hohage, G. Bacher, D. Reuter, and A. D. Wieck,\nAppl. Phys. Lett. 89, 231101 (2006).\n[91] Z. G. Yu, S. Krishnamurthy, M. Van Schilfgaarde, and\nN. Newman, Phys, Rev. B 71, 245312 (2005).\n[92] M. D. Mower, G. Vignale, and I. V. Tokatly, Phys, Rev.\nB83, 155205 (2011).\n[93] G. Marchetti, M. Hodgson, J. McHugh, R. Chantrell,\nand I. D’Amico, Materials 7, 2795 (2014).\n[94] M. Drogeler, C. Franzen, F. Volmer, T. Pohlmann,\nL. Banszerus, M. Wolter, K. Watanabe, T. Taniguchi,\nC. Stampfer, and B. Beschoten, Nano Lett. 16, 3533\n(2016).\n[95] C. R. Dean, A. F. Young, I. Meric, C. Lee, L. Wang,\nS. Sorgenfrei, K. Watanabe, T. Taniguchi, P. Kim, K. L.\nShepard, and J. Hone, Nat. Nanotechnol. 5, 722 (2010).\n[96] W. Han and R. K. Kawakami, Phys. Rev. Lett. 107,\n047207 (2011).\n[97] M. H. Guimar˜ aes, P. J. Zomer, J. Ingla-Ayn´ es, J. C.\nBrant, N. Tombros, and B. J. van Wees, Phys. Rev.Lett.113, 086602 (2014).\n[98] B. Raes, J. E. Scheerder, M. V. Costache, F. Bonell,\nJ. F. Sierra, J. Cuppens, J. Van de Vondel, and S. O.\nValenzuela, Nat. Commun. 7, 11444 (2016).\n[99] M. H. Guimar˜ aes, A. Veligura, P. Zomer, T. Maassen,\nI. Vera-Marun, N. Tombros, and B. Van Wees, Nano\nLetters 12, 3512 (2012).\n[100] A. W. Cummings, J. H. Garcia, J. Fabian, and\nS. Roche, Phys. Rev. Lett. 119, 206601 (2017).\n[101] P. J. Zomer, M. H. D. Guimar˜ aes, N. Tombros, and\nB. J. van Wees, Phys. Rev. B 86, 161416(R) (2012).\n[102] M. V. Kamalakar, C. Groenveld, A. Dankert, and S. P.\nDash, Nat. Commun. 6, 1 (2015).\n[103] K. Zollner, M. Gmitra, and J. Fabian, Phys. Rev. B\n99, 125151 (2019).\n[104] F. Paleari and A. Marini, Physical Review B 106,\n125403 (2022).\n[105] Y.-H. Chan, D. Y. Qiu, F. H. da Jornada, and S. G.\nLouie, Proceedings of the National Academy of Sciences\n118, e1906938118 (2021).\n[106] C. Guo, J. Xu, and Y. Ping, Journal of Physics: Con-\ndensed Matter 33, 234001 (2021).\n[107] F. Wu, D. Rocca, and Y. Ping, J. Mater. Chem. C 7,\n12891 (2019).\n[108] T. J. Smart, K. Li, J. Xu, and Y. Ping, npj Computa-\ntional Materials (2021).\n[109] G. Antonius and S. G. Louie, Phys. Rev. B 105, 085111\n(2022).\n[110] H.-Y. Chen, D. Sangalli, and M. Bernardi, Physical\nReview Letters 125, 107401 (2020).\n[111] H. Watanabe and Y. Yanase, Phys. Rev. X 11, 011001\n(2021).\n[112] F. de Juan, A. G. Grushin, T. Morimoto, and J. E.\nMoore, Nat. Commun. 8, 15995 (2017).\n[113] O. Hellman, P. Steneteg, I. A. Abrikosov, and S. I.\nSimak, Physical Review B 87, 104111 (2013).\n[114] F. Macheda, P. Barone, and F. Mauri, Physical Review\nLetters 129, 185902 (2022).\n[115] J. Towns, T. Cockerill, M. Dahan, I. Foster, K. Gaither,\nA. Grimshaw, V. Hazlewood, S. Lathrop, D. Lifka, G. D.\nPeterson, R. Roskies, J. R. Scott, and N. Wilkins-Diehr,\nComput. Sci. Eng. 16, 62 (2014)."    },    {        "title": "0909.3406v3.GHz_Spin_Noise_Spectroscopy_in_n_Doped_Bulk_GaAs.pdf",        "content": "arXiv:0909.3406v3  [cond-mat.mes-hall]  4 Feb 2010GHz Spin Noise Spectroscopy in n-Doped Bulk GaAs\nGeorg M. M¨ uller,∗Michael R¨ omer, Jens H¨ ubner, and Michael Oestreich\nInstitut f¨ ur Festk¨ orperphysik, Leibniz Universit¨ at Ha nnover, Appelstraße 2, D-30167 Hannover, Germany\n(Dated: October 31, 2018)\nWe advance spin noise spectroscopy to an ultrafast tool to re solve high frequency spin dynamics in\nsemiconductors. The optical non-demolition experiment re veals the genuine origin of the inhomoge-\nneous spin dephasing in n-doped GaAs wafers at densities at the metal-to-insulator t ransition. The\nmeasurements prove in conjunction with depth resolved spin noise measurements that the broaden-\ning of the spin dephasing rate does not result from thermal flu ctuations or spin-phonon interaction,\nas suggested previously, but from surface electron depleti on.\nPACS numbers: 72.25.Rb, 72.70.+m, 78.47.p, 78.55.Cr, 85.7 5.-d\nSpin noise spectroscopy (SNS) has developed into a\nuniversal tool to study the spin dynamics in quantum\noptics and solid state physics. The technique utilizes the\never present spin fluctuations in electronic spin ensem-\nbles at thermal equilibrium and probes the spin fluc-\ntuation dynamics by absorption-free Faraday rotation.\nFirst introduced by Aleksandrov and Zapassky in a gas\nof sodium atoms [1], the technique has been success-\nfully transferred by Oestreich et al.[2] to semiconduc-\ntor physics. This transfer is of special interest in the\ncontext of semiconductor spintronics since SNS is - in\ncontrast to the manifold Hanle [3], pump-probe Fara-\nday rotation [4, 15], and time- and polarization resolved\nphotoluminescence experiments [5] - a weakly perturbing\ntechnique that minimizes optical transitions from the va-\nlence to the conduction band [6]. SNS thereby avoids (a)\nheating of the carrier system which changes the spin in-\nteractions and the spin dephasing rates, (b) creation of\nfree electrons which inevitably interact and change the\nspin dynamics of localized electrons, and (c) the cre-\nation of holes which drives the electron spin dephasing\nby the Bir Aronov Pikus mechanism, i.e., in several cases\nonly SNS yields the intrinsic, unperturbed spin dephas-\ning rates [7, 8]. At the same time, SNS uniquely allows\nthree dimensional spatial mapping of the spin dynamics\n[9] and promises even single spin detection without de-\nstroying the spin quantum state. However, up to now\nthese advantages could not be used to its full extend,\nsince the temporal dynamics detected by SNS has been\ntechnically limited by the available bandwidth of the de-\ntection setup. Due to the in many cases low ratio of\nspin noise power to background noise, efficient data av-\neraging via analog-to-digital conversion and subsequent\nFFT as well as laser noise suppression via balanced pho-\ntodetection is necessary. Therefore, highly sensitive SNS\nis technically limited by the bandwidth of commercially\navailabledigitizers and lownoise balanced photoreceivers\nto frequencies of about 1 GHz [10].\nIn thisletter, we demonstratein n-dopedGaAs–prob-\nably the most prominent model system in semiconductor\nspintronics – that ultrafast, highly sensitive SNS [11] be-\ncomes feasible by sampling the temporal dynamics of theelectronspinwiththelaserrepetitionrateofapicosecond\nTi:Sapphire laser oscillator. In the first part, we present\nthe technique and argue that ultrafast SNS is applicable\nup to THz frequencies. In the second part, we combine\nultrafastand depth resolvedSNS measurements[9] to ex-\nplore the physical origin of the inhomogeneous spin de-\nphasing in n-doped GaAs close to the metal-to-insulator\ntransition (MIT) at high magnetic fields and study the\nfield dependence of the electron Land´ e g-factor.\n10 20 30 40 50 60 70 3530 3540 3550 3560 3570 3580 3590spin noise powerspin noise power\nfrequency f (MHz) frequency f (MHz)n-GaAs\nWollaston\nprism\nbalanced\nphotoreceivercryostat\n/c108/4/c108/4\n/c108/2/c108/2linear□polarizermagnetic□field\npulsed\nlaser\n80□MHz-\nto□spectral\nanalysis/c108/2a)\nb) c)PBS\nFIG. 1: (Color online) (a) Experimental setup for GHz SNS.\nThe repetition rate of the pulsed laser is doubledvia a Miche l-\nson interferometer like setup. (b, c) Spin noise spectra ac-\nquired from sample A at 25 K; the peaks correspond to 6.0\nand 1.0 mT (b) and to 594.7 and 588.3 mT (c), respectively.\nThe spectra are corrected for frequency dependent amplifica -\ntion via division by the amplification curve. The light solid\nlines are Lorentzian line fits to the data.\nFigure 1(a) schematically depicts the high frequency\nSNS experimental setup. The repetition rate of an ac-\ntively mode-locked ps-Ti:Sapphire laser ( frep= 80 MHz)\nis doubled to f′\nrep= 2frepvia a Michelson interferometer\nlike setup with different arm lengths. The linear polar-\nized ps laser pulses are transmitted through a n-doped\nGaAs sample where the wavelength of the laser is tuned2\nbelowthe GaAs bandgapto avoidabsorption. The mean\nspin polarization of the electron ensemble in the GaAs is\nzero at thermal equilibrium but the root mean square de-\nviation is unequal to zero and fluctuates with the trans-\nverse spin lifetime T∗\n2. The spin fluctuations are mapped\nvia Faraday rotation onto the direction of the linear po-\nlarization of the laser light which is measured by a po-\nlarization bridge. The resulting electrical signal from the\nbalanced photoreceiver is amplified by a low noise am-\nplifier, frequency filtered by a 70 MHz low pass filter,\nseamlessly digitized in the time domain (sampling rate\n180 MHz) and spectrally analyzed in real time via fast\nFourier transform. An external magnetic field Bin Voigt\ngeometry modulates the spin fluctuations with the Lar-\nmor precession frequency fL=g∗µBB/h, whereg∗is the\nelectron Land´ e g-factor. By sampling the spin dynam-\nics with the stroboscopic laser light, the spin noise spec-\ntrumS(f) centered around fLevolvesinto a sum of noise\npeaks/summationtext\nnS(f−nf′\nrep). The limit of the summation in-\ndexnis given by the ratio of the inverse pulse length to\nthe repetition rate. Due to the doubling of the original\nlaser repetition rate, only one summand of this sum falls\ninto the detection window. In principle, utilizing a GHz\ndigitizer card and a fs laser system with GHz repetition\nrate, both commercially available, increases the band-\nwidth of the detection and the detectable line width to a\nGHz and the detectable Larmor frequencies to the THz\nregime. We want to point out that this sampling with\npulsed laser light is not a nonlinear process like electri-\ncal frequency mixing and thereby does not introduce any\nadditional noise.\nThe two investigated samples AandBare commer-\ncial Si-doped GaAs wafers with doping densities right\nat the metal-to-insulator transition (MIT) ( nA\nd= 1.8·\n1016cm−3) and about 4 times above the MIT ( nB\nd=\n8.8·1016cm−3)andthicknessesof340 µmand370 µm, re-\nspectively. Both samples are anti-reflection coated to in-\ncrease the light transmittance. The samples are mounted\ninaHegasflowcryostatwith asuperconductingsplit coil\nmagnetinVoigtgeometry. Themagneticfieldiscarefully\ncalibrated by means of a Hall probe and the magnet sys-\ntem is always degaussed before starting a new series of\nmeasurement at lower magnetic fields to avoid magnetic\nhysteresis. The laser wavelength is set to 840 (845) nm\nfor measurements concerning sample A (B) if not stated\notherwise.\nFigure 1(b) and (c) depict two typical spin noise spec-\ntra of sample A at 25 K. Electronic and optical shot\nnoiseiseliminated in these spectraby subtractingin each\ncase two noise spectra at slightly different magnetic fields\nwhich results in a spin noise spectrum with one positive\nand one negative spin noise peak. The width and posi-\ntion of these two peaks are determined by fitting a sum\nof Lorentz lines to the experimental spectra. The re-\nsulting full width at half maximum of each Lorentz peak\n∆fFWHMand the peak position yield the spin dephasingrate Γ s= 1/T∗\n2=π∆fFWHMandg∗, respectively. The\nacquisitiontimesofbothspectraarelessthan20minutes.\nThe excellent signal to noise ratio of the high frequency\nspectrum (Fig. 1(c)) matches the signal to noise ratio of\nthe low frequency spectrum (Fig. 1(b)) proving that the\nultrafast SNS comes along without loss of sensitivity.\nNext, we employ high frequency SNS and measure the\ntransversemagnetic field dependence ofΓ sandg∗in bulk\nn-GaAs. Optical pumping techniques clearly influence\nboth Γ s[12] and g∗[13]. We focus on the doping regime\nclose to the MIT where an intricate interplay between lo-\ncalized and free electrons leads to a maximum in the low\ntemperature spin lifetime [14]. Many theoretical and ex-\nperimental groups have already investigated this doping\nregime but the physical origin of the experimentally ob-\nserved homogeneous or inhomogeneous broadening of Γ s\nwith increasing transverse magnetic fields was up to now\nunsolved. Kikkawa and Awschalom for example mea-\nsured via resonant spin amplification a plateau of the\nspin quality factor Q=g∗µBBT∗\n2/hat around 80 for a\ntemperature of 5 K [15]. They attributed this maximal\nQ-factor – which is a measure of the reduction of the\nspin lifetime in a transverse magnetic field – to the ther-\nmal electron distribution and the energy dependence of\ng∗yielding an inhomogeneous broadening. Puttika and\nJoynt gave a different explanation for the same exper-\nimental data and suggested a spin-phonon mechanism,\nwhich effectively leads to a homogeneous broadening of\nthe spin dephasing rate for localized electrons [12]. Fig-\nure 2(a) shows the Q-factor of a similar sample (sam-\nple A) measured via ultrafast SNS at a lattice temper-\nature of 25 K [29]. We measure a Q-factor that is de-\nspite a five times higher temperature about three times\nlarger than the value reported in Ref. [15]. This find-\ning directly disproves that the Qmeasured by Kikkawa\nand Awschalom is limited by a g-factor spread resulting\nfrom a thermally broadened electron energy distribution.\nSuch a thermally broadened g-factor spread is negligible\non these timescales due to a motional narrowing type\naveraging in the electron energy [16], which, addition-\nally, leads to a quadratic dependence of the broaden-\ning on the magnetic field and, hence, would not result\nin the formation of a Q-factor plateau. The ultrafast\nSNS measurements also discourage the theory of ther-\nmally activated lattice vibrations, since the line shape of\nthe SNS spectrum shows a clear crossover from a homo-\ngeneously broadened Lorentzian peak to an inhomoge-\nneously broadened Gaussian type peak [30].\nFor further investigations of the origin of the inhomo-\ngeneousbroadening, we use a high aperture cw spin noise\nsetup (see Ref. [9] for details) in orderto check for spatial\nelectron density variations within sample A that could,\ne.g., come from the Czochralski growth method. The\nunique spatial resolution strength of SNS originates from\nthe fact that the relative noise power becomes large for\nsmall electron ensembles so that most of the SNS signal3\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53 /s51/s46/s48/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48/s50/s53/s48/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53 /s51/s46/s48\n/s48/s46/s48/s48/s46/s53/s49/s46/s48/s45/s54/s48 /s45/s51/s48 /s48 /s51/s48 /s54/s48\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48/s48/s53/s49/s48/s49/s53/s50/s48/s102/s111/s99/s117/s115/s32/s112/s111/s115/s105/s116/s105/s111/s110/s32/s122/s32/s40 /s109/s41\n/s115/s97/s109/s112/s108/s101/s32/s65\n/s115/s97/s109/s112/s108/s101/s32/s66/s84/s32/s61/s32/s50/s53/s32/s75\n/s109/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32/s66/s32/s40/s84/s41\n/s32/s32/s115/s112/s105/s110/s32/s113/s117/s97/s108/s105/s116/s121/s32/s102/s97/s99/s116/s111/s114/s32/s81/s32/s84/s32/s61/s32/s50/s48/s32/s75\n/s99/s41\n/s116/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s84/s32/s40/s75/s41\n/s32/s32\n/s32/s115/s112/s97/s116/s105/s97/s108/s32/s103/s45/s102/s97/s99/s116/s111/s114/s32\n/s118/s97/s114/s105/s97/s116/s105/s111/s110/s32 /s103/s42/s47/s103/s42/s32/s40/s37/s41/s98/s41/s109/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32/s66/s32/s40/s84/s41\n/s32/s32/s105/s110/s116/s46/s32/s115/s112/s105/s110/s32/s110/s111/s105/s115/s101\n/s32/s112/s111/s119/s101/s114/s32/s40/s97/s114/s98/s46/s32/s117/s46/s41/s32/s97/s41\nFIG. 2: (Color online) (a) Spin quality factor as a function o f\nmagnetic field. In sample B, the spin noise power drops for\nhigh fields which we attribute to Pauli blocking of spin noise\ndue to filling of the Landau levels. (b) High aperture SNS\nmeasurement: Shift of the electron g-factor in dependence of\nthe position of the laser focus relative to the sample (indi-\ncated by the shaded area). The line is a guide to the eyes\nonly. The sample appears compressed by the refractive index\nof GaAs. Note that the average g-factor variation with the\nlaser focus outside the sample is lower than the average valu e\nwith the focus inside. This indicates that the spin noise in t he\nsample centercontributes stronger totheGHz spin noise mea -\nsurements (large Rayleigh range) than the spin noise from th e\nsample surfaces. (c) Integrated spin noise power as a functi on\nof temperature (corrected for different sample thickness; l aser\nwavelength: 840 nm, magnetic field: 6.75 mT). The lines are\nlinear fits.\nis acquired within the Rayleigh range of the laser focus.\nThe depth resolved measurements are carried out in a\nmicro-cryostat at a lattice temperature of 20 K, a laser\nwavelength of 830 nm, a Rayleigh range in vacuum of\n11µm and accordingly a depth resolution in the sample\nof about 80 µm, and a magnetic field of about 9 mT.\nFigure 2(b) shows the measured variation of the g-factor\n∆g∗/g∗versusfocusposition. Thefocuspositionisswept\nin this experiment from negative to positive z-position\nwhere the z-axis is chosen along the direction of light\npropagation and we define ∆ g∗/g∗= 0 atz= 0µm. The\nmeasurements reveal that the absolute value of g∗has\na pronounced maximum close to the front and the back\nsurface of the sample. Such a maximum of |g∗|is directly\nlinked at this doping concentration via the energy depen-\ndence of the g-factor to a minimum in electron density\n[22]. The width of the two measured peaks is equal to\nthe depth resolution of the laser which indicates that the\nspatialvariationof g∗issmallerthanthedepthresolution\noftheexperiment. Conductivitymeasurementsinsample\nA evince hopping behavior [8]. This g-factor variation is\nnot observed at higher temperatures where the conduc-\ntivity significantly increases and most electrons reside in\nthe conduction band.We model the spatial variation of the measured g-\nfactorbyasimplephenomenologicalmodelassumingpar-\ntial electron depletion at the surface, which results in a\ndecrease of the g-factor, the spin lifetime, and the noise-\npower at the sample surface, and get qualitative agree-\nment with the measured SNS profile.Other reasons for\nthe observed spatial g-factor broadening could involve\nstress in the sample or specifics of the sample growth\nwe are not aware of. A quantitative description is diffi-\ncult due to the intricate interplay between localized and\nfree electrons, Fermi level pinning, the influence of hop-\nping and diffusion on the depletion zone, and the elec-\ntric field induced change of g∗, the band gap, and the\ninterband transition probability [31]. Nevertheless, the\nspatially resolved g∗-variation directly entails the inho-\nmogeneous broadening measured by ultrafast SNS where\na Rayleigh range much larger than the sample thick-\nness is employed. The much smaller Q-factor reported\nin Ref. [15] most likely originates from the lower conduc-\ntivity at 5 K and the slightly lower doping concentration\nin accordance with our reasoning above.\nFigure 2(b) shows as a second feature a spatial asym-\nmetryofthe SNSsignal, i.e., therightpeakishigherthan\nthe left peak and ∆ g∗/g∗is larger if the laser focus is be-\nhind the sample ( z >46µm) compared to the laser focus\nin front of the sample ( z <−46µm). We have carefully\nchecked that this increase does not result from a doping\ngradient in the sample but from the finite measurement\ntime, i.e., ∆ g∗/g∗increaseswith laboratorytime andsat-\nurates after about one hour. To account for this effect,\nexperimental data presented in Fig. 2(a) and 3 are ac-\nquired after the saturation of this effect. We attribute\nthis behavior to the existence of deep centers acting as\nan electron trap, like the EL2 deep center, which has a\nmetastable state with a large lattice relaxation and is ex-\ncited by below band gap excitation [17–19]. Such deep\ncenters are the origin of various persistent photo effects\nin liquid phase grown GaAs. We exclude nuclear effects\nas origin since the spatial measurements were carried out\nat very low magnetic fields and since the integrated spin\nnoise power also changes with time. Within the mea-\nsurement error, the spin lifetime is not affected by this\neffect.\nFigure 2(a) also shows the Q-factor for sample B at\n25 K. Within the examined field range the Q-factor does\nnot level off as in sample A. The doping density of sam-\nple B of nB\nd= 8.8·1016cm−3is well above the MIT and\nthe electrons should therefore be delocalized. In order\nto directly prove the delocalized nature of the electrons\nin sample B, the integrated spin noise power is plotted\nin Fig. 2(c) for both samples as a function of tempera-\nture. The extrapolation to zero temperature yields for\nsample A a finite spin noise power proving partial local-\nization of electrons. The same extrapolation yields for\nsample B within the error bars zero spin noise power\nwhich is consistent with delocalized electrons and Pauli4\n/s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53 /s51/s46/s48/s45/s48/s46/s52/s51/s50/s45/s48/s46/s52/s51/s49/s45/s48/s46/s52/s51/s48/s45/s48/s46/s52/s50/s57/s45/s48/s46/s52/s50/s56\n/s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53 /s51/s46/s48/s45/s48/s46/s52/s49/s51/s45/s48/s46/s52/s49/s50/s45/s48/s46/s52/s49/s49/s45/s48/s46/s52/s49/s48/s45/s48/s46/s52/s48/s57\n/s56 /s49/s54 /s50/s52/s45/s48/s46/s52/s51/s50/s45/s48/s46/s52/s50/s56\n/s56 /s49/s54 /s50/s52/s45/s48/s46/s52/s49/s54/s45/s48/s46/s52/s48/s56/s83/s97/s109/s112/s108/s101/s32/s66\n/s50/s53/s32/s75\n/s32/s32/s101/s108/s101/s99/s116/s114/s111/s110/s32/s76/s97/s110/s100/s233/s32/s102/s97/s99/s116/s111/s114/s32/s103/s42\n/s109/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32/s66/s32/s40/s84/s41/s83/s97/s109/s112/s108/s101/s32/s65/s32\n/s32/s50/s53/s32/s75\n/s32/s32\n/s109/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32/s66/s32/s40/s84/s41/s32 /s32/s32\n/s116/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s84/s32/s40/s75/s41\n/s32 /s32/s32\n/s116/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s84/s32/s40/s75/s41\nFIG. 3: (Color online) Electron Land´ e g-factor as a function\nof magnetic field for sample A andB at 25 K and as afunction\nof thesample temperature at amagnetic field around850 mT.\nThe error bars of the magnetic field dependence include the\ninaccuracy of the magnetic field and the repetition rate f′\nrep\nand the fitting errors.\nblocking of the spin noise [32]. To our knowledge, there\nis no effective mechanism of inhomogeneous broadening\nof the spin dephasing rate for free electrons in this sam-\nple system. Hence, a strictly monotonic increase of the\nQ-factor with the magnetic field is expected in sample B.\nLast, we study the magnetic field and temperature de-\npendence of g∗. To our knowledge, similar investigations\nwere only carried out for undoped or very low n-doped\nbulk GaAs samples [20, 21]. Figure 3 showsthe magnetic\nfield dependence of g∗for sample A and B at 25 K. A\nlinear fit to the magnetic field dependence of g∗between\n0.5 T and 3 T yields a very small magnetic field depen-\ndence of g∗\nA=−0.4292(2)+0 .0003(1)T−1·Bfor sample\nA andg∗\nB=−0.4100(2)+0 .0003(1)T−1·Bfor sample B\natT= 25K. The gradient dg∗/dBis by more than one\norder of magnitude smaller than for free electrons in un-\ndoped GaAs [20] or for donor-bound electrons at higher\nmagnetic fields [21], where g∗(B) increases in both cases\nby 0.005T−1due to the energyshift ofthe lowestLandau\nlevel which is occupied by all electrons. At the doping\ndensities examined in this work, either Landau quanti-\nzation is suppressed because of momentum scattering or\nseveralLandaucylinders areoccupied since the cyclotron\nenergy is rather low compared to the Fermi energy. The\ndifference of g∗between the two samples extrapolated\ntoB= 0 is about 0 .02 which equates to a difference\nin Fermi energy of about 3 meV if we adapt the exper-\nimentally and theoretically reported energy dependence\nofg∗in GaAs of 6.3 eV−1(see Ref. [22] and references\ntherein). This difference in the Fermi energy is reason-\nable, i.e., the calculated difference in the Fermi level is\nsmaller assuming only an impurity band [23] and larger\nassuming only free electrons in bulk GaAs. The same\nenergy dependence of 6.3 eV−1explains quantitatively\nthe absolute value and the temperature dependence of\ng∗for sample B. The solid line in the right inset of Fig. 3depictsg∗(T) calculated by\ng∗(T) =/integraltext∞\n0DOS(E)f(E,T)(1−f(E,T))g∗(E)dE/integraltext∞\n0DOS(E)f(E,T)(1−f(E,T))dE,\nwhereDOS(E) is the threedimensionaldensityofstates,\nf(E,T) is the Fermi distribution, g∗(E) =g∗\n0+6.3eV−1·\nE, andg∗\n0is the electron Land´ e g-factor at the conduc-\ntion band minimum at T= 0 which is set to g∗\n0=−0.481\nin agreement within the error bars of the high precession\nmeasurements of Ref. [24]. The three dimensional den-\nsity of states is a good approximation for highly doped\nsamples. For sample A, the measured temperature de-\npendence of g∗is slightly larger than in sample B and\nis more difficult to calculate since donors at the MIT\nform an impurity band and ionization of electrons into\nthe conduction band has to be included.\nIn conclusion, we have overcome the major limita-\ntion of conventional spin noise spectroscopy and demon-\nstrated ultrafast SNS without any loss of sensitivity. We\nhaveappliedthetechniqueto n-dopedGaAsatandabove\nthe metal-to-insulator transition and proved in conjunc-\ntion with depth-resolved SNS that in the sample at the\nMIT surface depletion changesthe g-factorlocally result-\ning in inhomogeneous spin dephasing at high magnetic\nfields. This mechanism is of extrinsic nature and sam-\nples at the MIT with a spatially flat Fermi level should\nyield significantly higher Q-factors. Measurements on\nthe temperature dependence of the spin noise power ver-\nify the (de)localization of the electrons in the sample at\n(above) the MIT and ultrafast spin noise measurements\nyield the magnetic field and temperature dependence of\ng∗for thermalized electrons in the impurity and in the\nconduction band. Most importantly, we want to point\nout that ultrafast SNS is not limited to semiconductor\nspintronics but promises access to ultrafast spin dynam-\nics, e.g., in atomic ensembles, magnetically ordered sys-\ntems, like ferrimagnetic garnets where recently the Bose-\nEinstein condensation of magnons was discovered[25], or\nin superconductors at the phase transition [26].\nThis work was supported by the German Science\nFoundation (DFG priority program 1285 ‘Semiconductor\nSpintronics’), the Federal Ministry for Education and\nResearch (BMBF NanoQUIT), and Centre for Quantum\nEngineering and Space-Time Research in Hannover\n(QUEST). G.M.M. acknowledges support from the\nEvangelisches Studienwerk.\n∗Electronic mail: mueller@nano.uni-hannover.de\n[1] E. B. Aleksandrov and V. S. Zapasski ˘i, Sov. Phys.- JETP\n54, 64 (1981).\n[2] M. Oestreich, M. R¨ omer, R. J. Haug, and D. H¨ agele,\nPhys. Rev. Lett. 95, 216603 (2005).5\n[3] R. R. Parsons, Phys. Rev. Lett. 23, 1152 (1969).\n[4] J. J Baumberg, S. A. Crooker, D. D. Awschalom, N.\nSamarth, H. Luo, and J. K. Furdyna, Phys. Rev. B 50,\n7689 (1969).\n[5] A. P. Heberle, W. W. R¨ uhle, and K. Ploog, Phys. Rev.\nLett.72, 3887 (1994).\n[6] M. R¨ omer, J. H¨ ubner, and M. Oestreich, Rev. Sci. In-\nstrum.78, 103903 (2007).\n[7] G. M. M¨ uller, M. R¨ omer, D. Schuh, W. Wegscheider,\nJ. H¨ ubner, and M. Oestreich, Phys. Rev. Lett. 101,\n206601 (2008).\n[8] M. R¨ omer, G. M. M¨ uller, H. Bernien, J. H¨ ubner, and\nM. Oestreich, to be published.\n[9] M. R¨ omer, J. H¨ ubner, and M. Oestreich, Appl. Phys.\nLett.94, 112105 (2009).\n[10] S. A. Crooker, J. Brandt, C. Sandfort, A. Greilich, D. R.\nYakovlev, D. Reuter, A. D. Wieck, and M. Bayer, Phys.\nRev. Lett. 104, 036601 (2010).\n[11] S. Starosielec and D. H¨ agele, Appl. Phys. Lett. 93,\n051116 (2008).\n[12] W. O. Putikka and R. Joynt, Phys. Rev. B 70, 113201\n(2004).\n[13] T. Lai, L. Teng, Z. Jiao, H. Xu, L. Lei, J. Wen, and\nW. Lin, Appl. Phys. Lett. 91, 062110 (2007).\n[14] R. I. Dzhioev, K. V. Kavokin, V. L. Korenev, M. V.\nLazarev, B. Y. Meltser, M. N. Stepanova, B. P. Za-\nkharchenya, D. Gammon, and D. S. Katzer, Phys. Rev.\nB66, 245204 (2002).\n[15] J. M. Kikkawa and D. D. Awschalom, Phys. Rev. Lett.\n80, 4313 (1998).\n[16] I. Zutic, J. Fabian, and S. D. Sarma, Rev. Mod. Phys.\n76, 323 (2004).\n[17] A. Mitonneau and A. Mircea, Solid State Commun. 30,\n157 (1979).\n[18] G. M. Martin, Appl. Phys. Lett. 39, 747 (1981).\n[19] G. Vincent, D. Bois, and A. Chantre, J. Appl. Phys. 53,\n3643 (1982).\n[20] M. Oestreich, S. Hallstein, A. P. Heberle, K. Eberl,\nE. Bauser, and W. W. R¨ uhle, Phys. Rev. B 53, 7911\n(1996).\n[21] M. Seck, M. Potemski, and P. Wyder, Phys. Rev. B 56,\n7422 (1997).\n[22] M. J. Yang, R. J. Wagner, B. V. Shanabrook, J. R. Wa-\nterman, and W. J. Moore, Phys. Rev. B 47, 6807 (1993).[23] B. Shklovskii and A. Efros, Electronic Properties of\nDoped Semiconductors (Springer series in solid-state sci-\nences)(Springer, 1984).\n[24] J. H¨ ubner, S. D¨ ohrmann, D. H¨ agele, and M. Oestreich,\nPhys. Rev. B 79, 193307 (2009).\n[25] S. O. Demokritov, V. E. Demidov, O. Dzyapko, G. A.\nMelkov, A. A. Serga, B. Hillebrands, and A. N. Slavin,\nNature443, 430 (2006).\n[26] R. Prozorov, A. F. Fidler, J. R. Hoberg, and P. C. Can-\nfield, Nat. Phys. 4, 327 (2008).\n[27] K. V. Kavokin, Semicond. Sci. Technol. 23, 114009\n(2008).\n[28] S. A. Crooker, L. Cheng, and D. L. Smith, Phys. Rev. B\n79, 035208 (2009).\n[29] The experimental values given in this letter correspon d\nto the most symmetric measured spin noise peaks and,\nhence, to the longest measured spin lifetimes, since some\ntraces show an asymmetric behavior from unknown ori-\ngin. However, no systematic dependence on the angle be-\ntween sample surface or magnetic field direction can be\nreported.\n[30] At moderate fields, a physical correct fit function would\nbase upon Voigt profiles which have too many free pa-\nrameters for practical purposes. Nevertheless, fittingwit h\nGaussian profiles yields very similar peak widths as fit-\nting Lorentzian profiles. At small magnetic fields and\ntemperatures below 50 K, the temperature dependence\nof Γsscales with the temperature dependence of the con-\nductivity [8] suggesting a spin dephasing mechanism that\ndepends on motional narrowing and should therefore be\nless efficient at higher B[27].\n[31] In our simple model, we have to assume a width of the\ndepletion zone of some ten microns in order to mimic\nthe measured results. A more sophisticated model would\nprobably yield a different width. Nevertheless, we are not\naware of any theoretical description of depletion zones at\nthe MIT. Additionally, spin diffusion and electron hop-\nping also lead to smoothing of the measured g-factor pro-\nfile which is not accounted for in our modelling.\n[32] Crooker and co-workers reported for a lower doping den-\nsity of 7.1·1016cm−3still a finite fraction of spin noise\npower at extrapolation to zero temperature [28]."    },    {        "title": "2108.12769v1.Spin_accumulation_and_dissipation_excited_by_an_ultrafast_laser_pulse.pdf",        "content": "Spin accumulation and dissipation excited by an ultrafast laser pulse\nWen-Tian Lu and Zhe Yuan\u0003\nCenter for Advanced Quantum Studies and Department of Physics,\nBeijing Normal University, Beijing 100875, China\n(Dated: August 31, 2021)\nAn ultrafast spin current can be induced by femtosecond laser excitation in a ferromagnetic (FM)\nthin \flm in contact with a nonmagnetic (NM) metal. The propagation of an ultrafast spin current\ninto NM metal has recently been found in experiments to generate transient spin accumulation.\nUnlike spin accumulation in equilibrium NM metals that occurs due to spin transport at the Fermi\nenergy, transient spin accumulation involves highly nonequilibrium hot electrons well above the\nFermi level. To date, the di\u000busion and dissipation of this transient spin accumulation has not been\nwell studied. Using the superdi\u000busive spin transport model, we demonstrate how spin accumulation\nis generated in NM metals after laser excitation in an FM jNM bilayer. The spin accumulation shows\nan exponential decay from the FM jNM interface, with the decay length increasing to the maximum\nvalue and then decreasing until saturation. By analyzing the ultrafast dynamics of laser-excited hot\nelectrons, the \\e\u000bective mean free path\", which can be characterized by the averaged product of the\ngroup velocity and lifetime of hot electrons, is found to play a key role. The interface re\rectivity\nhas little in\ruence on the spin accumulation in NM metals. Our calculated results are in qualitative\nagreement with recent experiments.\nI. INTRODUCTION\nSince the discovery of femtosecond laser-induced ultra-\nfast demagnetization in nickel observed by the magneto-\noptical Kerr experiment (MOKE),1various related phys-\nical phenomena have attracted much attention and have\nbeen extensively studied. The physical mechanism of ul-\ntrafast demagnetization is under debate for two possi-\nbilities: the local dissipation of angular momentum2{10\nand nonlocal spin transport.11{14Beaurepaire et al.1ex-\nplained ultrafast demagnetization by considering the en-\nergy coupling between electrons, spins, and phonon baths\nvia the so-called three-temperature model. Later, sev-\neral mechanisms based on electron-electron,2electron-\nmagnon,3,4electron-phonon,5{7and electron-photon8{10\ninteractions were proposed. Unlike the local spin-\rip pic-\nture, Battiato et al.11,12proposed that the superdi\u000busive\nspin transport arising from the spin-dependent hot elec-\ntrons excited by a laser pulse played a major role, which\nwas subsequently found to have an application for broad-\nband terahertz emission.15,16\nThe study of hot electron transport can be traced back\nto 1987, when Brorson et al.17observed ultrafast electron\ntransport after femtosecond laser excitation of a gold thin\n\flm. In particular, the velocity of the heat transport was\nfound to be the same order of magnitude as the Fermi\nvelocity of Au, i.e., \u0018106m/s, indicating that the heat\ntransport was not due to electron-phonon relaxation but\noccurred due to the transport of nonequilibrium hot elec-\ntrons well above the Fermi energy. Malinowski et al.18\n\frst demonstrated that the transport of spin-polarized\nhot electrons was associated with ultrafast demagnetiza-\ntion in a magnetic multilayer excited by a laser pulse.\nThis pioneering work stimulated many related studies in\nspintronics involving ultrafast hot electron transport in\nthe following decade. An FM jNM bilayer was found to\nserve as a terahertz emitter,19,20in which a picosecondpulse of the spin-polarized current due to the excited hot\nelectrons from the FM metal was injected into the NM\nmetal. This picosecond time scale was intrinsically de-\ntermined because of the di\u000berence in the spin-dependent\nvelocity of the hot electrons. Owing to the large spin-\norbit coupling of the NM metal, the spin-polarized cur-\nrent was then converted into a transverse charge current\nvia the inverse spin Hall e\u000bect (ISHE). Then, the pi-\ncosecond pulse of the transverse charge current produced\nelectromagnetic radiation, whose frequency was naturally\nlocated in the terahertz region. Eschenlohr et al. did\nnot directly excite the FM metal using the femtosecond\nlaser. Instead, the NM metal Au in contact with FM Ni\nwas excited and ultrafast demagnetization was observed\nin the Ni induced by the hot electrons that were trans-\nported into the FM metal.21This experiment demon-\nstrated that the direct absorption of a femtosecond laser\nwas not the necessary condition for ultrafast demagneti-\nzation. Rudolf et al. found in a NijRujFe trilayer with\nparallel magnetization for the two FM metals that laser-\ninduced demagnetization in Ni transiently enhanced the\nmagnetization of the Fe layer, indicating spin transfer\ncarried by hot electrons.22Later, a spin transfer arising\nfrom femtosecond laser-induced spin-polarized hot elec-\ntrons was found to exert a torque on an FM metal, driv-\ning its magnetization dynamics within a subpicosecond\ntimescale.23Such ultrafast magnetization dynamics can\neven be realized without using an FM metal as the spin\ncurrent source. For example, in a Pt/Au/GdFeCo mul-\ntilayer, Wilson et al. made laser excitation in Pt and\ndemonstrated that the ferrimagnetic GdFeCo layer was\nswitched by injected hot electrons.24These \fndings sug-\ngest that hot electron transport is not only important in\nthe fundamental physics of ultrafast spin transport and\ndynamics but also has potential application in femtosec-\nond spintronics devices.25\nThere is not a consensus in the understanding of ultra-arXiv:2108.12769v1  [cond-mat.mes-hall]  29 Aug 20212\nfast hot electron transport. Melnikov et al. noted that\nthere could be a competition of ballistic and di\u000busive\npropagation for hot electron transport, and the measured\nspin relaxation time of the hot electrons was approxi-\nmately 1 ps.26Later, they experimentally demonstrated\nthat the ultrashort spin current pulses in Fe jAujFe epi-\ntaxial multilayers were related to the existence of a non-\nthermal spin-dependent Seebeck e\u000bect corresponding to\nballistic transport of spin-polarized electrons in Au.27\nFurther evidence was provided by an extremely long de-\ncay length on the order of 100 nm in Au for nonther-\nmalized electrons,28which were excited optically in the\nadjacent Fe thin \flm and injected through the Fe jAu in-\nterface as a spin \flter. Bergeard et al. reported a linear\ndependence of the onset of demagnetization time on the\nthickness of the attached Cu layer and attributed the\nobservation to the ballistic transport of hot electrons.29\nOn the other hand, Hofherr et al. found that the de-\nmagnetization dynamics of Ni jAu were mainly caused by\noptically generated spin currents, which exhibited a tran-\nsition from ballistic to di\u000busive transport.30B uhlmann et\nal.observed that the decay time of the spin polarization\ninjected from a laser-induced ferromagnet into a thin Au\nlayer increased with Au thickness.31Salvatella et al. pre-\nsented that the characteristic demagnetization times as a\nfunction of the Al thickness had a discontinuity at dAl=30\nnm, indicating di\u000busive heat transport of hot electrons.32\nTransient spin accumulation was observed in Cu, Ag,\nPt, and Au during the laser-induced ultrafast demagne-\ntization of an attached FM multilayer.33{35Such tran-\nsient spin accumulation is signi\fcantly di\u000berent from\nthe steady-state spin accumulation in equilibrium NM\nmetals36that is usually produced by spin pumping,37,38\nthe spin Hall e\u000bect39and the spin Seebeck e\u000bect.40Tran-\nsient spin accumulation involves highly nonequilibrium\nhot electrons and is expected to have a very short decay\ntime or length. However, the dynamics of ultrafast spin\naccumulation and spin relaxation remain unclear.41In\naddition, it is unknown to what extent the well-developed\nspin di\u000busion theory responsible for the steady-state spin\ncurrent is still applicable for the ultrafast, photoexcited\nspin current.42\nIn this article, we apply the superdi\u000busive spin trans-\nport theory and systematically study spin accumulation\nin an FejNM bilayer excited by a femtosecond laser pulse;\nsee Fig. 1. Au, Al and Pt are selected as typical exam-\nples of NM metals. The spin accumulation is found to\nexponentially decay from the Fe jNM interface, and the\nextracted spin di\u000busion length lsexhibits a nonmono-\ntonic dependence on time: increasing to the maximum\nvalue and then decreasing slightly until saturation. We\n\fnd that this behavior is mainly caused by the movement\nof the center of gravity of the excited hot electrons inside\nthe NM metal. By combining the numerical calculation\nand analytical analysis, we demonstrate that both the\nsaturated value of lsand the time to reach its maximum\nare proportional to the \\e\u000bective mean free path\" of the\nhot electrons in the NM metal. The in\ruence of interface\nFigure1\nNM\nSpin accumulationFMjsLaser Pump\nCaption :Schematic diagram ofsuperdiffusive spin transport and accumulation inanFM|NM bilayer .The hotelectrons intheFMmetal\nareexcited byafemtosecond laser pulse .With spin-dependent hotelectron mobility and interface reflection, theultrafast spin current\nisgenerated intheNM layer .Because thelifetime ofhotelectrons and theduration ofspin current pulse areboth inthefemtosecond\ntime scale, thespin current willdissipate during itstransmission, that isthespin accumulation caused byinelastic scattering .FIG. 1. Schematic diagram of ultrafast spin transport and\naccumulation in an FM jNM bilayer. The hot electrons in the\nFM metal are excited by a femtosecond laser pulse. Owing\nto the spin-dependent mobility and interface re\rection, an ul-\ntrafast spin current is injected into the NM layer. The elastic\nand inelastic scattering of the ultrafast spin current results in\nspin accumulation in the NM metal, which is illustrated by\nthe red shadow.\nre\rection on spin accumulation is also discussed. The\nrest of this paper is organized as follows. The theoretical\nmethods and computational details are brie\ry introduced\nin Sec. II. The calculated results for spin accumulation in\nFejNM bilayers are presented in Sec. III. This is followed\nby discussions about lsfor its nonmonotonic variation in\ntime (Sec. III A), the NM metal dependence (Sec. III B)\nand the e\u000bect of interface re\rection (Sec. III C). A com-\nparison with recent experiments is provided in Sec. III D.\nConclusions are given in Sec. IV.\nII. THEORETICAL METHODS AND\nCOMPUTATIONAL DETAILS\nThe superdi\u000busive spin transport model has been suc-\ncessfully applied to describe the experimental observation\nof ultrafast demagnetization resulting from excitation by\na femtosecond laser pulse. In this theory, a laser pulse\ncreates many nonequilibrium hot electrons, which are ex-\ncited from the occupied states below the Fermi energy EF\nto the energy bands above EF. The nonequilibrium elec-\ntron characteristics, e.g., group velocities and lifetimes,\ndepend on their spins in magnetic metals. For example, a\nphoton with an energy of 1.5 eV can excite the majority-\nspin 3delectrons in Fe to the unoccupied 4 sband, while\nthe minority-spin 3 delectrons are excited to other unoc-\ncupied 3dbands. Therefore, the excited nonequilibrium\nelectrons in Fe have spin-dependent group velocities lead-\ning to a spin-polarized current density. Meanwhile, there\nare holes in the 3 dbands below EFthat are neglected in\nthe superdi\u000busive spin transport model because of their\nrelatively low mobility.43\nNonequilibrium hot electrons above EFpropagate with\ntheir own group velocity and recombine with the holes be-\nlowEFduring propagation due to a \fnite lifetime. More-\nover, hot electrons are scattered by phonons, impurities,\nand/or other electrons. Here, we simultaneously con-\nsider the elastic scattering that conserves the scattered3\nelectron energy while changing their momentum and the\ninelastic scattering that allows the scattered electrons to\nlose energy. The latter may transfer the released en-\nergy via electron-hole recombination and excite another\nelectron to a higher energy, which produces a cascade of\nelectrons that contribute to transport.\nThe laser-excited hot electrons can transport across\nan FMjNM interface and enter the attached NM metal.\nThen, the spin angular momentum is transferred from the\nFM material to enhance the ultrafast demagnetization.\nInside the NM metal, the injected nonequilibrium spin-\npolarized hot electrons are scattered, which leads to spin\naccumulation. In this work, we explicitly calculate the\ntime- and spatial-dependent spin accumulation in NM\nmetals after laser excitation in FM metal, while numeri-\ncal techniques can be found in the previously published\nliterature.12,44,45\nSince the laser spot is much larger than the mean free\npath of the excited hot electrons, transport can be re-\nduced to one dimension along the interface normal of the\nmetallic multilayers ( z-axis). The key equation in the\nsuperdi\u000busive spin transport model reads12\n@n\u001b(E;z;t )\n@t+n\u001b(E;z;t )\n\u001c\u001b(E;z)=\u0012\n\u0000@\n@z^\u001e+^I\u0013\nSe\u000b\n\u001b(E;z;t );\n(1)\nwheren\u001b(E;z;t ) is the density of nonequilibrium hot\nelectrons with spin \u001bat energyE, positionzand time\nt.\u001c\u001b(E;z) is the lifetime of hot electrons with spin \u001b\nat energyE, and depends on the material via its po-\nsitionz. Thus, the second term on the left-hand side\nrepresents the dissipation of the nonequilibrium hot elec-\ntrons.Se\u000b\n\u001b(E;z;t ) in Eq. (1) is the e\u000bective source term\nfor hot electrons, including scattered and newly excited\nelectrons. ^\u001eand^Iare the electron \rux and identity op-\nerators, respectively. The electron \rux operator ^\u001eacting\non the source term S(z;t) can be explicitly expressed as\n^\u001eS(z;t) =Z+1\n\u00001dz0Zt\n\u00001dt0S(z0;t0)\u001e(z;tjz0;t0);(2)\nwhere the spin and energy indices are omitted for simplic-\nity. The \rux kernel \u001e(z;tjz0;t0) represents the electron\ndensity \rux at a given position zand timetresulting from\nan electron that is excited at z0andt0. The superdi\u000bu-\nsive spin transport equation (1) is nonlocal in space and\ntime and is solved iteratively.44\nSpin accumulation is de\fned by the di\u000berence in the\nmajority- and minority-spin electron density, which in-\ncludes the hot electrons above the Fermi energy and the\noccupied electrons below EF, i.e.,\nM(z;t) =Z\ndE[n\"(E;z;t )\u0000n#(E;z;t )]\n+nocc\n\"(z;t)\u0000nocc\n#(z;t): (3)\nHere,nocc\n\u001b(z;t) is the electron density below the Fermi\nlevel with spin \u001b, which is the summation of the electrons\nthat are not excited and the electrons that drop to a lower\nenergy than EFdue to inelastic scattering.In this work, we consider three types of FM jNM bi-\nlayers, namely, Fe jAu, FejAl and FejPt, where the thick-\nness of Fe is always \fxed at 10 nm. The thickness of\nthe NM metals is set as 1500 nm, 600 nm, and 200 nm\nfor Au, Al and Pt, respectively. This is to maintain a\nNM metal thickness that is much larger than the length\nscale of the spin accumulation to avoid the \fnite thick-\nness e\u000bect. The spin-dependent lifetimes and velocities\nof excited electrons are determined using \frst-principles\nmany-body calculations.46,47The excitation laser pulse\nis modeled using a Gaussian function with a wavelength\nof 780 nm (1.5 eV) and a full width at half maximum\n(FWHM) of 50 fs. An equal number of spin-up and spin-\ndown electrons in Fe are excited by the laser pulse, as in\nthe previous calculation.12The nonequilibrium hot elec-\ntrons between EFandEF+ 1:5 eV are discretized in\nenergy with an interval of 0.125 eV. The transition prob-\nability of electron scattering is set to be the same as that\nin the literature.12Numerically, a spatial grid of 1 nm\nand a time step of 1 fs are employed.\nIt is important to note that we do not explicitly in-\nclude phonons in the calculation, but phonon scattering\nis implicitly imposed in the elastic scattering rate to vary\nthe momenta of the nonequilibrium hot electrons.12In\naddition, the spin-phonon relaxation time and electron-\nphonon relaxation time are on the order of several\npicoseconds,6,48which is longer than or comparable to\nthe total time of our calculation. Therefore, electron\nthermalization due to electron-phonon and spin-phonon\ninteractions is neglected. This approximation is valid un-\nless the magnetization dynamics on a larger timescale\nneed to be studied.\nIII. RESULTS AND DISCUSSIONS\nUsing the superdi\u000busive spin transport model, the\nmagnetization dynamics in the Fe jNM bilayer are calcu-\nlated immediately after excitation by a femtosecond laser\npulse, which has the maximum amplitude at t= 300 fs.\nAs an example, the spin accumulation that is de\fned by\nEq. (3) in Pt is plotted at several times, t= 300, 500,\nand 2000 fs in Fig. 2(a). At every t, the calculated spin\naccumulation decreases from the Fe jPt interface, which\ncan be reproduced by an exponential decay, as shown by\nthe solid lines. Analogous to the semiclassical di\u000busion\ntheory of a spin current,49we can de\fne the decay length\nas the e\u000bective \\spin di\u000busion length\" ls.\nAtt= 300 fs, the laser pulse just reaches the maximum\nvalue, and only a small part of the spin angular momen-\ntum carried by the excited hot electrons has transferred\ninto Pt. The spin accumulation in Pt is relatively weak,\nand the \ftted lsis short. At t= 500 fs, hot electrons\nare su\u000eciently excited and transported across the Fe jPt\ninterface, resulting in a signi\fcant increase in spin ac-\ncumulation in Pt; see the red symbols in Fig. 2(a). At\nt= 2000 fs, the spin accumulation near the Fe jPt in-\nterface decreases, while increasing in the interior Pt at4\n100020003000400050000501001501\n389 fs1520 fsls (nm)T\nime (fs) Au \nAl \nPt2202 fs02 04 06 08 00123(\nb)(a)z\nPt (nm)300 fsM (Arb. Units)5\n00 fs2000 fs\nFIG. 2. (a) Calculated spin accumulation in Pt at several\ntimes after excitation by a laser pulse in Fe. z= 0 is de\fned\nat the Fe jAu interface. The solid lines are the exponential \fts\nto the calculated spin accumulation to extract the e\u000bective\nspin di\u000busion length ls. The data are o\u000bset for clarity. (b)\nThe extracted lsfor the three NM elements (NM=Au, Al, and\nPt) in the Fe jNM bilayer as a function of time. The arrows\ndenote the time at which lsreaches its maximum value. The\nlight red line illustrates the pro\fle of the laser pulse.\nzPt= 10\u001850 nm. This is because an increasing number\nof hot electrons move into Pt from the excitation side\nand are transported to a deeper position.\nTo better understand the behavior of spin accumula-\ntion in the NM layer, we plot the extracted spin di\u000busion\nlengthlsas a function of time in Fig. 2(b). The pro\fle of\nthe Gaussian laser pulse with an FWHM of 50 fs is also\nplotted as a light red line for reference. For all three NM\nmetals,lsincreases rapidly immediately after the excita-\ntion of the laser pulse. Then, lsreaches the maximum\nvalue at a certain time tmax, as denoted by the arrows\nat 2202 fs (Au), 1520 fs (Al), and 1389 fs (Pt). Next, lsslightly decreases and saturates to a constant value in the\ncalculated range of time. This nonmonotonic behavior is\nuniversal for the three NM metals, and among them, Au\nis found to have the largest saturation value of lsand the\nlatesttmax. The maximum value and saturated value\noflsin Al are both less than the corresponding values in\nAu. This is di\u000berent from the conventional spin-\rip di\u000bu-\nsion length for steady-state spin transport near the Fermi\nlevel, where due to the large spin-orbit interaction, Au\nwould have a shorter spin-\rip di\u000busion length than Al.50\nOn the other hand, the saturated values of lsare still\nmuch shorter than the conventional spin di\u000busion length\nbecause the calculated time is only several picoseconds.\nThis time scale is much less than the time needed to bring\nthe excited system back to equilibrium, which is usually\nhundreds of picoseconds or nanoseconds.41\nA. Nonmonotonic behavior of ls\nThe unexpected nonmonotonic dependence of lson\ntime is analyzed by separating the contributions from the\nhot electrons above EFand the occupied electronic states\nbelow the Fermi energy, i.e., the \frst and the second lines\nin Eq. (3). Since the laser does not excite the electrons\nin the NM metal, the occupied electrons have equal num-\nbers of majority and minority spins at a small t. After the\nexcited hot electrons from Fe move into the NM metal,\nthe occupied electrons are excited by the energy released\nfrom the inelastic scattering of hot electrons leaving holes\nbelowEFin the NM metal. These holes can then be re-\ncombined with hot electrons, and spin accumulation will\noccur because there are more hot electrons with majority\nspin than minority-spin electrons.\nThe calculated time-dependent spin accumulation at\nthe FejNM interface is plotted in Fig. 3 (a), (c), and (e)\nfor NM=Au, Al and Pt, respectively. The black, red and\nblue lines represent the total spin accumulation, the con-\ntribution from hot electrons above EF, and the contri-\nbution from the occupied states below EF, respectively.\nAfter laser excitation, the spin accumulation increases\nrapidly due to the excited majority-spin hot electrons\nthat are quickly injected into the NM metal. Spin accu-\nmulation soon reaches the maximum value and then de-\ncreases slowly. The hot electrons have a relatively short\nlifetime, so the red lines decay within 1 ps. Meanwhile,\nhot electrons lose their energy to other electrons via in-\nelastic scattering and drop below EF. Then, the spin\ndensity that occurs as a result of the occupied states\nincreases gradually and tends to saturate; see the blue\nline. The decay time is determined by the lifetime of the\nhot electrons. Au has the slowest decay time due to the\nlongest lifetime of hot electrons in Au, while the decay\ntime is the fastest in Pt due to its shortest lifetime. The\namplitude of the spin accumulation is larger in Pt than\nin Au or Al because the group velocity of hot electrons\nin Pt is small and the spin density cannot move quickly\neither into the interior Pt or back to Fe. In contrast, the5\n024AlM (Arb. Units)012(\ne)Au \nTotal \nHot \nOccupied(a)( b)(\nc)5\n001 0001 5002 0000510P tT\nime (fs)100020003000400050000.00.51.0 \n  5 nm \n10 nm \n15 nm \n20 nm \n25 nm \n30 nm1389 fs0.00.51.0M\n (Normalized) \n  20 nm \n  40 nm \n  60 nm \n  80 nm \n100 nm \n120 nm \n140 nm1520 fs(d)(\nf)0.00.51.0 \n  50 nm \n100 nm \n150 nm \n200 nm \n250 nm \n300 nm \n350 nm \n400 nm \n450 nm \n500 nm \n550 nm2202 fs\nFIG. 3. Calculated spin accumulation as a function of time\nfor NM=Au [(a) and (b)], Al [(c) and (d)], and Pt [(e) and\n(f)]. (a), (c) and (e) show the calculated data at the Fe jNM\ninterface, i.e., z= 0. The black, red and blue lines repre-\nsent the total spin accumulation, the contribution from hot\nelectrons above EFand from the occupied state below EF,\nrespectively. (b), (d) and (f) show the normalized total spin\naccumulation at di\u000berent positions inside the NM metal. The\nstars illustrate the maximum value in every curve. The verti-\ncal dashed line represents the time tmaxwhen the maximum\nlsoccurs, as marked by the arrows in Fig. 2(b).\nhot electrons in Au have a large velocity, and the spin\naccumulation at the Fe jAu interface has a small ampli-\ntude.\nAfter understanding the time-dependent spin accumu-\nlation at the FejNM interface, we plot the normalized\nspin accumulation at di\u000berent positions in the NM metal\nin Fig. 3(b), (d) and (f). At every time, the value at\nz= 0 is normalized to be unity. Then, the overall fea-\nture at every position zis very similar to the pro\fle at\nz= 0, while the maximum value is reached at a later time\nthan at the FejNM interface, as indicated by the stars.With increasing z, the maximum value of the spin accu-\nmulation decreases, and the time to reach the maximum\nvalue is also delayed. This is consistent with the physi-\ncal picture that spin-polarized hot electrons are injected\ninto the NM metal and their center of gravity moves to-\nwards the interior of the NM metal with a \fnite rate of\ndissipation.\nBased on the calculated curves in Fig. 3(b), (d) and\n(f), we can understand the nonmonotonic lsshown in\nFig. 2(b) as follows. At small t, the spin accumulation\nnear the FejNM interface rapidly increases to the max-\nimum value, while at a large zonly increases slightly.\nThe large di\u000berence in spin accumulation near and far\nfrom the interface results in a relatively small ls. At an\nintermediate time, the spin accumulation near the inter-\nface decreases and tends to saturate, whereas that far\nfrom the interface approaches the maximum. The rel-\natively small di\u000berence leads to a larger ls. Long after\nlaser excitation, the spin accumulation near the interface\nhas already saturated, and Maway from the interface\nalso decreases and tends to saturate. Then, the position-\ndependent di\u000berence in Mincreases again, resulting in\na reducedls. The time for the maximum lsto occur is\ndenoted by the vertical dashed lines in Fig. 3(b), (d) and\n(f). All the vertical lines are located at the time with M\nnear the interface already tending to saturate, while Min\nthe interior NM metal has not yet reached its maximum.\nThe location con\frms the above explanation.\nB. Material dependence\nThe nonmonotonic time dependence of lsis universal\nfor all three NM metals, but both the saturated value\noflsin the long time limit and the time tmaxwhen the\nmaximum of lsoccurs depend on the speci\fc material. To\nunderstand the di\u000berence in the three representative NM\nmetals, we \frst analytically derive the dynamics of the\n\frst generation hot electrons, i.e., the electrons excited\ndirectly by the laser pulse. In this approximation, we\nassume that the electrons fall below the Fermi energy as\nsoon as they are scattered for the \frst time.\nBecause the velocities and lifetimes for both spin chan-\nnels are the same in NM metals, we only consider a single\nspin case without loss of generality. The \frst-generation\nelectron density \rux \u001e\u001b(z;t) as a result of electrons that\nare excited at position z0and timet0(or, for an NM\nmetal, injected into the NM metal at z= 0 and at time\nt0) can be generally expressed as12\n\u001e\u001b(z;t) =Z\u0019\n2\n0sin\u0012\n2exp\u0014\n\u00001\ncos\u0012Zz\nz0dz0\n\u001c(z0)v(z0)\u0015\n\u0002\u0014\nt\u0000t0\u00001\ncos\u0012Zz\nz0dz0\nv(z0)\u0015\n\u0002(t\u0000t0)d\u0012; (4)\nwhere\u0012is de\fned as the polar angle which considers the xandycomponents of the velocity and \u0002( t) is a step\nfunction with a value of 1 for t>0 and a value of 0 for t\u00140. The total electron density that has not passed through6\npositionzcan be obtained accordingly,\n~n\u001b(z;t) =Z\u0019\n2\n0sin\u0012\n2\u001a\n1\u0000exp\u0014\n\u00001\ncos\u0012Zz\nz0dz0\n\u001c(z0)v(z0)\u0015\u001b\n\u0002\u0014\nt\u0000t0\u00001\ncos\u0012Zz\nz0dz0\nv(z0)\u0015\n\u0002(t\u0000t0)d\u0012: (5)\nInside a homogeneous NM metal, where vand\u001care both constant, the above equation can be simpli\fed as\n~n\u001b(z;t) =Z\u0019\n2\n0sin\u0012\n2\u0014\n1\u0000exp\u0012\n\u0000z\u0000z0\n\u001cvcos\u0012\u0013\u0015\n\u0002\u0012\nt\u0000t0\u0000z\u0000z0\nvcos\u0012\u0013\n\u0002(t\u0000t0)d\u0012: (6)\nTo obtain the local electron density at position z, we only need to di\u000berentiate Eq. (6) with respect to position as\nn\u001b(z;t) =@~n\u001b(z;t)\n@z=Z\u0019\n2\n0sin\u0012\n2\u001cvcos\u0012exp\u0012\n\u0000z\u0000z0\n\u001cvcos\u0012\u0013\n\u0002\u0012\nt\u0000t0\u0000z\u0000z0\nvcos\u0012\u0013\n\u0002(t\u0000t0)d\u0012: (7)\nThe above integral can be analytically carried out in the\nlimit oft!1 ,\nn\u001b(z;t!1 ) =1\n2v\u001c\u00000\u0012z\u0000z0\nv\u001c\u0013\n; (8)\nwhere the upper incomplete \u0000-function is de\fned by\n\u00000(x) =Z1\nxexp(\u0000t)\ntdt: (9)\nNote that Eq. (8) is the local distribution of the \frst\ngeneration electron density in the long time limit, which\ndecays exponentially with the length scale v\u001c. Similarly,\nthe other spin also exhibits exponential decay with the\nsame decay length. Therefore, the di\u000berence in the two\nspins, i.e., the spin accumulation Mat a time long af-\nter the excitation, has the same decay length v\u001c, which\ndetermines the saturated ls. To verify this analysis, we\n0 10 20 30 40\nvτ (nm)050100150200Saturated ls (nm)\n0 50 100 150\nSaturated ls (nm)150020002500tmax (fs)\nAu\nAl\nPt\nFIG. 4. Saturated lsin the calculation for the three NM\nmetals as a function of the product v\u001c, which is averaged in\nthe energy range from EFto 1.5 eV above EF. The red solid\nline illustrates the proportionality. Inset: the time tmaxwhen\nthe maximum lsoccurs as a function of the saturated ls. The\norange dashed line shows the linear relationship.plot the saturated lsfor Au, Al and Pt in Fig. 4 as a\nfunction of the product v\u001cthat is averaged for the hot\nelectrons in the energy range from EFto 1.5 eV above\nEF.lsis found to be larger than the averaged product\nv\u001c, but a perfect proportionality is illustrated by the red\nsolid line. In reality, multiple generations of excited hot\nelectrons extend the propagation length, but scaling with\nthe material parameter v\u001cstill holds.\nThe change in spin angular momentum exhibits\nElliott-Yafet-like behavior because it is only caused by\nthe decay and reexcitation of hot electrons. The aver-\naged product v\u001ccan be regarded as the \\e\u000bective mean\nfree path\" of the hot electrons. Then, the injected spin\ncan propagate to a deeper position in an NM metal with\na largerv\u001c, as found in Fig. 2(b). For the time tmaxwhen\nthe maximum lsappears, a similar explanation is found.\nNear the FejNM interface, the time to reach the maxi-\nmum spin accumulation is relatively fast within 500 fs\ndespite the slight di\u000berence among the three NM met-\nals. On the other hand, with a large \\e\u000bective mean\nfree path\", the spin accumulation away from the inter-\nface keeps increasing and therefore postpones the time\ntmax. Both the saturated values of lsandtmaxare quan-\ntities used to characterize the propagation capability of\nthe superdi\u000busive spin current, and are expected to have\na positive correlation. As plotted in the inset of Fig. 4,\nthe linear dependence of tmaxon the saturated lsfor the\nthree NM metals is shown.\nC. E\u000bect of interface re\rectivity\nAt an actual FejNM interface, the potentials seen by\nthe laser-excited hot electrons are signi\fcantly di\u000berent\non each side, resulting in a \fnite transmission probability\nof the incoming electrons. In particular, the transmission\nor re\rection probability depends on the energy and spin\nof the hot electrons.45In the above calculations, we ne-\nglect the interfacial re\rectivity and assume a transparent\ninterface for all electrons. In the following, the Fe jPt sys-\ntem is used as an example to examine the in\ruence of the7\n204 06 08 00242\n000 fs700 fsM (Arb. Units)z\nPt (nm) Transparent \nReal4 00 fs\nFIG. 5. Calculated spin accumulation in Pt for transparent\n(black line) and real (red line) interfaces at t= 400 fs, 700 fs,\n2000 fs as a function of the distance from the Fe jPt interface.\nFor the real interface, the spin- and energy-dependent re-\n\rectivity obtained from \frst-principles transport calculations\nhas been incorporated into the superdi\u000busive spin transport\nmodel.45The curves are o\u000bset for clarity.\ninterface re\rectivity on the spin accumulation and dissi-\npation in NM metals.\nHere, we follow our previous work,45where the FMjNM\ninterface resistance was calculated in the energy range\nof 0\u00181.5 eV above the Fermi level. The spin-down\nelectrons were found to have a higher re\rectivity than\nthe spin-up electrons, leading to the spin \fltering e\u000bect\nin both directions. The calculated interfacial re\rectivity\nhas been incorporated into the superdi\u000busive spin trans-\nport model to calculate the laser-induced magnetization\ndynamics and terahertz emission of the Fe jNM bilayer.45\nUsing this scheme, we calculate the spin accumulation\nin Pt after laser excitation in the attached Fe layer. In\nFig. 5, we plot the calculated spin accumulation with the\nreal interface re\rectivity (red lines) and with the trans-\nparent interface (black lines) at three times for a quan-\ntitative comparison. At t= 400 fs, i.e., 100 fs after the\nmaximum intensity of the laser pulse, the spin accumu-\nlations of the real and transparent interfaces are nearly\nidentical.Monly has a slightly lower amplitude at the\nreal interface. This is because of the spin \fltering ef-\nfect of the FejPt interface: the spin-up electrons have a\nhigher probability of passing through the interface than\nthe spin-down electrons. Although the spin \fltering ef-\nfect increases the nonequilibrium majority spins injected\ninto Pt, the accumulated spin at the interface also has a\nhigher probability of \rowing back to Fe. Eventually, the\ncalculatedMis slightly lower only at the Fe jPt interface\nafter consideration of the real interface re\rectivity.\nAt a later time, t= 700 fs and t= 2000 fs, the di\u000ber-\nence between the real and transparent interfaces becomesslightly larger but only within 5 nm from the interface.\nIn a more interior region of Pt, the interface re\rectivity\ndoes not change the numerical results of the spin accu-\nmulation. Therefore, our calculations and discussions in\nthe previous sections are not a\u000bected by the spin \fltering\ne\u000bect of the interface.\nD. Comparison with experiments\nOur calculated results are in qualitative agreement\nwith some recent experimental observations. By exclud-\ning optical interference, Melnikov et al. probed the spin\npolarization at the Au surface in the Fe jAu multilayer\nexcited by a laser pulse.27Since the majority-spin elec-\ntrons in Fe have a higher velocity, they were injected into\nAu \frst and then minority-spin electrons were injected.\nTherefore, the injected spin current exhibits a positive\nspin polarization in the \frst half and a negative polar-\nization later. However, the measured spin polarization\nwas always positive, although it increased \frst and then\ndecreased, forming a nonmonotonic behavior. The ex-\nperimental phenomena are in good agreement with the\ncalculated positive spin accumulation in the NM metal\nshown in Fig. 2 and Fig. 3, which is mainly contributed\nto by the majority-spin hot electrons because the mo-\nbility of the minority-spin carriers is much weaker and\nusually only has a minor e\u000bect on spin accumulation.\nThe estimated decay length of hot electrons in Au is ap-\nproximately 100 nm,28comparable with the saturated\nvalue ofls= 157:7 nm in our calculation. In addi-\ntion, the measured decay time of the spin polarization\nin the Au layer injected from the attached Fe in the ex-\nperiment increases with the thickness of the Au layer,\nsuggesting that hot electron transport is not purely bal-\nlistic in Au.31In our calculation, we could not repro-\nduce the quantitative lsusing the \frst-generation hot\nelectrons due to the signi\fcant contribution of the cas-\ncade of electrons, indicating the superdi\u000busive nature of\nthe ultrafast spin current. The multiple generations of\nhot electrons essentially exhibit a transition from ballis-\ntic to di\u000busive transport.12Note that we only consider\nthe electronic and magnetization dynamics in a few pi-\ncoseconds, which is much smaller than the time needed\nfor the system to return to equilibrium. To study the pro-\ncess forlsrecovering the spin-\rip di\u000busion length in equi-\nlibrium, a simulation must be performed with several or-\nders of magnitude longer time and include additional re-\nlaxation mechanisms,30such as the electron-phonon and\nspin-phonon interaction.\nIV. CONCLUSIONS\nWe systematically studied the spin accumulation and\ndissipation in NM metals with NM=Au, Al and Pt using\nthe superdi\u000busive spin transport model for Fe jNM bilay-\ners. The spin accumulation shows an exponential decay8\nwith the distance from the Fe jNM interface, which can\nbe characterized by an \\e\u000bective spin di\u000busion length\" ls.\nUnlike the constant spin-\rip di\u000busion length in equilib-\nrium, the calculated ls\frst increases and then decreases\nwith time. This nonmonotonic behavior can be under-\nstood by analyzing the hot electron dynamics excited by a\nlaser pulse. Speci\fcally, the ultrafast spin current carried\nby hot electrons is injected from the Fe jNM interface, and\ngradually decay while moving towards the interior region\nin the NM metal. The competition of spin accumulation\nnear the interface and in the interior region leads to the\ntime-dependent variation of ls. The saturated value of\nlsin the long-time limit is proportional to the averaged\nproduct of the electron velocity vand lifetime \u001c, the \\ef-\nfective mean free path\" of the hot electrons. The time\ntmaxfor the maximum lsto occur is also proportional.Using FejPt as an example, we demonstrate that the hot\nelectron re\rection at the Fe jNM interface only slightly\nreduces the spin accumulation near the interface but has\nvery little in\ruence on regions far from the interface. Our\ncalculated results are in good agreement with recent ex-\nperiments.\nACKNOWLEDGMENTS\nThis work was partly supported by the National Nat-\nural Science Foundation of China (Grants No. 61774018,\nNo. 12174028 and No. 11734004), the Recruitment Pro-\ngram of Global Youth Experts, and the Fundamental Re-\nsearch Funds for the Central Universities (Grants No.\n2018EYT03 and No. 2018STUD03).\n\u0003Corresponding author: zyuan@bnu.edu.cn\n1E. Beaurepaire, J.-C. Merle, A. Daunois, and J.-Y. Bigot,\nUltrafast Spin Dynamics in Ferromagnetic Nickel, Phys.\nRev. Lett. 76, 4250 (1996).\n2M. Krau\u0019, T. Roth, S. Alebrand, D. Steil, M. Cinchetti,\nM. Aeschlimann, and H. C. Schneider, Ultrafast demagne-\ntization of ferromagnetic transition metals: The role of the\nCoulomb interaction, Phys. Rev. B 80, 180407(R) (2009).\n3E. Carpene, E. Mancini, C. Dallera, M. Brenna, E. Puppin,\nand S. De Silvestri, Dynamics of electron-magnon interac-\ntion and ultrafast demagnetization in thin iron \flms, Phys.\nRev. B 78, 174422 (2008).\n4E. G. Tveten, A. Brataas, and Y. Tserkovnyak, Electron-\nmagnon scattering in magnetic heterostructures far out of\nequilibrium, Phys. Rev. B 92, 180412(R) (2015).\n5B. Koopmans, J. J. M. Ruigrok, F. Dalla Longa, and W. J.\nM. de Jonge, Unifying Ultrafast Magnetization Dynamics,\nPhys. Rev. Lett. 95, 267207 (2005).\n6B. Koopmans, G. Malinowski, F. Dalla Longa, D. Steiauf,\nM. F ahnle, T. Roth, M. Cinchetti, and M. Aeschlimann,\nExplaining the paradoxical diversity of ultrafast laser-\ninduced demagnetization, Nat. Mater. 9, 259 (2010).\n7Alireza Qaiumzadeh, Gerrit E. W. Bauer, and Arne\nBrataas, Manipulation of ferromagnets via the spin-\nselective optical Stark e\u000bect, Phys. Rev. B 88, 064416\n(2013).\n8G. P. Zhang and W. H ubner, Laser-Induced Ultrafast De-\nmagnetization in Ferromagnetic Metals, Phys. Rev. Lett.\n85, 3025 (2000).\n9J.-Y. Bigot, M. Vomir, and E. Beaurepaire, Coherent ultra-\nfast magnetism induced by femtosecond laser pulses, Nat.\nPhys. 5, 515 (2009).\n10F. Freimuth, S. Bl ugel, and Y. Mokrousov, Laser-induced\ntorques in spin spirals, Phys. Rev. B 103, 054403 (2021).\n11M. Battiato, K. Carva, and P. M. Oppeneer, Superdi\u000busive\nSpin Transport as a Mechanism of Ultrafast Demagnetiza-\ntion, Phys. Rev. Lett. 105, 027203 (2010).\n12M. Battiato, K. Carva, and P. M. Oppeneer, Theory of\nlaser-induced ultrafast superdi\u000busive spin transport in lay-\nered heterostructures, Phys. Rev. B 86, 024404 (2012).\n13D. M. Nenno, S. Kaltenborn, and H. C. Schneider, Boltz-\nmann transport calculation of collinear spin transport onshort timescales, Phys. Rev. B 94, 115102 (2016).\n14D. M. Nenno, B. Rethfeld, and H. C. Schneider, Particle-\nin-cell simulation of ultrafast hot-carrier transport in\nFe/Au heterostructures, Phys. Rev. B 98, 224416 (2018).\n15T. Seifert, S. Jaiswal, U. Martens, J. Hannegan, L. Braun,\nP. Maldonado, F. Freimuth, A. Kronenberg, J. Henrizi, I.\nRadu, E. Beaurepaire, Y. Mokrousov, P. M. Oppeneer, M.\nJourdan, G. Jakob, D. Turchinovich, L. M. Hayden, M.\nWolf, M. M unzenberg, M. Kl aui, and T. Kampfrath, E\u000e-\ncient metallic spintronic emitters of ultrabroadband tera-\nhertz radiation, Nat. Photonics 10, 483 (2016).\n16S. Zhang, Z. Jin, Z. Zhu, W. Zhu, Z. Zhang, G. Ma, and J.\nYao, Bursts of e\u000ecient terahertz radiation with saturation\ne\u000bect from metal-based ferromagnetic heterostructures, J.\nPhys. D: Appl. Phys. 51, 034001 (2018).\n17S. D. Brorson, J. G. Fujimoto, and E. P. Ippen, Femtosec-\nond electronic heat-transport dynamics in thin gold \flms,\nPhys. Rev. Lett. 59, 1962 (1987).\n18G. Malinowski, F. Dalla Longa, J. H. H. Rietjens, P. V.\nPaluskar, R. Huijink, H. J. M. Swagten, and B. Koopmans,\nControl of speed and e\u000eciency of ultrafast demagnetiza-\ntion by direct transfer of spin angular momentum, Nat.\nPhys. 4, 855 (2008).\n19T. Kampfrath, M. Battiato, P. Maldonado, G. Eilers,\nJ. N otzold, S. M ahrlein, V. Zbarsky, F. Freimuth, Y.\nMokrousov, S. Bl ugel, M. Wolf, I. Radu, P. M. Oppeneer,\nand M. M unzenberg, Terahertz spin current pulses con-\ntrolled by magnetic heterostructures, Nat. Nanotech. 8,\n256 (2013).\n20Z. Feng, H. Qiu, D. Wang, C. Zhang, S. Sun, B. Jin, and\nW. Tan, Spintronic terahertz emitter, J. Appl. Phys. 129,\n010901 (2021).\n21A. Eschenlohr, M. Battiato, P. Maldonado, N. Pontius, T.\nKachel, K. Holldack, R. Mitzner, A. F ohlisch, P. M. Op-\npeneer, and C. Stamm, Ultrafast spin transport as key to\nfemtosecond demagnetization, Nat. Mater. 12, 332 (2013).\n22D. Rudolf, C. La-O-Vorakiat, M. Battiato, R. Adam, J.\nM. Shaw, E. Turgut, P. Maldonado, S. Mathias, P. Grych-\ntol, H. T. Nembach, T. J. Silva, M. Aeschlimann, H. C.\nKapteyn, M. M. Murnane, C. M. Schneider, and P. M.\nOppeneer, Ultrafast magnetization enhancement in metal-\nlic multilayers driven by superdi\u000busive spin current, Nat.9\nCommun. 3, 1037 (2012).\n23A.J. Schellekens, K.C. Kuiper, R. R. J. C. de Wit, and B.\nKoopmans, Ultrafast spin-transfer torque driven by fem-\ntosecond pulsed-laser excitation, Nat. Commun. 5, 4333\n(2014).\n24R. B. Wilson, J. Gorchon, Y. Yang, C.-H. Lambert, S.\nSalahuddin, and J. Bokor, Ultrafast magnetic switching of\nGdFeCo with electronic heat currents, Phys. Rev. B 95,\n180409(R) (2017).\n25Z. Bai, Y. Liu, R. Kong, T. Nie, Y. Sun, H. Li, T. Sun, C.\nPandey, Y. Wang, H. Zhang, Q. Song, G. Liu, M. Kraft,\nW. Zhao, X. Wu, and L. Wen, Near-\feld Terahertz Sensing\nof HeLa Cells and Pseudomonas Based on Monolithic Inte-\ngrated Metamaterials with a Spintronic Terahertz Emitter,\nACS Appl. Mater. Interfaces 12, 35895 (2020).\n26A. Melnikov, I. Razdolski, T. O. Wehling, E. Th.\nPapaioannou, V. Roddatis, P. Fumagalli, O. Akt-\nsipetrov, A. I. Lichtenstein, and U. Bovensiepen, Ultra-\nfast Transport of Laser-Excited Spin-Polarized Carriers in\nAu/Fe/MgO(001), Phys. Rev. Lett. 107, 076601 (2011).\n27A. Alekhin, I. Razdolski, N. Ilin, J. P. Meyburg, D. Diesing,\nV. Roddatis, I. Rungger, M. Stamenova, S. Sanvito, U.\nBovensiepen, and A. Melnikov, Femtosecond Spin Cur-\nrent Pulses Generated by the Nonthermal Spin-Dependent\nSeebeck E\u000bect and Interacting with Ferromagnets in Spin\nValves, Phys. Rev. Lett. 119, 017202 (2017).\n28A. Alekhin, I. Razdolski, M. Berritta, D. B urstel, V. Tem-\nnov, D. Diesing, U. Bovensiepen, G. Woltersdorf, P. M.\nOppeneer, and A. Melnikov, Magneto-optical properties\nof Au upon the injection of hot spin-polarized electrons\nacross Fe/Au(001) interfaces, J. Phys.: Condens. Matter\n31, 124002 (2019).\n29N. Bergeard, M. Hehn, S. Mangin, G. Lengaigne, F. Mon-\ntaigne, M. L. M. Lalieu, B. Koopmans, and G. Malinowski,\nHot-Electron-Induced Ultrafast Demagnetization in Co/Pt\nMultilayers, Phys. Rev. Lett. 117, 147203 (2016).\n30M. Hofherr, P. Maldonado, O. Schmitt, M. Berritta, U.\nBierbrauer, S. Sadashivaiah, A. J. Schellekens, B. Koop-\nmans, D. Steil, M. Cinchetti, B. Stadtm uller, P. M. Op-\npeneer, S. Mathias, and M. Aeschlimann, Speed and ef-\n\fciency of femtosecond spin current injection into a non-\nmagnetic material, Phys. Rev. B 96, 100403(R) (2017).\n31K. B uhlmann, G. Saerens, A. Vaterlaus, and Y. Acremann,\nDetection of femtosecond spin injection into a thin gold\nlayer by time and spin resolved photoemission, Sci. Rep.\n10, 12632 (2020).\n32G. Salvatella, R. Gort, K. B uhlmann, S. D aster, A. Vater-\nlaus, and Y. Acremann, Ultrafast demagnetization by hot\nelectrons: Di\u000busion or super-di\u000busion? Struct. Dyn. 3,\n055101 (2016).\n33G.-M. Choi and D. G. Cahill, Kerr rotation in Cu, Ag, and\nAu driven by spin accumulation and spin-orbit coupling,\nPhys. Rev. B 90, 214432 (2014).\n34G.-M. Choi, B.-C. Min, K.-J. Lee and D. G. Cahill, Spin\ncurrent generated by thermally driven ultrafast demagne-\ntization, Nat. Commun. 5, 4334 (2014).\n35G.-M. Choi, Magneto-Optical Kerr E\u000bect Driven by Spin\nAccumulation on Cu, Au, and Pt, Appl. Sci. 8, 1378(2018).\n36T. Kimura, J. Hamrle, and Y. Otani, Estimation of spin-\ndi\u000busion length from the magnitude of spin-current ab-\nsorption: Multiterminal ferromagnetic/nonferromagnetic\nhybrid structures, Phys. Rev. B 72, 014461 (2005).\n37Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, En-\nhanced Gilbert Damping in Thin Ferromagnetic Films,\nPhys. Rev. Lett. 88, 117601 (2002).\n38E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Conver-\nsion of spin current into charge current at room tempera-\nture: Inverse spin-Hall e\u000bect, Appl. Phys. Lett. 88, 182509\n(2006).\n39J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back,\nand T. Jungwirth, Spin Hall e\u000bects, Rev. Mod. Phys. 87,\n1213 (2015).\n40K. Uchida, J. Xiao, H. Adachi, J. Ohe, S. Takahashi, J.\nIeda, T. Ota, Y. Kajiwara, H. Umezawa, H. Kawai, G.\nE. W. Bauer, S. Maekawa, and E. Saitoh, Spin Seebeck\ninsulator, Nat. Mater. 9, 894 (2010).\n41G. Malinowski, N. Bergeard, M. Hehn, and S. Mangin,\nHot-electron transport and ultrafast magnetization dy-\nnamics in magnetic multilayers and nanostructures follow-\ning femtosecond laser pulse excitation, Eur. Phys. J. B 91,\n98 (2018).\n42K.-H. Ko, and G.-M. Choi, Optical method of determining\nthe spin di\u000busion length of ferromagnetic metals, J. Magn.\nMagn. Mater. 510, 166945 (2020).\n43J. Ma, and S. Gao, Plasmon-Induced Electron-Hole Sepa-\nration at the Ag/TiO 2(110) Interface, ACS nano 13, 13658\n(2019).\n44M. Battiato, P. Maldonado, and P. M. Oppeneer, Treat-\ning the e\u000bect of interface re\rections on superdi\u000busive spin\ntransport in multilayer samples (invited), J. Appl. Phys.\n115, 172611 (2014).\n45W.-T. Lu, Y. Zhao, M. Battiato, Y. Wu, and Z. Yuan, In-\nterface re\rectivity of a superdi\u000busive spin current in ultra-\nfast demagnetization and terahertz emission, Phys. Rev. B\n101, 014435 (2020).\n46V. P. Zhukov, E. V. Chulkov, and P. M. Echenique, Life-\ntimes and inelastic mean free path of low-energy excited\nelectrons in Fe, Ni, Pt, and Au: Ab initio GW+T calcula-\ntions, Phys. Rev. B 73, 125105 (2006).\n47V. P. Zhukov, E. V. Chulkov, and P. M. Echenique,\nGW+Ttheory of excited electron lifetimes in metals,\nPhys. Rev. B 72, 155109 (2005).\n48J.-Y. Bigot, V. Halt\u0013 e, J.-C. Merle, and A. Daunois, Elec-\ntron dynamics in metallic nanoparticles, Chem. Phys. 251,\n181 (2000).\n49T. Valet, and A. Fert, Theory of the perpendicular mag-\nnetoresistance in magnetic multilayers, Phys. Rev. B 48,\n7099 (1993).\n50J. Bass, and W. P Pratt Jr, Spin-di\u000busion lengths in met-\nals and alloys, and spin-\ripping at metal/metal interfaces:\nan experimentalist's critical review, J. Phys.: Condens.\nMatter 19, 183201 (2007)."    },    {        "title": "1409.4045v2.Generation_and_Detection_of_Spin_Currents_in_Semiconductor_Nanostructures_with_Strong_Spin_Orbit_Interaction.pdf",        "content": "Generation and Detection of Spin Currents in Semiconductor Nanostructures with\nStrong Spin-Orbit Interaction\nFabrizio Nichele,1,\u0003Szymon Hennel,1Patrick Pietsch,1Werner Wegscheider,1\nPeter Stano,2, 3Philippe Jacquod,4Thomas Ihn,1and Klaus Ensslin1\n1Solid State Physics Laboratory, ETH Z urich, 8093 Z urich, Switzerland\n2RIKEN Center for Emergent Matter Science, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan\n3Institute of Physics, Slovak Academy of Sciences, Dubravska cesta 9, 84511 Bratislava, Slovakia\n4HES-SO, Haute Ecole d'Ing\u0013 enierie, 1950 Sion, Switzerland\n(Dated: March 1, 2022)\nStoring, transmitting, and manipulating information using the electron spin resides at the heart\nof spintronics. Fundamental for future spintronics applications is the ability to control spin currents\nin solid state systems. Among the di\u000berent platforms proposed so far, semiconductors with strong\nspin-orbit interaction are especially attractive as they promise fast and scalable spin control with\nall-electrical protocols. Here we demonstrate both the generation and measurement of pure spin\ncurrents in semiconductor nanostructures. Generation is purely electrical and mediated by the\nspin dynamics in materials with a strong spin-orbit \feld. Measurement is accomplished using a\nspin-to-charge conversion technique, based on the magnetic \feld symmetry of easily measurable\nelectrical quantities. Calibrating the spin-to-charge conversion via the conductance of a quantum\npoint contact, we quantitatively measure the mesoscopic spin Hall e\u000bect in a multiterminal GaAs\ndot. We report spin currents of 174 pA, corresponding to a spin Hall angle of 34%.\nThe generation and detection of spin currents in nanos-\ntructures is the central challenge of semiconductor spin-\ntronics. On the one hand, spin injection cannot be easily\nachieved by coupling semiconductors to ferromagnets [1]\nbecause of the lack of control over material interfaces [2].\nOn the other hand, magnetoelectric alternatives exploit-\ning the celebrated spin Hall e\u000bect (SHE) [3, 4], have deliv-\nered only qualitative measurement protocols in transport\nexperiments [5]. Alternatively to all-electrical setups,\nspin polarizing the current through a quantum point con-\ntact (QPC) with a magnetic \feld allows a quantitative\ncontrol over spin current generation and detection at the\nnanoscale [6{8]. The latter approach typically requires\nsuch high magnetic \felds (6 \u00008 Tesla) that, as a draw-\nback, the desired magnetoelectric e\u000bects are either sup-\npressed or totally altered.\nThis Letter reports two major advances of nanoscale\nsemiconductor spintronics. Namely, we develop novel ex-\nperimental methods to electrically generate andquanti-\ntatively measure spin currents in a two-dimensional semi-\nconductor nanostructure.\nIt is predicted that charge currents \rowing through\nspin-orbit interaction (SOI)-coupled nanostructures are\ngenerically accompanied by spin currents, if the spin-\norbit time is shorter than the electron dwell time [9{12].\nThis spin current generation mechanism is purely elec-\ntrical and based on the mesoscopic SHE (MSHE) [9, 10],\nwhere the electronic orbital dynamics in chaotic nanos-\ntructures cooperates with the SOI to make transport\nspin dependent. We will consider an open three-terminal\nquantum dot as represented in Fig. 1(a), where each lead\niis a QPC carrying Nispin degenerate modes. Running\na charge current Ibetween terminals 1 and 2, a spin cur-\nrent in all terminals, including 3, is expected due to theMSHE.\nFor a weak SOI, the spin currents' amplitude \ructu-\nates from sample to sample with zero average. For cav-\nities with a strong SOI, geometric correlations between\nthe spin and the orbital electronic dynamics lead to spin\ncurrents with large, predictable nonzero average values\n[13]. In the latter case, the spin currents' amplitude is\ndetermined by the nanostructure geometry and the SOI\nstrength. This particular mechanism renders spin cur-\nrents robust against decoherence and allows us to di\u000ber-\nentiate them from mesoscopic \ructuations. This is es-\nsential for spintronics applications, where spin currents\nmust be reproducible regardless of the microscopic details\nof the sample.\nTo detect and measure the spin currents described\nabove, we employ the scheme of Ref. [14], based on\nthe magnetic \feld parity of the voltage behind a QPC.\nWith reference to Fig. 1(a), we are interested in the spin\ncurrentI(S)\n3emitted from QPC 3. The energy depen-\ndent transmission probability of QPC 3,T(s)\n33, is shown\nin Fig. 1(b). At zero \feld, QPC 3is tuned to a con-\nductance of e2=hby a suitable gate voltage, correspond-\ning to a spin-independent transmission probability of one\nhalf. A weak in-plane magnetic \feld Bmodi\fes the elec-\ntrons' kinetic energy via Zeeman coupling, selectively in-\ncreasing or decreasing the transmission probability ac-\ncording to the spin eigenstate and magnetic \feld sign.\nFor simplicity, we assume I(S)\n3to be a pure spin cur-\nrent atB= 0, arising as two counterpropagating and\ninversely spin-polarized charge currents of equal magni-\ntude, as schematically shown in Fig. 1(c), where arrows\nindicate current amplitude and colors spin polarization.\nA magnetic \feld a\u000bects the QPC spin dependent trans-\nmission probability, enhancing one of the two charge cur-arXiv:1409.4045v2  [cond-mat.mes-hall]  18 May 20152\nI=03B=0\nI<03B>0\nI>03B<0B<0\nB=0\nB>0\nEnergy E01\n0.5\n0II(s)\nI3\n123\nBSpin 1\nSpin 2(a) (b)\n(c)\n(Spin 1)\nT33\nEZE=gμB Z B\nFIG. 1. (a) Schematic of the system used to generate spin\ncurrents. Charge currents are depicted as black lines, spin\ncurrents are depicted as red and blue lines. (b) Energy de-\npendent, spin sensitive, transmission probability of QPC 3. At\nzero \feld the transmission is tuned to 1 =2. A positive (nega-\ntive) \feld suppresses (enhances) the transmission probability\nof one spin eigenstate. (c) Representation of spin and charge\ncurrents in QPC 3as a function of magnetic \feld. The net\ncharge current in QPC 3varies with the magnetic \feld.\nrents and suppressing the other. The result is the \row\nof a net charge current I3in the QPC. The sign of I3re-\nverses by reversing the magnetic \feld sign. Note that the\nnet spin current remains constant in the three situations\ndepicted in Fig. 1(c).\nBy operating QPC 3as a \roating probe, the charge\ncurrentI3is constantly \fxed to zero. In this case, the\npresence of a spin current I(S)\n3in the QPC re\rects itself\nin an antisymmetric component of the voltage behind\nit:V3(B)6=V3(\u0000B). Remarkably, theory predicts that\nthe zero-\feld derivative of the measured voltage, @BV3,\nis proportional to the spin current I(S)\n3polarized along\nthe applied magnetic \feld. The proportionality coe\u000e-\ncient between the spin-to-charge signal @BV3andI(S)\n3is\ngiven by the QPC gfactor and its energy sensitivity \u0016 h!\nmeasured at N3= 0:5 [14]:\nI(S)\n3=e2\nh2\u0016h!\n\u0019g\u0016B@BV3: (1)\nMore generally, for N3\u00141 the presence of the spin\ncurrent still re\rects itself in a \fnite spin-to-charge signal\nwhose amplitude is directly proportional to the detector\nnormalized transconductance: @BV3/@VG3=G3. For\nN3= 0:5 it results in @VG3=G3=\u0000\u0019=\u0016h!. Details about\nthe derivation of this proportionality and Eq. (1) are re-\nported in the Supplemental Material [15]. Equation (1)\nnot only allows us to detect the presence of a spin cur-\nrent \rowing in QPC 3, but also to quantitatively measure\nand express it in units of ampere, giving the di\u000berence in\ncurrents carried by electrons with opposite polarization.\nGiven the large SOI of our system, the measurement pro-\ncess requires only weak magnetic \felds that do not a\u000bect\nVg4,g5 (V)B (T)\n  \n−0.8 −0.6 −0.4 −0.2 0−3−1.501.53\n6.26.36.4\n−3 −1.5 0 1.5 33.33.43.53.6\nB (T)V3/I (kΩ)−3 −1.5 0 1.5 366.16.26.36.46.5\nB (T)VC/I (kΩ)\nVg4,g5 (V)B (T)\n  \n−0.8 −0.6 −0.4 −0.2 0−3−1.501.53\n3.33.43.53.6−0.8 −0.6 −0.4 −0.2 00123\nVg4,g5 (V)G3 (2e2/h)(a)\n(c)(b)\n(d)\n(f) (e)12 T0 T\n109 mK\n585 mK\n109 mK\n585 mKV3/I (kΩ)VC/I (kΩ)\ng1\ng2g3g4g5\nQPC1QPC2\nIV3\nQPC3VCFIG. 2. (a) Atomic force micrograph of our sample, where\ndark lines indicate insulating trenches. The frame size is 5 \u0002\n5\u0016m2. (b) QPC 3conductance as a function of the voltage\napplied to g 4and g 5, for di\u000berent values of magnetic \feld. (c)\nCavity four-terminal resistance as a function of the voltage\napplied to g 4and g 5and magnetic \feld. (d) Linecut of (c)\nalong the dashed line for di\u000berent temperatures. (e) V3=Ias\na function of the voltage applied to g 4and g 5and magnetic\n\feld. (f) Linecut of (e) along the dashed line for di\u000berent\ntemperatures.\nthe generated spin currents. We note that our approach\nis restricted to the linear response regime, and to termi-\nnal 3 being a weak probe, i.e., N3\u00141\u001cN1;N2. The\nmeasurement protocol described here is independent of\nthe spin current generation mechanism. In particular,\nthe detection method can be used to measure spin cur-\nrents of other origin.\nMotivated by the theory above, we study a three-\nterminal chaotic cavity embedded in a p-type GaAs two-\ndimensional hole gas (2DHG) with a strong Rashba SOI.\nOur sample, shown in Fig. 2(a), lacks any spatial sym-\nmetry and consists of three leads and \fve in-plane gates,\nnamed QPCiand g jrespectively. The gates g 4and g 5,\ncolored in blue, tune the conductance of QPC 3with little\nin\ruence on the dot average conductance. The gates g 1,\ng2and g 3, depicted in red, tune the conductance of QPC1\nand QPC2, and also a\u000bect the dot shape. The lateral ex-\ntent of the cavity is about 2 \u0016m, the hole mean free path\nle= 4:8\u0016m and the spin-orbit length lSO= 36 nm [15].\nSpin rotational symmetry is then completely broken and,3\n−0.4 −0.2 0−20020406080\nVg4,g5 (V)∂B V3/I (Ω T−1)\n00.51\n∂VG3/G3 (a.u.)\n−0.6 −0.4 −0.2−20020406080\nVg4,g5 (V)∂B V3/I (Ω T−1)\n00.51\n∂VG3/G3 (a.u.)−0.4 −0.2 0−20020406080\nVg4,g5 (V)∂B V3/I (Ω T−1)\n00.51\n∂VG3/G3 (a.u.)\n−0.4 −0.2 0−20020406080\nVg4,g5 (V)∂B V3/I (Ω T−1)\n00.51\n∂VG3/G3 (a.u.)(a)\n(c)(b)\n(d) N1=4   N2=4 N1>10   N2>10N1=5   N2=4 N1=4   N2=5\nFIG. 3. (a)-(d) Comparison between @BV3=I(markers) and\n@VG3=G3(solid line) as a function of side gate voltage for\ndi\u000berent number of modes in QPC 1and QPC 2, as indicated in\neach sub\fgure. Dots and squares in (a) represent two identical\nmeasurements performed in di\u000berent cooldowns [16].\nwith such a strong SOI, our cavity is in the so-called spin\nchaos regime [13]. Unless di\u000berently stated, a charge cur-\nrentI\rows from terminal 1 to terminal 2, while terminal\n3 is connected to a high impedance voltage ampli\fer and\nis used to measure spin currents.\nTo measure the spin current in terminal 3, we \frst\nextract the detector electric and magnetic sensitivity via\na standard QPC characterization. Figure 2(b) shows the\ndetector conductance G3as a function of side gate voltage\nfor di\u000berent values of a magnetic \feld aligned with the\nQPC 3axis. The three well-developed plateaus visible at\nzero \feld split at \fnite \feld. The zero \feld slope and\nthe \fnite \feld splitting give, respectively, @VG3and the\ngfactor [15].\nAfter the detector calibration, QPC 3is operated as\na voltage probe. The spin current measurement is per-\nformed running an AC current ( I= 4 nA unless stated\notherwise) between terminals 1 and 2, and measuring the\nmagnetic dependence of the voltages VCandV3as de\fned\nin Fig. 2(a). The magnetic \feld is applied in-plane, to\nminimize its orbital e\u000bects, and aligned with the detec-\ntor axis (unless stated otherwise). The \fnite zero \feld\nderivative@BV3, and its correlation with @VG3=G3, indi-\ncates the presence of the spin current I(S)\n3.\nFigures 2(c) and (e) show the resistances RC=VC=I\nandV3=Ias a function of Band the voltage applied to\ng4and g 5. Panels (d) and (f) show line cuts along the\ndashed lines of Fig. 2(c) and (e), respectively, for di\u000ber-\nent temperatures. These line cuts are taken for N3= 0:5.\nOn top of a smooth background, higher frequency con-\nductance \ructuations (CFs) appear at low temperatures.\nThe cavity resistance RCis symmetric upon magnetic\feld reversal both in the slowly varying background and\nin the CFs, as expected from a two-terminal measure-\nment in the linear regime [17]. V3is, on the contrary,\nstrongly asymmetric. We \frst address the slowly varying\nbackground of V3(B). We will discuss the nature of the\nCFs below.\nLarge CFs do not allow us to measure meaningful volt-\nage asymmetries at small magnetic \felds. We therefore\naverage them out by a linear regression of V3(B) in a\nmagnetic \feld range between \u00001 T and 1 T. The chosen\nrange is an optimal compromise between not su\u000ecient\nCFs averaging at small \felds and unwanted changes of\nthe spin current at large \felds. This interval includes at\nleast six CFs, and we carefully checked that the averaged\nvoltage asymmetry only weakly depends on the precise\nmagnetic \feld range used for the analysis.\nThe detector voltage asymmetry extracted in this way\nis plotted in Fig. 3 (markers) together with the detec-\ntor transconductance. We normalize the detector voltage\nby the constant cavity current Iand show the detector\ntransconductance in arbitrary units. Panels (a)-(d) rep-\nresent di\u000berent cavity con\fgurations, with di\u000berent N1\nandN2as indicated in every sub\fgure. Despite the large\nerror bars introduced by the CFs, in Figs. 3(a), 3(b) and\n3(c) we observe what theory anticipates: a correlation of\nthe two quantities below the last detector conductance\nstep (right-hand side of each sub\fgure) and disappear-\nance of this correlation beyond the \frst plateau (left-\nhand side of each sub\fgure). This is the key observation\nfrom which we conclude that we measure a spin current.\nWhere the detector is most energy sensitive, we observe\nuseful signals with @BV3=I\u001970 \nT\u00001, with a typical\nbackground of 20 \nT\u00001. The latter value was also typ-\nically observed for N1;N2>6, as shown in Fig. 3(d),\nwhere the voltage asymmetry is uncorrelated with the\ntransconductance. This is expected due to suppression\nof spin currents by energy averaging when many modes\ncontribute to transport. In the following we give fur-\nther evidence supporting the spin current origin of the\nobserved voltage asymmetry.\nTo con\frm the spin related nature of our signal, we\nexploit a key ingredient for the spin-to-charge conver-\nsion: the magnetic \feld sensitivity of the detector QPC.\nA detector with zero gfactor should result in a vanish-\ning voltage asymmetry, regardless of the spin current in-\ntensity. We con\frmed this prediction in the two ways\nshown in Fig. 4. In Fig. 4(a) we modi\fed the measure-\nment scheme such that the current \rows from terminal\n1 to 3, while terminal 2 is used as the detector. The lat-\nter is characterized in Ref. [18], and the \frst mode shows\ng= 0. As expected, we observe a vanishing voltage asym-\nmetry forN2!0 for two di\u000berent cavity shapes (dots\nand squares respectively). As yet an additional test, we\nkept the measurement scheme as in Fig. 2(a), but ro-\ntated the sample by 90\u000ein the 2DHG plane, to have the\nmagnetic \feld perpendicular to QPC 3. Along this direc-4\n0.8 1 1.2−40−20020406080\nVg3 (V)∂B V2/I (Ω T−1)\n−0.500.51\n∂VG2/G2 (a.u.)\n−0.6 −0.4 −0.2−40−20020406080\nVg4,g5 (V)∂B V3/I (Ω T−1)\n−0.500.51\n∂VG3/G3 (a.u.)\n−0.6 −0.4 −0.2−40−20020406080\nVg4,g5 (V)∂B V3/I (Ω T−1)\n−0.500.51\n∂VG3/G3 (a.u.)\n−0.6 −0.4 −0.2−40−20020406080\nVg4,g5 (V)∂B V3/I (Ω T−1)\n−0.500.51\n∂VG3/G3 (a.u.)(a)\n(c)(b)\n(d) N1=4  N2=5 N1=4  N2=4N1=4  N3=4 N1=5  N2=4\nFIG. 4. (a) Comparison between @BV2=Iacquired with two\ndi\u000berent cavity shapes (dots and squares) both having N1=\nN3= 4 and@VG2=G2(solid line) as a function of the voltage\napplied to g 3. The electrical setup was modi\fed to use QPC 2\nas detector. (b), (c) and (d) as in Fig. 3 but with the magnetic\n\feld aligned perpendicularly to QPC 3.\ntion, thegfactor vanishes for all modes [15], which is\na well-known anisotropy of p-type QPCs [18, 19]. The\nlatter is a particularly powerful approach as it leaves the\nspin current unaltered and only suppresses the spin-to-\ncharge conversion e\u000eciency. In Figs. 4(b), 4(c) and 4(d)\nwe show three of such measurements for the same mode\ncon\fgurations as in Figs. 3(a), . 3(b) and . 3(c). In all\nthe three cases, the voltage asymmetry is comparable to\nthe background level of Fig. 3 and uncorrelated to the\ndetector transconductance, proving the importance of a\nmagnetic \feld sensitive measurement lead for observing\nan asymmetric voltage signal.\nThe spin-to-charge signal shown in Fig. 3 re\rects a ro-\nbust property of the system and is not linked to CFs. The\nCFs are phase coherent e\u000bects originating from electro-\nstatic cavity shape distortion and magnetic \rux penetra-\ntion [20], or from a purely in-plane \feld as a consequence\nof the asymmetric and \fnite-width con\fnement poten-\ntial [21]. We carefully checked that our results do not\ndepend on the CFs' pattern \frst by changing g 1, g2and\ng3while keeping N1andN2constant, second by apply-\ning a strong voltage asymmetry on g 4and g 5. In both\ncases, the CFs are completely changed without a signi\f-\ncant modi\fcation of the voltage asymmetry extracted by\naveraging. Additionally, Fig. 3(a) includes a measure-\nment performed during a di\u000berent cooldown (squares)\n[16]. Despite the completely di\u000berent CFs' \fngerprint,\nan identical voltage asymmetry is obtained, proving the\nrobustness of the measured e\u000bect. As visible in Fig. 2(f),\ncoherent contributions are almost entirely suppressed at\nT= 530 mK, which is a standard temperature scale forthe disappearance of coherent e\u000bects in quantum dots\nwith few open channels [22]. The average signal is, on\nthe other hand, more resistant to temperature increases\nbecause of its di\u000busionlike origin [13]. We performed\nadditional analysis of the temperature and charge cur-\nrent amplitude dependence of the spin-to-charge signal.\nThese measurements, reported in the Supplemental Ma-\nterial [15], con\frm the distinct nature of CFs and the\nspin-to-charge signal, as well as the fact that the spin-to-\ncharge signal is a linear e\u000bect.\nSo far, we discussed the presence of a robust spin\ncurrent in QPC 3visible from the slowing varying back-\nground of@BV3=I. As discussed in the context of the\nMSHE, CFs might also re\rect the presence of mesoscopic\nspin CFs [9{12]. Although the CFs occasionally show a\n\fnite zero \feld slope [see Fig. 2(f)], it was not possible to\nunivocally assign them to spin related or orbital e\u000bects,\nnot considered in Ref. [14]. In particular, we could not\ntest the QPC 3transmission dependence of @BV3=Ifor\nsingle CFs due to the in\ruence of g 4and g 5on the cavity\nshape.\nWe now evaluate the spin current amplitude for N3=\n0:5. With the measured detector sensitivity \u0016 h!=\n0:46 meV, its gfactorg= 0:27 and the typical volt-\nage asymmetry of @BV3=I= 60\nT\u00001, Eq. (1) gives\nI(S)\n3= 174 pA. We compare this value with theoreti-\ncal predictions on geometrical correlation induced spin\ncurrents. The spin transmission of QPC 3, calculated for\na three-terminal cavity in the spin chaos regime, is [13]:\nhT(S)\n13i=C1 + 2\u0018\n2lSOkFN1N3\nN1+N2+N3\u00190:137\u0002C:(2)\nTo evaluate this expression we used \u0018= 1, appropriate\nfor a ballistic dot, N1=N2= 4,N3= 0:5.Cis a system\nspeci\fc prefactor, of order unity. Neglecting spin \rips\ncaused by QPC 3itself, the expected spin current is\nI(S)\n3\u0019e2\nhhT(S)\n13i(V1\u0000V2) =C\u0002134 pA (3)\nfor a charge current of 4 nA. The very good agreement\nwith our measurement further supports the interpreta-\ntion that our signal goes beyond a mere spin current de-\ntection, but provides a quantitatively reliable magnitude.\nAs shown by Eq. 2, the spin conductance depends on N3,\nallowing larger spin currents to be generated by opening\nQPC3further. However, since the detection scheme re-\nquiresN3= 0:5, we could not probe this scenario.\nSimilarly to bulk materials, the spin-to-charge con-\nversion e\u000eciency of the cavity can be expressed via\nthe spin Hall angle \u0002, de\fned as the ratio between\nspin and charge current densities. In our case we get\n\u0002 = (I(S)\n3=N3)=(I=N 2)\u001934%, independent on N3for\nN3\u001cN1;N2. This e\u000eciency is substantially higher than\nany reported for semiconductors [4], making this system\ninteresting for future semiconductor spintronics applica-\ntions.5\nSUPPLEMENTARY MATERIAL\nIn this Supplemental Material section we provide ad-\nditional information on the wafer structure used for the\nexperiment and on the techniques adopted to measure\nthe sample and characterize the detector QPC. We fur-\nthermore show the current and temperature dependence\nof the spin-to-charge signal discussed in the main text.\nWe provide an analytical treatment of the spin-to-charge\nconversion e\u000bect directly applicable to our experiment\nand we discuss its dependence on the detector QPC con-\nductance.\nMaterials and Methods\nThe two-dimensional hole gas (2DHG) in use consists\nof a 15 nm GaAs quantum well placed 45 nm below the\nsurface. The sample is grown along the [001] direction\nand remotely doped with carbon. The wafer has been\nextensively characterized by magnetotransport measure-\nments in Ref. [23]. The total hole density is n= 3:0\u0002\n1015m\u00002and the strong Rashba spin-orbit interaction\n(SOI) results in a splitting of the dispersion relation in\nthe two subbands with di\u000berent total angular momentum\nprojection along the growth direction. The densities of\nthe two spin-orbit split subbands have been derived from\nthe periodicity of the Shubnikov-de Haas oscillations as\nn1= 1:05\u00021015m\u00002andn2= 1:95\u00021015m\u00002result-\ning in a cubic Rashba parameter \f= 1:4\u000210\u000028eVm3\n[24]. The spin-orbit length, de\fned as the length scale\nover which the electron spin rotates by 2 \u0019is calculated,\nas in Ref. [13], as lSO=vF\u001cSO= (\u0016hkF=m\u0003)(2\u0016h=\u0001SO).\n\u0001SO= 2\fRk3\nFis the spin-orbit energy splitting, m\u0003the\nhole e\u000bective mass and kFthe Fermi wavevector. Be-\ncause of the large di\u000berence in m\u0003andkFbetween the\ntwo spin-orbit split subbands [23], we use their density-\naveraged values m\u0003= 0:71meandkF= 1:43\u0002108m\u00001.\nTwo nominally identical cavities were de\fned by elec-\ntron beam lithography and wet etching (see Fig. 1(a)\nof the main text). The samples were measured in a di-\nlution refrigerator with a base temperature of 110 mK\nusing standard low-frequency lock-in techniques. The\ntilt angle between 2DHG and magnetic \feld could be\ntuned in-situ with an accuracy of less than 0 :05\u000e. The\nasymmetry in V3(B) and the conductance \ructuations\n(CFs) visible in Fig. 2(f) of the main text are genuine ef-\nfects due to the in-plane magnetic \feld only. Tilting the\n2DHG angle with respect to the external magnetic \feld\nbetween\u00000:25\u000eand 0:25\u000eleavesV3(B) una\u000bected, prov-\ning that an eventual residual out-of-plane component of\nthe magnetic \feld is negligible for the e\u000bects under dis-\ncussion. The voltage measurements are performed in a\nlongitudinal con\fguration, so that an out-of-plane mag-\nnetic \feld would not result in the appearance of a Hall\nslope. Changing the orientation of the device with re-\ng1\ng2g3g4g5\nQPC1QPC2QPC3I V4tV/2in\n-I-V/2inFIG. S.1. False color atomic force micrograph of the sample\nunder study, together with a schematic of the electrical setup\nused to characterize QPC 3.\nspect to the in-plane component of the magnetic \feld\nrequired warming up the sample and manually changing\nits bonding con\fguration. The electronic properties of\nthe sample and the characteristics of the QPC remained\nlargely unchanged by this carefully done procedure. The\ntwo devices showed quantitatively comparable behavior,\nreproducible over multiple cool-downs. In the main text\nwe show data from a single sample, characterized by a\nlarger tunability of the three QPCs used as leads.\nCharacterization of the detector QPC\nIn the following we describe in more detail the char-\nacterization measurements performed on QPC 3. The\ndata presented allows the extraction of the QPC g-factor\nand energy sensitivity, used to quantify the spin current\nintensity from the spin-to-charge signal. Furthermore,\nwe show the existence of a pronounced Zeeman splitting\nanisotropy that allowed us to obtain the data presented in\nFig. 4 (b), (c) and (d) of the main text. A similar proce-\ndure has been discussed in greater detail in Ref. [18, 19].\nFig. S.1 shows the electrical setup used for the char-\nacterization of QPC 3. A low-frequency AC bias Vin=\n15\u0016V was symmetrically applied to the cavity leads 2\nand 3 and the current \rowing in the structure was mea-\nsured as a function of the voltage applied to g4andg5.\nAt the same time, the voltage drop between terminal 1\nand 3 was recorded, allowing to extract a four terminal\nQPC conductance that does not depend on QPC 1and\nQPC 2. During the characterization of QPC 3, QPC 1and\nQPC 2were set to a very high conductance by negatively\nbiasingg1,g2andg3. We carefully checked that the pre-\ncise voltages applied to g1,g2andg3have no e\u000bects on\nthe presented results.\nVia \fnite bias measurements, we extracted the voltage\ndependent lever arm of g4andg5. The lever arm al-\nlows one to convert the gate voltage axis into energy and6\nFIG. S.2.QPC 3transconductance (numerical derivative of the linear conductance with respect to the gate voltage axis) as a\nfunction of gate voltage and magnetic \feld for di\u000berent \feld orientations. Dark lines indicate high transconductance, bright\nregions indicate low transconductance. (a) Magnetic \feld applied perpendicular to the plane of the 2DHG. (b) Magnetic\n\feld applied in the plane of the 2DHG and along the QPC axis. (c) Magnetic \feld applied in the plane of the 2DHG and\nperpendicular to the QPC axis.\nextract quantities such as the g-factor and the energy\nsensitivity \u0016h!from a conductance measurement. For the\nexperiment under consideration, the determination of the\nlever arm is irrelevant if both \u0016 h!and theg-factor are ex-\ntracted from the same conductance plot and in a narrow\ngate voltage range, as we do here. In fact, as shown by\nEq. (1) of the main text, the spin-to-charge conversion\namplitude is only given by the ratio \u0016 h!=g that does not\ndepend on the gate lever arm.\nClose to a conductance G3=e2=h, the relevant regime\nfor the e\u000bects presented in the main text, the lever arm\nwas measured to be 4 meVV\u00001. We determined \u0016 h!, i.e.\nthe curvature of the harmonic potential, by \ftting a sad-\ndle point model [25] to the QPC conductance [26]. Since\nwe do know the lever arm in this case, we can convert\nthe curvature into energy units by \ftting the equation:\nG(E) =2e2\nhN\u00121\n1 + exp (2\u0019(E\u0000E0)=(\u0016h!))\u0013\n;(S.1)\nwith the \ftting parameters being a constant energy shift\nE0and the potential curvature \u0016 h!.Nis the mode\nnumber of the plateau under consideration, in this case\nN= 1.\nThe QPC is characterized by a strong g-factor\nanisotropy, typical for QPCs embedded in 2DHGs grown\nalong the [001] crystallographic direction [18, 19]. The\nQPC transconductance (numerical derivative with re-\nspect to gate voltage) is shown in Fig. S.2 for three dif-\nferent magnetic \feld orientations. A \fnite spin splitting\nis present when the magnetic \feld is oriented perpen-\ndicularly to the plane of the 2DHG (Fig. S.2(a)) or in\nthe plane of the 2DHG and aligned along the QPC axis\n(Fig. S.2(c)). No spin splitting up to 12 T is visible when\nthe magnetic \feld is aligned in the plane of the 2DHG\nand perpendicular to the QPC axis (Fig. S.2(b)). In order\nto use the QPC to detect a spin current via the spin-to-\ncharge conversion scheme, it is necessary to have a \fnite\ng-factor (see Eq. (1) of the main text). In the present\ncase, this is possible by performing the experiment withthe magnetic \feld oriented as in Fig. S.2(b). The g-factor\nis obtained from Fig. S.2(b) by tracking the separation in\ngate voltage between spin split plateaus as a function of\nan in-plane magnetic \feld. The gate voltage separation\nis then converted into Zeeman energy using the gate de-\npendent lever arm. For the \frst mode we \fnd g= 0:27.\nIn order to suppress the spin-to-charge conversion e\u000e-\nciency, the magnetic \feld must be turned by 90\u000ein the\nplane of the 2DHG, obtained in the situation shown in\nFig. S.2(c). In this case all the modes are characterized\nbyg= 0.\nAs visible in Fig. S.2(b), the levels splitting as a func-\ntion of an in-plane magnetic \feld is not symmetric in\ngate voltage. For each pair of spin-split levels, the left\nbranch rises in energy (more negative values of gate\nvoltage) faster then the right branch. The anomalous\nmagnetic \feld dependence of the QPC levels in an in-\nplane magnetic \feld was recently studied in Ref. [18] and\nfound to be caused by a quadratic SOI peculiar for hole\ngases. This anomalous spin-splitting makes the points of\nstrongest magnetic \feld dependence of G3not to corre-\nspond to the points of strongest gate-voltage dependence,\nas assumed in Ref. [14], but to be slightly shifted to more\nnegative gate voltage. This e\u000bect might explain the small\nhorizontal o\u000bset between @BV3=Iand@VG3=G3seen in\nFig. 3(a), (b) and (c) of the main text.\nCurrent and temperature dependence of the\nspin-to-charge signal\nThe spin current detection method we employ is lim-\nited to the linear transport regime [27], estimated to\nbreak-down in our samples for currents of a few nA. In the\nlinear regime, @BV3=Ishould be independent of I. We\nplot this current dependence in Fig. S.3(a) for the same\ncon\fguration as Fig. 2(a) of the main text and N3= 0:5.\nRemarkably, the value of @BV3=Iis independent of Ifor\nI < 5 nA, con\frming the hypothesis that the slope in7\n−0.5 −0.4 −0.3 −0.2 −0.1 0−40−20020406080\nVg3,g4 (V)∂B V3/I (Ω T−1)\n  \nT = 109 mK\nT = 250 mK\nT = 530 mK\n0.5 5 5010204080\nI (nA)∂B V3/I (Ω T−1)(a) (b)\nI = 4 nAT = 109 mK\nVg3,g4 = −0.1 15 V\nFIG. S.3. (a) @BV3=Ias in Fig. 3(a) of the main text and\nN3= 0:5 for di\u000berent current intensities. (b) @BV3=Ias in\nFig. 3(a) of the main text for di\u000berent temperatures.\nV3(B) is a linear e\u000bect. For currents higher than 5 nA,\nin addition to sample heating, we enter the non-linear\ntransport regime where the detector voltage asymmetry\nand transconductance are not necessarily linked.\nFig S.3(b) shows the temperature dependence of the\nextracted@BV3=Ias in Fig. 3(a) of the main text, for\nthree di\u000berent temperatures. At T= 530 mK we still\nrecover the same gate dependence of @BV3as shown in\nFig. 3(a)of the main text, but with a reduction in peak-\nto-peak height by a factor of 15. The signal eventually\ndisappears entirely at higher temperature because of the\nensuing energy-averaging, a\u000becting also the geometric\ncorrelation corrections. Furthermore, for T > 600 mK,\nG3loses the step-like behavior (hence its energy sensitiv-\nity), becoming a smooth function of the gate voltage.\nSpin-to-charge conversion\nWe review here the theoretical basis for the spin-to-\ncharge conversion e\u000bect in a three-terminal mesoscopic\ncavity as the one depicted in Fig. 1(a) of the main text.\nWe extend the theory presented in Ref. [14] to prove the\nproportionality between @BV3=Iand@VG3=G3.\nThe cavity under consideration has three contacts (la-\nbeled 1,2 and 3), and each of them is at a voltage Vi\nand carries Nispin degenerate modes. We consider the\nsituation where N3\u00141 andN1;N2\u001d1. By applying\na voltage bias V2\u0000V1between leads 1 and 2, a charge\ncurrentIwill \row in the cavity. If contact 3 is grounded,\na charge current I(0)\n3will \row in it. In the following we\nare interested in the situation in which no charge current\n\rows in contact 3 at zero magnetic \feld. This situation\ncan be achieved either by applying a voltage V3that sets\nI(0)\n3to zero at zero magnetic \feld, or by leaving it \roating\nand connected to a volt meter that measures V3. In the\n\frst case, the application of a magnetic \feld will make\nthe current vary from zero, in the second case the cur-\nrent will always remain zero and the measured V3will\nvary. As we will show in the following, the zero \feld\nspin current I(S)\n3in contact 3 is directly proportional to\nthe zero-\feld derivative of I3, orV3, with respect to thein-plane magnetic \feld:\nI(S)\n3=\u0016h!\n\u0019\u0016@BI(0)\n3jB=0; (S.2)\nI(S)\n3=\u0016h!\n\u0019\u0016e2\nh\u0010\n2\u0000T(0)\n33\u0011\n@BV3jB=0; (S.3)\nwhere \u0016h!is the magnetic \feld energy sensitivity, \u0016=\ng\u0016B=2 is the electron magnetic moment and (2 \u0000T(0)\n33)\nis the charge transmission coe\u000ecient of contact 3. These\nQPC parameters are easily measured experimentally.\nIn the following theoretical treatment we do not in-\nclude the presence of any orbital e\u000bect. The latter as-\nsumption is well justi\fed if no out-of-plane magnetic \feld\nis applied. An in-plane \feld can give rise to other or-\nbital e\u000bects [21] which will not be accounted for in this\nsection, assuming that these are small enough. We will\nalways consider that the voltages applied to the system\nare within the linear response regime (i.e. small com-\npared to other energy scales). The generic current I(\u000b)\ni\nat a contact ican be calculated using Landauer-B uttiker\nformalism, resulting in:\nI\u000b\ni=X\nj\u0010\n2Nj\u000eij\u000e\u000b0\u0000T(\u000b)\nij\u0011\nVj; (S.4)\nwherei= 1;2;3 denotes the leads and \u000b= 0;x;y;z\ndenotes the spin polarization of the current. The generic\ntransmission coe\u000ecient is:\nT(\u000b)\nij=X\nm2i;n2jTr\u0010\nty\nmn\u001b(\u000b)tmn\u0011\n; (S.5)\nwhere\u001b(\u000b)are spin matrices, with \u001b(0)the identity ma-\ntrix. The 2\u00022 matricestmnindicate the probability of an\nelectron entering the cavity from the n-th mode of QPC\njto exit the cavity from the m-th mode of QPC i, their\nelementst\u001b\u001b0\nm;ntake into account spin \ripping. It can be\nshown that the transmission coe\u000ecients for charge and\nspin are:\n\u001c(0)\nij=X\n\u001b\u001b0\u001c\u001b\u001b0\nij (S.6)\n;\u001c(S)\nij=X\n\u001b\u001b0\u001b\u001c\u001b\u001b0\nij: (S.7)\nThe transmission probabilities from contact 1 or 2 to con-\ntact 3 are assumed to take the form:\nT\u001b\u001b0\n3i(B) =\u001c\u001b\u001b0\n3i(B)\u0000 (EF\u0000\u001b\u0016B); (S.8)\nhence they can be separated into a spin dependent part\nand an energy dependent part. The spin a\u000bects the sec-\nond term only via the Zeeman energy. In Eq. (S.8) it8\nwas assumed that the QPC has high energy sensitiv-\nity, hence \u0000 ( EF\u0000\u001b\u0016B) varies faster than \u001c\u001b\u001b0\n3i(B) with\nB.\u001c\u001b\u001b0\n3i(B) are phenomenological parameters, describing\nthe spin transmission of the QPC when it is fully open.\nEq. (S.8) is valid only in the limit when an electron re-\n\rected back in the cavity from contact 3 has a negligible\nprobability to come back to contact 3 again. This limit\nis achieved when N1;N2\u001dN3.\nUsing Landauer-B uttiker expressions for charge and\nspin current through contact 3 one has:\nI(0)\n3=e2\n\u0016h\u0010\nT(0)\n31(V3\u0000V1) +T(0)\n32(V3\u0000V2)\u0011\n;(S.9)\nI(S)\n3=e2\n\u0016h\u0010\nT(S)\n31(V3\u0000V1) +T(S)\n32(V3\u0000V2)\u0011\n:(S.10)\nImposingI3(0) = 0 allows to \fnd an analytical form for\nthe spin current:\nV3=\u0010\nT(0)\n31V1T(0)\n32V2\u0011\n=\u0010\nT(0)\n31+T(0)\n32\u0011\n; (S.11)\nI(S)\n3=e2\n\u0016h\u0010\nT(S)\n31(V3\u0000V1) +T(S)\n32(V3\u0000V2)\u0011\n:(S.12)\nEquations (S.2) and (S.3) are obtained by evaluating\n@BI3(0)jB=0with constant V3and@BV3jB=0forI(0)\n3= 0,\nrespectively. For obtaining a simpler analytical form of\nthese expressions, we will make an assumption of the spe-\nci\fc form of the QPC transmission \u0000, though the results\nof this section do not qualitatively depend on this choice.\nWe choose the saddle point potential model, which gives\nthe QPC energy dependent transmission probability as\n[25]:\n\u0000(EF;Vg;B) =1\n1 + exp (\u00002\u0019(EF\u0000\u000bVg\u0000\u001b\u0016B)=\u0016h!):\n(S.13)The partial derivative of the QPC transmission with re-\nspect to magnetic \feld is:\n@B\u0000(EF;Vg;B) =\u001b\u0016@Vg\u0000(EF;Vg;B): (S.14)\nThis allows us to write the magnetic \feld derivative of\nthe charge transmission coe\u000ecients T(0)\n3iin terms of spin\ntransmission coe\u000ecients T(S)\n3i:\n@BT(0)\n3i=@BX\n\u001b\u001b0T\u001b\u001b0\nijjB=0 (S.15)\n=X\n\u001b\u001b0\u001c\u001b\u001b0\nij@B\u0000(EF;Vg;0)jB=0 (S.16)\n=X\n\u001b\u001b0\u001c\u001b\u001b0\nij\u001b\u0016@Vg\u0000(EF;Vg;0)jB=0 (S.17)\n=X\n\u001b\u001b0\u001c\u001b\u001b0\nij\u001b\u0016\u0000(EF;Vg;0)@Vg\u0000(EF;Vg;0)jB=0\n\u0000(EF;Vg;0)\n(S.18)\n=\u0016T(S)\n3i@Vg\u0000(EF;Vg;0)jB=0\n\u0000(EF;Vg;0): (S.19)\nIn the middle of the \frst slope in the QPC conductance,\nthe energy sensitivity is maximal and it gives ( Vg=EF):\n@Vg\u0000(EF;Vg;0)jB=0\n\u0000(EF;Vg;0)=\u0000\u0019\n\u0016h!: (S.20)\nWith this, the zero-\feld derivative of the charge current\nis (V1,V2,V3are constant):\n@BI(0)\n3jB=0=\u0000e2\n\u0016h\u0010\n@BT(0)\n31(V3\u0000V1) +@BT(0)\n32(V3\u0000V2)\u0011\n(S.21)\n=\u0000e2\n\u0016h\u0012\n\u0016@Vg\u0000(EF;Vg;0)jB=0\n\u0000(EF;Vg;0)\u0013\u0010\nT(S)\n31(V3\u0000V1) +T(S)\n32(V3\u0000V2)\u0011\n(S.22)\n=\u0000\u0016@Vg\u0000(EF;Vg;0)jB=0\n\u0000(EF;Vg;0)I(S)\n3 (S.23)\n=\u0019\u0016\n\u0016h!I(S)\n3: (S.24)\nFinally, Eq (S.2) is obtained solving Eq. (S.24) for I(S)\n3. Similarly, the zero \feld derivative of V3is:\n@BV3jB=0=\u0000\u0016@Vg\u0000(EF;Vg;0)jB=0\n\u0000(EF;Vg;0)1\n2\u0000T(0)\n33h\ne2I(S)\n3: (S.25)\nIn the point of highest energy sensitivity we have:\n@BV3jB=0=h\ne2\u0019\u0016\n\u0016h!1\n2\u0000T(0)\n33I(S)\n3; (S.26)which results in Eq. (S.3) upon solving for I(S)\n3.9\nIt is interesting, in the light of our experimental re-\nsults, to investigate the behavior of @BV3jB=0around the\npoint of highest energy sensitivity. 2 \u0000T(0)\n33is the exactexpression for the charge current transmission coe\u000ecient\nof contact 3. It can be approximated as (see Eq. (S.8)):\n2\u0000T(0)\n33=T(0)\n31+T(0)\n32=X\n\u001b\u001b0\u0010\n\u001c\u001b\u001b0\n31+\u001c\u001b\u001b0\n32\u0011\n\u0000(EF;Vg;B ): (S.27)\nWe can apply the approximation from Eq. (S.8) to Eq. (S.10), \fnding a relation between I(S)\n3and \u0000:\nI(S)\n3=\u0000e2\nh X\n\u001b\u001b0\u001c\u001b\u001b0\n31(V1\u0000V3) +X\n\u001b\u001b0\u001c\u001b\u001b0\n32(V2\u0000V3)!\n\u0000(EF;Vg;B): (S.28)\nCombining the last three equations we get the expression:\n@BV3jB=0=\u0016C@Vg\u0000(EF;Vg;0)jB=0\n\u0000(EF;Vg;0); (S.29)\nwhereCis a prefactor containing the voltages Viand\nthe coe\u000ecients \u001c\u001b\u001b0\n3i. It is assumed to be constant with\nrespect to gate voltage. The results obtained here can be\nsummarized with the following proportionality relation:\n@BV3jB=0/@Vg\u0000(EF;Vg;0)jB=0\n\u0000(EF;Vg;0): (S.30)\nIt is worth reminding that the last proportionality is valid\nin the limit N1;N2\u001dN3andN3\u00141 and the coe\u000ecients\n\u001c\u001b\u001b0\n3iare supposed to weakly depend on magnetic \feld.\nIn the absence of SOI, reversing the magnetic \feld di-\nrection reverses the sign of the spin polarization S. In\nthis case\u001c(0)\n3i(B) =\u001c(0)\n3i(\u0000B) and\u001c(S)\n3i(B) =\u0000\u001c(S)\n3i(\u0000B)\n(see Eq. (S.6) and (S.7)). Since \u0000( EF;Vg;B) is an even\nfunction of B, it results that T\u001b\u001b0\n3i(B) =\u0000T\u001b\u001b0\n3i(\u0000B), and\nIS\n3(0) = 0 (see Eq. (S.12)): in the absence of SOI, both\n@BI(0)\n3and@BV3vanish.10\nWe acknowledge Christian Gerl for growing the wafer\nstructure. The authors wish to thank the Swiss National\nScience Foundation via NCCR QSIT \\Quantum Science\nand Technology\" for \fnancial support.\n\u0003Present Address: Center for Quantum Devices, Niels\nBohr Institute, University of Copenhagen, 2100 Copen-\nhagen, Denmark; email: fnichele@phys.ethz.ch; home-\npage: www.nanophys.ethz.ch\n[1] G. Schmidt, D. Ferrand, L. W. Molenkamp, A. T. Filip,\nand B. J. van Wees, Phys. Rev. B 62, R4790 (2000).\n[2] S. Sharma, A. Spiesser, S. P. Dash, S. Iba, S. Watanabe,\nB. J. van Wees, H. Saito, S. Yuasa, and R. Jansen, Phys.\nRev. B 89, 075301 (2014).\n[3] M. I. D'yakonov and V. I. Perel', JETP Lett. 13, 467\n(1971).\n[4] T. Jungwirth, J. Wunderlich, and K. Olejnik, Nat Mater\n11, 382 (2012).\n[5] C. Br une, A. Roth, E. G. Novik, M. Konig, H. Buh-\nmann, E. M. Hankiewicz, W. Hanke, J. Sinova, and\nL. W. Molenkamp, Nat Phys 6, 448 (2010).\n[6] R. M. Potok, J. A. Folk, C. M. Marcus, and V. Umansky,\nPhys. Rev. Lett. 89, 266602 (2002).\n[7] S. K. Watson, R. M. Potok, C. M. Marcus, and V. Uman-\nsky, Phys. Rev. Lett. 91, 258301 (2003).\n[8] S. M. Frolov, A. Venkatesan, W. Yu, J. A. Folk, and\nW. Wegscheider, Phys. Rev. Lett. 102, 116802 (2009).\n[9] W. Ren, Z. Qiao, J. Wang, Q. Sun, and H. Guo, Phys.\nRev. Lett. 97, 066603 (2006).\n[10] J. H. Bardarson, I. Adagideli, and P. Jacquod, Phys.\nRev. Lett. 98, 196601 (2007).\n[11] Y. V. Nazarov, New Journal of Physics 9, 352 (2007).\n[12] J. J. Krich and B. I. Halperin, Phys. Rev. B 78, 035338\n(2008).\n[13] I. Adagideli, P. Jacquod, M. Scheid, M. Duckheim,\nD. Loss, and K. Richter, Phys. Rev. Lett. 105, 246807\n(2010).[14] P. Stano and P. Jacquod, Phys. Rev. Lett. 106, 206602\n(2011).\n[15] See Supplemental Material for material and methods,\ntemperature and current dependence studies and a theo-\nretical treatment of the spin-to-charge conversion e\u000bect.\n[16] For a direct comparison, we horizontally shifted the red\ncurve to account for the di\u000berent pinch-o\u000b voltage of\nQPC 3in this cool-down.\n[17] H. B. G. Casimir, Rev. Mod. Phys. 17, 343 (1945).\n[18] F. Nichele, S. Chesi, S. Hennel, A. Wittmann, C. Gerl,\nW. Wegscheider, D. Loss, T. Ihn, and K. Ensslin, Phys.\nRev. Lett. 113, 046801 (2014).\n[19] A. Srinivasan, L. A. Yeoh, O. Klochan, T. P. Martin,\nJ. C. H. Chen, A. P. Micolich, A. R. Hamilton, D. Reuter,\nand A. D. Wieck, Nano Letters , Nano Lett. 13, 148\n(2013).\n[20] C. Beenakker and H. van Houten, in Semiconductor Het-\nerostructures and Nanostructures , Vol. Volume 44, edited\nby H. Ehrenreich and D. Turnbull (Academic Press,\n1991) pp. 1{228.\n[21] D. M. Zumb uhl, J. B. Miller, C. M. Marcus, V. I. Fal'ko,\nT. Jungwirth, and J. S. Harris, Phys. Rev. B 69, 121305\n(2004).\n[22] A. G. Huibers, J. A. Folk, S. R. Patel, C. M. Marcus,\nC. I. Duru oz, and J. S. Harris, Phys. Rev. Lett. 83,\n5090 (1999).\n[23] F. Nichele, A. N. Pal, R. Winkler, C. Gerl, W. Wegschei-\nder, T. Ihn, and K. Ensslin, Phys. Rev. B 89, 081306\n(2014).\n[24] R. Winkler, Spin-Orbit Coupling E\u000bects in Two-\nDimensional Electron and Hole Systems , Springer Tracts\nin Modern Physics, Vol. 191 (Springer-Verlag, Berlin,\n2003).\n[25] M. B uttiker, Phys. Rev. B 41, 7906 (1990).\n[26] C. R ossler, S. Baer, E. de Wiljes, P.-L. Ardelt, T. Ihn,\nK. Ensslin, C. Reichl, and W. Wegscheider, New Journal\nof Physics 13, 113006 (2011).\n[27] P. Stano, J. Fabian, and P. Jacquod, Phys. Rev. B 85,\n241301 (2012)."    },    {        "title": "2403.01307v1.Negative_Temperature_in_Spin_Dynamics_Simulations.pdf",        "content": "Negative Temperature in Spin Dynamics Simulations\nPui-Wai Ma∗and Sergei L. Dudarev†\nUnited Kingdom Atomic Energy Authority, Culham Science Centre,\nAbingdon, Oxfordshire, OX14 3DB, United Kingdom\nA simple and computationally efficient algorithm enables implementing negative temperature val-\nues in a spin dynamics simulation. The algorithm uses a Langevin spin dynamics thermostat with\na negative damping parameter, enabling the thermalization of an arbitrary interacting spin system\nto the Gibbs energy distribution with a given negative temperature value. Canonical spin dynamics\nsimulations at a negative temperature are as robust as conventional positive spin temperature sim-\nulations, providing a tool for quantitative dynamic studies of the physics of highly excited magnetic\nstates. Two simulation case studies describing spin systems with antiferromagnetic and ferromag-\nnetic ground states are explored. The phase transitions occurring in the negative temperature range\ndo not necessarily exhibit similarities with their positive temperature counterparts. The transition\ntemperatures and the character of spin alignment vary depending on the spatial range and strength\nof spin-spin interactions.\nNegative temperature is a well established concept in\nstatistical mechanics [1–5]. The occurrence of the neg-\native temperature phenomenon was first demonstrated\nexperimentally by Purcell and Pound [6] for the nuclear\nspin sybsystem of a LiF crystal. Similar studies were\nlater performed using other materials [7–10], achieving\nnegative temperatures on the nano-Kelvin scale.\nNegative temperature is a general notion not restricted\nto a spin ensemble. Negative temperatures are encoun-\ntered in the context of plasma states [11] and optical lat-\ntices [12, 13]. A negative temperature value is associated\nwith the formation of an ordered low entropy high en-\nergy state in an ensemble where the spectrum of energy\neigenstates is bounded from above [1–5]. From the ther-\nmodynamic definition of temperature, 1 /T=∂S(E)/∂E,\nwhere S(E) is the entropy of a macroscopic system at en-\nergy E, negative temperature corresponds to a range of\nstatistical configurations where entropy Sis a decreasing\nfunction of energy E. At the point where the derivative\nofSchanges sign and function S(E) changes from mono-\ntonically increasing to monotonically decreasing, we find\nthat 1 /T= 0, or in other words, Tchanges from T=∞+\ntoT=∞−. A system with negative temperature has a\nhigher energy than a system at T=∞+, hence necessar-\nily requiring that the spectrum of energy eigenstates is\nbounded from above. The highest attainable total energy\nstate corresponds to T= 0−. Negative temperatures are\nnot encountered in atomistic molecular dynamics simula-\ntions, since for particles moving in real space with kinetic\nenergy mv2/2, velocities are not bounded and a higher\naverage energy Eimplies higher positive temperature.\nIf a system is at equilibrium, the probability Wnof\nfinding it in an eigenstate {n}with energy Enis given\nby the Gibbs distribution\nWn=1\nZexp\u0012\n−En\nkBT\u0013\n, (1)\n∗leoma55@hotmail.com\n†Sergei.Dudarev@ukaea.uk; sergei.dudarev@linacre.ox.ac.ukwhere Z is the partition function Z =P\nnexp (−En/kBT). Equation (1) imposes no constraint\non the sign of T, which can be positive or negative,\nas long as the partition function can be computed.\nMonte Carlo simulations provide a convenient means\nfor studying statistical mechanical properties of systems\naccessible to simulations at negative temperature, but\nsuch studies are limited [11].\nIn a particular example of nuclear spins [6–10], since\nthe strength of spin-spin interactions is low, a negative\ntemperature state can be created by applying a strong\nexternal magnetic field Hext. In this strong magnetic\nfield limit, the nuclear spin Hamiltonian is dominated\nby the Zeeman term HZ=−I·Hext, where Iis the\nnuclear spin vector. In accord with Eq. (1), once the\ndirection of external field is instantaneously reversed, a\nmetastable short-lived population of inverted spin orien-\ntations is formed, characterized by a negative tempera-\nture value.\nThe external field reversal technique is what has so\nfar been used for generating negative temperature states,\nboth experimentally and computationally [6–10, 12]. The\nproblem with this technique is that temperature remains\nundefined during the equilibration process, when the sys-\ntem dynamically relaxes from the initial highly excited\nstate into a state of thermodynamic equilibrium at a pos-\nitive temperature. Besides, the technique only applies to\nsystems that can be manipulated by macroscopic spa-\ntially homogeneous external fields. For example, a spin\nsystem with a ferromagnetically ordered ground state\ncannot be transformed into a negative temperature con-\nfiguration using an applied external magnetic field tech-\nnique since the external field would simply change the\noverall orientation of the spins in response to any attempt\nto change the spin order.\nIn this study, we propose an algorithm that enables\nperforming a controlled computationally efficient ther-\nmalization of a dynamic spin ensemble to a negative tem-\nperature. The algorithm, involving the use of a Langevin\nspin thermostat developed earlier [14–16], enables per-\nforming a detailed quantitative exploration of equilib-arXiv:2403.01307v1  [cond-mat.mtrl-sci]  2 Mar 20242\nrium physics at negative temperatures. The approach is\ngeneral and applies to any interacting or non-interacting\ndynamic statistical mechanics spin ensemble. Dynamic\nspin systems exhibiting anti-ferromagnetic and ferromag-\nnetic ground states are explored, providing computa-\ntional examples where the Curie-Weiss phase transition\nlaw is observed in the negative temperature range.\nConsider a system of interacting spins described by a\ngeneral type Hamiltonian H(S1,S2, . . . ,SN). The cor-\nresponding Langevin equations of motion for individual\nspins Siare [14, 16–19]\ndSi\ndt=1\n¯h[Si×(Hi+hi)−γSi×(Si×Hi)], (2)\nwhere Hi=−∂H/∂Siis the effective field acting\noni-th spin, γis a damping parameter and hiis a\ndelta-correlated fluctuating field, satisfying conditions\n⟨hi(t)⟩= 0 and ⟨hiα(t)hjβ(t′)⟩=µδijδαβδ(t−t′) [14–16].\nSubscripts αandβdenote the Cartesian components of\na vector.\nAccording to the fluctuation-dissipation theorem [20–\n22], one can prove that in a thermal equilibrium, by iden-\ntifying the energy distribution with the Gibbs distribu-\ntion (Eq. 1), parameters µandγare related through\ntemperature Tas [15, 18, 23–25]:\nµ= 2γ¯hkBT. (3)\nIfTis negative and γis positive, a seemingly straight-\nforward idea is to assign a negative value to µ. However,\nhithen becomes a complex number vector. If one plugs\nsuch a complex fluctuating field into Eq. (2), this makes\nSicomplex-valued, a clearly unnecessary complication in\nthe context of a classical spin dynamics considered here.\nAn alternative, simple yet computationally efficient\nand robust, solution is proposed below. We assign a neg-\native value to the parameter γinstead, with µremaining\npositive. If we now examine the corresponding Fokker-\nPlanck equation [25] corresponding to Langevin equation\n(2), we see that this imposes no constraint on the sign\nofγand still enables attaining a Gibbs distribution with\nan arbitrary chosen T.\nIn terms of microscopic fluctuating spin dynamics, we\nnote that if γis negative, the dissipation term in Eq. (2)\npushes the spin vector Siaway from it being aligned with\nthe effective field Hiacting on it. This necessarily injects\nextra energy into the spin system, in full agreement with\nthe notion that a negative temperature corresponds to a\nhigh-energy excited thermodynamic state of a spin en-\nsemble. We note that if the method were to be applied\nto a system where the spectrum of energy states is not\nbounded from above, the Langevin term would continue\npumping energy into the system ad infinitum .\nA remarkable feature of the Langevin algorithm involv-\ning a negative friction parameter γand negative temper-\nature Tis that it is now the friction term that delivers\nenergy to the statistical system. This implies that the\nfluctuation term acts as effective damping, reversing theconventional meaning of the two (fluctuation and dissipa-\ntion) terms involved in a Langevin dynamics simulation\n[22, 26, 27].\nIn what follows, we consider two case studies illustrat-\ning applications of the algorithm. Both cases involve an\nensemble of interacting spins described by a general non-\ncollinear Heisenberg Hamiltonian\nH=−1\n2X\ni,jJijSi·Sj (4)\nwhere Si=Mi/gµBis an atomic spin vector, Miis the\ncorresponding atomic magnetic moment, g= 2.0023 is\nthe electronic g-factor, µBis the Bohr magneton and Jij\nis the exchange coupling parameter [28] between spins i\nandj. Although there are no literature data describing\nnegative temperature states involving atomic spin config-\nurations, it does not appear impossible that an applica-\ntion of advanced experimental techniques of spin flipping\nusing high energy laser pulses [29] could generate highy\nexcited non-collinear negative temperature spin states in\nan atomic ensemble. We note that the Ruderman-Kittel\ninteraction for nuclear spins [7] adopts the same func-\ntional form for the Hamiltonian. Therefore, the following\ndiscussion is equally applicable to nuclear as well as to\natomic spins. Also, since a similar Hamiltonian formal-\nism applies also to alloys [30, 31], we note the general\napplicability of the negative temperature notion to alloy\nconfigurations.\nIn the first case study, we assume that Jij̸= 0 only\nfor the first nearest neighbours, choosing the Heisenberg\nparameter and magnetic moment values of J1=−69.8\nmeV and |Mi|= 0.89µB. In the second case study, we\nassume J1= 22.52 meV, J2= 17.99 meV and |Mi|= 2.2\nµB. Both types of interaction are realized on a BCC lat-\ntice. The two sets of parameters selected above refer to\npure chromium and pure iron that have collinear anti-\nferromagnetic and collinear ferromagnetic ground states,\nrespectively [32]. We denote them as model 1 and model\n2.\nThe simulation cells explored in the study involved\n16,000 dynamic spins. Initial configurations were as-\nsumed to be the respective ground states, corresponding\ntoT= 0+K. Spin dynamics simulations were performed\nusing the coupled dynamic equations of motion, Eq. 2,\nsee Refs. [14–16]. The temperature of the system during\na simulation was monitored using an explicit expression\nderived in Ref. [25] for a dynamic spin ensemble. As was\nnoted in the derivation [25], the resulting value of the\ndynamic spin temperature can be positive or negative.\nT=P\ni|Si×Hi|2\n2kBP\niSi·Hi. (5)\nSince the numerator in the fraction is positive definite,\nthe value of Tcomputed using this formula is negative\nif the denominator is negative, i.e. if the direction of a\nspinSiis on average anti-parallel to the direction of the\neffective field Hiacting on it.3\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s45/s54/s48/s48/s45/s52/s48/s48/s45/s50/s48/s48/s48/s50/s48/s48/s52/s48/s48/s54/s48/s48/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s101/s108/s118/s105/s110/s41\n/s84/s105/s109/s101/s32/s40/s112/s115/s41/s32/s77/s111 /s100/s101 /s108/s32/s49/s32/s40/s67/s114/s41\n/s32/s77/s111 /s100/s101 /s108/s32/s50/s32/s40/s70/s101 /s41\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s45/s48/s46/s48/s49/s50/s45/s48/s46/s48/s48/s56/s45/s48/s46/s48/s48/s52/s48/s46/s48/s48/s48/s48/s46/s48/s48/s52/s48/s46/s48/s48/s56/s48/s46/s48/s49/s50/s45/s49/s47/s84\n/s84/s105/s109/s101/s32/s40/s112/s115/s41/s32/s77/s111 /s100/s101 /s108/s32/s49/s32/s40/s67/s114/s41\n/s32/s77/s111 /s100/s101 /s108/s32/s50/s32/s40/s70/s101 /s41/s72/s111 /s116/s116/s101 /s114\n/s67/s111 /s108/s100/s101 /s114\nFIG. 1. Process of thermalization of dynamic spin ensembles\nwith 16,000 spins from their respective ground states to −100\nK using a Langevin thermostat with γ=−0.05. For model\n1,J1=−69.8 meV and |Mi|= 0.89µB. For model 2, J1=\n22.52 meV, J2= 17 .99 meV and |Mi|= 2.2µB. These\nparameters approximately describe chromium and iron with\ncollinear antiferromagnetic and ferromagnetic ground states,\nrespectively [31, 32]. (top) Evolution illustrated using the\nconventional Kelvin temperature scale. (bottom) Evolution\nillustrated using a plot involving an alternative temperature\nscale [1–5], where the vertical axis shows values of −1/Tin\nK−1units.\nIn Fig. 1a, we illustrate the dynamics of thermalization\nprocess for models 1 and 2 as a function of time from\n0+Kto−100K, assuming γ=−0.05. Temperatures\nare computed on the fly using Eq. (5), and are shown to\napproach the prescribed values at the end of a simulation.\nEnergy is pumped into the spin system by a heat bath at\n−100K. Simulations show that temperatures first climb\nto∞+and then come back from ∞−.\nIndeed, a spin system at a negative temperature is hot-\nterthan the same system at any positive temperature.\nOne can adopt an alternative temperature scale [1–5] us-\ning a parameter −1/T, which offers a better description\nof how hota statistical system is. In Fig. 1b, we plotted\ntemperatures during the thermalization process again,\nbut now on a new temperature scale. Now the curves\n/s45/s57/s48/s48 /s45/s54/s48/s48 /s45/s51/s48/s48 /s48 /s51/s48/s48 /s54/s48/s48 /s57/s48/s48 /s49/s50/s48/s48 /s49/s53/s48/s48/s45/s48/s46/s49/s54/s45/s48/s46/s49/s50/s45/s48/s46/s48/s56/s45/s48/s46/s48/s52/s48/s46/s48/s48/s48/s46/s48/s52/s48/s46/s48/s56/s69/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s101/s108/s118/s105/s110/s41/s32/s77/s111 /s100/s101 /s108/s32/s49/s32/s40/s67/s114/s41\n/s32/s77/s111 /s100/s101 /s108/s32/s50/s32/s40/s70/s101 /s41\n/s45/s57/s48/s48 /s45/s54/s48/s48 /s45/s51/s48/s48 /s48 /s51/s48/s48 /s54/s48/s48 /s57/s48/s48 /s49/s50/s48/s48 /s49/s53/s48/s48/s48/s46/s48/s48/s46/s52/s48/s46/s56/s49/s46/s50/s49/s46/s54/s50/s46/s48/s50/s46/s52/s50/s46/s56/s51/s46/s50/s67\n/s83 /s32/s40/s107\n/s66 /s41\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s101/s108/s118/s105/s110/s41/s32/s77/s111 /s100/s101 /s108/s32/s49/s32/s40/s67/s114/s41\n/s32/s77/s111 /s100/s101 /s108/s32/s50/s32/s40/s70/s101 /s41FIG. 2. (top) The equilibrium energy per spin as a function\nof temperature. (bottom) The specific heat CSis obtained by\nperforming numerical differentiation. Peaks can be clearly ob-\nserved in the negative temperature region. They correspond\nto negative Curie and N´ eel temperature for model 1 and 2\nrespectively. The asymmetric peaks for model 2 are due to\nthe competition of J1andJ2in an interacting spin system.\nare continuous and the functional dependence is mono-\ntonic. The average energies of the spin systems are the\nsame at T=∞+and∞−.\nBy thermalizing the two model systems at various tem-\nperatures to equilibrium, we find the average energy per\nspin as a function of temperature, plotted in Fig. 2a. In\nFig. 2b we also plot the specific heat CScalculated by\nperforming the numerical differentiation of data in Fig.\n2a. Peaks of specific heat are observed in Fig. 2b in both\npositive and negative temperature regions. These peaks\ncorrespond to magnetic phase transitions and exhibit the\ncharacteristic shape commonly referred to as the Curie-\nWeiss law [33].\nIn the positive temperature region, the occurrence of\nmagnetic phase transitions is well known. The peaks\nof the specific heat correspond to the N´ eel temperature\nfor model 1 and to the Curie temperature for model 2,\nrespectively. At a point where the specific heat diverges,4\nFIG. 3. (top) Schematic picture for antiferromagnetic type-I.\n(bottom) Schematic picture for antiferromagnetic type-II.\nFIG. 4. Atomic spins of model 2 (Fe) at T=−1 K.\nthe spin systems undergoes a phase transition from a\nmagnetically ordered antiferromagnetic or magnetically\nordered ferromagnetic state to a paramagnetic state with\nno long-range spin orientational order.\nIn the negative temperature region, model 1 also ex-\nhibits a peak of the specific heat, illustrated in Fig. 2b,\nthat is symmetric with respect to the one in the positive\ntemperature region. The occurrence of this peak can be\ninterpreted as follows. Since the exchange coupling Jijis\nnon-zero only for the 1st nearest neighbours, if we trans-form J1→ −J1, the ground state of model 1 changes from\nan antiferromagnetic type-I (Fig. 3) to the ferromagnetic\nconfiguration. However, the effective field for any partic-\nular spin does not change, and as a result the absolute\nvalue of the transition temperature stays the same. Al-\nternately, if we make the transformation T→ −T, while\nnot changing J1, this produces the same energy distribu-\ntion (Eq. 1) as the reversal of the sign of J1. Model 1\nthen exhibits a phase transition from a paramagnetic to\na ferromagnetic state at a negative Curie temperature.\nFor model 2, the peaks are asymmetric with respect to\nthe origin of the temperature axis. Since the exchange\ncoupling constants Jijin model 2 extend to the 2nd near-\nest neighbour distance, by following the same logic as\nabove for model 1, we discover that J1andJ2are now in\ncompetition with each other at a negative temperature.\nThere is no longer an obvious preferred relative arrange-\nment of spin orientations that any pair of spins adopts\nat a negative temperature.\nDynamic simulations suggest that model 2 becomes an\nantiferromagnetic type-II (Fig. 3) system when the tem-\nperature is higher than the negative N´ eel temperature.\nFig. 4 illustrates the configuration of atomic spins at\nT=−1 K. The direction of alignment of spins shown\nin the Figure is arbitrary, and does not correspond to\nthe orientation of any Cartesian axis since the Hamilto-\nnian is invariant with respect to the rotation of a spin\nconfiguration as a whole.\nThe occurrence of type-II antiferromagnetic ordering\ncan be rationalized from the energetic perspective. If\nmodel 2 adopts an antiferromagnetic type-I ordering, the\nenergy of a magnetic moment at T= 0−is 8J1−6J2=\n72.22 meV. If it adopts type-II ordering, the energy per\nmoment is 6 J2= 107 .94 meV. Therefore, type-II ordering\ncorresponds to higher energy, which is preferable at a neg-\native temperature. The difference in the strength of the\neffective field in the positive and negative temperature\nregions is the reason for the lack of mirror symmetry in\nthe peak positions in the plot for CS. This result follows\nnaturally from spin dynamics simulations but cannot be\nreadily obtained if we consider a conventional mean-field\napproximation for the ferromagnetic and type-I antifer-\nromagnetic cases.\nThis highlights the fact that phase transitions in the\nnegative temperature region do not necessarily mirror\ntheir counterparts at positive temperatures. The nature\nof phase transitions and the transition temperatures de-\npend on the range and strength of spin-spin interactions.\nIn conclusion, in this study we develop and describe a\nsimple and robust algorithm for the thermalization of a\ndynamic spin system to a negative temperature using a\nLangevin spin dynamic thermostat. The algorithm en-\nables thermalizing a spin ensemble to any exact desired\nvalue of temperature, irrespectively of whether it is posi-\ntive or negative. There is no need to introduce an exter-\nnal field to flip the directions of selected spins, and the\nenergy distribution is not perturbed by the Zeeman term.\nThe case studies explored above suggest that the systems5\nwith antiferromagnetic and ferromagnetic ground states\nexhibit phase transitions both in the positive andnega-\ntive temperature regions. Negative Curie and N´ eel tem-\nperatures were found in dynamic spin simulations. Fi-\nnally, we note that although the examples considered in\nthis study only refer to dynamic spin ensembles, the same\nalgorithm should enable studying any dynamic statistical\nensemble obeying the fluctuation-dissipation theorem.\nACKNOWLEDGMENTS\nThis work has been carried out within the framework\nof the EUROfusion Consortium, funded by the Euro-pean Union via the Euratom Research and Training Pro-\ngramme (Grant Agreement No 101052200 — EUROfu-\nsion) and was partially supported by the Broader Ap-\nproach Phase II agreement under the PA of IFERC2-\nT2PA02. This work was also funded by the EPSRC En-\nergy Programme (grant number EP/W006839/1). To\nobtain further information on the data and models\nunderlying the paper please contact PublicationsMan-\nager@ukaea.uk. Views and opinions expressed are how-\never those of the authors only and do not necessarily re-\nflect those of the European Union or the European Com-\nmission. Neither the European Union nor the European\nCommission can be held responsible for them.\n[1] M. Goldman, Overview of spin temperature, thermal\nmixing and dynamic nuclear polarization, Applied Mag-\nnetic Resonance 34, 219 (2008).\n[2] A. Abragam and M. Goldman, Principles of dynamic nu-\nclear polarisation, Reports on Progress in Physics 41, 395\n(1978).\n[3] N. F. Ramsey, Thermodynamics and statistical mechan-\nics at negative absolute temperatures, Phys. Rev. 103,\n20 (1956).\n[4] M. J. Klein, Negative absolute temperatures, Phys. Rev.\n104, 589 (1956).\n[5] P. T. Landsberg, Negative temperatures, Phys. Rev. 115,\n518 (1959).\n[6] E. M. Purcell and R. V. Pound, A nuclear spin system\nat negative temperature, Phys. Rev. 81, 279 (1951).\n[7] A. S. Oja and O. V. Lounasmaa, Nuclear magnetic order-\ning in simple metals at positive and negative nanokelvin\ntemperatures, Rev. Mod. Phys. 69, 1 (1997).\n[8] P. Hakonen and S. Yin, Studies of nuclear magnetism in\nsilver at positive and negative nanokelvin temperatures,\nPhysica B: Condensed Matter 165-166 , 785 (1990), pro-\nceedings of the 19th International Conference on Low\nTemperature Physics.\n[9] P. J. Hakonen, K. K. Nummila, and R. T. Vuorinen, Spin\ndynamics in highly polarized silver at negative absolute\ntemperatures, Phys. Rev. B 45, 2196 (1992).\n[10] P. Hakonen, R. Vuorinen, and J. Martikainen, Anoma-\nlous spin-lattice relaxation in dilute RhFe at positive and\nnegative nanokelvin temperatures, Physica B: Condensed\nMatter 194-196 , 341 (1994).\n[11] J. A. Krommes and S. Rath, Monte Carlo sampling of\nnegative-temperature plasma states, Phys. Rev. E 67,\n066402 (2003).\n[12] A. S. Sørensen, E. Altman, M. Gullans, J. V. Porto,\nM. D. Lukin, and E. Demler, Adiabatic preparation of\nmany-body states in optical lattices, Phys. Rev. A 81,\n061603 (2010).\n[13] A. Rapp, S. Mandt, and A. Rosch, Equilibration rates\nand negative absolute temperatures for ultracold atoms\nin optical lattices, Phys. Rev. Lett. 105, 220405 (2010).\n[14] P.-W. Ma and S. L. Dudarev, Langevin spin dynamics,\nPhys. Rev. B 83, 134418 (2011).\n[15] P.-W. Ma and S. L. Dudarev, Longitudinal magnetic fluc-\ntuations in Langevin spin dynamics, Phys. Rev. B 86,054416 (2012).\n[16] P.-W. Ma and S. L. Dudarev, Dynamic magnetocaloric\neffect in bcc iron and hcp gadolinium, Phys. Rev. B 90,\n024425 (2014).\n[17] V. P. Antropov, S. V. Tretyakov, and B. N. Harmon,\nSpin dynamics in magnets: Quantum effects and numer-\nical simulations (invited), Journal of Applied Physics 81,\n3961 (1997).\n[18] P.-W. Ma, C. H. Woo, and S. L. Dudarev, Large-scale\nsimulation of the spin-lattice dynamics in ferromagnetic\niron, Phys. Rev. B 78, 024434 (2008).\n[19] D. P. Landau and K. Binder, A Guide to Monte Carlo\nSimulations in Statistical Physics , 3rd ed. (Cambridge\nUniversity Press, 2009).\n[20] S. Chandrasekhar, Stochastic problems in physics and\nastronomy, Rev. Mod. Phys. 15, 1 (1943).\n[21] R. Kubo, The fluctuation-dissipation theorem, Reports\non Progress in Physics 29, 255 (1966).\n[22] W. T. Coffey, Y. P. Kalmykov, and J. T. Waldron, The\nLangevin Equation , 2nd ed. (World Scientific, 2004).\n[23] W. F. Brown, Jr., Thermal fluctuations of a single-\ndomain particle, Phys. Rev. 130, 1677 (1963).\n[24] J. L. Garc´ ıa-Palacios and F. J. L´ azaro, Langevin-\ndynamics study of the dynamical properties of small mag-\nnetic particles, Phys. Rev. B 58, 14937 (1998).\n[25] P.-W. Ma, S. L. Dudarev, A. A. Semenov, and C. H.\nWoo, Temperature for a dynamic spin ensemble, Phys.\nRev. E 82, 031111 (2010).\n[26] R. M. Mazo, Brownian Motion (Clarendon Press, 2002).\n[27] S. L. Dudarev, The non-Arrhenius migration of intersti-\ntial defects in bcc transition metals, Comptes Rendus\nPhysique 9, 409 (2008).\n[28] I. Turek, J. Kudrnovsk´ y, V. Drchal, and P. Bruno, Ex-\nchange interactions, spin waves, and transition temper-\natures in itinerant magnets, Philosophical Magazine 86,\n1713 (2006).\n[29] B. Koopmans, G. Malinowski, F. Dalla Longa, D. Steiauf,\nM. F¨ ahnle, T. Roth, M. Cinchetti, and M. Aeschli-\nmann, Explaining the paradoxical diversity of ultrafast\nlaser-induced demagnetization, Nature Materials 9, 259\n(2010).\n[30] D. Nguyen-Manh, M. Y. Lavrentiev, and S. L. Dudarev,\nThe Fe–Cr system: atomistic modelling of thermody-\nnamics and kinetics of phase transformations, Comptes6\nRendus Physique 9, 379 (2008).\n[31] M. Y. Lavrentiev, J. S. Wr´ obel, D. Nguyen-Manh, S. L.\nDudarev, and M. G. Ganchenkova, Magnetic cluster ex-\npansion model for random and ordered magnetic face-\ncentered cubic Fe-Ni-Cr alloy, Journal of Applied Physics\n120, 043902 (2016).\n[32] M. Y. Lavrentiev, R. Soulairol, C.-C. Fu, D. Nguyen-Manh, and S. L. Dudarev, Noncollinear magnetism at\ninterfaces in iron-chromium alloys: The ground states\nand finite-temperature configurations, Physical Review\nB84, 144203 (2011).\n[33] P. D. Desai, Thermodynamic properties of iron and sil-\nicon, Journal of Physical and Chemical Reference Data\n15, 967 (1986)."    },    {        "title": "2311.13769v2.Spin_dynamics_in_intermediate_energy_heavy_ion_collisions_with_rigorous_angular_momentum_conservation.pdf",        "content": "Spin dynamics in intermediate-energy heavy-ion collisions with rigorous angular\nmomentum conservation\nRong-Jun Liu1, 2and Jun Xu3, 1,∗\n1Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201800, China\n2University of Chinese Academy of Sciences, Beijing 100049, China\n3School of Physics Science and Engineering, Tongji University, Shanghai 200092, China\nWe have revisited the spin dynamics in intermediate-energy heavy-ion collisions based on the\nimproved spin- and isospin-dependent Boltzmann-Uehling-Uhlenbeck transport model, particularly\nwith the constraint of rigorous angular momentum conservation incorporated. We have studied\nthe spin polarization of free nucleons and tritons/3He as well as the spin alignment of deuterons,\nand predicted the flow splittings for their different spin states. We have also demonstrated that the\nspin-dependent potential may enhance dissipations and thus have a non-negligible effect on the spin-\naveraged transverse flow at low collision energies. When rigorous angular momentum conservation\nin each spin-dependent nucleon-nucleon collision is incorporated, it affects the overall dynamics, the\nflow, and also the spin polarization, while the effects of the spin-orbit potential on the spin-related\nobservables are still appreciable. The well-developed SIBUU model could be further extended to\ninclude hyperons or vector mesons, or used as a hadronic afterburner for spin-related studies in\nrelativistic heavy-ion collisions, with more inelastic channels incorporated in the future.\nI. INTRODUCTION\nSpin dynamics is of general interest in various fields.\nIn the nuclear physics community, the spin and chiral dy-\nnamics in relativistic heavy-ion collisions have been un-\nder extensive investigations in the past fifteen years [1–\n3]. Due to the high energy density reached in relativistic\nheavy-ion collisions, the chiral symmetry is restored for\nboth quarks and antiquarks, and they can be considered\nas massless particles, whose spins are coupled with their\nmomenta depending on their helicities. Under the strong\nmagnetic field and vorticity field produced in relativistic\nheavy-ion collisions, the dynamics of these partons ex-\nhibits various chiral anomalies, which have been proposed\ntheoretically [4–6] and investigated experimentally [7–\n10]. More recently, the spin polarizations of Λ and ¯Λ hy-\nperons [11] and the spin alignment of vector mesons [11]\nhave been measured through the angular distribution of\ntheir decays, and these have attacked considerable atten-\ntion and led to various related studies, e.g., effects of the\nhadronic evolution on the spin observables [12, 13].\nCompared to the extensive investigations in high-\nenergy nuclear physics community, the spin dynamics in\nlow- and intermediate-energy heavy-ion collisions have\nnot attracted much attention. Since nucleons are mas-\nsive particles, their spins and momenta are decoupled,\nand their spin dynamics is mostly dominated by the nu-\nclear spin-orbit interaction and to some extent affected\nby the nuclear tensor force (see, e.g., Ref. [14] for a\nreview). Both the spin-orbit interaction and the ten-\nsor force are under hot investigations in nuclear struc-\nture studies and may largely affect the shell evolution\nof finite nuclei [15–20]. In low-energy reactions, the\nspin-orbit nuclear interaction may lead to the internal\n∗Correspond to junxu@tongji.edu.cnspin excitation of the colliding nuclei, enhance the dis-\nsipation, increase the fusion threshold, and affect sig-\nnificantly the fusion cross section based on the time-\ndependent Hartree-Fock approach [21–23], while the ten-\nsor force may have subdominant effects [24, 25]. In\nintermediate-energy heavy-ion collisions, we have devel-\noped a spin- and isospin-dependent Boltzmann-Uehling-\nUhlenbeck (SIBUU) transport model [26, 27] by incor-\nporating spin degree of freedom into the IBUU transport\nmodel. We have observed different collective flows for nu-\ncleons of different spin states with respect to ydirection\nperpendicular to the reaction plane [26, 28]. We have\nalso observed global and local spin polarizations [29] in\nydirection and along the beam direction, respectively.\nThese phenomena originate from different forces on nu-\ncleons of different spins, a general feature called the spin-\nHall effect [30] for particle transport under the spin-orbit\ninteraction.\nIn the present study, we have further improved the\nSIBUU transport model by incorporating the constraint\nof rigorous angular momentum conservation. We have\nalso made minor improvements on the treatment of\nspin-dependent nucleon-nucleon collisions and the spin-\ndependent coalescence procedure for the production of\nlight clusters with different spins. While the spin dynam-\nics remains qualitatively similar compared to our previ-\nous studies, we pay more attention to some new features\nin the present work. First of all, we find that the spin\npolarization as well as the splitting of the collective flows\nfor free nucleons can be affected by the constraint of an-\ngular momentum conservation. Second, besides the spin\npolarization of nucleons and tritons/3He with spin1\n2, the\nspin alignment of deuterons with spin 1 can also be ob-\nserved. Third, we find that the spin-orbit potential has a\nnon-negligible effect on the spin-averaged transverse flow\nat low collision energies.\nThe rest part of the paper is organized as follows. We\ndescribe the details of the SIBUU transport model asarXiv:2311.13769v2  [nucl-th]  24 Dec 20232\nwell as the improvements, especially the incorporation of\nthe angular momentum conservation, in Sec. II. We in-\nvestigate the spin polarization or spin alignment of free\nnucleons and light clusters, as well as the splitting of\ncollective flows for their different spin states, and com-\npare the spin-averaged transverse flow with and without\nspin-orbit potential, in Sec. III. We conclude and give an\noutlook in Sec. IV.\nII. THEORETICAL FRAMEWORK\nIn this section, we first review the basic framework of\nthe SIBUU transport model. Next, we describe in detail\nhow we incorporate the constraint of rigorous angular\nmomentum conservation in each nucleon-nucleon colli-\nsion and in the mean-field evolution. We also present the\nimproved treatments for spin-dependent nucleon-nucleon\ncollisions and coalescence procedures.\nA. Basic framework of SIBUU\nThe SIBUU transport model solves the spin-dependent\nBUU equation written as [31, 32]\n∂ˆf\n∂t+i\n̵h[ˆε,ˆf]+1\n2(∂ˆε\n∂⃗p⋅∂ˆf\n∂⃗r+∂ˆf\n∂⃗r⋅∂ˆε\n∂⃗p)\n−1\n2(∂ˆε\n∂⃗r⋅∂ˆf\n∂⃗p+∂ˆf\n∂⃗p⋅∂ˆε\n∂⃗r)=Ic, (1)\nwhere the single-particle energy ˆ εand the phase-space\ndistribution ˆfare 2×2 matrics\nˆε(⃗r,⃗p)=ε(⃗r,⃗p)ˆI+⃗h(⃗r,⃗p)⋅⃗σ, (2)\nˆf(⃗r,⃗p)=f0(⃗r,⃗p)ˆI+⃗g(⃗r,⃗p)⋅⃗σ, (3)\nwith⃗σ=(σx, σy, σz)andˆIbeing respectively the Pauli\nmatrices and the 2 ×2 unit matrix. In the above, ε(⃗r,⃗p)\nandf0(⃗r,⃗p)are the spin-averaged single-particle energy\nand phase-space distribution, and ⃗h(⃗r,⃗p)and⃗g(⃗r,⃗p)rep-\nresent their spin-dependent contributions, respectively.\nIcrepresents the collision term.\nWe start from the Skyrme-type spin-orbit nuclear in-\nteraction between nucleons at ⃗r1and⃗r2expressed as [33]\nvso=iW0(⃗σ1+⃗σ2)⋅⃗k′×δ(⃗r1−⃗r2)⃗k, (4)\nwhere W0is the strength of the spin-orbit coupling\nwhose default value is set to be 133 MeVfm5[34] in this\nstudy,⃗σ1(2)is the Pauli matrices, ⃗k=(⃗p1−⃗p2)/2 is the\nrelative momentum operator acting on the right with\n⃗p=−i∇, and⃗k′is the complex conjugate of ⃗k. Based\non the Hartree-Fock method [35], the contribution to theenergy-density functional can be expressed as\nVso=V0\nso+∑\nτVτ\nso\n=−W0\n2[ρ∇⋅⃗J+⃗s⋅∇×⃗j\n+∑\nτ(ρτ∇⋅⃗Jτ+⃗sτ⋅∇×⃗jτ)], (5)\nwhere both time-even and time-odd contributions are in-\ncluded. From the variational principle and in the semi-\nclassical approximation, the contribution of the spin-\norbit interaction to the single-particle energy can be writ-\nten as\nˆhso=εso+⃗h⋅⃗σ, (6)\nwith\nεso=−W0\n2∇⋅(⃗J+⃗Jτ)−W0\n2[⃗p⋅(∇×(⃗s+⃗sτ))],(7)\n⃗h=−W0\n2∇×(⃗j+⃗jτ)+W0\n2[∇(ρ+ρτ)×⃗p]. (8)\nIn the above, τ=n, pis the isospin index, ρ=∑τρτ,\n⃗s=∑τ⃗sτ,⃗j=∑τ⃗jτ, and ⃗J=∑τ⃗Jτare the num-\nber, spin, momentum, and spin-current densities, respec-\ntively. Omitting the isospin index, these densities can be\nexpressed in terms of f0and⃗gin Eq. (3) as [27]\nρ(⃗r)=2∫d3pf0(⃗r,⃗p), (9)\n⃗s(⃗r)=2∫d3p⃗g(⃗r,⃗p), (10)\n⃗j(⃗r)=2∫d3p⃗p\n̵hf0(⃗r,⃗p), (11)\n⃗J(⃗r)=2∫d3p⃗p\n̵h×⃗g(⃗r,⃗p). (12)\nThe spin-averaged energy density in Eq. (2) is\nε=p2\n2m+UMID+εso (13)\nwith\nUMID=a(ρ\nρ0)+b(ρ\nρ0)c\n±2Epot\nsym(ρ\nρ0)γsym\n(ρn−ρp\nρ)\n(14)\nbeing the momentum-independent mean-field potential,\nwith the “+(−)” sign for neutrons (protons), and param-\neters a=−209.2 MeV, b=156.4 MeV, c=1.35,Epot\nsym=18\nMeV, and γsym=2/3 that reproduce the empirical nu-\nclear matter properties.\nAs shown in Ref. [27], the left-hand side of the spin-\ndependent BUU equation can be solved if the time evo-\nlution of the ith test particle is simulated according to\nd⃗ri\ndt=∂\n∂⃗pi(ε+⃗h⋅⃗σi), (15)\nd⃗pi\ndt=−∂\n∂⃗ri(ε+⃗h⋅⃗σi), (16)\nd⃗σi\ndt=2⃗h×⃗σi. (17)3\nIn the above, ⃗σiis now a unit vector describing the ex-\npectation direction of the nucleon spin in the semiclassi-\ncal case, and the third equation describes the precession\nof⃗σiin the spin-dependent mean-field potential. Dif-\nferent forces on nucleons of different spins can be seen\nfrom Eq. (16). In the present study, we extend the above\nframework by using the lattice Hamiltonian method [36],\nso that the average number, spin, momentum, and spin-\ncurrent densities at the sites of a three-dimensional cubic\nlattice are calculated respectively as [29]\nρL(⃗rα)=∑\niS(⃗rα−⃗ri), (18)\n⃗sL(⃗rα)=∑\ni⃗σiS(⃗rα−⃗ri), (19)\n⃗jL(⃗rα)=∑\ni⃗piS(⃗rα−⃗ri), (20)\n⃗JL(⃗rα)=∑\ni(⃗pi×⃗σi)S(⃗rα−⃗ri). (21)\nIn the above, αis a site index and ⃗rαis the position of site\nα, and Sis the shape function describing the contribution\nof a test particle at ⃗rito the average density at ⃗rα, i.e.,\nS(⃗r)=1\nN(nl)6g(x)g(y)g(z) (22)\nwith\ng(q)=(nl−∣q∣)Θ(nl−∣q∣). (23)\nNis the number of parallel events, lis the lattice spac-\ning,ndetermines the range of S, and Θ is the Heaviside\nfunction. We adopt N=400,l=1 fm, and n=2 in the\npresent study. The Hamiltonian of the whole system is\nH=∑\ni⃗p2\ni\n2m+Nl3∑\nα[VMID(⃗rα)+Vso(⃗rα)] (24)\nwhere VMID is the corresponding potential energy den-\nsity from UMID, and Vsois the potential energy density\nfrom the spin-orbit interaction [Eq. (5)]. The canonical\nequations of motion in the lattice Hamiltonian framework\nnow become\nd⃗ri\ndt=∂H\n∂⃗pi, (25)\nd⃗pi\ndt=−∂H\n∂⃗ri, (26)\nd⃗σi\ndt=1\ni[⃗σi, H], (27)\nwhich are similar to Eqs. (15-17), but with a smearing of\nthe test-particle size and more accurate numerical deriva-\ntives.\nB. Spin-dependent nucleon-nucleon collisions\nThe above part describes the spin-dependent mean-\nfield evolution. For the collision term Icin Eq. (1), wealso treat it as spin-dependent in SIBUU. We only con-\nsider elastic nucleon-nucleon collisions, and all inelastic\ncollisions are turned off in the present study.\nWe use different spin-singlet and spin-triplet neutron-\nneutron (proton-proton) and neutron-proton elastic scat-\ntering cross sections as parameterized in Ref. [37], which\nwere extracted from the phase-shift analyses of nucleon-\nnucleon scatterings in free space [38]. For two nucleons\nwith their expectation spin directions at\n⃗σ1=(sinθ1cosϕ1,sinθ1sinϕ1,cosθ1),\n⃗σ2=(sinθ2cosϕ2,sinθ2sinϕ2,cosθ2), (28)\nwhere θ1(2)andϕ1(2)are the polar angle and azimuthal\nangle, respectively, for the spin direction of nucleon 1 (2),\nthe probability to form a spin-singlet state is\n∣⟨χ0,0∣Ψ⟩∣2=1\n4[1−cosθ1cosθ2−sinθ1sinθ2cos(ϕ1−ϕ2)],\nwith χ0,0and Ψ defined in Appendix A. The probabil-\nity to form a single-triplet state is thus 1 −∣⟨χ0,0∣Ψ⟩∣2.\nWe have also used the spin- and isospin-dependent Pauli\nblocking, with the local phase-space cell more specific for\nnucleons with different spin and isospin states. While\nisospin has only two states, the expectation direction of\nthe spin can be of any direction ( ϕ,θ). We count the\nnumber of nucleons with their spins in each bin (∆ ϕ,\n∆θ) in the solid angle, and the occupation probability\nfor a certain phase-space cell with a particular spin state\nis then evaluated from the projection contribution of all\nbins.\nHow the spins of nucleons change after a successful\ncollision is largely unknown. In the present study, we\nassume that during the collision the colliding nucleons\ninteract with each other through the short-range spin-\norbit interaction vLS=vr\nso(r)⃗LCM\nr⋅⃗S, where ⃗S=⃗σ1+⃗σ2\nis the total spin of the colliding nucleons, ⃗LCM\nris the\norbital angular momentum of the colliding nucleons with\nrespective to the center of mass (C.M.) of the collision.\nIt is known that the zero-range Skyrme-type spin-orbit\ninteraction is a reduced form of the finite-range spin-orbit\ninteraction [33, 39]. Here we take vr\nso(r)as that fitted by\nproton-proton scatterings [40], i.e.,\nvr\nso(r)=IT2\nµ(r)+[P+Q(µr)+R(µr)2]W(r),(29)\nwith\nTµ(r)=[1+3\nµr+3\n(µr)2]e−µr\nµr(1−e−cr2)2\n, (30)\nand\nW(r)=[1+e(r−r0)/a]−1. (31)\nIn the above, µis the pion mass, and a=0.2 fm, r0=0.5\nfm, and c=2.1 fm−2are the shape parameters. For the\nspin-singlet state ( S=0) of nucleon pairs, vr\nso(r)van-\nishes. For the spin-triplet state ( S=1) of nucleon pairs,4\nTABLE I. Parameters of the finite-range spin-orbit interac-\ntion in MeV for different isospin states ( T) of nucleon pairs.\nT I P Q R\n1-0.62697 -570.5571 -309.3605 819.1222\n00.10180 86.0658 46.6655 -356.5175\nthe values of I,P,Q, and Rfor different isospin states\n(T) are given in Table I. While the above short-range\nspin-orbit interaction is that in vacuum, the correspond-\ning in-medium interaction is expected to have the sim-\nilar magnitude, by comparing, e.g., the spin-dependent\ncross sections from nuclear interaction in free space and\nthat in nuclear medium based on the Dirac-Brueckner\napproach [41].\nIn each time step for the collision process, the short-\nrange spin-orbit interaction may lead to the procession\nof the nucleon spin according to\nd\ndt⃗σ1,2=vr\nso(r)⃗σ1,2×⃗LCM\nr. (32)\nIn the present study with spin degree of freedom, ⃗LCM\nris\nnot a conserved quantity, while the total angular momen-\ntum⃗LCM=⃗LCM\nr+⃗Sis conserved. Defining ∆ ⃗σ=⃗σ1−⃗σ2,\nthe above equations can be rewritten as\nd\ndt⃗S=vr\nso(r)⃗S×⃗LCM, (33)\nd\ndt∆⃗σ=vr\nso(r)∆⃗σ×(⃗LCM−⃗S). (34)\nThe above coupled equations can be solved numerically.\nActually, it turns out that the spin procession angle from\nthe above equations is rather small, which means that\nthe spin direction is largely unchanged after a successful\nnucleon-nucleon collision.\nC. Angular momentum conservation for each\ncollision\nThe angular momentum conservation for nucleon-\nnucleon collisions in transport simulation was first in-\nvestigated in Ref. [42] and recently revisited in Ref. [43].\nWithout considering nucleon spin, the conservation of\nthe orbital angular momentum requires in-plane nucleon-\nnucleon collisions. While there are different prescriptions\nfor such in-plane collisions, considerable effects on the\noverall dynamics due to the constraint of angular momen-\ntum conservation have been observed [43]. In the follow-\ning, we present an improved treatment in the presence of\nnucleon spins compared to that discussed in Ref. [43].\nThe total angular momentum conservation for a colli-\nsion between nucleon 1 and nucleon 2 with coordinates\n⃗r1(2), momenta ⃗p1(2), and spins ⃗σ1(2)requires\n⃗r1×⃗p1+⃗r2×⃗p2+⃗σ1+⃗σ2=⃗r′\n1×⃗p′\n1+⃗r′\n2×⃗p′\n2+⃗σ′\n1+⃗σ′\n2.(35)From now on, we take the convention that quantities with\n(without) “ ′” represent those after (before) the nucleon-\nnucleon collision, and quantities with (without) an aster-\nisk represent those in the C.M. (lab) frame of the colli-\nsion. Defining the central and relative coordinates and\nmomenta as\n⃗R=(⃗r1+⃗r2)/2,⃗r=(⃗r1−⃗r2)/2,\n⃗P=⃗p1+⃗p2,⃗p=⃗p1−⃗p2,\nEq. (35) is identical to\n⃗R×⃗P+⃗r×⃗p+⃗σ1+⃗σ2=⃗R′×⃗P′+⃗r′×⃗p′+⃗σ′\n1+⃗σ′\n2.\nSince⃗R=⃗R′and⃗P=⃗P′are satisfied, the task is to\nfind proper ⃗r′and⃗p′which lead to a given orbital angular\nmomentum ⃗L′\nr=⃗r′×⃗p′once⃗σ′\n1+⃗σ′\n2is determined. To\ndo that, first we need to select properly the direction of\nthe C.M. momentum ⃗p∗′\n1in such a way that the relative\nmomentum ⃗p′is perpendicular to the angular momentum\n⃗L′\nr, i.e.,⃗p′⋅⃗L′\nr=0. Through Lorentz transformation, ⃗p′⋅⃗L′\nr\ncan be further expressed as\n⃗p′⋅⃗L′\nr=⃗p∗′\n1⋅(2⃗L′\nr+2γ2\nγ+1⃗β⋅⃗L′\nr⃗β)+γ∆e∗⃗β⋅⃗L′\nr.\nIn the above, ⃗βis the velocity of the C.M. frame of\nthe nucleon-nucleon collision with respect to the lab\nframe, γ=1/√\n1−β2is the Lorentz factor, and ∆ e∗=√\np∗′\n12+m2\n1−√\np∗′\n22+m2\n2is the energy difference between\ntwo particles after collisions in the C.M. frame, which\nvanishes for elastic collisions with m1=m2=m. The\nabove equation shows that the direction of ⃗p∗′\n1should be\nperpendicular to\n⃗q=2⃗L′\nr+2γ2\nγ+1⃗β⋅⃗L′\nr⃗β. (36)\nIn the collision treatment, we first determine the polar\nangle θaccording to the differential cross section. After-\nwards, we select the direction of ⃗p∗′\n1perpendicular to ⃗q\nby setting\n⃗p∗′\n1=p∗′\n1(cosφ⃗ˆe1+sinφ⃗ˆe2), (37)\nwith the unit vectors ⃗ˆe1and⃗ˆe2written as\n⃗ˆe1=⃗q×⃗p∗\n1\n∣⃗q×⃗p∗\n1∣,⃗ˆe2=⃗q×⃗p∗\n1×⃗q\n∣⃗q×⃗p∗\n1×⃗q∣. (38)\nIt is easy to see that ⃗ˆe1and⃗ˆe2form a plane perpendicular\nto⃗q, and the angle φcharacterizes the direction of ⃗p∗′\n1\nin that plane. By substituting Eq. (37) into ⃗p∗′\n1⋅⃗p∗\n1=\np∗′\n1p∗\n1cosθ, we get\nsinφ=cosθ\n⃗ˆe2⋅⃗p∗\n1. (39)5\nSince we have the constraint ∣sinφ∣≤1, the range of cos θ\nis also limited to ∣⃗ˆe2⋅⃗p∗\n1∣≥∣cosθ∣, and this is equivalent\nto\n−∣⃗p∗\n1×⃗q∣\nqp∗\n1≤cosθ≤∣⃗p∗\n1×⃗q∣\nqp∗\n1. (40)\nNow the direction of ⃗r′×⃗p′is the same as ⃗L′\nr. To\nensure that they have the same magnitude, the relative\ncoordinates ⃗r′of the colliding nucleons must satisfy\n⃗r′=A⃗ˆe⃗p′+L′\nr\np′⃗ˆe⃗p′×⃗L′\nr, (41)\nwhere⃗ˆe⃗p′and⃗ˆe⃗p′×⃗L′\nrare the unit vector in the direction\nof⃗p′and⃗p′×⃗L′\nr, respectively. Acan be any real number,\nand its value is determined by minimizing ∣⃗r′−⃗r∣, which\nleads to [42]\n⃗r′=(⃗r⋅⃗ˆe⃗p′)⃗ˆe⃗p′+L′\nr\np′⃗ˆe⃗p′×⃗L′\nr. (42)\nIt can be verified that the total angular momentum is\nnow conserved with a minimum ∣⃗r′−⃗r∣.\nD. Angular momentum conservation for mean-field\nevolution\nBesides nucleon-nucleon collisions, the mean-field evo-\nlution may also violate the conservation of the total an-\ngular momentum, which includes the contribution from\nthe orbital angular momentum and the nucleon spin. To\ncorrect for that, we use the following method in the simi-\nlar spirit as in Ref. [44]. At each time step, we correct the\nmomentum of each particle with a small amount ∆ ⃗pito\nensure the conservation of the total angular momentum,\nwith its each component expressed as\n∆Jc=∑\niεabcri,a∆pi,b, (43)\nwhere εabcis the Levi-Civita symbol, a,b, and crepresent\nsymbols of the Cartesian coordinate, and double symbols\nimply summation. To make the global correction small\nenough, we define a function\nL=∑\ni∆pi,a∆pi,a+λc(∑\niεabcri,a∆pi,b−∆Jc),(44)\nand the values of ∆ pi,aandλccan be obtained by taking\nthe partial derivatives of this function\n{∂L\n∂∆pi,a=0\n∂L\n∂λc=0⇒{∆pi,a=1\n2εabcri,bλc\nλc∑iri,ari,a−λa∑iri,ari,c+2∆Jc=0.\nIn this way, the correction of the momentum ∆ pi,acan\nbe obtained once we solve a ternary system of equationsabout λc, which can help us to conserve the total angu-\nlar momentum in the mean-field evolution. The resulting\n∆piis about a few keV/c for each nucleon. Since there\nare many test particles used in the study, we note that\nthe violations of the total energy and the total momen-\ntum are of the order 0 .0001%, smaller than the numerical\nfluctuation.\nE. Spin-dependent coalescence model\nWe use a spin-dependent coalescence model to study\nthe production of light clusters with different spins, with\nminor improvement compared to that in Ref. [45]. The\nyields of deuterons ( d) as well as tritons ( t) and3He from\nfree nucleons are expressed by the following Wigner func-\ntion forms [46, 47]\nfd=8gdexp(−ρ2\nσ2\nd−p2\nρσ2\nd), (45)\nft/3He=82gt/3Heexp⎛\n⎝−ρ2+λ2\nσ2\nt/3He−(p2\nρ+p2\nλ)σ2\nt/3He⎞\n⎠.(46)\nIn the above,\n⃗ρ=1√\n2(⃗r1−⃗r2),⃗pρ=1√\n2(⃗p1−⃗p2),\n⃗λ=1√\n6(⃗r1+⃗r2−2⃗r3),⃗pλ=1√\n6(⃗p1+⃗p2−2⃗p3)\nare relative coordinates and momenta, and σd=2.26 fm,\nσt=1.59 fm, and σ3He=1.76 fm are the width param-\neters for the Wigner functions that reproduce the root-\nmean-square radii of corresponding light clusters [48]. In\nthe spin-averaged case, the statistical factors have the\nvalues gd=3/4 and gt/3He=1/4. With the information\nof nucleon spin explicitly available, the statistic factor\ngdfor deuterons with a neutron and a proton forming a\nspin-triplet state can be calculated from\ngd=∣⟨χ1,1∣Ψ⟩∣2+∣⟨χ1,0∣Ψ⟩∣2+∣⟨χ1,−1∣Ψ⟩∣2\n=1−∣⟨χ0,0∣Ψ⟩∣2\n=1\n4[3+cosθ1cosθ2+sinθ1sinθ2cos(ϕ1−ϕ2)],\n(47)\nwhere θ1(2)andϕ1(2)are the azimuthal angle and po-\nlar angle for the spin direction of nucleon 1(2), and the\ndefinitions of χ1,1,χ1,0,χ1,−1,χ0,0, and Ψ can be found\nin Appendix A. The ρ0,0term of the spin arrangement\nmatrix characterizing the spin alignment can be written\nas\nρ0,0=∣⟨χ1,0∣Ψ⟩∣2\n1−∣⟨χ0,0∣Ψ⟩∣2\n=1−cosθ1cosθ2+sinθ1sinθ2cos(ϕ1−ϕ2)\n3+cosθ1cosθ2+sinθ1sinθ2cos(ϕ1−ϕ2),(48)6\nwhere the sin θ1sinθ2cos(ϕ1−ϕ2)term is a small correc-\ntion term compared to that in Ref. [49]. The spin state\nof tritons (3He) is determined by the additional neutron\n(proton), and the rest neutron-proton pair form a spin-\nsinglet state, so the statistic factor can be calculated from\ngt/3He=∣⟨χ0,0∣Ψ⟩∣2\n=1\n4[1−cosθ1cosθ2−sinθ1sinθ2cos(ϕ1−ϕ2)].\n(49)\nIII. RESULTS AND DISCUSSIONS\nWe explore the spin dynamics in intermediate-energy\nheavy-ion collisions based on the improved SIBUU trans-\nport model, and compare extensively the spin-dependent\nobservables with and without the constraint of rigor-\nous angular momentum conservation (AMC). These spin-\ndependent observables include the global and local spin\npolarizations as well as transverse and elliptic flows for\nnucleons and light clusters of different spin states. Com-\npared with traditional IBUU calculations, we will also il-\nlustrate the effect of the spin-dependent potential on the\nspin-averaged transverse flow. According to our previous\nstudies, the spin polarization as well as the spin splitting\nof the collective flow are largest at the beam energy of\naround 100 AMeV [28, 29]. To have an appreciable angu-\nlar momentum and bulk medium to illustrate the effect\nof the spin-orbit potential, the following discussions will\nbe mainly focused on midcentral Au+Au collisions at the\nbeam energy of 100 AMeV.\nA. Spin dynamics and polarization\nWe begin with showing contours of nucleon number\ndensity and spin density in the reaction plane (x-o-z\nplane) in the left window of Fig. 1, for the typical mid-\ncentral Au+Au collisions at Elab=100 AMeV. Here we\nonly show the density evolution with AMC, which leads\nto a slightly lower density compared to the calculation\nwithout AMC (see Fig. 5 of Ref. [43]). Generally, the\nglobal spin polarization, shown by the ycomponent of\nthe spin density sy, is generated by the dominating con-\ntribution of the time-odd term (∇×⃗j)yover that of the\ntime-even term (∇ρ×⃗p)y, as demonstrated in Ref. [29].\nThe lower density with AMC thus generally leads to a\nweaker syin both participant and spectator regions. The\nright window of Fig. 1 displays the rapidity distribution\nof the global spin polarization of free nucleons, with the\nlatter identified by their low densities ρ<ρ0/8 in SIBUU\nsimulations. The global spin polarization in ydirection\nperpendicular to the reaction plane is defined as\nPy=Nsy=+1\n2−Nsy=−1\n2\nNsy=+1\n2+Nsy=−1\n2, (50)where Nsy=±1\n2is the number of free nucleons at spin state\nsy=±1\n2. Such spin polarization was measured for Λ hy-\nperons in relativistic heavy-ion collisions [50], and here\nwe are discussing the spin polarization for free nucleons in\nintermediate-energy heavy-ion collisions. The global spin\npolarization is generally stronger at midrapidities, where\nit is dominated by the spin density syof the participant,\nand weaker at large rapidities, where it is affected by the\nspin density syof the spectators. It is seen that SIBUU\nwith AMC predicts a slightly weaker global spin polariza-\ntion compared to that without AMC. This is consistent\nwith the weaker sywith the constraint of AMC shown in\nthe left window of Fig. 1.\nThe density contours in the transverse plane (x-o-y\nplane) may provide us with additional information of the\nspin dynamics in a different point of view, and they are\ndisplayed in the left window of Fig. 2. While the general\nfeatures of the evolution of nucleon number density as\nwell as the yandzcomponents of the spin densities sy\nandszrespectively are similar to those shown in Ref. [29],\nweaker syandszare observed with AMC compared to\nthat without AMC in both participant and spectator re-\ngions. This is again due to the slightly lower density\nreached with AMC. We also display the azimuthal angu-\nlar distribution of the spin polarization for free nucleons\ninyandzdirections in the right window of Fig. 2, with\nthe azimuthal angle calculated from ϕ=atan2(py, px).\nHere ϕ=0 and±πcorrespond to nucleons that emit in\n±xdirections, and ϕ=±π\n2corresponds to nucleons that\nemit in±ydirections. It is seen from panel (a) that the\nPyof total free nucleons, which include those from both\nparticipant and spectator regions, has a flat ϕdistribu-\ntion. The ϕdependence becomes stronger for nucleons\nwith larger transverse momenta pT, since they emit at\nearly stages and their Pyreflect the different syin the\nparticipant and spectator regions. It is seen that the Py\nare affected differently by the constraint of AMC for low-\nand high- pTnucleons. The spin polarization Pzinzdi-\nrection, which is defined similarly as Eq. (50), is called\nthe local spin polarization, and for Λ hyperons its az-\nimuthal angular dependence measured by STAR Collab-\noration [51] has attracted considerable attentions [52–55].\nWithout the constraint of AMC, the ϕdistribution of Pz\nis consistent with szshown in the left window, and it\nhas an opposite sign compared to that for Λ hyperons at\nrelativistic energies [51]. With the constraint of AMC,\nwhile the ϕdependence is qualitatively similar, the Pz\nhas a smaller magnitude.\nThe spin polarization of nucleons may affect the pro-\nduction of nuclear clusters in different spin states, and\nfor light clusters this can be studied through a spin-\ndependent coalescence method as discussed in Sec. II E.\nIn Fig. 3 (a), we display the rapidity dependence of\nthe triangular components of the spin arrangement ma-\ntrix for deuterons, and here ρ1,1,ρ0,0, and ρ−1,−1corre-\nspond to the relative fractions of deuterons in spin states\nSy=+1, 0, and −1, respectively. It is seen that ρ1,1\nandρ−1,−1have oppositive rapidity dependence, while7\n-20020x (fm)t = 90 fm/ct  = 70 fm/ct  = 50 fm/ct  = 30 fm/cz\n (fm)0.120.300.450.600.750.901.051.20-\n200t = 10 fm/cw\nithout AMCwith AMC-\n8.0-4.00.04.08.0ρ/ρ0s\ny (10-3fm-3)with AMC-\n200-200-\n200-200-200-20020\n-1.0- 0.50 .00 .51 .00123Py (%)y\nr/ybeamr\n with AMC \nwithout AMCEbeam = 100 AMeV, b = 8 fm\nFIG. 1. Left: Contours of the reduced density ρ/ρ0(first row), and the ycomponent of the spin density with (second row)\nand without (third row) angular momentum conservation in the x-o-z plane, in Au+Au collisions at the beam energy of 100\nAMeV and impact parameter b =8 fm. Right: Spin polarization of free nucleons in ydirection as a function of reduced rapidity\nyr/ybeam\nr in the same reaction of the left window.\n-200200\n.120.300.450.600.750.901.051.20ρ/ρ0-\n200-\n6.0-3.00.03.06.0sy (10-3fm-3)-\n200-\n3.0-1.00.01.03.0t = 90 fm/ct  = 70 fm/ct  = 50 fm/ct  = 30 fm/ct  = 10 fm/cs\nz (10-3fm-3)y\n (fm)x (fm)-\n200-\n200-200-\n200-200-200-20020without AMCwithout AMCw\nith AMCwith AMCwith AMC\n-4-2024Py (%)Ebeam = 100 AMeV, b = 8 fm, |yr/ybeamr\n| < 0.5(\na)-\n3-2-10123-101Pz (%)φ\n (rad)(b)    pT>0 GeV/c \n   pT>0.2 GeV/c \n   pT>0.3 GeV/cwith without  AMC\nFIG. 2. Left: Contours of the reduced density ρ/ρ0(first row), the ycomponent of the spin density with (second row) and\nwithout (third row) angular momentum conservation, and the zcomponent of the spin density with (fourth row) and without\n(fifth row) angular momentum conservation in the x-o-y plane, in Au+Au collisions at the beam energy of 100 AMeV and\nimpact parameter b =8 fm. Right: Spin polarization of mid-rapidity free nucleons at different transverse momenta pTiny(a)\nandz(b) directions as a function of azimuthal angle ϕin the same reaction of the left window.\nρ0,0has almost no rapidity dependence and is slightly\nsmaller than 1 /3. The deviation of ρ0,0from 1/3, which\nis called the spin alignment, has attracted considerable\ninterest [56, 57]. Referencing the situation here, in rel-\nativistic heavy-ion collisions one expects that ρ1,1and\nρ−1,−1of vector mesons could be more sensitive to the\nspin polarization of the quark-gluon plasma, while ρ0,0is\nmore easily measurable [11, 49]. For tritons, the neutron-\nproton pair form a spin-singlet state, so the triton spin\nis determined by the residue neutron. In Fig. 3 (b) it is\nseen that the spin polarization of tritons is similar to thatof free nucleons. The result is similar for3He as shown\nin Fig. 3 (c), except that the Pydistribution is a little\nnarrower and the Pywith AMC is larger compared to\nthat of tritons, since now the residue proton dominates\nthe spin of3He. In Fig. 3 one observes a weaker global\nspin polarization for light clusters with the constraint of\nAMC, consistent with the weaker global spin polarization\nof nucleons shown above. For the local spin polarization,\nit is rather weak for light clusters.\nWe devote Fig. 4 to the global spin polarizations for\nfree nucleons at different beam energies and impact pa-8\n0.300.320.340.36 \nρ-1,-1ρ1,1ρyb\nlack: with AMCr\ned: without AMCρ0,0 \n   Ebeam = 100 AMeVb\n = 8 fmdeuterons(a)0\n24(\nc)Py (%)y\nr/ybeamr\ntritons(b)-\n1.0-0.50 .00 .51 .00243HePy (%)\nFIG. 3. Triangular components of the spin arrangement ma-\ntrix for deuterons (a) as well as spin polarization of tritons\n(b) and3He (c) in ydirection as a function of reduced ra-\npidity yr/ybeam\nr in Au+Au collisions at the beam energy of\n100 AMeV and impact parameter b =8 fm with and without\nangular momentum conservation.\nrameters. As already observed in Ref. [29], Pyfor midra-\npidity nucleons is larger at about 100 AMeV but smaller\nat lower or higher energies, and this is due to a weaker\nvorticity characterized by the ∇×⃗jterm at lower ener-\ngies, and due to more free nucleons at higher energies.\nThe Pyis also seen to be stronger in more peripheral\ncollisions, since the spin-orbit potential depends on the\ndensity gradient and is stronger near the surface of the\nnuclear matter. While a weaker sywith AMC compared\nto that without AMC is observed in each scenario, the\nresulting Pyof free nucleons may depend on various com-\npetition effects. Generally, one sees that the constraint\nof AMC has some influence on the spin polarizations,\nwhile the resulting Pyare still appreciable at midrapidi-\nties in most cases. At larger rapidities and especially at\nsmaller beam energies Pybecomes negative, since it is\ndominated by the spectator matter with a negative sy\n(see the left window of Fig. 1). Compared to the results\nwithout AMC, the weaker spin-orbit potential with AMC\nleads to a less negative syof the spectator matter, and\nthus a less negative Pyat larger rapidities especially at\nsmaller beam energies.\n0123 \n  with AMC \n  without AMCPy (%)y\nr/ybeamr\nb = 8 fm5\n0    100   400 AMeV-\n1.0-0.50.00 .51 .002468E\nbeam = 1\n00 AMeV  with without AMC \n   b=4 fm \n       8 fm \n     12 fm(a)(\nb)FIG. 4. Comparison of the spin polarization of free nucleons\ninydirection as a function of reduced rapidity yr/ybeam\nr in\nAu+Au collisions at different beam energies but a fixed im-\npact parameter b =8 fm (a) and at a fixed beam energy of\n100 AMeV and different impact parameters b =4, 8, and 12\nfm (b) with and without angular momentum conservation.\nB. Spin-dependent flows\nWe now turn to the flows of nucleons in different spin\nstates. Due to the coupling between the nucleon spin\nand the angular momentum in ydirection, nucleons with\ndifferent spins sy=1\n2σyare expected to be affected by\ndifferent spin-orbit potentials, and this is called the spin-\nHall effect [30]. As shown in the upper panel of Fig. 5,\nnucleons with sy=−1\n2(+1\n2)are affected by a more repul-\nsive (attractive) spin-orbit potential, and they thus have\na stronger (weaker) transverse flow. We can define the\nspin up-down differential transverse flow as [26]\nFud(yr)=1\nN(yr)N(yr)\n∑\ni=1(σy)i(px)i, (51)\nwhere N(yr)is the nucleon number at rapidity yr, and\n(σy)iand(px)iare the ycomponent of the spin and\nthe momentum in xdirection for the ith nucleon. As\nshown in the middle panel of Fig. 5, the spin up-down\ndifferential transverse flow has an appreciable magnitude,\nand it actually characterizes the strength of the spin-\norbit potential [26, 45]. The elliptic flow of free nucleons\nat midrapidities, as shown in the lower panel of Fig. 5,\nalso demonstrates the spin splitting, with nucleons of\nspinsy=−1\n2(+1\n2)affected by a more repulsive (attrac-9\n-1.0- 0.50 .00 .51 .0-20-1001020\n<px/A> (MeV/c)y\nr/ybeamr\nEbeam = 100 AMeVb\n = 8 fmw\nith without  AMC \n   y = +1/2  \n   y = -1/2  (a)\n-1.0- 0.50 .00 .51 .0-6-4-20246(b)Fud (MeV/c)y\nr/ybeamr\n with AMC \nwithout AMCEbeam = 100 AMeVb\n = 8 fm\n01 002 003 004 00-4-20246\nw\nith without  AMC \n    y = +1/2 \n    y = -1/2(c)|\nyr/ybeamr\n|<0.5v2 (%)p\nT (MeV/c)Ebeam = 100 AMeV b\n = 8 fm\nFIG. 5. Transverse flows of spin-up ( sy=+1\n2) and spin-\ndown ( sy=−1\n2) free nucleons (a) and spin up-down differ-\nential transverse flows (b) as a function of reduced rapidity\nyr/ybeam\nr, and elliptic flow of spin-up and spin-down free nu-\ncleons at midrapidities (c) as a function of transverse mo-\nmentum pT, in Au+Au collisions at the beam energy of 100\nAMeV and impact parameter b =8 fm with and without an-\ngular momentum conservation.\ntive) spin-orbit potential and thus leading to a stronger\n(weaker) v2. Incorporating the constraint of AMC, a\nlower density is reached, and this generally leads to a\nweaker transverse flow as shown in Fig. 5 (a), and also\nweakens slightly the spin up-down differential transverse\nflow in Fig. 5 (b). The preferred in-plane collisions withAMC generally enhances the in-plane flow and thus v2\nat lower pT, but weakens the squeeze-out flow in ±ydi-\nrection and thus leads to less negative v2at higher pT.\nThe spin-dependent flows of free nucleons may also\nlead to the spin splitting of flows for light clusters at\ndifferent spin states, and this is displayed in Fig. 6 where\nthe productions of deuterons and tritons are calculated\nfrom the spin-dependent coalescence method. Note here\nthe transverse flow is divided by the constituent nucleon\nnumber, i.e., 2 for deuterons and 3 for tritons. It is seen\nthat the spin splittings of both transverse flows and ellip-\ntic flows are stronger for deuterons, compared with those\nfor tritons. This is understandable, since deuterons at\nspin states Sy=±1 and 0 are formed by respectively\nthe spin-triplet and spin-singlet state of neutron-proton\npairs, so the spin effect are doubled or cancelled, while the\nspin effect for tritons at sy=+1\n2and−1\n2is determined by\nthe residue neutron apart from the spin-singlet neutron-\nproton pair. The results are similar for3He. Again, the\nconstraint of AMC leads to systematically weaker trans-\nverse flows and stronger elliptic flows, at least for larger\npTvalues.\nC. Spin-averaged flows\nWhile the spin-dependent potential does leads to the\nspin polarization and the spin-dependent flows, one may\nargue that the spin-averaged observables predicted by\ntraditional transport models are still valid. This is only\npartially true when the collision energy is relatively high.\nAt lower collision energies, the dissipation effect from the\nspin-dependent potential is non-negligible. As seen in\nfusion reactions, the incorporation of the spin-dependent\npotential affects significantly the fusion threshold and the\nfusion cross section [21–23]. Here we compare the spin-\naveraged transverse flow at Elab=50 AMeV in Fig. 7.\nWith the spin-dependent potential ( W0≠0), the stronger\ndissipation effect leads to a stronger transverse flow for\nfree nucleons, compared to the case without the spin-\ndependent potential ( W0=0). The effect on the slope\nof the transverse flow is further enhanced when we con-\nsider deuterons from the coalescence of nucleons. Com-\nparable to the effect of the spin-dependent potential, the\nincorporation of the AMC constraint leads to a system-\natical reduction of the transverse flow. It is thus seen\nthat one should incorporate properly the spin-dependent\nmean-field potential and the constraint of AMC to pre-\ndict accurately the transverse flow.\nIV. CONCLUSION AND OUTLOOK\nWe have improved the framework of the spin-\nand isospin-dependent Boltzmann-Uehling-Uhlenbeck\n(SIBUU) transport model, in the aspects of spin-\ndependent nucleon-nucleon collisions as well as the co-\nalescence method, and particularly by incorporating the10\n-1.0- 0.50 .00 .51 .0-30-20-100102030\n<px/A> (MeV/c)y\nr/ybeamr\n    Ebeam = 100 AMeVb\n = 8 fmw\nith without  AMC \n    y = +1 \n    y = 0 \n    y = -1(\na)deuterons\n-1.0- 0.50 .00 .51 .0-30-20-100102030\n \n    y = +1/2 \n    y = -1/2y\nr/ybeamr\nEbeam = 100 AMeV  \n     b = 8 fm(\nc)tritonsw\nith without  AMC\n01 00200300400500-4048\nd\neuterons(\nb)     y = +1 \n    y = 0 \n    y = -1with without  AMCv2 (%)p\nT (MeV/c)Ebeam = 100 AMeVb\n = 8 fm, |yr/ybeamr\n| < 0.5\n01 00200300400500-4048\n(\nd)tritons \n    y = +1/2 \n    y = -1/2p\nT (MeV/c)Ebeam = 100 AMeVb\n = 8 fm, |yr/ybeamr\n| < 0.5w\nith without  AMC\nFIG. 6. Transverse flow as a function of reduced rapidity yr/ybeam\nr (upper) and elliptic flow as a function of transverse\nmomentum pT(lower) for deuterons at spin states Sy=+1, 0, and −1 (left) and for spin-up ( sy=+1\n2) and spin-down ( sy=−1\n2)\ntritons (right) in Au+Au collisions at the beam energy of 100 AMeV and impact parameter b =8 fm with and without angular\nmomentum conservation.\nconstraint of rigorous angular momentum conservation.\nWe have revisited the global and local spin polariza-\ntion in intermediate-energy heavy-ion collisions, as well\nas the spin-dependent transverse flows and elliptic flows,\nfor spin-half nucleons, spin-one deuterons, and spin-half\ntritons/3He. While the constraint of rigorous angular\nmomentum conservation generally leads to weaker spin\npolarization and have some influence on the flows, the\neffects of the spin-orbit potential on the spin dynamics\nand spin-related observables are still appreciable. We\nhave further demonstrated that the spin-orbit potential\nhas a non-negligible effect on the spin-averaged trans-\nverse flows of nucleons and light clusters at low collision\nenergies.\nExperimentally, it is difficult to identify different spin\nstates for nucleons or light clusters. On the other hand,\nthe spin polarizations of hyperons or the spin alignment\nof vector mesons are measurable through the angular dis-\ntribution of their decays. It is thus of great interest to\nincorporate the relevant inelastic collision channels for\nthe production of these particles, and see whether their\nspins will be affected by the nucleon spin dynamics. Thewell-developed SIBUU model in the present study could\nthen be used to study of spin dynamics in heavy-ion col-\nlisions at the beam energy of a few AGeV, or even used\nas a hadronic afterburner for the spin-relevant studies in\nrelativistic heavy-ion collisions.\nACKNOWLEDGMENTS\nThis work is supported by the Strategic Priority Re-\nsearch Program of the Chinese Academy of Sciences un-\nder Grant No. XDB34030000, the National Natural Sci-\nence Foundation of China under Grant Nos. 12375125,\n11922514, and 11475243, and the Fundamental Research\nFunds for the Central Universities.11\n-4-2024n\nucleons(\nb)<px/A> (MeV/c)Ebeam = 50 AMeV \n     b = 8 fm(\na)y\nr/ybeamr\n-1.0-0.50.00 .51 .0-4-2024w\nith without AMC \n W0 ≠ 0 \n W0 = 0deuterons\nFIG. 7. Spin-averaged transverse flows of free nucleons (a)\nand deuterons (b) as a function of reduced rapidity yr/ybeam\nr\nin Au+Au collisions at the beam energy of 50 AMeV and im-\npact parameter b =8 fm with and without angular momentum\nconservation and/or spin-dependent mean-field potential.\nAppendix A: Definition of spin states\nThe expectation value of a nucleon spin is determined\nby the azimuthal angle ϕand the polar angle θ\n⃗σ=(sinθcosϕ,sinθsinϕ,cosθ), (A1)\nand its spin state can be written as\nχ=⎛\n⎝e−iϕ\n2cosθ\n2\neiϕ\n2sinθ\n2⎞\n⎠. (A2)\nThe expectation values of the spin in x,y, and zdirec-\ntions are\nσx=χ+σxχ=sinθcosϕ, (A3)\nσy=χ+σyχ=sinθsinϕ, (A4)\nσz=χ+σzχ=cosθ. (A5)We can define the two-nucleon spin state as the direct\nproduct of the spin state of each nucleon\nΨ=χ1⊗χ2=⎛\n⎜⎜⎜⎜⎜\n⎝e−i(ϕ1+ϕ2)\n2cosθ1\n2cosθ2\n2\ne−i(ϕ1−ϕ2)\n2cosθ1\n2sinθ2\n2\nei(ϕ1−ϕ2)\n2sinθ1\n2cosθ2\n2\nei(ϕ1+ϕ2)\n2sinθ1\n2sinθ2\n2⎞\n⎟⎟⎟⎟⎟\n⎠. (A6)\nIn this way, the expectation values of the spin in the\na=x,y, and zdirections for nucleon 1 and nucleon 2 can\nbe expressed as\n(σa)1=χ+\n1σaχ1=Ψ+(σa)1Ψ, (A7)\n(σa)2=χ+\n2σaχ2=Ψ+(σa)2Ψ, (A8)\nwith\n(σa)1=σa⊗I2×2, (A9)\n(σa)2=I2×2⊗σa. (A10)\nWith the spin-up and spin-down states for each nucleon\npair defined as\n∣↑↑⟩=⎛\n⎜⎜⎜\n⎝1\n0\n0\n0⎞\n⎟⎟⎟\n⎠,∣↑↓⟩=⎛\n⎜⎜⎜\n⎝0\n1\n0\n0⎞\n⎟⎟⎟\n⎠,∣↓↑⟩=⎛\n⎜⎜⎜\n⎝0\n0\n1\n0⎞\n⎟⎟⎟\n⎠,∣↓↓⟩=⎛\n⎜⎜⎜\n⎝0\n0\n0\n1⎞\n⎟⎟⎟\n⎠,\nthe spin-singlet and the spin-triplet states of two nucleons\ncan be expressed respectively as\nχ0,0=⎛\n⎜⎜⎜⎜\n⎝0\n1√\n2\n−1√\n2\n0⎞\n⎟⎟⎟⎟\n⎠(A11)\nand\nχ1,1=⎛\n⎜⎜⎜\n⎝1\n0\n0\n0⎞\n⎟⎟⎟\n⎠, χ1,0=⎛\n⎜⎜⎜⎜\n⎝0\n1√\n21√\n2\n0⎞\n⎟⎟⎟⎟\n⎠, χ1,−1=⎛\n⎜⎜⎜\n⎝0\n0\n0\n1⎞\n⎟⎟⎟\n⎠. (A12)\nBy taking the product of Eq. (A6) with Eqs. (A11) and\n(A12), we can calculate the probability of a nucleon pair\nin the spin-singlet state and in the spin-triplet state, and\nthe normalization condition\n∣⟨χ0,0∣Ψ⟩∣2+∣⟨χ1,1∣Ψ⟩∣2+∣⟨χ1,0∣Ψ⟩∣2+∣⟨χ1,−1∣Ψ⟩∣2=1\nis satisfied. The above definition of the spin state is gen-\neral, since we can rotate the polar angle θand the az-\nimuthal angle ϕin Eq. (A1) so that the projection of the\nspin can be performed in an arbitrary direction.\n[1] D. E. Kharzeev, J. Liao, S. A. Voloshin, and G. Wang,\n“Chiral magnetic and vortical effects in high-energynuclear collisions—A status report,” Prog. Part. Nucl.12\nPhys. 88, 1 (2016).\n[2] X. G. Huang, “Electromagnetic fields and anomalous\ntransports in heavy-ion collisions — A pedagogical re-\nview,” Rept. Prog. Phys. 79, 076302 (2016).\n[3] F. Becattini and M. A. Lisa, “Polarization and Vorticity\nin the Quark–Gluon Plasma,” Ann. Rev. Nucl. Part. Sci.\n70, 395 (2020).\n[4] K. Fukushima, D. E. Kharzeev, and H. J. Warringa,\n“The Chiral Magnetic Effect,” Phys. Rev. D 78, 074033\n(2008).\n[5] Y. Burnier, D. E. Kharzeev, J. Liao, and H. U. Yee, “Chi-\nral magnetic wave at finite baryon density and the electric\nquadrupole moment of quark-gluon plasma in heavy ion\ncollisions,” Phys. Rev. Lett. 107, 052303 (2011).\n[6] D. Kharzeev and A. Zhitnitsky, “Charge separation in-\nduced by P-odd bubbles in QCD matter,” Nucl. Phys. A\n797, 67 (2007).\n[7] B. I. Abelev et al. (STAR), “Azimuthal Charged-Particle\nCorrelations and Possible Local Strong Parity Violation,”\nPhys. Rev. Lett. 103, 251601 (2009).\n[8] B. Abelev et al. (ALICE), “Charge separation relative to\nthe reaction plane in Pb-Pb collisions at√sNN=2.76\nTeV,” Phys. Rev. Lett. 110, 012301 (2013).\n[9] L. Adamczyk et al. (STAR), “Observation of charge\nasymmetry dependence of pion elliptic flow and the pos-\nsible chiral magnetic wave in heavy-ion collisions,” Phys.\nRev. Lett. 114, 252302 (2015).\n[10] F. Zhao (STAR), “Λ (K0\nS)−h±and Λ −pazimuthal corre-\nlations with respect to event plane and search for chiral\nmagnetic and vortical effects,” Nucl. Phys. A 931, 746\n(2014).\n[11] M. S. Abdallah et al. (STAR), “Pattern of global spin\nalignment of ϕand K∗0mesons in heavy-ion collisions,”\nNature 614, 244 (2023).\n[12] F. Becattini, G. Cao, and E. Speranza, “Polarization\ntransfer in hyperon decays and its effect in relativistic\nnuclear collisions,” Eur. Phys. J. C 79, 741 (2019).\n[13] X. L. Xia, H. Li, X. G. Huang, and H. Z. Huang, “Feed-\ndown effect on Λ spin polarization,” Phys. Rev. C 100,\n014913 (2019).\n[14] J. Xu, B. A. Li, W. Q. Shen, and Y. Xia, “Dynam-\nical effects of spin-dependent interactions in low- and\nintermediate-energy heavy-ion reactions,” Front. Phys.\n(Beijing) 10, 102501 (2015).\n[15] M. G. Mayer, “On Closed Shells in Nuclei,” Phys. Rev.\n74, 235 (1948).\n[16] M. G. Mayer, “On Closed Shells in Nuclei. 2,” Phys. Rev.\n75, 1969 (1949).\n[17] O. Haxel, J. H. D. Jensen, and H. E. Suess, “On the\n“Magic Numbers” in Nuclear Structure,” Phys. Rev. 75,\n1766 (1949).\n[18] T. Otsuka, T. Suzuki, R. Fujimoto, H. Grawe, and\nY. Akaishi, “Evolution of Nuclear Shells due to the Ten-\nsor Force,” Phys. Rev. Lett. 95, 232502 (2005).\n[19] T. Otsuka, T. Matsuo, and D. Abe, “Mean Field with\nTensor Force and Shell Structure of Exotic Nuclei,” Phys.\nRev. Lett. 97, 162501 (2006).\n[20] T. Otsuka, T. Suzuki, M. Honma, Y. Utsuno, N. Tsun-\noda, K. Tsukiyama, and M. Hjorth-Jensen, “Novel fea-\ntures of nuclear forces and shell evolution in exotic nu-\nclei,” Phys. Rev. Lett. 104, 012501 (2010).\n[21] A. S. Umar, M. R. Strayer, and P. G. Reinhard, “Res-\nolution of the fusion window anomaly in heavy-ion colli-\nsions,” Phys. Rev. Lett. 56, 2793 (1986).[22] J. A. Maruhn, P. G. Reinhard, P. D. Stevenson, and\nM. R. Strayer, “Spin-excitation mechanisms in Skyrme-\nforce time-dependent Hartree-Fock,” Phys. Rev. C 74,\n027601 (2006).\n[23] P. G. Reinhard, A. S. Umar, K. Thomas R. Davies, M. R.\nStrayer, and S. J. Lee, “Dissipation and forces in time-\ndependent Hartree-Fock calculations,” Phys. Rev. C 37,\n1026 (1988).\n[24] P. D. Stevenson, E. B. Suckling, S. Fracasso, M. C. Bar-\nton, and A. S. Umar, “Skyrme tensor force in heavy ion\ncollisions,” Phys. Rev. C 93, 054617 (2016).\n[25] K. Godbey, L. Guo, and A. S. Umar, “Influence of the\ntensor interaction on heavy-ion fusion cross sections,”\nPhys. Rev. C 100, 054612 (2019).\n[26] J. Xu and B. A. Li, “Probing in-medium spin-orbit poten-\ntial with intermediate-energy heavy-ion collisions,” Phys.\nLett. B 724, 346 (2013).\n[27] Y. Xia, J. Xu, B. A. Li, and W. Q. Shen, “Equations of\nmotion of test particles for solving the spin-dependent\nBoltzmann–Vlasov equation,” Phys. Lett. B 759, 596\n(2016).\n[28] Y. Xia, J. Xu, B. A. Li, and W. Q. Shen, “Spin-orbit\ncoupling and the up-down differential transverse flow in\nintermediate-energy heavy-ion collisions,” Phys. Rev. C\n89, 064606 (2014).\n[29] Y. Xia and J. Xu, “Nucleon spin polarization in\nintermediate-energy heavy-ion collisions,” Phys. Lett. B\n800, 135130 (2020).\n[30] J. E. Hirsch, “Spin hall effect,” Phys. Rev. Lett. 83, 1834\n(1999).\n[31] P. Ring and P. Schuck, The Nuclear Many-Body Problem\n(Springer, Berlin, 1980).\n[32] H. Smith and H.H. Jensen, Transport Phenomena (Ox-\nford University Press, Oxford, 1989).\n[33] D. Vautherin and D. M. Brink, “Hartree-Fock calcu-\nlations with Skyrme’s interaction. 1. Spherical nuclei,”\nPhys. Rev. C 5, 626 (1972).\n[34] L. W. Chen, C. M. Ko, B. A. Li, and J. Xu, “Density\nslope of the nuclear symmetry energy from the neutron\nskin thickness of heavy nuclei,” Phys. Rev. C 82, 024321\n(2010).\n[35] Y. M. Engel, D. M. Brink, K. Goeke, S. J. Krieger, and\nD. Vautherin, “Time-dependent hartree-fock theory with\nSkyrme’s interaction,” Nucl. Phys. A 249, 215 (1975).\n[36] R. J. Lenk and V. R. Pandharipande, “Nuclear mean field\ndynamics in the lattice Hamiltonian Vlasov method,”\nPhys. Rev. C 39, 2242 (1989).\n[37] Y. Xia, J. Xu, B. A. Li, and W. Q. Shen, “Simulat-\ning spin dynamics with spin-dependent cross sections in\nheavy-ion collisions,” Phys. Rev. C 96, 044618 (2017).\n[38] R. A. Arndt, R. H. Hackman, and L. D. Roper,\n“Nucleon-nucleon scattering analyses. ii. neutron-proton\nscattering from 0 to 425 mev and proton-proton scatter-\ning from 1 to 500 mev,” Phys. Rev. C 15, 1002 (1977).\n[39] J. S. Bell and T. H. R. Skyrme, “Cviii. the nuclear spin-\norbit coupling,” Philosophical Magazine 1, 1055 (1956).\n[40] R. B. Wiringa, V. G. J. Stoks, and R. Schiavilla, “An\nAccurate nucleon-nucleon potential with charge indepen-\ndence breaking,” Phys. Rev. C 51, 38 (1995).\n[41] F. Sammarruca, E. J. Stephenson, K. Jiang, J. Liu,\nC. Olmer, A. K. Opper, and S. W. Wissink, “Test-\ning microscopic medium effects on nucleons and mesons\nusing polarization observables in high-spin, unnatural-13\nparity (→p,p′)reactions at 200 mev,” Phys. Rev. C 61,\n014309 (1999).\n[42] C. Gale and S. Das Gupta, “Conservation laws and nu-\nclear transport models,” Phys. Rev. C 42, 1577 (1990).\n[43] R. J. Liu and J. Xu, “Revisiting angular momentum con-\nservation in transport simulations of intermediate-energy\nheavy-ion collisions,” Universe 9, 36 (2023).\n[44] M. Papa, G. Giuliani, and A. Bonasera, “Constrained\nmolecular dynamics II: A N-body approach to nuclear\nsystems,” J. Comput. Phys. 208, 403 (2005).\n[45] Y. Xia, J. Xu, B. A. Li, and W. Q. Shen, “Spin transport\nin intermediate-energy heavy-ion collisions as a probe of\nin-medium spin–orbit interactions,” Nucl. Phys. A 955,\n41 (2016).\n[46] L. W. Chen, C. M. Ko, and B. A. Li, “Light cluster\nproduction in intermediate-energy heavy ion collisions in-\nduced by neutron rich nuclei,” Nucl. Phys. A 729, 809\n(2003).\n[47] K. J. Sun and L. W. Chen, “Analytical coalescence for-\nmula for particle production in relativistic heavy-ion col-\nlisions,” Phys. Rev. C 95, 044905 (2017).\n[48] G. Ropke, “Light nuclei quasiparticle energy shift in hot\nand dense nuclear matter,” Phys. Rev. C 79, 014002\n(2009).\n[49] Z. T. Liang and X. N. Wang, “Spin alignment of vec-\ntor mesons in non-central A+A collisions,” Phys. Lett. B629, 20 (2005).\n[50] L. Adamczyk et al. (STAR), “Global Λ hyperon polariza-\ntion in nuclear collisions: evidence for the most vortical\nfluid,” Nature 548, 62 (2017).\n[51] J. Adam et al. (STAR Collaboration), “Polarization of\nΛ (Λ) hyperons along the beam direction in Au +Au\ncollisions at√sNN=200 GeV,” Phys. Rev. Lett. 123,\n132301 (2019).\n[52] F. Becattini, M. Buzzegoli, and A. Palermo, “Spin-\nthermal shear coupling in a relativistic fluid,” Phys. Lett.\nB820, 136519 (2021).\n[53] B. Fu, S. Y. F. Liu, L. Pang, H. Song, and Y. Yin,\n“Shear-Induced Spin Polarization in Heavy-Ion Colli-\nsions,” Phys. Rev. Lett. 127, 142301 (2021).\n[54] F. Becattini and Iu. Karpenko, “Collective longitudinal\npolarization in relativistic heavy-ion collisions at very\nhigh energy,” Phys. Rev. Lett. 120, 012302 (2018).\n[55] S. Y. F. Liu, Y. Sun, and C. M. Ko, “Spin polariza-\ntions in a covariant angular-momentum-conserved chiral\ntransport model,” Phys. Rev. Lett. 125, 062301 (2020).\n[56] X. L. Sheng, L. Oliva, and Q. Wang, “What can we learn\nfrom the global spin alignment of ϕmesons in heavy-ion\ncollisions?” Phys. Rev. D 101, 096005 (2020), [Erratum:\nPhys.Rev.D 105, 099903 (2022)].\n[57] X. L. Sheng, L. Oliva, Z. T. Liang, Q. Wang, and X. N.\nWang, “Spin Alignment of Vector Mesons in Heavy-Ion\nCollisions,” Phys. Rev. Lett. 131, 042304 (2023)."    },    {        "title": "1003.5901v1.Disentangling_phonons_from_spins_in_ion_trap_based_quantum_spin_simulators.pdf",        "content": "arXiv:1003.5901v1  [cond-mat.quant-gas]  30 Mar 2010Disentangling phonons from spins in ion-trap-based quantu m spin simulators\nC.-C. Joseph Wang and James Freericks1,∗\n1Department of Physics, Georgetown University, 37th and O St s. NW, Washington, DC 20057, USA\n(Dated: October 19, 2021)\nWe compute how phonon creation affects the fidelity of the quan tum spin dynamics in trapped ion\nsimulators. A rigorous treatment of the quantum dynamics is made by employing an exact operator\nfactorization of the evolution operator. Although it is oft en assumed that phonon creation modifies\nthe dynamics of the spin evolution, for an Ising spin-spin in teraction in an external magnetic field,\nphonons have no effect on the probabilities of spin product states measured in the d irection of\nthe Ising model axis. Phonons play a much more important role in influencing the effective spin\ndynamics for Heisenberg or XY model spin simulators or for ot her observables, like witness operators\nin the Ising model.\nPACS numbers: 03.67.Pp, 03.67.Ac, 03.67.Lx, 37.10.Ty\nIn the early 1980’s, Richard Feynman proposed that\none could used controlled quantum-mechanical systems\nto simulate the many-body problem [1]. In recent years,\nthere has been significant success in trying to achieve\nthis goal [2]. We focus here on one platform for per-\nforming analog quantum computation, the simulation of\ninteracting quantum spins in a linear Paul trap with a\nfinite number of ions [3–6]. In these systems, the clock\nstates of the ions are the pseudospin states, which can\nbe manipulated independently by a spin-dependent force\ndriven by laser beams. The lasers couple the pseudospin\nstates to the lattice vibrations of the trapped ions, which\nleads to effective spin-spin interactions when the phonon\ndegrees of freedom are integrated out [4, 7, 8]. Recently,\nion traps have been used to simulate the transverse-field\nIsing model with two [5] and three [3] ions in a trap.\nExperiments are usually run in the Lamb-Dicke regime\nwhere spontaneous phonon creation is low, but it is gen-\nerally believed that phonon creation, especially as one\ntunes the driving force close to resonance with one of\nthe vibrational mode frequencies, will act to dephase and\ndecohere the quantum dynamics leading to errors in the\nanalogsimulatorsthat will build up asa function oftime.\nSurprisingly, for the simplest class of emulators (namely\nthe transverse-field Ising model) one can show that there\nisno dephasing or decoherence by the phonons to lowest\norder in the Lamb-Dicke parameter if one measures the\nprobabilities of product states of spins where the spin\nquantization axis coincides with the Ising model axis.\nFor more complicated models, like the XY or Heisen-\nberg models, phonon dephasing and decoherence cannot\nbe removed, and hence this will create more stringent\nrestrictions on the fidelities of quantum simulators for\nthose models when simulated in ion traps. Moreover,\nwhen measuring entanglement witness operators, even in\nthe transverse field Ising model, the phonons should af-\nfect the results.\nQuantum spin models are of interest for a number of\nreasons. Currently, there is a significant focus on the\nphysics of frustration [9], and how that can lead to theformation of a so-called spin liquid [10]. These spin liq-\nuid states are complicated quantum many-body systems\nthat exhibit significant entanglement of their wavefunc-\ntions, and could also exhibit emergent phenomena within\ntheir low-energy excitation spectra. Conventional com-\nputation fails to describe these systems well. Exact di-\nagonalization studies are limited to small size lattices,\nwhile quantum Monte Carlo simulations suffer from the\nsign problem and cannot reach low temperatures. Hence,\ntheyformanimportanttestcaseforquantum simulators,\nand can allow the simulators to display interesting new\nphysical phenomena.\nWhenNions are placed in a (harmonic) linear Paul\ntrap [11], they form a nonuniform lattice, with increas-\ning interparticle spacing as one moves from the cen-\nter to the edge of the chain. The ions vibrate in all\nthree spatial dimensions about these equilibrium posi-\ntions [12] and are described by the phonon Hamiltonian\nHph=/summationtext\nαν¯hωαν(a†\nανaαν+ 1/2), in which a†\nανis the\nphonon creation operator of the normal mode νalong\nthe spatial direction α∈x,y,z. Theαth spatial compo-\nnentofthejth ion’sdisplacementoperator δˆRα\njis related\nto theαth phonon normal mode amplitude (eigenvector\nof the dynamical matrix) bαν\njand theαth phonon oper-\nator viaδˆRα\nj=/summationtext\nνbαν\nj[aαν+a†\nαν]/radicalbig\n¯h/2MωανwithM\nthe mass of the ion.\nThe two clock states used to describe the quantum\nspins are split by an energy difference ¯ hω0. A spin-\ndependent force on the ions is generated by using light to\ncouple these states to a third state via stimulated Raman\ntransitions [13]. The light that couples to this state is de-\ntuned by a small frequency µcalled the detuning. The\nrotating wave approximation is used to map this three-\nstate system onto a two-state system in the regime where\nω0≫µ(these differ by at least 3 orders of magnitude\nin typical experiments). The two-level system is then\nrepresented by an effective laser-ion Hamiltonian [13–15]\nHLI(t) =−¯hN/summationdisplay\nj=1Ωj(δk·δˆRj)σx\njsin(µt) (1)2\nwhereδkis the wavevector difference of the two inter-\nfering laser beams that generate the Raman coupling.\nHere the Lamb-Dicke limit δk|δˆRj(t)| ≪1 is assumed so\nthat the expansion for exp( −iδk·δˆRj)≈1−iδk·δˆRj\nis taken. The Roman index jdenotes the different ions,\nΩjis the Rabi frequency due to the Raman coupling at\nthejth ion site, and the spin operator σx\njis the Pauli\nspin matrix in the x-direction. While one can couple to\nany direction by tuning the phase difference of the Ra-\nman beams, for simplicity, and without loss of generality,\nwe assume we have the lasers tuned to create a coupling\nto the spin component in the x-direction only here. For\nexample, we consider the ions crystallized along the z-\naxis of the trap, so the x- andy-directions correspond to\ntransverse phonons. Then coupling the Raman lasers in\nthe transverse direction minimizes effects of ion heating\nand allows for an identical spin axis for each ion [8].\nIn experiments, one uses adiabatic quantum state evo-\nlution to evolve the ground state from an easily prepared\ninitial state to the desired complex quantum state that\nwill be studied. For spin models generatedin an ion trap,\nit is easy to create a fully polarized ferromagnetic state\nvia optical pumping, and then use a spin rotation to ori-\nent the ferromagneticstate in any direction. Then, if one\nintroduces a Hamiltonian with a magnetic field in the di-\nrection of the polarized state, it is in the ground state\nof the system. By slowly reducing the magnitude of the\nfield and turning on the spin Hamiltonian of interest, we\ncan reach the ground state of the Hamiltonian of interest\n(even in zero field). Hence, we need to add an addi-\ntional Zeeman term HB(t) =/summationtextN\nj=1B(t)·ˆσjwith a time-\ndependent effective magnetic field B(t) (the coupling is\ntothedifferentPaulispinmatrices). Forexample, amag-\nneticfieldinthe y-directionismadebydrivingaresonant\nradio-frequency field with frequency ω0between the two\nhyperfine states to implement the spin flips [15]. The full\nHamiltonian is then H(t) =Hph+HLI(t)+HB(t).\nQuantum dynamics for a time-dependent Hamiltonian\narefound bycalculatingthe evolutionoperatorasatime-\nordered product U(t,t0) =Ttexp[−i/integraltextt\nt0dt′H(t′)/¯h] and\noperating it on the initial quantum state |ψ(t0)/an}bracketri}ht. For\nthe adiabatic evolution of the ground state, we start\nour system in a state with the spins aligned along the\nmagnetic field and no phonons at time t0:|ψ(t0)/an}bracketri}ht=\n|0/an}bracketri}htph⊗ | ↑y↑y...↑y/an}bracketri}ht. In time-dependent perturbation\ntheory, one rewrites the evolution operator in the inter-\naction picture with respect to the time-independent part\noftheHamiltonian. Thisprocedureproducesafactorized\nevolution operator\nU(t,t0) =e−i\n¯hHph(t−t0)UI(t,t0) (2)\nwhich is the first step in our factorization procedure (the\nfirst factor on the left is called the phonon evolution\noperator). The second factor on the right is the evo-\nlution operator in the interaction picture which satis-fies an equation of motion given by i¯h∂UI(t,t0)/∂t=\n[VI(t) +HB(t)]UI(t,t0), withVI(t) = exp[iHph(t−\nt0)/¯h]HLI(t)exp[−iHph(t−t0)/¯h] since [HB(t),Hph] =\n0. The only difference between HLI(t) andVI(t) is\nthat the phonon operators are replaced by their inter-\naction picture values: aαν→aανexp[−iωαν(t−t0)] and\na†\nαν→a†\nανexp[iωαν(t−t0)].\nWe now work to factorize the evolution operator fur-\nther. Motivated by the classic problem on the driven\nharmonic oscillator in Landau and Lifshitz’s Quantum\nMechanics [16] and similar discussions in Gottfried’s\nbook [17], we factorize the interaction picture evolution\noperator further UI(t,t0) = exp[−iWI(t)/¯h]¯U(t,t0) with\nWI(t) defined by WI(t) =/integraltextt\nt0dt′VI(t′) (we call the factor\non the right the phonon-spin evolution operator). One\nimportant result we use is that the multiple commutator\nsatisfies [[WI(t),VI(t′)],VI(t′′)] = 0, which greatly sim-\nplifies the analysis below.\nThe equation of motion for the spin evolution operator\n¯U(t,t0) satisfies\ni¯h∂\n∂t¯U(t,t0) =¯H(t)¯U(t,t0), (3)\nin which the operator ¯H(t) is given by the expression\n¯H(t) =ei\n¯hWI(t)[−i¯h∂t+HB(t)+VI(t)]e−i\n¯hWI(t).(4)\nThe operator ¯H(t) can then be expanded order by order\nas\nei\n¯hWI(t)VI(t)e−i\n¯hWI(t)=VI(t)+i\n¯h[WI(t),VI(t)] (5)\nei\n¯hWI(t)HB(t)e−i\n¯hWI(t)=N/summationdisplay\nj=1/braceleftBig\nB(t)·ˆσj (6)\n+i\n¯h[WI(t),B(t)·ˆσj]+.../bracerightBig\n,\nei\n¯hWI(t)i¯h∂te−i\n¯hWI(t)=VI(t)+1\n2i\n¯h[WI(t),VI(t)],(7)\nwhere we used the facts that ∂WI(t)/∂t=VI(t) and that\n[[WI(t),VI(t′)],VI(t′′)] = 0.\nExplicit calculations then yield\nVI(t) =−N/summationdisplay\nj=1N/summationdisplay\nν=1/summationdisplay\nα¯hΩjηανbαν\nj(aανe−iωανt+a†\nανeiωανt)\n×sin(µt)σx\nj, (8)\nWI(t) =−N/summationdisplay\nj=1N/summationdisplay\nν=1/summationdisplay\nα¯hΩjηανbαν\nj\nω2αν−µ2σx\nj (9)\n×/braceleftBig/bracketleftBig\ne−iωανt(iωανsinµt+µcosµt)\n−e−iωανt0(iωανsinµt0+µcosµt0)/bracketrightBig\naαν+h.c./bracerightBig\n,\n(10)3\nin whichηαν=δαxδkα/radicalbig\n¯h/2Mωανis the Lamb-Dicke\nparameter for the phonon mode ανwith theαth compo-\nnent of the laser momentum δkα.\nWe choose the magnetic field in the y-direction. Then\nthe termi\n¯h[WI(t′),HB(t′)]isnonzeroandproportionalto\nspin operators in the z-direction times phonon operators.\nBut it is also proportional to the Lamb-Dicke parameter\nηαν≪1 which is small in currentexperiments. Hence, tolowest-order in the Lamb-Dicke regime, we drop all com-\nmutatortermswiththemagnetic-fieldpieceoftheHamil-\ntonian (recallthe magneticfield is a similarmagnitude to\nthe spin exchange parameters, so the additional Lamb-\nDicke parameter multiplying the magnetic field makes\nit a less important term). The equation of motion can\nnow be directly integrated. The spin evolution operator\n¯U(t,t0) becomes\n¯U(t,t0)≈ Ttexp\n−i\n¯h/integraldisplayt\nt0dt′\nN/summationdisplay\nj,j′=1Jjj′(t′)σx\njσx\nj′+B(t′)N/summationdisplay\nj=1σy\nj\n\n, (11)\nwhich is the third factor for the evolution operator\nof the Ising model in a transverse field. The spin\nexchange terms Jjj′(t) =J0\njj′+ ∆Jjj′(t) arise from\nthei[WI(t),VI(t)]/2¯hcommutator and include a time-\nindependent exchange interaction between two ions J0\njj′\nand a time-dependent exchange interaction ∆ Jjj′(t).\nThetime-independent termdeterminesthe effectivespin-\nspin Hamiltonian that is being simulated, while the time-\ndependent terms can be thought of as diabatic correc-\ntions, which are often small in current experimental set-\nups, but need not be neglected. For simplicity, we set\nthe initial time t0= 0. Then the expression for the spin\nexchange interaction Jjj′(t) is\nJjj′(t) =¯h\n2N/summationdisplay\nν=1ΩjΩj′η2\nxνbxν\njbxν\nj′\nµ2−ω2xν(12)\n×[ωxν−ωxνcos2µt−2µsinωxνtsinµt],\nwhich can be antiferromagnetic Jjj′(t)>0 or ferromag-\nneticJjj′(t)<0 depending on the laser detuning µandthe detailed phonon mode properties ωxν,bxν\nj, andbxν\nj′.\nWe need to evaluate one more identity before\narriving at our main result. We further fac-\ntorize the entangled phonon-spin evolution operator\nexp[−iWI(t)/¯h] into the product exp[ −i/summationtext\nνΓ∗\nxν(t)a†\nxν]\nexp[−i/summationtext\nνΓxν(t)axν]exp[−/summationtext\nνΓxν(t)Γ∗\nxν(t)/2] with the\nspin operator defined to be Γ∗\nxν(t) =/summationtext\njγxν\nj(t)σx\njand\nits complex conjugate is Γ xν(t), in which the func-\ntionγxν\nj(t) satisfiesγxν\nj(t) = Ω jηxνbxν\nj/(µ2−ω2\nxν)×\n[exp{−iωxνt}(iωxνsinµt+µcosµt)−µ].\nWe are now ready to show our main result that\nphonons have no effect on the probabilities to observe\nany of the 2Nproduct states with a quantization axis\nalong the Ising axis |β/an}bracketri}ht=| ↑xor↓x/an}bracketri}ht⊗| ↑ xor↓x/an}bracketri}ht ⊗...\nfor theNionic spins. Using the fundamental axiom of\nquantum mechanics, the probability Pβ(t) to observe a\nproduct spin state |β/an}bracketri}htstarting initially from the phonon\nground state (for all phonon modes) |0/an}bracketri}htand not measur-\ning any of the final phonon states involves the trace over\nall possible final phonon configurations\nPβ(t) =∞/summationdisplay\nnx1=0···∞/summationdisplay\nnxν=0···∞/summationdisplay\nnxN=0|/an}bracketle{tβ|⊗ph/an}bracketle{tnx1,...nxν,...nxN|U(t,t0)|0,0,...,0/an}bracketri}htph⊗|Φ(0)/an}bracketri}ht|2(13)\nwhere|Φ(0)/an}bracketri}htis any initial spin state (it need not be a\nproduct state). Note that since the pure phonon fac-\ntor of the evolution operator exp[ −iHpht/¯h] is a phase\nfactor, it has no effect on the probabilities when eval-\nuated in the phonon number operator basis, so we can\ndrop that factor. Next, the term exp[ −i/summationtext\nνΓxν(t)axν]\ngives 1 when operating on the phonon vacuum to the\nright, so it can be dropped. We are thus left with\nthree factors in the evolution operator. One involvesexponentials of the phonon creation operator multiplied\nby spin operators, one involves products of spin opera-\ntors that resulted from the factorization of the coupled\nphonon-spin evolution operator factor, and one is the\npure spin evolution factor ¯U. The two factors that ap-\npear on the left involve only Ising spin operators, and\nhence the product state basis is an eigenbasis for those\noperators. This fact allows us to directly evaluate the\nexpression in Eq. (13). We expand the evolution of the4\ninitial state at time tin terms of the product-state ba-\nsis|Φ(t)/an}bracketri}ht=¯U(t,t0= 0)|Φ0\nS/an}bracketri}ht=/summationtext\nβ′cβ′(t)|β′/an}bracketri}htwith|β′/an}bracketri}ht\ndenoting each of the 2Nproduct state basis vectors and\ncβ′(t) is a number. Using the fact that the product statessatisfy the eigenvalue equation σx\nj|β′/an}bracketri}ht=mx\njβ′|β′/an}bracketri}htwith\neigenvalues mx\njβ′= +1 (for | ↑x/an}bracketri}ht) or−1 (for| ↓x/an}bracketri}ht), we\narrive at the expression for the probability\nPβ(t) =|cβ(t)|2exp\n−/summationdisplay\nνjj′γxν\nj(t)γxν∗\nj′(t)mx\njβmx\nj′β\n∞/summationdisplay\nnx1=0···∞/summationdisplay\nnxN=0N/productdisplay\nν=1\n1\nnxν!\n\n/summationdisplay\njj′γxν\nj(t)γxν∗\nxj′(t)mx\njβmx\nj′β\n\nnxν\n.\n(14)\nWe used the matrix element\nph/an}bracketle{tnx1,...,n xν,...nxN|e−i/summationtext\nνΓ∗\nxν(t)a†\nxν|0,...0/an}bracketri}htph(15)\n=(−i)nx1+...nxν+..nxN√nx1!..nxν!..nxN!Γ∗\nx1(t)nx1...Γ∗\nxν(t)nxν...Γ∗\nxN(t)nxN\nin the derivation. The summations become exponentials,\nwhich exactly cancel the exponential term and finally\nyieldPβ(t) =|cβ(t)|2, whichiswhatwewouldhavefound\nif we evaluated the evolution of the spins using just the\nspin evolution operator ¯U. Hence, phonons have no ob-\nservable effects on the probability of product states for the\ntransverse-field quantum Ising model . If we do not mea-\nsure the probability of product states, then the terms\nfrom the coupled phonon-spin evolution operator remain\nspin operators, and one can show that the probabilities\nare changed by the phonons by simply evaluating the re-\nsult in Eq. (13) directly. In other words, it is because the\nΓ operators are diagonal in the product space basis along\nthe Ising axis that allows us to disentangle the phonon\nand spin dynamics. In cases where this cannot be done,\nwe expect the phonon and spin dynamics to generically\nremain entangled. In particular, when one measures a\ntypical entanglement witness operator, one measures the\nprobabilities for the state to be projected onto product\nstates that do not lie along the Ising axis. In this case,\nthe phonons and spins will remain entangled, and the\nphonons will affect the expectation value of the witness\noperator.\nThe simplest way to set up XY or Heisenberg model is\nto introduce additional laser beams with perpendicular\nδkvalues so one gets pure spin-spin interactions along\neach of the different coordinate axes (if one cannot in-\ntroduce perpendicular beams, then other methods can\nbe tried, but they require strategies to reduce unwanted\ncross terms like σxσy). For the XY or Heisenberg model,\nwe generate additional entangled spin-phonon evolution\noperator factors for the different coordinate directions\n(xandyfor the XY model and all for the Heisenberg).\nBut, since one cannot find a basis of quantum states that\nsimultaneously diagonalizes the Pauli spin operators in\nmore than one coordinate direction, the above derivation\ncannot be completed, and so the phonons will generically\nmodify the probabilities for the spin states.In conclusion, we have explicitly shown that phonons\ndo not affect the probability of observing spin prod-\nuct states in ion trap simulators of the transverse-field\nIsing model to lowest order in the Lamb-Dicke parame-\nter. This implies that simulations of the transverse-field\nIsing model have the phonon motion and the spin evolu-\ntion disentangled, and hence will be much cleaner than\nsimulations of the XY or Heisenberg model. Our results\ndo not depend on the detuning lying far enough from res-\nonance, unless the proximity to resonance moves the sys-\ntem out of the Lamb-Dicke regime so higher-order terms\nneed to be included, in which case phonon entanglement\neffects are likely to affect the results.\nACKNOWLEDGEMENTS\nWe acknowledge useful discussions with Kihwan Kim,\nChrisMonroe, andL.-M Duan. This workwassupported\nunder ARO grant number W911NF0710576 with funds\nfrom the DARPA OLE Program.\n∗Electronic address: joseph@physics.georgetown.edu\n[1] R. P. Feynman, Int. J. Theor. Phys. 21, 467 (1981).\n[2] L. Buluta and F. Nori, Science 326, 108 (2009).\n[3] K. Kim, M.-S. Chang, S. Korenblit, R. Islam, E. E. Ed-\nward, J. K. Freericks, G.-D. Lin, L.-M. Duan and C.\nMonroe, submitted to Nature (2010).\n[4] K. Kim, M.-S.Chang, R.Islam, S.Korenblit, L.-M. Duan\nand C. Monroe, Phy. Rev. Lett. 103, 120502 (2009).\n[5] A. Friedenauer, H. Schmitz, J. T. Glueckert, D. Porras,\nand T. Schaetz, Nat. Phys. 4, 757 (2008).\n[6] C. Monroe, D. M. Meekhof, B. E. King, W. M. Itano,\nand D. J. Wineland, Phys. Rev. Lett. 75, 4714 (1995).\n[7] D. Porras and J. I. Cirac, Phys. Rev. Lett. 92, 207901\n(2004).\n[8] S.-L. Zhu, C. Monroe, and L.-M. Duan, Phys. Rev. Lett.\n97, 050505 (2006).\n[9] G. Misguich and C. Lhuillier, “Two-Dimensional Quan-\ntum Antiferromagnets,” in Frustrated Spin Systems ,\nedited byH. T. Diep, (WorldScientific, Singapore, 2005);\nL. Balents, Nature (London) 464, 199 (2010).5\n[10] G. Misguich in Exact Methods in Low-dimensional Statis-\ntical Physics and Quantum Computing , edited by J. Ja-\ncobsen, S. Ouvry, V. Pasquier, D. Serban and L. Cuglian-\ndolo, Lecture Notes of the Les Houches Summer School:\n89, July, 2008 (Oxford University Press, Oxford, 2010).\n[11] M. Johanning, A. F. Varon, and Ch. Wunderlich,\narXiv:0905.0118 (2009).\n[12] D. F. V. James, App. Phys. B 66, 181 (1998); C. Mar-\nquet, F. Schmidt-Kaler and D. F. V. James, App. Phys.\nB76, 199 (2003).\n[13] P. J. Lee, K.-A. Brickman, L. Deslauriers, P. C. Haljan,\nL.-M. Duan, and C. Monroe, J. Opt. B 7, S371 (2005).[14] P. Lambropoulos and D. Petrosyan, Fundamentals of\nQuantum Optics and Quantum Information (Springer-\nVerlag, Berlin, 2007).\n[15] D. J. Wineland, C. Monroe, W. M. Itano, D. Leibfried,\nB. E. King, and D. M. Meekhof, J. Res. Natl. Inst. Stand.\nTechnol. 103, 259 (1998).\n[16] L. D. Landau and E. M. Lifshitz, Quantum Mechan-\nics: Nonrelativistic Theory , third edition (Butterworth-\nHeinemann, New York, 1981).\n[17] K. Gottfried, Quantum Mechanics, Vol. 1: Fundamentals\n(Benjamin, New York, 1966)."    },    {        "title": "0909.0060v2.Phase_Sensitive_Probes_of_Nuclear_Polarization_in_Spin_Blockaded_Transport.pdf",        "content": "arXiv:0909.0060v2  [cond-mat.mes-hall]  20 Aug 2010Phase-Sensitive Probes of Nuclear Polarization in Spin-Bl ockaded Transport\nM. S. Rudner1, I. Neder1, L. S. Levitov2, B. I. Halperin1\n(1)Department of Physics, Harvard University, 17 Oxford St., C ambridge, MA 02138\n(2)Department of Physics, Massachusetts Institute of Technol ogy, 77 Massachusetts Ave, Cambridge, MA 02139\nSpin-blockaded quantum dots provide a unique setting for st udying nuclear-spin dynamics in a\nnanoscale system. Despite recent experimental progress, o bserving phase-sensitive phenomena in\nnuclear spin dynamics remains challenging. Here we point ou t that such a possibility opens up in\nthe regime where hyperfine exchange directly competes with a purely electronic spin-flip mechanism\nsuch as the spin-orbital interaction. Interference betwee n the two spin-flip processes, resulting\nfrom long-lived coherence of the nuclear-spin bath, modula tes the electron-spin-flip rate, making it\nsensitive to the transverse component of nuclear polarizat ion. In a system repeatedly swept through\na singlet-triplet avoided crossing, nuclear precession is manifested in oscillations and sign reversal of\nthe nuclear-spin pumping rate as a function of the waiting ti me between sweeps. This constitutes a\npurely electrical method for the detection of coherent nucl ear-spin dynamics.\nDue to their long coherence times, nuclear spins offer\nunique opportunities to study quantum dynamics. While\nconventional techniques using RF fields probe macro-\nscopic groups of spins, currently there is wide interest in\nthe local dynamics of nanoscale groups of spins. In par-\nticular, semiconducting quantum dots have emerged as a\nplatform for investigating the intriguing quantum many-\nbody dynamics of coupled electron and nuclear spins [1–\n8]. Many experimental achievements in this field rely\non the phenomenon of spin blockade, in which electrons\nmust flip their spins in order to pass through the sys-\ntem in accordance with the Pauli exclusion principle [9].\nSuch spin-blockaded transport is sensitive to the nuclear-\npolarization-dependent Overhauser field, which controls\nthe energy splittings between electronic spin states. This\nsensitivity affords an exciting opportunity to electrically\ncontrol and detect the states of nuclear spins[10, 11].\nIn this Rapid Communication, we propose a method\nwhich can be used to electrically probe coherent nuclear\nphenomena, such as Larmor precession or Rabi oscilla-\ntions, which are not easily probed by the Overhauser ef-\nfect alone. We identify a regime where transport is sen-\nsitive to all three vector-components of the local nuclear\npolarization, due to coherentinterplaybetween hyperfine\ncoupling and an electron-only spin-coupling such as the\nspin-orbit interaction[12, 13]. As we show, systems with\nsuch sensitivity host a variety of new phenomena aris-\ning from coherent nuclear dynamics. Recently observed\noscillations of the nuclear pumping rate controlled by nu-\nclear Larmorprecession[14] indicate that this unexplored\nregime in now within experimental reach.\nHere we investigate these phenomena in a two-electron\ndouble quantum dot in a uniform magnetic field. The\ntwo-electron singlet and m=±1 triplet states are cou-\npledviathedifferenceofhyperfinefieldsduetotransverse\nnuclearpolarizationin thetwodots, ∆ Bnuc,⊥[10,11,15].\nWithout an additional coupling, the electron spin-flip\nrate depends only on |∆Bnuc,⊥|, and not on the orien-\ntation of ∆ Bnuc,⊥in the XY plane [15].\nThe situation becomes considerably more interesting\nFIG. 1: (Color online) Coherent interplay of hyperfine and\nspin-orbit mediated transitions. a) Interdot tunneling in the\ntriplet state is suppressed by Pauli exclusion, but can be me -\ndiated by the hyperfine and spin-orbit interactions which do\nnot conserve electron spin. b) The singlet and triplet lev-\nels|S/angbracketrightand|T+/angbracketrightexhibit an avoided crossing with split-\nting|vθ|=|vSO+vHFeiθ|when exchange energy compen-\nsates Zeeman energy. c) Behavior near the S−T+cross-\ning, controlled by |vθ|, is sensitive to the relative phase θ\nbetween spin-orbit and hyperfine matrix elements. d) Phase-\ndependent transition probability, Eq.(3), with vSO= 0.4√/planckover2pi1β,\nandvHF= 0.6√/planckover2pi1β.\nwhen singlet-triplet transitions can occur due to either\nthe spin-orbit or the hyperfine interaction (see Fig.1). In\nthis case, we find that the probability of electron spin-\nflip depends on the angleθof the nuclear polarization\nmeasured in the XY plane relative to a fixed axis deter-\nmined by the spin-orbit interaction. Such a dependence\nmakeselectron transportsensitive to the phase ofnuclear\nspin precession. Below, we analyze this phenomenon\nfor a Landau-Zener-type process, occurring when the\nelectronic system is swept through a singlet-triplet level\ncrossing, as employed e.g. in Refs.[14, 16] and depicted\nin Fig.1b. To make contact with experiment, we focus\nin particular on resulting signatures in nuclear spin po-\nlarization. Strikingly, we find robust oscillations in the2\nnuclear spin pumping rate which result from nuclear pre-\ncession and survive even after averaging over the distri-\nbution of random initial nuclear spin states. Note that\nothertypesofelectron-onlycouplingsuchasZeemancou-\npling to the nonuniform field of a micromagnet[17] can\nproduce analogous effects.\nThe origin of the spin-angle dependence can be un-\nderstood heuristically by analyzing the avoided crossing\nthat opens near the degeneracy of the triplet and singlet\nlevels|T+∝an}bracketri}htand|S∝an}bracketri}htin nonzero magnetic field, circled\nin Fig.1b. Due to the spin-orbit interaction, tunneling\nbetween dots is accompanied by a rotation of electron\nspin [18] which introduces a nonzero spin-flip amplitude.\nIn addition, the hyperfine interaction between electron\nand nuclear spins allows electronic transitions accom-\npanied by nuclear spin flips, described by the effective\nHamiltonian HHF=A+|S∝an}bracketri}ht∝an}bracketle{tT+|+A−|T+∝an}bracketri}ht∝an}bracketle{tS|, with\nA±=/summationtextNnuc\ni=1giI±\ni. The magnitude and sign of each cou-\npling constant gidepends on the location of nucleus i\n(positive in dot 1 and negative in dot 2).\nTaking advantage of the fact that the number of nu-\nclei interacting with the electrons is very large, Nnuc≈\nO(106), we note that the commutator [ A+,A−] is typ-\nically smaller than A+A−by a factor of order 1 /N1/2\nnuc.\nThis allows us to treat A+as a nearly classical variable,\nA+≈vHFeiθ.HerevHFis proportional to the magnitude\nof the transverse hyperfine difference field ∆ Bnuc,⊥, and\nθdescribes its orientation in the XY plane.\nTogether, the spin-orbit and hyperfine pieces provide\na matrix element between the singlet and triplet states:\nvθ≡ ∝an}bracketle{tS|H|T+∝an}bracketri}ht=vSO+vHFeiθ. (1)\nEvolution in the {|S∝an}bracketri}ht,|T+∝an}bracketri}ht}subspace near the level\ncrossing is described by the 2 ×2 Hamiltonian\nHST+=/parenleftbigg1\n2∆v∗\nθ\nvθ−1\n2∆/parenrightbigg\n,∆(t) =εT+(t)−εS(t),(2)\nwhereεT+andεSaretheenergiesofthe diabatic |S∝an}bracketri}htand\n|T+∝an}bracketri}htstates. The level detuning ∆( t) can be controlled\nby electrostatic gates and/or magnetic field.\nWe consider the case where the system is initialized to\nthe “(0,2)” singlet state |S∝an}bracketri}htwith both electrons resid-\ning in the right dot (large positive ∆), and then swept\nthroughtheavoidedcrossingtothe“(1 ,1)”chargeregime\nwith one electron on each dot (see Fig.1b). For a con-\nstant sweep rate β=|d∆/dt|, the electron spin flip in\nsuch a model is interpreted as a Landau-Zener transition\noccurring with probability (see Fig.1d)\nPLZ= 1−exp/parenleftbig\n−2π|vSO+vHFeiθ|2//planckover2pi1β/parenrightbig\n,(3)\nwherePLZis the probability to remain in the ground\nstate. The explicit dependence on the phase θof nuclear\npolarization, which enters through the matrix element\nvθ, shows the singlet/triplet transition probability’s sen-\nsitivity to the transverse nuclear polarization vector.Model (2) providesa useful heuristic for understanding\nelectron spin dynamics, in particular for situations where\nthe nuclear spin state is characterized by a well-defined\nazimuthal angle θ. However, to understand the behav-\nior with more general initial states and to account for\nthe effects of back-action on the nuclei, we must examine\nthe quantum many-body dynamics of coherently coupled\nelectron and nuclear spins. By solving this problem be-\nlow, we will further justify the form of Eq.(1) and will\nobtain the electron and nuclear spin-flip rates.\nAkeynewfeaturehereistherelaxedselectionrulegov-\nerning nuclear pumping: due to spin-orbit coupling, elec-\ntron spin flips may occur withorwithouta compensating\nnuclear spin flip. Although these two processes appar-\nently lead to orthogonal final states, long-lived coherence\nof the nuclear spin bath allows interference between dif-\nferenttransition sequences (seeFig.2). Moreover,a single\nelectron can change the total nuclear polarization by an\namount ∆mthat can be larger than 1, and of either sign .\nAt this point it is convenient to map the problem\nonto the bipartite 1-dimensional quantum walk shown\nin Fig.2a, where each unit cell is labeled by m, thez-\nprojection of the total nuclear spin Iz=/summationtext\niIz\ni. In doing\nso, [Iz,A±] remains nonzero. Intracell hopping between\ninternal states TandS, characterized by ∆ m= 0, de-\nscribes a spin-orbit transition occurring with amplitude\nvSO. Intercell hopping, characterized by ∆ m=±1, de-\nscribes a hyperfine transition that occurs with the ampli-\ntudevHF=∝an}bracketle{tT,m−1|HHF|S,m∝an}bracketri}ht. Initially, weconsider\nsweeps which are fast compared to the nuclear Larmor\nperiod, and so neglect the nuclear Zeeman energy.\nBecauseNnucislarge, the valueof vHFwill changevery\nlittle during the course of a few sweeps, and we treat it\nhere as a constant. However, the initial values of vHF\nmay differ from one run to another, with a statistical\ndistribution of the form p(v)∝ve−v2/s2, wheresis a\nconstant. Additionally, becausethetypicalvaluesoftotal\nnuclear spin will be large, of order√Nnuc, we may take\nthe ladder of allowed values of mto be infinite while the\ntotal number of sweeps is not too large.\nThe state of the system evolves according to\ni/planckover2pi1˙ψT\nm=vSOψS\nm+vHFψS\nm+1+1\n2∆(t)ψT\nm\ni/planckover2pi1˙ψS\nm=vSOψT\nm+vHFψT\nm−1−1\n2∆(t)ψS\nm.(4)\nAt timet0, the system is initialized to the singlet state\nwith polarization m=m0:ψS\nm=δm,m0,ψT\nm= 0. The\nexpected change of polarization,\n∝an}bracketle{t∆m∝an}bracketri}ht=/summationdisplay\nm(m−m0)Pm;Pm=|ψS\nm|2+|ψT\nm|2,(5)\nis determined by the probabilities {Pm}to find the nu-\nclear spin with z-projection min the final state at time\ntFafter the sweep. For convenience we take the (non-\nclassical) initial nuclear state to be an eigenstate of Iz,\nbuttheframeworkcanbeappliedformoregeneralstates.3\nFIG. 2: (Color online) Quantum walk model of electron-\nnuclear spin dynamics near the S−T+crossing, Eq.(4). a)\nUnit cells are labeled by m, thez-component of total nu-\nclear spin. Intracell and intercell hopping correspond to s pin-\norbit and hyperfine transitions, with matrix elements vSOand\nvHF, respectively. b) Expected change of nuclear polarization\n/angbracketleft∆m/angbracketright, Eq.(9). The star indicates the optimal sweep rate β∗\nalong the dashed line.\nIn the Fourier representation |ψm∝an}bracketri}ht=\n1\n2π/contintegraltextdθe−imθ|ψθ∝an}bracketri}ht, where |ψm∝an}bracketri}htand|ψθ∝an}bracketri}htare two-\ncomponent spinors ( ψT\nm, ψS\nm)Tand (ψT\nθ, ψS\nθ)T, the\ndynamics for different θcomponents decouple. We\nobtain\ni/planckover2pi1d\ndt|ψθ∝an}bracketri}ht=HST+|ψθ∝an}bracketri}ht, (6)\nwhereHST+is given in Eq.(2). Here the parameter θ\nplays the role of the azimuthal angle of ∆ Bnuc,⊥in the\nclassicalproblem,Eq.(1). Indeed, astate ψS,T\nθ∝δ(θ−θ0)\nis a coherent state concentrated near θ0.\nTo calculate ∝an}bracketle{t∆m∝an}bracketri}ht, we make use of the relation\nm|ψm∝an}bracketri}ht=i\n2π/contintegraltextdθd\ndθ/parenleftbig\ne−imθ/parenrightbig\n|ψθ∝an}bracketri}ht. Integrating by parts\nto move the derivative onto |ψθ∝an}bracketri}ht, we find\n∝an}bracketle{t∆m∝an}bracketri}ht=1\n2πi/contintegraldisplay\ndθ∝an}bracketle{tψθ|d\ndθ|ψθ∝an}bracketri}ht, (7)\nwhere without loss of generality we have taken m0= 0.\nTo evaluate expression (7), we must solve for the two-\nlevel dynamics of |ψθ∝an}bracketri}htunder the time dependent Hamil-\ntonian (2). We can write the evolution operator as\nU(θ) =/parenleftbiggaθbθeiϕ0(θ)\n−b∗\nθe−iϕ0(θ)a∗\nθ/parenrightbigg\n,|aθ|2+|bθ|2= 1,(8)\nwhereϕ0(θ) = arg[ vθ] (see Fig.1c). With this\nparametrization, we have aθ=a−θ, andbθ=b−θ.\nThe initial state |ψ(0)\nθ∝an}bracketri}ht= (0,1)Tevolves into |ψθ∝an}bracketri}ht=\n(bθeiϕ0(θ), a∗\nθ)T. Becauseaθandbθareevenfunctions of\nθ, the only term that contributes to ∝an}bracketle{t∆m∝an}bracketri}htis that gener-\nated when the derivative acts on eiϕ0(θ)in Eq.(7), giving\n∝an}bracketle{t∆m∝an}bracketri}ht=−1\n2π/contintegraldisplay\ndθ/parenleftbiggdϕ0\ndθ/parenrightbigg\n|bθ|2, (9)\nwhere|bθ|2is the singlet-triplet transition probability in\ntheθchannel. Comparing this result to that for the net\nelectron spin-flip probability, given by Pel=1\n2π/contintegraltext\n|bθ|2dθ,we note that, unlike the situation where only the hyper-\nfine interaction is present, here there is no simple rela-\ntionship between the electron and nuclear spin-flip rates.\nThe analysis leading to Eq. (9) is valid for arbitrary\ntime dependence ∆( t), and in particular can even de-\nscribe cases involving multiple S−Tcrossings. Remark-\nably, for very slow sweeps the change in polarization\nbecomes sharply quantized. Indeed, since |bθ|2→1 in\nthe adiabatic limit, the integral (9) is equal to a winding\nnumber:∝an}bracketle{t∆m∝an}bracketri}ht=−1\n2π/contintegraltext\ndϕ0. Thus∝an}bracketle{t∆m∝an}bracketri}ht=−1 or 0 if\nvθdoes or does not wind around the origin as θtraverses\nthe Brillouin zone, depending on the relative magnitudes\nofvHFandvSO[19]. The sign is negative because elec-\ntron transitions from StoT+flip nuclear spins from up\nto down.\nThe quantity ∝an}bracketle{t∆m∝an}bracketri}htexhibits interesting dependence on\nthe sweep speed, which can be analyzed most straightfor-\nwardly for linear sweeps ∆( t) =−βt, when|bθ|2is given\nby the Landau-Zener formula (3). Due to the absence of\ndynamics for very fast sweeps, and since ∝an}bracketle{t∆m∝an}bracketri}ht= 0 for\nvery slow sweeps in the region |vSO|>|vHF|, the polar-\nization efficiency is a non-monotonic function of sweep\nrate (see Fig.2b). In the limit of weak hyperfine interac-\ntion,|vHF| ≪ |vSO|, expanding the exponential in Eq.(3),\nwe find\n|∝an}bracketle{t∆m∝an}bracketri}ht| ≈(vHF/vSO)2λe−λ, λ= 2πv2\nSO//planckover2pi1β.(10)\nThis expression attains its maximum value |∝an}bracketle{t∆m∝an}bracketri}ht|max=\nv2\nHF/(ev2\nSO) at the optimal sweep rate β∗= 2πv2\nSO//planckover2pi1.\nNext we consider a generalization to multiple sweeps\nand show that dynamical polarization is sensitive to nu-\nclear Larmor precession between sweeps. We focus on a\nsequenceoftwoidenticalgatesweeps,whereeachindivid-\nualsweepisshortonthescaleofthenuclearLarmortime,\nbut with a waiting time ∆ tbetween sweeps long enough\nto allow nuclear spin precession. Individual sweeps may\npass through the S−Tcrossing one or more times. As-\nsuming that only one nuclear species contributes to the\nhyperfine field, the change in nuclear polarization due to\nthe two sweeps is given by\n∝an}bracketle{t∆m∝an}bracketri}ht=/contintegraldisplaydθ\n2πi∝an}bracketle{tU−1∂θU∝an}bracketri}ht,U=U(θ+∆θ)WU(θ),(11)\nwhereU(θ) is the evolution matrix for a single sweep,\ngiven by (8), Wis the electronic evolution operator for\nthe waiting interval ∆ t, the expectation value is taken\nover the state |ψ(0)\nθ∝an}bracketri}ht= (0,1)T, and ∆θ=ωL∆tis the\nprecession angle, with ωLthe nuclear Larmor frequency.\nDuetothe periodicityof U(θ) inθ, Eq.(8), ∝an}bracketle{t∆m∝an}bracketri}htexhibits\nperiodic oscillations in ∆ θwhich are a direct manifesta-\ntion of coherent nuclear polarization dynamics.\nThe analysis is simple when the detuning |∆|is very\nlarge in the interval between sweeps. Then W≈\nexp(iχσ3), where the phase χis large and very sensi-\ntive to any fluctuations in the waiting time ∆ tor other4\nparameters of the system. The resulting randomness\nofχcompletely suppresses coherence in the electronic\nstate between sweeps, while nuclei remain coherent. For\ndemonstration, below we choose the particular protocol\n∆(t) =β(tn−t), for|t−tn|<τ, and ∆ = ∞otherwise,\nwheren= 1,2, andτ≪∆t=t2−t1witht1andt2the\ntimes of passage through the avoided crossing.\nFor this experimentally relevant case, it is helpful to\nthink of the matrix-products defining UandU−1in\nEq.(11) as sums over amplitudes associated with all pos-\nsible histories of S−Ttransitions. Fast electron decoher-\nencebetweensweepssuppressesthe contributionofterms\narisingfrom non-identicalhistories in UandU−1, leaving\nbehind a sum of classical probabilities of all trajectories.\nStarting with the singlet initial state and summing over\nall final states, we find\n∝an}bracketle{t∆m∝an}bracketri}ht=/bracketleftbig\n−(1,1)∂η/contintegraltextdθ\n2πM(θ+∆θ)M(θ)(0,1)T/bracketrightbig\nη=0\nM(θ)≡/parenleftBigg\n1−p(θ)p(θ)eηdϕ\ndθ\np(θ)e−ηdϕ\ndθ1−p(θ)/parenrightBigg\n(12)\nwhereηis an auxiliary variable introduced for book-\nkeeping, and p(θ) =|bθ|2is the transition probability\nin theθchannel for a single sweep. Here the phase\nϕ(θ) is the sum of the geometric part ϕ0(θ) = arg[vθ],\nthe Stokes phase ϕs(θ), see e.g. Ref.[20], and the dy-\nnamical phase ϕad(θ) =1\n/planckover2pi1/integraltextτ\n−τE(βt,θ)dtassociated with\nadiabatic evolution on a single branch of the two-level\nspectrum of Hamiltonian HST+, Eq.(2), with E(∆,θ) =\n−/radicalbig\n(∆/2)2+|vθ|2. Equation (12) then gives:\n∝an}bracketle{t∆m∝an}bracketri}ht=−/contintegraldisplaydθ\n2π/braceleftbiggdϕ\ndθp(θ)+dϕ′\ndθ[1−2p(θ)]p′(θ)/bracerightbigg\n,(13)\nwherep′(θ) =p(θ+∆θ),ϕ′(θ) =ϕ(θ+∆θ).\nThe two terms count the contributions from the two\nsweeps. The factor 1 −2p(θ) accounts for the fact that\na transition from StoTon the first sweep reverses the\ndirection of nuclear spin flips in the second sweep. Os-\ncillations and sign reversal of ∝an}bracketle{t∆m∝an}bracketri}htas a function of the\nprecessionangle ∆ θareclearly visible in Fig. 3, where we\nhave used direct numerical integration to find the transi-\ntion amplitudes b(θ) needed to evaluate expression (13).\nThe dynamical phase contributes to ∝an}bracketle{t∆m∝an}bracketri}htthrough\ndϕad\ndθ=−/integraldisplayτ\n−τdt\n/planckover2pi1∂E(βt,θ)\n∂θ=4vSOvHF\n/planckover2pi1βsinθsinh−1|∆0|\n2|vθ|,\n(14)\nwheredϕad/dθgrows logarithmically with the range of\ndetunings 2∆ 0spanned by the sweep. Note that ∂E/∂θ\nis the group velocity in the Bloch band corresponding to\nthe 1D quantum walk, Fig. 2a.\nThis contributiondescribesDNP buildup at times long\nafter the level crossing. The in-plane component of elec-\ntron spin maintains a nonzero average value due to the\npresence of the spin-orbit interaction. While this rem-\nnant electron spin polarization can be small, its influenceFIG. 3: (Color online) Polarization due to a round trip\nthrough the S−T+crossing with vHF= 0.6,vSO= 0.4,∆0=\n50, and precession ∆ θ=ωL∆tbetween forward and return\nsweeps, Eq.(13). Here ωLis the nuclear Larmor frequency\nand ∆tis the waiting time. Dashed lines show /angbracketleft∆m/angbracketrightcalcu-\nlated using ϕ0(θ) and the dynamical phase, Eq.(14), neglect-\ningStokesphase. The oscillation phase shifts by π/2for faster\nsweeps wheredϕad\ndθis small (see inset). Units are chosen with\n/planckover2pi1= 1.\non nuclear polarization accumulates over a long time, as\nindicated by the 1 /βdependence. As a result, this con-\ntribution to DNP typically dominates over that of the\ngeometric phase ϕ0(θ), see Fig.3.\nEquation (11) can be readily extended to arbitrary se-\nries of non-identical sweeps. If we take into account clas-\nsical noise in the detuning ∆( t) during each sweep, the\nprimary effect is to flatten the function p(θ), causing it\nto saturate toward the value 1/2 for all θ. Because the\nqualitativeformof p(θ)ispreserved,however,oscillations\nresulting from the geometric phase contribution dϕ0/dθ\nsurvive under quite general circumstances.\nResonant effects at the nuclear Larmor frequency also\ncan be seen directly in the time dependence of electron\ntransport properties. As the transverse polarization pre-\ncesses in time, the matrix element vθtraces out the\ndashed circle shown in Fig.1c, leading to a modulation of\n|vθ|at the nuclear Larmor frequency. This effect can be\ndetected as an ac modulation of conductance (current)\nin the system, or as a correlation between the electron\nspin flip probabilities on successive sweeps through the\navoided crossing. Such measurements would constitute a\npurely electrical detection of nuclear spin dynamics.\nIn summary, we have shown that the coherent evolu-\ntion of nuclear spins in quantum dots can be observed\nthrough oscillations and sign reversal of the nuclear spin\npumping rate, which occur due to the coherent inter-\nplay between hyperfine and spin-orbit couplings. Recent\nobservations indicate that this interesting regime is now\nwithin experimental reach, opening new possibilities to\nexplore many-body spin dynamics in the solid state.\nWe thank H. Bluhm, H.-A. Engel, S. Foletti, M. B.\nHastings, J. J. Krich, E. I. Rashba, and A. Yacoby for\nhelpful discussions, and acknowledge support from W.\nM. Keck Foundation Center for Extreme Quantum In-\nformation Theory, and NSF grants DMR-0906475 and\nDMR-0757145.5\n[1] R. Hanson et al., Rev. Mod. Phys. 79, 1217 (2007).\n[2] A. V. Khaetskii, D. Loss, and L. Glazman, Phys. Rev.\nLett.88, 186802 (2002); Phys. Rev. B 67, 195329 (2003).\n[3] J. M. Taylor, C. M. Marcus, and M. D. Lukin, Phys. Rev.\nLett.90, 206803 (2003).\n[4] W. A. Coish and D. Loss, Phys. Rev. B 70, 195340 (2004)\n[5] S. I. Erlingsson and Y. V. Nazarov, Phys. Rev. B 70,\n205327 (2004).\n[6] W. M. Witzel, R. de Sousa, S. Das Sarma, Phys. Rev. B\n72, 161306(R) (2005).\n[7] W. Yao, R-B. Liu, L. J. Sham, Phys. Rev. B 74, 195301\n(2006).\n[8] G. Chen, D. L. Bergman, L. Balents, Phys. Rev. B 76,\n045312 (2007).\n[9] K. Ono et al.Science297, 1313 (2002).[10] J. R. Petta et al.Science309, 2180 (2005).\n[11] F. H. L. Koppens et al.Science309, 1346 (2005).\n[12] K. C. Nowack et al.Science 318, 1430 (2007).\n[13] A. Pfund, I. Shorubalko, K. Ensslin, R. Leturcq, Phys.\nRev. B79, 121306(R) (2009).\n[14] S. Foletti, et al.arXiv:0801.3613.\n[15] O. N. Jouravlev, Y. V. Nazarov, Phys. Rev. Lett. 96,\n176804 (2006).\n[16] J. R. Petta et al.Phys. Rev. Lett. 100, 067601 (2008).\n[17] M. Pioro-Ladri` ere et al.Nat. Phys. 4, 776 (2008).\n[18] Y. Meir, Y. Gefen, O. Entin-Wohlman, Phys. Rev. Lett.\n63, 798 (1989)\n[19] M. S. Rudner and L. S. Levitov, Phys. Rev. Lett. 102,\n065703 (2009).\n[20] Y. Kayanuma, Phys. Rev. A 55, R2495 (1997)."    },    {        "title": "0707.1131v2.Universal_dynamics_of_quantum_spin_decoherence_in_a_spin_bath.pdf",        "content": "arXiv:0707.1131v2  [cond-mat.mes-hall]  27 Jun 2008Universal dynamics of quantum spin decoherence in a nuclear spin bath\nYi-Ya Tian, Pochung Chen and Daw-Wei Wang\nPhysics Department, National Tsing-Hua University, Hsinc hu, Taiwan, ROC\n(Dated: August 5, 2021)\nWe systematically investigate the universal spin decohere nce dynamics of a localized electron\nin an arbitrary nuclear spin bath, which can be even far away f rom equilibrium due to the weak\nnuclear-lattice interaction. We show that the electron spi n relaxation dynamics (as well as spin pure\ndephasing and Hahn echo decay) can alwayshave a universal behavior as long as the initial state is\ncomposed of a sufficiently large amount of spin eigenstates. F or a given system, the pattern of the\nuniversal dynamics depends on the complicated initial cond ition only via a singleparameter, which\nmeasures the amount of phase coherence between different spi n eigenstates in the initial state. Our\nresults apply even when the number of the involved nuclei is n ot large, and therefore provide a solid\nfoundation in the comparison of theoretical/numerical res ults with the experimental measurement.\nAs an example, we also show numerical results for systems of n oninteracting spin bath in zero\nmagnetic field regime, and discuss the features of universal decoherent dynamics.\nI. INTRODUCTION\nLocalized spins in solid-state systems are one of the\nmost promising candidates for realizing the quantum\ncomputation due to its long coherence time [1] and the\npossible scalability [2]. Recently, quantum control of sin-\nglelocalizedspinbecomesexperimentallyfeasible[3], but\nthe nuclear spin bath induced decoherence, which is the\ndominant decoherent mechanism in the low temperature\nregime, is still hindering the further developments. To\nstudy the effects of nuclear spin bath, both analytical ap-\nproaches[4,5,6]and numericalsimulations[7,8]aredevel-\noped fordifferent parameterregime. Theeffect ofdipolar\ninteractionbetweennuclearspinsarealsostudiedinRefs.\n[9]. However, to the best of our knowledge, an uncon-\ntested conclusion about the spin decoherence dynamics\nand its relation to the experimental measurement is still\nunavailable, eventhoughthe deleteriouseffectsofnuclear\nspin have been verified in recent experiments [10,11].\nFrom experimental side, the most crucial limitation\nresults from the fact that the initial nuclear spin con-\nfiguration is very little known nor controllable. This is\na highly nontrivial problem, because even if a thermal-\nized spin bath is assumed in the beginning (as done in\nmost theoretical work [5,6,9]), any quantum measure-\nment or manipulation of electron spin can just destroy\nthe equilibrium and lead to a highly non-equilibrium nu-\nclear spin dynamics. The coherent time of nuclear spin\nbath is known to be extremely long (can exceed 1s in\nGaAs quantum well [12] and 25s in29Si isotope [13]) and\ntherefore it is very questionable if the nuclear spin bath\ncould be well-thermalizedfor the next quantum measure-\nment/manipulation in a short time during the quantum\ncomputation process. In orderto haveameaningful com-\nparison between theoretical results and the experimental\nmeasurement, the first and the most important question\none should ask is if there could be any universal dynam-\nics in such a system, which is insensitive to the details of\ninitial nuclear spin configuration.\nFrom theoretical side, answering above question is also\nvery difficult because the spin dynamics of one configura-tion can be very different from the other [7] even though\ntheir initial configurations are similar. Moreover, in a\ntypical quantum dot system, the number of nuclei can be\nvery huge ( N∼103−5), and hence it is also a significant\nchallenge for ordinary numerical simulation to explore\nsuch huge phase space. These challenges are fundamen-\ntally important to the understandingof the spin decoher-\nence mechanism and to its future application in quantum\ncomputation. However, to the best of our knowledge,\nthere has no systematically study in the literature to this\nimportant issue.\nIn this paper we address this issue by rigorously prove\nthe existence of a generic and universal electron spin de-\ncoherent dynamics in an arbitrary nuclear spin bath. By\n“universaldynamics” we mean an electron spin evolution\nwhich is of zero standard deviation over different initial\ncondition in the whole phase space. More precisely, we\nshow that (1) the universality of spin decoherence always\nexists if only the initial state is composed of sufficient\nlarge amount of spin eigenstates, and (2) for a given sys-\ntem, such universal dynamics depends on the initial con-\nfiguration only through a singleparameter, which mea-\nsures the amount of phase coherence between spin eigen-\nstates of the initial wavefunction. (3) The universality\nis ensured by the large amount of phase space rather\nthan the large value of nuclear number, N, and there-\nfore numerical simulation for a small size system (say\nN∼10−20) can be still good enough to compare with a\nrealistic system of much more nuclei [14]. Finally, (4) the\nuniversality of spin dynamics applies to the decoherence\nof the diagonal part ( Sz) as well as the off-diagonal part\n(Sx) of electron spin, no matter in a free induction decay\n(FID) or in a Hahn echo decay. Therefore our results\nresolves the fundamental problems in the comparison of\na theoretical calculation and an experimental measure-\nment, and provide a new direction for the future study\nofthe spindecoherence. We alsostudy thespin dynamics\nfor systems of different electron/nuclear spins, and find\nthat the spin dynamics is mainly determined by the ge-\nometric structure of the system density of states and is\ntherefore insensitive to the magnitude of nuclear spin.2\nThis paper is organized as fellows: In section II we de-\nscribe the system Hamiltonian and the initial wavefunc-\ntions in our study. In section III we show the universal\ndynamics of electron spin relaxation. We study the uni-\nversal dynamics by using both numerical and analytical\nmethods. In section IV we discuss the microscopic origin\nof the universality. In section V, we generalize our con-\nsideration to other spin systems. We conclude in section\nVI.\nII. SPIN EIGENSTATES AND PHASE SPACE\nA general spin decoherence due to nuclear spin bath is\ndescribed by the following Hamiltonian:\nˆH=ˆS·N/summationdisplay\niAiˆIi+ˆHn−n+ˆHZ, (1)\nwhereˆSandˆIiarerespectivelythedimensionlessspinop-\nerators (/planckover2pi1≡1) of the localized electron and the nucleus\nat lattice site i.Aiis the hyperfine coupling strength, de-\npending on the wavefunction profile of the localized elec-\ntron (Fig. 1(a)), and we use Ai=A0e−(3i/N)2for our\nnumerical calculation with Nbeing the number of nu-\nclei.We note that changing Nwill not change the\nshape of the electron wavefunction, and therefore\nincreasing number of nuclear just reduce the stan-\ndard deviation of the spin dynamics instead of its\naverage value. ˆHn−nandˆHZarerespectivelytheinter-\naction between nuclear spin and the Zeeman term due to\nexternal magnetic field. Even in the simplest case, where\nbothˆHZandˆHn−nare zero or neglected, the result-\ning dynamics due to the electron-nuclearcoupling only is\nstill quite complex, because it involves a huge amount of\neigenstatesin the Hilbert space. In orderto haveamean-\ningful comparison between theoretical(numerical) results\nand the experimental observation, the first question one\nshould ask is if there could be any universal dynamics in\nsuch spin system, which is insensitive to a general initial\ncondition of the system and therefore can be observed\nand repeatable in a realistic experiment. After all, it is\nvery difficult to control and/or manipulate the spin con-\nfiguration of nuclei in solid state systems. This question,\nto the best of our knowledge, has not been answered or\neven not addressed yet in the literature.\nIt is convenient to use spin eigenstate, |S∝an}bracketri}hte⊗ |j∝an}bracketri}htn≡\n|S∝an}bracketri}hte⊗ |{I1,z,I2,z,···,IN,z}j∝an}bracketri}htn, as the basis of calcula-\ntion, where |S∝an}bracketri}hteis electron spin eigenstate along certain\ndirection (will be specified below) and ˆIi,zis the nuclear\nspin eigenvalue along the direction of magnetic field (ˆ z)\nat theith site. For simplicity, in this paper, we assume\nthe electron spin is initially polarized only along zorx\naxis, and therefore a general initial wavefunction can be\nwritten to be: |ψ0∝an}bracketri}htx,z=|Sx,z∝an}bracketri}hte⊗/summationtextMΩ\nj=1aj|j∝an}bracketri}htn, where\naj=rjeiϕjis the coefficient of the jth spin eigenstate\nwith phase ϕjand amplitude rj. Here |Sz∝an}bracketri}hte≡ |+∝an}bracketri}hte(b) (c)(a)φ\nFIG. 1: (a) Schematic pictures for electron spin (white cir-\ncle) coupled to a nuclear spin bath (black dots). The elec-\ntron is assumed to be described by the orbital envelope wave\nfunction φ, and interacts with the nuclear spins (located at\n/vector ri) via a hyperfine interaction Ai=A0|φ(/vector ri)|2whereA0is\nthe coupling strength. (b) and (c) are two spin eigenstates\nwith maximum/minimum energies in zero magnetic field and\nJz= 0 case.\nand|Sx∝an}bracketri}hte≡1√\n2(|+∝an}bracketri}hte+|−∝an}bracketri}hte), andMΩis the size of the\nHilbert space of nuclear bath. Using above expression,\nwe consider the following three spin dynamics, which\nare related to the spin relaxation, spin pure dephasing,\nand Hahn echo decay respectively. The first two can\nbe expressed as ∝an}bracketle{tSz,x(t)∝an}bracketri}ht ≡ z,x∝an}bracketle{tψ0|ˆSz,x(t)|ψ0∝an}bracketri}htz,x=/summationtextMΩ\nj=1r2\njSz,x\nj,j(t) +/summationtextMΩ\nj/negationslash=lrjrle−i(ϕj−ϕl)Sz,x\nj,l(t), where\nSz,x\nj,l(t)≡n∝an}bracketle{tj|⊗e∝an}bracketle{tSz,x|ˆSz,x(t)|Sz,x∝an}bracketri}hte⊗|l∝an}bracketri}htnis the matrix\nelement. ∝an}bracketle{tSz(t)∝an}bracketri}htcan be very different from ∝an}bracketle{tSx(t)∝an}bracketri}htif\nthe nuclear spin is polarized by external magnetic field\nor with finite total angular momentum in a certain\ndirection. Similarly the Hahn echo decay is given by\nρH\n+−(τ)≡e∝an}bracketle{t+|ˆρH(τ)|−∝an}bracketri}hte, where the Hahn echo den-\nsity matrix ˆ ρH(τ)≡Trn{U(τ)|ψ0∝an}bracketri}htxx∝an}bracketle{tψ0|U(τ)†}=/summationtext\njn∝an}bracketle{tj|U(τ)|ψ0∝an}bracketri}htxx∝an}bracketle{tψ0|U(τ)†|j∝an}bracketri}htnandU(τ)≡\ne−iHτσxe−iHτ[9,15]. The characteristic time scale\nT2of pure dephasing is related to the single spin FID,\nwhile Hahn echo decay[15] is usually used to extract\nsingle spin behavior from an ensemble measurement.\nIII. UNIVERSAL DYNAMICS\nIn this section we show that the universality of spin\ndecoherence always exists if only the initial state is com-\nposed of sufficient large amount of spin eigenstates, and\nfor a given system, such universal dynamics depends on\nthe initial configuration only through a single parame-\nter, which measures the amount of phase coherence be-\ntween spin eigenstates of the initial wavefunction. We\nfirst show the numerical results for spin relaxation, spin\npure dephasing, and Hahn echo decay respectively. Then\nwe rigorously give the proof of the universality.\nA. Numerical study\nIn order to explore the spin dynamics from different\ninitial conditions in the whole phase space, in this pa-\nper we allow both the amplitude, {rj}, and the phase,3\n{ϕj}, to be independent variables and randomly chosen\naccording to distribution functions Pr(rj) andPϕ(ϕj)\nrespectively. The ensemble-averaged spin dynamics for\n∝an}bracketle{tSz(t)∝an}bracketri}htbecomes [ ∝an}bracketle{tSz(t)∝an}bracketri}ht]≡[/angbracketleftSz(t)/angbracketright]r,ϕ\n[/angbracketleftSz(0)/angbracketright]r,ϕ, where [f(r)]r≡\n/integraltext1\n0Pr(r)f(r)drdenotes the average of a function f(r),\nandsimilarly[ f(ϕ)]ϕ≡/integraltext2π\n0Pϕ(ϕ)f(ϕ)dϕ. [∝an}bracketle{tSz(0)∝an}bracketri}ht]r,ϕin\nthe denominator is for normalization. At the same time,\nthe associated normalized standard deviation (NSD) is\ndefined as follows:\nσ(t)≡/radicalBigg\n[∝an}bracketle{tˆSz(t)∝an}bracketri}ht2]r,ϕ−[∝an}bracketle{tˆSz(t)∝an}bracketri}ht]2r,ϕ\n[∝an}bracketle{tSz(0)∝an}bracketri}ht]2r,ϕ.(2)\nSimilar definition of averaged dynamics as well as the\nNSD for ∝an}bracketle{tSx(t)∝an}bracketri}htandρH\n+−(t) can be obtained easily. We\nnote that, if the NSD of the averagedspin dynamics goes\nto zero in the limit of infinite phase space, the averaged\ndynamics is also “the most probable” dynamics with al-\nmost zero probability in the other time-evolution behav-\nior. As a result, we can define it as a universal dynam-\nics of the given system, independent (in the probability\nsense) of the details of initial nuclear spin configuration.\nOn the other hand, the system has no universal dynam-\nics if the NSD is of the order of one, since the average\nvalue could not represent the characteristic dynamics of\na general initial condition.\nBefore analytically studying the universality of spin\ndynamics in a general system, it is more instructive\nto show some numerical results of the simplest sys-\ntem without magnetic field and nuclear spin interaction\n(ˆHZ=ˆHn−n= 0). We will first present the result\nfor∝an}bracketle{tSz(t)∝an}bracketri}htthen the results for ∝an}bracketle{tSx(t)∝an}bracketri}htandρH\n+−(t). For\nthe convenience of later discussion, we restrict the cal-\nculation inside a subspace, Γ, where the total angular\nmomentum, Jz=Sz+/summationtextN\ni=1Ii,z, is zero, and choose\nPr(r) =γ+(1−γ)δ(r) andPϕ(ϕ) = (1−ξ)/2π+ξδ(ϕ).\nHereγ∈[0,1] can be understood as the probability to\nhave a nonzero contribution in the subspace, Γ, while\nξ∈[0,1] is the probability to have a phase coherence at\na given value (set to be zero). They satisfys the normal-\nization condition:/integraltext1\n0Pr(r)dr=/integraltext2π\n0Pϕ(ϕ)dϕ= 1 for all\nξandγ.\nIn Fig. 2(a)-(c) we show the averaged electron spin\n(S=1\n2) relaxation [ ∝an}bracketle{tSz(t)∝an}bracketri}ht], in a noninteracting spin\nbath (I=1\n2) with zero magnetic field for ξ= 0, 0.5, and\n1 with two different values of γ. We observe that when\nγis small (γ= 0.016), meaning only a few spin eigen-\nstates are involved in |ψ0∝an}bracketri}htz, the NSDs are very large,\ni.e., no universal dynamics. This explains why in the\nliterature different initial states can result in very differ-\nent time-evolution patterns [7]. When γbecomes larger\n(γ= 0.32), the NSD decreases in all figures (see also\nFig. 2(d)), while the averaged dynamics also begin to\nshow different behavior for different value of ξ: A two-\ndecay curves for ξ→0 and a single mode oscillation for\nξ→1. In fact, as we will show later, the NSD always\ndecrease to zero even γis finite, as long as the size of/s48 /s50/s48 /s52/s48 /s54/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s48 /s50/s48 /s52/s48 /s54/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s48 /s50/s48 /s52/s48 /s54/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s48/s46/s48/s48/s48/s46/s48/s53/s48/s46/s49/s48/s48/s46/s49/s53/s48/s46/s50/s48/s40/s100/s41 /s40/s99/s41/s40/s98/s41/s40/s97/s41\n/s32/s32/s91/s60/s83\n/s122/s40/s116/s41/s62/s93\n/s116/s32/s91/s49/s47/s65 /s48/s93\n/s32/s32/s91/s60/s83\n/s122/s40/s116/s41/s62/s93\n/s116/s32/s91/s49/s47/s65 /s48/s93/s32/s32/s91/s60/s83\n/s122/s40/s116/s41/s62/s93\n/s116/s32/s91/s49/s47/s65 /s48/s93\n/s32/s32\nFIG. 2: (Color online) (a)-(c) Averaged electron spin re-\nlaxation (symboled lines) for ξ=0, 0.5, and 1. respectively.\nDashed and dotted lines are the uncertain range ([ /angbracketleftS(t)/angbracketright]±\nσ(t)) forγ= 0.016 and 0.32 respectively. (d) Time averaged\nNSD (σ≡limT→∞RT\n0σ(t)dt/T)v.s.γ.N= 11 in all figures.\nphase space becomes large enough. These results indi-\ncate that a universal dynamics can always be expected if\nthe initial state is composed of asufficiently largeportion\nof spin eigenstates in the phase space, simply due to the\nstrong quantum interference effects. We could also show\nthat the two decay time scale of Fig. 2(a) is due the the\nstructure of the system density of states, and wil discuss\nthat in more details in the latter section.\nIn Fig. 3(a) and (b) we respectively plot pure dephas-\ning(∝an}bracketle{tSx(t)∝an}bracketri}ht) andHahn echodecay( ρH\n+−(τ)) asafunction\nof time for fully coherent ( ξ= 1, filled circle) and fully\nincoherent ( ξ= 0, open circle) choice of initial condi-\ntion. To simplify the numerical calculation, we choose\nan initial wavefunction in a subspace of/summationtextN\ni=1II,z= 0\nfor the pure dephasing ( ∝an}bracketle{tSx(t)∝an}bracketri}ht, (a) ), and Jz= 0 (same\nas∝an}bracketle{tSz(t)∝an}bracketri}ht) for the Hahn echo decay ( ρH\n+−(τ), (b)). The\nelectron for the former case is initially polarized in the x\ndirection so that the average total angular momentum in\nzdirection, ∝an}bracketle{tSz(t)∝an}bracketri}ht, is still zero. As a result, the dynam-\nics of dephasing, ∝an}bracketle{tSx(t)∝an}bracketri}ht, is different from the relaxation,\n∝an}bracketle{tSz(t)∝an}bracketri}htdue to the different choice of subspace where the\ninitial wavefunction is defined. We believe such convec-\ntion is justified and will not affect any of our conclusion,\nbecause here we just use this numerical results as an ex-\nample to understand the general properties of the uni-\nversal dynamics. Full numerical results for any realistic\nsituation will need a much larger phase space and much\nlonger time. Within this subspace, different initial wave-\nfunctionsstill resultindifferent dynamics( notshowhere\n). However, when the initial wavefunction is composed of\nsufficient large amount of eigenstates in the subspace, we\nagain find a universal dynamics with almost zero NSD.\nIn Fig 3(a) and (b), we show results for γ= 1 for pure\ndephasing and Hahn echo decay in the two subspace de-\nscribed above. In (c), the NSD of the Hahn echo decay4\n/s48 /s50/s48 /s52/s48 /s54/s48/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s48/s46/s48/s48/s48/s46/s48/s50/s48/s46/s48/s52/s48/s46/s48/s54/s48/s46/s48/s56/s48/s46/s49/s48/s48/s46/s49/s50/s48 /s50/s48 /s52/s48 /s54/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48/s49/s46/s50\n/s48 /s50/s48 /s52/s48 /s54/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48/s49/s46/s50\n/s40/s98/s41 /s40/s97/s41\n/s40/s99/s41/s40/s100/s41\n/s32/s32/s83/s122 /s100\n/s40/s116/s41/s44/s32/s83/s122 /s111/s40/s116/s41\n/s116/s32/s91/s49/s47/s65 /s48/s93/s32/s32\n/s32/s32/s91/s60/s83\n/s120/s40/s116/s41/s62/s93\n/s116/s32/s91/s49/s47/s65 /s48/s93/s32 /s61/s49\n/s32 /s61/s48/s32 /s61/s49\n/s32 /s61/s48\n/s32/s32/s45/s43/s40 /s41\n/s32/s91/s49/s47/s65 /s48/s93\nFIG. 3: (Color online) (a) [ /angbracketleftSx(t)/angbracketright] and (b) ρH\n+−(τ) as a func-\ntion of time tandτrespectively (see text). Lines with filled\nsymbols and open symbols are for ξ= 1 and ξ= 0. (c) The\nNSD at τ= 10 for Hahn echo decay. (d) Sz\nd(t) (open circle)\nandSz\no(t) (filled circle) for the spin relaxation dynamics in\nEq. (3).\nis plotted as function of γatτ= 10. From these results,\nwe find similar single mode oscillation for ξ= 1, while\na two-decay curves for ξ= 0 in all the three dynamics\n(∝an}bracketle{tSz(t)∝an}bracketri}ht,∝an}bracketle{tSx(t)∝an}bracketri}htandρH\n+−(τ)).\nB. Analytical study\nTo analytically study the universal spin dynamics, we\nhave to do the ensemble-average first so that\n[∝an}bracketle{tSz(t)∝an}bracketri}ht] =Sd(t)+[r]2\nr\n[r2]r/vextendsingle/vextendsingle[e−iϕ]ϕ/vextendsingle/vextendsingle2So(t) (3)\nwhere we have used [ ∝an}bracketle{tSz(0)∝an}bracketri}ht]r,ϕ= [r2]r/summationtext\nj∝an}bracketle{tj|ˆSz|j∝an}bracketri}ht=\n[r2]rMΓin the normalization; Sd(t)≡M−1\nΓ/summationtext\njSj,j(t)\nandSo(t)≡M−1\nΓ/summationtext\nj1/negationslash=j2Sj1,j2(t) are the diagonal and\noff-diagonal maxtrix element of electron spin. We note\nthat Eq. (3) indicates that the averaged spin dynamics\ndepends on the initial condition only via a singleparam-\neter,β≡/parenleftbig\n[r]2\nr/[r2]r/parenrightbig/vextendsingle/vextendsingle[e−iϕ]ϕ/vextendsingle/vextendsingle2, which depends on the\nphase distribution function, Pϕ, much more significantly\nthan on the amplitude distribution function, Pr, since\n[r]2\nr∼[r2]rfor the usual function of Prandr≥0. We\nnote that although the experimental preparation of a co-\nherent nuclear spin bath (i.e. finite value of β) is not\neasy at the present stage, it has been realized how to\ncontrol the coherent electron spin dynamics via interac-\ntion with a singlenuclear spin in diamond [11]. There-\nfore, at least in a small quantum dot system, a coherent\npreparation and control of a few nuclear spin can be still\nrealized. In Fig. 3(d) we show the time-evolution of both\nSd(t) andSo(t) of spin matrix element. Not surprisingly,\nthey are of very different properties: the diagonal part(Sd(t)) shows a clear two-decay process: with a fast de-\ncay in short time and a slow decay in long time. How-\never, the off-diagonal part ( So(t)) does not decay at all,\nand shows a single mode oscillation. It is easy to see\nthat the numerical results shown in Fig. 2(a)-(c) can be\nobtained as a superposition of Sd(t) andSo(t), just as\nsuggested by Eq. (3). Numerical comparison between\nthese two approaches (ensemble average before and after\ntime-revolution) agree excellently well (not shown here),\nshowing that only a single parameter, β, is necessary to\nreproduce all the ensemble averaged spin relaxation dy-\nnamics.\nIn order to exam if the ensemble-averaged results of\nEq. (3) is a universal dynamics, we need to calcu-\nlate the fluctuation (NSD, Eq. (2)) about this aver-\nage. For simplicity, we first study the case in the com-\npletely random phase limit, say [ eiϕ]ϕ= 0. We then have\n[∝an}bracketle{tˆSz(t)∝an}bracketri}ht]2\nr,ϕ=M2\nΓ[r2]2Sd(t)2according to Eq. (3). After\nsome algebra we can derive: [ ∝an}bracketle{tˆSz(t)∝an}bracketri}ht2]r,ϕ−[∝an}bracketle{tˆSz(t)∝an}bracketri}ht]2\nr,ϕ=\n([r4]r−2[r2]2\nr)/summationtext\njSj,j(t)2+[r2]2\nr/summationtext\nj,lSj,lSl,j. Since we\nare interested in the upper bound of the NSD, we\nmay use/summationtext\nj,lSj,l(t)Sl,j(t) =/summationtext\nj∝an}bracketle{tj|ˆSz(t)ˆPΓˆSz(t)|j∝an}bracketri}ht ≤/summationtext\nj∝an}bracketle{tj|ˆSz(t)2|j∝an}bracketri}ht ≤S(S+1)/summationtext\nj∝an}bracketle{tj|j∝an}bracketri}ht=S(S+1)MΓ, where\nˆPΓ≡/summationtext\nl|l∝an}bracketri}ht∝an}bracketle{tl|is to project a state onto the subspace Γ\nwith electron spin eigenvalue Sz=S. Here we have used\nthefactthatforanystate, the expectationvalueof ˆSz(t)2\nmust be equal or smaller the expectation value of total\nelectron spin S2, which is, however, a conserved quan-\ntity of our system (see Eq. (1)). Similarly we also have/summationtext\njSj,j(t)2≤/summationtext\ni,j|Si,j(t)|2≤S(S+1)MΓ, andtherefore\n[∝an}bracketle{tˆSz(t)∝an}bracketri}ht2]r,ϕ−[∝an}bracketle{tˆSz(t)∝an}bracketri}ht]2\nr,ϕ∼ O(MΓ). Inotherwords,after\ndevided by [ ∝an}bracketle{tSz(0)∝an}bracketri}ht]2\nr,ϕ∝M2\nΓ, we findσ(t)∝M−1/2\nΓand\ntherefore goes to zero in the limit of N≫1 orMΓ→ ∞.\nWe can also apply similar method to study the NSD\nof a phase coherent initial state, i.e. [ eiϕ]ϕ∝ne}ationslash= 0.\nAfter some algebra, the expansion of [ ∝an}bracketle{tˆSz(t)∝an}bracketri}ht2]r,ϕ−\n[∝an}bracketle{tˆSz(t)∝an}bracketri}ht]2\nr,ϕwill have two additional summations (be-\nsides of the two shown above) with nonuniver-\nsal prefactors: first, we have/summationtext\ni,j,lSi,l(t)Sl,j(t) =\n∝an}bracketle{tV|ˆSz(t)ˆPΓˆSz(t)|V∝an}bracketri}ht ≤ ∝an}bracketle{tV|ˆSz(t)ˆSz(t)|V∝an}bracketri}ht ≤S(S+\n1)∝an}bracketle{tV|V∝an}bracketri}ht=S(S+ 1)/summationtext\ni,j∝an}bracketle{ti|j∝an}bracketri}ht=S(S+1)MΓ, where we\ndefine|V∝an}bracketri}ht ≡/summationtext\nj|j∝an}bracketri}htas an auxiliary state. Secondly,\n/summationtext\nl,jSl,l(t)Sl,j(t)≤/radicalBig/summationtext\nlSl,l(t)2/summationtext\nl|/summationtext\njSl,j(t)|2≤\nM1/2\nΓ/radicalBig\n∝an}bracketle{tV|ˆSz(t)ˆPΓˆSz(t)|V∝an}bracketri}ht ≤M1/2\nΓ/radicalBig\n∝an}bracketle{tV|ˆSz(t)2|V∝an}bracketri}ht ≤/radicalbig\nS(S+1)MΓ, where we have used the fact that the\ninner product of two vectors must be equal or smaller\nthan the product of their length. Therefore, after renor-\nmalized by the initial spin average, we find σ(t)∝M−1\nΓ\nand becomes to zero in large system size just as for the\ncomplete random phase case ( ξ= 0). From the above\nresults, we conclude that no matter how much phase co-\nherence between spin eigenstates of the initial wavefunc-\ntion, the spin relaxation dynamics can be always univer-\nsal (with zero NSD) in the limit of infinitely large phase5\n/s48 /s50/s48 /s52/s48 /s54/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s48 /s50/s48 /s52/s48/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s40/s98/s41/s40/s97/s41\n/s32/s91/s60/s83\n/s122/s40/s116/s41/s62/s93/s32/s91/s60/s83\n/s122/s40/s116/s41/s62/s93\n/s32/s32\n/s116/s32/s91/s49/s47/s65\n/s48/s93\n/s32/s32\n/s116/s32/s91/s49/s47/s65\n/s48/s93\nFIG. 4: (a) Comparison between the calculated /angbracketleftSz(t)/angbracketright(open\ncircle) with ξ= 0 and γ= 1 and /angbracketleftSz(t)/angbracketrightDOS(filled circle)\nobtained from density of states only. (b) Comparison betwee n\nthe calculated /angbracketleftSz(t)/angbracketrightwithξ= 1 and γ= 1 and the result\nobtained by including |Emax/min/angbracketrightonly (see text).\nspace (MΓ≫1). Similar derivation for dynamics of pure\ndephasing ( ∝an}bracketle{tSx(t)∝an}bracketri}ht) and Hahn echo decay ( ρH\n+−(τ)) can\nbe obtained straightforwardly.\nIV. MICROSCOPIC ORIGIN OF\nUNIVERSALITY\nAfter concluding the universality of the most general\nspin relaxation system (Eq. (1)), in the rest of this paper\nwe return to a less general case, zero magnetic field and\nnoninteracting spin bath ( ˆHZ=ˆHn−n= 0), to study the\nmicroscopic origin of the universal spin relaxation curves\nshown in Fig. 2(a)-(c). We first rewrite ∝an}bracketle{tˆSz(t)∝an}bracketri}htin terms\nof energy eigenstates, |E∝an}bracketri}ht:\n∝an}bracketle{tˆSs(t)∝an}bracketri}ht=/integraldisplay\ndED(E)/integraldisplay\ndE′D(E′)CEC∗\nE′\n×∝an}bracketle{tE|ˆSz|E′∝an}bracketri}hte−i(E−E′)t, (4)\nwhereD(E) is the density of states (DOS) of the system\nandCE≡ ∝an}bracketle{tψ0|E∝an}bracketri}ht. According to our numerical cal-\nculation, we observe that the matrix element, ∝an}bracketle{tE|ˆSz|E′∝an}bracketri}ht,\nvariesalmostrandomlyfordifferentenergies. Since now\nwe are interested in the simplestpossibleexplana-\ntion for the features of electron spin dynamics, we\ncan first neglect such structureless random ma-\ntrix element for simplcity. As we will see later,\nit turns out that this simplication does bring a\nvery useful understanding of the universal spin\ndynamics.\nFor the case when initial wavefunction |ψ0∝an}bracketri}ht, is totally\nrandomly distributed in the phase space, Γ, i.e. γ= 1\nandξ= 0, we can further assume that ∝an}bracketle{tψ0|E∝an}bracketri}htis also\nindependent of energy Ein above equation. As a re-\nsult, Eq. (4) can be approximated by ∝an}bracketle{tSz(t)∝an}bracketri}htDOS≡/vextendsingle/vextendsingle/integraltext\ndED(E)e−iEt/vextendsingle/vextendsingle2, whichisjustpowerspectrumtheden-\nsity of state. In Fig. 4(a) we show the full numerical\nresult of ∝an}bracketle{tSz(t)∝an}bracketri}htforξ= 0 compared to ∝an}bracketle{tSz(t)∝an}bracketri}htDOSgiven\nabove. One can see that the later can qualitatively re-\nproduce all the important structure of the full numerical\nresults. This agreement helps us to conclude that the\ndecay time of ∝an}bracketle{tSz(t)∝an}bracketri}htis mainly determined by the width/s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48/s49/s46/s50/s49/s46/s52\n/s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48/s49/s46/s50\n/s48 /s50/s48 /s52/s48 /s54/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s48 /s50/s48 /s52/s48 /s54/s48/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s32/s83/s61/s49/s32/s73/s61/s49/s47/s50\n/s32/s83/s61/s49/s32/s73/s61/s49/s32/s83/s61/s49/s47/s50/s32/s73/s61/s49/s47/s50\n/s32/s83/s61/s49/s47/s50/s32/s73/s61/s49/s40/s98/s41\n/s40/s97/s41\n/s32/s32/s68/s40/s69/s41\n/s69/s32/s91/s65\n/s48/s93\n/s32/s32/s68/s40/s69/s41\n/s69/s32/s91/s65\n/s48/s93\n/s32/s83/s61/s49/s32/s73/s61/s49/s47/s50\n/s32/s83/s61/s49/s32/s73/s61/s49/s32/s83/s61/s49/s32/s73/s61/s49/s47/s50\n/s32/s83/s61/s49/s32/s73/s61/s49/s40/s100/s41/s32/s32/s91/s60/s83\n/s122/s40/s116/s41/s62/s93\n/s116/s32/s91/s49/s47/s65\n/s48/s93/s40/s99/s41\n/s32/s32/s91/s60/s83\n/s122/s40/s116/s41/s62/s93\n/s116/s32/s91/s49/s47/s65\n/s48/s93\nFIG. 5: (Color online)(a) and (b) are the density of states\nof systems with different electron/nuclear spins. Note that\nin both figures, the energy axis for I= 1 cases (filled circle)\nhave been rescaled by a factor 1 /2 in order to fit the same\nscale as I= 1/2 case. (c) and (d) are ensemble-averaged\nspin relaxation curves for S= 1 and γ= 1 case, with ξ= 0\nandξ= 1 respectively. For comparison, results of different\nnuclear spins are shown together after rescaling the time ax is\n(see above).\nof DOS peaks (see Fig. 5(a)), while the time scale of the\nsecond peak of ∝an}bracketle{tSz(t)∝an}bracketri}htis given by the energy separation\nbetween the two peaks in DOS.\nAs for the single mode oscillation shown in Fig. 2(c)\nfor full spin coherent initial state ( ξ= 1), we can ap-\nply similar study but notice that the coefficient, CE,\nis not a constant for all eigenstate energy any longer.\nOur numerical calculation shows that |CE|is very small\n(∝M−1/2\nΓ) for all energy except near E=Emax/min,\nwhere|Emax/min∝an}bracketri}htis the eigenstatesof the top(maximum)\nand bottom(minimum) of the eigenenergy band. This is\nbecause|Emax/min∝an}bracketri}hthas a large overlap with some partic-\nular spin eigenstate (as shown in Fig. 1(a) and (b)) and\nthe overlapping coefficients are not cancelled out due to\nthesamephaseinafullspincoherentinitialstate( ξ= 1).\nIn Fig. 4(b), we compare the numerical result of the uni-\nversal dynamics ( γ= 1) of a full coherent initial state\n(ξ= 1) and the result calculated by using |Emax/min∝an}bracketri}ht\nonly (with proper nomalization). We find the agreement\nisexcellent, predictingthesameoscillationfrequencyand\neven the same phase. The agreement justifies the\napproximations used in the derivations after Eq.\n(4) and also shows that the universal behavior of the\nspin relaxation dynamics can be simply explained by the\nstructure of density of states and the two special spin\nconfigurations as shown in Fig. 1(a) and (b). As for re-\nsults with 0 < ξ <1, it can be also explained well by a\nlinear combination of above two results, as suggested by\nEq. (3).6\nV. RESULTS FOR DIFFERENT SPINS\nAfter systematically investigating the spin relaxation\ndynamics for an spin-half electron inside a spin-half\nnucearspinbath, herewefurtherextendthe studyofuni-\nversality to systems of different electron/nuclear spins.\nIn Fig. 5(a) and (b), we show the density of states\nfor (S,I) = (1\n2,1\n2), (S,I) = (1\n2,1), (S,I) = (1,1\n2), and\n(S,I) = (1,1) in different curves. For the convenience of\ncomparison, we rescale the energy scale in each plot and\nnormalize the hieght of DOS by the total size of phase\nspace, Γ. Surpisingly we find that the DOS structure is\nalmost the same for different nuclear spins Ias long as\nthe electron spin Sis the same. This reflects the fact\nthat the total Hilbert space of the nuclear spin bath has\nbeen large enough due to the number of nucluei so that\nthe spin degrees of freedom does play very little role in\nthe structure of energy spectrum. Analysing the energy\neigenstate configuration, we find the spin configuration\nnear the degeneracy regime (position of the peaks) are\nrelated to if the the central nuclei spin configuration is\npolarized and parallel (or anti-parallel) to the electron\nspin. Similar observation also applys to the triple peak\nstructure in Fig. 5(b) for S= 1: In Fig. 5(c) and (d),\nwe show the spin relaxation curves for S= 1 with spin\nphase random ( ξ= 0) and spin phase coherent ( ξ= 1)\ninitial wavefunctions respectively, after properly rescal-\ning the horizontal axis. One can see that results in (c)\nare very similar to the spin half case (Fig. 2(a)), while\nit shows a beating oscillation for a coherent initial wave-\nfunction (d). The rescaled time-evolution for I=1\n2and\nI= 1 are very similar, except for a small phase twist.\nWe then conclude that the the spin relaxation dynam-\nics is insensitive to the nuclear spin degrees of freedom,\nconsistent with our earlier statement that the universal\nspin dynamics is independent of the nuclear spin config-\nuration. Our results for S=I= 1 can be also applied to\nthe study ofspin dynamicsin the mixtures ofspinful cold\natoms in all-optical trap, where the localized “electron”\nand the “nucluei” can be prepared easily by using optical\nlattice with properwavelengthdifference. The advantage\nfor cold atom system is that the initial spin configura-\ntion can be prepared easily and the coupling strength,\nA0, can be tuned via optical Feshbach resonance and/or\nother method.\nVI. CONCLUSIONS\nIn this paper we rigorouslyprovethat the electronspin\ndecoherence due to nuclear spin bath can be always uni-\nversal if only coupled by sufficient large amount of spin\neignestates. There are several features about the uni-\nversal dynamics that we want to emphasize: First, in\nthe derivation above, we do not rely on any particular\nform of the distribution function ( PrandPϕ), hence\nthe universality of spin dynamics is independent of the\nnuclear spin configuration. However, if the initial stateis composed of only finite numbers of spin eigenstates\n(as done in the literature), our derivation will fail since\n[r2]r→0 in the denominator of σ(t), i.e., no univer-\nsal dynamics can be expected. Secondly, the univer-\nsality does not rely on any particular Hamiltonian, so\nour conclusion also applies to systems which include nu-\nclear spin interaction, finite magnetic field, or any other\nmore complicated system. Different system Hamiltoni-\nans just bring different averaged results of spin dynam-\nics, but the huge phase space ( notnecessary the huge nu-\nclear number) can alwaysensure it to be the most prob-\nable one regardless of the details in the initial condition.\nSuch important results lead to another conclusion that\na numerical simulation of a much smaller system (say\nN∼10−20) can still have large enough phase space\n(MΩ= (2S+ 1)×(2I+ 1)N∼103∼6) and hence gives\nsimilar results as given by macroscopic number of nu-\nclei [14]. Excellent agreement between our results\nof small size calculation (Fig. 2(a) with N= 11)\nand a meanfield type calculation of a much larger\nsystem (for example, Fig. 4 of Ref. [8] with\nN= 2000) ensures the existence of such scale-\nindepedent universal dynsmics. Our results there-\nfore make a realistic comparison between a theoretical\ncalculation and experimental data possible, leaving only\na single unknown parameter, β, as a fitting parameter.\n(For example, in the Fig. 4 of Ref. [8], β= 0is\nexpected due to the thermalized initial bath.) Fi-\nnally, ourderivationreliesonthe factthatelectronspinis\na conserved quantity with an upper-bounded eigenvalue\n(not scaled with system size). This mayexplain why spin\neigenstate can be a special basis for studying universal\nphysics and restricts a naive application of our results to\nthe relaxation dynamics of other physical quantities.\nIt is also worthy to note that the universal dy-\nnamics may bear a close relationship with the\nquantum central limit theorem (QCLT). QCLT\nhas been used to study the quantum state esti-\nmation without using a large ensemble [16] and to\nexplain why quantum and classical random walks\npossess different behaviors [17]. It is nature to\nconjecture that the existence of the universal dy-\nnamics and the reason why a small size system\ncan already capture the behavior of the macro-\nscopic system can be understood in the context\nof QCLT. For example, in Ref [16], it was pointed\nout that a quantum state estimation with small\nerror using small size ensemble is possible. This is\nclearly resemble to our work, where a small size\nsystem can capture the universal dynamics of a\nmacroscopic system. However, such a connection\nis not at all transparent in the details of the ap-\nproach. Further study about such an interesting\nconnection is beyound the scope of this paper but\ncould be very interesting for furure investigation.\nWe appreciate fruitful discussion with S. Das sarma,\nX. Hu, S. K. Saikin, L. J. Sham, and J. M. Taylor. This\nwork is supported by NSC Taiwan.7\n1A.V.KhastskiiandY.V.Nazarov, Phys.Rev.B 61, 12639\n(2000).\n2D.LossandD.P.DiVincenzo, Phys.Rev.A57, 120(1998);\nB. E. Kane, Nature (London) 393, 133 (1998).\n3Atature, M. et al., Science 312, 551 (2006); Hanson, R.\net al, Phys. Rev. Lett. 94, 196802 (2005); Koppens et al.,\nNature,442, 766 (2006).\n4A.V.Khaetskii, D.Loss, andL. Glazman, Phys.Rev.Lett.\n88, 186802 (2002);\n5I.A. Merkulov, Al. L. Efros and M. Rosen, Phys. Rev. B\n65, 205309, (2002);\n6C. Deng and X. Hu, Phys. Rev. B 73, 241303(R) (2006).\n7J. Schliemann, A.V. Khaetskii, and D. Loss et al., Phys.\nRev. B66, 245303 (2002).\n8K. A. Al-Hassanieh et al., Phys. Rev. Lett. 97, 037204,\n(2006);\n9R. de Sousa and S. Das Sarma, Phys. Rev. B 68, 193207\n(2003); W. M. Witzel and Rogerio de Sousa and S. Das\nSarma, Phys. Rev. B 72161306 (2005); W. M. Witzel andS. Das Sarma, Phys. Rev B 74, 035322 (2006); W. Yao,\nR.-B. Liu, and L.J. Sham, Phys. Rev. B 74, 195301 (2006).\n10Petta,et al., Science 309, 2180 (2005); A. C. Johnson, et\nal., Nature 435, 925 (2005);\n11L. Childress, et al., Science, 314, 281 (2006).\n12A. Berg, et al, Phys. Rev. Lett. 64, 2563 (1990). J. M.\nTaylor, C. M. Marcus, and M. D. Lukin, Phys. Rev. Lett.\n90, 206803 (2003).\n13T. D. Ladd, et al, Phys. Rev. B. 71, 014401 (2005).\n14We find similar observation has also been mentioned in V.\nV. Dobrovitski and J. M. Taylor and M. D. Lukin, Phys.\nRev. B73, 245318 (2006). But they did not give any phys-\nical understanding or explanation for such result.\n15E. L. Hahn, Phys. Rev. 80, 580 (1950).\n16M. Hayashi, quant-ph/060819.\n17C. D. Cushen and R. L. Hudson, J. Appl. Prob. 8, 454\n(1971); G. Grimmett, Phys. Rev. E . 69, 026119 (2004);\nN. Konno, ibid.72, 0626113 (2005);"    },    {        "title": "2402.19141v1.Inertial_spin_waves_in_spin_spirals.pdf",        "content": "Inertial spin waves in spin spirals\nMikhail Cherkasskii,1,∗Ritwik Mondal,2and Levente Rózsa3, 4\n1Institute for Theoretical Solid State Physics, RWTH Aachen University, DE-52074 Aachen, Germany\n2Department of Physics, Indian Institute of Technology (ISM) Dhanbad, IN-826004, Dhanbad, India\n3Department of Theoretical Solid State Physics, Institute of Solid State Physics and Optics,\nHUN-REN Wigner Research Centre for Physics, H-1525 Budapest, Hungary\n4Department of Theoretical Physics, Budapest University of Technology and Economics, H-1111 Budapest, Hungary\nInertial effects in spin dynamics emerge on picosecond time scales, giving rise to nutational\nexcitations at THz frequencies. Here, we describe a general framework for investigating the\nprecessional and nutational excitations in any type of spin structure within linear spin-wave theory.\nWe consider the particular cases of planar and conical spin spirals in detail. We observe a change in\nthe sign of the curvature of the high-frequency nutational spin-wave band as the spiral period\nis decreased when passing from the ferromagnetic to the antiferromagnetic limit. We identify\nconditions for the interaction parameters where the curvature changes sign and asymptotical flat\nbands are formed.\nI. INTRODUCTION\nSpin waves or magnons are collective excitations in\nmagnetically ordered solids that can propagate through a\nmaterial without the movement of charge, making them\nhighly promising for information storage and manipulation\nin low-loss spintronic devices [1, 2]. The magnonic band\nstructure is possible to detect experimentally using spin-\npolarized neutron or electron scattering, and Brillouin light\nscattering [3–6]. The measured band structure can be used\nto identify the spin structure and to extract microscopic\nspin-model parameters.\nThe competition between different interactions leads to\na spatial variation of the spin orientation in so-called\nnon-collinear spin structures. Spin spirals exhibit full\nrotations of the magnetic moments periodically along one\nspatial direction, with the spins rotating either in a plane\nor, often after applying an external field, on the surface\nof a cone. Such types of magnetic ordering have been\nobserved in a wide array of materials [7], e.g., in rare-\nearthmetals, multiferroics[8], bulkchiralmagnets[9,10]or\nultrathin magnetic films on heavy-metal substrates [11–13].\nThe excitations of these complex magnetic configurations\noften exhibit intriguing phenomena like nonreciprocal spin-\nwave propagation, which have significant implications for\nspintronic devices [14].\nRecent advances in the sub-picosecond manipulation of\nspins has drawn attention to spin inertia. This concept can\nbe described by including an inertial term in the Landau-\nLifshitz-Gilbert (LLG) equation, leading to a separation\nbetweenthedirectionsofthemagneticmomentandangular\nmomentumandtherebygivingrisetospinnutation[15–20].\nSeveral theoretical proposals have been set forth to account\nfor the microscopic origin of spin inertia within the LLG\nequation [21–27]. Theoretical and experimental studies\nhave suggested that the inertial relaxation time ranges\nfrom a few femtoseconds to several hundred femtoseconds,\nestablishing the characteristic time scale on which nutation\ncan be observed [17, 19, 28, 29]. The spin inertia results\nin a nutational resonance in linear response that typically\n∗macherkasskii@hotmail.comfalls in the THz regime [30–35], meaning that it is orders\nof magnitudes higher than the precessional resonance\nfrequency in ferromagnets. The nutational resonance has\nbeen experimenatlly verified in CoFeB and NiFe thin\nfilms [28] and in epitaxial Co [29], in which materials an\ninertial relaxation time of about 300 fs was measured.\nThe inertial dynamics gives rise to nutational spin waves\npropagating in the materials. Previous theoretical works\non nutational spin waves focused on collinear magnetic\nstructures, i.e., ferromagnets and antiferromagnets [27, 33,\n36–39]. These investigations predicted that the nutational\nspin waves appear at a higher frequency, while the\ninertia decreases the precessional spin-wave frequencies.\nThis is due to the hybridization between the nutation\nresonance and precession resonance. The nutational spin-\nwave dispersion follows the familiar parabolic wave-vector\ndependence of precessional spin waves in ferromagnets,\nalthough with a constant frequency shift. In contrast,\nin two-sublattice collinear antiferromagnets the nutational\nspin-wave dispersion was found to exhibit a small negative\ncurvatureatlowwavevectors, i.e., thegroupvelocitypoints\noppositely to the wave vector in this case [39].\nAntiferromagnets can be considered as a periodic spin\nstructure with modulation wave vectors at the edge of the\nBrillouin zone, while ferromagnets in this description are\nfound at the Γpoint. Spin spirals with wave vectors along\nthe line connecting these points can be used to interpolate\nbetween these two limits, and to shed light on the\ndifferences between their nutational spin-wave dispersions.\nHere, we delve into the dynamics of inertial spin waves in\nspin spirals. First, we derive the equation for linear spin-\nwave theory in the inertial regime in general non-collinear\nspin structures. We apply this method to spin spirals to\nfindoutatwhichpointthecurvatureofthenutationalspin-\nwavebandisinvertedandthebandbecomesasymptotically\nflat. Flat bands of single-particle excitations are known to\ngive rise to correlated ordered phases when interactions are\ntaken into account, including flat-band ferromagnetism in\nfermionic systems [40], magnon crystals in quantum spin\nmodels [41], Wigner crystals in optical lattices of bosons\nand fermions [42], and topological order in the form of the\nfractional quantum Hall effect [43–45]. It is expected that\ncorrectionsbeyondlinearspin-wavetheorywillsignificantly\ninfluence the inertial spin excitations in such flat bands.arXiv:2402.19141v1  [cond-mat.mes-hall]  29 Feb 20242\nThis paper is organized as follows. In Sec. II, we discuss\nthe formalism for inertial spin waves in non-collinear spin\nstructures. In Sec. III, we specify the calculations for a one-\ndimensional conical spin spiral with nearest-neighbour and\nnext-nearest-neighbour interactions. In Sec. IV, we present\nthe calculated spin-wave dispersion relations. In Sec. V,\nwe derive the conditions for obtaining the flat band. We\nsummarize the results in Sec. VI.\nII. LINEAR SPIN-WAVE THEORY\nWe adopt the classical Heisenberg model, whose\nHamiltonian reads\nH=−1\n2X\ni,j,α,βSα\niJα,β\ni,jSβ\nj−X\ni,α,βSα\niKα,β\ni,iSβ\ni\n−X\ni,αMiBα\niSα\ni,(1)\nwhere αandβare the Cartesian coordinates {x, y, z},\nSiis the spin unit vector at site i,Jα,β\ni,jstands for the\nelements of the tensorial exchange interaction between sites\niandj,Kα,β\ni,iis the single-site anisotropy tensor, Miis the\nmagnitude of the magnetic moment, and Bis the external\nmagnetic field.\nThe exchange tensor Ji,jcan be further decomposed\nas [46]\nJi,j=1\n3Tr (Ji,j)1+ \nJi,j+JT\ni,j\n2−1\n3Tr(Ji,j)1!\n+Ji,j−JT\ni,j\n2\n=Ji,j+Ki,j+Di,j.\nHere, the first term Ji,jdescribes the isotropic exchange,\nthe coefficient of the scalar product Si·Sjof the\nspins in the Hamiltonian. The second term Ki,jis the\nsymmetric, traceless two-site anisotropy tensor, which\nhas five independent elements similarly to the single-site\nanisotropy tensor Ki,i. The last term Di,jstands for\nDzyaloshinskii–Moriya interaction (DMI) [47, 48], which\nmay be rewritten as ST\niDi,jSj=Di,j(Si×Sj)using the\nDzyaloshinsky–Moriya vector Di,j.\nThe dynamics of the spins is described by the inertial\nLandau-Lifshitz-Gilbert (ILLG) equation, which reads\n∂tSi=Si×(−γBeff,i+α∂tSi+η∂ttSi),(2)\nwhere γis the absolute value of the gyromagnetic ratio, αis\nthe Gilbert damping, ηis the inertial relaxation time, and\nthe effective magnetic field acting on each spin is defined\nas\nBeff,i=−1\nMi∂H\n∂Si. (3)\nIf all spins are parallel to the local effective magnetic field,\nS(0)\ni×Beff,i=0, (4)then the system is in equilibrium, ∂tSi=0. Moreover,\nwe only consider equilibrium states which are local minima\nof the Hamiltonian in Eq. (1). To describe non-collinear\nspin structures, we introduce local coordinate systems,\nwhere the quantities will be denoted with a tilde. The\nlocal systems at each site are chosen such a way that the\nequilibrium orientations of the spins are along the z-axes,\n˜S(0)\ni=ez. Therefore, we define the rotation matrix Riat\nsiteias\nRi˜S(0)\ni=S(0)\ni. (5)\nIntroducing the exchange tensor, single-site anisotropy\ntensor and magnetic field in the local coordinate system\nas\n˜Bi=BiRi, (6)\n˜Ji,j=RT\niJi,jRj, (7)\n˜Ki,i=RT\niKi,iRi, (8)\nthe Hamiltonian in Eq. (1) may be rewritten as\nH=−1\n2X\ni,j,α,β˜Sα\ni˜Jα,β\ni,j˜Sβ\nj−X\ni,α,β˜Sα\ni˜Kα,β\ni,i˜Sβ\ni(9)\n−X\ni,αMi˜Bα\ni˜Sα\ni. (10)\nIn linear spin-wave theory, we assume that the spins\ndemonstrate small deviation from the equilibrium direction\nin the local coordinates, justifying the approximation\n˜Sz\ni≈1−1\n2\u0014\u0010\n˜Sx\ni\u00112\n+\u0010\n˜Sy\ni\u00112\u0015\n. (11)\nSubstituting Eq. (11) into Eq. (10) and keeping terms up\nto second order in the small quantities ˜Sx\ni,˜Sy\niyields\nH=E0+HSW, (12)\nwhere\nE0=−1\n2X\ni,j˜Jz,z\ni,j−X\ni˜Kz,z\ni,i−X\niMi˜Bz\ni(13)\nis the energy of the equilibrium state, and\nHSW=1\n2X\ni,j\u0002˜Sx\ni˜Sy\ni\u0003\u0014\nA1,i,jA2,i,j\nA2,j,iA3,i,j\u0015\u0014˜Sx\nj\n˜Sy\nj\u0015\n(14)\nis the harmonic spin-wave Hamiltonian, with\nA1,i,j=−˜Jx,x\ni,j+δij X\nk˜Jz,z\ni,k−2˜Kx,x\ni,i\n+ 2˜Kz,z\ni,i+Mi˜Bz\ni!\n, (15)\nA2,i,j=−˜Jx,y\ni,j−δij2˜Kx,y\ni,i, (16)\nA3,i,j=−˜Jy,y\ni,j+δij X\nk˜Jz,z\ni,k−2˜Ky,y\ni,i\n+ 2˜Kz,z\ni,i+Mi˜Bz\ni!\n. (17)3\nNote that the linear terms in ˜Sx\ni,˜Sy\niwill vanish due to\nthe equilibrium condition Eq. (4), and that the spin-\nwave Hamiltonian is described by a positive-definite matrix\naroundenergyminima. Introducingthecircularlypolarized\ncoordinates ˜S±\ni=˜Sx\ni±i˜Sy\ni, the spin-wave Hamiltonian\ntransforms into\nHSW=1\n4X\ni,j\u0002˜S+\ni˜S−\ni\u0003˜HSW,i,j\"˜S−\nj\n˜S+\nj#\n,(18)\nwhere\n˜HSW=1\n2\nA1+A3+i\u0010\nA2−AT\n2\u0011\nA1−A3−i\u0010\nA2+AT\n2\u0011\nA1−A3+i\u0010\nA2+AT\n2\u0011\nA1+A3−i\u0010\nA2−AT\n2\u0011\n.\n(19)\nIntroducing the spin velocities ˜Vi, the ILLG equation\n(2) may be rewritten as a system of first-order differential\nequations [35],\n∂t˜Si=˜Vi, (20)\n∂t˜Vi=−1\nη˜SiטVi−γ\nη˜Si\u0010\n˜Si·˜Beff,i\u0011\n+γ\nη˜Beff,i\n−α\nη˜Vi−˜Si˜V2\ni. (21)\nCalculating the effective field ˜Beff,ifrom the spin-wave\nHamiltonian Eq. (18) and linearizing Eq. (21) in the small\nvariables ˜S±\niand ˜Viresults in\n∂t\"\n˜S⊥\n˜V⊥#\n=\u00140 1\n−γη−1M−1˜HSWiη−1σz−αη−11\u0015\"\n˜S⊥\n˜V⊥#\n,\n(22)\nwhere ˜S⊥is a vector over the lattice sites of components\n˜S⊥\ni=h\n˜S−\ni,˜S+\nii\n,˜V⊥is defined analogously, Mis a\ndiagonal matrix containing the magnetic moments Mi, and\nσzis a Pauli matrix in the subspace of the ±components.\nNote that in the linear approximation ∂t˜Sz\ni=˜Vz\ni= 0,\nwhich is consistent with ˜Vz\nivanishing from the equations\ncontaining four components per lattice site.\nThe system of homogeneous linear differential equations\nin Eq. (22) may be solved by assuming a harmonic time\ndependence, and replacing ∂tbyiω. This yields an\neigenvalue equation for the spin-wave frequencies ωand\nthe eigenmodes. Using the formula for expressing the\ndeterminant of 2×2block matrices with the determinants\nof the blocks yields the condition\ndeth\n−ηω21+ω(σz+iα1) +γM−1˜HSWi\n= 0(23)\nfor the eigenvalues. This agrees with the equations\nderived for the inertial spin-wave modes in ferromagnets\nand antiferromagnets in Ref. [39]. The structure of ˜HSW\nensures that the particle-hole constraint of linear spin-wave\ntheory [49] still holds in the inertial case, meaning that the\neigenvaluesarefoundinpairsof ωand−ω∗. Intheclassical\nlimit discussed here, this simply ensures that the Sx\niand\nSy\nispin components can always chosen to be real as a linear\nFIG. 1. Sketch of a conical spin spiral along the y-axis with\nSy= 0.5and the lattice wave vector κ=−π/(8a), where ais\nthelatticeconstant. Panels(a)and(b)showthesamestructure,\nbut drawn in different projections. Light blue cones in panel (a)\nare shown as a guide to the eye.\ncombination of complex time-dependent solutions eiωtand\ne−iω∗t. Furthermore, taking the limit η→0transforms\nEq. (23) to\ndeth\nω(σz+iα1) +γM−1˜HSWi\n= 0,(24)\nthe well-known generalized eigenvalue equation of spin\nwaves in the non-inertial case [14, 50].\nIII. CONICAL SPIN SPIRAL\nAs an application of the general theory, we discuss the\nspin-wave modes of a conical spin spiral which can be\nexpressed analytically. We consider a one-dimensional\nchainofatomsalongthe y-axis. IntheHamiltonianEq.(1),\nwe only take into account isotropic exchange interactions\nJi,j=Ji,j1as defined in Eq. (2), Dzyaloshinsky–Moriya\nvectors along the y-direction Dz,x\ni,j=−Dx,z\ni,j=Dy\ni,j, a single-\nsite hard-axis anisotropy along the chain Ky,y\ni,i=K>0,\nand a magnetic field Byoriented parallel to the chain.\nWe consider the conical spin spiral\nS(0)\ni=\nsin (κri)q\n1−(Sy)2\nSy\ncos (κri)q\n1−(Sy)2\n, (25)\nwhere ridenotes the position along the chain. This\nstructure is illustrated in Fig. 1. As can be seen in4\nthe figure, κdetermines the period of the harmonic\nmodulation of the spins, and the parameter Sygoverns\nthe opening angle of the cone. The value κ= 0\ndescribes a ferromagnetic state, while κ=π/a, Sy=\n0corresponds to the antiferromagnetic state, enabling\na continuous transformation between the two limits by\nchanging the wave number κ. Substituting Eq. (25) into\nthe Hamiltonian, the energy of the spiral is found to be\nE=−N\n2h\nJ0(Sy)2+\u0010\n1−(Sy)2\u0011\n˜Jκ\n+2K(Sy)2+ 2MBySyi\n, (26)\nwhere Nis the number of lattice sites, and we introduced\nthe Fourier transforms\nJκ=X\nri−rjJi,je−iκ(ri−rj)=X\nd>02Jdcos (κd),(27)\nDy\nκ=X\nri−rjDy\ni,je−iκ(ri−rj)=X\nd>02iDy\ndsin (κd),(28)\n˜Jκ=Jκ+iDy\nκ, (29)\nwith d=rj−rithe distance between two sites, and\nwe took the symmetry of the isotropic exchange and the\nantisymmetry of the Dzyaloshinsky–Moriya interaction in\nsite indices into account. Minimizing Eq. (26) with respect\nto the parameters κandSyresults in the conditions\nX\nd>0(2Jdsin (κd) + 2Dy\ndcos (κd)) = 0 (30)\nand\nSy=MBy\n˜Jκ− J 0−2K. (31)\nNote that these equations are applicable as long as the\nright-hand side of Eq. (31) is not larger than 1in absolute\nvalue, otherwise the solution is the collinear state with\nSy= 1, and κdoes not influence the energy. It can be\nshown that if the parameters minimize the energy, then\nEq. (25) is also an equilibrium state satisfying Eq. (4). For\nfurther details of the calculation and the generalization to\nhigher-dimensional lattices, see Ref. [51].\nThe rotation matrices in Eq. (5) required for\ntransforming to the local coordinate system read\nRi=R(2)\niR(1)\ni, (32)\nwith\nR(1)\ni=\n1 0 0\n0q\n1−(Sy)2Sy\n0 −Syq\n1−(Sy)2\n(33)\na uniform rotation in the yz-plane followed by\nR(2)\ni=\ncos (κri) 0 sin ( κri)\n0 1 0\n−sin (κri) 0 cos ( κri)\n,(34)a modulated rotation in the xz-plane creating the spin\nspiral. Because the spin spiral is harmonic, the linear spin-\nwave Hamiltonian in Eq. (18) may be brought into a block-\ndiagonal form after Fourier transformation over the atomic\nsites,\n˜HSW,ky=\u0014dkya∗\n−ky\nakyd∗\n−ky\u0015\n, (35)\nwhere\naky=−1\n2h\n1−(Sy)2i\u0014\nJky+ 2K −1\n2\u0010\n˜Jκ−ky+˜Jκ+ky\u0011\u0015\n,\n(36)\ndky=aky+˜Jκ−1\n2(1 +Sy)˜Jκ−ky−1\n2(1−Sy)˜Jκ+ky.\n(37)\nHere we used the energy-minimum condition Eq. (31) to\neliminate Byfrom the expressions. Note that aky=a∗\n−ky,\nbutdky̸=d∗\n−kyifSy̸= 0.\nEquation (23) determining the magnon modes simplifies\nto calculating the determinant of 2×2matrices for each k\nvalue. It may be explicitly written as\nη2ω4−2iαηω3−ω2\u0010\nηγM−1dky+ηγM−1d∗\n−ky+α2+ 1\u0011\n−ωγM−1h\n(1−iα)dky−(1 +iα)d∗\n−kyi\n+γ2M−2\u0010\ndkyd∗\n−ky−akya∗\n−ky\u0011\n= 0. (38)\nDue to the particle-hole constraint mentioned in Sec. II, of\nthe four solutions of the secular equation it is sufficient to\ntreat the two eigenfrequencies with Re ω > 0. We will\ndenote the solution with the the lower real part of the\nfrequency the precessional spin wave ωp, while the higher\nfrequency corresponds to the nutational spin wave ωn. In\nthe non-dissipative case α= 0, these may be expressed as\nωp=1\n2√\na+ 2W−1\n2s\n−3a−2W−2b√\na+ 2W,(39)\nand\nωn=1\n2√\na+ 2W+1\n2s\n−3a−2W−2b√\na+ 2W,(40)\nrespectively. Here, we used the following notations:\nW=−5a\n6+U+V, (41)\nV=−P\n3U, (42)\nU=\"\n−Q\n2−\u0012P3\n27+Q2\n4\u00131/2#1/3\n,(43)\nQ=ac\n3−a3\n108−b2\n8, (44)\nP=−a\n12−c, (45)5\na=−η−2\u0010\nηγM−1d∗\n−ky+ηγM−1dky+ 1\u0011\n,(46)\nb=η−2γM−1\u0010\nd∗\n−ky−dky\u0011\n, (47)\nc=η−2γ2M−2\u0010\ndkyd∗\n−ky−akya∗\n−ky\u0011\n. (48)\nSince the solutions can be indexed with the wave number\nkytaking values between −π/aandπ/a, for an easier\ncomparison we will show the dispersion relation in this\natomic or extended Brillouin zone [14] for all values of the\nmagnetic period κ.\nIV. SPIN-WAVE DISPERSIONS\nWe numerically investigated the dependence of the\nprecession and nutation bands on the spin-spiral period κ\nand the opening angle of the cone characterized by Sy.\nWe restricted the interactions to nearest and next-nearest\nneighbors as J1=MJ′,J2=MρJJ′,Dy\n1=MD′y,\nDy\n2=MρDD′yandK=MK′, where the interaction\ncoefficients denoted with a prime are in units of magnetic\nfield and ρJ, ρDare dimensionless ratios of the interactions\nbetween second and first neighbors. We selected the values\nρJ,ρD, andK′. In order to keep the total frequency range\nof the dispersion the same as the period is varied, we also\nfixed the parameter\nE′2=J′2+ (D′y)2. (49)\nWe chose the κandSyvalues, and determined the spin-\nmodel parameters J′,D′y, and Bybased on the conditions\nEq. (30) and (31), i.e., we determined the Hamiltonian\nsuch a way that it is minimized for the selected period and\ncone angle. In particular, J′andD′ywere expressed from\nEq. (49) and\nD′y\nJ′=−sin (κa) + 2ρJsin (2 κa)\ncos (κa) + 2ρDcos (2 κa). (50)\nThe chosen parameter values are summarized in Table I\nand the numerically calculated spectra are shown in Fig. 2.\nThe parameters are chosen such a way as to highlight\nthe various possible changes in the dispersion relation;\nthey were not selected to describe a specific material.\nFigure 2(a) displays the familiar ungapped parabolic\nprecessional dispersion relation of the ferromagnetic state\nfor nearest-neighbor and next-nearest-neighbor couplings.\nThe nutational band is shifted by a constant η−1to\nhigher frequencies [39]. The spectrum is identical when\nthe ferromagnetic direction is rotated out from the z-axis\ntowards the y-axis (Fig. 2(i)) since the system is invariant\nunderglobalspinrotations. Thedispersionslightlychanges\nquantitatively when the next-nearest-neighbor interactions\nare turned off (Fig. 2(m)). A qualitative difference can\nonly be observed if a hard-axis anisotropy K′is introduced\nperpendiculartotheplane(Fig.2(e)): thistermcannotgap\nout the Goldstone mode of the precessional band, since the\nsystem remains rotationally invariant around the y-axis,\nbut the nutational frequency at zero wave vector is slightly\nincreased compared to η−1, and the two branches are no\nlonger shifted by only a constant value with respect to each\nother.The dispersion relations for the κa=πtwo-sublattice\nstructures are displayed in the bottom row of Fig. 2. In\nthe isotropic antiferromagnet (Fig. 2(d)), the dispersion\nis periodic with kya=πin the extended Brillouin zone;\nthis would correspond to a double degeneracy inside the\nmagnetic Brillouin zone when the dispersion is folded\nback at kya=±π/2. Note that while the precession\nband is linear around kya= 0with a finite group\nvelocity, the nutational band is parabolic with a negative\ncurvature [39]. The hard-axis anisotropy perpendicular to\nthe spins (Fig. 2(h)) does not lift the Goldstone mode at\nzero wave vector, but it introduces a gap at kya=π,\nmeaning that the spectrum would no longer be degenerate\nin the magnetic Brillouin zone. The nutational band is\nless affected by the anisotropy. Tilting the spins towards\nthey-axis by an external magnetic field (Fig. 2(l) and (p))\ndecreases the angle between the two sublattices, leading to\na spin-flop state. While the Goldstone mode is preserved,\nthis also increases the frequencies at higher wave vectors,\nsince the ferromagnetic limit has to be reached at Sy= 1.\nThis already leads to a local maximum at kya=πin\nFig. 2(l). The reduced cone angle considerably flattens\nthe nutational band around kya= 0, particularly in\nFig. 2(p). This can again be understood by approaching\nthe ferromagnetic configuration, since at that point the\ncurvature of the dispersion has to revert back to positive.\nThe transformation between the ferromagnetic and the\nantiferromagnetic state in the dispersion can also be\nobserved when changing the spiral wave vector κin\nFig. 2(b), (c), (f), (g), (j), (k), (n), and (o). The\nGoldstone mode associated with a global shift of the phase\nof the spiral is observable in all panels. The largest\ninfluence of increasing the wave vector can be seen in the\nnutational band around kya= 0where the minimum is\nturned into a local maximum, and in the precessional band\naround kya=πwhere the maximum is turned into a\nminimum. The effect of the hard-axis anisotropy is best\nvisible in increasing the precessional frequencies at kya=π\n(Fig. 2(f) and (g)). Increasing the collinear component\nSyis qualitatively similar to decreasing the wave vector,\nsince both decrease the angle between the neighboring\nspins. In Fig. 2(j) and (k), a non-reciprocity can be\nobserved in the dispersion relation, i.e., the frequencies\nbetween positive and negative kyvalues differ. In the\nconsidered system, the non-reciprocity has three necessary\ncriteria: (i) a finite spiral wave vector κa /∈ {0, π}, (ii)\na finite collinear component Sy, and (iii) a finite next-\nnearest-neighbour interaction ρJ(cf. Fig. 2(n) and (o)\nwhere the non-reciprocity vanishes). Note that (ii) is\ngenerally necessary for breaking the effective time-reversal\nsymmetry of the spirals that protects the reciprocity of the\ndispersion. However, condition (i) may be weakened and\nthe non-reciprocity observed in ferromagnetic states when\nthe Dzyaloshinsky–Moriya interaction is pointing along\nthe ferromagnetic orientation in a different geometry [52].\nCondition (iii) is only required because of the special\nmathematical structure of the nearest-neighbor model [51].\nWe present a comparison between the dispersion\ncharacteristics of precessional spin waves both in the\npresence and the absence of inertia in Fig. 3. The\nfrequencies are decreased by the presence of inertia in6\n0123\n0123\n0123\n-π-π\n20π\n2π0123-π-π\n20π\n2π-π-π\n20π\n2π-π-π\n20π\n2π\nFIG. 2. Dispersion relations of precession (blue) and nutation (red) spin waves in (a)-(d) planar spiral with next-nearest-neighbor\ncoupling for K′= 0T, (e)-(h) planar spiral with next-nearest-neighbor coupling for K′=−1T, (i)-(l) conical spiral with next-\nnearest-neighbor coupling for K′= 0T, and (m)-(p) conical spiral with only nearest-neighbor coupling for K′= 0T. The period\nof the spiral is varied between the rows of panels. The purple line shows the characteristic inertial frequency η−1. We used the\nparameters: γ= 2π×28GHz/T, α= 0,η= 1ps,E′=8T and the varied values listed in Table I.\nρJ=e−2, ρD=e−2ρJ=e−2, ρD=e−2ρJ=e−2, ρD=e−2ρJ= 0, ρD= 0\nSy= 0,K′= 0T Sy= 0,K′=−1T Sy= 0.4,K′= 0T Sy= 0.4,K′= 0T\nκa J′(T)D′y(T)κa J′(T)D′y(T)κa J′(T)D′y(T)κa J′(T)D′y(T)\n(a)0 8.00 0.00(e)0 8.00 0.00(i)0 8.00 0.00(m) 0 8.00 0.00\n(b)π/3 1.63 7.83(f)π/3 1.63 7.83(j)π/3 1.63 7.83(n)π/3 4 6.93\n(c)2π/3−6.78 4.24(g)2π/3−6.78 4.24(k)2π/3−6.78 4.24(o)2π/3−4 6.93\n(d)π −8.00 0.00(h)π −8.00 0.00(l)π −8.00 0.00(p)π −8.00 0.00\nTABLE I. Parameters used for the numerical calculations shown in Fig. 2.\nall cases. The ratio of the frequencies in the presence\nand in the absence of inertia was found in Ref. [39]\nto show a pronounced dependence on the wave vector\nin ferromagnets, while in antiferromagnets it is almost\nindependent of the wave vector. In spin spirals, the\ndependence on the wave vector is most obvious in the\ncase of the non-reciprocal dispersions (Fig. 3(j) and (k)),\nwhere the positions of the minima and maxima are also\nshifted. However, it is not possible to directly compare the\nprecessional dispersions with and without the inertial term\nin experiments, and the “bare” values of the interactionparameters determining the shape of the dispersion are\nnot known, either. Therefore, the most convincing\nexperimental evidence of inertial dynamics would have to\nrely on the detection of the nutational spin waves.\nV. CONDITION FOR THE FLAT BAND\nWe observed in Sec. IV that the curvature of\nthe nutational band at the center of the Brillouin\nzone has different signs in the ferromagnetic and the7\n0246\n0246\n0246\n-π-π\n20π\n2π0246-π-π\n20π\n2π-π-π\n20π\n2π-π-π\n20π\n2π\nFIG. 3. Dispersion relations of inertial (blue curves at η= 1ps) and non-inertial (orange curves at η= 0ps) precessional spin waves.\nThe nutational bands are not shown. Panels (a)-(d) show a planar spiral with next-nearest-neighbor coupling for K′= 0T, (e)-(h)\na planar spiral with next-nearest-neighbor coupling for K′=−1T, (i)-(l) a conical spiral with next-nearest-neighbor coupling for\nK′= 0T, and (m)-(p) a conical spiral with only nearest-neighbor coupling for K′= 0T. The period of the spiral is varied between\nthe rows of panels. The parameters are the same as in Fig. 2: γ= 2π×28GHz/T, α= 0,E′=8T, and the values given in Table I.\nantiferromagnetic limits, and the sign may be inverted by\nchanging the wave vector of the spiral or the cone angle.\nAt the point where the curvature vanishes, the nutational\nbandbecomesrelativelyflatinawiderangeofwavevectors.\nWe will derive a condition on the model parameters where\nthis inversion of the curvature occurs. We will consider\nthe case Sy= 0, because then the dispersion is reciprocal,\nmeaning that there is always a maximum or a minimum at\nkya= 0. In this case, the matrix elements of the spin-wave\nHamiltonian in Eqs. (36) and (37) simplify to\naky=1\n2\u00141\n2\u0010\n˜Jκ−ky+˜Jκ+ky\u0011\n− Jky−2K\u0015\n=a∗\n−ky,(51)\ndky=1\n2\u0014\n2˜Jκ−1\n2\u0010\n˜Jκ−ky+˜Jκ+ky\u0011\n− Jky−2K\u0015\n=d∗\n−ky,\n(52)\nand the spin-wave frequencies may be written in a closed\nform asω=\u001a1\n2η2\u0002\n1 + 2 ηγM−1dky\u0003\n±1\n2η2h\n1 + 4 ηγM−1dky+ 4\u0000\nηγM−1aky\u00012i1\n2\u001b1\n2\n,(53)\nwhere the negative and positive signs pertain to\nprecessional and nutational bands, respectively. The point\nwhere the curvature inverts can be obtained by expanding\nthe nutational frequency in kyaround kya= 0, and\ndetermining where the quadratic term vanishes. Note\nthat the linear and cubic terms in kyvanish due to the\nreciprocity of the dispersion, leaving the quartic terms\nas the leading correction at this point. This yields the\ncondition\nηγM−1\u0010\n∂2\nky˜Jκ+∂2\nkyJ0\u0011\n+\u0000\nηγM−1\u00012\u0010\n˜Jκ− J 0−2K\u0011\n∂2\nkyJ0= 0.(54)\nConsidering only nearest-neighbor interactions and using\nthe parameter E′from Eq. (49), the wave vector κfor which8\n-15 -10 -5 0π\n28π\n1517π\n30-π-π\n20π\n2π0123\n-π-π\n20π\n2π0123-π-π\n20π\n2π0123\nFIG. 4. (a) The relation between single-site anisotropy K′and\nwave vector κgiven by Eq. (55), at which the curvature of the\nnutational band vanishes. (b-d) display the precessional (blue)\nand nutational (red) bands for the parameters (b) K′= 0T,\nJ′=−1.79T,D′y= 7.79T,κa=π/2 + 0.07π; (c)K′=−5T,\nJ′=−1.32T,D′y= 7.88T,κa=π/2 + 0 .052π; and (d)\nK′=−10T,J′=−1.04T,D′y= 7.93T,κa=π/2 + 0 .041π.\nThe other parameters are E′= 8T,γ= 2π×28GHz/T, α= 0,\nη= 1ps,Sy= 0,ρJ= 0,ρD= 0in all panels.\nthis condition is satisfied may be given in a closed form:\ncos (κa) =1\n4ηγE′[1 + 2 ηγ(E′− K′)]\n−1\n4ηγE′q\n[1 + 2 ηγ(E′− K′)]2+ 8ηγE′.(55)\nNote that E′>0andK′<0, meaning that cos (κa)<0,\ni.e., the inversion of the curvature happens between κa=\nπ/2andκa=π.\nBased on Eq. (55), we illustrate the relation between\nK′andκain Fig. 4(a). Increasing the magnitude of the\nanisotropy K′shifts κacloser to π/2. It can be seen from\nEq. (55) that κaapproaches π, i.e., the antiferromagnetic\nconfiguration, as the inertial parameter ηis decreased.\nAs can be seen in Fig. 4(b)-(d), at the point where the\ncondition is satisfied the nutational band appears flat in a\nrather large range approximately between −π/2< kya <\nπ/2. The total width of the nutational band decreases as\nthemagnitudeof K′isincreased. Notethattheprecessional\nband appears flat in the vicinity of the point kya=π,\nwhere the curvature also inverts as mentioned in Sec. IV.\nHowever, the inversion of the curvature of the precessional\nband happens at a numerically different value than that of\nthe nutational band, and the precessional band becomes\nless flat at the point where Eq. (55) is satisfied as the\nmagnitude of K′is increased.VI. CONCLUSION\nIn this study, we discussed precessional and nutational\nspin waves in non-collinear spin configurations based on\nthe linearization of the inertial Landau–Lifshitz–Gilbert\nequation. We derived the general formula for determining\nthe spin-wave frequencies and eigenmodes, and discussed\nhow it transforms in the inertial-free limit. We applied\nthe method to calculate the dispersion relations for conical\nspin spirals in one dimension. We found the requirements\nfor observing a non-reciprocal dispersion in the considered\nmodel, and observed that the curvature of the nutational\nband is inverted around kya= 0when passing from the\nferromagnetic to the antiferromagnetic limit, while in the\nprecessional band the inversion occurs around kya=πin\nthe extended Brillouin zone. We derived the condition for\nthe inversion of the curvature of the nutational band, and\ndemonstrated that the band becomes relatively flat in a\nwide range of wave vectors.\nAlthough the observed nutational bands are not flat in\nthe whole extended Brillouin zone, the nutational band\nwill split into multiple bands at the boundary of the\nmagnetic Brillouin zone at ±κ/2when the spin spiral\nbecomes anharmonic and the spin waves hybridize with\neach other. Such flat bands in reduced magnetic Brillouin\nzones, also known as spin-wave Landau levels, have been\ninvestigated in detail in non-collinear spin structures in\nthe non-inertial limit in Ref. [53]. Based on the analogy\nwith electronic systems, it is also expected that due to the\nhigh density of states in the flat bands, linear spin-wave\ntheory may no longer accurately describe the nature of the\nexcitations, and the interaction between the spin waves\nplays a more pronounced role. The formalism presented\nhere may stimulate further theoretical and experimental\ninvestigations of inertial spin waves in various kinds of non-\ncollinear spin structures.\nNote:Thegeneralformulaeforthespin-waveeigenmodes\nof the inertial Landau–Lifshitz–Gilbert equation has been\nrecently derived in Ref. [54] independently of our work.\nACKNOWLEDGMENTS\nWe are grateful to Ulrich Nowak, Tobias Dannegger and\nDavidAngsterforfruitfuldiscussions. Financialsupportby\nthe faculty research scheme at IIT (ISM) Dhanbad, India\nunder Project No. FRS(196)/2023-2024/PHYSICS, by the\nScience and Engineering Research Board (SERB), India\nunder Project No. SRG/2023/000612, by the National\nResearch, Development, and Innovation Office (NRDI) of\nHungary under Project Nos. K131938 and FK142601, by\nthe Ministry of Culture and Innovation and the National\nResearch, Development and Innovation Office within the\nQuantum Information National Laboratory of Hungary\n(Grant No. 2022-2.1.1-NL-2022-00004), by the Hungarian\nAcademy of Sciences via a János Bolyai Research Grant\n(GrantNo. BO/00178/23/11)aregratefullyacknowledged.\n[1] A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and\nB. Hillebrands, Nature Physics 11, 453 (2015).[2] A. Hirohata and K. Takanashi, Journal of Physics D:\nApplied Physics 47, 193001 (2014).9\n[3] J. O. Vasseur, L. Dobrzynski, B. Djafari-Rouhani, and\nH. Puszkarski, Phys. Rev. B 54, 1043 (1996).\n[4] R. Chisnell, J. S. Helton, D. E. Freedman, D. K. Singh,\nR. I. Bewley, D. G. Nocera, and Y. S. Lee, Phys. Rev. Lett.\n115, 147201 (2015).\n[5] O. Nikotin, P. A. Lindgård, and O. W. Dietrich, Journal of\nPhysics C: Solid State Physics 2, 1168 (1969).\n[6] H. Ehrenberg, H. Weitzel, H. Fuess, and B. Hennion,\nJournal of Physics: Condensed Matter 11, 2649 (1999).\n[7] Y. A. Izyumov, Soviet Physics Uspekhi 27, 845 (1984).\n[8] I. Sosnowska, T. P. Neumaier, and E. Steichele, Journal of\nPhysics C: Solid State Physics 15, 4835 (1982).\n[9] M. Uchida, Y. Onose, Y. Matsui, and Y. Tokura, Science\n311, 359 (2006).\n[10] A.BauerandC.Pfleiderer,Phys.Rev.B 85,214418(2012).\n[11] M. Bode, M. Heide, K. von Bergmann, P. Ferriani,\nS. Heinze, G. Bihlmayer, A. Kubetzka, O. Pietzsch,\nS. Blügel, and R. Wiesendanger, Nature 447, 190 (2007).\n[12] Y. Yoshida, S. Schröder, P. Ferriani, D. Serrate,\nA. Kubetzka, K. von Bergmann, S. Heinze, and\nR. Wiesendanger, Phys. Rev. Lett. 108, 087205 (2012).\n[13] K. von Bergmann, A. Kubetzka, O. Pietzsch, and\nR. Wiesendanger, Journal of Physics: Condensed Matter\n26, 394002 (2014).\n[14] M. Garst, J. Waizner, and D. Grundler, Journal of Physics\nD: Applied Physics 50, 293002 (2017).\n[15] E. Olive, Y. Lansac, and J.-E. Wegrowe, Appl. Phys. Lett.\n100, 192407 (2012).\n[16] E. Olive, Y. Lansac, M. Meyer, M. Hayoun, and J.-E.\nWegrowe, J. Appl. Phys. 117, 213904 (2015).\n[17] S. Bhattacharjee, L. Nordström, and J. Fransson, Phys.\nRev. Lett. 108, 057204 (2012).\n[18] H. Suhl, IEEE Trans. Magn. 34, 1834 (1998).\n[19] D. Böttcher and J. Henk, Phys. Rev. B 86, 020404(R)\n(2012).\n[20] R. Mondal, L. Rózsa, M. Farle, P. M. Oppeneer, U. Nowak,\nand M. Cherkasskii, J. Magn. Magn. Mater. 579, 170830\n(2023).\n[21] E. Olive, Y. Lansac, M. Meyer, M. Hayoun, and J.-E.\nWegrowe, J. Appl. Phys. 117, 213904 (2015).\n[22] J.-E. Wegrowe and E. Olive, J. Phys.: Condens. Matter 28,\n106001 (2016).\n[23] R. Mondal, M. Berritta, A. K. Nandy, and P. M. Oppeneer,\nPhys. Rev. B 96, 024425 (2017).\n[24] R. Mondal, M. Berritta, and P. M. Oppeneer, J. Phys.:\nCondens. Matter 30, 265801 (2018).\n[25] M. Fähnle, D. Steiauf, and C. Illg, Phys. Rev. B 84, 172403\n(2011).\n[26] M. Fähnle, D. Steiauf, and C. Illg, Phys. Rev. B 88,\n219905(E) (2013).\n[27] T. Kikuchi and G. Tatara, Phys. Rev. B 92, 184410 (2015).\n[28] K. Neeraj, N. Awari, S. Kovalev, D. Polley,\nN. Zhou Hagström, S. S. P. K. Arekapudi, A. Semisalova,\nK. Lenz, B. Green, J.-C. Deinert, I. Ilyakov, M. Chen,M. Bawatna, V. Scalera, M. d’Aquino, C. Serpico,\nO. Hellwig, J.-E. Wegrowe, M. Gensch, and S. Bonetti,\nNat. Phys. 17, 245 (2021).\n[29] V. Unikandanunni, R. Medapalli, M. Asa, E. Albisetti,\nD.Petti, R.Bertacco, E.E.Fullerton,andS.Bonetti,Phys.\nRev. Lett. 129, 237201 (2022).\n[30] M.-C. Ciornei, J. M. Rubí, and J.-E. Wegrowe, Phys. Rev.\nB83, 020410(R) (2011).\n[31] M.-C. Ciornei, Ph.D. thesis, École Polytechnique,\nUniversidad de Barcelona, 2010.\n[32] S. V. Titov, W. T. Coffey, Y. P. Kalmykov, and\nM. Zarifakis, Phys. Rev. B 103, 214444 (2021).\n[33] M. Cherkasskii, M. Farle, and A. Semisalova, Phys. Rev. B\n103, 174435 (2021).\n[34] M. Cherkasskii, M. Farle, and A. Semisalova, Phys. Rev. B\n102, 184432 (2020).\n[35] R. Mondal, S. Großenbach, L. Rózsa, and U. Nowak, Phys.\nRev. B 103, 104404 (2021).\n[36] I. Makhfudz, E. Olive, and S. Nicolis, Appl. Phys. Lett.\n117, 132403 (2020).\n[37] S. V. Titov, W. J. Dowling, Y. P. Kalmykov, and\nM. Cherkasskii, Phys. Rev. B 105, 214414 (2022).\n[38] A.M.Lomonosov, V.V.Temnov,andJ.-E.Wegrowe,Phys.\nRev. B 104, 054425 (2021).\n[39] R. Mondal and L. Rózsa, Phys. Rev. B 106, 134422 (2022).\n[40] H. Tasaki, Progress of Theoretical Physics 99, 489 (1998).\n[41] M. E. Zhitomirsky and H. Tsunetsugu, Phys. Rev. B 70,\n100403(R) (2004).\n[42] C. Wu, D. Bergman, L. Balents, and S. Das Sarma, Phys.\nRev. Lett. 99, 070401 (2007).\n[43] K. Sun, Z. Gu, H. Katsura, and S. Das Sarma, Phys. Rev.\nLett.106, 236803 (2011).\n[44] E. Tang, J.-W. Mei, and X.-G. Wen, Phys. Rev. Lett. 106,\n236802 (2011).\n[45] T. Neupert, L. Santos, C. Chamon, and C. Mudry, Phys.\nRev. Lett. 106, 236804 (2011).\n[46] A. Lászlóffy, L. Rózsa, K. Palotás, L. Udvardi, and\nL. Szunyogh, Phys. Rev. B 99, 184430 (2019).\n[47] I. Dzyaloshinsky, J. Phys. Chem. Solids 4, 241 (1958).\n[48] T. Moriya, Phys. Rev. Lett. 4, 228 (1960).\n[49] V. P. Flynn, E. Cobanera, and L. Viola, New J. Phys. 22,\n083004 (2020).\n[50] L. Rózsa, J. Hagemeister, E. Y. Vedmedenko, and\nR. Wiesendanger, Phys. Rev. B 98, 100404(R) (2018).\n[51] D. Wuhrer, L. Rózsa, U. Nowak, and W. Belzig, Phys. Rev.\nRes.5, 043124 (2023).\n[52] L. Udvardi and L. Szunyogh, Phys. Rev. Lett. 102, 207204\n(2009).\n[53] T. Weber, D. M. Fobes, J. Waizner, P. Steffens, G. S.\nTucker, M. Böhm, L. Beddrich, C. Franz, H. Gabold,\nR. Bewley, D. Voneshen, M. Skoulatos, R. Georgii,\nG. Ehlers, A. Bauer, C. Pfleiderer, P. Böni, M. Janoschek,\nand M. Garst, Science 375, 1025 (2022).\n[54] T. Dannegger, D. Angster, and U. Nowak, manuscript in\npreparation."    },    {        "title": "2106.11377v1.Spin_structure_factors_of_doped_monolayer_Germanene_in_the_presence_of_spin_orbit_coupling.pdf",        "content": "arXiv:2106.11377v1  [cond-mat.str-el]  21 Jun 2021Spin structure factors of doped monolayer\nGermanene in the presence of spin-orbit\ncoupling\nFarshad Azizi and Hamed Rezania∗\nDepartment of Physics, Razi University, Kermanshah, Iran\nCorresponding author’s email: rezania.hamed@gmail.com\nAbstract\nIn this paper, we present a Kane-Mele model in the presence of magnetic field and\nnext nearest neighbors hopping amplitudes for investigati ons of the spin susceptibilities of\nGermanene layer. Green’s function approach has been implem ented to find the behavior\nof dynamical spin susceptibilities of Germanene layer with in linear response theoryand in\nthe presence of magnetic field and spin-orbit coupling at fini te temperature. Our results\nshow the magnetic excitation mode for both longitudinal and transverse components of\nspintends tohigherfrequencies withspin-orbit couplings trength. Moreover thefrequency\npositions of sharp peaks in longitudinal dynamical spin sus ceptibility are not affected by\nvariation of magnetic field while the peaks in transverse dyn amical susceptibility moves\n∗Corresponding author. Tel./fax: +98 831 427 4569., Tel: +98 831 42 7 4569\n1to lower frequencies with magnetic field. The effects of electr on doping on frequency\nbehaviors of spin susceptibilities have been addressed in d etails. Finally the tempera-\nture dependence of static spin structure factors due to the e ffects of spin-orbit coupling,\nmagnetic field and chemical potential has been studied.\nKeywords: Germanene; Green’s function; Optical absorption\n1 Introduction\nA lot of theoretical and experimental studies have been performe d on Graphene as a one-\natom-thick layer of graphite since it’s fabrication[1]. The low energy lin ear dispersion and\nchiral property of carbon structure leads to map the nearest ne ighbor hopping tight binding\nhamiltonian which at low energy to a relativistic Dirac Hamiltonian for mas sless fermions with\nFermi velocity vF. Novel electronic properties have been exhibited by Graphene laye r with a\nzero band gap which compared to materials with a non-zero energy g ap. These materials have\nintriguing physical properties and numerous potential practical a pplications in optoelectronics\nand sensors[2].\nRecently, the hybrid systems consisting of Graphene and various t wo-dimensional materials\nhave been studied extensively both experimentally and theoretically [3, 4, 5]. Also, 2D materi-\nals could be used for a extensive applications in nanotechnology [6, 7 ] and memory technology\n[8]. While the research interest in Graphene-based superlattices is g rowing rapidly, people have\nstarted to question whether the Graphene could be replaced by its close relatives, such as 2 di-\nmensional hexagonal crystal of Germanene. This material shows a zero gap semiconductor with\n2massless fermion charge carriers since their πandπ∗bands are also linear at the Fermi level[9].\nGermanene as counterpart of Graphene, is predicted to have a ge ometry with low-buckled hon-\neycomb structure for its most stable structures in contrast to t he Graphene monolayer [9, 10].\nSuch small buckling as vertical distance between two planes of atom s for Germanene comes\nfrom the mixing of sp2andsp3hybridization[11, 12]. The behavior of Germanene electronic\nstructure shows a linear dispersion close to K and K’ points of the fir st Brillouin zone. However\nab initio calculations indicated that spin-orbit coupling in Germanene causes t o small band gap\nopening at the Dirac point and thus the Germanene has massive Dirac fermions[10, 13]. Also\nthe band gap due to the spin orbit coupling in Germanene is more remar kable rather than that\nin Graphene[14]. The intrinsic carrier mobility of Germanene is higher th an Graphene[15]. The\ndifferent dopants within the Germanene layer gives arise to the sizab le band gap opening at\nthe Dirac point an the electronic properties of this material are affe cted by that[16, 17]. In a\ntheoretical work, the structural and electronic properties of s uperlattices made with alternate\nstacking ofGermanenelayer aresystematically investigated byusin gadensity functionaltheory\nwith the van der Waals correction[18]. It was predicted that spin orb it coupling and exchange\nfield together open a nontrivial bulk gap in Graphene like structures leading to the quantum\nspin hall effect [19, 20]. The topological phase transitions in the 2D cr ystals can be understood\nbased on intrinsic spin orbit coupling which arises due to perpendicular electric field or interac-\ntion with a substrate. Kane and Mele[21] applied a model Hamiltonian to describe topological\ninsulators. Such model consists of a hopping and an intrinsic spin-or bit term on the Graphene\nlike structures. The Kane-Mele model essentially includes two copies with different sign for\nup and down spins of a model introduced earlier by Haldane[22]. Such m icroscopic model was\n3originally proposed to describe the quantum spin Hall effect in Graphe ne[21]. Subsequent band\nstructure calculations showed, however, that the spin orbit gap in Graphene is so small[23, 24]\nthat the quantum spin Hall effect in Graphene like structures is beyo nd experimental relevance.\nIn-plane magnetic field affects the magneto conductivity of honeyc omb structures so that\nthe results show the negative for intrinsic gapless Graphene. Howe ver the magneto-resistance\nof gapless Graphene presents a positive value for fields lower than t he critical magnetic field\nand negative above the critical magnetic[25]. Moreover, microwave magneto transport in doped\nGraphene is an open problem[26].\nThe many body effects such as Coulomb interaction and its dynamical screening present the\nnovel features for any electronic material. The collective spectru m and quasiparticle properties\nofelectronicsystems aredeterminedformdynamicalspinstructu refactors. Theseareappliedto\nimplytheopticalpropertiesofthesystem. Moreoveralotofstudie shavebeendoneoncollective\nmodes of monolayer Graphene both theoretically [27] and experimen tally[28]. However there\nare no the extensive theoretical studies on doped bilayer systems .\nThe frequency dependence of dynamical spin susceptibility has bee n studied and the re-\nsults causes to find the collective magnetic excitation spectrum of m any body system. It is\nworthwhile to explain the experimental interpretation of imaginary p art of dynamical spin sus-\nceptibilities. Slow neutrons scatter from solids via magnetic dipole inte raction in which the\nmagnetic moment of the neutron interacts with the spin magnetic mo ment of electrons in the\nsolid[29]. We can readily express the inelastic cross-section of scatt ering of neutron beam from\na magnetic system based on correlation functions between spin den sity operators. In other\nwords the differential inelastic cross section d2σ/dΩdωcorresponds to imaginary part of spin\n4susceptibilities. ωdescribes the energy loss of neutron beam which is defined as the diff erence\nbetween incident and scattered neutron energies. Ω introduces s olid angle of scattered neu-\ntrons. The spin excitation modes of the magnetic system have been found via the frequency\nposition of peaks in d2σ/dΩdω. Depending on component of spin magnetic moment of electrons\nthat interacts with spin of neutrons, transverse and longitudinal spin susceptibility behaviors\nhave been investigated. The imaginary and real part of non-intera cting change susceptibilities\nof Graphene within an analytical approach have been calculated A th eoretical work has been\nperformedforcalculating both[30]. Theresults ofthisstudy show s thereisno remarkableangle\ndependence for imaginary part of polarizability around the van Hove singularity, i.e, ¯ hω/t= 2.0\nwheretimplies nearest neighbor hopping integral.\nIt is worthwhile to add few comments regarding the comparison Germ anene and Graphene\nlike structure. As we have mentioned, the most important differenc e between Germanene struc-\nture ad Graphene one arises from the nonzero overlap function be tween nearest neighbor atoms\nin Germanene structure. Response functions of Graphene in the p resence of spin-orbit cou-\npling have been studied recently[31, 32]. In this references, the op tical absorption of Graphene\nstructure in the presence of spin-orbit coupling and magnetic field h as been theoretically stud-\nied. Optical absorption rate corresponds to the charge transitio n rate of electrons between\nenergy levels. However in our work, we have investigated the dynam ical spin susceptibility of\nGermanene structure due to spin-orbit coupling. In other words w e have specially studied the\ntransition rate of magnetic degrees freedom of electrons. Such s tudy is a novelty of our work so\nthat there is no the theoretical work on the study of magnetic exc itation modes in Germanene\nstructure due to spin-orbit coupling. These studies have been per formed for Graphene like\n5structures and the most important difference between our result s and the spin susceptibilities\nof Graphene structure is the effects of overlap function in German ene structure compared to\nGraphene lattice. In fact overlap function has considerable impact on the frequency position of\nexcitation modeandalso onthe intensity ofscattered neutron bea mfromGermanene structure.\nThe purpose of this paper is to provide a Kane Mele model including intr insic spin-orbit\ninteraction for studying frequency behavior of dynamical spin sus ceptibility of Germanene layer\ninthepresence ofmagnetic fieldperpendicular totheplane. Using th esuitable hopping integral\nand on site parameter values, the band dispersion of electrons has been calculated. Full band\ncalculation beyond Dirac approximation has been implemented to deriv e both transverse and\nlongitudinal dynamical spin susceptibilities. We have exploited Green’s function approach to\ncalculate the spin susceptibility, i.e. the time ordered spin operator c orrelation. The effects of\nelectron doping, magnetic field and spin-orbit coupling on the spin str ucture factors have been\nstudied. Also we discuss and analyze to show how spin-orbit coupling a ffects the frequency\nbehavior of the longitudinal and transverse spin susceptibilities. Als o we study the frequency\nbehavior of dynamical spin susceptibility of Germanene due to variat ion of chemical potential\nand magnetic field. Also the effects of spin-orbit coupling constant a nd magnetic field on\ntemperature dependence of both transverse and longitudinal st atic spin susceptibilities have\nbeen investigated in details.\n2 Model Hamiltonian and formalism\nThe crystal structure of Germanene has been shown in Fig.(1). Th e unit cell of Germanene\nstructure is similar to Graphene layer and this honeycomb lattice dep icted in Fig.(2). The\n6primitive unit cell vectors of honeycomb lattice have been shown by a1anda2. In the presence\nof longitudinal magnetic field, the Kane-Mele model[21] ( H) for Germanene structure includes\nthe tight binding model ( HTB), the intrinsic spin-orbit coupling ( HISOC) and the Zeeman term\n(HZeeman) due to the coupling of spin degrees of freedom of electrons with ex ternal longitudinal\nmagnetic field B\nH=HTB+HISOC+HZeeman. (1)\nThe tight binding part of model Hamiltonian consists of three parts; nearest neighbor hop-\nping, next nearest neighbor (2NN) hopping and next next nearest neighbor (3NN) hopping\nterms. The tight binding part, the spin orbit coupling term and the Ze eman part of the model\nHamiltonian on the honeycomb lattice are given by\nHTB=−t/summationdisplay\ni,∆,σ/parenleftig\naσ†\nj=i+∆bσ\ni+h.c./parenrightig\n−t′/summationdisplay\ni,∆′,σ/parenleftig\naσ†\nj=i+∆′aσ\ni+bσ†\ni+∆′bσ\ni/parenrightig\n−t”/summationdisplay\ni,∆”,σ/parenleftig\naσ†\nj=i+∆”bσ\ni+h.c./parenrightig\n−/summationdisplay\ni,σµ/parenleftig\na†σ\niaσ\ni+b†σ\nibσ\ni/parenrightig\n,\nHISOC=iλ/summationdisplay\ni,∆′,σ/summationdisplay\nα=A,B/parenleftig\nνa\ni+∆′,ia†σ\ni+∆′σz\nσσ′aσ′\ni+νb\ni+∆′,ib†σ\ni+∆′σz\nσσ′bσ′\ni/parenrightig\n,\nHZeeman=−/summationdisplay\ni,σσgµBB/parenleftig\na†σ\niaσ\ni+b†σ\nibσ\ni/parenrightig\n. (2)\nHereaσ\ni(bσ\ni) is an annihilation operator of electron with spin σon sublattice A(B) in unit cell\nindexi. The operators fulfill the fermionic standard anti commutation re lations{aσ\ni,aσ′†\nj}=\nδijδσσ′. As usual t,t′,t′′denote the nearest neighbor, next nearest neighbor and next ne xt\nnearest neighbor hopping integral amplitudes, respectively. The p arameterλintroduces the\nspin-orbit coupling strength. Also Brefers to strength of applied magnetic field. gandµB\nintroduce the gyromagnetic and Bohr magneton constants, resp ectively.σzis the third Pauli\nmatrix, and νa(b)\nji=±1 as discussed below. Based on Fig.(2), a1anda2are the primitive\n7vectors of unit cell and the length of them is assumed to be unit. The symbol∆=0,∆1,∆2\nimplies the indexes of lattice vectors connecting the unit cells including nearest neighbor lattice\nsites. The translational vectors ∆1,∆2connecting neighbor unit cells are given by\n∆1=i√\n3\n2+j1\n2,∆2=i√\n3\n2−j1\n2. (3)\nAlso index ∆′=∆1,∆2,−∆1,−∆2,j,−jimplies the characters of lattice vectors connecting\nthe unit cells including next nearest neighbor lattice sites. Moreover index∆” =√\n3i,j,−j\ndenotes the characters of lattice vectors connecting the unit ce lls including next next nearest\nneighbor lattice sites. We consider the intrinsic spin-orbit term[21] of the KM Hamiltonian in\nEq.(2). The expression νa(b)\njigives±1 depending on the orientation of the sites. A standard\ndefinition for να\njiin each sublattice α=A,Bisνα\nji=/parenleftigdα\nj×dα\ni\n|dα\nj×dα\ni|/parenrightig\n.ez=±1 where dα\njanddα\ni\nare the two unit vectors along the nearest neighbor bonds connec ting siteito its next-nearest\nneighborj. Moreover ezimplies the unit vector perpendicular to the plane. Because of two\nsublattice atoms, the band wave function ψσ\nn(k,r) can be expanded in terms of Bloch functions\nΦσ\nα(k,r). The index αimplies two inequivalent sublattice atoms A,Bin the unit cell, rdenotes\nthe position vector of electron, kis the wave function belonging in the first Brillouin zone of\nhoneycomb structure. Such band wave function can be written as\nψσ\nn(k,r) =/summationdisplay\nα=A,BCσ\nnα(k)Φσ\nα(k,r), (4)\nwhereCσ\nnα(k) is the expansion coefficients and n=c,vrefers to condition and valence bands.\nAlso we expand the Bloch wave function in terms of Wannier wave func tion as\nΦσ\nα(k,r) =1√\nN/summationdisplay\nRieik.Riφσ\nα(r−Ri), (5)\n8so thatRiimplies the position vector of ith unit cell in the crystal and φαis the Wannier wave\nfunction of electron in the vicinity of atom in ith unit cell on sublattice index α. By inverting\nthe expansion Eq.(4), we can expand the Bloch wave functions in ter ms of band wave function\nas following relation\nΦσ\nα(k,r) =/summationdisplay\nn=c,vDσ\nαn(k)ψσ\nn(k,r), (6)\nwhereDσ\nαn(k) is the expansion coefficients and we explain these coefficients in the f ollowing.\nThesmallBucklinginGermanenecausestotheconsiderablevaluefor 2NNand3NNhopping\namplitude. Moreover we have considerable values for overlap param eters of electron wave\nfunctions between 2NN and 3NN atoms. The band structures of ele ctrons with spin σof\nGermanene described by model Hamiltonian in Eq.(2) are obtained by u sing the matrix form\nof Schrodinger as follows\nHσ(k)Cσ(k) =Eσ\nn(k)Sσ(k)Cσ(k),\nHσ(k) =\nHσ\nAA(k)Hσ\nAB(k)\nHσ\nBA(k)Hσ\nBB(k)\n,Cσ(k) =\nCσ\nnA(k)\nCσ\nnB(k)\n,\nSσ(k) =\nSσ\nAA(k)Sσ\nAB(k)\nSσ\nBA(k)Sσ\nBB(k)\n. (7)\nUsing the Bloch wave functions, i.e. Φ α(k), the matrix elements of HandSare given by\nHσ\nαβ(k) =/an}bracketle{tΦσ\nα(k)|H|Φσ\nβ(k)/an}bracketri}ht, Sσ\nαβ(k) =/an}bracketle{tΦσ\nα(k)|Φσ\nβ(k)/an}bracketri}ht. (8)\nThe matrix elements of Hσ\nαβandSσ\nαβare expressed based on hopping amplitude and spin-orbit\ncoupling between two neighbor atoms on lattice sites and can be expa nded in terms of hopping\namplitudes t,t′,t”, spin orbit coupling λand overlap parameters. The diagonal elements of\n9matrixes Hin Eq.(7) arise from hopping amplitude of electrons between next nea rest neighbor\natoms on the same sublattice and spin-orbit coupling. Also the off diag onal matrix elements\nwith spin channel σ, i.e.Hσ\nAB,Hσ\nBA, raise from hopping amplitude of electrons between nearest\nneighbor atomsandnext next nearest neighboratomsonthediffer ent sublattices. These matrix\nelements are obtained as\nHσ\nAB(k) =t/parenleftig\n1+eik.∆1+eik.∆2/parenrightig\n+t”/parenleftig\n2cos(ky)+e−i√\n3kx/parenrightig\n=t/parenleftig\n1+2cos(ky/2)e−i√\n3kx/2/parenrightig\n+t”/parenleftig\n2cos(ky)+e−i√\n3kx/parenrightig\n,\nHσ\nAA(k) = 2t′/parenleftig\ncos(√\n3kx/2+ky/2)+cos(√\n3kx/2+ky/2)+cos(ky/2)/parenrightig\n−2λ/parenleftig\nsin/parenleftig1\n2ky/parenrightig\n−sin/parenleftig√\n3\n2kx+1\n2ky/parenrightig\n−sin/parenleftig√\n3\n2kx−1\n2ky/parenrightig/parenrightig\n−µ−σgµBB,\nHσ\nBB(k) =−2t′/parenleftig\ncos(√\n3kx/2+ky/2)+cos(√\n3kx/2+ky/2)+cos(ky/2)/parenrightig\n+ 2λ/parenleftig\nsin/parenleftig1\n2ky/parenrightig\n−sin/parenleftig√\n3\n2kx+1\n2ky/parenrightig\n−sin/parenleftig√\n3\n2kx−1\n2ky/parenrightig/parenrightig\n−µ−σgµBB,\nHσ\nBA(k) =H∗\nAB(k). (9)\nBased on matrix elements Hσ\nαβ(k), the model Hamiltonian in Eq.(2) is written in terms of\nFourier transformation of creation and annihilation fermionic opera tors as\nH=/summationdisplay\nk,σ/bracketleftig\nHσ\nAA(k)aσ†\nkaσ\nk+HAB(k)aσ†\nkbσ\nk+HBA(k)bσ†\nkaσ\nk+Hσ\nBB(k)bσ†\nkbσ\nk/bracketrightig\n, (10)\nso that the operator aσ\nk(bσ\nk) annihilates an electron at wave vector kwith spin index σon\nsublattice A(B) and has the following relation as\naσ\nk=1√\nN/summationdisplay\nkeik·Riaσ\ni, bσ\nk=1√\nN/summationdisplay\nkeik·Ribσ\ni. (11)\nThe matrix elements of S(k), i.e.SAA(k) ,SAB(k),SBA(k) andSBB(k) are expressed as\nSAB(k) =s/parenleftig\n1+eik.∆1+eik.∆2/parenrightig\n+s”/parenleftig\n2cos(ky)+e−i√\n3kx/parenrightig\n10=s/parenleftig\n1+2cos(ky/2)e−i√\n3kx/2/parenrightig\n+s”/parenleftig\n2cos(ky)+e−i√\n3kx/parenrightig\nSAA(k) = 1+2 s′/parenleftig\ncos(√\n3kx/2+ky/2)+cos(√\n3kx/2+ky/2)+cos(ky/2)/parenrightig\nSBB(k) =SAA(k), SBA(k) =S∗\nAB(k), (12)\nso thatsis theoverlap between orbital wave functionof electron respect t o the nearest neighbor\natoms,s′denotes the overlap between orbital wave function of electron re spect to the next\nnearest neighbor atoms and s” implies the overlap between orbital wave function of electron\nrespect to the next next nearest neighbor atoms. The density fu nctional theory and ab initio\ncalculations has been determined the hopping amplitudes and overlap valuess,s′,s” as[18]t=\n−1.163,t′=−0.055,t” =−0.0836,s= 0.01207,s′= 0.0128,s” = 0.048. Using the Hamiltonian\nand overlap matrix forms in Eqs.(9,12), the band structure of elect rons, i.e.Eσ\nη(k) has been\nfound by solving equation det/parenleftig\nH(k)−E(k)S(k)/parenrightig\n= 0. Moreover the matrix elements of C(k)\ncanbefoundbasedoneigenvalueequationinEq.(7). Eq.(7)canbere writtenasmatrixequation\nas follows\n\nψc(k,r)\nψv(k,r)\n=\nCσ\ncA(k)Cσ\ncB(k)\nCσ\nvA(k)Cσ\nvB(k)\n\nΦσ\nA(k,r)\nΦσ\nB(k,r)\n. (13)\nIn a similar way, we can express the matrix from for Eq.(6)\n\nΦσ\nA(k,r)\nΦσ\nB(k,r)\n=\nDσ\nAc(k)Dσ\nAv(k)\nDσ\nBc(k)Dσ\nBv(k)\n\nψσ\nc(k,r)\nψσ\nv(k,r)\n,\n\nDσ\nAc(k)Dσ\nAv(k)\nDσ\nBc(k)Dσ\nBv(k)\n=\nCσ\ncA(k)Cσ\ncB(k)\nCσ\nvA(k)Cσ\nvB(k)\n−1\n(14)\nThe final results for band structure and expansion coefficients, i.e .Cσ\nnαandDσ\nαn, are lengthy\nand are not given here. The valence and condition bands of electron s have been presented by\n11Eσ\nv(k) andEσ\nc(k) respectively. In the second quantization representation, we ca n rewrite the\nEq.(14) as\n\naσ†\nk\nbσ†\nk\n=\nDσ\nAc(k)Dσ\nAv(k)\nDσ\nBc(k)Dσ\nBv(k)\n\ncσ†\nc,k\ncσ†\nv,k\n, (15)\nUsing band energy spectrum, the Hamiltonian in Eq.(2) can be rewritt en by\nH=/summationdisplay\nk,σ,η=c,vEσ\nη(k)cσ†\nη,kcσ\nη,k, (16)\nwherecσ\nη,kdefines the creation operator of electron with spin σin band index ηat wave vector\nk. Since longitudinal magnetic field has been applied perpendicular to th e Germanene layer,\nthe electronic Green’s function depends on the spin index σ=↑,↓. According to the model\nHamiltonian introduced in Eq.(2), the elements of spin resolved Matsu bara Green’s function\nare introduced as the following forms\nGσ\nAA(k,τ) =−/an}bracketle{tT(ak,σ(τ)a†\nk,σ(0))/an}bracketri}ht, Gσ\nAB(k,τ) =−/an}bracketle{tT(ak,σ(τ)b†\nk,σ(0))/an}bracketri}ht,\nGσ\nBA(k,τ) =−/an}bracketle{tT(bk,σ(τ)a†\nk,σ(0))/an}bracketri}ht, Gσ\nBB(k,τ) =−/an}bracketle{tT(bk,σ(τ)b†\nk,σ(0))/an}bracketri}ht.(17)\nTintroduces the time ordering operator and arranges the creation and annihilation operators\nin terms of time of them without attention to the their algebra. The F ourier transformation of\neach Green’s function element is obtained by\nGσ\nαβ(k,iωm) =/integraldisplay1/kBT\n0dτeiωmτGσ\nαβ(k,τ), α,β=A,B, (18)\nwhereωm= (2m+1)πkBTdenotes the Fermionic Matsubara frequency. After some algebra ic\ncalculation, the following expression is obtained for Green’s function s in Fourier presentation\nGσ\nαβ(k,iωm) =/summationdisplay\nη=c,vDσ∗\nαη(k)Dσ\nβη(k)\niωm−Eσ\nη(k), (19)\n12whereα,βrefer to the each atomic basis of honeycomb lattice and Eσ\nη(k) is the band structure\nof Germanene layer in the presence of magnetic field and spin-orbit c oupling. For determining\nthe chemical potential, µσ, we use the relation between concentration of electrons ( ne) and\nchemical potential. This relation is given by\nne=1\n4N/summationdisplay\nk,η,σ1\neEση(k)/kBT+1. (20)\nInfactthediagonalmatrixelements oftheHamiltonianinEq.(9) depe ndsonchemical potential\nµ. Thus eigenvalues, i.e. Eσ\nη(k) includes the factor µ. Therefore the right hand of Eq.(20)\ndepends on chemical potential µ. With an initial guess for chemical potential µ, we can solve\nthe algebraic equation 20 so that we can find the chemical potential value for each amount\nfor electronic concentration ne. These statements have been added to the manuscript after\nEq.(20). Based on the values of electronic concentration ne, the chemical potential, µ, can be\nobtained by means Eq.(20). In order to obtain the magnetic excitat ion spectrum of Germanene\nstructure both transverse and longitudinal dynamical spin susce ptibilities have been presented\nusing Green’s function method in the following section.\n3 Dynamical and static spin structure factors\nThe correlation function between spin components of itinerant elec trons in Germanene layer\nat different times can be expressed in terms of one particle Green’s f unctions. The frequency\nFourier transformation of this correlation function produces the dynamical spin susceptibility.\nThe frequency position of peaks in dynamical spin susceptibility are a ssociated with collective\nexcitation of electronic gas described by Kane-Mele model Hamiltonia n in the presence of\n13magnetic field. These excitations are related to the spin excitation s pectrum of electrons on\nHoneycomb structure. In the view point of experimental interpre tation, the dynamical spin\nsusceptibility of the localized electrons of the system is proportiona l to inelastic cross-section\nfor magnetic neutron scattering from a magnetic system that can be expressed in terms of spin\ndensity correlationfunctions ofthe system. Inother words thed ifferential inelastic crosssection\nd2σ/dΩdωis proportionalto imaginary partof spin susceptibilities. ωdenotes the energy loss of\nneutronbeamwhichisdefinedasthedifferencebetweenincidentand scatteredneutronenergies.\nThe solid angle Ω implies the orientation of wave vector of scattered n eutrons from the localized\nelectronsofthesample. Wecanassumethewave vectorofincident neutronsisalong zdirection.\nThe solidangle Ωdepends onthe polaranglebetween wave vector of s cattered neutrons andthe\nwave vector of the incident neutrons. The frequency position of p eaks ind2σ/dΩdωdetermines\nthe spin excitation spectrum of the magnetic system[29]. In order t o study the general spin\nexcitation spectrum of the localized electron of the systems, both transverse and longitudinal\ndynamical spin-spin correlation functions have been calculated. Lin ear response theory gives us\nthe dynamical spin response functions based on the correlation fu nction between components\nof spin operators. We introduce χ+−as transverse spin susceptibility and its relation is given\nby\nχ+−(q,ω) =i/integraldisplay+∞\n−∞dteiωt/an}bracketle{t[S+(q,t),S−(−q,0)]/an}bracketri}ht\n= lim\niΩn−→ω+i0+/integraldisplay1/(kBT)\n0dτeiΩnτ/an}bracketle{tTS+(q,τ)S−(−q,0)/an}bracketri}ht\n=χ+−(q,iΩn−→ω+i0+), (21)\nin which Ω n= 2nπkBTis the bosonic Matsubara frequency and Timplies the time order op-\nerator. Also the wave vector qin Eq.(21) implies the difference between incident and scattered\n14neutron wave vectors. The Fourier transformations of transve rse components of spin density\noperators, ( S+(−)), in terms of fermionic operators is given by\nS+(q) =/summationdisplay\nk/parenleftig\na†↑\nk+qa↓\nk+b†↑\nk+qb↓\nk/parenrightig\n, S−(q) =/summationdisplay\nk/parenleftig\na†↓\nk+qa↑\nk+b†↓\nk+qb↑\nk/parenrightig\n. (22)\nSubstituting the operator form of S+andS−into definition of transverse spin susceptibil-\nity in Eq.(21), we arrive the following expression for transverse dyn amical spin susceptibility\n(χ+−(q,iΩn))\nχ+−(q,iΩn) =/integraldisplay1/(kBT)\n0dτeiΩnτ1\nN2/summationdisplay\nk,k′/angbracketleftig\nT/parenleftig\na†↑\nk+q(τ)a↓\nk(τ)+b†↑\nk+q(τ)b↓\nk(τ)/parenrightig\n×/parenleftig\na†↓\nk+q(0)a↑\nk(0)+b†↓\nk+q(0)b↑\nk(0)/parenrightig/angbracketrightig\n, (23)\nthatNis the number of unit cells in Germanene structure. In order to calcu late the correlation\nfunction in Eq.(23), one particle spin dependent Green’s function ma trix elements presented in\nEq.(19) should be exploited. After applying Wick’s theorem andtaking Fourier transformation,\nwe can transverse susceptibility in terms of one particle spin depend ent Green’s function\nχ+−(q,iΩn) =−kBT\nN/summationdisplay\nk/summationdisplay\nα,β/summationdisplay\nmG↑\nβα(k,iωm)G↓\nαβ(k+q,iΩn+iωm). (24)\nThe applied magnetic field to the Germanene layer causes to the spin d ependent property for\none particle Green’s function. In order to perform summation over Matsubara frequency ωm\nin Eq.(24), the Matsubara Green’s function elements should be writt en in terms of imaginary\npart of retarded Green’s function matrix elements using Lehman eq uation[33] as\nGσ\nαβ(k,iωm) =/integraldisplay+∞\n−∞dǫ\n2π−2ImGσ\nαβ(k,iωm−→ǫ+i0+)\niωm−ǫ, (25)\nwhereGσ\nαβ(k,iωm−→ǫ+i0+) denotes the retarded Green’s function matrix element. By\nreplacing Lehman representation for Matsubara Green’s function matrix elements into Eq.(24)\n15andtaking summation over fermionic Matsubara frequency ωm, we obtaindynamical transverse\nspin susceptibility of electrons on Germanene structure as following form\nχ+−(q,iΩn) =−1\nN/summationdisplay\nk/summationdisplay\nα,β/integraldisplay+∞\n−∞/integraldisplay+∞\n−∞dǫdǫ′\nπ2ImG↑\nβα(k,ǫ+i0+)ImG↓\nαβ(k+q,ǫ′+i0+)\n×nF(ǫ)−nF(ǫ′)\niΩn+ǫ−ǫ′, (26)\nwherenF(x) =1\nex/kBT+1implies well known Fermi-Dirac distortion function. The Fourier\ntransformation of longitudinal component of the spin, i.e. Sz(q), is given in terms of fermionic\noperators as\nSz(q) =/summationdisplay\nk,σσ/parenleftig\na†σ\nk+qaσ\nk+b†σ\nk+qbσ\nk/parenrightig\n(27)\nAlsoχzzis introduced as longitudinal spin susceptibility and its relation can be e xpressed in\nterms of correlation function between zcomponent of spin operators as\nχzz(q,ω) =i/integraldisplay+∞\n−∞dteiωt/an}bracketle{t[Sz(q,t),Sz(−q,0)]/an}bracketri}ht\n= lim\niΩn−→ω+i0+/integraldisplay1/(kBT)\n0dτeiΩnτ/an}bracketle{tTSz(q,τ)Sz(−q,0)/an}bracketri}ht\n=χzz(q,iΩn−→ω+i0+). (28)\nAfter some algebraic calculations similar to transverse spin suscept ibility case, we arrive te final\nresults for Matsubara representation of longitudinal dynamical s pin susceptibility as\nχzz(q,iΩn) =−1\nN/summationdisplay\nk/summationdisplay\nα,β,σ/integraldisplay+∞\n−∞/integraldisplay+∞\n−∞dǫdǫ′\nπ2ImGσ\nβα(k,ǫ+i0+)ImGσ\nαβ(k+q,ǫ′+i0+)\n×nF(ǫ)−nF(ǫ′)\niΩn+ǫ−ǫ′, (29)\nThe dynamical spin structure factor for both longitudinal and tra nsverse spin directions are\nobtained based on retarded presentation of susceptibilities as\nχzz(q,ω) =χzz(q,iΩn−→ω+i0+), χ+−(q,ω) =χ+−(q,iΩn−→ω+i0+).(30)\n16so thatχzzandχ+−are retarded dynamical spin structure factors for longitudinal a nd trans-\nverse components of spins, respectively. The imaginary part of re tarded dynamical spin struc-\nture factor, i.e. Im χ+−(q,ω),Imχzz(q,ω), is proportional to the contribution of localized spins\nin the neutron differential cross-section. For each qthe dynamical structure factor has peaks at\ncertain energies which represent collective excitations spectrum o f the system. The imaginary\nparts of both retarded transverse and longitudinal spin structu re factors are given by\nImχ+−(q,ω) =1\nN/summationdisplay\nk/summationdisplay\nα,β/integraldisplay+∞\n−∞dǫ\nπImG��\nβα(k,ǫ+i0+)ImG↓\nαβ(k+q,ǫ+ω+i0+)\n×/parenleftig\nnF(ǫ)−nF(ǫ+ω)/parenrightig\n,\nImχzz(q,ω) =1\nN/summationdisplay\nk/summationdisplay\nα,β,σ/integraldisplay+∞\n−∞dǫ\nπImGσ\nβα(k,ǫ+i0+)ImGσ\nαβ(k+q,ǫ+ω+i0+)\n×/parenleftig\nnF(ǫ)−nF(ǫ+ω)/parenrightig\n. (31)\nThe frequencies of collective magnetic excitation modes are determ ined via finding the position\nof peaks in imaginary part of of imaginary part of dynamical spin susc eptibilities.\nStatic transverse spin structure factor ( s+−(q)) which is a measure of magnetic long range\nordering for spin components along the plane, i.e. transverse direc tion, can be related to\nimaginary part of retarded dynamical spin susceptibility using followin g relation\ns+−(q) =/angbracketleftig\nS+(q)S−(−q)/angbracketrightig\n=kBT/summationdisplay\nn1\n2π/integraldisplay∞\n−∞dω−2Imχ+−(q,iωn−→ω+i0+)\niωn−ω\n=/integraldisplay+∞\n−∞dωnB(ω)\nπImχ+−(q,iωn−→ω+i0+). (32)\nMoreover canfindthe staticlongitudinal spin structure factor, i.e .s+−(q), by using Im χzz(q,ω)\nas\nszz(q) =/an}bracketle{tSz(q)Sz(−q)/an}bracketri}ht=kBT/summationdisplay\nn1\n2π/integraldisplay∞\n−∞dω−2Imχzz(q,iωn−→ω+i0+)\niωn−ω\n=/integraldisplay+∞\n−∞dωnB(ω)\nπImχzz(q,iωn−→ω+i0+). (33)\n17In the next section, the numerical results of dynamical spin struc ture and static spin structures\nof Germanene layer have been presented for various magnetic field and spin-orbit coupling\nstrength.\n4 Numerical Results and Discussions\nWeturntoapresentationofourmainnumericalresultsofimaginary partofdynamicalstructure\nfactorsof Germanene layer atfinite temperature inthe presence of magnetic field andspin-orbit\ncoupling. Alsothetemperaturedependence ofstaticstructuref actorshasbeenaddressedinthis\nsection. Using the electronic band structure, the matrix elements of Fourier transformations\nof spin dependent Green’s function are calculated according to Eq.( 19). The imaginary part\nof both transverse and longitudinal dynamical spin susceptibilities is made by substituting the\nGreen’s function matrix elements into Eq.(31). In the following, the f requency behavior of\nimaginary part of dynamical spin susceptibilities is studied at fixed wav e number q= (0,4π\n3)\nin the Brillouin zone where the length of unit cell vector of honeycomb lattice is taken to be\nunit. Furthermore the static structure factors have been obta ined by using Eqs.(32,33).\nThe frequency behaviors of both the transverse and longitudinal dynamical spin suscepti-\nbilities have been addressed in this present study. Also the spin stru cture factors behaviors\nhave been investigated for Germanene structure.\nThe optimized atomic structure of the Germanene with primitive unit c ell vector length\na= 1 is shown in Fig.(1). The primitive unit cell include two Ge atoms.\nIn Fig.(3), we depict the frequency dependence of imaginary part o f longitudinal dynamical\nspin susceptibility, Imχzz(q0,ω), of undoped Germanene layer for different values of spin-\n18orbit coupling, namely λ/t= 0.08,0.12,0.16,0.2, in the absence of magnetic field by setting\nkBT/t= 0.05. In fact the effects of spin-orbit coupling strength on frequen cy dependence of\nImχzz(q,ω) have been studied in this figure. As shown in Fig.(3), the frequency positions\nof sharp peaks in Imχzz(q0,ω), that imply spin excitation mode for longitudinal components\nof spins, moves to higher frequencies with increase of spin-orbit co upling. This fact can be\nunderstood from this point that the increase of spin-orbit coupling leads to enhance band gap\nin density of states and consequently the excitation mode appears in higher frequency. Note\nthat this figure shows the inelastic cross section neutron particles from itinerant electrons of\nthe system due to longitudinal component along zdirection of magnetic moment of electrons\nand neutron beam. Another novel feature in Fig.(3) is the increase of intensity of sharp peak\nwithλ. For frequencies above normalized value 1.5, there is no collective ma gnetic excitation\nmode for longitudinal components of electron spins as shown in Fig.(3 ).\nThe frequency dependence of imaginary part of longitudinal dynam ical spin structure factor\nof undoped Germanene layer in the presence of spin polarization for different magnetic field\nvaluesgµBB/thas been shown in Fig.(4) at fixed λ/t= 0.6 by setting kBT/t= 0.05. A\nnovel feature has been pronounced in this figure. It is clearly obse rved that all curves for\ndifferent magnetic field indicates two clear magnetic excitation collect ive modes at frequencies\nω/t≈0.45 andω/t≈2.1. The frequency positions of magnetic excitation mode is independe nt\nof magnetic field. The intensity of sharp peaks in ω/t≈2.1 in imaginary part of longitudinal\nsusceptibility, i.e. Imχzz(q0,ω), decreases with magnetic field. However the height of sharp\npeaks in frequency position ω/t≈0.45 enhances with increase of spin-orbit coupling as shown\nin Fig.(4).\n19In Fig.(5), the imaginary part of longitudinal dynamical spin structu re factor of Germanene\nlayer has been plotted for different values of chemical potential, na melyµ/t= 0.2,0.3,0.4,0.5,\nat fixed spin-orbit coupling λ/t= 0.3 by setting kBT/t= 0.05 in the absence of magnetic\nfield. This figure implies the frequency position and intensity of collect ive excitation mode in\nω/t≈1.5 has no dependence on chemical potential. Although the intensity o f sharp peak in\nImχzz(q0,ω) at frequency position ω/t≈0.4 increases with chemical potential. There is no\nconsiderable change forfrequency positionin Imχzz(q0,ω)atω/t≈0.4 withchemical potential\naccording to Fig.(5). Also the intensity of low energy magnetic excita tion mode reduces with\ndecreasing chemical potential value µ.\nThe behavior of longitudinal static spin structure factor szz(q0) of undoped Germanene\nlayer in terms of normalized temperature kBT/tfor different values of λ/thas been presented\nin Fig.(6). The applied magnetic field is assumed to be zero. This functio n is a measure for\nthe tendency to magnetic long range ordering for the longitudinal c omponents of spins in the\nitinerant electrons. This figure implies static spin structure for spin components perpendicular\nto the Germanene plane includes a peak for each value of λ/t. The temperature position of\npeak in longitudinal spin static structure factor is the same for all s pin-orbit coupling strengths.\nAlthough the height of peak increases with λ/twhich justifies the long range ordering for\nzcomponents of spins improves with spin-orbit coupling. Another nov el feature is the non\nzero value for static structure factor szz(q0) at zero temperature. However the increase of\ntemperature up to peak position leads to raise magnetic long range o rdering. Upon more\nincreasing temperature above normalized value 0.25, the thermal fl uctuations causes to reduce\nszz(q0) so that magnetic long range ordering of the electrons decays as s hown in Fig.(6).\n20Thetemperaturedependenceofstaticlongitudinalspinstructur efactorofdopedGermanene\nlayer for various chemical potential has been studied in Fig.(7) for gµBB/t= 0.0 by setting\nλ/t= 0.4. In contrast to the undoped case in Fig.(6), it is clearly observed t he longitudinal\nspin structure factor for each value finite chemical potential get s zero value at zero temperature\naccording to Fig.(7). In fact the quantum fluctuations at zero tem perature for doped case leads\nto destroy any magnetic long range ordering. Moreover the peak in temperature dependence of\nszz(q0) tends to higher temperature upon increasing electron doping. Als o the height of peak\n,as a measure of magnetic long range ordering for longitudinal compo nents of spins, reduces\nwith chemical potential according to Fig.(7). Upon increasing tempe rature above normalized\nvalue 1.5,szz(q0) increases with µ/tand consequently the increase of electron doping improves\nthe long range ordering in temperature region above normalized valu e 1.5.\nThe effect of longitudinal magnetic field on the behavior of szz(q0) in terms of normalized\ntemperature kBT/tin undoped case for different values of magnetic field, namely gµBB/t=\n0.3,0.4,0.5,0.6,0.7 has been plotted in Fig.(8). The static structure is considerably aff ected by\nmagnetic field at low temperatures below normalized amount 0.5 where the quantum effects\nare more remarkable. In addition, at fixed values of temperatures above normalized value 0.5,\nszz(q0) is independent of magnetic field and all curves fall on each other on the whole range\nof temperature in this temperature region. Also temperature pos ition of peak in longitudinal\nstatic spin structure factormoves to lower temperature with incr easing magnetic field according\nto Fig.(8). Moreover the height of peak enhances with magnetic field . It can be understood\nfrom the fact that applying magnetic field along zdirection perpendicular to the plane causes\nlong range ordering of zcomponents of spins of electrons which increases the longitudinal s tatic\n21structure factor at low temperatures below 0.5.\nWe have also studied the dynamical and static transverse spin stru cture factors of Ger-\nmanene layer in the presence of magnetic field and spin-orbit coupling . Our main results for\nimaginary part of dynamical transverse spin susceptibility of undop ed Germanene layer for\ndifferent spin-orbit coupling strengths at fixed temperature kBT/t= 0.05 in the absence of\nmagnetic field are summarized in Fig.(9). The collective magnetic excita tion modes for spin\ncomponents parallel to the plane tends to higher values with λ/taccording to Fig.(9). This\nfeature arises from the increase of band gap with spin-orbit couplin g so that collective mode ap-\npears at higher frequencies as shown in Fig.(9). Also the height of pe ak in Imχ+−(q0,ω), which\nis proportional to intensity of scattered neutron beam from itiner ant electrons of Germanene\nstructure, reduces with decreasing spin-orbit coupling strength .\nIn Fig.(10) we plot the numerical results of Im χ+−(q,ω) of undoped Germanene layer as a\nfunctionofnormalizedfrequency ω/tforvariousmagneticfield,namely gµBB/t= 0.0,0.2,0.4,0.6\nby settingkBT/t= 0.05. It is clearly observed that the frequency position of collective m ag-\nnetic excitation mode moves to lower values with magnetic field. It can be understood from the\nfact that the applying magnetic field gives rise to reduce the band ga p and consequently the\nexcitation mode takes place at lower frequencies. Moreover the int ensity of collective excita-\ntion mode for transverse components of spins is clearly independen t of magnetic field strength\naccording to Fig.(10).\nFig.(11) presents the effect of electron doping on the frequency d ependence of Im χ+−(q0,ω)\nofGermanenelayerbysetting λ/t= 0.3atfixedvalueoftemperatureintheabsenceofmagnetic\nfield. There are two collective magnetic excitation mode for each valu e chemical potential. The\n22frequency positions of excitation mode are the same for all values o f chemical potential. The\nintensity of low frequency peak increases with chemical potential h owever the intensity of high\nfrequency peak reduces with electron doping as shown in Fig.(11). T he increase of electron\ndoping leads to decrease transition rate of electron from valence b and to conduction one and\nconsequently the intensity of excitation mode decreases.\nThe behavior of transverse static spin structure factor s+−(q0) of undoped Germanene\nlayer as a function of normalized temperature kBT/tfor different values of λ/tin the absence\nof magnetic field has been presented in Fig.(12). This function is a mea sure for the tendency to\nmagnetic long range ordering for the transverse components of s pins in the itinerant electrons.\nA peak ins+−(q0) is clearly observed for each value of λ/t. The peak is located at 0.25 for all\nvalues of spin-orbit coupling strengths. Although the height of pea k increases with λ/twhich\njustifies the long range ordering for transverse components of s pins improves with spin-orbit\ncoupling. Another novel feature is the non zero value for static st ructure factor s+−(q0) at zero\ntemperature. In temperature region below peak position, the incr ease of temperature leads to\nraise magnetic long range ordering. Upon more increasing temperat ure above normalized value\n0.25, the thermal fluctuations causes to reduce szz(q0) and magnetic long range ordering of the\nelectrons as shown in fig.(12).\nThe temperature dependence of static transverse spin structu re factor of doped Germanene\nlayer for various chemical potential in the absence of magnetic field has been studied in Fig.(13)\nby settingλ/t= 0.3. In contrast to the undoped case in Fig.(12), it is clearly observed the\ntransverse spin structure factor for each value finite chemical p otential gets zero value at zero\ntemperature according to Fig.(13). In fact the quantum fluctuat ions at zero temperature for\n23doped case leads to destroy any magnetic long range ordering. Mor eover the temperature\nposition of peak in s+−(q0) appears in kBT/t= 0.5 for all amounts of chemical potential. Also\nthe height of peak ,as a measure of magnetic long range ordering for transverse components\nof spins, reduces with chemical potential according to Fig.(13). In other words the increase of\nelectron doping leads to decrease magnetic long range ordering for transverse components of\nspins.\nThe effect of longitudinal magnetic field on the behavior of s+−(q0) in terms of normalized\ntemperature kBT/tin undoped case for different values of magnetic field, namely gµBB/t=\n0.4,0.45,0.5,0.55,0.6 has been plotted in Fig.(14). The static transverse spin structur e factor\nis considerably affected by magnetic field at low temperatures below n ormalized amount 0.25\nwhere the quantum effects are more remarkable. In addition, at fix ed values of temperatures\nabove normalized value 0.25, s+−(q0) is independent of magnetic field and all curves fall on\neach other on the whole range of temperature in this temperature region. Also temperature\nposition of peak in longitudinal static spin structure factor moves t o lower temperature with\nincreasing magnetic field. Moreover the height of peak enhances wit h magnetic field. It can be\nunderstood from the fact that applying magnetic field along zdirection perpendicular to the\nplane causes long range ordering of components of spins parallel to the plane of electrons which\nincreases the longitudinal static structure factor at low tempera tures below 0.25.\n5 Conflict Of interest Statement\nThere is no conflict of interest statement in this manuscript.\n24Figure 1: Crystal structure of Germanene\nFigure 2: The structure of honeycomb structure is shown. The ligh t dashed lines denote the\nBravais lattice unit cell. Each cell includes two nonequivalent sites, wh ich are indicated by A\nand B.a1anda2are the primitive vectors of unit cell.\n250 0.5 1 1.5 2 2.5 300.050.10.150.2\n0.08\n0.12\n0.16\n0.2\nω/tλ/t\nχ  (   ,ω)q\n0zzIm\nFigure 3: The imaginary part of dynamical longitudinal spin susceptib ility Imχzz(q0,ω) of\nundoped Germanene layer versus normalized frequency ω/tfor different values of normalized\nspin-orbit coupling λ/tat fixed temperature kBT/t= 0.05. The magnetic field is considered to\nbe zero.\n260 0.5 1 1.5 2 2.5 300.511.52\n0.6\n0.7\n0.8\n0.9\nω/tB/tµBg\nImχ  (   ,ω)q0zz\nFigure 4: The imaginary part of dynamical longitudinal spin susceptib ility Imχzz(q0,ω) of\nundoped Germanene layer versus normalized frequency ω/tfor different values of normalized\nmagnetic field gµBB/tat fixed spin-orbit coupling λ/t= 0.6. The normalized temperature is\nconsidered to be kBT/t= 0.05.\n270 0.5 1 1.5 2 2.5 300.10.20.30.40.5\n0.2\n0.3\n0.4\n0.5µ/\ntω/t\nImχ  (   ,ω)0qzz\nFigure 5: The imaginary part of dynamical longitudinal spin susceptib ility Imχzz(q0,ω) of\ndoped Germanene layer versus normalized frequency ω/tfor different values of normalized\nchemical potential µ/tat fixed spin-orbit coupling λ/t= 0.3. The magnetic field is assumed to\nbe zero. The normalized temperature is considered to be kBT/t= 0.05.\n280 1 2 3 4 5012345\n0.2\n0.3\n0.4\n0.5\n0.6\ns  (q  , T)zz0λ/t\nk T/tB\nFigure 6: The longitudinal static spin structure factor szz(q0,T) of undoped Germanene layer\nversus normalized temperature kBT/tfor different values of spin-orbit coupling strength λ/tin\nthe absence of magnetic field.\n290 1 2 3 4 500.511.52\n1.0\n1.2\n1.4\n1.6\n1.8\n2.0\ns  (   ,T)q0µ/t\nk  T/tBzz\nFigure 7: The longitudinal static spin structure factor szz(q0,T)of undoped Germanene layer\nversus normalized temperature kBT/tfor different values of chemical potential µ/tin the ab-\nsence of magnetic field. Spin-orbit coupling strength has been fixed atλ/t= 0.4.\n300 1 2 3 4 502468 0.3\n0.4\n0.5\n0.6\n0.7\ns  (   ,T)q0zz\nk  T/tBB/tµ\nBg\nFigure 8: The longitudinal static spin structure factor szz(q0,T) of undoped Germanene layer\nversus normalized temperature kBT/tfor different values of magnetic field gµBB/tat fixed\nspin-orbit coupling strength λ/t= 0.4.\n310 1 2 3 4 501234\n0.3\n0.4\n0.5\n0.6λ/t\nImχ  (   ,ω)q0+-\nω/t\nFigure 9: The imaginary part of dynamical transverse spin suscept ibility Imχ+−(q0,ω) of\nundoped Germanene layer versus normalized frequency ω/tfor different values of normalized\nspin-orbit coupling λ/tat fixed temperature kBT/t= 0.05. The magnetic field is considered to\nbe zero.\n320 1 2 3 4 501234\n0.0\n0.2\n0.4\n0.6\nImχ  (   ,ω)+-B/tµg\nB\nω/tq0\nFigure 10: The imaginary part of dynamical transverse spin suscep tibility Imχ+−(q0,ω) of\nundoped Germanene layer versus normalized frequency ω/tfor different values of normalized\nmagnetic field gµBB/tat fixed spin-orbit coupling λ/t= 0.6. The normalized temperature is\nconsidered to be kBT/t= 0.05.\n330 1 2 3 4 5012345\n0.7\n0.75\n0.8\n0.85\n0.9µ/t\nImχ  (   ,ω)+-q\n0\nω/t\nFigure 11: The imaginary part of dynamical transverse spin suscep tibility Imχ+−(q0,ω) of\ndoped Germanene layer versus normalized frequency ω/tfor different values of normalized\nchemical potential µ/tat fixed spin-orbit coupling λ/t= 0.3. The magnetic field is assumed to\nbe zero. The normalized temperature is considered to be kBT/t= 0.05.\n340 1 2 3 4 501234\n0.1\n0.3\n0.5\n0.7\n0.9\ns  (q )+-λ/t\nT/t kB\nFigure 12: The transverse static spin structure factor s+−(q0,T) of undoped Germanene layer\nversus normalized temperature kBT/tfor different values of spin-orbit coupling strength λ/tin\nthe absence of magnetic field.\n350 1 2 3 4 500.20.40.60.8 0.1\n0.2\n0.3\n0.4\ns  (q )+-0µ/\nT/tBkt\nFigure 13: The transverse static spin structure factor s+−(q0,T) of doped Germanene layer\nversusnormalizedtemperature kBT/tfordifferentvaluesofchemicalpotential µ/tintheabsence\nof magnetic field with spin-orbit coupling λ/t= 0.3.\n360 0.5 1 1.5 2 2.5 3012345\n0.4\n0.45\n0.5\n0.55\n0.6\ns  (q )+-\nT/tkBgµBB/t\nFigure 14: The transverse static spin structure factor s+−(q0,T)of undoped Germanene layer\nversus normalized temperature kBT/tfor different values of magnetic field gµBB/t. Spin-orbit\ncoupling strength has been fixed at λ/t= 0.4.\n37References\n[1] K. S. Novoselov, A. K. Geim, S. V. Morosov, D. Jiang, Y. Zhang, S . V. Dubonos, I. V.\nGrigorieva and A. A. Firsov, Electric field effect in atomically thin carbo n films, Science\n306, 666 (2004)\n[2] X. -L. Wang, X. S. Dou and C. Zhang, zero gap materials for futu re spintronics, NPG Asia\nMaterials 2, 31 (2010)\n[3] Y.Xu, Y.Liu, H.Chen, X.Lin, B.Yu, J.Luo, Anabinitiostudyofen ergy-bandmodulation\nin Graphene-based two dimensional layered superlattices, J. Mat. Chem 22, 23821 (2012)\n[4] K. Chang and W. X. Chen, In situ synthesis of MoS 2/Graphene nanosheet composites\nwith extraordinarily high electrochemical performance for lithium ion batteries, Chem.\nCommunication 47, 4252 (2011)\n[5] K. Chang and W. X. Chen, I-Cysteine-Assisted synthersis of La yered MoS 2/ Graphene\ncomposites with excellent electrochemical performance for lithium io n batteries, Acs Nano\n5, 4720 (2011)\n[6] C. R. Dean, etal, boron nitride substrates for high-quality Graphene electonics, N at. Nan-\notechnol 5, 722 (2010)\n[7] K. S. Novoselov, D. Jiang, F. Schedin, T. J. Booth, V. V Khotkev ich, S. V. Mosorov, A.\nK. Geim, Proc, Natl, Two dimensional atomic crystals, Acad. Sci. USA 102, 10541 (2005)\n[8] S. Bertolazzi, D. Krasnozhon and A. Kis, Nonvolatile memory cells b ased on\nMos2/Graphene heterostructures, Acs Nano 7, 3246 (2013)\n38[9] S. Cahangirov, M. Topsakal, E. Akturk, H. Sahin and S. Ciraci, tw o-and one-dimensional\nhoneycomb structures of silicon and Germanene, Phys. Rev. Lett 102, 236804 (2009)\n[10] C. C. Liu, W. Feng and Y. Yao, Quantum spin hall effect in Silicene an d two-dimensional\nGermanium, Phys. Rev. Lett 107, 076802 (2011)\n[11] J. E. Padilha and R. B. Pontes, free-standding bilayer silicene: T he effect of stacking order\non structural, electronic, and transport properties, The Journ al of Chemistry C 119, 3818\n(2015)\n[12] S. Chowdhury and D. Jana, A theoretical review on electonic, m agnetic and optical prop-\nerties of silicene, Reports on Progress in Physics 79, 126501 (2016 )\n[13] T. P. Kaloni, Tuning the structural, electronic, and magnetic pr operties of Germanene by\nthe adsorption of 3d transition metal atoms, The Journal of Phys ical Chemistry C 118,\n25200 (2014)\n[14] C. -C. Liu, H. Jiang and Y. Yao, Low energy effective Hamiltonian in volving spin-orbit\ncoupling in silicene and two dimensional germanium and Tin, Phys. Rev .B 8 4, 195430\n(2011)\n[15] X. -S. Ye, etal, intrinsic carrier mobility of Germanene is larger than Graphene’s: fir st\nprinciple calculations, Rsc Advances 4, 21216 (2016)\n[16] M. M. Monshi, S. M. Aghaei and I. Calizo, DFT study of adsorptio n behavior NO, Co,\nNO2 nad NH3 molecules on Graphene like BC3: A search for highly sensit ive molecular\nsensor, Surface Science 665, 96 (2017)\n39[17] M. Sun, Magnetism in transition-metal-doped Germanene: a firs t principles study, Com-\nputational Materials Science 118, 112 (2016)\n[18] X. Li, S. Wu, S. Zhouand Z. Zhu, structural and electronic pro perties of Germanene/MoS 2\nmonolayer and silicene/MoS 2monolayer superlattices, Nanoscale Research Letters 9, 110\n(2014)\n[19] Z. Qiao, etal, Quantum Anomalous hall effact in Graphene from Rashba and excha nge\neffects, Phys. Rev. B 82, 161414 (R) (2010)\n[20] W. -K. Tse, etal, Quantum anomalous hall effect in single-layer and bilayer Graphene,\nPhys. Rev. B 83, 155447 (2011)\n[21] C. L. Kane and E. J. Mele, Z2 Topological order and the quantum spin hall effect, Phys.\nRev. Lett 95, 146802 (2005)\n[22] F. D. M. Haldane, Model for a quantum hall effect without landau levels: condensed\nmatter realization of the ”parity anomaly” Phys. Rev. Lett 61, 201 5 (1988)\n[23] H. Min, J. E. Hill, N.A. Sinitsyn, B. R.Sahu, L. Kleinman, andA. H.M a Donald, Intrinsic\nand rashba spin-orbit interactions in Graphene sheets, Phys. Rev . B 74, 165310 (2006)\n[24] Y. Yao, F. Ye, X. -L. Qi, S. -C. Zhang, and Z. Fang, Spin orbit ga p of Graphene: First\nprinciples calculations, Phys. Rev. B 75, 041401 (2007)\n[25] E. H. Hwang and S. Das Sarma, Graphene magnetoresistance in a parallel magnetic field\nspin polarization effect, Phys. Rev. B 80, 075417 (2009)\n40[26] R. G. Mani, J. H. Smet, K. von Klitzing, V. Narayanamurti, W. B. J ohnson and V. Uman-\nsky, Zero-resistance staes induced by electromagnetic-wave ex citation in GaAs/AlGaAs,\nNature (London) 420, 646 (2002)\n[27] E. H. Hwang and S. Das Sarma, Phys. Rev. B 75, 205418 (2007) ; B. Wunsch etalNew J.\nPhy 8, 318 (2006)\n[28] Y. Liu, R. F. Willis, K. V. Emtsev, and T. Seyller, Phy. Rev. B 78, 20 1403 (2008)\n[29] S. Doniach and E. H. Sondheimer, Green’s functions for solid sta te physicists (Imperial\nCollege Press, London, 1999)\n[30] T. Stauber, J. Schliemann and N M Peres, Phys. Rev. B 81, 0854 09 (2010)\n[31] M. Inglot, V. K. Dugaev, E. Ya. Sherman and J. Barnas, Phys. Rev. B 91, 195428 (2015)\n[32] M. Inglot, V. K. Dugaev, E. Ya. Sherman and J. Barnas, Phys. Rev. B 89, 155411 (2014)\n[33] G. D. Mahan, Many Particle Physics, Plenumn Press, New York, 1 993\n41"    },    {        "title": "2205.07901v1.Emergent_tracer_dynamics_in_constrained_quantum_systems.pdf",        "content": "Emergent tracer dynamics in constrained quantum systems\nJohannes Feldmeier,1, 2William Witczak-Krempa,3, 4, 5and Michael Knap1, 2\n1Department of Physics, Technical University of Munich, 85748 Garching, Germany\n2Munich Center for Quantum Science and Technology (MCQST), Schellingstr. 4, D-80799 M unchen, Germany\n3Universit\u0013 e de Montr\u0013 eal, C.P. 6128, Succursale Centre-ville, Montr\u0013 eal, QC, Canada, HC3 3J7\n4Institut Courtois, Universit\u0013 e de Montr\u0013 eal, Montr\u0013 eal (Qu\u0013 ebec), H2V 0B3, Canada\n5Centre de Recherches Math\u0013 ematiques, Universit\u0013 e de Montr\u0013 eal, Montr\u0013 eal, QC, Canada, HC3 3J7\n(Dated: May 18, 2022)\nWe show how the tracer motion of tagged, distinguishable particles can e\u000bectively describe trans-\nport in various homogeneous quantum many-body systems with constraints. We consider systems\nof spinful particles on a one-dimensional lattice subjected to constrained spin interactions, such\nthat some or even all multipole moments of the e\u000bective spin pattern formed by the particles are\nconserved. On the one hand, when all moments|and thus the entire spin pattern|are conserved,\ndynamical spin correlations reduce to tracer motion identically, generically yielding a subdi\u000busive\ndynamical exponent z= 4. This provides a common framework to understand the dynamics of\nseveral constrained lattice models, including models with XNOR or tJz{ constraints. We consider\nrandom unitary circuit dynamics with such a conserved spin pattern and use the tracer picture\nto obtain exact expressions for their late-time dynamical correlations. Our results can also be ex-\ntended to integrable quantum many-body systems that feature a conserved spin pattern but whose\ndynamics is insensitive to the pattern, which includes for example the folded XXZ spin chain. On\nthe other hand, when only a \fnite number of moments of the pattern are conserved, the dynamics\nis described by a convolution of the internal hydrodynamics of the spin pattern with a tracer distri-\nbution function. As a consequence, we \fnd that the tracer universality is robust in generic systems\nif at least the quadrupole moment of the pattern remains conserved. In cases where only total\nmagnetization and dipole moment of the pattern are constant, we uncover an intriguing coexistence\nof two processes with equal dynamical exponent but di\u000berent scaling functions, which we relate to\nphase coexistence at a \frst order transition.\nCONTENTS\nI. Introduction 1\nII. Models and spin correlations 3\nA. Constrained tJ{ like systems 3\nB. Generic structure of dynamical spin\ncorrelations 3\nIII. Random unitary circuits with conserved pattern 5\nA. Random circuit tJz{ model 5\nB. Random circuit XNOR model 6\n1. Conserved pattern: spin picture 7\n2. Conserved pattern: domain wall picture 8\nC. Random circuit dimer model 10\nIV. Integrable quantum systems with conserved\npattern 10\nA. Integrable tJz{ model 11\nB. Folded XXZ spin chain 11\nV. Broken pattern conservation 12\nA.m= 0: Di\u000busion 13\nB.m\u00152: Tracer di\u000busion 13\nC.m= 1: Hydrodynamic phase coexistence 13\nVI. Conclusion & Outlook 15\nA. Strength of staggered oscillations 16References 17\nI. INTRODUCTION\nRecent years have seen rapid progress in quantum sim-\nulation technology, with increasing capacity to directly\nprobe the out-of-equilbrium properties of many-body\nquantum systems. These advances have led to immense\ntheoretical and experimental interest in the thermaliza-\ntion of closed, interacting quantum many-body systems\ntowards translationally invariant equilibrium states [1{6].\nA breakthrough has been the key realization that the late\ntime dynamics in such systems can be understood via an\nemergent, e\u000bectively classical hydrodynamic description.\nThis includes the di\u000busive transport of local densities in\nsystems with global conserved charges [7{10], as well as\nthe dynamics of entanglement [11{15] and quantum in-\nformation [16{21].\nThe remarkable emergence of classical hydrodynam-\nics from a closed quantum time evolution is currently\nalso being explored in many-body systems featuring more\nexotic conservation laws|or constraints |such as gauge\ntheories and fractonic quantum matter [22{33]. Cru-\ncially, constraints generally have a qualitative impact\non the thermalization process of many-body systems to-\nwards equilibrium. While in some instances the presence\nof fractonic constraints can be as severe as precluding\nthermalization altogether [34{37], in many others they\nlead to novel subdi\u000busive universality classes of (emer-arXiv:2205.07901v1  [cond-mat.str-el]  16 May 20222\ngent) hydrodynamic relaxation of the non-equilibrium\nevolution at late times [38{52].\nIn this work, we study the emergence of another classi-\ncal process in the dynamics of interacting quantum many-\nbody systems: the tracer motion of tagged particles.\nWhile at \frst sight the notion of a tagged particle ap-\npears to be at odds with the indistinguishability of quan-\ntum particles in many-body systems, here we show how\nthe e\u000bects of kinetic constraints can nonetheless lead to\nthe emergence of such tracer motion. For this purpose we\nfocus on the dynamics of one-dimensional systems with\na conserved pattern of e\u000bective spins or charges through-\nout much of this work, see Fig. 1 for an illustration.\nThis setup is similar to certain nearest-neighbor simple\nexclusion processes in classical two-component systems,\nwhere tracer motion describes the local component im-\nbalance [53]. Similar constraints have recently also been\ndiscussed in the context of fractonic quantum systems\nin terms of \\Statistically Localized Integrals of Motion\"\n(SLIOMs) [36], which can be interpreted as an e\u000bective\nconserved pattern.\nMore generally, we investigate the dynamics of local\nspin correlations in one-dimensional systems featuring a\nconserved number of spinful particles. The setup is sim-\nilar to thetJ{ model, which consists of spinful fermions\nwith the condition of no double occupancies. In our case,\nthe usual Heisenberg spin exchange is substituted by con-\nstrained spin interactions: We require that some or even\nall multipole moments of the spin pattern formed by the\nparticles are conserved. For much of this work we focus\non random unitary circuits that satisfy these constraints.\nWe will therefore call the systems studied in this work\n`tJ{ like'. We \fnd that the anomalously slow tracer\ndi\u000busion of hard core particles in one dimension plays a\nvital role in describing their dynamical spin correlations.\nThe mapping between spin correlations and tracer dy-\nnamics becomes exact for systems with an exactly con-\nserved spin pattern, where the tracer motion gives rise to\na subdi\u000busive dynamical exponent z= 4. Such systems\nare similar in structure to the tJz{ model, where spin\ninteractions diagonal in the z-basis preserve the spin pat-\ntern. We thus call such systems ` tJz{ like'. This frame-\nwork yields a unifying picture to understand the dynam-\nics of constrained lattice models studied in recent works\nthat can be mapped|either directly or e\u000bectively|to a\ntJz{ like structure [36, 43, 46, 54, 55]. We use this pic-\nture to derive the full long-time pro\fle of the dynamical\nspin correlations in a random unitary tJz{ circuit model\nand a random XNOR circuit [46].\nAlthough our main focus is on the dynamics of generic\nsystems, we demonstrate that the tracer picture is appli-\ncable also to certain integrable quantum systems. These\nfeature an e\u000bective conserved spin pattern but their dy-\nnamics per se is insensitive to this pattern. As examples\nwe consider the integrable Jz!0 limit of the tJz{ model\nand the folded XXZ chain [56{59]. Through the tracer\npicture we are able to reproduce their spin di\u000busion con-\nstants at in\fnite temperature and predict the full pro\fle\n/uni2191/uni2193/uni2191 /uni2193/uni2191 /uni2193/uni2191/uni2191/uni2193/uni2193 /uni2191 /uni2191/uni2191 /uni2193/uni2191 /uni2193/uni2191 /uni2193/uni2191/uni2191/uni2193/uni2193 /uni2191 /uni2191/uni2191 /uni2193/uni2191 /uni2193/uni2191 /uni2193/uni2191/uni2191/uni2193/uni2193/uni2191/uni2191 /uni2193/uni2191 /uni2193/uni2191 /uni2193/uni2191/uni2191/uni2193/uni2193/uni2191\n/uni2191/uni2191time\nspace+ − +\ntime\nspaceintegrable\nmodel (z= 2)\n<latexit 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Tracer di\u000busion in constrained quantum sys-\ntems. a) We consider one-dimensional systems with a pat-\ntern of e\u000bective excitations (blue and red squares) which is\nconserved during the time evolution. In\fnite temperature\ndynamical correlation functions in such an ensemble map di-\nrectly on the tracer probability distribution of hard core ran-\ndom walkers. The fundamental objects of tracer di\u000busion are\nthe e\u000bective excitations of the conserved pattern. b)Solving\nthe tracer problem provides us with quantitatively accurate\ndescriptions of transport in a number of generic random uni-\ntary circuit models as well as integrable quantum systems.\nThe e\u000bective conserved patterns can assume a complex struc-\nture as in the quasi one-dimensional dimer model studied in\nRef. [43].\nof their spin correlations at late time, in agreement with\nour numerical simulations.\nWe then consider models in which only a \fnite number\nof moments of the spin pattern are conserved. The re-\nsulting spin correlations are given by a convolution of the\ntracer motion and the internal dynamics of the pattern.\nAs a consequence, we \fnd that the tracer-motion univer-\nsality is robust to breaking the pattern conservation if all\nmoments up to at least the quadrupole moment of the\npattern are conserved. In addition, for dipole-conserving\nspin interactions we uncover a competition between two\nhydrodynamic processes that both have dynamical expo-\nnentz= 4 but that exhibit di\u000berent scaling functions.\nThe long-time pro\fle of the spin correlations is then de-\nscribed by a non-universal mixture of these two scaling\nfunctions. We argue that this intriguing situation is rem-\niniscent to phase coexistence at a \frst order transition\nbetween a Gaussian and a non-Gaussian hydrodynamic\nphase.\nThe remainder of this paper is structured as follows:\nIn Sec. II we introduce the tJ{ like models studied in\nthis work and derive a general expression for their spin\ncorrelations at late times. We apply these results to spe-\nci\fc random unitary circuit examples in Sec. III, treating3\nin detail the random XNOR model [46]. We consider two\nintegrable models in Sec. IV and discuss cases where only\na \fnite number of multipole moments of the pattern are\nconserved in Sec. V.\nII. MODELS AND SPIN CORRELATIONS\nWe introduce a novel class of tJ{ like many-body\nsystems of spinful particles in one dimension with con-\nstrained spin interaction terms. The constraints are such\nthat either the entire spin pattern or a \fnite number\nof multipole moments of the pattern are conserved. We\nderive a general expression for the in\fnite temperature\ndynamical spin correlations at late times in such systems.\nA. Constrained tJ{ like systems\nWe are interested in conservation laws inherent to\nmodels of the form\n^Hm=\u0000tL=2\u00001X\nx=\u0000L=2X\n\u001b(^~cy\nx+1;\u001b^~cx;\u001b+h:c:) +^HS;m\n[^HS;m;X\nxxn^Sz\nx] = 08n\u0014m:(1)\nEq. (1) describes a constant number of spinful fermions\nwith nearest-neighbor hopping on a one-dimenional lat-\ntice, with the usual tJ{ constraint of no double occu-\npancies (indicated by the tilde over the fermion opera-\ntors; the fermionic nature of the particles is not essential\nhere). A spin pattern is then formed by the fermions\nin a squeezed space where all empty sites are removed.\nThe spin interaction ^HS;mbetween the fermions is gen-\neralized to not only conserve the total magnetization but\npotentially higher moments of this spin pattern (all up\nto themth moment) as well. Examples for ^HS;minclude\n^HS;0=JX\nx^Sx\u0001^Sx+1+:::\n^HS;1=JX\nx(^S+\nx^S\u0000\nx+1^S\u0000\nx+2^S+\nx+3+h:c:) +:::\n:::\n^HS;1=JX\nx^Sz\nx^Sz\nx+1+:::;(2)\nwhere `...' refers to diagonal terms in the z\u0000basis or to\nlonger-range o\u000b-diagonal terms that ful\fll the conserva-\ntion law of Eq. (1). We note that ^H0=^HtJis a con-\nventionaltJ{ model while ^H1=^HtJzis atJz{ model\nin which the entire spin pattern is a constant of motion.\nLattice spin models such as Eq. (2) provide a novel way\nof interpolating between these two limiting cases; one can\nconstruct such models recursively [39].The Hamiltonians of Eqs. (1,2) serve as our starting\nmotivation and we can qualitatively determine their uni-\nversal late-time dynamics at high energies by consider-\ninggeneric many-body systems with the same Hilbert\nspace structure and conserved quantities. To intro-\nduce a model-independent notation we expand any state\nj iwith a \fxed number Nfof particles asj i=P\nx;\u001b (x;\u001b)jx;\u001biin terms of the basis states\njx;\u001bi; x 1<:::<xNf; \u001bi2f\u0006 1g: (3)\nHere,xlabels the positions of the particles on the chain\nfrom left to right and \u001btheir respective spins in the\nz\u0000basis. The time evolution represented by the unitary\n^Um(t) should then ful\fll\n[^Um(t);NfX\nj=1jn^\u001bj] = 08n\u0014m; (4)\nand can be Hamiltonian, such as in Eq. (1), or generic,\nsuch as in random unitary quantum circuits or classical\nstochastic lattice gases; either case is expected to exhibit\nthe same universal dynamical behavior. We emphasize\nthat the conservation of moments in Eq. (4) applies to\nthe squeezed-space variables of the pattern, which are\nrelated to the original spins non-locally. In particular,\nthe momentsP\nxxn^Sz\nxin the original spin space are not\nconserved due to the hopping part of Eq. (1). We will\nmake use of the non-local property Eq. (4) throughout\nour work and show that various constrained models stud-\nied recently are part of this e\u000bective description.\nB. Generic structure of dynamical spin correlations\nWe derive a general expression for the dynamical spin\ncorrelations in a system described by Eqs. (3,4) under the\nassumption of chaotic, thermalizing dynamics at in\fnite\ntemperature. Integrable dynamics will be considered in\nSec. IV. We will assume open boundary conditions in\na system of length Lcontaining a \fxed number of Nf\nparticles, i.e. density \u001a=Nf=L. The spin operator ^Sz\nr\nat sitercan then be expressed in terms of the pattern\nspin operators ^ \u001bjvia\n^Sz\nr=NfX\nj=1\u000e^xj;r^\u001bj: (5)\nThe time evolution ^Um(t) applied to a basis state jx;\u001bi\nis given by\n^Um(t)jx;\u001bi=X\nx0;\u001b0a(x0\u001b0jx\u001b;t)jx0;\u001b0i; (6)\nwith the matrix elements a(x0\u001b0jx\u001b;t) normalized toP\nx0;\u001b0ja(x0\u001b0jx\u001b;t)j2= 1. Using Eqs. (5,6), the dy-4\nnamical spin correlations read\nC(r;t) :=h^Sz\nr(t)^Sz\n0(0)i=\n=1\nNX\nx;\u001b;i\nx0;\u001b0;j\u001b0\nj\u001bi\u000ex0\nj;r\u000exi;0ja(x0\u001b0jx\u001b;t)j2;(7)\nwhere the normalization N=NnNsis given by the num-\nber of di\u000berent particle position Nn=\u0000L\nNf\u0001\non the lat-\ntice and the number of di\u000berent spin patterns Ns= 2Nf.\nThe expectation value h\u0001iis taken with respect to an `in-\n\fnite temperature' ensemble over all basis states. Under\n^Um(t), both the number of particles as well as the total\nmagnetization of the spin pattern are conserved and we\nexpect both of their local densities to contribute a hydro-\ndynamic mode at long length scales and late times. In\ngeneral, the precise transport coe\u000ecients of the particle\nmode and the spin pattern mode are determined by a\nmode-coupled Ansatz and the details of the microscopic\ntime evolution. Nonetheless, we expect that qualitatively ,\nwe can describe the hydrodynamic behavior of the two\nmodes independently at late times in thermalizing sys-\ntems. We therefore make the approximation to set\nja(x0\u001b0jx\u001b;t)j2'pn(x0jx;t)ps(\u001b0j\u001b;t) (8)\nin Eq. (7), where we introduced the particle and spin\npath distributions pn(x0jx;t) andps(\u001b0j\u001b;t); they ful\fllP\nx0pn(x0jx;t) = 1 =P\n\u001b0ps(\u001b0j\u001b;t). The spin correla-\ntions of Eq. (7) are thus determined by the following two\nexpressions which describe spin pattern dynamics and\nparticle dynamics, respectively:\nF(j\u0000i;t) :=1\nNsX\n\u001b;\u001b0\u001b0\nj\u001bips(\u001b0j\u001b;t)\nK(j\u0000i;r;t) :=1\nNnX\nx;x0\u000ex0\nj;r\u000exi;0pn(x0jx;t) =\n=P(j;rji;0;t)P(i;0):(9)\nIn the last step we introduced the probability P(i;0) to\n\fnd theith particle (counted from the left) at site x= 0,\nas well as the probability P(j;rji;0;t) to \fnd the jth\nparticle at site x=rat timetgiven that particle iwas\nlocated at site x= 0 at time 0. We can rewrite the latter\nprobability as\nP(j;rji;0;t) =X\n`P(j;rji;`; 0)P(i;`ji;0;t): (10)\nWe notice that P(j;rji;`; 0) in Eq. (10) is simply the\nprobability to \fnd particle jatrgiven that particle i\nis at`at the same time. It has the exact expression ( \u0012(\u0001)is the Heaviside theta function)\nP(j;rji;`; 0) =\u000ei\u0000j;0\u000er\u0000`;0+\n+\u0012(j\u0000i\u00001)\u0012(r\u0000`\u00001)\u0000r\u0000`\u00001\nj\u0000i\u00001\u0001\u0000L\u0000r+`\u00001\nN\u0000j+i\u00001\u0001\n\u0000L\nN\u0001 +\n+\u0012(i\u0000j\u00001)\u0012(`\u0000r\u00001)\u0000`\u0000r\u00001\ni\u0000j\u00001\u0001\u0000L\u0000`+r\u00001\nN\u0000i+j\u00001\u0001\n\u0000L\nN\u0001\nji\u0000jj\u001d1\u00191q\n\u0019\u00001\n\u001a2\u00001\n\u001a\u0001\njj\u0000ijexp(\n\u0000\u0002\nr\u0000`\u0000j\u0000i\n\u001a\u00032\n\u00001\n\u001a2\u00001\n\u001a\u0001\njj\u0000ij)\n;\n(11)\nwhere in the last line we made an approximation for large\nji\u0000jj\u001d1, leading to a Gaussian centered around r\u0000\n`\u0000j\u0000i\n\u001a= 0 with width proportional top\nji\u0000jj.1For\nthe second expression on the right hand side of Eq. (10)\nwe de\fne\nP(i;`ji;0;t) =:Gtr(`;t); (12)\nsinceP(i;`ji;0;t) traces the motion of particle i, which\nwe assume to be in the bulk of the spin pattern. Gtr(`;t)\nthus corresponds to the time dependent tracer probabil-\nity distribution of a bulk particle. Using Eqs. (11,12) in\nEqs. (9,10) we obtain\nK(j\u0000i;r;t) =P(i;0)Z\nd`P(j;rji;`; 0)Gtr(`;t)'\n'P(i;0)Gtr\u0010\nr\u0000j\u0000i\n\u001a;t\u0011\n:\n(13)\nThe last line follows since Gtr(`;t) is in general a prob-\nability distribution whose width increases in time while\nthe width of P(j;rji;`; 0) is a constant of orderp\njj\u0000ij.\nTherefore, at late times Gtr(`;t) is much broader and\nwe can substitute the approximation P(j;rji;`; 0)'\n\u000e(r\u0000`\u0000j\u0000i\n\u001a) into Eq. (13). Finally, inserting Eq. (13)\nand Eq. (9) into Eq. (7) we \fnd\nC(r;t) =Z\ndjF(j;t)Gtr(r\u0000j=\u001a;t ) = [ ~F ? Gtr](r;t);\n(14)\nwith ~F(j;t) =\u001aF(\u001aj;t). The dynamical spin correla-\ntionsC(r;t) at late times are thus given quite generally\nas a convolution between the internal dynamics of the\nspin pattern and the tracer distribution of a distinguish-\nable particle on the lattice. Since Gtr(`;t) follows the\ntrajectory of the ith particle (with some iin the bulk)\ncounted from the left, the relevant tracer problem is one\n1The last line of Eq. (11) actually follows in a grand canonical\nsetting with average density \u001aof particles. Nonetheless, the lo-\ncation of the center and the scaling of the width remain valid for\n\fxed particle number.5\nof hard core interacting particles that can never swap rel-\native positions. We note that in reciprocal space Eq. (14)\nrepresents two independent decay processes of long wave-\nlengthk-modes.\nIII. RANDOM UNITARY CIRCUITS WITH\nCONSERVED PATTERN\nWe use the result of Eq. (14) to study a number of\ntJz{ like models with a spin pattern that is a constant\nof motion. We remark that the results of this section\nshould apply very generally to models featuring recently\nintroduced \\Statistically Localized Integrals of Motion\"\n(SLIOMs) [36], which can be interpreted as a conserved\npattern.\nIf the entire pattern is constant, the spin dynamics\nbecomes trivial, F(j;t) =\u000e(j) for all times in Eq. (14)\nand thus\nC(r;t)'Gtr(r;t) (15)\nmaps directly to a tracer problem. Due to the trivial pat-\ntern dynamics, our initial approximation Eq. (8) simply\nbecomesja(x0\u001bjx\u001b;t)j2'pn(x0jx;t), i.e., the matrix\nelements of the time evolution can be considered approx-\nimately independent of the underlying spin pattern when\ninserted into Eq. (7). In fact, with a conserved pattern\nthe correlations of Eq. (7) can be recast as\nC(r;t) =Gtr(r;t) +R(r;t); (16)\nwhereR(r;t) is the di\u000berence between the exact correla-\ntions and the tracer distribution. It reads explicitly\nR(r;t) =1\nNX\nx;x0;\u001b\ni6=j\u001bj\u001bi\u000ex0\nj;r\u000exi;0ja(x0\u001bjx\u001b;t)j2;(17)\nand captures contributions to C(r;t) due to spins j6=i\nmoving to site rat timet, given that spin istarted at\nsitex= 0 initially. Due to the summation over the spin\nvalues\u001bj,R(r;t) acquires both a positive and negative\ncontribution from \u001bjand\u0000\u001bj, respectively. We then ex-\npect generically that contributions to R(r;t) from spins\njwithjj\u0000ij\u001d1 vanish approximately due to cancel-\nlation of positive and negative contributions, justifying\nEq. (15). In this section, we will consider generic systems\nwhereR(r;t) = 0 exactly upon averaging over the ran-\ndom time evolution, and in Sec. IV integrable quantum\nsystems in which R(r;t) = 0 exactly since the time evo-\nlution is indeed independent of the underlying pattern,\nhence in both cases ja(x0\u001bjx\u001b;t)j2=pn(x0jx;t). In ei-\nther case, since C(r;t) =Gtr(r;t) exactly, we will be able\nto use existing results from the theory of tracer dynamics\nto obtain full long-time spin correlation pro\fles.\nHere, we \frst consider systems subject to a random\ntime evolution, such as a classical stochastic lattice gas\nor random unitary quantum circuits. Averaging over therandom evolution (denoted by \u0001\u0001\u0001) we are interested in\nthe associated averaged correlations C(r;t). For certain\nmodels that we consider, R(r;t) = 0 (see below) and thus\nthe mapping to tracer dynamics is exact upon averaging\nover the random evolution, C(r;t) =Gtr(r;t). The re-\nsulting tracer problem we have to solve is one of particles\nhopping randomly on a one-dimensional lattice subject to\na hard core exclusion principle. The hard core property\nis a direct consequence of the pattern conservation. Of\nparticular interest to us is the nearest neighbor simple\nexclusion process in one dimension, for which the long\ntime tracer distribution function is known to be [60{63]:\nGtr(`;t)!G(nI)\ntr(`;t) :=1\n(16Dt\u00192)1=4expn\n\u0000`2\n4p\nDto\n;\n(18)\nwhere the superscript ( nI) indicates that we are consider-\ning generic, non-integrable systems.G(nI)\ntr(`;t) takes the\nform of a Gaussian that broadens subdi\u000busively slowly,\nh\u0001`(t)2i= 2p\nDt: (19)\nThe generalized di\u000busion constant Dis determined via\nthe density \u001aof particles on the chain and the bare hop-\nping rate \u0000 per time step of an inividual particle,\nD=\u0000\n\u0019\u0000\n\u001a\u00001\u00001\u00012: (20)\nWe consider two examples in detail in the following, the\nrandomtJz{ model and the random XNOR model, for\nwhich Eqs. (18,20) will provide us with the exact long\ntime spin correlations after identifying conserved spin\npatterns in the appropriate variables.\nA. Random circuit tJz{ model\nOur \frst example is a direct implementation of a ran-\ndom version of the tJz{ model. We consider a chain with\nlocal Hilbert space spanned by the states jqi=j\u00001i,j0i,\nj1i. Each basis state can be written as jqi=jx;\u001biand\nwe demand that the pattern \u001bof\u00061-spins be a constant\nof motion. We then consider a random unitary time evo-\nlution given by\n^U(t) =tLY\n`=1^U`; (21)\nwhere individual two-site gates ^U`are arranged spatially\nas shown in Fig. 2. Each of the ^U`is given by\n^U`=X\ns^Ps^Us^Ps; (22)\nwhereslabels the symmetry sectors of the two-site lo-\ncal Hilbert space that are connected under the con-\nstraint of keeping \u001bconstant. Speci\fcally, there are6\ntime<latexit 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t101MSD s2(t)circuitq\nt\np\n°25 0 25\ndistance r0.000.020.040.06spin correlation C(r,t)\ncircuit\ntracerb) c)\nFIG. 2. RandomtJz{ circuit. (a) In the random tJz\n{ circuit we consider a three-state local Hilbert space and\ntwo-site local random unitary gates. The local gates connect\nstates with spins hopping between neighboring lattice sites.\n(b)The mean squared displacement (MSD) associated to the\nspin correlations C(r;t) agrees with the tracer prediction of\nEq. (25) for particle density \u001a= 2=3 and single particle hop-\nping rate \u0000 = 1. (c)The form of the correlations C(r;t) at\ntimet= 7000 of the circuit evolution. It assumes a Gaussian\nshape as expected from the tracer distribution. We obtain\nthese numerical results by sampling the discrete stochastic\nMarkov process of Eq. (23). The data was averaged over 1000\nrandomly chosen product initial states of the Markov process\nin a system of length L= 5000.\n\fve sectors that contain only a single local con\fgura-\ntion,fj\u00001\u00001ig,fj\u000011ig,fj1\u00001ig,fj11ig, andfj00ig,\nas well as two sectors that contain two states each,\nfj0;1i;j1;0ig,fj0;\u00001i;j\u00001;0ig.^Psis a projector onto\nthese connected sectors. The unitary operators ^Usact-\ning within each sector are then chosen randomly from the\nHaar measure.\nAveraging the time evolution over the random gates,\nthe associated circuit-averaged probabilities required to\ncompute the spin correlations C(r;t) are given by a\nclassical discrete Markov process. Speci\fcally, we fol-\nlow Ref. [46] in introducing the notation jx;\u001b) :=\njx;\u001bihx;\u001bjfor the projector onto the state jx;\u001bi, as\nwell as an associated inner product ( ^Aj^B) := Tr\u0002^A^By\u0003\nfor operators. The matrix elements for the time evolu-\ntion are then given by [46]\nja(x0\u001bjx\u001b;t)j2= (x0;\u001bj^Ttjx;\u001b); (23)\ntime<latexit 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↓ ↑ ↑FIG. 3. Random XNOR circuit. In the random XNOR\ncircuit we consider a two-state local Hilbert space and four-\nsite local random unitary gates. The allowed moves under\nthese gates connect local con\fgurations by exchanging near-\nest neighbor states, conditioned on the two surrounding sites\nbeing in the same state.\nwith a transfer matrix ^Tgiven by\n^T=LO\n`=1^T`\n^T`=X\ns1\ndsX\ns1;s22sjs1)(s2j;(24)\nwheredsis the size of the local two-site symmetry sector\ns. We see that Eqs. (23,24) describe the averaged proba-\nbilitiesja(x0\u001bjx\u001b;t)j2in terms of a stochastic lattice gas:\nFor each applied gate a particle hops with probability 1 =2\nto an empty neighboring site and stays at its position if\nthe neighboring site is occupied by another particle. In\nparticular,ja(x0\u001bjx\u001b;t)j2=ja(x0jx;t)j2isindependent\nof the spin pattern \u001b. Thus, the contribution R(x;t) of\nequation Eq. (17) vanishes due to cancellation of positive\nand negative spin contributions. The long-time mean\nsquared displacement of the tracer process is in turn ex-\nactly described by the simple nearest-neighbor exclusion\nprocess through Eqs. (18-20). In our case, the density\nof particles is given by \u001a= 2=3 at in\fnite temperature.\nFurthermore, for a single time step consisting of two lay-\ners as shown in Fig. 2 (a) there are two attempted moves\nat rate 1=2 per particle, such that we can e\u000bectively set\n\u0000 = 1. This yields D= 1=4\u0019for the random circuit tJz\n{ model, see also Fig. 1. The theory of tracer di\u000busion\nof hard core particles in one dimension thus predicts a\nmean squared displacement\n\u001b2(t) :=X\nrr2C(r;t) =p\nt=\u0019 (25)\nat long times twith a Gaussian shape of the averaged\ncorrelations C(r;t). The mean squared displacement thus\ngrows subdi\u000busively \u0018p\ntas opposed to conventional dif-\nfusive growth\u0018t. We con\frm this prediction by numer-\nically sampling the stochastic Markov process Eq. (23)\nwhich yields the spin correlations in Fig. 2 (b+c).\nB. Random circuit XNOR model\nWe consider a second example of generic unitary quan-\ntum dynamics where we can use the tracer formulae7\nEqs. (18-20) to derive the long-time behavior of local\nspin correlations, the random XNOR circuit [46]. The\nmodel is an e\u000bective spin S= 1=2 system with Hilbert\nspace spanned by the local states j\"i;j#i. The local\nunitaries ^U`that generate the time evolution are four-\nsite gates that conserve both the total magnetization\n^M=P\nx^Sz\nxas well as the number of Ising domain walls\n^D=P\nx(^Sz\nx+1\u0000^Sz\nx)2. Therefore, ^U`can exchange the\ncentral two spins only if the outer two spins have the\nsame value, see Fig. 3. Writing ^U`=P\ns^Ps^Us^Psas\nin Eq. (22), the only symmetry sectors sthat contain\nmore than a single state are fj\";\";#;\"i;j\";#;\";\"igand\nfj#;\";#;#i;j#;#;\";#ig. We refer to this system as the\nrandom XNOR model following Ref. [46], which estab-\nlished that spin correlations in this model show subdi\u000bu-\nsive transport with z= 4. While the dynamical exponent\nis in agreement with the tracer picture, the tJz{ like exis-\ntence of a conserved pattern is not immediately apparent\nin the random XNOR model. We \frst describe the map-\nping to such a conserved pattern using the original spin\nvariablesSz\nx2f\";#g, see also Refs. [54, 64]. We then\nconstruct an equivalent mapping using domain wall vari-\nables ^Sz\nx+1\u0000^Sz\nx, see also Refs. [55, 58, 65]. Combining\nboth pictures, we will be able to explain the full form of\nthe spin correlations C(r;t) at long times.\n1. Conserved pattern: spin picture\nLet us consider a state jsiwithsx2f\";#gin a system\nof lengthL. We map this state (bijectively up to bound-\nary terms) to a state j\u001ciof e\u000bective `superspin' degrees\nof freedom \u001ci2f*;0;+gon a chain of length ~L(s) which\nexplicitly depends on the state jsiin the original spin-1 =2\npicture, see also Refs. [54, 64]. We start at the left end\nx=\u0000L=2 of the original chain and consider the bond\nbetween the \frst two spins s\u0000L=2;s\u0000L=2+1. There are\ntwo possibilities:\n1. Ifs\u0000L=2=s\u0000L=2+1, we add\u001c=*and\u001c=+\nto the superspin con\fguration, for s\u0000L=2=\"and\ns\u0000L=2=#, respectively.\n2. Ifs\u0000L=26=s\u0000L=2+1, consider the next bond be-\ntweens\u0000L=2+1;s\u0000L=2+2: Ifs\u0000L=2+1=s\u0000L=2+2, we\nagain accordingly add \u001c=*or\u001c=+to the super-\nspin con\fguration for s\u0000L=2+1=\"ands\u0000L=2+1=#,\nrespectively. On the other hand, if also s\u0000L=2+16=\ns\u0000L=2+2, we add the superspin \u001c= 0.\nThe above steps determine the \frst element (from the\nleft) of the superspin con\fguration. The next superspin\nis determined by moving to the next bond between two\nspins and repeating the above steps. This process is re-\niterated until all bonds in the original picture have been\naccounted for, yielding j\u001ci=j\u001c(s)i. An example of this\nmapping is illustrated in Fig. 4 (a).\nWith the superspin description j\u001ciwe are back to a tJ\n{ like Hilbert space structure. 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/uni2191/uni2193/uni2191/uni2191\n/uni2193/uni2191/uni2193/uni2193 /uni2191 /uni2193/uni2191/uni2193/uni2193 /uni2191FIG. 4. Conserved charge pattern in the random\nXNOR circuit. a) Mapping between original spin-1 =2 de-\ngrees of freedom and an e\u000bective conserved superspin pat-\ntern. Going from left to right, two neighboring aligned spins\nare mapped to a corresponding superspin *or+, while two\nneighboring domain walls map to a vacant site 0. The pattern\nof non-zero superspins is conserved under random XNOR dy-\nnamics. In the domain wall picture (empty and \flled circles),\nthe mobile vacancies correspond to domain wall pairs ( ).\nFurthermore, in the domain wall picture the objects that form\na conserved pattern are given by single bonds of aligned spins\n() as well as single domain walls paired up with a neighbor-\ning aligned bond ( ). This domain wall conserved pattern\nis obtained by removing all mobile domain wall pairs ( )\nsee also Fig. 5. b)+c) Elementary XNOR moves within the\nabove mapping. In superspin language, vacancies 0 and su-\nperspins*=+exchange positions. In domain wall language,\na mobile domain wall pair exchanges positions with one of the\nobjects contributing to the domain wall pattern (i.e. or\n).\ndynamics described above (see also Fig. 4) the number\nand the pattern of non-zero superspins are indeed con-\nserved: On the one hand, every bond of aligned spins\ncontributes a non-zero superspin and the total number of\naligned nearest neighbor spins is constant due to domain\nwall conservation. On the other hand, two opposite su-\nperspins located next to each other, e.g. j:::*+:::i, trans-\nlate into a local con\fguration of four spins, j:::\"\"##:::i,\non which the XNOR gates of Fig. 3 can only act triv-\nially. Hence,*and+can never exchange relative po-\nsitions. If we write j\u001ci=jx;\u001bias before, the pattern\n\u001bis conserved. Through this mapping we are led back\nto the random circuit tJz{ constraints considered in the\nprevious section, accounting for the dynamical exponent\nz= 4.\nIn addition, we also analyze in the following how the\nconserved superspin pattern translates quantitatively into\nthe correlations of the original spin variables. To this end,8\nwe de\fne the quantities\n^\u0014x:=1\n2(^Sz\nx+^Sz\nx+1) (26)\nwithin the spin-1 =2 picture, which detect whether the\nbond between the spins at x;x+ 1 contributes a non-\nzero superspin to j\u001ci. The dynamic correlation function\nh^\u00142r(t)^\u00140(0)ithen probes how a non-zero superspin ex-\ncitation initially located between sites 0 ;1 spreads to the\nbond between 2 r;2r+ 1. Crucially, according to the ran-\ndom XNOR gates depicted in Fig. 4, the non-zero super-\nspins*;+only move by steps of length two with respect\nto the original lattice. At the same time they move only\nby a distance rwithin the compressed superspin pattern\n\u001c. Since the superspin description reduces to the random\ntJz{ model analyzed above we can write\nh^\u00142r(t)^\u00140(0)i=1\n2G(nI)\ntr(r;t): (27)\nHere, the prefactor 1 =2 corresponds to the probability of\n\fnding a non-zero superspin between sites 0 ;1. The hop-\nping rate of superspins entering Eq. (27) through Eq. (20)\ncan again e\u000bectively be set to \u0000 = 1 for the circuit geom-\netry of Fig. 3. On the other hand, care needs to be taken\nto determine the density \u001aof non-zero superspins, as the\nin\fnite temperature average in the original spin variables\nsxdoes not transfer directly to an in\fnite temperature\naverage in the superspin picture. We will derive this den-\nsity below in the domain wall picture, see Eq. (39); for\nnow we quote only the obtained result \u001a= 3=4 entering\nEq. (27).\nInserting the de\fnition Eq. (26) of ^ \u0014xback into Eq. (27)\nwe obtain\n1\n2G(nI)\ntr(r;t) =1\n4\u0010\nh^Sz\n2r\u00001(t)^Sz\n0(0)i+\n+ 2h^Sz\n2r(t)^Sz\n0(0)i+h^Sz\n2r+1(t)^Sz\n0(0)i\u0011\n;(28)\nwhere we made use of translational invariance in the bulk.\nWe could have performed an equivalent calculation for\nthe correlationh^\u00142r+1(t)^\u00140(0)iand thus\n1\n2G(nI)\ntr(r;t) =1\n4\u0010\nh^Sz\n2r(t)^Sz\n0(0)i+\n+ 2h^Sz\n2r+1(t)^Sz\n0(0)i+h^Sz\n2r+2(t)^Sz\n0(0)i\u0011\n;(29)\nwhich will be relevant for resolving the A=B-sublattice\nstructure further below. From Eq. (28) we then obtain\nthe mean squared displacement at long times,\n\u001b2(t) =X\nrr2h^Szr(t)^Sz\n0(0)i\nt\u001d1\u0000\u0000\u0000!1\n2X\nr(2r)2n\nh^Sz\n2r\u00001(t)^Sz\n0(0)i+\n+ 2h^Sz\n2r(t)^Sz\n0(0)i+h^Sz\n2r+1(t)^Sz\n0(0)io\n=\n=X\nr(2r)2G(nI)\ntr(r;t) =X\nrr2G(nI)\ntr(r;16t) =8\n3p\u0019p\nt;\n(30)\n/uni2193 /uni2191 /uni2193 /uni2191 /uni2193 /uni2193 /uni2191 /uni2191\n− − + −+/uni2193 /uni2193\nA sublattice B A AB\n/uni2193 /uni2191 /uni2193/uni2191 /uni2193 /uni2193 /uni2191 /uni2191/uni2193 /uni2193XNOR dynamicsconserved \npattern\nconserved patternsublattice B B A A A\n− + +− −−\nB−+\nA A(˜x)2\n<latexit 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= const .\n<latexit sha1_base64=\"xqzxfvuxcL472cbbXlpNIeA2tOE=\">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</latexit>A/BFIG. 5. Sublattice symmetry. In the domain wall pic-\nture, we obtain a conserved pattern by removing all mobile\ndomain wall pairs (illustrated as ). The resulting pat-\ntern is formed by single aligned bonds ( ) and pairs of a\nsingle domain wall with an aligned bond to its right ( ).\nThe pattern is subject to an exclusion constraint of nearest\nneighbor domain walls, i.e. every \flled circle necessarily has\nat least two empty circle neighbors (one to the left and one\nto the right). The sublattice charge of all charged domain\nwalls ^~\u0014that are not part of a mobile pair is then conserved.\nThis is due to the mobile pairs having a spatial extension of\nlength two, such that single domain walls exchanging their\nposition with such pairs only move by steps of length two at\na time, hence preserving their sublattice. They also preserve\ntheir charge value since the domain wall charges are perfectly\nanticorrelated globally.\nwhere we used Eqs. (19,20) with \u001a= 3=4 and \u0000 = 1.\nWe can absorb the factor of 16 in G(nI)\ntr(r;16t) into the\ne\u000bective (sub)di\u000busion constant to obtain D= 16=9\u0019,\nsee also Fig. 1. Again the dynamics is subdi\u000busive with\nz= 4.\nTo verify this prediction we simulate the circuit of\nFig. 3 numerically, again using the mapping to a clas-\nsical Markov process. The derivation of the associated\ntransfer matrix ^Tproceeds in full analogy to the tJz{\ncase. Fig. 6 (b) demonstrates the validity of Eq. (30). In\naddition, Eq. (28) predicts a Gaussian enveloping shape\nof the charge correlations, which we numerically verify\nin Fig. 6 (a). Intriguingly however, Eq. (28) in principle\nallows for additional sublattice structure. We indeed \fnd\nsizeable staggered oscillations on top of the Gaussian in\nFig. 6 (a). These oscillations do not decay at large times\nand thus hint at additional structure in the model. In\norder to explain and quantitatively describe these short-\ndistance oscillations, we will switch to a domain wall pic-\nture in the following.\n2. Conserved pattern: domain wall picture\nAn alternative description of the Hilbert space struc-\nture can be given in terms of the domain wall charge\nvariables\n^~\u0014x:=1\n2(^Sz\nx+1\u0000^Sz\nx); (31)9\nwhich ascribes a sign depending on whether the local\ncon\fguration isj:::#\":::iorj:::\"#:::i. After \fxing the\nleftmost spin, a complete description of a spin con\fgura-\ntion is also given in terms of the locations of its domain\nwalls, (~\u0014x)2, regardless of their sign. We can construct a\ndomain wall version of the conserved charge pattern, see\nalso Refs. [55, 58, 65]: Starting from the left of the sys-\ntem, neighboring domain walls are paired up into mobile\npairs , see Fig. 4 (a). The remaining single domain\nwalls are paired up with their corresponding right neigh-\nbor bond that connects two aligned spins, . De\fned\nin this way, the dynamics in the system is generated by\nmobile domain wall pairs ( ) moving through the\nsystem, exchanging positions with single bonds of aligned\nspins ( ) and with the pairs of domain walls and aligned\nbonds ( ). The elementary dynamical processes are\ndepicted in Fig. 4 (b+c). By removing all mobile domain\nwall pairs, see e.g. Fig. 5, we obtain a conserved pattern\nin the domain wall description formed by single aligned\nbonds and the pairs of domain wall and aligned\nbond. The conserved pattern thus exhibits a blockade of\nnearest neighbor domain walls. The number of conserved\npatterns with such a blockade grows as a Fibonacci se-\nquence in system size. This Fibonacci number then also\ncorresponds to the number of disconnected subsectors in\nthe Hilbert space [54, 55].\nDue to the two-site spatial extension of the domain\nwall pairs, the total A=B sublattice charge of all domain\nwalls ^~\u0014xthat are not part of a mobile pair is a conserved\nquantity, see Fig. 5. Formally, we can express\n^~\u0014x=^~\u0014(single )\nx +^~\u0014(pair)\nx; (32)\nwhere ^~\u0014(single )\nx is the domain wall charge operator for sin-\ngledomain walls while ^~\u0014(pair)\nx is the operator for domain\nwalls that are part of mobile pairs . The two sublattice\ncharges\n^QA=B=X\nx2A=B^~\u0014(single )\nx = const: (33)\nare conserved quantities. The operator ^~\u0014xthus separates\ninto a part that has overlap with the sublattice conserva-\ntion laws of Eq. (33) and a part not associated to any such\nconservation law (P\nx2A=B^~\u0014(pair)\nx6= const:). The domain\nwall charge correlation function on the even sublattice at\nlate times will thus be dominated by the transport asso-\nciated to the conserved quantity ^QA, i.e.\nh^~\u00142r(t)^~\u00140(0)it\u001d1=h^~\u0014(single )\n2r (t)^~\u0014(single )\n0 (0)i: (34)\nCorrections to Eq. (34) are expected to decay quickly\n(generically exponentially fast) in time. Since the sin-\ngle domain wall charges are part of a conserved pattern\nas described above their dynamical correlations are again\ngiven by the tracer distribution,\nh^~\u00142r(t)^~\u00140(0)it\u001d1=C0G(nI)\ntr(r;t); (35)with constant prefactor C0to be determined. The den-\nsity\u001aand the hopping rate \u0000 entering Eq. (35) are the\nsame as in the spin picture above.\nTo compute C0we equate Eqs. (34,35) and take the\nsum overr,\nC0=X\nrh^~\u0014(single )\n2r^~\u0014(single )\n0i; (36)\nresulting in a simple static (notice the absence of the cir-\ncuit average) correlation function at in\fnite temperature.\nWe can express Eq. (36) as\nC0=D\u0000^~\u0014(single )\n0\u00012EX\nrD\n^~\u0014(single )\n2rE0\n; (37)\nwhereh\u0001i0denotes a modi\fed in\fnite temperature aver-\nage with a single positive domain wall \fxed to sit at site 0.\nThe factorD\u0000^~\u0014(single )\n0\u00012E\nis determined within the map-\nping to a conserved pattern from Fig. 4:D\u0000^~\u0014(single )\n0\u00012E\n=\n#( )=Lcorresponds to the density of single domain\nwalls paired up with a neighboring aligned spin bond.\nUsing that the overall density of domain walls is 1 =2 and\nthat by symmetry #( )=L= #( )=L, we obtain\nL\n2= 2\u0001#( ) + #( ) = 3\u0001#( )\n!D\u0000^~\u0014(single )\n0\u00012E\n= #( )=L= 1=6:(38)\nWith this result we can also compute the density \u001aof\nhard core particles (corresponding to the density of non-\nzero superspins of the previous section) that we used in\nEq. (30):\n\u001a= 1\u0000#( )\n#( ) + #( )= 1\u0000L=6\nL=6 +L=2= 3=4:(39)\nThe remaining correlation functionD\n^~\u0014(single )\n2rE0\nin\nEq. (37) refers only to single domain walls and can thus\nbe computed within the ensemble of possible conserved\ndomain wall patterns, see Fig. 5. Making use of this prop-\nerty we derive the exact value of this correlation function\nin Appendix A and quote here our \fnal result\nC0=1\n12\u0000\n'\u00001\u0001\n(40)\nfor the constant C0, where'= (1 +p\n5)=2 is the golden\nratio. Then, inserting the de\fnition Eq. (31) in Eq. (35)\nyields\nC0G(nI)\ntr(r;t) =1\n4\u0010\n2h^Sz\n2r(t)^Sz\n0(0)i\n\u0000h^Sz\n2r\u00001(t)^Sz\n0(0)i\u0000h^Sz\n2r+1(t)^Sz\n0(0)i\u0011\n:(41)\nPerforming the equivalent derivation for the correlations\nh^~\u00142r+1(t)^~\u00140(0)igives us the additional relation\n\u0000C0G(nI)\ntr(r;t) =1\n4\u0010\n2h^Sz\n2r+1(t)^Sz\n0(0)i\n\u0000h^Sz\n2r(t)^Sz\n0(0)i\u0000h^Sz\n2r+2(t)^Sz\n0(0)i\u0011\n:(42)10\n°50 0 50\ndistance r0.000.050.100.150.20|C(r,t)°G(nI)\ntr(r,16t)|\ncircuit\ntracer°40 °20 0 20 40\ndistance r0.000.010.020.03spin correlation hˆSzr(t)ˆSz\n0(0)icircuit\ntracer\n102103104105\ntime t102MSD s2(t)circuit\n8\n3pppta) b)\n°50 0 50\ndistance r0.00.10.20.30.4|C(r,t)°G(nI)\ntr(r,16t)|\ncircuit\ntracer[10\u00002]\n<latexit sha1_base64=\"rc0+UMLA+W1g6r8cnD48PeKaqWU=\">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</latexit><latexit sha1_base64=\"rc0+UMLA+W1g6r8cnD48PeKaqWU=\">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</latexit><latexit sha1_base64=\"rc0+UMLA+W1g6r8cnD48PeKaqWU=\">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</latexit><latexit sha1_base64=\"rc0+UMLA+W1g6r8cnD48PeKaqWU=\">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</latexit>c)\nFIG. 6. Random XNOR model: Numerical results. a) Spatial shape of the spin correlation function at time t= 104of\nthe random circuit shown in Fig. 3. The black dashed line corresponds to the Gaussian of the tracer probability distribution\npredicted by Eq. (44). b)The mean squared displacement \u001b2(t) of the spin correlations agrees with the late time prediction\nof Eq. (30). c)The absolute value\f\fC(r;t)\u0000G(nI)\ntr(x;16t)\f\fof the oscillations on top of the Gaussian shape as seen in a). The\nblack dashed line corresponds to the prediction of Eq. (44). The results demonstrate the quantitative accuracy of the tracer\ndescription. The data was averaged over 12800 random initial states of the associated stochastic Markov process in a system\nof lengthL= 5000.\nAdding Eqs. (28,41) as well as Eqs. (29,42) \fnally yields\nthe long time correlations\nh^Sz\n2r(t)^Sz\n0(0)i=1\n2(1 + 2C0)G(nI)\ntr(r;t)\nh^Sz\n2r+1(t)^Sz\n0(0)i=1\n2(1\u00002C0)G(nI)\ntr(r;t);(43)\nwhich we can rewrite as\nC(r;t) =h^Szr(t)^Sz\n0(0)i= [1 + 2C0(\u00001)r]G(nI)\ntr(r;16t):\n(44)\nEq. (44) explains the staggered oscillations observed nu-\nmerically in Fig. 6 (a). To check that our quantitative\ndescription (i.e. the constant C0of Eq. (40)) is accu-\nrate, we show in Fig. 6 (c) that the quantity\f\fC(r;t)\u0000\nG(nI)\ntr(x;16t)\f\fagrees with the prediction of Eq. (44). We\nemphasize that the staggering of the correlations de-\ncribed by Eq. (44) persists up to in\fnite time.\nC. Random circuit dimer model\nWe mention only brie\ry a third example of tJz{ like\ndynamics in a dimer model on a bilayer square lattice\ngeometry as introduced in Ref. [43]. The system has a\nshort direction along which periodic boundaries are cho-\nsen, making the geometry quasi one-dimensional, and the\ntime evolution is generated by local plaquette \rips of par-\nallel dimers,\n^H=\u0000JX\np\u0010\njihj+h:c:\u0011\n: (45)\nThe model is equivalent to a model of closed directed\nloops and site-local charges on a square lattice cylinder\nwith short circumference. The existence of a conserved\ncharge pattern and its role in inducing a dynamical ex-\nponentz= 4 due to the emergence of a hard core tracerproblem has been discussed already in Ref. [43]. Intu-\nitively, the site-local charges in the model are dimers that\ngo in between the two layers, positive or negative depend-\ning on which sublattice they occupy. When a charge is\nenclosed by a loop, it cannot escape the loop under the\ndynamics of Eq. (45). In the presence of a \fnite den-\nsity of loops that wind across the circumference of the\ncylinder (with the same chirality), a conserved pattern\nis formed by the summing up charges always in between\ntwo such loops, see Fig. 1 and Ref. [43] for details. The\ntracer prediction of z= 4 and the Gaussian shape of the\ndynamical charge correlations have been veri\fed numer-\nically in Ref. [43]. Only the precise values of e\u000bective\n(sub)di\u000busion constants are unknown since the mapping\nto a conserved pattern is an e\u000bective one.\nIV. INTEGRABLE QUANTUM SYSTEMS\nWITH CONSERVED PATTERN\nThe presence of a conserved charge pattern can also\nbe of use in integrabletJz{ like quantum systems and,\nin certain cases, provides an alternative route to deter-\nmine the long time pro\fle and di\u000busion constant of their\nspin correlations at in\fnite temperature. The associated\ntracer motion relevant for the correlations C(r;t) from\nEq. (14) is one with ballistically instead of di\u000busively\nmoving particles. As single particles move ballistically,\nthe many-body tracer distribution will be di\u000busive. A\nsimilar situation was considered in Refs. [66, 67], where\nthe spin di\u000busion constant of an interacting determinis-\ntic classical cellular automaton with invariant spin pat-\ntern and pattern-independent dynamics was computed\nexactly. According to Eq. (14), this spin di\u000busion is di-\nrectly associated with a Gaussian tracer distribution with11\nthe same di\u000busion constant,\nGtr(`;t)!G(I)\ntr(`;t) =1p\n4\u0019Dtexpn\n\u0000`2\n4Dto\n:(46)\nThe superscript ( I) indicates an integrable process which\nimplies a broadening of Gtras\u0018p\nt(instead of\u0018t1=4\nwhich we have obtained for generic systems in Sec. III).\nUsing the result of Ref. [66], the di\u000busion constant Dof\nEq. (46) is determined by the density \u001aof particles along\nthe chain as well as their e\u000bective velocity ve\u000b,\nh\u0001x2i= 2Dt\nD=ve\u000b\n2(\u001a\u00001\u00001):(47)\nIn the discrete time cellular automaton considered in\nRef. [66], all particles have a \fxed velocity of ve\u000b= 2.\nA. Integrable tJz{ model\nThe above connection can be put to direct use in the\nintegrableJz!0 limit of the tJz{ model [68],\n^Ht=\u0000X\nx;\u001b(^~cy\nx+1;\u001b^~cx;\u001b+h:c:); (48)\nwhich features only the hopping of spinful fermions\nwith forbidden double occupancy. The matrix elements\na(x0\u001bjx\u001b;t) =a(x0jx;t) of the time evolution ob-\ntained from Eq. (48) do not depend on the pattern \u001b,\nsoR(r;t) = 0 in Eq. (17), and the mapping to a tracer\nproblem is exact at all times. At in\fnite temperature, the\ndensity of particles is \u001a= 2=3 and we can determine the\ne\u000bective velocity ve\u000bby noting that ^Htcan be mapped\nto a problem of spinless free fermions [68],\n^Ht=\u0000X\nkcos(k)^fy\nk^fk: (49)\nWe give an intuitive argument for why this is possible:\nSince the dynamics is oblivious to the conserved spin pat-\ntern, for a given inital basis state we can simply i)`write\ndown' the invariant spin pattern for bookkeeping pur-\nposes, ii)remove the fermions' spins, iii)perform the\ntime evolution with a Hamiltonian of spinless fermion\nhopping (of same hopping strength), and then iv)rein-\ntroduce the pattern afterwards. Steps i)and iv)in this\nmapping are of course highly non-local.\nAt in\fnite temperature we then expect all momentum\nmodes of Eq. (49) to be occupied with equal probabil-\nity. While Ref. [66] derived Eq. (47) for an automaton in\nwhich every particle has the same absolute velocity, here\nwe need to consider a distribution of di\u000berent momen-\ntum modes. We expect that the mean displacement of\nthe equivalent tracer process depends only on the aver-\nage of the absolute velocity and thus predict the e\u000bective\n0 10 20 30\nspace r0.000.050.100.15spin correlation/angbracketleftˆSz\nr(t)ˆSz\n0(0)/angbracketright\n∆→∞\n∆=4\ntracerFIG. 7. Folded XXZ chain. Pro\fle of the dynamical spin\ncorrelation function for the folded XXZ chain of Eq. (53) (blue\ncurve, \u0001!1 ) at timet= 18 in a system of length L= 30.\nThe black dashed line shows the prediction of Eq. (55), there\nare no \ft parameters. The red line shows the pro\fle obtained\nfor the XXZ chain Eq. (52) at anisotropy \u0001 = 4, illustrating\ngood agreement with Eq. (55) at intermediate times, even for\nmoderate values of anisotropy.\nvelocity for Eq. (47) to be given by the average absolute\ngroup velocity of Eq. (49):\nve\u000b=1\n2\u0019Z\u0019\n\u0000\u0019dkj@kcos(k)j=2\n\u0019: (50)\nThis leads to the in\fnite temperature spin di\u000busion con-\nstant\nDt=1\n2\u0019(51)\nfor the quantum model ^Htof Eq. (48). Eq. (47) also yields\nthe di\u000busion constant at in\fnite temperature but with\n\fxed density \u001a6= 2=3 or chemical potential \u0016such that\n\f\u0016= const:as\f!1 .\nB. Folded XXZ spin chain\nAs a second example we consider an integrable version\nof the random XNOR model, the folded XXZ chain. It\nis obtained from the integrable XXZ model\n^HXXZ=\u0000X\ni1\n2(^S+\ni^S\u0000\ni+1+^S\u0000\ni^S+\ni+1) + \u0001 ^Sz\ni^Sz\ni+1(52)\nin the limit of large anisotropy \u0001 !1 , which using a\nSchrie\u000ber-Wol\u000b transformation yields\n^HfXXZ =\u0000X\ni(1\n4+^Sz\ni\u00001^Sz\ni+2)(^S+\ni^S\u0000\ni+1+^S\u0000\ni^S+\ni+1):(53)\n^HfXXZ remains integrable and many of its thermody-\nnamic and dynamic properties have been analyzed in re-\ncent works [56{59]. We note that ^HfXXZ consists of four-\nsite spin- and domain wall-conserving terms and thus fea-\ntures the same e\u000bective conserved pattern of superspins12\nas the random XNOR model above. In particular, it can\nbe demonstrated that in the superspin picture, the folded\nXXZ chain becomes equivalent to the integrable limit of\nthetJz{ model from Eq. (48), ^HfXXZ!^Ht[64]. As\na consequence, the spin di\u000busion constant DfXXZ of the\nfolded XXZ chain can be obtained from the di\u000busion con-\nstantDof superspins determined by the Hamiltonian ^Ht\nof Eq. (48). In order to relate the two, we recall that the\nin\fnite temperature average in the original spin picture\nimplies a density \u001a= 3=4 of superspins in the associated\nintegrabletJz{ model. Furthermore, in the original lat-\ntice superspins always move by two sites. The variance\nh\u0001x2i= 2DfXXZtof the original spin correlation pro\fle\nis thus determined by\n2DfXXZt\n4=h\u0001x2i\n4= 2Dt\n!DfXXZ =4\n3\u0019;(54)\nwhere we have used that D= 1=3\u0019for\u001a= 3=4 and\nve\u000b= 2=\u0019in Eq. (47). The value of DfXXZ in Eq. (54)\nis in agreement with previous analytical results [69], ob-\ntained using generalized hydrodynamics [70, 71], as well\nas numerical results for the XXZ chain [72, 73]. In addi-\ntion, following the analysis of the sublattice domain wall\ncharge in the XNOR model, see Fig. 5, we predict the\nfull long time pro\fle of the dynamical spin correlations\nfor the folded XXZ chain to be\nC(r;t) = [1 + 2C0(\u00001)r]G(I)\ntr(r;t): (55)\nHere,G(I)\ntr(r;t) is from Eq. (46) with D=DfXXZ. In par-\nticular, the long time pro\fle features characteristic stag-\ngered oscillations of strength 2 C0withC0from Eq. (40).\nThese oscillations also lead to a distinct contribution\nto the spin conductivity \u001b(k;!) =\f\n2hj(k;!)j(\u0000k;\u0000!)i,\nwith spin current j(k;!). Using the continuity equation\n@tSz\nx(t) +@xj(x;t) = 0, we relate \u001b(k;!) =\f!2\n2k2C(k;!)\nand obtain\n\u001b(k=\u0019;!!0) =C0\f\n2DfXXZ: (56)\nThe \fnite momentum conductivity of the folded XXZ\nchain thus exhibits a \fnite contribution at k=\u0019.\nWe verify Eq. (55) numerically for the folded XXZ\nchain of Eq. (53) in systems of \fnite size and at inter-\nmediate time scale in Fig. 7. We consider a chain of size\nL= 30 spins by making use of the conserved pattern and\nemploying sparse matrix evolution. The in\fnite temper-\nature average for the spin correlations was approximated\nby averaging over 105randomly chosen product initial\nstates for the time evolution. We \fnd good agreement\nwith Eq. (55) already at a time t= 18 in Fig. 7. Further-\nmore, we note that the anisotropic XXZ chain of Eq. (52)\nhas recently been implemented in quantum simulation\nexperiments [74]. In a simple perturbative argument, the\nHamiltonian ^HfXXZ should provide a good estimate forthe time evolution of ^HXXZ up to times t\u0018\u00012in the\nanisotropy. Signatures of Eq. (55) should thus be visi-\nble at intermediate times already at moderate anisotropy\nstrength. We demonstrate this numerically in Fig. 7 by\ntime-evolving an in\fnite temperature density matrix per-\nturbed by a spin excitation in the center of the chain. We\nuse matrix product state methods with a bond dimen-\nsion\u001f= 1200 for the time evolution with the anisotropic\nXXZ Hamiltonian of Eq. (52) [75]. We \fnd good agree-\nment with the folded limit and the expression in Eq. (55)\nalready for \u0001 = 4 at a time t= 18 in Fig. 7.\nWe conclude this section with the following remark: As\ndiscussed previously, the folded XXZ Hamiltonian ^HfXXZ\nmaps to the integrable limit ^Htof thetJz{ model in the\nsuperspin picture. Similarly, one can apply the reverse\nof this mapping to the tJz{ like deterministic cellular\nautomaton studied in Refs. [66, 67], which is e\u000bectively\na local automaton for the superspins. Under the reverse\nmapping we then obtain an automaton for spin-1 =2 vari-\nables that is able to mimic the long-time dynamics of\nthe folded XXZ chain. A di\u000berent cellular automaton\nmimicking the folded XXZ dynamics, which exhibits dif-\nferent left- and right-mover velocities, has recently been\nstudied in Ref. [76]. In our prescription outlined above\nthe automaton obtained from Ref. [66] by inverting the\nmapping from superspins to spin-1 =2 is non-local in the\nspin-1=2 variables and features symmetric left- and right-\nmovers of equal speed instead.\nV. BROKEN PATTERN CONSERVATION\nIn this section, we return to random unitary circuit\nmodels but relax the condition of an exactly conserved\nspin pattern. In particular, we consider constrained\n`tJm{ like' models in which only a certain number m\nof moments of the spin pattern remains constant, see\nEqs. (1,3). Remarkably, breaking the pattern conserva-\ntion does not immediately imply conventional di\u000busion\nbut the resulting dynamics sensitively depends on the\nnumber of conserved moments. To see this, we note that\nthe pattern-internal spin dynamics in the presence of m\nconserved multipole moments is governed by the follow-\ning hydrodynamic equation for the coarse-grained spin\ndensityh^\u001bx(t)i[38, 39],\n@th^\u001bxi+ (\u00001)mDs@2m+2\nxh^\u001bxi= 0: (57)\nEq. (57) describes (sub)di\u000busive dynamics with dynami-\ncal exponent z= 2m+ 2 and its fundamental solution,\nwhich corresponds to the spin part of Eq. (9) via linear\nresponse, reads in momentum space:\nF(k;t) =1p\n2\u0019exp(\u0000Dsk2m+2t); (58)\nnormalized such thatR\ndxF(x;t) = 1. Using Eq. (58)\nin Eq. (14) we obtain the spin correlations in momentum13\ntime<latexit 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/uni2193/uni2191/uni2193/uni2191/uni2193 /uni2191/uni2193/uni2191/uni2193/uni03B3\n101102103104\ntime t100101102103MSD s2(t)m=0\nm=2a)\nb) /uni223Ct\n/uni223C t(normal diffusion)\n(tracer diffusion)\nFIG. 8. Moment-conserving random circuit evolution.\na)We consider tJm{ like random circuits with spin terms\nthat conserve a given number mof multipole moments (in\nthe depicted example the dipole moment m= 1) of the spin\npattern. The spin terms are applied with some probability \r,\nthe particle hopping terms with probability 1 \u0000\r.b)The dy-\nnamics of random circuits with multipole moment conserving\nspin pattern depends on the number mof conserved moments.\nIf only the total spin of the pattern is conserved ( m= 0), the\nmean-squared displacement \u001b2(t)\u0018tfollows conventional dif-\nfusion. If all moments up to the quadrupole moment ( m= 2)\nor higher are conserved, \u001b2(t)\u0018p\ntremains dominated by\nanomalously slow hard core tracer motion.\nspace,\nC(k;t) =Gtr(k;t)F(k;t) =\n=1\n2\u0019expn\n\u0000Dsk2m+2t\u0000p\nDtk2o\n;(59)\nwhich we will analyze for di\u000berent values of min the\nfollowing. While m= 0 leads to conventional di\u000busion,\nm\u00152 preserves the tracer mapping at long times. The\ncasem= 1 turns out to be special, with a competition\nbetween two processes that have the same dynamical ex-\nponent but di\u000berent scaling functions.A.m= 0: Di\u000busion\nForm= 0, under rescaling space and time in Eq. (59)\naccording to k!k=\u0015,t!\u0015zt=\u00152t, we obtain\nC(k;t)\u0018expn\n\u0000Dsk2t\u0000p\nDtk2=\u0015o\n\u0015!1\u0000\u0000\u0000\u0000! expn\n\u0000Dsk2to\n:(60)\nThis implies that the dynamical exponent is z= 2.\nTherefore, if only the total spin of the pattern is con-\nserved the correlations C(k;t) are described by conven-\ntional di\u000busion as expected. We verify this result numer-\nically intJ{ like random unitary circuits with charge-\nconserving two-spin gates, see Fig. 8 (a). The mean\nsquared displacement \u001b2(t) of the resulting real space\ncorrelations C(r;t) indeed scales di\u000busively \u001b2(t)\u0018tat\nlong times as shown in Fig. 8 (b).\nB.m\u00152: Tracer di\u000busion\nOn the other hand, rescaling k!k=\u0015,t!\u0015zt=\u00154t\nform\u00152 in Eq. (59) yields\nC(k;t)\u0018expn\n\u0000Dsk2m+2t=\u00152m\u00002\u0000p\nDtk2o\n\u0015!1\u0000\u0000\u0000\u0000! expn\n\u0000p\nDtk2o\n;(61)\nand the dynamical exponent is z= 4. Thus, if all mo-\nments of the spin pattern up to at least the quadrupole\nmoment are conserved the long-time correlations remain\ndominated by the anomalously slow tracer motion of\nEq. (18). Again, we verify this numerically by computing\nthe mean-squared displacement \u001b2(t) =P\nrr2C(r;t) of\na random time evolution with m\u0000pole conserving spin\ninteractions, see Fig. 8 (b). In practice, we have to make\nsure to avoid localization of the spin pattern dynam-\nics due to a strong fragmentation of the Hilbert space\ninto disconnected subsectors, which can occur for all\nm\u00151 [34{36]. This is achieved by choosing spin-gates\nof su\u000ecient range, which ensures ergodicity of the spin\ndynamics. As expected, the system is described by z= 4\nsubdi\u000busive tracer dynamics in the case of quadrupole\nconservation m= 2, i.e.\u001b2(t)\u0018p\nt, see Fig. 8 (b).\nC.m= 1: Hydrodynamic phase coexistence\nFor the special case of m= 1, both terms in the ex-\nponent ofC(k;t) are equally relevant under the rescaling\nk!k=\u0015,t!\u00154t,\nC(k;t)\u0015!1\u0000\u0000\u0000\u0000! expn\n\u0000Dsk4t\u0000p\nDtk2o\n: (62)\nThe correlation function C(k;t) is thus subject to a com-\npetition between two inequivalent dynamical processes14\n0 20 40 60\ndistance r0.000.020.040.06charge correlation C(r,t)\n°20 0 20\nr/t1/40.000.050.10t1/4C(r,t)\n0 1 2\ndistance r/t1/40.000.250.500.751.00t1/4C(r,t)\n0.0 0.5 1.0\nmomentum t1/4k012|logC(k,t)|\nt1t2t3 <<0 1<</uni03B3/uni223Ck4\n/uni223Ck23£10°16£10°11004£10°1\nmomentum t1/4k10°1100|logC(k,t)|\n/uni223Ck2/uni223Ck4\n/uni03B3/uni03B3\na) b) c)\nFIG. 9. Hydrodynamic phase mixing. a) The pro\fle of the spin correlations at di\u000berent times for a probability \r6= 0 of\nrandom circuit updates that conserve the dipole-moment of the spin pattern. Inset : A scaling collapse of the pro\fles determines\nthe dynamical exponent z= 4.b)The scaling function associated to some 0 \u0014\r\u00141 is neither the Gaussian, that is associated\nto hard core tracer di\u000busion, nor the fundamental solution of the dipole-conserving hydrodynamic equation Eq. (57). It is\ninstead given by a convolution of the two. Here, the scaling functions are normalized such that their value at r= 0 is equal to\none.c)Logarithm of the Fourier transform C(k;t) of the real space correlations C(r;t). The momentum axis has been rescaled\nindependently for di\u000berent \r, de\fned in such a way that all curves coincide at t1=4k= 1,jlogC(k;t)j= 1. This allows us to\ndirectly compare the relative strengths of the ( kt1=4)2{ and (kt1=4)4{ contributions to jlogC(k;t)jfor di\u000berent \r. We see that\nincreasing the probability \rof dipole-conserving spin updates leads to an increasing weight of the ( t1=4k)4-term in Eq. (64).\nInset: On a double logarithmic scale the crossover from ( kt1=4)2{ to (kt1=4)4{ behavior becomes visible.\nthat both have z= 4 but that have di\u000berent forms\nof their respective scaling function. Notably, although\nC(k;t) =C(kt1=4) is a function of kt1=4, it can not be\nwritten in terms of some universal scaling function K(\u0001)\nthat is independent of microscopic details. Instead,\nC(k;t) =K\u0010\nk(Dt)1=4;Ds=p\nD\u0011\n; (63)\ni.e. the form of C(kt1=4) depends non-trivially on the\nratioDs=p\nDwhich determines the mixture of the two\nuniversal processes. Speci\fcally, we can express\nlogC(k;t) =\u0000(Ds+p\nD)h\n\u0016(kt1=4)4+ (1\u0000\u0016) (kt1=4)2i\n;\n(64)\nwhere\u0016=\u0016(Ds=p\nD) =Dsp\nD(1 +Dsp\nD)\u00001and 0\u0014\u0016\u00141.\nThe speci\fc mixture, and thus the long time and length\nscale pro\fle of the spin correlations, is sensitive to micro-\nscopic details of the time evolution, reminiscent of UV-\nIR-mixing [77{83]. This results in a continuously varying\nhydrodynamic universality class controlled by the micro-\nscopic mixing parameter \u0016. Here, we identify a hydrody-\nnamic universality class with both the dynamical expo-\nnent and the scaling function.\nWe con\frm these theoretical considerations numer-\nically in Fig. 9, where we consider a random unitary\ntime evolution with dipole-conserving dynamics within\nthe spin pattern, see Fig. 8 (a). To ensure ergodicity,\nwe use dipole-conserving spin updates ranging over eight\nsites. For any given probability \rat which non-trivial\nspin pattern rearrangements occur the dynamical expo-\nnent isz= 4, as demonstrated by the scaling collapse\nofC(r;t) evaluated at di\u000berent times in Fig. 9 (a). How-\never, varying this probability \re\u000bectively controls the\nratioDs=p\nDand leads to di\u000berent scaling functions asshown in Fig. 9 (b). In particular, the limiting distribu-\ntions are a Gaussian for \r= 0 and the dipole-conserving\nhydrodynamic scaling function of Eq. (58) (for m= 1) as\n\r!1. In addition, we numerically compute the Fourier\ntransformC(k;t) of the correlation pro\fle to verify the\nprediction of Eq. (64). Fig. 9 (c) shows that increasing\nthe rate\rof the spin dynamics leads to an increasing\ncontribution of the ( kt1=4)4\u0000term to log C(k;t).\nThe arbitrary mixing of two distinct dynamical scal-\ning functions as in Eq. (64) can be viewed as phase co-\nexistence of two hydrodynamic phases, in analogy with\nmore conventional phase coexistence occuring at \frst or-\nder equilibrium transitions. To make this analogy more\ntangible, let us imagine a situation in which the dynamics\nof the spin pattern is given by\nF\u000b(k;t) =1p\n2\u0019exp(\u0000Dsjkj\u000bt); (65)\nnow with some general exponent 0 <\u000b<1whose value\nis governed by some underlying model (e.g. through\nthe power-law decay of a long-ranged spin term in the\nconstrained tJ{ like models). Using that C\u000b(k;t) =\nF\u000b(k;t)Gtr(k;t), we obtain the dynamical exponent zas\na function of \u000b:\nz(\u000b) =(\n\u000b;for\u000b<4\n4;for\u000b\u00154: (66)\nIn particular, \u000b>4 corresponds to Gaussian tracer mo-\ntion while\u000b<4 is associated with non-Gaussian scaling\nfunctions. In addition, for \u000b6= 4 we can always write the\nreal space correlations C\u000b(r;t) as\nC\u000b(r;t) = (Dt)\u00001=4F\u000b\u0000\nr(Dt)\u00001=4\u0001\n; (67)15\n1 2 3 4 5\nexponent a0.000.010.02order parameter h(a)\nFIG. 10. Hydrodynamic order parameter. The order pa-\nrameterh(\u000b) of Eq. (68) quanti\fes the deviation of the charge\ncorrelation pro\fle from a Gaussian. \u000blabels the exponent\nof the spin pattern's internal dynamics, see Eq. (65). There\nis a clear discontinuity at \u000b= 4, where long-time dynami-\ncal correlations switch between Gaussian tracer motion and\na non-Gaussian dipole-conserving pro\fle. As a consequence,\n\u000b= 4 can be interpreted as a \frst order dynamical transition,\nwhich allows for phase coexistence of distinct hydrodynamic\nuniversality classes.\nwith a normalized universal scaling functionR\ndxF\u000b(x) = 1. If we thus consider \u000b= 4 to sep-\narate a Gaussian and a non-Gaussian dynamical phase,\nwe can accordingly de\fne an order parameter that\nquanti\fes the non-Gaussianity of the scaling function\nfor a given \u000bvia\nh(\u000b) := min\n\u0015>0Z\ndx\u0010\n\u0015F\u000b(\u0015x)\u00001p\u0019exp(\u0000x2)\u00112\n=\n=8\n<\n:1\n2\u0019min\n\u0015>0R\ndk\u0010\ne\u0000(jkj=\u0015)\u000b\u0000e\u0000k2=4\u00112\n; \u000b< 4\n0; \u000b> 4:\n(68)\nWe have evaluated h(\u000b) numerically in Fig. 10, where we\nsee a clear discontinuity at \u000b= 4. The central prop-\nerty \u0001h(\u000b= 4)>0 can also be demonstrated analyti-\ncally. In particular, the variance of the \u000b!4\u0000scaling\nfunction vanishes, as opposed to a Gaussian [39]. This\njump in the order parameter suggests that we can in-\ndeed interpret the point \u000b= 4 as a \frst order dynamical\ntransition, with Eq. (64) describing the Gaussian/non-\nGaussian phase mixture.\nVI. CONCLUSION & OUTLOOK\nIn this work, we investigated the emergent hydrody-\nnamics oftJ{ like many-body systems in one dimension\nwith constrained spin interactions. We found that for\nchaotic, thermalizing systems the dynamical spin corre-\nlation function at in\fnite temperature is given by a con-\nvolution of the dynamics of the underlying spin pattern\nand the tracer motion of hard core particles. In tJz{\nlike systems all multipole moments of the spin patternare constants of motion and spin correlations are given\nby tracer dynamics alone. This allowed us to demon-\nstrate the emergence of subdi\u000busion with dynamical ex-\nponentz= 4 in several random circuit lattice models\nthat feature a (e\u000bective) constant spin pattern. Using\nresults from the theory of tracer motion we provided ex-\npressions for the full long-time pro\fle of dynamical spin\ncorrelations in these models. It will be interesting to see\nin the future whether additional one-dimensional systems\nfall under this dynamical universality class. We also re-\nmark that although we mostly focused on situations with\na conserved \u001bi=\u00061 spin pattern, our results generalize\nto patterns of higher e\u000bective spin \u001bi=\u0000jSj;:::;jSj. In\nsuch an instance, all odd power spin density correlations\nof the form\nD\u0000^Sz\nx(t)\u00012n+1\u0000^Sz\n0(0)\u00012n+1E\n; (69)\nwith integer n, reduce to the same tracer process up to\na global prefactor. We emphasize that our results apply\nto in\fnite temperature correlations. Whether extensions\nto \fnite temperatures are possible depends sensitively on\nwhether the average charge of the pattern vanishes also\nat \fnite temperatures for a given model.\nWe further established a connection to integrable tJz{\nlike quantum systems which feature conserved spin pat-\nterns and a time evolution that is independent of the\npattern. By mapping to a tracer problem of ballistically\nmoving particles, we were able to reproduce the spin dif-\nfusion constant of the folded XXZ chain and provided\nits full long-time correlation pro\fle. The characteristic\nstaggered oscillations of the resulting correlation pro\fle\ncan be veri\fed in quantum simulation experiments on\nXXZ chains already at moderate anisotropy. We further\npoint out a connection to anomalous current correlations\nrecently reported for the XXZ chain, the XNOR circuit,\nand for a deterministic classical automaton, Refs. [84, 85].\nRef. [84] provides a picture in which such correlations are\ne\u000bectively due to the tracer motion of a single domain\nwall. We expect that the same arguments, and thus the\npresence of anomalous correlations, apply in any system\nwith a conserved pattern, which includes Ref. [85].\nMoreover, it is interesting to study non-integrable\nHamiltonian quantum systems with conserved patterns,\ne.g. via the tJz{ model at \fnite Jzor via adding diago-\nnal interactions to the folded XXZ model. For example,\na stochastic spin chain closely related to the folded XXZ\nmodel is the so-called Ising-Kawasaki model [86, 87],\ntaken in a particular limit: the Hamiltonian HfXXZ\nis supplemented by nearest-neighbor (NN) and next-\nnearest-neighbor (NNN) Ising terms. The name de-\nrives from the fact that the quantum Hamiltonian is ob-\ntained from the Markov operator of a classical Ising chain\nundergoing Kawasaki magnetization-conserving dynam-\nics. The NNN Ising coupling does not commute with\nHfXXZ, leading to integrability breaking in many sec-\ntors [55] while preserving the pattern conservation de-\nscribed above. Such systems should fall under the same16\nuniversality class as the generic random circuits consid-\nered in this work and we expect them to exhibit z= 4\nsubdi\u000busive hard core tracer dynamics. A version of the\nfolded XXZ model in which integrability is broken by\nnoise was recently studied in Ref. [88], concluding z= 4\nas well.\nIn addition, we investigated the dynamics of generic\nsystems in which only a \fnite number of multipole\nmoments of the spin pattern remains conserved. We\nfound that the characteristics of tracer dynamics survive\nif at least the multipole moments up to and including\nthe quadrupole moment are constant. Intriguingly, if\nonly the moments up to the dipole are conserved there\nemerges a special scenario in which spin correlations are\nsubject to a competition between two hydrodynamic\nprocesses with dynamic exponent z= 4 but di\u000berent\nscaling functions. The resulting shape of the long-time\ncorrelations are then susceptible to microscopic details\nof the time evolution and we \fnd the situation to be\nreminiscent of the coexistence of di\u000berent hydrodynamic\nphases at a \frst order transition. How such a scenario\nmay qualitatively arise in systems other than the ones\nconsidered here is an interesting open question for future\nresearch.\nAcknowledgments. { We thank Alvise Bastianello,\nSarang Gopalakrishnan, Oliver Hart, Gabriel Longpr\u0013 e,\nFrank Pollmann, Tibor Rakovszky, Pablo Sala, St\u0013 ephane\nVinet and Philip Zechmann for insightful discus-\nsions. We acknowledge support from the Deutsche\nForschungsgemeinschaft (DFG, German Research Foun-\ndation) under Germany's Excellence Strategy-EXC-\n2111-390814868, TRR80 and DFG grant No. KN1254/2-\n1, No. KN1254/1-2, the European Research Council\n(ERC) under the European Union's Horizon 2020 re-\nsearch and innovation programme (grant agreement No.\n851161), as well as the Munich Quantum Valley, which is\nsupported by the Bavarian state government with funds\nfrom the Hightech Agenda Bayern Plus. We thank the\nNanosystems Initiative Munich (NIM) funded by the\nGerman Excellence Initiative and the Leibniz Supercom-\nputing Centre for access to their computational resources.\nW.W.-K. was funded by a Discovery Grant from NSERC,\na Canada Research Chair, and a grant from the Fonda-\ntion Courtois. Matrix product state simulations were\nperformed using the TeNPy package [75].\nData and materials availability. { Data analysis\nand simulation codes are available on Zenodo upon rea-\nsonable request [89].\nAppendix A: Strength of staggered oscillations\nIn this appendix we derive the exact analytical ex-\npression Eq. (40) for the constant C0de\fned in Eq. (35).\nThis constant determines the strength of the staggered\noscillations on top of the Gaussian enveloping shape\nfor the dynamical spin correlation pro\fle in both therandom XNOR circuit and the folded XXZ model, cf.\nEqs. (44,55). We recall that according to Eq. (37), C0\ncan be written in the domain wall picture as\nC0=\u001c\u0010\n^~\u0014(single )\n0\u00112\u001dX\nrD\n^~\u0014(single )\n2rE0\n; (A1)\nwhere the expectation value h\u0001i0is taken with respect to\nan ensemble where a domain wall with positive charge\n~\u0014(single )\n0 = 1 is \fxed to sit at bond 0. We have already\nevaluated\u001c\u0010\n^~\u0014(single )\n0\u00112\u001d\n= 1=6 in the main text and so\nwe focus on the correlation function\nD\n^QAE0\n:=X\nrD\n^~\u0014(single )\n2rE0\n: (A2)\nD\n^QAE0\ncounts the A-sublattice charge of single domain\nwalls provided a positive domain wall is located at the ori-\ngin. The sublattice charge Eq. (A2) refers only to single\ndomain walls and is conserved in the time evolution. In\nparticular,D\n^QAE0\ncan be calculated entirely within the\nensemble of all possible conserved domain wall patterns,\nsee Fig. 5, where mobile domain wall pairs have already\nbeen removed. This is our approach in the following.\nWe \frst recall that the ensemble of possible conserved\npatterns in the domain wall picture is subject to an ex-\nclusion principle of nearest-neighbor domain walls, see\nFig. 5. That is, a bond with a domain wall must have\ntwo aligned bonds as its neighbors. For simplicity and\nwithout loss of generality we can always assume the left-\nmost domain wall in the system to have positive charge,\n\fxing a Z2degree of freedom for the staggering. The ex-\nclusion property then gives rise to a Fibonacci sequence\nfor the number Np(`) of possible conserved pattern con-\n\fgurations for a number `of bonds:\nNp(`) =Np(`\u00001) +Np(`\u00002): (A3)\nGiven that a (positive) domain wall is \fxed to sit at the\norigin, let us move to the right of the origin and derive\nthe probability p(r) to encounter the next domain wall\nexactly at a distance r. We \fnd p(1) = 0 due to the\nexclusion principle and\np(r\u00152) =Np(`\u0000r)\nNp(`)`!1\u0000\u0000\u0000!'\u0000r; (A4)\nwith the golden ratio '=1+p\n5\n2. As required,P1\nr=1p(r) = 1=('2\u0000') = 1. Let us then further de-\nrive the probability p(AjA) that as we go to right from a\ndomain wall located on the Asublattice, the next domain\nwall we encounter is again located on the Asublattice,\np(AjA) =1X\nn=1p(2n) ='\u000021X\nn=0'\u00002n=1\n'2\u00001=1\n';\n(A5)17\nwhere we used the de\fning equation of the golden ratio\nin the last equality. From this result, we also obtain the\nprobability p(BjA) = 1\u0000p(AjA) to \fnd the next domain\nwall we encounter on the Bsublattice, given that the\nprevious one is located on the Asublattice. Similarly, we\nhavep(BjB) =p(AjA) andp(AjB) =p(BjA).\nWe now take into account that the charges of the do-\nmain walls have to be perfectly anticorrelated, i.e., if\nthere is a positive domain wall at the origin, the next\ndomain wall we encounter to the right must have neg-\native charge. Therefore, moving to the right from the\npositive domain wall at the origin, we can determine the\ncharge of the next domain wall that we \fnd at an Asub-\nlattice bond by counting the number of Bsublattice do-\nmain walls in between. The probability p(A;nB= 0;A)\nto \fnd no other domain wall between two consecutive A\ndomain walls is p(A;nB= 0;A) =p(AjA). The proba-\nbility to \fnd a number nB\u00151 ofBsublattice domain\nwalls between two consecutive Adomain walls is given\nby\np(A;nB\u00151;A) =p(AjB)\u0002\np(BjB))\u0003nB\u00001p(BjA) =\n=\u0002\n1\u0000p(AjA)\u00032\u0002\np(AjA)\u0003nB\u00001:\n(A6)\nIf the number of Bsublattice domain walls between two\nconsecutive Adomain walls is even, the two Adomain\nwalls have opposite charge. If the number of Bsublattice\ndomain walls between two consecutive Adomain walls is\nodd, they have equal charge. Therefore, the probability\nto \fnd two consecutive Asublattice domain walls with\nopposite charge is given by\nP\u0000=1X\nn=0p(A;nB= 2n;A)\n=p(AjA) +\u0002\n1\u0000p(AjA)\u00032p(AjA)1X\nn=0\u0002\np(AjA)\u00032n\n=p(AjA) +\u0002\n1\u0000p(AjA)\u00032p(AjA)\n1\u0000\u0002\np(AjA)\u00032=2\n'2;\n(A7)\nwhere in the last equality we inserted Eq. (A5). Accord-\ningly, the probability to \fnd two consecutive Asublattice\ndomain walls with equal charge is given by\nP+= 1\u0000P\u0000= 1\u00002\n'2: (A8)We note that P\u0000\u00190:764>0:5, i.e. two consecutive A\ndomain wall charges are anticorrelated. This is a `re-\nmainder' of the perfect anticorrelation between two con-\nsecutive domain walls irrespective of the sublattice.\nLet us \fnally take a randomly chosen conserved do-\nmain wall pattern con\fguration with a positive domain\nwall \fxed at the origin and consider the r-thAsublattice\ndomain wall to the right from the origin. The charge of\nther-thAdomain wall is then determined by a sequence\nof exactlyrpairs of two consecutive Adomain walls. The\nprobability to \fnd r0\u0014ranticorrelated consecutive A\ndomain wall pairs in this sequence is\u0000r\nr0\u0001\n(P\u0000)r0(P+)r\u0000r0,\nwhere the binomial coe\u000ecient accounts for the reordering\nof the anticorrelated pairs in the sequence. Since the do-\nmain wall at the origin is positive, a sequence containing\nr0anticorrelated consecutive domain wall pairs implies a\ncharge of (\u00001)r0for ther-thAdomain wall. Perfoming\na sum over the possible number 0 \u0014r0\u0014rof anticorre-\nlated pairs in the sequence and summing over the charge\ncontributions from all Asublattice domain walls we ob-\ntain\nD\n^QAE0\n= 1 + 21X\nr=1rX\nr0=0\u0012r\nr0\u0013\n(\u00001)r0(P\u0000)r0(P+)r\u0000r0=\n= 1 + 21X\nr=1(P+\u0000P\u0000)r=P+\nP\u0000='2\n2\u00001:\n(A9)\nIn the \frst line of the above equation, the \frst term of\nunity is due to the positive contribution of the positive\ncharge \fxed at the origin, while the factor of two is due\nto symmetric contributions from right and left of the ori-\ngin. According to Eq. (A2), Eq. (A9) yields the corre-\nlation functionP\nrD\n^~\u0014(single )\n2rE0\n. Inserting into Eq. (A1)\n\fnally we obtain\nC0=1\n6\u0010'2\n2\u00001\u0011\n=1\n12\u0000\n'\u00001\u0001\n; (A10)\ncompleting our proof.\n[1] J. M. Deutsch, Quantum statistical mechanics in a closed\nsystem, Phys. Rev. A 43, 2046 (1991).\n[2] M. Srednicki, Chaos and quantum thermalization, Phys.\nRev. E 50, 888 (1994).\n[3] M. Rigol, V. Dunjko, and M. Olshanii, Thermalization\nand its mechanism for generic isolated quantum systems,Nature 452, 854 (2008).\n[4] L. D'Alessio, Y. Kafri, A. Polkovnikov, and M. Rigol,\nFrom quantum chaos and eigenstate thermalization to\nstatistical mechanics and thermodynamics, Advances in\nPhysics 65, 239 (2016).\n[5] A. M. Kaufman, M. E. Tai, A. Lukin, M. Rispoli,18\nR. Schittko, P. M. Preiss, and M. Greiner, Quantum ther-\nmalization through entanglement in an isolated many-\nbody system, Science 353, 794 (2016).\n[6] T. Brydges, A. Elben, P. Jurcevic, B. Vermersch,\nC. Maier, B. P. Lanyon, P. Zoller, R. Blatt, and C. F.\nRoos, Probing R\u0013 enyi entanglement entropy via random-\nized measurements, Science 364, 260 (2019).\n[7] P. M. Chaikin and T. C. Lubensky, Principles of Con-\ndensed Matter Physics (Cambridge University Press,\n1995).\n[8] S. Mukerjee, V. Oganesyan, and D. Huse, Statistical the-\nory of transport by strongly interacting lattice fermions,\nPhys. Rev. B 73, 035113 (2006).\n[9] J. Lux, J. M uller, A. Mitra, and A. Rosch, Hydrodynamic\nlong-time tails after a quantum quench, Phys. Rev. A 89,\n053608 (2014).\n[10] A. Bohrdt, C. B. Mendl, M. Endres, and M. Knap,\nScrambling and thermalization in a di\u000busive quantum\nmany-body system, New Journal of Physics 19, 063001\n(2017).\n[11] A. Nahum, J. Ruhman, S. Vijay, and J. Haah, Quantum\nEntanglement Growth under Random Unitary Dynam-\nics, Phys. Rev. X 7, 031016 (2017).\n[12] C. Jonay, D. A. Huse, and A. Nahum, Coarse-grained\ndynamics of operator and state entanglement (2018),\narXiv:1803.00089.\n[13] M. Knap, Entanglement production and information\nscrambling in a noisy spin system, Phys. Rev. B 98,\n184416 (2018).\n[14] T. Rakovszky, F. Pollmann, and C. W. von Keyserlingk,\nSub-ballistic Growth of R\u0013 enyi Entropies due to Di\u000busion,\nPhys. Rev. Lett. 122, 250602 (2019).\n[15] T. Rakovszky, C. W. von Keyserlingk, and F. Pollmann,\nEntanglement growth after inhomogenous quenches,\nPhys. Rev. B 100, 125139 (2019).\n[16] A. Nahum, S. Vijay, and J. Haah, Operator Spreading\nin Random Unitary Circuits, Phys. Rev. X 8, 021014\n(2018).\n[17] C. W. von Keyserlingk, T. Rakovszky, F. Pollmann, and\nS. L. Sondhi, Operator Hydrodynamics, OTOCs, and\nEntanglement Growth in Systems without Conservation\nLaws, Phys. Rev. X 8, 021013 (2018).\n[18] V. Khemani, A. Vishwanath, and D. A. Huse, Oper-\nator Spreading and the Emergence of Dissipative Hy-\ndrodynamics under Unitary Evolution with Conservation\nLaws, Phys. Rev. X 8, 031057 (2018).\n[19] T. Rakovszky, F. Pollmann, and C. W. von Keyserlingk,\nDi\u000busive Hydrodynamics of Out-of-Time-Ordered Corre-\nlators with Charge Conservation, Phys. Rev. X 8, 031058\n(2018).\n[20] A. Nahum, J. Ruhman, and D. A. Huse, Dynamics of\nentanglement and transport in one-dimensional systems\nwith quenched randomness, Phys. Rev. B 98, 035118\n(2018).\n[21] D. E. Parker, X. Cao, A. Avdoshkin, T. Sca\u000edi, and\nE. Altman, A Universal Operator Growth Hypothesis,\nPhys. Rev. X 9, 041017 (2019).\n[22] R. M. Nandkishore and M. Hermele, Fractons, Annual\nReview of Condensed Matter Physics 10, 295 (2019).\n[23] M. Pretko, X. Chen, and Y. You, Fracton phases of\nmatter, International Journal of Modern Physics A 35,\n2030003 (2020).\n[24] C. Chamon, Quantum Glassiness in Strongly Correlated\nClean Systems: An Example of Topological Overprotec-tion, Phys. Rev. Lett. 94, 040402 (2005).\n[25] J. Haah, Local stabilizer codes in three dimensions with-\nout string logical operators, Phys. Rev. A 83, 042330\n(2011).\n[26] B. Yoshida, Exotic topological order in fractal spin liq-\nuids, Phys. Rev. B 88, 125122 (2013).\n[27] S. Vijay, J. Haah, and L. Fu, A new kind of topological\nquantum order: A dimensional hierarchy of quasiparti-\ncles built from stationary excitations, Phys. Rev. B 92,\n235136 (2015).\n[28] S. Vijay, J. Haah, and L. Fu, Fracton topological order,\ngeneralized lattice gauge theory, and duality, Phys. Rev.\nB94, 235157 (2016).\n[29] M. Pretko and L. Radzihovsky, Fracton-Elasticity Dual-\nity, Phys. Rev. Lett. 120, 195301 (2018).\n[30] M. Pretko, Subdimensional particle structure of higher\nrankU(1) spin liquids, Phys. Rev. B 95, 115139 (2017).\n[31] M. Pretko, The fracton gauge principle, Phys. Rev. B 98,\n115134 (2018).\n[32] M. Pretko, Higher-spin Witten e\u000bect and two-\ndimensional fracton phases, Phys. Rev. B 96, 125151\n(2017).\n[33] D. J. Williamson, Z. Bi, and M. Cheng, Fractonic matter\nin symmetry-enriched U(1) gauge theory, Phys. Rev. B\n100, 125150 (2019).\n[34] P. Sala, T. Rakovszky, R. Verresen, M. Knap, and\nF. Pollmann, Ergodicity Breaking Arising from Hilbert\nSpace Fragmentation in Dipole-Conserving Hamiltoni-\nans, Phys. Rev. X 10, 011047 (2020).\n[35] V. Khemani, M. Hermele, and R. Nandkishore, Local-\nization from Hilbert space shattering: From theory to\nphysical realizations, Phys. Rev. B 101, 174204 (2020).\n[36] T. Rakovszky, P. Sala, R. Verresen, M. Knap, and F. Poll-\nmann, Statistical localization: From strong fragmenta-\ntion to strong edge modes, Phys. Rev. B 101, 125126\n(2020).\n[37] S. Scherg, T. Kohlert, P. Sala, F. Pollmann,\nB. Hebbe Madhusudhana, I. Bloch, and M. Aidelsburger,\nObserving non-ergodicity due to kinetic constraints in\ntilted Fermi-Hubbard chains, Nature Communications\n12, 4490 (2021).\n[38] A. Gromov, A. Lucas, and R. M. Nandkishore, Fracton\nhydrodynamics, Phys. Rev. Research 2, 033124 (2020).\n[39] J. Feldmeier, P. Sala, G. De Tomasi, F. Pollmann, and\nM. Knap, Anomalous Di\u000busion in Dipole- and Higher-\nMoment-Conserving Systems, Phys. Rev. Lett. 125,\n245303 (2020).\n[40] A. Morningstar, V. Khemani, and D. A. Huse, Kinetically\nconstrained freezing transition in a dipole-conserving sys-\ntem, Phys. Rev. B 101, 214205 (2020).\n[41] P. Zhang, Subdi\u000busion in strongly tilted lattice systems,\nPhys. Rev. Research 2, 033129 (2020).\n[42] J. Iaconis, S. Vijay, and R. Nandkishore, Anomalous sub-\ndi\u000busion from subsystem symmetries, Phys. Rev. B 100,\n214301 (2019).\n[43] J. Feldmeier, F. Pollmann, and M. Knap, Emergent frac-\nton dynamics in a nonplanar dimer model, Phys. Rev. B\n103, 094303 (2021).\n[44] S. Moudgalya, A. Prem, D. A. Huse, and A. Chan, Spec-\ntral statistics in constrained many-body quantum chaotic\nsystems, Phys. Rev. Research 3, 023176 (2021).\n[45] E. Guardado-Sanchez, A. Morningstar, B. M. Spar, P. T.\nBrown, D. A. Huse, and W. S. Bakr, Subdi\u000busion and\nHeat Transport in a Tilted Two-Dimensional Fermi-19\nHubbard System, Phys. Rev. X 10, 011042 (2020).\n[46] Singh, H. and Ware, B. A. and Vasseur, R. and Friedman,\nA. J., Subdi\u000busion and Many-Body Quantum Chaos\nwith Kinetic Constraints, Phys. Rev. Lett. 127, 230602\n(2021).\n[47] J. Iaconis, A. Lucas, and R. Nandkishore, Multipole con-\nservation laws and subdi\u000busion in any dimension, Phys.\nRev. E 103, 022142 (2021).\n[48] P. Glorioso, J. Guo, J. F. Rodriguez-Nieva, and A. Lucas,\nBreakdown of hydrodynamics below four dimensions in\na fracton \ruid (2021), arXiv:2105.13365.\n[49] K. T. Grosvenor, C. Hoyos, F. Pe~ na Benitez, and\nP. Sur\u0013 owka, Hydrodynamics of ideal fracton \ruids, Phys.\nRev. Research 3, 043186 (2021).\n[50] A. Osborne and A. Lucas, In\fnite families of fracton\n\ruids with momentum conservation, Phys. Rev. B 105,\n024311 (2022).\n[51] J. Feldmeier and M. Knap, Critically Slow Operator Dy-\nnamics in Constrained Many-Body Systems, Phys. Rev.\nLett. 127, 235301 (2021).\n[52] A. G. Burchards, J. Feldmeier, A. Schuckert, and\nM. Knap, Coupled Hydrodynamics in Dipole-Conserving\nQuantum Systems (2022), arXiv:2201.08852.\n[53] H. Spohn, Large scale dynamics of interacting particles\n(Springer Science & Business Media, 2012).\n[54] G. De Tomasi, D. Hetterich, P. Sala, and F. Pollmann,\nDynamics of strongly interacting systems: From Fock-\nspace fragmentation to Many-Body Localization (2019),\narXiv:1909.03073.\n[55] Z.-C. Yang, F. Liu, A. V. Gorshkov, and T. Iadecola,\nHilbert-Space Fragmentation from Strict Con\fnement,\nPhys. Rev. Lett. 124, 207602 (2020).\n[56] L. Zadnik and M. Fagotti, The Folded Spin-1/2 XXZ\nModel: I. Diagonalisation, Jamming, and Ground State\nProperties, SciPost Phys. Core 4, 10 (2021).\n[57] L. Zadnik, K. Bidzhiev, and M. Fagotti, The Folded Spin-\n1/2 XXZ Model: II. Thermodynamics and Hydrodynam-\nics with a Minimal Set of Charges, SciPost Phys. 10, 99\n(2021).\n[58] B. Pozsgay, T. Gombor, A. Hutsalyuk, Y. Jiang,\nL. Pristy\u0013 ak, and E. Vernier, Integrable spin chain with\nhilbert space fragmentation and solvable real-time dy-\nnamics, Phys. Rev. E 104, 044106 (2021).\n[59] K. Bidzhiev, M. Fagotti, and L. Zadnik, Macroscopic Ef-\nfects of Localized Measurements in Jammed States of\nQuantum Spin Chains, Phys. Rev. Lett. 128, 130603\n(2022).\n[60] T. E. Harris, Di\u000busion with \"Collisions\" between Parti-\ncles, Journal of Applied Probability 2, 323 (1965).\n[61] D. G. Levitt, Dynamics of a Single-File Pore: Non-\nFickian Behavior, Phys. Rev. A 8, 3050 (1973).\n[62] S. Alexander and P. Pincus, Di\u000busion of labeled particles\non one-dimensional chains, Phys. Rev. B 18, 2011 (1978).\n[63] H. van Beijeren, K. W. Kehr, and R. Kutner, Di\u000busion\nin concentrated lattice gases. III. Tracer di\u000busion on a\none-dimensional lattice, Phys. Rev. B 28, 5711 (1983).\n[64] R. G. Dias, Exact solution of the strong coupling t\u0000V\nmodel with twisted boundary conditions, Phys. Rev. B\n62, 7791 (2000).\n[65] G. I. Menon, M. Barma, and D. Dhar, Conservation laws\nand integrability of a one-dimensional model of di\u000busing\ndimers, Journal of Statistical Physics 86, 1237 (1997).\n[66] M. Medenjak, K. Klobas, and T. Prosen, Di\u000busion in De-\nterministic Interacting Lattice Systems, Phys. Rev. Lett.119, 110603 (2017).\n[67] K. Klobas, M. Medenjak, and T. Prosen, Exactly solvable\ndeterministic lattice model of crossover between ballistic\nand di\u000busive transport, Journal of Statistical Mechanics:\nTheory and Experiment 2018 , 123202 (2018).\n[68] M. Kotrla, Energy spectrum of the hubbard model with\nu=1, Physics Letters A 145, 33 (1990).\n[69] S. Gopalakrishnan and R. Vasseur, Kinetic Theory of\nSpin Di\u000busion and Superdi\u000busion in XXZ Spin Chains,\nPhys. Rev. Lett. 122, 127202 (2019).\n[70] O. A. Castro-Alvaredo, B. Doyon, and T. Yoshimura,\nEmergent Hydrodynamics in Integrable Quantum Sys-\ntems Out of Equilibrium, Phys. Rev. X 6, 041065 (2016).\n[71] B. Bertini, M. Collura, J. De Nardis, and M. Fagotti,\nTransport in Out-of-Equilibrium XXZ Chains: Exact\nPro\fles of Charges and Currents, Phys. Rev. Lett. 117,\n207201 (2016).\n[72] C. Karrasch, J. E. Moore, and F. Heidrich-Meisner, Real-\ntime and real-space spin and energy dynamics in one-\ndimensional spin-1\n2systems induced by local quantum\nquenches at \fnite temperatures, Phys. Rev. B 89, 075139\n(2014).\n[73] C. Karrasch, Hubbard-to-heisenberg crossover (and e\u000e-\ncient computation) of drude weights at low temperatures,\nNew Journal of Physics 19, 033027 (2017).\n[74] P. N. Jepsen, J. Amato-Grill, I. Dimitrova, W. W. Ho,\nE. Demler, and W. Ketterle, Spin transport in a tunable\nHeisenberg model realized with ultracold atoms, Nature\n588, 403 (2020).\n[75] J. Hauschild and F. Pollmann, E\u000ecient numerical sim-\nulations with Tensor Networks: Tensor Network Python\n(TeNPy), SciPost Phys. Lect. Notes , 5 (2018).\n[76] B. Pozsgay, A yang{baxter integrable cellular automa-\nton with a four site update rule, Journal of Physics A:\nMathematical and Theoretical 54, 384001 (2021).\n[77] Y. You, J. Bibo, F. Pollmann, and T. L. Hughes, Fracton\nCritical Point in Higher-Order Topological Phase Tran-\nsition (2020), arXiv:2008.01746.\n[78] N. Seiberg and S.-H. Shao, Exotic U(1) Symmetries, Du-\nality, and Fractons in 3+1-Dimensional Quantum Field\nTheory, SciPost Phys. 9, 46 (2020).\n[79] P. Gorantla, H. T. Lam, N. Seiberg, and S.-H. Shao, Low-\nenergy limit of some exotic lattice theories and UV/IR\nmixing, Phys. Rev. B 104, 235116 (2021).\n[80] Y. You, J. Bibo, T. L. Hughes, and F. Pollmann, Frac-\ntonic critical point proximate to a higher-order topo-\nlogical insulator: How does UV blend with IR? (2021),\narXiv:2101.01724.\n[81] O. Hart and R. Nandkishore, Experimental signatures of\ngapless type ii fracton phases (2021), arXiv:2106.15631.\n[82] P. Sala, J. Lehmann, T. Rakovszky, and F. Pollmann,\nDynamics in systems with modulated symmetries (2021),\narXiv:2110.08302.\n[83] O. Hart, A. Lucas, and R. Nandkishore, Hidden quasi-\nconservation laws in fracton hydrodynamics, Phys. Rev.\nE105, 044103 (2022).\n[84] S. Gopalakrishnan, A. Morningstar, R. Vasseur, and\nV. Khemani, Theory of anomalous full counting statis-\ntics in anisotropic spin chains (2022), arXiv:2203.09526.\n[85] i. c. v. Krajnik, J. Schmidt, V. Pasquier, E. Ilievski, and\nT. Prosen, Exact anomalous current \ructuations in a\ndeterministic interacting model, Phys. Rev. Lett. 128,\n160601 (2022).\n[86] M. D. Grynberg, Revisiting Kawasaki dynamics in20\none dimension, Phys. Rev. E 82, 051121 (2010),\narXiv:1009.4511 [cond-mat.stat-mech].\n[87] S. Vinet, G. Longpr\u0013 e, and W. Witczak-Krempa,\nExcitations and ergodicity of critical quantum spin\nchains from non-equilibrium classical dynamics (2021),\narXiv:2107.04615.[88] J. De Nardis, S. Gopalakrishnan, R. Vasseur, and\nB. Ware, Subdi\u000busive hydrodynamics of nearly-\nintegrable anisotropic spin chains, arXiv:2109.13251\n(2021).\n[89] All data and simulation codes are available upon reason-\nable request at 10.5281/zenodo.6553436."    },    {        "title": "1310.4811v1.Entanglement_dynamics_in_a_system_attached_to_self_interacting_spinbaths.pdf",        "content": "Entanglement dynamics in a system attached to self-interacting\nspinbaths\nNageswaran Rajendran\u0003\nFachbereich Physik, Technische Universit at Dortmund, 44221 Dortmund, Germany.\n(Dated: June 8, 2021)\nDynamics of quantum entanglement shared between system spins which are con-\nnected to thermal equilibrium baths is studied. Central spin system comprises of\nthe entangled spins, and is connected to baths and one of the bath has strong intra-\nenvironmental coupling. The dynamics between the system and baths are studied\nand inferred that that intra-envirnomental coupling guards against the system from\nthe e\u000bects of the spin baths.\nI. INTRODUCTION\nWide ranges of open quantum dynamics are being studied in the implementations of\nquantum computing systems [1, 2]. Decoherence is a natural quantum phenomena occurs\nwhen the system interacts with environments, depending on the coherence time, decoherence\nmight lead to irreversible loss of information. Basically, the environments can be indepen-\ndent harmonic oscillator baths [3], and spinbath systems [2]. Hence, we need to suppress\nthe decoherence during the dynamics of the system, there are various methods, such as\nquantum error corrections [4], dynamical decoupling [5], decoherence free subspaces [6] can\nbe used in various implementation schemes. However Tessieri and Wilkie suggested a model\nof central systems attached to a bath of spins interacting within themselves. Tessieri-Wilkie\nmodel suggests that strong intracoupling between the spins in the baths can suppress the\ndecoherence [7], which is a necessity for the quantum computation and information imple-\nmentations. With the similar e\u000bect, entanglement sharing suppress the decoherence in a\nsingle spin self-interacting with the thermal(initially) baths [8] and strong decoupling e\u000bects\non the disentanglement in the system against the variations in coupling constants between\nthe baths [9]. The model consists of a central system of spins, which is connected with N\n\u0003nageswaran@chaos.physik.tu-dortmund.dearXiv:1310.4811v1  [quant-ph]  17 Oct 20132\nnumber of spin baths with an intra-coupling strength \u0015. Description of this type of open\nquantum system can be represented by the Hamiltonian.\nII. MODEL\nA pair of spin-1/2 system is taken as in an entangled state j\t0i, initially, and it is\nconnected to the two system of baths, in which \frst bath has a coupled spin pair and\nanother bath has one spin. Dynamics of the system is studied under the conditions of\nferromagnetic and antiferromagnetic interaction shared among the spins.\nH=HS+HB+HSB (1)\nWhere,HS,HBandHSBare denoted as the Hamiltonian system, bath and system-bath\ndescription, respectively. The Hamiltonian of the kthspin in the central system is given as\nHk\nS=~!S\n2\u001bk\nz+\f\u001bk\nx (2)\nwhere!kis the frequency of the system and \u001bx;y;zare Pauli operators; \frepresent reducibility\nof the Hamiltonian, in other words, spin ups and downs which form a even and odd subspaces\nformed by the basis states of the system,\n\f8\n><\n>:6= 0;Hamiltonian becomes irreducible;\n= 0;reducible Hamiltonian.(3)\nUsing the natural units, ~,kBare set to unity. Each of the bath description is represented\nas\nHl\nB=X\ni!i\n2\u001bi\nz+\fX\ni\u001bi\nx+\u0015X\ni;j\ni6=j\u001bi\nx\u001bj\nx (4)\n\u0015=8\n><\n>:[0;n]; n2R;for antiferromagnetic case\n[\u0000n;0];ferromagnetic case.(5)3\n\u0015is the spin-spin coupling in each of its bath. And the system-bath description is represented\nas\nHSB=\u00150X\nl\u001bSk\nx\u001bBl\nx (6)\nThe initial state of the open system can be described by \u001a(0) =\u001aS(0)\n\u001aB(0), the initial\nstate of the bath is in thermal equilibrium and it is in the form,\n\u001aB(0) =exp(\u0000HB=kT)\nTr(exp(\u0000HB=kT))(7)\nEvolution of the entire system under its Hamiltonian is represented by the interaction\npicture.\n\u001a(t) =e\u0000iHt\u001a(0)eiHt(8)\nIII. ENTANGLEMENT EVOLUTION\nAs an entanglement measure concurrence has been used for this case, which built over the\nentanglement of formation, and may be the generalized measure for the class of arbitrary\nmixed entangled states [10]. Concurrence of an entangled state \u001ais given by\nC[\u001a] = maxf0;\u00151\u00003X\ni=2\u0015ig (9)\nAs,\u0015iis the eigenvalue of the matrix \u001a~\u001a, and ~\u001ais de\fned as ( \u001by\n\u001by)\u001a(\u001by\n\u001by), and\u0015i\nare arranged in decreasing order.\nThe system has two spins, which are taken as an arbitrary entangled pair j\ti=\nexp (\u0000i\u000e=2) cos (\u000b=2)j00i+ exp (i\u000e=2) sin (\u000b=2)j11i, as our state. With the di\u000berent set\nof cases are calculated for the initial states\nj\t(0)i=8\n>>>><\n>>>>:1p\n2j00i+1p\n2j11i\np\n3\n2e\u0000i\u0019=8j00i+1\n2ei\u0019=8j11i\np\n3\n2e\u0000i\u0019=8j00i+1\n2ei\u0019=8j11i+\u000fj01i:(10)4\nThe system is connected to all the bath through the coupling with the strength of \u00150.\nThe dynamics of the system coupled to spin baths, which are also inter connected through\nthe (anti-)ferromagnetic coupling. The system is connected to two baths, which are inde-\npendent, and \frst bath has two spins interacting within themselves with a coupling strength\n\u0015which varies from \u00152[\u00002;2] and in some cases stronger coupling strength is considered\nfor comparative purposes. And the second bath has a single spin.\nFrom the equations 2, 4 and 6, the system Hamiltonian is de\fned as,\nHS=\u0010!s\n2\u001b(1)\nz+\f\u001b(1)\nx\u0011\n\n\u0010!s\n2\u001b(2)\nz+\f\u001b(2)\nx\u0011\n: (11)\nHamiltonian of the \frst spin bath with the intra-bath coupling \u0015is given by\nH(1)\nB=!(1)\nb1\n2\u001b(1)\nz+!(2)\nb1\n2\u001b(2)\nz+\f\u001b(1)\nx+\f\u001b(2)\nx+\u0015\u001b(1)\nx\u001b(2)\nx; (12)\nand also the Hamiltonian of the second bath with a single spin is given by\nH(2)\nB=!(1)\nb2\n2\u001b(1b2)\nz+\f\u001b(1b2)\nx: (13)\nThe system and bath Hamiltonian with the coupling strength \u00150is written as\nHSB=\u00150(\u001bs\nx\u001bB1\nx+\u001bs\nx\u001bB2\nx) (14)\nFor the sake of brevity, let us assign the values !s= 0:7, as the system has lesser energy\nas the system is in low temperature. And !i\nB= 1 and\f= 0:01 and considering the system\nlies in the low temperature kBT= 0:02\nThe density matrices of the system and \frst and second bath can be represented as\n\u001a0=\u001a(1;2)\ns\n\u001a(3;4)\nB1\n\u001a(5)\nB2(15)\n\u001a(t) = exp(\u0000iHt)\u001a0exp(iHt) (16)\n\u001a(t) is traced over the baths, which gives Tr3;4;5(\u001a(t)) =\u001a1;2\ns, the reduced density matrices\nof the system.\nThe quantity of entanglement shared between the system spins is a\u000bected through system-\nbath interaction and intra-bath coupling among the spins in the bath. Therefore, the con-\ncurrence of the evolved state may be viewed as a function of coupling parameter among the5\nspins in the bath and the evolution of the state with respect to time. Hence the concurrence\nof the state of the system C(\u001a1;2\ns) is calculated using the 9 numerically, for the di\u000berent\ninitial states as mentioned in 10.\nA. Entangled state\nA maximally entangled state is taken as the initial state for the \frst case. Hence the\nconcurrence for the entangled state must read from unity. Both at Antiferromagnetic and\nferromagenetic regions, the system tend to behave similar manner, while comparing the\nconcurrence of the system. When the intra-bath coupling goes to zero, the system, When\nthe intra-environmental coupling gets stronger, e\u000bects on the eigenstates of the systems\nlesser compared to the weaker coupling cases. From the \fgIII A(a) and (b), we could see\nfor the higher intra-bath coupling the, the system tries to revoke its entanglement as the\nsystem and bath evolves.\n(a)\n (b)\nFIG. 1. Concurrence plotted for the state j\ti=1p\n2(j00i+j11i) various coupling strength \u0015=\n0;1;2;10 and the system bath coupling \u00150= 0:1 \fg.(a) and \u00150= 1 \fg(b)\nSystem-bath coupling has additional e\u000bects when that \ructuate between 0.1 and 1. For\nthe stronger \u00150= 1 system-bath coupling, the eigenstates of the spinbaths gets coupled to\nthe system, which decreases the entanglement of the state of the system. Nevertheless, the\nevolution of the system tells that the decoherence may be e\u000bectively handled.6\nB. Partially entangled state\nIn the case of partially entangled state as the input state, j\ti=p\n3\n2e\u0000i\u0019=8j00i+1\n2ei\u0019=8j11i,\nthe concurrence of the initial state is 0 :6, and the entanglement dynamics follows the same\npattern as it happened in maximally entangled state during the evolution.\nEntanglement measure for the system at the initial state is calculated as 0.6.\n(a)\n (b)\nFIG. 2. Concurrence plotted for the state j\ti=p\n3\n2e\u0000i\u0019=8j00i+1\n2ei\u0019=8j11ivarious coupling strength\n\u0015= 0;1;2;10 and the system bath coupling \u00150= 0:1 and\u00150= 1\nThe partially entangled state with additional elements added to the entangled state j\ti=\np\n3\n2e\u0000i\u0019=8j00i+1\n2ei\u0019=8j11i+\u000f(j01i+j10i):\n(a)\n (b)\nFIG. 3. Concurrence plotted for the state j\ti=p\n3\n2e\u0000i\u0019=8j00i+1\n2ei\u0019=8j11i+\u000fj01ivarious coupling\nstrength\u0015= 0;1;2;10 and the system bath coupling \u00150= 0:1 and\u00150= 1\nSimilar to the maximally entangled cases, the system retrieves its concurrence during the7\nevolution; even for the partially entangled and entangled mixed states, the system seems to\nbe robust against its interaction with two of its baths.\nHowever, e\u000bect of the system-bath coupling, for \u00150= 1 seems to a\u000bect the system heavily.\nTo see the e\u000bects of quantum Zeno dynamics during the evolution for the higher system-bath\ncoupling could be interesting.\nIV. FREQUENT MEASUREMENTS\nQuantum Zeno e\u000bects can be viewed quantum systems by doing frequent measurements\n[11]. Applying frequent measurements on the system decouples the system from its en-\nvironment. If the initial state is in \u001a(1;2)=j (1)ih (2)j, which undergoes the evolution\nU(tk) = expiHtk, wheretKis the survival time, t>tk, the state will start to decay. Hence\nfrequent projective measurement within the interval tkwill lead the state stay in the state.\nTherefore, the survival probability is given by,\nPzeno=kh 0j (tN)ik2= NY\nk=1D\n 0\f\f\fe({\n~tk(H))\f\f\f 0E!2\n(17)\nWhereNis the iterated measurements. When the selective projective measurement\nis made the probability is proportional to cos(2N)(\u000e=2). In our case, jTr(e(\u0000iHtk)\u001as)j2\ncalculated to compute the Zeno probability.\nWhen the system has lesser coupling strength with the baths, the entanglement of the\nsystem slowly decreases during the free evolution. To suppress the decay of the state of\nthe system, number of frequent measurements are applied during the evolution tk= 2\u0019=\u0015 0\nduring the free evolution.\nWith the Zeno measurements, we can see 4 the survival probability of the state increases\nas the number of measurements increases even in the strong system-bath coupling.\nV. CONCLUSION\nBy having the spinbaths with strong intra-bath coupling strength, robustness of the state\nof the system may be preserved. The entanglement measure of the system goes down when\nthe system gets interacted to the baths, which are in thermal equilibrium initially, and the8\n(a)\n (b)\nFIG. 4. Survival probability of the state during the evolution for the \u00150= 0:1;1\ndecreasing e\u000bect may be due to `inertia' of the baths, as we could also see when intra-bath\ncoupling nears to zero, the rate of change of concurrence is relatively faster. The system\nretrieves its initial amount of entanglement, as the system progresses to evolve under its\nHamiltonian.\nAs we have seen the entanglement goes down, when the system undergoes relatively\nsmaller interaction, i.e, weakly coupled with the baths. The system has been set to undergo\ninto the frequent measurements to view the robustness of the system. During the frequent\nmeasurements with various number of iterations.\nThe interaction between the system and bath is playing an important role in the system-\nbath dynamics, apart from the strong coupling between the intra-bath spins. This could be\nan interesting inference for the experimental implementations.\n[1] W. H. Zurek, Rev. Mod. Phys. 75, 715 (2003).\n[2] H.-P. Breuer and F. Petruccione, The Theory of open quantum systems (Oxford University\nPress, 2002).\n[3] A. O. Caldeira and A. J. Leggett, Ann. Phys. (N. Y). 149, 374 (1983).\n[4] E. Knill, R. La\ramme, and L. Viola, Phys. Rev. Lett. 84, 2525 (2000).\n[5] G. S. Uhrig, Phys. Rev. Lett. 98, 100504 (2007).\n[6] D. A. Lidar, I. L. Chuang, and K. B. Whaley, Phys. Rev. Lett. 81, 2594 (1998).\n[7] L. Tessieri and J. Wilkie, J. Phys. A. Math. Gen. 36, 12305 (2003).9\n[8] C. M. Dawson, A. P. Hines, R. H. McKenzie, and G. J. Milburn, Phys. Rev. A 71, 52321\n(2005).\n[9] F. F. Olsen, A. Olaya-Castro, and N. F. Johnson, J. Phys. Conf. Ser. 84, 12006 (2007).\n[10] W. K. Wootters, Phys. Rev. Lett. 80, 2245 (1998).\n[11] B. Misra and E. C. G. Sudarshan, J. Math. Phys. 18, 756 (1977)."    },    {        "title": "1209.5667v1.Spin_noise_spectroscopy_of_donor_bound_electrons_in_ZnO.pdf",        "content": "Spin noise spectroscopy of donor bound electrons in ZnO\nH. Horn,1A. Balocchi,2X. Marie,2A. Bakin,3A. Waag,3M. Oestreich,1and J. H ubner1,\u0003\n1Institut f ur Festk orperphysik, Leibniz Universit at Hannover, Appelstr. 2, 30167 Hannover, Germany\n2INSA-CNRS-UPS, LPCNO, Universit\u0013 e de Toulouse, 135 Av. de Rangueil, 31077 Toulouse, France\n3Institute for Semiconductor Technology, Technische Universit at Braunschweig,\nHans-Sommer-Stra\u0019e 66, 38106 Braunschweig, Germany\n(Dated: November 6, 2018)\nWe investigate the intrinsic spin dynamics of electrons bound to Al impurities in bulk ZnO by\noptical spin noise spectroscopy. Spin noise spectroscopy enables us to investigate the longitudinal\nand transverse spin relaxation time with respect to nuclear and external magnetic \felds in a single\nspectrum. On one hand, the spin dynamic is dominated by the intrinsic hyper\fne interaction with\nthe nuclear spins of the naturally occurring67Zn isotope. We measure a typical spin dephasing time\nof 23 ns in agreement with the expected theoretical values. On the other hand, we measure a third,\nvery high spin dephasing rate which is attributed to a high defect density of the investigated ZnO\nmaterial. Measurements of the spin dynamics under the in\ruence of transverse as well as longitudinal\nexternal magnetic \felds unambiguously reveal the intriguing connections of the electron spin with\nits nuclear and structural environment.\nPACS numbers: 42.50.Lc, 42.62.Fi, 68.55.Ln, 78.67.Rb\nAmong the oxide based II-VI semiconductors, ZnO has\nbeen devoted a continuously high attention in what con-\ncerns its optoelectronic properties [1] and possible ap-\nplication in semiconductor spintronics [2, 3]. Especially\nthe long electron spin coherence times at room temper-\natures which results from the rather weak in\ruence of\nthe spin orbit splitting onto the conduction band states\n[4, 5] makes ZnO and ZnO nanostructures [6] a promis-\ning material in semiconductor spintronics and spin based\nquantum-optronics [7]. The semiconductor material ZnO\nis easily available with a high abundance and nowadays\ncomes with a vast selection of growth and structuring\nmethods [8]. Furthermore, ZnO bears a plethora of inter-\nesting spin physics due to its intriguing exciton dynamics\n[9] and scalability of its nuclear spin properties [10].\nIn this work we investigate the intrinsic spin dynamics\nof donor bound electrons in the wide band gap material\nZnO by optical spin noise spectroscopy via below band\ngap Faraday rotation. Spin noise spectroscopy measures\nthe omnipresent \ructuations of the spin degree of free-\ndom [11{15] and has developed into a powerful tool for\nthe investigation of the intrinsic spin dynamics in semi-\nconductors [16] since it generally avoids optical excita-\ntion [17{19]. In the measurements presented here, SNS\nreveals the transverse and longitudinal spin relaxation\nof donor bound electrons due to hyper\fne interaction\nwhereat only a low fraction of the host nuclei carry a nu-\nclear spin. Both, transverse and longitudinal times are\nacquired in a single spin noise spectrum by SNS which\nallows the straightforward extraction of the in\ruence of\nnuclear and external magnetic \felds. Furthermore, we\nobserve an additional, very short spin dephasing time\nwhich we attribute to the increased interaction with de-\nfects located inside the e\u000bective donor volume.\nThe sample is a thin \flm of predominantly bulk,\n(0001)-grown ZnO with a nominal thickness of 450 nmdeposited on a sapphire substrate by MBE growth. In-\nvestigations by scanning electron microscopy reveal a\nnanoporous structure with a granularity varying between\n10 and 20 nm at the surface. Henceforth, the investi-\ngated material most likely exhibits a high defect density\ninduced by surface states which is con\frmed by a short\nphotoluminescence lifetime at low temperatures which\ndrops even further at elevated temperatures. Figure 1a\ndepicts the time resolved photoluminescence spectrum\nof the sample recorded with a synchroscan streak cam-\nera system under above band gap excitation with a fre-\nquency doubled, picosecond Ti:sapphire laser oscillator\n[20]. The donor-bound exciton transition D0X and its\nLO phonon replicas are clearly visible. The inset de-\npicts the time transient of the D0X transition decaying\nmono-exponentially with a lifetime of \u001cl= 60 ps mea-\nsured atT= 4 K. The donor-bound exciton transition\nis the dominant optical transition. The spin dynamics\nof the donor bound electron is explored in the spin noise\nmeasurements presented later. For a better characteriza-\ntion of the donor-bound exciton transition, we performed\ncontinuous-wave (cw)-photoluminescence and transmis-\nsion measurements which are shown in Fig. 1b. From\nthe photoluminescence spectra we attribute the neutral\nD0X (3.360 eV) and ionized D+X (3.364) transition to\nAl impurities besides an unidenti\fed background PL sig-\nnal around the D+X transition [21]. The shaded area in\nFig. 1b indicates the spectral region where SNS measure-\nments are performed in dependence of the probe photon\nenergy detuning relative to the D0X transition. These\nmeasurements are described further below.\nFigure 2 depicts the experimental setup for the mea-\nsurements of the spin dynamics in transmission by spin\nnoise spectroscopy. The light source is a frequency dou-\nbled, cw-Ti:sapphire ring laser with a spectral width of\n0:3 neV. The linear polarized light is focused down to aarXiv:1209.5667v1  [cond-mat.mes-hall]  25 Sep 20122\nintensity (arb. units)\n10100100010000\nenergy (eV)3.15 3.2 3.25 3.3 3.35 3.4 3.4540 ps\n80 ps\n101001000\ntime (ps) 0 100 200\n2LO-D X0LO-D X0D X0\nTES/\nDBXintensity (arb. units)\ntransmission (arb. units)\nenergy (eV)\nD X0\nD X+\nTES/ \nDBXa)\nb)\n101001000\n00.20.4\n3.3 3.32 3.34 3.36 3.38\nFIG. 1. (Color online) a) Photoluminescence spectra recorded\nat 40 ps and 80 ps, respectively, after excitation with a pi-\ncosecond laser pulse with a photon energy of 3.54 eV and a\nsample temperature of T=4 K. The inset shows the PL decay\ntransient of the donor bound D0X exciton. b) CW-PL (black\nline) recorded with a higher resolution. The labels D0X and\nD+X mark the spectral positions of the neutral and ionized\ndonor-bound exciton transition, respectively. The red dots\nare transmission measurements performed with a spectrally\nnarrow laser.\nspot size with a typical diameter of 3 \u0016m and an inten-\nsity of 132\u0016W=\u0016m2and transmitted through the sample.\nThe sample is mounted in a Helium cold \fnger cryostat\nand cooled to a temperature of 4 K. The transmitted\nprobe beam acquires the stochastic spin dynamics of the\ndonor electron spin ensemble in the ZnO sample via be-\nlow band gap Faraday rotation. The time-dependent,\n\ructuating Faraday rotation angle is analyzed by a po-\nlarization bridge consisting of a polarizing beam splitter\nand a balanced photo receiver. Finally, the time-domain\nFaraday rotation data is analyzed via fast Fourier trans-\nformation (FFT) spectral analysis in the radio frequency\nregime on a standard personal computer.\nThe spin noise signal is low compared to the strong\nbackground optical shot noise and hence a background\nnoise spectrum is recorded at a transverse magnetic \feld\nofB?= 10 mT which is then subtracted from the lower\nθ(t)\nsample\nin \nmicro-\ncryostatTi:Sapphire \nring laser\nA/D conversion via digitizer card & FFT spectrum analysisZnO in transverse magnetic field\npolarization \nbridgebalanced\nphotoreceiver\nx\nyzlens lens\nω    2ω\nBxFIG. 2. (Color online) Experimental spin noise setup. The\nlight source is a frequency doubled cw-Ti:Sapphire laser. The\nlinear polarized probe light is focussed through the sample\nwhich is mounted in a Helium cold-\fnger cryostat. The rota-\ntion of the linear polarized light is measured by a polarization\nbridge and a balanced receiver. The electrically ampli\fed dif-\nference signal is digitized in the time-domain and spectrally\nanalyzed by a computer. External magnetic \feld can be ap-\nplied in transverse and longitudinal (not shown) direction.\n\feld spectrum. At B?= 10 mT, the spin noise signal is\ncompletely shifted out of the measurable spectrum and\nhence only optical shot and electrical noise is recorded.\nFurthermore, the obtained noise spectrum is divided by\nthe optical shot noise spectrum [22] in order to eliminate\nthe non-uniform ampli\fcation of the balanced detector.\nThe result is a normalized noise spectrum with only pos-\nitive values (see Fig. 3a). A typical mono-exponential,\ni.e., homogeneous decay of the spin orientation with a de-\ncay constant \u001csresults in a Lorentzian shaped spin noise\ncontribution in the noise spectra with a full width given\nby\rh= 1=(\u0019\u001cs). Inhomogeneities in the sample broaden\nthe spin noise signal due to, e.g., g-factor variations [23]\nor hyper\fne interaction [24] and lead to a frequency spec-\ntrum following a normal distribution with a standard de-\nviation\u001bs. The inhomogeneous spin dephasing rate \riis\n\fnally given by 2 \u001bs=\ri=\u0019[25].\nThe spin noise spectra with applied transverse mag-\nnetic \feld B?are shown in Fig. 3a and reveal three dis-\ntinct contributions. First, a very low homogeneous spin\ndephasing rate \r(1)\nhappearing as a Lorentzian like peak\ncentered at zero frequency. We attribute this contribu-\ntion to the longitudinal spin relaxation rate of localized\nelectrons with respect to the stochastic nuclear \feld ori-\nentation BNat the respective donor site. Second, an in-\nhomogeneous spin noise contribution \rioriginating from\nthe same localized electrons but which follows the ap-\nplied transverse magnetic \feld with the respective Lar-\nmor frequency \u0017L=g\u0003\u0016BB=h. The magnitude of \ri\nis given by the dispersion of BNwithin the ensemble\nof localized electrons. A third spin noise contribution\nwith a very high homogeneous spin dephasing rate \r(2)\nh\nand centered as well at the Larmor frequency \u0017Lis at-3\nspin noise (shot noise units)\n00.0050.01\nfrequency (MHz)0 20 40 60 800 mT\n0.5 mT\n1 mT\n2 mT\n4 mT\n10 mT\nLarmor freuqency (MHz)02040\ntransverse magnetic field (mT)0 1 2data\nlinear fit\n00.010.02\nfrequency (MHz)0 20 40 60 80(Bz = 0.34 mT) - (B = 0 mT)\n(Bz = 0.68 mT) - (B = 0 mT)\n(Bz = 1.36 mT) - (B = 0 mT)\n(Bz = 2.72 mT) - (B = 0 mT)rate (MHz)\n024152025\nlongitudinal magnetic field (mT)0 1 2 3γi\nγspin noise (shot noise units)b)a)\nh(1)\nFIG. 3. (Color online) a) Spin noise spectra recorded with\ndi\u000berent transverse magnetic \felds. The background spec-\ntrum has been acquired at B?= 10 mT and the probe laser\nphoton energy is 3.355 eV. The data is \ftted by a model ac-\ncording to the description in the text. The inset depicts the\nchange in Larmor frequency for the inhomogeneous spin noise\ncontribution. b) Spin noise di\u000berence spectra for di\u000berent lon-\ngitudinal magnetic \felds. The spectra are normalized to an\naverage spin noise power density in the range of 50 to 80 MHz,\nin order to compensate for experimental drifts. The data is\n\ftted according the model described in the text. The inset\ncompares the homogeneous longitudinal ( \r(1)\nh) and inhomo-\ngeneous transverse ( \ri) spin relaxation rates in dependence\nofBk. The probe laser photon energy is 3.359 eV for these\nmeasurements.\ntributed to donor-bound electrons interacting with one\nor more defects within the e\u000bective donor-volume. This\nis discussed in more detail later. The dependence of the\nLarmor frequency \u0017Lwith magnetic \feld is shown in the\ninset of Fig. 3 and yields an e\u000bective Land\u0013 e g-factor of\ng\u0003= 1:97(9) which \fts very well to the Land\u0013 e g-factor\nof the Al-donor bound electron as measured by electron\nparamagnetic resonance spectroscopy [26]. This assign-\nment is consistent with the spectral identi\fcation of the\nD0X transition as elucidated in the previous section. The\nwidth of the inhomogeneously broadened spin noise spec-\ntrum, i.e., \riis determined by the \ructuating nuclear\n\feld sampled by the localized electron wave function [24].In ZnO only1\n2\u00024:1% of all lattice ions interact via hy-\nper\fne interaction with the donor electrons due the nat-\nural abundance of the67Zn isotope with a nuclear spin\nofIN= 5=2. The electron wave function of an electron\nbound to an Al-donor has an e\u000bective Bohr radius of\nrB= 1:93 nm and thus experiences an interaction with\n\u00185\u0002103nuclei. Taking the natural abundance of67Zn\ninto account together with a Fermi-contact hyper\fne in-\nteraction strength of 3 :7\u0016eV [10] one obtains a standard\ndeviation of the local magnetic \feld of \u0001 BN= 0:22 mT\nwhich corresponds to a theoretical limit for the inhomo-\ngeneous spin dephasing time of ( \u0019\ri)\u00001= 26:3 ns. The\nextracted inhomogeneous spin dephasing times from the\ndata presented in Fig. 3a yield T\u0003\n2= 23(\u00062:5) ns match-\ning very well to the theoretical expected times limited by\nthe hyper\fne interaction.\nThe longitudinal spin relaxation rate \r(1)\nhis de\fned\nwith respect to the e\u000bective magnetic \feld axis, i.e.,\nBN+B?. The corresponding spin noise contribution\nappears as a narrow Lorentzian like centered at zero fre-\nquency. The spin noise power density at very low frequen-\ncies is usually superimposed by 1 =felectrical noise which\nmakes a clear assignment of \r(1)\nhdi\u000ecult. However, a lon-\ngitudinal spin relaxation time ( \u0019\r(1)\nh)\u00001>\u0018200 ns can be\nextracted even without taking into account the frequency\nrange from DC to 500 kHz in the data evaluation. Fur-\nthermore, the observed spin noise power associated with\n\r(1)\nhdecreases with increasing transverse magnetic \feld\nsince the average projection of BN+B?onto the direc-\ntion of observation decreases [27].\nThe situation changes for external magnetic \felds ap-\nplied in longitudinal , i.e.,z-direction. Here an increasing\nmagnetic \feld quenches the in\ruence of the stochastic\nnature of the hyper\fne interaction [19]. Figure 3b shows\nspin noise spectra obtained by measuring the di\u000berence\nbetween spin noise recorded with an applied longitudinal\nBkand zero magnetic \feld. Clearly visible is that spin\nnoise power from the inhomogeneous transverse part is\nredistributed to the homogeneous longitudinal spin noise\ncontribution with increasing Bk. The total spin noise\npower is constant in thermal equilibrium, i.e., the ex-\nternal magnetic \feld is small compared to kBT=\u0016 Band\npolarization e\u000bects are negligible. The data presented\nin Fig. 3b is \ftted by a single Lorentzian and Gaussian\nfunction centered both at zero frequency. We could not\nresolve the Maxwell-type distribution of the nuclear \feld\norientation [27]. The inset of Fig. 3b shows the spin de-\nphasing rates \riand\r(1)\nh. The longitudinal spin dephas-\ning rate is slightly higher compared to the measurements\nwithB?due to a smaller detuning from the optical res-\nonance of the D0X in the corresponding measurements.\nThe homogenous spin contribution with the very high\nspin dephasing rate \r(2)\nhcancels out in the di\u000berence mea-\nsurements with longitudinal magnetic \feld due to the\nstrongly dispersed spin noise power density.4\na)\nb)\n050100\nenergy (eV)3.355 3.36 3.3650.0010.010.11\n00.20.4energy (eV)\n3.355 3.36 3.365\nγh\nγi γh  − noise power\nγi − noise power  spin noise power (shot noise units) rate (MHz)\ntransmission (arb. units)(2)\n(2)\nFIG. 4. (Color online) Dependency of a) the total spin noise\npower and b) spin dephasing rate on the detuning relative to\nthe D0X transition. The spectral position of the maximum\nspin noise power at 3.360 eV indicates that the major spin\nnoise contribution originate from Al-donor bound electrons.\nIn order to unveil the origin of the third, very fast ho-\nmogeneous spin noise contribution \r(2)\nhwe perform spin\nnoise measurements with a varying detuning with respect\nto the D0X transition. Figure 4 compares a) the spin\nnoise power and b) the spin dephasing rate of the \r(2)\nh\nspin noise contribution with the inhomogeneous \ricontri-\nbution. The spectral position of the maximum spin noise\npower at 3.360 eV indicates that the major spin noise\ncontribution originate for both from Al-donor bound elec-\ntrons. However, the total spin noise power of the \r(2)\nh\ncontribution is about one order of magnitude larger than\nthe inhomogeneous contribution. The relative spin noise\npower scales with the contributing densities of spins such\nthat we conclude, that roughly 90% of all donor-bound\nelectrons undergo a fast spin-relaxation via defect medi-\nated spin-scattering in conjunction with spin orbit split-\nting. The exact mechanism is not clear at this point, but\nwe assume an Elliot-Yafet like spin dephasing mechanism\nwhere a high scattering rate leads to a higher probabil-\nity of an electron spin \rip [28, 29]. Under the assump-\ntion that at least one defect within the donor volume\nalready leads to a fast spin decay we estimate a defect\ndensityndby relating the donor volume VAlto the aver-\nage defect volume Vd=n\u00001\nd=VAl=0:9 which results in\nnd= 3\u00021019cm\u00003. The high defect density is consis-tent with the nanocrystalline structure as measured by\nscanning electron microscopy. Furthermore, the spin de-\nphasing rate \r(2)\nhis about one order of magnitude larger\ncompared to \ri. In fact it is even higher than \u0017Lfor all\nmeasured magnetic \felds (excluding the background ac-\nquisition). This in turn explains the non-Gaussian form\nof the spin noise contribution associated with \r(2)\nhsince\nthe spin relaxation follows that of an overdamped oscil-\nlator, which can be approximated by an exponential, i.e.,\nhomogenous decay in the time domain.\nIn conclusion, we present the \frst all optical spin noise\nmeasurements on the wide-band gap semiconductor ma-\nterial ZnO. All measured spin noise contributions are\nidenti\fed to originate from the Al-donor bound electron\nin thermal equilibrium. The presented spin noise mea-\nsurements unveil the rich physics of spin dynamics gov-\nerned by the Overhauser nuclear \feld with the extracted\ninhomogeneous spin dephasing times being in accordance\nwith the peculiar natural ZnO isotope composition. Most\ninterestingly, we found additionally a strong defect me-\ndiated spin noise signal giving rise to very short spin life-\ntimes possibly originating from an Elliot-Yafet like spin\n\rip mechanism but which certainly deserves more exten-\nsive future investigations.\nWe thank H. Schmidt for taking the SEM pictures\nand M. Mansur-Al-Suleiman for the helpful discussion.\nWe gratefully acknowledge \fnancial support by the NTH\nschool Contacts in Nanosystems , the BMBF project\nQuaHLRep and by the Deutsche Forschungsgemeinschaft\nin the framework of the priority program SPP1285 Semi-\nconductor Spintronics .\n\u0003jhuebner@nano.uni-hannover.de\n[1]U.Ozg ur, Journal of Applied Physics 98, 041301 (2005).\n[2] I. Zutic, J. Fabian, and S. Das Sarma, Review of Modern\nPhysics 76, 323 (2004).\n[3] T. Dietl, H. Ohno, F. Matsukura, J. Cibert, and D. Fer-\nrand, Science 287, 1019 (2000).\n[4] S. Ghosh, V. Sih, W. H. Lau, D. D. Awschalom, S.-Y.\nBae, S. Wang, S. Vaidya, and G. Chapline, Applied\nPhysics Letters 86, 232507 (2005).\n[5] N. J. Harmon, W. O. Putikka, and R. Joynt, Physical\nReview B 79, 115204 (2009).\n[6] N. Jan\u0019en, K. M. Whitaker, D. R. Gamelin, and\nR. Bratschitsch, Nano Letters 8, 1991 (2008).\n[7] W. K. Liu, K. M. Whitaker, A. L. Smith, K. R. Kit-\ntilstved, B. H. Robinson, and D. R. Gamelin, Physical\nReview Letters 98, 186804 (2007).\n[8] A. Janotti and C. G. van de Walle, Reports on Progress\nin Physics 72, 126501 (2009).\n[9] D. Lagarde, A. Balocchi, P. Renucci, H. Carr\u0012 ere, F. Zhao,\nT. Amand, X. Marie, Z. X. Mei, X. L. Du, and Q. K.\nXue, Physical Review B 78, 033203 (2008).\n[10] K. M. Whitaker, S. T. Ochsenbein, A. L. Smith, D. C.\nEchodu, B. H. Robinson, and D. R. Gamelin, The Jour-\nnal of Physical Chemistry C 114, 14467 (2010).5\n[11] F. Bloch, Physical Review 70, 460 (1946).\n[12] E. B. Aleksandrov and V. S. Zapasskii, Soviet Physics -\nJETP 54, 64 (1981).\n[13] G. E. Katsoprinakis, A. T. Dellis, and I. K. Kominis,\nPhysical Review A 75, 042502 (2007).\n[14] S. A. Crooker, D. G. Rickel, A. V. Balatsky, and D. L.\nSmith, Nature 431, 49 (2004).\n[15] H. Horn, G. M. M uller, E. M. Rasel, L. Santos, J. H ubner,\nand M. Oestreich, Physical Review A 84, 043851 (2011).\n[16] G. M. M uller, M. Oestreich, M. R omer, and J. H ubner,\nPhysica E: Low-dimensional Systems and Nanostructures\n43, 569 (2010).\n[17] G. M. M uller, M. R omer, D. Schuh, W. Wegscheider,\nJ. H ubner, and M. Oestreich, Physical Review Letters\n101, 206601 (2008).\n[18] M. R omer, H. Bernien, G. M uller, D. Schuh, J. H ubner,\nand M. Oestreich, Physical Review B 81, 075216 (2010).\n[19] R. Dahbashi, J. H ubner, F. Berski, J. Wiegand, X. Marie,\nK. Pierz, H. W. Schumacher, and M. Oestreich, Applied\nPhysics Letters 100, 031906 (2012).\n[20] J. H ubner and M. Oestreich, in Spin physics in semi-\nconductors , edited by M. I. Dyakonov (Springer, Berlin,\n2008) pp. 115{134.[21] C. F. Klingshirn, A. Waag, A. Ho\u000bmann, and J. Geurts,\nZinc Oxide: From fundamental properties towards novel\napplications (Springer, Berlin Heidelberg, 2010).\n[22] The subtraction of the background noise power spectrum\nwith the noise spectrum of the unilluminated detector\nyields the optical shot noise spectrum. Hereby the re-\nsulting spin noise power spectrum is expressed in units\nof optical shot noise power.\n[23] F. Berski, H. Kuhn, J. G. Lonnemann, J. H ubner, and\nM. Oestreich, ArXiv e-prints , arXiv:1207.0081v1 (2012).\n[24] I. A. Merkulov, A. L. Efros, and M. Rosen, Physical\nReview B 65, 205309 (2002).\n[25] D. Pines and C. P. Slichter, Physical Review 100, 1014\n(1955).\n[26] S. B. Orlinskii, J. Schmidt, P. G. Baranov, V. Lorrmann,\nI. Riedel, D. Rauh, and V. Dyakonov, Physical Review\nB77, 115334 (2008).\n[27] M. M. Glazov and E. L. Ivchenko, ArXiv e-prints ,\narXiv:1206.3479 (2012).\n[28] R. J. Elliott, Physical Review 96, 266 (1954).\n[29] Y. Yafet, R. W. Keyes, and E. N. Adams, J. Phys. Chem.\nSolids 1, 137 (1956)."    },    {        "title": "2204.07563v2.Geometric_integration_of_classical_spin_dynamics_via_a_mean_field_Schrödinger_equation.pdf",        "content": "arXiv:2204.07563v2  [cond-mat.str-el]  22 Jul 2022Geometric integration of classical spin dynamics via a mean -field Schrödinger equation\nDavid Dahlbom,1Hao Zhang,1,2Cole Miles,3Xiaojian Bai,4Cristian D. Batista,1,5and Kipton Barros6,∗\n1Department of Physics and Astronomy, The University of Tenn essee, Knoxville, Tennessee 37996, USA\n2Materials Science and Technology Division, Oak Ridge Natio nal Laboratory, Oak Ridge, Tennessee 37831, USA\n3Department of Physics, Cornell University, Ithaca, New Yor k 14850, USA\n4Neutron Scattering Division, Oak Ridge National Laborator y, Oak Ridge, TN 37831, USA\n5Quantum Condensed Matter Division and Shull-Wollan Center ,\nOak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA\n6Theoretical Division and CNLS, Los Alamos National Laborat ory, Los Alamos, New Mexico 87545, USA\nThe Landau-Lifshitz equation describes the time-evolutio n of magnetic dipoles, and can be derived\nby taking the classical limit of a quantum mechanical spin Ha miltonian. To take this limit, one\nconstrains the many-body quantum state to a tensor product o f coherent states, thereby neglecting\nentanglement between sites. Expectation values of the quan tum spin operators produce the usual\nclassical spin dipoles. One may also consider expectation v alues of polynomials of the spin operators,\nleading to quadrupole and higher-order spin moments, which satisfy a dynamical equation of motion\nthat generalizes the Landau-Lifshitz dynamics [Zhang and B atista, Phys. Rev. B 104, 104409\n(2021)]. Here, we reformulate the dynamics of these N2−1generalized spin components as a mean-\nfield Schrödinger equation on the N-dimensional coherent state. This viewpoint suggests effici ent\nintegration methods that respect the local symplectic stru cture of the classical spin dynamics.\nI. INTRODUCTION\nThe Landau-Lifshitz dynamics (LLD),\ndsi\ndt=−si×∂H\n∂si, (1)\ndescribes the time evolution of classical spins siwith con-\nserved Hamiltonian H(s1,...sL). In the special case of\na time-invariant effective field Bi=−∂H/∂si, each spin\nsiwould simply precess around Bi.\nLLD is one possible classical limit of a quantum me-\nchanical spin system. Effectively, each spin operator ˆSα\ni\nis replaced by its expectation value, sα\ni=∝an}bracketle{tZ|ˆSα\ni|Z∝an}bracketri}ht, rep-\nresenting the spin angular momentum on site i, measured\nin units of /planckover2pi1= 1. The quantum state is approximated\nas a tensor product, |Z∝an}bracketri}ht=|Z1∝an}bracketri}ht⊗···⊗|ZL∝an}bracketri}ht, thereby ne-\nglecting entanglement between sites. Spin operators ˆSα\ni\nfor distinct sites i∝ne}ationslash=jcommute. In each local Hilbert\nspacei, spin operators act as generators for the Lie group\nSU(2), which satisfy the commutation relation\n/bracketleftBig\nˆSα\ni,ˆSβ\ni/bracketrightBig\n= iǫαβγˆSγ\ni, (2)\nwhere we use the convention of summation over repeated\nGreek indices (here, γ= 1,2,3). The fully antisymmetric\nLevi-Civita symbol ǫαβγappearing in this commutator\nis the underlying source of the vector cross product ap-\npearing in the LLD. An explicit construction of the spin\noperators ˆSα\ni, in an arbitrary spin- Srepresentation, is\npresented in Appendix A.\nOur interest is a generalized spin dynamics (GSD) that\nbetter approximates local quantum spin states.1Unlike\nLLD, which describes only the expected spin dipole si,\nGSD describes the evolution of a full set of quantum ex-\npectation values nifor each local Hilbert space i. Thisgeneralization is strictly necessary to model large classe s\nof magnets with effective spins S >1/2and strong single-\nion anisotropy induced by the combination of spin-orbit\ncoupling and crystal field effects, such as 4d–5dand4f–\n5f-electron materials as well as several 3dmagnets.2–4\nThe generalization is also necessary to describe mag-\nnets comprising weakly-coupled entangled units, such as\ndimers,5trimers6and tetrahedra.7Both LLD and GSD\nare classical approximations that neglect entanglement\nbetween different local Hilbert spaces.\nLet us now define the generalized spin vector ni. Local\nquantum states |Zi∝an}bracketri}hthave dimension N= 2S+1for spins\nof magnitude S. Such states evolve under special unitary\ntransformations, i.e., the Lie group SU( N). This group is\ngenerated by the traceless Hermitian operators, i.e., the\nLie algebra su(N), with dimension N2−1. An operator\nbasisˆTα\niforsu(N)spans all local physical observables\nfor sitei. Given an underlying quantum state |Zi∝an}bracketri}ht, we\ndefine generalized spin components to be the expectation\nvalues,nα\ni=∝an}bracketle{tZi|ˆTα\ni|Zi∝an}bracketri}ht. Without loss of generality, we\ntakeˆSα\nito be a subset of ˆTα\ni, such that the three com-\nponents of the spin dipole sα\niare a subset of the N2−1\ncomponents of nα\ni.\nGeneralized spins evolve according to the GSD,1\ndnα\ni\ndt=fαβγ∂H\n∂nβ\ninγ\ni, (3)\nwhere the classical Hamiltonian H(n1,...nL)is given by\nthe quantum Hamiltonian ˆHunder the substitution rule\nˆTα\ni→nα\ni(cf. Appendix B). The structure constants fαβγ\nare defined by the commutation relation for generators,\n/bracketleftBig\nˆTα\ni,ˆTβ\ni/bracketrightBig\n= ifαβγˆTγ\ni. (4)\nIn the special case of N= 2, one finds fαβγ=ǫαβγ,\nthereby recovering the LLD of Eq. (1).2\nThis paper is concerned with the efficient numerical\nintegration of Eq. (3), in a way that respects the under-\nlying geometric structure. We will reformulate GSD as a\nmean-field Schrödinger equation,\nd\ndtZi=−iHiZi, (5)\nwhereZi∈CNrepresents components of |Zi∝an}bracketri}htin some\nbasis, and\nHi=∂H\n∂nα\niTα(6)\nmay be interpreted as an effective local Hamiltonian ma-\ntrix that acts on Zi. Without loss of generality, we have\nassumed a basis for each local Hilbert space such that\nthe matrix representations Tαof the quantum operators\nˆTα\niare independent of site i. The generalized spin com-\nponents are then\nnα\ni=Z†\niTαZi. (7)\nThe derivation of Eq. (5) will be presented in Sec. II.\nThere are great practical advantages to reformulating\nthe dynamics of expectation values ni(t)as a dynamics\nof underlying coherent states Zi(t). WhenNis large,\nit is much preferred to work with the 2Nreal compo-\nnents of the complex vector Zirather than the N2−1\nreal components of ni. We emphasize that both objects\ncarry the same physical information. The state Zimaps\nto local physical observables nivia Eq. (7). Conversely,\nniis physically valid if and only if there exists a corre-\nsponding Zithat satisfies Eq. (7). Note that an overall\ncomplex phase factor, Zi→e−iφZi, is irrelevant to ob-\nservables ni. Also note that the magnitude |Zi|is invari-\nant under the unitary evolution of Eq. (5). The remaining\nphysical states Zilive on a 2(N−1)-dimensional man-\nifold, known mathematically as the complex projective\nspace CP(N−1). The corresponding space of physically\nallowed spins ni, embedded within RN2−1, is therefore\nhighly constrained. A simple way to enforce these con-\nstraints is to work with Zidirectly, as in Eq. (5).\nAs a concrete example, consider the case of a sin-\ngle spin-1/2 site, with N= 2quantum levels (spin up\nand spin down), and a physical manifold of dimension\n2(N−1) = 2 . This manifold can be understood through\nthe Bloch sphere construction, which yields normalized\ndipoles. The Landau-Lifshitz equation defines a dynam-\nics directly on these dipoles, which inherently conserves\ndipole magnitude. The Schrödinger dynamics, Eq. (5),\ndefines an equivalent dynamics via the evolution of the\nquantum state Z∈C2, with conserved magnitude, up\nto an irrelevant complex phase factor, yielding again two\nreal degrees of freedom. In the mathematics literature,\nthe equivalence between these two dynamics is known as\nthe momentum map.8,9\nEquation (3), for general N, can be understood as a\nspecial type of Lie-Poisson system.8The geometric mean-\ning of a Lie-Poisson system (for antisymmetric structureconstantsfαβγ) is perhaps best understood through the\nmatrixni=nα\niTα, known in high-energy physics as the\ncolor field. As we will review in Sec. II, this matrix\nevolves dynamically as dni/dt= i[ni,Hi], and its eigen-\nvalues are constants of motion. Numerical methods have\nrecently been designed to exactly respect this isospectral\nflow.10,11In our specific context, there is a unique great-\nest eigenvalue of ni, and the associated eigenvector is Zi,\nup to an irrelevant scaling factor. All other eigenvalues\nofniare degenerate. The isospectral flow condition be-\ncomes equivalent to the constraints implicit in Eq. (7).\nOur approach therefore reformulates the full matrix dy-\nnamics of nias the dynamics of the single eigenvector Zi.\nThe final scheme, which we call the Schrödinger midpoint\nmethod, will be presented in Sec. III.\nAn important property of Eq. (5) is that it has a canon-\nical Hamiltonian structure for any N. Specifically, the\nreal and imaginary components of Ziact as canonical\npositions and momenta, and obey Hamilton’s equations\nof motion. The Schrödinger midpoint method exactly\nrespects this symplectic structure and therefore enables\ndynamical integration over arbitrarily long time-scales\nwithout numerical drift.\nGiven a specific quantum Hamiltonian ˆH, the numeri-\ncal implementation of the Schrödinger midpoint method\nis relatively straightforward. The main task is to build\nthe matrix Hi(n1,...nL)for each site i. Using the\nframework of Appendix B, the classical Hamiltonian\nH(n1,...nL)will be at most linear in each spin com-\nponentnα\ni. ThenHican be viewed as a mean-field ap-\nproximation to the quantum Hamiltonian ˆHunder the\nsubstitution ˆTα\nj→nα\njfor all sites j∝ne}ationslash=i, up to an irrele-\nvant constant shift. The linear combination of generators\nTαappearing in Eq. (6) becomes a simple polynomial of\nspin operators, directly reflecting the definition of ˆH. We\nwill demonstrate this procedure through explicit exam-\nples in Sec. IV.\nII. CLASSICAL DYNAMICS IN THE\nSCHRÖDINGER PICTURE\nA. Unitary evolution of expectation values\nEquation (3) is a Lie-Poisson system, and describes co-\nadjoint orbits on the dual Lie algebra.8,12We will make\nthis statement concrete using ordinary matrix language.\nLetTαbe generators for SU( N) in the defining rep-\nresentation. That is, Tαare a basis for the Lie algebra\nsu(N), the space of traceless, Hermitian, N×Nmatrices.\nThe matrix commutator is,\n[Tα,Tβ] = ifαβγTγ, (8)\ninherited from that of the quantum operators, Eq. (4).\nWe will require that the basis satisfies an orthonormal-\nity condition,3\ntrTαTβ=τδαβ. (9)\nThis condition makes it possible to interpret Tαas a basis\nalso for the dual Lie algebra, su∗(N). Orthonormality is\nequivalent to antisymmetry of fαβγin all indices\nfαβγ=−fβαγ=−fαγβ. (10)\nOur convention is to select a basis Tαthat includes\nthe three spin matrices Sαas a subset. Substitution of\nTα→Sαinto Eq. (9) determines τ, as given in Eq. (A7)\nof Appendix A.\nEquation (3) defines the dynamics of spin components\nnα\ni. Equivalently, we may consider the time evolution of\nthe matrix\nni=nα\niTα, (11)\ninterpreted as an element of dual Lie algebra su∗(N).\nIt will be convenient to identify the energy gradient\n∂H/∂nα\niwith the matrix Hidefined in Eq. (6), inter-\npreted as an element of su(N). Using this notation,\nEq. (3) is compactly expressed as\ndni\ndt= i [ni,Hi], (12)\nwhich follows from the antisymmetry of fαβγ, and the\nmatrix commutators in Eq. (8). The dynamics may also\nbe written\nni(t) =Ui(t)ni(0)U−1\ni(t), (13)\nwhereU(t)∈SU(N)satisfies\nd\ndtUi(t) =−iHi(t)Ui(t), (14)\nwith initial condition U(0) =I, as may be verified by\nexplicit differentiation.\nB. Schrödinger dynamics of coherent states\nNow we will reformulate the unitary evolution of the\nmatrixnias a dynamics of the vector Zithat gives rise to\nexpectation values nivia Eq. (7). To derive this dynam-\nics in a way that builds physical intuition, we introduce\nthe outer product,\nρi=ZiZ†\ni, (15)\nin analogy with a pure density matrix for site i.\nSpin components, Eq. (7), may be calculated in two\ndifferent ways,\ntrρiTα=nα\ni= trniTα/τ. (16)\nFor the second equality, we used the definition of Eq. (11)\nand orthonormality, Eq. (9).The matrix ρiis Hermitian, and the generators Tα\nspan allN×NHermitian, traceless matrices. From this\nwe deduce\nρi=ni/τ+cI, (17)\nwhereIis the identity matrix, and c=|Zi|2/N.\nBecauseIcommutes with any matrix, the matrices ni\nandρishare the same dynamical equation\ndρi\ndt= i [ρi,Hi], (18)\nor equivalently,\nρi(t) =Ui(t)ρi(0)U−1\ni(t). (19)\nThis dynamics may be interpreted as the von Neumann\nevolution of the density matrix. Referring to Eq. (15), it\nfollows that the coherent states must evolve as,\nZi(t) =Ui(t)Zi(0). (20)\nTaking the time derivative of both sides, and substituting\nfrom Eq. (14) yields\ndZi/dt=−iHiZi,\nwhich confirms our claim that Eq. (5) is a reformulation\nof the GSD defined in Eq. (3).\nThe results of this section may be restated in a more\nabstract and general mathematical language. Equa-\ntion (12) can be viewed as an isospectral flow dW/dt=\n[B,W], withW=niandB= (iHi)†. The eigenvalues of\nWare constants of motion. Each eigenvector vofWsat-\nisfies the dynamical equation dv/dt=Bv, corresponding\nto our Schrödinger equation. Although we derived this\nresult in the context of the Lie-Poisson system on su(N),\nit generalizes to a much broader class of so-called reduc-\ntiveLie algebras.10,11Note that most classical Lie alge-\nbras are reductive, including gl(N,C),gl(N,R),so(N),\nandsp(N), in addition to our working example of su(N).\nIn our application to spin dynamics, we benefit from the\nfact that a single eigenvector, v=Zi, fully describes the\nmatrixW=ni. More generally, if the initial condition\nW(0)has low rank (up to some constant shift), then the\ntime-evolved state W(t)will continue to have low rank,\nand modeling the dynamics through the time-evolving\neigenvectors becomes beneficial.\nC. Conservation laws\nLie-Poisson systems such as Eq. (3) satisfy a number of\nconservation laws. Some of these are associated with the\ngeometric structure of the phase space, and are indepen-\ndent of the choice of Hamiltonian. One may verify that\nany function C(ni)that satisfies (∂C/∂nα\ni)fαβγnγ\ni= 0is\na constant of motion. Such functions arise from Casimirs\nof the Lie algebra, i.e., symmetric, homogeneous poly-\nnomials of the basis matrices Tαthat commute with4\nall algebra elements. There are N−1Casimirs of\nSU(N). The simplest example is the quadratic Casimir\n|ni|2=/summationtext\nα(nα\ni)2which, for spin-1\n2systems, reduces to\nthe dipole magnitude squared.\nA second class of conservation laws arise from sym-\nmetries of the Hamiltonian. For example, energy is a\nconstant of motion provided that the Hamiltonian has\nno explicit time-dependence. One may verify dH/dt= 0\ndirectly by contracting ∂H/∂nα\nion both sides of Eq. (3),\nand using the antisymmetry of fαβγ.\nThe Schrödinger dynamics, Eq. (5), is equivalent to the\ngeneralized spin dynamics, Eq. (3), and therefore shares\nthese conservation laws.\nEquation (3) has the form of a Lie-Poisson system.\nThe Darboux-Lie theorem states that Lie-Poisson sys-\ntems have a local Hamiltonian structure.8,13That is, in\nthe neighborhood of each spin configuration nα\ni, there\nexists a change of coordinates that gives rise to canoni-\ncal variables (pi,qi)that satisfy Hamilton’s equations of\nmotion locally. Interestingly, the Schrödinger equation\ngives rise to a Hamiltonian dynamics that is valid glob-\nally. Specifically, in Appendix C we demonstrate that the\nreal and imaginary components of the coherent state,\nZi= (pi−iqi)/√\n2, (21)\nsatisfy Hamilton’s equations of motion,\ndpi\ndt=−∂H\n∂qi,dqi\ndt= +∂H\n∂pi. (22)\nThe canonical Hamiltonian structure of the Schrödinger\nequation ensures conservation of the symplectic 2-form/summationtext\ni,adpi,a∧dqi,a.\nD. State normalization\nThe magnitude of Ziis a conserved quantity in the\nSchrödinger dynamics, Eq. (5). Absent other knowledge,\nthe normalization convention |Zi|= 1is a natural choice,\nand emphasizes the interpretation of Zias a quantum\nmechanical coherent spin state. Rescaling Zican be use-\nful, however, to adjust the overall magnitude of the clas-\nsical spin components. A carefully selected rescaling can\nstrongly enhance the agreement between an approximate\nclassical model and the true quantum mechanical sys-\ntem.14\nConsider, first, the LLD of spin dipoles, Eq. (1). This\ncoincides with Eq. (3) on the Lie algebra su(2), such that\nthe matrices Tαappearing in Eq. (6) are the three gener-\natorsSαof SU(2). Although we have so far interpreted\nTαas generators in the fundamental representation of\nSU(N), this assumption is unnecessary. In particular, we\ncan faithfully describe the LLD using generators Tαin\nany irreducible representation of SU(2), labeled by spin\nS∈ {1\n2,1,3\n2...}. Each dipole magnitude |si|is a con-\nserved quantity under LLD, and takes the value S|Zi|2in the spin- Srepresentation. To model a different dipole\nmagnitude |si|=s0, we should normalize Zisuch that\n|Zi|2=s0/S (23)\nConsider, second, the GSD, Eq. (3), interpreted as the\nevolution of N2−1spin components nα\niin the fun-\ndamental representation of SU( N). Equivalently, the\nSchrödinger picture describes this dynamics as the uni-\ntary evolution of coherent spin states Zi. Our conven-\ntion to take the spin matrices Sαas a subset of the\northonormal generators Tαimposes the normalization\n|ni|=S|Zi|2, whereS= (N−1)/2. One may wish\nto select the normalization of Ziaccording to a quantum\nmechanical sum rule derived from the SU( N) quadratic\nCasimir.\nIII. NUMERICAL METHODS\nGeometric integration aims to approximate the flow of\na dynamical system while exactly satisfying geometrical\nconstraints.13,15,16For example, Eq. (19) suggests that\none integration time-step should take the form12\nρi(t)→ρi(t+∆t) =Utρi(t)U−1\nt. (24)\nTo obtain an exactly unitary matrix Utthat approxi-\nmates the integral of Eq. (14), one may use, e.g., the\nmethod of Runge-Kutte Munthe-Kaas.17This unitary\nevolution ensures conservation of Casimirs, but conser-\nvation of other geometrical properties, such as the local\nsymplectic 2-form, is not automatically guaranteed.\nDesigning efficient symplectic integration schemes for\nLie-Poisson systems is a topic of considerable inter-\nest.18–21A common strategy is to employ operator split-\nting and the fact that the composition of symplectic maps\nis again symplectic. For example, in the context of LLD,\none may partition the spins into non-interacting groups,\nand cycle through symplectic updates for each of these\ngroups using symmetric Strang splitting (i.e., a Suzuki-\nTrotter type decomposition).22–24\nAlternatively, one may seek symplectic integrators for\nLie-Poisson systems that update all dynamical variables\nsimultaneously, and in a symmetric way. The spheri-\ncal midpoint method, designed specifically for LLD of\ndipoles, is one example.25Quite recently, Modin and\nViviani introduced the class of Isospectral Symplectic\nRunge–Kutta methods (IsoSyRK),10which applies to\nany Lie-Poisson system on a reductive Lie algebra; this\ncovers most Lie-Poisson systems of practical interest, in-\ncluding generalized spin dynamics on su(N), and the\nLLD as a special case. The Isospectral Minimal Mid-\npoint (IMM) method is a particularly elegant variant of\nIsoSyRK.11\nThe standard implicit midpoint method is known to be\nsymplectic when applied to canonical Hamiltonian sys-\ntems.13We will next demonstrate that the implicit mid-\npoint method applied to the Schrödinger equation (5)5\ncoincides with IMM method applied to the equivalent\nLie-Poisson system, Eq. (3).\nA. Schrödinger midpoint\nTo integrate Eq. (5) over one time-step Zi→Z′\ni, the\nimplicit midpoint method has the symmetric form\nZ′\ni−Zi\n∆t=−i˜Hi˜Zi, (25)\nThe right-hand side involves the midpoint state,\n˜Zi=Z′\ni+Zi\n2. (26)\nThe symbol ˜Hidenotes the local Hamiltonian of Eq. (6),\nevaluated as a function of ˜Zjat all sitesj.\nThe self-consistent value of Z′\nimay be calculated nu-\nmerically as follows. Starting with an initial guess Z′\ni;0=\nZi, we iteratively calculate\n˜Zi;k=1\n2/parenleftbig\nZ′\ni;k+Zi/parenrightbig\n, (27)\nZ′\ni;k+1=Zi−i∆t˜Hi;k˜Zi;k, (28)\nwith˜Hi;kdefined in the natural way. Iterations terminate\nwhenZ′\ni;khas converged within numerical tolerance. For\nexample, at 64-bit floating point precision, we may re-\nquire|Z′\ni;k+1−Z′\ni;k|<10−14. This condition is typically\nsatisfied in k/lessorsimilar10iterations, given a reasonably small\nstep size ∆t.\nThe Schrödinger equation is a canonical Hamiltonian\nsystem, via Eqs. (21) and (22). For such systems, the\nimplicit midpoint method is known to be a symplectic\nintegrator.13\nThe Schrödinger midpoint method is norm preserving.\nTo see this, we left-multiply both sides of Eq. (25) by ˜Z†\ni,\n1\n2∆t(Z′\ni+Zi)†(Z′\ni−Zi) =−i˜Z†\ni˜Hi˜Zi.(29)\nThe right-hand side is purely imaginary, since ˜Hiis Her-\nmitian. Setting the real terms on the left-hand side to\nzero, we find |Zi|2=|Z′\ni|2.\nWe will now demonstrate that the Schrödinger mid-\npoint method is an instance of the IMM method.10,11The\ndensity matrix ρi=ZiZ†\nievolves according to Eq. (18).\nThis dynamics can be understood as an isospectral flow\ndW/dt= [B(W),W]whereB=−iHiandW=ρi. One\ntime-stepW→W′of the IMM method is defined as,\nW=/parenleftbigg\nI−∆t\n2B(˜W)/parenrightbigg\n˜W/parenleftbigg\nI+∆t\n2B(˜W)/parenrightbigg\n(30)\nW′=/parenleftbigg\nI+∆t\n2B(˜W)/parenrightbigg\n˜W/parenleftbigg\nI−∆t\n2B(˜W)/parenrightbigg\n.(31)\nwhere the midpoint state ˜Wand final state W′are to be\nsolved self-consistently.Equations (25) and (26) may be rewritten as\nZi=/parenleftbigg\nI+i∆t\n2˜Hi/parenrightbigg\n˜Zi (32)\nZ′\ni=/parenleftbigg\nI−i∆t\n2˜Hi/parenrightbigg\n˜Zi. (33)\nIntuitively, this says that the midpoint state ˜Zican be\nobtained either by integrating forward from Zi, or back-\nward from Z′\ni. Calculating the outer products W=ZiZ†\ni\nandW′=Z′\niZ′†\ni, we exactly reproduce the IMM equa-\ntions. Note that Eq. (26) defines ˜Zias a simple vector\naverage of the initial and final states, whereas ˜W=˜Zi˜Z†\ni\ncannot be expressed that way.\nB. Schrödinger midpoint applied to the LLD\nThe LLD, Eq. (1), is a special case of the GSD, Eq. (3).\nIt can therefore be formulated as a Schrödinger equa-\ntion on an SU(2) representation, and integrated using\nthe midpoint method, Eqs. (25) and (26). In the spe-\ncial cases of spin S=1\n2andS= 1representations, the\nSchrödinger midpoint method may be equivalently re-\nformulated as an update rule operating directly on spin\ndipoles,si→s′\ni. The final result, derived in Appendix D,\nis\nsi=˜si+∆t\n2˜si×∂H\n∂˜si−∆t2\n4f(˜si) (34)\ns′\ni=˜si−∆t\n2˜si×∂H\n∂˜si−∆t2\n4f(˜si), (35)\nwhere the quadratic correction,\nf(˜si) =\n\n1\n2∂H\n∂˜si/parenleftBig\n∂H\n∂˜si·˜si/parenrightBig\n−1\n4/vextendsingle/vextendsingle/vextendsingle∂H\n∂˜si/vextendsingle/vextendsingle/vextendsingle2\n˜si(spin-1\n2)\n∂H\n∂˜si/parenleftBig\n∂H\n∂˜si·˜si/parenrightBig\n(spin-1),(36)\ndepends on the choice of SU(2) representation. The mid-\npoint state ˜sican be solved self-consistently from the\ninitial state siusing Eq. (34) alone. We emphasize that\n˜siisnota simple average of siands′\ni. Once˜siis known,\nthe final state s′\nican be solved directly using Eq. (35).\nThe spin-1\n2variant of the Schrödinger dynamics coin-\ncides with the framework presented in Ref. 9, albeit in a\ndifferent language. In our notation, the expected dipole\nissα\ni=Z†\niσαZi/2, whereσαare the Pauli spin matri-\nces. In the math literature, this functional dependence\nsi(Zi)is known as the momentum map .8The functional\ndependence H(Zi)is called the collective Hamiltonian .\nThe spin-1 variant of Eqs. (34) and (35) was previously\nderived in Ref. 11 by applying the IMM method to the\nLie-Poisson system on so(3). Recall that the generators\nof SO(3) in its defining representation are also generators\nof SU(2) in its spin-1 representation.6\nC. Spherical midpoint\nThe spherical midpoint method is a powerful sym-\nplectic integrator for LLD.25One integration time-step\ns→s′is defined by the update rule\ns′\ni−si\n∆t=−¯si×∂H\n∂¯si, (37)\ninvolving the normalized midpoint dipole,\n¯si=s′\ni+si\n|s′\ni+si|. (38)\nThe classical Hamiltonian on the right-hand side\nH(¯s1,...¯sL)is evaluated at the midpoint spin configu-\nration.\nThe new spin s′can be solved self-consistently using it-\nerations analogous to Eqs. (27) and (28). Here, however,\nnormalization of the midpoint spin ¯sis employed, which\nis crucial to the good properties of the method. Various\nproofs of the symplectic structure have been given, and\nare somewhat involved.26\nD. Heun-projected (HeunP)\nAll methods above are symplectic. For purposes of\nbenchmarking, we will also compare with the Heun\nmethod, a non-symplectic Runge-Kutta scheme of second\norder. Applied to Eq. (5), one full integration time-step\nis\nZ(1)\ni=Zi−i∆tHiZi (39)\nZ(2)\ni=Zi−i∆t\n2/parenleftBig\nHiZi+H(1)\niZ(1)\ni/parenrightBig\n, (40)\nZ′\ni=Z(2)\ni/|Z(2)\ni|. (41)\nThe first step can be interpreted as an explicit Euler pre-\ndictor for the update. The second, corrector step involves\nthe local Hamiltonian H(1)\nievaluated at Z(1)\njfor allj. Fi-\nnally, the output Z′\niis normalized using a projection step.\nWe use the name HeunP in reference to prior work that\nemployed the same scheme to integrate LLD, where the\ndipoles were the dynamical variables.27,28\nIV. NUMERICAL BENCHMARKS\nA. Model definitions\nFor our numerical examples, we consider the 1D\nHeisenberg spin chain with an easy-axis anisotropy. We\nstart from the quantum Hamiltonian,\nˆH=JL/summationdisplay\ni=1ˆSi·ˆSi+1+DL/summationdisplay\ni=1(ˆSz\ni)2, (42)whereˆSαdenote spin operators in the spin-1 representa-\ntion. We will study this model in two classical limits.\nThe LLD, Eq. (1), retains only the spin dipole degrees\nof freedom. Its classical Hamiltonian\nHLLD=JL/summationdisplay\ni=1si·si+1+DL/summationdisplay\ni=1(sz\ni)2, (43)\nis obtained from ˆHunder the substitution rule ˆSi→si.\nWe will employ the normalization convention |si|= 1.\nThe LLD can be formulated as a Schrödinger dynamics\nin some spin- Srepresentation of SU(2). Per Eq. (23), the\nproper normalization of states is |Zi|2= 1/S.\nThe GSD of Eq. (3) describes an alternative classical\nlimit. This approach allows modeling each spin-1 state\nin a more physically correct way, and is especially im-\nportant when there is strong single-ion anisotropy (here,\nlargeD). Each spin dipole sigeneralizes to an eight com-\nponent vector nithat includes the original dipole si, as\nwell as five additional quadrupole components. The local\ncoherent state Zi∈C3carries information equivalent to\nthe generalized spin ni∈R8.\nFollowing the arguments of Appendix B, the classical\nHamiltonian is obtained from ˆHunder the substitution\nˆTα\ni→nα\ni, with the result,\nHGSD=JL/summationdisplay\ni=1si·si+1+DL/summationdisplay\ni=1cαnα\ni, (44)\nup to an irrelevant constant shift. We have used the\ngeneric expansion,\n(Sz)2=c0I+cαTα. (45)\nwhere the spin-1 matrices Sαare a subset of the gener-\natorsTαof SU(3) in the defining representation. Any\nsingle-ion anisotropy term could be expanded in this\nmanner. The coefficients cαmay be calculated explic-\nitly using Eq. (9), but doing so is not needed for the\nnumerics.\nSubstitution of HGSDinto Eq. (3) defines the gen-\neralized spin dynamics. In practice, we will use the\nSchrödinger formulation, involving the local Hamiltonian\nof Eq. (6),\nHGSD\ni=∂HGSD\n∂nα\niTα(46)\n=J(sα\ni−1+sα\ni+1)Sα+DcαTα. (47)\nIn a numerical implementation, it is convenient to now\nundo the expansion of Eq. (45),\nHGSD\ni=J(sα\ni−1+sα\ni+1)Sα+D[(Sz)2−c0I].(48)\nThis equation, and the definition sα\ni=Z†\niSαZi, are suf-\nficient to close the Schrödinger dynamics, Eq. (5). Note,\nin particular, that we need not explicitly select all SU(3)\ngeneratorsTα; the spin matrices Sαdefined in Eq. (A2)7\nFigure 1. Numerical accuracy of integrated dipole trajecto ries\nfor a single-spin Hamiltonian with an easy-axis anisotropy ,\nD=−1. A large time step, ∆t= 0.5, is selected to emphasize\nerror.(a)LLD using the Schrödinger midpoint method in the\nspin-1\n2representation. (b)The quantum mechanically correct\nGSD, using the Schrödinger midpoint method applied to an\nS= 1spin (i.e., N= 3). Dashed black lines indicate reference\ntrajectories given in Eqs. (50) and Eq. (51).\nare sufficient. We will use the normalization |Zi|2= 1\nappropriate to coherent states.\nThec0term in (48) ensures trHGSD\ni= 0, as expected\nfor an element of su(N). Under the Schrödinger dynam-\nics, this constant shift has the effect of rescaling Ziby\na physically irrelevant pure phase. Note, however, that\nthis constant shift may meaningfully affect the result of\nnumerical integration.\nB. Dynamics of a single spin\nAs a first test, we will consider the dynamics of a single\nspin, setting J= 0. Here, GSD is exactly faithful to the\ntrue quantum dynamics, whereas LLD is only approxi-\nmate.\nFor simplicity, we take the initial condition to be a\npure dipole,\ns(t= 0) =\nsx\n0\nsy\n0\nsz\n0\n=\nsin(θ)\n0\ncos(θ)\n (49)\nwhereθdenotes the polar angle relative to the z-axis.\nThe LLD result is simple precession about the z-axis,\nsLLD(t) =\nsx\n0cos(ωLLDt)\nsx\n0sin(ωLLDt)\nsz\n0\n, (50)\nwhereωLLD= 2Dsz\n0is the angular frequency of preces-\nsion.\nIn the single-site limit, the GSD describes the exact\nquantum evolution of expectation values. The trajectory\nfor a single S= 1spin has been given analytically inRef. 1. For our pure dipole initial condition, the dipole\npart of the trajectory is\nsGSD(t) =\nsx\n0cos(ωGSDt)\nsx\n0sz\n0sin(ωGSDt)\nsz\n0\n. (51)\nLike the dipole-only approximation, we again find preces-\nsion, but here the frequency ωGSD=Dis independent\nof the initial angle θ. They-component of the dipole is\nreduced by the factor sz\n0, such that the total spin dipole\nmagnitude is no longer a constant of motion. This im-\nplies that weight oscillates between the three dipole and\nfive quadrupole components of generalized spin n(t), and\nthis effect is enhanced when the z-component of the spin\nis small. Recall that the quadratic Casimir |n(t)|2is a\nconstant of motion.\nFigure 1 illustrates the numerical integration of the\nLLD and GSD trajectories using the Schrödinger mid-\npoint method for spin-1\n2and spin-1 respectively, and\nθ= 2π/5. A fairly large integration time-step ∆t= 0.5\nwas selected to emphasize numerical error. For LLD,\nthe spherical midpoint method26is also applicable, and\nwe observed its error to be about 2 times smaller (not\nshown). In all cases tested, energy conservation appears\nto be numerically exact under the Schrödinger midpoint\nintegration scheme.\nC. Dynamics of a spin chain\nNext, we study a chain of L= 100 spins with periodic\nboundary conditions, and a ferromagnetic Heisenberg in-\nteractionJ=−1. We follow Ref. 11 and select an initial\nstate consisting of smoothly varying pure spin dipoles,\nsi=\ncos(2πx2\ni)sin(2πx3\ni)\nsin(2πx2\ni)sin(2πx3\ni)\ncos(2πx3\ni)\n (52)\nwherexi= (i−1)/99for indicesi= 1...100.\n1. Dynamics of a pure Heisenberg chain\nIn the absence of the anisotropy term, D= 0, the clas-\nsical Hamiltonian is purely a function of the spin dipole,\nand the LLD and GSD coincide.\nFigure 2 illustrates the approximate conservation of en-\nergy for the various integration schemes applicable. Note\nthat the Schrödinger midpoint method and the spherical\nmidpoint method are both symplectic, and therefore pre-\nvent energy drift over large time-scales. In contrast, the\nHeunP scheme is non-symplectic, and exhibits a large,\nunphysical energy drift. Finally, we remark that the\nSchrödinger midpoint result with spin-1 precisely repro-\nduces the curve shown in the top panel of the fourth figure\nin Ref. 11. To reproduce the bottom panel, we needed to\nintegrate the spherical midpoint trajectory backwards in\ntime.8\nFigure 2. Energy fluctuations ∆H=H(t)−H(0)for the\nnumerically integrated LLD of a pure Heisenberg spin chain,\nJ=−1, absent anisotropy, D= 0. We compare the follow-\ning integration methods: the HeunP method applied to the\nSchrödinger equation, the Schrödinger midpoint method in\nthe spin-1\n2and spin-1 representations, and the spherical mid-\npoint method. The integration time-step is ∆t= 0.1. The\nsame data is plotted on three different energy scales to illus -\ntrate the huge variation between the integration schemes fo r\nthis special model.\nFigure 3. Energy fluctuations for the numerically integrate d\nLLD of a Heisenberg spin chain with J=−1, now including\nan easy-axis anisotropy, D=−1. The numerical integration\nschemes are the same as in Fig. 2, but here we use a much\nsmaller integration time-step, ∆t= 0.02.\n2. LLD including easy-axis anisotropy\nWe now include an easy-axis anisotropy, D=−1, in\nthe Heisenberg chain. With this additional term, the\nLLD and GSD classical limits deviate. First we will con-\nsider the LLD case.\nFigure 3 compares accuracy of four integration\nschemes. Relative to the D= 0case in Fig. 2, we ob-\nserve much larger energy fluctuations, despite a signifi-\ncantly smaller time-step of ∆t= 0.02(down from ∆t=\n0.1). Energy fluctuations observed from the Schrödinger\nmidpoint and spherical midpoint numerical integrationFigure 4. (a)Energy fluctuations for the numerically inte-\ngrated GSD of a Heisenberg spin chain with S= 1,J=−1\nand easy-axis anisotropy, D=−1. We use the Schrödinger\nmidpoint method with varying time-step ∆t.(b)Time evo-\nlution of the dipole magnitude |si|for site index i= 60.\nschemes are now of the same order. In all cases tested,\nwhen applying the Schrödinger midpoint method to the\nLLD, the spin-1\n2representation is preferred over the spin-\n1 representation. There remains a tremendous advantage\nin using a symplectic integration scheme—the energy of\nthe HeunP method continues to drift strongly.\n3. GSD including easy-axis anisotropy\nOur final numerical benchmark is the Heisenberg spin\nchain with easy-axis anisotropy D=−1using the gener-\nalized spin dynamics. Here we must work with all three\nlevels of the S= 1spins, which give rise to both dipole\nand quadrupole moments, and the traditional LLD nu-\nmerical integration schemes do not apply.\nFigure 4 shows time integration using the Schrödinger\nmidpoint method. The top panel illustrates that the en-\nergy fluctuations decrease approximately quadratically\nwith time-step ∆t, consistent with the second order ac-\ncuracy of the implicit midpoint method.13The bottom\npanel shows the time evolution of the dipole magnitude\n|si|for site index i= 60. Fluctuations in the dipole\nmagnitude are possible because weight can be trans-\nferred to the quadrupole part of the generalized spins\nni. All spins in the initial configuration, Eq. (52), are\npure dipoles, with a relatively slow variation along the\nspin chain. Therefore the initial dynamics is reasonably\nwell approximated by the single spin limit, J≈0, pre-\nviously considered in Fig. 1. At times t/greaterorsimilar10, however,\nhigh-frequency spatial variations in the spin chain prop-\nagate to site i= 60. At this point, chaotic dynamics can\nbe observed, and the three trajectories integrated using\ndifferent ∆tquickly separate. As expected for a symplec-\ntic integrator, no significant energy drift is observed over\narbitrarily long trajectory lengths.9\nV. CONCLUSIONS\nWe have presented a numerical integration scheme,\nthe Schrödinger midpoint method, that applies to the\ngeneralized spin dynamics, Eq. (3). In the special case\nof the Landau-Lifshitz dynamics, Eq. (1), this method\nreduces to previously known symplectic integrators.9,11\nThe Schrödinger midpoint method can be viewed as a\nspecial case of the Isospectral Midpoint Method,10,11\nwhich applies to general Lie-Poisson systems. Compared\nto IMM, our approach is specialized to Eq. (3), which\narises as a classical limit of quantum mechanics, and\nhas the numerical advantage of describing the evolution\nof a single eigenvector vector (the coherent spin state)\nrather than that of a full matrix. The method exactly\nrespects the local symplectic structure of the Lie-Poisson\nsystem, or equivalently, the global symplectic structure\nof the Schrödinger equation, which may be understood\nas a canonical Hamiltonian system. We anticipate that\nthis method will be of broad applicability to the study of\nspinS >1\n2systems with strong single-ion anisotropy, for\nwhich the spin quadrupole (and perhaps higher-order)\nmoments cannot be ignored.1,4,29–32ACKNOWLEDGMENTS\nWe thank Martin Mourigal and Ying Wai Li for in-\nsightful discussions. This work was supported by the\nU.S. Department of Energy, Office of Science, Office of\nBasic Energy Sciences, under Award No. DE-SC0022311.\nCODE AVAILABILITY\nCode to reproduce all plots is available online at\nhttps://github.com/SunnySuite/SchrodingerMidpoint.j l.\nThe methods we present are also implemented in Sunny ,\nan open-source code for simulating general spin sys-\ntems.33\nAppendix A: Spin operators as representations of\nSU(2)\nHere we review some properties of the irreducible rep-\nresentations of SU(2), which serve as quantum spin op-\nerators. The SU(2) irreps are conventionally labeled\nby a spin index S∈ {1\n2,1,3\n2,...}, and have dimension\nN= 2S+1.\nFor spinS= 1/2, a conventional basis for the spin\nmatrices is Sα=σα/2, whereσαare the Pauli matrices,\nSx=1\n2/bracketleftbigg\n0 1\n1 0/bracketrightbigg\n, Sy=1\n2/bracketleftbigg\n0−i\ni 0/bracketrightbigg\n, Sz=1\n2/bracketleftbigg\n1 0\n0−1/bracketrightbigg\n. (A1)\nMore generally, it is convenient to work in a basis with Szdiagonal. For spin S= 1,\nSx=1√\n2\n0 1 0\n1 0 1\n0 1 0\n,Sy=1√\n2\n0−i 0\ni 0−i\n0 i 0\n, Sz=\n1 0 0\n0 0 0\n0 0−1\n. (A2)\nThis pattern generalizes to arbitrary spin S,\nSx=\n0a1\na10...\n......aN−1\naN−10\n, Sy=\n0−ia1\nia10...\n......−iaN−1\niaN−10\n,Sz=\nS\nS−1\n...\n−S\n, (A3)\nwhere the off-diagonal elements, aj=1\n2/radicalbig\n2(S+1)j−j(j+1),satisfy the symmetry aj=aN−j. These coefficients\nalso enter into raising and lowering operators, defined as S±=Sx±iSy.\nThe spin matrices Sαdefine the action of the quan-\ntum spin operators. The many-body spin operator ˆSα\niis\ndefined to act only on the ith local Hilbert space,\nˆSα\ni=I⊗···⊗ˆSα\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nith term⊗···⊗I.\nThe local operator ˆSαis defined by its action∝an}bracketle{tea|ˆSα|eb∝an}bracketri}ht= (Sα)abin some basis |e1∝an}bracketri}ht,...,|eN∝an}bracketri}ht. The\noperators ˆSα\nishare all mathematical properties of the\nmatricesSα, to be listed below.\nBy construction, the spin matrices satisfy the commu-\ntation relations,\n[Sα,Sβ] = iǫαβγSγ, (A4)10\nwhich gives rise to the same commutation relations,\nEq. (2), for the quantum spin operators.\nTheNeigenvalues for each spin matrix Sαrun from\nm=−S,...,S . The total angular momentum for each\nspiniis scalar, i.e.,\n|S|2= (Sx)2+(Sy)2+(Sz)2=S(S+1)I, (A5)\nwithIthe identity matrix. This polynomial is the sole\nCasimir of su(2).\nThe spin matrices satisfy the orthonormality condition\ntrSαSβ=τδαβ, (A6)\nwhich is a special case of Eq. (9). The normalization\nconstant is the sum of squares of eigenvalues,\nτ=S/summationdisplay\nm=−Sm2=2\n3S/parenleftbigg\nS+1\n2/parenrightbigg\n(S+1).(A7)\nAppendix B: Review of generalized spin dynamics\nFollowing Ref. 1, we here review how Eq. (3) emerges\nas a classical limit of a many-body quantum system.\nConsider a many-body quantum spin Hamiltonian ˆH,\nsuch as the spin chain model of Eq. (42). This Hamilto-\nnian may involve spin operators ˆSαwith spin represen-\ntationS, corresponding to local Hilbert space dimension\nN= 2S+ 1. The choice S >1/2would be appropri-\nate to model, e.g., the effective spin angular momentum\nassociated with a collection of spins that have aligned\naccording to Hund’s rules.\nForN >2, the Hamiltonian may include local\nanisotropy terms, such as (ˆSz\ni)2. Such terms are nota\nlinear combination of the ˆSα\ni, i.e., do not generate usual\nrotations of the local spin state. Importantly, however,\nany local physical operator canbe decomposed as a lin-\near combination of the N2−1generators ˆTα\niof SU(N)\nin the fundamental representation, plus a constant shift.\nUsing the completeness of the generators ˆTα\niin each\nlocal Hilbert space, we can write a general Hamiltonian\nas a polynomial expansion,\nˆH=/summationdisplay\niJ(1)\n(i,α)ˆTα\ni+1\n2/summationdisplay\ni,jJ(2)\n(i,α),(j,β)ˆTα\niˆTβ\nj+..., (B1)\nup to an irrelevant constant shift. Recall that summation\nover repeated Greek indices is implied.\nAny coefficient J(n)\n[...]that couples a site with itself (e.g.,\na single-ion anisotropy term) can be effectively absorbed\ninto lower order coefficients J(n−1)\n[...]by the completeness\nof the generators ˆTα\nifor each site i. Therefore, without\nloss of generality, we require that the coefficients couple\nonly distinct sites, e.g.,\nJ(2)\n(i,α),(j,β)∝(1−δij). (B2)SinceˆTα\niandˆTβ\njcommute for i∝ne}ationslash=j, we also have freedom\nto symmetrize the coefficients, e.g.,\nJ(2)\n(i,α),(j,β)=J(2)\n(j,β),(i,α). (B3)\nThe Hamiltonian ˆHdetermines the evolution of a gen-\neral quantum state,\nd\ndt|ψ∝an}bracketri}ht=e−itˆH|ψ∝an}bracketri}ht, (B4)\nwhere we take /planckover2pi1= 1. The time evolution of an arbitrary\nexpectation value ∝an}bracketle{tˆA∝an}bracketri}ht=∝an}bracketle{tψ|ˆA|ψ∝an}bracketri}htfollows,\nid\ndt∝an}bracketle{tˆA∝an}bracketri}ht=∝an}bracketle{t[ˆA,ˆH]∝an}bracketri}ht. (B5)\nTo take the classical limit, we will ignore quantum\nentanglement and approximate the time-evolving many-\nbody state |ψ∝an}bracketri}htas a tensor product of coherent states of\na given algebra,\n|ψ∝an}bracketri}ht ≈ |Z∝an}bracketri}ht=L/circlemultiplydisplay\ni=1|Zi∝an}bracketri}ht. (B6)\nIn the remainder of this section, our purpose is to demon-\nstrate the following: This approximation yields a self-\nconsistent and closed dynamics on local expectation val-\nues, namely, Eq. (3).\nProduct states allow the factorization of expectation\nvalues over distinct sites,\n∝an}bracketle{tZ|ˆTα\niˆTβ\nj|Z∝an}bracketri}ht=nα\ninβ\nj. (B7)\nIt follows that, under the assumption of Eq. (B6), the\nexpected energy H=∝an}bracketle{tˆH∝an}bracketri}htis\nH=/summationdisplay\niJ(1)\n(i,α)nα\ni+1\n2/summationdisplay\ni,jJ(2)\n(i,α),(j,β)nα\ninβ\nj+... (B8)\nIn other words, substituting ˆTα\ni→nα\niin the quantum\nHamiltonian ˆHyields the classical Hamiltonian H.\nInserting the general Hamiltonian of Eq. (B1) into the\ndynamics Eq. (B5) for a local operator ˆA=ˆAkyields,\nid\ndt∝an}bracketle{tˆAk∝an}bracketri}ht=/summationdisplay\niJ(1)\n(i,α)∝an}bracketle{t[ˆAk,ˆTα\ni]∝an}bracketri}ht\n+1\n2/summationdisplay\ni,jJ(2)\n(i,α),(j,β)∝an}bracketle{t[ˆAk,ˆTα\niˆTβ\nj]∝an}bracketri}ht+...(B9)\nFor the first term, we use\n/summationdisplay\niJ(1)\n(i,α)[ˆAk,ˆTα\ni] =J(1)\n(k,α)[ˆAk,ˆTα\nk]. (B10)11\nFor the second term, Eq. (B2) ensures i∝ne}ationslash=j, and we\nrequire either k=iork=j. It follows,\n1\n2/summationdisplay\ni,jJ(2)\n(i,α),(j,β)[ˆAk,ˆTα\niˆTβ\nj]\n=1\n2[ˆAk,ˆTα\nk]/summationdisplay\nj/parenleftBig\nJ(2)\n(k,α),(j,β)ˆTβ\nj+J(2)\n(j,α),(k,β)ˆTα\nj/parenrightBig\n= [ˆAk,ˆTα\nk]/summationdisplay\njJ(2)\n(k,α),(j,β)ˆTβ\nj. (B11)\nIn the second step, we used the symmetrization conven-\ntion of Eq. (B3). Combining results, and using again\nthe approximation Eq. (B6) to factorize the expectation\nvalues on distinct sites, we find\nid\ndt∝an}bracketle{tˆAk∝an}bracketri}ht=∝an}bracketle{t[ˆAk,ˆTα\nk]∝an}bracketri}ht/parenleftBig\nJ(1)\n(k,α)+/summationdisplay\njJ(2)\n(k,α),(j,β)∝an}bracketle{tˆTβ\nj∝an}bracketri}ht+.../parenrightBig\n.\n(B12)\nThe second factor on the right-hand side is ∂H/∂nα\nk,\nand this result holds to all expansion orders. Relabel-\ning(k,α)→(i,β), the result is\nid\ndt∝an}bracketle{tˆAi∝an}bracketri}ht=∝an}bracketle{t[ˆAi,ˆTβ\ni]∝an}bracketri}ht∂H\n∂nβ\ni, (B13)\nwhich is valid for any local operator ˆAi. Selecting ˆAi=\nˆTα\niand using Eq. (4), we reproduce the generalized spin\ndynamics, Eq. (3),\nd\ndtnα\ni=fα,β,γ∂H\n∂nβ\ninγ\ni. (B14)\nAlthough we motivated the form of ˆHin Eq. (B1) us-\ning the language of spin systems, this Hamiltonian is in\nfact fully general. We could have taken each local Hilbert\nspaceito represent, e.g., a tensor product space of mul-\ntiplequantum spins. The time evolution of classical ex-\npectation values, Eq. (B14), would then capture quantum\nentanglement effects within each local Hilbert space.\nAppendix C: Hamiltonian structure of the\nSchrödinger equation\nEachsu(N)generator can be decomposed into its\npurely real and imaginary parts,\nTα=Aα+iBα. (C1)\nBecauseTαis Hermitian, it follows that AαandBαare\nsymmetric and antisymmetric, respectively.\nSubstituting this decomposition into the Schrödinger\nequation of Eq. (5), we find\nd\ndtZi=∂H\n∂nα\ni(−iAα+Bα)Zi. (C2)\nDecomposing the state vector into real and imaginary\nparts,Zi=1√\n2(pi−iqi), (C3)\nallows formulation of the Schrödinger equation as an en-\ntirely real dynamics,\ndpi\ndt=∂H\n∂nα\ni(−Aαqi+Bαpi) (C4)\ndqi\ndt=∂H\n∂nα\ni(Aαpi+Bαqi). (C5)\nExpectation values nα\ni=Z†\niTαZimay be written,\nnα\ni=1\n2(pi−iqi)†Tα(pi−iqi). (C6)\nExpanding the right-hand side, and decomposing Tαinto\nits symmetric and antisymmetric parts, we find\nnα\ni=1\n2pT\niAαpi+1\n2qT\niAαqi\n−1\n2qT\niBαpi+1\n2pT\niBαqi. (C7)\nDifferentiation yields\n∂nα\ni\n∂pi=Aαpi+Bαqi (C8)\n∂nα\ni\n∂qi=Aαqi−Bαpi. (C9)\nEnergy derivatives can now be evaluated. Using the chain\nrule,\n∂H\n∂pi=∂H\n∂nα\ni(Aαpi+Bαqi) (C10)\n∂H\n∂qi=∂H\n∂nα\ni(Aαqi−Bαpi). (C11)\nInserting these results into Eqs. (C4) and (C5), we find\nthat the Schrödinger equation is described by Hamilton’s\nequations of motion,\ndpi\ndt=−∂H\n∂qi,dqi\ndt= +∂H\n∂pi. (C12)\nAppendix D: Derivation of the Schrödinger\nmidpoint formulas for LLD\nIn this Appendix, we derive the results stated in\nSec. III B. The LLD of spin dipoles, Eq. (1), may be\nformulated as a Schrödinger dynamics, Eq. (5), involving\ngeneratorsSαof SU(2) in some representation. The mid-\npoint method applied to the Schrödinger dynamics can\nbe interpreted as two half time-steps, Eq. (32) and (33).\nConsider the former, from which we determine\nsα\ni=˜Z†\ni/parenleftbigg\nSα+i∆t\n2[Sα,˜Hi]−∆t2\n4˜HiSα˜Hi/parenrightbigg\n˜Zi,(D1)12\nwheresα\ni=Z†\niSαZi.\nThe first term is an expectation value, ˜sα\ni=˜Z†\niSα˜Zi.\nThe second term may be expanded using Eqs. (6)\nand (A4),\ni[Sα,˜H] = i∂H\n∂˜sβ\ni[Sα,Sβ] =−ǫαβγ∂H\n∂˜sβ\niSγ. (D2)\nIn vector notation,\ni˜Z†\ni[S,˜H]˜Zi=˜si×∂H\n∂˜si. (D3)\nTo evaluate the third term, we must incorporate more\ninformation about the matrix representation of the gen-eratorsSα. Let us introduce the notation,\n˜Hi=∂H\n∂˜sβ\niSβ=a·S (D4)\nIn the special cases of the spin-1\n2and spin-1 represen-\ntations, the matrix spin operators satisfy\n(a·S)S(a·S) =/braceleftBigg\na(a·S)\n2−|a|2S\n4(spin-1\n2)\na(a·S) (spin-1),(D5)\nvalid for any a∈R3.\nIt follows,\n˜HS˜H=\n\n1\n2∂H\n∂˜si/parenleftBig\n∂H\n∂˜si·S/parenrightBig\n−1\n4/vextendsingle/vextendsingle/vextendsingle∂H\n∂˜si/vextendsingle/vextendsingle/vextendsingle2\nS(spin-1\n2)\n∂H\n∂˜si/parenleftBig\n∂H\n∂˜si·S/parenrightBig\n(spin-1).(D6)\nInserting these results into Eq. (D1) yields Eq. (34). A\nsimilar argument yields Eq. (35). Combined, these equa-\ntions provide a closed-form update rule entirely in terms\nof the spin dipoles.\n∗kbarros@lanl.gov\n1H. Zhang and C. D. Batista,\nPhys. Rev. B 104, 104409 (2021).\n2V. S. Zapf, D. Zocco, B. R. Hansen, M. Jaime,\nN. Harrison, C. D. Batista, M. Kenzelmann, C. Nie-\ndermayer, A. Lacerda, and A. Paduan-Filho,\nPhys. Rev. Lett. 96, 077204 (2006).\n3S.-H. Do, H. Zhang, T. J. Williams, T. Hong,\nV. O. Garlea, J. Rodriguez-Rivera, T.-H. Jang, S.-\nW. Cheong, J.-H. Park, C. D. Batista, et al. ,\nNature communications 12, 1 (2021).\n4X. Bai, S.-S. Zhang, Z. Dun, H. Zhang, Q. Huang, H. Zhou,\nM. B. Stone, A. I. Kolesnikov, F. Ye, C. D. Batista, and\nM. Mourigal, Nat. Phys. 17, 467 (2021).\n5M. Jaime, V. F. Correa, N. Harrison, C. D. Batista,\nN. Kawashima, Y. Kazuma, G. A. Jorge, R. Stern, I. Hein-\nmaa, S. A. Zvyagin, Y. Sasago, and K. Uchinokura,\nPhys. Rev. Lett. 93, 087203 (2004).\n6Y. Qiu, C. Broholm, S. Ishiwata, M. Azuma,\nM. Takano, R. Bewley, and W. J. L. Buyers,\nPhys. Rev. B 71, 214439 (2005).\n7Y. Okamoto, G. J. Nilsen, J. P. Attfield, and Z. Hiroi,\nPhys. Rev. Lett. 110, 097203 (2013).\n8J. E. Marsden and T. S. Ratiu, Introduction to Mechanics\nand Symmetry, A Basic Exposition of Classical Mechanical\nSystems , Texts in Applied Mathematic, Vol. 17 (Springer\nBerlin Heidelberg, 1999).\n9R. I. McLachlan, K. Modin, and O. Verdier,\nIMA Journal of Numerical Analysis 35, 546 (2015).\n10K. Modin and M. Viviani,\nFound Comput Math 20, 889 (2020).\n11M. Viviani, Bit Numer Math 60, 741 (2020).\n12K. Engø and S. Faltinsen,\nSIAM J. Numer. Anal. 39, 128 (2001).\n13E. Hairer, C. Lubich, and G. Wanner, Geometric Numeri-cal Integration: Structure-Preserving Algorithms for Ord i-\nnary Differential Equations , Springer Series in Computa-\ntional Mathematics, Vol. 31 (Springer Berlin Heidelberg,\n2006).\n14T. Huberman, D. A. Tennant, R. A.\nCowley, R. Coldea, and C. D. Frost,\nJournal of Statistical Mechanics: Theory and Experiment 2008, P05017 (2008).\n15A. Iserles, H. Z. Munthe-Kaas, S. P. Nørsett, and\nA. Zanna, Acta Numerica 9, 215 (2000).\n16A. Iserles and G. R. W. Quispel, arXiv:1602.07755 [math]\n(2016), arXiv:1602.07755 [math].\n17H. Munthe-Kaas, Bit Numer Math 38, 92 (1998).\n18G. Zhong and J. E. Marsden,\nPhysics Letters A 133, 134 (1988).\n19P. J. Channell and J. C. Scovel,\nPhysica D: Nonlinear Phenomena 50, 80 (1991).\n20R. I. McLachlan, Phys. Rev. Lett. 71, 3043 (1993).\n21R. I. McLachlan and C. Scovel,\nJ Nonlinear Sci 5, 233 (1995).\n22M. Krech, A. Bunker, and D. P. Landau,\nComputer Physics Communications 111, 1 (1998).\n23I. P. Omelyan, I. M. Mryglod, and R. Folk,\nPhys. Rev. Lett. 86, 898 (2001).\n24J. Tranchida, S. J. Plimpton,\nP. Thibaudeau, and A. P. Thompson,\nJournal of Computational Physics 372, 406 (2018).\n25R. I. McLachlan, K. Modin, and O. Verdier,\nPhys. Rev. E 89, 061301 (2014).\n26R. McLachlan, K. Modin, and O. Verdier,\nMath. Comp. 86, 2325 (2017).\n27B. Skubic, J. Hellsvik, L. Nordström, and O. Eriksson,\nJ. Phys.: Condens. Matter 20, 315203 (2008).\n28J. H. Mentink, M. V. Tretyakov, A. Fa-\nsolino, M. I. Katsnelson, and T. Rasing,\nJ. Phys.: Condens. Matter 22, 176001 (2010).13\n29Y. Akagi, Y. Amari, N. Sawado, and Y. Shnir,\nPhys. Rev. D 103, 065008 (2021), arXiv:2101.10566.\n30K. Remund, R. Pohle, Y. Akagi, J. Romhányi,\nand N. Shannon, arXiv:2203.09819 [cond-mat] (2022),\narXiv:2203.09819 [cond-mat].\n31Y. Amari, Y. Akagi, S. B. Gudnason, M. Nitta, andY. Shnir, arXiv:2204.01476 [cond-mat, physics:hep-th]\n(2022), arXiv:2204.01476 [cond-mat, physics:hep-th].\n32H. Zhang, Z. Wang, D. Dahlbom, K. Barros, and\nC. D. Batista, arXiv:2203.15248 [cond-mat] (2022),\narXiv:2203.15248 [cond-mat].\n33(2022),https://github.com/SunnySuite/Sunny.jl/ ."    },    {        "title": "2209.13629v2.Spin_Berry_curvature_of_the_Haldane_model.pdf",        "content": "Spin Berry curvature of the Haldane model\nSimon Michel1and Michael Pottho\u000b1, 2\n1I. Institute of Theoretical Physics, Department of Physics,\nUniversity of Hamburg, Notkestra\u0019e 9-11, 22607 Hamburg, Germany\n2The Hamburg Centre for Ultrafast Imaging, Luruper Chaussee 149, 22761 Hamburg, Germany\nThe feedback of the geometrical Berry phase, accumulated in an electron system, on the slow\ndynamics of classical degrees of freedom is governed by the Berry curvature. Here, we study local\nmagnetic moments, modelled as classical spins, which are locally exchange coupled to the (spinful)\nHaldane model for a Chern insulator. In the emergent equations of motion for the slow classical-\nspin dynamics there is a an additional anomalous geometrical spin torque, which originates from the\ncorresponding spin-Berry curvature. Due to the explicitly broken time-reversal symmetry, this is\nnonzero but usually small in a condensed-matter system. We develop the general theory and compute\nthe spin-Berry curvature, mainly in the limit of weak exchange coupling, in various parameter\nregimes of the Haldane model, particularly close to a topological phase transition and for spins\ncoupled to sites at the zigzag edge of the model in a ribbon geometry. The spatial structure of\nthe spin-Berry curvature tensor, its symmetry properties, the distance dependence of its nonlocal\nelements and further properties are discussed in detail. For the case of two classical spins, the\ne\u000bect of the geometrical spin torque leads to an anomalous non-Hamiltonian spin dynamics. It is\ndemonstrated that the magnitude of the spin-Berry curvature is decisively controlled by the size of\nthe insulating gap, the system size and the strength of local exchange coupling.\nI. INTRODUCTION\nThe time evolution of a quantum system with a non-\ndegenerate and gapped ground state, steered by external\ntime-dependent classical degrees of freedom, is governed\nby the adiabatic theorem [1, 2]: If prepared at time t= 0\nin its ground state, the system state at t >0 is given\nby its instantaneous ground state for the then existing\ncon\fguration of classical degrees of freedom. Roughly,\nthe adiabatic theorem applies, if the typical time scale\n\u001cof the classical time evolution is large compared to\nthe inverse of the gap \u0001 Ebetween the ground and the\n\frst excited state. Besides the dynamical phase, the sys-\ntem accumulates a geometrical phase during the adia-\nbatic time evolution, which cannot be gauged away in\ncase of a cyclic motion in the classical state space [3{5].\nThis Berry phase and related phenomena, such as the\nmolecular Aharonov-Bohm e\u000bect, have been studied ex-\ntensively in molecular physics [6, 7], assuming that the\ncoordinatesRmof the nuclei can be treated as classical\nobservables.\nThe Berry-phase physics of the quantum system also\nfeeds back to the dynamical state of the classical observ-\nables as has been pointed out early [8{10]. In molecular\nsystems, for example, this leads to an additional geomet-\nricforce term in the classical Newtonian set of equations\nof motion, which has the form of a Lorentz force despite\nthe absence of a physical magnetic \feld.\nThe geometric force plays an important role also in\nother contexts as, for example, in the semiclassical the-\nory of electron dynamics in crystals, where the wave vec-\ntorkis treated as a dynamical classical variable [7, 11].\nIts pendant on the side of the quantum system is the k-\nspace Berry phase, an important quantity in topological\nband theory [12{14]. In fact, Berry phase and geometri-\ncal force are rather general concepts that can be appliedto con\fguration spaces of various kinds of classical de-\ngrees of freedom.\nHence, a similar situation arises for another type of\nquantum-classical hybrid systems as well, namely for a\nquantum-mechanical electron system with an exchange\ncoupling to the local magnetic moments Smof magnetic\natoms, which are modelled as classical spins, i.e., vectors\nof \fxed length S2\nm= 1. The typical (picosecond) time\nscale for classical spin dynamics is much longer than that\nof the fast (femtosecond) electron quantum dynamics,\nsuch that the electron system can adiabatically follow\nthe classical spin con\fguration and accumulate a \fnite\nspin-Berry phase [15]. The feedback of the Berry physics\non the classical system, however, is di\u000berent and given\nby a geometrical spin torque rather than a force, and can\nthereby strongly a\u000bect the precessional spin dynamics\n[16].\nThis spin torque is given in terms of the gauge invariant\nspin-Berry curvature [16], which must be distinguished\nfrom thek-space Berry curvature and the Rm-space or\natomic-nuclei-Berry curvature of molecular dynamics. A\nsimpli\fed variant of the approach presented in Ref. 16\nhas been used to discuss the e\u000bect of the geometrical spin\ntorque on the spin-wave spectrum of magnetic systems\n[17, 18].\nIn the recent years, the spin-Berry curvature and the\ngeometrical spin torque have been studied more inten-\nsively. A numerical study for an exactly solvable one-\ndimensional model of local magnetic moments has been\ncarried out to quantitatively disentangle the geometrical\nspin torque from other contributions to the spin dynam-\nics [19]. This work also emphasizes the close relation to\ngeometric-friction e\u000bects [20, 21], i.e., Gilbert spin damp-\ning, which can be derived within linear-response theory\n[22{26] or adiabatic-response theory [27]. To go beyond\nthe strict adiabatic limit, a generalized geometrical spinarXiv:2209.13629v2  [cond-mat.mes-hall]  27 Dec 20222\ntorque originating from the non-Abelian spin-Berry cur-\nvature has been proposed recently [28], in the spirit of\nRef. [5]. E\u000bects of the geometrical spin torque in purely\nclassical spin systems, where a single slow spin [29] or\nseveral slow spins [30] interact with an exchange-coupled\nsystem of fast spins, have been studied within a formal\nframework largely analogous to the quantum-classical hy-\nbrid case. This goes back to earlier work that drew at-\ntention to holonomy e\u000bects in purely classical systems\n[31].\nThe main subject of this paper is to study the spin-\nBerry curvature for a model for local magnetic moments\n(classical spins) coupled to an insulator. The latter is de-\nscribed by a two-band tight-binding model for electrons\nhopping on a two-dimensional lattice. With the (spin-\nful) Haldane model on the honeycomb lattice [32, 33] we\nchoose a prototypical model for a Chern insulator.\nOur motivation for this choice is twofold: First, the\nHaldane model stands at the origin of the development\ntopological band theory [12{14, 34] and represents the\n\frst model for a quantum anomalous Hall insulator. It\ncan be experimentally realized in a setup with ultracold\nfermions [35]. Furthermore, there is generally a rapidly\nincreasing interest in the study of magnetic impurities\nin topological insulators and at the boundaries of topo-\nlogical materials in particular [36{48] as well as in their\nmutual interaction [49{53].\nSecond, if the Hamiltonian of the quantum system\nis time-reversal symmetric, there may be a \fnite Berry\nphase originating from a singularity in the classical con-\n\fguration space, while the Berry curvature vanishes [54].\nA \fnite Berry curvature ( k-,Rm- or spin-Berry curva-\nture), however, requires an explicit breaking of time-\nreversal symmetry. In case of the spin-Berry curva-\nture in a model including an exchange coupling to the\nclassical spins, time-reversal symmetry is already bro-\nken intrinsically, since the classical spins act like local\nsymmetry-breaking magnetic \felds. This e\u000bect, how-\never, becomes irrelevant in the (physical) limit of weak\nexchange-interaction strength J. As is shown later, the\nspin-Berry curvature generally vanishes in perturbation\ntheory up toO(J2), if the quantum system itself is time-\nreversal symmetric. The Haldane model is constructed\nsuch that the symmetry is broken due to a \\magnetic\n\feld\" coupling to the orbital degrees of freedom rather\nthan to the local electron spin (note that the original\nmodel is spinless anyway), which averages to zero in a\nunit cell. The mechanism by which the Haldane model\nleads to a nonzero Chern number can be realized in ac-\ntual materials, where the spin-orbit coupling acts as a\nsource of the orbital magnetic \feld [55, 56].\nThe spinful Haldane model is interesting, as it allows\nus to study the impact of the spin-Berry curvature on\nthe e\u000bective interaction between local magnetic moments\nand thus on the resulting e\u000bective spin dynamics. The\nstandard indirect magnetic RKKY exchange is of the\nBloembergen-Rowland type [57] in case of an insulator,\nand its strength depends exponentially on the insulatinggap \u0001E, opposed to RKKY theory for metallic phases\n[58, 59]. A central question is thus if the geometric spin\ntorque is able to \\boost\" the indirect coupling between\ntwo classical spins.\nTo this end, we generalize the theory of Ref. [16] to an\narbitrary number of classical spins and derive the non-\nlocal geometrical spin torque that emerges in the adia-\nbatic limit, see Sec. II. This requires to deduce explicit\nexpressions for the spin-Berry curvature, which can be\nevaluated numerically, and an analysis of time-reversal\nand spatial symmetry transformations. In particular, we\nshow that the spin-Berry curvature is closely related to\nthe spin susceptibility of the electron system. For two\nclassical spins, we explicitly demonstrate that the result-\ning adiabatic spin dynamics is anomalous and cannot be\nderived from an e\u000bective Hamiltonian.\nOn this basis, we numerically analyze the dependence\nof the bulk spin-Berry curvature on the positions and\non the distance between two sites, at which two clas-\nsical spins are exchange coupled to the electron sys-\ntem, see Sec. III. Furthermore, we study its dependence\non the parameters of the Haldane model in the weak-\nJregime both, for topologically trivial and nontrivial\nphases, and especially in the parametric vicinity of a\ntopological phase transition. Invoking a phase-space ar-\ngument, one can show that the spin-Berry curvature is in\nfact continuous across the transition. It diverges, how-\never, for the Haldane model in a ribbon geometry, if the\nclassical spins couple to sites at the zigzag edge.\nII. THEORY\nA. E\u000bective low-energy theory\nWe consider a quantum-classical hybrid system con-\nsisting ofMclassical spins S\u0011(S1;:::;SM) of \fxed\nlengthSm=jSmj= 1, the intrinsic dynamics of which\nare governed by a classical Hamilton function Hcl(S),\nandNelectrons with corresponding quantum tight-\nbinding Hamiltonian ^Hquconstructed with the help of\ncreation and annihilation operators cy\ni\u001bandci\u001b. Herei\nrefers to a site of a given lattice and \u001b=\";#to the elec-\ntron spin projection. The orthonormal states ji\u001bispan\nthe one-particle Hilbert space. To develop the general\ntheory, it is not yet necessary to further specify Hcl(S)\nand ^Hqu. Concrete calculations will be performed for\nHcl(S) = 0 and for ^Hqugiven by the Haldane model\n[32, 33], but trivially generalized for spinful electrons,\nsee Sec. II G. Spins and electrons interact via a local ex-\nchange\n^Hint(S) =JMX\nm=1Smsim; (1)\nbetween the mth classical spin Smand the local spin\nsimof the electron system at the site imgiven bysi=3\n1\n2P\n\u001b\u001b0cy\ni\u001b\u001c\u001b\u001b0ci\u001b0, where\u001c= (\u001cx;\u001cy;\u001cz)Tis the vector\nof Pauli matrices. With J >0 the coupling is antiferro-\nmagnetic. The total Hamiltonian is\n^H(S) =^Hqu+Hcl(S) +^Hint(S): (2)\nThe coupled equations of motion for the spin con\fgura-\ntionS=S(t) and the pure N-electron statej\ti=j\t(t)i\nare given by:\ni@tj\t(t)i=^H(S)j\t(t)i;\n_Sm(t) =@h^H(S)i\n@Sm\u0002Sm(t): (3)\nHere,h\u0001\u0001\u0001i is the expectation value with the state j\t(t)i,\nand the dot denotes the time derivative. The equations\ncan be derived from the Hamiltonian ^H(S) or, equiv-\nalently, via the action principle, from the Lagrangian\nL=L(S;_S;j\ti;_j\ti;h\tj;_h\tj):\nL=X\nmA(Sm)_Sm+h\tji@tj\ti\u0000h\tj^H(S)j\ti:(4)\nHere,\nA(Sm) =\u00001\nS2me\u0002Sm\n1 +eSm=Sm; (5)\nwith a \fxed unit vector e, which is usually chosen as\ne=ez[60]. We have\nr\u0002A(Sm) =\u0000Sm=S3\nm; (6)\nand thusA(Sm) can be interpreted as the vector poten-\ntial of a unit magnetic monopole located at Sm= 0. For\nthe derivation of Eq. (3) from Eq. (4) see the supplemen-\ntal material of Ref. [29].\nThe Lagrangian formulation is convenient to imple-\nment the central constraint\nj\t(t)i=j\t0(S(t))i (7)\nof adiabatic spin dynamics (ASD) theory [16, 29, 30].\nThis is a holonomic constraint without explicit time de-\npendence. We thereby assume that the fast electron dy-\nnamics is perfectly constrained to the manifold of in-\nstantaneous ground states j\t0(S)iof^H(S), and that\nthe ground states are non-degenerate. The smooth map\nS7!j\t0(S)iis characterized by the spin-Berry connec-\ntion:\nCm(S) =ih\t0(S)j@\n@Smj\t0(S)i: (8)\nUnter a local ( S-dependent) gauge transformation\nj\t0(S)i7!j \t0\n0(S)i=ei\u001f(S)j\t0(S)i (9)\ngiven by a smooth function \u001f(S), the spin-Berry connec-\ntion transforms as\nCm(S)7!C0\nm(S) =Cm(S)\u0000@\u001f(S)\n@Sm; (10)while the spin-Berry curvature\n\nm\u000b;m0\u000b0(S) =@\n@Sm\u000bCm0\u000b0(S)\u0000@\n@Sm0\u000b0Cm\u000b(S) (11)\n(\u000b;\u000b0=x;y;z ) is gauge invariant.\nUsing the constraint Eq. (7), we can eliminate the elec-\ntron degrees of freedom in the Lagrangian (4) so that a\nspin-only low-energy theory is obtained. The e\u000bective\nLagrangian depends on the spin degrees of freedom only\nand is given by:\nLe\u000b(S;_S) =X\nmA(Sm)_Sm+h\t0(S)ji@tj\t0(S)i\n\u0000h\t0(S)j^H(S)j\t0(S)i\u0000X\nm\u0015mS2\nm:(12)\nThe last term, with Lagrange multipliers \u0015mtakes care\nof the normalization condition Sm= 1. The Euler-\nLagrange equations are straightforwardly derived. We\n\fnd\n0 =d\ndt@Le\u000b\n@_Sm\u0000@Le\u000b\n@Sm=_Sm\u0002Sm\nS3m+@h^H(S)i\n@Sm\n\u0000X\n\u000bX\nm0\u000b0\nm\u000bm0\u000b0_Sm0\u000b0e\u000b+ 2\u0015mSm; (13)\nwheree\u000bis the\u000bth canonical unit vector, h\u0001\u0001\u0001i is the ex-\npectation value in the ground state j\t0(S)i, and where\nwe have made use of Eqs. (6), (8), (11) and the antisym-\nmetry of the spin-Berry curvature\n\nm\u000b;m0\u000b0(S) =\u0000\nm0\u000b0;m\u000b(S): (14)\nScalar multiplication of Eq. (13) with Smyields an ex-\npression for \u0015m. Taking the cross product with Smfrom\nthe right and using S2\nm= 1, we \fnd the e\u000bective equation\nof motion\n_Sm=@h^H(S)i\n@Sm\u0002Sm+Tm\u0002Sm; (15)\nwhere the last term, Tm\u0002Sm, with\nTm=Tm(S;_S) =X\n\u000bX\nm0\u000b0\nm0\u000b0m\u000b(S)_Sm0\u000b0e\u000b(16)\nis the geometric spin torque. We note that for M= 1\nthis reproduces the result given in Ref. [16].\nThe spin-Berry curvature represents the feedback of\nthe geometrical Berry physics of the quantum system on\nthe classical spin dynamics in the deep adiabatic limit.\nIn addition to the \frst term in Eq. (15), it gives rise to an\nanomalous spin torque. The strength of this geometrical\nspin torque is determined by the elements of the spin-\nBerry curvature tensor.4\nB. Spin-Berry curvature\nThere are various useful general representations of the spin-Berry curvature. First, starting from the de\fnition Eq.\n(11) and using Eq. (8), we get\n\nm\u000b;m0\u000b0(S) =i\u0012@h\t0j\n@Sm\u000b@j\t0i\n@Sm0\u000b0\u0000@h\t0j\n@Sm0\u000b0@j\t0i\n@Sm\u000b\u0013\n=\u00002 Im\u0012@h\t0j\n@Sm\u000b@j\t0i\n@Sm0\u000b0\u0013\n; (17)\nwherej\t0i=j\t0(S)ihas been written for short.\nSecond, letj\tn(S)iforn > 0 be the (possibly degenerate) nth excited eigenstate of ^H(S), i.e., ^H(S)j\tn(S)i=\nEn(S)j\tn(S)i. Di\u000berentiating with respect to Sm\u000byields the following identity, valid for n6= 0:\nh\t0(S)jrm\u000bj\tn(S)i=h\t0(S)jrm\u000b^H(S)j\tn(S)i\nEn\u0000E0; (18)\nwhererm\u000b\u0011@=@Sm\u000b. Inserting a resolution of the unity with a local orthonormal basis of eigenstates\nfj\tn(S)ign=0;1;:::in Eq. (17) and using Eq. (18), we \fnd:\n\nm\u000b;m0\u000b0(S) =\u00002 ImX\nn6=0h\t0(S)jrm\u000b^H(S)j\tn(S)ih\tn(S)jrm0\u000b0^H(S)j\t0(S)i\n(E0(S)\u0000En(S))2; (19)\nand with Eqs. (1) and (2),\n\nm\u000b;m0\u000b0(S) =\u00002J2ImX\nn6=0h\t0(S)jsim\u000bj\tn(S)ih\tn(S)jsim0\u000b0j\t0(S)i\n(E0(S)\u0000En(S))2: (20)\nNote that a possibly present classical-spin-classical-spin coupling in Hcl(S) does not contribute here.\nThird, we note that the energy eigenstates for J= 0 are trivially independent of the spin con\fguration S. Expanding\nj\tn(S)i=j\t(0)\nni+O(J) andEn=E(0)\nn+O(J) and noting the explicit J2prefactor in Eq. (20), the spin-Berry\ncurvature takes the form\n\nm\u000b;m0\u000b0(S) = \nm\u000b;m0\u000b0=\u00002J2ImX\nn6=0h\t(0)\n0jsim\u000bj\t(0)\nnih\t(0)\nnjsim0\u000b0j\t(0)\n0i\n\u0000\nE(0)\n0\u0000E(0)\nn\u00012+O(J3) (21)\nin the weak- Jlimit. We see that the spin-Berry curvature becomes independent of the classical spin con\fguration in\nthis case.\nFourth, in the weak- Jlimit, there is a close relation with the retarded magnetic spin susceptibility of the unperturbed\n(J= 0) electron system, which is de\fned as\n\u001fi\u000b;i0\u000b0(t) =\u0000i\u0002(t)e\u0000\u0011th[si\u000b(t);si0\u000b0(0)]i(0): (22)\nHere, \u0002 is the step function, \u0011is a positive in\fnitesimal, and h\u0001\u0001\u0001i(0)denotes the expectation value with the ground\nstate of the J= 0 electron system. Furthermore, si\u000b(t) =ei^H0tsi\u000be\u0000i^H0twith ^H0=^HjJ=0. Expanding the\ncommutator and inserting a resolution of the unity with eigenstates of ^H0yields the Lehmann representation via\nFourier transformation:\n\u001fi\u000b;i0\u000b0(!) =Z\ndtei!t\u001fi\u000b;i0\u000b0(t) =X\nn6=0 \nh\t(0)\n0jsi\u000bj\t(0)\nnih\t(0)\nnjsi0\u000b0j\t(0)\n0i\n!+i\u0011\u0000(E(0)\nn\u0000E(0)\n0)\u0000h\t(0)\n0jsi0\u000b0j\t(0)\nnih\t(0)\nnjsi\u000bj\t(0)\n0i\n!+i\u0011\u0000(E(0)\n0\u0000E(0)\nn)!\n:(23)\nThe susceptibility at != 0 and its !-derivative at != 0 are given by:\n\u001fi\u000b;i0\u000b0(0) =\u00002 ReX\nn6=0h\t(0)\n0jsi\u000bj\t(0)\nnih\t(0)\nnjsi0\u000b0j\t(0)\n0i\nE(0)\nn\u0000E(0)\n0;@\n@!\u001fi\u000b;i0\u000b0(0) =\u00002iImX\nn6=0h\t(0)\n0jsi\u000bj\t(0)\nnih\t(0)\nnjsi0\u000b0j\t(0)\n0i\n\u0000\nE(0)\nn\u0000E(0)\n0\u00012:\n(24)\nHere, for a gapped system, we can disregard the in\fnitesimal i\u0011. Comparing with Eq. (21), we conclude:\n\nm\u000b;m0\u000b0(S) = \nm\u000b;m0\u000b0=\u0000iJ2@\n@!\u001fim\u000b;i0m\u000b0(!)\f\f\f\n!=0+O(J3): (25)5\nWe see that, in the weak-coupling limit, the spin-Berry curvature describes the linear magnetic response of the electron\nsystem due to a slow time-dependent perturbation and in fact represents the \frst nontrivial correction to a static\nperturbation.\nC. Time reversal and spin rotation\nEquations (15) and (16) tell us that anomalous adi-\nabatic spin dynamics under a geometrical spin torque\nrequires a \fnite spin-Berry curvature. In this context, it\nis remarkable that the spin-Berry curvature vanishes in\nthe weak-Jregime, if the underlying quantum system is\ntime-reversal symmetric.\nThis is easily seen as follows: Let \u0002 be the usual anti-\nunitary representation of time reversal in the Fock space.\nWe have \u0002sim\u0002y=\u0000sim, and thus \u0002 ^Hint\u0002y=\u0000^Hint.\nObviously, the interaction term Eq. (1), involving classi-\ncal spins that may be seen as local magnetic \felds, ex-\nplicitly breaks time-reversal symmetry. However, this is\nirrelevant in the weak- Jlimit, see Eqs. (24) and (25),\nsince here the spin-Berry curvature is a property of the\nunperturbed ( J= 0) electron system only. The un-\nperturbed electron system is time-reversal symmetric if\n[^Hqu;\u0002] = 0. The case of a non-degenerate ground state,\nas considered here, implies that there is no Kramers de-\ngeneracy and that the total number of spin-1 =2 elec-\ntrons must be even. Hence, the time-reversal operator\nsquares to unity, \u00022= +1, and we can choose an or-\nthonormal basis of time-reversal-symmetric eigenstates\nj\t(0)\nniof^Hqu, i.e., we have \u0002 j\t(0)\nni=j\t(0)\nnifor the\nground state ( n= 0) and all excited states ( n > 0).\nTherewith, we have h\t(0)\n0jsi\u000bj\t(0)\nni=h\u0002\t(0)\n0jsi\u000bj\u0002\t(0)\nni\nfor the matrix element in Eq. (24). Anti-linearity of \u0002\nimpliesh\u0002\t(0)\n0jsi\u000bj\u0002\t(0)\nni=h\t(0)\n0j\u0002ysi\u000b\u0002j\t(0)\nni\u0003, and\nwith \u0002ysi\u000b\u0002 =\u0000si\u000band \u0002y\u0002 = 1, we eventually\n\fndh\t(0)\n0jsi\u000bj\t(0)\nni=\u0000h\t(0)\n0jsi\u000bj\t(0)\nni\u0003. We conclude\nthat time-reversal symmetry enforces that the matrix el-\nements are purely imaginary and that\n@\n@!\u001fi\u000b;i0\u000b0(0) = 0: (26)\nThe spin-Berry curvature vanishes in the weak- Jlimit\nfor a time-reversal symmetric electron system. As we\nare particularly interested in the geometrical spin torque,\nthis motivates us to study systems with broken time-\nreversal symmetry already at J= 0. BeyondO(J2),\ntime-reversal symmetry is broken explicitly.\nNext, we consider the usual unitary representation\nU=U(n;') = exp(\u0000istotn') of spin rotations on\nthe Fock space, where the unit vector nde\fnes the\nrotation axis and 'the rotation angle. The gener-\nators are given by the components of the total spin\nstot=P\nisi. Invariance of ^Hquunder spin ro-\ntations, [ ^Hqu;stot] = 0, implies that its eigenstates\nfj\tnigcan be simultaneously chosen as eigenstates of\nU. Assimis a vector operator, we have Uysim\u000bU=P\n\fR\u000b\fsim\f, whereR=R(n;') is the de\fning SO(3)matrix representation. With this, the left matrix el-\nement in the expression for \n m\u000b;m0\u000b0[see Eqs. (24)\nand (25)] can be written as h\t(0)\n0jUUysim\u000bUUyj\t(0)\nni=P\n\fR\u000b\fei\u001e0h\t(0)\n0jsim\fj\t(0)\nnie\u0000i\u001en, where the phase fac-\ntorsei\u001enare the corresponding eigenvalues of U.\nThese cancel with those obtained from the right\nmatrix element and, hence, we \fnd \n m\u000b;m0\u000b0=P\n\f\f0R\u000b\f\nm\f;m0\f0RT\n\f0\u000b0. Irreducibility of Rand Schur's\nlemma thus imply \n m\u000b;m0\u000b0= \nmm0\u000e\u000b\u000b0. Furthermore,\nthe antisymmetry condition Eq. (14) requires that the\nmatrix \n with elements \n mm0is skew-symmetric. In case\nof two impurity spins ( M= 2) this means\n(\nm\u000b;m0\u000b0) = \n\n1= \n\u0012\n0 1\n\u00001 0\u0013\n\n1; (27)\ni.e., the spin-Berry curvature is fully determined by a\nsingle real number \n. Note that spin-rotation symmetry\nrequires \n = 0 in the case of a single impurity spin ( M=\n1), i.e., a \fnite spin-Berry curvature is obtained beyond\nthe weak-Jlimit only.\nD. RKKY interaction\nThe static ( != 0) magnetic susceptibility \u001fi\u000b;i0\u000b0(0) is\ngenerally nonzero, even if ^Hquis time-reversal symmet-\nric and the matrix elements are purely imaginary. This\nquantity just describes the linear-in- Jresponse of the\nelectron system at time t. In the adiabatic limit, the re-\nsponse of the electron system is in fact static: The local\nmagnetic moment hsimiat siteimin the (instantaneous)\nground statej\t0(S(t))iat timet, which is induced by\nthe classical perturbations JSm0(t) that couple to sim0,\nis given by:\nhsim\u000bi=X\nm0\u000b0\u001fim\u000b;im0\u000b0(0)JSm0\u000b0(t) +O(J2):(28)\nInvariance of ^Hquunder SU(2) spin rotations implies that\n\u001fim\u000b;im0\u000b0(0) =\u000e\u000b\u000b0\u001fim;im0(0)\u0011\u000e\u000b\u000b0\u001fmm0(0), so that\nwe can writehsimi=P\nm0J\u001fm;m0(0)Sm0(t) up to linear\norder inJ. With the Hellmann-Feynman theorem we\nhave@h^H(S)i=@Sm=Jhsimi+@Hcl(S)=@Sm. Assum-\ning thatHcl\u00110, for the sake of simplicity, and using\nEqs. (15) and (28), we \fnd\n_Sm=X\nm0J2\u001fm;m0(0)Sm0\u0002Sm+Tm(S;_S)\u0002Sm:(29)\nFor a time-reversal-symmetric Hamiltonian, the second,\ngeometrical term vanishes while the \frst is just the stan-\ndard classical spin dynamics described by the e\u000bective6\nclassical Hamiltonian HRKKY =P\nmm0JRKKY\nmm0SmSm0\nwith e\u000bective RKKY [59] exchange coupling JRKKY\nmm0=\nJ2\u001fmm0(!= 0). We note, that both spin torques in Eq.\n(29), the RKKY torque and the geometric torque, come\nwith the same prefactor J2.\nE. E\u000bective dynamics of two classical spins\nEquation (29) is an implicit nonlinear system of di\u000ber-\nential equations. It can be further simpli\fed in the case\nof two classical spins. With Eqs. (15) and (16) we get for\nM= 2:\n_S1=JRKKYS2\u0002S1\u0000\n_S2\u0002S1;\n_S2=JRKKYS1\u0002S2+ \n_S1\u0002S2: (30)\nThese equations hold in the weak- Jlimit withJRKKY=\nJ2\u001f12(!= 0) =J2\u001f21(0), where we can explicitly\nmake use of Eq. (27). Note that for \fnite \n they\nare not forminvariant under exchange of the two spins\n1$2. This is interesting, since any Hamilton function\nHcl(S1;S2) for identical classical spins would be sym-\nmetric,Hcl(S1;S2) =Hcl(S2;S1), and would lead to\nforminvariant equations of motion. In fact, the asymme-\ntry of the second term /\n demonstrates the anomalous,\nnon-Hamiltonian character of the resulting spin dynam-\nics. A similar discussion has been given earlier [30] in a\ndi\u000berent (purely classical) context.\nForm= 1;2 we introduce real and antisymmetric 3 \u00023\nmatrices\nAm=0\n@0\u0000SmzSmy\nSmz 0\u0000Smx\n\u0000SmySmx 01\nA; (31)\nsuch that we can replace the cross product by a matrix-\nvector multiplication, Sm\u0002(:::) =Am(:::), and write\n_S1=JRKKYS2\u0002S1+ \nA1_S2;\n_S2=JRKKYS1\u0002S2\u0000\nA2_S1: (32)\nUsing this notation, we can formally solve for _S1and _S2,\n\u0012_S1\n_S2\u0013\n=JRKKYM\u00001\u0012\nA20\n0A1\u0013\u0012\nS1\nS2\u0013\n; (33)\nwhere the 6\u00026 matrixMis de\fned as\nM\u0011 1\u0000\n\u0012\n0A1\n\u0000A20\u0013\n=\u0012\n1\u0000\nA1\n\nA21\u0013\n:(34)\nTherewith we have rewritten the equations of motion as\nexplicit di\u000berential equations. We note that the determi-\nnant of the block matrix is\ndetM= det( 1+ \n2A2A1): (35)\nA straightforward calculation yields the simple result\ndetM= \n4(S1S2)2\u00002\n2S1S2+ 1; (36)i.e., matrix inversion is possible unless det M= 0 or\nS1S2\n2= 1: (37)\nThe inversion can be done analytically:\nM\u00001=1\n1\u0000\n2S1S2\u0012\n1\u0000\n2S2\nS1 \nA1\n\u0000\nA2 1\u0000\n2S1\nS2\u0013\n;\n(38)\nwhereS1\nS2is the outer (dyadic) product of S1with\nS2. Using the identity S1\nS2\u0001S1\u0002S2= 0 and com-\nbining the results, we \fnd\n_S1=JRKKY\n1\u0000\n2S1S2[S2\u0002S1+ \nS1\u0002(S1\u0002S2)];\n_S2=JRKKY\n1\u0000\n2S1S2[S1\u0002S2\u0000\nS2\u0002(S2\u0002S1)]:\n(39)\nWe immediately see that jS1j,jS2j, and the scalar prod-\nuctS1S2are conserved. The total impurity spin S1+S2\nis not conserved for \n 6= 0. This is compensated by a re-\nspective dynamics of the total quantum spin, see Ref. [30]\nfor a discussion in the purely classical case.\nFor \n = 0 we recover the standard RKKY dynam-\nics governed by the RKKY coupling constant JRKKY\n12 =\nJRKKY\n21 =J2\u001f12(!= 0). Here,S1andS2precess around\nS1+S2= const with frequency !L=JRKKYjS1+S2j.\nThe spin dynamics in that case is a Hamiltonian dynam-\nics and derives from the e\u000bective RKKY Hamiltonian\nHRKKY =JRKKYS1S2.\nFor \fnite \n, we can distinguish between two additional\ne\u000bects: First, there is an overall renormalization of the\nRKKY coupling JRKKY\n127!JRKKY\n12=(1\u0000\n2S1S2), which\ndepends on the initial spin con\fguration. Second, there\nis an additional (non-Hamiltonian) coupling between the\nspins, the relative strength of which is given by \n. We\nnote that for \n ! 1 the additional coupling cannot\noutweigh the renormalization e\u000bect, see Eq. (39), and\nthere is no spin dynamics at all in this limit.\nEquation (37) shows that the theory must break down\nfor model parameters, where \n = \n c\u00111=(S1S2) (re-\ncall thatS1S2is conserved). For a spin-Berry curvature\nright at the critical value \n c, the dynamics of the two\nspins gets arbitrarily fast, such that the adiabatic theo-\nrem does not apply. On the other hand, Eq. (37) or Eqs.\n(39) show that the strongest renormalization e\u000bect due\nto the geometrical spin torque occurs for model parame-\nters, where \n is close to \n c, while the exchange coupling\nJat the same time must satisfy a condition\n1\n\u001c\u0011J2\u001f12(0)\n1\u0000S1S2\n2\u001c\u0001E (40)\nto ensure the applicability of the adiabatic theorem. As\na rule of thumb, for M= 2 spins and for a generic value\nofS1S2in the initial state, the value of the (with ~\u00111)\ndimensionless quantity \n should be \n = O(1) to get a\nstrong e\u000bect of the geometric torque.7\nIntroducing the dimensionless time scale t0=t=\u001c, the\ne\u000bective equations of motion can be rewritten in a form\nthat is independent of all model parameters, except for\nthe spin-Berry curvature:\ndS1\ndt0=S2\u0002S1+ \nS1\u0002(S1\u0002S2);\ndS2\ndt0=S1\u0002S2\u0000\nS2\u0002(S2\u0002S1): (41)\nLet us emphasize once more that these equations cannot\nbe derived from a Hamilton function H=H(S1;S2).\nThe nonlinear system of equations (41) is easily solved\nanalytically. We de\fne\n\u0006=S1+S2+ \nS2\u0002S1: (42)\nUsing Eq. (41), d\u0006=dt= 0 is obtained straightforwardly.\nFurthermore, since d(S1S2)=dt= 0, the enclosed angle\n'is conserved. Both, S1andS2undergo a precession\naround \u0006, i.e., we have _S1=\u0006\u0002S1and _S2=\u0006\u0002S2\nwith frequency !prec=p\n\u00062. WithjS1j=jS2j= 1, we\n\fnd\n!prec=q\n4 cos2('=2) + \n2sin2': (43)\nFor \n = 0 the conventional RKKY dynamics is recovered.F. Computation of the spin-Berry curvature\nThe actual calculations of the spin Berry curvature\nare performed for a gapped tight-binding model of non-\ninteracting electrons with Hamiltonian\n^Hqu=X\nii0;\u001btii0cy\ni\u001bci0\u001b=X\nIr;I0r0;\u001btIr;I0r0cy\nIr\u001bcI0r0\u001b\n=X\nktrr0(k)cy\nkr\u001bckr0\u001b=X\nk\u0017\"\u0017(k)cy\nk\u0017\u001bck\u0017\u001b:\n(44)\nHere,tii0is the (spin-independent) hopping matrix, and\nwe writei= (I;r), whereIare the lattice translation\nvectors and rlabels the sites within a unit cell. For the\nhoneycomb lattice considered below, r=A;B, corre-\nsponding to A- or B-sublattice sites. The hopping matrix\nis diagonalized with repect to I;I0by a unitary trans-\nformation of the form cy\nkr\u001b=P\nIU(r)\nIkcy\nIr\u001b, such that\ntIr;I0r0=P\nkU(r)\nIktrr0(k)U(r0)y\nkI0. In case of a transla-\ntionally symmetric lattice with periodic boundary con-\nditions,kruns over the discrete set of Lwave vectors\nin the \frst Brillouin zone, where Lis the number of\nunit cells, and U(r)\nIk=UIk=L\u00001=2eik\u0001I. In a sec-\nond step, the k-dependent diagonalization of trr0(k) =P\n\u0017Ur\u0017(k)\"\u0017(k)Uy\n\u0017r0(k) is achieved by the unitary trans-\nformationcy\nk\u0017\u001b=P\nrUr\u0017(k)cy\nkr\u001bfor eachk. Here,\u0017is\nthe band index, if ^His translationally symmetric. We\nhavetIr;I0r0=P\nkU(r)\nIkUr\u0017(k)\"\u0017(k)Uy\n\u0017r0(k)U(r0)y\nkI0. With\nthe combined transformation\nSIr;k\u0017=U(r)\nIkUr\u0017(k); (45)\nthis reads as tIr;I0r0=P\nk\u0017SIr;k\u0017\"\u0017(k)Sy\nk\u0017;I0r0.\nEvaluating the matrix elements in Eq. (20) for N-electron Slater determinants j\t(0)\nni=Q\nk\u0017\u001bcy\nk\u0017\u001bjvaciis straight-\nforward. We \fnd\n\nm\u000b;m0\u000b0=\u0000J2\u000e\u000b\u000b0Imocc:X\nk\u0017unocc:X\nk0\u00170X\nrr0Sy\nk\u0017;IrSIr;k0\u00170Sy\nk0\u00170;I0r0SI0r0;k\u0017\n(\"\u00170(k0)\u0000\"\u0017(k))2; (46)\nwhere (I;r) =imand (I0;r0) =im0. This representation applies to the translationally symmetric model ^H=^Hqu\nforJ= 0 and can be used for the case, where Jis treated perturbatively. Beyond the weak- Jregime, it applies for\neach spin con\fguration Sin^H=^Hqu+^Hint(S). But as translational symmetry is broken in this case, SIr;k\u0017does\nno longer contain the Fourier factor L\u00001=2eikIand must be computed numerically and explicitly include the \u000bindex.\nG. Spinful Haldane model\nFor generic parameters the half-\flled Haldane model\n[32] is a prototypical Chern insulator with broken time-\nreversal symmetry. Numerical calculations have been\nperformed for the spinful Haldane model at half-\flling,\nwhich consists of two identical copies of the Haldane\nmodel for electrons with spin projection \u001b=\"and\u001b=#,respectively. The spinful Haldane model is trivially in-\nvariant under SU(2) spin rotations. Rotation symmetry\nis broken in the full model Eq. (2), where the two copies\nare coupled by the interaction term Eq. (1).\nThe corresponding tight-binding quantum Hamilto-8\n2e+i\n2ei\nM\n+M\nunit cell\nFIG. 1. Haldane model on the honeycomb lattice. A sites in-\ndicated by blue dots with on-site potential + M. B sites: red\ndots, potential\u0000M. Nearest-neighbor hopping with ampli-\ntude\u001c1: dashed lines. Next-nearest-neighbor hopping \u001c2with\nadditional Peierls factor e\u0000i\u0018(ei\u0018) for hopping in clockwise\n(counterclockwise) direction: purple (light-blue) arrows.\nnian in Eq. (2) is given by\n^Hqu=MX\ni\u001bzicy\ni\u001bci\u001b\u0000\u001c1X\nhii0i;\u001bcy\ni\u001bci0\u001b\n\u0000\u001c2X\nhhii0ii;\u001bei\u0018ii0cy\ni\u001bci0\u001b: (47)\nHere,i,i0run over the sites of the two-dimensional bipar-\ntite honeycomb lattice, see Fig. 1. Mis the strength of a\nstaggered on-site potential, where the sign factor zi= +1\nfor a siteiin the A sublattice ( r=A) andzi=\u00001 foriin\nthe B sublattice ( r=B). ForM6= 0 the potential term\nbreaks particle-hole symmetry. The amplitude for hop-\nping between nearest-neighbor sites on the lattice, hii0i,\nis denoted by \u001c1. We set\u001c1= 1 to \fx the energy scale\n(and with ~= 1) also the time scale. The next-nearest-\nneighbor hopping amplitude ei\u0018ii0\u001c2with real\u001c2includes\na phase factor with \u0018ii0=\u0000\u0018for hopping from i0toiin\nclockwise direction (see purple lines in Fig. 1), and with\n\u0018ii0=\u0018in counterclockwise direction (light blue). This\nensures that the total \rux of the corresponding orbital\nmagnetic \feld through a unit cell vanishes. Time-reversal\nsymmetry is broken for \u001c26= 0 and\u00186= 0;\u0006\u0019.\nThe band structure of the spinful Haldane model con-\nsists of two bands with trivial twofold spin degeneracy\neach. The band structure is gapped for all Mand all\u0018,\nifj\u001c2=\u001c1j<1=3, except for parameter values satisfying\nthe condition [32] jM=\u001c 2j= 3p\n3jsin\u0018j, or, in terms of\nthe band gap given by\n\u0001E=jM\u00003p\n3\u001c2sin\u0018j!= 0; (48)\nforM;\u001c 2>0 and 0< \u0018 < \u0019 . In theM=\u001c 2vs.\u0018phase\ndiagram, see Fig. 2, this condition de\fnes lines of topo-\nlogical phase transitions (see red and blue lines in the\n2\n0\n2\n33\n033\nM\n2\nC=1\n C=+1C=0\nE(K)=0\n E(K/prime)=0\nFIG. 2. Topological phase diagram of the Haldane model in\nthe\u0018-M=\u001c 2plane.Cdenotes the \frst Chern number. The\nnearest-neighbor hopping \u001c1= 1 sets the energy scale.\n\fgure). At a phase transition, there is a band closure in\nthe \frst Brillouin zone, either at K= 2\u0019(1=3p\n3;1=3),\nwhenM=\u00003p\n3\u001c2sin(\u0018), orK0= 2\u0019(2=3p\n3;0), when\nM= 3p\n3\u001c2sin(\u0018) [33].\nIn the Altland-Zirnbauer classi\fcation [61], the model\nbelongs to symmetry class A. For parameters inside the\nboundary in Fig. 2, i.e., in the orange or green regions\nforjM=\u001c 2j<3p\n3jsin\u0018j, the model represents a Chern\ninsulator with \fnite Chern number C=\u00001 (\u0018 <0) and\nC= +1 (\u0018 >0). Outside the boundary, we have C= 0,\nand the system is a topologically trivial band insulator.\nThe Haldane model ^Hquis invariant under SU(2)\nspin rotations, i.e., we can make use of Eq. (27). Fur-\nthermore, it is invariant under the discrete C3rota-\ntions of the lattice around a \fxed site, and thus \n is\nC3invariant. Time-reversal symmetry, on the other\nhand, is broken explicitly. Under time reversal, we have\nh\t(0)\n0jsi\u000bj\t(0)\nni7!\u0000h \t(0)\n0jsi\u000bj\t(0)\nni\u0003. With Eq. (21), this\nimplies \n m\u000b;m0\u000b07!\nm0\u000b0;m\u000b, and using Eq. (27) we \fnd\n\n7!\u0000\n. Under a re\rection at a mirror symmetry axis of\nthe hexagonal lattice, the Hamiltonian transforms in the\nsame way as under time reversal, H7!\u0002H\u0002y. The \frst\ntwo terms on the right-hand side of Eq. (47) are invari-\nant. Re\rection of the hopping term for two next-nearest-\nneighbor sites i;jis represented by complex conjugation,\nsince\u0018ij=\u0000\u0018ji. Hence, we have\n\n7!\u0000\n under time reversal or re\rection :(49)\nThe Hamiltonian and thus \n is invariant under the com-\nbined transformation.\nH. Spin-Chern number\nLet us emphasize that the di\u000berent topological phases\nof the Haldane model are characterized by the \frst Chern9\nnumberC, which is obtained by integrating the conven-\ntionalk-space Berry curvature F(k) over the entire BZ,\ni.e., the torus T2. The spin-Berry curvature is a di\u000ber-\nent concept, motivated by the geometrical spin torque\ncontribution to the adiabatic impurity-spin dynamics.\nStill it can be employed to de\fne a topological invari-\nant: The Hamiltonian of the quantum system smoothly\ndepends on the parameters S= (S1;:::;SM)2 S,\nwhereSis the Cartesian product of two-spheres (recall\njSrj= 1). The parameter manifold Sis closed, i.e., has\nno boundary, and is 2 M-dimensional. Hence, according\nto the general theory (see Ref. [62], for example), the\nMth spin-Chern number C(S)\nMis given by\nC(S)\nM=iM\n(2\u0019)M1\nM!I\nStr!M; (50)\nwhere!=dA+A2is the spin-Berry-curvature two-form\nderived from the one-form A, the spin-Berry connection.\nIf the ground state of the Hamiltonian is nondegenerate\non the entire manifold S, the spin-Chern number C(S)\nMis\nwell-de\fned and quantized.\nThe computation of C(S)\nMis a highly nontrivial task.\nHere we note that C(S)\nM= 0 in the weak- Jlimit, on\nwhich we concentrate in the present study. This is easily\nveri\fed, since for J= 0 the spin-Berry curvature is inde-\npendent ofS(see Eq. (21)) and hence C(S)\nM(J= 0) = 0.\nQuantization of the spin-Chern number and continuity\nwith respect Jthen imply C(S)\nM(J) = 0 +O(J2) = 0.\nOn the other hand, in the strong-coupling limit, the in-\nteraction term, Eq. (1), will dominate the physics. In the\ncase of a single classical spin, the \frst Chern number of\nthe \\atomic\" model, ^H=^Hint(S) =JSsisC(S)\n1=\u00061\n[16, 33]. Generally, it is thus plausible that there is a\nnonzero spin-Chern number C(S)\nMforJ!1 . Hence,\nas a function of Jwe expect a topological phase transi-\ntion and thus an accompanying gap closure, which, due\nto the vanishing energy denominator in Eq. (20), may\nhave a substantial e\u000bect on the magnitude of the spin-\nBerry curvature close to transition. Interestingly, our\ndata presented below (see Sec. III F) indeed demonstrate\na qualitative di\u000berence between the weak and the strong-\ncoupling limit but do not hint towards singular behavior\nof the spin-Berry curvature. Further research along this\nline is in progress.\nIII. NUMERICAL RESULTS\nThe spin-Berry curvature of the Haldane model can\nbe computed numerically using the representation Eq.\n(46), if the exchange coupling Jis weak. As mentioned\nabove in the discussion of Eq. (46), a slightly generalized\nformula applies to the general, non-perturbative case. As\ntranslational symmetry is broken in this case, however,\nthe accessible system sizes are considerably smaller. Wewill, therefore, start with a discussion of results, obtained\nfor a bulk system in the weak- Jlimit.\nA. Spatial structure of the spin-Berry curvature\nThem;m0element of the spin-Berry curvature tensor\n\nm\u000b;m0\u000b0= \nmm0\u000e\u000b\u000b0describes the strength of the mu-\ntual geometric spin torque of two classical spins Smand\nSm0, locally exchange coupled to local spins of the elec-\ntron system at two sites imandim0of the lattice. Since\n\nmm0is antisymmetric, it is su\u000ecient to specify a single\nreal number \n for each pair of lattice sites, see Eq. (27).\nWe consider a translationally invariant system with pe-\nriodic boundary conditions, \fx the position of one site,\nand compute the spin-Berry curvature \n as a function of\nthe position of the second site, using Eq. (46).\nIn Fig. 3, the \fxed site in the center of the plot is\nmarked with green color, and the J-independent quan-\ntity \n=J2is given by the color code on each of the re-\nmaining lattice sites. Sites of the A sublattice are indi-\ncated by circles, those of the B sublattice sites by squares.\nA large lattice is chosen with L= 39\u000239 unit cells,\neach consisting of two sites, i.e., with a total number of\n2L= 3042 sites. Calculations have been done for pa-\nrameters\u001c2= 0:1 and\u0018=\u0019=4, where the system is in\nthe topologically nontrivial and the trivial phase, with\nM= 0:8Mcrit(left plot) and M= 1:2Mcrit(right plot),\nrespectively. For this parameter set we have Mcrit\u00190:37.\nNote that the parameters are chosen such that the gap\nsize \u0001Eis the same for both, the topologically nontrivial\n(left) and the trivial phase (right).\nOne can nicely see the invariance of \n under discrete\n2\u0019=3 rotations of the lattice around the \fxed site. This\nsetC3of spatial rotations forms in fact a symmetry group\nof the Haldane model. Furthermore, consistent with Eq.\n(27), \n changes sign if imandim0, i.e.,mandm0are\nexchanged. In the \fgure, where site imis kept \fxed,\nthis is seen be comparing \n for A-sublattice sites im0\nat opposite positions Imand\u0000Im0. Finally, consistent\nwith Eq. (49), \n changes its sign under a re\rection at\nthe horizontal axis through the central site imand at the\naxes rotated by 2 \u0019=3 and\u00002\u0019=3 against the horizontal.\nThis also implies that directly on these mirror axes \nvanishes.\nConcerning the distance dependence, we can distin-\nguish between two di\u000berent ranges. For small distances,\nup to about 3 unit cells, \n has an oscillatory distance\ndependence. In this close range , the spatial structure\nis rather complicated generally. On the other hand, in\nthefar range \n does not change sign and monotonically\ndecreases with increasing distance between imandim0\nalong any spatial direction.\nWe recall that, in the weak- Jlimit, the spin-Berry\ncurvature \n is purely a property of the spinful Haldane\nmodel. However, it is not directly related to its band\ntopology, as the weight factors in Eq. (46) are constructed\nfrom the same Bloch states but in a di\u000berent way as com-10\n/J2\n103\n104\n105\n106\n107\n108\n103\n104\n105\n106\n107\nFIG. 3. Nonzero element \n of the spin-Berry curvature \n m\u000b;m0\u000b0[see Eq. 27] for various positions imandim0of the two\nimpurity sites, to which the classical spins are coupled. The \frst spin resides at the central site i1(see green dot), and the\nsecond one is varied over the remaining 2 L\u00001 sites of the hexagonal lattice. Calculations for a lattice with L= 39\u000239 unit\ncells and periodic boundary conditions. Model parameters: \u001c2= 0:1,\u0018=\u0019=4.\u001c1= 1 sets the energy scale. Left:M= 0:8Mcrit,\nsuch that the system is in a topologically nontrivial phase. Right:M= 1:2Mcrit, topologically trivial phase. Results for the\nweak-Jregime. The color code indicates the value of \n =J2.\npared to the k-space Berry curvature. Nevertheless, the\nsign structure of \n in the far range is quite universal, and\nit is di\u000berent for the topologically nontrivial and the triv-\nial phase, as can be seen in Fig. 3, although the gap \u0001 E\nis the same. This implies that, to some degree, the spin\ndynamics is sensitive to the respective topological phase.\nIn this sense, the spin-Berry curvature can be seen as a\nmarker for the topological properties of the model.\nDi\u000berences between the two phases become more pro-\nnounced in the case of a smaller gap \u0001 E. Fig. 4 displays\nresults for a next-nearest-neighbor hopping amplitude \u001c2,\nwhich is smaller by two orders of magnitude as compared\nto Fig. 3. Choosing M= 0:8McritandM= 1:2Mcrit, re-\nspectively, as in Fig. 3, a small \u001c2leads an overall small\ngap \u0001Ein the entire phase diagram, as can be inferred\nfrom Eq. (48). Again, the gap \u0001 Eis the same for both,\nthe topologically nontrivial and the trivial phase.\nBut now, for small \u001c2, the spatial structure of \n islargely di\u000berent for both phases. In the nontrivial phase\n(left plot), the close range now spreads over the entire\n(\fnite) system, and there is hardly any visible decrease\nof \n with increasing distance. In the trivial phase (right),\non the contrary, the close range is limited to distances of a\nfew unit cells only, while \n is clearly reduced in size in the\nfar range. For still larger distances from the central site,\n\n increases again. This is due to \fnite-size e\u000bects, which\nare more pronounced for small \u0001 E. Since, as compared\nto Fig. 3, the gap is smaller by two orders of magnitude,\nthis is not unexpected.\nB. Finite systems in the small \u001c2regime\nInterestingly, the absolute value of \n in Fig. 4 is much\nlarger in the nontrival case, where the far range extends\nover the whole lattice. Quite generally, \fnite systems11\n/J2\n103\n104\n105\n106\n107\n108\n103\n104\n105\n106\n107\nFIG. 4. The same as Fig. 3 but for \u001c2= 0:001. Left:\ntopologically nontrivial phase at M= 0:8Mcrit.Right: trivial\nphase,M= 1:2Mcrit.\n(with periodic boundary conditions) in the regime of\nsmall next-nearest-neighbor hopping \u001c2are interesting as\nhere the spin-Berry curvature can be extremely large.\nThis is seen in Fig. 5, where \n is plotted as a function\nof\u001c2. Here, we have set M= 0, such that the model is\ntopologically nontrivial for all values of \u001c2. Exactly at\n\u001c2= 0, however, the model is a time-reversal symmetric\nsemimetal, such that the Chern number is zero. Hence,\nin the thermodynamical limit L!1 , there is a phase\ntransition at \u001c2= 0.\nFrom the log-log plot in Fig. 5 we can infer that\n\n/1=\u001c2\n2for\u001c2!0. This has interesting consequences,\nas has already been discussed above in Sec. II E: Namely,\nfor\u001c2!0, the spin dynamics is entirely dominated by\nthe anomalous contribution from the geometrical spin\n105\n104\n103\n102\n101\n2/1\n104\n103\n102\n101\n100101102103/J2\nl = 15\nl = 30\nl = 45\nl = 60\nl = 90\nl = 120\nl = 150\nl = 180\nl = 210\nFIG. 5. \n for next-nearest-neighbor sites as function of \u001c2for\n\u0018=\u0019=4 andM= 0. Note the log-log scale. Calculations for\n\fnite systems with periodic boundary conditions of di\u000berent\nsizeL=l\u0002las indicated.torque. The adiabatic spin dynamics slows down, and\nthe system in this limiting case ultimately shows no dy-\nnamics at all. At intermediate values for \u001c2, however,\ndepending on the value for J, the value of \n can be of\norder one. This is exactly the range, where dynamic ef-\nfects of the geometrical spin torque are most pronounced,\nas argued in Sec. II E.\nFig. 5 also shows that the 1 =\u001c2\n2behavior is realized for\nstrictly \fnite systems only. Comparing the results for\ndi\u000berentLwith linear extension lup tol= 210 (88200\nsites), we see that rather a linear dependence \n( \u001c2)/\u001c2\nis obtained in the thermodynamical limit. Note that this\nis just the \u001c2dependence that must be expected, when\nexpanding \n( \u001c2) around\u001c2= 0, where the model is time-\nreversal symmetric and thus \n = 0, up to linear order in\n\u001c2. We conclude that in the thermodynamical limit \n is\ncontinuous at the phase transition, when this is steered\nvia\u001c2, while \n diverges as 1 =\u001c2\n2for any \fnite system.\nThe underlying mechanism is the following: For small\nsystems, where the k-space is strongly discretized, the\nrelative contribution to \n from regions in the Brillouin\nzone close to KorK0is comparatively large, so that this\nbecomes the dominating contribution to \n, for model\nparameters close to the transition. For larger systems,\nhowever, the relative contribution diminishes, as can be\nseen in Fig. 5. If, for a \fnite system, \u001c2is su\u000eciently\nsmall, only the contributions from Kor fromK0are rel-\nevant in the double sum over kandk0in Eq. (46). In\nthis case, the \u001c2dependence of the spin-Berry curvature\nis essentially given via the energy denominator in Eq.\n(46) only, and since \u0001 E/\u001c2, see Eq. (48), we have the\nscaling \n/1=\u001c2\n2.\nThe di\u000berent scalings \n /\u001c2and \n/1=\u001c2\n2, dis-\ntinguish between asymptotic behavior in the thermody-\nnamic limit and for a \fnite-size system. Fig. 4 (left) for\nthe nontrivial phase is in fact representative for a sys-\ntem, where \fnite-size e\u000bects dominate, while in the case\nof Fig. 3 the system size is su\u000eciently large to re\rect the\nspin-Berry curvature in the thermodynamic limit (for not\ntoo large distances).\nC. Distance dependence\nFor an analysis of the distance dependence of \n, we\nrevert to the same parameters \u001c2= 0:1 and\u0018=\u0019=4,\nas underlying Fig. 3, but choose a larger system with\nL= 150\u0002150 unit cells. Fig. 6 shows the dependence\nof the spin-Berry curvature \n on the distance dbetween\nthe two impurity sites. dis de\fned as the Euclidean\ndistance between the sites imandim0on the hexagonal\nlattice in units of the nearest-neighbor distance. The\ndistance of a site to all of its six next-nearest neighbors,\nfor example, is given by d=p\n3. For distances 40 .\nd.100, we \fnd a nearly linear dependence of ln \n on\nd, i.e., \n/exp(\u0000d=\u0015) with\u0015 > 0, while for too large\ndistancesd&100, the linear trend is disturbed by \fnite-\nsize e\u000bects. Note that for the larger distances only the12\n0 20 40 60 80 100 120\ndistance1012\n1010\n108\n106\n104\n||/J2\nM/Mcrit: 1.2\nM/Mcrit: 0.8\nFIG. 6. \n=J2as a function of the distance dbetween the two\nimpurity sites for the topologically nontrivial ( M= 0:8Mcrit,\nblue diamonds) and the trivial phase ( M= 1:2Mcrit, orange\nsquares) along a path including A-sublattice sites, starting\nfrom the central (green) site in Fig. 3 in vertical direction.\nFilled symbols: positive sign of \n =J2. Hollow symbols: neg-\native sign. Further parameters as in Fig. 3, but for a larger\nlattice with L= 150\u0002150 unit cells (periodic boundary con-\nditions).\nsign of \n depends on Mbut not its absolute value, if\nthe gap is the same (as is the case for M= 1:2Mcritand\nM= 0:8Mcrit).\nPerforming calculations for di\u000berent Mto vary the gap\n\u0001E, we can extract the \u0001 Edependence of the slope\n\u00001=\u0015. This is plotted in Fig. 7. We \fnd a nearly lin-\near dependence 1 =\u0015/\u0001E. This means that the spin-\nBerry curvature has an exponential ddependence for\nlarged, which is controlled by the bulk band gap \u0001 E:\n\n/exp(\u0000\u0001Ed).\nThis behavior is reminiscent of the exponential decay\n0.2 0.3 0.4 0.5 0.6 0.7\nE\n0.20.30.40.51\nC=0\nC=1\nFIG. 7. 1=\u0015as a function of the gap \u0001 E, as obtained from\ncalculations for systems with L=l\u0002lwithl= 150 unit cells\nfor di\u000berent M. Calculations for the nontrivial and the trivial\nphase with the same gap size. See text for discussion.\n0 20 40 60 80 100 120\ndistance1011\n109\n107\n105\n103\n|JRKKY|/J2M/Mcrit: 1.2\nM/Mcrit: 0.8FIG. 8. The same as Fig. 6 but for JRKKY=J2as a function\nof the distance d. Filled symbols: positive sign of JRKKY=J2.\nHollow symbols: negative sign.\nof the RKKY exchange interaction with dfor insulating\nsystems [57]. Fig. 8 gives an example. Here, JRKKY=J2is\nplotted as function of dfor the same model parameters\nas in Fig. 6, and exponential behavior is found for the\nRKKY coupling in the same range 40 .d.100.\nThe range of distances with an exponential dependence\nof \n orJRKKY exactly corresponds to the far range seen\nin Fig. 3, where there is a comparatively smooth depen-\ndence of \n on the position of the second impurity spin.\nIn the close range, for d.20, the distance dependence\nof \n is less regular, and there are sign changes of \n in\naddition. This is somewhat reminiscent of the oscillatory\ndistance dependence of the RKKY exchange for a metal\nor semi-metal [49{53, 58, 59].\nAs can be seen in Figs. 6, 7 and 8, di\u000berences between\ntopologically nontrivial and trivial case are not very pro-\nnounced as concerns the distance dependence. This is\ngoverned by the \fnite gap \u0001 E, which has always been\nchosen to be the same when comparing both phases.\nQualitatively, the exponential decay of both, the spin-\nBerry curvature and the RKKY exchange, can be un-\nderstood easily: In the weak- Jregime the coupling of\ntwo impurity spins results from virtual second-order-in-\nJprocesses involving (de-)excitations of electron across\nthe gap \u0001E. This is not only the cause of the expo-\nnential distance dependence but also explains the small\nabsolute values of \n =J2andJRKKY=J2, which do not\nexceed values of the order of 10\u00002.\nD. Parameter dependence of the spin-Berry\ncurvature\nThe discussion of M= 2 spin dynamics in Sec. II E has\nshown that a value of \n close to unity is required for a\nsubstantial impact of the geometrical spin torque. As we\nhave seen, this is in fact possible in case of \fnite systems\nwith periodic boundary conditions at small values for \u001c2\n(see Figs. 4 and 5). We now focus on large systems again13\n1.0\n 0.5\n 0.0 0.5 1.0\n/\n6\n4\n2\n0246M/2\n102\n103\n104\n103\n102\n/J2\nFIG. 9. Thick gray lines: phase diagram of the Haldane\nmodel in the \u0018-M=\u001c 2plane (see Fig. 2 for comparison). The\ncolor codes the spin-Berry curvature \n =J2for next-nearest-\nneighbor sites im,im0. Calculations for a system with 27 \u000227\nunit cells with periodic boundary conditions at \u001c2= 10\u00001.\nand study the model for small distances between imand\nim0at\u001c2= 0:1, where \fnite-size e\u000bects can be neglected\nsafely.\nFig. 9 gives an overview over the dependence of the\nnonzero element \n of the spin-Berry curvature on the\nparameters Mand\u0018for next-nearest neighbors im,im0.\nThe boundaries of the topological phase transitions are\nsuperimposed in the \fgure (see thick gray lines). We \fnd\na somewhat larger absolute value of \n within a topo-\nlogically nontrivial phase. Furthermore, we note that\n\n!\u0000 \n under a sign change of \u0018. As a sign change of\nphase\u0018has the same e\u000bect for the Hamiltonian of the\nHaldane model Eq. (47) as a re\rection at a mirror sym-\nmetry axis of the hexagonal lattice, this observation is\neasily explained with Eq. (49). Otherwise the parame-\nter dependence is more or less featureless. Absolute val-\nues for \n do not exceed \u001810\u00003in the entire parameter\n1.0\n 0.5\n 0.0 0.5 1.0\n/\n6\n4\n2\n0246M/2\n103\n105\n107\n105\n103\n/J2\nFIG. 10. The same as Fig. 9 but for sites imandim0with\nEuclidean distance d= 5p\n3 (in units of the nearest-neighbor\ndistance).regime.\nFor larger distances between the sites imandim0, ab-\nsolute values for \n are smaller. But its parameter de-\npendence can be much more complicated. An example\nis given with Fig. 10, which displays results for sites at\na distanced=kim\u0000im0k= 5p\n3. This must be traced\nback to the matrix elements in Eq. (46).\nIn all cases we \fnd that the parameter dependence of \nis smooth, including parameter ranges where the model is\nclose to or right at a topological phase transition. This is\nworth mentioning since the squared energy denominator\nin Eq. (46), ( \"unocc:(k0)\u0000\"occ:(k))2, suggests that the\ncontributions of wave vectors k;k0at (or close to) the\ncritical point KorK0lead to a diverging or at least\nlarge spin-Berry curvature.\nE. Spin-Berry curvature close to a topological\nphase transition\nClose to a transition, however, a careful analysis of\nthe \fnite-size e\u000bects is necessary. For systems with a\n\fnite number of units cells L=l\u0002l, the spin-Berry\ncurvature is in fact discontinuous at a topological phase\ntransition (actually, the latter is well de\fned for L=1\nonly). Fig. 11 displays results for the Mdependence of\n\n. A \fnite jump \u0001\n at the critical point M=Mcrit=\n3p\n3\u001c2sin\u0018is found for various l. Atl= 15, the relative\njump \u0001\n=\n is considerable. With increasing L, however,\nit monotonically but slowly decreases with land is about\nan order of magnitude smaller at l= 51.\nThe data are consistent with the proposition that \n is\ncontinuous at M=Mcritin the limit L!1 . A numer-\nical proof, however, is di\u000ecult to achieve, since besides\ntheL!1 limit, the \u0001 M=jM\u0000Mcritj!0 limit must\nbe taken simultaneously. Fig. 12 displays the jump \u0001\nas function of lfor various \u0001 M=M crit. For comparatively\nlarge \u0001M=M crit= 0:1, we see that \u0001\n decreases but ap-\nproaches a \fnite value for l!1 . At smaller relative dis-\ntances \u0001M=M critto the transition point, however, this\nassumed convergence with leventually becomes invisible\nfor the largest feasible system sizes (see red line). Never-\ntheless, continuity of \n at a topological phase transition\nappears highly plausible.\nIn fact, an analytical argument can be given: Paramet-\nrically close to a topological phase transition and in the\nvicinity of the band-closure points KorK0in the \frst\nBrillouin zone, the band structure of the Haldane model\ntakes the form of a relativistic Dirac theory [32, 33]. If\n\u0014= (\u0001kx;\u0001ky) and\u00140= (\u0001k0\nx;\u0001k0\ny) denote the wave\nvectors relative toKorK0, i.e., if\u0014;\u00140= 0 refer to a\nband-closure point, we have\n\"\u0017(\u0014) =\"\u0006(\u0014)/\u0006q\n\u00142x+\u00142y+m2: (51)\nThe \\mass\" mis linearly related to the insulating gap:\nm/\u0001. We can analytically check for a possible diver-\ngence of the spin-Berry curvature in the thermodynami-\ncal limitL!1 on approaching the phase transition via14\nm!0 with help of Eq. (46). To this end, we compute the\ncontribution Ibulkof wave vectors in a su\u000eciently small\nballBwith radius \u0003 around \u0014= 0. Up to a constant\nfactor, we \fnd\nIbulk/Z\nBd2\u0014Z\nBd2\u00140 1\n\u0000p\n\u00142+m2+p\n\u001402+m2\u00012;\n(52)\nand the spin-Berry curvature is continuous at m= 0, if\nlimm!0Ibulkexists.\nNote that to realize the thermodynamical limit, the\nwave-vector sums in Eq. (46) have been replaced by in-\ntegrations. Furthermore, \u0017=\u0000(occupied) and \u00170= +\n(unoccupied), see Eq. (51). The \u0014dependence of the ma-\ntrix elements SIr;\u0014\u0017in the numerator in Eq. (46) can be\ndisregarded for small \u0003: The \frst factor of each matrix\nelement is a Fourier factor with a smooth \u0014dependence,\nsee Eq. (45). For each \u0017, the second factor is given by\nthe two-component eigenstate of the Dirac Hamiltonian\n[32, 33]\nHD/\u0014y\u001bx\u0000\u0014x\u001by+m\u001bz; (53)\nfrom which the dispersion Eq. (51) is derived. Rewriting\nthe two-dimensional \u0014integration (and analogously for\nthe\u00140integration) with the help of polar coordinates\n(\u0014;'), i.e.,R\nd2\u0014=R\nd\u0014\u0014R\nd', the\u0014-dependent part of\nthis factor is of the form e\u0006i'and thus cannot lead to a\ndivergence.\nThe remaining two-dimensional integral\nIbulk=Z\u0003\n0Z\u0003\n0d\u0014d\u00140 \u0014\u00140\n\u0000p\n\u00142+m2+p\n\u001402+m2\u00012(54)\ncan be computed analytically and turns out to stay \f-\nnite in the limit m= 0 with an additive lowest-order\ncorrection of the form \u0000m2lnm!0 form!0. This\nimplies that the spin-Berry curvature, at a topological\nphase transition, is a continuous function of the model\nparameters.\n0.0 0.5 1.0 1.5 2.0 2.5 3.0\nM/Mcrit0.0020\n0.0018\n0.0016\n0.0014\n0.0012\n0.0010\n0.0008\n0.0006\n0.0004\n/J2\nl=15\nl=21\nl=27\nl=33\nl=39\nl=45\nl=51\nFIG. 11. \n for next-nearest-neighbor sites as function of\nM=M critat\u001c2= 0:1 and\u0018= 0:1\u0019, whereMcritrefers to the\ncriticalMvalue. Calculations for system size L=l\u0002lwith\nvarious linear extensions las indicated.\n20 30 40 50 60 70\nL5105\n104\n5104\n/J2\nM/Mcrit=101\nM/Mcrit=102\nM/Mcrit=103\nM/Mcrit=106\nFIG. 12. The jump \u0001\n = j\ntop\u0000\ntrivj, obtained by com-\nputing the spin-Berry curvature at M=Mcrit\u0006\u0001M, as func-\ntion oflfor various \u0001 M=M crit. Calculations for next-nearest\nneighbors,\u001c2= 0:1,\u0018=\u0019=2.\nF. Strong exchange coupling\nThe representation Eq. (46) for the spin-Berry curva-\nture holds in case of weak exchange coupling J. For cou-\npling strengths beyond the perturbative regime, we must\nresort to Eq. (20), where the matrix elements are de\fned\nwith eigenstates carrying a nonvanishing dependence on\nthe spin con\fguration. The numerical evaluation can be\nperformed as described by the text below Eq. (46).\nFig. 13 displays the spin-Berry curvature as a function\nofJ. Comparing the results obtained from the full theory\n(blue data, left scale) with those of the perturbative-in-\nJapproach (orange), we see that perturbation theory\n101\n100101102\nJ0.0020\n0.0015\n0.0010\n0.0005\n0.00001,2/J2\nfull\nperturbative\n0.2\n0.1\n0.00.10.20.30.40.5\n1,1\n1x,1y\n1x,1z\n1y,1z\nFIG. 13. Spin-Berry curvature as function of Jfor anL=\n60\u000260 lattice at M= 0,\u001c2= 0:1,\u0018= 0:25\u0019, and for\na \fxed classical spin con\fguration Sm= (0;1;0),Sm0=\n(\u00001;0;0), whereimandim0are next-nearest neighbors. Blue\ndata (see left scale): O\u000b-site m;m0element \n = \n m\u000b;m0\u000b0\n(and\u000b=x,\u000b0=y), divided by J2, to be compared with\nthe corresponding perturbative result (orange data). Violet,\ngreen, red data (see right scale): On-site elements \n m\u000b;m\u000b0\nat the position im.15\napplies to coupling strengths J.0:1 and is still a good\napproximation up to J.1.\nFor even stronger Jbeyond the perturbative\nregime, the structure of the spin-Berry-curvature tensor\n\nm\u000b;m0\u000b0changes qualitatively. In the J! 1 limit,\nonly on-site elements \n m\u000b;m\u000b0are nonzero, and the blue\ncurve (left scale) approaches zero in this limit. These\non-site elements are given by the spin-Berry curvature of\nthe e\u000bective two-spin model H2\u0000spin=JsimSmatim,\nwheresimcan be treated as an s= 1=2 quantum spin.\nThe two-spin model is easily solved analytically [16], and\n\nm\u0011(1=2)P\n\u000b\u000b0\u000b00\"\u000b\u000b0\u000b00\nm\u000b;m\u000b0e\u000b00is given by the\n\\magnetic-monopole \feld\"\n\nm=\u00001\n2Sm\njSmj3=\u00001\n2Sm; (55)\nsincejSmj= 1 andJ >0. AsSm= (0;1;0) has been as-\nsumed to point in ydirection, we \fnd that only \n mx;mz =\n\u0000\nm;y= +1=2 remains nonzero (green data, right scale)\nforJ!1 . On the contrary, the perturbative approach,\nsee Eq. (27), yields \n m\u000b;m0\u000b0= \nmm0\u000e\u000b\u000b0= 0 for the\non-site elements m=m0due to the antisymmetry of the\ntensor.\nIn the intermediate- Jregime, see J\u001910 in Fig. 13,\nthere is still a \fnite o\u000b-site element (blue, left scale).\nThe on-site element \n mx;mz =\u0000\nm;y(green, right scale)\nis still far from its asymptotic value for J!1 , while\n\nmy;mz = \nm;xhas a \fnite negative value due to the\nproximity ofSm0pointing into\u0000xdirection. Finally, for\nthe chosen classical spin con\fguration, \n mx;my = \nm;z=\n0 in the entire Jrange.\nThe data in the intermediate-coupling regime ( J\u0018\nO(10)) demonstrate that the elements of the spin-Berry\ncurvature tensor may well assume values of the order of\none. They appear to be limited, however, by their J!1\nvaluesj\nm;\u000bj \u0014 1=2. Nevertheless, one would have a\nstrongly anomalous spin dynamics with it.\nG. Results for a ribbon geometry\nWe have seen that the spin-Berry curvature stays \fnite\nat a topological phase transition, since the gap merely\ncloses at a single critical point, KorK0in the two-\ndimensional Brillouin zone. This is a too weak singu-\nlarity to have a signi\fcant impact on the k;k0sums in\nEq. (46). For the model on a one-dimensional lattice,\nhowever, this is qualitatively di\u000berent.\nHere, we consider the Haldane model on a one-\ndimensional ribbon with a large number of unit cells Lx\nand periodic boundary conditions in the xdirection, and\nwith a \fnite number of unit cells Lyand open bound-\nary conditions in the ydirection, so that the ribbon is\nbounded by zigzag edges in the ydirection. Equation\n(46) applies accordingly for one-dimensional wave vec-\ntorskx;k0\nxand forI;I0= 1;:::;Ly.\nIn a topologically nontrivial phase with C=\u00061\nand disregarding the trivial spin degeneracy, the bulk-\n0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00\nkx/\n3\n2\n1\n0123(kx)\nFIG. 14. Band structure of a Haldane ribbon with zigzag\nedges. Calculations for a ribbon of \fnite thickness with Ly=\n50 unit cells in ydirection. Periodic boundary conditiosn are\nassumed along the xdirection. Parameters: \u0018=\u0019=2,\u001c2= 0:1\nandM= 0.\nboundary correspondence principle requires the existence\nof two gapless chiral eigenstates of ^Hquexponentially lo-\ncalized at the opposite edges. For \fnite but large Ly,\nthe band dispersions \"\u0017(kx) for\u0017= 1;:::;2Lyform two\nquasi-continua of bulk states in the one-dimensional Bril-\nlouin zone separated by the bulk band gap \u0001 E. Within\nthe bulk band gap, the edge-state dispersions \"1(kx) and\n\"2(kx) take the form of an avoided crossing at low ener-\ngies. With \u0014\u0011\u0001kxwe have\n\"1;2(\u0014)/\u0006p\n\u00142+m2; (56)\nwith a gap/2mthat is exponentially small in Lyfor\nlargeLy. ForLy!1 the energy spectrum is gapless\n(m= 0), and at low excitation energies the edge-state\ndispersions are linear with positive and negative slope,\nrespectively:\n\"1(\u0014)/\u0014; \" 2(\u0014)/\u0000\u0014: (57)\nAs an example, Fig. 14 shows the ribbon band structure\nfor model parameters, where the spectrum is particle-hole\nsymmetric. Fixing the Fermi energy at zero, \"F= 0, we\nhave a half-\flled system.\nAddressing the weak- Jlimit, we use Eq. (46) to com-\npute the contribution Iribto the spin-Berry curvature,\nfor sitesim;im0at one of the edges and for Lx;Ly!1 ,\ndue to relative wave vectors \u0014in a su\u000eciently small range\naround\u0014= 0. This is given by lim \"!0Irib(\") for positive\n\", where\nIrib(\") =Z\u0003\n\"d\u0014Z\u0003\n\"d\u00140 1\n(\u0014+\u00140)2: (58)\nIt is straightforward to see that the integral diverges for\n\"!0 asIrib(\")\u0018\u0000ln\", contrary to the two-dimensional\ncase, see Eq. (54). We conclude that the spin-Berry cur-\nvature is weakly (logarithmically) divergent in the topo-\nlogically non-trivial phase due to the presence of edge\nmodes.16\n101102103104\nLx0.450.500.550.600.650.700.75/J2\nLy:5\nLy:10\nLy:15\nLy:30\nLy:45\nFIG. 15. \n as function of nano-ribbon length Lx(with\nperiodic boundary conditions in xdirection) for various rib-\nbon widths Lyas indicated. Note the logarithmic scale for\nLx. The classical impurity spins are located at next-nearest-\nneighbor sites at the same zigzag edge. Calculations for the\nparticle-hole symmetric case with M= 0 and with Fermi en-\nergy\"F= 0. Further parameters: \u001c2= 0:1,\u0018= 0:1\u0019.\nThere are two sources for a \fnite gap regularizing the\nspin-Berry curvature. First, for a ribbon with a large but\n\fniteLx<1, thekx-space discretization, \u000ekx= 2\u0019=Lx,\nwill regularize the divergence, such that \n \u0018lnLx. Sec-\nond, forLx!1 , but for a \fnite ribbon width Ly, a\ngap/m < \u0001Eis produced by overlapping edge wave\nfunctions, see Eq. (56). In this case, \n \u0018\u0000lnm.\nFig. 15 displays the nonzero element \n of the spin-\nBerry curvature for two next-nearest-neighbor sites at\nthe same edge as function of Lxand for various \fnite\nLy. For \fxed Lxand with increasing Ly, the overlap\nbetween the edge states localized at di\u000berent edges and\nthus the gap parameter mdecrease exponentially fast\nwithLy, such that one expects that the gap due to k-\nspace discretization \u000ekx= 2\u0019=Lxquickly becomes the\ndominating factor. In fact, for any Lx, the spin-Berry\ncurvature \n as a function of Lyquickly converges to a\n\fnite value. On the scale of the plot, there is no visible\ndi\u000berence between the results for Ly= 30 and Ly=\n45 (see also the red and violet data in the inset). For\nthe largest Ly, one notes that \n still increases with Lx\nat values of the order of Lx=O(104), see inset. The\ndependence of \n on log Lxis close to linear, as expected,\nbut with a small slope.\nImportantly, the absolute values of \n =J2in the wide\nLx-Lyrange considered are of the order of unity. Accord-\ning to Eq. (37) and if Jis of the order of unity as well,\nthis is just the range, where the strongest e\u000bects of the\ngeometrical spin torque can be expected. We conclude\nthat, compared with the situation of the two-dimensional\nbulk system close to a topological phase transition, the\nphase-space reduction in the one-dimensional case is very\nfavorable for a large spin-Berry curvature.IV. SUMMARY AND OUTLOOK\nThe typical time scale of the dynamics of local mag-\nnetic moments, exchange coupled to an electron system,\nis controlled by the strength of the exchange coupling J.\nSince this is usually one or even several orders of mag-\nnitudes smaller than the characteristic energy scales of\nthe electron system, the local-moment dynamics is slow\nas compared to the characteristic electronic time scale.\nIn particular, if J\u001c\u0001E, where \u0001Eis size of the gap\nof an insulating electron system, the adiabatic theorem\napplies, and the ground state of the electron system at\nan instant of time tis well approximated by the instanta-\nneous ground state corresponding to the con\fguration of\nthe local moments at time t. Modelling the magnetic mo-\nments as classical spins of \fxed length, the spin dynamics\nis described by an e\u000bective low-energy classical theory,\nwhich is much simpler than the full coupled semiclassi-\ncal electron-spin dynamics, since it involves ground-state\nquantities of the electron system only.\nWe have developed this e\u000bective spin-only theory\nwithin a Lagrange formalism. Besides the torque result-\ning from well-known indirect magnetic exchange, there\nis is an additional spin torque that is given in terms of\nthe spin-Berry curvature. This geometrical spin torque\nis analogous to the geometrical force discussed in molec-\nular dynamics and is nonzero in case of electron systems\nwith broken time-reversal symmetry. As we have demon-\nstrated explicitly for the simple case of two classical spins,\nthe emergent dynamics is highly unconventional and dif-\nfers from the dynamics of any spin-only Hamiltonian. Fu-\nture studies may address this non-Hamiltonian spin dy-\nnamics for systems like the semi-classical Kondo- lattice\nmodel with a large number of spins.\nFor our present study, which aims at an in-principle\ndemonstration, we have considered the (spinful) Haldane\nmodel as a prototypical system, where time-reversal sym-\nmetry is broken explicitly. This choice has the additional\nbene\ft that the spin-Berry curvature can be compared\nwithk-space Berry curvature, which plays a central role\nin topological band theory. There is in fact a (weak) re-\nlation between both, as their Lehmann-type representa-\ntions involve exactly the same energy eigenstates, albeit\ndi\u000berent operators must be considered in the matrix el-\nements. This explains the sensitivity to the topological\nphase that has been observed in the study of the spatial\nstructure of the spin-Berry curvature tensor, particularly\nat the zigzag edges of the lattice. For forthcoming work,\nit will be interesting to consider other condensed-matter\nsystems with broken time reversal symmetry, such as sys-\ntems with spontaneous magnetic order.\nIt has been shown that the spin-Berry curvature ten-\nsor, in the weak- Jlimit, is closely related, namely by the\nfrequency derivative at != 0, to the magnetic suscep-\ntibility. This insight opens the possibility to make close\ncontact to well-known properties of the indirect magnetic\nexchange, i.e., to RKKY theory and to its pendant for in-\nsulators and semimetals, e.g., as concerns their distance17\ndependence.\nWe have systematically studied the magnitude of the\nspin-Berry curvature. As the example for two classical\nspins has shown, its e\u000bects, namely an overall renormal-\nization of the indirect magnetic exchange and an addi-\ntional coupling between the spins, are most pronounced,\nif this is of the order of unity. Apart from the strong- J\nregime, however, this is not easily achieved in case of the\nHaldane model. The main reason is that virtual second-\norder-in-Jprocesses are exponentially damped by the \f-\nnite gap size \u0001 E, similar to the exponential \u0001 Edepen-\ndence of the indirect magnetic exchange. This suggests\nthat strong e\u000bects should be expected for systems in close\nparametrical distance to a topological transition with a\nband closure. However, we could argue that the spin-\nBerry curvature is \fnite and continuous at a topological\ntransition of the in\fnite bulk system. On the contrary,\nan arbitrarily large curvature can be observed for \fnite\nsystems (with periodic boundaries). This underpins theview that a phase-space mechanism is at work: A large\nspin-Berry curvature of a condensed-matter system in\nthe thermodynamical limit requires parametric vicinity\nto a band closure on a D\u00001-dimensional submanifold\nof theD-dimensional Brillouin zone. In fact, we could\nobserve a logarithmically diverging spin-Berry curvature\nfor the Haldane model at the D= 1 zigzag edge, where a\n(D= 0 dimensional) gap closure is enforced via the bulk-\nboundary correspondence. This insight will be useful for\nforthcoming studies.\nACKNOWLEDGMENTS\nThis work was supported by the Deutsche Forschungs-\ngemeinschaft (DFG, German Research Foundation)\nthrough the SFB 925 (project B5), project ID 170620586,\nand through the research unit QUAST, FOR 5249\n(project P8), project ID 449872909.\n[1] A. Messiah, Quantum mechanics , Vol. II (Elsevier, Am-\nsterdam, 1961).\n[2] S. Jansen, M.-B. Ruskai, and R. Seiler, J. Math. Phys.\n48, 102111 (2007).\n[3] M. V. Berry, Proc. R. Soc. London A 392, 45 (1984).\n[4] B. Simon, Phys. Rev. Lett. 51, 2167 (1983).\n[5] F. Wilczek and A. Zee, Phys. Rev. Lett. 52, 2111 (1984).\n[6] C. A. Mead, Rev. Mod. Phys. 64, 51 (1992).\n[7] R. Resta, J. Phys.: Condens. Matter 12, R107 (2000).\n[8] H. Kuratsuji and S. Iida, Prog. Theor. Phys. 74, 439\n(1985).\n[9] J. Moody, A. Shapere, and F. Wilczek, Phys. Rev. Lett.\n56, 893 (1986).\n[10] B. Zygelman, Phys. Lett. A 125, 476 (1987).\n[11] A. Bohm, A. Mostafazadeh, H. Koizumi, Q. Niu, and\nJ. Zwanziger, The Geometric Phase in Quantum Systems\n(Springer, Berlin, 2003).\n[12] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045\n(2010).\n[13] X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057\n(2011).\n[14] C.-K. Chiu, J. C. Y. Teo, A. P. Schnyder, and S. Ryu,\nRev. Mod. Phys. 88, 035005 (2016).\n[15] G. Tatara, Physica E 106, 208 (2019).\n[16] C. Stahl and M. Pottho\u000b, Phys. Rev. Lett. 119, 227203\n(2017).\n[17] Q. Niu and L. Kleinman, Phys. Rev. Lett. 80, 2205\n(1998).\n[18] Q. Niu, X. Wang, L. Kleinman, W. Liu, D. M. C. Nichol-\nson, and G. M. Stocks, Phys. Rev. Lett. 83, 207 (1999).\n[19] U. Bajpai and B. K. Nikolic, Phys. Rev. Lett. 125,\n187202 (2020).\n[20] M. Head-Gordon and J. C. Tully, J. Chem. Phys. 103,\n10137 (1995).\n[21] M. Berry and J. Robbins, Proc. R. Soc. London A 442,\n659 (1993).\n[22] M. Onoda and N. Nagaosa, Phys. Rev. Lett. 96, 066603\n(2006).[23] S. Bhattacharjee, L. Nordstr om, and J. Fransson, Phys.\nRev. Lett. 108, 057204 (2012).\n[24] M. Sayad and M. Pottho\u000b, New J. Phys. 17, 113058\n(2015).\n[25] M. Sayad, R. Rausch, and M. Pottho\u000b, Phys. Rev. Lett.\n117, 127201 (2016).\n[26] U. Bajpai and B. K. Nikolic, Phys. Rev. B 99, 134409\n(2019).\n[27] M. Campisi, S. Denisov, and P. H anggi, Phys. Rev. A\n86, 032114 (2012).\n[28] N. Lenzing, A. I. Lichtenstein, and M. Pottho\u000b, Phys.\nRev. B 106, 094433 (2022).\n[29] M. Elbracht, S. Michel, and M. Pottho\u000b, Phys. Rev.\nLett. 124, 197202 (2020).\n[30] S. Michel and M. Pottho\u000b, Phys. Rev. B 103, 024449\n(2021).\n[31] J. H. Hannay, J. Phys. A 18, 221 (1985).\n[32] F. D. M. Haldane, Phys. Rev. Lett. 61, 2015 (1988).\n[33] B. A. Bernevig, Topological insulators and topological\nsuperconductors (Princeton University Press, Princeton,\n2013).\n[34] D. J. Thouless, M. Kohmoto, M. P. Nightingale, and\nM. den Nijs, Phys. Rev. Lett. 49, 405 (1982).\n[35] G. Jotzu, M. Messer, R. Desbuquois, M. Lebrat,\nT. Uehlinger, D. Greif, and T. Esslinger, Nature (Lon-\ndon)515, 237 (2014).\n[36] L. A. Wray, S.-Y. Xu, Y. Xia, D. Hsieh, A. V. Fedorov,\nY. S. Hor, R. J. Cava, A. Bansil, H. Lin, and M. Z.\nHasan, Nat. Physics 7, 32 (2011).\n[37] J. Honolka, A. A. Khajetoorians, V. Sessi, T. O. Wehling,\nS. Stepanow, J.-L. Mi, B. B. Iversen, T. Schlenk,\nJ. Wiebe, N. B. Brookes, A. I. Lichtenstein, P. Hofmann,\nK. Kern, and R. Wiesendanger, Phys. Rev. Lett. 108,\n256811 (2012).\n[38] M. R. Scholz, J. S\u0013 anchez-Barriga, D. Marchenko,\nA. Varykhalov, A. Volykhov, L. V. Yashina, and\nO. Rader, Phys. Rev. Lett. 108, 256810 (2012).\n[39] T. Valla, Z.-H. Pan, D. Gardner, Y. S. Lee, and S. Chu,\nPhys. Rev. Lett. 108, 117601 (2012).18\n[40] F. Goth, D. J. Luitz, and F. F. Assaad, Phys. Rev. B\n88, 075110 (2013).\n[41] T. Schlenk, M. Bianchi, M. Koleini, A. Eich, O. Piet-\nzsch, T. O. Wehling, T. Frauenheim, A. Balatsky, J.-\nL. Mi, B. B. Iversen, J. Wiebe, A. A. Khajetoorians,\nP. Hofmann, and R. Wiesendanger, Phys. Rev. Lett.\n110, 126804 (2013).\n[42] T. Eelbo, M. Wa\u0013 sniowska, M. Sikora, M. Dobrza\u0013 nski,\nA. Koz lowski, A. Pulkin, G. Aut\u0012 es, I. Miotkowski, O. V.\nYazyev, and R. Wiesendanger, Phys. Rev. B 89, 104424\n(2014).\n[43] Y. Li, X. Zou, J. Li, and G. Zhou, J. Chem. Phys. 140,\n124704 (2014).\n[44] Y. Jiang, C. Song, Z. Li, M. Chen, R. L. Greene, K. He,\nL. Wang, X. Chen, X. Ma, and Q.-K. Xue, Phys. Rev.\nB92, 195418 (2015).\n[45] C.-C. Chen, M. L. Teague, L. He, X. Kou, M. Lang,\nW. Fan, N. Woodward, K.-L. Wang, and N.-C. Yeh,\nNew J. Phys. 17, 113042 (2015).\n[46] A. Pieper and H. Fehske, Phys. Rev. B 93, 035123 (2016).\n[47] P. R u\u0019mann, S. K. Mahatha, P. Sessi, M. A. Valbuena,\nT. Bathon, K. Fauth, S. Godey, A. Mugarza, K. A.\nKokh, O. E. Tereshchenko, P. Gargiani, M. Valvidares,\nE. Jimenez, N. B. Brookes, M. Bode, G. Bihlmayer,\nS. Bl ugel, P. Mavropoulos, C. Carbone, and A. Barla, J.\nPhys. Mater. 1, 015002 (2018).\n[48] K. Sumida, M. Kakoki, J. Reimann, M. Nurmamat,\nS. Goto, Y. Takeda, Y. Saitoh, K. A. Kokh, O. E.\nTereshchenko, J. G udde, U. H ofer, and A. Kimura, NewJ. Phys. 21, 093006 (2019).\n[49] Q. Liu, C.-X. Liu, C. Xu, X.-L. Qi, and S.-C. Zhang,\nPhys. Rev. Lett. 102, 156603 (2009).\n[50] J. Gao, W. Chen, X. C. Xie, and F.-c. Zhang, Phys. Rev.\nB80, 241302 (2009).\n[51] V. D. Kurilovich, P. D. Kurilovich, and I. S. Burmistrov,\nPhys. Rev. B 95, 115430 (2017).\n[52] M. V. Hosseini, Z. Karimi, and J. Davoodi, J. Phys.:\nCondens. Matter 33, 085801 (2020).\n[53] O. M. Yevtushenko and V. I. Yudson, Phys. Rev. Lett.\n120, 147201 (2018).\n[54] J. Ihm, Phys. Rev. Lett. 67, 251 (1991).\n[55] Y. Ren, Z. Qiao, and Q. Niu, Rep. Prog. Phys. 79,\n066501 (2016).\n[56] Y. Mokrousov, \\Anomalous Hall E\u000bect,\" in Topology in\nMagnetism (Spinger, 2018) Chap. 6, p. 177.\n[57] N. Bloembergen and T. J. Rowland, Phys. Rev. 97, 1679\n(1955).\n[58] C. Kittel, in Solid State Physics, edited by F. Seitz,\nD. Turnbull, and H. Ehrenreich (Academic, New York,\n1968), Vol. 22, p. 1.\n[59] M. A. Ruderman and C. Kittel, Phys. Rev. 96, 99 (1954);\nT. Kasuya, Prog. Theor. Phys. 16, 45 (1956); K. Yosida,\nPhys. Rev. 106, 893 (1957).\n[60] P. A. M. Dirac, Proc. R. Soc. London A 133, 60 (1931).\n[61] A. Altland and M. R. Zirnbauer, Phys. Rev. B 55, 1142\n(1997).\n[62] M. Nakahara, Geometry, topology, and physics (Inst. of\nPhysics Publ., Bristol, 1998)."    },    {        "title": "1809.09090v1.Quantum_spin_transfer_torque_induced_nonclassical_magnetization_dynamics_and_electron_magnetization_entanglement.pdf",        "content": "Quantum spin-transfer torque induced nonclassical magnetization dynamics and\nelectron-magnetization entanglement\nPriyanka Mondal,1Marko D. Petrovi\u0013 c,2Petr Plech\u0013 a\u0014 c,2and Branislav K. Nikoli\u0013 c1,\u0003\n1Department of Physics and Astronomy, University of Delaware, Newark, DE 19716, USA\n2Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA\nThe standard spin-transfer torque (STT)|where spin-polarized current drives dynamics of mag-\nnetization viewed as a classical vector|requires noncollinearity between electron spins carried by\nthe current and magnetization of a ferromagnetic layer. However, recent experiments [A. Zholud\net al. , Phys. Rev. Lett. 119, 257201 (2017)] observing magnetization dynamics in spin valves at\ncryogenic temperatures, even when electron spin is collinear to magnetization, point at overlooked\nquantum e\u000bects in STT which can lead to highly nonclassical magnetization states. Using fully quan-\ntum many-body treatment, where an electron injected as spin-polarized wave packet interacts with\nlocal spins comprising the anisotropic quantum Heisenberg ferromagnetic chain, we de\fne quantum\nSTT as any time evolution of local spins due to initial many-body state not being an eigenstate of\nelectron+local-spins system . For time evolution caused by injected spin- #electron scattering o\u000b local\n\"-spins, entanglement between electron subsystem and local spins subsystem takes place leading to\ndecoherence and, therefore, shrinking of the total magnetization but without rotation from its initial\norientation which explains the experiments. Furthermore, the same processes|entanglement and\nthereby induced decoherence|are present also in standard noncollinear geometry, together with the\nusual magnetization rotation. This is because STT in quantum many-body picture is caused only\nby electron spin-#factor state, and the only di\u000berence between collinear and noncollinear geometries\nis in relative size of the contribution of the initial separable state containing such factor state to\nsuperpositions of separable many-body quantum states generated during time evolution.\nThe standard spin-transfer torque (STT) [1], predicted\nin the seminal work of Slonczewski [2] and Berger [3], is\na phenomenon where a \rux of spin-polarized electrons\ninjected into a ferromagnetic metal (FM) layer drives its\nmagnetization dynamics. The origin of STT is transfer\nof spin angular momentum from electrons to local mag-\nnetic moments of the FM layer, so it is fundamentally a\nnonequilibrium quantum many-body physics e\u000bect. Nev-\nertheless, local magnetic moments are typically treated\nasclassical vectors of \fxed length [1, 4] whose dynamics\nis governed by the Landau-Lifshitz-Gilbert (LLG) equa-\ntion [5] extended by adding the STT term [6{8]\nT/h^sei\u0002S(r): (1)\nThus, the nonequilibrium spin density h^seicaused by\n\rowing electrons must be noncollinear to the direction\nof local spin S(r) [i.e., to the local magnetization propor-\ntional to local spin], to drive magnetization dynamics in\nsuch a classical picture. The dynamics can include oscil-\nlations or complete reversal, whose conversion into resis-\ntance variations has emerged as a key resource for next\ngeneration spintronic technologies, such as nonvolatile\nmagnetic random access memories, microwave oscillators,\nmicrowave detectors, spin-wave emitters, memristors and\narti\fcial neural networks [9{11].\nFor example, passing current through a spin valve tri-\nlayer \fxed-FM/normal-metal/free-FM, as employed in\nearly experiments on standard STT [12, 13], causes \frst\nFM layer with \fxed magnetization to spin-polarize the\ncurrent which then impinges onto the second FM layer\nwith free magnetization that \ructuates in the classical\npicture due to a random magnetic \feld caused by thermalmotion. When impinging spins and \ructuating magne-\ntization become noncollinear, standard STT can either\namplify such \ructuations (for \fxed-to-free spin current\ndirection) or reduce them (for free-to-\fxed spin current\ndirection), as con\frmed theoretically [14] and experimen-\ntally [15] at room temperature.\nHowever, this well-established picture cannot explain\nvery recent experiments [16] on collinear spin valves at\ncryogenic temperatures .3 K, where resistance mea-\nsurements have revealed magnetization dynamics even\nthough thermal \ructuations that could introduce non-\ncollinearity between the free and \fxed magnetizations\nare suppressed. This implies a mechanism where stan-\ndard STT is zero, T\u00110 in Eq. (1), so that magneti-\nzation does not rotate from the the initial con\fguration.\nNevertheless, it changes its length, thereby signaling gen-\neration of highly nonclassical magnetization states [16].\nHowever, the proposed intuitive picture [16] where such\nmechanism would amplify quantum spin \ructuations, for\nboth \fxed-to-free and free-to-\fxed spin current direc-\ntions, cannot be rigorously justi\fed. That is, although\nquantum \ructuations of the local spin operators [17] (or,\nequivalently, zero-point energy of magnons as bosonic\nparticles to which spin operators can be mapped) play an\nimportant role in lowering the energy of classical ground\nstates of antiferromagnets [18] or noncollinear spin tex-\ntures [19], they vanish in a FM with uniaxial anisotropy\nbecause the collinear state of local magnetic moments is\nalso a ground eigenstate of the exact Hamiltonian [20].\nAside from few disparate attempts [21{23], a general\nframework for describing current-driven quantum dynam-\nicsof magnetization is lacking. Note that quantum trans-arXiv:1809.09090v1  [cond-mat.str-el]  24 Sep 20182\nJsd J, Jzor\nFIG. 1. Schematic view of a quantum many-body system,\nexhibiting quantum STT, which consists of a FM layer whose\nlocal spins comprise the XXZ quantum Heisenberg ferromag-\nnetic chain with anisotropic exchange interactions JandJz,\nand are attached to 1D TB chain where electron hops with\nparameter\r. The spin-polarized electron wave packet is in-\njected along the TB chain, with its spin pointing in the \u0000z- or\n+x-direction which is collinear (a case where standard STT\nof Slonczewski [2] and Berger [3] is absent) or noncollinear,\nrespectively, to local spins pointing initially along the + z-\ndirection. The spin of the wave packet interacts with local\nspins vias-dexchange interaction.\nport theories, such as the nonequilibrium Green func-\ntion formalism [7, 8, 24, 25] or the scattering matrix ap-\nproach [26, 27], are routinely used to compute h^seiin\nEq. (1) for a given single-particle Hamiltonian, but this\nserves only as an input [7, 8] for the LLG equation de-\nscribing classical dynamics of magnetization. The LLG\nequation can be justi\fed under the assumptions [5] of\nlarge spin S! 1 ,~!0 (whileS\u0002~!1) and\nin the absence of entanglement . The latter assump-\ntion means that local spins comprising the total mag-\nnetization should remain in a separable quantum state,\njS1i\njS2i\n\u0001\u0001\u0001\njSNi, as exempli\fed by the ground\nstate of FM,j\"i\nj\"i\n\u0001\u0001\u0001\nj\"i .\nInstead of classical micromagnetics [4, 14] or quantum-\nclassical [7, 8] description of standard-STT-induced mag-\nnetization dynamics, here we introduce a quantum many-\nbody picture of both\rowing-electron-spin{local-spins in-\nteractions and the ensuing time evolution of local spins\nat zero temperature. For this purpose, we employ a sys-\ntem depicted in Fig. 1 where spin-polarized electron wave\npacket, assumed to originate from a \fxed FM layer, is\ninjected along one-dimensional (1D) tight-binding (TB)\nchain whose sites in the middle host local spins compris-\ning a quantum Heisenberg ferromagnetic chain modeling\nthe free FM layer. The states of such composite quantum\nsystem electron+local-spins reside in the Hilbert space\nH=He\norb\nHe\nspin\nH1\nspin\u0001\u0001\u0001\nHN\nspin; (2)\nwhich is the tensor product of orbital electron subspace\nHe\norb(of \fnite dimension equal to the number of sites\nof the TB chain); two-dimensional subspace He\nspinfor\nelectron spin; and Hn\nspinas two-dimensional subspaces\nforn= 1;\u0001\u0001\u0001;Nlocal spins assumed to be spin-1\n2as\n−0.4−0.2 0.0 0.2 0.4 0.6 0.80246(a)\n−2−1 0 1 2\nEnergy Eigenvalue E n(eV)0510Degeneracy(b)(c)\nxzS1S2S3S4S5 se\n1\n2\n3\n4\n5\n6\n7FIG. 2. (a) Eigenspectrum of the XXZ quantum Heisenberg\nferromagnetic chain whose N= 5 local spins in Fig. 1 do not\ninteract with electron spin ( Jsd= 0). (b) Eigenspectrum of\nmany-body Hamiltonian in Eq. (3) whose local spins interact\nvias-dinteraction ( Jsd= 0:1 eV) with electron spin on TB\nsitei. (c) Expectation value of electron spin (\frst column)\nand local spins, extracted from their subsystem density ma-\ntrices via Eq. (6), in the degenerate ground state of the lowest\nenergy in panel (a) [red arrows] or (b) [blue arrows].\nwell. The system Hamiltonian acting in His\n^H=\u0000\rX\nhijijiihjj\u0000JsdX\nijiihij\n^se\u0001^Si(t)\n\u0000X\nhijih\nJ(^Sx\ni\u0001^Sx\nj+^Sy\ni\u0001^Sy\nj) +Jz^Sz\ni\u0001^Sz\nji\n;(3)\nwherejiiis electron orbital centered on site i;\n\r= 1 eV is hopping between nearest-neighbor sites; and\nJsd= 0:1 eV is the strength of s-dexchange interac-\ntion between electron and local spins. The exchange\ninteraction between the nearest neighbor local spins is\nJ= 0:1 eV andJz= 0:1005 eV, which are slightly dif-\nferent in order to include the uniaxial anisotropy. The\nthird term in Eq. (3) is standardly denoted as the XXZ\nquantum Heisenberg ferromagnetic chain [28, 29]. The\nspin operators in Eq. (3) are constructed as ^se=^I\n^\u001b\n^I\n\u0001\u0001\u0001\n ^Ifor electron spin; ^S1=^I\n^I\n^\u001b\n^I\n\u0001\u0001\u0001\n ^I\nfor \frst local spins and analogously for all other local\nspins, where ^\u001b= (^\u001bx;^\u001by;^\u001bz) is the vector of the Pauli\nmatrices and ^Iis the unit operator. The eigenspectrum\nof an isolated XXZ chain (i.e., of the third term in Eq. (3)\nalone) is shown in Fig. 2(a), while the eigenspectrum of\nthe whole many-body Hamiltonian in Eq. (3) is shown in\nFig. 2(b). The ground state in the former (latter) case\nhas degeneracy six (seven), as expected for coupled sys-\ntem of \fve (six) spin-1\n2[28].\nAtt= 0, the many-body quantum state is a separable\none\nhxj\t(t= 0)i=Ceikxx\u0000\u000ek2\nxx2=4\n\u001fe\n\u001f1\n\u0001\u0001\u0001\n\u001fN:(4)\nIts \frst factor is electron orbital quantum state in He\norb\nchosen as a Gaussian wave packet with momentum along\nthe +x-direction and centered on the left edge of TB3\nFIG. 3. Time dependence of the expectation value of spin\n(in units ~=2) obtained from spin-1\n2density matrix in Eq. (6)\nfor: (a) spin of injected electron wave packet in Fig. 1 which\natt= 0 points in the \u0000z-direction that is collinear and an-\ntiparallel to local spins pointing in the + z-direction; and (c)\n\frst local spin in Fig. 1 [time dependence of expectation value\nof local spins n=2{5 is nearly identical to (c)]. (b) Purity\nde\fned in Eq. (7) of the subsystem composed of electron de-\ngrees of freedom (orbital and spin) or of the subsystem com-\nposed of all local spins. (d) Probability in Eq. (8) to \fnd\nelectron-spin+local-spins subsystem in many-body quantum\nstatej\u001be;\u001b1\u001b2\u001b3\u001b4\u001b5i.\nchain, as illustrated in Fig. 1, where Cis the normal-\nization constant. To mimic current of electrons at the\nFermi level which interact with the ground state of free\nFM layer within a spin valve, we use kxa= 0:1 and\n\u000ekxa= 0:2 (ais the lattice spacing) which specify av-\nerage energy E=\u00002:36 eV and its standard deviation\n\u000eE= 0:054 eV for the wave packet to be close to the\nground state eigenenergy E0=\u00002:43 eV [Fig. 2(b)] of\nthe Hamiltonian in Eq. (3). In the ground state, all local\nspins are aligned with the anisotropy z-axis, as shown in\nFig. 2(c), so we choose \u001fn=\u00121\n0\u0013\nforn= 1;:::;N .\nTo mimic minority electrons in spin valve with collinear\nmagnetizations impinging on the free FM layer, we se-\nlect initial spin polarization of the wave packet in the\n\u0000z-direction, as described by the spinor \u001fe=\u0012\n0\n1\u0013\n.\nFor standard STT setup with noncollinear magnetiza-\ntions of the FM layers, we use spin polarization in the\n+x-direction,\u001fe=1p\n2\u00121\n1\u0013\n.\nAlthough many-body quantum system depicted in\nFig. 1 could be evolved via time-dependent density ma-\ntrix renormalization group [30] for large number of local\nspins\u0018100, for transparency of our discussion, operat-\ning with small number of excited states of the XXZ chain\nFIG. 4. Panels (a){(d) plot the same information as panels\n(a){(d), respectively, in Fig. 3 but for injected electron wave\npacket which at t= 0 is spin-polarized in the + x-direction,\ni.e.,noncollinear to local spins pointing in the + z-direction.\nwhich can be analyzed one by one, we employ N= 5 lo-\ncal spins. The chosen length Lx= 400 of the TB chain\nensures that wave packet does not re\rect from its bound-\naries within the time frame considered in Figs. 3 and 4.\nThe numerically exact j\t(t)i, governed by the Schrdinger\nequationi~@j\t(t)i=@t=^Hj\t(t)i, is obtained by using\nthe Crank-Nicolson algorithm [31].\nFigure 2(c) shows that degenerate ground state has\nelectron and local spins parallel to each other due to\ns-dinteraction between them acting to align them.\nThus, when an electron with spin- #along the\u0000z-\ndirection is injected, its spin is collinear to local spins\nbutj\t(t= 0)i=jGi\nj#e;\"1\u0001\u0001\u0001\"Ni(this form is used\nbelow for economy of notation) at t= 0 is not an eigen-\nstate of the Hamiltonian in Eq. (3). This causes time evo-\nlution of the electron-spin+local-spins subsystem, which\nrigorously de\fnes quantum STT even in situation where\nstandard STT in Eq. (1)is identically zero . In the course\nof time evolution, j\t(t)ibecomes an entangled state due\nto linear superpositions of separable states being gener-\nated fort>0. The entanglement entails that each sub-\nsystem must be described using the appropriate reduced\ndensity matrix [32]\n^\u001asub= Tr otherj\t(t)ih\t(t)j; (5)\nobtained via partial trace applied to the pure state\ndensity matrixj\t(t)ih\t(t)j. For example, tracing over\nthe states in the subspace He\norb\nHe\nspinH2\nspin\u0001\u0001\u0001\nHN\nspin\nyields the density matrix of \frst local spin\n^\u001a1(t) =1\n2h\n^I+S1(t)\u0001^\u001bi\n; (6)\nwhere S1(t) = Tr[^\u001a1(t)^\u001b] is the spin expectation value4\n(in units of ~=2), also denoted as the polarization (or\nBloch) vector [32]. Pure (or fully coherent) quantum\nstates of spin-1\n2are characterized by jS1j= 1, while\n0<jS1j<1 signi\fes their decoherence [32, 33] toward\nmixed (or partially coherent [34]) states. Figure 3(c)\nshows that \frst local spin has Sz\n1<1,Sx\n1=Sy\n1\u00110\nand, therefore,jS1j<1. The electron spin also exhibits\ndecoherence,jsej<1, in Fig. 3(a). Virtually the same\ntime-dependences as in Fig. 3(c) are obtained for other\nlocal spins i= 2;:::5, and, therefore, for total magne-\ntization as the sum of local spins. Thus, this could be\nprecisely the highly nonclassical state of magnetization\nconjectured from the measurement of the spin valve re-\nsistance [16], which increases /1\u0000Mzdue to magneti-\nzationMz=g\u0016BP\niSz\nishrinking without rotation (i.e.,\nMx=My= 0) away from its initial orientation.\nTo explain the origin of magnetization decoherence,\nor, equivalently, of the subsystem comprised of alllo-\ncal spins, we view multipartite [due to N+ 2 factors in\nEq. (2)] total system as a bipartite one, i.e., as being\ncomposed of the electron subsystem with states residing\ninHe\norb\nHe\nspinand the subsystem of local spins. The\npurity of the former is de\fned as [32, 33]\nPlocal\nspins(t) = Tr\b\n[^\u001alocal\nspins(t)]2\t\n; (7)\nwhere density matrix ^ \u001alocal\nspins(t) is obtained via Eq. (5)\nby tracing over the states in the subspace He\norb\nHe\nspin.\nThe decay ofPlocal\nspins(t) below one in Fig. 3(b) quan-\nti\fes \\true decoherence\" [33] of initially pure state\nj\"i\nj\"i\n\u0001\u0001\u0001\nj\"i as the decay [32, 33] of the o\u000b-\ndiagonal elements of ^ \u001alocal\nspins(t) caused by entanglement\nwith the electron subsystem. The purity of decohered\nelectron subsystem in Fig. 3(b) is identical to that of the\nlocal spin subsystem, as expected for entanglement in\nbipartite quantum systems [32, 33].\nTo understand the states of electron-spin+local-spins\nsubsystem which are excited during time evolution initi-\nated by injection of a single spin-polarized electron, we\ncompute the density matrix ^ \u001ae+local\nspins (t) of this subsystem\nobtained by partial trace in Eq. (5) performed over the\nstates inHe\norb. The probability to \fnd this subsystem in\nstatej\u001be;\u001b1:::\u001bNiat timet\nprobe+local\nspins (t) =h\u001be;\u001b1:::\u001bNj^\u001ae+local\nspins (t)j\u001be;\u001b1:::\u001bNi;\n(8)\nis shown in Fig. 3(d) for electron injected with spin along\nthe\u0000z-direction. The subspace of Hwhose states can\ngenerate nonzero probe+local\nspins (t) is restricted by energy\nbands in Fig. 2(b) [caused by anisotropy and bound-\naries [29]] and symmetries, such as that total spin in the\nz-direction has to be conserved in time evolution due to\noperator ^Sz\ntot= ^sz+^Sz\n1+:::^Sz\nNcommuting with the\nHamiltonian in Eq. (3), [ ^H;^Sz\ntot] = 0. Because of the\nlatter requirement, all states j\u001be;\u001b1:::\u001bNiparticipating\nin time evolution must have the same number of \"-spins,so that one \fnds in Fig. 3(d) progressive excitation of\nstates with \ripped spin of electron and one \ripped local\nspin with transfer of angular momentum of 1 \u0002~. How-\never, the initial state j#;\":::\"imaintains its probability\nclose to one, and other states with \ripped electron spin\nand one \ripped local spin have much smaller and nonuni-\nform probability. Such peculiar quantum superposition\nof separable many-body states, with large contribution\nfrom the initial state, leads to local spins maintaining\ntheir direction along the z-axis in Fig. 3(c). This can be\ncontrasted with na \u0010ve (i.e., not taking into account super-\npositions) intuition [22, 35] where spin- #electron simply\n\rips \frst local spin|the \rip then propagates to displace\ntransversally other local spins away from the anisotropy\naxis, eventually exciting white spectrum [22] of lowest\nspin waves modes [28] (where total spin is lower than\nthat of the ground state by 1 \u0002~) of the free FM layer.\nThe same e\u000bects|entanglement of electron state and\nstate of all local spins [Fig. 4(b)]; thereby induced de-\ncoherence of electron spin [Fig. 4(a)] and local spins\n[Fig. 4(c)]; and high probability [Fig. 4(d)] to \fnd ini-\ntial state of electron-spin+local-spins subsystem in the\ncourse of time evolution|is present also in standard\nSTT geometry with noncollinearity between spin of in-\njected electron and local spins. Furthermore, proba-\nbilities probe+local\nspins (t) in Fig. 4(d) to excite states of\ntypej\";\":::#:::\"iare simply half of those obtained\nfor collinear geometry in Fig. 3(d) since spin of in-\njected electron along the + x-direction used in Fig. 4\nmeansj!ei=1p\n2(j\"ei+j#ei) where only1p\n2j#eiterm,\nentering as a factor of the initial many-body state\njGi\n1p\n2(j\"ei+j#ei)\nj\" 1:::\"Ni, induces time evolu-\ntion of local spins and transfer of angular momentum.\nOn the other hand,1p\n2j\"ei\nj\" 1:::\"Niterm in the\ninitial many-body state is an eigenstate [Fig. 2(c)] of\nelectron-spin+local-spins subsystem and, therefore, has\nprobe+local\nspins (t) = 1=2 which does not evolve in time in\nFig. 4(d). Thus, identical pro\fle of curves in Figs. 3(d)\nand 4(d) reveal that in fully quantum many-body picture\nthere is no di\u000berence between standard STT and quantum\nSTT|both requirej#eifactor state in the initial many-\nbody state, where such factor is due to electron spin state\n(in the collinear case) or a term in the superposition of\nelectron spin states (in the noncollinear case).\nNote added. |During the completion of this work, we\nbecame aware of two studies [36, 37] where magnetization\ndynamics in collinear spin valves at cryogenic temper-\natures is attributed to current-enhanced quantum spin\n\ructuations. However, as discussed above, such \ruc-\ntuations are forbidden in the ground state of ferromag-\nnets [20], and if generated as random magnetic \feld [38]\nby the nonequilibrium spin shot noise [36], they would\nlead to rotation of magnetization away from the initial\ncollinear direction that was excluded from the experi-\nments of Ref. [16]. We propose to clarify the role of5\nspin shot noise by using a sequence of Lorentzian voltage\npulses to inject a train of levitons into a collinear spin\nvalve, where leviton as a minimal collective many-body\nexcitation above the Fermi sea carrying a single electron\ncharge, is free of particle-hole pairs and has vanishing\ncurrent noise [39, 40].\nWe are grateful to S. Urazhdin for instructive discus-\nsions. P. M. and B. K. N. were supported by NSF Grant\nNo. ECCS 150909. M. P. and P. P. were supported by\nARO MURI Award No. W911NF-14-0247.\n\u0003bnikolic@udel.edu\n[1] D. Ralph and M. Stiles, Spin transfer torques, J. Magn.\nMagn. Mater. 320, 1190 (2008).\n[2] J. C. Slonczewski, Current-driven excitation of magnetic\nmultilayers, J. Magn. Magn. Mater. 159, L1 1996.\n[3] L. Berger, Emission of spin waves by a magnetic mul-\ntilayer traversed by a current, Phys. Rev. B 54, 9353\n(1996).\n[4] D. V. Berkov and J. Miltat, Spin-torque driven magne-\ntization dynamics: Micromagnetic modeling, J. Magn.\nMagn. Mater. 320, 1238 (2008).\n[5] R. Wieser, Description of a dissipative quantum spin dy-\nnamics with a Landau-Lifshitz-Gilbert like damping and\ncomplete derivation of the classical Landau-Lifshitz equa-\ntion, Euro. Phys. J. B 88, 77 (2015).\n[6] A. Manchon, N. Ryzhanova, A. Vedyayev, M. Chschiev,\nand B. Dieny, Description of current-driven torques in\nmagnetic tunnel junctions, J. Phys.: Condens. Matter\n20, 145208 (2008).\n[7] M. D. Petrovi\u0013 c, B. S. Popescu, U. Bajpai, P. Plech\u0013 a\u0014 c,\nand B. K. Nikoli\u0013 c, Spin and charge pumping by steady\nor pulse current-driven magnetic domain wall: A self-\nconsistent multiscale time-dependent-quantum/time-\ndependent-classical approach, arXiv:1802.05682 (2018).\n[8] M. O. A. Ellis, M. Stamenova, and S. Sanvito, Multiscale\nmodeling of current-induced switching in magnetic tun-\nnel junctions using ab initio spin-transfer torques, Phys.\nRev. B 96, 224410 (2017).\n[9] N. Locatelli, V. Cros, and J. Grollier, Spin-torque build-\ning blocks, Nat. Mater. 13, 11 (2014).\n[10] A. D. Kent and D. C. Worledge, A new spin on magnetic\nmemories, Nat. Nanotech. 10, 187 (2015).\n[11] W. A. Borders, H. Akima, S. Fukami, S. Moriya, S. Kuri-\nhara, Y. Horio, S. Sato, and H. Ohno, Analogue spinorbit\ntorque device for arti\fcial-neural-network-based associa-\ntive memory operation, Appl. Phys. Expr. 10, 013007\n(2017).\n[12] M. Tsoi, A. G. M. Jansen, J. Bass, W. C. Chiang,\nV. Tsoi, and P. Wyder, Generation and detection of\nphase-coherent current-driven magnons in magnetic mul-\ntilayers, Nature 406, 46 (2000).\n[13] J. A. Katine, F. J. Albert, R. A. Buhrman, E. B. Myers,\nand D. C. Ralph, Current-driven magnetization rever-\nsal and spin-wave excitations in Co =Cu=Co pillars, Phys.\nRev. Lett. 84, 3149 (2000).\n[14] Z. Li and S. Zhang, Thermally assisted magnetization\nreversal in the presence of a spin-transfer torque, Phys.\nRev. B 69, 134416 (2004).[15] R. H. Koch, J. A. Katine, and J. Z. Sun, Time-resolved\nreversal of spin-transfer switching in a nanomagnet,\nPhys. Rev. Lett. 92, 088302 (2004).\n[16] A. Zholud, R. Freeman, R. Cao, A. Srivastava, and\nS. Urazhdin, Spin transfer due to quantum magnetiza-\ntion \ructuations, Phys. Rev. Lett. 119, 257201 (2017).\n[17] Y. Takahashi, Spin Fluctuation Theory of Itinerant Elec-\ntron Magnetism (Springer, Berlin, 2013).\n[18] A. Singh and Z. Te\u0014 sanovi\u0013 c, Quantum spin \ructuations\nin an itinerant antiferromagnet, Phys. Rev. B 41, 11457\n(1990).\n[19] A. Rold\u0013 an-Molina, M. J. Santander, A. S. N\u0013 u~ nez, and\nJ. Fern\u0013 andez-Rossier, Quantum \ructuations stabilize\nskyrmion textures, Phys. Rev. B 92, 245436 (2015).\n[20] K. Yosida, Theory of Magnetism (Springer, Berlin, 1996).\n[21] Y. Wang and L. J. Sham, Quantum dynamics of a nano-\nmagnet driven by spin-polarized current, Phys. Rev. B\n85, 092403 (2012); Y. Wang, and L. J. Sham, Quan-\ntum approach of mesoscopic magnet dynamics with spin\ntransfer torque, Phys. Rev. B 87, 174433 (2013); T. Tay\nand L. J. Sham, Theory of atomistic simulation of spin-\ntransfer torque in nanomagnets, Phys. Rev. B 87, 174407\n(2013).\n[22] J. P. Gauyacq and N. Lorente, Excitation of spin waves\nby tunneling electrons in ferromagnetic and antiferro-\nmagnetic spin-1\n2Heisenberg chains, Phys. Rev. B 83,\n035418 (2011); J. P. Gauyacq and N. and Lorente, Mag-\nnetic reversal of a quantum nanoferromagnet, Phys. Rev.\nB87, 195402 (2013).\n[23] F. Mahfouzi and N. Kioussis, Current-induced damping\nof nanosized quantum moments in the presence of spin-\norbit interaction, Phys. Rev. B 95, 184417 (2017).\n[24] P. M. Haney, D. Waldron, R. A. Duine, A. S. N\u0013 u~ nez,\nH. Guo, and A. H. MacDonald, Current-induced or-\nder parameter dynamics: Microscopic theory applied to\nCo/Cu/Co spin valves, Phys. Rev. B, 76, 024404 (2007).\n[25] B. K. Nikoli\u0013 c, K. Dolui, M. Petrovi, P. Plech\u0013 a\u0014 c,\nT. Markussen, and K. Stokbro, First-principles quan-\ntum transport modeling of spin-transfer and spin-\norbit torques in magnetic multilayers, arXiv:1801.05793\n(2018).\n[26] M. D. Stiles and A. Zangwill, Anatomy of spin-transfer\ntorque, Phys. Rev. B 66, 014407 (2002).\n[27] S. Wang, Y. Xu, and K. Xia, First-principles study of\nspin-transfer torques in layered systems with noncollinear\nmagnetization, Phys. Rev. B 77, 184430 (2008).\n[28] J. B. Parkinson and D. J. J. Farnell, An Introduction to\nQuantum Spin Systems (Springer, Berlin, 2010).\n[29] K. Joel, D. Kollmar, and L. F. Santos, An introduction\nto the spectrum, symmetries, and dynamics of spin-1 =2\nHeisenberg chains, Am. J. Phys. 81, 450 (2013).\n[30] U. Schollw ock, The density-matrix renormalization\ngroup, Rev. Mod. Phys. 77, 259 (2005).\n[31] J. E. Bay\feld, Quantum Evolution: An Introduction to\nTime-Dependent Quantum Mechanics (Wiley, New York,\n1999).\n[32] L. E. Ballentine, Quantum Mechanics: A Modern Devel-\nopment (World Scienti\fc, Singapore, 2014).\n[33] E. Joos, H. D. Zeh, C. Kiefer, D. J. W. Giulini, J. Kupsch,\nand I. Stamatescu, Decoherence and the Appearance of a\nClassical World in Quantum Theory (Springer, Berlin,\n2003).\n[34] B. K. Nikoli\u0013 c and S. Souma, Decoherence of transported\nspin in multichannel spin-orbit-coupled spintronic de-6\nvices: Scattering approach to spin-density matrix from\nthe ballistic to the localized regime, Phys. Rev. B 71,\n195328 (2005).\n[35] T. Balashov, A. F. Tak\u0013 acs, M. D ane, A. Ernst, P. Bruno,\nand W. Wulfhekel, Inelastic electron-magnon interaction\nand spin transfer torque, Phys. Rev. B 78, 174404 (2008).\n[36] A. Qaiumzadeh and A. Brataas, Quantum population of\nmagnons via spin shot noise, arXiv:1808.02907 (2018).\n[37] S. A. Bender, R. A. Duine, and Y. Tserkovnyak, Quan-\ntum spin-transfer torque and magnon-assisted transport\nin nanostructures, arXiv:1808.00777 (2018).[38] A. L. Chudnovskiy, J. Swiebodzinski, and A. Kamenev,\nSpin-torque shot noise in magnetic tunnel junctions,\nPhys. Rev. Lett. 101, 066601 (2008).\n[39] J. Keeling, I. Klich, and L. S. Levitov, Minimal excitation\nstates of electrons in one-dimensional wires, Phys. Rev.\nLett. 97, 116403 (2006).\n[40] J. Dubois, T. Jullien, F. Portier, P. Roche, A. Cavanna,\nY. Jin, W. Wegscheider, P. Roulleau, and D. C. Glattli,\nMinimal-excitation states for electron quantum optics us-\ning levitons, Nature 502, 659 (2013)."    },    {        "title": "1812.08403v3.Continuous_dynamical_decoupling_of_spin_chains__modulating_the_spin_environment_and_spin_spin_interactions.pdf",        "content": "arXiv:1812.08403v3  [quant-ph]  28 Aug 2019Continuous dynamical decoupling of spin chains: modulatin g the spin-environment\nand spin-spin interactions\nSharoon Austin,1Muhammad Qasim Khan,1Maryam Mudassar,1and Adam Zaman Chaudhry1,∗\n1School of Science & Engineering, Lahore University of Manag ement\nSciences (LUMS), Opposite Sector U, D.H.A, Lahore 54792, Pa kistan\nFor spins chains to be useful for quantum information proces sing tasks, the interaction between\nthe spin chain and its environment generally needs to be supp ressed. In this paper, we propose\nthe use of strong static and oscillating control fields in ord er to effectively remove the spin chain-\nenvironment interaction. We find that our control fields can a lso effectively transform the spin\nchain Hamiltonian. In particular, interaction terms which are absent in the original spin chain\nHamiltonian appear in the time-averaged effective Hamilton ian once the control fields are applied,\nimplying that spin-spin interactions can be engineered via the application of static and oscillating\ncontrol fields. This transformation of the spin chain can the n potentially be used to improve the\nperformance of the spin chain for quantum information proce ssing tasks. For example, our control\nfields can be used to achieve almost perfect quantum state tra nsfer across a spin chain even in the\npresence of noise. As another example, we show how the use of p articular static and oscillating\ncontrol fields not only suppresses the effect of the environme nt, but can also improve the generation\nof two-spin entanglement in the spin chain.\nPACS numbers: 03.65.Yz, 75.10.Pq, 03.67.Pp, 42.50.Dv\nI. INTRODUCTION\nSpin chains have been a subject of constant study for\nmany years now in diverse areas. For example, spin\nchains have been used to study phase transitions [1, 2],\nquantum chaos [3], high-temperature superconductivity\n[4], and Anderson localization [5]. On the experimental\nfront, the physical realization of spin chains ranges from\ntrapped ions [6] to optical lattices [7], solid state setups\n[8], and photonic systems [9]. In the context of quantum\ninformation and computation, spin chains have been, for\nexample, extensively studied to achieve perfect quantum\nstate transfer from one site to another [10–12], and to\ngenerate and distribute entanglement [13–17]. However,\none of the major hurdles towards the use of spin chains\nin such quantum information tasks is the inevitable cou-\npling of the spin chain to its environment [18, 19], which\nresults in the rapid decoherence of the fragile, gener-\nally many-body entangled, quantum spin chain state. As\nsuch, it is worthwhile studying ways in which the quan-\ntum spin chain can be effectively protected from its en-\nvironment.\nOne promising method of protecting the quantum spin\nchain is to use dynamical decoupling [20–30]. In dynam-\nical decoupling, control fields are applied rapidly on the\nquantum system that needs to be protected. The usual\napproach is to consider different pulse sequences applied\nto the system [20, 22, 28, 31, 32] which effectively modu-\nlate the system-environment interaction, thereby greatly\nextending the decoherence timescale. However, one can\nenvisage applying instead strong static and oscillating\ncontrol fields to dynamically decouple the spin chain, as\n∗adam.zaman@lums.edu.pkhas been done for a single qubit [33, 34], two qubits [35–\n37], and an effective large spin system [38]. This scheme\nhas the advantage that one need not worry about the\ntiming of the different fields; one simply turns on the\nrequired fields to achieve effective decoupling of the sys-\ntem from its environment. However, at the same time,\nthe spins in the spin chain are also interacting, and, as a\nresult of the control fields applied, this interaction is also\nmodulated. Consequently, the interactions between the\nspins are also changed due to the static and oscillating\ncontrol fields. Instead of considering the change of the\nspin-spin interactions as a nuisance, we can think about\nusingthischangetoouradvantage. Thatis, canweapply\nsimple static and oscillating fields to the spin chain such\nthat not only is the spin chain effectively decoupled from\nits environment (at least to lowest order), but the spin\nchain Hamiltonian is also changed in such a way that,\nfor example, state transfer fidelity improves or the per-\nformance of the spin chain in generating entanglement\nincreases? This is the question that we intend to answer\nin this paper. We note that control fields, in the form of\npulse sequences, have been used to engineer spin chain\nHamiltonians[39–42]; however,thesepulsesequencescan\nbe rather complicated. Our static and oscillating control\nfields can be used in conjunction with schemes based on\npulses, thereby realizing hybrid Hamiltonian engineering\ntechniques.\nWe start by considering the Hamiltonian of a general\none-dimensional spin chain which is an anisotropic ver-\nsion of the usual XYZ Hamiltonian [43]. We assume that\neach spin in this spin chain is coupled ‘locally’ to its en-\nvironment [35, 36]. In such a situation, we first find suit-\nable static and oscillating control fields that, when ap-\nplied to the spin chain, are able to dynamically decouple\nthe spin chain, at least to lowest order. The nice fea-\nture of these control fields is that the same field needs to2\nbe applied to each spin. We then proceed to investigate\nhow the spin chain interactions are modulated by these\ncontinuous dynamical decoupling control fields. We find\nthat the spin-spin interactions fundamentally change de-\npending on the control fields, and the effective spin chain\nHamiltonian contains interactions that are not present in\nthe original spin chain Hamiltonian. Interestingly, for a\nspecial set of control fields, even more additional interac-\ntion terms, similar to those in the Dzaolyshinskii-Moriya\ninteraction [44–48], can be generated. Our aim then is\nto analyze the spin chain with these control fields. We\nfirst look at the possibility of achieving perfect quantum\nstate transfer by removing the effect of the environment\nand, atthesametime, suitablyengineeringthespinchain\nHamiltonian. We then investigate the entanglement gen-\nerated between two spins of the spin chain via the spin\nchain interactions. To this end, we present numerical\nsimulations that first showthat the controlfields are able\nto effectively dynamically decouple the spin chain. Sec-\nond, the simulations show that the dynamics of the spin\nchain in the presenceof the controlfields is captured very\nwell by the effective time-averaged Hamiltonian which,\nin general, contains additional interaction terms. Third,\nwe show that for special control fields, the generation of\nentanglement can be enhanced even more due to the ad-\nditional interaction terms. After these numerical simula-\ntions, we subsequently endeavor to analytically solve the\ndynamicsofthetime-averagedeffectivespinchainHamil-\ntonian. We show that that if we impose a condition on\nthe coupling coefficients in the spin chain, we can trans-\nform ourproblem to asystem ofnon-interactingfermions\nviatheJordan-Wignertransformation[43]. Withthisap-\nproach, we are able to significantly reduce the computa-\ntional complexity of the problem. We then demonstrate\nthat our special control fields are able to enhance the\nentanglement generation, even for larger spin chains.\nThis paper is organized as follows. In Sec. II, we\npresent the static and oscillating control fields we use\nto dynamically decouple the spin chain from its envi-\nronment, and derive the effective spin chain Hamilto-\nnian in the presence of these control fields. The use of\nthese control fields towards obtaining perfect quantum\nstate transfer is investigated in Sec. III. The performance\nof the control fields in entanglement generation is then\nnumerically analyzed in Sec. IV. In Sec. V, we demon-\nstrate results for entanglement generation with relatively\nlarger spin chains, obtained after diagonalizing the ef-\nfective Hamiltonian via the Jordan-Wigner transforma-\ntion. Finally, we conclude in Sec. VI. This is followed by\na series of Appendices. In Appendix A, we present the\ntheory behind our dynamical decoupling method and the\neffective Hamiltonian approach. In Appendix B, we show\nhow effective transverse fields can be included, at least\nin principle, in the time-averaged Hamiltonian by adding\nmore control fields. Our method of simulating the ef-\nfect of noise via Ornstein-Uhlenbeck processesis outlined\nin Appendix C, while Appendix D shows how a single\nspin operation, such as a spin flip, can be executed, atleastinprinciple, withextremelyhighfidelityviasuitable\ncontinuous control fields. Details of the Jordan-Wigner\ntransformation are presented in Appendix E. Finally, in\nAppendix F, we investigate the degree of fine tuning re-\nquired in the special control fields in order to generate\nsignificant amounts of entanglement.\nII. THE FORMALISM\nWe start by considering the usual XYZ Hamiltonian\nwhich describes a one-dimensional spin chain. Consid-\nering only nearest-neighbor coupling, the Hamiltonian,\nwithzeromagneticfield, canbe writtenas(wetake¯ h= 1\nthroughout)\nH0=N−1/summationdisplay\nj=13/summationdisplay\nk=1λjkσ(j)\nkσ(j+1)\nk. (1)\nHereλjkare the coupling strengths between the spins,\njlabels the sites, and k= 1,2,3 denotesx,yandz\nrespectively. As usual, [ σ(p)\nl,σ(q)\nm] = 2iδpqεlmnσ(p)\nn, and\nnote that we are not using cyclic boundary conditions.\nWe want to dynamically decouple the spin chain from\nits environment. To this end, we first need to model the\nspinchain-environmentinteraction. Weassumethateach\nspin interacts ‘locally’ with the environment so that the\ninteraction between the spin chain and its environment\nis given by\nHSB=N−1/summationdisplay\nj=1B(j)\nxσ(j)\nx+B(j)\nyσ(j)\ny+B(j)\nzσ(j)\nz.(2)\nHereB(j)\nkare arbitrary environment operators (or ran-\ndomly fluctuating noise terms for a classical bath). Our\nbasic strategy is to apply periodic control fields to the\nspin chain to modulate the interaction between the spin\nchain and its environment in such a way that the spin\nchain becomes effectively decoupled from its environ-\nment, at least to lowest order. Corresponding to these\ncontinuouscontrolfields, thereisaunitaryoperator Uc(t)\nsuchthati∂Uc(t)\n∂t=Hc(t)Uc(t), whereHc(t) isthe Hamil-\ntonian describing the action of the control fields on the\nspin chain. Since we are considering periodic control\nfields,Uc(t+tc) =Uc(t). Furthermore, in order to decou-\nple the spin chain from the environment to lowest order,\nwe have the condition [33, 34, 36, 37]\n(3)/integraldisplaytc\n0dtU†\nc(t)HSBUc(t) = 0.\nFor completeness, the reasoning behind this condition is\nshown in Appendix A. Keeping the form of HSBin mind,\nwe guess that\n(4) Uc(t) =N/productdisplay\ni=1eiωnxσ(i)\nxteiωnyσ(i)\nyt,3\nwhereω= 2π/tcandnxandnyare integers, is one\npossible choice that can dynamically decouple the spin\nchain from its environment. Our task then is to check\nthat this is indeed the case. It is trivial to check that\nUc(t+tc) =Uc(t). We next define, for convenience,\nhj,k(t) =U†\nc(t)σ(j)\nkUc(t),\nwithσ(j)\n1=σ(j)\nx,σ(j)\n2=σ(j)\ny, andσ(j)\n3=σ(j)\nz. We find\nthat\nhj,1(t) = cos(2ωnyt)σ(j)\nx−sin(2ωnyt)σ(j)\nz,\nhj,2(t) = sin(2ωnxt)sin(2ωnyt)σ(j)\nx+\ncos(2ωnxt)σ(j)\ny+sin(2ωnxt)cos(2ωnyt)σ(j)\nz,\nhj,3(t) = cos(2ωnxt)sin(2ωnyt)σ(j)\nx−\nsin(2ωnxt)σ(j)\ny+cos(2ωnxt)cos(2ωnyt)σ(j)\nz.\nWith these expressions, it is straightforward to see that\nas long as nx/negationslash=ny, we meet the condition given by\nEq. (3). The corresponding control field Hamiltonian is\n(5)Hc(t) =N/summationdisplay\ni=1/braceleftig\nωny[sin(2ωnxt)σ(i)\nz−cos(2ωnxt)σ(i)\ny]\n−ωnxσ(i)\nx/bracerightig\n,\nwithnx/negationslash=ny. We emphasize that our decoupling scheme\nworks provided that tc≪τ, whereτis the environment\ncorrelation time, with exact decoupling achieved in the\nlimittc\nτ→0 (see Appendix A for more details). In other\nwords, providedthat ωislargeenough, weareableto dy-\nnamically decouple the spin chain from its environment,\nat least to lowest order, by using two oscillating fields in\ntheyandzdirections and a static field in the xdirection.\nThen, as long as each spin interacts ‘locally’ with the en-\nvironment [see Eq. (2)], independent of the detailed form\nof the spin chain-environment interaction, the spin chain\ncan be effectively decoupled from the environment.\nWe now observe that the control fields not only serve\nto dynamically decouple the spin chain, but they also\nmodify the spin chain Hamiltonian itself. Provided that\nthe control fields are strong enough and oscillating fast\nenough, the effective spin chain Hamiltonian in the pres-\nence of the control fields is [34, 36]\n¯H=1\ntc/integraldisplaytc\n0dtU†\nc(t)H0Uc(t).\nFor completeness, this relation is also derived in Ap-\npendix A. In particular, our effective Hamiltonian ap-\nproach is valid if λj,ktc≪1, with¯Hable to capture the\ndynamics perfectly in the limit λj,ktc→0. In our case,\nthe effective Hamiltonian becomes\n¯H=1\ntcN−1/summationdisplay\nj=1/integraldisplaytc\n0dt3/summationdisplay\nk=1λjkhj,k(t)hj+1,k(t).We now define\nI(j)\n1=1\ntc/integraldisplaytc\n0hj,1(t)hj+1,1(t)dt,\nI(j)\n2=1\ntc/integraldisplaytc\n0hj,2(t)hj+1,2(t)dt,\nI(j)\n3=1\ntc/integraldisplaytc\n0hj,3(t)hj+1,3(t)dt.\nThe effective Hamiltonian is then\n¯H=N−1/summationdisplay\nj=1/bracketleftig\nλj1I(j)\n1+λj2I(j)\n2+λj3I(j)\n3/bracketrightig\n.\nThe remainingtask is to evaluatethe integrals. Recalling\nthatnxandnyare integers with nx/negationslash=ny(since we want\nto dynamically decouple the spin chain from its environ-\nment), we find that, if ny/negationslash= 2nx,\nI(j)\n1=1\n2[σ(j)\nxσ(j+1)\nx+σ(j)\nzσ(j+1)\nz],\nI(j)\n2=1\n4[σ(j)\nxσ(j+1)\nx+2σ(j)\nyσ(j+1)\ny+σ(j)\nzσ(j+1)\nz],\nandI(j)\n3=I(j)\n2. This leads to\n¯H1=N−1/summationdisplay\nj=1/braceleftbiggλj1\n2/bracketleftig\nσ(j)\nxσ(j+1)\nx+σ(j)\nzσ(j+1)\nz/bracketrightig\n+(λj2\n+λj3)/bracketleftbigg1\n4(σ(j)\nxσ(j+1)\nx+2σ(j)\nyσ(j+1)\ny+σ(j)\nzσ(j+1)\nz)/bracketrightbigg/bracerightbigg\n,\n(6)\nfornx/negationslash=nyandny/negationslash= 2nx. Thus, by applying local con-\ntrol fields, the spin chain is dynamically decoupled from\nits environment, and the interactions between the spins\nare also transformed. While the transformed spin chain\nHamiltonian may be more complicated than the origi-\nnal spin chain Hamiltonian, this may not always be the\ncase. For example, one can check that the fully isotropic\nHeisenberg Hamiltonian, also known as the Heisenberg\nXXX model, remains unchanged. Also, the modified\nHamiltonian may itself be a very well-known and un-\nderstood model - for instance, the quantum Ising model\ntransforms to the XX model (also known as the isotropic\nXY model). Even if the effective spin chain Hamiltonian\nis relativelycomplicated, it is still tractablefor smallspin\nchains; moreover, the effective Hamiltonian can also be\nstudied for larger spin chains in some special cases (see\nSection V). Throughout the paper, our focus will be on\nshowinghowthemodified interactionscanimprovequan-\ntum state transfer and entanglement generation.\nWe now notice that if we use control fields such that4\nny= 2nx,I(j)\n1is the same as before, but now\nI(j)\n2=1\n4[σ(j)\nxσ(j+1)\nx+2σ(j)\nyσ(j+1)\ny+σ(j)\nxσ(j+1)\ny\n+σ(j)\nyσ(j+1)\nx+σ(j)\nzσ(j+1)\nz],\nI(j)\n3=1\n4[σ(j)\nxσ(j+1)\nx+2σ(j)\nyσ(j+1)\ny−σ(j)\nxσ(j+1)\ny\n−σ(j)\nyσ(j+1)\nx+σ(j)\nzσ(j+1)\nz].\nIn this case, we can then write the effective Hamiltonian\nas\n¯H2=N−1/summationdisplay\nj=1λj1\n2/bracketleftig\nσ(j)\nxσ(j+1)\nx+σ(j)\nzσ(j+1)\nz/bracketrightig\n+λj2\n4/bracketleftig\nσ(j)\nxσ(j+1)\nx\n+2σ(j)\nyσ(j+1)\ny+σ(j)\nxσ(j+1)\ny+σ(j)\nyσ(j+1)\nx+σ(j)\nzσ(j+1)\nz/bracketrightig\n+λj3\n4/bracketleftig\nσ(j)\nxσ(j+1)\nx+2σ(j)\nyσ(j+1)\ny−σ(j)\nxσ(j+1)\ny\n−σ(j)\nyσ(j+1)\nx+σ(j)\nzσ(j+1)\nz/bracketrightig\n.\n(7)\nThis case is even more interesting due to the additional\npresence of the ‘cross-interactions’ such as σ(j)\nxσ(j+1)\ny.\nSuch ‘cross-interactions’ arise in spin chains when one\nstudies Dzyaloshinskii-Moriya interactions in spin chains\n(although the signs of our additional terms differ). How-\never, in our case, these interactions are simply an effec-\ntive result of applying control fields to each spin. As we\nwill show, these additional interactions can significantly\nimprove entanglement generation.\nLet us now also note that the spin chain Hamiltonian\nthat we started from [see Eq. (1)] does not contain any\ntransverse fields which would contribute/summationtextN\nj=1bj1σ(j)\nx+\nbj2σ(j)\ny+bj3σ(j)\nzto the Hamiltonian. This is simply be-\ncause our dynamical decoupling fields, at least to lowest\norder, remove the effect of these terms, provided that\nthe time-dependence of the fields bj1,bj2, andbj3is slow\ncompared to tc. However, if these additional transverse\nfields arealsooscillatingwith frequencycomparableto ω,\nthen the effective Hamiltonian can, at least in principle,\ninclude the effect of static fields as well. Further details\nare presented in Appendix B [in particular, see Eq. (B2)\nfor the additional control fields that lead to additional\nterms in the effective Hamiltonian given by Eq. (B3) or\nEq. (B4)].\nIII. QUANTUM STATE TRANSFER\nAsafirstexampleofourformalism,westudythetrans-\nfer of a quantum state from one end of a quantum spin\nchain to the other [10, 49–55]. The most commonly stud-\nied scenario involves the quantum XX model\nHXX=N−1/summationdisplay\nj=1λj/bracketleftig\nσ(j)\nxσ(j+1)\nx+σ(j)\nyσ(j+1)\ny/bracketrightig\n+N/summationdisplay\nj=1Bjσ(j)\nz.The idea is that an arbitrary state for the spin chain can\nbe transferred to the other end. Writing the eigenstates\nof theσzoperator as |0/angbracketrightand|1/angbracketrightwithσz|s/angbracketright= (−1)s|s/angbracketright,\nit has been found that if the initial state of the spin\nchain is |Ψ(0)/angbracketright= (a|0/angbracketright+b|1/angbracketright)⊗ |0/angbracketright ⊗...⊗ |0/angbracketright, the\nstate of the spin chain after some time Tis|Ψ(T)/angbracketright=\n|0/angbracketright ⊗...⊗ |0/angbracketright ⊗(a|0/angbracketright+eiφb|1/angbracketright), whereφis known so\nthat the phase can be corrected at the end of the state\ntransfer. One way to achieve perfect state transfer is\nthat we set λj=/radicalbig\nj(N−j) [56, 57], which is optimal\nin terms of the transfer time [58]. Perfect state transfer\nis then achieved after time T=π/2, with the phase fac-\ntoreiφ= (−i)N−1that can be removed [59]. Practically\nspeaking, however, such perfect state transfer is difficult\ndue to the unwanted influences of the environment. To\nremove this detrimental effect, pulse sequences [60] have\nbeen considered and the direct modulation of the spin-\nspin coupling has also been investigated [61]. With our\nscheme, as we have discussed, local noise terms can be\neliminated to lowest order by applying a static as well as\noscillating control fields. These control fields also modify\nthe spin chain Hamiltonian. In particular, it is clear that\nthe quantum XX spin chain does not remain the quan-\ntum XX spin chain in the presence of the control fields.\nTo get around this, we note that if we originally have the\nquantum Ising model (with zero magnetic field),\nH0=N−1/summationdisplay\nj=1λjσ(j)\nxσ(j+1)\nx, (8)\nthe correspondingtime-averagedeffectiveHamiltonian in\nthe presence of the control fields is\n¯H1=N−1/summationdisplay\nj=1λj\n2/bracketleftig\nσ(j)\nxσ(j+1)\nx+σ(j)\nzσ(j+1)\nz/bracketrightig\n.(9)\nIn this case, it turns out that ¯H2=¯H1=¯H, sony=\n2nxandny/negationslash= 2nxlead to the same result. Notice that\nthe effective Hamiltonian contains σ(j)\nzσ(j+1)\nzinteractions\nwhichareabsentintheoriginalIsingchain. Inparticular,\nthe effective Hamiltonian is simply a rotated version of\nthe XX model. Defining the rotation operator UR=\nΠje−iπσ(j)\nx/4, we find that ¯HR=UR¯H1U†\nR=HXXwith\nBj= 0. In other words, if we start from the Ising spin\nchain, and apply the control fields given by\nHR\nc(t) =N/summationdisplay\ni=1/braceleftig\nωnz[sin(2ωnxt)σ(i)\ny+cos(2ωnxt)σ(i)\nz]\n−ωnxσ(i)\nx/bracerightig\n,\n(10)\nwherenxandnzare integers with nx/negationslash=nz, we can\nachieve excellent state transfer ( a|0/angbracketright+b|1/angbracketright)⊗|0/angbracketright⊗...⊗\n|0/angbracketright → |0/angbracketright⊗...⊗|0/angbracketright⊗(a|0/angbracketright+b|1/angbracketright) even in the presence\nof the local noise terms.5\n02468100.00.20.40.60.81.0\ntF(t)\nFIG. 1. (Color online) Dynamics of the fidelity of state trans -\nfer starting from the quantum Ising model with N= 4. We\nhave used λj=/radicalbig\nj(N−j), andny= 2,nx= 1; the initial\nstate is |1000/angbracketright. The dot-dashed, magenta curve shows the\nfidelity with the full Hamiltonian H0+HSB+HR\nc(t) numer-\nically solved [ H0andHR\nc(t) are given by Eqs. (8) and (10)\nrespectively], while the solid, black curve shows the dynam -\nics using the time-averaged effective Hamiltonian ¯HR. The\ndashed blue curve shows the fidelity if we simply use only the\nXX spin chain HXXin the presence of noise. Throughout the\npaper, we work in dimensionless units with ¯ hset equal to one\nandtc= 0.01. Also, as explained in the main text, the noise\nis modeled via Ornstein-Uhlenbeck processes with zero mean ,\ncorrelation time τ= 0.5, and standard deviation σ= 2.0.\nWe take an average over 20 noise realizations throughout the\npaper. In liquid NMR implementations of spin chains [54],\nλjare typically of the order of 100 Hz. tc= 0.01 then leads\nto control fields with frequencies in the 10 kHz regime and\namplitudes around 1 microtesla, and t= 1 is on the order of\n10 ms. With superconducting qubits [8], λjis on the order of\n10 MHz, meaning that tc= 0.01 corresponds to control field\nfrequencies in the GHz regime and amplitudes around 0 .01\ntesla. Smaller values of tccorrespond to higher frequencies\nand stronger fields, thereby leading to even better decoupli ng\nof the spin chain and the effective Hamiltonian approximatin g\nthe exact dynamics even more closely.\n02468100.00.20.40.60.81.0\ntF(t)\nFIG. 2. (Color online) Same as Fig. 1, except that we now\nhaveN= 10. The initial state is |100...0/angbracketright. The dashed-blue\nline essentially overlaps with the horizontal axis - the fide lity\nwithout control fields is very small.\nWe now numerically check our claims. Since the\nHamiltonian ¯HRpreserves the number of excitations,we restrict ourselves to studying the transfer |ψi/angbracketright=\n|1/angbracketright ⊗ |0/angbracketright ⊗...⊗ |0/angbracketright → |ψf/angbracketright=|0/angbracketright ⊗...⊗ |1/angbracketrightfor sim-\nplicity. To quantify the quality of the state transfer, we\nuse the fidelity F(t) =/angbracketleftψf|ρ(t)|ψj/angbracketrightwhereρ(t) is the\nspin state density matrix at time t. We model the ef-\nfect of the environment on the spins via classical noise\nfields acting on each spin. For simplicity, we assume\nthatB(j)\nkis the same for every j.B(j)\nx,B(j)\ny, andB(j)\nz\nare then generated via independent Ornstein-Uhlenbeck\nprocesses, each with zero mean, correlation time τ= 0.5,\nand standard deviation σ= 2.0 (for more details, see\nAppendix C and Ref. [62]). Let us also note that our\nspin state transfer depends on the reliable implementa-\ntion of a single spin operation in order to prepare the\ninitial state |1/angbracketright ⊗ |0/angbracketright ⊗...⊗ |0/angbracketright. As illustrated in Ap-\npendix D, our dynamical decoupling control fields can,\nat least in principle, be extended in order to implement\nsingle spin operations with fidelity very close to one.\nIn Fig. 1, we illustrate state transfer for N= 4. First,\nthe fidelity is captured verywell by the effective Hamilto-\nnian since the solid black curve essentially overlaps with\nthe dot-dashed magenta curve. Second, the effect of the\nnoise is effectively removed. Third, if we use the XX\nmodel with no control fields and noise present, the fi-\ndelity of the state transfer is significantly lower as shown\nbythe dashedblue curve. Similarresultsareobtained for\nlarger spin chains as illustrated in Fig. 2. Since the effec-\ntive Hamiltonian is the XX model, and it is known that\n(with a properchoice of the spin-spin coupling strengths)\ntheXXmodelleadstoperfect statetransfer[52], ourpro-\nposed dynamical decoupling fields should lead to near\nperfect state transfer for even larger spin chains. Thus,\nwe have demonstrated that if we use the quantum Ising\nmodel to begin with, we can obtain excellent quantum\nstate transfer simply by the use of static and oscillating\ncontrol fields even in the presence of noise.\nIV. NUMERICAL RESULTS FOR\nENTANGLEMENT GENERATION\nWe now quantitatively analyze the results of applying\nthecontrolfieldsforentanglementgeneration. Todothis,\nwe look at the concurrence [63] between two spins in the\nspin chain. Our strategy is simple. We consider the spin\nchain in the presence of local noise fields. To begin, we\ndo not apply any control fields, and examine the behav-\nior of the concurrence between two spins as a function\nof time. Thereafter, we apply our strong and rapidly os-\ncillating control fields. We find the concurrence between\ntwo spins in the presence of the noise fields by solving\nthe Schrodinger equation, thereby showing the effective-\nness of the control fields in dynamically decoupling the\nspin chain. We also show that the dynamical behavior is\ncaptured very well by the time-averaged effective Hamil-\ntonian approach. Finally, we compare the performance\nof the control fields with ny/negationslash= 2nxandny= 2nx. The\ninitial state we consider is either the fully polarized spin6\nstate, that is, |00...0/angbracketright(or|11...1/angbracketright), or the fully polar-\nized state with the first spin flipped, that is, |100...0/angbracketright\n(or|011...1/angbracketright). The fully polarized spin state can be re-\nalized experimentally by, for instance, applying a large\nmagnetic field at low temperatures (the coordinate sys-\ntem is set up such that the z-axis is aligned along the\nmagnetic field), and a π-pulse can be applied to the first\nspin to realize the spin flip. Indeed, these states are the\ninitial states most commonly used in studies of entangle-\nment dynamics (see, for example, Refs. [64] and [65]).\n02468100.00.10.20.30.40.5\ntC(t)\nFIG. 3. (Color online) Dynamics of the concurrence between\nspins1and4startingfromthequantumIsingmodelwith N=\n4. We have used λ1= 2 (λ2=λ3= 0), and ny= 2,nx= 1.\nThe dynamics for the concurrence without any control fields\nis shown by the dashed, blue curve, while the dot-dashed,\nmagenta curve shows the dynamics with the full Hamiltonian\nH0+HSB+Hc(t) numerically solved [see Eqs. (8) and (5) for\nH0andHc(t) respectively]. The solid, black curve shows the\ndynamics using the time-averaged effective Hamiltonian. Th e\ninitial state of the spin chain used here is |0000/angbracketright. As before,\nwe are working in dimensionless units with ¯ hset equal to\none and tc= 0.01. The dashed, blue line overlaps with the\nhorizontal axis.\nA. The quantum Ising model\nIn this case, the spin chain Hamiltonian (with zero\nmagnetic field) is given by Eq. (8), while the correspond-\ning effective Hamiltonian in the presence of the control\nfields is shownin Eq. (9). Fromnow on, forsimplicity, we\nwillbe assumingthatthe couplingstrengthsarethe same\nthroughout the chain, that is, λjkis independent of j. As\noutlined before, we aim to find the concurrence between\ntwo spins in the spin chain as a function of time. We\nfind the density matrix as a function of time, and then\ntake the partial trace over all the spins other than the\ntwo spins whose concurrence we are interested in. Hav-\ning found this two-spin density matrix ρ2(t) as a function\nof time, we find the concurrence [63] by first finding\nR=/radicalig√ρ2/tildewideρ2√ρ2,\nwith\n/tildewideρ2= (σy⊗σy)ρ∗\n2(σy⊗σy).The concurrence Cis then given by\nC(ρ) = max(0,λ1−λ2−λ3−λ4)\nwhereλiare the eigenvalues of Rin descending order.\n02468100.00.20.40.60.81.0\ntC(t)\nFIG. 4. (Color online) Same as Fig. 3, but now we have\nthe control fields given by Eq. (10). That is, the dot-dashed\nmagenta curve shows the dynamics with the Hamiltonian\nH0+HSB+HR\nc(t), the dynamics without any control fields is\nshown by the dashed-blue curve, while the solid, black curve\nshows the dynamics with the effective Hamiltonian. The ini-\ntial state is |1000/angbracketright. We have checked that when the concur-\nrence is approximately one, the purity of the state for spins\n1 and 4 is very close to one with the two-spin state being ap-\nproximately the fully entangled state1√\n2(|01/angbracketright+i|10/angbracketright). Once\nagain, the dashed, blue line overlaps with the horizontal ax is.\nLet us now present our results for the quantum Ising\nmodel. In Fig. 3, we illustrate three points. First, as can\nbe seen by comparing the solid, black curve with the dot-\ndashed, magenta curve, the time-averaged Hamiltonian\nreproducesthe exactnumericalresultsverywell. Second,\nthe concurrence in the presence of the noise is very low -\nthe dashed, blue curve overlaps with the horizontal axis.\nThird, the control fields are able to average out the ef-\nfect of the noise fields. Given the close relation between\nentanglement and quantum state transfer [52], we have\nalso found the entanglement with the initial state |1000/angbracketright\nand the control fields given by Eq. (10) [as in Fig. 1].\nThe results are shown in Fig. 4. We can see that we can\nnot only protect the spin chain against the environment,\nbut also generate almost perfect entanglement between\nthe ends of the spin chain, at least for N= 4.\nAs we have seen, if we start from the quantum Ising\nmodel, there is no difference between ¯H1and¯H2. In\nordertoinvestigatehowthecondition ny= 2nxcanmake\na difference, we now look at the XY model.\nB. XY model\nFor the XY model, the spin chain Hamiltonian, with\nzero magnetic field, is\nH0=N−1/summationdisplay\nj=1/bracketleftig\nλ1σ(j)\nxσ(j+1)\nx+λ2σ(j)\nyσ(j+1)\ny/bracketrightig\n.7\nIfλ1=λ2, we have the isotropic XY model, which we\nhave referred to as the XX model. In the presence of the\ncontrol fields, the effective Hamiltonian is\n¯H1=N−1/summationdisplay\nj=1/braceleftbiggλ1\n2/bracketleftig\nσ(j)\nxσ(j+1)\nx+σ(j)\nzσ(j+1)\nz/bracketrightig\n+λ2\n4/bracketleftig\nσ(j)\nxσ(j+1)\nx+2σ(j)\nyσ(j+1)\ny+σ(j)\nzσ(j+1)\nz/bracketrightig/bracerightbigg\n,\n(11)\nifny/negationslash= 2nx. Once again, note the presence of the ad-\nditional spin-spin interactions. On the other hand, if\nny= 2nx, the effective Hamiltonian is\n(12)¯H2=N−1/summationdisplay\nj=1/braceleftbiggλ1\n2/bracketleftig\nσ(j)\nxσ(j+1)\nx+σ(j)\nzσ(j+1)\nz/bracketrightig\n+λ2\n4/bracketleftig\nσ(j)\nxσ(j+1)\nx+2σ(j)\nyσ(j+1)\ny+σ(j)\nxσ(j+1)\ny\n+σ(j)\nyσ(j+1)\nx+σ(j)\nzσ(j+1)\nz/bracketrightig/bracerightbigg\n.\nThere are now even more additional spin-spin interac-\ntions;¯H2contains ‘cross-interaction’ terms σ(j)\nxσ(j+1)\ny\nandσ(j)\nyσ(j+1)\nxabsent in ¯H1.\nWe now present numerical simulations illustrating the\neffect of these additional terms. First, Fig. 5 shows the\nconcurrence between the first and last spins of the spin\nchain with N= 4, starting from the initial state |0000/angbracketright.\nThe dashed, blue curve (which is on top of the horizontal\naxis) illustrates that, in the absence of the control fields,\nentanglement generation is negligible. However, in the\npresence of the control fields with ny/negationslash= 2nx, significant\nentanglement can be generated, as evidenced by the dot-\ndashed, magenta curve. Moreover, the solid black curve,\nwhich lies essentially on top of the magenta curve, shows\nthat the dynamics are captured very well by the time-\naveragedeffective Hamiltonian ¯H1. Moving on, in Fig. 6,\nwe have again shown the dynamics of the entanglement\nbetween spins 1 and 4, but we have now used the spe-\ncial control fields with ny= 2nx. The initial state is\nagain|0000/angbracketright. As before, considerable entanglement is\ngenerated in the presence of the control fields, and the\ndynamics is captured very well by the effective Hamilto-\nnian (which is ¯H2now). Moreover, comparing Figures 5\nand 6, we see that the special choice ny= 2nxis able\nto generate more entanglement (at least between spins 1\nand 4). If we look instead at spins 2 and 3 of the spin\nchain [see Figures 7 and 8], we reach a similar conclu-\nsion. Thus, while fields with any nyandnxcan decouple\nthe spin chain (as long as ny/negationslash=nx), making the special\nchoiceny= 2nxcan be a much better strategy in the\nsense that the generation of a valuable quantum resource\nsuchasquantumentanglementisimproved. Thisresultis\nfurtherreinforcedin Fig.9wherewehaveusedadifferent\ninitial state, namely |0111/angbracketright. Once again, ¯H2can generate\nsignificantly more entanglement between the ends of thespin chain as compared to ¯H1. Similar conclusions hold\ntrue if we use the initial states |1000/angbracketrightand|1111/angbracketright.\n02468100.00.20.40.60.81.0\ntC(t)\nFIG. 5. (Color online) Plot of the concurrence between spins\n1 and 4 starting from the quantum XX model with N=\n4. Here we have ny/negationslash= 2nx(ny= 3,nx= 1), and λ1=\nλ2= 1, while λ3= 0. We have used tc= 0.01. We have\nshown the dynamics without any control fields (dashed, blue\ncurve which is overlapping with the horizontal axis), with\ncontrol fields using the total Hamiltonian H0+HSB+Hc(t)\n(dot-dashed, magenta curve), and using the time-averaged\nHamitonian ¯H1(solid, black curve). The initial state of the\nspin chain is |0000/angbracketright. No entanglement can be generated even\nin the absence of noise since the state |0000/angbracketrightis an eigenstate\nof the XX model; the control fields not only eliminate the\nnoise, but also modify the spin chain Hamiltonian such that\nentanglement can be generated.\n02468100.00.20.40.60.81.0\ntC(t)\nFIG. 6. (Color online) Same as Fig. 5, except that we are\nnow using control fields with ny= 2nx(ny= 2,nx= 1), and\nthe solid, black curve shows the entanglement dynamics due\nto the effective Hamiltonian ¯H2. The initial state is |0000/angbracketright.\nC. XYZ model\nWe now look at the full XYZ spin chain Hamiltonian,\ngiven by\nH0=3/summationdisplay\nk=1λkN−1/summationdisplay\nj=1σ(j)\nkσ(j+1)\nk.8\n02468100.00.20.40.60.81.0\ntC(t)\nFIG. 7. (Color online) Same as Fig. 5, except that we are\nnow showing the concurrence between spins 2 and 3 of the\nspin chain. Once again, the initial state is |0000/angbracketright.\n02468100.00.20.40.60.81.0\ntC(t)\nFIG. 8. (Color online) Same as Fig. 5, except that we are\nnow using control fields with ny= 2nx(ny= 2,nx= 1), and\nwe have shown the concurrence between spins 2 and 3 of the\nspin chain. The solid, black curve shows the entanglement\ndynamics due to the effective Hamiltonian ¯H2. The initial\nstate is again |0000/angbracketright.\n◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆\n◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆\n◆\n◆\n◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆02468100.00.20.40.60.81.0\ntC(t)\nFIG. 9. (Color online) Plot of the concurrence between spins\n1 and 4 for the quantum XY model with N= 4. Here we\nhaveny= 2nx(ny= 2,nx= 1), and λ1=λ2= 1, while λ3=\n0. We have shown the dynamics without any control fields\n(solid, dashed blue line), with control fields using the tota l\nHamiltonian H0+HSB+Hc(t) (dot-dashed, magenta curve),\nand using the time-averaged Hamitonians ¯H1(red diamonds)\nand¯H2(solid, black curve). The initial state is now |0111/angbracketright.One important comment is in order. If the spin chain\nis fully isotropic, that is, λ1=λ2=λ3, then the con-\ntrol fields cannot alter the spin chain Hamiltonian. The\nreason is simple - for the fully isotropic case, the Hamil-\ntonian can be written as an inner product of a vector\nconsisting of the Pauli matrices with itself, and this in-\nner product is of course invariant under unitary opera-\ntions. Therefore, we will focus on the anisotropic case\nwhere the coupling strengths are not all equal to each\nother. The time-averaged Hamiltonian is given by ¯H1\nifny/negationslash= 2nx[see Eq. (6)] and by ¯H2[see Eq. (7)] if\nny= 2nx. As shown in Fig. 10, with the initial state\n|0000/angbracketright, the dynamics are captured very well by our effec-\ntiveHamiltoniansincethe dot-dashedmagentaandsolid,\nblackcurvesoverlap. Itisalsoclearthatthecontrolfields\nwithny= 2nxare better at generating entanglement for\nthe given values ofthe interaction strengths. Once again,\nthe choice of the control fields can affect the interactions,\nand thereby the generation of a quantum resource such\nas entanglement, to a very large degree. Similar con-\nclusions hold true if we consider the initial state to be\n|0111/angbracketright,|1000/angbracketright, or|1111/angbracketrightinstead.\n◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆\n02468100.00.20.40.60.81.0\ntC(t)\nFIG. 10. (Color online) Dynamics of entanglement between\nspins 1 and 4 for the XYZ model with N= 4. We have\nusedλ1= 0.5,λ2= 1,λ3= 0.25. We have shown the dy-\nnamics without any control fields (dashed, blue line), with\ncontrol fields using the total Hamiltonian H0+HSB+Hc(t)\n(dot-dashed, magenta curve), and using the time-averaged\nHamitonians ¯H1(red diamonds) and ¯H2(solid, black curve).\nThe initial state of the spin chain is |0000/angbracketright. The dashed, blue\ncurve again overlaps with the horizontal axis.\nV. COMPARING THE EFFECTIVE\nHAMILTONIANS ¯H1AND¯H2FOR LARGER N\nHaving demonstrated that if we apply sufficiently\nstrong and rapidly oscillating control fields, the spin\nchain Hamiltonian can be approximated by ¯H1ifny/negationslash=\n2nxand by¯H2ifny= 2nx, we now aim to cast ¯H1and\n¯H2in diagonal form so that the concurrence for larger\nspin chains can be worked out easily. A commonly used\ntool in such a calculation is the Jordan-Wigner trans-\nformation which allows one to transform the problem9\nof interacting spins to a problem of spinless fermions\n[66, 67]. Unfortunately, the presence of the σ(j)\nzσ(j+1)\nzin-\nteractions in ¯H1and¯H2means that the spinless fermions\nare interacting. However, if the original spin chain is\nsuch that 2 λ1+λ2+λ3= 0, then the Jordan-Wigner\ntransformation allows us to transform both ¯H1and¯H2\nto non-interacting fermions, thereby making the prob-\nlem tractable and greatly reducing the computational\ncomplexity. We largely follow the approach presented\nin Refs. [66, 68], although we must emphasize that the\nspin chain Hamiltonian ¯H2we are solving is different.\n012345670.00.20.40.60.81.0\ntC(t)\nFIG. 11. (Color online) Dynamics of entanglement between\nspins 1 and 4 for the XYZ model with N= 4. We have used\nλ1= 2,λ2= 1,λ3=−5. The solid, black curve shows the\nentanglement with ¯H2, while the dashed magenta curve is for\n¯H1. The dashed magenta curve overlaps with the horizontal\naxis. The initial state of the spin chain is |1111/angbracketright. We have\nchecked that when the concurrence is approximately one, the\npurity of the state for spins 1 and 4 is very close to one with\nthe two-spin state being approximately the entangled state\n0.75|00/angbracketright+(−0.55+0.37i)|11/angbracketright.\n012345670.00.20.40.60.81.0\ntC(t)\nFIG. 12. (Color online) Dynamics of the entanglement be-\ntween spins 1 and 12 for the XYZ model with N= 12. We\nhave used λ1= 2,λ2= 1, and λ3=−5. The solid black\ncurve shows the concurrence with the time-averaged Hamilto -\nnian¯H2, while the dashed magenta curve shows the dynamics\nwith¯H1. The dashed magenta curve again overlaps with the\nhorizontal axis. The initial state of the spin chain is |11...1/angbracketright.\nWith the condition λ3=−2λ1−λ2to suppress the\ninteraction terms σ(j)\nzσ(j+1)\nz, we find that the effective012345670.00.20.40.60.81.0\ntC(t)\nFIG. 13. (Color online) Same as Fig. 12, except that we are\nnow looking at the concurrence between spins 6 and 7.\nHamiltonians are\n¯H1=−λ1N−1/summationdisplay\nj=1σ(j)\nyσ(j+1)\ny,\nwhile\n¯H2=N−1/summationdisplay\nj=1/braceleftbigg(λ1+λ2)\n2/bracketleftig\nσ(j)\nxσ(j+1)\ny+σ(j)\nyσ(j+1)\nx/bracketrightig\n−λ1/bracketleftig\nσ(j)\nyσ(j+1)\ny/bracketrightig/bracerightbigg\n.\nThe details of finding the dynamics with these effective\nHamiltonians are given in Appendix E. In summary, to\nfind the entanglement between any two spins of the spin\nchain, we first need to find the two-spin density matrix.\nIn order to find the two-spin density matrix, we calculate\nthe correlation functions. These correlation functions\ncan be expressed in terms of Pfaffians as discussed in\nAppendix E. Using this approach, we have checked that\nfor small spin chains, the results produced are the same\nas those obtained numerically. For example, the results\nshown in Fig. 11 were reproduced using the approach\nemploying the Jordan-Wigner transformation. Interest-\ningly, in this case, with the initial state |1111/angbracketright, we can\nachieve almost perfect entanglement between the first\nand last spins with ¯H2, while¯H1generates no entangle-\nment at all. In Appendix F, we investigate how the en-\ntanglement generated changes as the difference between\nnyand 2nxchanges. We then used the Jordan-Wigner\ntransformation approach to obtain the concurrence for\nlarger spin chains. As an example, in Fig. 12 we have\nshown the dynamics of the entanglement between the\nends of a spin chain with N= 12 for both ¯H1and¯H2\nwith the initial state |11...1/angbracketright. It is clear that the en-\ntanglement generated if the special control fields with\nny= 2nxare used is considerable, while no entangle-\nment is generated when ny/negationslash= 2nx. As the size of the\nspin chain is increased further, we found that, with ¯H2,\nthe entanglement between the ends of the spin chain de-\ncreases and the time at which this maximum is obtained10\nincreases, while no entanglement is generated with ¯H1.\nFor example, with N= 24, the maximum concurrence\nobtained between the ends with ¯H2is approximately0 .65\nfort= 8.7. It should also be kept in mind that with\nlarge spin chains, as the number of spins is increased, we\nshould considera rescaledconcurrencethat takesinto ac-\ncount the number of spins. For example, in Ref. [69], the\nrescaled concurrencehas been defined as CR= (N−1)C.\nIt is then clear that the entanglement generated by the\nspin chain dynamics is very significant if control fields\nwithny= 2nxare used. Moreover, if we look instead at,\nfor instance, spins 6 and 7 of the spin chain, we again\nobserve that ¯H2is better at generating entanglement as\ncompared to ¯H1(see Fig. 13).\nVI. CONCLUSION\nInsummary,wehaveshownthatapplyingsuitablecon-\ntrol fields to a spin chain can, at least to a large extent,\neliminate the interaction between the spin chain and its\nenvironment. Moreover, we have also shown that the ap-\nplication of the control fields modulates the spin-spin in-\nteraction in ways that can effectively generate spin-spin\ninteractions that are absent in the original spin chain\nHamiltonian. As an example of the constructive use\nof this modification, we have shown how, starting from\nthe quantum Ising chain, perfect quantum state transfer\ncan be achieved provided that control fields of sufficient\nstrength and frequency are applied. We have also pre-\nsented numerical simulations which show that two-spin\nentanglement generation in the spin chain can be im-\nproved by using particular control fields. Moreover, we\nhave also diagonalized the effective spin chain Hamilto-\nnian, at least for special coupling strengths, to show how\nthe effect of the control fields can be analyzed for larger\nspin chains. Due to the great theoretical and experi-\nmental interest in spin chains, especially in the context\nof quantum computation and information, our results\nshould be interesting not only from the perspective of\neffectively isolating spin chains from their environment,\nbut also for engineering spin-spin interactions via simple\nstatic and oscillating control fields.\nACKNOWLEDGEMENTS\nThe authors acknowledge support from the LUMS\nFIF Grant FIF-413. A. Z. C. is also grateful\nfor support from HEC under grant No 5917/Pun-\njab/NRPU/R&D/HEC/2016. Support from the Na-\ntionalCenterforNanoscienceandNanotechnologyis also\nacknowledged.Appendix A: Eliminating the spin\nchain-environment interaction and transformation of\nthe spin chain Hamiltonian\nLet us write the Hamiltonian for the spin chain in the\npresence of the control fields as\nHtot=H0+Hc(t)+HB+HSB\n=H′+HB+Hc(t).\nHereHc(t) is the controlfield Hamiltonian (acting on the\nspin chain), HBis the Hamiltonian of the environment,\nandHSBis the interaction between the spin chain and its\nenvironment. It is interesting to note that the environ-\nment of a quantum system itself has been modeled as a\nspin chain (see, for instance, Refs. [70–72] and references\ntherein). For future convenience, we have defined\n(A1) H′=H0+HSB.\nOur goal is to see how a state evolves under the action\nof the total Hamiltonian. To this end, let us first rotate\nthe basis by Uc(t)UB(t), whereUB(t) =e−iHBt. In this\nframe, the unitary time-evolution operator for the spin\nchain and the environment as a whole is\n˜Utot(t) =Texp/bracketleftbigg\n−i/integraldisplayt\n0˜H′(s)ds/bracketrightbigg\n,\nwhere˜H′(s) =U†\nB(s)U†\nc(s)H′Uc(s)UB(s). At time t=\nNtc(Nis a positive integer), due to the periodicity of\nthe control fields,\n˜Utot(t) =/bracketleftig\n˜Utot(tc)/bracketrightigN\n,\nand\n˜Utot(tc) =Texp/bracketleftbigg\n−i/integraldisplaytc\n0˜H′(s)ds/bracketrightbigg\n.\nNow comes the key step. We use the Magnus expansion\nto write\n˜Utot(tc) = exp[−itc(˜H(0)+˜H(1)+···)],\nwhere\n˜H(0)=1\ntc/integraldisplaytc\n0˜H′(t)dt,\nand\n˜H(1)=−i\n2tc/integraldisplaytc\n0dt1/integraldisplayt1\n0dt2[˜H′(t1),˜H′(t2)].\n˜H(0)is independent of tc, while˜H(1)increases with in-\ncreasingtc, suggesting that for small tc, only˜H(0)can be\nconsidered. Now,\n˜H(0)=1\ntc/integraldisplaytc\n0U†\nc(t)H0Uc(t)dt\n+1\ntc/integraldisplaytc\n0U†\nc(t)U†\nB(t)HSBUB(t)Uc(t)dt.11\nWritingHSB=/summationtext\nαFα⊗Bα, the latter term can be writ-\nten as\n1\ntc/integraldisplaytc\n0/summationdisplay\nαU†\nc(t)FαUc(t)U†\nB(t)BαUB(t)dt.\nIftcis much smaller than the environment correlation\ntimeτ, that istc\nτ≪1, then the time dependence of\nthe environment operators over the timescale tccan be\nneglected, leading to\n/integraldisplaytc\n0/summationdisplay\nαU†\nc(t)FαUc(t)Bαdt=/integraldisplaytc\n0U†\nc(t)HSBUc(t)dt.\nIf we now impose the condition\n/integraldisplaytc\n0U†\nc(t)HSBUc(t)dt= 0,\nwe find that\n˜H(0)=1\ntc/integraldisplaytc\n0U†\nc(t)H0Uc(t)dt.\nConsidering only the first term in the Magnus expansion,\n˜Utot(t)≈[e−itc˜H(0)]t/tc=e−it¯H,\nwith\n¯H=1\ntc/integraldisplaytc\n0U†\nc(t)H0Uc(t)dt.\nTransforming it back to the original frame, we find that\nthe unitary evolution operator is\n(A2) Utot(t)≈Uc(t)UB(t)e−it¯H.\nThus, the spin chain and its environment have been ef-\nfectively decoupled, at least to lowest order. Note that\ntchas to be much smaller than the environment correla-\ntion timeτfor our scheme to work (a similar result holds\nwhen pulses are used - see, for instance, Ref. [73]). As\nan illustration, in Fig. 14 we have shown the concurrence\nobtained with tc= 0.01 andtc= 0.5, and the environ-\nment correlation time is τ= 0.5. It is clear that with\ntc= 0.5, the control fields cannot protect the spin chain\nagainst the effect of the environment.\nIt is alsoworthexaminingthe next term in the Magnus\nexpansion. The part of ˜H(1)concerned with the dynam-\nics of the spin chain only is\n−i\n2tc/integraldisplaytc\n0dt1/integraldisplayt1\n0dt2[˜H0(t1),˜H0(t2)].\nThis is proportional to λ2\njktc, while¯His proportional to\nλjk. Thus, the correction to the effective Hamiltonian is\nnegligible if λjktc≪1, with our results becoming exact▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆\n◆\n◆◆◆◆◆◆◆◆012345670.00.20.40.60.81.0\ntC(t)\nFIG. 14. (Color online) Dynamics of entanglement between\nspins 1 and 4 for the XY model with N= 4 in the pres-\nence of control fields. We have used λ1= 1,λ2= 1, and\nλ3= 0. The solid, black curve shows the entanglement with\nthe effective Hamiltonian ¯H2, while the red diamonds show\nthe concurrence with the full time-dependent Hamiltonian\n[H0+Hc(t) +HSB(t)] withtc= 0.01. The blue triangles\nshows the dynamics of the full time-dependent Hamiltonian\nwithtc= 0.5. Here ny= 2 and nx= 1. The solid, black\ncurve and the red diamonds overlap, while the blue triangles\nlargely overlap with the horizontal axis. As always, the effe ct\nof the environment is modeled by Ornstein-Uhlenbeck pro-\ncesses with zero mean, correlation time τ= 0.5, and standard\ndeviation σ= 2.0. The initial state used here is |1000/angbracketright.\nin the limit λjktc→0. To see this more concretely,\nconsiderH0=λσ(1)\nxσ(2)\nx. Now,\nU†\nc(t)σ(1)\nxσ(2)\nxUc(t) = cos2(2ωnyt)σ(1)\nxσ(2)\nx+\n1\n2sin(4ωnyt)σ(1)\nxσ(2)\ny+1\n2sin(4ωnyt)σ(1)\nyσ(2)\nx+\nsin2(2ωnyt)σ(1)\nyσ(2)\ny.\nWe can then work out\n−i\n2tc/integraltexttc\n0dt1/integraltextt1\n0dt2[˜H0(t1),˜H0(t2)]. Although the\nfull expression is long and cumbersome, it is clear that\none of the terms is\n−i\n2tc/integraldisplaytc\n0dt1/integraldisplayt1\n0dt2λ2\n2cos2(2ωnyt1)sin(4ωnyt2)×\n[σ(1)\nxσ(2)\nx,σ(1)\nxσ(2)\ny].\nForny= 2, this is equal toλ2tc\n128πσ(2)\nz, thus illustrating\nour claim that ˜H(1)contains terms proportional to tc.\nIn short, the spin chain is effectively decoupled from the\nenvironment and its dynamics can be obtained from the\neffective Hamiltonian iftc\nτ≪1 andλj,ktc≪1. The lat-\nter condition is similar to what has been obtained before\nwhen pulses are applied to the spin chain [42]. In all the\nsimulations demonstrating the usefulness of our control\nfields, these conditions are satisfied.12\nAppendix B: Including effective magnetic fields in\nthe effective Hamiltonian\nConsider the total Hamiltonian\nH=H0+Hc(t)+HSB+Hd(t),(B1)\nwhereH0,HSB, andHc(t) given by Eqs. (1), (2), and (5)\nrespectively. Hd(t) is an additional Hamiltonian of the\nform\nHd(t) =/summationtextN\nj=1/bracketleftig\nbj1cos(2ωnyt)σ(j)\nx+bj2cos(2ωnxt)σ(j)\ny\n+bj3sin(2ωnzt)σ(j)\nz/bracketrightig\n,\n(B2)\nwherenxandnyare the same integers as in Hc(t), while\nnzis also an integer. Once again transforming to the\nframe of the control fields, we find that the effective\nHamiltonian is now\n¯H=1\ntc/integraldisplaytc\n0dtU†\nc(t)H0Uc(t)+1\ntc/integraldisplaytc\n0dtU†\nc(t)Hd(t)Uc(t).\nThe first term leads to the same effective Hamiltonian\nas before. Thus, an additional term has been added to\nthe effective time-independent Hamiltonian. Denoting\n¯Hd=1\ntc/integraltexttc\n0dtU†\nc(t)Hd(t)Uc(t), we find in a straightfor-\nward manner that\n(B3) ¯Hd=N/summationdisplay\nj=1/bracketleftbiggbj1\n2σ(j)\nx+bj2\n2σ(j)\ny+bj3\n4σ(j)\nz/bracketrightbigg\n,\nforny/negationslash= 2nxandnz=ny−nx. If, on the other hand,\nny= 2nxandnz=ny−nx, we find that\n(B4) ¯Hd=N/summationdisplay\nj=1/bracketleftbiggbj1\n2σ(j)\nx+bj2\n4σ(j)\nx+bj2\n2σ(j)\ny+bj3\n4σ(j)\nz/bracketrightbigg\n.\nIn this way, by applying additional oscillating fields with\nfrequencies similar to the control fields in Hc(t), we\ncan also effectively add static magnetic fields to the\nspin chain Hamiltonian. Similar to our prior treat-\nment, we expect our effective Hamiltonian approach to\nbe valid if bjktc≪1. As an example, if we choose\nbj1=bj3= 0, and we start from the quantum Ising\nmodelH0=/summationtextN−1\nj=1λjσ(j)\nxσ(j+1)\nx, by applying the control\nfields as well as additional oscillating fields described by\nHd(t) =/summationtextN\nj=1bjcos(2ωnxt)σ(j)\ny, weendupwitheffective\nHamiltonian (taking ny/negationslash= 2nx)\n(B5) ¯H=N−1/summationdisplay\nj=1λj\n2/bracketleftig\nσ(j)\nxσ(j+1)\nx+σ(j)\nzσ(j+1)\nz/bracketrightig\n+N/summationdisplay\nj=1bj\n2σ(j)\ny.\nThis is effectively the quantum XX model with a\ntransverse magnetic field. As another example, we can\nstart from the anisotropic Heisenberg Hamiltonian H0=/summationtextN−1\nj=1/bracketleftig\nλ1σ(j)\nxσ(j+1)\nx+λ2σ(j)\nyσ(j+1)\ny+λ3σ(j)\nzσ(j+1)\nz/bracketrightig\n.\nWith the application of the control Hamiltonian as\nwell as the additional oscillating field described by\nHd=/summationtextN\nj=1bsin(2ωnzt)σ(j)\nz, the effective Hamiltonian\nis (withny= 2nxandnz=ny−nx)\n¯H=N−1/summationdisplay\nj=1/braceleftbiggλ1\n2/bracketleftig\nσ(j)\nxσ(j+1)\nx+σ(j)\nzσ(j+1)\nz/bracketrightig\n+λj\n4/bracketleftig\nσ(j)\nxσ(j+1)\nx\n+2σ(j)\nyσ(j+1)\ny+σ(j)\nxσ(j+1)\ny+σ(j)\nyσ(j+1)\nx+σ(j)\nzσ(j+1)\nz/bracketrightig\n+λ3\n4/bracketleftig\nσ(j)\nxσ(j+1)\nx+2σ(j)\nyσ(j+1)\ny−σ(j)\nxσ(j+1)\ny\n−σ(j)\nyσ(j+1)\nx+σ(j)\nzσ(j+1)\nz/bracketrightig/bracerightbigg\n+N/summationdisplay\nj=1b\n4σ(j)\nz.\n(B6)\nToillustrateourresults, inFig.15below, wehaveplotted\ntheconcurrencebetweenthefirstandlastspinsfor N= 4\nusing the effective Hamiltonian Eq. (B6) and using the\ntotal Hamiltonian Eq. (B1) with bj1=bj2= 0, while\nbj3=bfor allj. It is clear that the effective Hamiltonian\ncaptures the exact dynamics exceedingly well.\n05 1 \u0000 \u0001 \u0002200.000.05\n0 \u0003 \u0004 \u0005\n\u0006 \u0007 \b \t0.20\ntC(t)\nFIG. 15. (Color online) Dynamics of entanglement between\nspins 1 and 4 for the XYZ model with N= 4 in the presence\nof an additional oscillating field. We have used λ1= 0.2,λ2=\n0.4,λ3= 0.3, andbj1=bj2= 0, while bj3= 1 for all j. The\nsolid, black curve shows the entanglement with the effective\nHamiltonian ¯Hgiven by Eq. (B6), while the dashed, purple\ncurve shows the concurrence with the full Hamiltonian given\nin Eq. (B1). Here ny= 2 and nx= 1. The curves overlap.\nThe initial state is |0000/angbracketright.\nAppendix C: Simulating the effect of noise\nIn the numerical simulations, the effect of the environ-\nment on the spin chain is modeled by the Hamiltonian\nHSB=N/summationdisplay\nj=1Bxσ(j)\nx+Byσ(j)\ny+Bzσ(j)\nz,13\nwhereBx,By, andBzare independent random variables\nobtainedbysolvingtheOrnstein-Uhlenbeckequation[62]\ncast in the form\ndBm=−(Bm−µ)\nτdt+σ/radicalbigg\n2\nτdW.\nHerem=x,y,z,µis the mean, σis the standard devi-\nation, andτis the correlation time. Wis the standard\nWiener process. Note that µandσhave the same di-\nmensions as Bm, and since this is a stochastic differential\nequation, (dW)2=dt. Throughout the paper, we have\nusedσ= 2.0 (which is comparable to the spin-spin cou-\npling strengths; see Fig. 1 caption), µ= 0 (that is, the\nmean is negligible compared to the spin-spin coupling\nstrengths), and τ= 0.5 (since we are using tc= 0.01\nthroughout, this means thattc\nτ= 0.02, thus fulfilling our\ndynamical decoupling condition). It should be kept in\nmind that in our system of units with ¯ h= 1,Bm(and\nthusσ) and the spin-spin coupling strengths have the\nsame units [see Eqs. (1) and (2)].\nAppendix D: Implementing single-spin operations\nThe dynamical decoupling approach can be adapted\nto implement single qubit operations (see, for exam-\nple. Refs. [36, 74]), which we now do in the context\nof spin chains for continuous control fields. Without\nloss of generality, let us suppose that we require a de-\nsired unitary operation to be implemented on the first\nspin in the spin chain (similar considerations apply if\nthe unitary operation is to be applied on some other\nspin in the spin chain). To implement such a single-\nspin operation, we first find continuous fields that not\nonly remove the effect of HSB, but also remove the effect\nof the spin-spin coupling between the first spin and its\nnearest neighbor. That is, we need to find Uc(t) such\nthat not only/integraltexttc\n0dtU†\nc(t)HSBUc(t) = 0, but also that/integraltexttc\n0dtU†\nc(t)σ(1)\nxσ(2)\nxUc(t) =/integraltexttc\n0dtU†\nc(t)σ(1)\nyσ(2)\nyUc(t) =/integraltexttc\n0dtU†\nc(t)σ(1)\nzσ(2)\nzUc(t) = 0. To achieve this, we mod-\nifyUc(t) to\nUc(t) =eiωn(1)\nxσ(1)\nxteiωn(1)\nyσ(1)\nytN/productdisplay\ni=2eiωn(2)\nxσ(i)\nxteiωn(2)\nyσ(2)\nyt.\n(D1)\nThis means that we apply different fields to the first spin\nas compared to all the other spins, that is,\n(D2)Hc(t) =N/summationdisplay\ni=1/braceleftig\nωn(i)\ny[sin(2ωn(i)\nxt)σ(i)\nz\n−cos(2ωn(i)\nxt)σ(i)\ny]−ωn(i)\nxσ(i)\nx/bracerightig\n,\nwithn(2)\nx=n(3)\nx=...=n(N)\nx, andn(2)\ny=n(3)\ny=...=\nn(N)\ny. From the condition/integraltexttc\n0dtU†\nc(t)HSBUc(t) = 0,we find that n(1)\nx/negationslash=n(1)\nyandn(2)\nx/negationslash=n(2)\ny. Next,/integraltexttc\n0dtU†\nc(t)σ(1)\nxσ(2)\nxUc(t) = 0 ifn(2)\ny/negationslash=n(1)\ny. The re-\nquirement that/integraltexttc\n0dtU†\nc(t)σ(1)\nyσ(2)\nyUc(t) = 0 leads to the\nfollowing conditions on the control fields:\nn(1)\nx/negationslash=n(2)\nx,\nn(1)\nx+n(2)\nx/negationslash=n(2)\ny,\nn(1)\nx−n(2)\nx/negationslash=n(2)\ny,\nn(2)\nx−n(1)\nx/negationslash=n(2)\ny,\nn(1)\nx+n(2)\nx/negationslash=n(1)\ny,\nn(1)\nx−n(2)\nx/negationslash=n(1)\ny,\nn(2)\nx−n(1)\nx/negationslash=n(1)\ny,\nn(1)\nx−n(2)\nx+n(1)\ny+n(2)\ny/negationslash= 0,\nn(1)\nx−n(2)\nx−n(1)\ny−n(2)\ny/negationslash= 0,\nn(1)\nx−n(2)\nx+n(1)\ny−n(2)\ny/negationslash= 0,\nn(1)\nx−n(2)\nx−n(1)\ny+n(2)\ny/negationslash= 0,\nn(1)\nx+n(2)\nx−n(1)\ny−n(2)\ny/negationslash= 0,\nn(1)\nx+n(2)\nx+n(1)\ny−n(2)\ny/negationslash= 0,\nn(1)\nx+n(2)\nx−n(1)\ny+n(2)\ny/negationslash= 0.\nSetting/integraltexttc\n0dtU†\nc(t)σ(1)\nzσ(2)\nzUc(t) = 0 leads to the same\nconditions. It is then straightforward to find integers\nn(1)\nx,n(1)\ny,n(2)\nx, andn(2)\nythat satisfy these requirements.\nFor example, n(1)\nx= 4,n(1)\ny= 8,n(2)\nx= 1, andn(2)\ny= 2\ndothe job. We illustratethe performanceofthese control\nfields in protecting the state of the first spin in Fig. 16.\nIt is clear that our control fields are able to preserve the\nstate of the first spin.\nWe now implement the single spin operation on the\nfirst spin. As an example, consider the unitary opera-\ntioneiπ\n2tgσ(1)\nxt. This transforms the state |0/angbracketrightfor the first\nspin to the state |1/angbracketrightafter timetg. To implement this\noperation, we first transform to the frame of the control\nfields. Then, we implement the single spin operation in\nthis frame, and thereafter transform back to the original\nframe. The net result is that the unitary operator that\nneeds to be implemented is given by\nUgate(t) =Uc(t)eiπ\n2tgσ(1)\nxt,\nwithUc(t) given by Eq. (D1). The corresponding Hamil-\ntonianHgate(t) is obtained via Hgate(t) =i∂Ugate\n∂tU†\ngate.\nA simple calculation shows that\n(D3)Hgate(t) =Hc(t)−π\n2tg/bracketleftig\nσ(1)\nxcos(2ωn(1)\nyt)\n+σ(1)\nzsin(2ωn(1)\nyt)cos(2ωn(1)\nxt)\n+σ(1)\nysin(2ωn(1)\nyt)sin(2ωn(1)\nxt)/bracketrightig\n,\nwithHc(t) given by Eq. (D2). With these albeit compli-\ncated control fields, we are able to implement, at least in14\n02468100.00.20.40.60.81.0\ntF(t)\nFIG. 16. (Color online) Dynamics of the fidelity of the first\nspin with N= 4. We have used λ1= 0.2,λ2= 0.4, and\nλ3= 0.5. The solid, black curve shows the fidelity in the\npresence of the control fields given by Eq. (D2) ( n(1)\nx= 4,\nn(1)\ny= 8,n(2)\nx= 1, and n(2)\ny= 2 with tc= 0.01), while the\ndashed, purple curve shows the fidelity for the first spin in\nthe absence of any control fields. The initial state of the spi n\nchain is |0000/angbracketright. As before, the effect of the environment is\nmodeled by Ornstein-Uhlenbeck processes with σ= 2.0 and\nτ= 0.5.\nprinciple, a single spin operation with high fidelity. The\nperformance of such a single spin protected gate is illus-\ntrated in Fig. 17, fromwhich it is clearthat high fidelities\ncan indeed be achieved.\n0.00.20.40.60.81.00.00.20.40.60.81.0\ntF(t)\nFIG. 17. (Color online) Dynamics of the fidelity of the first\nspin with N= 4. We have used λ1= 0.2,λ2= 0.4, and\nλ3= 0.5. The solid, black curve shows the fidelity in the\npresence of the control fields given by Eq. (D3) with n(1)\nx= 4,\nn(1)\ny= 8,n(2)\nx= 1,n(2)\ny= 2, and tc= 0.01. Here we have\nusedtg= 1. The dashed, purple curve shows the fidelity for\nthe first spin if the single spin operation is implemented via\nthe simple control field −π\n2tgσ(1)\nxin the presence of noise. The\ninitial state of the spin chain is |0000/angbracketright.Appendix E: The Jordan-Wigner Transformation\nOur objective is to find the dynamics with the effective\nHamiltonians\n¯H1=−λ1N−1/summationdisplay\nj=1σ(j)\nyσ(j+1)\ny,\nwhile\n¯H2=N−1/summationdisplay\nj=1/braceleftbigg(λ1+λ2)\n2/bracketleftig\nσ(j)\nxσ(j+1)\ny+σ(j)\nyσ(j+1)\nx/bracketrightig\n−λ1/bracketleftig\nσ(j)\nyσ(j+1)\ny/bracketrightig/bracerightbigg\n.\nAs mentioned before, we largely follow the approach pre-\nsented in Refs. [66, 68]. First, we introduce the rais-\ning and lowering operators a†\ni=1\n2(σ(i)\nx+iσ(i)\ny) and\nai=1\n2(σ(i)\nx−iσ(i)\ny). Thereafter, the fermionic operators\nciandc†\niare defined as\nci= exp\nπii−1/summationdisplay\nj=1a†\njaj\nai,\nc†\ni=a†\niexp\n−πii−1/summationdisplay\nj=1a†\njaj\n.\nUsing the Jordan-Wigner transformation for ¯H2, we get\n(E1) ¯H2=λ1/summationdisplay\ni,j/bracketleftbigg\nc†\niJijcj+1\n2(e−iφc†\niKijc†\nj+h.c.)/bracketrightbigg\n,\nwhere\nJ=−\n0 1 0\n1 0 1 0\n0 1 0 1 0\n. . . . . .\n. . . . .\n. . . . .\n0 1 0 1\n0 1 0\n,\nK=γ\n0 1 0\n−1 0 1 0\n0−1 0 1 0\n. . . . . .\n. . . . .\n. . . . .\n0−1 0 1\n0−1 0\n,15\nand\nγ=/radicalbig\nλ2\n1+(λ1+λ2)2\nλ1,\nφ= arctan[(λ1+λ2)/λ1].\nThe form of ¯H1after the Jordan-Wigner transformation\nis found to be the same as that in Eq. (E1), except that\nnowφ= 0 andγ= 1.\nFollowing Ref. [66], we now look for a linear transfor-\nmation of the form\n(E2) ηk=/summationdisplay\nigkici+hkic†\ni,\n(E3) η†\nk=/summationdisplay\nig∗\nkic†\ni+h∗\nkici,\nsuch that ¯H2becomes\n(E4) ¯H2=/summationdisplay\nkΛkη†\nkηk+ constant.\nThe constant term is unimportant. Our central task in\nfinding the dynamics is to find the eigenvalues Λ k. To do\nthis, we note that if Eq. (E4) is true, then\n(E5) [ηk,H]−Λkηk= 0\nSubstituting Eq. (E2) in Eq. (E5), we get\n(E6)/summationdisplay\nm[gkmJmj−eiφhkmKmj] =Λk\nλ1gkj,\n(E7)/summationdisplay\nm[gkmKmje−iφ−hkmJmj] =Λk\nλ1hkj.\nThese are further simplified by introducing the linear\ncombinations\nΦki=gki+eiφhki,\nΨki=gki−eiφhki.\nEqs. (E6) and (E7) can be cast in the form of matrix\nequations as\n(E8) Φk(J−K) =Λk\nλ1Ψk,\n(E9) Ψk(J+K) =Λk\nλ1Φk,\nin where Ψ kand Φ kdenote the kthrow of matrices Φ\n(whose matrix elements are given by Φ ki) and Ψ (whosematrix elements are given by Ψ ki) respectively. Elimi-\nnating Ψ k, we get\n(E10) Φk(J−K)(J+K) =/parenleftbiggΛk\nλ1/parenrightbigg2\nΦk.\nWethenviewEq.(E10)asaneigenvalueproblemtosolve\nforΛk. However,for ¯H1, it turns outthat Λ kcanbe zero,\ntherefore Φ kand Ψ kare solved using Eqs. (E6) and (E7)\nas a null space problem.\nWe now aim to find the concurrence for any two spins\nin our spin chain. Since the Pauli matrices form a basis,\nwe can write the two-spin state as\nρ(ij)(t) = Tr ij[ρtot(t)]\n=1\n4/summationdisplay\nαβΘi,j\nαβ(t)σ(i)\nα⊗σ(j)\nβ,\nwherewehaveintroducedthetime-dependentcorrelation\nfunctions Θi,j\nαβ=/angbracketleftσ(i)\nασ(j)\nβ/angbracketright= Trij[ρ(ij)(t)σ(i)\nασ(j)\nβ], and\nρtot(t) is the density matrix for the complete spin chain.\nOnce we can figure out these correlation functions, we\nhave the relevant two-spin density matrix, and thereby\nthe concurrence. To calculate the correlation functions,\nwepush the time dependence to the operators. We define\nci(t) =ei¯H2tcie−i¯H2t,\nc†\ni(t) =ei¯H2tc†\nie−i¯H2t.\nThe dynamics for ¯H1can be found analogously. To find\nthetime-evolvingoperators ci(t)andc†\ni(t), thestrategyis\nto first transform the operators ciandc†\nito the operators\nηkandη†\nk, since the Hamiltonian is diagonal in terms of\nηkandη†\nk. We then transform back to the operators ci\nandc†\ni. The result is that we can write\nci(t) =/summationdisplay\nlAil(t)cl+Bil(t)c†\nl,\nand\nc†\ni(t) =/summationdisplay\nlA∗\nil(t)c†\nl+B∗\nil(t)cl,\nwhere the matrices AandBare defined as\nA(t) =g′(t)g+h′(t)h∗,\nB(t) =g′(t)h+h′(t)g∗.\nHereg′(t) =g′e−itHandh′(t) =h′eitH, with\nH=λ1\nΛ1. .\n0 Λ2.\n0 0 Λ 30\n. . . . . .\n. . . . .\n. . . . .\n0 0 Λ N−10\n0 0 Λ N\n.16\nThe matrices gandhare the transformation matrices\ngiven in Eqs. (E2) and (E3), while g′andh′are the\ninverse transformation matrices, that is,\nci=/summationdisplay\njg′\nijηj+h′\nijη†\nj,\nc†\ni=/summationdisplay\njg′∗\nijη†\nj+h′∗\nijηj.\nWith the matrices AandBat hand, we calculate thecorrelation functions. For example,\nΘl,m\nxx(t)\n=/angbracketleftψ|Ql(t)Pl+1(t)Ql+1(t)...Pm−1(t)Qm−1(t)Pm(t)|ψ/angbracketright,\n(E11)\nwherePl(t) =c†\nl(t)+cl(t) andQl(t) =c†\nl(t)−cl(t). We\nnow choose our spin chain state to be |11...1/angbracketright. Just like\nthe results in Refs. [68, 75] for the standard XY model,\nΘl,m\nxx(t) can be expressed in Pfaffian form, that is,\nΘl,m\nxx(t) = pf\n0Fl,l+1Sl,l+1···Sl,m−1Fl,m\n0Wl+1,l+1···Wl+1,m−1Tl+1,m\n··· · ·\nWm−1,m−1Tm−1,m\nFm−1,m\n0\n, (E12)\nSimilarly, we also obtain\nΘl,m\nyy= (−1)m−lpf\n0Wl,l+1Tl,l+1···Tl,m−1Wl,m\n0Fl+1,l+1···Fl+1,m−1Sl+1,m\n··· · ·\nFm−1,m−1Sm−1,m\nWm−1,m\n0\n, (E13)\nΘl,m\nxy(t) =−ipf\n0Fl,l+1Sl,l+1···Sl,m−1Sl,m\n0Wl+1,l+1···Wl+1,m−1Wl+1,m\n··· · ·\nWm−1,m−1Wm−1,m\nSm−1,m\n0\n, (E14)\nΘl,m\nyx(t) =−ipf\n0Tl,l+1Wl,l+1···Wl,m−1Tl,m\n0Wl+1,l+1···Wl+1,m−1Tl+1,m\n··· · ·\nWm−1,m−1Tm−1,m\nFm−1,m\n0\n, (E15)\nΘl,m\nzz(t) = pf\n0Wl,lTl,mWl,m\n0Fl,mSl,m\n0Wm,m\n0\n, (E16)\nwhere\nFi,j=/angbracketleftQi(t)Pj(t)/angbracketright,\nSi,j=/angbracketleftQi(t)Qj(t)/angbracketright,\nTi,j=/angbracketleftPi(t)Pj(t)/angbracketright,\nWi,j=/angbracketleftPi(t)Qj(t)/angbracketright.Since the initial state is |11...1/angbracketright,\n/angbracketleftQi(t)Pj(t)/angbracketright= (Y(t)X†(t))i,j, (E17)\n/angbracketleftQi(t)Qj(t)/angbracketright=−(Y(t)Y†(t))i,j,(E18)\n/angbracketleftPi(t)Pj(t)/angbracketright= (X(t)X†(t))i,j, (E19)\n/angbracketleftPi(t)Qj(t)/angbracketright=−(X(t)Y†(t))i,j.(E20)17\nHereX(t) andY(t) are calculated in terms of A(t) and\nB(t) as\nX(t) =A(t)+B(t)∗\nY(t) =B(t)∗−A(t)\nWe also find that Θl,m\nz,0(t) =/angbracketleftσ(l)\nz/angbracketright=−Wl,l. All the other\ncorrelation functions are zero [68]. With the correlation\nfunctions now known, we can find the two-spin density\nmatrix and hence the concurrence.\nAppendix F: What if nyis not exactly equal to 2nx?\nWehaveshownthatchoosingspecialcontrolfieldssuch\nthatny= 2nxcan lead to better performance. A natural\nquestion that then arises is to investigate how stringently\nthis condition needs to be met. To check this, we have\nconsideredaspin chain N= 4andnumericallysolvedthe\nSchrodinger equation in the presence of the control fields\nwithnynotnecessarilyequalto2 nxtoseehowcloselythe\ndynamicsgeneratedby ¯H2arereproduced. As illustrated\nin Fig. 18, we have found that as nyapproaches 2 nx, the\ndynamics with the full time-dependent Hamiltonian ap-\nproach the dynamics with the effective Hamiltonian ¯H2.\nMoreover,nyneeds to very close to 2 nxfor the exact\ndynamics to be effectively the same as those generated\nby¯H2. That is, if the frequencies used are in the GHzregime [see Fig. 1 caption], then the error in the frequen-\ncies needs to be less than around 10 kHz. However, even\nif the condition ny= 2nxis not met so stringently, the\nentanglement generated can be significant as illustrated\nby the red diamonds and the blue circles in Fig. 18. We\nobtain similar results for N= 8.\n▲▲ ▲ ▲▲ ▲▲ ▲▲ ▲▲▲▲▲ ▲▲ ▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ ▲▲ ▲▲▲▲▲▲▲▲▲▲▲▲ ▲▲ ▲▲▲▲▲ ▲▲ ▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲\n▲\n▲\n▲\n▲\n▲▲▲▲▲▲ ▲▲ ▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲\n▲\n▲\n▲\n▲▲ ▲▲ ▲▲▲▲▲ ▲▲ ▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲\n▲\n▲\n▲\n▲\n▲\n▲\n▲\n▲\n▲\n▲ ▲▲ ▲▲▲▲▲ ▲▲ ▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲\n▲\n▲\n▲\n▲\n▲\n▲\n▲\n▲ ▲▲▲▲▲ ▲▲ ▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲\n▲\n▲\n▲\n▲▲▲▲▲ ▲▲ ▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲\n▲\n▲\n▲\n▲\n▲\n▲\n▲\n▲\n▲▲ ▲▲▲ ▲▲ ▲▲ ▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲\n▲\n▲\n▲\n▲\n▲\n▲\n▲\n▲ ▲▲ ▲▲ ▲▲▲ ▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲\n▲\n▲\n▲\n▲\n▲\n▲\n▲\n▲\n▲\n▲\n▲▲ ▲▲ ▲▲ ▲▲▲ ▲▲ ▲▲ ▲▲▲ ▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲\n▲\n▲\n▲\n▲\n▲\n▲\n▲\n▲▲▲ ▲▲ ▲▲▲ ▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲\n▲\n▲▲▲▲ ▲▲ ▲▲ ▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲\n▲\n▲\n▲\n▲\n▲\n▲\n▲▲ ▲▲ ▲▲ ▲▲ ▲ ▲▲ ▲▲ ▲▲ ▲ ▲▲▲ ◆◆ ◆ ◆◆ ◆◆ ◆◆ ◆◆◆◆◆ ◆◆ ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ ◆◆ ◆◆◆◆◆◆◆◆◆◆◆◆ ◆◆ ◆◆◆◆◆ ◆◆ ◆◆ ◆◆◆◆◆◆◆◆◆◆◆◆◆◆\n◆\n◆\n◆\n◆\n◆◆◆ ◆◆◆ ◆◆ ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆\n◆\n◆\n◆\n◆\n◆ ◆◆ ◆◆ ◆◆◆ ◆◆ ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆\n◆\n◆\n◆\n◆\n◆\n◆\n◆\n◆\n◆◆◆ ◆◆ ◆◆◆ ◆◆ ◆◆ ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆\n◆\n◆\n◆\n◆\n◆\n◆\n◆\n◆◆ ◆◆◆ ◆◆ ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆\n◆\n◆\n◆\n◆\n◆◆◆◆ ◆◆ ◆◆ ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆\n◆\n◆\n◆\n◆\n◆\n◆\n◆\n◆\n◆\n◆◆◆ ◆◆ ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆\n◆\n◆\n◆\n◆\n◆\n◆\n◆\n◆\n◆ ◆◆ ◆◆ ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆\n◆\n◆\n◆\n◆\n◆\n◆◆\n◆◆◆◆◆◆◆ ◆◆ ◆◆◆ ◆◆ ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆\n◆\n◆\n◆\n◆\n◆\n◆\n◆\n◆◆◆ ◆◆ ◆◆ ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ ◆◆ ◆◆ ◆◆◆ ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆\n◆\n◆\n◆\n◆\n◆\n◆\n◆◆ ◆◆◆ ◆◆ ◆◆ ◆◆ ◆◆◆◆◆◆◆◆◆◆◆\n● ●● ● ●●●● ●● ●●●●● ●● ●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●\n●\n●\n●\n●● ●●●●● ●● ●●●●● ●● ●●●●● ●● ●●●●●●●●●●●● ●● ●●●●● ●● ●●●●●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●●●●●●●●●●●\n●\n●●●●● ●● ●●●●● ●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●● ●●●●● ●● ●●●●● ●●●●●●● ●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●● ●●●●● ●● ●●●●● ●● ●●●●● ●● ●●●●● ●●●●●●●●●●●●●��●●●●●●●●● ●●●●● ●● ●●●●●●●●●●●●●●●●●●● ●● ●●●●● ●● ●●●●● ●● ●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●\n●●\n●\n●\n●\n●\n●\n●\n●●● ●● ●●●●● ●● ●●●●● ●●●●●●●●●●\n●\n●●● ●● ●●●●●●●●●●●\n●\n●●●●●●● ●● ●●●●● ●● ●●●● ● ●● ●● ●● ● ● ✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶\n012345670.00.20.40.60.81.0\ntC(t)\nFIG. 18. (Color online) Dynamics of the entanglement be-\ntween spins 1 and 4 for the XYZ model with N= 4. We have\nusedλ1= 2,λ2= 1, and λ3=−5 (same as Fig. 11). We have\nsetnx= 1, and vary the value of ny. We show the dynamics\nwithny= 2.00001 (orange triangles), ny= 2.0001 (red dia-\nmonds), ny= 2.01 (blue circles), and ny= 2.1 (green stars).\nThe solid, black line shows the dynamics with the effective\nHamiltonian ¯H2, while ¯H1(not shown) leads to no entan-\nglement. The green stars largely overlap with the horizonta l\naxis. The initial state is |1111/angbracketright.\n[1] P. Pfeuty, The one-dimensional Ising model with a trans-\nverse field, Ann. Phys., NY 57, 79 (1970).\n[2] A. C. M. Carollo and J. K. Pachos, Geomet-\nric phases and criticality in spin-chain systems,\nPhys. Rev. Lett. 95, 157203 (2005).\n[3] A. Gubin and L. F. Santos, Quantum chaos: an intro-\nduction via chains of interacting spins-1/2, Am. J. Phys.\n80, 246 (2012).\n[4] T. Vuleti´ c, B. Korin-Hamzi´ c, T. Ivek, S. Tomi´ c, B. Gor -\nshunov, M. Dressel, and J. Akimitsu, The spin-ladder\nand spin-chain system (La,Y,Sr,Ca) 14Cu24O41: Elec-\ntronicphases, charge andspindynamics, Phys.Rep. 428,\n169 (2006).\n[5] P. W. Anderson, Absence of diffusion in certain random\nlattices, Phys. Rev. 109, 1492 (1958).\n[6] R. Islam, E. E. Edwards, K. Kim, S. Korenblit,\nC. Noh, H. Carmichael, G. D. Lin, L. M. Duan, C. C.\nJoseph Wang, J. K. Freericks, and C. Monroe, Onset of\na quantum phase transition with a trapped ion quantum\nsimulator, Nat. Commun. 2, 377 (2011).\n[7] J. K. Pachos and P. L. Knight, Quantum com-\nputation with a one-dimensional optical lattice,\nPhys. Rev. Lett. 91, 107902 (2003).\n[8] J. Majer, J. M. Chow, J. M. Gambetta, J. Koch, B. R.\nJohnson, J. A. Schreier, L. Frunzio, D. I. Schuster, A. A.\nHouck, A. Wallraff, A. Blais, M. H. Devoret, S. M.\nGirvin, and R. J. Schoelkopf, Coupling superconductingqubits via a cavity bus, Nature 449, 443 (2007).\n[9] M. Grafe, R. Heilmann, A. Perez-Leija, R. Keil,\nF. Dreisow, M. Heinrich, H. Moya-Cessa, S. Nolte, D. N.\nChristodoulides, and A. Szameit, On-chip generation of\nhigh-order single-photon W-states, Nat. Photon. 8, 791\n(2014).\n[10] S. Bose, Quantum communication through an unmodu-\nlated spin chain, Phys. Rev. Lett. 91, 207901 (2003).\n[11] A. Kay, Perfect state transfer: Beyond nearest-neighb or\ncouplings, Phys. Rev. A 73, 032306 (2006).\n[12] C. Godsil, S. Kirkland, S. Severini, and J. Smith,\nNumber-theoretic nature of communication in quantum\nspin systems, Phys. Rev. Lett. 109, 050502 (2012).\n[13] S. R. Clark, C. M. Alves, and D. Jaksch, Efficient gener-\nation of graph states for quantum computation, New J.\nPhys.7, 124 (2005).\n[14] T. P. Spiller, I. D’Amico, and B. W. Lovett, Entan-\nglement distribution for a practical quantum-dot-based\nquantum processor architecture, New J. Phys. 9, 20\n(2007).\n[15] L. Banchi, A. Bayat, P. Verrucchi, and S. Bose,\nNonperturbative entangling gates between dis-\ntant qubits using uniform cold atom chains,\nPhys. Rev. Lett. 106, 140501 (2011).\n[16] S. Sahling, G. Remenyi, C. Paulsen, P. Monceau,\nV. Saligrama, C. Marin, A. Revcolevschi, L. P.\nRegnault, S. Raymond, and J. E. Lorenzo, Experi-18\nmental realization of long-distance entanglement be-\ntween spins in antiferromagnetic quantum spin chains,\nNat. Phys. 11, 255 (2015).\n[17] M. P. Estarellas, I. D’Amico, and T. P. Spiller,\nRobust quantum entanglement generation and\ngeneration-plus-storage protocols with spin chains,\nPhys. Rev. A 95, 042335 (2017).\n[18] H.-P. Breuer and F. Petruccione, The Theory of Open\nQuantum Systems (Oxford University Press, Oxford,\n2007).\n[19] U. Weiss, Quantum dissipative systems (World Scientific,\nSingapore, 2008).\n[20] L. Viola and S. Lloyd, Dynamical suppression\nof decoherence in two-state quantum systems,\nPhys. Rev. A 58, 2733 (1998).\n[21] L. Viola, E. Knill, and S. Lloyd, Dynam-\nical decoupling of open quantum systems,\nPhys. Rev. Lett. 82, 2417 (1999).\n[22] G. S. Uhrig, Keeping a quantum bit alive by optimized\nπ-pulse sequences, Phys. Rev. Lett. 98, 100504 (2007).\n[23] K. Khodjasteh, D. A. Lidar, and L. Viola, Arbitrarily\naccurate dynamical control in open quantum systems,\nPhys. Rev. Lett. 104, 090501 (2010).\n[24] J. R. West, B. H. Fong, and D. A. Lidar,\nNear-optimal dynamical decoupling of a qubit,\nPhys. Rev. Lett. 104, 130501 (2010).\n[25] G. de Lange, Z. H. Wang, D. Rist` e, V. V. Do-\nbrovitski, and R. Hanson, Universal dynamical decou-\npling of a single solid-state spin from a spin bath,\nScience330, 60 (2010).\n[26] L. Jiang and A. Imambekov, Universal dynamical\ndecoupling of multiqubit states from environment,\nPhys. Rev. A 84, 060302(R) (2011).\n[27] Y. Wang, X. Rong, P. Feng, W. Xu, B. Chong, J.-H.\nSu, J. Gong, and J. Du, Preservation of bipartite pseu-\ndoentanglement in solids using dynamical decoupling,\nPhys. Rev. Lett. 106, 040501 (2011).\n[28] A. Z. Chaudhry, Utilizing nitrogen-vacancy cen-\nters to measure oscillating magnetic fields,\nPhys. Rev. A 90, 042104 (2014).\n[29] T. Manovitz, A. Rotem, R. Shaniv, I. Cohen, Y. Shapira,\nN. Akerman, A. Retzker, and R. Ozeri, Fast dynami-\ncal decoupling of the Mølmer-Sørensen entangling gate,\nPhys. Rev. Lett. 119, 220505 (2017).\n[30] B. Pokharel, N. Anand, B. Fortman, and D. A.\nLidar, Demonstration of fidelity improvement using\ndynamical decoupling with superconducting qubits,\nPhys. Rev. Lett. 121, 220502 (2018).\n[31] H. Y. Carr and E. M. Purcell, Effects of diffusion on free\nprecession in nuclear magnetic resonance experiments,\nPhys. Rev. 94, 630 (1954).\n[32] A. Z. Chaudhry, Detecting the presence of weak\nmagnetic fields using nitrogen-vacancy centers,\nPhys. Rev. A 91, 062111 (2015).\n[33] F. F. Fanchini, J. E. M. Hornos, and R. d. J. Napoli-\ntano, Continuously decoupling single-qubit operations\nfrom a perturbing thermal bath of scalar bosons,\nPhys. Rev. A 75, 022329 (2007).\n[34] A. Z. Chaudhry and J. Gong, Amplification and suppres-\nsion of system-bath-correlation effects in an open many-\nbody system, Phys. Rev. A 87, 012129 (2013).\n[35] F. F. Fanchini and R. d. J. Napolitano, Continuous\ndynamical protection of two-qubit entanglement from\nuncorrelated dephasing, bit flipping, and dissipation,Phys. Rev. A 76, 062306 (2007).\n[36] A. Z. Chaudhry and J. Gong, Decoherence con-\ntrol: Universal protection of two-qubit states and\ntwo-qubit gates using continuous driving fields,\nPhys. Rev. A 85, 012315 (2012).\n[37] F. F. Fanchini, R. d. J. Napolitano, B. Cakmak, and\nA. O. Caldeira, Protecting the√swapoperation from\ngeneral and residual errors by continuous dynamical de-\ncoupling, Phys. Rev. A 91, 042325 (2015).\n[38] A. Z. Chaudhry and J. Gong, Protecting and enhanc-\ning spin squeezing via continuous dynamical decoupling,\nPhys. Rev. A 86, 012311 (2012).\n[39] A. Ajoy and P. Cappellaro, Quantum simulation\nvia filtered Hamiltonian engineering: Application\nto perfect quantum transport in spin networks,\nPhys. Rev. Lett. 110, 220503 (2013).\n[40] H.Frydrych, G.Alber, andP. Baˇ zant, Constructingpau li\npulse schemes for decoupling and quantum simulation,\nPhys. Rev. A 89, 022320 (2014).\n[41] D. Hayes, S. T. Flammia, and M. J. Biercuk, Pro-\ngrammable quantumsimulation bydynamicHamiltonian\nengineering, New. J. Phys. 16, 083027 (2014).\n[42] S. Choi, N. Y. Yao, and M. D. Lukin, Dynam-\nical engineering of interactions in qudit ensembles,\nPhys. Rev. Lett. 119, 183603 (2017).\n[43] J. B. Parkinson and D. J. J. Farnell, An Introduction to\nQuantum Spin Systems (Springer, Berlin, 2010).\n[44] I. Dzyaloshinsky, A thermodynamic theory of “weak”\nferromagnetism of antiferromagnetics, J. Phys. Chem.\nSolids4, 241 (1958).\n[45] T. Moriya, Anisotropic superexchange interaction and\nweak ferromagnetism, Phys. Rev. 120, 91 (1960).\n[46] M. Kargarian, R. Jafari, and A. Langari, Dzyaloshinski i-\nMoriya interaction and anisotropy effects on\nthe entanglement of the Heisenberg model,\nPhys. Rev. A 79, 042319 (2009).\n[47] R. Jafari, M. Kargarian, A. Langari, and M. Sia-\nhatgar, Phase diagram and entanglement of the\nIsing model with Dzyaloshinskii-Moriya interaction,\nPhys. Rev. B 78, 214414 (2008).\n[48] E. Mehran, S. Mahdavifar, and R. Jafari, Induced ef-\nfects of the Dzyaloshinskii-Moriya interaction on the\nthermal entanglement in spin-1/2 Heisenberg chains,\nPhys. Rev. A 89, 042306 (2014).\n[49] G. M. Nikolopoulos, D. Petrosyan, and P. Lambropou-\nlos, Electron wavepacket propagation in a chain of cou-\npled quantum dots, J. Phys.: Condens. Matter 16, 4991\n(2004).\n[50] L. Campos Venuti, S. M. Giampaolo, F. Illumi-\nnati, and P. Zanardi, Long-distance entanglement\nand quantum teleportation in xxspin chains,\nPhys. Rev. A 76, 052328 (2007).\n[51] C. Di Franco, M. Paternostro, and M. S. Kim, Perfect\nstate transfer on a spin chain without state initialization ,\nPhys. Rev. Lett. 101, 230502 (2008).\n[52] A. Kay, A review of perfect state transfer and its appli-\ncation as a constructive tool, Int. J. Quantum Inf. 8, 641\n(2010).\n[53] Y. Wang, F. Shuang, and H. Rabitz, All possible cou-\npling schemes in XYspin chains for perfect state trans-\nfer, Phys. Rev. A 84, 012307 (2011).\n[54] G. M. Nikolopoulos and I. Jex, Quantum state transfer\nand network engineering (Springer, 2013).19\n[55] K. Korzekwa, P. Machnikowski, and P. Horodecki,\nQuantum-state transfer in spin chains\nvia isolated resonance of terminal spins,\nPhys. Rev. A 89, 062301 (2014).\n[56] M. Christandl, N. Datta, A. Ekert, and A. J. Lan-\ndahl, Perfect state transfer in quantum spin networks,\nPhys. Rev. Lett. 92, 187902 (2004).\n[57] G. M. Nikolopoulos, D. Petrosyan, and P. Lambropoulos,\nCoherent electron wavepacket propagation and entangle-\nment in array of coupled quantum dots, Europhys. Lett.\n65, 297 (2004).\n[58] M.-H. Yung, Quantum speed limit for\nperfect state transfer in one dimension,\nPhys. Rev. A 74, 030303(R) (2006).\n[59] D. Petrosyan, G. M. Nikolopoulos, and P. Lambropoulos,\nState transfer in static and dynamic spin chains with\ndisorder, Phys. Rev. A 81, 042307 (2010).\n[60] H. Frydrych, A. Hoskovec, I. Jex, and G. Alber, Selec-\ntive dynamical decoupling for quantum state transfer, J.\nPhys. B: At. Mol. Opt. Phys. 48, 025501 (2014).\n[61] A. Zwick, G. A. Alvarez, G. Bensky, and G. Kurizki,\nOptimized dynamical control of state transfer through\nnoisy spin chains, New. J. Phys. 16, 065021 (2014).\n[62] K. Jacobs, Stochastic Processes for Physicists: Under-\nstanding Noisy Systems (Cambridge University Press,\n2010).\n[63] W. K. Wootters, Entanglement of forma-\ntion of an arbitrary state of two qubits,\nPhys. Rev. Lett. 80, 2245 (1998).\n[64] F. Galve, D. Zueco, S. Kohler, E. Lutz, and\nP. H¨ anggi, Entanglement resonance in driven spin chains,\nPhys. Rev. A 79, 032332 (2009).[65] X. Wang, Entanglement in the quantum heisenberg XY\nmodel, Phys. Rev. A 64, 012313 (2001).\n[66] E. Lieb, T. Schultz, and D. Mattis, Two soluble models\nof an antiferromagnetic chain, Ann.Phys. (Paris) 16, 407\n(1961).\n[67] E. Barouch, B. M. McCoy, and M. Dres-\nden, Statistical mechanics of the XY model. i,\nPhys. Rev. A 2, 1075 (1970).\n[68] L. Amico and A. Osterloh, Out of equilibrium correla-\ntion functions of quantum anisotropic XY models: one-\nparticle excitations, J. Phys. A: Math. Gen. 37, 291\n(2004).\n[69] J. Vidal, R. Mosseri, and J. Dukelsky, Entan-\nglement in a first-order quantum phase transition,\nPhys. Rev. A 69, 054101 (2004).\n[70] F. M. Cucchietti, J. P. Paz, and W. H.\nZurek, Decoherence from spin environments,\nPhys. Rev. A 72, 052113 (2005).\n[71] R. Jafari and H. Johannesson, Decoherence from spin\nenvironments: Loschmidt echo and quasiparticle excita-\ntions, Phys. Rev. B 96, 224302 (2017).\n[72] M. Majeed and A. Z. Chaudhry, Effect of initial system-\nenvironment correlations with spin environments, Eur. J.\nPhys. D 73, 16 (2019).\n[73] D. D. Bhaktavatsala Rao and G. Kurizki, From Zeno\nto anti-Zeno regime: Decoherence-control dependence\non the quantum statistics of the bath, Phys Rev A 83,\n032105 (2011).\n[74] K. Khodjasteh and L. Viola, Dynamically error-\ncorrected gates for universal quantum computation,\nPhys. Rev. Lett. 102, 080501 (2009).\n[75] E. Barouch and B. M. McCoy, Statistical mechan-\nics of the XY model. ii. spin-correlation functions,\nPhys. Rev. A 3, 786 (1971)."    },    {        "title": "2207.11978v1.Spin_Coulomb_drag_by_non_equilibrium_magnetic_textures.pdf",        "content": "arXiv:2207.11978v1  [cond-mat.mes-hall]  25 Jul 2022Spin Coulomb drag by non-equilibrium magnetic textures\nIgor Lyapilin\nInstitute of Metal Physics, UD RAS,\nEkaterinburg, 620137, Russia\nUral Federal University, Ekaterinburg\n(Dated: July 26, 2022)\nInteraction between local magnetization and conduction el ectrons is responsible for a variety of\nphenomena in magnetic materials. We have shown that the spin -dependent motive force induced\nby magnetization dynamics in a conducting ferromagnet lead to the spin Coulomb drag effect. The\nspin Coulomb drag an intrinsic friction mechanism which ope rates whenever the average velocities\nof up-spin and down-spin electrons differ.\nIntroduction: The spatial-temporal dynamics of mag-\nnetization /vectorM(/vector r,t) in a ferromagnetic system is described\nby the phenomenological Landau-Lifshitz-Gilbert (LLG)\nequation [1]. Along with this, there is also an inverse\nprocess when the spatial-temporal change in magneti-\nzation induces additional electromagnetic fields acting\non conduction electrons. The spin-driving force (SDF)\ncaused by these fields was first predicted by Berger [2]\nand has been recently formulated by generalizing Fara-\nday’s law [3]. These fields are non-conservative and spin-\ndependent. Explicit expressions for induced spin electro-\nmagnetic fields can be represented as [4–6]:\nE↑↓\ni=±¯h\n2e/vector m·(∂t/vector m×∂i/vector m),\nB↑↓\ni=∓¯h\n2eǫijk/vector m·(∂j/vector m×∂k/vector m), (1)\nHere/vector m=/vectorM/|/vectorM|isaunitvectordirectedalongthemag-\nnetization /vectorM(/vector r,t). Arrows ( ↑,↓) characterize the elec-\ntron spin directions.\nThe fields in (1) generate a spin-dependent Lorentz\nforce/vectorF↑↓\ns=e(/vectorEs+/vector v×/vectorB↑↓\ns). The manifestation of a\nspin-driving force is a universal phenomenon in magnetic\nmetals and can be understood through the gauge field\ntheory, the equation of motion, and Berry’s spin phase\n[7, 8]. A spin electric field was experimentally observed\nin the case of a moving domain wall [9].\nUnlike conventional electromagnetic fields, spin elec-\ntric and magnetic fields /vectorEs,and/vectorBsdramatically af-\nfect the kinetics of conduction electrons. In the field\n/vectorEs, electrons with spins ↑, or↓drift in opposite direc-\ntions; thereby inducing a spin-polarized current, /vectorjs=\n(n↑−n↓)e/vector vs.Heree, and/vector vs,are the charge of an elec-\ntron and the drift velocity of the carriers due to the spin\nelectric field Es, respectively. When emerging, the spin\nmagnetic field associated with the Berry phase produces\nthe spin Hall effect that is detected as an anomalous Hall\neffect[10]. TheLorentzforcecausedbythespinmagnetic\nfield deflects free charge carriers in opposite directions,\ndepending on the orientation of the spin ( ↑,↓), thereby\ninducing a Hall voltage and a spin current if the electron\nconcentrations n↑/negationslash=n↓.Here special attention should be drawn to one more\ncharacteristic feature of spin electromagnetic fields. As\ncan be seen, the electric and magnetic components of the\nspin fields act as a separator, selecting free charge carri-\ners in the spin direction ( ↑,↓). Consequently, the system\nofconductionelectronscanbe treatedasaset oftwospin\nsubsystems,eachofwhichischaracterizedbyitsownspin\ndirection. Since the electric field governs the drift veloc-\nity of charge carriers, it is obvious that the drift velocity\nof each of the subsystems is spin-dependent. It can be\nassumed that the spin-dependent fields create conditions\nunder which the manifestations of effects associated with\ntheCoulombinteractionoftwospinsubsystemsofcharge\ncarriers become possible.\nElectron-electron (e-e) interactions play a critical role\nin various phenomena, ranging from high-temperature\nsuperconductivity, the fractional quantum Hall effect, a\nWigner crystal, a Mott transition, etc. However, both\nthe magnitude and the influence of this interaction on\nthe kinetic properties of crystals are difficult to measure.\nOne of the methods that proved to be effective in mea-\nsuring scatteringrates directly due to the Coulomb inter-\naction is the Coulomb drag effect of charge carriers. The\nessence of the effect is to arise a response as a generated\nvoltage or electric current in a conductive system when\ncurrentpassesthroughanotherconductivefilm separated\nfrom the first one by a dielectric layer [11] (see literature\nthere). The effect is underlain by the interlayer Coulomb\ninteraction of conduction electrons separated by a dielec-\ntric.\nThe drag effect of charge carriers due to the Coulomb\ninteraction exhibits itself also in a system of spin-\npolarized charge carriers. Consider the conductivity of\ncarriers with different spins within a two-channel model\n(conductivity along two channels with different spin ori-\nentations ( ↑,↓) in each channel). Let the drift veloc-\nitiesv↑, v↓of carriers in each of the spin subsystems\nbe different. If other electron scattering sources lack,\nthe Coulomb interaction preservesthe system’s total mo-\nmentum and redistributes it between charge carriers in\ndifferent ( ↑) or (↓) channels. In this case, faster carri-\ners drag slow ones through transferring a part of their\nmomentum to implement the spin Coulomb drag effect2\n(SCD) [12]. Then, it is obvious that the charging current\nje=e(n↑+n↓)ve(veisthedriftvelocityofelectrons)will\nbeunchangedagainstthespincurrent js= (n↑v↑−n↓v↓).\nWhen the drift velocities of carriers in the spin channels\nare equalized, v↑=v↓, the spin current is js= 0, if\nn↑=n↓.\nIt should be underscored that for the spin Coulomb\ndrag effect to be observed, the spin system of conduc-\ntion electrons divides into two subsystems not physically\nbut through only the spin orientation ( ↑,↓). Such a rep-\nresentation of the electron system for the manifestation\nof the SCD effect is fundamental. Some of the possible\nscenarios for such a representation are considered in [12].\nNote that the concomitant spin electric and magnetic\nfields produced in the case of a spatial-temporal inhomo-\ngeneousmagneticstructurenaturallylead tospin separa-\ntion of charge carriers to provide the condition necessary\nfor SCD to arise as the difference in the drift velocities\nof electrons in spin channels.\nThe quantitative measure of SCD is the transresistiv-\nityρ↑↓; it is defined as the ratio between the electro-\nchemical potential gradient ζ↑for electrons with spin ↑\nand the electric current j↓,(j↓= 0).This definition is\nsimilar to that of the Coulomb drag effect in two spa-\ntially separated layers. The results of calculating ρ↑↓for\na three-dimensionalelectrongas in the random-phaseap-\nproximation are presented in [11, 13].\nNext, we demonstrate that the separation of charge\ncarriers by their spin state, associated with the concomi-\ntant spin electric and magnetic fields, when taking into\naccount the Coulomb interaction between charge carri-\ners, leads to the effect of spin Coulomb drag of charge\ncarriers.\nThe spin Coulomb drag effect can be calculated apply-\ning the Kubo method, Green’s functions, or the Boltz-\nmann kinetic equation. However, it is easier to under-\nstand SCD at the phenomenological level.\nSo, let us look into an electric current flowing through\na non-equilibrium inhomogeneous magnetic system; the\nlatter’s magnetization /vectorM(/vector r,t) has a spatial-temporal in-\nhomogeneity. The Hamiltonian of the system at hand\nis:\nH=He+HeE+Hsd+Hee+Hev+Hv,\nwhereHe=Hk+Hsis the Hamiltonian of free conduc-\ntionelectrons, HeEisinteractionwithanexternalelectric\nfield/vector e0.Hsdis exchange interaction between free and lo-\ncalized electrons. Heeis Coulomb interaction between\nconduction electrons. Hevdescribes the interaction with\nscattering centers. Hvis the lattice Hamiltonian.\nIt is assumed that upon performing the Hamiltonian’s\ncanonical transformation that diagonalizes the exchange\ninteraction, we arrive at the realization of concomitant\nspin electric Esand magnetic Bsfields. As a conse-\nquence, the divisionofthe systemofconductionelectrons\ninto two spin subsystems ( ↑,↓) takes place.To start with, we explore the SCD effect purely phe-\nnomenologically. For this, we resort to the equation of\nmotionm(d/vector υ/dt) =/summationtext\ni/vectorFi. The drift velocities of elec-\ntrons/vector υ↑, /vector υ↓are formed by forces acting on electrons,\ninduced by electric fields and the Coulomb interaction\nbetween electrons:\n/vector υ↑↓=<(/vectorF/vectorE∗+/vectorFc)τ\nm>\nHere/vectorF/vectorE∗+/vectorFcare the forces acting on the electrons from\nthe electric E∗=/vectorE0+Esfields and are also due to the\nCoulombinteraction /vectorFc. Timeτcharacterizesdissipative\nprocesses.\nFor simplicity, we restrict ourselves to investigation of\na spin subsystem of conduction electrons (for example,\nwith spin α=↑,↓). Letυαbe the speed of the mass-\ncenter,nαis the number ofsuch chargecarriers. We write\ndown an equation of motion for the electrons of the spin\nsubsystem with spin ( α=↑,↓) as follows:\n˙pα=−enα/vectorE∗+Fα,−α\nc+ ˙pα\nev, (2)\nwhere/vectorE∗=/vectorE0+/vectorEα./vectorE0,/vectorEαare the external and spin\nelectric fields. The first summand on the right side of\nthe equation involves the interaction of electrons with\nelectric fields. The second summand is responsible for\nthe Coulomb interaction between electrons with differ-\nent spin orientations (different spin subsystems). At the\nsame time, it is obviously that F↑↓\nc=−F↓↑\nc. Preserv-\ning the total momentum of the electronic system, the\nCoulomb interaction of electrons redistributes it between\nthe spin ( ↑,↓) subsystems of electrons (see figure). Then\n|n↑/vector vc|=|n↓/vector vc|, where /vector vcis the momentum lost (ac-\nquired) in a single electron-scattering act. The last term\n˙p↑\nev= (i¯h)−1[p↑,Hev] as relaxational one can be written\nin the relaxation time approximation.\nBeforeproceedingto the microscopicdescription ofthe\nSCD effect, consider it phenomenologically. For this, we\nwrite down expressionsfor the total charge jeand spin js\ncurrents, believing that each of the forces acting on the\nconduction electrons contributes to forming the electron\ndrift velocity. Besides, the momentum is assumed to be\ntransferred from the subsystem ↑to the subsystem ↓due\nto the Coulomb interaction. An expressionfor the charge\ncurrent is given as\nje=e[n↑(/vector υ0+/vector υs−/vector υc)+n↓(/vector υ0−/vector υs+/vector υc)],\nwhere/vector υ0,/vector υs,/vector υcare the components of the drift velocity\nof electrons, due to the action of the electric fields E0,Es\nand Coulomb interaction\nIf the charge current is spin-polarized ( n↑/negationslash=n↓, n=\nn↑+n↓), then\n/vectorje=e[n/vector υ0+(n↑−n↓)(/vector υs−/vector υc)] =en/vector υ0,\ne.g. the drag effect contributes to the spin-polarized cur-\nrent until /vector υs/negationslash=/vector υc. When /vector υs=/vector υc, the contribution to3\nthe spin-polarized current induced by the spin drag effect\nvanishes and /vectorje=e[(n↑+n↓)]/vector υ0.\nConsider an expression for the spin current. We have\n/vectorjs=e[n↑(/vector υ0+/vector υs−/vector υc)−n↓(/vector υ0−/vector υs+/vector υc)].\nor\n/vectorjs=e[(n↑−n↓)/vector υ0+n(/vector υs−/vector υc).\nAs can be seen, at n↑=n↓, the spin current is nonzeroas\nlong as there is a possibility of pumping the momentum\nbetween spin subsystems and /vectorjs= 0 when /vector υs=/vector υc. If\nn↑/negationslash=n↓, the SCD effect in the spin current manifests\nitself until pumping the electron momentum between the\nspin subsystems is possible to happen. When /vector υs=/vector υc,\ntheeffectoftheSCDturnstozeroand /vectorjs=e[(n↑−n↓)/vector υ0.\nUsing the mass-center approximation, the expression\nthat determines the Coulomb part of the force /vectorFccan be\nwritten as [12]\n/vectorFc≡/vectorF↑↓=−dmn↑(n↓/n)(/vector υ↑−/vector υ↓),(3)\nwheren=n↑+n↓. The coefficient dcontrols the\nspin drag. An explicit expression for the coefficient\ndcomes from the expression for the resistance matrix\n/vectorE↑=/summationtext\n↓ρ↑↓/vectorj↓. Then\nd=ne2\nmρ↑↓. (4)\nIn the Borninteractionapproximation,the expressionfor\ntransresistivity ρ↑↓can be deduced by various methods,\nfor example, such as the formalism of the Green’s func-\ntion method, the method of projection operators [14], or\nwithin the Boltzmann kinetic equation [16]. The micro-\nscopic theory of calculating the quantity ρ↑↓is similar\nto calculating the conventional Coulomb resistance be-\ntween parallel two-dimensional layers of an electron or\nhole gas [11]. However, it differs in some important as-\npects. Firstly, chargecarrierswithoppositespinsinteract\nwith the same set of scattering centers. Secondly, the en-\nergy spectrum of chargecarriersbecomes spin-dependent\nas a result of the canonical transformation of the original\nHamiltonian.\nIn keeping with [14, 15], we write down a general ex-\npression for ρ↑↓:\nρ↑↓=π¯h2\n4n↑n↓m2/summationdisplay\n/vector p′,/vector p,/vector q∞/integraldisplay\n−∞dωq2|V(q)|2f′\n1(ε/vector p,↑)\n×(f2(ε/vector p′,↓)−f2(ε/vector p′+/vector q,↓))/braceleftbiggf1(ε/vector p−/vector q,↑)\ne∆−1−1−f1(ε/vector p−/vector q,↑)\n1−e−∆]/bracerightbigg\nδ(ε/vector p−/vector q,↑−ε/vector p,↑+¯hω)δ(ε/vector p′+/vector q,↓−ε/vector p′,↓−¯hω).(5)\nHere ∆ = ¯ hω/kBT,kBis the Boltzmann constant, Vq=\n4π2/q2ǫisFouriercomponentoftheinteractionpotential,\nǫis the dielectric constant of material, fi(ε/vector p,σ),(i= 1,2)are the Fermi-Diracdistribution functions for conduction\nelectrons.\nGoingoverfromthesummationovervectors /vector p, /vector p′toin-\ntegrationandevaluating the integrals, we arriveat an ex-\npression for the degenerate statistics of conduction elec-\ntrons and ¯ hω≪kBT:\nρ↑↓=β\n6πn↑n↓e2V/summationdisplay\n/vector qq2∞/integraldisplay\n0dω×\n|V(q)|2χ′′\n0↑(q,ω)χ′′\n0↓(q,−ω)\n|ǫ(q,ω)|2sinh2(βω/2),(6)\nwhereβ= 1/kBT.Vis the volume of the system.\nχ0α(q,ω) is the noninteracting spin resolved density-\ndensity responsefunction, and ε(q,ω) = 1−Vqχ0↑(q,ω)−\nVqχ0↓(qω) is the random-phase approximation dielectric\nfunction.\nFurther calculations of the transresistivity ρ↑↓, as well\nas its various approximations, accounting for dissipation\nprocesses and possible methods for detecting the spin\nCoulomb drag effect face no difficulties (see, for example,\n[13, 17]).\nFinally, we note that the presence of a spin-dependent\nmagnetic field B↑↓acting differently on charged parti-\ncles with spin ( ↑,↓) (shifts them in opposite directions),\nalso affects the formation of drift velocities of charge car-\nriers, and, consequently, the charge and spin currents,\ndepending on the direction of their spin. Thus, the spin-\ndependent magnetic field will also manifest itself in the\nspin Coulomb drag effect.\nThe research was carried out within the state assign-\nment of Ministry of Science and Higher Education of the\nRussian Federation (theme ”Spin” No. 122021000036-3).\n[1] Y.Tserkovnyak A.Brataas,G.E.W Bauer,\nPhys.Rev.Lett. 88, 117601-4 (2002).\n[2] L.Berger,Phys. Rev. B. 33, 1572 (1986).\n[3] S. E.Barnes and S.Maekawa, Phys. Rev. Lett. 98,\n246601-4 (2007).\n[4] Y. Yamane, J. Ieda, J. Ohe, et all, J. Appl. Phys 109,\n07C735-3 (2011).\n[5] J. Ohe, S.Maekawa, J. Appl.Phys 105, 07C706-3 (2009).\n[6] G. Tatara, Physica E: Low-dimensional Systems and\nNanostructures 106, 208 (2019).\n[7] M. V. Berry, Proc. R. Soc. Lond. A 392, 45 (1984)\n[8] A. Stern, Phys. Rev. Lett. 68, 1022 (1992).\n[9] S. A. Yang, G. S. D. Beach, C. Knutson, et all, Phys.\nRev. Lett. 102, 067201-4 (2009).\n[10] N.Nagaosa, Rev. Mod. Phys. 82, 1539-1593 (2010) .\n[11] A. G. Rojo,J. Phys. Condens. Matter 11, R31-R52\n(1999).\n[12] I. D’Amico, G. Vignale, Europhys. Lett. 55, 566 (2001).\n[13] D’Amico and G. Vignale, Phys. Rev. B 62, 4853 (2000)\n[14] I.I. Lyapilin, H.M. Bikkin, Fiz TverdTela 45, 339 (2003).4\n[15] Antti-Pekka Jauho, H. Smith, Phys. Rev. B 47, 4420\n(1993)\n[16] D’Amico and G. Vignale, Phys. Rev. B 65, 085109-12\n(2002). Phys. Rev. B 33, 1572 (1986).[17] B. N. Narozhny, A. Levchenko, Rev. Mod Phys. 88,\n025003-55 (2016)."    },    {        "title": "1603.06471v1.Rashba_semiconductor_as_spin_Hall_material__Experimental_demonstration_of_spin_pumping_in_wurtzite__n__GaN_Si.pdf",        "content": "arXiv:1603.06471v1  [cond-mat.mes-hall]  21 Mar 2016Rashba semiconductor as spin Hall material: Experimental d emonstration of spin\npumping in wurtzite n-GaN:Si\nR. Adhikari,1,∗M. Matzer,1A. Tarazaga Mart´ ın-Luengo,1and A. Bonanni1,†\n1Institut f¨ ur Halbleiter-und-Festk¨ orperphysik, Johann es Kepler University, Altenbergerstr. 69, A-4040 Linz, Aus tria\n(Dated: May 7, 2018)\nAbstract\nPure spin currents in semiconductors are essential for impl ementation in the next generation of spintronic elements. H et-\nerostructures of III- nitride semiconductors are currentl y employed as central building-blocks for lighting and high -power devices.\nMoreover, the long relaxation times and the spin-orbit coup ling (SOC) in these materials indicate them as privileged ho sts\nfor spin currents and related phenomena. Spin pumping is an e fficient mechanism for the inception of spin current and its\nconversion into charge current in non-magnetic metals and s emiconductors with Rashba SOC viaspin Hall effects. We report\non the generation in n-GaN:Si – at room temperature and through spin pumping – of pu re spin current, fundamental for the\nunderstanding of the spin dynamics in these non-centrosymm etric Rashba systems. We find for n-GaN:Si a spin Hall angle\nθSH=3.03×10−3, exceeding by one order of magnitude those reported for othe r semiconductors, pointing at III-nitrides as\nparticularly efficient spin current generators.\nPACS numbers: 72.25.Dc, 72.25.Mk, 76.50.+g, 85.75.-d\n1In the emerging field of spin-orbitronics1–4spin-orbit coupling (SOC) is employed in both magnetic and non-\nmagnetic materials to generate, exploit and detect spin currents. Spin currents hold the key for the realization\nand implementation of the next generation of spin based nanoelectr onic devices with properties like non-volatility,\nlow power consumption and dissipation. While in magnetic materials the S OC is employed to create new classes\nof topological objects like magnetic skyrmions or Dzyalonshinskii-Mo riya domain walls1,5, spin-orbitronics in non-\nmagnetic materials mostly addresses the spin-to-charge convers ion through the spin Hall effect (SHE)6and the\nRashba-Edelstein effect7,8. The concept of SHE is borrowed from the anomalous Hall effect (AH E) where, due to the\nrelativistic SOC, asymmetric deflection of charge carriers takes pla ce depending on their spin orientations9. While\nAHE is studied in magnetic systems, SHE is mostly observed in non-mag netic ones. Based on the concept of spin\ndependent Mott scattering10, the SHE was predicted nearly four decades ago by D’yakonov and P erel’11and was\nproposed to be the effective process for producing pure spin curr ents in solid state systems. However, it was not until\nthe theoretical work of Hirsch etal.12and Zhang etal.13, that the extrinsic SHE received renewed attention. The\npossibility of an intrinsic mechanism was proposed by various theoret ical groups14–16. The intrinsic SHE depends\nonly on the electronic band structure of the material. This effect ar ises from the nonequilibrium dynamics of the\nBloch electrons as they undergo spin precession due to an induced k–dependent effective magnetic field, such as the\nRashba field. The extrinsic SHE, on the other hand, is the mechanism in which the spins acquire transverse velocity\ndue to SOC during scattering of electrons6,9,17. The extrinsic SHE is classified according to two different underlying\nmechanisms viz.the skew scattering and the side jump mechanism. Skew scattering is the asymmetric scattering\nof spin, within the scattering plane, due to an effective magnetic field gradient that arises as an effect of the SOC.\nThe scattering plane defines the spin polarization direction of the re sulting spin current. The side jump mechanism\nis the velocity integrated over time of deflection of electrons in oppo site directions by the electric fields experienced\nwhen approaching or withdrawing an impurity. This phenomenon resu lts in an effective transverse displacement of\nthe electrons upon multiple scattering events. At low carrier mobilitie s∼(10−2-102)cm2V−1s−1, the intrinsic and\nextrinsicsidejump mechanismscontributemostlytothe SHE, whilefo rcarriermobilitiesgreaterthan102cm2V−1s−1,\nextrinsic skewscattering is the dominant process6,17. Moreover,for mobilites exceeding 103cm2V−1s−1, spin Coulomb\ndrag becomes the dominant mechanism17and compels the spin Hall conductivity towards a saturation value. T he\nlack of direct electrical signals proved to be a major challenge in the o bservation of this effect, so that the initial\nexperimental efforts were mostly accomplished using optical means18–20.\nIn a series of seminal publications, Tserkovnyak, Brataas and Bau er21–23suggested a method for obtaining pure\nspin current in non-magnetic (NM) metals and semiconductors with n on-negligible SOC. They proposed a spin\nbattery23based on adiabatic pumping of spins from a ferromagnetic metal or in sulator (FM) grown in a FM/NM\nbilayer configuration when the system is driven to resonance under microwave irradiation. Such a battery leads to\nthe dynamic generation of pure spin current in a NM with non trifling SO Cviaferromagnetic resonance (FMR) of\nthe FM24. The magnetization dynamics of ferromagnets is well described by t he phenomenological Landau-Lifshitz-\nGilbert (LLG) equation25:\nd/vectorM(t)\ndt=−γ/vectorM(t)×/vectorHeff+α\nMs/vectorM(t)×d/vectorM(t)\ndt(1)\nwhere/vectorM(t)/Msis the unit vector of magnetization, γthe gyromagnetic ratio and /vectorHeffthe effective magnetic field\nexpressed as /vectorHeff=/vectorH+/vectorHM(t) +/vectorh(t), with /vectorHthe external magnetic field, /vectorHM(t) the dynamic demagnetizing field\nand/vectorhMW(t) the ac field due to microwave radiation. The magnetization /vectorM(t), is expressed as a sum of the static and\ndynamic components, i.e./vectorM(t) =/vectorM+/vector m(t) and the dimensionless coefficient αis the Gilbert damping parameter.\nThe first term on the right hand side of Eq.( 1) is the precession term, while the second term represents a dampin g\ncomponent that impels the precession of magnetization /vectorM(t) to spiral down to a static magnetization axis due to\nthe Gilbert damping parameter α. In a FM/NM hybrid bilayer, αis enhanced due to the transfer of spin angular\nmomentum from the FM to the NM through a dynamical process of ad iabatic spin pumping22,24,26,27. The spins\npumped in the NM are scattered by the effective spin-orbit field i.e.the Rashba field and a spin accumulation is\nachieved in the NM, leading to a spin current in the NM through SHE. An enhanced αin a FM/NM bilyer is a\nsignature of spin pumping and a fingerprint of the generation of pur e spin current in the NM. The spin current is\nconverted into charge current, through a reciprocal process c alled the inverse spin Hall effect (ISHE) viathe relation:\n/vectorJc=θSH/vectorJs×/vector σ, whereθSHis the spin Hall angle representing the efficiency of spin-to-charge c onversion of a material.\nThe charge current in the NM with SOC induces an electromotive forc e (emf), whose direction is perpendicular to\nthe spin current /vectorJsand to the spin polarization vector /vector σ. Spin pumping is an efficient mechanism to convert spin\ncurrent into charge current without any applied bias. Over the last few years several experimental works have been\npublished demonstrating spin pumping in heavy metals like Pt[ 26,28–31], Ta[32and33], Pd[31and34], Au[35], in\nsemiconductors like GaAs [ 36and37], Si[34and38], Ge[39] and ZnO[ 40and41] and recently also in 3D topological\n2insulators [ 42] and ferroelectric Rashba semiconductors [ 43]. Apart from ferromagnetic spin pumping from a metallic\nFM like permalloy (Py), it was also shown that spin currents can be rea lised in bilayers like Pt/YIG[ 44–46] based on\nferromagnetic insulators. Recent reports also point to spin pumpin g from paramagnets47and antiferromagnets48,49.\nThe successful generation and control of spin current in semicon ductors using the mechanism of spin pumping would\nopen wide perspectives for the integration of spin functionalities in s tate-of-the-art electronic and optoelectronic\ndevices based on semiconductors like Si, Ge, III-nitrides, III-ars enidesetc.\nAmongst the conventional III-V semiconductors, GaN and its alloy s AlGaN and InGaN have emerged as strategic\nmaterials for optoelectronic and electronic applications, primarily du e to their tunable wide bandgap and structure\ninducedpolarization. Furthermore,transitionmetaldopedIII-n itrideshavebeenstudiedextensivelyinthepastdecade\nas workbench magnetic semiconductors50,51. Another fundamental aspect, namely the presence of Rashba s pin orbit\ncoupling (RSOC)52,53, was recently demonstrated in degenerate wurtzite (wz) n-GaN:Si [ 54]. Using magnetotransport\nmeasurements it was shown that the Rashba parameter, αR, linear in kand accounting for the spin splitting of the\nconduction band in wz-GaN, has a quantitative value measured for n-GaN:Si to be ∼(4.5±1.0)meV·˚Ai.e.the same\nmagnitude found for a 2-dimensional electron gas (2DEG) formed a t the AlGaN/GaN interface54. In this work it was\nshown that in polar wz-GaN, the inversion asymmetry associated wit h the wurtzite crystal structure dominates over\nthe interfacial electric field in the conduction band of GaN. The RSOC in degenerate n-GaN:Si makes this material\na spin Hall system for prospective spin-orbitronic applications base d on III-nitride semiconductors. Here we report\non the generation of pure spin currents in n-GaN:Si using an adiabatic spin pumping technique and we estimate the\nspin Hall angle for n-GaN:Si. The evaluation of the spin-to-charge conversion efficiency inn-GaN:Si opens up the\npossibility to design and implement high performance spin based III-n itride nanoelectronic and optoelectronic devices.\nI. EXPERIMENTAL DETAILS\nThe experiments have been performed on a 1.9 µm thick degenerately doped n-GaN:Si film with electron concen-\ntration (1 .2×1019)cm−3and carrier mobility, µ∼180cm2V−1s−1, grown by metal organic vapour phase epitaxy\n(MOVPE) on c-Al2O3. A 10 nm permalloy (Py = Ni 80Fe20) film is used as FM layer and source of spins. The Py\nfilm is passivated with a 6 nm AlO xfilm which protects the FM from oxidation in order to avoid the detrime ntal\neffects of oxidised Py on the spin pumping efficiency55. The Py/ n-GaN:Si bilayer is driven to resonance conditions\nat room temperature under an X-band microwave excitation of (9 .44±0.01) GHz with an external magnetic field in\na Bruker Elexsys E580 electron paramagnetic resonance spectro meter. The permalloy–being a soft FM–has a small\nmagnetocrystalline anisotropy, so that /vectorM(t) in Py is aligned along the film plane when an in-plane magnetic field /vectorH\nis applied. Ohmic contacts to n-GaN:Si are fabricated by e-beam evaporation of Ti/Au/Al/Ti/Au as a metallic stack.\nThe quartz sample holder for the FMR measurements is provided with two high conducting copper wires and\nconnected to a Keithley 2700 DMM for measuring the generated dc v oltage. The dc voltage due to inverse spin\nHall-, thermal- and galvanomagnetic-effects is measured and the va lue of the component due to inverse spin Hall\neffect is employed to calculate the spin Hall angle in n-GaN:Si. Control experiments are also performed on contacted\nn-GaN:Si without Py, on Py/ c-Al2O3, on bare c-Al2O3substrates and on the wired sample holder solely. However, no\nmeasurable voltage or FMR absorption have been detected from th ese test samples and control experiments, ruling\nout experimental artefacts affecting the observed results.\nII. DETECTION OF SPIN HALL EFFECT IN n-GaN:Si\nA schematic representation of the spin pumping mechanism and of th e geometry employed for the detection of the\ngeneratedspin current in the NM areshown in Figs. 1(a) and1(b), respectively. The sketchesdepict the magnetization\nprecession in the FM for an applied magnetic field /vectorHat resonance condition under microwave excitation, leading to\nthe pumping of spin angular momentum and subsequent generation o f spin and charge currents in the NM through\nthe SHE and ISHE. Measurements are carried out with the magnetic field applied in the in-plane and out-of-plane\nconfiguration, respectively.\nSpin pumping is quenched for a perpendicular magnetic field and the an gle dependent measurement of the emf\nis essential to rule out secondary effects or experimental artefa cts. The direction of the applied magnetic field w.r.t\nthe sample plane is indicated in Fig. 1(c), while the first derivative of the FMR signal for Py/ c-Al2O3and Py/ n-\nGaN:Si/c-Al2O3is provided in Fig. 1(d). The dotted and solid lines represent the FMR line-shapes for Py /c-Al2O3\nand for the Py/ n-GaN:Si bilayer, respectively. The broadening of the FMR signal for the Py/GaN:Si bilayer w.r.t.\nthe reference Py/ c-Al2O3layer is the evidence of adiabatic spin pumping from the ferromagnet ic Py into the Rashba\nsemiconductor n-GaN:Si under resonance conditions24,26,39. The FMR signal and the electric potential difference\nbetween the electrodes attached to the n-GaN:Si layer are measured to detect the ISHE.\n3/c113H=0°/c113H=90°/c113H=180°\nHHH(c)(a)\n (b)\nh/c119\nNM\nFM\nSpin current\n-20 0 10 20 -10\nH H- (mT)FMRdI(H)/dH (arb. units)0\n(d)\nT = 300 K (RT)\nFigure 1. (a) Spin pumping mechanism; (b) sample structure a nd schematic illustration of the physical quantities descr ibing\nthe spin pumping and spin current generation and detection t hrough the Ohmic contacts on n-GaN:Si; (c) orientations of the\napplied magnetic field w.r.t. the sample surface and (d) /vectorHdependence of the FMR signals dI(H)/dH measured at T = 300 K\nfor Py/c-Al2O3and Py/n-GaN:Si bilayer at θH= 0◦.\nThe dynamics of the magnetization /vectorM(t) in Py under an effective magnetic field /vectorHeffis described by the LLG\nEq.(1). In FMR regime, the spin pumping driven by dynamical exchange inte raction, pumps into n-GaN:Si pure spin\ncurrent, which gets converted into charge current viaISHE, according to the relation /vectorJc=θSH/vectorJs×/vector σ, as discussed\nearlier. The charge current in the Rashba semiconductor leads to a n emf proportional to the generated spin current\nand whose amplitude is proportional to the microwave absorption an d maximized at the resonance field HFMR.\nThe emf VHgenerated in the NM n-GaN:Si is detected simultaneously with the FMR and at the resonanc e field\na peak is observed in the measured voltage. The experimental emf is plotted as a function of the applied magnetic\nfield/vectorHand reported in Figs. 2(a) and2(b) respectively for θH= 0◦andθH= 180◦. The angles 0◦and 180◦define\nthe directions of the applied in-plane magnetic field in accordance to t he geometry provided in Fig. 1(c). Since the\nRashba-Edelstein effect is observed in 2DEGs and in materials with gian t Rashba SOC56, and given that the system\ntreated here belongs to neither of these two classes, the observ ed emf is attributed to the interplay between SHE\nand ISHE. The experimental emf VHis a superposition of the voltage due to ISHE in the n-GaN:Si layer26,57and of\nspurious voltages originating from galvanomagnetic effects like ordin ary Hall effect (OHE) in n-GaN:Si, anomalous\nHall effect (AHE), planar Hall effect (PHE) in the Py layerand therma l heating effects due to microwaveirradiation58.\nThe voltage originating from these spurious mechanisms can be sepa rated from the one due to ISHE by fitting the\nexperimental data with the function26:\nVH=VSymΓ2\n(H−HFMR)2+Γ2+VAsym−2Γ(H−HFMR)\n(H−HFMR)2+Γ2(2)\nwhereΓis the half line width of the FMR line-shape and HFMRis the resonance field. The symmetric part of\nthe total voltage function, VSymΓ2//bracketleftbig\n(H−HFMR)2+Γ2/bracketrightbig\nwith an absorption line-shape is the contribution of the\n4Fitting\nVsymMeas.\n024\n-4-20\nVH(µV)\n010 20 -10 -20\nH H- (mT)FMR(a)\n(b)VAsym\nFitting\nVsymMeas.\nVAsymVH(µV)\nFigure 2. Voltage across the electrodes on n:GaN:Si layer for in-plane magnetic field orientations at (a )θH= 0◦and (b):\nθH= 180◦forPMW= 200 mW. Dots: experimental data; solid lines: fitting accor ding to Eq. ( 2).\nISHE voltage, VISHE, developed in n-GaN:Si due to the adiabatic spin pumping and to heating effects. On t he\nother hand, the asymmetric part of the function, VAsym[−2Γ(H−HFMR)]//bracketleftbig\n(H−HFMR)2+Γ2/bracketrightbig\nwith a dysonian\ndispersion line-shape is a consequence ofthe contributionsfrom AH E and OHE, as discussed earlier. The Hall voltages\nVAsymchange sign across HFMR, whileVISHE–being proportional to the integrated microwave absorption inten sity–\nis symmetric across HFMR, as expected from the fundamental spin pumping model25,26. A fitting of the measured\nemf at a microwave power of 200 mW with Eq.( 2) forθH= 0◦yieldsVSym= 2.80µVandVAsym=−0.453µV.\nThe ratio VSym/VAsym∼6 indicates that the major contribution to the measured voltage is p rovided by the efficient\nconversion of spin-to-charge current due to an interplay of spin p umping and direct and inverse spin Hall effects. The\nspurious heating effects from the microwave irradiation can be furt her eliminated by averaging the symmetric voltage\nfor parallel and antiparallel orientation of the applied field Haccording to:\nVISHE(θH) =VSym(θH)−VSym(θH+180◦)\n2(3)\nThis approach to the treatment of the data is justified, since VISHEchanges sign upon reversal of the magnetic field\nowing to a change of sign of the spin polarization vector /vector σ, not occurring for the voltage due to the microwave heating\neffects. Thus, for spin pumping experiments, the measured emf VHincludes contributions from both symmetric and\nasymmetric voltages, and the actual ISHE voltage VISHEis estimated by a proper treatment of the measured data.\nIt was reported30,36,59,60, that the symmetric voltage can also include contributions from PHE and galvanomagnetic\neffects in the metallic FM layer due to the electric and magnetic field com ponents of the microwave. However, the\nangular dependence of the emf w.r.t. the external magnetic field sh owed that the contribution of the ISHE to the\nsymmetric voltages at least five times greater than the ones from t he PHE60. In the above mentioned works, the\nthickness of the NM layerwas of the order of the spin diffusion length λN, while the thickness of n-GaN:Si studied here\nis 1.9µm,i.e.much greater than λN∼80nm in n-GaN. The fact that the thickness of the NM is orders of magnitude\ngreater than λNsuppresses the backflow spin current into the metallic FM, further reducing the contributions to the\nemf from galvanomagnetic effects. Careful identification and estim ation of the various components of the generated\n5emf in the NM layer in a spin pumping experiment is therefore essential for the estimation of the spin-to-charge\nconversion efficiency of a material. Here, for the in-plane magnetic fi eld, after eliminating the heating effects, a\nVISHE= 3.25µVis obtained and is exploited for the quantitative evaluation of the spin Hall angle for n-GaN:Si.\n0 100 200023\nV VSym Asym, (µV) 1\n-20\n-1\n-3\n-4\nPMW(mW)02\nVH(µV)\n-4-2022068100168200 mW VSym\nVAsym(a) (b)\n(c) (d)\n22068100168200 mW\nVSym\nVAsym V VSym Asym, (µV)\n0 10 20 -10 -20\nH H- (mT)FMRVH(µV)\nFigure 3. Dependence of the measured voltage on the applied m icrowave power for in-plane magnetic field orientations at\n(a)θH= 0◦and (c)θH= 180◦. Dependence of the VSymandVAsym for different microwave powers for (b) θH= 0◦and (d)\nθH= 180◦.\nThe dependence of the measured emf on the applied microwave powe r under FMR conditions is shown in Figs. 3(a)\nand3(c) fororientationsofthe magnetic field θH= 0◦andθH= 180◦, respectively. For a microwavepowerof 200mW,\nthe measured voltages at FMR conditions are +2.849 µV forθH= 0◦and -3.840 µV forθH= 180◦. It is to be noted\nhere that the voltagesreported in Figs. 3(a) and3(c) contain both the symmetric and asymmetric contributions. Aft er\nseparating the symmetric and asymmetric voltage components, th e symmetric part of the voltage, VSymis plotted\nas a function of PMWand shown in Figs. 3(b) and3(d) forθH= 0◦andθH= 180◦, respectively. Now, being VISHE\nproportional to the square of the microwave magnetic field ( hMW), it is expected to be linearly proportional to the\nmicrowave power PMW[39and57]. This is validated in Figs. 3(b) and3(d) forθH= 0◦andθH= 180◦, respectively.\nThe reversal of /vectorHcauses/vector σto change sign, which in turn induces the change in sign of the ISHE ele ctric field /vectorEISHE,\nas evidenced in Figs. 3(a) and3(c). It can be concluded that the fundamental relation for the IS HE,/vectorJc=θSH/vectorJs×/vector σ\noperates in the studied Py/ n-GaN:Si bilayer system. For an applied magnetic field perpendicular to the sample plane,\ni.e. θH= 90◦, the amplitude of VISHEis quenched even though a FMR signal for the Py is detected at the r esonance\nfield of 1245 mT. This result provides the necessary and sufficient co nfirmation of spin pumping through SHE and\nISHE in n-GaN:Si.\nIII. ESTIMATION OF SPIN HALL ANGLE IN n-GaN:Si\nThe magnetization precession described by the LLG Eq.( 1) drives the spin pumping in the Py/ n-GaN:Si film.\nWithin the fundamental model of spin pumping proposed by Tserkov nyaketal.21, the dc component of the generated\nspin current density j0\nsat the interface between Py and n-GaN:Si is:\n6j0\ns=ω\n2π/integraldisplay2π\n0¯h\n4πg↑↓\nr1\nM2s/bracketleftBigg\n/vectorM(t)×d/vectorM(t)\ndt/bracketrightBigg\nzdt (4)\nwhere ¯handg↑↓\nrare the Dirac constant and real part of the spin mixing conductanc e, while/bracketleftBig\n/vectorM(t)×d/vectorM(t)/dt/bracketrightBig\nz\nis thezcomponent of/bracketleftBig\n/vectorM(t)×d/vectorM(t)/dt/bracketrightBig\n. Now,g↑↓\nris proportional to the reflection and transmission coefficients\nof the majority and minority spins of the NM electrons, which in turn d epend on the transparency of the FM/NM\ninterface61,62. Thus, a transparent interface i.e.one with negligible spin scattering potential centres, would enhance\ng↑↓\nr, by augmenting the spin current density in the NM. The resonance c ondition obtained as a solution of the LLG\nequation is given by:\n/parenleftbiggω\nγ/parenrightbigg2\n=HFMR(HFMR+4πMs) (5)\nwhereHFMRis the resonance field, ωthe frequency of the microwave radiation and γis the gyromagnetic ratio.\nUsing Eqs.( 4) and (5) and the dynamic components of the magnetization /vectorM(t) obtained as a solution of Eq.( 1) with\nthe components of /vectorHeffas discussed above28, the spin current density j0\nsis:\nj0\ns=g↑↓\nrγ2h2\nMW¯h/bracketleftBig\n4πMsγ+/radicalbig\n(4πMs)2γ2)+4ω2/bracketrightBig\n8πα2[(4πMs)2γ2+4ω2](6)\nThe spin Hall angle θSHis related to the spin current j0\nsand to the inverse spin Hall voltage VISHEas:\nθSH=/parenleftbigg¯h\n2e/parenrightbiggVISHE(dNσN+dFσF)\nwσNtanh/parenleftBig\ndN\n2λN/parenrightBig\nj0s(7)\nwheredN,σNandλNare the thickness, conductivity and spin diffusion length of the n-GaN:Si layer, respectively,\nwhiledFandσFarethethicknessandconductivityoftheferromagnetPy. There alpartofthespinmixingconductance\ng↑↓\nris given by25,28,29,63:\ng↑↓\nr=4πMs√\n3γdF\n2gµBµ0ω/parenleftbig\nWFM/NM−WNM/parenrightbig\n(8)\nwheregandµBare the Land´ e gfactor and the Bohr magneton, WFM/NMandWNMthe spectral width of the\nPy/GaN:Si and Py layers, as shown in Fig. 1(d). With g= 2.12, 4πMs= 0.938T,dF= (1×10−8)m,µB=(9.27×\n10−24)JT−1,ω=(5.931×1010)s−1andµ0=(4π×10−7)H/m, we calculate the spin mixing conductance g↑↓\nrto be\n(1.38×1018)m−2. UsingEqs.( 6)and(7)andwiththeparameters γ=(1.86×1011)T−1s−1,hMW= 0.15mT, ¯h=(1.054×\n10−34)Js,λN= 80nm, dN= 1.9µm,w= 3mm, dF= 10nm, σN=(3.5587×104)Ω−1m−1,σF=(1.6×106)Ω−1m−1,\nj0\ns=(1.2254×10−10)Jm−2andVISHE= 3.25µV we find the spin Hall angle for n-GaN:Si to be θSH= 3.03×10−3, at\nleast one order of magnitude higher than those reported for Si [ 38], Ge[64], ZnO [41] andn-GaAs [6and65]. The spin\nHall angle θSHdepends on both the side jump and the skew scattering mechanisms . Theoretically29the side jump\ncontribution to θSHis given by (3 /8)1/2(kFλN)−1, which corresponds to 1 .08×10−2in the case under consideration\n. The theoretical value is at least one order of magnitude higher tha n the one obtained experimentally. Defining the\nevolutionofSHEin termsofmobilityofthe systemin questionaccordin gtoVignale etal.17, withµ∼180cm2V−1s−1,\nn-GaN:Si falls in the limit of clean-ultraclean regime, which is dominated by the skew scattering. Considering the\nabove mentioned overestimation of the magnitude of the side jump c ontribution and the consequent discrepancy\nbetween the theoretical and experimental values of θSH, one can infer that SHE in n-GaN:Si is dominated by skew\nscattering.\nThe experiments discussed above for the 1 .9µm thickn-GaN:Si layer have been carried out also on a 150nm thick\nn-GaN:Si film. Like in the case of the 1 .9µmn-GaN:Si sample, an AlO xpassivated 10 nm Py film is used as the\nsource of spin. The spin Hall voltage measured in the 150 nm thick n-GaN:Si is reported in Fig. 4. Here, the intensity\nof the asymmetric component of the voltage is more than one order of magnitude higher than the one observed for\n70 10 20 -10 -20\nH H- (mT)FMR0510\n-5\n-10V(µV)(a)\n0 100 200\nPMW(mW)0246\n50 150(b)\nVSym\nVAsym\nV VSym Asym, (µV)VSymVAsym\nFigure 4. (a) Voltage across the electrodes in the 150 nm n-GaN:Si control sample under a microwave excitation of powe r\n200 mW. Dots: experimental data; solid lines: fitting accord ing to Eq. ( 2). (b) Measured VISHE andVAsym as a function of the\nmicrowave power. Lines: linear fits.\nthe 1.9µmn-GaN:Si sample, as shown in Figs. 4(a) and4(b). For a NM layer thinner than its spin diffusion length\nλN, the spin backflow current Ibffrom the NM to FM exceeds the forward spin current due to spin pum pingIsp[25\nand66]. In such a scenario, VISHEdue to the inverse spin Hall effect cannot be detected in the NM. How ever, the\nasymmetric voltages can still be seen, being independent of spin pum ping. In the present case of n-GaN:Si, the spin\ndiffusion length is λN∼80 nm. It was reported that for NM metal systems, the asymmetr y voltage contribution for a\nfilm thickness of the order of λNis close to 100% of the total emf measured. This is due to the dominan ce ofIbfover\nIsp[67]. However, with a greater thickness of the NM film, the asymmetric c ontribution diminishes and approaches\na minimum saturation value for a film thickness ≫λN. Furthermore, it was reported by Flovik etal.68that this\nstrongVAsymcould be assigned to Eddy current effects and it was shown that in th e case of a NM layer of Pt, the\nasymmetry decreases considerably with increasing the thickness o f the NM layer. In the systems considered here, for\na thin layer of n-GaN:Si the spin backflow is likely to be the dominant mechanism that lea ds to the suppression of the\nVsymsignal, since the Oersted fields induce a significant distortion of the F MR lineshape, not observed for the 1 .9µm\nthickn-GaN:Si sample. For the case of ZnO, also a wide band gap wurtzite se miconductor like GaN, D’Ambrosio\netal.[40] reported the measured emf from the ZnO layer in a Py/ZnO bilayer u nder FMR conditions, as due to PHE\nin the Py layer. In our case of n-GaN:Si, the control experiments – as previously mentioned –did not reveal any\nmeasurable voltage due to PHE in the Py layer. Moreover, the thickn ess of the ZnO film used in case of D’Ambrosio\netal.was 200 nm, making spin backflow a likely reason for the observed dom inant contribution from PHE. On the\nother hand, in our case for calculating the spin Hall angle θSHwe carry out measurements on a 1 .9µm thickn-GaN:Si\nlayer, which can be considered as a bulk system where the role of spin backflow is largely suppressed – as discussed\npreviously in detail – upon comparison with the 150 nm n-GaN:Si layer. A detailed investigation of the dependence\nof spin pumping on the thickness and carrier concentration of n-GaN:Si is needed for an in-depth understanding of\nthis behaviour.\nIV. SUMMARY\nWe have provided experimental demonstration of spin pumping induc ed spin current generation and its detection\nat room temperature using the ISHE in the Rashba semiconductor n-GaN:Si. From the fundamental model of\n8spin pumping, a spin mixing conductance of (1 .38×1018)m−2for the Py/ n-GaN:Si interface and a spin Hall angle\nθSH= 3.03×10−3for wzn-GaN:Si are found. The value obtained for θSHis at least one order of magnitude higher\nthan those of other semiconductors like Si, Ge, ZnO and n-GaAs. The experimental demonstration of generation of\npure spin current in n-GaN:Si and its enhanced spin-charge conversion efficiency over ot her functional semiconductors\npoints at III-nitrides as model systems for studies on spin-relate d phenomena in non-centrosymmetricsemiconductors.\nMoreover, this work paves the way to the realization of nitride-bas ed low power optoelectronic, non-volatile and low\ndissipative spin devices like e.g.spin batteries.\nACKNOWLEDGEMENTS\nThis work was supported by the Austrian Science Foundation – FWF ( P22477 and P24471 and P26830), by the\nNATO ScienceforPeaceProgramme(ProjectNo. 984735)andbyt he EU7thFrameworkProgrammes: CAPACITIES\nproject REGPOT-CT-2013-316014 (EAgLE), EIT+ (NanoMat P2I G.01.01.02-02-002/08) and FunDMS Advanced\nGrant of the European Research Council (ERC grant No.227690) .\n∗rajdeep.adhikari@jku.at\n†alberta.bonanni@jku.at\n1A. Hoffmann and S. D. Bader, Phys. Rev. Applied 4, 047001 (2015) .\n2T. Kuschel and G. Reiss, Nat. Nanotechnol. 10, 22 (2015) .\n3W. Zhang, M. B. Jungfleisch, W. Jiang, J. Sklenar, F. Y. Fradin , J. E. Pearson, J. B. Ketterson, and A. Hoffmann,\nJ. Appl. Phys. 117, 172610 (2015) .\n4A. Manchon, Nat. Phys. 10, 340 (2014).\n5A. Fert, V. Cros, and J. Sampaio, Nat. Nanotechnol. 8, 152 (2013) .\n6J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back, and T. Jungwirth, Rev. Mod. Phys. 87, 1213 (2015) .\n7V. Edelstein, Solid State Commun. 73, 233 (1990) .\n8G. Vignale and I. V. Tokatly, Phys. Rev. B 93, 035310 (2016) .\n9N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, and N. P. Ong ,Rev. Mod. Phys. 82, 1539 (2010) .\n10N. F. Mott, Proc. R. Soc. A 124, 425 (1929) .\n11M. I. D’yakonov and V. I. Perel’, JETP Lett. 13, 467 (1971).\n12J. E. Hirsch, Phys. Rev. Lett. 83, 1834 (1999) .\n13S. Zhang, Phys. Rev. Lett. 85, 393 (2000) .\n14J. Sinova, D. Culcer, Q. Niu, N. A. Sinitsyn, T. Jungwirth, an d A. H. MacDonald, Phys. Rev. Lett. 92, 126603(1) (2004) .\n15S. Murakami, N. Nagaosa, and S.-C. Zhang, Science301, 1348 (2003) .\n16S. Murakami, “Advances in solid state physics,” (Springer B erlin Heidelberg, 2006) pp. 197–209.\n17G. Vignale, J. Supercond. Nov. Magn. 23, 3 (2009) .\n18Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom, Science306, 1910 (2004) .\n19J. Wunderlich, B. Kaestner, J. Sinova, and T. Jungwirth, Phys. Rev. Lett. 94, 047204 (2005) .\n20T. Jungwirth, J. Wunderlich, and K. Olejnik, Nat. Mater. 11, 382 (2012) .\n21Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. Lett. 88, 117601 (2002) .\n22Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. B 66, 224403 (2002) .\n23A. Brataas, Y. Tserkovnyak, G. E. W. Bauer, and B. I. Halperin ,Phys. Rev. B 66, 060404 (2002) .\n24K. Ando, Semicond. Sci. Tech. 29, 043002(1) (2014) .\n25Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I. Halperin ,Rev. Mod. Phys. 77, 1375 (2005) .\n26E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Appl. Phys. Lett. 88, 182509 (2006) .\n27Y. Pu, P. M. Odenthal, R. Adur, J. Beardsley, A. G. Swartz, D. V . Pelekhov, M. E. Flatt´ e, R. K. Kawakami, J. Pelz, P. C.\nHammel, and E. Johnston-Halperin, Phys. Rev. Lett. 115, 246602 (2015) .\n28K. Ando, Y. Kajiwara, S. Takahashi, S. Maekawa, K. Takemoto, M. Takatsu, and E. Saitoh, Phys. Rev. B 78, 014413 (2008) .\n29K. Ando, S. Takahashi, J. Ieda, Y. Kajiwara, H. Nakayama, T. Y oshino, K. Harii, Y. Fujikawa, M. Matsuo, S. Maekawa,\nand E. Saitoh, J. Appl. Phys. 109, 103913 (2011) .\n30A. Azevedo, L. H. Vilela-Le˜ ao, R. L. Rodr´ ıguez-Su´ arez, A . F. Lacerda Santos, and S. M. Rezende,\nPhys. Rev. B 83, 144402 (2011) .\n31C. T. Boone, J. M. Shaw, H. T. Nembach, and T. J. Silva, J. Appl. Phys. 117, 223910 (2015) .\n32G. Allen, S. Manipatruni, D. E. Nikonov, M. Doczy, and I. A. Yo ung, Phys. Rev. B 91, 144412 (2015) .\n33C. O. Avci, K. Garello, C. Nistor, S. Godey, B. Ballesteros, A . Mugarza, A. Barla, M. Valvidares, E. Pellegrin, A. Ghosh,\nI. M. Miron, O. Boulle, S. Auffret, G. Gaudin, and P. Gambardel la,Phys. Rev. B 89, 214419 (2014) .\n34Y. Ando, K. Ichiba, S. Yamada, E. Shikoh, T. Shinjo, K. Hamaya , and M. Shiraishi, Phys. Rev. B 88, 140406 (2013) .\n35B. Heinrich, C. Burrowes, E. Montoya, B. Kardasz, E. Girt, Y. -Y. Song, Y. Sun, and M. Wu,\nPhys. Rev. Lett. 107, 066604 (2011) .\n936L. Chen, F. Matsukura, and H. Ohno, Nat. Commun. 4, 2055(1) (2013) .\n37C. Liu, S. J. Patel, T. A. Peterson, C. C. Geppert, K. D. Christ ie, G. Stecklein, C. J. Palmstrom, and P. A. Crowell,\nNat Commun 7, 10296(1) (2016) .\n38K. Ando and E. Saitoh, Nat. Commun. 3, 629(1) (2012) .\n39S. Dushenko, M. Koike, Y. Ando, T. Shinjo, M. Myronov, and M. S hiraishi, Phys. Rev. Lett. 114, 196602(1) (2015) .\n40S. D’Ambrosio, L. Chen, H. Nakayama, F. Matsukura, T. Dietl, and H. Ohno, Jpn. J. Appl. Phys. 54, 093001(1) (2015) .\n41J.-C. Lee, L.-W. Huang, D.-S. Hung, T.-H. Chiang, J. C. A. Hua ng, J.-Z. Liang, and S.-F. Lee,\nAppl. Phys. Lett. 104, 052401(1)052401(4) (2014) .\n42M. Jamali, J. S. Lee, J. S. Jeong, F. Mahfouzi, Y. Lv, Z. Zhao, B . K. Nikoli, K. A. Mkhoyan, N. Samarth, and J.-P. Wang,\nNano Letters 15, 7126 (2015) .\n43C. Rinaldi, J. C. Rojas-Snchez, R. N. Wang, Y. Fu, S. Oyarzun, L. Vila, S. Bertoli, M. Asa, L. Baldrati, M. Cantoni, J.-M.\nGeorge, R. Calarco, A. Fert, and R. Bertacco, APL Mat. 4, 032501 (2016) .\n44Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida, M. Miz uguchi, H. Umezawa, H. Kawai, K. Ando, K. Takanashi,\nS. Maekawa, and E. Saitoh, Nature464, 262 (2010) .\n45C. W. Sandweg, Y. Kajiwara, A. V. Chumak, A. A. Serga, V. I. Vas yuchka, M. B. Jungfleisch, E. Saitoh, and B. Hillebrands,\nPhys. Rev. Lett. 106, 216601(1) (2011) .\n46L. J. Cornelissen, J. Liu, R. A. Duine, J. B. Youssef, and B. J. van Wees, Nat. Phys. 11, 1022 (2015).\n47Y. Shiomi and E. Saitoh, Phys. Rev. Lett. 113, 266602(1) (2014) .\n48L. Frangou, S. Oyarz´ un, S. Auffret, L. Vila, S. Gambarelli, a nd V. Baltz, Phys. Rev. Lett. 116, 077203(1) (2016) .\n49R. Cheng, J. Xiao, Q. Niu, and A. Brataas, Phys. Rev. Lett. 113, 057601(1) (2014) .\n50A. Bonanni, M. Sawicki, T. Devillers, W. Stefanowicz, B. Fai na, T. Li, T. E. Winkler, D. Sztenkiel, A. Navarro-Quezada,\nM. Rovezzi, R. Jakie/suppress la, A. Grois, M. Wegscheider, W. Jantsc h, J. Suffczy´ nski, F. D’Acapito, A. Meingast, G. Kothleitne r,\nand T. Dietl, Phys. Rev. B 84, 035206 (2011) .\n51T. Dietl, K. Sato, T. Fukushima, A. Bonanni, M. Jamet, A. Bars ki, S. Kuroda, M. Tanaka, P. N. Hai, and H. Katayama-\nYoshida, Rev. Mod. Phys. 87, 1311 (2015) .\n52E. I. Rashba, Fizika Tverd. Tela, 2, 1224 (1960).\n53E. I. Rashba, Sov. Phys. Usp. 7, 823 (1965).\n54W. Stefanowicz, R. Adhikari, T. Andrearczyk, B. Faina, M. Sa wicki, J. A. Majewski, T. Dietl, and A. Bonanni,\nPhys. Rev. B 89, 205201 (2014) .\n55S.-I. Kim, M.-S. Seo, J.-H. Seo, H. J. Yun, J. Lee, Y. S. Choi, a nd S. young Park, Curr. Appl. Phys. 14, 1743 (2014) .\n56J.-C. Rojas-S´ anchez, L. Vila, G. Desfonds, S. Gambarelli, J. P. Attan, J. M. De Teresa, C. Magn, and A. Fert,\nNat Commun. 4, 3944 (2013) .\n57K. Ando, S. Takahashi, J. Ieda, H. Kurebayashi, T. Trypiniot is, C. H. W. Barnes, S. Maekawa, and E. Saitoh,\nNat. Mater. 10, 655 (2011) .\n58H. J. Juretschke, J. Appl. Phys. 31, 1401 (1960) .\n59L. Chen, S. Ikeda, F. Matsukura, and H. Ohno, App. Phys. Express 7, 013002 (2014) .\n60J. Lustikova, Y. Shiomi, and E. Saitoh, Phys. Rev. B 92, 224436 (2015) .\n61W. Zhang, W. Han, X. Jiang, S.-H. Yang, and S. S. P. Parkin, Nat. Phys. 11, 496 (2015) .\n62C.-F. Pai, Y. Ou, L. H. Vilela-Le˜ ao, D. C. Ralph, and R. A. Buh rman, Phys. Rev. B 92, 064426 (2015) .\n63P. Deorani and H. Yang, Appl. Phys. Lett. 103, 232408 (2013) .\n64M. Koike, E. Shikoh, Y. Ando, T. Shinjo, S. Yamada, K. Hamaya, and M. Shiraishi, App. Phys. Express 6, 023001 (2013) .\n65M. Ehlert, C. Song, M. Ciorga, M. Utz, D. Schuh, D. Bougeard, a nd D. Weiss, Phys. Rev. B 86, 205204 (2012) .\n66A. Hoffmann, IEEE T. Magn. 49, 5172 (2013) .\n67V. Vlaminck, J. E. Pearson, S. D. Bader, and A. Hoffmann, Phys. Rev. B 88, 064414 (2013) .\n68V. Flovik, F. Maci, A. D. Kent, and E. Wahlstrm, J. Appl. Phys. 117, 143902 (2015) .\n10"    },    {        "title": "1201.4306v1.Josephson_dynamics_of_a_spin_orbit_coupled_Bose_Einstein_condensate_in_a_double_well_potential.pdf",        "content": "arXiv:1201.4306v1  [cond-mat.quant-gas]  20 Jan 2012Josephson dynamics of a spin-orbit coupled Bose-Einstein c ondensate\nin a double well potential\nDan-Wei Zhang,1Li-Bin Fu,2Z. D. Wang,3and Shi-Liang Zhu1,4,∗\n1Laboratory of Quantum Information Technology and SPTE,\nSouth China Normal University, Guangzhou, China\n2Science and Technology Computation Physics Laboratory,\nInstitute of Applied Physics and Computational Mathematic s, Beijing 100088, China\n3Department of Physics and Center of Theoretical and Computa tional Physics,\nThe University of Hong Kong, Pokfulam Road, Hong Kong, China\n4Center for Quantum Information, IIIS, Tsinghua University\nWe investigate the quantum dynamics of an experimentally re alized spin-orbit coupled Bose-\nEinstein condensate in a double well potential. The spin-or bit coupling can significantly enhance\nthe atomic inter-well tunneling. We find the coexistence of i nternal and external Josephson effects\nin the system, which are moreover inherently coupled in a com plicated form even in the absence of\ninteratomic interactions. Moreover, we show that the spin- dependent tunneling between two wells\ncan induce a net atomic spin current referred as spin Josephs on effects. Such novel spin Josephson\neffects can be observable for realistically experimental co nditions.\nPACS numbers: 03.75.Lm, 67.85.Hj\nI. INTRODUCTION\nBased on the Berry phase effect [1, 2] and its non-\nAbeliangeneralization[3], thecreationofsyntheticgauge\nfields in neutral atoms by controlling atom-light inter-\naction has attracted great interest in recent theoretical\nstudies [4–18], and has been realized in spinor Bose-\nEinstein condensates (BECs) in the pioneering experi-\nments of the NIST group [19, 20] and also in several sub-\nsequent experiments of other groups [21–23]. The neu-\ntral atoms in the generated effective Abelian and non-\nAbelian gauge fields behave like electrons in an electro-\nmagnetic field [19, 21, 22] or electrons with spin-orbit\n(SO) coupling [20]. Different from electrons that are\nfermions, the atoms with the synthetic SO coupling can\nbe bosons and typically BECs. This bosonic counterpart\nof the SO coupled materials has no direct analogin solid-\nstate systems and thus has received increasing attention\n[24–44] for different types of SO coupling, different inter-\nnal atomic structures (pseudospin-1/2, spin-1 and spin-2\nbosons, etc.), and different external conditions (homoge-\nnous, trapped and rotated). These theoretical investiga-\ntions focus mainly on the static properties of SO coupled\nBECsand haverevealrichphase diagramsofthe ground-\nstates [25–28, 30, 31] and exotic vortex structures [33–\n35, 39–41]. However, to our knowledge, their dynamics\nhas been less studied [24, 42–44], where the SO coupled\nBECs are demonstrated to exhibit interesting relativistic\ndynamics, such as analogs of self-localization [42], Zit-\nterbewegung [43] and Klein tunneling [44] under certain\nconditions.\nOn the other hand, quantum dynamics of a BEC in\na double well potential has been widely investigated. In\n∗Electronic address: slzhu@scnu.edu.cn\n) (xV\nx 0 b\u0010 b\u000eL R pL\nnRnLpR\n2:\n2:ppJ\nnnJ(a)(b)\nnpJ\npnJ\nFIG. 1: (Color online) A schematic representation of (a) a SO\ncoupled BEC in a double well trap. (b) The dynamic process\nof the system, where the blue solid and dashed lines repre-\nsent the inter-well tunneling without and with spin-flippin g,\nrespectively, the green circles represent the Raman coupli ng.\nThe inter-well spin-flipping tunneling induced by the Raman\ncoupling is negligible under current experiment condition s\n[20, 50]. The atomic interaction terms are not shown in the\nfigure.\nparticular, the coherent atomic tunneling between two\nwells results in oscillatory exchange of the BEC, which\nis analogous to the Josephson effects (JEs) for neutral\natoms [45–48]. The weakly interacting BECs provide a\nfurther context [46–48] for JEs in superconductor sys-\ntems because they display a nonlinear generalization of\ntypical d.c. and a.c. JEs and macroscopic quantum self-\ntrapping, all of which have been observed in experiments\n[49–51]. Apartfromtheconventionalsingle-speciesBECs\n[45–48], theJosephsondynamicsoftwo-speciesBECs[52]\nandspinorBECswithout SOcoupling[53] havealsobeen\nstudied [54–57]; however, the dynamics of SO coupled\nBECs is yet to be explored.\nIn this paper, we investigate the dynamics of a specific\nSO coupled BEC, which was realized in the experiment\nof the NIST group, in a double well trapping potential.2\nWe find that the SO coupling in the system contributes\nto and increases the atomic tunneling to a large extent,\nwhich can significantly enhance the atomic JEs. The full\ndynamics of the system contains both internal and exter-\nnal JEs, which are moreoverinherently coupled in a com-\nplicated form even in the absence of interatomic interac-\ntions. We further demonstrate that the spin-dependent\nJosephsontunnelingcanleadtoanetatomicspincurrent\nby varying conditions, which we refer to as spin Joseph-\nson effects. The predicted spin-Josephson currents are\nrobustagainsttheparameteradjustmentandvaryingini-\ntial conditions, and can be observable in the SO coupled\nBECs under realistic experimental conditions.\nThe paper is organized as follows. In Sec. II we con-\nstruct a model that can be used to study the quantum\ndynamics of a SO coupled BEC in a double well poten-\ntial. Then, in Sec. III, the Josephson dynamics of the\nconstructed system is investigated, with the complicat-\nedly coupled internal and external JEs being addressed.\nIn Sec. IV we demonstrate that the spin JEs exhibit in\nthe system under realistic conditions. Brief discussions\nand a short conclusion are given in Sec. V.\nII. MODEL\nIn a very recent experiment, the NIST group realized\na synthetic SO coupling in the87Rb BEC, in which a\npairofRamanlasersgenerateamomentum-sensitivecou-\npling between two internal atomic states [20]. In the bare\npseudo-spin basis | ↑/an}bracketri}htb=|mF= 0/an}bracketri}htand| ↓/an}bracketri}htb=|mF=\n−1/an}bracketri}ht, the SO coupling is described by the single particle\nHamiltonian given by [20]\nˆh=p2\n2mˆI+1\n2/parenleftbigg\nδΩe2ikLx\nΩe−2ikLx−δ/parenrightbigg\n,(1)\nwherepis the atomic momentum in the xyplane,mis\nthe atomic mass, δis the tunable detuning behaved as a\nZeeman filed, kLis the wave number of the Raman laser,\nand Ω is the Raman coupling strength. Such kind of\nSO coupling is equivalent to that of an electronic system\nwithequalcontributionfromRashbaandDresselhausSO\ncoupling, and thus it is effective just in one-dimension\n(1D). So we restrict our discussions in 1D and focus on\nthe motion of atoms along xaxis by freezing their yand\nzdegrees of freedom.\nTo proceed further, we introduce the dressed pseudo-\nspins| ↑/an}bracketri}ht=e−ikLx| ↑/an}bracketri}htband| ↓/an}bracketri}ht=eikLx| ↓/an}bracketri}htb[20, 41], then\nthe single-particle Hamiltonian in 1D (along xaxis) can\nbe written as\nˆh0=¯h2ˆk2\nx\n2m+2αˆkxσz+Ω\n2σx+δ\n2σz,(2)\nwhereˆkxis the atomic wave vector operator, and α=\nEr/kLis the SO coupling strength with Er= ¯h2k2\nL/2m\nbeing the single-photon recoil energy. The dispersion re-\nlation of the single particle Hamiltonian (2) with δ= 0isE±(kx) =¯h2k2\nx\n2m±/radicalbig\n4α2k2x+Ω2/4, which exhibits a\nstructure of two branches. We are interested in the\nlower energy one E−(kx). There is only one minimum\ninkx= 0 for large Raman coupling Ω >4Er, where\nthe atoms of both atomic levels condense. However,\nthe lower branch for Ω <4Erpresents two minima for\ncondensation of dressed pseudo-spin-up (left one) atoms\nand dressed pseudo-spin-down (right one) atoms, respec-\ntively. The Raman coupling and a small δmodulate the\npopulation of atoms in these two states [20]. Here we fo-\ncus on the later regime, i.e. Ω <4Er, because such BEC\nwith spin-separated and non-zero central momentum is\nmore interesting in contrast to a regular BEC with zero\ncentral momentum.\nTo be clearer, we can rewrite the Hamiltonian (2) as\nˆh0=/parenleftbigg\nH↑Ω/2\nΩ/2H↓/parenrightbigg\n, (3)\nwhereH↑=¯h2\n2m(ˆk2\nx+ 2kLˆkx) +δ\n2andH↓=¯h2\n2m(ˆk2\nx−\n2kLˆkx)−δ\n2. Since it is more straightforward to describe\nthe system in terms of dressed pseudo-spin states com-\npared with using bare ones, we will work in the dressed\npseudo-spin space and simply refer to dressed pseudo-\nspin as spin for convenience hereafter. We also note that\nthe parameters kL, Ω andδin the single-particle Hamil-\ntonian can be tuned independently in a wide range [20],\nmaking the SO coupled BEC a suitable platform for in-\nvestigatingthe Josephsondynamicsin thepresenceofSO\ncoupling.\nNow we turn to consider such a SO coupled BEC in a\ndouble well potential denoted by V(x) as shown in Fig.\n1(a). Note that the double well potential here is assumed\nto be spin-independent. To investigate the dynamics of\nthe system, we adopt the two-mode approximation [45–\n48] with the field operator\nˆΨσ(x)≃ˆaLσψLσ(x)+ˆaRσψRσ(x), (4)\nwhereψjσ(x) is the ground state wave function of the j\nwell (j=L,R) with spin σ(σ=↑,↓), and ˆajσis the an-\nnihilation operator for spin σin thejwell, satisfying the\nbosonic commutation relationship [ˆ ajσ,ˆa†\nkσ′] =δjkδσσ′.\nThe validity of the two-mode approximation holds under\ntwoconditions: the weakatomicinteractionandsmallef-\nfectiveZeemansplitting, astheatomscannotbepumped\noutoftheloweststateofeachwellinthis case. Inthe sec-\nond quantization formalism, the total Hamiltonian reads\nH=/integraldisplay\ndxˆΨ†(x)/bracketleftig\nˆh0+V(x)+ˆhint/bracketrightig\nˆΨ(x),(5)\nwhere the two-component field operator ˆΨ = (ˆΨ↑,ˆΨ↓)T,\nand the interaction Hamiltonian ˆhintwill be specified\nbelow. Substituting Eq. (4) into Eq. (5), one can rewrite3\nthe total Hamiltonian as\nH=/summationdisplay\nj,σεjσˆa†\njσˆajσ+/summationdisplay\nσσ′/parenleftig\nJσσ′ˆa†\nLσˆaRσ′+h.c./parenrightig\n+Ω\n2/summationdisplay\nj/parenleftig\nˆa†\nj↑ˆaj↓+h.c./parenrightig\n+δ\n2/summationdisplay\nj/parenleftig\nˆa†\nj↑ˆaj↑−ˆa†\nj↓ˆaj↓/parenrightig\n+Hint,(6)\nwhereεjσ=/integraltext\ndxψ∗\njσ(x)[Hσ+V(x)]ψjσ(x)≈1\n2¯hωj−\nEris the single-particle ground state energy in the\njwell withωjbeing the harmonic frequency of this\nwell,Jσσ=/integraltextdxψ∗\nLσ(x)[Hσ+V(x)]ψRσ(x) andJσ¯σ=/integraltextdxψ∗\nLσ(x)Ω\n2ψR¯σ(x) (withσand ¯σreferring to different\nspins) are the tunneling terms shown in Fig. 1(b). In\naddition, the interaction Hamiltonian is given by\nHint=1\n2/summationdisplay\nj/parenleftig\ng(j)\n↑↑ˆa†\nj↑ˆa†\nj↑ˆaj↑ˆaj↑+g(j)\n↓↓ˆa†\nj↓ˆa†\nj↓ˆaj↓ˆaj↓\n+2g(j)\n↑↓ˆa†\nj↑ˆa†\nj↑ˆaj↓ˆaj↓/parenrightig\n,\n(7)\nwhereg(j)\nσσ′=2¯h2aσσ′\nml2\n⊥/integraltext\ndx|ψjσ(x)|2|ψjσ′(x)|2is the effec-\ntive 1D interacting strength with aσσ′being thes-wave\nscattering length between spin σandσ′andl⊥being the\noscillator length associated to a harmonic vertical con-\nfinement. Note that here we have ignored the inter-well\natomic interactions because the s-wave scattering length\n(which is on the order of nanometers) is much smaller\nthan the inter-well distance (which is on the order of\nmicrometers). We have also dropped the inter-well cou-\npling since its strength is exponentially smaller than the\nintra-well counterpart. The Hamiltonian (6) describes\nthe dynamic process of the system schematically shown\nin Fig. 1(b).\nFor simplicity, we assume the double well potential to\nbe symmetric as shown in Fig. 1(a), with each well hav-\ning the same harmonic trapping frequency ω. Thus we\nhaveεL=εRandg(L)\nσσ′=g(R)\nσσ′. Such kind of double well\npotential can be generated in experiments [51] with the\nform\nV(x) =a(x2−b2)2, (8)\nwhere the parameters aandbare both tunable in the\nexperiments [51]. Expanding V(x) nearx=±b, one ob-\ntains its harmonic form as V(2)(x).=1\n2mω2(x±b)2and\nthusa=mω2/8b2. The ground state wavefunctions of\nthe BEC in each well potential with each spin can be ap-\nproximately represented by its corresponding lowest en-\nergy single-particle wavefunction, which can be worked\nout by solving the equations [ Hσ+1\n2mω2(x±b)2]ψjσ=\nεjσψjσ(here±are forj=L,R, respectively). The re-\nsults are [24]\nψL↑=ϕ(L)\n0(x)e−ikLx,\nψL↓=ϕ(L)\n0(x)eikLx,\nψR↑=ϕ(R)\n0(x)e−ikLx,\nψR↓=ϕ(R)\n0(x)eikLx,(9)1 2 3 4−0.4−0.200.20.40.60.8J(η) / Er\n1 2 3 4−0.200.20.40.60.8\n1 2 3 4−0.200.20.40.60.8\nb / µmJ(η) / Er\n1 2 3 4−0.200.20.40.60.8\nb / µm  100J(T)\n100J(V)\n|J(SO)|\n100J(R)\n(c)  I0 = 3 µm(a)  I0 = 1 µm (b)  I0 = 2 µm\n(d)  I0 = 4 µm\nFIG. 2: (Color online) The energy scales of tunneling terms\nJ(η)as a function of bfor (a)l0= 1µm, (b)l0= 2µm, (c)\nl0= 3µm, and (d) l0= 4µm, respectively. In (a)-(d) we set\nΩ/¯h=Er/¯h= 22.5 kHz.\nwhereϕ(L)\n0(x) =1√\nl0√πe−(x+d)2/2l2\n0, andϕ(R)\n0(x) =\n1√\nl0√πe−(x−d)2/2l2\n0withl0=/radicalbig\n¯h/mωbeing the oscillator\nlength. Substituting Eq. (9) into the expressions of Jσσ′,\none can obtain\nJ↑↑=J(T)+J(SO)+J(V)+J(Z),\nJ↓↓=J(T)+J(SO)+J(V)−J(Z),\nJ↑↓=J↓↑=J(R),(10)\nwhere the terms J(T)=−¯h2\n2m/integraltext\ndxϕ(L)\n0ϕ′′(R)\n0,J(SO)=\n−Er/integraltext\ndxϕ(L)\n0ϕ(R)\n0,J(V)=/integraltext\ndxϕ(L)\n0V(x)ϕ(R)\n0,J(Z)=\nδ\n2/integraltext\ndxϕ(L)\n0ϕ(R)\n0, andJ(R)=Ω\n2/integraltext\ndxϕ(L)\n0e−2ikLxϕ(R)\n0.\nComparedwith theatomictunnelingofaregularBEC,\nthe SO coupled BEC in this system exhibits two addi-\ntional tunneling channels, the SO coupling induced tun-\nneling term J(SO)and the Raman coupling induced one\nJ(R). Toclarifythe effectsofthese termsinthe tunneling\nprocesses,weneedtoworkoutandtocomparetheenergy\nscales of all the terms J(η), whereη={T,V,Z,SO,R }.\nSubstituting Eqs. (9) and (8) into Eq. (10), we can ob-\ntain the following analytical solutions\nJ(η)=ξηe−b2/l2\n0, (11)\nwhereξT=¯h2\n2ml2\n0(1\n2−b2\nl2\n0),ξV=¯h2\n8mb2l4\n0(3\n4l4\n0−b2l2\n0+b4),\nξZ=δ/2,ξSO=−Er, andξR= Ωe−k2\nLl2\n0. Since the Zee-\nman filedδis independently tunable to the double-well\nstructure and should be small, we here further assume\nδ≪Erand thus we focus on the comparison among\nJ(SO,R,T,V ). TheeffectsofZeeman-splittinginducedtun-\nneling will be specified in the Sec. IV.\nForl0∼√\n2b, we haveξT∼0 andξV∼¯h2\n4mb2, and\nthus|ξSO|\n|ξV|= 8π2(b\nλL)2>∼100. Here we have assumed4\nthe same wavelength λL= 2π/kL= 0.8µm and recoil\nfrequencyEr/¯h= 22.5 kHz as those in the experiments\n[20],bandl0to be on the order of micrometers [50, 51].\nIn fact, in the regime of b2/l2\n0∼[0.5,2], we find that\n|ξSO|>∼100max{|ξT|,|ξV|}. (12)\nBesides, one can check that ξR∼Ωe−64forl0∼1µm\nandλL= 0.8µm, and thus the Raman-coupling induced\ntunneling is negligible in this system. The comparisons\namongJ(η)forsometypicalparametersareshownin Fig.\n2. In otherwords,wefind that underrealisticexperiment\nconditions [20, 50], the spin-flipping tunneling induced\nby Raman coupling is negligible but the SO coupling in-\nduced tunneling term J(SO)dominates and moreover it\ngreatly enhances atomic tunneling in this system. Thus\nwe may rewrite the tunneling terms as\nJ↑↓=J↓↑≈0,\nJ↑↑≈J↓↓≈J(SO)=−γEr,(13)\nwhereγ= exp(−b2/l2\n0)∼[0.1,0.6]. It is worthwhile to\nnote that the new tunneling terms J(SO)andJ(R)in\nthis SO coupled system are both tunable, enabling us to\nstudy the interesting effects of SO coupling in the atomic\ninter-well tunneling. For instance, one can decrease the\neffective wave number in xaxis to the scale kL∼1/l0so\nthatJ(R)∼0.37Ωand then the Raman-couplinginduced\ntunneling can revive. This can be achieved by adjusting\nthe angle between the applying Raman lasers and the\ntrapping potential or alternatively by using lasers with\nlarger wavelength. In addition, in the same way one can\ntune the recoil energy Erto identify the enhancement\nof atomic tunneling due to the SO coupling (i.e. the\neffect ofJ(SO)) in experiments. As a first step to inves-\ntigate the system under current experiment conditions,\nwe here concentrate on the tunneling regime governed by\nEq. (13).\nIII. FULL DYNAMICS OF THE SYSTEM\nWe are now in the position to investigate the quan-\ntum dynamics of the system constructed in the previous\nsection. We first address the non-interacting case, i.e.Hint= 0 in Eq. (6), in which the single-particle Hamil-\ntonian is given by\nH0≃J↑↑/parenleftig\nˆa†\nL↑ˆaR↑+ˆaL↑ˆa†\nR↑/parenrightig\n+J↓↓/parenleftig\nˆa†\nL↓ˆaR↓+ˆaL↓ˆa†\nR↓/parenrightig\n+Ω\n2/parenleftig\nˆa†\nL↑ˆaL↓+ˆaL↑ˆa†\nL↓+ˆa†\nR↑ˆaR↓+ˆaR↑ˆa†\nR↓/parenrightig\n+δ\n2/parenleftig\nˆa†\nL↑ˆaL↑−ˆa†\nL↓ˆaL↓+ˆa†\nR↑ˆaR↑−ˆa†\nR↓ˆaR↓/parenrightig\n.\n(14)\nHere we have dropped the tunneling terms Jσ¯σsince\nthese spin-flipping tunneling progresses can be negligible\nin the current experiment conditions [20, 50]. In order\nto study the dynamic properties of the system, we need\nto work with the equation of motion. The corresponding\nHeisenberg equations read\ni¯hd\ndtˆajσ= [ˆajσ,H0] =Jσσˆa¯jσ+Ω\n2ˆaj¯σ+(−1)pδ\n2ˆajσ,\n(15)\nwhereσand ¯σrefer to different spin, while jand¯j\nto different wells, and p= 0,1 are forσ=↑,↓, re-\nspectively. Using the mean-field approximation, one has\nˆajσ≃ /an}bracketle{tˆajσ/an}bracketri}ht ≡ajσwithajσbeingcnumbers. Thus we\ncan rewrite the equations of motion as\ni¯h˙ajσ=Jσσa¯jσ+Ω\n2aj¯σ+(−1)pδ\n2ajσ.(16)\nBy defining a four-component wavefunction Φ =\n(aL↑,aL↓,aR↑,aR↓)T, Eq. (16) is rewritten as i¯hd\ndtΦ =\nHMΦ, where the Hamiltonian of the system is given by\nHM=\nδ\n2Ω\n2J↑↑0\nΩ\n2−δ\n20J↓↓\nJ↑↑0δ\n2Ω\n2\n0J↓↓Ω\n2−δ\n2\n. (17)\nWe now look into the JEs in this system. Let us fur-\nther express ajσasajσ=/radicalbig\nNjσeiθjσ, where the particle\nnumbersNjσand phases θjσare all time-dependent in\ngeneral. According to Eq. (16), we can obtain\ni¯h˙Njσ\n2−¯hNjσ˙θjσ=Jσσ/radicalbigNjσN¯jσei(θ¯jσ−θjσ)\n+Ω\n2/radicalbig\nNjσNj¯σei(θj¯σ−θjσ)+(−1)pδ\n2Njσ.(18)\nSeparating the image and real parts of Eq. (18) yields\ntwo groups of equations as\n˙Njσ=2Jσσ\n¯h/radicalbigNjσN¯jσsin(θ¯jσ−θjσ)+Ω\n¯h/radicalbig\nNjσNj¯σsin(θj¯σ−θjσ),\n˙θjσ=Jσσ\n¯h/radicalig\nN¯jσ\nNjσcos(θ¯jσ−θjσ)+Ω\n2¯h/radicalig\nNj¯σ\nNjσcos(θj¯σ−θjσ)+(−1)pδ\n2¯h.(19)\nEq. (19) consists actually of eight coupled equations. To\nsimplify these equations, we introduce φσ=θRσ−θLσ\nandρσ=NRσ−NLσfor the phase and particle number\ndifferences between two wells with the same spin σ, andφj=θj↓−θj↑andρj=Nj↓−Nj↑for the phase and\nparticlenumberdifferencesbetweentwospinsinthesame5\nwellj, respectively. Thus we can obtain\n˙ρσ=L1sinφσ+1\n2/summationdisplay\nj(−1)qL2sinφj,\n˙ρj=1\n2/summationdisplay\nσ(−1)pL1sinφσ+L2sinφj,(20)\nwhereL1=−4Jσσ\n¯h√NRσNLσ,L2=−2Ω\n¯h/radicalbigNj↑Nj↓, and\nq= 0,1 forj=R,L, respectively ( p= 0,1 forσ=↑,↓,\nrespectively). From the above Eq. (20), we find the\ncoexistence of internal JE related to ˙ ρj(φj) and external\nJErelatedto ˙ ρσ(φσ). Moreover,theinternalandexternal\nJEs are inherently coupled in a more complicated form.\nBefore ending this section, we briefly discuss the\nweakly interacting cases, which have been assumed to\nmeet the requirement of two-mode approximation. In\nthis regime, the mean-field analysis still works well, and\nthe dropped term Hintcan be taken into count within\nthe previous discussions. This leads to two additional\nterms related to interactions into Eq. (16), and now the\nequations of motion are given by\ni¯h˙ajσ=Jσσa¯jσ+Ω\n2aj¯σ+(−1)pδ\n2ajσ\n+gσσ|ajσ|2ajσ+gσ¯σ|aj¯σ|2ajσ,(21)\nwhere the interacting strength gσσ′can be found as\ngσσ′=√\n2¯h2aσσ′√πml2\n⊥l0. The estimation of the interaction en-ergyand the Josephsondynamics in the presence ofweak\ninteractions will be presented in the next section.\nIV. JOSEPHSON EFFECTS IN WEAK RAMAN\nCOUPLING REGIMES\nIn the preceding section, we have shown that the SO\ncoupled BEC in a double well potential exhibits the com-\nplicated coupled external and internal Josephson dynam-\nics. We, in this section, consider a specific dynamic pro-\ncess of the system in the weak Raman coupling regime\n(i.e. Ω/Er≪1), where the external Josephson dynamic\ndominates. Infact, themanipulationanddetectionofthe\nSO coupled BECs in this regime have been performed in\nexperiments [20].\nFor the weak Raman coupling, we find that the ra-\ntioν≡ |Jσσ|/Ω can reach several hundreds from Eq.\n(13). Thus within the time scale τ∼¯h/Ω≃45 ms\nfor Ω = 0.001Er, one can ignore the effects of the spin-\nflipping tunneling, which leads to two external Josephson\ntunneling processes for different spins. The spins in this\nregime are conserved and then the total particle number\nof spinσ Nσt=NLσ+NRσare time-independent con-\nstants. We assume N↑t=N↓t=Ntfor simplicity. The\nequations of motion (21) in this case can be rewritten as\ni¯hd\ndt/parenleftbigg\naLσ\naRσ/parenrightbigg\n=/parenleftbigg\n(−1)pδ\n2+gσσ|aLσ|2+gσ¯σ|aL¯σ|2Jσσ\nJσσ (−1)pδ\n2+gσσ|aRσ|2+gσ¯σ|aR¯σ|2/parenrightbigg/parenleftbigg\naLσ\naRσ/parenrightbigg\n.(22)\nBydefiningthenormalizedinter-wellparticlenumberdif-\nference for spin σasZσ= [NRσ−NLσ]/Nt(−1≤ Zσ≤\n1), the equations of motion (19) become rather simple\nin this case (similar to those for the regular two species\nBECs [54]), which are given by\n˙Zσ=−2Jσσ\n¯h√1−Zσsinφσ,\n˙φσ=Jσσ\n¯hZσ√\n1−Z2σcosφσ+Uσσ\n¯hZσ+Uσ¯σ\n¯hZ¯σ+(−1)pδ\n2¯h.\n(23)\nThe spin-dependent atomic density current is given by\nIσ=Nt·˙Zσ. (24)\nFrom Eq.(24), we can define the net spin current as\nIs=I↑−I↓, (25)\nand the total atomic current as\nIa=I↑+I↓. (26)\nWe first considerthe JEs ofthe system in the noninter-\nacting limit, i.e. Uσσ=Uσ¯σ= 0 in Eq. (23), which canbe realizedbyFeshbachresonance[58]. Under this condi-\ntion, the two external Josephson tunneling processes for\ndifferent spins are decoupled. We numerically calculate\nEqs. (23), with some typical results of the time evolu-\ntion ofZσfor different initial conditions being shown in\nFig. 3. In the calculations, we have assumed the zero\nZeeman filed δ= 0 in Fig. 3(a) and (b), and small Zee-\nman fieldδ= 0.01Erin (c) and (d). Compared with\nzero Zeeman field cases, a small Zeeman field results in\na deviation in Josephson tunneling strengths Jσσand in\ntime-cumulative phases δ/2¯hfor different spins. Here\nZ↑(t) andZ↓(t) demonstrate the oscillatory Josephson\ntunnelings which are similar to the early results in Ref.\n[47]. As shown in Fig. 3(a-d), they are spin-dependent\nand the dynamic evolution of each one depends on its\nown tunneling strength, phase and initial conditions. We\nalso calculate Iσ,IsandIain this regime with typical\nresults being shown in Fig. 4. It is interesting to see\nthat the spin-dependent atomic density currents due to\nthe spin-related Josephson tunnelings give rise to a net\nspin current (cf. Fig. 4), and moreover in some certain6\n0246810−1−0.500.51Zσ\n0246810−1−0.500.51\n0246810−1−0.500.51\nt / msZσ\n  \n0246810−1−0.500.51\nt / msZ↑\nZ↓\n(d)(b) (a)\n(c)\nFIG. 3: (Color online) The time evolution of Zσin nonin-\nteracting limitation. In (a) and (b) we have δ= 0 and\nJ↑↑=J↓↓=−0.1Er; In (c) and (d) we have δ= 0.01Er,\nJ↑↑=−0.905Er, andJ↓↓=−0.105Er. The initial conditions\nareZ↑(0) =−Z↓(0) = 0.3,φ↑(0) = 0.5φ↓(0) =π/4 in (a)\nand (c); and Z↑(0) =Z↓(0) = 0, φ↑(0) =φ↓(0) =π/4 in (b)\nand (d).\n0 2 4 6 8 10−4000−2000020004000(Iσ, Is, Ia) / Nt\n0 2 4 6 8 10−4000−2000020004000\nt / ms(Iσ, Is, Ia) / Nt\n  I↑\nI↓\nIs\nIa(b)(a)\nFIG. 4: (color online) Josephson currents in the noninter-\nacting limit. The time evolution of spin-dependent atomic\ncurrents Iσ, a net spin current Isand a total atomic current\nIain (a) for the same conditions in Fig. 3(a); and in (b) for\nthe some conditions in Fig. 3(b).\ninitial conditions the total atomic current can be zero,\nwhich leads to a new interesting pure spin currents (cf.\nFig. 4(b)). We call such new JEs as spin Josephson\neffects, which can be observable in experiments by mea-\nsuring the time-evolution of spin-dependent population\nimbalance of the atomic gas [59].\nFor weakly interacting cases, we have to estimate0246810−1−0.500.51Zσ\n0246810−1−0.500.51\n0246810−1−0.500.51\nt / msZσ\n  \n0246810−1−0.500.51\nt / msZ↑\nZ↓(b)(a)\n(c) (d)\nFIG. 5: (Color online) The time evolution of Zσin the weakly\ninteracting regime with Uσσ= 0.01ErandUσσ′= 0.011Er.\nOther parameters and initial conditions in (a)-(d) are the\nsame with those in Fig. 3(a)-(d), respectively.\nthe interaction energy UσσandUσ¯σ, which should be\nUσσ,Uσ¯σ≪¯hωdue to the two-mode approximation.\nThis requirement results in Uσσ≈Uσ¯σ≪0.1Er. In\nthis regime, the two spin-Josephson tunneling processes\nare coupled via atomic interactions. To understand the\neffects of the interaction, we show in Fig. 5 some typical\nresults of the time evolution of Zσfor the same initial\nconditions and parameters in Fig. 3. It clearly demon-\nstrate that the modification of Zσ(t) due to atomic in-\nteractions is not significant and even very minor in some\ncases (such as the case for Zσ(0) = 0 and δ= 0 in Fig.\n3(b), 5(b)) since the interactionenergyis smallcompared\nwith the tunneling energy. Therefore the spin Josephson\ndynamics still exhibit a similar oscillatory feature in this\nregime.\nTo see more clearly the oscillatory properties of the\nspin JEs, we have numerically calculated the frequency\nspectra of the net spin currents Isfor various conditions\n(such as those in Fig. 3 and Fig. 5). We find that the\nspectra for different cases exhibit a single peak centered\nat the slightly shifted frequency, as seen in Fig. 6. The\nsingle-peak feature shown in Fig. 6 implies that the spin\ncurrentIs(t) can well be described by a sin-function,\nwhile the weak interatomic interactions or the small Zee-\nman field can merely modify the period and amplitude\nof the current slightly. Thus we conclude that the spin\nJEs in this system are robust against the parameter ad-\njustment and initial conditions.7\n   \n0.0 0.2 0.4 0.6 0.8 1.03(a)\n3(c)\n5(a)\n5(c)Spectrum of Is\nFrequency  (kHz)\nFIG. 6: (Color online) Spectra of the net spin currents Isfor\nthe cases in Fig. 3(a,c) and in Fig. 5 (a,c). A single peak in\nthe spectrum of each case implies that the spin current Is(t)\nis well described by a sin-function.\nV. DISCUSSION AND CONCLUSION\nBefore concluding, we briefly discuss another specific\ndynamic process of the system in the relatively strong\nRaman coupling regime, Ω ≫ |Jσσ|, which can be real-\nizedsuchasbytuning Jσσ∼ −0.1ErandΩ∼Er. Inthis\nregime, within the time scale ¯ h/|Jσσ|one can ignore theatomicinter-welltunneling andconsideronlythe internal\ndynamics in each single well. The atomic tunneling be-\ntween two spins refers to spin-flipping is induced by the\nRaman coupling, and thus such Josephson tunneling is\nin the spin space. Considering the atomic gas condenses\nwith a finite but opposite momentum for different spins,\nthe internal JEs connect interesting quantum tunneling\nin the momentum space.\nIn summary, we have investigated the quantum dy-\nnamics of a SO coupled BEC in a symmetric double\nwell potential. The SO coupling contributes to atomic\ntunneling between wells and significantly enhances JEs\nfor realistic conditions. We have predicted a novel spin\nJosephsoneffectswhich canbe observedinapracticalex-\nperiment since all the required ingredients, including the\nSO coupled BECs and the tunable double well potential,\nhave already been achieved in the previous experiments.\nAcknowledgments\nThis work was supported by the NBRPC\n(No.2011CBA00302), the SKPBRC (Nos.2011CB922104\nand 2011CB921503), the NSF of China (Grant Nos.\n10974059, 11075020 and 11125417), and the GRF\n(HKU7058/11P) and CRF (HKU8/CRF/11G) of the\nRGC of Hong Kong.\n[1] M. V. Berry, Proc. Roy. Soc. London A 392, 45 (1984).\n[2] C. P. Sun and M. L. Ge, Phys. Rev. D 41, 1349 (1990);\nS. L. Zhu and Z. D. Wang, Phys. Rev. Lett. 85, 1076\n(2000); S. L. Zhu, Z. D. Wang, and Y. D. Zhang, Phys.\nRev. B61, 1142 (2000).\n[3] F. Wilczek and A. Zee, Phys. Rev. Lett. 52, 2111 (1984);\nA. Zee, Phys. Rev. A 38, 1 (1988).\n[4] For a review, see J. Dalibard, F. Gerbier, G. Juzeli¨ unas ,\nand P.¨Ohberg, Rev. Mod. Phys. 83, 1523 (2011).\n[5] R. Dum and M. Olshanii, Phys. Rev. Lett. 76, 1788\n(1996).\n[6] G.Juzeli¯ unas andP. ¨Ohberg, Phys.Rev.Lett. 93, 033602\n(2004).\n[7] J. Ruseckas, G. Juzeli¯ unas, P. ¨Ohberg, and M. Fleis-\nchhauer, Phys. Rev. Lett. 95, 010404 (2005)\n[8] G. Juzeli¯ unas, J. Ruseckas, P. ¨Ohberg, and M. Fleis-\nchhauer, Phys. Rev. A 73, 025602 (2006).\n[9] S. L. Zhu, H. Fu, C. J. Wu, S. C. Zhang, and L. M. Duan,\nPhys. Rev. Lett. 97, 240401 (2006).\n[10] X. J. Liu, X. Liu, L. C. Kwek, and C. H. Oh, Phys. Rev.\nLett.98, 026602 (2007).\n[11] T. D. Stanescu, C. Zhang, and V. Galitski, Phys. Rev.\nLett.99, 110403 (2007).\n[12] K. J. G¨ unter, M. Cheneau, T. Yefsah, S. P. Rath, and J.\nDalibard, Phys. Rev. A 79, 011604(R) (2009).\n[13] G. Juzeli¯ unas, J. Ruseckas, and J. Dalibard, Phys. Rev .A81, 053403 (2010).\n[14] N. R. Cooper, Phys. Rev. Lett. 106, 175301 (2011).\n[15] J. D. Sau, R. Sensarma, S. Powell, I. B. Spielman, and\nS. Das Sarma, Phys. Rev. B 83, 140510(R) (2011).\n[16] D. L. Campbell1, G. Juzeli¯ unas, and I. B. Spielman,\nPhys. Rev. A 84, 025602 (2011).\n[17] Z. F. Xu and L. You, arXiv: 1110.5705.\n[18] S. L. Zhu, L.-B. Shao, Z. D. Wang, and L.-M. Duan,\nPhys. Rev. Lett. 106, 100404 (2011); L. Jiang, T. Kita-\ngawa, J. Alicea, A. R. Akhmerov, D. Pekker, G. Refael,\nJ. I. Cirac, E. Demler, M. D. Lukin, and P. Zoller, ibid.\n106, 220402 (2011); S. L. Zhu, D. W. Zhang, and Z. D.\nWang,ibid.102, 210403 (2009).\n[19] Y. J. Lin, R. L. Compton, A. R. Perry, W. D. Phillips,\nJ. V. Porto, and I. B. Spielman, Phys. Rev. Lett. 102,\n130401 (2009); Y. J. Lin, R. L. Compton, K. Jim´ enez-\nGarc´ ıa, J. V. Porto, and I. B. Spielman, Nature 462, 628\n(2009); Y. J Lin, R. L. Compton, K. Jim´ enez-Garc´ ıa,\nW. D. Phillips, J. V. Porto, and I. B. Spielman, Nature\nPhys.,7, 531, (2011).\n[20] Y.-J. Lin, K. Jim´ enez-Garc´ ıa, and I. B. Spielman, Na-\nture,471, 83 (2011).\n[21] Z. Fu , P. Wang, S. Chai, L. Huang, and J. Zhang, Phys.\nRev. A84, 043609 (2011).\n[22] M. Aidelsburger, M. Atala, S. Nascimbne, S. Trotzky, Y. -\nA. Chen, I. Bloch, Phys. Rev. Lett. 107, 255301 (2011).\n[23] J. Struck, C. Olschlager, R. Le Targat, P. Soltan-Panah i,8\nA. Eckardt, M. Lewenstein, P. Windpassinger, and K.\nSengstock, Science 333, 996 (2011).\n[24] T. D. Stanescu, B. Anderson, andV. Galitski, Phys. Rev.\nA78, 023616 (2008).\n[25] C. Wang, C. Gao, C. M. Jian, and H. Zhai, Phys. Rev.\nLett.105, 160403 (2010).\n[26] C. M. Jian and H. Zhai, Phys. Rev. B 84, 060508 (2011).\n[27] T. L. Ho and S. Zhang, Phys. Rev. Lett. 107, 150403\n(2011).\n[28] C. J. Wu, I. Mondragon-Shem, and X. F. Zhou, Chin.\nPhys. Lett. 28, 097102 (2011).\n[29] S. K. Yip, Phys. Rev. A 83, 043616 (2011).\n[30] Z. F. Xu, R. L¨ u, and L. You, Phys. Rev. A 83, 053602\n(2011).\n[31] T. Kawakami, T. Mizushima, and K. Machida, Phys.\nRev. A84, 011607 (2011).\n[32] E. van der Bijl and R.A. Duine, Phys. Rev. Lett. 107,\n195302 (2011).\n[33] S. Gopalakrishnan, A. Lamacraft, and P. M. Goldbart,\nPhys. Rev. A 84, 061604(R) (2011).\n[34] S. Sinha, R. Nath, and L. Santos, Phys. Rev. Lett. 107,\n270401 (2011).\n[35] H. Hu, B. Ramachandhran, H. Pu, and X. J. Liu, Phys.\nRev. Lett. 108, 010402 (2012).\n[36] Q. Zhu, C. Zhang, and B. Wu, arXiv: 1109.5811.\n[37] R. Barnett, S. Powell, T. Grass, M. Lewenstein, and S.\nDas Sarma, arXiv: 1109.4945.\n[38] Y. Deng, J. Cheng, H. Jing, C.-P. Sun, and S. Yi, arXiv:\n1110.0558.\n[39] X.-Q. Xu and J. H. Han, Phys. Rev. Lett. 107, 200401\n(2011).\n[40] X. F. Zhou, J. Zhou, and C. J. Wu, Phys. Rev. A 84,\n063624 (2011) .\n[41] J. Radi´ c, T. Sedrakyan, I. Spielman, and V. Galitski,\nPhys. Rev. A 84, 063604 (2011).\n[42] M. Merkl, A. Jacob, F. E. Zimmer, P. ¨Ohberg, and L.\nSantos, Phys. Rev. Lett. 104, 073603 (2010).\n[43] Y. Zhang, L. Mao, and C. Zhang, arXiv: 1102.4045[Phys. Rev. Lett. (to be published)].\n[44] D. W. Zhang, Z. Y. Xue, H. Yan, Z. D. Wang, and S. L.\nZhu, arXiv: 1104.0444 [Phys. Rev. A (to be published)].\n[45] J. Javanainen, Phys. Rev. Lett. 57, 3164 (1986).\n[46] G. J. Milburn, J. Corney, E. M. Wright, and D. F. Walls,\nPhys. Rev. A 55, 4318 (1997).\n[47] A. Smerzi, S. Fantoni, S. Giovanazzi, and S. R. Shenoy,\nPhys. Rev. Lett. 79, 4950 (1997); S. Raghavan, A.\nSmerzi, S. Fantoni, and S. R. Shenoy, Phys. Rev. A 59,\n620 (1999).\n[48] S. Giovanazzi, A. Smerzi, and S. Fantoni, Phys. Rev.\nLett.84, 4521 (2000).\n[49] S. Levy, E. Lahoud, I. Shomroni, and J. Steinhauer, Na-\nture449, 579 (2007).\n[50] M. Albiez, R. Gati, J. F¨ olling, S. Hunsmann, M. Cris-\ntiani, and M, K. Oberthaler, Phys. Rev. Lett. 95, 010402\n(2005).\n[51] L. J. LeBlanc, A. B. Bardon, J. McKeever, M. H. T.\nExtavour, D. Jervis, J. H. Thywissen, F. Piazza, and A.\nSmerzi, Phys. Rev. Lett. 106, 025302 (2011).\n[52] G. Thalhammer, G. Barontini, L. De Sarlo, J. Catani, F.\nMinardi, and M. Inguscio, Phys. Rev. Lett. 100, 210402\n(2008).\n[53] M.-S. Chang, Q. Qin, W. Zhang, L.You, andM. S.Chap-\nman, Nat. Phys. 1, 111 (2005).\n[54] X.-Q. Xu, L.-H. Lu, and Y.-Q. Li, Phys. Rev. A 78,\n043609 (2009).\n[55] B. Sun and M. S. Pindzola, Phys. Rev. A 80, 033616\n(2009).\n[56] H. Pu, W. P. Zhang, and P. Meystre, Phys. Rev. Lett.\n89, 090401 (2002).\n[57] R. Qi, X.-L. Yu, Z. B. Li, and W. M. Liu, Phys. Rev.\nLett.102, 185301 (2009).\n[58] C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, Rev.\nMod. Phys. 82, 1225 (2010).\n[59] Y.-A Chen, S. Nascimb` ene, M. Aidelsburger, M. Atala,\nS. Trotzky, I.Bloch, Phys.Rev.Lett. 107, 210405 (2011)."    },    {        "title": "1212.3325v1.Spin_dynamics_in_tunneling_decay_of_a_metastable_state.pdf",        "content": "arXiv:1212.3325v1  [quant-ph]  13 Dec 2012Spin dynamics in tunneling decay of a metastable state\nYue Ban1and E. Ya. Sherman1,2\n1Department of Physical Chemistry, The University of the Bas que Country UPV/EHU, 48080 Bilbao, Spain\n2IKERBASQUE Basque Foundation for Science, Bilbao, 48011 Bi zkaia, Spain\n(Dated: October 30, 2018)\nWe analyze spin dynamics in the tunneling decay of a metastab le localized state in the presence\nof spin-orbit coupling. We find that the spin polarization at short time scales is affected by the\ninitial state while at long time scales both the probability - and the spin density exhibit diffraction-\nin-time phenomenon. We find that in addition to the tunneling time the tunneling in general\ncan be characterized by a new parameter, the tunneling lengt h. Although the tunneling length is\nindependent on the spin-orbit coupling, it can be accessed b y the spin rotation measurement.\nPACS numbers: 03.65.Xp, 03.75.-b, 71.70.Ej\nI. INTRODUCTION\nSpin-orbit(SO)coupling, interactionofparticlespinor\npseudospin with the orbital motion, provides an efficient\nway to control and manipulate spin, charge, and mass\ntransport. Two different kinds of SO coupled systems at-\ntractagreatdealofinterestduetotheknownandyetun-\nexploredvarietyofphenomenathey can demonstrateand\ndue to their possible applications in quantum technolo-\ngies. Oneclassofsystems, investigatedformorethan five\ndecades by now, is semiconductors and semiconductor-\nbased nanostructures [1–7]. The other class, lavished at-\ntention only recently, is cold atoms and Bose-Einstein\ncondensates, where by engineering external optical fields\nonecancoupleorbitalmotiontothe pseudospindegreeof\nfreedom [8–12] and cause Dresselhaus and Rashba types\nof SO coupling similar to that in solids.\nIn quantum systems of interest the tunneling plays\nan important role and either completely determines or\nstronglyinfluencesthe particledynamics. Atcertaincon-\nditions the tunneling rate depends on the spin of the\nparticle [2, 3, 13, 14]. The understanding of the tun-\nneling is the key for the understanding of the transient\nprocesses in a broad variety of systems, including, e.g.\ncharge transport in molecular nanostructures [15]. One\nof the key issues in the tunneling theory is the evalua-\ntion of the time spent by the particle in the classically\nforbidden regions. Similar to the scattering problem of\npropagating wave packets, the question of the tunnel-\ning time for the decay of a metastable system [16–20]\nmay also be posed. The spin-dependent effects based on\nthe Larmor clock concept for potential barriers [21, 22],\nHartman effect in graphene [23], and effective exchange\nfields in semiconductors [24] can provide a measure of\nthis time. In classically forbidden regions there is not\nonly precession but also a rotation of the magnetic mo-\nment into the magnetic field directionand the time ofthe\ninteraction with the barrieris closely related to this rota-\ntion [25, 26]. The effects of SO coupling on the tunneling\nthrough semiconductor quantum-well structures with a\nlateral potential barrier [27, 28] provide another tool to\nutilize electron spin modifying the charge transport innonmagnetic systems.\nHere we investigate the spin-dependent tunneling of\na state initially localized in the potential at short and\nlong compared to the state lifetime time scales [29]. This\npaper is organized as follows. In Sec. II we introduce\nthe model Hamiltonian and formulate the physical ob-\nservables of interest. We use the SU(2) spin rotation to\ngaugeoutspin-orbitcouplingand restoreits effectsin the\ncalculation of the observables by inverse transformation.\nIn Sec. III we study the dynamics at short time scale,\nconcentratingonspinoscillations,decay,andescapefrom\nthe localizing potential. In Sec. IV we study long-term\ndynamics in the far-field zone. The diffraction in time in\nspindensityisobservedatalongdistance, whereSOcou-\npling forms a precursor in the propagating density. We\nshow that the tunneling can be characterizednot only by\ntime, but also by a certain length parameter, which can\nbe accessed by detecting the spin precession due to the\nSO coupling. Conclusions summarize the results.\nII. MODEL FOR SPIN-DEPENDENT\nTUNNELING\nAs a model we consider shownin Fig.1 time-dependent\npotentialU(x,t),infinite at x <0, with the time-\ndependence:\nU(x,t) =U1(x) (t<0), U(x,t) =U2(x) (t>0).(1)\nAtt≤0the potentialholdsbound stateswith wavefunc-\ntionsϕj(x) andenergy Ej; itchangesat t>0fromastep\nto a barrier to allow the tunneling. Such time-dependent\npotential can be produced by a recently developed tech-\nnique [30] where a moving laser beam “paints” a broad\nvariety of coordinate- and time-dependent potentials.\nWith SO coupling taken into account, the total one-\ndimensional Hamiltonian is\nˆH=1\n2m/parenleftBig\n/planckover2pi1ˆkx+ˆAx/parenrightBig2\n+U(x)−p2\nso\n2m,(2)\nwithˆkx=−i∂/∂x, the vector potential ˆAx=psoˆσx,\npso≡mα//planckover2pi1, whereαis the Dresselhaus SO coupling2\nFIG. 1. Time-dependent potential U(x):U1(x) att <0 and\nU2(x) whent >0.U1(x) is a step in the positive halfplane,\nthat is zero from x= 0 tox=a1andU0fromx=a1to\ninfinity, while U2(x) is a barrier of the height U0extended\nfromx=a1tox=a2. The potential is always infinite in the\nnegativehalfplane. Thebarrierwidthisdefinedas d≡a2−a1.\nconstant, and ˆ σxis the corresponding Pauli matrix. The\nspatial scale of spin precession is characterized by the\nlength 2ξ=/planckover2pi1/pso. The typical values of ξfor differ-\nent systems with SO coupling can be of interest. For (i)\nelectrons in semiconductor GaAs nanostructures [1] with\nm≈6×10−29g,α//planckover2pi1∼0.5×106cm/s, (ii) for6Li atoms\n[9] withm≈10−23g,α//planckover2pi1∼10 cm/s, and (iii) for87Rb\natoms [12] with m≈1.5×10−22g,α//planckover2pi1∼0.3 cm/s, we\nobtainξ∼10−5cm. It is interesting to mention that\nalthough masses and coupling constants for electrons in\nnanostructures and cold atoms are very different, the re-\nsulting precession length is on the same order of magni-\ntude for all these systems. Since typical localizationscale\nof electrons in nanostructures a1is on the order of 10−6\ncm, they are in the weak-coupling regime with ξ/a1≫1.\nCold atoms, however, are localized on the scale of the or-\nder visible light wavelength, that is of 10−4cm, and the\nstrong SO coupling regime with ξ < a1can be achieved\nthere. The characteristic timescale corresponding to the\nparticle motion inside the potential, ma2\n1//planckover2pi1, is on the or-\nder of 0.1-1 ps for electrons in nanostructures and 0.1-1\nms for cold atoms.\nFrom now on we use the system of units with /planckover2pi1≡1\nand particle effective mass m≡1. The wave function\ncorresponding to ˆHisψ(x,t) with the initial state set as\nψ(x,0) =χϕj(x), (3)\nand the spinor χ= [1,0]Tcorresponds to the z-axis ori-\nentation. Such a function can be produced by applying\na magnetic field along the z-axis for an electron or by\na special design of optical field in the case of a neutral\nbosonic atom [12]. At t >0, neither the orbital wave\nfunctionϕjis the eigenstate, nor χis the eigenspinor\nof Hamiltonian ˆH. The initial state begins to evolve at\nt>0 with spin precession and the probability to find the\nparticle inside the potential decreases simultaneously.\nWe use a gauge transformationˆ˜H=SˆHS−1with a\nunitary matrix S= exp(iˆσxx/2ξ) to gauge away the SO\ncoupling. The gauge transformation shifts ˆkxby ˆσx/2ξand turns SO coupling into a constant, with the wave\nfunction evolving in time transformed as:\n˜ψ(x,t) =Sψ(x,t) =/bracketleftbigg˜ψ1(x,t)\n˜ψ−1(x,t)/bracketrightbigg\n,(4)\nwith the upper and lower components\n˜ψσ(x,t) =/integraldisplay∞\n0Gσ(k)φk(x)exp/parenleftbigg\n−ik2t\n2/parenrightbigg\ndk,(5)\nand the coefficients are\nGσ(k) =/integraldisplay∞\n0˜ψσ(x,0)φk(x)dx. (6)\nHere,φk(x) is the eigenstate of the Hamiltonian corre-\nsponding to U2(x):\nφk(x) =\n\nC(k)sin(kx), (0<x<a 1)\nD(k)e−κkx+F(k)eκkx,(a1<x<a 2)\n/radicalbig\n2/πsin[kx+θ(k)],(x>a2)(7)\nnormalizedas /angbracketleftφk′|φk/angbracketright=δ(k−k′). The coefficients C(k),\nD(k),F(k) and the phase θ(k) satisfy the boundary con-\nditions of the potential U2(x), andκk=√2U0−k2.\nIn the tunneling regime k <√2U0, while in the prop-\nagating regime k >√2U0, andiκkis substituted by\nq=√\nk2−2U0.\nThe initial wave function corresponding to Hamilto-\nnianˆ˜Hcan be expressed as\n˜ψ(x,0) =ϕj(x)/bracketleftbigg\ncos(x/2ξ)\nisin(x/2ξ)/bracketrightbigg\n. (8)\nThe coefficients Gσ(k) become\nG1(k) =/integraldisplay∞\n0cos(x/2ξ)ϕj(x)φk(x)dx, (9)\nG−1(k) =i/integraldisplay∞\n0sin(x/2ξ)ϕj(x)φk(x)dx.(10)\nIn what follows, we investigate the particle motion by\ncalculating the physical observables such as probability\ndensity, spin density, and spin polarization defined as:\nρ(x,t) =ψ†(x,t)ψ(x,t), (11)\nσi(x,t) =ψ(x,t)†ˆσiψ(x,t), (12)\npi(x,t) =ˆσi(x,t)\nρ(x,t), (13)\nrespectively. As the spin rotates around the effective\nSO coupling field along the x-direction, the integral of\nσx(x,t)-component over the x>0 half axis is conserved.\nFor this reason, we investigate spin density and spin po-\nlarization in the more informative y−component.3\nIII. SPIN DYNAMICS AT SHORT TIMES\nTo address the short-term dynamics on the time less\nthan the lifetime of the initial bound state [29], we study\ntwo relevant quantities. First quantity is σy(a2,t), the\nspin density at the exit of the barrier a2. The other one,\np[w]\ny(t), defined as\np[w]\ny(t) =/integraldisplaya1\n0ψ†(x,t)σyψ(x,t)dx\n/integraldisplaya1\n0ψ†(x,t)ψ(x,t)dx,(14)\nbeing the spin polarization ycomponent in the poten-\ntial, represents the integrated spin dynamics inside the\npotential. The denominator of Eq.(14) is the probability\nto find the particle inside the potential, which decays to\n1/eof its initial value at the lifetime of the metastable\nstate.\nWe considerthe evolutionofthreeinitial orbitalstates,\nas shown in Fig.2 for the spin component at the edge.\nWe choose U0= 16 so that there are two bound states,\nthe ground state ϕ0and the first excited state ϕ1, and\nuse dimensionless a1≡1. Here, the initial state has\na strong impact on the time-dependent σy(a2,t). It is\ndemonstrated in Fig.2 that it takes very short time for\nσy(a2,t) to develop into the minimum from zero, irre-\nspective of the initial state. This behavior is similar to\nthe fast development of a plateau in the outgoing flux af-\nter the potential change [29]. However, spin density with\nϕ1decays faster than that with ϕ0. The linear combi-\nnation of these two bound states ϕcom= (ϕ0+ϕ1)/√\n2\nshows strong oscillations due to the interference between\nthem. As both spin density and probability density in\nthe potential decay with time, spin polarization tends to\na constant at large time.\n0 1 2 3 4\nt-0.15-0.1-0.050y(a2, t)σ\nFIG. 2. (Color online) Comparison of time evolution of\nσy(a2,t) at the right edge of the barrier a2with different ini-\ntial states, the ground state ϕ0(solid), the first excited state\nϕ1(dashed) and the linear combination ϕcom= (ϕ0+ϕ1)/√\n2\n(dot-dashed), provided by the barrier d= 0.4,U0= 16, and\nξ= 0.5. Negative σy(a2,t) is related to the direction of spin\nprecession determined by the sign of coupling constant α.\nThe tunneling resulted from various initial states has\na strong effect on spin polarization, especially in short-time scales, as shown in Fig.3. Spin polarization inside\nthe potential oscillates between boundaries. The oscilla-\ntion rate is fast and determined by the energy difference\nof the initially bound states. With the time, the con-\ntribution of the excited state rapidly decays due to the\nfast tunneling, and the spin remains in the state achieved\nby the time of the decay of the upper bound state. The\namplitude of resonances for ϕ1is larger than ϕ0, as the\nformer possesses larger momentum. As for the linear\ncombination of these two bound states, the spin polar-\nization is greatly enhanced because of interferences. On\nthe other hand, strength of SO coupling strongly influ-\nences the spin polarization in the potential as illustrated\nbymakingcomparisonsfor ξ= 0.5andξ= 1parameters.\n00.020.040.06py[w](t)\n00.10.2py[w](t)\n0 2 4 6 8\n t-0.6-0.4-0.20py[w](t)ϕ0(a)\n(b)ϕ1\n(c)ϕcom\nFIG. 3. (Color online) Time evolution of p[w]\ny(t) in the poten-\ntial from 0 to a1, caused by SO coupling with ξ= 0.5 (solid)\nandξ= 1 (dashed), with different initial coordinate states:\n(a)ϕ0, (b)ϕ1, and (c) ϕcom= (ϕ0+ϕ1)/√\n2, provided by the\nsame barrier as in Fig.2.\nHaving illustrated the evolution of spin density at the\nboundary,σy(a2,t), and averaged spin polarization, we\ncan address the details of the spin distribution inside the\npotential. As shown in Fig.4, this distribution oscillates\ndue to the interferences between the ground state and\nother eigenstates. The amplitude of oscillations becomes4\nweaker with time as the state decays.\n0.00.20.40.60.81.00.00.51.01.52.02.53.0\nxt0.87\n/MinuΣ0.84Σy/LParen1x, t/RParen1\nFIG. 4. (Color online) Spin density ( ycomponent) in the\npotential vs time and position for ξ= 0.5 andψ(x,t= 0) =\nχϕ0. The parameters of the barrier are the same as in Fig.2.\nIV. SPIN DYNAMICS IN THE FAR-FIELD\nZONE\nHere we investigate the long-term spin dynamics, by\nconsidering the probability density ρandycomponent\nof spin density σy(X,t) which can be detected at a given\nlong distance X≫1. The density evolution for ρ(X,t)\natX= 10πis shown in Fig.5 for different SO coupling\nstrength. Under different ξ, the time evolutions of prob-\nability density show a strong sharp peak followed by os-\ncillations due to the diffraction-in-time process [31–33].\nHowever, unlike the case without SO coupling [29, 34],\nfor strong couplings (small ξ), there exist some oscilla-\ntions before the sharppeak, where the interferenceoftwo\nvelocities of spin up and spin down components are re-\nmarkable. A very interesting feature is the precursor of\nthe main peak corresponding to the opposite spin. The\nprecursor becomes stronger with the increase in the SO\ncoupling. The time dependence of σy(X,t) atX= 10π\nis shown in Fig.6. Different values of ξresult in differ-\nent spin rotation angles and corresponding spin density,\nprovided that σy(x,t) is detected at the same position.\nSpin precession in the ( y,z)-plane is described by the\nclassical precession angle, β[so]\n0= Ωt, where the preces-\nsion frequency is Ω ≈2αk0=k0/ξand the wave packet\nis detected at the position Xat the time instant X/v,\nthe velocity v=k0. Therefore, the rotation angle is\nβ[so]\n0=X/ξ, independent on time t, provided that the\nwave packet is considered classically. Ideally, for a free\nparticle propagating from the origin x(t= 0) = 0, spin\npolarizations components are px= 0,py=−sinβ[so]\n0,\npz= cosβ[so]\n0. As a result, at X=nπξ, the classical spin\npolarization should be py= 0.However, in the tunneling\nproblem we consider, the value of pyis not zero even at\nX=nπξ.00.010.020.03\n51015202530\nt00.020.040.060.086780.008\n6780.008ρ(X, t)(a)\n(b)(X, t)ρ\nFIG. 5. (Color online) (a) Time-dependent density at the\nobservationpoint X= 10πfor systemparameter U0= 16,d=\n0.4, andξ= 0.5. Inset shows the precursor dynamics on a\nshorter time scale. Dashed line corresponds to U0= 16,d=\n0.4, andξ= 1. (b) Solid line corresponds to U0= 16,d= 0.4,\nandξ= 0.5, dot-dashed line corresponds to U0= 8,d= 0.4,\nandξ= 0.5. Inset shows the precursor dynamics on a shorter\ntime scale.\n51015202530\nt00.010.02y(X, t)σ\nFIG. 6. Spin density in the observation point X= 10π\nforU0= 16,d= 0.4,ψ(x,t= 0) =χϕ0andξ= 0.5. The\nprecursor is made by the contribution of the opposite spin.\nOther parameters are the same as in Fig.2.\nFor the weak SO coupling ξ≫1, where |ψ−1/ψ1| ≪1,\nthe spin polarization can be calculated as:\npy≈ −sinx\nξ+2cosx\nξℑψ−1\nψ1. (15)\nSecond term in Eq.(15) can be viewed as a result of a\ncorrection to the particle displacement in the form py=5\n0.511.522.53\nd-1.8-1.6-1.4-1.2-1-0.8-0.6xδ\nκ\nFIG. 7. (Color online) Parameter of additional displacemen t\nδxas the function of the barrier transparency. Filled circles\ncorrespond to U0= 16 and variable width d, the squares\ncorrespond to d= 0.4 with variable U0, andξ= 10.\n−sin(x/ξ+δx/ξ), where, according to Eqs.(9),(10):\nδx=−2 lim\nξ→∞ξℑ/integraldisplay∞\n0G−1(k)exp(−ik2t/2)φk(x)dk\n/integraldisplay∞\n0G1(k)exp(−ik2t/2)φk(x)dk.(16)\nThis deviation of pyfrom zero at X=nπξdemonstrates\nthat the real precession angle is β[so]=β[so]\n0+δβ[so],\nwith the correction δβ[so]can be viewed as a result of ad-\nditionaldisplacementoftheparticle δxwithδx=ξδβ[so].\nFrom Eq.(16), we can see that δxis independent of the\nSO coupling, while the parameters of the barrier play an\nimportant role, as the eigenfunctions φk(x) strongly de-\npend on them. Numerical results also show that δxis\nindependent of coordinate and time if measured at dis-\ntanceX≫a2at timest>X/v. The tunneling length δx\nis presented in Fig.7 for two different barriers. This fig-\nureshowsaclearcrossoverfromtheclassical(transparent\nbarrier) to the tunneling (opaque barrier) regimes, where\nδxshows a saturation, and the additional displacementis universal. The saturation of the tunneling length for\nopaque barriers seems similar to the Hartman effect [35]\non the group delay in a scattering case [36]. However,\nafter making comparisons of δt≡δx/vand calculated\ngroupdelayforasinglebarrierwithparameterspresented\nin Fig.7, we cannot draw the unambiguous conclusion on\nthe relevance of these two quantities.\nV. CONCLUSIONS\nIn conclusion, we investigated the spin dynamics of\na tunneling particle initially localized in a potential in\nthe presence of SO coupling. It is shown that at short-\ntime scales initial states play an important role in spin\npolarization while the spin density and probability den-\nsity possess diffraction-in-time phenomenon at long-time\nscales. We showed that in addition to the time, tunnel-\ning can be characterized by a characteristic length. We\nuse the rotation angle to identify the tunneling length in\nthe presence of weak coupling. The tunneling length de-\npends on the barrier parameters, being independent on\nthe strength of the spin-orbit coupling. These effects can\nbe observed in experiments with cold atoms, where the\nSO coupling is strong enough to cause spin precession on\na relatively short spatial scale.\nACKNOWLEDGEMENT\nThis work of EYS was supported by the MCINN of\nSpain Grant FIS2009-12773-C02-01,by “Grupos Consol-\nidadosUPV/EHUdel GobiernoVasco”GrantIT-472-10,\nand by the UPV/EHU under program UFI 11/55. Y. B.\nacknowledges financial support from the Basque Govern-\nment (Grant No. BFI-2010-255). We are grateful to J.G.\nMuga and D. Sokolovski for valuable discussions.\n[1] I.ˇZuti` c, J. Fabian and S. Das Sarma, Rev. Mod. Phys.\n76, 323 (2004).\n[2] S. Amasha, K. MacLean, I. P. Radu, D. M. Zumb¨ uhl,\nM. A. Kastner, M. P. Hanson, and A. C. Gossard, Phys.\nRev. B78, 041306(R) (2008).\n[3] P. Stano and P. Jacquod, Phys. Rev. B 82, 125309\n(2010).\n[4] P. Stano and J. Fabian, Phys. Rev. B 72, 155410 (2005).\n[5] R. Romo and S. E. Ulloa, Phys. Rev. B 72, 121305(R)\n(2005).\n[6] I. Malajovich, J. M. Kikkawa, and D. D. Awschalom, J.J.\nBerry and N. Samarth, Phys. Rev. Lett. 84, 1015 (2000).\n[7] D. D. Awschalom and M. E. Flatt´ e, Nat. Phys. 3, 153\n(2007).\n[8] T. D. Stanescu, B. Anderson, andV. Galitski, Phys. Rev.\nA78, 023616 (2008).[9] X.-J. Liu, M. F. Borunda, X. Liu, and J. Sinova, Phys.\nRev. Lett. 102, 046402 (2009).\n[10] C. Wang, C. Gao, C.-M. Jian, and H. Zhai, Phys. Rev.\nLett.105, 160403 (2010).\n[11] Y.-J. Lin, K. Jimenez-Garcia, and I. B. Spielman, Natur e\n471, 83 (2011).\n[12] D. L. Campbell, G. Juzeli˜ unas, and I. B. Spielman, Phys .\nRev. A84, 025602 (2011)\n[13] M. M. Glazov, P. S. Alekseev, M. A. Odnoblyudov, V. M.\nChistyakov, S. A. Tarasenko, and I. N. Yassievich, Phys.\nRev. B71, 155313 (2005).\n[14] D. Khomitsky and E. Ya. Sherman, EPL 90, 27010\n(2010).\n[15] X. Zheng, F. Wang, C.-Y. Yam, Y. Mo, and G.H. Chen,\nPhys. Rev. B 75, 195127 (2007), Y. Wang, C.-Y. Yam,\nG.H. Chen, T. Frauenheim, and T.A. Niehaus, Chem.\nPhys.391, 69 (2011), C.-Y. Yam, X. Zheng, G.H. Chen,6\nY. Wang, Th. Frauenheim, and T.A. Niehaus, Phys. Rev.\nB83, 245448 (2011).\n[16] A. Marchewka and E. Granot, Phys. Rev. A 79, 012106\n(2009).\n[17] G. K¨ albermann, Phys. Rev. C 79, 024613 (2009); 77,\n041601 (2008).\n[18] N. G. Kelkar, H.M. Casta˜ neda, and M. Nowakowski, Eu-\nrophys. Lett. 85, 20006 (2009).\n[19] S. Cordero, G. Garc´ ıa-Calder´ on, R. Romo, and J. Villa v-\nicencio, Phys. Rev. A 84, 042118 (2011).\n[20] M. Pons, D. Sokolovski, and A. del Campo, Phys. Rev.\nA85, 022107 (2012).\n[21] A. I. Baz’, Yad. Fiz. 5, 229 (1967).\n[22] V. F. Rybachenko, Yad. Fiz. 5, 895 (1967).\n[23] R. A. Sepkhanov, M. V. Medvedyeva, and C. W. J.\nBeenakker, Phys. Rev. B 80, 245433 (2009).\n[24] B.HuangandI.Appelbaum, Phys.Rev.B 82, 241202(R)\n(2010).\n[25] M. B¨ uttiker and R. Landauer, Phys. Rev. Lett. 49, 1739\n(1982).\n[26] M. B¨ uttiker, Phys. Rev. B 27, 6178 (1983).\n[27] V. A. Sablikov and Y. Ya. Tkach, Phys. Rev. B 76,\n245321 (2007).[28] V. I. Perel’, S. A. Tarasenko, I. N. Yassievich, S. D.\nGanichev, V. V. Bel’kov, and W. Prettl, Phys. Rev. B\n67, 201304 (2003).\n[29] Y. Ban, E. Ya. Sherman, J. G. Muga, and M. B¨ uttiker,\nPhys. Rev. A 82, 062121 (2010).\n[30] K. Henderson, C. Ryu, C. MacCormick and M. G.\nBoshie, New J. Phys. 11, 043030 (2009).\n[31] M. Moshinsky, Phys. Rev. 88, 625 (1952). The result\nof this paper can be understood in terms of diffraction\nof perfectly coherent propagating waves. For decay of a\nmetastable state, the diffraction can be seen as produced\nby partially coherent waves: M. Born and E. Wolf, Prin-\nciples of Optics , Chapter X, 1999 (7th Ed.), see also in\nRef. [34].\n[32] A. Steane, P. Szriftgiser, P. Desbiolles, and J. Daliba rd,\nPhys. Rev. Lett. 74, 4972 (1995).\n[33] A. del Campo, G. Garc´ ıa-Calder´ on, and J.G. Muga,\nPhys. Rep. 476, 1 (2009).\n[34] E. Torrontegui, J. Mu˜ noz, Y.Ban, and J. G. Muga, Phys.\nRev. A83, 043608 (2011).\n[35] T. E. Hartman, J. Appl. Phys. 33, 3427 (1962).\n[36] E. P. Wigner, Phys. Rev. 98, 145 (1955)."    },    {        "title": "2305.12385v2.Patterns__spin_spin_correlations_and_competing_instabilities_in_driven_quasi_two_dimensional_spin_1_Bose_Einstein_condensates.pdf",        "content": "Patterns, spin-spin correlations and competing instabilities in driven quasi-two-dimensional spin-1\nBose-Einstein condensates\nSandra M Jose,1Komal Sah,1, 2and Rejish Nath1\n1Department of Physics, Indian Institute of Science Education and Research, Pune 411 008, India\n2Department of Physics, University of California, Davis 95616, USA\nWe analyze the formation of transient patterns and spin-spin correlations in quasi-two-dimensional spin-1\nhomogeneous Bose-Einstein condensates subjected to parametric driving of s-wave scattering lengths. The\ndynamics for an initial ferromagnetic phase is identical to that of a scalar condensate. In contrast, intriguing\ndynamics emerges for an initial polar state. For instance, we show that competition exists between density\npatterns and spin-mixing dynamics. Dominant spin-mixing dynamics lead to a gas of polar core vortices and\nanti-vortices of di fferent spin textures. The density modes of the Bogoliubov spectrum govern the wavenumber\nselection of Faraday patterns. The spin modes determine the vortex density and the spatial dependence of\nspin-spin correlation functions. When the density patterns outgrow the spin-mixing dynamics, the spin-spin\ncorrelations decay exponentially with a correlation length of the order a spin healing length; otherwise, they\nexhibit a Bessel function dependence. Strikingly, competing instabilities within density and spin modes emerge\nwhen both scattering lengths are modulated at di fferent frequencies and appropriate modulation amplitudes.\nThe competing instability leads to a superposition of density patterns or correlation functions of two distinct\nwavelengths. Our studies reveal that fine control over the driven dynamics can be attained by tuning interaction\nstrengths, quadratic Zeeman field, driving frequencies, and amplitudes.\nI. INTRODUCTION\nPeriodically driven Bose-Einstein condensates (BECs) have\nbeen a playground for studying various phenomena such as\nFaraday patterns [1–17], dark-soliton lattice [18], dynamical\nlocalization [19–21], spin freezing [22, 23], matter-wave jets\n[24–27], bright solitons [28, 29], parametric instability [30],\ngenerating higher harmonics [31] etc. Faraday patterns are\nobserved experimentally in elongated condensates by modu-\nlating either the trap frequencies [2, 16] or the interactions\n[32, 33]. Such patterns o ffer critical insights into the ele-\nmentary excitations of condensates because the pattern size\nis determined by the Bogoliubov mode resonant with half of\nthe driving frequency. A non-monotonous excitation spectrum\ncan make the wavenumber selection non-trivial [4, 10].\nBecause of the spin degrees of freedom, spinor condensates\nare ideal for exploring spin textures and magnetic phases [34–\n36]. Interesting phenomena such as dynamical stabilization\n[23], spin-squeezing [37, 38], Shapiro resonances [39, 40],\nparametric resonances [41] and the quantum walk in momen-\ntum space [42, 43] have been reported in driven spinor con-\ndensates. In this paper, we analyze the formation of tran-\nsient density and spin patterns, and spin-spin correlations in\na quasi-two-dimensional (Q2D) spin-1 homogeneous conden-\nsate subjected to the parametric driving of s-wave scattering\nlengths a0anda2. We consider three cases where either a0or\na2is modulated individually, and both are modulated simul-\ntaneously. Modulating either a0ora2leads to modulations\nin both spin-independent and spin-dependent interactions, af-\nfecting dynamics of both spatial and spin degrees of freedom.\nAs we show, the dynamics depends critically on the initial\nstates and in particular, we consider ferromagnetic and polar\nphases. A ferromagnetic phase is immune to the modulation\nofa0[44], whereas a2modulation results in dynamics simi-\nlar to that of a driven scalar condensate [1, 2]. In contrast, an\ninitial polar condensate exhibits non-trivial dynamics. Unsta-ble Bogoliubov modes lead to both Faraday patterns and spin-\nmixing dynamics. There exists an implicit competition be-\ntween density modulations and spin-mixing dynamics. When\nthe spin-mixing becomes dominant, a gas of polar core vor-\ntices (PCVs) and anti-vortices of di fferent spin textures are\nformed. Previous studies on PCVs in Q2D spin-1 condensates\nare based on the Kibble-Zurek mechanism via quenching the\nquadratic Zeeman field from non-polar phases [37, 45, 46].\nThe unstable momentum of the density mode quantifies the\nwavenumber of the Faraday pattern, whereas that of the spin\nmode determines the vortex density. At longer times, spin-\nmixing disrupts the selection of higher harmonics, which is in\nhigh contrast to the case of a scalar condensate. If the spin-\ndependent interactions are stronger than the spin-independent\nones, the Faraday patterns outgrow the spin dynamics, and the\nspin-spin correlations decay exponentially over distance with\na correlation length of the order a spin healing length. In con-\ntrast, when the spin dynamics are dominant, the spin-spin cor-\nrelations are governed by a Bessel function with an argument\ndepending on the momentum of the unstable spin mode.\nInterestingly, when both scattering lengths are modulated\nsimultaneously with the same frequency, the modulation\namplitudes can be chosen such that parametric driving is\npresent only in spin-independent or spin-dependent interac-\ntions. Hence, it is possible to excite the density or spin mode\nalone during the initial stage of the dynamics. When the mod-\nulation frequencies di ffer, a fascinating scenario of competing\ninstabilities appears. For instance, two momenta from either\ndensity or spin mode become equally unstable, causing com-\npetition between the two wavelengths. In a classical fluid, two\npatterns with di fferent symmetries coexist in the onset of com-\npeting instabilities [47]. In our case, the competing instability\nresults in a superposition of Faraday patterns or correlation\nfunction with di fferent wavelengths whose amplitudes vary in\ntime. To conclude, our studies reveal that the nature of driven\ndynamics in spinor condensates can be controlled by tuning\nthe interaction strengths, quadratic Zeeman field, modulationarXiv:2305.12385v2  [cond-mat.quant-gas]  1 Aug 20232\nfrequencies, and amplitudes.\nThe paper is structured as follows. In Sec. II, we discuss the\nsetup and the meanfield equations describing a spin-1 conden-\nsate. The Mathieu-like equations governing the wavenumber\nselection of a driven spin-1 condensate is derived in Sec. III.\nThe dynamics of driven ferromagnetic and polar phases, in-\ncluding the spin-spin correlations is discussed in Sec. IV and\nSec. V, respectively. Competing instability of di fferent den-\nsity or spin modes is discussed in Sec. V C 1. The experimen-\ntal possibilities are discussed in Sec. VI. Finally, we summa-\nrize and provide an outlook in Sec. VII.\nII. SETUP AND MODEL\nWe consider a Q2D spin-1 homogeneous Bose gas in the\npresence of a quadratic Zeeman field q. The Hamiltonian de-\nscribing the system is\nˆH=Z\ndρX\nm=0,±1ˆψ†\nm(ρ) \n−ℏ2\n2M∇2\nρ+qm2!\nˆψm(ρ)+ˆHZ+ˆVI,\n(1)\nwhere ˆψmis the field operator which annihilates a boson in the\nmth Zeeman state, Mis the mass of a boson and ρ=(x,y).\nThe quadratic Zeeman Hamiltonian is,\nˆHZ=qZ\ndρX\nm1,m2ˆψ†\nm1(ρ)\u0010ˆF2\nz\u0011\nm1,m2ˆψm2(ρ), (2)\nand the interaction operator is\nˆVI=1\n2Z\ndρh\n˜c0: ˆn2(ρ) :+˜c1:ˆF2(ρ) :i\n, (3)\nwhere ˜ c0,1=c0,1/√\n2πlzwith lz=p\nℏ/mωzbeing the trans-\nverse width of the condensate provided by the harmonic po-\ntential Vt(z)=mω2\nzz2/2, ˆn(ρ)=Pf\nm=−fˆψ†\nm(ρ)ˆψm(ρ) is the\ntotal density operator. The symbol : : denotes the normal\nordering that places annihilation operators to the right of the\ncreation operators. The components of the spin density oper-\nator are\nˆFν∈x,y,z(ρ)=X\nm,m′(fν)mm′ˆψ†\nm(ρ)ˆψm′(ρ), (4)\nwith fνbeing theνth component of the spin-1 matrices. The\nspin-independent and spin-dependent interaction constants\narec0=(g0+2g2)/3>0 and c1=(g2−g0)/3, respectively\nwith gF=4πℏ2aF/mrelated to the scattering length aF=0,2\nof the total spin-Fchannel.\nAt very low temperatures the system is described by\nthe coupled non-linear Gross-Pitaevskii equations (NLGPEs),where ˆψm(ρ) is replaced by a c-number ψm(ρ),\niℏ∂ψ1\n∂t=−ℏ2∇2\nρ\n2M+q+˜c0n+˜c1Fzψ1+˜c1F−√\n2ψ0(5)\niℏ∂ψ0\n∂t=−ℏ2∇2\nρ\n2M+˜c0nψ0+˜c1√\n2F+ψ1+˜c1F−√\n2ψ−1(6)\niℏ∂ψ−1\n∂t=−ℏ2∇2\nρ\n2M+q+˜c0n−˜c1Fzψ−1+˜c1F+√\n2ψ0,(7)\nwhere Fν=P\nm,m′ψ∗\nm(fν)mm′ψm′,n(ρ,t)=P\nm|ψm(ρ,t)|2and\nF±=Fx±iFy. To study the modulation-induced dynamics, we\nsolve Eqs. (5)-(7) numerically, starting from a homogeneous\ndensity embedded with a small noise in all three components\n[37, 46].\nIII. TIME MODULATION OF s-WA VE SCATTERING\nLENGTHS\nWe consider a time dependent aj(t)=¯aj[1+2αjcos(2ωjt)],\nwhere ¯ ajis the mean scattering length, αjis the modula-\ntion amplitude and 2 ωjis the driving frequency. Scattering\nlengths can be periodically modulated by Feshbach resonance\n[32, 33, 48], or using radio frequency or microwave fields [49–\n52]. We consider three cases: (i) a0time-dependent and a2\nconstant, (ii) a0constant and a2time-dependent, (iii) both a0\nanda2are time-dependent. These cases can be implemented\nby independently controlling the two scattering lengths [53].\nThe interaction coe fficients ˜ c0,1for the three cases are\n( i ) ˜ c0(t)=¯c0+(2α0¯g0/3) cos(2ω0t),\n˜c1(t)=¯c1−(2α0¯g0/3) cos(2ω0t),\n( ii ) ˜ c0(t)=¯c0+(4α2¯g2/3) cos(2ω2t),\n˜c1(t)=¯c1+(2α2¯g2/3) cos(2ω2t),\nand\n(iii) ˜ c0(t)=¯c0+(2α0¯g0/3) cos(2ω0t)+(4α2¯g2/3) cos(2ω2t),\n˜c1(t)=¯c1−(2α0¯g0/3) cos(2ω0t)+(2α2¯g2/3) cos(2ω2t),\nwhere ¯ c0=(¯g0+2¯g2)/3 and ¯ c1=(¯g2−¯g0)/3 with ¯ gj=\n4πℏ2¯aj/(M√\n2πlz).\nThe homogeneous solution in the presence of modulation is\nψ(t)=√\n¯nζexp(−iθ(t)/ℏ) where ψ(t)=(ψ1,ψ0,ψ−1)Tand\nθ(t)=Zt\n0\u0002¯n˜c0(t′)+A˜c1(t′)\u0003dt′+Bt (8)\nwhere A=h\n2n0(n1+n−1)+F2\nz+4n0√n1n−1i\n/¯nandB=\nq(n1+n−1)/¯nwith Fz=n1−n−1andnm=¯nζ2\nm. Now, we\nintroduce\nψ(ρ,t)=h√\n¯nζ+w(t) cos( k·ρ)i\ne−iθ(t)/ℏ, (9)\nin Eqs. (5-7), where w(t)=(w1,w0,w−1)Tis the amplitude of\nmodulations and linearize in w(t). Writing w(t)=u(t)+iv(t)3\nFigure 1. (color online). Wavenumber selection of an initial fer-\nromagnetic homogeneous phase for a2modulation, ¯ c0¯n=0.3ℏωz,\n¯c1¯n=−0.1ℏωz,q=−0.3ℏωz, andα2=0.4. (a) The most unstable\nmomentum kuas a function of driving frequency ω2. The insets show\nthe numerical results of Faraday patterns, and (b) and (c) show the\ncorresponding condensate momentum density for ω2/ωz=0.3 and\nω2/ωz=0.5 atωzt=330 andωzt=250, respectively. C.D. stands\nfor condensate density, and in (b) and (c), the central peak at k=0 is\nremoved for the visibility of the momentum rings.\nwhere u=(u1,u0,u−1)Tandv=(v1,v0,v−1)Tare real-\nvalued vectors, we obtain the two first-order coupled di ffer-\nential equations:\n−ℏdv\ndt=(EkI+M1+M2)u (10)\nℏdu\ndt=(EkI+M1+M3)v (11)\nwhere k=|k|,Ek=ℏ2k2/2M,Iis a 3×3 identity matrix,\nM1is a time-independent diagonal matrix depending only on\nthe Zeeman field with elements ( M1)11=(M1)33=q−Band\n(M1)22=−B. WhereasM2,3depends on the initial spinor\nψm, and are real, interaction and time-dependent matrices.\nEquations (10) and (11) can be combined into a Mathieu-like\nsecond-order di fferential equation depending on the system\nparameters.IV . FERROMAGNETIC PHASE\nFor an initial ferromagnetic phase with ζ=(1,0,0)T,\nthe matrixM2is diagonal in form with elements ( M2)11=\n2¯n[˜c0(t)+˜c1(t)], (M2)22=0 and (M2)33=−2¯n˜c1(t), and\nM3has only one non-zero element ( M3)33=−2¯n˜c1(t). The\nrelevant Mathieu equation is,\nd2u1\ndt2+1\nℏ2h\nϵ2\nk,1+4¯nEkα2¯g2cos(2ω2t)i\nu1=0, (12)\nwhere\nϵk,1=p\nEk[Ek+2(¯c0+¯c1)¯n], (13)\nis the Bogoliubov dispersion describing the density excita-\ntions [35]. The dynamical stability of the uniform ferromag-\nnetic phase demands ¯ c0+¯c1>0 (realϵk,1). Since Eq. (12)\nis independent of α0, the ferromagnetic state is immune to the\nparametric driving of a0. It is because periodic modulation\nofa0causes equal and out-of-phase oscillations in ˜ c0and ˜c1,\nwhich cancel each other.\nAccording to the Floquet theorem, the solutions of Eq. (12)\nareu1(t)=b(t) exp(σt), where b(t)=b(t+π/ω 2) and\nσ(k,ω2,α) is called the Floquet exponent. If Re( σ)>0, the\nferromagnetic state is dynamically unstable against the forma-\ntion of transient density modulations or Faraday patterns [see\ninsets of Fig. 1(a)]. The pattern size is determined by the most\nunstable momentum ku, i.e., momentum for which σis the\nlargest. For vanishing modulation amplitude ( α2→0),kuis\ndetermined by the resonance ϵk,1=ℏω2and the correspond-\ning Floquet exponent is σ≃¯nEkuα2¯g2/ℏ2ω2[3]. Sinceϵk,1is\na monotonously increasing function of k,kuincreases with ω2\n(see the unstable momentum rings in Figs. 1(b) and 1(c) for\ntwo di fferent frequencies). It implies that the pattern size de-\ncreases monotonously with increasing driving frequency ω2.\nAt longer times, the higher harmonics ( ϵk,1=jℏω2with\nj=2,3,...) become relevant, causing the emergence of other\nrings of higher|k|in the momentum density, thus heating and\ndestroying the condensate [54, 55].\nV . POLAR PHASE\nFor the polar phase ζP=(0,1,0)T, the chemical potential\nisµ2D=¯c0¯nand we have\nM2=¯n˜c1(t) 0 ˜ c1(t)\n0 2˜ c0(t) 0\n˜c1(t) 0 ˜ c1(t);M3=¯n˜c1(t)1 0−1\n0 0 0\n−1 0 1.\nWe get the Mathieu-like equations\nd2u0\ndt2+1\nℏ2Ek[Ek+2¯n˜c0(t)]u0=0 (14)\nd2u+\ndt2+1\nℏ2(Ek+q)\u0002Ek+q+2¯n˜c1(t)\u0003u+=0, (15)4\nwhere u+=u1+u−1. Unlike the ferromagnetic case, both the\ndensity and spin modes,\nϵk,0=p\nEk(Ek+2¯c0¯n) (16)\nϵk,±1=p\n(Ek+q)(Ek+q+2¯c1¯n), (17)\nbecome important for the dynamics. ϵk,0corresponds to\nthe density modulations (phonons) and the degenerate ϵk,±1\nmodes associated with the elementary process of ( 0,0)+\n(0,0)↔(k,±1)+(−k,∓1). When q=0 and ¯ c0=¯c1, all three\nmodes are degenerate. The dynamical stability of the homo-\ngeneous polar phase demands ¯ c0>0 and q(q+2¯c1¯n)≥0 and\nq+¯c1¯n≥0. Below we consider the three cases of modulation.\nIn the numerical calculations of Eqs. (5)-(7), the initial polar\nphase is embedded with a noise field δ(ρ) populating the vac-\nuum modes in the limit q→∞ , based on the truncated Wigner\nprescription [46, 56],\nδ(ρ)=1√\nVX\nkα+1\nkexp(ik·ρ)\nα0\nkukexp(ik·ρ)+α0∗\nkvkexp(−ik·ρ)\nα−1\nkexp(ik·ρ),(18)\nwhere Vis the volume, αm\nkare complex Gaussian random vari-\nables with zero mean and satisfiesD\nαm∗\nkαm′\nk′E\n=(1/2)δmm′δkk′\nand the amplitudes are given by,\nuk=s\nEk+c0¯n\n2√Ek(Ek+2c0¯n)−1\n2(19)\nvk=q\n1−u2\nk. (20)\nWe also found qualitatively similar results for an initial state\nψ(ρ,t=0)=p+1exp(iθ1)\n[1−p2\n+1−p2\n−1] exp( iθ0)\np−1exp(iθ−1)(21)\nwith a noise from a uniform distribution, where pαandθαare\nrandom numbers with p±1≪1.\nA. a0Modulation\nFora0modulation, the Mathieu Eqs. (14) and (15) become\nd2u0\ndt2+1\nℏ2\"\nϵ2\nk,0+4¯nEkα0¯g0\n3cos(2ω0t)#\nu0=0 (22)\nd2u+\ndt2+1\nℏ2\"\nϵ2\nk,±1−4¯n(Ek+q)α0¯g0\n3cos(2ω0t)#\nu+=0.(23)\nFor a given ω0and in the limit α0→0, the resonances\nϵk,0=ℏω0andϵk,±1=ℏω0provide us two unstable momenta\nk(0)\nuandk(+)\nuwith Floquet exponents σ(0)≃¯nEk(0)\nuα0¯g0/3ℏ2ω0\nandσ(+)≃¯n(Ek(+)\nu+q)α0¯g0/3ℏ2ω0. Using k(0)\nuandk(+)\nuob-\ntained from the resonance conditions, we rewrite the Floquet\nexponents as\nσ(0)=α0¯n¯g0\n3ℏ2ω0\u0014q\n(¯n¯c0)2+ℏ2ω2\n0−¯n¯c0\u0015\n, (24)\nσ(+)=α0¯n¯g0\n3ℏ2ω0\u0014q\n(¯n¯c1)2+ℏ2ω2\n0−¯n¯c1\u0015\n. (25)\n−80 0 80\nx/lz−80080 y/lz\nC. D.\n(a)\n−80 0 80\nx/lz−80080×10−2C. D.\n(b)\n0 2.3 4.6 6.9 9.2\n 0.71 0.84 0.96 1.09 1.22Figure 2. Density patterns for a0modulation on an initial polar phase.\nThe parameters are q=0, ¯c0¯n=¯c1¯n=0.2ℏωz,α0=0.4, and\nω0/ωz=0.2 atωzt=600. The density pattern in (b) m=±1 differs\nfrom that of (a) m=0. C.D. stands for condensate density.\nFigure 3. Population dynamics for a0modulation and q=0. Other\nparameters are ¯ c0¯n=¯c1¯n=0.2ℏωz,α0=0.4, andω0/ωz=0.2.\nThe population in each component is Nm(t)=R\n|ψm(x,y,t)|2dxdy ,\nthe total population N=P\nmNm, the population with k,0 inm=0\nisN0,k,0=R\nk,0dkxdky|˜ψ0(kx,ky,t)|2and the zero-momentum popula-\ntion in m=0 isN0,k=0. The dashed-dotted lines show the exponential\nfit determined by the Floquet exponent, and the solid lines are from\nthe numerical calculations of NLGPEs.\nWhen ¯ c0=¯c1,σ(0)=σ(+)and interestingly, σ(+)is indepen-\ndent of the quadratic Zeeman field q, which has interesting\nconsequences as we discuss later.\n1. Degenerate modes\nWhen all three modes are degenerate (when q=0 and\n¯c1=¯c0),k(0)\nu=k(+)\nu=kuandσ(0)=σ(+). It implies that\nboth density and spin modes get populated simultaneously by\nperiodic driving. The unstable density mode ( ϵk,0) leads to\nthe formation of Faraday patterns [see Fig. 2(a)] or equiva-\nlently the exponential growth of atoms with finite momenta\ninm=0 component, shown as N0,k,0in Fig. 3. The unsta-\nbleϵk,±1modes initiate the population transfer from m=05\nFigure 4. The momentum density of m=0 component at (a) ωzt=\n300 and (b) ωzt=650 for ¯ c0¯n=¯c1¯n=0.2ℏωz,α0=0.4, and\nω0/ωz=0.2. The momentum ring in (a) and the primary one in (b)\nis given by the resonance ϵk,0=ℏω0. Two higher harmonics can also\nbe seen in (b). C.D. stands for the condensate density, and the central\npeak at k=0 is removed for the visibility of the momentum rings.\ntom=±1, leading to the spin dynamics seen in Fig. 3. At\nthe initial stage, the population in m=±1 grows exponen-\ntially at a rate determined by σ(+)and exhibits oscillations of\nfrequency 2 ω0. Eventually, a density pattern also develops in\nm=±1 components [see Fig. 2(b)].\nThe momentum density of m=0 is shown in Fig. 4 at two\ndifferent instances. At the early stages of the pattern forma-\ntion, the most unstable momenta are given by the resonance\nconditionϵk,0=ℏω0, which we marked by kuin Fig.4(a). Al-\nthough the next leading unstable momenta come from the sec-\nond harmonics ϵk,0=2ℏω0, there are also contributions from\nprocesses in which a pair of atoms, each from m= +1 and\nm=−1 with momenta of magnitudes ( ku,−ku) or (±ku,±ku)\nscatter into ( +k,−k) or (k,k′) with|k+k′|=2ku. All such k\nandk′lead to a disk-shaped pattern in the momentum space\nof the condensate wavefunction in addition to the harmonics,\nbut with lesser amplitudes, as shown in Fig. 4(b). At longer\ntimes, the condensate gets destroyed by heating.\nMagnetization dynamics : The initial polar condensate has\na null spin density vector F. The spin-mixing dynamics\nin Fig. 3 leads to the emergence of spin textures shown in\nFig. 5(a). Strikingly, the velocity field of transverse spin den-\nsity vector F⊥(x,y) in Fig. 5(b) reveals the formation of polar-\ncore vortices (PCVs) and anti-vortices [34, 57, 58]. They are\nmarked by circles in Figs. 5(a) and 5(b). The core of a PCV is\nfilled with m=0 atoms with no vorticity, and the surrounding\nm= +1 and m=−1 atoms have opposite vorticity. Fig-\nure 5(c) shows the time dependence of vortex density Nv/L2,\nwhere L2is the area of the numerical box we use. The vortex\nnumber Nvis determined by computing the phase winding at\nthe smallest loops defined by the grid size. At t=0, the initial\nrandom noise in m=±1 components contributes to the vortex\nnumber Nv, which eventually decays over time as the PCVs\nmaterialize. Nvdecreases until the number of PCVs reaches a\nsteady value N(PCV)\nv and the decay rate is determined by σ(+).\nSince the amplitude of spin texture oscillates in time with\nthe driving frequency, Nvexhibits the same. When the spin-\n−15 0 15\nx/lz−15015y/lz(a)\n−15 0 15\nx/lz−15015y/lz(b)\n0 5 10 15\nσ(+)t0.00.51.01.52.0Nv/L2(c)\nω0= 0.1ωz\nω0= 0.3ωz\nω0= 0.5ωz\n0.10.20.30.40.5\nω0/ωz0.000.030.060.090.12\n(d)\nN(PCV)\nv/L2\nconst.×k2\nuFigure 5. Spin textures for a0modulation and q=0. (a) The trans-\nverse spin density F⊥(x,y) and (b) the velocity field of F⊥(x,y)=\n(Fx,Fy), i.e.,⃗∇ϕ/|⃗∇ϕ|, whereϕ=arctan( Fx/Fy) atωzt=670 or\nσ(+)t=14.74. Other parameters are the same as in Fig. 3. Both\nvortices and antivortices are marked in (b). (c) The dynamics of the\nvortex density Nv/L2, where Lis the numerical box size, is obtained\nby averaging over ten realizations of noises. It decreases in time\nand eventually saturates. The saturated vortex density (before the\ncondensate is destroyed, taken around σ(+)t=15) as a function of\ndriving frequency ω0is shown in (d), which exhibits a quadratic de-\npendence on the most unstable momentum. The latter is expected,\nconsidering the Q2D nature of the condensate.6\ntexture amplitude is tiny, the Nvbecomes large from the noise\ncontribution. If the amplitude is su fficiently large, the noise\nis overshadowed. Once Nvreaches a steady value, the min-\nima shown in the inset of Fig. 5(c)] approximately provide\nus with the number of PCVs. Interestingly, N(PCV)\nv is deter-\nmined by the unstable momentum kuas shown in Fig. 5(c)]\nand as expected it exhibits k2\nubehaviour. It implies that the\nlarger the modulation frequency, the denser the spin-vortex\ngas. At longer times ( σ(+)t>16), we observed that m=0 ho-\nmogeneous condensate gets significantly depleted, and PCVs\ndisintegrated into independent gases of vortices in m=1 and\nm=−1 components.\nFurther, we analyze the scaled spin-spin correlations,\nCα(ρ,t)=1\nNα\"\ndρ′Fα(ρ+ρ′,t)Fα(ρ′,t), (26)\nwhereα∈ {x,y,z}andNα=!\ndρFα(ρ)2.Cx(ρ,t) at an\ninstant well before the condensate is destroyed is shown in\nFig. 6(a). The radial correlations along the transverse and lon-\ngitudinal magnetization densities Cα(ρ)=(1/2π)R2π\n0Cα(ρ)dθ\nare found to be governed by Bessel function J0(kuρ), where\nρ=|ρ|and in particular, Cx,y(ρ)∝J0(kuρ) and Cz(ρ)∝\nJ2\n0(kuρ) [see Fig. 6(b)]. Similar Bessel correlations are pre-\ndicted in spinor condensates subjected to quantum quenches\n[45, 56, 59].\n2. Non-degenerate modes\nForq,0 or ¯ c1,¯c0, the degeneracy between the den-\nsity (ϵk,0) and the spin ( ϵk,±1) modes is lifted and consequently,\nk(0)\nu,k(+)\nu. First, we consider q=0 and ¯ c1,¯c0. Ifσ(0)and\nσ(+)are comparable, the dynamics remains qualitatively the\nsame as in the case of degenerate modes. So we consider the\ntwo extreme scenarios: ¯ c0≪¯c1and ¯c1≪¯c0. For ¯ c0≪¯c1,\nthe density mode is the soft one and consequently σ(0)≫σ(+).\nIn that case, the Faraday pattern in m=0 forms well before\nthe spin-mixing dynamics takes place [see the dynamics of\nN0,k,0andN±1in Fig. 7(a)]. The initial growth of N0,k,0is\ndetermined by σ(0)[dotted-dashed line in Fig. 7(a)]. By the\ntime the population transfer to m=±1 takes place, the ho-\nmogeneous condensate in m=0 is significantly altered and\ndepleted, making our linear stability analysis invalid. In con-\ntrast, for ¯ c0≫¯c1,N±1out-grows N0,k,0[see Fig. 7(b)] and\nthe initial dynamics is governed by σ(+). The spin-dynamics\nleads to spin textures and the formation of PCVs. The spin-\nspin correlations have di fferent behavior for the two extreme\ncases. For ¯ c0≪¯c1,Cx,y,z(ρ) decays exponentially [see Fig. 8]\nwith a correlation length of the order of a spin healing length,\nwhich is proportional to ∝1/√¯c1¯n, whereas, for ¯ c1≪¯c0, they\nexhibit a Bessel function dependence of Cx,y(ρ)=J0(k(+)\nuρ)\nandCz(ρ)=J2\n0(k(+)\nuρ).\nNon-zero q . The nature of dynamics also depends criti-\ncally on the quadratic Zeeman field q. A finite qnot only\nlifts the degeneracy of the modes but also introduces a gap,\n∆ =p\nq(q+2¯c1¯n) in the spin modes [see Fig. 9(a)]. It means\nthat there is no spin-mixing dynamics if the driving frequency\nFigure 6. Spin-spin correlations for a0modulation and q=0. (a) The\ntransverse magnetization correlation Cx(ρ,t) in the xy-plane and (b)\nthe radial correlation function Cx,z(ρ) atωzt=500. Other parameters\nare the same as in Fig. 3. We see that Cx,y(ρ)∝J0(kuρ) and Cz(ρ)∝\nJ2\n0(kuρ). The plots are obtained by taking an average of results from\nten di fferent realizations of initial noise.\nlies below the gap ( ω0<∆), and the periodic modulation only\nleads to forming the Faraday pattern in m=0. In contrast,\nforω0≥∆, both spin and density modes contribute to the\ndynamics. Since the Mathieu exponents are independent of\nq[see Eqs. (24) and (25)], both modes are unstable simulta-\nneously for ¯ c1=¯c0. In that case, the dynamics is identical\nto the case of degenerate modes, but the momenta governing\nthe Faraday pattern and the vortex density are di fferent, i.e.,\nk(0)\nu,k(+)\nu. The spatial dependence of spin-spin correlations\nis governed by k(+)\nuvia the Bessel function.\nEven though the Mathieu exponents are independent of q,\nthe above results indicate an implicit dependence of qon the\ndynamics for a fixed ω0. For instance, for su fficiently small\nqsuch that ∆< ω 0, both modes are unstable simultaneously,\nwhereas for su fficiently large qsuch that ∆> ω 0, only the\ndensity mode is unstable. A similar scenario, as shown in\nFig. 9(b), also emerges for ¯ c1≪¯c0. For small q, the spin\ndynamics dominates ( σ(+)≫σ(0)) and for large q, only the\nFaraday pattern is formed. When the spin-mixing dynamics7\nFigure 7. The population dynamics for a0modulation and q=0.\n(a) ¯c0¯n=0.2ℏωz, ¯c1¯n=4ℏωz,α0=0.03 andω0/ωz=0.3. (b)\n¯c0¯n=0.4ℏωz, ¯c1¯n=0.05ℏωz,α0=0.25 andω0/ωz=0.1. In\n(a)N0,k,0outgrows N±1and vice versa in (b). The solid lines are\nthe numerical results, and dotted-dashed lines are the exponential fit\nprovided by σ(0).\nFigure 8. The radial transverese spin-spin correlation function Cx(ρ)\nfor ¯c0¯n=0.2ℏωz, ¯c1¯n=4ℏωz,α0=0.03 andω0/ωz=0.3, exhibiting\nexponential decay at an instant ωzt=240. The dots are obtained\nfrom average of results from ten di fferent realizations of initial noise.\nThe solid line is an exponential fit. The corresponding population\ndynamics is shown in Fig. 7(a).\nFigure 9. (a) The Bogoliubov spectrum for ¯ c0¯n=¯c1¯n=0.2ℏωz,\nω0/ωz=0.3, and q=0.1ℏωz. The unstable momenta as a function of\nω0map the spectrum in the limit α→0. (b) The Mathieu exponents\nof density and spin modes as a function of qfor ¯c0¯n=0.4ℏωz, ¯c1¯n=\n0.05ℏωz,ω0/ωz=0.3 andα0=0.3.∆is the gap of the spin modes\natk=0. In (b), the primary resonance associated with the spin mode\nchanges from ϵk,±1=ℏω0toϵk,±1=2ℏω0as a function of q, leading\nto a significant decrease in the Mathieu exponent σ(+).\nFigure 10. The results for a2modulation. σ(0)/σ(+)as a function of\nω2for ¯c0¯n=0.2���ωzandq=0 with di fferent ¯ c1/¯c0. For ¯ c0=¯c1, we\ngetσ(0)/σ(+)=2 (dashed line), and for ¯ c1<¯c0(solid line), the value\nofσ(0)/σ(+)crosses one, which indicates there is a change in the\nbehavior of dynamics above and below a critical driving frequency\n(ω2≈0.18ωz).\ndominates (¯ c1≪¯c0), the spin-spin correlations are again gov-\nerned by the Bessel functions as before.\nB.a2Modulation\nFora2modulation, the Mathieu Eqs. (14) and (15) become\nd2u0\ndt2+1\nℏ2\"\nϵ2\nk,0+8¯nEkα2¯g2\n3cos(2ω2t)#\nu0=0 (27)\nd2u+\ndt2+1\nℏ2\"\nϵ2\nk,±1+4¯n(Ek+q)α2¯g2\n3cos(2ω2t)#\nu+=0.(28)8\nThe Mathieu exponents for the density and spin modes are\nobtained as,\nσ(0)=2α2¯n¯g2\n3ℏ2ω2\u0014q\n(¯n¯c0)2+ℏ2ω2\n2−¯n¯c0\u0015\n, (29)\nσ(+)=α2¯n¯g2\n3ℏ2ω2\u0014q\n(¯n¯c1)2+ℏ2ω2\n2−¯n¯c1\u0015\n. (30)\nComparing to the case of a0-modulation [Eq. (24)], an addi-\ntional factor of 2 appears in Eq. (29). Therefore we expect\ndifferent dynamics for a2-modulation for a given set of inter-\naction parameters. The results of a2-modulation for q=0\nare summarized in Fig. 10 where we show the ratio of Math-\nieu exponents associated with the density and spin modes as a\nfunction of the driving frequency. For degenerate modes, i.e.,\nwhen ¯ c1=¯c0(dashed line in Fig. 9), σ(0)=2σ(+), that means\nFaraday pattern emerges well before the spin-mixing occurs.\nThat is also the case for ¯ c1>¯c0(dotted-dashed line in Fig. 9).\nInterestingly, for ¯ c1<¯c0/2, the nature of dynamics depends\nonω2. For smaller ω2,σ(0)/σ(+)<1, i.e., spin-mixing domi-\nnates the density modulations and vice versa for large values\nofω2. Making qnon-zero would lead to an explicit depen-\ndence ofω2on the dynamics for any value of ¯ c0and ¯c1, but\nqualitative features remain the same.\nC. Modulation of both a0anda2\nAs we have seen in the previous cases, an implicit com-\npetition exists between the density modulations (Faraday pat-\nterns) and spin-mixing. By controlling interaction strengths,\nquadratic Zeeman field, or the driving frequencies, we can ac-\ncess di fferent scenarios in which one dominates, or both co-\noccur in the dynamics. Further, we show greater controllabil-\nity is achieved by simultaneously modulating a0anda2. To do\nso, we assume a phase di fferenceϕbetween a0anda2modula-\ntions, i.e., a2(t)=¯a2[1+2α2cos(2ω2t+ϕ)]. Strikingly, taking\nω0=ω2,α2=α0¯g0/¯g2andϕ=0 is equivalent to modulating\n¯c0while keeping ¯ c1constant. In that case, Eqs. (14) and (15)\nbecome,\nd2u0\ndt2+1\nℏ2h\nϵ2\nk,0+4¯nEkα0¯g0cos(2ω0t)i\nu0=0 (31)\nd2u+\ndt2+1\nℏ2ϵ2\nk,±1u+=0. (32)\nThe above equations convey that Faraday patterns are formed\ninm=0, but spin mixing does not occur. The numerical\ncalculations of NLGPEs also confirm this. If we take ϕ=\nπandα2=α0¯g0/2¯g2, ¯c1becomes periodic in time, and ¯ c0\nremains a constant. The corresponding equations of motion\nare,\nd2u0\ndt2+1\nℏ2ϵ2\nk,0u0=0 (33)\nd2u+\ndt2+1\nℏ2h\nϵ2\nk,±1+2¯n(Ek+q)α0¯g0cos(2ω0t)i\nu+=0.(34)\nIn this case, the spin mode is unstable, leading to the pop-\nulation transfer from m=0 to m=±1. The latter causes\n−40 0 40\nx/lz−40040y/lz(a)C. D.\n−40 0 40\nx/lz−40040×10−2C. D.\n(b)\n−1 0 1\nkxlz−101kylz(c)C. D.\n−1 0 1\nkxlz−101\nC. D.\n(d)\n0 0.0 0.0\n 0.86 0.96 1.07\n0 104107\n0 102105Figure 11. Results for simultaneous modulation of a0anda2with\na phase di fference ofϕ=π. Real-space condensate density of (a)\nm=0 and (b) m=±1 for q=0, ¯c0¯n=¯c1¯n=0.2ℏωz,α0=0.2,α2=\n0.05, andω0/ωz=ω2/ωz=0.2 atωzt=800. The corresponding\nmomentum densities are shown in (c) and (d).\nlocal depletions in the homogeneous density of m=0 and\nrandom density peaks emerging in m=±1 components [see\nFigs. 11(a) and 11(b)]. The momentum ring in the momentum\ndensity of m=±1 [see Fig. 11(d)] arises from the spin-mixing\nprocess, ( 0,0)+(0,0)↔(k,±1)+(−k,∓1). In contrast, the\nprocess in which a pair of atoms, each in m= +1 and m=−1\nwith momenta ( ku,−ku) or (±ku,±ku) scatter into ( +k,−k) or\n(k,k′) with|k+k′|=2kupopulate the non-zero momenta in\nm=0 [see Fig. 11(c)] at a later time. It is starkly di fferent\nfrom Fig. 4(b) of the pure a0modulation where the primary\nunstable momenta come from the resonance with the density\nmode.\n1. Competing Instabilities\nWe show that by modulating a0anda2withω0,ω2and\ncarefully choosing the modulation amplitudes α0andα2, the\nintriguing scenario of competing instabilities emerges. In par-\nticular, two distinct momenta of density or spin mode com-\npete. To see the competing instability among the density\nmodes, we take ¯ c0≪¯c1such thatσ(0)≫σ(+). The primary\nresonances emerge from modulating a0anda2areϵk,0=ℏω0\nandϵk,0=ℏω2and let the corresponding unstable momenta\nand Mathieu exponents be ( k(0)\nu0,k(0)\nu2) and (σ(0)\n0,σ(0)\n2). The rel-9\nFigure 12. Competing instability among the density modes. (a) The\ntwo primary peaks in Mathieu exponent vs kfrom the simultaneous\nmodulation of a0anda2for ¯c0¯n=0.2��ωz, ¯c1¯n=4ℏωz,q=0,\nα0=0.024,ω0=0.1ωz,α2=0.01, andω2=0.3ωz. The solid\nline is for a0, and the dashed line is for a2modulations. (b) and (c)\nshow the density of m=0 component at ωzt=420 andωzt=460,\nrespectively, exhibiting the change in wavelength of patterns in time.\n(b) has a wavelnegth of 1 /k(0)\nu0and (c) has a wavelength of 1 /k(0)\nu2with\nk(0)\nu0<k(0)\nu2.\nevant equation of motion is\nd2u0\ndt2+1\nℏ2\"\nϵ2\nk,0+4Ek¯n\n3(α0¯g0cos 2ω0t+2α2¯g2cos 2ω2t)#\nu0=0,\n(35)\nwhich is generally a quasi-periodic Mathieu equation. The\nstability regions of Eq. (35) studied using di fferent approxi-\nmation methods reveal a very complex structure [60]. When\nω0/ω2is a rational number, Eq. (35) exhibits an overall pe-\nriodicity, and the Floquet theorem becomes valid. Then, the\ninstability regions are just a union of those arising from the\nindependent modulations of a0anda2. In Fig. 12(a), we show\nthe two primary instability tongues associated with a0anda2\nmodulations for ω0/ω2=1/3 andω2=0.3ωz. The modula-\ntion amplitudes are taken such that the peak of σ(0)\n0andσ(0)\n2\nare approximately the same. In the dynamics, we observe an\noscillation between the patterns of two di fferent wavelengths\n1/k(0)\nu0and 1/k(0)\nu2with k(0)\nu0<k(0)\nu2[see Figs. 12(b) and 12(c)].\nSimilarly, for ¯ c1≪¯c0, the competing instability arises\namong the two di fferent spin-mode momenta. In that case,\nFigure 13. Competing instability among the spin modes. (a) The\ntwo primary peaks in Mathieu exponent vs kfrom the simultaneous\nmodulation of a0anda2for ¯c0¯n=0.4ℏωz, ¯c1¯n=0.05ℏωz,q=0,\nα0=0.205,ω0=0.1ωz,α2=0.1, andω2=0.3ωz. The solid line\nis for a0, and the dashed line is for a2modulations. (b) shows the\nspin-spin correlation functions Cx(ρ) and Cz(ρ). Fit-1 and Fit-2 are\nrespectively the Eqs. (37) and (38). The numerical results in (b) are\nobtained by taking the average over ten realizations of noises.\nthe relevant Mathieu equation is\nd2u+\ndt2+1\nℏ2\u0014\nϵ2\nk,±1−4(Ek+q)¯n\n3\n(α0¯g0cos 2ω0t−α2¯g2cos 2ω2t)\u0015\nu+=0. (36)\nAgain, for a rational ratio of ω0/ω2, we obtain the instability\ntongues from the union of instabilities as shown in Fig.13(a).\nThe e ffect of competing instabilities is directly visible in the\nbehavior of the spin-spin correlations functions during the\ntransient stage, and they are of the form\nCx,y(ρ,t)=D(t)J0\u0010\nk(+)\nu0ρ\u0011\n+[1−D(t)]J0\u0010\nk(+)\nu2ρ\u0011\n, (37)\nCz(ρ,t)=J0\u0010\nk(+)\nu0ρ\u0011\nJ0\u0010\nk(+)\nu2ρ\u0011\n(38)\nwhere k(+)\nu0andk(+)\nu2are unstable momenta of the spin mode\nfrom two modulation frequencies ω0andω2andD(t) is the\ntime-dependent amplitude.10\nVI. EXPERIMENTAL CONSIDERATIONS\nNow we briefly examine the experimental possibilities. As\ndiscussed above, the emergent spin textures leading to PCVs\ncan be observed except when ¯ c0≪ |¯c1|for which the spin\nmode is completely outplayed by the density mode in the in-\nstability dynamics. In the state-of-the-art experimental setups\nof spin-1 condensates, for instance, in23Na,87Rb and7Li, the\nratio ¯ c1/¯c0is, respectively, 0.036, -0.004 [36] and -0.46 [61],\nwhich supports the formation of spin textures and PCVs. For\n87Rb and7Li, since the spin-dependent interactions are ferro-\nmagnetic, preparing the initial polar phase requires a quadratic\nZeeman field. We numerically verified all three cases and con-\nfirmed the formation of spin textures and PCVs identical to the\ncase of degenerate modes. In a given atomic setup, Feshbach\nresonances are required to access di fferent regimes of inter-\naction strengths we consider, and in particular, independent\ncontrol of a0anda2is needed. The latter has been proposed\nto achieve via combining magnetic and rf-field-induced Fes-\nhbach resonances [53]. Longitudinal and transverse spin-spin\ncorrelations are computed once the corresponding magnetiza-\ntions are measured, as demonstrated in Ref. [62].\nVII. SUMMARY AND OUTLOOK\nIn summary, we analyzed the density patterns and spin tex-\ntures in a parametrically driven Q2D spin-1 condensate for\ntwo initial phases: ferromagnetic and polar. An initial fer-\nromagnetic condensate is immune to periodic modulation of\na0, whereas, for a2modulation, it exhibits similar dynamics\nto that of a scalar condensate. An initial polar phase revealed\ninteresting dynamics; for instance, a gas of polar core vortices\nand anti-vortices is seen with its density determined by the\nmomentum of the unstable spin mode. Also, there is com-petition between Faraday patterns and spin-mixing dynamics,\nwhich can be controlled by tuning the interaction strengths,\nquadratic Zeeman field, or driving frequencies. When spin-\nmixing dynamics dominates, the spin-spin correlation func-\ntions exhibit a Bessel function behavior as a function of rel-\native distance. Otherwise, they decay exponentially with a\ncorrelation length of the order of a spin healing length. Mod-\nulating both scattering lengths creates an exciting scenario of\ncompeting instabilities among density or spin modes. It pro-\nduces the superposition of Faraday patterns or spin correlation\nfunctions of two distinct wavelengths.\nOur studies open up several perspectives for future studies.\nFor instance, one could select an appropriate initial state to\nengineer exotic spin textures or vortices via periodic modula-\ntion. The same analyses can be extended to condensates of\nhigher spin where the availability of three or more scattering\nlengths may lead to complex scenarios. Another exciting as-\npect is to analyze the e ffect of harmonic confinement and the\nrole of transverse excitations.\nVIII. ACKNOWLEDGEMENTS\nWe acknowledge Chinmayee Mishra for the discussions\nduring the initial stages of the work. We thank National Su-\npercomputing Mission (NSM) for providing computing re-\nsources of ”PARAM Brahma” at IISER Pune, which is imple-\nmented by C-DAC and supported by the Ministry of Electron-\nics and Information Technology (MeitY) and Department of\nScience and Technology (DST), Government of India. R.N.\nfurther acknowledges DST-SERB for Swarnajayanti fellow-\nship File No. SB /SJF/2020-21 /19, and the MATRICS grant\n(MTR /2022/000454) from SERB, Government of India and\nNational Mission on Interdisciplinary Cyber-Physical Sys-\ntems (NM-ICPS) of the Department of Science and Technol-\nogy, Government of India, through the I-HUB Quantum Tech-\nnology Foundation, Pune, India.\n[1] K. Staliunas, S. Longhi, and G. J. de Valc ´arcel, Phys. Rev. Lett.\n89, 210406 (2002).\n[2] P. Engels, C. Atherton, and M. A. Hoefer, Phys. Rev. Lett. 98,\n095301 (2007).\n[3] A. I. Nicolin, R. Carretero-Gonz ´alez, and P. G. Kevrekidis,\nPhys. Rev. A 76, 063609 (2007).\n[4] R. Nath and L. Santos, Phys. Rev. A 81, 033626 (2010).\n[5] K. Staliunas, S. Longhi, and G. J. de Valc ´arcel, Phys. Rev. A\n70, 011601 (2004).\n[6] M. Modugno, C. Tozzo, and F. Dalfovo, Phys. Rev. A 74,\n061601 (2006).\n[7] N. Katz and O. Agam, New J. Phys. 12, 073020 (2010).\n[8] A. I. Nicolin, Phys. Rev. E 84, 056202 (2011).\n[9] P. Capuzzi, M. Gattobigio, and P. Vignolo, Phys. Rev. A 83,\n013603 (2011).\n[10] K. Łakomy, R. Nath, and L. Santos, Phys. Rev. A 86, 023620\n(2012).\n[11] A. Bala ˇz and A. I. Nicolin, Phys. Rev. A 85, 023613 (2012).\n[12] A. Bala ˇz, R. Paun, A. I. Nicolin, S. Balasubramanian, and\nR. Ramaswamy, Phys. Rev. A 89, 023609 (2014).[13] B. K. Turmanov, B. B. Baizakov, and F. K. Abdullaev, Phys.\nRev. A 101, 053616 (2020).\n[14] A. B. Bhattacherjee, Physi. Scr. 78, 045009 (2008).\n[15] J. B. Sudharsan, R. Radha, M. C. Raportaru, A. I. Nicolin, and\nA. Bala ˇz, J. Phys. B: At. Mol. Opt. Phys. 49, 165303 (2016).\n[16] R. Cominotti, A. Berti, A. Farolfi, A. Zenesini, G. Lamporesi,\nI. Carusotto, A. Recati, and G. Ferrari, Phys. Rev. Lett. 128,\n210401 (2022).\n[17] H. Zhang, S. Liu, and Y .-S. Zhang, Phys. Rev. A 105, 063319\n(2022).\n[18] G. Verma, U. D. Rapol, and R. Nath, Phys. Rev. A 95, 043618\n(2017).\n[19] H. Lignier, C. Sias, D. Ciampini, Y . Singh, A. Zenesini,\nO. Morsch, and E. Arimondo, Phys. Rev. Lett. 99, 220403\n(2007).\n[20] A. Eckardt, T. Jinasundera, C. Weiss, and M. Holthaus, Phys.\nRev. Lett. 95, 200401 (2005).\n[21] A. Zenesini, H. Lignier, D. Ciampini, O. Morsch, and E. Ari-\nmondo, Phys. Rev. Lett. 102, 100403 (2009).\n[22] W. Zhang, B. Sun, M. S. Chapman, and L. You, Phys. Rev. A\n81, 033602 (2010).11\n[23] T. M. Hoang, C. S. Gerving, B. J. Land, M. Anquez, C. D.\nHamley, and M. S. Chapman, Phys. Rev. Lett. 111, 090403\n(2013).\n[24] L. W. Clark, A. Gaj, L. Feng, and C. Chin, Nature 551, 356\n(2017).\n[25] H. Fu, L. Feng, B. M. Anderson, L. W. Clark, J. Hu, J. W.\nAndrade, C. Chin, and K. Levin, Phys. Rev. Lett. 121, 243001\n(2018).\n[26] Z. Wu and H. Zhai, Phys. Rev. A 99, 063624 (2019).\n[27] H. Fu, Z. Zhang, K.-X. Yao, L. Feng, J. Yoo, L. W. Clark,\nK. Levin, and C. Chin, Phys. Rev. Lett. 125, 183003 (2020).\n[28] H. Saito and M. Ueda, Phys. Rev. Lett. 90, 040403 (2003).\n[29] F. K. Abdullaev, J. G. Caputo, R. A. Kraenkel, and B. A. Mal-\nomed, Phys. Rev. A 67, 013605 (2003).\n[30] S. Lellouch, M. Bukov, E. Demler, and N. Goldman, Phys. Rev.\nX7, 021015 (2017).\n[31] J. Bera, A. Q. Batin, S. Ghosh, B. Malomed, and U. Roy, Phil.\nTrans. R. Soc. A 381, 20220075 (2023).\n[32] J. H. V . Nguyen, M. C. Tsatsos, D. Luo, A. U. J. Lode, G. D.\nTelles, V . S. Bagnato, and R. G. Hulet, Phys. Rev. X 9, 011052\n(2019).\n[33] Z. Zhang, K.-X. Yao, L. Feng, J. Hu, and C. Chin, Nat. Phys.\n16, 652 (2020).\n[34] L. E. Sadler, J. M. Higbie, S. R. Leslie, M. Vengalattore, and\nD. M. Stamper-Kurn, Nature 443, 312 EP (2006).\n[35] D. M. Stamper-Kurn and M. Ueda, Rev. Mod. Phys. 85, 1191\n(2013).\n[36] Y . Kawaguchi and M. Ueda, Phys. Rep. 520, 253 (2012).\n[37] H. Saito and H. Hyuga, Phys. Rev. A 78, 033605 (2008).\n[38] T. M. Hoang, M. Anquez, B. A. Robbins, X. Y . Yang, B. J.\nLand, C. D. Hamley, and M. S. Chapman, Nat. Commun. 7,\n11233 (2016).\n[39] B. Evrard, A. Qu, K. Jim ´enez-Garc ´ıa, J. Dalibard, and F. Ger-\nbier, Phys. Rev. A 100, 023604 (2019).\n[40] Y . Imaeda, K. Fujimoto, and Y . Kawaguchi, Phys. Rev. Re-\nsearch 3, 043090 (2021).\n[41] P. Xu and W. Zhang, Phys. Rev. A 104, 023324 (2021).\n[42] S. Dadras, A. Gresch, C. Groiseau, S. Wimberger, and G. S.\nSummy, Phys. Rev. Lett. 121, 070402 (2018).\n[43] S. Dadras, A. Gresch, C. Groiseau, S. Wimberger, and G. S.\nSummy, Phys. Rev. A 99, 043617 (2019).[44] K. Sah, Faraday Patterns in Spinor Spin-1 Bose-Einstein Con-\ndensates , Master’s thesis, Indian Institute of Science Education\nand Research Pune (2019).\n[45] H. Saito, Y . Kawaguchi, and M. Ueda, Phys. Rev. A 76, 043613\n(2007).\n[46] L. A. Williamson and P. B. Blakie, Phys. Rev. Lett. 116, 025301\n(2016).\n[47] K. Kumar and K. M. S. Bajaj, Phys. Rev. E 52, R4606 (1995).\n[48] C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, Rev. Mod.\nPhys. 82, 1225 (2010).\n[49] T. V . Tscherbul, T. Calarco, I. Lesanovsky, R. V . Krems, A. Dal-\ngarno, and J. Schmiedmayer, Phys. Rev. A 81, 050701 (2010).\n[50] Y . Ding, J. P. D’Incao, and C. H. Greene, Phys. Rev. A 95,\n022709 (2017).\n[51] T. M. Hanna, E. Tiesinga, and P. S. Julienne, New J. Phys. 12,\n083031 (2010).\n[52] D. J. Papoular, G. V . Shlyapnikov, and J. Dalibard, Phys. Rev.\nA81, 041603 (2010).\n[53] P. Zhang, P. Naidon, and M. Ueda, Phys. Rev. Lett. 103, 133202\n(2009).\n[54] T. Boulier, J. Maslek, M. Bukov, C. Bracamontes, E. Magnan,\nS. Lellouch, E. Demler, N. Goldman, and J. V . Porto, Phys.\nRev. X 9, 011047 (2019).\n[55] P. Zhang and Y . Gu, SciPost Physics 9, 079 (2020).\n[56] R. Barnett, A. Polkovnikov, and M. Vengalattore, Phys. Rev. A\n84, 023606 (2011).\n[57] T. Isoshima, K. Machida, and T. Ohmi, J. Phys. Soc. Jpn. 70,\n1604 (2001).\n[58] T. Mizushima, N. Kobayashi, and K. Machida, Phys. Rev. A\n70, 043613 (2004).\n[59] A. Lamacraft, Phys. Rev. Lett. 98, 160404 (2007).\n[60] I. Kovacic, R. Rand, and S. Mohamed Sah, Applied Mechanics\nReviews 70, 020802 (2018).\n[61] S. Huh, K. Kim, K. Kwon, and J.-y. Choi, Phys. Rev. Res. 2,\n033471 (2020).\n[62] S. Huh, K. Mukherjee, K. Kwon, J. Seo, S. I. Mistakidis, H. R.\nSadeghpour, and J. yoon Choi, arXiv.2303.05230 (2023)."    },    {        "title": "1904.05863v3.Spin_dynamics_of_the_antiferromagnetic_Heisenberg_model_on_a_kagome_bilayer.pdf",        "content": "Spin dynamics of the antiferromagnetic Heisenberg model on a kagome bilayer\nPreetha Saha,1Depei Zhang,1Seung-Hun Lee,1and Gia-Wei Chern1\n1Department of Physics, University of Virginia, Charlottesville, VA 22904, USA\n(Dated: May 19, 2021)\nWe study the spin dynamics of classical Heisenberg antiferromagnet with nearest neighbor inter-\nactions on a quasi-two-dimensional kagome bilayer. This geometrically frustrated lattice consists of\ntwo kagome layers connected by a triangular-lattice linking layer. By combining Monte Carlo with\nprecessional spin dynamics simulations, we compute the dynamical structure factor of the classi-\ncal spin liquid in kagome bilayer and investigate the thermal and dilution e\u000bects. While the low\nfrequency and long wavelength dynamics of the cooperative paramagnetic phase is dominated by\nspin di\u000busion, weak magnon excitations persist at higher energies, giving rise the half moon pattern\nin the dynamical structure factor. In the presence of spin vacancies, the dynamical properties of\nthe diluted system can be understood within the two population picture. The spin di\u000busion of the\n\\correlated\" spin clusters is mainly driven by the zero-energy weather-van modes, giving rise to\nan autocorrelation function that decays exponentially with time. On the other hand, the di\u000busive\ndynamics of the quasi-free \\orphan\" spins leads to a distinctive longer time power-law tail in the\nautocorrelation function. We discuss the implications of our work for the glassy behaviors observed\nin the archetypal frustrated magnet SrCr 9pGa12\u00009pO19(SCGO).\nI. INTRODUCTION\nThe SCGO is one of the most intensely studied frus-\ntrated magnets [1{14]. Thermodynamically, SCGO does\nnot exhibit any signs of magnetic ordering down to tem-\nperaturesTg= 3.5{7 K, depending weakly on the va-\ncancy concentration x= 1\u0000p. BelowTg, the magnet\nenters an unconventional spin-glass phase. A coopera-\ntive paramagnetic regime, also known as a classical spin\nliquid, emerges at temperatures below the Curie-Weiss\nconstant \u0002 CW\u0019500 K. Geometrically, SCGO belongs to\na class of frustrated Heisenberg antiferromagnets on the\nso-called bi-simplex lattices [15{18]. These are networks\nof corner-sharing simplexes such as triangles and tetrahe-\ndra. Canonical examples include the pyrochlore [15, 16]\nand kagome [19{22] antiferromagnets. In SCGO, the\nCr3+ions with spin S= 3=2 reside on a two-dimensional\nlattice consisting of corner-sharing tetrahedra and trian-\ngles, known as the kagome bilayer or pyrochlore slab, as\nshown in Fig. 1. The strong short-range spin correla-\ntions in the low-temperature liquid phase result from the\nconstraints of zero total spin in every simplex, a condi-\ntion that minimizes the nearest-neighbor exchange inter-\nactions on such unit.\nConsiderable experimental e\u000borts have been devoted to\nunderstanding the unusual spin glass phase in SCGO [2{\n6, 11{14]. Despite the characteristic \feld-cooled and\nzero-\feld-cooled hysteresis in the bulk susceptibility, sev-\neral dynamical properties of its glassy phase are dis-\ntinctly di\u000berent from those of conventional spin glasses.\nThese include the quadratic T2behavior of the speci\fc\nheat [2, 3], the linear !-dependent dynamical susceptibil-\nity\u001f00[6], and a signi\fcantly weaker memory e\u000bect [23].\nTaken together, these features suggest that SCGO be-\nlongs to a new state of glassy magnets, dubbed the spin\njam [14, 23], that include several other magnetic com-\npounds [24, 25]. The source of this unusual dynamical\nphase in SCGO, however, remains to be clari\fed. Oneplausible scenario is that quantum \ructuations transform\nthe macroscopic degeneracy associated with the classical\nspin liquid of the kagome bilayer into the rugged energy\nlandscape of spin jam [14, 26]. It remains to be shown\nhow the unusual glassy behaviors of the spin jam evolve\nfrom the spin dynamics of the cooperative paramagnet.\nToward this goal, we present in this paper the \frst sys-\ntematic study of the dynamical properties of the bilayer-\nkagome classical spin liquid. By combining Monte Carlo\nsimulations with energy-conserving Landau-Lifshitz dy-\nnamics, we compute the dynamical structure factor of\nthe liquid regime. At the energy scales of the exchange\ninteraction, we \fnd signals of spin-wave excitations in the\nform of half moon pattern, replacing the pinch-point sin-\ngularity of the static structure factor. On the other hand,\nthe low-energy dynamics is dominated by spin-di\u000busion\ndriven mostly by the zero-energy modes. The di\u000busion\nconstant is found to depend weakly on temperature, but\ndecrease signi\fcantly with increasing vacancy densities.\nOur results will also serve as an important bench-\nmark against which dynamical behaviors induced by\nother perturbations can be compared. Of particular in-\nterest are those perturbations, such as quantum order-\nby-disorder, that give rise to glassy dynamics character-\nistic of either the conventional spin-glass or the exotic\nspin-jam states. It is also worth noting that the dynam-\nical properties of classical spin liquid has been exten-\nsively studied for Heisenberg antiferromagnets on both\npyrochlore [15, 16, 27] and kagome lattices [28{31]. An-\nother aim of this paper is thus to compare the spin dy-\nnamics of bilayer kagome against these two well studied\nbi-simplex frustrated magnets.\nThe rest of the paper is organized as follows. In sec-\ntion II, we discuss the ground-state manifold of Heisen-\nberg antiferromagnet on the kagome bilayer. We also\noutline the numerical framework that combines Monte\nCarlo simulation with energy-preserving Landau-Lifshitz\ndynamics method for computing the dynamical structurearXiv:1904.05863v3  [cond-mat.str-el]  18 May 20212\nfactor of a classical spin liquid. Magnetic excitations re-\nvealed from the dynamical structure factor are discussed\nin Sec. III. In particular, half-moon features, which are\nthe dynamical manifestation of the famous pinch-point\nstructure at \fnite energies, are highlighted. Systematic\nanalysis of the low-energy spin dynamics, which is dom-\ninated by di\u000busive modes, is presented in Sec. IV. We\npresent in Sec. V dynamical features due to quenched\ndisorder introduced by vacancies. Of particular interest\nis the emergence of quasi-free orphan spins that inter-\nact with each other through a week e\u000bective interaction\nmediated by the background spin-liquid. We conclude\nin Sec. VI with a brief summary and outlook on future\nwork.\nII. MODEL AND METHOD\nWe consider the classical Heisenberg model with near-\nest neighbor interactions on the kagome bilayer\nH=JX\nhijiSi\u0001Sj (1)\nHereJ > 0 is the antiferromagnetic exchange, hijide-\nnotes nearest-neighbor pairs, and the classical spins Si\nare unit vectors. The kagome bilayer has a Bravais tri-\nangular lattice with a unit cell consisting of two corner\nsharing tetrahedra of opposite orientation. The bases of\nthe tetrahedra in the two kagome layers are connected by\ntriangle units; see Fig. 1. The triangle and tetrahedron\nare the regular simplexes with q= 3 andq= 4 corners,\nrespectively. Importantly, because of this corner-sharing\nsimplex structure, the exchange interaction can also be\nexpressed as a sum of the squared total spin of both types\nof simplexes\nH=J\n2X\n\u0002L2\n\u0002+X\n4L2\n4+ const. (2)\nHereL\u0002=P\ni2\u0002Sidenotes total spin in a tetrahedron,\nL4=P\ni24Sidenotes total spins of a triangle, andP\n\u0002\nandP\n4indicate summation over tetrahedra and trian-\ngles, respectively, in the kagome-bilayer lattice. One can\nimmediately see that the exchange energy is minimized\nby the condition that total spin of every simplexes is zero:\nL\u0002=L4= 0; (3)\nThe ground-state condition is con\frmed by our Monte\nCarlo simulations. The fact that a macroscopic num-\nber of spin con\fgurations satisfy the minimum energy\ncondition leads to a classical spin liquid regime at tem-\nperaturesT.J. Indeed, our Monte Carlo simulations\nshow no signs of phase transition down to temperatures\nT\u00190:001J, consistent with previous studies [32{35]. In-\nstead, a spin-disordered phase with strong short-range\ncorrelation is obtained at low temperatures.\n<latexit sha1_base64=\"94RZZHhO2ku4eJ8F+Wydckl8Gyo=\">AAAB6nicbVA9TwJBEJ3DL8Qv1NJmIzHBhtxRqCXRxhKjIAlcyN6yBxv29i67cybkwk+wsdAYW3+Rnf/GBa5Q8CWTvLw3k5l5QSKFQdf9dgpr6xubW8Xt0s7u3v5B+fCobeJUM95isYx1J6CGS6F4CwVK3kk0p1Eg+WMwvpn5j09cGxGrB5wk3I/oUIlQMIpWuq/S83654tbcOcgq8XJSgRzNfvmrN4hZGnGFTFJjup6boJ9RjYJJPi31UsMTysZ0yLuWKhpx42fzU6fkzCoDEsbalkIyV39PZDQyZhIFtjOiODLL3kz8z+umGF75mVBJilyxxaIwlQRjMvubDITmDOXEEsq0sLcSNqKaMrTplGwI3vLLq6Rdr3kXtfpdvdK4zuMowgmcQhU8uIQG3EITWsBgCM/wCm+OdF6cd+dj0Vpw8plj+APn8weKVY1P</latexit>(a)<latexit sha1_base64=\"e5MifZ84H8qc2xUX3eb4jAAxuMw=\">AAAB6nicbVA9TwJBEJ3DL8Qv1NJmIzHBhtxRqCXRxhKjIAlcyN6yBxv29i67cybkwk+wsdAYW3+Rnf/GBa5Q8CWTvLw3k5l5QSKFQdf9dgpr6xubW8Xt0s7u3v5B+fCobeJUM95isYx1J6CGS6F4CwVK3kk0p1Eg+WMwvpn5j09cGxGrB5wk3I/oUIlQMIpWuq8G5/1yxa25c5BV4uWkAjma/fJXbxCzNOIKmaTGdD03QT+jGgWTfFrqpYYnlI3pkHctVTTixs/mp07JmVUGJIy1LYVkrv6eyGhkzCQKbGdEcWSWvZn4n9dNMbzyM6GSFLlii0VhKgnGZPY3GQjNGcqJJZRpYW8lbEQ1ZWjTKdkQvOWXV0m7XvMuavW7eqVxncdRhBM4hSp4cAkNuIUmtIDBEJ7hFd4c6bw4787HorXg5DPH8AfO5w+L2o1Q</latexit>(b)FIG. 1: (Color online) Top: Lattice structure of kagome\nbilayer. It can be viewed as a quasi-two-dimensional network\nof corner-sharing simplexes. There are two kinds of simplexes:\ntetrahedron and triangles, both comes with two (opposite)\norientations, as shown in the bottom panel. Spins in the\nground states satisfy the constraint that total spin in both\ntypes of simplexes is zero: L\u0002=L4= 0.\nIn general, there are two types of spin dynamics in the\nliquid regime. At short time scales, or high frequencies\n(!\u0018J), there are spin-wave excitations corresponding to\nsmall amplitude deviations from the ground-state mani-\nfold. These excitations are similar to the magnons in un-\nfrustrated magnets. On the other hand, the macroscopic\nnumber of zero modes, or the weather-vane modes, that\nconnect di\u000berent ground states dominate the long-time\ndynamical behaviors of the frustrated bi-simplex antifer-\nromagnet. The resultant drifting of the system in the\nground-state manifold gives rise to spin-di\u000busion behav-\niors and an exponential decaying spin autocorrelation.\nIn the following, we discuss our simulation results within\nthis general picture.\nThe equation of motion for classical spins is given by\nthe Landau-Lifshitz equation\ndSi\ndt=\u0000Si\u0002@H\n@Si=\u0000JX\nj0Si\u0002Sj; (4)\nwhere the prime indicates summation is restricted to the\nnearest neighbors of the i-th spin. Here we numerically\nintegrate the Landau-Lifshitz equation to compute the\ndynamical structure factor of the classical spin liquid.\nLow temperature Monte Carlo simulations are \frst used\nto obtain spin con\fgurations in equilibrium of a speci\fed\ntemperature. These are then used as the initial states for\nthe energy-conserving precession dynamics simulations.\nAn e\u000ecient semi-implicit integration algorithm [36] is\nemployed to integrate the above Landau-Lifshitz equa-\ntion. The high e\u000eciency of the algorithm comes from fact\nthat it preserves the spin length at every time step and\nthe energy values are well conserved with time irrespec-\ntive of the step size or the time span of the simulation.\nFrom the numerically obtained spin trajectories Si(t), we3\n(a)(b)\n(c)(d)!ΓKMXY\n3210!\nΓKMXY\n3210\n(a)(b)\n(c)(d)\n<latexit sha1_base64=\"KvxRniss9Kg+Z5mbwoKY5paS4ik=\">AAAB8HicbVDLSgMxFL1TX7W+qi7dBIvgqswUqS6LblxWsA9ph5JJM21okhmTjFCGfoUbF4q49XPc+Tdm2llo64HA4Zx7ybkniDnTxnW/ncLa+sbmVnG7tLO7t39QPjxq6yhRhLZIxCPVDbCmnEnaMsxw2o0VxSLgtBNMbjK/80SVZpG8N9OY+gKPJAsZwcZKD32BzTgI0eOgXHGr7hxolXg5qUCO5qD81R9GJBFUGsKx1j3PjY2fYmUY4XRW6ieaxphM8Ij2LJVYUO2n88AzdGaVIQojZZ80aK7+3kix0HoqAjuZBdTLXib+5/USE175KZNxYqgki4/ChCMToex6NGSKEsOnlmCimM2KyBgrTIztqGRL8JZPXiXtWtWrV2t3F5XGdV5HEU7gFM7Bg0towC00oQUEBDzDK7w5ynlx3p2PxWjByXeO4Q+czx+EaJA6</latexit>q<latexit sha1_base64=\"KvxRniss9Kg+Z5mbwoKY5paS4ik=\">AAAB8HicbVDLSgMxFL1TX7W+qi7dBIvgqswUqS6LblxWsA9ph5JJM21okhmTjFCGfoUbF4q49XPc+Tdm2llo64HA4Zx7ybkniDnTxnW/ncLa+sbmVnG7tLO7t39QPjxq6yhRhLZIxCPVDbCmnEnaMsxw2o0VxSLgtBNMbjK/80SVZpG8N9OY+gKPJAsZwcZKD32BzTgI0eOgXHGr7hxolXg5qUCO5qD81R9GJBFUGsKx1j3PjY2fYmUY4XRW6ieaxphM8Ij2LJVYUO2n88AzdGaVIQojZZ80aK7+3kix0HoqAjuZBdTLXib+5/USE175KZNxYqgki4/ChCMToex6NGSKEsOnlmCimM2KyBgrTIztqGRL8JZPXiXtWtWrV2t3F5XGdV5HEU7gFM7Bg0towC00oQUEBDzDK7w5ynlx3p2PxWjByXeO4Q+czx+EaJA6</latexit>q<latexit sha1_base64=\"gYJa+h4eRi6XDaD289+2EUHPjMA=\">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mKqMeiF48t2FpoQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgpr6xubW8Xt0s7u3v5B+fCoreNUMWyxWMSqE1CNgktsGW4EdhKFNAoEPgTj25n/8IRK81jem0mCfkSHkoecUWOlptsvV9yqOwdZJV5OKpCj0S9/9QYxSyOUhgmqdddzE+NnVBnOBE5LvVRjQtmYDrFrqaQRaj+bHzolZ1YZkDBWtqQhc/X3REYjrSdRYDsjakZ62ZuJ/3nd1ITXfsZlkhqUbLEoTAUxMZl9TQZcITNiYgllittbCRtRRZmx2ZRsCN7yy6ukXat6l9Va86JSv8njKMIJnMI5eHAFdbiDBrSAAcIzvMKb8+i8OO/Ox6K14OQzx/AHzucPe2OMuw==</latexit>0<latexit sha1_base64=\"KSvMflgKifuNDya4zl1AGMrrYiM=\">AAAB63icbVDLSsNAFL2pr1pfVZduBovgKiTF17LoxmUF+4A2lMl00g6dmYSZiVBCf8GNC0Xc+kPu/BsnbRbaeuDC4Zx7ufeeMOFMG8/7dkpr6xubW+Xtys7u3v5B9fCoreNUEdoiMY9VN8SaciZpyzDDaTdRFIuQ0044ucv9zhNVmsXy0UwTGgg8kixiBJtc8tz65aBa81xvDrRK/ILUoEBzUP3qD2OSCioN4Vjrnu8lJsiwMoxwOqv0U00TTCZ4RHuWSiyoDrL5rTN0ZpUhimJlSxo0V39PZFhoPRWh7RTYjPWyl4v/eb3URDdBxmSSGirJYlGUcmRilD+OhkxRYvjUEkwUs7ciMsYKE2PjqdgQ/OWXV0m77vpXbv3hota4LeIowwmcwjn4cA0NuIcmtIDAGJ7hFd4c4bw4787HorXkFDPH8AfO5w/NFo1u</latexit>0.25\n<latexit sha1_base64=\"LJq5OFVVV8wT2e7Wg5z97v7iRaQ=\">AAAB6nicbVDLSgNBEOz1GeMr6tHLYBA8LbvB1zHoxWNE84BkCbOT3mTI7OwyMyuEkE/w4kERr36RN//GSbIHTSxoKKq66e4KU8G18bxvZ2V1bX1js7BV3N7Z3dsvHRw2dJIphnWWiES1QqpRcIl1w43AVqqQxqHAZji8nfrNJ1SaJ/LRjFIMYtqXPOKMGis9eO5Ft1T2XG8Gskz8nJQhR61b+ur0EpbFKA0TVOu276UmGFNlOBM4KXYyjSllQ9rHtqWSxqiD8ezUCTm1So9EibIlDZmpvyfGNNZ6FIe2M6ZmoBe9qfif185MdB2MuUwzg5LNF0WZICYh079JjytkRowsoUxxeythA6ooMzadog3BX3x5mTQqrn/pVu7Py9WbPI4CHMMJnIEPV1CFO6hBHRj04Rle4c0Rzovz7nzMW1ecfOYI/sD5/AFb1o0y</latexit>0.5\n<latexit sha1_base64=\"/wzyRuSORETK3pDGad/RmsE/dw0=\">AAAB63icbVBNS8NAEJ34WetX1aOXxSJ4CklR67HoxWMF+wFtKJvtpl26uwm7G6GE/gUvHhTx6h/y5r9x0+agrQ8GHu/NMDMvTDjTxvO+nbX1jc2t7dJOeXdv/+CwcnTc1nGqCG2RmMeqG2JNOZO0ZZjhtJsoikXIaSec3OV+54kqzWL5aKYJDQQeSRYxgk0ueW79alCpeq43B1olfkGqUKA5qHz1hzFJBZWGcKx1z/cSE2RYGUY4nZX7qaYJJhM8oj1LJRZUB9n81hk6t8oQRbGyJQ2aq78nMiy0norQdgpsxnrZy8X/vF5qopsgYzJJDZVksShKOTIxyh9HQ6YoMXxqCSaK2VsRGWOFibHxlG0I/vLLq6Rdc/1rt/ZwWW3cFnGU4BTO4AJ8qEMD7qEJLSAwhmd4hTdHOC/Ou/OxaF1zipkT+APn8wfUr41z</latexit>0.75\n<latexit sha1_base64=\"SgsVJBJrdi+AePRU+CIe+9NgDFo=\">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0iKqMeiF48V7Qe0oWy2m3bpZhN2J0Ip/QlePCji1V/kzX/jts1BWx8MPN6bYWZemEph0PO+ncLa+sbmVnG7tLO7t39QPjxqmiTTjDdYIhPdDqnhUijeQIGSt1PNaRxK3gpHtzO/9cS1EYl6xHHKg5gOlIgEo2ilB9/1euWK53pzkFXi56QCOeq98le3n7As5gqZpMZ0fC/FYEI1Cib5tNTNDE8pG9EB71iqaMxNMJmfOiVnVumTKNG2FJK5+ntiQmNjxnFoO2OKQ7PszcT/vE6G0XUwESrNkCu2WBRlkmBCZn+TvtCcoRxbQpkW9lbChlRThjadkg3BX355lTSrrn/pVu8vKrWbPI4inMApnIMPV1CDO6hDAxgM4Ble4c2Rzovz7nwsWgtOPnMMf+B8/gBVyI0u</latexit>1.0\nFIG. 2: (Color online) Temperature-scaled dynamical struc-\nture factor \fS(q;!) of the classical spin liquid in the bilayer\nkagome antiferromagnet at four di\u000berent temperatures: (a)\nT=J = 0:01, (b) 0.05, (c) 0.1, and (d) 0.6; here \f= 1=T. The\nlinear size of the simulated lattice is L= 30, with number of\nspinsN= 7\u0002L2.\ncompute the dynamical correlation function S(q;t)\nS(q;t) =hSq(t)\u0001S\u0003\nq(0)i; (5)\nwhere Sq(t)\u0011P\niSi(t) exp(iq\u0001ri)=p\nNis the spatial\nFourier transform of the instantaneous spin con\fgura-\ntion, andh\u0001\u0001\u0001i denotes the ensemble average over inde-\npendent initial states of a given temperature. The dy-\nnamical structure factor is then given by\nS(q;!) =Z\nS(q;t)e\u0000i!tdt\n=1\nNX\nijZ\ndthSi(t)\u0001Sj(0)ie\u0000i!tdt; (6)\nwhich is essentially the space-time Fourier transform of\nthe spin-spin correlator Cij(t)\u0011hSi(t)\u0001Sj(0)i.\nIII. MAGNONS AND HALF MOON PATTERNS\nThe intensity plot of the scaled dynamical structure\nfactor\fS(q;!), where\f= 1=T, is shown in Fig. 2\nfor four di\u000berent temperatures. The spin excitations are\nclearly dominated by the low-energy quasi-static \ructu-\nations that extend over most of the Brillouin zone; also\nsee Fig. 3 for the density plots of S(q;!) in the recipro-\ncal space at constant energies. At low temperatures, the\nsimilar patterns of the quasi-static excitations indicating\nnontrivial scaling behaviors to be discussed below. More-\nover, the relatively weak excitations at higher energies\n!&Jresult from the magnon \ructuations in the vicin-\nity of an instantaneous ground state. Contrary to the\nkagome antiferromagnets [29, 30], no sharp propagating\nmodes can be seen in the dynamical structure factor of\nthe bilayer kagome.\n0.2 0.4 0.6 0.8 10.2 0.4 0.6 0.80.2 0.4 0.6 0.8qx\n<latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit>qy\n<latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit>qx\n<latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit>qx\n<latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit>(a)(b)(c)\n!=2J!=J!=0\n⇥2·10\u00004⇥10\u00004FIG. 3: (Color online) (Color online) Density plots of the\ndynamical structure factor S(q;!) atT= 0:005Jin the re-\nciprocal space: (a) != 0, (b)!=J, and (c) 2J. The system\nsize isL= 30. The dashed circles indicate the pinch point at\n!= 0, and the half moon pattern at higher energies.\nThe static structure factor, corresponding to Fig. 3(a)\nwith!= 0, exhibits sharp pinch points which are a hall-\nmark of highly correlated spin liquid in bi-simplex frus-\ntrated magnets. The source of these singularities can be\nattributed to the ground-state constraints Eq. (3), which\ntranslates into a solenoid condition r\u0001B= 0 for an emer-\ngent \\magnetic\" or \rux \feld that is a coarse-grained rep-\nresentation of the spin con\fguration. This in turn gives\nrise to an anisotropic dipolar-like correlation of the \rux\n\feld, which manifests itself as the pinch-point singularity\nin the reciprocal space [20, 37, 38].\nAt \fnite temperatures, the width of the pinch-point is\nroughly proportional top\nT[39]. Interestingly, the pinch\npoint is also smeared with increasing !, and is replaced\nby the so-called half moon pattern at !&J, as shown in\nFig. 3 (b) and (c). Similar features, called the \\excitation\nrings\" have been observed in the \fnite-energy dynami-\ncal structure factor of the coplanar spin liquid phase of\nkagome [29]. It has been pointed out that the half-moon\ncan be viewed as the pinch-point with a dispersive dy-\nnamical \rux \feld [40]. These crescent patterns at high\nenergies are the remnants of the propagating magnons\nmentioned above. Compared with the coplanar phase in\nkagome, the half-moon feature is much weaker in the liq-\nuid phase of bilayer kagome, indicating less rigid local\nstructures in the instantaneous ground state.\nIV. SPIN DIFFUSION\nThe relatively weak half moon excitations also indicate\na dominating spin di\u000busive dynamics in bilayer kagome.\nIn general, spin di\u000busion dominates the excitation spec-\ntrum of disordered Heisenberg systems in the hydrody-\nnamic limit [41, 42]. In frustrated magnets, this di\u000bu-\nsion results from the macroscopic number of zero-energy4\ntA(t)     \n 0.01 0.1 10.050.10.20.40.8\nT⌧\u00001(a)(b)10-210-11\n04080120160     T=0.01\n<latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit>0.035\n<latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit>0.08\n<latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit>0.2\n<latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit>0.6\n<latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit>\nFIG. 4: (Color online) (a) The ensemble averaged spin auto-\ncorrelation function A(t) =P\nihSi(t)\u0001Si(0)i=Non aL= 30\nlattice for varying temperatures. (b) Extracted relaxation\ntime\u001cofA(t) = exp(\u0000t=\u001c) as a function of temperature. The\ndashed line shows the power-law \u001c\u0018T\u00000:924dependence.\nmodes in the instantaneous ground state, causing the sys-\ntem to wander around the degenerate manifold. One par-\nticular manifestation of this di\u000busion is the decay of the\nspin autocorrelation function\nA(t) =1\nNX\nihSi(t)\u0001Si(0)i=X\nqS(q;t); (7)\nwhere againh\u0001\u0001\u0001i is the thermal average, which is\nachieved through averaging over independent initial\nstates from Monte Carlo simulations. Fig. 4(a) shows\nA(t) as a function of time for various temperatures ob-\ntained from a L= 30 system. The decay of the auto-\ncorrelation function is found to be exponential A(t)\u0018\nexp(\u0000t=\u001c) in the low temperature regime, and the nu-\nmerically extracted time constant \u001cis shown in Fig. 4(b)\nas a function of temperature.\nThe nearly linear segment in the log-log plot suggests\na power-law dependence \u001c\u0018T\u0000\u0010, where the numeri-\ncally obtained exponent \u0010= 0:924\u00060:015, which is close\nto 1 as predicted by a soft-spin Langevin dynamics model\nfor frustrated magnets with macroscopic ground-state de-\ngeneracy [27]. The exponential decay with \u001c\u00181=Tis\nconsistent with the zero-mode driven spin-di\u000busion sce-\nnario [15, 27], since the zero modes have no intrinsic\nenergy scales, and the only relevant one is set by the\ninverse temperature. This result is also in stark con-\ntrast to the high- Tconventional paramagnet in which\nthe spin-di\u000busion is shown to produce a power-law tail\nin the autocorrelation function [42{46].\nWhile the microscopic mechanisms of spin-di\u000busion\ncould be thermal or quantum \ructuations, or the large\nnumber of zero modes in frustrated systems, fundamen-\ntally the di\u000busive spin dynamics is related to the fact that\nthe total spin density m=P\niSi=Nis a constant of the\nequation of motion. By combining the continuity equa-\ntion@m=@t+r\u0001j= 0 with a phenomenological Fick's\nlaw for local spin current j=\u0000Drm, one arrives at the\nfamiliar di\u000busion equation for the magnetization density.\nIn the hydrodynamic regime, this introduces a di\u000busion\ntimescale\u001cd= 1=Dq2for perturbations characterized by\n 0 0.2 0.4 0.6 0.8 1\n0100200300400t\nDq210.80.60.40.200100200300400 0.1 1 0.01 0.1 1 1010.10.010.11⌧\u00001d\n|q|\n<latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit>|q|=\n<latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit>(a)(b)S(q,t)FIG. 5: (Color online) (a) Time dependence of the normal-\nized dynamical correlation function S(q;t)=S(q;0) atT=\n0:5Jfor various wave vectors close to the zone center. The\nsimulated system size is L= 60. (b) The inverse relaxation\ntime\u001c\u00001\ndextracted from panel (a) as a function of jqj. For\neach wave vector the data is \ftted to an exponentially decay-\ning function.\nwave vector q. This is indeed con\frmed by our dynamical\nsimulations. Fig. 5(a) shows the time dependence of the\ndynamical correlation function S(q;t) for various wave\nvectors. Each curve is obtained after averaging over 500\nindependent initial states from Monte Carlo simulations.\nThe correlation function is found to decay exponentially\nwith time:S(q;t)\u0018exp(\u0000t=\u001cd), where the numerically\nextracted relaxation time, shown in Fig. 5(b), is isotropic\nin the reciprocal space and follows nicely the expected be-\nhavior\u001c\u00001\nd=Dq2for wave vectors close to the Brillouin\nzone center.\nMore generally, here we try to understand our re-\nsults using the hydrodynamic theory of the paramag-\nnetic state, which suggests a generalized dynamical sus-\nceptibility: \u001f(q;!) =\u0000\u001f(q)Dq2=(Dq2\u0000i!) [41, 47],\nwhere\u001f(q) is the static susceptibility at wave vector\nqandDis the spin di\u000busion coe\u000ecient. The dynami-\ncal structural factor is obtained through the \ructuation-\ndissipation theorem: S(q;!)\u00192[nB(!) + 1] Im\u001f(q;!),\nwherenB(!) = 1=(e\f!\u00001). In the!\u001cTregime, as-\nsuming\u001f(q)\u0019\u001fis a constant for small q, the dynamical\nstructure factor can be expressed in a scaling form\n\fq2S(q;!) =\u001f2D\n(!=q2)2+D2; (8)\nA similar result can be obtained from the Langevin soft-\nspin model [27]. By plotting \fq2Sversus!=q2, we \fnd\nnice data collapsing from curves of di\u000berent wave vec-\ntors, as shown in Fig. 6 (a) and (b), indicating a static\nsusceptibility that indeed weakly depends on qfor wave\nvectors close to zone center. On the other hand, we \fnd\nthat the collapsing of data points from di\u000berent tempera-\ntures is not very satisfactory. Instead, we \ft the collapsed\ndata points from each temperature with the Lorentzian\nscaling function in Eq. (8) and extract both the spin dif-\nfusion coe\u000ecient Dand static susceptibility \u001f. The tem-\nperature dependence of these two quantities are shown in\nFig. 6(c) and (d). The spin-di\u000busion coe\u000ecient decreases5\n-1010.20.40.60.814008000120016002000010002000 481216\n-4-2 0 2 4!/q2\nTT(a)(b)\n(c)(d)D\u000048121620S(q,!)\u0000q2S(q,!)\n!q\n 0 0.04 0.080.090.110.13 0.3 0.4 0.5\n 0 0.04 0.08\nFIG. 6: (Color online) (a) Dynamical structure factor S(q;!)\nas a function of !at varying wave vectors qat a tempera-\ntureT= 0:01J. (b) Scaling collapse according to Eq. (8)\nfor data points from di\u000berent wave vectors shown in panel\n(a). These curves are well approximated by a Lorentzian cen-\ntered on!= 0. By \ftting the collapsed data-points to the\nscaling function, the numerically extracted spin di\u000busion co-\ne\u000ecientDand static susceptibility \u001f(normalized to the value\natT= 0:01) are shown in panels (c) and (d), respectively, as\nfunctions of temperature.\nquite appreciably with temperature, while the suscepti-\nbility remains roughly the same within the error bars.\nV. DILUTION EFFECTS\nWe next investigate the e\u000bect of dilution on the spin\ndynamics of the liquid phase. Previous studies have indi-\ncated that dilution with non-magnetic vacancies does not\ninduce the spin-glass behavior of SCGO [17, 48]. In fact,\nthe condition Eq. (3) is satis\fed for every simplex, for\nboth tetrahedron and trinagle, in the ground states even\nfor strong dilution [17]. Consequently, a macroscopic de-\ngeneracy remains and the low- Tphase seems well ap-\nproximated by a Coulombic classical spin liquid. To\ndemonstrate this, we compute the dynamical structure\nfactor of the diluted kagome bilayer using a combination\nof Monte Carlo simulations with the energy-conserving\nLandau-Lifshitz dynamics simulations. Fig. 7 shows the\ncomputedS(q;!) atT= 0:01Jfor four di\u000berent vacancy\nconcentrations. In addition to the thermal average over\nindependent initial states, the S(q;!) of the diluted sys-\ntem is computed with a further average over the disor-\nder, or di\u000berent vacancy con\fgurations. Interestingly, we\n(aΓKMXY\n3210ΓKMXY(a) x=0\n3210\n!\n!(b) x=0.1\n(c) x=0.3(d) x=0.5\n0.00.51.0\nqqFIG. 7: (Color online) Dynamical structure factor S(q;!)\nof the classical spin liquid in the diluted bilayer kagome an-\ntiferromagnet at T= 0:01 for four di\u000berent vacancy concen-\ntrations: (a) x= 0 (no dilution), (b) 0.1, (c) 0.3, and (d) 0.5.\nThe linear size of the simulated lattice is L= 30. The density\nplots for diluted systems ( x6= 0) are further averaged over 50\ndi\u000berent disorder con\fgurations.\n\fnd no dramatic change to the calculated S(q;!) even\nfor vacancy density as high as x= 0:5. The quasi-static\nexcitations show similar patterns for all concentrations,\nalthough both the energy of spin-wave-like excitations at\nlarge!and the bandwidth of the quasi-static excitations\nare slightly reduced with increasing vacancy concentra-\ntions.\nFocusing on the small- !andqregime, we found that\nthe dynamical structure factor is still well approximated\nby the scaling function of Eq. (8), as shown in Fig. 8(a)\nand (b), indicating a dominating spin di\u000busion behav-\nior. The di\u000busion coe\u000ecient Dextracted from the data-\npoint collapsing is plotted in Fig. 8(c) as a function of x.\nThe reduced di\u000busivity with increasing vacancy concen-\ntration indicates a longer relaxation time \u001cd= 1=Dq2,\nor a slower dynamics, caused by the disorder, although\nthe system remains liquid-like. Fig. 8(d) shows the ex-\ntracted static susceptibility \u001fin theq!0 limit versus\nvacancy concentration x. This trend is consistent with\nthe two population picture since the quasi-free orphan\nspins dominate the low- Tstatic susceptibility, hence \u001f\nincreases with the vacancy concentration.\nSince the presence of vacancies does not change the\nliquid nature or the frustrated spin-interactions in the\nkagome bilayer, it is unclear whether the non-magnetic\nimpurities introduce any new dynamical e\u000bect. On the\nother hand, the so-called orphan spins due to the dilution\nare known to induce nontrivial e\u000bects on the equilibrium\nproperties of the kagome bilayer [49]. The orphan spin\ncorresponds to defect triangular simplex with only one\nsurviving spin and two non-magnetic sites [17, 50]. An\nexample of the orphan spin is shown in Fig. 9. One can\nalso think of orphan spin as connecting a q= 3 triangular\nsimplex and a q= 1 point simplex, which is the spin it-6\n 0 0.2 0.4 0.6\n 0 0.1 0.2 0.3 1.5 3 4.5 0 0.1 0.2 0.3!/q2\u0000(a)(b)\n(c)(d)01.53.04.5\n-0.6-0.3 0 0.3 0.601.53.04.56.07.5\n-0.4-0.2 0 0.2 0.4!/q2\nxxD\u0000q2S(q,!)<latexit sha1_base64=\"rsoZXVuMwzrPSi6LpgcqI+uDZEo=\">AAACEXicbZA9SwNBEIb3/IzxK2ppsxiECBLuRFA70cYyojGB3BnmNnPJkr0Pd/eEcOQv2PhXbCwUsbWz89+4iVdo4gsLD+/MsDOvnwiutG1/WTOzc/MLi4Wl4vLK6tp6aWPzRsWpZFhnsYhl0weFgkdY11wLbCYSIfQFNvz++ajeuEepeBxd60GCXgjdiAecgTZWu1RxfdRA724PqBuC7jEQ2dWwMmY/oHf71I1D7MJeu1S2q/ZYdBqcHMokV61d+nQ7MUtDjDQToFTLsRPtZSA1ZwKHRTdVmADrQxdbBiMIUXnZ+KIh3TVOhwaxNC/SdOz+nsggVGoQ+qZztKmarI3M/2qtVAfHXsajJNUYsZ+PglRQHdNRPLTDJTItBgaASW52pawHEpg2IRZNCM7kydNwc1B1Dqsnl4fl07M8jgLZJjukQhxyRE7JBamROmHkgTyRF/JqPVrP1pv1/tM6Y+UzW+SPrI9vzK2cYg==</latexit>\n\u0000q2S(q,!)<latexit sha1_base64=\"rsoZXVuMwzrPSi6LpgcqI+uDZEo=\">AAACEXicbZA9SwNBEIb3/IzxK2ppsxiECBLuRFA70cYyojGB3BnmNnPJkr0Pd/eEcOQv2PhXbCwUsbWz89+4iVdo4gsLD+/MsDOvnwiutG1/WTOzc/MLi4Wl4vLK6tp6aWPzRsWpZFhnsYhl0weFgkdY11wLbCYSIfQFNvz++ajeuEepeBxd60GCXgjdiAecgTZWu1RxfdRA724PqBuC7jEQ2dWwMmY/oHf71I1D7MJeu1S2q/ZYdBqcHMokV61d+nQ7MUtDjDQToFTLsRPtZSA1ZwKHRTdVmADrQxdbBiMIUXnZ+KIh3TVOhwaxNC/SdOz+nsggVGoQ+qZztKmarI3M/2qtVAfHXsajJNUYsZ+PglRQHdNRPLTDJTItBgaASW52pawHEpg2IRZNCM7kydNwc1B1Dqsnl4fl07M8jgLZJjukQhxyRE7JBamROmHkgTyRF/JqPVrP1pv1/tM6Y+UzW+SPrI9vzK2cYg==</latexit>\nFIG. 8: (Color online) Data points collapsing of the scaled\ndynamical structure factor \fq2Sversus!=q2for diluted bi-\nlayer kagome with (a) x= 0:2 and (b)x= 0:3 vacancy con-\ncentrations at a temperature T= 0:01J. The dashed lines\ncorrespond to the Lorentzian scaling function in Eq. (8). The\nextracted di\u000busion coe\u000ecient Dand static susceptibility \u001f\n(normalized with respect to x= 0) as functions of the vacancy\nconcentration are shown in panels (c) and (d), respectively.\nself. The orphan spin behaves as a quasi-free spin with a\nfractionalized length S=2 when perturbed by a magnetic\n\feld [34, 35]. Experimentally, these seemingly isolated\nfree spins in diluted SCGO produce a Curie-like compo-\nnent in the static susceptibility even at temperatures well\nbelow \u0002 CW[49, 50]. Detailed Monte Carlo simulations\nuncover a complex spin texture surrounding the defect\nsimplex whose total spins indeed sum to S=2 [34, 35]. The\nfractionalized spin-texture also provides a natural expla-\nnation for the short-range oscillating signal observed in\nnuclear magnetic resonance [51].\nAn intuitive argument for the fractionalized S=2 or-\nphan spins was originally given by Henley from the view-\npoint of bi-simplex structure [17]. Because each spin is\nshared by two simplexes in the bi-simplex lattices such\nas the kagome-bilayer, the total magnetization can be\nwritten as Mtot=1\n2P\n\u000bL\u000b, where\u000bnow runs over\ntetrahedral, triangular, and q= 1 simplexes in the pres-\nence of orphan spins. In the ground states, total spin\nof each tetrahedron and triangle simplex remains zero,\nas evidenced by Monte Carlo simulations [17]. As a\nresult, the total magnetization of the system becomes\nMtot=1\n2Pq=1\n\u000bL\u000b, where now the summation is re-\nstricted toq= 1 single-point simplexes. As shown above,\nsuchq= 1 simplex is just the orphan spin itself, so we\nhaveMtot=1\n2P\ni2orphanSi, which also means that each\norphan spin can be viewed as a quasi-free spin with a\nfractionalized length S=2 when in a magnetic \feld [17].\nFIG. 9: (Color online) Orphan spin (red arrow) induced by\nvacancies in the kagome bilayer. An orphan spin resides in\ndefective triangular simplex, in which two of the spins are\nremoved, in either of the kagome layers. From the viewpoint\nof simplex, two adjacent vacancies in a triangle removes one\ntriangle simplex , but produces a q= 1 (point) simplex, which\nis the orphan spin itself and transforming two neighboring\ntetrahedral into triangular simplexes.\nA natural question then is what is the dynamical man-\nifestation of these orphan spins. To this end, we exam-\nine the spin-spin autocorrelation function A(t) de\fned\nin Eq. (7). Fig. 10 shows the semi-log plot of autocor-\nrelation functions with and without vacancies obtained\nfrom our dynamical simulations. In both cases, the ini-\ntial decay of the autocorrelation can be well described by\nan exponential function, i.e. A(t)\u0018e\u0000t=\u001cfor smallt.\nHowever, while the exponential decay persists to longer\ntime-scales in the non-diluted system, the autocorrela-\ntion function of the diluted magnet exhibits a long-time\ntail, indicating a signi\fcantly reduced decline rate of the\nspin-autocorrelation.\nThis two-stage relaxation of the autocorrelation func-\ntion can be understood in the framework of the two popu-\nlation picture [49] discussed previously, namely, the clas-\nsical spin liquid of kagome bilayer can be viewed of con-\nsisting of the \\correlated\" population which forms mo-\nmentless clusters ( L\u0002=L4= 0) and the population of\nquasi-free \\orphan\" spins that weakly interact with each\nother [35]. Of course, at very strong dilution, the set\nof free spins also include those completely isolated mag-\nnetic ions [17]. Dynamically, these two populations of\nspins are expected to behave di\u000berently. As discussed in\nSec. IV, spin di\u000busion in the classical spin liquid, which\nis mainly driven by the zero energy modes, results in an\nautocorrelation function A(t) which decays exponentially\nwith time. On the other hand, since the vacancy-induced\norphan spins can be viewed as nearly free spins, one ex-\npects their dynamical behavior to be similar to that of\nuncorrelated paramagnet. Earlier works have shown that\ndi\u000busion of Heisenberg spins in an uncorrelated param-\nagnet leads to a power-law tail in the autocorrelation\nfunction, i.e. A(t)\u00181=t\u000b[42{46], where the exponent \u000b\ndepends strongly on the dimensionality. For 2D Heisen-\nberg magnet, it is estimated to be \u000b\u00191:05\u00060:025 [42].\nTo verify the above picture, we present detailed exam-\nination of both the short-time and long-time behaviors7\nA(t)\n<latexit sha1_base64=\"bbEjPiL4jbm2zjX6vbSc2dX9+FA=\">AAAB63icbVBNSwMxEJ2tX7V+VT16CRahXsquFNRb1YvHCvYD2qVk02wbmmSXJCuUpX/BiwdFvPqHvPlvzLZ70NYHA4/3ZpiZF8ScaeO6305hbX1jc6u4XdrZ3ds/KB8etXWUKEJbJOKR6gZYU84kbRlmOO3GimIRcNoJJneZ33miSrNIPpppTH2BR5KFjGCTSTdVcz4oV9yaOwdaJV5OKpCjOSh/9YcRSQSVhnCsdc9zY+OnWBlGOJ2V+ommMSYTPKI9SyUWVPvp/NYZOrPKEIWRsiUNmqu/J1IstJ6KwHYKbMZ62cvE/7xeYsIrP2UyTgyVZLEoTDgyEcoeR0OmKDF8agkmitlbERljhYmx8ZRsCN7yy6ukfVHz6rXrh3qlcZvHUYQTOIUqeHAJDbiHJrSAwBie4RXeHOG8OO/Ox6K14OQzx/AHzucPMaiNtA==</latexit>\nt<latexit sha1_base64=\"J0cIIe9QtxxvaLVWr4GJH/cYRmQ=\">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mkoN6KXjy2YGuhDWWz3bRrN5uwOxFK6C/w4kERr/4kb/4bt20O2vpg4PHeDDPzgkQKg6777RTW1jc2t4rbpZ3dvf2D8uFR28SpZrzFYhnrTkANl0LxFgqUvJNoTqNA8odgfDvzH564NiJW9zhJuB/RoRKhYBSt1MR+ueJW3TnIKvFyUoEcjX75qzeIWRpxhUxSY7qem6CfUY2CST4t9VLDE8rGdMi7lioaceNn80On5MwqAxLG2pZCMld/T2Q0MmYSBbYzojgyy95M/M/rphhe+ZlQSYpcscWiMJUEYzL7mgyE5gzlxBLKtLC3EjaimjK02ZRsCN7yy6ukfVH1atXrZq1Sv8njKMIJnMI5eHAJdbiDBrSAAYdneIU359F5cd6dj0VrwclnjuEPnM8f5AmNBA==</latexit>\nx=0\n<latexit sha1_base64=\"pMjxebIcMpCFdri8OO6xOifDKjg=\">AAAB7HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUA9C0YvHCqYttKFstpN26WYTdjdiKf0NXjwo4tUf5M1/47bNQVsfDDzem2FmXpgKro3rfjuFldW19Y3iZmlre2d3r7x/0NBJphj6LBGJaoVUo+ASfcONwFaqkMahwGY4vJ36zUdUmifywYxSDGLalzzijBor+U/kmrjdcsWtujOQZeLlpAI56t3yV6eXsCxGaZigWrc9NzXBmCrDmcBJqZNpTCkb0j62LZU0Rh2MZ8dOyIlVeiRKlC1pyEz9PTGmsdajOLSdMTUDvehNxf+8dmaiy2DMZZoZlGy+KMoEMQmZfk56XCEzYmQJZYrbWwkbUEWZsfmUbAje4svLpHFW9c6rV/fnldpNHkcRjuAYTsGDC6jBHdTBBwYcnuEV3hzpvDjvzse8teDkM4fwB87nD4bhjd0=</latexit>\nx=0.3\n<latexit sha1_base64=\"ewDtzBiiXM8MG33MCz2c1x+V1XA=\">AAAB7nicbVBNSwMxEJ3Ur1q/qh69BIvgadnVgnoQil48VrAf0C4lm2bb0Gx2SbJiWfojvHhQxKu/x5v/xrTdg7Y+GHi8N8PMvCARXBvX/UaFldW19Y3iZmlre2d3r7x/0NRxqihr0FjEqh0QzQSXrGG4EaydKEaiQLBWMLqd+q1HpjSP5YMZJ8yPyEDykFNirNR6wtfYdc575YrruDPgZeLlpAI56r3yV7cf0zRi0lBBtO54bmL8jCjDqWCTUjfVLCF0RAasY6kkEdN+Njt3gk+s0sdhrGxJg2fq74mMRFqPo8B2RsQM9aI3Ff/zOqkJL/2MyyQ1TNL5ojAV2MR4+jvuc8WoEWNLCFXc3orpkChCjU2oZEPwFl9eJs0zx6s6V/fVSu0mj6MIR3AMp+DBBdTgDurQAAojeIZXeEMJekHv6GPeWkD5zCH8Afr8AWZGjlI=</latexit>\n<latexit sha1_base64=\"O88zXii9Vtr3VYMwkjcaJav0nuM=\">AAAB7HicbVBNS8NAEJ34WetX1aOXxSJ4CkkR9SIUvXisYNpCG8pmu2mXbnbD7kYsob/BiwdFvPqDvPlv3LY5aOuDgcd7M8zMi1LOtPG8b2dldW19Y7O0Vd7e2d3brxwcNrXMFKEBkVyqdoQ15UzQwDDDaTtVFCcRp61odDv1W49UaSbFgxmnNEzwQLCYEWysFDxde26tV6l6rjcDWiZ+QapQoNGrfHX7kmQJFYZwrHXH91IT5lgZRjidlLuZpikmIzygHUsFTqgO89mxE3RqlT6KpbIlDJqpvydynGg9TiLbmWAz1IveVPzP62QmvgpzJtLMUEHmi+KMIyPR9HPUZ4oSw8eWYKKYvRWRIVaYGJtP2YbgL768TJo1179wa/fn1fpNEUcJjuEEzsCHS6jDHTQgAAIMnuEV3hzhvDjvzse8dcUpZo7gD5zPH7YWjfg=</latexit>x=0.2\n10-410-310-210-11\n 0 200 400 600 800 1000   \nFIG. 10: (Color online) Semi-log plot of autocorrelation func-\ntionA(t) atT= 0:01 without vacancies (red) and with 20%\n(green) and 30% vacancies (blue), obtained from Landau-\nLifshitz dynamics simulations of a L= 48 system. The black\ndashed line indicates an exponential decay A(t)\u0019e\u0000t=\u001cat\nsmallt.\nof spin autocorrelation function for a diluted kagome bi-\nlayer with a vacancy density x= 0:3. We performed\nextensive Monte Carlo and Landau-Lifshitz dynamics\nsimulations on a L= 48 lattice (with total number\nof spinsN= 7\u0002L2= 16128). Over 50 indepen-\ndent realizations of the disorder were constructed, and\nfor each vacancy con\fguration, 100 independent initial\nspin states are prepared at the simulation temperatures.\nFig. 11(a) shows semi-log plot of spin-autocorrelation\nat various temperatures. At short time scale, the de-\ncrease ofA(t) can be reasonably approximated by an\nexponential decay A(t)\u0018e\u0000t=\u001c(T), similar to the undi-\nluted case, with a temperature-dependent decay time\nconstant\u001c. The numerically extracted relaxation time\n\u001c, shown in Fig. 12(a), again exhibits a power-law de-\npendence on temperature \u001c\u00181=T\u0010, with an expo-\nnent\u0010\u00180:952\u00060:017, which is similar to the undiluted\ncase. As discussed in the previous section, the spherical\napproximation for the classical spin liquid predicts an ex-\nponent\u0010= 1. It is unclear whether the deviation here is\ndue to \fnite-size e\u000bect or the soft-spin approximation.\nAt longer time scales, the decay of the autocorrela-\ntion function slows down and turns into a power-law\ntail,A(t)\u0018A1+C(T)=t\u000b, with the same exponent\n\u000bfor di\u000berent temperatures. Interestingly, as shown in\nFig. 12(b), the amplitude of this power-law tail also ex-\nhibits a power-law dependence C\u00181=T\u0011, with an expo-\nnent\u0011= 2:123\u00060:021. It is also worth noting that the\ndecay of spin-autocorrelation saturates to a small but\nnon-zero value at large times, as shown in Fig. 12(a).\nSimilar results, which can be attributed to \fnite-size ef-\nfect, have been reported in the spin-dynamics of uncor-\nrelated classical Heisenberg chains [45, 46].\nTo better understand the power-law decay and the ori-\ngin of the nonzero asymptotic A1in the diluted systems,\nwe consider the dynamics of an orphan spin. At any \fnite\ntemperatures, the total spin of individual simplex does\nnot vanish identically, hence the ground-state condition\nt(a)\n(b)A(t)\n10-310-210-11\n 0.1 1 10 100 1000      T=0.010.020.040.060.100.20\nT=0.010.020.040.060.100.2010-210-110100200300400      <latexit sha1_base64=\"Yk5Q2UDdaoTG4q/mn1LZSSO0Fqo=\">AAAB+HicbVBNS8NAEN3Ur1o/WvXoZbEI9WBJiqjHVi8eK9gPaEPYbDft0s0m7E6EWPpLvHhQxKs/xZv/xm2bg7Y+GHi8N8PMPD8WXINtf1u5tfWNza38dmFnd2+/WDo4bOsoUZS1aCQi1fWJZoJL1gIOgnVjxUjoC9bxx7czv/PIlOaRfIA0Zm5IhpIHnBIwklcqNipwhs9xw+tzGUDqlcp21Z4DrxInI2WUoemVvvqDiCYhk0AF0brn2DG4E6KAU8GmhX6iWUzomAxZz1BJQqbdyfzwKT41ygAHkTIlAc/V3xMTEmqdhr7pDAmM9LI3E//zegkE1+6EyzgBJuliUZAIDBGepYAHXDEKIjWEUMXNrZiOiCIUTFYFE4Kz/PIqadeqzmW1dn9Rrt9kceTRMTpBFeSgK1RHd6iJWoiiBD2jV/RmPVkv1rv1sWjNWdnMEfoD6/MHwoqR4Q==</latexit>A(t)\u0000A1FIG. 11: (Color online) (a) Semi-log plot of the spin auto-\ncorrelation function A(t) =P\nihSi(t)\u0001Si(0)i=NofL= 48\nkagome-bilayer with 30% vacancy at varying temperatures.\nThe dashed lines correspond to the initial exponential decay\nof the autocorrelation function, i.e. A(t)\u0018exp(\u0000t=\u001c) for\nsmallt. (b) The log-log plot of same autocorrelation func-\ntions, with the asymptotic value at large time subtracted, at\nvarying temperatures. The dashed lines indicate power-law\nlong-tailsA(t)\u0018C=t\u000b, with an exponent \u000b= 2:18.\nEq. (3) is not strictly satis\fed. Indeed, the \ructuation of\nsimplex magnetization is given by hL2\n\u000bi\u0018T=J [15, 35].\nThis also indicates a non-zero coupling between orphan\nspins and the background correlated spin liquid. This\nresidual coupling leads to incoherent precession of orphan\nspins and the exponential decay of the orphan-spin au-\ntocorrelation function. On the other hand, as shown in\nRef. [35], there is an emergent e\u000bective interaction be-\ntween the orphan spins. At low enough temperatures,\ntheir collective dynamics induced by this residual inter-\naction thus slows down the exponential decay of auto-\ncorrelation function that is caused by coupling to the\nbackground spin liquid, and turns it into a power-law de-\ncay, similar to the anomalous spin di\u000busion in classical\nHeisenberg magnets at high temperatures [42].\nInterestingly, the exponent \u000b\u00192:1 obtained from our\nnumerical \ftting is signi\fcantly di\u000berent from that of the\n2D paramagnet. This unusual result could be attributed\nto the complex interaction between the orphan spins. As\ndemonstrated in Ref. 35, there is an emergent Heisenberg\nexchange interaction between the orphan spins, that is\ndetermined by the charge-charge correlator of the under-\nlying Coulomb spin liquid. Moreover, the sign (FM vs\nAFM) depends on whether the two orphan spins belong\nto the same kagome layer or not. It has been speculated8\n 0.01 0.1110102103\n10-210-11\n 0.01 0.1TT⌧\u00001C\n(a)(b)\nFIG. 12: (Color online) (a) The decay-time constant \u001cas\na function of temperature Tin log-log plot. The solid line\ncorresponds to a power-law dependence \u001c\u00181=T0:953. (b)\nThe amplitudeCof the power-law long-tail versus the tem-\nperature. The straight line of the log-log plot indicates a\npower-law relationship C\u00181=T2:123.\nwhether this complex and potentially frustrated interac-\ntion might lead to glassy dynamics at low temperatures.\nIndeed, although it is believed that the autocorrelation\nfunction of spin-glass exhibits an stretched exponential\ndecay at temperatures above the glass transition Tg, gen-\neral scaling rules near the glass transition point imply\na cuto\u000b power-law [52{56], such as the Ogielski form\nA(t)\u0018t\u0000\u000bexp[\u0000(\u0015t)\f], where the parameter \u0015!0\nasT!Tg. If the kagome-bilayer can be viewed as ex-\nhibiting a glass transition at T= 0, as conventional 2D\nspin-glasses, the autocorrelation function might be domi-\nnated by a power-law behavior at intermediate time scale\nbefore it is cut o\u000b by the stretched exponential. Further\nlarger scale simulations are required to investigate this\nscenario.\nThe power-law tail and the associated collective behav-\niors also depend strongly on the density xof orphan spins.\nThe e\u000bective interaction between two such defects sepa-\nrated by a distance ris given by Je\u000b(r)\u0018TJ(r=\u0018(T)),\nwhere\u0018\u00181=p\nTis the temperature-dependent correla-\ntion length of the background spin liquid, and the func-\ntionJ(r)\u0018exp(\u0000r) decays exponentially at large dis-\ntances [35]. The T-linear prefactor here indicates the en-\ntropic origin of the e\u000bective interaction, namely Je\u000barises\nfrom conformational entropy of the \ructuating back-\nground correlated spins. Since the average distance be-\ntween orphan spins scales as `\u00181=px, one thus obtains\nan average interaction Je\u000b\u0018Je\u000b(`)\u0018Texp(\u0000p\nT=x),\nwhich becomes exponentially weak at small vacancy per-\ncentages. Despite this weakened interaction, the collec-\ntive behavior of orphan spins would set in at a tempera-\nture that is of the order of the e\u000bective interaction. This\ngives the condition: T\u0003\u0018Je\u000b(T\u0003). Using the expression\nforJe\u000babove, one thus obtain a characteristic temper-\natureT\u0003\u0018xthat decreases linearly with the reduced\ndefect density. Physically, the thermal correlation length\nat thisT\u0003is comparable to inter-orphan-spin distance.VI. DISCUSSION AND OUTLOOK\nTo summarize, we have extensively characterized the\nspin dynamics in the liquid phase of Heisenberg antifer-\nromagnet on the kagome bilayer, which is relevant for\nthe frustrated magnet SCGO. By computing the dynam-\nical structure factor at di\u000berent temperatures and dilu-\ntions, we show that the spin excitations are dominated by\nspin di\u000busion in the low energy, long time regime. The\nspin di\u000busion constant depends weakly on temperature,\nbut decreases with dilution. Another interesting result is\nthe half moon pattern of the dynamical structure factor\nwith energy !&J. Similar features have recently been\nobserved in some pyrochlore compounds, it remains to\nbe seen whether these remnants of the propagating spin\nwaves can be observed in SCGO. Our simulations on di-\nluted bilayer kagome shows that spin di\u000busion remains\nthe dominant process in the presence of site disorder.\nThis result further con\frms, from the dynamical view-\npoint, that site-disorder itself does not immediately cause\nglassy behaviors in the classical spin liquid, although the\ndi\u000busion relaxation time becomes longer with increasing\ndisorder. However, for disorder due to non-magnetic va-\ncancies, the presence of so-called orphan spins results in\nan intriguing power-law tail in the spin-autocorrelation\nfunction. This power-law slow dynamics indicates that\nthe system might be on the verge of a glass transition,\nwhich could be induced by other perturbations.\nAs discussed above, our work o\u000bers important bench-\nmark for future dynamics studies of kagome bilayer that\ninclude other perturbations. Of particular interest are\nthose perturbations that might transform the classical\nspin liquid into either the conventional spin glass or the\nmore exotic spin jam. Indeed, since the di\u000busive spin dy-\nnamics in highly frustrated magnets is mainly driven by\nthe zero-energy modes, one expects a diminishing di\u000bu-\nsivity when the number of such zero modes is signi\fcantly\nreduced. For example, the entropic barrier in the copla-\nnar phase of kagome reduces the continuous weather-van\nmodes to discrete zero modes de\fned on closed loops.\nIt has been proposed that the much slower relaxation of\nthese discrete loops might give rise to glassiness without\nintrinsic disorder in kagome [57, 58]. However, the copla-\nnar phase induced by thermal order-by-disorder seems to\nremain a classical spin liquid [30]. A transition into the\nglassy regime might still occur at a lower temperature\nwhen the dynamics is dominated by quantum tunneling\nof loops [57].\nContrary to kagome Heisenberg antiferromagnets,\nthere is no thermal induced coplanar or collinear phase in\nkagome bilayer. On the other hand, it has been proposed\nby one of us and co-authors in Ref. [26] that a coplanar\nregime, in which spins in each tetrahedron are collinear,\ncan be induced by quantum \ructuations. Moreover, dif-\nferent coplanar ground states can be mapped to discrete\nhexagonal tilings. Importantly, there is no continuous\nweather-van modes in this coplanar regime, and the only\nzero-energy modes are system-wide extended strings [26].9\nAs jamming transition often occurs in such constrained\ndiscrete models, the resultant coplanar phase is dubbed\nthe spin jam [14, 26]. It is argued that quantum \ruc-\ntuations transform the degenerate classical ground-state\nmanifold into a rugged landscape that is di\u000berent from\nthat of conventional spin glass. While this spin-jam\npicture seem to explain some properties of SCGO and\nother similar glassy magnets, such as the much weaker\nmemory e\u000bect [23, 25], an important open question is to\nsee how dynamical behaviors characteristic to spin-jam\nevolve from the classical spin liquid, which will be left for\nfuture study.Acknowledgments\nWe thank Anjana Samarakoon and Israel Klich for use-\nful discussions and for collaborations on related projects.\nDZ and SHL were supported by the U.S. Department of\nEnergy, O\u000ece of Science, O\u000ece of Basic Energy Sciences\nunder Award Number DE-SC0016144. The author also\nacknowledge the support of Advanced Research Comput-\ning Services at the University of Virginia.\n[1] X. Obradors, A. Labarta, A. Isalgu, J. Tejada, J. Ro-\ndriguez, and M. Pernet, Magnetic frustration and lattice\ndimensionality in SrCr 8Ga4O19, Solid State Commun.\n65, 189 (1988).\n[2] A. P. Ramirez, G. P. Espinosa, and A. S. Cooper, Strong\nfrustration and dilution-enhanced order in a quasi-2D\nspin glass, Phys. Rev. Lett. 64, 2070 (1990).\n[3] A. P. Ramirez, G. P. Espinosa, and A. S. Cooper,\nElementary excitations in a diluted antiferromagnetic\nKagome lattice, Phys. Rev. B 45, 2505 (1992).\n[4] C. Broholm, G. Aeppli, G. P. Espinosa, and A. S. Cooper,\nAntiferromagnetic \ructuations and short-range order in\na kagome lattice, Phys. Rev. Lett. 65, 3173 (1990).\n[5] Y. J. Uemura, A. Keren, K. Kojima, L. P. Le, G. M.\nLuke, W. D. Wu, Y. Ajiro, T. Asano, Y. Kuriyama, M.\nMekata, H. Kikuchi, and K. Kakurai, Spin \ructuations\nin frustrated Kagom\u0013 e lattice system SrCr 8Ga4O19stud-\nied by muon spin relaxation, Phys. Rev. Lett. 73, 3306\n(1994).\n[6] S.-H. Lee, C. Broholm, G. Aeppli, A. P. Ramirez, T. G.\nPerring, C. J. Carlile, M. Adams, T. J. L. Jones, and\nB. Hessen, Spin-glass and non-spin-glass features of a\ngeometrically frustrated magnet, Europhys. Lett. 35, 127\n(1996).\n[7] S.-H. Lee, C. Broholm, G. Aeppli, T. G. Perring,\nB. Hessen, and A. Taylor, Isolated spin pairs and\ntwo-dimensional magnetism in SrCr 9pGa12\u00009pO19, Phys.\nRev. Lett. 76, 4424 (1996).\n[8] P. Schi\u000ber, A. P. Ramirez, K. N. Franklin, and S.-\nW. Cheong, Interaction-induced spin coplanarity in a\nkagome magnet: SrCr 9pGa12\u00009pO19, Phys. Rev. Lett.\n77, 2085 (1996).\n[9] A. P. Ramirez, B. Hessen, and M. Winklemann, Entropy\nbalance and evidence for local spin singlets in a kagome-\nlike magnet, Phys. Rev. Lett. 84, 2957 (2000).\n[10] K. Iida, S.-H. Lee, and S.-W. Cheong, Coexisting or-\nder and disorder hidden in a quasi-two-dimensional frus-\ntrated magnet, Phys. Rev. Lett. 108, 217207 (2012).\n[11] A. Keren, Y. J. Uemura, G. Luke, P. Mendels, M.\nMekata, and T. Asano, Magnetic dilution in the geomet-\nrically frustrated SrCr 9pGa12\u00009pO19and the role of lo-\ncal dynamics: A muon spin relaxation study, Phys. Rev.\nLett. 84, 3450 (2000).\n[12] P. Mendels, A. Keren, L. Limot, M. Mekata, G. Collin,\nand M. Horvati\u0013 c, Ga NMR study of the local suscep-\ntibility in kagome-based SrCr 8Ga4O19: pseudogap andparamagnetic defects, Phys. Rev. Lett. 85, 3496 (2000).\n[13] D. Bono, L. Limot, P. Mendels, G. Collin, N. Blanchard,\nCorrelations, spin dynamics, defects: the highly frus-\ntrated kagome bilayer, Low Temp. Phys. 31704 (2005).\n[14] J. Yang, A. Samarakoon, S. Dissanayake, H. Ueda, I.\nKlich, K. Iida, D. Pajerowski, N. P. Butch, Q. Huang,\nJ. R. D. Copley, and S.-H. Lee, Spin jam induced by\nquantum \ructuations in a frustrated magnet, Proc. Nat.\nAcad. Sci. 112, 11519 (2015).\n[15] R. Moessner and J. T. Chalker, Properties of a classical\nspin liquid: The Heisenberg pyrochlore antiferromagnet,\nPhys. Rev. Lett. 80, 2929 (1998).\n[16] R. Moessner and J. T. Chalker, Low-temperature prop-\nerties of classical geometrically frustrated antiferromag-\nnets, Phys. Rev. B 58, 12049 (1998).\n[17] C. L. Henley, E\u000bective Hamiltonians and dilution e\u000bects\nin Kagome and related anti-ferromagnets, Can. J. Phys.\n79, 1307 (2001).\n[18] B. Canals and C. Lacroix, Pyrochlore antiferromagnet: A\nthree-dimensional quantum spin liquid, Phys. Rev. Lett.\n80, 2933 (1998).\n[19] J. T. Chalker, P. C. W. Holdsworth, and E. F. Shender,\nHidden order in a frustrated system: Properties of the\nHeisenberg Kagome antiferromagnet, Phys. Rev. Lett.\n68, 855 (1992).\n[20] D. A. Garanin and B. Canals, Classical spin liquid: Ex-\nact solution for the ini\fnite-component antiferromagnetic\nmodel on the kagome lattice, Phys. Rev. B 59, 443\n(1999).\n[21] C. L. Henley, Long-range order in the classical kagome\nantiferromagnet: E\u000bective Hamiltonian approach, Phys.\nRev. B 80, 180401(R) (2009).\n[22] G.-W. Chern and R. Moessner, Dipolar order by disor-\nder in the classical Heisenberg antiferromagnet on the\nkagome Lattice, Phys. Rev. Lett. 110, 077201 (2013).\n[23] A. M. Samarakoon, T. J. Sato, T. Chen, G.-W. Chern,\nJ. Yang, I. Klich, R. Sinclair, H. Zhou, and S.-H. Lee,\nAging, memory, and nonhierarchical energy landscape of\nspin jam, Proc. Nat. Acad. Sci. 113, 11806 (2016).\n[24] J. Yang, A. M. Samarakoon, K. W. Hong, J. R. D. Cop-\nley, Q. Huang, A. Tennant, T. J. Sato, and S.-H. Lee,\nGlassy behavior and isolated spin dimers in a new frus-\ntrated magnet BaCr 9pGa12\u00009pO19, J. Phys. Soc. Jpn. 85,\n094712 (2016).\n[25] A. M. Samarakoon, M. Takahashi, D. Zhang, J. Yang,\nN. Katayama, R. Sinclair, H. D. Zhou, S. O. Diallo, G.10\nEhlers, D. A. Tennant, S. Wakimoto, K. Yamada, G.-\nW. Chern, T. J. Sato, S.-H. Lee, Scaling of Memories and\nCrossover in Glassy Magnets, Sci. Rep. 7, 12053 (2017).\n[26] I. Klich, S.-H. Lee, and K. Iida. Glassiness and exotic\nentropy scaling induced by quantum \ructuations in a\ndisorder-free frustrated magnet. Nat. Commun. 5: 3497\n(2014).\n[27] P. H. Conlon and J. T. Chalker, Spin dynamics in py-\nrochlore Heisenberg antiferromagnets, Phys. Rev. Lett.\n102, 237206 (2009).\n[28] A. Keren, Dynamical simulation of spins on kagome and\nsquare lattices, Phys. Rev. Lett. 72, 3254 (1994).\n[29] J. Robert, B. Canals, V. Simonet, and R. Ballou, Prop-\nagation and ghosts in the classical kagome antiferromag-\nnet, Phys. Rev. Lett. 101, 117207 (2008).\n[30] M. Taillefumier, J. Robert, C. L. Henley, R. Moessner,\nand B. Canals, Semiclassical spin dynamics of the anti-\nferromagnetic Heisenberg model on the kagome lattice,\nPhys. Rev. B 90, 064419 (2014).\n[31] T. Bilitewski, M. E. Zhitomirsky, and R. Moessner, Dy-\nnamics and energy landscape of the jammed spin liquid,\nPhys. Rev. B 99, 054416 (2019).\n[32] T. Arimori and H. Kawamura, Ordering of the antiferro-\nmagnetic Heisenberg model on a pyrochlore slab, J. Phys.\nSoc. Jpn. 70, 3695 (2001).\n[33] H. Kawamura and T. Arimori, Chiral Kosterlitz-Thouless\ntransition in the frustrated Heisenberg antiferromagnet\non a pyrochlore slab, Phys. Rev. Lett. 88, 077202 (2002).\n[34] A. Sen, K. Damle, and R. Moessner, Fractional spin tex-\ntures in the frustrated magnet SrCr 9pGa12\u00009pO19, Phys.\nRev. Lett. 106, 127203 (2011).\n[35] A. Sen, K. Damle, and R. Moessner, Vacancy-induced\nspin textures and their interactions in a classical spin\nliquid, Phys. Rev. B 86, 205134 (2012).\n[36] J. H. Mentink, M. V. Tretyakov, A. Fasolino, M. I. Kat-\nsnelson, and Th. Rasing, Stable and fast semi-implicit\nintegration of the stochastic Landau-Lifshitz equation,\nJ. Phys.: Condens. Matter 22, 176001 (2010)\n[37] S. V. Isakov, K. Gregor, R. Moessner, and S. L. Sondhi,\nDipolar spin correlations in classical pyrochlore magnets,\nPhys. Rev. Lett. 93, 167204 (2004).\n[38] C. L. Henley, Power-law spin correlations in pyrochlore\nantiferromagnets, Phys. Rev. B 71, 014424 (2005).\n[39] P. H. Conlon and J. T. Chalker, Absent pinch points and\nemergent clusters: Further neighbor interactions in the\npyrochlore Heisenberg antiferromagnet, Phys. Rev. B 81,\n224413 (2010).\n[40] H. Yan, R. Pohle, and N. Shannon, Half moons are\npinch points with dispersion, Phys. Rev. B 98, 140402(R)\n(2018).\n[41] B.I. Halperin and P.C. Hohenberg, Scaling laws for dy-\nnamic critical phenomena, Phys. Rev. 177, 952 (1969).\n[42] G. M uller, Anomalous spin di\u000busion in classical Heisen-\nberg magnets, Phys. Rev. Lett. 60, 2785 (1988).[43] R. W. Gerling and D. P. Landau, Spin-dynamics study\nof the classical ferromagnetic XY chain, Phys. Rev. B\n41, 7139 (1990).\n[44] R. W. Gerling and D. P. Landau, Time-dependent behav-\nior of classical spin chains at in\fnite temperature, Phys.\nRev. B 42, 8214 (1990).\n[45] V. Constantoudis and N. Theodorakopoulos, Nonlinear\ndynamics of classical Heisenberg chains, Phys. Rev. E\n55, 7612 (1997).\n[46] D. Bagchi, Spin di\u000busion in the one-dimensional classical\nHeisenberg model, Phys. Rev. B 87, 075133 (2013).\n[47] D. Forster, Hydrodynamic Fluctuations, Broken Sym-\nmetry, and Correlation Functions (Benjamin, Reading,\n1975);\n[48] E. F. Shender, V. B. Cherepanov, P. C. W. Holdsworth,\nand A. J. Berlinsky, Kagome antiferromagnet with de-\nfects: satisfaction, frustration, and spin folding in a ran-\ndom spin system, Phys. Rev. Lett. 70, 3812 (1993).\n[49] P. Schi\u000ber and I. Daruka, Two-population model for\nanomalous low-temperature magnetism in geometrically\nfrustrated magnets, Phys. Rev. B 56, 13712 (1997).\n[50] R. Moessner and A. J. Berlinsky, Magnetic susceptibility\nof diluted pyrochlore and SrCr 9\u00009xGa3+9xO19antiferro-\nmagnets, Phys. Rev. Lett. 83, 3293 (1999).\n[51] L. Limot, P. Mendels, G. Collin, C. Mondelli, B. Oulad-\ndiaf, H. Mutka, N. Blanchard, M. Mekata, Suscepti-\nbility and dilution e\u000bects of the kagome bilayer geo-\nmetrically frustrated network: A Ga NMR study of\nSrCr 9pGa12\u00009pO19, Phys. Rev. B 65, 144447 (2002).\n[52] R. G. Palmer, D. L. Stein, E. Abrahams, and P. W. An-\nderson, Models of Hierarchically Constrained Dynamics\nfor Glassy Relaxation, Phys. Rev. Lett. 53, 958 (1984).\n[53] H. Sompolinsky and A. Zippelius, Relaxational dynam-\nics of the Edwards-Anderson model and the mean-\feld\ntheory of spin-glasses, Phys. Rev. B 25, 6860 (1982).\n[54] Andrew T. Ogielski, Dynamics of three-dimensional Ising\nspin glasses in thermal equilibrium, Phys. Rev. B 32,\n7384 (1985).\n[55] H. Pinkvos, A. Kalk, and Ch. Schwink, Zero-\feld \u0016SR\nmeasurements in CuMn and AuMn spin glasses inter-\npreted in the frame of a fractal cluster model, Phys. Rev.\nB41, 590 (1990).\n[56] A. Keren, P. Mendels, I. A. Campbell, and J. Lord, Prob-\ning the Spin-Spin Dynamical Autocorrelation Function in\na Spin Glass above Tgvia Muon Spin Relaxation, Phys.\nRev. Lett. 77, 1386 (1996).\n[57] O. C\u0013 epas and B. Canals, Heterogeneous freezing in a geo-\nmetrically frustrated spin model without disorder: Spon-\ntaneous generation of two time scales, Phys Rev. B 86,\n024434 (2012).\n[58] O. C\u0013 epas, Multiple time scales from hard local con-\nstraints: Glassiness without disorder, Phys. Rev. B 90,\n064404 (2014)."    },    {        "title": "1301.5452v2.Decoherence_of_a_single_ion_qubit_immersed_in_a_spin_polarized_atomic_bath.pdf",        "content": "arXiv:1301.5452v2  [quant-ph]  25 Mar 2013Decoherence of a single-ion qubit immersed in a spin-polari zed atomic bath\nL. Ratschbacher1, C. Sias1,2, L. Carcagni1, J. M. Silver1, C. Zipkes1, M. K¨ ohl1,3\n1CavendishLaboratory,University of Cambridge, JJ Thomson Avenue, Cambridge CB30HE, United Kingdom\n2Istituto Nazionale di Ottica, 50019 Sesto Fiorentino, Ital y\n3Physikalisches Institut, University of Bonn, Wegelerstra sse 8, 53115 Bonn, Germany\nWe report on the immersion of a spin-qubit encoded in a single trapped ion into a spin-polarized\nneutralatom environment, which possesses bothcontinuous (motional) and discrete (spin)degrees of\nfreedom. The environment offers the possibility of a precise microscopic description, which allows us\nto understand dynamics and decoherence from first principle s. We observe the spin dynamics of the\nqubit and measure the decoherence times ( T1andT2), which are determined by the spin-exchange\ninteraction as well as by an unexpectedly strong spin-nonco nserving coupling mechanism.\nPACS numbers: 03.67.-a 03.65.Yz 37.10.Ty\nA spin-1/2 system represents the most fundamental\nquantum mechanical object. Its spin-dynamics and -\ndecoherence when interacting with an environment de-\ntermine its potential use as a qubit and are responsible\nfor a multitude of impurity effects encountered in the\nsolid state. While an extensive amount of theoretical\nwork on this problem exists (for reviews see [1, 2]), ex-\nperiments with well-controlled and adjustable environ-\nments are scarce. The necessary sensitivity to probe\nsingle spins, ideally with a single-shot readout, is often\nincompatible with the ability to adjust the properties\nof the environment. Among the few examples of con-\ntrolled decoherence processes are the quantum measure-\nment process [3], the decoherence of motional quantum\nstates of trapped ions by noisy classical electric fields [4],\nand the effects of spontaneous emission [5–7]. These are\nrelated to spin-boson physics, where the environment is\nformed by a set of harmonic oscillator modes [1]. On\nthe contrary, the decoherence of a localized spin impu-\nrity inside an environment of other spins lies at the heart\nof the so-called central-spin problem [2]. Here, the oc-\ncurring decoherence mechanisms are different from spin-\nboson physics since the spectrum of the bath is discrete\nand spin-spin or spin-orbit interactions play a role. The\ncentral-spin model has been often applied as a simplified\nand approximate description of the interaction of semi-\nconductor quantum dots [8] and color centres in solids [9]\nwith their environment.\nHere we investigate the model system of a single local-\nized spin-1/2 coupled to an spin-polarized environment\nof tunable density. Specifically, we embed a trapped sin-\ngle Yb+ion, initially laser-cooled to Doppler tempera-\nture, into a spin-polarized ultracold neutral bath of87Rb\natoms and study the resulting decoherence of the ion’s\ninternal spin state. We observe an intricate decoherence\nmechanism, which is spin-nonconservingand involvesthe\ncoupling of the orbital degrees of freedom with the spin\ndegrees of freedom. We measure the longitudinal ( T1)\nandthetransverse( T2)coherencetimes, alsowithrespect\nto the energy separation, and identify the time scales for\nZeeman- and hyperfine-state relaxation.\n171 Yb + 174 Yb + a b\nc 87 Rb 12.6 \nGHz \n6.8 GHz F = 2 \nF = 1 \n-2 -1 0 1 2mFF = 1 \nF = 0 \n-1 0 1mF-1/2 1/2 mJ\nFIG. 1. (Color online) aIllustration of the trapped single\nion spin coupled to a spin-polarized neutral atom cloud. b\nLevel structure of the Yb+electronic ground state in a weak\nmagnetic field. To implement the spin-1/2 system, we use\neither the Zeeman qubit |mJ=±1/2/angbracketrightin the isotope174Yb+\nor the magnetic field insensitive hyperfinequbit |F= 0,mF=\n0/angbracketrightand|1,0/angbracketrightin the isotope171Yb+.cThe cloud of neutral\n87Rb atoms is prepared in one of the four atomic spin states\n|F= 2,mF= 2/angbracketrighta,|2,−2/angbracketrighta,|1,1/angbracketrightaor|1,−1/angbracketrighta.\nThe interaction between an ion and a neutral atom is,\nto leading order, via a central potential Vσ(r) where r\nis the internuclear separation. Asymptotically ( r→ ∞),\nthe central potential is determined by the polarization\ninteraction −C4/2r4and is spin-independent [5, 11, 12].\nFor small r, collisions in the electronic singlet and triplet\nchannels exhibit different potentials VS(r) andVT(r) giv-\ning rise to a spin-exchange interaction, which is spin-\nconserving for the total spin of atom and ion. Langevin\ncollisions in the central potential, i.e. collisions with en-\nergies above the centrifugal barrier, occur at an energy-\nindependent rate γL= 2π/radicalbig\nC4/µna, whereµis the re-\nduced mass and nais the neutral atom density. In our\nexperiment we have γL/na= 2.1×10−15m3/s and we\nuse typical densities of na= 1018m−3. Even though the\nLangevin rate is significantly smaller than the total col-\nlision rate γc(see Supplementary Material), in previous\nexperiments the Langevin process has dominated cold\ninelastic ion-atom collisions [2, 3, 13, 16–18]. For spin-\ndependent processes additionally the anisotropic mag-2\n0 2 4 6 8 10 0.00.20.40.60.81.0\n0 2 4 6 8 10 |2,-2> a0.40.50.6\n|2,2>ab pBright \nt/tL|1,1>aa\n \nt/tLcp↑, \n8  \nRb state |2,2> a|1,-1> a|1,1> a\nFIG. 2. (Color online) Spin-relaxation in the174Yb+Zeeman\nqubit. The probability pBrightof the ion to occupy the bright\n| ↑/angbracketrightstate after preparation in | ↑/angbracketright(full symbols) or | ↓/angbracketright(open\nsymbols) in a bath of (a)|1,1/angbracketrightaor(b)|2,2/angbracketrightaatoms. Error\nbars denote one standard deviation uncertainty intervals r e-\nsultingfrom approximately3000measurementsperspinstat e.\ncThe equilibrium spin states (corrected for detection efficie n-\ncies) for all four atomic bath configurations |2,−2/angbracketrighta,|1,−1/angbracketrighta,\n|1,1/angbracketrightaand|2,2/angbracketrighta.\nnetic dipole-dipole interaction and the second-orderspin-\norbit coupling can play important roles, the latter in\nparticular for heavy atoms [19]. They induce coupling\nbetween the spin and the orbital motion and break the\nconservation of the total electronic spin. In this work we\ndetermine which of these mechanisms lead to spin deco-\nherence.\nFirst, we study spin relaxation of the Zeeman qubit of\nthe isotope174Yb+, which represents an ideal two-level\nsystem (see Figure 1). We prepare the |mJ= 1/2/angbracketright ≡ | ↑/angbracketright\nor the|mJ=−1/2/angbracketright ≡ | ↓/angbracketright qubit state in an adjustable\nmagnetic field inside the neutral atom cloud. During\nthe interaction period, the ion undergoes binary colli-\nsions with the atoms, and we normalize interaction times\nby the Langevin time constant tL= 1/γL. We detect\nthe population of the qubit state by electronic shelving\nand determining the probability of the ion being in a\nbright (pBright) or dark ( pDark) state using light scat-\ntering on the S1/2−P1/2transition (see Supplemen-\ntary Material). Measurements of the T1time are per-\nformed for varying density and neutral atom spin com-\nposition. The data for the two different bath config-\nurations |F= 1,mF= 1/angbracketrightaand|2,2/angbracketrightaare shown in\nFigures 2a and 2b, respectively. We find that the ini-\ntially polarized ion spin relaxes into a mixed steady-state\nwithin a few Langevin collision times. This result cannot\nbe understood from spin-exchange collisions, for which,\nfor example, the doubly spin-polarized combination | ↑/angbracketright\nand|2,2/angbracketrightais protected from spin-changing collisions due\nto spin conservation. From our steady-state population\ndata for atoms in the |2,2/angbracketrightastate and the measured\nT1= (2.50±0.39)tL, we determine the spin-exchange\nand spin-relaxation rates to be γ↓,SE= (0.22±0.03)/T1\nandγ↑,SR=γ↓,SR= (0.39±0.02)/T1(see Supplemen-\ntaryMaterial). An analogousresult holdsforthe |2,−2/angbracketrighta\nstate. Notably, the measured spin-relaxing collision rate\nis approximately 5 orders of magnitude higher than thecharge-exchange collision rate [3, 4]. All measurements\nwere performed at an energy splitting of the ion Zee-\nman qubit of 37.5MHz, however the same behaviour was\nfound for different Zeeman splittings in the range from\n0.39MHz to 139MHz in a |2,2/angbracketrightaenvironment.\nThe observed spin dynamics of the Yb+ion in a spin\npolarized cloud of87Rb atoms is inconsistent with the\npicture of dominant spin-exchange and negligible spin-\nrelaxation, that has previously been reported in He++\nCs collisions [21], in neutral atom-atom collisions even\nat room temperature [22], and in semiconductor quan-\ntum dots [23]. Moreover, investigations in both ultracold\ngases [24] and optically pumped vapor cells [25] have re-\nvealed that even for heavy alkali-metal collisions the hi-\nerarchyγc≈γSE≫γSRis fulfilled. In such a situation,\nthe dominant spin-exchange interaction leads to a steady\nstate near a perfect polarization of spin and environment\nboth when the environment is initially polarized but the\nspin is unpolarized and when the single spin is contin-\nuously pumped but the environment is initially unpo-\nlarized. The strong spin-relaxation observed here could\nbe aided by a level crossing of both the incoming sin-\ngletA1Σ+and triplet a3Σ+channels to a3Π channel\nat short internuclear separation [26, 27], which poten-\ntially provides a mechanism for spin-orbit coupling. A\nconceptually similar competition between spin-orbit in-\nduced relaxation and spin-conserving processes seems to\nplay a role in the ultrafast relaxation of magnetization in\nferromagnets [28].\nThe dependence of the steady-state spin population of\nthe ion on the atomic bath configuration has been deter-\nmined for the |2,−2/angbracketrighta,|1,−1/angbracketrighta,|1,1/angbracketrighta, and|2,2/angbracketrightastates\nandisshowninFigure2c. Thesteady-statevaluesforthe\n|1,−1/angbracketrightaand|1,1/angbracketrightastatescanbequalitativelyunderstood\nconsideringtheir projections into the electron singlet and\ntriplet bases and taking into account that spin-exchange\ncollisions leading to a transition of the atomic hyperfine\nstate from the |F= 1/angbracketrightato the|F= 2/angbracketrightamanifold are\nenergetically suppressed.\nNext, we study the spin relaxation of the magnetic\nfield insensitive hyperfine qubit of the electronic ground\nstate of171Yb+(see Figure 1 and Supplementary Mate-\nrial). The spin detection is performed by resonant light\nscattering on the S1/2|F= 1/angbracketright →P1/2|F= 0/angbracketrighttransition\nand measuring the probability pBright(pDark) of the ion\nbeing in a bright (dark) state |F= 1/angbracketright(|F= 0/angbracketright). If the\nneutral atoms are prepared in the |1,−1/angbracketrightastate (see Fig-\nure 3a), the ion relaxes towards the spin ground state,\nas expected for amplitude decoherence, and we observe\nan exponential decay with a time constant T1of a few\nLangevin collision times.\nIn contrast, if the neutral atoms are prepared in the\n|2,2/angbracketrightastate (Figure 3b), the steady state probability to\nmeasure the ion spin in the |F= 1/angbracketrightmanifold is 0.16(1).\nWe interpret this non-zero steady-statevalue as resulting\nfrom the intake of hyperfine energy from the atoms. In3\n0 2 4 6 8 100.00.20.40.60.81.0\n0 2 4 6 8 10 120.100.150.200.250.300.35\n0 2 4 6 8 10 120.00.20.40.60.81.0\n0 2 4 6 8 10 120.100.150.200.250.300.35pDark e d cpBright a\n|1,-1>a\nt/tLt/tL t/tL0.00.20.40.60.81.0\n|2,2>apBright b\nt/tL\n> 1 , 1 | > 0 , 1 | > 1 - , 1 |\nFIG. 3. (Color online) Hyperfine spin relaxation in171Yb+.\naThe probability pBrightafter preparation in the |1,0/angbracketright(full\nsymbols) or the |0,0/angbracketright(open symbols) state vs. the interaction\ntime for atoms in the |1,−1/angbracketrightastate. Error bars denote one\nstandard deviation uncertainty intervals resulting from a to-\ntal of 5000 measurements. The fit values at t= 0 are limited\nby detection errors. bSimilar data for collisions with87Rb\natoms in the |2,2/angbracketrightastate and a total of 19000 measurements.\nc-eZeeman-resolved detection within the F= 1 manifold af-\nter preparation in |1,0/angbracketright(atoms in |2,2/angbracketrighta) with 1200 measure-\nments per Zeeman state. The measurements are performed\nby applying resonant πpulses, exchanging the population of\nthe dark state |0,0/angbracketrightwith|1,−1/angbracketright,|1,0/angbracketrightor|1,1/angbracketrightimmediately\nbefore the detection of the probability of the ion being in th e\ndark state pDark.\norder to quantitatively assess this effect, we assign the\nsteady-state distribution of the ion’s hyperfine spin state\na “spin-temperature” Tsbyp1\np0= 3exp/parenleftBig\n−EHFS\nkBTs/parenrightBig\nresult-\ning inTs= 200mK. Here p1=/summationtext\nmp|1,m/angbracketrightis the proba-\nbilitypBrightcorrectedfor detection efficiencies (see Sup-\nplementary Material) and EHFS=h×12.6GHz is the\nhyperfine energy of the ion. In comparison, we estimate\nthe kinetic energy Ekinfrom the balance of elastic and\ninelastic processes, assuming to be not dominated by mi-\ncromotionheatingeffectsatthesetemperatures[29]. The\naverage kinetic energy intake of the ion per Langevin\ncollision due to a hyperfine flip in the neutral atom is\nδEheat=ǫEHFS\nama\nma+mi. Here, mais the mass of the\natom,mithe mass of the ion, 0 ≤ǫ≤1 describes the\nprobability that an atomic hyperfine flip occurs during a\nLangevin collision, and EHFS\na=h×6.8GHz is the in-\nternal hyperfine energy of the atom. The average ion\nenergy loss by elastic ion-atom momentum transfer is\nδEcool=−2mami\n(ma+mi)2Ekinper Langevin collision. As aresult, the steady-state average kinetic energy of the ion\nis/angbracketleftEkin/angbracketright=ǫEHFS\nama+mi\n2mi=ǫkB×240mK. For ǫnear\nunity, we hence obtain /angbracketleftEkin/angbracketright ≈kBTs, which signals an\nequidistribution of energy between kinetic and spin de-\ngrees of freedom. In contrast, if the atomic bath is pre-\npared in its hyperfine ground state |F= 1/angbracketrighta, no energy-\nreleasing hyperfine changes can occur and the temper-\nature of ion is expected to be limited by micromotion\nheating, which is on the order of 20mK [21, 29].\nAs an additional degree of freedom in the case of\n171Yb+, we study the spin transfer within the |F= 1/angbracketright\nmanifold of171Yb+when starting from |1,0/angbracketright. The data\nare shown in Figure 3c-e, together with the results of a\nfour-level rate equation model, which involvesspin trans-\nfer within the |F= 1/angbracketrightmanifold as well as decay into the\n|F= 0/angbracketrightstate. Most strikingly, the decay out of the ini-\ntiallyprepared |1,0/angbracketrightstateexhibitstwodifferenttimecon-\nstants(seeFigure3d). Thefastinitial decayis associated\nwith the population of the |1,±1/angbracketrightstates, where the occu-\npation correspondingly rises (see Figure 3c and e). Even-\ntually, the |F= 1/angbracketrightstates decayto their steady-statepop-\nulations. The significant build-up of population in the\n|1,−1/angbracketrightstate and the relaxation from |1,1/angbracketright → |0,0/angbracketrightare\nboth forbidden by spin conservation, however, so as for\ntheZeemanqubitweobserveastrongspin-nonconserving\nprocess.\nQuantum mechanically, an isolated spin-1/2 can exist\nin superposition states of spin-up and spin-down that are\nnot describable by classical physics. Coupling to an en-\nvironment affects the quantum mechanical superposition\nand, eventually, leads to decoherence as the information\nof the quantum correlations gets lost and the quantum\nmechanicalsuperpositionstransforminto probabilitydis-\ntributions of occupation numbers. In the problem dis-\ncussed here, it is important to understand whether this\ndecoherence results from inelastic spin-relaxation pro-\ncesses,i.e. amplitudedecoherence,orwhetheralsoelastic\n(forward) scattering contributes to additional phase de-\ncoherence of superposition states. In the hydrogen maser\n[30] spin-exchange processes have been identified as a\nleading source of both line shifts and broadening.\nWe characterize the influence of ion-atom collisions on\nthe hyperfine clock transition in171Yb+by measuring\nits coherence time and line shift using the Ramsey tech-\nnique. To this end, we prepare the ion in the |0,0/angbracketright\nhyperfine ground state and apply a π/2-pulse on the\n12.6-GHz clock transition, creating a superposition state\n(|0,0/angbracketright+i|1,0/angbracketright)/√\n2. After a waiting time of 27ms we\napply a second π/2-pulse, followed by a readout of the\n|F= 1/angbracketrightspin state. Figure 4 displays our results. In pan-\nels a-d we show sample traces of our Ramsey fringes as\na function of the detuning of the microwave pulse from\nthe clock transition with increasing density of neutral\natoms during the interaction time. Even without neu-\ntral atoms, the visibility of the Ramsey fringes is limited\nto (55±2)% owing to magnetic field fluctuations of the4\nFrequency offset [Hz] t/t LFrequency shift [Hz] Visibility p(|0,0>) \nFIG. 4. (Color online) Spin coherence of the hyperfine clock\ntransition |0,0/angbracketright ↔ |1,0/angbracketrightin171Yb+.a-dRamsey fringes\nrecorded for increasing atomic density. eRamsey contrast\nas a function of t/tLdisplays an exponential decay with a\ntime constant of T2= (1.4±0.2)tL.fNo frequency shift of\nthe clock transition is observed within the experimental er -\nrors. Error bars denote one standard deviation uncertainty\nintervals.\nbias field. We find that the spin coherence decays within\nT2= (1.4±0.2)tL(see Figure 4e). This is on the time\nscaleofthepopulationrelaxationofthe |1,0/angbracketrightstate,which\nis driven by the spin-exchange and spin-orbit interaction.\nThe measurement hence identifies spin-relaxation as the\nleading mechanism of spin decoherence in our system\nwith anon-detectable contributionfrom elastic (forward)\nscattering. Finally, we note that the frequency shift ∆ ν\nof the clock transition is below the resolution set by the\nspin relaxation rate and is ∆ ν≤4×10−11EHFS\ni/hfor\nour densities (see Figure 4f).\nOur measurements realize a novel spin-bath with addi-\ntionalcontinuousdegreesoffreedominwhichwehaveob-\nserved an unexpectedly strong coupling between internal\n(spin) and external (orbital) degrees of freedom. Future\nexperiments with a driven impurity could reveal whether\nthis coupling mechanism could lead to similar quantum\nphase transitions as predicted for the spin-boson model\n[1] and how the physics of the single ion in the spin-\npolarized bath relates to the physics of polarons [31–34].\nWe thank A. Imamoglu and C. Kollath for dis-\ncussions. The work has been supported by EPSRC\n(EP/H005676/1), ERC (grant 240335), the Leverhulme\nTrust (CS), the Royal Society, and the Wolfson Founda-\ntion.\n[1] A. J. Leggett, S. Chakravarty, A. T. Dorsey,\nM. P. A. Fisher, A. Garg, and W. Zwerger,\nReviews of Modern Physics 59, 1 (1987).\n[2] N. V. Prokofev and P. C. E. Stamp, Report on Progressin Physics 63, 669 (2000).\n[3] M. Brune, E. Hagley, J. Dreyer, X. Maˆ ıtre, A. Maali,\nC. Wunderlich, J. M. Raimond, and S. Haroche,\nPhys. Rev. Lett. 77, 4887 (1996).\n[4] C. J. Myatt, B. E. King, Q. A. Turchette, C. A. Sack-\nett, D. Kielpinski, W. M. Itano, C. Monroe, and D. J.\nWineland, Nature 403, 269 (2000).\n[5] C. F. Roos, G. P. T. Lancaster, M. Riebe, H. H¨ affner,\nW. H¨ ansel, S. Gulde, C. Becher, J. Eschner, F. Schmidt-\nKaler, and R. Blatt, Phys. Rev. Lett. 92, 220402 (2004).\n[6] R. Ozeri, C. Langer, J. D. Jost, B. DeMarco, A. Ben-Kish,\nB. R. Blakestad, J. Britton, J. Chiaverini, W. M. Itano,\nD. B. Hume, D. Leibfried, T. Rosenband, P. O. Schmidt,\nand D. J. Wineland, Phys. Rev. Lett. 95, 030403 (2005).\n[7] J. T. Barreiro, M. M¨ uller, P. Schindler, D. Nigg, T. Monz ,\nM. Chwalla, M. Hennrich, C. F. Roos, P. Zoller, and\nR. Blatt, Nature 470, 486 (2011).\n[8] R. Hanson, B. Witkamp, L. M. K. Vandersypen, L. H. W.\nvan Beveren, J. M. Elzerman, and L. P. Kouwenhoven,\nPhys. Rev. Lett. 91, 196802 (2003).\n[9] R. Hanson, V. V. Dobrovitski, A. E. Feiguin, O. Gywat,\nand D. D. Awschalom, Science 320, 352 (2008).\n[5] R. Cˆ ot´ e and A. Dalgarno,\nPhys. Rev. A 62, 012709 (2000).\n[11] O. P. Makarov, R. Cˆ ot´ e, H. Michels, and W. W. Smith,\nPhys. Rev. A 67, 042705 (2003).\n[12] Z. Idziaszek, T. Calarco, P. S. Julienne, and A. Simoni,\nPhys. Rev. A 79, 010702 (2009).\n[13] A. T. Grier, M. Cetina, F. Oruˇ cevi´ c, and V. Vuleti´ c,\nPhys. Rev. Lett. 102, 223201 (2009).\n[2] C. Zipkes, S. Palzer, C. Sias, and M. K¨ ohl, Nature 464,\n388 (2010).\n[3] C. Zipkes, S. Palzer, L. Ratschbacher, C. Sias, and\nM. K¨ ohl, Phys. Rev. Lett. 105, 133201 (2010).\n[16] S. Schmid, A. H¨ arter, and J. Hecker-Denschlag, Phys.\nRev. Lett. 105, 133202 (2010).\n[17] F. H. J. Hall, M. Aymar, N. Bouloufa-Maafa, O. Dulieu,\nand S. Willitsch, Phys. Rev. Lett. 107, 243202 (2011).\n[18] S. T. Sullivan, W. G. Rellergert, S. Kotochigova, and\nE. R. Hudson, Phys. Rev. Lett. 109, 223002 (2012).\n[19] F. H. Mies, C. J. Williams, P. S. Julienne, and\nM. Krauss, J. Res. Natl. Inst. Stand. Technol. 101, 521\n(1996).\n[4] L. Ratschbacher, C. Zipkes, C. Sias, and M. K¨ ohl, Nature\nPhysics8, 649 (2012).\n[21] F. G. Major and H. G. Dehmelt,\nPhys. Rev. 170, 91 (1968).\n[22] A. Ben-Amar Baranga, S. Appelt, M. V. Romalis, C. J.\nErickson, A. R. Young, G. D. Cates, and W. Happer,\nPhys. Rev. Lett. 80, 2801 (1998).\n[23] C. Latta, A. Srivastava, and A. Imamo˘ glu,\nPhys. Rev. Lett. 107, 167401 (2011).\n[24] J. S¨ oding, D. Gu´ ery-Odelin, P. Desbiolles, G. Ferrar i,\nand J. Dalibard, Phys. Rev. Lett. 80, 1869 (1998).\n[25] S. Kadlecek, L. W. Anderson, C. J. Erickson, and T. G.\nWalker, Phys. Rev. A 64, 052717 (2001).\n[26] O. Dulieu, private communication (2012).\n[27] A. A. Buchachenko and A. K. Belyaev, private commu-\nnication (2012).\n[28] B. Koopmans, G. Malinowski, F. D. Longa, D. Steiauf,\nM. F¨ ahnle, T. Roth, M. Cinchetti, and M. Aeschlimann,\nNature Materials 9, 259 (2010).\n[29] C. Zipkes, L. Ratschbacher, C. Sias, and M. K¨ ohl, New\nJournal of Physics 13, 053020 (2011).5\n[30] J. M. V. A. Koelman, S. B. Crampton, H. T. C.\nStoof, O. J. Luiten, and B. J. Verhaar,\nPhys. Rev. A 38, 3535 (1988).\n[31] A. Schirotzek, C.-H. Wu, A. Sommer, and M. W. Zwier-\nlein, Phys. Rev. Lett. 102, 230402 (2009).\n[32] F. Chevy and C. Mora, Reports on Progress in Physics\n73, 112401 (2010).\n[33] C. Kohstall, M. Zaccanti, M. Jag, A. Trenkwalder,\nP. Massignan, G. M. Bruun, F. Schreck, and R. Grimm,\nNature485, 615618 (2012).\n[34] M. Koschorreck, D. Pertot, E. Vogt, B. Fr¨ ohlich, M. Fel d,\nand M. K¨ ohl, Nature 485, 619 (2012).SUPPLEMENTARY MATERIAL\nExperimental setup\nWe prepare up to 8 ×105neutral87Rb atoms in the\n|F= 2,mF= 2/angbracketrightahyperfine state of the electronic\nground state at temperatures down to T≈200nK in\na harmonic magnetic trap of characteristic frequencies\n[1] (ωx,ωy,ωz) = 2π×(8,26,27)Hz. When prepared\nin the|F= 1,mF=−1/angbracketrightastate the trap frequencies\nare (ωx,ωy,ωz) = 2π×(5,10,15)Hz. Alternatively, the\natoms can be transferred into an optical dipole trap\nformed by two crossed laser beams at 1064nm. Here,\nthe atoms can also be trapped in the hyperfine ground\nstate|1,1/angbracketrightaor the|2,−2/angbracketrightastate. At the same location,\nwe trap a single Yb+ion in a radio-frequency Paul trap\nwith secular trap frequencies of ω⊥= 2π×150kHz ra-\ndially and ωax= 2π×55kHz axially [2, 3]. The atomic\ndensity at the location of the ion in the centre of the neu-\ntral bath is determined for magnetically trapped clouds\nfrom atom number and temperature measured in absorp-\ntion imaging and the harmonic trap frequencies. For op-\ntically trapped clouds, the atomic density is determined\nby measuring the inelastic loss rate of the ion in its elec-\ntronically excited D3/2-state [4]. In both cases we esti-\nmate the uncertainty of the absolute atomic densities to\nbe about 40%. In order to avoid losses of the ion due\nto inelastic collisions in the highly reactive electronically\nexcited states [4], the atoms are removed from the posi-\ntion of the ion before detection of the174Yb+qubit. For\nsteady-state spin measurements the same atomic cloud\nhas been used for up to eight consecutive measurements\nby displacing and returning the atoms by moving the po-\nsition of the optical trap beams.\nIon-atom interaction\nThe interaction between an ion and a neutral atom\nat long distances is dominated by the attractive polar-\nization interaction potential, which is of the form of a\ncentral potential V(r) =−C4\n2r4. Here,C4=α0q2/(4πǫ0)2\nis proportional to the neutral particle polarizability α0,\nqis the charge of the ion, ǫ0is the vacuum permittiv-\nity, andris the internuclear separation. The total colli-\nsion rate between atoms and ions in the central potential\nis given by γc=na√\n2π(1 +π2/16)(C4//planckover2pi1)2/3(E/µ)1/6\nand includes quantum mechanical forward-scattering [5].\nHere,Eisthecollisionenergy. Inourexperimentwehave\nγL/na= 2.1×10−15m3/s andγc/na= 2.5×10−14m3/s\nfor a collision energy of E=kB×100mK.6\nIon qubit preparation and detection\nUltraviolet light at 370nm, resonantwith S1/2→P1/2\ntransition, is used for laser cooling and spin state prepa-\nration, while laser light at 935nm clears out the2D3/2\nstate. State preparation in either Zeeman state of the\nS1/2ground state of the174Yb+isotope is achieved by\noptical pumping with σ+−orσ−−polarized light on the\nS1/2→P1/2transition. State detection is performed\nby frequency-selective electron shelving in the F7/2state\nvia theD5/2state using a laser at 411nm and subse-\nquent resonant light scattering on the S1/2→P1/2tran-\nsition [4]. Imperfections in the detection scheme limit\nthe direct correspondence of the detection of a bright\nor dark quantum state, as determined from the statis-\ntics of photon events, and the occupation of a specific\nupper or lower spin quantum state. The probability\nof spin occupation p↓is determined by p↓= (pDark−\nηDark,↑)/(ηDark,↓−ηDark,↑) from the detected distribu-\ntion of dark states pDarkand the independently deter-\nmined detection errors (1 −ηDark,↓) andηDark,↑. For\nmeasurements with87Rb|F= 2/angbracketrighta, the detection errors\nare (1−ηDark,↓) = 0.19±0.01andηDark,↑= 0.03±0.01,\nandformeasurementsinvolvingthe |F= 1/angbracketrightamanifoldwe\nfind (1−ηDark,↓) = 0.10±0.01andηDark,↑= 0.00+0.01.\nThe largerdetection errorsofthe Zeeman qubit when the\natoms are in the |F= 2/angbracketrightastate result from the increased\nion kinetic energy due to release of atomic hyperfine en-\nergy.\nFor the isotope171Yb+, state preparation in the |0,0/angbracketright\nstate is achieved by a short (250 µs) laser pulse reso-\nnant with the S1/2|F= 1/angbracketright →P1/2|F= 1/angbracketrighttransition\n[6]. Microwave radiation resonant with the 12 .6-GHz\nhyperfine transitions is used for coherent qubit manip-\nulation on the clock transition, for Rabi π-flips on the\n|0,0/angbracketright → |1,−1/angbracketrightand|0,0/angbracketright → |1,1/angbracketrighttransitions as well\nas for repumping during Doppler cooling. The mag-\nnetic bias field strength depends on the atom trap con-\nfiguration and ranges from 5 .64G in the optical trap to\n7.56G in the magnetic trap of the87Rb|1,−1/angbracketrightaatoms.\nThe spin detection probes fluorescence of the ion on the\nS1/2|F= 1/angbracketright →P1/2|F= 0/angbracketrighttransition and we use astandard statistical analysis of photon detection events\nfor state discrimination [7]. The fluorescence detection\ncan distinguish the states |0,0/angbracketrightand|F= 1/angbracketrightand the\ndetection errors are (1 −ηDark,↓) = 0.02±0.01 for the\nhyperfine ground state |F= 0/angbracketrightandηDark,↑= 0.07±0.03\nfor the|F= 1/angbracketrighthyperfine states.\nDetermination of spin-exchange and spin-relaxation\nrates\nThe spin relaxation behaviour is well described by\nsolutions of the general two-level rate equation model\np↑(t) =p↑,0e−t/T1+p↑,∞(1−e−t/T1) with the lon-\ngitudinal coherence time T1= 1/(γ↑+γ↓) and the\nequilibrium spin state p↑,∞=γ↑/(γ↑+γ↓). The de-\ncay rates for the spin states γ↑=γ↑,SE+γ↑,SRand\nγ↓=γ↓,SE+γ↓,SRare decomposed into a spin-exchange\npart depending on the atomic bath spin and a spin-\nrelaxation part including spin-orbit coupling. For col-\nlision with the maximally polarized bath state |2,2/angbracketrightaat\nenergies much larger than the Zeeman splittings, the re-\nlationsγ↑,SE= 0 and γ↓,SR=γ↑,SRhold true and re-\nsult inγ↓,SE= (2p↑,∞−1)/T1= (0.22±0.03)/T1and\nγ↑,SR=γ↓,SR= (1−p↑,∞)/T1= (0.39±0.02)/T1.\n[1] S. Palzer, C. Zipkes, C. Sias, and M. K¨ ohl, Phys. Rev.\nLett.103, 150601 (2009).\n[2] C. Zipkes, S. Palzer, C. Sias, and M. K¨ ohl, Nature 464,\n388 (2010).\n[3] C. Zipkes, S. Palzer, L. Ratschbacher, C. Sias, and\nM. K¨ ohl, Phys. Rev. Lett. 105, 133201 (2010).\n[4] L. Ratschbacher, C. Zipkes, C. Sias, and M. K¨ ohl, Nature\nPhysics8, 649 (2012).\n[5] R. Cˆ ot´ e and A. Dalgarno,\nPhys. Rev. A 62, 012709 (2000).\n[6] S. Olmschenk, K. C. Younge, D. L. Moehring,\nD. N. Matsukevich, P. Maunz, and C. Monroe,\nPhys. Rev. A 76, 052314 (2007).\n[7] M. Acton, K.-A. Brickman, P. C. Haljan, P. J. Lee,\nL. Deslauriers, and C. R. Monroe, Quantum Information\n& Computation 6, 465 (2006).7\nIon isotope Atom state T1/tLp↑,∞/summationtextp|1,x/angbracketright,∞T2/tL\n171Yb+|1,−1/angbracketrighta1.73±0.17 0.000+0.005\n171Yb+|2,2/angbracketrighta3.39±0.16 0.163±0.0131.4±0.2\n174Yb+|2,2/angbracketrighta2.50±0.390.609±0.015\n174Yb+|2,−2/angbracketrighta 0.423±0.026\n174Yb+|1,1/angbracketrighta1.60±0.240.563±0.017\n174Yb+|1,−1/angbracketrighta 0.457±0.021\nTABLE I. Summary of the decoherence times ( T1andT2) and steady-state spin distributions of all spin-bath comb inations. In\naddition to the statistical errors quoted, all times have sy stematic uncertainties of 40% due to the determination of th e absolute\natomic densities at the location of the ion. All values are co mpensated for detection efficiencies."    },    {        "title": "1006.5608v1.Spin_dynamics_of_current_driven_single_magnetic_adatoms_and_molecules.pdf",        "content": "arXiv:1006.5608v1  [cond-mat.mes-hall]  29 Jun 2010Spin dynamics of current driven single magnetic adatoms and molecules\nF. Delgado, and J. Fern´ andez-Rossier\nDepartamento de F´ ısica Aplicada, Universidad de Alicante , San Vicente del Raspeig, 03690 Spain\n(Dated: November 6, 2021)\nA scanning tunneling microscope can probe the inelastic spi n excitations of a single magnetic\natom in a surface via spin-flip assisted tunneling in which tr ansport electrons exchange spin and\nenergy with the atomic spin. If the inelastic transport time , defined as the average time elapsed\nbetween two inelastic spin flip events, is shorter than the at om spin relaxation time, the STM\ncurrent can drive the spin out of equilibrium. Here we model t his process using rate equations and\na model Hamiltonian that describes successfully spin flip as sisted tunneling experiments, including\na single Mn atom, a Mn dimer and Fe Phthalocyanine molecules. When the STM current is not\nspin polarized, the non-equilibrium spin dynamics of the ma gnetic atom results in non-monotonic\ndI/dVcurves. In the case of spin polarized STM current, the spin or ientation of the magnetic atom\ncan be controlled parallel or anti-parallel to the magnetic moment of the tip. Thus, spin polarized\nSTM tips can be used both to probe and to control the magnetic m oment of a single atom.\nPACS numbers:\nI. INTRODUCTION\nA single magnetic atom is arguably the smallest sys-\ntem where the spin can be used to store classical and/or\nquantum information. Therefore, there is great inter-\nest in probing and manipulating the spin state of a sin-\ngle atom or a single molecule in a solid state environ-\nment. Examples of this are single Phosphorous donors in\nSilicon,1nitrogen-vacancy centers in diamonds,2–5single\nMn atomsinII-VI6,7andIII-V8semiconductors,and sin-\ngle magnetic adatoms in surfaces9–20. Whereas in most\ncases the spin of the single atom is probed by optical\nmeans, the possibility of coupling the spin of as single\natom to an electrical circuit is particularly appealing.\nTremendous recent experimental progress has made it\npossible to probe the spin of a single and a few atoms\ndeposited in conducting surfaces by means of scanning\ntunneling microscopes.9–20Therearetwocomplementary\ntechniques that afford this: spin polarized STM and spin\nflip inelastic electron tunnel spectroscopy (IETS). The\nworking principle of spin polarized STM is spin depen-\ndent magneto resistance,21similar to that of tunnel mag-\nneto resistance junction: tunneling between two spin po-\nlarized conductors depends on the relative orientation of\ntheir magnetic moments. Control of the spin orientation\nof either the tip or the substrate affords spin contrast\nSTM imaging.17\nIn the case of spin flip IETS, electrons tunnel from\nthe STM tip to the surface (or vice versa), and exchange\ntheir spin with the atom, so that they produce a spin\ntransition, whose energy is provided by the bias volt-\nage. Thus, a new conduction channel opens when the\nbias voltage is made larger than a given spin transition\n[see Fig. 1(c),1(d)]. This results in a step in the con-\nductance as a function of bias and permits to determine\nthe energy of the spin excitations, and how they evolve\nas a function of an applied magnetic field.9–12,15,16,18,22\nWhen the atom is weakly coupled to its environment, the\nspin is quantized,23,24the spin transitions have sharplydefined energies which can be described with a single ion\nspin Hamiltonian whose parameters can be inferred from\nthe experiments.9–12,16This is the case of Mn, Co and Fe\natoms9–11,16aswellasFeandCoPhthalocyanines,12,15,18\nall of them deposited on a insulating monolayeron top of\na metal. Remarkably, spin flip IETS does not require a\nspin polarized tip to extract information about the spin\ndynamics.\nBothIETSandspinpolarizedSTM arebasedupon the\nfact that the spin state of the atom affects the transport\nelectrons,yieldingaspin-dependentconductance. There-\nfore, we must expect that the transport electrons do af-\nfect the spin of the atom. This is the main theme of this\npaper. In the case of spin flip IETS, there are two rele-\nvanttimescales. Ononeside, theinelastictransporttime\norcharge time Tq, which is defined as the average time\nelapsed between two inelastic spin flip events . On the\nother side, the magnetic atom spin relaxation time ,T1.\nIn theTq>> T1regime, the transport electrons always\ninteract with a atomic spin in equilibrium with the envi-\nronment. As a result, the occupation of the spin states\nis bias independent and the conductance is expected to\nhave flat plateaus in between the inelastic steps. In the\nTq<< T1regime this is no longer the case, the current\ndrives the atomic spin out of equilibrium, so that the oc-\ncupations of the spin states are bias dependent. As we\nshowed in a previous work,25for the case of a single Mn\natom, this results in a modified conductance line-shape,\nwith non-monotonic behaviour in between steps. In this\nwork we give an extended account of these effects, and\nconsider also the case of Mn dimers and FePc.\nNon-equilibrium effects become particularly appealing\nwhen either the tip or the substrate are also spin polar-\nized. In this case we have current flow between two mag-\nnetic objects, which is expected to result in spin transfer\ntorque.26It has been proposed theoretically,25and inde-\npendently verified experimentally,27that the spin orien-\ntation of a single Mn atom can be controlled with a a\nspin polarized tip. In this paper we provide a thorough2\nFIG. 1: (Color online). (a) Scheme of the proposed setup:\na magnetic STM tip and a magnetic adatom on a insulating\nmonolayer deposited on a metal. A current flows through the\nadatom when a bias voltage is applied between tip and sur-\nface. (b) Energy level diagram of tip, atom and surface and\ntypical microscopic process that gives rise to spin-flip tun nel-\ning. (c) Equivalent circuit that accounts for the increase o f\nthe conductance schematically shown in (d).\nanalysis of this effect and extend our study to include the\neffect of an external magnetic field, as in the case of the\nexperiments.27\nThe results presented in this paper are based on\na phenomenological spin-depedent tunneling Hamilto-\nnian22,25,28–31and are, in most instances, in agree-\nment with existing experiments. In the case of a sin-\ngle magnetic atom with spin 1 /2 it is possible to de-\nrive our spin-dependent Hamiltonian from a single or-\nbital Anderson model, by means of a Schrieffer-Wolff\ntransformation.32,33Within this picture, the spin-flip as-\nsisted tunneling events would correspond to inelastic co-\ntunneling in the Anderson model. In the single orbital\nAnderson model the spin-flip channel can dominate the\nelastic channel, which can be even zero in the so called\nsymmetric case. Further work34is in progress to gener-\nalize this picture to the higher spin case relevant to this\npaper.\nThe spin dynamics of current driven nanomagnets in\nthe Coulomb Blockade regime has been thoroughly stud-\nied from the theory standpoint25,35–44. The systems\nstudied include magnetic grains41,45, semiconductor39\nand Mn doped quantum dots36,40, molecular magnets\nand magnetic molecules35,42,43,46. In contrast to most\nof the theory work published to date, here we can\nmodel the current driven dynamics of a quantum spin\nwhose Hamiltonian parameters are accurately known\nfrom experiments10, making it possible to compare suc-\ncessfully theory and experiment.\nThe rest of the manuscript is organized as follows. In\nSec. II we present the model Hamiltonian for the mag-\nnetic atom(s), the transport electrons, and their cou-pling, which accounts both for spin-assisted tunneling\nand Korringalike atomic spin relaxationdue to exchange\ncoupling with the electrodes. The transition rates and\nnon-equilibrium dynamics leading to the current are an-\nalyzed in Sec. IIC. In Sec. III we present the results of\ncurrent driven spin dynamics under the influence of non-\nmagnetic tip in three cases: the single Mn adatom, the\nMn dimer and the FePC molecule. In Sec. IV we discuss\nthe case of a spin polarized tip and analyze in detail the\ncase of a single Mn adatom. In Sec. V we present our\nmain conclusions and discuss open questions.\nII. THEORY\nA. Hamiltonian\nInthissectionwepresentthephenomenologicalHamil-\ntonian, its microscopicjustification, the rateequationap-\nproach for the atom spin dynamics, including both spin\nrelaxation and spin driving terms, and the calculation\nof the current. The system of interest is shown in Fig.\n1(a). We use a model Hamiltonian which describes the\nsystem of interest split in 3 parts: tip, substrate and the\nmagnetic atom(s)25\nH=HT+HS+HSpin+V. (1)\nThe first two terms describe the tip and surface surface:\nHT+HS=/summationdisplay\nk,σ,ηǫση(k)c†\nkσηckση, (2)\nwherec†\nkσηcreates and electron in electrode η=T,S,\nwith momentum kand spin σdefined along the spin\nquantization axis, /vector n. Unless stated otherwise, we take\n/vector nparallel to the magnetization of the tip, which is a\nstatic vector in our theory. Since we consider a non-\nmagnetic surface, we have ǫσ,S(k) =ǫS(k). All the re-\nsults of this paper are trivially generalized to the case of\na non-magnetic tip and a magnetic surface.\nThe spin of the magnetic adatom(s) is (are) de-\nscribed with a single ion Hamiltonian, exchange cou-\npled to other magnetic adatoms and to the transport\nelectrons.9–12,22,25\nHSpin=/summationdisplay\ni/bracketleftBig\nDS′2\nz(i)+E/parenleftBig\nS′2\nx(i)−S′2\ny(i)/parenrightBig/bracketrightBig\n+1\n2/summationdisplay\ni,j,aJi,jS′\na(i).S′\na(j)+gµB/summationdisplay\ni/vectorS′(i)./vectorB.(3)\nThe first term describes the single ion magneto-\ncrystalline anisotropy, the second describes the inter-\natomic exchange couplings and the third corresponds to\nthe Zeeman splitting term under an applied magnetic\nfield/vectorB. Here the prime denotes that the spin quantiza-\ntion axis is chosen with z′along the easy axis of the sys-\ntem, not along the magnetic moment of the tip, /vector n. This3\nmakes necessary to rotate HSpinwhen/vector nis not parallel to\nthe easy axis. The value of the local spin S(i), the mag-\nnetic anisotropy coefficients DandE, and the exchange\ncoupling between atoms in the chain Ji,j, change from\natom to atom and also depend on the substrate.9–11,22,47\nIn the following, we denote the eigenvalues and eigenvec-\ntors ofHSpinasEMand|M∝angbracketrightrespectively.\nWe model the coupling of the magnetic chain\nwith the reservoirs with the following Kondo-like\nHamiltonian:22,25,28,29,31\nV=/summationdisplay\nα,λ,λ′,σ,σ′,iTλ,λ′,α(i)τ(α)\nσσ′\n2ˆSα(i)c†\nλ,σcλ′σ′,(4)\nwhereilabels the magnetic atoms in the surface, λ=\n(k,η) labels the single particle quantum numbers of the\ntransport electrons (other than their spin σ), and the in-\ndexαruns over 4 values, a=x,y,z, andα= 0. We\nuseτ(a)andˆSafor the Pauli matrices and the spin op-\nerators in the /vector nframe, while ˆS0=τ(0)is the identity\nmatrix.Tλ,λ′,αforα=x,y,zis the exchange-tunneling\ninteraction between the localized spin and the transport\nelectrons, and potential scattering for α= 0. Attend-\ning to the nature of the initial and final electrode, Eq.\n(4) describes four types of exchange interaction, two of\nwhich contribute to the current, the other two are crucial\nto account for the atom spin relaxation.\nB. Justification of the Hamiltonian\nThe phenomenological spin models of Eqs. (3) and (4)\ncapture most of the experimental results, as we show be-\nlow. These models imply that that the magnetic atom is\nin a well defined charge state except for classically for-\nbidden fluctuations that enable tunneling from the tip to\nthe surface. Fig. 1(b) shows a typical level alignment in\nwhich the spin model can be applied. The basic condi-\ntion is therefore, that the chemical potential of the elec-\ntrodes must be far enough from the chemical potential of\nthecentral-quantizedregion. Inthis way,chargeaddition\nand chargeremovalareclassicallyforbidden. Although it\nis outside the scope of this work, we claim that the quan-\ntum charge fluctuations that give rise to spin dependent\ntunneling are due to inelastic cotunneling. In the case of\nspin 1/2, the equivalence between the spin model, origi-\nnally proposed by Appelbaum28and a single site Ander-\nsonmodelwasrigorouslyshownbyAnderson32generaliz-\ning the Schrieffer and Wolff transformation33to the case\nof a single site coupled to two reservoirs. Within this pic-\nture, the atomic spin is exchanged coupled to the trans-\nportelectronsandthe magnitudeoftheexchangeisgiven\nbyTλ,λ′,α=x,y,z≃δ−1VλVλ′, whereVkηis the hybridiza-\ntion between the Anderson site and the single particle\nstatekin the electrode η, andδis the energy difference\nbetween the Anderson level and the electrode Fermi en-\nergy. ThisisthesocalledKineticexchange. Importantly,both electrode conserving and electrode non-conserving\nprocesses are included and their strengths are not inde-\npendent, since they both depend on hopping matrix ele-\nmentsVλbetween the localized orbital in the atom and\nthe extended orbitals in either the tip or the sample. In-\nterestingly,theSchriefferandWolfftransformation33also\nyields a spin-independent tunneling term which would\nyield the α= 0 contribution in Eq. (4), and it corre-\nsponds to the elastic tunneling contribution. Within this\nAnderson-Kondo picture, the strength of the ( α= 0)\nelastic channel and that of the spin dependent channel\n(α=x,y,z) are comparable and, in the so called sym-\nmetric case, the elastic term vanishes identically. Thus,\nfor spin 1 /2 case, this picture can account for the large\nstrength of the inelastic signal. The generalization to\nhigher spin case, relevant for the experiments,9–12,16,27\nwill be published elsewhere.34\nKeeping these considerations in mind, and follow-\ning Anderson,32we assume that Hamiltonian (4) arises\nfrom kinetic exchange. The momentum dependence of\nTλ,λ′,α(i) can have important consequences in the con-\nductance profile48in an energy scale of eV, but it can be\nsafely neglected in IETS. We thus parametrize\nTη,η′,α(i) =vη(i)vη′(i)Tα, (5)\nwherevS(i) andvT(i) aredimensionless factorsthat scale\nas the surface-adatom and tip-adatom hopping integrals.\nBecause kinetic exchange is spin rotational invariant we\nhaveTx=Ty=Tz≡ |T |. Thus, Eq. (4) implies that\nthe spin-assisted tunneling and the atomic spin relax-\nations are both due to kinetic exchange, and Eq. (5) im-\nplies that their amplitudes depend on the tip-atom and\nsurface-atom tunneling matrix elements.\nC. Rates and Master equation\nOur primary goal is to study transport and spin dy-\nnamics. This is done considering Vas a perturbation to\nthe otherwise uncoupled magnetic atom and transport\nelectrons. The quantum spin dynamics is described by\nmeans of a master equation for the diagonal elements of\nthe density matrix, PM, described in the basis of eigen-\nstates|M∝angbracketrightofHSpin. The master equation is derived us-\ningthe standardsystemplusreservoirtechnique,49where\nthe transport electrons act as a reservoir for the atomic\nspin(s). The master equation49reads\ndPM\ndt=/summationdisplay\nMPM′WM′,M−PM/summationdisplay\nM′WM,M′,(6)\nwhereWM,M′arethetransitionratesbetweenthe atomic\nspin state MandM′. These rates can be written as\nWM,M′=/summationtext\nη,η′Wη→η′\nM,M′, whereWη→η′\nM,M′are the scattering\nrates from an atomic spin state MtoM′in which a\nquasiparticle electron goes from electrode ηtoη′as a4\nresult of exchange process. They are given by:\nWη→η′\nM,M′=/summationdisplay\nkk′,σσ′Γη→η′\nkσM,k′σ′M′fη(ǫkσ)[1−fη′(ǫk′σ′)] (7)\nwherefη(ǫ) = 1/(1+exp[ −β(ǫ−µη)]) is the occupation\nprobability in electrode ηfor electrons in equilibrium at\nchemical potential µηand temperature T= 1/(kBβ).\nΓη→η′\nkσM,k′σ′M′istherateatwhichanelectroninlead ηwith\nwavenumber kand spin σis scattered into a lead η′with\nwavenumber k′and spin σ′, with the impurity spin un-\ndergoing a transition between states MandM′. Quan-\ntum rates Γ’s are calculated at the lowest order in the\nelectrode-chain coupling using Fermi Golden rule with\nthe perturbation given by V(see appendix A for details):\nΓη→η′\nkσM,k′σ′M′=2π\n/planckover2pi1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay\nα,iTαvη(i)vη′(i)SM,M′\nα(i)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n×δ(ǫση(k)+EM−ǫσ′η′(k′)−EM′),(8)\nwhere we have defined the matrix elements\nSM,M′\nα(i) =∝angbracketleftM|Sα(i)|M′∝angbracketright. (9)\nThe rates in Eq. (7) describe 3 types of processes:\n1. Elastic processes, Wη→η′\nM,M, in which the state of\natomic spin remains unchanged and a transport\nelectronistransferedfromoneelectrodetoanother.\nThese processes are responsible for the elastic cur-\nrent and have no effect on the spin dynamics. The\nrates of the elastic processes scale with v2\nTv2\nST2\n0.\n2. Spin transitions Wη→η\nM,M′. In these, a spin transition\nin the atomic spin is produced due to the creation\nor annihilation of an electron hole pair either in\nthe tip or in the surface. These processes do not\ncontribute to the current. At very small tempera-\nture, the fastest process of this type is atomic spin\nrelaxation: a spin transition from an excited state\nEMtoalowerenergystate EM′whichresultsinthe\nexcitationofanelectronhole pairin oneofthe elec-\ntrodes. This spin relaxation process is very similar\nto the nuclear spin relaxation due to hyperfine cou-\npling to conduction electrons in metals and to Mn\nspin relaxation in diluted magnetic semiconductors\ndue to itinerant carriers6. At zero bias these pro-\ncessesdominate the atomic spin relaxationtime T1.\nThe rates of the spin transition processes scale like\nv4\nTT2andv4\nST2.\n3. Spin flip assisted tunneling Wη→η′\nM,M′. In these pro-\ncesses, which contribute both to inelastic current\nand to the dynamics of the atomic spin, a trans-\nport electron goes from electrode ηtoη′inducing\na spin transition from state MtoM′. The rates of\nthe spin flip assisted tunneling processes scale like\nv2\nTv2\nST2.The steady state solutions of Eq.(6) depend, in gen-\neral, on the Hamiltonian parameters, the temperature\nand the bias voltage. We refer to the steady state so-\nlutions as PM(V). At zero bias, the steady state solu-\ntions are those of thermal equilibrium. At finite bias, the\nPM(V)candepartsignificantlyfromequilibriumdepend-\ningontherelativeefficiency ofthetransportassistedspin\nexcitations and the spin relaxation. Eq. (6) does not in-\nclude spin coherences. This approximation is good if the\nspin decoherence is faster than spin relaxation, which is\nknown to be the case due to hyperfine coupling50in Mn\natom. However, future work should address this point\nmore carefully, in particular when the magnetization of\nthe tip/vector nis not parallel to the single ion easy axis.\nD. Relevant parameters\nThe behaviour of the system is characterized by the\nrates in Eq.(7), which depend on a number of physical\nquantities like the temperature, the bias voltage V, den-\nsity of states at the Fermi Energy of tip and surface, ρT\nandρS, the tip-atom vT(i) and surface-atom vS(i) hop-\npings, the spin independent T0and spin dependent T\ncouplings. Inthisworkweattempttogrouptheunknown\nparameters either in terms of dimensionless numbers or\nas experimentally accessible quantities. For that matter,\nwe define the zero bias elastic conductance\ng0≡π2\n4G0ρTρS|T0|2χ2, (10)\nwhereG0= 2e2/his the quantum of conductance and\nχ=/summationdisplay\nivT(i)vS(i) (11)\nis a parameter that quantifies the tip-surface transmis-\nsion through the magnetic atoms. The density of states\nat the Fermi energy for spin σin the electrode ηare de-\nnoted by ρησ. We define the spin assisted conductance\ngsas\ngS=ζ2g0 (12)\nwhere\nζ=T\nT0. (13)\nis the ratio of the spin-flip assisted and elastic tunnel\nmatrix elements. We shall use spin polarization of the\ntip, defined as\nPT=ρT↑−ρT↓\nρT↑+ρT↓. (14)\nAnother important parameter is the ratio\nr(i)≡vT(i)\nvS(i), (15)5\nwhich decreases as the tip is retracted from the surface.\nIn most instances, we shall have r(i)<1. As a gen-\neral rule, the processes that drive the magnetic adatom\nout of equilibrium are proportional to v2\nTv2\nSwhereas the\nprocessesthat cool the spin down(if kbT <|eV|) arepro-\nportional to v4\nT+v4\nS. Thus, the non-equilibrium effects\nare higher as R ≡r2/(1+r2) increases. Therefore, cur-\nrent will be increased without changing the applied bias\nV. In contrast, the inelastic ratio ζand the magnitude\nof the tip polarization PTare not so easy to control.\nE. Current\nThe calculation of the rates for a tunnel event in which\na transport electron goes from one electrode to the other,\ninducing a spin transition between states MandM′\n(whereMcould be equal to M′in the elastic channel),\npermits to obtain an expression for the current in terms\nof the steady state solutions of the master equation\nIS→T=e/summationdisplay\nMM′PM(V)/parenleftbig\nWS→T\nM,M′−WT→S\nM,M′/parenrightbig\n,(16)\nwhereWη→η′\nM,M′are the scattering rates from state Mto\nM′induced by interaction with a quasiparticle which is\ninitiallyinreservoir ηandendsupin η′, giveninequation\n(7). We adopt the convention that positive bias voltage\nV >0 means electrons flowing from tip to surface , see\nFig. 1(b). Thus, we have:\neV=µS−µT (17)\nwithethe value of the electron charge with its sign.\n1. Current for non-magnetic tips\nThe expression for the current in the case on non-\nmagnetic tip and substrate can be written as the sum\nof two terms, elastic and inelastic, I=I0+IINgiven by\nthe expressions:22\nI0=g0V, (18)\nwhereg0is given by Eq.(10) and\nIIN=gS\nG0/summationdisplay\nM,M′,ai−(∆M,M′+eV)/vextendsingle/vextendsingle/vextendsingleSM,M′\na,TS/vextendsingle/vextendsingle/vextendsingle2\nPM(V)(19)\nwith\nSM,M′\na,ηη′≡1\nχ/summationdisplay\nivη(i)vη′(i)∝angbracketleftM|Sa(i)|M′∝angbracketright.(20)\nHere we have introduced the current associated to a sin-\ngle channel with energy ∆ M,M′=EM−EM′, and bias\nV\ni±(∆+eV) =G0\ne(G(∆+eV)±G(∆−eV)),(21)withG(ω)≡ω/parenleftbig\n1−e−βω/parenrightbig−1. The curve i−, is odd in the\nbias, whereas i+, relevantinthe caseofmagnetictips dis-\ncussed below, is even. In contrast, di−/dVis even and\ndi+/dVis odd. In this non-magnetic tip case, the elas-\ntic current provides no information about the spin state\nwhatsoever. In contrast, the inelastic steps in conduc-\ntance arise from Eq. (19) and permit to extract informa-\ntion about the spin transition energies, ∆ M,M′and spin\nmatrix elements SM,M′\na,TS. The basic effects of the elastic\nandinelastictermsinthe conductancecanbe understood\nin terms of an equivalent electric circuit schematically\nshown in Fig.1(c) and (d). For low enough voltage, the\nonly channels that can conduct current are the elastic\nones, while the inelastic channels remains close. In this\nsituationthe (inelastic) switch isopen. When the voltage\nis increased such as inelastic channels are open (switch\nis closed), these new channels contributes to the current,\nleading to a smaller resistance, see Fig. 1(d).\n2. Current for magnetic tips\nIn the case of a magnetic tip, the current has 3 con-\ntributions, I=I0+IMR+IIN. This result is different\nfrom the non-magnetic case on two counts. First, the\nelastic case has a magnetoresistive term, so that current\nis now proportional to the relative orientation of the av-\nerage adatom spin and the magnetic moment in the tip\n/vector n:\nI0+IMR=g0[1+2ζ∝angbracketleftSz,TS∝angbracketrightPT]V,(22)\nwhere\n∝angbracketleftSz,TS∝angbracketright=/summationdisplay\nM,iPM(V)vT(i)vS(i)∝angbracketleftM|Sa(i)|M∝angbracketright(23)\nis the average magnetization along the zaxis, that we\ntake parallel to the magnetic moment of the tip. As we\nshow both below and in Ref. 25, both PM(V) and the\naverage atom magnetization depend on voltage. Impor-\ntantly, the magnetoresistive contribution to the elastic\ncurrent makes it possible to track changes in the single\natom magnetization experimentally.\nThe second difference with the non-magnetic tip arises\nin the inelastic current, which is now given by the\nexpression:25,29\nIIN=gS\nG0/summationdisplay\nM,M′/bracketleftBig\ni−(∆M,M′+eV)/summationdisplay\na/vextendsingle/vextendsingle/vextendsingleSM,M′\na,TS/vextendsingle/vextendsingle/vextendsingle2\n+PTi+(∆M,M′+eV)Ξxy(M,M′)/bracketrightBig\nPM(V).(24)\nThe new term in the second line involves the matrix ele-\nments\nΞxy(M,M′) = 2Im/bracketleftBig\nSM,M′\nx,TSSM′,M\ny,T,S/bracketrightBig\n.(25)\nAs opposed to the standard inelastic current [Eq. (19)],\nwhich gives rise to steps of equal height for positive and6\nnegative bias, the PTdependent term of the inelastic\nconductance, proportional to i+, yields steps at the exci-\ntation energies of opposite sign as the polarity of the bias\nis reversed. Both the elastic and inelastic term propor-\ntional to PTcan produce a dI/dVwhich is not an even\nfunction of bias.\nIII. NON MAGNETIC TIP\nIn this section analyze the implications of a non-\nequilibrium population distribution when a finite bias\nis applied between tip and surface. These effects will\nbe more relevant when the current through the sys-\ntem increases (by increasing the coupling to the elec-\ntrodes). Next, we will study these effects in three differ-\nent systems: the Mn monomer, Sec. IIIA, the Mn dimer,\nSec. IIIB both deposited on a Cu 2N surface and the iron\nPhthalocianine molecule, FePc, deposited on an oxidized\nCu(1110) surface, Sec. IIIC.\nA. Mn monomer\nLet us consider first the case of a single Mn adatom in\nCu2N, which has been widely studied experimentally9,27\nand theoretically.22,29,37,51–55The spin of the Mn atom\nin this environment is S= 5/2. The parameters of the\nsingle ion spin Hamiltonian have been determined exper-\nimentally to be D=−0.039 meV, E= 0.007 meV,10and\ng= 1.98.9SinceE <<|D|we can limit our qualitative\ndiscussion to the case E= 0, so that the eigenstates of\nHSpinare also eigenstates of Sz(numerical simulations\nwill be done with E= 0.007 meV and do not change\nqualitatively). In the absence of applied magnetic field\nand at temperatures much smaller than the zero field\nsplitting 4 |D|, the equilibrium distribution is such that\nthe two ground states, Sz=±5/2, are equally likely and\nthe average magnetization is zero. At an energy of 4 |D|\nabove the ground state level, we find a couple of degen-\nerate excited states, with Sz=±3/2. Finally, the two\nstates with Sz=±1/2 are found at 6 |D|.\nFrom the experiments, performed at low current,9the\nexperimental dI/dVlineshape is piecewise constant with\ntwo steps symmetrically located eV=±4D. This is ac-\ncounted for by the equilibrium theory.22As we show in\nFig. 2, and also in our previous work,25non-equilibrium\neffects modify the dI/dVlineshape. In particular, the\ndI/dVcurve is not flat after the inelastic step and it has\na small decay for |eV|larger than the inelastic thresh-\nold. This non-equilibrium effect has been already ob-\nserved experimentally.11,27. Using the same theory with\nasmallertip-atomcoupling(smaller vT) resultsin dI/dV\nlineshapes identical to those equilibrium calculations22.\nThe non-monotonic dI/dVcan be explained as fol-\nlows. As the bias goes across the inelastic threshold,\n|eV|= 4|D|, there is a population transfer from the\nground state doublet ( Sz=±5/2) to the first excited-2 -1 0 1 2V (mV)11,11,21,3G/g0\n-2 -1 0 1 2\nV (mV)00,20,4\nPM(a) (b)\n±5/2\n±3/2\n±1/2\nFIG. 2: (Color online) (a) dI/dV, in units of g0, for single\nMn in Cu 2N surface probed with a non-polarized tip for a\nfixed temperature kBT= 0.5K andB= 0.(b)Steady state\npopulations of each eigenstate versus applied bias. Here ζ=\n1,vS= 1 and vT= 1.\n0.1 0.2 0.3 0.4\n0 0.5 1 1.5 2\nV (mV)0 0.5 1 1.5 2\nV (mV)10−310−210−11001011/T (ns−1)\n1/Tq\n1/T11/T11/Tq(a) (b)\nFIG. 3: Inverse of the relaxation time of the spin state Sz=\n+5/2 (black dashed lines) and charging time T−1\nq≡IIN/e\n(blue solid lines), versus the applied bias for two different\ncouplings with the tip: (a) vT= 0.1, and (b) vT= 1. In both\ncases,T′= 0.5K,vS= 1 and ζ= 1.\nstate doublet Sz=±3/2. This can bee seen in Fig. 2(b).\nAs soon as the population transfer to the first excited\ndoublet takes place, a second inelastic channel opens:\nthe transition from the first to the second excited state\ndoublet ( Sz=±1/2, whose energy is 2 |D|, smaller than\nthe first step). It turns out the intensity of the primary\ninelastic step (∆ = 4 |D|), given by the matrix element\n|∝angbracketleft±5/2|S±|∓3/2∝angbracketright|2, islargerthantheintensityofthesec-\nondary transition (∆ = 2 |D|). Thus, the depletion of the\nprimary transition in favor of the secondary one results\nin a decrease of the conductance. In the case of FePc\nmolecules, discussed below, the secondary transition is\nstronger than the first one, resulting in an increase of the\nconductance after the first step.\nThenon-equilibrium occupationscan be understoodas\nthe balance between two driving forces. Spin-flip assisted\ntunneling events heat the atomic spin, delivering energy7\n0 5 10\nBz (T)0246(E−EG)/J\nS=0S=1S=2S=3\nδ1δ2δ3\nFIG. 4: (a) Energy spectra corresponding to Hamiltonian (3)\nfor a Mn dimer over a Cu 2N surface versus applied magnetic\nfield. Spectrum is referred to the ground state energy and\ngiven in units of the exchange coupling ( J= 5.9 meV). The\nmagnetic field is applied in the surface plane forming a 55\nangle with the Cu-N direction.\nof the order of eVat a pace set by the inelastic current.\nThe steadystate isreachedwhen the heating poweris ex-\nactly compensated by dissipation. The later occurs via\natomic spin relaxation due to exchange coupling to the\ntip and surface electrons. This process is enabled even\nat zero bias. Interestingly, the steady state occupations\ncan differ enormously from the zero bias thermal equi-\nlibrium. At eV= 2meV, the occupation of the ground\nstate doublet is half of the one in equilibrium and barely\ntwice the one of the higher energy spin levels, which are\nalmost empty at zero bias.\nThe inverse of the lifetimes of the two competing pro-\ncesses are shown in Fig. 3. There we show the relaxation\nrate 1/T1of a magnetic spin state, i.e. Sz= +5/2, as a\nfunction of V. When the tip is fully decoupled (no cur-\nrent through the system, R= 0), the spin relaxes in a\ntime scale T1which is independent of the applied bias.\nFor a small coupling, Fig. 3(a), the relaxation rate in-\ncreases by several orders of magnitude when the bias is\nincreased. This effect is even more dramatic when the\nratiorapproaches 1 [ R= 1/2, see Fig. 3(b)]. In the\nweak coupling curve we can easily see the crossover from\nthe equilibrium regime at low bias, where T1<< Tq, to\nthe non-equilibrium regime, for which T1>> Tq. In the\nother case the crossover occurs at a much lower voltage.\nTo plot these curves we take a zero bias conductance\nσ0= 0.75µS for the vT= 1 case.−4 −2 0 2 4\neV/J0.81.21.60246G/g00510(a)\n(b)\n(c)δ1δ2δ3\nFIG. 5: dI/dVcurve for the Mn dimer over a Cu 2N surface\nproved with a non-polarized tip. Each panel corresponds to a\npair of{vT(1),vT(2)}values: (a) {2,1.2}, (b){0.4,0.1}and\n(d){0.01,0.001}.T= 0.6K,ζ= 1,vS(i) = 0.25.\nB. Dimer\n1. Non equilibrium effects\nFrom the discussion above, the non-monotonic line-\nshape observed for the Mn monomer is related to non-\nequilibrium effects. Interestingly, correlation Kondo-like\neffects could also modify the lineshape31. Given the fact\nthat Kondo effect occurs in the case of a Cobalt atom\ndeposited in the same surface,11this type of effect can\nnot be ruled out in the Mn monomer. In contrast, the\nMn dimer has a S= 0 ground state9,22,53and provides\nan ideal system to test the non-equilibrium physics27.\nThe Mn dimer was studied experimentally under low\ncurrent conditions by Hirjibehedin et al.9and, more re-\ncently, under high current conditions by Loth et al.27\nThey have observed a dramatic modification of the line-\nshape, which can be accounted for by our theory, as we\nshow here. The Mn-Mn exchange interaction in this sys-\ntem is antiferromagnetic. The fitting9of the experimen-\ntal results to the Hamiltonian model, Eq. (3), gives a\nJ1,2=J= 5.9, while D, Eandgare kept as for the\nmonomer.\nFig.4showsthelowestenergyspectraoftheMndimer.\nSinceJ >>|D|,E, the total spin Sis a good quantum\nnumber at zero order in |D|/J. Thus, the ground state\nisS= 0, the first excited state S= 1 and energy J, the\nsecondS= 2 and energy 3 Jand the third S= 3 and\nenergy 6J, all energies measured with respect to that of\nthe ground state . The 2 S+1 degeneracy of the S >0\nmultiplets is weakly lifted by the small anisotropy terms\nDandE. The allowed transitions induced by the ex-\nchange coupling (4), when the tip is more coupled to one\nof the two atoms, satisfy ∆ S=±1. The lowest energy\ntransitions are marked in Fig. 4 with vertical arrows at8\nenergies δi, withi= 1,2,3.\nThe experimental results of the IETS show very dif-\nferent profiles as the current through the system is\nchanged.27For low currents only the transition at energy\nδ1≈Jis observed and flat plateaus appear in the dI/dV\nspectra before and after the inelastic step. This pri-\nmary step corresponds to the transition from the S= 0\nground state to the first excited state S= 1. As the\ncurrent is increased, by reducing the tip-atom distance,\nadditional steps appear at higher energies, correspond-\ning to the transitions between the S= 1 and S= 2,\nand theS= 2 and S= 3 states, with energies δ2≈2J\nandδ3≈3J. In addition, the dI/dVline shapes are\nnot flat away from the steps either. These results are\nreproduced by our non-equilibrium theory. Fig. 5 shows\nthe theoretical dI/dVcurve for three different couplings\nwiththetip. Whenthetipisweaklycoupledtothechain,\nFig. 5(a), the step correspondingto the S= 0→S= 1 is\nclearly visible, while excitations from the S= 1→S= 2\nare quenched since the S= 1 is only slightly populated.\nWhen the coupling vTis increased(higher current), tran-\nsitionsS= 1→S= 2 and S= 2→S= 3 become\npossible for bias |eV|>2Jand|eV|>3Jrespectively (\nsee Fig. 5(b)). The new transitions are possible at high\ncurrent due to a significant current induced occupation\nof the excited states S= 1 and S= 2 in the Mn dimer.\nIn contrast with the Mn monomer, the the excited state\nspin flip transitions energies δ2andδ3are largerthan the\nprimary spin transition, resulting in in new steps in the\nspectra. These experimental results, together with the\ntheoretical interpretation, provide strong evidence of the\ncapability of the STM current to drive the spins of the\nmagnetic adatoms.\n2. The case of symmetric coupling\nWhereas the results above are in very good agreement\nwith the experimental data,27it is worth pointing out\nthat this is only so if we assume that the exchange as-\nsisted tunneling is stronger through one of the atoms.\nHowever, it vanishes identically in the symmetric cou-\npling case, vT(1) =vT(2). In Fig. 6 we plot the height of\nthe inelastic step, given by AS→T=/summationtext\na/summationtext\nM′/vextendsingle/vextendsingle/vextendsingleSG,M′\na,TS/vextendsingle/vextendsingle/vextendsingle2\n,\nas a function of the lateral position of the tip across the\ndimer axis, as modeled by the ratio vT(1)/vT(2). The\ninelastic step cancels identically when the tip is in the\nmiddle. This prediction of the model is at odds with un-\npublished experimental data, which do not show a strong\ndependence of the inelastic current as the tip is moved\nalong the Mn dimer axis. From the theoretical point of\nview, the cancellation of the exchange assisted tunneling\nin the case of the Mn dimer symmetrically coupled to the\ntip arises from the fact that, in this particular case, the\noperator in the transition matrix element (20) is the to-\ntal spin of the dimer, and then the eigenstates of S2and\nSzare also eigenstates of V. As a result, the coupling0 0.2 0.4 0.6 0.8 1\nvT(1)/vT(2)00.10.20.3AT−>S\nFIG. 6: Transition amplitude AS→Tversus the ratio\nvT(1)/vT(2).T= 0.6K,ζ= 1,vS(i) = 0.25.\nHamiltonian is diagonal, and no transitions are possible.\nNotice that this problem is specific of the dimer. In the\ncase of the monomer the observed spin transitions occur\nwithin states with the same S= 5/2. In the case of the\ntrimer and longer chains the tip can not be coupled iden-\ntically to all the atoms and the theory accounts for the\ndata.22\nThereareseveralspininteractionsotherthantheinter-\natomicexchangethatbreakthespinrotationalinvariance\nand could, in principle, solve the problem: the single ion\nanisotropy terms, DandE, the hyperfine coupling with\nthe nuclear spin of the Mn, I= 5/2, and the direct mag-\nnetic dipolar coupling. We have included them in our\ncalculations, but they are much weaker than the domi-\nnant exchange, so that they do not change qualitatively\nthe curve (6). Thus, even in spite of the apparent success\nof the perturbative approach using Eq. (4), this partic-\nular result indicates the presence of additional terms in\nthe Hamiltonian or the need to go beyond lowest order\nin perturbation theory . Further work, going beyond the\nphenomenological theory is under way.34\nC. Magnetic molecules\nAs a final example of our non-equilibrium theory\nwith non-magnetic electrodes, we consider the case of\nIETS through Iron Phthalocyanine(FePc) molecules, de-\nposited on oxidized Cu surface.15FePc are flat organic\nmolecules with C4symmetry with a core made of a single\nFe2+ion surrounded by 4 Nitrogen atoms embedded in\nBenzene groups. In gas phase, the crystal field of the lig-\nands is high enough as to reduce the spin of Fe2+from\nS= 2 (high spin) to S= 1 (intermediate spin). Be-\ncause of the C4symmetry of the gas phase, the single\nspin Hamiltonian of the molecule has E= 0.\nAccording to the IETS data,15the symmetry is re-\nduced when deposited on the oxidized surface. In partic-\nular, two adsorbed states ( αandβ) were experimentally\nobserved with different spin excitations.15In both cases\nthe spin excitations of the FePc could be assigned to S=9\n−10 −5 0 5 10\nV (mV)00.511.52G/g0α−FePc\n−15 −10 −5 0 5 10 15\nV (mV)β−FePc\n(a) (b)\n0T5T10T15T\n0T5T10T15T\nFIG. 7: Differential conductance versus applied bias at zero\nmagnetic field for the α(a) andβ(b) FePc on Cu(110)(2 ×\n1)−Oas a function of applied bias for different magnetic field\napplied perpendicular to the sample surface, as in Ref. 15.\nCurves have been shifted by 0 .5 units for clarity. T= 0.5K,\nPT= 0,vS= 1,vT= 0.2 andζ= 1.\n1buttheanisotropyparameters,determinedfromtheex-\nperimental differential conductance curves, varied in the\ntwo cases. For the α(β) configuration, D=−3.8±0.04\nmeV (D=−6.9±0.04 meV), E= 1.0±0.01 meV\n(E= 2.1±0.04meV) and g= 2.3±0.02(g= 2.4±0.05).\nThe origin of this drastic change in anisotropy deserves\nfurther theory work.\nTheS= 1 single spin model can be solved analytically\n(see for instance appendix in Ref. 56). With D <0,\nandE= 0, the ground state would be the Sz=±1\ndoublet with energy −Dbelow the Sz= 0 excited state.\nAt finite Ethe ground state doublet splits, in bonding\nand anti-bonding combination of the states Sz=±1.\nThe splitting is 2 E. Thus, there are two spin transitions.\nThe low energy one, with ∆ = 2 Eand ∆Sz= 0, and the\nhigh energy one, with energy |D|+Eand ∆Sz=±1.\nThe magnetic field along the zaxis competes with the E\ninduced splitting ofthe groundstate. As Bincreases, the\nEinduced mixing of the Sz=±1 components decreases,\nand so it does the primary transition, which occurs via\n∆Sz= 0 events.\nFig. 7 shows our non-equilibrium theoretical dI/dV\nresultsusingthevaluesof DandEgivenabove. Ourthe-\nory reproduces not only the evolution of the steps with\nthe magnetic field, but also the mild non-equilibrium fea-\ntures reported in Ref. 15. After the first (second) step\nthe conductance has a small positive (negative) slope. In\ncontrast, the equilibrium theory,57yields flat steps. The\nsignofthe non-equilibriumslopesdepends onthe relative\nvalue of the inelastic channel strengths of the primaryand secondary transitions. At eV= 2E, when the first\nexcited state is populated, the secondary transition, with\nenergy|D| −Ebecomes possible, at finite temperature.\nSince this transition between excited states has a larger\nquantum yield, the overall conductance increases. The\nopposite scenario occurs in the second step.\nIV. SPIN POLARIZED TIP\nThe results of the previous section give very strong\nsupport to the notion that tunneling electrons can drive\nthe spin of the magnetic adatoms far from equilibrium.\nSince we have been considering spin unpolarized tunnel-\ning electrons, these non-equilibrium effects can not result\nin a net spin transfer. From the theory standpoint, this\nshould change dramatically in the case of spin-polarized\ntransport electrons. As it was shown in a seminal work\nby Slonczewski,26the back action of transport electrons\non a magnetic moment can be used to rotate the magne-\ntization direction. This effect, known as spin-transfer\ntorque, have been observed in nano-pillars of tens of\nnanometers58down to tiny nanomagnets made of 100\natoms,13but still in the semiclassical domain. In a pre-\nvious paper25we modeled the spin dynamics of a single\nMn atom under the influence of spin polarized current.\nWe found that the if the tip was spin polarized, the spin\npolarized current would result in a net spin magnetiza-\ntionofthe magneticadatomwhoseorientationrelativeto\nthe tip moment would depend on the polarity of the bias,\nquite in agreement with the macroscopic spin transfer\ntorque. In parallel to our work, S. Loth et. al.27demon-\nstrated experimentally the single atom spin transfer.\nIn our work in Ref. 25 the origin of the tip magnetiza-\ntion was ferromagnetic order. In the experiment of Loth\net al., the spin polarized current is achieved by sticking\na single Mn atom into the tip and applying a magnetic\nfieldtofreezeitsspinfluctuations. Theexternalmagnetic\nfield affects also the surface atom. The very different role\nplayed by the Mn in the tip and the Mn in the surface\nunderlines the important role played by spin isolation.\nWhereas it is still possible to model the spin of the Mn\nin the surface as a quantized spin weakly coupled to the\nsurfaceelectrons,thispictureseemstobreakdownforthe\ncase of the Mn in the tip, due to a combination of charge\ntransfer, Kondo coupling and very reduced spin lifetime.\nThus, the Mn in the tip acts as a spin filter for the trans-\nport electrons. Whereas this picture works qualitatively,\nwe believe this issue deserves further work.59\nA. Current induced spin switching\nThe flow of spin polarized current through a single\nmagnetic atom is expected to result in a transfer of a\nnet spin into the atom. In the case of a single or a few\nmagnetic atoms, where time reversal symmetry is not\nspontaneouslybroken at zero magnetic field , the equilib-10\nrium occupation of states with opposite Szis the same,\nresulting in a null average magnetization. Spin polar-\nized current changes this situation via spin-flip inelastic\ntunnel.25The mechanism is the following. The dominant\ninelastic transitions in the case of the Mn monomer are:\n•Spin increasing (SI) transition, for which the Mn\nspin goes from Sz=−5\n2toSz=−3\n2and the trans-\nport electron goes from the high energy electrode\nwith spin ↑to the low energy electrode with spin ↓.\n•Spin decreasing (SD) transition, for which the Mn\nspin goes from Sz= +5\n2toSz= +3\n2and the trans-\nport electron goes from the high energy electrode\nwith spin ↓to the low energy electrode with spin ↑.\nIn the case of spin unpolarized current, these two pro-\ncesses are equally likely and result in the depletion of the\ntwostatesofthegroundstatedoubletshowninFig. 2(b).\nIn the case of a spin polarized tip, the two processes are\nno longer equally likely, resulting in a net spin transfer\nfrom the spin current to the atomic spin. Let us con-\nsider the case where there are more ↓than↑electrons\nin the tip. This means negative tip spin average (i.e.,\nPT<0) and positive tip magnetization. When electrons\ngo from the tip to the surface ( V >0) , theSDprocesses\nare dominant, as a result of which the positive Szstates\nare depleted and a negative ∝angbracketleftSz∝angbracketrightis expected. Thus, we\nexpect that at positive bias (electrons going from tip to\nsurface) the current co-polarizes the spin of the atom.\nWe now consider electrons going from surface to tip.\nSince the density of states of spin ↓electrons is higher,\ntheSIprocess is now more likely than the STone. As\na result, the negative Szstates should be depleted, re-\nsulting in a positive atomic spin. Thus, we expect that\nV <0 (electrons going from surface to tip) the current\ncounter-polarizes the spin of the atom.\nOur simulations confirm this scenario. We only con-\nsider the simplest case in which the tip polarization is\nassumed parallel to the Mn easy axis -perpendicular to\nthe Cu 2N surface. We considerfirst the case of zeromag-\nnetic field. We choose PT<0, because is convenient for\nthe discussion at finite positive field below. In Fig. 8(a)\nwe show the average atomic spin moment along the easy\naxis, as a function of the applied bias. It vanishes at\nzero bias, reflecting the absence of spontaneous time re-\nversal symmetry breaking of such a small system. At\nfinite bias the magnetic moment aligns with that of the\ntip when electrons flow from tip to surface ( V >0), and\ndo exactly the opposite when the electrons flow from the\nsurface to the tip ( V <0). Interestingly, the average\natomic spin is finite even when |eV|<4|D|, the exci-\ntation energy. This is due to the existence of thermally\nexcited quasiparticles. However, the time necessary to\ndrive the spin of the atom increases exponentially when\nfor|eV|<4|D|.25\nThe average atomic spin increases both with the ap-\nplied voltageand the spin polarizationofthe tip, asgiven\nbyPT. The effect is null at PT= 0, as it should, and it−2 −1 0 1 2\nV (mV)−2−1012<Sz>(a)\n−1 −0.5 0 0.5 1\nPT−2−1012<Sz>\n−2 −1 0 1 2\nV (mV)00.51\nPM(b) (c)\n+5/2 −5/2\nFIG. 8: (Color online) (a) Average magnetization < Sz>for\nthe Mn monomer in Cu 2N surface probed with two differently\npolarized tips, PT=−1/3 (red-dashed line) and PT=−1\n(solid-blue line) versus applied bias. (b) “Universal curv e” of\nthe magnetization versus tip polarization (for positive ap plied\nbias). (c) Steady state populations of each eigenstate vers us\napplied bias. Solid lines for energy levels with Sz<0 and\ndashed lines for Sz>0. Here ζ= 0.5,vS= 1,T= 0.5K,\nB= 0. and vT= 0.7.\nis maximal for half-metallic tips PT=±1. For a fixed\ntip polarization the effect saturates at a certain voltage.\nInterestingly, the saturation magnetic moment depends\nonly on the value of PTand is quite independent of tem-\nperature and other parameters in the calculation. We\ndiscuss this universal behavior in Sec. IVC. The non-\nzero atomic spin polarization reflects the bias induced\nbreaking asymmetry of the steady state occupation of\nthe two states of the ground state doublet Sz=±5/2,\nas shown in Fig. 8(c). Notice the striking difference with\nthecaseofequilibrium, forwhichtheoccupationsofthese\ntwo degenerate states are identical. The steady state is\nreached thanks to the competition between bias induced\nspin-transfer and exchange induced spin relaxation dis-\ncussed in the previous section.\nB. Effects of spin polarization on transport\nImportantly, the current induced polarization of the\natomic spin can be detected through its influence on the\nconductance of the system. The simplest effect comes11\nfromthe elasticmagnetoresistance: conductanceislarger\nwhen spin polarization of tip and magnetic atom are par-\nallel. In the case discussed above, this results in a larger\nconductance at large positive bias than at large negative\nbias. At small bias, there are several competing effects.\nFor simplicity let us consider the case of a single mag-\nnetic adatom. We can write the differential conductance\nas\nG=G1+G2+G3+G4 (26)\nwhere the different terms are obtained by deriving I\nin Eqs. (22-24) with respect to bias and are shown in\nFig. 9(b):\nG1=g0(1+2ζ∝angbracketleftSz∝angbracketrightPT),\nG2= 2g0VζPTd∝angbracketleftSz∝angbracketright\ndV,\nG3=gS\nG0/summationdisplay\nM,M′PM(V)/bracketleftBigg/summationdisplay\na/vextendsingle/vextendsingle/vextendsingleSM,M′\na/vextendsingle/vextendsingle/vextendsingle2\n×i′\n−(∆M,M′+eV)+PTΞxy(M,M′)i′\n+(∆M,M′+eV)/bracketrightBigg\n,\nG4=gS\nG0/summationdisplay\nM,M′/bracketleftBigg\ni−(∆M,M′+eV)/summationdisplay\na/vextendsingle/vextendsingle/vextendsingleSM,M′\na/vextendsingle/vextendsingle/vextendsingle2\n+PTi+(∆M,M′+eV)Ξxy(M,M′)/bracketrightBigg\ndPM(V)\ndV,(27)\nwithi′\n±≡di±/dV. Ξxy(M,M′) is defined as in Eq. (25)\nbut with without the weighting factors. G1andG2(G3\nandG4) correspondto the elastic (inelastic) contribution\nof the current. G1gives the dominant magnetoresistive\ncontribution at large bias discussed above. G2gives a\nsmaller contribution associated to the change of the av-\nerage adatom spin as a function of bias. This term is\nresponsible of the non-monotonic decay of the conduc-\ntance after the pronounced change induced by G1. In\nthe extreme case shown in Fig. 9, corresponding to a half\nmetallic tip, G1is the dominant contribution. Finally,\nthe two inelastic contributions, G3andG4peak close to\nthe transition energies ±4|D|.G3corresponds to the in-\nelastic conductance, as if the occupations PM(V) where\nbiasindependent, and G4isthecontributioncomingfrom\nthe fact that PMdo depend on the bias. In turn, both\nG3andG4have twocontributions, one that is present for\nnon-magnetic tip and another one proportionalto the tip\nspin polarization PT.\nExperimentally it might be hard to disentangle G2,\nG3andG4, but not G1which provides a direct way to\nquantify the atomic spin at large bias, denoted by ±V∞:\nG1(+V∞)−G1(−V∞)\nG(V= 0)= 4ζPT∝angbracketleftSz∝angbracketright(+V∞) (28)\nwhereG(V= 0) = g0is the zero bias conductance\nand we have used the fact that, at zero magnetic field,\n∝angbracketleftSz∝angbracketright(+V∞) =−∝angbracketleftSz∝angbracketright(−V∞). Since G1can be the domi-\nnant contribution, replacing G1by the total Gin equa-−1 0 1\nV (mV)00.511.52G/g0\n−1 0 1\nV (mV)00.51(a) (b)\nG1\nG2G3\nG4\nFIG. 9: (Color online) (a) Differential conductance for theM n\nmonomer versus the applied bias for two different tip polar-\nizations: PT=−1/3 (dashed line) and PT=−1 (solid line).\n(b) Each of the contributions to the dI/dVforPT=−1/3:\nG1(thin-black line), G2(thick-blue line), G3(thin-dashed\nline) and G4(thick-green line) versus applied bias. T= 0.5K,\nζ= 0.5 andvT= 0.7. Magnetization direction was fixparallel\nto the easy axis and no magnetic field was applied.\n−20 −10 0 10 20\nV (mV)−2−101<Sz>\nFIG. 10: (Color online) Average magnetization /an}bracketle{tSz/an}bracketri}htfor the\nMn monomer versus applied bias for B= 7T corresponding\nto the low current (blue line) and hight current (black line)\nregimes. The tip magnetization direction was fix parallel to\nthe easy axis and to the applied field. PT=−0.31,T= 0.5K\nandvS= 1.\ntion (28) can give a rough estimate of the quantities in\nthe right hand side of that equation.\n1. Finite magnetic field\nWe now analyze how the magnetic field used in the ex-\nperimental setup27changes the picture discussed above.\nIn these experiments, the tip polarization is achieved by\nattaching a single Mn atom at the tip apex and applying\na very intense magnetic field (7T) perpendicular to the\nsurface. In the experiments with spin polarized tips,2712\n−20 −10 0 10 20\nV (mV)11.11.2G/g0\n−20 −10 0 10 20\nV (mV)00.51(a) (b)\nG1\nG3\nG4\nG2\nFIG. 11: (Color online) (a) Differential conductance for B=\n7T versus applied bias for the low current regime. (b) Each\nof the contributions to the dI/dV:G1(thin-black line), G2\n(thick-blue line), G3(thin-dashed line) and G2(thick-green\nline) versus applied bias. T= 0.5K,ζ= 0.5 andvT= 0.7.\nMore detail in the text.\nthedI/dVcurve changes radically from low current to\nhigh current . In both cases, the dI/dVcurves are not\neven with respect to bias, as expected from the discus-\nsion in the previous section. However, the lineshapes\ndiffer significantly from the ones described in our pre-\nvious work25and in the previous section (Fig. 9). The\norigin of this discrepancy can be traced back to the ef-\nfect of the magnetic field on the surface adatom, absent\nin our previouscalculation. Since kbT << gµ BB, the ap-\nplied field already polarizes completely the atomic spin\nat zero bias. In particular, this means that the occupa-\ntion of the atomic spin state Sz=−5/2 is very close to 1\nand at zero bias the average spin of the adatom is finite,\nnegative, and parallel to the tip spin polarization, which\nmaximizes theG1term in the conductance.\nWhen bias is applied, the occupation of the ground\nstateSz=−5/2 is depleted, which degrades the elastic\ncontributions to the conductance. This is reflected in the\nevolution of the averageatomic spin as a function of bias,\nshown in Fig. 10 for two cases: low current ( vT= 0.08,\nζ= 0.44) and high current( vT= 0.7,ζ= 0.5). The val-\nues ofvTandζ, as well as the value of PT= 0.31, where\nchosen to reproduce the experimental conductance data\nof Lothet al.27. It is apparent that in both cases the de-\npletion of the average spin is larger for negative bias, for\nwhich transport electrons tend to counter-polarize the\ntip, than for positive bias, for which the depletion can\nbe interpreted as non-equilibrium heating of the atomic\nspin. In the large current case, the current is able to re-\nverse the average atomic spin from the equilibrium −2.5\nat zero bias to +1 at -20 mV. The corresponding change\nintheelasticcomponentsofconductance, G1andG2, due\nto spin contrast, is shown in figures (11b) for low current\nand in (12b) for high current. In the high current case,\nthese contribution are dominant, and explain the exper-\nimental observation that the conductance is smaller at\nnegative voltage. Thus, in the high current case the fact\nthatG(+20)< G(−20) can be linked to the reversal of\nthe atomic spin at negative bias and the resulting reduc-−20 −10 0 10 20\nV (mV)0.811.2G/g0\n−20 −10 0 10 20\nV (mV)00.511.5\n(a) (b)\nG1\nG3G2\nG4\nFIG. 12: (Color online) (a) Differential conductance for B=\n7T versus applied bias for the hight current regime. (b) Each\nof the contributions to the dI/dV:G1(thin-black line), G2\n(thick-blue line), G3(thin-dashed line) and G2(thick-green\nline) versus applied bias. T= 0.5K,ζ= 0.5 andvT= 0.7.\nMore detail in the text.\ntionof the conductance, compared to zero bias.\nIn the low current case the small changes in the atomic\nspin make G1andG2, the elastic magnetoresistive con-\ntributions, quite independent of bias, so that change as a\nfunction bias so that they simply offset the total conduc-\ntance. This results in a conductance which is minimal\nat zero bias and the larger at large negative bias than\nat large positive bias. As opposed to the high current\ncase, where the asymmetry is dominated by G1andG2,\nreflecting the reversal of the average atomic spin, in the\nlow current case the asymmetry comes mostly from the\nasymmetry in the inelastic step G3, associated to the\nterm proportional to PT.\nC. Universal magnetization profile\nIn Fig. 8(c) we plot the saturation magnetization,\nreached after a large enough bias is applied, as a function\noftip polarization. We have verifiedthat this curveturns\nout to be independent of all transport parameters, ζ,r,\nvη... and it depends only on the spin of the magnetic\natomSand anisotropy parameters DandE. In fact, it\ndoes not depends either on the temperature, as long as\nour approach of neglecting the phonon contribution re-\nmains valid. When combined with eq. (28), this could\nbe used to determine the tip polarization.\nThe universality of the saturation atomic magnetiza-\ntion comes from the fact that, at large bias, i±(∆M,M′+\neV)≈ ±eVso that the equation (A3) for the rates can\nbe simplified to:\nWM,M′∝V/bracketleftbigg\n|SM,M′\nz|2+ρT↑\nρT|SM,M′\n−|2+ρT↓\nρT|SM,M′\n+|2/bracketrightbigg\n.\n(29)\nMaking use of Eq. (29), the master equation for the13\noccupation of the spin states in steady state reads\n′/summationdisplay\nM′/bracketleftbigg\n|SM,M′\nz|2+PT+1\n2|SM,M′\n−|2+1−PT\n2|SM,M′\n+|2/bracketrightbigg\n×(PM′−PM) = 0,∀M, (30)\nwhere the prime indicates sum is done over M′∝negationslash=M.\nEq. (30) shows that, in the large bias limit, the atomic\nspin steady state occupations PM, and consequently the\naverage magnetization ∝angbracketleftSz∝angbracketright, depend only on the matrix\nelements of the spin operators, and the polarization of\nthe tip, and do not depend on the coupling strength to\nthe tip and surface.\nV. SUMMARY AND CONCLUSIONS\nWe have studied the mutual influence of non-\nequilibrium transport electrons and the spin of one and\ntwo magnetic adatoms in STM configuration. Our re-\nsults indicate that non-equilibrium effects are essen-\ntial to understand present IETS STM spectra of mag-\nnetic adatoms. Our theory is able to describe correctly\nthe experimental observations of IETS on single Mn\natoms, both with non-magnetic10and magnetic27tips,\non Mn dimers at low and high current27, and on FePc\nmolecules15. Our theory is based on a phenomenological\nspin model to describe both the atomic spin states and\ntheir coupling to the transport electrons. Current is cal-\nculated to lowest order in the tunneling Hamiltonian and\ndepends on the occupation of the spin states PM. The\nspin dynamics is described with a master equation for\nthePMthat includes the effect of carrier induced spin\nrelaxation and the spin pumping due inelastic spin-flip\nassisted tunneling.\nWhen the time elapsed between inelastic spin flip\nevents,Tqis much longer than the atomic spin relax-\nation time, T1, the atomic spin remains in equilibrium\nandPMdoes not depend on bias. In that limit, the\ndifferential conductance dI/dVis piecewise constant22\nexcept for the steps when the bias voltage matches the\nenergy of the spin excitations. When the Tqis compara-\nble orsmallerthan T1, the occupationsofthe atomicspin\nstates are driven awayfrom equilibrium and they depend\non the bias voltage. Whereas non-monotonic dI/dVhad\nbeen already observed experimentally, the recent results\nreported by Loth et al.27have confirmed this scenario by\ncontrolling the tip-adatom distance. In addition, the use\nof spin-polarized tips amplifies the changes in the dI/dV\ncurves as the conductance is increased.\nThe results of Loth et al.27also indicate that, in the\ncase of magnetic tips, the orientation of the average\natomic spin can be switched at will from parallel ( V >0)\nto antiparallel ( V <0) with respect to the magnetic\ntip. The control of the spin of a single atom and a sin-\ngle magnetic molecule had been predicted by theory22,46\nThe control of atomic spin with non-equilibrium spin\npolarized carriers is similar to that obtained by optical\npumping50.The main conclusions are the following:\n1. The dynamics of the atomic spin under the influ-\nence of tunneling electrons is governed by two in-\ntrinsic time scales, the inelastic transport time Tq\nand the atomic spin relaxation time T1. When the\ncurrentinducedspinflipsoccurmoreoftenthanthe\ntime it takes to the atomic spin to relax, i.e., when\nTq< T1, non-equilibrium effects build-up. This\nmakes the occupation of the spin states different\nfrom that of equilibrium.\n2. The conductance lineshape is sensitive to the oc-\ncupation of the atomic spin states. This might be\nused toperformtransport-detectedsingleatomres-\nonance experiments. Our calculationsindicate that\nnon-equilibriumeffectshavebeenobservedinsingle\nMn atoms, in Mn dimers and in FePc molecules.\n3. A rough estimate of the quality factor of the spin\nexcitation with energy ∆ = /planckover2pi1ω, defined as Q=\nωT1, can be obtained from the transport experi-\nments. We assume that the current at which the\nlow temperature conductance line shape starts to\ndeviate from a piecewise constant function yields\nTq≃T1. Let ∆GinandVbe the height and bias\nof the primary inelastic step. Then, the inelastic\ncurrent is IIN= ∆G0V=e/Tq. Then we get\nωT1≃G0\n∆G. (31)\n4. Spin polarized STM can be used to magnetize the\natomicspinbothparallelorantiparalleltothemag-\nnetic tip moment. When electrons tunnel through\nthe magnetic atom from the magnetic tip to the\nsurface, the atomic is magnetized parallel to the\ntip. Reversing the bias results in an opposite spin\npolarization.\n5. The bias induced adatom spin polarization results\nin asymmetric conductance lineshapes due, in most\npart, to the dependence of conductance on the rel-\native orientation of the adatom and tip magnetiza-\ntions.\n6. The saturation atomic spin magnetization, ob-\ntained at large bias, is only a function of the impu-\nrity spin, the anisotropy parameters and the polar-\nization of the tip.\nFuture work should address open problems, like the\norigin of the spin assisted tunneling Hamiltonian for spin\nlargerthan1 /2,theeffectofatomicspincoherence,which\nwe haveneglected in the master equation(6) and the fact\nthat the observed inelastic steps in the Mn dimer do not\ndepend on the lateral position of the tip, in contrast with\nour theory.\nACKNOWLEDGMENT14\nWe acknowledge fruitful discussions with A. S. N´ u˜ nez,\nJ. J. Palacios, C. F. Hirjibehedin and S. Loth.\nThis work has been financially supported by MEC-\nSpain (Grants JCI-2008-01885,MAT07-67845and CON-\nSOLIDER CSD2007-00010).\nAppendix A: Equation for the rates\nInthisappendixwederivethegeneralexpressionofthe\ntransition rates. Applying the Fermi Golden rule using\nthe tunneling Hamiltonian (4) as perturbation, one gets\nΓη→η′\nkσM,k′σ′M′=2π\n/planckover2pi1δ(ǫση(k)+EM−ǫσ′η′(k′)−EM′)\n×/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay\nα,σσ′Tατ(α)\nσσ′\n2SM,M′\nα,ηη′/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n. (A1)\nThe modulus square in Eq. (A1) can be expanded to\nobtain\n/summationdisplay\nα,β,σσ′TαTβτ(α)\nσσ′\n2τ(β)\nσ′σ\n2SM,M′\nα,ηη′SM′,M\nβ,ηη′.(A2)\nConsidering the explicit form of the Pauli matrix ele-\nments, the sum over σandσ′can be done, with just a\nfew non-zero contributions. Using the definition of the\ntransition rates Wη→η′\nM,M′, Eq. (7), we obtain after some\nalgebra\nWη→η′\nM,M′= =2πT2\n0χ2\n/planckover2pi1G(∆M,M′+µη−µη′)Σηη′\nM,M′,(A3)\nwhere\nΣηη′\nM,M′=1\n4δMM′/parenleftBigg\nR+(ηη′)+2ζR−(ηη′)/summationdisplay\naSM,M\na,ηη′/parenrightBigg\n+ζ2/bracketleftBigg/vextendsingle/vextendsingle/vextendsingleSM,M′\n+,ηη′/vextendsingle/vextendsingle/vextendsingle2\nρη↓ρη′↑+/vextendsingle/vextendsingle/vextendsingleSM,M′\n−,ηη′/vextendsingle/vextendsingle/vextendsingle2\nρη↑ρη′↓+R+(ηη′)/vextendsingle/vextendsingle/vextendsingleSM,M′\nz,ηη′/vextendsingle/vextendsingle/vextendsingle2/bracketrightBigg\n. (A4)\nHere we have introduced R±(ηη′) =ρη↑ρη′↑±ρη↓ρη′↓\nand the operators S±=Sx±iSy. Notice that expression\n(A4) is valid for all states M, M′, in contrast to Eq. (5)\nof Ref. 25 where only the inelastic matrix elements were\nexplicitly written.\nAppendix B: Equations for the current\nHere we shall derive expressions (18) and (19). Let\nus start with the expression for the current, Eq. (16).\nDividing the contribution into its elastic and inelastic\npart, with the help of Eq. (A3) and (A4), we can write\nI0+IMR=2πeT2\n0\n/planckover2pi1χ2ρSρT\n8[1+2ζ∝angbracketleftSz,T,S∝angbracketrightPT]\n×(G(eV)+G(−eV)), (B1)\nor, using the definition of g0andi+\nI0+IMR=g0\ne[1+2ζ∝angbracketleftSz,T,S∝angbracketrightPT]i−(eV),\nwhich is the result of Eq. (18). For the inelastic contri-\nbution (M∝negationslash=M′), the difference WS→T\nMM′−WT→S\nM,M′can be\nwritten with the help of Eqs. (A3) and (A4) as\nWT→S\nMM′−WS→T\nM,M′=ζ22πχ2ρSρTT2\n0\n8/planckover2pi1/parenleftBig\ni−(∆M,M′+eV)\n×/summationdisplay\na/vextendsingle/vextendsingle/vextendsingleSM,M′\na,T,S/vextendsingle/vextendsingle/vextendsingle2\n+PTi+(∆M,M′+eV)\n×2Im/bracketleftBig\nSM,M′\nx,T,SSM′,M\ny,T,S/bracketrightBig/parenrightBig\n.\nUsing the definitions of gsandΞxy(M,M′), expression\n(19) is recovered.\n1B. Kane, Nature 393, 133 (1998).\n2F. Jelezko, T. Gaebel, I. Popa, M. Domhan, A. Gruber,\nand J. Wrachtrup, Phys. Rev. Lett. 93, 130501 (2004).\n3L. Childress, M. V. Gurudev Dutt, J. M. Taylor, A. S.\nZibrov, F.Jelezko, J. Wrachtrup,P.R.Hemmer, andM.D.\nLukin, Science 314, 281 (2006).\n4R. Hanson, V. V. Dobrovitski, A. E. Feiguin, O. Gywat,\nand D. D. Awschalom, Science 320, 352 (2008).\n5P. Neumann, N. Mizuochi, F. Rempp, P. Hemmer,\nH. Watanabe, S. Yamasaki, V. Jacques, T. Gaebel,\nF. Jelezko, and J. Wrachtrup, Science 320, 1326 (2008).\n6L. Besombes, Y. Leger, J. Bernos, H. Boukari, H. Mari-\nette, J. P. Poizat, T. Clement, J. Fern´ andez-Rossier, and\nR. Aguado, Phys. Rev. B 78, 125324 (2008).\n7Y. L´ eger, L. Besombes, J. Fern´ andez-Rossier, L. Main-gault, andH.Mariette, Phys.Rev.Lett. 97, 107401 (2006).\n8A. Kudelski, A. Lemaˆ ıtre, A. Miard, P. Voisin, T. C. M.\nGraham, R. J. Warburton, and O. Krebs, Phys. Rev. Lett.\n99, 247209 (2007).\n9C. F. Hirjibehedin, C. P. Lutz, and A. J. Heinrich, Science\n312, 1021 (2006).\n10C. Hirjibehedin, C.-Y. Lin, A. Otte, M. Ternes, C. P. Lutz,\nB. A. Jones, and A. J. Heinrich, Science 317, 1199 (2007).\n11A. F. Otte, M. Ternes, K. von Bergmann, S. Loth,\nH. Brune, C. P. Lutz, C. F. Hirjibehedin, , and A. J. Hein-\nrich, Nature Physics 4, 847 (2008).\n12X. Chen, Y.-S. Fu, S.-H. Ji, T. Zhang, P. Cheng, X.-C. Ma,\nX.-L. Zou, W.-H. Duan, J.-F. Jia, and Q.-K. Xue, Phys.\nRev. Lett. 101, 197208 (2008).\n13S. Krause, L. Berbil-Bautista, G. Herzog, M. Bode, and15\nR. Wiesendanger, Science 317, 1537 (2007).\n14F. Meier, L. Zhou, J. Wiebe, andR.Wiesendanger, Science\n320, 82 (2008).\n15N. Tsukahara, K. Noto, M. Ohara, S. Shiraki, N. Takagi,\nY. Takata, J. Miyawaki, M. Taguchi, A. Chainani, S. Shin,\net al., Phys. Rev. Lett. 102, 167203 (2009).\n16A. J. Heinrich, J. A. Gupta, C. P. Lutz, and D. M. Eigler,\nScience306, 466 (2004).\n17R. Wiesendanger, Rev. Mod. Phys. 81, 1495 (2009).\n18Y. Fu, T. Zhang, S. H. Ji, X. Chen, X. C. Ma, J. F. Jia,\nand Q. K. Xue, Phys. Rev. Lett. 103, 257202 (2009).\n19H. Brune and P. Gambardella, Surface Science 603, 1812\n(2009).\n20L. Zhou, J. Wiebe, S. Lounis, E. Vedmedenko, F. Meier,\nS. Bl¨ ugel, P. H. Dederichs, and R. Wiesendanger, Nature\nPhys.6, 187 (2010).\n21J. Slonczewski, Phys. Rev. B 39, 6996 (1989).\n22J. Fern´ andez-Rossier, Phys. Rev.Lett. 102, 256802 (2009).\n23C. M. Canali and A. H. MacDonald, Phys. Rev. Lett. 85,\n5623 (2000).\n24T. O. Strandberg, C. M. Canali, and A. H. MacDonald,\nNature Materials 6, 648 (2007).\n25F. Delgado, J. J. Palacios, and J. Fern´ andez-Rossier, Phys .\nRev. Lett. 104, 026601 (2010).\n26J. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996).\n27S. Loth, K. von Bergmann, M. Ternes, A. F. Otte, C. P.\nLutz, and A. J. Heinrich, Nature Physics 10, 1038 (2010).\n28J. A. Appelbaum, Phys. Rev. 154, 633 (1967).\n29J. Fransson, Nano Lett. 9, 2414 (2009).\n30J. Fransson, O. Eriksson, and A. V. Balatsky, Phys. Rev.\nB81, 115454 (2010).\n31R. Zitko and T. Pruschke, arXiv:1002.4082. To appear in\nNew J. of Phys.\n32P. W. Anderson, Phys. Rev. Lett. 17, 95 (1966).\n33J. R. Schrieffer and P. A. Wolff, Phys. Rev. 149, 491\n(1966).\n34F. Delgado and J. Fern´ andez-Rossier, in preparation.\n35F. Elste and C. Timm, Phys. Rev. B 75, 195341 (2007).\n36J. Fern´ andez-Rossier and R. Aguado, Phys. Rev. Lett. 98,\n106805 (2007).\n37J. Fransson, Phys. Rev. B 77, 205316 (2008).38A. S. N´ u˜ nez and R. A. Duine, Phys. Rev. B 77, 054401\n(2008).\n39M. Braun, J. K¨ onig, and J. Martinek, Phys. Rev. B 70,\n195345 (2004).\n40A. L. Efros, E. I. Rashba, and M. Rosen, Phys. Rev. Lett.\n87, 206601 (2001).\n41X. Waintal and O. Parcollet, Phys. Rev. Lett. 94, 247206\n(2005).\n42C. Romeike, M. R. Wegewijs, and H. Schoeller, Phys. Rev.\nLett.96, 196805 (2006).\n43J. Lehmann and D. Loss, Phys. Rev. Lett. 98, 117203\n(2007).\n44J.-i. InoueandA.Brataas, Phys.Rev.B 70, 140406(2004).\n45L. Michalak, C. M. Canali, and V. G. Benza, Phys. Rev.\nLett.97, 096804 (2006).\n46M. Misiorny and J. Barna´ s, Phys. Rev. B 76, 054448\n(2007).\n47M. A. Barral, R. Weht, G. Lozano, and A. M. Llois, Phys-\nica B398, 369371 (2007).\n48J. Merino and O. Gunnarsson, Phys. Rev. B 69, 115404\n(2004).\n49C. Cohen-Tannoudji, G. Grynberg, and J. Dupont-Roc,\nAtom-Photon Interactions (WILEY-VCH Verlag GmbH\nand Co. KGaA, 1998).\n50C.L.Gall, L.Besombes, H.Boukari, R.Kolodka, J.Cibert,\nand H. Mariette, Phys. Rev. Lett. 102, 127402 (2009).\n51N. Lorente and J.-P. Gauyacq, Phys. Rev. Lett. 103,\n176601 (2009).\n52B. Sothmann and J. K¨ onig, arXiv:1003.3794.\n53A. N. Rudenko, V. V. Mazurenko, V. I. Anisimov, and\nA. I. Lichtenstein, Phys. Rev. B 79, 144418 (2009).\n54C.-Y. Lin and B. A. Jones, arXiv::1003.4841.\n55M. Persson, Phys. Rev. Lett. 103, 050801 (2009).\n56J. van Bree, P. M. Koenraad, and J. Fern´ andez-Rossier,\nPhys. Rev. B 78, 165414 (2008).\n57J.-P. Gauyacq, F. D. Novaes, and N. Lorente, Phys. Rev.\nB81, 165423 (2010).\n58E. B. Myers, D. C. Ralph, J. A. Katine, R. N. Louie, and\nR. A. Buhrman, Science 285, 867 (1999).\n59We acknowledge A. S. N´ u˜ nez for this remark"    },    {        "title": "1910.03904v2.Transverse_spin_dynamics_of_light__the_generalized_spin_momentum_locking_for_structured_guided_modes.pdf",        "content": "Transverse spin dynamics of light: the generalized spin -momentum locking \nfor structured guided modes  \nPeng Shi1,#, Luping Du1,# ,*, Congcong Li1, Anatoly V. Zayats2,†, Xiaocong Yuan,1,‡  \n1Nanophotonics Research Centre, Shenzhen Key Laboratory of Micro -Scale Optical Information Technology , Shenzhen \nUniversity, 518060, China  \n2Department of Physics and London Centre for Nanotechnology, King’s College London, Strand, London, WC2R 2LS, \nUnited Kingdom  \nABSTRACT : Spin-momentum locking, a manifestation of topological properties that govern the behavior of surface \nstates, was studied intensively in condensed matter physics resulting in the discovery of topological insulators. The \nphotonic spin -momentum locking was introduced for surface plane -waves which intrinsically carry transverse optical \nspin, leading to many intriguing phenomena and applications in unidirectional waveguiding, metrology and quantum \ntechnologies . In addition to spin, optical waves can exhibit complex properties of vectorial electromagnetic fields, \nassociated with orbital angular momentum or nonuniform intensity variations. Here, by considering both spin and \nangular momentum, we demonstrate a spin -momentum relationship that governs vectorial properties of guided \nelectromagnetic waves, extending photonic spin -momentum locking to two -dimensional vector field of structured \nguided waves. This generalized spin -momentum locking was proven both theoretica lly and experimentally with four \nkinds of surface structured waves. We further formulate a set of spin -momentum equations which are analogous to the \nMaxwell’s equations. These fundamental equations enable us to understand the underlying physics of optical spin-orbit \ncoupling in guided waves,  showing practical importance in spin optics, topological photonics and optical spin -based \ndevices and techniques. \nI. INTRODUCTION  \nSpin-momentum locking (SML), characterized by \nunidirectional surface spin states, has given rise to extensive \nstudies in topological insulators  [1], superconductor  [2], \nmagnon  [3], cold-atom systems  [4] and Bose –Einstein \ncondensates  [5]. The photonic analogy of unidirectional \nsurface spin states  was demonstrated with the pseudo -spin by \nengineering the ‘extrinsic’ spin -orbit interaction and \nbreaking the time -reversal symmetry in artificial photonic \nstructures with importance for applications  [6-8]. On the \nother hand, due to an ‘intrinsic’ spin -orbit coupling governed \nby the Maxwell’s field theory, the SML of light was reported \nand linked to modes with the evanescent fields, such as \nsurface waves or waveguided modes  [9-11]. For example, \nsurface plasmon polaritons (SPPs) as surface modes \npropagating a t an insulator/metal interface [12], exhibit \nfeatures of SML that are analogous to the behavior of surface \nstate on a topological insulator [6-8]. Although  photons are \nbosons with integer spin and surface and waveguided \nelectromagnetic (EM) modes subjected  to backscattering  \n[13], in contrast to the helical fermion  behavior of surface \nDirac modes , they possess the topological ℤ 4 invariant and \nhence can transport spin unidirectionally  [9]. This intrinsic \noptical SML lays the foundation for many intriguing \nphenomena such as spin-controlled unidirectional excitation \nof surface and waveguided modes, and offers potential \napplications in  photonic integrated circuits, polarisation \nmanipulation, me trology and quantum technologies for \ngenerating polarisation entangled states [14-20]. Optical transverse spin [11] plays a key role in the intrinsic \nSML  effect in evanescent waves.  In contrast to the \nconventional longitudinal spin of light with the vector \nparallel to the propagating direction, the orientation of \ntransverse spin lies in the plane perpendicular to the \npropagating direction, enabling many intriguing phenomena \nand applic ations  [21]. The optical transverse spin has been \nstudied extensively in the past few years and been observed \nin many optical configurations, including evanescent waves \n[22, 23] , waveguided modes [24], interference fields [25], \nfocused beams [26], special structured beams [27], and most \nrecently bulk EM waves in bianisotropic media [28]. These \ntransverse spins in the past were generally defined with \nrespect to the wave -vector k, i.e., by calculating the spin \nangular momentum (SAM) and comparing the spin \norientation to the wave -vector. This kind of empirical \nperspective provide an  intuitive way to identify the  optical \ntransverse spin in various optical configurations, particularly \nfor the surface plane waves which resulted in the discovery \nof SML in analogy t o electronic topological insulators. T his \napproach however cannot be generalized to a structured \nsurface wave with an arbitrary trajectory . Although one can \ndefine a “local” wave vector  [29], which is related to the \norbital energy flow density ( Po), it cannot describe \nquantitatively optical transverse spin associated with a \nstructured vector wave for which the spin part of the Poynting \nvector ( Ps) is also important.  \nHere, we propose a framework to consider the optical \ntransverse spin from the pers pective of energy flow density  \n(Poynting vector, P). For a scalar field, the photon \nmomentum p = ℏk, where k is the wave vector and ℏ is the reduced Plank constant, determines the energy flow density \nvia P=c2p, where c is the speed of light ( see Supplementary \nText I ). Therefore, the wave vector,  momentum and energy \nflow density, which has in general both spin and orbital \ncomponents (P = Ps + Po), are closely  interlinked  [29]. The \nenergy flow density can be represented through a current \ndensity  term in the Hertz potential ( see Supplementary Text \nV). Therefore, the proposed treatment allows also extending \nthe concepts of the dynamics of transverse spin from EM \nwaves to fluid, acoustic and gravitational waves [30-32]. \nUsing the Poynting vector con siderations,  we derive d an \nintrinsic spin -momentum curl-relation ship, which, on the one \nhand, reveals the optical transverse spin dynamics in EM \nfields and, on the other hand, extends the understanding of \nthe SML from plane evanescent waves to a 2D chiral spin \nswirl associated with the structured guided modes, therefore, \ngeneralizing the optical SML to arbitrary evanescent vector \nfields. We  formul ate four equations relating spin and \nmomentum of the electromagnetic wave that are analogous \nto the Maxwell’s equations for EM fields. The results are \nimportant for understanding  the spin dynamics and spin -orbit \ncoupling in EM waves from RF to UV spectral ranges and \nfor applications in  spin optics, topological photonics, \npolarisation measurements, metrology, the development of \nrobust optical spin -based devices and techniques for quantum \ntechnologies.  \n \nFigure 1 | Generalization of SML for structured guided \nmodes.  (a) In unstructured, plane guided wave, optical SML \nresults in the transverse spin ( S) uniformly distributed and \nparallel to the interface. The spin vector direction is \nperpendicular to the wave vector k and flips if the \npropagation direction flipped from + k to − k. (b) In an \narbitrary structured guided wave, the optical spin is related to \nthe vorticit y of the energy flow  density:  P. The transverse \nspin vector varies from the ‘up’ state to the ‘down’ state  \naround the energy flow, remaining perpendicular to the local \nwave vector. This forms a chiral swirl of the 2D transverse \nspin which is locked to the energy propagating direction and \nfulfills a right handed rule. The direction of the local \ntransverse spin vector flips if the energy flow flipped from \nforward (‘+ P’) to backward (‘− P’). \nII. RESULTS  \nFor an arbitrary EM wave, the curl of the energy flow \ndensity can be presented as (see Supplementary Text II)  \n\n2\n2 1Re4c\n \n   \n           Ρ p\nE H E H\nS\nH E H E\n,  (1)  where S is the spin angular momentum, ω represents the \nangular frequency of an EM field,  E and H indicate the \nelectric and magnetic field, respectively. Here, the symbol \n⨂ indicates a dyad vector and * denotes the complex \nconjugate.  The second part on the right -hand side of Eq. (1) \nhas a same structure as the quantum 2 -form [33] that \ngenerates the Berry phase associated with a circuit, which \nindicates a spin -orbit interaction in the optical system  (see \nSupplementary Text III  for more discussion) . In particular, \nfor EM waves with an evanescent field, such as surface or \nguided waves, an intrinsic  spin-momentum relationship can \nbe derived from  the Maxwell’s theory:  \n        \n21\n2 SP .              (2)  \nSince curl of a vector field can be regarded as its current \nvortices, Eq. (2) reveals that the optical spin of an evanescent \nfield is associated with the local vorticity  of the EM energy \nflow and is source -less (∇·S=0). The SAM in this case is \nrelated to the transverse gradient of the energy flow density. \nAt the same time, the longitudinal optical spin does not fulfill  \nthe above spin -momentum relationship. For example, a  \nmonochromatic circularly polarized plane -wave bears the \nSAM aligned parallel to the wave vector, while the curl of \nthe Poynting vector vanishes because of the uniformity of the \nenergy flow density over the space.  Therefore, the spin -\nmomentum law in Eq. (2) only describes the dynamics of \noptical transverse spin present in the evanescent waves.  It \nalso reveals that, in add ition to the optical spin oriented along \nthe surface (in -plane transverse SAM), which has been \nrecently studied intensively, there exists another category of \nthe transverse spin of an evanescent field oriented out of the \nsurface plane. This SAM can be indu ced by the in -plane \nenergy flow  density  of the structured guided or surface wave, \nwhile the in -plane transverse spin is due to the gradient of \nenergy flow  density  normal to the inte rface. The appearance \nof a transverse spin indicates to the rotation of pol arization \nand hence the phase difference between all th e field \ncomponents of the wave.  \nThe SML in an evanescent  plane wave [Fig. 1(a) ] as \ndemonstrated in previous works  [9] is a special, one -\ndimensional case  of SML with the SAM vector aligned along \nthe interface. Assuming the guided mode  propagating along \n𝑦-direction and evanescently decaying along 𝑧-direction, \none can deduce the Poynting vector 𝐏=\n𝐲̂𝜔𝜀(2𝛽)⁄𝑒−2𝑘𝑧𝑧 and the SAM 𝐒=𝐱̂𝜀𝑘𝑧(2𝜔𝛽)⁄𝑒−2𝑘𝑧𝑧, \nwhere ε and μ denote the permittivity and permeability of the \nmedium, respectively, and β and ikz stands for the in -plane \nand out -of-plane wave vector components,  respectively. The \nenergy flow  density  and SAM of the evanescent  plane wave \nare connected through the generalized spin-momentum \nrelation: 𝐒=𝛁×𝐏2𝜔2⁄=−𝐱̂(𝜕𝑃𝑦/𝜕𝑧)/2𝜔2. For \nstructured evanescent modes with spatially varying intensity \ndistribution, the inhomogeneity of energy flow density can \ninduce several SAM com ponents in different directions.  The \nvariation of the energy flow density in z-direction induces an \nin-plane component of the SAM, while its in -plane variations \ninduce a z-component. Both are perpendicular to the local \nenergy propagation direction. The rel ationship between the \ntwo components leads to a chiral spin texture with spin \nvectors swirling around the energy flow  [Fig. 1 (b)]. More \nimportantly,  its tendency of directional variation (i.e., the \nchirality) is locked to the momentum . This is a manifestat ion \nof the generalized optical SML associa ted with an EM \nevanescent wave.\n \nFigure 2 | Spin -momentum locking in various surface structured waves. (a)-(d), The spatial distributions of the energy \nflow density for different structured surface waves: (a), surface  Cosine beam, (b), surface  Bessel beam with topological charge \nl=± 1; (c), surface Weber beam; and (d), surface Airy beam.  These beams can either propagate in the forward ( labelled “+ P”) \nor backward ( labelled “ –P”) directions.  (e)-(h), Transverse SAM  components Sz and Sx and the cross -sections of the energy \nflow distributions along the green dashed lines in (a)-(d) for the beams propagating in direction indicated with the arrow \nlabelled “+ P”. (i)-(l), The same as (e)-(h) for the beams propagating in the opposite direction indicated with the arrow labelled \n“–P” (c.f. Fig. 1). The inserts at the top of the panels (e)-(l) show the local transverse spin vector orientations. The spin vectors \nare swirling around the energy flow  and their local orientations vary from the ‘up’ to the ‘down’ states (fulfilling the right \nhanded rule). These orientations are inverted for the waves with the opposite direction of the energy propagating. Note that for \nthe beams with curved trajectory, t he spin variation is considered in the plane perpendicular to the local tangential direction of \nthe energy flow. The distance unit is the wavelength of light in vacuum.  \nIt should be noted that the transverse spin discussed here \nis different from  the “spins” in conventional topological \nphotonics, typically called a “pseudo -spin”. For a pseudo -\nspin, the spin -momentum locking is achieved by engineering \nthe spin -orbit interaction in artificial photonic structures  in \norder to break the  time-reversal symmetry  [8]. In the case of \nthe optical transverse spin of an evanescent wave, the \ngeneralized spin -momentum locking is an ‘intrinsic’ feature \nof the spin -orbit interaction governed solely by the \nMaxwell’s theory. Th e nonzero spin Chern number for the \nstructured waves (Supplementary Text III) implies the \nexistence of nontrivial helical states of electromagnetic \nwaves which are strictly locked to the energy propagating \ndirection. However, since the topological ℤ 2 invari ant of \nthese states vanishes owing to the time -reversal symmetry of \nthe Maxwell’s equations, there is no protection against \n(back)scattering. Although the transformation of the two \nhelical states of evanescent waves are not topologically \nprotected against scattering, the SML and the induced \nunidirectional excitation and propagation are the intrinsic \nfeature of the Maxwell’s theory and are topological nontrivial \npossessing the ℤ 4 topological invariant.  To demonstrate the \nSML features described by Eq. (2), fo ur types of the EM modes exhibiting evanescent field with inhomogeneous \nspatial energy distribution were investigated, including the \nsolutions of a wave equation in Cartesian coordinate (Cosine \nbeam) [34], in cylindrical coordinate (Bessel beam) [35], in \nparabolic coordinate (Weber beam) [36], and in Cartesian \ncoordinate but with a parabolic path (Airy beam) [37] (see \nSupplementary Text IV for the details of the calculations) . \nThe magnitudes of their energy flow densities are shown in \nFig. 2  (top panels), while the beams’ propagation directions \ncan either be forward (‘+ P’) or backward (‘ –P’). The \ncorresponding cross -section distributions along the dashed \nlines are shown in the middle and bottom panels for the \nbeams with opposite propagation d irections, together with \nthe SAM distributions and the spin vector variation patterns. \nFor all four different types of the beams, the orientation of \nphoton  spin vectors varies progressively from the ‘up’ state \nto the ‘down’ state when their photon energies  propagate \nalong the forward  direction ( Fig. 2 , middle panels). The \nintrinsic SML present in evanescent waves ensures the \ntopological protection in terms of spin vector swirl being \ncompletely determined by the energy flow density. Thus, to \nobserve the reve rsal of the spin swirling from the ‘down’ \nstate to the ‘up’ state, the propagation direction must be \nreversed ( Fig. 2 , bottom panels). This SML is preserved even \nfor surface modes suffering from the Ohmic losses [12], \nwhich influence only the intensity of the wave but not the \norientation of photonic spin vector. Note that the spin vector \nhas orientation along the interface at the maxima of the energy flow density, and are normal to it at the nodes. \nTherefore, a period of spin variation can be defined betwee n \nthe two adjacent nodes of energy flow density which exhibits \na similar feature to a topological soliton  [38-42]. \n \nFigure 3 | Experimental validation of the SML. The measured out -of-plane SAM components ( Sz) for (a), (e), (i) surface \nCosine beam, (b), (f), (j) surface Bessel beam, (c), (g), (k) surface Weber beam, and (d), (h), (l) surface Airy beam: the spatial \ndistributions of Sz spin components for the beams with (a)-(d) forward (+ P) and (e)-(h) opposite (− P) energy propaga ting \ndirection, (i)-(l) the cross -sections of (a)-(h). The direction of the out -of-plane transverse SAM is inverted for the waves \npropagating in opposite directions. The distance unit is the wavelength of light in vacuum.  \nIn orde r to experimentally observe the spin -momentum \nlocking features associated with the structured surface waves \nand out -of-plane transverse SAM, the experiments were \nperformed on the example of surface plasmon polaritons \n(SPPs). SPPs were excited under the con dition of a total \ninternal reflection using a microscope objective with high \nnumerical aperture NA=1.49. Spatial light modulator and \namplitude masks were employed to modulate the phase and \nwavevector of the excited SPPs to generate the desired \nplasmonic mo des (see Supplementary Text VI for the \ndetailed description of the experiment). A scanning near field \noptical microscope, which employs a dielectric nanosphere \nto scatter the SPPs to the far field, and a combination of a \nquarter waveplate and a polarizer to extract the two circular \npolarization components of the far -field signal, were used to \nmeasure the out -of-plane SAM component 𝑆𝑧=\n𝜀𝛽24𝜔𝑘𝑧2⁄(𝐼𝑅𝐶𝑃−𝐼𝐿𝐶𝑃). The corresponding in -plane spin \ncomponents were also reconstructed from the measurements \n(see Supplementary Text VII). The measured distributions of \nthe SAM components are shown in Fig. 3  and Figs. S15 -S18 \nfor the four types of structured SPP waves propagating in the \nforward and backward directions. All the predicted SAM and \nSML features are observed experimentally: (i) the variation \nof SAM from the positive/negative state to the \nnegative/positive across the beam profile and (ii) the reversal \nof spin variation when inversing the beam propagation \ndirection.  III. DISCUSSION  \nSince  the optical energy flow density can be divided into \nthe spin ( Ps) and orbital ( Po) parts: P=Ps+Po, where \nPs=c2∇×S/2 and obey the spin -momentum relationship (Eq. \n2), we can formulate a set of the Maxwell -like equations \nlinking the transverse spin and the Poynting vector of \nevanescent EM field s (Table 1). This formulation  provides  \ncomprehensive understanding of the spin -momentum \ndynamics in guided waves . The same as variations of E field \ninduces H field in the Maxwell’s equations, equation ∇×P \n=2ω2S indicates that the spatial variations of the energy flow \ndensity induces the transverse spin angular momentum. In \nthe same manner, equation  ∇×S=2Ps/ c2=2(P-Po) /c2 tells us \nthat the spin variation in turn contributes to the energy flux \ndensity, with the remainder provided from the orbital part ( Po) \nof the Poynting vector. Consolidating spin -momentum \nequations results in an analogue of a Helmholtz equation \n∇2S+4k2S= 2∇×Po/c2, which describes spin -orbit interaction \nin evanescent waves, linking transverse spin and orbital part \nof the Poynting vector. In both the Helmholtz equation and \nthe last Maxwell’s equation, current J is an external source \nof magnetic field; similarly, in the corresponding spin -\nmomentum equations, Po, which determines the orbital \nangular momentum , influences the spin.  Since an EM wave \nin a source -free and homogenous medium can be described \nwith Hertz p otential (Ψ) satisfying the Helmholtz equation , \nand the Poynting vector can be calculated from the Hertz \npotential as 𝐏 ∝ (Ψ*∇Ψ−Ψ∇Ψ*) [43], one can obtain the \nspin and orbital properties of the EM guided waves directly \nfrom the spin -momentum equations without any knowledge \non the electric and magnetic fields (Supplementary Text V). \nTable 1 | Spin/momentum equations  and the analogy to  \nthe Maxwell’s equations.  \nSpin/momentum equations  Maxwell’s equations  \n0 P\n \n0 E  \n0 S\n \n0 H \n22 PS\n \ni EH  \n22oc   S P P\n \ni   H J E  \nHelmholtz equations  \n2 2 242o kc   S S P\n \n22k  H H J\n \nIV. CONCLUSION  \nWe have demonstrated an intrinsic spin -momentum law \nwhich governs the transverse spin dynamics of evanescent \nEM waves. It was shown that the 1D uniform spin of surface \nplane wave can be extended to a 2D chiral spin swirl for \nstructured guided modes, providing  a manifestation of the \ngeneralized photonic spin -momentum locking. Four different \ntypes of structured surface waves, including the Cosine beam, \nBessel beam, Weber beam and Airy beam, have been \ninvestigated both theoretically and experimentally, to \ndemonst rate the concept of the generalized SML. \nFurthermore, starting from this relation, we obtained a set of \nspin/momentum related equations that are analogous to the \nMaxwell’s equations. This new optical spin framework can \nbe used to evaluate the spin -orbit co upling in the EM guided \nwaves and for designing specific transverse spin structures, \nwithout a priory  information on the electric and magnetic \nfields. The generalized intrinsic spin -momentum features \ncould also appear in other types of waves with evanescen t \nfield, such as fluid, surface elastic, acoustic and gravitational \nwaves. The effect could be of importance to the development \nof spin optics for quantum technologies and topological \nphotonics.  \nACKNOWLEDGES  \nThis work was supported, in part, by the Nationa l Natural \nScience Foundation of China grants 61427819, 61490712, \n61622504, 61905163 and 61705135 , the leadership of \nGuangdong province program grant 00201505, the Natural \nScience Foundation of Guangdong Province grant \n2016A030312010, the Science and Techno logy Innovation \nCommission of Shenzhen grants KQTD2015071016560101, KQTD2017033011044403, KQTD2018041218324255, \nJCYJ20180507182035270, KQJSCX20170727100838364 \nand ZDSYS201703031605029, the EPSRC (UK) and the \nERC iCOMM project (789340). L.D. acknowledges th e \nsupport given by the Guangdong Special Support Program. \nThe authors thank Prof. Michael Somekh for proof -reading \nthe manuscript.  \n \nCOMPETING INTERESTS  \nAuthors declare no competing interests.  \n \n \n#These authors contributed equally to the work;  \n \nCORRESPONDING AUTHOR : \n*lpdu@szu.edu.cn  \n†a.zayats@kcl.ac.uk  \n‡xcyuan@szu.edu.cn  \nREFERENCES  \n[1] C. L. Kane, and E. J. Mele, Phys. Rev. Lett. 95, 146802 \n(2005).  \n[2] K. Gotlieb, C.-Y. Lin, M. Serbyn, W. Zhang, C. L. \nSmallwood, C. Jozwiak, H. Eisaki, Z. Hussain, A. \nVishwanath, A. Lanzara, Science 362, 1271 (2018).  \n[3] N. Okuma, Phys. Rev. Lett. 119, 107205 (2017).  \n[4] M. Aidelsburger, Science 354, 35 (2016).  \n[5] Y.-J. Lin, K. Jime´ nez -Garcı´a, and I. B. Spielman, \nNature 471, 83 (2011).  \n[6] F. D. M. Haldane  and S. Raghu, Phys. Rev. Lett. 100, \n013904 (2008).  \n[7] Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, \nNature 461, 772 (2009).  \n[8] T. Ozawa, Hannah M. Price, A. Amo, N. Goldman, M. \nHafezi, L. Lu, Mikael C. Rechtsman, D. Schuster, J. \nSimon, O. Zilberberg, and I. Carusotto, Review of \nModern Physics 91, 015006 (2019).  \n[9] K. Y. Bliokh, D. Smirnova, and F. Nori, Science 348, \n1448 (2015).  \n[10] T. Van Mechelen, and Z. Jacob, Optica 3, 118 (2016).  \n[11] A. Aiello, P. Banzer, M. Neugebauer, and G. Leuchs, \nNature Photonics 9, 789  (2015).  \n[12] A. V. Zayats, Igor I. Smolyaninov, Alexei A. Maradudin, \nPhysics Reports 408, 131 (2005).  \n[13] X. Chen , Z.-C. Gu, Z.-X. Liu, X.-G. Wen, Science 338, \n1604 (2012).  \n[14] P. Lodahl, S. Mahmoodian, S. Stobbe, A. \nRauschenbeutel, P. Schneeweiss, J. Volz,  H. Pichler, and \nP. Zoller, Nature 541, 473 (2017).  \n[15] Francisco J. Rodrí guez -Fortuñ o, G. Marino, P. Ginzburg, \nD. O’Connor, A. Martí nez, Gregory A. Wurtz, and \nAnatoly V. Zayats, Science 340, 328 (2013).  \n[16] M. Neugebauer, P. Woźniak, A. Bag, G. Leuchs, and P. \nBanzer, Nature Communications 7, 11286 (2016).  \n[17] Michela F. Picardi, Anatoly V. Zayats, and Francisco J. \nRodrí guez -Fortuñ o, Rev. Lett. 120, 117402 (2018).  \n[18] A. Bag, M. Neugebauer, P. Woźniak , G. Leuchs, and P. \nBanzer, Phys. Rev. Lett. 121, 193902 (2018).  [19] F. J. Rodrí guez -Fortuñ o, N. Engheta, A. Martí nez, and \nAnatoly V. Zayats, Nature Communications 6, 8799 \n(2015).  \n[20] T. V. Mechelen, and Z. Jacob, Nanophotonics 8(8), 1399 \n(2019).  \n[21] K. Y. Bliokh, F. J. Rodrí guez -Fortuñ o, F. Nori, and A. \nV. Zayats, Nature Photonics 9, 796 (2015).  \n[22] K. Y. Bliokh, and F. Nori, Phys. Rev. A 85, 061801(R) \n(2012).  \n[23] K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, Nature \nCommunications 5, 3300 (2014).  \n[24] F. Khosravi, C. L. Cortes, and Z. Jacob, Optics Express \n27(11), 15846 (2019).  \n[25] A. Y. Bekshaev, K. Y. Bliokh, and F. Nori, Phys. Rev. \nX 5, 011039 (2015).  \n[26] K. Y. Bliokh, and F. Nori, Physics Reports 592, 1 (2015).  \n[27] A. Aiello and P. Banzer, J. Opt. 18, 085605 (2016).  \n[28] L. Peng, L. Duan, K. Wang, F. Gao, L. Zhang, G. Wang, \nY. Yang, H. Chen , and S. Zhang, Nature Photonics 13, \n8782 (2019).  \n[29] M. V. Berry, J. Opt. A: Pure Appl. Opt. 11, 094001 \n(2009).  \n[30] K. Y. Bliokh  and F. Nori, Phys. Rev. B 99, 174310 \n(2019).  \n[31] Y. Long, J. Ren, and H. Chen, Proc. Natl. Acad. Sci. \nUSA. 115(40), 9951 (2018).  \n[32] S. Golat, Eugene A. Lim, and Francisco J. Rodrí guez -\nFortuñ o, Phys. Rev. D 101, 084046 (2020).  \n[33] M. V. Berry, Proc. R. Soc. A 392, 45 (1984).  \n[34] J. Lin, J. Dellinger, P. Genevet, B. Cluzel, F. de Fornel, \nand F. Capasso, Phys. Rev. Lett. 109, 093904 (2012).  \n[35] L.-P. Du, A.-P. Yang, Anatoly V. Zayats , and X.-C. \nYuan, Nature Physics 15, 650 (2019).  \n[36] Zhang, P., Hu, Y., Li, T., Cannan, D., Yin, X., \nMorandotti, R., Chen, Z., and Zhang, X., Nonparaxial \nMathieu and Weber accelerating beams. Phys. Rev. Lett. \n109, 193901 (2012).  \n[37] A. Minovich, Angela E. Klein, N. Janunts, T. Pertsch, \nDragomir N. Neshev, and Yuri S. Kivshar,  Phys. Rev. \nLett. 107, 116802 (2011).  \n[38] M. Gilles, P.-Y. Bony, J. Garnier, A. Picozzi, M. \nGuasoni, and J. Fatome, Nature Photonics 11, 102 (2017).  \n[39] S. Tsesses, E. Ostrovsky, K. Cohen, B. Gjonaj, N. H. \nLindner, and G. Bartal, Scie nce 361, 993 (2018).  \n[40] Timothy J. Davis, D. Janoschka, P. Dreher, B. Frank, \nFrank -J. Meyer zu Heringdorf, H. Giessen, Science 368, \neaba6415 (2020).  \n[41] Y. Dai, Z. Zhou, A. Ghosh, K. Kapoor, M. Dąbrowski, \nA. Kubo, C.-B. Huang, and H. Petek, arXiv:1912.03826 \n(2019).  \n[42] S. Tsesses, K. Cohen, E. Ostrovsky, B. Gjonaj, and G. \nBartal, Nano Lett. 19, 4010 (2019).  \n[43] E. Wolf, Proc. Phys. Soc. 74, 269 (1959).  \n \n \n \n \n \n \n \n \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n Transverse spin dynamics of light: the generalized spin -momentum locking for structured guided modes  \nPeng Shi1,#, Luping Du1,# ,*, Congcong Li1, Anatoly V. Zayats2,†, Xiaocong Yuan,1,‡  \n1Nanophotonics Research Centre, Shenzhen Key Laboratory of Micro -Scale Optical Information Technology, Shenzhen \nUniversity, 518060, China  \n2Department of Physics and London Centre for Nanotechnology, King’s College London, Strand, London, WC2R 2LS, \nUnited Kingdom  \n#These authors contributed equally to the work  \nCorresponding Author: *lpdu@szu.edu.cn; †a.zayats@kcl.ac.uk; ‡xcyuan@szu.edu.cn  \n \n \n \nContents:  \nI. Spin and orbit decomposition of an arbitrary electromagnetic field  \nII. Spin-momentum relation for electromagnetic guided waves  \nIII. Discussion on the spin -orbit interaction and spin topological properties for electromagnetic guided waves  \nIV. Validation of the spin -momentum relation for various surface waves  \nV. Discussion on Maxwell -like spin -momentum equations  \nVI. Experimental setup and methods  \nVII. Reconstruction of the in -plane spin angular momentum components  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n I. Spin and orbit decomposition of an arbitrary electromagnetic field  \nAll the definitions of physical quantities in the manuscript and Supplementary Materials are derived from the Maxwell’s theory \n[s1] and the Riemann –Silberstein (R -S) vector representation for the electromagnetic (EM) fields [ s2-s4]. In the following, we \nonly consider the Cartesian coordinates ( x, y, z) with directional unit vector (𝐱̂,𝐲̂,𝐳̂), while the complex beams in other \ncoordinate systems can also be derived with the same procedure.  \nThe energy flow density (EFD), also known as the Poynting vector ( P) of an optical wave can be expressed as [ s5]:  \n1Re2P E H\n,                                        (S1) \nwhere E and H represent the electric and magnetic fields, respectively. The superscript * denotes the complex conjugate. For \na time -harmonic, monochromatic EM wave in the homogeneous medi um, the Poynting vector can also be expressed as the \nmomentum density of the field  \n  21Im4\nemElectric magneticc\n\n               ppPp E E H H\n.                        (S2) \nwhere ε, μ denote the permittivity and permeability of the medium, c is the speed of the light and ω is the  angular frequency \nof wave. The momentum density can be represented as the contributions from the electric ( pe) and magnetic ( pm) field \ncomponents.  \nBy considering the Gaussian’s law of the electric and magnetic fields in a passive and lossless medium, the momentum density \nof the field can be expressed as [ 29] \n \n 11Im Im48x x y y z z\nx x y y z zE E E E E E\nH H H H H H\n  \n\n      \n             p E E H H\n.            (S3) \nThis expression can be simplified introducing the 6 -vector in the homogeneous medium [ s2-s4] \n1\n2i\n\nE\nH\n,                                          (S4) \nresulting in  \n 3311ˆ ˆ ˆ\n2i      p p r p r S\n,                              (S5) \nwhere 𝐩̂3(𝐫) is the Hermitian local momentum operator in the position representation: 𝐩̂3(𝐫)=\n(𝛿(𝐫̂−𝐫)𝐩̂3+𝐩̂3𝛿(𝐫̂−𝐫))2⁄ with 𝐩̂3=−iℏ∇ being the momentum operator, 𝐒̂ is the spin -1 matrix in SO(3) space [ 29] \nand ℏ is the reduced Planck constant.  Physically, the first and second terms in Eq. ( S5) correspond to the orbital and spin parts \nof the momentum density, respectively. Therefore, the orbital momentum density (OMD)  and spin momentum density (SMD) \ncan be expressed as   s1Im8     p E E H H,                                (S6a)  \n \n o1Im4x x y y z z\nx x y y z zE E E E E E\nH H H H H H\n  \n      \n     p\n,                            ( S6b) \nrespectively. The orbital momentum density \n3ˆo    p p r\n  is proportional to the local momentum vector. From Eq. \n(S6a) and the second term in Eq. ( S5), the spin angular momentum (SAM) is  \n1Im4   S E E H H\n.                                    (S7)  \nII. Spin -momentum relation for electromagnetic guided waves  \ni. General case of the spin -momentum relationship  \nBy employing Eq. ( S6b), the curl of the OMD can be calculated as  \n \n o1Im4x x y y z z\nx x y y z zE E E E E E\nH H H H H H\n  \n       \n       p\n.                       (S8) \nOn the other hand, from the relationship between the electric and magnetic fields within the Maxwell’s theory, the curl of the \nSMD can be evaluated as  \n2\ns\n21Im8\n1Im4x x y y z z\nk\nx y y y z z\n\n\n\n\n      \n                                p E E H H\nE E E E E E\nS\nH H H H H H\n,                       (S9) \nwhere k=ω/c is the wave vector of the field in the vacuum. By introducing a Dyad’s vector given by [ s5] \n1 2 1 2 1 2\n1 2 1 2 1 2\n12\n1 2 1 2 1 2i i j i k i\ni j j j k j\ni k j k k kx x x x x x\nx x x x x x\nx x x x x x\n\nrr\n,                                      (S10)  \nwhere the two vectors are 𝐫1=(𝑥𝑖1,𝑥𝑖1,𝑥𝑖1)T and 𝐫2=(𝑥𝑖2,𝑥𝑖2,𝑥𝑖2)T, the curl of the momentum density can be expressed as  \n  2Re4k                   p S E H E H H E H E\n.              (S11)  \nAccordingly, the curl of the energy flow density is  \n  2\n22Re2ssvv          Ρ p S E H H E\n,                    (S12)  \nwhere \n   1 2 1 2 1 2 2s    r r r r r r  [s5]. Note that the second  part in the right -hand side of Eq. ( S12) has a same structure \nas the quantum 2 -form [ 33] that generates the Berry phase associated with a circuit, which indicates a spin -orbit interaction in \nthe optical system (the relation between this quantum 2 -form and  Berry phase will be discussed in Section III).  ii.     The spin -momentum relation for the electromagnetic guided waves  \nThe existence of an interface between media with different relative permittivity and permeability breaks the dual symmetry \nbetween the e lectric and magnetic features and the intrinsic connection between the spin and energy flow  densities  should be \nconsidered individually for TM and TE guided modes. We first consider the situation of the guided waves on the example of \na transverse magnetic (TM) surface EM wave propagating in xy-plane ( Hz = 0). In this case, the evanescent field exponentially \ndecaying in the z-direction can be expressed as 𝐹(𝑥,𝑦)𝑒−𝑘𝑧𝑧, where ikz is the normal to the interface component of the wave \nvector (Fig. 1). By employing the Maxwell’s equations and the Hertz potential theory, the relation between the electric and \nmagnetic field components can be expressed as [ s6] \n22\n22z z z\nxx\nz z z\nyyk E i EEHxy\nk E i EEHyx\n\n\n \n \n.                                 (S13)  \nwhere 𝛽=√𝑘𝑥2+𝑘𝑦2 represents the in -plane component of the wave vector and is related to kz by 𝛽2+(𝑖𝑘𝑧)2=𝑘2 After \ntedious calculations, the curl of the energy flow density can be expressed as  \n \n \n  221Re 24y z y z\nz x z x\nx y x y x y x yi E E E E\ni E E E E\ni E E E E i H H H H\n  \n \n\n    \n     \n   Ρ SS\n.                   (S14)  \nSimilarly, for the TE evanescent wave ( Ez = 0), e.g., Bloch surface wave [ s7], the field components fulfill the following \nconditions [ s6]: \n22\n22z z z\nxx\nz z z\nyyi H k HEHyx\ni H k HEHxy\n\n\n \n \n                                   (S15)  \nand the curl of  the energy flow density also is  \n \n \n  22 1Re 24y z z y\nz x x z\nx y y x x y y xH H H H\ni H H H H\nH H H H E E E E\n   \n\n\n              Ρ SS\n.                  (S16)  \nAs the result, for both the TM and TE guided modes, the spin angular momentum and energy flow density (or momentum \ndensity) fulfill the relationship   \n2211\n22 k    S P p\n.                                     (S17)  \nIt worth noting that the orbital angular momentum of optical vortex results from the rotation of photon momentum ( L = r×Po, \nwhere r is the position vector and Po is the orbital flow density). It is associated to the global phase structure of a beam. \nWhereas in our case, the transverse optical spin of an EM field originates from the curl/vorticity of energy flow density ( Po). It is associated to the ‘local’ rotatio n of polarization states in the photon transportation.  \nIII. Discussion on the spin -orbit interaction and spin topological properties for electromagnetic guided waves   \ni. Optical Dirac equation and spin -orbit interaction  \nThe Dirac equation is originally derived for a spin -1/2 particle [ s8]: \n 2 ˆ i c mct    Hαpβ\n.                                  (S18)  \nHere, 𝐇̂ denotes the Hamiltonian operator , Φ is the electric wave function and the four Dirac matrices can be expressed in \nterms of the Pauli matrices:  \n0\n0\n00\n0\n0\n0i\ni x i\ni\nz\n\n\nα\nβ\n,                                      (S19)  \nwhere i = x, y, z and the Pauli matrices are  \n01 0 0 1 0 1 0\n0 1 1 0 0 0 1x y zi\ni                             \n. \nFor the time -harmonic electromagnetic field propagating in the homogeneous medium, where is in the absence of charges and \ncurrents, the Maxwell’s equation can be written as  \n\n\n0\n00\n000\n0\n1\n1r\nr\nrt t t\nt t t  \n   \n     \n  \n          \n          E\nHH\nH H HE\nE E EH\n,                            (S20)  \nwhere εr and μr are the relative permittivity and permeability, respectively.  Here, with the identity for arbitrary two vectors A \nand B: \nˆ i  A B A S B  [29], where 𝐒̂ is the spin -1 matrix in SO(3) expressed as:  \n0 0 0 0 0 0 0\nˆ ˆˆˆ, , 0 0 , 0 0 0 , 0 0\n0 0 0 0 0 0 0x y zii\nS S S i i\nii       \n                       S\n,                     (S21)  \nthe curl operator can be rewritten as  \n1ˆ ˆˆ i    S S p\n.                                   (S22)  \nFirstly, if we consider the electromagnetic field in the homogeneous space, the latter two equations of Eq. ( S20) can be written \nas  \nˆˆ ˆˆcit       0SHp\nS0\n.                              (S23)  The Hamiltonian  operator 𝐇̂ is \nˆˆ ˆ ˆ ˆˆcc\n    0SHτpp\nS0\n,                                   (S24)  \nwhere \nˆvτ denotes the energy flow density operator (the momentum density operator is \nˆcτ ) and the corresponding energy \nflow density can be expressed as  \n2ˆ1ˆ Reˆ 2c c c\n         0SP p E H τ\nS0\n.                    (S25)  \nInterestingly, the 1st-order partial derivative of position operator is  \nˆ0ˆ ˆ ,ˆ0icc   Sr H r τ\nS\n,                                  (S26)  \nwhich indicates  energy flow density operator \nˆvτ  has the property of a velocity operator and can  describe the photon trajectory  \nin accord with the momentum operator. Moreover, the SAM operator of electromagnetic wave is  \nˆ0ˆ\nˆ0\n\nSΣ\nS\n,                                          (S27)  \nand the corresponding  SAM can be expressed as  \n ˆ0 1 1 1ˆ Imˆ 4 0  \n          \nSS E E H H Σ\nS\n.               (S28)  \nThus, we can calculate that  \nˆ ˆˆ ˆˆ ,ic   Σ HΣτp\n and \nˆ ˆˆ ˆˆ ,ic  L H L τp\n ,                        (S29)  \nwhere the OAM operator is  \nˆˆˆL r p . These equations show that the SAM and OAM are not conserved individually , and the \nevolution of SAM and OAM is relative to \nˆˆvτp , which is similar with the curl of energy flow density. However, the total \nangul ar momentum operator \nˆˆˆJLΣ  is conservative owing to  \nˆ ˆˆ ˆˆ ˆˆ , , , 0i i i             J H J H L H Σ\n.                             (S30)  \nThe conservative properties of total angular momentum are critical in the analysis of the Chern number and spin -orbit \ninteraction for structured electromagnetic waves.  \nii. Photonic spin Chern number for the structured electromagnetic waves  \nThe photonic spin Chern number for a plane wave is introduced in Ref. [ 9]. For a structured light field, the electric/magnetic \nfield can be expressed in a plane -wave basis [ s9]  \n 2\n21\n2\n1\n2i\nk\ni\nked\ned\n\n\n\n\n\nkr\nk\nkr\nkE r E k k\nH r H k k\n,                                 (S31)  \nwhere k = ω/c = | k| is the wave number. Transversality constraints for plane waves given by the Maxwell’s equations  ∇·E = \n∇·H = 0, which result in the requirement s 𝐤∙𝐄̃(𝐤)=𝐤∙𝐇̃(𝐤)=0 for each single plane wave in the basis , relates the field \nvector to the wave vector and, therefore, reduces the full 3D vector space of the electromagnetic field components to the 2D \nsubspace of the  components tangential to a sphere in the k space. Owing to  the conservation of total OAM Eq. ( S30), this \nsubspace is invariant for the total angular operator \nˆJ , and one can divide it into two parts consistent with the transversality \ncondition [s10]: \nˆˆˆJLΣ\n, \nˆˆ ˆ    Lκκ L Σ  and \nˆ ˆκκΣ Σ . \nwhere κ = k/k and the modified OAM and SAM operators \nˆL  and \nˆΣ  can be regarded as projections of the operators \nˆL  \nand \nˆΣ  onto the transversality subspace.   \nFollowing Ref. [29], we choose an auxiliary vector e0 and define two unit vectors as  \n0\n2 1 2\n0  kee e eeκ\neκ\n.                                   (S32)  \nThe vectors (e1, e2, κ) form a Cartesian frame in which vectors 𝐄̃(𝐤) and 𝐇̃(𝐤) lie in the transverse plane (e1, e2). Next, we \nintroduce the circular polarization basis:  \n1 2 1 211\n22ii    e k e e e k e e\n,                        (S33)  \nin which the plane -wave components of the field can be represented as  \n E k C k C k\n                                 (S34)  \nwith 𝐂𝜎(𝐤)=𝐶𝜎(𝐤)𝐞𝜎(𝐤), where σ=± 1 and Cσ(k) are the scalar amplitudes of the circularly polarized components. For the \nstructured waves, the OAM of the field is then  \n    \n2 ˆ ˆ\nso d        J r p p r L ΣLΣ ,                           (S35)  \nwhere the modulated  SAM and OAM components are  \n \n 2 2 222\n211ˆ\n22\n1ˆ\n2kk\nkddcc\nidc\n\n\n\n\n\n   \n      \nkk\nkΣ C k k C k C k k\nL C k k Α k C k kκ κ\nk\n.               (S36)  \nIt is worth noting that the transformation to the helicity basis is associated with the transition to the local coordinate fr ame \nwith the z axis attached to the k-vector, which induces pure gauge Coriolis -type potential [ s11] \n† ˆ ˆ ˆ iU U k Α\n.                                       (S37)  This is the Berry gauge field (connection), which corresponds to the monopole curvature \n3 ˆ ˆ ˆk  k F Α k  with \n ˆdiag 1, 1,0\n [s12]. By using the Dirac representation  and electric –magnetic duality, it can be rewritten as  \n\n ˆ\nˆ ˆ\n  \n      Σ kκ k\nL k L k k Α kk\n.                           (S38)  \nFrom this analysis, the topological Chern numbers for the two helical states can be defined as  \n 2 1ˆ 222Cd       k F k k k k\n,                       (S39)  \nwhere the normalization condition, which has the meaning of the number of photons in the wave packet, has a form of \n |1 N    kk\n [s10]. Therefore, we can obtain the total Chern number to be  \n10tCC\n\n.                                      (S40)  \nThe vanish ed total Chern number reflects the time -reversal symmetry of non-magnetic  Maxwell surface modes.  \nOn the other hand, the spin Chern number is  \n14spinCC\n\n\n.                                     (S41)  \nThis nonzero spin Chern number implies that the nontrivial helical states of electromagnetic waves indeed exist and are stric tly \nlocked to the energy propagation direction. Despite the existence of such nontrivial helical states at the interface go verned by \nthe spin -momentum locking, the topological ℤ2 invariant of these states vanishes  \nmod 2 02spinC\n                                       (S42)  \nowing to the time -symmetry of the Maxwell’s equations. Thus, the spin -momentum locking of optical transverse spin \ndiscussed here is different from the “pseudo -spin” [ 6-8] in artificial photonic structures which is engineered to break the time -\nreversal symmetry, therefore, possessing protection against back -scattering. Although the transformatio n of the two helical \nstates of evanescent waves are not topologically protected against scattering, the SML and the induced unidirectional excitat ion \nand transportation of photons are the intrinsic feature of the Maxwell’s theory and are topological nontri vial ( possess ℤ4 \ntopological invariant ). \niii. Berry phase and spin texture for surface electromagnetic waves  \nFor the general case, the curl of energy flow given in Eq. ( S12) can be abbreviated as \n  2\n2Re2ssv        Ρ S E H H E\n. \nThe second part of this equation  has a same structure as the quantum 2 -form [ 33] that generates the Berry phase associated \nwith a circuit, which indicates a spin -orbit interaction in the optical system. In the case of guided modes, this part can be \nrewritten as    2Re 2\nss       E H H E S.                            (S43)  \nThus , the phase change can be obtained as  \n  2Re 2\nss CCC d d          E H H E a S a\n,                    (S44)  \nwhere C is a two -dimensional  connected region  of the interface.  By applying the general spin -momentum locking equation \nand the Stokes' theorem to Eq. ( S44), the phase change can be rewritten as  \n22\nCC d d d        aS a P a P r\n.                            (S45)  \nTherefore, the phase change  is determined by the optical traject ory around a connected space, which is analogous to the concept \nof geometric phase in condensed matter physics. In addition, this phase change, determined by the optical trajectory, can als o \nbe seen in the optical spin -Hall effect [ s13] and optical Magnus effect [ s12], which result in the separation of the two helical \nstates and the helicity -depended unidirectional propagation. Thus, the momentum -locked chiral spin texture is also related to \nthe optical -trajectory -determined Berry phase and spin -orbit inter action.  \nIf we consider a region where the energy flow density vanishes at the boundary, the integral also vanishes. Thus, the orienta tion \nof SAM should be reversed at the two sides of the extreme point of the energy flow density. This can explain the chira l property \nof spin texture for the structured guided waves, which is one of the key points of our observations (Fig. 2 of the main text) . In \naddition, the topological number of the chiral spin texture can be +1/−1, which denote the spin vector varies from ‘up’ state to \n‘down’ state or ‘down’ state to ‘up’ state accordingly.  \nFinally, if we consider the integral of the SAM over a total two -dimensional plane, we obtain  \n2102dd     aaS a P r\n                                 (S46)  \nowing to the disappearance  of energy flow at infinity for a finite size optical beam. Equation (S46) confirms the “local” \nproperty of the transverse SAM of surface structured waves [ 26], in contrast to the longitudinal SAM, for which the integral \ndoes not vanish.  \nIV. Validation of the spin -momentum relation for various surface waves  \nWe will now verify the above spin -momentum relation for various TM -polarised surface waves. Note that the TE mode \nevanescent waves can be verified in a same way by exchanging the electric field component s with magnetic field components.  \ni. Surface plane wave  \nFor a monochromatic TM mode with an evanescent field decaying in z-direction, the 𝐸𝑧 field component satisfies the \nHelmholtz equation:  \n220zzE k E  \n,                                     (S47)  where k = ω/c is the wave vector of the wave. Assuming that the surface wave propagates along y-axis, the electric and \nmagnetic fields can be written as [ 12] \n\nˆ ˆexp\nˆexpz\nz\nzki i y k z\ni y k z\n  \nE z y\nHx\n,                                 (S48)  \nwhere 𝐱̂,𝐲̂,𝐳̂ are the unit direction vectors. Here, β = ky is the propagation constant. Thus, the energy flow density of the \nevanescent wave is:  \n 1ˆ Re exp 222zkz\n   P E H y\n,                              (S49)  \nand the SAM can be calculated to be:  \n 211ˆ Im exp 24 2 2y z\nzP kkzz        S E E H H x\n.                (S50)  \nBy examining Eq. ( S20) and Eq. ( S21), one can find that the spin -momentum relationship  for the surface plane wave is satisfied.  \nii.     Surface Cosine beam  \nThe same as above, assuming the beam propagates alon g y-direction, the z-component of the electric field can be expresses as \n[34]: \n cos expz x y zE k x ik y k z   \n,                               (S51)  \nwhere A is a complex constant.  \nBy employing Eq. ( S13), the other field components can be calculated as  \n \n 22\n22sin exp cos exp\ncos exp sin expy xz\nx x y z x x y z\nyz x\ny x y z y x y zk kkE k x ik y k z H k x ik y k z\nik k ikE k x ik y k z H k x ik y k z\n\n\n          \n          \n.         (S52)  \nThe Poynting vector and the curl can be calculated as  \n 2 2\n21ˆ Re cos exp 222y\nxzkk x k z\n    P E H y\n                        (S53)  \nand \n  2 2\n2ˆ ˆ 2 cos sin 2 exp 22y\nz x x x zkk k x k k x k z\n    P x z\n.                     (S54)  \nOn the other hand, the SAM can be deduced to be : \n  2 2\n221ˆ ˆ 2 cos sin 2 exp 242y\nz x x x zkk k x k k x k z\n      S x z P\n,                 (S55)  \nwhich satisfies Eq. ( S17). The Poynting vector and SAM distributions for the Cosine beams with forward (+ y direction) and \nbackward ( –y direction) propagation directions are summarized in Fig. S1, for the special case when kx= ky =β·sin(π /4).  \nFig. S1 | The Poynting vector and spin properties for the surface Cosine beams. a -c, The energy flow density and SAM \ndistributions for the surface Cosine beam propagating along the +y direction. d, The normalized spin vector pattern of the \nbeam along the dashed lines in a-c. e-g, The energy flow density and SAM distributions for the surfac e Cosine beam \npropagating along the -y direction. h, the normalized spin vector pattern of the beam along the dashed lines in e-g. The arrows \nin (a) and ( e) show the propagating direction of the beams. The scale bar and the distance unit are the wavelength  of light in \nvacuum.  \niii.    Surface Bessel beam  \nSurface Bessel beams are the solutions of the Maxwell’s equations in the cylindrical coordinate ( r,\n,z). The general form of \nthe z-component electric field can be expressed as [ s14]: \n \n2zkz il\nzlE J r e e ,                                   (S56)  \nwhere Jl stands for the Bessel function of the first kind with order l.  \nBy employing Eq. ( S13), the other field components can be calculated as:  \n \n \n20zz\nzz\nzk z k z il il\nr z l r l\nk z k z il il z\nll\nkz il\nz l zlE k J r e e H J r e er\nilkE J r e e H i J r e er\nE J r e e H\n\n\n  \n  \n\n\n  \n  \n \n,                        (S57)  \nwhere \nlJr  stands for the first order derivative of the Bessel function.  \nThe Poynting vector and the curl can, therefore, be calculated as  \n 2 22 1ˆ Re exp 222lzlJ r k zr    P E H e\n                          (S58)  \nand \n  2\n2 2ˆ ˆ exp 2r z l z l l zlk J r J r J r k zr         P e e.                     (S59)  \nHere, 𝐞̂𝑟,𝐞̂𝜙,𝐞̂𝑧 are the unit direction vectors of the cylindrical coordinate system. On the other hand, the SAM can be \nrepresented as  \n  2\n2 2\n21ˆ ˆ exp 222r z l z l l zlk J r J r J r k zr         S e e P\n,                 (S60)  \nwhich satisfies Eq. ( S17). The Poynting vector and SAM distributions for the surface Bessel beams with topological charge of \nl= +/−1 are given in Fig. S2.  \n \nFig. S2 | The Poynting vector and spin properties for the surface Bessel beams. a -c, The energy flow density and SAM \ndistributions for the surface Bessel beam with topological charge of +1, where the energy flow is counter -clockwise (as \nindicated by arrow in a). d, the normalized spin vector pattern of the beam along the dashed lines a-c. e-g, The energy flow \ndensity and SAM distributions for the surface Bessel beam with topological charge of -1, where the energy flow propagates \nclockwise (as indicated by arrow in e). h, the normalized spin vector pattern of the beam along the dashed lines in e-g. The \nscale bar and the distance unit are the wavelength of light in vacuum.  \n \niv.    Surface Weber beam  \nSurface Weber beams are the solutions of the Maxwell’s equations in the cylinder parabolic coordinates with a trial solution \nof 𝐸𝑧=𝐹(𝜎)𝐺(𝜏)𝑒−𝑘𝑧∙𝑧, where F and G are the separation functions of 𝜎 and 𝜏. In the parabolic coordinates (𝜎,𝜏,𝑧), where \nσ∈(−∞,∞);𝜏∈[0,∞);𝑧∈(0,∞), the coordinates are related to those in the Cartesian coordinates by: x+iy=(σ+iτ)2/2 and \nz = z. \n \nFig. S3 | The Poynting vector and spin properties for the surface Weber beams. a -f, The energy flow density (a,b) and \nSAM distributions (d -f) for the surface Weber beam propagating along the +y direction. c, The normalized spin vector pattern \nof the beam a long the dashed lines in f. g-l, The energy flow density (g,h) and SAM distributions (j -l) for the surface Weber \nbeam propagating along the -y direction. i, the normalized spin vector pattern of the beam along the dashed lines in l. The \narrows in ( b) and ( h) show the propagation direction of the Weber beams. The scale bar and the distance unit are the wavelength \nof light in vacuum.  \nBy substituting the trial solution into the Helmholtz equation, one can get a characteristic formula for the surface Weber \nbeam  as: \n22\n2 2 2 2\n22110FGFGFG                    \n.                       (S61)  \nSince the two terms are functions of independent variables σ and τ, they can be separated as : \n \n2\n22\n2\n2\n22\n220\n20FaF\nGaG   \n     \n  \n,                                  (S62)  \nwhere 2𝛽𝑎 is the separation constant. Eq. (S66) demonstrates a two -dimensional propagation -invariant Weber beams in a \nparabolic cylindrical coordinate [ s14].  \nAfter solving the equation and transferring back to the Cartesian coordinates and using the relationship: 𝑥=𝜎𝜏, 𝑦=\n(𝜎2−𝜏2)2⁄, the general form of the propagating -wave solution is  \n \n2222\n13\n2 22\n1 2 1 2 1\n2\n2 22\n3 2 1 2 11F G 2 F G\n2\n1 1 1 1F ; ; F ; ;2 4 2 2 4 2 1\n3 3 3 3 22 F ; ; F ; ;2 4 2 2 4 2z\nzkz\nz e e o o\ni k zE i e\nia iaii\ne\nia iai i i     \n\n \n  \n\n   \n                               \n,            (S63)                                                                        \nwhere Γ1=Γ[𝑖𝑎2⁄+14⁄], Γ3=Γ[𝑖𝑎2⁄+34⁄] and Γ[x] is the complex Gamma function. Here, F 1 is the confluent \nhypergeometric function of the first kind. The Poynting vector and SAM distributions for the evanescent Weber beams \npropagating along +/− y directions are illustrated in Fig. S3  for the beam  parameter  a=40. Note that we use the numerical \ncalculation to solve the field components of a Weber beam with Eq. ( S13). \nv. Surface Airy wave  \nBy employing the trial solution of 𝐸𝑧=A(𝑥,𝑦)𝑒𝑖𝑘𝑦𝑦−𝑘𝑧𝑧, the Helmholtz equation can be simplified to \n2\n220ixy  \n , \nwhen the small quantity \n2\n2y\n\n representing the variation of the field along the propagating direction can be ignored under the \nparaxial approximation √ln2\n(𝑎𝑤𝑚)2≪𝛽. The solution of the Airy function \n2\n220ixy  \n  can be expressed as [ s15] \n2 3 2 1 1 1 2\n2 12 2, Ai4m m m m m i x a y y a x y\nm\nmmyx y x iay e e              \n\n,                        (S64)  \nwhere Ai indicates the Airy function of the first kind, 𝑥𝑚=𝑥𝑤𝑚⁄  and 𝑦𝑚=𝑦𝛽𝑤𝑚2⁄  are the modulated coordinates, a is a \nparameter defining the exponen tial apodization of the Airy beam, and 2 wm is the width of the main lobe.  \nThe Poynting vector and SAM distributions for the surface Airy beams propagating along the +/−y -direction are summarized \nin Fig. S4 for the beam parameters 𝑎=0.01 and 𝑤𝑚=1.1𝜋𝛽⁄.  \nFig. S4 | The Poynting vector and spin properties for the surface Airy beams. a -f, The energy flow density (a,b) and SAM \ndistributions (d -f) for the surface Airy beam propagating in the +y direction. c, the normalized spin vector pattern of the beam \nalong the dashed lines in f. g-l, The energy flow density (g,h) and SAM distributions (j -l) for the surface Airy beam propagating \nin the −y direction. i, the normalized spin vector pattern of the beam along the dashed lines in l. The arrows in ( b) and ( h) \nshow  the propagating direction of the Airy beams. The scale bar and the distance unit are the wavelength of light in vacuum.  \nvi.    Summary of structured wave solutions  \nThe summary for the above four special types of surface structured beams are shown in Table . S1. Note that we use the \nnumerical simulations to solve the field components of the Weber beams with Eq. (S13), as the derivations of the confluent \nhypergeometric function do not exist. Thus, the analytical expressions of the energy flow density and the SAM for the Weber \nbeams are not provided in Table S1. It was demonstrated that all of the surface structured beams considered above fulfill the  \ngeneralized spin -momentum relationship.  \nNevertheless, there are also many surface mode solutions of the Maxwell’ s equations which cannot be solved explicitly and, \ntherefore, the spin -momentum relationship cannot be verified analytically. Fortunately, from the classical field theory and \nquantum field theory [ s16], arbitrary EM waves can be expanded by the plane -wave solutions. One can verify that the \nrelationship between the SAM and energy flow density for the superposition of two plane evanescent waves are indeed fulfilled  \n[s17], which indicates that the generalized spin -momentum locking (three -dimensional spin -momen tum locking relationship) \nwould be present in an arbitra ry structured evanescent field.  \nTable. S1. Summarization of 𝑬𝒛, EFD and SAM for surface Cosine, Bessel, Weber and Airy beams.  \nSum  Cosine beam  Bessel beam  Weber beam  Airy beam  \nHelmholtz \nequation  ∇2𝐸𝑧+𝑘2𝐸𝑧=0 \nTrial \nsolution  A(𝑥,𝑦)𝑒−𝑘𝑧𝑧 A(𝑟,𝜙)𝑒−𝑘𝑧𝑧 𝐹(𝜎)𝐺(𝜏)𝑒−𝑘𝑧∙𝑧 A(𝑥,𝑦)𝑒𝑖𝑘𝑦𝑦−𝑘𝑧𝑧 \n𝑬𝒛 𝑐𝑜𝑠(𝑘𝑥𝑥)𝑒𝑖𝑘𝑦𝑦−𝑘𝑧𝑧 𝑘𝑟2𝐽𝑙(𝑘𝑟𝑟)𝑒𝑖𝑙𝜙−𝑘𝑧𝑧 1\n√2𝜋[|Γ1|2𝐹𝑒(𝜎)𝐺𝑒(𝜏)\n+2𝑖|Γ3|2𝐹𝑜(𝜎)𝐺𝑜(𝜏)]𝑒−𝑘𝑧∙𝑧 1\n𝜀Ã(𝑥,𝑦)𝑒𝑖𝑘𝑦𝑦−𝑘𝑧𝑧 \nEFDs  𝐲̂𝐶𝑐𝑜𝑠2(𝑘𝑥𝑥) 𝛟̂𝐶𝐽𝑙2(𝑘𝑟𝑟) 1\n2{−𝑅𝑒(𝐸𝑧𝑑∗��𝑦𝑑)𝐱̂\n+𝑅𝑒(𝐸𝑧𝑑∗𝐻𝑥𝑑)𝐲̂} {    𝐱̂𝐶Im(Ã𝜕Ã∗\n𝜕𝑥)\n+𝐲̂𝐶[Im(Ã𝜕Ã∗\n𝜕𝑦)−𝑖𝑘𝑟|Ã|2]\n}    \n \nSAMs  𝐶\n2𝜔2{−𝐳̂𝑘𝑥𝑠𝑖𝑛(2𝑘𝑥𝑥)\n+𝐱̂2𝑘𝑧𝑐𝑜𝑠2(𝑘𝑥𝑥)} 𝐶\n𝜔2{𝐫̂𝑘𝑧𝐽𝑙2(𝑘𝑟𝑟)\n+𝐳̂𝑘𝑟𝐽𝑙(𝑘𝑟𝑟)𝐽′𝑙(𝑘𝑟𝑟)} 𝑖\n2𝜔{𝐼𝑚(𝜀𝐸𝑧𝑑∗𝐸𝑦𝑑)𝐱̂\n𝐼𝑚(𝜀𝐸𝑧𝑑𝐸𝑥𝑑∗)𝐲̂\n𝐼𝑚[𝜀𝐸𝑦𝑑∗𝐸𝑥𝑑+𝜇𝐻𝑥𝑑𝐻𝑦𝑑∗]𝐳̂} 𝐶\n𝜔2\n{      𝐱̂𝑘𝑧[Im(Ã𝜕Ã∗\n𝜕𝑦)−𝑖𝑘𝑟|Ã|2]\n−𝐲̂𝑘𝑧Im(Ã∗𝜕Ã\n𝜕𝑥)\n+𝐳̂[Im(𝜕Ã\n𝜕𝑥𝜕Ã∗\n𝜕𝑦)+𝑖𝑘𝑟Re(Ã𝜕Ã∗\n𝜕𝑥)]}      \n \nParameter  𝐶=𝜔𝜀𝑘𝑦\n2𝑘𝑟2𝑒−2𝑘𝑧𝑧 𝐶=𝜔𝜀𝑙\n2𝑟𝑘𝑟2𝑒−2𝑘𝑧𝑧 𝑦+𝑖𝑥=(𝜎+𝑖𝜏)22⁄ \nσ∈(−∞,∞);𝜏∈[0,∞) C=𝑖𝜔\n2𝜀𝑘𝑟2𝑒−2𝑘𝑧𝑧 \nV. Discussion on spin/momentum locking feature for surface electromagnetic waves   \nThe demonstrated spin -momentum curl relation exhibits the intrinsic locking property between the SAM and optical EFD, and \nextends the spin -momentum locking to an arbitrary structured guided wave. This is one of the interesting physical effects as \ndemonstrated in the main text. Moreover, sta rting from this relationship and noting that  the Poynting vector of the field can be \ndivided into the spin ( Ps) and orbital part ( Po): P=Ps+Po, where Ps=c2∇× S/2 , we can obtain a set of spin/momentum equations \nthat are analogous to the Maxwell equations (Table. 1 in the main text).  As mentioned in the main text, one can obtain the spin \nand orbital properties of the guided EM waves directly from the spin/momentum equations without any knowledge about the \nelectric and magnetic fields.  In a traditional manne r, the spin and orbital angular momentum properties of an EM field is obtained by firstly calculating the \nelectric and magnetic fields. For a time -harmonic monochromatic  EM wave  in a source free, homogeneous and linear isotropic \nmedium, the Hertz's wave eq uation independent of the coordinate system is [ s6] \n220 k  ΠΠ\n.                                    (S65)  \nEq. ( S65) has two types of independent solutions: Πe and Πm, where the subscript “ e” and “ m’ denote for the TM and TE mode \nEM waves, respectively. These result in independent sets of TM waves  \n2\nee\nek\ni  \n EΠ Π\nH Π\n,                                  (S66)  \nand TE waves  \n2m\nmmi\nk \n  E Π\nHΠ Π\n.                                  (S67)  \nAssuming the optical axis along the z -direction so  that 𝚷 can be expressed as 𝚷=Ψ𝐳̂, where Ψ is the model of vector Hertz \npotential. Obviously, the Hertz potential satisfies the Helmholtz equation:  \n220 k  \n.                                      (S68)  \nFor the TM waves, the electric and magnetic fields can be calculated as:  \n220x z x\ny z y\nz z zE k H ixy\nE k H iyx\nE k k H\n  \n  \n   \n.                               (S69)  \nand for the TE waves, they are:  \n220x x z\ny y z\nz z zE i H kyx\nE i H kxy\nE H k k\n  \n  \n   \n,                              (S70)  \nAfter obtaining the EM field, one can calculate the Poynting vector, spin and orbital angular momentum by the classic \ndefinition as in Eq. ( S1) and Eq. ( S7).   \nFig. S5 | The orbital and spin flow density for the surface Cosine mode.  a and b show the Po and Ps for the surface Cosine \nmode propagating in +y direction with kx =β·sin(π/4). c and d give the Po and Ps for the surface Cosine mode propagating in \n−y direction with kx =β·sin(π/4). All quantities are normalized by the maximum of energy flow density.  \n \nFig. S6 | The orbital and spin flow density for the surface Bessel mode.  a and b show the Po and Ps for the surface Bessel \nmode propagating in +φ direction . c and d indicate the Po and Ps for the surface Bessel mode propagating in −φ direction . All \nquantit ies are normalized by the maximum of energy flow density.  \n \n \nFig. S7 | The orbital and spin flow density for the surface Weber mode.  a, b show the x and y -components of Po and c, d \nexhibit the x and y -components of Ps for the surface Weber mode propagating in +y direction with a = 40. e, f show the x and \ny-components of Po and g, h exhibit the x and y -components of Ps for the surface Weber mode propagating in −y direction \nwith a = 40. All quantities are normalized by t he maximum of energy flow density.  \n \nNoting that the Poynting vector can be calculated from the Hertz potential  directly  by 𝐏 ∝ (Ψ*∇Ψ−Ψ∇Ψ*), we can obtain the \nspin and orbital properties of the guided EM waves directly from the Maxwell’s equations without any knowledge about the \nelectric and magnetic fields. It is worth to mention that the classification of the EM field to the TM and TE modes i n this case, \nbecomes unnecessary because they have the same spin and orbital properties. In this way, we can obtain the orbital flow density \nPo = P+ΔP/4k2 and spin flow density Ps = −ΔP/4k2, where Δ is the Laplace operator. The spatial distributions of Po and Ps for \nthe four surface structured waves are shown  in Fig s. S5-S8, respectively.  \n \nFig. S8 | The orbital and spin flow density for the surface Airy mode.  a, b show the x and y -components of Po and c, d \nexhibit the x and y -components of Ps for the surface Airy mode propagating in +y direction with a = 0.01. e, f show the x and \ny-components of Po and g, h exhibit the x and y -components of Ps for the surface Airy mode propagating in −y direction with \na = 0.01. All quantities are normalized by  the maximum of energy flow density.  \nVI. Experimental setup and methods  \nThe experimental setup is shown in Fig. S9 . The experiment was performed on the example of surface plasmon polaritons \n(SPPs), which are TM mode evanescent waves supported at a metal - dielectric interface. A He -Ne laser beam with a \nwavelength of 632.8nm was used as a light source. After a telescope system to expand the beam, a combination of linear \npolarizer (LP), half -wave plates (HWP), quarter -wave plates (QWPs) and vortex wave plates  (VWPs) was employed to \nmodulate the polarization of the laser beam. A  spatial light modulator (SLM) was then utilized to modulate the phase of the \nbeam.  \n \nFig. S9 |  The experimental setup for excitation and mapping of the structured SPP waves . a, Schemati c diagram. b-c, \nThe designed opaque masks employed in the experiment for generation of a SPP Cosine beams with opposite propagation \ndirections. Incident light was blocked except for the two open segments with the angle spread of 8o. d, The designed opaque \nmask for generation of SPP Weber and Airy beams. Note that a spatial light modulator (SLM) was also employed in order to \ngenerate the desired Weber beam and Airy beam. e, the mask complementary of ( d) to generate the corresponding beams with \nopposite propa gation directions to those generated with (d). LP: linear polarizer; HWP: half -wave plate; QWP: quarter -wave \nplate; VWP: vortex wave plate; RM: reflector mirror; PMT: photo -multiplier tube; BS: non -polarizing beam -splitter.  \nThe structured beam was then tightly focused by an oil -immersion objective (Olympus, NA=1.49, 100× ) onto the sample \nconsisting of a thin silver film (45 -nm thickness) deposited on a cover slip, to form the desired SPP beams at the air/silver \ninterface. A polystyrene nanosphere  of 160 nm radius  was immobilized on the silver film surface, as a near -field probe to \nscatter the SPPs to the far field. The preparation of the sample can be found elsewhere [s18]. The sample was fixed on a Piezo \nscanning stage (Phys ik Instrumente, P -545) providing resolution down to 1 nm. A low NA objective (Olympus, NA=0.7, 60× ) \nwas employed to collect the scattering radiation from the nanosphere. A combination of quarter wave plate (QWP) and linear \npolarizer was used to extract the  right -handed (RCP) and left -handed (LCP) circular polarization components of the collected \nsignals. Finally, the intensities of RCP and LCP  components are measured by a photo -multiplier tube (PMT, Hamamatsu \nR12829). As the piezo scanning stage raster scan ned the near -field region, the distributions of RCP and LCP components can \nbe mapped and used to reconstruct the longitudinal SAM component: 𝑆𝑧=(𝐼𝑅𝐶𝑃−𝐼𝐿𝐶𝑃)𝜀𝛽24𝜔𝑘𝑧2⁄ . \ni. Surface Cosine wave  \n \nFig. S10 | Experimental results for the SPP Cosine be ams.  a, The image of the reflected beam captured at the back focal \nplane  of the excitation objective lens when the opaque mask shown in Fig. S9 (b) was used. In this case, an SPP Cosine beam \npropagating along the +y direction was generated. b-d, The measured intensity distributions of the RCP and LCP components \nand the retrieved distribution of the longitudinal SAM ( Sz) component of the SPP  Cosine be am. e-h, The same as b -d for the \nopaque mask shown in Fig. S9 (c) which generates an SPP Cosine beam pro pagating along the –y direction. The scanning step \nsize in the experiment was 25 nm. The reversal of the optical spin for the two Cosine beams with opposite propagation \ndirections is clearly observed. The arrows in ( a) and ( e) show the propagating directio n of the generated Cosine beams. i-k \nand l-n, The corresponding theoretical calculation results obtained with the vectorial diffraction theory for b -d and e -h, \nrespectively. The distance units is the wavelength of light in vacuum.  \nTo generate the SPP Cosin e beams in the experiment, a pair of opaque masks were designed and put right below the objective \nlens in order to filter the wave vectors in the incident plane of the objective. Incident light was, thus, blocked except for  the \ntwo open angles with angle s pread of 8o (Fig. S9 (b-c)). \nThe experimental results are shown in Fig. S10 (a-h) together with the simulation results in Fig. S10(i-n) obtained with the \nvectorial diffraction theory [ s19]. As can be seen, the two opaque masks generate the SPP Cosine beams with opposite \npropagation directions. The theoretical and experimental results match well and the spin -momentum locking property of the \nSPP Cosine beams is clearly demonstrated.  \nii. Surface Bessel wave  \n \nFig. S11 | Experimental results for the SPP Bessel b eams.  a, The back focal plane image of the reflected beam when LCP \nlight was used to the excite the SPP with topological charge of +1 and the energy propagating counter -clockwise. b-d, The \nmapped IRCP and ILCP distributions and the retrieved distribution o f the S z component of the SPP  Bessel be am. e-h, The same \nas b-d for RCP light used to excite the SPP Cosine beam propagating clockwise. The scanning step size in the experiment was \n20 nm. It clearly demonstrates the reversal of the optical spin for the two  Bessel beams with opposite propagation directions. \nThe arrows in ( a) and ( e) indicate the propagating directions of the generated Bessel beams. i-k and l-n, The corresponding \ntheoretical calculation results obtained with the vectorial diffraction theory f or b-d and e -h, respectively. The distance units is \nthe wavelength of light in vacuum.  \nThe method to generate the SPP Bessel beams can be found elsewhere [ 35]. Here, we use the left -handed and right -handed CP \nlights to generate the SPP Bessel beams with to pological charge of +1 and –1, respectively. The experimental results are shown \nin Fig. S11 , along with the theoretical results for comparison. The spin -momentum locking property for the SPP Bessel beams \nwas clearly demonstrated.  \niii. Surface Weber and Air y beams  \nThe SPP Weber and Airy beams were generated by the vectorial Fourier integral method [ s15]. In the experiment, we utilize \nthe opaque masks as shown in Figs. S9 (d-e) to adjust the spatial frequencies of incident light and employ the SLM to code the \nphases into the incident beam . The phase for the generation of SPP Weber wave can be expressed as  \n1ln tan cos2yr\nWeber y rkkia k k ia \n                               (S71)  \nwith a parameter a=10 was used for the experiment. The phase diagram shown in Fig. S12 (a), together with the opaque mask \nshown in Fig. S9 (d), was employed to generate the SPP Weber beam propagating in +y direction. Similarly, the phase diagram \nshown in Fig. S12 (b) and the mask shown in Fig. S9 (e) were employed to generate the Weber beam pro pagating in –y direction. \nThe experimental results along with the theoretical simulations are shown in Fig. S13 . \n \nFig. S12 |  Phase diagrams produced with the SLM for generation of the SPP Weber and Airy beams . a-b, the phases \nused for generation of the SP P Weber beams with opposite propagating directions: a for the +y direction and b for the –y \ndirection. c-d, the phases for generation of the SPP Airy beams with opposite propagating directions: c for the +y direction \nand d for the –y direction.  \n \nThe method  for generating the SPP Airy beams can be found in [ s15]. The cubic phase  \n \n     3\n11arcsin 3 arcsinAiry y y i a k b k                             (S72)  \nwas encoded into the SLM for generation of the surface Airy beams with parameters a1=0.1 and b1= 15. The phase diagram \nshown in Fig. S12 (c), together with the opaque mask shown in Fig. S9 (d) were employed to generate the SPP Airy beam \npropagating in + y direction. Similarly, the phase diagram shown in Fig. S12 (d) and the opaque mask shown in Fig. S9 (e) was \nemployed to generate the Airy beam propagating in –y direction. The experimental results along with the theoretical \nsimulations are shown in Fig. S14 . \n \nFig. S13 | Experimental results for the SPP Weber beams.  a, The back focal plane image of the reflected beam generated \nwith the mask in Fig. S9 (d) and the phase diagram in Fig. S12 (a). SPP Weber beam propagates in +y direction. b-d, The \nmeasured IRCP and ILCP distributions and the retrieved distribution of the S z component of the SPP  Weber be am. e-h, The same \nas b-d for the SPP Weber beam propagating along the –y direction (generated with the mask shown in Fig. S9 (e) the phase \ndiagram shown in Fig. S12 (b)). The arrows in a and e indicate the propagating directions  of the generated Weber beams. The \nscanning step size in the experiment was 25 nm. i-k and l-n, The corresponding theoretical calculation results obtained with \nthe vectorial diffraction theory for b-d and e-h, respectively. The distance units is the wavele ngth of light in vacuum.  \n \nFig. S14 | Experimental results for the SPP Airy beams.  a, The back focal plane image of the reflected beam generated with \nthe mask in Fig. S9 (d) and the phase diagram in Fig. S12 (c). SPP Airy beam propagates in the +y direction. b-d, The measured \nIRCP and ILCP distributions and the retrieved distribution of the Sz component of the generated SPP  Airy be am. e-h, The same \nas b-d for an SPP Airy beam propagating in the –y direction (g enerated with the in Fig. S9 (e) and the phase diagram in Fig. \nS12(d)). The scanning step size in the experiment was 25 nm. The arrows in a and e indicate the propagating directions of the \ngenerated Airy beams. i-k and l-n, The corresponding theoretical cal culation results obtained with the vectorial diffraction \ntheory for b-d and e-h, respectively. The distance units is the wavelength of light in vacuum.  \nVII. Reconstruction of the in-plane  spin angular momentum components  \nFor transverse magnetic (TM) evanescent modes, using the Maxwell’s equation, the relation between the transverse electric, \nmagnetic field components and longitudinal electric field component ( Ez) can be expressed by Eq. ( S13). Using the above \nrelationships, the spin angular momentum c omponents can be calculated to be:  \n2 Im\n2 Im\n2 Imz\nx z z\nz\ny z z\nzz\nzES k Ey\nES k Ex\nEESxy\n\n   \n   \n  ,                                   (S.73)  \nwhere K=𝜀4ωβ2⁄  is a physical constant. By carefully examining the expression of spin vectors, we find that  \n22\n222 Im 2 Imx z z z z\nz z z zS E E E Ek E k Ey y y y y\n                        \n,                       (S.74a)  \nand \n22\n222 Im 2 Imy z z z z\nz z z zS E E E Ek E k Ex x x x x                        \n.                      (S.74b)  \nOn the other hand, the z-component electric field Ez satisfies Helmholtz equation:  \n22\n2 2 2\n220zz\nz z zEEE k E Exy     \n.                                 (S.75)  \nTherefore, by  employing the Eq. ( S75), the Eq. ( S74a ) can be converted into  \n22\n2\n222 Im 2 Imy x z z\nz z z z z zS S E Ek E E E k Ey x x x\n                       \n.                    (S.76)  \nBy employing the conservation law of the spin vectors ( ∇∙𝐒=0 which can be deduced from Eq. (S.7), one can obtain a pair \nof linear ly partial differential equations for transverse spin vectors:  \n22\n222x x z\nzS S Skx y x    \n,                                      (S.77a)  \n22\n222yy z\nzSS Skx y y   \n.                                      (S.77b)  \nThe transverse SAM  component  Sx can be expressed by the Fourier expansion  \nsin cos sin cosx m m n n\nnm x x y ym x m x n y n yS A B C DL L L L   \n       \n,                  (S.78)  \nwhere  Lx and Ly are boundary condition parameters. Am, Bm, Cm and Dm are constants to be determined. From Eq. ( S.77a ) and \nEq. ( S.78), we can get that:  \nsin cos sin cos 2z\nmn m m n n z\nnm x x y ym x m x n y n y SA B C D kL L L L x   \n           \n,               (S.79)  \nwhere 𝜆𝑚𝑛=(𝑚𝜋\n𝐿𝑥)2+(𝑛𝜋\n𝐿𝑦)2 is the eigenvalue.  The expansion coefficients can be determined through the Fourier integral of \nequation ( S.79). Omitting th e complex mathematical process, we obtain that Sx can be solved to be ： 00sin sin sin cos\ncos sin cos cos\n12 sin sin4\n12 sin cos4xymm\nx y x y\nx\nnm\nmm\nx y x y\nLLz\nmz\nx y mn x y\nz\nmz\nx y mn x ym x n y m x n yABL L L L\nSm x n y m x n yCDL L L L\nS m x n yA k dxdyL L x L L\nS m x n yB k dxdyL L x L L   \n   \n\n\n\n\n  \n\n\n\n\n\n00\n00\n0012 cos sin4\n12 cos cos4xy\nxy\nxyLL\nLLz\nmz\nx y mn x y\nLLz\nmz\nx y mn x yS m x n yC k dxdyL L x L L\nS m x n yD k dxdyL L x L L\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n \n  \n\n.                   (S.80)  \nIn a similar manner,  from Eq. ( S.77b ), Sy can be expressed as:  \n  \n00sin sin sin cos\ncos sin cos cos\n12 sin sin4\n12 sin cos4xymm\nx y x y\ny\nnm\nmm\nx y x y\nLLz\nmz\nx y mn x y\nz\nmz\nx y mn xm x n y m x n yABL L L L\nSm x n y m x n yCDL L L L\nS m x n yA k dxdyL L y L L\nS m x n yBkL L y L   \n   \n\n\n\n\n   \n\n \n\n\n\n00\n00\n0012 cos sin4\n12 cos cos4xy\nxy\nxyLL\ny\nLLz\nmz\nx y mn x y\nLLz\nmz\nx y mn x ydxdyL\nS m x n yC k dxdyL L y L L\nS m x n yD k dxdyL L y L L\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n \n  \n\n .                   (S.81)  \nThrough the measured longitudinal SAM  component Sz, the transverse SAM  components  can be obtained  from  Eq. (S.80) and \nEq. (S.81), and a complete photonic spin vector can be constructed .  \nIn addition, we can employ the symmetry of surface electromagnetic modes to simplify the calculation further. Taking the \nsurface Bessel mode for example, Sz is mirror symmetric with respect to x, y-axes, which can be expressed as \nˆx z zm S S  and \nˆy z zm S S\n, where \nˆim  is the mirror operator  with respect to i-axis in Cartesian coordinates. Noted that the i-coordinate is \nmirror antisymmetric with respect to i-axis is mirror symmetric with respect to another axis, so that we have \nˆxm x x  and \nˆym x x\n. Thus, we can get the following relationships:  \n \nˆ ˆz z z z\nxyS S S Smmx x x x        ,                                 (S.82)  \nwhich can be utilized to simplify the Eqs. (S.80) and (S.81):  \n00sin cos\n12 sin cos4xyxm\nnm xy\nLLz\nmz\nx y mn x ym x n ySALL\nS m x n yA k dxdyL L x L L\n\n\n  \n  \n \n\n,                      (S.83a)  00cos sin\n12 cos sin4xyym\nnm xy\nLLz\nmz\nx y mn x ym x n ySBLL\nS m x n yB k dxdyL L y L L\n\n\n  \n  \n \n.                      (S.83b)  \nThrough this method, we reconstructed from the experimental measurements the in-plane  SAM components for the four \nstructured waves as shown in Figs. S15 -18, where the results  obtained from the experiment match well with the theoretical \nsimulations.  \n \nFig. S15 | Measured and simulated SAM components for the SPP Cosine beams. a -c and g-i, experimentally obtained and \nd-f and j-l theoretical SAM components a,d,g,j Sz, b,e,h,k Sx and c,f,i,l Sy  for the SPP Cosine beams propagating in a-f +y  \nand g-l −y direction. The scale bar is the wavelength of light in vacuum.   \n \nFig. S16 | Measured and simulated SAM components for the SPP Bessel beams. a -c and g-i, experimentally obtained and \nd-f and j-l theoretical SAM components a,d,g,j Sz, b,e,h,k Sx and c,f,i,l Sy  for the SPP Bessel beams with vortex topological \ncharge a-f +1 and g-l −1. The scale bar is the wavelength of light in vacuum.   \n \n \nFig. S17 | Measured and simulated SAM components for the SPP Weber beams. a -c and g-i, experimentally obtained and \nd-f and j-l theoretical SAM components a,d,g,j Sz, b,e,h,k Sx and c,f,i,l Sy  for the SPP Weber beams propagating in  a-f +P \nand g-l −P direction . The scale bar is the wavelength of light in vacuum.   \n \nFig. S18 | Measured and simulated SAM components for the SPP Airy beams. a -c and g-i, experimentally obtained and \nd-f and j-l theoretical SAM components a,d,g,j Sz, b,e,h,k Sx and c,f,i,l Sy  for the SPP Airy beams propagating in  a-f +P and \ng-l −P direction . The scale bar is the wavelength of light in vacuum.   \n \n \n \n \n \n \n \n \n \n \n \nReferences and links:  \n[s1] Jackson, J. D., Classical Electrodynamics, New York: Wiley, 1999.  \n[s2] Barnett, Stephen M., Optical Dirac equation. New Journal of Physics 16, 093008 (2014).  \n[s3] Bialynicki -Birula, I., and Bialynicka -Birula, Z., The role of the Riemann –Silberstein vector in classical and quantum \ntheories of electromagnetism. J. Phys. A:  Math. Theor. 46, 053001 (2013).  \n[s4] Elbistan, M., Horvá thy, P. A., Zhang, P. -M., Duality and helicity: the photon wave function approach, Physics Letters A \n381, 2375 –2379 (2017).  \n[s5] Auld,  B. A., Acoustic fields and waves in solids Volume I, New York: J ohn Wiley & Sons , INC., 1973.  \n[s6] Mustafa, E., and Cohen, J. M., Hertz and Debye potentials and electromagnetic fields in general relativity. Class. Quantum  \nGrav. 4, 1623 (1987).  \n[s7] Wang, R., et al., Bloch surface waves confined in one dimension with a single polymeric nanofibre, Nature \nCommunications 8, 14330  (2017).  \n[s8] Shen, Shun -Qing, Topological Insulators: Dirac Equation in Condensed Matters, Springer -Verlag Berlin Heidelberg, 2012.  \n[s9] Li, Chun -Fang, Spin and orbital angular momentum of a class of nonparaxial light beams having a globally defined \npolarization. Phys. Rev. A 80, 063814 (2009).  \n[s10] Bliokh, K. Y., Alonso, Miguel A., Ostrovskaya, Elena A., and Aiello, A., Angular momen ta and spin -orbit interaction \nof nonparaxial light in free space. Phys. Rev. A 82, 063825 (2010).  \n[s11] Bliokh, K. Y.,  Gorodetski, Y., Kleiner, V., and Hasman, E., Coriolis Effect in Optics: Unified Geometric Phase and \nSpin-Hall Effect. Phys. Rev. Lett.  101, 030404 (2008).  \n[s12] Bliokh, K. Y., Niv, A., Kleiner, V., and Hasman, E., Geometrodynamics of spinning light, Nature Photonics 2, 748 -753 \n(2008).  \n[s13] Duval, C., Horvá th, Z., Horvá thy, P.A., Geometrical spinoptics and the optical Hall effect. Journal of Geometry and \nPhysics 57, 925 –941 (2007).  \n[s14] Durnin, J., and Miceli Jr, J. J., and Eberly, J. H., Diffraction -Free Beams, Phys. Rev. Lett. 58, 1499 -1501 (1987).  \n[s15] Kou, Shan Shan, Yuan, Guanghui, Wang, Qian, Du, Luping, Balaur, E., Zhang, D., Tang, D., Abbey, B., Yuan, X. -C., \nand Lin, Jiao, On -chip photonic Fourier transform with surface plasmon polaritons, Light: Science & Applications 5, \ne16034 (2016).  \n[s16] Sakurai, J. J., Advanced quantum mechanics, New Jersey: Addison Wesley, 1967.  \n[s17] Bekshaev, Aleksandr Y., Bliokh, Konstantin Y., and Nori, Franco, Mie scattering and optical forces from evanescent \nfields: A complex -angle approach, Optics Express 21, 7082 -7095 (2013).  \n[s18] Yang, A. -P., Du, L. -P., Meng, F. -F., and Yuan, X. -C., Optical transverse spin coupling through a plasmonic nanoparticle \nfor particle -identifcation and feld -mapping. Nanoscale 10, 9286 –9291 (2018).  \n[s19] Novotny, L., Hecht, B., and  Keller, O., Principles of nano -optics, Ca mbridge University Press, 2012.  \n "    },    {        "title": "2110.01128v1.Giant_Spin_Lifetime_Anisotropy_and_Spin_Valley_Locking_in_Silicene_and_Germanene_from_First_Principles_Density_Matrix_Dynamics.pdf",        "content": "Giant Spin Lifetime Anisotropy and Spin-Valley\nLocking in Silicene and Germanene from\nFirst-Principles Density-Matrix Dynamics\nJunqing Xu,yHiroyuki Takenaka,yAdela Habib,zRavishankar Sundararaman,\u0003,{\nand Yuan Ping\u0003,y\nyDepartment of Chemistry and Biochemistry, University of California, Santa Cruz, CA\n95064, USA\nzDepartment of Physics, Applied Physics and Astronomy, Rensselaer Polytechnic Institute,\n110 8th Street, Troy, New York 12180, USA\n{Department of Materials Science and Engineering, Rensselaer Polytechnic Institute, 110\n8th Street, Troy, New York 12180, USA\nE-mail: sundar@rpi.edu; yuanping@ucsc.edu\nAbstract\nThrough First-Principles real-time Density-Matrix (FPDM) dynamics simulations, we in-\nvestigate spin relaxation due to electron-phonon and electron-impurity scatterings with spin-\norbit coupling in two-dimensional Dirac materials - silicene and germanene, at \fnite temper-\natures and under external \felds. We discussed the applicability of conventional descriptions\nof spin relaxation mechanisms by Elliott-Yafet (EY) and D'yakonov-Perel' (DP) compared to\nour FPDM method, which is determined by a complex interplay of intrinsic spin-orbit cou-\npling, external \felds, and electron-phonon coupling strength, beyond crystal symmetry. For\nexample, the electric \feld dependence of spin relaxation time is close to DP mechanism for\nsilicene at room temperature, but rather similar to EY mechanism for germanene. Due to\nits stronger spin-orbit coupling strength and buckled structure in sharp contrast to graphene,\ngermanene has a giant spin lifetime anisotropy and spin valley locking e\u000bect under nonzero\nEzand relatively low temperature. More importantly, germanene has extremely long spin\nlifetime (\u0018100 ns at 50 K) and ultrahigh carrier mobility, which makes it advantageous for\nspin-valleytronic applications.\nIntroduction\nShortly after the discovery of graphene, sig-\nni\fcant advances have been made in the \feld\nof spintronics, exploiting spin transport in-\nstead of charge transport, with much less dis-\nsipation and unprecedented potentials for low-\npower electronics. Several properties are key\nparameters for optimizing spin transport, such\nas long spin lifetime and di\u000busion length, highelectronic mobility, spin-valley locking e\u000bect.\nLong spin lifetime and di\u000busion length ensure\nrobust spin state during propagation in a de-\nvice. The spin-valley locking e\u000bect is valuable\nfor the emerging \feld of research - \"valleytron-\nics\",1{3which utilizes the valley-pseudospin de-\ngree of freedom as basic unit for quantum in-\nformation technology.\nGraphene is a very promising spintronic ma-\nterial4, in particular, with the extremely high\nelectronic mobility5and the longest known spin\n1arXiv:2110.01128v1  [cond-mat.mtrl-sci]  3 Oct 2021di\u000busion length at room temperature6. How-\never, due to its weak spin-orbit coupling (SOC),\nspin-valley locking e\u000bect may be realized only\nthrough external e\u000bects, e.g., through prox-\nimity e\u000bect - by interfacing with large SOC\nmaterials such as transition metal dichalco-\ngenides (TMDs)7,8. Other 2D materials9in-\ncluding TMDs10have also shown exciting prop-\nerties for spin-/valley-tronics.1{3For instance,\nin Ref. 10, the ultralong spin/valley lifetime is\nobserved (e.g. 2 \u0016s in p-type monolayer WSe 2\nat 5 K) and the spin lifetime is insensitive to\nin-plane magnetic \felds, which is a signature of\nthe spin-valley locking e\u000bects. However, in gen-\neral, TMDs have much lower electric mobility\nthan graphene11,12, which is not ideal for elec-\ntronic device applications.\nAs the counterparts of graphene, silicene and\ngermanene have attracted signi\fcant attention\nthroughout a decade due to their resemblance\nand distinction from graphene13{17. They pos-\nsess several remarkable properties, including\ngate-tunable carrier concentration, high elec-\ntric mobility18,19(Fig. S8), quantum spin hall\ne\u000bects20,21, etc. Moreover, due to the buck-\nled geometry, their SOC strength is highly en-\nhanced and they will also have the advantages\nof tunable band structures by applying perpen-\ndicular electric \felds Ezand high spin polar-\nizations, which are promising for spintronic ap-\nplications14. Their electronic structure under\n\fniteEzhas similarity to those of TMDs: the\nsigns of spins in KandK0valleys being oppo-\nsite which is imposed by time-reversal symme-\ntry, band splittings induced by SOC and highly\npolarized states. Therefore, silicene and ger-\nmanene can combine the advantages of both\ngraphene and TMDs but avoid some shortcom-\nings such as limited spin lifetimes ( \u0014tens of ns)\nin graphene samples,22,23the absence of spin-\nvalley locking in graphene, and low mobilities\nin TMDs11,12.\nDespite of their promising properties, the po-\ntential for spin-based information technologies\nby silicene and germanene has yet to be demon-\nstrated. Understanding spin dynamics and\ntransport in materials is of key importance for\nspintronics and spin-based quantum informa-\ntion science, and one key metric of useful spindynamics is spin lifetime \u001cs. Compared with\ngraphene, of which \u001cshas been extensively stud-\nied experimentally and theoretically6,25,26,\u001csof\nsilicene and germanene have not been measured\nand the existing few theoretical studies were\ndone based on relatively simple models27,28,\nwhich do not include realistic interactions with\nphonons and impurities. Recently we developed\na First-Principles Density-Matrix (FPDM) ap-\nproach with quantum descriptions of scatter-\ning processes between electron-phonon (e-ph),\nelectron-impurities (e-i) and electron-electron,\nto simulate spin-orbit mediated spin dynam-\nics in general solid-state systems with arbitrary\nsymmetry29,30. By employing this new method,\nwe will be able to predict \u001csof silicene and ger-\nmanene at \fnite temperatures with realistic in-\nteractions with environment without introduc-\ning any simpli\fed model or empirical parame-\nters, for ps to \u0016s timescale simulation.\nResults and discussions\nElectronic structure. We \frst show elec-\ntronic quantities of silicene and germanene un-\nder di\u000berent Ez, which are closely related to\nspin dynamics and essential for understanding\nspin relaxation mechanisms.\nFig. 1(a) describes a schematic picture of\nband structures under Ez. DFT results of band\ngaps and the average of band splittings h\u0001i\nbetween two conduction/valence bands (bro-\nken Kramers' degeneracy under Ez\feld) are\nshown in Fig. 1(c). ' hi' represents taking av-\nerage of electronic quantity Aby24hAi=P\nknf0(\u000fkn)Akn=P\nknf0(\u000fkn), wheref0is the\nderivative of Fermi-Dirac distribution function,\nandkandnare k-point and band indices re-\nspectively. At Ez= 0, due to time-reversal and\ninversion symmetries, every two bands form a\nKramers degenerate pair.31A \fniteEzsplits a\nKramers pair into spin-up and spin-down bands\ndue to broken inversion symmetry and the split-\nting increases with Ez(see Fig.1(c)). As the\nearlier model Hamiltonian study13presented,\nthe phase transition from topological insulators\nto band insulators happens at the critical elec-\ntric \feldEcrwith the schematic band struc-\n2Figure 1: Electronic quantities of silicene and germanene under di\u000berent perpendicular electric\n\feld (Ez). (a) The schematic diagrams of band structures of silicene and germanene under di\u000berent\nEz(see calculated band structures in Fig. S3-S5), where Ecris the critical electric \feld leading\nto zero band gap. (b) The schematic diagrams of internal magnetic \felds Bin(blue arrows) at a\nFermi circle near one Dirac cone of planar silicene, (buckled) silicene, (buckled) germanene induced\nby breaking inversion symmetry under \fnite Ez.Bin\nkn= 2\u0001knSexp\nkn=(ge\u0016B), wherekandnare\nk-point and band indices, respectively. ge\u0016Bis the electron spin gyromagnetic ratio. \u0001 knis energy\ndi\u000berence between two conduction (valence) bands when nis a conduction (valence) band. Sexpis\na vector and Sexp\u0011\u0000\nSexp\nx;Sexp\ny;Sexp\nz\u0001\n, whereSexp\niis spin expectation value along direction iand is\nthe diagonal element of spin matrix siin Bloch basis. (c) Band gaps (black lines) and the averaged\nband splitting energies h\u0001i(red lines) between two conduction/valence bands. (d) The averaged\nout-of-plane ( Bin\nz) and in-plane internal magnetic \felds ( Bin\nx). Throughout this work, hAimeans the\naverage24of electronic quantity AandhAi=P\nknf0(\"kn)Akn=P\nknf0(\"kn).f0is the derivative of\nFermi-Dirac distribution function. In this \fgure, for averaging, T= 300 K and chemical potential\n\u0016is set in the middle of the band gap.\ntures shown in Fig. 1(a). Based on our DFT\ncalculations, the band gaps close in silicene and\ngermanene (black solid and dashed lines in Fig.\n1(c)) at 0.2 and 2.5 V/nm respectively, and the\ngaps open again above Ecr.\nThe band splittings between spin-up and\ndown under a \fnite Ezare e\u000bectively induced\nbyk- and band-dependent \\internal\" magnetic\n\felds Bin, which are SOC \felds induced by\nbroken inversion symmetry.31Binis de\fned as\n2\u0001knSexp\nkn=(ge\u0016B),ge\u0016Bis the electron spin gy-\nromagnetic ratio. Sexp\u0011\u0000\nSexp\nx;Sexp\ny;Sexp\nz\u0001\n,\nwhereSexp\niis spin expectation value along di-\nrectioniand is the diagonal element of spin\nmatrixsiin Bloch basis.\nIn Fig. 1(b), we depict schematic diagrams\nofBin(blue arrows) on a Fermi circle near\none Dirac cone of planar silicene, buckled sil-\nicene, and buckled germanene induced by \f-\nniteEz. In planar silicene, Binis purely in-\nplane, known as Rashba Spin-Orbit \felds Bin\nR.Binin buckled silicene is analogous to an ad-\nmixture of Bin\nRand out-of-plane \feld Bin\nz, be-\ncause buckled geometry results in the presence\nofBin\nz. Di\u000berent from silicene, Binin buckled\ngermanene is nearly fully out-of-plane. This\nindicates that stronger intrinsic SOC in ger-\nmanene signi\fcantly increases out-of-plane Bin\nz\nand, as a result, the proportion of in-plane com-\nponent diminishes. We examine the averaged\nBinalong z,\nBin\nz\u000b\n, and along x,\nBin\nx\u000b\n, underEz\nshown in Fig.1(d). While\nBin\nx\u000b\nfor silicene and\ngermanene increase slowly with similar values,\nBin\nz\u000b\n, for germanene due to stronger intrinsic\nSOC, rises much more rapidly than that for sil-\nicene as a function of Ez.Binand its anisotropy\nbetween z and x are very important for spin re-\nlaxation.\nSpin relaxation under \fnite Ez.Spin life-\ntime, and even spin relaxation mechanism, can\nbe tuned by applying electric \felds.26,32Under-\n3standing the e\u000bect of electric \feld is critical for\nthe control and manipulation of spin relaxation.\nwe start our theoretical studies of spin relax-\nation from its electric \feld dependence.\nAs we perform FPDM calculations of spin\nlifetimes in this work, it is important to un-\nderstand the connection and distinction be-\ntween FPDM method and previous theoreti-\ncal models. Previously, spin relaxation mecha-\nnisms are often analyzed based on phenomeno-\nlogical models, such as Elliott{Yafet (EY) and\nD'yakonov-Perel' (DP) mechanisms.31EY rep-\nresents the spin relaxation pathway due to spin-\n\rip scattering. DP is activated when inver-\nsion symmetry is broken which results in ran-\ndom spin precession between adjacent scatter-\ning events. We denote their corresponding spin\nlifetime with \u001cEY\nsand\u001cDP\nsrespectively. They\nare often approximated by some simpli\fed re-\nlations:31,33,34(i)\u0000\n\u001cEY\ns;i\u0001\u00001\u00194hb2\nii\n\u001c\u00001\np\u000b\n(EY\nrelation) along direction i, where\u001cpis carrier\nlifetime.b2\ni= 1\u00002Sexp\niis the degree of mix-\nture of spin-up and spin-down states, so called\n\"spin mixing\"31,35, and is calculated at Ez= 0.\n(ii)\u0000\n\u001cDP\ns;i\u0001\u00001\u0019\n\u001c\u00001\np\u000b\u00001h\n2\u0000\n2\nii(DP rela-\ntion), where \n i=ge\u0016BBin\niis Larmor preces-\nsion frequency31withBin\nide\fned earlier. An-\nother qualitative estimation of the total spin\nrelaxation rate by taking into account both\nmechanisms is using36\u0000\n\u001cE+D\ns\u0001\u00001=\u0000\n\u001cEY\ns\u0001\u00001+\u0000\n\u001cDP\ns\u0001\u00001. In the following, we will compare\nFPDM calculations and these phenomenologi-\ncal models with \frst-principles input parame-\nters including \u001cp,b2\niand \ni.\nWe \frst investigate out-of-plane and in-plane\nspin lifetime \u001cs;zand\u001cs;x, respectively, and their\nanisotropy ( \u001cs;z=\u001cs;x) atEz= 0 and 300K. From\nFig.2 (a) and (b), we \fnd that (i) \u001cs;zand\u001cs;x\nof silicene are much longer than germanene;\n(ii) large anisotropy (10-100) of \u001csis observed\nfor both materials, e.g., 47 and 14 for silicene\nand germanene at zero Ezrespectively, much\ngreater than 0.5 for graphene7,26. Both phe-\nnomena may be qualitatively understood based\non EY relation (typically dominant in inversion\nsymmetric systems) as discussed in the follow-\ning. The comparison between FPDM calcula-\ntions and EY relation with \frst-principles in-\nFigure 2: Spin lifetimes \u001cs(lefty-axis) and its\nanisotropy (right y-axis, red lines) of (a) in-\ntrinsic silicene and (c) intrinsic germanene as\na function of Ezat 300 K. (b) and (d) are \u001cs;z\nobtained by di\u000berent methods of intrinsic sil-\nicene and germanene, respectively. \\FPDM\" -\n\frst-principles real-time density-matrix calcu-\nlations. \\EY\" and \\DP\" correspond to \u001cEY\ns;z\nand\u001cDP\ns;zevaluated by EY and DP relations,\nrespectively. \\E+D\" corresponds to \u001cE+D\ns =\n1=\u0002\n1=\u001cEY\ns+ 1=\u001cDP\ns\u0003\n.\nputs is shown in Fig.2 (c) and (d), and they\ngive the same order of magnitude of spin life-\ntime. Roughly speaking, the larger spin life-\ntime of silicene is mainly from the smaller spin\nmixingb2compared with germanene based on\ntheir intrinsic SOC strength. While the large\nspin anisotropy for both systems is a result of\nlarge ratio of b2\nz=b2\nx(Fig. S6(a)).\nWe then discuss Ezdependence of spin relax-\nation. From Fig. 2(a) and 2(b), \u001cs;xof silicene\nand germanene rapidly reduces with increasing\nEz. This trend can be qualitatively understood\nas follows. Finite Ezbreaks inversion symme-\ntry and splits Kramers degeneracy. This in-\nduces Binwith a rapidly increased zcompo-\nnent as shown in Fig. 1(d), and thus leads to\nfast in-plane spin relaxation (perpendicular to\nBin\nz) and reduces in-plane spin lifetime \u001cs;x.\nUnlike\u001cs;x, out-of-plane spin lifetime \u001cs;zof\ngermanene is insensitive to Ezin Fig. 2(c)\nalthough\u001cs;zof silicene decreases fast with\nEz(Fig. 2(a)). To better understand \u001cs;zde-\npendence on Ez, we compare FPDM \u001cs;zwith\n4the model calculations based on EY ( \u001cEY\ns;z), DP\n(\u001cDP\ns;z) and EY+DP ( \u001cE+D\ns;z) mechanisms as in-\ntroduced earlier, in Fig. 2(b) and 2(d). From\nFig. 2(b) for silicene, we show that \u001cE+D\ns;z and\n\u001cDP\ns;zapproximately agree with FPDM \u001cs;zin\ntrends. For germanene, however, from Fig.\n2(d) we \fnd that \u001cEY\ns;zis in good agreement\nwith the FPDM \u001cs;z, but neither \u001cE+D\ns;z nor\u001cDP\ns;z\ncapture the qualitative trend. Therefore, z-\ndirection spin relaxation in germanene should\nbe mostly driven by EY mechanism, insensitive\ntoEz. The suppression of DP mechanism in\ngermanene under \fnite Ezmay be due to the\nhugeBinanisotropyBin\nz=Bin\nx(see Fig. 1(d) and\nFig. S6(b)): as Bin\nzis so strong, any in-plane\nspins will be quickly relaxed and all spins are\npinned along z; thus the total spin can only de-\ncay through direct spin-\rip processes but not\nthrough spin precession driven by Binwhich\nchanges spin direction gradually.\nTemperature dependence of \u001cs;zand spin{\nvalley locking. It is important to understand\nthe sensitivity of spin lifetime to temperature\nand determine the optimal operating tempera-\nture.30,38,39Therefore, we show temperature de-\npendence of \u001cs;zwithoutEzand withEz= 2\nand 5 V/nm (higher than Ecrby\u00182 V/nm)\nin intrinsic silicene and germanene respectively\nin Fig. 3(a) and (b). Without Ez,\u001cs;zof both\nsilicene and germanene increase fast on cool-\ning. This is the usual behavior of EY spin\nlifetime simply due to weaker e-ph scattering\nwith lowering temperature. Since phonon oc-\ncupation is smaller at a lower temperature, \u001cp\nis longer (Fig. S8), so \u001csis longer (\u001cs/\u001cpwith\nEY mechanism). Under \fnite Ez,\u001cs;zof ger-\nmanene are similar to the values under zero Ez\nas shown in Fig. 2(b), which are expected from\nthe discussions on Ezdependence of germanene\nabove. In sharp contrast, \fnite Ezsigni\fcantly\nreduces\u001cs;zof silicene and modi\fes the temper-\nature dependence. To interpret such complex\ntemperature dependence, in Fig. 3(c), we com-\npare FPDM \u001cs;zand the model ones \u001cEY\ns;z,\u001cDP\ns;z\nand\u001cE+D\ns;z for silicene. All model relations fail\nto reproduce the temperature dependence un-\nder \fniteEzfor silicene. The failure of the DP\nrelation in particular below 200 K is probably\nFigure 3: Temperature dependent \u001cs;zof intrin-\nsic (a) silicene and (b) germanene under Ez= 0\nandEz6= 0 (2 and 5 V/nm for silicene and ger-\nmanene, respectively). (c) \u001cs;zof intrinsic sil-\nicene under Ez6= 0 obtained by di\u000berent meth-\nods. The labels of the curves have the same\nmeanings as in Fig. 2. (d) Relative interval-\nley spin relaxation contribution \u0011of germanene\nunderEz= 0 andEz6= 0.\u0011is de\fned as\n\u0011=(\u001cinter\ns;z)\u00001\n(\u001cinters;z)\u00001+w(\u001cintras;z)\u00001, where\u001cinter\ns;zand\u001cintra\ns;z\nare intervalley and intravalley spin lifetimes,\ncorresponding to scattering processes between\nKandK0valleys and within a single Kor\nK0valley, respectively. \u0011being close to 1 or 0\ncorresponds to intervalley or intravalley scatter-\ning dominant spin relaxation, respectively. See\nmore details of \u0011in Ref. 37.\nbecause it is inapplicable for weak scattering\nregime (weak e-ph scattering at low temper-\natures).31Other relations have been proposed\nfor weak scattering31,34,40, e.g.,\u001cs\u0018j\nj\u00001and\n\u001cs\u00182\u001cp, but none can capture the correct tem-\nperature dependence of \u001cs;zof silicene under\n\fniteEz, e.g.,\u001cs\u0018 j\nj\u00001predicts\u001cs;z<20\nps at all temperatures investigated here. Our\ntheoretical studies highlight the importance of\nsimulating spin lifetime using FPDM method\nfor reliable prediction of spin lifetimes for large\nvariations of external conditions.\nSince the bands near the Fermi energy are\ncomposed of the Dirac cone electrons around\nKandK0valleys in silicene and germanene,\nspin relaxation process mostly arises from in-\n5Figure 4: Spin relaxation in germanene at 50 K under 0 and 5 V/nm. (a) \u001cs;zand (b) relative\nintervalley spin relaxation contribution \u0011of germanene as a function of excess carrier density n,\nwhich is electron density neminus hole density nh, and controlled by chemical potential \u0016. Positive\ncarrier density corresponds to electron doping. (c) \u001cs;zof germanene as a function of in-plane\nmagnetic \feld - Bx. (d)\u001cs;zof germanene as a function of impurity density niof neutral Ge\nvacancy (with neandnhkept the same as intrinsic germanene). \u0011being close to 1 or 0 correspond\nto spin relaxation being dominated by intervalley or intravalley scattering, respectively. The blue\nand green dotted vertical lines correspond to \u0016at the minima of \frst and second conduction bands\nrespectively. The black and red dotted lines are \u001cs;zin the zero nilimit under Ez= 0 and 5 V/nm,\nrespectively.\ntervalley and intravalley scatterings. To scruti-\nnize contributions of intervalley and intravalley\nscatterings to spin lifetime, we examine relative\nintervalley spin relaxation contribution \u0011(see\nits de\fnition in the caption of Fig. 3). \u0011be-\ning close to 1 or 0 correspond to spin relaxation\nbeing dominated by intervalley or intravalley\nscattering, respectively. From Fig. 3(d), it is\nfound that for germanene under Ez=5 V/nm,\nthe\u0011becomes close to 1 at T\u001470 K. This in-\ndicates that under these conditions spin relax-\nation in germanene is dominated by intervalley\nprocess, which is a signature of so-called spin-\nvalley locking. In addition, the long \u001cs;zreaches\naround 100 ns at 50K and Ez=5 V/nm and in-\ndicates stability of spin states against spin re-\nlaxation. Di\u000berent from germanene, in silicene,\nsince the relatively small SOC splitting even at\nEz=2 V/nm as shown in Fig. 1(b), the spin re-\nlaxation is mostly through intravalley scatter-\ning (not shown), which indicates silicene will\nhardly show the spin-valley locking property.\nFormally the terminology \\spin-valley lock-\ning\" means that spin index (spin-up and spin-\ndown) becomes locked with the valley index\n(e.g.,KorK0).41This implies that (i) two val-leys (e.g.,KandK0) exist with opposite spin\npolarizations. (ii) Within one valley, spins are\nall highly polarized along one direction, and\ncarriers should have the same sign of spin. This\nneeds large SOC splitting, i.e., the spin-up and\ndown bands being largely separated.\nPoint (ii) will cause highly suppressed\nintravalley spin relaxation (i.e., relaxation\nthrough scattering processes within one val-\nley). We explain the reason using germanene\nas an example: Under 5 V/nm, the SOC split-\nting for germanene is large, 23 meV at K. To\nhave spins with the same sign, most of carriers\nshould be located around the band edges (which\nrequire low temperatures and low carrier den-\nsities). Then the intravalley spin-\rip transition\nbetween an occupied state at band edges and an\nempty state at the second conduction/valence\nband will be rather weak at relatively low tem-\nperatures, since phonon occupation becomes\nnegligible at the corresponding phonon energy\n(comparable to SOC splitting).\nOn the other hand, for intervalley e-ph pro-\ncesses, the corresponding phonon wavevectors\nare away from \u0000 (e.g., around K) and the min-\nimum phonon frequency is \fnite (e.g., 7 meV\n6for germanene, see Fig. S7). Therefore, with\nspin-valley locking, spin relaxation through e-\nph scattering (mostly intervalley) will be highly\nsuppressed leading to long spin lifetime, be-\ncause of small phonon occupations at relatively\nlow temperatures.\nThe existence of spin-valley locking not only\nleads to long spin lifetime but also allows the\nutilization of previously developed valleytronic\ntechnologies to design germanene-based de-\nvices1,3. Our calculations provide guidance on\nthe necessary conditions to realize spin-valley\nlocking in germanene.\nCarrier density and magnetic \feld depen-\ndence of\u001cs;zfor Germanene at low T. As\nintervalley scattering was shown being domi-\nnant in germanene at 50 K with a very long\nspin lifetime, we further investigate its spin re-\nlaxation as a function of carrier density, which\ncan be easily tuned by electrical gate experi-\nmentally. We will also determine the range of\ncarrier density where spin-valley locking hap-\npens.\nAs shown in Fig. 4(a), we plot \u001cs;zof ger-\nmanene as a function of excess carrier density n\n(which is electron density neminus hole density\nnh, and controlled by chemical potential \u0016). We\nfound it is very sensitive to nat the degenerate\ndoping range ( n&4\u00021010cm\u00002, corresponding\nto\u0016above the conduction band minimum). As\nthe carriers contributing to spin relaxation have\nhigher energies when nincreases, the strong n\ndependence of \u001cs;zshould be mainly a result of\nthe strength of the e-ph scattering being en-\nhanced at higher energies (Fig. S10).\nMoreover, we \fnd that the relative inter-\nvalley spin relaxation contribution \u0011 >0.9 at\nn64\u00021010cm\u00002underEz=5 V/nm in Fig.\n4(b). This indicates spin relaxation being dom-\ninated by intervalley processes and the presence\nof spin-valley locking at the corresponding con-\ndition, consistent with our above discussions\nabout spin-valley locking.\nWe then investigate e\u000bects of spin-valley lock-\ning under in-plane magnetic-\feld Bx. From\nFig. 4(c) at 50K for germanene, Bxhas weak\ne\u000bects on spin lifetime of germanene under \f-\nniteEz. This is because the applied externalmagnetic \feld is too weak compared with inter-\nnal B \feld Binunder \fnite Ez. This weak Bx\ndependence of spin lifetime is often used as an\nexperimental evidence of spin-valley locking.10\nImpurity e\u000bects and spin di\u000busion length.\nFinally, we investigate the e\u000bects of the\nelectron-impurity scattering. As an initial the-\noretical investigation, we will consider only\none common neutral defect in germanene in\nthis work - single Ge atom vacancy.42From\nFig. 4(d), we observe that \u001cs;zare reduced\nby introducing impurities and the reduction\nbecomes signi\fcant when impurity density ni\napproaches 1012cm\u00002. Another interesting ob-\nservation is that under \fnite Ez= 5 V/nm, \u001cs;z\nreduces much less than the one with Ez= 0.\nOur simulations suggest that if nican be con-\ntrolled below 1012cm\u00002, especially under a\n\fniteEz, germanene can exhibit long spin life-\ntime over 100 ns at or below 50 K.\nAt the end, we compute in-plane spin di\u000bu-\nsion length ljj;szforz-direction (out-of-plane)\nspin polarization of germanene using the rela-\ntion31ljj;sz=p\nD\u001cs;z, whereDis di\u000busion co-\ne\u000ecient.Dcan be estimated using the gen-\neral form of Einstein relation43D=\u0016c(ne+\nnh)=d(ne+nh)\nd\u0016, where\u0016cis the average of elec-\ntron and hole mobility, which are obtained from\nsolving Boltzmann equation as detailed in the\nmethod section. From Table 1, we can see ljj;sz\nof germanene at 300 K is 2-3 \u0016m, shorter than\nthe longest measured value of graphene sam-\nples,\u001812\u0016m.6At 50 K, as mobilities are higher\nand\u001cs;zare longer, ljj;szbecome 400 \u0016m with-\nout impurities and 120 \u0016m withni=1011cm\u00002,\nwhich are much longer than experimental val-\nues of graphene at di\u000berent temperatures rang-\ning from 1 to 40 \u0016m.9\nConclusions\nBy employing our newly developed \frst-\nprinciples density-matrix dynamics approach,\nwe computed spin lifetime of two Dirac 2D ma-\nterials - silicene and germanene as a function\nof temperature, electrical doping and neutral\nimpurities, as well as applied electric and mag-\n7Table 1: Spin dynamic and transport properties of intrinsic germanene under Ez=5 V/nm without\nand with neutral impurities (with impurity density ni).\u0016cis the average of electron and hole\nmobility. The method of calculating mobility is given in Supporting Information. The theoretical\nresults of electron and hole mobility are given in Fig. S8. Dis di\u000busion coe\u000ecient. ljj;szis spin\ndi\u000busion length of z-direction spin. The formula computing Dandljj;szare given in the main text.\nT (K)ni(cm\u00002)\u0016c(cm2/V/s)D(cm2/s)\u001cs;z(ns)ljj;sz(\u0016m)\n300 0 3.2\u0002104830 0.1 2.9\n300 10112.5\u0002104620 0.1 2.5\n50 0 3.8\u000210616700 97 400\n50 10114.5\u00021052000 76 120\n50 10125.8\u0002104250 26 25\nnetic \felds. We \fnd silicene and germanene\nhave qualitative di\u000berent spin relaxation mech-\nanisms under \fnite Ez\felds. We did systematic\ncomparisons between our FPDM \u001csand those\nestimated by phenomenological models with\n\frst-principles input parameters. We \fnd that\ngermanene out-of-plane spin relaxation can be\nqualitatively understood by EY relation, re-\ngardless of the one with and without E \feld.\nOn the other hand, spin relaxation in silicene is\nmore complicated: although at room temper-\nature, the trends of Ezdependence of silicene\n\u001cs;zcan be captured by a combination of EY\nand DP relation, the temperature dependence\nof\u001cs;zunder \fnite Ezcannot be explained by\nany simpli\fed model relations.\nWe demonstrated giant spin lifetime\nanisotropy (two order of magnitude higher than\ngraphene) and provided the condition for spin-\nvalley locking with long spin lifetime in ger-\nmanene. Speci\fcally, we show that at a low T\n- 50 K,\u001csof germanene can reach 100 ns and\nlscan exceed 100 \u0016m (longer than graphene\nsamples), if impurity density is controlled low\n(\u00141011cm\u00002). This is very promising because\nthe spin-valley locking property has only been\nrealized in either TMDs which usually have\nmuch lower carrier mobility, or graphene on\nsubstrates which have complexity of interfacial\nengineering. The realization of spin-valley lock-\ning in single materials with long spin lifetime\nand ultrahigh mobility opens up highly promis-ing pathways for spin-valleytronic applications.\nMethods\nTo predict spin relaxation time from \frst prin-\nciples, we employ our newly developed ab initio\ndensity-matrix dynamics approach, which in-\ncludes quantum descriptions of various scatter-\ning processes and is applicable to general solid-\nstate systems.29,30The density matrix master\nequation due to the e-ph and e-i scattering in\ninteraction picture reads:\nd\u001a12(t)\ndt=1\n2X\n3458\n>>>>>><\n>>>>>>:[I\u0000\u001a(t)]13\u001a45(t)\u0002h\nPe\u0000ph\n32;45(t) +Pe\u0000i\n32;45(t)i\n\u0000[I\u0000\u001a(t)]45\u001a32(t)\u0002h\nPe\u0000ph\n45;13(t) +Pe\u0000i\n45;13(t)i\u00039\n>>>>>>=\n>>>>>>;\n+H:C:; (1)\nwhere\u001ais density matrix. H.C. is Hermitian\nconjugate. The subindex, e.g., \\1\" is the com-\nbined index of k-point and band. The weights\nof k points must be considered when doing sum\nover k points. Pe\u0000phandPe\u0000iare the gen-\neralized scattering-rate matrices for the e-ph\nand e-i scattering respectively. Note that Pc\nwithcbeing a scattering channel is related\nto its value PS;cin the Schrodinger picture\nasPc\n1234(t) =PS;c\n1234exp [i(\"1\u0000\"2\u0000\"3+\"4)t]:\nPS;cis time independent and is computed from\n8corresponding e-ph or e-i matrix elements and\nelectron and phonon energies.\nAll energies and matrix elements are calcu-\nlated on coarse kandqmeshes using the DFT\nsoftware JDFTx,44and are then interpolated to\nextremely \fne meshes in a basis of maximally\nlocalized Wannier functions.45{47Starting from\nat an initial state with a net spin, we evolve the\ndensity matrix \u001a(t) through the master equa-\ntion Eq. 1 for a long enough simulation time,\ntypically from ns to \u0016s. We then obtain the\nevolution of spin observable Si(t) (i=x;y;z )\nfrom\u001a(t) (Eq. S1). At the end, spin lifetime\n\u001cs;iis obtained by \ftting Si(t) to an exponential\ndecay curve with decay constant \u001cs;i.\nMore details are given in Supporting Informa-\ntion Sec. SI and SII and Ref. 29.\nUsing the same \frst-principles electron and\nphonon energies and matrix elements on \fne\nmeshes, we calculate the carrier mobility by\nsolving the linearized Boltzmann equation us-\ning a full-band relaxation-time approximation12\n(Supporting Information Sec. SVI).\nAuthor contributions\nJ.X. and H.T. performed the ab initio calcula-\ntions and analyses. R.S. and Y.P. designed and\nsupervised all aspects of the study. All authors\ncontribute to the writing of the manuscript.\nAcknowledgements\nWe thank Mani Chandra for helpful discussions.\nThis work is supported by the Air Force Of-\n\fce of Scienti\fc Research under AFOSR Award\nNo. FA9550-YR-1-XYZQ and National Science\nFoundation under grant No. DMR-1956015.\nA. H. acknowledges support from the Ameri-\ncan Association of University Women(AAUW)\nfellowship program. This research used re-\nsources of the Center for Functional Nanomate-\nrials, which is a US DOE O\u000ece of Science Fa-\ncility, and the Scienti\fc Data and Computing\ncenter, a component of the Computational Sci-\nence Initiative, at Brookhaven National Labo-\nratory under Contract No. DE-SC0012704, thelux supercomputer at UC Santa Cruz, funded\nby NSF MRI grant AST 1828315, the National\nEnergy Research Scienti\fc Computing Center\n(NERSC) a U.S. Department of Energy O\u000ece\nof Science User Facility operated under Con-\ntract No. DE-AC02-05CH11231, the Extreme\nScience and Engineering Discovery Environ-\nment (XSEDE) which is supported by National\nScience Foundation Grant No. ACI-154856248,\nand resources at the Center for Computational\nInnovations at Rensselaer Polytechnic Institute.\nReferences\n(1) Schaibley, J. R.; Yu, H.; Clark, G.;\nRivera, P.; Ross, J. S.; Seyler, K. L.;\nYao, W.; Xu, X. Valleytronics in 2D ma-\nterials. Nat. Rev. Mater. 2016 ,1, 1{15.\n(2) Zhang, L.; Gong, K.; Chen, J.; Liu, L.;\nZhu, Y.; Xiao, D.; Guo, H. Generation\nand transport of valley-polarized current\nin transition-metal dichalcogenides. Phys.\nRev. B 2014 ,90, 195428.\n(3) Tao, L.; Naeemi, A.; Tsymbal, E. Y.\nValley-spin logic gates. Phys. Rev. Appl.\n2020 ,13, 054043.\n(4) Han, W.; Kawakami, R. K.; Gmitra, M.;\nFabian, J. Graphene spintronics. Nat.\nNanotechnol. 2014 ,9, 794{807.\n(5) Das Sarma, S.; Adam, S.; Hwang, E. H.;\nRossi, E. Electronic transport in two-\ndimensional graphene. Rev. Mod. Phys.\n2011 ,83, 407{470.\n(6) Drogeler, M.; Franzen, C.; Volmer, F.;\nPohlmann, T.; Banszerus, L.; Wolter, M.;\nWatanabe, K.; Taniguchi, T.;\nStampfer, C.; Beschoten, B. Spin lifetimes\nexceeding 12 ns in graphene nonlocal\nspin valve devices. Nano Lett. 2016 ,16,\n3533{3539.\n(7) Cummings, A. W.; Garcia, J. H.;\nFabian, J.; Roche, S. Giant Spin Lifetime\nAnisotropy in Graphene Induced by Prox-\nimity E\u000bects. Phys. Rev. Lett. 2017 ,119,\n206601.\n9(8) Gmitra, M.; Fabian, J. Proximity E\u000bects\nin Bilayer Graphene on Monolayer WSe 2:\nField-E\u000bect Spin Valley Locking, Spin-\nOrbit Valve, and Spin Transistor. Phys.\nRev. Lett. 2017 ,119, 146401.\n(9) Avsar, A.; Ochoa, H.; Guinea, F.;\nOzyilmaz, B.; van Wees, B. J.; Vera-\nMarun, I. J. Colloquium: Spintronics\nin Graphene and Other Two-Dimensional\nMaterials. Rev. Mod. Phys. 2020 ,92,\n021003.\n(10) Dey, P.; Yang, L.; Robert, C.; Wang, G.;\nUrbaszek, B.; Marie, X.; Crooker, S. A.\nGate-Controlled Spin-Valley Locking of\nResident Carriers in WSe2 Monolayers.\nPhys. Rev. Lett. 2017 ,119, 137401.\n(11) Cheng, L.; Liu, Y. What limits the intrin-\nsic mobility of electrons and holes in two\ndimensional metal dichalcogenides? J.\nAm. Chem. Soc. 2018 ,140, 17895{17900.\n(12) Ciccarino, C. J.; Christensen, T.; Sun-\ndararaman, R.; Narang, P. Dynamics and\nspin-valley locking e\u000bects in monolayer\ntransition metal dichalcogenides. Nano\nLett.2018 ,18, 5709{5715.\n(13) Ezawa, M. Valley-polarized metals and\nquantum anomalous Hall e\u000bect in silicene.\nPhys. Rev. Lett. 2012 ,109, 055502.\n(14) Tsai, W.-F.; Huang, C.-Y.; Chang, T.-R.;\nLin, H.; Jeng, H.-T.; Bansil, A. Gated sil-\nicene as a tunable source of nearly 100%\nspin-polarized electrons. Nat. Commun.\n2013 ,4, 1{6.\n(15) D\u0013 avila, M. E.; Xian, L.; Cahangirov, S.;\nRubio, A.; Le Lay, G. Germanene:\na novel two-dimensional germanium al-\nlotrope akin to graphene and silicene. New\nJ. Phys. 2014 ,16, 095002.\n(16) Zhang, L.; Bampoulis, P.; Rudenko, A.;\nYao, Q. v.; Van Houselt, A.; Poelsema, B.;\nKatsnelson, M.; Zandvliet, H. Structural\nand electronic properties of germanene on\nMoS2. Phys. Rev. Lett. 2016 ,116, 256804.(17) Zhao, J.; Liu, H.; Yu, Z.; Quhe, R.;\nZhou, S.; Wang, Y.; Liu, C. C.; Zhong, H.;\nHan, N.; Lu, J.; et al., Rise of silicene:\nA competitive 2D material. Prog. Mater.\nSci.2016 ,83, 24{151.\n(18) Shao, Z.-G.; Ye, X.-S.; Yang, L.;\nWang, C.-L. First-principles calculation\nof intrinsic carrier mobility of silicene. J.\nAppl. Phys. 2013 ,114, 093712.\n(19) Ye, X.-S.; Shao, Z.-G.; Zhao, H.; Yang, L.;\nWang, C.-L. Intrinsic carrier mobility of\ngermanene is larger than graphene's: \frst-\nprinciple calculations. RSC Adv. 2014 ,4,\n21216{21220.\n(20) Liu, C.-C.; Feng, W.; Yao, Y. Quan-\ntum Spin Hall E\u000bect in Silicene and Two-\nDimensional Germanium. Phys. Rev. Lett.\n2011 ,107, 076802.\n(21) Acun, A.; Zhang, L.; Bampoulis, P.;\nFarmanbar, M.; van Houselt, A.;\nRudenko, A. N.; Lingenfelder, M.;\nBrocks, G.; Poelsema, B.; Katsnel-\nson, M. I.; Zandvliet, H. J. W. Ger-\nmanene: the germanium analogue of\ngraphene. J. Phys. Condens. Matter\n2015 ,27, 443002.\n(22) Han, W.; Kawakami, R. K. Spin relax-\nation in single-layer and bilayer graphene.\nPhys. Rev. Lett. 2011 ,107, 047207.\n(23) Kamalakar, M. V.; Groenveld, C.;\nDankert, A.; Dash, S. P. Long distance\nspin communication in chemical vapour\ndeposited graphene. Nat. Commun. 2015 ,\n6, 1{8.\n(24) Restrepo, O. D.; Windl, W. Full \frst-\nprinciples theory of spin relaxation in\ngroup-IV materials. Phys. Rev. Lett.\n2012 ,109, 166604.\n(25) Cummings, A. W.; Roche, S. E\u000bects of de-\nphasing on spin lifetime in ballistic spin-\norbit materials. Phys. Rev. Lett. 2016 ,\n116, 086602.\n10(26) Habib, A.; Xu, J.; Ping, Y.; Sun-\ndararaman, R. Electric Field and Sub-\nstrate E\u000bects Dominate Spin-Phonon Re-\nlaxation in Graphene. arXiv preprint\narXiv:2012.11550 2020 ,\n(27) Bishnoi, B.; Ghosh, B. Spin transport in\nsilicene and germanene. RSC Adv. 2013 ,\n3, 26153{26159.\n(28) Babaee Touski, S. Spin transport in arm-\nchair silicene nanoribbon on the substrate:\nThe role of charged impurity. Phys. Status\nSolidi B 2019 ,256, 1900082.\n(29) Xu, J.; Habib, A.; Sundararamany, R.;\nPing, Y. Ab initio Ultrafast Spin\nDynamics in Solids. arXiv preprint\narXiv:2012.08711 2020 ,\n(30) Xu, J.; Habib, A.; Kumar, S.; Wu, F.;\nSundararaman, R.; Ping, Y. Spin-Phonon\nRelaxation from a Universal Ab Initio\nDensity-Matrix Approach. Nat. Commun.\n2020 ,11, 2780.\n(31) \u0014Zuti\u0013 c, I.; Fabian, J.; Sarma, S. D. Spin-\ntronics: Fundamentals and Applications.\nRev. Mod. Phys. 2004 ,76, 323.\n(32) Guimar~ aes, M. H.; Zomer, P. J.; Ingla-\nAyn\u0013 es, J.; Brant, J. C.; Tombros, N.; van\nWees, B. J. Controlling spin relaxation\nin hexagonal BN-encapsulated graphene\nwith a transverse electric \feld. Phys. Rev.\nLett.2014 ,113, 086602.\n(33) Fabian, J.; Sarma, S. D. Spin relaxation of\nconduction electrons in polyvalent metals:\nTheory and a realistic calculation. Phys.\nRev. Lett. 1998 ,81, 5624.\n(34) Leyland, W.; Harley, R.; Henini, M.;\nShields, A.; Farrer, I.; Ritchie, D. Os-\ncillatory Dyakonov-Perel spin dynamics\nin two-dimensional electron gases. Phys.\nRev. B 2007 ,76, 195305.\n(35) Kurpas, M.; Junior, P. E. F.; Gmitra, M.;\nFabian, J. Spin-orbit coupling in elemental\ntwo-dimensional materials. Phys. Rev. B\n2019 ,100, 125422.(36) Bronold, F. X.; Martin, I.; Saxena, A.;\nSmith, D. L. Magnetic-\feld dependence of\nelectron spin relaxation in n-type semicon-\nductors. Phys. Rev. B 2002 ,66, 233206.\n(37)wis a weight factor related to how many\npercent of total Szcan be relaxed out by\nintravalley scattering itself. wbeing close\nto 0 and 1 correspond to intravalley scat-\ntering can only relax a small part and most\nof excess spin respectively. In Supporting\nInformation Sec. SVIII and Fig. S11, we\ngive more details about w. .\n(38) Yang, L.; Sinitsyn, N. A.; Chen, W.;\nYuan, J.; Zhang, J.; Lou, J.;\nCrooker, S. A. Long-Lived Nanosec-\nond Spin Relaxation and Spin Coherence\nof Electrons in Monolayer MoS2 and\nWS2. Nat. Phys. 2015 ,11, 830{834.\n(39) Kikkawa, J. M.; Awschalom, D. D. Reso-\nnant Spin Ampli\fcation in n-Type GaAs.\nPhys. Rev. Lett. 1998 ,80, 4313.\n(40) Wu, M.; Jiang, J.; Weng, M. Spin dynam-\nics in semiconductors. Phys. Rep. 2010 ,\n493, 61{236.\n(41) Xu, X.; Yao, W.; Xiao, D.; Heinz, T. F.\nSpin and pseudospins in layered transition\nmetal dichalcogenides. Nat. Phys. 2014 ,\n10, 343{350.\n(42) Hastuti, D. P.; Nurwantoro, P., et al. Sta-\nbility study of germanene vacancies: The\n\frst-principles calculations. Mater. Today\nCommun. 2019 ,19, 459{463.\n(43) Kubo, R. The \ructuation-dissipation the-\norem. Rep. Prog. Phys. 1966 ,29, 255.\n(44) Sundararaman, R.; Letchworth-\nWeaver, K.; Schwarz, K. A.; Gunceler, D.;\nOzhabes, Y.; Arias, T. A. JDFTx: Soft-\nware for Joint Density-Functional Theory.\nSoftwareX 2017 ,6, 278{284.\n(45) Marzari, N.; Vanderbilt, D. Maximally Lo-\ncalized Generalized Wannier Functions for\nComposite Energy Bands. Phys. Rev. B\n1997 ,56, 12847.\n11(46) Brown, A. M.; Sundararaman, R.;\nNarang, P.; Goddard, W. A.; Atwa-\nter, H. A. Nonradiative Plasmon Decay\nand Hot Carrier Dynamics: E\u000bects of\nPhonons, Surfaces, and Geometry. ACS\nNano 2016 ,10, 957{966.\n(47) Habib, A.; Florio, R.; Sundararaman, R.\nHot Carrier Dynamics in Plasmonic Tran-\nsition Metal Nitrides. J. Opt. 2018 ,20,\n064001.\n(48) Towns, J.; Cockerill, T.; Dahan, M.; Fos-\nter, I.; Gaither, K.; Grimshaw, A.; Ha-\nzlewood, V.; Lathrop, S.; Lifka, D.; Pe-\nterson, G. D.; Roskies, R.; Scott, J. R.;\nWilkins-Diehr, N. XSEDE: Accelerating\nScienti\fc Discovery. Comput. Sci. Eng.\n2014 ,16, 62{74.\n12"    },    {        "title": "0710.5260v1.On_the_nature_of_steady_states_of_spin_distributions_in_the_presence_of_spin_orbit_interactions.pdf",        "content": "arXiv:0710.5260v1  [cond-mat.mes-hall]  28 Oct 2007On the nature of steady states of spin distributions in the pr esence of spin-orbit\ninteractions\nDimitrie Culcer and R. Winkler\nAdvanced Photon Source, Argonne National Laboratory, Argo nne, IL 60439. and\nNorthern Illinois University, De Kalb, IL 60115.\n(Dated: May 26, 2018)\nIn the presence of spin-orbit interactions, the steady stat e established for spin distributions in an\nelectric field is qualitatively different from the steady sta te for charge distributions. This is primarily\nbecause the steady state established for spin distribution s involves spin precession due to spin-orbit\ncoupling. We demonstrate in this work that the spin density m atrix in an external electric field\nacquires two corrections with different dependencies on the characteristic momentum scattering\ntime. One part is associated with conserved spins, diverges in the clean limit and is responsible\nfor the establishment of a steady-state spin density in elec tric fields. Another part is associated\nwith precessing spins, is finite in the clean limit and is resp onsible for the establishment of spin\ncurrents in electric fields. Scattering between these distr ibutions has important consequences for\nspin dynamics and spin-related effects in general, and expla ins some recent puzzling observations,\nwhich are captured by our unified theory.\nI. INTRODUCTION\nSpin-orbit interactions are frequently the most impor-\ntant factor determining spin dynamics in solids. From a\ntechnologicalperspective, novelphysicalphenomenathat\nmay lead to improved memory devices and advances in\nquantum information processing have been shown to be\nintimately related to spin-orbit interactions.1Practical\napplications usually rely on generating and maintaining\na spin polarization, and in this context spin-orbit\ninteractions can play a constructive or a destructive role.\nIn an external electric field spin precession gives rise to\nsteady-state spin densities2,3,4,5,6,7,8,9,10,11,12and spin\ncurrents,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29, 30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46, 47,48,49,50\nyet spin precession is often the leading cause of spin po-\nlarization decay.51,52,53,54These facts suggest that spin\nprecession plays a nontrivial role in the establishment of\na steady state for spin distributions in electric fields.\nSpin-orbit interactions are present in the band struc-\nture and in potentials due to impurity distributions.\nSpin-orbit coupling is in principle always present in im-\npurity potentials and gives rise to skew scattering, which\nleads to the extrinsic anomalous Hall effect and the ex-\ntrinsic spin-Hall effect.55Band structure spin-orbit cou-\npling may arise from the inversion asymmetry of the\nunderlying crystal lattice56(bulk inversion asymmetry),\nfrom the inversion asymmetry of the confining poten-\ntial in two dimensions57(structure inversion asymme-\ntry), and may be present also in inversion symmetric\nsystems.58Band structure spin-orbit coupling gives rise\nto spin precession, is the cause of magnetic anisotropy\nin magnetic semiconductors,59and causes spin flips even\nin the course of elastic scattering by spin-independent\npotentials.53,60Band structure spin-orbit interactions in\nspin-1/2 electron systems can always be represented by\na Zeeman-like Hamiltonian H= (1/2)σ·Ωkdescrib-\ning the interaction of the spin with an effective wave\nvector-dependent magnetic field Ωk. An electron spin\nat wave vector kprecesses about this field with fre-quency Ω k//planckover2pi1≡ |Ωk|//planckover2pi1and is scattered to a different\nwave vector within a characteristic momentum scatter-\ning time τp. Throughout this paper we assume that\nεFτp//planckover2pi1≫1, where εFis the Fermi energy, which is tan-\ntamount to assuming that the carriers’ mean free path is\nmuch larger that their de Broglie wavelength. Within\nthis range, the relative magnitude of the spin preces-\nsion frequency Ω kand inverse scattering time 1 /τpde-\nfine three qualitatively different regimes. In the ballis-\ntic (clean) regime no scattering occurs and the temper-\nature tends to absolute zero, so that εFτp→ ∞and\nΩkτp//planckover2pi1→ ∞. The weak scattering regime is character-\nized by fast spin precession and little momentum scatter-\ning due to, e.g., a slight increase in temperature, yielding\nεFτp//planckover2pi1≫Ωkτp//planckover2pi1≫1. In the strong momentum scat-\ntering regime εFτp//planckover2pi1≫1≫Ωkτp//planckover2pi1.\nIn this paper, we will be concerned with the inter-\nplay of external fields, scattering processes, and band\nstructure spin-orbit interactions in establishing a steady\nstate for spin distributions. We will concentrate on\nelectrons in uniform electric fields. In charge transport,\nthe steady state is characterized by a nonequilibrium\ncorrection to the density matrix that is divergent in\nthe clean limit, indicating a competition between the\nelectric field, accelerating charge carriers, and scattering,\nwhich inhibits their forward motion. On the other hand,\nnonequilibrium corrections that arise as a result of band\nstructure spin-orbit coupling in crystal Hamiltonians re-\npresent a different kind of interplay between the electric\nfield and scattering processes. Firstly, the spin-orbit\nsplitting of the bands gives rise to spin-dependent scat-\ntering even from spin-independent potentials. Secondly,\nthe presence of spin precession causes the steady state\nestablished for spin distributions to be highly nontrivial.\nWe will demonstrate that the density matrix contains\na contribution due to precessing spins and one due to\nconserved spins, and the steady states established for\neach of these are qualitatively different. Steady state\ncorrections ∝τp, which diverge in the clean limit, are2\nassociated with the absenceof spin precession and give\nrise to spin densities in external fields.2,3,4,5,6,7,8,9,10,11,12\nSteady state corrections independent of τp, which\nare finite in the clean limit, are associated with spin\nprecession and give rise to spin currents in external\nfields.13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29, 30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46, 47,48,49,50\nScattering between these two distributions induces sig-\nnificant corrections to steady-state spin densities and\nspin currents.\nIn research on electrical generation of spin den-\nsities and currents two rather different approaches\nare employed. Linear response theories based on\nGreen’s functions4,7,32,33,34,35,36,37,38,46,55provide a dia-\ngrammatic interpretation of spin-related phenomena in\nelectric fields. Such theories usually provide reliable re-\nsults, but the physical picture is not always evident.\nTheories based on a kinetic equation for the density\nmatrix3,5,8,39,40,41,42tend to be transparent. However,\nit is often difficult, in such theories, to determine from\nthe outset all the terms that play an important role in\nthe processes under study. Correspondences between the\ntwo approaches mentioned were identified in a recent pa-\nper bySinitsyn et al.61The formalismused in the present\npaper, although relyingon a density matrix, is equivalent\nto Kubo linear response theory.\nWe consider large, uniform systems, working in mo-\nmentum space without making semiclassical approxima-\ntions (by which we understand approximations pertain-\ning to simultaneous consideration of a particle’s position\nand momentum.) The role of scattering in spin-related\neffects is an issue that is currently under intense inves-\ntigation. In our general formalism, the terms responsi-\nble for scattering are derived rigorously beginning with\nthe quantum Liouville equation for the density operator.\nOne particular issue at the center of current debate is\nthe physical interpretation of vertex corrections in spin-\nrelated phenomena. In charge transport, it is well known\nthat vertex correctionsdecreasethe weight ofsmall-angle\nscattering. We demonstrate that, in spin transport, scat-\ntering phenomena have two main consequences. Firstly,\nthe driving term in the equation for the spin density ac-\nquires a contribution due to spin-dependent scattering.\nSecondly, scattering between the distributions of con-\nserved and precessingspins enters the equations for these\ntwo distributions. Our work suggests that both of these\nprocesses are contained in vertex corrections.\nIn a recent related article,62we focused on the spin\ncurrent response of a semiconductor to an electric field,\nshowing that spin currents are not restricted to the spin-\nHalleffect anddwellingbrieflyonthe distinctionbetween\nnonequilibrium spin currents and spin densities excited\nby electric fields. In this article, we concentrate on the\nsteady state and on the difference between the steady\nstate established for spin distributions and that estab-\nlished for charge distributions. We investigate aspects\nof the steady state for spin distributions that arise as a\nresult of spin precession and have no analog in charge\ndistributions. We provide a detailed analysis of the dis-tributionsofconservedandprecessingspins, andattempt\nto shed light on the physical content of vertex corrections\nin systems with spin-orbit interactions. In the process,\nwe give a detailed exposition of the underlying theory,\nwhich was also employed in our recent work.62\nThe outline of this paper is as follows. In the next\nsection we start from the quantum Liouville equation\nand derive an equation describing the time evolution of\nthe electron density matrix including the full scattering\nterm in the first Born approximation. Section III is de-\nvotedto the complexcaseofsteady statesin the presence\nof band structure spin-orbit coupling, demonstrating the\nexistence of two spin distributions and their subtle inter-\nplay. Analytical expressions are given for general scat-\ntering as far as possible, as well as exact solutions for\nshort-range impurity scattering. We close with a sum-\nmary of our findings.\nII. TIME EVOLUTION OF THE DENSITY\nOPERATOR\nWe consider a system of non-interacting spin-1/2 elec-\ntrons. The electrons are represented by a one-particle\ndensity operator ˆ ρ. The expectation value of an observ-\nable represented by a Hermitian operator ˆOis given by\ntr(ˆρˆO), which motivates us to study the density opera-\ntor ˆρin detail. The dynamics of ˆ ρare described by the\nquantum Liouville equation, which is projected onto a\nset of states of definite wave vector (in which the matrix\nelements of ˆ ρform the density matrix), and an equation\nis obtained for the time evolution of the density matrix.\nThisequationisvalidforanyelasticscatteringinthefirst\nBorn approximation. In this article we discuss systems\nwith long mean free paths and do not consider diffusion\nterms explicitly. Consequently, semiclassical approxima-\ntions are not necessary, and the method employed in the\npresent work is equivalent to the Kubo linear response\nformalism.\nA. Quantum Liouville equation\nThe quantum Liouville equation satisfied by ˆ ρis\ndˆρ\ndt+i\n/planckover2pi1[ˆH+ˆU,ˆρ] = 0. (1)\nThe Hamiltonian ˆH, considered here in the framework\nof the envelope function approximation (EFA), contains\ncontributions due to the kinetic energy and spin-orbit\ncoupling. The effect of the lattice-periodic potential of\nthe ions is taken into account through a replacement of\nthe carrier mass by the effective mass. We shall hence-\nforth refer to ˆHas the EFA Hamiltonian. Scattering\nis introduced into the system through the potential ˆU,\nwhich may be due to impurities, phonons, surface rough-\nness, or other perturbations. In this article we focus on\nimpurity scattering, as the effects that are discussed are3\nfrequently observed at very low temperatures, where the\nrole of phonon scattering may be neglected.\nThe Liouville equation is projected onto a set of time-\nindependent states of definite wave vector {|ks∝an}bracketri}ht}, which\nare not assumed to be eigenstates of the EFA Hamilto-\nnianˆH. The matrix elements of ˆ ρin this basis will be\nwritten as ρkk′≡ρss′\nkk′=∝an}bracketle{tks|ˆρ|k′s′∝an}bracketri}ht, with correspond-\ning notations for the matrix elements of ˆHandˆU. Spin\nindices will not be shown explicitly in our subsequent\nderivation, the quantities ρkk′,Hkk′, andUkk′being\ntreated as matrices in spin space. ρkk′is referred to as\nthe density matrix. With our choice of basis functions of\ndefinite wave vector, matrix elements of the Hamiltonian\nHkk′=Hkδkk′are diagonal in k. However, if the EFA\nHamiltonian contains spin-orbit coupling terms, the ma-\ntrix elements Hkare generally off-diagonal in spin space.\nMatrix elements of the scattering potential Ukk′are off-\ndiagonal in k. Matrix elements diagonal in kin the scat-\ntering potential would lead to a redefinition of Hk, which\nis analogous,in Green’sfunction formalisms, to the offset\nintroduced by the real part of the self energy. Scattering\nis assumed elastic and, given that we will work in the\nfirst Born approximation, the matrix Ukk′is assumed\ndiagonal in spin. (We thus do not take into account\nso-called skew scattering terms, which require terms of\nthird order in Ukk′as well as explicit inclusion of spin-\norbitcoupling in the scatteringpotential, and whichwere\nstudied in a recent work.42) As mentioned above, we fo-\ncus in this article on impurity scattering. The impurities\nare assumed uncorrelated and our normalization is such\nthat∝an}bracketle{tks|ˆU|k′s′∝an}bracketri}ht∝an}bracketle{tk′s′|ˆU|ks∝an}bracketri}ht=ni|Ukk′|2δss′, whereniis\nthe impurity density. Ukk′refers therefore to the matrix\nelements of the potential due to a single impurity. Ex-\nplicit expressionsforthese matrixelements for ascreened\nCoulomb potential in two and three dimensions are given\nin Appendix A.\nρkk′is divided into a part diagonal in kand a part\noff-diagonal in k, given by ρkk′=fkδkk′+gkk′, where,\ningkk′, it is understood that k∝ne}ationslash=k′. The quantum\nLiouville equation can be broken down into equations for\nfkandgkk′\ndfk\ndt+i\n/planckover2pi1[Hk,fk] =−i\n/planckover2pi1[ˆU,ˆg]kk, (2a)\ndgkk′\ndt+i\n/planckover2pi1[ˆH,ˆg]kk′=−i\n/planckover2pi1[ˆU,ˆf+ ˆg]kk′.(2b)In the first Born approximation the solution to Eq. (2b)\ncan be written as\ngkk′=−i\n/planckover2pi1/integraldisplay∞\n0dt′e−iˆHt′/bracketleftBig\nˆU,ˆf(t−t′)/bracketrightBig\neiˆHt′|kk′.(3)\nIn order to shorten the equations, factors of /planckover2pi1that ap-\npear in the time evolution operators will be omitted, i.e.,\neiˆHt′≡eiˆHt′//planckover2pi1. These factors will be restored in the fi-\nnal results. Since εFτp//planckover2pi1≫1, we shall expand ˆf(t−t′)\nin the time integral around tand, noting that terms be-\nyondˆf(t) are of higher order in the scattering potential,\nwe shall only retain the first term, ˆf(t). The equation\nforfkthen becomes\ndfk\ndt+i\n/planckover2pi1[Hk,fk]+ˆJ(fk) = 0, (4a)\nin which the scattering term ˆJ(fk) is given by\nˆJ(fk) =1\n/planckover2pi12/integraldisplay∞\n0dt′/bracketleftBig\nˆU,e−iˆHt′/bracketleftBig\nˆU,ˆf(t)/bracketrightBig\neiˆHt′/bracketrightBig\nkk.(4b)\nThe integral over time in Eq. (4b) can be performed by\ninserting a regularizing factor e−��t′and letting η→0\nsubsequently. We remark that, for potentials diagonal in\nspin space and spin-degenerate bands, Eq. (4b) simpli-\nfies to the customary expression for Fermi’s golden rule.\nTherefore, Eq. (4) can be viewed as a generalization of\nFermi’s golden rule that explicitly takes into account the\nspin degree of freedom. The commutator present in Eq.\n(4a) is a commutator in spin space that represents the\neffect of spin precession due to spin-orbit interactions.\nB. Scattering term\nThe scattering term ˆJ(fk) will now be evaluated for\nan electron system with spin-orbit interactions. After\ninserting a complete set of states, ˆJ(fk) becomes66\nˆJ(fk) =ni\n/planckover2pi12lim\nη→0/integraldisplay∞\n0dt′e−ηt′/summationdisplay\nk′/bracketleftBig\nUkk′e−iHk′t′(Uk′kfk−fk′Uk′k)eiHkt′−e−iHkt′(Ukk′fk′−fkUkk′)eiHk′t′Uk′k/bracketrightBig\n.\n(5)\nThe EFA Hamiltonian contains a kinetic energy term and a spin-orbit co upling term, Hk=Hkin\nk+Hso\nk. In spin-1/2\nelectron systems, band structure spin-orbit coupling can alwaysb e representedas a Zeeman-like interactionof the spin\nwith a wave vector-dependent effective magnetic field Ωk, thusHso\nk= (1/2)σ·Ωk. Common examples of effective\nfields are the Rashba spin-orbit interaction,57which is often dominant in quantum wells with inversion asymmetry,\nand the Dresselhaus spin-orbit interaction,56which is due to the inversionasymmetry of the underlying crystal lat tice.4\nThe spin-orbit interaction for electrons is usually much smaller than t he kinetic energy at typical Fermi energies, with\nthe result that terms that are second order in the ratio of the two ,Hso\nk/Hkin\nk, can usually be ignored. In the basis in\nspin space spanned by spin eigenstates | ↑∝an}bracketri}htand| ↓∝an}bracketri}ht(commonly referred to as the Pauli basis), taking into account\nthe fact that Ukk′is diagonal in spin space, the scattering term simplifies to\nˆJ(fk) =niVd\n/planckover2pi12lim\nη→0/integraldisplay∞\n0dt′e−ηt′/integraldisplayddk′\n(2π)dWkk′/bracketleftBig\ne−iˆHk′t′/parenleftbig\nfk−fk′/parenrightbig\neiHkt′+e−iHkt′/parenleftbig\nfk−fk′/parenrightbig\neiHk′t′/bracketrightBig\n.(6)\nHereWkk′=|Ukk′|2is the transition rate, and sums over wave vector have been conve rted into integrals following\nthe standard procedure/summationtext\nk′→Vd/integraltextddk′/(2πd), where dis the dimensionality of the system and the normalization\nvolumeVdis chosen to be the d-dimensional unit cell volume. The density matrix fkis decomposed into a scalar part\nand a spin-dependent part, fk=nk11+Sk. Performing the time integral, after a series of lengthy but straigh tforward\ncalculations, which are summarized in Appendix B, the scattering ter m can be expressed in the form ( ˆJ0+ˆJs)(nk)+\nˆJ0(Sk), with\nˆJ0(Xk) =πniVd\n2/planckover2pi1/integraldisplayddk′\n(2π)dWkk′(Xk−Xk′)/bracketleftbig\nδ(ǫk+−ǫ′\nk+)+δ(ǫk−−ǫ′\nk−)+δ(ǫk+−ǫ′\nk−)+δ(ǫk−−ǫ′\nk+)/bracketrightbig\n,(7a)\nˆJs(Xk) =πniVd\n2/planckover2pi1/integraldisplayddk′\n(2π)dWkk′(Xk−Xk′)σ·(ˆΩk+ˆΩk′)/bracketleftbig\nδ(ǫk+−ǫ′\nk+)−δ(ǫk−−ǫ′\nk−)/bracketrightbig\n+πniVd\n2/planckover2pi1/integraldisplayddk′\n(2π)dWkk′(Xk−Xk′)σ·(ˆΩk−ˆΩk′)/bracketleftbig\nδ(ǫk+−ǫ′\nk−)−δ(ǫk−−ǫ′\nk+)/bracketrightbig\n. (7b)\nXkrepresents either nkorSk, andˆΩkis a unit vector\nalongΩk. The energies ǫk±represent the eigenvalues\nofHk, and are given by ǫk±=ε0k±Ωk/2, where the\nkineticenergy ε0k=/planckover2pi12k2/2m∗andΩk=|Ωk|. Theterm\nˆJs(Xk) in Eq. (7b) illustrates the fact that, when spin-\norbit interactions are present in the band structure, even\na spin-independent scattering potential usually gives rise\nto spin-dependent scattering in the scattering integral.\nC. Time evolution of the density matrix in an\nexternal electric field\nIn the presence of a constant uniform electric field E,\nk=q−eEt//planckover2pi1is the gauge-invariant crystal momentum\n(withqthe canonical momentum.) The states |ks∝an}bracketri}htare\nchosentohavethe form |ks∝an}bracketri}ht=eiq·r|uks∝an}bracketri}ht, where|uks∝an}bracketri}htare\nlattice-periodic functions. We subdivide fk=f0k+fEk,\nwhere the equilibrium density matrix f0kis given by the\nFermi-Dirac distribution, and the correction fEkis due\nto the electric field E. To first order in E, the correction\nfEksatisfies\n∂fEk\n∂t+i\n/planckover2pi1[H,fEk]+ˆJ(fEk) =eE\n/planckover2pi1·∂f0k\n∂k.(8)\nThe matrix fEkis in turn divided, as above, into a scalar\npart and a spin-dependent part, fEk=nEk11 +SEk.\nTo first order in Hso\nk/Hkin\nk, the scattering term can be\nexpressed as ˆJ(fEk) = (ˆJ0+ˆJs)(nEk)+ˆJ0(SEk), where\nthe scattering operators have been defined in Eq. (7).\nThe driving term arising from the electric field in Eq.\n(8) is (eE//planckover2pi1)·∂f0k/∂k. The equilibrium density matrix,f0k, is subdivided as f0k=n0k11 +S0k, with a corre-\nsponding subdivision for the driving term. The equation\nfornEkis\n∂nEk\n∂t+ˆJ0(nEk) =eE\n/planckover2pi1·∂n0k\n∂k. (9)\nThe solution of this equation is given by the well-known\nexpression\nnEk=eEτp\n/planckover2pi1·∂n0k\n∂k, (10)\nin other words, nEkdescribes the shift of the Fermi\nsphere in the presence of the electric field E. The ex-\npression for the momentum relaxation time τpis a little\ndifferent depending on the dimensionality of the system.\nIn three dimensions, using γto denote the relative angle\nbetween kandk′,\n1\nτd=3p=mkVd=3\n2π/planckover2pi13/integraldisplayπ\n0dγsinγWkk′(1−cosγ).(11a)\nIn two dimensions, with the same notation for γ,\n1\nτd=2p=mVd=2\n2π/planckover2pi13/integraldisplay2π\n0dγWkk′(1−cosγ).(11b)\nThe spin-dependent part of the nonequilibrium correc-\ntion to the density matrix SEkis interpreted as the spin\ndensity induced by E. The equation governing the time\nevolution of SEkis\n∂SEk\n∂t+i\n/planckover2pi1[Hk,SEk]+ˆJ0(SEk) =eE\n/planckover2pi1·∂S0k\n∂k−ˆJs(nEk).\n(12)5\nIt is seen from Eq. (12) that spin-dependent scattering\ngives rise to a renormalization of the driving term in the\nequation for SEk. Evidently, this renormalization has no\nanalog in charge transport, see Eq. (9).\nSo far, our work has not been restricted to the steady\nstate. In this context, we remark briefly that Eq. (12)\ncan be used to describe spin relaxation, and is valid\nboth in the presence and in the absence of external elec-\ntric fields.54(In the presence of electric fields SEkis an\nelectric-field-induced nonequilibrium correction, whereas\nin their absence it is to be interpreted more generally as\na nonequilibrium correction.)\nIII. STEADY STATES FOR CONSERVED AND\nPRECESSING SPINS\nIn the presence of band structure spin-orbit interac-\ntions, an electron spin at wave vector kprecesses about\nan effective magnetic field Ωk. The spin can be resolved\ninto components parallel and perpendicular to Ωk. In\nthe course of spin precession the component of the spin\nparallel to Ωkis conserved, while the perpendicular com-\nponent is continually changing. It will proveuseful in our\nanalysis to divide the spin distribution into a part repre-\nsenting conservedspin and a part representingprecessing\nspin. This is accomplished below.\nA. Distributions of conserved and precessing spins\nFirstly, the effectivesourceterm, whichenterstheRHS\nof Eq. (12), is divided into two parts, ( eE//planckover2pi1)·∂S0k/∂k−\nˆJs(nEk) = ΣEk/bardbl+ΣEk⊥. Here,Σ Ek/bardblcommuteswiththe\nspin-orbit Hamiltonian and is given by Σ Ek/bardbl=αkHso\nk,\nwhere\nαk=tr/braceleftbigg/bracketleftbiggeE\n/planckover2pi1·∂S0k\n∂k−ˆJs(nEk)/bracketrightbigg\nHso\nk/bracerightbigg\ntr[(Hso\nk)2],(13)\nwhile Σ Ek⊥is the remainder. In matrix language Σ Ek⊥\nisorthogonal to the spin-orbit Hamiltonian and thus\ntr(ΣEk⊥Hso\nk) = 0. Projections onto and orthogonal to\nHso\nkare most easily carried out by defining projectors\nP/bardblandP⊥. The actions of these projectors on the basis\nmatrices σiare given by\nP/bardblσi=2ΩkiHso\nk\nΩ2\nk, (14a)\nP⊥σx=(Ω2\nky+Ω2\nkz)σx−ΩkxΩkyσy−ΩkxΩkzσz\nΩ2\nk,\n(14b)\nand the actions of P⊥on the remaining basis matrices\nare obtained by cyclic permutations.\nSecondly, SEkislikewisedivided intotwoterms: SEk/bardbl,\ncommuting with the spin-orbit Hamiltonian and SEk⊥,orthogonal to it. It is helpful to think of SEk/bardblas the\ndistribution of conserved spins. SEk⊥can be thought of\nas the distribution of precessing spins. Equation (12) is\ndivided into separate equations for SEk/bardblandSEk⊥:\n∂SEk/bardbl\n∂t+P/bardblˆJ0(SEk) = ΣEk/bardbl, (15a)\n∂SEk⊥\n∂t+i\n/planckover2pi1[Hk,SEk⊥]+P⊥ˆJ0(SEk) = ΣEk⊥.(15b)\nThe absence of the commutator [ Hk,SEk/bardbl] = 0 in Eq.\n(15a) indicates the absence of spin precession, while the\ncommutator [ Hk,SEk⊥] in Eq. (15b) represents spin pre-\ncession. In order to solve Eqs. (15a) and (15b) for ar-\nbitrary scattering, it is necessary to expand SEk/bardbland\nSEk⊥in the transition rate Wkk′, as\nSEk/bardbl=S(−1)\nEk/bardbl+S(0)\nEk/bardbl+S(1)\nEk/bardbl+O(W2\nkk′),(16a)\nSEk⊥=S(0)\nEk⊥+S(1)\nEk⊥+O(W2\nkk′). (16b)\nThis expansion is indeed an expansion in the parameter\n/planckover2pi1/(Ωkτp), a fact that can be most clearly seen by exam-\nining Eq. (16b) and noting that the calculation of each\nterm involves integration over time, which brings in a\nfactor of 1 /Ωk, and the action of ˆJ0. This expansion is\ntherefore most suited to systems in the weak scattering\nregime. The expansion of SEk/bardblstarts at order −1, a fact\nwhich can be understood by inspecting Eq. (15a). In the\nsteady state the time derivative drops out, and the oper-\natorˆJ0is first order in Wkk′, while the right-hand side\nis independent of Wkk′. As a result, the expansion of\nthe solution must start at order −1. We examine next\nEq. (15b) for SEk⊥. SinceHkis independent of Wkk′,\nand the right hand side is also independent of Wkk′, the\nexpansion of SEk⊥must start at order zero.\nHaving divided the nonequilibrium correction to\nthe spin density matrix into a part due to con-\nserved spin and one due to precessing spin, we\nwish to determine the contributions these parts\nmake to spin densities3,4,5,6,7,8,9,10,11,12and spin\ncurrents13,23,24,26,27,28,29,30,31,32,33,34,35,36,37,38,39, 40,41,42,43,44,45,46,47,48\nin the steady state. The steady state density of spin\ncomponent σis found by taking the trace tr(ˆ sσSEk),\nwhere ˆsσ= (/planckover2pi1/2)σσ. The steady-state spin cur-\nrent is found by taking the trace tr( ˆJσ\niSEk), where\nˆJσ\ni=/planckover2pi1kjsσ/m∗+ (1/4/planckover2pi1)∂Ωσ/∂kj11. (The scalar term\nhas zero expectation value.) Henceforth, for clarity\nand definiteness, integrals over wave vectors will be\nrepresented as two-dimensional. In the integrals below\nθ′refers to the polar angle of k′. The extension to three\ndimensions is straightforward.\nB. Steady state for conserved spins\nWe have shown that the expansion of SEk/bardblbegins at\norder−1, and we wish to find the first term in this ex-\npansion. In the steady state, Eq. (15a) for the first term6\nin this expansion can be written as\nP/bardblˆJ0(S(−1)\nEk) = ΣEk/bardbl. (17)\nThis equation can be recast as\nS(−1)\nEk/bardbl\nτ0−P/bardblˆJ′\n0(S(−1)\nEk/bardbl) = ΣEk/bardbl, (18)\nwhere we have introduced\nτ0=m∗\n2π/planckover2pi13/integraldisplay\ndθ′Wkk′, (19a)\nˆJ′\n0S(−1)\nEk/bardbl=m∗\n2π/planckover2pi13/integraldisplay\ndθ′Wkk′S(−1)\nEk′/bardbl.(19b)\nτ0is the quantum lifetime of the charge carriers, i.e., the\ntime between twoconsecutive scatteringevents. It differs\nfromthemomentumscatteringtimeofEqs.(11)because,\nfor nonisotropic scattering mechanisms, the information\nabout the initial momentum is not lost after time τ0.\nEquation (18) can be solved iteratively for any scattering\nS(−1)\nEk/bardbl= ΣEk/bardblτ0+P/bardblˆJ′\n0(ΣEk/bardbl)τ2\n0\n+P/bardblˆJ′\n0[P/bardblˆJ′\n0(ΣEk/bardbl)]τ3\n0+...(20)\nThe equations for higher orders in Wkk′are easily de-\nduced. However, the term of order −1 is by far the dom-\ninant one in the weak momentum scattering regime and\nis expected to be dominantovera wide rangeofstrengths\nof the scattering potential.\nWe examinemorecloselythe natureofthe steadystate\nestablished for conserved spins. It is evident that this\nsteady state involves no spin precession, and that the\ncorrection SEk/bardbldepends explicitly onthe nonequilibrium\nshift in the Fermi surface and diverges in the ballistic\nregime, as τ0→ ∞. In addition, it is important to\nnote that scattering terms contain only the even func-\ntionWkk′. As a result, the correction SEk/bardbldoes not give\nrise to a spin current. Inspection of Eq. (20) shows that\nintegrals of the form\n/integraldisplay\ndθˆJσ\niSEk/bardbl (21)\ncontain an odd number of powers of kand are therefore\nzero. Consequently, in the absence of impurity spin-orbit\ninteractions, the distribution of conserved spins can give\nno spin current. It can, however, give rise to a nonequi-\nlibrium spin density2,3,4,5,6,7,8,9,10,11,12, since integrals of\nthe form\n/integraldisplay\ndθˆsσSEk/bardbl (22)\ncontain an even number of powers of kand may be\nnonzero.C. Steady state for precessing spins\nThe equations for the contributions to SEk⊥of orders\nzero and one in Wkk′are\n∂S(0)\nEk⊥\n∂t+i\n/planckover2pi1[Hk,S(0)\nEk⊥] = Σ Ek⊥+P⊥ˆJ′\n0(SEk/bardbl),(23a)\n∂S(1)\nEk⊥\n∂t+i\n/planckover2pi1[Hk,S(1)\nEk⊥] =P⊥ˆJ′\n0(S(0)\nEk⊥). (23b)\nTo solve the equation for S(0)\nEk⊥it is easiest to go into the\ninteraction picture, obtain an expression for S(0)\nEk⊥, and\nthen transform back to the Schr¨ odinger picture. This\nprocedure yields for S(0)\nEk⊥\nS(0)\nEk⊥=1\n2ˆΩk×[ΣEk⊥+P⊥ˆJ′\n0(SEk/bardbl)]·σ\nΩk//planckover2pi1,(24)\nwhere we have written Σ Ek⊥= (1/2)ΣEk⊥·σand\nSEk/bardbl= (1/2)SEk/bardbl·σ. The result expressed by Eq.\n(24) is valid for any elastic scattering. Since it does\nnot depend explicitly on the form of the impurity po-\ntential, this term is usually regarded as intrinsic. Terms\nof higher order in Wkk′are regarded as extrinsic because\nthey depend explicitly on the form of the impurity po-\ntential. Nevertheless, it is evident from our work that, if\nΣEk⊥+P⊥ˆJ′\n0(SEk/bardbl) vanishes, then S(0)\nEk⊥vanishes and\nall the terms in SEkof higher order in Wkk′also vanish.\nThe steady state established for precessing spins is fi-\nnite in the clean limit. An argumentsimilar to that given\nabove for the distribution of conserved spin SEk/bardblshows\nthatS(0)\nEk⊥cannot lead to a nonequilibrium spin density\n(although, as will be shown below, higher-order terms in\nSEk⊥can contribute to the spin density). For, taking\nthe expectation value of the spin operator, one arrives\nat integrals of the form/integraltextdθˆsσS(0)\nEk⊥, which involve odd\nnumbers of powersof kand are thereforezero. This term\nin the distribution of precessing spin does, however, give\nrise to nonzero spin currents, since integrals if the form/integraltext\ndθˆJσ\niSEk⊥contain an even numbers of powers of k\nand may be nonzero. Consequently, in the absence of\nspin-orbit coupling in the scattering potential, nonequi-\nlibrium spin currents arise from spin precession.\nIn orderto investigateterms ofhigher orderin Wkk′, it\nis easiest to examine a concrete case. Although the gen-\neral conclusions apply to any elastic, spin-independent\nscattering, this analysis will be done in the next section\nin the context of short-range impurity scattering, where\nan exact solution is possible, which reveals an interesting\nphysical picture.\nD. Short-range impurities\nAn enlightening closed-form solution can be found for\nSEk/bardblandSEk⊥for short-range impurities. In this case7\nWkk′≡Wis a constant and ˆJ0SEk= (SEk−¯SEk)/τp,\nwhere the scattering time τp=/planckover2pi13/(mW) and the bar\nrepresents averaging over directions in k. Equation (20)\nyields a closed-form solution for SEk/bardbl\nSEk/bardbl= ΣEk/bardblτp+P/bardbl(1−¯P/bardbl)−1¯ΣEk/bardblτp.(25)\nThis solution enters Eq. (24) for S(0)\nEk⊥. The equations\nfor the contributions to SEk⊥of higher orders in Wcan\nbe easily determined\nS(1)\nEk⊥=1\n2ˆΩk×{ˆΩk×[ΣEk⊥−ˆJ0(SEk/bardbl)]}·σ\nΩ2\nkτp,(26a)\nS(2)\nEk⊥=1\n2ˆΩk×{ˆΩk×{ˆΩk×[ΣEk⊥−ˆJ0(SEk/bardbl)]}}·σ\nΩ3\nkτ2p,\n(26b)\nand so on. Since Σ Ek⊥is orthogonal to the spin-orbit\nHamiltonian Hso\nk, we have tr(Σ Ek⊥Hso\nk) = 0, which tells\nus immediately that ΣEk⊥⊥ˆΩk. As a result, ˆΩk׈Ωk×\nΣEk⊥=−ΣEk⊥, and\nS(1)\nEk⊥=−1\n2[ΣEk⊥−ˆJ0(SEk/bardbl)]·σ\nΩ2\nkτp//planckover2pi12,(27a)\nS(2)\nEk⊥=−1\n2ˆΩk×[ΣEk⊥−ˆJ0(SEk/bardbl)]·σ\nΩ3\nkτ2p//planckover2pi13.(27b)\nIt is evident that the terms S(odd)\nEk⊥andS(even)\nEk⊥give two\nseparate geometric progressions. These progressions are\neasily summed to give for SEk⊥\nSEk⊥=Ωk×(ΣEk⊥τp+P⊥¯SEk/bardbl)·στp\n2/planckover2pi1(1+Ω2\nkτ2p//planckover2pi12)\n−(ΣEk⊥τp+P⊥¯SEk/bardbl)\n1+Ω2\nkτ2p//planckover2pi12. (28)\nOnce again, if Σ Ek⊥τp+P⊥¯SEk/bardblvanishes, then all the\ncorrections to SEk⊥of order zero and higher also vanish.\nLet us analyze the two terms in SEk⊥. By identifying\nterms in SEk⊥even and odd in k, as was done above,\nit is evident that the first term on the RHS of Eq. (28)\nleads to a spin current, but no spin density. The second\nterm does not lead to a spin current, but it does produce\na spin density. Nevertheless, this term tends to zero in\nthe ballistic regime as well as in the strong momentum\nscattering regime, and it is not expected to be dominant.\nThe closed-form solution found in this section shows\nthat, in the absence of spin-orbit interactions in the im-\npurity potential, there is only onespin current, which in\nthe weak momentum-scattering limit is independent of\nτpand in the strong momentum-scattering limit is ∝τ2\np.\nBearing in mind that if S(0)\nEk⊥vanishes all corrections of\nhigherorderalsovanish, we concludethat, in the absence\nof spin-orbit interactions in the impurity potential, the\ndistinction between intrinsic (disorder-independent) and\nextrinsic (disorder-dependent)spin currentsis not useful.E. Steady state spin densities and currents\nOur work helps to understand the origins of nonequi-\nlibrium spin densities and spin currents in electric fields.\nThe preceding sections illustrate the fact that nonequi-\nlibrium spin densities have two origins. The first, giving\nthe dominant contribution, arises from the absence of\nprecession. As charge carriers are accelerated, a fraction\nof their spin is conserved and produces a steady-state\nspin density,2,3,4,5,6,7,8,9,10,11,12a process which has no\nanalog in charge transport. Thus the dominant contri-\nbution to the nonequilibrium spin density in an electric\nfieldexistsbecauseinthe courseofspinprecessionacom-\nponent of each individual spin is preserved. For an elec-\ntron with wave vector k, this spin component is parallel\ntoΩk. In equilibrium the average of these conserved\ncomponents is zero. However, when an electric field is\napplied, the Fermi surface is shifted, and the average of\nthe conserved spin components may be nonzero, as illus-\ntrated in Fig. 1. This intuitive physical argument, to our\nknowledge absent to date from the literature, explains\nwhy the nonequilibrium spin density ∝τ−1\npandrequires\nscattering to balance the drift of the Fermi surface. It\nis interesting to note, also, that, although spin densities\nin electric fields require the presence of band structure\nspin-orbit interactions and therefore spin precession, the\ndominant contribution arises as a result of the absence\nof spin precession.\nAn additional contribution, which vanishes in both the\nballistic and the weak momentum scattering regimes, is\nassociated with spin precession. This contribution arises\nfrom the term on the last line of Eq. (28). The origin of\nthis term can be understood by noting that, in the pres-\nence of an electric field, the effective magnetic field about\nwhich a spin precesses changes slowly.23(This is true in\nbetween scattering events.) This change is contained in\nthe gauge-invariant crystal wave vector k=q−eEt//planckover2pi1.\nThe fact that the effective magnetic field is changing\nslowly causes the spin to acquire a small component\nin the direction in which the effective magnetic field is\nchanging. This component is proportional to the rate of\nchangeoftheeffectivemagneticfieldandtherefore,inour\ncase, toE. It is associated with the ‘flow’ of Ωkaround\nthe Fermi surface discussed by Shytov et al.41This argu-\nment explains why this term in the spin density vanishes\nin the clean limit as well as in the strong momentum\nscattering limit. For, in the absence of scattering, as k\nchanges,the effectivemagneticfieldwill circlearoundthe\nFermi surface, and the component of the spin following it\nwill average to zero. In the strong momentum scattering\nlimit, on the other hand, the spin will not have time to\nacquire a component in the direction in which the effec-\ntive magnetic field is changing, due to the high frequency\nof scattering events.\nFurthermore, in the absence of spin-orbit interactions\nin the impurity potentials, spin currents are associated\nwith displacement of spins. The relation between spin\ncurrents and spin precession was made explicit in the8\nkxky\nkxky\n0 00 0E=0 E>0 (b) (a)\nFIG. 1: Effective field Ωkat the Fermi energy in the Rashba\nmodel57(a) without63(E= 0) and (b) with an external elec-\ntric field ( E >0).\nwork of Sinova et al.23\nWe remarkthat both SEk/bardblandSEk⊥are invariant un-\nder time-reversal. As a result, the tensor characterizing\nthe response of spin currents to electric fields is invariant\nunder time reversal, whereas the tensor characterizing\nthe response of spin densities to electric fields changes\nsign under time reversal, as expected in both cases.24,64\nF. Interplay of conserved and precessing spin\ndensities\nIt is enlightening to compare the results obtained in\nthe absence of scattering (i.e., the clean limit) with the\nresults obtained when scattering is present. The aim is\nto obtain an understanding of the way scattering pro-\ncesses affect steady-state spin distributions in electric\nfields. This is done by comparing results obtained using\nthe approach outlined in this paper with results obtained\npreviously using Green’s functions approaches, both for\nthe case when scattering is not included and for the case\nin which scattering is taken into account. This process\nwill aid us in identifying the information contained in\nvertex correctionsto spin-related quantities in the frame-\nwork of Green’s functions-based theories of systems with\nspin-orbit interactions. The nature of this information is\nby no means obvious, and we will show that it has no\nanalog in charge transport.\nIn the absence of scattering, Eq. (12) takes the form\n∂SEk\n∂t+i\n/planckover2pi1[Hk,SEk] =eE\n/planckover2pi1·∂S0k\n∂k.(29)\nComparison of Eqs. (12) and (29) shows that the driving\nterm in the equation for the nonequilibrium spin distri-\nbutionSEkis renormalized by the term ˆJs(nEk), which\naccounts for spin-dependent scattering.\nIn addition, Eq. (23a) shows that scattering mixes the\ndistributions of conserved and precessing spins. This is\nso because when one spin at wave vector kand precess-\ning about Ωkis scattered to wave vector k′and pre-\ncesses about Ωk′, its conserved component changes, a\nprocess which alters the distributions of conserved and\nprecessing spin. Consequently, scattering processes insystems with spin-orbit interactions cause a renormaliza-\ntion of the driving term for the spin distribution, con-\ntained in Σ Ek⊥, as well as scattering between the con-\nserved and precessing spin distributions, described by\nP⊥ˆJ′\n0(SEk/bardbl). Our analysis suggests that contributions\ndue to these two processes are contained in vertex cor-\nrections to spin-dependent quantities found in Green’s\nfunctions formalisms.\nIntwodimensions, forHamiltonianslinearinwavevec-\ntor, we find that the renormalization term ˆJs(nEk) does\nnot contribute to the spin current for any elastic scat-\ntering. Therefore, the vertex correction to spin currents,\nfound in otherwork,7,34,35,39,40,46representsonlyscatter-\ning between conserved and precessing spin distributions.\nBy noting that in Eq. (28) the correction to the source\nterm in the equation for the precessing spin distribution\nhas the form P⊥¯SEk/bardbland thus depends on the steady\nstate spin density, it becomes evident that the existence\nof a nonzero nonequilibrium spin density does affect the\nspin current.\nFurthermore, in three dimensions, for electrons in\nzincblende crystals, which are described by the k3-\nDresselhaus model, we find that, for short range impu-\nrities, the spin current obtained after inclusion of scat-\ntering is the same as when scattering is not included. It\nis known that the vertex correction to spin currents also\nvanishes41in these systems. Noting that the steady-state\nspin density in zincblende crystals vanishes by symme-\ntry, and more generally vanishes in any non-gyrotropic\nmedium,2these observations reinforce our conclusion re-\nlating to the connection between vertex corrections to\nspin currentsand the presence of a steady-statespin den-\nsity. We therefore expect vertex corrections to spin cur-\nrents to vanish in non-gyrotropic materials, in which no\nsteady-state spin density is possible in an electric field.2\nG. Comparison with previous work\nOur calculations for known cases give results in agree-\nment with previous work. Firstly, our results agree with\nprevious calculations of nonequilibrium spin densities.4,7\nFurthermore, in two dimensions, for Hamiltonians lin-\near in wave vector, the spin current vanishes for short-\nrange impurities,7,34,35,39,40,46as well as for small-angle\nscattering.40,41(We find that in fact it vanishes for any\nelastic scattering.50) For spin-orbit Hamiltonians charac-\nterized solely by one angular Fourier component N(Ref.\n41) the spin current ∝N.\nSpin currents in systems in which band structure spin-\norbit interactions are negligible were studied by En-\ngelet al.,42whodemonstratedthatspincurrentsinthose\ncircumstances are due to skew scattering. Skew scatter-\ning appears as a term of third order in the scattering\npotential Ukk′(which must include spin-orbit coupling\nexplicitly), whereas in this work we have restricted our\ndiscussion to terms of second order in Ukk′, and we have\nnot considered higher-order skew-scattering effects.9\nH. Observable effects\nSpin densities and spin currents excited by electric\nfields giverise toobservableeffects. In materialsin which\na nonequilibrium spin density is excited by an electric\nfield, the presence of this spin density can be observed,\nfor example, by means of magnetic circular dichroism65,\nwhich has long been used as a characterization tool for\nmagnetic materials. Similarly, a spin current flowing\ntransversely to the direction of the charge current (i.e.,\nthe spin-Hall effect) will give rise to a spin accumula-\ntion at the edge of the sample. A spin current flow-\ning parallel to the direction of the charge current could\nbe used as a means of spin injection from one semicon-\nductor into another, as discussed in our recent work.62\nSpin accumulation as a result of a spin-Hall current, or\na spin density injected by means of a longitudinal spin\ncurrent, can in turn be measured using magnetic circular\ndichroism65techniques developed recently. We note that\nalternative techniques can be used to observe nonequilib-\nrium spin densities,9,10,11,12spin currents,19,20and edge\nspin accumulations.16,17,18\nIV. SUMMARY\nWe have demonstrated that, in the presence of band\nstructure spin-orbit interactions, the steady state estab-\nlished for the carrier spin distribution contains two qual-\nitatively distinct contributions, corresponding to con-servedand precessingspin. Thedistribution ofconserved\nspin acquires a nonequilibrium correction that diverges\nin the ballistic regime. This correction is responsible for\nthe establishment of the dominant nonequilibrium spin\ndensities in electric fields. The distribution of precessing\nspin acquires a nonequilibrium correction that is finite in\nthe ballistic regime. This correctionis responsible for the\nestablishment of nonequilibrium spin currents in electric\nfieldsandasmallnonequilibriumspinpolarization,which\nvanishes in the ballistic and strongmomentum scattering\nregimes. We havedemonstratedthat, when spin-orbitin-\nteractions are present in the band structure and absent\nfrom the impurity potential, there is only one contribu-\ntion to the spin current, which appears independent of\ndisorderin the ballistic regimebut dependent on disorder\nin the strong momentum scattering regime. Moreover,\nwe have also shown that scattering processes in systems\nwith spin-orbit interactions give rise to a renormalization\nof the driving term in the equation for the spin distri-\nbution, as well as scattering between the conserved and\nprecessing spin distributions, which sheds light on the\nnature of vertex corrections in these systems.\nThe authors would like to acknowledge enlightening\ndiscussions with E. Rashba, Q. Niu, A. H. MacDon-\nald, J. Sinova, D. L. Smith, J. Shi, H. A. Engel, and\nR. A Duine. The research at Argonne National Labo-\nratory was supported by the US Department of Energy,\nOffice of Science, Office of Basic Energy Sciences, under\nContract No. DE-AC02-06CH11357.\n1I.ˇZuti´ cet al., Rev. Mod. Phys. 76, 323 (2004).\n2E.L.IvchenkoandG.E.Pikus, JETP Lett. 27, 604(1978).\n3L. S. Levitov et al., Sov. Phys. JETP 61(1), 133 (1985).\n4V. M. Edelstein, Solid State Comm. 73, 233 (1990).\n5A. G. Aronov et al., Sov. Phys. JETP 73, 537 (1991).\n6L. I. Magarill et al., Semiconductors 35(9), 1081 (2001).\n7J. Inoue, G. E. W. Bauer, and L. W. Molenkamp, Phys.\nRev. B67, 033104 (2003).\n8H.-A. Engel, E. I. Rashba, B. I. Halperin, Phys. Rev. Lett.\n98, 036602 (2007).\n9L. E. Vorob’ev et al., JETP Lett. 29, 441 (1979).\n10Y. Kato et al., Phys. Rev. Lett. 93, 176601 (2004).\n11S. D. Ganichev, S. N. Danilov, P. Schneider, V. V. Bel’kov,\nL. E. Golub, W. Wegscheider, D. Weiss, and W. Prettl,\ncond-mat/0403641 (2004).\n12A. Y. Silov, P. A. Blajnov, J. H. Wolter, R. Hey,\nK. H. Ploog, and N. S. Averkiev, Appl. Phys. Lett. 85,\n5929 (2004).\n13M. I. D’yakonov and V. I. Perel’, Zh. Eksp. Teor. Fiz 60,\n1954 (1971).\n14J. E. Hirsch, Phys. Rev. Lett. 83, 1834 (1999).\n15S. Murakami et al., Science 301, 1348 (2003).\n16Y. K. Kato et al., Science 306, 1910 (2004).\n17J. Wunderlich et al., Phys. Rev. Lett. 94, 047204 (2005).\n18V. Sihet al., Nature Phys. 1, 31-35 (2005).\n19S.O.ValenzuelaandM. Tinkham, Nature 442, 176(2006).20B. Liuet al., cond-mat/0610150 (2006).\n21N. P. Stern et al., Phys. Rev. Lett. 97, 126603 (2006).\n22S. D. Ganichev et al., Phys. Rev. Lett. 86, 004358 (2001).\n23J. Sinova et al., Phys. Rev. Lett. 92, 126603 (2004).\n24H. A. Engel et al., cond-mat/0603306\n25R. Winkler, cond-mat/0605390.\n26I. Adagideli and G. E. W. Bauer, Phys. Rev. Lett. 95,\n256602 (2005); X.-D. Cui et al., cond-mat/0608546.\n27J. Shiet al., Phys. Rev. Lett. 96, 076604 (2006).\n28E. I. Rashba, Phys. Rev. B 70, 201309 (2004).\n29S. Zhang and Z. Yang, Phys. Rev. Lett. 84, 066602 (2005).\n30Y. Wang et al., Phys. Rev. Lett. 96, 066601 (2006).\n31L. Sheng et al., Phys. Rev. Lett. 95, 136602 (2005).\n32T. Damker et al., Phys. Rev. B 69, 205327 (2004).\n33E. M. Hankiewicz andG. Vignale, Phys. Rev.B 73, 115339\n(2006).\n34J. Inoue, G. E. W. Bauer, and L. W. Molenkamp,\nPhys. Rev. B 70, 041303 (2004).\n35O. V. Dimitrova, Phys. Rev. B 71, 245327 (2005).\n36A.G.Mal’shukovandK.A.Chao, Phys.Rev.B 71, 121308\n(2005).\n37B. K. Nikolic et al., Phys. Rev. B 73, 075303 (2006).\n38O. Bleibaum and S. Wachsmuth, Phys. Rev. B 74, 195330\n(2006).\n39E. G. Mishchenko et al., Phys. Rev. Lett. 93, 226602\n(2004).10\n40A. Khaetskii, Phys. Rev. Lett. 96, 056602 (2006).\n41A. V. Shytov et al., Phys. Rev. B 73, 075316 (2006).\n42H. -A. Engel et al., Phys. Rev. Lett. 95, 166605 (2005).\n43J. Schliemann and D. Loss, Phys. Rev. B 71, 085308\n(2005).\n44P. -Q. Jin and Y. -Q. Li, Phys. Rev. B 74, 085315 (2006).\n45J. Liet al., Phys. Rev. B 71, 241305 (2005).\n46P. Schwab and R. Raimondi, Eur. Phys. J. B 25, 483\n(2002).\n47P. L. Krotkov and S. Das Sarma, Phys. Rev. B 73, 195307\n(2006).\n48M. Zarea. and S. Ulloa, Phys. Rev. B. 73, 165306 (2006).\n49N. A. Sinitsyn et al., Phys. Rev. B 70, 245211 (2004).\n50N. Sugimoto, S. Onoda, S. Murakami, and N. Nagaosa,\nPhys. Rev. B 73, 113305 (2006).\n51M. I. D’yakonov and V. I. Perel’, Sov. Phys.–Solid State\n13, 3023 (1972).\n52M. I. D’yakonov and V. Y. Kachorovski˘ ı, Sov. Phys. Semi-\ncond.20, 110 (1986).\n53G. E. Pikus and A. N. Titkov, in Optical Orientation ,\nedited by F. Meier and B. P. Zakharchenya (Elsevier, Am-\nsterdam, 1984), pp. 73–131.\n54N.S.Averkiev et al., J.Phys.:Cond.Mat. 14, R271(2002).\n55S. Onoda, N. Sugimoto, and N. Nagaosa,\nProg. Theor. Phys. 116, 61 (2006).\n56G. Dresselhaus, Phys. Rev. 100, 580 (1955).\n57Y.A.BychkovandE. I.Rashba, JETP Lett. 39, 66(1984).\n58J. M. Luttinger, Phys. Rev. 102, 1030 (1956).\n59T. Jungwirth et al., Rev. Mod. Phys. 78, 809 (2006).\n60R. J. Elliott, Phys. Rev. 96, 266 (1954); Y. Yafet in Solid\nState Physics , edited by F. Seitz and D. Turnbull, Vol. 14\n(Academic Press, New York 1963), p. 2.\n61N. A. Sinitsyn et al., Phys. Rev. B 75, 045315 (2007).\n62D. Culcer andR.Winkler, cond-mat/0708.4009, submittedto Phys. Rev. Lett.\n63R. Winkler, Phys. Rev. B 69, 045317 (2004).\n64D. Culcer, Y. Yao, A. H. MacDonald, and Q. Niu,\nPhys. Rev. B 72, 045215 (2005).\n65G. Srajer et al., J. Mag. Mag. Mat. 307, 1 (2006).\n66In order to obtain Eq. (5) in the work of Averkiev et. al.,54\nit is sufficient to replace the EFA Hamiltonian Hkby the\nkinetic energy term only, thus Hk=/planckover2pi12k2/2m≡ε0, and\ncarry out the integration over time. We note that a scat-\ntering term equivalent to that derived here was found\nusing the Keldysh formalism by E. L. Ivchenko et al.,\nSov. Phys. JETP 71(3), 551 (1990).\nAPPENDIX A: MATRIX ELEMENTS OF A\nSCREENED COULOMB POTENTIAL\nIn two dimensions, the matrix element Ukk′of a\nscreenedCoulomb potential between plane wavesis given\nby\nUkk′=−Ze2\nǫ0Vd=21/radicalbig\n|k−k′|2+1/L2s,(A1)\nwhereZis the ionic charge, Vd=2corresponds to the unit\ncellarea, and Lsis the screeninglength. The correspond-\ning expression in three dimensions is\nUkk′=−Ze2\nǫ0Vd=31\n|k−k′|2+1/L2s,(A2)\nwhere now Vd=3is the unit cell volume.\nAPPENDIX B: INTEGRALS OVER TIME\nFor a Hamiltonian given by Hk=ε0k11+(1/2)σ·Ωk, the product of two time evolution operators e−iˆHk′t′eiˆHkt′\ncan be written as\ne−iˆHk′teiˆHkt′=ei(ε0−ε′\n0)t′/bracketleftbigg\ncosΩkt′\n2cosΩ′\nkt′\n2−iσ·ˆΩk′cosΩkt′\n2sinΩ′\nkt′\n2+iσ·ˆΩksinΩkt′\n2cosΩ′\nkt′\n2\n+(ˆΩk·ˆΩk′+iσ·ˆΩk′׈Ωk) sinΩkt′\n2sinΩ′\nkt′\n2/bracketrightbigg\n, (B1)\nwith a similar expression holding for the Hermitian conjugate of this pr oduct. In the time integrals the trigonometric\nfunctions are expressed as complex exponentials, the integrals ar e evaluated, and the results are replaced by their\nprincipal parts, yielding\n/integraldisplay∞\n0dt′ei(ε0−ε′\n0)t′cosΩkt′\n2cosΩ′\nkt′\n2=π/planckover2pi1\n4[δ(ǫk+−ǫ′\nk+)+δ(ǫk−−ǫ′\nk−)+δ(ǫk+−ǫ′\nk−)+δ(ǫk−−ǫ′\nk+)] (B2a)\n/integraldisplay∞\n0dt′ei(ε0−ε′\n0)t′cosΩkt′\n2sinΩ′\nkt′\n2=π/planckover2pi1\n4i[δ(ǫk+−ǫ′\nk−)+δ(ǫk−−ǫ′\nk−)−δ(ǫk+−ǫ′\nk+)−δ(ǫk−−ǫ′\nk+)] (B2b)\n/integraldisplay∞\n0dt′ei(ε0−ε′\n0)t′sinΩkt′\n2cosΩ′\nkt′\n2=π/planckover2pi1\n4i[δ(ǫk+−ǫ′\nk+)+δ(ǫk+−ǫ′\nk−)−δ(ǫk−−ǫ′\nk−)−δ(ǫk−−ǫ′\nk+)] (B2c)\n/integraldisplay∞\n0dt′ei(ε0−ε′\n0)t′sinΩkt′\n2sinΩ′\nkt′\n2=π/planckover2pi1\n4[δ(ǫk+−ǫ′\nk−)+δ(ǫk−−ǫ′\nk+)−δ(ǫk−−ǫ′\nk−)−δ(ǫk+−ǫ′\nk+)].(B2d)\nEvaluation of all time integrals in this manner leads to Eq. (7) for the s cattering term."    },    {        "title": "0909.5565v1.A_quantum_jump_description_for_the_non_Markovian_dynamics_of_the_spin_boson_model.pdf",        "content": "arXiv:0909.5565v1  [quant-ph]  30 Sep 2009A quantum jump description for the non-Markovian\ndynamics of the spin-boson model\nE.-M. Laine\nDepartment of Physics and Astronomy, University of Turku, FI-2 0014 Turku, Finland\nE-mail:emelai@utu.fi\nAbstract. We derive a time-convolutionless master equation for the spin-boso n\nmodel in the weak coupling limit. The temporarily negative decay rates in the\nmaster equation indicate short time memory effects in the dynamics w hich is explicitly\nrevealedwhen the dynamicsis studied using the non-Markovianjump description. The\napproach gives new insight into the memory effects influencing the sp in dynamics and\ndemonstrates, how for the spin-boson model the the co-operat ive action of different\nchannels complicates the detection of memory effects in the dynamic s.Non-Markovian dynamics of the spin-boson model 2\n1. Introduction\nThe spin-boson model describes a two-level system interacting wit h a bosonic\nenvironment. It is widely studied, mostly because of its many applicat ions in the field\nof quantum information theory and chemistry. The model can desc ribe for example\na simple two-well potential [1], double quantum dots [2] and different biomolecular\nsystems [3,4]. Becausethespin-bosonmodelisnotexactlysolvable manyapproximative\napproaches have been used in order to unravel the dynamics of th e spin-boson system.\nThe model has been studied using perturbation theory in the syste m-environment\ncoupling with the Markov approximation [5, 6] and projection operat or methods [7]. In\nreference [1] an approximative scheme called the interacting blip app roximation (NIBA)\nhas been introduced. It is a perturbative approach in the tunneling matrix element\nbased on path integral methods. There also exists an exact appro ach to the problem\nusing Feynman-Vernon pathintegral method [8, 9]. However, this presents only aformal\nsolution.\nUnderstanding the short-time dynamics of the spin-boson model is crucial in\nunderstanding quantum computing systems [10, 11]. The Markovian approximation\nis valid only for long times and thus the non-Markovian description of t he dynamics is\nneeded. NIBA fails for large tunneling matrix element and cannot the refore be applied\nto biomolecular systems [3, 4]. Many projection operator methods a nd path integral\nmethods are difficult to treat and do not necessarily give much physic al insight to the\nproblem. The main motivation of this paper is to write down a time-conv olutionless\nmaster equation describing the dynamics of the spin-boson model a nd to study the\nequation via non-Markovian quantum jumps. This description gives a clear picture of\nthe processes involved in the dynamics and reveals the role of memor y effects in the\nevolution of the spin.\nIn section 2 we derive a time-convolutionless master equation descr ibing the\nreduced dynamics of the spin-boson model by using the time-convo lutionless projection\noperator method [12]. The master equation is derived in second orde r and the secular\napproximation is performed. The derived master equation is then st udied in section 3\nviathenon-Markovianquantumjump(NMQJ)approach[13,14,15] andtheconclusions\nare presented in section 4.\n2. Derivation of the master equation\nThe Hamiltonian describing the spin-boson model is\nH=ǫ\n2σz+∆\n2σx+/summationdisplay\nnωna†\nnan+σz\n2/summationdisplay\nnλn(a†\nn+an). (1)\nHere,σz=|e/an}bracketri}ht/an}bracketle{te|−|g/an}bracketri}ht/an}bracketle{tg|,σx=|e/an}bracketri}ht/an}bracketle{tg|+|g/an}bracketri}ht/an}bracketle{te|in the eigenbasis of the two-state\nsystem{|e/an}bracketri}ht,|g/an}bracketri}ht},ǫis the energy bias between the two states, ∆ is tunneling amplitude\nbetween the states, and ωnandanare the energies and annihilation operators of the\ncorresponding modes of the bosonic environment respectively. Th e tunneling amplitudeNon-Markovian dynamics of the spin-boson model 3\n∆ is assumed to be a real number. The parameter λndescribes the coupling strength\nbetween the two-state system and the nth mode of the reservoir. The coupling of\nthe system to the environment is a bilinear coupling through operato rσz,i.e., the\nenvironment is sensitive to the value of the operator σz.\nWe start the derivation of the master equation by writing down a gen eral form of\nthe second order time-convolutionless master equation [12]\nd\ndt˜ρS(t) =−/integraldisplayt\n0trE[˜HI(t),[˜HI(t′),˜ρS(t)⊗ρE]]dt′. (2)\nHere, thenotation ˜A(t)standsforanoperator Awrittenintheinteractionpicture. Now,\nif one writes a decomposition for the interaction Hamiltonian of the fo rmHI=S⊗E\nin terms of the system eigenoperators Sω, i.e.˜HI(t) =/summationtext\nωe−iωtSω⊗˜E(t), and performs\nthe secular approximation, (2) may be written in the form\n˙˜ρS(t) =−i[HLS(t),˜ρS(t)]+D(t)(˜ρS(t)), (3)\nwith\nD(t)(˜ρS(t)) =/summationdisplay\nωγω(t)/bracketleftbigg\nSω˜ρS(t)S†\nω−1\n2/braceleftig\nS†\nωSω,˜ρS(t)/bracerightig/bracketrightbigg\n(4)\nand\nHLS(t) =/summationdisplay\nωλω(t)S†\nωSω. (5)\nThe time-dependent coefficients γω(t) andλω(t) in (4) and (5) are determined from the\nexpression\nΓω(t)≡/integraldisplayt\n0dt′eiωt′trE/braceleftig˜E†(t)˜E(t−t′)ρE/bracerightig\n=/integraldisplayt\n0dt′eiωt′trE/braceleftig˜E†(t′)˜E(0)ρE/bracerightig\n(6)\nby the relation Γ ω(t) =1\n2γω(t)+iλω(t).\nThe system eigenoperators for the spin-boson model written in th e system\neigenbasis{ψ+,ψ−}are\nS0=ǫ\n2ω0σz, Sω0=−∆\n2ω0σ−, S−ω0=−∆\n2ω0σ+, (7)\nwhereσz=|ψ+/an}bracketri}ht/an}bracketle{tψ+|−|ψ−/an}bracketri}ht/an}bracketle{tψ−|,σ−=|ψ−/an}bracketri}ht/an}bracketle{tψ+|,σ+=|ψ+/an}bracketri}ht/an}bracketle{tψ−|andω0=√\nǫ2+∆2\nis the system eigenenergy. In the limit of continuous spectrum of th e environment and\nfor zero temperature one can write the coefficients γω(t) in (4) as\nγω(t) = 2/integraldisplayt\n0dt′/integraldisplay∞\n0dω′J(ω′)cos[(ω−ω′)t′], (8)\nwhereJ(ω) is the spectral density. For an Ohmic spectral density J(ω) =α\n2ωe−ω/ωc\n[12], where ωcis the cutoff frequency, the coefficients are\nγ±ω0(t) =αωc\n1+ω2\nct2[ωctcos(ω0t)∓sin(ω0t)]\n+αω0e∓ω0/ωc/bracketleftbigg\nℜ[Si(z)]∓ℑ[Ci(z)]±π\n2/bracketrightbigg\nγ0(t) =αω2\nct\n1+ω2\nct2, (9)Non-Markovian dynamics of the spin-boson model 4\nwhere Ci(z) is the cosine integral, Si( z) is the sine integral and z=ω0t+iω0\nωc. The decay\nratesγω0(t) andγ−ω0(t) have oscillatory behavior and they get temporarily negative\nvalues resembling memory effects in the reduced dynamics, while the d ecay rateγ0(t)\nremains always positive. For simplicity we ignore the Lamb shift (5) and restrict to the\nregimeωc≪ω0, where the secular approximation is valid in the non-Markovian regime .\nThe solution to the master equation presented in (3), (7)and (9) c an be written in\nterms of the density matrix elements as\nρ(t) =M(t)ρ(0), (10)\nwith\nM(t) =\ng(t) 0 0 f(t)\n0e−ζ(t)0 0\n0 0 e−ζ(t)0\n1−g(t) 0 0 1 −f(t)\n. (11)\nHere,\ng(t) =f(t)+e−η(t),\nf(t) =e−η(t)ξ(t),\nη(t) =∆2\n4ω2\n0/integraldisplay\n(γω0(t)+γ−ω0(t))dt,\nξ(t) =∆2\n4ω2\n0/integraldisplay\nγ−ω0(t)e−η(t)dt,\nζ(t) =/integraldisplay\n(ǫ2\n2ω2\n0γ0(t)+∆2\n8ω2\n0(γω0(t)+γ−ω0(t)))dt. (12)\nThe matrix in equation (10) acts on ρ(0) as on a column vector consisting of the density\nmatrix elements.\n3. Jump description of the dynamics\nThe master equation derived in the previous section can be studied w ith the NMQJ-\nmethod [13], which is a generalization of the Monte Carlo wave function method [16] to\nthe non-Markovian regime. The key increment in this description is th at whenever the\ncoefficients γω(t) get negative values the direction of the jump process is reversed for\neach wave function realization in the corresponding channel. The de nsity matrix for an\nensemble of wave functions with Nmembers at time tis\nρS(t) =/summationdisplay\nαNα(t)\nN|φα(t)/an}bracketri}ht/an}bracketle{tφα(t)|, (13)\nwhereNα(t) is the number of ensemble members in state |φα(t)/an}bracketri}htat timet. When\nthe decay rates are positive the jumps taking place in a single wave fu nction history\ncan be determined by the Monte Carlo wave function method [16]. Whe n a decay\nrate becomes negative, the reversed jump operator in the corre sponding channel jis\nDj\nα→α′(t) =|φα′(t)/an}bracketri}ht/an}bracketle{tφα(t)|, with|φα(t)/an}bracketri}ht=Cj−|φ′\nα(t)/an}bracketri}ht/||Cj−|φ′\nα(t)/an}bracketri}ht||. The reversedNon-Markovian dynamics of the spin-boson model 5\njumps can take the wave functions to superposition states destr oyed earlier by jumps\nand they therefore resemble the memory effects in the dynamics. T he memory effects,\nby recreation of superpositions, allow a reversal of the decohere nce process which we call\nhere recoherence. The reversed transition occurs during a time s tepδtwith probability\nPj\nα→α′(t) =Nα′(t)\nNα(t)|γj(t)|δt/an}bracketle{tφα′(t)|C†\njCj|φα′(t)/an}bracketri}ht. (14)\nLet us study the dynamics of the spin-boson model via the jump des cription.\nWe concentrate here on giving a clear picture of the effect of the re versed jumps on\nthe dynamics and will not perform the actual unraveling of the mast er equation (3).\nThe intention is to give insight into the origin of memory effects by desc ribing the\nsystem dynamics in terms of non-Markovian jumps. The plots descr ibing the dynamics\npresented in this section are obtained by numerically evaluating (11) .\nThe spin-boson model involves three jump operators C1=σ−,C2=σ+and\nC3=σz. The operators C1andC2generate jumps that affect the populations of\nthe system eigenstates and the operator C3produces phase flips. The corresponding\ndecay rates are γ1(t) =∆2\n4ω2\n0γω0(t),γ2(t) =∆2\n4ω2\n0γ−ω0(t) andγ3(t) =ǫ2\n4ω2\n0γ0(t) (c.f. (7) and\n(9)), which are plotted in figures 1,2 and 3. The first two decay rate s have temporarily\nnegative values while the third decay rate remains always positive. Th is indicates that\nin channels C1=σ−andC2=σ+there occurs reversed jumps resulting into memory\neffects, but that the phase flip channel C3=σzlacks memory.\n0246810/Minus0.0002/Minus0.00010.00000.00010.00020.0003\nΩctΓ1/LParen1t/RParen1\nFigure 1: The decay rate γ1(t) in units of ωcforα= 0.01,ω0/ωc= 10 and\nǫ/∆ = 1/(2√\n3).\n0246810/Minus0.00015/Minus0.00010/Minus0.000050.000000.000050.000100.000150.00020\nΩctΓ2/LParen1t/RParen1\nFigure 2: The decay rate γ2(t) in units of ωcforα= 0.01,ω0/ωc= 10 and\nǫ/∆ = 1/(2√\n3).Non-Markovian dynamics of the spin-boson model 6\n02468100.0000.0020.0040.0060.008\nΩctΓ3/LParen1t/RParen1\nFigure 3: The decay rate γ3(t) in units of ωcforα= 0.01,ω0/ωc= 10 and\nǫ/∆ = 1/(2√\n3).\nAssume now that the system is initially in a pure state |ψ0(0)/an}bracketri}ht=1√\n2(|ψ+/an}bracketri}ht+|ψ−/an}bracketri}ht),\ni.e., all the ensemble members are initially in the same state. We use the n otation|ψ0/an}bracketri}ht\nfor the state vector exposed to only deterministic evolution. When ǫ/ne}ationslash= 0, there exists\na phase flipped counter part |ψPH\n0/an}bracketri}htfor the state|ψ0/an}bracketri}ht, into which the single realization\ncan jump through channel C3=σz. The possible states and jumps are demonstrated\nin figure 4.\n/VertBar1Ψ0/Greater /VertBar1Ψ0PH/Greater\n/VertBar1Ψ/Plus/Greater /VertBar1Ψ/Minus/Greater\nFigure 4: The states involved in the effective ensemble (13). The pos sible jumps in the\ndynamics are presented as arrows between the states. Here, |ψ0/an}bracketri}ht=α0|ψ+/an}bracketri}ht+β0|ψ−/an}bracketri}ht\nand|ψPH\n0/an}bracketri}ht=α0|ψ+/an}bracketri}ht−β0|ψ−/an}bracketri}ht.\nIn figure 5 the coherence is plotted for a fixed ratio ω0/ωcand varying values of\nǫ/∆. One can see that as ǫ/∆ varies, the behavior of the coherence changes in such\nway that for small values of ǫ/∆ there is recoherence but as the ratio gets larger the\nsystem decoheres with no revivals of coherence. Whether the dec ay ratesγ1(t) and\nγ2(t) get negative values depends only on the ratio ω0/ωc, so for large values of ǫ/∆ the\nsystem will decohere although there are reversed jumps in the pro cess. Let us study the\nreal part of the coherence Re[ ρ±] =1\nN/summationtext\niαiβi, where|ψi\n0/an}bracketri}ht=αi|ψ+/an}bracketri}ht+βi|ψ−/an}bracketri}ht. Jumps\n|ψ0/an}bracketri}ht→|ψ+/an}bracketri}ht,|ψ0/an}bracketri}ht→|ψ−/an}bracketri}ht,|ψPH\n0/an}bracketri}ht→|ψ+/an}bracketri}ht,|ψPH\n0/an}bracketri}ht→|ψ−/an}bracketri}htand phase flips reduce the\nnumber of terms in the sum presenting the coherence and therefo re induce decoherence.\nReversed jumps|ψ0/an}bracketri}ht←|ψ+/an}bracketri}ht,|ψ0/an}bracketri}ht←|ψ−/an}bracketri}htand|ψPH\n0/an}bracketri}ht←|ψ+/an}bracketri}ht,|ψPH\n0/an}bracketri}ht←|ψ−/an}bracketri}htproduce\nnew terms into the sum, but if there is an equal amount of reversed jumps to|ψ0/an}bracketri}htand\n|ψPH\n0/an}bracketri}htthe new terms in the sum cancel each other and no recoherence ta kes place. So\nthe appearance of recoherence does not only depend on the numb er of reversed jumps,Non-Markovian dynamics of the spin-boson model 7\n0.00.20.40.60.81.01.21.40.499960.5\nΩctRe/LBracket1Ρ/Plus/Minus/RBracket1\nFigure 5: The coherence forω0\nωc= 10 andǫ\n∆=1\n2√\n7(black line),ω0\nωc= 10 andǫ\n∆= 0.255\n(dark gray line),ω0\nωc= 10 andǫ\n∆=1\n2√\n3(light gray line). Here, the coupling constant is\nα= 0.01 and the initial state is |ψ0(0)/an}bracketri}ht=1√\n2(|ψ+/an}bracketri}ht+|ψ−/an}bracketri}ht).\nbut also on the proportion of the number of jumps into different tar get states.\nTo find out for which valuesǫ\n∆there is recoherence let us study the quantity\nN0(t)−NPH\n0(t). Here,N0(t) is the number of ensemble members in state |ψ0/an}bracketri}htand\nNPH\n0(t) the number of elements in state |ψPH\n0/an}bracketri}ht. The real part of the coherence is an\nincreasing function, when N0(t)−NPH\n0(t) is increasing, and a decreasing function when\nN0(t)−NPH\n0(t) is decreasing. The regions, where there is recoherence for a fixe d value\nofω0/ωccan be seen in figure 6. One can see that as the energy bias ǫbecomes larger\ncompared with the tunneling amplitude ∆ the probability of a phase flip in creases. As\nmore phase flips occur a reversed jump to |ψPH\n0/an}bracketri}htgets as probable as a reversed jump to\nstate|ψ0/an}bracketri}ht, which then causes the system to decohere.\n0.0 0.5 1.0 1.501\n270.2551\n23\nΩctΕ\n/CapDΕlta\nFigure 6: The time regions for recoherence for different values ofǫ\n∆. In light brown\nareas there is recoherence. Here,ω0\nωc= 10,α= 0.01 and|ψ0(0)/an}bracketri}ht=1√\n2(|ψ+/an}bracketri}ht+|ψ−/an}bracketri}ht). The\nvalues ofǫ\n∆used in figure 5 are marked in the plot.\nThestudyaboveillustrateshowforthespin-bosonmodeltherear ecaseswherethere\nare no revivals of coherence, but there occurs reversed jumps in dicating memory effects\non the level of single wave function realizations. There are no reviva ls of coherences due\nto the co-operative action of the phase flip channel and the energ y exchanging channels.\nI.e, although there is information about the phase returning to the system via reversed\njumps in the energy exchanging channels, the overall information fl ow about the phase\nis from the system to the environment due to the phase flips occurr ing in the dynamics.Non-Markovian dynamics of the spin-boson model 8\nThis example illustrates, how in complex quantum dynamics it requires s ubtlety to\ndetermine whether memory effects play a role in the system dynamics .\nThe NMQJ-approach gives a scheme to detect whether there are m emory effects\nin the system dynamics although there are no revivals of coherence in the level of the\ndensity matrix. In [17] the concept of non-Markovianity was clarifie d by defining a\nmeasure for non-Markovianity. It measures the amount of inform ation flowing from the\nenvironment to thesystem. For thespin-boson model this measur e isindependent of the\nparameterǫ, i.e., the maximum value for non-Markovianity is obtained when there a re\nno phase flips occurring in the dynamics. The NMQJ-description thus gives a similar\npicture of the memory effects in the dynamics of the spin-boson mod el as the measure\nof non-Markovianity.\n4. Conclusions\nWe derived a time-convolutionless master equation describing the sp in-boson model in\nthe weak coupling limit. The equation describes the short time memory effects in the\ndynamics manifesting themselves through negative decay rates in t he master equation.\nThe memory effects are studied via the non-Markovian quantum jum p approach.\nThis description shows memory effects on the level of singe wave fun ction realizations\nalthough there are no revivals of coherence in the density matrix de scription. This\nbehavior demonstrates, how in complex quantum dynamics the co-o perative action of\ndifferent channels complicates the detection of memory effects in th e dynamics.\nAcknowledgments\nThe author is grateful to Jyrki Piilo for helpful discussions and com ments.\nReferences\n[1] Leggett A J, Chakravarty S, Dorsey A T, Fisher M, Garg A and Zw erger W 1987 Rev. Mod. Phys.\n591\n[2] Brandes T 2005 Phys. Rep. 408315\n[3] Garg A, Onuchic J N and Ambegaokar V J 1985 J. Chem. Phys. 834491\n[4] Gilmore J and McKenzie R H 2008 J. Phys. Chem. A1122162\n[5] Cheche T O and Lin S H 2001 Phys. Rev. E64061103\n[6] Wilhelm F K, Kleff S and von Delft J 2003 Chem. Phys. 296345\n[7] DiVincenzo D P and Loss D 2005 Phys. Rev. B71035318\n[8] Grifoni M, Paladino E and Weiss U 1999 Eur. Phys. J. B10719\n[9] Grifoni M, Winterstetter M and Weiss U 1997 Phys. Rev. E56334\n[10] Privman V 2002 Mod. Phys. Lett. B16459\n[11] Privman V 2003 J. Stat. Phys. 110957\n[12] Breuer H-P and Petruccione F 2002 The theory of Open Quantum Systems (Oxford: Oxford\nUniversity Press)\n[13] Piilo J, Maniscalco S, H¨ ark¨ onen K and Suominen K-A 2008 Phys. Rev. Lett. 100180402\n[14] Piilo J, H¨ ark¨ onen K, Maniscalco S and Suominen K-A 2009 Phys. Rev. A79062112Non-Markovian dynamics of the spin-boson model 9\n[15] Breuer H-P and Piilo J 2009 EPL8550004.\n[16] Dalibard J, Castin Y and Mølmer K 1992 Phys. Rev. Lett. 68580\n[17] Breuer H-P, Laine E-M, Piilo J 2009 Measure for the Non-Markov ianity of Quantum Processes\nPreprint quant-ph/0908.0238"    },    {        "title": "1012.3552v2.Magnetic_dynamics_driven_by_the_spin_current_generated_via_spin_Seebeck_effect.pdf",        "content": "arXiv:1012.3552v2  [cond-mat.mes-hall]  3 May 2011Magnetic dynamics driven by the spin-current generated via spin-Seebeck effect\nChenglong Jia1,2and Jamal Berakdar1\n1. Institut f¨ ur Physik, Martin-Luther Universit¨ at Halle -Wittenberg, 06099 Halle, Germany\n2. Key Laboratory for Magnetism and Magnetic Materials of th e\nMinistry of Education, Lanzhou University, Lanzhou 730000 , China\nWe consider the spin-current driven dynamics of a magnetic n anostructure in a conductive mag-\nnetic wire under a heat gradient in an open circuit, spin Seeb eck effect geometry. It is shown that\nthe spin-current scattering results in a spin-current torq ue acting on the nanostructure and leading\ntoprecession and displacement. The scattering leads also t o a redistribution of the spin electrochem-\nical potential along the wire resulting in a break of the pola rity-reversal symmetry of the inverse\nspin Hall effect voltage with respect to the heat gradient inv ersion.\nIntroduction.- The discovery in the 1820s by T.J. See-\nbeck that due to a temperature gradient an electric volt-\nage emerges along the temperature drop, revealed the\nrelationship between heat and charge currents. The re-\nversal of Seebeck’s effect, i.e. the appearance of a tem-\nperature gradient upon an applied voltage, was shortly\nthereafter confirmed by J.-C. Peltier in 1834. In addition\nto other thermo-electric phenomena such as the Joule\nheating, in magnetic fields new thermo-magnetic effects\narise: A resistive conductor with a temperature gradient\n∇Tplaced in a magnetic field Bperpedicular to ∇T\ndevelops a potential drop normal to both ∇TandB.\nThis phenomenon is termed the Ettingshausen effect and\nits reverse is the Nernst effect. In a magnetic material\nthe anomalous Nernst effect occurs (i.e. the Nernst effect\ndue to the spontaneous magnetization) which was first\nobserved for Ni and Ni-Cu alloy1,2. Recently, in Refs.3–6\nmeasurements of the planar and the anomalous Nernst\neffect were reported for a variety of materials includ-\ning magnetic semiconductor, ferromagnetic metals, pure\ntransition metals, oxides, and chalcogenides. A quali-\ntatively new phenomena, the Spin-Seebeck effect (SSE),\nwas discovered 2008 by Uchida et al.7showing that in a\nferromagnetic material (a mm-size Ni 81Fe19sample) and\nin an open-circuit geometry (which is also the geometry\nstudied in this work, cf. Fig. 1) a heat current results\nin a spin current, i.e. a flow of spin angular momen-\ntum and hence a spin voltage, even if ∇Tis parallel to\nB(where the Nernst-Ettingshausen effect does not con-\ntribute). The spin voltageis reflected by a chargevoltage\nthat emerges (due the inverse spin Hall effect (ISHE)) in\na Pt strip deposited on the sample perpendicular to ∇T\n(cf. Fig.1). Further experiments on resistive conductors\n(8for Ni 81Fe19), insulating ferrimagnets (LaY 2Fe5O12\nin9), and for ferromagnetic semiconductors (GaMnAs10)\nunderline the generality of SSE. These fascinating effects\nare not only of fundamental importance; thermo-electric\nelements are already indispensable for temperature sens-\ning and control and for current-heat conversion. SSE\nopens the way for thermo-electric spintronic devices with\nqualitatively new tools for energy-consumption reduc-\ntion. It is highly desirable to explorewhether SSE can be\nutilized to steer localized magnetic textures, as problem\naddressed here. Theoretically, the reciprocity betweenthe dynamics in the magnetic order and the heat gradi-\nents is governed by the Onsager relations. The Onsager\nmatrix were discussed from a general point of view in\nRef.11with a focus on the transport of charge, magneti-\nzation, and heat. In Refs.12,13a thermo-magnetic meso-\nscopic circuit theory was put forward. Ref.14pointed\nout the occurrence of a thermally excited spin current\nin resistive conductors with an embedded ferromagnetic\nnanoclusters. Otherrecentworks12,13addressedthether-\nmally induced spin-transfer torque in a spin valves struc-\ntureswhereasthephenomenologicalstudy15isfocusedon\nthe spin-transfer torques in quasi one-dimensional mag-\nnetic domain walls (DWs) by introducing a viscous term\ninto the Landau-Lifshitz-Gilbert equation (LLG).\nSpin current- Themicroscopicmechanismfortheappear-\nance of the spin current in SSE is not yet completely\nunderstood, it is however an experimental fact that in\nthe geometry of Fig.1 the thermal gradient ∇Tgener-\nates a steady state spin current Js\nTwithout a charge\ncurrent7–10. The amplitude of Js\nTis found to be deter-\nmined by7–10\nJs\nT=−κ∇T (1)\nwhereκis a temperature-independent transport coeffi-\ncient whose properties are discussed in7–10; no charge\ncurrent is generated. The purpose of this work is to in-\nspect the dynamics triggered by Js\nT(eq.(1)) for the case\nwhere localized magnetic texture M16is present in the\nFIG. 1. A localized magnetic structure Min a ferromagnetic\nconductor of length 2 Lsubject to the constant temperature\ngradient ∇T.Bis a saturation magnetic field along ∇T.\nThe pure spin current Js\nxis signaled by the inverse spin Hall\nvoltageVISHEmeasured in a Pt strip. xandzaxes are\nindicted.2\nferromagnetic (FM) conductor (cf. Fig.1), a problem of\ngreat importance and has not been addressed so far. As\nshow below, the quantum-mechanical scattering of Js\nT\nfromMacts in effect with a spin-current torque on M\nwhich results in an oscillatoryand a displacement motion\nofM. Upon scattering Js\nTalso changes. This leads to a\nredistribution of the spin electrochemical potential which\ncan be measured via ISHE.\nThe system under consideration is illustrated in Fig.\n1. Two thermal reservoirs with different temperatures T\nandT+∆Tcreate along the xaxis in a FM conductive\nwire of length 2 La steadyT-gradient ∇Tand hence a\nsteady-state spin current Js\nx. This means, without know-\ning the detail of the operators associated with SSE, these\nproject the system onto a chargeless eigenstate ψ(x) of\nthe spin current operator js\nµ(k). Generally, such a state\ncan be written as19\nψ(x) =1\n2/bracketleftbigg\neikx/parenleftbiggeiφ\ne−iφ/parenrightbigg\n+e−ikx/parenleftbiggeiθ\ne−iθ/parenrightbigg/bracketrightbigg\n.(2)\nThexexpectation value of the charge current je(k) =\ni/planckover2pi1\n2m[(∇ψ†(x))ψ(x)−ψ†(x)∇ψ(x)] vanishes, i.e. je\nx(k)≡0\n(heremis the effective mass). In contrast, for the spin\ncurrentjs\nµ(k) =i/planckover2pi1\n2m[(∇ψ†(x))σµψ(x)−ψ†(x)σµ∇ψ(x)]\nwe infer\njs\nx(k) =/planckover2pi1k\n2m(cos2φ−cos2θ), (3)\njs\ny(k) =/planckover2pi1k\n2m(sin2θ−sin2φ). (4)\nIn general, the thermal transport is ballistic20but with\ndiffusive spins, i.e., upon creating (2) the spin coherence\nis lifted by scattering events that randomize φandθ\nwithin [0,2π]. Hence, the expectation value of the spin\ncurrent vanishes on the scale of the spin-flip diffusion\nlength21,22,i.e.,Js\nµ(k) =/contintegraltextjs\nµdθdφ/(2π)2= 0. However,\nwhen the wire is magnetically polarized and driven to\nsaturation by the magnetic field B7–10, we find /angb∇acketleftσx/angb∇acket∇ight /negationslash= 0\nbut/angb∇acketleftσy/angb∇acket∇ight= 0. Eq.(2) reads then for an exchange-split\nconductor\nψB(x) =1\n2/bracketleftbigg\neikx/parenleftbigg1\n1/parenrightbigg\n+e−ikx/parenleftbiggeiθ\ne−iθ/parenrightbigg/bracketrightbigg\n.(5)\nHereθ∈[0,2π] still appears due to the residual spin\nprecession and diffusion. Then we have Je≡0,Js\ny(k) =/contintegraltext\njs\nydθ/2π= 0, whereas Js\nx(k) =Js\n0(k) =/planckover2pi1k/2min line\nwith the experimental observation7–10.\nThe main purpose of the present work is to investigate\nthe influence of a localized magnetic, non-diffusive scat-\ntererM(wherex= 0 is taken as its central position (see\nFig.1)).Mhasauniaxialanisotropyalonganaxischosen\nto bez. IfM(x) has an internal structure, e.g. a non-\ncollinearity, which varies on a scale larger than λ= 2π/k\n(the variation scale of (5)), one can unitary transform to\nalign locally with Mwhich introduces a weak gauge po-\ntential that can be dealt23with in a perturbative way us-\ning the Green’s function constructed from (5) (similarly\nas done in24,25). We find Mhas a stronger influence ifits range of variation is comparable to λ. In this case\nthe magnetic texture acts in effect as M(x) =M0δ(x),\nwhere the magnetic moment M0derives from an average\nofM(x) over its extension w. The model is realizable\nfor10rather then for metals. The interaction between M\nand the electron spin σreads26,\nHint=gM(x)·σ (6)\nwheregis a local coupling constant and M0is large\nenough to be treated classically. For M0aligned with\nzaxis, as in Fig.1, we derive using eqs.(5,6) the expres-\nsionforthe spinorwavefunction in the presenceof M(x),\nnamely\nψs(x) =/braceleftBigg\neikx\n2/parenleftbig1\n1/parenrightbig\n+e−ikx\n2/parenleftbigr\nr⋆/parenrightbig\n+e−ikx\n2/parenleftbigteiθ\nt⋆e−iθ/parenrightbig\nforx<0,\ne−ikx\n2/parenleftbigeiθ\ne−iθ/parenrightbig\n+eikx\n2/parenleftbigreiθ\nr⋆e−iθ/parenrightbig\n+eikx\n2/parenleftbigt\nt⋆/parenrightbig\nforx>0.\n(7)\nThe scattering state ψs(x) describes the spinor wave\nfunction in the original spin channel, which is partially\nreflected into the original-spin and the spin-flip channels,\nand also partially transmitted into theses two channel,\nwhich gives the complex spin reflection and transmission\ncoefficients randt,\nr=−iα\n1+iα, t=1\n1+iα, α=gM0m/k/planckover2pi12.(8)\nThe magnetic scattering gives rise to a short pseudo-\ncircuit to the charge channels, for we find\nje\nx=2αsinθ\n1+α2. (9)\n/angb∇acketleftje\nx/angb∇acket∇ightvanishes however beyond the spin diffusion length\nafter averaging over θ. Counterparts, i.e. a pure spin\ncurrent generated by a charge current when scattered off\na magnetic structure, are well known, e.g.26–28. The spin\ncurrent carried by (5) is modified upon scattering and a\n/Minus1.0/Minus0.5 0.0 0.5 1.0/Minus1.0/Minus0.50.00.51.0\nx/Slash1LVISHE/LParen1x/RParen1/Slash1VISHE0/LParen1L/RParen1\nΑF/EquΑl0.2ΑF/EquΑl0.1ΑF/EquΑl0.0\nFIG. 2. The electric voltage V ISHEgenerated by the inverse\nspin-Hall effect in the Pt layer as a function of the mag-\nnetic scattering strength αF=gM0m/kF/planckover2pi12withkFbeing\nthe Fermi wave vector. The relaxation rate is Γ /EF= 0.01.3\nnone-zeroJs\nyemerges\nJs\nx/Js\n0=/braceleftBigg1+3α2\n(1+α2)2forx<0,\n1−α2\n(1+α2)2forx>0,(10)\nJs\ny/Js\n0=/braceleftBigg\n2α3\n(1+α2)2forx<0,\n2α\n(1+α2)2forx<0.(11)\nWithin linear response, the spin voltage along the\nwire isµs(x) =ξsx/angb∇acketleftJs\nx(α)/angb∇acket∇ightwhereξsis a function\nof the elementary charge, the spin-dependent electric\nconductivity, the spin-dependent Seebeck coefficient,\nand the spin-Seebeck coefficient of the FM wire29.\nThe quantum-mechanically averaged spin current is\n/angb∇acketleftJs\nx(α)/angb∇acket∇ight=Tr(Js\nx(α)ρ)30. The single electron density ma-\ntrix isρ=−1\nπIm/summationtext\nkψsψ†\ns\nEF−Ek+iΓwhereEFis the Fermi\nenergy,Ek=/planckover2pi12k2/2m+/planckover2pi12k2\n/bardbl/2mwithk/bardblbeing the\ntransverse wave vector, and Γ is a Lorenzian relaxation\nrate due to disorder31. Depositing a conductive strip\nwith a strong spin-orbit coupling, e.g. Pt, as shown in\nFig.1,µs(x) can be imaged via the electric voltage VISHE\ngenerated by the inverse spin-Hall effect in Pt using the\nrelation\nVISHE(x)\nV0\nISHE(L)=/angb∇acketleftJs\nx(α)/angb∇acket∇ight\n/angb∇acketleftJsx(0)/angb∇acket∇ightx\nL(12)\nwhere V0\nISHE(x) is the electric voltage measured in\nPt in absence of the magnetic scatterer M. Explic-\nitly, V0\nISHE(x) =γξsx/angb∇acketleftJs\nx(0)/angb∇acket∇ightwithγbeing a system-\ndependent parameter7–10, determined by the spin-Hall\nangle in Pt, the spin-injection efficiency across the\nFM/Pt interface, and the length and the thickness of\nPt wire. Due to the spin current scattering off Mthe\nHall voltage V ISHElosses its odd symmetry with respect\nto a reflection at x= 0, i.e. we deduce −VISHE(−x)/negationslash=\nVISHE(x). AsshowninFig.2theamountofthesymmetry\nbreak depends on α, and can be taken in the experiment\nas an indicator of the presence of magnetic scattering\ncenters.\nMagnetization dynamics. - In as much as Js\nxis modified\nby the presence of M, the scattering triggers a dynam-\nics ofMwhich is usually much slower than the carrier\nscattering dynamics and can be classically treated ( M0\nis assumed large). Js\nxacts onMwith a torque Tµthat\nfollows from the jump in the spin current at the point\nx= 0,Tµ=Js\nµ(0−)−Js\nµ(0+). Hence,Tµderives from\nour quantum mechanical calculations as\nTx=Js\n04α2\n(1+α2)2, Ty=−Js\n02α(1−α2)\n(1+α2)2.(13)\nBoth components are transversal. Tytends to align Mto\nthe direction of the FM magnetization, while Txtries to\nrotate the moment Maround the axis ˆ ex. Equivalently,\nwithin our model, the spin-current torque Tµis obtained\nfrom the spin density Sµ(x) accumulated at the localized\nmoment (due to interference of incoming and reflectedm|| \nmy\nXvx (g/ /uni210F.15514)m|| \nmyag=0 & α=0.1 ag=0.05 & α=0.2 \n-5 05-1.0 -0.5 00.5 1.0 \nt( /uni210F.15514/g) 0 2 4\nt( /uni210F.15514/g) 0 10 20 30 40 \nFIG. 3. Precession of the magnetic moment M(eq.(16)) for\ndifferent spin-current scattering strength α(cf. eq.8) and\nGilbert damping ag. Herevx=∂X/∂tandDz/g= 5.\nm|| \nmy\nvx (gw/ /uni210F.15514)\nx(w/10) vx (gw/ /uni210F.15514)\nx(w/10) m|| \nmyag=0 & α=0.1 ag=0.1 & α=0.5 \n-5 05-0.3 00.3 0.6 0.9 \nt( /uni210F.15514/g) 0 0.02 0.04 0.06 0.08 \nt( /uni210F.15514/g) 0 0.1 0.2 0.3 0.4 \nFIG. 4. Displacement of the magnetic moment (given by\neq.(17)) vs. time for the same parameters of Fig.3. vx=\n∂x/∂t.\nwaves) as\nTµ=−gM0\n/planckover2pi1[n×S(x= 0)] (14)\nwherenis the unit vector along M, and the spin den-\nsity we obtain from Sµ(x) =ψ†(x)σµψ(x). SinceM0is\nassumed large (say ≥5/2µB) the spin current-induced\nmagnetizationdynamicscanbetreatedwiththemodified\nLLG equation32,33\n∂n\n∂t=Dz\n/planckover2pi1[n׈ez]+ag\n/planckover2pi1n×∂n\n∂t−g\n/planckover2pi1[n×S(0)] (15)\nwhereDzis the anisotropy energy and agis the Gilbert\ndamping parameter34. Two motion types of Moccur:\nPrecession- Introducing the following magnetization\ndistribution\nn= [m/bardbl(t)sinX(t),my(t),m/bardbl(t)cosX(t)] (16)4\nand propagating with the LLG equation (Eq.15) starting\nfromm/bardbl(0) = 1, my(0) = 0,andX(0) = 0, we cal-\nculate the time dependence of Mshown in Fig.3. The\noscillations of X(t) results in small xandycomponents\nof the magnetization. The magnetic moment precesses\nwith a velocity vx(t) =∂X(t)/∂tin the presence of the\nSSE-generated spin current ( vx(0)/negationslash= 0). We note that\nthe maximum deflection angle Xmaxdepends implicitly\non the spin current dynamics through the parameter α,\nas determined by eq. (8). Also the Fermi energy enters\nthrough the kdependence of α. The dynamics is a mix-\nture of anisotropy-dominated precession and damping.\nDisplacement- Letusconsidertheinitiallocalizedmag-\nnetic moment distribution\nn= [m/bardbl(t)sinζ,my(t),m/bardbl(t)cosζ]\nζ= cos−1/bracketleftbig\ntanh2(x/w)/bracketrightbig\n(17)\nwherewstands for the extension ofthe localized moment\nandm/bardbl(0) = 1, my(0) = 0, x(0) = 0.As concluded from\nFig.4, the moment is set in motion when subjected to\nthe spin current. The velocity changes from positive to\nnegative, which is different from the motion of a single\nN´ eel wall36(the velocity decreases to zero in a fractionof a nanosecond, and the DWs stops completely.).\nRemarks and conclusions.- Our main result is that the\nSSE generated spin current in a wire may scatter from a\nlocalized magnetic structure setting it in an precessional\nand a displacement motion. The scattering also leads\nto a redistribution of the spin current in the wire and\nhence changes the ISHE signal. From these results con-\nclusion can be made on the influence of a collection of\nnon-interacting localized moments but no statement can\nbe made when they interact or even form clusters. We\nalso note that the present conclusions do not apply to a\ndomain wall (DW), (except for very close non-resonant\n(transversal)180◦DW pair, e.g. asin37). In fact, ourini-\ntial finding23is that a single sharp DW is less affected by\nthe spin current because the spin-current torques acting\nfrom left and right ofthe DWpartially compensate. This\nis not so for an adiabatic or asymmetric DW because the\nT-gradientmodifies the DW along ∇T. As forthe exper-\nimental observation of the magnetic moment dynamics,\nit should be noted that the temperature gradient has to\nbe sustained on the time-scale of this dynamics; a fast\n(e.g., femtosecond) strong heat pulse is inappropriate for\nour (constant ∇T, linear response) study and may cause\nlocally a longitudinal dynamics of M.\n1V.W. Rindner and K. M. Koch, Z. Naturforsch. 13a, 26\n(1958).\n2V. G. Nentwich, Z. Naturforsch. 19a, 1137 (1964).\n3Y. Onose, Y. Shiomi, Y. Tokura, Phys. Rev. Lett. 100,\n016601 (2008).\n4Y. Pu, E. Johnston-Halperin, D.D. Awschalom, J. Shi,\nPhys. Rev. Lett. 97, 036601 (2006).\n5Y. Pu, D. Chiba, F. Matsukura, H. Ohno, J. Shi, Phys.\nRev. Lett. 101, 117208 (2008).\n6T. Miyasato, it al., Phys. Rev. Lett. 99, 086602 (2007).\n7K. Uchida, it al., Nature 455, 346 (2008).\n8K. Uchida, it al., Solid State Commun. 150, 524 (2010).\n9K. Uchida, it al., Nature Mat. 9, 894 (2010).\n10C. M. Jaworski, it al., Nature Mat. 9, 898 (2010).\n11M. Johnson, R.H. Silsbee, Phys. Rev. B 35, 4959 (1987).\n12M. Hatami, G.E.W. Bauer, Q. Zhang, P.J. Kelly, Phys.\nRev. B79, 174426 (2009).\n13M. Hatami, G.E.W. Bauer, Q. Zhang, P.J. Kelly, Phys.\nRev. Lett. 99, 066603 (2007).\n14O. Tsyplyatyev, O. Kashuba, V.I. Falko, Phys. Rev. B 74,\n132403 (2006).\n15A.A. Kovalev, Y. Tserkovnyak, arXiv:0906.1002v2.\n16As an example, we take a single-molecule magnets Mn 12\n[Mn12O12(O2C-C6H4-SAc) 16(H2O)4] as the local mag-\nnetic scatterer M. The total spin and diameter of Mn 12\nareM0= 10 and w= 3nm, respectively17. The local ex-\nchange coupling between the carriers and the moment M\nis given as gM0= 1meV18.\n17H. B. Heersche, Z. de Groot, J. A. Folk, and H.S. J. van\nder Zant, Phys. Rev. Lett. 96, 206801 (2006).\n18R.Q. Wang, L. Sheng, R. Shen, B. Wang, and D.Y. Xing,\nPhys. Rev. Lett. 105, 057202 (2010).\n19We employ a continuous model; a discretized treatmentbased on Heisenberg spins is straightforward and does not\nalter qualitatively the amplitude of the spin/charge cur-\nrent.\n20X. Zotos, F. Naef, and P. Prelovˇ sek, Phys. Rev. B 55,\n11029 (1997); A. V. Sologubenko, it al., Phys. Rev. B 62,\nR6108 (2000).\n21M. Hatami, G. E. W. Bauer, S. Takahashi, and S.\nMaekawa, Solid State Commun. 150, 480 (2010).\n22J. Xiao, G. E. W. Bauer, K.-C. Uchida, E. Saitoh, and S.\nMaekawa, Phys. Rev. B 81, 214418 (2010).\n23C.L. Jia, J. Berakdar, unpublished.\n24N. Sedlmayr, V.K. Dugaev, andJ. Berakdar, Phys.Rev. B\n79, 174422 (2009); Phys. Status Solidi B 247, 2603 (2010).\n25V. K. Dugaev, J. Barnas and J. Berakdar, J. Phys. A 36,\n9263 (2003).\n26V. K. Dugaev, V. R. Vieira, P. D. Sacramento, J. Barna´ s,\nM.A.N.Ara´ ujo, andJ.Berakdar, Phys.Rev.B 74, 054403\n(2006).\n27A. Vedyayev, M. Chschiev, and B. Dieny, J. Phys.:\nCondens. Matter 20, 145208 (2008); A. Manchon, N.\nRyzhanova, A. Vedyayev, M. Chschiev, and B. Dieny, J.\nPhys.: Condens. Matter 20, 145208 (2008).\n28M. Araujo, V. Dugaev, V. Veira, J. Berakdar, and J. Bar-\nnas, Phys. Rev. B 74, 224429 (2006).\n29K. Uchida, it al., J. Appl. Phys. 105, 07C908 (2010).\n30V. K. Dugaev, J. Berakdar, and J. Barna´ s, Phys. Rev. B\n68, 104434 (2003).\n31O. E. Dial, R. C. Ashoori, L. N. Pfeiffer and K. W. West,\nNature(London) 448, 176 (2007).\n32Ya. B. Bazaliy, B. A. Jones, and S.-C. Zhang, Phys. Rev.\nB57, R3213 (1998).\n33J. Fern´ andez-Rossier, M. Braun, A. S. N´ u nez, and A. H.\nMacDonald, Phys. Rev. B 69, 174412 (2004).5\n34Temperature effects can be included in eq.(15) as in35. by\nassuming Mto be in a local thermal equilibrium at the\ntemperature T(x= 0) (assuming ∇T=constant ) an ap-\nplying the fluctuation-dissipation theorem which yields a\nstochastic field that adds to the effective field in eq.(15).\nIn35it is shown that at low Tthermal fluctuations do notalter qualitatively the LLG dynamics.\n35A. Sukhov, J. Berakdar, Phys. Rev. Lett. 102, 057204\n(2009).\n36Z. Li and S. Zhang, Phys. Rev. Lett. 92, 207203 (2004).\n37V. Dugaev, J. Berakdar, and J. Barnas, Phys. Rev. Lett.\n96, 047208 (2006)."    },    {        "title": "2106.11127v1.Resonant_Measurement_of_Non_Reorientable_Spin_Orbit_Torque_from_a_Ferromagnetic_Source_Layer_Accounting_for_Dynamic_Spin_Pumping.pdf",        "content": "Resonant Measurement of Non-Reorientable Spin-Orbit Torque from a\nFerromagnetic Source Layer Accounting for Dynamic Spin Pumping\nJoseph A. Mittelstaedt1and Daniel C. Ralph1, 2\n1Cornell University, Ithaca, NY 14850, USA\n2Kavli Institute at Cornell, Ithaca, NY 14853, USA\n(Dated: June 22, 2021)\nUsing a multilayer structure containing (cobalt detector layer)/(copper spacer)/(Permalloy source\nlayer), we show experimentally how the non-reorientable spin-orbit torque generated by the Permal-\nloy source layer (the component of spin-orbit torque that does not change when the Permalloy\nmagnetization is rotated) can be measured using spin-torque ferromagnetic resonance (ST-FMR)\nwith lineshape analysis. We \fnd that dynamic spin pumping between the magnetic layers exerts\ntorques on the magnetic layers as large or larger than the spin-orbit torques, so that if dynamic spin\npumping is neglected the result would be a large overestimate of the spin-orbit torque. Nevertheless,\nthe two e\u000bects can be separated by performing ST-FMR as a function of frequency. We measure a\nnon-reorientable spin torque ratio \u0018Py= 0:04\u00060:01 for spin current \row from Permalloy through an\n8 nm Cu spacer to the Co, and a strength of dynamic spin pumping that is consistent with previous\nmeasurements by conventional ferromagnetic resonance.\nI. INTRODUCTION\nCurrent-induced spin-orbit torques o\u000ber the potential\nfor e\u000ecient manipulation of nanoscale magnets for mem-\nory applications [1{3]. Among the families of materi-\nals that are of interest as potential sources of spin-orbit\ntorque are metallic ferromagnets [4{19]. They are pre-\ndicted to generate current-induced spin currents with\nboth a reorientable component in which the spin is al-\nways parallel to the source-layer magnetization, plus a\nnon-reorientable component (with a \fxed spin direction\nin-plane and perpendicular to the charge current) that,\nremarkably, is not dephased within the source magnet\neven though the spin is not aligned with the magnetiza-\ntion [14]. The non-reorientable component has the same\nsymmetries as the torque generated by the spin Hall ef-\nfect from a non-magnetic source material.\nPreviously, spin-orbit torques from metallic ferromag-\nnets have been probed indirectly by exciting nonlocal\nspin currents through magnetic insulators [10, 20{23],\nand the reciprocal process of spin-to-charge conversion\nhas been measured for spin currents injected from an in-\nsulating ferromagnetic into a metallic ferromagnet [24{\n29]. However, it is more challenging to make direct mea-\nsurements of spin-orbit torques generated by ferromag-\nnets in magnetic trilayer structures that would be nec-\nessary for memory applications (i.e., ferromagnetic spin-\nsource layer / non-magnetic spacer / ferromagnetic free\nlayer to be manipulated). Ideally, one would like to apply\na current and make quantitative measurements of just\nthe de\rection of the free-layer magnetization. However,\nin structures with two magnetic layers an applied cur-\nrent will exert torques on both layers and their dynamics\nwill also be coupled by interlayer interactions, making it\nchallenging to isolate just the strength of the spin-orbit\ntorque acting on the free layer. For the reorientable com-\nponent of the spin torque, one approach to address this\nchallenge is to perform spin-torque ferromagnetic reso-\nnance (ST-FMR) experiments and analyze the depen-\ndence on dc current of the separate linewidths for eachmagnetic layer [7, 11, 15, 17{19], although care should\nbe taken in avoiding artifacts that can a\u000bect this method\n[30, 31]. Here, we demonstrate that ST-FMR can also be\nused to achieve quantitative measurements of the non-\nreorientable (independent of the source-layer magneti-\nzation) component of spin-orbit torque generated by a\nferromagnetic metal layer in a magnetic trilayer, by an-\nalyzing the resonant amplitudes and lineshapes. For a\ncorrect analysis of trilayers containing two metallic mag-\nnets (e.g., Co and Permalloy (Py)), it is critical to take\ninto account that dynamic spin pumping [32, 33] cou-\nples the two magnetic layers and alters their resonance\namplitudes. Still, the e\u000bects of spin currents associated\nwith spin-orbit torques and those arising from dynamic\nspin pumping can be separated by analyzing the fre-\nquency dependence of the ST-FMR signals. Therefore,\nST-FMR performed on magnetic trilayers can provide\nquantitative measurements of both the non-reorientable\nspin-orbit torque produced by a magnetic source layer\nand dynamic spin pumping. Our approach is similar to\nthat of Yang et al. [34], except that their method does\nnot account for dynamic spin pumping.\nII. SAMPLE FABRICATION AND\nMEASUREMENTS\nWe used DC magnetron sputtering onto thermally ox-\nidized high-resistivity silicon wafers to grow heterostruc-\ntures of containing both a Co \\detector\" layer, whose\nresonance dynamics we analyze, and a Ni 81Fe19(Py)\n\\source\" layer that generates a spin current to act on the\nCo. Speci\fcally, the sample layer structure is SiO x/Ta\n(1)/Cu (8)/Co (8)/Cu (8)/Py ( tPy)/Ta (1) with num-\nbers in parentheses indicating thickness in nm. The two\nCu spacer layers have the same thickness, and are de-\nsigned so that the in-plane component of the Oersted\n\feld that they generate will cancel within the Co de-\ntector, and consequently the only net in-plane Oersted\n\feld acting on the Co layer will be due to current \rowingarXiv:2106.11127v1  [cond-mat.mes-hall]  21 Jun 20212\nwithin the Py layer. This will simplify our analysis of the\nspin-orbit torque acting on the Co layer. Cu was chosen\nfor the spacer layers because of its long spin di\u000busion\nlength and negligible spin Hall e\u000bect [35]. The Ta lay-\ners provide layer-smoothing and protection against ox-\nidation, and have su\u000eciently large resistivity that they\ncarry negligible current.\nDuring the measurement, we apply an RF current to-\ngether with an in-plane external magnetic \feld at an\nangle\u001erelative to the axis of current \row, and sweep\nthe external \feld strength to tune through the magnetic\nresonance (Figure 1(b)). Each sample is studied using\nseveral di\u000berent RF frequencies and \feld angles \u001e, and\nall measurements are performed at room temperature.\nWe detect a DC signal that arises from mixing between\nthe applied RF current and the resistance variations due\nto the angle-dependent magnetoresistance in both mag-\nnetic layers, originating from both anisotropic magne-\ntoresistance within the layers and spin Hall magnetore-\nsistance [36]. (We work at su\u000eciently large applied mag-\nnetic \felds that the magnetizations of the Co and Py\nlayers are saturated parallel in equilibrium, so contri-\nbutions from giant magnetoresistance are second order\nin the precession angles, and negligible for the measure-\nment.) The circuitry of the measurement is described\nelsewhere [37{39]. The two magnetic layers produce two\nresonances, with the lower resonant \feld being primarily\nfrom the Co and the higher resonant \feld primarily from\nthe Py. We \ft these resonances to two sets of symmet-\nric and antisymmetric lorentzians to capture the signal\nvoltages from each of the magnetic layers.\nThe dependence on the angle \u001efor the symmetric and\nantisymmetric resonance amplitudes of the Co detector\nlayer are shown in Figure 1(c) and (d): both follow a\nsin(2\u001e) cos(\u001e) dependence. This is the form expected\nfrom \\conventional\" current-induced torques { from the\nin-plane component of the Oersted \feld and from spin-\norbit torques arising from a spin current with polariza-\ntion in-plane and perpendicular to the current. It is also\nthe form expected in our geometry for torques arising\nfrom interfacial spin rotation as discussed in refs. [6, 8],\nand based on the analysis below the same angular de-\npendence will be produced by the torques from dynamic\nspin pumping. We observe no signi\fcant contribution\nfrom unconventional current-induced torques that would\nlead to an altered angular dependence [40]. We note,\nhowever, that since both magnetizations have the same\nequilibrium orientation for our geometry, our measure-\nments are not sensitive to the reorientable torques arising\nfrom the spin anomalous Hall e\u000bect or planar Hall e\u000bect\n[4, 11] which create spin currents whose polarization is\nparallel to the magnetization.\nIII. ANALYSIS: EFFECT OF DYNAMIC SPIN\nPUMPING\nWe can account for the resonant dynamics of the\nmagnetic layers, including the e\u000bects of spin pumpingbetween the layers, using a modi\fed Landau-Lifshitz-\nGilbert-Slonczewski (LLGS) equation [32, 33]\n_mi=\u000bimi\u0002_mi+\u001cneq;i+\u001ceq;i\u0000\u000b0\nimj\u0002_mj(1)\nwhereiandjidentify the magnetic layers, \u001ceq;iare\nthe equilibrium torques acting on layer icoming from\nthe applied \feld and anisotropy, and \u001cneq;iare the non-\nequilibrium torques acting on layer icoming from spin-\norbit torques and current-generated Oersted \felds. \u000bi\nis the e\u000bective Gilbert damping parameter for layer\ni, including both the intrinsic damping and the time-\naveraged e\u000bect of the dynamically-pumped spins emerg-\ning from layer i, while the term containing \u000b0\nidescribes\nthe e\u000bect of the dynamically-pumped spins from layer j\nimpinging on layer ito exert a time-dependent torque.\nDipole coupling of the form mi\u0001mjis generally also\nrelevant in systems with more than one ferromagnetic\nlayer, but we designed our Cu spacers to be su\u000eciently\nthick to minimize this e\u000bect, and found by measuring the\nshift of the minor loop in hysteresis measurements that\nit is indeed negligible in our system (see Supplementary\nInformation).\nOur treatment of the torques due to dynamic spin\npumping follow references [32, 33, 41]. Compared to the\nmore-familiar time-averaged DC component of the spin-\npumped spin current that generates DC voltage signals\nin inverse spin Hall experiments [42{44] and modi\fes the\nmagnetic damping [45, 46], the time-varying spin current\nassociated with dynamic spin pumping is much larger. In\nsystems with just a single ferromagnetic layer the e\u000bects\nfrom the dynamic portion of the pumped spins are gener-\nally time-averaged to zero, but with a second ferromag-\nnetic layer present the pumped spins will induce dynamic\ncoupling between the magnetic layers [32, 33, 47, 48].\nBecause the spins pumped by layer jare oscillating and\nare generally out-of-phase with the precession in layer i,\nthey exert a nontrivial torque on layer iwhich depends\non the phase di\u000berence between the oscillations of the\ntwo layers, introducing \feld and frequency dependence\nto the interaction. Based on this \feld and frequency de-\npendence, we will show that it is possible to separate the\ne\u000bects of the spin-pumping-induced interaction between\nthe layers from the direct e\u000bects of the non-reorientable\nspin-orbit torque from the source layer.\nTo model the resonance lineshapes, we consider the\ncase that the resonance \feld is much larger than the co-\nercive \felds of both magnetic layers so that the equilib-\nrium magnetizations of both layers are saturated in the\nsame direction. We then solve for in-plane ( x) and out-\nof-plane (z) de\rections of each layer, and only keep terms\nto \frst order in these de\rections from equilibrium. We\ncan \fnd the de\rections of each layer by solving Eq. (1)\nfollowing the same procedure as [49], noting that we can\nthink of the dynamic spin pumping as e\u000bectively mod-\nifying the torques on each layer. After decoupling the\nin-plane and out-of-plane de\rections of each layer, the3\nFIG. 1. (a) Schematic of our device near the Co resonance. The Oersted \feld from the symmetric spacer layers around the\nCo will cancel inside of the Co layer. Even at the Co resonance condition the Py precession is still signi\fcant. (b) example\nresonance signal of the tPy= 5:7 nm sample showing the Co (left, lower resonant \feld) and Py (right, higher resonant \feld)\nresonances and their symmetric and antisymmetric components for an applied microwave frequency f= 8 GHz. The dashed\nvertical line represents the Co resonance \feld. The antisymmetric part of the Py resonance is still substantial there. (c,d) The\nangular variation in the Co symmetric (c) and antisymmetric (d) resonance components, showing the sin(2 \u001e) cos(\u001e) angular\ndependence expected from SHE-like torques and in-plane Oersted torques, respectively.\nde\rections of the detector layer are\nmdx=Ld(\u0000!d2[\u001cdz\u0000i!\u000b0\ndmsx]\n+i![\u001cdx+i!\u000b0\ndmsz])(2)\nmdz=Ld(!d1[\u001cdx+i!\u000b0\ndmsz]\n+i![\u001cdz+i!\u000b0\ndmsx])(3)\nwhere the d subscript denotes parameters\nassociated with the detection layer, Ld =\u0002\u0000\n!2\u0000!2\nd0\u0001\n+i!!+\nd\u000bd\u0003\u00001,\u001cdxand\u001cdzthe in-plane and out-of-plane non-equilibrium torques,\n!d1=\rB,!d2=\r(B+\u00160Me\u000b;d),!2\nd0=!d1!d2,\nand\u00160Me\u000b;d= 1:82\u00060:09 T for the Co layer based\non the precession frequency. The expressions for the\nsource-layer de\rections are the same as the detec-\ntor layer's but with sanddinterchanged and with\n\u00160Me\u000b;s= 0:93\u00060:05 T for the Py.\nWe can then completely decouple this system by ig-\nnoring any terms which are above \frst order in the \u000b's\nand\u000b0's. The result for the in-plane de\rection of the de-\ntector layer, which generates the signal we are primarily\nconcerned with, is4\nmdx=Ld\u0000\n\u0000!d2\u001cdz+Ls\u000b0\nd\u0002\n(!2\ns0\u0000!2)(!s1+!d2)!2\u001csx\u0000!2!+\ns\u000bs(!s2!d2+!2)\u001csz\u0003\n+i!\u0000\n\u001cdx+Ls\u000b0\nd\u0002\n(!2\ns0\u0000!2)(!s2!d2+!2)\u001csz+!2!+\ns\u000bs(!s1+!d2)\u001csx\u0003\u0001\u0001 (4)\nwithLs=\u0002\n(!2\u0000!2\ns0)2+!2!+2\ns\u000b2\ns\u0003\u00001and!+\ni=!i1+\n!i2. The real part of the expression within the large\nparentheses corresponds to the antisymmetric compo-\nnent of the detector-layer resonance and the imaginary\npart to the symmetric component. The terms pro-\nportional to Ls\u000b0\ndrepresent modi\fcations of the usual\nexpression for the ST-FMR resonance caused by spin\npumping from the source layer. Through these terms,\ntorques acting on the source layer ( \u001csx,\u001csz) produce a\nmodi\fcation of the torques acting on the detector layer,\nwith the strength of the e\u000bect depending on the am-\nplitude of the source-layer resonance through Ls. The\nterms in Eq. (4) inside the square brackets containing\n(!2\ns0\u0000!2) correspond to the antisymmetric component\nof the source-layer resonance while the other terms cor-\nrespond to the symmetric component.\nThis expression can be simpli\fed signi\fcantly for ap-\nplication to our measurements. First, the symmetric\npart of our source (Py) resonance is essentially zero near\nthe detector (Co) resonance \feld as seen in Figure 1(b),\nso we can ignore all terms in Eq. (4) which correspond\nto the symmetric part of the source resonance. Second,\nthe dampinglike torque acting on the source layer, \u001csx,\nis quite small compared to the Oersted torque on the\ndetector layer, \u001cdz, as illustrated in Figure 1(b) by the\nfact that the symmetric part of the Py resonance is tiny\ncompared to the antisymmetric part of the Co resonance.\nThis allows us to ignore all terms in Eq. (4) which in-\nclude\u001csxas well. After these simpli\fcations, the in-plane\noscillation of the detector layer takes the form\nmdx=Ld(\u0000!d2\u001cdz\n+i!\u0000\n\u001cdx+Ls\u000b0\nd(!2\ns0\u0000!2)(!s2!d2+!2)\u001csz\u0001\u0001\n:\n(5)\nThe amplitude of the DC mixing signal we detect has\nthe form\nVmix=I\n2\u0000\nRAMR\nd Re [mdx] +RAMR\ns Re [msx]\u0001\nsin(2\u001e);\n(6)\nwhereIis the total current \rowing in the multilayerandRAMR\ni sin(2\u001e) is the\u001e-derivative of the resistance\nchange of the whole heterostructure due to the angle-\ndependent magnetoresistance of layer i. In general, it is\nalso necessary to include an additional voltage signal due\nthe DC component of the pumped spin current and the\ninverse spin Hall e\u000bect [42{44], but we calculate that this\ncontribution is negligible for our samples (see Appendix\nA).\nEven at the position of the detector-layer resonance\nit is not correct to ignore the contribution proportional\nto the source-layer magnetoresistance (the second term\nin Eq. (6)), because precession of the detector layer will\nproduce dynamic spin pumping that drives precession of\nthe source layer. Since this precession of the source layer\nis driven by spin pumping from the detector layer, near\nthe detector layer's resonance condition the source-layer\nprecession has the same resonant \feld and linewidth as\nthe detector-layer resonance. This means that when we\nmeasure the lineshape which we attribute to the detector\nlayer, some of this signal is actually due to the source-\nlayer precession and the mixing voltage which results.\nWe can take into account the source-layer oscillations\nby writing their in-plane component in the form\nmsx=Ls(\u0000!s2\u001csz+i!\u001csx)\n+LdLs(!2\ns0\u0000!2)\u0002\n!2(!d1+!s2)\u001cdx\u000b0\ns\n+i!(!d2!s2+!2)\u001cdz\u000b0\ns\u0003(7)\nwhere the \frst term is the source-layer resonance in the\nabsence of any dynamic spin pumping coupling and the\nLdterm the one producing a resonance in the source\nlayer of the same shape as the detector layer. Near the\ndetector-layer resonance condition, the \frst term is es-\nsentially constant and hence will not play a signi\fcant\nrole in our analysis.\nCombining Eqs. (5), (6) and (7), the voltage am-\nplitudes we measure upon \ftting the detector-layer\nresonance as a sum of symmetric and antisymmetric\nLorentzians is\nVdS=IRAMR\nd sin(2\u001e)\n21\n\u000bd!+\nd\u0002\n\u001cdx+LsA(!s2!d2+!2)\u001csz\u000b0\nd\u0003\n+IRAMR\ns sin(2\u001e)\n21\n\u000bd!+\ndLsA(!s2!d2+!2)\u001cdz\u000b0\ns(8)\nVdA=IRAMR\nd sin(2\u001e)\n21\n\u000bd!+\nd!d2\n!\u001cdz (9)\nwhereLsA=Ls(!2\ns0\u0000!2). The \frst part of each mixing\nvoltage comes from the detector layer and the secondpart is due to the source layer. There is also a small\ncontribution of the source-layer resonance to VdAthat is5\nFIG. 2. (a,b) Measured values of \u0018FMR for a sample with 5.7 nm Py with a \ft to Eq. (12). In (a) we plot the data versus\nfrequency and in (b) we plot the same data versus calculated values of R(!) to show that the data vary linearly with R(!).\nThe horizontal line is the extracted value of \u0018DL, which is the asymptote of (a)/the intercept of (b). (c) The e\u000bect of Py\nthickness on \u0018DL. (d) The dependence of the dynamic spin pumping parameter on the source Py layer thickness.\nnot included in Eq. (9) because it contributes no more\nthan 1 % of signal from the detector layer for our system,\nso we can ignore it in our analysis.\nWe are left with two signi\fcant modi\fcations to the\nST-FMR signals caused by dynamic spin pumping, both\nof which modify the measured symmetric lineshape am-\nplitude of the detector layer. The \frst is due to spins\npumped from the source layer to the detector layer,\nthereby altering the detector layer resonance, an ef-\nfect which is proportional to the Oersted torque on the\nsource layer, \u001csz, with an amplitude proportional to\nthe antisymmetric component of the source-layer reso-\nnance. This contribution adds directly to the damp-\ninglike torque on the detector layer, \u001cdx. The second\nis due to spins pumped from the detector layer to the\nsource layer, causing the source layer to oscillate with\nthe same general lineshape as the detector-layer reso-\nnance. The size of this term proportional to the Oersted\ntorque on the detector layer \u001csz, and it also has an am-plitude proportional to the antisymmetric component of\nthe source-layer resonance. From Eqs. (8) and (9) it is\nclear that the spin-pumping terms are not expected to al-\nter the usual angular dependence of the ST-FMR signals\n(/sin(2\u001e) cos(\u001e)) as long as the current-induced torques\nare proportional to cos( \u001e), the dependence expected for\nboth conventional-symmetry spin-orbit torques and Oer-\nsted torques [40].\nThe e\u000eciency of the antidamping spin-orbit torque\ngenerated by the Py source layer acting on the Co detec-\ntor layer can be characterized by the antidamping torque\ne\u000eciency, de\fned as\n\u0018DL=\u001cdxeMsat,dtd\n\u0016BJc;s; (10)\nwhereeis the electron charge, Msat;d= 1:19\u00060:08\u0002106\nA/m for the Co layer, td= 8 nm for the Co layer, and\nJc;sthe charge current density within the source mag-\nnetic layer. This e\u000eciency, and also the in\ruence of the6\ndynamic spin pumping, can be determined by analyzing\nhow the ratio of the symmetric and antisymmetric res-\nonance amplitudes depends on measurement frequency.\nFollowing ref. [50], we \frst de\fne an intermediate pa-\nrameter\n\u0018FMR =e\u00160Msat;dtstd\n~r!d2\n!d1VdS\nVdA(11)\nwherets=tPy. The values of \u0018FMR extracted for a\nsample with ts= 5:7 nm as a function of di\u000berent applied\nRF frequencies are shown by the circles in Figure 2(a).\nWe \fnd that \u0018FMR increases by more than a factor of 3 as\nthe frequency is decreased from 14 GHz to 6 GHz, a much\nstronger dependence than is found in samples with non-\nmagnetic source layers [37]. We will show that this can\nbe explained by dynamic spin pumping, in that for lower\nRF frequencies the resonance \felds of the two magnetic\nlayers move closer together, increasing the source layer\nresonance amplitude, and hence the value of LsA, at the\ndetector-layer resonance condition which enhances the\ne\u000bects of the dynamic spin pumping.\nFor magnetic layers as thick as those we employ,\n\u00155 nm, and since the detector layer's interfaces are\nsymmetric and are with the light metal Cu, Oersted\ntorques should dominate over any weak interfacial \feld-\nlike torques for both the source and detector layers [50{\n52]. We therefore assume that \u001cdz=\r\u00160Jc;sts=2 (only\nthe current density within the source layer contributes to\nthe Oersted \feld acting on the detector layer because the\ncontributions from the Cu spacer layers cancel and the\nTa layers have high resistivity and negligible current den-\nsities) and \u001csz=\r\u00160(Jc;dtd+Jc;CutCu)=2 (the current\ncreating the Oersted \feld in the source layer comes from\nthe detector layer and the Cu layers, which comprise the\nrest of the conductive layers in the device). The ratio\nof these torques can be written \u001csz=\u001cdz=\u0000(1\u0000xs)=xs\nwherexs, the fraction of the total RF current \rowing\nwithin the source layer, is in the range of 0.06 to 0.12\ndepending on the Py thickness and is calculated from\nthe layer resistivities using a parallel-conduction model\n(see Supplementary Information). It then follows that\n\u0018FMR =\u0018DL+R(!)\u000b0\ne\u000b (12)\nwith\nR(!) =\u0000e\u00160Msat;dtstd\n~LsA(!s2!d2+!2)1\u0000xs\nxs(13)\n\u000b0\ne\u000b=\u000b0\nd\u0000xs\n1\u0000xsRAMR\ns\nRAMR\nd\u000b0\ns: (14)\nAll of the parameters that are frequency dependent are\ncontained within R(!). Furthermore, all of the param-\neters that enter R(!) are independently measurable, so\nwe can use Eq. (12) to \ft to the data in Figure 2(a),\nusing just two adjustable \ft parameters, \u0018DLand\u000b0\ne\u000b.\nThe \ft is shown both in Figure 2(a) and in Figure 2(b),\nwhich display the same data points plotted as a func-\ntion of measurement frequency in (a) and as a function\nofR(!) in (b). For the tPy= 5:7 nm sample, we \fnd\u0018DL= 0:05\u00060:01 and\u000b0\ne\u000b= 4:0\u00060:4\u000210\u00003. The dom-\ninant source of uncertainty comes from the determina-\ntion ofMe\u000b;dwhich we measure by \ftting the frequency\ndependence of the detector-layer resonant \feld to the\nKittel equation. We note in samples without dynamic\nspin pumping, and in which like our samples the \feld-\nlike spin-orbit torque is negligible relative to the Oersted\ntorque,\u0018DL=\u0018FMR. Therefore, Figure 2(a) illustrates\nthat the neglect of spin-pumping in our analysis would\nlead to a large overestimate of \u0018DL.\nWe have performed the same analysis on samples in\nwhich the Py layers vary in thickness from 5 nm to 12\nnm. The results for \u0018DLand\u000b0\ne\u000bare shown in Fig-\nure 2(c) and (d). The overall dampinglike torque e\u000e-\nciency\u0018DLof the magnetization-independent spin-orbit\ntorque from Py through the 8 nm Cu spacer to the Py\nlayer is 0:04\u00060:01, and it does not have a signi\fcant\ndependence on the Py thickness within our experimen-\ntal uncertainty. The value we determine for \u000b0\ne\u000bdoes\ndepend on the Py thickness, changing by more than a\nfactor of 2 between Py thicknesses of 5 and 12 nm. The\nparameter\u000b0\ndshould not depend on the Py thickness, so\nthe changes in \u000b0\ne\u000b(tPy) indicate that the second term in\nEq. (14) is important as well as the \frst { that the sig-\nnal at the detector resonance is a\u000bected by spin pumping\nfrom the detector layer to the source layer that excites\nprecession in the source (Py) layer. We are not able to\nmake a separate determination of the parameters \u000b0\ndand\n\u000b0\nsin Eq. (14), however, because we are not able to make\nan accurate calibration of the ratio RAMR\ns=RAMR\nd. We\ncan calibrate the anisotropic magnetoresistance of indi-\nvidual Co and Py layers, but we \fnd that when they are\ncombined within a trilayer there is also a signi\fcant ad-\nditional contribution from spin Hall magnetoresistance\nthat cannot be calibrated without separate control over\nthe magnetization angles in the two layers.\nIV. DISCUSSION\nWe can compare our results to previous measure-\nments. E\u000borts to determine the non-reorientable spin\nHall e\u000bect in ferromagnetic layers via spin Seebeck and\nspin pumping experiments are complicated by di\u000ecul-\nties in determining the interface spin mixing conduc-\ntance [53] and the spin di\u000busion length. Furthermore,\nmany of these experiments do not provide su\u000ecient in-\nformation to distinguish between the reorientable and\nnon-reorientable components of the spin current gener-\nated by Py. The two measurements with direct quan-\ntitative claims we have found in this class of experi-\nments give a value for the non-reorientable spin Hall\nratio within Py of \u0012SH(Py) = 0.02 by spin pumping in\nYIG/Cu/Py [25, 26]. The quantity we measure, the spin-\norbit torque e\u000eciency, \u0018DL, is related to the spin Hall\nratio as\u0018DL=\u0012SH(Py)TintwhereTintis an interfacial\nspin transfer coe\u000ecient less than or equal to 1. There-\nfore our measurements establish a lower bound for the\nnon-reorientable spin Hall e\u000bect for the Py in our sam-7\nples,\u0012SH(Py)\u00150:04\u00060:01. We suspect the di\u000berence\nis that the spin mixing conductance may be overesti-\nmated [53] in refs. [25, 26], leading to an underestimate\nof\u0012SH(Py). Miao et al. [24] and Wu et al. [27] studied\nYIG/Py samples and quoted values of \u0012SH(Py) relative\nto values for similar samples made with Pt instead of\nPy:\u0012SH(Py)=\u0012SH(Pt) = 0.38 [24] and 0.98 [27]. These\nestimates rely on assumptions that the interfacial spin\ntransparency and spin di\u000busion lengths are similar for\nYIG/Py and YIG/Pt.\nDas et al. [10, 20] detected both reorientable and non-\nreorientable components of spin-orbit torque generated\nby Py using nonlocal measurement scheme where a spin\ncurrent generated in Py was converted to magnons in\nYIG which then were converted back into a spin cur-\nrent in Pt or Py and transduced into a voltage through\nthe inverse spin Hall e\u000bect. Their value of \u0012SH(Py)\nis only given in relation to that of Pt, they quote\n[GPy\u0012SH(Py)]=[GPt\u0012SH(Pt)] = 0:09 whereGiis a param-\neter describing the spin current to magnon conversion in\neach of the materials. Yang et al. [34] attempted to mea-\nsure the non-reorientable component of the spin curernt\nalone from Py using lineshape analysis of ST-FMR sig-\nnals for Py/YIG and CoFeB/YIG samples (both with no\nspacer layer), but without accounting for dynamics spin\npumping. They claimed values of the non-reorientable\nspin-orbit torque e\u000eciencies \u0018DL= 0:009 for Py acting\non YIG and\u00000:0014 for CoFeB acting on YIG; values\nmuch lower in magnitude than our result. We suspect\nthat these low values may be due primarily to poor inter-\nfacial spin transmission from the Py into the YIG. Low\nspin transparency has also been a general feature found\nin spin-orbit torque measurements from heavy metals\nacting on iron garnets (see the comparisons by Gupta et\nal. [54]).\nBased on measurements of conventional FMR in mag-\nnet/spacer/magnet samples, the parameter \u000b0\ne\u000bis ex-\npected to have a scale comparable to the intrinsic Gilbert\ndamping in magnetic layers, \u000b[32]. This is true for our\nsamples. The values of \u000b0\ne\u000bplotted in Fig. 2(d) are about\nhalf the damping parameters in our Co and Py layers de-\ntermined by the ST-FMR \fts.V. CONCLUSION\nWe have measured and analyzed spin-torque ferromag-\nnetic resonance (ST-FMR) signals for samples contain-\ning the layer structure Py source layer/Cu spacer/Co\ndetector layer. We \fnd that the resonance amplitudes\nand lineshapes are determined by a combination of di-\nrect current-induced torques and dynamic spin pumping\nbetween the magnetic layers. In fact, the strength of the\ntorque created by dynamic spin pumping can be substan-\ntially larger than the direct current-induced spin-orbit\ntorque, so the contribution from dynamics spin pump-\ning should be considered whenever analyzing resonant\nmeasurements in samples with more than one ferromag-\nnetic layer [13, 34]. The frequency dependence of the\nST-FMR signals allows these two e\u000bects to be separated,\nand therefore our experiment provides independent mea-\nsurements of the non-reorientable spin-orbit torque gen-\nerated by the Py layer (the portion of the spin-orbit\ntorque that does not depend on the orientation of the\nPy magnetization) and the strength of dynamical spin\npumping. We \fnd an e\u000eciency for the non-reorientable\nspin-orbit torque generated by Py (and acting through\na 8 nm Cu spacer) of 0 :04\u00060:01. The strength of the\ndynamic spin pumping, as characterized by the parame-\nter\u000b0\ne\u000b, appears consistent with previous measurements\nusing conventional ferromagnetic resonance [32].\nVI. ACKNOWLEDGEMENTS\nWe thank Bob Buhrman for discussions. This research\nwas supported in part by Task 2776.047 of ASCENT,\none of six centers in JUMP, a Semiconductor Research\nCorporation program sponsored by DARPA, and by the\nNSF (DMR-1708499). The devices were fabricated us-\ning the shared facilities of the Cornell NanoScale Facility,\na member of the National Nanotechnology Coordinated\nInfrastructure (NNCI), and the Cornell Center for Mate-\nrials Research, both of which are supported by the NSF\n(Grants No. NNCI-1542081 and No. DMR-1719875).\nAppendix A: DC Spin Pumping Contribution\nThe DC component of the pumped spin current from\nthe Co is converted into a DC charge current through\nthe inverse spin Hall e\u000bect in the Py layer, as described\nin ref. [49]. The generated DC spin pumping voltage\nfollows the form\nVDC\nSP=\u00002e\n~\u0012SHRtotWsin\u001e~\n4\u0019g\"#\ne\u000b\u0015sd,s\u0014!d1\u001c2\ndx+!d2\u001c2\ndz\n(\u000bd!+\nd)2LdS(B)\u0015\ntanh\u0012ts\n2\u0015sd;s\u0013\n(A1)\nwhereRtotis the total resistance of the heterostructure, Wis the width of the device, \u0015sd;sis the spin di\u000busion8\nFIG. 3. Comparison of the DC spin pumping contribution to\n\u0018FMR to the total signal for 5.7 nm Py. The contribution is\nso small as to be negligible.\nlength of the source, and LdS(B) is the symmetric line-\nshape of the detector layer. We have either measured\nor can estimate with reasonable certainty all quantities\nin this expression. For the ts= 5:7 nm Py sample,\nRtot= 12:1 \n,W= 20\u0016m,\u000bd= 0:014,\u0012SH\u00190:05,we approximate g\"#\ne\u000b\u00198 nm\u00002[53] and\u0015sd;s\u001910 nm\n[55] as reasonable estimates compared to other systems,\nwe approximate 1.5 mA of RF current is \rowing in the\nPy layer using the input microwave power after account-\ning for loss in the cabling and current shunting, which\nleads to\u001cdx\u00194\u0002106s\u00001and\u001cdz\u00198\u0002106s\u00001, and\n!d1; !d2and!+\nd, which are evaluated at the Co reso-\nnant \feld and hence depend on frequency, are on the\norder of 10 GHz, 300 GHz and 300 GHz, respectively.\nTo compare this more directly to what we have mea-\nsured before, we utilize the fact that this will only add\nto the symmetric component of the resonance. Since we\nlook at the ratio of the symmetric and antisymmetric\nresonance amplitudes, we compute VDC\nSP=VAwhereVA\nis the amplitude of the antisymmetric component of the\nLorentzians. We then compare this to VS=VAin Fig-\nure 3, where we have included constants to convert the\nvoltage ratios to torque e\u000eciencies. This contribution\nends up being negligibly small, compared to both the\ntotal signal due to dynamic spin pumping and the sig-\nnal due to the dampinglike torque from the source layer,\nnever rising above 5 % of the measured symmetric signal\nfor the thickest Py layers and at the highest frequencies\nbut is less than 1 % for most thicknesses and frequencies.\nWe therefore are justi\fed in ignoring this contribution in\nour main analysis.\n[1] A. Brataas, A. D. Kent, and H. Ohno, Current-induced\ntorques in magnetic materials, Nature Materials 11, 372\n(2012).\n[2] K. L. Wang, J. G. Alzate, and P. K. Amiri, Low-power\nnon-volatile spintronic memory: STT-RAM and beyond,\nJournal of Physics D: Applied Physics 46, 074003 (2013).\n[3] F. Oboril, R. Bishnoi, M. Ebrahimi, and M. B. Tahoori,\nEvaluation of hybrid memory technologies using SOT-\nMRAM for on-chip cache hierarchy, IEEE Transactions\non Computer-Aided Design of Integrated Circuits and\nSystems 34, 367 (2015).\n[4] T. Taniguchi, J. Grollier, and M. Stiles, Spin-transfer\ntorques generated by the anomalous hall e\u000bect and\nanisotropic magnetoresistance, Physical Review Applied\n3, 044001 (2015).\n[5] A. M. Humphries, T. Wang, E. R. J. Edwards, S. R.\nAllen, J. M. Shaw, H. T. Nembach, J. Q. Xiao, T. J.\nSilva, and X. Fan, Observation of spin-orbit e\u000bects with\nspin rotation symmetry, Nature Communications 8, 911\n(2017).\n[6] V. Amin, J. Zemen, and M. Stiles, Interface-generated\nspin currents, Physical Review Letters 121, 136805\n(2018).\n[7] S. Iihama, T. Taniguchi, K. Yakushiji, A. Fukushima,\nY. Shiota, S. Tsunegi, R. Hiramatsu, S. Yuasa,\nY. Suzuki, and H. Kubota, Spin-transfer torque induced\nby the spin anomalous hall e\u000bect, Nature Electronics 1,\n120 (2018).\n[8] S.-H. C. Baek, V. P. Amin, Y.-W. Oh, G. Go, S.-J. Lee,\nG.-H. Lee, K.-J. Kim, M. D. Stiles, B.-G. Park, and K.-J.\nLee, Spin currents and spin{orbit torques in ferromag-\nnetic trilayers, Nature Materials 17, 509 (2018).[9] J. D. Gibbons, D. MacNeill, R. A. Buhrman, and D. C.\nRalph, Reorientable spin direction for spin current pro-\nduced by the anomalous hall e\u000bect, Physical Review Ap-\nplied 9, 064033 (2018).\n[10] K. S. Das, J. Liu, B. J. van Wees, and I. J. Vera-Marun,\nE\u000ecient injection and detection of out-of-plane spins via\nthe anomalous spin hall e\u000bect in permalloy nanowires,\nNano Letters 18, 5633 (2018).\n[11] C. Safranski, E. A. Montoya, and I. N. Krivorotov,\nSpin{orbit torque driven by a planar hall current, Na-\nture Nanotechnology 14, 27 (2018).\n[12] Y. Ou, Z. Wang, C. S. Chang, H. P. Nair, H. Paik,\nN. Reynolds, D. C. Ralph, D. A. Muller, D. G. Schlom,\nand R. A. Buhrman, Exceptionally high, strongly tem-\nperature dependent, spin hall conductivity of SrRuO3,\nNano Letters 19, 3663 (2019).\n[13] M. W. Keller, K. S. Gerace, M. Arora, E. K. Delczeg-\nCzirjak, J. M. Shaw, and T. J. Silva, Near-unity spin hall\nratio in Ni xCu1\u0000xalloys, Physical Review B 99, 214411\n(2019).\n[14] V. P. Amin, J. Li, M. D. Stiles, and P. M. Haney, Intrin-\nsic spin currents in ferromagnets, Physical Review B 99,\n220405(R) (2019).\n[15] T. Seki, S. Iihama, T. Taniguchi, and K. Takanashi,\nLarge spin anomalous hall e\u000bect in L10-FePt : Sym-\nmetry and magnetization switching, Physical Review B\n100, 144427 (2019).\n[16] T. Y. Ma, C. H. Wan, X. Wang, W. L. Yang, C. Y. Guo,\nC. Fang, M. K. Zhao, J. Dong, Y. Zhang, and X. F.\nHan, Evidence of magnetization switching by anomalous\nspin hall torque in NiFe, Physical Review B 101, 134417\n(2020).9\n[17] C. Safranski, J. Z. Sun, J.-W. Xu, and A. D.\nKent, Planar hall driven torque in a ferromag-\nnet/nonmagnet/ferromagnet system, Physical Review\nLetters 124, 197204 (2020).\n[18] Y. Hibino, K. Yakushiji, A. Fukushima, H. Kubota, and\nS. Yuasa, Spin-orbit torque generated from perpendicu-\nlarly magnetized Co/Ni multilayers, Physical Review B\n101, 174441 (2020).\n[19] Y. Koike, S. Iihama, and S. Mizukami, Composition de-\npendence of the spin-anomalous hall e\u000bect in a ferromag-\nnetic Fe{Co alloy, Japanese Journal of Applied Physics\n59, 090907 (2020).\n[20] K. S. Das, W. Y. Schoemaker, B. J. van Wees, and I. J.\nVera-Marun, Spin injection and detection via the anoma-\nlous spin hall e\u000bect of a ferromagnetic metal, Physical\nReview B 96, 220408(R) (2017).\n[21] J. Cramer, A. Ross, S. Jaiswal, L. Baldrati, R. Lebrun,\nand M. Kl aui, Orientation-dependent direct and inverse\nspin hall e\u000bects in Co60Fe20B20, Physical Review B 99,\n104414 (2019).\n[22] Y. Omori, E. Sagasta, Y. Niimi, M. Gradhand, L. E.\nHueso, F. Casanova, and Y. Otani, Relation between\nspin hall e\u000bect and anomalous hall e\u000bect in 3d ferro-\nmagnetic metals, Physical Review B 99, 014403 (2019).\n[23] T. Wimmer, B. Coester, S. Gepr ags, R. Gross, S. T. B.\nGoennenwein, H. Huebl, and M. Althammer, Anomalous\nspin hall angle of a metallic ferromagnet determined by\na multiterminal spin injection/detection device, Applied\nPhysics Letters 115, 092404 (2019).\n[24] B. F. Miao, S. Y. Huang, D. Qu, and C. L. Chien, Inverse\nspin hall e\u000bect in a ferromagnetic metal, Physical Review\nLetters 111, 066602 (2013).\n[25] H. Wang, C. Du, P. C. Hammel, and F. Yang, Spin cur-\nrent and inverse spin hall e\u000bect in ferromagnetic met-\nals probed by Y3Fe5O12-based spin pumping, Applied\nPhysics Letters 104, 202405 (2014).\n[26] C. Du, H. Wang, F. Yang, and P. C. Hammel, System-\natic variation of spin-orbit coupling withd-orbital \flling:\nLarge inverse spin hall e\u000bect in 3d transition metals,\nPhysical Review B 90, 140407 (2014).\n[27] H. Wu, C. H. Wan, Z. H. Yuan, X. Zhang, J. Jiang, Q. T.\nZhang, Z. C. Wen, and X. F. Han, Observation of pure\ninverse spin hall e\u000bect in ferromagnetic metals via fer-\nromagnetic/antiferromagnetic exchange-bias structures,\nPhysical Review B 92, 054404 (2015).\n[28] M. Wahler, N. Homonnay, T. Richter, A. M uller,\nC. Eisenschmidt, B. Fuhrmann, and G. Schmidt, Inverse\nspin hall e\u000bect in a complex ferromagnetic oxide het-\nerostructure, Scienti\fc Reports 6, 28727 (2016).\n[29] D. Tian, Y. Li, D. Qu, S. Y. Huang, X. Jin, and C. L.\nChien, Manipulation of pure spin current in ferromag-\nnetic metals independent of magnetization, Physical Re-\nview B 94, 020403(R) (2016).\n[30] J.-W. Xu and A. D. Kent, Charge-to-spin conversion\ne\u000eciency in ferromagnetic nanowires by spin torque\nferromagnetic resonance: Reconciling lineshape and\nlinewidth analysis methods, Physical Review Applied\n14, 014012 (2020).\n[31] S. Karimeddiny and D. C. Ralph, Resolving dis-\ncrepancies in spin-torque ferromagnetic resonance\nmeasurements: Lineshape vs. linewidth analyses,\narXiv:2103.04172 [cond-mat.mes-hall] (2021), dC bias.\n[32] B. Heinrich, Y. Tserkovnyak, G. Woltersdorf,\nA. Brataas, R. Urban, and G. E. W. Bauer, Dy-\nnamic exchange coupling in magnetic bilayers, PhysicalReview Letters 90, 187601 (2003).\n[33] Y. Pogoryelov, M. Pereiro, S. Jana, A. Kumar,\nS. Akansel, M. Ranjbar, D. Thonig, D. Primetzhofer,\nP. Svedlindh, J. \u0017Akerman, O. Eriksson, O. Karis, and\nD. A. Arena, Nonreciprocal spin pumping damping in\nasymmetric magnetic trilayers, Physical Review B 101,\n054401 (2020).\n[34] W. L. Yang, J. W. Wei, C. H. Wan, Y. W. Xing, Z. R.\nYan, X. Wang, C. Fang, C. Y. Guo, G. Q. Yu, and X. F.\nHan, Determining spin-torque e\u000eciency in ferromagnetic\nmetals via spin-torque ferromagnetic resonance, Physical\nReview B 101, 064412 (2020).\n[35] J. Bass and W. P. Pratt, Spin-di\u000busion lengths in metals\nand alloys, and spin-\ripping at metal/metal interfaces:\nan experimentalist's critical review, Journal of Physics:\nCondensed Matter 19, 183201 (2007).\n[36] H. Nakayama, M. Althammer, Y.-T. Chen, K. Uchida,\nY. Kajiwara, D. Kikuchi, T. Ohtani, S. Gepr ags,\nM. Opel, S. Takahashi, R. Gross, G. E. W. Bauer,\nS. T. B. Goennenwein, and E. Saitoh, Spin hall mag-\nnetoresistance induced by a nonequilibrium proximity\ne\u000bect, Physical Review Letters 110, 206601 (2013).\n[37] L. Liu, T. Moriyama, D. C. Ralph, and R. A. Buhrman,\nSpin-torque ferromagnetic resonance induced by the spin\nhall e\u000bect, Physical Review Letters 106, 036601 (2011).\n[38] C.-F. Pai, L. Liu, Y. Li, H. W. Tseng, D. C. Ralph, and\nR. A. Buhrman, Spin transfer torque devices utilizing\nthe giant spin hall e\u000bect of tungsten, Applied Physics\nLetters 101, 122404 (2012).\n[39] C. He, G. Yu, C. Grezes, J. Feng, Z. Zhao, S. A. Razavi,\nQ. Shao, A. Navabi, X. Li, Q. L. He, M. Li, J. Zhang,\nK. L. Wong, D. Wei, G. Zhang, X. Han, P. K. Amiri,\nand K. L. Wang, Spin-torque ferromagnetic resonance in\nW/Co-Fe-B/W/Co-Fe-B/MgO stacks, Physical Review\nApplied 10, 034067 (2018).\n[40] D. MacNeill, G. M. Stiehl, M. H. D. Guimaraes,\nR. A. Buhrman, J. Park, and D. C. Ralph, Con-\ntrol of spin{orbit torques through crystal symmetry\nin WTe2/ferromagnet bilayers, Nature Physics 13, 300\n(2016).\n[41] Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I.\nHalperin, Nonlocal magnetization dynamics in ferromag-\nnetic heterostructures, Reviews of Modern Physics 77,\n1375 (2005).\n[42] E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Con-\nversion of spin current into charge current at room tem-\nperature: Inverse spin-hall e\u000bect, Applied Physics Let-\nters88, 182509 (2006).\n[43] O. Mosendz, V. Vlaminck, J. E. Pearson, F. Y. Fradin,\nG. E. W. Bauer, S. D. Bader, and A. Ho\u000bmann, De-\ntection and quanti\fcation of inverse spin hall e\u000bect from\nspin pumping in permalloy/normal metal bilayers, Phys-\nical Review B 82, 214403 (2010).\n[44] A. Azevedo, L. H. Vilela-Le~ ao, R. L. Rodr\u0013 \u0010guez-Su\u0013 arez,\nA. F. L. Santos, and S. M. Rezende, Spin pumping and\nanisotropic magnetoresistance voltages in magnetic bi-\nlayers: Theory and experiment, Physical Review B 83,\n144402 (2011).\n[45] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer,\nEnhanced gilbert damping in thin ferromagnetic \flms,\nPhysical Review Letters 88, 034067 (2002).\n[46] S. Mizukami, Y. Ando, and T. Miyazaki, Magnetic relax-\nation of normal-metal (NM)/80NiFe/NM \flms, Journal\nof Magnetism and Magnetic Materials 239, 42 (2002).\n[47] D. Richardson, K. Srinivasan, A. Kalitsov, A. Ajan,10\nS. Jain, S. Katz, and M. Wu, Interlayer exchange cou-\npling in magnetic hard-soft bilayered structures, Physical\nReview Applied 11, 044016 (2019).\n[48] R. Medwal, S. Gupta, R. S. Rawat, A. Subramanian, and\nY. Fukuma, Spin pumping in asymmetric Fe 50 Pt 50\n/Cu/Fe 20 Ni 80 trilayer structure, physica status solidi\n(RRL) { Rapid Research Letters 13, 1900267 (2019).\n[49] S. Karimeddiny, J. A. Mittelstaedt, R. A. Buhrman, and\nD. C. Ralph, Transverse and longitudinal spin-torque\nferromagnetic resonance for improved measurement of\nspin-orbit torque, Physical Review Applied 14, 024024\n(2020).\n[50] C.-F. Pai, Y. Ou, L. H. Vilela-Le~ ao, D. C. Ralph, and\nR. A. Buhrman, Dependence of the e\u000eciency of spin hall\ntorque on the transparency of Pt/ferromagnetic layer in-\nterfaces, Physical Review B 92, 064426 (2015).\n[51] I. M. Miron, K. Garello, G. Gaudin, P.-J. Zermatten,\nM. V. Costache, S. Au\u000bret, S. Bandiera, B. Rodmacq,\nA. Schuhl, and P. Gambardella, Perpendicular switch-\ning of a single ferromagnetic layer induced by in-planecurrent injection, Nature 476, 189 (2011).\n[52] M. Jiang, H. Asahara, S. Sato, S. Ohya, and M. Tanaka,\nSuppression of the \feld-like torque for e\u000ecient magne-\ntization switching in a spin{orbit ferromagnet, Nature\nElectronics 3, 751 (2020).\n[53] L. Zhu, D. C. Ralph, and R. A. Buhrman, E\u000bective spin-\nmixing conductance of heavy-metal{ferromagnet inter-\nfaces, Physical Review Letters 123, 057203 (2019).\n[54] V. Gupta, T. M. Cham, G. M. Stiehl, A. Bose, J. A.\nMittelstaedt, K. Kang, S. Jiang, K. F. Mak, J. Shan,\nR. A. Buhrman, and D. C. Ralph, Manipulation of the\nvan der waals magnet Cr2Ge2Te6 by spin{orbit torques,\nNano Letters 20, 7482 (2020).\n[55] J. Bass, CPP magnetoresistance of magnetic multilayers:\nA critical review, Journal of Magnetism and Magnetic\nMaterials 408, 244 (2016).\n[56] F. Warkusz, Size e\u000bects in metallic \flms, ElectroCom-\nponent Science and Technology 5, 99 (1978).\n[57] D. Gall, Electron mean free path in elemental metals,\nJournal of Applied Physics 119, 085101 (2016).\nAppendix B: Evidence that Bilinear Coupling is Negligible in Our Samples\nTo test whether we have any signi\fcant dipole or other bilinear coupling of the form m1\u0001m2between the two\nmagnetic layers, we conducted vibrating sample magnetometry (VSM) measurements of the magnetization hysteresis\nloop for the bilayer. We performed a total of three hysteresis sweeps for each sample. The \frst covers a \feld range\nlarger than the coercive \feld for both magnetic layers and allows us to read the full hyseresis curve of the bilayer.\nThe other two are minor loops which only switch the magnetization of one layer while keeping the magnetization\nof the other layer constant. The results of these hysteresis sweeps on a device with a 6.7 nm Py layer are shown in\nSupplementary Fig. 4.\nThe di\u000berence in the center of the minor hysteresis loops should be equal to the bilinear coupling e\u000bective \feld\nbetween the two layers, since the magnet with the larger coercive \feld has a di\u000berent orientation and hence would\ninduce opposite directions of the bilinear exchange \feld in the two minor loops. The di\u000berence in the minor loop\ncenters is only 2 Oe, and is similar for all of the other Py thicknesses. This is su\u000eciently weak coupling that it can\nbe neglected in our analysis relative to the coupling produced by dynamic spin pumping.\nFIG. 4. (a) The full hysteresis curve with the two minor loops superimposed for the 6.7 nm Py sample. The vertical lines\nindicate the center of the minor hysteresis loops. (b) The minor hysteresis loops superimposed. The centers of the minor loops\nare 2 Oe apart, indicating that there is not signi\fcant bilinear coupling.11\nFIG. 5. Conductance of (a) Co and (b) Py devices with a \ft to the Fuchs-Sondheimer model to determine the resistivity of\neach layer.\nAppendix C: Resistivities of the Ferromagnetic Layers\nTo measure the resistance of the ferromagnetic layers in our devices, we use a separate series of samples with the\nstructure Ta (1) / Cu (3) / FM ( tFM) / Ta (1), which were created so that there would be less shunting from the\nferromagnetic layers leading to a better signal to noise ratio. By investigating the dependence of the heterostructure's\nresistance on the ferromagnetic layer's thickness, we are able to estimate the resistivity of the ferromagnetic layer.\nIn a simple parallel conductor model, we would expect the heterostructure resistance to have a linear dependence\non the inverse ferromagnetic layer thickness, but we found that since the thickness of our ferromagnetic layers are\ngenerally comparable to the mean free path of the ferromagnets, a full Fuchs-Sondheimer model [56] was needed to\nget reasonable results. Using electron mean free paths from the literature [57] for Co and Ni (which we used as an\napproximation for the mean free path of Py), we were able to \fnd the bulk resistivities of the Co and Py layers using\nthe Fuchs-Sondheimer model to \ft the thickness dependence of the device conductivity as shown in Supplementary\nFigure 5. We found the bulk resistivity of Co to be 16 :2\u00060:2\u0016\n-cm and the bulk resistivity of Py to be 24 :9\u00060:4\n\u0016\n-cm. When calculating the shunting factors for the main text, we calculated the resistances of the various layers\nusing the full Fuchs-Sondheimer model for that particular thickness of the ferromagnetic layer.\nAppendix D: Demonstration that Both Anisotropic Magnetoresistance and Spin Hall Magnetoresistance\nContribute to the Angle-Dependent Magnetoresistance of the Magnetic Trilayers\nTo measure the anisotropic magnetoresistance (AMR) in our devices, we use a separate series of samples with\nthe structure Ta (1) / Cu (3) / FM ( tFM) / Ta (1), which were created so that there would be less shunting from\nthe ferromagnetic layers leading to a better signal-to-noise ratio. We measure the resistance of our device with a\nnanovoltmeter while rotating a magnetic \feld in the plane. We measure the resistance change of the whole device,\nbut we would like to measure the resistance change of just the ferromagnetic layer. To get this quantity, we use a\nparallel resistor model and assume that the AMR resistance change is small compared to the resistance of the device.\nWe de\fne\u001eas the angle between the current and external magnetic \feld, Rtot(\u001e) the total device resistance, RNM\nthe resistance of the non-magnetic layer, and RFM(\u001e) =R0+RAMR ;0cos2\u001ethe ferromagnet resistance with R0the12\nFIG. 6. Device conductance as a function of applied \feld angle for (a) the 10 nm Py device and (b) the 8 nm Co device.\nbase resistance and RAMR ;0the AMR resistance. Expanding, we \fnd that\n1\nRtot(\u001e)=1\nRNM+1\nRFM(\u001e)(D1)\n=1\nRNM+1\nR01\n1 +RAMR ;0=R0cos2\u001e(D2)\n1\nRtot(\u001e)\u00191\nRNM+1\nR0\u0000RAMR ;0\nR2\n0cos2\u001e (D3)\nwhich provides a way for us to extract the desired quantity from \ftting the total conductance as a function of applied\n\feld angle. Fits for the devices with 10 nm Py and 8 nm Co are shown in Supplementary Fig. 6.\nWe have also measured the AMR resistance change of a full trilayer structure, Cu (8) / Co (8) / Cu (8) / Py\n(tPy=10) from the main text. Using the values we have measured above for the 8 nm Co and 10 nm Py single layers,\nwe would expect the conductance of this device to vary by 75 \u0016S due to the AMR by adding the contributions\nfrom each of the ferromagnetic layers. However, we actually measure a conductance variation of 23 \u0016S, which is\nover three times smaller than what we expect from the single-layer measurements. We attribute this discrepancy to\nthe presence of spin Hall magnetoresistance in the full trilayer samples, which only exists when we have layers with\nnon-negligible spin Hall e\u000bect. In the single layers, the only other conductive layer was the Cu which has a negligible\nspin Hall e\u000bect. This has precluded a simple measurement of the appropriate RAMR\ni to use in the expressions in the\nmain text.\nAppendix E: Derivation of the E\u000bects of Dynamic Spin Pumping\nTo derive the e\u000bect which the dynamic spin pumping has on the resonances of our layers, we begin with the\nmodi\fed LLGS equation proposed by Heinrich [32]\n_mi=\u000b0\nimi\u0002_mi+\u001cneq;i+\u001ceq;i+\u000b0\ni(mi\u0002_mi\u0000mj\u0002_mj) (E1)\nwhere \u001cneq;iare the nonequilibrium torques which oscillate at the RF frequency such as the Oersted torque or the\nspin-orbit torque, \u001ceq;iare the equilibrium torques which do not oscillate such as those from the static external \feld\nor from anisotropy, \u000b0\niis the intrinsic damping parameter and \u000b0\niis a parameter describing the e\u000bect of spin pumping\nthat both layers have on layer i. We can combine the e\u000bect of damping with the spin pumping out of layer ito arrive\nat a modi\fed damping parameter \u000bi=\u000b0\ni+\u000b0\ni\n_mi=\u000bimi\u0002_mi+\u001cneq;i+\u001ceq;i\u0000\u000b0\nimj\u0002_mj: (E2)\nFirst we assume that both layers are saturated in the same direction and have the same equilibrium magnetization\ndirection. We use a coordinate system where ^ yis along the equilibrium direction of the magnetization and ^ zis normal13\nto the \flm. The magnetization is essentially constant along the ^ yaxis, so our problem reduces to a 2D problem in\nthex\u0000zplane. The torques we need to consider are\n\u001ceq;i=\u0012\nmiz!i2\n\u0000mix!i1\u0013\n;\u001cneq;i=\u0012\n\u001cix\n\u001ciz\u0013\n;mi\u0002_mi=\u0012\n_miz\n\u0000_mix\u0013\n(E3)\nwith!i1=\rB,!i2=\r(B+\u00160Me\u000b;i),\rthe gyromagnetic ratio and Me\u000b;ithe e\u000bective magnetization of layer i.\nPutting this all into the modi\fed LLGS, arrive at\n0\nB@_msx\n_msz\n_mdx\n_mdz1\nCA=0\nB@\u000bs_msz+\u001csx+!s2msz\u0000\u000b0\ns_mdz\n\u0000\u000bs_msx+\u001csz\u0000!s1msx+\u000b0\ns_mdx\n\u000bd_mdz+\u001cdx+!d2mdz\u0000\u000b0\nd_msz\n\u0000\u000bd_mdx+\u001cdz\u0000!d1mdx+\u000b0\nd_msx1\nCA: (E4)\nTo solve this, we \frst consider the system for the source and detector layers independently, treating the coupling\nterms between the source and detector layers as modi\fcations to the nonequilibrium torques\n\u0012\n_msx\n_msz\u0013\n=\u0012\n\u000bs_msz+!s2msz+\u001csx\u0000\u000b0\ns_mdz\n\u0000\u000bs_msx\u0000!s1msx+\u001csz+\u000b0\ns_mdx\u0013\n(E5)\n\u0012\n_mdx\n_mdz\u0013\n=\u0012\n\u000bd_mdz+!d2mdz+\u001cdx\u0000\u000b0\nd_msz\n\u0000\u000bd_mdx\u0000!d1mdx+\u001cdz+\u000b0\nd_msx\u0013\n(E6)\nand use the ansatz\nmi(t) =mie\u0000i!t(E7)\nto arrive at the linearized form of the LLGS\n\u0012\n\u0000i!m sx\n\u0000i!m sz\u0013\n=\u0012\n\u0000msz(i!\u000bs\u0000!s2) +\u001csx+i!\u000b0\nsmdz\nmsx(i!\u000bs\u0000!s1) +\u001csz\u0000i!\u000b0\nsmdx\u0013\n(E8)\n\u0012\n\u0000i!m dx\n\u0000i!m dz\u0013\n=\u0012\n\u0000mdz(i!\u000bd\u0000!d2) +\u001cdx+i!\u000b0\ndmsz\nmdx(i!\u000bd\u0000!d1) +\u001cdz\u0000i!\u000b0\ndmsx\u0013\n(E9)\nWe can solve for the dynamics of each subsystem without decoupling as usual [49], which results in\nmsx=Ls(\u0000!s2[\u001csz\u0000i!\u000b0\nsmdx] +i![\u001csx+i!\u000b0\nsmdz])\nmsz=Ls(!s1[\u001csx+i!\u000b0\nsmdz] +i![\u001csz+i!\u000b0\nsmdx])\nmdx=Ld(\u0000!d2[\u001cdz\u0000i!\u000b0\ndmsx] +i![\u001cdx+i!\u000b0\ndmsz])\nmdz=Ld(!d1[\u001cdx+i!\u000b0\ndmsz] +i![\u001cdz+i!\u000b0\ndmsx])\nwithLi=\u0002\u0000\n!2\u0000!2\ni0\u0001\n+i!!+\ni\u000bi\u0003\u00001,!2\ni0=!i1!i2and!+\ni=!i1+!i2. We now need to decouple the source and\ndetector layer resonances. If we ignore any terms greater than \frst order in the damping parameters, this is relatively\neasy to do via substitution. We will only solve for the detector layer magnetizations as this system is symmetric\nunder interchanging s$d. The in-plane de\rection becomes\nmdx=Ld\u0000\n\u0000!d2\u001cdz+i!\u001cdx+\u000b0\ndLs\u0002\n\u0000!2\u001csx(!s1+!d2)\u0000i!\u001csz(!s2!d2+!2)\u0003\u0001\n(E10)\nand the out-of-plane de\rection becomes\nmdz=Ld\u0000\n\u0000!d1\u001cdx+i!\u001cdz+\u000b0\ndLs\u0002\n!2\u001csz(!s2\u0000!d1) +i!\u001csx(!d1!s1\u0000!2)\u0003\u0001\n: (E11)\nWe now separate the dynamic spin pumping term into real and imaginary parts. It will be convenient to de\fne\nLi=(!2\u0000!2\ni0)\u0000i!!+\ni\u000bi\n(!2\u0000!2\ni0)2+!2!+2\ni\u000b2\ni\u0011Li\u0000\n(!2\u0000!2\ni0)\u0000i!!+\ni\u000bi\u0001\n(E12)\nso that the in-plane de\rections become\nmdx=Ld\u0000\n\u0000!d2\u001cdz+Ls\u000b0\nd\u0002\n(!2\ns0\u0000!2)!2\u001csx(!s1+!d2)\u0000!2!+\ns\u000bs\u001csz(!s2!d2+!2)\u0003\n+i!\u0000\n\u001cdx+Ls\u000b0\nd\u0002\n(!2\ns0\u0000!2)\u001csz(!s2!d2+!2) +!2!+\ns\u000bs\u001csx(!s1+!d2)\u0003\u0001\u0001 (E13)14\nand the out-of-plane de\rections become\nmdz=Ld\u0000\n!d1\u001cdx+Ls\u000b0\nd\u0002\n(!2\ns0\u0000!2)!2\u001csz(!d1\u0000!s2) +!2!+\ns\u000bs\u001csx(!d1!s1\u0000!2)\u0003\n+i!\u0000\n\u001cdz+Ls\u000b0\nd\u0002\n(!2\ns0\u0000!2)\u001csx(!2\u0000!d1!s1) +!2!+\ns\u000bs\u001csz(!d1\u0000!s2)\u0003\u0001\u0001\n:(E14)\nBefore converting to \feld units, we make two approximations. First, the latter part of the dynamic spin pumping\ncontributions to Eq. (E13) and Eq. (E14) correspond to the symmetric part of the source layer Lorentzian. This\ncomponent falls o\u000b quickly away from the source resonant \feld and will be nearly zero near the detector layer\nresonance \feld, and are thus insigni\fcant for our analysis. Also for our system, the dampinglike torque on the source\nlayer\u001csxis quite small, as is evidenced by the symmetric part of the source layer resonances being small compared\nwith all other resonance components, including \u001cdzin particular. This justi\fes ignoring the occurrences of this torque\nappearing in the antisymmetric source layer Lorentzian in Eq. (E13) and Eq. (E14). This leaves only one part of the\ndynamic spin pumping having a signi\fcant e\u000bect, and the in-plane and out-of-plane de\rections can be written as\nmdx=Ld\u0000\n\u0000!d2\u001cdz+i!\u0000\n\u001cdx+Ls(!2\ns0\u0000!2)\u000b0\nd\u001csz(!s2!d2+!2)\u0001\u0001\n(E15)\nand\nmdz=Ld\u0000\n!d1\u001cdx+Ls(!2\ns0\u0000!2)\u000b0\nd!2\u001csz(!d1\u0000!s2) +i!\u001cdz\u0001\n: (E16)\nWe now wish to write the source resonance in \feld coordinates. To do so we expand the resonant frequency about\nthe source resonant \feld:\n!2\ns0(B) =!2\ns0jB=Bs0+(B\u0000Bs0)d!2\ns0\ndB\f\f\f\f\nB=Bs0(E17)\n=!2+ (B\u0000Bs0) \n!s1d!s2\ndB\f\f\f\f\nB=Bs0+!s2d!s1\ndB\f\f\f\f\nB=Bs0!\n(E18)\n=!2+\r(B\u0000Bs0)!+\nsjBs0 (E19)\n!2\ns0\u0000!2=\r(B\u0000Bs0)!+\nsjBs0 (E20)\nso in \feld coordinates\nLs(!2\ns0\u0000!2) =\r!+\nsjBs0(B\u0000Bs0)\n\r2!+2sjBs0(B\u0000Bs0)2+!2!+2s\u000b2s=1\n\u000bs!+sjBs01\n!(B\u0000Bs0)\u0001s\n(B\u0000Bs0)2+ \u00012s\u0011LsA (E21)\nwhere we have de\fned \u0001 i\u0011!\u000bi=\rwhich is the linewidth of the resonance associated with layer i.\nWe can extract the symmetric and antisymmetric detector-layer resonance amplitudes by taking the real part of\nEq. (E15) and Eq. (E16) combined with Eq. (E21) and identifying the prefactors on the symmetric and antisymmetric\nLorentzian shapes, as done in more detail in [49]. Doing so, we \fnd that\nAdx=1\n\u000bd!+\nd1\n!!d2\u001cdz (E22)\nSdx=1\n\u000bd!+\nd\u0000\n\u001cdx+LsA(!s2!d2+!2)\u001csz\u000b0\nd\u0001\n(E23)\nAdz=\u00001\n\u000bd!+\nd1\n!\u0000\n!d1\u001cdx+LsA!2(!d1\u0000!s2)\u001csz\u000b0\nd\u0001\n(E24)\nSdz=1\n\u000bd!+\nd\u001cdz (E25)\nwithSandArepresenting the symmetric and antisymmetric amplitudes and the subscript xandzrepresenting that\nthose quantities deal with the in-plane and out-of-plane magnetization de\rection amplitudes, respectively. Everything\nwith a \feld dependence is evaluated at Bd0, except where noted in LsA, since all of these resonance amplitudes are\nmeasured close to the detector layer resonance \feld. The e\u000bect of dynamic spin pumping captured in Eqs. (E23)\nand (E24) comes from spins pumped by the source layer which travel to and exert torques on the detector layer.\nIn the next section, we deal with another slightly more subtle e\u000bect which the dynamic spin pumping has on our\nmeasurements.15\n1. Dynamic Spin Pumping from Detector Layer to Source Layer\nHere we show spins pumped from the detector layer to the source layer also end up having an e\u000bect on our\nmeasurement of the detector-layer resonance lineshape. To see that, we will start at the equivalent of Eq. (E10) for\nthe source layer magnetization (simply replacing d$s) and rearrange some terms\nmsx=Ls\u0000\n\u0000!s2\u001csz+i!\u001csx+\u000b0\nsLd\u0002\n\u0000!2\u001cdx(!d1+!s2)\u0000i!\u001cdz(!d2!s2+!2)\u0003\u0001\n(E26)\n=Ls(\u0000!s2\u001csz+i!\u001csx) +\u000b0\nsLdLs\u0002\n\u0000!2\u001cdx(!d1+!s2)\u0000i!\u001cdz(!d2!s2+!2)\u0003\n(E27)\n=Ls(\u0000!s2\u001csz+i!\u001csx) +\u000b0\nsLdLs\u0002\n(!2\ns0\u0000!2)!2(!d1+!s2)\u001cdx\u0000!2!+\ns(!d2!s2+!2)\u000bs\u001cdz\n+i!\u0000\n(!2\ns0\u0000!2)(!d2!s2+!2)\u001cdz+!2!+\ns(!d1+!s2)\u000bs\u001cdx\u0001\u0003\n:(E28)\nThe \frst part of Eq. (E28) is the usual equation for the source layer resonance, with the second part being due to\nthe dynamic spin pumping. Near the detector resonance, the \frst term in Eq. (E28) is essentially a constant o\u000bset,\nbut theLdterm means that the source layer magnetization will resonate at the deteciton layer's resonance \feld and\nwith the detector layer's linewidth. The voltage we measure is still due to recti\fcation from the soure layer AMR\nmixing with the applied current, but since we measure the voltage across the whole heterostructure this signal will\nadd to the resonance we measure at the detector-layer's resonant \feld and linewidth.\nTo see how this a\u000bects things, we can make a few simpli\fcations again. If we want to look near the detector layer\nresonance, the terms related to the symmetric part of the source layer resonance (on the right in the square brackets\nin Eq. (E28) will be negligible (same argument as above), so we can ignore them, letting the source layer oscillation\nnear the detector layer resonance look like\nmsx=Ls(\u0000!s2\u001csz+i!\u001csx) +Ld\u0002\nLsA!2(!d1+!s2)\u001cdx\u000b0\ns+i!LsA(!d2!s2+!2)\u001cdz\u000b0\ns\u0003\n: (E29)\nSince we are only concerned with this signal near the detector-layer resonance, and the \frst term in Eq. (E29) is\nessentially a constant o\u000bset, we ignore it in the following analysis. We can decompose the signal from the second\npart into symmetric and antisymmetric parts as above. We \fnd that\n~Ad=\u00001\n\u000bd!+\nd1\n!LsA!2(!d1+!s2)\u001cdx\u000b0\ns (E30)\n~Sd=1\n\u000bd!+\ndLsA(!d2!s2+!2)\u001cdz\u000b0\ns (E31)\nwhere the\u0018signi\fes that this appears as additions to the detector-layer resonance since these lorentzians have the\nsame resonant \feld and lineshape, but actually come from the source layer.\n2. Overall E\u000bect on Measured Detector-Layer Resonance\nCombining Eqs. (E22) and(E30), and Eqs. (E23) and (E31), keeping note of which layer gives rise to the recti\fcation\nproducing the voltage we measure, we \fnd the expression for the voltage we measure when \ftting the resonance\nwhich has the resonant \feld and linewidth of the detector layer:\nVdS=IRAMR\nd sin(2\u001e)\n21\n\u000bd!+\nd\u0002\n\u001cdx+LsA(!s2!d2+!2)\u001csz\u000b0\nd\u0003\n(E32)\n+IRAMR\ns sin(2\u001e)\n21\n\u000bd!+\ndLsA(!s2!d2+!2)\u001cdz\u000b0\ns (E33)\nVdA=IRAMR\nd sin(2\u001e)\n21\n\u000bd!+\nd1\n!!d2\u001cdz\u0000IRAMR\ns sin(2\u001e)\n21\n\u000bd!+\nd1\n!\u000b0\nsLsA!2(!d1+!s2)\u001cdx: (E34)\nwithIthe total current in the heterostructure and and RAMR\ni the resistance change of the whole stack due to the\nAMR of layer i.\nIn the conventional ST-FMR setting with only one ferromagnetic layer, we could look at the ratio of VdSandVdA\nto \fnd the charge-to-spin conversion e\u000eciency, but with the added terms this ratio is now harder to interpret, so\nwe will work to simplify this expression a bit and see what information we can get from this ratio. We \frst try to\nsimplify our expression for VdA. We can write it as\nVdA=IdRAMR\nd sin(2\u001e)\n21\n\u000bd!+\nd1\n!!d2\u001cdz\u0012\n1\u0000RAMR\ns\nRAMR\ndLsA!2(!d1+!s2)\n!d2\u001cdx\n\u001cdz\u000b0\ns\u0013\n(E35)16\nFIG. 7. Source layer thickness dependence of the fraction of current \rowing in the Py layer, xs.\nand will now show that the term subtracting from 1 is small. For the Co/Py system, all of the terms subtracting\n1 are known or can be reasonably approximated. First, LsA!2(!d1+!s2)=!d2\u0019\u00001 which we can \fnd from \ftting\nthe resonances. We estimate the torque ratio to be in the range 0 :2\u00000:5. The AMR ratio is close to 1 from our\nAMR measurements in Section D. The spin pumping coupling term \u000b0\nsshould be on the same order as damping in\nPy, so estimating 0 :01 should be reasonable. Putting all of this together, the whole term to the right of the 1 inside\nthe parentheses comes out to no more than 0 :01, so this should be no more than a 1% correction to VdA. Therefore,\nwe ignore the e\u000bect of dynamic spin pumping on VdA, which will simplify our analysis.\nAfter simplifying VdA, theVdS=VdAratio becomes\nVdS\nVdA=!\n!d2\u0014\u001cdx\n\u001cdz+LsA(!s2!d2+!2)\u001csz\n\u001cdz\u0012\n\u000b0\nd+\u001cdz\n\u001cszRAMR\ns\nRAMR\nd\u000b0\ns\u0013\u0015\n(E36)\nand we can then de\fne\n\u0018FMR =\u0018DL+\u0011LsA(!s2!d2+!2)\u001csz\n\u001cdz\u0012\n\u000b0\nd+\u001cdz\n\u001cszRAMR\ns\nRAMR\nd\u000b0\ns\u0013\n(E37)\nwhich is the expression that appears in the main text.\nAs discussed in the main text, the torque ratio can be expressed as \u001csz=\u001cdz=\u0000(1\u0000xs)=xswhere the shunt factor\nxsis the fraction of the total RF current \rowing within the source layer. Based on a parallel-conduction model\nusing the layer resistances determined as described above, the value of xsas a function of Py thickness is shown in\nSupplementary Fig. 7."    },    {        "title": "1202.1406v2.Prediction_of_Giant_Spin_Motive_Force_due_to_Rashba_Spin_Orbit_Coupling.pdf",        "content": "arXiv:1202.1406v2  [cond-mat.mes-hall]  14 May 2012Prediction of Giant Spin Motive Force due to Rashba Spin-Orb it Coupling\nKyoung-Whan Kim1, Jung-Hwan Moon2, Kyung-Jin Lee2, and Hyun-Woo Lee1\n1PCTP and Department of Physics, Pohang University of Scienc e and Technology, Pohang, 790-784, Korea\n2Department of Materials Science and Engineering, Korea Uni versity, Seoul, 136-701, Korea\n(Dated: April 7, 2018)\nMagnetization dynamics in a ferromagnet can induce a spin-d ependent electric field through spin\nmotive force. Spin current generated by the spin-dependent electric field can in turn modify the\nmagnetization dynamics through spin-transfer torque. Whi le this feedback effect is usually weak and\nthus ignored, we predict that in Rashba spin-orbit coupling systems with large Rashba parameter\nαR, the coupling generates the spin-dependent electric field [ ±(αRme/e/planckover2pi1)(ˆz×∂m/∂t)], which can\nbe large enough to modify the magnetization dynamics signifi cantly. This effect should be relevant\nfor device applications based on ultrathin magnetic layers with strong Rashba spin-orbit coupling.\nSimilar to the mutual induction between electric and\nmagnetic fields through the Faraday and Maxwell’s laws,\nspin current and magnetization induce the dynamics of\neach other in magnetic systems through spin-transfer\ntorque (STT) [1–4] and spin motive force (SMF) [5–7].\nThrough many years of extensive study [3, 4], it has\nbeen demonstrated that STT is a powerful tool to induce\nthe magnetization dynamics in ferromagnetic nanostruc-\ntures. In contrast, SMF has received much less atten-\ntion [8–10] since SMF is very weak and thus inefficient to\ngenerate the spin current.\nIn this Letter, we demonstrate that SMF can be orders\nof magnitude enhanced in systems with strong Rashba\nspin-orbit coupling (RSOC). RSOC arises generically\nwhen structural inversion symmetry is broken [11]. Re-\ncent experiments on ultrathin ( ∼1 nm) ferromagnetic\nlayers with the strong structural inversion asymmetry\n(such as Pt/Co/AlO x[12, 13]) observed large effective\nmagnetic fields predicted by RSOC theories [14, 15]. A\nthin ferromagnetic layer in contact with topological insu-\nlators [16, 17] may also have strong RSOC. In such mag-\nnetic systems, magnetization dynamics can induce large\nspin current through SMF, and this spin current can in\nturn modify the magnetization dynamics through STT.\nThus even for purely magnetic-field-driven magnetization\ndynamics, STT can have sizable magnitude because of\nlarge spin current generated through SMF. We also pro-\npose a SMF-based method to measure the strength of\nRSOC in magnetic systems. This method allows for un-\nambiguous distinction between RSOC effects and other\neffects [18], which may be difficult to distinguish through\nother methods. Thus SMF can also be a useful tool to\nquantify RSOC.\nRSOC can be described by the Rashba Hamiltonian\nHR[11],\nHR=αR\n/planckover2pi1(σ×p)·ˆz=αR\n/planckover2pi1(σxpy−σypx),(1)\nwhere the vectors σandpare the Pauli matrix and\nthe momentum, respectively. ˆzis the unit vector along\nthe inversion symmetry breaking direction (perpendicu-\nlar to ferromagnetic layer), and αRis the Rashba con-\nstant. The total Hamiltonian Hof a conduction electronthen becomes H0+HR, whereH0=p2/2me−Jexσ·m.\nHeremeis effective mass of conduction electrons, Jex\n(<0) is the exchange coupling energy, and mis the\nunit vector along the local magnetization, which is in\ngeneral time- and position-dependent. An insight into\nthe RSOC effect on SMF can be gained from the ve-\nlocity operator v= [r,H]/i/planckover2pi1=p/me+van, where the\nanomalous velocity van= [r,HR]/i/planckover2pi1=αRˆz×σ//planckover2pi1arises\ndue to RSOC. When the exchange energy is sufficiently\nlarger than RSOC, σfor majority (minority) electrons\nwill be almost anti-parallel (parallel) to m. It is then\nevident that the magnetization dynamics ∂m/∂tinduces\nthe acceleration dvan/dtand the effective electric field\n−(me/e)dvan/dt. Thisheuristiccalculationresultsinthe\nRSOC contribution to the spin-dependent electric field\nERSOC\n±,\nERSOC\n±=±αRme\ne/planckover2pi1/parenleftbigg\nˆz×∂m\n∂t/parenrightbigg\n, (2)\nwhere±applies to the majority and minority electrons,\nrespectively. The total spin-dependent electric field E′\n±\nthen becomes E±+ERSOC\n±, where\nE±=±/summationdisplay\niˆxi/planckover2pi1\n2e/parenleftbigg∂m\n∂t×∂m\n∂xi/parenrightbigg\n·m (3)\nrepresents the conventional spin-dependent electric field\nexamined in previous literatures [7, 19] in the absence of\nRSOC. For more rigorous derivation of Eq. (2), we may\nadopt the calculation scheme in Ref. [6]. One first diag-\nonalizes the exchange coupling term −Jexσ·mby intro-\nducing the unitary operator U†=eiθσy/2eiφσz/2, where\nthe Euler angles ( θ,φ) are defined by spatio-temporal\nprofile of m= (sinθcosφ,sinθsinφ,cosθ). The trans-\nformed Hamiltonian H′≡U†HU−i/planckover2pi1U†∂tUbecomes\nH′= (p+eA′)2/2me−Jexσz−eA′\n0, where mis ro-\ntated to (0 ,0,1) and the exchange coupling term is di-\nagonal,−Jexσz. Information on SMF is now stored\nin vector and scalar potentials A′andA′\n0given by\nA′=−(i/planckover2pi1/e)U†∇U−(αRme/e/planckover2pi1)U†(σ׈z)UandA′\n0=\n(i/planckover2pi1/e)U†∂tU+α2\nRme/e/planckover2pi12. Recalling that spins tend to2\nFIG. 1: (color online) Voltage produced by a precessing DW\nin a nanowire. For simplicity, the DW motion along the\nnanowire is assumed to be suppressed by a notch in the\nnanowire. (a) Schematic structure of DW. DW is assumed\nto precess and change its structure periodically between th e\nBloch DW ( ψ= 0 orπ) and the Neel DW ( ψ=π/2 or\n3π/2) withω=dψ/dt= 2π×100 MHz. (b) Voltage be-\ntween the two ends of the nanowire due to E±. This voltage\ndoes not depend on time tin this particular case since the\nt-dependences of m,∂m/∂t, and∂m/∂ximutually cancel in\nEq. (3). The voltage magnitude is comparable to the value\nreported in Ref. [8]. (c) Voltage due to ERSOC\n±, which oscil-\nlates witht. HereαR= 10−10eV·m [12],P= 1, and the DW\nwidth of 20 nm are assumed.\nbe parallel or anti-parallel to m= (0,0,1), we may re-\ntain only diagonal components of the potentials for lead-\ning order calculation. Then the conventional formula\n−∂tA′−∇A′\n0determines the total electric field E′\n+(E′\n−)\nfor the majority σz=−1 (minority σz= +1) electrons.\nThis calculation reproduces Eqs. (2) and (3).\nWe estimate the relative magnitude of ERSOC\n±with re-\nspecttoE±;ERSOC\n±isoftheorderof( αRme/e/planckover2pi1)ω,where\nωis the characteristic angular frequency of the magneti-\nzation dynamics, and E±is of the order of ( /planckover2pi1/e)(ω/L),\nwhereLis the characteristic length of a magnetic struc-\nture [such as domain wall (DW) width] producing E±.\nThe relative ratio becomes αRmeL//planckover2pi12≈αRL×1038\n(kg/J2·s2) forme= 9.1×10−31kg. For the value\nαR= 10−10eV·m reported for Pt/Co(0.6 nm)/AlO x[12]\nand forL= 20 nm, this ratio becomes about 30. Thus\nERSOC\n±can be more than an order of magnitude larger\nthanE±.\nFor magnetic materials with nonzero spin polariza-\ntionP, spin-dependent electric fields can be measured\nthrough conventional electric voltage measurement [9].Figure 1 compares electrical voltages produced by a pre-\ncessing DW in a nanowire. The voltage due to ERSOC\n±\n[Fig. 1(c)] exhibits time-dependence different from that\ndue toE±[Fig. 1(b)] because of the functional form dif-\nference between Eqs. (2) and (3). Note that the oscilla-\ntion amplitude in Fig. 1(c) is about 80 times larger than\nthe dc value in Fig. 1(b).\nIn regardto the value of αR, 10−10eV·m for Pt/Co(0.6\nnm)/AlO x[12] is not exceptional. The αRvalue in the\nrange (0 .4−3)×10−10eV·m was reported by photo-\nelectron spectroscopy measurements [20, 21] for a thin\nnonmagnetic metal layer in contact with a heavy atomic\nelement layer. Since magnetism does not suppress αR,\nits value can be in a similar range for proper com-\nbinations of a thin magnetic layer in contact with a\nheavy atomic element layer. A recent effective field mea-\nsurement result [22] for Ta/CoFeB(1 nm)/MgO implies\nαR≈0.2×10−10eV·m, which falls close to the com-\nmon range. For this particular multilayer, it was sug-\ngested [22] that the contact with the oxide layer MgO\nmay also be an important source of αR. Still another\ncandidate system is a thin magnetic layerin contact with\ntopological insulators. For instance, BiTeI has a large\nbulk RSOC with αR= 3.8×10−10eV·m [16] and was\npredicted [17] to become a topological insulator under\npressure.\nLarge spin-dependent electric field implies large spin\ncurrent. The spin current density generated by SMF be-\ncomesJSMF\ns=σ↑(E++ERSOC\n+)−σ↓(E−+ERSOC\n−) =\nσc(E++ERSOC\n+) [23], where σ↑(↓)is the longitudinal elec-\ntrical conductivity of majority (minority) electrons and\nσc=σ↑+σ↓. SinceERSOC\n+is much larger than E+and\nERSOC\n±is perpendicular to ˆz[Eq. (2)], JSMF\nsflowswithin\na magnetic layer. Then for a typical in-plane charge con-\nductivity σc= 107Ω−1·m−1of metallic ferromagnetic\nlayersand αR= (0.2−3)×10−10eV·m,JSMF\nshas magni-\ntude (1.8−27)τ−1A·s/m2, whereτis the characteristic\ntime scale of magnetization dynamics. For fast DW mo-\ntion with speed 400 m/s [24], we find that |JSMF\ns|goes\nup to (0.18−2.7)×1011A/m2near the DW center of\nwidth 20 nm. Thus RSOC allows SMF to generate large\nspin current density for fast magnetization dynamics.\nLargeJSMF\nsgenerated by magnetization dynamics im-\nplies that magnetization dynamics itselfcan be signifi-\ncantly modified by JSMF\nsthrough STT. Next we exam-\nine this feedback effect. In conventional situations where\nSMF is negligible, the magnetization dynamics is de-\nscribed by the Landau-Lifshitz-Gilbert (LLG) equation,\n∂m/∂t=−γm×Heff+αGm×∂m/∂t+T(Js), where T\nrepresents the STT and depends on externally supplied\nspin current density Js. Hereγis the gyromagnetic ra-\ntio,αGis the Gilbert damping parameter, and Heffis the\nsum of an external magnetic field and effective magnetic\nfields due to magnetic anisotropy and magnetic exchange\nenergy. To examine the feedback effect, one simply needs3\nto replace T(Js) byT(Js+JSMF\ns) =T(Js)+T(JSMF\ns) to\nobtain the modified LLG equation,\n∂m\n∂t=−γm×Heff+αGm×∂m\n∂t+T(Js)+T(JSMF\ns).(4)\nNote that STT Tnow has two spin current sources, Js\nandJSMF\ns. Thus even when there is no externally sup-\nplied spin current ( Js= 0), STT still affects the magne-\ntization dynamics as long as JSMF\nsis not zero. Therefore\nSTT becomes relevant even for purely field-driven mag-\nnetization dynamics.\nTo gain an insight into roles of the feedback STT\nT(JSMF\ns), we express it in the following form,\nT(JSMF\ns) =m×D ·∂m\n∂t. (5)\nThis form is natural since Tis orthogonal to m[thus\nT(JSMF\ns) =m×(function of JSMF\ns)] andJSMF\nsis propor-\ntional to ∂m/∂t[thus (function of JSMF\ns) =D ·∂m/∂t].\nHereDwill depend on mand be a 3 ×3 matrix in gen-\neral. Although Dis not a constant, the structural sim-\nilarity between Eq. (5) and the Gilbert damping torque\nαGm×∂m/∂timplies that Dmay be interpreted as a\ncorrection to the Gilbert damping parameter αG. Thus\nroles of the feedback STT can be analyzed by using this\ndamping correction picture.\nAfter explicit calculation, we find that the matrix ele-\nments of the 3 ×3 matrix Dare given by\nDij=η/summationdisplay\nk(Xki+ ˜αRǫ3ki)(Xkj+ ˜αRǫ3kj),(6)\nwhereXki= (m×∂m/∂xk)i,η=µB/planckover2pi1σc/2e2Ms, ˜αR=\n2αRme//planckover2pi12,µB(>0) is the Bohr magneton, Msis the sat-\nuration magnetization, and ǫijkis the Levi-Civita per-\nmutation symbol ( ǫ123= 1). To derive Eq. (6), we\nsimply combine the spin-dependent electric field equa-\ntions [Eqs. (2) and (3)] with known contributions to\nSTT. To be specific, the adiabatic STT contribution\n[∝(Js· ∇)m] and the field-like STT contribution [ ∝\nαRm×(ˆz×Js)] [14, 15] are taken into account in the\ncalculation whereas the nonadiabatic STT contribution\nand recently discovered Slonczewski-like STT contribu-\ntion [25–27] areignoredsince the latter twocontributions\nare smaller than the former two.\nThe importance of the feedback STT can be estimated\nfrom the relative magnitude of Dwith respect to αG.\nHereαGincludes all contributions other than the SMF\ncontribution. The intrinsic bulk Gilbert damping pa-\nrameter is of the order of 0.01 for typical metallic fer-\nromagnets. In a thin magnetic layer, this intrinsic value\nis enhanced by the conventional spin pumping mecha-\nnism [28, 29] and the enhanced value is estimated to\n0.1/dF(nm), which is0.1forthin magneticlayerofthick-\nnessdF= 1 nm. Thus for the feedback effect to be\na relevant factor for the magnetization dynamics of athin magnetic layer, Dshould be comparable to or larger\nthan 0.1. When RSOC is absent (˜ αR= 0),Dreduces to\nthe result in Ref. [19] and is of the order of η/L2. For\nσc= 107Ω−1·m−1,Ms= 106A/m and L= 20 nm, η\nis 0.2 nm2andη/L2is 0.0005. Thus except for special\nsituations with L/lessorsimilar1 nm,Dbecomes much smaller than\n0.1 and the feedback effect is negligible. When RSOC\nis strong, on the other hand, the largest contribution to\nDarises from the term η˜α2\nR/summationtext\nkǫ3kiǫ3kjand thus Dbe-\ncomesofthe orderof α′\nG≡η˜α2\nR[30]. For αR= 10−10eV·\nm [12], ˜αRis 2.6 nm−1andα′\nGis 1.4. Since α′\nG≫αG,\none of evident effects of the feedback effect is to enhance\nthe effective damping significantly.\nFor deeper understanding of the feedback effect, how-\never, one should address the detailed structure of the\ndamping matrix Din Eq. (6). Extensive discussion\non implications of the damping matrix structure will\nbe presented elsewhere. Here we present one sim-\nple implication contained in the largest contribution\nα′\nG/summationtext\nkǫ3kiǫ3kj, which reduces to α′\nG(δij−δ3iδ3j) af-\nter simple algebra. This contribution makes the effec-\ntive damping anisotropic since/summationtext\nj(δij−δ3iδ3j)(∂m/∂t)j\nbecomes ( ∂m/∂t)iwhen∂m/∂tis perpendicular to ˆ z-\ndirection, and vanishes when ∂m/∂tis parallel to ˆ z-\ndirection. For a magnetic layer with the perpendicu-\nlar magnetic anisotropy, this damping anisotropy can be\ntested in experiments by measuring the effective damp-\ning in two different ways. When the effective damp-\ning is measured by ferromagnetic resonance, ∂m/∂tre-\nmains essentially perpendicular to ˆ z-direction and thus\nthe effective damping becomes αG+α′\nG. The effec-\ntive damping may be measured alternatively by using\nthe relation that the field-driven DW velocity below the\nWalker-breakdown threshold is inversely proportional to\nthe effective damping. This measurement should pro-\nduceasmallereffectivedampingvaluethantheferromag-\nnetic resonance since ∂m/∂tis along the ±ˆz-direction\nat the DW center. From a simple calculation based\non the Thiele’s collective coordinate description [31] of\nthe DW configuration in terms of the DW position and\nDW tilting angle, we find that the effective damping\nbecomes αG+α′\nG/3. Thus in strong RSOC systems,\nwhereα′\nG≫αG, we predict that the effective damping\nfor the ferromagnetic resonance is about factor 3 larger\nthan that for the DW motion. By the way, Ref. [12] de-\ntermined the effective damping for Pt/Co(0.6 nm)/AlO x\nfromthefield-drivenDWmotionmeasurementandfound\n0.5, which is in reasonable agreement with our prediction\nαG+α′\nG/3≈α′\nG/3≈0.5 for the system.\nFigure 2 shows schematically an experimental setup\nto test the RSOC-enhanced SMF directly. When a uni-\nformlymagnetized magnetic domain precesses due to ex-\nternal magnetic field B(t) =Bsinωt,ERSOC\n±will gener-\nate electric voltages VxandVy. SinceERSOC\n±is position-\nindependent [Eq. (2)], these voltages should be propor-\ntional to the spacing dbetween electrodes. Also since the4\ndB(t) = B sin ωt\nd\nFIG. 2: (color online) Measurement scheme of the RSOC ef-\nfect on the SMF for a uniformly magnetized layer with the\nperpendicular magnetic anisotropy.\ndirection of ERSOC\n±rotates within xy-plane as ∂m/∂tro-\ntates within xy-plane (due to the perpendicular magnetic\nanisotropy), the voltages will be oscillatory with the os-\ncillation phases of VxandVydifferent from each other by\nabout 90◦. Thesefeaturesarequalitativelydifferent from\nother effects that may affect this measurement. The con-\nventional spin pumping [28] may also contribute to the\nelectric voltages if the two electrical contacts with the\nelectrodes are not identical, as demonstrated in Ref. [32].\nThese contribution is however independent of dand thus\ndistinguishable from the RSOC-enhanced SMF contri-\nbution. Spin Hall effect in contacts or the neighbor-\ning heavy metal layer can also generate a contribution.\nThis spin Hall contribution is however different from the\nRSOC-enhanced SMF contribution since ˆz×mdeter-\nmines the direction of the voltage gradient in case of the\nspin Hall contribution whereas ˆz×∂m/∂tdoes in case of\nthe RSOC-enhanced SMF contribution. Recalling that\nmand∂m/∂tare orthogonal to each other, these two\ncontributions differ by the oscillation phase of 90◦. A\nrecent experiment [18] demonstrated that in certain ex-\nperimentalsituations[33], thespinHalleffect canbe very\nsimilar to the RSOC effect. The electrical voltage mea-\nsurement in Fig. 2 can be a very decisive experiment to\nverify the RSOC-enhanced SMF and also RSOC itself\nfree from such ambiguities. The Rashba constant αRcan\nbedeterminedfromtheproportionalityconstantbetween\nthe voltages and d.\nLastly two remarks are in order. Firstly, temperature\nmay affect the predicted phenomena. The value of αR\nmay depend on temperature. But recalling that the re-\nported value of αR= 10−10eV·m for Pt/Co/AlO x[12]\nwas obtained at room temperature, we expect that αR\ncan stay large even at room temperature. Thermal fluc-\ntuation of mis another possible source of the tempera-\nture dependence of SMF. Since the fluctuation is not cor-\nrelated in space and time, it will not affect the SMF volt-\nage measurements in Figs. 1 and 2 significantly. Thus we\nexpect that temperature effects do not modify SMF ef-\nfects qualitatively. Secondly, though not elaboratedhere,\nRSOC gives rise to not only ERSOC\n±[Eq. (2)] but also aspin-dependent magnetic field. Our preliminary calcu-\nlation indicates that certain types of magnetization dy-\nnamics such as DW motion may be accelerated by this\neffective magnetic field. Further research is required to\nunderstand its effects.\nIn conclusion, we demonstrated that SMF can be or-\nders of magnitudes enhanced in strong RSOC systems.\nThe enhanced SMF is strong enough to modify the mag-\nnetization dynamics significantly. RSOC-enhanced SMF\nmay affect performance of various device applications\nbased on ultrathin magnetic layers [24, 33, 34] since\nRSOC effects tend to get larger in thinner systems. Thus\nSMF is not a weak effect of purely scientific concern any\nmore but instead a technologically relevant effect that\nshould be taken into account in future studies. As a final\nremark, during the revision of this manuscript, we were\ninformed that two other groups [35, 36] also obtained\nEq. (2).\nWe gratefully acknowledge M. Stiles for critical com-\nments on the manuscript. We also acknowledge B.-\nC. Min, S.-Y. Park, and Y. Jo for stimulating dis-\ncussion. This work was financially supported by the\nNRF (2009-0084542, 2010-0014109, 2010-0023798, 2011-\n0028163, 2011-0030784) and BK21. KWK acknowledges\nthe financial support by the NRF (2011-0009278)and TJ\nPark.\n[1] J. C. Slonczewski, J. Magn. Mag. Mater. 159, L1 (1996).\n[2] L. Berger, Phys. Rev. B 54, 9353 (1996).\n[3] S. I. Kiselev et al., Nature 425, 380 (2003).\n[4] K.-J. Lee, A. Deac, O. Redon, J.-P. Nozi´ eres, and B.\nDieny, Nature Mater. 3, 877 (2004).\n[5] L. Berger, Phys. Rev. B 33, 1572 (1986).\n[6] G. E. Volovik, J. Phys. C 20, L83 (1987).\n[7] S. E. Barnes and S. Maekawa, Phys. Rev. Lett. 98,\n246601 (2007).\n[8] S. A. Yang et al., Phys. Rev. Lett. 102, 067201 (2009).\n[9] J. Ohe, S. E. Barnes, H.-W. Lee, and S. Maekawa, Appl.\nPhys. Lett. 95, 123110 (2009).\n[10] Y. Yamane et al., Phys. Rev. Lett. 107, 236602 (2011).\n[11] Y. A. Bychkov and E. I. Rashba, J. Exp. Theor. Phys.\nLett.39, 78 (1984).\n[12] I. M. Miron et al., Nature Mater. 9, 230 (2010).\n[13] U. H. Pi et al., Appl. Phys. Lett. 97, 162507 (2010).\n[14] K.ObataandG.Tatara, Phys.Rev.B 77, 214429 (2008).\n[15] A. Manchon and S. Zhang, Phys. Rev. B 78, 212405\n(2008) ; A. Matos-Abiague and R. L. Rodriguez-Suarez,\nPhys. Rev. B 80, 094424 (2009).\n[16] K. Ishizaka et al., Nature Mater. 10, 521 (2011).\n[17] M. S. Bahramy, B.-J. Yang, R. Arita, and N. Nagaosa,\nNature Commun. 3, 679 (2012).\n[18] L. Liu, O. J. Lee, T. J. Gudmundsen, D. C. Ralph, and\nR. A. Buhrman, arXiv:1110.6846.\n[19] S. Zhang and S.-L. Zhang, Phys. Rev. Lett. 102, 086601\n(2009).\n[20] J. Henk, M. Hoesch, J. Osterwalder, A. Ernst, and P.\nBruno, J. Phys.: Condens. Matter 16, 7581 (2004).5\n[21] C. R. Ast et al., Phys. Rev. Lett. 98, 186807 (2007).\n[22] T. Suzuki et al., Appl. Phys. Lett. 98, 142505 (2011).\n[23] The Hall conductivity is ignored since it is orders of\nmagnitude smaller than the longitudinal conductivity.\nThe diffusion contribution is also ignored, which coarse-\ngrains the spatial-dependence of the spin current density\nbut does not cause qualitative changes [K.-W. Kim, J.-\nH. Moon, K.-J. Lee, and H.-W. Lee, Phys. Rev. B 84,\n054462 (2011); J. Shibata and H. Kohno, Phys. Rev. B\n84, 184408 (2011)].\n[24] I. M. Miron et al., Nature Mater. 10, 419 (2011).\n[25] X. Wang and A. Manchon, arXiv:1111.1216;\narXiv:1111.5466.\n[26] K.-W. Kim, S.-M. Seo, J. Ryu, K.-J. Lee, and H.-W. Lee,\narXiv:1111.3422v2.\n[27] D. A. Pesin and A. H. MacDonald, arXiv:1201.0990.\n[28] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys.\nRev. Lett. 88, 117601 (2002).\n[29] Whereastheconventional spinpumpingeffect arises fro mthe induced spin current perpendicular to a magnetic\nlayer, the RSOC-enhanced SMF induces in-plane spin\ncurrent. Thus their contributions to the Gilbert damp-\ning enhancement are independent and should be treated\nseparately.\n[30] When a ferromagnet is a good conductor in the diffusive\nregime, this expression for α′\nGagrees with Eq. (2) in M.\nHankiewicz, G. Vignale, and Y. Tserkovnyak, Phys. Rev.\nB75, 174434 (2007).\n[31] A. A. Thiele, Phys. Rev. Lett. 30, 230 (1973).\n[32] M. V. Costache, M. Sladkov, S. M. Watts, C. H. van der\nWal, and B. J. van Wees, Phys. Rev. Lett. 97, 216603\n(2006).\n[33] I. M. Miron et al., Nature 476, 189 (2011).\n[34] S. Ikeda et al., Nature Mater. 9, 721 (2010).\n[35] Private communications with A. Sakai and H. Kohno.\n[36] Private communications with Y. Yamane, J. Ieda, and S.\nMaekawa."    },    {        "title": "1004.1966v2.Influence_of_the_particle_number_on_the_spin_dynamics_of_ultracold_atoms.pdf",        "content": "Influence of the particle number on the spin dynamics of ultracold atoms\nJannes Heinze,1,\u0003Frank Deuretzbacher,2,yand Daniela Pfannkuche1\n1I. Institut f ¨ur Theoretische Physik, Universit ¨at Hamburg, Jungiusstrasse 9, 20355 Hamburg, Germany\n2Institut f ¨ur Theoretische Physik, Leibniz Universit ¨at Hannover, Appelstr. 2, 30167, Hannover, Germany\nWe study the dependency of the quantum spin dynamics on the particle number in a system of ultracold\nspin-1 atoms within the single-spatial-mode approximation. We find, for all strengths of the spin-dependent\ninteraction, convergence towards the mean-field dynamics in the thermodynamic limit. The convergence is,\nhowever, particularly slow when the spin-changing collisional energy and the quadratic Zeeman energy are\nequal, i. e. deviations between quantum and mean-field spin dynamics may be extremely large under these\nconditions. Our estimates show, that quantum corrections to the mean-field dynamics may play a relevant role\nin experiments with spinor Bose-Einstein condensates. This is especially the case in the regime of few atoms,\nwhich may be accessible in optical lattices. Here, spin dynamics is modulated by a beat note at large magnetic\nfields due to the significant influence of correlated many-body spin states.\nPACS numbers: 67.85.Fg, 67.85.De, 67.85.Hj, 75.50.Mm\nI. INTRODUCTION\nThe spin degree of freedom of spinor Bose-Einstein con-\ndensates (BECs) alows one to explore magnetism of ultra-\ncold quantum fluids [1], which exhibit intriguing phenom-\nena, like the formation of coreless vortices [2] and other\nspin textures [3–5], and spontaneous symmetry breaking af-\nter a quench [6]. Beside the investigation of the ground-state\nphases [7–10], the main focus of research is on the study of\nspin dynamics [11–15]. So far most experiments confirmed\nthe mean-field (MF) description, including a nonlinear res-\nonance phenomenon near a critical magnetic field [16–19],\nwhich is caused by the interplay of spin-changing collisions\nand quadratic Zeeman shift. Interestingly, the Hamiltonian\nof the spin-dependent interatomic interactions appears also in\nnonlinear quantum optics [9]. As a result, spinor BECs allow\none to explore, e. g., four-wave mixing [18, 20] and parametric\ndown conversion [21–23] with matter waves. Moreover, spin-\nchanging collisions are responsible for the dynamical evolu-\ntion of squeezed collective spin states and entanglement from\nuncorrelated (product) states [24–28], which provides a way\nto overcome the standart quantum limit in precision measure-\nments with matter waves. Those beyond-mean-field correla-\ntions can be described by means of the Bogoliubov approxi-\nmation as long as the quantum fluctuations of the spinor field\noperator are small [21–23, 29].\nCorrelations limit the validity of the MF approximation.\nHence, knowledge of the boundaries of the Gross-Pitaevskii\nequation (GPE) is of great interest. The validity of the GPE\nhas been proven for weakly interacting spinless bosons in the\nlimitN!1 withNafixed [30, 31], where Nis the num-\nber of particles and ais the scattering length. We show in\nthis article by means of a numerically exact diagonalization of\nthe effective spin Hamiltonian [9, 10, 29, 32], that the quan-\ntum spin dynamics in the single-mode approximation (SMA)\nconverges towards the MF dynamics in the thermodynamic\n\u0003Electronic address: jheinze@physnet.uni-hamburg.de\nyElectronic address: fdeuretz@itp.uni-hannover.delimit (TDL). This is, interestingly, the case for all strengths of\nthe spin-dependent interaction (within the SMA). The conver-\ngence is, however, particularly slow in a regime, where the\nspin-changing collisional energy and the quadratic Zeeman\nenergy are equal. We determine the validity time of the ini-\ntial MF dynamics under these conditions, which grows loga-\nrithmically with the number of particles. From this we expect\nthat quantum corrections to the MF dynamics may play an im-\nportant role in experiments with spinor BECs. Additionally,\nwe discuss a beat-note phenomenon in the spin dynamics of\nfew atoms (N\u001810), which may be observed in deep optical\nlattices.\nThe paper is organized as follows. In Sec. II we outline the\nmethod to calculate the quantum spin dynamics. Afterwards,\nin Sec. III, we discuss few particles. First, in Subsec. III A we\napply the formalism to two particles to make the method clear\nand to discuss similarities with the MF dynamics. Then, we\ndiscuss the spin dynamics of three particles in Subsec. III B,\nwhich is modulated by a beat note at large magnetic fields.\nA similar beat-note phenomenon is recovered for few atoms,\nwhich is shown in Subsec. III C. In Sec. IV we turn to the\ncomparison of the N-particle quantum dynamics with the MF\ndynamics. This is done for two typical initial states. First, in\nSubsec. IV A, we study the initial state, where all atoms are\nin them= 0Zeeman sublevel, and then, in Subsec. IV B, we\nanalyze the initial dynamics of the transversely magnetized\nstate. We finally summarize our results in Sec. V.\nII. METHODS\nA. Effective spin Hamiltonian\nThe two-body interaction between ultracold spin-1 atoms is\nmodeled by a spin-dependent \u000epotential [7]\nVint.(~ r1\u0000~ r2) =\u000e(~ r1\u0000~ r2)\u0010\n~c0+~c2~f1\u0001~f2\u0011\nwith the interaction strengths c0andc2and the dimensionless\nspin-1 matrices ~fiof atomi= 1;2. Typically,c0is one or twoarXiv:1004.1966v2  [cond-mat.quant-gas]  18 May 20102\norders of magnitude larger than c2. The atoms are confined by\na spin-independent trapping potential Vtrap(~ r). Additionally, a\nhomogeneous magnetic field along the z-direction generates\nthe potential\nVZ=\u0000~pfz\u0000~q\u0000\n4\u0000f2\nz\u0001\n;\nwherep/Bandq/B2are the coefficients of the linear\nand quadratic Zeeman energy, respectively. The many-body\nHamiltonian is then\nH00=X\ni\u0014\n\u0000~2\n2m\u0001i+Vtrap(~ ri)\u0015\n+X\ni<j~c0\u000e(~ ri\u0000~ rj)\n\u0000X\nih\n~pfz;i+~q\u0000\n4\u0000f2\nz;i\u0001i\n+X\ni<j~c2\u000e(~ ri\u0000~ rj)~fi\u0001~fj:\nThe first line contains the spin-independent part of H00, which\nacts only in position space, and the second line contains the\nZeeman Hamiltonian, which acts only in spin space. Only the\nspin-dependent interaction in the third line couples the spin to\nthe motional degrees of freedom.\nIn the absence of the spin-dependent interaction (c2= 0)\nthe states of the ground-state multiplet are of the form\n 0(~ r1;:::;~ rN)\nj\u001fsi\nwith 0being the totally symmetric (nondegenerate) ground\nstate of the spinless problem and with j\u001fsibeing an arbitrary\ntotally symmetric N-particle spin function [33]. In many ex-\nperimental situations one can restrict the description to these\nstates [14, 15, 18, 19, 34, 35], since jc2j\u001cc0. As a conse-\nquence, the motion of the atoms is frozen in the ground state\n 0and the system is essentially zero-dimensional.\nLet us assume that we have found the spatial ground state\n 0and the corresponding energy E0. An integration over the\nspatial degrees of freedom leads to the effective spin Hamil-\ntonian (the diagonal offset E0\u00004N~qis neglected)\nH0=\u0000~pX\nifz;i+~qX\nif2\nz;i+~gsX\ni<j~fi\u0001~fj\nwith\ngs=c2Z\nd~ rd~ r 3:::~ rN\f\f 0(~ r;~ r;~ r 3;:::;~ rN)\f\f2:\nThe integral is the averaged local pair correlation function g(2)\nof the ground state  0. It can be viewed as the inverse volume\nof the ground state g(2)\u00111=V.\nUsing the projection operators Nm=P\nijmiihmji;which\ncount the number of particles with polarization m= +;0;\u0000;\nand the relation\nX\ni<j~fi\u0001~fj=1\n2\u0000~F2\u00002N\u0001\n;where~F2=\u0000~f1+:::+~fN\u00012is the square of the total spin,\nwe obtain a more convenient form of the Hamiltonian\nH0=\u0000~p(N+\u0000N\u0000) +~q(N++N\u0000) +~gs\n2\u0000~F2\u00002N\u0001\n:\nThez-component of the total spin Fz=N+\u0000N\u0000commutes\nwith the occupation number operators Nmand the Hamilto-\nnianH0, which leads to a decomposition of the dynamics into\nsubspaces with different Fz=M=\u0000N;:::;N; i. e.\nhNmi=X\nMhNmiM:\nTherefore, the linear Zeeman energy, which is a constant in\neach subspace, does not influence the dynamics. In the fol-\nlowing we neglect the linear Zeeman energy and set p= 0.\nThe remaining Hamiltonian\nH=~q(N++N\u0000) +~gs\n2\u0000~F2\u00002N\u0001\n(1)\nhas a spin-flip symmetry\nHM=H\u0000M;\nsince~F2is not changed by a rotation of 180\u000earound thex-\naxis and since N++N\u0000is unaffected if N+$N\u0000.\nB. Matrix representation\nThe matrix elements of the Hamiltonian (1) are most con-\nveniently calculated within the second quantization formal-\nism. We express the totally symmetric N-particle spin func-\ntionsj\u001fsiby linear combinations of occupation number basis\nstatesjN+;N0;N\u0000i, which are eigenstates of the occupation\nnumber operators Nm=ay\nmam. The bosonic creation and\nannihilation operators ay\nm,amact on these states in the usual\nway and obey the commutation relations\u0002\nam;ay\nm0\u0003\n=\u000emm0\nand zero else.\nBefore we proceed, we change the labeling of the occupa-\ntion number basis states jN+;N0;N\u0000iin order to simplify\nthe following formulas. We use the set of quantum numbers\n(\u0011;M;N )instead of (N+;N0;N\u0000). Both labels are related\nto each other through \u0011=N++N\u0000,M=N+\u0000N\u0000and\nN=N++N0+N\u0000. We will not explicitly refer to the\nnumber of particles N, i. e.jN+;N0;N\u0000i=j\u0011;Mi.\nThe first summand of (1), the quadratic Zeeman Hamilto-\nnianHq=~q(N++N\u0000), is diagonal in the occupation num-\nber basis and given by\nh\u0011;MjHqj\u0011;Mi=\u0011~q: (2)\nThe second summand, the spin-dependent interaction Hamil-\ntonianHs=~gs\n2\u0000~F2\u00002N\u0001\n, is tridiagonal. In order to calcu-\nlateHswe use the formula\n~F2=F2\nz+1\n2\u0010\nF+F\u0000+F\u0000F+\u00113\nwith the angular momentum creation and annihilation opera-\ntorsF\u0006=Fx\u0006 iFy, which are given by\nF\u0006=p\n2\u0010\nay\n\u0006a0+ay\n0a\u0007\u0011\nin dimensionless units. The diagonal elements of Hsare\nh\u0011;MjHsj\u0011;Mi=~gs\n2\u0002\nM2\u00002\u00112+\u0011(2N\u00001)\u0003\n(3)\nand the secondary diagonal elements are\nh\u0011+ 2;MjHsj\u0011;Mi=h\u0011;MjHsj\u0011+ 2;Mi=\n~gs\n2p\n(N\u0000\u0011\u00001)(N\u0000\u0011)(\u0011+M+ 2)(\u0011\u0000M+ 2):(4)\nAs discussed before, Hdecomposes into subblocks, which\ncan be diagonalized independently for each eigenvalue of the\ntotal magnetization M, sinceHcommutes with Fz. The basis\nstates of one subblock are given by\n\f\fjMj;M\u000b\n;\f\fjMj+ 2;M\u000b\n;:::;\f\f\u0011max;M\u000b\n(5)\nwith\u0011max=N\u00001orN, leading to a total dimension of\ndim(N;M ) =\u0016N\u0000jMj\n2\u0017\n+ 1 (6)\nwith the common floor function.\nC. Population dynamics\nOnce we have determined the eigenstates j\u000fiand eigenfre-\nquencies!\u000fofH, we can calculate the time evolution of the\nsystem, if the initial state j iihas been specified. The initial\nstatej iievolves according to\nj (t)i= exp\u0000\n\u0000 iHt=~\u0001\nj ii;\nwhere the spectral representation of the time evolution opera-\ntor is given by\nexp\u0000\n\u0000 iHt=~\u0001\n=X\n\u000fj\u000fih\u000fjexp\u0000\n\u0000 i!\u000ft\u0001\n: (7)\nThe time evolution of the relative population n0=N0=Nis\nthus given by\nn0(t) =h ijeiHt=~ay\n0a0e\u0000 iHt=~j ii=N: (8)\nThe population of the other spin components is completely de-\ntermined by the conservation of the total number of particles\nNand of the total magnetization hFzivia\nn\u0006(t) =1\n2h\n1\u0006hFzi=N\u0000n0(t)i\n:\nFor the initial states chosen here, hFzi= 0. In the following\ndiscussion we present only the time evolution of n0and we\nneglect any contributions, which are constant in time, since\nthey are determined by the initial state j ii.D. Initial states\nWe will discuss the time evolution of the number state\nj\u0012Ni=j0;N;0i, where all the atoms are in the m= 0 Zee-\nman state, and the transversely magnetized state\nj\u0010Ni=1p\nN!\u00121\n2ay\n++1p\n2ay\n0+1\n2ay\n\u0000\u0013N\nj0;0;0i;\nwhere all the spins are pointing into the positive x-direction.\nThis state is a superposition of number states from all sub-\nspaces\nj\u0010Ni=NX\nM=\u0000N\u0011maxX\n\u0011=jMj;\u0001\u0011=2\u001fM\n\u0011j\u0011;Mi\nwith the coefficients\n\u001fM\n\u0011=\u00121\n2\u0013\u0011+N\n2s\u0012N\n\u0011\u0013\u0012\u0011\n\u0011+M\n2\u0013\n: (9)\nE. Dimensionless coupling parameter\nThe essential parameter, which characterizes the interplay\nbetween the quadratic Zeeman energy and the spin-dependent\ninteraction energy, is given by\nK=2q\n(2N\u00001)gs!q\nc2\u001afor largeN;\nwhere\u001a=N=V is the particle density. Kcan be positive or\nnegative depending on the sign of qandc2.\nIII. FEW-ATOM DYNAMICS\nA. Two atoms\nWe begin with two atoms to illustrate the method and\nsince typical features of the two-atom dynamics occur also for\nlarger particle numbers and in the MF limit. Two-atom spin\ndynamics was experimentally investigated in Refs. [14, 15]\nfor a system being initially in the number state.\nLet us start with the calculation of the matrix elements of\nthe Hamiltonian (1). The matrix Hdecomposes into 5 subma-\ntrices with total magnetization M= 2;1;0;\u00001;\u00002. Accord-\ning to Eq. (6) the subspace with M= 0has dimension 2 and\nthe others have dimension 1. If one writes down Eq. (8) for\nan arbitrary initial state, one sees, that only energy differences\nwithin the same subspace lead to a sinusoidal oscillation with\nfrequency!ij=j!i\u0000!jj. Thus, only the M= 0 subspace\ncontributes to the time evolution with exactly one frequency\n!and the others lead to constant offset amplitudes, which we\nwill neglect in the following. According to Eq. (5) the two ba-\nsis states of the M= 0 subspace arej0;0iandj2;0i. Using\nEqs. (2–4) we obtain the 2\u00022matrix\nH0=~\u0012\n0p\n2gsp\n2gs2q\u0000gs\u0013\n: (10)4\n0 1 2 3 4 5\ncoupling parameter |K|00.10.20.30.40.5\nA∝1/K2amplitude A\nFIG. 1: (color online). Two-atom amplitude as a function of jKjfor\nthe initially prepared number state, K < 0(red) andK > 0(blue).\nLet us consider an arbitrary initial state, which is given by\nj ii=\u000bj0;0i+\fj2;0i+other terms: (11)\n(The other terms are the components of the irrelevant sub-\nspaces, like\rj1;1i+\u000ej2;2i+:::and so on. Thus, in general\nj\u000bj2+j\fj261.)From the diagonalization of the matrix (10)\nwe obtain the spectral representation of the time evolution op-\nerator (7), which we insert into Eq. (8) together with the initial\nstate (11). The result is an ordinary cosine oscillation\nn0\n0(t) =Acos(!t) (12)\nwith the amplitude\nA=2\n!2n\n2p\n2<(\u000b\f\u0003)qgs+g2\ns\u0002\n2(j\u000bj2\u0000j\fj2)\u0000p\n2<(\u000b\f\u0003)\u0003o\n;\n(13)\nwhere<(\r)denotes the real part of \rand\r\u0003is its complex\nconjugate, and the frequency\n!=p\n(2q\u0000gs)2+ 8g2s: (14)\nTime-independent terms have been neglected in Eq. (12), i. e.\nn0\n0(t) =n0(t)\u0000n0;const..\nLet us first discuss the initial state j\u00122i=j0;2;0i=j0;0i,\ni. e.(\u000b= 1;\f= 0) . In that case, the system undergoes Rabi\noscillations [14, 15] and the amplitude (13) becomes\nA=4g2\ns\n(2q\u0000gs)2+ 8g2s=4\n9\u00006K+ 9K2: (15)\nFig. 1 shows the amplitude Aas a function of the coupling\nparameterjKjfor the two cases K < 0(red) andK > 0\n(blue). For smalljKj, the Hamiltonian is dominated by the\nspin-dependent interaction Hs=~gs\n2\u0000~F2\u00002N\u0001\n, and since\nthe number state is not an eigenstate of Hs, the system under-\ngoes large oscillations. In the opposite limit of large jKj, the\nHamiltonian is approximately equal to the quadratic Zeeman\nenergyHq=~q(N++N\u0000). Here, the oscillation amplitude\n0 1 2 3 4 500.020.040.060.080.10.120.14\ncoupling parameter |K|0 1 2 3 4 500.050.10.150.20.25\nMF\ntwo atoms\nA∝1/K\nA∝KK|A|absolute value of the amplitude |A|FIG. 2: (color online). Absolute value of the two-atom amplitude jAj\nas a function ofjKjfor the transversely magnetized state, K < 0\n(red) andK > 0(blue). Inset: comparison with the MF limit.\nconverges to zero on the one hand, since the number state is\nan eigenstate of Hq, and on the other hand, since the occupa-\ntion number operator n0(t) =ay\n0a0=Ncommutes with Hq,\nand thus becomes a constant of motion.\nFor the transversely magnetized state j\u00102iwe calculate the\ncoefficients (\u000b= 1=2;\f=p\n2=4)using Eq. (9). The ampli-\ntude (13) takes the form\nA=qgs\n(2q\u0000gs)2+ 8g2s=K\n6\u00004K+ 6K2:\nAis negative for K < 0, which corresponds to a phase shift of\n\u0019in the cosine function of Eq. (12), and is not in contradiction\nto the requirement n0(t)>0, sincen0;const.>jAj.\nFig. 2 showsjAjas a function ofjKjforK < 0(red)\nandK > 0(blue). The transversely magnetized state is an\neigenstate of the spin-dependent interaction Hsand thus the\noscillation amplitude is small for small jKj. Again, the ampli-\ntude drops down in the opposite limit of large jKj, since the\noccupation number operator n0(t) =ay\n0a0=Ncommutes with\nHqand becomes a constant of motion. Between these limit-\ning cases the amplitude has a maximum, which is located at\njKj= 1. The MF amplitude (App. A) has a maximum at the\nsame position and shows the same limiting behavior, which\nis clear, since the above arguments were independent of the\nnumber of particles (inset of Fig. 2).\nB. Three atoms\nIn the case of three atoms, there are 3 subspaces of dimen-\nsion 2 (those with M= 0;\u00061), which contribute to the dy-\nnamics. However, since His symmetric under polarization\nreflectionM$\u0000M, theM=\u00061subspaces yield the same\nfrequency. The corresponding Hamiltonians are given by\nH0=~\u0012\n0p\n6gsp\n6gs2q+gs\u0013\n; H\u0006=~\u0012\n0 2gs\n2gs2q\u00003gs\u0013\n;5\n0 0 .1 0 .2 0 .3 0 .4 0 .50.50.520.540.560.580.6\ntime t[2π/ |gs|]0 1 2 3 4 5K0 1 2 3 4 5K00.020.040.060.080.10.120.14\n051015202530amplitudes A0/1\nω0/1[gs]A0\nA1ω0ω14gs (a) (b)\n(c)relative population of\nm= 0 Zeeman state n0(t)\nFIG. 3: (color online). Evolution of three atoms in the transversely\nmagnetized state. (a) A0=1as a function of K. (b)!0=1as a function\nofK. (c) Dynamics for K=\u00005(red) compared to the MF evolution\n(blue). The oscillation of the m= 0 population is modulated by a\nbeat frequency of 2gs, which is absent in the MF limit (App. A).\nwhere inH\u0006we have subtracted the offset ~(q+ 2gs)from\nthe diagonal. For the initially prepared number state, the dy-\nnamics is restricted to the M= 0 subspace, which leads to\nsimilar results as in the previously discussed two-atom case.\nBy contrast, the transversely magnetized state j\u00103iis dis-\ntributed over all subspaces. Since all subspaces evolve inde-\npendently, they can be evaluated in the same way as before\nand added thereafter. The time evolution of the m= 0popu-\nlation is now given by a sum of two cosines\nn0\n0(t) =A0cos(!0t) +A1cos(!1t)\n(constant offsets are neglected) with different amplitudes\nA0=1=qgs=!2\n0=1\nand unequal frequencies\n!0=p\n(2q+gs)2+ 24g2s; ! 1=p\n(2q\u00003gs)2+ 16g2s:\nThe amplitudes A0andA1as a function of Kshow the same\nbehavior as discussed for two atoms [see Fig. 3(a)]. The fre-\nquencies!0and!1as a function of Kare shown in Fig. 3(b).\nAt largeKthey differ by 4gsso that the oscillation dynamics\nis modulated by a beat frequency of 2gs:\nn0\n0(t)\u00191\n10Kn\ncos\u0002\n(2q+gs)t\u0003\n+ cos\u0002\n(2q\u00003gs)t\u0003o\n\u00191\n5Kcos\u0002\n(2q\u0000gs)t\u0003\ncos(2gst): (16)\nFig. 3(c) shows the three-atom dynamics (red) compared to\nthe MF dynamics (blue) at K=\u00005.\n0 0 .1 0 .2 0 .3 0 .4 0 .50.50.510.520.530.540.550.560.570.580.59\ntime t[2π/ |gs|]relative population of\nm= 0 Zeeman state n0(t)FIG. 4: (color online). Evolution of seven atoms in the transversely\nmagnetized state for K=\u00005. The exact numerical calculation\n(blue) is compared to the first-order perturbative evolution (17) (red).\nThe envelope function of (17) is drawn as a dashed line.\nC. Dynamics of few atoms for large K\nFor intermediate values of K, the dynamics of few atoms\nlooks rather chaotic, since many frequencies contribute to the\nevolution. For large K, the transversely magnetized initial\nstate shows a fast oscillation, which is modulated by a beat\nfrequency, as in the three-atom case. The limiting dynamics\nis obtained from a perturbative calculation: For 1=K\u00190, the\nquadratic Zeeman energy is the dominant part of the Hamilto-\nnian and thus we choose the number states jN+;N0;N\u0000ias a\nbasis. The interaction energy is a small perturbation. In a first\nstep we approximate the eigenstates and energies up to first\norder in\u0015=N=(8K). This is separately done for subspaces\nofHwith different magnetization Fz=M, since their evo-\nlution is decoupled. The result and the representation of the\ninitial statej\u0010Ni(9) is inserted into Eq. (8). After neglecting\nterms of order \u00152;\u00153;:::one obtains the time evolution\nn0\n0(t) =N\u00001\n2K(2N\u00001)cos\u0002\n(2q\u0000gs)t\u0003\u0002\ncos(2gst)\u0003N\u00002:\n(17)\nThe amplitude of the oscillation is proportional to 1=K, as\nin the two-particle and MF limiting cases (Fig. 2). The fre-\nquency of the fast oscillation, which is approximately given\nby\u00192q, is determined by the level spacing of Hq, which co-\nincides again with the two-particle (14) and MF results (A3).\nBut different from these two limiting cases, the fast oscillation\nis modulated by a beat frequency of 2gs.\nFig. 4 shows the time evolution of seven atoms at K=\u00005.\nThe beat note is a clear signature that correlated spin states\ncontribute to the dynamics of few atoms. We believe, that the\nbeat frequency can be observed in deep optical lattices with\nfew atoms at each lattice site, similar to the measurements of\nRefs. [14, 15]. The observation is facilitated by the fact, that\nthe beat frequency is independent of the number of particles.\nEq. (17) is valid for rather large particle numbers. We found\nexcellent agreement with the initial evolution of .300parti-\ncles forK=\u00005from a comparison with the numerical re-6\n2000\n1000\n500\n250MF limit\n0 2 4 6 8 100.30.40.50.60.70.80.91\ntime t[TTDL = 2π/(|gs|N)]50200\n10\n0 0 .2 0 .4 0 .6 0 .8 100.20.40.60.81\ntime t[2π/ |gs|]relative population of\nm= 0 Zeeman state n0(t)relative population of\nm= 0 Zeeman state n0(t)\nFIG. 5: (color online). ( top) Evolution of the number state with N=\n10;50;200bosons atK= 0. The time scale is 2\u0019=jgsj. (bottom )\nEvolution for N= 250;500;1000;2000 bosons atK= 0. The time\nscale is now TTDL = 2\u0019=(jgsjN) = 2\u0019=(jc2j\u001a). In the MF limit,\nthe number state is a steady state, and n0(t) = 1 . With increasing\nparticle number Nthe population n0(t)decreases more slowly on\nthe time scale TTDLuntil it stays constant in the thermodynamic limit.\nsults, although \u0015\u001d1for these parameters.\nIV .N-PARTICLE QUANTUM DYNAMICS VS.\nMEAN-FIELD DYNAMICS\nOne can show by means of the large- Nmethod [36] that the\nMF description of the spin system considered here becomes\nexact in the thermodynamic limit (TDL) [37]. That means\nthat the time evolution of a coherent initial state is described\nby the MF equations of motion in the TDL. Further, one of the\nground states of the system becomes a coherent state and its\nenergy can be calculated by minimizing the expectation value\nofHon the MF phase space. Product states (A1) become\nidentical with coherent states in the limit N!1 and thus the\nN-particle quantum dynamics of the initial states considered\nhere converges towards the MF solution in the TDL. In the\nfollowing we study the influence of the finite particle number\nNon the quantum corrections to the MF dynamics.\nA. Number state\nThe number state is a steady state of the MF equations of\nmotion for all values of the coupling strength K(App. A).\nThat makes it particularly useful to study corrections, which\n0 0 .5 1 1 .5 2\ncoupling parameter ( −K)00.11000\n2000\n4000\n8000\n16000fraction of atoms in m=±1\naftert= 0.5TTDL\n32000\n0.020.040.060.080.120.140.160.18FIG. 6: (color online). Relative population of m=\u00061Zeeman\nstates,n\u0006= 1\u0000n0, aftert= 0:5TTDLas a function of (\u0000K).\nBeyond-MF corrections are largest around K=\u00001and decrease\nwith increasing particle number.\ngo beyond the conventional MF dynamics, since quantum spin\nfluctuations are strongly amplified in this initial state [22, 23,\n29].\n1. Time evolution for zero K\nAtK= 0, the Hamiltonian consists only of the interac-\ntionHs, which has the total-spin states jF;Mias eigenstates.\nTo calculate the N-particle quantum dynamics, one needs the\nrepresentation of the occupation number operator n0and the\ninitial statej\u0012Niin the total-spin basis fjF;Mig. The calcu-\nlation is done in App. B. The evolution is given by\nn0(t) =X\nFCFcos\u0002\ngs(2F+ 3)t\u0003\n; (18)\nwhere the coefficients CFare approximated by\nCF\u0019F\nNexp\u0012\n\u00001\n2F2\nN\u0013\n(19)\nfor large particle numbers N(see App. B). The dynamics is\nperiodic with 2\u0019=jgsj, since all the frequencies in (18) are\nmultiples of gs. Fig. 5( top) shows a rapid oscillation on the\ntime scale 2\u0019=jgsj, which seems to be in contradiction with\nthe MF result, since the number state is a steady state of the\nMF equations of motion for all K. The time scale 2\u0019=jgsj,\nhowever, becomes infinite in the TDL, since 2\u0019V=jc2j!1\nforV!1 . Thus, we need a time scale, which stays constant\nin the TDL. A proper choice is given by\nTTDL= 2\u0019=(jgsjN) = 2\u0019=(jc2j\u001a);\nwhich is inversely proportional to the spin-dependent inter-\naction energy. By plotting the initial N-particle quantum dy-\nnamics as a function of TTDL[Fig. 5( bottom )] one sees conver-\ngence towards the MF limiting behavior with increasing parti-\ncle numberN. The numerical result is confirmed by the ana-\nlytical formulas: The distribution (19) has a maximum around7\n0 1 2 3 4 50.50.60.70.80.91\ntime t[TTDL]relative population of\nm= 0 Zeeman state n0(t)\nFIG. 7: (color online). Evolution of the m= 0 population for 2000\nparticles in the transversely magnetized state for different coupling\nstrengths:K=\u00000:1(red),K=\u00001(green) andK=\u00002(blue).\nThe corresponding MF evolution is drawn as a dashed line.\nF=p\nNand a width proportional top\nN. Thus, the only\ncosine oscillation, which contributes to the dynamics (18) has\nzero frequency in the TDL, since\n2p\nNgs= 2c2\u001a=p\nN!0 ( forN!1 ):\n2. Time evolution for large K\nA perturbative calculation with the small parameter \u0015=\n1=(2K)leads to the first-order result\nn0\n0(t) =4(N\u00001)\nK2(2N\u00001)2cos(\n2qt\u0014\n1 +2N\u00003\nK(2N\u00001)\u0015)\n:\nAs in the case of two particles the amplitude drops down pro-\nportional to 1=K2[compare with Eq. (15)] and the frequency\nof the oscillation converges to 2q. Moreover, the amplitude is\nproportional to 1=Nand thus becomes zero in the TDL.\n3. Beyond-mean-field corrections in dependence of K\nWe finally analyze the corrections to the MF dynamics in\ndependence of the coupling parameter K. For that reason,\nwe have numerically calculated the fraction of atoms in the\nm=\u00061Zeeman states, n\u0006(t) = 1\u0000n0(t), after a fixed time\nt= 0:5TTDLas a function of (\u0000K). Fig. 6 shows a resonant\nenhancement of the m=\u00061population around K=\u00001. For\nN= 1000 the relative population of the m=\u00061Zeeman\nstates is largest and the resonance maximum is slightly below\njKj= 1. With increasing particle number Nthe resonance\nmaximum decreases and its position converges to K=\u00001.\nBeyond-MF corrections were analyzed in a similar way in\na recent experiment [22, 23] and explained within the Bo-\ngoliubov theory. Different from here, two resonances have\nbeen observed there, which is due to the restricted motion of\n0.6 0.7 0.8 0 .9 1 1 .1 1 .2 1 .3 1 .400.20.40.60.811.21.4oscillation period [ TTDL]MF limit\n2000\n300\n50\n20\n2\ncoupling parameter ( −K)FIG. 8: (color online). Oscillation period of the initial evolution in\ndependence of (\u0000K). For small and large jKj, theN-particle and\nMF period are in good agreement. In the vicinity of K=\u00001, the\nperiod is strongly enhanced in systems with large particle numbers.\nThe enhancement of the oscillation period is absent in small systems\nwith less than 20 particles.\nthe atoms to the ground state in our approach. However, on\nthe first resonance, the corresponding Bogoliubov mode has\na similar shape than the MF ground state so that our method\nis applicable there. Indeed, in a large trap, both approaches\nlead to the same value for the position of the first resonance,\nnamelyjq=(c2\u001a)j= 1.\nB. Transversely magnetized state\n1. Time evolution for three different values of K\nThe transversely magnetized state shows a rich dynamics\nin the MF limit [18, 34]. Fig. 7 shows the initial evolution of\nN= 2000 atoms for three different coupling strengths. We\ncompare the exact N-particle quantum dynamics (solid line)\nto the corresponding MF evolution (dashed line).\nAt smalljKj(red) one sees no difference between the two\nsolutions within the time 5TTDL. The evolution is an ordinary\ncosine oscillation with amplitude A=K=4and frequency\n!= 2c2\u001a(!periodTTDL=2), which is determined by the\nspin-dependent interaction energy (App. A).\nIn the opposite limit of large jKj, the MF evolution (blue\ndashed) is given by a cosine oscillation with amplitude A=\n1=(4K)and frequency != 2q[!periodTTDL=(2K)]. The\nexactN-particle quantum dynamics (blue solid) oscillates\nwith the same fast frequency of 2q. Moreover, the fast os-\ncillation is modulated by the envelope function of Eq. (17).\nThe beat note vanishes in the TDL, since\n\u0002\ncos(2gst)\u0003N\u00002\u00191\u00002(c2\u001at)2\nN:\nDespite the beat note, both oscillations coincide quite well for\ntimes, which are smaller than 2TTDL.8\n0 0 .2 0 .4 0 .6 0 .8 10.50.60.70.80.91\ntime t[TTDL]MF limit\n50\n200500\n2000100 1000relative population of\nm= 0 Zeeman state n0(t)\nFIG. 9: (color online). Initial evolution of the relative population\nof them= 0 Zeeman state on resonance (K=\u00001)for different\nparticle numbers Ncompared to the MF limit.\nAtK=\u00001, the relative population of the m= 0 Zeeman\nstate converges towards 1 in the MF limit (green dashed). The\namplitude becomes maximal with A= 1=4and the oscillation\nperiod diverges. The period of the exact N-particle quantum\nevolution (green solid) is enhanced, but still finite. Moreover,\nthe green solid curve does not converge towards 1, but rather\noscillates around 0.82. Note, that the evolution is coherent and\nexhibits a revival although the initial dynamics seems to be\nstrongly damped and to converge towards a constant value. At\nK=\u00001, one observes the largest deviations from the MF dy-\nnamics. The initial evolutions coincide only for times, which\nare smaller than TTDL=2. A similar observation was made in\nthe experiment [18, 19].\n2. Beyond-mean-field corrections in dependence of K\nFig. 8 shows plots of the oscillation period against the cou-\npling strength (\u0000K)for different particle numbers and the\nMF limit. In the MF limit, the oscillation period was obtained\nfrom the analytical solution (App. A), while in the other cases,\nthe period is 2 times the position of the first maximum of the\ninitial oscillation.\nOne sees that the N-particle oscillation period coincides\nquite well with the MF period in the limit of small and large\njKj, for all particle numbers. In the resonance region around\nK=\u00001, the deviations are quite large for small particle num-\nbers; see the curves for N= 2;20and 50. With increasing\nparticle number, the N-particle period coincides almost ev-\nerywhere with the MF limit, except in a small region around\nK=\u00001, which shrinks to zero in the TDL; see the curves\nforN= 300 and 2000. However, exactly on resonance at\nK=\u00001, the deviation from the MF limit is always infinitely\nlarge for finite particle numbers N.\nThe enhancement of the oscillation period is a typical fea-\nture of the spin systems with large particle numbers. One\nclearly sees the first occurence of a weak maximum for N=\n20, which becomes rather pronounced for N= 2000 . For\n1.5 2 2 .5 3 3 .5\nlog10(N)K=−1K= 1\n0.250.30.350.40.450.5position of first oscillation maximum [ TTDL ]FIG. 10: (color online). Position of the first oscillation maximum\nagainst the number of particles Non resonance (jKj= 1) .\nparticle numbers smaller than \u001920, the maximum is ab-\nsent. One further sees that the position of the resonance max-\nimum rapidly approaches 1 from below with increasing parti-\ncle numberN.\nThe enhancement of the oscillation period has been exper-\nimentally observed in an antiferromagnetic spin-287Rb BEC\n[18] and, for a similar initial state, in a spin-123Na BEC [19].\nThe large differences between the exact N-particle quantum\ndynamics and the MF evolution close to the critical coupling\nstrength atjKj= 1 due to finite- Ncorrections may be rele-\nvant for studies of quantum chaos [38, 39].\n3. Limit for the validity of the mean-field dynamics\nWe conclude from the previous discussion, that the conver-\ngence to the MF limiting dynamics in systems with a finite\nnumber of particles is slowest at jKj= 1; see Figs. 6–8.\nHence, we study now, how fast the oscillation period con-\nverges towards infinity at the critical value of jKj= 1, since\nthis worst-case scenario provides us with the smallest upper\nbound for the validity of the MF approximation.\nFig. 9 shows the initial evolution of the relative population\nof them= 0 Zeeman state at K=\u00001forN= 50\u00002000\nparticles. As expected, the initial evolution coincides with the\nMF dynamics for longer times, when the particle number is\nincreased. One sees that the N-particle quantum dynamics is\nclose to the MF evolution for times, which are smaller than\nthe position of the first oscillation maximum.\nHence, we plot in Fig. 10 the position of the first oscilla-\ntion maximum against the number of particles on a double\nlogarithmic scale. One finds a logarithmic dependency of the\nvalidity time TK(N)of the MF approximation on the number\nof particles\nTK=\u00001(N) =TTDL[0:13 + 0:09 log10(N)]\nTK=1(N) =TTDL[0:18 + 0:09 log10(N)]:\nThus, for a spin-1 BEC of 106atoms and a negative coupling9\nparameter,K < 0, one finds the MF dynamics to be valid for\ntimes, which are smaller than \u00190:67TTDL.\nV . CONCLUSIONS AND OUTLOOK\nWe studied the quantum spin dynamics of ultracold spin-1\natoms within the single-mode approximation. This was done\non the basis of an exact diagonalization of the effective many-\nbody spin Hamiltonian [9]. The chosen method allowed us\nto discuss the crossover from few atoms to small condensates\nwithin one framework.\nOur numerical calculations showed convergence of the\nquantum spin dynamics towards the mean-field dynamics\nin the thermodynamic limit for all strengths of the spin-\ndependent interaction. Moreover we showed that quantum\ncorrections to the mean-field dynamics are particularly large\nat the critical value of the coupling parameter, where the spin-\nchanging collisional energy equals the quadratic Zeeman en-\nergy. For the state, where all the atoms are in the m= 0sub-\nlevel, we compared the initial quantum spin dynamics to re-\nsults obtained from a Bogoliubov approximation [22, 23, 29].\nFor the transversely magnetized state we estimated the valid-\nity time of the initial mean-field dynamics at the critical value\nof the coupling parameter, which grows logarithmically with\nthe number of particles. From this estimate we conclude that\nquantum corrections to the mean-field dynamics may play an\nimportant role in experiments with spinor BECs. It would be\ninteresting to study systematically the dependency of the spin\ndynamics on the particle number in future experiments.\nQuantum corrections to the mean-field spin dynamics are\nparticularly large in the regime of few atoms. Here, many-\nbody spin correlations lead to a beat-note phenomenon at large\nmagnetic fields. The regime of few atoms (N\u001810)should\nbe accessible in deep optical lattices.\nOur results may be relevant for the study of quantum chaos.\nIt was shown [38] that the strong nonlinear behavior of the\nmean-field equations at the critical value of the coupling pa-\nrameter leads to classical chaos in the spin dynamics of spin-2\natoms. But exactly at this point, the finite- Nquantum dynam-\nics largely deviates from the classical limit. It would be inter-\nesting to study this aspect systematically, similar to the work\nin Ref. [39] on a periodically driven double-well system.\nAcknowledgments\nThe authors wish to thank K. Sengstock, K. Bongs and\nF. Lechermann for fruitful discussions. F. D. acknowledges\nfunding by the DFG (SFB407, QUEST) and the ESF (EU-\nROQUASAR), and is grateful for the hospitality of S. M.\nReimann at the Division of Mathematical Physics, LTH, Lund\nUniversity, Sweden. This work has been performed within\nthe Excellence Cluster Frontiers in Quantum Photon science,\nwhich is supported by the Joachim Hertz Stiftung.Appendix A: Mean-field dynamics\nOne assumes in the MF approach, that the system is in a\nproduct state of the form\n\u0000\n\u000b+j+i+\u000b0j0i+\u000b\u0000j\u0000i\u0001\nN(A1)\nwith\u000bmbeing complex numbers, which are normalized ac-\ncording toP\nmj\u000bmj2= 1. For the states (A1) and the Hamil-\ntonian (1), one derives the MF equations of motion [34]\ni@t\u000b+=gsN\u0010\nA\u0003\u000b0+hfzi\u000b+\u0011\n+q\u000b+\ni@t\u000b0=gsN\u0010\nA\u000b++A\u0003\u000b\u0000\u0011\n(A2)\ni@t\u000b\u0000=gsN\u0010\nA\u000b0\u0000hfzi\u000b\u0000\u0011\n+q\u000b\u0000;\nwhere we have defined A=hf+i=p\n2 = (\u000b\u0003\n+\u000b0+\u000b\u0003\n0\u000b\u0000)\nandhfzi= (j\u000b+j2\u0000j\u000b\u0000j2).\nThe number state j0;N;0i=j0i\nNis a steady state of\n(A2), since\n\u000b0(t) = 1; \u000b\u0006(t) = 0\nis a solution of the MF equations of motion (A2) for all values\nofgsandq. Thus, this state shows no population dynamics in\nthe MF limit and n0(t) =j\u000b0(t)j2= 1.\nThe time evolution of the transversely magnetized state\nj\u0010y\nNi=\u0000\n\u00001=2j+i\u0000 i=p\n2j0i+ 1=2j\u0000i\u0001\nN\nis calculated in Refs. [34, 35]. In this state, all the spins are\npointing into the positive y-direction. The solution of the MF\nEqs. (A2) is given in terms of Jacobi elliptic functions [40]\n\u000b\u0006(t) =\u0007s\n2\"\ncnk(qt\n2)dnk(qt\n2)\n1\u0000ksn2\nk(qt\n2)\u0000i(1 +k)snk(qt\n2)\n1 +ksn2\nk(qt\n2)#\n;\n\u000b0(t) =sp\n2\"\n(1\u0000k)snk(qt\n2)\n1\u0000ksn2\nk(qt\n2)\u0000icnk(qt\n2)dnk(qt\n2)\n1 +ksn2\nk(qt\n2)#\n;\nwheres= exp(\u0000 i(gsN\u0000q)t=2)andk= 1=K. For the spin\npopulations nm(t) =j\u000bm(t)j2the solution simplifies to\nn0(t) =\u0002\n1\u0000ksn2\nk(qt)\u0003\n=2; (A3)\nn\u0006(t) =\u0002\n1 +ksn2\nk(qt)\u0003\n=4:\nThese solutions are also valid for the initial state j\u0010Ni, which\nis used here, since the evolution of the relative populations nm\nis unaffected by rotations around the z-axis.\nFor smallk= 1=K, one can approximate sn k(x)\u0019sin(x)\nand Eq. (A3) becomes\nn0(t)\u0019\u00121\n2\u00001\n4K\u0013\n+1\n4Kcos(2qt) ( largeK):\nThat means, the evolution is a cosine oscillation with ampli-\ntudeA= 1=(4K)and frequency != 2q.10\nFor largek= 1=K, we approximate sn k(x)\u0019sin(kx)=k,\nwhich leads to\nn0(t)\u0019\u00121\n2\u0000K\n4\u0013\n+K\n4cos(2c2\u001at) ( smallK):\nThat means, in the interaction dominated regime, the oscilla-\ntion amplitude is A=K=4and the frequency is != 2c2\u001a.\nAtjKj= 1, the evolution becomes aperiodic and the rel-\nativem= 0 population asymptotically approaches 1, i. e.\nn0(t)!1fort!1 . Here, the dynamics exhibits a maxi-\nmum of the amplitude and the oscillation period diverges.\nAppendix B: Quantum dynamics of the number state at zero K\nIn the following, we derive the population dynamics of the\nnumber state at zero K. In this limiting regime, the Hamilto-\nnian consists only of the interaction Hs, which has the eigen-\nbasisfjF;Mig. The occupation number operator N0and the\ninitial statej\u0012Nineed to be expressed in this basis to calculate\nthe dynamics. The expansion of j\u0012Niis given by\nj\u0012Ni=j0;N;0i=NX\nF=Fmin;\u0001F=2\u001fFjF;0i: (B1)\nSincej\u0012Nihas theFzeigenvalueM= 0, it is a superposition\nof the statesjF;M = 0i. Due to symmetry reasons, the sum-\nmation runs over F=Fmin;Fmin+ 2;:::;N withFmin= 0\nor 1 ifNis even or odd, respectively [33]. The coefficients\n\u001fFare determined later. By inserting the expansion (B1) into\nEq. (8) the evolution becomes\nn0(t) =X\nF;F0\u001fF\u001fF0hF;0jN0jF0;0icos\u0002\n(!F\u0000!F0)t\u0003\n=N\n(B2)\nwith the frequencies !F=gs\u0002\nF(F+ 1)\u00002N\u0003\n. The matrix\nelementshF;0jN0jF0;0iare calculated in the following.\nThe occupation number operator N0can be written in terms\nof spherical tensor operators\nN0=1\n3N\u0000r\n2\n3T(2)\n0;\nwhereT(2)\n0is the zeroth component of the one-particle spher-\nical tensor operator T(2)\nqof rank 2. Its five components are\nT(2)\n\u00062=ay\n\u0006a\u0007 (B3)\nT(2)\n\u00061=1p\n2\u0010\nay\n0a\u0007\u0000ay\n\u0006a0\u0011\nT(2)\n0=1p\n6\u0010\nay\n+a+\u00002ay\n0a0+ay\n\u0000a\u0000\u0011\n: (B4)\nNis proportional to the identity matrix and thus its matrix\nelements are\nhF;0jNjF0;0i=N\u000eFF0:From the Wigner-Eckart theorem one finds, that the matrix\nelementshF0;M0jT(2)\n0jF;Mivanish forjF\u0000F0j>2. The\nnonzero matrix elements of T(2)\n0are\nhF;0jT(2)\n0jF;0iandhF+ 2;0jT(2)\n0jF;0i:\nThe Wigner-Eckart theorem allows one to calculate the matrix\nelements of T(2)\n0from a special class of matrix elements\nhF+q;0jT(2)\n0jF;0i=hF;2; 0;0jF+q;0i\nhF;2;\u0000F;\u0000qjF+q;\u0000F\u0000qi\n\u0002hF+q;\u0000F\u0000qjT(2)\n\u0000qjF;\u0000Fi (B5)\nforq= 0;2. The bracketshf1;f2;m1;m2jf0;m0iare the\nClebsch-Gordan coefficients (CGC). Eq. (B5) simplifies the\ncalculation, since the states jF;\u0000Fihave a rather simple rep-\nresentation in the occupation number basis\njF;\u0000Fi=cF\u0000\nay\n\u0000\u0001Fh\nay\n+ay\n\u0000\u0000\u0000\nay\n0\u00012=2iN\u0000F\n2j0;0;0i(B6)\nwith the normalization constant\n1\nc2\nF=N\u0000F\n2X\nk=0\u0012N\u0000F\n2\nk\u00132k!(k+F)!(N\u0000F\u00002k)!\n2N\u0000F\u00002k:(B7)\nAfter inserting Eqs. (B3), (B4), (B6) and the required CGCs\ninto Eq. (B5), a lengthy calculation leads to the nonzero ma-\ntrix elements\nhF;0jT(2)\n0jF;0i=1p\n6\u0012\n3F\u00002N+ 6c2\nF\nd2\nF\u0013F+ 1\n2F\u00001(B8)\nand\nhF+ 2;0jT(2)\n0jF;0i=\nr\n3\n2cF\ncF+2N\u0000F\n2s\n(F+ 1)(F+ 2)\n(2F+ 1)(2F+ 3); (B9)\nwhere\n1\nd2\nF=N\u0000F\n2X\nk=0k\u0012N\u0000F\n2\nk\u00132k!(k+F)!(N\u0000F\u00002k)!\n2N\u0000F\u00002k:(B10)\nThe coefficients \u001fFof the expansion (B1) can be calculated\nby means of Eq. (B6). Using\njF;0i= 1=p\n(2F)!\u0000\nF+\u0001FjF;\u0000Fi (B11)\nwe obtain\n\u001fF=hF;0j0;N;0i=hF;\u0000Fj\u0000\nF\u0000\u0001Fj0;N;0i=p\n(2F)!\n=cF\u0012\n\u00001\n2\u0013N\u00002F\n2s\nN!\u00122F\nF\u0013\u00001\n: (B12)11\n0 50 100 150 20000.0040.0080.012\ntotal spin Fcoefficient CF\nFIG. 11: (color online). Exact amplitudes CF[(B13), blue crosses],\nof 2000 atoms compared to the approximation [(B24), red line].\nThe evolution is obtained by inserting Eqs. (B8)–(B12) into\nEq. (B2). Only terms with jF\u0000F0j= 2lead to non-constant\ncontributions to the evolution:\nn0(t)0=N\u00002X\nF=Fmin;\u0001F=2CFcos\u0002\ngs(2F+ 3)t\u0003\n;\nwith frequencies gs(2F+ 3) =!F+2\u0000!Fand amplitudes\nCF= (N\u0000F)c2\nF(N\u00001)!\u00101\n2\u0011N\u00002F\n\u0002s\u00122F\nF\u0013\u00001\u00122F+ 4\nF+ 2\u0013\u00001(F+ 1)(F+ 2)\n(2F+ 1)(2F+ 3):(B13)\nThe amplitudes (B13) can be approximated for large N.\nThe most involved part is the approximation of cF. The fac-\ntorials in (B7) can be written as binomial coefficients.\nk!(k+F)!(N\u0000F\u00002k)! =N!\u0012\u0012N\n#\u0013\u0012#\n#\u0000F\n2\u0013\u0013\u00001\n;\nwhere#= 2k+F. We obtain from (B7)\n1\nc2\nF=NX\n#=F;\u0001#=2N!\n2N\u0000#\u0012N\u0000F\n2\n#\u0000F\n2\u00132\u0012\u0012N\n#\u0013\u0012#\n#\u0000F\n2\u0013\u0013\u00001\n:\n(B14)\nThe terms with #\u0019Ndominate the sum in (B14). The ap-\nproximation\n1\n2n\u0012n\nk\u0013\n\u00191p\n2\u0019p\nn=4exp\u0014\n\u00001\n2(k\u0000n=2)2\nn=4\u0015\n(B15)\nholds for large n. Applying this to\u0000#\n(#\u0000F)=2\u0001\ngives\n1\nc2\nF\u0019NX\n#=FN!\n2Nr\n\u0019#\n2\u0012N\u0000F\n2\n#\u0000F\n2\u00132\u0012N\n#\u0013\u00001\nexp\u0012F2\n2#\u0013\n:\n(B16)\nFurther, we approximate #=Nin the exponential and the\nsquare root of (B16). The exponential dependence on F2shows, that cFis negligible for F\u001dN. Thus, we assume\nF\u001cNin the following. The binomials in (B16) are ex-\npanded into factorials and approximated using (n\u0000k)!\u0019\nn!=nkforn\u001dk:\n\u0012N\u0000F\n2\n#\u0000F\n2\u00132\u0012N\n#\u0013\u00001\n\u0019\u0012N\u0000F\n2N\u0013N\u0000#\u0012N\u0000#\nN\u0000#\n2\u0013\n:(B17)\nWe apply (B15) to the right-hand side of (B17)\n\u00101\n2\u0011N\u0000#\u0012N\u0000#\nN\u0000#\n2\u0013\n\u0019s\n2\n\u0019(N\u0000#): (B18)\nFurther, forF\u001cN, one gets to first order\n\u0010N\u0000F\nN\u0011N\u0000#\n\u0019exp\u0012\n\u0000F(N\u0000#)\nN\u0013\n: (B19)\nWe insert the approximations (B17)–(B19) into (B16) and ap-\nproximate the sum by an integral\n1\nc2\nF\u0019e\u0000FeF2=(2N)N!\n2N1\n2ZN\nFd#r\nN\nN\u0000#eF#\nN;\nwhere an additional factor 1=2accounts for the step size of 2\nin the sum. The integral has the value\nZN\nFd#r\nN\nN\u0000#eF#\nN=\u0000Nr\u0019\nFeFerf r\nF(N\u0000#)\nN!\n(B20)\nwith the error function erf (x). Inserting the upper limit Nof\nthe integral into (B20) leads to erf (0) = 0 . WithF\u001cN, the\nlower limitFgives erf (p\nF)\u00191forF\u00155and thus\n1\nc2\nF\u0019N!\n2Np\u0019\n2Np\nFexp\u00121\n2F2\nN\u0013\n: (B21)\nThe maximum of (B21) is at F=p\nN=2. The Gaussian\nexp(\u0000F2=N)has a width ofp\nN=2, which is an upper bound\nfor the width of cF. Now we approximate the other terms in\n(B13). Since F\u0019p\nNfor the dominant contributions,\ns\u00122F\nF\u0013\u00122F+ 4\nF+ 2\u0013\n\u0019\u00122F\nF\u0013\n\u001922Fr\n1\n\u0019F; (B22)\nwhere we used (B15) in the last step. For 1\u001cF\u001cN, the\nremaining factors become\nN\u0000F\nN\u0001s\n(F+ 1)(F+ 2)\n(2F+ 1)(2F+ 3)\u00191\n2: (B23)\nInserting (B21)–(B23) into (B13) leads to\nCF\u0019F\nNexp\u0012\n\u00001\n2F2\nN\u0013\n: (B24)\nThe approximation (B24) and the exact amplitude (B13) are\ncompared for 2000 atoms in Fig. 11.12\n[1] J. Stenger, S. Inouye, D. M. Stamper-Kurn, H.-J. Miesner, A. P.\nChikkatur, and W. Ketterle, Nature 396, 345 (1998).\n[2] L. E. Sadler, J. M. Higbie, S. R. Leslie, M. Vengalattore, and D.\nM. Stamper-Kurn, Nature 443, 312 (2006).\n[3] M. Vengalattore, S. R. Leslie, J. Guzman, and D. M. Stamper-\nKurn, Phys. Rev. Lett. 100, 170403 (2008).\n[4] M. Vengalattore, J. Guzman, S. R. Leslie, F. Serwane, and D.\nM. Stamper-Kurn, arXiv:0901.3800.\n[5] J. Kronj ¨ager, C. Becker, P. Soltan-Panahi, K. Bongs, and K.\nSengstock, arXiv:0904.2339.\n[6] A. Lamacraft, Phys. Rev. Lett. 98, 160404 (2007).\n[7] T.-L. Ho, Phys. Rev. Lett. 81, 742 (1998).\n[8] T. Ohmi and K. Machida, J. Phys. Soc. Jpn. 67, 1822 (1998).\n[9] C. K. Law, H. Pu, and N. P. Bigelow, Phys. Rev. Lett. 81, 5257\n(1998).\n[10] M. Koashi and M. Ueda, Phys. Rev. Lett. 84, 1066 (2000).\n[11] H. Schmaljohann, M. Erhard, J. Kronj ¨ager, M. Kottke, S. van\nStaa, L. Cacciapuoti, J. J. Arlt, K. Bongs, and K. Sengstock,\nPhys. Rev. Lett. 92, 040402 (2004).\n[12] M.-S. Chang, C. D. Hamley, M. D. Barrett, J. A. Sauer, K. M.\nFortier, W. Zhang, L. You, and M. S. Chapman, Phys. Rev. Lett.\n92, 140403 (2004).\n[13] T. Kuwamoto, K. Araki,T. Eno, and T. Hirano, Phys. Rev. A 69,\n063604 (2004).\n[14] A. Widera, F. Gerbier, S. F ¨olling, T. Gericke, O. Mandel, and I.\nBloch, Phys. Rev. Lett. 95, 190405 (2005).\n[15] A. Widera, F. Gerbier, S. F ¨olling, T. Gericke, O. Mandel, and I.\nBloch, New. J. Phys. 8, 152 (2006).\n[16] M.-S. Chang, Q. Qin, W. Zhang, L. You, and M. S. Chapman,\nNature Phys. 1, 111 (2005).\n[17] W. Zhang, D. L. Zhou, M.-S. Chang, M. S. Chapman, and L.\nYou, Phys. Rev. A 72, 013602 (2005).\n[18] J. Kronj ¨ager, C. Becker, P. Navez, K. Bongs, and K. Sengstock,\nPhys. Rev. Lett. 97, 110404 (2006).\n[19] A. T. Black, E. Gomez, L. D. Turner, S. Jung, and P. D. Lett,\nPhys. Rev. Lett. 99, 070403 (2007).\n[20] E. V . Goldstein and P. Meystre, Phys. Rev. A 59, 3896 (1999).\n[21] S. R. Leslie, J. Guzman, M. Vengalattore, J. D. Sau, M. L. Co-\nhen, and D. M. Stamper-Kurn, Phys. Rev. A 79, 043631 (2009).[22] C. Klempt, O. Topic, G. Gebreyesus, M. Scherer, T. Henninger,\nP. Hyllus, W. Ertmer, L. Santos, and J. J. Arlt, Phys. Rev. Lett.\n103, 195302 (2009).\n[23] C. Klempt, O. Topic, G. Gebreyesus, M. Scherer, T. Henninger,\nP. Hyllus, W. Ertmer, L. Santos, and J. J. Arlt, Phys. Rev. Lett.\n104, 195303 (2010).\n[24] A. Sørensen, L.-M. Duan, J. I. Cirac, and P. Zoller, Nature 409,\n63 (2001).\n[25] H. Pu and P. Meystre, Phys. Rev. Lett. 85, 3987 (2000).\n[26] L.-M. Duan, A. Sørensen, J. I. Cirac, and P. Zoller, Phys. Rev.\nLett. 85, 3991 (2000).\n[27] L.-M. Duan, J. I. Cirac, and P. Zoller, Phys. Rev. A 65, 033619,\n(2002).\n[28] ¨O. E. M ¨ustecaplıo ˘glu, M. Zhang, and L. You, Phys. Rev. A 66,\n033611 (2002).\n[29] X. Cui, Y . Wang, and F. Zhou, Phys. Rev. A 78, 050701(R)\n(2008).\n[30] E. H. Lieb, R. Seiringer, and J. Yngvason, Phys. Rev. A 61,\n043602 (2000).\n[31] L. Erd ˝os, B. Schlein, and H.-T. Yau, Phys. Rev. Lett. 98, 040404\n(2007).\n[32] Z. Chen, C. Bao, and Z. Li, J. Phys. Soc. Jpn. 78, 114002\n(2009).\n[33] E. Eisenberg and E. H. Lieb, Phys. Rev. Lett. 89, 220403\n(2002).\n[34] J. Kronj ¨ager, C. Becker, M. Brinkmann, R. Walser, P. Navez,\nK. Bongs, and K. Sengstock, Phys. Rev. A 72, 063619 (2005).\n[35] J. Kronj ¨ager, Ph.D. thesis, University of Hamburg (2007); avail-\nable online: urn:nbn:de:gbv:18-34281.\n[36] L. G. Yaffe, Rev. Mod. Phys. 54, 407 (1982).\n[37] J. Heinze, diploma thesis, University of Hamburg (2009); avail-\nable online.\n[38] J. Kronj ¨ager, K. Sengstock, and K. Bongs, New. J. Phys. 10,\n045028 (2008).\n[39] C. Weiss and N. Teichmann, Phys. Rev. Lett. 100, 140408\n(2008).\n[40] I. N. Bronstein, K. A. Semendjajew, G. Musiol, and H. M ¨uhlig,\nTaschenbuch der Mathematik (Verlag Harri Deutsch, 2001)."    },    {        "title": "1106.4389v2.Effect_of_spin_diffusion_on_current_generated_by_spin_motive_force.pdf",        "content": "arXiv:1106.4389v2  [cond-mat.mes-hall]  21 Aug 2011Effect of spin diffusion on current generated by spin motive fo rce\nKyoung-Whan Kim1, Jung-Hwan Moon2, Kyung-Jin Lee2, and Hyun-Woo Lee1\n1PCTP and Department of Physics, Pohang University of Scienc e and Technology, Pohang, 790-784, Korea\n2Department of Materials Science and Engineering, Korea Uni versity, Seoul 136-701, Korea\n(Dated: November 16, 2018)\nSpin motive force is a spin-dependent force on conduction el ectrons induced by magnetization\ndynamics. In order to examine its effects on magnetization dy namics, it is indispensable to take\ninto account spin accumulation, spin diffusion, and spin-fli p scattering since the spin motive force is\nin general nonuniform. We examine the effects of all these on t he way the spin motive force generates\nthechargeandspincurrentsinconventionalsituations, wh ere theconductionelectron spinrelaxation\ndynamics is much faster than the magnetization dynamics. Wh en the spin-dependent electric field\nis spatially localized, which is common in experimental sit uations, we find that the conservative part\nof the spin motive force is unable to generate the charge curr ent due to the cancelation effect by the\ndiffusion current. We also find that the spin current is a nonlo cal function of the spin motive force\nand can be effectively expressed in terms of nonlocal Gilbert damping tensor. It turns out that any\nspin independent potential such as Coulomb potential does n ot affect our principal results. At the\nlast part of this paper, we apply our theory to current-induc ed domain wall motion.\nPACS numbers:\nI. INTRODUCTION\nIn a ferromagnetic system, dynamics of space-time de-\npendent magnetizationvector M(x,t) is describedby the\nfollowing phenomenological equation [1–4]\n∂M\n∂t=−γM×Heff+α\nMsM×∂M\n∂t\n+µB\neMs(js·∇)M−βµB\neM2sM×(js·∇)M.(1)\nThis is the Landau-Lifshitz-Gilbert (LLG) equation gen-\neralized to include the spin-transfer torque terms (last\ntwo terms). Here, Heffis the functional derivative of en-\nergy with respect to M,αis Gilbert damping constant,\nMsis saturation magnetization, µBis Bohr magneton,\neis electron charge, βis nonadiabaticity, and jsis spin\npolarizedelectriccurrentgivenbydifferencebetweencur-\nrent by spin-up and spin-down electrons. LLG equation\ndescribes the dynamics of magnetization under applied\nelectromagnetic fields. On the other hand, there exists\na reciprocal process; temporal and spatial variation of\nmagnetization induces additional electromagnetic fields\non conduction electrons. These fields are spin-dependent\nand in general nonconservative. Thus the resulting spin-\ndependentmotiveforceiscalledspinmotiveforce(SMF).\nSMF is firstly predicted by Berger [5] and recently for-\nmulated by generalizing Faraday’s law [6]. It is also sug-\ngested [7] that SMF is also described by spin pumping\neffect[8]. Recently, SMFanditseffectisintensivelystud-\nied [7, 9–20] in this field.\nWithout any other perturbation, the explicit expres-\nsionsoftheinducedspinelectromagneticfieldsareknownas [7, 10–12, 19–21]\nE↑↓\ni=±/planckover2pi1\n2eM3s(∂tM×∂iM)·M, (2)\nB↑↓\ni=∓/planckover2pi1\n2eM3sǫijk(∂jM×∂kM)·M,(3)\nwhere↑and↓stand for spin-up and down electrons.\nThe fields generate spin-dependent Lorentz force F↑↓\ns=\ne(E↑↓\ni+v↑↓×B↑↓\ns). While the term “motive force” refers\ntoquantitiesofvoltagedimension,thetermSMFissome-\ntimes used to denote F↑↓\ns. In this paper, we adopt the\nlatter terminology. It is easily noticed that SMF is spin\ndependent, nonconservative, spatially varying, and local-\nized in general situations. As one can see from Eqs. (2)\nand (3), SMF is usually too small to be measured di-\nrectly. Another way is to study the effect of SMF on\nmagnetization dynamics. Since SMF induces additional\nspin current, additional spin-transfer torque arises and\nit changes LLG equation. Consequently, LLG equation\n(without applied spin current) is modified as [7]\n∂M\n∂t=−γM×Heff+1\nMsM×/parenleftbigg\nD·∂M\n∂t/parenrightbigg\n,(4)\nwhereDis modified damping tensor given by\nDij=αδij+η\nM4s/summationdisplay\nk(M×∂kM)i(M×∂kM)j,(5)\nandη=µB/planckover2pi1σ/2e2Ms. Here,σis electrical conductivity.\nHowever, the previous work has a crucial limitation\nthat spin density has been considered to be constant\nwhile spin density in reality is nonuniform since SMF is\nspatially varying. The main consequence of the nonuni-\nform spin density n↑↓(x,t) is diffusion current propor-\ntional to −∇n↑↓, which suppresses the effect of SMF.\nTherefore, a realistic model should take into account2\nspin accumulation (nonuniform spin density), diffusion\nand spin-flip scattering. The purpose of this paper is to\nfind the spin density n↑↓, diffusion current, total current\ninduced by SMF, and their effect on magnetization dy-\nnamics in the presence of spin accumulation, diffusion,\nand spin-flip scattering. As one shall see in Sec. III, the\nsolution of n↑↓in the most general situation is too com-\nplicated to study the effect on magnetization dynamics.\nTo obtain simple analytic expressions, we take an ap-\nproximation that spin-flip time is much shorter than the\ntime scale of magnetization dynamics. As a final com-\nment, our result does not assume any specific form of\nSMF. Thus, it remains valid for the modified SMF due\nto, for instance, nonadiabaticity [10], spin-orbit coupling\n[20], and other kinds of spin dependent electric field [22].\nSeveral previous works are closely related to our work.\nSpin drift-diffusion equation, which has similar form to\nour theory is suggested in Ref. [10]. And, the effect on\nspin and charge current is investigated from Boltzmann\nequation in Ref. [19]. We set our starting point as the\nequation of motion of conduction electrons in Ref. [1]\nto make our analysis consistent with previous theories in\nthis field. Different from the previous theories focusing\non 1D, we successfully generalized our result to 3D, and\nfound that nonconservative part of SMF plays a crucial\nrole in current in a higher dimensional system. In Sec.\nIII, we compare our result with the previous theory qual-\nitatively and quantitatively. In addition, we investigated\nthe effect of charge neutrality on our results. It turns\nout that charge neutrality potential does not change our\nprincipal results, charge current and spin current, even\nthough it changes charge density and spin density. Fur-\nthermore, we show that any spin independent potential\ncannot alter our principal results, either.\nThis paper is organized as follows. In Sec. II, we\nconstruct the spin drift-diffusion equation and introduce\nvariables. Then, we solve the equation in Sec. III, and\ndiscuss various implications. In Sec. IV, we apply our\nresult to current-induced domain wall (DW) motion and\nbriefly discuss the effect of spin diffusion. In Sec. V, we\ngeneralize our theory for general boundary condition and\nfor general spin indendendent potentials. Finally, in Sec.\nVI, there are concluding remarks. Technical details are\nin Appendices.\nII. MODEL\nA. Spin drift-diffusion equation\nTo construct the equation of n↑↓(x,t), we take the\nstarting point as the equation of spin density mof con-\nduction electrons [1],\n∂m\n∂t+∇·J=−1\nτexMsm×M−∝angb∇acketleftΓ∝angb∇acket∇ight.(6)\nHere,Jis spin current tensor, τex=/planckover2pi1/SJex, andSde-\nnotes the magnitude of spin of local magnetization. Theleft-hand side is based on the continuity equation. The\nfirst term on the right-hand side is the precession term\ndue to the exchange coupling between conduction elec-\ntrons and magnetization. ∝angb∇acketleftΓ∝angb∇acket∇ightincludes the effect of spin\nscattering processes. Here, the second rank tensor Jis\ndefined by\nJ=−µB\nejs⊗m\n|m|. (7)\nThe effect of the perpendicular component to Mof Eq.\n(6) is already investigated by Zhang and Li [1], and they\nfound the nonadiabatic term of LLG equation. In the\nabsence of spin accumulation, the magnitude of mis\nconstant, so it suffices to solve only the perpendicular\ncomponent of the equation. However, in the presence\nof the spin accumulation, the magnitude variation of m\nshould be also studied. We define spin number density\nns≡ |m|/µB. Taking care of the fact that nshas space-\ntime dependence, the parallel component of Eq. (6) to\nˆm≡m/|m|results in\n∂ns\n∂t+1\nµBˆm·∝angb∇acketleftΓ∝angb∇acket∇ight=1\ne∇·js. (8)\nIt is convenient to separate the variables to that of up\nand down electrons. ns=n↑−n↓andjs=j↑−j↓. Here,\nn↑↓andj↑↓denote spin number density of spin-up/down\nelectrons and charge current density generated by spin-\nup/down electrons, respectively. Equation (8) is nothing\nbut the continuity equation containing spin nonconserv-\ning processes described by Γ. To obtain independent\nequations of spin-up/down electrons, we use the follow-\ning continuity equation of total electron number density\n∂ne\n∂t=1\ne∇·je, (9)\nwherene=n↑+n↓is electron number density and je=\nj↑+j↓is charge current density. Combining Eqs. (8) and\n(9), one obtains\n∂n↑↓\n∂t±1\n2µBˆm·∝angb∇acketleftΓ∝angb∇acket∇ight=1\ne∇·j↑↓. (10)\nNote that Γrepresents spin-flip scattering processes. As\na simple model, we take the well-known form of spin-flip\nscattering,\n1\n2µBˆm·∝angb∇acketleftΓ∝angb∇acket∇ight=n↑\nτ↑−n↓\nτ↓, (11)\nwhereτ↑is characteristic time of the spin-flip scattering\nprocess from spin-up to -down state, and τ↓is similarly\ndefined. Then,\n∂n↑↓\n∂t+n↑↓\nτ↑↓−n↓↑\nτ↓↑=1\ne∇·j↑↓, (12)\nwhich is the spin drift-diffusion equation. Similar form\nof Eq. (12) was also suggested in Ref. [10].3\nFor simplicity, we may assume without losing general-\nitythat the SMF isturned onat t= 0 andthat, for t <0,\nthe system is in equilibrium. We set n↑↓(x,t= 0) =n↑↓\n0,\nwheren↑↓\n0is equilibrium electron density of spin up and\ndown at t <0. By definition, the equilibrium density\nn↑↓\n0is the equilibrium solution of Eq. (12) for t <0.\nInserting n↑↓(x,t <0) =n↑↓\n0to Eq. (12), one ob-\ntains an important constraint n↑\n0/τ↑=n↓\n0/τ↓. With\nthe help of this constraint, four variables n↑\n0,n↓\n0,τ↑and\nτ↓can be described by three variables, n↑\n0,n↓\n0, andτsf\n(τ−1\nsf=τ↑−1+τ↓−1). Then, Eq. (12) is rewritten with\nonly one spin-flip time τsf. In addition, current j↑↓can\nbe written as σ↑↓E↑↓\ns+eD↑↓∇n↑↓, whereσ↑↓andE↑↓\nsare\nrespectively the conductivity and SMF (divided by e) for\nspin-up/down electrons. Then, one straightforwardlyob-\ntains the final form of the equation of our model.\n∂n↑↓\n∂t−D↑↓∇2n↑↓+n↑\n0n↓\n0\nn↑\n0+n↓\n01\nτsf/parenleftigg\nn↑↓\nn↑↓\n0−n↓↑\nn↓↑\n0/parenrightigg\n=σ↑↓\ne∇·E↑↓\ns. (13)\nAs mentioned in Sec. I, we treat E↑↓\nsas nonconserva-\ntive, spatially varying fields. In addition, it is assumed\nthat spin dependence of Esis given by E↑\ns=−E↓\ns≡Es.\nSlight generalization of our theory at the final step al-\nlows to investigate the formula for E↑\ns∝negationslash=−E↓\ns. No other\nrestriction of E↑↓\nsis not assumed in order to obtain max-\nimally generalized result.\nAs suggested in Ref. [10], in realistic systems,\nthe Coulomb interaction should be taken into account.\nHence, oneintroducesCoulombpotential Vcandadditto\nthe spin motive force as E↑↓\ns→E↑↓\ns−∇Vc. The Coulomb\ninteraction strongly suppresses the charge accumulation.\nMathematically the interaction may thus be handled by\nimposing the charge neutrality constraint. We show in\nSec. V that chargeneutrality constraintchanges electron\ndensities, but not currents. Hence, the LLG equation is\nhardly affected by charge neutrality potential. For this\nreason, we do not take into account the Coulomb inter-\naction until Sec. V in order to show simple logical flow.\nNote that all variables in Eq. (13) are not indepen-\ndent. Einstein’s relation is given by σ↑↓=e2D↑↓N↑↓\nwhereN↑↓is density of states of spin-up/down elec-\ntrons at Fermi energy. Since N↑↓∝n↑↓\n0, one obtains\nσ↑/D↑n↑\n0=σ↓/D↓n↓\n0. This is one of the key constraints\nof our model.\nThe solution of Eq. (13) is very complicated as one\nshall see in Sec. III. To gain an insight, it is illustrative\nto assume that τsfis much smaller than the time scale\nof magnetization dynamics so Esis almost constant in\ntime scale within τsf. We found that, in this limit, the\nsolution is much simpler and it is easier to catch physical\nmeanings.B. Variable definitions and relations\nIn Sec. III, there appear several variables and quan-\ntities which have not been defined yet. To help readers,\nwe present definitions of them here, rather than Sec. III.\nSince Eq. (13) is coupled, it is convenient to solve it in\nmatrix form. Hence, we define spin accumulation vector,\nwhich is a column vector defined by\nN=/parenleftbigg\nn↑\nn↓/parenrightbigg\n. (14)\nSimilarly, we define current density and SMF vector.\nJ=/parenleftbigg\nj↑\nj↓/parenrightbigg\n, (15)\nE=/parenleftbigg\nE↑\ns\nE↓\ns/parenrightbigg\n=Es/parenleftbigg\n1\n−1/parenrightbigg\n. (16)\nEquations (14)-(16) are related by the following relation.\nJ=/parenleftbigg\nσ↑0\n0σ↓/parenrightbigg\nE+e/parenleftbigg\nD↑0\n0D↓/parenrightbigg\n∇N.(17)\nThe first term in right-hand side corresponds to conven-\ntional electrical current and the second term corresponds\nto diffusion current.\nInstead of diffusion constants, it is more physical and\nintuitive to express results in terms of spin-flip length\nwhich is defined by λ↑↓\nsf=/radicalbig\nD↑↓τsf. The averaged spin\ndiffusion length is also defined by the conventional way\nλ−2\nsf=n↑\n0λ↑2\nsf+n↓\n0λ↓2\nsf\n(n↑\n0+n↓\n0)λ↑2\nsfλ↓2\nsf. (18)\nBy Einstein’s relation, Eq. (18) is equivalent to\nλ2\nsf=σ↑λ↓2\nsf+σ↓λ↑2\nsf\nσ↑+σ↓. (19)\nCombining Eqs. (18) and (19), λ↑↓\nsfis represented in\nterms of λsf.\nλ↑↓2\nsf=λ2\nsfσ\nσ↓↑n↓↑\n0\nn↑\n0+n↓\n0, (20)\nwhereσ=σ↑+σ↓is total electrical conductivity.\nConductivity polarization Pand density polarization\nPnare defined by\nP=σ↑−σ↓\nσ, (21)\nPn=n↑\n0−n↓\n0\nn↑\n0+n↓\n0. (22)\nWith these polarizations, σ↑↓andn↑↓\n0are represented\nin terms of σand (n↑\n0+n↓\n0) asσ↑↓= (1±P)σ/2 and\nn↑↓\n0= (1±Pn)(n↑\n0+n↓\n0)/2.4\nLastly, we use a mathematical convention that ˜A(k) is\nthe Fourier transform of a position dependent function\nA(x) with respect to x. That is,\n˜A(k)≡ F[A(x)](k) =1\n(2π)d/2/integraldisplay\nA(x)e−ik·xddx,(23)\nfor ad-dimensional system.\nIII. CHARGE AND SPIN CURRENTS IN THE\nPRESENCE OF SPIN DIFFUSION\nA. Solution of the spin drift-diffusion equation for\nlocalized electric field\nBefore solving Eq. (13) for general cases, we first solve\nthe equation for localized EssinceEsis localized in most\ncases. In Sec. V, we generalize our theory to include\nspatially extended Es.\nSinceEsislocalized,itispossibletotakeFouriertrans-\nform with respect to position. Then, ˜Eand˜Nare well-\ndefined localized functions of kexcept initial condition\npart. In addition, localized Esimplies that the bound-\nary condition is given by n↑↓(|x| → ∞,t) =n↑↓\n0because\nEs(|x| → ∞,t) = 0 does not affect spin density. After\nFourier transform, Eq. (13) is written as, in matrix form,\n∂˜N\n∂t+Ω\nτsf˜N=1\ne/parenleftbigg\nσ↑0\n0σ↓/parenrightbigg\nik·˜E,(24)\nwhere\nΩ =\nλ↑2\nsfk2+n↓\n0\nn↑\n0+n↓\n0−n↑\n0\nn↑\n0+n↓\n0\n−n↓\n0\nn↑\n0+n↓\n0λ↓2\nsfk2+n↑\n0\nn↑\n0+n↓\n0\n.(25)\nEquation (24) is a first order ordinary differential equa-\ntion with respect to tand the initial condition is given\nby˜N(k,t= 0) = (2 π)d/2δ(k)(n↑\n0n↓\n0)T. The solution is\nsimply given by\n˜N(k,t) =e−Ωt/τsf˜N(k,0) (26)\n+/integraldisplayt\n0e−Ω(t−t′)/τsf/parenleftigg\nσ↑\ne0\n0σ↓\ne/parenrightigg\nik·˜E(k,t′)dt′.\nSince the first term of Eq. (26) represents the time vari-\nation of equilibrium number density, one can realize that\nthe term should be given by e−Ωt/τsf˜N(k,0) =˜N(k,0).\nMathematical derivation of this argument is given in Ap-\npendix A. The second term of Eq. (26) is almost impos-\nsible to take inverse Fourier transform. Hence, as men-\ntioned, we use an approximation that τsfis very small.\nIn this limit, Appendix B shows that\ne−Ωt/τsfΘ(t)≈τsfΩ−1δ(t), (27)\nwhere Θ( t) is Heaviside step function. By this approxi-\nmation, solution of the spin drift-diffusion equation Eq.(26) becomes\n˜N(k,t) =˜N(k,0)+τsf\neΩ−1/parenleftbigg\nσ↑0\n0σ↓/parenrightbigg\nik·˜E(k,t).(28)\nThe inverse of Ω is explicitly given by\nΩ−1=1\ndetΩ\nλ↓2\nsfk2+n↑\n0\nn↑\n0+n↓\n0n↑\n0\nn↑\n0+n↓\n0\nn↓\n0\nn↑\n0+n↓\n0λ↑2\nsfk2+n↓\n0\nn↑\n0+n↓\n0\n,(29)\ndetΩ = λ↑2\nsfλ↓2\nsfk2/parenleftig\nk2+λ−2\nsf/parenrightig\n. (30)\nNow, excited charge density ∆ ne≡(n↑+n↓)−(n↑\n0+n↓\n0)\nand excited spin density ∆ ns≡(n↑−n↓)−(n↑\n0−n↓\n0) are\ngiven by,\n∆˜ne(k,t) =/parenleftbig1 1/parenrightbig˜N(k,t)−(2π)d/2δ(k)(n↑\n0+n↓\n0)\n=στsf\neλ2\nsf1−P2\n1−P2nPnk2+Pλ−2\nsf\nk2+λ−2\nsfik·˜Es(k,t)\nk2,(31)\n∆˜ns(k,t) =/parenleftbig1−1/parenrightbig˜N(k,t)−(2π)d/2δ(k)(n↑\n0−n↓\n0)\n=στsf\neλ2\nsf1−P2\n1−P2nk2+PnPλ−2\nsf\nk2+λ−2\nsfik·˜Es(k,t)\nk2(32)\nink-space.\nAt this stage, thereis no need to showcomplicatedreal\nspace expressions of the densities, because what affects\nLLG equation mainly is spin current. In the next subsec-\ntion, we find the expressions of charge and spin currents\nin bithk-space and real space.\nB. Charge and spin currents\nCharge and spin currents in the absence of spin diffu-\nsion are given by\nje=σ↑E↑\ns+σ↓E↑\ns=PσEs, (33)\njs=σ↑E↑\ns−σ↓E↑\ns=σEs. (34)\nIn this subsection, how the spin current and charge cur-\nrent generated by SMF is changed by spin diffusion from\nEqs. (33) and (34) is examined with the help of Eqs.\n(17) and (28). After some algebra,\n˜J=/parenleftbigg\nσ↑0\n0σ↓/parenrightbigg\n˜E+e/parenleftbigg\nD↑0\n0D↓/parenrightbigg\nik˜N\n=˜Es/parenleftbigg\nσ↑\n−σ↓/parenrightbigg\n+ik(ik·˜Es)\nk2/parenleftig\nk2+λ−2\nsf/parenrightig/parenleftbiggσ↑k2+Pσ↑λ−2\nsf\n−σ↓k2+Pσ↓λ−2\nsf/parenrightbigg\n.(35)5\nNow, the expressions of charge current ˜je=˜j↑+˜j↓and\nspin current ˜js=˜j↑−˜j↓are straightforward.\n˜je=Pσ/bracketleftigg\n˜Es+ik(ik·˜Es)\nk2/bracketrightigg\n, (36)\n˜js=P˜je+(1−P2)σ/parenleftigg\n˜Es+ik(ik·˜Es)\nk2+λ−2\nsf/parenrightigg\n.(37)\nWe present jsin terms of jein order for one to see easily\njs=jeforP=±1 ; for perfectly polarized electrons\nwithout spin-flip, the spin current should be the same\namount of the charge current.\nEquations(36)and (37) arethe principalresultsofthis\npaper. One can obtain d-dimensional real space expres-\nsions by taking inverse Fourier transform. This is one\nof the key advantages of our theory. Our result is easily\ngeneralizable to d-dimensional result. As one shall see,\nit turns out that the nonconservative part of Esplays a\ncrucial role in a higher dimensional system.\nFirst of all, we present 1D result. 1D real space ex-\npression of Eqs. (36) and (37) are\nje(x,t) = 0, (38)\njs(x,t) = (1−P2)/integraldisplay\ndx′e−|x−x′|/λsf\n2λsfσEs(x′,t).(39)\nOne can notice that the 1D charge current is perfectly\ncanceled by diffusion current for small spin-flip time\nlimit. This is natural in the sense that, for small spin-\nflip time, the system tends to be in equilibrium at each\ntimet. At equilibrium, the gradientofchemicalpotential\nvanishes, so does charge current. However, spin current\ndoes not vanish by this reason because of spin noncon-\nserving process. Due to spin diffusion, it is natural that\nthespincurrentbecomesnonlocalwith integrationkernel\nwidthλsf. Here, the factor (1 −P2) seems unexpected.\nThis factor comes from spin diffusion effect, and should\nexist regardless of diffusion length scale. It yields more\ncancelation for more polarized electrons. Eventually, for\nP=±1, spin current also vanishes, which is actually\nrequired since js=jein the limit P=±1.\nEquation(39) behavesquite differentlyfortwolimiting\ncases. Let λbe the characteristic length scale (such as\nDW width) of localized Es. Ifλsf≪λ,e|x−x′|/λsf≈\n2λsfδ(x−x′). Then,\njs(x,t)≈(1−P2)σEs(x,t), (40)\nwhich is local. Very short diffusion length cannot make\nthe spin current nonlocal. It is very interesting that (1 −\nP2)factordoesnotdisappeareventhoughdiffusioneffect\nis very small. The existence of diffusion makes (1 −P2)\nfactor regardless of how the effect is strong or weak. For\nλsf≫λ,\njs(x,t)≈σEs(1−P2)λ\n2λsfe−|x−X(t)|/λsf,(41)-50 0 500.00.51.01.52.02.53.0\n-100 -50 0 50 10001234(a)  (b)  \nFIG. 1: (color online) Results of micromagnetic simulation\nfor (a) charge current and (b) spin current induced by SMF\ngenerated from 1D DW oscillator, which is fixed at x= 0, but\nrotating with ω= 10GHz, in the presence of spin diffusion\nwithλsf= 0.5nm (black rectangle), 5 .0nm (red circle), and\n50nm (blue triangle). Solid line represents currents witho ut\ndiffusion. Here, DW width is set to be 10nm.\nwhereEsis averaged SMF and X(t) is the position of\nlocalized Es(such as DW position). One can see that\nthe current is highly suppressed by the factor λ/λsf. In\nthis highly diffusive regime, spin current is also highly\nsuppressed.\nThe main features of our result is similar to those of\nRef. [19], except for vanishing charge current. They\nclaim that charge current can exist in general, while we\nfind that Einstein’s relation prevents the existence of\ncharge current in 1D.\nWe confirmed our analytic result by comparing it with\nmicromagnetic simulation for a 1D DW oscillator. The\nresult is in Fig. 1. General features are the same as an-\nalytic result ; i) charge and spin currents are highly sup-\npressed by spin diffusion, ii) spin current becomes non-\nlocal, iii) charge current almost vanish independently of\ndiffusion length, and iv) spin current is more suppressed\nby larger diffusion length.\nNow, we generalize the results to higher dimension. In\n2D and 3D real spaces, Eqs. (36) and (37) are converted\nto\nj2D\ne=PσEs−Pσ\n2π∇/integraldisplay\nd2x′ln|x−x′|\nλsf∇′·Es,(42a)\nj3D\ne=PσEs+Pσ\n4π∇/integraldisplay\nd3x′∇′·Es\n|x−x′|, (42b)\nand\nj2D\ns=Pj2D\ne+(1−P2)σEs\n+(1−P2)σ\n2π∇/integraldisplay\nd2x′K0/parenleftbigg|x−x′|\nλsf/parenrightbigg\n∇′·Es,(43a)\nj3D\ns=Pj3D\ne+(1−P2)σEs\n+(1−P2)σ\n4π∇/integraldisplay\nd3x′e−|x−x′|/λsf\n|x−x′|∇′·Es,(43b)\nrespectively. Here, K0(x) is the zeroth order modified\nBessel function of the second kind. One can be uncom-\nfortable because j2D\neseems dependent on λsfwhile Eq.\n(36) is not. Here λsfis included because the argument6\nof logarithmic function should be dimensionless. In fact,\none can easily check that j2D\nedoes not depend on λsfby\nusing ln|x−x′|/λ′\nsf= ln|x−x′|/λsf+(constant) for any\npositiveλ′\nsf.\nFrom now on, we present only 3D expressionsbut omit\n2D, for simplicity. One can easily obtain 2D expressions\nby replacing the integral kernels 1 /|x−x′| → −2ln|x−\nx′|/λsfand exp( −|x−x′|/λsf)/|x−x′| →2K0/parenleftig\n|x−x′|\nλsf/parenrightig\n.\nThe reason why the expressions depend on dimension\nis nothing but the fact that the inverse Fourier trans-\nforms, which give integration kernels, depend on dimen-\nsion. Hence, essential physics are the same for 2D and\n3D except the mathematical expressions of the integra-\ntion kernels.\nOverallfeaturesofthe spincurrentinhigherdimension\nis similar to 1D, except for the existence of nonvanishing\nPjeterm. However, the main feature of jeis completely\ndifferent from 1D case. First of all, jedoes not vanish.\nWe argued qualitatively why the charge current vanishes\nin a 1D system by using chemical potential argument.\nHowever, in a higher dimensional system, chemical po-\ntential cannot be defined in general since Esis noncon-\nservative. Note that diffusion current D↑↓∇n↑↓is con-\nservative. Note also that nonconservative field cannot be\ncanceled by conservative field. This is why the charge\ncurrent exists in a higher dimensional system. The non-\nlocal term in Eq. (42) can be interpreted as Coulomb\npotential under charge density −∇ ·Es. The canceled\npart of the charge current is nothing but conservative\nCoulomb part of Es. Secondly, it is very interesting that\nEq. (42) is converted after some algebra to\nje=Pσ\n4π/integraldisplay\nd3x′∇′×(∇′×Es)\n|x−x′|. (44)\nNow one can notice the importance of nonconservative\npart ofEs(nonvanishing ∇ ×Es) and the dependence\nof this on the charge current. If ∇ ×Eshappens to be\nzero, the charge current also vanishes, and this is consis-\ntent to the chemical potential argument. Lastly, it is also\ninteresting that charge current does not depend on dif-\nfusion length. This is qualitatively understandable from\nthe fact that the effect of Esis maximally canceled by\ndiffusion current in small spin-flip time regime, regard-\nless of spin diffusion length. Here, maximal cancelation\nis slightly different from perfect cancelation in 1D case in\nthe sense that the (conservative) diffusion current cannot\ncancelPσEsperfectly in principle.\nIt might be ambiguous what the “conservative part”\nof a vector field is mathematically. Helmholtz’s theorem\nguarantees that a spatially localized vector field can be\nuniquely decomposed into conservative (curl-free) part\nand solenoidal (divergence-free) part. Note that the sec-\nondterminEq. (42)isexactlythe sameastheformulaof\n(negative of) conservative part of the Helmholtz decom-\nposition. Note also that the resulting total current [Eq.\n(44)] is divergence-free. Therefore, the charge diffusion\ncurrent and total charge current are respectively given\nFIG. 2: (color online) Results of the micromagnetic simu-\nlation for vortex resonant oscillation in 2D thin film. Here,\nthe vortex core size is about 5nm, the resonant frequency is\n605.5MHz, and P=Pn= 0.7. The size of arrows are log-\nscaled (size ∝ln(1+norm)). (a) Magnetization profile at the\ntime when all calculation was performed. The blue arrow\ndenotes the direction of vortex core motion. (b) SMF. (c)\nAccumulated charge density −e∆ne(divided by στsf/λ2\nsf).\nThe maximum value (white) is 1 .39×10−5V and the mini-\nmum value (black) is −1.62×10−5V. (d)∇ ·Es. The max-\nimum value (white) is 8 .43×1010V/m2and the minimum\nvalue (black) is −7.42×1010V/m2. (e) Charge diffusion cur-\nrent, which is the conservative part of SMF. (f) Total charge\ncurrent, which is the solenoidal part of SMF.\nby conservative part and solenoidal part of Helmholtz\ndecomposition. One shall see in Sec. V that this claim is\ngenerally valid for arbitrary boundary conditions.\nTo understand the maximal cancelation of the charge\ncurrent qualitatively, it would be very helpful to visualize\nthe Helmholtz decomposition of the charge current. We\nperformed a micromagnetic simulation for vortex reso-\nnant oscillation in 2D thin film. When the vortex DW\nwall [Fig. 2(a)] moves along the blue arrow, SMF is gen-\nerated as shown in Fig. 2(b). The spatial dependence of\nSMF induces charge accumulation as shown in Fig. 2(c).\nDue to the charge accumulation, charge diffusion current\nis generated as shown in Fig. 2(e). Figure 2(e) can be7\nqualitatively understood by Fig. 2(d). Recall that the\ndiffusion current is given by Coulomb field generated by\ncharge density −∇ ·Es. Thus, dipole-like nature of the\nchargedensity −∇·Es[Fig. 2(d)] implies dipole-like field\n[Fig. 2(e)]. Summing up Figs. 2(b) and 2(e), one obtains\ntotal current as Fig. 2(f). Note that the total current is\ndefinitely nonconservative (has finite curl). It is interest-\ningthatFig. 2(f)issimilartomagneticfieldgeneratedby\ntwo separate conducting wires. This infers the solenoidal\nnature of the total charge current. It is very interesting\nthat total chargecurrentbehavesasmagneticfield rather\nthan electric field.\nC. Magnetization dynamics and nonlocal damping\ntensor\nIt is also important to see how the magnetization dy-\nnamics is changed by our result. By analogue of Ref.\n[1], it is obvious that the modified LLG equation is de-\nscribed by Eq. (1), in which Eq. (43) are added to js\nterms, rather than Eq. (34). However, there exist two\nnontrivial features in the modified LLG equation.\nThe first one is spatial dependence of γandα. It is\nimportant to notice that γandαin Eq. (1) are renor-\nmalized parameters [1],\nγ=γ0/parenleftbigg\n1+nsµB\nMs1\n1+β2/parenrightbigg−1\n, (45)\nα=γ0\nγ/parenleftbigg\nα0+βnsµB\nMs1\n1+β2/parenrightbigg\n,(46)\nwhereγ0andα0are original parameters of the system.\nThere has not been any problem of this renormalization\nwithout spin accumulation, but nsis no longer constant\nin the presence of spin accumulation. Hence, γandα\ncannot be a simple constant in principle. At this stage, it\nis more convenient to write down LLG equation without\nparameter renormalization,\n/parenleftbigg\n1+1\n1+β2nsµB\nMs/parenrightbigg∂M\n∂t=−γ0M×Heff\n+/parenleftbigg\n1+β\nα01\n1+β2nsµB\nMs/parenrightbiggα0\nMsM×∂M\n∂t\n+µB\neMs(js·∇)M−βµB\neM2sM×(js·∇)M.(47)\nIn fact, the effect of nsvariation is small. Note that\nnsµB/Ms= (n↑\n0−n↓\n0)µB/Ms×∆ns/(n↑\n0−n↓\n0). Here, the\nfirst factor is of the order of 10−2and the second factor\nis first order in SMF. Then, the effect should be very\nsmall compared to ordinary first order effect of SMF. In\naddition, one shall see in Sec. IV, that symmetry can\nreduce the effect of ∆ nsin collective coordinate level. In\nthe case, the effect of ∆ nsvanishes through odd function\nintegration.\nThe next one is effective damping constant. Without\napplied electric field, the modified LLG equation is ob-\ntained by taking Gilbert damping as a damping tensorlike Eq. (4). Then, it is interesting to see how the damp-\ning tensor is generalized by spin diffusion effect. In the\npresence of spin diffusion, local damping tensor Dijbe-\ncomes nonlocal damping tensor Dij(x,x′). By using Eq.\n(43), it turns out that the damping tensor in Eq. (4) is\nmodified as\nDij(x,x′) =αδijδ(x−x′)+(1−P2)η\nM4se−|x−x′|/λsf\n2λsf\n×(M×∂xM)i(M′×∂xM′)j,(48)\nfor 1D and\nDij(x,x′) =αδijδ(x−x′)+η\nσM4sSkl(x,x′)\n×(M×∂kM)i(M′×∂lM′)j,(49)\nSij(x,x′) =σδijδ(x−x′)+P2σ\n4π∂i∂j1\n|x−x′|\n+(1−P2)σ\n4π∂i∂je−|x−x′|/λsf\n|x−x′|.(50)\nfor 3D. Here, M′=M(x′,t) and·is redefined as inner\nproduct with respectto both coordinatebasisand spatial\nbasis,\n[D ·f(x)]i=/summationdisplay\nj/integraldisplay\nddx′Dij(x,x′)fj(x′),(51)\nfor a vector field f. As a passing remark, the tensor Dis\nindeed a damping tensor in the sense that it decreasesto-\ntal magnetic energy. It can be demonstrated by showing\nenergy dissipation is negative,\ndE\ndt=−/integraldisplay/parenleftbigg\nHeff·∂M\n∂t/parenrightbigg\nddx <0.(52)\nThe mathematical details related to Eqs. (48)-(52) are\nin Appendix C.\nIV. EXAMPLE : DOMAIN WALL MOTION\nIn this section, we apply our theory to 1D DW mo-\ntion. We find equationofmotion ofcollectivecoordinates\n(X(t),φ(t)) of tail-to-tail transverse DW. Here, X(t) is\nDW position and φ(t) is tilting angle. Mathematical de-\ntails of obtaining the collective coordinate equation is\ndescribed in Appendix D.\nWithout SMF, DW motion is described by [3, 23–28]\n−∂φ\n∂t+α\nλ∂X\n∂t=−βb0\nJ\nλ, (53)\n1\nλ∂X\n∂t+α∂φ\n∂t=−b0\nJ\nλ−γKd\nMssin2φ,(54)\nwhereb0\nJ=PjµB/eMsfor applied current j,Kdrepre-\nsents dipole field integration, and λis DW width. In the8\npresence of SMF, Eqs. (53) and (54) are modified as [29]\n−∂φ\n∂t+α\nλ∂X\n∂t=−βb0\nJ\nλ−2βη\n3λ2∂φ\n∂t, (55)\n1\nλ∂X\n∂t+α∂φ\n∂t=−b0\nJ\nλ−2η\n3λ2∂φ\n∂t−γKd\nMssin2φ.(56)\nOne can expect that the effect of ηwill be suppressed by\nspin diffusion, so a renormalized parameter ˜ ηwill replace\nη. Thus, the expected equations of motion are\n−∂φ\n∂t+α\nλ∂X\n∂t=−βb0\nJ\nλ−2β˜η\n3λ2∂φ\n∂t, (57)\n1\nλ∂X\n∂t+α∂φ\n∂t=−b0\nJ\nλ−2˜η\n3λ2∂φ\n∂t−γKd\nMssin2φ.(58)\nAs described in Appendix D, it turns out that Eqs. (57)\nand (58) are indeed valid, and the renormalized SMF\nparameter is given by\n˜η= (1−P2)F(ζ)η, (59)\nF(ζ) =−3\n2ζ−3\n2ζ2+3\n4ζ3/parenleftbiggΓ′(z)\nΓ(z)/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nz=ζ\n2,(60)\nwhereζ=λ/λsfand Γ(z) is the Gamma function.\nThe renormalization is largely dependent on relative\nmagnitude of DW width and spin diffusion length. The\nasymptotic behavior of the function Fis given by\nF(ζ) =/braceleftigg\n3\n2ζ−3\n2ζ2+π2ζ3\n8+O(ζ4)ζ≪1,\n1−4\n5ζ2+O(ζ−4)ζ≫1.(61)\nForλ≪λsf, Eqs. (55) and (56) are reproduced except\n(1−P2) factor which should exist as already discussed.\nForλ≫λsf, the effect is highly suppressed by spin dif-\nfusion, so the overall effect of SMF goes as1\nλλsfrather\nthan1\nλ2.\nV. FURTHER GENERALIZATIONS\nA. Extended electric field - general boundary\ncondition\nIn the case that the magnetization dynamics is gen-\nerated by applied electric field, the electric field is no\nlonger localized. Hence, it is necessary to generalize our\ntheory to non-localized electric field ; Esdoes not vanish\nat|x| → ∞. In this case, ˜Edoes not give well-defined\nFouriertransform,butincludesdeltafunctionparts. Fur-\nthermore, in order to obtain Eq. (44), one obtains ad-\nditional boundary terms when integrating Eq. (42) by\nparts. Hence, the charge current may not be canceled by\nthe diffusion current even if the electric field is conserva-\ntive.\nThe problem can be treated by Green’s function\nmethod with given boundary condition, as described inthe last part of this section. However, it is hard to catch\nthe physical meaning, so we presentmore intuitive analy-\nsis for this case. It is very convenient to use the linearity\nof our theory. Suppose E↑↓can be decomposed into two\ncomponents E↑↓=E↑↓\n1+E↑↓\n2. Then, since Eqs. (13) and\n(17) are linear, the current j↑↓can also be decomposed\ninto two components j↑↓=j↑↓\n1+j↑↓\n2, where j↑↓\niis the\ncurrent generated by Ei.\nTo take advantage of this linearity, we decompose E\ninto three components as follows; any vector field E\ncan be uniquely decomposed into irrotational part E1,\nsolenoidal part E2, and boundary part E3,\nE↑↓=E↑↓\n1+E↑↓\n2+E↑↓\n3, (62)\n∇·E↑↓\n1∝negationslash= 0,∇×E↑↓\n1= 0,n·E↑↓\n1|x∈∂= 0,(63)\n∇·E↑↓\n2= 0,∇×E↑↓\n2∝negationslash= 0,n·E↑↓\n2|x∈∂= 0,(64)\n∇·E↑↓\n3= 0,∇×E↑↓\n3= 0,n·E↑↓\n3|x∈∂∝negationslash= 0.(65)\nHere,∂means boundary and ndenotes a normal unit\nvector perpendicular to the boundary. From given E↑↓,\none can obtain E↑↓\n3by solving Laplace equation with\nNeumann boundary condition and E↑↓\n1,2by Helmholtz’s\ntheorem. As previously mentioned, our result [Eq. (44)]\nindicates that the conservative source part E↑↓\n1cannot\ncontribute to the charge current while the nonconserva-\ntive solenoidal part E↑↓\n2can give rise to a nonvanishing\ncontribution σ↑E↑\n2+σ↓E↓\n2. [Eq. (42) or Eq. (77) for\ngeneral boundary condition]\nIn the presence of nonvanishing boundary condition, it\nis important to investigate the effect of E↑↓\n3to the spin\naccumulation. Fortunately, the source term of Eq. (13)\ndepends on the divergence of E↑↓\nsonly. Hence, E↑↓\n3can-\nnotcontribute tothe spin accumulationsince ∇·E↑↓\n3= 0.\nTherefore, E↑↓\n3canonlycontributetothecurrentsviathe\nfirst term in Eq. (17). Consequently, the expressions of\ncharge and spin current should be σ↑E↑\n3±σ↓E↓\n3added to\nEqs. (42) and (43) (or Eqs. (38) and (39)).\nIt can be a good example that a constant spin inde-\npendent electric field Eapp=Eappˆxis applied in a 1D\nwire. In this case, E↑↓\n3=Eappˆx. Hence, the charge and\nspin current should be σ↑Eapp±σ↓Eappadded to Eqs.\n(38) and (39), so\nje=σEapp, (66)\njs=PσEapp+(1−P2)/integraldisplay\ndx′e−|x−x′|/λsf\n2λsfσEs(x′,t).(67)\nIn this case, the charge current is not canceled by diffu-\nsion current, but only charge current generated by SMF\n(which is spatially localized) is canceled.\nB. Charge neutrality and other spin independent\npotentials\nThere is another important physical consequence of\nspin accumulation we have ignored. This is Coulomb9\npotential of accumulated electrons. Since Coulomb po-\ntential is in general strong, electron tends to make local\ncharge neutral. Hence, giving a charge neutrality con-\nstraint,\n∆n↑+∆n↓= 0 (68)\nis a good approximation. To do this, it is convenient\nto introduce charge neutrality potential Vcand replace\nE↑↓\ns→E↑↓\ns− ∇Vcas suggested in Ref. [10]. Then, the\nsolution of the spin drift-diffusion equation [Eq. (28)] is\nmodified by\n/parenleftbigg\n∆˜n↑\n∆˜n↓/parenrightbigg\n=τsf\neΩ−1/parenleftbigg\nσ↑0\n0σ↓/parenrightbigg\nik·/parenleftbigg˜Es−ik˜Vc\n−˜Es−ik˜Vc/parenrightbigg\n,\n(69)\nwhereVcis found self-consistently to satisfy Eq. (68).\nAfter straightforward algebra, one obtains charge neu-\ntrality potential in terms of Es,\n˜Vc(k,t) =−Pik·˜Es\nk2−(Pn−P)ik·˜Es\nk2+λ−2\nsf,(70)\nVc(x,t) =−P\n4π/integraldisplay\nd3x′1\n|x−x′|∇·Es\n−Pn−P\n4π/integraldisplaye−|x−x′|/λsf\n|x−x′|��·Es.(71)\nThe effect of this potential is twofold. Firstly, Vcgives\nadditional electrical current −σ↑↓∇Vc. Ink-space,\n∆˜j↑\nc(k,t) =−σ↑ikVc, (72)\n∆˜j↓\nc(k,t) =−σ↓ikVc. (73)\nSecondly, Vcaffects diffusion current through ∇n↑↓. Af-\nter some algebra, one obtains\n/parenleftbigg\n∆˜j↑\nD\n∆˜j↓\nD/parenrightbigg\n=ik/parenleftigg\nλ↑2\nsf0\n0λ↓2\nsf/parenrightigg\nΩ−1/parenleftbigg\nσ↑0\n0σ↓/parenrightbigg/parenleftbigg\nk2Vc\nk2Vc/parenrightbigg\n=ikVc/parenleftbigg\nσ↑\nσ↓/parenrightbigg\n, (74)\nwhich exactly cancels Eqs. (72) and (73). Hence, there\nare no additional charge and spin currents attributed to\nVs.\nIt is very interesting that one does not need to assume\nanyspecific formof VctoreachEq. (74). The mathemat-\nical origin of the exact cancelation is that the additional\nforce is conservative (described by ik×(scalar)) and spin\nindependent. A spin independent potential generates ad-\nditional current, but spin accumulation adjusts fast to\nmake diffusion current cancel it. Of course, the modi-\nfied spin accumulation n↑↓affects LLG equation by Eq.\n(47), but weclaimedthatthisisignorable. Consequently,\nany spin independent potential cannot modify the main\nfeatures of our result. As another side remark, this can-\ncelation can be verified for exact Coulomb potential even\nwithout using the charge neutrality approximation Eqs.\n(68)-(71).C. Arbitrary boundary\nSince SMF is usually strongly localized, we have con-\nsidered an infinite boundary problem. However, for a\nfinite system, the expression of the charge and spin cur-\nrents should be slightly modified. Note that the key part\nof our theory is to find real space expression of Ω−1. (See\nEq. (28)). In real space, Ω is a differential operator,\nΩreal=\n−λ↑2\nsf∇2+n↓\n0\nn↑\n0+n↓\n0−n↑\n0\nn↑\n0+n↓\n0\n−n↓\n0\nn↑\n0+n↓\n0−λ↓2\nsf∇2+n↑\n0\nn↑\n0+n↓\n0\n.\n(75)\nHence, the problem is to find the inverse operator, i.e.,\nthe Green’s function corresponding Ω realat the given\ngeometry.\nIn order to find the Green’s function of Ω real, one\ncan get a hint from Eq. (29). Firstly, let GL(x,x′)\nandGH(x,x′) be respectively the Green’s function cor-\nresponding Laplacian ∇2and modified Helmholtz op-\nerator∇2−λ−2\nsffor the given geometry. In order to\nobtain the Green’s function, it is plausible to replace\n1/k2→ −GL(x,x′) and 1/(k2+λ−2\nsf)→ −GH(x,x′) in\nEq. (29). Then, one obtains\nGΩ(x,x′) =−1\nσ\nσ↑GL+σ↓GH\nλ↑2\nsfσ↑GL−σ↑GH\nλ↑2\nsf\nσ↓GL−σ↓GH\nλ↓2\nsfσ↓GL+σ↑GH\nλ↓2\nsf\n.(76)\nOnecanverifythis isindeedthe Green’sfunctionofΩ real\nby showing Ω realGΩ=δ(x−x′). Then, one straight-\nforwardly concludes that the generalized expressions of\ncharge and spin current [Eqs. (42)-(43)] are given by\nje=PσEs−Pσ∇/integraldisplay\nddx′GL(x,x′)∇′·Es,(77)\njs=Pje+(1−P2)σEs\n−(1−P2)σ∇/integraldisplay\nddx′GH(x,x′)∇′·Es.(78)\nForinfiniteboundarywithvanishingboundarycondition,\n4πGL=−1/|x−x′|and 4πGH=−e−|x−x′|/λsf/|x−x′|,\nso Eqs. (42)-(43) are reproduced.\nVI. CONCLUSION\nBy constructing spin drift-diffusion equation from the\nequation of motion of conduction electrons, we studied\nthe effect of SMF in the presence of spin accumulation,\nspin diffusion and spin flip scattering, whichwereignored\n[7] or considered only in 1D [19] in previous theories.\nIt turns out that, in realistic regime, the conservative\npart of charge current is canceled by diffusion current,\nand spin current becomes nonlocal. Consequently, the\nmagnetization dynamics is affected by nonlocal Gilbert\ndamping tensor, instead of the previously reported (lo-\ncal)Gilbert damping tensor. By calculatingspin-transfer10\ntorque generated by nonlocal spin current, we obtained\nthe explicit expressions of the nonlocal Gilbert damping\ntensor.\nDifferent from the previous work focusing on 1D [19],\nwe also obtained the results for 2D and 3D as well as 1D.\nIn a 1D system, the results of the previous theory are\nreproduced. It turnsout that Einstein’srelationprevents\nthe existence of charge current, but for parameter sets\nwhich do not satisfy the Einstein’s relation, the charge\ncurrent can be induced by the spin motive force. After\ngeneralizing the results to higher dimension, we find that\nthe nonconservative part of SMF plays an important role\nin charge and spin currents.\nAs an illustration of suppression of the effect of SMF,\nwe demonstrated equations of motion of collective coor-\ndinate of1Dcurrent-inducedDWmotion. We found that\nspin diffusion renormalizes SMF depending on the rela-\ntive magnitude of DW width and spin diffusion length.\nWe also investigated the system under spatially ex-\ntended (non-localized) electric field. In this case, it turns\nout that the spatially extended part of the electric field\ncan contribute to the charge current even though it is\nconservative. However, the existence of spatially ex-\ntended part of the electric field cannot alter the result\nthatthechargecurrent generated by (localized) SMF can-\nnot include conservative part.\nOur result is solid in the sense that our principal re-\nsults are not changed by any spin independent potential.\nA spin independent potential modifies current via addi-\ntional electric field, but spin density rapidly adjusts to\nmake diffusion current cancel it. Consequently, a spin\nindependent potential can modify spin density, but not\ncurrent.\nAcknowledgments\nThis work is financially supported by the NRF (2009-\n0084542, 2010-0014109, 2010-0023798), KRF (KRF-\n2009-013-C00019) and BK21. KWK acknowledges the\nfinancial support by TJ Park.\nAppendix A: Absence of Time Variation of\nEquilibrium Number Density\nIn this section, we show that e−Ωt/τsf˜N(k,0) does not\nhave time dependence, where Ω is given by Eq. (25) and\n˜N(k,0) = (2π)d/2δ(k)(n↑\n0n↓\n0)T. It suffices to calculate\ne−Ωt/τsfatk= 0 because of δ(k) factor. It is easy to\nshow that Ω is idempotent for k= 0. In other words,\nΩn|k=0= Ω|k=0. Then, for k= 0,\ne−Ωt/τsf=∞/summationdisplay\nn=0Ωn\nn!/parenleftbigg\n−t\nτsf/parenrightbiggn\n=I+∞/summationdisplay\nn=1Ω\nn!/parenleftbigg\n−t\nτsf/parenrightbiggn\n,\n=I+Ω(e−t/τsf−1). (A1)whereIis the identity matrix. Since\nΩ˜N(k,0) = (2π)d/2δ(k)Ω/parenleftbigg\nn↑\n0\nn↓\n0/parenrightbigg\n= 0 (A2)\nthe second term of Eq. (A1) vanishes after applied to\n˜N(k,0). Finally, one obtains e−Ωt/τsf˜N(k,0) =˜N(k,0).\nNote that the result is obtained without the approxi-\nmation that τsfis small.\nAppendix B: Derivation of Eq. (27)\nThe idea is based on the delta sequence\nlim\nn→∞ne−ntΘ(t) =δ(t), (B1)\nfor natural number n. This can be generalized as\nlim\nu→∞ne−uztΘ(t) =δ(t), (B2)\nwherezis a complex number satisfying ℜ[z]>0. This\ngeneralization is obvious in that uze−uztis localized at\nt= 0 and/integraltext∞\n−∞uze−uztΘ(t)dt= 1.\nWe now generalize this relation to matrices. For a\ndiagonalizable matrix Mwith eigenvalues λisatisfying\nℜ[λi]>0, we claim that\nlim\nu→∞uMe−uMtΘ(t) =δ(t)I. (B3)\nThe proof is simple. Since Mis diagonalizable, one can\nwrite\nuM=Q\nuλ10···\n0uλ2···\n.........\nQ−1,(B4)\nfor some Q. Then\nuMe−uMxΘ(x)\n=Q\nuλ1e−uλ1Θ(x) 0 ···\n0 uλ2e−uλ2Θ(x)···\n.........\nQ−1\n→Q\nδ(x) 0···\n0δ(x)···\n.........\nQ−1=δ(x)I, (B5)\nasu→ ∞.\nBy taking M= Ω and u= 1/τsf,\nlim\nτsf→+0Ω\nτsfe−Ωt/τsfΘ(t) =δ(t)I, (B6)\nso, for small τsf,\nΩ\nτsfe−Ωt/τsfΘ(t)≈δ(t)I, (B7)11\nwhich is exactly Eq. (27).\nAs a passing remark, one should check that the real\nparts of eigenvalues of Ω are really positive. It is guar-\nanteed by Tr Ω >0 and detΩ >0 for any nonzero k.\nHowever, the approximation Eq. (27) makes singularity\natk= 0. Fortunately, this singularity is removed by ik\nfactor in Eq. (26).\nAppendix C: Nonlocal damping tensor and energy\ndissipation\n1. 1D damping tensor\nStarting from\njs= (1−P2)/integraldisplay\ndx′e−|x−x′|/λsf\n2λsfσEs(x′,t),(C1)\nEs=/planckover2pi1\n2eM3s∂tM·(∂xM×M), (C2)\nspin-transfer torque driven by SMF is given by\nµB\neMsjs∂xM= (1−P2)η\nM3s∂xM\n×/integraldisplay\ndx′e−|x−x′|/λsf\n2λsf∂tM′·(∂xM′×M′)\n=M\nMs×/integraldisplay\ndx′/bracketleftigg\n(1−P2)η\nM4se−|x−x′|/λsf\n2λsf\n(M×∂xM)(M′×∂xM′)]·∂tM′.(C3)\nThen, comparing with Eqs. (4) and (51),\nDij(x,x′) =αδijδ(x−x′)+(1−P2)η\nM4se−|x−x′|/λsf\n2λsf\n×(M×∂xM)i(M′×∂xM′)j.(C4)\nIt is interesting to note that\nlim\nλsf→0Dij(x−x′) =δ(x−x′)/bracketleftbigg\nαδij+(1−P2)η\nM4s\n×(M×∂xM)i(M×∂xM)j].(C5)\nThis isexactlythe previousresultEq. (5) except (1 −P2)\nfactor, whichshouldexistregardlessofdiffusionstrength.\nNow, the remaining step is to calculate energy dissipa-\ntion. Energy dissipation is calculated by the integration\nenergy density dissipation −Heff·∂tM. From now on,\nthe subscription effis dropped until this section ends.\nEnergy dissipation by a nonlocal damping tensor Disgiven by\ndE\ndt=−/integraldisplay\ndxH·/bracketleftbiggM\nMs×D ·∂tM/bracketrightbigg\n=1\nMs/integraldisplay\ndx(M×H)·D ·∂tM\n≈ −γ\nMs/integraldisplay\ndx(M×H)·D·(M×H)\n=−γ\nMs/integraldisplay/integraldisplay\ndxdx′(M×H)iDij(x,x′)(M′×H′)j,(C6)\nup to first order. Here, H′=H(x′,t). The first term\nin Eq. (C4) gives definitely negative dE/dt. The second\nterm, which comes from SMF, gives\ndESMF\ndt=−(1−P2)ηγ\nMs/integraldisplay/integraldisplay\ndxdx′(H·∂xM)\n×e−|x−x′|/λsf\n2λsf(H′·∂xM′). (C7)\nIn order to show dE/dt < 0, it is sufficient to show that/integraltext/integraltext\ndxdx′f(x)e−a|x−x′|f(x′) is positive for real function\nf. This is verified by Parseval’s relation and convolution\ntheorem of Fourier transform.\n/integraldisplay/integraldisplay\ndxdx′f(x)e−a|x−x′|f(x′)\n=/integraldisplay\ndxf(x)∗/bracketleftbigg/integraldisplay\ndx′e−a|x−x′|f(x′)/bracketrightbigg\n=/integraldisplay\ndkF[f(x)](k)∗F/bracketleftbigg/integraldisplay\ndx′e−a|x−x′|f(x′)/bracketrightbigg\n(k)\n=√\n2π/integraldisplay\ndkF[f(x)](k)∗F[e−a|x|](k)F[f(x)](k)\n= 2a/integraldisplay\ndk|F[f(x)](k)|2\nk2+a2>0. (C8)\nThis implies dE/dt < 0.\n2. 3D damping tensor\nFrom now on, Einstein’s convention is used. Compo-\nnentwise expressions of spin current and SMF for a 3D\nsystem are\njs,i=σEs,i+P2σ\n4π∂i/integraldisplay\nd3x′∂′\njEs,j\n|x−x′|\n+(1−P2)σ\n4π∂i/integraldisplay\nd3x′e−|x−x′|/λsf\n|x−x′|∂′\njEs,j,(C9)\nEs,i=/planckover2pi1\n2eM3s∂tM·(∂iM×M). (C10)12\nBy integrating by parts in order to remove derivatives in\nfront ofEs,j,\njs,i=σEs,i−P2σ\n4π∂i/integraldisplay\nd3x′Es,j(x′)∂′\nj1\n|x−x′|\n−(1−P2)σ\n4π∂i/integraldisplay\nd3x′Es,j(x′)∂′\nje−|x−x′|/λsf\n|x−x′|,\n=σEs,i+P2σ\n4π/integraldisplay\nd3x′Es,j(x′)∂′\ni∂′\nj1\n|x−x′|\n+(1−P2)σ\n4π/integraldisplay\nd3x′Es,j(x′)∂′\ni∂′\nje−|x−x′|/λsf\n|x−x′|.(C11)\nHere, it is convenient to introduce nonlocal conductivity\ntensorS,\njs,i=/integraldisplay\nd3x′Sij(x,x′)Es,j(x′),(C12)\nSij(x,x′) =σδijδ(x−x′)+P2σ\n4π∂′\ni∂′\nj1\n|x−x′|\n+(1−P2)σ\n4π∂′\ni∂′\nje−|x−x′|/λsf\n|x−x′|.(C13)\nNow, spin-transfer torque driven by SMF is given by\nµB\neMs(js·∇)M\n=µB\neMs∂iM/integraldisplay\nd3x′·Sij(x,x′)Es,j(x′)\n=M\nMs×/bracketleftbiggη\nσM4s(M×∂iM)\n×/integraldisplay\nd3x′·Sij(x,x′)(M′×∂jM′)·∂tM′/bracketrightbigg\n.(C14)\nHence, the nonlocal damping tensor for 3D is given by\nDij(x,x′) =αδijδ(x−x′)+η\nσM4sSkl(x,x′)\n×(M×∂kM)i(M′×∂lM′)j.(C15)\nNow, we calculate energy dissipation. This is an ana-\nlogue of the previous section. After some algebra,\ndESMF\ndt=−γη\nσMs/integraldisplay/integraldisplay\nd3xd3x′Skl(x,x′)\n×(H·∂kM)(H′·∂lM′).(C16)\nIn order to show dESMF/dt <0, one should verify\n/integraldisplay/integraldisplay\nd3xd3x′(H·∂iM)Sij(x,x′)(H′·∂jM′)>0.(C17)\nNote that Sij(x,x′) is a function of ( x−x′). It is con-\nvenient to write Sij=Sij(x−x′) at this stage. Similar\nto the previous subsection, by using Parseval’s theorem\nand convolution theorem of Fourier transform,\n/integraldisplay/integraldisplay\nd3xd3x′(H·∂iM)Sij(x−x′)(H′·∂jM′)\n=/integraldisplay\nd3kF[H·∂iM](k)∗˜Sij(k)F[H·∂jM](k).(C18)Here,˜Sij(k) is given by\n˜Sij(k) =σ\n(2π)3/2/bracketleftigg\nδij−P2kikj\nk2−(1−P2)kikj\nk2+λ−2\nsf/bracketrightigg\n.\n(C19)\nNote that the integrand in Eq. (C18) is an expecta-\ntion value of 3 ×3 matrix ˜Sij(k) with respect to vector\nF[H·∂iM](k). Since ˜Sij(k) is a Hermitian matrix, the\nintegrand is always positive if the eigenvalues of ˜Sij(k)\nare positive. Recall that the eigenvalues of matrix kikj\nis given by (0 ,0,k2). The corresponding eigenvectors are\ndefinitely the eigenvectors of ˜Sij(k), so the eigenvalues\nof˜Sij(k) is given by/parenleftbigg\nσ\n(2π)3/2,σ\n(2π)3/2,σ\n(2π)3/2λ−2\nsf(1−P2)\nk2+λ−2\nsf/parenrightbigg\n,\nwhich are all positive. This proves dE/dt < 0.\nAppendix D: 1D DW motion in the presence of spin\ndiffusion\n1. Collective coordinate equation of 1D DW for\nspace-time dependent spin current and spin density\nThe starting equation is 1D version of Eq. (47).\n/parenleftbigg\n1+1\n1+β2nsµB\nMs/parenrightbigg∂M\n∂t=−γ0M×Heff\n+/parenleftbigg\n1+β\nα01\n1+β2nsµB\nMs/parenrightbiggα0\nMsM×∂M\n∂t\n+bJ∂M\n∂x−βbJM\nMs×∂M\n∂x, (D1)\nwherebJ=µBjs/eMs. The main difference from the\nconventional approach is that bJandnsare space-time\ndependent. The equation is rewritten as\nγ0M×Htot= 0, (D2)\nwhere\nγ0Htot=−/parenleftbigg\n1+1\n1+β2nsµB\nMs/parenrightbiggM\nM2s×∂M\n∂t+γ0Heff\n−/parenleftbigg\nα0+β\n1+β2nsµB\nMs/parenrightbigg1\nMs∂M\n∂t\n+bJM\nM2s×∂M\n∂x+βbJ\nMs∂M\n∂x. (D3)\nWe set the effective magnetic field as Heff=2A\nM2s∂2M\n∂x2+\nHKMx\nMsˆx+Hd, whereAandKdrepresents exchange cou-\npling and anisotropy, and Hdis dipole field. In addition,\nthe magnetization profile is set to be tail-to-tail trans-13\nverse wall,\nM=Ms(cosθ,sinθcosφ,sinθsinφ),(D4)\nsinθ= sech/parenleftbiggx−X(t)\nλ/parenrightbigg\n, (D5)\ncosθ= tanh/parenleftbiggx−X(t)\nλ/parenrightbigg\n, (D6)\nφ=φ(t), (D7)\nwhereX(t) is DW position and φ(t) is tilting angle.\nNote that Eq. (D2) implies Htot=aMfor some a.\nThen, one can define force density f=−Htot·∂M\n∂Xand\ntorque density τ=−Htot·∂M\n∂φ, which are identically\nzero. Then, collective coordinate equation is given by\ncalculating total force and total torque,\nF≡/integraldisplay\nfdx= 0, (D8)\nT≡/integraldisplay\nτdx= 0. (D9)\nEach equation implies respectively,\n−∂φ\n∂t+α\nλ∂X\n∂t+1\n1+β2/parenleftbigg\n−∂φ\n∂t+β\nλ∂X\n∂t/parenrightbigg\n×/integraldisplay∆nsµB\nMssech2/parenleftbiggx−X\nλ/parenrightbiggdx\n2λ\n=−βb0\nJ\nλ−β\nλ/integraldisplay\nbs\nJsech2/parenleftbiggx−X\nλ/parenrightbiggdx\n2λ,(D10)\n1\nλ∂X\n∂t+α∂φ\n∂t+1\n1+β2/parenleftbigg1\nλ∂X\n∂t+β∂φ\n∂t/parenrightbigg\n×/integraldisplay∆nsµB\nMssech2/parenleftbiggx−X\nλ/parenrightbiggdx\n2λ\n=−b0\nJ\nλ−/integraldisplaybs\nJ\nλsech2/parenleftbiggx−X\nλ/parenrightbiggdx\n2λ−γKd\nMssin2φ.(D11)\nwhereb0\nJis space-time independent part of bJwhich\ncomes from applied spin current, bs\nJ=bJ−b0\nJ,Kdcor-\nresponds integration of dipole field. Here, αandγare\nrenormalized parameter by the same way with n↑↓\n0. One\ncan check that Eqs. (D10) and (D11) reproduces Eqs.\n(53) and (54) if ∆ ns= 0 and bs\nJ= 0.\n2. 1D DW motion in the presence of SMF and spin\ndiffusion\nNow, we apply\nbs\nj=µBσ\neMs1−P2\n2λsf/integraldisplay\ndx′e−|x−x′|/λsfEs(x′,t),(D12)\nEs=/planckover2pi1\n2eλsech2/parenleftbiggx−X\nλ/parenrightbigg∂φ\n∂t, (D13)\n∆˜ns=στsf\neλ2\nsf1−P2\n1−P2nk2+PnPλ−2\nsf\nk2+λ−2\nsfik˜Es\nk2.(D14)to Eqs. (D10) and (D11).\nFirst, we calculate the integral/integraltext∆nssech2/parenleftbigx−X\nλ/parenrightbig\ndx\nwhichcorrespondstotheeffectofspindensity. Notethat\n∆nsis an odd function of x−X. Hence, the integrand\nis an odd function so the integral vanishes.\nThe next step is to calculate/integraltextbs\nJ\nλsech2/parenleftbigx−X\nλ/parenrightbigdx\n2λ. Af-\nter some algebra,\n/integraldisplaybs\nJ\nλsech2/parenleftbiggx−X\nλ/parenrightbiggdx\n2λ\n=(1−P2)ηζ\n4λ2∂φ\n∂t/integraldisplay/integraldisplay\ne−ζ|u−u′|sech2usech2u′dudu′\n=(1−P2)ηζ\n4λ2∂φ\n∂t/integraldisplayπζk2csch2/parenleftbigkπ\n2/parenrightbig\nk2+ζ2dk\n=(1−P2)ηζ\n4λ2∂φ\n∂t/integraldisplay∞\n02πζk2csch2/parenleftbigkπ\n2/parenrightbig\nk2+ζ2dk(D15)\nwhereζ=λ/λsf,u= (x−X)/λ, andu′= (x′−X)/λ.\nParseval’s relation and convolution theorem of Fourier\ntransform is used at the third step. By using the identity\ncsch2/parenleftbiggkπ\n2/parenrightbigg\n= 4∞/summationdisplay\nn=1ne−nπk,(D16)\nthe integral can be expressed by Laplace transform L.\n/integraldisplay∞\n02πζk2csch2/parenleftbigkπ\n2/parenrightbig\nk2+ζ2dk=∞/summationdisplay\nn=1L/bracketleftbigg8πζnk2\nk2+ζ2/bracketrightbigg\n(nπ)\n=∞/summationdisplay\nn=1/integraldisplay∞\n016\nnπsinnπζu\n(u+1)3du.(D17)\nRecall the Fourier series of a sawtooth function\n∞/summationdisplay\nn=116\nnπsinnπx= 8(2n+1−x),(D18)\nfor 2n < x < 2n+2. Then the integral is given by\n∞/summationdisplay\nn=1/integraldisplay∞\n016\nnπsinnπζu\n(u+1)3du\n=∞/summationdisplay\nn=0/integraldisplay2n+2\nζ\n2n\nζ8(2n+1−uζ)\n(1+u)3du\n=−4−4ζ+2ζ2∞/summationdisplay\nn=01\n(n+ζ/2)2.(D19)\nBy using the relation\ndn\ndznΓ′(z)\nΓ(z)= (−1)n+1n!∞/summationdisplay\nk=11\n(z+k)n+1,(D20)14\none obtains the integral in closed form as\n8\n3F(ζ)\nζ≡/integraldisplay∞\n02πζk2csch2/parenleftbigkπ\n2/parenrightbig\nk2+ζ2dk\n=−4−4ζ+2ζ2/parenleftbiggΓ′(z)\nΓ(z)/parenrightbigg′/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nz=ζ\n2\n=−4−4ζ+2ζ2Γ′′/parenleftig\nζ\n2/parenrightig\nΓ/parenleftig\nζ\n2/parenrightig\n−Γ′/parenleftig\nζ\n2/parenrightig2\n/parenleftig\nζ\n2/parenrightig2.(D21)\nThe prefactor 8 /3ζis to make lim λsf→0F(ζ) = 1. There-\nfore, we finally obtain\n/integraldisplaybs\nJ\nλsech2/parenleftbiggx−X\n2λ/parenrightbiggdx\n2λ= (1−P2)2η\n3λ2F(ζ)∂φ\n∂t,\n(D22)and, consequently,\n−∂φ\n∂t+α\nλ∂X\n∂t=−βb0\nJ\nλ−2β˜η\n3λ2∂φ\n∂t, (D23)\n1\nλ∂X\n∂t+α∂φ\n∂t=−b0\nJ\nλ−2˜η\n3λ2∂φ\n∂t−γKd\nMssin2φ,(D24)\nwhereηis renormalized parameter by spin diffusion de-\nfined as ˜η= (1−P2)F(ζ)η.\n[1] S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 (2004).\n[2] A. Thiaville, Y. Nakatani, J. Miltat, and Y. Suzuki, Eu-\nrophys. Lett. 69, 990 (2005).\n[3] G. Tatara, T. Takayama, H. Kohno, J. Shibata, Y.\nNakatani, and H. Fukuyama, J. Phys. Soc. Jpn. 75,\n064708 (2006).\n[4] M. D. Stiles, W. M. Saslow, M. J. Donahue, and A. Zang-\nwill, Phys. Rev. B 75, 214423 (2007).\n[5] L. Berger, Phys. Rev. B 33, 1572 (1986).\n[6] S. E. Barnes and S. Maekawa, Phys. Rev. Lett. 98,\n246601 (2007).\n[7] S. Zhang and S. S.-L. Zhang, Phys. Rev. Lett. 102,\n086601 (2009).\n[8] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys.\nRev. Lett. 88, 117601 (2002).\n[9] W. M. Saslow, Phys. Rev. B 76, 184434 (2007).\n[10] Y. Tserkovnyak and M. Mecklenburg, Phys. Rev. B 77,\n134407 (2008).\n[11] J. Shibata and H. Kohno, Phys. Rev. Lett. 102, 086603\n(2009).\n[12] Y. Tserkovnyak and C. H. Wong, Phys. Rev. B 79,\n014402 (2009).\n[13] R. A. Duine, Phys. Rev. B 79, 014407 (2009).\n[14] J. Ohe, S. E. Barnes, H.-W. Lee, and S. Maekawa, Appl.\nPhys. Lett. 95, 123110 (2009).\n[15] J. Ohe and S. Maekawa, J. Appl. Phys 105, 07C706\n(2009).\n[16] S. A. Yang, G. S. D. Beach, C. Knutson, D. Xiao, Q.Niu, M. Tsoi, and J. L. Erskine, Phys. Rev. Lett. 102,\n067201 (2009).\n[17] P. N. Hai, S. Ohya, M. Tanaka, S. E. Barnes, and S.\nMaekawa, Nature 458, 489 (2009).\n[18] C. H. Wong and Y. Tserkovnyak, Phys. Rev. B 81,\n060404(R) (2010).\n[19] S. S.-L. Zhang and S. Zhang, Phys. Rev. B 82, 184423\n(2010).\n[20] K.-W. Kim, J.-H. Moon, K.-J. Lee, and H.-W. Lee (un-\npublished).\n[21] G. E. Volovik, J. Phys. C 20, L83 (1987).\n[22] E. Ya. Sherman, A. Najmaie, H. M. van Driel, A. L.\nSmirl, and J. E. Sipe, e-print arXiv:cond-mat/0606725.\n[23] A. A. Thiele, Phys. Rev. Lett. 30, 230 (1973).\n[24] N. L. Schryer and L. R. Walker, J. Appl. Phys. 45, 5406\n(1974).\n[25] A. P. Malozemoff and J. C. Slonczewski, Magnetic Do-\nmains Walls in Bubble Materials (Academic, New York,\n1979).\n[26] G. Tatara and H. Kohno, Phys. Rev. Lett. 92, 086601\n(2004).\n[27] S.-W. Jung, W. Kim, T.-D. Lee, K.-J. Lee, and H.-W.\nLee, Appl. Phys. Lett. 92, 202508 (2008)\n[28] J. Ryu and H.-W. Lee, J. Appl. Phys. 105, 093929\n(2009).\n[29] S.-I. Kim, J.-H. Moon, W. Kim, and K.-J. Lee, Curr.\nAppl. Phys. 11, 61 (2011)."    },    {        "title": "2208.09575v1.Many_body_theory_of_phonon_induced_spin_relaxation_and_decoherence.pdf",        "content": "Many-body theory of phonon-induced spin relaxation and decoherence\nJinsoo Park,1Yao Luo,1Jin-Jian Zhou,2and Marco Bernardi1,\u0003\n1Department of Applied Physics and Materials Science,\nCalifornia Institute of Technology, Pasadena, CA 91125, USA.\n2School of Physics, Beijing Institute of Technology, Beijing 100081, China.\nFirst-principles calculations enable accurate predictions of electronic interactions and dynamics.\nHowever, computing the electron spin dynamics remains challenging. The spin-orbit interaction\ncauses various dynamical phenomena that couple with phonons, such as spin precession and spin-\n\ripe-ph scattering, which are di\u000ecult to describe with current \frst-principles calculations. In this\nwork, we show a rigorous framework to study phonon-induced spin relaxation and decoherence, by\ncomputing the spin-spin correlation function and its vertex corrections due to e-ph interactions.\nWe apply this approach to a model system and develop corresponding \frst-principles calculations\nof spin relaxation in GaAs. Our vertex-correction formalism is shown to capture the Elliott-Yafet,\nDyakonov-Perel, and strong-precession mechanisms \u0000three independent spin decoherence regimes\nwith distinct physical origins { thereby unifying their theoretical treatment and calculation. Our\nmethod is general and enables quantitative studies of spin relaxation, decoherence, and transport\nin a wide range of materials and devices.\nI. INTRODUCTION\nLinear response theory provides a microscopic un-\nderstanding of the response of a system to external\nperturbations and computes the associated correlation\nfunctions [1{6]. First-principles calculations of elec-\ntronic interactions [7{14] complement this formalism,\nenabling precise predictions of materials properties\nand transport coe\u000ecients without resorting to em-\npirical models or \ftting parameters. In this context,\nelectron-phonon ( e-ph) interactions are particularly\nimportant as they govern a wide range of phenomena\nsuch as charge transport [15], superconductivity [16],\nspin transport [17{19] and spin decoherence [20{22].\nThe Boltzmann transport equation (BTE) is widely\nused to study the response to an external electric\n\feld [23, 24]. The \feld drives the electronic populations\nfnk, for states with band nand crystal momentum k,\naway from the equilibrium Fermi-Dirac distribution f0\nnk,\nwhile thee-ph interactions dissipate electron energy and\nact to restore equilibrium, resulting in a steady-state\ncurrente(fnk\u0000f0\nnk)vnk, whereeis the electron charge\nandvnkis the band velocity [23]. In the many-body\nformalism, the BTE at low electric \feld is formally\nequivalent to the ladder vertex-correction to the dc\nconductivity [4]. In that framework, one determines the\ncurrent-current correlation function, with vertex correc-\ntions from the e-ph interactions obtained by summing\nover ladder diagrams, and computes the conductivity\nfrom the dissipative part of the susceptibility. A key\nfactor making this approach equivalent to the BTE is\nthat the electron velocity is band-diagonal in the Bloch\nbasis,hmkj^vjnki=\u000enm@kEnk=~[4].\nHowever, studying the response to an external \feld\nof an arbitrary operator that couples with phonons\nis more di\u000ecult. The matrix representation of an\n\u0003Corresponding author: bmarco@caltech.eduoperator ^Ais in general nondiagonal in the Bloch\nbasis, ^Anmk=hmkj^Ajnki, and can mix states in\ndi\u000berent bands. The BTE cannot be applied in this\ncase because due to its population-based formalism\nit neglects such o\u000b-diagonal (inter-band) components.\nA framework treating the response of non-diagonal\noperators coupled with e-ph interactions is still missing.\nAn important example is spin relaxation and de-\ncoherence, where spin-orbit coupling (SOC) makes\nthe spin operators non-diagonal in the band index,\nand phonons can change the electron spin through\ne-ph interactions [19]. Theories of spin decoherence\nfocus on two distinct models \u0000the Elliott-Yafet (EY)\nmechanism [25, 26], where e-ph collisions rotate the spin\ndirection, and the Dyakonov-Perel (DP) mechanism [27],\nwhere spin precession in the SOC \feld induces a mo-\ntional narrowing of the spin. The dominant mechanism\ndepends on the system \u0000typically, EY dominates\nin centrosymmetric and DP in non-centrosymmetric\nmaterials. Spin relaxation exhibits opposite trends\nin these two mechanisms, with spin relaxation times\nproportional to the e-ph relaxation times in EY, and\ninversely proportional in DP. We have recently shown\nthat EY spin relaxation can be computed from \frst-\nprinciples in the spin relaxation time approximation\n(sRTA) [28]\u0000the spin counterpart of the transport RTA\nfor charge transport [1, 4] \u0000but spin precession and the\nDP mechanism are neglected in the sRTA.\nHere we show a many-body approach to compute\nthe susceptibility for an arbitrary non-diagonal oper-\nator coupled to e-ph interactions. Our diagrammatic\napproach, based on the Kubo formula with vertex\ncorrections to the susceptibility in an external injection\n\feld, calculates an e\u000bective phonon-dressed operator\nand its renormalized dynamics. We derive a Bethe-\nSalpeter equation (BSE) for the vertex corrections, and\nspecializing to the spin operator, we use the vertex\ncorrections to compute spin relaxation and precession.\nWe show that the vertex corrections can capture spinarXiv:2208.09575v1  [cond-mat.mtrl-sci]  20 Aug 20222\ndecoherence due to both the EY and DP mechanisms\nand can also model the strong-precession regime, a third\nmechanism distinct from EY and DP. We \fnd these\nthree mechanisms in the exact solution of a two-level\nsystem, and also identify them in a real material,\nGaAs, using \frst-principles calculations. Combined with\n\frst-principles e-ph calculations, our method is poised to\nadvance microscopic understanding of phonon-induced\nspin decoherence [29], with applications ranging from\nsolid-state qubits to quantum materials with spin Hall\ne\u000bect, valley-dependent spin physics, and Rashba e\u000bect.\nThe paper is organized as follows: In Sec. II, we\nderive the BSE for the phonon-dressed vertex, discuss its\nphysical interpretation, and calculate the susceptibility\nin response to an injection \feld. In Sec. III \u0000IV, we\napply this formalism to study spin dynamics in a model\ntwo-level system and in a real material, GaAs, discussing\nspin relaxation due to the EY, DP, and strong-precession\nmechanisms.\nII. THEORY\nWe derive a self-consistent BSE for the vertex correc-\ntion to the susceptibility due to e-ph interactions, focus-\ning on a general vector observable ^A. We then present\na physical interpretation of the vertex corrections and\nthe renormalized dynamics of the operator. We employ\natomic units and set ~= 1.\nA. Interacting Green's function\nWe consider an unperturbed Hamiltonian H0diagonal\nin the Bloch basis, hn0kjH0jnki=\"nk\u000enn0. The inter-\nacting imaginary-time Green's function G(i!a) is written\nusing the Dyson equation as [1]\nG(i!a)\u00001=G(0)(i!a)\u00001\u0000\u0006(i!a); (1)\nwhere!aare fermionic Matsubara frequencies, G(0)(i!a)\nis the non-interacting Green's function, and \u0006( i!a) is the\nlowest order (Fan-Migdal) e-ph self-energy [1, 15, 30],\nwhose band- and k-dependent expression is\n\u0006nn0k(i!a) =\u00001\n\fNqVucX\nmm0q\u0017;iqc[gn0m0\u0017(k;q)]\u0003gnm\u0017(k;q)\n\u0002D\u0017q(iqc)Gmm0k+q(i!a+iqc):\n(2)\nHere,\f= 1=kBTat temperature T,Nqis the num-\nber of q-points in the summation, Vucis the unit cell\nvolume,qcis the bosonic Matsubara frequency of the\nphonon, andD\u0017q(iqc) = 2!\u0017q=((iqc)2\u0000!2\n\u0017q) is the non-\ninteracting phonon Green's function for a phonon with\nmode index \u0017, wave-vector q, and energy !\u0017q. Thee-ph\nmatrix elements gnm\u0017(k;q) quantify the probability am-\nplitude for an electron in a Bloch state j nki, with band\n(a)(b)(c)\n=+FIG. 1. (a) Bare bubble diagram without the vertex cor-\nrection. (b) Bubble diagram including the vertex correction.\n(c) Bethe-Salpeter equation for the vertex corrections \u0003 from\nelectron-phonon interactions within the ladder approxima-\ntion. The wavy line is the phonon propagator and the red\ndots are the e-ph matrix elements gnm\u0017(k;q).\nindexnand crystal momentum k, to scatter into a \fnal\nstatej mk+qiby emitting or absorbing a phonon [15, 31],\ngnm\u0017(k;q)=h mk+qj@\u0017q^Vj nki; (3)\nwhere@\u0017q^Vis the perturbation to the potential acting\non an electron due to a given phonon mode ( \u0017;q).\nB. Kubo formula and correlation function\nWe consider a complex vector operator ^A, with ma-\ntrix elements in the direction \u000bwritten as A\u000b\nnmk=\nhmkj^A\u000bjnki. We derive the ^A\u0000^Acorrelation function\nwith a procedure analogous to the derivation of the dc\nconductivity in the ladder approximation [4]. Here, the\noperator ^Ais in general non-diagonal in the band index,\nleading to matrix elements A\u000b\nnmk, so the derivation for\nthe diagonal case given in Ref. [4] needs to be extended\nto non-diagonal operators and vertex corrections.\nWe \frst derive the correlation function in imaginary\ntime and frequency, and then extend it to real frequen-\ncies via analytic continuation. The retarded correlation\nfunction for the operator ^Acan be obtained from the\nKubo formula [1]\n\u001f\u000b\f(p;i\u0017b) =Z\f\n0d\u001cei\u0017b\u001cD\nT\u001c^A\u000b(p;\u001c)^A\f(\u0000p;0)E\n;(4)\nwhere pis a wave-vector, \u0017bis a bosonic Matsubara fre-\nquency,\u001cis imaginary time ranging from 0 to \f= 1=kBT\nat temperature T, andT\u001cis the imaginary time-ordering\noperator. Here we focus on the p!0 limit, so we drop\npfrom the equations. This correlation function can be\nexpressed as a sum of bubble diagrams Pas [1]\n\u001f\u000b\f(i\u0017b) =1\n\fX\ni!aP(i!a;i!a+i\u0017b): (5)3\nLet us consider the bare bubble diagram that includes\nthe electron self-energy only in the electron propagator\nG, as shown in Fig. 1(a):\n\u001f\u000b\f(i\u0017b) =1\n\fVucX\ni!aTr\u0002\nG(i!a)^A\u000bG(i!a+i\u0017b)^A\f\u0003\n;(6)\nwhere the trace is evaluated over the band and momen-\ntum indices. In this expression, the operator ^Acan be\nregarded as the bare vertex of the correlation function.\nFor the velocity operator, Eq. (6) leads to the well-known\nDrude conductivity [1, 4].\nIn this work, the corrections to the vertex originate\nfrom thee-ph interactions, which couple electronic states\nwith di\u000berent band and crystal momenta. Figure 1(b)\nshows the correlation function including the vertex cor-\nrection \u0003,\n\u001f\u000b\f(i\u0017b) =1\n\fVucX\ni!aTr\u0002\nG(i!a)^A\u000bG(i!a+i\u0017b)\n\u0002^A\f\u0003\f(i!a;i!a+i\u0017b)\u0003\n;(7)\nwhere ^A\f\u0003\f(i!a;i!a+i\u0017b) is the phonon-dressed ver-\ntex for the operator ^Ain the Cartesian direction \f. Note\nthat the vertex correction \u0003\f(i!a;i!a+i\u0017b) is a complex-\nvalued vector that contains information about the oper-\nator dynamics renormalized by the e-ph interactions.\nC. Bethe-Salpeter equation for the phonon-dressed\nvertex\nThe leading correction to the vertex is obtained by\nsumming over ladder diagrams, which can be viewed as\nan abstract form of charge conservation in the presence\nofe-ph scattering [1, 4]. The vertex correction \u0003\u000b\nnn0k\nsatis\fes the self-consistent BSE, shown diagrammatically\nin Fig. 1(c) and written as\nA\u000b\nnn0k\u0003\u000b\nnn0k(i!a;i!a+i\u0017b) =A\u000b\nnn0k\n\u00001\n\fNqVucX\nmm0ll0q\u0017;iqc[gn0m0\u0017(k;q)]\u0003gnm\u0017(k;q)D\u0017q(iqc)\n\u0002Gmlk+q(i!a+iqc)Gl0m0k+q(i!a+i\u0017b+iqc)\n\u0002A\u000b\nll0k\u0003\u000b\nll0k+q(i!a+iqc;i!a+i\u0017b+iqc):\n(8)\nThe kernel of this BSE [32] is the e-ph interaction\n[gn0m0\u0017(k;q)]\u0003gnm\u0017(k;q)D\u0017q(iqc).\nFollowing Mahan [1] and Ref. [4], we \frst sum over\nthe bosonic Matsubara frequency iqcin Eq. (8). This\nsummation, de\fned as S(i!a;i!a+i\u0017b), reads:\nS(i!a;i!a+i\u0017b) =X\nll0Sll0(i!a;i!a+i\u0017b)\n=1\n\fX\nll0iqcD\u0017q(iqc) \u0003\u000b\nll0k+q(i!a+iqc;i!a+i\u0017b+iqc)\n\u0002Gmlk+q(i!a+iqc)Gl0m0k+q(i!a+i\u0017b+iqc):(9)As usual, the summation is done by constructing a con-\ntour integral along a circle at in\fnity:\nIdz\n2\u0019inB(z)D\u0017q(z) \u0003\u000b\nll0k+q(i!a+z;i!a+i\u0017b+z)\n\u0002Gmlk+q(i!a+z)Gl0m0k+q(i!a+i\u0017b+z);(10)\nwherenBare Bose-Einstein occupations. The integrand\nhas poles at z=iqc,z=\u0006!\u0017q, and branch cuts along\nz=\u0000i!aandz=\u0000i!a\u0000i\u0017b[1, 4]. Employing Cauchy's\nresidue theorem, we obtain\nSll0(i!a;i!a+i\u0017b)\n=\u0000N\u0017q\u0003\u000b\nll0k+q(i!a+!\u0017q;i!a+i\u0017b+!\u0017q)\n\u0002Gmlk+q(i!a+!\u0017q)Gl0m0k+q(i!a+i\u0017b+!\u0017q)\n\u0000[N\u0017q+ 1]\u0003\u000b\nll0k+q(i!a\u0000!\u0017q;i!a+i\u0017b\u0000!\u0017q)\n\u0002Gmlk+q(i!a\u0000!\u0017q)Gl0m0k+q(i!a+i\u0017b\u0000!\u0017q)\n\u0000Zd\"0\n2\u0019if(\"0)2!\u0017q\n(\"0\u0000i!a)2\u0000!2\u0017qGl0m0k+q(\"0+i\u0017b)\n\u0002[\u0003\u000b\nll0k+q(\"0+i\u0011;\"0+i\u0017b)Gmlk+q(\"0+i\u0011)\n\u0000\u0003\u000b\nll0k+q(\"0\u0000i\u0011;\"0+i\u0017b)Gmlk+q(\"0\u0000i\u0011)]\n\u0000Zd\"0\n2\u0019if(\"0)2!\u0017q\n(\"0\u0000i!a\u0000i\u0017b)2\u0000!2\u0017qGmlk+q(\"0\u0000i\u0017b)\n\u0002[\u0003\u000b\nll0k+q(\"0\u0000i\u0017b;\"0+i\u0011)Gl0m0k+q(\"0+i\u0011)\n\u0000\u0003\u000b\nll0k+q(\"0\u0000i\u0017b;\"0\u0000i\u0011)Gl0m0k+q(\"0\u0000i\u0011)];\n(11)\nwhereN\u0017q=nB(!\u0017q) are temperature dependent phonon\noccupations, f(\") is the Fermi-Dirac distribution func-\ntion, and\u0011is a positive in\fnitesimal.\nThe leading contribution to Sll0(i!a;i!a+i\u0017b) comes\nfrom the combination of retarded and advanced Green's\nfunctions,GRandGA, while terms of O([GR]2;[GA]2)\ncan be neglected at low electron density [1, 4]. There-\nfore, after the analytic continuations i!a!\"\u0000i\u0011\nandi!a+i\u0017b!\"+\u0017+i\u0011, and using the identity\n1\nx+i\u0011=P1\nx\u0000i\u0019\u000e(x), we obtain Sll0(\"\u0000i\u0011;\" +i\u0011) in\nlimit of\u0017!0,\nSll0(\"\u0000i\u0011;\"+i\u0011)\n=\u0000[N\u0017q+f(\"+!\u0017q)]\u0003\u000b\nll0k+q(\"+!\u0017q)\n\u0002GR\nmlk+q(\"+!\u0017q)GA\nl0m0k+q(\"+!\u0017q)\n\u0000[N\u0017q+ 1\u0000f(\"\u0000!\u0017q)]\u0003\u000b\nll0k+q(\"\u0000!\u0017q)\n\u0002GR\nmlk+q(\"\u0000!\u0017q)GA\nl0m0k+q(\"\u0000!\u0017q);(12)\nwhere the index A (R) stands for advanced (retarded)\nfunction, and \u0003\u000b(\")\u0011\u0003\u000b(\"\u0000i\u0011;\"+i\u0011).\nUsing this result, we write the self-consistent BSE for4\nthe phonon-dressed vertex ^A\u0003 at energy \"as:\nA\u000b\nnn0k\u0003\u000b\nnn0k(\") =A\u000b\nnn0k\n+1\nNqVucX\nmm0ll0q\u0017[gn0m0\u0017(k;q)]\u0003gnm\u0017(k;q)A\u000b\nll0k+q\n\u0002\u0014\n(N\u0017q+f(\"+!\u0017q))\u0003\u000b\nll0k+q(\"+!\u0017q)\n\u0002GR\nmlk+q(\"+!\u0017q)GA\nl0m0k+q(\"+!\u0017q)\n+ (N\u0017q+ 1\u0000f(\"\u0000!\u0017q))\u0003\u000b\nll0k+q(\"\u0000!\u0017q)\n\u0002GR\nmlk+q(\"\u0000!\u0017q)GA\nl0m0k+q(\"\u0000!\u0017q)\u0015\n:\n(13)\nBy solving Eq. (13), we obtain the phonon-dressed ver-\ntexA\u000b\nnn0k\u0003\u000b\nnn0k(\") and its dependence on band, crystal\nmomentum and energy.\nIn the weak scattering regime, where the electron spec-tral function has a well-de\fned quasiparticle peak [5] and\nthe o\u000b-diagonal self-energy can be neglected [33, 34], the\nGreen's function becomes band-diagonal and the self-\nenergies can be evaluated on-shell. Then the product\nof the retarded and advanced Green's functions, GRGA,\ncan be approximated as [35]\nGR\nmk+q(\")GA\nm0k+q(\")\n=GA\nm0k+q(\")\u0000GR\nmk+q(\")\nGR\nmk+q(\")\u00001\u0000GA\nm0k+q(\")\u00001\n\u0019\u0019\u000e(\"\u0000\"m0k+q)+\u0019\u000e(\"\u0000\"mk+q)\u0000iP1\n\"\u0000\"m0k+q+iP1\n\"\u0000\"mk+q\ni(\u0006R\nmk+q\u0000\u0006A\nm0k+q) +i(\"mk+q\u0000\"m0k+q);\n(14)\na function that is strongly peaked at electron energies\n\"=\"mk+qand\"=\"m0k+q. Therefore, we can further\nsimplify the full-frequency BSE in Eq. (13) to a double-\npole ansatz, which evaluates the vertex corrections only\nat these two energies:\nA\u000b\nnn0k\u0003\u000b\nnn0k(\") =A\u000b\nnn0k+2\u0019\nNqVucX\nmm0q\u0017[gn0m0\u0017(k;q)]\u0003gnm\u0017(k;q)\n\u00021\n2\u0014\nf(N\u0017q+fmk+q)(\u000e(\"+!\u0017q\u0000\"mk+q)\u0000i\n\u0019P1\n\"+!\u0017q\u0000\"m0k+q)\n+ (N\u0017q+ 1\u0000fmk+q)(\u000e(\"\u0000!\u0017q\u0000\"mk+q)\u0000i\n\u0019P1\n\"\u0000!\u0017q\u0000\"m0k+q)g\u0002A\u000b\nmm0k+q\u0003\u000b\nmm0k+q(\"mk+q)\ni(\u0006R\nmk+q\u0000\u0006A\nm0k+q) +i(\"mk+q\u0000\"m0k+q)\n+f(N\u0017q+fm0k+q)(\u000e(\"+!\u0017q\u0000\"m0k+q) +i\n\u0019P1\n\"+!\u0017q\u0000\"mk+q)\n+ (N\u0017q+ 1\u0000fm0k+q)(\u000e(\"\u0000!\u0017q\u0000\"m0k+q) +i\n\u0019P1\n\"\u0000!\u0017q\u0000\"mk+q)g\u0002A\u000b\nmm0k+q\u0003\u000b\nmm0k+q(\"m0k+q)\ni(\u0006R\nmk+q\u0000\u0006A\nm0k+q) +i(\"mk+q\u0000\"m0k+q)\u0015\n;\n(15)\nwhere\"equals\"nkor\"n0k, andfmk+q\u0011f(\"mk+q).\nWe have tested the consistency of this theory by deriv-\ning a Ward identity [1, 4, 36] relating the self-energy and\nvertex corrections (see Appendix A). This result guaran-\ntees thate-ph diagrams are taken into account consis-\ntently in the self-energy and in our BSE.\nD. The dressed vertex and its interpretation\nWe focus on the dressed operator divided by the band\nenergy di\u000berence, a key term in Eq. (15):\nA\u000b\nmm0k+q\u0003\u000b\nmm0k+q(\"m0k+q)\ni(\u0006R\nmk+q\u0000\u0006A\nm0k+q)+i(\"mk+q\u0000\"m0k+q): (16)\nThis ratio describes the renormalized dynamics associ-\nated with the operator ^Ain the presence of e-ph interac-\ntions. This dynamics is obtained by dividing Eq. (16) bythe bare operator expectation value A\u000b\nmm0k+q, obtaining\n\u0003\u000b\nmm0k+q(\"m0k+q)\ni(\u0006R\nmk+q\u0000\u0006A\nm0k+q)+i(\"mk+q\u0000\"m0k+q): (17)\nThe physical meaning of this ratio can be understood\nby analyzing the simple case of the velocity operator.\nAs the velocity operator is band-diagonal and satis\fes\nv\u000b\nmm0k+q=v\u000b\nmk+q\u000emm0, the band energy di\u000berence in\nthe denominator vanishes, so the denominator is purely\nreal because \u0006A\nm0k+q= (\u0006R\nm0k+q)\u0003. Thus Eq. (16) for the\nvelocity operator becomes\nv\u000b\nmk+q\u0003\u000b\nmmk+q(\"mk+q)\ni(\u0006R\nmk+q\u0000\u0006A\nmk+q)=v\u000b\nmk+q\u001ce-ph\nmk+q\u0003\u000b\nmmk+q(\"mk+q);\n(18)\nwhere we used \u001ce-ph\nmk+q= 1=j2 Im \u0006mk+qjfor thee-ph col-\nlision time. This equation gives the renormalized e-ph5\nTABLE I. Summary of the formalism for charge transport and spin decoherence.\nCharge transport (Ref. [4]) Spin decoherence\nOperator vnk(diagonal) snmk(non-diagonal)\nExternal \feldF Vector potential ( A) Magnetic \feld ( B)\nInjection \feld _F E(\u0017) =\u0000i\u0017A(\u0017) _B(\u0017) =\u0000i\u0017B(\u0017)\nVertex correction \u0003 \u0003\u000b\nnk(\"nk) \u0003\u000b\nnn0k(\"nk);\u0003\u000b\nnn0k(\"n0k)\nRenormalized dynamics \u001c,! \u001c(tr)\u000b\nnk=\u001ce-ph\nnk\u0003\u000b\nnk(\u000fnk)1\n1\n\u001c\u000b\nnn 0k(\"nk)+i!\u000b\nnn 0k(\"nk)=\u0003\u000b\nnn 0k(\"nk)\ni(\u0006R\nnk\u0000\u0006A\nn0k)+i(\"nk\u0000\"n0k)\nmean free path, and dividing by the bare velocity we ob-\ntain the renormalized relaxation time, also known as the\ntransport relaxation time [4],\n\u001c\u000b(tr)\nmk+q\u0011\u001ce-ph\nmk+q\u0003\u000b\nmmk+q(\"mk+q) =\u0003\u000b\nmmk+q(\"mk+q)\ni(\u0006R\nmk+q\u0000\u0006A\nmk+q):\n(19)\nFor a non-diagonal operator, both the vertex correc-\ntion and the operator expectation value are complex, so\nthe ratio in Eq. (17) cannot be represented by a single\nreal quantity with units of time as in Eq. (19). To ex-\ntend the vertex correction to non-diagonal operators, we\ngeneralize this formalism by de\fning the renormalized\nmicroscopic relaxation times \u001c\u000b\nmm0k+q(\") and introduc-\ning the precession frequencies !\u000b\nmm0k+q(\"):\n1\n1\n\u001c\u000b\nmm 0k+q(\")+i!\u000b\nmm0k+q(\")\n\u0011\u0003\u000b\nmm0k+q(\")\ni(\u0006R\nmk+q\u0000\u0006A\nm0k+q)+i(\"mk+q\u0000\"m0k+q);(20)\nwhere\"equals\"mk+qor\"m0k+q. This way, with-\nout the vertex correction, the renormalized relaxation\ntime reduces to the (non-diagonal) e-ph collision time,\n\u001ce-ph\nmm0k+q= 1=jIm \u0006mk+q+ Im \u0006m0k+qj, and the renor-\nmalized precession frequency reduces to the bare oper-\nator rotation frequency, !B= (\"mk+q+ Re \u0006mk+q)\u0000\n(\"m0k+q+ Re \u0006m0k+q), withA\u000b\nmm0k+q(t)/ei!Bt.\nE. Vertex correction to the susceptibility\nWe derive the vertex-corrected susceptibility in re-\nsponse to an external \feld for the generic observable\n^A. Suppose that the complex operator ^A\u000bcouples to\na vector \feldF\u000b, with perturbation Hamiltonian H0=\n\u0000^A\u000bF\u000b. The susceptibility is de\fned as the response\nfunction in\nh^A\u000b(\u0017)i=\u001f\u000b\f(\u0017)F\f(\u0017); (21)\nwhereFis the external \feld along the direction \f, and\nh^A\u000b(\u0017)iis the response of the system along \u000bat fre-\nquency\u0017due to the applied \feld.To study relaxation and dissipation, we rewrite the re-\nsponse of the system as\nh^A\u000b(\u0017)i=\u001b\u000b\f(\u0017)_F\f(\u0017); (22)\nthus expressing it in terms of the susceptibility \u001b\u000b\fto the\n\\injection \feld\" at frequency \u0017, and _F\f(\u0017) =\u0000i\u0017F\f(\u0017).\nThe injection \feld produces a nonequilibrium electron\ndistribution with an injection rate equal to the inverse\nrelaxation time of ^A[37]. From Eqs. (21)-(22), we obtain\n\u001b\u000b\f(\u0017) =\u001f\u000b\f(\u0017)\n\u0000i\u0017: (23)\nWhenFis the vector potential A, the injection \feld\nbecomes the electric \feld E(\u0017) =\u0000i\u0017A(\u0017), the observ-\nable of interest is the current operator A\u000b\nnmk=e\u000enmv\u000b\nnk,\nand\u001b\u000b\f(\u0017) is the frequency-dependent conductivity ten-\nsor. WhenFis the magnetic \feld B, the injection \feld\nis its time derivative, _B(\u0017) =\u0000i\u0017B(\u0017), and the observ-\nable is the electron magnetic moment A\u000b\nnmk=g\u0016Bs\u000b\nnmk,\nwhich is proportional to the spin matrix s\u000b\nnmk[37, 38].\nThese results are summarized in Table I.\nWe write the correlation function with vertex correc-\ntion [see Eq. (7)] as a contour integral along a circle at\nin\fnity [1, 4],\n\u001f\u000b\f(i\u0017b) =\u00001\nVucIdz\n2\u0019if(z) Tr\u0002\nG(z)^A\u000bG(z+i\u0017b)^A\f\u0003\f(z;z+i\u0017b)\u0003\n;(24)\nwhich has branch cuts along z=\u0000i\u0017bandz= 0, and\npoles atz=i!a, and thus\n\u001f\u000b\f(i\u0017b) =1\nVucZd\"\n2\u0019if(\") Tr\u0014\n\u0000G(\"+i\u0011)^A\u000bG(\"+i\u0017b)^A\f\u0003\f(\"+i\u0011;\"+i\u0017b)\n+G(\"\u0000i\u0011)^A\u000bG(\"+i\u0017b)^A\f\u0003\f(\"\u0000i\u0011;\"+i\u0017b)\n\u0000G(\"\u0000i\u0017b)^A\u000bG(\"+i\u0011)^A\f\u0003\f(\"\u0000i\u0017b;\"+i\u0011)\n+G(\"\u0000i\u0017b)^A\u000bG(\"\u0000i\u0011)^A\f\u0003\f(\"\u0000i\u0017b;\"\u0000i\u0011)\u0015\n:(25)\nAfter the analytic continuation i\u0017b!\u0017+i\u0011, we obtain\nthe retarded correlation function to leading order by ne-6\nglecting the terms GRGRandGAGA[1, 4]:\n\u001f\u000b\f(\u0017)\n=1\nVucZd\"\n2\u0019i(f(\")\u0000f(\"+\u0017)) Tr\u0002\nGR(\")^A\u000bGA(\"+\u0017)^A\f\u0003\f(i!a\u0000i\u0011;i!a+i\u0017b+i\u0011)\u0003\n\u00191\nNkVucX\nnmkZd\"\n2\u0019i(f(\")\u0000f(\"+\u0017))A\u000b\nnmkA\f\nmnk\n\u0002\u0003\f\nmnk(\"\u0000i\u0011;\"+\u0017+i\u0011)\u0019\u000e(\"\u0000\"nk)+\u0019\u000e(\"+\u0017\u0000\"mk)\ni(\u0006R\nmk\u0000\u0006A\nnk)+i(\"mk+\u0017\u0000\"nk);\n(26)\nwhereNkis the number of k-points, and we used Eq. (14)\nin the last equality. This equation characterizes the\nfrequency-dependent response of the system [39].\nWe focus on the dc limit \u0017!0, where the driving \feld\nis static. The susceptibility with respect to the injection\n\feld becomes\nlim\n\u0017!0\u001b\u000b\f(\u0017) =\u0000lim\n\u0017!01\n\u0017Im\u001f\u000b\f(\u0017)\n=1\nNkVucReX\nnmkA\u000b\nnmkA\f\nmnk\n\u00021\n2[(\u0000dfnk\nd\")\u0003\f\nmnk(\"nk) + (\u0000dfmk\nd\")\u0003\f\nmnk(\"mk)]\ni(\u0006R\nmk\u0000\u0006A\nnk) +i(\"mk\u0000\"nk):\n(27)\nThis static susceptibility has both band-diagonal ( n=\nm) and o\u000b-diagonal ( n6=m) contributions. The band-\ndiagonal contribution\n\u001b(d)\n\u000b\f(0) =1\nNkVucX\nnkA\u000b\nnnkA\f\nnnk\u001ce-ph\nnk\u0003\f\nnnk(\"nk)(\u0000dfnk\nd\")\n(28)\nis the only contribution to the static susceptibility for\na band-diagonal operator. For example, for the velocity\noperator, Eq. (28) becomes the well-known electrical con-\nductivity tensor within the BTE, which has been studied\nextensively using both empirical and \frst-principles cal-\nculations [23, 24, 31, 40, 41] (note that solving exactly the\nBTE is equivalent to computing the velocity vertex cor-\nrection [4]). For non-diagonal operators, our formalism\nintroduces an o\u000b-diagonal contribution in Eq. (27):\n\u001b(nd)\n\u000b\f(0) =1\nNkVucReX\nn6=mkA\u000b\nnmkA\f\nmnk\n\u00021\n2[(\u0000dfnk\nd\")\u0003\f\nmnk(\"nk) + (\u0000dfmk\nd\")\u0003\f\nmnk(\"mk)]\ni(\u0006R\nmk\u0000\u0006A\nnk) +i(\"mk\u0000\"nk):\n(29)For the velocity operator, this term enables studies of\ncharge transport in the presence of inter-band coher-\nence [42]; for the spin operator, this contribution is\nessential to describe how e-ph interactions modify spin\nprecession.\nF. Renormalized relaxation time\nWe derive an expression for the renormalized macro-\nscopic relaxation time of an operator ^Adue toe-ph in-\nteractions. Using Eq. (28), the average relaxation time\nfor the band-diagonal components A\u000b\nnnkis\n\u001c\u000b\f=P\nnkA\u000b\nnnkA\f\nnnk\u001ce-ph\nnk\u0003\f\nnnk(\"nk)(\u0000dfnk\nd\")\nP\nnkA\u000b\nnnkA\f\nnnk\u0000\n\u0000dfnk\nd\"\u0001:(30)\nFor the velocity operator, this equation gives the\nwell-known Drude dc electrical conductivity, while\nfor the spin operator one obtains the phonon-dressed\nmacroscopic spin relaxation time, as discussed below.\nThese results generalize the linear response treatment\nfor band-diagonal operators presented in Ref. [4] and\nextend it to non-diagonal operators.\nIII. SPIN RELAXATION AND DECOHERENCE\nWe now specialize to the non-diagonal spin operator,\nand apply our formalism to study phonon-induced spin\nrelaxation and decoherence. The BSE for the phonon-\ndressed spin vertex \u0000called hereafter spin-phonon BSE\n\u0000is a key result obtained from Eq. (13) by replacing\nA\u000b\nnn0kwith the spin operator s\u000b\nnn0k. In matrix form and\nusing a compact notation, the spin-phonon BSE can be\nwritten in a way that clearly matches the diagram in\nFig. 1(c):\ns\u0003k(\")=sk+1\nNqVucX\n\u0017q\u0006gy\n\u0017kq\u0002\nGAs\u0003GR\u0003\nk+q;\n\"\u0006!\u0017qg\u0017kqF\u0006(T);\n(31)\nwhere s\u0003k(\") = snn0k\u0003nn0k(\") is the phonon-dressed\nspin vertex, F\u0006(T) =N\u0017q+1\n2\u0006[f(\"\u0006!\u0017q)\u00001\n2] is a ther-\nmal occupation factor at temperature T, and [ g\u0017kq]nm=\ngnm\u0017(k;q) aree-ph matrix elements [31].\nIn the weak scattering regime, this spin-phonon BSE\ncan be rewritten using the double-pole ansatz discussed\nabove (where \"equals\"nkor\"n0k):7\ns\u000b\nnn0k\u0003\u000b\nnn0k(\") =s\u000b\nnn0k+2\u0019\nNqVucX\nmm0q\u0017[gn0m0\u0017(k;q)]\u0003gnm\u0017(k;q)\n\u00021\n2\u0014\nf(N\u0017q+fmk+q)(\u000e(\"+!\u0017q\u0000\"mk+q)\u0000i\n\u0019P1\n\"+!\u0017q\u0000\"m0k+q)\n+ (N\u0017q+ 1\u0000fmk+q)(\u000e(\"\u0000!\u0017q\u0000\"mk+q)\u0000i\n\u0019P1\n\"\u0000!\u0017q\u0000\"m0k+q)g\u0002s\u000b\nmm0k+q\u0003\u000b\nmm0k+q(\"mk+q)\ni(\u0006R\nmk+q\u0000\u0006A\nm0k+q) +i(\"mk+q\u0000\"m0k+q)\n+f(N\u0017q+fm0k+q)(\u000e(\"+!\u0017q\u0000\"m0k+q) +i\n\u0019P1\n\"+!\u0017q\u0000\"mk+q)\n+ (N\u0017q+ 1\u0000fm0k+q)(\u000e(\"\u0000!\u0017q\u0000\"m0k+q) +i\n\u0019P1\n\"\u0000!\u0017q\u0000\"mk+q)g\u0002s\u000b\nmm0k+q\u0003\u000b\nmm0k+q(\"m0k+q)\ni(\u0006R\nmk+q\u0000\u0006A\nm0k+q) +i(\"mk+q\u0000\"m0k+q)\u0015\n:\n(32)\nThis BSE for the phonon-dressed spin vertex, used in\nthis work to study spin dynamics, should not be con-\nfused with the widely used BSE for excitons and optical\nspectra [2], which is entirely unrelated.\nThe vertex corrections \u0003\u000b\nnn0kobtained by solving the\nBSE govern spin dynamics as they renormalize spin re-\nlaxation and precession [29]. The macroscopic spin re-\nlaxation times are obtained using the thermal average in\nEq. (30),\n\u001c(s)\n\u000b\f=P\nnks\u000b\nnnks\f\nnnk\u001ce-ph\nnk\u0003\f\nnnk(\"nk)(\u0000dfnk\nd\")\nP\nnks\u000b\nnnks\f\nnnk\u0000\n\u0000dfnk\nd\"\u0001:(33)\nFor\u000b=\falong the external magnetic \feld, Eq. (33)\ngives the longitudinal spin relaxation time, usually called\nT1, along the direction \u000b, while for a perpendicular mag-\nnetic \feld one obtains the transverse spin relaxation time,\nT2[43]. The renormalized microscopic spin relaxation\ntimes (\u001c\u000b\nnn0k) and spin precession rates ( !\u000b\nnn0k), which\nare matrices in Bloch basis, are computed from the ver-\ntex corrections \u0003\u000b\nnn0kusing\n1\n1\n\u001c\u000b\nnn 0k(\")+i!\u000b\nnn0k(\")\u0011\u0003\u000b\nnn0k(\")\ni(\u0006R\nnk\u0000\u0006A\nn0k) +i(\"nk\u0000\"n0k):\n(34)\nThe diagonal components with n=n0give the renor-\nmalized microscopic spin relaxation times, \u001c\f\nnnk=\n\u001ce-ph\nnk\u0003\f\nnnk(\"nk), entering Eq. (33).IV. RESULTS\nWe apply our formalism to study spin relaxation and\ndecoherence. We \frst present analytic results for a two-\nlevel model system, and then focus on \frst-principles cal-\nculations on a real material, GaAs. Application to a\nwider range of materials is presented in our companion\npaper [29].\nA. Two-level system with optical phonon\nscattering\nWe study spin dynamics in a two-level system to un-\nderstand di\u000berent phonon-induced spin relaxation mech-\nanisms. In our model, the electron spins undergo phonon-\ninduced spin-\rip transitions together with spin precession\nin the SOC \feld modi\fed by the e-ph interactions. We\nsolve the spin-phonon BSE for this system and derive\nanalytic expressions for the vertex corrections and spin\nrelaxation times. Our analysis sheds light on phonon-\ndressed operators and their renormalized dynamics, pro-\nviding a starting point to understand phonon-induced\nspin relaxation in real materials with complex band struc-\ntures, phonon dispersions, and e-ph interactions.\nConsider a periodic two-level system where each level\nis spin degenerate (see Fig. 2). The Hilbert space con-\nsists of four Bloch states, which are eigenstates of the\nHamiltonian:\nHjni=\"njni (35)\nwherejniis then-th energy eigenstate. The two lowest-\nenergy statesj1iandj2iare degenerate ( \"1=\"2) and\ndi\u000ber only in their spin part. The other two states, j3i\nandj4i, are higher in energy by !Oand are perturbed\nby an internal magnetic \feld along ^ xdue to SOC. This\n\feld causes a small Zeeman splitting, \u0001 \u001c!O, such that\n\"3;4=\"1+!O\u0006\u0001\n2.\nWe separate the space-dependent part j niand the\nspin-dependent part j\u001fniof the two eigenstates as jni=\nj ni\nj\u001fni. The two lowest states have an identical space-8\ndependent partj iand are spin polarized along ^ z:\nj1i=j i\n \n1\n0!\n;j2i=j i\n \n0\n1!\n; (36)\nwith the following spin matrix elements along z:\nh1j^szj1i=\u0000h2j^szj2i=1\n2;\nh1j^szj2i= 0:(37)\nAbove, ^s= ^\u001b=2 is the spin operator and ^ \u001bare Pauli\nmatrices. The two upper bands have an identical space-\ndependent partj\u001eiand are spin polarized along ^ x:\nj3i=j\u001ei\n1p\n2 \n1\n1!\n;j4i=j\u001ei\n1p\n2 \n1\n\u00001!\n;(38)\nwith spin matrix elements\nh3j^szj3i=h4j^szj4i= 0;\nh3j^szj4i=1\n2:(39)\nThe space-dependent part of the two lower states is or-\nthogonal to that of the upper states, and the spin matrix\nelements between the two sets of states are zero.\nIn our model, an electron can scatter between the lower\nand upper levels by emitting or absorbing an optical\nphonon. These transitions are associated with e-ph ma-\ntrix elements gnm=hmj\u0001^Vjni, where \u0001 ^Vis the pertur-\nbation potential due to the optical phonon. We assume\nthat this perturbation potential has the form\n\u0001^V= \u0001^V(r)\n \na b\nb a!\n; (40)\nwhere ^V(r) is the space-dependent part and\u0000a b\nb a\u0001\nthe\nspin-dependent part of the perturbation, with aandb\nreal numbers. Due to the presence of SOC, the spin-\ndependent part\u0000a b\nb a\u0001\nis di\u000berent from the identity ma-\ntrix. We consider a small spin-mixing b, wherea2+b2=\n1, so that each phonon collision has a small probability\nb2\u001c1 to \rip the z-component of the spin [25]. The\ne-ph matrix elements become gnm=g0h\u001fmj\u0000a b\nb a\u0001\nj\u001fni,\nwhereg0=h\u001ej\u0001^V(r)j i. Thus the spin-dependent e-ph\nmatrix elements are\ng13=g23=1p\n2g0(a+b);\ng14=\u0000g24=1p\n2g0(a\u0000b):(41)\nB. Spin-phonon BSE and spin relaxation times\nWe construct the spin-phonon BSE for our two-level\nsystem. There are four non-zero spin matrix elements in\nFIG. 2. Schematic of the energy levels, scattering rates, and\nspin orientations of the four Bloch states in our model.\nour model: sz\n11,sz\n22,sz\n34, andsz\n43, wheresz\nnm=hmj^szjni.\nUsing Eq. (32), the diagonal matrix elements have only\none vertex correction, while the o\u000b-diagonal matrix ele-\nments have two energy-dependent vertex corrections. As-\nsuming a small Zeeman splitting \u0001, we can neglect the\nenergy dependence of the o\u000b-diagonal vertex corrections,\nand de\fne \u0003z\nnm= \u0003z\nnm(\"n). We thus have four unknown\nvertex corrections: \u0003z\n11, \u0003z\n22, \u0003z\n34, and \u0003z\n43. Taking the\ne-ph scattering rates of the two lower and upper states to\nbe \u0000 1and \u0000 3, respectively, the spin-phonon BSE becomes\n\u0003z\n11= 1 +\u00121\n2\u0000b2\u0013\u0012\u00001\u0003z\n34\n\u0000i\u0001 + \u0000 3+\u00001\u0003z\n43\ni\u0001 + \u0000 3\u0013\n\u0000\u0003z\n22=\u00001\u0000\u00121\n2\u0000b2\u0013\u0012\u00001\u0003z\n34\n\u0000i\u0001 + \u0000 3+\u00001\u0003z\n43\ni\u0001 + \u0000 3\u0013\n\u0003z\n34= 1 +\u00121\n2\u0000b2\u0013\u0012\u00003\u0003z\n11\n\u00001+\u00003\u0003z\n22\n\u00001\u0013\n\u0003z\n43= 1 +\u00121\n2\u0000b2\u0013\u0012\u00003\u0003z\n11\n\u00001+\u00003\u0003z\n22\n\u00001\u0013\n:(42)\nNote that this equation treats the phonon-induced spin\n\rips and the spin precessional dynamics self-consistently.\nThe \frst two lines in Eq. (42) give \u0003z\n11= \u0003z\n22, while\nfrom the third and fourth lines we obtain \u0003z\n34= \u0003z\n43.\nTherefore, the solution of the spin-phonon BSE is\n\u0003z\n11= \u0003z\n22=\u00002\n3+ \u00012+ \u00003\u00001(1\u00002b2)\n\u00002\n34b2(1\u0000b2) + \u00012\n\u0003z\n34= \u0003z\n43=(\u00002\n3+ \u00012)(\u00001+ \u00003(1\u00002b2))\n(\u00002\n34b2(1\u0000b2) + \u00012)\u00001:(43)\nThe state-dependent spin relaxation times \u001cz\n11,\u001cz\n22,\u001cz\n34,\nand\u001cz\n43are obtained using Eq. (20):\n\u001cz\n11=\u001cz\n22=\u00002\n3+ \u00012+ \u00003\u00001(1\u00002b2)\n(\u00002\n34b2(1\u0000b2) + \u00012)\u00001\n\u001cz\n34=\u001cz\n43=(\u00002\n3+ \u00012)(\u00001+ \u00003(1\u00002b2))\n(\u00002\n34b2(1\u0000b2) + \u00012)\u00001\u00003:(44)\nFigure 3(a) shows the vertex correction for the lowest\nstate, \u0003z\n11in Eq. (43), and Fig. 3(b) shows the spin relax-\nation time\u001cz\n11from Eq. (44), both plotted as a function9\n0.0010 .010 .11 1 011010010000\n.0010 .010 .11 1 0110100\nVertex correction Λ11 \nSpin-phonon BSE \nElliott-Yafet \nDyakonov-Perel \nStrong precession(a)\nS pin relaxation time (Units of 1/Δ)e\n-ph collision time (Units of 1/Δ)(b)D\nyakonov-Perelr\negimeElliott-Yafetr\negimeStrong precessionr\negimeStrong precessionr\negimeDyakonov-Perelr\negimeElliott-Yafetr\negime\nFIG. 3. (a) The vertex correction \u0003z\n11as a function of the e-ph\ncollision time \u001c1for the model system in Fig. 2. (b) Spin re-\nlaxation time as a function of the e-ph collision time \u001c1for our\nmodel system. We show the full solution of the spin-phonon\nBSE in Eqs. (43)-(44)) (black curve) and approximate results\nfor the Elliott-Yafet (blue, Eqs. (45)-(46)), Dyakonov-Perel\n(red, Eqs. (49)-(50)), and strong-precession regimes (green,\nEqs. (51)-(52)). These results are obtained by setting \u001c3=\u001c1\nandb= 0:02.\nof thee-ph collision time \u001c1= 1=\u00001. These quantities\nshow three distinct regimes with a qualitatively di\u000berent\ndependence on the e-ph collision time, which correspond\nto the EY, DP and strong-precession regimes. Our for-\nmalism encompasses these three regimes [29] because it\ncan capture both spin-\rip scattering and spin precession.\nThe physics of these regimes is discussed below.\nC. Elliott-Yafet regime\nWe \frst focus on the EY regime, where spin relaxation\noccurs primarily through spin-\rip transitions. In this\nregime, the spin-\rip probability is small since b\u001c1,\nand the spin precession rate is much smaller than the\nscattering rate, \u0001 \u001c2\u00003b. Using these conditions inEq. (43), we obtain the following vertex corrections in\nthe EY regime:\n\u0003z\n11= \u0003z\n22=\u001c1+\u001c3\n4b21\n\u001c1\n\u0003z\n34= \u0003z\n43=\u001c1+\u001c3\n4b21\n\u001c3;(45)\nwhere\u001c1= 1=\u00001and\u001c3= 1=\u00003are state-dependent e-ph\ncollision times. These vertex corrections determine the\nspin relaxation times via Eq. (34):\n\u001cz\n11=\u001cz\n22=\u001cz\n34=\u001cz\n43=\u001c1+\u001c3\n4b2: (46)\nThese results, shown in Fig. 3, approximate well the\nsolution of the spin-phonon BSE in the EY regime.\nIn the conventional theory of EY spin relaxation, the\nspin relaxation times are proportional to 1 =b2and to\nthee-ph collision times (here, \u001c1and\u001c3) [28, 44, 45].\nOur results in Eq. (46) are consistent with that trend,\nalthough the spin relaxation times are proportional to\nthe average of the e-ph collision times of the two levels,\n(\u001c1+\u001c3)=2, and not to their individual values. This\ndi\u000berence is a result of considering both forward- and\nback-scattering processes between the two electronic lev-\nels in the spin-phonon BSE [4], di\u000berent from the simpler\nsRTA [28, 44, 45] which neglects electron back-scattering.\nD. Dyakonov-Perel regime\nNext, we discuss the regime where spin precession is\nimportant and governs spin relaxation. Here, the spin-\n\rip probability is still small ( b\u001c1) but the internal\nmagnetic \feld due to SOC is signi\fcant, such that \u0001 \u001d\n2\u00003b. Inserting these conditions in Eqs. (43)-(44), the\nvertex corrections become\n\u0003z\n11= \u0003z\n22=\u00002\n3+ \u00003\u00001+ \u00012\n\u00012;\n\u0003z\n34= \u0003z\n43=(\u00002\n3+ \u00012)(\u00001+ \u00003)\n\u00012\u00001;(47)\nand for the spin relaxation times we obtain\n\u001cz\n11=\u001cz\n22=\u00002\n3+ \u00003\u00001+ \u00012\n\u00012\u00001;\n\u001cz\n34=\u001cz\n43=(\u00002\n3+ \u00012)(\u00001+ \u00003)\n\u00012\u00001\u00003:(48)\nThese results describe spin relaxation governed by spin\nprecession and renormalized by the e-ph interactions.\nIn the DP regime, the e-ph scattering rates are much\ngreater than the bare spin precession rate \u0001, and thus\n\u00001;3\u001d\u0001. The vertex corrections in the DP regime be-\ncome\n\u0003z\n11= \u0003z\n22=\u00002\n3+ \u00003\u00001\n\u00012\u00001\u00001\n\u0003z\n34= \u0003z\n43=\u00002\n3+ \u00003\u00001\n\u00012\u00001\u00003;(49)10\nwhile for the spin relaxation times we obtain\n\u001cz\n11=\u001cz\n22=\u001cz\n34=\u001cz\n43=\u00002\n3+ \u00003\u00001\n\u00012\u00001: (50)\nThese results, shown in Fig. 3, are an excellent approxi-\nmation to the BSE solution in the DP regime.\nA hallmark of DP relaxation is the inverse propor-\ntionality between the spin relaxation and e-ph collision\ntimes [19], a trend captured by our treatment of the DP\nregime [see Eq. (50)]. Our formalism shows in Eq. (49)\nthat this trend originates from the inverse-square scaling\nof the vertex corrections with the e-ph collision times,\n\u0003\u0018\u00002\n1;3\u00181=\u001c2\n1;3. Note that in the EY regime, rescaling\nthee-ph collision times by a constant factor has no ef-\nfect on the vertex corrections, which depend only on the\nratio\u001c1=\u001c3, and thus the EY spin relaxation times are\nproportional to the e-ph collision times. The situation is\ndi\u000berent in the DP regime, where rescaling the e-ph col-\nlision times changes the vertex corrections. These scal-\ning trends can be employed in real materials to identify\nthe dominant microscopic mechanisms for spin relaxation\nand decoherence [29].\nE. Strong-precession regime\nA third, distinct regime is realized when the Zeeman\nsplitting is much greater than the e-ph scattering rates,\n\u0001\u001d\u00001;3, such that the spins precess much faster than\nthe rate of e-ph collisions. In this strong-precession\nregime [43, 46, 47], the vertex corrections become\n\u0003z\n11= \u0003z\n22= 1;\n\u0003z\n34= \u0003z\n43= 1 +\u00003\n\u00001(51)\nand can be approximated as \u0003 \u00191. Therefore, the ver-\ntex corrections are less important than in the EY or DP\nregimes. It follows that in the strong-precession regime\nthe spin relaxation times are proportional to the e-ph\ncollision times:\n\u001cz\n11=\u001cz\n22=1\n\u00001;\n\u001cz\n34=\u001cz\n43=1\n\u00001+1\n\u00003:(52)\nThese results, shown in Fig. 3, approximate well the BSE\nsolution in the strong-precession regime. Their physi-\ncal interpretation is interesting: in the strong-precession\nregime, the spins precess for many full cycles between\nphonon collisions, randomizing the spin direction. Conse-\nquently, the spin relaxation times become equal to the e-\nph collision times, and thus the vertex corrections \u0003 \u00191\ncan be neglected.F. Uncovering the three regimes in GaAs\nTo identify the three spin relaxation mechanisms in a\nreal material, we study spin relaxation in GaAs using\nour \frst-principles implementation of the spin-phonon\nBSE [29]. We focus on spin relaxation for conduction\nband electrons in GaAs at room temperature (300 K),\nand investigate how the spin relaxation times depend on\nthee-ph collision times.\nWe compute the ground state and band structures\nof GaAs with density functional theory (DFT), using a\nplane-wave basis in the Quantum ESPRESSO code [48]\nand employing the HSE06 hybrid functional [49] to ob-\ntain an accurate band gap and electronic structure. We\nuse fully relativistic norm-conserving pseudopotentials,\ngenerated in the local-density approximation (LDA) with\nPseudo Dojo [50], together with a kinetic energy cut-\no\u000b of 72 Ry and a relaxed lattice constant of 5.60 \u0017A.\nThe phonon energies and perturbation potentials are\ncomputed using density functional perturbation theory\n(DFPT) [7]. Using our perturbo code [31], we compute\nthee-ph matrix elements on coarse Brillouin zone grids\nwith 8\u00028\u00028kandqpoints, following which we generate\nspinor Wannier functions with the Wannier90 code [51]\nand use them in perturbo [31], with a method we devel-\noped in Ref. [28], to jointly interpolate the e-ph matrix\nelements and spin matrices. The long-range quadrupole\ne-ph interactions [10, 11, 52, 53] are included to fully ac-\ncount fore-ph coupling with long-wavelength phonons.\nWe interpolate the e-ph matrix elements to \fne Brillouin\nzone grids with up to 200 \u0002200\u0002200kandqpoints, and a\n10 meV Gaussian broadening for the energy-conserving\ndelta functions in the e-ph scattering rates [41]. The\nspin relaxation times in Eq. (30) are computed using the\ntetrahedron integration method [54], assuming a non-\ndegenerate electron concentration of 1016cm\u00003. The\nspin-phonon BSE in Eq. (32) is solved with an augmented\niterative approach described in Ref. [29].\nUsing our \frst-principles spin-phonon BSE, we com-\npute the spin relaxation time for electron spins in GaAs,\nand obtain a value 51 ps at 300 K in excellent agreement\nwith the experimental value of 42 ps [55]. We also obtain\nan average e-ph collision time of 410 fs, computing using\nh\u001ce-phi=P\nnk\u001ce-ph\nnnk(\"nk)(\u0000dfnk\nd\")\nP\nnk\u0000\n\u0000dfnk\nd\"\u0001 (53)\nand an average vertex correction h\u0003zi\u0019171, computed\nfrom\nh\u0003zi=P\nnkjsz\nnnkj2\u0003z\nnnk(\"nk)(\u0000dfnk\nd\")\nP\nnkjsz\nnnkj2\u0000\n\u0000dfnk\nd\"\u0001: (54)\nThese \\real-material\" values for GaAs are shown with a\ndot in Fig. 4(a) and (b), where we also show spin relax-\nation times and vertex corrections obtained by arti\fcially\nvarying the average e-ph collision time (by rescaling the\ne-ph matrix elements). The resulting trends show that\nthe spin relaxation times are inversely proportional to11\n0.010 .11 1 011010010000\n.010 .11 1 010100Vertex correction Spin-phonon BSE(a)S pin relaxation time (ps)e\n-ph collision time (ps)(b)D\nyakonov-Perelr\negimeElliott-Yafetr\negimeStrong precessionr\negimeStrong precessionr\negimeDyakonov-Perelr\negimeElliott-Yafetr\negimeG\naAs\nFIG. 4. First-principles calculations of (a) vertex corrections\nand (b) spin relaxation times for electron spins in GaAs, com-\nputed at room temperature and plotted as a function of the\naveragee-ph collision time. The arrows indicate the values\nfor the real material, GaAs (black dot), while the other val-\nues are obtained by rescaling the e-ph interaction strength, as\nexplained in the text. The vertical dotted lines are placed at\nthe in\rection points of the spin relaxation times, and separate\nthe EY, DP, and strong-precession regimes.\nthee-ph collision time, placing GaAs in the DP spin-\nrelaxation regime.\nUpon decreasing the average e-ph collision time h\u001ce-phi\nin Fig. 4(a), the system evolves from the DP to the EY\nregime, and the vertex correction \frst increases and then\nsaturates to a maximal value of \u00181900 in the short e-ph\ncollision time limit. As a consequence, the spin relax-\nation time in Fig. 4(b) peaks at h\u001ce-phi\u00190:09 ps and\nthen decreases linearly at shorter e-ph collision times.\nThis crossover from the DP to the EY mechanism, ob-\ntained here by arti\fcially tuning the e-ph collision time\nin GaAs, is consistent with the results from our model\nsystem.\nConversely, when the e-ph collision time is increased,\nthe system transitions from the DP to the strong-\nprecession regime: Fig. 4(a) shows that the vertex cor-\nrection \frst decreases as h\u001ce-phi\u00002and then plateaus to aTABLE II. Summary of the Elliott-Yafet, Dyakonov-Perel,\nand strong-precession regimes.\nEY DP Strong-precession\nSpin-\rip Spin precession Spin precession\n\u0003\u00181=b2\u00181=(\u001ce-ph)2\u00181\n\u001cs\u0018\u001ce-ph=b2\u00181=\u001ce-ph\u0018\u001ce-ph\nminimal value close to unity for long e-ph collision times.\nAccordingly, the spin relaxation time in Fig. 4(b) reaches\na minimum forh\u001ce-phi\u00193 ps and then increases linearly\nat longere-ph collision times.\nIn summary, real materials exhibit the same spin re-\nlaxation mechanisms and vertex correction trends as our\ntwo-level system. These regimes for phonon-induced spin\ndynamics are summarized in Table II. The presence of\nthree distinct spin-relaxation regimes is general, and we\nexpect it to be valid beyond the case of a simple semi-\nconductor (GaAs) studied here.\nV. DISCUSSION\nOur formalism can capture all three of the EY, DP\nand strong-precession regimes in a uni\fed framework.\nThe reason can be inferred from the diagrammatic rep-\nresentation of the spin-phonon BSE in Fig. 1(c). In the\ndiagrams, the EY relaxation is due to e-ph scattering\nin the presence of spin mixing, which originates from\nthee-ph interactions [ gn0m0\u0017(k;q)]\u0003andgnm\u0017(k;q) and\nthe wiggly line in the kernel of the BSE. The DP and\nstrong-precession mechanisms are included by virtue of\nthe electron propagators with two di\u000berent band indices,\nGmlk+qGl0m0k+q, placed between the wiggly line and the\nspin vertex. These propagators take into account spin\nprecession (due to the SOC \feld) between e-ph colli-\nsions. This elegant formalism captures a wide range of\nspin physics in a single diagram.\nAlthough our discussion has focused on T1spin relax-\nation times, the spin decoherence times T2at \fnite mag-\nnetic \felds can also be computed, as we plan to show\nin future work. Finally, the approach presented in this\nwork for the phonon-dressed vertex is general and goes\nbeyond spin relaxation. It can be employed to study the\ndynamics of any observable that couples with phonons,\nfor which we also expect to \fnd the three phonon-induced\nrelaxation regimes discussed above for spin dynamics.\nVI. CONCLUSION\nWe have formulated a theory for the vertex cor-\nrections from e-ph interactions to the susceptibility\nof a non-diagonal operator. The key result is a self-\nconsistent BSE to calculate the phonon-dressed vertex,\nwhich encodes the dynamics of the operator coupling\nwith phonons. When applied to spin, this approach12\nenables quantitative calculations of spin relaxation\nand decoherence [29]. We have shown that our spin-\nphonon BSE captures both spin-\rip transitions and spin\nprecession, unifying the treatment of three spin deco-\nherence mechanisms (EY, DP, and strong-precession)\nconventionally treated with separate heuristic models.\nBy leveraging e\u000ecient work\rows for \frst-principles\ne-ph calculations [31], our method enables quantitative\nstudies of spin relaxation and decoherence in a wide\nrange of bulk and two-dimensional materials, as we\nshow in the companion paper [29]. These advances\nopen new avenues for understanding spin relaxation and\ndecoherence in spintronics, magnetism, multiferroics,\nquantum materials and quantum technologies.\nACKNOWLEDGMENTS\nThis work was supported by the National Science\nFoundation under Grants No. DMR-1750613 and QII-\nTAQS 1936350, which provided for method development,\nand Grant No. OAC-2209262, which provided for code\ndevelopment. J.P. acknowledges support by the Korea\nFoundation for Advanced Studies.\nAppendix A: Ward identity\nWe derive a Ward identity for our BSE in Eq. (13). The\nWard identity relates the vertex corrections with the elec-\ntron self-energy [1, 4, 36], and guarantees that diagrams\nare taken into account consistently in the self-energy and\nin the BSE for the vertex [1, 4, 36].\nFor a system with Hamiltonian H, the operator ^Ais\nrelated to the the negative derivative of the Hamilto-\nnian with respect to the external \feld F,^A=\u0000rFH.\nWe compute the change of the Fan-Migdal self-energy in\nEq. (2) with respect to F, by taking the derivative\nrF\u0006nn0k(i!a)\n=\u00001\n\fNqVucX\nmm0ll0q\u0017;iqc[gn0m0\u0017(k;q)]\u0003gnm\u0017(k;q)\n\u0002D\u0017q(iqc) (rFGmm0k(i!a+iqc)):\n(A1)\nEmploying the matrix identity\nrFGmm0k=\u0000X\nll0Gmlk\u0002\nrF(G\u00001)\u0003\nll0kGl0m0k\n=\u0000X\nll0Gmlk(\u0000rFHll0k\u0000rF\u0006ll0k)Gl0m0k;\n(A2)we obtain a self-consistent equation for the self-energy\nderivatives:\nrF\u0006nn0k(i!a)\n=\u00001\n\fNqVucX\nmm0ll0q\u0017;iqc[gn0m0\u0017(k;q)]\u0003gnm\u0017(k;q)\n\u0002D\u0017q(iqc)Gmlk+q(i!a+iqc)Gl0m0k+q(i!a+iqc)\n\u0002(rFHll0k+rF\u0006ll0k(i!a+iqc)):\n(A3)\nComparing Eq. (A3) and Eq. (8) in the i\u0017b!0 limit,\nwe discover the Ward identity expressed in terms of the\nHamiltonian, vertex corrections, and self-energy:\n(rFH)\u0003(i!a;i!a) =rFH+rF\u0006(i!a); (A4)\nwhere ( rFH)\u0003(i!a;i!a) = (@Hnn 0k\n@F\u000b)\u0003\u000b\nnn0k(i!a;i!a).\nEquation (A4) can be equivalently expressed in terms\nof the operator matrix elements A\u000b\nnn0k,\nA\u000b\nnn0k\u0003\u000b\nnn0k(i!a;i!a) =A\u000b\nnn0k\u0000@\u0006nn0k(i!a)\n@F\u000b:(A5)\nOur expression for the Ward identity in Eqs. (A4)-\n(A5) is consistent with the Ward identity for the velocity\noperator derived in Ref. [4]:\nv\u000b\nnnk\u0003\u000b\nnnk(i!a;i!a) =v\u000b\nnnk+@\u0006nnk(i!a)\n@k\u000b; (A6)\nas the velocity operator is de\fned as ^ v=rFHwith\nF=k. This result further validates our BSE for the\nphonon-dressed vertex.\nREFERENCES\n[1] G. D. Mahan, Many-Particle Physics , 3rd ed., Physics of\nSolids and Liquids (2000).\n[2] M. Rohl\fng and S. G. Louie, Phys. Rev. B 62, 4927\n(2000).\n[3] A. Sekine, D. Culcer, and A. H. MacDonald, Phys. Rev.\nB96, 235134 (2017).\n[4] S. Kim, S. Woo, and H. Min, Phys. Rev. B 99, 165107\n(2019).\n[5] J.-J. Zhou and M. Bernardi, Phys. Rev. Res. 1, 033138\n(2019).\n[6] D. C. Desai, B. Zviazhynski, J.-J. Zhou, and M. Bernardi,\nPhys. Rev. B 103, L161103 (2021).\n[7] S. Baroni, S. de Gironcoli, A. Dal Corso, and P. Gian-\nnozzi, Rev. Mod. Phys. 73, 515 (2001).\n[8] F. Giustino, M. L. Cohen, and S. G. Louie, Phys. Rev.\nB76, 165108 (2007).\n[9] L. A. Agapito and M. Bernardi, Phys. Rev. B 97, 235146\n(2018).\n[10] V. A. Jhalani, J.-J. Zhou, J. Park, C. E. Dreyer, and\nM. Bernardi, Phys. Rev. Lett. 125, 136602 (2020).13\n[11] G. Brunin, H. P. C. Miranda, M. Giantomassi, M. Royo,\nM. Stengel, M. J. Verstraete, X. Gonze, G.-M. Rignanese,\nand G. Hautier, Phys. Rev. Lett. 125, 136601 (2020).\n[12] I.-T. Lu, J. Park, J.-J. Zhou, and M. Bernardi, npj Com-\nput. Mater. 6, 1 (2020).\n[13] I.-T. Lu, J.-J. Zhou, J. Park, and M. Bernardi, Phys.\nRev. Mater. 6, L010801 (2022).\n[14] J.-J. Zhou, J. Park, I. Timrov, A. Floris, M. Cococ-\ncioni, N. Marzari, and M. Bernardi, Phys. Rev. Lett. 127,\n126404 (2021).\n[15] M. Bernardi, Eur. Phys. J. B 89, 239 (2016).\n[16] W. E. Pickett, Rev. Mod. Phys. 61, 433 (1989).\n[17] G. Balasubramanian, I. Y. Chan, R. Kolesov, M. Al-\nHmoud, J. Tisler, C. Shin, C. Kim, A. Wojcik, P. R. Hem-\nmer, A. Krueger, T. Hanke, A. Leitenstorfer, R. Brats-\nchitsch, F. Jelezko, and J. Wrachtrup, Nature 455, 648\n(2008).\n[18] R. Jansen, Nat. Mater. 11, 400 (2012).\n[19] I. \u0014Zuti\u0013 c, J. Fabian, and S. Das Sarma, Rev. Mod. Phys.\n76, 323 (2004).\n[20] M. Veldhorst, C. H. Yang, J. C. C. Hwang, W. Huang,\nJ. P. Dehollain, J. T. Muhonen, S. Simmons, A. Laucht,\nF. E. Hudson, K. M. Itoh, A. Morello, and A. S. Dzurak,\nNature 526, 410 (2015).\n[21] A. Noiri, K. Takeda, T. Nakajima, T. Kobayashi, A. Sam-\nmak, G. Scappucci, and S. Tarucha, Nature 601, 338\n(2022).\n[22] L. Petit, H. G. J. Eenink, M. Russ, W. I. L. Lawrie,\nN. W. Hendrickx, S. G. J. Philips, J. S. Clarke, L. M. K.\nVandersypen, and M. Veldhorst, Nature 580, 355 (2020).\n[23] J. M. Ziman, Electrons and Phonons: The Theory\nof Transport Phenomena in Solids (Oxford University\nPress, New York, 2001).\n[24] M. Lundstrom, Fundamentals of Carrier Transport\n(Cambridge University Press, New York, 2009).\n[25] R. J. Elliott, Phys. Rev. 96, 266 (1954).\n[26] Y. Yafet, in Solid State Physics , Vol. 14, edited by\nF. Seitz and D. Turnbull (1963) pp. 1{98.\n[27] M. Dyakonov and V. Perel, Sov. Phys. Solid State USSR\n13, 3023 (1972).\n[28] J. Park, J.-J. Zhou, and M. Bernardi, Phys. Rev. B 101,\n045202 (2020).\n[29] J. Park, J.-J. Zhou, and M. Bernardi arXiv:2203.06401\n(2022).\n[30] M. Cardona, Phys. Status Solidi A 188, 1209 (2001).\n[31] J.-J. Zhou, J. Park, I.-T. Lu, I. Maliyov, X. Tong,\nand M. Bernardi, Comput. Phys. Commun. 264, 107970\n(2021).\n[32] G. Stefanucci and R. van Leeuwen, Nonequilibrium\nMany-Body Theory of Quantum Systems: A Modern In-\ntroduction (Cambridge University Press, 2013).\n[33] J.-M. Lihm and C.-H. Park, Phys. Rev. B 101, 121102(R)\n(2020).\n[34] P. B. Allen and V. Heine, J. Phys. C: Solid State Phys.\n9, 2305 (1976).[35] D. Vollhardt and P. W ol\re, Phys. Rev. B 22, 4666 (1980).\n[36] J. C. Ward, Phys. Rev. 78, 182 (1950).\n[37] K. Shen, G. Vignale, and R. Raimondi, Phys. Rev. Lett.\n112, 096601 (2014).\n[38] M. O\u000edani, R. Raimondi, and A. Ferreira, Condens.\nMatter 3, 18 (2018).\n[39] P. Coleman, Introduction to Many-Body Physics (Cam-\nbridge University Press, 2015).\n[40] W. Li, Phys. Rev. B 92, 075405 (2015).\n[41] J.-J. Zhou and M. Bernardi, Phys. Rev. B 94, 201201(R)\n(2016).\n[42] D. Culcer, A. Sekine, and A. H. MacDonald, Phys. Rev.\nB96, 035106 (2017).\n[43] A. A. Burkov and L. Balents, Phys. Rev. B 69, 245312\n(2004).\n[44] A. Baral, S. Vollmar, S. Kaltenborn, and H. C. Schneider,\nNew J. Phys. 18, 023012 (2016).\n[45] S. Vollmar, D. J. Hilton, and H. C. Schneider, Phys. Rev.\nB96, 075203 (2017).\n[46] L. Szolnoki, A. Kiss, B. D\u0013 ora, and F. Simon, Sci. Rep. 7,\n9949 (2017).\n[47] G. Cs} osz, L. Szolnoki, A. Kiss, B. D\u0013 ora, and F. Simon,\nPhys. Rev. Research 2, 033058 (2020).\n[48] P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car,\nC. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococ-\ncioni, I. Dabo, A. D. Corso, S. de Gironcoli, S. Fabris,\nG. Fratesi, R. Gebauer, U. Gerstmann, C. Gougoussis,\nA. Kokalj, M. Lazzeri, L. Martin-Samos, N. Marzari,\nF. Mauri, R. Mazzarello, S. Paolini, A. Pasquarello,\nL. Paulatto, C. Sbraccia, S. Scandolo, G. Sclauzero, A. P.\nSeitsonen, A. Smogunov, P. Umari, and R. M. Wentzcov-\nitch, J. Phys.: Condens. Matter 21, 395502 (2009).\n[49] J. Heyd, G. E. Scuseria, and M. Ernzerhof, J. Chem.\nPhys. 118, 8207 (2003).\n[50] M. J. van Setten, M. Giantomassi, E. Bousquet, M. J.\nVerstraete, D. R. Hamann, X. Gonze, and G. M. Rig-\nnanese, Comput. Phys. Commun. 226, 39 (2018).\n[51] G. Pizzi, V. Vitale, R. Arita, S. Bl ugel, F. Freimuth,\nG. G\u0013 eranton, M. Gibertini, D. Gresch, C. Johnson,\nT. Koretsune, J. Iba~ nez-Azpiroz, H. Lee, J.-M. Lihm,\nD. Marchand, A. Marrazzo, Y. Mokrousov, J. I. Mustafa,\nY. Nohara, Y. Nomura, L. Paulatto, S. Ponc\u0013 e, T. Pon-\nweiser, J. Qiao, F. Th ole, S. S. Tsirkin, M. Wierzbowska,\nN. Marzari, D. Vanderbilt, I. Souza, A. A. Mosto\f, and\nJ. R. Yates, J. Phys. Condens. Matter 32, 165902 (2020).\n[52] J. Park, J.-J. Zhou, V. A. Jhalani, C. E. Dreyer, and\nM. Bernardi, Phys. Rev. B 102, 125203 (2020).\n[53] G. Brunin, H. P. C. Miranda, M. Giantomassi, M. Royo,\nM. Stengel, M. J. Verstraete, X. Gonze, G.-M. Rignanese,\nand G. Hautier, Phys. Rev. B 102, 094308 (2020).\n[54] P. E. Bl ochl, O. Jepsen, and O. K. Andersen, Phys. Rev.\nB49, 16223 (1994).\n[55] S. Oertel, J. H ubner, and M. Oestreich, Appl. Phys. Lett.\n93, 132112 (2008)."    },    {        "title": "0706.2463v1.Frequency_dependence_of_induced_spin_polarization_and_spin_current_in_quantum_wells.pdf",        "content": "arXiv:0706.2463v1  [cond-mat.mes-hall]  17 Jun 2007Frequency dependence of induced spin polarization and spin current in quantum wells\nO. E. Raichev∗\nInstitute of Semiconductor Physics, National Academy of Sc iences of Ukraine, Prospekt Nauki 45, 03028, Kiev, Ukraine\n(Dated: April 20, 2022)\nDynamic response of two-dimensional electron systems with spin-orbit interaction is studied theo-\nretically, on the basis of quantum kinetic equation taking i nto account elastic scattering of electrons.\nThe spin polarization and spin current induced by the applie d electric field are calculated for the\nwhole class of electron systems described by p-linear spin-orbit Hamiltonians. The absence of\nnon-equilibrium intrinsic static spin currents is confirme d for these systems with arbitrary (non-\nparabolic) electron energy spectrum. Relations between th e spin polarization, spin current, and\nelectric current are established. The general results are a pplied to the quantum wells grown in\n[001] and [110] crystallographic directions, with both Ras hba and Dresselhaus types of spin-orbit\ncoupling. It is shown that the existence of the fixed (momentu m-independent) precession axes in\n[001]-grown wells with equal Rashba and Dresselhaus spin ve locities or in symmetric [110]-grown\nwells leads to vanishing spin polarizability at arbitrary f requency ωof the applied electric field.\nThis property is explained by the absence of Dyakonov-Perel -Kachorovskii spin relaxation for the\nspins polarized along these precession axes. As a result, a c onsiderable frequency dispersion of spin\npolarization at very low ωin the vicinity of the fixed precession axes is predicted. Pos sible effects\nof extrinsic spin-orbit coupling on the obtained results ar e discussed.\nPACS numbers: 73.63.Hs, 72.25.Pn, 73.50.Mx\nI. INTRODUCTION\nThepresenceofspin-orbitinteractioninsolidsprovides\na natural way of manipulating spin states of electrons by\npurely electrical means, without application of a mag-\nnetic field. This property is a subject of interest for the\nnovel and rapidly developing field of spintronics.1Some\nimportant manifestations of spin manipulation are the\ngeneration of non-equilibrium spin polarization of elec-\ntrons and excitation of spin currents by a driving elec-\ntric field which also leads to the usual charge current.\nThough the problem of field-induced spin polarization\nis an old one,2,3,4,5it has been recently set at the fo-\ncus of attention. The main reason for this is the ap-\npearance of experimental works6,7,8,9,10,11which demon-\nstrate the spin polarization generated under the current\nflow both in bulk and two-dimensional (2D) semiconduc-\ntor layers with spin-orbit interaction. Another reason is\nthe rapidly growing interest to the related phenomenon,\nthe intrinsic spin-Hall effect, when the electric current\nin the presence of spin-orbit coupling leads to a non-\nequilibrium spin current in perpendicular direction (note\nthat weak spin currents can exist even in equilibrium12).\nAfter the theoretical proposal13of the intrinsic spin-Hall\neffect based on the Rashba model representing spin-orbit\ninteraction for 2D electrons in quantum wells,14it has\nbeen realized15,16,17,18,19,20,21that this effect is absent\nin the infinite 2D system in the static (zero-frequency)\nlimit and exists at non-zero frequency or in finite-size\nsamples. The absence of the static intrinsic spin-Hall ef-\nfect is not a general property, since it is related to the\n∗Electronic address: raichev@isp.kiev.uaspecific form of the spin-orbit Hamiltonian. It has been\nshown20,22,23,24,25,26that the static spin-Hall effect ex-\nists for more complicated models involving higher-order\n(cubic) terms in the momentum dependence of the spin-\norbit Hamiltonian. The experimental observation of the\nspin-Hall effect for the 2D hole system27described by the\nspin-orbit Hamiltonian of this kind has confirmed this\nconclusion. The edge spin accumulation due to the spin-\nHall effect has been also observed8,11,28for 2D electron\nsystems. Recently, it has been suggested29,30that devi-\nation of electron band dispersion from the parabolic one\ncan lead to non-zero intrinsic static spin-Hall effect. This\npoint, however, remains controversial, since the analysis\ngiven in Ref. 17 predicts zero static spin-Hall effect even\nin this case. The calculations presentedin this paper also\nshowthe absence ofstatic spin currentsfornon-parabolic\nelectron band dispersion.\nIn contrast to the static regime, the frequency-\ndependent induced spin polarization and spin current\nhave not been extensively studied. Theoretical calcula-\ntions of the frequency-dependent spin-Hall current based\non the Rashba model have been done in Refs. 16, 18\nand 31 by using the methods of non-equilibrium Green’s\nfunctions, Kubo-Greenwood linear response theory, and\nquantum kinetic equation, respectively, with the same\nresult. The authors of Ref. 31 also calculated the fre-\nquency dependence of the induced spin polarization. A\nmore complicated case of frequency-dependent response,\nwhen both Rashba and Dresselhaus (linear in momen-\ntum) terms are included in the spin-orbit Hamiltonian,\nhas been studied in Refs. 32 and 33 in the dynamical\n(collisionless) regime. A comparative numerical study of\nthe frequency dependence ofthe spin-Hall effect has been\ndone in Ref. 23 for linear, cubic, and modified Rashba\nmodels. The resonancesin frequencydependence ofspin-\nHallconductivityinmagneticfieldhavebeendescribedin2\nRef. 34forthe Rashbamodel. Nevertheless, asystematic\ninvestigation of the frequency-dependent problem of the\ninducedspinpolarizationandspincurrentisstillmissing.\nAsteptowardssystematicdescriptionofthefrequency-\ndependent spin response is undertaken in this paper by\nconsidering the important class of spin-orbit Hamiltoni-\nans:\nˆhp= ¯hΩp·ˆσ,¯hΩα\np=ξpµαβpβ, (1)\nwhereˆσis the vector of Pauli matrices, p= (px,py) is\nthe 2D momentum of electrons, and µαβis the matrix\nof spin velocities. Next, ξpis an arbitrary function of\nthe absolute value of electron momentum. This func-\ntion describes possible isotropic corrections to spin-orbit\ninteraction, which may have the same origin as the non-\nparabolicity of the band spectrum. The case ξp= 1 cor-\nresponds to the p-linear spin-orbit Hamiltonian of the\ngeneral form. The p-linear spin-orbit coupling terms\nappear in quantum wells due to both the structural in-\nversion asymmetry14(Rashba term) and bulk inversion\nasymmetry (Dresselhaus term), the latter contribution is\nsensitive to orientation of the quantum well with respect\nto crystallographic axes. By solving the quantum kinetic\nequationforthe matrixdistributionfunction ofelectrons,\nwith taking into account elastic scattering, the spin po-\nlarizationandspin currentsarefound on anequal footing\nand a relation between them is established. The calcu-\nlations not only provide analytical expressions for these\nquantities, but also demonstrate a need to reconsider the\nknownresults5,35forstatic spinpolarizationbasedonthe\nHamiltonian (1) in special cases, when the symmetry of\nµαβallows zero spin precession for some chosen direc-\ntions of the spin vector. The examples of this kind are\nthe quantum wells grown in [001] crystallographic direc-\ntion in the case of equal Rashba and Dresselhaus spin\nvelocities and symmetric quantum wells (only the Dres-\nselhaus term is present) grown in [110] crystallographic\ndirection. In particular, it is shown that the induced spin\npolarization in these systems remains zero even when the\nfrequency of the applied field goes to zero. The calcula-\ntionsalsoshowthatthe staticspin currentinthe electron\nsystemsdescribed bythe Hamiltonian (1) is zeroeven ifa\nnon-parabolicity of electron band spectrum is taken into\naccount. The presented theory neglects the spin-orbit\ncorrections to the scattering potential, which means that\nthe effects of extrinsic spin-orbit coupling, such as the\nextrinsic spin currents and Elliot-Yafet spin relaxation,\nare not included in the calculations.\nThe paper is organized as follows. In Sec. II we con-\nsider the quantum kinetic equation and present its an-\nalytical solutions. The expressions for the induced spin\ndensity vector and spin current tensor, obtained on the\nbasis of these solutions, are given in Sec. III. In that\nsection we also establish a general relation between these\nquantities and present a detailed analysis of two impor-\ntant cases, [001]-grown and [110]-grown quantum wells\nwith both Rashba and Dresselhaus spin-orbit coupling.\nA relation between the induced spin density and electriccurrent is derived and analyzed in Sec. IV. The obtained\nresults and the limits of their applicability are discussed\nin Sec. V.\nII. GENERAL CONSIDERATION\nThe Hamiltonian of the problem is written in the form\nˆHp+ˆhp+Vr+ˆHext\nt, whereˆHpis the Hamiltonian of free\nelectrons in the absence of spin-orbit interaction, ˆhpis\nthe spin-orbitHamiltionian, Vris the potential ofimpuri-\ntiesorotherstaticinhomogeneities(thespin-orbitcorrec-\ntions to this potential are neglected, so only the intrinsic\nspin-orbit coupling is considered), and ˆHext\nt=−eEt·ris\nthe Hamiltonian of the external perturbation due to the\napplied time-dependent electric field Et(heree=−|e|\nis the electron charge). The calculations are based on\nthe quantum kineticequation forthe Wignerdistribution\nfunction ˆ ρpt, which is a 2 ×2 matrix over the spin indices\n(see, for example, Refs. 36 and 37). For the spatially-\nhomogeneous problem considered below, this equation is\nwritten in the form\n∂ˆρpt\n∂t+i\n¯h/bracketleftBig\nˆhp,ˆρpt/bracketrightBig\n+eEt·∂ˆρpt\n∂p=/hatwideJ(ˆρ|pt),(2)\nwhere/hatwideJis the collision integral describing the elastic\nscattering. This integral is written below in the Marko-\nvian approximation and under the assumptions ¯ hω≪ε\nand ¯h/τ≪ε, whereωis the frequency of the applied\nperturbation, εis the mean kinetic energy of electrons\nandτis the characteristic scattering time. One has (see\nRef. 37, problem 13.10)\n/hatwideJ(ˆρ|pt) =1\n¯h2/integraldisplaydp′\n(2π¯h)2W(|p−p′|)/integraldisplay0\n−∞dt′eλt′(3)\n×/bracketleftBig\nei(εp+ˆhp)t′/¯h(ˆρp′t−ˆρpt)e−i(εp′+ˆhp′)t′/¯h−(p′↔p)/bracketrightBig\n,\nwhereW(|p|) =/integraltext\nd∆re−ip·∆r/¯h/an}bracketle{t/an}bracketle{tVr+∆rVr/an}bracketri}ht/an}bracketri}htis the spa-\ntial Fourier transform of the correlation function of the\nscattering potential, εpis the kinetic energy of electron\nin the absence of spin-orbit interaction (the energy spec-\ntrum is isotropic but not necessarily parabolic), λ→+0,\nand (p′↔p) denotes the term obtained from the pre-\nceding one in the square brackets by permutation of mo-\nmenta. The integration over t′in Eq. (3) is carried out\nelementary, but the resulting expression is rather lengthy\nand, for this reason, is not presented here.\nSearching for the linear response to the Fourier com-\nponentEe−iωtof the applied electric field, we repre-\nsent the matrix distribution function in the form ˆ ρpt=\nˆf(eq)\np+ˆfpe−iωt, where ˆfpis the Fourier component of\nthe non-equilibrium part of the distribution function and\nˆf(eq)\npis the equilibrium distribution function,\nˆf(eq)\np=1\n2[fεp+¯hΩp+fεp−¯hΩp]\n+Ωp·ˆσ\n2Ωp[fεp+¯hΩp−fεp−¯hΩp], (4)3\nwhich is expressed through the Fermi distribution fε.\nHere and below, Ω p≡ |Ωp|. The function (4) does not\nlead to spin polarization of electron system because its\nmatrix part is antisymmetric in momentum. Since a sub-\nstitution of the function (4) into Eq. (2) makes both the\ncommutator and the collision integral (3) equal to zero,\none has a closed integral equation for ˆfp:\n−iωˆfp+i\n¯h/bracketleftBig\nˆhp,ˆfp/bracketrightBig\n+eE·∂ˆf(eq)\np\n∂p=/hatwideJ(ˆf|p),(5)\nwhich determines the linear response of the electron sys-\ntem.\nThe induced spin density is defined as\ns=1\n2/integraldisplaydp\n(2π¯h)2Tr(ˆσˆfp), (6)\nand the induced (non-equilibrium) spin current density\nis given by the tensor\nqα\nγ=1\n2/integraldisplaydp\n(2π¯h)2Tr({ˆσα,ˆuγ(p)}ˆfp),(7)\nwhereˆu(p) =∂(εp+ˆhp)/∂pisthegroup-velocitymatrix,\nTr denotes the matrix trace, and {ˆa,ˆb}= (ˆaˆb+ˆbˆa)/2\ndenotesthesymmetrizedmatrixproduct. Theexpression\n(7) describes the flow of the spin polarized along αin the\ndirection γ. Onemayalsointroducetheaveragespin S=\ns/n2D, where n2D= (2π¯h)−2/integraltext\ndpTrˆρpis the electron\ndensity. The tensors of spin polarizability, χαβ, and spin\nconductivity, Σα\nγβ, are introduced according to\nsα=χαβ(ω)Eβ, qα\nγ= Σα\nγβ(ω)Eβ.(8)\nIt is assumed in the following that the spin-splitting\nenergy 2¯hΩpis small in comparison to the mean energy\nof electrons. Then it is convenient to apply an efficient\nmethod of solution of Eq. (5) based on the expansion of\nthe collision integral in series with respect to the small\nparameter ¯ hΩp/ε; see Refs. 5, 25, 31, and problem 13.11\nin the book 37. Using the spin-vector representation\nˆfp= f0\np+ˆσ·fpand retaining only the terms of the\nfirst order in Ωpunder the collision integral, we obtain\ncoupled equations for scalar and vector parts of the dis-\ntribution function:\n−iωf0\np+1\n2eE·∂[fεp+¯hΩp+fεp−¯hΩp]\n∂p\n=2π\n¯h/integraldisplaydp′\n(2π¯h)2W(|p−p′|)/bracketleftbig\n(f0\np′−f0\np)δ(εp′−εp)\n−¯h(Ωp−Ωp′)·(fp′−fp)∂δ(εp′−εp)\n∂εp′/bracketrightbigg\n(9)\nand\n−iωfp+eE·∂\n∂pΩp[fεp+¯hΩp−fεp−¯hΩp]\n2Ωp\n−2[Ωp×fp] =2π\n¯h/integraldisplaydp′\n(2π¯h)2W(|p−p′|)[(fp′−fp)\n×δ(εp′−εp)−¯h(Ωp−Ωp′)(f0\np′−f0\np)∂δ(εp′−εp)\n∂εp′/bracketrightbigg\n.(10)The expressions containing formal derivatives of the δ-\nfunctions under the integrals should be evaluated using\nintegration by parts. The Fermi distribution functions\nstanding in the field terms also can be expanded in series\nof Ωp. Then one can see that the iterational expansions\nof f0\npandfpstart with the terms of zero and first order\nin Ωp, respectively. For this reason, the last term under\nthe collision integral in Eq. (9) can be neglected, and the\nsolution of this equation is\nf0\np≃ −eE·vp\nν(1)\np−iω∂fεp\n∂εp, (11)\nwherevp=∂εp/∂pis the group velocity of electron in\ntheabsenceofspin-orbitinteraction. Hereandbelow, the\nrelaxation rates appearing in the problem are defined as\nν(n)\np=2π\n¯h/integraldisplaydp′\n(2π¯h)2W(|p−p′|)[1−cos(nθ)]δ(εp′−εp),\n(12)\nwhereθdenotes the scattering angle /hatwiderpp′. The rate ν(n)\np\ndescribes relaxation of the n-th angular harmonic of the\ndistribution function.\nEquation(11)describestheDrude responseofthe elec-\ntron system. The next correction to f0\npis of the order\nof (¯hΩp/ε)2. This correction is essential for calculation\nof the frequency-dependent conductivity and dielectric\nfunction of2Delectronswith spin-orbitsplitting38,39(see\nSec. IV), but it is not important for calculation of the\ninduced spin polarization and spin current. After sub-\nstituting the expression (11) into the last term of the\ncollision integral in Eq. (10), this term is unified with\nthe field term on the left-hand side of Eq. (10). As a\nresult, one gets a closed equation for the vector-function\nfp:\n−iωfp−2[Ωp×fp]+Fp(ω)\n=2π\n¯h/integraldisplaydp′\n(2π¯h)2W(|p−p′|)(fp′−fp)δ(εp′−εp).(13)\nThevector Fcontainsbothisotropicandanisotropiccon-\ntributions:\nFp(ω) =eΩEF(i)\np(ω)−e(E·p)ΩpF(a)\np(ω),(14)\nF(i)\np(ω) =Rp(ω)(/tildewideνp−iω)−p2∂Rp(ω)\n∂p2ν(2)\np,(15)\nF(a)\np(ω) =2¯h\np2ξp/bracketleftbigg\nRp(ω)/tildewideνp−p2∂Rp(ω)\n∂p2(ν(2)\np−iω)/bracketrightbigg\n,(16)\nwhereΩEis the constant vector with components Ωα\nE=\nµαβEβ,\nRp(ω) =ξp\nν(1)\np−iω∂fεp\n∂εp, (17)\nand\n/tildewideνp=ν(1)\np−ν(2)\np−p2\n2∂\n∂εp/parenleftbigg\nν(2)\np∂εp\n∂p2/parenrightbigg\n(18)4\nis a relaxationrate. Note that /tildewideνpgoes to zero in the limit\nof short-range scattering potential.\nIt is convenient to search for the solution of Eq. (13)\nin the form\nfp=e(E·p)ΩpF(a)\np(ω)\nν(2)\np−iω+gp. (19)\nSince the first term of this expression is proportional to\nΩp, it does not contribute to the vector product in Eq.\n(13), thereby representing a non-precessing part of the\nsolution. Note that the angular average of this term is\ndirected along ΩE. The substitution (19) leads to the\nfollowing equation for gp:\n−iωgp−2[Ωp×gp]+iωGp(ω)\n=mp\n¯h3/integraldisplay2π\n0dϕ′\n2π[W(|p−p′|)(gp′−gp)]|p′|=|p|,(20)\nwheremp=1\n2(∂p2/∂εp)isthep-dependenteffectivemass\nand\nGp(ω) =−ΩEeξp(ν(2)\np+/tildewideνp−iω)\n(ν(1)\np−iω)(ν(2)\np−iω)∂fεp\n∂εp.(21)\nThe collision integral in Eq. (20) is already reduced,\nby means of integration over the absolute value of p′,\nto the integral over the angle ϕ′of the vector p′. Ow-\ning to the substitution (19), the inhomogeneous (field-\ndependent) term of Eq. (20), iωGp(ω), is proportional\nto the frequency ωand isotropic in the momentum space.\nWith the aid of the definition (21), it is convenient to\nwrite the angular-averaged distribution function fp≡\n(2π)−1/integraltext2π\n0dϕfpas\nfp=−Gp(ω)+Qp(ω)+gp, (22)\nQp(ω) =−eΩE∂[p2Rp(ω)]\n∂p2.\nWe also point out the exact relation\nfp=−2\niω[Ωp×gp]+Qp(ω), (23)\nwhich is obtained by applying the procedure of angular\naveraging to Eq. (20) and by using Eq. (22).\nIn spite of the isotropy of the term (21), Eq. (20)\nrequires a numerical solution. The physical reason for\nthis is the effect of precession in the presence of angular-\ndependent scattering. The angular dependence of the\nvector product [ Ωp×gp], in the general case, is different\nfrom that of gpstanding there, and the standard method\nof solution, based on expansion of the distribution func-\ntionin seriesofangularharmonics,leadstoaninfinite set\nof coupled equations. There are, however, a number of\nimportantsituationswhen Eq. (20)canbe solvedanalyt-\nically. These situations are described in the subsections\nbelow.A. Short-range scattering potential\nLet us consider the limit of short-range scattering po-\ntential, when W(|p−p′|) is replaced by a constant W\nandν(n)\np=νp=mpW/¯h3for any number n. The mo-\nmentum dependence of the scattering rate is associated\nwith possible non-parabolicity of the band spectrum and\nhas to be ignored in the parabolic approximation. Since\nthe right-hand side of Eq. (20) is reduced in this case to\nνp(gp−gp), a regular way of solving exists. The solution\nis\ngp=−iω\nνp(νp−iω)∆2p∂fεp\n∂εp/braceleftbig\n(νp−iω)2Ap\n+2(νp−iω)[Ωp×Ap]+4Ωp(Ωp·Ap)},(24)\nwhere ∆2\np= (νp−iω)2+4Ω2\npandApis the vector with\ncomponents\nAα\np=eξp(ˆT−1\np)αβΩβ\nE. (25)\nHereˆT−1\npdenotes the matrix inverse of the symmetric\n3×3 matrix\nTαβ\np= 4/bracketleftBig\n(ΩαpΩβ\np−δαβΩ2p)/∆2p/bracketrightBig\n+δαβiω\nνp(26)\nobtained as a result of angular averaging. The whole\nsolution, according to Eqs. (16)-(19), is\nfp=−e(E·p)Ωp2¯h\nξp∂\n∂p2/bracketleftbiggξp\nνp−iω∂fεp\n∂εp/bracketrightbigg\n+gp,(27)\nand its angular average is written as\nfp=1\nνp∂fεp\n∂εp/parenleftbigg\neξpΩE−iω\nνpAp/parenrightbigg\n+Qp(ω).(28)\nThis vector determines the magnitude and the direction\nof the induced spin polarization.\nB. Isotropic spin splitting\nThe next exactly solvable situation is realized when\nthe energy spectrum of electrons remains isotropic in the\npresence of spin-orbit coupling. In other words, Ω p= Ωp\ndepends only on the absolute value of p. This imposes\ncertain constraints on the matrix µαβ:\nµ2\nxx+µ2\nyx+µ2\nzx=µ2\nxy+µ2\nyy+µ2\nzy,\nµxxµxy+µyxµyy+µzxµzy= 0.(29)\nThis situation is realized, for example, in [001]-grown\nquantum wells ( x/bardbl[100],y/bardbl[010],z/bardbl[001]) with only\nRashba or only Dresselhaus type of spin-orbit coupling\nandin [111]-grownquantumwells( x/bardbl[112],y/bardbl[110],z/bardbl5\n[111]) with both types of coupling. The solution is writ-\nten in the form similar to that of Eq. (24). By introduc-\ningtheisotropicquantity /tildewide∆2\np= (ν(1)\np−iω)(ν(2)\np−iω)+4Ω2\np\nand the vector\nBα\np=eξp(ˆU−1\np)αβΩβ\nE, (30)\nwhereˆU−1\npdenotes the matrix inverse of\nUαβ\np= 4(ΩαpΩβ\np−δαβΩ2\np)//tildewide∆2\np+δαβiω\nν(2)\np,(31)\nwe find\ngp=−iω(ν(2)\np+/tildewideνp−iω)\nν(2)\np(ν(1)\np−iω)(ν(2)\np−iω)/tildewide∆2p∂fεp\n∂εp\n×/braceleftBig\n(ν(1)\np−iω)(ν(2)\np−iω)Bp(32)\n+2(ν(2)\np−iω)[Ωp×Bp]+4Ωp(Ωp·Bp)/bracerightBig\n.\nIt is easy to see that the results (24) and (32) become\nequivalent if one assumes isotropic spin splitting in Eq.\n(24) and short-range scattering in Eq. (32). The angular\naverage of the whole solution is\nfp=(ν(2)\np+/tildewideνp−iω)\nν(2)\np(ν(1)\np−iω)∂fεp\n∂εp/parenleftBigg\neξpΩE−iω\nν(2)\npBp/parenrightBigg\n+Qp(ω).\n(33)\nC. Fixed precession axis\nThere is also a special case, when Eq. (20) is solved\nin the most simple way. This happens when the vector\nproduct [ Ωp×ΩE] is zerofor arbitrary pandE. In other\nwords, the symmetry of the matrix µαβshould allow ex-\nistence of a fixed (momentum-independent) precession\naxis. This imposes the following constraints:\nµxyµyx=µxxµyy, µxyµzx=µxxµzy.(34)\nUnder these conditions, both ΩpandΩEare directed\nalong the fixed precession axis, without regard to direc-\ntions ofpandE. The examplesare[001]-grownquantum\nwells (x/bardbl[100],y/bardbl[010],z/bardbl[001]) with equal absolute\nvalues of Rashba and Dresselhaus velocities, when the\nprecession axis is in the quantum well plane at the angle\nofπ/4 or−π/4 with respect to the main crystallographic\naxes, and [110]-grownwells ( x/bardbl[110],y/bardbl[001],z/bardbl[110])\nwith only Dresselhaus type of coupling, when the preces-\nsionaxisisperpendiculartothequantumwellplane. The\nsolution of Eq. (20) in this case is non-precessing (makes\nthe vector product equal to zero) and isotropic (makes\nthe collision integral equal to zero):\ngp=Gp(ω). (35)\nThis vector is directed along ΩE. The averaged whole\nsolution fpis also directed along ΩE:\nfp=Qp(ω). (36)D. Static limit\nIf the frequency ωgoes to zero, the solution of Eq.\n(20) is trivial, gp= 0. Therefore, the function fpfrom\nEq. (19) with gp= 0 and ω= 0 describes the static\nspin-dependent response in the general case. The only\nexception is the special case considered in the previous\nsubsection, when there exists a non-zero solution, gp=\nGp(0). The averaged distribution function for the static\nlimit is\nfp=ΩEeξp(ν(2)\np+/tildewideνp)\nν(1)\npν(2)\np∂fεp\n∂εp+Qp(0) (37)\nforthe generalcase. In the specialcase onlythe lastterm\nof this expression remains.\nIII. SPIN RESPONSE\nTo describe the spin response, one should calculate the\nintegrals over momentum pin Eqs. (6) and (7). It is\nconvenient to separate the angular averaging from the\nintegration over the squared absolute value of momen-\ntum,p2, according to dp=1\n2dp2dϕ. Then, after using\nthe representation ˆfp= f0\np+ˆσ·fpand taking the ma-\ntrix trace, the density of the induced spin polarization is\ngiven by\ns=/integraldisplay∞\n0dp2\n4π¯h2fp, (38)\nand the density of non-equilibrium spin current is\nqα\nγ=/integraldisplay∞\n0dp2\n4π¯h2[m−1\npfαppγ\n+µαβf0p(δβγξp+2(∂ξp/∂p2)pβpγ)].(39)\nThe second term, which appears in the expression (39)\nowing to the spin-orbit correction to the group velocity,\ngives zero contribution because, according to Eq. (11),\nthescalarpartofthedistributionfunctionisantisymmet-\nric in momentum. If the frequency ωis zero, the vector\npart of the distribution function is symmetric in momen-\ntum, so the first term of the expression (39) also gives\nzero contribution. Therefore, the non-equilibrium static\nspin currents do not exist for the model described by the\nspin-orbit Hamiltonian (1). At non-zero frequency, the\nspin currents are associated with g-contribution to the\ndistribution function, because the first term of Eq. (19)\nis symmetric in p:\nqγ=/integraldisplay∞\n0dp2\n4π¯h2mpgppγ, (40)\nwhereqγ= (qx\nγ,qy\nγ,qz\nγ). Looking at the expressions (24)\nand (32), one can conclude that it is the second terms in\nthe braces of these expressions that are responsible for\nthe spin currents.6\nOn the other hand, the induced spin density is deter-\nmined by the angular-averagedsymmetric part of fpand\nexists in the static regime as well. Applying either the\nexact relations (22) and (23) or the expressions(28),(33),\n(36),and(37)describingdifferentphysicalsituations,one\nshould always ignore the term Qp(ω) which represents a\nfull derivative over p2and, for this reason, does not con-\ntribute to the integral (38). The static spin polarization\nobtained in this way is\ns=eΩE\n4π¯h/integraldisplay∞\n0dp2ξp∂fεp\n∂εp/bracketleftBigg\n1\nν(2)\np\n−p2\n2ν(1)\npν(2)\np∂\n∂εp/parenleftbigg\nν(2)\np∂εp\n∂p2/parenrightbigg/bracketrightBigg\n. (41)\nIntheparabolicapproximation εp=p2/2mandatξp= 1\nthis expression is rewritten as\ns=ΩEem\n2π¯h/integraldisplay∞\n0dε∂fε\n∂ε/bracketleftBigg\nτ(2)\nε+τ(1)\nε1\n2∂lnτ(2)\nε\n∂lnε/bracketrightBigg\n,(42)\nwhereτ(n)\nεp= 1/ν(n)\npare the relaxation times. For degen-\nerate electron gas, when ( ∂fε/∂ε)≃ −δ(ε−εF) andεF\nis the Fermi energy, the integral over energy is taken in a\nstraightforwardway. Inthe caseofshort-rangescattering\npotential ( τ(n)\nε=τ= 1/ν) one arrives at the well-known\nresult35sα=χαβEβwithχαβ=−µαβ[emτ/2π¯h2]\nvalid for any linear spin-orbit Hamiltonian including the\ncase of Rashba spin-orbit coupling studied in the early\npapers.2,4\nIt is important that Eqs. (41) and (42) are not valid\nin the special situation when a fixed precession axis ex-\nists; see subsection C ofSec. II. In fact, astraightforward\nsubstitutionofEq. (36)intoEq. (38)showsthat, atarbi-\ntrary frequency of the applied field, the spin polarization\ndoes not appear in this special situation. The physical\nexplanation of this remarkable property is based on the\nfacts that in the caseof afixed precessionaxis(a) there is\nno spin relaxation40by the Dyakonov-Perel-Kachorovskii\n(DPK) mechanism41forthe spinsdirected alongthis axis\nand (b) the induced spin polarization can, in principle,\nappear only along this axis, without regard to the direc-\ntion of the applied electric field. Imaginethat the electric\nfield is abruptly turned on. The anisotropic distribution\nof electrons over momenta, which determines the electric\ncurrent, is established during the momentum relaxation\ntime 1/ν(1)\np. However, the distribution of electrons over\nspins, which determines the spin density, is established\nduring the spin relaxation time, and the corresponding\ntransient process becomes infinitely slow if the spin re-\nlaxation is absent. Therefore, if a periodic alternating\nfield acts on electrons in a sample with a fixed preces-\nsion axis, the spin density cannot react to the field at\nany frequency ω, and the spin polarizability is zero. The\nabsence of the static spin polarization in the case of a\nfixed precession axis cannot be revealed by considera-\ntion of the static spin response alone. From the formalpoint of view, this paradox is related to the fact that\nin the static limit the additional part of the distribution\nfunction, gp=Gp(0), which cancels the contribution of\nthe first term in the expression (19), still exists, onlyin\nthe case of a fixed precession axis. One can say that\nthe dependence of the induced spin polarization on the\nparameters (components µαβ) of the spin-orbit Hamilto-\nnian is non-analytic in the region of parameters where\nthe fixed precession axis appears. This means that the\nresult for the spin polarization depends on the order of\nlimiting transitions. If first ωis aimed to zero and then\nthe precession axis is fixed, the polarization is finite. If\nfirst the precession axis is fixed and then ωis aimed to\nzero, the polarization is zero. More details on the issue\nof non-analyticity will be given below in this section, by\nconsidering concrete examples. It is important to state\nthat the consideration given above does not provide spin\nrelaxation mechanisms other than the DPK mechanism.\nInclusionofthe Elliot-Yafetmechanism (seeRefs. 42and\n43 for the 2D case) can lead to a finite relaxation of the\nspins oriented along the fixed precession axis and, there-\nfore, to a finite induced spin polarization for this special\ncase; see Sec. V for more discussion.\nUsing Eq. (23) together with Eqs. (20), (38), and\n(40), one can obtain an exact relation connecting the\ninduced spin polarization and spin current. From Eqs.\n(38) and (23) one has s=−(2πiω)−1/integraltext∞\n0dp2[Ωp×gp].\nSince [Ωp×gp] =ξp[µβ×gp]pβ, whereµβis the vector\nwith components µαβ, we find, for the case of parabolic\nband and ξp= 1,\ns=−2m\ni¯hω[µβ×qβ]. (43)\nThis relation is remarkable in the sense that it does not\ncontain any parameters describing scattering. The re-\nlation (43) is also valid for a non-parabolic model pro-\nvided that the momentum dependence of 1 /mpandξp\nis the same; in this case the effective mass mshould be\nreplaced by the momentum-independent quantity mpξp.\nUsing Eq. (43), one can directly relate the low-frequency\nbehavior of the spin current to the staticinduced spin\npolarization. Equation (43) can be derived from the\ncontinuity equation (for the case of Rashba model see\nRef. 44 and Sec. 64 of Ref. 37). In detail, when\nEq. (2) is summed over the momentum p, the collision-\nintegral contribution and the electric-field term vanish,\nand Eq. (43) directly follows from the relation between\nthe momentum-averaged first and second terms on the\nleft-hand side of Eq. (2). This observation also demon-\nstrates that the validity of Eq. (43) is not restricted\nby the assumption of linear response used in this paper:\nthe non-equilibrium part of the distribution function ˆ ρ\nis not necessarily small. On the other hand, Eq. (43) is\nspecific for the chosen form (1) of the spin-orbit Hamil-\ntonian. Applying Eq. (43) to the model including both\nRashba and Dresselhaus linear spin-orbit coupling terms\nin [001]-grown quantum wells (see subsection A below),\none obtains the relations between the components of the7\nspindensityandspincurrentrecentlyderived18,32forthis\nmodel. The corresponding relations37,44for the Rashba\nmodel also follow from Eq. (43).\nSince it is established that the non-parabolic correc-\ntions to the band spectrum and the corrections to the\nspin-orbit Hamiltonian do not lead to qualitative modifi-\ncations of the induced spin polarization and spin current,\nwe neglect these corrections in the following, by assum-\ningεp=p2/2mandξp= 1. Let us consider first the fre-\nquencybehavioroftheinducedspinpolarizationandspin\ncurrent in the quantum wells described by the Rashba\nmodel. The non-zero components of the spin-velocity\nmatrix are µxy=−µyx=vR, wherevRis the Rashba\nvelocity. The spin splitting described by the Hamilto-\nnian (1) is isotropic in this situation. Using Eq. (33) and\ntaking into account that the matrix (31) is diagonal, we\nobtain\ns=−ΩEem\n2π¯hT(ω,vR), (44)\nwhere the frequency-dependent function\nT(ω,v) =/integraldisplay∞\n0dεp/parenleftbigg\n−∂fεp\n∂εp/parenrightbigg/parenleftBigg\n1−εp\n2∂ν(2)\np/∂εp\nν(1)\np−iω/parenrightBigg\n×2(vp/¯h)2\n2(vp/¯h)2ν(2)\np−iω[(ν(1)\np−iω)(ν(2)\np−iω)+4(vp/¯h)2](45)\nhas dimensionality of time. Equations (44) and (45) gen-\neralizethe result ofRef. 31obtained in the limit ofshort-\nrangescatteringpotentialtothe caseofarbitraryscatter-\ning potential. They also describe [111]-grown quantum\nwells with both Rashba and Dresselhaus spin-orbit cou-\npling, where µxy=−µyx=vR+2vD/√\n3 andvDis the\nDresselhaus velocity. In this case, vRin Eq. (44) should\nbe merely replaced by vR+2vD/√\n3 and the direction of\nthe spin polarization vector swith respect to Eis the\nsame as for the pure Rashba coupling. In particular,\ns= [emvT(ω,v)/2π¯h2](−Ey,Ex,0) where vis either vR\norvR+2vD/√\n3, respectively. Next, in symmetric [001]-\ngrown quantum wells, where only the Dresselhaus spin-\norbit coupling is present and the non-zero components\nareµyy=−µxx=vD, Eq. (44) with T(ω,vD) is also\nvalid, leading to s= [emvDT(ω,vD)/2π¯h2](Ex,−Ey,0).\nIn all these cases, the spin polarization is in the quantum\nwell plane ( sz= 0), and the direction of this polariza-\ntion is frequency-independent. The spin currents can be\neither calculated directly or extracted from Eq. (43).\nThere exist only the currents of z-polarized spins, and\nthese currents are directed perpendicular to the applied\nfield:\n/parenleftbiggqz\nx\nqz\ny/parenrightbigg\n=ieωT(ω,v)\n4π¯h/parenleftbigg\n−Ey\nEx/parenrightbigg\n, (46)\nwhereT(ω,v) is given by Eq. (45) with v=vRfor the\npure Rashba coupling and v=vR+ 2vD/√\n3 for [111]-\ngrown quantum wells. For symmetric [001]-grown quan-\ntum wells one should use v=vDand change the sign ofthe right-hand side. Therefore, the symmetry properties\nand frequency dependence of the spin currents remain\nthe same for all important cases of isotropic spin split-\nting considered in this paragraph. In the limit of short-\nrange scattering potential and in the case of degenerate\nelectrons, Eqs. (46) and (45) with v=vRgive the result\nobtained previously in Ref. 16. The universal behavior13\nof the spin-Hall conductivity Σz\nxy=−Σz\nyxexists in the\nlimitvpF/¯h≫ω≫ν(2)\npF, wherepFis the Fermi momen-\ntum.\nNow we turn to more complicated situations when the\nanisotropy of spin splitting is essential due to combined\neffect of both Rashba and Dresselhaus spin-orbit cou-\npling. These cases are considered in the following sub-\nsections in the limit of short-range scattering potential,\nwhen the expressions obtained in subsection A of Sec. II\nare valid.\nA. [001]-grown quantum wells\nLet us study the case of [001]-grown quantum wells.\nIf the Cartesian coordinate axes are chosen along the\nprincipal crystallographic directions, there are four com-\nponents of the spin-velocity tensor:\nµxy=−µyx=vR, µyy=−µxx=vD.(47)\nThe spin splitting depends on the angle ϕof the vector p\naccordingtoΩ2\np= (v2\nR+v2\nD)(p/¯h)2−2vRvD(p/¯h)2sin(2ϕ)\nand ∆2\np=a−bsin(2ϕ), where\na= (ν−iω)2+4(v2\nR+v2\nD)(p/¯h)2, b= 8vRvD(p/¯h)2.\n(48)\nTheangularaveraginginEq. (26)resultsin Txx\np=Tyy\np=\n(ν−iω)2/2r−1/2 +iω/νandTyx\np=Txy\np=b/2r−\n(a/r−1)(v2\nR+v2\nD)/4vRvD, wherer=√\na2−b2. After\nsometransformations, weobtain the relationdefining the\ntensors of spin polarizability and spin conductivity:\n/parenleftbigg\nsx\nsy/parenrightbigg\n=em\n2π¯h2ν/integraldisplay∞\n0dεp\nDp(ω)/parenleftbigg\n−∂fεp\n∂εp/parenrightbigg\n×/bracketleftbigg\nκp(ω)−/tildewideκp(ω)\n/tildewideκp(ω)−κp(ω)/bracketrightbigg/parenleftbigg\nEx\nEy/parenrightbigg\n,(49)\nand\n/parenleftbiggqz\nx\nqz\ny/parenrightbigg\n=ieω\n4π¯hνv2\nR−v2\nD\n4vRvD/integraldisplay∞\n0dεp\nDp(ω)/parenleftbigg\n−∂fεp\n∂εp/parenrightbigg\n×/bracketleftbigg\nθp(ω)−/tildewideθp(ω)\n/tildewideθp(ω)−θp(ω)/bracketrightbigg/parenleftbigg\nEx\nEy/parenrightbigg\n,(50)\nwhere the denominator is given by\nDp(ω) =(v2\nR−v2\nD)2\n8v2\nRv2\nD/parenleftBiga\nr−1/parenrightBig\n+iω\nν/parenleftbigg(ν−iω)2\nr−1/parenrightbigg\n+/parenleftbiggiω\nν/parenrightbigg2\n. (51)8\nThe elements of the matrix in Eq. (49) are\nκp(ω) =v2\nR−v2\nD\n4vD/braceleftbigg/bracketleftbiggv2\nR−v2\nD\n2v2\nR−iω\nν/bracketrightbigg\n×/parenleftBiga\nr−1/parenrightBig\n+iω\nνvD\nvRb\nr/bracerightbigg\n,(52)\nand/tildewideκp(ω) is obtained from this expression by the per-\nmutations vR↔vD. The elements of the matrix in Eq.\n(50) are given by\nθp(ω) =/parenleftbigg\n1−iω\nν/parenrightbigg/parenleftBiga\nr−1/parenrightBig\n(53)\nand\n/tildewideθp(ω) =v2\nR+v2\nD\n2vRvD/parenleftBiga\nr−1/parenrightBig\n−iω\nνb\nr.(54)\nEquations (49) and (50) demonstrate the symmetry re-\nlationsχxx=−χyy,χxy=−χyx, Σz\nxx=−Σz\nyy, and\nΣz\nxy=−Σz\nyx.\nThough the matrix in Eq. (49) retains the symme-\ntry ofµαβ, the ratio/tildewideκp(ω)/κp(ω) is not equal to vR/vD.\nFor this reason, the direction of spin polarization in the\nplane is different from the direction of ΩEand depends\non the frequency. The direction of the spin current is\nnot perpendicular to the direction of the field and is also\nfrequency-dependent. The components of the spin den-\nsity and spin current are related according to Eq. (43),\nwhich can be written, for this particular case, in the\nform18,32qz\nx= (i¯hω/2m)(vRsx+vDsy)/(v2\nR−v2\nD) and\nqz\ny= (i¯hω/2m)(vDsx+vRsy)/(v2\nR−v2\nD).\nIn the collisionless limit, ν→0, Eqs. (49) and (50) are\nreduced to the results obtained in Ref. 32. Note that the\nformal substitution ν→0 makes the spin currents finite\natω→0 because the denominator Dp(ω) is reduced\nto−(ω/ν)2, while the functions (53) and (54) become\nproportional to ω/ν. The static spin currents for the\nsystems described by the spin-velocity matrix (47) have\nbeen also studied in the collisionless limit in Refs. 45\nand 46. All these studies show that, as the ratio vR/vD\nis varied, the spin current reverses its sign going through\nzero atv2\nR=v2\nD. This general property, also reflected in\nEq. (50), follows from the Berry phase analysis.45\nIn thecase v2\nR=v2\nD, asalreadymentioned, thereexists\na fixed precession axis directed at the angle of π/4 (or\n−π/4) in the quantum well plane. Therefore, not only\nthe spin current, but also the induced spin density goes\nto zero in this case. This behavior is demonstrated in\nFigs. 1-3, where the calculated real parts of the compo-\nnents ofspin polarizabilityand spin conductivitytensors,\nReχαβand ReΣz\nαβ, are plotted as functions of the ratio\nof Rashba and Dresselhaus velocities. The case of de-\ngenerate electron gas is assumed. The spin polarizability\nis expressed in the units of static polarizability for sym-\nmetric [001]-grownquantum wells, χ0=emvD/2π¯h2ν. If\nthe frequency decreases, both χxxandχyx, as expected,\napproach their static values χ0and (vR/vD)χ0, respec-\ntively. However, this never happens at vR=vD, when/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48/s49/s46/s50/s49/s46/s52/s49/s46/s54\n/s49/s48/s45/s49/s49/s48/s45/s50/s47 /s61/s49/s48/s45/s51\n/s32/s32/s82/s101\n/s120/s120/s32/s47\n/s48/s32/s44/s32/s82/s101\n/s121/s120/s32/s47\n/s48\n/s118\n/s82/s47/s118\n/s68/s32/s82/s101\n/s120/s120\n/s32/s82/s101\n/s121/s120\nFIG. 1: Dependence of the real part of spin polarizabil-\nity in [001]-grown quantum wells on the ratio vR/vDat\nvDpF/¯hν= 1 for several frequencies. The polarizability is ex-\npressed in the units of χ0=emvD/2π¯h2ν. The diagonal and\noff-diagonal components are shown by the solid and dashed\nlines, respectively.\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48/s45/s49/s46/s48/s45/s48/s46/s56/s45/s48/s46/s54/s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s47 /s61/s49/s48/s45/s50\n/s47 /s61/s49/s48/s45/s49/s47 /s61/s49\n/s32/s32/s82/s101/s122\n/s120/s120/s32/s44/s32/s82/s101/s122\n/s121/s120/s32/s32/s40/s117/s110/s105/s116/s115/s32/s111/s102/s32/s32/s101/s47/s56 /s41\n/s118\n/s82/s47/s118\n/s68/s32/s82/s101/s122\n/s120/s120\n/s32/s82/s101/s122\n/s121/s120\nFIG. 2: Dependence of the real part of spin conductivity in\n[001]-grown quantum wells on the ratio vR/vDatvDpF/¯hν=\n1 for several frequencies. The spin conductivity is express ed\nin the universal units. The diagonal and off-diagonal (Hall)\ncomponents are shown by the solid and dashed lines, respec-\ntively. The difference between them is negligible at small ω.\nthe polarizability remains exactly zero at arbitrary ω.\nThe real part of the spin conductivity (Fig. 2) shows\nprominent peaks near the point vR=vDat small ω,\nthough its behavior is analytic. The imaginary parts\nofχxxandχyx(not shown) also have sharp peaks in\nthe vicinity of vR=vDat small ω. The depression\nof the spin polarizability in the region vR≃vDis ex-\ntended with the increase of the disorder, as shown in\nFig. 3. This also means that, especially for the ”dirty”\ncasevDpF≪¯hν, the frequency dispersion of the spin po-\nlarizability remains significant at very small frequencies\nifvRis close enough to vD. The corresponding behavior\nis illustrated in Fig. 4 and is explained by the reduc-9\ntion and disappearance of the DPK spin relaxation as vR\napproaches to vD.\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48/s49/s46/s50/s49/s46/s52/s49/s46/s54\n/s34/s100/s105/s114/s116/s121/s34/s34/s99/s108/s101/s97/s110/s34/s47 /s61/s49/s48/s45/s50\n/s32/s32/s82/s101\n/s120/s120/s32/s47\n/s48/s32/s44/s32/s82/s101\n/s121/s120/s32/s47\n/s48\n/s118\n/s82/s47/s118\n/s68/s32/s82/s101\n/s120/s120\n/s32/s82/s101\n/s121/s120\nFIG. 3: Effect of disorder on the real part of spin polar-\nizability. The ”clean” and the ”dirty” cases correspond to\nvDpF/¯hν= 10 and vDpF/¯hν= 0.1, respectively.\n/s48/s46/s48\n/s53/s46/s48/s120/s49/s48/s45/s52\n/s49/s46/s48/s120/s49/s48/s45/s51/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s118\n/s82/s47/s118\n/s68/s61/s48/s46/s57/s53\n/s48/s46/s49/s48/s46/s51/s118\n/s68/s112\n/s70/s47 /s61/s49\n/s32/s32/s82/s101\n/s120/s120/s32/s47\n/s48\n/s47\nFIG. 4: Low-frequency dispersion of the real part of spin\npolarizability in [001]-grown quantum wells in the vicinit y of\nthe fixed precession axis ( vRis close to vD).\nB. [110]-grown quantum wells\nThe next case we consider is [110]-grown quantum\nwells. If the Cartesian coordinate axes are chosen with\nOZperpendicular to the quantum well plane and OY\nalong the principal crystallographic direction ( x/bardbl[110],\ny/bardbl[001],z/bardbl[110]), there are three components of the\nspin-velocity tensor:\nµxy=−µyx=vR, µzx=vD/2.(55)\nOne has Ω2\np= (v2\nR+v2\nD/8)(p/¯h)2+(v2\nD/8)(p/¯h)2cos(2ϕ)\nand ∆2\np=c+dcos(2ϕ), where\nc= (ν−iω)2+(4v2\nR+v2\nD/2)(p/¯h)2, d=vD(p/¯h)2/2.\n(56)Applying the equations listed in subsection A of Sec.\nII, we find that the field Excan induce both y- and\nz-polarized spins, while the field Eyinduces only x-\npolarized spins:\nsx=−emvREy\n2π¯h2ν/integraldisplay∞\n0dεp/parenleftbigg\n−∂fεp\n∂εp/parenrightbigg/parenleftbigg\n1−iω/ν\nKxp(ω)/parenrightbigg\n,\n(57)\nsy=emvREx\n2π¯h2ν/integraldisplay∞\n0dεp/parenleftbigg\n−∂fεp\n∂εp/parenrightbigg/parenleftbigg\n1−iω/ν\nKy\np(ω)/parenrightbigg\n,(58)\nandsz=−(vD/2vR)sy, where\nKx\np(ω) =/parenleftbigg4v2\nR\nv2\nD+1/parenrightbigg/parenleftBigg/radicalbigg\nc−d\nc+d−1/parenrightBigg\n+iω\nν,(59)\nand\nKy\np(ω) =−4v2\nR\nv2\nD/parenleftBigg/radicalbigg\nc+d\nc−d−1/parenrightBigg\n+iω\nν.(60)\nThe component szis a symmetric function of vR, while\nsxandsyare antisymmetric functions of vR.\nThe spin currents appear for y- andz-polarized spins\nand flow in the direction perpendicular to the applied\nfield:\nqy\nx=i¯hω\nmvD(4v2\nR/v2\nD+1)sx, qy\ny=i¯hωvD\n4mv2\nRsy,(61)\nand\nqz\nx= (2vR/vD)qy\nx, qz\ny= (2vR/vD)qy\ny.(62)\nThe relations (61) and (62) satisfy the general require-\nment (43). The components qz\nxandqz\nyare symmetric\nfunctions of vR, whileqy\nxandqy\nyare antisymmetric func-\ntions ofvR.\nFor symmetric quantum wells, where vR= 0, both the\npolarizationand the spin currentsdisappearforarbitrary\nω. If both vRandωgo to zero, the z-component of the\nspin polarizability tensor, χzx(ω), is described by a sim-\nple formula applied to the case of degenerate electrons:\nχzx≃ −χ0\n2(2vR/vD)2γF\n(2vR/vD)2γF−iω, (63)\nwhereγF=/radicalbig\nν2+(vDpF/¯h)2−ν. Equation (63) illus-\ntratesthe non-analyticbehaviorofthe spin polarizability\ninthe vicinityofthefixedprecessionaxis. Thecalculated\ndependence of the real part of χzxon the ratio vR/vD,\nshown in Fig. 5 for the case of degenerate electron gas,\nis similar to the dependence of χxxshown in Fig. 1 for\n[001]-grown wells. The region of depression near vR= 0\nis extended in the ”dirty” limit vDpF≪¯hν, as it is seen\ndirectly from Eq. (63). The behavior of the component\nχyxremains analytic at vR→0 andω→0, though it\nis strongly affected in the vicinity of vR= 0 at low fre-\nquencies. In contrast, the component χxy(not shown in10\nFig. 5) stays close to its static value −(vR/vD)χ0up to\nthe frequency region ω∼ν. The real part of the spin\nconductivity Σz\nyxshows peaks in the vicinity of vR= 0\nat low frequencies, while the low-frequency behavior of\nΣz\nxyis monotonic in this region, see Fig. 6. According\nto Eq. (63), the frequency dispersion of the spin po-\nlarizability remains significant at very low frequencies if\nvRis small enough. It is remarkable that in the ”dirty”\nlimit the characteristic rate describing this dispersion,\n(2vR/vD)2γF, is equal to 2( vRpF/¯h)2/ν, which is the\nDPK spin relaxation rate for the Rashba model.\n/s45/s48/s46/s53/s48 /s45/s48/s46/s50/s53 /s48/s46/s48/s48 /s48/s46/s50/s53 /s48/s46/s53/s48/s45/s48/s46/s54/s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50\n/s49/s48/s45/s49\n/s49/s48/s45/s50\n/s48/s46/s51\n/s49/s48/s45/s49\n/s49/s48/s45/s50\n/s47 /s61/s49/s48/s45/s51\n/s32/s32/s82/s101\n/s121/s120/s32/s47\n/s48/s32/s44/s32/s32/s82/s101\n/s122/s120/s32/s47\n/s48\n/s118\n/s82/s47/s118\n/s68/s32/s82/s101\n/s121/s120\n/s32/s82/s101\n/s122/s120\nFIG. 5: Dependence of the real part of spin polarizability in\n[110]-grown quantum wells on the ratio vR/vDatvDpF/¯hν=\n1 for several frequencies. The polarizability is expressed in\nthe units of χ0=emvD/2π¯h2ν.\n/s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52 /s48/s46/s53/s45/s48/s46/s52/s45/s48/s46/s51/s45/s48/s46/s50/s45/s48/s46/s49/s48/s46/s48/s48/s46/s49/s48/s46/s50\n/s48/s46/s51\n/s48/s46/s49\n/s48/s46/s51/s48/s46/s49/s47 /s61/s48/s46/s48/s51\n/s32/s32/s82/s101/s122\n/s120/s121/s32/s44/s32/s82/s101/s122\n/s121/s120/s32/s32/s40/s117/s110/s105/s116/s115/s32/s111/s102/s32/s32/s101/s47/s56 /s41\n/s118\n/s82/s47/s118\n/s68/s32/s82/s101/s122\n/s120/s121\n/s32/s82/s101/s122\n/s121/s120\nFIG. 6: Dependence of the real part of spin conductivity in\n[110]-grown quantum wells on the ratio vR/vDatvDpF/¯hν=\n1 for several frequencies.\nIV. CURRENT RESPONSE\nIt is important to relate the behavior of spin polar-\nization and spin currents studied in the previous sectionto the frequency dispersion of the conductivity (or di-\nelectric function) of 2D electron layers with spin-orbit\ninteraction. Some relations of this kind have been estab-\nlished previously33,39in the collisionless approximation.\nIn this section the corresponding relations are obtained\nand analyzed for the general p-linear model of spin-orbit\ncoupling described by the Hamiltonian (1), with taking\ninto account the electron-impurity interaction. The cal-\nculation is based on the kinetic equation(2). The electric\ncurrent density is defined as\nj=e/integraldisplaydp\n(2π¯h)2Tr(ˆu(p)ˆfp). (64)\nBelowwe neglectthe non-parabolicityofthe energyspec-\ntrum and the deviation of the Hamiltonian (1) from the\nlinearity, by putting mp=mandξp= 1. Let us multi-\nply Eq. (2) by ep/m, sum it over p, and take the matrix\ntrace. UsingEqs. (6) and (64), onehas the exactrelation\n−iω(jβ−2eµαβsα)−e2Eβn2D\nm=e/integraldisplaydp\n(2π¯h)2pβ\nmTr/hatwideJ(ˆf|p).\n(65)\nThe right-hand side of this equation should be set at\nzero in the collisionless approximation. To consider the\ncollision-induced contribution in the general case, one\nshould calculate the distribution function ˆfpwith the\naccuracy up to the terms ∼(¯hΩp/ε)2. Instead of doing\nthis, we consider the limit of short-range scattering po-\ntential, when the right-hand side of Eq. (65) is exactly\ntransformed to −νjβ. Therefore,\njβ=e2n2D\nm(ν−iω)Eβ−2iω\nν−iωesαµαβ.(66)\nEquation (66) contains the usual Drude term and the\nterm induced by the spin-orbit interaction.38,39Together\nwith Eq. (43), this equation establishes the relationship\nbetween the electric current, spin current, and induced\nspin density. The validity of both Eq. (43) and Eq. (66)\nis not restricted by the assumption of linear response.\nExpressingthecurrentsthroughtheelectricconductiv-\nityσβλ(ω) and spin conductivity Σα\nβλ(ω), one can write\nthe equation that relates these tensors:\nσβλ=δβλe2n2D\nm(ν−iω)+σ(SO)\nβλ,\nσ(SO)\nβλ=4me\n¯h(ν−iω)[µβ×µκ]δΣδ\nκλ. (67)\nIn general, the spin-orbit term σ(SO)\nβλbrings non-diagonal\ncontributions to σβλ. This should lead to a weak Hall\neffect in the absence of magnetic field at finite frequen-\ncies. For the Rashba model, when only the components\nµxy=−µyx=vRexist, the conductivity is diagonal and\nisotropic,47σ(SO)\nβλ=σ(SO)δβλ, where\nσ(SO)=−4mev2\nR\n¯h(ν−iω)Σz\nxy. (68)11\nIn the collisionless limit, this equation gives a relation39\nbetween the imaginary part of σ(SO)and the real part of\nthe spin-Hall conductivity Σz\nxy.\nIn the case of [001]-grown quantum wells with both\nRashba and Dresselhaus types of coupling,\n/parenleftBigg\nσ(SO)\nyλ\nσ(SO)\nxλ/parenrightBigg\n=4me(v2\nR−v2\nD)\n¯h(ν−iω)/parenleftbigg−Σz\nxλ\nΣz\nyλ/parenrightbigg\n.(69)\nThe non-diagonal part of σ(SO)\nβλis related to the diagonal\ncomponents of the spin conductivity, while the diago-\nnal part is expressed through the spin-Hall conductivity\n(such an expression has been recently established33in\nthe collisionless regime). The tensor σ(SO)\nβλcan be diago-\nnalized by in-plane rotation of the Cartesian coordinate\naxesOXandOY. In contrast to the spin polarization,\nthe quantity σ(SO)\nβλis analytic at v2\nR=v2\nD.\nIn the case of [110]-grown quantum wells the conduc-\ntivity is diagonal (in the chosen coordinate system) but\nanisotropic:\n/parenleftBigg\nσ(SO)\nyy\nσ(SO)\nxx/parenrightBigg\n=4me(v2\nR+v2\nD/4)\n¯h(ν−iω)/parenleftbigg−Σz\nxy\nΣz\nyx/parenrightbigg\n.(70)\nThe expressions for the components Σz\nαβentering Eqs.\n(69) and (70) are obtained from the expressions for spin\ncurrentspresentedin subsectionsAand B ofthe previous\nsection.\nAlthough the contribution σ(SO)\nβλis small as (¯ hΩp/ε)2\nwith respect to the Drude conductivity, its frequency de-\npendence has qualitatively new features. If Ω p≫ν,\nso that the spin-split states are well-defined, the term\nσ(SO)\nβλdescribes resonance absorption of electromagnetic\nradiation, typically in the THz region, associated with\ntransitions between these states. In the case of isotropic\nspin splitting48(the Rashba model is considered below)\nand degenerate electron gas, the resonance takes place38\natω≃ωr≡2|vR|pF/¯h. Substituting in Eq. (68) the de-\ntailed expression for the spin conductivity, see Eq. (46)\nand Eq. (45) with ν(1)\np=ν(2)\np=ν, one has an equation\nσ(SO)=e2mv2\nRiω\nπ¯h2(ν−iω)ω2\nr\nω2rν−2iω[(ν−iω)2+ω2r],(71)\nwhich describes the resonance under consideration at\nωr≫ν. Of course, this resonance also exists in\nthe frequency dependence of the spin conductivity and\nspinpolarizability.31Anotherimportantfeaturefollowing\nfrom Eq. (71) is the presence of low-frequency dispersion\nunder the opposite condition, ωr≪ν. This dispersion\nappears when ωis comparable to the DPK spin relax-\nation rate. Since this rate, for in-plane spin polarization,\nis given by νDP= 2(vRpF/¯h)2/ν, Eq. (71) in the limits\nωr≪νandω≪νgives\nReσ(SO)≃ −e2\nπ¯hmv2\nRνDP\n¯hν2ω2\nω2+ν2\nDP.(72)Therefore, σ(SO)essentially depends on ωin the region\nof frequencies ω≪ν, when the Drude conductivity still\nremains frequency-independent. Another contribution\nto the frequency dependence of the conductivity in this\nregion exists owing to weak localization.49Though the\nweak-localization correction is larger in magnitude than\nReσ(SO), its frequency dependence is slow (logarithmic),\nand can be distinguished from the dependence given by\nEq. (72).\nV. SUMMARY AND DISCUSSION\nIn this paper, the electric-field-induced spin density\nsand intrinsic spin current qβin 2D electron layers\ndescribed by the general p-linear spin-orbit interaction\nHamiltonian ˆ σαµαβpβ≡ˆσ·µβpβare studied in the\nclassical region of frequencies ω. The consideration is\ndone for macroscopic systems and at zero magnetic field.\nThe quantities sandqβare closely related to each other\nthrough Eq. (43) following from the balance equation\nfor spin density. To find them, a careful analysis of the\nquantum kinetic equation is carried out taking into ac-\ncount interaction of electrons with impurities or other\nstatic inhomogeneities. The presented results are valid\nfor arbitrary correlation between the spin splitting en-\nergy 2¯hΩp, disorder-induced broadening ¯ hν, and energy\n¯hω, under condition that all these energies are small\nin comparison with the characteristic kinetic energy of\nelectrons. A complete analytical solution of the linear-\nresponse problem for the matrix distribution function is\ngiven in the limit of short-range scattering potential (see\nsubsection A of Sec. II). This solution is applied to [001]-\nand [110]-grown quantum wells with both Rashba and\nDresselhaus types of spin-orbit coupling (subsections A\nand B of Sec. III). All other situations when analyt-\nical solutions exist are described is subsections B, C,\nand D of Sec. II. The theory also takes into account\nthe isotropic energy-dependent corrections to the effec-\ntive mass m(non-parabolicity effect) and to the spin-\nvelocitymatrix µαβ. Thisisreflectedbythesubstitutions\nm→mpandµαβ→ξpµαβassumed from the beginning\nof the consideration. Since these weak corrections do not\nlead to qualitative effects (in particular, it is shown that\nthe static spin currents remain equal to zero in the pres-\nence of these corrections), they are ignored in the most\npart of the applications, starting from Eq. (42).\nThe main approximation of the present consideration\nis the neglect of the spin-dependent contribution to the\nscattering potential. This contribution, also caused by\nthe spin-orbit interaction, is often referred to as the ex-\ntrinsic spin-orbit coupling. In the first order with re-\nspect to the extrinsic spin-orbit coupling, there appear\nspin currents which are not equal to zero in the static\nlimit. This leads to the extrinsic spin-Hall effect,50,51,52\nwhich is beyond the scope of the present paper. The ex-\ntrinsic spin-orbit coupling also leads to an additional in-\nduced spin polarization, which is considered, in the limit12\nω≫ν≫Ωp, in Ref. 53. In the second order with re-\nspecttotheextrinsicspin-orbitcouplingthereappearad-\nditional relaxation terms in the kinetic equation. These\nterms are responsible for the Elliot-Yafet spin relaxation\nof 2D electrons.42,43The corresponding spin relaxation\nrate is many orders of magnitude smaller that the mo-\nmentum relaxation rate and often can be neglected.\nThe other important approximation is the spatial ho-\nmogeneity of the problem, which implies the neglect\nof the gradient terms in the kinetic equation (2). For\nthis reason, the results of this paper cannot be di-\nrectlyappliedfordescriptionofthespatialdistributionof\nspin density in 100 µm wide 2D layers recently studied\nexperimentally.8,28From the theoretical point of view,\nthe problem of finite-size samples is difficult, not only\nbecause of the presence of gradient terms, but also be-\ncause of the need to derive the boundary conditions for\nthe matrix Wigner distribution function. In the general\ncase, the requiredboundarycondition should be a matrix\nequation.54Some simple forms of boundary conditions\nused in theoreticaldescriptionofthe spin-Hall effect (see,\nforexample, Ref. 16)havebeenwritten withoutaderiva-\ntion. Therefore, the problem of spatially inhomogeneous\ndistribution of the spin density still remains topical.\nThe main qualitative result of this study is the pre-\ndiction of vanishing spin polarization at arbitrary fre-\nquency of the applied electric field for the special situ-\nations when fixed (momentum-independent) precession\naxes exist. The well-known examples of these situations\nare asymmetric [001]-grown quantum wells with equal\nRashba and Dresselhaus velocities ( v2\nR=v2\nD) and sym-\nmetric [110]-grown quantum wells ( vR= 0). This re-\nmarkable property, which does not follow from the con-\nsideration of the static response, is explained by the ab-\nsence of DPK spin relaxation for these special situations.\nAs a consequence, in the close vicinity of the fixed pre-\ncession axis, which means that v2\nRis close to v2\nDfor [001]-\ngrownquantumwellsor vRisclosetozerofor[110]-grown\nquantum wells, the frequency dispersion of spin polariza-\ntion is significant in the low-frequency region, when the\npolarization can be excited by applying an alternating\nelectrical bias to the 2D sample. For [110]-grown quan-\ntum wells this dispersion is given by Eq. (63). Since\nthe Rashba velocity is variable by modifying the shape\nof the quantum well via a bias applied to an external\n(top) gate, these phenomena can be investigated exper-\nimentally. As concerns possible technological use, these\nphenomena are of interest for the purposes of frequency\nfiltering and amplification in future spintronics devices.\nIt should be noted that the predicted behavior in the\nvicinity of the fixed precession axes can be influenced by\ntheextrinsicspin-orbitcoupling. Fromthispointofview,\nthe [001]- and [110]-grownquantum wells appear to be in\nquite different positions. In [001]-grown quantum wells\nwithv2\nR=v2\nD, where the fixed precession axis lies in\nthe plane of motion, a coupling of z-polarized spins with\nin-plane polarized spins occurs when the spin-dependent\ncontribution to the scattering potential is taken into ac-count. This means that the applied electric field can ex-\ncite the spin density perpendicular to the precession axis\nas a combined effect of extrinsic spin-Hall current exci-\ntation and precession. Also, as mentioned above, there\nappears the Elliot-Yafet relaxation of the in-plane spin\ndensity. As a result, finite spin polarizations will ap-\npear for the directions both perpendicular and parallel\nto the precession axis, and the non-analyticity with re-\nspect to the order of limiting transitions v2\nR→v2\nDand\nω→0 will be removed. Still, the frequency dispersion\nof the spin polarizability should persist down to the fre-\nquenciescomparablewith the Elliot-Yafetrelaxationrate\nνEY. Using the estimate νEY/ν∼10−5given in Ref. 42\nand taking into account that ν∼1012s−1, one finds\nνEY∼107s−1, which means that the frequency dis-\npersion can be principally observed in the region above\n10 MHz. On the other hand, in [110]-grown symmetric\nquantum wells, where the fixed precession axis is per-\npendicular to the plane of motion, the applied electric\nfield cannot excite the in-plane polarized spins even in\nthe presence of the extrinsic spin-orbit coupling. Exci-\ntation of the current of z-polarized spins (the extrinsic\nspin-Hall current) does not lead to spin polarization in z\ndirection far from the boundaries of the sample. Finally,\nthe Elliot-Yafet relaxation of z-polarized spin density is\nabsent in [110]-grown symmetric quantum wells. In con-\nclusion, the extrinsic spin-orbit coupling does not lead to\na finite spin polarization in [110]-grownsymmetric quan-\ntum wells and cannot modify Eq. (63). Modification\nof the low-frequency behavior described by this equation\nmayoccurowingtonon-Markovianmemoryeffects55and\neffects of spatial inhomogeneity due to finite sample size.\nThe other important result is Eq. (66), which es-\ntablishes a simple relation between the electric current\nand induced spin polarization. Though the validity of\nthis equation is restricted by the approximation of short-\nrange scattering potential, Eq. (66) is applicable to the\nwhole class of the systems described by the p-linear spin-\norbit Hamiltonians and does not require the linear re-\nsponse approximation. It can be useful in applications\nand in analysis of experiments where both spin polariza-\ntion and electric conductivity are measured. Equation\n(66) also shows that the spin-orbit term in the electric\nconductivity is of purely dynamic origin, this term van-\nishes at ω→0. It is not clear whether this property\nis specific for the short-range scattering potential or re-\nmains valid for arbitrary scattering potential. The cor-\nresponding calculations are now in progress. In combi-\nnation with Eq. (43), the result (66) relates the elec-\ntric current with the spin current and leads to a unified\ndescription of spin and charge response to the applied\nelectric field.\nThe results for frequency dispersion of the spin po-\nlarization and spin current obtained in this paper can\nbe directly reformulated for description of transient spin\nresponse,18,29,31which is investigated experimentally by\nthe time-resolved spectroscopy.6,11,56If the spin-split\nelectron states are well-defined, the transient process13\nshows coherent oscillations. If the spin splitting is\nsuppressed by collisional broadening, there should ap-\npear long-time transients associated with the DPK spin\nrelaxation.18,31Under conditions close to the appearance\nof the fixed precession axis, for example, when vRis close\nto zero in [110]-grownquantum wells, the duration of the\ntransient process, according to Eq. (63), is expected to\nincrease dramatically.Acknowledgments\nTheauthorisgratefultoSergeyTarasenkoforahelpful\ndiscussion.\n[1]Semiconductor Spintronics and Quantum Computation ,\nEds. D. D. Awschalom, D. Loss, and N. Samarth\n(Springer-Verlag, Berlin, 2002).\n[2] F. T. Vasko and N. A. Prima, Fiz. Tverd. Tela\n(Leningrad) 21, 1734 (1979) [Sov. Phys. - Solid State\n21, 994 (1979)].\n[3] A. G. Aronov, and Yu. B. Lyanda-Geller, Pisma Zh.\nEksp. Teor. Fiz. 50, 398 (1989) [JETP Lett. 50, 431\n(1989)].\n[4] V. M. Edelstein, Solid State Commun. 73, 233 (1990).\n[5] A. G. Aronov, Yu. B. Lyanda-Geller, and G. E. Pikus,\nZh. Eksp. Teor. Fiz. 100, 973 (1991) [JETP, 73, 537\n(1991)].\n[6] Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D.\nAwschalom, Phys. Rev. Lett. 93, 176601 (2004).\n[7] A. Yu. Silov, P. A. Blajnov, J. H. Wolter, R. Hey, K. H.\nPloog, and N. S. Averkiev, Appl. Phys. Lett. 85, 5929\n(2004).\n[8] V. Sih, R. C. Myers, Y. K. Kato, W. H. Lau, A. C.\nGossard, and D. D. Awschalom, Nature Physics 1, 31\n(2005).\n[9] S. D. Ganichev, S. N. Danilov, P. Schneider, V. V.\nBel’kov, L. E. Golub, W. Wegscheider, D. Weiss, W.\nPrettl, J. Magnetism and Magnetic Materials 300, 127\n(2006).\n[10] C. L. Yang, H. T. He, L. Ding, L. J. Cui, Y. P. Zeng,\nJ. N. Wang, and W. K. Ge, Phys. Rev. Lett. 96, 186605\n(2006).\n[11] N. P. Stern, S. Ghosh, G. Xiang, M. Zhu, N. Samarth,\nand D. D. Awschalom, Phys. Rev. Lett. 97, 126603\n(2006).\n[12] E. I. Rashba, Phys. Rev. B 68, 241315(R) (2003).\n[13] J. Sinova, D. Culcer, Q. Niu, N. A. Sinitsyn, T. Jung-\nwirth, andA.H.MacDonald, Phys.Rev.Lett. 92, 126603\n(2004).\n[14] E. I. Rashba, Fiz. Tverd. Tela 2, 1224 (1960) [Sov. Phys.\n- Solid State 2, 1109 (1960)]; Y. A. Bychkov and E. I.\nRashba, J. Phys. C 17, 6039 (1984); F.T. Vasko, Pis’ma\nZh. Eksp. Teor. Fiz. 30, 574 (1979) [JETP Lett. 30, 541\n(1979)].\n[15] J. I. Inoue, G. E. W. Bauer, andL. W. Molenkamp, Phys.\nRev. B70, 041303(R) (2004).\n[16] E. G. Mishchenko, A. V. Shytov , and B. I. Halperin,\nPhys. Rev. Lett. 93, 226602 (2004).\n[17] O. V. Dimitrova, Phys. Rev. B 71, 245327 (2005).\n[18] O. Chalaev and D. Loss, Phys. Rev. B 71, 245318 (2005).\n[19] A. Khaetskii, Phys. Rev. Lett. 96, 056602 (2006).\n[20] A. Khaetskii, Phys. Rev. B 73, 115323 (2006).\n[21] S. Y. Liu and X. L. Lei, Phys. Rev. B 73, 205327 (2006).\n[22] S. Murakami, Phys. Rev. B 69, 241202(R) (2004).\n[23] K. Nomura, J. Sinova, N. A. Sinitsyn, and A. H. Mac-Donald Phys. Rev. B 72, 165316 (2005).\n[24] S. Y. Liu and X. L. Lei, Phys. Rev. B 72, 155314 (2005).\n[25] A. V. Shytov, E. G. Mishchenko, H.-A. Engel, and B. I.\nHalperin, Phys. Rev. B 73, 075316 (2006).\n[26] O. Bleibaum and S. Wachsmuth, Phys. Rev. B 74,\n195330 (2006).\n[27] J. Wunderlich, B. Kaestner, J. Sinova, and T. Jungwirth ,\nPhys. Rev. Lett. 94, 047204 (2005); K. Nomura, J. Wun-\nderlich, J. Sinova, B. Kaestner, A. H. MacDonald, and\nT. Jungwirth, Phys. Rev. B 72, 245330 (2005).\n[28] V. Sih, W. H. Lau, R. C. Myers, V. R. Horowitz, A.\nC. Gossard, and D. D. Awschalom, Phys. Rev. Lett. 97,\n096605 (2006).\n[29] C. M. Wang, X. L. Lei, and S. Y. Liu, Phys Rev B 73,\n113314 (2006).\n[30] P.L.KrotkovandS.Das Sarma, Phys.Rev.B 73, 195307\n(2006).\n[31] V. V. Bryksin and P. Kleinert, Phys. Rev. B 73, 165313\n(2006).\n[32] S. I. Erlingsson, J. Schliemann, and D. Loss, Phys. Rev.\nB71, 035319 (2005).\n[33] J. A. Maytorena, C. Lopez-Bastidas, and F. Mireles,\nPhys. Rev. B 74, 235313 (2006).\n[34] X. F.WangandP.Vasilopoulos, Phys. Rev.B 75, 075331\n(2007).\n[35] A. V. Chaplik, M. V. Entin, and L. I. Magarill, Physica\nE13, 744 (2002).\n[36] M. I. Dyakonovand A. V. Khaetskii, Zh. Eksp. Teor. Fiz.\n86, 1843 (1984) [JETP 59, 1072 (1984)].\n[37] F. T. Vasko and O. E. Raichev, Quantum Kinetic Theory\nand Applications (Springer, New York, 2005).\n[38] L. I. Magarill, A. V. Chaplik, and M. V. Entin, Zh. Eksp.\nTeor. Fiz. 119, 175 (2001) [JETP 92, 153 (2001)].\n[39] E. I. Rashba, Phys. Rev. B 70, 161201(R) (2004).\n[40] N. S. Averkiev and L. E. Golub, Phys. Rev. B 60, 15582\n(1999).\n[41] M. I. Dyakonov and V. I. Perel, Fiz. Tverd. Tela 13,\n3581 (1971) [Sov. Phys. - Solid State 13, 3023 (1972)];\nM. I. Dyakonov and V. Yu. Kachorovskii, Fiz. Tekh.\nPoluprovod. 20, 178 (1986) [Sov. Phys. - Semicond. 20,\n110 (1986)].\n[42] N. S. Averkiev, L. E. Golub, and M. Willander, J. Phys.:\nCondens. Matter 14, R271 (2002).\n[43] E. L. Ivchenko and S. A. Tarasenko, Zh. Eksp. Teor. Fiz.\n126, 426 (2004) [JETP 99, 379 (2004)].\n[44] A. A. Burkov, A. S. Nunez, and A. H. MacDonald, Phys.\nRev. B70, 155308 (2004).\n[45] S.-Q. Shen, Phys. Rev. B 70, 081311(R) (2004).\n[46] N. A. Sinitsyn, E. M. Hankiewicz, W. Teizer, and J.\nSinova, Phys. Rev. B 70, 081312(R) (2004).\n[47] This statement is in disagreement with the result of C.14\nZhang and Z. Ma, Phys. Rev. B 71, 121307(R) (2005),\nwho obtained non-diagonal resistivity for the Rashba\nmodel.\n[48] If the conductivity tensor is isotropic, the absorptio n co-\nefficient for the electromagnetic wave normally incident\non a 2D layer is equal to 4 πReσ(ω)/c√ǫ, wherecis the\nvelocity of light and ǫis the dielectric permittivity.\n[49] For 2D systems with spin-orbit coupling, see L. E. Golub ,\nPhys. Rev. B 71, 235310 (2005) and references therein.\n[50] J. E. Hirsch, Phys. Rev. Lett. 83, 1834 (1999).\n[51] H.-A. Engel, B. I. Halperin, and E. I. Rashba, Phys. Rev.\nLett.95, 166605 (2005).[52] W-K. Tse and S. Das Sarma, Phys. Rev.Lett. 96, 056601\n(2006).\n[53] S. A. Tarasenko, Phys. Rev. B 73, 115317 (2006).\n[54] O. E. Raichev and F. T. Vasko, Phys. Rev. B 57, 12350\n(1998).\n[55] I. S. Lyubinskiy and V. Yu. Kachorovskii, Phys. Rev. B\n73, 041301(R) (2006).\n[56] V. Sih, W. H. Lau, R. C. Myers, A. C. Gossard, M.\nE. Flatte, and D. D. Awschalom, Phys. Rev. B 70,\n161313(R) (2004)."    },    {        "title": "2309.03749v1.Coherent_spin_dynamics_between_electron_and_nucleus_within_a_single_atom.pdf",        "content": "Coherent spin dynamics between electron and nucleus within a single atom  Lukas M. Veldman1, Evert W. Stolte1, Mark P. Canavan1, Rik Broekhoven1, Philip Willke2, Laëtitia Farinacci1, Sander Otte1*   1Department of Quantum Nanoscience, Kavli Institute of Nanoscience, Delft University of Technology, 2628 CJ Delft, The Netherlands. 2Physikalisches Institut, Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany. *e-mail: a.f.otte@tudelft.nl   The nuclear spin, being much more isolated from the environment than its electronic counterpart, enables quantum experiments with prolonged coherence times and presents a gateway towards uncovering the intricate dynamics within an atom. These qualities have been demonstrated in a variety of nuclear spin qubit architectures1-3, albeit with limited control over the direct environment of the nuclei. As a contrasting approach, the combination of electron spin resonance (ESR) and scanning tunnelling microscopy (STM)4 provides a bottom-up platform to study the fundamental properties of nuclear spins of single atoms on a surface5,6. However, access to the time evolution of these nuclear spins, as was recently demonstrated for electron spins7,8, remained a challenge. Here, we present an experiment resolving the nanosecond coherent dynamics of a hyperfine-driven flip-flop interaction between the spin of an individual nucleus and that of an orbiting electron. We use the unique local controllability of the magnetic field emanating from the STM probe tip9 to bring the electron and nuclear spins in tune, as evidenced by a set of avoided level crossings in ESR-STM. Subsequently, we polarize both spins through scattering of tunnelling electrons and measure the resulting free evolution of the coupled spin system using a DC pump-probe scheme. The latter reveals a complex pattern of multiple interfering coherent oscillations, providing unique insight into the atom’s hyperfine physics. The ability to trace the coherent hyperfine dynamics with atomic-scale structural control adds a new dimension to the study of on-surface spins, offering a pathway towards dynamic quantum simulation of low-dimensional magnonics.  Nuclear spins have shown great promise as building blocks for quantum information in molecular spin qubits2,10, NV centers3,11, and donors in silicon1. They also are an excellent resource for quantum simulation12, magnetic sensing13,14 and spintronics15, and are potentially scalable via engineered molecular and atomic networks16,17. Their key advantage arises from their longer coherence times compared to their electron spin counterpart18, though the intricacies of the decoherence channels depend on the exact interaction with the environment and may be difficult to describe in detail. Scanning tunneling microscopy (STM) constitutes an excellent means of investigation here, since it not only permits to address individual electron spin states in electron spin resonance (ESR) experiments with sub-nanometer resolution4,19, but also allows for an atomically precise control of their environment20,21. Up to now, interactions involving the nuclear spin could be measured indirectly by probing the hyperfine coupling in ESR-STM between the nucleus and the surrounding electrons5. In addition, the nuclear spin of individual copper atoms could be polarized via spin pumping induced by the spin-polarized tunneling current6. However, accessing the coherent dynamics involving the nucleus remained challenging, due to its weak coupling to the tunneling electrons. In this work, we show the free, coherent evolution between the nuclear spin and the electron spin in a single hydrogenated titanium atom. By fine-tuning the electronic Zeeman energy using the local field of the probe tip, we identify a parameter space where electronic and nuclear spin states hybridize. In a second step, we probe the free coherent evolution of the coupled system by electric DC pump-probe experiments. Here, we reveal an emerging beating pattern, that originates from multiple quantum oscillations with different frequencies at the points of hybridization.  We use a commercial low-temperature STM equipped with high frequency cabling to send both RF signals and nanosecond DC pulses down to the tip. The sample system consists of Ti atoms deposited on bilayer MgO islands grown on Ag(100), that become hydrogenated by residual hydrogen22,23. For all measurements, we use spin-polarized tips that are created by picking up co-deposited Fe atoms onto the tip apex. We study individual Ti adsorbed onto the oxygen sites of MgO – well-isolated from neighboring spins using atom manipulation (see Fig. 1a) – which exhibit an effective electron spin S = ½ 22 with an anisotropic g-factor g 23. Throughout this work, we focus on 47Ti isotopes, which carry a nuclear spin I = 5/2. Along the principal axes of the crystal field, the system is described by the following Hamiltonian:  ℋ\"=$%𝜇!𝑔\"%𝐵#$%,\"+𝐵%'(,\"*𝑆,\"+𝐴\"𝑆,\"𝐼,\"+𝑄\"𝐼,\")*\"*+,,,-(1)  With the anisotropic hyperfine and quadrupole contributions, respectively, Α = [10, 10, 130] ΜΗz and Q = [1.5, 1.5, –3] MHz (see Methods)5. The first term describes the Zeeman energy of the electron spin with contributions from both the external Bext and the tip-induced magnetic field Btip. We neglect the effect of either of these fields on the nuclear spin, since their contributions are small compared to the other terms. As shown in Fig. 1b, for small field values this Hamiltonian results in a number of avoided crossings between states with equal total angular momentum.   State initialization via spin pumping We start our investigation by applying a magnetic field of 1.5 T, which is large compared to the hyperfine interaction, in order to drive ESR transitions between the individual spin states of a 47Ti atom. Similar to measurements of Ti on a bridge binding site of MgO 24,25, we find a large anisotropy in the hyperfine coupling ranging from 10 MHz in-plane to 130 MHz out-of-plane (see Methods). Since we aim for a regime in which the hyperfine interaction competes with the Zeeman splitting of the electron, the experiments are performed with an out-of-plane magnetic field.  For certain magnetic tips with sufficient spin-polarization, we observe that the hyperfine-split ESR peaks have different intensities, which indicates a strong polarization of the nuclear spin. Such nuclear polarization has been observed for Cu atoms on MgO 6 and was modeled by taking into account inelastic spin scattering events between the tunneling electrons and electron spin. As shown in Fig. 1c, we find that the polarization is strongly dependent on the applied bias voltage while measuring at constant current. We believe that this may be due to the bias-dependent efficiency of the spin scattering channels involved, but a more complex mechanism involving the Ti orbital excitation23 or Pauli spin blockade26 may be at play. We find that the effective temperature of the nuclear spin population drops below 10 mK at voltages larger than 100 mV, more than two orders of magnitude lower than the actual experimental temperature of 1.5 K.   Tuning electron-nuclear spin entanglement This highly efficient spin pumping allows us to overcome a major limitation: in previous ESR-STM experiments, the frequency ranges accessible for a given temperature were limited to the spin contrast set by the Boltzmann distribution. Here, owing to the nuclear polarization, we can investigate a much lower frequency regime, in which the level of entanglement between the electron and nuclear spins can be tuned. In Fig. 2a, we show the different contributions to the energy diagram of a 47Ti in a low-field regime. When the total electronic Zeeman energy – due to the external and tip magnetic field – is comparable to the hyperfine splitting, multiple avoided level crossings occur in the spectrum. The number of avoided crossings has increased compared to Fig. 1b as a result of taking into account a finite angle between the external field and the tip field. In particular, an additional avoided crossing appeared around Btip = 23 mT, involving a superposition of states that differ only in the electron spin, not the nuclear spin. This electron-only hybridization will be of importance below.  We identify these tuning points in our experiment by performing ESR measurements in the low field regime using an external field of merely 20 mT (Fig. 2b). Here, in order to fine-tune the coupled spin system, we vary the tip-induced magnetic field, which we adjust by the tunnel conductance G of the junction. At large tip fields (G ≥ 20 pS) multiple ESR peaks are visible in addition to several very sharp (~3 MHz) NMR type resonances around 60 MHz. The uneven splitting of the NMR type resonances is here caused by the quadrupole interaction5. Below G ≈ 20 pS, the ESR and NMR transitions start to mix and overlap, accompanied by a redistribution of their intensities as shown in Fig. 2c. This is consistent with the presence of avoided level crossings between the energy levels, as expected from Fig. 2a and modelled in Figs. 2d and e. For the simulations we consider both ESR and NMR transitions with separately scaled intensities (see Methods).   Probing coherent spin dynamics Having identified the appropriate tip-atom distances for inducing superposition states, we perform DC pump-probe experiments to explore the coupled spin dynamics. We use a two-pulse sequence to initialize both electron and nucleus spins to state |↓,−5/2⟩ via inelastic scattering27 (See Fig. 3a and Methods). Then, after a free, varied time evolution, the final state of the system is probed by a 5 ns probe pulse.  The observed spin dynamics following this pulse sequence, shown in Fig. 3a, depend on the tip magnetic field: when the STM tip is close (i.e., at large conductance values) we observe fast, low-amplitude oscillations that become slower and stronger as the tip is retracted. This is the expected behavior when the system moves through an avoided crossing8. However, around G = 17 pS a beating pattern appears due to interference with a second oscillation. Upon further retraction of the tip, below G = 15 pS, no spin dynamics are detected anymore. Figs. 3b and c show the simulated time evolution of the Sz and Iz expectation values for the electron and the nuclear spin, respectively, calculated using Lindblad formalism starting from the |↓,−5/2⟩ state28.  We find excellent agreement between the data and calculations, with in particular a beating pattern that arises when the electron and nucleus states are entangled. While the electron shows an interference pattern, the nuclear spin is dominated by a ~40 ns oscillation.  To understand the origin of these different oscillations, we focus on the region of interest marked in the energy spectrum of Fig. 2a: in Fig. 4a, we identify a combination of three eigenstates that form a pair of avoided level crossings, which we assign to the observed dynamics. When we tune the tip field to the second avoided level crossing at 23.7 mT, we find that the population is evenly split between state |5⟩ and |6⟩ corresponding to the lowest energy ESR resonance observed in Fig. 2. Here, the nucleus remains unaffected and only the electron spin forms a superposition between up and down: |↑,−5/2⟩±|↓,−5/2⟩. Accordingly, we can fit in Fig. 4b a trace from the pump-probe data with a single sinusoid obtaining a frequency of roughly 75 MHz, matching the expected energy splitting between |5⟩ and |6⟩. The transition between states |4⟩ and |5⟩, on the other hand, corresponds to a flip-flop between the electron and nuclear spin. When these states hybridize they form the superposition state |↑,−5/2⟩±|↓,−3/2⟩. Thus, when the tip field is at 20.5 mT, right in middle of the two tuning points, the dynamics are expected to be a mixture between the electron oscillation discussed above and an additional flip-flop dynamic between electron and nucleus. In Fig. 4b, we observe this as a clear interference pattern in the time evolution, which we fit using three sinusoids with frequencies of 65 MHz, 20 MHz and their sum. This matches the calculated energy splitting at this tip field. Correspondingly, we can attribute the 65 MHz oscillation to a Larmor precession of the electron spin due to the in-plane component of the tip field while the 20 MHz oscillation is a flip-flop between the electron and the nucleus driven by the hyperfine interaction.  The reduced coupling of the nuclear spin to the environment is expected to result in an enhanced coherence time when the nuclear spin is involved, compared to the dynamics of only the electron spin14,29,30. Indeed, the data shown in Fig. 4b still shows observable dynamics up to 120 ns, whereas the oscillation in Fig. 4c has decayed already after 80 ns. We point out, however, that the increased coherence time may also in part result from a decreased sensitivity to magnetic tip field noise since the energy levels in Fig. 4a diverge less at 21 mT than at 23 mT, akin to a clock-transition31,32.   Conclusion Developing single atom quantum information processing requires thorough understanding of the underlying electron and nuclear spin dynamics. This demands initialization, tuning and readout tailored on the atomic length scale. Using pump-probe spectroscopy, we revealed the collective coherent dynamics of the internal spin dynamics inside a single atom. The magnetized STM tip functioned in this work as a control knob to locally tune the nature of these dynamics. This technique has the potential to be extended to any on-surface atomic or molecular spin system, yielding a great variety of phenomena to explore in the future. Moreover, the prospect of STM for engineering bottom-up atomic designer assemblies can provide an integral atomic-scale understanding into the fundamentals of complex coherent spin dynamics.  References 1 Pla, J. J. et al. High-fidelity readout and control of a nuclear spin qubit in silicon. Nature 496, 334-338, (2013). 2 Vincent, R., Klyatskaya, S., Ruben, M., Wernsdorfer, W. & Balestro, F. Electronic read-out of a single nuclear spin using a molecular spin transistor. Nature 488, 357-360, (2012). 3 Neumann, P. et al. Single-Shot Readout of a Single Nuclear Spin. Science 329, 542-544, (2010). 4 Baumann, S. et al. Electron paramagnetic resonance of individual atoms on a surface. Science 350, 417-420, (2015). 5 Willke, P. et al. Hyperfine interaction of individual atoms on a surface. Science 362, 336-339, (2018). 6 Yang, K. et al. Electrically controlled nuclear polarization of individual atoms. Nat Nanotechnol 13, 1120-1125, (2018). 7 Yang, K. et al. Coherent spin manipulation of individual atoms on a surface. Science 366, 509-512, (2019). 8 Veldman, L. M. et al. Free coherent evolution of a coupled atomic spin system initialized by electron scattering. Science 372, 964-968, (2021). 9 Yang, K. et al. Tuning the Exchange Bias on a Single Atom from 1 mT to 10 T. Phys Rev Lett 122, 227203, (2019). 10 Thiele, S. et al. Electrically driven nuclear spin resonance in single-molecule magnets. Science 344, 1135-1138, (2014). 11 Fuchs, G. D., Burkard, G., Klimov, P. V. & Awschalom, D. D. A quantum memory intrinsic to single nitrogen-vacancy centres in diamond. Nat Phys 7, 789-793, (2011). 12 Randall, J. et al. Many-body-localized discrete time crystal with a programmable spin-based quantum simulator. Science 374, 1474-1478, (2021). 13 Maze, J. R. et al. Nanoscale magnetic sensing with an individual electronic spin in diamond. Nature 455, 644-647, (2008). 14 Zaiser, S. et al. Enhancing quantum sensing sensitivity by a quantum memory. Nat Commun 7, 12279, (2016). 15 Zutic, I., Fabian, J. & Das Sarma, S. Spintronics: Fundamentals and applications. Rev Mod Phys 76, 323-410, (2004). 16 Wernsdorfer, W. & Ruben, M. Synthetic Hilbert Space Engineering of Molecular Qudits: Isotopologue Chemistry. Adv Mater 31, 1806687, (2019). 17 Kane, B. E. A silicon-based nuclear spin quantum computer. Nature 393, 133-137, (1998). 18 Yang, S. et al. High-fidelity transfer and storage of photon states in a single nuclear spin. Nat Photonics 10, 507-511, (2016). 19 Willke, P., Yang, K., Bae, Y., Heinrich, A. J. & Lutz, C. P. Magnetic resonance imaging of single atoms on a surface. Nat Phys 15, 1005-1010, (2019). 20 Khajetoorians, A. A., Wiebe, J., Chilian, B. & Wiesendanger, R. Realizing All-Spin-Based Logic Operations Atom by Atom. Science 332, 1062-1064, (2011). 21 Wang, Y. et al. Universal quantum control of an atomic spin qubit on a surface. Npj Quantum Inform 9, 48, (2023). 22 Yang, K. et al. Engineering the Eigenstates of Coupled Spin-1/2 Atoms on a Surface. Phys Rev Lett 119, 227206, (2017). 23 Steinbrecher, M. et al. Quantifying the interplay between fine structure and geometry of an individual molecule on a surface. Phys Rev B 103, 155405, (2021). 24 Farinacci, L., Veldman, L. M., Willke, P. & Otte, S. Experimental Determination of a Single Atom Ground State Orbital through Hyperfine Anisotropy. Nano Lett 22, 8470-8474, (2022). 25 Kim, J. et al. Anisotropic Hyperfine Interaction of Surface-Adsorbed Single Atoms. Nano Lett 22, 9766-9772, (2022). 26 McMillan, S. R., Harmon, N. J. & Flatte, M. E. Image of Dynamic Local Exchange Interactions in the dc Magnetoresistance of Spin-Polarized Current through a Dopant. Phys Rev Lett 125, 257203, (2020). 27 Natterer, F. D. Waveform-sequencing for scanning tunneling microscopy based pump-probe spectroscopy and pulsed-ESR. Methodsx 6, 1279-1285, (2019). 28 Johansson, J. R., Nation, P. D. & Nori, F. QuTiP: An open-source Python framework for the dynamics of open quantum systems. Comput Phys Commun 183, 1760-1772, (2012). 29 Degen, C. L., Reinhard, F. & Cappellaro, P. Quantum sensing. Rev Mod Phys 89, 035002, (2017). 30 Savytskyy, R. et al. An electrically driven single-atom \"flip-flop\" qubit. Sci Adv 9, eadd9408, (2023). 31 Bae, Y. et al. Enhanced quantum coherence in exchange coupled spins via singlet-triplet transitions. Sci Adv 4, eaau4159, (2018). 32 Shiddiq, M. et al. Enhancing coherence in molecular spin qubits via atomic clock transitions. Nature 531, 348-351, (2016).   Fig. 1 | Single atom nuclear polarization.  a, STM topography of the single 47TiH studied in this work. A schematic drawing shows the magnetic STM tip above the electron spin (blue) and nuclear spin (red) of the single atom. b, Energy diagram of the spin states of a single 47TiH. In the high field regime, the eigenstates resemble Zeeman product states. ESR transitions (purple arrows) can be driven between states with equal nuclear spin. c, ESR-STM measurements at different applied DC bias (T = 1.5 K, Bext = 1.5 T, VRF = 25 mV, Iset = 2.5 pA, f0 = 11.5 – 12.56 GHz). Line traces at 35 mV, 80 mV and 120 mV are shown with fits using six Fano lineshapes scaled by the Boltzmann factor in order to extract an effective temperature. \n  Fig. 2 | ESR and NMR-type measurements in the low-field regime. a, Energy diagram of the atomic eigenstates as a function of hyperfine coupling, quadrupole moment, external and tip-induced magnetic field. b, ESR-STM measurements (T = 400 mK, Bext = 20 mT, VRF = 40 mV, Iset = 2 pA) as function of tip-sample conductance. As indicated with icons, tip-sample separation decreases towards the right side of the plot, corresponding to an increasing tip-induced magnetic field. Both ESR and NMR-type transitions are observed and indicated as such. The bottom close-up is a separate dataset showing the splitting of the NMR transitions and a curve upwards of the bottom ESR transition signaling the avoided level crossing. c, Zoom-in to marked region of (b). d, Simulations of the ESR-STM measurements (see Methods), where we consider the tip field to make an 8° angle with the out-of-plane external field. e, Zoom-in to marked region of (d). \n  Fig. 3 | Free evolution measurements and Lindblad simulations. a, Pump-probe data for different tip-atom distances set by the junction conductance (Vset = 130 mV, T = 400 mK, B = 15 mT, for details on the pulse scheme, see Methods). b, Lindblad simulation of the free time evolution of the electron spin when initialized to |↓,−5/2⟩. c, Corresponding Lindblad simulation of the free time evolution of the nuclear spin. The calculations also show the onset of an additional oscillation in the nuclear spin at around 13 mT. However, since the period is an order of magnitude longer than the coherence time of the electron spin, it is not visible in our measurements. \n  Fig. 4 | Origin of the beating pattern. a, Zoom-in on the relevant avoided level crossings of Fig. 2a. The Bloch spheres illustrate the dominating dynamics arising from the superpositions of the corresponding states. b, Fit to a line trace from the pump-probe data from Fig. 3a showing a beating pattern. The pattern arises from two dominating frequencies and their sum frequency. c, Fit to a line trace from the pump-probe data from Fig. 3a showing a single frequency oscillation.    \nMethods  Sample preparation The sample was prepared in-situ in a separate vacuum chamber with a residual pressure of 10-10 mbar. The surface of the Ag(100) crystal was cleaned by two cycles of Ar+ sputtering and annealing to 650°C. Subsequently, bilayer MgO islands were grown by evaporating Mg in an oxygen environment of 10-6 mbar while the sample was annealed to ~500°C. We deposited Fe and Ti atoms on the surface by e-beam evaporation. Magnetic tips were created by picking up 3 to 15 Fe atoms with a tungsten wire STM tip.    Equipment The measurements were conducted using a commercial Unisoku USM-1300 low-temperature STM. All DC bias values reported are with respect to the sample. For ESR measurements, we used a Rohde & Schwarz SMB100A to generate radiofrequency voltage signals and a Tektronix PSPL5542 bias-tee to combine the RF with the DC bias before sending the combined output to the STM tip. The RF signal was chopped at 271 Hz and we used a Stanford Research SR830 lock-in amplifier for detection. The ESR measurements were performed in constant-current mode. A wiring schematic of the components used during ESR measurements is shown in Extended Data Fig. 1a.  For pump-probe measurements, we used a Keysight AWG M8195a to generate the DC pulses. For details on the pulse trains, see Supplementary Section 4. The DC bias was sent through the DC input of the Tektronix PSPL5542 bias-tee and combined with the pulse train via a Mini-Circuits ZFRSC-183-S+ splitter before being sent to the STM tip.  A wiring schematic of the components used during ESR measurements is shown in Extended Data Fig. 1b.  Hyperfine anisotropy measurements In order to correctly calculate the eigenenergies of the Hamiltonian presented in Equation (1), we need to determine the hyperfine vector A. Due to the C4v symmetry of the crystal structure of the oxygen binding site, the two in-plane directions Ax and Ay are equal. Therefore, we only need two measurements to characterize A fully: the hyperfine splitting in-plane and out-of-plane. Consequently, we sweep the angle of the external field from in-plane to out-of-plane while measuring the magnitude of the hyperfine splitting in a 47Ti atom. Results are shown in Extended Data Fig. 2: we find an in-plane splitting of 10 MHz, in accordance with previous measurements5 and an out-of-plane splitting of 130 MHz.   Simulations of ESR results ESR simulations were done by generating Fano functions for each possible spin transition between two eigenstates. The amplitude Inm of a transition between states |𝑚⟩ and |𝑛⟩ was calculated by:   𝐼./∝|𝑉./|)∆𝑃./F𝐁%'(∙%I𝑛J𝐒LJ𝑛M−I𝑚J𝐒LJ𝑚M*N(2)   This amplitude consists of three components. The part between curly brackets is the readout strength, which scales with the difference in expectation value of each spin between the two states, projected onto the measurement axis. The first part corresponds to the driving amplitude between the two states. To achieve a result that matches the experiment we distinguish here between ESR and NMR type transitions and add them with separate scaling factors. The two different driving terms are modelled by:  𝑉./012=I𝑛J𝐁%'(∙𝐒LJ𝑚M𝑉./342=I𝑛J𝐁%'(∙𝐈,J𝑚M(3)  We found the simulations to match the experimental results best when using a ratio 𝐼./012:𝐼./342=1:100. This difference in driving amplitudes may be explained by the nuclear spin having significant longer relaxation and coherence times than the electron spin.  The population difference ΔPnm is calculated by finding the steady state solution to the Bloch-Redfield equations for the coupled spin system connected to a spin-polarized bath (tip, t) and a spin-averaged bath (sample, s). In a general form this equation is written as:   d𝜌./(𝑡)d𝑡=−𝑖𝜔𝜌./(𝑡)+$𝑅./56(𝑡)𝜌56(𝑡)56(4)  Here, the first term is the unitary von Neumann evolution of the density matrix 𝜌X and the second term describes the influence of the environment via the Redfield tensor element Rnmkl. When a bias is applied between the two baths the resulting tunnelling electrons scatter with the atomic electron spin via Kondo interaction:  ℋ\"789:8=$𝐽𝐒L∙𝐬X\"\"*%,;(5)  Where 𝐒L is the surface electron spin and 𝐬X\" the spin density of the tip and sample bath and J the coupling between the bath and surface spin. The relevant Redfield tensor that gives rise to the spin pumping of the atomic electron spin due to the spin-polarized tunnelling electrons is given by:  𝑅./56=1ℏ)𝐽%𝐽;$⎝⎜⎜⎜⎛−𝛿/6$I𝑛J𝑆,<J𝑝MI𝑝J𝑆,=J𝑘M𝑔%→;<=%𝜔?5*?−𝛿.5$I𝑙J𝑆,<J𝑝MI𝑝J𝑆,=J𝑚Mc𝑔%→;<=%𝜔?6*d∗?+I𝑛J𝑆,=J𝑘MI𝑙J𝑆,<J𝑚Mc𝑔%→;<=(𝜔.5)+c𝑔%→;<=(𝜔/6)d∗d⎠⎟⎟⎟⎞<,=*+,,,-(6)  Here, the ω are the energy differences between different eigenstates. The correlation function between tip and sample baths is given by:  𝑔%→;<=(𝜔./)=ℏ𝜋𝜚%𝜚;$𝜏A!!A<𝜏AA!=4A,A!,A!!𝜂A!!A!%𝜂AA;l𝑓(𝜀%)𝑓%1−𝑓(𝜀;)*𝛿(𝜀%−𝜀;−𝜀./−𝑉)d𝜀%d𝜀;(7)  Where 𝜚% and 𝜚; are the densities of states of the tip and sample baths, 𝜂̂; is the spin-averaged density matrix of the sample bath and 𝜂̂%=\"#(𝕀L+𝐧∙𝝉X) is the spin-polarized density matrix of the tip bath with polarization vector 𝐧, in which 𝝉X=(𝜏̂+,𝜏̂,,𝜏̂-) is a vector composed of Pauli matrices. The spin polarized electrons tunnelling between tip and sample result in an effective spin pumping of the electron spin which also affects the nucleus through the hyperfine coupling. We use the following coupling strengths between atomic electron spin and the tip and sample baths: 𝜚%𝐽% = 0.011 and 𝜚;𝐽;\t= 0.05. Coupling to each bath individually is also taken into account and leads to relaxation processes due to spin scattering.  Technical details on pump-probe measurements The pump-probe measurements were performed in constant-height mode after tracking the center of the atom for ~1 hour and while linearly compensating for drift due to piezo creep. For the pump-probe data shown in the main text the tip was placed ~300 fm away from the center in order to minimize the in-plane field component of the tip.  To pump both electron and nuclear spins we start the pulse train with a 400 ns positive bias pulse. We then add an extra 5 ns negative pump pulse that we empirically found to increase the signal intensity. We hypothesize that the addition of this pulse brings the atomic electron spin in a superposition state close to |↓,−5/2⟩ which maximizes the amplitude of the resulting spin dynamics. In Extended Data Fig. 3 pulse sequences with and without this extra pulse are compared. To increase the signal detected by the lock-in amplifier we switch the polarity of the 5 ns probe pulse in the B-cycle27. All pulses went through a 5 GHz low pass filter before being sent to the STM to minimize artefacts. The pulses were calibrated to have an amplitude of 110 mV at the tip sample junction.  Lindblad simulations To simulate the expected time dynamics of the coupled electron-nuclear spin system we solve the Lindblad equations for the Hamiltonian presented in the main text. Since there are no electrons tunnelling during the free evolution time, it is not necessary to use the Bloch-Redfield model that takes into account the spin polarisation of the tip bath. We only calculate the unitary von Neumann evolution and add some effective relaxation and decoherence processes in the form of Lindblad collapse operators that interact with the electron spin. In the general form the Lindblad equation is written as:  d𝜌X(𝑡)d𝑡=−𝑖ℏuℋ\",𝜌X(𝑡)v+$12w2𝐿L.𝜌X(𝑡)𝐿L.B−𝜌X(𝑡)𝐿L.B𝐿L.−𝐿L.B𝐿L.𝜌X(𝑡)y.(8)  Where the Lindblad collapse operators are given by 𝐿L.={𝛾.𝐴,. with 𝐴,. the operators coupling the spin to the environment and γn the rates at which these processes happen. For our simulations we use the following specific Lindblad operators to simulate effective relaxation and decoherence processes occurring on the electron spin due to coupling with the tip and sample baths.  𝐿LC=1{𝑇D1{1+𝑒CℏF5$G⁄𝑆,C𝐿LI=1{𝑇D1{1+𝑒ℏF5$G⁄𝑆,I𝐿L-=2𝑇D−𝑇)4𝑇D𝑇)𝑆,-(9)  We use effective relaxation and decoherence times T1, T2 = 300 ns and a temperature T = 400 mK. We separately take into account decoherence from the mechanical vibration of the tip, leading to magnetic field noise, by applying a 2 mT Gaussian filter over the magnetic field axis. These Lindblad calculations were done using a python package called QuTiP28.    Tip dependence To show the influence of the microtip, we perform ESR-STM measurements with a different microtip on the same 47Ti atom bound on the oxygen site. The results are shown in Extended Data Fig. 4 for two different Bext values. Compared to the data taken with the tip of the main text (Fig. 2) we find a number of differences. First, at both external field values, the lowest two ESR resonances are closer together at large conductance values and are split apart when the tip is retracted (G < 40 pS). Second, the splitting between NMR resonances is so small that they are observed as a single peak that shifts in energy with applied tip field. We get a good agreement in the simulations by adjusting a single parameter compared to the simulations shown in Fig. 2: the angle of the magnetic tip field with the external field. While we found a very small angle of 8° for the tip used in Fig. 2, we find an angle of ∼80° for the tip used in Extended Data Fig. 4. This means that when we approach the atom with the latter tip, we effectively add more in-plane field than out-of-plane field. Combined with the anisotropy of the hyperfine tensor, this results in qualitatively different behavior of the resonances with increasing tip field: the splitting between ESR resonances becomes smaller (comparable to Extended Data Fig. 2) and the NMR resonances, which are a direct measurement of this energy difference, shift to lower frequency.   Data availability All data presented in this paper are publicly available at https://doi.org/10.5281/zenodo.8316339.  Acknowledgements This work was supported by the Dutch Research Council (NWO Vici grant VI.C.182.016). P.W. acknowledges funding from the Emmy Noether Programme of the DFG (WI5486/1-1) and the Daimler and Benz Foundation.  Author contributions LMV, LF and SO conceived the experiment. LMV, MPC and EWS performed the measurements. LMV, MPC and RB performed the calculations. All authors analyzed and discussed the results. LMV, LF, PW and SO wrote the manuscript, with input from all authors. SO supervised the project.  Competing interests The authors declare no competing interests.     Extended Data Fig. 1 | Wiring diagrams. a, Schematic of electronics configuration during experiments in ESR-STM mode. b, Schematic of electronics configuration during experiments in pump-probe mode.    \n  Extended Data Fig. 2 | Hyperfine anisotropy measurements. ESR-STM measurements on 47Ti at angles varying from in-plane (0°) to out-of-plane (90°) with fits of 6 equidistant Fano functions. Traces are offset for clarity. Inset: magnitude of hyperfine splitting obtained from fitting the entire dataset. All fitting parameters, except the hyperfine splitting magnitude and peak amplitude, were determined by first fitting an identical data set taken on a Ti isotope without nuclear spin. Experimental parameters: Iset = 3 pA, Vset = 60 mV, VRF = 25 mV, T = 1.5 K, |Bext| = 0.5 –1.5 T.    \n  Extended Data Fig. 3 | Pump-Probe pulse scheme. a, Pulse schemes in the lock-in A and B cycles, without (top) and with (bottom) additional pump pulse. b, Comparison between measurements obtained using different widths tadd for the additional pump pulse (Iset = 4 pA, Vset = 130 mV). Curves were shifted vertically for clarity. The free evolution time on the horizontal axis corresponds to the time t indicated in (a). Measurements shown in the main paper were all performed using tadd = 5 ns.      \n  Extended Data Fig. 4 | Different microtip. a, ESR-STM measurements acquired on the same 47Ti atom as Fig. 2, but with a different microtip. Data taken at Bext = 40 mT, T = 400 mK, VRF = 40 mV, Iset = 2 pA. b, Same as (a), but taken at Bext = 25 mT. c, Simulation corresponding to (a). d, Simulation corresponding to (b). For the simulations we consider the tip field to make an 80° angle with the out-of-plane external field.  \n"    },    {        "title": "1605.00710v2.Electrical_Detection_of_Magnetization_Dynamics_via_Spin_Rectification_Effects.pdf",        "content": "Electrical Detection of Magnetization Dynamics via Spin Rectification Effects\nMichael Harder, Yongsheng Gui, Can-Ming Hu\u0003\nDepartment of Physics and Astronomy, University of Manitoba, Winnipeg, Canada R3T 2N2\nAbstract\nThe purpose of this article is to review the current status of a frontier in dynamic spintronics and contem-\nporary magnetism, in which much progress has been made in the past decade, based on the creation of a\nvariety of micro- and nano-structured devices that enable electrical detection of magnetization dynamics.\nThe primary focus is on the physics of spin rectification effects, which are well suited for studying magneti-\nzation dynamics and spin transport in a variety of magnetic materials and spintronic devices. Intended to be\nintelligible to a broad audience, the paper begins with a pedagogical introduction, comparing the methods\nof electrical detection of charge and spin dynamics in semiconductors and magnetic materials respectively.\nAfter that it provides a comprehensive account of the theoretical study of both the angular dependence\nand line shape of electrically detected ferromagnetic resonance (FMR), which is summarized in a handbook\nformate easy to be used for analyzing experimental data. We then review and examine the similarity and\ndifferences of various spin rectification effects found in ferromagnetic films, magnetic bilayers and magnetic\ntunnel junctions, including a discussion of how to properly distinguish spin rectification from the spin pump-\ning/inverse spin Hall effect generated voltage. After this we review the broad applications of rectification\neffects for studying spin waves, nonlinear dynamics, domain wall dynamics, spin current, and microwave\nimaging. We also discuss spin rectification in ferromagnetic semiconductors. The paper concludes with both\nhistorical and future perspectives, by summarizing and comparing three generations of FMR spectroscopy\nwhich have been developed for studying magnetization dynamics.\nKeywords: spin rectification, ferromagnetic resonance, magnetization dynamics, magnetoresistance, spin\ntorque, spin pumping, dynamic spintronics, contemporary magnetism\nContents\n1 Introduction 2\n2 The Ingredients of Spin Rectification 4\n2.1 Magnetotransport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5\n2.1.1 Two-Current Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6\n2.1.2 Anisotropic Magnetoresistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7\n2.1.3 Giant Magnetoresistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8\n2.1.4 Tunnel Magnetoresistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11\n2.1.5 Spin Hall Magnetoresistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12\n2.2 Magnetization Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14\n2.2.1 Field Torque Induced Magnetization Dynamics . . . . . . . . . . . . . . . . . . . . . . 14\n2.3 The Method of Analyzing Spin Rectification: Basic Examples . . . . . . . . . . . . . . . . . . 16\n\u0003Corresponding author at: Dynamic Spintronics Group, University of Manitoba, http://www.physics.umanitoba.ca/ \u0018hu/\nEmail addresses: michael.harder@umanitoba.ca (Michael Harder), ysgui@physics.umanitoba.ca (Yongsheng Gui),\nhu@physics.umanitoba.ca (Can-Ming Hu)\nPreprint submitted to Physics Reports August 19, 2016arXiv:1605.00710v2  [cond-mat.mtrl-sci]  18 Aug 20162.3.1 In-Plane Angular Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17\n2.3.2 Out-of-Plane Angular Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19\n3 Spin Rectification Effects in Magnetic Structures 20\n3.1 Ferromagnetic Films: Early Studies Using Pulsed Microwave Sources . . . . . . . . . . . . . . 20\n3.2 Micro-Structured Monolayers: The Spin Dynamo . . . . . . . . . . . . . . . . . . . . . . . . . 22\n3.2.1 Method and Device Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24\n3.2.2 Lineshape Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25\n3.2.3 Angular Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28\n3.3 Magnetic Bilayers: The Spin Battery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28\n3.3.1 The Basic Physics of Spin Pumping and Spin Hall Effects . . . . . . . . . . . . . . . . 29\n3.3.2 Spin Battery and \"Longitudinal\" Spin Pumping Voltage . . . . . . . . . . . . . . . . . 32\n3.3.3 Inverse Spin Hall Effect and \"Transverse\" Spin Pumping Voltage . . . . . . . . . . . . 33\n3.3.4 Spin Rectification vs Spin Pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35\n3.3.5 Spin-Transfer Torque Rectification in Bilayers . . . . . . . . . . . . . . . . . . . . . . . 36\n3.3.6 Spin Hall Magnetoresistance Rectification . . . . . . . . . . . . . . . . . . . . . . . . . 38\n3.4 Magnetic Tunnel Junctions: The Spin Diode . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39\n3.4.1 Spin-Transfer Torque Rectification in Magnetic Tunnel Junctions . . . . . . . . . . . . 39\n3.4.2 Voltage Torque Rectification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42\n4 Applications of Spin Rectification 43\n4.1 Electrical Detection of Spin Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43\n4.1.1 Review of Spin Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44\n4.1.2 Detection of Spin Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46\n4.2 Electrical Detection of Nonlinear Magnetization Dynamics . . . . . . . . . . . . . . . . . . . . 48\n4.2.1 Review of Nonlinear Magnetization Dynamics . . . . . . . . . . . . . . . . . . . . . . . 48\n4.2.2 Detection of Nonlinear Magnetization Dynamics . . . . . . . . . . . . . . . . . . . . . 49\n4.3 Electrical Detection of ac Spin Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50\n4.4 Electrical Detection of Magnetization Dynamics in Ferromagnetic Semiconductors . . . . . . 51\n4.5 Electrical Detection of Domain Wall Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 53\n4.6 Spintronic Microwave Sensing and Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55\n4.7 Spintronic Rectifier for Electromagnetic Energy Harvesting . . . . . . . . . . . . . . . . . . . 57\n5 Summary and Outlook: Three Generations of FMR Spectroscopy 58\n1. Introduction\nMagnetization dynamics is a venerable subject in solid state physics, originating with the discovery of\nferromagnetic resonance (FMR) absorption [1–6] in the first half of the 20thcentury. Yet still today, well into\nthe 21stcentury, the study of magnetization dynamics continues to be an active and fast paced research field\n[7–9]. Early understanding of magnetization dynamics was based solely on the phenomenological Landau-\nLifshitz-Gilbert (LLG) equation that does not consider magneto-transport effects, with experimental studies\nemploying microwave absorption measurements. However this has changed recently. The application of\nmodern material growth and nano-lithographic techniques, which was extended from the semiconductor to\nmagnetism community, has revealed a wealth of new and rich physics demonstrating that charge transport\nand spin dynamics are strongly correlated in magnetic materials. On one hand charge transport can be\ncontrolled through magnetization dynamics (or magnetization configurations) through, for example, giant\nmagnetoresistance (GMR) in magnetic multilayer devices [10, 11]. On the other hand, the magnetization in\nnano-structured magnetic multilayers patterned with a sub-micrometer lateral size can be reversed by a dc\ncurrent [12, 13]. Further studies have even found that dc currents and dc voltages can be generated by the\nrectification effect of high-frequency magnetization dynamics [14–48]. This so-called spin rectification effect\nprovides a novel technique for the study of magnetization dynamics – electrical detection.\n2Electrical detection techniques are not new to the study of dynamic material properties, having long\nbeen used in semiconductors and photovoltaics. For example, it is well known that the dynamic interaction\nbetweenphotonsandelectronchargecangeneratenon-thermalelectrondiffusion, thusconvertingsomeofthe\nphoton energy into electric energy when an electromagnetic wave is incident upon a material surface. This\nso-called photovoltaic effect was first observed by French physicist Edmond Becquerel in 1839 [49] using a\npair of platinum electrodes dipped in an electrolyte solution [50]. Nowadays, exploitation of the photovoltaic\neffect in solar cells provides the third most important renewable energy source after hydro and wind power.\nThe key element of a solar cell is a semiconducting device in which electrons in the valence band can absorb\nphoton energy, jump into the conduction band, and then generated electron-hole pairs which are separated\nby a p-n junction and collected by different electrodes [51]. In addition to the photovoltage produced in this\nmanner, the generation of electron-hole pairs also results in photoconductivity [52] – the enhanced electrical\nconductivity of semiconducting materials. Hence this effect is widely used in photodetectors over a broad\nfrequency range, from infrared up to gamma radiation [53–56].\nThe high conversion efficiency from photon energy to an electric signal in semiconductors also makes\nphotovoltageandphotoconductivityahighlysensitivedetectionmethodtostudychargedynamics(especially\nin two-dimensional electron systems). These techniques have provided insight into, for example, inter-\nband transitions [57], inter-subband transitions [58], magneto-plasmonics [59] and spin-flip excitations [60].\nSuch success naturally begs the question, could electrical detection provide insight into spin dynamics in\nmagneticallyorderedmaterials? Onthesurfaceitseemslikethetransitiontospindynamicsinferromagnetic\nmaterials would be difficult. First, the excitation of spin dynamics occurs in d-shell electrons, which have\nless influence on the electric response of a material. Early work on spin-induced current and voltage in\nsemiconductors [61] and ferromagnetic metal films [62] indicated that the conversion efficiency between\nphoton energy and electricity through spin excitations was very poor, often requiring high power (up to\nkW) GHz and THz sources. Second, the conductivity of ferromagnetic metals is at least four orders of\nmagnitude larger than that of semiconductors. Hence the impact of microwave photons on the conductivity\nof ferromagnetic metals was expected to be negligible.\nHowever, starting from the middle 2000s, triggered by the rapid development of micro/nano-structuring\ntechniques applied to ferromagnetic devices, there has been a surge of research interest in the study of spin\ndynamics in ferromagnetic materials by means of electrical detection. A vast number of experimental and\ntheoretical investigations have been carried out, not only on all aspects of traditional spin waves in ferro-\nmagnetic monolayers, but also on novel spin phenomena such as spin pumping (SP) and spin-transfer torque\n(STT) in ferromagnetic multilayers. Through this surge of research, which was sometimes accompanied by\nintense scientific debate, a comprehensive understanding of one of the key mechanisms responsible for dc\nvoltage generation in a variety of magnetic devices, the spin rectification effect (SRE), has been developed.\nThe simplest way to understand the SRE is to highlight the close analogy between spin rectification\nand the well known optical rectification which occurs in nonlinear media with large second-order suscep-\ntibility [63, 64]. In optical rectification the nonlinear optical response of the time-dependent electric field\ne0cos (!t)is governed by the trigonometric relation cos2(!t) = [1 + cos (2 !t)]=2, and hence results in op-\ntical rectification and second harmonic generation. Similarly, spin rectification is the generation of a dc\nvoltage/current due to the nonlinear coupling between an oscillating current and an oscillating resistance in\nmagnetic structures. Despite the similarities, including that both optical and spin rectification are linearly\ndependent on the electromagnetic power, spin rectification has several unique features. First, the dc signal\nis linearly proportionally to the resistance of the sample, which makes it a powerful tool to investigate spin\ndynamics in monolayer nano-structured samples, which are notoriously difficult to study using traditional\ntransmission/reflection measurements since such a signal is proportional to sample volume. Second, the os-\ncillating resistance caused by the dynamic magnetization can be driven by a time dependent magnetic field,\nh0cos (!t+\u001e), where\u001eis the relative phase between microwave eandhfields and plays an important role\nin the curious line shape of electrically detected ferromagnetic resonance. Due to its high sensitivity and fea-\nsibility of implementation, spin rectification has significant impact on the study of spin dynamics. Over the\npast decade, and with the great effort of many research groups, the SRE has been widely employed to study\nferromagnetic resonance (FMR), spin wave resonance, and domain wall resonance in various ferromagnetic\nmaterials including ferromagnetic metals, ferromagnetic semiconductors, ferrimagnetic insulators as well as\n3multilayer structures. Furthermore, properly analyzing the SRE has been found essential for studying novel\nspin dynamics including phase-resolved FMR, spin-transfer torque, spin pumping, and the spin Hall effect.\nParallel to such a path of basic research on spintronics, the SRE has also found its way into novel microwave\napplications, such as spintronic microwave imaging and wireless microwave energy harvesting.\nThis article provides a review of the rectification effects in magnetic structures and their applications. We\nwill examine the physical basis of electrical detection techniques as they apply to magnetization dynamics,\nand examine a variety of applications in which such techniques have been employed. In Sec. 2 we begin by\nexamining the “ingredients\" of spin rectification, namely magnetoresistance, magnetization dynamics and\ntheir nonlinear coupling through the generalized Ohm’s law. In Sec. 3 we turn to various device structures,\nstarting with early pulsed microwave studies followed by an examination of rectification in ferromagnetic\nmonolayers, bilayers and magnetic tunnel junctions. Along the way we discuss spin pumping and spin Hall\neffects and the important role of line shape and angular analyses in distinguishing the many competing\nvoltage producing effects in magnetic devices. Having explained the basic techniques of electrical detection\nof magnetization dynamics, in Sec. 4 we turn to several important applications. On the side of fundamental\nphysics we look at the detection of spin waves and nonlinear magnetization dynamics, dc electrical detection\nof ac spin current, magnetization dynamics in ferromagnetic semiconductors and studies of domain wall\ndynamics. On the applied side, we examine the role of spin rectification in novel microwave sensing, imaging,\nand wireless energy harvesting technologies. The article is concluded in Sec. 5 including both historical\nand future perspectives. By reviewing the historical evolution of three generations of FMR spectroscopy,\nincluding cavity transmission, flip chip absorption, and electrical detection techniques, we also glimpse into\nthe future of FMR spectroscopy, where the venerable science of magnetism is merging with the modern\nphysics of cavity QED, leading us to a new field of cavity spintronics [65].\n2. The Ingredients of Spin Rectification\nSpin rectification (SR) is the production of a dc voltage Vdcin a ferromagnetic structure due to the\nnonlinear coupling between a dynamic resistance R[H(t)]and a dynamic current I(t). To understand the\norigin of this effect, consider a ferromagnetic structure excited by a microwave field at angular frequency\n!. The electric and magnetic fields inside the structure therefore take the form e(t) =e0e\u0000i!tandh(t) =\nh0e\u0000i(!t\u0000\b), where \bis a phase shift associated with losses in the system [66]. The e(t)field will drive a\ncurrent, I(t) =I0e\u0000i!t=\u001be(t)and the h(t)field will drive magnetization precession, m(t) =\u001fh(t), where\n\u001band\u001fare the high frequency material response functions, the conductivity and susceptibility respectively.\nIf the structure is a simple monolayer, h(t)alone will drive the magnetization (Recently it has actually\nbeen found that spin-orbit torques may also be present in single layer devices [67]. We will discuss spin-\norbit torques briefly in Sec. 3.3.5, however for the purpose of this review we will focus on field driven\nmagnetization dynamics in single layer devices). In bilayer devices or magnetic tunnel junctions where a\nspin current js(t)is present, an additional spin torque will contribute to the magnetization precession so that\nm(t) =\u001fh(t)+\u001fsjs(t), where\u001fsis the frequency dependent effective susceptibility due to the spin current.\nDue to the magnetoresistance (MR) of the heterostructure, the effect of the magnetization precession is to\nproduce a dynamic resistance, R(t) =R[H(t)], dependent on the magnetic field H(t) =H0+h(t). As a\nconsequence, a nonzero time averaged voltage can be measured along the current direction,\nV=hRefI(t)gRefR[H(t)]gi=1\n2I0h0\u0001rR(H0) cos \b: (2.1)\nHere we have expanded the resistance in a Taylor series around the dc field H0to first order in the rf field\nh(t),R(H(t)) =R(H0) +h(t)\u0001rR(H0)(rR(H0) =@R=@ HjH0is the field gradient of the resistance\nevaluated at H0), and taken the time average over one period, denoted by hi. Based on this discussion\nwe see that spin rectification requires three ingredients: 1) Some form of magnetoresistance, whereby the\nresistance of the heterostructure depends on the orientation of the magnetization. 2) A torque which drives\nthe rf magnetization, resulting in a time varying magnetoresistance. 3) An rf current which can couple to\nthe rf resistance and produce a dc voltage. These three requirements are summarized in Fig. 1.\n4Figure 1:The ingredients of spin rectification. Magnetoresistance effects result in a magnetization orientation dependence in\nthe resistance of the ferromagnetic heterostructure. The origin of the magnetoresistance will depend on the device structure\nand various forms of magnetoresistance are discussed in Sec. 2.1. By driving the magnetization with microwave frequency\nfields the resulting precession will produce an rf resistance. The magnetization may be driven by a field torque produced by\nan rf magnetic field, hrf, or from a spin torque produced by a spin polarized current js. The rf resistance can then couple\nnonlinearly to an rf current and produce a rectified voltage.\nIn this section we provide a general review of the ingredients of spin rectification: Sec. 2.1 contains a\ndiscussion of the relevant magnetoresistance effects while in Sec. 2.2 we review magnetization dynamics and\nthe Landau-Lifshitz-Gilbert equation. For simplicity we only concern ourselves with field torques in Sec.\n2.2, leaving the treatment of spin torques to Sec. 3.3.5 and Sec. 3.4 where we focus on bilayer devices and\nmagnetic tunnel junctions (MTJs). Finally in Sec. 2.3 we present the simplest example of spin rectification,\nthe voltage produced in a ferromagnetic monolayer, which will serve as an illustrative example which sets\nthe stage for the detailed description of SR in more complex structures presented in Sec. 3.\n2.1. Magnetotransport\nMagnetotransport – the modification of transport properties due to magnetic fields – can depend on\nboth the charge and spin of the carrier and as such is responsible for a myriad of phenomena in metals and\nsemiconductors [68]. However the magnetotransport effects which result in spin rectification are those which\ncausemagnetoresistance, wheretheelectricalresistanceofthematerialdependsonanappliedmagneticfield,\nR=R(H). Although even non-magnetic metals display so called ordinary magnetoresistance (OMR) due to\ncharged-inducedmagnetotransport(theLorentzforce), themaximummagnetoresistanceratioofsucheffects,\nMR = [R(H)\u0000R(0)]=R(0), is too small to be of technological interest. On the other hand in magnetic\nheterostructures large magnetoresistance due to spin-dependent transport dominates and as a result MR\neffects in magnetic materials have a rich history of applications. The anisotropic magnetoresistance (AMR),\nfirst discovered by William Thomson in 1856 [69], was the first MR effect used by IBM to build hard disk\ndrive read heads in 1991. The chief limitation of AMR for memory applications was the low MR ratio of\n\u0018a few %. For this reason the celebrated discovery of giant magnetoresistance (GMR), independently by\nFert [10] and Grünberg [11], with room temperature MR ratios up to \u0018100%, revolutionized the magnetic\nrecording industry. By 1997 IBM had used spin valve sensors based on GMR to replace the AMR sensors in\nmagnetoresistive hard disk drive read heads, providing an increase in areal density of up to 100% per year.\nWhile GMR in multilayer structures has become important for memory applications, it is important to note\nthat from a physics perspective GMR is not limited to multilayer structures but is more generally due to\ninhomogeneities and can be realized e.g. in granular magnetic solids [70].\nEvenfurtherincreasesintheMRratiobecamepossiblewithadvancesinthefabricationoftunnelbarriers\nmade of crystalline magnesium oxide (MgO) which made tunnel magnetoresistance (TMR) an attractive\npossibility for next generation memory devices. TMR, originally discovered in 1975 by Jullière in Fe/Ge-\nO/Co junctions at 4.2 K [71], has recently been used to achieve MR ratios of 600% at room temperature\nand over 1100% at 4.2 K in CoFeB/MgO/CoFeB magnetic tunnel junctions (MTJs)[72]. The large MR\nratio of MTJs allows the creation of non volatile, high endurance memories with fast random access making\nmagnetic random access memory (MRAM) a good candidate for a ‘universal memory’ [73].\nAll of these memory applications we have described rely on the staticproperties of MR, which are sum-\nmarized in Table 1 and which we will discuss further in this section as the first ingredient of spin rectification.\n5Table 1:Summary of static magnetoresistance effects discussed in detail in Sec. 2.1. The magnetoresistance effects for\nferromagnetic monolayers are considered for an in-plane Hfield and in-plane magnetization M.\u0012mdenotes the angle between\nthe magnetization and the current flow, \u0012is the angle between the magnetization of the two ferromagnetic layers, and \u0012sis the\nangle between the magnetization and the spin polarization.\nMR\nEffectOriginDevice\nStructureAngular Dependence MR ratio\nAMR SO CouplingFM mono-\nlayerRAMR (\u0012m) =R(0)\u0000\u0001Rsin2\u0012m\u0018a few %\nPHE SO couplingFM mono-\nlayerRPHE(\u0012m) =1\n2\u0001Rsin 2\u0012m \u0018a few %\nAHE SO couplingFM mono-\nlayerRAHE(\u0012m) = 0 \u0018a few %\nGMRSpin accumula-\ntionFM/NM\nmultilayersGGMR (\u0012) =GP+ \u0001Gsin2(\u0012=2)or, if GMR\nratio is small, RGMR (\u0012) =RP+\u0001Rsin2(\u0012=2)\u0018100 %\nNonaligned FM\ninclusionsGranular\nmagnetic\nsolids\u001810 %\nTMRSpin dependent\ntunnellingFM/Insulator\nmultilayer\n(MTJ)Same as GMR \u0018600 %\nSMRSHE and spin\nrelaxation\nthrough STTFMI/NM\nbilayerRSMR(\u0012s) =R(0) + \u0001Rsin2\u0012s \u00180.01 %\nHowever, when combined with the second ingredient, magnetization dynamics, spin rectification opens the\nbroader possibility of novel applications utilizing the dynamic properties of magnetoresistance.\n2.1.1. Two-Current Model\nThe two-current model, which attributes the resistivity in ferromagnetic materials to two independent\nspin channels, provides the basis for understanding OMR, AMR, GMR and TMR [74]. This model is based\non three key properties of 3d ferromagnets. First, as noted by Mott [75], the low effective mass of the\nvalencespband electrons, compared to the high effective mass of the dband electrons, means that sp\nelectrons carry most of the electric current. Second, because spin-flip scattering in ferromagnetic materials\nis negligible [76] the current can be modelled as two independent spin channels – spin-up (majority) and\nspin-down (minority) electrons. And finally, due to exchange splitting, the density of states (DOS) of d\nband electrons is greater for spin-up than spin-down electrons. Since the scattering rate is proportional to\nthe DOS, this final observation means that the scattering of up spins will be greater, and therefore the up\nspin channel will provide the greatest resistance contribution (hence the term majority electrons). Although\noriginally described for collinear systems, spin rectification often occurs under non-collinear conditions and\nthe two-current model has also been extended to such situations [77, 78].\nIn the case of OMR the electron current corresponds to the spband electrons while the hole current\ncorresponds to the dband holes and it is sufficient to only consider the effect of the Lorentz force on these\ntwo currents, ignoring the spin degree of freedom. However for AMR, GMR and TMR the spin dependence\nof the two currents is necessary. Insight into the origin of this spin dependence can be gained by considering\n6the Drude conductivity per spin for free electrons [74]\n\u001b=e2\nhk2\nF\n3\u0019\u0015: (2.2)\nHerekFis the Fermi momentum and \u0015is the mean free path which depends on the Fermi velocity and the\nrelaxation time \u001c,\u0015=vF\u001c. The relaxation time can be estimated by Fermi’s golden rule\n\u001c\u00001=2\u0019\n\u0016hhV2\nscatiN(\u000fF) (2.3)\nwherehV2\nscatiis the average value of the scattering potential and N(\u000fF)is the DOS at the Fermi energy.\nFrom Eqs. 2.2 and 2.3 we see that the conductivity has an intrinsic spin dependence caused by the spin\ndependence of kF;vFandN(\u000fF)which is due to the band structure of the material, but there is also a\ncontribution from the scattering potential which is not an intrinsic property of the ferromagnetic material\nand can be due to e.g. defects, impurities or lattice vibrations. In the case of AMR the scattering potential,\ndue to the spin-orbit interaction, is fundamental, however in GMR and TMR the effect of the scattering\npotential may be averaged out by interface effects in the multilayer structure, leaving the band structure as\nthe key contribution to the spin dependent conductivity [74].\n2.1.2. Anisotropic Magnetoresistance\nThe earliest discovered magnetoresistance effect in ferromagnetic metals was AMR [69]. The key char-\nacteristic of AMR is the angular dependence,\nR(H) =R(0)\u0000\u0001Rsin2(\u0012M) (2.4)\nwhere\u0012Mis the angle between the current and magnetization direction and \u0001R=Rk\u0000R?is the difference\nin resistance between the current and magnetization aligned ( Rk) and orthogonal ( R?). It should be noted\nthat Eq. 2.4 may not be appropriate for single crystalline films due to the band structure influence on AMR\n[79]. In such samples AMR can exhibit additional four-fold symmetry [80–83], asymmetric behaviour [84],\nand strong dependence on the angle between the current and crystalline axis [80–85]. In both poly and single\ncrystalline films typically \u0001R> 0[86], although there are exceptions [79, 87], and while the effect of AMR\nis small compared to GMR or TMR in multilayer structures, in monolayers AMR is the dominate effect and\nis typically as large as a few percent, although this depends on, amongst other factors, the material, sample\ngeometry and temperature [79], for example as the film thickness decreases, AMR decreases [88].\nExperimentally \u0012Mis controlled by an applied magnetic field Hand depends on the anisotropy fields in\nthe sample [43]. Therefore the magnetic field range over which AMR takes place is determined by the field\nneeded to change the direction of the magnetization, typically on the order of 10 mT. The exact relationship\nFigure 2:(a) AMR measurement configuration. A dc current is sent through the ferromagnetic sample and the voltage is\nmeasured as a function of the angle \u0012between the magnetization and current direction. Typically \u0012is controlled by changing\nthe field strength with \u0012H= 90\u000efixed. With zero applied field the magnetization will lie along the easy axis of the sample,\nin the current direction, and the resistance will be large. As the magnetic field strength is increased toward saturation, the\nmagnetization rotates to align with the field and the resistance decreases. (b) AMR of \u00180:4%in a permalloy (Py) microstrip\nat\u0012H= 90\u000e. The arrows denote the anisotropic field \u00160HA=\u00160Nx0M0= 4:0mT. The open circles are experimental data\nand the solid curve is a fit using R(0) = 112:66 \n;\u0001R= 0:47 \nandHA= 4:0mT.Source :Adapted from Ref. [43].\n7between\u0012MandHcan be determined by minimizing the free energy, F/H\u0001Mwith respect to \u0012M, taking\ninto account the anisotropy fields. For an in-plane magnetization with only shape anisotropy \u0012Msatisfies\nHsin (\u0012H\u0000\u0012M)\u0000Mcos\u0012Msin\u0012M(Nx0\u0000Nz0) = 0:\nHere\u0012His the angle of Hwith respect to the current direction, and Nx0andNz0are the demagnetization\nfactors in the coordinate system defined in Fig. 2 (a), where ( bx0;by;bz0) are fixed with the sample length\nalongbz0and the sample width along bx0and the measurement coordinate system ( bx;by;bz) rotates with the\nHdirection along bz. Typically the dimension of the sample in the bx0direction is much less than in the\nbz0direction, so the Nz0factor can be ignored. Fig. 2 (b) shows the AMR measured in a Py microstrip of\ndimension 300 \u0016m\u000220\u0016m\u000250 nm at an angle \u0012H= 90\u000ein which case sin\u0012M=H=Nx0M. The measured\ndata (symbols) has been fit using R(0) = 112:66 \n;\u0001R= 0:47 \n;\u00160HA= 4:0mT andNx0= 0:004and the\nAMR ratio is found to be \u0001R=R(0)\u00180:4%.\nThe physical origin of AMR is spin-dependent scattering in ferromagnetic metals due to their band\nstructure and the spin-orbit interaction [79, 86, 89]. This means that to understand the magnitude and sign\nof the effect a detailed knowledge of the band structure is required. However the angular dependence can\nbe determined either by symmetry arguments [79] or through the generalized Ohm’s law [90, 91],\nE=\u001a?J+\u0001\u001aO\nH2(J\u0001H)H+\u0001\u001a\nM2(J\u0001M)M\u0000\u001aH\njHjJ\u0002H\u0000\u001aAHE\njMjJ\u0002M (2.5)\nwhich is a phenomenological modification of the usually linear Ohm’s law to include the nonlinear effects\nintroduced by the presence of magnetic fields and a nonzero magnetization. The second and third terms\ndescribe OMR and AMR respectively while the third and fourth terms describe the ordinary Hall effect\n(OHE) and the anomalous Hall effect (AHE). OMR and the OHE are included here for completeness.\nAlthough the angular dependence of OMR is the same as AMR, as we will see, the spin rectified voltage\nis proportional to \u0001Rwhich in ferromagnetic monolayers is much smaller for OMR than it is for AMR.\nAlso, as illustrated in Fig. 2 (b) AMR is measured by sweeping the magnetic field and therefore in such\nmeasurements OMR would just contribute a constant background since \u0012His fixed. Likewise the AHE\ndominates over the OHE. We note that recently additional galvanometric effects have been predicted [92].\nThe resistivity along a general direction bnis defined as\n\u001an=bn\u0001E\njJj: (2.6)\nUsing this definition the generalized Ohm’s law immediately leads to the angular dependence of AMR\ndescribedbyEq. 2.4. Thein-planeresistancemeasuredtransversetothecurrentdirectioncanbedetermined\nby takingbnalong the width of the microstrip so that bn\u0001J= 0. This transverse AMR is know as the planar\nHall effect (PHE) and has an angular dependence\nRPHE (\u0012M) =\u0001R\n2sin (2\u0012M):\nTheAHEwillnotcontributewhenthevoltageismeasuredalongthestripduetothecrossproduct. However,\nalthough there is no static transverse contribution either, the AHE will contribute to the rectified transverse\nvoltage.\n2.1.3. Giant Magnetoresistance\nGMRisaspinaccumulationeffectpresentinmultilayerstructuresconsistingofalternatingferromagnetic\n(FM) and normal metal (NM) layers and as the name suggests results in much larger magnetoresistance\nratios than AMR, up to nearly 100%. The simplest description of GMR is the resistor model [93], based\non the idea of the two band model discussed above, that the current is carried by two non-interacting\nspin channels. If the magnetizations of the alternating FM layers are aligned, the majority electrons will\n8Figure 3:Resistor model of GMR. Due to the small probability of spin flip scattering the majority (up) and minority (down)\nelectrons can be treated as two independent conduction channels. (a) When the magnetization of the FM layers is aligned, the\nmajority electrons with spin along the magnetization direction can move with little scattering and therefore have an overall low\nresistance, while the minority electrons are strongly scattered in each FM layer and have a resulting high resistance. Adding\nthe resistance from the two channels in parallel leads to a low resistance state. (b) When the magnetization of the FM layers\nis antiparallel, the minority and majority electrons are scattered equally after passing through two FM layers. Adding the\nresistance for the two channels in parallel leads to a high resistance state. Source :Adapted from Ref. [74].\nexperience a resistance R\"in each layer, whereas the minority electrons which are more strongly scattered\nwill have resistance R#>R\". Adding these resistances in parallel, the P state has a resistance\nRP=2R\"R#\nR\"+R#: (2.7)\nOntheotherhand, ifthemagnetizationsareantiparallel, theminorityandmajorityelectronswillexperience\nthe same resistance after travelling through the two layers, so that the AP state has resistance\nRAP=R\"+R#\n2(2.8)\nand therefore the GMR ratio is\nGMR =\u0001R\nR=RAP\u0000RP\nRP=\u0000\nR#\u0000R\"\u00012\n4R#R\"=(\u000b\u00001)2\n4\u000b(2.9)\nwhere\u000b=R#=R\"=\u001a#=\u001a\"is the spin-asymmetry parameter. Therefore in order to achieve GMR the\nmultilayer structure must be capable of achieving two resistance states – the parallel, P and antiparallel,\nAP, states where the magnetization of the alternating FM layers is aligned or anti aligned respectively. The\nP state can always be achieved by applying sufficiently large magnetic fields, however the AP state must be\nachieved by sample design. Generally RP> R AP, although there are exceptions for multilayers comprised\nof different FM layers [94]. One limitation of this resistor model is that surface scattering at the FM/NM\ninterface has been ignored [95]. We should note that here we have adopted the most optimistic definition of\nthe GMR ratio. An alternative definition, GMR = (RAP\u0000RP)=(RAP+RP)is also used by many authors.\nInitial studies of GMR achieved the AP state using antiferromagnetic (AFM) coupling between the FM\nlayers [96] at H= 0. Adjusting the thickness of the NM layer produces oscillations in the exchange coupling\nbetween layers [97–100] and therefore it is possible to choose a thickness such that the layers are AFM\ncoupled. The disadvantage of the AFM coupled multilayer structures is that it takes a large Hto switch\nbetween the AP and P states. Fortunately the AP states can be achieved in other ways. Fig. 4 summarizes\nthe different structures that give GMR. One option is to have a [FM/NM] nmultilayer repeating nidentical\nFM and NM layers. The AP state can then be realized by AFM exchange coupling between layers or by\ndipolar coupling, which can be made the dominant coupling effect by increasing the thickness of the NM\nlayer to reduce the exchange interactions. Alternatively a FM/NM/FM multilayer can be used. This so\ncalled pseudo spin valve contains both a soft FM layer with small coercivity field, Hc, and a hard FM\n9Figure 4:Sample structures which display GMR. The resistance state in all structures is controlled with an in-plane magnetic\nfield. (a) A multilayer structure composed of alternating ferromagnetic (FM) and nonmagnetic (NM) layers. When the\nmagnetization of all FM layers are aligned (all solid arrows) the structure will be in the low resistance P-state, while the high\nresistance AP-state will occur when alternating FM layers point in opposite directions (dashed arrows). The FM layers may or\nmay not be exchange locked. Also shown are the CIP and CPP current directions. In both cases the resistance is measured in\nthe current direction. (b) A pseudo spin valve composed of two FM layers of high (hard) and low (soft) coercivities separated\nby a NM layer. (c) A spin valve where a FM layer is pinned by exchange coupling to an antiferromagnetic (AF) layer. The\nthickness of the NM layer is adjusted so that the magnetization of the free FM layer can still be adjusted. Source :Adapted\nfrom Ref. [74].\nlayer [101–103]. An applied field will switch the soft layer first, providing the AP state. A true spin valve\nstructure, AF/FM/NM/FM contains a free FM layer that can be switched in a small field and a FM layer\nwhich is pinned by an antiferromagnetic layer and therefore only reverses under a large field [104].\nAs shown in Fig. 4 (a), GMR experiments are normally performed with the current in the layer plane\n(CIP) or current perpendicular to the layer plane (CPP), with the resistance measured in the current\ndirection. However experiments have been performed with the current at an angle to the layer plane (CAP)\n[105, 106]. For CIP GMR the relevant length scale is the mean free path, \u0015of the NM and FM, however for\nCPP GMR due to spin dependent accumulation effects the longer (compared to \u0015) spin diffusion length is\nimportant [107]. Although CPP GMR >CIP GMR for a sample with fixed lateral dimension and thickness,\nthe CPP resistance will be as much as 8 orders of magnitude less than the CIP resistance [108]. Therefore\ninitiallyGMRexperimentswereperformedusingtheCIPgeometryduetothe0.01-1 \nresistanceswhichcan\neasily be achieved. However improvements in fabrication techniques have lead to CPP becoming standard,\nwith a best MR ratio at room temperature of 80% as reported by Jung et al. [109], which is even higher\nFigure 5:(a) GMR measurement setup, which is similar to that used for AMR measurements. The key difference is that\nfor a multilayer sample the resistance could be measured using a current applied in the sample plane (shown) or the current\ncould be applied perpendicularly. Typically the magnetoresistance ratio for the CPP geometry will be greater than for the CIP\ngeometry, however the absolute resistance measured will be less. (b) GMR curve for Fe/Cr and Co/Cu multilayer structures\n[115]. The MR ratio is up to 2 orders of magnitude greater than AMR. (c) The angular dependence of GMR. While the\nconductance is linear in (1\u0000cos\u0012)the resistance has slight deviations from linearity, however these variations are higher order\nin the GMR ratio [116].\n10than TMR in Al-O based MTJs. For a good review of experimental techniques see Ref. [95].\nThe angular dependence of GMR has been studied in both CIP [104, 110] and CPP geometries [111] and\nwas found to follow\nR(\u0012) =RP+1\n2(RAP\u0000RP) (1\u0000cos\u0012) =RP+ \u0001Rsin2(\u0012=2) (2.10)\nwhere\u0012is the angle between the magnetization of the two layers and \u0001R=RAP\u0000RP. Different approaches\nto a theoretical analysis of the angular dependence predict that the conductance, not the resistance, varies\nascos\u0012[112–114]. Although the deviations of the resistance from the cos\u0012dependence is second order in\nthe GMR ratio they are noticeable in the data shown in Fig. 5 (c). This deviation is more pronounced in\nthe CPP geometry.\n2.1.4. Tunnel Magnetoresistance\nSimilar to GMR, TMR is due to a resistance difference between P and AP configurations in a multilayer\nstructure. However in TMR the FM layers are separated by an insulator and therefore TMR is a result\nof spin polarized tunnelling, rather than spin accumulation effects. A typical schematic structure of a spin\nvalve magnetic tunnel junction (MTJ) in which TMR is observed is shown in Fig. 6. Although a pseudo spin\nvalve may also be used, the spin valve is preferred because the resistance change occurs near zero magnetic\nfield and the spin valve has greater magnetic stability [118]. The ability to control the tunnelling probability\nof the majority and minority electrons through high quality interfaces and a choice of electrode and barrier\nmaterials is what allows the high MR ratios observed in MTJs.\nEarly observations of small TMR ratios ( \u001814%at 4.2 K) were made as early as 1975 [71, 119], however\nas these were not reproducible due to the difficulty of the fabrication process, little research was done on the\neffect until the discovery of GMR provided a renewed interest in the possibility of large MR ratios. The first\nreproducible experiments used a pseudo spin valve MTJ with an amorphous aluminum oxide tunnel barrier\nand achieved MR of 20%at room temperature [120, 121]. Further optimization of fabrication techniques\nled to TMR ratios of \u001870%in Al-O MTJs, however this was not high enough for modern technological\napplications. To increase TMR the amorphous tunnel barrier had to be replaced with a crystalline barrier\nsuch as MgO which is capable of symmetry selection. Improvements in the structure quality of MgO based\nMTJs quickly led to the giant TMR results of up to 200% in 2004 [117, 122] and modern devices are capable\nof achieving MR ratios of over 500% at RT [72, 123].\nThe increased TMR due to an MgO tunnel barrier can be understood, at least qualitatively, by consider-\ning the phenomenological model of Jullière [71] where the TMR is due to spin-dependent electron tunnelling.\nIn Jullière’s model spin is assumed to be conserved during tunnelling so that the two-current model can be\napplied and the tunnelling probability will be proportional to the spin dependent DOS of the two electrodes.\nFigure 6:(a) Schematic cross section of a typical spin valve MTJ. Exchange coupling with the AF layer pins the magnetization\nof the bottom FM electrode and a tunnel barrier separates the free and pinned layers. (b) Typical TMR data taken from a\nFe(001)/MgO(001)/Fe(001) MTJ [117]. The resistance is largest in the antiparallel state and a MR ratio of nearly 200% is\nfound at room temperature.\n11Also the tunnelling probability is assumed to be spin independent, which means that spin dependent tun-\nnelling rates will only be a result of the spin dependent DOS. Since the conductance Gis proportional to\nthe DOS,Ni\nj(EF), wherej=L;R indicates the left and right electrodes and i=\";#indicates the majority\nand minority electrons in the electrode, we have\nGP/N\"\nLN\"\nR+N#\nLN#\nR; G AP/N\"\nLN#\nR+N#\nLN\"\nR\nand so the TMR ratio is\nTMR =GP\u0000GAP\nGAP=RAP\u0000RP\nRP=2PLPR\n1\u0000PLPR(2.11)\nwhere\nPj=N\"\nj\u0000N#\nj\nN\"\nj+N\"\nj; j=L;R:\nTMR ratios estimated from measured spin polarizations using Jullière’s model agree with the measured\nTMR ratios fairly well. Therefore using the experimental results for polarization in 3d ferromagnetic metals\nand their alloys, which typically range from 0< P < 0:6below 4.2 K [73, 124], Eq. 2.11 can be used to\nestimate a maximum TMR of \u0018100%. This value is further reduced at room temperature due to thermal\nspin fluctuations.\nOnewaytoincreasethepolarizationoftheelectrodewouldbetousehalf-metallicferromagneticmaterials\nwhich act as metals for one spin direction and insulators for the other, and therefore theoretically have\npolarizations of 100% [125, 126]. Half metal electrodes have been made using e.g. CrO 2or certain Heusler\nalloys, howeverlargeTMRratioshaveonlybeenachievedatlowtemperatures[127]. Anotherwaytoincrease\nthe polarization of the electrodes would be to exploit another limitation of Jullière’s model – the assumption\nthat the tunnelling is completely incoherent. This assumption is incorrect even for amorphous insulating\nlayers and is clearly incorrect for crystalline tunnel barriers such as MgO where the tunnelling states have\nspecific symmetries. The different symmetry states experience different decay rates determined by their\nsymmetry matching with the lattice, while all states tunnel at the same rate through the amorphous Al-O.\nThischaracteristiccanbeexploitedtoallowonlyhighlyspinpolarizedstatestotunnelthrough, dramatically\nincreasing the TMR ratio.\nThe magnitude of TMR depends on the electrode and barrier materials, the fabrication technique and\nquality of interfaces. These details will not be discussed in detail here, however for more information see\ne.g. [127–129]. It is useful to mention though that for device applications the Fe/MgO/Fe MTJs we have\ndiscussed cannot be used. This is because the pinned layer used in spin valve MTJs uses a synthetic\nferrimagnetic tri-layer that has an fcc(111) orientation on which the bcc(001) Fe/MgO/Fe structure cannot\nbe grown. The reliability of the standard fcc(111) pinned layer is essential and it is therefore easier to use a\nnew MTJ structure. The standard alternative to the Fe/MgO/Fe MTJ is a CoFeB/MgO/CoFeB structure\n[130]. The amorphous CoFeB electrode layer can be grown on the synthetic pinning layer and also allows\nthe growth of a MgO(001) tunnel barrier.\nThe angular dependence for small TMR ratios is the same as Eq. 2.10 which was already discussed for\nGMR [131]. However for larger TMR ratios, the deviations from cos\u0012dependence may become pronounced\n[132]andthecorrectangulardependenceismosteasilyexpressedintermsoftheconductance, Gas[133,134]\nG(\u0012) =R(\u0012)\u00001=GP+GAP\n2+GP\u0000GAP\n2cos\u0012=GP+ \u0001Gsin2(\u0012=2): (2.12)\nHereGP=G(\u0012= 0) =R\u00001\nP,GAP=G(\u0012=\u0019) =R\u00001\nAPand\u0001G=GAP\u0000GP. Eq. 2.12 and Eq. 2.10 agree\ntoO\u0000\nTMR2\u0001\n.\n2.1.5. Spin Hall Magnetoresistance\nThe MR effects discussed so far all require an electrical current to pass through the FM material in which\nthe resistance change is observed, meaning these effects cannot be used to study the magnetic properties of\nferromagnetic insulators (FMIs). However the discovery of spin Hall magnetoresistance (SMR) in FMI/NM\n12hybrid structures [135–148] can be used to study FMI since this effect produces a magnetoresistance in the\nNM layer through the influence of the magnetization in the FMI. This is dramatically different than the\nMR effects already discussed where the magnetoresistance occurs in the FM layer. The key distinguishing\nfeatures of SMR, compared to AMR, is the greater perpendicular resistance, RT\u0018Rk> R?for SMR\nwhileRk> R?\u0018RTfor AMR, where RTis the perpendicular MR with M?IandMperpendicular to\nthe sample plane [135]. Most studies have focussed on Yttrium-Iron-Garnet (YIG)/Pt structures and the\nremoval of Pt or the use of a non magnetic insulator is found to remove the effect.\nFigure 7:(a) - (d) Systematic measurements of the SMR in a Pt jYIG bilayer showing the \u000b;\fand\r(defined in (f)-(h)\nrespectively) dependence of the magnetoresistance [135]. Black circles are experimental data while red and blue curves are\nthe expected curves due to SMR and AMR respectively. (b) Schematic illustration of spin Hall magnetoresistance in a\nFMI(green)/NM(blue) hybrid structure. The charge current in the NM generates a spin current through the SHE. If the\nmagnetization in the FMI is perpendicular to the spin polarization at the FMI/NM interface, a spin-transfer torque will act on\nthe magnetization, reducing the current in the NM and increasing the resistance. When the spin polarization and magnetization\nare aligned, the spins are reflected and the resistance decreases [149]. (a) shows the \u000bdependence of the longitudinal MR, (b)\nshows the\u000bdependence of the transverse MR, (c) shows the \fdependence of the longitudinal MR. For these cases the angular\ndependence of SMR and AMR is the same. (d) shows the \rdependence of the longitudinal MR which shows the key difference\nbetween SMR and AMR. Source: Data from Ref. [135]. Schematic adapted from Ref. [149].\nSMR is due to a combination of the spin Hall effect (SHE) [150–153] and spin dependent scattering at the\nFMI/NM interface. A charge current in the NM will generate a spin current through the SHE. This results\nin an accumulation of spin at the NM/FMI interface. If the spin polarization is parallel to the magnetization,\nthen the spin currents do not enter the FMI and are reflected back at the interface. The inverse spin Hall\neffect(ISHE)thengeneratesanelectriccurrentwhichisparalleltotheoriginalcurrent, increasingthecurrent\nin the NM, resulting in a decreased resistance. On the other hand when the polarization of the spin current is\nperpendicular to the magnetization, spin diffusion into the FMI can occur via spin-transfer torque (discussed\nmore in Secs. 3.3.5 and 3.4). When spin current does cross the interface, spin accumulation is reduced and\nthe charge current in the NM is reduced, increasing the resistance. Therefore the net result of the spin Hall\neffect in the metal, and spin dependent scattering at the metal-insulator interface is that the resistance will\nbe high (low) when the spin current polarization is perpendicular (parallel) to the magnetization of the FMI\nand the resistance change may be written as\nR=R0+ \u0001Rmaxsin2\u0012s\nwhere\u0012sis the angle between the magnetization and the spin polarization [138]. \u0001Rmaxwill depend on the\ndirection of magnetization rotation as shown in Fig. 7 (a) - (d). Since SMR is due in part to spin diffusion\nat the FMI/NM interface, the sample size must be comparable to the spin diffusion length.\n13SMR is much smaller than AMR, \u00180:01%in YIG/Pt [135], though this value depends on both tem-\nperature [144] and the Pt thickness [140]. SMR has been used to measure the spin mixing conductance in\nYIG/Pt [142] and comparative studies of spin pumping, the spin Seebeck effect and SHR have been per-\nformed [141, 147]. Since the initial work on YIG/Pt samples, different FMI/NM hybrid structures have also\nbeen studied using for example Fe 3O4, NiFe 2O4[139] and CoFe 2O4[145] as FMI and Au, Cu [139] and Ta\n[138] as NM. It has been found that SMR is enhanced in trilayer (FMI/NM/FMI) structures and it exhibits\na different angular dependence from AMR [137]. However, it should be noted that for thin ferromagnetic\npolycrystalline films the angular dependent magnetoresistance is much more complex than that generally\naccepted for isotropic bulk samples [79, 154–157]. In particular, single layer polycrystalline Fe thin films\nsandwiched between insulating nonmagnetic layers [158] have recently been found to reproduce all AMR\ncorrelations previously reported in polycrystalline films, including the ones that resembles SMR. These find-\nings suggest that caution should be taken when using angular dependent MR to ascertain the prescence of\nSMR.\nTo rule out the possibility that the MR was caused by known proximity effects, which cause NM layers\nto acquire FM properties when deposited on a FM layer [159, 160], studies have been performed with Cu\ninserted between the NM and FMI. Since the proximity effect is not present in Cu and exchange forces will\nnot be effective over the thick Cu layer used, the observation of MR in Pt/Cu/YIG confirms the explanation\ndue to SMR, since the long diffusion length of copper would still enable SMR. Nevertheless proximity effects\nmay still be important [136] and in fact both effects may be responsible [143].\n2.2. Magnetization Dynamics\nThe second requirement of spin rectification is some form of magnetization dynamics which can be\ncontrolled to produce a dynamic magnetoresistance. Magnetization dynamics may be produced by either\nafield torque , where an applied rf magnetic field interacts with the magnetization through a Zeeman type\ninteraction, or by a spin torque , where a spin polarized current influences the magnetization motion through\nan exchange interaction. Depending on device structure either or both of these torques may be present\nand it is necessary to know the response function of the magnetization motion to the field or spin torque.\nDetermining such a response function in the case of field torques will be the focus of this section, with spin\ntorques discussed in Secs. 3.3.5 and 3.4.\n2.2.1. Field Torque Induced Magnetization Dynamics\nOne way to excite a dynamic magnetization is by applying a microwave field which will cause the mag-\nnetization to precess. This precessional motion is described by the phenomological Landau-Lifshitz-Gilbert\n(LLG) equation. The Landau-Lifshitz (LL) equation [161] without damping follows from the Heisenberg\nequation of motion for the spin operator Sin the presence of a magnetic field Hiwith a Zeeman type\ninteraction [7],\ndM\ndt=\u0000\rM\u0002Hi (2.13)\nwhere\r=\u00160gjqj=2m > 0is the gyromagnetic ratio of the material undergoing FMR. gis the Landé g-\nfactor which is approximately 2 for a free electron and in general depends on the ferromagnetic material\nand the frequency of the applied microwave field, typically ranging from 1.7 to 2.3. Hiis the internal\nmagnetic field which contains contributions from the external applied field, H, the anisotropy fields and the\ndemagnetization fields. Together these magnetic fields produce a torque which causes the magnetization to\nprecess around Hi.\nWithout damping the precession described by the LL equation will continue regardless of the magnitude\nofH. Of course this behaviour is not observed experimentally and instead hysteresis curves show that for\nlarge enough Hthe magnetization should saturate and align with the magnetic field. This behaviour can be\nintroduced by the addition of a damping term that will produce a torque towards the magnetic field. Such\ndamping can be introduced in various ways. When it is necessary to distinguish the spin-lattice relaxation\nfrom the spin-spin relaxation Bloch damping may be used [162]. However the simplest way to introduce\n14Figure 8:(a) Without damping the magnetization experiences a torque which causes it to precess at a constant cone angle\naroundthe Hfielddirection. (b)Dampingproducesaninwardtorquewhichreducestheconeangleandcausesthemagnetization\nto align with H. (c) The Lorentz and dispersive line shape components which both contribute to the susceptibility determined\nby solving the LLG equation. Lis symmetric around the resonance field, while Dis antisymmetric. (d) The spin resonance\nphase, \u0002, which is another key characteristic (along with the line shape components) of the magnetization dynamics. Near\nresonance the spin resonance phase changes from \u0019to 0 within a range on the order of the line width \u0001H.\ndamping is via the Gilbert damping parameter \u000b[163],\ndM\ndt=\u0000\rM\u0002Hi+\u000b\nM\u0012\nM\u0002dM\ndt\u0013\n: (2.14)\nThe Gilbert damping term produces a torque which causes the magnetization to move inward and align with\nthe field and is applicable over a wide power range. Since this torque is perpendicular to the magnetization\ndirection the LLG equation of Eq. 2.14 describes precession and damping that keep the magnitude of M\nconstant. Note that, although the Gilbert damping is introduced phenomenologically, it has a microscopic\norigin in the spin-orbit interaction [164].\nThe internal magnetic field Hihas both a dc and rf contribution, Hi=H0i+hie\u0000i!t. As mentioned\nthe internal field consists of the externally applied field and any anisotropy fields which are present. If the\ndominant effect is the shape anisotropy, then the internal field can be related to the applied field and the\nmagnetization through the demagnetization factors, H0ik=Hk\u0000NkM0kandhik=hk\u0000Nkmk. Here the\nksubscript denotes the kthcomponent of the respective fields, H0iis the internal dc field, His the applied\ndc field,hiis the internal rf field, his the applied rf field, Nkis the demagnetization factor in the kth\ndirection, and M=M0+m(t)whereM0andm(t)are the equilibrium and non-equilibrium magnetizations\nrespectively. The demagnetization factors, which relate the internal magnetic fields to the externally applied\nmagnetic fields, are geometry dependent and obey the sum rule Nx+Ny+Nz= 1[165]. We define our\ncoordinate system so that the external magnetic field is always along the zaxis. By doing so both the dc field\nin and perpendicular to the plane can be described by the same solution to the LLG equation by choosing\nthe correct demagnetization factors. M0will also be along bzand sinceMis constant m(t)will only have\ndynamical components along bxandby(actually for elliptical precession mwill have a 2!dynamic component\nalongbz, however we will not consider this higher order term here). Using the dc and rf components in the\nLLG equation and keeping only linear terms in mandh(motivated by the low microwave power used by\ntypical experiments, but see Sec. 4.2) the Polder tensor \u001f, which relates the rf magnetization to the rf\n15magnetic field, can be determined,\nm=\u001fh=0\n@\u001fxxi\u001fxy0\n\u0000i\u001fxy\u001fyy 0\n0 0 01\nAh=0\n@j\u001fxxj j\u001fxyjei\u0019=20\nj\u001fxyje\u0000i\u0019=2j\u001fyyj 0\n0 0 01\nAhei\u0002(2.15)\nwhere (\u001fxx;\u001fxy;\u001fyy) = (D+iL) (Axx;Axy;Ayy)with [24]\nAxx=\rM0[M0Ny+ (H\u0000NzM0)]\n\u000b![2 (H\u0000NzM0) +M0(Nz+Ny)]\nAxy=\u0000M0\n\u000b[2 (H\u0000NzM0) +M0(Nz+Ny)]\nAyy=\rM0[M0Nx+ (H\u0000NzM0)]\n\u000b![2 (H\u0000NzM0) +M0(Nz+Ny)]\nand\nL=\u0001H2\n(H\u0000Hr)2+ \u0001H2; D =\u0001H(H\u0000Hr)\n(H\u0000Hr)2+ \u0001H2:\nHereH=jHjandM0=jM0j. Also note that the rf magnetic field hcontains a possible phase shift with\nrespect to the electric field which will drive the rf current in the sample. Here this phase shift is kept implicit\nfor simplicity, but we will have to include it explicitly in the next section when we analyze spin rectification.\nHris the resonance field which depends on !and the sample configuration according to the Kittel formula\n!2=\r2[Hr+M0(Ny\u0000Nz)] [Hr+M0(Nx\u0000Nz)]; (2.16)\nand\u0001His a measure of the resonance line width (for the symmetric Lorentz function L,\u0001His the half\nwidth half maximum) which in general depends on !;Hand the sample configuration,\n\u0001H=2 (H\u0000NzM0) +M0(Nx+Ny)\nH+Hr+M0(Nx+Ny\u00002Nz)\u000b!\n\r\nbut reduces to \u0001H\u0019\u000b!=\rnear resonance. The form of the line width which follows directly from the\nLLG equation does not take into account the effect of magnetization inhomogeneities and for comparison to\nexperimental results the expression \u0001H= \u0001Hin+\u000b!=\ris typically used, where \u0001Hinis the inhomogeneous\nline width broadening. \u0002is the spin resonance phase which describes the phase shift between the rf driving\nforce and the dynamic magnetization response. It is defined by tan \u0002 = \u0001H=(H\u0000Hr) =L=Dso that\ncos \u0002 =D=p\nL2+D2andsin \u0002 =L=p\nL2+D2. Near FMR \u0002changes sign over a field range of order \u0001H\nfrom\u0019forH <Hrto 0 forH >Hras shown in Fig. 8 (d).\nAn important feature of \u001fis that each element contains both a Lorentz and dispersive component which\nhave opposite symmetries around HrwithL(Hr+\u000eH) =L(Hr\u0000\u000eH)andD(Hr+\u000eH) =\u0000D(Hr\u0000\u000eH)\nwhere\u000eHis the field detuning. This symmetry which is illustrated in Fig. 8 (c) is strongly dependent on\nthe spin resonance phase, as we can see from the fact that for any element of \u001f,j\u001fijj2/L2+D2=L. In\naddition, both the Lorentz and dispersive components are symmetric under a field rotation of \u0019,H!\u0000H\n(note that under this rotation Hr!\u0000Hr).\n2.3. The Method of Analyzing Spin Rectification: Basic Examples\nHaving discussed magnetoresistance and magnetization dynamics we are now in a position to consider\nanalyzing spin rectification [24, 43, 166]. The dc voltage produced by SR results from the nonlinear coupling\nbetween a time dependent resistance caused by magnetoresistance and a time dependent current driven by\nan rfefield. SR is nonlinear in the sense that only contributions of O\u0000\nh2;j2;hj\u0001\nwill contribute to the time\naverage which is taken over one period. In this section we present the simplest example of spin rectification,\n16whichoccursinferromagneticmonolayerswherethemagnetoresistanceiscausedbymagnetizationprecession\ndrivenbyatorquefromanrf hfield. Thereforewewillnamethiseffect field torque rectification todistinguish\nit from SR dueto spintorque which willbe discussed inSecs. 3.3and 3.4. This simplest caseof SRillustrates\nthe key ideas that can be used to investigate SR in more complex device structures and may be analyzed\nusing Eq. 2.1. However if the voltage is not measured parallel to the rf current in general the planar,\nordinary and extraordinary Hall effects will also contribute to the measured voltage. It is therefore easier\nto determine the voltage line shape directly from the generalized Ohm’s law of Eq. 2.5. This method also\nprovides another way to see the nonlinearity of the SR effect since the rf current generates corrections to\nthe electric field due to AMR and the Hall effects. We will consider only the AMR and AHE contributions\nin Eq. 2.5 which are dominant in ferromagnetic metals (the OMR and OHE could be treated analogously).\nAn important feature of the voltage line shape is its angular dependence, which allows the different\nphysical contributions and the different driving fields to be separated. In FM monolayers both the static\nbehaviour,summarizedinTable1,andtherfbehaviour,characterizedbyacoordinatechange,willcontribute\nto the angular dependence. Even before going through a detailed derivation of the line shape it is possible\nto note the key angular dependencies due to the static behaviour from the magnetoresistance summarized\nin Table 1. For AMR, RAMR (\u0012m) =R(0)\u0000\u0001Rsin2\u0012mso ifM0andHare collinear, VAMR/@R=@\u0012H=\nsin 2\u0012H. Proceeding analogously we expect VPHE/cos 2\u0012Hand thatVAHEwill have no angular dependence\nfrom the static behaviour. Each of these effects will also have an angular contribution which is unique to\neach component of the driving rf field, which must be determined by an analysis of the generalized Ohm’s\nlaw.\n2.3.1. In-Plane Angular Dependence\nWefirstconsiderthecasewhere Hisappliedinthesampleplane. Thecoordinatesystemusedisshownin\nFig. 9 (a). (bx0;by;bz0)are the sample coordinates, with bz0fixed along the long axis of the sample, bx0along the\nwidth andbynormal to the sample plane. (bx;by;bz)rotate with Hso thatH=Hbzmakes an angle \u0012Hwithbz0.\nNote that since we always consider field rotations in the x0\u0000z0plane, thebydirection will remain fixed in all\ncoordinate systems and there is no need to define both sample and field bydirections. The in-plane coordinate\nsystems are related by (bx;by;bz) = (cos\u0012Hbx0\u0000sin\u0012Hbz0;by;sin\u0012Hbx0+ cos\u0012Hbz0), which can be conveniently\nwritten as~x=U~x0withUbeing a rotation about byby\u0012H. If the sample length (along bz0) is much greater\nthan its width (along bx0), the current will flow along bz0. If in addition there is no applied dc current then\nJ=jtbz0=Re(jz0e\u0000i!t)bz0(a dc current will lead to a photoresistance [24] which we do not discuss here).\nSplitting the magnetization into its dc and rf components, M=M0bz+mt=M0bz+Re\u0000\nme\u0000i!t\u0001\n, the time\naverage of the electric field is\nFigure 9:(a) Coordinate systems for an in-plane static magnetic field H. Thez0andx0axes are fixed along the sample’s length\nand width respectively with the yaxis normal to the sample, and current is assumed to flow along z0. Thezaxis is along the\nstatic field Hand static magnetization M0which is rotated by an angle \u0012Hwith respect to the z0axis. (b) Coordinate systems\nfor the first out-of-plane rotation where the field is rotated along the sample length. The x0axis is now along the length of the\nsample, the yaxis is along the width of the sample and the z0axis is normal to the sample, with current along x0. Again the\nstatic field and magnetization are along the zaxis which is rotated by an angle \u001eHwith respect to z0. (c) Coordinate systems\nfor the second out-of-plane rotation where the field is rotated along the sample width. The yaxis is now along the length of\nthe sample, the x0axis is along the width of the sample and the z0axis is normal to the sample and the current is along y.\nAgain the static field and magnetization are along the zaxis which is rotated by an angle  Hwith respect to z0.\n17EMW=\u0000\u0001\u001a\nM2\n0hjt\u0001M0mt+jt\u0001mtM0i+RAHEhjt\u0002mti: (2.17)\nHere we have used the fact that to lowest order jMj=M0is constant and therefore m\u0001M= 0. From\nEq. 2.17 we can see that, due to the cross product, only a magnetization component perpendicular to the\nplane will produce a signal due to the AHE. This is why, when His in-plane, there is no static angular\ndependenceassociatedwiththeAHE,andalsowhytheAHEwillnotcontributewhenthevoltageismeasured\nlongitudinally i.e. along the length of the sample. We also see that only mxwill contribute to longitudinal\nmeasurements. These expectations are realized when we calculate the longitudinal voltage by integrating\nEMWalong the length of the strip,\n(VSR)l\n\u0012=\u0001R\nM0sin (2\u0012H)hIt\nz0mt\nxi=\u0001R\n2M0Iz0sin (2\u0012H)Re(mx): (2.18)\nHere \u0001R=\u001al=AandIz0=jz0AwhereAandlarethecrosssectionalareaandlengthofthestriprespectively.\nAs expected the sin (2\u0012H)dependence is the key characteristic of the AMR induced SR.\nEven though only mxcontributes to the SR voltage, since mxcan be related to the rf hfield using the\nsusceptibility in Eq. 2.15 both hxandhycan drivemx. Furthermore hmust be written in the primed\ncoordinates, m=\u001fh=\u001fUh0which will introduce an additional \u0012Hdependence which is different for each\ncomponent of h0and will also allow hz0to contribute. Using the susceptibility the photovoltage is\n(VSR)l\n\u0012=\u0001R\n2M0Iz0\u0002\nA\u0012x\nLL+A\u0012x\nDD\u0003\n(2.19)\nwhere\nA\u0012x\nL= sin (2\u0012H) [\u0000Axxhx0cos\u0012Hsin \bx0\u0000Axyhycos \by+Axxhz0sin\u0012Hsin \bz0];\nA\u0012x\nD= sin (2\u0012H) [Axxhx0cos\u0012Hcos \bx0\u0000Axyhysin \by\u0000Axxhz0sin\u0012Hcos \bz0]:\nHere we have introduced the notation Aij\nLandAij\nDfor the amplitudes of the Lorentz and dispersive con-\ntributions respectively where i=\u0012;\u001e; corresponding to the \u0012H,\u001eHand Hrotations as defined in Fig.\n9 andj=x;yindicating from which component of the dynamic magnetization, mxormythis amplitude\narises. We will use this same notation when discussing out-of-plane rotations in the next section. We also\ndenote the voltage as (VSR)j\niagain withi=\u0012;\u001e; andj=l;tindicating that it is a voltage measured along\nthe length or width of the sample respectively. Similar notation will be used when we discuss spin pumping.\nThe relative phase \bbetween the rf electric and magnetic fields, which may be different in the bx0;byand\nbz0directions, is now explicit. The amplitudes A\u0012x\nLandA\u0012x\nDof the Lorentz and dispersive line shapes show\nthe key characteristics of the AMR SR: V/sin (2\u0012H) cos\u0012Hfor precession driven by hx0,V/sin (2\u0012H)\nfor precession driven by hyandV/sin (2\u0012H) sin\u0012Hfor precession driven by hz0. The different angular\ndependence allows the contribution from each h0component to be separated, and the ratio of Lorentz and\ndispersive amplitudes for each component, A\u0012x\nL=A\u0012x\nD, allows the relative phase to be determined for each h0\ncomponent.\nFor transverse measurements, across the width win thebx0direction, both the AMR and AHE will\ncontribute,\n(VSR)t\n\u0012=\u0000\u0001R\n2M0Iz0cos (2\u0012H)Re(mx)\u0000wRAHE\n2jz0Re(my): (2.20)\nAgain only mxcontributes to the SR due to AMR. On the other hand only mycontributes to the AHE\nas expected. Again the magnetization is related to hthrough the susceptibility and an additional angular\ndependence is introduced by changing from htoh0,\n(VSR)t\n\u0012=\u0000\u0001R\n2M0Iz0cos (2\u0012H)\nsin (2\u0012H)\u0002\nAtx\nLL+Atx\nDD\u0003\n\u0000wRAHE\n2jz0\u0002\nAty\nLL+Aty\nDD\u0003\n18where\nA\u0012y\nL=Axyhx0cos\u0012Hcos \bx0\u0000Ayyhysin \by\u0000Axyhz0sin\u0012Hcos \bz0;\nA\u0012y\nD=Axyhx0cos\u0012Hsin \bx0+Ayyhycos \by\u0000Axyhz0sin\u0012Hsin \bz0\n\u0001R= \u0001\u001aw=AandIz0=jz0A. The characteristic of the SR due to the PHE is the cos (2\u0012H)dependence,\nwhereas the SR due to AHE has no angular dependence, other than that due to the coordinate change of h.\n2.3.2. Out-of-Plane Angular Dependence\nFor an out-of-plane magnetic field we now have two planes of rotation to consider, as shown in Fig. 9\n(b) and (c), however the treatment is almost identical to the in-plane case. For both out-of-plane rotations\nwe choose a coordinate system where again byis parallel to by0,bz0is normal to the sample plane, and bzis\ndirected along the external magnetic field direction. The latter condition allows us to use the same solution\nof the LLG equation as the in-plane case (though the demagnetization factors will change). This choice of\ncoordinates means that for a rotation along the long axis of the sample, as shown in Fig. 9 (b), bx0is along\nthe long axis of the sample, byis along the width of the sample and we choose \u001eHto denote the angle between\nbzandbz0. For a rotation along the short axis of the sample, as shown in Fig. 9 (c) bx0is now along the width\nof the sample, byis along the length and we choose  Hto denote the angle between bzandbz0. In both cases\nthexandx0coordinates are related by ~x=Uy~x0(with\u0012Hreplaced by \u001eHor Has appropriate).\nProceeding as previously for the in-plane case, the photo voltage for the out-of-plane rotation by \u001eH\nmeasured along the strip in the bx0direction is\n(VSR)l\n\u001e=\u0001R\n2M0Ix0sin (2\u001eH)Re(mx) =\u0001R\n2M0Ix0h\nA\u001ex\nLL+A\u001ex\nDDi\n(2.21)\nwhere\nA\u001ex\nL= sin (2\u001eH) [\u0000Axxhx0cos\u001eHsin \bx0\u0000Axyhycos \by\u0000Axxhz0sin\u001eHsin \bz0];\nA\u001ex\nD= sin (2\u001eH) [Axxhx0cos\u001eHcos \bx0\u0000Axyhysin \by+Axxhz0sin\u001eHcos \bz0]:\nAgain for measurements made along the width of the sample, now in the bydirection, both the AMR and\nAHE will contribute. The form of the photo voltage is\n(VSR)t\n\u001e=\u0001R\n2M0Ix0sin\u001eHRe(my)\u0000wRAHE\n2jx0sin\u001eHRe(mx)\n=\u0001R\n2M0Ix0h\nA\u001ey\nLL+A\u001ey\nDDi\n\u0000wRAHE\n4jx0sec\u001eHh\nA\u001ex\nLL+A\u001ex\nDDi\nwhere\nA\u001ey\nL= sin\u001eH[Axyhx0cos\u001eHcos \bx0\u0000Ayyhysin \by+Axyhz0sin\u001eHcos \bz0];\nA\u001ey\nD= sin\u001eH[Axyhx0cos\u001eHsin \bx0+Ayyhycos \by+Axyhz0sin\u001eHsin \bz0]:\nFor the other out-of-plane rotation by angle  Halong the width of the strip, the longitudinal voltage\nmeasured along the strip length in the bywill vanish, (VSR)l\n = 0, since the static magnetization has no com-\nponent along the measurement direction and therefore the average dynamic magnetization in this direction\nwill be zero. This fact can be used to separate spin pumping and spin rectification as we will discuss in Sec.\n3.3.4. On the other hand, in the traverse direction along bx0the voltage will be\n(VSR)t\n =\u0001R\n2M0Iysin ( H)Re(my) +wRAHE\n2jysin HRe(mx)\n=\u0001R\n2M0Iyh\nA y\nLL+A y\nDDi\n+wRAHE\n2jysec H\u0010\nA x\nLL+A x\nDD\u0011\n(2.22)\n19where\nA x\nL= sin (2 H) [\u0000Axxhx0cos Hsin \bx0\u0000Axyhycos \by\u0000Axxhz0sin Hsin \bz0];\nA x\nD= sin (2 H) [Axxhx0cos Hcos \bx0\u0000Axyhysin \by+Axxhz0sin Hcos \bz0];\nA y\nL= sin H[Axyhx0cos Hcos \bx0\u0000Ayyhysin \by+Axyhz0sin Hcos \bz0];\nA y\nD= sin H[Axyhx0cos Hsin \bx0+Ayyhycos \by+Axyhz0sin Hsin \bz0]:\nThe angular dependence of the rectified voltage in all six measurement/rotation configurations is sum-\nmarized in Table 4 for both AMR and AHE. In general when VSRis measured as a function of angle,\nboth Lorentz and dispersive amplitudes will contribute, but angular fitting will allow the hx0;hyandhz0\ncomponents as well as the AMR and AHE contributions to be separated. The line shape also has distinct\nsymmetries under the rotation H!\u0000Hwhich are summarized in Table 3 in Sec. 3.3 after spin pumping\nhas been discussed.\nIn closing our detailed discussion of field torque spin rectification we should note that our analysis here\nassumes that the static magnetization is fully saturated and aligned with the magnetic field. If the sample\nhas large anisotropy fields, there will be a regime where this is not the case. This is particularly true in the\nout-of-plane configurations, although non collinear dynamics can also be observed for an in-plane static field\n[167]. Such non collinear behaviour can clearly be observed in !\u0000Hdispersion curves as deviations from\nthe normal Kittel-like behaviour. Despite this assumption, any formulas determined above will be exact\nif we simply replace \u0012H!\u0012M. The resulting expression can then be related to experimentally accessible\nparameters by determining the relationship, \u0012M(\u0012H). For a detailed discussion of such an analysis see Ref.\n[167]. It should also be mentioned that the effect of eddy currents has recently been investigated in FM/NM\nbilayers with thin NM layers, and can influence the line shape, as well as induce screening effects in such\nsystems [168, 169].\n3. Spin Rectification Effects in Magnetic Structures\nIn Section 2.1 we discussed the physical origins of various magnetoresistance effects which occur in\nferromagnetic heterostructures. Since SR depends on both the underlying MR effect and the cause of\nmagnetization dynamics, the origin and characteristics of spin rectification will also vary between different\ndevice structures. Spin rectification in ferromagnetic monolayers, bilayer devices and MTJs will be the focus\nof this section. Initial studies of field torque induced SR in ferromagnetic monolayers began as early as 1960\nusing pulsed microwave sources and will be the first topic of discussion. These initial studies only became\nnoticed by the community when advances in device fabrication techniques allowed the direct integration\nof ferromagnetic heterostructures and coplanar wave guides (CPWs). For monolayer devices this led to\nthe so called spin dynamo which will be considered next. Parallel to the rediscovery of SR via the spin\ndynamo, SR became known by the spin pumping (SP) community since both effects will contribute to\nvoltage measurements made on FM/NM bilayer samples. Our next topic of discussion will therefore be the\ngeneration of dc voltages in bilayer samples, due to SR, SP and spin-transfer torque (STT). Finally STT in\nMTJs and the so called spin diode will be considered.\n3.1. Ferromagnetic Films: Early Studies Using Pulsed Microwave Sources\nThe pioneering work in the study of SR was performed by Juretschke in the early 1960’s using pulsed\nmicrowave sources [62, 170] with a theoretical basis similar to the one discussed in Section 2.3 [90] (but\nusing the Bloch equations). A systematic study of the power, field strength and angular dependence of\nthe rectified signal confirmed that the dc voltage accompanying FMR in thin FM films arose from the\nnonlinear coupling between the rf current and field torque induced magnetoresistance – the dc signal was\nlinearly dependent on the microwave power, see Fig. 10 (c), and the angular dependence followed V(\u0012H) =\n(V1cos 2\u0012H+V2) cos\u0012H, see Fig. 10 (d). This angular dependence is consistent with the discussion in Sec.\n2.3 of a transverse measurement driven by a single field component, hx0with \bx0= 0, where both AMR,\n20Figure 10:(a) The experimental setup used for early pulsed microwave studies. A Ni thin film was placed at the end of\na rectangular waveguide. The sample holder was made of the same material as the film to simplify the field profile in the\nwaveguide and a recessed shorting block allowed control over the distance Dbetween the short and the sample. (b) Typical\nvoltage curves measured for several values of D, showing both dispersive and Lorentz contributions to the line shape. (c)\nTypical pulsed microwave power sensitivity of \u00181 nV/mW illustrating the high powers required to obtain a measurable voltage\nsignal. (d) The angular dependence of the transverse rectified voltage, corresponding to magnetization precession driven by a\nsinglehfield component ( hx0) and containing both AMR and AHE contributions. Source :Adapted from Ref. [62].\nV1, and the AHE, V2, contribute and the line shape consists of both Lorentz and dispersive contributions.\nThe typical measurement setup and data for pulsed microwave studies is shown in Fig. 10. This early work\nestablished a novel way to study the GHz frequency properties of thin FM materials – the initial samples\nstudied varied in thickness from \u001810 - 100 nm and had lateral dimensions \u0018cm\u0002cm.\nWhile the experimental design itself was sufficiently simple to facilitate adoption within the community,\nJuretschke’s ideas did not become widespread until some 40 years later, due to the following experimental\nand theoretical limitations:\n1. The experimental configuration was not optimal. In a transverse measurement, both AMR and the\nAHE contribute to the measured voltage, leading to a more complex signal. Recent works have allowed\nfor the independent study of AMR and the AHE using either a longitudinal configuration where only\nAMR contributes [24] or special samples where the AHE dominates and AMR is ignorable in the\ntransverse direction [171]. Also the precise control of the field configuration required to justify a\ntheoretical description with a single hfield component was difficult to achieve experimentally. This\nissue has been resolved through the development of a full theoretical description including all hfield\ncomponents[24]andthedevelopmentofnovelonchipdevicestructureswithintegratedCPWs,allowing\nfor precise field orientation control.\n2. Very high microwave powers were required. Due to the low power sensitivities of \u00181 nV/mW shown\nin Fig. 10 (c), large microwave powers of up to 5 kW were required to produce measurable voltage\nsignals of\u0018mV. This has been greatly improved in recent years due to sample design/fabrication,\nwhich allows the FM material to be directly integrated into a CPW chip, leading to typical power\nsensitivities of\u0018\u0016V/mW. Sensitive detection techniques, such as lock-in amplification, capable of\nmeasuring signals down to the nV scale have also helped the situation.\n3. The line shape was not fully understood. Fig. 10 (b) shows the complex line shape observed in these\nearly experiments. At the time the role of the relative phase in controlling such a line shape (discussed\n21in Sec. 2.3) had not been discovered and therefore the precise meaning of the Lorentz and dispersive\ncontributions to the voltage line shape was unknown.\nDespite these difficulties there was a small amount of early work performed using high power pulsed\nmicrowavesfollowingeitherJuretschke’swaveguidetechnique[172]orananalogousapproachwithmicrowave\ncavities[173](whichallowedmVvoltagesatslightlylowerpower). Thisveryearlyworkisbrieflysummarized\nin Ref. [170]. Pulsed microwave techniques were also used to study spin wave resonances [174] and applied\nto FM semiconductors, including in the study of the AHE [175–177] and also used to study the AHE in\nparamagnetic metals [178].\nHowever, asidefromthefewworksmentioned, Juretschke’sworkwentlargelyunappreciateduntiltherec-\ntificationeffectwasrediscoveredfourdecadeslater[18,23]. Twoverydifferentpathsledtotheseexperiments\n[18, 23]. One path came from the semiconductor community studying electrical detection of spin dynam-\nics, which led to the extension of such powerful experimental methods from semiconductor [60, 61, 179] to\nferromagnetic materials [16, 23]. The other path emerged from theoretical progress in studying magnetism,\nmotivated by the prediction of a dc voltage induced by spin pumping in magnetic bilayers (see section 3.3).\nThe spin pumping community, having observed the voltage produced in bilayer samples at FMR, needed to\nensure the source of such voltage was in fact spin pumping, which required careful analysis of the voltage\nproduced by rectification effects [18]. In 2006, these two paths crossed during the development of dynamic\nspintronic devices which integrated micro-structured ferromagnetic monolayers with microwave CPWs.\n3.2. Micro-Structured Monolayers: The Spin Dynamo\nIn this section we consider spin rectification in ferromagnetic monolayers. In such devices the merger of\nadvanced fabrication techniques and integration with coplanar wave guides allows for enhanced control of\nthe driving magnetic field and increased power sensitivity compared to pulsed microwave studies. The first\nobservation of SR in such devices was made independently by Costache et al. [18] and Gui et al. [23] using\npermalloy (Py) in a longitudinal configuration where the rectification is due to AMR. AMR based SR has\nalso been observed in ferromagnetic semiconductors where it allows for broadband material characterization\n[31]. Although SR in single ferromagnetic devices is most commonly due to field torque driven magnetization\ndynamics and AMR, in certain single layer devices, such as FM nanowires and notched FM rings where large\nspatial variations of the magnetization exist [25, 27], or homogenous ferromagnets with certain crystalline\nsymmetries [67], as it is also possible to observe spin torque induced rectification. Also, in certain FM\nmaterials (such as Co 90Zr10) AMR may be suppressed allowing studies of pure AHE SR [171]. Some of\nthe key experimental work related to SR in single ferromagnetic devices is summarized in Table 2. Our\n\u0002\u0003\u0004\u0005\u0006\n\u0007\u0003\b\u0005\u0006\u0002\u0002\u0003\u0004\u0005\u0002\u0003\u0004\u0005\u0006\n\u0007\u0003\b\u0005\u0006\u0002\u0003\u0004\u0002\u0003\u0006\u0005\n\u0003\u0007\u0005\n\t\n\n\u000b\b\b\n\t\u0003\n\u0005\f\t\f\t\n\t\n\n\u000b\r\u000b\u000e\n\u000e\u0002\nFigure 11:A striking analogy exists between Faraday’s dynamo and the spin dynamo. (a) In Faraday’s dynamo a revolving\ncopper disk converts energy from rotation to an electrical current. (b) Meanwhile a spin dynamo, with a FM strip, converts\nenergy from spin precession to a bipolar current of electricity, which can be rectified into a dc voltage. Source :Adapted from\nRef. 23. (c) A schematic illustration and (d) micrograph of the CPW structure used to provide the driving rf magnetic field\nin the spin dynamo.\n22discussion here will focus on ferromagnetic monolayers where SR is due to AMR and field torque driven\nmagnetization dynamics, with spin torque discussed later in the context of bilayer devices and MTJs.\nTable 2:A summary of key experimental results using single ferromagnetic devices. In Refs. [18, 23, 31] the current in the\nFM material was due to capacitive and/or inductive coupling with the current in the CPW, while in Ref. [171] the rf magnetic\nfield was due to the current in the Py.\nSpin Rectification due to Field Torque\nReference Device Structure Result\nCostache et al. [18]\nMeasurement of dc voltage due to SR\nin Py\nGui et al. [23]\nMeasurement of bipolar dc current due\nto SR in Py\nWirthmann et al. [31]\nMeasurement of SR in ferromagnetic\nsemiconductor GaMnAs\nChen et al. [171]\nMeasurement of SR due to AHE in\nCo90Zr10which has suppressed AMR\nSpin Rectification due to Spin-Transfer Torque\nReference Device Structure Result\nYamaguchi et al. [25]\nMeasurement of dc voltage due to SR\nin Py nanowire\nBedau et al. [27]\nDetection of domain wall resonance\nusing SR\n233.2.1. Method and Device Structure\nA spin dynamo [23, 24] is an on-chip device consisting of a FM microstrip, such as Py, placed in the\nmicrowave field of a ground-signal-ground (GSG) CPW, so named due to its analogy with a Faraday dynamo\nas shown in Fig. 11. A first and second generation spin dynamo are shown in Fig. 12 (a) and (d). In the\nfirst generation device Py microstrips are placed either beside the short of the CPW or between the GS lines.\nAs shown in Fig. 12 (a) two FM strips may be located on one device. The key difference between the first\nand second generation devices is: 1) a different component of the microwave field will drive precession ( hy\nvshx0for a first and second generation device respectively) and 2) the field intensity in the FM layer will\nbe greater in the second generation device since it can be located closer to the CPW – a second generation\nspin dynamo has \u0018200 nm spacing between the CPW and microstrip, compared to \u001840\u0016m in the first\ngeneration structure, which results in an increase of the in-plane hfield by two orders of magnitude.\nFig. 12 (b) shows the field dependent voltage measured using a first generation spin dynamo consisting\nof a Cu/Cr CPW fabricated beside a Py microstrip of dimensions 300 \u0016m\u000220\u0016m\u000250 nm on a SiO 2/Si\nsubstrate. The line shape is measured at an in-plane angle of \u0012H= 120\u000eat a frequency of !=2\u0019= 5:56\nGHz. The line shape follows Eq. 2.19 and since the relative phase is nearly zero it has a dominant Lorentz\ncontribution, following the symmetry expected from Table 3. In comparison, Fig. 12 (e) shows the voltage\nmeasured in a second generation device where the Py is above the CPW short, being electrically isolated\nby a 200 nm thick SiO 2layer. Again the in-plane line shape in Fig. 12 (e) is nearly Lorentz, but as can be\nseen from Eq. 2.19, for an hx0driving field a Lorentz line shape means that the relative phase is nearly 90\u000e.\nFig. 12 (c) shows a typical linear power sensitivity curve for a first generation device. In this device\nthe power sensitivity is 41 \u0016V/mW – a three order of magnitude improvement over the power sensitivity\nof pulsed microwave studies. The magnitude of this improvement is typical of spin dynamos, whose power\nsensitivity typically ranges from 10 - 100 \u0016V/mW [23]. This large power sensitivity has enabled studies\nacross a large range of powers, allowing electrical detection of very weak modes due to spin waves, as well\nas the study of nonlinear effects [32, 180] (see Sec. 4.1 and 4.2 respectively).\nFigure 12:(a) and (d) Schematic diagrams of a first and second generation spin dynamo respectively. Circulating arrows\nindicate the direction of the Oersted field. In the first generation device the ferromagnetic microstrip is placed beside the short\nof the CPW, therefore the dominant rf magnetic field in the FM is in the ydirection. In the second generation spin dynamo\nthe FM is placed above (or below) the CPW (with an electrically insulating layer in-between) and therefore the dominant rf\nmagnetic field in the FM is in the x0direction. (b) and (e) The line shape measured at an in-plane angle of \u0012H= 120\u000eat a\nfrequency of !=2\u0019= 5:56GHz for the first generation and !=2\u0019= 8GHz for the second generation spin dynamo respectively\n[43]. The first generation device used a Py microstrip of dimension 300 \u0016m\u000220\u0016m\u000250 nm while the second generation\ndevice used a Py strip of dimension 300 \u0016m\u00027\u0016m\u0002100 nm with a 200 nm thick SiO 2insulating layer. In both cases\nthe line shape is nearly perfect Lorentz, which means for the first generation dynamo that the relative phase is nearly zero,\nwhile for the second generation dynamo the relative phase is nearly 90\u000e. The fits (black curves) are done using Eq. 2.19 with\n(b)\u00160\u0001H= 3:6mT and\u00160Hr= 39:1mT and (e) \u00160\u0001H= 6:0mT and\u00160Hr= 76:5mT. In both cases the line shapes\nfollow the theoretical expectation summarized in Table 3. (c) Linear dependence of the photo voltage on microwave power in\na first generation spin dynamo. The power sensitivity for this device is 41 \u0016V/mW, typical for microstructured spin dynamos.\nSource :Adapted from Refs. [43] and [166].\n243.2.2. Lineshape Analysis\nAn analysis of spin rectification experiments requires understanding both the line shape and the angular\ndependence of the rectified voltage. Experimentally the line shape is studied by controlling the magnitude\nof the applied static magnetic field, while the angular dependence of course involves changing the direction\nof the field. The importance of a careful line shape analysis is emphasized by considering the subtlety of the\nrelative phase dependence of the line shape derived in Eq. 2.19 and shown experimentally in Fig. 12 (b)\nand (e), where it is clear that even though the line shape is nearly Lorentz, there is still a small dispersive\ncomponent. The fact that the spin rectification may have both Lorentz and dispersive contributions is\nespecially important when other voltage producing effects, such as spin pumping, need to be distinguished.\nIn early studies it was assumed that the rectification line shape was dispersive – a fact that was used to\ndistinguish the effect from spin pumping [37, 181]. This assumption is correct when the rf currents are\ndominated by capacitative coupling, e.g. when the sample is in close proximity to the waveguide and FMR\nis driven by the hx0field [152]. Due to the importance of this special case, the line shape symmetry for\n\bx0= \by= \bz0= 0is summarized in Table 3. The SP line shape is also summarized in this table and will\nbe discussed more in Section 3.3. The key features summarized in Table 3 are the Lorentzian and dispersive\nnature of the voltage line shape as well as the symmetry of the line shape under the change H!\u0000H.\nThis latter symmetry is determined both from the fact that L(H;Hr) =L(\u0000H;\u0000Hr)andD(H;Hr) =\n\u0000D(\u0000H;\u0000Hr)and from the angular dependence of the line shape and its symmetry under the change\n\u0012H!\u0012H+\u0019. Despite the importance of this simplified case, in general both dispersive and Lorentz\ncontributions will be present except in certain carefully designed devices and both should be considered\nwhen performing a line shape analysis [42, 43]. This means that the relative phase, which in general\ndepends in a complicated way on e.g. the waveguide, coaxial cables, bonding wires and sample holder, must\nbe calibrated. This can be done by first separating the field components by fitting the angular dependence\n(see Sec. 3.2.3), and then using these results to determine the LandDfitting.\nThe importance of both Lorentz and dispersive contributions to the spin rectification voltage has been\ndemonstratedbysystematicallycontrollingthe relativephasebetweentherfcurrent andmagneticfieldusing\na phase shifter inserted in one path through the technique of spintronic Michelson interferometry [33, 43].\nFig. 13 shows representative data from this study, which demonstrates the frequency dependence of the\nrelative phase in the first generation spin dynamo shown in Fig. 12 (a). The line shape clearly changes from\nalmost purely Lorentz, at !=2\u0019= 5:5GHz where \bx0= 14\u000eto almost purely dispersive at !=2\u0019= 5:0GHz\nwhere \bx0= 87\u000e. This indicates the device dependent nature of the SR line shape, as the relative phase\nFigure 13:A demonstration of the frequency dependence of the relative phase. The data was collected using the first generation\nspin dynamo shown in Fig. 12 (a). The left panel shows FMR spectra at \u0012H= 120\u000efor several frequencies ranging from 5.0\nto 5.5 GHz. The line shape fits (black lines) are performed using Eq. 2.19. The right panel shows the Lorentz (squares) and\ndispersive (circles) amplitudes of the voltage as a function of \u0012H. The solid curves are a fit to sin 2\u0012H. Both panels clearly\nshow the line shape changing between dominantly Lorentz at 5.5 GHz and dispersive at 5.0 GHz. Source :Adapted from Ref.\n[43].\n25will generally be a complicated function of device material, cabling setup, sample holder geometry etc. The\nfact that the line shape depends on the relative phase has been exploited to perform microwave imaging by\nTable 3:Voltage line shapes of SR and SP. Theoretical voltage curves for SR are calculated under the simplification \bx0=\n\by= \bz0= 0and illustrate whether the line shape is Lorentz or dispersive, as well as the symmetry under a 180\u000erotation\nH!\u0000H.His directed at any angle in the indicated rotation plane, other than integer multiples of 90\u000e; at these high\nsymmetry points the voltage may vanish. For SP the line shape in all rotation planes is identical: Lorentz and symmetric in\nH, however the signal may vanish in certain configurations as summarized in Table 4.\nSP\n26Table 4:The angular dependence of SR for in-plane and both out-of-plane configurations. The only contribution to longitudinal\nmeasurements comes from AMR, however both AMR and the AHE contribute to the transverse voltage.\nAMR sin 2\u0012H sin 2\u0012Hcos\u0012Hsin 2\u0012Hsin\u0012H cos 2\u0012H cos 2\u0012Hcos\u0012Hcos 2\u0012Hsin\u0012H\nAHE 0 0 0 constant cos\u0012H sin\u0012H\nAMR sin 2\u001eHsin\u001eHsin 2\u001eH sin 2\u001eHcos\u001eHsin2\u001eH sin\u001eH sin\u001eHcos\u001eH\nAHE 0 0 0 sin2\u001eH sin\u001eH sin\u001eHcos\u001eH\nAMR 0 0 0 sin2 H sin Hcos H sin H\nAHE 0 0 0 sin2 H sin Hcos H sin H\n27studying the phase shift produced when an object to be imaged is placed between the microwave source and\nthe spin dynamo [182, 183]. This approach has also been used to perform dielectric measurements [184] (see\nSection 4.6).\n3.2.3. Angular Dependence\nIn Sec. 3.2.2 it was pointed out that in order to characterize the relative phase, which may be different\nfor eachhcomponent, the voltage must be separated by driving field. This requirement is most clearly\nemphasized by the experimental data shown in Fig. 12 (b) and (e) which were taken for a first and second\ngeneration spin dynamo respectively. Notice that while the line shape is nearly Lorentz in both cases,\nthe relative phase is very different – for these devices \by=\u00009\u000eand\bx0=\u0000102\u000ehave been measured\nrespectively [43]. This is in agreement with the line shape analysis – from the summary provided in Table 3\nwith zero relative phase the hydriven FMR will be Lorentz, however the hx0driven FMR will be dispersive.\nThis means that the hx0driven FMR must have a relative phase near 90\u000eif the line shape will be Lorentzian.\nThis highlights the fact that for \bto be correctly determined, the hcomponent driving FMR must first be\nidentified, which can be done by performing an analysis of the angular dependence.\nThe key angular dependencies, based on the analysis of Sec. 2.3, are summarized in Table 4. The AMR\nand AHE will have different angular dependencies for each hcomponent in each measurement configuration,\nand therefore by measuring the rectified voltage as a function of the static field orientation, the driving field\ncanbedetermined, whichthenallowsonetoproceedwithalineshapeanalysis. Thereforetocharacterizethe\nspinrectificationbothlineshapeandangularexperimentsarenecessary, whicharecarriedoutbyvarying jHj\nandthefielddirectionrespectively. Table4summarizestheangulardependenceforallmeasurement/angular\nconfigurations discussed in Sec. 2.3 however it should be noted that in single crystalline films, where the\nangular dependence of AMR may differ from Eq. 2.4, the angular dependence of SR may differ from Table\n4 [79–85].\n3.3. Magnetic Bilayers: The Spin Battery\nIn ferromagnetic/normal metal (FM/NM) bilayers a dc voltage may still be generated by field torque\nrectification, however an additional signal due to spin-transfer torque rectification and also spin pumping\nmay be present. SP and STT require a conversion between charge and spin currents within the device\nand therefore rely on the spin Hall effects [153] (although the recently studied spin torques arising from\nRashba/Dresselhaus spin-orbit interactions will not require the spin Hall effect, and will be discussed in\nSec. 3.3.5, for simplicity here we generally exclude spin-orbit torques from our definition of spin torque).\nAs a result, the magnitude of the ST and SP voltage will depend on the strength of the spin-orbit coupling\nin the normal metal layer, which is the physical origin of the spin Hall effect. Therefore to observe large\nspin-transfer torque rectification strong spin-orbit coupling is required, which makes platinum (Pt) a popular\nchoice for the NM layer in such devices. In Sec. 3.3.5 we will discuss recent studies which have shown that\ncurrent driven spin-torque may arise from the Rashba/Dresselhaus spin-orbit interactions.\nThe three voltage producing effects in bilayer devices are shown in Fig. 14. In rectification (both\nfield and spin-transfer torque induced) the role of the magnetization dynamics is to generate a dynamic\nmagnetoresistance which can nonlinearly couple to an rf current and produce a dc voltage. This process\nis shown in Fig. 14 (a) for field induced magnetization dynamics. Note that since the rf hfield directly\nexcites FMR in the FM layer the NM layer is not active in this process. However for both STT and SP\nthe NM is necessary since it is where the spin/charge current conversion takes place. In spin pumping,\nschematically illustrated in Fig. 14 (b), the role of magnetization precession its very different from its\npurpose in rectification. In SP the role of the microwave excited FMR is to set up a non-equilibrium\nspin distribution at the FM/NM interface. Via a diffusive process this spin distribution can equilibrate by\ninjecting a spin current from the FM into the NM. The inverse spin Hall effect will then convert this spin\ncurrent into a charge current and the dc component will then produce a dc voltage. Finally STT rectification\nis shown in Fig. 14 (c). An rf current flowing in the NM layer is converted into an ac spin current via the spin\nHall effect. The ac spin current then flows into the FM layer and produces a torque on the magnetization\ndue to the exchange interaction. This torque drives magnetization precession, which will generate an rf\n28Figure 14:In FM/NM bilayer structures the dc voltage may be due to (a) field induced spin rectification, (b) spin pumping\nor (c) spin-transfer torque induced rectification. (a) In field induced rectification a microwave hfield excites FMR in the\nferromagnetic layer. The magnetization precession generates an rf magnetoresistance which couples nonlinearly to the rf\ncurrent flowing in the FM layer, producing a dc voltage which can have both Lorentz and dispersive components and depends\non both the relative and spin resonance phase. (b) In spin pumping the microwave induced magnetization precession generates\na non-equilibrium spin distribution at the FM/NM interface which equilibrates by sending a spin current into the NM. Via\nthe inverse spin Hall effect this spin current is converted into a charge current, and the dc contribution then generates a dc\nvoltage which is purely Lorentzian. (c) In spin-transfer torque rectification an rf charge current in the NM is converted into an\nac spin current through the spin Hall effect and enters the FM layer. The misalignment between the FM magnetization and\nthe spin current polarization produces a torque on the magnetization, resulting in an rf magnetoresistance which can produce\na dc voltage by coupling to the rf charge current already present in the FM.\nmagnetoresistance which can couple to the rf current and produce a dc voltage. Since the line shape of SP\nis determined by the form of the spin current generation, js\u0018h2, there is no relative phase dependence and\nthe SP line shape will be purely Lorentzian. Similarly, for STT rectification since it is jswhich drives FMR,\nand noth, there is no relative phase dependence. The presence of these three different physical effects in\nbilayer devices makes their study intriguing from both a practical and fundamental physics viewpoint and\nas a result there has been a large number of studies performed on bilayers in the past decade, which we will\ndiscuss in detail in the next five sections. We start our discussion with a brief review of spin pumping and\nspin Hall effects, followed by a summary of experimental observations of spin pumping in both the \"vertical\"\nand \"transverse\" configurations. We then discuss the separation of spin rectification and spin pumping and\nlook at spin-transfer torque rectification. To provide some guidance to the relevant literature a summary of\nseveral key studies is provided in Table 5.\n3.3.1. The Basic Physics of Spin Pumping and Spin Hall Effects\nThe kernel of spintronics is the generation and manipulation of spin currents, making experimental\ntechniques which reliably generate spin currents of the utmost importance. One of the widely used methods\nfor spin current generation is the transport of non-equilibrium magnetization pumped by FMR, known as\nspin pumping (see e.g. Ref. [185, 186] for comprehensive reviews). In a FM/NM bilayer, spin pumping\ncan be understood schematically using Fig. 15: At FMR, the precessional magnetization M(t)in the\nferromagnet differs from the saturation magnetization M0by an amount \u0001M(t), which corresponds to\nthe non-equilibrium magnetization generated by FMR. Inside the FM layer the precessing magnetization\nrelaxes towards its equilibrium position via magnetization damping, leading to the partial dissipation of\nthe non-equilibrium magnetization \u0001M(t). At the interface of the FM/NM bilayer, the non-equilibrium\nFigure 15:A schematic illustration of spin pumping in a FM/NM bilayer. The magnetization, M, of the FM layer precesses\naround the static field direction H. This precession \"pumps\" a spin current into the normal metal layer due to a non-equilibrium\nspin accumulation generated at the FM/NM interface.\n29Table 5:A summary of key experimental results using bilayer (FM/NM) devices. In Refs. [20, 42, 187] a microwave cavity is\nused to provide the rf magnetic field, while in Ref. [188] a CPW is used.\nSpin Pumping and Spin Rectification\nReference Device Structure Result\nCostache et al. [19]\nElectrical detection of SP in \"vertical\"\nconfiguration (without ISHE) using\nPt/Py/Al device\nSaitoh et al. [20]\nElectrical detection of SP in\n\"transverse\" configuration through the\nISHE using Py/Pt bilayer\nMosendz et al. [181]\nSeparation of SP and SR voltage\nsignals using line shape symmetry (L\nvs D respectively) using Py/Pt bilayer\nAzevedo et al. [42]\nSeparation of SP and SR voltage\nsignals using line shape symmetry and\nangular dependence using Py/Pt\nbilayer\nChen et al. [187]\nSeparation of SP and SR voltage\nsignals using line shape symmetry and\nangular dependence in\nGaMnAs/p-GaAs bilayer\nBai et al. [188]\nUniversal method of SP and SR\nseparation based on symmetry\nconsiderations\nspin-transfer Torque\nReference Device Structure Result\nLiu et al. [41]\nElectrical detection of SR due to STT\nin Py/Pt bilayer\n30magnetization diffuses from the FM into the NM as a flow of spin current, which contributes to additional\ndissipation of \u0001M(t). Magnetization precession, magnetization damping, and magnetization diffusion, these\nare the three key ingredients of spin pumping. As shown in Fig. 15, since \u0001M(t)involves both a static and\na precessional component, the spin pumping effect simultaneously generates both dc and ac spin current in\nthe FM/NM bilayer.\nAlthough the concept and full theory of spin pumping [189, 190] were not developed until 2002, histori-\ncally, the fact that the three key ingredients of magnetization precession, damping, and diffusion would lead\nto the generation of a spin current in FM/NM bilayers was first revealed in the transmission-electron-spin\nresonance (TESR) experiments performed in the late 1970’s [191, 192]. In such microwave absorption ex-\nperiments, Silsbee et al. found that the TESR of copper foils was greatly enhanced in the presence of the\nferromagnetic films (such as permalloy, iron, and nickel) deposited on one surface of the copper foil [192]. By\ndeveloping a phenomenological theory using the Bloch equations to describe magnetization precession and\ndamping, coupled by the transport of non-equilibrium magnetization across the FM/NM interface, Silsbee\net al. revealed the key physics of spin pumping, i.e, the generation of a spin current via the interplay of\nmagnetization precession, damping, and diffusion in FM/NM bilayers [192].\nMicroscopically, spin pumping is a consequence of spin dependent reflectivity and transmission param-\neters of NM electrons at the FM/NM interface. The microscopic theory of spin pumping was derived by\nTserkovnyak et al. [189, 190], using the spin mixing conductance as the main parameter to rigorously de-\nscribe the spin current. According to this theory the spin current injection into the normal metal relaxes\nover a characteristic length scale \u0015SD, the spin diffusion length, and the spin current density decays away\nfrom the interface as [189, 190]\njs(y) =js(0)sinh [(tNM\u0000n)=\u0015SD]\nsinh [tNM=\u0015SD];js(0) =\u0016hGr\n4\u0019M2\n0M\u0002dM\ndt: (3.1)\nHereGris the real part of the spin mixing conductance, G\"#=Gr+iGi,\u0016his the reduced Planck constant,\ntNMis the thickness of the NM layer, nis the distance normal to the FM/NM interface (see Fig. 15) and\nMandM0are the full and dc magnetization of the FM layer respectively, just as in Sec. 2.3. Only the spin\npolarization direction b\u001bis determined by the direction of jsin Eq. 3.1, with the spatial direction normal to\nthe interface (since the current diffuses away from the interface). The process of spin pumping is also closely\nrelated to the earlier idea of a voltage generated by spin flip scattering at the FM/NM interface [193] and\ndepends on the spin mixing conductance in the same way as the spin Hall magnetoresistance discussed in\nSec. 2.1.5 [194].\nIn addition to TESR experiments that can measure the spin current, other methods have been demon-\nstrated to detect spin pumping. Comparing the spin current of Eq. 3.1 to the LLG equation in Eq. 2.14,\nthe spin current has the same form as the Gilbert damping term and therefore from the perspective of the\nFM the flow of spin current will result in an increased damping due to the transfer of angular momentum\ninto the NM [189]. Indeed such a signature of spin pumping enhanced magnetization damping was observed\nin Refs. [195–199]. However, owing to the simplicity of charge transport measurements, the most common\nway to detect spin currents is through their conversion into a charge current via the inverse spin Hall effect\n(ISHE).\nSimilartospinrectificationandspinpumping, inwhichthebasicphysicsconceptsweredevelopeddecades\nbefore modern nanotechnologies made them practically useful, spin Hall effects (SHEs) were initially studied\nin the 1970’s [150] but they have only become the subject of intense interest recently after their theoretical\nrediscovery [151]. For an excellent review on spin Hall effects see Ref. [153]. Fig. 16 schematically illustrates\nboth the SHE and ISHE which physically result from spin dependent scattering of charge carriers due to\nthe spin-orbit interaction. In the spin Hall effect, a change current is converted to a spin current, and the\nresulting spin accumulation at the sample boundaries can be detected e.g. via Kerr rotation microscopy\n[200] or polarized electroluminescence [201]. The inverse spin Hall effect is the inverse process, where a spin\ncurrent is converted into a charge current according to\nJc=e\u0012SH\n\u0016hJs\u0002b\u001b: (3.2)\n31Figure 16:Schematic illustrations of (a) the spin Hall effect and (b) inverse spin Hall effect. J cand J sdenote the charge\ncurrent and the spatial direction of the spin current respectively. The spin polarization direction of the spin current is in the\nup-spin direction.\nHere\u0012SHis the spin Hall angle which characterizes the efficiency of spin/charge current conversion and is\ndependent on the strength of the spin-orbit interaction in a material. Due to its ability to convert spin to\ncharge currents, for spin pumping experiments the inverse spin Hall effect can be exploited to convert the\nspin current generated through spin pumping into a charge current which can be electrically measured.\n3.3.2. Spin Battery and \"Longitudinal\" Spin Pumping Voltage\nBefore the dc spin pumping voltage was measured via the inverse spin Hall effect, the first measurement\nof such a voltage [19, 202] used a device called the spin battery [190], which was the spintronics analog\nof solar cells. As was known from the spin injection experiment [203] without microwave irradiation, spin\ndiffusion at the FM/NM interface leads to spin accumulation and depletion near the interface. From the\nviewpoint of the two-channel model for spin transport (see section 2.1), the physics of spin accumulation\nand depletion near the interface of a FM/NM is analogous to the charge accumulation and depletion in a\nsemiconductor P/N junction. Comparing the two, spin pumping at FMR in a FM/NM bilayer is analogous\nto a solar cell made of a P/N junction where, instead of the interband electrical dipole transition that\nconverts energy of sunlight to the electrical energy of a charge current, the magnetic dipole transition of\nFMR converts energy from microwaves to produce spin current.\nFigure 17:(a) A schematic diagram of the device used to observe longitudinal spin pumping. A 25 nm thick Py strip with\nlateral dimensions 0.3 \u0016m\u00023\u0016m is placed at the short end of a CPW with 30 nm thick Pt and Al contacts. Py FMR is\ndriven by an hyfield and a voltage is measured between the Pt and Al contacts using lock-in amplification. (b) The dc voltage\nmeasured due to longitudinal spin pumping. Here flowandfhighare the two frequencies of the rf field used during the lock-in\nmeasurement. The peaks and dips correspond to the resonance at fhighandflowrespectively. Source :Panel (b) from Ref.\n[19].\nIn the spin battery, when the spin injection rate (via spin pumping) is lower than the spin relaxation\nrate, the NM acts as a pure spin sink. However if the spin injection rate exceeds the spin relaxation rate the\nspin accumulation at the interface may allow a back flow of spin current into the FM. Due to spin dependent\n32conductivities of the FM this back flow results in the production of a dc voltage. To observe this effect\nCostache et al. [19] used the device structure shown in Fig. 17 (a). FMR was generated in a Py strip by\nthe Oersted field of a CPW. The Py strip was connected at both ends to normal metals. To measure the dc\nvoltage that was longitudinally generated across the interface by spin pumping, two different contact metals\nwere used. Pt, which is an excellent spin sink, will not generate a voltage, whereas Al, with its small spin\nflip relaxation rate, will produce a dc voltage. Such an asymmetry between the two contacts is expected to\nproduce a net voltage across the Py strip. Typical dc voltage curves measured on such a Pt/Py/Al device\nare plotted in Fig. 17 (b), showing the measured voltage of about \u0018200 nV. In contrast, measurements on\nreference samples made of Pt/Py/Pt structure exhibit only weak signals up to 20 nV. Such an experimental\ncontrast, combined with a detailed theoretical investigation [202], supported the case that the dc voltage\nlongitudinally generated across the Py/Al interface was observed in the Pt/Py/Al device [19]. However, it\nshould be mentioned that in 2006 when such a \"longitudinal\" spin pumping experiment was performed [19],\nthe method for line shape analysis of the rectification voltage was not yet established [24, 33, 42, 43, 181].\nHence, the asymmetric line shape shown in Fig. 17 (b), which indicates a contribution to the voltage from\nspin rectification, was not analyzed. In 2007, the issue of how to distinguish the dc voltage signal caused by\nspin pumping [19] and spin rectification [23] was raised and discussed via private communications among\nthe groups of Refs. [19, 23, 202]. In section 3.3.4 we will address this previously controversial subject and\nreview the methods and solutions developed in the past decade to solve this problem.\n3.3.3. Inverse Spin Hall Effect and \"Transverse\" Spin Pumping Voltage\nCompared to the \"longitudinal\" spin pumping configuration, the use of a \"transverse\" configuration\nusing the inverse spin Hall effect, as introduced by Saitoh et al. [20], has several practical advantages, such\nas the ease of charge current detection and larger dc voltages, typically several \u0016V even at low microwave\npowers. Hence, in contrast to the \"longitudinal\" spin pumping voltage, the \"transverse\" spin pumping\nvoltage induced by the inverse spin Hall effect has been studied by many different groups. Fig. 18 shows\na typical experimental setup which may be used to detect spin pumping via the inverse spin Hall effect.\nNormally a FM/NM bilayer is placed on top of a CPW, allowing the hx0field generated by the current in the\nCPW to drive magnetization precession in the FM. This magnetization precession produces a spin current\nvia the spin pumping mechanism which is then converted into a charge current in the NM via the inverse\nspin Hall effect and is detected as a voltage across the bilayer in the z0direction. Due to its large spin-orbit\nFigure 18:(a) A schematic diagram of a typical bilayer device. The device is similar to the spin dynamo, with the microstrip\nreplaced by the FM/NM bilayer. (b) Experimental data for a Py/Pt bilayer [181]. The lateral dimensions of the bilayers are\n2.92 mm\u000220\u0016m and they are each 15 nm thick. The insulating layer separating the 30 \u0016m wide, 200 nm thick Au CPW\nfrom the bilayer was 100 nm thick MgO. The FMR is driven by an hx0field and due to the capacitative coupling the relative\nphase is zero, which means that the AMR in Py will be purely dispersive, as shown with the black triangles in the data. The\nblue circles show the signal from the Py/Pt which contains both the Lorentz (dotted-dashed line) and dispersive (dotted line)\ncomponents.\ncoupling Pt has a large spin Hall angle, \u0012SH\u00180:05[188] (although the exact value has been the subject\nof some controversy [204], and care must be taken to either separate spin pumping and spin rectification\neffects as discussed in Sec. 3.3.4, or to eliminate AMR induced SR altogether [205].) and therefore Pt is a\ncommon NM used in spin pumping experiments.\n33With the charge current determined by the ISHE in Eq. 3.2, the spin pumping voltage can be found by\nintegrating the time averaged charge current density,\nVSP=RNMZ\nhjc(n;t)\u0001bqidA=RNMe\u0012SH\n\u0016h\u0014Z\nhjs(n;t)idA\u0015\n(bn\u0002b\u001b)\u0001bq\nwherebqis the measurement direction (either along the length or width of the sample – in the configuration\nshown in Fig. 18 bq=bz0) anddA=dndwwithdwalong the width of the sample. To determine VSP\nwe will use the coordinate systems in Fig. 9. In either the in-plane or out-of-plane configurations, taking\nM=\u0000\nmt\nx;mt\ny;M0\u0001\nwill yield the same expression for the spin current density\njs(n;t) =\u0016hGr\n4\u0019M2\n0sinh [(tNM\u0000n)=\u0015SD]\nsinh [tNM=\u0015SD]!Im(m\u0003\nxmy)\nand therefore\nVSP=\u0014Im\u0012m\u0003\nxmy\nM2\n0\u0013\n(bn\u0002b\u001b)\u0001bq (3.3)\nwhere the proportionality constant \u0014=\u0012SH\u0015SD\n\u001bNM\u0000e!\n4\u0019\u0001Gr\ntNMwtanh\u0010\ntNM\n2\u0015SD\u0011\ndepends on material and device\nspecific parameters, such as the normal metal conductivity \u001bNM, but is not important for a line shape\nanalysis. We can immediately see that because the voltage depends on m\u0003\nxmythere will be no dependence\non the spin resonance phase and the line shape will be completely Lorentzian. Using the solution to the\nLLG equation found in Eq. 2.15, VSPfor in-plane magnetic fields is\n(VSP)l\n\u0012=\u0014\u0000\nAxxAxyh2\nx0sin\u0012Hcos2\u0012H+AyyAxyh2\nysin\u0012H+AxxAxyh2\nz0sin3\u0012H\u0001\nL;\n(VSP)t\n\u0012=\u0000\u0014\u0000\nAxxAxyh2\nx0cos3\u0012H+AyyAxyh2\nycos\u0012H+AxxAxyh2\nz0cos\u0012Hsin2\u0012H\u0001\nL:\nFor the two out-of-plane rotations the spin pumping voltage is\n(VSP)l\n\u001e= 0;\n(VSP)t\n\u001e=\u0014\u0000\nAxxAxysin\u001eHcos2\u001eHh2\nx0+AxyAyysin\u001eHh2\ny+AxxAxysin3\u001eHh2\nz0\u0001\nL;\n(VSP)l\n =\u0014\u0000\nAxxAxysin Hcos2 Hh2\nx0+AxyAyysin Hh2\ny+AxxAxysin3 Hh2\nz0\u0001\nL;\n(VSP)t\n = 0:\nThe angular dependence of spin pumping is summarized in Table 6 for each measurement/rotation config-\nuration.\nIn the analysis presented here we have directly used the rf magnetization determined from the LLG\nequation. However it should be noted that is also possible to replace the Im (m\u0003\nxmy)term in Eq. 3.3 with\nsin2\u0012cwhere\u0012cis the cone angle. The resulting expression assumes that the ellipticity of the precession is\nsmall, but provides a convenient way to characterize the behaviour at low microwave powers [181].\nReturning to the typical spin pumping experiment shown in Fig. 18, since the microwave magnetic field\nis in thehx0direction, when the relative phase is zero the line shape due the spin rectification will be purely\ndispersive. This is what is observed in the experimental data shown in Fig. 12 (b). The voltage in a Py\nmonolayer is seen to be dispersive, while the voltage in the Py/Pt bilayer has both a dispersive (calculated\nwith the dotted line) and a Lorentz (calculated with the dashed-dotted line) component. Therefore the line\nshape difference may be used to separate spin rectification from spin pumping in this case. However as we\nsaw in Sec. 2.3, when the relative phase is nonzero the SR voltage will have both dispersive and Lorentz\ncontributions, andthereforedistinguishingSPfromSRbecomesamorecomplex. Sincethefirstexperimental\nobservations of spin pumping [19, 20, 206] were made almost concurrently with the first observations of spin\nrectification in microstructured devices, methods to separate the two effects were not yet fully developed.\nInstead these initial studies attempted to reduce the effect of spin rectification by placing the sample at a\nposition of minimal rf electric field so that there would not be any directly applied rf current. However more\n34Table 6:The angular dependence of spin pumping.\nsin\u0012H sin\u0012Hcos2\u0012H sin3\u0012H cos\u0012H cos3\u0012H cos\u0012Hsin2\u0012H\n0 0 0 sin3\u0012H sin\u0012H sin\u0012Hcos2\u0012H\nsin3 H sin Hcos2 H sin H 0 0 0\nrobust methods that can actually separate SP from SR are desirable so that SP in various device structures\ncan be reliably studied. This is a necessary step in order to exploit spin pumping as a direct source of spin\npolarized current. Accurately measuring spin pumping is also needed to reliably determine the spin Hall\nangle, which is of practical interest in the development of spintronic devices. Therefore shortly after the\nfirst spin pumping measurements it was realized that a robust method to separate spin pumping and spin\nrectification signals was necessary.\n3.3.4. Spin Rectification vs Spin Pumping\nSeveral methods have been developed to separate spin pumping and spin rectification in FM/NM bilayers\n(for a concise review see the supplementary material of Ref. [188]). The first method developed relies on\na line shape analysis [181]. This method is effective when the spin rectification signal is purely dispersive\nso that it can clearly be distinguished from the Lorentzian spin pumping. However this method becomes\nproblematic when there is a relative phase shift between rf electric and magnetic fields. Another method\nusing the angular dependence has also been used [42]. As we see from Tables 4 and 6, for the case of an in-\nplane static field, when FMR is driven by a transverse hfield both spin rectification and spin pumping have\nasin\u0012Hcos2\u0012Hangular dependence. However when a normal hfield produces the voltage, spin pumping\nfollowsa sin\u0012Hwhilespinrectificationfollowsa sin\u0012Hcos\u0012Hangulardependence, andthereforeitispossible\nto distinguish the two effects. Using such a special measurement configuration, a pure spin pumping signal\nhas been detected in the Py/Pt bilayer [44].\nA method combining line shape analysis with the even and odd contributions of the voltage signal with\nrespect to the static magnetic field has also been developed [207]. However this method also cannot be used\nwhen the microwave field is parallel to the sample, which is the same limitation encountered when using\nthe angular dependence to separate the effects. More recently a universal method based on the magnetic\n35field symmetries has been implemented [188]. This method exploits the symmetries of VSPandVSRwhen\na magnetic field is applied nearly perpendicular to the sample plane, enabling a reliable measurement\nconfiguration in which onlyspin pumping contributes to the voltage signal. As a result the separation step\ncan effectively be performed duringthe experiment, reducing the need for complex line shape and angular\nanalyses. From Tables 4 and 6 we can see that for an out-of-plane magnetic field and longitudinal voltage\nmeasurements\nat H= 0\u000ethere isno spin pumping andVSR(\u001eH;H) =VSR(\u001eH;\u0000H) =\u0000VSR(\u0000\u001eH;H)(3.4)\nand\nat\u001eH= 0\u000ethere isno spin rectification andVSP( H;H) =\u0000VSP( H;\u0000H) =\u0000VSP(\u0000 H;H):(3.5)\nTherefore as shown in Fig. 19, VSPandVSRcan be distinguished experimentally based on their Hfield\nsymmetries. This method enables an accurate determination of the spin Hall angle and the characterization\nof a materials spin/current conversion efficiency. Since the work of Bai et al. in 2013 [188] a method for\nthe spin pumping and spin rectification separation based on the difference in symmetry dependence of VSP\nandVSRon the spin diffusion direction was used by Zhang et al. [208], which allows separation of the\neffects without line shape fitting. It is also worth noting that in addition to the methods discussed here, the\nrectification signal in a Schottky diode (due to tunnelling anisotropic magnetoresistance) [209, 210] may be\ndistinguished from spin pumping based on its bias dependence [211].\nFigure 19:(a) - (d) dc voltage measurements on a Py/Pt bilayer as a function of the magnetic field Happlied at angles (a)\n\u001eH=\u00001:5\u000e; H= 0\u000e(b)\u001eH= 0\u000e; H=\u00000:65\u000e(c)\u001eH= 1:5\u000e; H= 0\u000e(d)\u001eH= 0\u000e; H= 0:65\u000e. When H= 0\u000ethe\nsignal is purely spin rectification, while at \u001eH= 0\u000ethe signal is purely spin pumping (e) - (h) dc voltage measurements on a Py\nmonolayer at angles (e) \u001eH=\u00001:3\u000e; H= 0\u000e(f)\u001eH= 0\u000e; H=\u00000:2\u000e(g)\u001eH= 1:3\u000e; H= 0\u000e(h)\u001eH= 0\u000e; H= 0:2\u000e.for\ncomparison. At  H= 0\u000ethe spin rectification is still present. However at \u001eH= 0\u000ethere is no voltage signal since there is no\nspin pumping in a monolayer. Source :Adapted from Ref. [188].\n3.3.5. Spin-Transfer Torque Rectification in Bilayers\nThe final voltage producing effect observed in FM/NM bilayers is spin-transfer torque induced rectifi-\ncation. As shown in Fig. 14 (c) STT induced rectification requires the conversion of an rf charge current\nin the NM into an ac spin current via the spin Hall effect. This ac spin current then produces a torque on\nthe magnetization of the FM via the exchange interaction, subsequently generating a dynamic resistance\nand coupling to the rf charge current to create a dc voltage. The key ingredient in all spin torque induced\neffects is the generation of a spin polarized current, and while the focus of this section will be on spin Hall\ninduced STT, it is important to note that there are other physical mechanism which may generate spin\npolarized currents. First, a spin polarized current may result from the flow of an electric current between\nnon collinear magnetic structures. This form of spin-transfer torque may occur within domain wall struc-\ntures [25, 27] and is the basis of spin rectification in MTJs which will be discussed further in Sec. 3.4.\nSecond, in crystalline structures lacking inversion symmetry, spin-orbit coupling may induce a polarization\n36of the conduction electrons in the presence of an electric current [212]. This so-called spin-orbit torque has\nrecently attracted much attention from the spintronics community [67, 213–225] and results in an effective\nwave vector dependent magnetic field. The spin-orbit torque may arise due to either Rashba (interface)\nor Dresselhaus (bulk) spin-orbit interactions in certain FM/NM devices and may produce a field-like term\nwhich can even oppose the field-like torque [218]. In such situations the spin-torque line shape may contain\na dispersive contribution, unlike the case of pure spin Hall induced spin torque, and care must be taken\nto separate the effects [219, 223]. Here we will not discuss the recent and developing subject of spin-orbit\ntorques in great detail, choosing to focus on the spin Hall induced STT and its distinction from SR and SP.\nA typical measurement setup used to detect the spin-transfer torque rectification via the spin Hall effect\nis shown in Fig. 20 (a). Unlike in the detection of field torque or spin pumping, a typical STT rectification\nmeasurement setup will apply the rf current directly to the bilayer sample, avoiding the use of a CPW or\ncavity to apply an rf field and instead uses a bias tee to allow the direct application of both rf and dc currents\nas well as the measurement of the dc voltage across the sample. As we discuss below this means that the\ncontribution due to spin-transfer torque rectification will be purely symmetric. Typical experimental data,\nmeasured on Pt(15 nm)/Py(15 nm) and Pt(6 nm)/Py(4 nm) bilayers at 8 GHz, is shown in Fig. 20 (b). As\nwould be expected the line shape contains both symmetric and dispersive contributions due to both field\nand spin-transfer torque rectification effects.\nFigure 20:(a) A schematic diagram of a typical spin-transfer torque rectification measurement setup. A bias tee is used to\nprovide both an rf and dc current as well as measure the dc voltage produced in the bilayer. (b) Typical experimental data for\nPy/Pt bilayers of different thicknesses (shown in nm) [41]. The voltage is made of a mixture of dispersive and symmetric line\nshapes due to the field and spin-transfer torques respectively. Source: Adapted from Ref. [41].\nThe idea of current induced magnetization dynamics was proposed in the works of Berger [226] and\nSlonczewski [227] and was first seen in Co/Cu multilayers by observing the magnetoresistance changes when\nlarge current densities ( \u0018108A/cm2) were injected through point contacts [228, 229]. For a detailed review\nof spin-transfer torque see e.g. Ref. [186, 230]. The effect of a spin polarized current on the magnetization\ndynamics can be described by modifying the LLG equation to include an extra torque term [227, 231] which\nmay be written as [41],\ndM\ndt=\u0000\rM\u0002H+\u000b\nM0M\u0002dM\ndt\u0000\u0000jsM\u0002(b\u001b\u0002M) (3.6)\nwhere \u0000 =\u0000(\r\u0016h)=\u0000\n2e\u00160M2\n0tFM\u0001\nwithtFMbeing the thickness of the FM layer. The first two terms are just\nthe unmodified LLG equation describing precession about Hand Gilbert damping respectively, and the last\nterm is the modification due to a the spin-transfer torque resulting from a current with spin polarization\nalongb\u001b. Here we have neglected the field-like term arising from the spin-orbit torque. For a discussion of\nthe LLG modification due to spin-orbit torques, see e.g. Ref. [223].\nThe STT term provides an additional torque in the M\u0000b\u001bplane. Since the magnitude of Mis taken\nto be constant, there is no spin-transfer torque component along M, although in principle there may be a\ncomponent perpendicular to the M\u0000b\u001bplane which would act as a field-like torque. However in FM/NM\nbilayers the out-of-plane spin-transfer torque may be neglected [41, 48] (see also the discussion in [186] and\n[232–235]).\nThe voltage generated along the length of the FM can now be determined by solving the LLG equation\nwith the STT term in analogy with the approach taken in Sec. 2.3. When the field torque is due to hx0the\n37voltage is given by\nVmix=1\n2\u0001R\nM0Iz0sin 2\u0012Hcos\u0012H(Axxhx0D+SjsL) (3.7)\nwhereS=\u0000\n\u0000M2\n0\u0001\n=(2\u0001H(d!r=dH)jH=Hr)andVmixindicates that the voltage is due to both the field\ntorque and STT rectifications. In a FM/NM bilayer AMR provides the magnetoresistance which produces\nthe dc signal, however a more general expression may be obtained by replacing \u0000\u0001Rsin 2\u0012HwithdR=d\u0012\n[41], which is obtained by expanding the voltage as a function of \u0012(\u0012may be\u0012Has it is for AMR, however\nit may also be another relevant angle, depending on the system in question, see Table 1 and Refs. [22, 28]).\nThe first term in Eq. 3.7 is just the field torque rectification and has a dispersive line shape. However the\npresence of a spin-transfer torque also has a residual effect on this term by increasing the line width (for the\nsame reasons as the spin pumping induced line width enhancement). The lack of a symmetric component to\nthe field torque line shape is due to the fact that there is no phase shift between the rf current, jrfand the\nOersted field. This is expected, since the Oersted field is produced by the current flowing in the NM layer,\nwhich is exactly the same current injected into the FM. This situation differs from the measurement of the\nfield torque rectification using a CPW, where the Oersted field is generated by the current flowing in the\nCPW which is electrically isolated from the FM microstrip and may therefore have a phase shift compared\ntojrf. On the other hand the second term, which is the voltage due to spin-transfer torque rectification, is\npurely symmetric as expected, since the spin current, js, generated through the spin Hall effect will be in\nphase with the rf current with which it couples to produce the dc signal. This is the same situation as that\nobserved for spin pumping.\n3.3.6. Spin Hall Magnetoresistance Rectification\nThus far we have examined in detail the phenomena of spin rectification in FM/NM bilayers, taking\ncare to highlight the distinction between spin pumping and field or spin-transfer torque driven rectification.\nIn such systems the magnetoresistance required for rectification results from the current flow in the FM\nlayer, however, as discussed in Sec. 2.1.5, small magnetoresistance effects can also be observed in FMI/NM\nbilayers due to spin dependent scattering at the interface, which means that rectification effects can also play\na role in FMI/NM bilayers. Both theoretical [236, 237] and experimental [238–242] investigations of SMR\nrectification have been performed, revealing new information about the magnetization dynamics of magnetic\ninsulators. The first studies focussed on the direct rectification of the ac spin Hall effect by means of SMR\n[238, 239], with theoretical descriptions including the effect of a relative phase shift between rf microwave\nfield and current as well as multiple driving fields [238]. In these studies the angular dependence was used to\ndistinguish SMR rectification from spin pumping, however due to extremely small SMR magnetoresistance\nratios, the electrical signal due to the ac spin Hall effect was overwhelmed by the dc spin Hall effect signal\ninduced by spin pumping.\nIn order to more easily observe SMR rectification, another approach is to use the spin-torque FMR\nin a FMI/NM bilayer. The theoretical description of such systems was provided by Chiba et al. using a\ndrift-diffusion model with quantum mechanical boundary conditions at the FMI/NM interface. They found\na rectified voltage consisting of both symmetric and asymmetric contributions,\nVSMR=\u0000\u0001R\n2Iz0[C(HSTT+\u000bh)L+C+hD] sin\u0012Hcos2\u0012H: (3.8)\nHere\nC=\r\n\u000bq\n(2\u0019M0\r)2+!2; C+=\r\n\u000b![1 + 2\u0019M0\u000bC];\nHSTTis an effective field accounting for the spin-transfer torque,\nHSTT=\u0016h\u0012SHJc\n2eM0tFMIRe(\u0011); \u0011=2\u0015SD\u001aG\"#tanh\u0010\ntNM\n2\u0015SD\u0011\n1 + 2\u0015SD\u001aG\"#coth\u0010\ntNM\n\u0015SD\u0011;\n38and the magnetoresistance ratio, \u0001R= \u0001\u001al=A =\u001a\u00122\nSH(2\u0015SD=tNM) tanh (tNM=2\u0015SD)Re(\u0011=2), depends on\nthe normal metal thickness, tNM. Note that due to the interface nature of SMR, the ratio of the Lorentz\nand dispersive contributions is strongly dependent on sample thickness, with large dispersive contributions\npresent for small thickness, and also that, unlike other rectification effects we have considered which are\nproportional to the microwave power P, SMR rectification is proportional top\nP.\nThe experimental setup for STT SMR experiments is shown in Fig. 21 (a), with a calculation according\nto Eq. 3.8 including the effect of spin pumping in panel (b) for tFMI= 4nm. For simplicity, the treatment\nleadingtoEq. 3.8includesonlyan x0componentoftherfdrivingfield(wehavedroppedthesubscriptsothat\ncomparing to our previous descriptions of spin pumping and spin rectification h=hx0) and also has assumed\nthat the relative phase \b = 0. From Table 6 we see that the spin pumping voltage in this configuration\nwill also have a sin\u0012Hcos2\u0012Hangular dependence, however the contributions can be distinguished based\non a line shape analysis, assuming the relative phase shift is 0. Based on such line shape differences recent\nexperiments have provided evidence of SMR rectification [240–242], taking into account the effect of non-zero\nrelative phase shifts and finding good agreement with the predicted thickness dependence of the dispersive\ncontribution.\nFigure 21:(a) The experimental setup used to observe SMR rectification via spin-transfer torque driven magnetization dynam-\nics. (b) Calculated SMR and SP contributions to the voltage signal. While the VSPis symmetric, VSMRhas a large dispersive\ncomponent. Source: Adapted from Ref. [236].\n3.4. Magnetic Tunnel Junctions: The Spin Diode\n3.4.1. Spin-Transfer Torque Rectification in Magnetic Tunnel Junctions\nspin-transfer torque is not only relevant in bilayer devices but also plays an important role in the voltage\nproduced in spin valves and magnetic tunnel junctions [17, 21, 22, 28, 230, 243]. When a current flows\nthrough the pinned layer of an MTJ it develops a spin polarization due to the exchange interaction. The\ntransfer of spin angular momentum carried by this spin polarized current will then exert a torque on the\nfree layer magnetization, as we have just discussed for bilayers, leading to magnetization dynamics and\nrectification. This rectification effect can be interpreted as a form of spin diode, where high and low\nresistance states are controlled by the direction of current flow (which changes the spin polarization of the\nspin current and therefore modifies the spin-transfer torque) and high switching rates are standard [17].\nAlthough the initial power sensitivity of such diodes was only 1.4 mV/mW\u00001(still well below the 3 800\nmV/mW\u00001of semiconductor diodes) recent breakthroughs in MTJ design have enabled power sensitivities\nof 25 000 mV/mW\u00001at low temperatures by Cheng et al. [246], and at room temperature 12 000 mV/mW\u00001\nby Miwa et al. [244], and 75 400mV/mW\u00001by Fang et al. [245]. While Fang’s method does not require a\nbias magnetic field, Miwa’s method has a better signal-to noise ratio. Such high power sensitivities opens\nthe door to enhanced microwave imaging and sensing techniques which will be discussed in Sec. 4.6. It\nshould be noted that an analogous diode effect due to GMR has also been observed [247–249].\nTable 7 summarizes some of the key studies in the development of the spin transfer based spin diode and\nMTJ rectification and Fig. 22 shows a typical spin diode experiment in detail. Similar to the spin-transfer\n39Figure 22:(a) A typical MTJ spin-transfer torque measurement configuration. A bias tee allows the application of both an rf\nand dc current as well as the measurement of the voltage across the MTJ. The applied current becomes spin polarized by the\npinned layer and the spin polarized current can then apply a torque to the free layer magnetization. (b) Typical experimental\nvoltage signal as a function of frequency for several magnetic field strengths [17]. The line shape has both dispersive and\nsymmetric contributions (from the \fFTand\fSTTterms in Eq. 3.10 respectively).\nTable 7:A summary of key experimental results using magnetic tunnel junctions (spin diodes).\nReference Device Structure Result\nTulapurkar et al. [17]\nElectrical detection of SR due to STT\nin a CoFeB/MgO/CoFeB MTJ\nSankey et al. [28]\nUse of SR to measure bias and angular\ndependence of STT vector in\nCoFeB/MgO/CoFeB MTJ\nMiwa et al. [244]\nFang et al. [245]\nRoom temperature spin diode with\nlarge power sensitivity due to\noptimization of MTJ structure\n(CoFeB/MgO/FeB and\nCoFe/Ru/CoFeB MTJs).\ntorque measurement in bilayer systems, a bias tee is used to allow the application of both an rf and dc\ncurrent/voltage as well as to measure the voltage across the device.\nThe current becomes spin polarized after passing through the fixed layer and can then act as a torque\non the free layer. Typical experimental voltage curves as a function of the current frequency are shown in\nFig. 22 (b) for several magnetic field strengths. This data was collected from an MTJ with a pinned layer\nmade of CoFeB and CoFe layers antiferromagnetically coupled across a Ru spacing layer. The free CoFeB\nlayer was separated from the pinned layer by an MgO tunnel barrier and a 0.55 mA rf current was applied\nwith no bias current/voltage. The line shape is seen to have both Lorentz and dispersive components as\nwould be expected from Eq. 3.10.\nSince the spin current polarization will be along the direction of the pinned layer, it is convenient to\ndescribe the magnetization dynamics in an MTJ by writing the STT modified LLG equation as\ndM\ndt=\u0000\rM\u0002Hi+\u000b\nM0\u0012\nM\u0002dM\ndt\u0013\n+\r\fSTTI\nM0M\u0002(M\u0002Mp) +\r\fFTIM\u0002Mp:(3.9)\nHereMis the magnetization of the free layer, Mpis the magnetization direction of the pinned layer, which\n40polarizes the current and Iis the amplitude of the rf current applied to the MTJ. Finally \fSTTand\fFTare\nthe spin-transfer torque and field-like torque amplitudes per unit current respectively. The \fSTTterm is the\nsame as the in-plane modification made to the LLG equation in Eq. 3.6 while the \fFTterm is an explicit\nout-of-plane torque which is necessary for MTJs [134, 250].\nIn the bilayer system, where the rectification was due to AMR, the voltage could be determined using\nthe generalized Ohm’s law. In an MTJ where the magnetoresistance is due to TMR, the voltage must be\nfound by solving the modified LLG equation to determine the angle \u0012between the pinned and free layer\nmagnetizations and then expanding the magnetoresistance as a function of this angle. Following the initial\nstudy by Tulapurkar et al. [17], here we will analyze in detail the case of zero bias voltage/current. We take\ntheyaxis perpendicular to the easy axis plane and the magnetizations in the x\u0000zplane with the zaxis\nalong the easy axis of the MTJ. The demagnetization fields will be along the yaxis since the thickness of the\nMTJ is small compared to its lateral dimensions. We take Malongbzwhich is at an angle \u0012to the pinned\nlayer magnetization along bz0. If we assume the magnetization of Mis constant, then when we expand M\naround its equilibrium value M0it can have two components perpendicular to M0. It is convenient to write\nthese two directions as M\u0002MpandM\u0002(M\u0002Mp)and expand Mas\nM=M0+ae\u0000i!tM0\u0002Mp\njM0\u0002Mpj+be\u0000i!tM0\u0002(M0\u0002Mp)\njM0\u0002(M0\u0002Mp)j:\nTakingI=I0e\u0000i!tthe LLG equation can be solved for the coefficients aandbin analogy with how mxand\nmywere found for the field torque rectification. In the case of the spin dynamo, since we were measuring a\nvoltage along the current direction, only mxcontributed to the rectified voltage, but mxhad a contribution\nfrom bothhxandhydue to the matrix form of the susceptibility. By analogy, since the resistance depends\noncM\u0001Mp, onlybwill contribute to the spin diode voltage, but will depend on both the amplitudes \fSTT\nand\fFTby the solution to Eq. 3.9. We can write the resistance given by Eq. 2.10 as\nR=RP+\u0001R\n2\u0010\n1\u0000cM\u0001MP\u0011\n=RP+\u0001R\n2[1\u0000cos\u0012+jsin\u0012j(Re(b) cos (!t) +Im(b) sin (!t))]:\nTherefore the voltage is\nV=\u0001R\n2I0jsin\u0012jRe(b)hsin2!ti=\u0001R\n4\rI2\n0sin2\u0012Re\u0014~!\fFT+i!\fSTT=M0\n!2\u0000!2r\u0000i!\u0001\u0015\n(3.10)\nwhere!r=\rp\nH(H+M0)is the resonance frequency which follows the Kittel formula, \u0001 =\u000b\r(2H+M0)\nand~!=\r(H+M0). For more discussion of the zero bias line shape see [21, 251, 252].\nHere we have assumed that the spin-transfer torque coefficients \fare independent of the angle between\nthe magnetization directions of the pinned and free layers. In general this may not be true [253] with\nangular dependence appearing explicitly in the conductance and also in non-collinear systems which require\na rigorous generalization of the two-current model [252] (which is consistent with a scattering approach\n[77]). However it has been suggested [254] that the angular dependence of the spin transfer efficiency and\nthe electric conductance cancel, leaving the coefficients independent of the angle \u0012. However the same\ncancellation does not occur for the bias voltage dependence since the conductivities of the spin channels\nhave different bias voltage dependencies [134, 255] and therefore the \fcoefficients will generally have a Vbias\ndependence. Kubota et al. [22] have performed a systematic study of the bias voltage dependence of the\nspin-transfer torque rectification in MTJs which have also been studied in Ref. [28]. In the analysis of\nthe biased samples, the torque dependence on the bias current dominates and the angular dependence is\nignored in comparison. Spin-transfer torque in spin valves is analogous [256]. We should note that most\nSTT experiments, including those discussed here have employed amplitude modulation. However in 2013\nGonçalves et al. [257] demonstrated that magnetic field modulation, often used in conventional FMR, could\nimprove the sensitivity of STT-FMR.\nIn addition to the STT signal in MTJs in principle a field torque may also exist if an rf magnetic field\nis applied. The voltage produced by field torque rectification in the free layer would then be the same as\n41we have already discussed in Sec. 2.3. Finally, apart from the spin-torque and field-like torque, a voltage-\ninduced torque can also play an important role in driving FMR in the ultrathin ferromagnetic structures\nwhich may be present in MTJs.\n3.4.2. Voltage Torque Rectification\nThe origin of voltage torque rectification in MTJs is voltage-controlled magnetic anisotropy, and as\nthis effect does not rely on the microwave current, it can dominant in MTJs with thick tunnel barriers.\nIn 2008 and 2009 theoretical work indicated that an electric field can substantially alter the interfacial\nmagnetic anisotropy energy and even induce magnetization reversal in 3d transition ferromagnets [258–260].\nLater, in 2011, both Wang et. al [261] and Shiota et. al [262] experimentally realized the direct resistance\nswitching induced by a dc voltage in MgO-based MTJs. Since the estimated power consumption for single\nswitchingiswellbelowthatrequiredforthespin-current-injectionswitchingprocess, theseresultsopenanew\navenue for exploring voltage-controlled spintronic devices analogous to those which are already ubiquitous\nin semiconductor technologies.\nFigure 23:Magnetization dynamics in MTJs can also be driven by a voltage-torque due to electric field induced interfacial\nmagnetic anisotropy changes. (a) Application of a dc voltage can switch the easy axis from the in-plane to out-of-plane\ndirections. (b) Application of an rf voltage can excite magnetization precession about an external magnetic field direction. The\nyellow arrow indicates the effective rf magnetic field induced by the anisotropy control. Source: Adapted from Ref. 263.\nA consequence of the voltage-controlled magnetic anisotropy is that magnetization oscillations in an\nMTJ device can also be excited by an applied microwave voltage, resulting in resistance oscillations. Fig.\n23 shows the basic concept of voltage-induced FMR in an MTJ with an ultrathin ferromagnetic layer, where\nswitching of the magnetic easy axis between the in-plane and the out-of-plane directions is demonstrated by\ncontrolling the sign of the dc voltage bias. As the tunnelling resistance depends on the relative magnetization\nconfiguration, the excited FMR dynamics (Fig. 23 (b)) alter the configuration and generate an oscillating\nresistance. Mixing with the microwave current, a dc voltage is produced [263, 264]. As expected, the\nobserved rectification voltage is linearly proportional to the square of the input microwave voltage and\nshown in Fig. 24 (b). In MgO-based MTJs the magnitude of high-frequency spin-torque and voltage-torque\ncan be similar, and thus quantitative descriptions of voltage-driven magnetization dynamics in MTJs should\ngenerallyincludebothtorqueterms[264]. Ithasalsobeenshownthatvoltage-controlledmagneticanisotropy\ncan be used in spin diode rf detectors, such as to improve sensitivity [264–266].\nIn this section we have discussed the dc voltage produced in monolayer devices with integrated CPWs,\nknown as spin dynamos, in FM/NM bilayers, where field torque, spin pumping and spin-transfer torque\nall contribute, and in MTJs where the voltage signal is due to spin-transfer torque caused by the pinned\nmagnetization. These effects are summarized in Table 8.\n42Figure 24:(a) Radiofrequency power dependence of the voltage spectra measured under a constant external magnetic field of\n0.05 T at\u001eH= 45\u000e. (b) Intensity of the peak voltage signal, which is proportional to the square of the input rf voltage Vrf, as\nexpected. The FMR precession angle is shown on the right hand axis. Source: Adapted from Ref. 263.\nTable 8:Summary of the dc voltage observed in ferromagnetic thin films, FM/NM bilayers and MTJs.\nDevice Effect Voltage Lineshape\nThin Film Field Torque V=Vh\nSR(hjrfhrfi)Lorentz and dispersive, con-\ntrolled by relative phase and\ndriving field component\nBilayerField Torque,\nSpin-transfer\ntorque, Spin\nPumpingV=Vh\nSR(hjrfhrfi)\n+VSTT\nSR(hjrfjs\nrfi)\n+VSP\u0000\nhh2\nrfi\u0001Lorentz and dispersive from field\ntorque, Lorentz from spin pump-\ning, Lorentz from spin-transfer\ntorque (if field-like spin-transfer\ntorque is negligible)\nMTJSpin-Transfer\nTorque, Field\nTorque, Voltage\nTorqueV=Vh\nSR(hjrfhrfi)\n+VSTT\nSR(hjrfjs\nrfi)Lorentz and dispersive from\nfield torque, Lorentz from spin-\ntransfer torque and dispersive\nfrom field-like spin torque (Volt-\nage torque behaviour is analo-\ngous to spin-transfer torque)\n4. Applications of Spin Rectification\nHaving described the physical mechanisms of spin rectification and its characteristics in various device\nstructures, in this section we turn to the applications of spin rectification, both in the study and under-\nstanding of basic physics and in the development of imaging and material characterization techniques. As\nwe will see, spin rectification has proven to be a versatile technique which is used in a wide variety of studies\ndue to its high sensitivity, phase sensitivity and ease of use.\n4.1. Electrical Detection of Spin Waves\nSpin waves are collective excitations which occur in magnetic lattices due to the short-range exchange\nintegrationand/orthelong-rangedipole-dipoleinteraction. Unlikethe uniformprecessionofFMR,spinwave\nexcitations are characterized by a spatially varying spin precession phase and therefore have a propagation\ndirection and a finite wavelength. From a fundamental physics perspective, since a set of discrete spin\nwaves can easily be excited under microwave radiation (with FMR being the lowest order mode), spin wave\nspectroscopy provides a powerful tool for understanding both the exchange and dipole-diple interactions.\n43On the application side, the generation of spin waves is an important energy loss mechanism in modern\nspintronic devices operated at high frequencies and the inverse of the lowest spin wave frequency determines\nthe switching timescale of devices.\n4.1.1. Review of Spin Waves\nWhen determining the nature of spin waves, the boundary conditions at the sample interface are of\nutmost importance. In a thin film these pinning conditions will, roughly speaking, cause the spin waves\nto form standing waves. Therefore the wavelength of spin waves with wave vector perpendicular to the\nfilm will be small and these so called perpendicular standing spin waves (PSSWs) will be dominated by\nthe exchange interaction. Conversely the comparatively large wavelength in the film plane results in dipole-\ndipoleinteractiondominatedspinwaves, socalledmagnetostaticmodes. Thein-planemagnetostaticmodes\ncan be conveniently classified based on the orientation of the wave vector with respect to the magnetization.\nWhen the wave vector is perpendicular to an in-plane magnetization, Damon-Eshbach (DE) spin waves (also\nknown as surface spin waves) will form, while if the wave vector is parallel to the in-plane magnetization\nback ward volume modes (BVM) will be produced. On the other hand when the magnetization is normal\nto the film plane, due to the in-plane symmetry there is only one type of in-plane spin wave, the forward\nvolume modes (FVM). The different types of spin waves in thin magnetic films are sketched in Fig. 25,\nwith panels (a) and (b) illustrating the various types of spin waves classified by the magnetization and wave\nvector orientation, and panel (c) schematically showing the spin wave dispersions.\nFigure 25:Schematic illustration of spin waves in a thin film in the case that exchange and dipole-dipole interactions can\nbe treated separately. For both in-plane and out-of-plane magnetizations the short-range exchange interaction will produce\nperpendicular standing spin waves normal to the thin film. On the other hand the short-range dipole-dipole interaction is\nresponsible for spin waves with an in-plane wave vector. (a) For the case of in-plane magnetization the spin waves due to the\ndipole-dipole interaction will either be Damon-Eshbach modes, with in-plane wave vector perpendicular to M, or backward\nvolume modes, with in-plane wave vector parallel to M. (b) For the perpendicularly magnetized thin film, due to the in-plane\nsymmetry, there are only one type of in-plane modes, the forward volume spin waves. (c) Schematic spin-wave dispersion.\nIn the long wavelength regime ( k\u0018\u0016m\u00001) the Damon-Eshbach (red) and backward volume modes (blue) are shown, with\nthe grey shaded area indicating arbitrary magnetization orientation. In the nanometer wavelength regime the dispersion is\ncosine-like. The red shaded area indicates a region of heavy spin wave damping. Source: Panel (c) adapted from Ref. [267].\nIgnoring the effects of damping, the dispersion of the resonance frequency of PSSWs can be determined\nfrom the LLG equation by including an additional exchange interaction term [268], leading to the PSSW\ndispersion relations\nPSSW\u0000M0in-plane:!2\nr=\r2\u0000\nH+M0+ 2Ak2\nz=\u00160M0\u0001\u0000\nH+ 2Ak2\nz=\u00160M0\u0001\n;\nPSSW\u0000M0out-of-plane: !r=\r\u0012\nH\u0000M0+2Ak2\nz\n\u00160M0\u0013\n:\nwhereAis the exchange stiffness constant characterizing the strength of the exchange interaction [269].\n44The spin wave dispersion relations for the DE, BVM and FVM modes can be determined using a magneto\nstatic approximation [270], leading to\nBVM:!2\nr=\r2H\u0014\nH+M0\u00121\u0000e\u0000kzd\nkzd\u0013\u0015\n;\nDE:!2\nr=\r2\"\nH(H+M0) +\u0012M0\n2\u00132\u0000\n1\u0000e\u00002kxd\u0001#\n;\nFVM:!2\nr=\r2\u0014\n(H\u0000M0)2+M0(H\u0000M0)\u0012\n1\u0000e\u0000kTd\nkTd\u0013\u0015\n:\nwherekT=q\nk2x+k2yis the in-plane wave vector. Notice that all of the spin wave modes, except for\nBVM modes, are shifted to higher frequencies than the FMR and therefore require more energy to excite.\nInterestingly, due to the nature of the dipole-dipole interaction, the BVM modes actually require less energy\nthan FMR due to their negative group velocity.\nThe spin wave discussed above consider the limits where either exchange or dipolar interactions are\ndominant. Of course in general both interactions will contribute to the formation of spin waves, leading to\nso called dipole exchange spin waves (DESW) [271–273] which follow the dispersion [273],\n!2\nr=\r2\u0012\nHi+2Ak2\n\u00160M0\u0013\u0012\nHi+2Ak2\n\u00160M0+M0Fp\u0013\n(4.1a)\nwhere\nFp=Pp+ sin2\u001eH \n1\u0000Pp+M0Pp(1\u0000Pp)\nHi+2Ak2\n\u00160M0!\n; (4.1b)\nand\nPp=ky\n2Zd\n0Zd\n0 p(z) p(z0) exp (\u0000kyjz\u0000z0j)dzdz0: (4.1c)\nThe geometry used here is that of Fig. 25 (b) with the magnetization tilted an angle \u001eHfrom thezaxis\nand we denote the total internal field, including demagnetization factors, by Hi. Herek2=k2\nz+k2\nywith\nky=n\u0019=w(wbeing the width of the thin film) and kz= (p\u0000\u0001p)\u0019=d. The discrete number pis the\ninteger number of half wavelengths along the zdirection, normal to the sample with \u0001pa correction factor\ndetermined by the pinning condition at the interfaces z= 0andz=d,\n\u0012\n2A@ p\n@z\u0000Ks p\u0013\f\f\f\f\nz=0;d= 0: (4.2)\nFinally pis an eigenfunction of the PSSW with the form  p=Cp(\u000bsinkzz+\fcoskzz)whereCpis\na normalization constant and \u000band\fare constraints determined by the surface anisotropy Ksand the\nexchange stiffness. In the limit \u000b=\f!1the spins at the surface of the material are completely pinned and\n\u0001p= 0, while if\u000b=\f!0, the surface spins are completely free and \u0001p= 1, thereforepis constrained by\n0\u0014p\u00141.\nEq. 4.1a describes DESW that depend on p(\u0001p) andn. However this expression may be used to describe\nmagneto static modes and PSSW as well by taking appropriate limits. Setting the exchange stiffness, A= 0\nthe perpendicular profile of the spin waves becomes trivial and the DESW reduce to magneto static modes\nwith quantization number nand the exact form (BVM, FVM, DE) depending on the measurement geometry.\nAlternatively, setting n= 0the dispersion of Eq. 4.1a reduces to the case of PSSW characterized by p\u0000\u0001p.\nTherefore the FMR ( p\u0000\u0001p= 0;n= 0), DESW ( p\u0000\u0001p6= 0;n6= 0), PSSW (p\u0000\u0001p6= 0;n= 0) and\nmagneto static modes ( p\u0000\u0001p= 0;n6= 0) can all be described by Eq. 4.1a, which makes Eq. 4.1a useful\nin the experimental identification of the spin wave modes across the whole Hrange. For more in depth\ndiscussion of spin waves see e.g. Refs. [7, 274, 275].\n454.1.2. Detection of Spin Waves\nExperimentally there are many techniques which have been adapted to study spin waves. Some of these\ntechniques are summarized in Table 9. Generally these techniques fall into the categories of time-domain\nmeasurements, or field and frequency domain measurements. Time domain techniques measure the temporal\nevolution of the magnetization typically using pulse-inductive microwave magnetometry (PIMM) or time\nand spatially resolved magneto-optic Kerr effect (MOKE) microscopy. PIMM uses a pulsed excitation to\ngenerate spin waves, and the inductance induced voltage is measured in a stripline [276]. On the other\nhand MOKE techniques employ a femtosecond laser to carry out pump probe measurements. As both the\npolarization and intensity of light reflected from a magnetized surface will change based on its magnetic\nproperties, the location of a spin wave resonance can be directly measured using time-resolved scanning Kerr\nmicroscopy as a local spectroscopic probe. More recently x-ray detected magnetic resonance (XDMR) which\nuses x-ray magnetic circular dichroism (XMCD) to probe the local magnetization in a microwave pump field\nhas been used to investigate resonance processes with element specificity [282, 283].\nAlternatively, field and frequency domain techniques have also proven successful in the study of spin\nwaves. Such techniques include both direct measurements of the !(k)spin wave dispersion using Brillouin\nlight scattering (BLS) [297–299, 309, 310] as well as broadband measurements of the microwave reflec-\ntion/transmission/absorption as a function of frequency and external magnetic fields [291–293, 311, 312]. In\nBLS the Stokes and anti-Stokes shifts produced by the creation and absorption of magnons are measured us-\ningasensitivetandemFabry-Pérotinterferometer, whichcanaccessboththespatialandtemporalproperties\nof spin wave packets, and simultaneously various frequencies in a wide frequency range [299]. More recently,\nferromagnetic resonance force microscopy (FMRFM) of confined spin-wave modes with improved, 100 nm\nresolution, has been developed [303]. Ferromagnetic resonance spectra in Py disks (diameters ranging from\nFigure 26:Typical voltage spectra focusing on different Hfield ranges, measured under a nearly perpendicular applied\nmagnetic field. All voltage spectra are normalized and vertically offset. (a) Typical spectra over the whole Hfield range at\nvarious frequencies (4.5 to 5.5 GHz in 0.1 GHz steps). Arrows indicate the positions of FMR and perpendicular standing spin\nwaves forp= 2(S2, S20) andp= 3(S3, S30). (b) Spectra of forward volume modes found near FMR. The inset shows the\ndetection and magnetization field static field configuration. The rf field was provided by a CPW (not shown). (c) Spectra of\nthe dipole-exchange spin waves found near the S2perpendicular standing spin wave. Note that in (b) and (c) the 0 of the\nfield axis is centred at the FMR and S2 spin wave position respectively, with \u0001Hindicating the deviation from this position.\nSource :Adapted from Ref. [26].\n46Table 9:A summary of experimental techniques used for the investigation of linear spin waves and nonlinear magnetization\ndynamics. The schematic setups and non exhaustive list of references given here are meant only as a brief overview and starting\npoint for further reading.\nTechniqueExperimental\nSetupLinear Spin Wave\nStudiesNonlinear Studies\nTime-Domain Techniques\nPulse-Inductive\nMicrowave Magnetometer\nSampling Oscilloscope Pulse \n[276–279] [280, 281]\nTime Resolved XMCD\nX-Rays Microwaves [282–284] [285]\nTime and Spatially\nResolved MOKE\nMicroscopy\n[286–289] [290]\nFrequency and Field Domain\nBroadband Spin Wave\nSpectrometer\nVNA \n[291–293] [294–296]\nBrillouin Light Scattering\n [297–299] [300–302]\nFerromagnetic Resonance\nForce Microscopy\nCantilever [303] [304]\nElectrical Detection via\nSpin Rectification\nV Microwave Generator [26, 305–308] [32]\n47100 to 750 nm) has shown multiple distinguishable edge modes.\nAlso falling into the field and frequency domain category is the use of spin rectification [26, 305–308]. A\nkey difference in SR based spin wave detection is that the device itself acts as the detector. This enables\nsimple implementation, simplified data analysis and, increased sensitivity due to the lock-in techniques used.\nOne of the key results arising from SR detection of spin waves is the unambiguous observation of DESW\n[26, 308]. Although spin waves were experimentally studied as early as the 1950’s, for many years only\nmagneto static modes or PSSWs could be observed [286, 298, 313–316] with the detection of DESW and\nthe related question of the boundary condition dependence of spin dynamics [317–320] remaining elusive.\nThis initial inability to observe the DESW was mainly due to the weak signals compared to FMR and\nPSSW and the fact that the spacing between DESW resonances is only a few mT, which means they are\neasily washed out by Gilbert damping broadening of the resonance peak. The high sensitivity of electrical\ndetection based on SR allowed the DESW to be observed for the first time [26] and subsequent studies have\nfurther elucidated the nature of DESW, e.g. the pinning conditions in nanowires [308].\nFig. 26 nicely illustrates the versatility of spin rectification in the study of spin waves, showing PSSW in\nFig. 26 (a), FVM modes in Fig. 26 (b) and DESW in Fig. 26 (c). This sensitive data can be ready used to\nassign the order of each spin wave mode [26]. For a more detailed review of spin-wave detection techniques\nsee e.g. [267].\n4.2. Electrical Detection of Nonlinear Magnetization Dynamics\nThe onset of nonlinear dynamics produces a rich array of new physics compared to the linear regime, such\nas amplitude dependent resonance frequency, foldover effects and bistability. These nonlinear fingerprints\nare found throughout nature, from mechanical [321] to magnetic systems [322, 323]. In magnetic systems\nthe key effects of nonlinear dynamics are 1) power dependent resonance position shifts of both the FMR\nand spin wave modes and 2) line width broadening. The former is due to a nonlinearity of the driving force\nwhile the later is due to additional nonlinear damping. The onset of the foldover and bistability effects also\noccur in magnetic systems when the resonance shift reaches a critical value which was initially investigated\nby Anderson and Suhl [322]. However in the experimental search for foldover effects in magnetic systems,\nlarge discrepancies from the Anderson-Suhl model have been found [324–336]. These discrepancies were\npartially due to the presence of spin wave instabilities and thermal effects, which made the interpretation\nof early experiments difficult, but the primary reason that the critical frequency shift could not be reached\nis due to the effect of nonlinear damping, which effectively pushes the power and frequency required to\nobserve foldover effects beyond the range initially investigated. This later complication was identified by\nspin rectification studies which were subsequently used to study the foldover FMR [32, 180].\n4.2.1. Review of Nonlinear Magnetization Dynamics\nThe effects of resonance shift and line width broadening can be described by determining the cone\nangle of magnetization precession in the nonlinear regime. Physically the shift in resonance frequency\noccurs due to the decrease in Mzas the magnetization precession is tilted to larger cone angles by a high\ndriving power. However as the resonance frequency is shifted, nonlinear damping becomes more important.\nBased on the analysis in Sec. 2.2 the cone angle \u0012cis determined by \u00122\nc\u0018\u0000\nm2\nx+m2\ny\u0001\n=M2\n0, where we\nhave assumed that M2\n0\u001d\u0000\nm2\nx+m2\ny\u0001\n. The solution for the LLG equation can then be used to relate\nthe cone angle to the driving rf field. For example when His applied nearly perpendicular to the sample\nplane\u00122\nc=h2=h\n(H\u0000Hr)2+ \u0001H2i\n, whereh=hx=hy. However to be consistent with this higher order\nexpansion, the nonlinear effects on the resonance field and the line width must also be taken into account,\nwhich can be done by making the replacements [32, 180]\n\u0001H!\u0001H0+\fM0\u00122\nc= \u0001Hin+\u000b!=\r +\fM0\u00122\nc;\nHr!H0\nr\u00001\n2M0\u00122\nc:\nHere \u0001H0= \u0001Hin+\u000b!=\randH0\nrare the line width and resonance position without any nonlinear\nmodifications, respectively, and \fis the nonlinear damping coefficient.\n48Figure 27:(a) - (c) A discontinuity in \u00122\ncoccurs when the rf driving field exceeds a threshold hth. (d) Theh\u0000!phase diagram.\nIn the region below hc, which is the critical field in the absence of nonlinear damping, no foldover is observed. Even with\nnonlinear damping and high powers, below the threshold hthno foldover is observed. Source :Adapted from Ref. [180].\nFor sufficiently large microwave power P, the microwave field h/p\nPexceeds a critical field strength\nhthwhere the FMR response shows a discontinuity in \u0012c, as shown in Fig. 27 (a) - (c). This discontinuity is\nthe foldover effect. In Fig. 27 (a) - (c) the nonlinear damping term is ignored (i.e. \f!0) in which case the\nabove treatment reduces to that of Anderson and Suhl [322]. However the introduction of nonlinear damping\nwill result in a high power region where the foldover effect is suppressed as shown in Fig. 27 (d). Below\nthe critical field in the absence of nonlinear damping, hcno foldover will be observed. For hth> h > hc\nnonlinear damping will suppress the onset of foldover effects, while for h>hthfoldover can now be observed.\n4.2.2. Detection of Nonlinear Magnetization Dynamics\nTable 9 provides a brief summary of experimental techniques that have been used to study nonlinear\nmagnetization dynamics. The schematics shown here indicate the use of CPWs to deliver a driving rf field,\nwhich enables the application of high powers due the enhanced field confinement and close proximity to\nsamples under investigation. However early studies of nonlinear effects in magnetic materials often used\nmicrowave cavities, such as those in a pulsed microwave reflection cavity spectrometer system [294–296]. In\nsuch systems the peak power levels with a pulse width of about 50 \u0016s can be as high as 1 kW, allowing\nthe measurements of spin wave instabilities in both ferrimagnetic (YIG) [294] and ferromagnetic (Py) films\n[296].\nTo investigate the wave vector dependence of nonlinear effects, which is necessary to fully understand\nFMR at high microwave powers [337], combined BLS and microwave pumping techniques have also been\nused [300–302]. More recently micro focussed BLS has enabled sensitive local probing of micro- and/or\nnano-structured devices. For example, Edwards et al. reported an experimental observation of parametric\nexcitation of magnetization oscillations in a Py micro disk adjacent to a Pt layer [302], demonstrating control\nof nonlinear dynamic magnetic phenomena in microscopic structures via pure spin currents. Additionally\nthe longitudinal magneto-optical Kerr effect has been used to investigate large precession angles of up to\n\u0012c= 20\u000e[290],demonstratingtheSuhlthresholdeffectforparametricspinwavegeneration[337]. Meanwhile,\nferromagnetic resonance force microscopy [304] has also proven to be a useful tool for nonlinear studies,\nallowing simplified sample fabrication of single magnetic nano-structures by removing the requirement of\nelectrical contacts. Finally the phase resolved ability of XMCD measurements has allowed new nonlinear\nphenomena to be observed at low magnetic bias fields resulting in the generalization of spin-wave turbulence\ntheories [285].\n49Electricaldetectiontechniquesbasedonspinrectificationhavealsoprovenusefulinthestudyofnonlinear\nmagnetization dynamics in a variety of magnetic structures [32, 180, 338–340]. For example, nonlinear FMR\nin a micro-structured MTJ has been reported [338] and the large precession angle magnetization dynamics\nin a nanoscale MTJ have been used to measure the electric-field modulation ratio of magnetic anisotropy\nenergy density [339]. Zhou et al. have also studied nonlinear effects in YIG/Pt bilayers where foldover has\nbeen observed and the power dependent cone angle due to spin pumping can be explained by accounting\nfor nonlinear damping [340]. Finally nonlinear phenomena has also been investigated in a Py strip using\na second generation spin dynamo, as shown in Fig. 28. Electrical detection enables several key nonlinear\nbehaviours to be easily observed, such as the foldover effect, shown in Fig. 28 (a) at high powers (25\ndBm) with the blue (red) curves measured while sweeping the field up (down), and the onset of nonlinear\nbehaviour as a function of microwave power, seen in panel (b), where, in addition to FMR (dark blue),\nseveral perpendicular standing spin wave modes can be observed at lower field.\nFigure 28:Nonlinear magnetization dynamics detected using spin rectification. (a) Observation of the foldover effect at high\npower (25 dBm). The green curve shows the rectification signal at low power for comparison. (b) Linear to nonlinear transition\nobserved in the normalized spin rectification voltage over a power range, 1 mW <P < 100mW. FMR (dark blue) and several\nperpendicular standing spin wave modes (at lower field) can be observed. Source :Adapted from Ref. [32].\n4.3. Electrical Detection of ac Spin Current\nSpin pumping is perhaps the most common technique used to generate pure spin currents and has been\nemployedinNM/FMbilayers[20],NM/ferromagneticsemiconductorbilayers[187]andevenNM/ferromagnetic\ninsulator (FMI) bilayers [39]. The use of FMI is especially intriguing since it explicitly demonstrates the\nflow of a pure spin current in the absence of charge current (see Ref. [341] for an in depth discussion of\nspin currents in FMI). In the treatment of spin pumping presented in Sec 3.3.3 we tacitly assumed that the\nspin polarization \u001bwas time independent and directed along M0. This assumption of a dc spin polarization\nresults in a dc spin current which, after being converted to a charge current via the inverse spin Hall effect,\ncan be directly detected as a dc voltage. To date, most work on spin pumping and spin currents has focused\non this dc effect. However the assumption that \u001bis constant neglects the fact that the direction of spin\npolarization is actually determined by the full time dependent magnetization, and therefore \u001bis actually\ntime dependent. This means that in addition to a dc component directed along the external field direction,\nthere is also a transverse ac spin current. Recently spin pumping theory has been extended to include such\nan ac spin current [342] and proposals for an ac spin current source have been put forth [343]. A sketch\nof the dc and ac spin currents is shown in Fig. 29 (a). As we already know, the precessing magnetization\nwill pump a spin current into an adjacent material layer due to a non-equilibrium spin distribution at the\ninterface. Here the dc (yellow) and ac (red) component of this spin current are explicitly shown. In addition\nto the obvious dynamical difference between the dc voltage, Vdc, generated through the ISHE and the ac\nvoltage,Vac, generated by the ac spin current, the other key differences between these two signals are: 1)\nVacis an order of magnitude larger than Vdc[342, 345]. This is due to the fact that, as we saw in Sec. 3.3.3,\nthe dc spin pumping voltage is proportional to sin2\u0012c, while the ac spin pumping voltage is proportional\n50Figure 29:(a) The spin polarization of the pumped spin current has a small dc component (red) along the static field direction\nand a larger transverse ac component (blue). The dc voltage can be measured in-plane transverse to the static field while the ac\nvoltage will be along the field direction. The dc spin pumping has been studied more intensively since it can easily be measured\nthrough the inverse spin Hall effect. (b) The experimental setup used to detect the ac spin current through the simultaneous\nenhancement of spin rectification and spin pumping. (c) By tuning the angle of the external field, the Py and YIG resonance\ncan be made to occur simultaneously, showing enhanced Py rectification and (d) enhanced YIG spin pumping at the mutual\nresonance condition. Source :Panels (c) and (d) adapted from Ref. [344].\ntosin\u0012ccos\u0012c. Therefore the dc signal is maximized at \u0012c= 90\u000ewhile the ac signal is largest at \u0012c= 45\u000e.\n2) The ac voltage scales as the root of the microwave power,p\nP, while the dc spin pumping is linearly\nproportional to P[345]. 3) As shown in Fig. 29 (a), Vdcis measured transverse to the applied static field\ndirection, whereas the Vacwill be generated parallel to the static magnetic field.\nIn general detection of ac spin pumping is more difficult than measuring the dc effect, since the ac\nsignal will have the same high frequency as the microwave field and parasitic effects such as hfield induced\neddy currents and the FMR excitation at the same frequency produce large backgrounds which must be\nseparated. Nevertheless a variety of approaches have succeeded in measuring the ac spin current, using for\nexample rectifying diodes [138, 345], a vector network analysis [346] or more recently direct detection via\nx-ray techniques [347]. However it turns out that the ac spin current can also be measured through electrical\ndetection by exploiting the spin torque exerted by an ac spin current [344]. The schematic setup for such an\napproach is shown in Fig. 29 (b). In a conventional dc spin pumping experiment the spin current flowing\nfrom the FMI (or FM metal) to the NM will increase the FMI (FM) damping [198]. However if the NM is\nreplaced by a FM layer the ac spin current generated in the FMI can act as a spin torque on the FM, causing\nthe precession amplitude to increase and pump an ac spin current back into the FMI, thus compensating\nthe initially increased damping of the FMI (the dc spin current cannot produce this effect since it will be\nparallel to the static field direction, and therefore will not produce a torque on the FM). Therefore when\nboth the FMI and FM are undergoing resonance simultaneously, the damping will be reduced and both\nthe spin rectification signal from the FM and the spin pumping from the FMI will be enhanced. As shown\nin Fig. 29 (c) the simultaneous resonance condition can be achieved by tuning the angle of the external\nmagnetic field as shown. An enhancement of both spin rectification and spin pumping is seen at  H\u001812\u000e.\n(The second, smaller peak, near  H\u001815\u000eresults from an additional weak resonance signal in the YIG\ndue to a spin wave mode). In addition to the ac spin current detection via SP and SR enhancement, this\ntechnique also shows that the ac spin current can be used as a spin torque and could therefore be applied\nto magnetization switching techniques.\n4.4. Electrical Detection of Magnetization Dynamics in Ferromagnetic Semiconductors\nIn a 1931 letter to Rudolf Perierls (who pioneered the “hole\" concept in semiconductors), Wolfgang\nPauli offered the following advice regarding the sensitivity of semiconductor properties to impurities: “One\nshouldn’t work on semiconductors, that is a filthy mess; who knows whether any semiconductors exist\" (En-\nglish translation of the original German: “Über Halbleiter soll man nicht arbeiten, das ist eine Schweinerei,\nwer weiß, ob es überhaupt Halbleiter gibt)? [348]. Today the deliberate introduction of impurities into\nthe crystal structure of a semiconductor (“doping\"), which alters the conducting properties and permits the\ncreation of semiconductor junctions between differently-doped regions of the crystal, plays a fundamental\n51role in modern semiconductor technology and related IT industries. Interestingly, when the paramagnetic\nelement Mn is substituted for Ga in a GaAs lattice, it not only acts as an acceptor, making it a p-type\nmaterial, but also the produced holes mediate a ferromagnetic interaction between the local moments of the\nopendshells in the Mn atoms, resulting in ferromagnetic properties at low temperatures [349] (for a review\nof ferromagnetic semiconductors see e.g. Refs. [350–354]). Therefore ferromagnetic semiconductors, such\nas GaMnAs, offer a unique combination of material properties, which are amenable to both data process-\ning (due to p-type conducting behaviours) and data storage (due to hysteretic magnetization) in a single\nmaterial system.\nFigure 30:(a) SR voltage signal at different frequencies measured on a GaMnAs Hall bar placed between the ground and signal\nlines of a CPW. The inset shows the linear frequency dependence of the FMR line width. (b) . Source :Panel (a) adapted\nfrom Ref. [31]. Panel (b) adapted from Ref. [355].\nSimilartoferromagneticmetals, SRalsoexistsinferromagneticsemiconductorsandjustasinmetalsmay\nhave several origins, e.g. AMR [31] and spin-orbit torques [355]. As in other material systems, SR allows\nsimple, broadband FMR measurements enabling, for example, determination of damping characteristics.\nFig. 30 (a) shows a typical voltage signal to due to AMR induced SR in a GaMnAs Hall bar as shown\nin Table 2. By fitting the line shape to Eq. 2.21 both the resonance position (shown by crosses) and the\nline width (shown in the inset) can be determined just as they are for SR in a ferromagnetic metal. In\ngeneral, a better understanding of damping in ferromagnetic semiconductors is necessary for applications to\ndata storage technologies. An interesting step in this direction was made in 2011 by Fang et al. [355] who\nfabricated a nanostructured GaMnAs device and introduced a form of spin-orbit-driven FMR originating\nfrom the effective magnetic field in the magnetic material induced by an rf electric current. Due to the\nimproved homogeneity of the nano-structured sample they found an exceptionally low extrinsic damping,\nimplying that the GaMnAs could be used for data storage if further homogeneity refinements could be made.\nTypical voltage measurements from such a device are shown in Fig. 30 (b) where the inset displays the\nfrequency dependent line width from which the Gilbert damping can be determined.\nNotonlyareAMRandspin-orbittorqueinducedspinrectificationobservedinferromagneticsemiconduc-\ntors, additionally in 2013, Chen et al. [187] reported the observation of spin pumping in a GaMnAs/p-GaAs\nbilayer structure, again using electrical detection of FMR. Just like the case for the FM/NM bilayers dis-\ncussed in Sec. 3.3 the precession of the magnetization in the ferromagnetic layer, GaMnAs, produces a\npure spin current which is pumped into the adjacent non-magnetic GaAs layer. This spin current is subse-\nquently converted to a change current by the ISHE and hence generates a voltage drop across the device. Of\ncourse spin rectification is also present and therefore the techniques of line shape and angular dependence\nanalysis must be used to separate the two effects. Fig. 31 shows the angular dependent measurements\nmade in this ferromagnetic semiconductor system. The angular dependence shown in panels (d) and (e)\nfollow the expectations based on our discussion in Sec. 2.3 (with deviations from the expected sinusoidal\nbehaviour due to normalization by the microwave absorption, I, which also has an angular dependence).\nMore recently Nakayama et al. [356] investigated the rectified voltage in a Pt/GaMnAs bilayer structure\n52using a similar analysis method. Interestingly they found an exceptionally large spin mixing conductance\nat the Pt/GaMnAs interface, 6.2 \u00021019cm\u00002which is ten times greater than that of the GaMnAs/p-GaAs\ninterface. This result indicates that hybrid structures with ferromagnetic semiconductors and metals may be\nof interest in the investigation of spin-current related phenomena due to their large efficiency of spin-current\ngeneration at the interface.\nFigure 31:Measurements of the microwave absorption and spin rectification/spin pumping voltage in a GaMnAs/p-GaAs\nbilayer. (a) Device structure. Voltage measurements are made in the x0direction. (b) Derivative of the microwave absorption\nand (c) dc voltage for a magnetic field nearly perpendicular to the bilayer. (d) The Lorentz and (e) dispersive amplitude of\nthe voltage signal as a function o \u001eHnormalized to the microwave absorption I. In (d) the dotted and dashed lines show\nthe contribution of spin pumping and spin rectification respectively, with the solid fitted curve the total voltage due to both\neffects. In (e) the solid curve is the calculated voltage due only to spin rectification (as spin pumping does not contribute\nto the dispersive component). The theoretical angular dependence follows from our discussion in Sec. 2.3, with deviations\nfrom the expected sinusoidal dependence the result of normalization by the microwave absorption, which also has an angular\ndependence. Source :Adapted from Ref. [187].\n4.5. Electrical Detection of Domain Wall Dynamics\nA magnetic domain is a spatial transition region between different magnetic moments with an angular\ndisplacement of 90\u000eor180\u000e[357]. The spatial dimensions of such a region depends strongly on the material,\nand can be a few to thousands of atoms in soft materials. Micro magnetic calculations based on the\nLLG equation have revealed two types of domain walls which can be classified by the canting direction of\nmagnetic moments: in a Néel wall the moments rotate in a direction parallel to i.e. within the plane of,\nthe domain wall while for a Bloch wall they rotate in a direction perpendicular to i.e. through the plane of,\nthe domain wall. In thin films without perpendicular magnetic anisotropy, so that the easy axis lies in the\nfilm plane, this means that a Néel wall has moments which rotate in-plane, while a Bloch wall has moments\nwhich rotate out-of-plane. In thin films with low anisotropies Néel walls are energetically favourable [358]\nand vortex structures may form when two Néel walls intersect. From a fundamental physics perspective\nstudying magnetic domains can teach us about the link between the microscopic and macroscopic properties\nof ferromagnetic materials [357]. On the other hand the ability to directly introduce domain walls through\nthe design of pinning sites and to control domain wall motion lays the foundation for magnetic domain wall\nlogic [359–363] which promises a non-volatile memory device with the high performance and reliability of\nconventional solid-state memory, but at the low cost of conventional magnetic disk drive storage.\nIn order to study domain wall properties, e.g. inertia and resonance frequency, and to manipulate\ndomain walls for magnetic logic purposes, it is important to study domain wall motion. For this purpose rf\nexcitations have been found to be exceptionally efficient, moving domain walls at far lower current densities\n53Figure 32:(a) Device used to measure the domain wall resonance frequency and depinning field via spin rectification. A\ndomain wall is pinned near the constriction sight formed by notches in a Py ring. (b) Measurements of the pinning field (blue)\nof a vortex domain wall and the voltage (red) due to AMR rectification. (c) The experimental setup to perform a magnetic\nfingerprinting experiment of a ferromagnetic nanowire. (d) The voltage spectrum as a function of the rf current frequency\nfor Py semicircular (S2, S3, S4) and “boomerang\" shaped (S1) nanowires. The black curves are measured in the absence of\na domain wall, while the red curves are measured after a domain wall has been introduced by saturating the magnetization\nperpendicular to the easy axis along the strip. Source :Adapted from Refs. [27] and [364].\nthan pulsed voltages [365–369]. Since magnetoresistance occurs during the domain wall motion, which is\nefficiently driven by rf currents, spin rectification effects will be present during domain wall motion and have\nbecome a useful tool to study the domain dynamics in nano-structured magnets, such as the determination\nof the nonlinear pining potential of a domain wall [370].\nIn general rectification effects and the study of domain walls intersect in three key ways. First, more\nindirectly related to rectification but directly related to our discussion of magnetoresistance in Sec. 2.1, is\nthe direct use of magnetoresistance effects, such as GMR or AMR, to study domain wall properties. For\nexample, GMR has been used to study the time dependence of the domain wall position [371, 372] and to\ndetermine the domain wall mass [365], while domain wall pinning conditions have been investigated through\nAMR [373], and time resolved AMR measurements have provided insight into the effect of current and\nspin-transfer torque on domain wall motion [374].\nSecond, domain walls display their own magnetoresistance, which can yield rectification effects and can\nbe manipulated by controlling the spin structure at the pinning site [375]. In analogy with GMR, the\nswitching of domains will produce a magnetoresistance due to a mistracking effect, where the orientation\nof the electron spins comprising the current lag behind the orientation of the local magnetization inside\nthe domain wall [376–379] resulting in a spin accumulation effect [379]. Direct measurements of domain\nmagnetoresistance have been used to study e.g. spin current asymmetry [380].\nFinally, and most relevant to our discussion of spin rectification, due to their general effect on magneti-\nzation dynamics, domain walls will influence AMR rectification in ferromagnetic nano structures. Fig. 32\n(a) shows a Py ring structure where constriction sites, introduced by notches in the ring, will lead to domain\nwall formation. A bias T allows the injection of an rf current and detection of the dc voltage produced\nthrough AMR rectification. Fig. 32 (b) shows the measurement results with the depinning field (the field\nrequired to move the domain wall past the pinning site) shown in blue and the rectification voltage shown\nin red. Thus SR measurements can be used to determine the domain wall resonance of both transverse and\n54vortex domain walls and have even been used to observe multiple resonances within a magnetic domain\n[27, 381].\nAnotherinterestinguseofrectificationinnanostructureswithdomainwallsisin“magneticfingerprinting\"\nof a sample. When a current is passed through a ferromagnetic nanowire a voltage will be generated due to\nSTT as discussed in Sec. 3.3.5. In the presence of a domain wall the voltage spectrum will strongly depend\non the internal spin structure of the wall, and therefore can be used as a fingerprint of the sample [25]. An\nexample of such “fingerprints\" is shown in Fig. 32 (d). The voltages are measured using either a semicircular\nnanowire of varying thicknesses (S2, S3, S4) or a “boomerang\" structured nanowire (S1), as shown in Fig.\n32 (a). The red (black) curves show the rectification voltage in the presence (absence) of a domain wall.\n4.6. Spintronic Microwave Sensing and Imaging\nIn addition to applications for magnetic storage and logic devices which we have already discussed,\nspin rectification is playing an increasingly important role in the development of novel microwave sensing\nand imaging techniques. High speed telecommunications require highly sensitive microwave devices that\nare operational at room temperatures. In particular the rectifying behaviour of semiconductor diodes is\nof key importance, making conventional diodes a foundation of modern electronics. However, although\nsemiconductor diodes satisfy the high sensitivity and room temperature requirements, they have poor signal-\nto-noise ratios which means they cannot operate at low input power. Spin rectification on the other hand\nbehaves remarkably different from semiconductor rectification and has been used to develop the spin-transfer\ntorque diode, which can form the basis of nano meter scale radio frequency detectors for telecommunications.\nFig. 33 (a) illustrates a semiconductor p-n junction diode. When current flows from the n to p side the\nspace charge region is enlarged resulting in a high resistance state. On the other hand when the current\nflows from the p to n side the space charge region is reduced, producing a low resistance state. For the MTJ\nbased spin-transfer torque diode shown in Fig. 33 (b) the varying resistance states are produced by the\nTMR of an MTJ. For a positive current the angle between the fixed and free layers is maximized, resulting\nin a high resistance state, while for a negative current the angle is minimized, producing a low resistance\nstate. Due to this oscillating resistance a rectified voltage is produced.\nSince the dc voltage generated, and hence the efficiency of microwave rectification, is proportional to\nthe magnetoresistance ratio of the device, an MTJ is logically the best choice for microwave detection and\nin 2005, using a conventional CoFeB/MGO/CoFeB MTJ, Tulapurkar et al. [17] first demonstrated the\nconcept of a spin-transfer torque diode. However the sensitivity achieved at the time, up to 1.4 mV/mW,\nwas still roughly three orders of magnitude lower than commercially available semiconductor Schottky diode\nFigure 33:(a) The behaviour of a semiconductor p-n junction diode compared to (b) an MTJ based spin-transfer torque diode.\nSource :Adapted from Ref. [17].\n55detectors (3 800 mV/mW). More recently, using an improved understanding of spin dynamics in such devices\nto improve device fabrication, highly sensitive nanoscale spin-transfer torque diodes with sensitivity up to\n75 400 mV/mW have been developed [244–246, 382]. This order of magnitude sensitivity enhancement over\nsemiconductor devices, while still meeting the required functionalities, indicates the exciting potential of\nspintronic and spin rectification based devices for microwave sensing. One downside to MTJ based sensing\nis the complex device fabrication required. In this regard the recently realized GMR based diode effect may\nprove advantageous [247, 248].\nMicrowave detection also has important applications for novel imaging techniques. A key advantage\nof microwave imaging compared to conventional optical imaging is the increased penetration depth of mi-\ncrowaves, enabling imaging behind barriers/obstacles and the potential to “see the invisible\". However unlike\noptical imaging there are no microwave lenses, which necessitates the use of both amplitude andphasein-\nformation in order to produce accurate images. The simplest way to obtain the required phase information\nis through an interferometry based technique, known as spintronic Michelson interferometry [33]. The basis\nof this technique is illustrated in Fig. 34. By using two coherently split microwave paths to drive the\ncurrent and field, respectively, in a spintronic device, any change in path length, due to either a purposely\nintroduced phase shift or changes in dielectric path length introduced by obstacles, can be observed. The\nspintronic device used as a sensor can be either a spin dynamo [33] or an MTJ [382, 383]. Other techniques,\nsuch as XMCD, have also been successful in phase resolved measurements [384–387], however an impor-\ntant advantage of spintronic Michelson interferometry is its simplicity. When compared to other microwave\nimaging techniques spintronic Michelson interferometry is inexpensive and avoids complex reconstruction\nalgorithms.\nAs illustrated in Fig. 34, the rectified voltage in the spintronic Michelson interferometer depends directly\non the phase difference between the CPW and Py signals. Since microwaves have the ability to penetrate\ndielectric objects, and interact with their internal structures and defects, this means that an object inserted\ninto path-2 will change the microwave phase and thus influence the measured voltage. Since microwaves\nare non ionizing and, at the power levels required for imaging, have not been shown to cause any long term\ndamage to human tissue, microwave imaging techniques have great potential potential for use in medical\nimaging technologies, particularly for breast cancer imaging due to the good dielectric contrast of breast\ntissue [388–391]. Additionally the fact that working in the near field allows the Abbe diffraction limit to be\novercome [392], means that microwaves have the capability to detect structures even smaller than their \u0018\ncm wavelength.\nFollowing the proof of concept for such near-field spin rectification based microwave imaging systems\n[33, 182, 183] these techniques have been used for a variety of imaging purposes, including breast cancer\n[393] and ground penetrating radar detection [394]. In addition to the actual imaging of objects, rectification\nFigure 34:The rectification voltage measured by adjusting an rf phase shifter inserted into (a) one of the two microwave paths,\nand (b) only one microwave path which is connected to either the CPW or (c) the Py strip of a spin dynamo. The strong\nsinusoidal dependence of V(\b)is observed in (a) when both paths are coupled in the spintronic device, demonstrating the\ncontrollable influence of the relative electromagnetic phase on the rectified voltage. Source :Adapted from Ref. [33].\n56based sensors have been used in a variety of other contexts. For example, it has been demonstrated that\nspin rectification can be used as an effective method to detect the full hfield vector polarization [29, 395].\nPhase resolved techniques have also enabled the identification of non resonant rectification [396], and have\nbeen used to develop non destructive dielectric characterization techniques [184]. Also, imaging has been\nperformed using Seebeck induced rectification [397] in MTJs [398, 399].\n4.7. Spintronic Rectifier for Electromagnetic Energy Harvesting\nIn the move towards renewable energy a multifaceted approach is necessary to provide both sufficient\nenergy levels and sustainability. Well known green energy techniques such as solar, wind, geothermal and\nocean energy are of course important contributors to this vision, however with the ubiquitous presence of\nWiFi and cell phone signals, the use of energy harvesting from ambient electromagnetic fields is also poised\nto be a key contributor in future green energy technologies. The feasibility of energy harvesting relies on\nthe efficient operation of rectifiers, which use a nonlinear voltage-current relationship to convert microwave\nfrequency fields into a dc power source. While there are a variety of potential devices, such as the Schottky\ndiode, semiconductor tunnel junction, metal-insulator-metal (MIM) diode and spin diode based on a MTJ,\nthe spin diode is believe to have the most long term potential [400].\nUndoubtedly the Schottky diode will play an important role in the next decade, as it is currently the\nmost commonly used device in rectifiers, with the most matured technology. Unfortunately, as summarized\nin Fig. 35, the Schottky diode already reached its maximum responsivity of 19.4 A/W (at 300 K) in\nthe 1970s, with fundamental limitations set by the laws of thermionic emission which govern the electric\ncurrent flow. In contrast, the current flow due to quantum tunnelling in MTJs, created by separating two\nconductors with a very thin insulator, are mainly limited by the barrier thickness, and still have much\nroom for growth. Although the MTJ fabrication process is very complex, it has benefited from the push\nfor magnetoresistive random-access memory (MRAM) motivated by the huge consumer electronics market.\nIn the coming years this push will likely result in a maturation of fabrication techniques analogous to that\nexperienced by Schottky diodes in the mid 20thcentury. As a consequence, spin diode technology represents\nthe most promising device platform in the development of effective ambient energy harvesting technology\n[400].\nFig. 35 shows the evolution of the responsivity in different rectifier devices. Despite the fact that the\nfirst realization of spin diodes in 2005 achieved only 0.01 A/W responsivity at zero current bias [17], by 2014\nthis value had been improved by two orders of magnitude to 2.1 A/W [244]. Just as semiconductor tunnel\njunctions have already surpassed the Schottky diode limitations, the trend in Fig. 35 suggests that in the\nnear future spin diodes may do the same. Although it is still too early to conclusively predict which kind\nof tunnelling device will dominate future recitifier technology, certainly the performance and feasibility of\nmicrowave energy harvesting devices will greatly benefit from this technological competition.\nFigure 35:Improvement of the rectification capabilities of various diode technologies over the years. Source :Adapted from\nRef. [400].\n575. Summary and Outlook: Three Generations of FMR Spectroscopy\nIn this review we have examined the role of electrical detection in the study of magnetization dynamics.\nWe saw that, due to spin rectification, which produces a dc voltage via the nonlinear coupling of a dynamic\nmagnetoresistance with a dynamic microwave current, the magnetization dynamics of magnetic materials,\nincluding metals, semiconductors and even insulators, can be probed through electrical detection. Although\nthe examination of rectification spectra may be complicated by extraneous effects, such as spin pumping,\nwell established techniques such as angular and line shape analyses enable a wealth of information to be\nFigure 36:The three generations of FMR spectroscopy techniques. (a) Early investigations of spin dynamics were based on\nferromagnetic resonance cavities. The microwave signal through the cavity was detected electrically using a rectifying diode.\n(b) The second generation of FMR spectroscopy arose with the invention of the CPW and VNA. The CPW provides an\neasy way to generate localized, on chip microwave fields, while a VNA can directly measure microwave transmission/reflection\nproperties. (c) Finally, the third and most recent generation is based on electrical detection of spin rectification using lock-in\ntechniques.\nextracted from the rectified voltage, making such techniques powerful for both fundamental physics studies\nand novel spintronic applications.\nFrom a historical perspective, electrical detection represent the third generation in a long history of FMR\nspectroscopy, dating back to the early 20thcentury [1, 3]. The first generation of ferromagnetic resonance\nspectroscopy made use of microwave cavities, typically waveguides, as shown in Fig. 36 (a). With the\nferromagnetic sample placed at the position of strongest microwave magnetic field (usually the centre of\nthe cavity) the microwave signal through the cavity can be detected using a rectifying semiconductor diode.\nThis technology was commercially developed by the German company Bruker and is actually most widely\nused today in electron paramagnetic resonance studies, although it is also a powerful tool in studies of mag-\nnetization dynamics in bulk ferromagnetic materials and ��lms [198]. However, since microwave cavities work\n58in a very narrow frequency range, measurements of the dispersion relation are difficult and time consum-\ning. Additionally as the absorption of microwaves is proportional to the volume of ferromagnetic materials,\nthe technology is not suitable for the study of micro- and nano-structured samples, due to extremely weak\nsignals.\nAs a result of advances in microwave transmission and detection technology, the second generation\nof FMR spectroscopy was born, combining the broadband capabilities of vector network analyzers and\ncoplanar waveguides. In 1969, Cheng Wen invented the CPW at RCA’s Sarnoff Laboratories in New\nJersey [401] (interestingly CPW is also the acronym of his English name, Cheng P. Wen). This invention\ncompactified traditional three-dimensional microwave devices into two-dimensions while still maintaining\nhigh performance. This not only greatly reduced the size of microwave devices but also allowed the on chip\nintegration of microwave devices and other electronics. The invention of the CPW, which allowed improved\nmicrowave transmission, was complemented by the development of the scalar network analyzer by Hewlett-\nPackard in 1973, which enabled the direct measurement of microwave amplitudes over a frequency range\nof 15 MHz to 18 GHz. The rapid development of the telecommunications industry at the end of the 20th\ncentury further demanded improved high-frequency measurement systems, leading to the commercialization\nof the modern vector network analyzer, which can not only measure the amplitude but also the phase\nof broadband microwaves. As shown in Fig. 36 (b), the second generation of broadband ferromagnetic\nresonance measurement technology combines the CPW and VNA [402, 403]. The magnetic sample is placed\non the signal (S) strip of a GSG CPW and the microwave field carried by the CPW drives FMR precession.\nBy fixing the magnetic field the S-parameter can be measured by the VNA as a function of microwave\nfrequency producing an FMR spectra. This technique is widely used to study thin magnetic films, especially\nultra-thin films [404]. In addition, a related technique combining a modulated magnetic field and broadband\nstripline with pulsed inductive microwave magnetometry (PIMM) has been developed in some laboratories\nto measure FMR [276, 405].\nThe first two generations of FMR spectroscopy indirectly measure the energy loss due to FMR, and\ncontinuetobeusedaspowerfultoolstostudymagnetizationdynamics. Howeverduetotheinevitableenergy\nloss in a transmission line, which is very sensitive to the microwave frequency as well as the measurement\nenvironment, these techniques require careful calibration. Additionally for many applications the expense of\na VNA may be prohibitive. This lead to the development of the third generation of FMR spectroscopy in the\nbeginning of the 21stcentury, electrical detection, as shown in Fig. 36 (c). Electrical detection has proven\nto be a powerful tool in the study of spin excitations in micro- and nano-structured magnetic samples [14–\n28]. In this third generation of FMR spectroscopy, microwave fields are still produced by a CPW, however\nlock-in amplification is used to measure the spin rectification voltage resulting from the spin dynamics of\nthe sample. Since this technique 1) has a high microwave field density due to the CPW, 2) has a voltage\nsignal which is independent of sample volume, and is instead dependent on sample resistance which can be\ntailored by careful device fabrication, and 3) uses the highly sensitive measurements of lock-in amplification\nwhich can detect voltages down to the several nV scale, electrical detection through spin rectification has\nbeen used to study a variety of nanostructured samples from FM nanowires [25] to MTJs [17]. In addition,\ndue to its high sensitivity and ease of use, spin rectification has been used to study a wide range of materials\n– from FM metals, to FM semiconductors, to FM insulators – a wide range of physics – from magnetization\ndynamics, to spin pumping and the spin Hall effect – and to develop new approaches to microwave sensing\nand imaging.\nWhile at the time of writing, in 2015, the third generation of FMR spectroscopy has become a well\nestablished technique for the study of microstructured magnetic devices, a new generation of FMR spec-\ntroscopy is just forming. Known as cavity spintronics [65], this emerging field incorporates elements of first\nand third generation FMR spectroscopy techniques to perform both microwave transmission and electrical\ndetection of high quality ferromagnetic samples in low loss microwave cavities/resonators, and promises to\nreveal important insights into the physics of strongly coupled spin-photon systems. As predicted by Soykal\nand Flatteé [406, 407] high quality YIG samples, combined with high quality microwave cavities can ex-\nhibit coherent spin-photon interactions with exceptionally long decoherence times. Such systems were first\nstudied experimentally by Huebl et al. [408] using microwave transmission and various foundational studies\nhave confirmed key predictions about these strongly coupled systems [409, 410], which can be accurately\n59Figure 37:(a) A schematic picture of the cavity spintronic system, consisting of a YIG/Pt bilayer inside of a microwave\ncavity. The strong electrodynamic coupling between the magnetization dynamics, governed by the LLG equation, and the\nelectrodynamics, governed by Maxwell’s equations, lead to the creation of a new quasiparticle, the cavity-magnon-polariton.\n(b) The CMP will modify the spin current generated in the YIG/Pt bilayer, allowing characteristic coupling signatures, such\nas a large avoided crossing, to be electrically detected. The horizontal and diagonal dashed lines indicate the uncoupled cavity\nand FMR dispersions respectively. Source :Panel (a) adapted from Ref. [411].\ndescribed by coupling the Maxwell and LLG equations to form a cavity-magnon-polariton (CMP) [411–415].\nAn exciting possibility for future spintronic technologies is the electrical detection of the CMP [411, 416],\nwhich indicates the influence of strong coupling on the spin current in bilayer devices and could enable the\ncoherent control of spin current. With the theoretical and experimental foundation in place, a flurry of\nrecent applications have been proposed e.g. magnon dark mode memories [417], magnon qubit coupling\n[418, 419], and the coupling of optical and microwave frequency regimes through whispering gallery modes\n[420–423]. On this new frontier of strong magnon-photon coupling the electrical detection of magnetization\ndynamics is poised to play a key role due to its versatile capabilities as we have reviewed in this article.\nIn this perspective, the twilight of the fourth generation of FMR spectroscopy, with its focus on cavity\nspintronics and quantum magnetism, is on the horizon.\nAcknowledgements\nThis review article has been based on a decade of collaboration and communication with many colleagues\nas identified in the reference section. We thank all members and alumni of the Dynamic Spintronics Group\nat the University of Manitoba, especially N. Mecking, A. Wirthmann, X.L. Fan, L.H. Bai, Y. Hou, B.M.\nYao, L. Fu, Z.H. Zhang, and P. Hyde for their contributions. We are also thankful for the encouragement\nand/or feedback provided during the public screening of our preprint by W.E. Bailey, G.E.W. Bauer, J.W.\nCai, C.L. Chien, M. Costache, V. Flovik, B. Heinrich, T. Jungwirth, M. Kläui, O. Klein, A. Kovalev, I.\nKrivorotov, G.J. Schmidt, D. Stancil, T. Stobiecki, Y. Tserkovnyak, Y.Z. Wu, J.Q. Xiao, H. Yang, S. Yuasa,\nand W.X. Zhang. Specially, the authors would like to thank the pioneer of this field, Hellmut Juretschke\nat the age of 92, for surprising us by sending an inspiring note. This work was supported by NSERC, CFI,\nCBCF, UofM, and NSFC (No. 11429401) Grants.\n60References\nReferences\n[1] V. K. Arkad’yev, J. Russ. Phys.-Chem. Soc. 44 (1912) 165.\n[2] J. Dorfmann, Some remarks on the knowledge of the mechanism of magnetic phenomena, Zeitschrift für Phys. 17 (1923)\n98.\n[3] Griffiths J. H. E., Anomalous High-frequency Resistance of Ferromagnetic Metals, Nature 158 (1946) 670. doi:10.1038/\n158670a0 .\n[4] C. Kittel, Interpretation of anomalous larmor frequencies in ferromagnetic resonance experiment, Phys. Rev. 71 (1947)\n270. doi:10.1103/PhysRev.71.270.2 .\n[5] C. Kittel, On the theory of ferromagnetic resonance absorption, Phys. Rev. 73 (1948) 155. doi:10.1103/PhysRev.73.155 .\n[6] S. V. Vonsovskii, Magnetism, John Wiley & Sons, Inc., 1974.\n[7] B. Hillebrands, K. Ounadjela (Eds.), Spin Dynamics in Confined Magnetic Structures I, Topics in Applied Physics,\nSpringer, 2002. doi:10.1007/3-540-40907-6 .\n[8] B. Hillebrands, K. Ounadjela (Eds.), Spin Dynamics in Confined Magnetic Structures II, Springer, 2002. doi:10.1007/\n3-540-46097-7 .\n[9] B. Hillebrands, A. Thiaville (Eds.), Spin Dynamics in Confined Magnetic Structures III, Springer, 2006. doi:10.1007/\n10938171_4 .\n[10] M. N. Baibich, J. M. Broto, A. Fert, F. N. Van Dau, F. Petroff, Giant Magnetoresistance of (001)Fe/(001)Cr Magnetic\nSuperlattices, Phys. Rev. Lett. 61 (1988) 2472. doi:10.1103/PhysRevLett.61.2472 .\n[11] G. Binasch, P. Grünberg, F. Saurenbach, W. Zinn, Enhanced magnetoresistance in layered magnetic structures with\nantiferromagnetic interlayer exchange, Phys. Rev. B 39 (1989) 4828. doi:10.1103/PhysRevB.39.4828 .\n[12] J. Katine, F. Albert, R. Buhrman, E. Myers, D. C. Ralph, Current-Driven Magnetization Reversal and Spin-Wave Exci-\ntations in Co /Cu /Co Pillars, Phys. Rev. Lett. 84 (2000) 3149. arXiv:cond-mat/9908231v1 ,doi:10.1103/PhysRevLett.\n84.3149.\n[13] Y. Huai, F. Albert, P. Nguyen, M. Pakala, T. Valet, Observation of spin-transfer switching in deep submicron-sized and\nlow-resistance magnetic tunnel junctions, Appl. Phys. Lett. 84 (2004) 3118. arXiv:cond-mat/0504486 ,doi:10.1063/1.\n1707228.\n[14] M. Tsoi, A. G. M. Jansen, J. Bass, W. Chiang, V. Tsoi, P. Wyder, Generation and detection of phase-coherent current-\ndriven magnons in magnetic multilayers, Nature 406 (2000) 46.\n[15] S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Emley, R. J. Schoelkopf, R. A. Buhrman, D. C. Ralph, Microwave\noscillations of a nanomagnet driven by a spin-polarized current., Nature 425 (2003) 380. arXiv:cond-mat/0306259 ,\ndoi:10.1038/nature01967 .\n[16] Y. S. Gui, S. Holland, N. Mecking, C. Hu, Resonances in Ferromagnetic Gratings Detected by Microwave Photoconduc-\ntivity, Phys. Rev. Lett. 95 (2005) 056807. doi:10.1103/PhysRevLett.95.056807 .\n[17] A. A. Tulapurkar, Y. Suzuki, A. Fukushima, H. Kubota, H. Maehara, K. Tsunekawa, D. D. Djayaprawira, N. Watanabe,\nS. Yuasa, Spin-torque diode effect in magnetic tunnel junctions, Nature 438 (2005) 339. doi:10.1038/nature04207 .\n[18] M. V. Costache, S. M. Watts, M. Sladkov, C. H. Van Der Wal, B. J. Van Wees, Large cone angle magnetization\nprecession of an individual nanopatterned ferromagnet with dc electrical detection, Appl. Phys. Lett. 89 (2006) 232115.\narXiv:cond-mat/0609190 ,doi:10.1063/1.2400058 .\n[19] M. V. Costache, M. Sladkov, S. M. Watts, C. H. Van Der Wal, B. J. Van Wees, Electrical detection of spin pumping due\nto the precessing magnetization of a single ferromagnet, Phys. Rev. Lett. 97 (2006) 216603. arXiv:cond-mat/0609089 ,\ndoi:10.1103/PhysRevLett.97.216603 .\n[20] E. Saitoh, M. Ueda, H. Miyajima, G. Tatara, Conversion of spin current into charge current at room temperature: Inverse\nspin-Hall effect, Appl. Phys. Lett. 88 (2006) 182509. doi:10.1063/1.2199473 .\n[21] J. C. Sankey, P. M. Braganca, A. G. F. Garcia, I. N. Krivorotov, R. A. Buhrman, D. C. Ralph, Spin-transfer-driven\nferromagnetic resonance of individual nanomagnets, Phys. Rev. Lett. 96 (2006) 227601. arXiv:cond-mat/0602105 ,doi:\n10.1103/PhysRevLett.96.227601 .\n[22] H. Kubota, A. Fukushima, K. Yakushiji, T. Nagahama, S. Yuasa, K. Ando, H. Maehara, Y. Nagamine, K. Tsunekawa,\nD. D. Djayaprawira, N. Watanabe, Y. Suzuki, Quantitative measurement of voltage dependence of spin-transfer torque\nin MgO-based magnetic tunnel junctions, Nat. Phys. 4 (2008) 37. doi:10.1038/nphys784 .\n[23] Y. S. Gui, N. Mecking, X. Zhou, G. Williams, C.-M. Hu, Realization of a room-temperature spin dynamo: The spin\nrectification effect, Phys. Rev. Lett. 98 (2007) 107602. arXiv:cond-mat/0701013 ,doi:10.1103/PhysRevLett.98.107602 .\n[24] N. Mecking, Y. S. Gui, C.-M. Hu, Microwave photovoltage and photoresistance effects in ferromagnetic microstrips, Phys.\nRev. B 76 (22) (2007) 224430. arXiv:cond-mat/0710.1974 ,doi:10.1103/PhysRevB.76.224430 .\n[25] A. Yamaguchi, H. Miyajima, T. Ono, Y. Suzuki, S. Yuasa, A. Tulapurkar, Y. Nakatani, Rectification of radio frequency\ncurrent in ferromagnetic nanowire, Appl. Phys. Lett. 90 (2007) 182507. arXiv:cond-mat/0606305 ,doi:10.1063/1.\n2737134.\n[26] Y. S. Gui, N. Mecking, C.-M. Hu, Quantized spin excitations in a ferromagnetic microstrip from microwave photovoltage\nmeasurements, Phys. Rev. Lett. 98 (2007) 217603. arXiv:cond-mat/0704.3397 ,doi:10.1103/PhysRevLett.98.217603 .\n[27] D. Bedau, M. Kläui, S. Krzyk, U. Rüdiger, G. Faini, L. Vila, Detection of current-induced resonance of geometrically\nconfined domain walls, Phys. Rev. Lett. 99 (2007) 146601. doi:10.1103/PhysRevLett.99.146601 .\n[28] J. C. Sankey, Y. T. Cui, R. A. Buhrman, D. C. Ralph, J. Z. Sun, J. C. Slonczewski, Measurement of the Spin-Transfer-\nTorque Vector in Magnetic Tunnel Junctions, Nat. Phys. 4 (2007) 67. arXiv:0705.4207 ,doi:10.1038/nphys783 .\n61[29] L. H. Bai, Y. S. Gui, A. Wirthmann, E. Recksiedler, N. Mecking, C.-M. Hu, Z. H. Chen, S. C. Shen, The rf magnetic-field\nvector detector based on the spin rectification effect, Appl. Phys. Lett. 92 (2008) 032504. doi:10.1063/1.2837180 .\n[30] X.Hui, A.Wirthmann, Y.S.Gui, Y.Tian, X.F.Jin, Z.H.Chen, S.C.Shen, C.-M.Hu, Electricdetectionofferromagnetic\nresonance in single crystal iron film, Appl. Phys. Lett. 93 (2008) 232502. doi:10.1063/1.3029744 .\n[31] A. Wirthmann, X. Hui, N. Mecking, Y. S. Gui, T. Chakraborty, C.-M. Hu, M. Reinwald, C. Schüller, W. Wegscheider,\nBroadband electrical detection of spin excitations in Ga 0:98Mn0:02As using a photovoltage technique, Appl. Phys. Lett.\n92 (2008) 232106. arXiv:0806.0785 ,doi:10.1063/1.2943280 .\n[32] Y. S. Gui, A. Wirthmann, N. Mecking, C.-M. Hu, Direct measurement of nonlinear ferromagnetic damping via the\nintrinsic foldover effect, Phys. Rev. B 80 (2009) 060402(R). doi:10.1103/PhysRevB.80.060402 .\n[33] A. Wirthmann, X. Fan, Y. S. Gui, K. Martens, G. Williams, J. Dietrich, G. E. Bridges, C.-M. Hu, Direct phase probing\nand mapping via spintronic michelson interferometry, Phys. Rev. Lett. 105 (2010) 017202. doi:10.1103/PhysRevLett.\n105.017202 .\n[34] C. T. Boone, J. A. Katine, J. R. Childress, V. Tiberkevich, A. Slavin, J. Zhu, X. Cheng, I. N. Krivorotov, Resonant\nNonlinear Damping of Quantized Spin Waves in Ferromagnetic Nanowires: A Spin Torque Ferromagnetic Resonance\nStudy, Phys. Rev. Lett. 103 (2009) 167601. arXiv:0905.4699 ,doi:10.1103/PhysRevLett.103.167601 .\n[35] S. Bonetti, V. Tiberkevich, G. Consolo, G. Finocchio, P. Muduli, F. Mancoff, A. Slavin, J. Åkerman, Experimental\nEvidence of Self-Localized and Propagating Spin Wave Modes in Obliquely Magnetized Current-Driven Nanocontacts,\nPhys. Rev. Lett. 105 (2010) 217204. arXiv:0909.3331 ,doi:10.1103/PhysRevLett.105.217204 .\n[36] V.A.Atsarkin, V.V.Demidov, L.V.Levkin, A.M.Petrzhik, Effectofresonancemicrowaveirradiationonmanganitefilm\nconductivity around the ferromagnetic transition, Phys. Rev. B 82 (2010) 144414. doi:10.1103/PhysRevB.82.144414 .\n[37] O. Mosendz, V. Vlaminck, J. E. Pearson, F. Y. Fradin, G. E. W. Bauer, S. D. Bader, A. Hoffmann, Detection and\nquantification of inverse spin Hall effect from spin pumping in permalloy/normal metal bilayers, Phys. Rev. B 82 (2010)\n214403. arXiv:1009.5089 ,doi:10.1103/PhysRevB.82.214403 .\n[38] P. Saraiva, A. Nogaret, J. C. Portal, H. E. Beere, D. A. Ritchie, Dipolar spin waves of lateral magnetic superlattices,\nPhys. Rev. B 82 (2010) 224417. doi:10.1103/PhysRevB.82.224417 .\n[39] Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida, M. Mizuguchi, H. Umezawa, H. Kawai, K. Ando, K. Takanashi,\nS. Maekawa, E. Saitoh, Transmission of electrical signals by spin-wave interconversion in a magnetic insulator, Nature\n464 (2010) 262. doi:10.1038/nature08876 .\n[40] C. W. Sandweg, Y. Kajiwara, K. Ando, E. Saitoh, B. Hillebrands, Enhancement of the spin pumping efficiency by spin\nwave mode selection, Appl. Phys. Lett. 97 (2010) 252504. arXiv:1008.4714v1 ,doi:10.1063/1.3528207 .\n[41] L. Liu, T. Moriyama, D. C. Ralph, R. A. Buhrman, Spin-torque ferromagnetic resonance induced by the spin Hall effect,\nPhys. Rev. Lett. 106 (2011) 036601. arXiv:1011.2788 ,doi:10.1103/PhysRevLett.106.036601 .\n[42] A. Azevedo, L. H. Vilela-Leão, R. L. Rodríguez-Suárez, A. F. Lacerda Santos, S. M. Rezende, Spin pumping and\nanisotropic magnetoresistance voltages in magnetic bilayers: Theory and experiment, Phys. Rev. B 83 (2011) 144402.\ndoi:10.1103/PhysRevB.83.144402 .\n[43] M. Harder, Z. X. Cao, Y. S. Gui, X. L. Fan, C.-M. Hu, Analysis of the line shape of electrically detected ferromagnetic\nresonance, Phys. Rev. B 84 (2011) 054423. arXiv:1105.3236 ,doi:10.1103/PhysRevB.84.054423 .\n[44] Z. Feng, J. Hu, L. Sun, B. You, D. Wu, J. Du, W. Zhang, A. Hu, Y. Yang, D. M. Tang, B. S. Zhang, H. F. Ding,\nSpin Hall angle quantification from spin pumping and microwave photoresistance, Phys. Rev. B 85 (21) (2012) 214423.\ndoi:10.1103/PhysRevB.85.214423 .\n[45] H. L. Wang, C. H. Du, Y. Pu, R. Adur, P. C. Hammel, F. Y. Yang, Large spin pumping from epitaxial Y 3Fe5O12thin\nfilms to Pt and W layers, Phys. Rev. B 88 (2013) 100406. doi:10.1103/PhysRevB.88.100406 .\n[46] P. Deorani, J. Son, K. Banerjee, N. Koirala, M. Brahlek, S. Oh, H. Yang, Observation of inverse spin Hall effect in\nbismuth selenide, Phys. Rev. B 90 (2014) 094403. arXiv:1404.1146 ,doi:10.1103/PhysRevB.90.094403 .\n[47] W. T. Soh, B. Peng, C. K. Ong, An angular analysis to separate spin pumping-induced inverse spin Hall effect from spin\nrectification in a Py/Pt bilayer, J. Phys. D. Appl. Phys. 47 (2014) 285001. doi:10.1088/0022-3727/47/28/285001 .\n[48] Y. Wang, P. Deorani, X. Qiu, J. H. Kwon, H. Yang, Determination of intrinsic spin Hall angle in Pt, Appl. Phys. Lett.\n105 (2014) 152412. arXiv:1410.1601 ,doi:10.1063/1.4898593 .\n[49] E. Becquerel, Recherches sur les effets de la radiation chimique de la lumière solaire, au moyen des courant électriques,\nC. R. Acad. Sci. 9 (1839) 145.\n[50] R. Williams, Bequerel photovoltaic effect in binary compounds, J. Chem. Phys. 32 (1960) 1505. doi:10.1063/1.1730950 .\n[51] M. Grätzel, Photoelectochemical cells, Nature 414 (2001) 338. doi:10.1038/35104607 .\n[52] R. L. Petritz, Theory of photoconductivity in semiconductor films, Phys. Rev. 104 (1956) 1508. doi:10.1103/PhysRev.\n104.1508 .\n[53] A. Rogalski, Infrared detectors: An overview, Infrared Phys. Technol. 43 (2002) 187. doi:10.1016/S1350-4495(02)\n00140-8.\n[54] G. Konstantatos, I. Howard, A. Fischer, S. Hoogland, J. Clifford, E. Klem, L. Levina, E. H. Sargent, Ultrasensitive\nsolution-cast quantum dot photodetectors., Nature 442 (2006) 180. doi:10.1038/nature04855 .\n[55] C. Soci, A. Zhang, B. Xiang, S. A. Dayeh, D. P. R. Aplin, J. Park, X. Y. Bao, Y. H. Lo, D. Wang, ZnO nanowire UV\nphotodetectors with high internal gain, Nano Lett. 7 (2007) 1003. doi:10.1021/nl070111x .\n[56] G. F. Knoll, Radiation Detection and Measurement, 4th Edition, John Wiley & Sons, Inc., 2010.\n[57] J. C. Maan, T. Englert, D. C. Tsui, A. C. Gossard, Observation of cyclotron resonance in the photoconductivity of\ntwo-dimensional electrons, Appl. Phys. Lett. 40 (1982) 609. doi:10.1063/1.93197 .\n[58] H. Schneider, F. Fuchs, B. Dischler, J. D. Ralston, P. Koidl, Intersubband absorption and infrared photodetection at 3.5\nand 4.2\u0016m in GaAs quantum wells, Appl. Phys. Lett. 58 (1991) 2234. doi:10.1063/1.104936 .\n62[59] S.Holland,K.Bittkau,C.-M.Hu,C.Heyn,D.Heitmann,Filling-factor-dependentoscillationsofthecoupledintersubband\nplasmon-LOphononmodeinatunnelingcoupled-bilayersystem, Phys.Rev.B66(2002)233302. doi:10.1103/PhysRevB.\n66.233302 .\n[60] C.-M. Hu, C. Zehnder, C. Heyn, D. Heitmann, THz collective spin-flip excitation of a two-dimensional electron system,\nPhys. Rev. B 67 (2003) 201302. doi:10.1103/PhysRevB.67.201302 .\n[61] S. D. Ganichev, E. L. Ivchenko, V. V. Bel’kov, S. A. Tarasenko, M. Sollinger, D. Weiss, W. Wegscheider, W. Prettl,\nSpin-galvanic effect, Nature 417 (2002) 153. doi:10.1038/417153a .\n[62] W. G. Egan, H. J. Juretschke, DC Detection of Ferromagnetic Resonance in Thin Nickel Films, J. Appl. Phys. 34 (1963)\n1477. doi:10.1063/1.1729604 .\n[63] M. Bass, P. A. Franken, F. Ward, G. Weinreich, Optical Rectification, Phys. Rev. Lett. 9 (1962) 446. doi:10.1103/\nPhysRevLett.9.446 .\n[64] Y. R. Shen, The Principles of Nonlinear Optics, John Wiley & Sons, Inc., 2002.\n[65] C.-M. Hu, Dawn of Cavity Spintronics, Phys. in Can. 72 (2016) 76. arXiv:1508.01966 .\n[66] J. D. Jackson, Classical Electrodynamics, 3rd Edition, John Wiley & Sons, Inc., 1999.\n[67] H.Kurebayashi, J.Sinova, D.Fang, A.C.Irvine, T.D.Skinner, J.Wunderlich, V.Novák, R.P.Campion, B.L.Gallagher,\nE. K. Vehstedt, L. P. Zârbo, K. Výborný, A. J. Ferguson, T. Jungwirth, An antidamping spin-orbit torque originating\nfrom the Berry curvature, Nat. Nanotechnol. 9 (2014) 211. arXiv:1306.1893 ,doi:10.1038/nnano.2014.15 .\n[68] J. Stöhr, H. C. Siegmann, Magnetism From Fundamentals to Nanoscale Dynamics, Springer, 2006. doi:10.1007/\n978-3-540-30283-4 .\n[69] W. Thomson, On the Electro-Dynamic Qualities of Metals:–Effects of Magnetization on the Electric Conductivity of\nNickel and of Iron, Proc. R. Soc. London 8 (1856) 546. doi:10.1098/rspl.1856.0144 .\n[70] J. Xiao, J. Jiang, C. Chien, Giant magnetoresistance in nonmultilayer magnetic systems, Phys. Rev. Lett. 68 (1992)\n3749. doi:10.1103/PhysRevLett.68.3749 .\n[71] M. Jullière, Tunneling between ferromagnetic films, Phys. Lett. A 54 (1975) 225. doi:10.1016/0375-9601(75)90174-7 .\n[72] S. Ikeda, J. Hayakawa, Y. Ashizawa, Y. M. Lee, K. Miura, H. Hasegawa, M. Tsunoda, F. Matsukura, H. Ohno, Tunnel\nmagnetoresistance of 604% at 300 K by suppression of Ta diffusion in CoFeB/MgO/CoFeB pseudo-spin-valves annealed\nat high temperature, Appl. Phys. Lett. 93 (2008) 082508. doi:10.1063/1.2976435 .\n[73] S. Parkin, C. Kaiser, A. Panchula, K. Roche, M. Samant, Magnetically engineered spintronic sensors and memory, Proc.\nIEEE 91 (2003) 661. doi:10.1109/JPROC.2003.811807 .\n[74] E. Tsymbal, D. G. Pettifor, Perspectives of giant magnetoresistance, in: H. Ehrenreich, F. Spaepen (Eds.), Solid State\nPhys., Vol. 56, Academic Press, 2001, pp. 113–237. doi:10.1016/s0081-1947(01)80019-9 .\n[75] N. F. Mott, The Resistance and Thermoelectric Properties of the Transition Metals, Proc. R. Soc. A Math. Phys. Eng.\nSci. 156 (888) (1936) 368. doi:10.1098/rspa.1936.0154 .\n[76] A. Fert, Historical Overview: From Electron Transport in Magnetic Materials to Spintronics, in: E. Y. Tsymbal, I. Žutić\n(Eds.), Handb. Spin Transp. Magn., CRC Press, Boca Raton, 2011, pp. 3–17. doi:10.1201/b11086 .\n[77] A. A. Kovalev, A. Brataas, G. E. W. Bauer, Spin transfer in diffusive ferromagnet-normal metal systems with spin-flip\nscattering, Phys. Rev. B 66 (2002) 224424. arXiv:cond-mat/0207458 ,doi:10.1103/PhysRevB.66.224424 .\n[78] J. Barnaś, A. Fert, M. Gmitra, I. Weymann, V. K. Dugaev, From giant magnetoresistance to current-induced switching\nby spin transfer, Phys. Rev. B 72 (2005) 024426. doi:10.1103/PhysRevB.72.024426 .\n[79] T. McGuire, R. Potter, Anisotropic magnetoresistance in ferromagnetic 3d alloys, IEEE Trans. Magn. 11 (1975) 1018.\ndoi:10.1109/TMAG.1975.1058782 .\n[80] R.Ramos, S.K.Arora, I.V.Shvets, AnomalousanisotropicmagnetoresistanceinepitaxialFe 3O4thinfilmsonMgO(001),\nPhys. Rev. B 78 (2008) 214402. doi:10.1103/PhysRevB.78.214402 .\n[81] C. R. Hu, J. Zhu, G. Chen, J. X. Li, Y. Z. Wu, Direct comparison of anisotropic magnetoresistance and planar Hall effect\nin epitaxial Fe 3O4thin films, Phys. Lett. A 376 (2012) 3317. doi:10.1016/j.physleta.2012.08.009 .\n[82] Z. Ding, J. X. Li, J. Zhu, T. P. Ma, C. Won, Y. Z. Wu, Three-dimensional mapping of the anisotropic magnetoresistance\nin Fe 3O4single crystal thin films, J. Appl. Phys. 113 (2013) 17B103. doi:10.1063/1.4796178 .\n[83] X. Xiao, J. X. Li, Z. Ding, Y. Z. Wu, Four-fold symmetric anisotropic magnetoresistance of single-crystalline Ni(001)\nfilm, J. Appl. Phys. 118 (2015) 203905. doi:10.1063/1.4936175 .\n[84] B. Chen, Z. Ding, D. Ma, J. Li, X. Xiao, Y. Wu, Asymmetric anisotropic magnetoresistance in epitaxial Fe 3O4thin\nfilms, Phys. Lett. A 379 (2015) 1780. doi:10.1016/j.physleta.2015.05.006 .\n[85] X. Xiao, J. H. Liang, B. L. Chen, J. X. Li, D. H. Ma, Z. Ding, Y. Z. Wu, Current-direction dependence of the transport\npropertiesinsingle-crystallineface-centered-cubiccobaltfilms,J.Appl.Phys.118(2015)043908. doi:10.1063/1.4927620 .\n[86] R. Potter, Magnetoresistance anisotropy in ferromagnetic NiCu alloys, Phys. Rev. B 10 (1974) 4626. doi:10.1103/\nPhysRevB.10.4626 .\n[87] O. Jaoul, I. Campbell, A. Fert, Spontaneous resistivity anisotropy in Ni alloys, J. Magn. Magn. Mater. 5 (1977) 23.\ndoi:10.1016/0304-8853(77)90193-7 .\n[88] A. F. Mayadas, Resistivity of Permalloy thin films, J. Appl. Phys. 45 (1974) 2780. doi:10.1063/1.1663668 .\n[89] J. Smit, Magnetoresistance of ferromagnetic metals and alloys at low temperatures, Physica 17 (1951) 612. doi:10.1016/\n0031-8914(51)90117-6 .\n[90] H. J. Juretschke, Electromagnetic Theory of dc Effects in Ferromagnetic Resonance, J. Appl. Phys. 31 (1960) 1401.\ndoi:10.1063/1.1735851 .\n[91] J.-P. Jan, Galvanomagnetic and Thermomagnetic Effects in Metals, Solid State Phys. 5 (1956) 1–96. doi:10.1016/\ns0081-1947(08)60101-0 .\n[92] Y. Zhang, X. R. Wang, H. W. Zhang, Extraordinary galvanomagnetic effects in polycrystalline magnetic films, Europhys.\n63Lett. 113 (2016) 47003. doi:10.1209/0295-5075/113/47003 .\n[93] D. Edwards, J. Mathon, R. Muniz, A resistor network theory of the giant magnetoresistance in magnetic superlattices,\nIEEE Trans. Magn. 27 (1991) 3548. doi:10.1109/20.102926 .\n[94] J. George, L. Pereira, A. Barthélémy, F. Petroff, L. Steren, J. Duvail, A. Fert, R. Loloee, P. Holody, P. Schroeder,\nInverse spin-valve-type magnetoresistance in spin engineered multilayered structures, Phys. Rev. Lett. 72 (1994) 408.\ndoi:10.1103/PhysRevLett.72.408 .\n[95] J. Bass, Giant Magnetoresistance: Experiment, in: E. Y. Tsymbal, I. Žutić (Eds.), Handb. Spin Transp. Magn., CRC\nPress, Boca Raton, 2012, pp. 69–93. doi:10.1201/b11086-7 .\n[96] P. Grünberg, R. Schreiber, Y. Pang, Layered Magnetic Structures: Evidence for Antiferromagnetic Coupling of Fe Layers\nacross Cr Interlayers, Phys. Rev. Lett. 57 (1986) 2442. doi:10.1103/PhysRevLett.57.2442 .\n[97] S. S. P. Parkin, N. More, K. P. Roche, Oscillations in exchange coupling and magnetoresistance in metallic superlattice\nstructures: Co/Ru, Co/Cr, and Fe/Cr, Phys. Rev. Lett. 64 (1990) 2304. doi:10.1103/PhysRevLett.64.2304 .\n[98] S. S. P. Parkin, Systematic variation of the strength and oscillation period of indirect magnetic exchange coupling through\nthe 3d, 4d, and 5d transition metals, Phys. Rev. Lett. 67 (1991) 3598. doi:10.1103/PhysRevLett.67.3598 .\n[99] S. S. P. Parkin, R. Bhadra, K. Roche, Oscillatory magnetic exchange coupling through thin copper layers, Phys. Rev.\nLett. 66 (1991) 2152. doi:10.1103/PhysRevLett.66.2152 .\n[100] D. Mosca, F. Petroff, A. Fert, P. Schroeder, W. Pratt, R. Laloee, Oscillatory interlayer coupling and giant magnetoresis-\ntance in Co/Cu multilayers, J. Magn. Magn. Mater. 94 (1991) L1. doi:10.1016/0304-8853(91)90102-G .\n[101] J. Barnaś, A. Fuss, R. Camley, P. Grünberg, W. Zinn, Novel magnetoresistance effect in layered magnetic structures:\nTheory and experiment, Phys. Rev. B 42 (1990) 8110. doi:10.1103/PhysRevB.42.8110 .\n[102] T. Shinjo, H. Yamamoto, Large Magnetoresistance of Field-Induced Giant Ferrimagnetic Multilayers, J. Phys. Soc. Japan\n59 (1990) 3061. doi:10.1143/JPSJ.59.3061 .\n[103] C. Dupas, P. Beauvillain, C. Chappert, J. P. Renard, F. Trigui, P. Veillet, E. Vélu, D. Renard, Very large magne-\ntoresistance effects induced by antiparallel magnetization in two ultrathin cobalt films, J. Appl. Phys. 67 (1990) 5680.\ndoi:10.1063/1.345925 .\n[104] B. Dieny, V. Speriosu, S. Parkin, B. Gurney, D. Wilhoit, D. Mauri, Giant magnetoresistive in soft ferromagnetic multi-\nlayers, Phys. Rev. B 43 (1991) 1297. doi:10.1103/PhysRevB.43.1297 .\n[105] T. Ono, T. Shinjo, Magnetoresistance of Multilayers Prepared on Microstructured Substrates, J. Phys. Soc. Japan 64\n(1995) 363. doi:10.1143/JPSJ.64.363 .\n[106] M. A. M. Gijs, M. T. Johnson, A. Reinders, P. E. Huisman, R. J. M. van de Veerdonk, S. K. J. Lenczowski, R. M. J.\nvan Gansewinkel, Perpendicular giant magnetoresistance of Co/Cu multilayers deposited under an angle on grooved\nsubstrates, Appl. Phys. Lett. 66 (1995) 1839. doi:10.1063/1.113339 .\n[107] T. Valet, A. Fert, Theory of the perpendicular magnetoresistance in magnetic multilayers, Phys. Rev. B 48 (1993) 7099.\ndoi:10.1103/PhysRevB.48.7099 .\n[108] W. Pratt, S.-F. Lee, J. Slaughter, R. Loloee, P. Schroeder, J. Bass, Perpendicular giant magnetoresistances of Ag/Co\nmultilayers, Phys. Rev. Lett. 66 (1991) 3060. doi:10.1103/PhysRevLett.66.3060 .\n[109] J. W. Jung, Y. Sakuraba, T. T. Sasaki, Y. Miura, K. Hono, Enhancement of magnetoresistance by inserting thin NiAl\nlayers at the interfaces in Co 2FeGa 0:5Ge0:5/Ag/Co 2FeGa 0:5Ge0:5current-perpendicular-to-plane pseudo spin valves,\nAppl. Phys. Lett. 108 (2016) 102408. doi:10.1063/1.4943640 .\n[110] A. Chaiken, G. A. Prinz, J. J. Krebs, Magnetotransport study of Fe-Cr-Fe sandwiches grown on ZnSe(100), J. Appl.\nPhys. 67 (1990) 4892. doi:10.1063/1.344722 .\n[111] P. Dauguet, P. Gandit, J. Chaussy, S. Lee, A. Fert, P. Holody, Angular dependence of the perpendicular giant magne-\ntoresistance of multilayers, Phys. Rev. B 54 (1996) 1083. doi:10.1103/PhysRevB.54.1083 .\n[112] A. Vedyayev, B. Dieny, N. Ryzhanova, J. B. Genin, C. Cowache, Angular Dependence of Giant Magnetoresistance in\nMagnetic Multilayered Structures, Europhys. Lett. 25 (1994) 465. doi:10.1209/0295-5075/25/6/012 .\n[113] A. Vedyayev, N. Ryzhanova, B. Dieny, P. Dauguet, P. Gandit, J. Chaussy, Angular variation of giant magnetoresistance\nfor current perpendicular to the plane of the layers, Phys. Rev. B 55 (1997) 3728. doi:10.1103/PhysRevB.55.3728 .\n[114] C. Blaas, P. Weinberger, L. Szunyogh, J. Kudrnovský, V. Drchal, P. M. Levy, C. Sommers, On the orientational depen-\ndence of giant magnetoresistance, Eur. Phys. J. B 9 (1999) 245. doi:10.1007/s100510050763 .\n[115] S. S. P. Parkin, Giant Magnetoresistance in Magnetic Nanostructures, Annu. Rev. Mater. Sci. 25 (1995) 357. doi:\n10.1146/annurev.ms.25.080195.002041 .\n[116] L. B. Steren, R. Morel, A. Barthélémy, F. Petroff, A. Fert, P. Holody, R. Loloee, P. Schroeder, Giant magnetoresistance\nin hybrid magnetic nanostructures, J. Magn. Magn. Mater. 140 (1995) 495. doi:10.1016/0304-8853(94)01529-5 .\n[117] S. Yuasa, T. Nagahama, A. Fukushima, Y. Suzuki, K. Ando, Giant room-temperature magnetoresistance in single-crystal\nFe/MgO/Fe magnetic tunnel junctions, Nat. Mater. 3 (2004) 868. doi:10.1038/nmat1257 .\n[118] S. Gider, B.-U. Runge, A. C. Marley, S. S. P. Parkin, The Magnetic Stability of Spin-Dependent Tunneling Devices,\nScience 281 (1998) 797. doi:10.1126/science.281.5378.797 .\n[119] S. Maekawa, U. Gafvert, Electron tunneling between ferromagnetic films, IEEE Trans. Magn. 18 (1982) 707. doi:\n10.1109/TMAG.1982.1061834 .\n[120] J. S. Moodera, L. R. Kinder, T. M. Wong, R. Meservey, Large Magnetoresistance at Room Temperature in Ferromagnetic\nThin Film Tunnel Junctions, Phys. Rev. Lett. 74 (1995) 3273. doi:10.1103/PhysRevLett.74.3273 .\n[121] T. Miyazaki, Giant magnetic tunneling effect in Fe/Al 2O3/Fe junction, J. Magn. Magn. Mater. 139 (1995) L231. doi:\n10.1016/0304-8853(94)01648-8 .\n[122] S. S. P. Parkin, C. Kaiser, A. Panchula, P. M. Rice, B. Hughes, M. Samant, S.-H. Yang, Giant tunnelling magnetoresis-\ntance at room temperature with MgO (100) tunnel barriers, Nat. Mater. 3 (2004) 862. doi:10.1038/nmat1256 .\n64[123] Y.M.Lee,J.Hayakawa,S.Ikeda,F.Matsukura,H.Ohno,Effectofelectrodecompositiononthetunnelmagnetoresistance\nof pseudo-spin-valve magnetic tunnel junction with a MgO tunnel barrier, Appl. Phys. Lett. 90 (2007) 212507. doi:\n10.1063/1.2742576 .\n[124] R. Meservey, P. Tedrow, Spin-polarized electron tunneling, Phys. Rep. 238 (1994) 173. doi:10.1016/0370-1573(94)\n90105-8.\n[125] M. Bowen, M. Bibes, A. Barthélémy, J. P. Contour, A. Anane, Y. Lemaître, A. Fert, Nearly total spin polarization\nin La 2=3Sr1=3MnO 3from tunneling experiments, Appl. Phys. Lett. 82 (2003) 233. arXiv:cond-mat/0211158 ,doi:\n10.1063/1.1534619 .\n[126] T. Ishikawa, T. Marukame, H. Kijima, K. I. Matsuda, T. Uemura, M. Arita, M. Yamamoto, Spin-dependent tunneling\ncharacteristics of fully epitaxial magnetic tunneling junctions with a full-Heusler alloy Co 2MnSi thin film and a MgO\ntunnel barrier, Appl. Phys. Lett. 89 (2006) 192505. doi:10.1063/1.2378397 .\n[127] S. Yuasa, Tunneling Magnetoresistance: Experiment (MgO Magnetic Tunnel Junctions), in: E. Y. Tsymbal, I. Žutić\n(Eds.), Handb. Spin Transp. Magn., CRC Press, Boca Raton, 2012, pp. 217–231. doi:10.1201/b11086-15 .\n[128] P. R. LeClair, J. S. Moodera, Tunneling Magnetoresistance: Experiment (Non-MgO Magnetic Tunnel Junctions), in:\nE. Y. Tsymbal, I. Žutić (Eds.), Handb. Spin Transp. Magn., CRC Press, Boca Raton, 2012, pp. 197–216. doi:10.1201/\nb11086-14 .\n[129] K. D. Belashchenko, E. Y. Tsymbal, Tunneling Magnetoresistance: Theory, in: E. Y. Tsymbal, I. Žutić (Eds.), Handb.\nSpin Transp. Magn., CRC Press, Boca Raton, 2012, pp. 233–250. doi:10.1201/b11086-16 .\n[130] D. D. Djayaprawira, K. Tsunekawa, M. Nagai, H. Maehara, S. Yamagata, N. Watanabe, S. Yuasa, Y. Suzuki, K. Ando,\n230% room-temperature magnetoresistance in CoFeB/MgO/CoFeB magnetic tunnel junctions, Appl. Phys. Lett. 86\n(2005) 092502. doi:10.1063/1.1871344 .\n[131] J. S. Moodera, L. R. Kinder, Ferromagnetic-insulator-ferromagnetic tunneling: Spin-dependent tunneling and large\nmagnetoresistance in trilayer junctions, J. Appl. Phys. 79 (1996) 4724. doi:10.1063/1.361653 .\n[132] H. Jaffrès, D. Lacour, F. Nguyen Van Dau, J. Briatico, F. Petroff, A. Vaurès, Angular dependence of the tunnel magne-\ntoresistance in transition-metal-based junctions, Phys. Rev. B 64 (2001) 064427. doi:10.1103/PhysRevB.64.064427 .\n[133] J. C. Slonczewski, Conductance and exchange coupling of two ferromagnets separated by a tunneling barrier, Phys. Rev.\nB 39 (1989) 6995. doi:10.1103/PhysRevB.39.6995 .\n[134] Y. Suzuki, H. Kubota, Spin-torque diode effect and its application, J. Phys. Soc. Japan 77 (2008) 031002. doi:10.1143/\nJPSJ.77.031002 .\n[135] H. Nakayama, M. Althammer, Y.-T. Chen, K. Uchida, Y. Kajiwara, D. Kikuchi, T. Ohtani, S. Geprägs, M. Opel,\nS. Takahashi, R. Gross, G. E. W. Bauer, S. T. B. Goennenwein, E. Saitoh, Spin Hall Magnetoresistance Induced by\na Nonequilibrium Proximity Effect, Phys. Rev. Lett. 110 (2013) 206601. arXiv:1211.0098 ,doi:10.1103/PhysRevLett.\n110.206601 .\n[136] Y. M. Lu, J. W. Cai, S. Y. Huang, D. Qu, B. F. Miao, C. L. Chien, Hybrid magnetoresistance in the proximity of a\nferromagnet, Phys. Rev. B 87 (2013) 220409(R). doi:10.1103/PhysRevB.87.220409 .\n[137] Y.-T. Chen, S. Takahashi, H. Nakayama, M. Althammer, S. T. B. Goennenwein, E. Saitoh, G. Bauer, Theory of spin\nHall magnetoresistance, Phys. Rev. B 87 (2013) 144411. arXiv:1302.1352 ,doi:10.1103/PhysRevB.87.144411 .\n[138] C. Hahn, G. de Loubens, O. Klein, M. Viret, V. V. Naletov, J. Ben Youssef, Comparative measurements of inverse\nspin Hall effects and magnetoresistance in YIG/Pt and YIG/Ta, Phys. Rev. B 87 (2013) 174417. arXiv:1302.4416 ,\ndoi:10.1103/PhysRevB.87.174417 .\n[139] M. Althammer, S. Meyer, H. Nakayama, M. Schreier, S. Altmannshofer, M. Weiler, H. Huebl, S. Geprägs, M. Opel,\nR. Gross, D. Meier, C. Klewe, T. Kuschel, J.-M. Schmalhorst, G. Reiss, L. Shen, A. Gupta, Y.-T. Chen, G. E. W. Bauer,\nE.Saitoh, S.T.B.Goennenwein, QuantitativestudyofthespinHallmagnetoresistanceinferromagneticinsulator/normal\nmetal hybrids, Phys. Rev. B 87 (2013) 224401. arXiv:1304.6151 ,doi:10.1103/PhysRevB.87.224401 .\n[140] N. Vlietstra, J. Shan, V. Castel, B. J. van Wees, J. Ben Youssef, Spin-Hall magnetoresistance in platinum on yttrium\niron garnet: Dependence on platinum thickness and in-plane/out-of-plane magnetization, Phys. Rev. B 87 (2013) 184421.\narXiv:1301.3266 ,doi:10.1103/PhysRevB.87.184421 .\n[141] M. Weiler, M. Althammer, M. Schreier, J. Lotze, M. Pernpeintner, S. Meyer, H. Huebl, R. Gross, A. Kamra, J. Xiao,\nY. T. Chen, H. Jiao, G. E. W. Bauer, S. T. B. Goennenwein, Experimental test of the spin mixing interface conductivity\nconcept, Phys. Rev. Lett. 111 (2013) 176601. arXiv:1306.5012 ,doi:10.1103/PhysRevLett.111.176601 .\n[142] N. Vlietstra, J. Shan, V. Castel, J. Ben Youssef, G. E. W. Bauer, B. J. van Wees, Exchange magnetic field torques in\nYIG/Pt bilayers observed by the spin-Hall magnetoresistance, Appl. Phys. Lett. 103 (2013) 032401. arXiv:1305.3117 ,\ndoi:10.1063/1.4813760 .\n[143] B. F. Miao, S. Y. Huang, D. Qu, C. L. Chien, Physical Origins of the New Magnetoresistance in Pt/YIG, Phys. Rev.\nLett. 112 (2014) 236601. doi:10.1103/PhysRevLett.112.236601 .\n[144] S.R.Marmion,M.Ali,M.McLaren,D.A.Williams,B.J.Hickey,TemperaturedependenceofspinHallmagnetoresistance\nin thin YIG/Pt films, Phys. Rev. B 89 (2014) 220404(R). doi:10.1103/PhysRevB.89.220404 .\n[145] M. Isasa, A. Bedoya-Pinto, S. Vélez, F. Golmar, F. Sánchez, L. E. Hueso, J. Fontcuberta, F. Casanova, Spin Hall\nmagnetoresistance at Pt/CoFe 2O4interfaces and texture effects, Appl. Phys. Lett. 105 (2014) 142402. doi:10.1063/1.\n4897544.\n[146] V. L. Grigoryan, W. Guo, G. E. W. Bauer, J. Xiao, Intrinsic magnetoresistance in metal films on ferromagnetic insulators,\nPhys. Rev. B 90 (2014) 161412. arXiv:1407.3571 ,doi:10.1103/PhysRevB.90.161412 .\n[147] N. Vlietstra, J. Shan, B. J. van Wees, M. Isasa, F. Casanova, J. Ben Youssef, Simultaneous detection of the spin-Hall\nmagnetoresistance and the spin-Seebeck effect in platinum and tantalum on yttrium iron garnet, Phys. Rev. B 90 (2014)\n174436. arXiv:1410.0551 ,doi:10.1103/PhysRevB.90.174436 .\n65[148] T. Lin, C. Tang, H. M. Alyahayaei, J. Shi, Experimental Investigation of the Nature of the Magnetoresistance Effects in\nPd-YIG Hybrid Structures, Phys. Rev. Lett. 113 (2014) 037203. doi:10.1103/PhysRevLett.113.037203 .\n[149] A. Brataas, Insulating Magnets Control Neighbor’s Conduction, Physics (College. Park. Md). 6 (2013) 56. doi:10.1103/\nPhysics.6.56 .\n[150] M. I. D’yakonov, V. I. Perel, Possibility of orienting electron spins with current, JETP Lett. 13 (1971) 467.\n[151] J. E. Hirsch, Spin Hall effect, Phys. Rev. Lett 83 (1999) 1834. arXiv:cond-mat/9906160 ,doi:10.1103/PhysRevLett.83.\n1834.\n[152] A. Hoffmann, Spin Hall Effects in Metals, IEEE Trans. Magn. 49 (2013) 5172. doi:10.1109/TMAG.2013.2262947 .\n[153] J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back, T. Jungwirth, Spin Hall effects, Rev. Mod. Phys. 87 (2015) 1213.\narXiv:1411.3249v1 ,doi:10.1103/RevModPhys.87.1213 .\n[154] I. A. Campbell, A. Fert, Transport Properties of Ferromagnets, in: E. P. Wohlfarth (Ed.), Handb. Ferromagn. Mater.,\nVol. 3, 1982, Ch. 9, pp. 747–804. doi:10.1016/s1567-2719(82)03012-1 .\n[155] T. Rijks, S. Lenczowski, R. Coehoorn, W. de Jonge, In-plane and out-of-plane anisotropic magnetoresistance in Ni 80Fe20\nthin films, Phys. Rev. B 56 (1997) 362. doi:10.1103/PhysRevB.56.362 .\n[156] W. Gil, D. Görlitz, M. Horisberger, J. Kötzler, Magnetoresistance anisotropy of polycrystalline cobalt films: Geometrical-\nsize and domain effects, Phys. Rev. B 72 (2005) 134401. arXiv:cond-mat/0506232 ,doi:10.1103/PhysRevB.72.134401 .\n[157] A. Kobs, S. Hee, W. Kreuzpaintner, G. Winkler, D. Lott, P. Weinberger, A. Schreyer, H. P. Oepen, Anisotropic Interface\nMagnetoresistance in Pt/Co/Pt Sandwiches, Phys. Rev. Lett. 106 (2011) 217207. doi:10.1103/PhysRevLett.106.217207 .\n[158] L. K. Zou, Y. Zhang, L. Gu, J. W. Cai, L. Sun, Tunable angular-dependent magnetoresistance correlations in magnetic\nfilms and their implications for spin Hall magnetoresistance analysis, Phys. Rev. B 93 (2016) 075309. doi:10.1103/\nPhysRevB.93.075309 .\n[159] Y. M. Lu, Y. Choi, C. M. Ortega, X. M. Cheng, J. W. Cai, S. Y. Huang, L. Sun, C. L. Chien, Pt magnetic polarization\non Y 3Fe5O12and magnetotransport characteristics, Phys. Rev. Lett. 110 (2013) 147207. doi:10.1103/PhysRevLett.\n110.147207 .\n[160] S. Y. Huang, X. Fan, D. Qu, Y. P. Chen, W. G. Wang, J. Wu, T. Y. Chen, J. Q. Xiao, C. L. Chien, Transport magnetic\nproximity effects in platinum, Phys. Rev. Lett. 109 (2012) 107204. doi:10.1103/PhysRevLett.109.107204 .\n[161] L. Landau, E. Lifshitz, On the Theory of the disperison of magnetic permeability in ferromagnetic bodies, Phys. Z.\nSowjetunion 169 (1935) 14.\n[162] F. Bloch, Nuclear induction, Phys. Rev. 70 (1946) 460. doi:10.1103/PhysRev.70.460 .\n[163] T. Gilbert, A Phenomenological Theory of Damping in Ferromagnetic Materials, IEEE Trans. Magn. 40 (2004) 3443.\ndoi:10.1109/TMAG.2004.836740 .\n[164] M. C. Hickey, J. S. Moodera, Origin of intrinsic Gilbert damping, Phys. Rev. Lett. 102 (2009) 137601. arXiv:0812.3184 ,\ndoi:10.1103/PhysRevLett.102.137601 .\n[165] M. S. Sodha, N. C. Srivastava, Microwave Propagation in Ferrimagnetics, Springer, 1981. doi:10.1007/\n978-1-4757-5839-9 .\n[166] Y. S. Gui, L. Bai, C.-M. Hu, The physics of spin rectification and its application, Sci. China Physics, Mech. Astron. 56\n(2012) 124. doi:10.1007/s11433-012-4956-6 .\n[167] Y. Huo, L. H. Bai, P. Hyde, Y. Z. Wu, C.-M. Hu, Spin rectification for collinear and noncollinear magnetization and\nexternal magnetic field configurations, Phys. Rev. B 91 (2015) 174430. arXiv:1502.06859v1 ,doi:10.1103/PhysRevB.91.\n174430.\n[168] V. Flovik, F. Macià, A. D. Kent, E. Wahlström, Eddy current interactions in a ferromagnet-normal metal bilayer\nstructure, and its impact on ferromagnetic resonance lineshapes, J. Appl. Phys. 117 (2015) 143902. arXiv:1412.1385 ,\ndoi:10.1063/1.4917285 .\n[169] V. Flovik, B. H. Pettersen, E. Wahlström, Eddy-current effects on ferromagnetic resonance: Spin wave excitations and\nmicrowave screening effects, J. Appl. Phys. 119 (2016) 163903. arXiv:1602.07463 ,doi:10.1063/1.4948302 .\n[170] H. J. Juretschke, DC detection of spin resonance in thin metallic films, J. Appl. Phys. 34 (1963) 1223. doi:10.1063/1.\n1729445.\n[171] H. Chen, X. Fan, H. Zhou, W. Wang, Y. S. Gui, C.-M. Hu, D. Xue, Spin rectification enabled by anomalous Hall effect,\nJ. Appl. Phys. 113 (2013) 17C732. doi:10.1063/1.4799248 .\n[172] M. H. Seavey, Galvanomagnetic effects in ferromagnetic resonance, J. Appl. Phys. 31 (1960) 216S. doi:10.1063/1.\n1984668.\n[173] W. Heinz, L. Silber, dc Effects in Ferromagnetic Resonance in Thin Ferrite Films, J. Appl. Phys. 33 (1962) 1306.\ndoi:10.1063/1.1728705 .\n[174] W. M. Moller, H. J. Juretschke, Determination of spin-wave boundary conditions by dc effects in spin-wave resonance,\nPhys. Rev. B 2 (1970) 2651. doi:10.1103/PhysRevB.2.2651 .\n[175] M. Toda, Transport Properties of HgCr 2Se4at Ferromagnetic Resonance, Appl. Phys. Lett. 17 (1970) 1. doi:10.1063/\n1.1653231 .\n[176] J. N. Chazalviel, I. Solomon, Experimental evidence of the anomalous Hall effect in a nonmagnetic semiconductor, Phys.\nRev. Lett. 29 (1972) 1676. doi:10.1103/PhysRevLett.29.1676 .\n[177] J. N. Chazalviel, Spin-dependent Hall effect in semiconductors, Phys. Rev. B 11 (1975) 3918. doi:10.1103/PhysRevB.\n11.3918.\n[178] F. J. Wald, H. J. Juretschke, Extraordinary Hall effect in paramagnetic gadolinium, Phys. Rev. B 3 (1971) 2412. doi:\n10.1103/PhysRevB.3.2412 .\n[179] C. L. Yang, H. T. He, L. Ding, L. J. Cui, Y. P. Zeng, J. N. Wang, W. K. Ge, Spectral Dependence of Spin Photocurrent\nand Current-Induced Spin Polarization in an InGaAs/ InAlAs Two-Dimensional Electron Gas, Phys. Rev. Lett. 96 (2006)\n66186605. doi:10.1103/PhysRevLett.96.186605 .\n[180] Y. S. Gui, A. Wirthmann, C.-M. Hu, Foldover ferromagnetic resonance and damping in permalloy microstrips, Phys.\nRev. B 80 (2009) 184422. doi:10.1103/PhysRevB.80.184422 .\n[181] O. Mosendz, J. E. Pearson, F. Y. Fradin, G. E. W. Bauer, S. D. Bader, A. Hoffmann, Quantifying spin Hall angles\nfrom spin pumping: Experiments and theory, Phys. Rev. Lett. 104 (2010) 046601. arXiv:0911.2725 ,doi:10.1103/\nPhysRevLett.104.046601 .\n[182] X. Zhu, M. Harder, A. Wirthmann, B. Zhang, W. Lu, Y. S. Gui, C.-M. Hu, Near-field microwave phase imaging by a\nspintronic sensor, in: J. Chu, Z. Wang (Eds.), Proc. SPIE, Seventh Int. Conf. Thin Film Phys. Appl., Vol. 7995, 2010,\np. 79951X. doi:10.1117/12.888266 .\n[183] Z. X. Cao, M. Harder, L. Fu, B. Zhang, W. Lu, G. E. Bridges, Y. S. Gui, C. M. Hu, Nondestructive two-dimensional phase\nimaging of embedded defects via on-chip spintronic sensor, Appl. Phys. Lett. 100 (2012) 252406. doi:10.1063/1.4729785 .\n[184] X. F. Zhu, M. Harder, A. Wirthmann, B. Zhang, W. Lu, Y. S. Gui, C.-M. Hu, Dielectric measurements via a phase-\nresolved spintronic technique, Phys. Rev. B 83 (2011) 104407. doi:10.1103/PhysRevB.83.104407 .\n[185] Y. Tserkovnyak, A. Brataas, G. Bauer, B. Halperin, Nonlocal magnetization dynamics in ferromagnetic heterostructures,\nRev. Mod. Phys. 77 (2005) 1375. arXiv:0409242 ,doi:10.1103/RevModPhys.77.1375 .\n[186] A. Brataas, Y. Tserkovnyak, G. E. W. Bauer, P. J. Kelly, Spin Pumping and Spin Transfer, in: S. Maekawa, S. O.\nValenzuela, E. Saitoh, T. Kimura (Eds.), Spin Curr., Oxford University Press, 2012, Ch. 8, pp. 87–135. arXiv:1108.0385 .\n[187] L. Chen, F. Matsukura, H. Ohno, Direct-current voltages in (Ga,Mn)As structures induced by ferromagnetic resonance.,\nNat. Commun. 4 (2013) 2055. doi:10.1038/ncomms3055 .\n[188] L. Bai, P. Hyde, Y. S. Gui, C.-M. Hu, V. Vlaminck, J. E. Pearson, S. D. Bader, A. Hoffmann, Universal Method for\nSeparating Spin Pumping from Spin Rectification Voltage of Ferromagnetic Resonance, Phys. Rev. Lett. 111 (2013)\n217602. arXiv:1308.4041 ,doi:10.1103/PhysRevLett.111.217602 .\n[189] Y. Tserkovnyak, A. Brataas, G. Bauer, Enhanced Gilbert Damping in Thin Ferromagnetic Films, Phys. Rev. Lett. 88\n(2002) 117601. arXiv:cond-mat/0110247 ,doi:10.1103/PhysRevLett.88.117601 .\n[190] A. Brataas, Y. Tserkovnyak, G. E. W. Bauer, B. I. Halperin, Spin battery operated by ferromagnetic resonance, Phys.\nRev. B. 66 (2002) 060404(R). arXiv:cond-mat/0205028 ,doi:10.1103/PhysRevB.66.060404 .\n[191] A. Janossy, P. Monod, Spin waves for single electrons in paramagnetic metals, Phys. Rev. Lett. 37 (1976) 612. doi:\n10.1103/PhysRevLett.37.612 .\n[192] R. H. Silsbee, A. Janossy, P. Monod, Coupling between ferromagnetic and conduction-spin-resonance modes at a\nferromagnetic-normal-metal interface, Phys. Rev. B 19 (1979) 4382. doi:10.1103/PhysRevB.19.4382 .\n[193] L. Berger, Generation of dc voltages by a magnetic multilayer undergoing ferromagnetic resonance, Phys. Rev. B 59\n(1999) 11465. doi:10.1103/PhysRevB.59.11465 .\n[194] M.Weiler, G.Woltersdorf, M.Althammer, H.Huebl, S.T.B.Goennenwein, SpinPumpingandSpinCurrentsinMagnetic\nInsulators, Solid State Phys. 64 (2013) 123–156. doi:10.1016/B978-0-12-408130-7.00005-8 .\n[195] S. Mizukami, Y. Ando, T. Miyazaki, Ferromagnetic resonance linewidth for NM/80NiFe/NM films (NM=Cu, Ta, Pd and\nPt), J. Magn. Magn. Mater. 226 (2001) 1640. doi:10.1016/S0304-8853(00)01097-0 .\n[196] S. Mizukami, Y. Ando, T. Miyazaki, Effect of spin diffusion on Gilbert damping for a very thin permalloy layer in\nCu/permalloy/Cu/Pt films, Phys. Rev. B 66 (2002) 104413. doi:10.1103/PhysRevB.66.104413 .\n[197] R. Urban, G. Woltersdorf, B. Heinrich, Gilbert damping in single and multilayer ultrathin films: role of interfaces in\nnonlocal spin dynamics, Phys. Rev. Lett. 87 (2001) 217204. doi:10.1103/PhysRevLett.87.217204 .\n[198] B. Heinrich, Y. Tserkovnyak, G. Woltersdorf, A. Brataas, R. Urban, G. E. W. Bauer, Dynamic exchange coupling in\nmagnetic bilayers, Phys. Rev. Lett. 90 (2003) 187601. arXiv:cond-mat/0210588 ,doi:10.1103/PhysRevLett.90.187601 .\n[199] K. Lenz, T. Toliński, J. Lindner, E. Kosubek, K. Baberschke, Evidence of spin-pumping effect in the ferromagnetic\nresonance of coupled trilayers, Phys. Rev. B 69 (2004) 144422. doi:10.1103/PhysRevB.69.144422 .\n[200] Y. K. Kato, R. C. Myers, A. C. Gossard, D. D. Awschalom, Observation of the spin Hall effect in semiconductors., Science\n306 (2004) 1910. doi:10.1126/science.1105514 .\n[201] J. Wunderlich, B. Kaestner, J. Sinova, T. Jungwirth, Experimental Observation of the Spin-Hall Effect in a Two-\nDimensional Spin-Orbit Coupled Semiconductor System, Phys. Rev. Lett. 94 (2005) 047204. arXiv:cond-mat/0410295 ,\ndoi:10.1103/PhysRevLett.94.047204 .\n[202] X. Wang, G. E. W. Bauer, B. J. Van Wees, A. Brataas, Y. Tserkovnyak, Voltage generation by ferromagnetic resonance\nat a nonmagnet to ferromagnet contact, Phys. Rev. Lett. 97 (2006) 216602. doi:10.1103/PhysRevLett.97.216602 .\n[203] M. Johnson, R. H. Silsbee, Interfacial charge-spin coupling: Injection and detection of spin magnetization in metals,\nPhys. Rev. Lett. 55 (1985) 1790. doi:10.1103/PhysRevLett.55.1790 .\n[204] L. Liu, R. A. Buhrman, D. C. Ralph, Review and Analysis of Measurements of the Spin Hall Effect in Platinum,\narXiv:1111.3702 .\n[205] M. Obstbaum, M. Härtinger, H. G. Bauer, T. Meier, F. Swientek, C. H. Back, G. Woltersdorf, Inverse spin Hall effect in\nNi81Fe19/normal-metal bilayers, Phys. Rev. B 89 (2014) 060407(R). doi:10.1103/PhysRevB.89.060407 .\n[206] A. Azevedo, L. H. Vilela Leão, R. L. Rodriguez-Suarez, A. B. Oliveira, S. M. Rezende, Dc effect in ferromagnetic\nresonance: Evidence of the spin-pumping effect?, J. Appl. Phys. 97 (2005) 10C715. doi:10.1063/1.1855251 .\n[207] O. Rousseau, M. Viret, Interaction between ferromagnetic resonance and spin currents in nanostructures, Phys. Rev. B\n85 (2012) 144413. doi:10.1103/PhysRevB.85.144413 .\n[208] X. Zhang, C. Zou, L. Jiang, H. X. Tang, Superstrong coupling of thin film magnetostatic waves with microwave cavity,\nJ. Appl. Phys. 119 (2016) 023905. doi:10.1063/1.4939134 .\n[209] C. Gould, C. Rüster, T. Jungwirth, E. Girgis, G. M. Schott, R. Giraud, K. Brunner, G. Schmidt, L. W. Molenkamp,\nTunneling Anisotropic Magnetoresistance: A Spin-Valve-Like Tunnel Magnetoresistance Using a Single Magnetic Layer,\n67Phys. Rev. Lett. 93 (2004) 117203. arXiv:cond-mat/0407735 ,doi:10.1103/PhysRevLett.93.117203 .\n[210] A. Matos-Abiague, J. Fabian, Anisotropic tunneling magnetoresistance and tunneling anisotropic magnetoresistance:\nSpin-orbit coupling in magnetic tunnel junctions, Phys. Rev. B 79 (2009) 155303. arXiv:0807.3564 ,doi:10.1103/\nPhysRevB.79.155303 .\n[211] C. Liu, Y. Boyko, C. C. Geppert, K. D. Christie, G. Stecklein, S. J. Patel, C. J. Palmstrøm, P. A. Crowell, Electrical\ndetectionofferromagneticresonanceinferromagnet/n-GaAsheterostructuresbytunnelinganisotropicmagnetoresistance,\nAppl. Phys. Lett. 105 (2014) 212401. arXiv:1408.5842v1 ,doi:10.1063/1.4902088 .\n[212] P. Gambardella, I. M. Miron, Current-induced spin-orbit torques, Philos. Trans. R. Soc. A 369 (2011) 3175. doi:\n10.1098/rsta.2010.0336 .\n[213] J. Kim, J. Sinha, M. Hayashi, M. Yamanouchi, S. Fukami, T. Suzuki, S. Mitani, H. Ohno, Layer thickness dependence\nof the current-induced effective field vector in Ta|CoFeB|MgO., Nat. Mater. 12 (2013) 240. arXiv:1207.2521 ,doi:\n10.1038/nmat3522 .\n[214] X. Fan, J. Wu, Y. Chen, M. J. Jerry, H. Zhang, J. Q. Xiao, Observation of the nonlocal spin-orbital effective field, Nat.\nCommun. 4 (2013) 1799. doi:10.1038/ncomms2709 .\n[215] K. Garello, I. M. Miron, C. O. Avci, F. Freimuth, Y. Mokrousov, S. Blügel, S. Auffret, O. Boulle, G. Gaudin, P. Gam-\nbardella, Symmetry and magnitude of spin-orbit torques in ferromagnetic heterostructures., Nat. Nanotechnol. 8 (2013)\n587. arXiv:1301.3573 ,doi:10.1038/nnano.2013.145 .\n[216] M. Jamali, K. Narayanapillai, X. Qiu, L. M. Loong, A. Manchon, H. Yang, Spin-Orbit Torques in Co/Pd Multilayer\nNanowires, Phys. Rev. Lett. 111 (2013) 246602. arXiv:1311.3033 ,doi:10.1103/PhysRevLett.111.246602 .\n[217] X. Fan, W. Wang, Y. Wang, H. Zhou, J. Rao, X. Zhao, C. Gao, Y. S. Gui, C.-M. Hu, D. Xue, Rapid characterizing of\nferromagnetic materials using spin rectification, Appl. Phys. Lett. 105 (2014) 262404. doi:10.1063/1.4905358 .\n[218] T. D. Skinner, M. Wang, A. T. Hindmarch, A. W. Rushforth, A. C. Irvine, D. Heiss, H. Kurebayashi, A. J. Ferguson,\nSpin-orbit torque opposing the Oersted torque in ultrathin Co/Pt bilayers, Appl. Phys. Lett. 104 (2014) 062401. arXiv:\n1312.2409 ,doi:10.1063/1.4864399 .\n[219] A. R. Mellnik, J. S. Lee, A. Richardella, J. L. Grab, P. J. Mintun, M. H. Fischer, A. Vaezi, A. Manchon, E.-A. Kim,\nN. Samarth, D. C. Ralph, Spin-transfer torque generated by a topological insulator, Nature 511 (2014) 449. arXiv:\n1402.1124 ,doi:10.1038/nature13534 .\n[220] C. Ciccarelli, K. M. D. Hals, A. Irvine, V. Novak, Y. Tserkovnyak, H. Kurebayashi, A. Brataas, A. Ferguson, Magnonic\nchargepumpingviaspin-orbitcoupling,Nat.Nanotechnol.10(2014)50. arXiv:1411.2779 ,doi:10.1038/nnano.2014.252 .\n[221] T. D. Skinner, K. Olejník, L. K. Cunningham, H. Kurebayashi, R. P. Campion, B. L. Gallagher, T. Jungwirth, A. J.\nFerguson, Complementary spin-Hall and inverse spin-galvanic effect torques in a ferromagnet/semiconductor bilayer, Nat.\nCommun. 6 (2015) 6730. arXiv:1502.0287 ,doi:10.1038/ncomms7730 .\n[222] Y. Wang, P. Deorani, K. Banerjee, N. Koirala, M. Brahlek, S. Oh, H. Yang, Topological Surface States Originated Spin-\nOrbit Torques in Bi 2Se3, Phys. Rev. Lett. 114 (2015) 257202. arXiv:1505.07937 ,doi:10.1103/PhysRevLett.114.257202 .\n[223] C.-F. Pai, Y. Ou, L. H. Vilela-Leão, D. C. Ralph, R. A. Buhrman, Dependence of the efficiency of spin Hall torque on\nthe transparency of Pt/ferromagnetic layer interfaces, Phys. Rev. B 92 (2015) 064426. arXiv:1411.3379 ,doi:10.1103/\nPhysRevB.92.064426 .\n[224] C. Ciccarelli, L. Anderson, V. Tshitoyan, A. J. Ferguson, F. Gerhard, C. Gould, L. W. Molenkamp, J. Gayles, J. Železný,\nL. Šmejkal, Z. Yuan, J. Sinova, F. Freimuth, T. Jungwirth, Room-temperature spin-orbit torque in NiMnSb, Nat. Phys.\narXiv:1510.03356 ,doi:10.1038/nphys3772 .\n[225] T. Nan, S. Emori, C. T. Boone, X. Wang, T. M. Oxholm, J. G. Jones, B. M. Howe, G. J. Brown, N. X. Sun, Comparison\nof spin-orbit torques and spin pumping across NiFe/Pt and NiFe/Cu/Pt interfaces, Phys. Rev. B 91 (2015) 214416.\narXiv:1503.04104 ,doi:10.1103/PhysRevB.91.214416 .\n[226] L. Berger, Emission of spin waves by a magnetic multilayer traversed by a current, Phys. Rev. B 54 (1996) 9353.\ndoi:10.1103/PhysRevB.54.9353 .\n[227] J. Slonczewski, Current-driven excitation of magnetic multilayers, J. Magn. Magn. Mater. 159 (1996) L1. doi:10.1016/\n0304-8853(96)00062-5 .\n[228] M. Tsoi, A. Jansen, J. Bass, W.-C. Chiang, M. Seck, V. Tsoi, P. Wyder, Excitation of a Magnetic Multilayer by an\nElectric Current, Phys. Rev. Lett. 80 (1998) 4281. doi:10.1103/PhysRevLett.80.4281 .\n[229] E. B. Myers, Current-Induced Switching of Domains in Magnetic Multilayer Devices, Science 285 (1999) 867. doi:\n10.1126/science.285.5429.867 .\n[230] D. C. Ralph, M. D. Stiles, Spin transfer torques, J. Magn. Magn. Mater. 320 (2008) 1190. arXiv:0711.4608 ,doi:\n10.1016/j.jmmm.2007.12.019 .\n[231] S. Zhang, P. M. Levy, A. Fert, Mechanisms of spin-polarized current-driven magnetization switching, Phys. Rev. Lett.\n88 (2002) 236601. arXiv:cond-mat/0202363 ,doi:10.1103/PhysRevLett.88.236601 .\n[232] X. G. Zhang, W. H. Butler, Large magnetoresistance in bcc Co/MgO/Co and FeCo/MgO/FeCo tunnel junctions, Phys.\nRev. B 70 (2004) 172407. arXiv:cond-mat/0409155 ,doi:10.1103/PhysRevB.70.172407 .\n[233] Z. Li, S. Zhang, Domain-wall dynamics driven by adiabatic spin-transfer torques, Phys. Rev. B 70 (2004) 024417.\narXiv:cond-mat/0407064 ,doi:10.1103/PhysRevB.70.024417 .\n[234] Z. Li, S. Zhang, Domain-wall dynamics and spin-wave excitations with spin-transfer torques, Phys. Rev. Lett. 92 (2004)\n207203. doi:10.1103/PhysRevLett.92.207203 .\n[235] J. He, Z. Li, S. Zhang, Current-driven vortex domain wall dynamics by micromagnetic simulations, Phys. Rev. B 73\n(2006) 184408. arXiv:cond-mat/0605125 ,doi:10.1103/PhysRevB.73.184408 .\n[236] T. Chiba, G. E. W. Bauer, S. Takahashi, Current-Induced Spin-Torque Resonance of Magnetic Insulators, Phys. Rev.\nAppl. 2 (2014) 034003. arXiv:1404.2360 ,doi:10.1103/PhysRevApplied.2.034003 .\n68[237] T. Chiba, M. Schreier, G. E. W. Bauer, S. Takahashi, Current-induced spin torque resonance of magnetic insulators\naffected by field-like spin-orbit torques and out-of-plane magnetizations, J. Appl. Phys. 117 (2015) 17C715. arXiv:\n1409.8406 ,doi:10.1063/1.4913632 .\n[238] R. Iguchi, K. Sato, D. Hirobe, S. Daimon, E. Saitoh, Effect of spin Hall magnetoresistance on spin pumping measurements\nin insulating magnet/metal systems, Appl. Phys. Express 7 (2014) 013003. doi:10.7567/APEX.7.013003 .\n[239] J. Rao, X. Fan, L. Ma, H. Zhou, X. Zhao, J. Zhao, F. Zhang, S. Zhou, D. Xue, Observation of spin rectification in\nPt/yttrium iron garnet bilayer, J. Appl. Phys. 117 (2015) 17C725. doi:10.1063/1.4914962 .\n[240] M. Schreier, T. Chiba, A. Niedermayr, J. Lotze, H. Huebl, S. Geprägs, S. Takahashi, G. E. W. Bauer, R. Gross,\nS. T. B. Goennenwein, Current-induced spin torque resonance of a magnetic insulator, Phys. Rev. B 92 (2015) 144411.\narXiv:1412.7460 ,doi:10.1103/PhysRevB.92.144411 .\n[241] J. Sklenar, W. Zhang, M. B. Jungfleisch, W. Jiang, H. Chang, J. E. Pearson, M. Wu, J. B. Ketterson, A. Hoffmann,\nDriving and detecting ferromagnetic resonance in insulators with the spin Hall effect, Phys. Rev. B 92 (2015) 174406.\narXiv:1505.07791v1 ,doi:10.1103/PhysRevB.92.174406 .\n[242] M. B. Jungfleisch, W. Zhang, J. Sklenar, J. Ding, W. Jiang, H. Chang, F. Y. Fradin, J. E. Pearson, J. B. Ketterson,\nV. Novosad, M. Wu, A. Hoffmann, Large Spin-Wave Bullet in a Ferrimagnetic Insulator Driven by the Spin Hall Effect,\nPhys. Rev. Lett. 116 (2016) 057601. arXiv:1508.0142 ,doi:10.1103/PhysRevLett.116.057601 .\n[243] T. Moriyama, R. Cao, X. Fan, G. Xuan, B. K. Nikolić, Y. Tserkovnyak, J. Kolodzey, J. Q. Xiao, Tunnel Barrier\nEnhanced Voltage Signal Generated by Magnetization Precession of a Single Ferromagnetic Layer, Phys. Rev. Lett. 100\n(2008) 067602. arXiv:0802.3646 ,doi:10.1103/PhysRevLett.100.067602 .\n[244] S. Miwa, S. Ishibashi, H. Tomita, T. Nozaki, E. Tamura, K. Ando, N. Mizuochi, T. Saruya, H. Kubota, K. Yakushiji,\nT. Taniguchi, H. Imamura, A. Fukushima, S. Yuasa, Y. Suzuki, Highly sensitive nanoscale spin-torque diode., Nat. Mater.\n13 (2014) 50. doi:10.1038/nmat3778 .\n[245] B. Fang, M. Carpentieri, X. Hao, H. Jiang, J. A. Katine, I. N. Krivorotov, B. Ocker, J. Langer, K. L. Wang, B. Zhang,\nB. Azzerboni, P. K. Amiri, G. Finocchio, Z. Zeng, Giant spin-torque diode sensitivity in the absence of bias magnetic\nfield, Nat. Commun. 7 (2016) 11259. arXiv:1410.4958 ,doi:10.1038/ncomms11259 .\n[246] X. Cheng, J. A. Katine, G. E. Rowlands, I. N. Krivorotov, Nonlinear ferromagnetic resonance induced by spin torque in\nnanoscale magnetic tunnel junctions, Appl. Phys. Lett. 103 (2013) 082402. doi:10.1063/1.4819179 .\n[247] J. Kleinlein, B. Ocker, G. Schmidt, Using giant magneto resistance stripes to efficiently generate direct voltage signals\nfrom alternating current excitations, Appl. Phys. Lett. 104 (2014) 153507. doi:10.1063/1.4871383 .\n[248] S. Ziętek, P. Ogrodnik, M. Frankowski, J. Chęciński, P. Wiśniowski, W. Skowroński, J. Wrona, T. Stobiecki, A. Żywczak,\nJ. Barnaś, Rectification of radio-frequency current in a giant-magnetoresistance spin valve, Phys. Rev. B 91 (2015)\n014430. doi:10.1103/PhysRevB.91.014430 .\n[249] S.Zi¸ etek, P.Ogrodnik, W.Skowroński, P.Wiśniowski, M.Czapkiewicz, T.Stobiecki, J.Barnaś, Theinfluenceofinterlayer\nexchange coupling in giant-magnetoresistive devices on spin diode effect in wide frequency range, Appl. Phys. Lett. 107\n(2015) 122410. doi:10.1063/1.4931771 .\n[250] Z. Diao, Z. Li, S. Wang, Y. Ding, A. Panchula, E. Chen, L.-C. Wang, Y. Huai, Spin-transfer torque switching in\nmagnetic tunnel junctions and spin-transfer torque random access memory, J. Phys. Condens. Matter 19 (2007) 165209.\ndoi:10.1088/0953-8984/19/16/165209 .\n[251] J. N. Kupferschmidt, S. Adam, P. W. Brouwer, Theory of the spin-torque-driven ferromagnetic resonance in a\nferromagnet/normal-metal/ferromagnet structure, Phys. Rev. B 74 (2006) 134416. arXiv:cond-mat/0607145 ,doi:\n10.1103/PhysRevB.74.134416 .\n[252] A. A. Kovalev, G. E. W. Bauer, A. Brataas, Current-driven ferromagnetic resonance, mechanical torques, and rotary\nmotion in magnetic nanostructures, Phys. Rev. B 75 (2007) 014430. arXiv:cond-mat/0609258 ,doi:10.1103/PhysRevB.\n75.014430 .\n[253] O. Boulle, V. Cros, J. Grollier, L. G. Pereira, C. Deranlot, F. Petroff, G. Faini, J. Barnas, A. Fert, Shaped angular\ndependence of the spin transfer torque and microwave generation without magnetic field, Nat. Phys. 3 (2007) 492.\narXiv:0704.0370 ,doi:10.1038/nphys618 .\n[254] J. C. Slonczewski, Currents, torques, and polarization factors in magnetic tunnel junctions, Phys. Rev. B 71 (2005)\n024411. arXiv:cond-mat/0404210 ,doi:10.1103/PhysRevB.71.024411 .\n[255] I. Theodonis, N. Kioussis, A. Kalitsov, M. Chshiev, W. H. Butler, Anomalous bias dependence of spin torque in magnetic\ntunnel junctions, Phys. Rev. Lett. 97 (2006) 237205. arXiv:cond-mat/0606080 ,doi:10.1103/PhysRevLett.97.237205 .\n[256] A. Manchon, S. Zhang, Spin Torque Effects in Magnetic Systems: Theory, in: E. Y. Tsymbal, I. Žutić (Eds.), Handb.\nSpin Transp. Magn., CRC Press, Boca Raton, 2012, pp. 157–178. doi:10.1201/b11086-11 .\n[257] A. M. Gonçalves, I. Barsukov, Y.-J. Chen, L. Yang, J. A. Katine, I. N. Krivorotov, Spin torque ferromagnetic resonance\nwith magnetic field modulation, Appl. Phys. Lett. 103 (2013) 172406. arXiv:1310.7996 ,doi:10.1063/1.4826927 .\n[258] C.-G. Duan, J. P. Velev, R. F. Sabirianov, Z. Zhu, J. Chu, S. S. Jaswal, E. Y. Tsymbal, Surface Magnetoelectric Effect\nin Ferromagnetic Metal Films, Phys. Rev. Lett. 101 (2008) 137201. arXiv:0808.0841 ,doi:10.1103/PhysRevLett.101.\n137201.\n[259] K. Nakamura, R. Shimabukuro, Y. Fujiwara, T. Akiyama, T. Ito, A. J. Freeman, Giant Modification of the Magnetocrys-\ntalline Anisotropy in Transition-Metal Monolayers by an External Electric Field, Phys. Rev. Lett. 102 (2009) 187201.\ndoi:10.1103/PhysRevLett.102.187201 .\n[260] M. Tsujikawa, T. Oda, Finite Electric Field Effects in the Large Perpendicular Magnetic Anisotropy Surface\nPt/Fe/Pt(001): A First-Principles Study, Phys. Rev. Lett. 102 (2009) 247203. arXiv:0905.0060 ,doi:10.1103/\nPhysRevLett.102.247203 .\n[261] W.-G. Wang, M. Li, S. Hageman, C. L. Chien, Electric-field-assisted switching in magnetic tunnel junctions, Nat. Mater.\n6911 (2011) 64. doi:10.1038/nmat3171 .\n[262] Y. Shiota, T. Nozaki, F. Bonell, S. Murakami, T. Shinjo, Y. Suzuki, Induction of coherent magnetization switching in a\nfew atomic layers of FeCo using voltage pulses, Nat. Mater. 11 (2011) 39. doi:10.1038/nmat3172 .\n[263] T. Nozaki, Y. Shiota, S. Miwa, S. Murakami, F. Bonell, S. Ishibashi, H. Kubota, K. Yakushiji, T. Saruya, A. Fukushima,\nS. Yuasa, T. Shinjo, Y. Suzuki, Electric-field-induced ferromagnetic resonance excitation in an ultrathin ferromagnetic\nmetal layer, Nat. Phys. 8 (2012) 492. doi:10.1038/nphys2298 .\n[264] J. Zhu, J. A. Katine, G. E. Rowlands, Y.-J. Chen, Z. Duan, J. G. Alzate, P. Upadhyaya, J. Langer, P. K. Amiri, K. L.\nWang, I. N. Krivorotov, Voltage-Induced Ferromagnetic Resonance in Magnetic Tunnel Junctions, Phys. Rev. Lett. 108\n(2012) 197203. arXiv:1205.2835 ,doi:10.1103/PhysRevLett.108.197203 .\n[265] Y. Shiota, S. Miwa, S. Tamaru, T. Nozaki, H. Kubota, A. Fukushima, Y. Suzuki, S. Yuasa, High-output microwave\ndetector using voltage-induced ferromagnetic resonance, Appl. Phys. Lett. 105 (2014) 192408. doi:10.1063/1.4902025 .\n[266] W. Skowroński, T. Nozaki, Y. Shiota, S. Tamaru, K. Yakushiji, H. Kubota, A. Fukushima, S. Yuasa, Y. Suzuki, Per-\npendicular magnetic anisotropy of Ir/CoFeB/MgO trilayer system tuned by electric fields, Appl. Phys. Express 8 (2015)\n053003. doi:10.7567/APEX.8.053003 .\n[267] B. Lenk, H. Ulrichs, F. Garbs, M. Münzenberg, The building blocks of magnonics, Phys. Rep. 507 (2011) 107. arXiv:\n1101.0479 ,doi:10.1016/j.physrep.2011.06.003 .\n[268] C. Herring, C. Kittel, On the theory of spin waves in ferromagnetic media, Phys. Rev. 81 (1951) 869. doi:10.1103/\nPhysRev.81.869 .\n[269] D. S. Schmool, H. Kachkachi, Single-Particle Phenomena in Magnetic Nanostructures, Elsevier, 2015. doi:10.1016/bs.\nssp.2015.06.001 .\n[270] L. R. Walker, Magnetostatic modes in ferromagnetic resonance, Phys. Rev. 105 (1957) 390. doi:10.1103/PhysRev.105.\n390.\n[271] M. Sparks, Ferromagnetic resonance in thin films. I. theory of normal-mode frequencies, Phys. Rev. B 1 (1970) 3831.\ndoi:10.1103/PhysRevB.1.3831 .\n[272] M. Sparks, Effect of exchange on magnetostatic modes, Phys. Rev. Lett. 24 (1970) 1178. doi:10.1103/PhysRevLett.24.\n1178.\n[273] B. A. Kalinkos, A. N. Slavin, Theory of dipole-exchange spin wave spectrum for ferromagnetic films with mixed exchange\nboundary conditions, J. Phys. C Solid State Phys. 19 (1986) 7013. doi:10.1088/0022-3719/19/35/014 .\n[274] S. O. Demokritov, Spin Wave Confinement, Pan Stanford Publishing, 2009. doi:10.1142/9789814241205 .\n[275] D. D. Stancil, A. Prabhakar, Spin Waves: Theory and Applications, Springer, 2009. doi:10.1007/978-0-387-77865-5 .\n[276] T. J. Silva, C. S. Lee, T. M. Crawford, C. T. Rogers, Inductive measurement of ultrafast magnetization dynamics in\nthin-film Permalloy, J. Appl. Phys. 85 (1999) 7849. doi:10.1063/1.370596 .\n[277] M. Covington, T. M. Crawford, G. J. Parker, Time-resolved measurement of propagating spin waves in ferromagnetic\nthin films, Phys. Rev. Lett. 89 (2002) 237202. doi:10.1103/PhysRevLett.89.237202 .\n[278] G. Counil, J. V. Kim, T. Devolder, C. Chappert, K. Shigeto, Y. Otani, Spin wave contributions to the high-frequency\nmagnetic response of thin films obtained with inductive methods, J. Appl. Phys. 95 (2004) 5646. doi:10.1063/1.1697641 .\n[279] R. Bonin, M. L. Schneider, T. J. Silva, J. P. Nibarger, Dependence of magnetization dynamics on magnetostriction in\nNiFe alloys, J. Appl. Phys. 98 (2005) 123904. doi:10.1063/1.2143121 .\n[280] J. P. Nibarger, R. Lopusnik, T. J. Silva, Damping as a function of pulsed field amplitude and bias field in thin film\nPermalloy, Appl. Phys. Lett. 82 (2003) 2112. doi:10.1063/1.1564866 .\n[281] T. Gerrits, M. Schneider, A. Kos, T. Silva, Large-angle magnetization dynamics measured by time-resolved ferromagnetic\nresonance, Phys. Rev. B 73 (2006) 094454. arXiv:cond-mat/0508708 ,doi:10.1103/PhysRevB.73.094454 .\n[282] J. Goulon, A. Rogalev, F. Wilhelm, N. Jaouen, C. Goulon-Ginet, C. Brouder, X-ray detected ferromagnetic resonance in\nthin films, Eur. Phys. J. B 53 (2006) 169. doi:10.1140/epjb/e2006-00367-6 .\n[283] G. Boero, S. Rusponi, J. Kavich, A. L. Rizzini, C. Piamonteze, F. Nolting, C. Tieg, J.-U. Thiele, P. Gambardella,\nLongitudinal detection of ferromagnetic resonance using x-ray transmission measurements, Rev. Sci. Instrum. 80 (2009)\n123902. doi:10.1063/1.3267192 .\n[284] R. Salikhov, R. Abrudan, F. Bruüssing, S. Buschhorn, M. Ewerlin, D. Mishra, F. Radu, I. A. Garifullin, H. Zabel,\nPrecessional dynamics and damping in Co/Cu/Py spin valves, Appl. Phys. Lett. 99 (2011) 092509. doi:10.1063/1.\n3633115.\n[285] H. G. Bauer, P. Majchrak, T. Kachel, C. H. Back, G. Woltersdorf, Nonlinear spin-wave excitations at low magnetic bias\nfields, Nat. Commun. 6 (2015) 8274. doi:10.1038/ncomms9274 .\n[286] J. P. Park, P. Eames, D. M. Engebretson, J. Berezovsky, P. A. Crowell, Spatially resolved dynamics of localized spin-wave\nmodes in ferromagnetic wires, Phys. Rev. Lett. 89 (2002) 277201. arXiv:cond-mat/0207022 ,doi:10.1103/PhysRevLett.\n89.277201 .\n[287] S. Tamaru, J. A. Bain, R. J. M. Van De Veerdonk, T. M. Crawford, M. Covington, M. H. Kryder, Imaging of quantized\nmagnetostatic modes using spatially resolved ferromagnetic resonance, J. Appl. Phys. 91 (2002) 8034. doi:10.1063/1.\n1450815.\n[288] M. J. Pechan, C. Yu, R. L. Compton, J. P. Park, P. A. Crowell, Direct measurement of spatially localized ferromagnetic-\nresonance modes in an antidot lattice (invited), J. Appl. Phys. 97 (2005) 10J903. doi:10.1063/1.1857412 .\n[289] S. Neusser, G. Duerr, H. G. Bauer, S. Tacchi, M. Madami, G. Woltersdorf, G. Gubbiotti, C. H. Back, D. Grundler,\nAnisotropic Propagation and Damping of Spin Waves in a Nanopatterned Antidot Lattice, Phys. Rev. Lett. 105 (2010)\n067208. doi:10.1103/PhysRevLett.105.067208 .\n[290] T.Gerrits, P.Krivosik, M.L.Schneider, C.E.Patton, T.J.Silva, DirectDetectionofNonlinearFerromagneticResonance\nin Thin Films by the Magneto-Optical Kerr Effect, Phys. Rev. Lett. 98 (2007) 207602. doi:10.1103/PhysRevLett.98.\n70207602.\n[291] M. Bailleul, D. Olligs, C. Fermon, S. O. Demokritov, Spin waves propagation and confinement in conducting films at the\nmicrometer scale, Europhys. Lett. 56 (2001) 741. doi:10.1209/epl/i2001-00583-2 .\n[292] I. Neudecker, G. Woltersdorf, B. Heinrich, T. Okuno, G. Gubbiotti, C. H. Back, Comparison of frequency, field, and time\ndomain ferromagnetic resonance methods, J. Magn. Magn. Mater. 307 (2006) 148. doi:10.1016/j.jmmm.2006.03.060 .\n[293] J. Podbielski, F. Giesen, D. Grundler, Spin-Wave Interference in Microscopic Rings, Phys. Rev. Lett. 96 (2006) 167207.\ndoi:10.1103/PhysRevLett.96.167207 .\n[294] Y. T. Zhang, C. E. Patton, G. Srinivasan, G. O. White, C. J. Brower, C. A. Edmondson, Measurement of the spin-wave\ninstability threshold h critin microwave ferrites, Rev. Sci. Instrum. 58 (1987) 620. doi:10.1063/1.1139226 .\n[295] R. G. Cox, C. E. Patton, M. A. Wittenauer, P. Kabos, L. Chen, Spin wave instability in single crystal Zn-Y hexagonal\nferrite at 8.93 GHz, J. Appl. Phys. 89 (2001) 4454. doi:10.1063/1.1352689 .\n[296] S. Y. An, P. Krivosik, M. A. Kraemer, H. M. Olson, A. V. Nazarov, C. E. Patton, High power ferromagnetic resonance\nand spin wave instability processes in Permalloy thin films, J. Appl. Phys. 96 (2004) 1572. doi:10.1063/1.1763996 .\n[297] S. O. Demokritov, B. Hillebrands, A. N. Slavin, Brillouin Light Scattering Studies of Confined Spin Waves: Linear and\nNonlinear Confinement, Phys. Rep. 348 (2001) 441. doi:10.1016/S0370-1573(00)00116-2 .\n[298] J. Jorzick, S. O. Demokritov, B. Hillebrands, M. Bailleul, C. Fermon, K. Y. Guslienko, A. N. Slavin, D. V. Berkov,\nN. L. Gorn, Spin wave wells in nonellipsoidal micrometer size magnetic elements, Phys. Rev. Lett. 88 (2002) 047204.\ndoi:10.1103/PhysRevLett.88.047204 .\n[299] T. Sebastian, K. Schultheiss, B. Obry, B. Hillebrands, H. Schultheiss, Micro-focused Brillouin light scattering: imaging\nspin waves at the nanoscale, Front. Phys. 3 (2015) 1. doi:10.3389/fphy.2015.00035 .\n[300] W. Wettling, W. D. Wilber, P. Kabos, C. E. Patton, Light Scattering from Parallel-Pump Instabilities in Yttrium Iron\nGarnet, Phys. Rev. Lett. 51 (1983) 1680. doi:10.1103/PhysRevLett.51.1680 .\n[301] P. Kabos, C. E. Patton, G. Wiese, A. D. Sullins, E. S. Wright, L. Chen, Butterfly curves and critical modes for second-\norder spin-wave instability processes in yttrium iron garnet films, J. Appl. Phys. 80 (1996) 3962. doi:10.1063/1.363354 .\n[302] E. R. J. Edwards, H. Ulrichs, V. E. Demidov, S. O. Demokritov, S. Urazhdin, Parametric excitation of magnetization\noscillations controlled by pure spin current, Phys. Rev. B 86 (2012) 134420. doi:10.1103/PhysRevB.86.134420 .\n[303] F. Guo, L. M. Belova, R. D. McMichael, Spectroscopy and Imaging of Edge Modes in Permalloy Nanodisks, Phys. Rev.\nLett. 110 (2013) 017601. doi:10.1103/PhysRevLett.110.017601 .\n[304] F. Guo, L. M. Belova, R. D. McMichael, Nonlinear ferromagnetic resonance shift in submicron Permalloy ellipses, Phys.\nRev. B 91 (2015) 064426. doi:10.1103/PhysRevB.91.064426 .\n[305] M. Bailleul, R. Höllinger, K. Perzlmaier, C. Fermon, Microwave spectrum of square permalloy dots: Multidomain state,\nPhys. Rev. B 76 (2007) 224401. doi:10.1103/PhysRevB.76.224401 .\n[306] A. Yamaguchi, K. Motoi, H. Miyajima, Y. Nakatani, Magnetic field dependence of rectification radio frequency current\nflowing through a single layered ferromagnetic wire, J. Appl. Phys. 105 (2009) 07D301. doi:10.1063/1.3056143 .\n[307] A.Yamaguchi, K.Motoi, H.Miyajima, T.Uchiyama, Y.Utsumi, High-orderstandingspinwavemodesinFe 19Ni81micron\nwire observed by homodyne method, J. Phys. Conf. Ser. 266 (2011) 012113. doi:10.1088/1742-6596/266/1/012113 .\n[308] Z. Duan, I. N. Krivorotov, R. E. Arias, N. Reckers, S. Stienen, J. Lindner, Spin wave eigenmodes in transversely\nmagnetized thin film ferromagnetic wires, Phys. Rev. B 92 (2015) 104424. doi:10.1103/PhysRevB.92.104424 .\n[309] A. Ercole, A. Adeyeye, J. Bland, D. Hasko, Size-dependent spin-wave frequency in ferromagnetic wire-array structures,\nPhys. Rev. B 58 (1998) 345. doi:10.1103/PhysRevB.58.345 .\n[310] C. Bayer, J. Jorzick, S. O. Demokritov, A. N. Slavin, K. Y. Guslienko, D. V. Berkov, N. L. Gorn, M. P. Kostylev,\nB. Hillebrands, Spin-wave excitations in finite rectangular elements, Top. Appl. Phys. 101 (2006) 57. doi:10.1007/\n10938171_2 .\n[311] C. Kittel, Excitation of Spin Waves in a Ferromagnet by a Uniform rf Field, Phys. Rev. 110 (1958) 1213. doi:10.1103/\nPhysRev.110.1295 .\n[312] Y. Zhai, J. Shi, X. Y. Zhang, L. Shi, Y. X. Xu, H. B. Huang, Z. H. Lu, H. R. Zhai, A study of the non-uniform effect\non the shape anisotropy in patterned NiFe films of ferromagnetic resonance, J. Phys. Condens. Matter 14 (2002) 7865.\ndoi:10.1088/0953-8984/14/34/306 .\n[313] R. L. White, I. H. Solt, Multiple ferromagnetic resonance in ferrite spheres, Phys. Rev. 104 (1956) 56. doi:10.1103/\nPhysRev.104.56 .\n[314] J. F. Dillon, Magnetostatic modes in ferrimagnetic spheres, Phys. Rev. 112 (1958) 59. doi:10.1103/PhysRev.112.59 .\n[315] J. Jorzick, S. Demokritov, C. Mathieu, B. Hillebrands, B. Bartenlian, C. Chappert, F. Rousseaux, A. Slavin, Brillouin\nlight scattering from quantized spin waves in micron-size magnetic wires, Phys. Rev. B 60 (1999) 15194. doi:10.1103/\nPhysRevB.60.15194 .\n[316] Z. K. Wang, M. H. Kuok, S. C. Ng, D. J. Lockwood, M. G. Cottam, K. Nielsch, R. B. Wehrspohn, U. Gösele, Spin-\nwave quantization in ferromagnetic nickel nanowires., Phys. Rev. Lett. 89 (2002) 027201. doi:10.1103/PhysRevLett.89.\n027201.\n[317] G. Rado, J. Weertman, Spin-wave resonance in a ferromagnetic metal, J. Phys. Chem. Solids 11 (1959) 315. doi:\n10.1016/0022-3697(59)90233-1 .\n[318] P. Pincus, Excitation of spin waves in ferromagnets: eddy current and boundary condition effects, Phys. Rev. 118 (1960)\n658. doi:10.1103/PhysRev.118.658 .\n[319] R. F. Soohoo, Excitation and Boundary Effects in Spin-Wave Resonance, J. Appl. Phys. 32 (1961) 1959. doi:10.1063/\n1.2000381 .\n[320] R. F. Soohoo, General exchange boundary condition and surface anisotropy energy of a ferromagnet, Phys. Rev. 131\n(1963) 594. doi:10.1103/PhysRev.131.594 .\n71[321] L. D. Landau, E. M. Lifshitz, Mechanics, 3rd Edition, Elsevier, 1976.\n[322] P. W. Anderson, H. Suhl, Instability in the motion of ferromagnets at high microwave power levels, Phys. Rev. 100 (1955)\n1788. doi:10.1103/PhysRev.100.1788 .\n[323] G. Bertotti, I. D. Mayergoyz, C. Serpico, Spin-wave instabilities in large-scale nonlinear magnetization dynamics., Phys.\nRev. Lett. 87 (2001) 217203. doi:10.1103/PhysRevLett.87.217203 .\n[324] H. Suhl, “Foldover\" effects caused by spin wave interactions in ferromagnetic resonance, J. Appl. Phys. 31 (1960) 935.\ndoi:10.1063/1.1735723 .\n[325] P. Gottlieb, Nonlinear effects of crystalline anisotropy on ferrimagnetic resonance, J. Appl. Phys. 31 (1960) 2059. doi:\n10.1063/1.1735497 .\n[326] J. I. Masters, Instability of magnetic resonance in single crystal spheres of yttrium iron garnet, J. Appl. Phys. 31 (1960)\n41S. doi:10.1063/1.1984596 .\n[327] I. B. Goldberg, Low-power nonlinear effects in the ferromagnetic resonance of spheres of Ba 3Zn2Fe24O41, J. Appl. Phys.\n51 (1980) 1857. doi:10.1063/1.327760 .\n[328] L. M. Silber, C. E. Patton, H. F. Naqvi, Low power nonlinear effects in the ferromagnetic resonance of Zn 2Y and MnZnY\nhexagonal ferrites, J. Appl. Phys. 54 (1983) 4071. doi:10.1063/1.332537 .\n[329] D. J. Seagle, S. H. Charap, J. O. Artman, Foldover in YIG, J. Appl. Phys. 57 (1985) 3706. doi:10.1063/1.334997 .\n[330] K. D. McKinstry, C. E. Patton, M. Kogekar, Low power nonlinear effects in the ferromagnetic resonance of yttrium iron\ngarnet, J. Appl. Phys. 58 (1985) 925. doi:10.1063/1.336167 .\n[331] Y. Zhang, C. Patton, M. Kogekar, Ferromagentic resonance foldover in single crystal YIG films–Sample heating or Suhl\ninstability, IEEE Trans. Magn. 22 (1986) 993. doi:10.1109/TMAG.1986.1064446 .\n[332] K. Gnatzig, H. Dötsch, M. Ye, A. Brockmeyer, Ferrimagnetic resonance in garnet films at large precession angles, J.\nAppl. Phys. 62 (1987) 4839. doi:10.1063/1.338988 .\n[333] Y. T. Zhang, C. E. Patton, G. Srinivasan, The second-order spin-wave instability threshold in single-crystal yttrium-\niron-garnet films under perpendicular pumping, J. Appl. Phys. 63 (1988) 5433. doi:10.1063/1.340364 .\n[334] M. Chen, C. E. Patton, G. Srinivasan, Y. T. Zhang, Ferromagnetic resonance foldover and spin-wave instability in\nsingle-crystal YIG films, IEEE Trans. Magn. 25 (1989) 3485. doi:10.1109/20.42343 .\n[335] Y. Fetisov, C. Patton, V. Synogach, Nonlinear ferromagnetic resonance and foldover in yttrium iron garnet thin films-\ninadequacy of the classical model, IEEE Trans. Magn. 35 (1999) 4511. doi:10.1109/20.809144 .\n[336] Y. K. Fetisov, C. E. Patton, Thermal microwave foldover and bistability in ferromagnetic resonance, IEEE Trans. Magn.\n40 (2004) 473. doi:10.1109/TMAG.2004.824132 .\n[337] H. Suhl, The theory of ferromagnetic resonance at high signal powers, J. Phys. Chem. Solids 1 (1957) 209. doi:\n10.1016/0022-3697(57)90010-0 .\n[338] C. Bi, X. Fan, L. Pan, X. Kou, J. Wu, Q. Yang, H. Zhang, J. Q. Xiao, Electrical detection of nonlinear ferromagnetic\nresonance in single elliptical permalloy thin film using a magnetic tunnel junction, Appl. Phys. Lett. 99 (2011) 232506.\ndoi:10.1063/1.3670302 .\n[339] E. Hirayama, S. Kanai, J. Ohe, H. Sato, F. Matsukura, H. Ohno, Electric-field induced nonlinear ferromagnetic resonance\nin a CoFeB/MgO magnetic tunnel junction, Appl. Phys. Lett. 107 (2015) 132404. doi:10.1063/1.4932092 .\n[340] H. Zhou, X. Fan, L. Ma, L. Cui, C. Jia, S. Zhou, Y. S. Gui, C.-M. Hu, D. Xue, Spin pumping in the nonlinear dynamic\nregime of a Pt/Y 3Fe5O12heterostructure, Appl. Phys. Lett. 108 (2016) 192408. doi:10.1063/1.4949555 .\n[341] M. W. Wu, A. Hoffmann (Eds.), Recent Advances in Magnetic Insulators - From Spintronics to Microwave Applications,\nVol. 64, Elsevier, 2013.\n[342] H. Jiao, G. E. W. Bauer, Spin backflow and ac voltage generation by spin pumping and the inverse spin hall effect, Phys.\nRev. Lett. 110 (2013) 217602. arXiv:1210.0724 ,doi:10.1103/PhysRevLett.110.217602 .\n[343] P. P. Hofer, H. Aramberri, C. Schenke, P. A. L. Delplace, Proposal for an ac spin current source, Europhys. Lett. 107\n(2014) 27003. arXiv:1403.6674 ,doi:10.1209/0295-5075/107/27003 .\n[344] P. Hyde, L. Bai, D. M. J. Kumar, B. W. Southern, C.-M. Hu, S. Y. Huang, B. F. Miao, C. L. Chien, Electrical detection\nof direct and alternating spin current injected from a ferromagnetic insulator into a ferromagnetic metal, Phys. Rev. B\n89 (2014) 180404(R). arXiv:1310.4840 ,doi:10.1103/PhysRevB.89.180404 .\n[345] D. Wei, M. Obstbaum, M. Ribow, C. H. Back, G. Woltersdorf, Spin Hall voltages from a.c. and d.c. spin currents, Nat\nCommun 5 (2014) 3768. doi:10.1038/ncomms4768 .\n[346] M. Weiler, J. M. Shaw, H. T. Nembach, T. J. Silva, Phase-sensitive detection of spin pumping via the ac inverse spin\nHall effect, Phys. Rev. Lett. 113 (2014) 157204. arXiv:1401.6469 ,doi:10.1103/PhysRevLett.113.157204 .\n[347] J. Li, L. R. Shelford, P. Shafer, A. Tan, J. X. Deng, P. S. Keatley, C. Hwang, E. Arenholz, G. van der Laan, R. J. Hicken,\nZ. Q. Qiu, Direct Detection of Pure ac Spin Current by X-Ray Pump-Probe Measurements, Phys. Rev. Lett. 117 (2016)\n076602. doi:10.1103/PhysRevLett.117.076602 .\n[348] W. Pauli, Wolfgang Pauli Wissenschaftlicher Briefwechsel mit Bohr, Einstein, Heisenberg u.a., Vol. 6 of Sources in the\nHistory of Mathematics and Physical Sciences, Springer, 1985. doi:10.1007/978-3-540-78801-0 .\n[349] H. Ohno, H. Munekata, T. Penney, S. Vonmolnar, L. L. Chang, Magnetotransport Properties of P-Type (In,Mn)As\nDiluted Magnetic III-V Semiconductors, Phys. Rev. Lett. 68 (1992) 2664. doi:10.1103/PhysRevLett.68.2664 .\n[350] H. Ohno, Making Nonmagnetic Semiconductors Ferromagnetic, Science 281 (1998) 951. doi:10.1126/science.281.5379.\n951.\n[351] A. H. MacDonald, P. Schiffer, N. Samarth, Ferromagnetic semiconductors: moving beyond (Ga,Mn)As, Nat. Mater. 4\n(2005) 195.\n[352] T. Jungwirth, J. Sinova, J. Mašek, J. Kučera, A. H. MacDonald, Theory of ferromagnetic (III,Mn)V semiconductors,\nRev. Mod. Phys. 78 (2006) 809. arXiv:cond-mat/0603380 ,doi:10.1103/RevModPhys.78.809 .\n72[353] T. Dietl, H. Ohno, Dilute ferromagnetic semiconductors: Physics and spintronic structures, Rev. Mod. Phys. 86 (2014)\n187. arXiv:1307.3429 ,doi:10.1103/RevModPhys.86.187 .\n[354] T. Jungwirth, J. Wunderlich, V. Novák, K. Olejník, B. L. Gallagher, R. P. Campion, K. W. Edmonds, A. W. Rushforth,\nA. J. Ferguson, P. Němec, Spin-dependent phenomena and device concepts explored in (Ga,Mn)As, Rev. Mod. Phys. 86\n(2014) 855. arXiv:1310.1944 ,doi:10.1103/RevModPhys.86.855 .\n[355] D. Fang, H. Kurebayashi, J. Wunderlich, K. Výborný, L. P. Zârbo, R. P. Campion, A. Casiraghi, B. L. Gallagher,\nT. Jungwirth, A. J. Ferguson, Spin-orbit-driven ferromagnetic resonance, Nat. Nanotechnol. 6 (2011) 413. doi:10.1038/\nnnano.2011.68 .\n[356] H. Nakayama, L. Chen, H. W. Chang, H. Ohno, F. Matsukura, Inverse spin Hall effect in Pt/(Ga,Mn)As, Appl. Phys.\nLett. 106 (2015) 222405. doi:10.1063/1.4922197 .\n[357] A. Hubert, R. Schäfer, Magnetic Domains: The Analysis of Magnetic Microstructures, Springer, 2009. doi:10.1007/\n978-3-540-85054-0 .\n[358] R. McMichael, M. Donahue, Head to head domain wall structures in thin magnetic strips, IEEE Trans. Magn. 33 (1997)\n4167. doi:10.1109/20.619698 .\n[359] D. A. Allwood, Magnetic Domain-Wall Logic, Science 309 (2005) 1688. doi:10.1126/science.1108813 .\n[360] S. S. P. Parkin, M. Hayashi, L. Thomas, Magnetic Domain-Wall Racetrack Memory, Science 320 (2008) 190. doi:\n10.1126/science.1145799 .\n[361] G. Malinowski, O. Boulle, M. Kläui, Current-induced domain wall motion in nanoscale ferromagnetic elements, J. Phys.\nD 44 (2011) 384005. doi:10.1088/0022-3727/44/38/384005 .\n[362] O. Boulle, G. Malinowski, M. Kläui, Current-induced domain wall motion in nanoscale ferromagnetic elements, Mater.\nSci. Eng. R Reports 72 (2011) 159. doi:10.1016/j.mser.2011.04.001 .\n[363] C. T. Boone, J. A. Katine, M. Carey, J. R. Childress, X. Cheng, I. N. Krivorotov, Rapid Domain Wall Motion in\nPermalloy Nanowires Excited by a Spin-Polarized Current Applied Perpendicular to the Nanowire, Phys. Rev. Lett. 104\n(2010) 097203. arXiv:0909.1822 ,doi:10.1103/PhysRevLett.104.097203 .\n[364] A. Yamaguchi, H. Miyajima, T. Ono, Y. Suzuki, S. Yuasa, The rectification of radio-frequency signal by magnetic domain\nwall in a single-layered ferromagnetic nanowire, Appl. Phys. Lett. 91 (2007) 132509. doi:10.1063/1.2793620 .\n[365] E. Saitoh, H. Miyajima, T. Yamaoka, G. Tatara, Current-induced resonance and mass determination of a single magnetic\ndomain wall, Nature 432 (2004) 203. doi:10.1038/nature03119.1.\n[366] A. Yamaguchi, T. Ono, S. Nasu, K. Miyake, K. Mibu, T. Shinjo, Real-Space Observation of Current-Driven Domain Wall\nMotion in Submicron Magnetic Wires, Phys. Rev. Lett. 92 (2004) 77201. doi:10.1103/PhysRevLett.92.077205 .\n[367] K. S. Buchanan, P. E. Roy, F. Y. Fradin, K. Y. Guslienko, M. Grimsditch, S. D. Bader, V. Novosad, Vortex dynamics in\npatterned ferromagnetic ellipses, J. Appl. Phys. 1 (2006) 172. arXiv:cond-mat/0602509 ,doi:10.1063/1.2175602 .\n[368] I. Neudecker, M. Kläui, K. Perzlmaier, D. Backes, L. J. Heyderman, C. A. F. Vaz, J. A. C. Bland, U. Rüdiger, C. H.\nBack, Spatially resolved dynamic eigenmode spectrum of Co rings, Phys. Rev. Lett. 96 (2006) 057207. doi:10.1103/\nPhysRevLett.96.057207 .\n[369] B. Van Waeyenberge, A. Puzic, H. Stoll, K. W. Chou, T. Tyliszczak, R. Hertel, M. Fähnle, H. Brückl, K. Rott, G. Reiss,\nI. Neudecker, D. Weiss, C. H. Back, G. Schütz, Magnetic vortex core reversal by excitation with short bursts of an\nalternating field., Nature 444 (2006) 461. doi:10.1038/nature05240 .\n[370] D. Bedau, M. Kläui, M. T. Hua, S. Krzyk, U. Rüdiger, G. Faini, L. Vila, Quantitative Determination of the Nonlinear\nPinning Potential for a Magnetic Domain Wall, Phys. Rev. Lett. 101 (2008) 256602. doi:10.1103/PhysRevLett.101.\n256602.\n[371] T. Ono, H. Miyajimia, K. Shigeto, K. Mibu, N. Hosoito, T. Shinjo, Propagation of a Magnetic Domain Wall in a\nSubmicrometer Magnetic Wire, Science 284 (1999) 468. doi:10.1126/science.284.5413.468 .\n[372] T. Ono, Y. Ooka, S. Kasai, H. Miyajima, N. Nakatani, N. Hayashi, K. Shigeto, K. Mibu, T. Shinjo, Magnetization reversal\nand electric transport in ferromagnetic nanowires, Mater. Sci. Eng. B 84 (2001) 126. doi:10.1016/S0921-5107(01)\n00585-2.\n[373] M. Kläui, C. A. F. Vaz, J. Rothman, J. A. C. Bland, W. Wernsdorfer, G. Faini, E. Cambril, Domain wall pinning in\nnarrow ferromagnetic ring structures probed by magnetoresistance measurements, Phys. Rev. Lett. 90 (2003) 097202.\ndoi:10.1103/PhysRevLett.90.097202 .\n[374] M. Hayashi, L. Thomas, Y. B. Bazaliy, C. Rettner, R. Moriya, X. Jiang, S. S. P. Parkin, Influence of current on field-\ndriven domain wall motion in permalloy nanowires from time resolved measurements of anisotropic magnetoresistance,\nPhys. Rev. Lett. 96 (2006) 197207. doi:10.1103/PhysRevLett.96.197207 .\n[375] A. von Bieren, A. K. Patra, S. Krzyk, J. Rhensius, R. M. Reeve, L. J. Heyderman, R. Hoffmann-Vogel, M. Kläui,\nDomain-wall induced large magnetoresistance effects at zero applied field in ballistic nanocontacts, Phys. Rev. Lett. 110\n(2013) 067203. arXiv:1208.4985v1 ,doi:10.1103/PhysRevLett.110.067203 .\n[376] M. Viret, D. Vignoles, D. Cole, J. Coey, W. Allen, D. Daniel, J. Gregg, Spin scattering in ferromagnetic thin films, Phys.\nRev. B 53 (1996) 8464. doi:10.1103/PhysRevB.53.8464 .\n[377] J. F. Gregg, W. Allen, K. Ounadjela, M. Viret, M. Hehn, S. M. Thompson, J. M. D. Coey, Giant magnetoresistive effects\nin a single element magnetic thin film, Phys. Rev. Lett. 77 (1996) 1580. doi:10.1103/PhysRevLett.77.1580 .\n[378] P. M. Levy, S. Zhang, Resistivity due to Domain Wall Scattering, Phys. Rev. Lett. 79 (1997) 5110. doi:10.1103/\nPhysRevLett.79.5110 .\n[379] U. Ebels, A. Radulescu, Y. Henry, L. Piraux, K. Ounadjela, Spin Accumulation and Domain Wall Magnetoresistance in\n35 nm Co Wires, Phys. Rev. Lett. 84 (2000) 983. doi:10.1103/PhysRevLett.84.983 .\n[380] C. H. Marrows, B. C. Dalton, Spin mixing and spin-current asymmetry measured by domain wall magnetoresistance,\nPhys. Rev. Lett. 92 (2004) 097206. doi:10.1103/PhysRevLett.92.097206 .\n73[381] S. Sangiao, M. Viret, Electrical detection of internal dynamical properties of domain walls, Phys. Rev. B 89 (2014)\n104412. doi:10.1103/PhysRevB.89.104412 .\n[382] X. Fan, Yunpeng Chen, Chong Bi, Yunsong Xie, J. Kolodzey, J. D. Wilson, R. N. Simons, H. Zhang, J. Q. Xiao, Magnetic\ntunnel junction-based on-chip microwave phase and spectrum analyzer, in: 2014 IEEE MTT-S Int. Microw. Symp., Vol. 1,\n2014, pp. 1–4. doi:10.1109/MWSYM.2014.6848560 .\n[383] B. M. Yao, L. Fu, X. S. Chen, W. Lu, L. H. Bai, Y. S. Gui, C.-M. Hu, Rapid microwave phase detection based on a solid\nstate spintronic device, Appl. Phys. Lett. 104 (2014) 062408. arXiv:1309.3572 ,doi:10.1063/1.4865374 .\n[384] Y. Guan, W. E. Bailey, E. Vescovo, C. C. Kao, D. A. Arena, Phase and amplitude of element-specific moment precession\nin Ni 81Fe19, J. Magn. Magn. Mater. 312 (2007) 374. doi:10.1016/j.jmmm.2006.10.1111 .\n[385] T.Martin, G.Woltersdorf, C.Stamm, H.A.Duerr, R.Mattheis, C.H.Back, G.Bayreuther, Layerresolvedmagnetization\ndynamics in interlayer exchange coupled Ni 81Fe19/Ru/Co 90Fe10by time resolved x-ray magnetic circular dichroism, J.\nAppl. Phys. 103 (2008) 07B112. doi:10.1063/1.2836340 .\n[386] D. A. Arena, Y. Ding, E. Vescovo, S. Zohar, Y. Guan, W. E. Bailey, A compact apparatus for studies of element and\nphase-resolved ferromagnetic resonance, Rev. Sci. Instrum. 80 (2009) 083903. arXiv:0905.2353 ,doi:10.1063/1.3190402 .\n[387] W. E. Bailey, C. Cheng, R. Knut, O. Karis, S. Auffret, S. Zohar, D. Keavney, P. Warnicke, J.-S. Lee, D. a. Arena,\nDetection of microwave phase variation in nanometre-scale magnetic heterostructures., Nat. Commun. 4 (2013) 2025.\ndoi:10.1038/ncomms3025 .\n[388] E. C. Fear, S. C. Hagness, P. M. Meaney, M. Okoniewski, M. A. Stuchly, Enhancing breast tumor detection with near-field\nimaging, Microw. Mag. IEEE 3 (2002) 48. doi:10.1109/6668.990683 .\n[389] E. C. Fear, P. M. Meaney, M. A. Stuchly, Microwaves for breast cancer detection?, Potentials, IEEE 22 (2003) 12.\ndoi:10.1109/MP.2003.1180933 .\n[390] R. K. Amineh, M. Ravan, A. Trehan, N. K. Nikolova, Near-field microwave imaging based on aperture raster scanning\nwith TEM horn antennas, IEEE Trans. Antennas Propag. 59 (2011) 928. doi:10.1109/TAP.2010.2103009 .\n[391] N. Nikolova, Microwave Imaging for Breast Cancer, IEEE Microw. Mag. 12 (2011) 78. doi:10.1109/MMM.2011.942702 .\n[392] E. A. Ash, G. Nicholls, Super-resolution aperture scanning microscope., Nature 237 (1972) 510. doi:10.1038/237510a0 .\n[393] L. Fu, W. Lu, D. Rodriguez Herrera, D. Flores Tapia, Y. S. Gui, S. Pistorius, C.-M. Hu, Microwave radar imaging using\na solid state spintronic microwave sensor, Appl. Phys. Lett. 105 (2014) 122406. doi:10.1063/1.4896691 .\n[394] B. M. Yao, L. Fu, X. S. Chen, W. Lu, H. Guo, Y. S. Gui, C.-M. Hu, Ground penetrating detection using miniaturized\nradar system based on solid state microwave sensor, Rev. Sci. Instrum. 84 (2013) 124702. doi:10.1063/1.4838662 .\n[395] Z. Wang, X. Zhu, W. Lu, Passive detection of polarization and relative phases for L-band microwaves, in: 2013 Int. Conf.\nElectromagn. Adv. Appl., 2013, p. 1386. doi:10.1109/ICEAA.2013.6632478 .\n[396] X. F. Zhu, M. Harder, J. Tayler, A. Wirthmann, B. Zhang, W. Lu, Y. S. Gui, C.-M. Hu, Nonresonant spin rectification\nin the absence of an external applied magnetic field, Phys. Rev. B 83 (2011) 140402. doi:10.1103/PhysRevB.83.140402 .\n[397] Z. H. Zhang, Y. S. Gui, L. Fu, X. L. Fan, J. W. Cao, D. S. Xue, P. P. Freitas, D. Houssameddine, S. Hemour, K. Wu,\nC.-M. Hu, Seebeck rectification enabled by intrinsic thermoelectrical coupling in magnetic tunneling junctions, Phys.\nRev. Lett. 109 (2012) 037206. doi:10.1103/PhysRevLett.109.037206 .\n[398] L. Fu, Z. X. Cao, S. Hemour, K. Wu, D. Houssameddine, W. Lu, S. Pistorius, Y. S. Gui, C.-M. Hu, Microwave reflection\nimaging using a magnetic tunnel junction based spintronic microwave sensor, Appl. Phys. Lett. 101 (2012) 232406.\ndoi:10.1063/1.4769837 .\n[399] Y. Gui, A. M. Mehrabani, D. Flores-Tapia, L. Fu, L. Bai, S. Pistorius, L. Shafai, C.-M. Hu, New horizons for microwave\napplications using spin caloritronics, Solid State Commun. 198 (2014) 45. doi:10.1016/j.ssc.2014.07.026 .\n[400] S.Hemour, K.Wu, Radio-frequencyrectifierforelectromagneticenergyharvesting: Developmentpathandfutureoutlook,\nProc. IEEE 102 (2014) 1667. doi:10.1109/JPROC.2014.2358691 .\n[401] C. P. Wen, C. P. Wen, A Surface Nonreciprocal Applications, IEEE Trans. Microw. Theory Tech. 17 (1969) 1087.\ndoi:10.1109/TMTT.1969.1127105 .\n[402] F. Giesen, J. Podbielski, B. Botters, D. Grundler, Vortex circulation control in large arrays of asymmetric magnetic rings,\nPhys. Rev. B 75 (2007) 184428. doi:10.1103/PhysRevB.75.184428 .\n[403] N. Kuhlmann, A. Vogel, G. Meier, Magnetization dynamics and cone angle precession in permalloy rectangles, Phys.\nRev. B 85 (2012) 014410. doi:10.1103/PhysRevB.85.014410 .\n[404] N. H. G. Grenda, P. A. Hyde, Y. S. Gui, M. P. Wismayer, J. D. A. Jung, C.-M. Hu, B. W. Southern, K.-W. Lin, Angular\ndependence of ferromagnetic resonance measurements in exchange coupled Ni 80Fe20/NiO bilayers, J. Phys. D 46 (2013)\n205002. doi:10.1088/0022-3727/46/20/205002 .\n[405] S. S. Kalarickal, P. Krivosik, M. Wu, C. E. Patton, M. L. Schneider, P. Kabos, T. J. Silva, J. P. Nibarger, Ferromagnetic\nresonance linewidth in metallic thin films: Comparison of measurement methods, J. Appl. Phys. 99 (2006) 093909.\ndoi:10.1063/1.2197087 .\n[406] Ö. O. Soykal, M. E. Flatté, Strong Field Interactions between a Nanomagnet and a Photonic Cavity, Phys. Rev. Lett.\n104 (2010) 077202. arXiv:0907.3926 ,doi:10.1103/PhysRevLett.104.077202 .\n[407] Ö. O. Soykal, M. E. Flatté, Size dependence of strong coupling between nanomagnets and photonic cavities, Phys. Rev.\nB 82 (10) (2010) 104413. arXiv:1005.3068 ,doi:10.1103/PhysRevB.82.104413 .\n[408] H. Huebl, C. W. Zollitsch, J. Lotze, F. Hocke, M. Greifenstein, A. Marx, R. Gross, S. T. B. Goennenwein, High\ncooperativity in coupled microwave resonator ferrimagnetic insulator hybrids, Phys. Rev. Lett. 111 (2013) 127003. arXiv:\n1207.6039 ,doi:10.1103/PhysRevLett.111.127003 .\n[409] Y. Tabuchi, S. Ishino, A. Noguchi, T. Ishikawa, R. Yamazaki, K. Usami, Y. Nakamura, Coherent coupling between\nferromagnetic magnon and superconducting qubit arXiv:1410.3781 .\n[410] X. Zhang, C.-L. Zou, L. Jiang, H. X. Tang, Strongly Coupled Magnons and Cavity Microwave Photons, Phys. Rev. Lett.\n74113 (15) (2014) 156401. arXiv:1405.7062 ,doi:10.1103/PhysRevLett.113.156401 .\n[411] L. Bai, M. Harder, Y. P. Chen, X. Fan, J. Q. Xiao, C.-M. Hu, Spin Pumping in Electrodynamically Coupled Magnon-\nPhoton Systems, Phys. Rev. Lett. 114 (2015) 227201. arXiv:1504.01335 ,doi:10.1103/physrevlett.114.227201 .\n[412] Y. Cao, P. Yan, H. Huebl, S. T. B. Goennenwein, G. E. W. Bauer, Exchange magnon-polaritons in microwave cavities,\nPhys. Rev. B 91 (2015) 094423. arXiv:1412.5809 ,doi:10.1103/PhysRevB.91.094423 .\n[413] B. Zare Rameshti, Y. Cao, G. E. W. Bauer, Magnetic spheres in microwave cavities, Phys. Rev. B 91 (2015) 214430.\narXiv:1503.02419v1 ,doi:10.1103/PhysRevB.91.214430 .\n[414] B. M. Yao, Y. S. Gui, Y. Xiao, H. Guo, X. S. Chen, W. Lu, C. L. Chien, C.-M. Hu Theory and experiment on\ncavity magnon-polariton in the one-dimensional configuration, Phys. Rev. B 92 (2015) 184407. arXiv:1509.05804 ,\ndoi:10.1103/PhysRevB.92.184407 .\n[415] M. Harder, L. Bai, C. Match, J. Sirker, C.-M. Hu, Study of the cavity-magnon-polariton transmission line shape arXiv:\n1601.06049 .\n[416] H. Maier-Flaig, M. Harder, R. Gross, H. Huebl, S. Goennenwein, Cavity spin pumping in different coupling regimes\narXiv:1601.05681 .\n[417] X. Zhang, C.-L. Zou, N. Zhu, F. Marquardt, L. Jiang, H. X. Tang, Magnon dark modes and gradient memory, Nat.\nCommun. 6 (2015) 8914. arXiv:1507.02791v1 ,doi:10.1038/ncomms9914 .\n[418] Y.Tabuchi, S.Ishino, A.Noguchi, T.Ishikawa, R.Yamazaki, K.Usami, Y.Nakamura, Quantummagnonics: Themagnon\nmeets the superconducting qubit, Comptes Rendus Phys. arXiv:1508.05290 ,doi:10.1016/j.crhy.2016.07.009 .\n[419] Y. Tabuchi, S. Ishino, A. Noguchi, T. Ishikawa, R. Yamazaki, K. Usami, Y. Nakamura, Coherent coupling between a\nferromagnetic magnon and a superconducting qubit, Science 349 (2015) 405. arXiv:1410.3781 ,doi:10.1126/science.\naaa3693.\n[420] A. Osada, R. Hisatomi, A. Noguchi, Y. Tabuchi, R. Yamazaki, K. Usami, M. Sadgrove, R. Yalla, M. Nomura, Y. Naka-\nmura, Cavity Optomagnonics with Spin-Orbit Coupled Photons, Phys. Rev. Lett. 116 (2016) 223601. arXiv:1510.01837 ,\ndoi:10.1103/PhysRevLett.116.223601 .\n[421] X. Zhang, N. Zhu, C.-L. Zou, H. X. Tang, Optomagnonic whispering gallery microresonators arXiv:1510.03545 .\n[422] J. A. Haigh, S. Langenfeld, N. J. Lambert, J. J. Baumberg, A. J. Ramsay, A. Nunnenkamp, A. J. Ferguson, Magneto-\noptical coupling in whispering-gallery-mode resonators, Phys. Rev. A 92 (2015) 063845. arXiv:1510.06661 ,doi:10.\n1103/PhysRevA.92.063845 .\n[423] J. Bourhill, N. Kostylev, M. Goryachev, D. L. Creedon, M. E. Tobar, Ultrahigh cooperativity interactions between\nmagnons and resonant photons in a YIG sphere, Phys. Rev. B 93 (2016) 144420. arXiv:1512.07773 ,doi:10.1103/\nPhysRevB.93.144420 .\n75"    },    {        "title": "1006.1202v3.Duality_of_the_spin_and_density_dynamics_for_two_dimensional_electrons_with_a_spin_orbit_coupling.pdf",        "content": "arXiv:1006.1202v3  [cond-mat.mes-hall]  19 Oct 2010Duality of the spin and density dynamics for two-dimensiona l electrons with a\nspin-orbit coupling\nI. V. Tokatly1,3and E. Ya. Sherman2,3\n1Nano-bio Spectroscopy group and ETSF Scientific Developmen t Centre,\nDpto. F´ ısica de Materiales, Universidad del Pa´ ıs Vasco,\nCentro de F´ ısica de Materiale CSIC-UPV/EHU-MPC, E-20018 S an Sebastian, Spain\n2Department of Physical Chemistry, Universidad del Pa´ ıs Va sco UPV-EHU, 48080 Bilbao, Spain\n3Basque Foundation for Science IKERBASQUE, 48011, Bilbao, S pain\nWe study spin dynamics in a two-dimensional electron gas wit h a pure gauge non-Abelian spin-\norbit field, for which systems with balanced Rashba and Dress elhaus spin-orbit couplings, and the\n(110)-axis grown GaAs quantum wells are typical examples. W e demonstrate the duality of the\nspin evolution and the electron density dynamics in a system without spin-orbit coupling, which\nconsiderably simplifies and deepens the analysis of spin-de pendent processes. This duality opens\na venue for the understanding of this class of systems, highl y interesting for their applications in\nspintronics, through known properties of the systems witho ut spin-orbit coupling.\nPACS numbers: 72.25.-b\nThe understanding of spin dynamics in a two-\ndimensional (2D) electron gas with spin-orbit (SO) in-\nteraction is highly important both for theoretical and\napplied spintronics, [1–4] including the design of devices\nwith controlled spin transport. In many physically in-\nteresting situations the SO coupling can be elegantly de-\nscribed as an effective non-Abelian vector potential. [5–\n17]Thereexistsaclassofsystems, wheretheSOcoupling\ncorrespondsto apure gaugenon-Abelian field. Therefore\nit can be gauged away and the behavior of a physical\nsystem should map to that of a system without SO cou-\npling. In practice, there are at least two systems of this\nsort widely investigated from the applied point of view.\nThese are 2D electrons with balanced Rashba and Dres-\nselhaus couplings, and the electron gas in the (110)-axis\ngrown GaAs quantum wells.\nThe coupled spin-charge dynamics is commonly de-\nscribed using the diffusion approximation, where the rate\nof the spin precession is much less than the momentum\nscatteringrate.[18,19]Currently,high-mobility2Dstruc-\ntures, where the time scale of momentum relaxation is\nlonger than the spin rotation time,[20, 21] became avail-\nable. Since here spins can make several turns between\ncollisions with impurities, the conventional Dyakonov-\nPerel mechanism [18] is not applicable,[22] and another\ntype of analysisis required. In the present paper we solve\nthis problem for systems with a pure gauge SO coupling,\nincluding quantum effects due to the weak localization.\nWe show for this general class of systems the existence\nof a duality of observables allowing the spin dynamics\nto be fully mapped to the density dynamics in systems\nwithout SO coupling. As a result, several regimes, in-\ncluding magnetic field dependence of the spin dynamics,\ncan easily be explored using a single formula.\nWerepresenttheHamiltonianofa2Delectrongaswith\nSO coupling as follows [16, 17] (the system of units with¯h= 1 is employed)\nH=1\n2m/integraldisplay\nd2ρΨ+(i∂i+Ai)2Ψ+W/bracketleftbig\nΨ+,Ψ/bracketrightbig\n,(1)\nwhere Ψ( ρ) is a spinor field operator, and the func-\ntionalW[Ψ+,Ψ] contains all spin-independent contribu-\ntions, including the external potential, electron-electron\ninteractions, and, possibly, the electron-phononcoupling.\nHeremis the electron effective mass, and Aiare 2×2\nmatrix-valuedcomponentsofanon-Abelian SU(2) gauge\nfield describing the SO coupling. In the broad class of\nsystems of interest, Aiis a pure gauge, that is it can be\nremoved by a local SU(2) transformation. The general\nform of a pure gauge vector potential,\nAi=mαi(h·σ), (2)\ncorresponds to the following SO Hamiltonian\nHso=α(h·σ)(k·ν), (3)\nwhereσis a vector of Pauli matrices, αis the SO\ncoupling constant, his a three-component unit vec-\ntor for the SO field direction, νis a 2D vector in the\n(x,y) plane, and αi=ανi.The two practically impor-\ntant systems described by the Hamiltonian of Eq. (3)\nare: (i) the balanced Rashba-Dresselhaus system [23, 24]\nwithh=(±1,±1,0)/√\n2,ν= (±1,±1)/√\n2,and (ii)\nthe (110)-axis grown GaAs quantum well,[25–28] where\nh= (0,0,1),ν= (1,0) with the well axes chosen with\nrespect to the crystal axes as x/bardbl[110], y/bardbl[001],andz/bardbl\n[110]. Both systems are expected to demonstrate highly\nanisotropic spin relaxation times with the spin compo-\nnent along the h-axis having a very low relaxation rate,\narising only due to a spin-dependent disorder.[29] Spin\ncurrents in the thermodynamical equilibrium state,[30]\nbeing common for 2Delectron systems with SO coupling,\nare absent [16] in the structures described by the Hamil-\ntonian in Eq. (3).2\nA localSU(2) transformation, which gauges away the\nabove type of SO coupling is ˜Ψ(ρ) =UAΨ(ρ), where\nUA= exp[imα(h·σ)(ρ·ν)]. (4)\nThe transformation (4) renders invariant all spin-\nindependent quantities, such as the charge and current\ndensities, while the spin density transforms covariantly:\n/tildewideS=1\n2tr{σU−1\nA(S·σ)UA}. (5)\nWhen the SO coupling is gauged away the dynamics\nof the transformed spin density /tildewideS(ρ,t) reduces to the\nspin dynamics in the electron gas without SO interac-\ntion, which significantly simplifies the analysis. Then,\nthe physical spin density S(ρ,t) is restored by\nS=1\n2tr{σUA(/tildewideS·σ)U−1\nA}, (6)\nto obtain measurable results. Here we follow this guide-\nline and show that this approach allows to describe all\nregimes of spin dynamics on the same footing.\nWe consider a 2D electron gas with a SO coupling of\nEq. (1) and, initially, a uniform spin density Sproduced,\nfor example, by a static magnetic field B. Att= 0\nthe magnetic field is released and the spin relaxes due to\na disorder potential and other interactions. To describe\nthis process we first eliminate the SO by the gauge trans-\nformation of Eq. (4). Using Eq. (5) we find that the ini-\ntial uniform physical spin density is mapped to the spin\ntexture/tildewideS(ρ,0) =/tildewideS/bardbl(ρ,0)+/tildewideS⊥(ρ,0),where the term\n/tildewideS/bardbl(ρ,0) =h(S·h), (7)\nbeing parallel to h, is untouched by the transformation\nand remains uniform, while the orthogonal to hpart\ntransforms into the helix structure [31, 32]\n/tildewideS⊥(ρ,0) = [S−h(S·h)]cos(Qhx·ρ)−(S×h)sin(Qhx·ρ),\n(8)\nwhereQhx= 2mανis the helix wave vector.\nSince in the transformed system there is no SO cou-\npling, the uniform part of the initial spin distribution,\nEq. (7), is constant in time. A nontrivial dynamics oc-\ncurs in the orthogonal channel due to a diffusional decay\nof the initial helix spin texture described by Eq. (8):\n/tildewideSβ1\n⊥(ρ,t) =/integraldisplay\nDβ1β2(ρ−ρ′,t)/tildewideSβ2\n⊥(ρ′,0)d2ρ′,(9)\nwhereDβ1β2(ρ−ρ′,t) is the exact spin diffusion Green’s\nfunction of a 2D electron gas, which takes into ac-\ncountthedisorder,electron-electronandelectron-phonon\ninteractions. To proceed further we note that in a\nnonmagnetic system without SO coupling the spin dif-\nfusion Green’s function is diagonal in spin subspace\nDβ1β2(ρ,t) =δβ1β2D(ρ,t). Hence Eq. (9) simplifies as\n/tildewideS⊥(ρ,t) =/tildewideS⊥(ρ,0)D(Qhx,t), (10)whereD(q,t) is a Fourier component of the spin diffusion\nGreen’s function\nD(q,t) =/integraldisplay\nd2ρe−i(q·ρ)D(ρ,t), (11)\nand we have taken into account that only the Fourier\ncomponents of D(ρ,t) with the modulus of the wave vec-\ntorq=Qhxcontribute to the dynamics of the helix in\nEq. (8). Since the time-dependent factor in Eq. (10)\nis scalar, the transformation back to the physical spin,\nEq. (6), simply reduces to removing tildas and the coor-\ndinatedependenceinEq.(10). Thus,wegetthefollowing\nexactresult for the observable spin evolution\nS⊥(t) =S⊥(0)/integraldisplaydω\n2πD(Qhx,ω)e−iωt.(12)\nIn Eq. (12) we represented D(q,t) via the Fourier inte-\ngral because in the ω-domain there is a simple expression\nof the spin diffusion Green’s function D(q,ω) in terms\nof the spin-spin correlator (the spin response function)\nχ[S]\nββ(q,ω), where β= (x,y,z) is the Cartesian index cor-\nresponding to the spin component\nD(Qhx,ω) =1\niω/bracketleftBigg\nχ[S]\nββ(Qhx,ω)\nχ[S]\nββ(Qhx,0)−1/bracketrightBigg\n.(13)\nThis equation can be derived by considering a linear re-\nsponse on a time-dependent magnetic field that is adi-\nabatically switched on at t=−∞, and then suddenly\nswitched off at t= 0, i. e. B(t) =eδtθ(−t)Bwithδ→0\n(see, e. g., Ref. [33] for similar calculations). Usually the\nSO coupling is weak on the Fermi energy scale, which\nimpliesQhx≪kF, wherekFis the Fermi momentum.\nTherefore in most situations one can safely replace the\nstaticω= 0 response function in Eq. (13) by the macro-\nscopic Pauli spin susceptibility, χ[S]\nββ(Qhx,0) =χP.\nEquations (12) and (13) are the result of the spin-\ndensity dynamics duality and give the exactevolution\nof the uniform spin density. The problem is solved\nby mapping the spin relaxation in the physical system\nto the “washing out” an inhomogeneous spin texture\nin a dual system without SO coupling. The real spin\nrelaxes because of SO-induced precession and random-\nness introduced by disorder, phonons, and interelectron\ninteractions.[34] For the transformed spin, it is the evolu-\ntionofthenonuniformspindensitydistributions, againin\nthe presence of the disorder and interactions between the\ncarriers. An interesting exact feature of the pure gauge\nSO coupling (in addition to the well known anisotropy)\nis the absence of the spin precession – the vector S⊥(t)\nis always collinear to its initial direction. In the trans-\nformed picture this is related to the diagonal structure of\nthe spin response in a nonmagnetic electron gas. For the\nreal system this translates to the fact that spins of elec-\ntrons with opposite momenta precess around the h-axis\nin the opposite directions with the same rate.3\nAt the level of the random phase approximation the\nspin response function χ[S]\nββ(q,ω) is equal to the den-\nsity response function χ(q,ω) of a noninteracting, but\npossibly disordered and/or coupled to phonons electron\ngas, while the Pauli susceptibility χPis proportional\nto the compressibility ∂n/∂µ, withnandµbeing the\nelectron concentration and chemical potential, respec-\ntively. Hence the spin diffusion Green’s function entering\nEq. (12) reduces to\nD(Qhx,ω) =1\niω/bracketleftbiggχ(Qhx,ω)\n∂n/∂µ−1/bracketrightbigg\n,(14)\nwhich is exactly the density diffusion Green’s function.\nTherefore in this physically important case the spin re-\nlaxation is mapped to the ordinary density diffusion.\nNow we apply Eqs. (14) and (12) to a noninteracting\ndisordered 2D electron gas with a momentum relaxation\ntimeτ, andstudypossibleregimesofspindynamics. The\ndensity-density correlator χ(Qhx,ω) can be obtained ei-\nther diagrammaticallyor by solving the kinetic equation.\nIn the semiclassicalregime, correspondingto the summa-\ntion of ladder diagrams, one obtains:\nD(Qhx,ω) =K(Qhx,ω)\n1−K(Qhx,ω), (15)\nK(Qhx,ω) =1\n2π/integraldisplaydθ\n1−iωτ+iΩsoτcosθ,(16)\nwhere the only SO-dependent parameter in the problem\nΩso≡QhxvF(vFis the Fermi velocity) is the maximum\nspin precession rate, and Ω soτ=ℓQhx(electron mean\nfree path ℓ=vFτ) characterizes the relaxation regime.\nWe begin with Ω soτ≪1 regime, which coresponds to\na pure diffusion, studied in the coordinate representation\ninRef. [17]. HerethediffusionGreen’sfunction, Eq.(15),\nreduces to\nD(Qhx,ω) =1\nDQ2\nhx−iω, (17)\nwhereD=v2\nFτ/2 is the diffusion coefficient. Inserting\nD(Qhx,ω) of Eq. (17) into Eq. (12) we obtain:\nS⊥(t) =S⊥(0)exp/parenleftbig\n−DQ2\nhxt/parenrightbig\n, (18)\nwhich exactly corresponds to the Dyakonov-Perel’ mech-\nanism with the spin relaxation rate Γ s=DQ2\nhx. More-\nover, the factor 1 /2 in the definition of Dacquires an\ninteresting physical meaning in terms of the spin pre-\ncession: it corresponds to the angular averaging of the\nprecession rate/angbracketleftbig\nΩ2\nso(k)/angbracketrightbig\n= Ω2\nso/2.\nThe opposite, clean limit Ω soτ≫1, in terms of the\ndual (transformed) system corresponds to a reversible,\npurely ballistic washing out the helix texture. In this\nregimeD(Qhx,ω)≈ K(Qhx,ω) (see Ref.[35]) and the\nintegration in Eqs. (12) and (16) yields:\nS⊥(t) =S⊥(0)J0(Ωsot) =S⊥(0)J0(QhxvFt),(19)\nFIG. 1: (Color online.) Time dependence of the spin for\ndifferent parameters of SO coupling, shown near the plots,\nwiths(t) defined as S⊥(t)≡s(t)S⊥(0)/S⊥(0).\nwhereJ0(Ωsot) is the Bessel function. The same result\ncan be derived directly from the microscopic spin preces-\nsion with a k-dependent rate Ω so(k) = Ωsocosφ, where\nφis the angle between kandν. Indeed, the net result of\nthe inhomogeneous precession reproduces Eq. (19),\nS⊥(t) =S⊥(0)/integraldisplay\ncos(Ωsotcosφ)dφ\n2π=S⊥(0)J0(Ωsot).\n(20)\nIn contrast, in the systems with pure Rashba or Dressel-\nhaus coupling, spin precession rate does not depend on\nthe direction of momentum. As a result, in the ballistic\nregime, the z-component of the total spin demonstrates\ncosine-like rather than the Bessel function time depen-\ndence.\nAn intermediate regime of Ω soτ∼1, can be investi-\ngated numerically. The results are presented in Fig.1 for\ndifferentparameters. Onecanclearlyseeacrossoverfrom\nthe oscillating Bessel function-like behavior to the expo-\nnential Dyakonov-Perel’ decay. At short times, the be-\nhavior of spin is universal: S⊥(t) =S⊥(0)/parenleftbig\n1−Ω2\nsot2/2/parenrightbig\ndue to the unperturbed precession of the spins. For the\ndensity dynamics the universal short-time behavior is a\ndirect consequence of the f-sum rule,[36] as can be seen\nby expanding Eq. (12) at t→0.\nAnalysis of Eqs. (12),(15), and (16) shows that Ω soτ=\n1 is a critical point. With the decrease in Ω soτin the\nrange Ω soτ >1,the first zero of S⊥(t) rapidly shifts to\nlarger times, with negative value regions becoming very\nshallow. At Ω soτ <1 zeroes of S⊥(t) disappear and the\ndynamics is a pure decay.\nThe gauge transformation approach allows to analyze\nsystems, where a direct treatment of the SO coupling\nwould cause difficulties. The first effect we consider is\nthe influence of the orbital motion in a nonquantizing\nmagnetic field Balong the z-axis on the spin dynamics.\nWe assume that due to a small g−factor of electron, the\nZeeman coupling to the magnetic field does not cause a\nrelevant spin precession. If Ω soτ≪1, the density evolu-\ntion atB= 0 is diffusive, and the electron mobility de-4\ncreases with Bdue to the Lorentz force as/parenleftbig\n1+ω2\ncτ2/parenrightbig−1,\nwhereωc=|e|B/mcis the cyclotron frequency. By\nthe Einstein relation, the diffusion coefficient is renor-\nmalized by the same factor D(B) =D(0)//parenleftbig\n1+ω2\ncτ2/parenrightbig\n.\nHence the spin relaxation rate in Eq. (18) decreases as\nΓs(B) = Γs(0)//parenleftbig\n1+ω2\ncτ2/parenrightbig\n, which reproduces the results\nof the direct quantum kinetic theory.[39] For illustration,\nwe consider the limit ωcτ≫1 at short times t≪τin\na more detail. Here the trajectories of electrons are very\ncloseto circlesand the spin-independent kernelin Eq.(9)\ncan be represented as (cf. Ref.[35])\nD(ρ′−ρ,t) =1\n2πd(t)δ(|ρ′−ρ|−d(t)),(21)\nwith the displacement d(t) = 2Rc|sin(ωct/2)|,where\nRc=vF/ωcis the cyclotron radius, and RcQhx=\nΩso/ωc. Straightforward integration in Eq. (9) yields (cf.\nEq.(19))\nS⊥(t) =S⊥(0)J0(Qhxd(t)), (22)\nwhereQhxd(t) = 2Ω so|sin(ωct/2)|/ωc. For a very weak\nfield (in a very clean system) ωc≪Ωso, the result re-\nproduces Eq. (19), as expected. In the opposite limit\nωc≫Ωso, no relaxation occurs. In terms of spin pre-\ncession this can be understood[22, 37, 38] as very fast\nchanges at the frequency ωcin the direction of the SO\nfield, keeping the total spin out of relaxation. In terms\nof the nonuniform density dynamics, this implies that\nthe electrons, forced to circulate around small radius cy-\nclotron orbits, can not spread out to destroy large scale\n∼1/Qhx≫Rcdensity variations. At long times, the dif-\nfusion behavior takes over, and the relaxation becomes\nexponential.\nAs a second example we briefly discuss the effect of\nweak localization on the spin relaxation [40, 41] by con-\nsideringω-dependent renormalization of the momentum\nrelaxation rate with the correction:[42–44]\nδτ−1\nwl(ω) =2\nτ1\n∂n/∂µ/integraldisplayd2q\n(2π)21\nDq2−iω.(23)\nThis correction arises due to the enhanced return prob-\nability, which slows down of the density dynamics and\neventually leads to the algebraic tail in spin relaxation at\nlong time as S⊥(t)∼1/t.[41] In terms of spin preces-\nsion, the enhanced backscattering slows down the spin\nrelaxation because upon the return to the initial point\nthe direction of the electron spin remains the same.\nTo conclude, we have shown that in a wide class of\nsystems the non-Abelian gauge field description of SO\ncoupling reveals the duality of experimental observables\nand ensures the exact mapping of the spin dynamics\nto the density evolution. The evolution is described in\nterms of the responce to an external perturbation with\nthe wavevector equal to the spin helix wavevector Qhx.We presentedexplicit resultsforaweakSOcouplingwith\nQhx≪kFvalidforallsystemsofinterest(thisrestriction\nis not required in general). This exact mapping opens a\nvenueforunderstandingthe wholeclassofpracticallyim-\nportant systems through better studied, and more simple\nproperties of the systems without SO coupling.\nIVT acknowledges funding by the Spanish MEC\n(FIS2007-65702-C02-01), ”Grupos Consolidados\nUPV/EHU del Gobierno Vasco” (IT-319-07), and\nthe European Community through e-I3 ETSF project\n(Grant Agreement: 211956). This work of EYS was sup-\nported by the University of Basque Country UPV/EHU\ngrant GIU07/40, MCI of Spain grant FIS2009-12773-\nC02-01, and ”Grupos Consolidados UPV/EHU del\nGobierno Vasco” grant IT-472-10.\n[1] R. Winkler, Spin-orbit Coupling Effects in Two-\nDimensional Electron and Hole Systems , Springer Tracts\nin Modern Physics (2003).\n[2] I. Zutic, J. Fabian, and S. Das Sarma, Rev. Mod. Phys.\n76, 323 (2004); J. Fabian, A. Matos-Abiague, C. Ertler,\nP. Stano, and I. Zutic, Acta Physica Slovaca 57, 565\n(2007).\n[3]Spin Physics in Semiconductors , Springer Series in Solid-\nState Sciences, Ed. by M.I. Dyakonov, Springer (2008)\n[4] M.W. Wu, J.H. Jiang, and M.Q. Weng, Physics Reports\n493, 61 (2010).\n[5] V. P. Mineev and G. E. Volovik, Journal of Low Temper-\nature Physics 89, 823 (1992).\n[6] J. Fr¨ ohlich and U. M. Studer, Rev. Mod Phys. 65, 733\n(1993).\n[7] I. L. Aleiner and V. I. Fal’ko, Phys. Rev. Lett. 87, 256801\n(2001).\n[8] L. S. Levitov and E. I. Rashba, Phys. Rev. B 67, 115324\n(2003).\n[9] Y. Lyanda-Geller, Phys. Rev. Lett. 80, 4273 (1998), Y.\nLyanda-GellerandA.D.Mirlin, Phys.Rev.Lett. 72.1894\n(1994), J. B. Miller, D. M. Zumb¨ uhl, C. M. Marcus, Y.\nB. Lyanda-Geller, D. Goldhaber-Gordon, K. Campman,\nand A. C. Gossard, Phys. Rev. Lett. 90, 076807 (2003).\n[10] S.-R. Eric Yang and N. Y. Hwang, Phys. Rev. B 73,\n125330 (2006).\n[11] Q. Liu, T. Ma, and S.-C. Zhang, Phys. Rev. B 76, 233409\n(2007).\n[12] N. Hatano, R. Shirasaki, and H. Nakamura, Phys. Rev.\nA75, 032107 (2007).\n[13] J.-S. Yang, X.-G. He, S.-H. Chen, and C.-R. Chang,\nPhys. Rev. B 78, 085312 (2008).\n[14] B.W.A. Leurs, Z. Nazario, D.I. Santiago, and J. Zaanen,\nAnnals of Physics 323, 907 (2008).\n[15] R. Raimondi and P. Schwab, Europhys. Lett. 87, 37008\n(2009); M. Milletar, R. Raimondi, and P. Schwab, Euro-\nphys. Lett. 82, 67005 (2008).\n[16] I. V. Tokatly, Phys. Rev. Lett. 101, 106601 (2008).\n[17] I. V. Tokatly and E.Ya. Sherman, Annals of Physics 325,\n1104 (2010).\n[18] M.I. Dyakonovand V.I. Perel’, Sov. Phys. Solid State 13,\n3023 (1972).5\n[19] P. Kleinert and V. V. Bryksin, Phys. Rev. B 79, 045317\n(2009); T. D. Stanescu and V. Galitski, Phys. Rev. B 75,\n125307 (2007).\n[20] M. Griesbeck, M. M. Glazov, T. Korn, E. Ya. Sherman,\nD. Waller, C. Reichl, D. Schuh, W. Wegscheider, and C.\nSch¨ uller, Phys. Rev. B 80, 241314 (2009)\n[21] T. Korn, Physics Reports 494415 (2010) and references\ntherein.\n[22] See e.g., D. Culcer and R. Winkler, Phys. Rev. B 76,\n195204 (2007).\n[23] N. S. Averkiev and L. E. Golub, Phys. Rev. B 60, 15582\n(1999).\n[24] J. Schliemann, J. C. Egues, and D. Loss, Phys. Rev. Lett.\n90, 146801 (2003).\n[25] M.I. Dyakonov and V.Yu. Kachorovskii, Fiz. Tekh.\nPoluprovodn. (St.-Petersburg) 20, 178 (1986), [Sov.\nPhys. Semicond. 20, 110 (1986)].\n[26] G. M. M¨ uller, M. R¨ omer, D. Schuh, W. Wegscheider, J.\nH¨ ubner, and M. Oestreich, Phys. Rev. Lett. 101, 206601\n(2008).\n[27] V. V. Bel’kov, P. Olbrich, S. A. Tarasenko, D. Schuh, W.\nWegscheider, T. Korn, C. Sch¨ uller, D. Weiss, W. Prettl,\nandS.D. Ganichev, Phys.Rev.Lett. 100, 176806 (2008).\n[28] S. A. Tarasenko, Phys. Rev. B 80, 165317 (2009); Y.\nZhou and M. Wu, EPL 89, 57001 (2010).\n[29] M. M. Glazov and E. Ya. Sherman, Phys. Rev. B 71,\n241312(R) (2005); V. K. Dugaev, E. Ya. Sherman, V. I.\nIvanov, and J. Barna´ s, Phys. Rev. B 80, 081301 (2009);\nM. M. Glazov, M. A. Semina, and E. Ya. Sherman Phys.\nRev. B81, 115332 (2010); M. M. Glazov, E. Ya. Sher-\nman, and V. K. Dugaev, Physica E (2010) in print.\n[30] E. I. Rashba, Phys. Rev. B 68, 241315 (2003).\n[31] B. A. Bernevig, J. Orenstein, and S.-C. Zhang, Phys.\nRev. Lett. 97, 236601 (2006).\n[32] J. D. Koralek, C. P. Weber, J. Orenstein, B. A. Bernevig,Shou-Cheng Zhang, S. Mack, and D. D. Awschalom, Na-\nture458, 610 (2009).\n[33] D. Belitz and T. R. Kirkpatrick, Rev. Mod. Phys. 66,\n261 (1994).\n[34] M. M. Glazov and E. L. Ivchenko, Pis’ma Zh. Eksp.\nTeor. Fiz. 75476 (2002) [JETP Lett. 75, 403 (2002)];\nM. M. Glazov and E. L. Ivchenko, JETP 991279 (2004);\nD. Stich, J. Zhou, T. Korn, R. Schulz, D. Schuh, W.\nWegscheider, M. W. Wu, and C. Sch¨ uller, Phys. Rev.\nLett.98, 176401 (2007).\n[35] In the time-coordinate representation in this limit D(ρ′−\nρ,t) =δ(|ρ′−ρ|−vFt)/(2πvFt).\n[36] G. Giuliani and G. Vignale, Quantum Theory of the\nElectron Liquid , Cambridge University Press (2005), 798\npages.\n[37] M. M. Glazov, Solid State Commun. 142, 531 (2007).\n[38] C.H. Chang, J. Tsai, H.-F. Lo, and A. G. Malshukov,\nPhys. Rev. B 79, 125310, (2009).\n[39] E.L. Ivchenko, Fiz. Tverd. Tela 15, (1973) 1566 [Sov.\nPhys. Solid State 15, (1973) 1048]. The Green’s function\nformalism: A. A. Burkov and L. Balents, Phys. Rev. B\n69, 245312 (2004), in the presence of SO coupling leads\nto the same effect.\n[40] A. G. Mal’shukov, K. A. Chao, and M. Willander, Phys.\nRev. Lett. 76, 3794 (1996).\n[41] I. S. Lyubinskiy and V. Yu. Kachorovskii, Phys. Rev. B\n70, 205335 (2004).\n[42] S. Hershfield and V. Ambegaokar, Phys. Rev. B 34, 2147\n(1986).\n[43] G. Strinati, C. Castellani, and C. Di Castro, Phys. Rev.\nB40, 12237 (1989).\n[44] V.K. Dugaev and D.E. Khmel’nitskii, Zh. Eksp. Teor.\nFiz.901871 (1986) [JETP Lett. 63, 1097 (1986)]."    },    {        "title": "1804.08634v2.Microscopic_linear_response_theory_of_spin_relaxation_and_relativistic_transport_phenomena_in_graphene.pdf",        "content": "Microscopic linear response theory of spin relaxation and\nrelativistic transport phenomena in graphene\nManuel Offidani∗, Roberto Raimondi†and Aires Ferreira∗\n∗Department of Physics, University of York, York YO10 5DD, United Kingdom\n†Dipartimento di Matematica e Fisica, Università Roma Tre, 00146 Rome, Italy\nAbstract\nWe present a unified theoretical framework for the study of spin dynamics and relativistic transport phenomena in disordered\ntwo-dimensionalDiracsystemswithpseudospin--spincoupling. Theformalismisappliedtotheparadigmaticcaseofgraphene\nwith uniform Bychkov--Rashba interaction and shown to capture spin relaxation processes and associated charge-to-spin\ninterconversion phenomena in response to generic external perturbations, including spin density fluctuations and electric\nfields. A controlled diagrammatic evaluation of the generalized spin susceptibility in the diffusive regime of weak spin-orbit\ninteraction allows us to show that the spin and momentum lifetimes satisfy the standard Dyakonov-Perel relation for both\nweak (Gaussian) and resonant (unitary) nonmagnetic disorder. Finally, we demonstrate that the spin relaxation rate can be\nderived in the zero-frequency limit by exploiting the SU(2) covariant conservation laws for the spin observables. Our results\nset the stage for a fully quantum--mechanical description of spin relaxation in both pristine graphene samples with weak\nspin--orbit fields and in graphene heterostructures with enhanced spin--orbital effects currently attracting much attention.\n1arXiv:1804.08634v2  [cond-mat.mes-hall]  10 Jul 20181 Introduction 2\n1 Introduction\n1.1 Spin Relaxation in Graphene\nGraphene is considered a promising material for spintronics applications due to its negligible hyperfine interactions\nand low spin–orbit coupling (SOC) [1, 2]. Early theoretical estimates hinted at ultra-long spin lifetime ( τs≈1–\n100µs) [3], whereas experiments found τsto be limited to a few nanoseconds [4]. The microscopic mechanisms\nresponsible for the relatively fast spin relaxation in high-mobility graphene samples remain controversial [5], but\nrecent findings indicate that spinful scatterers, such as magnetic adatoms, are the primary cause of spin relaxation\n[6, 7, 9, 8].\nThe spin dynamics in graphene is conventionally probed by means of nonlocal transport measurements [10, 11].\nIn this approach, a spin current is injected from ferromagnetic electrodes into the graphene channel and allowed\nto diffuse under the effect of a perpendicular magnetic field. The Larmor precession of the electron’s spin about\nthe external field modulates the average spin accumulation detected away from the injection point (Hanle curve),\nresulting in a bona fide spin signal from which τscan be deduced. Such Hanle precession measurements found a\nlarge spread in τsfrom tens of picoseconds up to a few nanoseconds [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22],\nreflecting the different sample preparation and device fabrication methods. Theoretical studies have revealed a\nnumber of possible spin relaxation sources, including magnetic impurities, spin–orbit active adatoms, ripples and\nother substrate effects [23, 24, 25, 26, 27, 28, 29, 30]. Numerical approaches have provided further insight into\nthe relaxation mechanisms, enriching the scenario to include the impact of electron-hole puddles, pseudospin-spin\ncoherence and ballistic effects [31, 32, 33]. Despite the relatively short τsin of clean samples, the high charge carrier\nmobility allows spins to diffuse over extremely long distances up to 13 µm at room temperature [34, 35, 36].\nThe paradigmatic model for studies of spin relaxation in graphene is the two-dimensional (2D) Hamiltonian of\nmassless Dirac fermions supplemented with a (uniform or random) Bychkov-Rashba interaction [37]. This type of\nSOC has its origin in perturbations breaking the inversion symmetry, which include substrate-induced electric fields,\nadatoms, and ripple-induced gauge fields [3, 4]. The Bychkov-Rashba interaction in graphene (hereafter referred to\nas Rashba SOC) can be seen as a non-Abelian gauge field that couples to the intrinsic pseudospin of Dirac fermions,\nenabling spin relaxation upon impurity scattering e.g.,via the familiar Dyakanov-Perel (DP) mechanism [38].\nGraphene with random Rashba SOC has been recently shown to host novel charge-to-spin conversion effects\nby means of a quantum extension of the Boltzmann transport theory [39, 40]. Previous theoretical descriptions\nof spin relaxation in such 2D Dirac-Rashba models were based on semiclassical approximations [41, 42]. On the\nother hand, a fully quantum-mechanical theory of spin–orbit-coupled transport in the static (DC) limit has been\nformulated recently by the authors [43, 44]. Analogously to the 2D electron gas (2DEG) case [45, 46, 47], it was\nshown that impurity scattering corrections exactly balance the intrinsic generation of spin Hall current for spin-\nindependent disorder, /angbracketleftJSH/angbracketrightE= 0, where Eis an external DC electric field [43]. The vanishing of the spin Hall effect\nin this model is connected to the establishment of a robust nonequilibrium in-plane spin polarization /angbracketleftS/angbracketrightE/negationslash= 0with\nS⊥E, known as inverse spin-Galvanic effect (ISGE) [44]. However, a time-dependent framework able to unveil how\nthe steady state is reached within the 2D Dirac-Rashba model is yet to be developed. In this paper, we address\nthis problem. We derive the coupled spin-charge drift-diffusion equations for nonmagnetic disorder and generic\nhomogeneous perturbations by means of the diagrammatic technique for disordered electrons. A similar approach\nhas been adopted very recently in the context of 2DEGs with both Bychkov-Rashba and Dresselhaus interactions\n[48], whereitwasshownperfectagreementbetweentheKubodiagrammaticformalismandtheKeldyshSU(2)gauge\ntheory [49]. In this work, we extend the standard quantum diagrammatic formalism to accommodate the enlarged\n2(spin)⊗2 (pseudospin) Clifford structure of the 2D Dirac–Rashba model leading to a 16-dimensional diffuson\noperator in the absence of intervalley scattering. We find that the typical DP relation connecting the spin relaxation\ntime (SRT) and the momentum lifetime in the weak SOC regime, that is τs∝τ−1forλτ/lessmuch1, whereλis the SOC\nstrength, holds at all orders in the scattering potential strength. The meaning and interpretation of our results for\nthe SRTs can be also clarified by the SU(2) covariant conservation laws inherent to the diagrammatic (perturbative)\nstructure, whose usage allows us derive the DP relation even in the zero-frequency limit. In particular, we provide\nthe analytical expression of τsin the unitary limit of very strong potential scattering.\n1.2 Dirac–Rashba model\nThe effective low-energy Hamiltonian describing the electronic properties of 2D Dirac fermions subject to a uniform\nRashba interaction around the Kpoint reads as [50]\nH=/integraldisplay\ndxΨ†(x) [vσ·p+λ(σ×s)·ˆz+V(x)] Ψ(x), (1)1 Introduction 3\nFig. 1:(a) Energy dispersion around the Kpoint. The splitting of the Dirac bands leads to a spin gap or pseudogap .\n(b) Tangential winding of the spin texture in regimes I and II.\nwherevis the bare velocity of massless Dirac fermions, p=−ı∇is the 2D kinematic momentum operator, λis the\nSOC strength and σi,si(i=x,y,z) are Pauli matrices associated with sublattice (pseudospin) and spin degrees of\nfreedom, respectively. Here, V(x)is a disorder potential describing elastic scattering from nonmagnetic short-range\nimpurities. For simplicity, in this work we neglect intervalley scattering processes, which in the pure Rashba model\ncan renormalize the momentum lifetime but are not expected to impact fundamentally the spin dynamics [43]. Thus\nit suffices to consider the low-energy dynamics around the Kpoint.\nThe energy dispersion relation of the free Hamiltonian H0=H−Vin Eq.(1) is\n/epsilon1µν(k) =µλ+ν/radicalbig\nλ2+v2|k|2, (2)\nwhereµ,ν=±1labels the various subbands (Fig.1(a)).\nThe Rashba interaction aligns the electron spin at right angles to the wavevector, the so-called spin–momentum\nlocking configuration (Fig.1(b)) [51, 52]. For Fermi energy |/epsilon1|>2|λ|(region II), the split Fermi surface displays\ncounter-rotatingspintexturesreminiscentof(nonchiral)2DEGswithRashbainteraction[37]. Aregime(pseudogap,\nregion I) where the Fermi energy intersects a single subband, with electronic states having well-defined spin helicity,\nextends for energies |/epsilon1|<2|λ|. In the conventional 2DEG this circumstance only happens at a single point i.e., the\nintersectionbetweentheparabolicbands[53]. Importantly, thespintextureofenergybandsinthe2DDirac–Rashba\nmodel is modulated by the band velocity i.e.,\n/angbracketlefts/angbracketrightµνk=−µ/angbracketleftσ/angbracketrightµνk׈ z, (3)\nwhere/angbracketleftσ/angbracketrightµνk= (1/v)∇k/epsilon1µν(k)is the pseudospin polarization vector. The entanglement between pseudospin and\nspin degrees of freedom in the model is responsible for a rich energy dependence of transport coefficients [43, 44].\nFor brevity of notation, we assume /epsilon1,λ> 0in the remainder of the work.\n1.3 Disorder effects\nThe random potential in Eq. (1) affects the spin dynamics by inducing elastic transitions between electronic states\n(µνk)→(µ/primeν/primek/prime)associated with different effective Larmor fields, Ωµνk=λ/angbracketlefts/angbracketrightµνk≈−µνλˆk׈zfor/epsilon1/greatermuchλ.\nThis random change in the spin precession axis is responsible for the irreversible loss of spin information. To\ndescribe the effects of disorder, we employ standard many-body perturbation theory methods. We work within the\nzero-temperature Green’s function formalism.\nThe retarded(R)/advanced(A) single-particle Green’s function ( a=A,R≡−,+) is\nGa(x,x/prime;t−t/prime) =∓ı/angbracketleftbig\n0|T/bracketleftbig\nΨ(x,t),Ψ†(x/prime,t/prime)/bracketrightbig\n|0/angbracketrightbig\nθ(±t∓t/prime), (4)\nwhereTis the time-ordering symbol and θ(.)is the Heaviside step function. Changing to the energy domain, one\nobtains\nGa(x,x/prime;/epsilon1) =/angbracketleftx/prime|1\n[Ga\n0(/epsilon1)]−1−V|x/angbracketright, (5)\nwhereGa\n0(/epsilon1) = (/epsilon1+ıvσ·∇+λ(σ×s)·ˆz±ı0+)−1is the Green’s function of free 2D Dirac–Rashba fermions.\nThe central quantity in our approach is the disorder averaged Green’s function, Ga(x−x/prime,/epsilon1) =Ga(x,x/prime;/epsilon1),\nwhere the bar·denotes the average over all impurity configurations (Fig. 2(a)). Its momentum representation is\nGa\nk(/epsilon1) =1\n[Ga\n0k(/epsilon1)]−1−Σa\nk(/epsilon1), (6)1 Introduction 4\n(b)(a)\n(c)\nFig. 2:(a)DysonequationforthedisorderedaveragedGreen’sfunction. (b-c)Approximationschemesforevaluation\nof the self energy: Gaussian (b) and T-matrix approximation (TMA) (c). Box shows Feynman rules for the\ndisorder potential insertions (dashed lines) and impurity density insertion (red crosses).\nwhereGa\n0k(/epsilon1)is the Fourier transform of Ga\n0(x−x/prime;/epsilon1)and\nΣa\nk(/epsilon1) =/integraldisplay\nd(x−x/prime)e−ık(x−x/prime)/angbracketleftx/prime|V1\n1−Ga\n0(/epsilon1)V|x/angbracketright, (7)\nis the disordered averaged self energy within the noncrossing approximation. The latter neglects coherent multiple\nimpurity scattering corrections, which is justified in the diffusive regime with /epsilon1τ/greatermuch1[54]. The self-energy induced\nby short-range impurities is k-independent, Σa\nk(/epsilon1)≡Σa(/epsilon1), and hence we drop this index in what follows.\nTo account for the characteristic resonant (unitary) scattering regime of graphene with relaxation time τ∝/epsilon1\n[55, 56], we adopt a T-matrix approach by evaluating the self energy Σa(/epsilon1)at all orders in V. We obtain\nΣa(/epsilon1) =niu0\n1−u0ga\n0(/epsilon1)+O(n2\ni) =niTa(/epsilon1), (8)\nwhereu0parameterizes the scattering strength of the spin-transparent (scalar) impurities, niis the impurity areal\ndensity and Ta(/epsilon1)is the single-impurity T-matrix. Note that multiple impurity scattering diagrams ∝O(n2\ni)can\nbe neglected in the limit /epsilon1τ/greatermuch1i.e., away from the Dirac point (refer to Sec. 2.4 for a brief discussion of the spin\nrelaxation within the full noncrossing approximation). We have also introduced\nga\n0(/epsilon1) =ga\n0,0(/epsilon1)γ0+ga\n0,zz(/epsilon1)γzz+ga\n0,r(/epsilon1)γr, (9)\nas the momentum integrated Green’s function of the clean system [cf. Eq.(75) of Appendix A], where γ0≡σ0s0is\nthe4×4identity matrix, γzz=σzsz,γr= (σ×s)zand\nga\n0,0(/epsilon1) =−1\n8πv2[/epsilon1(LII(/epsilon1) +aıπθ II(/epsilon1)) +λ(LI(/epsilon1) +aıπθ I(/epsilon1))], (10)\nga\n0,zz(/epsilon1) =−λ\n8πv2(LI(/epsilon1) +aıπθ I(/epsilon1)), (11)\nga\n0,r(/epsilon1) = +/epsilon1\n16πv2(LI(/epsilon1) +aıπθ I(/epsilon1)). (12)\nIn the above, θI(II)(/epsilon1) =θ(/epsilon1+ 2λ)∓θ(/epsilon1−2λ)selects the energy regime and\nLI(II)(/epsilon1) = log/vextendsingle/vextendsingle/vextendsingle/vextendsingleΛ2\n/epsilon1(/epsilon1+ 2λ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle∓log/vextendsingle/vextendsingle/vextendsingle/vextendsingleΛ2\n/epsilon1(/epsilon1−2λ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle, (13)\nwith Λdenoting the ultraviolet cutoff of the low-energy theory [55].\nThe self energy simplifies in two important limiting cases: (i) weak Gaussian disorder ( |u0|/lessmuch|ga\n0|−1) and (ii)\nunitary disorder ( u0→±∞). In the weak scattering regime, it suffices to only take into account the ’rainbow’\ndiagram with two impurity lines in the Dyson expansion; see Fig.2(b). For scalar disorder this approximation is\nequivalent to assuming that the= disorder potential satisfies white-noise statistics [54]\n/angbracketleftV(x)/angbracketright= 0, (14)\n/angbracketleftV(x)V(x/prime)/angbracketright=niu2\n0δ(x−x/prime). (15)2 Microscopic linear response theory for spin relaxation 5\nIn this case we have\nΣa(/epsilon1)|Gauss.=niu2\n0ga\n0(/epsilon1). (16)\nThe real part of the self-energy provides a parametrically small renormalization of the band structure, which can\nbe safely neglected in the diffusive regime of interest [43]. We thus find\nΣR/A=∓ıni(η0γ0+ηrγr+ηzzγzz), (17)\nwhere the functions η0,ηr,ηzz, proportional to the imaginary parts of Eqs.(10)-(12), have different forms depending\non the Fermi level position. In this work, we will restrict the analysis to diffusive systems with weak SOC λτ/lessmuch1\nand/epsilon1/greatermuchλ. It is thus convenient to express the various quantities in Σa(/epsilon1)in terms of the quasiparticle broadening\nin regime II, i.e.,\n1\n2τ≡niη0|/epsilon1>2λ. (18)\nExplicitly, we have\n1\n2τ/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nGauss.=niu2\n0/epsilon1\n4v2, ηzz= 0, ηr= 0. (19)\nFor a typical choice of parameters, say, ni= 1012cm−2,u0= 1eV (u0is in units of eV·nm−2) and/epsilon1= 50meV, one\nfindsτ|Gauss/similarequal1.14ps, which is representative of clean graphene samples [55].\nWithin the T-matrix formalism, the nondiagonal part of /IfracturΣa(/epsilon1)acquires a finite value. However, in the unitary\nlimit of strong potential scattering ( u0→∞), we have Σa(/epsilon1) =−ni/ga\n0,0(/epsilon1)and we recover a scalar self-energy, with\n1\n2τ/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nTMA;u→∞=ni\n/epsilon14π2v2\nπ2+L2\nII(/epsilon1), ηzz= 0, ηr= 0. (20)\nIn this case, considering λ= 10meV, Λ = 10eV andni,/epsilon1as above one obtains a substantially shorter scattering\ntimeτ|TMA= 0.08ps. The unitary result captures the typical energy dependence τ∝/epsilon1observed in high-\nmobilitygraphenesamples[55], wherethechargecarriermobilityislikelylimitedbyshort-rangescatterers, including\nadsorbates, short-range ripples and vacancies [57, 58, 59, 60].\n2 Microscopic linear response theory for spin relaxation\n2.1 General formalism\nWe consider the long-wavelength spin dynamics generated by a generic external perturbation\nHext\nαβ(x,t) =−JαβAαβ(x,t), (21)\nwhereJαβ∝σαsβ(α,β= 0,i) is the current density operator ( α=x,y) or density operator ( α= 0,z) andAαβis\na generalized vector potential [43]. We will consider in detail two important cases: (i) an electric field perturbation\ne.g.,Hext\nx0(x,t) =−vσxs0Ax(x,t)and (ii) a spin density fluctuation Hext\n0i(x,t) =−1\n2σ0siBi(x,t). The induced spin\npolarization density\nSi(x,t) =1\n2/angbracketleftΨ†(x,t)σ0siΨ(x,t)/angbracketright, (22)\nis evaluated within the framework of linear response theory. This approach has been applied to derive charge–spin\ndiffusion equations describing spin dynamics and magnetoelectric effects in 2DEGs [48, 61, 62]. As shown below, a\nsuitable extension of this approach to accommodate the enlarged (spin ⊗pseudospin) Clifford algebra γαβ=σαsβ\nwill allow us to obtain a rigorous microscopic theory of diffusive transport and spin relaxation for 2D Dirac systems.\nThe linear response of the ˆi-component of the spin polarization vector at zero temperature reads as\nSi(x,t) =−/integraldisplay\ndx/prime/integraldisplay∞\n−∞dt/primeχi,αβ(x−x/prime,t−t/prime)∂t/primeAαβ(x/prime,t/prime), (23)\nwhereχi,αβ(x−x/prime,t−t/prime)is the generalized spin susceptibility associated to the external perturbation i.e., an electric\nfieldEx(x,t) =−∂tAx(x,t)or a ’spin injection field’ Φi(x,t) =−∂tBi(x,t)[63]. Expressing the above equation in\nterms of the Fourier transform χi,αβ(q,ω)in the long-wavelength limit q→0we have\nχi,αβ(0,ω) =κ\n2Tr/angbracketleftbig\nγ0iGR(x,x/prime;/epsilon1+ω)γαβGA(x/prime,x;/epsilon1)/angbracketrightbig\n, (24)2 Microscopic linear response theory for spin relaxation 6\n(a)\n(b)\n(c)\nFig. 3:Diagrammatic technique for evaluation of generalized spin susceptibilities. (a) Two-particle ladder diagram.\n(b) BS equation for the vertex renormalization. (c) Skeleton expansion of the ladder diagram in terms of an\ninfinite series of two-particle, noncrossing diagrams. Full (open) square denotes a T(T†) matrix insertion.\nwhereκ=v(κ= 1/2) for a electric (spin injection) field and Tr is the trace over all degrees of freedom. Terms\ninvolving products of Green’s functions on the same sector (RR and AA) are smaller by a factor of (/epsilon1τ)−1and thus\ncan be safely neglected.\nThe disorder average in Eq.(24) is evaluated by means of the diagrammatic technique (Fig.3). For brevity of\nnotation, we first present the formalism within the Gaussian approximation for the self-energy, Eq.(16). In Sec. 2.3,\nwe provide the connection with the full T-matrix result.\nA summation of noncrossing two-particle (ladder) diagrams leads to\nχ(NC)\ni,αβ(0,ω) =κ\n2/summationdisplay\nktr/braceleftbig\nγ0iGR\nk(/epsilon1+ω) ˜γαβ(ω)GA\nk(/epsilon1)/bracerightbig\n, (25)\nwhere tr is the trace over internal degrees of freedom (spin and sublattice). The dressed vertex ˜γαβsatisfies the\nBethe-Salpeter (BS) equation\n˜γαβ(ω) =γαβ+4\n2πτN 0/summationdisplay\nkGR\nk(/epsilon1+ω) ˜γαβ(ω)GA\nk(/epsilon1). (26)\nwhereN0≡/epsilon1/πv2(for theT-matrix extension see Eq. (59) and text therein). Projecting onto the elements of the\nClifford algebra\n˜γαβ/rho1ς(ω) =1\n4tr[˜γαβ(ω)σ/rho1sς], (27)\nwe recast the BS equation into the form\n˜γαβ/rho1ς(ω) =δα/rho1δβς+/summationdisplay\nµ,ν=0,x,y,zMµν/rho1ς(ω)˜γαβµν(ω), (28)\nwhere\nMµν/rho1ς(ω) =1\n2πτN 0/summationdisplay\nktr/bracketleftbig\nGR\nk(/epsilon1+ω)γµνGA\nk(/epsilon1)γ/rho1ς/bracketrightbig\n. (29)\nIntroducing the 16-dimensional vectors ˜γαβ(ω) = (˜γαβ00(ω),...,˜γαβzz(ω))tandγαβ= (0,0,....,γαβαβ,...,0)ta more\ncompact matrix form for Eq.(26) is given in terms of the diffuson operator D−1as\nD−1˜γαβ(ω)≡(116×16−Mt(ω))˜γαβ(ω) =γαβ. (30)2 Microscopic linear response theory for spin relaxation 7\nThe spin relaxation rates are simply identified as the poles of the generalized susceptibility in the complex ω-plane.\nThe determination of the SRTs is thus reduced to the analysis of the behavior of D−1=D−1(ω)[64].\nThe formal result Eq. (30) deserves a few comments. Firstly, D−1spans in principle the entire Clifford algebra,\nwhich physically encodes the coupled dynamics of spin and other observables associated with the elements γαβ.\nHowever, by exploiting symmetries, D−1can be reduced into block diagonal form, such that only some observables\nare coupled to the spin polarizations along the three spatial directions. Secondly, a distinct feature of Dirac systems\nis that spin densities are coupled to charge currents even in the case (considered here) of spatially uniform external\nperturbations q=0. The linear Dirac dispersion of graphene is reflected in the form of the charge current Ji=vσi\nand spin currentJa\ni=vσisa/2vertices, which do not depend explicitly on momentum; by virtue of that they can be\ndirectly identified (apart from constants) as elements of the Clifford algebra. Therefore all the relevant information\nabout coupling between currents and densities is built-in on the 16×16diffusion operator Eq. (30) in our formalism.\nThis will allows us to obtain a unified description of spin relaxation processes and relativistic transport phenomena\n(e.g., charge-to-spin conversion) within our q=0 formalism. We analyze the implications below.\nThe coupling of the electrons’ spin to currents or other observables in the long wavelength limit also suggests two\nequivalent scenarios to study spin relaxation. The first natural choice is to consider spin injection and investigate\nthe relaxation of the spin density profile (density–density response); alternatively, one can probe the spin response\nindirectly by exciting an observable coupled to the spin density through D−1. For instance, as we will see in the\nfollowing, one can drive a charge current via application of an electric field to obtain a in-plane spin polarization of\ncarriers (ISGE). In that case, the information about the in-plane SRTs is readily accessible by examining how the\nsteady state (Edelstein) polarization is achieved (density–current response).\nBefore moving on, let us stress that within the Gaussian approximation, a useful relation can be derived con-\nnecting the generalized susceptibility Eq.(25) and the renormalized vertex:\nχ(NC)\ni,αβ(0,ω) =κ\n2α/summationdisplay\nµνMµν0i(ω)˜γαβµν(ω) =κ\n2α(˜γαβ0i(ω)−δα0δβi), (31)\nwhereα≡(2πτN 0)−1and we have used Eq.(28). The above equation states the spin response can be solely\nobtained from the associated component of the renormalized vertex ˜γαβ0i. A similar relation holds for other\nresponse functions. For example the AC longitudinal (Drude) conductivity is written as\nσxx(ω) =v2/summationdisplay\nktr/braceleftbig\nγx0GR\nk(/epsilon1+ω) ˜γx0(/epsilon1,ω)GA\nk(/epsilon1)/bracerightbig\n=v2\nα(˜γx0x0(ω)−1). (32)\nThereforeEq.(31)andsimilarrelationsallowtoidentifythecomponentsofarenormalizedvertexwiththeassociated\nobservables, and will turn useful in the following.\nLet us now determine the allowed couplings to Sx,y,zby exploring symmetry. The model of Eq.(1) is invariant\nunder the group C∞v, which is an emergent symmetry of the continuum (long-wavelength) theory. As rotations in\nthe continuum do not describe the sublattice symmetry A↔Bof the graphene system, a representation Ufor the\nrelevant set of discrete operations has to be considered. Relevant to us are C2, the rotation of πaround the ˆz-axis\nexchanging sublattice (and valleys), and Rx, the reflection over the ˆx-axis. We have\nU(C2) =τxsz, (33)\nU(Rx) =τzσxsyrx. (34)\nwithrx: (x,y)→(x,−y)andτi=x,y,zare Pauli matrices acting on the valley degree of freedom. We also make use\nof isospin (valley) rotations Λx,y,z[65, 66]\nΛx,y=τx,yσz, (35)\nΛz=τz. (36)\nFor scalar disorder it suffices to examine the form of the clean-system susceptibility at ω= 0\nχRA,clean\ni,αβ≡1\n4Tr/bracketleftbig\nγ0iGR\n0(/epsilon1)γαβGA\n0(/epsilon1)/bracketrightbig\n. (37)\nFor any of the symmetries Slisted above, we have S−1GR/A\n0S=GR/A\n0, and inserting resolutions of the identity in\nthe formS†Sinto Eq.(37) we find\nχRA,clean\ni,αβ=pαβp0i\n4Tr/bracketleftbig\nγ0iGR\n0(/epsilon1)γαβGA\n0(/epsilon1)/bracketrightbig\n=pαβp0iχRA,clean\ni,αβ, (38)2 Microscopic linear response theory for spin relaxation 8\nwherepαβ(p0i) =±1is the parity of γαβ(γ0i)underS. From this result, we see that a nonzero response requires\nthe operator γαβto have the same parity of the spin vertex under the action of any of S. The allowed couplings\nand parities under Sare shown in the Tab.1. As anticipated above, the in-plane components Sx(y)are coupled\nto orthogonal charge currents σy(x), as well as spin Hall currents γxz(γyz)and staggered magnetizations γzy(γzx)\n[43, 44]. The out-of-plane component Szis instead coupled to a mass term σzand in-plane spin currents γxx,γyy.\nPolarization C2RxΛx,y,zCouplings\nSx -1-1+1σy, γxz, γzy\nSy -1+1+1σx, γyz, γzx\nSz +1-1+1σz, γxx,γyy\nTab. 1:Table summarizing the allowed couplings to the spin polarizations in the 2D Dirac–Rashba model with\nnonmagnetic scalar disorder.\n2.2 Diffusive equations and SRTs\nIn the following, we choose to consider the in-plane spin response to an AC electric field Hext\n/bardbl=−vσiAi(ω) =\n−(iω)−1vσiEi(ω),i=x,y. This choice, as discussed above, is equivalent to consider in-plane spin injection, but has\nthe advantage to allow for a unified description of spin dynamics and charge-spin interconversion, e.g. to capture the\nISGE or other similar effects [40, 67, 68]. For the out-of-plane spin dynamics, we take a spin-density perturbation\nHext\n⊥=1\n2szBz(ω)(see Tab. 1).\n2.2.1 In-plane spin dynamics\nWithout loss of generality, let us consider the dynamics of the ˆypolarization. According to Tab.1, syis coupled\nto three operators: σx, σyszandσzsx. However, leading terms in the (/epsilon1τ)−1expansion are only contained in the\nsy/σxsub-block. Hence, to capture the SRTs it suffices to restrict to this 2×2algebra. As anticipated above, we\nconsider here the response to an AC electric field Ex(ω), associated with the vertex κγx0=vσx≡vx. (Details of\ncalculation and full form of the 4×4diffuson operator is given in Appendices C and D.) To capture purely diffusive\nprocesses, we expand D−1(ω)in the low-frequency and small SOC limits, ωτ/lessmuch1andλτ/lessmuch1, respectively. In this\nregime, Eq.(30) is written then as\n/parenleftbigg1\n2(1−ıωτ)λ\n/epsilon1Γs(1 + 3ıωτ)\nλ\n/epsilon1Γs(1 + 3ıωτ) Γs−ıωτ/parenrightbigg/parenleftbigg˜γx0x0\n˜γx00y/parenrightbigg\n=/parenleftbigg1\n0/parenrightbigg\n, (39)\nwhere Γs=τ/τs= 2λ2τ. In the light of previous discussions (cf. Eqs.(31) and (32)), ˜vx0and˜v0yare connected by\na linear transformation to the steady-state charge current and the spin polarization (Appendix D).\nOff-diagonal elements of Eq.(39) carry in relation to diagonal ones an extra order of smallness λ//epsilon1, suggesting\nspin and charge to be weakly coupled in this limit. Their inclusion however encodes charge-to-spin interconversion\nand it is essential to get a correct physical description. The eigenvalues −ıω±are found as\n−ıω+/similarequal1\nτ/parenleftbigg\n1 + 16Γ3\ns\n/epsilon12τ/parenrightbigg\n, (40)\n−ıω−/similarequal1\nτs/parenleftbigg\n1−Γ3\ns\n/epsilon12τ/parenrightbigg\n, (41)\nand can be associated with charge current and spin relaxation times, respectively. We see then the SRT can be\nidentified as the mass ( ω= 0) term of the spin-spin part of the diffuson\n1\nτs≡1\nτ/bardbl\ns/similarequal1−M0y0y(ω= 0)/similarequal2λ2τ. (42)2 Microscopic linear response theory for spin relaxation 9\nInverting Eq.(39), we find\n˜γx0x0/similarequal1\nτ2\n−ıω+1\nτ, (43)\n˜γx00y/similarequal2λ\n/epsilon11\nτΓs\n−ıω+Γs\nτ, (44)\nfrom which, by using Eqs.(31) and (32) is it possible, upon Fourier transform, to derive the diffusive equation of\nmotion for coupled charge-spin dynamics as\n∂tJx(t) =−1\n2τ(Jx(t)−J0\nx(t)), (45)\n∂tSy(t) =−1\nτ/bardbl\ns(Sy(t)−S0\ny(t)), (46)\nwhereJ0\nx(t)≡2v2Ex(t)/αandS0\ny(t)≡ −λEx(t)//epsilon1α. Note that charge current relaxation is regulated by the\ntransport time τtr≡2τ, indicating the absence of backscattering [43, 54, 55].\n2.2.2 Out-of-plane spin dynamics\nFor the out-of-plane spin dynamics we consider the renormalized vertex κ˜γ0z=1\n2˜sz. The off diagonal components\nof the associated 4×4diffuson block contains sub-leading terms in the (/epsilon1τ)−1expansion (Appendix C), such that\nthe out-of-plane SRTs can be calculated similarly to Eq.(42) as\n1\nτ⊥s/similarequal1−M0z0z(ω)/similarequal4λ2τ. (47)\nThe generalization of the equations of motion Eqs.(45),(46) in this case is written as\n∂tSz(t) =−1\nτ⊥s(Sz(t)−S0\nz(t)), (48)\nwhereS0\nz(t) = ˙Bz(t)/4αis the effect of the external perturbation (spin-injection field). The in-plane and out-of-\nplane SRTs are in the following relation\n1\nτ/bardbl\ns=1\n21\nτ⊥s, (49)\nwhich is nothing but the well-known DP ratio for 2DEGs [61]. The above result has also been obtained for graphene\nwithin the time-dependent perturbation theory for the density matrix [42]. The agreement between graphene and\nthe Rashba 2DEG results is expected at high electronic density /epsilon1/greatermuchλ.\n2.3 SRT from the conservation laws in the DC limit\nIn this section, we demonstrate how the SRTs we have obtained above can be equivalently extracted in the static\nlimitω= 0.This remarkable result is rooted in the conservation laws associated to the disordered Dirac–Rashba\nHamiltonian Eq.(1) [43]. The first step is to write the Heisenberg equation of motion for the spin polarizations\n∂tSi=ı[H,Si] =2λ\nv/epsilon1lj/epsilon1c\nliJc\nj, (50)\nwhere/epsilon1lj,/epsilon1c\nliare the second and third rank Levi-Civita tensors and Jc\nj=/angbracketleftJc\nj/angbracketrightis the ˆj-component of the pure spin\ncurrent (with polarization ”c”). As before, we consider an electric field applied along the ˆxdirection. We find\n∂tSy=2λ\nvJz\ny, (51)\nwhereJz\nyis identified as the spin Hall current according to the chosen geometry. The spin Hall current is written\nin response to the electric field\nJz\ny=σz\nyxEx, (52)2 Microscopic linear response theory for spin relaxation 10\nwhereσz\nyxis the DC spin Hall conductivity calculated according to Eq.(25) with ˜γ0y→v˜γyz. As for now no\nassumption has been made for the self-energy approximation associated to the scalar impurities field. Let us start\nfromthemoretransparentGaussiancase. UsingthecorrespondingversionofEq.(31)for σz\nyx, togetherwithEq.(28)\nwe have\nσz\nyx=v2\n2α˜γx0yz=v2\n2α(Mx0yz˜γx0x0+M0yyz˜γx00y). (53)\nIn the latter we have neglected the terms MyzyzandMzxyzwhich, as said above, provide higher order corrections\nin the (/epsilon1τ)−1expansion. Multiplying both sides of Eq.(53) by the electric field Ex, and using Sy=χy,x0Extogether\nwith Eq.(31), we find\nJz\ny=v2\n2αMx0yz˜γx0x0Ex+vM0yyzSy. (54)\nDespite the Dirac character of fermions, the steady-state case of the continuity equation Eq.50 imposes the latter\nspin Hall current to vanish, analogously to the 2DEG case [43]. This implies the establishment of the out-of-\nequilibrium value for the spin polarization as\nS0\ny=−˜γx0x0\n2αvMx0yz\nM0yyzEx. (55)\nEvaluating the above quantities explicitly ˜γx0x0= 2, Mx0yz/M0yyz=λ//epsilon1and we recover the ISGE obtained in\nRef.[44]. Using Eq.(51) we finally arrive at\nJz\ny=vM0yyz/parenleftbig\nSy−S0\ny/parenrightbig\n(56)\nand therefore\n∂tSy≡−1\nτ/bardbl\ns/parenleftbig\nSy−S0\ny/parenrightbig\n, (57)\nwhere we have identified the spin relaxation time\n1\nτ/bardbl\ns=−2λM 0yyz= 2λ2τ, (58)\nin perfect accordance with the result obtained above, Eq.(42). The bubble M0yyzis therefore what completely\ndetermines the in-plane spin relaxation.\nWe now ask how the above result is modified when treating the self-energy in the T-matrix approximation. The\nBethe Salpeter equation Eq.(26) now reads\n˜γx0(/epsilon1) =γx0+ni/summationdisplay\nkTR(/epsilon1)GR\nk(/epsilon1) ˜γx0(/epsilon1)GA\nk(/epsilon1)TA(/epsilon1), (59)\nwhereTR/A(/epsilon1)is the single-impurity T-matrix in the R/A sectors introduced in Eq.(8). Projecting onto the Clifford\nalgebra, similarly to Eq.(28), we have\n˜γx0/rho1ς=δx/rho1δ0ς+/summationdisplay\nµνζξ =0,x,y,zY/rho1ςζξNµνζξ˜γx0µν, (60)\nwhere we have defined\nNµνζξ=ni\n4/summationdisplay\nktr(GR\nkγµνGA\nkγζξ), (61)\nY/rho1ςζξ=1\n4tr[TAγ/rho1ςTRγζξ]. (62)\nRecasting Eq.(60) in vector notation, in the same spirit of Eq.(30), we have\n˜γx0=γx0+Y Nt˜γx0, (63)\nand consequently\nY−1(˜γx0−γx0) =Nt˜γx0. (64)2 Microscopic linear response theory for spin relaxation 11\nThe latter equation allows again to find a connection with the observables. For example, the generalization of\nEq.(31) is written as\nχy,x0=2v\nni/summationdisplay\nµνNµν0y˜γx0µν=2v\nni/summationdisplay\nµνY−1\n0yµν˜γx0µν. (65)\nThe spin Hall conductivity instead is found as\nσz\nyx=2v2\nni/summationdisplay\nµνY−1\nyzµν˜γx0µν. (66)\nDifferentlytotheGaussiancase, wherewewereabletorelatetheresponseofanobservableuniquelytotheassociated\ncomponent of the renormalized vertex, in the T-matrix limit in principle all components of ˜γx0would contribute,\neach of them with weight given by Y−1. In the limiting case of unitary limit u0→∞, where limu0→∞TR/A=\n−1\ngR/A\n0,we find a simplification as\nY−1\n/rho1ςζξ=|gR\n0,0|2δ/rho1ζδςξ. (67)\nThis implies that for Eq.(66) a relation similar to the Gaussian case is obtained\nJz\ny=σz\nyxEx=2v2\nni|gR\n0,0|2˜γx0yzEx=2v2\nniNx0yz˜γx0x0Ex+vN00yz|gR\n0,0|−2Sy, (68)\nwhere we have restricted ourselves again to the dominant subspace σx/sy. After standard algebra, we arrive at\n∂tSy=2λ\nvσz\nyxEx=2λ\nvvN0yyz(Sy−S0\ny), (69)\nand the SRT defined as\n1\nτ/bardbl\ns= 2λ|gR\n0,0|−2N00yz= 2λ1\n/epsilon1216π2v4\nπ2+L2\nIIN00yz= 2λ2τ, (70)\nwhere we have used the definition of τin the unitary limit, Eq.(20). We conclude that the the formal expression\nconnecting τsandτ(the DP relation) is the same as found in the Gaussian limit for the self-energy. However, given\nthe different dependence of τon the Fermi level in the two approximations—cf. Eq.(18) and Eq.(20)—one has\nτ(/epsilon1)\nτ/bardbl\ns(/epsilon1)=\n\n2λ2\n/epsilon12/parenleftBig\n2v2\nniu2\n0/parenrightBig2\nGaussian,\n/epsilon12λ2\n2/parenleftBig\nπ2+L2\nII\n4π2niv2/parenrightBig2\nUnitary.(71)\nThe SRT associated to the out-of-plane component can be derived along the same lines. The relevant Heisenberg\nequation now reads\n∂tSz=−2λ\nv(Jx\nx+Jy\ny), (72)\nand a similar reasoning that lead to Eq,(58), allows us to conclude\n1\nτ⊥s= 2λ(M0zxx+M0zyy) = 4λ2τ, (73)\nin the Gaussian limit, and a similar relation for the unitary limit.\n2.4 Discussion\nHere, we discuss the DP relation obtained in Eq.(71) within the Gaussian and unitary limits of potential scattering.\nThe energy dependence of the spin lifetime for fixed impurity concentration is shown in Fig.4. Away from the Dirac\npoint, within the Gaussian approximation, the spin lifetime increases linearly since τ∝/epsilon1−1(see Eq.(18)). In\nthe unitary limit, instead, one has a linear dependence τ∝/epsilon1(see Eq.(20)), leading to vanishing spin lifetime at\nhigh electron doping. On the other hand, near the Dirac point, the noncrossing approximation breaks down. It\nis not surprising that the spin lifetime dependences are found to be nonphysical as /epsilon1→0: vanishingly small for\nthe Gaussian limit and divergent for the unitary limit. To overcome this limitation one needs to evaluate crossing3 Conclusions 12\n  \n  \n  \n020406080100\n−100 −50 0 50 100\nFig. 4:DP in-plane spin relaxation time calculated according different schemes for the self-energy: SCBA (red line),\nGaussian (blue line) and unitary limit (green line). The most important feature obtained within the SCBA\nis a strong renormalization of τ/bardbl\nsin the vicinity of the Dirac point, reflecting a disorder-induced finite density\nof states in that region. In the plot λ= 1meV and Γ = 60meV.\ndiagrams encoding quantum coherent processes, which includes weak localization corrections and diffractive skew-\nscattering from two or more impurities [54, 69, 70]. However, here we are mostly interested in the diffusive regime\naway from the Dirac point /epsilon1τ/greatermuch1, thus neglecting interference effects that can correct the standard DP relation\n[32, 33, 71]. However, an important refinement is possible within the noncrossing formalism used here by evaluating\ntheO(n2\ni) terms in Eq. (8). Such higher-order terms encode the strong renormalization of the single-particle\npropagators by incoherent multiple scattering approaching the Dirac point. To show this, it suffices to resum the\ninfinite class of ’rainbow’ diagrams, a scheme known as self-consistent Born approximation (SCBA). The SCBA\nself-energy is given by the solution of the following self-consistent equation [66]\n1\n2τ/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nSCBA=−/IfracturΣSCBA (/epsilon1) =−/Ifractur/bracketleftbiggni\n4πv2(/epsilon1−ΣSCBA (/epsilon1)) log/parenleftbigg−Λ2\n(/epsilon1−ΣSCBA (/epsilon1))2/parenrightbigg/bracketrightbigg\n. (74)\nIn Fig.4 we show that the SCBA provides a physical (finite, nonzero) τsapproaching the Dirac point. To obtain\na representative curve, we take λ= 1meV and we choose the impurity density and the scattering strength such\nthat the SCBA nonperturbative energy scale Γ = Λe−2πv2/(niu2\n0)[66] is a few tens meV. The in-plane SRT is then\nfound to lie in the range 50-100 ps. Concerning the magnitude of τ/bardbl\nswe note the result is compatible with previous\nreports where the (uniform or random) Rashba SOC is treated by semiclassical or numerical approaches [32, 42].\n3 Conclusions\nInthepresentwork, welaidthefoundationsofageneralmicroscopictheoryofdiffusivetransportandspinrelaxation\nin 2D Dirac systems subject to spin–orbit interactions. Our work represents the logical extension of the previously-\ndeveloped diagrammatic treatments [61, 62] to all orders in the scattering potential, for disordered electron systems\nwith an enlarged pseudospin ⊗spin Clifford algebra [43, 44]. We applied the formalism to the paradigmatic case\nof 2D Dirac fermions with Rashba spin–orbit coupling considering the purely diffusive regime λτ/lessmuch1/lessmuch/epsilon1τ. We\ndemonstrated how the Dyakonov-Perel relation between momentum and spin lifetime τ∝τ−1\nsholds in both the\nGaussian (weak short-range scatterers) and the unitary (strong short-range scatterers) limits, despite the drastic\ndifferent dependence momentum scattering times τ=τ(/epsilon1)in the two regimes. We derived the same result both by\ndirect diagrammatic resummation (in the noncrossing approximation) and by exploiting the conservation laws of the\ntheory in the zero-frequency limit. Under the diffusive regime λτ/lessmuch1/lessmuch/epsilon1τis not possible to study the dynamics\nin the region of Fermi energies comparable to the Rashba pseudogap region /epsilon1∼2λ, which was recently predicted to\ndisplay interesting out-of-equilibrium phenomena [44]. The strong spin-momentum locking approaching this regime\nlets us infer a modification of the relation between τsandτtowards the Elliot-Yafet type τs∝τ. Our theory sets\nthe stage to study the spin dynamics in that regime. This topic has become of renewed great interest due to recent4 Acknowledgements 13\nprogresses in graphene-based heterostructures, where the spin relaxation anisotropy has been recognized as a viable\ntool to estimate the induced large spin-orbital effects [72, 73, 74, 75].\n4 Acknowledgements\nThe authors are grateful to Ignacio Wilson-Rae for stimulating discussions. M. O. and A. F. acknowledge funding\nfrom EPSRC (Grant No. EP/N004817/1). Data availability statement (EPSRC).—No new data were created\nduring this study.A Clean Green’s Function 14\nAppendix\nA Clean Green’s Function\nThe explicit form of the clean single-particle Green’s function is\nGa\n0k(/epsilon1) =−1\n2/summationdisplay\nµ=±1La\n0µ/bracketleftBig\n(λ+µ/epsilon1)γ0+vσ·k−µ/epsilon1\n2γr+v(s×k)z+λγzz+δM2φk/bracketrightBig\n, (75)\nwhere\nLA(R)\n0µ=µ\nv2k2−/epsilon12−2µλ/epsilon1±ı0+sign(/epsilon1−µλ), (76)\nδM2φk=−1\n2(/epsilon1+ 2µλ) [(σysy−σxsx) sin 2φk+ (σxsy+σysx) cos 2φk], (77)\nandφkis the angle formed by the wavevector with ˆkx.\nB Integrals and expansion\nThe current work makes extensive use of momentum integrations involving products of two renormalized Green’s\nfunctions with analyticity in opposite halves of the complex plane, see e.g. Eqs.(24),(26). The retarded function is\ndisplaced in energy by the amount ω. Similarly to the bare Green’s function decomposition Eqs.(75) and (77), we\nwrite the renormalized (disorder averaged) propagators as\nGa\nk(/epsilon1) =Ma\n1k(/epsilon1)La\n1k(/epsilon1) +Ma\n2k(/epsilon1)La\n2k(/epsilon1), (78)\nwhereMa\nik(/epsilon1) =Ma(0)\ni+v2k2Ma(2)\ni, i={1,2}are matrix coefficients and the kernels La\nµ=La\nikare obtained in\nthe Gaussian limit from the functions L0µof Eq.(76) by analytical continuation /epsilon1→/epsilon1+aı\n2τ. In theTmatrix\napproach, the analytical continuation has to be performed as to include the other matrix structure of the self-energy\n∝γr,γKM[44]. We can generically write\nLa\nik(/epsilon1) =1\nv2k2−za\ni(/epsilon1), (79)\nwhereza\ni(/epsilon1)are complex quantities. Given the decomposition in Eq.(78), the integrals we need to solve are reduced\nto product of two kernels in different combinations, accompanied or not by a factor v2k2. Terms proportional to\nv4k4can be shown to vanish upon angular integration/integraltext\ndφk. We write below an exactsolution and then expand\nat linear order in ω. For simplicity we show the results for the Gaussian approximation. The first type of integrals\nis\nΓij=/integraldisplay∞\n0dkk\n2πLR\nik(/epsilon1+ω)LA\njk(/epsilon1) =/integraldisplay∞\n0dkk\n2π1\nv2k2−zR\nik(/epsilon1+ω)1\nv2k2−zA\njk(/epsilon1)(80)\n=1\nzR\nik(/epsilon1+ω)−zA\njk(/epsilon1)/integraldisplay∞\n0dkk\n2π/parenleftBigg\n1\nv2k2−zR\nik(/epsilon1+ω)−1\nv2k2−zA\njk(/epsilon1)/parenrightBigg\n(81)\n=1\n4πv21\nzR\nik(/epsilon1+ω)−zA\njk(/epsilon1)× (82)\n/bracketleftbigg\n−log/parenleftbig\n−zR\nik(/epsilon1+ω)/parenrightbig\n+ log/parenleftbig\n−zA\njk(/epsilon1)/parenrightbig\n−/parenleftbigg1\nzR\nik(/epsilon1)∂/epsilon1zR\nik(/epsilon1)/parenrightbigg\nω/bracketrightbigg\n, (83)\nwhere the principal branch of the log function has been chosen. Note za\n1(λ)→za\n2(−λ). Thus, Γ11(λ) = Γ 22(−λ)\nandΓ12(λ) = Γ 21(−λ). At linear order in ωwe find\nΓ11=1\n4πv2/bracketleftbiggπ\n/epsilon1+λ−1\n/epsilon1(/epsilon1+ 2λ)+ıπωττ\n/epsilon1+λ/bracketrightbigg\n, (84)\nΓ22= Γ11(λ→−λ), (85)\nΓ12=1\n4πv2/bracketleftbigg2ıπ\n4/epsilon1λ−ω/epsilon1+λ\n2/epsilon12λ(/epsilon1+ 2λ)/parenleftbigg\n1 +ıπ/epsilon1+ 2λ\n2λ/parenrightbigg/bracketrightbigg\n, (86)\nΓ21= Γ12(λ→−λ), (87)C Full form of the diffuson 15\nwhere we have retained leading order terms in (/epsilon1τ)−1. The other class of integrals we need to solve is\nΓ(3)\nij=/integraldisplayΛ/v\n0dkk3\n2πLR\nik(/epsilon1+ω)LA\njk(/epsilon1) =/integraldisplayΛ/v\n0dkk3\n2π1\nv2k2−zR\nik(/epsilon1+ω)1\nv2k2−zA\njk(/epsilon1)(88)\n=1\n4πv2zR\nik(/epsilon1+ω)−zA\njk(/epsilon1)/bracketleftbigg\n2ıIm/parenleftbigg\nzR\nik(/epsilon1) log/parenleftbiggΛ2\n−zR\nik(/epsilon1)/parenrightbigg/parenrightbigg\n−∂/epsilon1zR\nik(/epsilon1)/parenleftbigg\n1−log/parenleftbiggΛ2\n−zR\nik(/epsilon1)/parenrightbigg/parenrightbigg\nω/bracketrightbigg\n,(89)\nwhere the ultraviolet cutoff Λ/v/greatermuchkFhas been introduced to regularize the integrals. A careful evaluation yields\nΓ(3)\n11=1\n4πv2/bracketleftbiggπ/epsilon1(/epsilon1+ 2λ)τ\n/epsilon1+λ(1 +ıωτ) + log/vextendsingle/vextendsingle/vextendsingle/vextendsingleΛ2\n/epsilon12+ 2/epsilon1λ/vextendsingle/vextendsingle/vextendsingle/vextendsingle−1−ωπ/epsilon1λτ\n2(/epsilon1+λ)2/bracketrightbigg\n, (90)\nΓ(3)\n12=1\n4πv2/bracketleftbigg2ıπ/epsilon1+ 2λLII\n4λ−ω/epsilon1+λ\n2/epsilon1λ/parenleftbigg\n1 + 2π/epsilon1+ 2λ\n4λ/parenrightbigg/bracketrightbigg\n, (91)\nand the expressions for 1↔2again obtainable with the replacement λ→−λ.\nC Full form of the diffuson\nHere we report the full form of the two relevant blocks of the diffuson, involving Sy,z. The expressions are provided\nat leading order in the expansions for ωτ/lessmuchλτ/lessmuch1/lessmuch/epsilon1τ.\n•Subspaceσx, sy, σysz, σzsx\nD−1/vextendsingle/vextendsingle\nsy=\n1\n2(1−ıωτ)2λ3τ2\n/epsilon1(1 + 3ıωτ)−λ2τ\n/epsilon1(1 + 2ıωτ)λ3τ\n/epsilon12(1 + 2ıωτ)\n2λ3τ2\n/epsilon1(1 + 3ıωτ) 2λ2τ2−ıωτ−λτ(1 + 2ıωτ)λ2τ\n/epsilon1(1 + 2ıωτ)\nλ2τ\n/epsilon1(1 + 2ıωτ)λτ(1 + 2ıωτ)1\n2(1−ıωτ)λ\n2/epsilon1(1 +ıωτ)\n−λ3τ\n/epsilon12(1 + 2ıωτ)−λ2τ\n/epsilon1(1 + 2ıωτ)λ\n2/epsilon1(1 +ıωτ) 1−ıωτλ2\n2/epsilon12\n,(92)\n•Subspacesz, σxsx, σysy, σz\nD−1/vextendsingle/vextendsingle\nsz=\n4λ2τ2−ıωτ λτ (1 + 2ıωτ)λτ(1 + 2ıωτ)−λ\nπτ/epsilon12+O[(/epsilon1τ)−4]\n−λτ(1 + 2ıωτ)1\n2(1−ıωτ)λ2τ2\n2(1 + 3ıωτ)O[(/epsilon1τ)−4]\n−λτ(1 + 2ıωτ)λ2τ2\n2(1 + 3ıωτ)1\n2(1−ıωτ)O[(/epsilon1τ)−4]\n−λ\nπτ/epsilon12+O[(/epsilon1τ)−4]O[(/epsilon1τ)−4]O[(/epsilon1τ)−4] 1 +ıω\n4/epsilon12τ\n.(93)\nD Equation for operators instead of vertices\nIn the main text, we have written equations of motion for the renormalized vertices, rather than for the observables\nthemselves. As an example, here we report the diffusive matrix D−1for the observables, in the relevant sub-block\nsy/σxfor the in-plane spin dynamics. Also here we consider the response to an external electric field Ex. To this\naim we recall in the Gaussian approximation (cf. Eq.(31) and (32))\nJx=σxxEx=v2\nα(˜γx0x0−1)Ex, (94)\nSy=vχy,0xEx=v\n2α˜γx00yEx. (95)\nManipulating Eq.(30) we have\nD−1˜γx0=γx0=⇒ C˜γx0=CDγx0, (96)D Equation for operators instead of vertices 16\nwhere we have defined the matrix\nC=vEx\nαdiag(v,1\n2). (97)\nConsequently by subtracting to both sides v2Exγx0/αwe have\nvEx\nα/bracketleftbigg/parenleftbiggv˜γx0x0\n˜γx0x0\n2/parenrightbigg\n−/parenleftbiggv\n0/parenrightbigg/bracketrightbigg\n≡/parenleftbiggJx\nSy/parenrightbigg\n= (CD−v2\nαEx1)γx0. (98)\nWe conclude the diffusive matrix for the observables is\nD−1\nobs= (CD−v2\nαEx1)−1. (99)\nDirect inspection shows that D−1\nobsandD−1share the same pole structure, justifying the approach in the main text.\nReferences\n[1] D. Huertas-Hernando, F. Guinea, and A. Brataas, Phys. Rev. B 74, 155426 (2006).\n[2] S. Konschuh, M. Gmitra, and J. Fabian. Phys. Rev. B 82, 245412 (2010).\n[3] D. Pesin, and A. H. MacDonald, Nature Materials 11, 409-416 (2012).\n[4] W. Han, R. Kawakami, M. Gmitra, and J. Fabian, Nature Nanotechnology 9, 794-807 (2014).\n[5] S. Roche, and S. O Valenzuela, Journal of Physics D: Applied Physics 47, 094011 (2014).\n[6] S. L.-Avila et al., Phys. Rev. Lett. 115, 106602 (2015).\n[7] M. B. Lundeberg, R. Yang, J. Renard, and J. A. Folk. Phys. Rev. Lett. 110,156601 (2013).\n[8] B. Raes et al., Nat. Comm. 711444 (2016).\n[9] S. Omar, M. H. D. Guimarães, A. Kaverzin, B. J. van Wees, and I. J. Vera-Marun. Phys. Rev. B 95, 081403(R)\n(2017).\n[10] M. Johnson, and R. H. Silsbee. Phys. Rev. Lett. 55, 1790 (1985).\n[11] F. J. Jedema, A. T. Filip and B. J. van Wees. Nature 410, 345 (2001).\n[12] N. Tombros et al.,Nature 448, 571-574 (2007).\n[13] C. Josza, et al., Phys. Rev. B 80, 241403(R) (2009).\n[14] M. Popinciuc et al., Phys. Rev. B 80, 214427 (2009).\n[15] W. Han, et al., Phys. Rev. Lett. 105, 167202 (2010).\n[16] T. Y. Yang et al., Phys. Rev. Lett. 107, 047206 (2011).\n[17] W. Han, and R. Kawakami, Phys. Rev. Lett. 107, 047207 (2011).\n[18] K. Pi, et al., Phys. Rev. Lett. 104, 187201 (2011).\n[19] S. Jo, et al., Phys. Rev. B 84, 075453 (2011).\n[20] P. J. Zomer, M. H. D. Guimaraes, N. Tombros, and B. J. Van Wees, Phys. Rev. B 86, 161416 (2012).\n[21] M. Drögeler et al., Nano Lett. 14, 6050 (2014).\n[22] M. Drögeler et al., Nano Lett. 16, 3533 (2016).\n[23] A. H. Castro Neto and F. Guinea. Phys. Rev. Lett. 103, 026804 (2009).\n[24] D. Huertas-Hernando, F. Guinea, and A. Brataas, Phys. Lett 103, 075453 (2009).D Equation for operators instead of vertices 17\n[25] C. Ertler, S. Konschuh, M. Gmitra, and J. Fabian, Phys. Rev. B 80, 041405(R) (2009).\n[26] D. Kochan, M. Gmitra, and J. Fabian, Phys. Rev. Lett. 112, 116602 (2014).\n[27] T. Maasen et al., Phys. Rev. B 83, 115410 (2011).\n[28] F. Volmer et al., Phys. Rev. B 88, 161405(R) (2013).\n[29] D. V. Fedorov et al., Phys. Rev. Lett. 110, 156602 (2013).\n[30] D. Soriano et al., 2D Materials 2, 022002 (2015).\n[31] D. V. Tuan, F. Ortmann, D. Soriano, S. O. Valenzuela, and S. Roche. Nat. Physics 10, 857 (2014).\n[32] D. V. Tuan, et al., Scientific Reports 6, 21046 (2016).\n[33] A. W. Cummings, and S. Roche, Phys. Rev. Lett. 116, 086602 (2016).\n[34] M. Wojtaszek, I. J. Vera-Marun, T. Maassen, and B. J. van Wees. Phys. Rev. B 87, 081402 (2013).\n[35] M. H. Guimarães et al., Nano Lett. 12, 3512 (2012).\n[36] J. I.-Aynés, M. H. D. Guimarães, R. J. Meijerink, P. J. Zomer, and B. J. van Wees. Phys. Rev. B 92, 201410(R)\n(2015).\n[37] Y. A. Bychkov, and E. I. Rashba, JEPT Lett. 39, 78 (1984).\n[38] M. W. Wu, J. H. Jiang, M. Q. Weng. Physics Reports 493, 61 (2010).\n[39] C. Huang. Y. D. Chong, and M. A. Cazalilla, Phys. Rev. B 94, 085414 (2016).\n[40] C. Huang, Y. D. Chong, and M. A. Cazalilla, Phys. Rev. Lett. 119, 136804 (2017).\n[41] H. Ochoa, A. H. Castro Neto, and F. Guinea. Phys. Rev. Lett. 108, 206808 (2012).\n[42] P. Zhang, and M. W. Wu, New Journal of Physics 14, 033015 (2012).\n[43] M. Milletarì, M. Offidani, A. Ferreira, and R. Raimondi, Phys. Rev. Lett. 119, 246801 (2017).\n[44] M. Offidani, M. Milletarì, R. Raimondi, and A. Ferreira, Phys. Rev. Lett. 119, 196801 (2017).\n[45] O. V. Dimitrova, Phys. Rev. B 71, 245327 (2005).\n[46] R. Raimondi, C. Gorini, P. Schwab, and M. Dzierzawa, Phys. Rev. B 74, 035430 (2006).\n[47] R. Raimondi, P. Schwab, C. Gorini, and G. Vignale, Annalen der Physik 524, 153-162 (2012).\n[48] A. Maleki, and R. Raimondi, Condensed Matter MDPI 17, 2(2) (2017).\n[49] C. Gorini et al., Phys. Rev. B 95, 205424 (2017).\n[50] The ordering of the basis is (A,B)t, where A= (A↑,A↓)includes up and down spin states on sublattice site\nA(similarly for B). We use natural units where ~≡1≡e, unless stated otherwise.\n[51] P. Schwab, R. Raimondi, and C. Gorini, EPL 93, 67004 (2011).\n[52] D. Hsieh et al. Nature 460, 1101 (2009).\n[53] V. Brosco, L. Benfatto, E. Cappelluti, and C. Grimaldi, Phys. Rev. Lett. 116, 166602 (2016).\n[54] M. Milletarì, and A. Ferreira, Phys. Rev. B 94, 134202 (2016).\n[55] A. Ferreira et al., Phys. Rev. B 83, 165402 (2011).\n[56] A. Ferreira, T. G, Rappoport, M. A. Cazalilla, and A. H. Castro Neto, Phys. Rev. Lett. 112, 066601 (2014).\n[57] J. H. Chen, W. G. Cullen, C. Jang, M. S. Fuhrer, and E. D. Williams, Phys. Rev. Lett. 102, 236805 (2009).D Equation for operators instead of vertices 18\n[58] M. Monteverde et al., Phys. Rev. Lett. 104, 126801 (2010).\n[59] Z. H. Ni et al.,Nano Letters 10, 3868–3872 (2010).\n[60] J. Katoch et al., Phys. Rev. B 82, 081417(R) (2010).\n[61] A. A. Burkov, A. S. Nunez, and A. H. MacDonald, PRB 70, 155308 (2004).\n[62] A. A. Burkov, and L. Balents, Phys. Rev. B69, 245312 (2004).\n[63] K. Shen, G. Vignale, and R. Raimondi, Phys. Rev. Lett. 112, 096601 (2014).\n[64] J. Rammer, Quantum Transport Theory, Taylor & Francis Inc, (1998).\n[65] E. McCann, et al., Phys. Rev. Lett. 97, 146805 (2006).\n[66] P. M. Ostrovsky, I. V. Gornyi, and A. D. Mirlin. Phys. Rev. B 74, 235443 (2006).\n[67] Attention must be paid on this point when considering space-dependent dynamics. In fact, while spin-injection\ncan be performed locally, electric field lines extending in space can generate spin nonlocally, i.e. far away from\nthe source. This is commonly taken into account to model spin-valve configurations for non-local Hanle spin\nprecession schemes [68, 40]. However this circumstance is irrelevant for our purpose of obtaining the SRTs.\n[68] D. A. Abanin, A. V. Shytov, L. S. Levitov, and B. I. Halperin, Phys. Rev. B 79, 03504 (2009).\n[69] I. A. Ado, I. A. Dmitriev, P. M. Ostrovsky and M. Titov. EPL 111, 37004 (2015).\n[70] M. Milletarì, and A. Ferreira, Phys. Rev. B 94, 201402(R) (2016).\n[71] E. McCann, and V. Falko, Phys. Rev. Lett 108, 166606 (2012).\n[72] A.W. Cummings, J.H. Garcia, J. Fabian, and S. Roche, Phys. Rev. Lett. 119, 206601 (2017).\n[73] T. S. Ghiasi, J. Ingla-Aynes, A. A. Kaverzin, and B. J. Van Wees, Nano Lett. 17, 7528-7532 (2017).\n[74] L. A. Benitez et al.,Nat. Physics 14, 303 (2017).\n[75] T. Wakamura, F. Reale, P. Palczynski, S. Guéron, C. Mattevi, and H. Bouchiat. Phys. Rev. Lett. 120, 106802\n(2018)."    },    {        "title": "1911.11157v2.Spin_liquids_in_geometrically_perfect_triangular_antiferromagnets.pdf",        "content": "Spin liquids in geometrically perfect triangular antiferromagnets\nYuesheng Li,1, 2Philipp Gegenwart,1and Alexander A. Tsirlin1,\u0003\n1Experimental Physics VI, Center for Electronic Correlations and Magnetism,\nUniversity of Augsburg, 86159 Augsburg, Germany\n2Wuhan National High Magnetic Field Center and School of Physics,\nHuazhong University of Science and Technology, 430074 Wuhan, China\nThe cradle of quantum spin liquids, triangular antiferromagnets show strong proclivity to magnetic\norder and require deliberate tuning to stabilize a spin-liquid state. In this brief review, we juxtapose\nrecent theoretical developments that trace the parameter regime of the spin-liquid phase, with\nexperimental results for Co-based and Yb-based triangular antiferromagnets. Unconventional spin\ndynamics arising from both ordered and disordered ground states is discussed, and the notion of a\ngeometrically perfect triangular system is scrutinized to demonstrate non-trivial imperfections that\nmay assist magnetic frustration in stabilizing dynamic spin states with peculiar excitations.\nI. INTRODUCTION\nFrustrated magnets entail competing exchange inter-\nactions and behave di\u000berently from their non-frustrated\ncounterparts, where all couplings act to stabilize a mag-\nnetic order. It is then natural that in the presence of\nfrustration any ordering e\u000bects are impeded, and a non-\nordered, paramagnetic-like state survives down to much\nlower temperatures than in conventional magnets. Even-\ntually, frustrated spin systems may not show any long-\nrange magnetic order at all, and enter instead a pecu-\nliar low-temperature state known as spin liquid . On the\nmost basic level, this state can be matched with ordi-\nnary liquids in the sense that spins develop short-range\ncorrelations but lack any long-range magnetic order, and\nthe system does not undergo symmetry breaking upon\ncooling.\nAn exact de\fnition of a spin liquid is, however, more\ncomplex than that and involves a fair amount of ambigu-\nity or even contention. Key ingredients of the spin-liquid\nstate are the absence of long-range magnetic order and\nthe presence of persistent spin dynamics down to zero\ntemperature. These two aspects essentially distinguish\nthe spin liquid from magnetically ordered states with\nsymmetry breaking, and from spin glasses, where spin\n\ructuations slow down upon cooling, so that spins even-\ntually become static. On the other hand, without pos-\ntulating any microscopic aspects of this disordered and\ndynamic (liquid-like) state, such a de\fnition allows fun-\ndamentally di\u000berent systems with similar phenomenology\nto be classi\fed as spin liquids.\nMicroscopically, one distinguishes between quantum\nspin liquids where spins are quantum-mechanically en-\ntangled, and classical spin liquids that can be naively\nseen as a \\soup\" of di\u000berent magnetic orders, all hav-\ning the same energy. This classical scenario leaves spins\nto \ructuate as long as the temperature is high enough\nto overcome transition barriers between di\u000berent ordered\nstates. At very low temperatures, classical spin liquid\n\u0003altsirlin@gmail.comis expected to freeze, whereas its quantum counterpart\nremains dynamic by virtue of quantum \ructuations. Ex-\nperimentally, this presence or absence of spin freezing\nserves as a useful diagnostic tool along with magnetic\nexcitations that bear signatures of many-body entangle-\nment in quantum spin liquids although may be exotic in\nclassical spin liquids too [1, 2].\nThe gap between these two de\fnitions { phenomeno-\nlogical and microscopic { has led, and still leads to a\ncommon confusion of whether a given material should be\ninterpreted as a spin liquid. If it is, how to juxtapose\nthe experimental magnetic response with theory, and if\nit is not, is there still a room for the spin-liquid physics?\nIt is exactly these controversial but pertinent issues that\nwe seek to address in the present brief review using tri-\nangular antiferromagnets as an example. For a more de-\ntailed introduction into the physics of spin liquids beyond\nthe triangular systems we refer readers to excellent sum-\nmaries [3, 4], as well as more technical and elaborate\noverviews of the \feld [5, 6] that were recently published.\nHistorically, triangular antiferromagnets were \frst sys-\ntems where magnetic frustration was encountered, and\nearly ideas of quantum spin liquid state were estab-\nlished [7]. On the experimental side, the \feld has seen\nseveral revivals related to active studies of organic charge-\ntransfer salts and, more recently, Co2+and Yb3+oxide\ncompounds that are the main topic of our present review.\nFor practical purposes, we restrict this review to geomet-\nrically perfect spin-1\n2triangular antiferromagnets, where\nmagnetic ions form regular triangular framework, and ex-\nclude systems with higher spin as well as systems with\nthree-fold frustrated loops comprising non-equivalent ex-\nchange pathways (an excellent overview of all those can\nbe found in Ref. 8). In particular, organic charge-transfer\nsalts that are often discussed in the context of spin-liquid\nphysics [9] are beyond the scope of our present review,\nbecause their spin lattices entail distorted triangles, and\nlow-temperature structural instabilities abound.\nAs simple and natural as it seems, the de\fnition of the\ngeometrically perfect material also appears to be contro-\nversial { perhaps even more controversial than the de\f-\nnition of the spin liquid itself. The natural developmentarXiv:1911.11157v2  [cond-mat.str-el]  12 Mar 20202\nFIG. 1. Spin states in triangular antiferromagnets. (a) Valence-bond solid with valence bonds between nearest neighbors;\na superposition of all such states makes the nearest-neighbor RVB state. (b) 120\u000eand stripe orders. (c) Magnetic phase\ndiagram [10] with lines corresponding to di\u000berent \u0001 values in Eq. (2); SL stands for the spin liquid. Panel (c) is reprinted with\npermission from Ref. 10, c\rAmerican Physical Society, 2018.\nof the \feld requires that every new spin-liquid candidate\nis claimed to be more perfect and ideal than its predeces-\nsors, which raises a question of whether any material is\ntruly \\geometrically perfect\". We shall discuss this issue\nat some length to show non-trivial departures from the\n\\perfection\", and argue that they may strongly in\ruence\nthe physics, sometimes in a very interesting way.\nII. THEORY\nA. Resonating valence bonds\nEarly ideas of the quantum spin liquid appeared in the\ncontext of Heisenberg spins on the triangular lattice. An-\nderson and Fazekas [7, 11] conjectured that long-range-\nordered 120\u000eN\u0013 eel state (Fig. 1b) obtained by classical\nminimization is not the lowest-energy state of a quan-\ntum system, where spins gain additional energy by form-\ning pairs, quantum-mechanical singlets. These singlet\npairs were termed valence bonds and treated using for-\nmalism initially developed by Rumer for molecules [12{\n14]. The intuitive chemical analogy was further backed\nby the fact that valence bonds not only occur as spin-\npairs but can also form resonant con\fgurations akin to\nPauling's idea of resonating bonds in molecules [15].\nThe presence or absence of the resonance e\u000bect distin-\nguishes valence-bond solid (VBS), the combination of\nstatic and non-resonating valence bonds on a lattice, from\nthe resonating-valence-bond (RVB) states obtained by a\nsuperposition of di\u000berent VBS states. It is this RVB\nstate with singlet pairs restricted to nearest neighbors\n(Fig. 1a), that was proposed as the lowest-energy, liquid-\nlike ground state of triangular antiferromagnets [7, 11].\nBoth VBS and RVB states would fall under the most\ngeneral de\fnition of a spin liquid in the sense that they\nlack long-range magnetic order and show persistent spin\ndynamics. However, the VBS state involves in fact some\nsymmetry breaking, although it now relates to singletpairs and not to individual spins. In real materials,\nthis order becomes even more tangible, because VBS\nstates are intertwined with lattice distortions that sta-\nbilize singlet pairs on bonds with stronger exchange in-\nteractions [9].\nAnother, and more crucial di\u000berence is that only the\nRVB states show fractional, fermionic (spin-1\n2) statis-\ntics of magnetic excitations that can be contrasted with\nthe bosonic (spin-1) statistics of magnons in long-range-\nordered (anti)ferromagnets. Excitation brings spin dimer\ninto a triplet state, which can be thought as a superpo-\nsition of classical states with two parallel spins. In the\nRVB states, these spins can separate from each other\nand move through the crystal independently, because\nthe ground-state wavefunction embraces all possible con-\n\fgurations of valence bonds. In this case, the spin-1\nexcitations break down (fractionalize) into two spinons\n(Fig. 2) that become elementary excitations of an RVB\nstate [16, 17]. Emergent spinon excitations re\rect the\nhighly non-trivial, entangled nature of RVB quantum\nspin liquids and bear direct relation to charge fraction-\nalization of fractional quantum Hall state in electronic\nsystems with quantum entanglement [18{20]. Topologi-\ncal aspects behind this relation are elaborated in a recent\nreview article [21].\nThe possibility of spinon and other fractionalized ex-\ncitations triggered major interest in theories of quan-\ntum spin liquids. Experimental work inspired by their\npredictions initially focused on the search for materials\nthat evade magnetic order as best candidates for real-\nworld hosts of spinon excitations, although some of the\nrecent results suggest that even long-range-ordered mag-\nnets may show non-trivial spectral features [22, 23] po-\ntentially related to fractionalization. The question of\nwhether these complex excitations are truly fractional-\nized is, however, much more subtle [24], and we shall see\nthis below.\nDi\u000berent types of spin liquids exist under the umbrella\nof RVB physics. In triangular systems, short-ranged va-3\nFIG. 2. Spinon excitation in the nearest-neighbor RVB state of triangular antiferromagnets. The excitation is created by\nbreaking one of the valence bonds. Two unpaired spins can propagate independently and constitute spin-1\n2(spinon) excitations.\nlence bonds like those shown in Fig. 1a give rise to a\ngappedZ2quantum spin liquid [25, 26] characterized by\na topological order [27] with two types of excitations.\nThose at higher energies are spinons due to the breaking\nof valence bonds, whereas low-energy excitations are vi-\nsons related to the mutual arrangement of the valence\nbonds and topological order therein [27, 28]. Among\nmany peculiar properties of these excitations, we note\ntheir potential usage in topological quantum comput-\ning [29].\nWhen valence bonds occur between all atoms in the\ncrystal, including distant neighbors, gapless spin liquid\ncharacterized by purely spinon excitations is formed [17].\nSpinons develop a Fermi surface and interact with the\nU(1) gauge \feld [30]. The main lure of this U(1) or,\ncolloquially, \\spinon-metal\" state is associated with high-\ntemperature superconductivity, because fractionalization\nseparates spin and charge and may allow charge propa-\ngate unhindered by magnetic e\u000bects. A comparatively\nrecent review of this topic can be found in Ref. 31.\nExperimentally, both gapless and gapped quantum\nspin liquids are characterized by a continuum of spinon\nexcitations conveniently probed by inelastic neutron scat-\ntering. Thermodynamic and transport measurements of-\nfer additional diagnostic tools. The \\short-ranged\" RVB\nstates are gapped and should give rise to an exponen-\ntial behavior of the magnetic susceptibility and speci\fc\nheat. In contrast, the gapless U(1) quantum spin liq-\nuid in a triangular system is identi\fed by the sub-linear\nT2\n3power-law behavior of the speci\fc heat [32], although\nlinear or even exponential (gapped) behavior may occur\ntoo when spinons interact, eventually reducing the gauge\nsymmetry to Z2[33, 34].\nPredictions for the magnetic susceptibility [35] as well\nas optical [36] and thermal [37] conductivities of spin liq-\nuids with spinon Fermi surfaces are available too. How-\never, one has to be aware that by no means these pre-\ndictions exhaust all possible experimental responses, nor\nthe family of RVB states embraces all possible instances\nof quantum spin liquids. Many other types of spin-liquid\nphases have been envisaged by theory [5, 6], and their ex-\nperimental identi\fcation should be taken pragmatically.\nAny unconventional behavior, such as linear or sublinear\nevolution of the magnetic speci\fc heat, coupled with the\nabsence of magnetic order and presence of an excitationcontinuum may be a strong, if not compelling signature\nof the spin-liquid physics.\nB. Spin Hamiltonians\nFurther evidence for the spin-liquid behavior may come\nfrom the fact that material in question lies close to the\ninteraction regime where theory predicts breakdown of\nmagnetic order with the formation of a spin liquid. For\nthe purpose of the current review, we restrict ourselves\nto spin models with pair-wise interactions typical for in-\nsulating systems. Materials like organic charge-transfer\nsalts approach or even undergo metal-insulator transi-\ntions and require more complex spin models that also in-\nclude multi-spin terms, most notably ring exchange, on\ntop of the pair-wise interactions. These additional terms\nplay crucial role in stabilizing quantum spin liquids [9].\nAnisotropic spin Hamiltonian of triangular antiferro-\nmagnets comprises several terms,\nH=X\nm\u0002\nHXXZ\nm+H\u0006\u0006\nm+Hz\u0006\nm\u0003\n; (1)\nwherem= 1 describes nearest-neighbor interactions,\nm= 2 describes second-neighbor interactions, etc. The\n\frst term of Eq. (1) stands for the XXZ Hamiltonian [38],\nHXXZ\nm=JmX\nhiji(Sx\niSx\nj+Sy\niSy\nj+ \u0001Sz\niSz\nj); (2)\nand \u0001 is the extent of the XXZ anisotropy in the notation\nof Refs. 10 and 39.\nThe second term describes diagonal components of the\nexchange beyond the XXZ model,\nH\u0006\u0006\nm=X\nhiji2J\u0006\u0006\nm[(Sx\niSx\nj\u0000Sy\niSy\nj) cos'\u000b\u0000\n\u0000(Sx\niSy\nj+Sy\niSx\nj) sin'\u000b];(3)\nwhereas the third term stands for the o\u000b-diagonal\nanisotropy,\nHz\u0006\nm=X\nhijiJz\u0006\nm[(Sy\niSz\nj+Sz\niSy\nj)] cos'\u000b\u0000\n\u0000(Sx\niSz\nj+Sz\niSx\nj) sin'\u000b]: (4)4\nHere,'\u000b= 0;\u00062\u0019=3 is the bond-dependent pre-factor\nthat re\rects the rotation of the local coordinate frame\nunder three-fold symmetry of the lattice.\nThe \u0001 = 1 and J\u0006\u0006\nm=Jz\u0006\nm= 0 regime implies\nisotropic (Heisenberg) interactions. In systems of our\ninterest, departures from this regime can be caused by\nthe easy-plane anisotropy (\u0001 <1), anisotropy in the\nxyplane (J\u0006\u0006\nm6= 0), or the o\u000b-diagonal anisotropy\n(Jz\u0006\nm6= 0).\nC. Survey of magnetic ground states\nContrary to Anderson's conjecture, Heisenberg anti-\nferromagnets with purely nearest-neighbor interactions\nfeature the long-range-ordered 120\u000eground state [40, 41]\nthat also survives (sometimes with weak modi\fcations) in\nthe presence of easy-axis (\u0001 >1) or easy-plane (\u0001 <1)\nanisotropies [42{44]. Only in the pure Ising limit will a\npartially disordered state occur [45]. This state is, how-\never, not a quantum spin liquid, because Ising spins are\nclassical. Large residual entropy indicative of a classical\nspin liquid is found indeed [45].\nQuantum spin liquid can be stabilized by interactions\nbeyond nearest neighbors. The J1\u0000J2model of Heisen-\nberg spins on the triangular lattice received ample at-\ntention and was shown to host a spin-liquid phase at\nJ2=J1'0:07\u00000:15 [46{50], although the nature of this\nphase remains vividly debated, with both gapless [51{54]\nand gapped [46, 55] scenarios being likely proposals. At\nhigherJ2=J1, a collinear stripe order (Fig. 1b) becomes\nstable [56{58].\nExchange anisotropy o\u000bers another route to the spin-\nliquid state(s). The e\u000bect of the J\u0006\u0006\n1andJz\u0006\n1terms of\nEqs. (3) and (4) largely resembles that of J2, because\nthey stabilize stripe order too [59, 60]. A region of the\nquantum spin liquid phase separates these stripe states\nfrom the 120\u000eorder [10, 61] and may thus appear even\nin the absence of interactions beyond nearest neighbors\n(Fig. 1c). Interestingly, this quantum spin liquid of the\nanisotropic J1-only model is connected to the correspond-\ning regime of the isotropic J1\u0000J2model [10, 61], indi-\ncating the same (conceivably, Dirac [54, 61]) type of a\nquantum spin liquid anticipated in triangular antiferro-\nmagnets if either of the J2,J\u0006\u0006\n1, orJz\u0006\n1are properly\ntuned { a positive message for the experiment. Another\n(dual) spin-liquid region has been identi\fed in the limit of\nlargeJz\u0006\n1[39] but may be harder to reach in real materi-\nals, whereJ\u0006\u0006\n1andJ\u0006\nzare usually smaller than J1. It is\nalso worth noting that none of these putative spin-liquid\nphases is directly related to the RVB states discussed\npreviously.\nIII. CO-BASED MATERIALS\nMany triangular antiferromagnets were studied over\nthe years, but only a handful of them show close rela-tion to the aforementioned physics, while others entail\ndi\u000berent types of structural deformations. We \frst dis-\ncuss Co-compounds that ful\fll our criterion of geometri-\ncal perfection, do not yet enter the spin-liquid state, but\ncan be described by model Hamiltonians in the vein of\nEq. (1) and reveal highly non-trivial excitations.\nA. Single-ion physics\nCo2+proved convenient for studying spin-1\n2antifer-\nromagnets and especially triangular systems. In the\noctahedral environment, Co2+(3d7) features S=3\n2,\nbut its orbital moment remains unquenched. Spin-\norbit coupling then splits this manifold and separates\nthe lower-lying Kramers doublet that becomes predom-\ninantly occupied at low temperatures, acting as an ef-\nfective spin-1\n2[62]. Orbital degeneracy is thus lifted by\nthe spin-orbit coupling without lowering the symmetry,\nand a regular atomic arrangement with the strong geo-\nmetrical frustration can be preserved, in stark contrast\nto other spin-1\n2ions. For example, Cu2+, Ti3+, and V4+\nare all subject to strong Jahn-Teller e\u000bects that distort\ntriangular frameworks or even lead to their defragmenta-\ntion [63, 64].\nThe typical splitting between the ground-state\n(je\u000b=1\n2) Kramers doublet and lowest excited state is of\nthe order of 10\u000020 meV [65], suggesting that at least be-\nlow 50 K the magnetism is purely spin-1\n2. In the trigonal\nand hexagonal symmetries typical for triangular antifer-\nromagnets, Co2+acts as an anisotropic magnetic ion with\ndi\u000berentg-values and di\u000berent exchange interactions for\nthe in-plane and out-of-plane directions [62]. The ex-\ntent of this XXZ exchange anisotropy is relatively weak,\nthough (Table I).\nMany of the Co2+-based triangular materials are non-\nfrustrated, because closely spaced Co2+ions feature fer-\nromagnetic interactions arising from nearly 90\u000esuperex-\nchange pathways, as in CoCl 2and CoBr 2[66] or\f-\nCo(OH) 2[67]. Robust antiferromagnetism is only pos-\nsible in materials with large Co{Co separations that\nnaturally eliminate any couplings beyond J1, because\nsecond-neighbor Co{Co distances of at least 10 \u0017A are pro-\nhibitively large for the superexchange. With negligible\nJ\u0006\u0006andJz\u0006terms, Co2+triangular antiferromagnets\nare doomed to remain in the 120\u000e-ordered state, but can\nbe used to probe its interesting dynamics.\nB. Hexagonal perovskites and magnetic excitations\nBest material prototypes of Co-based triangular an-\ntiferromagnets are found among hexagonal perovskites\nof the 6H-Ba 3CoX 2O9family with X = Sb [68{70],\nNb [71, 72], and Ta [73, 74] (Fig. 3a). They share many\nsimilarities, but only the Sb compound was so far studied\nin detail thanks to the availability of large single crys-\ntals [75]. It behaves as a typical easy-plane triangular5\nFIG. 3. Excitations of the 120\u000eordered state. (a) Crys-\ntal structure of 6H-Ba 3CoSb 2O9with the triangular layers of\nCo2+ions in theabplane. (b) Excitation spectrum probed by\ninelastic neutron scattering, with the white lines showing one-\nmagnon dispersion from linear spin-wave theory [84]. Note\nthe excitation continuum that appears above 1 meV at the\nK-point and above 1.8 meV at the M-point, while at lower\nenergies quasiparticle bands repelled by the continuum [85]\nare observed [76, 84]. Panel (a) was prepared using the VESTA\nsoftware [86], and Ba atoms were omitted for clarity. Panel\n(b) is reprinted from Ref. 84, c\rCC-BY-4.0.\nantiferromagnet with the 120\u000emagnetic order in zero\n\feld [68, 70, 76] and anisotropic magnetization process\nrevealing the1\n3-plateau at 10\u000015 T for in-plane \felds, but\nno plateau when the \feld is applied perpendicular to the\neasy plane [77{79]. High-\feld behavior of Ba 3CoSb 2O9\nwas of signi\fcant interest, because the coplanar V-phase\nstabilized by quantum \ructuations [80, 81] could be de-\ntected experimentally for the \frst time in a spin-1\n2mag-\nnet [82, 83]. However, the main draw of this material\nrelates to its non-trivial magnetic excitations that were\nrecently juxtaposed with theoretical predictions for tri-\nangular antiferromagnets.\nAlready the \frst theoretical studies of spin dynamics\nexposed salient deviations from non-interacting magnon\nscenario of linear spin-wave theory. Not only the spin\nwaves are renormalized, sometimes changing their shape\nto produce roton minima [87] around the M-points [88{\n90], but also the spectrum is washed out into a broad\ncontinuum at energies ~! > J 1[90{93]. These \fndings\nwere initially interpreted as signatures of spinon excita-\ntions { an idea inspired by the work on square-lattice\nHeisenberg antiferromagnets.\nSpin-1\n2antiferromagnets, especially in 2D, show re-\nduced ordered moments indicative of spin \ructuations\nin the ordered state. This \ructuating component can\nbe represented by an RVB state and held responsible for\nvarious spectral features [94], including the broadening\nof spectral lines at high energies interpreted as a spinon\ncontinuum [22, 95]. While a similar physics could be\nenvisaged in the triangular case [89], and spinons may\nindeed account for a large part of the calculated spec-\ntral weight [92], non-linear spin-wave theory o\u000bered an\nalternative explanation in terms of interacting magnons.\nThe roton minima can be reproduced by including the\n1=Scorrections [96, 97] that are crucial in this case,TABLE I. Microscopic parameters of Co-based triangular\nantiferromagnets. All N\u0013 eel temperatures TNand exchange\nconstantsJare given in Kelvin. Two parameter sets for\nBa3CoSb 2O9are obtained from \fts to the magnetization\ndata [77] and to the inelastic neutron scattering data in the\n1\n3-plateau phase [103], respectively.\nJ1\u0001TNTN=Jzgkg? Ref.\nBa3CoSb 2O9 19.5 0.95 3.8 0.21 3.87 3.84 [77]\n20.3 0.86 3.8 0.22 3.95 [103]\nBa2La2CoTe 2O12 22 3.8 0.17 3.5 4.5 [104]\nBa8CoNb 6O24 1.7 1.0 0.1? { 3.84 [105]\ngiven the nearly 60 % reduction of the ordered moment\nwith respect to its classical value [98, 99]. Moreover,\nnon-collinear spin arrangement facilitates magnon de-\ncays [100, 101] that eventually account for the continuum\nat~!>J 1[102].\nTwo aforementioned scenarios { exotic fractionalized\nexcitations vs. interacting magnons { are in fact encoun-\ntered in several materials of current interest, such as the\nKitaev candidate \u000b-RuCl 3[23, 24]. On the experimental\nside this leads to a large deal of ambiguity, because even\nthe observation of an excitation continuum, the main \fn-\ngerprint and ultimate signature of the spin-liquid physics,\nappears to be inconclusive when it comes to the question\nof whether spinons occur or magnons decay.\nExperiments on Ba 3CoSb 2O9con\frmed that only at\nlow energies, ~!=J1\u00141:6 meV, do the excitations re-\nsemble magnons (Fig. 3b). At higher energies, a broad\ncontinuum is observed [70, 76, 84]. Magnons are strongly\nrenormalized and damped [76], in agreement with non-\nlinear spin-wave calculations. As for the continuum part\nof the spectrum, the bulk of the spectral weight is ob-\nserved below 4 meV [84] corresponding to ~! <2:5J1in\nfair agreement with both spinon [92] and magnon [102]\nscenarios. The aspect missing in both is the double-band\nstructure around the M-point at 1:3\u00001:6 meV, (Fig. 3b)\nthe energy range where excitation continuum already ap-\npears in other parts of the Brillouin zone [84]. Recent\ntheory work ascribed this feature, avoided quasiparticle\ndecay, to an interaction between the one-magnon band\nand continuum [85]. Instead of smearing out the for-\nmer, strong interaction separates the two. The contin-\nuum shifts to higher energies, while the \\one-magnon\"\n(quasiparticle) band is pushed down in agreement with\nthe experimental observations.\nThis strong-interaction scenario relies on the presence\nof an excitation continuum, while making no assump-\ntions regarding its magnon or spinon origin. Indeed,\nwhereas magnons are a good starting point for under-\nstanding all spectral features of large-spin triangular an-\ntiferromagnets [106], the spin-1\n2case of Ba 3CoSb 2O9is\nmore involved. First, vestiges of the continuum are seen\nup to much higher energies ( ~!'6J1) than any theory\nwould predict. Second, exact parametrization obtained\nby inelastic neutron scattering in the1\n3-plateau phase of6\nFIG. 4. Crystal structures of triangular Co-based antiferro-\nmagnets. The caxis is along the vertical direction, whereas\ntriangular layers are in the abplane. VESTA software [86] was\nused for crystal structure visualization.\nBa3CoSb 2O9does not allow an adequate quantitative de-\nscription of the zero-\feld spectra even on the level of non-\nlinear spin-wave theory [103]. This leaves the problem of\ncalculating spectral properties of 120\u000e-ordered quantum\nantiferromagnet largely open, and the magnon-spinon di-\nchotomy unresolved.\nC. New materials\nZero-\feld 120\u000emagnetic order generic to the 6H-\nBa3CoX 2O9materials appears to be one obstacle in the\nrealization of a spin-liquid phase. This motivated several\nattempts to suppress magnetic order by increasing the\ndistance between the Co2+layers.\nFrom the structural standpoint, triangular layers of\nCo2+are separated by perovskite-type slabs with non-\nmagnetic ions. The thickness of these slabs can be in-\ncreased (in theory, arbitrarily) leading to compounds\nlike Ba 2La2CoX 2O12(X = Te, W) with the inter-\nlayer Co{Co distance of about 9.8 \u0017A and, eventually, to\nBa8CoNb 6O24, where the triangular planes of Co2+are\nas far as 19 \u0017A apart (Fig. 4). With the exception of the\nlatter, these compounds still develop long-range magnetic\norder [104, 107]. In fact, even the Mn-based (spin-5\n2) ana-\nlog of Ba 8CoNb 6O24reveals the 120\u000eorder [108], whereas\nBa8CoNb 6O24itself also shows a characteristic peak in\nthe spin-lattice relaxation rate at 0.1 K reminiscent of\nthe magnetic ordering transition, although the NQR line\ndoes not broaden below this temperature [109]. These\nobservations suggest that residual interlayer couplings\n(likely of dipolar nature) always remain in place and tend\nto induce magnetic order with a sizable TN=J= 0:1\u00000:2.\nDespite the increased interlayer spacing and reduced\nTN, even the most 2D materials show the same dynamics\nas their less 2D analogs [105]. They can be most natu-\nrally understood as systems approaching eventual long-\nrange order, which is indeed anticipated in any J1-onlyXXZ antiferromagnet. An important lesson from these\nmaterials is that not only the interlayer couplings but\nalsoJ1can become very small (Table I), thus shifting\nthe onset of spin-spin correlations and TNto very low\ntemperatures. Recently reported materials like glaserite-\ntype Na 2BaCo(PO 4)2lacking magnetic order down to\n50 mK [110] should also be scrutinized from this per-\nspective and do not necessarily feature spin-liquid be-\nhavior of any kind. Co2+vanadates of the same glaserite\nfamily entail ferromagnetic J1[111{113], suggesting the\npresence of a ferromagnetic exchange component that in\nphosphates may nearly cancel the antiferromagnetic one\nleading to a material with a very weak J1and only a\nweak frustration.\nIV. YBMGGAO 4\nThe Co2+compounds give access to only a limited part\nof the general parameter space of triangular antiferro-\nmagnets and remain in the region of the 120\u000emagnetic\norder. Other phases should be probed by systems with\nlarger second-neighbor couplings and/or with a more pro-\nnounced exchange anisotropy. Anisotropic exchange in-\nteractions with sizable o\u000b-diagonal terms occur between\n4fions [114], for example in spin-ice compounds with the\nthree-dimensional pyrochlore lattice [115]. The work on\n4fmagnets in 2D remained scarce until 2015 when syn-\nthesis and investigation of YbMgGaO 4[116] suggested\nthe possibility of a spin-liquid state and spurred interest\nin 4f-based triangular antiferromagnets. In contrast to\nthe Co2+compounds, where XXZ interaction regime is\nknown with all certainty, the 4 fions with their di\u000berent\nelectronic con\fgurations and crystal-\feld ground states\no\u000ber (at least potentially) a much broader diversity of mi-\ncroscopic scenarios. An inevitable drawback is that each\nmaterial features many exchange parameters leaving the\nmicroscopic parametrization ambiguous and the physical\nscenario controversial.\nMost of the 4 f-based triangular antiferromagnets re-\nported to date utilize Yb3+as the magnetic ion. We shall\ndiscuss its behavior in greater detail using the best stud-\nied material YbMgGaO 4as an example. A more detailed\nand technical overview of this material can be found in a\nrecent progress report [117].\nA. Nature of the Yb3+magnetism\nThe central part of YbMgGaO 4are Yb3+ions with\ntheir valence electrons residing in the 4 fshell, which is\nsubject to a strong spin-orbit coupling. While formally a\nJ=7\n2ion, Yb3+reveals an e\u000bective spin-1\n2physics at low\ntemperatures, because crystal electric \felds (CEFs) split\ntheJ=7\n2multiplet into four Kramers doublets, similar\nto the e\u000bect of spin-orbit coupling on Co2+(Sec. III A).\nOnly the lowest CEF level is relevant to cooperative mag-\nnetism observed at low temperatures. This level is a7\nFIG. 5. Determination of the exchange parameters in YbMgGaO 4[116]. The Curie-Weiss \ftting of the inverse susceptibility,\n\u001f\u00001(a), is performed in the temperature range of the Rln 2 plateau in the magnetic entropy, Smag(b), whereas the temperature-\nindependent van Vleck contribution \u001f0can be cross-checked by the slope of the magnetization isotherm, M(H), above the\nsaturation \feld (c). The saturated magnetization is Ms= \u0016g\u0016BS'1:6\u0016B/Yb3+compatible with S=1\n2and \u0016g'3:29, the\npowder-averaged g-value determined from ESR [118].\nKramers doublet and can be directly mapped onto a\nspin-1\n2problem. More precisely, one refers to the mag-\nnetic moment of Yb3+associated with this doublet as a\npseudospin-1\n2, because it combines strongly intertwined\nspin and orbital moments. Its principal feature is mag-\nnetic anisotropy caused by the complex nature of the\nground-state wavefunction and by the in\ruence of the\nhigher-lying CEF levels on the exchange.\nAnother crucial feature of Yb3+is the strong localiza-\ntion of its 4 felectrons. Despite this ultimate localiza-\ntion, magnetic interactions between the pseudospins are\nnot of purely dipolar nature and involve orbital overlap.\nFor example, in YbMgGaO 4with the Yb{Yb distance of\n3.85\u0017A, the dipolar interaction of 0.25 K makes only 14%\nof the total interaction J1'1:8 K determined experi-\nmentally from the Curie-Weiss temperature [118]. This\nputs forward superexchange as the main mechanism of\nmagnetic couplings in Yb-based triangular antiferromag-\nnets and even allows a microscopic evaluation of magnetic\ninteractions based on superexchange theory [119].\nThe superexchange is weak, though, so Yb3+oxides do\nnot reveal their interesting cooperative magnetism un-\nless cooled down to temperatures of the order of 1 K.\nMany of these compounds were known since decades but\ntraditionally described as simple paramagnets until mea-\nsured at low enough temperatures. This naturally sets a\nquestion of the relevant temperature scale. Which tem-\nperature is low enough, and how to decide whether the\nabsence of magnetic order down to a certain tempera-\nture is a signature of the spin-liquid behavior, or simply\na feature of very weak magnetic interactions that are eas-\nily overridden by thermal \ructuations within the selected\ntemperature range?\nB. Low-temperature behavior\nYbMgGaO 4serves as a good illustration of how this\ntemperature scale can be determined experimentally.\nThe strength of magnetic interactions is gauged by the\nCurie-Weiss temperature \u0012, but the Curie-Weiss \ft, \u001f=\nFIG. 6. Low-temperature properties of YbMgGaO 4. (a)\nPower-law scaling of the magnetic speci\fc heat Cmagmea-\nsured in zero \feld [116], and (b) thermal conductivity \u0014sup-\npressed with respect to the non-magnetic reference compound\nLuMgGaO 4[120]. Dotted lines are extrapolations that high-\nlight the absence of the linear contribution, which would be\nexpected in a gapless quantum spin liquid.\nC=(T\u0000\u0012)+\u001f0, must be done in a prudently chosen tem-\nperature range. At high temperatures (above 30 \u000050 K\nin Yb3+oxides), 1=\u001fdeviates from the linear behavior\ndue to CEF excitations (Fig. 5a), while at low temper-\natures, typically below 10 K, magnetic interactions come\ninto play. Therefore, both upper and lower limits of the\n\ft require a careful consideration.\nMagnetic entropy guides the choice. Between 10 and\nabout 30 K, it shows a plateau at Rln 2 corresponding to\nthe ground-state doublet (Fig. 5b). The plateau implies\nthat, on one hand, magnetic interactions have been over-\nridden by thermal \ructuations and, on the other hand,\nthe temperature is not high enough to trigger CEF ex-\ncitations, although the very presence of the higher-lying\nCEF levels causes the non-zero van Vleck term \u001f0in the\nsusceptibility. Fitting three parameters ( \u001f0,\u0012, and the\nCurie constant C) in such a narrow temperature win-\ndow certainly becomes ambiguous, but the estimate of\n\u001f0can be cross-checked by measuring \feld dependence\nof the magnetization, because above saturation M(H)\nstill shows a small linear slope caused by the \u001f0Hcontri-\nbution due to the CEF excitations (Fig. 5c). As for the\nCurie constant C, it is proportional to the square of the\ng-factor that, in turn, can be independently determined8\nFIG. 7. Excitation continuum in YbMgGaO 4at energies\nof 0.3 meV ( ~!= 2J1) and 1.5 meV ( ~!= 10J1) [122].\nRight panel shows the calculation based on the spinon-metal\nscenario. Reprinted from Ref. 122 with the permission, c\r\nMacmillan Publishers Limited, 2016.\nfrom electron spin resonance (ESR). This leaves \u0012as the\nonly independent \ftting parameter and eventually leads\nto the estimates of J1'1:8 K and \u0001'0:55 from the\ninverse susceptibility measured for di\u000berent \feld direc-\ntions [118].\nThe interaction strength of about 2 K also manifests\nitself in other physical quantities. Magnetic speci\fc heat\n(Fig. 6a) shows a maximum at 2.5 K due to short-range\norder [116], whereas muon relaxation rate gradually in-\ncreases around the same temperature, indicating the on-\nset of spin-spin correlations [121]. At temperatures well\nbelow 2 K, one expects that spin-spin correlations dom-\ninate over thermal \ructuations, and experimental re-\nsponse of the magnetic ground state can be probed. Only\nvia measurements in this low-temperature range can one\nassure that the given material hosts a spin-liquid state or\nat least bears relation to the spin-liquid physics.\nIn YbMgGaO 4, several experimental techniques\nprobed the nature of the ground state. First, mag-\nnetic speci\fc heat shows no anomalies down to at least\n50 mK [116], whereas muons reveal persistent spin dy-\nnamics [121]: two necessary conditions of the spin-liquid\nbehavior. Below 0.4 K, magnetic speci\fc heat follows the\nCm(T)\u0018T\rpower law with \r'0:7 [116], indicating\ngapless spin excitations (Fig. 6). Moreover, the \rvalue\nis compatible with2\n3expected for the U(1) quantum spin\nliquid in triangular antiferromagnets (Sec. II A).\nFractionalized (spinon) excitations of the U(1) state\nshould also manifest themselves in the magnetic response\nand thermal transport. These experiments did not ar-\nrive at a consistent picture, though. YbMgGaO 4shows\nno thermal transport due to spinons (Fig. 6b). More-\nover, its thermal conductivity is suppressed with respect\nto the non-magnetic Lu-based analog suggesting the ab-\nsence of mobile spinons [120]. ac-susceptibility measure-\nments even revealed signatures of spin freezing around\n100 mK [123], but with only a small magnitude of thecusp and without any associated peak in the speci\fc heat,\nalthough in spin glasses entropy change at the freezing\npoint is usually detectable [124]. Moreover, no splitting\nbetween \feld-cooled and zero-\feld-cooled curves mea-\nsured in dc-\feld has been observed [125]. Magnetic re-\nsponse of YbMgGaO 4is thus very di\u000berent from that\nof a canonical spin glass { perhaps not surprising, given\npersistent spin dynamics probed by muons even below\nthe alleged freezing point [121]. All these observations\nsuggest that YbMgGaO 4does show dynamic spins and\nhosts a spin liquid of some kind, whereas the peak in the\nac-susceptibility may re\rect only a minor frozen compo-\nnent.\nC. Spin dynamics\nMagnetic excitations of YbMgGaO 4do not contribute\nto heat transport. Nevertheless, they do form a contin-\nuum that has been extensively studied by neutron scat-\ntering [122, 125{128]. It extends to about 2 meV corre-\nsponding to energies as high as ~!'13J1. With a very\nsimilar momentum dependence of the spectral weight at\nlow (0.3 meV) and high (1.5 meV) energies [122], this\nspectrum (Fig. 7) is drastically di\u000berent from the typ-\nical response of 120\u000e-ordered Co-based antiferromagnets\n(Sec. III B), where the spectral weight is restricted to\nmuch lower energies and shows di\u000berent momentum de-\npendence at di\u000berent energies, even inside the continuum.\nIn YbMgGaO 4, the spectral weight accumulates at\nthe zone boundary and appears to be nearly absent at\nthe zone center, similar to the kagome spin liquid pin-\npointed experimentally in herbertsmithite [129]. Gap-\nless spinon excitations of the U(1) quantum spin liq-\nuid can explain this distribution of the spectral weight\ndown to at least 0.3 meV [122]. This scenario was fur-\nther supported by a V-shaped band splitting in the ap-\nplied magnetic \feld [126] predicted theoretically for non-\ninteracting spinons [130], although interactions between\nspinons change the scenario qualitatively [131].\nAn alternative scenario was o\u000bered in Refs. 125 and 128\nthat interpret the same broad continuum as excitations\nout of a generic valence-bond state. Neutron scatter-\ning experiments down to 0.07 meV [125], about 4 times\nlower energy than in Ref. 122, suggest a change in the\nwidth of the continuum with the threshold value around\n0.2 meV, which is comparable to the interaction strength\nJ1(Fig. 8). Above this energy, the excitations can be as-\nsigned to the breaking of nearest-neighbor valence bonds,\nwhile below 0.2 meV the spectral weight is described by\nprocesses that involve a re-arrangement of valence bonds\nand orphan spins, with valence bonds up to third neigh-\nbors included in the model [125].\nInterestingly, both pictures entail spin-1\n2excitations\nalbeit with a very di\u000berent physics behind them. The\nspinon-metal scenario of Ref. 122 postulates fractional-\nized nature of the excitations without explicating their\nmicroscopic origin. It can be better traced in the valence-9\nFIG. 8. Valence-bond model of the YbMgGaO 4excitations [125, 128]. (a) Cartoon representation of the excitation continuum\ncomprising two types of processes: (b) breaking of nearest-neighbor valence bonds, and (c) re-arrangement of valence bonds\nand orphan spins, where only nearest-neighbor valence bonds are shown for the sake of clarity, but valence bonds up to third\nneighbors are included in the actual model. (d) Momentum dependence of the spectral weight at two energies in di\u000berent parts\nof the continuum [125].\nbond scenario, although here not all spin-1\n2excitations\nare fractionalized. The continuum of high-energy excita-\ntions indicates that, upon breaking a valence bond, two\nunpaired spins can separate, similar to spinon excitations\nof an RVB state. These spinon-like excitations show a\ngap of about 0.2 meV, which is on the order of J1and\nin agreement with theory (Sec. II A). On the other hand,\nlower-energy excitations caused by the re-arrangement\nof valence bonds \fll this gap, extend to lowest energies,\nand stem from the presence of orphan spins in the ground\nstate. These excitations are fractional but not fraction-\nalized. Their presence { or, more precisely, the inelastic\nnature of the re-arrangement process { further indicates a\ndeparture from the simple RVB state, where interchang-\ning of unpaired spins and valence bonds should cost no\nenergy. The \fnite energy cost of this process may stem\nfrom a local non-equivalence of di\u000berent lattice bonds\nthat, in turn, has to be caused by structural inhomo-\ngeneities.\nD. Structural randomness\nWhether or not YbMgGaO 4is prone to structural in-\nhomogeneities has been a matter of signi\fcant discussion.\nOn one hand, this material was put forward as a geomet-\nrically perfect triangular antiferromagnet based on its ro-\nbust trigonal symmetry that renders all nearest-neighbor\nYb{Yb distances as well as relevant superexchange path-\nways equal [116]. The absence of any detectable magnetic\nimpurities [118] and the lack of spin freezing (beyond the\ne\u000bects discussed in Sec. IV B) would also imply that this\nmaterial is less likely to su\u000ber from inhomogeneities and\ndisorder than other spin-liquid candidates, such as her-\nbertsmithite [132]. On the other hand, YbMgGaO 4is\nstill imperfect in the sense that Mg and Ga are randomly\ndistributed in the non-magnetic slabs that separate the\ntriangular planes of Yb3+(Fig. 11). Although such a\nmixture of non-magnetic atoms between the magnetic\nplaces would normally have little e\u000bect on the magnetismwithin these planes, the YbMgGaO 4case appears to be\ndi\u000berent.\nUnequal charges of Mg2+and Ga3+prove to be cru-\ncial. Depending on the local arrangement of these non-\nmagnetic species, the Yb3+ions experience di\u000berent\nCEFs. Inelastic neutron scattering reveals three CEF-\nrelated peaks, each of them being much broader than the\ninstrumental resolution (Fig. 9a). Moreover, a shoulder\naround 87 meV would indicate a \\fourth\" CEF excitation\nforbidden for Yb3+, since only three Kramers doublets\nare available for excitations. Ref. 133 o\u000bered an atomistic\ninterpretation of this strange CEF spectrum. The local\narrangement of Mg2+and Ga3+creates an uneven charge\ndistribution and causes local displacements of both Yb3+\nand surrounding oxygens (Fig. 9c). The magnitude of\nthese displacements being as large as 0.1 \u0017A implies that\nthe CEF energies may change compared to the undis-\ntorted scenario. A superposition of several local con\fg-\nurations obtained within this approach leads to a decent\ndescription of the experimental spectrum (Fig. 9b).\nWhereas CEF excitation energies do not determine\nmagnetic interactions per se , they in\ruence the g-values\nthat are spread over \fnite ranges \u0001 g?=g?'0:1 and\n\u0001gk=gk'0:3, respectively. Moreover, local displace-\nments change the Yb{O{Yb angles that may not a\u000bect\nrelative values of the exchange parameters, but do change\ntheir absolute values. For example, from the superex-\nchange theory of Ref. 119 one expects that the absolute\nvalue ofJ1varies by about 50% throughout the crystal.\nOther experiments support not only the presence of\nthis structural randomness, but also its tangible e\u000bect\non the magnetism. First, absent magnetic contribution\nto the thermal conductivity [120] is naturally explained\nby the random exchange couplings that will cause local-\nization of magnetic excitations regardless of their exact\norigin. Second, inelastic scattering in the fully polarized\nstate above 7.5 T shows an abnormally broad distribution\nof the spectral weight and hardly resembles spin wave of\na ferromagnet [127, 133]. Third, the spin-liquid state of\nYbMgGaO 4is remarkably insensitive to pressure [134],10\nFIG. 9. CEF excitations of YbMgGaO 4[133]. (a) Inelastic neutron scattering reveals an additional feature around 87 meV\nincompatible with four Kramers doublets of Yb3+(momentum dependence and the data for LuMgGaO 4, the non-magnetic\nreference compound, exclude possible phonon origin of this excitation). (b) Simulation of the experimental spectrum combined\nfrom several local con\fgurations obtained with di\u000berent distributions of Mg2+and Ga3+around Yb3+; note that the 87 meV\nfeature appears as the side peak of the highest CEF excitation. (c) Sample local con\fguration with opposite displacements of\nYb3+and O2\u0000caused by the uneven charge distribution.\nsuggesting that structural inhomogeneities may be in-\nstrumental in destabilizing magnetic order and facilitat-\ning spin dynamics. All these observations serve as the\nmost direct evidence that structural randomness is cen-\ntral to the magnetism of YbMgGaO 4.\nE. Possible scenarios\nSeveral microscopic parameterizations reported for\nYbMgGaO 4are summarized in Table II. Whereas all\nstudies agree on the presence of easy-plane anisotropy\n(\u0001<1), more subtle (but crucial) details of the second-\nneighbor interactions and o\u000b-diagonal anisotropy remain\ncontroversial and illustrate challenges in the experimen-\ntal determination of the exchange parameters for 4 fmag-\nnetic ions. As many as four independent parameters have\nto be used for nearest-neighbor interactions, another four\nparameters can be envisaged for second-neighbor inter-\nactions, etc.\nBy disregarding J2, the authors of Ref. 118 used\nCurie-Weiss temperatures and electron-spin-resonance\n(ESR) linewidths to determine all four components of\nthe nearest-neighbor exchange tensor. This leads to a\nsizable XXZ anisotropy with the relatively weak but non-\nnegligible additional terms J\u0006\u0006\n1andJz\u0006\n1(Table II, set\nI). Fits to the inelastic neutron data [127] suggest a\nroughly similar interaction regime [137], but with a siz-\nable second-neighbor coupling J2=J1'0:2 that is as-\nsumed to be isotropic (set II). The primary reason for\nincludingJ2is the peak of the neutron spectral weight\nat theM-points (Fig. 10b), as typical for the stripe phase\natJ2=J1>0:15. Alternatively, this stripe phase can be\nstabilized by the J\u0006\u0006\n1andJz\u0006\n1terms (set IV) that de-\nscribe the neutron data even in the absence of J2[135].TABLE II. Exchange parameters for YbMgGaO 4estimated\nby \ftting di\u000berent sets of the experimental data: i) Curie-\nWeiss temperatures and ESR linewidths [118]; ii) inelastic\nneutron scattering in the fully polarized state and di\u000buse scat-\ntering in zero \feld modeled with [127] and without [135] J2;\niii) inelastic neutron scattering and THz spectra [136]. The\nexchange parameters are introduced in Sec. II B and given in\nKelvin with error bars where available.\nI [118] II [127] III [136] IV [135]\nJ1 1.8(2) 2.54(5) 1.98(7) 2.5\n\u0001 0.54(5) 0.58(2) 0.88(3) 0.76\nJ\u0006\u0006\n1=J1 0.09(1) 0.06 0.4(3) 0.26\nJz\u0006\n1=J1 0.02(4) 0 0.6(6) 0.45\nJ2=J1 0 0.22 0.18(7) 0\nOn the other hand, it was argued that these terms induce\na large magnon gap [138] that is incompatible with the\napparent gapless behavior [116].\nThe accuracy of the neutron-based parametrization\nwas improved by adding THz data that probe excita-\ntions at the zone center, as opposed to neutron scatter-\ning that is more sensitive to the zone boundary. This\ncombined approach [136] hardly improves estimates of\ntheJ\u0006\u0006\n1andJz\u0006\n1terms, but lends additional con\fdence\nin the \fnite- J2scenario (set III). The estimated value of\nJ2= 0:3\u00000:4 K largely exceeds the dipolar coupling of\n0.07 K and serves as the \frst experimental evidence of\nthe long-range superexchange in Yb3+magnets.\nWithJ2=J1'0:2, YbMgGaO 4may not be far from\nthe quantum spin liquid region of J1\u0000J2triangular an-\ntiferromagnets (Sec. II C). An optimistic scenario would\ndeem YbMgGaO 4the \frst material prototype of this\nquantum spin liquid, but several experimental observa-11\nFIG. 10. Neutron scattering from Yb-based triangular an-\ntiferromagnets. (a) Static structure factor obtained theoret-\nically (DMRG) for the spin-liquid phase of the anisotropic\nspin Hamiltonian, Eq. (1) [10]. (b) Momentum dependence\nof the spectral weight for YbMgGaO 4(single crystal) with\npeaks at the M-points [127]. (c) Momentum dependence of\nthe spectral weight for NaYbO 2(powder sample) with the\nhighest intensity at the K-point [140]. Here, Kstands for the\nzone corner and Mfor the midpoint of the zone edge. Panel\n(a) is reprinted with permission from Ref. 10, c\rAmerican\nPhysical Society, 2018.\ntions speak against such an interpretation. First, low-\nenergy spectral weight peaks at the M-point [127], while\ntheory expects the peak at K[10] (Fig. 10). Second, ab-\nsent magnetic contribution to the thermal transport [120]\nleaves little room for a genuine, macroscopically entan-\ngled quantum state. Third, the randomness e\u000bect ex-\nplicated in Sec. IV D raises serious doubts about inter-\npreting YbMgGaO 4within the framework of any model\nhaving uniform exchange parameters.\nThe peak of the spectral weight at the M-point led\nto an idea that the ground state of YbMgGaO 4may\nbe a melted stripe order, where stripes are stabilized by\nthe sizable J2, but their directions are random under a\nchange in the sign of J\u0006\u0006\n1[138] or a variation of all ex-\nchange parameters throughout the crystal [139], the lat-\nter scenario being consistent with the predictions of the\nsuperexchange theory [119]. Both types of disorder lead\nto a liquid-like classical phase, although it remains un-\nclear whether excitations of this random stripe state [138]\nmay be responsible for the peculiar spin dynamics ob-\nserved experimentally (Sec. IV C).\nIn fact, all ordered phases of triangular antiferro-\nmagnets are rather unstable toward randomness e\u000bects.\nModels with random distribution of nearest-neighbor ex-\nchange couplings were considered in the literature [141,\n142] even before YbMgGaO 4made a compelling exper-\nimental case for their relevance. Already weak random-\nness transforms the 120\u000eorder into a glassy state [143],\nbut the most interesting behavior is found in the limit of\nstrong randomness and/or \fnite J2, where a gapless spin-\nliquid-like phase distinct from either spin glass or valence-bond glass appears [144]. This phase can be represented\nas a valence-bond state with short-range spin singlets and\na small fraction of unpaired (orphan) spins [145]. Alter-\nnatively, by analyzing a valence-bond state with random\nbond strengths, one can show that it is intrinsically un-\nstable toward nucleation of spin-1\n2topological defects,\norphan or unpaired spins [146].\nThe above scenario is remarkably similar to the phe-\nnomenological model developed for the interpretation\nof magnetic excitations in Ref. 125. Indeed, a suit-\nable parameterization of the valence-bond state with or-\nphan spins allows quantitative description of the low-\ntemperature thermodynamics, including the peculiar T2\n3\npower law of the speci\fc heat [146] that was initially\nascribed to the U(1) quantum spin liquid. Ref. 145 de-\nscribes this new, randomness-induced phase as a \\many-\nbody localized RVB state\", but it clearly deviates from\nthe conventional RVB scenario, because gapped vison ex-\ncitations are preempted by the gapless low-energy dy-\nnamics of orphan spins. An interesting question is\nwhether the mixed state of orphan spins and valence\nbonds, while not being an RVB state in the original sense,\nstill shows quantum entanglement. First numerical re-\nsults suggest that this may be the case [144], and give cer-\ntain hope that the randomness-induced spin-liquid-like\nphase is not yet another case of the spin-liquid mimicry,\nbut on a longer run may disengage itself from the pre-\ncautionary \\-like\" ending.\nV. OTHER 4fMATERIALS\nThe complexity of YbMgGaO 4arises from the intri-\ncate combination of magnetic frustration and structural\nrandomness. The former is essential, and the latter un-\navoidable, but a crucial problem on the materials side\nis whether this structural randomness can be reduced to\nthe level that it does not change the physics qualitatively,\nleaving room for the \\genuine\" behavior of regular trian-\ngular antiferromagnets described in Sec. II C.\nClose-packed layers of Yb3+, the main building block\nof YbMgGaO 4, are common for many structure types,\nincluding the abundant family of delafossites, where the\npresence of only one sort of non-magnetic species should\neliminate the randomness e\u000bect (Fig. 11). The absence\nof randomness was indeed con\frmed in NaYbO 2[140,\n147, 148] by the observation of three resolution-limited\npeaks of the CEF excitations and the absence of extra\nfeatures in high-energy inelastic neutron spectra [140].\nNo signi\fcant randomness is expected in the isostruc-\ntural materials NaYbS 2[149], NaYbSe 2[150, 151], and\nCsYbSe 2[152] as well. Interestingly, all these compounds\nreveal main signatures of the spin-liquid behavior, absent\nmagnetic order [140, 150{152] and persistent spin dynam-\nics [140, 149, 153] down to low temperatures. Moreover,\nin NaYbO 2the spectral weight of the excitation contin-\nuum peaks at the K-point of the Brillouin zone [140, 148]\n(Fig. 10c) in agreement with theoretical results for the12\nFIG. 11. Crystal structures of Yb-based triangular anti-\nferromagnets: YbMgGaO 4, NaYbO 2as representative of the\nYb3+delafossites, and KBaYb(BO 3)2as representative of the\nYb3+borates. All structures are trigonal, with cchosen as\nthe vertical direction, whereas triangular layers are in the ab\nplane. VESTA software [86] was used for crystal structure vi-\nsualization.\nTABLE III. g-values and exchange parameters of the Yb3+\ndelafossites. The g-values are estimated by electron spin reso-\nnance [154], whereas Jand \u0001 are calculated from Curie-Weiss\ntemperatures obtained for two di\u000berent \feld directions and\nrepresented as \u0012=3\n2J, whereJis the cumulative (nearest-\nneighbor and next-nearest-neighbor) coupling between spins\npointing along this direction. Only the Jvalue is given for\nNaYbO 2due to the lack of single crystals for this compound.\ngkg?J \u0001 Ref.\nNaYbO 2 1.75 3.28 6.0 { 147\nNaYbS 2 0.57 3.19 9.0 0.13 149\nNaYbSe 2 1.01 3.13 4.7 0.49 151\nquantum-spin-liquid phase of triangular antiferromag-\nnets [10]. These promising observations suggest that the\nYb3+delafossites may give experimental access to the\nspin-liquid phase in the absence of structural disorder.\nSo far little is known about the microscopic regime\nof the Yb3+delafossites. Average exchange couplings\ngauged by their Curie-Weiss temperatures and saturation\n\felds are at least 2-3 times stronger than in YbMgGaO 4\n(Table III), which shifts the relevant temperature scale\ntoward higher temperatures, but simultaneously pushes\nthe fully polarized phase above 14 \u000016 T [151], the fea-\nsibility limit of neutron-scattering experiments. The g-\ntensors determined by electron spin resonance are indica-\ntive of an easy-plane anisotropy (Table III) that may be\neven more pronounced than in the case of YbMgGaO 4.\nAnother generic property of the Yb3+delafossites is\ntheir \feld-induced phase transition between the puta-\ntive spin-liquid phase in zero \feld and one or several\nstates with long-range magnetic order. In-plane mag-\nnetic \felds trigger the transition already around 2 T ac-\ncompanied by the1\n3magnetization plateau [151, 152],\nwhich is compatible with the up-up-down magnetic or-\nder con\frmed experimentally for NaYbO 2[148]. In con-trast, out-of-plane \felds cause magnetic order only above\n8 T with no plateau feature in the magnetization. This\ndrastic di\u000berence in the transition \felds re\rects the siz-\nable easy-plane anisotropy (Table III), whereas the pres-\nence (absence) of the magnetization plateau for the in-\nplane (out-of-plane) \felds resembles the response of Co-\nbased XXZ triangular antiferromagnets [77, 78], where\ninteractions are restricted to nearest neighbors. Indeed,\nin the fully isotropic (Heisenberg) case, the up-up-down\nphase and associated1\n3-plateau occur only at J2=J1<\n0:125 [155, 156], suggesting that in the Yb3+delafossites\ntheJ2=J1ratio should be lower than in YbMgGaO 4, in\nagreement with the fact that the absolute value of J1\nincreases.\nOther Yb-based triangular materials include the family\nof borate compounds ABaYb(BO 3)2(A = Na, K) [157{\n159], where YbO 6octahedra are not directly linked to\neach other, but connected via BO 3triangles, with the\nnearest-neighbor Yb{Yb distance increasing to 5 :3\u00005:4\u0017A\n(Fig. 11). These materials may serve as interesting ref-\nerence systems, where J2is e\u000bectively suppressed, giv-\ning way to the purely nearest-neighbor triangular model.\nThe main question at this juncture is whether at least J1\ncan be strong enough to produce any tangible magnetism.\nThe Curie-Weiss temperatures well below 0.3 K [158, 159]\nsuggest that an extremely cold environment would be\nneeded to access frustrated behavior of these triangu-\nlar antiferromagnets. With the shortest interlayer Yb{\nYb distance (5 :3\u00005:4\u0017A) approaching the intralayer one\n(6:6\u00006:7\u0017A), 2D nature of the magnetism also comes into\nquestion, whereas mixing of A+and Ba2+in the same\ncrystallographic site creates random electric \felds acting\non Yb3+, similar to the YbMgGaO 4case (Sec. IV D).\nBoth delafossites [160], YbMgGaO 4-type [161], and\nmixed-cation borate [158, 162] compounds exist for many\nif not all 4fions. The properties of these materials re-\nmain to be explored, but a few literature cases suggest\nthat at least Ce-based compounds will likely reveal an ul-\ntimate XY anisotropy reported for Ce3+in the trigonal\nCEF [163]. Only a moderate easy-plane anisotropy has\nbeen observed in Er3+selenides, AErSe 2(A = Na, K,\nCs) [164, 165], but in this case the \frst crystal-\feld exci-\ntation lies below 1 meV in the selenides [165] or around\n2 meV in the isostructural sulphide [166] and is likely to\nimpact the low-energy physics.\nTriangular magnetic layers are also featured by some\nof the non-chalcogenide 4 fcompounds. Most notably,\nCeCd 3P3shows signatures of quasi-2D magnetism and\na long-range-ordering transition around 0.4 K, although\nit undergoes a structural phase transition already at\n127 K [167]. YbAl 3C3belongs to the same structural\nfamily and is known to develop singlet ground state [168,\n169] as a consequence of a similar structural phase transi-\ntion [170, 171]. Metallicity of these compounds [167] may\nbe another important ingredient. Itinerant electrons will\ngenerally mediate long-range interactions, both within\nand between the triangular planes, thus rendering the\nmapping onto simple short-ranged spin models impossi-13\nble or at least ambiguous.\nNon-Kramers ions can be accommodated in the trian-\ngular geometry too, although they are known to produce\nIsing spins [172] that are quite interesting on their own\nright but leave little room for quantum \ructuations and\nspin-liquid behavior, at least in zero \feld. TmMgGaO 4is\nan example of a triangular Ising antiferromagnet, where\nstructural randomness plays a role as crucial as in the\nYb3+analog, and the nature of magnetism remains con-\ntroversial [173, 174].\nVI. SUMMARY AND OUTLOOK\nTriangular antiferromagnets are no longer a land\nof static non-collinear magnetic structures of ever-\nincreasing complexity [8]. Recent theory work charted\nstability regions of the quantum spin liquid (Sec. II C)\nand identi\fed unusual quantum features in the excitation\nspectra of the 120\u000e-ordered state (Sec. III B). Structural\nrandomness brought yet another dimension into this al-\nready rich phase diagram by making unexpected connec-\ntions to valence-bond states that were historically pro-\nposed for triangular antiferromagnets and led to impor-\ntant conceptual developments but have been discarded as\npossible ground states of any realistic spin Hamiltonian.\nOn the experimental side, the research reviewed in\nthis article leads us to reconsider the notion of the ge-\nometrically perfect spin-liquid material and the role that\nrandomness or structural disorder play therein. The\nYbMgGaO 4case shows that neither lattice symmetry nor\ncomplete site occupations are su\u000ecient conditions of a\n\\perfect material\", because even subtle structural e\u000bects\nfar away from the magnetic planes can strongly a\u000bect\nmagnetic interactions and spin dynamics. This imposes\nvery hard requirements for the material characterization,\nand renders probes like CEF excitations central to decid-\ning whether a given material is \\perfect\" or not. On the\nmore positive side, it also o\u000bers a convenient experimen-\ntal knob for tuning the material toward a dynamically\ndisordered state, potentially a spin liquid.\nThe quest for \\structurally perfect\" spin-liquid mate-\nrials was largely motivated by the common notion of im-\nperfections being detrimental for the genuine spin-liquid\nstate. A material with imperfections and without mag-\nnetic order was supposed to be a classical spin liquid\nthat will freeze at low enough temperatures and show\nmundane spin dynamics. YbMgGaO 4reveals this may\nnot be the case, and even signatures of spin freezing\n(Sec. IV B) do not preclude dynamic behavior for the\nmajority of spins that, in fact, show highly unusual exci-\ntations. Taken together, these ideas suggest that struc-\ntural imperfections should not always be avoided, but\ncan be congenially used to facilitate spin dynamics. The\ngeneral questions of which structural imperfections can\nbe used in this way, and how to identify their in\ruence\non magnetic interactions, remain interesting avenues for\nfuture research in spin-liquid materials.With the phase diagram of anisotropic J1\u0000J2triangu-\nlar antiferromagnets fully charted and disorder-free ma-\nterials like Yb3+delafossites already available, experi-\nmental access to the quantum spin liquid phase outlined\nin Sec. II C becomes imminent. It makes, however, only\none out of many interesting questions in the \feld. We\noutline a few others below.\nFirst, with the exception of Ba 3CoSb 2O9\feld-induced\nbehavior is largely unknown, and even theory studies\nof the anisotropic J1\u0000J2Hamiltonians in the applied\n\feld remain scarce [155, 156]. NaYbO 2shows a quite\nunusual transition between the spin-liquid phase in zero\n\feld and the collinear up-up-down phase in the applied\n\feld [147, 148] (a similar transformation was claimed\nin randomness-in\ruenced YbMgGaO 4too [175]). Both\nphases may be quantum in nature, and their evolution\nis opposite to the typical scenario of Kitaev materials,\nwhere magnetic order is suppressed by the applied \feld\ngiving way to a disordered state, possibly a spin liq-\nuid [176]. A quantum critical point separating the spin-\nliquid and up-up-down phases can be envisaged, with the\nYb3+delafossites o\u000bering direct experimental access to\nit.\nSecond, structural randomness could be tuned.\nYbMgGaO 4lies in the limit of strong randomness, where\nvalence bonds coexist with a signi\fcant fraction of or-\nphan spins. Rami\fcations of this coexistence remain to\nbe fully understood, but the most clear one is that abun-\ndant disorder renders magnetic excitations localized. An\nopposite limit of weak randomness, with valence bonds\nand only a minute fraction of orphan spins, may be, in\ncontrast, the closest experimental realization of an RVB\nstate. It can be conceived in Yb-based delafossites or\nother disorder-free triangular compounds that are suit-\nably doped to induce weak randomness.\nThird, all Yb-based triangular antiferromagnets stud-\nied so far are not magnetically ordered, which is even\nsurprising because in other classes of materials \fnding\nan ordered state is by far easier than achieving a spin\nliquid. Tuning one of these materials toward an ordered\nstate (or \fnding another Yb-based triangular compound\nwith long-range magnetic order) may be an interesting\nendeavor. The primary goal here will be to probe mag-\nnetic excitations at di\u000berent energy scales, keeping in\nmind the experience with Co-based compounds, where\nexact nature of the excitation continuum remains con-\ntroversial. The proclivity of Yb3+and other 4 fions\nto the anisotropic exchange implies strong tendency to\nmagnon breakdown with unusual spectral features that\nwill benchmark theoretical studies of quantum spin-1\n2tri-\nangular antiferromagnets.\nUsing theoretical map chart and appreciating the two-\nfaced role of structural randomness will undoubtedly lead\nto new discoveries in the \feld of spin liquids and trian-\ngular antiferromagnets.14\nACKNOWLEDGMENTS\nIntriguing and inspiring, triangular antiferromagnets\nand especially YbMgGaO 4have triggered many fruit-\nful collaborations and insightful discussions. Our spe-\ncial thanks goes to Sasha Chernyshev for his critical\nvision and comprehensive theoretical development of\nanisotropic triangular models, and to Sebastian Bachus\nand Yoshi Tokiwa for their impressive work in the milli-K\nlab. We are grateful to Devashibhai Adroja, Lei Ding,\nFranziska Gru\u0019ler, Pascal Manuel, Astrid Schneidewind,and Qingming Zhang for common work on triangular\nantiferromagnets, to Martin Mourigal for his insightful\nand critical remarks, to Shiyan Li for providing thermal-\nconductivity data of YbMgGaO 4for this review, and to\nMichael Baenitz for sharing unpublished results on the\nYb3+delafossites. Last but not least, we thank Ger-\nman Science Foundation (DFG) for supporting our re-\nsearch on frustrated magnets via Project No. 107745057\n(TRR80). AT further thanks Alexander von Humboldt\nFoundation for the \fnancial support via the Sofja Ko-\nvalevskaya Award.\n[1] L. Balents, Spin liquids in frustrated magnets, Nature\n464, 199 (2010).\n[2] M. Gingras and P. McClarty, Quantum spin ice: a\nsearch for gapless quantum spin liquids in pyrochlore\nmagnets, Rep. Prog. Phys. 77, 056501 (2014).\n[3] J. Knolle and R. Moessner, A \feld guide to spin liquids,\nAnn. Rev. Condens. Matter Phys. 10, 451 (2019).\n[4] C. Broholm, R. Cava, S. Kivelson, D. Nocera, M. Nor-\nman, and T. Senthil, Quantum spin liquids, Science 367,\neaay0668 (2020).\n[5] L. Savary and L. Balents, Quantum spin liquids: a re-\nview, Rep. Prog. Phys. 80, 016502 (2017).\n[6] Y. Zhou, K. Kanoda, and T.-K. Ng, Quantum spin liq-\nuid states, Rev. Mod. Phys. 89, 025003 (2017).\n[7] P. Anderson, Resonating valence bonds: A new kind of\ninsulator?, Mater. Res. Bull. 8, 153 (1973).\n[8] O. Starykh, Unusual ordered phases of highly frustrated\nmagnets: a review, Rep. Prog. Phys. 78, 052502 (2015).\n[9] B. Powell and R. McKenzie, Quantum frustration in\norganic Mott insulators: from spin liquids to uncon-\nventional superconductors, Rep. Prog. Phys. 74, 056501\n(2011).\n[10] Z. Zhu, P. Maksimov, S. White, and A. Chernyshev,\nTopography of spin liquids on a triangular lattice, Phys.\nRev. Lett. 120, 207203 (2018).\n[11] P. Fazekas and P. Anderson, On the ground state\nproperties of the anisotropic triangular antiferromagnet,\nPhil. Mag. 30, 423 (1974).\n[12] G. Rumer, Zur Theorie der Spinvalenz, Nachrichten\nvon der Gesellschaft der Wissenschaften zu G ottingen,\nMathematisch-Physikalische Klasse , 337 (1932).\n[13] G. Rumer, E. Teller, and H. Weyl, Eine f ur die Valen-\nztheorie geeignete Basis der bin aren Vektorinvarianten,\nNachrichten von der Gesellschaft der Wissenschaften\nzu G ottingen, Mathematisch-Physikalische Klasse , 499\n(1932).\n[14] L. Pauling, The calculation of matrix elements for Lewis\nelectronic structures of molecules, J. Chem. Phys. 1, 280\n(1933).\n[15] L. Pauling and G. Wheland, The nature of the chemical\nbond. V. The quantum-mechanical calculation of the\nresonance energy of benzene and naphthalene and the\nhydrocarbon free radicals, J. Chem. Phys. 1, 362 (1933).\n[16] S. Kivelson, D. Rokhsar, and J. Sethna, Topology of\nthe resonating valence-bond state: Solitons and high- Tc\nsuperconductivity, Phys. Rev. B 35, 8865(R) (1987).[17] P. Anderson, G. Baskaran, Z. Zou, and T. Hsu,\nResonating-valence-bond theory of phase transitions\nand superconductivity in La 2CuO 4-based compounds,\nPhys. Rev. Lett. 58, 2790 (1987).\n[18] V. Kalmeyer and R. Laughlin, Equivalence of the\nresonating-valence-bond and fractional quantum Hall\nstates, Phys. Rev. Lett. 59, 2095 (1987).\n[19] R. Laughlin, The relationship between high-\ntemperature superconductivity and the fractional\nquantum Hall e\u000bect, Science 242, 525 (1988).\n[20] V. Kalmeyer and R. Laughlin, Theory of the spin liquid\nstate of the Heisenberg antiferromagnet, Phys. Rev. B\n39, 11879 (1989).\n[21] X.-G. Wen, Choreographed entanglement dances: Topo-\nlogical states of quantum matter, Science 363, eaal3099\n(2019).\n[22] B. Dalla Piazza, M. Mourigal, N. Christensen,\nG. Nilsen, P. Tregenna-Piggott, T. Perring, M. Enderle,\nD. McMorrow, D. Ivanov, and H. R\u001cnnow, Fractional\nexcitations in the square-lattice quantum antiferromag-\nnet, Nature Phys. 11, 62 (2015).\n[23] A. Banerjee, J. Yan, J. Knolle, C. Bridges, M. Stone,\nM. Lumsden, D. Mandrus, D. Tennant, R. Moessner,\nand S. Nagler, Neutron scattering in the proximate\nquantum spin liquid \u000b-RuCl 3, Science 356, 1055 (2017).\n[24] S. Winter, K. Riedl, P. Maksimov, A. Chernyshev,\nA. Honecker, and R. Valent\u0013 \u0010, Breakdown of magnons in\na strongly spin-orbital coupled magnet, Nature Comm.\n8, 1152 (2017).\n[25] R. Moessner and S. Sondhi, Resonating valence bond\nphase in the triangular lattice quantum dimer model,\nPhys. Rev. Lett. 86, 1881 (2001).\n[26] R. Moessner and S. Sondhi, Resonating valence bond\nliquid physics on the triangular lattice, Prog. Theor.\nPhys. Supplement 145, 37 (2002).\n[27] A. Ioselevich, D. Ivanov, and M. Feigelman, Ground-\nstate properties of the Rokhsar-Kivelson dimer model on\nthe triangular lattice, Phys. Rev. B 66, 174405 (2002).\n[28] D. Ivanov, Vortexlike elementary excitations in the\nRokhsar-Kivelson dimer model on the triangular lattice,\nPhys. Rev. B 70, 094430 (2004).\n[29] L. Io\u000be, M. Feigel'man, A. Ioselevich, D. Ivanov,\nM. Troyer, and G. Blatter, Topologically protected\nquantum bits using Josephson junction arrays, Nature\n415, 503 (2002).\n[30] G. Baskaran and P. Anderson, Gauge theory of high-\ntemperature superconductors and strongly correlated15\nFermi systems, Phys. Rev. B 37, 580(R) (1988).\n[31] P. Lee, N. Nagaosa, and X.-G. Wen, Doping a Mott in-\nsulator: Physics of high-temperature superconductivity,\nRev. Mod. Phys. 78, 17 (2006).\n[32] O. Motrunich, Variational study of triangular lattice\nspin-1/2 model with ring exchanges and spin liquid state\nin\u0014-(ET) 2Cu2(CN) 3, Phys. Rev. B 72, 045105 (2005).\n[33] S.-S. Lee, P. Lee, and T. Senthil, Amperean pairing in-\nstability in the U(1) spin liquid state with Fermi surface\nand application to \u0014-(BEDT-TTF) 2Cu2(CN) 3, Phys.\nRev. Lett. 98, 067006 (2007).\n[34] V. Galitski and Y. Kim, Spin-triplet pairing instability\nof the spinon Fermi surface in a U(1) spin liquid, Phys.\nRev. Lett. 99, 266403 (2007).\n[35] C. Nave, S.-S. Lee, and P. Lee, Susceptibility of a spinon\nFermi surface coupled to a U(1) gauge \feld, Phys. Rev.\nB76, 165104 (2007).\n[36] T.-K. Ng and P. Lee, Power-law conductivity inside the\nMott gap: Application to \u0014-(BEDT-TTF) 2Cu2(CN) 3,\nPhys. Rev. Lett. 99, 156402 (2007).\n[37] C. Nave and P. Lee, Transport properties of a spinon\nFermi surface coupled to a U(1) gauge \feld, Phys. Rev.\nB76, 235124 (2007).\n[38] Same interaction terms are often written as Jz\nmSz\niSz\nj\nandJ\u0006\nm(S+\niS\u0000\nj+S\u0000\niS+\nj) [2], where in the isotropic case\n(\u0001 = 1)J\u0006\nmis twice smaller than Jz\nm. This di\u000berence\nshould be taken into account when comparing exchange\nparameters reported in di\u000berent publications.\n[39] P. Maksimov, Z. Zhu, S. White, and A. Chernyshev,\nAnisotropic-exchange magnets on a triangular lattice:\nSpin waves, accidental degeneracies, and dual spin liq-\nuids, Phys. Rev. X 9, 021017 (2019).\n[40] D. Huse and V. Elser, Simple variational wave func-\ntions for two-dimensional Heisenberg spin-1\n2antiferro-\nmagnets, Phys. Rev. Lett. 60, 2531 (1988).\n[41] T. Jolicoeur and J. Le Guillou, Spin-wave results for\nthe triangular Heisenberg antiferromagnet, Phys. Rev.\nB40, 2727(R) (1989).\n[42] B. Kleine, E. M uller-Hartmann, K. Frahm, and\nP. Fazekas, Spin-wave analysis of easy-axis quantum an-\ntiferromagnets on the triangular lattice, Z. Phys. B 87,\n103 (1992).\n[43] D. Yamamoto, G. Marmorini, and I. Danshita, Quan-\ntum phase diagram of the triangular-lattice XXZ model\nin a magnetic \feld, Phys. Rev. Lett. 112, 127203 (2014).\n[44] D. Sellmann, X.-F. Zhang, and S. Eggert, Phase dia-\ngram of the antiferromagnetic XXZ model on the trian-\ngular lattice, Phys. Rev. B 91, 081104(R) (2015).\n[45] G. H. Wannier, Antiferromagnetism. the triangular\nIsing net, Phys. Rev. 79, 357 (1950).\n[46] Z. Zhu and S. White, Spin liquid phase of the S= 1=2\nJ1\u0000J2Heisenberg model on the triangular lattice, Phys.\nRev. B 92, 041105(R) (2015).\n[47] P. Li, R. Bishop, and C. Campbell, Quasiclassical mag-\nnetic order and its loss in a spin-1\n2Heisenberg antifer-\nromagnet on a triangular lattice with competing bonds,\nPhys. Rev. B 91, 014426 (2015).\n[48] S. Saadatmand, B. Powell, and I. McCulloch, Phase di-\nagram of the spin-1\n2triangularJ1\u0000J2Heisenberg model\non a three-leg cylinder, Phys. Rev. B 91, 245119 (2015).\n[49] W.-J. Hu, S.-S. Gong, W. Zhu, and D. Sheng, Compet-\ning spin-liquid states in the spin-1\n2Heisenberg model\non the triangular lattice, Phys. Rev. B 92, 140403(R)(2015).\n[50] S. Saadatmand and I. McCulloch, Symmetry fractional-\nization in the topological phase of the spin-1\n2J1\u0000J2tri-\nangular Heisenberg model, Phys. Rev. B 94, 121111(R)\n(2016).\n[51] R. Kaneko, S. Morita, and M. Imada, Gapless spin-\nliquid phase in an extended spin-1/2 triangular Heisen-\nberg model, J. Phys. Soc. Jpn. 83, 093707 (2014).\n[52] Y. Iqbal, W.-J. Hu, R. Thomale, D. Poilblanc, and\nF. Becca, Spin liquid nature in the Heisenberg J1\u0000J2\ntriangular antiferromagnet, Phys. Rev. B 93, 144411\n(2016).\n[53] S.-S. Gong, W. Zheng, M. Lee, Y.-M. Lu, and D. Sheng,\nChiral spin liquid with spinon fermi surfaces in the\nspin-1\n2triangular Heisenberg model, Phys. Rev. B 100,\n241111(R) (2019).\n[54] S. Hu, W. Zhu, S. Eggert, and Y.-C. He, Dirac spin\nliquid on the spin-1/2 triangular Heisenberg antiferro-\nmagnet, Phys. Rev. Lett. 123, 207203 (2019).\n[55] D.-V. Bauer and J. Fj\u001arestad, Schwinger-boson mean-\n\feld study of the J1\u0000J2Heisenberg quantum antifer-\nromagnet on the triangular lattice, Phys. Rev. B 96,\n165141 (2017).\n[56] T. Jolicoeur, E. Dagotto, E. Gagliano, and S. Bacci,\nGround-state properties of the S= 1=2 Heisenberg an-\ntiferromagnet on a triangular lattice, Phys. Rev. B 42,\n4800 (1990).\n[57] A. Chubukov and T. Jolicoeur, Order-from-disorder\nphenomena in Heisenberg antiferromagnets on a trian-\ngular lattice, Phys. Rev. B 46, 11137 (1992).\n[58] P. Lecheminant, B. Bernu, C. Lhuillier, and L. Pierre,\nJ1\u0000J2quantum Heisenberg antiferromagnet on the tri-\nangular lattice: A group-symmetry analysis of order by\ndisorder, Phys. Rev. B 52, 6647 (1995).\n[59] Y.-D. Li, X. Wang, and G. Chen, Anisotropic spin\nmodel of strong spin-orbit-coupled triangular antiferro-\nmagnets, Phys. Rev. B 94, 035107 (2016).\n[60] Q. Luo, S. Hu, B. Xi, J. Zhao, and X. Wang, Ground-\nstate phase diagram of an anisotropic spin-1\n2model on\nthe triangular lattice, Phys. Rev. B 95, 165110 (2017).\n[61] J. Iaconis, C. Liu, G. Hal\u0013 asz, and L. Balents, Spin liq-\nuid versus spin orbit coupling on the triangular lattice,\nSciPost Phys. 4, 003 (2018).\n[62] A. Abragam and B. Bleaney, Electron paramagnetic res-\nonance of transition ions (Clarendon Press, Oxford,\n1970).\n[63] H. Pen, J. van den Brink, D. Khomskii, and\nG. Sawatzky, Orbital ordering in a two-dimensional tri-\nangular lattice, Phys. Rev. Lett 78, 1323 (1997).\n[64] T. McQueen, P. Stephens, Q. Huang, T. Klimczuk,\nF. Ronning, and R. Cava, Successive orbital ordering\ntransitions in NaVO 2, Phys. Rev. Lett. 101, 166402\n(2008).\n[65] R. Bo\u0015 ca, Zero-\feld splitting in metal complexes, Coord.\nChem. Rev. 248, 757 (2004).\n[66] M. Wilkinson, J. Cable, E. Wollan, and W. Koehler,\nNeutron di\u000braction investigations of the magnetic or-\ndering in FeBr 2, CoBr 2, FeCl 2, and CoCl 2, Phys. Rev.\n113, 497 (1959).\n[67] D. Hunt, G. Garbarino, J. Rodr\u0013 \u0010guez-Velamaz\u0013 an,\nV. Ferrari, M. Jobbagy, and D. Scherlis, The magnetic\nstructure of \f-cobalt hydroxide and the e\u000bect of spin-\norientation, Phys. Chem. Chem. Phys. 18, 30407 (2016).16\n[68] Y. Doi, Y. Hinatsu, and K. Ohoyama, Structural and\nmagnetic properties of pseudo-two-dimensional triangu-\nlar antiferromagnets Ba 3MSb 2O9(M = Mn, Co, and\nNi), J. Phys.: Condens. Matter 16, 8923 (2004).\n[69] Y. Shirata, H. Tanaka, A. Matsuo, and K. Kindo, Ex-\nperimental realization of a spin-1/2 triangular-lattice\nHeisenberg antiferromagnet, Phys. Rev. Lett. 108,\n057205 (2012).\n[70] H. Zhou, C. Xu, A. Hallas, H. Silverstein, C. Wiebe,\nI. Umegaki, J. Yan, T. Murphy, J.-H. Park, Y. Qiu,\nJ. Copley, J. Gardner, and Y. Takano, Successive\nphase transitions and extended spin-excitation contin-\nuum in the S=1\n2triangular-lattice antiferromagnet\nBa3CoSb 2O9, Phys. Rev. Lett. 109, 267206 (2012).\n[71] M. Lee, J. Hwang, E. Choi, J. Ma, C. Dela Cruz,\nM. Zhu, X. Ke, Z. Dun, and H. Zhou, Series of\nphase transitions and multiferroicity in the quasi-two-\ndimensional spin-1\n2triangular-lattice antiferromagnet\nBa3CoNb 2O9, Phys. Rev. B 89, 104420 (2014).\n[72] K. Yokota, N. Kurita, and H. Tanaka, Magnetic phase\ndiagram of the S=1\n2triangular-lattice Heisenberg an-\ntiferromagnet Ba 3CoNb 2O9, Phys. Rev. B 90, 014403\n(2014).\n[73] K. Ranjith, K. Brinda, U. Arjun, N. Hegde, and\nR. Nath, Double phase transition in the triangular an-\ntiferromagnet Ba 3CoTa 2O9, J. Phys.: Condens. Matter\n29, 115804 (2017).\n[74] M. Lee, E. Choi, J. Ma, R. Sinclair, C. Dela Cruz, and\nH. Zhou, Magnetic and electric properties of triangular\nlattice antiferromagnets Ba 3ATa 2O9(A = Ni and Co),\nMater. Res. Bull. 88, 308 (2017).\n[75] D. Prabhakaran and A. Boothroyd, Crystal growth of\nthe triangular-lattice antiferromagnet Ba 3CoSb 2O9, J.\nCrystal Growth 468, 345 (2017).\n[76] J. Ma, Y. Kamiya, T. Hong, H. Cao, G. Ehlers, W. Tian,\nC. Batista, Z. Dun, H. Zhou, and M. Matsuda, Static\nand dynamical properties of the spin-1/2 equilateral\ntriangular-lattice antiferromagnet Ba 3CoSb 2O9, Phys.\nRev. Lett. 116, 087201 (2016).\n[77] T. Susuki, N. Kurita, T. Tanaka, H. Nojiri, A. Mat-\nsuo, K. Kindo, and H. Tanaka, Magnetization process\nand collective excitations in the S= 1=2 triangular-\nlattice Heisenberg antiferromagnet Ba 3CoSb 2O9, Phys.\nRev. Lett. 110, 267201 (2013).\n[78] G. Quirion, M. Lapointe-Major, M. Poirier, J. Quil-\nliam, Z. Dun, and H. Zhou, Magnetic phase diagram\nof Ba 3CoSb 2O9as determined by ultrasound velocity\nmeasurements, Phys. Rev. B 92, 014414 (2015).\n[79] A. Sera, Y. Kousaka, J. Akimitsu, M. Sera, T. Kawa-\nmata, Y. Koike, and K. Inoue, s=1\n2triangular-lattice\nantiferromagnets Ba 3CoSb 2O9and CsCuCl 3: Role of\nspin-orbit coupling, crystalline electric \feld e\u000bect, and\nDzyaloshinskii-Moriya interaction, Phys. Rev. B 94,\n214408 (2016).\n[80] A. Chubukov and D. Golosov, Quantum theory of an\nantiferromagnet on a triangular lattice in a magnetic\n\feld, J. Phys.: Condens. Matter 3, 69 (1991).\n[81] C. Griset, S. Head, J. Alicea, and O. Starykh, Deformed\ntriangular lattice antiferromagnets in a magnetic \feld:\nRole of spatial anisotropy and Dzyaloshinskii-Moriya in-\nteractions, Phys. Rev. B 84, 245108 (2011).\n[82] G. Koutroulakis, T. Zhou, Y. Kamiya, J. Thompson,\nH. Zhou, C. Batista, and S. Brown, Quantum phase\ndiagram of the S=1\n2triangular-lattice antiferromagnetBa3CoSb 2O9, Phys. Rev. B 91, 024410 (2015).\n[83] X. Liu, O. Prokhnenko, D. Yamamoto, M. Bartkowiak,\nN. Kurita, and H. Tanaka, Microscopic evidence of a\nquantum magnetization process in the S=1\n2triangular-\nlattice Heisenberg-like antiferromagnet Ba 3CoSb 2O9,\nPhys. Rev. B 100, 094436 (2019).\n[84] S. Ito, N. Kurita, H. Tanaka, S. Ohira-Kawamura,\nK. Nakajima, S. Itoh, K. Kuwahara, and K. Kaku-\nrai, Structure of the magnetic excitations in the\nspin-1/2 triangular-lattice Heisenberg antiferromagnet\nBa3CoSb 2O9, Nature Comm. 8, 235 (2017).\n[85] R. Verresen, R. Moessner, and F. Pollmann, Avoided\nquasiparticle decay from strong quantum interactions,\nNature Phys. 15, 750 (2019).\n[86] K. Momma and F. Izumi, VESTA 3 for three-\ndimensional visualization of crystal, volumetric and\nmorphology data, J. Appl. Crystallogr. 44, 1272 (2011).\n[87] These roton minima strongly in\ruence thermodynamic\nproperties of triangular antiferromagnets. Their relation\nto rotons in super\ruid4He is discussed in Ref. 89.\n[88] W. Zheng, J. Fj\u001arestad, R. Singh, R. McKenzie, and\nR. Coldea, Anomalous excitation spectra of frustrated\nquantum antiferromagnets, Phys. Rev. Lett. 96, 057201\n(2006).\n[89] W. Zheng, J. Fj\u001arestad, R. Singh, R. McKenzie, and\nR. Coldea, Excitation spectra of the spin-1\n2triangular-\nlattice Heisenberg antiferromagnet, Phys. Rev. B 74,\n224420 (2006).\n[90] F. Ferrari and F. Becca, Dynamical structure factor of\ntheJ1\u0000J2Heisenberg model on the triangular lattice:\nMagnons, spinons, and gauge \felds, Phys. Rev. X 9,\n031026 (2019).\n[91] A. Mezio, C. Sposetti, L. Manuel, and A. Trumper,\nA test of the bosonic spinon theory for the triangular\nantiferromagnet spectrum, Europhys. Lett. 94, 47001\n(2011).\n[92] E. Ghioldi, A. Mezio, L. Manuel, R. Singh, J. Oitmaa,\nand A. Trumper, Magnons and excitation continuum in\nXXZ triangular antiferromagnetic model: Application\nto Ba 3CoSb 2O9, Phys. Rev. B 91, 134423 (2015).\n[93] E. Ghioldi, M. Gonzalez, S.-S. Zhang, Y. Kamiya,\nL. Manuel, A. Trumper, and C. Batista, Dynami-\ncal structure factor of the triangular antiferromagnet:\nSchwinger boson theory beyond mean \feld, Phys. Rev.\nB98, 184403 (2018).\n[94] C.-M. Ho, V. Muthukumar, M. Ogata, and P. Anderson,\nNature of spin excitations in two-dimensional Mott in-\nsulators: Undoped cuprates and other materials, Phys.\nRev. Lett. 86, 1626 (2001).\n[95] N. Christensen, H. R\u001cnnow, D. McMorrow, A. Harri-\nson, T. Perring, M. Enderle, R. Coldea, L. Regnault,\nand G. Aeppli, Quantum dynamics and entanglement\nof spins on a square lattice, Proc. Nat. Acad. Sci. 104,\n15264 (2007).\n[96] O. Starykh, A. Chubukov, and A. Abanov, Flat spin-\nwave dispersion in a triangular antiferromagnet, Phys.\nRev. B 74, 180403(R) (2006).\n[97] A. Chernyshev and M. Zhitomirsky, Spin waves in a\ntriangular lattice antiferromagnet: Decays, spectrum\nrenormalization, and singularities, Phys. Rev. B 79,\n144416 (2009).\n[98] L. Capriotti, A. Trumper, and S. Sorella, Long-range\nN\u0013 eel order in the triangular Heisenberg model, Phys.\nRev. Lett. 82, 3899 (1999).17\n[99] S. White and A. Chernyshev, Neel order in square and\ntriangular lattice Heisenberg models, Phys. Rev. Lett.\n99, 127004 (2007).\n[100] A. Chernyshev and M. Zhitomirsky, Magnon decay\nin noncollinear quantum antiferromagnets, Phys. Rev.\nLett. 97, 207202 (2006).\n[101] M. Zhitomirsky and A. Chernyshev, Colloquium: Spon-\ntaneous magnon decays, Rev. Mod. Phys. 85, 219\n(2013).\n[102] M. Mourigal, W. Fuhrman, A. Chernyshev, and\nM. Zhitomirsky, Dynamical structure factor of the\ntriangular-lattice antiferromagnet, Phys. Rev. B 88,\n094407 (2013).\n[103] Y. Kamiya, L. Ge, T. Hong, Y. Qiu, D. Quintero-Castro,\nZ. Lu, H. B. Cao, M. Matsuda, E. Choi, C. Batista,\nM. Mourigal, H. Zhou, and J. Ma, The nature of spin\nexcitations in the one-third magnetization plateau phase\nof Ba 3CoSb 2O9, Nature Comm. 9, 2666 (2018).\n[104] Y. Kojima, M. Watanabe, N. Kurita, H. Tanaka,\nA. Matsuo, K. Kindo, and M. Avdeev, Quantum mag-\nnetic properties of the spin-1\n2triangular-lattice antifer-\nromagnet Ba 2La2CoTe 2O12, Phys. Rev. B 98, 174406\n(2018).\n[105] R. Rawl, L. Ge, H. Agrawal, Y. Kamiya, C. Dela\nCruz, N. Butch, X. Sun, M. Lee, E. Choi, J. Oit-\nmaa, C. Batista, M. Mourigal, H. Zhou, and J. Ma,\nBa8CoNb 6O24: A spin-1\n2triangular-lattice Heisenberg\nantiferromagnet in the two-dimensional limit, Phys.\nRev. B 95, 060412(R) (2017).\n[106] T. Kim, K. Park, J. Leiner, and J.-G. Park, Hy-\nbridization and decay of magnetic excitations in\ntwo-dimensional triangular lattice antiferromagnets, J.\nPhys. Soc. Jpn. 88, 081003 (2019).\n[107] R. Rawl, M. Lee, E. Choi, G. Li, K. Chen, R. Baumbach,\nC. dela Cruz, J. Ma, and H. Zhou, Magnetic properties\nof the triangular lattice magnets A 4B'B2O12(A = Ba,\nSr, La; B' = Co, Ni, Mn; B = W, Re), Phys. Rev. B\n95, 174438 (2017).\n[108] R. Rawl, L. Ge, Z. Lu, Z. Evenson, C. Dela Cruz,\nQ. Huang, M. Lee, E. Choi, M. Mourigal, H. Zhou, and\nJ. Ma, Ba 8MnNb 6O24: A model two-dimensional spin-5\n2\ntriangular lattice antiferromagnet, Phys. Rev. Mater. 3,\n054412 (2019).\n[109] Y. Cui, J. Dai, P. Zhou, P. Wang, T. Li, W. Song,\nJ. Wang, L. Ma, Z. Zhang, S. Li, G. Luke, B. Nor-\nmand, T. Xiang, and W. Yu, Mermin-Wagner physics,\n(H;T) phase diagram, and candidate quantum spin-\nliquid phase in the spin-1\n2triangular-lattice antiferro-\nmagnet Ba 8CoNb 6O24, Phys. Rev. Materials 2, 044403\n(2018).\n[110] R. Zhong, S. Guo, G. Xu, Z. Xu, and R. Cava, Strong\nquantum \ructuations in a quantum spin liquid can-\ndidate with a Co-based triangular lattice, Proc. Nat.\nAcad. Sci. 116, 14505 (2019).\n[111] A. M oller, N. Amuneke, P. Daniel, B. Lorenz, C. de la\nCruz, M. Gooch, and P. Chu, AAg 2M[VO 4]2(A = Ba,\nSr; M = Co, Ni): a series of ferromagnetic insulators,\nPhys. Rev. B 85, 214422 (2012).\n[112] G. Nakayama, S. Hara, H. Sato, Y. Narumi, and H. No-\njiri, Synthesis and magnetic properties of a new series of\ntriangular-lattice magnets, Na 2BaMV 2O8(M = Ni, Co,\nand Mn), J. Phys.: Condens. Matter 25, 116003 (2013).\n[113] L. Sanjeewa, V. Garlea, M. McGuire, C. McMillen, and\nJ. Kolis, Magnetic ground state crossover in a seriesof glaserite systems with triangular magnetic lattices,\nInorg. Chem. 58, 2813 (2019).\n[114] W. Wolf, Anisotropic interactions between magnetic\nions, J. Phys. Colloques 32, C1 (1971).\n[115] J. Rau and M. Gingras, Frustrated quantum rare-earth\npyrochlores, Ann. Rev. Condens. Matter Phys. 10, 357\n(2019).\n[116] Y. Li, H. Liao, Z. Zhang, S. Li, F. Jin, L. Ling,\nL. Zhang, Y. Zou, L. Pi, Z. Yang, J. Wang, Z. Wu, and\nQ. Zhang, Gapless quantum spin liquid ground state in\nthe two-dimensional spin-1/2 triangular antiferromag-\nnet YbMgGaO 4, Sci. Reports 5, 16419 (2015).\n[117] Y. Li, YbMgGaO 4: A triangular-lattice quantum\nspin liquid candidate, Adv. Quantum Technologies 2,\n1900089 (2019).\n[118] Y. Li, G. Chen, W. Tong, L. Pi, J. Liu, Z. Yang,\nX. Wang, and Q. Zhang, Rare-earth triangular lattice\nspin liquid: A single-crystal study of YbMgGaO 4, Phys.\nRev. Lett. 115, 167203 (2015).\n[119] J. Rau and M. Gingras, Frustration and anisotropic ex-\nchange in ytterbium magnets with edge-shared octahe-\ndra, Phys. Rev. B 98, 054408 (2018).\n[120] Y. Xu, J. Zhang, Y. Li, Y. Yu, X. Hong, Q. Zhang, and\nS. Li, Absence of magnetic thermal conductivity in the\nquantum spin-liquid candidate YbMgGaO 4, Phys. Rev.\nLett. 117, 267202 (2016).\n[121] Y. Li, D. Adroja, P. K. Biswas, P. J. Baker, Q. Zhang,\nJ. Liu, A. Tsirlin, P. Gegenwart, and Q. Zhang, Muon\nspin relaxation evidence for the U(1) quantum spin-\nliquid ground state in the triangular antiferromagnet\nYbMgGaO 4, Phys. Rev. Lett. 117, 097201 (2016).\n[122] Y. Shen, Y.-D. Li, H. Wo, Y. Li, S. Shen, B. Pan,\nQ. Wang, H. Walker, P. Ste\u000bens, M. Boehm, Y. Hao,\nD. L. Quintero-Castro, L. Harriger, M. Frontzek,\nL. Hao, S. Meng, Q. Zhang, G. Chen, and J. Zhao, Ev-\nidence for a spinon Fermi surface in a triangular-lattice\nquantum-spin-liquid candidate, Nature 540, 559 (2016).\n[123] Z. Ma, J. Wang, Z.-Y. Dong, J. Zhang, S. Li, S.-H.\nZheng, Y. Yu, W. Wang, L. Che, K. Ran, S. Bao,\nZ. Cai, P. \u0015Cerm\u0013 ak, A. Schneidewind, S. Yano, J. S.\nGardner, X. Lu, S.-L. Yu, J.-M. Liu, S. Li, J.-X. Li, and\nJ. Wen, Spin-glass ground state in a triangular-lattice\ncompound YbZnGaO 4, Phys. Rev. Lett. 120, 087201\n(2018).\n[124] B. Schmidt, H. Aoki, T. Cichorek, J. Custers, P. Gegen-\nwart, M. Kohgi, M. Lang, C. Langhammer, A. Ochiai,\nS. Paschen, F. Steglich, T. Suzuki, P. Thalmeier,\nB. Wand, and A. Yaresko, Low-energy excitations of the\nsemimetallic one-dimensional S= 1=2 antiferromagnet\nYb4As3, Physica B 300, 121 (2001).\n[125] Y. Li, S. Bachus, B. Liu, I. Radelytskyi, A. Bertin,\nA. Schneidewind, Y. Tokiwa, A. Tsirlin, and P. Gegen-\nwart, Rearrangement of uncorrelated valence bonds ev-\nidenced by low-energy spin excitations in YbMgGaO 4,\nPhys. Rev. Lett. 122, 137201 (2019).\n[126] Y. Shen, Y.-D. Li, H. Walker, P. Ste\u000bens, M. Boehm,\nX. Zhang, S. Shen, H. Wo, G. Chen, and J. Zhao, Frac-\ntionalized excitations in the partially magnetized spin\nliquid candidate YbMgGaO 4, Nature Comm. 9, 4138\n(2018).\n[127] J. Paddison, M. Daum, Z. Dun, G. Ehlers, Y. Liu,\nM. Stone, H. Zhou, and M. Mourigal, Continuous ex-\ncitations of the triangular-lattice quantum spin liquid\nYbMgGaO 4, Nature Phys. 113, 117 (2017).18\n[128] Y. Li, D. Adroja, D. Voneshen, R. I. Bewley, Q. Zhang,\nA. Tsirlin, and P. Gegenwart, Nearest-neighbour res-\nonating valence bonds in YbMgGaO 4, Nature Comm.\n8, 15814 (2017).\n[129] T.-H. Han, J. Helton, S. Chu, D. Nocera, J. Rodriguez-\nRivera, C. Broholm, and Y. Lee, Fractionalized excita-\ntions in the spin-liquid state of a kagome-lattice antifer-\nromagnet, Nature 492, 406 (2012).\n[130] Y.-D. Li and G. Chen, Detecting spin fractionalization\nin a spinon Fermi surface spin liquid, Phys. Rev. B 96,\n075105 (2017).\n[131] L. Balents and O. Starykh, Collective spinon spin wave\nin a magnetized U(1) spin liquid, Phys. Rev. B 101,\n020401(R) (2020).\n[132] M. Norman, Colloquium: Herbertsmithite and the\nsearch for the quantum spin liquid, Rev. Mod. Phys.\n88, 041002 (2016).\n[133] Y. Li, D. Adroja, R. Bewley, D. Voneshen, A. Tsir-\nlin, P. Gegenwart, and Q. Zhang, Crystalline electric-\n\feld randomness in the triangular lattice spin-liquid\nYbMgGaO 4, Phys. Rev. Lett. 118, 107202 (2017).\n[134] M. Majumder, G. Simutis, I. Collings, J.-C. Orain,\nT. Dey, Y. Li, P. Gegenwart, and A. Tsirlin, Persistent\nspin dynamics in the pressurized spin-liquid candidate\nYbMgGaO 4, arXiv:1902.07749.\n[135] Y.-D. Li, Y. Shen, Y. Li, J. Zhao, and G. Chen, E\u000bect\nof spin-orbit coupling on the e\u000bective-spin correlation\nin YbMgGaO 4, Phys. Rev. B 97, 125105 (2018).\n[136] X. Zhang, F. Mahmood, M. Daum, Z. Dun, J. Paddi-\nson, N. Laurita, T. Hong, H. Zhou, N. Armitage, and\nM. Mourigal, Hierarchy of exchange interactions in the\ntriangular-lattice spin liquid YbMgGaO 4, Phys. Rev. X\n8, 031001 (2018).\n[137] Note that the data of Ref. 127 were rather insensitive to\ntheJ\u0006\u0006\n1andJz\u0006\n1terms, so the values of Ref. 118 were\nemployed there without further re\fnement.\n[138] Z. Zhu, P. Maksimov, S. White, and A. Cherny-\nshev, Disorder-induced mimicry of a spin liquid in\nYbMgGaO 4, Phys. Rev. Lett. 119, 157201 (2017).\n[139] E. Parker and L. Balents, Finite-temperature behavior\nof a classical spin-orbit-coupled model for YbMgGaO 4\nwith and without bond disorder, Phys. Rev. B 97,\n184413 (2018).\n[140] L. Ding, P. Manuel, S. Bachus, F. Gru\u0019ler, P. Gegen-\nwart, J. Singleton, R. Johnson, H. Walker, D. Adroja,\nA. Hillier, and A. Tsirlin, Gapless spin-liquid state in\nthe structurally disorder-free triangular antiferromag-\nnet NaYbO 2, Phys. Rev. B 100, 144432 (2019).\n[141] K. Watanabe, H. Kawamura, H. Nakano, and T. Sakai,\nQuantum spin-liquid behavior in the spin-1/2 random\nHeisenberg antiferromagnet on the triangular lattice, J.\nPhys. Soc. Jpn. 83, 034714 (2014).\n[142] T. Shimokawa, K. Watanabe, and H. Kawamura, Static\nand dynamical spin correlations of the S=1\n2random-\nbond antiferromagnetic Heisenberg model on the tri-\nangular and kagome lattices, Phys. Rev. B 92, 134407\n(2015).\n[143] S. Dey, E. Andrade, and M. Vojta, Destruction of long-\nrange order in noncollinear two-dimensional antiferro-\nmagnets by random-bond disorder, Phys. Rev. B 101,\n020411(R) (2020).\n[144] H.-Q. Wu, S.-S. Gong, and D. Sheng, Randomness-\ninduced spin-liquid-like phase in the spin-1\n2J1\u0000J2tri-\nangular Heisenberg model, Phys. Rev. B 99, 085141(2019).\n[145] H. Kawamura and K. Uematsu, Nature of the\nrandomness-induced quantum spin liquids in two di-\nmensions, J. Phys.: Condens. Matter 31, 504003 (2019).\n[146] I. Kimchi, A. Nahum, and T. Senthil, Valence bonds in\nrandom quantum magnets: Theory and application to\nYbMgGaO 4, Phys. Rev. X 8, 031028 (2018).\n[147] K. Ranjith, D. Dmytriieva, S. Khim, J. Sichelschmidt,\nS. Luther, D. Ehlers, H. Yasuoka, J. Wosnitza, A. Tsir-\nlin, H. K uhne, and M. Baenitz, Field-induced instability\nof the quantum spin liquid ground state in the Je\u000b=1\n2\ntriangular-lattice compound NaYbO 2, Phys. Rev. B 99,\n180401(R) (2019).\n[148] M. Bordelon, E. Kenney, C. Liu, T. Hogan,\nL. Posthuma, M. Kavand, Y. Lyu, M. Sherwin,\nN. Butch, C. Brown, M. Graf, L. Balents, and S. Wil-\nson, Field-tunable quantum disordered ground state in\nthe triangular-lattice antiferromagnet NaYbO 2, Nature\nPhys. 15, 1058 (2019).\n[149] M. Baenitz, P. Schlender, J. Sichelschmidt, Y. Onyki-\nienko, Z. Zangeneh, K. Ranjith, R. Sarkar, L. Hozoi,\nH. C. Walker, J.-C. Orain, H. Yasuoka, J. van den Brink,\nH. Klauss, D. Inosov, and T. Doert, NaYbS 2: A planar\nspin-1\n2triangular-lattice magnet and putative spin liq-\nuid, Phys. Rev. B 98, 220409(R) (2018).\n[150] W. Liu, Z. Zhang, J. Ji, Y. Liu, J. Li, X. Wang, H. Lei,\nG. Chen, and Q. Zhang, Rare-earth chalcogenides: A\nlarge family of triangular lattice spin liquid candidates,\nChin. Phys. Lett. 35, 117501 (2018).\n[151] K. Ranjith, S. Luther, T. Reimann, B. Schmidt,\nP. Schlender, J. Sichelschmidt, H. Yasuoka, A. Stry-\ndom, Y. Skourski, J. Wosnitza, H. K uhne, T. Doert, and\nM. Baenitz, Anisotropic \feld-induced ordering in the\ntriangular-lattice quantum spin liquid NaYbSe 2, Phys.\nRev. B 100, 224417 (2019).\n[152] J. Xing, L. Sanjeewa, J. Kim, G. Stewart,\nA. Podlesnyak, and A. Sefat, Field-induced mag-\nnetic transition and spin \ructuations in the quantum\nspin-liquid candidate CsYbSe 2, Phys. Rev. B 100,\n220407(R) (2019).\n[153] R. Sarkar, P. Schlender, V. Grinenko, E. Haeussler,\nP. Baker, T. Doert, and H.-H. Klauss, Quantum spin\nliquid ground state in the disorder free triangular lat-\ntice NaYbS 2, Phys. Rev. B 100, 241116(R) (2019).\n[154] J. Sichelschmidt, P. Schlender, B. Schmidt, M. Baenitz,\nand T. Doert, Electron spin resonance on the spin-1/2\ntriangular magnet NaYbS 2, J. Phys.: Condens. Matter\n31, 205601 (2019).\n[155] M. Ye and A. Chubukov, Half-magnetization plateau\nin a Heisenberg antiferromagnet on a triangular lattice,\nPhys. Rev. B 96, 140406(R) (2017).\n[156] M. Ye and A. Chubukov, Quantum phase transitions in\nthe Heisenberg J1\u0000J2triangular antiferromagnet in a\nmagnetic \feld, Phys. Rev. B 95, 014425 (2017).\n[157] M. Sanders, F. Cevallos, and R. Cava, Magnetism in\nthe KBaRE(BO 3)2(RE = Sm, Eu, Gd, Tb, Dy, Ho, Er,\nTm, Yb, Lu) series: materials with a triangular rare\nearth lattice, Mater. Res. Express 4, 036102 (2017).\n[158] S. Guo, T. Kong, F. Alex Cevallos, K. Stolze,\nand R. Cava, Crystal growth, crystal structure and\nanisotropic magnetic properties of KBaR(BO 3)2(R =\nY, Gd, Tb, Dy, Ho, Tm, Yb, and Lu) triangular lattice\nmaterials, J. Magn. Magn. Mater. 472, 104 (2019).19\n[159] S. Guo, A. Ghasemi, C. Broholm, and R. Cava, Mag-\nnetism on ideal triangular lattices in NaBaYb(BO 3)2,\nPhys. Rev. Materials 3, 094404 (2019).\n[160] J. Xing, L. Sanjeewa, J. Kim, G. Stewart, M.-H. Du,\nF. Reboredo, R. Custelcean, and A. Sefat, Crystal syn-\nthesis and frustrated magnetism in triangular lattice\nCsRESe 2(RE = La-Lu): Quantum spin liquid candi-\ndates CsCeSe 2and CsYbSe 2, ACS Materials Lett. 2, 71\n(2020).\n[161] F. Alex Cevallos, K. Stolze, and R. Cava, Structural dis-\norder and elementary magnetic properties of triangular\nlattice ErMgGaO 4single crystals, Solid State Comm.\n276, 5 (2018).\n[162] S. Guo, T. Kong, and R. Cava, NaBaR(BO 3)2(R =\nDy, Ho, Er and Tm): structurally perfect triangular\nlattice materials with two rare earth layers, Mater. Res.\nExpress 6, 106110 (2019).\n[163] J. Banda, B. Rai, H. Rosner, E. Morosan, C. Geibel, and\nM. Brando, Crystalline electric \feld of Ce in trigonal\nsymmetry: CeIr 3Ge7as a model case, Phys. Rev. B 98,\n195120 (2018).\n[164] J. Xing, L. Sanjeewa, J. Kim, W. Meier, A. May,\nQ. Zheng, R. Custelcean, G. Stewart, and A. Sefat, Syn-\nthesis, magnetization, and heat capacity of triangular\nlattice materials NaErSe 2and KErSe 2, Phys. Rev. Ma-\nterials 3, 114413 (2019).\n[165] A. Scheie, V. Garlea, L. Sanjeewa, J. Xing, and A. Sefat,\nCrystal \feld hamiltonian and anisotropy in KErSe 2and\nCsErSe 2, arXiv:2001.03175.\n[166] S. Gao, F. Xiao, K. Kamazawa, K. Ikeuchi, D. Biner,\nK. Kr amer, C. R uegg, and T. Arima, Crystal-electric-\n\feld excitations in a quantum-spin-liquid candidate\nKErS 2, arXiv:1911.10662.\n[167] J. Lee, A. Rabus, N. Lee-Hone, D. Broun, and E. Mun,\nThe two-dimensional metallic triangular lattice antifer-\nromagnet CeCd 3P3, Phys. Rev. B 99, 245159 (2019).\n[168] A. Ochiai, T. Inukai, T. Matsumura, A. Oyamada, and\nK. Katoh, Spin gap state of S= 1=2 Heisenberg an-tiferromagnet YbAl 3C3, J. Phys. Soc. Jpn. 76, 123703\n(2007).\n[169] Y. Kato, M. Kosaka, H. Nowatari, Y. Saiga, A. Yamada,\nT. Kobiyama, S. Katano, K. Ohoyama, H. Suzuki,\nN. Aso, and K. Iwasa, Spin-singlet ground state in the\ntwo-dimensional frustrated triangular lattice: YbAl 3C3,\nJ. Phys. Soc. Jpn. 77, 053701 (2008).\n[170] T. Matsumura, T. Inami, M. Kosaka, Y. Kato,\nT. Inukai, A. Ochiai, H. Nakao, Y. Murakami,\nS. Katano, and H. Suzuki, Structural phase transition\nin the spin gap system YbAl 3C3, J. Phys. Soc. Jpn. 77,\n103601 (2008).\n[171] D. Khalyavin, D. Adroja, P. Manuel, A. Daoud-Aladine,\nM. Kosaka, K. Kondo, K. McEwen, J. Pixley, and Q. Si,\nField-induced long-range magnetic order in the spin-\nsinglet ground-state system YbAl 3C3: Neutron di\u000brac-\ntion study, Phys. Rev. B 87, 220406(R) (2013).\n[172] V. Nekvasil and I. Veltrusk\u0013 y, E\u000bective Hamiltonian and\nground state properties of rare-earth ions in iron gar-\nnets, J. Magn. Magn. Mater. 86, 315 (1990).\n[173] Y. Shen, C. Liu, Y. Qin, S. Shen, Y.-D. Li, R. Bewley,\nA. Schneidewind, G. Chen, and J. Zhao, Intertwined\ndipolar and multipolar order in the triangular-lattice\nmagnet TmMgGaO 4, Nature Comm. 10, 4530 (2019).\n[174] Y. Li, S. Bachus, H. Deng, W. Schmidt, H. Thoma,\nV. Hutanu, Y. Tokiwa, A. Tsirlin, and P. Gegenwart,\nPartial up-up-down order with the continuously dis-\ntributed order parameter in the triangular antiferromag-\nnet TmMgGaO 4, Phys. Rev. X 10, 011007 (2020).\n[175] W. Steinhardt, Z. Shi, A. Samarakoon, S. Dissanayake,\nD. Graf, Y. Liu, W. Zhu, C. Marjerrison, C. Batista, and\nS. Haravifard, Field-induced transformation of the spin\nliquid state in triangular antiferromagnet YbMgGaO 4,\narXiv:1902.07825.\n[176] S. Winter, A. Tsirlin, M. Daghofer, J. van den Brink,\nY. Singh, P. Gegenwart, and R. Valent\u0013 \u0010, Models and\nmaterials for generalized Kitaev magnetism, J. Phys.:\nCondens. Matter 29, 493002 (2017)."    },    {        "title": "1110.0114v1.Spin_orbital_phase_synchronization_in_the_magnetic_field_driven_electron_dynamics_in_a_double_quantum_dot.pdf",        "content": "arXiv:1110.0114v1  [nlin.CD]  1 Oct 2011Spin-orbital phase synchronization in the magnetic field-d riven\nelectron dynamics in a double quantum dot.\nL. Chotorlishvili,1,4E.Ya. Sherman,2,3Z. Toklikishvili,5, A. Komnik,4and J. Berakdar1\n1Institut f¨ ur Physik, Martin-Luther Universit¨ at Halle-W ittenberg,\nHeinrich-Damerow-Str.4, 06120 Halle, Germany\n2Departamento de Quimica F´ ısica,\nUniversidad del Pa´ ıs Vasco UPV/EHU, 48080 Bilbao, Spain\n3IKERBASQUE, Basque Foundation for Science, E-48011 Bilbao , Spain\n4Institut f¨ ur Theoretische Physik, Universit¨ at Heidelbe rg,\nPhilosophenweg 19, D-69120 Heidelberg,Germany\n5Physics Department of the Tbilisi State University,\nChavchavadze av.3, 0128, Tbilisi, Georgia\nAbstract\nWe study the dynamics of an electron confined in a one-dimensi onal double quantum dot in the\npresenceofdrivingexternalmagneticfields. Theorbitalmo tionoftheelectroniscoupledtothespin\ndynamics by spin orbit interaction of the Dresselhaus type. We derive an effective time-dependent\nHamiltonian model for the orbital motion of the electron and obtain a synchronization condition\nbetween the orbital and the spin dynamics. From this model we deduce an analytical expression\nfor the Arnold tongue and propose an experimental scheme for realizing the synchronization of the\norbital and spin dynamics.\n1I. INTRODUCTION\nPhase synchronization and related phenomena are among the most fascinating effects of\nnonlinear dynamics. Besides the deep fundamental interest[1–7], p hase synchronization has\na broad range of applications in chemistry [8], ecology [9], astronomy [10], in the field of\ninformationtransferusingchaoticsignals[11], andforthecontrol ofhighfrequencyelectronic\ndevices [12]. In nonlinear dissipative systems, phase synchronizatio n occurs if the frequency\nof the external driving field is close to the eigenfrequency of the sy stem. In this case, for\na certain frequency interval of the driving field the oscillations of th e nonlinear dissipative\nsystem can be synchronized with the perturbing force. Usually, fo r stronger driving fields\nthe frequency interval for the synchronization becomes broade r and the synchronization\nprotocol is more efficient. The broad and growing interest in the pha se synchronization calls\nfor the analysis of new possible realizations of this phenomenon. A pa rticularly interesting\nissue is the application of the synchronization protocols for magnet ic nanostructures, which\nhave rich applications [11–16] and exhibit interesting nonlinear dyna mical properties that\ncan be exploited as a testing ground for dynamical systems [17, 18].\nA principal challenge in nanoscience is to find an efficient procedure fo r the manipulation\nof the systems states. A high level of accuracy on the state cont rol is required especially in\nsuch applications as quantum computing, where a precise tailoring of the entangled states\nis highly desirable [19, 20]. With this in mind several physical systems we re considered up\nto now, e. g. Josephson junction qubits and Rydberg atoms in a qua ntum cavities [19, 20],\nion traps [21], single molecular nanomagnets [22], and nanoelectrome chanical resonators\n[23, 24]. Among others, one of the most promising systems are elect ron spins confined\nin two-dimensional quantum dots [25–27] and in one-dimensional nan owires and nanowire-\nbased quantum dots [28–31]. The key element of the corresponding models is the spin orbit\n(SO) coupling term, which is linear in the electron momentum. Such mom entum-dependent\ncoupling offers a new way of manipulating the spin by changing the elect ron momentum via\na periodic electric field. This is the idea of the electric-dipole spin reson ance proposed by\nRashba and Efros for the electrons confined in nanostructures o n the scale of 10 nm [27].\nHowever, the external electric field can strongly affect the orbita l dynamics and thus the\nsystem is driven out of the linear regime [32]. The nonlinearities usually r esult in a complex\nbehavior of the affected systems and their dynamics might become c omplicated and even\n2unpredictable. On the other hand, the nonlinearity may also lead to a number of interesting\nphenomena. In spite of the huge interest in the systems with SO cou pling, the influence\nof the spin dynamics on the orbital motion was not addressed yet in f ull detail. With the\npresent work we would like to bridge this gap. Our goal is to investigat e the possibilities of\ncontrolling the orbital motion of the electron via external magnetic fields acting on its spin\nand via the spin-orbit coupling. That can be considered as opposite t o the electric-dipole\nspin resonance protocol proposed by Rashba and Efros [27]. We will demonstrate that: (i)\nby using an external driving field one can achieve a sufficient degree o f control over the\norbital motion, and (ii) as a result, one can devise a very efficient syn chronization protocol\nof the orbital motion and the spin dynamics based on the application o f a pulsed external\nmagnetic field.\nII. THEORETICAL MODEL\nWe consider a model system of a single electron confined in a double qu antum dot de-\nscribed by a potential of the form U(x) =U0/bracketleftbig\n−2(x/d)2+(x/d)4/bracketrightbig\n. HereU0is the energy\nbarrier separating two minima with 2 dbeing the distance between them. We assume the\nsystem is dissipative, and the dissipation is a thermal effect appearin g due to a coupling\nto environment. The dissipation, which impacts mainly on the orbital m otion, is essen-\ntial for the synchronization processes we are going to discuss late r in the text. For strong\ndriving magnetic fields, the influence of the dissipation on the spin dyn amics is negligibly\nsmall and can be ignored. In addition, we assume that the temperat ure is low enough to\nprevent the activated over-the-barrier motion. For the particu lar value of U0∼20 meV the\nlow-temperature regime means T <100 K. For the GaAs-based structure with the electron\neffective mass mbeing 0.067 of the free electron mass and d∼100 nm, the tunneling prob-\nability is small and a classical consideration is justified. To quantify th e SO interaction we\nuse a coupling term of the Dresselhaus type Hso=αPxσx, wherePxis the momentum of\nthe electron and σxis the Pauli matrix. Therefore, the Hamiltonian of the one dimensiona l\nsystem reads:\nH=P2\nx\n2m+U(x)+αPxσx+µBgBz(t)σz\n2+µBgBx(t)σx\n2, (1)\n3 \nFIG. 1. Schematic illustration of the infinite series of exte rnal magnetic field pulses applied to the\nsystem. A series of the short pulses Bz(t) =B0∞/summationtext\nt=0δτ(t−nT) with the pulse width τand time\ninterval between pulses T, is applied along the zaxis. Series of the pulses with the larger width T\nand a shorter interval between the pulses τ,Bx(t) =B0∞/summationtext\nt=0∆T(t−τn) is applied along the xaxis.\nThe amplitude of the pulses B0is the same in both cases.\nwhereµBistheBohrmagnetonand gistheelectronLand´ efactor. Here Bz(t) =B0∞/summationtext\nt=0δτ(t−\nnT) is an infinite series of external magnetic field pulses with the pulse st rengthB0, which\nis applied along the zaxis. The temporal width of the pulses applied along the z-axis is\nsmaller than the interval between the pulses τ≪T(in what follows we set T= 1). On the\nother hand, for the pulses along the x-axisBx(t) =B0∞/summationtext\nt=0∆T(t−τn) the pulse duration is\nlarger than interval between pulses T≪τ(cf. Fig. 1). A different route to the control of\nthe spin-dynamics in double quantum dots via electric field pulses is out lined in [33].\nIntroducing the characteristic maximum momentum of the electron Pmax\nx=√2mU0\nwe can estimate the maximal precession rate of the spin due to the S O coupling Ωmax\nso=\n(2α//planckover2pi1)√2mU0, while the magnetic field pulse of the amplitude B=B0induces a spin\nprecession with the rate Ω B=µB|g|B0//planckover2pi1. Therefore, if Ω B>Ωmax\nsowe can during the pulse\nneglect the spin rotation produced by the SO coupling and the spin is c ompletely controlled\nby the external driving fields. We need a protocol with two driving fie lds in order to fulfill\nthe synchronization requirements as discussed later in the text. N amely, for the control of\nthe spin dynamics via the external driving fields, the amplitudes of th e fields should belarge,\nB0>(2α/µB|g|)√2mU0. On the other hand, a strong constant magnetic field produces a\nhigh frequency precession of the spin Ω B=µB|g|B0//planckover2pi1, while for the synchronization we\nneed to tune the precession frequency up or down keeping fixed th e strong driving field\n4amplitude. Below we will show that the optimal conditions for the sync hronization are\nrealized using two types of the driving pulses. Applying short pulses a long thez-axisτ≪T,\nBz(t) =B0∞/summationtext\nt=0δτ(t−nT) and long pulses along the x-axisBx(t) =B0∞/summationtext\nt=0∆T(t−τn), we\ncan realize a spin precession with a frequency, which is inversely prop ortional to the time\ninterval between the short pulses Ω ∼1/Tindependently from the driving field strength\nB0. In what follows, for convenience we use dimensionless units via the t ransformations\nE→E/4U0,x→x/d,t→t/radicalbig\n4U0/m,Px→Px/√2mU0,ε→α/4U0.\nIII. DISSIPATIVE SYSTEM AND THE PROBLEM OF PHASE SYNCHRONIZA-\nTION BETWEEN ORBITAL AND SPIN MOTION\nA. Spin dynamics in pulsed magnetic fields\nAs was stated above the synchronization can occur if the frequen cy of the driving field\nis close to the eigenfrequency of the nonlinear dissipative system. I f this is the case, in the\nparticular frequency interval, the oscillations of the nonlinear dissip ative system and the\nfield can be synchronized. With the increase in the driving field amplitud e, the synchroniza-\ntion can occur in a broader frequency interval, and the synchroniz ation protocol becomes\nmore efficient. Our aim is to develop a method for the synchronization of the dynamics\nof the electron spin and the orbital motion, using an external drivin g magnetic field and\nSO coupling. Although the magnetic field is not coupled to the orbital m otion directly, a\nsufficiently strong field influences the orbital motion through the sp in dynamics if the SO\ncoupling is present. If SO term is relatively small Ωmax\nso<ΩB, that is\nΩB=µB|g|B0\n/planckover2pi1>Ωmax\nso=2α\n/planckover2pi1/radicalbig\n2mU0, (2)\nthe spin and, correspondingly, the orbital motion can be controlled externally. From Eq. (1)\nit is easy to see, that in between the short pulses the electron spin r otates around the x-axis\nand the equations of motion for the electron spin in this case read\n˙σx= 0,˙σy=−ΩBσz,˙σz= ΩBσy. (3)\nOn the other hand, during the short pulses we have\n˙σx=−ΩBσy,˙σy= ΩBσx,˙σz= 0. (4)\n5Considering the dynamics due to the pulse acting on the spin at the mo ment of time t=t0,\nwe can split the evolution operator ˆTevdefined as\nσ/parenleftBig\nt[−]\n0+T/parenrightBig\n=ˆTevσ/parenleftBig\nt[−]\n0/parenrightBig\n(5)\ninto two parts: ˆTev=ˆTR׈Tδ, where\nσ/parenleftBig\nt[+]\n0/parenrightBig\n=ˆTδσ/parenleftBig\nt[−]\n0/parenrightBig\n,\nσ/parenleftBig\nt[−]\n0+T/parenrightBig\n=ˆTRσ/parenleftBig\nt[+]\n0/parenrightBig\n. (6)\nHere we introduced the notations t[+]\n0≡t0+0 andt[−]\n0≡t0−0. The operator ˆTRdescribes\nthe rotationof the electron spin around the x-axis produced by the long pulse of the external\ndriving magnetic field Bx(t) =B0∞/summationtext\nt=0∆T(t−τn), which is applied along the x-axis and ˆTσ\ncorresponds to the evolution produced by the short pulses Bz(t) =B0∞/summationtext\nt=0δτ(t−nT) applied\nalong the z-axis. Integrating Eq. (4) for a short time interval t∈/parenleftBig\nt[−]\n0,t[+]\n0/parenrightBig\nwe obtain\nˆTδ(σx) =σx(t[−]\n0)cos(Ω Bτ)−σy(t[−]\n0)sin(Ω Bτ), (7)\nˆTδ(σy) =σx(t[−]\n0)sin(Ω Bτ)+σy(t[−]\n0)cos(Ω Bτ). (8)\nIntegrating Eq. (3) during the time interval t∈/parenleftBig\nt[+]\n0,t[−]\n0+T/parenrightBig\nof long applied pulse we find\nˆTR(σy) =/radicalbigg\n1−/parenleftBig\nσx/parenleftBig\nt[+]\n0/parenrightBig/parenrightBig2\ncos(ΩBT), (9)\nˆTR(σz) =/radicalbigg\n1−/parenleftBig\nσx/parenleftBig\nt[+]\n0/parenrightBig/parenrightBig2\nsin(ΩBT). (10)\nCombining Eq. (7) with Eq. (9) we finally can reconstruct the complet e picture of the full\ntime evolution of the electron spin:\n\n\nσy\nn+1=/radicalbig\n1−(σx\nn+1)2cos((n+1)Ω BT),\nσz\nn+1=/radicalbig\n1−(σx\nn+1)2sin((n+1)Ω BT),\nσx\nn+1=σx\nncos(ΩBτ)−/radicalbig\n1−(σxn)2sin(ΩBτ)cos(nΩBT).(11)\nThe recurrent relations Eq. (11) describe the spin dynamics. Accu racy of the employed\napproximations may be checked by the validity of the normalization co nditionσ2= 1.In\norder to identify, whether the nonlinear map (11) is chaotic or regu lar, we evaluate the\nLyapunov exponent for the spin system [34]. Taking into account th e peculiarity of the\n6 \n \nFIG. 2. Lyapunov exponent as a function of the initial spin co mponent σx\n0. After N= 1000\niterations theLyapunov exponent is negative λ(σx\n0)<0, meaningthat the spindynamics is regular.\nT= 1, Ω Bτ= 1, Ω BT= 20. At these conditions, the dependence is weak.\nsystem (11), which is the fact, that the equation for the xcomponent σx\nn+1is self-consistent\nσx\nn+1=f(σx\nn), we deduce for the Lyapunov exponent\nλ(σx\n0) = lim\nN→∞\nδσ→01\nNln/vextendsingle/vextendsingle/vextendsingle/vextendsinglefN(σx\n0+δσ)−fN(σx\n0)\nδσ/vextendsingle/vextendsingle/vextendsingle/vextendsingle= lim\nN→∞1\nNln/vextendsingle/vextendsingle/vextendsingle/vextendsingledfN(σx\n0)\ndσx\n0/vextendsingle/vextendsingle/vextendsingle/vextendsingle.(12)\nHere,σx\n0is the initial value of the spin projection and the small increment of th e initial\nvaluesσx\n0+δσquantifies the sensitivity of the recurrence relations (11) with res pect to the\nslight change in the initial conditions. After some algebra from Eqs. ( 11) and (12) we finally\nobtain:\nλ(σx\n0) = lim\nN→∞1\nNN−1/summationdisplay\nn=0ln/vextendsingle/vextendsingle/vextendsingle/vextendsingledf(σx\nn)\ndσx\nn/vextendsingle/vextendsingle/vextendsingle/vextendsingle=1\nNN−1/summationdisplay\nn=0ln/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglecos(ΩB)+σx\nn/radicalbig\n1−(σx\nn)2sin(ΩB)cos(nΩB)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle.\n(13)\nThe results of the numerical calculations are presented on Figs. 2 a nd 3. From Fig. 2 we\nsee that the Lyapunov exponent is negative, λ(σx\n0)<0 and therefore the spin dynamics\nis regular, since the initial distance between two neighboring trajec tories starting from the\ninitial points σx\n0andσx\n0+δσis not increasing asymptotically after an infinite number of\niterations δσeNλ(σ0). Therefore, from Figs. 2 and 3 we conclude, that the dynamics of\nthe electron spin is controlled by the magnetic field pulses, thus follow ing equation σx(t) =\nσx\n0cos(Ωt) we achieve the spin manipulation by magnetic fields. The spin rotation frequency\nis determined by the time interval between the short pulses Ω ≈1/T.\n7 \nFIG. 3. Time dependence of the spin projection σx\nn≡σx(n).T= 1, Ω Bτ= 1, Ω BT= 20. From\nFig. 3 we see that σx\nn=σx(t) is periodic in time and the frequency of oscillations Ω is in versely\nproportional to the time lapse between pulses Ω ≈1/T.\nB. Synchronization of the spin and the orbital motion.\nWith the spin dynamics discussed in the previous subsection, for the orbital motion of\nelectron we can write the following effective Hamiltonian assuming that σx(t)≈cos(Ωt):\nH=P2\nx\n2m+U(x)+αPxcos(Ωt). (14)\nThe equation of motion for the system (14) has the form:\n¨x+γ˙x−x+x3=−βsin(Ωt), (15)\nwhere two new dimensionless quantities are introduced: β=αΩm/4U0andγ→γ/4U0m.\nWe seek a solution of Eq. (15) using the following ansatz\nx(t) =1\n2A(t)eiΩt+1\n2A∗(t)e−iΩt. (16)\nAssuming that the amplitude A(t) in Eq. (16) is a slow variable the following condition\napplies\n˙A(t)eiΩt+˙A∗(t)e−iΩt= 0. (17)\n8 \nFIG. 4. Arnold tongue diagram in terms of the decay constant γand field frequency Ω, plotted\nusing the conditions (25). Shadowed regions define paramete r values for which the synchronization\nis possible.\nTaking into account Eqs. (16) and (17) from Eq. (15) we deduce:\niΩ˙A(t)eiΩt−/parenleftbiggΩ2\n2A(t)eiΩt+c.c./parenrightbigg\n+\n+γ/parenleftbiggiΩ\n2A(t)eiΩt+c.c./parenrightbigg\n−/parenleftbigg1\n2A(t)eiΩt++c.c./parenrightbigg\n+ (18)\n+1\n8/parenleftbig\nA3(t)e3iΩt+3|A(t)|2A(t)eiΩt+c.c./parenrightbig\n=β\n2i/parenleftbig\neiΩt−c.c./parenrightbig\n.\nMultiplying Eq. (18) by the exponent e−iΩtand averaging it over the fast phases we find:\n˙A(t)+(Ω2)+1)i\n2ΩA(t)+γ\n2A(t)−3i\n8Ω|A(t)|2A(t) =−β\n2Ω. (19)\nIntroducing the notations\nA(t) = 2√γz(t), τ=tγ\n2,∆ =Ω2+1\nγΩ, ε=β\n2Ωγ3/2, (20)\nwe can rewrite Eq. (19) in a more compact form\n˙z(τ)+i∆z(τ) =−z(τ)+3i\nΩ|z(τ)|2z(τ)+ε. (21)\nInserting z(τ) =R(τ)eiϕ(τ)into Eq. (21), for the real and imaginary parts we obtain\n\n\n˙R(τ) =−R+εcosϕ(τ),\n˙ϕ(τ)+∆ =3\nΩR2(τ)−ε\nR(τ)sinϕ(τ).(22)\n9 \nFIG. 5. Arnold tongue in terms of the parameters ∆ =/parenleftbig\nΩ2+1/parenrightbig\n/γΩ, (∆,ε) plotted using Eq. (26).\nPointbbelongs to the domain where a synchronization is not possibl e, while abelongs to the\nsynchronization domain.\nUsing Eq. (22) and setting ˙R= 0, ˙ϕ= 0 for the stationary solutions we obtain:\nf(ξ) =ξ+ξ/parenleftbigg3\nΩξ−∆/parenrightbigg2\n=ε2, (23)\nξ1,2=Ω\n9/parenleftBig\n2∆±√\n∆2−3/parenrightBig\n, (24)\nwhereR2=ξandξ1,2are rootsof the equation df(ξ)/dξ= 0. In order to identify the Arnold\ntongue [35], which shows the regions in which a synchronization is poss ible, we utilize the\nstandard condition df(ξ)/dξ= 0. From Eq. (24) it is not difficult to see that the roots of\nthe equation df(ξ)/dξ= 0 are real if ∆ >√\n3. Taking into account that ∆ = (Ω2+1)/γΩ\nwe can rewrite the inequality in the form Ω2+ 1> γΩ√\n3. Consequently we obtain the\nfollowing criteria for the synchronization\n0<Ω<1\n2(γ√\n3−/radicalbig\n3γ2−4),Ω>1\n2(γ√\n3+/radicalbig\n3γ2−4), γ >2√\n3(25)\nGraphical representation of the conditions (25) is shown in Fig. 4.\nEq. (25) defines the synchronization area in terms of the externa l field frequency Ω\nand the dissipation constant γ. The minima points of the function f(ξ), (23) does not\ndepend on the SO coupling constant α. Therefore criteria (25) is independent of the values\n10 \nFIG. 6. Orbital and spin dynamics plotted using the exact num erical integration of the Eq. (15).\nPanel a) corresponds to the values of parameters for which a s ynchronization is possible (see Fig. 5,\npoint a). In particular/parenleftBig\nε/√\nΩ = 4.41,∆ = 1.88/parenrightBig\n. The right panel b) corresponds to the point b\non Fig.5,/parenleftBig\nε/√\nΩ = 1.05,∆ = 3.03/parenrightBig\n, i.e. the dynamics occurs outside of the synchronization ar ea.\nWe see that in the former case the orbital and the spin dynamic s are in phase and in the latter\ncase the phase difference is about π/2 .\nof the SO coupling strength as well. Nevertheless, inserting the roo tsξ1,2of the equation\ndf(ξ)/dξ= 0 into Eq. (23) one can derive more illustrative and precise criteria in the form\nof the parametrical curve:\n2\n81(±3√\n∆2−3+9∆+∆3∓∆2√\n∆2−3)−/parenleftbiggε√\nΩ/parenrightbigg2\n= 0. (26)\nThe parametrical curve defined by Eq. (26) represents the bord er of the synchronization\ndomain, see Fig. 5. Taking into account that β=αΩm/4U0,ε=β/2Ωγ3/2we obtain\nε√\nΩ=α√\nΩm\n8U0γ3/2. (27)\nFrom Eq. (27) we see that the parameters of Fig. 5 depend on the o scillation frequency\nthat can be easily controlled by tuning the time interval between puls es Ω≈1/T. All\nother parameters in Eq. (27), such as the SO coupling constant α, barrier height U0, and\nthe electron effective mass mare internal characteristics of the system whereas the decay\nconstant γis related to the thermal effects. Using Eq. (26) and Fig. 5 one can s ynchronize\nthe electron orbital motion with its spin dynamics.\n11IV. CONCLUSIONS.\nWe have investigated the classical electron dynamics in a double dot p otential, with the\nspin of electron being controlled by external magnetic fields. We hav e shown that the orbital\nelectrondynamicscanbecontrolledveryeffectively bythefieldinthe presenceofaspin-orbit\ncoupling. Using the proposed protocol of magnetic field pulses of diff erent duration we have\nshown that it is possible to synchronize the spin and the orbital motio n of the electron. In\nparticular, if the driving field amplitude is large enough, B0>(2α/µB|g|)√2mU0, the spin\ndynamics is periodical in time. Then σx(t) =σx\n0cos(Ωt), where the oscillation frequency\nis inversely proportional to the time interval between pulses Ω ≈1/Tand can be tuned\nindependently from the amplitude of the pulses B0. As a consequence the orbital dynamics\ncanbestudiedwithreduced effective, time-dependent, one-dimen sional model. Byusing this\nmodel we found the synchronization condition between the orbital and the spin dynamics.\nFurthermore, we derived an analytical expression for the Arnold t ongue that defines the\nvalues of the parameters for which a synchronization is possible. Sin ce in the designed\nprotocol the spin precession rate is determined by the interval be tween the applied pulses\nwe believe that it can be realized in future experiments on semiconduc tor quantum dot\ndevices.\nAcknowledgments The financial support by the Deutsche Forschungsgemeinschaft\n(DFG) through SFB 762, and contract BE 2161/5-1, Grant No. KO -2235/3, and STCU\nGrant No. 5053 is gratefully acknowledged. EYS acknowledges supp ort of the MCI of Spain\ngrant FIS2009-12773-C02-01, and ”Grupos Consolidados UPV/E HU del Gobierno Vasco”\ngrant IT-472-10.\n[1] P. S. Landa and A. Rabinovitch, Phys. Rev. E 61, 1829 (2000).\n[2] L. Glass, Nature 410, 277 (2001).\n[3] V. S. Anishchenko and A. N. Pavlov, Phys. Rev. E 57, 2455 (1998)\n[4] N. F. Rulkov, M. A. Vorontsov, and L. Illing, Phys. Rev. Le tt.89, 277905 (2002).\n[5] A. Pikovsky, M. Rosenblum, J. Kurths, and S. Strogatz, Ph ysics Today 56, 47 (2003).\n[6] G. V. Osipov, A. Pikovsky, and J. Kurths, Phys. Rev. Lett. 88, 054102 (2002).\n[7] A. K. Kozlov, M. M. Sushchik, Ya. I. Molkov, and A. S. Kuzne tsov, Int. J. Bifurcation and\n12Chaos9, 2271 (1999).\n[8] L. Kocarev and U. Parlitz, Phys. Rev. Lett. 74, 5028 (1995).\n[9] J.H. Peng, E. J. Ding, M. Ding, and W. Yang, Phys. Rev. Lett .76, 904 (1996).\n[10] V. Anishchenko, S. Nikolaev, and J. Kurths, Phys. Rev. E 76, 046216 (2007).\n[11] A. L. Fradkov, B. Andrievsky, and R. Evans, Phys. Rev. E 73066209 (2006).\n[12] S. H. Strogatz, Nonlinear Dynamics and Chaos: with Appl ications to Physics, Biology, Chem-\nistry, and Engineering (Reading, Mass. Addison-Wesley, 19 94).\n[13] C. F. Hirjibehedin, C. P. Lutz, and A. J. Heinrich, Scien ce312, 1021 (2006).\n[14] S. Rusponi, T. Cren, N. Weiss, M. Epple, P. Buluschek, L. Claude, and H. Brune, Nat. Mater.\n2, 546 (2003).\n[15] T. Mirkovic, M. L. Foo, A. C. Arsenault, S. Fournier-Bid oz, N. S. Zacharia, and G. A. Ozin,\nNat. Nanotechnol. 2, 565 (2007).\n[16] J. A. Stroscio and R. J. Celotta, Science 306, 242 (2004).\n[17] L. Chotorlishvili, Z. Toklikishvili, A. Komnik, and J. Berakdar, Phys. Rev. B 83, 184405\n(2011).\n[18] L.Chotorlishvili, P.Schwab, Z.Toklikishvili, andJ. Berakdar, Phys.Rev.B 82, 014418 (2010).\n[19] J. Raimond, M. Brune, and S. Haroche, Rev. Mod. Phys. 73, 565 (2001).\n[20] S.N. Shevchenko, S. Ashhab, and F. Nori, Phys. Rep. 492, 1 (2010).\n[21] F. Zahringer, G. Kirchmair, R. Gerritsma, E. Solano, R. Blatt, and C.F. Roos, Phys. Rev.\nLett.104, 100503 (2010).\n[22] W. Wernsdorfer, N. Aliaga-Alcalde, D. N. Hendrickson, and G. Christou, Nature 416, 406\n(2002).\n[23] G. Heinrich and F. Marquardt, EPL 93, 18003 (2011).\n[24] R. B. Karabalin, M. C. Cross, and M. L. Roukes, Phys. Rev. B79, 165309 (2009).\n[25] M. Valin-Rodriguez, A. Puente, L. Serra, and E. Lippari ni, Phys. Rev. B 66, 235322 (2002).\n[26] L. S. Levitov and E. I. Rashba, Phys. Rev. B 67, 115324 (2003).\n[27] E. I. Rashba and Al. L. Efros, Phys. Rev. Lett. 91, 126405 (2003).\n[28] Y. V. Pershin, J. A. Nesteroff, and V. Privman, Phys. Rev. B 69, 121306 (2004).\n[29] C. L. Romano, P. I. Tamborenea, and S. E. Ulloa, Phys. Rev . B74, 155433 (2006).\n[30] C. L¨ u, U. Z¨ ulicke, and M. W. Wu, Phys. Rev. B 78, 165321 (2008).\n[31] M. Crisan, D. S´ anchez, R. L´ opez, L. Serra, and I. Grosu , Phys. Rev. B 79, 125319 (2009).\n13[32] D. V. Khomitsky and E. Ya. Sherman, Phys. Rev. B 79, 245321 (2009).\n[33] J. Waetzel, A. S. Moskalenko, and J. Berakdar, arXiv:11 03.5866.\n[34] H. G. Schuster, Deterministic Chaos, An Introduction ( VCH-Verlagsgesellschaft-Physik Ver-\nlag, Weinheim, 1985).\n[35] A.P. Kuznetsov, S.P.Kuznetsov, andN.M.Ryskin, Nonli nearOscillations (Moscow, Fizmatlit,\nin Russian, 2002).\n14"    },    {        "title": "2006.04431v1.Time_dependent_spintronic_anisotropy_in_magnetic_molecules.pdf",        "content": "Time-dependent spintronic anisotropy in magnetic molecules\nKacper Wrze\u0013 sniewski\u0003and Ireneusz Weymann\nFaculty of Physics, Adam Mickiewicz University,\nul. Uniwersytetu Pozna\u0013 nskiego 2, 61-614 Pozna\u0013 n, Poland\n(Dated: June 9, 2020)\nWe theoretically study the quench dynamics of induced anisotropy of a large-spin magnetic\nmolecule coupled to spin-polarized ferromagnetic leads. The real-time evolution is calculated by\nmeans of the time-dependent density-matrix numerical renormalization group method implemented\nwithin the matrix product states framework, which takes into account all correlations in very ac-\ncurate manner. We determine the system's response to a quench in the spin-dependent coupling\nto ferromagnetic leads. In particular, we focus on the transient dynamics associated with crossing\nfrom the weak to the strong coupling regime, where the Kondo correlations become important. The\ndynamics is examined by calculating the time-dependent expectation values of the spin-quadrupole\nmoment and the associated spin operators. We identify the relevant time scales describing the\nquench dynamics and determine the in\ruence of the molecule's e\u000bective exchange coupling and\nleads spin-polarization on the dynamical behavior of the system. Furthermore, the generalization\nof our predictions for large values of molecule's spin is considered. Finally, we analyze the e\u000bect\nof \fnite temperature and show that it gives rise to a reduction of magnetic anisotropy by strong\nsuppression of the time-dependent spin-quadrupole moment due to thermal \ructuations.\nI. INTRODUCTION\nMolecular magnetism is a rapidly developing area of\ntheoretical and experimental research, providing con-\ncepts for novel applications in spintronic devices and\nquantum technologies [1{10]. Single-molecule magnets\n(SMM), in particular those of large spin ( S>1), are\nespecially appealing due to their unique magnetic char-\nacteristics and a wide perspective of engineering and syn-\nthesizing new specimen with sought properties [3]. One\nprominent feature present in magnetic molecular systems\nis the uniaxial magnetic anisotropy, which leads to the\nmagnetic bistability and suppression of spin-reversal pro-\ncesses [11{15]. It is a property of crucial importance for\nthe memory storage and information processing appli-\ncations. Additionally, when transverse anisotropy com-\nponent is considerable, quantum tunneling of magneti-\nzation may occur [16{19]. The transport properties of\nmagnetic molecules have already been extensively stud-\nied [20{31], including the in\ruence of the Kondo e\u000bect\n[32{36] in the strong coupling regime [37{43]. However,\nwhen the physics of SMM systems incorporates spintron-\nics, new prominent e\u000bects are revealed, including switch-\ning with spin-polarized currents, Berry-phase blockade\nor spintronic anisotropy among many others [44{52]. In\nfact, the latter e\u000bect is of particular interest, as it allows\nfor generation of magnetic anisotropy in spin-isotropic\nmolecules [53].\nMoreover, the dynamics of molecular systems, an im-\nportant aspect of the on-going research in molecular mag-\nnetism, has recently gained a lot of attention and has\nbeen explored both experimentally [54{56] and theoret-\nically [57{60]. Thus, broadening further the knowledge\n\u0003wrzesniewski@amu.edu.pland understanding of transport and dynamical proper-\nties of large-spin molecules is important both because of\nexciting fundamental aspects as well as due to possible\napplications in modern nanoelectronics, spintronics and\nquantum information.\nMotivated by recent progress within this \feld, in this\npaper we investigate the dynamical behavior of a large-\nspin magnetic molecule attached to spin-polarized leads,\nwith an emphasis on the buildup of quadrupolar ex-\nchange \feld, referred to as spintronic anisotropy. Com-\nmonly, the intrinsic magnetic anisotropy arises from the\nspin-orbit interaction. However, it has been shown that\nthe ferromagnetic proximity e\u000bect [61{64] can generate\nsigni\fcant magnetic anisotropy in molecular systems in\nform of an e\u000bective exchange \feld [53]. The advantage\nof this approach is the possibility to electrically control\nboth the magnitude of the anisotropy and the spin state\nof the system. When the coupling strength to external\ncontacts is varied, a rapid change in the magnetic prop-\nerties of the system occurs [53, 65]. In particular, the\nmolecule's quadrupolar moment is signi\fcantly reduced,\nwhen the system is tuned from the weak to the strong\ncoupling regime. This rapid change of the moment is due\nto the Kondo screening of the molecule's orbital level spin\nand is a non-trivial many-body e\u000bect resulting from the\ninterplay of magnetism and the Kondo physics. It was\nalso shown experimentally that quadrupolar interaction\nin SMM systems has an important in\ruence on tunnel-\ning dynamics [66]. In a real setup, tuning between the\ndi\u000berent coupling regimes can be achieved by electrically\nshifting the tunnel barriers with respective gates.\nConsidering all the above, we focus on the theoret-\nical study of the quench dynamics of the spintronic\nquadrupole moment due to the Kondo correlations. In\nparticular, we identify the universal time scale for the\ndynamics describing the quench of spintronic anisotropy.\nMoreover, we examine the in\ruence of the magnitude ofarXiv:2006.04431v1  [cond-mat.mes-hall]  8 Jun 20202\ne\u000bective exchange coupling, leads spin-polarization and\ntotal spin of the molecule on discussed dynamical e\u000bects.\nLastly, we also analyze the in\ruence of \fnite tempera-\nture, showing that in certain range of temperatures, a\nstrong suppression of magnetic properties is predicted.\nIn pursuance of the precise analysis of the system's re-\nsponse to the considered quench in the strong coupling\nregime, we resort to the Wilson's numerical renormaliza-\ntion group (NRG) method [67{69]. We use the extended\nimplementation allowing for studying the time evolution\nof the system, namely, the time-dependent numerical\nrenormalization group (tNRG) [70{74]. This method al-\nlows for taking into account all the correlations in a fully\nnon-perturbative manner and, thus, generating reliable\npredictions for the dynamics of the system under inves-\ntigation.\nThis paper is structured as follows. Section II consists\nof the Hamiltonian description of the considered system,\nthe overview of the quench protocol and a summary of\nthe numerical renormalization group method used for cal-\nculations of time-dependent expectation values of local\nobservables. In Sec. III we present the numerical results\nand relevant analysis for the quantum quenches in the\ncoupling strength from the weak to the strong coupling\nregime. We also present and discuss the e\u000bects of \fnite\ntemperature on dynamical behavior. Finally, the work is\nconcluded in Sec. IV.\nII. THEORETICAL FRAMEWORK\nA. Hamiltonian\nThe e\u000bective spin Hamiltonian of a molecular magnet\nexpressed only with spin coordinates can be written as\nHe\u000b=BSz+DQzz: (1)\nHere,Szis thez-th component of the total spin S,\nQzz\u0011S2\nz\u0000S(S+ 1)=3 is thez-th diagonal tensor el-\nement of the spin-quadrupole moment, Bis a dipolar\n\feld corresponding to external magnetic \feld, Dis a\nquadrupolar \feld related to intrinsic spin-orbit interac-\ntion. This approximate approach is well-established for\nconvenient description of the system's spectrum and is\noften used to interpret the spectroscopic data [3]. In our\nconsiderations, however, the above-introduced quantities\nare generated purely by the spin-dependent coupling to\nferromagnetic leads [53, 62], which allows for tuning of\nbothBandDby electrical means|the property that\nmakes this approach advantageous from the application\npoint of view.\nIn order to analyze the dynamical behavior of the\nsystem with tNRG and, especially, capture all the\nferromagnetic-proximity induced e\u000bects, we model the\nentire system in the following way. Magnetic molecule\nis described by a single molecular level, through which\nthe electronic transport takes place, which is exchange-\ncoupled to magnetic core of the molecule speci\fed by the\nΓRSMC\nU\nJ\ns\nΓLσ σFIG. 1. Schematic of the considered system. A large-spin\nmolecule with a molecular level is coupled to external spin-\npolarized leads with the spin-dependent coupling strengths\n\u0000\u001b\nLand \u0000\u001b\nR, for the left and right lead. The molecular level is\nexchange-coupled to magnetic core spin of the molecule with\nstrength J. Coulomb correlations of the molecule are denoted\nbyU.\ne\u000bective spin SMC. Thus, the molecule can be expressed\nby the Hamiltonian\nHSMM =\"n+Un\"n#|{z}\nmolecularlevel\u0000JSMC\u0001s; (2)\nwhere the level occupation is expressed as, n=n\"+n#=\ndy\n\"d\"+dy\n#d#, withdy\n\u001b(d\u001b) being the fermionic creation\n(annihilation) operator for an electron with spin \u001b. The\nmolecular level energy is denoted by \"and the Coulomb\ncorrelations are described by U. We assume ferromag-\nnetic exchange interaction J >0 between the spin of the\nelectron on the orbital level sand the magnetic core spin\nSMC. The total spin is then expressed as S=SMC+s.\nThe molecule is coupled to left and right spin-polarized\nferromagnetic leads [75{78], see Fig. 1. Here, we exploit\nthe correspondence between the coupling to two leads at\nequilibrium with the magnetic moments arranged in the\nparallel con\fguration and the coupling to a single ferro-\nmagnetic lead. The equivalence can be shown by car-\nrying out an orthogonal transformation [33], after which\nthe central part of the system couples exclusively to even\nlinear combination of reservoir's operators with e\u000bective\ncoupling strength \u0000\u001b= \u0000\u001b\nL+ \u0000\u001b\nRand spin polarization p.\nConsequently, the leads can be described by an e\u000bective\nreservoir of noninteracting quasiparticles\nHLead=X\nk\u001b\"k\u001bcy\nk\u001bck\u001b; (3)\nwherecy\nk\u001b(ck\u001b) is the creation (annihilation) operator of\nan electron with momentum k, spin\u001band energy \"k\u001b,\nwhich is given by appropriate linear combination of elec-\ntron operators in the left and right leads. On the other\nhand, the spin-dependent coupling is speci\fed by the tun-\nneling term\nHTun=X\nk\u001bV\u001b(cy\nk\u001bd\u001b+ H:c:); (4)3\nwhereV\u001bare the e\u000bective tunnel matrix elements, as-\nsumed to be momentum independent.\nThe spin-dependent coupling between the molecule\nand the e\u000bective lead is expressed as, \u0000\u001b=\u0019\u001a\u001bjV\u001bj2,\nwith\u001a\u001bbeing the spin-dependent density of states of fer-\nromagnetic electrodes. By introducing the spin polariza-\ntion of the leads p, the coupling strength can be written\nin the following manner, \u0000\"(#)= \u0000(1\u0006p), with \u0000\"(#)de-\nnoting the coupling to the spin-up (spin-down) electron\nband of the ferromagnetic reservoir and \u0000 = (\u0000\"+\u0000#)=2.\nFinally, the full Hamiltonian of the considered system\nreads\nH=HSMM +HLead+HTun: (5)\nB. Quench protocol and NRG implementation\nThe time-dependent Hamiltonian describing the dy-\nnamics after a quantum quench has the following general\nform\nH(t) =\u0012(\u0000t)H0+\u0012(t)H; (6)\nwhere the Hamiltonian H0denotes the initial Hamilto-\nnian of the system. On the other hand, His the Hamil-\ntonian describing the time evolution after the sudden\nquench at time t= 0 and\u0012(t) is the Heaviside step func-\ntion. Both Hamiltonians have a form outlined in Eq. (5)\nwith appropriate parameters modi\fed according to the\nevaluated quench. The time-dependent expectation value\nof a given local operator O(t) can be calculated from\nO(t)\u0011hO (t)i= Tr\b\ne\u0000iHt\u001a0eiHtO\t\n: (7)\nHere,\u001a0is the initial density matrix of the system de-\nscribed by the Hamiltonian H0.\nLet us now brie\ry discuss the most important aspects\nconcerning the NRG implementation of the quench calcu-\nlations [67{69]. The essential part of the NRG procedure\nis the logarithmic discretization of the conduction band\nfollowed by mapping of the discretized Hamiltonian to\na one-dimensional tight-binding chain called the Wilson\nchain [68]. This is performed for both Hamiltonians H\nandH0. Subsequently, the two Hamiltonians are inde-\npendently solved in an iterative fashion using the NRG\nprocedure [69]. At each step of iteration, there are states\nthat are used to construct the state-space of the next it-\neration and the states that are discarded. The discarded\nstates are used to create the full many-body eigenbases\n[70]\nX\nnsejnseiD\n0D\n0hnsej= 1andX\nnsejnseiDDhnsej= 1;(8)\nof both Hamiltonians, H0andH, respectively and to\nconstruct the full density matrix \u001a0at temperature T\u0011\n1=\f[79]\n\u001a0=X\nnsee\u0000\fED\n0ns\nZjnseiD\n0D\n0hnsej; (9)where\nZ\u0011X\nnsee\u0000\fED\n0ns (10)\nis the partition function. Here, sdenotes a state at Wil-\nson siten, whileecorresponds to an environmental state\ndescribing the rest of the chain.\nThe time-dependent expectation value hO(t)iof an op-\neratorOcan be conveniently evaluated in the frequency\nspace and then Fourier-transformed to the time domain\n[80]. The frequency-dependent expectation value hO(!)i\nof a local operator Oexpressed in the corresponding\neigenstates of the two Hamiltonians is given by [81]\nhO(!)i=XX06=KKX\nnX\nn0X\nss0eXhnsejwn0\u001a0n0jns0eiX0\n\u0002X0hns0ejOjnseiX\u000e(!+EX\nns\u0000EX0\nns0):(11)\nwhereX=K(X=D) denotes a kept (discarded) state.\nHere,\u001a0n0is the contribution of the density matrix com-\ning from iteration n0andwn0is the corresponding weight.\nWe also use NRG to determine the linear-response con-\nductance between the two ferromagnetic leads from the\nfollowing formula [82]\nG=e2\nh\u0019\u0000Z\nd!\u0012\n\u0000@f\n@!\u0013\n[(1 +p)A\"(!) + (1\u0000p)A#(!)];\n(12)\nwhereA\u001b(!) is the molecular level's spectral function,\nde\fned asA\u001b(!) =\u0000(1=\u0019)Imhhd\u001bjdy\n\u001biiR\n!, withhhd\u001bjdy\n\u001biiR\n!\nbeing the Fourier transform of the retarded Green's func-\ntionhhd\u001bjdy\n\u001biiR\nt=\u0000i\u0012(t)hfd\u001b(t);dy\n\u001b(0)gi.\nFor the NRG calculations we used the discretization\nparameter 2 6\u000363, set the length of the Wilson chain\nto beN= 80 and kept at least NK= 4000 energeti-\ncally lowest-lying states at each iteration. In order to\nsuppress the band discretization e\u000bects, we also used the\nOliveira'sz-averaging [83] by performing calculations for\nNz= 4 di\u000berent discretizations. More details and tech-\nnicalities concerning the implementation of calculations\ncan be found in Ref. [81].\nIII. RESULTS AND DISCUSSION\nA. Static properties of the molecule\nIn order to obtain a better understanding of the mag-\nnetic correlations present in the considered system, let us\n\frst examine the static properties. As the main focus is\nput on the quadrupolar \feld, we tune the system to the\nparticle-hole symmetry point by setting the energy of the\norbital level to \"=\u0000U=2. As a result, the dipolar \feld\nvanishes [61] and only quadrupolar term is present in the\nsystem [second term of Eq. (1)]. This approach allows us\nto precisely describe the generated uniaxial anisotropy,\nquanti\fed by the amplitude D, see Eq. (1).4\n0.331.002.003.004.00/angbracketleftQzz/angbracketright,/angbracketleftS2\nz/angbracketright,/angbracketleftS2/angbracketright(a)\n0.00.20.40.60.81.0\nG/G 0\n/angbracketleftQzz/angbracketright\n/angbracketleftS2\nz/angbracketright\n/angbracketleftS2/angbracketright\nG/G 0\n10−210−1100\nΓ/U0.000.331.00/angbracketleftQzz/angbracketrightJ/U = 0.002\nJ/U = 0.004\nJ/U = 0.006\nJ/U = 0.008\nJ/U = 0.010\nJ/U = 0.0200.20.40.60.81.0\nG/G 0(b)\nFIG. 2. (a) The spin-quadrupole moment hQzziexpectation\nvalues of the corresponding spin operators, hS2iandhS2\nzi,\nand normalized linear-response conductance G, with G0=\n2e2=h, forS= 3=2 spin molecule as a function of the coupling\nstrength \u0000. The parameters are: U= 1=2,\"=U =\u00001=2,\nJ=U = 2\u000110\u00003, in units of band halfwidth W\u00111,p= 0:5\nand temperature T=U\u001810\u000018. (b) The spin-quadrupole\nmomenthQzzi(solid lines) and normalized linear-response\nconductance G(dashed lines) plotted versus \u0000 for di\u000berent\nvalues of exchange coupling J.\nIn Fig. 2(a) we present the spin-quadrupole mo-\nmenthQzziand expectation values of the corresponding\nspin operators,hS2iandhS2\nzi, as the coupling strength\n\u0000 is varied. The general behavior of spin-quadrupole\nmoment (dark-blue line) is that in the weak-coupling\nregime (\u0000=U.10\u00001), it acquires the value hQzzi=\nS(2S\u00001)=3 = 1, while in the strong-coupling regime\n(\u0000=U&10\u00001) this value is strongly reduced to hQzzi=\nSMC(2SMC\u00001)=3 = 1=3. The suppression of the mo-\nment in the strong-coupling regime is due to the pres-\nence of the Kondo correlations. The Kondo e\u000bect is ex-\nposed in the conductance dependence (yellow line), as it\nsaturates to unitary value G=G 0= 1 in the strong cou-\npling regime. In consequence, the spin of the molecular\nlevel is screened, and the total spin of the molecule is\nreduced from S= 3=2 toS=SMC= 1, leading even-\ntually tohQzzi= 1=3. To clearly show how the expec-\ntation values of spin operators in\ruence the value of the\nspin-quadrupole moment, we also plot hS2iandhS2\nzias\na function of the coupling strength. It is noteworthy\nthat both quantities in the Kondo regime do not achieve\nthe value, that would be expected when electron on the\norbital level was fully screened by the Kondo correla-\ntions. The expected values only approach this limit, i. e.hS2i!2,hS2\nzi!1, however, the resulting value of the\nspin-quadrupole moment is indeed hQzzi= 1=3.\nIn Fig. 2(b) we present the spin-quadrupole moment\nhQzzi(solid lines) and the linear-response conductance\nG(dashed lines) as a function of \u0000 for di\u000berent val-\nues of the exchange coupling J. As evident, when the\nmagnitude of Jis varied, the spin-quadrupole moment\nvalues in the weak and strong coupling regimes are con-\nserved, however, the increase of the exchange coupling\nextends the transitional range of coupling strength where\nthe crossover between the weak and Kondo regimes de-\nvelops. Although the considered model is an e\u000bective\none, we expect that in the case of molecules with strong\nexchange couplings between localized spins and those of\nitinerant electrons, the quench in the coupling strength\nneeds to be superior than in the case of systems with\nsmall magnitudes of the exchange couplings. The role of\nmagnitude of Jon the dynamics of the spin-quadrupole\nmoment is discussed in more detail in Sec. III C.\nB. Dynamics of quadrupole moment\nThe important alteration in the SMM's magnetic prop-\nerties is when the coupling strength \u0000 is switched from\nthe weak coupling regime, where the spin-quadrupole mo-\nment is saturated acquiring hQzzi=S(2S\u00001)=3 = 1, to\nthe strong coupling regime. In the latter case, the Kondo\ncorrelations are present and the moment is reduced to\nhQzzi=SMC(2SMC\u00001)=3 = 1=3.\nIn order to study the dynamics of this e\u000bect, we per-\nform a quantum quench of the initial Hamiltonian H0in\nthe coupling strength from \u0000 0=U= 0:002 to various \fnal\nvalues of \u0000=Uin the range of strong coupling where the\nKondo e\u000bect is well-established. In Fig. 3(a) we display\nthe resulting time-dependence of the spin-quadrupole\nmomentQzz(t). First of all, independently of \u0000, both\ninitialQzz(t= 0) and \fnalQzz(t!1 ) are in agree-\nment with thermal expectation values. As can be seen,\nthe long-term value is also independent of the coupling\nstrength \u0000 as long as the \fnal coupling strength is within\nthe strong coupling regime, here \u0000 =U&0:1, indicated by\nthe unitary conductance, see Fig. 2. However, the value\nof \u0000 clearly in\ruences the dynamics of the transition.\nIn the short-time limit, the initial value of hQzzi= 1\nholds for at least time t\u0001\u0000\u001910\u00001. The reduction of\nquadrupole moment takes place for times t\u0001\u0000&10\u00001,\nand the dependence of this process is in\ruenced by the\n\fnal coupling strength. In the range of 0 :4&\u0000=U&0:2,\nthe drop ofQzz(t) is observable as early as t\u0001\u0000\u00192\u000110\u00001\nand at middle rate approaches the long-time limit for\ntimest\u0001\u0000&101. However, when the quench is eval-\nuated for higher values of the \fnal coupling strength,\nthe time-dependence is more rapid. On one side, Qzz(t)\nstarts to drop at later times, t\u0001\u0000\u00195\u0001101for \u0000=U= 2,\nbut on the other side, it achieves long-time limit signif-\nicantly faster, i. e. Qzz(t= 2=\u0000)\u0019Qzz(t!1 ) also\nfor \u0000=U= 2. Therefore, by tuning the value of the \f-5\n10−210−1100101102\nt·Γ0.51.01.52.0Γ/UQzz(t)\n(a)\n0.330.400.500.600.700.800.901.00\n10−210−1100101102\nt·Γ0.40.60.81.0Qzz(t)(b)\nΓ/U= 0.20\nΓ/U= 0.24\nΓ/U= 0.28\nΓ/U= 0.32\nΓ/U= 0.36\nΓ/U= 0.40\nΓ/U= 0.60\nΓ/U= 0.80\nΓ/U= 1.00\n10−310−210−1100101\nt·Γ20.40.60.81.0Qzz(t)(c)\nΓ/U= 0.20\nΓ/U= 0.24\nΓ/U= 0.28\nΓ/U= 0.32\nΓ/U= 0.36\nΓ/U= 0.40\nΓ/U= 0.60\nΓ/U= 0.80\nΓ/U= 1.00\nFIG. 3. The time dependence of the spin-quadrupole\nmomentQzz(t) after quench in the coupling strength from\n\u00000=U= 0:002 to di\u000berent values of the \fnal coupling strength.\n(a) presents the density plot of Qzz(t), while (b) and (c) show\nQzz(t) as a function of t\u0001\u0000 and t\u0001\u00002, respectively. The other\nparameters are the same as in Fig. 2.\nnal coupling strength \u0000, the timescale of the considered\ntransition can be varied by an order of magnitude.\nTo clearly display the above behavior, in Fig. 3(b)\nwe show the time evolution of quadrupolar moment for\ndi\u000berent values of \u0000 chosen from the range, where the\ndynamics is most interesting. Finally, in Fig. 3(c) we\nshow the same results in the form of several curves on\na rescaled time axis in order to clearly indicate that the\nrelaxation is universally governed by t/1=\u00002. The long-\ntime limit is achieved at times t\u00192=\u00002, when one can\nsee that all the curves converge.\nC. Role of molecule's exchange coupling\nAn important impact on the magnetic and transport\nproperties of SMM systems has the magnitude of ex-\nchange coupling Jbetween the e\u000bective magnetic core\nspinSMCand the spin sof electrons occupying the or-\n10−210−1100101102103104105106\nt·Γ0.0050.0100.0150.020J/UQzz(t)\n0.30.40.50.60.70.80.9FIG. 4. The spin-quadrupole moment Qzz(t) after the\nquench from the weak to the strong coupling regime (\u0000 0=U=\n0:002 and \u0000 =U= 0:4) as a function of time and exchange in-\nteraction J=U. The other parameters are the same as in Fig.\n2.\nbital level. In Fig. 4 we show the time-dependent spin-\nquadrupole moment, considering similar quench as in\nprevious section, with \u0000 0=U= 0:002 and \u0000=U= 0:4,\nas a function of exchange coupling J.\nIt can be clearly seen that the magnitude of Jdoes\nnot have a substantial in\ruence on the short time evo-\nlution of spin-quadrupole moment for elapsed time up\ntot\u0001\u0000\u0019102. ForJ=U.0:002, the long-time limit\nhQzzi= 1=3 is achieved already around t\u0001\u0000\u0019101\nand there is no further dynamics as the time elapses.\nHowever, when the exchange coupling is increased above\nJ=U\u00190:002, further decrease of spin-quadrupole mo-\nment takes place at long times, i.e. for t\u0001\u0000&102.\nThe observed reduction is down to values well below\nhQzzi= 1=3, i.e.Qzz(t!1 ).0:3, and the dynamics\nof this process is strongly dependent on the magnitude\nofJ. When the time-dependence for larger values of J\nis evaluated, the new long-time limit is achieved at ear-\nlier times, while the value of spin-quadrupole moment is\naccordingly further suppressed.\nIn order to better understand the e\u000bect of exchange\ncoupling on the spin-quadrupole moment dynamics, we\nexamine the time-dependence of two spin operators,\nS2\nz(t) andS2(t)=3, which are the constituent compo-\nnents of the operator Qzz(t). The corresponding plots are\nshown in Fig. 5. The results were obtained for exchange\ncoupling set to J=U = 0:01, where the time dependence\nreveals two stages of magnetic moment's reduction.\nIn Fig. 5(a) we display the time-dependent spin-\nquadrupole moment after the quench in the coupling\nstrength, while in Fig. 5(b) we present the time-\ndependent expectation values of spin operators S2\nz(t) and\nS2(t)=3.\nTo highlight the most important stages of the dynam-\nics, we marked three time ranges with respective pale col-6\nt·Γ0.20.40.60.81.0Qzz(t)(a)Qzz(t)\n10−210−1100101102103104105106\nt·Γ1.01.52.0S2(t),S2\nz(t)/3(b)S2\nz(t)\nS2(t)/3\nFIG. 5. The time dependence of the spin-quadrupole mo-\nmentQzz(t) and the expectation values of S2\nz(t) and S2(t)=3\nafter quench in the coupling strength from \u0000 0=U= 0:002 to\n\u0000=U= 0:4. The exchange coupling is set to J=U = 0:01. The\nother parameters are the same as in Fig. 2.\nors. The red background contains short-time dynamics,\nwhen a rapid suppression of all expectation values takes\nplaces in a similar manner like in the case of weaker J\ncoupling. This dynamics is associated with the change\nof the ground state and the reduction of the total spin\nSfromS= 3=2 toS= 1 due to the Kondo screening\nof the orbital spin. This stage of time evolution ends\nup at times around t\u0001\u0000\u0019101. Consequently, the next\nsection is marked with purple background and contains\nthe further dynamics associated with in\ruence of the ex-\nchange interaction J. For times t\u0001\u0000&101, the time-\ndependence of the spin operators clearly shows that the\ntotal spin of the system Sremains intact and has already\nachieved the long-time limit. Further ongoing dynamics\ntakes place exclusively for S2\nz(t) in the form of second\nstage of slow suppression of the total spin-quadrupole\nmoment toQzz(t).1=3. The total time of this stage\nevolution greatly depends on the strength of J, as shown\nin Fig. 4, i.e. the system faster reaches equilibrium for\nhigher values of the exchange coupling J.\nD. In\ruence of leads' spin polarization\nThe ferromagnetism of the leads is of great importance\nfor the considered molecular system. In general, when a\n\fnite spin polarization of electrodes is assumed ( p >0)\nand their magnetic moments are aligned, the e\u000bective\ndipolar and quadrupolar exchange \felds are generated\n[53, 84]. The strengths of these \felds and their occur-rence strongly depend on the degree of spin polarization\np. In our analysis, we focus on the quadrupolar term by\ntuning the molecular level to the particle-hole symmetry\npoint\"=\u0000U=2. In such a con\fguration, the charge \ruc-\ntuations for both spin directions are equal and indepen-\ndently of the magnitude of pthe dipolar \feld is canceled.\nMoreover, it was shown that for large-spin molecules, a\nvery small spin polarization ( p= 0:01) of the leads can\ngive rise tohQzzi=S(2S\u00001)=3 = 1. In equilibrium, a\nsimilar dependence to that shown in Fig 2 is predicted\nfor wide range of p, with \fnite temperature suppressing\nthe spin-quadrupole moment in the regime of very weak\ncoupling strengths [65]. Therefore, let us now discuss the\nin\ruence of leads spin polarization pon the dynamics of\nthe spin-quadrupole moment after the quench from the\nweak to the strong coupling regime (\u0000 0=U= 0:002 and\n\u0000=U= 0:4). The results presenting Qzz(t) for a wide\nrange of spin polarizations pare shown in Fig. 6(a).\nIn the range of small and moderate spin polarizations\n(p.0:4), the time evolution of spin quadrupole moment\nhas both qualitatively and quantitatively similar depen-\ndence, with two stages of Qzz(t) = 1 reduction present,\nas discussed in previous section for the case of signif-\nicant exchange coupling J=U = 0:01. However, when\nthe spin polarization is increased further ( p&0:4), a\nnew step emerges right before the time t\u0001\u0000\u00191, when\nQzz(t)\u00193=4. This step is elongated as pis increased,\nreaching times up to t\u0001\u0000\u0019101forp= 0:9 and even-\ntually not fully relaxing to Qzz(t) = 1=3, whenp!1.\nThis behavior might seem to be counter-intuitive, as one\ncould expect that with increased spin-polarization of the\nleads, the dynamics of the quench should be faster along\nwith rapid suppression of the spin-quadrupole moment.\nHere, we predict quite opposite dependence, as the dy-\nnamics is faster and more straightforward when the spin-\npolarization of the leads has low-to-moderate values.\nThe mechanism responsible for the emergence of an ad-\nditional step inQzz(t) in the case of highly spin-polarized\nelectrodes is closely related to an enhanced di\u000berence\nbetween spin-dependent couplings \u0000\"= \u0000(1 +p) and\n\u0000#= \u0000(1\u0000p) whilstpis increased. The bottleneck of\nthe associated dynamics is governed by the coupling to\nthe minority band, which in result slows down the dy-\nnamics of the spin-quadrupole moment. To clearly an-\nalyze the discussed e\u000bect, we show the spin-quadrupole\nmomentQzz(t) in Fig. 6(b) and the square of molecular-\nlevel magnetization s2\nz(t) in Fig. 6(c) as a functions of\nt\u0001\u0000. Here, we recall that molecular-level magnetization\nis de\fned as sz(t) = (n\"(t)\u0000n#(t))=2. The discussed dy-\nnamics revealed for highly spin-polarized lead takes place\nat timest\u0001\u0000.1. The e\u000bect is evident for p= 0:9, where\nthe dependence of Qzz(t) exposes a short plateau with\nQzz(t)\u00193=4. The time-dependence of the local opera-\ntors2\nz(t) plays an important role here. As can be seen in\nthe \fgure, s2\nz(t) is a monotonically decreasing function of\ntime forp.0:7 in this time regime. Interestingly, fur-\nther enhancement of spin-polarization generates a subtle\noscillation in the time-dependence of s2\nz(t). In that time7\n10−210−1100101102103104\nt·Γ0.20.40.60.8pQzz(t)\n(a)\n0.40.60.8\n10−210−11001011020.250.500.751.00Qzz(t)(b)\np= 0.1\np= 0.3\np= 0.5\np= 0.7\np= 0.9\n10−210−1100101102\nt·Γ0.200.25s2\nz(t)(c)\nFIG. 6. (a) The spin-quadrupole moment Qzz(t) after a\nquench from the weak to the strong coupling regime as a func-\ntion of time and spin polarization p. (b) The spin-quadrupole\nmomentQzz(t) and (c) square of the molecular level's mag-\nnetization s2\nz(t) after the quench as a function of t\u0001\u0000 shown\nfor several values of p. The other parameters are the same as\nin Fig. 5.\nrange, \u0000\"= \u0000(1 +p) is responsible for a rapid drop of\ns2\nz(t) due to decrease of the spin-up component. However,\nthe time evolution is not balanced by the spin-down com-\nponent due to a signi\fcantly slower dynamics governed\nby \u0000#= \u0000(1\u0000p). As a result, the molecular level has\na minimum in s2\nz(t), which is below the long-time limit\nthermal value, at time t\u0001\u0000\u00196\u000110\u00001, see Fig. 6(c).\nSubsequently, a small increase takes places leading to\neventual relaxation. This non-monotonic dependence of\ns2\nz(t) is observed as a result of the interplay between the\nspin-dependent couplings and, in consequence, temporar-\nily pauses the reduction of Qzz(t). A similar dynamical\nbehavior was predicted for the magnetization of a single\nquantum dot system coupled to ferromagnetic lead [81].\nE. In\ruence of magnitude of molecule's spin\nIn order to generalize our analysis for magnetic\nmolecules of arbitrarily given total spin S, we exam-\nine the results of tNRG calculations for models with\ndi\u000berent values of magnetic core spin SMC. First of\nall, we would like to note that in equilibrium, qualita-\ntively a very similar dependence of hQzzion the cou-\n10−210−1100\nΓ/U01234/angbracketleftQzz/angbracketright(a) SMC= 2\nSMC= 3/2\nSMC= 1\nSMC= 1/2\n10−210−1100101102\nt·Γ01234Qzz(t)(b)FIG. 7. (a) The static value of spin-quadrupole moment Qzz\nfor di\u000berent values of SMCas a function of coupling \u0000 and (b)\nQzz(t) for several values of SMCas a function of time elapsed\nafter the quench in the coupling \u0000. The parameters are the\nsame as in Fig. 2.\npling strength \u0000 is expected independently of the value\nofSMC, see Fig. 7(a). As the spin of magnetic core\nis increased, the maximal value of the spin-quadrupole\nmoment in the weak coupling regime is increasing ac-\ncordingly withhQzzi=S2\nz\u0000S(S+ 1)=3. Further-\nmore, the transition range of coupling strengths preced-\ning the strong coupling regime is similar for all cases,\ni.e. 5\u000110\u00002.\u0000=U.2\u000110\u00001. Eventually, the impor-\ntant change in\ruenced by the total spin of the molecule\nis in the value, by which the spin-quadrupole moment\nis reduced, when the coupling regime is switched from\nthe weak to the strong one. This di\u000berence is enhanced,\nwhen the total spin number is increased.\nThe time-dependent spin-quadrupole moment after the\nquench as a function of time for several values of SMCis\nshown in Fig. 7(b). Here, one can see that the dynamics\nis still governed by the strength of coupling to electrodes,\nas the reduction of spin-quadrupolar moment starts and\nachieves long-time limit at similar moments on the time\naxis rescaled with \u0000. For all considered values of SMC,8\ntime-evolutions behave in a similar fashion with the main\ndistinction of initial and \fnal values of hQzziand the\nrate of reduction. Similar e\u000bects and dependencies due\nto the exchange interaction or spin-polarization of the\nleads as discussed in previous sections are predicted also\nfor molecules with even higher total spin number (not\nshown here). Therefore, our dynamical studies presented\nin this work have a general character and are valid for\nbroad range of SMM systems, in which the quadrupo-\nlar exchange \feld emerges from ferromagnetic proximity\ne\u000bect.\nF. Finite temperature e\u000bects\nLastly, we examine the in\ruence of \fnite temperature\nTon the spin-quadrupole moment and its time evolu-\ntion following the quench in the coupling. In Fig. 8 we\nshow the time-dependent expectation value of the spin-\nquadrupole moment as well as the corresponding spin\noperators for several values of temperatures.\nFor the regime of very low temperatures, T=U.10\u00008,\nboth the initial and \fnal values, as well as the quench\ndynamics of the spin-quadrupole moment remain simi-\nlar to the case of zero temperature. Further increase of\ntemperature, however, has a signi\fcant impact on the\nsystem's behavior. First of all, the initial value of the\nspin quadrupole moment in the weak coupling regime is\nsigni\fcantly suppressed as the temperature is increased.\nThe inset in Fig. 8(a) shows how the increase of temper-\nature reduces the static spin-quadrupole moment in the\nweak coupling regime. The vertical dashed line repre-\nsents the initial coupling strength \u0000 0=U= 0:002, we have\nused in the evaluation of the quench dynamics. From the\nabove considerations, it is evident that the \fne tuning\nof the coupling strength for the initial state is critical,\nin particular when the suppression of the maximal value\nof the moment is expected. Furthermore, the long-time\nlimit value of the time-dependent spin-quadrupole mo-\nment is also reduced by the temperature, however, not\nas strongly as the initial value. In consequence, as the\ntemperature is increased, the whole transient dynamics is\nexposing more moderate dependence, with almost com-\npletely \rat one for temperatures T=U&10\u00006. It is also\nimportant to note, that all the characteristic time scales\ndiscussed in earlier analysis for T= 0 are conserved for\nthe \fnite temperatures, where a considerable suppres-\nsion of spin-quadrupole moment is predicted. In par-\nticular, the rapid reduction associated with the change\nof the ground state that dominates the dynamics for\ntimest\u0001\u0000.101and further slower dynamics in\ruenced\nby the exchange coupling taking place for times up to\nt\u0001\u0000\u0019103are all still noticeable for temperatures up to\nT=U\u001910\u00007.\nThe inspection of the spin operators S2\nz(t) andS2(t),\nsee Figs. 8(b) and 8(c), brings the conclusion that the\ntemperature in\ruences spin-quadrupole moment only by\nreduction of the z-th component of the SMM's spin.\n0.00.51.0Qzz(t)(a)\n10−410−310−2\nΓ/U01/angbracketleftQzz/angbracketright\n1.01.52.0S2\nz(t)(b)T/U = 1·10−8\nT/U = 2·10−8\nT/U = 4·10−8\nT/U = 1·10−7\nT/U = 2·10−7\nT/U = 4·10−7\nT/U = 1·10−6\n10−210−1100101102103104\nt·Γ2.53.03.5S2(t)(c)FIG. 8. (a) The spin-quadrupole moment Qzz(t), (b) S2\nz(t)\nand (c) S2(t) for several values of temperature Tplotted as\na function of time elapsed after the quench in \u0000. The inset\nin (a) shows the in\ruence of temperature on the static value\nof the spin-quadrupole moment in the weak coupling regime.\nDashed vertical line indicates the initial value of the coupling\nstrength. The parameters are the same as in Fig. 5.\nMeanwhile the square value of the total spin remains un-\na\u000bected, as it corresponds to the change of the molecule's\nground state. This fact clearly indicates the reduction of\nthe molecule's magnetic anisotropy due to thermal \ruc-\ntuations.\nIV. CONCLUSIONS\nWe have analyzed the quench dynamics of large-spin\nmagnetic molecules attached to spin-polarized ferromag-\nnetic leads. The study was performed by using the time-\ndependent numerical renormalization group method. We\nfocused on the dynamics of the spin-quadrupole moment\nand a quench associated with switching the system from\nthe weak coupling regime to the strong coupling one, in\nwhich the Kondo correlations are present and the screen-\ning of the molecular level spin develops.\nIn general, we have shown that the time necessary\nto achieve the new thermal value in the strong cou-\npling regime is inversely proportional to squared coupling9\nstrength, i.e. t/1=\u00002. Furthermore, we examined the\nrole of ferromagnetic exchange coupling Jand showed\nthat when the magnitude of this interaction is consid-\nerable, additional step in the time-dependence emerges,\ncorresponding to slower dynamics mediated by J, which\nis an interesting case of interplay between the Kondo\ncorrelations and ferromagnetism exposed in our dynami-\ncal studies. The in\ruence of electrodes' spin-polarization\nwas also discussed, strongly indicating that the quench\ndynamics is faster and more straightforward for low-to-\naverage values. We also generalized our studies to sys-\ntems with higher total spin numbers to show that obser-\nvations and conclusions are valid for a wide class of SMM\nsystems with arbitrary total spin number.\nFinally, we studied the in\ruence of \fnite tempera-\nture on the spin-quadrupole moment. A strong suppres-sion of the time-dependent value is predicted in certain\nrange of temperatures, due to reduction of the molecule's\nanisotropy. The important role of the coupling strength\n\fne tuning is also indicated, which may have relevance\nfor the experiments.\nACKNOWLEDGMENTS\nThis work was supported by the Polish National Sci-\nence Centre from funds awarded through the decision\nNo. 2017/27/B/ST3/00621. The computing time at the\nPozna\u0013 n Supercomputing and Networking Center is ac-\nknowledged.\n[1] Roman Bo\u0014 ca, Theoretical Foundations of Molecular Mag-\nnetism (Elsevier, 1999).\n[2] J. Tejada, E. M. Chudnovsky, E. del Barco, J. M. Her-\nnandez, and T. P. Spiller, \\Magnetic qubits as hardware\nfor quantum computers,\" Nanotechnology 12, 181{186\n(2001).\n[3] D. Gatteschi, R. Sessoli, and J. Villain, Molecular Nano-\nmagnets (Oxford University Press, 2006).\n[4] Lapo Bogani and Wolfgang Wernsdorfer, \\Molecular\nspintronics using single-molecule magnets,\" Nat. Mater.\n7, 179{186 (2008).\n[5] J org Lehmann, Alejandro Gaita-Ari~ no, Eugenio Coro-\nnado, and Daniel Loss, \\Quantum computing with\nmolecular spin systems,\" J. Mater. Chem. 19, 1672{1677\n(2009).\n[6] Matteo Mannini, Francesco Pineider, Philippe Sainc-\ntavit, Chiara Danieli, Edwige Otero, Corrado Sciancale-\npore, Anna Maria Talarico, Marie-Anne Arrio, Andrea\nCornia, Dante Gatteschi, and Roberta Sessoli, \\Mag-\nnetic memory of a single-molecule quantum magnet wired\nto a gold surface,\" Nat. Mater. 8, 194{197 (2009).\n[7] Romain Vincent, Svetlana Klyatskaya, Mario Ruben,\nWolfgang Wernsdorfer, and Franck Balestro, \\Electronic\nread-out of a single nuclear spin using a molecular spin\ntransistor,\" Nature 488, 357{360 (2012).\n[8] Peter Jacobson, Tobias Herden, Matthias Muenks, Gen-\nnadii Laskin, Oleg Brovko, Valeri Stepanyuk, Markus\nTernes, and Klaus Kern, \\Quantum engineering of spin\nand anisotropy in magnetic molecular junctions,\" Nat.\nCommun. 6, 1{6 (2015).\n[9] Roberta Sessoli, \\Magnetic molecules back in the race,\"\nNature 548, 400{401 (2017).\n[10] Khadijeh Naja\f, Aleksander L. Wysocki, Kyungwha\nPark, Sophia E. Economou, and Edwin Barnes, \\Toward\nLong-Range Entanglement between Electrically Driven\nSingle-Molecule Magnets,\" J. Phys. Chem. Lett. 10,\n7347{7355 (2019).\n[11] P. Gambardella, S. Rusponi, M. Veronese, S. S. Dhesi,\nC. Grazioli, A. Dallmeyer, I. Cabria, R. Zeller, P. H.\nDederichs, K. Kern, C. Carbone, and H. Brune, \\Gi-\nant Magnetic Anisotropy of Single Cobalt Atoms and\nNanoparticles,\" Science 300, 1130{1133 (2003).[12] Cyrus F. Hirjibehedin, Chiung-Yuan Lin, Alexander F.\nOtte, Markus Ternes, Christopher P. Lutz, Barbara A.\nJones, and Andreas J. Heinrich, \\Large Magnetic\nAnisotropy of a Single Atomic Spin Embedded in a Sur-\nface Molecular Network,\" Science 317, 1199{1203 (2007).\n[13] Pietro Gambardella, Sebastian Stepanow, Alexandre\nDmitriev, Jan Honolka, Frank M. F. de Groot, Magal\u0013 \u0010\nLingenfelder, Subhra Sen Gupta, D. D. Sarma, Peter\nBencok, Stefan Stanescu, Sylvain Clair, St\u0013 ephane Pons,\nNian Lin, Ari P. Seitsonen, Harald Brune, Johannes V.\nBarth, and Klaus Kern, \\Supramolecular control of the\nmagnetic anisotropy in two-dimensional high-spin Fe ar-\nrays at a metal interface,\" Nat. Mater. 8, 189{193 (2009).\n[14] Lei Xu, Ziba Zangeneh, Ravi Yadav, Stanislav Av-\ndoshenko, Jeroen van den Brink, Anton Jesche, and\nLiviu Hozoi, \\Spin-reversal energy barriers of 305 K for\nFe2+ d6 ions with linear ligand coordination,\" Nanoscale\n9, 10596{10600 (2017).\n[15] Aleksander L. Wysocki and Kyungwha Park, \\Nature of\nHyper\fne Interactions in TbPc2 Single-Molecule Mag-\nnets: Multicon\fgurational Ab Initio Study,\" Inorg.\nChem. 59, 2771{2780 (2020).\n[16] Eugene M. Chudnovsky and Javier Tejada, \\Macroscopic\nQuantum Tunneling of the Magnetic Moment,\" Cam-\nbridge Core (1998), 10.1017/CBO9780511524219.\n[17] Euan K. Brechin, Colette Boskovic, Wolfgang Werns-\ndorfer, Jae Yoo, Akira Yamaguchi, E. Carolina Sa~ nudo,\nThomas R. Concolino, Arnold L. Rheingold, Hidehiko\nIshimoto, David N. Hendrickson, and George Chris-\ntou, \\Quantum Tunneling of Magnetization in a New\n[Mn18]2+ Single-Molecule Magnet with S = 13,\" J. Am.\nChem. Soc. 124, 9710{9711 (2002).\n[18] M. Misiorny and J. Barna\u0013 s, \\Quantum tunneling of mag-\nnetization in single molecular magnets coupled to ferro-\nmagnetic reservoirs,\" EPL 78, 27003 (2007).\n[19] M. Mannini, F. Pineider, C. Danieli, F. Totti, L. So-\nrace, Ph. Sainctavit, M.-A. Arrio, E. Otero, L. Joly, J. C.\nCezar, A. Cornia, and R. Sessoli, \\Quantum tunnelling\nof the magnetization in a monolayer of oriented single-\nmolecule magnets,\" Nature 468, 417{421 (2010).\n[20] Gwang-Hee Kim and Tae-Suk Kim, \\Electronic Trans-\nport in Single-Molecule Magnets on Metallic Surfaces,\"10\nPhys. Rev. Lett. 92, 137203 (2004).\n[21] Carsten Timm and Florian Elste, \\Spin ampli\fcation,\nreading, and writing in transport through anisotropic\nmagnetic molecules,\" Phys. Rev. B 73, 235304 (2006).\n[22] Maciej Misiorny, Ireneusz Weymann, and J\u0013 ozef Barna\u0013 s,\n\\Spin e\u000bects in transport through single-molecule mag-\nnets in the sequential and cotunneling regimes,\" Phys.\nRev. B 79, 224420 (2009).\n[23] M. Misiorny, I. Weymann, and J. Barna\u0013 s, \\Spin diode\nbehavior in transport through single-molecule magnets,\"\nEPL89, 18003 (2010).\n[24] Maciej Misiorny, Ireneusz Weymann, and J\u0013 ozef\nBarna\u0013 s, \\Temperature dependence of electronic trans-\nport through molecular magnets in the Kondo regime,\"\nPhys. Rev. B 86, 035417 (2012).\n[25] M. Misiorny, E. Burzur\u0013 \u0010, R. Gaudenzi, K. Park, M. Lei-\njnse, M. R. Wegewijs, J. Paaske, A. Cornia, and H. S. J.\nvan der Zant, \\Probing transverse magnetic anisotropy\nby electronic transport through a single-molecule mag-\nnet,\" Phys. Rev. B 91, 035442 (2015).\n[26] Anna P lomi\u0013 nska and Ireneusz Weymann, \\Pauli spin\nblockade in double molecular magnets,\" Phys. Rev. B\n94, 035422 (2016).\n[27] Anna P lomi\u0013 nska, Maciej Misiorny, and Ireneusz Wey-\nmann, \\Manipulating spins of magnetic molecules: Hys-\nteretic behavior with respect to bias voltage,\" EPL 121,\n38006 (2018).\n[28] F. Pawlicki and I. Weymann, \\Andreev transport\nthrough single-molecule magnets,\" Phys. Rev. B 98,\n085411 (2018).\n[29] A. Chiesa, E. Macaluso, P. Santini, S. Carretta, and\nE. Pavarini, \\First-principles many-body models for elec-\ntron transport through molecular nanomagnets,\" Phys.\nRev. B 99, 235145 (2019).\n[30] Silvia Gim\u0013 enez-Santamarina, Salvador Cardona-Serra,\nand Alejandro Gaita-Ari~ no, \\Theoretical insights on the\nimportance of anchoring vs molecular geometry in mag-\nnetic molecules acting as junctions,\" J. Magn. Magn.\nMater. 485, 212{216 (2019).\n[31] J. de Bruijckere, P. Gehring, M. Palacios-Corella,\nM. Clemente-Le\u0013 on, E. Coronado, J. Paaske, P. Hedeg\u0017 ard,\nand H. S. J. van der Zant, \\Ground-State Spin Blockade\nin a Single-Molecule Junction,\" Phys. Rev. Lett. 122,\n197701 (2019).\n[32] Jun Kondo, \\Resistance minimum in dilute magnetic al-\nloys,\" Progress of Theoretical Physics 32, 37{49 (1964).\n[33] L. I. Glazman and M.E. Raikh, \\Resonant kondo trans-\nparency of a barrier with quasilocal impurity states,\"\nJETP Lett. 47, 452 (1988).\n[34] A. C. Hewson, The Kondo problem to heavy fermions\n(Cambridge University Press, 1997).\n[35] D. Goldhaber-Gordon, Hadas Shtrikman, D. Mahalu,\nDavid Abusch-Magder, U. Meirav, and M. A. Kast-\nner, \\Kondo e\u000bect in a single-electron transistor,\" Nature\n391, 156 EP { (1998).\n[36] Sara M. Cronenwett, Tjerk H. Oosterkamp, and\nLeo P. Kouwenhoven, \\A tunable kondo e\u000bect\nin quantum dots,\" Science 281, 540{544 (1998),\nhttp://science.sciencemag.org/content/281/5376/540.full.pdf.\n[37] V. Madhavan, W. Chen, T. Jamneala, M. F. Crommie,\nand N. S. Wingreen, \\Tunneling into a Single Magnetic\nAtom: Spectroscopic Evidence of the Kondo Resonance,\"\nScience 280, 567{569 (1998).\n[38] C. Romeike, M. R. Wegewijs, and H. Schoeller, \\SpinQuantum Tunneling in Single Molecular Magnets: Fin-\ngerprints in Transport Spectroscopy of Current and\nNoise,\" Phys. Rev. Lett. 96, 196805 (2006).\n[39] Alexander F. Otte, Markus Ternes, Kirsten von\nBergmann, Sebastian Loth, Harald Brune, Christo-\npher P. Lutz, Cyrus F. Hirjibehedin, and Andreas J.\nHeinrich, \\The role of magnetic anisotropy in the Kondo\ne\u000bect,\" Nat. Phys. 4, 847{850 (2008).\n[40] J. J. Parks, A. R. Champagne, T. A. Costi, W. W. Shum,\nA. N. Pasupathy, E. Neuscamman, S. Flores-Torres, P. S.\nCornaglia, A. A. Aligia, C. A. Balseiro, G. K.-L. Chan,\nH. D. Abru~ na, and D. C. Ralph, \\Mechanical Control of\nSpin States in Spin-1 Molecules and the Underscreened\nKondo E\u000bect,\" Science 328, 1370{1373 (2010).\n[41] Maciej Misiorny, Ireneusz Weymann, and J\u0013 ozef Barna\u0013 s,\n\\Interplay of the Kondo E\u000bect and Spin-Polarized Trans-\nport in Magnetic Molecules, Adatoms, and Quantum\nDots,\" Phys. Rev. Lett. 106, 126602 (2011).\n[42] Maciej Misiorny, Ireneusz Weymann, and J\u0013 ozef Barna\u0013 s,\n\\Underscreened Kondo e\u000bect in S= 1 magnetic quantum\ndots: Exchange, anisotropy, and temperature e\u000bects,\"\nPhys. Rev. B 86, 245415 (2012).\n[43] Peter Zalom, Joeri de Bruijckere, Rocco Gaudenzi, Herre\nS. J. van der Zant, Tom\u0013 a\u0014 s Novotn\u0013 y, and Richard Ko-\nryt\u0013 ar, \\Magnetically Tuned Kondo E\u000bect in a Molecu-\nlar Double Quantum Dot: Role of the Anisotropic Ex-\nchange,\" J. Phys. Chem. C 123, 11917{11925 (2019).\n[44] Maciej Misiorny and J\u0013 ozef Barna\u0013 s, \\Magnetic switching\nof a single molecular magnet due to spin-polarized cur-\nrent,\" Phys. Rev. B 75, 134425 (2007).\n[45] Gabriel Gonz\u0013 alez and Michael N. Leuenberger, \\Berry-\nPhase Blockade in Single-Molecule Magnets,\" Phys. Rev.\nLett. 98, 256804 (2007).\n[46] Maciej Misiorny and J\u0013 ozef Barna\u0013 s, \\E\u000bects of intrinsic\nspin-relaxation in molecular magnets on current-induced\nmagnetic switching,\" Phys. Rev. B 77, 172414 (2008).\n[47] F. Delgado, J. J. Palacios, and J. Fern\u0013 andez-Rossier,\n\\Spin-Transfer Torque on a Single Magnetic Adatom,\"\nPhys. Rev. Lett. 104, 026601 (2010).\n[48] Maciej Misiorny and J\u0013 ozef Barna\u0013 s, \\E\u000bects of Transverse\nMagnetic Anisotropy on Current-Induced Spin Switch-\ning,\" Phys. Rev. Lett. 111, 046603 (2013).\n[49] Maciej Misiorny and Ireneusz Weymann, \\Transverse\nanisotropy e\u000bects on spin-resolved transport through\nlarge-spin molecules,\" Phys. Rev. B 90, 235409 (2014).\n[50] Pawan Tyagi and Edward Friebe, \\Large resistance\nchange on magnetic tunnel junction based molecular\nspintronics devices,\" J. Magn. Magn. Mater. 453, 186{\n192 (2018).\n[51] Emil Sierda, Micha Elsebach, Roland Wiesendanger, and\nMaciej Bazarnik, \\Probing Weakly Hybridized Magnetic\nMolecules by Single-Atom Magnetometry,\" Nano Lett.\n19, 9013{9018 (2019).\n[52] B. Verlhac, N. Bachellier, L. Garnier, M. Ormaza, P. Ab-\nufager, R. Robles, M.-L. Bocquet, M. Ternes, N. Lorente,\nand L. Limot, \\Atomic-scale spin sensing with a sin-\ngle molecule at the apex of a scanning tunneling micro-\nscope,\" Science 366, 623{627 (2019).\n[53] Maciej Misiorny, Michael Hell, and Maarten R.\nWegewijs, \\Spintronic magnetic anisotropy,\" Nat. Phys.\n9, 801{805 (2013).\n[54] J. O. Johansson, J.-W. Kim, E. Allwright, D. M. Rogers,\nN. Robertson, and J.-Y. Bigot, \\Directly probing spin\ndynamics in a molecular magnet with femtosecond time-11\nresolution,\" Chem. Sci. 7, 7061{7067 (2016).\n[55] Keiichi Katoh, Satoshi Yamashita, Nobuhiro Yasuda,\nYasutaka Kitagawa, Brian K. Breedlove, Yasuhiro\nNakazawa, and Masahiro Yamashita, \\Control of the\nSpin Dynamics of Single-Molecule Magnets by using a\nQuasi One-Dimensional Arrangement,\" Angew. Chem.\nInt. Ed. 57, 9262{9267 (2018).\n[56] Gheorghe Taran, Edgar Bonet, and Wolfgang Wernsdor-\nfer, \\Decoherence measurements in crystals of molecular\nmagnets,\" Phys. Rev. B 99, 180408(R) (2019).\n[57] David Roosen, Maarten R. Wegewijs, and Walter Hof-\nstetter, \\Nonequilibrium dynamics of anisotropic large\nspins in the kondo regime: Time-dependent numerical\nrenormalization group analysis,\" Phys. Rev. Lett. 100,\n087201 (2008).\n[58] Anna P lomi\u0013 nska, Maciej Misiorny, and Ireneusz Wey-\nmann, \\Spin-resolved dynamical conductance of a corre-\nlated large-spin magnetic molecule,\" Phys. Rev. B 95,\n155446 (2017).\n[59] H. Hammar and J. Fransson, \\Transient spin dynamics\nin a single-molecule magnet,\" Phys. Rev. B 96, 214401\n(2017).\n[60] Anna P lomi\u0013 nska, Ireneusz Weymann, and Maciej Mi-\nsiorny, \\Dynamical spin accumulation in large-spin mag-\nnetic molecules,\" Phys. Rev. B 97, 035415 (2018).\n[61] J. Martinek, Y. Utsumi, H. Imamura, J. Barna\u0013 s,\nS. Maekawa, J. K onig, and G. Sch on, \\Kondo E\u000bect in\nQuantum Dots Coupled to Ferromagnetic Leads,\" Phys.\nRev. Lett. 91, 127203 (2003).\n[62] J. Martinek, M. Sindel, L. Borda, J. Barna\u0013 s, R. Bulla,\nJ. K onig, G. Sch on, S. Maekawa, and J. von Delft,\n\\Gate-controlled spin splitting in quantum dots with fer-\nromagnetic leads in the Kondo regime,\" Phys. Rev. B 72,\n121302(R) (2005).\n[63] J. R. Hauptmann, J. Paaske, and P. E. Lindelof,\n\\Electric-\feld-controlled spin reversal in a quantum dot\nwith ferromagnetic contacts,\" Nat. Phys. 4, 373{376\n(2008).\n[64] M. Gaass, A. K. H uttel, K. Kang, I. Weymann, J. von\nDelft, and Ch. Strunk, \\Universality of the Kondo E\u000bect\nin Quantum Dots with Ferromagnetic Leads,\" Phys. Rev.\nLett. 107, 176808 (2011).\n[65] Krzysztof P. W\u0013 ojcik, Maciej Misiorny, and Ireneusz Wey-\nmann, \\Giant superconducting proximity e\u000bect on spin-\ntronic anisotropy,\" Phys. Rev. B 100, 045401 (2019).\n[66] Gheorghe Taran, Edgar Bonet, and Wolfgang Werns-\ndorfer, \\The role of the quadrupolar interaction in the\ntunneling dynamics of lanthanide molecular magnets,\"\nJ. Appl. Phys. 125, 142903 (2019).\n[67] Kenneth G. Wilson, \\The renormalization group: Criti-\ncal phenomena and the kondo problem,\" Rev. Mod. Phys.\n47, 773{840 (1975).\n[68] Ralf Bulla, Theo A. Costi, and Thomas Pruschke, \\Nu-\nmerical renormalization group method for quantum im-\npurity systems,\" Rev. Mod. Phys. 80, 395{450 (2008).\n[69] We used the open-access Budapest Flexible DM-NRG\ncode, http://www.phy.bme.hu/~dmnrg/; O. Legeza,\nC. P. Moca, A. I. T\u0013 oth, I. Weymann, G. Zar\u0013 and,\narXiv:0809.3143 (2008) (unpublished).\n[70] Frithjof B. Anders and Avraham Schiller, \\Real-time dy-namics in quantum-impurity systems: A time-dependent\nnumerical renormalization-group approach,\" Phys. Rev.\nLett. 95, 196801 (2005).\n[71] Frithjof B. Anders and Avraham Schiller, \\Spin preces-\nsion and real-time dynamics in the kondo model: Time-\ndependent numerical renormalization-group study,\"\nPhys. Rev. B 74, 245113 (2006).\n[72] H. T. M. Nghiem and T. A. Costi, \\Generalization of the\ntime-dependent numerical renormalization group method\nto \fnite temperatures and general pulses,\" Phys. Rev. B\n89, 075118 (2014).\n[73] H. T. M. Nghiem and T. A. Costi, \\Time-dependent\nnumerical renormalization group method for multiple\nquenches: Application to general pulses and periodic\ndriving,\" Phys. Rev. B 90, 035129 (2014).\n[74] H. T. M. Nghiem and T. A. Costi, \\Time-dependent\nnumerical renormalization group method for multiple\nquenches: towards exact results for the long time limit\nof thermodynamic observables and spectral functions,\"\nPhys. Rev. B 98, 155107 (2018).\n[75] J. Martinek, Y. Utsumi, H. Imamura, J. Barna\u0013 s,\nS. Maekawa, J. K onig, and G. Sch on, \\Kondo e\u000bect\nin quantum dots coupled to ferromagnetic leads,\" Phys.\nRev. Lett. 91, 127203 (2003).\n[76] Mahn-Soo Choi, David S\u0013 anchez, and Rosa L\u0013 opez,\n\\Kondo e\u000bect in a quantum dot coupled to ferromag-\nnetic leads: A numerical renormalization group analy-\nsis,\" Phys. Rev. Lett. 92, 056601 (2004).\n[77] J. Martinek, M. Sindel, L. Borda, J. Barna\u0013 s, R. Bulla,\nJ. K onig, G. Sch on, S. Maekawa, and J. von Delft, \\Gate-\ncontrolled spin splitting in quantum dots with ferromag-\nnetic leads in the kondo regime,\" Phys. Rev. B 72, 121302\n(2005).\n[78] M. Sindel, L. Borda, J. Martinek, R. Bulla, J. K onig,\nG. Sch on, S. Maekawa, and J. von Delft, \\Kondo\nquantum dot coupled to ferromagnetic leads: Numerical\nrenormalization group study,\" Phys. Rev. B 76, 045321\n(2007).\n[79] Andreas Weichselbaum and Jan von Delft, \\Sum-rule\nconserving spectral functions from the numerical renor-\nmalization group,\" Phys. Rev. Lett. 99, 076402 (2007).\n[80] Andreas Weichselbaum, \\Tensor networks and the nu-\nmerical renormalization group,\" Phys. Rev. B 86, 245124\n(2012).\n[81] Kacper Wrze\u0013 sniewski and Ireneusz Weymann, \\Quench\ndynamics of spin in quantum dots coupled to spin-\npolarized leads,\" Phys. Rev. B 100, 035404 (2019).\n[82] Yigal Meir and Ned S. Wingreen, \\Landauer formula for\nthe current through an interacting electron region,\" Phys.\nRev. Lett. 68, 2512{2515 (1992).\n[83] M. Yoshida, M. A. Whitaker, and L. N. Oliveira,\n\\Renormalization-group calculation of excitation prop-\nerties for impurity models,\" Phys. Rev. B 41, 9403{9414\n(1990).\n[84] J. Martinek, M. Sindel, L. Borda, J. Barna\u0013 s, R. Bulla,\nJ. K onig, G. Sch on, S. Maekawa, and J. von Delft, \\Gate-\ncontrolled spin splitting in quantum dots with ferromag-\nnetic leads in the kondo regime,\" Phys. Rev. B 72, 121302\n(2005)."    },    {        "title": "1703.08405v2.Gate_control_of_the_spin_mobility_through_the_modification_of_the_spin_orbit_interaction_in_two_dimensional_systems.pdf",        "content": "Gate control of the spin mobility through the modi\fcation of\nthe spin-orbit interaction in two-dimensional systems\nM. Luengo-Kovac,1F. C. D. Moraes,2G. J. Ferreira,3A. S. L. Ribeiro,2\nG. M. Gusev,2A. K. Bakarov,4V. Sih,1and F. G. G. Hernandez2,\u0003\n1Department of Physics, University of Michigan, Ann Arbor, Michigan 48109, United States\n2Instituto de F\u0013 \u0010sica, Universidade de S~ ao Paulo, S~ ao Paulo, SP 05508-090, Brazil\n3Instituto de F\u0013 \u0010sica, Universidade Federal de Uberl^ andia, Uberl^ andia, MG 38400-902, Brazil\n4Institute of Semiconductor Physics and Novosibirsk State University, Novosibirsk 630090, Russia\n(Dated: November 14, 2021)\nSpin drag measurements were performed in a two-dimensional electron system set close to the\ncrossed spin helix regime and coupled by strong intersubband scattering. In a sample with uncom-\nmon combination of long spin lifetime and high charge mobility, the drift transport allows us to\ndetermine the spin-orbit \feld and the spin mobility anisotropies. We used a random walk model\nto describe the system dynamics and found excellent agreement for the Rashba and Dresselhaus\ncouplings. The proposed two-subband system displays a large tuning lever arm for the Rashba\nconstant with gate voltage, which provides a new path towards a spin transistor. Furthermore,\nthe data shows large spin mobility controlled by the spin-orbit constants setting the \feld along the\ndirection perpendicular to the drift velocity. This work directly reveals the resistance experienced\nin the transport of a spin-polarized packet as a function of the strength of anisotropic spin-orbit\n\felds.\nThe pursuit for a new active electronic component\nbased on \row of spin, rather than that of charge, strongly\nmotivates research in semiconductor spintronics [1{5].\nSince the Datta-Das proposal for a ballistic spin tran-\nsistor, full electrical control of the spin state was sug-\ngested using the gate-tunable Rashba spin-orbit interac-\ntion (SOI) [6{10]. Further studies, including the Dressel-\nhaus SOI [11], were made to assure a nonballistic tran-\nsistor robust against spin-independent scattering [12{14].\nFor example, it has been demonstrated that SU(2) spin\nrotation symmetry, preserving the spin polarization, can\nbe obtained in the persistent spin helix (PSH) formed\nwhen the strengths of the Rashba and Dresselhaus SOI\nare equal (\u000b=\f) [15{19]. This is possible because the\nuniaxial alignment of the spin-orbit \feld suppresses the\nrelaxation mechanism when the spins precess about this\n\feld while experiencing momentum scattering [20]. Gate\ncontrol of this symmetry point was experimentally ob-\nserved [21{23] and allowed to produce a transition to the\nPSH\u0000(\u000b=\u0000\f) in the same subband [24]. Drift in those\nsystems showed surprising properties [25, 26] such as the\ncurrent-control of the temporal spin-precession frequency\n[27]. Although the helical spin-density texture could be\neven transported without dissipation under certain con-\nditions [15], the spin transport su\u000bers additional resis-\ntance from the spin Coulomb drag [28{32]. These fric-\ntional forces appear as a lower mobility for spins than for\ncharge and studies in new systems are still necessary to\nunderstand this important constraint for future devices.\nA two-dimensional electron gas (2DEG) hosted in a\nquantum well (QW) with two occupied subbands o\u000bers\nunexplored opportunities for the study of spin transport\n[33, 34]. Theoretically, the inter- and intra-subband spin-\norbit couplings (SOCs) have been extensively studied[35{38]. In terms of a random walk model (RWM) [39],\nthe spin drift and di\u000busion was recently developed for\nthese systems displaying two possible scenarios regarding\nthe intersubband scattering (ISS) rate [40]. The interplay\nbetween the two subbands may introduce new features to\nthe PSH dynamics, for example, a crossed persistent spin\nhelix [41] may arise when the subbands are set to orthog-\nonal PSHs (i.e., \u000b1=\f1and\u000b2=\u0000\f2) in the weak ISS\nlimit. In this report, we experimentally study spin drag\nin a system with the two-subbands individually set close\nto the PSH+and PSH\u0000, but with strong ISS, where the\ndynamics is given by the averaged SOCs of both sub-\nbands. The combination of long spin lifetime and high\ncharge mobility allows us to determine the spin mobility\nand the spin-orbit \feld anisotropies with the application\nof an accelerating in-plane voltage. We are able to con-\ntrol the SOCs in both subbands and to show a linear\ndependence for the sum of the Rashba constants with\ngate voltage. Finally, we determine an inverse relation\nfor the spin mobility dependence on the SOCs directly\nrevealing the resistance experienced in the transport of\na spin-polarized packet as a function of the strength of\nanisotropic spin-orbit \felds.\nThe sample consists of a single 45 nm wide GaAs QW\ngrown in the [001] ( z) direction and symmetrically doped.\nDue to the Coulomb repulsion of the electrons, the charge\ndistribution experiences a soft barrier inside the well.\nFigure 1(a) shows the calculated QW band pro\fle and\ncharge density for both subbands. The electronic system\nhas a con\fguration with symmetric and antisymmetric\nwave functions for the two lowest subbands with sub-\nband separation of \u0001 SAS = 2 meV. The subband den-\nsity (n 1= 3.7, n 2= 3.3\u00021011cm\u00002) was obtained from\nthe Shubnikov-de Hass (SdH) oscillations as shown inarXiv:1703.08405v2  [cond-mat.mes-hall]  18 May 20172\n0102030405060\n00.511.522.53\n00.511.522.53Rxx (:)Rxy (k:)\nBext (T)6789FFT f (T)n2n1246-15-10-50-0.100.10.2n (x1011 cm-2)Vg (V)Ez (V/Pm)n1n2n1+n2-20-100-0.100.10.2E (meV)Ez (V/Pm)H1H2H3-20020z (nm)(a)(b)\n(d)(c)\nFIG. 1. (a) Longitudinal ( Rxx) and Hall ( Rxy) magnetoresis-\ntance of the two-subband QW. From the SdH periodicity, one\ncan obtain the subbands density n\u0017in the lower inset. The\ntop inset shows the potential pro\fle and subbands charge den-\nsity calculated from the self-consistent solution of Schr odinger\nand Poisson equations for Ez=0. (b) Subband energy levels\nand (c) electron concentration dependence on V gandEz. (d)\nGeometry of the device and contacts con\fguration.\nFig. 1(a) and the low-temperature charge mobility was\n2.2\u0002106cm2/Vs [42]. A device was fabricated in a cross-\nshaped con\fguration with width of w=270\u0016m and chan-\nnels along the [1 \u001610] (x) and [110] (y) directions. Lateral\nOhmic contacts deposited l=500\u0016m apart were used to\napply an in-plane voltage ( Vip) in order to induce drift\ntransport. For the \fne tuning of the subband SOCs,\na semitransparent contact on top of the mesa structure\n(Vg) was used to modify structural symmetry and sub-\nband occupation. The e\u000bect of Vgon the subband energy\nlevels (\u000f\u0017) and densities ( n\u0017) is shown in Fig. 1(b) and\n(c) as a function of the out-of-plane electric \feld ( Ez).\nNote that the total density changes linearly with Vgand\nthatVg=0 corresponds to a built-in electric \feld of 0.15\nV/\u0016m. Figure 1(d) displays the experimental scheme\nwith the connection of VipandVg[43].\nTo describe the magnetization dynamics and the mea-\nsured SO \felds for our two-subband system, we combine\nthe calculated SOCs with RWM [39, 40, 44]. For a [001]\nGaAs 2DEG, the x and y components of the SO \felds\nfor each subband \u0017=f1;2gare\nBSO;\u0017(k) =2\ng\u0016B0\nB@\u0000\n+\u000b\u0017+\f1;\u0017+ 2\f3;\u0017k2\nx\u0000k2\ny\nk2\u0001\nky\n\u0000\n\u0000\u000b\u0017+\f1;\u0017\u00002\f3;\u0017k2\nx\u0000k2\ny\nk2\u0001\nkx1\nCA;\n(1)\nplus corrections due to the intersubband SOCs [23, 35{\n38, 41]. Above, g=\u00000:44 is the electron g-factor for\nGaAs and\u0016Bis the Bohr magneton. The SOCs are the\nusual Rashba \u000b\u0017and linear\f1;\u0017and cubic\f3;\u0017Dressel-\nhaus terms. Considering the strong intersubband scat-\ntering (ISS) regime of the RWM [40], the randomization\nof the momenta k(within the Fermi circle k=kF) and\nsubband\u0017is much faster than the spin precession. Con-\nsequently, the dynamics is governed by an averaged SOC\n-3-2-10123\n-0.3-0.2-0.100.10.20.3DX/EX\nEz (V/Pm)2ndsubband\n1stsubbandPSH+PSH--40-2002040-40-2002040x (Pm)y (Pm)-40-2002040-40-2002040x (Pm)y (Pm)\n-40-2002040-40-2002040\nx (Pm)y (Pm)-40-2002040-40-2002040\nx (Pm)y (Pm)-40-2002040-40-2002040\nx (Pm)y (Pm)-2-10123\n-0.3-0.1500.150.3SOC (meVA)D1D2K6DXEz (V/Pm)-1012\n-0.3-0.1500.150.3E1,2*E1,1Ez (V/Pm)0.40.60.81\n-0.3-0.1500.150.3Ez (V/Pm)E3,1E3,26E*X(a)(b)\n(d)(c)\n(e)1stPSH+2ndPSH-\nEz=0Ez=0.15Ez=0.3FIG. 2. (a)-(c) Calculated SOCs for the Rashba ( \u000b\u0017),\nlinear (\f1;\u0017) and cubic ( \f3;\u0017) Dresselhaus for each subband\n\u0017=f1;2g, as well as intersubband SOCs \u0011and \u0000 as a func-\ntion ofEz. The purple lines give the sum of \u000b\u0017and\f\u0003\n\u0017. (d)\nThe ratio\u000b\u0017=\f\u0017=\u00061 when the subband \u0017is set to the\nPSH\u0006regime. The insets show the single-subband magne-\ntization maps on the xyplane for the PSH\u0006regimes, and\nthe self-consistent potentials and subband densities for the\nrespectiveEz. (e) Two-subband magnetization maps in the\nstrong ISS regime for di\u000berent Ez. AtEz= 0 the well is sym-\nmetric (\u000b\u0017= 0) and the magnetization shows an isotropic\nBessel pattern. For \fnite Ezthe broken symmetry leads\nto the stripped PSH pattern in accordance with the pos-\nitive ratioP\u000b\u0017=P\f\u0017[purple line in panel (d)]. The ar-\nrows in the Fermi circle show the \frst harmonic component\nofPBSO;\u0017(k), illustrating the transition from isotropic to\nuniaxial \feld with increasing Ez. All the xy maps are frames\nof the spin pattern at t = 13 ns.\n\feldhBSOi= (hBx\nSOi;hBy\nSOi) transverse to the drift ve-\nlocity vdr= (vx\ndr;vy\ndr) Namely, the \feld components read\nhBx\nSOi=hm\n~g\u0016B2X\n\u0017=1(+\u000b\u0017+\f\u0003\n\u0017)i\nvy\ndr; (2)\nhBy\nSOi=hm\n~g\u0016B2X\n\u0017=1(\u0000\u000b\u0017+\f\u0003\n\u0017)i\nvx\ndr; (3)\nwhere\f\u0003\n\u0017=\f1;\u0017\u00002\f3;\u0017, andm= 0:067m0is the e\u000bec-\ntive electron mass for GaAs and ~is Planck's constant.\nSince Bx(y)\nSO/vy(x)\ndr, it is convenient to analyze the linear\ncoe\u000ecients bx(y)= By(x)\nSO=vx(y)\ndr, which are given by the3\nterms between square brackets above.\nThe intra- and intersubband SOCs are calculated\nwithin the self-consistent Hartree approximation [35{38]\nfor GaAs quantum wells tilted by Ez. The chemical\npotential is set to return the density n=n1+n2=\n7\u00021011cm\u00002forEz= 0, while it varies linearly for\n\fniteEzin Fig. 1(c). The SOCs are de\fned from\nthe matrix elements \u0011\u0017;\u00170=h\u0017j\u0011wV0+\u0011HV0\nHj\u00170iand\n\u0000\u0017;\u00170=\rh\u0017jk2\nzj\u00170i, wherej\u0017iis the eigenket for subband\n\u0017,\u0011w= 3:47\u0017A2and\u0011H= 5:28\u0017A2are bulk coe\u000ecients\n[35{38, 45], V0=@zV(z) andV0\nH=@zVH(z) are the\nderivatives of the heterostructure and Hartree potentials\nalongz,\r= 11 eV \u0017A3is the bulk Dresselhaus constant,\nandkzis the z-component of the momentum. The usual\nintrasubband Rashba and linear Dresselhaus SOCs are\n\u000b\u0017=\u0011\u0017;\u0017and\f1;\u0017= \u0000\u0017;\u0017. The non-diagonal terms are\nthe intersubband SOCs \u0011=\u001112and \u0000 = \u0000 12. The calcu-\nlated SOCs, plotted in Fig. 2(a)-(c) as a function of Ez,\nshow agreement with previous studies [46, 47]. The high-\ndensitynmakes the cubic Dresselhaus \f3;\u0017\u0019\r\u0019n\u0017=2\ncomparable with \f1;\u0017, strongly a\u000becting the PSH tuning\n[17]\u000b\u0017=\f\u0017, with\f\u0017=\f1;\u0017\u0000\f3;\u0017.\nNearEz\u00190:04 V/\u0016m, the SOCs reach almost simul-\ntaneously the balanced condition for the PSH+in the\n\frst subband ( \u000b1=\f1= +1) and for the PSH\u0000in the\nsecond subband ( \u000b2=\f2=\u00001), as shown by the ratio\n\u000b\u0017=\f\u0017in Fig. 2(d). The expected magnetization pat-\nterns for the single-subband PSH is shown in the inset\nof Fig. 2(d). The PSH\u0000shows more stripes than the\nPSH+due to the higher value of \u000b, which grows quickly\nwithin the Ezrange. However, the ratio of the aver-\naged SOCs (P\u000b\u0017)=(P\f\u0017) approaches the PSH regimes\nonly forjEzj>0:3 V/\u0016m. As we will see next, the\nexperimental data matches well the strong ISS regime\nof the RWM, therefore the dynamics is governed by the\naveraged SOCs. In this case, the expected magnetiza-\ntion patterns are shown in Fig. 2(e). With increasing Ez\nthe system transitions from isotropic ( Ez= 0) to uniax-\nial (Ez>0:3 V/\u0016m), as indicated by the formation of\nstripes and the orientation of the \frst harmonic compo-\nnent of the total \feldPBSO;\u0017(k) [arrows in Fig. 2(e)].\nWe are interested in the determination of the\nanisotropy for the coe\u000ecients bx(y), estimated in one or-\nder of magnitude in Fig. 3(a) and (b). We measured\nthe spin polarization using time-resolved Kerr rotation\nas function of the space and time separation of pump\nand probe beams. All optical measurements were per-\nformed at 10 K. A mode-locked Ti:Sapphire laser with a\nrepetition rate of 76 MHz tuned to 816.73 nm was split\ninto pump and probe pulses. The polarization of the\npump beam was controlled by a photoelastic modulator\nand the intensity of the probe beam was modulated by\nan optical chopper for cascaded lock-in detection. An\nelectromagnet was used to apply an external magnetic\n\feld in the plane of the QW. The spatial positioning of\nthe pump relative to the probe (d) was controlled using\n00.511.52\n-50-250255075A(d) (a.u.)d (Pm)dc = 27.1 Pm0255075vdr || yVip (mV)\n0123\n-40-30-20-10010203040Bext (mT)0IK (a.u.)255075vdr || yVip (mV)0246810\n0246810\n00.511.522.53i (mA)BSO (mT)vdr (Pm/ns) by=(3.0  0.3) mTns/Pmbx=(0.4  0.2) mTns/Pmvdr || yvdr || x00.511.522.53\n01020304050607080vdr (Pm/ns)Pys=(1.4  0.1) x105cm2/VsPxs=(2.2  0.1)x105cm2/VsEip (V/m)020406080100120140160\nVip (mV)(a)(b)\n(f)(e)±\n±\n±±(d)(c)bxbyFIG. 3. Calculated coe\u000ecients b with vdrparallel to (a) x\nand (b) y for each subband (colored) and total \feld (black).\n(c) Amplitude of the drifting spin polarization in space show-\ning, for example, the center of the packet d cfor 75 mV. (d)\nLinear dependence of v drwith the channel V ip. The slope\ngives the spin mobility along v drin x or y. (e) Field scan\nof\u001eKfor several V ipmeasured at d c. (f) By(x)\nSOas function\nof vx(y)\ndrand the current \rowing in that channel. The slopes\nbx(y)give the strength of the SOCs that generate the \feld\nalong y(x) for drift in x(y). The solid lines are gaussian (c)\nand linear (d and f) \fttings. Scans taken at t=13 ns.\na scanning mirror. We de\fned the spin injection point\nto be x=y=0 at t=0. The application of an in-plane\nelectric \feld (E ip=Vip/l), in the x or y-oriented channel,\nadds a drift velocity to the 2DEG electrons and allows\nus to determine the spin mobility and the spin-orbit \feld\ncomponents [48{50].\nThe sample was rotated such that each channel un-\nder study was oriented parallel to the external magnetic\n\feldBextkvdrfor all measurements reported here. From\nthe SOI form in k-space, we expected BSO?vdrimply-\ning that the observable BSOdirection will be BSO?Bext.\nConsidering this orientation, we can model the Kerr rota-\ntion signal as \u001eK(Bext;d) = A(d) cos ( !t) with the pre-\ncession frequency given by != (g\u0016B=~)p\nB2\next+B2\nSO,\nwhere A(d) is the amplitude at a given pump-probe spa-\ntial separation and B SOis the internal SO \feld compo-\nnent perpendicular to Bext(and to vdr).\nFigure 3 shows the results of the spin drag experiment\nwith the gate contact open. Scanning the pump-probe\nseparation in space at \fxed long time delay (13 ns), we\ndetermined the central position d cof the spin packet4\namplitude for several V ipin a given crystal orientation.\nFrom the values of d cin Fig. 3(c), we calculated the drift\nvelocity as v dr=dc/t and plotted it as a function of V ip\nin Fig. 3(d). The slope of the linear \ft give us spin mo-\nbilities (\u0016x;y\ns) in the range of 105cm2/Vs. Values in the\nsame order of magnitude have been measured by Doppler\nvelocimetry for the transport in single subband samples\n[32]. Nevertheless, in those systems the spin lifetimes\nwere restricted to the picosecond range and the trans-\nport was limited to the nanometer scale.\nFollowing the drifting spin packet in space, Fig. 3(e)\ndisplays a B extscan from where changes in the amplitude\nof zeroth resonance determined B SOstrength at d c. As\nexplained above, the data con\frmed the perpendicular\norientation between BSOandvdrand did not show a\ncomponent parallel to Bextwithin the experimental res-\nolution [51]. From the Lorenztian shape of the B extscan\n[52, 53], we evaluated a spin lifetime of 7 ns at V ip=0.\nThis experiment was only possible due to the nanosecond\nspin lifetime in our sample that extends the spin trans-\nport to several tens of micrometers [54, 55].\nFigure 3(f) shows the \ftted values of B SOfor sev-\neral Vipapplied along x and y. We observed highly\nanisotropic spin-orbit \felds in the range of several mT\nas expected from Fig 3(a) and (b). The B SOorientation\nwas aligned primary with the x axis in agreement with\nthe simulation in Fig 2(e). The slopes bx(y)=By(x)\nSO/vx(y)\ndr\ngive the strength of the SOCs that generate the \feld ac-\ncording to Eqs. 2 and 3. For this condition of the sample\nas-grown, we foundP\u000b\u0017= 0.57 meV \u0017A andP\f\u0003\n\u0017= 0.75\nmeV\u0017A.\nNote the inverse behaviour on V ipfor the mobility and\nfor BSOstrength in perpendicular directions. In Fig.\n3(c) and (d), the axis with the largest mobility is also\nthe axis with smaller spin-orbit \feld in the perpendic-\nular direction. This result may be related to the spin\nCoulomb drag observed previously in the transport of\nspin-polarized electrons [31, 32]. Next, we demonstrate\nthe direct control of the spin mobility through the gate\nmodi\fcation of the subband SOCs.\nFigure 4(a) shows that the magnitude and the orienta-\ntion with the largest \u0016scan be tuned by E z. BSOdisplays\nanisotropic components with Bx\nSObeing larger in all the\nstudied range, which con\frms the preferential alignment\ntowards the PSH+in Fig. 2(e). The variation of Bx\nSO\nhas a minimum (indicated by an arrow) close to position\nwhen the second subband attains the PSH\u0000(with BSO\nalong y). Dividing Fig. 4(a) panels, the values for b are\nplotted in Fig. 4(b). The lines plotted together with\nthe data are the expected values using Eq. 2 and 3 with\nthe SOCs from Fig. 2(a)-(c). When the QW approaches\nthe symmetric condition (E z=0), bx(y)decreases remov-\ning the anisotropy of B SOas simulated in Fig. 2(e).\nThe addition and subtraction of bxandbygive the sum\nof the Rashba and Dressselhaus SOCs displayed in Fig.\n4(c). Dashed lines corresponding to the purple curves in\n-0.200.20.40.60.81\n00.050.10.15SOC (meVA)6DXEz (V/Pm)6E*X00.511.522.53\n-0.0500.050.10.15b (mT/Pm/ns)vdr || yvdr || xEz (V/Pm)-4.4 eÅ235.0 eÅ20123456-0.0500.050.10.15BSO (mT)Ez (V/Pm)BSO || xBSO || y00.511.522.53-15-10-50Ps (x105cm2/Vs)Vg (V)vdr || yvdr || x(a)(b)\n(d)(c)\n01234\n00.20.40.60.811.26\u000bDX+E*X), 6\u000b\u0010DX+E*X) (meVA)Ps (x105cm2/Vs)vdr || yvdr || xP0s-1.1x105𝑐𝑚2𝑉𝑠𝑚𝑒𝑉Å-3.4x105 𝑐𝑚2𝑉𝑠𝑚𝑒𝑉ÅFIG. 4. (a) Spin mobility and B SOas a function of the\ngate-tunable E z. (b) Ratio bx(y)from (a), showing a crossing\nat Ez=0. (c) SOCs obtained from the addition and subtrac-\ntion ofbxandbyin (b). (d) Spin mobility as function of the\nSOCs that de\fne the B SOstrength along the direction per-\npendicular to v dr. The solid lines are linear \fttings and the\ndashed lines (b,c) are the theoretical results from the RWM\ncombined with the self-consistent calculation of the SOCs.\nFig.2(a) and (c) are plotted together displaying excellent\nagreement. The slope for the Rashba SOI indicates a tun-\ning lever arm of 35 e \u0017A2. This value is considerably larger\nthan those reported in recent studies for single subband\nsamples, typically below 10 e \u0017A2[17, 23]. Finally, Fig.\n4(d) presents \u0016x(y)\ns[from (a)] against the SOCs de\fn-\ning By(x)\nSO:P(\u0000\u000b\u0017+\f\u0003\n\u0017) andP(\u000b\u0017+\f\u0003\n\u0017), respectively.\nThis last plot illustrates the inverse dependence, with\nnegative slope, for the spin mobility and strength of the\nSOCs perpendicular to the drift direction. The di\u000berent\nslopes for x and y channels give us a hint that this e\u000bect\ndepends not only on how B SOchanges with v dr(given\nby the SOCs) but also in the magnitude of the \felds. A\ncommon maximum value \u00160\ns=3\u0002105cm2/Vs was found\nindependent of vdrorientation.\nIn conclusion, we have studied a 2DEG system with\ntwo subbands set close to the crossed PSH regime under\nstrong intersubband scattering and successfully described\nit using a random walk model. In the spin transport\nwith nanosecond lifetimes over micrometer distances, we\ndemonstrate the control of the subbands spin-orbit cou-\nplings with gate voltage and observed spin mobilities in\nthe range of 105cm2/Vs. Speci\fcally, the sum of the\nRashba SOCs presents a linear behaviour with remark-\nably large tunability lever arm with gate voltage. We5\ntailored the spin mobility by controlling the strength of\nthe spin-orbit interaction in the direction perpendicular\nto the drift velocity. Our \fndings provided evidence of\nthe rich physical phenomena behind multisubband sys-\ntems and experimentally demonstrated relevant proper-\nties required for the implementation of a nonballistic spin\ntransistor.\nThis work is a result of the collaboration initiative\nSPRINT No. 2016/50018-1 of the S~ ao Paulo Research\nFoundation (FAPESP) and the University of Michi-\ngan. F.G.G.H also acknowledges \fnancial support from\nFAPESP Grants No. 2009/15007-5, No. 2013/03450-\n7, and No. 2014/25981-7 and 2015/16191-5. G.J.F.\nacknowledges the \fnancial support from CNPq and\nFAPEMIG. The work at the University of Michigan is\nsupported by the National Science Foundation under\nGrant No. DMR-1607779.\n\u0003Corresponding author.\nElectronic address: felixggh@if.usp.br\n[1] S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J.\nM. Daughton, S. von Moln\u0013 ar, M. L. Roukes, A. Y.\nChtchelkanova, and D. M. Treger, Science 294, 1488\n(2001).\n[2] I. ^Zuti\u0013 c, J. Fabian, and S. Das Sarma, Rev. Mod. Phys.\n76, 323 (2004).\n[3] D. D. Awschalom and M. E. Flatt\u0013 e, Nat. Phys. 3, 153\n(2007).\n[4] J. Wunderlich, B.-G. Park, A. C. Irvine, L. P. Z^ arbo,\nE. Rozkotov\u0013 a, P. Nemec, V. Nov\u0013 ak, J. Sinova, and T.\nJungwirth, Science 330, 1801 (2010).\n[5] J. C. Egues, G. Burkard, and D. Loss, Appl. Phys. Lett.\n82, 2658 (2003).\n[6] E. I. Rashba, Sov. Phys. Solid State 2, 1109 (1960); Y. A.\nBychkov and E. I. Rashba, J. Phys. C 17, 6039 (1984);\nJETP Lett. 39, 78 (1984).\n[7] S. Datta and B. Das, Appl. Phys. Lett. 56, 665 (1990).\n[8] J. Nitta, T. Akazaki, H. Takayanagi, and T. Enoki, Phys.\nRev. Lett. 78, 1335 (1997).\n[9] V. Lechner, L. E. Golub, P. Olbrich, S. Stachel, D. Schuh,\nW. Wegscheider, V. V. Bel'kov and S. D. Ganichev, Appl.\nPhys. Lett. 94, 242109 (2009).\n[10] P. Chuang, S.-C. Ho, L. W. Smith, F. S\fgakis, M. Pep-\nper, C.-H. Chen, J.-C. Fan, J. P. Gri\u000eths, I. Farrer, H.\nE. Beere, G. A. C. Jones, D. A. Ritchie, and T.-M. Chen,\nNat. Nanotechnol. 10, 35 (2014).\n[11] G. Dresselhaus, Phys. Rev. 100, 580 (1955).\n[12] J. Schliemann, J. C. Egues, and D. Loss, Phys. Rev. Lett .\n90, 146801 (2003).\n[13] M. Ohno and K. Yoh, Phys. Rev. B 77, 045323 (2008).\n[14] Y. Kunihashi, M. Kohda, H. Sanada, H. Gotoh, T. So-\ngawa, and J. Nitta, Appl. Phys. Lett. 100, 113502 (2012).\n[15] B. A. Bernevig, J. Orenstein, and S.-C. Zhang, Phys.\nRev. Lett. 97, 236601 (2006).\n[16] J. D. Koralek, C. Weber, J. Orenstein, B. Bernevig, S.-\nC. Zhang, S. Mack, and D. Awschalom, Nature 458, 610\n(2009).\n[17] M. P. Walser, C. Reichl, W. Wegscheider, and G. Salis,Nat. Phys. 8, 757 (2012).\n[18] J. Schliemann, Rev. Mod. Phys. 89, 011001 (2017).\n[19] M. Kohda and G. Salis, Semiconductor Science and Tech-\nnology (2017). at press: https://doi.org/10.1088/1361-\n6641/aa5dd6\n[20] M. I. D'yakonov and V. I. Perel', Sov. Phys. Solid State\n13, 3023 (1972).\n[21] M. Kohda, V. Lechner, Y. Kunihashi, T. Dollinger, P.\nOlbrich, C. Sch onhuber, I. Caspers, V. V. Bel'kov, L.\nE. Golub, D. Weiss, K. Richter, J. Nitta, and S. D.\nGanichev, Phys. Rev. B 86, 081306(R) (2012).\n[22] J. Ishihara, Y. Ohno, and H. Ohno, Appl. Phys. Exp. 7,\n013001 (2014).\n[23] F. Dettwiler, J. Fu, S. Mack, P. J. Weigele, J. C. Egues,\nD. D. Awschalom, and D. M. Zumb uhl, arXiv:1702.05190\n[cond-mat.mes-hall].\n[24] K. Yoshizumi, A. Sasaki, M. Kohda, and J. Nitta, Appl.\nPhys. Lett. 108, 132402 (2016). Here, the PSH\u0000is la-\nbeled as inverse PSH (iPSH).\n[25] L. Yang, J. D. Koralek, J. Orenstein, D. R. Tibbetts, J.\nL. Reno, and M. P. Lilly Phys. Rev. Lett. 109, 246603\n(2012).\n[26] Y. Kunihashi, H. Sanada, H. Gotoh, K. Onomitsu, M.\nKohda, J. Nitta, and T. Sogawa, Nat. Commun. 7, 10722\n(2016).\n[27] P. Altmann, F. G. G. Hernandez, G. J. Ferreira, M. Ko-\nhda, C. Reichl, W. Wegscheider, and G. Salis, Phys. Rev.\nLett.116, 196802 (2016).\n[28] I. D'Amico and G. Vignale, Europhys. Lett. 55, 566\n(2001).\n[29] I. D'Amico and G. Vignale, Phys. Rev. B 68, 045307\n(2003).\n[30] W.-K. Tse and S. Das Sarma, Phys. Rev. B 75, 045333\n(2007).\n[31] C. P. Weber, N. Gedik, J. E. Moore, J. Orenstein,\nJ. Stephens, and D. D. Awschalom, Nature 437, 1330\n(2005).\n[32] L. Yang, J. D. Koralek, J. Orenstein, D. R. Tibbetts, J.\nL. Reno, and M. P. Lilly Nat. Phys. 8, 153 (2012).\n[33] F. G. G. Hernandez, L. M. Nunes, G. M. Gusev, and A.\nK. Bakarov, Phys. Rev. B 88, 161305(R) (2013).\n[34] F. G. G. Hernandez, G. M. Gusev, and A. K. Bakarov,\nPhys. Rev. B 90, 041302(R) (2014).\n[35] E. Bernardes, J. Schliemann, M. Lee, J. C. Egues, and\nD. Loss, Phys. Rev. Lett. 99, 076603 (2007).\n[36] E. Bernardes, J. Schliemann, C. Egues, and D. Loss,\nPhys. Status Solidi C 3, 4330 (2006).\n[37] R. S. Calsaverini, E. Bernardes, J. C. Egues, and D. Loss,\nPhys. Rev. B 78, 155313 (2008).\n[38] J. Fu and J. C. Egues, Phys. Rev. B 91, 075408 (2015).\n[39] L. Yang, J. Orenstein, and D.-H. Lee, Phys. Rev. B 82,\n155324 (2010).\n[40] G. J. Ferreira, F. G. G. Hernandez, P. Altmann, G. Salis,\nPhys. Rev. B 95, 125119 (2017).\n[41] J. Fu, P. H. Penteado, M. O. Hachiya, D. Loss, and J. C.\nEgues, Phys. Rev. Lett. 117, 226401 (2016).\n[42] The barriers were made of short-period AlAs/GaAs su-\nperlattices in order to shield the doping ionized impuri-\nties and e\u000eciently enhance the mobility, more details in\nK.-J. Friedland, R. Hey, H. Kostial, R. Klann, and K.\nPloog, Phys. Rev. Lett. 77, 4616 (1996).\n[43] The measured low-temperature resistance was 10 \n for\nthe sample and 100 \n for the lateral Ohmic contacts.\nThus, the e\u000bective V ipin the channel is about 10 times6\nsmaller than the applied voltage. The values of V ipused\nfor the plots and calculations are the e\u000bective ones.\n[44] The model neglects changes in the intersubband\nscattering-time that depends on the QW symmetry as\nstudied in: N. C. Mamani, G. M. Gusev, T. E. Lamas,\nand A. K. Bakarov, O. E. Raichev, Phys. Rev. B 77,\n205327 (2008); N. C. Mamani, G. M. Gusev, E. C. F. da\nSilva, O. E. Raichev, A. A. Quivy, and A. K. Bakarov,\nPhys. Rev. B 80, 085304 (2009).\n[45] R. Winkler, Spin-Orbit Coupling E\u000bects in Two-\nDimensional Electron and Hole Systems , Springer Tracts\nin Modern Physics (Springer, Berlin, 2003).\n[46] M. Studer, M. P. Walser, S. Baer, H. Rusterholz, S.\nSch on, D. Schuh, W. Wegscheider, K. Ensslin, and G.\nSalis Phys. Rev. B 82, 235320 (2010).\n[47] L. Meier, G. Salis, I. Shorubalko, E. Gini, S. Sch on and\nK. Ensslin, Nat. Phys. 3, 650 (2007).\n[48] J. M. Kikkawa and D. D. Awschalom, Nature 397, 139\n(1999).[49] Y. Kato, R. C. Myers, A. C. Gossard and D. D.\nAwschalom, Nature 427, 50 (2004).\n[50] B. M. Norman, C. J. Trowbridge, J. Stephens, A. C.\nGossard, D. D. Awschalom, and V. Sih, Phys. Rev. B\n82, 081304(R) (2010).\n[51] V. K. Kalevich and V. L. Korenev, Zh. Eksp. Teor. Fiz .\n52, 859 (1990) [JETP Lett. 52, 230 (1990)].\n[52] R. I. Dzhioev, K. V. Kavokin, V. L. Korenev, M. V.\nLazarev, B. Ya. Meltser, M. N. Stepanova, B. P. Za-\nkharchenya, D. Gammon, and D. S. Katzer, Phys. Rev.\nB66, 245204 (2002).\n[53] S. A. Crooker, M. Furis, X. Lou, P. A. Crowell, D. L.\nSmith, C. Adelmann, and C. J. Palmstr\u001cm, J. Appl.\nPhys. 101, 081716 (2007).\n[54] S. Ullah, G. M. Gusev, A. K. Bakarov, and F. G. G.\nHernandez, J. Appl. Phys. 119, 215701 (2016).\n[55] F. G. G. Hernandez, S. Ullah, G. J. Ferreira, N. M. Kawa-\nhala, G. M. Gusev, and A. K. Bakarov, Phys. Rev. B 94,\n045305 (2016)."    },    {        "title": "1411.5339v2.Dynamical_spin_injection_at_a_quasi_one_dimensional_ferromagnet_graphene_interface.pdf",        "content": "1 \n Dynamical  spin injection at a quasi -one-dimensional ferromagnet -graphene \ninterface  \nS. Singh1, A. Ahmadi1, C.T. Cherian2, 4, E. R. Mucciolo1, E. del Barco1, and \nB. Özyilmaz2,3,4,5  \n1Department of Physics, University of Central Florida, Orlando, Florida USA, 32816  \n2Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore 117542  \n3NanoCore, 4 Engineering Drive 3, National University of Singapore, Singapore 117576  \n4Graphene Research Center, National University of Singapore, Singapore 117542  \n5NUS Graduate School for Integrative Sciences and Engineering (NGS), National University of \nSingapore, Singapore 117456  \nAbstract:  We present a s tudy of dynamical spin injection  from a three -dimensional ferromagnet \ninto two-dimensional single -layer graphene.  Comparative f erromagnetic resonance (F MR) \nstudies of f erromagnet/g raphene strips buried underneath the central line of a coplanar \nwaveguide show that t he FMR line width  broadening is the largest  when the graphene layer \nprotrudes laterally  away from  the ferromagnet ic strip, indicating that the spin current is injected \ninto the graphene areas away from the area directly underneath the ferromagnet being excited. \nOur results confirm that the observed damping is indeed a signature of dynamical spin injec tion, \nwherein a pure spin current  is pumped into the single -layer graphene from the precessing \nmagnetization of the ferromagnet . The observed spin pumping efficiency is difficult to reconcile \nwith the expected backflow of spins according to  the standard sp in pumping theory and the \ncharacteristics of graphene, and constitutes a n enigma for spin pumping in two-dimensional \nstructures.  2 \n The efficient generation  of pure spin currents holds a great deal of prom ise for spintronics \napplications , with several existin g methods  already demonstrated, such as  the spin H all effect  1, \nelectrical spin injection  2, voltage -based spin pumping3 dynamical spin pumping  4,5, and optical \ngenerat ion of  spin packets6. Particularly,  the dynamical generation of spin current s carries special  \ninterest because no net charge current is involved in the process . In this method, spin a ngular \nmomentum is transferred from the precessional magnetization in a ferromagnet  (FM)  to an \nadjacent  non-magnetic  (NM)  system. This approach has already been experimentally \ndemonstrated in several FM/NM interfaces, including NM systems such as metals7, \nsemiconductors  5,8 or organic based materials9. A few advantages of this method are  that it does \nnot suffer from the impedance mismatch problem, it is scalable to large samples, it provides a \nhigh spin injection efficiency , and it is not based on the spin -orbit coupling to function, allowing \nits employment in sy stems without this interaction .  \nWithin the exciting cur rent trend to explore novel low -dimensional systems, the possibility to \ninject pure spin currents in graphene and other two -dimensional (2D) crystals has attracted \nconsiderable  attention in the past f ew years . In particular, graphene rises as  a prototype system to \nexplore this physics  due to its crystalline nature, excellent  electronic  properties10, tunable spin -\norbit co upling (i.e., via adatom engineering )11,12 and long spin relaxation lengths13. In addition, \ngraphene can act as a  high-fidelity channel for spin information  transfer  (due to its small intrinsic \nspin-orbit coupling and absence  of nuclear spins) , as well as provide the platform  for electrical \nmanipulat ion of the  spin polarization  (i.e., on-demand enhancement of spin -orbit coupling11,12). \nDynamical spin injection in FM/g raphene (FM/Gr) interfaces has been recently demonstrated  \n14,15. In the original work, we associated  the observed enhan cement in dynamical damping of \nextended FM/Gr films  to the generation of pure spin current in graphene  result ing from loss es of 3 \n spin angular mome ntum  in the ferro magnet  (i.e., spin pumping14). Subsequently, the dynamical \ninjection of spin currents in graphene was demonstrated by spin -charge conversio n \nmeasurements in a P d strip  placed laterally and in close proximity to a FM/Gr interface  \nundergoing FMR15. Although providing evidence for spin injection, none of these w orks shed \nlight into the real nature of spin pumping at the FM/Gr interface.  Estimates of the spin -mixing \nconductance obtained from the broadening of the FMR peaks resulted in surprisingly high values \n(e.g., \ng 5.261019 m-2 from Py/Gr  = Py/Gr - Py = 110-2), comparable to systems with high \nspin-orbit coupling7 (such as heavy metals  Pt and Pd). In addition, the direct deposition of the \nferromagnetic Permalloy (Py) film on top of the graphene layer could cause magneto -structural16 \nchanges in the Py surface  and a subsequent increase in straight fields altering the spin dynamics \nand accumulation in the semiconductor17, responsible for the observed change in damping when \ncomparing with films without graphene, i.e., deposited on the bare wafer.  A direct measuremen t \nof the spin Hall angle in FM/Gr interfaces has not been reported yet. A  recent work by R. \nOhshima et al.  18 published during the preparation of this manuscript , claims the observation of \nthe ISHE signal in single -layer graphene upon conversion of a spin current pumped from an \nextended insulating ferromagnet ( i.e., YIG) into a charge current. According to th at report, the \nspin current is injected perpendicularly to the YIG/G r interface , and the ISHE electric field \nmeasured within the graphene plane, as is the norm in other ferromagnet/conductor \nheterostructures. However, we find this interpretation rather questionable, since  conduction \nperpendicular to a single -layer graphene in not possible  simply due to the lack of a third \ndimension. Since conduction perpendicular to the graphene sheet is not possible, the geometry \nproposed by the authors, where the ISHE is measured along the graphene  plane, is impracticable.  4 \n The observations in that work  could be explained in terms of alternative physics, such as an \ninverse Rashba -Edelstein effect19 or similar, but never in terms of the ISHE.  \nIn this letter we present FMR experiments performed on different FM/Gr interfaces designed \nto systematically identify and eliminate  damping enhancement arising  from process es other than  \nspin pumping. In particular, a substantial enhancement of the Gilbert  damping observed in Py/Gr \nstrips when the graphene layer protrudes a few micrometers away from the edges of a narrow Py \nstrip is univocally a ssociated to spin pumping at the quasi -one-dimensional  interface between the \nPy edge and graphene , which shows lower spin -mixing conductance values (\ng 6.891018 \nm-2) than in extended films but still comparable to those obtained in Py/Pt interfaces  (e.g., \ng\n~1-41019 m-2). We also provide a theoretical analysis which  shows  the observed  spin \ninjection efficiency to lie well beyond that expected from the spin conduction channels provided \nby single -layer graphene, open ing fundamental  question s about the nature of spin injection in to \nthis 2D crystal . \nThe graphene  layers used in our experiments are  grown by the standard chemical vapor \ndeposition ( CVD ) method on thin Cu foil s20. Graphene is subsequently transferred onto the \nsubstrate  using a wet chemistry  process  and characterized by Raman spectroscopy . We use \n14 nm-thick films of Ni20Fe80 Permalloy  (deposited by e -beam evaporation in high vacuum \nconditions) as the ferromagnet for  all our studies. For FMR on extended films, we place  the \nsample  upside -down  on the central part of our broad -band -CPW  FMR sensor , which is  coated \nwith an insulating polymer  to prevent electrical contact with the sample14. For FMR on patterned \nfilms  in the shape of long and narrow  Py/Gr strips , the sample is buried directly underneath the \ncentral line of the CPW , isolated  from it by a thick  (~100 nm) insulating layer of oxide.    5 \n Before getting into the detailed discussion of the main results of this work , we want to briefly \ndiscuss a set of experiments designed to address the effect of magneto -structural changes in the \nsurface of the Py due to the immediate presence of graphene underneath, which  could  cause a \nnon-dynamical  broadening of the FMR 21. In the first experiment, the stacking order  of the Py \nand graphene layers has been reversed with respect to the original experiments14, where the Py \nwas deposited directly on top of the graphene layer  (i.e., a Py/Gr stacking) . In the present case, \nthe Py film is depo sited on a bare Si wafer coated with 300nm of thermally grown SiO 2 and \ngraphene transferred on top afterwards  (i.e., a Gr/Py stacking) , with the objective of maintaining \nthe Py film unaltered by the presence of graphene. A clear enhancement of the Gilbert damping \nis obtained when graphene is present  (i.e.,  Gr/Py   = Gr/Py  - Py = 3.410-3), resulting in a spin -\nmixing conductance of \ng 1.951019 m-2, i.e., three times lower than in previous Py/Gr \nsamples. In the second experiment, a 20  nm-thick Cu spacing l ayer was inserted in between the \nPy and graphene ( i.e., a Py/Cu/Gr stacking), and FMR results compared to those in Py and Py/Cu \nsamples. The rationale is to use the Cu layer as a structural spacer between the Py and the single -\nlayer graphene in order to ma intain the Py film unchanged. Note that a thin Cu film does not \ncontribute to the absorption/diffusion of the spin pumped away from the Py film, since Cu has a \nsubstantially larger spin diffusion length than the Cu  layer  thickness used in these experiments22. \nAgain, a clear enhancement of the G ilbert damping is observed when graphene is present ( i.e., \nPy/Cu/Gr   =  Py/Cu/Gr  -  Py/Cu = 4.210-3), resulting in a spin -mixing conductance of \ng\n2.381019 m-2, i.e., comparable to the values obtained in Gr/Py samples.   6 \n \n \nFigure 1: Sketches illustrating the strips used in the experiments. The Py strips are all the same \ndimensions, with a length of 3 mm and a width w  = 25m. a) Py strip. b) Py/Gr strip. c)  Py/Gr -prt strip, \nwith graphene protruding from the sides of the Py strip by d  = 20 m. \n \nThe set of experiments described above eliminate structu ral changes in the Py as a possible \ncause for the observed damping enhancement. However, the presence of the ferromagnet i n close \nproximity to the single -layer graphene , even in areas away from the FMR excitation , may \ninfluence the diffusion of the spins p umped away from the Py film, which can  still act as a spin \nsink since electrons can flow back into it . To avoid this situation , we patterned the Py/Gr film \ninto long  (l = 3mm)  and narrow  (w = 25 m) strips that  are placed directly underneath the central \nline of the -CPW, as shown in Fig.  1 and in the inset s to Fig. 2. Essentially, we prepare d three \ndifferent samples for this study : a) a Py strip  (Fig.  1a); b) a Py/Gr  strip (Fig.  1b); and,  c) a Py/Gr \nstrip with  the single -layer  graphene protruding  away  from  the Py strip  on both sides , which we \nshall call Py/Gr -prt henceforth (Fig.1c) . The upper inset to Fig. 2 shows a  scanning e lectron  \nmicroscope  image of a Py/Gr -prt strip, where  one can clearly see the cont inuous sheet of \ngraphene extending  away from the central ferromagnet  strip. Note that t he length by which \ngraphene protrudes  on each side of the ferromagnet strip , i.e., d ~12 m, is larger than the  spin \ndiffusion length of CVD graphene ( s ~2 m)23, in order to allow for a total relaxation of the spin \npumped away from the ferromagnet.  The devices are prepared by  transferr ing a single -layer 7 \n graphene  onto a GaAs (undoped) /SiO 2(100nm)  substrate , after which  unwanted graphene areas  \nare etch ed away using a photo resist mask  and standard optical lithography . Following etching  \nand e-beam  evaporation  of the Py strip , a 100  nm-thick layer of silicon oxide is grown atop to \ninsulate the device from the  central line of the -CPW , which ultimately cov ers the sample  (as \ndepicted in  the lower inset to Fig. 2). This geometry guarantees a homogeneous FMR excitation \nof the whole Py strip .  \n \nFigure 2: Field -derivative of the  FMR response for the three strips measured  (Py: Permalloy Gr: \ngraphene and Gr -prt: graphene protruding from the sides of the Py strip).  The upper inset shows an \nelectronic microscope image of the Py/Gr -prt stripe before being placed underneath the central line of \nthe -CPW sensor  (lower inset) . \n \nStandard  broadban d FMR measurements are performed on the samples described above to \nextract the FMR line width s. The corresponding field-derivative s, dM/dH, obtained at  an \nirradiation frequency of  12 GHz with the dc magnetic field applied in the plane of the Py strips \nare shown in F ig. 2. The FMR linewidth, defined as the peak -to-peak distance in the dM/dH data, \nis the largest for the sample with graphene protruding away from the Py strip ( i.e., the Py/Gr -prt 8 \n strip) and the smallest for the sample with Py only (i.e., the Py strip).  The frequency dependence \nof the in-plane excited FMR linewidth for the se samples is shown in Fig. 3a. The observed linear \nfrequency dependence of the linewidth can be explained by  means of the dynamical Gilbert \ndamping model , using the following ex pression:  \n                  \nf H H\n\n34\n0  ,                                (1)                                                                                        \nwhere \nh gB/  is the gyromagnetic ratio and α is the damping parameter, which is related to \nthe Gilbert damp ing through the expression \nSM G , with \nSM  being the saturation \nmagnetization . The two different contributions to the damping in Eqn. 1 are: a) a sample \ndependent i nhomogeneous part  (first term in the equation) , which can be calculated from the \nintercept at zero  extrapolated frequency and it does not depend on frequency ; and, b) dynamical \ndamping  (second term in the equation) , which scales  linearly with frequency  and from whose \nslope the Gilbert damping can be calculated . These extracted damping parameters  for in -plane \nFMR excitation (Fig.  3a) are as follows:   Py = 9.110-3 & GPy = 0.239  GHz; Py/Gr = 11.310-3 \n& GPy/Gr = 0.299  GHz ; Py/Gr-prt = 13.010-3 & GPy/Gr-prt = 0.333 GHz . There is a considerable \nenhancement in damping when going  from the Py -only strip t o the Py/Gr strip, where graphene \nis only present underneath the ferromagnet . This damping can not be attributed to spin pumping \ngiven the 2D nature of graphene, which  is located only underneath the Py strip and  does not \nprovide any conduction channel perpe ndicular to the interface.  Indeed , it has been shown that \ngraphene can act as an effective tunnel barrier for electrical spin injection into silicon due to the \nvery large resistivity of carriers across the graphene sheet24. Most likely, t he observed FMR \nbroadening is due to changes in the magnetic response of the Py due to surface changes induced \nby the graphene, such as an enhancement  of two-magnon scattering  processes . It has been 9 \n recently shown that the deposition of Co films  on graphene results  in magnetic variations a nd \nenhanced ma gnetic coercivity16. In our case, for example, we observe a slight  change (<10%) in \nthe magnetization saturation when graphene is present  (not shown here) . Out-of-plane excited \nFMR measurements on these strips seem to support this hypothesis . Two-magnon processes are \nsubstantially weaker when the precession of the magnet ization is excited wit h the dc field out of \nthe plane, which would explain the similar  H vs. f slopes for Py and Py/Gr strips in the out -of-\nplane excited FMR data of Fig.  3b (black and red data) .  \n \nFigure 3: a) In -plane and b) ou t-of-plane frequency dependences of the FMR linewidth of the three strips \nmeasured  (Py: Permalloy Gr: graphene and Gr -prt: graphene protruding from the sides of the Py strip) . \nThe insets show the FMR field excitation situations and the corresponding direct ions of propagation and \npolarization of the pumped spin currents  (Js) into the graphene area protruding away from the edges of \nthe Py strip.  \n \nThe central result of this work is the clear enhancement of the dynamical damping observed \nin Py/Gr -prt strips und er both in -plane and out -of-plane excited FMR (blue data in Figs.  3a and \n3b). Importantly, t his additional damping can only result from the relaxation of spins  in the area \nof graphene away from the Py strip, where an enhanced relaxation due to proximity ef fects as 10 \n discussed in the introduction for extended films is not an option . Comparing the FMR \nbroadening in the Py/Gr and Py/Gr -prt strips, and using Eqn.  (1) we extract a change in the \ndamping parameter Py/Gr -prt  = Py/Gr -prt - Py/Gr = 1.310-3, resul ting in a spin -mixing \nconductance of \ng 6.891018 m-2. These are a factor 2 -3 smaller than the values found in \nexperiments performed on extended films but still comparable to those found in Py/Pt or Py/Pd \nsamples.  \nWe now discuss the fundamental implications of the experiments described above. The \nobserved additional damping enhancement  provided by the protruding graphene  support s our \nassertion  that spin  pumping must occur across the quasi -one-dimensional  Py/Gr -prt interface at \nthe very edge of the Py strip. The picture we propose is the following. The proximity  of Py to \ngraphene induces a wea k equilibrium ferromagnetization in the latter25. The precessing \nmagnetization in the Py film pumps a spin current into the graphene layer  underneath the Py , \nthus creating an additional non-equilibrium spin accumulation in that layer. Part of the excess \nspin polarization is rel axed by local defects and impurities present on graphene (through  local -\nmoment scattering or spin -orbit coupling). When the graphene layer does not protrude away \nfrom the Py, t he remainder non -equilibrium spin accumulation creates a coherent backflow spin \ncurrent into the Py.  Thus, in steady state, there is no net spin current and the enhanced damping \nof the FMR is mainly due to spin relaxation in the graphene layer underneath the Py. However, \nwhen the graphene layer extends beyond the Py, the non -equilibri um spin accumulation causes a \nspin current to flow into the protruded graphene regions, reducing the amount of coherent \nbackflow into the Py and thus increasing the damping of the FMR due to losses of angular \nmomentum . In this case, it is standard to obtai n the spin -mixing conductance associated to the \npumping of spin into the extended graphene regions through  the expression26 11 \n  \n .                                                      (2) \nYet, this is only justified if the spin current relaxes much faster than the charge diffuses ( i.e., \nwhen the electronic motion in the extended graphene regions is ballistic or when very strong spin \nscattering is present, which are unlikely in our samples). We rather expect the charge to diffus e \nwith a relaxation time \n , where \n  is the spin relaxation time. In this case, a non -\nequilibrium spin population  builds up  on the protruding graphene  near the Py edge. This causes \nthe spin current  to partially diffuse  back into the graphene underneath the Py.  Thus, t he resulting  \nspin-mixing conductance is smaller than that obtained from  measuring the excess damping26 \n                                                               (3) \nwhere \n  is the backflow (dimensionless) parameter, with \n  denot ing the \ndensity of states of graphene at the Fermi level , \n being the length of the Py strip,  A being the \narea of the Py/graphene interface, and \n  representing the spin diffusion length (notice \nthat \n , where \n  = 20 m is the length of the protruding  graphene region). Assuming \ns, \n m2/s, and \n , with \n  meV and \nm/s (see Ref.13 ) we find \n  m2, which i s a much large r value (by several \norders of magnitude) than  the experimental value \n  m2. This implies  a 12 \n negative  value for \n  and therefore indicates that Eqn. (3) may not be directly applicable to \nour setup . We note that this analysi s may change in experiments  performed  in devices using h -\nBN substrates, for which the spin relaxation parameters would drastically change. H -BN has also \nbeen shown to decrease the conductance mismatch in electrical spin injection  and therefore may \naffect s pin reflection and relaxation at FM/graphene interfaces  27. We believe that the main \nproblem  in our analysis  is not in the est imate of the backflow parameter  β, since this follows \nstraightforwardly from reasonable estimates for the graphene parameters l, D, τs, and NF. Instead, \nwe believe that the problem lies on the assumption that spin currents pumped by the Py are fully \ninjected into the protruding graphene sheets. It may be possible that very close to the edge where \nthe protruding graphene meets the Py, there is a strongly enhanced spin relaxation, effectively \nmaking λs a much shorter length scale, rendering the backflow negligible. This relaxation could \nbe due to the Py -Gr edge acting approximately as a semiconducting p -n junction, regaining the \nconducta nce mismatch that were supposed to be eliminated by the spin pumping method, with \ngraphene underneath the Py being heavily p -doped, while protruding areas  are almost undoped . \nHowever, this interpretation requires further experimental verification and a mor e detailed \ntheoretical modeling.   \nIn conclusion, we have presented experimental evidence of a substantial increase in damping \nin Py/Gr strips when graphene is left to protrude from the sides of the ferromagnet. The \nsurprisingly high spin mixing conductance  obtained from the observations raises question s about \nthe physics of dynamical spin injection into two -dimensional structures such as graphene.  Our \nimmediate future objective is the direct measurement of the ISHE voltage generated in the \nprotruding graph ene region as a result of the  pumped spins from the ferromagnet, for which \nelectrodes will be placed at the opposite ends of the protruding graphene lines. However, YIG-13 \n based insulating ferromagnetic strips will be used for this purpose, since the low -resistivity Py \nstrip in the present configuration acts as an electrical shunt and prevents the observation of the \neffect.   \n \nAknowledgements  \nWe thank A. Brataas for valuable discussions. S.S. E.M. and E.d.B. acknowledge support from \nthe National Science Foundat ion (ECCS #1402990).  A.A., C.T.C. and B.O. acknowledge support \nfrom the Singapore National Research Foundation (Grant Nos. NRFRF2008 -07 and NUS -YIA \nR144 -000-283-101). \n \n \n \n \n \n \n \n \n \n 14 \n Supplemental Material  \nDynamical spin injection at a quasi -one-dimensional ferrom agnet -graphene \ninterface  \nS. Singh1, A. Ahmadi1, C.T. Cherian2, 4, E. R. Mucciolo1, E. del Barco1, and \nB. Özyilmaz2,3,4,5  \n1Department of Physics, University of Central Florida, Orlando, Florida USA, 32816  \n2Department of Physics, National University of Singa pore, 2 Science Drive 3, Singapore 117542  \n3NanoCore, 4 Engineering Drive 3, National University of Singapore, Singapore 117576  \n4Graphene Research Center, National University of Singapore, Singapore 117542  \n5NUS Graduate School for Integrative Sciences and E ngineering (NGS), National University of \nSingapore, Singapore 117456  \nIn this document we present a set of experiments designed to study and prevent changes of the \nmagnetic properties of the Py films due to structural changes associate to the adjacent singl e-\nlayer graphene and their possible effect in broadening the FMR linewidth of the ferromagnet.  \n1. Reversing the FM/Gr stacking order  \nWe first turn to experiments in which the graphene layer is transferred onto pre -deposited Py \nfilms, in contrast with our prev ious studies14, where the graphene was transferred onto the \nsubstrate ( i.e., GaAs) and subsequently Py was deposited atop. Since graphene is not atomically \nflat when deposited on most su bstrates28 (with exception of boron nitride), the growth of the \nferromagnet on top of the graphene surface can result in a rough interface and damping 15 \n contributions due to processes other than the spin pumping, such as two magnon scattering. In \naddition, the saturation magnetization of the film can also be modified. We have performed \nnanoscale tunneling microscopy on our graph ene films on SiO 2 substrates and observed the \nnanometer sized ripples of our graphene surfaces. To eliminate an enhancement of damping due \nto structural changes at the interface, we transfer single -layer graphene onto of Py films which \nhave been previously  deposited on SiO 2 wafers, so that the Py is unaffected, or weakly affected \n(see Fig.  SM1a).  \n \nFigure SM4: a) Sketch illustrating a Py film in where graphene is wet transferred on the top of \nthe film. b) Frequency dependence of th e FMR linewidth of Py and Gr/Py films.  \n \nWe performed FMR measurements in both Py and Gr/Py by applying the dc magnetic field \nin the plane of the films. The width of the resonance, in units of external dc field, is extracted by \ntaking the peak -to-peak dista nce of the FMR field -derivatives. The frequency dependences of \nFMR linewidth for both samples are shown in figure SM1b . The frequency -dependent width of a \nFMR resonance represents the effective dynamical damping of the precessing magnetization of \nthe ferro magnet. Any loss of spin angular momentum  (in this case due to the presence of 16 \n graphene)  will correspondingly result in an increase in linewidth. Since pure spin currents carry \naway spin angular momentum, we can theoretically associate t he damping enhancem ent to be a \ndirect consequence of dynamical spin pumping from the Py into the graphene layer .  \nThe damping parameters extracted for these samples are: Py = 9.610-3 & GPy = 0.257  GHz; \nPy/Gr = 13.010-3 & GPy/Gr = 0.339  GHz. This corresponds to a change i n the damping \nparameter ( Δα) of 35%, as compared to 88% change observed in our previous studies of films \nwith the Py deposited onto of the graphene layer14. The difference in the relative damping change \njust by reversing the stacking order of the Py and graphene layers is indicative that a good \nfraction of the FMR broadening in Py/Gr samples was likely due to structural changes in the \nsurface of the Py film, which has been deposited on top of the rough graphene s urface. It is \npossible that in our experiment the wet transfer of graphene into only one of the Py films causes \nsome degree of contamination ( e.g., oxidation of the Py surface) altering the magnetic properties \nof the ferromagnetic film or the spin polariza tion at the interface. Recent experiments have \nshown how contamination during the wet transfer process of two -dimensional crystals can \ndecrease spin polarization at the interface29. We do not observe any appreciable change in the \nsaturation magnetization, so such contamination is not drastically changing the \nmagnetostructural properties of the Py film.  Nevertheless, the substantial amount of damping \nobserved in the Gr/Py samples o f Fig.  SM1 indicates that spin pumping at the Gr/Py interface is \nlikely the main cause of FMR broadening.  \n2. Inserting a Cu spacing layer in between the Py film and graphene  \nIt was brought to our attention that while graphene is transferred onto evaporated Py  films \nfrom an aqueous solution there is a possibility that the Py is oxidized, causing an enhancement of 17 \n damping of structural origin. Although we find this possibility unlikely to explain the observed \nresults, we made controlled extended films by inserti ng a thin layer of Copper (Cu) in between \nthe Py film and graphene, and compared the FMR results with Py/Cu films without graphene \nunderneath. The actual samples that we developed are: a) bare Py (14nm); b) Py(14nm)/Cu \n(20nm); and, Py(14nm)/Cu(20nm)/Gr. Th e cartoons representing samples are shown in the inset \nto Fig. SM2a. We use 20nm -thick Cu because it is substantially less than the spin diffusion \nlength and one expects to see negligible spin relaxation and hence no enhanced damping due to \nthe presence of  Cu itself in the FMR experiments22. In addition, the effect of a graphene layer \n20nm away from the Py sample will be negligible and the Py/Cu interface will remain unaltered, \nwhether or not graphene is underneath the heterostructures.  \n \nFigure SM5: a) Illustrations of the Py, Cu and Gr heterostructures. b) Frequency dependence of \nthe FMR linewidth of Py , Py/Cu and Py/Cu/Gr  films . \n \nThe in -plane excited FMR linewidths for the above mentioned samples are presented as a \nfunction of frequency in Fig.  SM2b. As exp ected, there is no change in damping due to presence \nof the Cu film underneath the Py, since the pumped spins reflect coherently back into the 18 \n ferromagnet (no diffusion in the Cu film). One would need to increase the thickness of the Cu \nfilm beyond the spi n diffusion length for pumping -induced damping to be observable22. Now, in \nthe case of the Py/Cu/Gr stacking we do observe a clear change in the damping (i.e. the slope of \nthe frequency dependence of FMR linewidth changes). The extracted damping parameters for \nthese samples are: Py = 1210-3 & GPy = 0.315  GHz; Cu/Py = 11.910-3 & GCu/Py = 0.312  GHz ; \nand, Gr/Cu/Py = 16.210-3 & GGr/Cu/Py = 0.422  GHz . This is about 35 % change in damping due to \nthe presence of graphene underneath the Cu film, which is similar to what we observe in \nexperiments with graphene transferred atop of the Py films.  \n3. Comparison between different experiments  \nA first inclination is to attribute the observed damping enhancements to spin pumping into \nthe whole graphene area underneath the ferromagnet. However, one  needs to keep in mind that \ngraphene is a two dimensional material with no conduction channels perpendicular to the plane. \nFor standard 3D FM/NM heterostructures where spin pumping has been studied7, the spin \ngradient is perpendicular to the FM/NM interface, i.e. the spin polarization decays \nperpendicularly to the plane. In the case of graphene, the only d irection spins can decay is along \nthe plane of the graphene sheet, and this needs to be away from the microwave excitation area. In \nour above explained experiments on extended thin films only a small part (~20  um  1000  µm) \nof the total film (~5  mm  5 mm)  is excited by to the central line of the µ -CPW. Therefore, the \nspin density pumped from the excited part of the ferromagnet into the graphene directly \nunderneath can only decay in the horizontal graphene plane which is not under microwave \nexcitation. Unfo rtunately, in experiments with extended films (as those presented in this \nSupplemental Information document), the areas of graphene away from excitation are in close 19 \n proximity to the ferromagnet, which is covering the whole film. Given that graphene is hig hly \nsensitive to its environment30, the presence of the ferromagnet is expected to change the intrinsic \nelectronic and spin relaxation properties and result in an enhanced diffusion of the spin currents. \nThis asks for a device geometry in which the damping assoc iated with spin pumping can be \nsystematically differentiated, which is the purpose of the experiments discussed in the main text \nof this article.  \nFor the sake of clarity, we have compared the change in damping observed for different \nconfiguration samples i n Table SM1, including the results in Py/Gr -prt strips discussed in the \nmain text, where the enhanced spin diffusion due to the proximity of the ferromagnet in areas of \ngraphene away from the excitation has been eliminated. Although being the smallest, t he 11% \ndamping enhancement observed in the latter case can be univocally associated to spin pumping \ninto single -layer graphene and still stands comparable to values observed in heavy metals. It has  \nbeen shown that a small density of  adatoms ( e.g., atomic hyd rogen) on the graphene surface can \ninduce local deformations resulting in huge enhancement s of spin orbit coupling11. Recently, a \nlarge spin Hall Effect  has also been reported for CVD grown graphene (similar to what is used \nfor this work) du e to presence of residual Cu adatoms at the graphene surface. Although Cu itself \nhas low spin-orbit  interaction , it can induce local deformation in the graphene lattice which \nresults in an enhance ment of the spin -orbit interaction .  \n \n \n \n 20 \n  \nTable  SM1 : Comparison between damping enhancements for all the samples discussed in this \narticle.  \n \nReferences:  \n1 A. K. Patra, S. Singh, B. Barin, Y. Lee, J. -H. Ahn, E. del Barco, E. R. Mucciolo, and B. Özyilmaz,  \nApplied Physics Letters 101 (16) (2012).  \n2 Masa Ishigami, J. H. Chen, W. G. Cullen, M. S. Fuhrer, and E. D. Williams,  Nano Letters 7 (6), \n1643 (2007).  \n3 André Dankert, M. Venkata Kamalakar, Abdul Wajid, R. S. Patel, and SarojP Dash,  Nano Res., 1 \n(2014).  \n4 Th. Gerrits, M. L. Schneid er, and T. J. Silva,  Journal of Applied Physics 99 (2) (2006).  \n5 O. Mosendz, V. Vlaminck, J. E. Pearson, F. Y. Fradin, G. E. W. Bauer, S. D. Bader, and A. \nHoffmann,  Physical Review B 82 (21), 214403 (2010).  \n6 F. Schedin, A. K. Geim, S. V. Morozov, E. W. Hill, P. Blake, M. I. Katsnelson, and K. S. Novoselov,  \nNat Mater 6 (9), 652 (2007).  \n7 Jayakumar Balakrishnan, Gavin Kok Wai Koon, Manu Jaiswal, A. H. Castro Neto, and Barbaros \nOzyilmaz,  Nat Phys 9 (5), 284 (2013).  \n Sample  Change in da mping  \nΔα Possible damping contributions  \nPy/Gr/Substrate  88% Interface roughness damping + \nenhanced damping due to Py/Gr \nhybridization + spin pumping  \nGr/Py/Substrate  34% Enhanced damping due to  \nPy/Gr hybridization + spin \npumping  \nPy/Cu/Gr/Substrate  35% Enhanced damping due to  \nCu/Gr hybridization + spin \npumping  \nPy/Gr/Substrate strips  \n(graphene only under Py and \nhomogeneously irradiated)  24% Interface  roughness  damping  \nPy/Gr/Substrate strips  \n(graphene protruding out of Py)  15% Spin pumping only  21 \n  \n 22 \n  \nREFERENCES  \n1 Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom,  Science 306 (5703), 1910 (2004).  \n2 F. J. Jedema, A. T. Filip, and B. J. van Wees,  Nature 410 (6826), 345 (2001).  \n3 Eduardo R. Mucciolo, Claudio Chamon, and Charles M. Marcus,  Physical Review Letters 89 (14), \n146802 (2002);  Susan K. Watson, R. M. Potok, C. M. Marcus, and V. Umansky,  Physical Review \nLetters 91 (25), 258301 (2003).  \n4 K. Ando, T. Yoshino, and E. Saitoh ,  Applied Physics Letters 94 (15), 152509 (2009);  E. \nSaitoh, M. Ueda, H. Miyajima, and G. Tatara,  Applied Physics Letters 88 (18) (2006).  \n5 K. Ando, S. Takahashi, J. Ieda, H. Kurebayashi, T. Trypiniotis, C. H. W. Barnes, S. Maekawa, and E. \nSaitoh,  Nat Mater 10 (9), 655 (2011).  \n6 S. Kuhlen, K. Schmalbuch, M. Hagedorn, P. Schlammes, M. Patt, M. Lepsa, G. Güntherodt, and B. \nBeschoten,  Physical Review Letters 109 (14), 146603 (2012).  \n7 O. Mosendz, V. Vlaminck, J. E. Pearson, F. Y. Fradin, G. E. W. Bauer, S . D. Bader, and A. \nHoffmann,  Physical Review B 82 (21), 214403 (2010).  \n8 Kazuya Ando and Eiji Saitoh,  Nat Commun 3, 629 (2012).  \n9 Shun Watanabe, Kazuya Ando, Keehoon Kang, Sebastian Mooser, Yana Vaynzof, Hidekazu \nKurebayashi, Eiji Saitoh, and Henning Sir ringhaus,  Nat Phys 10 (4), 308 (2014).  \n10 A. K. Geim and K. S. Novoselov,  Nat Mater 6 (3), 183 (2007).  \n11 Jayakumar Balakrishnan, Gavin Kok Wai Koon, Manu Jaiswal, A. H. Castro Neto, and Barbaros \nOzyilmaz,  Nat Phys 9 (5), 284 (2013).  \n12 Jayakumar Balakr ishnan, Gavin Kok Wai Koon, Ahmet Avsar, Yuda Ho, Jong Hak Lee, Manu \nJaiswal, Seung -Jae Baeck, Jong -Hyun Ahn, Aires Ferreira, Miguel A. Cazalilla, Antonio H. Castro \nNeto, and Barbaros Özyilmaz,  Nat Commun 5 (2014).  \n13 Nikolaos Tombros, Csaba Jozsa, Mihait a Popinciuc, Harry T. Jonkman, and Bart J. van Wees,  \nNature 448 (7153), 571 (2007);  Wei Han and R. K. Kawakami,  Physical Review Letters 107 (4), \n047207 (2011);  T. Y. Yang, J. Balakrishnan, F. Volmer, A. Avsar, M. Jaiswal, J. Samm, S. R. Ali, A. \nPachoud , M. Zeng, M. Popinciuc, G. Güntherodt, B. Beschoten, and B. Özyilmaz,  Physical Review \nLetters 107 (4), 047206 (2011).  \n14 A. K. Patra, S. Singh, B. Barin, Y. Lee, J. -H. Ahn, E. del Barco, E. R. Mucciolo, and B. Özyilmaz,  \nApplied Physics Letters 101 (16) (2012).  \n15 Zhenyao Tang, Eiji Shikoh, Hiroki Ago, Kenji Kawahara, Yuichiro Ando, Teruya Shinjo, and \nMasashi Shiraishi,  Physical Review B 87 (14), 140401 (2013).  \n16 A. J. Berger, W. Amamou, S. P. White, R. Adur, Y. Pu, R. K. Kawakami, and P. C. Hammel,  Jo urnal \nof Applied Physics 115 (17) (2014).  \n17 S. P. Dash, S. Sharma, J. C. Le Breton, J. Peiro, H. Jaffrès, J. M. George, A. Lemaître, and R. Jansen,  \nPhysical Review B 84 (5), 054410 (2011).  \n18 Ryo Ohshima, Atsushi Sakai, Yuichiro Ando, Teruya Shinjo, Kenj i Kawahara, Hiroki Ago, and \nMasashi Shiraishi,  Applied Physics Letters 105 (16) (2014).  \n19 J. C. Rojas Sánchez, L. Vila, G. Desfonds, S. Gambarelli, J. P. Attané, J. M. De Teresa, C. Magén, \nand A. Fert,  Nat Commun 4 (2013).  \n20 Xuesong Li, Weiwei Cai, Jin ho An, Seyoung Kim, Junghyo Nah, Dongxing Yang, Richard Piner, \nAruna Velamakanni, Inhwa Jung, Emanuel Tutuc, Sanjay K. Banerjee, Luigi Colombo, and Rodney \nS. Ruoff,  Science 324 (5932), 1312 (2009).  23 \n 21 See supplementary material at [URL will be inserted by  AIP] for the experiemnts with different \nstacking order of Py and graphene.  \n22 Th. Gerrits, M. L. Schneider, and T. J. Silva,  Journal of Applied Physics 99 (2) (2006).  \n23 Ahmet Avsar, Tsung -Yeh Yang, Sukang Bae, Jayakumar Balakrishnan, Frank Volmer, Manu \nJaiswal, Zheng Yi, Syed Rizwan Ali, Gernot Güntherodt, Byung Hee Hong, Bernd Beschoten, and \nBarbaros Özyilmaz,  Nano Letters 11 (6), 2363 (2011).  \n24 O. M. J. van 't Erve, A. L. Friedman, CobasE, C. H. Li, J. T. Robinson, and B. T. Jonker,  Nat Nano 7 \n(11), 737 (2012).  \n25 P. Lazić, G. M. Sipahi, R. K. Kawakami, and Igor Žutić,  Physical Review B 90 (8), 085429 (2014).  \n26 Yaroslav Tserkovnyak, Arne Brataas, Gerrit E. W. Bauer, and Bertrand I. Halperin,  Reviews of \nModern Physics 77 (4), 1375 (2005).  \n27 M. V enkata Kamalakar, André Dankert, Johan Bergsten, Tommy Ive, and Saroj P. Dash,  Sci. Rep. \n4 (2014).  \n28 Masa Ishigami, J. H. Chen, W. G. Cullen, M. S. Fuhrer, and E. D. Williams,  Nano Letters 7 (6), \n1643 (2007).  \n29 André Dankert, M. Venkata Kamalakar, Abdu l Wajid, R. S. Patel, and SarojP Dash,  Nano Res., 1 \n(2014).  \n30 F. Schedin, A. K. Geim, S. V. Morozov, E. W. Hill, P. Blake, M. I. Katsnelson, and K. S. Novoselov,  \nNat Mater 6 (9), 652 (2007).  \n "    },    {        "title": "2012.08340v1.Dynamically_stabilized_spin_superfluidity_in_frustrated_magnets.pdf",        "content": "Dynamically stabilized spin super\ruidity in frustrated magnets\nRicardo Zarzuela,1Daniel Hill,2Jairo Sinova1;3and Yaroslav Tserkovnyak2\n1Institut f ur Physik, Johannes Gutenberg Universit at Mainz, D-55099 Mainz, Germany\n2Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA\n3Institute of Physics Academy of Sciences of the Czech Republic,\nCukrovarnick\u0013 a 10, 162 00 Praha 6, Czech Republic\nWe study the onset of spin super\ruidity, namely coherent spin transport mediated by a topolog-\nical spin texture, in frustrated exchange-dominated magnetic systems, engendered by an external\nmagnetic \feld. We show that for typical device geometries used in nonlocal magnetotransport ex-\nperiments, the magnetic \feld stabilizes a spin super\row against \ructuations, up to a critical current.\nFor a given current, the critical \feld depends on the precessional frequency of the texture, which\ncan be separately controlled. We contrast such dynamic stabilization of a spin super\ruid to the\nconventional approaches based on topological stabilization.\nIntroduction. | Low-dissipation spin-dependent trans-\nport o\u000bers pro mising perspectives for the design of high-\nspeed communication architectures in the next genera-\ntion of spintronic devices [1, 2]. A prominent example\nlies in the concept of spin super\ruidity [3{7], introduced\nin the late 70s by analogy with the phenomena of (charge)\nsuperconductivity and mass super\ruidity in4He, albeit\nnot exempt from dissipation in the form of spin relax-\nation/dephasing. One of the hallmarks of spin super-\n\ruidity is the algebraic decay of spin signals over long\ndistances, in contrast to the exponential suppression ob-\nserved for (incoherent) magnon-mediated transport [8, 9].\nExperimental signatures of super\ruid spin transport have\nbeen recently reported in (high-quality) antiferromag-\nnetic platforms for device geometries usually used in non-\nlocal magnetotransport measurements [10, 11]. However,\nparasitic anisotropies arising naturally in the fabrication\nprocess of spintronic (collinear) magnetic systems have\na detrimental e\u000bect on the spin super\ruid state, since\nthese break the underlying spin O(2) symmetry of the\n(anti)ferromagnetic host and, therefore, open a gap in\nthe excitation spectrum that lifts the Goldstone mode\nsustaining the spin super\row.\nMagnetic systems with frustrated interactions domi-\nnated by isotropic exchange o\u000ber an alternative route to\novercome these key challenges, by averaging the parasitic\nanisotropies out at the macroscopic level and restoring an\ne\u000bective SO(3) symmetry in the spin space. Furthermore,\nthe minimal description of these noncollinear magnetic\nplatforms is provided by the O(4)-nonlinear sigma model\n[12{14], whose excitation spectrum for bulk systems con-\nsists of three Goldstone branches. An external magnetic\n\feld lifts two of these soft (angular) modes [15]. The re-\nmaining linearly dispersing mode (corresponding to rota-\ntions along the direction of the magnetic \feld), in turn, is\nable to sustain a phase-coherent precessional state, which\ncan be triggered by (interfacial) spin-orbit torques [16].\nIn particular, for the nonlocal device geometries usually\nconsidered (i.e., thin \flms with an out-of-plane orienta-\ntion of the spin accumulations and the external magnetic\n\feld), the collective \rapping out of the basal plane initi-ates the unwinding (phase slips) of the order parameter.\nTherefore, gapping these angular modes by an external\nmagnetic \feld should impede the (topological) relaxation\nchannel for the spin super\row, enhancing its stability.\nIt is, therefore, vital to understand to what extent\nthe \ructuations of the SO(3)-valued order parameter are\ndetrimental to a collective spin \row in frustrated mag-\nnets, in the presence of the stabilizing e\u000bect of the mag-\nnetic \feld. In this Letter, we show that the in-plane (ro-\ntational) Goldstone mode can sustain spin supercurrents\nand study the corresponding steady-state solution. We\nexamine the robustness of the underlying phase-coherent\nspin con\fguration against collective \ructuations and de-\ntermine its stability threshold as a function of the mag-\nnetic \feld and the precessional frequency. In this regard,\nwe show that even though, under purely topological con-\nsiderations, there is a low-energy route for the collective\nrelaxation of the spin-super\ruid state, the analysis of its\ndynamics dictates a \fnite range of stable solutions. It is,\ntherefore, not topology, which is dictated by the free en-\nergy (as, e.g., in the case of easy-plane anisotropies [16]),\nthat is key here but a \feld-induced dynamic stabilization\nof the basal winding textures.\nEquations of motion. | At the macroscopic level, frus-\ntrated magnets are described by an SO(3) order parame-\nter,^R(~ r;t), which represents smooth and slowly varying\nrotations of the initial spin con\fguration [12, 17]. Dy-\nnamics of ^Rand the nonequilibrium spin density m(~ r;t)\nof the system are governed by the equations [15, 16]:\nm=\u001f(!+\rB); (1)\n@tm+\u000bs!=Ar\u0001~\n+\rm\u0002B; (2)\nwhere\u001f,\randAdenote the spin susceptibility, the gy-\nromagnetic ratio and the order-parameter sti\u000bness, re-\nspectively. Furthermore, \u000bparametrizes losses due to\ndissipative processes (Gilbert damping) in the bulk and\ns'~S=a3, withSandabeing the length of the mi-\ncroscopic spin operators and the lattice spacing, respec-\ntively.Bdenotes the external magnetic \feld, !\u0011\niTrh\n^R>^L@t^Ri\n=2 is the local precessional frequency andarXiv:2012.08340v1  [cond-mat.mes-hall]  15 Dec 20202\nthe spin \felds \nk\u0011iTrh\n^R>^L@k^Ri\n=2,k=x;y;z , de-\nscribe the spatial variations of the instantaneous state of\nthe spin texture, with [ ^L\u000b]\f\r\u0011\u0000i\u000f\u000b\f\rbeing the gener-\nators of the SO(3) group. Note that the dissipative term\n\u000bs!in Eq. (2) is the one responsible for the algebraic\ndecay of the spin signal when balanced with the appro-\npriate boundary conditions for the spin supercurrent, as\nwe will show in the next section.\nA convenient representation of the order parameter is\ngiven in terms of unit-norm quaternions, q= (w;v) [16].\nThe parametrization w= cos(\u001e=2) andv= sin(\u001e=2)n,\nwherenand\u001e(~ r;t) represent the rotation axis and the\nlocal rotation angle for spins, respectively, yields the well-\nknown Rodrigues' rotation formula: ^R\u000b\f= cos\u001e\u000e\u000b\f+\n(1\u0000cos\u001e)n\u000bn\f+ sin\u001e\u000f\u000b\r\fn\r. By applying this repre-\nsentation to Eqs. (1) and (2), we obtain the following\nequation of motion for the quaternion [19]:\n@2\ntq+1\nT@tq\u0000v2\nmr2q+\u0015(q)q (3)\n\u0000\r\u0000\nB?^@tq+1\n2@tB?^q\u0001\n= 0;\nwhereT\u0011\u001f=\u000bs de\fnes a characteristic relaxation time,\nvm\u0011p\nA=\u001fdenotes the speed of spin waves in the mag-\nnetic medium and \u0015(q)\u0011@tq\f@tq?\u0000v2\nm@kq\f@kq?+\n\r\n2!\u0001Bis a quadratic prefactor. Here, q?\u0011(w;\u0000v)\ndenotes the adjoint quaternion, B\u0011(0;B), the Hamil-\nton product is de\fned by q1^q2\u0011(w1w2\u0000v1\u0001\nv2;w1v2+w2v1+v1\u0002v2), and the scalar product reads\nq1\fq2\u00111\n2(q1^q?\n2+q2^q?\n1) =w1w2+v1\u0001v2.\nxyz\nIL\nIR\nL\nFIG. 1: Two-terminal geometry for the generation and de-\ntection of spin super\ruidity in frustrated magnetic platforms.\nThe spin precession (blue arrows) along the spin accumula-\ntions\u0016L;R(red arrows) is depicted as a rotating triad of vec-\ntors, which represents the internal spin frame of the texture.\nThe black arrow represents the rotation axis nand the local\nrotation angle \u001eis illustrated by the rotation of the green and\norange arrows of each triad within the plane perpendicular to\nn(which is oriented along the zaxis here).\nSpin super\ruid state. | Hereafter, we restrict ourselves\nto a quasi-one-dimensional geometry, by assuming trans-\nlational symmetry along the yandzspatial directions,\nand a \fnite length Lalong thexdirection, see Fig. 1.We also consider lateral contacts extending along the yz\nplane for spin injection via spin Hall physics [18] and as-\nsume that the spin accumulations \u0016L(left interface) and\n\u0016R(right interface) are parallel to the zaxis, which set\nthe (uniform) direction of n=^ezacross the sample. Fur-\nthermore, we apply the magnetic \feld collinear as well,\nB=Bn, to stabilize the super\row, and take \u001e(~ r;t) to be\nspatially smooth and slowly varying. As a result, Eq. (3)\nturns into the following dynamical equation for the rota-\ntion angle [19]:\n@2\nt\u001e+1\nT@t\u001e\u0000v2\nm@2\nx\u001e+\r@tB= 0: (4)\nThe precessional steady solution to the above equation is\nobtained by considering the ansatz \u001e(x;t) =X(x)+\u001c(t).\nThe boundary conditions are\n\u00002A@xq^q?jL=gL\n4\u0019\u0002\n\u0016L\u00002~@tq^q?jL\u0003\n;(5)\n2A@xq^q?jR=gR\n4\u0019\u0002\n\u0016R\u00002~@tq^q?jR\u0003\n;(6)\nwhich arise from the exchange of angular momentum be-\ntween the magnet and adjacent (heavy-)metal contacts\nin the form of ordinary exchange torques and enhanced\nGilbert damping [15, 16]. Here, gLandgRdenote the\nspin mixing conductance at the left ( x=\u0000L\n2) and right\n(x=L\n2) terminals, respectively [20]. In what follows, we\nassume the same spin mixing conductance at both inter-\nfaces,g\u0011gL=gR. The spin super\ruid state is therefore\ndescribed, under an external magnetic \feld, by [19]\n\u001es(x;t) =\u001e0+kx+1\n2\u0010\u000bs\nA\u0011\n!x2+!t; (7)\nwith\nk=g\n8\u0019A(\u0016R\u0000\u0016L); (8)\n!=g\n2\u0016L+\u0016R\n~g+ 2\u0019\u000bsL: (9)\nWe therefore conclude that the external magnetic \feld\nhas no e\u000bect on the spin texture sustaining the spin su-\nper\row [8, 9].\nFluctuations and stability. | We proceed next to study\nthe robustness of the spin super\ruid state, Eq. (7),\nagainst \ructuations of the order parameter. First, we\nintroduce the following orthonormal set fqs;\u00181;\u00182;\u00183g\nof quaternions, where qscorresponds to the super\ruid\nsolution given by Eq. (7), \u00181= (0;^ey),\u00182= (0;^ex), and\n\u00183= 2@\u001esqs=\u0000\n\u0000sin(\u001es=2);cos(\u001es=2)^ez\u0001\n. Note that\n\u00181;2represent\u0019rotations around the yandxaxes, re-\nspectively. We incorporate \ructuations into the order\nparameter via the following parametrization:\nq=qsq\n1\u00002j\ttj2\u0000\t2\nl+^e+\tt+^e\u0000\t\u0003\nt+\u00183\tl;(10)\nwhere \tt(complex valued) and \t l(real valued) rep-\nresent the transverse and longitudinal (with respect to3\nthe rotation axis) \ructuation modes of the (unit-norm)\nquaternion order parameter. Here, ^e\u0006\u00111p\n2(\u00181\u0006i\u00182)\nare chiral quaternions with the properties ^e2\n\u0006= 0 and\n^e+\f^e\u0000=^e\u0000\f^e+= 1. We note that the transverse\n\ructuation modes \t tcorrespond to the out-of-plane ro-\ntations along the x- andy-axis, which are gapped out\nby the magnetic \feld. By inserting this parametrization\ninto Eq. (3) and keeping terms up to \frst order in \t tand\n\tl, we obtain the following dynamical equations for the\n\ructuation \felds [19]:\n@2\nt\tl+1\nT@t\tl\u0000v2\nm@2\nx\tl= 0; (11)\n@2\nt\tt+\u00121\nT\u0000i\rB\u0013\n@t\tt\u0000v2\nm@2\nx\tt (12)\n+1\n4h\n(@t\u001es)2\u0000v2\nm(@x\u001es)2+ 2\rB@t\u001esi\n\tt= 0:\nStability analysis of Eq. (11) in Fourier space (with re-\nspect to the xcoordinate) yields the eigenvalues\n\u0015\u0006\nl(q) =1\n2T\"\n\u0006s\n1\u0000q2\nq2c\u00001#\n; (13)\nwhereqandqc\u00111=2vmTdenote the Fourier wavevec-\ntor and its critical value, respectively. For jqj\u0014qc, the\neigenvalues of the dynamical system are real valued and\nnegative,\u0015\u0006\nl\u00140. Forjqj> qcthe eigenvalues are\ncomplex valued, \u0015\u0006\nl(q) =\u00001\n2T\u0006i\n(q). Consequently,\n\u0016\tl/e\u0015\u0006\nlt\u0018e\u0000t=2Tdecays exponentially with time, so\nthat the spin super\ruid solution (7) is robust against lon-\ngitudinal \ructuations. This is in agreement with the fact\nthat rotations within the basal ( xy) plane cannot unwind\nthe order parameter.\nIn the case of transverse \ructuations, we suppose that\nthe winding of the super\ruid phase, described by @x\u001es,\nis changing smoothly across the magnet. As a result, we\nperform a local stability analysis of the \t tmodes sup-\nposing@x\u001es'kis approximately constant. Again, by\nFourier transforming Eq. (12) we obtain the dynamical\nsystem\n@2\nt\u0016\tt+\u00121\nT\u0000i\rB\u0013\n@t\u0016\tt (14)\n+h\n!2\n4+v2\nm\u0000\nq2\u0000k2\n4\u0001\n+\r\n2!Bi\n\u0016\tt= 0;\nwhere \u0016 (q;t)\u0011F[ (x;t)] denotes the Fourier transform\nof the \ructuation \feld. Stability analysis of the above\ndynamical system yields the eigenvalues\n\u0015\u0006\nt=1\n2\u001a\n\u00001\nT\u0006Re(Z) +i[\rB\u0006Im(Z)]\u001b\n; (15)\nwithZ\u0011q\n(i\rB\u00001\nT)2\u0000(!2+v2m(4q2\u0000k2) + 2\r!B).\nThe instability threshold arises when \u00001\nT\u0006Re(Z)>0,\n0.00.20.40.60.81.00246810B>Bc<latexit sha1_base64=\"9aJmMqX7/8YVkTip9O6w7I3SdHc=\">AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUC9S6sVjBfsBbSib7aZdutmE3YlQQn+EFw+KePX3ePPfuG1z0NYHA4/3ZpiZFyRSGHTdb6ewtr6xuVXcLu3s7u0flA+PWiZONeNNFstYdwJquBSKN1Gg5J1EcxoFkreD8d3Mbz9xbUSsHnGScD+iQyVCwShaqV2/rfczNu2XK27VnYOsEi8nFcjR6Je/eoOYpRFXyCQ1puu5CfoZ1SiY5NNSLzU8oWxMh7xrqaIRN342P3dKzqwyIGGsbSkkc/X3REYjYyZRYDsjiiOz7M3E/7xuiuG1nwmVpMgVWywKU0kwJrPfyUBozlBOLKFMC3srYSOqKUObUMmG4C2/vEpaF1XvsnrzcFmp1fM4inACp3AOHlxBDe6hAU1gMIZneIU3J3FenHfnY9FacPKZY/gD5/MH3BuPSA==</latexit>\n<latexit sha1_base64=\"ntrDEwu7cMXpfJZo5i6WYwgjLMg=\">AAACB3icbVDLSsNAFJ34rPXRqEs3g0VwVRNRdFl047KCfUAbwmQ6bYfOI8xMCiXNB/gNbnXtTtz6GS79E6dtFrb1wIXDOfdyLieKGdXG876dtfWNza3twk5xd2//oOQeHjW0TBQmdSyZVK0IacKoIHVDDSOtWBHEI0aa0fB+6jdHRGkqxZMZxyTgqC9oj2JkrBS6pY7kpI8uRmHKs8lwErplr+LNAFeJn5MyyFEL3Z9OV+KEE2EwQ1q3fS82QYqUoZiRrNhJNIkRHqI+aVsqECc6SGePZ/DMKl3Yk8qOMHCm/r1IEdd6zCO7yZEZ6GVvKv7ntRPTuw1SKuLEEIHnQb2EQSPhtAXYpYpgw8aWIKyo/RXiAVIIG9vVQkrEM9uJv9zAKmlcVvzrivd4Va7e5e0UwAk4BefABzegCh5ADdQBBgl4Aa/gzXl23p0P53O+uubkN8dgAc7XLzh+mgw=</latexit>!/vm|k|<latexit sha1_base64=\"JVDrsuIRDYWiSyUzLOV7Jra/Siw=\">AAACCXicbVC7TsMwFHV4lvJKYWSxqJCYSoJAMFZlYSwSfUhtFDmu01q1nch2iqo0X8A3sMLMhlj5Ckb+BLfNQFuOdKWjc+7VuTpBzKjSjvNtra1vbG5tF3aKu3v7B4d26aipokRi0sARi2Q7QIowKkhDU81IO5YE8YCRVjC8m/qtEZGKRuJRj2PicdQXNKQYaSP5dqnbR5wjWLsY+SnPJsOJb5edijMDXCVuTsogR923f7q9CCecCI0ZUqrjOrH2UiQ1xYxkxW6iSIzwEPVJx1CBOFFeOns9g2dG6cEwkmaEhjP170WKuFJjHphNjvRALXtT8T+vk+jw1kupiBNNBJ4HhQmDOoLTHmCPSoI1GxuCsKTmV4gHSCKsTVsLKQHPTCfucgOrpHlZca8rzsNVuVrL2ymAE3AKzoELbkAV3IM6aAAMnsALeAVv1rP1bn1Yn/PVNSu/OQYLsL5+ARurmnw=</latexit>\u0000B/vm|k|B<Bc<latexit sha1_base64=\"H6R8bsWltOHzrca6UfICv1rKXNE=\">AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUMFDqRePFewHtKFstpt26WYTdidCCf0RXjwo4tXf481/47bNQVsfDDzem2FmXpBIYdB1v53C2vrG5lZxu7Szu7d/UD48apk41Yw3WSxj3Qmo4VIo3kSBkncSzWkUSN4Oxnczv/3EtRGxesRJwv2IDpUIBaNopXb9tt7P2LRfrrhVdw6ySrycVCBHo1/+6g1ilkZcIZPUmK7nJuhnVKNgkk9LvdTwhLIxHfKupYpG3PjZ/NwpObPKgISxtqWQzNXfExmNjJlEge2MKI7MsjcT//O6KYbXfiZUkiJXbLEoTCXBmMx+JwOhOUM5sYQyLeythI2opgxtQiUbgrf88ippXVS9y+rNw2WlVs/jKMIJnMI5eHAFNbiHBjSBwRie4RXenMR5cd6dj0VrwclnjuEPnM8f2QmPRg==</latexit>\n12345|k| increases decreasesωFIG. 2: Stability diagram and critical threshold for the spin\nsuper\ruid state (7) parametrized by the normalized Larmor\n(magnetic \feld) and precessional ( !) frequencies. The phase-\ncoherent precessional state is stable for magnetic \felds above\nthe critical one (blue region). Red dots indicate the two spe-\ncial points of the diagram, namely (0 ;0) (unstable) and (0 ;1)\n(stable). Red and green lines display the paths within the\ndiagram parametrized by the increase of the wavevector jkj\n(for constant Band!) and the decrease of !(for \fxedBand\njkj), respectively.\nsince the transverse \ructuation will therefore blow up\nas time increases. Working out this condition leads to\nthe following critical value for the magnetic \feld, above\nwhich the phase-coherent precessional state (7) is robust\nagainst \ructuations of the order parameter:\n2\rBc=v2\nmk2\n!\u0000!; (16)\nwhich holds up to the frequency !c=vmjkj. Figure 2 de-\npicts the stability diagram of the spin super\ruid state (7)\nin terms of the precessional frequency and the external\nmagnetic \feld.\nDiscussion. | From Eq. (7) we can clearly extract\ntwo length scales for the spin-super\ruid phase, namely\n`p= 2\u0019=jkjand`nl=p\nA=\u000bs! . The former deter-\nmines the pitch of the magnetic spiral sustaining the spin-\ncarrying state, whereas the latter determines the spatial\nrate of change of the associated wave number (due to\ndamping). Our constant-pitch treatment for the instabil-\nity, based on the local wave number k, should work when\nkdoes not vary much on the lengthscale set by \ructua-\ntion wavelengths q\u00001. This translates into the condition\n`\u00001\nnl\u001cpkq. It thus needs to be veri\fed, for internal\nconsistency, that the corresponding lower bound for q4\n(which becomes progressively smaller as \u000b!0) does not\nsigni\fcantly a\u000bect the critical magnetic \feld [19]. Let us\nnow discuss in some detail the physics encapsulated in\nthe corresponding local-stability diagram, Fig. 2.\nFirst, there are two special points in Fig. 2, namely\npu= (0;0) and ps= (0;1). The former is always un-\nstable, where phase slips are triggered by \rapping out of\nthe basal plane. On the other hand, psis stable, since\nhere the phase velocity associated with the spin super-\n\row becomes faster than the spin waves and the super-\n\ruid cannot relax towards the uniform magnetic con\fg-\nuration. Second, keeping !andB\fxed, by increasing\nthe wave vectorjkjwe move along the straight path (red\nline) that converges asymptotically towards pu, leading\nto a stability-instability transition. This means, in par-\nticular, that when we have a single spin-current injector\n(e.g.,\u0016L6=0and\u0016R=0in the Fig. 1 setup), the local\nwave vector k(x) increases towards the injector, where\nthe instability thus sets in \frst. Third, by \fxing Band\njkjand decreasing the frequency ( !!0) we move along\nthe horizontal green line towards the unstable regime.\nThis case is relevant, for example, when we increases the\nchannel length Land/or Gilbert damping \u000b, which would\nreduce the frequency, according to Eq. (9).\nA question that naturally arises is what happens be-\nyond the stability regime. Since the onset of instabil-\nity occurs close to the injector, we speculate that spe-\ncial boundary solutions (such as the contact-soliton ones\nfound in conventional ferromagnetic platforms [24, 25])\nor chaotic dynamics may emerge near the injector, which\nwould suppress the overall spin-current injection. A lower\nbut \fnite spin current may then still propagate along\nthe channel, once the stable regime is reached at a low\nenough wave vector. We leave it as an open question\nto elucidate how the unstable boundary dynamics settle\ninto a stable steady \row in the bulk. Finally, as discussed\nin Ref. 16, the degradation of the spin super\row in the\nabsence of magnetic \feld and low frequencies occurs via\nphase slippage mediated by (Anderson-Toulouse) 4 \u0019vor-\ntices [26]. As a result, a mesoscopic residual spin current\nmay prevail,/jr\u001esj'4\u0019=L, which becomes negligible\nin macroscopic samples. A large enough magnetic \feld\nprecludes the nucleation of these topological textures by\ngapping out the out-of-plane rotations.\nThis research was supported in part by the Na-\ntional Science Foundation under Grant No. NSF PHY-\n1748958. RZ and JS acknowledge support by the Tran-\nsregional Collaborative Research Center (SFB/TRR) 173\nSPIN+X, the Dynamics and Topology Centre funded by\nthe State of Rhineland Palatinate and the Alexander von\nHumboldt Foundation. YT is supported by the NSF un-\nder Grant No. DMR-1742928 and the Alexander von\nHumboldt Foundation and is grateful for the hospitality\nof the University of Mainz, where most of this work has\nbeen carried out.[1] P. Upadhyaya, S.K. Kim and Y. Tserkovnyak, Phys. Rev.\nLett.118, 097201 (2017).\n[2] Y. Tserkovnyak and M. Kl aui. Phys. Rev. Lett. 119,\n187705 (2017).\n[3] E.B. Sonin, Solid. State Commun. 25, 253 (1978).\n[4] E.B. Sonin, Zh. Eksp. Teor. Fiz 74, 2097 (1978).\n[5] E.B. Sonin, Pis'ma Zh. Eksp. Teor. Fiz. 30, 697 (1979).\n[6] J. K onig, M.C. B\u001cnsager, and A.H. MacDonald. Phys.\nRev. Lett. 87, 187202 (2001).\n[7] E.B. Sonin, Advances in Physics 59, 181 (2010).\n[8] S. Takei and Y. Tserkovnyak, Phys. Rev. Lett. 112,\n227201 (2014).\n[9] S. Takei, B.I. Halperin, A. Yacoby and Y. Tserkovnyak,\nPhys. Rev. B 90, 094408 (2014).\n[10] W. Yuan, Q. Zhu, T. Su, Y. Yao, W. Xing, Y. Chen, Y.\nMa, X. Lin, J. Shi, R. Shindou, X.C. Xie and W. Han,\nSci. Adv. 4, eaat1098 (2018).\n[11] P. Stepanov, S. Che, D. Shcherbakov, J. Yang, K. Thila-\nhar, G. Voigt, M.W. Bockrath, D. Smirnov, K. Watan-\nabe, T. Taniguchi, R.K. Lake, Y. Barlas, A.H. MacDon-\nald, and C.N. Lau. Nat. Phys. 14, 907 (2018).\n[12] T. Dombre and N. Read, Phys. Rev. B 39, 6797 (1989).\n[13] P. Azaria, B. Delamotte, and D. Mouhanna, Phys. Rev.\nLett.68, 1762 (1992).\n[14] A. V. Chubukov, T. Senthil, and S. Sachdev, Phys. Rev.\nLett.72, 2089 (1994).\n[15] Y. Tserkovnyak and H. Ochoa, Phys. Rev. B 96, 100402\n(2017).\n[16] H. Ochoa, R. Zarzuela, and Y. Tserkovnyak, Phys. Rev.\nB98, 054424 (2018).\n[17] B. I. Halperin and W. M. Saslow, Phys. Rev. B 16, 2154\n(1977).\n[18] J. Sinova, S.O. Valenzuela, J. Wunderlich, C.H. Back,\nand T. Jungwirth, Rev. Mod. Phys. 87, 1213 (2015).\n[19] See Supplemental Material.\n[20] The concept of spin mixing conductance [21] admits a\nmicroscopic de\fnition for glassy systems in the regime of\nweak spin-orbit coupling [15, 22]. In this work we intro-\nduce the spin-mixing conductance as a phenomenological\nparameter that would ultimately relate the torque to the\ndriven electrical current (in conjunction with the spin\nHall phenomenology) [23], regardless of the strength of\nthe spin-orbit interaction.\n[21] Y. Tserkovnyak, A. Brataas, G.E.W. Bauer and B.I.\nHalperin, Rev. Mod. Phys. 77, 1375 (2005).\n[22] M.F. Jakobsen, A. Qaiumzadeh and A. Brataas, Phys.\nRev. B 100, 134431 (2019).\n[23] Y. Tserkovnyak and S. Bender, Phys. Rev. B 90, 014428\n(2014).\n[24] E. Iacocca and M.A. Hoefer, Phys. rev. B 99, 184402\n(2019).\n[25] T. Schneider, D. Hill, A. Kakay, K. Lenz, J. Lind-\nner, J. Fassbender, P. Upadhyaya, Y. Liu, K. wang,\nY. Tserkovnyak, I.N. Krivorotov, and I. Barsukov,\narXiv:1811.09369 (2018).\n[26] P.W. Anderson and G. Toulouse, Phys. Rev. Lett. 38,\n508 (1976).5\nSupplemental Material\nDERIVATION OF EQUATION (3)\nTo begin with, we incorporate Eq. (1) into Eq. (2) and thus obtain the equation\n@t!=v2\nmr\u0001~\n+\r(!\u0002B\u0000@tB)\u00001\nT!: (17)\nSecond, in the quaternion representation the identities != 2@tq^q?and\nk= 2@kq^q?,k=x;y;z hold. Third,\nwe apply the Hamilton product q?^\u0001to Eq. (17), so that we obtain\n@2\ntq?+1\nT@tq?\u0000v2\nmr2q?+\u0015(q)q?\u0000\r\u0000\n@tq?^B+1\n2q?^@tB\u0001\n= 0;\ntaking into account the expressions\nq?^@\u0016(@\u0016q^q?) =\u0000@2\n\u0016q?\u0000(@\u0016q^@\u0016q?)q?; (18)\nq?^(@\u0016q^q?) =\u0000@\u0016q?; (19)\nq?^[(@\u0016q^q?)\u0002B] =\u0000@\u0016q?^B+1\n2(!\u0001B)q?: (20)\nFinally, by applying the adjoint operator to the above equation and using the identity ( q1^q2)?=q?\n2^q?\n1, we derive\nEq. (3).\nPHASE-COHERENT PRECESSIONAL SOLUTION TO EQUATION (7)\nWe start by noting that, if \u001es(x;t) denotes the precessional angle of the spin super\ruid state, the identities @\u0016qs=\n1\n2(@\u0016\u001es)\u00183and@\u0016\u00183=\u00001\n2(@\u0016\u001es)qshold. Therefore, @2\n\u0016qs=1\n2(@2\n\u0016\u001es)\u00183\u00001\n4(@\u0016\u001es)2qsand!= (@t\u001es)n, so that\n(@2\nt\u0000v2\nmr2)qs=1\n2\u0000\n@2\nt\u001es\u0000v2\nm@2\nx\u001es\u0001\n\u00183\u00001\n4\u0002\n(@t\u001es)2\u0000v2\nm(@x\u001es)2\u0003\nqs; (21)\n\u0015(qs) =1\n4\u0002\n(@t\u001es)2\u0000v2\nm(@x\u001es)2\u0003\n+\r\n2B@t\u001es; (22)\nB?^@tq=1\n2(B@t\u001es)qs; (23)\n@tB?^q=\u0000(@tB)\u00183: (24)\nWith account of all these identities Eq. (3) becomes\n\u0014\n@2\nt\u001es+1\nT@t\u001es\u0000v2\nm@2\nx\u001es+\r@tB\u0015\n\u00183=0; (25)\nwhich, in turn, leads to Eq. (4) since \u001836=0. The separation of variables \u001es(x;t) =X(x) +\u001c(t) yields, for a uniform\nexternal magnetic \feld, the system of equations\n\u001c00(t) +1\nT\u001c0(t) +\r@tB=\u0016; (26)\nX00(x) =\u0016\nv2m; (27)\nwhere\u0016is a constant to be determined. The solutions to these ordinary di\u000berential equations read\n\u001c(t) =\u001c0+\u0016Tt\u0000\re\u0000t=TZt\n\u00001B(t0)et0=Tdt0; (28)\nX(x) =X0+kx+1\n2\u0016\nv2mx2; (29)\nwithX0,\u001c0andkbeing constants to be determined by imposing boundary conditions. The latter, as discussed in\nRefs. 15 and 16, are given by \u0000A^n\u0018\u0001~\nj\u0018=g\u0018\n4\u0019[\u0016\u0018\u0000~!\u0018], with\u0018=L;R indicating the terminals of the device.6\nFor the lateral con\fguration depicted in Fig. 1 of the main text, we have ^ nL=\u0000^nR= ^x. By accounting for the\nparametrization w= cos(\u001e=2),v= sin(\u001e=2)nof the order parameter, Eqs. (5) and (6) become\nA@x\u001e\u0000\n\u0000L\n2;t\u0001\n=gL\n4\u0019\u0002\n\u0000\u0016L+~@t\u001e\u0000\n\u0000L\n2;t\u0001\u0003\n; (30)\nA@x\u001e\u0000L\n2;t\u0001\n=gR\n4\u0019\u0002\n\u0016R\u0000~@t\u001e\u0000L\n2;t\u0001\u0003\n: (31)\nNext we consider the phase-coherent precessional ansatz \u001es, for which @x\u001es=k+\u0016x=v2\nmand@t\u001es=\u0016Tunder\nconstant magnetic \feld. As a result, we need to solve the following linear system of equations for kand\u0016:\nA\u0014\nk+\u0016\nv2m\u0012\n\u0000L\n2\u0013\u0015\n=gL\n4\u0019[\u0000\u0016L+~\u0016T]; (32)\nA\u0014\nk+\u0016\nv2m\u0012L\n2\u0013\u0015\n=gR\n4\u0019[\u0016R\u0000~\u0016T]; (33)\nfrom which the expressions (8) and (9) ( !=\u0016T) follow.\nDERIVATION OF EQUATIONS (11) AND (12)\nEquations (11) and (12) are derived by incorporating the parametrization (10) into Eq. (3) and expanding up to\n\frst order in the \ructuation \felds. The following intermediate results have been used:\n1)@2\n\u0016q'\u0000\u00021\n4(@\u0016\u001es)2+@\u0016\tl@\u0016\u001es+1\n2\tl@2\n\u0016\u001es\u0003\nqs+^e+@2\n\u0016\tt+^e\u0000@2\n\u0016\t\u0003\nt+\u0002\n@2\n\u0016\tl+1\n2@2\n\u0016\u001es\u00001\n4\tl(@\u0016\u001es)2\u0003\n\u00183;(34)\n2)@\u0016q'\u00001\n2\tl@\u0016\u001esqs+^e+@\u0016\tt+^e\u0000@\u0016\t\u0003\nt+\u0002\n@\u0016\tl+1\n2@\u0016\u001es\u0003\n\u00183; (35)\n3)\u0015(q)'\u00021\n4(@t\u001es)2\u00001\n4v2\nm(r\u001es)2+\r\n2@t\u001esB\u0003\n+\u0002\n@t\tl@t\u001es\u0000v2\nmr\tl\u0001r\u001es+\r@t\tlB\u0003\n; (36)\n4)B?^@tq'\u00001\n2(\tl@t\u001es)B?^qs+@t\ttB?^^e++@t\t\u0003\ntB?^^e\u0000+ (@t\tl+1\n2@t\u001es)B?^\u00183; (37)\n5)@tB?^q'@tB?^qs+ \tt@tB?^^e++ \t\u0003\nt@tB?^^e\u0000+ \tl@tB?^\u00183; (38)\n6)B?^qs=\u0000B\u00183;B?^\u00183=Bqs;B?^^e\u0006=\u0006iB^e\u0006: (39)\nSTABILITY ANALYSIS\nAs discussed in the main text, the phase coherent precessional state (7) is robust against longitudinal \ructua-\ntions \tl. Furthermore, transverse \ructuations proliferate when the exponent \u00001\nT\u0006Re(Z) is positive, which de-\ntermines the instability threshold for the spin super\ruid solution. Since Re( Z) =r1=2cos(\u0012=2), withrei\u0012\u0011Z2=\u00021\nT2\u0000\r2B2\u0000!2\u0000v2\nm(4q2\u0000k2)\u00002\r!B\u0003\n\u0000ih\n2\rB\nTi\n, this threshold can be recast as\nr+rcos\u0012\u00152\nT2; (40)\nwhich, after some algebra, yields the inequality v2\nmk2\u0015!2+ 4v2\nmq2+ 2\r!B. The highest magnetic \feld satisfying the\nlatter inequality occurs for q= 0 (see the \frst paragraph of the Discussion section in the man text, however), which\nleads to Eq. (16) in the main text."    },    {        "title": "2004.01957v1.Energy_and_momentum_conservation_in_spin_transfer.pdf",        "content": "Energy and momentum conservation in spin transfer\nAlexander Mitrofanov and Sergei Urazhdin\nDepartment of Physics, Emory University, Atlanta, GA, USA.\nWe utilize simulations of spin-polarized electron scattering by a chain of localized quantum spins\nto show that energy and linear momentum conservation laws impose strong constraints on the\nproperties of magnetic excitations induced by spin transfer. In turn, electron's orbital and spin\ndynamics depends on the dynamical characteristics of the local spins. Our results suggest the\npossibility to achieve precise control of spin transfer-driven magnetization dynamics by tailoring the\nspectral characteristics of the magnetic systems and the driving electrons.\nThe advent of spin transfer (ST) e\u000bect [1{3] has\ntransformed our understanding of nanomagnetism, and\nspurred multiple novel applications [4{8]. ST is caused\nby the interaction of spin currents carried by conduc-\ntion electrons with the magnetization of magnetic mate-\nrials, resulting in the absorption of electron's spin angu-\nlar momentum component non-collinear with the mag-\nnetization [1, 9, 10]. The absorbed angular momentum\ndrives magnetization dynamics, which can result in mag-\nnetization reversal [11, 12], precession [13, 14] and other\ndynamical e\u000bects [15, 16].\nEnergy and linear momentum conservation laws play\na central role in the dynamical processes in nature,\nbut their relevance to ST has remained virtually unex-\nplored. The threshold current for ST-driven magneti-\nzation dynamics was initially attributed to the require-\nment that spin accumulation must exceed the energy Em\nof the magnetic excitation quanta [magnons] generated\nby ST [17]. However, the energy of magnons associated\nwith quasi-uniform magnetization precession excited by\nST is small, and the threshold was identi\fed with the\ncompensation of the dynamical damping by ST [10, 11].\nRecent studies showed that ST can excite dynamical\nmodes throughout the magnon spectrum [18, 19], which\nspans frequencies fmfrom GHz to THz ranges for com-\nmon ferromagnets (Fs) [20]. Excitation of high-frequency\nmagnons may play a signi\fcant role in the interplay be-\ntween thermal phenomena and ST [21]. Nonlinear in-\nteractions among these high-frequency modes can also\nprofoundly in\ruence ST-induced dynamics [22, 23]. ST\ncan also drive magnetic dynamics in antiferromagnets\n(AFs) [24{26], where the lowest dynamical frequencies\nare typically in 100s of GHz or in the THz range [27, 28],\nwhich may enable ultrafast devices and THz oscillators\nbased on AFs driven by ST [29].\nThe energies Em=hfmof THz magnons are in the\nmeV range. If energy conservation plays a role in ST,\na large electrical bias may be required to provide en-\nergy su\u000ecient to generate such magnons. Likewise, linear\nmomentum conservation may impose strict requirements\non the momentum of the driving electrons in magnetic\nnanodevices envisioned to operate with short-wavelength\nmagnons generated by ST [5, 30]. However, both energy\nand momentum of magnons have been neglected in theanalyses of ST, which with a few exceptions [31{35] have\napproximated magnetization as a classical vector \feld.\nHere, we use simulations of spin-polarized electron\nscattering by a quantum spin chain to show that energy\nand momentum conservation laws impose signi\fcant con-\nstraints on the magnetic dynamics, as well as the elec-\ntron's orbital and spin dynamics resulting from ST. Our\nresults suggest the possibility to control the characteris-\ntics of magnetic excitations generated by ST by optimiz-\ning these constraints, which may provide a new route for\nthe development of e\u000ecient magnetic nanodevices.\nTo analyze ST, we consider scattering of an electron\nwavepacket by a ferromagnet modeled as a 1D spin-1/2\nchain. In the tight-binding approximation, this system\ncan be described by the Hamiltonian [34, 36]\n^H=\u0000X\nibjiihi+ 1j\n\u0000X\njJsdjjihjj\n^Sj\u0001^s+J^Sj\u0001^Sj+1+\u0016B^Sz\njB;(1)\nwhere indices i;jenumerate the tight-binding sites, ^s,^Sj\nare the spin operators of the electron and the local spins,\nbis the electron hopping parameter, Jdescribes the ex-\nchange sti\u000bness of the local spins, Jsd- their exchange\nwith the electron, B=\u0000Bzis the magnetic \feld, and\n\u0016Bis the Bohr magneton. We use periodic boundary\nconditions for both the electron and the spin chain, to\navoid spurious e\u000bects of re\rections at the boundaries.\nTo analyze ST, the system is initialized with the elec-\ntron forming a Gaussian wave packet spin-polarized along\nthe x-axis, while the local spins are in their ground state\naligned with the z-axis. The system is then evolved ac-\ncording to the Hamiltonian Eq. (1). The wavepacket is\npartially re\rected and partially transmitted by the local\nspins [Fig. 1(a)]. One can clearly identify the time inter-\nvals when the wavepacket is localized mostly outside or\ninside the spin chain, allowing us to analyze the e\u000bects\nof scattering by tracking the time evolution.\nTo analyze the evolution of each subsystem, we intro-\nduce the density matrices ^ \u001ae=Trm^\u001aand ^\u001am=Tre^\u001afor\nthe electron and the local spins, respectively, by tracing\nout the full density matrix ^ \u001awith respect to the other\nsubsystem [34]. The expectation value of an observablearXiv:2004.01957v1  [cond-mat.mes-hall]  4 Apr 20202\n0 20 400 20 40\n40 80 120 1602040\nSitest (fs)\n20\n19.50.5\n0\n< sz>< Sz> \nt (fs)local\nspinselectron< sx>, < Sx>\nt (fs)local\nspins\nelectron0.5\n0\n𝑣i \nIncidentReflected Transmitted\nxz(a)\n𝑣r 𝑣t (b)\n(c)\nFigure 1. (Color online) ST due to scattering of electron\nwavepacket by the chain of 40 spins, with , a= 0:2 nm,\nb= 1 eV, J=Jsd= 0:1 eV, B= 20 T. (a) Pseudocolor map of\nwavepacket intensity in the position-time coordinates, for the\ninitial wavepacket polarization along the x-axis. Schematics:\nthe wavepacket and the spin chain before and after scattering.\n(b),(c) Evolution of the expectation values of x (b) and z (c)\ncomponents of electron's and chain's spins.\n^Aassociated with the electron isD\n^AE\n=Tr(^A^\u001ae), while\nthe probability of its value aisPa=h aj^\u001aej ai, where\n ais the corresponding eigenstate. Similar relations hold\nfor the observables associated with the local spins.\nExchange interaction of the electron with the local\nspins results in the oscillation of its x spin component,\nwhich rapidly decays due to dephasing [Fig. 1(b)], consis-\ntent with the ST mechanisms [1, 10]. The x-component\nof the local spins mirrors this evolution, so that the x-\ncomponent of the total spin is conserved. The contribu-\ntion of the Zeeman term in Eq. (1) that breaks the spin\nconservation is negligible on the considered time scales.\nThe z-component of electron spin increases from zero\nto almost its maximum value 1 =2, with the local spins\nmirroring this evolution, Fig. 1(c). This transfer of the\nspin component collinear with the magnetization is con-\nsistent with the recently demonstrated nonclassical con-\ntribution to ST [33, 34, 36]. Since the constraints im-\nposed by energy and momentum conservation are ex-\npected to be general, we do not separate between the\ntwo contributions to ST in the analysis below.\nThe evolution of di\u000berent contributions to energy is\nillustrated in Fig. 2(a). The magnetic energy Emcom-\nprising the Zeeman and the exchange energies of the lo-\ncal spins increases due to their excitation by ST, while\nthe exchange energy Esdbetween the local spins and the\nelectron initially decreases due to the increase of the elec-\ntron's spin-up [majority] component [see Fig. 1(c)]. The\ntwo subsystems no longer interact after scattering, so Esd\nincreases back to zero. Since the Hamiltonian is time-\n03 06 0-0.050.000.050\n3 06 00.00.40.80\n3 06 0-3032\n.53.03.50.20.4t2t1E\ntotE\nhopEmE\nsd<E> (eV)t\n (fs)t\n2t1E\nlectron distributiont\n (fs)t0t1t2r\neflectedtransmittedr\neflected(d)( c)(b)( a)t\nransmitted<ke> (nm-1)t\n (fs)t2t2t1t\n1t0t\n0Ee (eV)<\nke> (nm-1)t0Figure 2. (Color online) (a) Evolution of di\u000berent contri-\nbutions to energy, as de\fned in the text. The curves are\nshifted by the t= 0 values for clarity. (b),(c) Majority (solid\ncurves) and minority (dashed) contributions to the transmit-\nted and re\rected wavepacket components (b), and the cor-\nresponding average wavevectors (c). The re\rected minority\ncomponent (not shown) is negligible. (d) Energy vs momen-\ntum for forward-propagating electron components at times t0,\nt1, and t2, as marked in (a),(b). Solid curves: spin-dependent\nelectron dispersion inside the spin chain, dashed curve - dis-\npersion outside the chain.\nindependent, the total energy of the system is conserved\n[dashed line in Fig. 2(a)]. The de\fcit of energy asso-\nciated with a \fnite Emafter scattering is made up by\nthe reduction of the electron's kinetic energy Ehop. This\nsuggests that the relation between the electron's kinetic\nenergy and the magnetic excitation spectrum plays an\nimportant role in ST, as con\frmed below.\nWe now analyze the momentum evolution. Before\nscattering, the wave packet contains only the forward-\npropagating component, with equal majority and minor-\nity spin contributions, Fig. 2(b). During scattering, the\nminority contribution decreases, while the majority con-\ntribution increases, consistent with the transfer of z spin\ncomponent shown in Fig. 1(c). Additionally, a majority-\nspin backward-propagating component emerges due to\nthe electron re\rection by the spin chain. The re\rected\nminority-spin component is negligible in the approxima-\ntion of the same electron hopping parameter inside and\noutside the spin chain, consistent with the mechanisms\nof electron-magnon scattering discussed below.\nThe momentum of the re\rected majority-spin com-\nponent is considerably smaller than that of the origi-\nnal wave packet [Fig. 2(c)], indicating that electron re-\n\rection by the chain involves a large transfer of energy.\nMeanwhile, the momentum of the majority-spin forward-3\n-3030.00.50.00.60\n5 0.00.20.000.06E\nbm\n(c)(\nd)E\nfm\nEe (eV)k\ne (nm-1)DistributionElectronM agnon(\nb)E\nfm\nE\nbm\nEm (eV)k\nm (nm-1)kfm\nkbm\n(a)kfm\nkbm\nDistribution\nFigure 3. (Color online) (a) Momentum distribution of\nthe generated magnons, at t=t1. (b) Magnon dispersion.\n(c) Momentum distribution of the majority-spin wave packet\ncomponent at t=t0(dashed curve) and at t=t1(solid\ncurve). (d) Electron dispersion Ee= 2b(1\u0000cos(kea)) out-\nside the spin chain (dashed curve), and majority-spin disper-\nsionEe;\"= 2b(1\u0000cos(kea))\u0000Jsdin the spin chain (solid\ncurve). The momenta kf\nm,kb\nmand the energies Ef\nm,Eb\nmof\nthe forward- and the backward-propagating magnon groups\nare indicated.\npropagating component increases, and that of the minor-\nity component decreases as the electron enters the chain,\nconsistent with the spin-splitting of the electronic band\nstructure inside the chain due to the sd exchange [see\nFig. 2(d)]. However, the di\u000berence between the momenta\nof the two spin components remains signi\fcant even af-\nter scattering, indicating that the electron experiences\nspin-dependent momentum and energy loss. This is con-\n\frmed by Fig. 2(d), which shows the average momentum\nand energy of the forward-propagating components cal-\nculated for instants t0,t1, andt2before, during, and after\nscattering, as marked in panels (a) and (b). The momen-\ntum and the energy of the majority-spin component are\nslightly reduced at t2relative to t0, while those of the\nminority-spin component are signi\fcantly reduced.\nThe variations of the electron's energy and momentum\n[Fig. 2] are inconsistent with quasi-elastic scattering, sug-\ngesting that the energy and the momentum of the gen-\nerated magnons play a signi\fcant role in the scattering\nprocess. This is con\frmed by the analysis of the rela-\ntion between the distribution of the generated magnons\nand the characteristics of the wave packet, Fig. 3. Two\ndistinct groups of magnons are generated: forward-\npropagating magnons with a large central momentum\nkf\nm, and backward-propagating magnons with a small\ncenteral momentum kb\nm[Fig. 3(a)]. We use the magnon\ndispersion relations Em= 4J(1\u0000cos(kma))) +S\u0016BB,\nwhereais the tight-binding site spacing, to determine the\ncorresponding magnon energies Ef\nmandEb\nm[Fig. 3(b)].\nThe relations between the momenta of the gener-\nated magnons and the characteristics of the electron\nwavepacket are illustrated in Fig. 3(c), which showsthe majority-spin momentum distributions of the wave\npacket att=t0and att1. The di\u000berence between the\ninitial central momentum ki\neof the wave packet and the\nmomentum kr\neof the re\rected component is equal to the\nmomentum kf\nmof the forward-propagating magnons gen-\nerated due to ST, while the corresponding di\u000berence for\nthe momentum kt\neof the transmitted wave packet com-\nponent is equal to the momentum kb\nmof the generated\nbackward-propagating magnons.\nBy analyzing the dispersion of the electron outside the\nspin chain, as well as the majority-spin dispersion of elec-\ntron inside the spin chain [Fig. 3(d)], we \fnd that the\nenergyEr\neof the re\rected component is reduced relative\nto the initial energy Ei\neby the energy Ef\nmof the forward-\npropagating magnons generated by scattering, while the\nenergyEt\neof the transmitted component is reduced by\nthe energy Eb\nmof the backward-propagating magnons.\nHere, the term \\energy\" refers to the expectation value of\nenergy of the corresponding quantum-mechanical state,\nrather than the net energy carried by the wave. Thus,\ngeneration of forward-propagating magnons is associated\nwith electron re\rection, while generation of backward-\npropagating magnons - with the forward scattering of\nelectrons, described by the relations\nEi\ne=Ef(b)\nm+Er(t)\ne; ki\ne=kf(b)\nm+kr(t)\ne (2)\nbetween the energies and the momenta of the quasipar-\nticles involved in the corresponding scattering processes.\nTo con\frm our interpretation, we solved these equations\nusing the magnon and the electron dispersions. For in-\nstance, the equation for the momentum kt\neof the trans-\nmitted electron is\nb\u0000S\u0016BB\u0000Jsd+ 4Jcos(ki\nea\u0000kt\nea)\n2(cos(ktea)\u0000cos(kiea))= 0: (3)\nIts numeric solution is consistent with Fig. 3(c) [36].\nEquation (2a) describes energy conservation, as ex-\npected for the time-independent Hamiltonian Eq. (1).\nHowever, its translation symmetry is broken by the spin\nchain, so the momentum needs not be conserved. In-\ndeed, the momentum of the forward-propagating major-\nity electron becomes reduced after scattering, as expected\nsince its energy is reduced due to magnon generation [see\nFig. 2(d)]. However, the generated magnon with momen-\ntumkb\nmpropagates backward, i.e. the total momentum\nis reduced in this process. Nevertheless, the momentum\nrelation Eq. (2b) is governed by the same spatial inter-\nference between the spin wave and the incident/scattered\nelectron wavefunctions as in the momentum-conserving\nprocesses, and therefore we for simplicity call it the mo-\nmentum conservation condition.\nElectron scattering described by Eq. (2) is governed by\nthe electron and magnon dispersions. Here, we demon-\nstrate one of the consequences - dependence of electron\nscattering and ST on the magnon dispersion - which is4\n02 04 002 04 0- 40 4 0\n5   0.1 eV \n0.3 eV \n1 eV00.5<  sz>t\n (fs) 0.1 eV \n0.3 eV \n 1 eV(\nd)(c)(\nb)(a)<\n sx> t\n (fs)kre\n0\n0.5Electron distributionk\ne (nm-1)0.1 eV0\n.3 eV1\n eVk\nfm\nMagnon distributionk\nm (nm-1)0.1 eV0\n.3 eV1\n eV\nFigure 4. (Color online) E\u000bects of magnon dispersion on ST.\n(a),(b) Electron (a) and magnon (b) momentum distributions\natt=t1, for the labeled values of J. (c),(d) Evolution of the\nx- (c) and the z-component of electron spin (d).\nnot captured by the models based on the classical ap-\nproximation for magnetization [36].\nFigure 4(a) shows the electron momentum distribu-\ntions at the instant t1, for three di\u000berent values of ex-\nchange sti\u000bness J. The transmitted component is not\nsigni\fcantly a\u000bected by the variations of J, as expected\nsince the energy of the magnons with a small momentum\nkb\nm, which are generated by the transmitted electrons,\nis almost independent of the exchange sti\u000bness. In con-\ntrast, the magnitude of the momentum of the re\rected\ncomponent rapidly decreases with increasing J, which is\nmirrored by the decrease of the momentum kf\nmof the gen-\nerated magnons [Fig. 4(b)]. This e\u000bect is consistent with\nthe increase of the energy Ef\nmof these large-momentum\nmagnons, resulting in a decrease of the scattered elec-\ntron's energy. At J= 1 eV, the momentum of the scat-\ntered electron becomes close to zero, i.e. all of its initial\nenergy is transferred to the generated magnon.\nThe evolution of both the x- and the z-components of\nthe electron spin is similar for J= 0:1 eV andJ= 0:3 eV,\n[Figs. 4(c),(d)]. However, for J= 1 eV, the transfer of\nboth the x- and the z- components of spin is reduced.\nThe electron's energy is no longer su\u000ecient to generate\nthe largest-momentum magnons, resulting in a reduced\ne\u000eciency of ST. In our simulation, electron can be scat-\ntered into any band states, so this e\u000bect of magnon dis-\npersion on ST becomes noticeable only at large J, when\nthe magnon energies become comparable to the electron\nband energy. In real systems, the available electron en-\nergy is much smaller, as de\fned by the occupied Fermi\nsurface. Consequently, a signi\fcant dependence of elec-tron scattering and spin dynamics on the magnon dis-\npersion can be expected even for modest variations of J\nor other parameters controlling the magnon dispersion,\nsuch as the magnetic anisotropy or \feld [36]. We leave\nanalysis of these e\u000bects to future studies.\nTo summarize, we have shown that energy and mo-\nmentum conservation laws de\fne the energies and the\nmomenta of magnons generated in the spin transfer pro-\ncess. As one of the consequences, the spectral distribu-\ntion of spin waves generated by spin transfer in tunnel\njunctions must signi\fcantly di\u000ber from those in metal-\nlic systems. The demonstrated relations may provide a\npath for the development of laser-like magnetic nanode-\nvices, where speci\fc magnetic modes are excited by spin\ntransfer due to the judicious optimization of constraints\nimposed by the conservation laws.\nThe demonstrated relations are relevant not only to\nspin transfer, but also to orbital and the spin dynamics\nof electrons scattered by the ferromagnets. For instance,\nelectron backscattering at magnetic interfaces, which in-\nvolves generation of large-momentum magnons, should\nstrongly depend on the available electron energy. The\nconstraints imposed on spin transfer by the conservation\nlaws are also particularly relevant for antiferromagnets,\nwhere the characteristic magnon energies are two orders\nof magnitude larger than in ferromagnets.\nThis work was supported by the U.S. Department of\nEnergy, O\u000ece of Science, Basic Energy Sciences, under\nAward # de-sc2218976.\n[1] J. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996).\n[2] L. Berger, Phys. Rev. B 54, 9353 (1996).\n[3] Y. B. Bazaliy, B. A. Jones, and S.-C. Zhang, Phys. Rev.\nB57, R3213 (1998).\n[4] A. D. Kent and D. C. Worledge, Nature Nano. 10, 187\n(2015).\n[5] B. Divinskiy, V. E. Demidov, S. O. Demokritov, A. B.\nRinkevich, and S. Urazhdin, Appl. Phys. Lett. 109,\n252401 (2016).\n[6] N. Locatelli, V. Cros, and J. Grollier, Nature Mater. 13,\n11 (2013).\n[7] J.-V. Kim, in Solid State Physics (Elsevier, 2012) pp.\n217{294.\n[8] T. Chen, R. K. Dumas, A. Eklund, P. K. Muduli,\nA. Houshang, A. A. Awad, P. Durrenfeld, B. G. Malm,\nA. Rusu, and J. Akerman, Proc. IEEE 104, 1919 (2016).\n[9] S. Zhang, P. M. Levy, and A. Fert, Phys. Rev. Lett. 88\n(2002).\n[10] D. Ralph and M. Stiles, J. Magn. Magn. Mater. 320,\n1190 (2008).\n[11] J. A. Katine, F. J. Albert, R. A. Buhrman, E. B. Myers,\nand D. C. Ralph, Phys. Rev. Lett. 84, 3149 (2000).\n[12] S. Mangin, D. Ravelosona, J. A. Katine, M. J. Carey,\nB. D. Terris, and E. E. Fullerton, Nature Mater. 5, 210\n(2006).\n[13] S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Emley,5\nR. J. Schoelkopf, R. A. Buhrman, and D. C. Ralph,\nNature 425, 380 (2003).\n[14] W. H. Rippard, M. R. Pufall, S. Kaka, S. E. Russek, and\nT. J. Silva, Phys. Rev. Lett. 92, 027201 (2004).\n[15] V. E. Demidov, S. Urazhdin, and S. O. Demokritov,\nNature Mater. 9, 984 (2010).\n[16] M. Madami, S. Bonetti, G. Consolo, S. Tacchi, G. Car-\nlotti, G. Gubbiotti, F. B. Manco\u000b, M. A. Yar, and\nJ.\u0017Akerman, Nature Nano. 6, 635 (2011).\n[17] M. Tsoi, A. G. M. Jansen, J. Bass, W.-C. Chiang,\nM. Seck, V. Tsoi, and P. Wyder, Phys. Rev. Lett. 80,\n4281 (1998).\n[18] K.-J. Lee, A. Deac, O. Redon, J.-P. Nozi\u0012 eres, and B. Di-\neny, Nature Mater. 3, 877 (2004).\n[19] M. L. Polianski and P. W. Brouwer, Phys. Rev. Lett. 92\n(2004).\n[20] F. Menzinger, G. Caglioti, G. Shirane, R. Nathans, S. J.\nPickart, and H. A. Alperin, J. Appl. Phys. 39, 455\n(1968).\n[21] G. E. W. Bauer, E. Saitoh, and B. J. van Wees, Nature\nMater. 11, 391 (2012).\n[22] V. E. Demidov, S. Urazhdin, E. R. J. Edwards, M. D.\nStiles, R. D. McMichael, and S. O. Demokritov, Phys.\nRev. Lett. 107(2011).\n[23] B. Divinskiy, S. Urazhdin, S. O. Demokritov, and V. E.Demidov, Nature Comm. 10(2019).\n[24] T. Jungwirth, X. Marti, P. Wadley, and J. Wunderlich,\nNature Nano. 11, 231 (2016).\n[25] J. \u0014Zelezn\u0013 y, P. Wadley, K. Olejn\u0013 \u0010k, A. Ho\u000bmann, and\nH. Ohno, Nature Phys. 14, 220 (2018).\n[26] T. Moriyama, K. Oda, T. Ohkochi, M. Kimata, and\nT. Ono, Sci. Rep. 8(2018).\n[27] J. van Kranendonk and J. H. van Vleck, Rev. Mod. Phys.\n30, 1 (1958).\n[28] T. Kampfrath, A. Sell, G. Klatt, A. Pashkin, S. M ahrlein,\nT. Dekorsy, M. Wolf, M. Fiebig, A. Leitenstorfer, and\nR. Huber, Nature Phot. 5, 31 (2010).\n[29] R. Khymyn, I. Lisenkov, V. Tiberkevich, B. A. Ivanov,\nand A. Slavin, Sci. Rep. 7(2017).\n[30] D. Grundler, Nature Nano. 11, 407 (2016).\n[31] S. Urazhdin, Phys. Rev. B 69, 134430 (2004).\n[32] Y. Wang and L. J. Sham, Phys. Rev. B 85, 092403 (2012).\n[33] A. Zholud, R. Freeman, R. Cao, A. Srivastava, and\nS. Urazhdin, Phys. Rev. Lett. 119, 257201 (2017).\n[34] P. Mondal, U. Bajpai, M. D. Petrovi\u0013 c, P. Plech\u0013 a\u0014 c, and\nB. K. Nikoli\u0013 c, Phys. Rev. B 99, 094431 (2019).\n[35] S. A. Bender, R. A. Duine, and Y. Tserkovnyak, Phys.\nRev. B 99, 024434 (2019).\n[36] See Supplemental Material for additional details on sim-\nulations and results supporting the analysis in the main\ntext."    },    {        "title": "1403.0378v1.Chaos_in_two_black_holes_with_next_to_leading_order_spin_spin_interactions.pdf",        "content": "arXiv:1403.0378v1  [gr-qc]  3 Mar 2014Eur. Phys.J.C manuscriptNo.\n(will beinsertedbytheeditor)\nChaos in two black holes with next-to-leading order spin-sp in\ninteractions\nGuoqingHuang1, XiaotingNi1,Xin Wua,1\n1Department ofPhysics, Nanchang University, Nanchang 3300 31, China\nReceived: date /Accepted: date\nAbstract We take intoaccountthe dynamicsofa complete\nthirdpost-NewtonianconservativeHamiltonianoftwospin -\nning black holes, where the orbital part arrives at the third\npost-Newtonian precision level and the spin-spin part with\nthe spin-orbit part includes the leading-order and next-to -\nleading-order contributions. It is shown through numerica l\nsimulationsthatthenext-to-leadingorderspin-spincoup lings\nplay an important role in chaos. A dynamical sensitivity to\nthevariationofsingleparameterisalsoinvestigated.Inp ar-\nticular,thereareanumberof observable orbitswhoseinitial\nradiiarelargeenoughandwhichbecomechaoticbeforeco-\nalescence.\nKeywords black hole ·post-Newtonian approximations ·\nchaos·Lyapunovexponents\nPACS04.25.dg·04.25.Nx ·05.45.-a·95.10.Fh\n1 Introduction\nMassivebinaryblack-holesystemsarelikelythemostpromi s-\ningsourcesforfuturegravitationalwavedetectors.Thesu c-\ncessful detection of the waveforms means using matched-\nfiltering techniques to best separate a faint signal from the\nnoise and requiresa very precise modelling of the expected\nwaveforms. Post-Newtonian (PN) approximations can sat-\nisfy this requirement. Up to now, high-precision PN tem-\nplates have already been known for the non-spin part up to\n3.5PN order ( i.e.the order 1 /c7in the formal expansion in\npowers of 1 /c2withcbeing the speed of light) [1,2], the\nspin-orbitpartupto3.5PNorderincludingtheleading-ord er\n(LO,1.5PN),next-to-leading-order(NLO,2.5PN)andnext-\nto-next-to-leading-order(NNLO, 3.5PN) interactions[3- 5],\nand the spin-spinpart up to 4PN orderconsisting of the LO\n(2PN),NLO (3PN)andNNLO(4PN)couplings[6-9].\nae-mail: xwu@ncu.edu.cnHowever, an extremely sensitive dependence on initial\nconditions as the basic feature of chaotic systems would\npose a challenge to the implementation of such matched\nfilters, since the number of filters required to detect these\nwaveforms is exponentially large with increasing detectio n\nsensitivity. This has led to some authors focusing on re-\nsearch of chaos in the orbits of two spinning black holes.\nChaoswasfirstlyfoundandconfirmedinthe2PNLagrangian\napproximation of comparable mass binaries with the LO\nspin-orbit and LO spin-spin effects [10]. Moreover, it was\nreported in [11] that the presence of chaos should be ruled\nout in these systems because no positive Lyapunov expo-\nnents could be found. As an answer to this claim, Refs.\n[12,13]obtainedsomepositiveLyapunovexponentsandpoin ted\nout these zero Lyapunov exponents of [11] due to the less\nrigorouscalculationoftheLyapunovexponentsoftwonearb y\norbits with unapt rescaling. In fact, the conflicting result s\non Lyapunovexponentsare because the two papers [11,12]\nused different methods to compute their Lyapunov expo-\nnents, as was mentioned in [14]. Ref. [11] computed the\nstabilizing limit values of Lyapunov exponents, and Ref.\n[12] worked out the slopes of fit lines. This is the so-called\ndoubt regarding different chaos indicators causing two dis -\ntinct claims on the chaotic behavior.Besides this, there wa s\nsecond doubt on different dynamical approximations mak-\ning the same physical system have distinct dynamical be-\nhaviors.The 2PN harmonic-coordinatesLagrangianformu-\nlation of the two-black hole system with the LO spin-orbit\ncouplings of one spinning body allows chaos [15], but the\n2PN ADM (Arnowitt-Deser-Misner) -coordinates Hamilto-\nnian does not [16,17]. Levin [18] thought that there is no\nformal conflict between them since the two approaches are\nnot exactly but approximately equal, and different dynam-\nical behaviors between the two approximately related sys-\ntems are permitted according to dynamical systems theory.\nSeen from the canonical, conjugate spin coordinates [19],2\nthe former non-integrability and the latter integrability are\nclearer. As extensions, both any PN conservative Hamilto-\nnian binary system with one spinning body and a conser-\nvativeHamiltonianof two spinningbodieswithoutthe con-\nstraint of equal mass or with the spin-orbit couplings not\nrestricted to the leading order are still integrable. Recen tly,\n[20,21]arguedtheintegrabilityofthe2PNHamiltonianwit h-\noutthespin-spincouplingsandwiththeNLOand/orNNLO\nspin-orbitcontributionsincluded.Onthecontrary,theco rre-\nspondingLagrangiancounterpartwithspineffectslimited to\nthe spin-orbit interactionsup to the NLO terms exhibitsthe\nstronger chaoticity [22].Third doubt relates to different de-\npendence of chaos on single dynamical parameter or initial\ncondition.Thedescriptionofthechaoticregionsandchaot ic\nparameter spaces in [15] are inconsistent with that in [23].\nThe differentclaims are regardedto be correct accordingto\nthestatementof[24]thatchaosdoesnotdependonlyonsin-\nglephysicalparameterorinitialconditionbutacomplicat ed\ncombinationofallparametersandinitial conditions.\nIt isworthemphasizingthatthe spin-spineffectsarethe\nmost important source for causing chaos in spinning com-\npact binaries, but they were only restricted to the LO term\nin the published paperson research of the chaotic behavior.\nIt should be significant to discuss the NLO spin-spin cou-\nplings included to a contribution of chaos. For the sake of\nthis, we shall considera complete3PN conservativeHamil-\ntonian of two spinning black holes, where the orbital part\nis up to the 3PN order and the spin-spin part as well as the\nspin-orbitpartincludestheLOandNLOinteractions.Inthi s\nway, we want to know whether the inclusion of the NLO\nspin-spin couplings have an effect on chaos, and whether\nthereischaosbeforecoalescenceofthe binaries.\n2 Third post-NewtonianorderHamiltonianapproach\nItistoodifficulttostrictlydescribethedynamicsofasyst em\noftwomasscomparablespinningblackholesingeneralrel-\nativity.Instead,thePNapproximationmethodisoftenused .\nSuppose that the two bodies have masses m1andm2with\nm1≤m2. Other mass parameters are the total mass M=\nm1+m2,thereducedmass µ=m1m2/M,themassratio β=\nm1/m2andthemassparameter η=µ/M=β/(1+β)2.As\ntootherspecifiednotations,a 3-dimensionalvector rrepre-\nsentstherelativepositionofbody1tobody2,itsunitradia l\nvectorisn=r/rwiththeradius r=|r|,andpstandsforthe\nmomentaofbody1relativetothecentre.Themomenta,dis-\ntancesandtime tarerespectivelymeasuredintermsof µ,M\nandM.Additionally,geometricunits c=G=1areadopted.\nThe two spin vectors are Si=SiˆSi(i=1,2) with unit vec-\ntorsˆSiandthespinmagnitudes Si=χim2\ni/M2(0≤χi≤1).\nInADMcoordinates,thesystemcanbeexpressedasthedi-mensionlessconservative3PN Hamiltonian\nH(r,p,S1,S2) =Ho(r,p)+Hso(r,p,S1,S2)\n+Hss(r,p,S1,S2). (1)\nIn the following, we write its detailed expressionsalthoug h\ntheyaretoolong.\nFortheconservativecase,theorbitalpart Hodoesnotin-\ncludethedissipative2.5PNterm(whichistheleadingorder\nradiationdampinglevel)buttheNewtonianterm HNandthe\nPNcontributions H1PN,H2PNandH3PN, thatis,\nHo=HN+H1PN+H2PN+H3PN. (2)\nAsgivenin [25],theyare\nHN=p2\n2−1\nr, (3)\nH1PN=1\n8(3η−1)p4−1\n2[(3+η)p2+η(n·p)2]1\nr\n+1\n2r2, (4)\nH2PN=1\n16(1−5η+5η2)p6+1\n8[(5−20η−3η2)p4\n−2η2(n·p)2p2−3η2(n·p)4]1\nr+1\n2[(5+8η)p2\n+3η(n·p)2]1\nr2−1\n4(1+3η)1\nr3, (5)\nH3PN=1\n128(−5+35η−70η2+35η3)p8+1\n16[(−7\n+42η−53η2−5η3)p6+(2−3η)η2(n·p)2\n×p4+3(1−η)η2(n·p)4p2−5η3(n·p)6]1\nr\n+[1\n16(−27+136η+109η2)p4+1\n16(17\n+30η)η(n·p)2p2+1\n12(5+43η)η(n·p)4]1\nr2\n+{[−25\n8+(1\n64π2−335\n48)η−23\n8η2]p2\n+(−85\n16−3\n64π2−7\n4η)η(n·p)2}1\nr3\n+[1\n8+(109\n12−21\n32π2)η]1\nr4. (6)\nThe spin-orbit part Hsois linear functions of the two\nspins. It is the sum of the LO spin-orbit term HLO\nsoand the\nNLOspin-orbitterm HNLO\nso,i.e.\nHso(r,p,S1,S2)=HLO\nso(r,p,S1,S2)+HNLO\nso(r,p,S1,S2).\n(7)\nRef.[5]gavetheirexpressions\nHso=1\nr3[g(r,p)S+g∗(r,p)S∗]·L, (8)3\nwheretherelatednotationsare\nS=S1+S2,S∗=1\nβS1+βS2,\ng(r,p) =2+[19\n8ηp2+3\n2η(n·p)2−(6+2η)1\nr],\ng∗(r,p) =3\n2+[−(5\n8+2η)p2+3\n4η(n·p)2\n−(5+2η)1\nr],\nand the Newtonian-lookingorbital angularmomentumvec-\ntoris\nL=r×p. (9)\nThe constant terms in gandg∗correspond to the LO part,\nandtheothers,theNLO part.\nSimilarly,thespin-spinHamiltonian Hssalsoconsistsof\ntheLOspin-spincouplingterm HLO\nssandtheNLOspin-spin\ncouplingterm HNLO\nss, namely,\nHss(r,p,S1,S2)=HLO\nss(r,S1,S2)+HNLO\nss(r,p,S1,S2).(10)\nThefirst sub-Hamiltonianreads[25]\nHLO\nss=1\n2r3[3(S0·n)2−S2\n0] (11)\nwithS0=S+S∗. The second sub-Hamiltonian is made of\nthreeparts,\nHNLO\nss=Hs2\n1p2+Hs2\n2p2+Hs1s2p2. (12)\nTheyarewrittenas[7,8]\nHs2\n1p2=η2\nβ2r3[1\n4(p1·S1)2+3\n8(p1·n)2S2\n1\n−3\n8p2\n1(S1·n)2−3\n4(p1·n)(S1·n)(p1·S1)]\n−η2\nr3[3\n4p2\n2S2\n1−9\n4p2\n2(S1·n)2]\n+η2\nr3β[3\n4(p1·p2)S2\n1−9\n4(p1·p2)(S1·n)2\n−3\n2(p1·n)(p2·S1)(S1·n)\n+3(p2·n)(p1·S1)(S1·n)\n+3\n4(p1·n)(p2·n)S2\n1\n−15\n4(p1·n)(p2·n)(S1·n)2], (13)\nHs2\n2p2=HS2\n1p2(1↔2), (14)Hs1s2p2=η2\n2r3{3\n2{[(p1×S1)·n][(p2×S2)·n]\n+6[(p2×S1)·n][(p1×S2)·n]\n−15(S1·n)(S2·n)(p1·n)(p2·n)\n−3(S1·n)(S2·n)(p1·p2)\n+3(S1·p2)(S2·n)(p1·n)\n+3(S2·p1)(S1·n)(p2·n)\n+3(S1·p1)(S2·n)(p2·n)\n+3(S2·p2)(S1·n)(p1·n)\n−1\n2(S1·p2)(S2·p1)+(S1·p1)(S2·p2)\n−3(S1·S2)(p1·n)(p2·n)\n+1\n2(S1·S2)(p1·p2)}\n+3η2\n2r3β{−[(p1×S1)·n][(p1×S2)·n]\n+(S1·S2)(p1·n)2\n−(S1·n)(S2·p1)(p1·n)}\n+3η2β\n2r3{−[(p2×S2)·n][(p2×S1)·n]\n+(S1·S2)(p2·n)2\n−(S2·n)(S1·p2)(p2·n)}\n+6η\nr4[(S1·S2)−2(S1·n)(S2·n)]. (15)\nHere,p1=−p2=p. In a word, the conservative Hamilto-\nnian (1) up to the 3PN order is not completely given until\nEq. (15) appears. Clearly, Hamiltonian (1) does not depend\nonanymassbutthemassratio.\nTheevolutionsofposition randmomentum psatisfythe\ncanonicalequationsofthe Hamiltonian(1):\ndr\ndt=∂H\n∂p,dp\ndt=−∂H\n∂r. (16)\nThespinvariablesvarywithtimeaccordingtothefollowing\nrelations\ndSi\ndt=∂H\n∂Si×Si. (17)\nBesides the two spin magnitudes, there are four con-\nserved quantities in the Hamiltonian (1), includingthe tot al\nenergyE=Handthreecomponentsofthetotalangularmo-\nmentumvector J=L+S.Afifthconstantofmotionisab-\nsent,sotheHamiltonian(1)isnon-integrable.1Itshighnon-\nlinearity seems to imply that it is a richer source for chaos.\nNext,we shall search forchaos, andparticularlyinvestiga te\n1Based on the idea of [19], the Hamiltonian (1) can be expresse d as a\ncompletely canonical Hamiltonian with a 10-dimensional ph ase space\nwhen the canonical, conjugate spin coordinates are used ins tead ofthe\noriginal spin variables. If the system is integrable, at lea st five inde-\npendent integrals of motion beyond the constant spin magnit udes are\nnecessary.4\ntheeffectoftheNLOspin-spininteractionsonthedynamics\nofthesystem.\n3 Detectionofchaosbeforecoalescence\nWith numerical simulations, we use some chaos indicators\nto describe dynamical differences between the NLO spin-\nspincouplingsexcludedandincluded.Theappropriateones\nof the indicators are selected to study dependence of chaos\non single parameter when the NLO spin-spin couplingsare\nincluded. Finally, we expect to find chaos before coales-\ncencebyestimatingtheLyapunovandinspiraldecaytimes.\n3.1 Comparisons\nNumerical methods are convenient to study nonlinear dy-\nnamics of the Hamiltonian (1). Symplectic integrators are\nefficientnumericaltoolssincetheyhavegoodgeometricand\nphysical properties, such as the symplectic structure con-\nservedandenergyerrorswithoutsecularchanges.However,\nthey cannot provide high enough accuracies, and the com-\nputationsare expensivewhenthemixedsymplecticintegra-\ntionalgorithms[21,26]withacompositeofthesecond-orde r\nexplicitleapfrogsymplecticintegratorandthesecond-or der\nimplicit midpoint rule are chosen. In this sense, we would\nprefer to adopt an 8(9) order Runge-Kutta-Fehlberg algo-\nrithm of variable time steps. In fact, it gives such high ac-\ncuracy to the energy error in the magnitude of about order\n10−13∼10−12whenintegrationtimereaches106,asshown\nin Fig. 1. Here, orbit 1 we consider has initial conditions\n(p(0);r(0))=(0,0.39,0;8.55,0,0),whichcorrespondtothe\ninitial eccentricity e0=0.30 and the initial semi-majoraxis\na0=12.2. Other parameters and initial spin angles are re-\nspectively β=0.79,χ1=χ2=χ=1.0,θi=78.46◦and\nφi=60◦, where polar angles θiand azimuthal angles φi\nsatisfy the relations ˆSi=(cosφisinθi,sinφisinθi,cosθi), as\ncommonly used in physics. The NLO spin-spin couplings\nare not included in Fig. 1(a), but in Fig. 1(b). It can be\nseen clearly that the inclusion of the NLO spin-spin cou-\nplings with a rather long expression decreases only slightl y\nthenumericalaccuracy.Therefore,ournumericalresultsa re\nshowntobereliablealthoughtheenergyerrorshavesecular\nchanges.\nWe apply several chaos indicators to compare dynam-\nical behaviors of orbit 1 according to the two cases with-\nout and with the NLO spin-spin couplings. The method of\nPoincarésurfaceofsectioncanprovideacleardescription of\nthe structureof phasespace toa conservativesystem whose\nphasespaceis4dimensions.Asapointtonote,itisnotsuit-\nable for such a higher dimensional system (1). Fortunately,\npower spectra, Lyapunov exponents and fast Lyapunov in-dicatorswould work well in finding chaos regardlessof the\ndimensionalityofphasespace.\n3.1.1 Powerspectrumanalysis\nPower spectrum analysis reveals a distribution of various\nfrequencies ωof a signal x(t). It is the Fourier transforma-\ntion\nX(ω)=/integraldisplay+∞\n−∞x(t)e−iωtdt, (18)\nwhereiis the imaginary unit. In general, the power spec-\ntraX(ω)are discrete for periodic and quasi periodic orbits\nbut continuous for chaotic orbits. That is to say, the classi -\nfication of orbits can be distinguished in terms of different\nfeaturesofthespectra.Onthebasisofthis,weknowthrough\nFig.2thattheorbitseemstoberegularwhentheNLOspin-\nspin couplingsare not included, but chaotic when the NLO\nspin-spin couplingsare included.Notice that the method of\npower spectra is only a rough estimation of the regularity\nand chaoticity of orbits. More reliable chaos indicators ar e\nstronglydesired.\n3.1.2 Lyapunovexponents\nThe maximum Lyapunov exponent is used to measure the\naverageseparationrateoftwoneighboringorbitsinthepha se\nspaceandgivesquantitativeanalysistothestrengthofcha os.\nIts calculations are usually based on the variational metho d\nandthetwo-particlemethod[27].Theformerneedssolving\nthevariationalequationsaswellastheequationsofmotion ,\nand the latter needs solving the equations of motion only.\nConsidering the difficulty in deriving the variational equa -\ntions of a complicated dynamical system, we pay attention\nto the application of the latter method. In the configuration\nspace,it is definedas[28]\nλ=lim\nt→∞1\ntln|Δr(t)|\n|Δr(0)|, (19)\nwhere|Δr(0)|and|Δr(t)|are the separations between the\ntwo neighboring orbits at times 0 and t, respectively. The\ninitialdistancecannotbetoobigortoosmall,and10−8isre-\ngardedasto its suitablechoice in the doubleprecision[27] .\nForthesakeoftheoverflowavoided,renormalizationsfrom\ntime to time are vital in the tangent space. A bounded orbit\nis chaotic if its Lyapunov exponent is positive, but regular\nwhen its Lyapunov exponent tends to zero. In this way, we\ncanknowfromFig.3thatorbit1isregularforthecasewith-\nout the NLO spin-spin couplings, but chaotic for the case\nwiththe NLO spin-spincouplings.Of course,it takesmuch\ncomputational cost to distinguish between the ordered and\nchaoticcases.5\n3.1.3 FastLyapunovindicators\nA quicker method to find chaos than the method of Lya-\npunov exponents is a fast Lyapunov indicator (FLI). This\nindicator that was originally considered to measure the ex-\npansion rate of a tangential vector [29] does not need any\nrenormalization,whileitsmodifiedversiondealingwithth e\nuseofthe two-particlemethod[30]does.Themodifiedver-\nsionisoftheform\nFLI(t)=log10|Δr(t)|\n|Δr(0)|. (20)\nItscomputationisbasedonthefollowingexpression:\nFLI=−k(1+log10|Δr(0)|)+log10|Δr(t)|\n|Δr(0)|, (21)\nwherekdenotes the sequential number of renormalization.\nThe FLI of Fig. 4(a) corresponding to Fig. 3(a) increases\nalgebraically with logarithmic time log10t, and that of Fig.\n4(b)correspondingtoFig.3(b)doesexponentiallywithlog -\narithmic time. The former indicates the character of order,\nbutthelatter,thefeatureofchaos.Onlywhentheintegrati on\ntime addsupto 1 ×105, canthe orderedandchaotic behav-\niorsbe identifiedclearly forthe use of FLI unlikethe appli-\ncation of Lyapunov exponent.There is a threshold value of\nthe FLIs between order and chaos, 5. Orbits whose FLI are\nlargerthan5arechaotic,whereasthosewhoseFLIsareless\nthan5areregular.\nThe above numerical comparisons seem to tell us that\nchaosbecomeseasierwhentheNLOspin-spintermsarein-\ncluded. This sounds reasonable. As claimed in [20,21], the\nsystem(1)isintegrableandnotatallchaoticwhenthespin-\nspin couplings are turned off. The occurrence of chaos is\ncompletely due to the spin-spin couplings, which include\nparticularly the NLO spin-spin contributions leading to a\nsharpincreaseinthestrengthofnonlinearity.Infact,wee m-\nploy FLIs to find that there are other orbits (such as orbits\n2-5in Table1),whichare notchaoticforthe absenceofthe\nNLO spin-spin couplings but for the presence of the NLO\nspin-spincouplings.Inaddition,the strengthofthechaot ic-\nity of orbits 6-8 increases. As a point to illustrate, the oth er\ninitial conditions beyond Table 1 are those of orbit 1; the\nstarting spin unit vectorsof orbit 2 are those of orbit 1, and\nthose of orbits 3-8 are θ1=84.26◦,φ1=60◦,θ2=84.26◦\nandφ2=45◦. Hereafter,onlythe dynamicsof the complete\nHamiltonian (1) with the NLO spin-spin effects included is\nfocusedon.\n3.2 Lyapunovandinspiraldecaytimes\nTakingβ=0.5,theinitialconditionsandtheinitialunitspin\nvectors of orbit 1 as reference, we start with the spin pa-\nrameterχat the value 0.2 that is increased in increments\nof 0.01 up to a final value of 1 and draw dependence ofFLI onχin Fig. 5(a). This makes it clear that chaos oc-\ncurs when χ≥0.7. Precisely speaking, the larger the spin\nmagnitudes get, the stronger the chaos gets. Note that this\ndependence of chaos on χrelies typically on the choice of\nthe initial conditions, the initial unit spin vectors and th e\nother parameters. As claimed in [24], there is different de-\npendenceof chaoson χif the choice changes. On the other\nhand,takingtheinitialspinanglesoforbit3,fixingthespi n\nparameter χ=0.90andtheinitialconditions (p(0);r(0))=\n(0,0.39,0;8.4,0,0), which correspond to the initial eccen-\ntricitye0=0.28 and the initial semi-major axis a0=11.6,\nwe study the range of the mass ratio βbeginningat 0.5 and\nending at 1 in increments of 0.01. At once, dependence of\nFLIonβcanbedescribedinFig.5(b).Thereischaoswhen\nβ≤0.86 and chaos seems easier for a smaller mass ratio.\nAsinthepanel(a),thisresultisgivenonlyunderthepresen t\ninitialconditions,initialunitspinvectorsandotherpar ame-\nters.\nDothe above-mentionedchaoticbehaviorsoccurbefore\nthemergerofthebinaries?Toanswerit,wehavetocompare\nthe Lyapunov time Tλ=1/λ(i.e.the inverse of the Lya-\npunov exponent)with the inspiral decay time Td, estimated\nby[31]\nTd=12\n19c4\n0\nγ/integraldisplaye0\n0e29/19[1+(121/304)e2]1181/2299\n(1−e2)3/2de,(22)\nwherethetwo parametersare\nc0=a0(1−e2\n0)e−12/19\n0(1+121\n304e2\n0)−870/2299(23)\nandγ=64m1m2M/5.WhenTλislessthan Td(orλTd>1),\nchaoswouldbeobserved.Because Tλ=3.0×103andTd=\n1.3×103for orbit 1, the chaoticity can not be seen before\nthe merger. Values of λTdfor orbits 2-8 are listed in Table\n1. Clearly, only chaotic orbit 8 is what we expect. Besides\nthese,we plottwo panels(a) and(b)of Fig. 6 regardingde-\npendenceofLyapunovexponentonsingleparameter,which\ncorrespondrespectively to Figs 5(a) and 5(b). Here are two\nfacts.First,theresultsinFig.6arethesameasthoseinFig .\n5. Second, lots of chaotic orbits whose Lyapunovtimes are\nmany times greater than the inspiral times should be ruled\nout, and there are only a small quantity of desired chaotic\norbitsleft.\nInordertomaketheaccuracyofthePNapproachbetter,\nweshouldchooseorbitswhoseinitialradiiarelargerenoug h\nthanroughly10 M.AllchaoticorbitsinTable2areexpected.\nNoticethattheotherinitialconditionsoftheseorbitsbey ond\nthis table are y=z=px=pz=0, and the starting spin an-\ngles are still the same as those of orbit 3. Althoughan orbit\nhasa largeinitialradius,it maystill bechaoticwhenitsin i-\ntial eccentricity is high enough. This supports the result o f\n[23]thata higheccentricorbitcaneasilyyieldchaos.6\nTable1ValuesofFLIsand λTdfordifferent orbits.FLIacorresponds totheNLOspin-spin couplingsturned off.FLIb, λ,λdandλTdcorrespond\ntothe NLO spin-spin couplings included.\nOrbit β χ x p ye0a0FLIa FLIb λ λ dλTd\n2 0.90 1.0 8.55 0.390 0.30 12.2 3.6 10.2 1.9E-4 8.1E-4 0.2\n3 0.50 0.93 18.6 0.195 0.29 14.4 4.5 9.0 1.7E-4 3.7E-4 0.5\n4 0.71 0.95 17.5 0.200 0.30 13.5 3.9 12.4 2.8E-4 5.3E-4 0.5\n5 0.65 0.90 35.4 0.100 0.65 21.5 4.2 9.8 2.1E-4 3.8E-4 0.7\n6 0.86 0.90 8.40 0.390 0.28 11.6 6.5 20.4 4.2E-4 9.3E-4 0.5\n7 0.50 0.83 19.5 0.185 0.33 14.6 7.0 11.5 2.5E-4 3.8E-4 0.7\n8 0.50 0.97 18.7 0.190 0.32 14.1 9.4 22.5 5.5E-4 4.3E-4 1.3\nTable 2Values of λTdfor chaotic orbitswithbig initialradii when the NLOspin-s pin contributions are included.\nOrbit χ β x p ye0a0λ λ dλTd\n9 0.95 0.50 14.5 0.240 0.16 12.5 6.8E-4 5.2E-4 1.3\n10 0.97 0.50 19.5 0.185 0.33 14.6 3.8E-4 3.7E-4 1.0\n11 0.93 0.50 20.5 0.175 0.37 14.9 4.4E-4 3.9E-4 1.1\n12 0.93 0.50 25.5 0.140 0.50 17.0 4.7E-4 3.8E-4 1.2\n13 0.90 0.48 35.4 0.100 0.65 21.5 4.2E-4 3.5E-4 1.2\n14 0.90 0.44 35.4 0.100 0.65 21.5 4.3E-4 3.4E-4 1.3\n4 Conclusions\nThis paper is devoted to studying the dynamicsof the com-\nplete3PNconservativeHamiltonianofspinningcompactbi-\nnaries in which the orbital part is accurate to the 3PN order\nand the spin-spinpart as well asthe spin-orbitpart include s\ntheLO andNLO contributions.Because ofthehighnonlin-\nearity,theNLOspin-spincouplingsincludedgiverisetoth e\noccurrenceof strong chaos in contrast with those excluded.\nBy scanning single parameter with the FLIs, we obtained\ndependence of chaos on the parameter. It was shown suffi-\nciently that chaos appears easier for larger spins or smalle r\nmass ratios under the present considered initial condition s,\nstarting unit spin vectors and other parameters. So does for\na smaller initial radius. In spite of this, an orbit with a lar ge\ninitialradiusisstill possiblychaoticifitsinitialecce ntricity\nishighenough.Aboveall,therearesome observable chaotic\norbits whose initial radii are suitably large and whose Lya-\npunovtimesarelessthanthecorrespondinginspiraltimes.\nAcknowledgements This research is supported by the Natural Sci-\nence Foundation ofChina under Grant Nos. 11173012 and 11178 002.\nReferences\n1. P.Jaranowski, G. Schäfer, Phys. Rev. D 55, 4712 (1997) .\n2. N.E.Pati and C. M.Will,Phys. Rev. D 65, 104008 (2002).\n3. T. Damour, P. Jaranowski, G. Schäfer, Phys. Rev. D 77, 064032\n(2008).\n4. T.Damour,P.JaranowskiandG.Schäfer,Phys.Rev.D 78,024009\n(2008).5. A. Nagar, Phys. Rev. D 84, 084028 (2011).\n6. J. Hartung, J.Steinhoff, Ann. Phys. 523, 919 (2011).\n7. J. Steinhoff, S. Hergt and G. Schäfer, Phys. Rev. D 77, 081501\n(2008).\n8. S. Hergt, G.Schäfer, Phys. Rev. D 78, 124004 (2008).\n9. M.Levi, Phys. Rev. D 85,064043 (2012).\n10. J. Levin, Phys. Rev. Lett. 84, 3515 (2000).\n11. J. D. Schnittman and F. A. Rasio, Phys. Rev. Lett. 87, 121101\n(2001).\n12. N. J.Cornish and J. Levin, Phys. Rev. Lett. 89,179001 (2002).\n13. N. J.Cornish and J. Levin, Phys. Rev. D 68, 024004 (2003).\n14. X. Wu and Y.Xie,Phys. Rev. D 76, 124004 (2007).\n15. J. Levin, Phys. Rev. D 67,044013 (2003).\n16. C. Königsdörffer and A. Gopakumar, Phys. Rev. D 71, 024039\n(2006).\n17. A.Gopakumar andC.Königsdörffer, Phys.Rev. D 72,121501(R)\n(2005).\n18. J. Levin, Phys. Rev. D 74,124027 (2006).\n19. X. Wu,Y.Xie, Phys. Rev. D 81, 084045 (2010).\n20. X. Wu and S. Y.Zhong, Gen. Relat.Gravit. 43,2185 (2011).\n21. L. Mei,M.Ju, X.Wu,and S.Liu, Mon.Not.R. Astron. Soc. 435,\n2246 (2013).\n22. Y. Wang and X. Wu, Classical Quantum Gravity 28, 025010\n(2011).\n23. M.D. Hartl,A.Buonanno, Phys. Rev. D 71, 024027 (2005).\n24. X. Wu and Y.Xie,Phys. Rev. D 77, 103012 (2008).\n25. A. Buonanno, Y. Chen, T. Damour, Phys. Rev. D 74, 104005\n(2006).\n26. L. Mei,X.Wu, and F. Liu,Eur. Phys. J. C 73, 2413 (2013).\n27. G.Tancredi, A.Sánchez and F.Roig,Astorn.J. 121, 1171 (2001).\n28. X. Wu and T.Y.Huang, Phys. Lett.A 313, 77 (2003).\n29. C. Froeschlé and E. Lega, Celest. Mech. Dyn. Astron. 78, 167\n(2000).\n30. X. Wu, T. Y. Huang, and H. Zhang, Phys. Rev. D 74, 083001\n(2006).\n31. P. C.Peters, Phys. Rev. B 136, 1224 (1964).7\n/s48 /s50 /s52 /s54 /s56 /s49/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53/s51/s46/s48/s51/s46/s53\n/s78/s111/s32/s32/s32/s32 /s32/s72\n/s115/s115/s78/s76/s79\n/s40/s97/s41\n/s32/s32/s69/s47/s49/s48/s45/s49/s51\n/s116/s32/s47/s49/s48/s53/s48 /s50 /s52 /s54 /s56 /s49/s48/s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52\n/s72\n/s115/s115/s78/s76/s79\n/s40/s98/s41\n/s32/s32/s69/s47/s49/s48/s45/s49/s51\n/s116/s47/s49/s48/s53\nFig.1Energy errors of orbit1. The NLO spin-spin couplings are not included inpanel (a) butin Panel (b).\n/s48 /s49 /s50 /s51 /s52 /s53/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s78/s111/s32/s32/s32/s32 /s72\n/s115/s115/s78/s76/s79\n/s40/s97/s41\n/s32/s32/s112/s111/s119/s101/s114/s32/s32/s115/s112/s101/s99/s116/s114/s117/s109\n/s102/s114/s101/s113/s117/s101/s110/s99/s121/s32/s47/s32/s49/s48/s45/s52/s48 /s49 /s50 /s51 /s52 /s53/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53/s48/s46/s54\n/s72\n/s115/s115/s78/s76/s79\n/s40/s98/s41\n/s32/s32/s112/s111/s119/s101/s114/s32/s32/s115/s112/s101/s99/s116/s114/s117/s109\n/s102/s114/s101/s113/s117/s101/s110/s99/s121/s32/s47/s32/s49/s48/s45/s52\nFig.2Power spectra corresponding toFig. 1.\n/s50/s46/s48 /s50/s46/s53 /s51/s46/s48 /s51/s46/s53 /s52/s46/s48 /s52/s46/s53 /s53/s46/s48 /s53/s46/s53 /s54/s46/s48/s45/s53/s46/s48/s45/s52/s46/s53/s45/s52/s46/s48/s45/s51/s46/s53/s45/s51/s46/s48/s45/s50/s46/s53/s45/s50/s46/s48/s45/s49/s46/s53\n/s78/s111/s32/s32/s32/s32 /s72\n/s115/s115/s78/s76/s79\n/s40/s97/s41\n/s32/s32/s108/s111/s103\n/s49/s48/s32/s124 /s124\n/s108/s111/s103\n/s49/s48/s32/s116/s50/s46/s48 /s50/s46/s53 /s51/s46/s48 /s51/s46/s53 /s52/s46/s48 /s52/s46/s53 /s53/s46/s48 /s53/s46/s53 /s54/s46/s48/s45/s52/s46/s48/s45/s51/s46/s53/s45/s51/s46/s48/s45/s50/s46/s53/s45/s50/s46/s48/s45/s49/s46/s53/s45/s49/s46/s48\n/s72\n/s115/s115/s78/s76/s79\n/s40/s98/s41\n/s32/s32/s108/s111/s103\n/s49/s48/s32/s124 /s124\n/s108/s111/s103\n/s49/s48/s32/s116\nFig.3The maximum Lyapunov exponents λcorresponding to Fig.1.8\n/s50/s46/s48 /s50/s46/s53 /s51/s46/s48 /s51/s46/s53 /s52/s46/s48 /s52/s46/s53 /s53/s46/s48/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53/s51/s46/s48/s51/s46/s53/s52/s46/s48/s52/s46/s53\n/s78/s111/s32/s32/s32/s32 /s72\n/s115/s115/s78/s76/s79\n/s40/s97/s41\n/s32/s32/s70/s76/s73\n/s108/s111/s103\n/s49/s48/s32/s116/s50/s46/s48 /s50/s46/s53 /s51/s46/s48 /s51/s46/s53 /s52/s46/s48 /s52/s46/s53 /s53/s46/s48/s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52/s49/s54\n/s72\n/s115/s115/s78/s76/s79\n/s40/s98/s41\n/s32/s32/s70/s76/s73\n/s108/s111/s103\n/s49/s48/s32/s116\nFig.4The fast Lyapunov indicators (FLIs) corresponding to Fig.1 .\n/s48/s46/s52 /s48/s46/s53 /s48/s46/s54 /s48/s46/s55 /s48/s46/s56 /s48/s46/s57 /s49/s46/s48/s48/s53/s49/s48/s49/s53/s50/s48/s50/s53/s51/s48/s51/s53/s52/s48\n/s32/s32/s70/s76/s73/s40/s97/s41\n/s48/s46/s53/s48 /s48/s46/s53/s53 /s48/s46/s54/s48 /s48/s46/s54/s53 /s48/s46/s55/s48 /s48/s46/s55/s53 /s48/s46/s56/s48 /s48/s46/s56/s53 /s48/s46/s57/s48 /s48/s46/s57/s53 /s49/s46/s48/s48/s48/s53/s49/s48/s49/s53/s50/s48/s50/s53/s51/s48/s51/s53/s52/s48/s52/s53/s53/s48/s53/s53/s54/s48\n/s32/s32/s70/s76/s73/s40/s98/s41\nFig.5(color online)The FLIs asa function of χorβwhen the NLO spin-spin interactions are included. AllFLIs l arger than 5 mean chaos.\n/s48/s46/s50 /s48/s46/s51 /s48/s46/s52 /s48/s46/s53 /s48/s46/s54 /s48/s46/s55 /s48/s46/s56 /s48/s46/s57 /s49/s46/s48/s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52/s49/s54\n/s99/s61/s49/s48/s45/s52/s100/s61/s55/s46/s49/s56/s42/s49/s48/s45/s52/s40/s97/s41\n/s32/s32/s47/s49/s48/s45/s52\n/s48/s46/s53/s48 /s48/s46/s53/s53 /s48/s46/s54/s48 /s48/s46/s54/s53 /s48/s46/s55/s48 /s48/s46/s55/s53 /s48/s46/s56/s48 /s48/s46/s56/s53 /s48/s46/s57/s48 /s48/s46/s57/s53 /s49/s46/s48/s48/s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52/s49/s54/s49/s56\n/s99/s61/s49/s48/s45/s52/s40/s98/s41\n/s100\n/s32/s32/s47/s49/s48/s45/s52\nFig. 6(color online) The maximum Lyapunov exponents λcorresponding to Fig. 5. Note that λ>λcmeans chaos, and λ>λdwithλd=1/Td\nindicatesthe occurrence of chaos before coalescence."    },    {        "title": "1408.4547v1.Coherent_transport_and_manipulation_of_spins_in_indirect_exciton_nanostructures.pdf",        "content": "arXiv:1408.4547v1  [cond-mat.mes-hall]  20 Aug 2014Coherent transport and manipulation of spins in indirect ex citon nanostructures\nA. Violante,1,∗R. Hey,1and P. V. Santos1\n1Paul-Drude-Institut f¨ ur Festk¨ orkperelektronik, Hausv ogteiplatz 5-7, 10117 Berlin, Germany\n(Dated: August 13, 2018)\nWe report on the coherent control and transport of indirect e xciton (IX) spins in GaAs double\nquantum well (DQW) nanostructures. The spin dynamics was in vestigated by optically generating\nspins using a focused, circularly polarized light spot and b y probing their spatial distribution using\nspatially and polarization resolved photoluminescence sp ectroscopy. Optically injected exciton spins\nprecess while diffusing over distances exceeding 20 µm from the excitation spot with a spatial\nprecessionfrequencythatdependsonthespintransportdir ectionas well asonthebiasappliedacross\nthe DQW structure. This behavior is attributed to the spin pr ecession in the effective magnetic field\ninduced by the spin-orbit interaction. From the dependence of the spin dynamics on the transport\ndirection, bias and external magnetic fields we directly det ermined the Dresselhaus and Rashba spin\nsplitting coefficients for the structure. The precession dyn amics is essentially independent on the IX\ndensity, thus indicating that the long spin lifetimes are no t associated with IX collective effects. The\nlatter, together with the negligible contribution of holes to the spin dynamics, are rather attributed\nto spatial separation of the electron and hole wave function s by the electric field, which reduces the\nelectron-hole exchange interaction. Coherent spin preces sion over long transport distances as well\nas the control of the spin vector using electric and magnetic field open the way for the application\nofIXspins in the quantum information processing.\nPACS numbers: 71.35.-y, 75.76.+j, 75,70,Tj, 73.21.Fg\nI. INTRODUCTION\nAn indirect (or dipolar) exciton ( IX) consists of a\nCoulomb-correlated electron-hole pair with the individ-\nual particles confined in two closely spaced quantum\nwells (QWs) in a double quantum well (DQW) structure\n(cf. Fig. 1) subjected to a transverse electric field Fz.\nFzcontrols via the quantum confined Stark effect both\ntheIXenergy and radiative lifetime, which can reach\nthe ms-range.1The electric field can be used to later-\nally confine IXgases in DQW plane using electrostatic\ngates2as well as to control the interconversion between\nIXand photons. In addition, the electrical dipoles aris-\ning from the charge separation give rise to strong repul-\nsive exciton-exciton interactions3. Finally, IXs are com-\nposite bosonsand, therefore, susceptible tothe formation\nof collective exciton phases, as evidenced by the experi-\nments presented in Refs. 4–6. These interesting IXprop-\nerties have been combined to demonstrate different de-\nvice functionalities including light storage cells, switches,\nand transistors7–9. The long IXlifetimes also enable the\nlong-range transport and coupling of IXpackets. Differ-\nent approaches have also been introduced for the trans-\nport ofIXs based on bare diffusion,10drift induced by\nrepulsive IX-IXinteractions,4or by using spatially vary-\ning electric fields, including electrostatic ramps11as well\nas moving electrostatic12and acoustic lattices13–15.\nRecently, the confinement of single IXs has been\ndemonstrated,2thus opening the way for applications of\nsingleIXs in quantum information.16Here, the manip-\nulation and transport of IXspins offers further exciting\nperspectives. IXs have several favorable properties for\nspin control and manipulation. The reduced overlap be-\ntween the electron and hole wave functions reduces notonly the radiative lifetime but also the exchange inter-\naction between the exciton-bound electron spins and the\nshort-livingholespins(the excitonicorBir-Aronov-Pikus\n(BAP) mechanism17,18), thus enhancing the exciton spin\nlifetime.19–21\nThe long lifetime also enables for spin control using\nthe spin-orbit (SO) coupling. Due to the Bulk Inver-\nsion Asymmetry (BIA) intrinsic to the noncentrosym-\nmetric III-V materials (such as GaAs),22,23moving spins\nfeel an effective magnetic field (the Dresselhaus field,24\nBD), which modifies the spin vector during precession.\nIn addition, the Structural Inversion Anisotropy (SIA)\ncreated by Fzinduces a tunable contribution to the SO\nmagnetic field (the Rashba field,25BR), thus providing\na powerful tool for the dynamic spin control during mo-\ntion.Fzalso affects spins by changing the g-factor.26\nThe SO fields can, however, also lead to the dephasing of\nexciton spin ensembles (the D’yakonov-Perel’ dephasing\nmechanism). The spin dephasing time can be electrically\ncontrolled by selecting Fzto modulate the total SO field\nBSO=BD+DR.27TheIXspin lifetime also enhances\nunder high particle densities. In analogy with the spin\ndynamicsofatomiccondensates,28the latterhasbeen at-\ntributedtotheformationofcollective IXphases.5,20,29–31\nIn particular, experiments carriedout in this regimehave\nlead to the observation of spin currents in IXgases with\na variety of spatially resolved polarization patterns,31,32\nwhich have been attributed to the SO magnetic field.\nMost of the previous IXspin studies have addressed\nthe spin dynamics very close to the spin excitation area\n(i.e., within distances <10µm), one exception being the\nself-organized extended (i.e., >20µm) spin patterns re-\nported for the collective regime at high IXdensities.5\nIn this work, we demonstrate that optically injected IX2\nspins can be transported over distances exceeding 20 µm\nfrom the injection spot. The spins coherently precess\nduring motion: the spatial precession period depends on\nthe motion direction and can be controlled by the ver-\ntical field Fzapplied across the DQW. This precession\nbehavior, which does not depend on IXdensity, is at-\ntributed to the precession of the electron spins in the\nanisotropic SO field. From the dependence of the spin\npolarization on Fz, motion direction, and external mag-\nnetic field Bextwe have determined the IXspin lifetime\nas well as the absolute magnitude and orientation of the\nSO fields in the DQW structures. In contrast to the\nspin patterns in collective regime addressed in Ref.5, we\nshow that transport and manipulation of IXensembles\nreportedhere takesplace in the drift-diffusive regime and\ncan be potentially extended down to the single spin level.\nIn the following, we first describe the sample structure\nand the photoluminescence (PL) technique used to probe\nthe transport of excitons spins. We then present exper-\nimental results demonstrating the transport and coher-\nent precession of IX spins in the SO field together with\ndetailed studies of the spin dynamics as a function of\npropagationdirection, bias, and external magnetic fields.\nThese results on spin transport and SO effects are com-\npared to the state-of-the art in the final sections of the\nmanuscript.\nII. EXPERIMENTAL DETAILS\nThe samples used in the experiment contain three\nsets of asymmetric GaAs DQWs grown by molecu-\nlar beam epitaxy on a n+-doped GaAs(001) substrate\n[cf. Figs. 1(a)-(b)]. Each DQW is formed by a dQW,1=\n14 nm (QW 1) and adQW,2= 17 nm-thick QW (QW 2)\nseparated by a thin (4 nm-thick) Al0.3Ga0.7Asbarrier.\nMultipleDQWswereusedinordertoenhancethephoton\nyield of the PL experiments. The different QW widths\nenable the selective generation of direct excitons in ei-\nther QW 1or QW 2by the appropriate selection of the\nexcitation wavelength33. The electric field Fzfor the\nformation of IXs was induced by a bias voltage VBIAS\napplied between the n-doped substrate and a thin (10\nnm-thick) semitransparent Ti electrode (STE) deposited\non the sample surface.\nThe spectroscopic experiments were carried out in a\nHe bath cryostat at 2 K. Excitons were generated using\na right circularly polarized beam from a cw Ti:Sapphire\nlaserresonant with the electron-heavyhole direct exciton\n(DX) transition of the wider (QW 2) QW. The laser was\nfocused ontoa spot with a diameterof2.5 µm using ami-\ncroscopeobjective. ThePLaroundthelaserspotwascol-\nlected by the same objective, analyzed regarding the cir-\ncular polarization, and then imaged on the entrance slit\nof a spectrometer connected to a cooled charge-coupled-\ndevice (CCD) camera. The detected CCD images con-\ntain information about the energy (direction perpendic-\nular to the slit) and spatial distribution of the PL inten-\nFIG. 1: (a) Double quantum well (DQW) structures for in-\ndirect exciton ( IX) spin transport. The IXs are optically\nexcited using a focused laser pulse in a DQW subject to an\nelectric field ( Fz) induced bya bias VBIASapplied between the\ndoped substrate and a semitransparent top electrode (STE).\nIXspintransport ismappedbyimaging theIXphotolumines-\ncence (PL) emitted along direction kwith polarization sen-\nsitivity. During motion along k, the spins precess under the\nspin-orbit ( BSO) and external magnetic field ( Bext). (b) En-\nergy band diagram of the GaAs/(Al,Ga)As DQW along the\nvertical ( z) direction showing the direct ( DX) andIXtransi-\ntions under the electric field Fz. (c) Spatially and spectrally\nresolved images of the PL with right ( σ+, upper panel) and\nleft (σ−, lower panel) circular polarizations recorded under\na biasVBIAS=−2 V and a laser power of 13 µW.DXand\nIXdenote the emission from direct and indirect excitons in\nthe DQW structure, while the line at 1.515 eV arises from\nexcitons in the GaAs substrate.\nsity (along the slit). The spatial (coordinate r) depen-\ndence of the spin polarization was calculated according\ntoρz(r) = [Iσ+(r)−Iσ−(r)]/[Iσ+(r)+Iσ−(r)], where\nIσ+(r) (Iσ−(r)) denote the PL intensity at rwith right\n(left) circular polarization.\nIII. RESULTS\nA. Coherent spin precession during transport\nFigure1(c) compares PL images with right ( σ+) and\nleft (σ−) polarization recorded with the spectrometer\nslit aligned with the sample crystallographic direction\nx′= [110]. The emission patterns at 1.515 eV and\n1.525 eV arise, respectively, from excitons in the GaAs\nsubstrate and from the low energy tail of DXs in QW 2.\nIn both cases, the emission is restricted to a region close\nto the excitation spot, thus indicating negligible diffusion\ndistancesfor these excitations. The line at 1.520eVis as-\nsociated with the emission of IXs in the DQW structure,\nwhich in this case was subject to a bias VBIAS=−2 V.\nThe emission extends to radii exceeding 40 µm - these3\nlong transport distances arise from the fact that the IXs\nare driven not only by diffusion but also by the drift\nforces induced by repulsive exciton-exciton dipolar in-\nteractions at the areas of high exciton density close to\nthe excitation area. This assertion is supported by the\nstrong dependence of the extension of the IXPL cloud\non the excitation density Pl, as displayed by the dashed\nand dash-dotted lines in Fig. 2(a). These curves dis-\nplayed the spatial dependence of the total PL intensity\n(i.e., [Iσ+(r) +Iσ−(r)] determined from PL images for\nlaser excitation powers of Pl= 13µW (as in Fig. 1(c))\nand 1.6 µW, respectively. The characteristic shape of\nthe spatial PL profiles are consistent with the observa-\ntions of Ref.4, which reported that the IXPL reduces in\nthe area of hot IXclose to the excitation spot. Further-\nmore, a closer look to Fig. 1(c) also reveals that the IX\nemissionenergyred-shiftsasonemovesawayfromtheex-\ncitation spot. In fact, the average IXemission energy at\nx′= 0µm is∼0.5 meV higher than at x′= 75µm. This\nenergetic blue-shift is attributed to the strong repulsive\nIX-IXinteraction at the high IXdensities (estimated34\nto be of 2 ×1010cm−2) close to x′= 0.\nThe intensity of the circularly polarized IXemission\nin Fig.1(c) is also spatially modulated with the maxima\nfor theσ+polarization corresponding to minima in the\nσ−polarization. The symbols in Fig. 2(a) display the\ncorresponding profiles for the spin polarization ρz. This\nspatial PL modulation is attributed to the precession of\ntheIXelectron spins in the effective SO magnetic field\nBSOduring their motion away from the excitation spot.\nTheρzoscillationsextend overarangeofradialdistances\nshorter than the radius of the emission cloud. While\nthe amplitude of the oscillations increases with Pl, their\nspatial period remains essentially constant. A similar\nincrease in spin polarization with excitation density was\npreviously reported in Ref.20and attributed to the onset\nof collective exciton effects. The fact that the spatial\nprecession period does not change with the excitation\ndensity indicates, however, that the spin precession rates\nunder the SO field does not depend on the IXdensity.\nIn addition, the larger ρzamplitudes in our case is at\nleast in part due to the shorter expansion times of the\nIXcloud resulting from IX-IXinteractions under high\nPl, which reducesthe impact ofspin scatteringprocesses.\nThe solid lines in Fig. 2(a) are fits of the experimental\nresults to damped oscillations with a spatial dependence\ngiven by:\nρs(r) =ρs(0)e−r/ℓscos(2πr/ℓp(ˆ r)).(1)\nHereℓp(ˆ r) is the spin spatial precession period for prop-\nagation along ˆ r, which is equal to the SO precession pe-\nriodℓsoin the absence of external magnetic fields. ℓs\ndenotes the effective spin transport decay length. Equa-\ntion (1) describes the spin dynamics in a non-degenerate\nQW at low temperatures (i.e., for thermal energies less\nthan the QW confinement energy), when spin precession\nrate Ω so(k) =µBBSO\n/planckover2pi1becomes proportional to the elec-tron wave vector k. Under this condition, the spin pre-\ncession period (for Bext= 0)ℓp=ℓso=me\n2π/planckover2pi12k\nΩso(k)\nonly depends on direction ˆ r. The spin decay length ℓs,\nin contrast, depends on temporal dynamics of the carrier\nmotion as well as on the details of the spin scattering\nprocesses. The fits carried out with a single value for\nℓp(i.e., independent of r) displayed in Eq. 1reproduces\nvery well the measured spin polarization profiles.\nFIG. 2: (a) Profiles for the spin polarization ( ρs, symbols)\nand for the average PL intensity ( ¯IPL, dashed and dot-\ndashed lines) for exciton transport along the x′||[110] (sym-\nbols) recorded for laser excitation densities Pℓ= 1.6µW\n(open symbols) and 13 µW (solid symbols). and [ ¯110] direc-\ntions(opensymbols). (b)Correspondingprofilesfortransp ort\nalongy′||[¯110] recorded for Pℓ= 6µW. The solid lines super-\nimposed on the symbols are fits of the experimental results to\nEq. 1. (c) In-plane dependence of the Dresselhaus ( BD) and\nRashba ( BR) contributions to the SO effective magnetic field\nBSO=BD+BR.\nB. Dependence on propagation direction\nThe wave vector ( k) dependence of the SO contribu-\ntionsBD(k) andBR(k) for a III-V QW is illustrated\nin Fig.2(c). Here, we use a reference frame defined\nby the axes x′||[110],y′||[¯110], and z||[001]. The associ-\nated (temporal) electron spin precession frequencies Ωi,\n(i=D,R)canbe writtenasafunction ofthewavevector\nkas:35\n/planckover2pi1ΩD(k) =γ/parenleftBigg\nπ\ndeff\nQW/parenrightBigg2\nky′\nkx′\n0\nand (2)4\n/planckover2pi1ΩR(k) = 2Fzr41\nky′\n−kx′\n0\n (3)\nwhereγandr41are the Dresselhaus and Rashba spin\nconstants,respectively. Equations 2and3applyforsmall\nwavevectors(i.e., |k′\nx|,|k′\nx|<<π\ndeff\nQW.deff\nQWis theeffective\nextension along zof the electronic wave function in the\nQW confining the electrons (QW 2in the present case, as\ndetermined by the sign of V BIAS).deff\nQWis determined by\nboth the QW width and by the penetration depth into\nthe barrier layers. The corresponding amplitudes of the\nspatial precession frequencies ℓ−1\nso,ican be derived from\nthe precession distance required for a spin rotation by\n2π. The following expressions are obtained:36\nℓ−1\nso,D=γme\n2π/planckover2pi13/parenleftBigg\nπ\ndeff\nQW/parenrightBigg2\nandℓ−1\nso,R=me\nπ/planckover2pi13r41Fz.\n(4)\nIfBD||BRthe resulting spatial precession frequency will\nbe givenby ℓ−1\nso=|ℓ−1\nso,D±ℓ−1\nso,R|, wherethe signisdefined\nby the relative orientation of the two fields.\nEquations 2and4predicts that the spatial precession\nperiodℓsodepends both on transport direction and on\nbias electric field Fz. In order to check the in-plane\nanisotropyof BSO, weshowin Fig. 2(b) spin polarization\nprofilesrecordedfor IXmotionalongthe [1 ¯10]surfacedi-\nrection(i.e., orthogonaltotheoneinFigs. 1(c) and2(b)).\nThe period of the precession oscillations is much larger\nthan for the motion along the [110] direction. The larger\nperiod, which in this case becomes comparable with the\nspin decay length ℓs, makes the amplitude of the oscilla-\ntions weaker than in Fig. 2(a). The dependence of ℓsoon\ntransport direction is in agreement with the diagrams of\nFig.2(c): while BDandBRadd for motion along [110],\nthey partially cancel each other for motion along [ ¯110],\nthus leading to a weaker SO field. Interestingly, while\na weaker BSOreduces spin scattering, it does not nec-\nessarily improve the visibility of the ρzoscillations since\none also has to satisfy the constraint ℓso< ℓs. On the\nother hand, a stronger BSOreduces the spatial preces-\nsion period. In this case, however, one needs to reduce\nthe excitation spot to dimensions at least a factor of 2\nsmaller than the oscillation period in order to measure\nthe oscillations.\nC. Electric spin control\nThe electric control of the spin precession frequency is\ndemonstrated by the series of profiles recorded for trans-\nportalong[110]in Fig. 3(a). Theamplitude Fzofelectric\nfield was determined from the applied bias VBIASusing\nthe procedure described in Ref.14. The spatial preces-\nsion period ℓsoreduces by over 15% with increasing Fz,\nFIG. 3: (a) Profiles for the spin polarization ( ρs, symbols)\nrecorded for increasing electric fields Fzapplied along the\ngrowth direction of the DQW structure. (b) Dependence\nof the spatial spin precession frequency 1 /ℓsoon the elec-\ntric field Fzfor transport along x′||[110] (solid symbols) and\ny′||[¯110] (opensymbol). The solid line is afitofthedataalong\n[110] to ℓ−1\nso=ℓ−1\nso,D+ℓ−1\nso,R(cf. Eq. 4) with γ= 17.9 eV˚A3,\nr41=−8.5 eV˚A2, anddeff\nQW= 21 nm. The corresponding in-\nverse spatial precession periods ℓ−1\nso,Dandℓ−1\nso,Rare illustrated\nby the dashed lines. The dot-dashed line shows the expected\ninverse precession period for motion along [1 ¯10].\nthus indicating an enhancement of the Rashba field BR\nas well as of the total SO field BSO. Larger changes can\nbe in principle be achieved by increasing the bias across\nthe DQWs. These, however,alsoreducethe PL yieldand\nmay lead to field-induced carrierleakage out of the DQW\nstructure, thus hampering an efficient spin detection.37\nThe measured spatial precession frequencies ( ℓ−1\nso) are\nplotted as a function of the transverse electric field in\nFig.3(b). The solid line is a fit of the data mea-\nsured along [110] to the expression: ℓ−1\nso=ℓ−1\nso,D+ℓ−1\nso,R\n(cf. Eq.4), which gives γ= 17.9 eV˚A3,r41=−8.5 eV˚A2,\nanddeff\nQW= 21 nm (the signs for γandr41will be de-\ntermined below. The corresponding inverse precession\nperiodsℓ−1\nso,Dandℓ−1\nso,Rfor propagation along this direc-5\ntion are illustrated by the dashed lines. For the applied\nbias, indirect excitons are created in the DQW struc-\nture with the electrons confined within the wider (17 nm\nwide) QW. The value for deff\nQWused in the fits is compat-\nible with an electronic wave function with a penetration\ndepth of 2 nm in the barriers. As will be further dis-\ncussed later, these SO parameters agree well with values\nreported in the literature.\nThe open symbol in Fig. 3(b) shows the inverse pre-\ncession period measured for spin propagation along y′,\nwhere the Dresselhaus contribution changes in sign rel-\native to the x′. The measured value agrees well with\nthe calculated inverse periods ℓ−1\nsofor this propagation\nalong this direction (dash-dotted line). Note that the\nchange in relative signs of the Rashba and Dresselhaus\neffective magnetic fields can in principle also be attained\nfor a fixed motion direction by simply reversing the bias\npolarity. For the asymmetric DQWs used in the present\nexperiments, however, a reversal in direction of Fzmoves\nthe electrons to the narrow QW, thus also changing the\nmagnitude of the Dresselhaus contribution. In addition,\nwhile the leakage current is very low for negative biases\n(a few nA), it increases considerably for high positive\nbiases, thus indicating enhanced carrier leakage.\nD. Magnetic field dependence\nAn external magnetic field Bextapplied along the\nDQW plane provides an additional degree of freedom to\ncontrolthespinvector. Theexperimentswerecarriedout\nundertheconfigurationillustratedinFig. 1(a), whereone\nmaps the spins moving along x′under an external field\nBext||y′and collinear with Bso. Figure 4(a) displays ρz\nprofiles along x′recorded under different external mag-\nnetic fields. Large magnetic fields ( |Bext|>6mT)\nmainly induce a reduction of the overall spin polariza-\ntion (Hanle effect), from which one can extract the spin\ndephasing time τs. For that purpose, we plot in Fig. 4(b)\nthe spin IX polarization at the excitation spot ¯ ρzas a\nfunction of Bext. The lines superimposed on the data\npoints are fits to the Hanle expression:\n¯ρz(Bext) =¯ρz(0)\n1+(ωLτ∗)2with/planckover2pi1ωL=geµBBext.(5)\nwhere ¯ρzis the degree of polarization at the generation\npoint,1\nτ∗=1\nτIX+1\nτswithτIXthe indirect exciton life-\ntime. By using the conduction band g-factor |ge|= 0.44\nand considering τIX>> τs, one extracts from the fits\nspin lifetimes of 3 and 6 ns for excitation powers of\nPl= 1µW and 13 µW, respectively. These lifetimes\nareconsiderablylongerthanthe typicalcarrierscattering\ntimes, thus indicating that the spin dynamics observed\nhere takesplacein diffusive (orstrongscattering)regime.\nWeak magnetic fields, in contrast, mainly affect the\nspin spatial precession frequency, as indicated by the\nsymbols in Fig. 4(c). In contrast to the SO field, which\nFIG. 4: (a) Spin polarization profiles as a function of an\nexternal magnetic field ( Bext). The curves are stacked for\nclarity. (b) ρzclose to the excitation spot as a function of\nBextmeasured for optical excitation densities Pl= 13µW\n(squares) and 1 µW (dots). The dashed lines display fits to\nEq. 5, from which one extracts spin lifetimes of 6 and 3 ns,\nrespectively. (c) Dependence of the inverse precession per iod\n(1/ℓp= 1/ℓso+1/ℓB, whereℓsoandℓBare the precession pe-\nriod around the SO and external fields, respectively) on Bext\n(symbols). The lines are the predictions from the diffusion\n(DIX) and drift ( vD) models discussed in the text.\nreverses with carrier momentum, the external field has a\nfixed direction and its action on the spin vector only de-\npends on time. These features can be exploited to deter-\nmine the absolute orientation of Bso. WhenBextis ori-\nented along + y′(upper curves for Bext>0in Fig.4(a)),\nthe spatial spin precession rate increases (reduces) for\nspins moving along + x′(−x′). The opposite behavior is\nobserved when the orientation of Bextreverses (curves\nforBext<0). From these results, together with the\nincrease of the spatial precession frequency along the x′\nwith|Fz|(cf. Fig. 3(a), we unambiguously establish that\nboth the Rashba and Dresselhaus spin orbit fields point\ntowards y′when spins move along x′. The Dresselhaus\nsplitting constant γin Eq.2is thus positive while the\nRashba parameter r14is negative.\nThe different natures of the two fields allows us to\nextract information about the precession dynamics dur-\ning motion and their impact on the precession period ℓp.\nHere, we first note that the different ℓpvalues in Fig. 4(a)\nalong +x′and−x′implies that both fields (i.e., Bsoand\nBext) must act simultaneously on the spins during the\nmotion. This becomes clear if one considers the hypo-\nthetical situation, where the spins initially move to the\nfinal position x′within a very short time (thereby pre-\ncessing only around Bso(k)) and then remain trapped\nthere until recombination (thus precessing only around\nBext). In this case, Bextwould simply add a fixed (i.e.,\nposition independent) phase to the spatial oscillations6\ndetermined by Bsoandℓpwould be the same for spin\nmotion along both the positive and negative x′axes, in\ncontrast to the results displayed in Fig. 4(a).\nSinceBextandBsoare collinear in Fig. 4(a), the in-\nverse precession period can be written as ℓ−1\np=ℓ−1\nso+\nℓ−1\nB(Bext), where ℓ−1\nB(Bext) is the contribution associ-\nated with the external field. In order to account for the\ndependence of ℓ−1\nponBextone needs information about\nthe spin position x′as a function of time ( t). We con-\nsider the following two extreme situations. First, if the\nmotion is mainly driven by the IX-IX repulsion forces\nat the excitation region, the carriers are expected to\nmove with an approximately constant velocity vD, yield-\ningℓ−1\nB(Bext) =ℓ−1\nB,vD=gµB\n2π/planckover2pi1Bext\nvD. This model predicts\nthe linear dependence of ℓ−1\npon field indicated by the\ndashed line in Fig. 4(c). The very small values used\nforvD= 80 m/s, which corresponds to only ∼1% of\nthe average exciton thermal velocity at 2 K, probably\narise from the fact that diffusion has been completely\nneglected. While the behavior for small fields is well ac-\ncounted for, the model fails to reproduce the deviations\nfrom linearity observed as the |Bext|increases. The sec-\nond extreme situation occurs if motion in purely driven\nby diffusion, in which case x′=√DIXt, whereDIXis\nthe IX diffusion coefficient. A straight-forward calcula-\ntion yields ℓ−1\nB(Bext) =ℓ−1\nB,D=/radicalBig\ngµB\n2π/planckover2pi1Bext\nDIX, leadingtothe\nbehavior depicted by the solid line in Fig. 4(c). The cal-\nculations were carried out using an exciton diffusion co-\nefficientDIX= 40×10−4cm2/s, which is approximately\na factor of 2 larger than the one expected from Ref.10for\ndiffusion in a 15 nm wide QW. Despite its simplicity, this\napproximation reproduces reasonably well the measured\nbehavior for large field amplitudes. A good fit requires,\nhowever, values for DIXhigher that those expected from\nRef.10. This becomes necessary to reduce the IX trans-\nport (and precession) time and, in that way, compensate\nfor the errors arising neglecting of drift effects. A more\nprecise description of the spin dynamics by taking into\naccount drift and diffusion is presently under study.\nIV. DISCUSSIONS\nThe experimental results of the previous section un-\nambiguously show the coherent precession of optically\nexcited spins in the effective magnetic field Bsoinduced\nby the SO interaction. The spin precession period de-\npends on the in-plane motion direction - it is essentially\nindependent of the IXdensity and can be controlled by\nexternal electric and magnetic fields. The measured spin\nlifetimes (of a few ns) indicate that transport takes place\nin the drift-diffusive regime. The spin dynamics is well-\ndescribed by taking into account the SO parameters as\nwell as the g-factor for electrons, thus showing that it\nis determined by the IXelectron spins. In fact, the SO\nparameter γ= 17.9 eV˚A3compares well with those de-\nterminedfromelectronspintransportin (100),38(110),39and (111) QWs37with comparable widths. The magni-\ntudeoftheRashbaparameter r41=−8.5eV˚A2, however,\nlies above the experimental values of 4 and 6 ±1 eV˚A2of\nRefs.40and37, respectively, aswellasthe calculatedvalue\nof 5.2±1 eV˚A2from Ref.36. While further investigations\nare required to clarify this difference, we speculate that\nit may be related to the asymmetric barriersin the DQW\nstructure, which make the electronic wave function more\nsusceptible to the applied electric fields.\nThecoherenttransportandprecessionofelectronspins\nin the SO field in the drift-diffusive transport regime has\nbeenpreviouslyreportedforbulkGaAslayers41,42aswell\nas QW structures.38,39,41,43,44In these experiments, elec-\ntron transport has been driven by an external electric (or\npiezoelectric field): in this way, the spins could be moved\nover microscopic distances before dephasing (assumed to\nbe due to the D’yakonov-Perel’mechanism45) sets in. To\nour knowledge, however, the coherent precession of spins\ninjected in an undoped QWs in the absence of an elec-\ntric driving field demonstrated here has so far not been\nreported.\nThe observation of coherent spin precession in the\nSO field requires small ratios ℓso/ℓs(cf. Eq. 1). Here,\nℓso∼aso/angbracketleftΩso/angbracketright(asois a proportionality constant and\nthe/angbracketleft/angbracketright’s denote thermal averagesover the transport path)\nonly depends on the strength of the SO interaction. The\nspin decay length ls, in contrast, depends on transport\ntime. Forpurelydiffusivemotionin theD’yakonov-Perel’\nregime,ls∼DIX\n2r1\nτc/angbracketleftΩ2so/angbracketright, where τcis the carrier scat-\ntering time and DIXthe diffusion coefficient, reduces\nwith transport distance r. As a result, the requirement\nℓso/ℓs∼2τc\nDIX/angbracketleftΩso/angbracketright\n/angbracketleftΩ2so/angbracketrightasor <1 becomes increasinglydifficult\ntobesatisfiedforlongtransportdistances r(aswellasfor\nstrong SO fields). In the present experiments, the expan-\nsion forces due to IX-IXrepulsive interactions appear to\nhave a fundamental role, since they considerably reduce\nthe transport time, thus increasing ℓsand enabling the\nobservation of precession oscillations.\nCoherentspin patterns resultingfrom precessionin the\nSO field have been reported for macroscopically ordered\nIXphases.5,31The long spin coherence lengths required\nfor their formation have been attributed to the reduced\nscattering in the ordered state, which enables the exciton\ntransport in the ballistic regime. The results presented\nin the previous section show that long spin lifetimes and\ntransport lengths, as well as coherent precession in the\nSO field can alsobe achievedin the drift-diffusive regime.\nThis behavior is followed occurs over a wide range of ex-\nciton densities, thus indicating that it is not associated\nwith collective effects. Interestingly, the spatial extent\nof the precession oscillations reported here is comparable\nto the ones for the self-organized spin patterns.31There\nare additional fundamental differences between the two\ncases. While the spin pattern observed in Ref.31results\nfrom self-organization and does not rely on the initial\nexciton polarization, the ones presented here appear as\na consequence of the coherent transport of optically in-\njected spins. In addition, the dynamics observed here is7\nassociated with electron spin precession, while Ref.31has\ninvoked the simultaneous evolution of hole and electron\nspins.\nFinally, as in most ofthe reportsofcoherentprecession\nof optically injected spins in the SO field, the absence of\nholespin contributionsto the spin dynamicsis attributed\nto the short hole spin coherence times as well as the spa-\ntial separation of the electron and hole wave function in\nthe DQW structure. Since we have excited and probed\nIXspins, an interesting question is whether the electrons\nand hole remain bound during transport. Although we\ncannot present a conclusive evidence for this behavior,\nwe mention that the IXspins have been excited with\nsmall excess energy (approx. 10 meV), thus reducing the\nprobability of exciton dissociation followed by the spa-\ntial separation of electrons and holes. In addition, the\nIXspatial profiles (cf., e.g., Fig. 1(c)) do not show the\ncharacteristic rings observed when electrons and holes\ndiffuse independently.\nV. CONCLUSIONS\nWe have presented experimental evidence for the co-\nherent precession of exciton spins during transport in\nthe effective magnetic field induced by the SO interac-\ntion. The precession dynamics is essentially independent\nof the IX density, but can be controlled by an external\nbias (via changes in the SO field) or magnetic field. Thespin dynamics are well explained by taking into account\nelectronspinprecession,withnonoticeableholespincon-\ntribution. The technique used by us allows for the con-\ntrol and discrimination of precession effects arising from\nthe Rashba and Dresselhaus SO contributions, as well as\nthose induced by an external magnetic field. In this way,\nwe were able to directly determine the absolute value\nand sign of the spin splitting parameters for the Rashba\nand Dresselhaus SO mechanisms. In addition, the spin\ndynamics in the presence of an external magnetic field\nyield information about the precession dynamics during\ntransport. The long spin transport distances as well as\nthe control of the spin vector open the way for the use of\nIX spins to encode and manipulate quantum information\nin opto-electronic semiconductor devices.\nVI. ACKNOWLEDGEMENTS\nWe thank Manfred Ramsteiner for discussions and\ncomments on the manuscript. We are indebted to S.\nRauwerdink, W. Seidel, S. Krauß, and A. Tahraoui for\nthe assistance in the preparation of the samples. We also\nacknowledge financial support from the German DFG\n(grant SA-598/9).\n∗e-mail address: violante@pdi-berlin.de\n1K. Sivalertporn, L. Mouchliadis, A. L. Ivanov, R. Philp,\nand E. A. Muljarov. Direct and indirect excitons in semi-\nconductor coupled quantum wells in an applied electric\nfield.Phys.Rev.B, 85:045207, Jan 2012.\n2G. J. Schinner, J. Repp, E. Schubert, A. K. Rai, D. Reuter,\nA. D. Wieck, A. O. Govorov, A. W. Holleitner, and J. P.\nKotthaus. Confinement and interaction of single indirect\nexcitons in a voltage-controlled trap formed inside double\nInGaAs quantum wells. Phys.Rev.Lett., 110:127403, Mar\n2013.\n3Kobi Cohen, Ronen Rapaport, and Paulo V. Santos. Re-\nmote dipolar interactions for objective density calibrati on\nand flow control of excitonic fluids. Phys.Rev.Lett.,\n106(12):126402, Mar 2011.\n4L. V. Butov, A.C. Gossard, and D. S.Chemla. Macroscop-\nically ordered state in an exciton system. Nature, 418:751,\n2002.\n5A. A. High, J. R. Leonard, A. T. Hammack, M. M. Fogler,\nL. V. Butov, A. V. Kavokin, K. L. Campman, and A. C.\nGossard. Spontaneous coherence in a cold exciton gas.\nNature, 483:584, March 2012.\n6Yehiel Shilo, Kobi Cohen, Boris Laikhtman, Ken West,\nLoren Pfeiffer, and Ronen Rapaport. Particle correlations\nand evidence for dark state condensation in a cold dipolar\nexciton fluid. NatureCommunications, 4:2335, 2013.\n7Alex A. High, Ekaterina E. Novitskaya, Leonid V. Butov,\nMicah Hanson, and Arthur C. Gossard. Control of excitonfluxes in an excitonic integrated circuit. Science, 321:229,\n2008.\n8AlexanderG. Winbow, Aaron T. Hammack, Leonid V.Bu-\ntov, and Arthur C. Gossard. Photon storage with nanosec-\nond switching in coupled quantum well nanostructures.\nNanoLett., 7(5):1349–1351, 2007.\n9G. Grosso, J. Graves, A. T. Hammack, A. High, L. V.\nButov, M. Hanson, and A. C. Gossard. Excitonic switches\noperating at around 100 k. Nat.Photonics, 3(10):577–580,\nOctober 2009.\n10Z. V¨ or¨ os, R. Balili, D.W. Snoke, L. Pfeiffer, and K. West.\nLong-distance diffusion of excitons in double quantumwell\nstructures. Phys.Rev.Lett., 94:226401, 2005.\n11A. G¨ artner, A. W. Holleitner, J. P. Kotthaus, and\nD. Schuh. Drift mobility of long-living excitons in cou-\npled GaAs quantum wells. Appl.Phys.Lett., 89:052108,\n2006.\n12A. G. Winbow, J. R. Leonard, M. Remeika, Y. Y.\nKuznetsova, A. A. High, A. T. Hammack, L. V. Butov,\nJ. Wilkes, A. A. Guenther, A. L. Ivanov, M. Hanson, and\nA. C. Gossard. Electrostatic conveyer for excitons. Phys.\nRev.Lett, 106:196806, 2011.\n13J. Rudolph, R. Hey, and P. V. Santos. Long-range exciton\ntransportbydynamicstrain fieldsinaGaAs quantumwell.\nPhys.Rev.Lett., 99:047602, 2007.\n14S. S. Lazi´ c, A. Violante, K. Cohen, R. Hey, R. Rapaport,\nand P. V. Santos. Scalable interconnections for remote\nindirectexcitonsystemsbasedonacoustic transport. Phys.8\nRev.B, 89:085313, Feb 2014.\n15A. Violante, K. Cohen, S. Lazi´ c, R. Hey, R. Rapaport,\nand P. V. Santos. Indirect exciton dynamics in moving\npotentials. NewJ.Phys., 89:085313, 2014.\n16I. Zutic, J. Fabian, and S. D. Sarma. Spintronics: Funda-\nmentals and applications. Rev.Mod.Phys., 76:323, 2004.\n17A.G. Aronov G.L. Bir and G.E. Pikus. Sov.Phys.JETP,\n42:705, 1975.\n18G. L. Bir and G. E. Pikus. Symmetry andStrain-Induced\nEffectsinSemiconductors. John Wiley & Sons, New York,\n1974.\n19AV Larionov, VB Timofeev, J Hvam, and C Soerensen.\nCollective behavior of interwell excitons in gaas/algaas\ndouble quantum wells. JETPLett., 71(3):117–122, 2000.\n20A. V. Larionov, V. E. Bisti, M. Bayer, J. Hvam, and K. So-\nerensen. Coherent spin dynamics of an interwell excitons\ngas in gaas/algaas coupled quantum wells. Phys.Rev.B,\n73:235329, 2006.\n21K. Kowalik-Seidl, X. P. V¨ ogele, B. N. Rimpfl, S. Manus,\nJ. P. Kotthaus, D. Schuh, W. Wegscheider, and A. W.\nHolleitner. Long exciton spin relaxation in coupled quan-\ntum wells. Appl.Phys.Lett., 97(1):011104, 2010.\n22M. I. D’yakonov and V. I. Perel’. Spin orientation of elec-\ntrons associated with the interband absorption of light in\nsemiconductors. SPJETP, 33:1053–1059, 1971.\n23Mikhail I. Dyakonov. Spinphysics insemiconductors.\nSpringer, 2008.\n24G. Dresselhaus. Spin-orbit coupling effects in zinc blende\nstructures. Phys.Rev., 100:580–586, Oct 1955.\n25E. I. Rashba. Sov.Phys.SolidState, 2:1109, 1960.\n26M. Poggio, G. M. Steeves, R. C. Myers, N. P. Stern,\nA. C. Gossard, and D. D. Awschalom. Spin transfer\nand coherence in coupled quantum wells. Phys.Rev.B,\n70:121305(R), 2004.\n27A.V. Larionov and L.E. Golub. Electric field control of\nspin-orbit splittings in gaas/algaas coupled quantum well s.\nPhys.Rev.B, 78:033302, 2008.\n28E. A. Burt, R. W. Ghrist, C. J. Myatt, M. J. Holland,\nE. A. Cornell, and C. E. Wieman. Coherence, correlations,\nand collisions: What one learns about bose-einstein con-\ndensates from their decay. Phys.Rev.Lett., 79:337–340,\n1997.\n29V. B. Timofeev, A. V. Larionov, M. Grassi Alessi,\nM. Capizzi, A. Frova, and J. M. Hvam. Charged exci-\ntonic complexes in gaas/al0.35ga0.65as p-i-n double quan-\ntum wells. Phys.Rev.B, 60:8897–8901, 1999.\n30J. R. Leonard, Y. Y. Kuznetsova, Sen Yang, L. V. Bu-\ntov, T. Ostatnick, A. Kavokin, and A. C. Gossard. Spin\ntransport of excitons. NanoLett., 9:4204–4208, 2009.\n31A. A. High, A. T. Hammack, J. R. Leonard, Sen Yang,\nL. V. Butov, T. Ostatnick´ y, M. Vladimirova, A. V. Ka-\nvokin, T. C. H. Liew, K. L. Campman, and A. C. Gossard.\nSpin currents in a coherent exciton gas. Phys.Rev.Lett.,110:246403, Jun 2013.\n32A. V. Kavokin, M. Vladimirova, B. Jouault, T. C. H. Liew,\nJ. R. Leonard, and L. V. Butov. Ballistic spin transport\nin exciton gases. Phys.Rev.B, 88:195309, Nov 2013.\n33M. Stern, V. Garmider, E. Segre, M. Rappaport, V. Uman-\nsky, Y. Levinson, and I. Bar-Joseph. Photoluminescence\nring formation in coupled quantum wells: Excitonic ver-\nsus ambipolar diffusion. Phys.Rev.Lett., 101(25):257402,\n2008.\n34Christoph Schindler Roland Zimmermann. Excitonexci-\nton interaction in coupled quantum wells. SolidState\nCommun., 144:395, 2007.\n35A. Hern´ andez-M´ ınguez, K. Biermann, R. Hey, and P. V.\nSantos. Spin transport and spin manipulation in (110)\nand (111) GaAs quantum wells. Phys.StatusSolidiB, in\nprint:–, 2013.\n36R. Winkler. Spin-Orbit Coupling Effects in\nTwo-Dimensional Electron andHoleSystems, volume 191.\nSpringer, Berlin, 2003.\n37A. Hern´ andez-M´ ınguez, K. Biermann, R. Hey, and P.V.\nSantos. Electrical suppression of spin relaxation in\nGaAs(111)B quantum wells. Phys.Rev.Lett., 109:266602,\n2012.\n38James A. H. Stotz, RudolfHey, Paulo V. Santos, and K. H.\nPloog. Coherentspintransport viadynamicquantumdots.\nNat.Mater., 4:585, 2005.\n39O. D. D. Couto, Jr., F. Iikawa, J. Rudolph, R. Hey, and\nP. V. Santos. Anisotropic spin transport in (110) GaAs\nquantum wells. Phys.Rev.Lett., 98:036603, 2007.\n40P. S. Eldridge, W. J. H. Leyland, P. G. Lagoudakis, R. T.\nHarley, R. T. Phillips, R. Winkler, M. Henini, and D. Tay-\nlor. Rashba spin-splitting of electrons in asymmetric quan -\ntum wells. Phys.Rev.B, 82(4):045317, Jul 2010.\n41Y. Kato, R. C. Myers, A. C. Gossard, and D. D.\nAwschalom. Coherentspinmanipulation withoutmagnetic\nfields in strained semiconductors. Nature, 427:50, 2004.\n42M. Beck, C. Metzner, S. Malzer, and G. H. Doehler. Spin\nlifetimes and strain-controlled spin precession of drifti ng\nelectrons in zinc blende type semiconductors. Europhys.\nLett., 75:597, 2006.\n43H. Sanada, T. Sogawa, H. Gotoh, K. Onomitsu, M. Kohda,\nJ. Nitta, and P. V. Santos. Acoustically induced spin-orbit\ninteractions revealed by two-dimensional imaging of spin\ntransport in GaAs. Phys.Rev.Lett., 106(21):216602, May\n2011.\n44H. Sanada, Y. Kunihashi, H. Gotoh, K. Onomitsu, M. Ko-\nhda, J. Nitta, P. V. Santos, and T. Sogawa. Manipulation\nof mobile spin coherence using magnetic-field-free electro n\nspin resonance. Nat.Phys., 9:280, March 2013.\n45M. I. D’yakonov and V. I. Perel’. Spin relaxation of con-\nduction electrons in noncentrosymmetric semiconductors.\nSov.Phys.SolidState, 13:3023–3026, 1972."    },    {        "title": "2402.04981v1.SWAP_algorithm_for_lattice_spin_models.pdf",        "content": "SWAP algorithm for lattice spin models\nGreivin Alfaro Miranda,1Leticia F. Cugliandolo,1, 2and Marco Tarzia3\n1Sorbonne Universit´ e, Laboratoire de Physique Th´ eorique et Hautes Energies,\nCNRS UMR 7589, 4 Place Jussieu, 75252 Paris Cedex 05, France\n2Institut Universitaire de France, 1 rue Descartes, 75005 Paris France\n3Sorbonne Universit´ e, Laboratoire de Physique Th´ eorique de la Mati` ere Condens´ ee,\nCNRS UMR 7600, 4 Place Jussieu, 75252 Paris Cedex 05, France\n(Dated: February 8, 2024)\nWe adapted the SWAP molecular dynamics algorithm for use in lattice Ising spin models. We\ndressed the spins with a randomly distributed length and we alternated long-range spin exchanges\nwith conventional single spin flip Monte Carlo updates, both accepted with a stochastic rule which\nrespects detailed balance. We show that this algorithm, when applied to the bidimensional Edwards-\nAnderson model, speeds up significantly the relaxation at low temperatures and manages to find\nground states with high efficiency and little computational cost. The exploration of spin models\nshould help in understanding why SWAP accelerates the evolution of particle systems and shed light\non relations between dynamics and free-energy landscapes.\nThe slow relaxation of spin and structural glasses [1–4]\nposes a computational challenge because it is not feasible\nto investigate their progression towards equilibrium us-\ning standard numerical simulations. Efforts to overcome\nthis slowdown have led to the development of specialized\ncomputers and numerical techniques.\nSpin-glasses are interesting physical systems which also\nattract theoretical attention because they map to a broad\nrange of hard combinatorial optimization problems. In\nthe context of these frustrated magnets, methods such as\nparallel tempering [5], as well as the recently introduced\nreplicated simulated annealing [6, 7], enable the equili-\nbration of larger sample sizes compared to conventional\nsingle spin flip Monte Carlo. The special purpose Janus\nmachine has allowed to equilibrate spin-glass models of\nrather large sizes, though only those with discrete vari-\nables and couplings [8]. More recently, deep reinforce-\nment learning methods have been explored to find the\nground states of finite dimensional spin-glasses [9–11].\nIn the context of structural glasses, Berthier et al. [12–\n14] made significant advancements over previous non-\nlocal particle exchange methods [15], achieving acceler-\nated equilibration of several glass-forming liquids. Their\nbreakthrough was first realized in polydisperse mixtures\nwhich still exhibit the characteristics of glass-forming liq-\nuids, and then generalized to many other models [13]\nThe method incorporated standard Molecular Dynam-\nics, periodically alternated with a step in which two ran-\ndomly selected particles with typically different size are\nexchanged using a Monte Carlo rule satisfying detailed\nbalance. This technique, referred to as SWAP, has proven\nsuccessful in equilibrating particle systems of unprece-\ndented size, all the way down to the glass transition tem-\nperature [16].\nThis spectacular effect was interpreted as direct ev-\nidence against a static, cooperative explanation of the\nglass transition such as the one offered by the ran-\ndom first-order transition theory (RFOT) [17]. Yet thisclaim was contested in [18] where the efficiency of SWAP\nwas explained in terms of its ability to avoid the slow-\ndown caused by numerous metastable local minima in\nthe (free-)energy landscape: the non-local particle ex-\nchanges thus might be overcoming barriers quickly, in\ncontrast to a strictly local dynamics that would ne-\ncessitate cooperative rearrangements to do it. Despite\nthe distinct (mean-field) free-energy landscapes of struc-\ntural and spin glasses, they both harbor a multitude of\nmetastable states. This suggests the potential success of\na similar algorithm in accelerating the dynamics of spin\nglasses, models in which dynamics facilitation [19] is ab-\nsent by construction.\nIn this Letter, we adapt the SWAP method for applica-\ntion to finite-dimensional spin models. In this reshaping\nwe introduce a bypass model in which we assign a length\nto the spin variables, akin to the role of particle diameters\nin the original SWAP implementation. The exchange of\nconstituents, in our case, the spins, will effectively mit-\nigate the local energy barriers created by the quenched\nrandomness, allowing us to explore configurations that\nwould otherwise remain inaccessible.\nWe have chosen to focus on two two-dimensional (2D)\nproblems for which we know the equilibrium phases and\nground states. In the Supplemental Material (SM) we\nvalidate the method with a study of a clean and un-\nfrustrated spin system with a finite-temperature second-\norder transition from a paramagnetic to a ferromagnetic\nphase. The core of this work is the study of the 2D\nEdwards-Anderson (EA) model, a random magnet with\nspin-glass properties only at zero temperature, but ex-\nceedingly long physical relaxation at low temperatures.\nThis 2D problem is not just a theoretical construct: thin\nfilm spin-glass materials have regained experimental in-\nterest in recent years [20]. Moreover, the ground state\nconfigurations can be identified exactly with special algo-\nrithms [21, 22]. This gives us a knowledge against which\nwe can confront the performance of our algorithm.arXiv:2402.04981v1  [cond-mat.stat-mech]  7 Feb 20242\nConcretely, we modify the Ising models by introduc-\ning associated “∆-models” in which the lengths of the\nspins are drawn initially from a pre-determined probabil-\nity distribution. We then perform a single-spin-flip evo-\nlution alternating at random time-steps with a non-local\nexchange of spins, the swaps. The distribution of the\nspin-lengths remains unchanged but local energy barriers\nare uplifted so that the acceptance of new configurations\nis propitiated. In the ∆-model with uniform ferromag-\nnetic couplings, SWAP does not improve over the stan-\ndard single-spin-flip evolution of the parent Ising model\n(see the SM). In contrast, in the frustrated case, SWAP\nmanages to take the ∆-model to equilibrium faster than\nusual implementations of parallel tempering do for the\nEA model at low temperatures. After instantaneous\nquenches, we find the ground state of the ∆ = 2-Model\naround 2 decades faster than with parallel tempering\nMonte Carlo [23, 24]. This is further improved by a tem-\nperature annealing, that enables us to find at least 99%\nof the ground states for ∆ ≥0.5.\nWe start from a finite-dimensional Ising Model (IM),\nH=−X\n⟨ij⟩Jijσiσj, σ i=±1, i= 1, . . . , N (1)\nwhere N=LD,⟨ij⟩indicates a sum over nearest neigh-\nbors on a D dimensional square lattice with linear size\nL(each pair added once) and periodic boundary con-\nditions. In a clean ferromagnetic model the coupling\nstrengths are all equal Jij=J > 0. In the EA model,\nthey are drawn from a symmetric probability distribu-\ntion centered at zero, and with variance J2. The 2DEA\nmodel has degenerate spin-glass ground states with a gap\nto the lowest excitations for bimodal couplings, and only\ntwo ground states related by symmetry with a continuous\nspectrum of excitations in the Gaussian case. Otherwise,\ns+s¡S¡S+\nFIG. 1: Sketch of a spin configuration of the modified 2DEA\nmodel. Two spins are singled out for analysis (surrounded\nby green bubbles). The neighboring up and down spins are\ncolored red and blue, respectively. The solid (red) and dashed\n(blue) links represent Jij>0 and Jij<0, respectively. The\nlength of the arrows are proportional to the length of the\nspins, that is, the local τivalues. They are here chosen to\ntake only two values, for simplicity.the model is paramagnetic at any non-vanishing temper-\nature. The Gaussian system undergoes domain growth at\nT= 0 [25]. In the bimodal case, clusters of spins main-\ntain their relative orientation in all ground states and\nform a backbone on which coarsening takes place while\nall other spins behave paramagnetically [26]. Single spin\nflip dynamics freezes at T≲0.2 (see Fig. SM20).\nWe will consider a variation of model (1) in which the\nspins have different amplitudes [27] or lengths,\nH=−X\n⟨ij⟩Jijsisj, s i=σiτi. (2)\nTheσis are Ising variables σi=±1 and the τiare in-\ndependently and initially drawn from a normalized box\ndistribution, pτ(τi),i.e.\nτi∈[1−∆/2,1 + ∆ /2], 0≤∆≤2. (3)\nThe average over the τidistribution is denoted [ ...]. The\nparameter ∆ controls the spin length variation: their\nmean and variance are [ τi] = 1 and [ τ2\ni]−[τi]2= ∆2/12.\n∆≤2 ensures that τi≥0, and the Ising case is recovered\nfor ∆ = 0. The space-varying τiinduce random interac-\ntions between the Ising spins even in the unfrustrated\ncase since (2) can be recast as\nH=−X\n⟨ij⟩Jijσiσj, Jij=Jijτiτj.(4)\nThe new continuously varying exchanges Jijfollow a\nsymmetric distribution function with a gap that closes\nfor ∆→2, see Fig. SM16, with local spatial correlations\nviatheτiwhich can be further enhanced by SWAP, see\nFig. 3(c).\nWe proved that, in mean-field, the only effect of ∆ is to\nmodify Tc(the calculations for the fully-connected and\nBethe lattice ∆-models are in the SM [28]). Additionally,\nthe real-space RG of the ∆-model on the hierarchical lat-\ntice yields the same fixed point as the one of the EA [29–\n31]. These results indicate that assigning a length to the\nspins does not change the physics of the problem.\nAt each Monte Carlo (MC) sweep, with probability p\nthe algorithm acts as follows:\np=\n\npswap 7→N(non-local) exchange attempts\nsi↔sj,\n1−pswap7→Nsingle spin flip attempts\nσi→ −σi.\nThe microscopic moves are accepted with the Metropo-\nlis acceptance probability pacc= min(1 , e−β∆E), where\nβ= 1/(kBT), and ∆ Eis the energy variation due to\nthei-th spin flip or the spin exchange between the i-th\nandj-th spins (more than one lattice spacing apart). In\nthe following, ⟨...⟩is an average over thermal MC noise\nand initial conditions of the {σi}and [ ...] stands for the\naverage over the couplings {Jij}and the diameters τi.3\nThe mechanism whereby the swaps accelerate the dy-\nnamics is exemplified in Fig. 1. For simplicity, in the\nsketch we use two spin lengths only, a large one (long\narrows and S±), and a smaller one (short arrows and\ns±), and bimodal Jij=±Jinteractions. The energy\nbarrier to flip the upper-left highlighted spin is ∆ E=\n4J(S2+Ss)>0 (as S > 0 and s >0); therefore, this\nspin is blocked at low temperatures. Instead, the energy\nvariation after an exchange of the two highlighted spins\nis ∆E=J(s2−S2)<0 (as s < S ). This non-local\nmove will take place since it is energetically favorable,\nand it may thus help unblocking the upper-left spin and\nits surroundings. This mechanism will be confirmed by\nthe analysis below.\nUnder single spin flips (i.e., pswap= 0), the couplings\nare quenched. Instead, the siexchanges partially an-\nneal the effective randomness Jijwith speed controlled\nbypswap. Frustrated plaquettes remain frustrated since\nneither the signs of JijnorJijchange. However, the\ndiffusion of the τican affect the magnitude of the local\nfrustration, quantified by fP≡Q\n⟨ij⟩∈PJij(t), where the\nproduct runs over the links of a plaquette P. The cumula-\ntive probability, defined as P<(fP< x) =Rx\n−∞pfP(y)dy\nfor negative fPi.e. frustrated plaquettes, is plotted for\n3 times reached with SWAP after a T= 0 quench in\nthe inset of Fig. 2. SWAP reduces the magnitude of the\nfrustration, as the probability of finding large negative\nvalues for fPdecreases with time, up until a constant\nfunctional form is reached. The two-time local correla-\ntion, CJ(t, tw) =P\ni,j[⟨Jij(t)Jij(tw)⟩]/P\ni,j[⟨J2\nij(tw)⟩],\ndisplayed in the main part of Fig. 2, becomes stationary\nand lim t≫1limtw≫1CJ(t, tw) = 1. Effectively quenched\nconfigurations of the effective couplings J∗\nij=Jij(tmax)\nare reached in each run after ∼105sweeps in a system\nwith L= 32. Importantly enough, not all quenches lead\nto the same final J∗\nijconfiguration. This is observed from\nthe calculation of the correlation of the τisampled in dif-\nFIG. 2: The two-time correlations of the couplings Jijin the\n∆ = 1 .5-model with L= 32 quenched to T= 0 evolved with\nSWAP ( pswap= 0.1). The waiting times tware given in the\nkey. Inset: the cumulative probability of local frustrations fP\nat three times after the quench.\nFIG. 3: (a) Asymptotic probability of reaching a ground\nstate after a T= 0 quench and a quadratic annealing, start-\ning from T0= 1.0 during tf≈107MCs. L= 32. Inset\n(b) Probability distributions of the ground state energy den-\nsity differences found after T= 0 quenches and annealing\nprotocols, Eq. (6), of models with different ∆. In all cases,\ntheJijand initial lengths {τi(t= 0)}are the same, and\nthe data are sampled over 103initial Ising spin conditions\n{σi(t= 0) = ±1}. (c) The overlap of the initial (left panel)\nand final (right panel) siconfiguration with the ground state\nof the model with couplings J∗\nij=Jij(tmax). The light bullets\nand triangles are located at frustrated plaquettes with local\nfrustration fPbeing greater or smaller than one, respectively.\nferent runs of the dynamics (according to the definition\nin Eq. (SM30), results not shown).\nWe now investigate the efficiency of the individual\nSWAP runs to reach the ground state of the 2DEA model\nwith interactions J∗\nij. With this aim we stored the cou-\nplings Jij(t) and the Ising spins σi(t). Concomitantly, we\nused the facility in Bonn [32] to find the unique (apart\nfrom global spin reversal) ground state σgs\niof a 2DEA\nmodel with the J∗\nijinteractions. Then, we calculated\nthe overlap and the probability of reaching the ground\nstate [23]\nq(t) =1\nNNX\ni=1σgs\niσi(t),P0(t) =1\nNrNrX\nα=1δ|qα(t)|,1,(5)\nwhere the index αruns over the simulation runs. Fig-\nure SM18 demonstrates that P∞\n0∼0.5 for ∆ = 1,4\nP∞\n0∼0.75 for ∆ = 1 .5 and P∞\n0saturates to one for\n∆ = 2, last blue bullet in Fig. 3. Optimization with re-\nspect to pswapshows no strong dependence on pswapfor\nvalues in between 0.05 and 0.5, say. We use pswap= 0.1\nhenceforth. However, an inconvenience resides in the fact\nthat even in the cases in which the ground states are\nreached, for the same realization of the Jijand initial\nconditions of the τi, different thermal evolutions produce\ndifferent final configurations with a broad energy density\ndistribution, inset in Fig. 3. Therefore, although being\nground states of the 2DEA model with couplings J∗\nij,\nthese configurations originate from metastable states of\ntheτivariables. Further optimization with respect to the\nτiis thus necessary.\nIn order to optimize both the τiandσi, we adopt the\nthermal protocol [33]\nT(t) =T0(1−t/tf)a, (6)\nwith T0= 1.0,tfthe total number of MC-sweeps, and\na= 1 (linear) or a= 2 (quadratic). SWAP is now able\nto find ground states 100% of the runs for almost all ∆\nvalues, pink triangles in Fig. 3. The spread of ground\nstate energies (Gaussian distributed) narrows consider-\nably with respect to the one of T= 0 quenches, inset\nin Fig. 3 (although it does not disappear completely).\nThe configurations reached are ground states of models\nwith only slightly different J∗\nij. This is confirmed by the\nevolution of the self correlation of the length variables in\ntwo different runs, τ(1)\niandτ(2)\ni: after an initial decorrela-\ntion which increases with ∆, these variables progressively\ncorrelate again to reduce frustration until becoming al-\nmost identical when T→0 at the final annealing time\n(Fig. SM23).\nFIG. 4: The two-time Ising spin self-correlation at three\nwaiting-times. Data for four kinds of T= 0 evolution of a\nsystem with L= 32 and ∆ = 1 .5 starting from random initial\nconditions: single spin flips of the ∆ model with Jij(i) and\nJ∗\nij(ii), SWAP with only local exchanges (iii) and, our main\ninterest, SWAP with non-local ones.\nLet us now compare the speed enhancement of the\nnon-local SWAP algorithm in relation to other methods.\nFIG. 5: Characteristic relaxation time ταextracted from the\ndecay of the σself-correlation after quenches to the target\ntemperatures, evolved with SWAP with non-local and only\nlocal spin exchanges with the same pswap= 0.1. In the inset,\ncomparison of the two relaxation times, as a function of tem-\nperature for two ∆-models.\nWe confront it to the evolution of random initial σiun-\nder single spin flips of the 2DEA with (i) Jijcouplings\n(in which the τi’s are quenched), (ii) the optimized J∗\nij\nfound with the quadratic annealing (in which the τi’s\nare quenched but close to the least frustrated pattern),\nand (iii) SWAP with only local exchanges (in which the\nτi’s are annealed). With this aim we calculated the self-\ncorrelation Cσ(t, tw) =N−1PN\ni=1[⟨σi(t)σi(tw)⟩] after a\nT= 0 quench for three waiting times. The results are\nplotted in Fig. 4 which shows that for single spin flips\nfreeze and the system is trapped for L= 32 systems.\nMoreover, non-local SWAP is more efficient in advancing\nthe evolution than just local exchanges.\nTo precisely quantify the speed up achieved by non-\nlocal moves in comparison to only local swaps, we com-\npared a characteristic relaxation time, τα, of both dy-\nnamics at rather high temperatures, to ensure equilib-\nrium. We define this ταas the time for which the self-\ncorrelation has become age independent (i.e. C(t, tw) =\nC(t−tw)) and has decayed to 20% of its original value\n(i.e.C(τα) = 0 .2C(t−tw= 0)). As can be seen in\nFig. 5 the non-local spin exchanges accelerates the dy-\nnamics as we decrease the temperature, improving with\nrespect to the local exchanges by one order of magnitude\natT= 0.9, both dynamics are indistinguishable at higher\ntemperatures (here, around T≈2.4).\nLet us summarize our results. We adapted the SWAP\nalgorithm to act on finite dimensional spin models. We\nshowed that it accelerates the evolution of a 2D disor-\ndered model at very low temperatures. The method al-\nlowed us to sample ground states of an Ising spin-glass\nwith little numerical effort. Other parameters to opti-\nmize, which we have not explored in depth yet, are Pτ(τi)\nand the annealing scheme. The knowledge gained from\nthe 2D models studied here will serve us to extend this\nstudy to the more interesting 3D cases with a finite tem-5\nperature phase transition.\nIn the disordered spin model, the sluggish dynamics\nat low temperatures is due to the intricate nature of\nthe (free-)energy landscape. Hence, the efficiency of the\nSWAP algorithm in accelerating the dynamics can only\nbe attributed to the partial smoothing of the landscape,\nwhich mitigates higher barriers as needed. This is fur-\nther evidenced by the lack of acceleration in the non-\ndisordered Ising case, characterized by a “simple” land-\nscape. However, the acceleration achieved in the spin\nmodel is modest compared to the one accomplished in\nstructural glasses, suggesting that facilitation [19] may\nalso be at work in the latter case. Besides, we observe\ncorrelated patterns of spin lengths, τi, at low tempera-\ntures, hinting at similar phenomena in structural glasses,\nwhich calls for further investigation.\nAcknowledgments. LFC acknowledges financial support\nfrom ANR-19-CE30-0014 and ANR-20- CE30-0031. We\nthank L. Berthier, G. Biroli, J. Kurchan, E. Marinari,\nF. Ricci-Tersenghi, F. Rom´ a, J. J. Ruiz-Lorenzo and F.\nZamponi for discusions and suggestions.\n[1] J.-P. Bouchaud, L. F. Cugliandolo, J. Kurchan, and\nM. M´ ezard, 12, 161 (1998).\n[2] L. Berthier and G. Biroli, Rev. Mod. Phys. 83, 587\n(2011).\n[3] L. Berthier, G. Biroli, J.-P. Bouchaud, L. Cipelletti, and\nW. van Saarloos, Dynamical Heterogeneities in Glasses,\nColloids, and Granular Media (Oxford University Press,\nOxford, 2011).\n[4] G. I. Menon and P. Ray, The physics of disordered sys-\ntems (Hindustan Book Agency, Gurgaon, 2012).\n[5] D. J. Earl and M. W. Deem, Phys. Chem. Chem. Phys.\n7, 3910 (2005).\n[6] C. Baldassi, C. Borgs, J. T. Chayes, A. Ingrosso, C. Lu-\ncibello, L. Saglietti, and R. Zecchina, Proc. Nat. Acad.\nSc.113, E7655 (2016).\n[7] M. C. Angelini and F. Ricci-Tersenghi, Phys. Rev. X 13,\n021011 (2023).\n[8] F. Belletti et al., Computing in Science & Engineering\n11, 48 (2009).\n[9] C. Fan, M. Shen, Z. Nussinov, Z. Liu, Y. Sun, and Y.-Y.\nLiu, Nat. Commun. 14, 725 (2023).\n[10] S. Boettcher, Nature Communications 14(2023).\n[11] C. Fan, M. Shen, Z. Nussinov, Z. Liu, Y. Sun, and Y.-Y.\nLiu, Nat. Commun. 14(2023).\n[12] L. Berthier, D. Coslovich, A. Ninarello, and M. Ozawa,\nPhys. Rev. Lett. 116, 238002 (2016).\n[13] A. Ninarello, L. Berthier, and D. Coslovich, Phys. Rev.\nX7, 021039 (2017).\n[14] L. Berthier, E. Flenner, C. J. Fullerton, C. Scalliet, and\nM. Singh, J. Stat. Mech. 2019 , 064004 (2019).\n[15] L. Berthier and D. R. Reichman, Nat. Rev. Phys. 5, 102\n(2023).\n[16] C. Scalliet, B. Guiselin, and L. Berthier, Phys. Rev. X\n12, 041028 (2022).\n[17] M. Wyart and M. E. Cates, Phys. Rev. Lett. 119, 195501(2017).\n[18] L. Berthier, G. Biroli, J.-P. Bouchaud, and G. Tarjus, J.\nChem. Phys. 150, 094501 (2019).\n[19] D. Chandler and J. P. Garrahan, Annual review of phys-\nical chemistry 61, 191 (2010).\n[20] S. Guchhait and R. Orbach, Phys. Rev. Lett. 112, 126401\n(2014).\n[21] A. K. Hartmann, J. Stat. Phys. 144, 519 (2011).\n[22] H. Khoshbakht and M. Weigel, Phys. Rev. B 97, 064410\n(2018).\n[23] F. Rom´ a, S. Risau-Gusman, A. J. Ram´ ırez-Pastor, F. Ni-\neto, and E. E. Vogel, Physica A 388, 2821 (2009).\n[24] W. Wang, J. Machta, and H. G. Katzgraber, Phys. Rev.\nE92, 013303 (2015).\n[25] J. Kisker, L. Santen, M. Schreckenberg, and H. Rieger,\nPhys. Rev. B 53, 6418 (1996).\n[26] F. Rom´ a, S. Bustingorry, and P. M. Gleiser, Phys. Rev.\nLett. 96, 167205 (2006).\n[27] M. Krasnytska, B. Berche, Y. Holovatch, and R. Kenna,\nJ. Phys.: Complexity 1, 035008 (2020).\n[28] M. Mezard and G. Parisi, Eur. Phys. J. B 20, 217 (2001).\n[29] B. W. Southern and A. P. Young, J. Phys. C: Solid State\nPhys. 10, 2179 (1977).\n[30] B. Drossel, H. Bokil, and M. A. Moore, Phys. Rev. E 62,\n7690 (2000).\n[31] B. Drossel and M. Moore, Eur. Phys. J. B 21, 589 (2001).\n[32] J. Charfreitag, M. J¨ unger, S. Mallach, and P. Mutzel, in\n2022 Proceedings of the Symposium on Algorithm Engi-\nneering and Experiments , edited by C. A. Phillips and\nB. Speckmann (2022), p. 54.\n[33] S. J. Rubin, N. Xu, and A. W. Sandvik, Phys. Rev. E\n95, 052133 (2017).\n[34] M. Krasnytska, B. Berche, Y. Holovatch, and R. Kenna,\nEntropy 23, 1175 (2021).\n[35] M. Dudka, M. Krasnytska, J. J. Ruiz-Lorenzo, and\nY. Holovatch, J. Magn. Magn. Mat. 575, 170718 (2023).\n[36] M. Henkel and M. Pleimling, Phys. Rev. B 78, 224419\n(2008).\n[37] A. Bray, Adv. Phys. 43, 357 (1994).\n[38] D. A. Huse and C. L. Henley, Phys. Rev. Lett. 54, 2708\n(1985).\n[39] A. R. Rivera, G. A. P´ erez Alc´ azar, and J. A. Plascak,\nPhys. Rev. B 41, 4774 (1990).\n[40] A. Sicilia, J. J. Arenzon, A. J. Bray, and L. F. Cuglian-\ndolo, EPL 82, 10001 (2008).\n[41] J. L. Iguain, S. Bustingorry, A. B. Kolton, and L. F.\nCugliandolo, Phys. Rev. B 80, 094201 (2009).\n[42] F. Corberi, E. Lippiello, A. Mukherjee, S. Puri, and\nM. Zannetti, J. Stat. Mech. 2011 , P03016 (2011).\n[43] F. Corberi, L. F. Cugliandolo, F. Insalata, and M. Picco,\nPhys. Rev. E 95, 022101 (2017).\n[44] R. Paul, S. Puri, and H. Rieger, EPL 68, 881 (2004).\n[45] R. N. Bhatt and A. P. Young, Phys. Rev. B 37, 5606\n(1988).\n[46] J.-S. Wang and R. H. Swendsen, Phys. Rev. B 38, 4840\n(1988).\n[47] K. Hukushima and K. Nemoto, J. Phys.: Condens. Mat-\nter5, 1389 (1993).\n[48] C. K. Thomas, D. A. Huse, and A. A. Middleton, Phys.\nRev. Lett. 107, 047203 (2011).\n[49] F. Barahona, R. Maynard, R. Rammal, and J. P. Uhry,\nJ. Phys. A: Math. Gen. 15, 673 (1982).\n[50] F. Barahona, J. Phys. A: Math. Gen. 15, 3241 (1982).\n[51] H. Takayama, K. Nemoto, and H. Matsukawa, J. Magn.6\nMagn. Mat. 31-34 , 1303 (1983).\n[52] D. A. Huse, Phys. Rev. B 43, 8673 (1991).\n[53] H. Rieger, B. Steckemetz, and M. Schreckenberg, Euro-\nphys. Lett. 27, 485 (1994).\n[54] C. K. Thomas and A. A. Middleton, Phys. Rev. E 87,\n043303 (2013).\n[55] N. Xu, K.-H. Wu, S. J. Rubin, Y.-J. Kao, and A. W.\nSandvik, Phys. Rev. E 96, 052102 (2017).\n[56] L. A. Fern´ andez, E. Marinari, V. Mart´ ın-Mayor,\nG. Parisi, and J. J. Ruiz-Lorenzo, J. Stat. Mech. 2018 ,\n103301 (2018).\n[57] H. Khoshbakht and M. Weigel, Phys. Rev. B 97, 064410\n(2018).\n[58] L. A. Fern´ andez, E. Marinari, V. Mart´ ın-Mayor, I. Paga,\nand J. J. Ruiz-Lorenzo, Phys. Rev. B 100, 184412 (2019).\n[59] L. A. Fern´ andez, E. Marinari, V. Mart´ ın-Mayor,\nG. Parisi, and J. J. Ruiz-Lorenzo, J. Phys. A: Math.\nTheor. 52, 224002 (2019).\n[60] H. Rieger, in Annual Reviews of Computational Physics ,\nedited by D. Stauffer (World Scientific, 1995), vol. 2, p.\n295.7\nSUPPLEMENTAL MATERIAL\nContents\nA. The ferromagnetic Model 7\nA.1 An equivalent Random Bond Ising Model 8\nA.2 Equilibrium properties 8\nA.2.1 The averaged magnetizations 8\nA.2.2 The phase transition 10\nA.3 Dynamical properties 11\nA.3.1 Instantaneous configurations 11\nA.3.2 The energy density 11\nA.3.3 The space-time correlation 11\nA.3.4 The growing length for the soft spins 13\nA.3.5 The growing length for the Ising spins 14\nA.4 Conclusions 14\nB The spin-glass Model 15\nB.1 An equivalent frustrated Ising model 15\nB.2 The mean-field critical temperature 15\nB.3 The probability of reaching the ground state after a T= 0 quench 17\nB.4 The overlap correlation and the spin-glass growing length 17\nB.5 The correlation of the τivariables 19\nB.6 The two-time correlation 19\nB.8 Measures of frustration 19\nB.7 Instantaneous configurations 21\nA. THE FERROMAGNETIC MODEL\nIn this Section we test the performance of the SWAP\nmethod when applied to a clean ferromagnetic (FM)\nmodel. Concretely, we compare the rate of approach to\nequilibrium of the SWAP algorithm applied to a ∆-model\nbuilt upon the standard 2DIM, to the one of single spin\nflips applied to the unfrustrated Ising case. In both cases\nwe work below their finite temperature critical points.\nWe start by generalizing the Hamiltonian of the 2DIM\nH=−JX\n⟨ij⟩sisj=−JX\n⟨ij⟩τiσiτjσj. (SM1)\nJ > 0 and, in the rest of this Section we rescale the\ninteractions so as to set J= 1. Concretely, we work with\na square lattice with periodic boundary conditions. We\nuse the summation convention in Eq. (SM1) such that the\ncritical temperature of the Ising model is TIM\nc= 2.27.\nA ferromagnetic model with spins with variable size\nhas been considered in [27, 34, 35]. One of the motiva-\ntion was to describe “structurally-disordered magnets”\nwith two (or more) chemically different magnetic compo-\nnents. Specificaly, the critical properties of model with\nvariable Ising spin lengths drawn from a bimodal distri-\nbution function, was studied in detail in [35], with specialemphasis on its critical properties. We will comment on\ntheir findings when discussing the critical properties of\nour model.\nAt the initial time of the simulation we need to choose\nthe orientation of the Ising spins σiand the lengths τi\nof the spins si. For the former we consider two cases,\nσi=±1 with probability a half, mimicking an infinite\ntemperature initial state, or σi= 1 for all irepresenting\na zero temperature one. For the latter, we draw the τi\nindependently from the box distribution (3).\nAs explained in the main text, we alternate conven-\ntional Monte Carlo (MC) updates and spin exchanges\nwith probability p. For this model, the energy vari-\nation employed in the acceptance probability pacc=\nmin(1 , e−β∆E), is\n∆E=\n\n2siX\nj∈∂isj σi→ −σi\n(si−sj)\nX\nk∈∂isk−X\nk∈∂jsk\nsi↔sj\nThe first line corresponds to the flipping of the i-th site\nand the second one to the exchange between the spins on\nthei-th and j-th sites, respectively, which are more than\none lattice spacing apart. The symbol ∂irepresents the8\nneighbors of the ith spin, i.e. the four nearest neighbors\non the square lattice. Unless pswap = 1, the algorithm\nproduces non-conserved order parameter kinetics.\nA.1 An equivalent Random Bond Ising Model\nAfter the introduction of si=τiσi, random interac-\ntions between the Ising spins σiemerge. It is straightfor-\nward to see that by separating the Ising degrees of free-\ndom and the length-ones, the product of the latter plays\nthe role of random couplings in the original Ising Hamil-\ntonian (SM1). A Random Bond Ising Model (RBIM) is\nthen recovered\nH=−X\n⟨ij⟩Jijσiσj with Jij=Jτiτj,(SM2)\nthat is, a specific and structured distribution of the cou-\nplings Jijinduced by the one of the τilengths. This\ndistribution is neither the box nor the bimodal one usu-\nally considered in the literature. Being the variables τi\nandτji.i.d. for i̸=j, then p(Jij) =pτ(τi)pτ(τj), so\nthat the probability distribution function (pdf) of the\nnew couplings can be found by means of a Mellin trans-\nform, yielding\np(Jij) =\n\n1\n∆2log\u0012Jij\nu2\n−\u0013\nu2\n−≤ Jij≤u−u+\n1\n∆2log\u0012u2\n+\nJij\u0013\nu−u+≤ Jij≤u2\n+\n0 elsewhere\nwith u−= 1−∆/2 and u+= 1 + ∆ /2.\nFIG. SM1: The probability distribution function of the cou-\npling strengths Jij=τiτj(J= 1) arising from the prod-\nuct of the spin-length variables. Several values of the length-\ncontrolling parameter ∆ given in the key are considered.\nThe mean and variance are\n[Jij] = 1 , [J2\nij]−[Jij]2=∆2\n144(∆2+ 24) .(SM3)A plot of p(Jij) for several values of the length\ncontrolling-parameter ∆ is shown in Fig. SM1, where\nit is clear that as we take ∆ →0 the pdf tends to a\nDirac-delta distribution centered at 1 as it is expected\nin the hard-spin limit. These new distributions maintain\nJij≥0 for all ∆, so frustration is avoided.\nHowever, notice that the Jijare not i.i.d. variables, as\nfor two bonds with a common spin, k, the exchanges are\ncorrelated\ni k jJik Jkj\n[JikJkj] = [τiτ2\nkτj] = [τi][τ2\nk][τj] = 1 +∆2\n12\n̸= [Jik][Jkj] = [τi][τk]2[τj] = 1 .(SM4)\nTherefore, the joint pdf of all couplings Jijis not just\nthe product of the individual pdfs p(Jij).\nWe will characterize the dynamics using both interpre-\ntations: firstly, as a ferromagnetic model with soft-spins\nand secondly, as a RBIM with the above kind of bonds.\nA.2 Equilibrium properties\nIn order to appropriately describe the coarsening dy-\nnamics we need to first locate the equilibrium critical\ntemperature. In particular, we have to establish its de-\npendence on the parameter ∆ and also characterize the\nequilibrium properties in the spontaneous symmetry bro-\nken phase and close to the critical point. Finally, we have\nto prove that the equilibrium properties do not depend\non the microscopic dynamic rules.\nSince the equilibrium configurations below the critical\npoint should be magnetized, for this study we initiate all\nsimulations in σ-ordered configurations, σi= 1 for all\ni. In this way we force a positive magnetization at low\ntemperatures. The length variables τiare drawn from the\nbox distribution initially. In this way we sample different\npositively magnetized initial configurations of the sisoft\nspins.\nA.2.1 The averaged magnetizations\nTwo magnetization densities can be defined,\nms=1\nNNX\ni=1[⟨si⟩], (SM5)\nmσ=1\nNNX\ni=1[⟨σi⟩], (SM6)\nwhere the angular brackets denote average over thermal\nnoises and the square brackets average over the distribu-\ntion of the τis. If one assumes that, in equilibrium, the9\nFIG. SM2: The temperature dependence of both equilibrium\nmagnetization densities (SM5) and (SM6) in the FM model\nwith L= 160 and ∆ = 1. A hundred zero temperature, com-\npletely ordered, initial conditions with different choices of the\nτivariables were evolved with both single-spin flip dynam-\nics and the SWAP method with pswap= 0.5 during T= 216\nMC-sweeps. After this time, considered to be sufficient for\nequilibration, we sampled the magnetization densities at 104\nMC-sweeps. The small difference between the σandsmag-\nnetization densities drops at high temperatures, as expected\nfor the convergence of mσ=msto zero in the paramagnetic\nphase. Here and in all other plots the errorbars are estimated\nfrom the standard deviations.\naverage [ ⟨τiσi⟩] factorizes as [ τi][⟨σi⟩], and using the fact\nthat [ τi] = 1 (for L→ ∞ ), then\nms=1\nNX\ni[τi][⟨σi⟩] =mσ∀T,∆. (SM7)\nThis hypothesis is put to the test in Fig. SM2 where\nwe plot both magnetization densities for an equilibrated\nL= 160 lattice and a not too strong disorder, ∆ = 1.\nThe data show a very small systematic deviation, with\nmσ≤ms, suggesting that there might be a weak corre-\nlation between the τiandσivariables along the evolu-\ntion, disappearing at high temperatures when the para-\nmagnetic disordered phase prevails with ms=mσ= 0.\nIndeed, the difference is due to the fact that thermal fluc-\ntuations tend to favor the reversal of shorter spins in the\nsoft-spin case, since these flips cost less energy than the\nones of longer spins, and hence msis slightly higher than\nmσin the ordered phase. (The same features are visible\nwithin the domains in the out of equilibrium snapshots\nin Fig. SM8.)\nIn Fig. SM2 we show equilibrium data obtained with\nthe two microscopic dynamics under study, pure single\nspin-flip and SWAP. The static equilibrium properties\nare blind to the choices pswap= 0.5 and pswap= 0. The\nequivalence between the datasets built with the two dy-\nnamic rules can be verified for other values of pswap in\nthe range 0 ≤pswap<1 and other values of ∆.\nThe distribution of the spin sivalues at three times,\nt= 0 and two subsequent ones, for evolutions that led to\na positive magnetized domain are shown in Fig. SM3.The t= 0 data are the distribution of the spin-\nlengths (3), multiplied by ±1. The evolution drives the\nsystem to positive magnetization and the weight of the\npdf progressively moves to the positive support. Note\nthat the two peaks are not symmetric around their mid-\npoints for t >0.\nFIG. SM3: Probability distribution of the spins siafter a\nquench to T= 0.77Tcfrom an infinite temperature ini-\ntial condition, with (left) single-spin-flip kinetics and (right)\nSWAP dynamics ( pswap= 0.5). Data were sampled using 50\nrealizations of the τis, with L= 128 and ∆ = 1.\nFIG. SM4: The ∆ dependence of the spin smagnetization\ndensity for L= 16 using both kinds of dynamics. The tem-\nperature is set to T= 1.67 in all cases which corresponds to\n0.74< T/T c<0.84 depending on the ∆ considered. The\nequilibration time was set to 218MC-steps and data points\nand error bars were calculated using initial states with σi= 1\nand 800 choices of the τi.\nThe relaxation time depends on the size of the sys-\ntem, temperature and the length-controlling parameter\n(or disorder-width), teq(L;T,∆). Therefore, by increas-\ningLto reduce finite size effects we are in turn increasing\nthe relaxation time, and this makes equilibrium harder\nto access. In order to test the equilibrium properties at\nlarge values of ∆ and, in particular, the fact that they\ndo not depend on the microscopic dynamics, it is con-\nvenient to use small L. In Fig. SM4 we compare the ∆\ndependence of the equilibrium magnetization obtained\nwith single spin flip and SWAP dynamics. Just a few\naveraged values show a deviation, with the SWAP ones\nbeing slightly below the single spin flip ones. This differ-10\nence is not systematic and in any case very weak so we\ndo not consider it relevant.\nA.2.2 The phase transition\nA second order ferromagnetic-paramagnetic transition\nseparates magnetized and paramagnetic phases at a ∆\ndependent Tc. The Binder Cumulant, defined as\ngs\nL= 1−\u0014\u001c\u0010PN\ni=1si\u00114\u001d\u0015\n3\u0014\u001c\u0010PN\ni=1si\u00112\u001d\u00152, (SM8)\nallows one to pin-down the critical point. It locates the\ncritical temperature where the gs\nLdata for different L\ncross, as displayed in Fig. SM5. We find Tc(∆ = 0) ∼\n2.27 in good agreement with TIM\nc. Then, Tc(∆ = 1) =\n2.17, a slightly lower value than TIM\nc. In Fig. SM6 we\nplot the full ∆ dependence of the critical temperatures\nestimated from the crossing of the Binder parameter.\nThe continuous line is a linear fit which represents the\ndata rather accurately. The range of variation of Tc,\n2.05−2.27, with the parameter ∆ ∈[0,2] is of the same\norder as the one found in other works for the conven-\ntional RBIM with a box distribution of couplings and\n[Jij] = 1 [36]. The analysis of the Binder cumulant of\nthe magnetization mσand the two equilibrium magneti-\nzations obtained with the SWAP method yield a Tc(∆)\nwhich is equivalent to this one within error bars.\nWe have also performed a high temperature series ex-\npansion analysis (not shown) that tends to confirm that\nTc(∆) decreases with increasing ∆.\nWith the critical temperature and its dependency on ∆\nassessed, we proceed to probe the equilibrium behavior\naround criticality at Tc(∆), and compare it to that of\nFIG. SM5: Binder cumulant’s temperature dependence for\nthe equilibrium spin smagnetization defined in Eq. (SM8)\nobtained with single spin flip dynamics and ∆ = 1. The\nresults are equivalent for equilibrium data obtained with the\nSWAP evolution with pswap= 0.5.\nFIG. SM6: The critical temperature Tcestimated with MC\nsimulation (datapoints) and a linear fit (curve) against ∆.\nThe 2DIM critical value is found at ∆ = 0. The variation of\nTcwith ∆ is quite weak.\nthe 2D pure ferromagnetic Ising universality class, for\nwhich the magnetization and correlation length critical\nexponents are\nβ= 0.125, ν = 1, (SM9)\nrespectively. In Fig. SM7 the equilibrium magnetizations\nmsof the model with ∆ = 1 and different system sizes\nare scaled using the Ising values (SM9). The data fall on\na single master curve, confirming that the critical prop-\nerties of the 2DIM are preserved in the ∆-model.\nFIG. SM7: Critical scaling of the equilibrium magnetization\ndensity ms, with t≡(T−Tc)/Tcthe reduced temperature.\n∆ = 1. We fixed the critical exponents to the values of the\nclean ferromagnetic 2DIM, β= 1/8 and ν= 1, and we used\nthe estimated value of Tcfor ∆ = 1.\nIn [35] the critical properties of this very same model\nwith a bimodal distribution of lengths was studied an-\nalytically and numerically. It was shown in this paper\nthat, apart from a special length distribution, in all other\ncases the criticality is the one of the 2D dilute Ising\nmodel. According to the Harris criterium, the 2D di-\nlute Ising model is marginal and the critical exponents11\n(apart from logs) are the same as the ones of the 2DIM.\nA.3 Dynamical properties\nHaving checked that the equilibrium properties of the\n∆-model do not deviate significantly from the ones of the\nclean ferromagnetic Ising universality class, we proceed\nto study the domain growth. We start the dynamics\nfrom an infinite temperature initial configuration and we\nperform an instantaneous sub-critical quench. Studying\nthe coarsening phenomena in this set-up will provide us\nwith a direct survey of the microscopic evolution, hence\nletting us contrast the efficiency of the SWAP method to\nthe one of the standard single-spin-flip kinetics.\nA.3.1 Instantaneous configurations\nWe first present snapshots of the system domains along\nthe Monte Carlo evolution. As with the magnetization\ndensities, there are two ways to analyze the domains: 1)\nto consider the sispins as a whole, keeping track of both\ntheir sign and length, or 2) to isolate the Ising depen-\ndence σi. Clearly, the former carries more information\nthan the latter.\nSnapshots of the sispins obtained with single spin\nflip and pswap = 0.5 SWAP dynamics, are displayed in\nFig. SM8 top and bottom, respectively. As we lost the\nbinary description of the spin variables, we add a color\nheat map to distinguish between the different local spin\nlengths.\nAs in any domain growth process the size of the do-\nmains increases with time. However, here, there are fluc-\ntuations within the domains with the same orientation\nbut smaller absolute value than the rest (green within\nblue, and yellow within red). The interfaces are typically\nconstituted by a bilayer with short spin length (green and\nyellow) located in between oriented regions with large\nspin length (blue and red). The domains built with\nSWAP (lower row) allow the spins with longer length to\nget together more easily and hence appear with darker\ncolor than the ones grown with single spin flip dynamics.\nTo quantify the rate at which domain growth occurs we\nneed to measure the time-dependent typical domain size,\nR(t), along the simulation. There are several numerical\nways to get an indirect or direct measurement of this\nquantity. In the following we extract it from the inverse\ndomain perimeter density and the space-time correlation.\nA.3.2 The energy density\nThe inverse domain perimeter density,\nR(t) =−eeq\ne(t)−eeq=−eeq\nδe(t), (SM10)gives a first estimate of the growing length. The denom-\ninator δe(t) is the distance between the time-dependent\naveraged energy density and the equilibrium value, eeq=\nlimt→∞e(t). For the clean Ising model, R(t)∼t1/zdwith\nzdthe dynamical exponent. This in turn fixes a decay\nexponent for the energy density difference that would go\nasδe∼t−1/zd.zd= 2 for non-conserved order parame-\nter dynamics while zd= 3 for the locally conserved order\nparameter ones [37]. Comparing the dynamic exponents\nshould be the simplest way to see whether one algorithm\ndrives the system towards equilibrium faster than the\nother.\nWe calculated the equilibrium energy density in the\nlong-time limit ( t > 218MC-sweeps) of runs initiated\ninσ-ordered initial conditions and averaged over 100\nrealizations of the τis. The time-dependent e(t) =\n[⟨H(t)⟩]/Nwas computed, instead, after quenches from\ncompletely disordered initial conditions, with parameters\nsuch that the equilibrium state is ordered. The decay of\nthe excess energy curves is shown in Fig. SM9 in double\nlogarithmic scale. A power law fit of the energy decays\nobtained with both methods leads to a time-dependent\neffective exponent zeffreported in the caption.\nAfter a short transient in which the effective exponent\nof the single spin flip dynamics is close to 2, the decay of\nthe excess energy slows down and the effective exponent\nincreases in time reaching a value close to 3 in the con-\nsidered time window (blue data points in the main panel\nand the inset). The reason for this is that the variable\nlength model has, from the point of view of the Ising\nspin variables σi, quenched random bonds. Therefore,\nthe single spin flip dynamics feels these randomness and\nits evolution is slowed down, with the effective exponent\ndeveloping a dependence on the disordered strength, as\npointed out in the literature [36, 38–44].\nThe SWAP method circumvents the slowing down in-\ntroduced by the variable length of the spins. The dynam-\nical exponent of single-spin-flip kinetics of the 2DIM [37],\nzd= 2, is recovered when one adds spin exchanges (or-\nange data points in the main panel and the inset).\nA.3.3 The space-time correlation\nWe now focus on the one-time spatial correlation func-\ntion which also carries information on the typical domain\nsize along the evolution. As with the magnetization den-\nsities, we either measure the space-time correlation of the\ncomplete spin-variables,\nNCs(r, t) =NX\nj=1NX\nk=1[⟨sj(t)sk(t)⟩]\f\f\f\n|⃗ rj−⃗ rk|=r\n=NX\nj=1NX\nk=1[⟨τjσj(t)τkσk(t)⟩]\f\f\f\n|⃗ rj−⃗ rk|=r(SM11)12\nt=0 t= 128 t= 16384 t= 1 3107 2\nsi\n-2 -1 0 1 2\nFIG. SM8: Instantaneous snapshots of the ∆-model with ∆ = 2 for quench at T= 0.8Tc. First row: single-spin-flip dynamics.\nSecond row: SWAP dynamics. The color scale binned as indicated in the bar shows the lengths of the local spins. Note that\nthe sizes of the domains with the same orientational order look very similar but the darkness within them different. With\nSWAP longer spins can get together more easily.\nwhere the double sum over indices jandkis restricted\nby the condition on the distance between the spins. We\nnormalize by the number of terms considered. Or else we\nmeasure the space-time correlations of the Ising variables,\nin direct correspondence with the RBIM interpretation,\nNCσ(r, t) =NX\nj=1NX\nk=1[⟨σj(t)σk(t)⟩]\f\f\f\n|⃗ rj−⃗ rk|=r.(SM12)\nIn both cases, ⟨. . .⟩is the average over σiinitial condi-\ntions and noise realizations of the dynamics on the one\nhand, and over the 2D equidistant lattice sites on the\nother. As stated in prior Sections, [ ...] represents a dis-\norder average over several realizations of the lengths {τi}.\nWe note that\nCs(r= 0, t) =1\nNX\nj[τ2\nj] = 1 +∆2\n12,(SM13)\nCσ(r= 0, t) = 1 , (SM14)\nat all times t.Dynamic scaling states that a single domain length\nR(t) should scale all correlation functions. We now study\nseparately the correlations of the siandσispins to con-\nfirm that this is indeed the case in this problem.\nNot too close to the critical temperature, where the\nequilibrium magnetizations are not too low, one can es-\ntimate the typical domain size Rs,σ(t) from the rsuch\nthat the correlations Cs,σ(r, t) decay to, say, 1 /eof their\nzero distance values. A dynamic scaling regime, in which\nCs,σ(r, t)∼f\u0012r\nRs,σ(t)\u0013\n(SM15)\nis expected for ξeq≪r≪Lwith ξeqthe equilibrium\ncorrelation length and Lthe linear system size. Numeri-\ncally, it is convenient to measure ralong any of the two\naxes of the 2D lattice with PBCs.\nThe decay and scaling of the space-time correlation\nCs(r, t) in a model with ∆ = 1 are studied in Figs. SM10,\nusing the SWAP method with pswap = 0.5 and single\nspin-flips. At different times, the curves differ, demon-\nstrating, once again, the out of equilibrium character of13\nFIG. SM9: Averaged energy density relaxation of the fer-\nromagnetic ∆-model in double logarithmic scale. Disordered\ninitial configurations prepared at T0=∞were evolved at\n0.55Tcusing single-spin flip and SWAP kinetics with pswap=\n0.5. Data correspond to ∆ = 1 and L= 128, and were aver-\naged over 100 different initial conditions for both the σis and\ntheτis. A power law fit over moving time windows including\nthree data points yields the effective exponent zeffreported\nin the inset.\nthe dynamics.\nThe (golden) curve, which approaches at far distances\na finite constant, corresponds to the correlation decay of\nan ordered initial configuration that we heated up to the\ndesired temperature T < T cand evolves in equilibrium.\nAt very large r, the two spins in Eqs. (SM11) and (SM12)\nare expected to become independent and the average fac-\ntorize, ⟨sjsk⟩ ∼ ⟨sj⟩⟨sk⟩, leading to\nlim\nr→∞Cs\neq(r) = [⟨si⟩2] =m2\ns, (SM16)\nlim\nr→∞Cσ\neq(r) = [⟨σi⟩2] =m2\nσ. (SM17)\nMoreover, in equilibrium the two magnetization densities\nare almost identical. The first of these limits is verified\nnumerically in Fig. SM10 and the other one as well.\nDynamic scaling is checked in the inset of Fig. SM10\nand it works equally fine for the σcorrelations, see\nFig. SM11.\nThe dynamical scaling master curves, f(x), for Cnot\ntoo close to zero coincide, as can be seen in Fig. SM11.\nThere are differences when the Cs get close to zero, with\noscillations for SWAP which are absent for single spin\nflip (somehow reminiscent of the oscillations also present\nwhen working with local-spin-exhanges as in the case of\nKawasaki dynamics).\nA.3.4 The growing length for the soft spins\nThe growing length Rsmeasured from the spins s\nspace-time correlations generated with the two updat-\ning rules are studied in Fig. SM12 and Fig. SM13. In the\nFIG. SM10: SWAP dynamics with pswap= 0.5 of an L= 2048\nferromagnetic system with ∆ = 1, quenched from T0=∞\ntoT≈0.77Tc. Main panel: space-time correlation of the\nsspins at several times given in the key. The solid (gold)\nline is the equilibrium correlation while the dashed green line\nism2\nsin equilibrium. Inset: test of dynamic scaling with\nthe typical domain size Rs(t) estimated from Cs(Rs(t), t) =\nCs(r= 0, t)/e, with Cs(r= 0, t) = 1 + ∆2/12. Averages are\nperformed over 10 runs.\nFIG. SM11: Scaling of the space-time correlations CsandCσ\nfor the single-spin flip ( pswap= 0) and SWAP dynamics with\npswap = 0.5. The same master curve describes both sets of\ndata. Same parameters as in Fig. SM10.\nformer, the clean ferromagnetic Ising model’s Rs=Ris\nanalyzed as a benchmark. In the latter, the Rsof the\nsoft spin model is studied. As stated before, the typical\ndomain sizes measured are fitted via a power law, with\nan effective exponent which depends on the width ∆,\nRs(t) =λ tz−1\neff(∆). (SM18)\nλis a non-universal parameter. The exponent zeffis mea-\nsured by taking averages over successive time-windows\nalong the domain growth, as we did in the previous Sec-\ntion when the inverse perimeter density was calculated.\nThe convergence of zefftowards zd= 2 in the clean case\nis verified in Fig. SM12 for single-spin-flip kinetics, while\nthe SWAP method is unable to accelerate the dynamics,14\nand produces a slightly slower convergence.\nFIG. SM12: The growing length of the clean Ising model, ∆ =\n0, with single spin flip and SWAP dynamics. The inset shows\nthe time dependence of the effective exponent zeff, measured\nby performing a fit of the data at (moving) six consecutive\ntimes.\nIn Fig. SM13 we study zefffor the soft-spin model with\n∆ = 1. When the system is evolved with single-spin-\nflip kinetics, we get zeff∼3, and these update rules are\nnot convenient. However, when we implement the SWAP\nmethod, the effective exponent decreases to 2 .125, a value\nthat is very close to the theoretical zd= 2 of the regular\n2DIM with non-conserved order parameter dynamics.\nFIG. SM13: The growing length Rs(estimated from the anal-\nysis of the sspins space-time correlation) of the ∆ = 1-model,\nwith single spin flip and SWAP dynamics. Sub-critical quench\ntoT≈0.74TcandL= 2048. In the inset, the time depen-\ndence of the effective exponent zeff, measured by performing\na fit of the data at (moving) six consecutive times.\nFIG. SM14: The growing length Rσ(estimated from the anal-\nysis of the Ising σspins space-time correlation). Sub-critical\nquench to T≈0.74TcandL= 2048. The inset shows the\neffective exponent zeffvariation in time, measured by per-\nforming a fit of the data at (moving) six consecutive times.\nA.3.5 The growing length for the Ising spins\nNow, we repeat the analysis above but this time mea-\nsuring Rσ(t). Tracking the domain growth of the Ising\nvariables in Fig. SM14, similar conclusions are reached.\nThe dynamic exponent remains close to 3 for the single-\nspin-flip updates, while zeffdecreases when SWAP is per-\nformed, approaching a value close to 2 .25 in the numerical\ninterval explored, which is slightly larger than zd= 2 for\nthe 2DIM with non-conserved order parameter [37].\nFinally, we study the disorder dependence, ∆, of the\nasymptotic value of zeff, which we call z∞\neff. We measure it\nin the last available time-interval. The results are plotted\nin Fig. SM15. There is a large increase of z∞\neffwith the\nwidth of the spin length distribution for the single spin\nflip dynamics, while there is none, apart from noise, in\nthe SWAP simulations.\nA.4 Conclusions\nWith the SWAP dynamics one is able to obtain the\nequilibrium phase diagram of the ∆-model, and charac-\nterize its second order phase transition and magnetized\nlow temperature phase. They do not vary considerably\nwith respect to the ones of the standard 2DIM.\nConcerning the out of equilibrium dynamics and fur-\nther approach to equilibrium following a quench from the\ndisordered phase, with the SWAP method one recovers\nthe dynamic exponent of the original 2DIM, z∞\neff∼2 for\nany ∆. There is no improvement compared to the stan-\ndard single spin flip dynamics of the original 2DIM in\nthis unfrustrated case but, at least, there is no deteri-\noration either. Therefore, we have validated the use of\nthe SWAP algorithm for this model, although we do not15\nFIG. SM15: The asymptotic dynamical exponent, z∞\neff, esti-\nmated from the effective exponent zeffof the spins sgrowing\nlength, in the latest time interval accessed by the simula-\ntion, against the disorder width ∆. Sub-critical quench to\nT≈0.74Tc, and L= 2048.\nimprove with respect to the standard single spin flip MC\nusing it. Instead, in cases with frustration, as we show in\nthe main text and in the next section, the use of SWAP\ndoes accelerate the approach to equilibrium considerably.\nB THE SPIN-GLASS MODEL\nIn this Section we provide more details and additional\nresults on the behavior of the by-pass ∆-model that mod-\nifies the ±J2DEA model. Concretely, in Eq. (2) we\nused quenched random interactions drawn from a bi-\nmodal symmetric distribution\np(Jij) =1\n2δ(Jij−J) +1\n2δ(Jij+J). (SM19)\nWe set J= 1 in the numerical applications. In equilib-\nrium this 2DEA model has no finite temperature phase\ntransition: it has spin-glass ground states and is para-\nmagnetic at non-vanishing temperatures [45–48]. Still,\nit presents a non-trivial slow relaxation towards the\nequilibrium paramagnetic state at low enough temper-\natures [21, 22, 26, 33, 49–59].\nThe Hamiltonian of the frustrated ∆-model takes the\nform\nH=−X\n⟨ij⟩Jijsisj=−X\n⟨ij⟩Jijτiτjσiσj. (SM20)\nFor pure Ising spins, τi= 1∀i, it boils down to the 2DEA\nmodel with the bimodal couplings in Eq. (SM19).\nB.1 An equivalent frustrated Ising model\nBy following the same recipe used to analyze the ferro-\nmagnetic model, one can re-express the Hamiltonian withrandomly chosen lengths τias an EA Ising spin-glass\nH=−X\n⟨ij⟩Jijσiσj, (SM21)\nwith an uncommon kind of couplings, defined as Jij=\nJijτiτj. The Jijare correlated through the site depen-\ndence of the {τi}, similarly to what happened with the\nones of the ferromagnetic model, see Eq. (SM4).\nFIG. SM16: The probability distribution function of the cou-\npling strengths Jij=Jijτiτj, arising from the product of the\nspin-lengths τiτjand the quenched couplings Jijtaking ±1\nvalues with probability a half. The dashed line represents a\nGaussian distribution in normal form for comparison.\nThe mean and variance of the new effective couplings\nJijare\n[Jij] = 0 , [J2\nij]−[Jij]2=J2\u0012\n1 +∆2\n12\u00132\n.(SM22)\nMoreover, there is a persistent quenched randomness,\ntheJij, that is unaffected by the choice of the dynamics,\nunlike the {τi}that remain locally unmodified only when\nthe dynamics do not involve spin exchanges.\nIn Fig. SM16 we show the distribution of the couplings\nJijinduced by the one of the τifor three values of ∆.\nThe usual Gaussian distribution with zero mean and unit\nvariance is also shown for reference. The distribution of\nthe couplings Jijis just two delta peaks at ±Jfor ∆→0\nand it progressively shrinks the gap for increasing ∆ until\nclosing it completely when ∆ = 2.\nAs explained in the main text, the frustration in the\nmodel is only determined by the Jijcouplings. The in-\ntroduction of the spin lengths and the insofar induced\nJijdo not remove it.\nB.2 The mean-field critical temperature\nIn mean-field (within the fully connected approxima-\ntion) the critical temperature of an Ising model with fixed\ncouplings Jij(i.e. when both the Jij’s and the τi’s are16\nquenched) in equilibrium at an inverse temperature β\ncan be estimated from the use of the TAP approach. Af-\nter having conveniently scaled the Jijwith Nto ensure\na good thermodynamic limit, one derives the following\nequations for the local magnetizations:\nmi= tanh\"\nβX\n∂iJijmj+βhi−β2X\n∂iJ2\nij(1−m2\nj)mi#\nThe sumsP\n∂iindicate the spins jconnected to i. As-\nsuming a continuous phase transition and taking hi∼0\nas well, the local magnetization should be mi∼0. If,\nmoreover, one replaces J2\nijby [J2\nij] in the Onsager reac-\ntion term,\nmi∼βX\n∂iJijmj+βhi−β2X\n∂i[J2\nij]mi\n∼βX\n∂iJijmj+βhi−β2J2\u0012\n1 +∆2\n12\u00132\nmi.\nThis equation can now be taken to the basis of eigen-\nvectors of the matrix with elements Jij. Calling ⃗ vµ\nthe eigenvector associated to the eigenvalue λµ, and\nmµ=⃗ m·⃗ vµ,\nmµ∼βλµmµ+βhµ−β2J2\u0012\n1 +∆2\n12\u00132\nmµ.\nThe linear susceptibilities are\nχµ=∂mµ\n∂hµ\f\f\f\f⃗h=0∼β\n1−βλµ+β2J2\u0012\n1 +∆2\n12\u00132.\nThe first susceptibility to diverge is the one associated to\nthe largest eigenvalue λmaxand this arises at\nβc=λmax±\"\nλ2\nmax−4J2\u0012\n1 +∆2\n12\u00132#1/2\n2J2\u0012\n1 +∆2\n12\u00132.(SM23)\nIn the Sherrington-Kirkpatrick model, the mean-field\nlimit of the EA model, ∆ →0 and λmax= 2J. Then,\nβc=J−1. If, in the ∆-model, λmax= 2[J2\nij]1/2, which\nseems reasonable, then\nβc∝[J2\nij]−1/2⇒ Tc=J\u0012\n1 +∆2\n12\u0013\n.(SM24)\nIn Fig. SM17 we plot, with orange data points, the\ncritical temperature Tc(∆) obtained from diagonalizing\nsymmetric matrices with such elements and linear size\nL= 32 ( J= 1). The datapoints are consistent with the\nquadratic dependence on ∆ linking Tc=Jat ∆ = 0 and\nTc= 4/3Jat ∆ = 2.\nFIG. SM17: The ∆ dependence of the critical temperature\nin mean-field. Orange data: results for the fully connected\nmodel with the TAP approach. Blue data: results for the\nBethe lattice model with connectivity k= 2. The trend is the\nsame and the range of variation of the critical temperature\nwith ∆ is very weak in both cases.\nAn alternative way to estimate the ∆ dependence of\nthe critical temperature with a mean-field approach is to\nplace the model on a Bethe lattice with connectivity k.\nBy defining the cavity field acting on site iproduced by\nthe effect of kneighboring spins (in the absence of its\nj-th neighbor), we obtain the recursive equations\nhi→j(τi) =\nX\nm∈∂i/j1\nβatanh [tanh ( βJimτiτm) tanh ( βhm→i(τm))].\n(SM25)\nThe sum excludes m=j. These equations admit\nhi→j= 0 as a solution on all sites, corresponding to the\nparamagnetic phase.\nSince we will be interested in the temperature regime\nin the vicinity of the critical point, T≲Tc, in which the\ncavity fields are small, we can expand the right-hand-side\nof Eq. (SM25):\nhi→j(τi)≃X\nm∈∂i/jtanh ( βJimτiτm)hm→i(τm).(SM26)\nDue to the randomness of the spin-amplitudes, these\nequations must be interpreted as a self-consistent inte-\ngral equation for the probability distribution of the local\ncavity fields\nP(h|τ) =\nZ kY\nm=1\"X\nJmdτmdhmpτ(τm)P(hm|τm)#\nδ(h−˜h),\nwith ˜h=P\nmtanh ( βJmττm)hm. Using the integral rep-\nresentation of the δ-function, one obtains the following17\nequation for the Fourier transform of the probability dis-\ntribution\nˆP(q|τ) =\"ZX\nJdτ′pτ(τ′)ˆP(qtanh ( βJττ′)|τ′)#k\n,\nAssuming that the fields follow a Gaussian distribution\nwith variance, σ2\nh=h(τ)2−h(τ)2, we have\nˆP(q|τ) = 1−iqh(τ)−q2\n2h2(τ) +. . .\nplugging this expression into the equation above, using\nthe fact that h(τ)n≪1 for T≲Tc, and expanding up\nto second order one finds\nh(τ) =kZX\nJdτ′pτ(τ′) tanh ( βJττ′)h(τ′),\nh2(τ) =kZX\nJdτ′pτ(τ′) tanh2(βJττ′)h2(τ′)\n−k−1\nkh(τ)2.\nForh(τ) = 0 we are interested in the second equa-\ntion, that defines a linear integral operator f(τ) =R\ndτ′Γ(τ′, τ)f(τ′), with f(τ) = h2(τ) and the (non-\nsymmetric) kernel\nΓ(τ′, τ) =kX\nJ=±Jpτ(τ′) tanh2(βJττ′). (SM28)\nTherefore a solution of Eq. (SM27) with a non-vanishing\nfunction h2(τ) only exists if such integral operator has\nan eigenvector with eigenvalue 1. For the specific case\nof the box distribution of width ∆ we have diagonalized\nthe integral operator numerically for J= 1 and k= 2,\nfor several values of βand ∆, on a grid of 2048 ×2048\nintervals. The results for the critical temperature are\nreported in blue Fig. SM17. Note that for ∆ →0 the\ncritical temperature tends to the Bethe lattice value for\nthe Ising spin-glass, βcJ= atanh(p\n1/k)∼0.88, that\nisTc= 1.13J. The increasing trend is the same as the\none derived in the fully connected model with the TAP\nmethod. The range of variation of the critical tempera-\nture with ∆ is very weak in both cases.\nB.3 The probability of reaching the ground state\nafter a T= 0quench\nWe investigate the efficiency of SWAP to reach the\nground states of the 2DEA model with the interactions\nJ∗\nijobtained at the latest time reached after a zero tem-\nperature quench with SWAP.\nThe main panel in Fig. SM18 displays P0(t) for three\nvalues of ∆ and pswap = 0.5. In all cases, after a fast\nincrease ending at t≲103MCs,P0(t) saturates to a P∞\n0\nFIG. SM18: The probability of reaching the ground state of\nthe ∆-model with J∗\nij=Jij(tmax), in systems with L= 8\nafter zero temperature quenches. Main panel: three ∆ values\ngiven in the right key and pswap = 0.5. For ∆ = 2, 96 .5%\nof the runs find a ground state. Lower inset: the asymptotic\nvalue against the parameter pswap for ∆ = 2. Upper inset:\nScaling with t⋆(L) = 1 .25L3.75of the data for ∆ = 2, pswap=\n0.1 and three system sizes specified in the left key.\nwhich increases with ∆ and gets very close to 1 for ∆ = 2.\nFurther optimization of the algorithm achieved by gaug-\ningpswap is studied in the lower inset which shows the\ndependence of P∞\n0onpswap in the ∆ = 2-model. For\nintermediate values, 0 .1≲pswap≲0.9,P∞\n0remains ap-\nproximately constant apart from numerical noise, and\nit decays to zero at the two extremes of either non-\nlocal spin exchanges ( pswap→1) or pure single-spin-flips\n(pswap→0). Finally, we checked the dependence on sys-\ntem size using ∆ = 2 and pswap= 0.1. The curves are\nsimilar and the percentage of ground states found is in-\ndependent of L. The upper inset shows the scaling of P0\nagainst t/t⋆(L) with t⋆(L)∼1.25L3.75.\nB.4 The overlap correlation and the spin-glass\ngrowing length\nThe overlap or four spin correlation is defined as\nNC4(r, t) =\nNX\ni,j=1\n|⃗ ri−⃗ rj|=rhD\nσ(1)\nj(t)σ(2)\nj(t)σ(1)\ni(t)σ(2)\ni(t)Ei\n(SM29)\nand the characteristic length over which it decays defines\nthe spin-glass ordering length. It is here estimated from\nRσ(t) = 2R∞\n0dr C 4(r, t), as defined in [53].\nThe Metropolis dynamics of the 2DEA at T≃0.5 yield\na super exponential decay, C4∼e−(r/Rσ(t))β, with β >1.\nThe spin-glass length scales as Rσ(t)∼(t/τ(T))1/zwith\nz∼7 and an Arrhenius characteristic time τ(T) at these\nrelatively high temperatures [56].\nThe spatial dependence of C4, at different times, is\nstudied in Fig. SM19 at T= 0.5 and T= 0.1. At high18\ntemperatures SWAP on the ∆ = 1 model yields equiva-\nlent results to the single-spin flip evolution of the original\n∆ = 0 model. At the longest time t∼106considered the\ngrowing length is of the order of 20, say, while the sys-\ntem’s linear size is L= 512.\nAt low temperatures instead SWAP is much more ef-\nficient in building long-range correlations, even between\ncopies that may have evolved towards different effective\nrandom couplings Jij, than the single-spin flip evolution\nof the original ∆ = 0 model. Still, the correlations ob-\ntained with SWAP at the same time t∼106decay faster\nthan at higher T, reaching a shorter Rσ(t).\nFIG. SM19: Overlap correlation C4(r, t), Eq. (SM29), versus\nthe Cartesian distance rfor several times given in the keys.\nL= 512 ±J2DEA (∆ = 0) model evolved with single-spin\nand ∆ = 1 model evolved with SWAP and pswap= 0.2. Data\nforT= 0.5 (above) and T= 0.1 (below). While single spin\nflips and SWAP yield equivalent results at high temperature,\nthe latter builds longer correlations at low temperature.\nFor single-spin-flip dynamics, the growing length of\nthe±J2DEA Ising model, which we simply call R(t) in\nFig. SM20, freezes at low enough temperatures ( T≤0.2),\nsaturating at a very short value [60]. The dynamic ex-\nponent, plotted in the inset of the same figure, diverges.\nBy increasing the temperature the plateau is surpassed\nand the evolution persists. The latter growth can be de-\nscribed with the power-law t1/zeff, as in Eq. (SM18), and\nzeffconverges to a value close to 8.5 at Tlarger than 0.3,\nsay.\nFinally, in Fig. SM21 we compare the performance of\nFIG. SM20: Typical growing length in the ±J2DEA model\nwith L= 512 evolved with single spin flip dynamics at several\ntemperatures displayed in the key. The inset shows the time\ndependence of the dynamical exponent zeff: it converges to\n∼8.5 after the saturation of the growing length at the value\nRp= 4.483±0.004 is superseded at high-enough temperatures\n(T >0.2).\nSWAP with different values of the parameter pswapto the\none of the single spin flip updates when applied to the\n∆ = 1 model at high temperature. The data in the figure\ndemonstrate that the two methods perform similarly.\nFIG. SM21: Typical growing length of the frustrated soft spin\nmodel with L= 512 and ∆ = 1, evolving with SWAP and\nsingle spin flip dynamics at a relatively high temperature,\nT= 0.5. The growth in the ±J2DEA with single spin flip\ndynamics is plotted for comparison. Several values of pswap\nwere used. The inset shows the time evolution of the dynam-\nical exponent. In all cases a similar long-time limit, close to\n8, is reached.\nWe henceforth focus on T < 0.3. In Fig. SM22(a) we\ndisplay the spin-glass length for the ∆ = 0 and ∆ = 1\nmodels quenched to T= 0.1, and evolved with single\nspin flip and SWAP dynamics. At such low tempera-\nture the single spin flip dynamics of the ∆ = 0 model\nfreeze and the spin-glass length saturates to Rσ∼5 (blue19\nFIG. SM22: Spin-glass growing length in systems with L= 32\n(triangles) and L= 512 (squares) quenched to T= 0.1. The\nthree curves compare the single spin flip evolution of the Ising\n(∆ = 0) and soft (∆ = 1) models to the SWAP one of the\nlatter. The inset displays the dynamical exponent zeffas a\nfunction of t, the center of the time interval over which the\nalgebraic fit was performed (with 12 data points, and L= 32).\nIn dashed black is the fit with zeff= 49.5(86) and amplitude\nA= 3.94(2).\ndata). Instead, both single spin flips (orange) and SWAP\n(green) of the ∆ = 1 model accelerate the dynamics at\nlong times, with the latter becoming more efficient in the\nlast four time decades. We estimated running effective\nexponents from fits over a moving window with 12 data\npoints (inset). Within numerical accuracy zeffconverges\ntoz∞\neff∼49. The dashed line is the algebraic law A t1/z∞\neff.\nHowever, the long-time configurations are not in equi-\nlibrium. First, the maximal length Rσ∼6 for L= 32 is\nfar from L/2. Second, the τivariables, which are also dy-\nnamical under SWAP, are still evolving. Hence, σ(1)\niand\nσ(2)\nimay be optimized with respect to different Jijand\nthus not really inform us about the performance of the\nalgorithm and measurement in taking one system close\nto equilibrium.\nB.5 The correlation of the τivariables\nIn Fig. SM23 we plot the time evolution of the corre-\nlation\nCτ(1)τ(2)(t) =\u00141\nNNP\ni=1τ(1)\ni(t)τ(2)\ni(t)\u0015\n−1\n\u0012\n1 +∆2\n12\u0013\n−1(SM30)\nwhere τ(1)\niandτ(2)\niare the values of the length vari-\nables in two runs of the same quench disordered model\n(same Jij) starting from the same initial condition of the\nIsing spins and length variables and evolved with differentMonte Carlo noises. At equal times Cτ(1)τ(2)(t= 0) = 1\nby definition since τ(1)\ni(0) = τ(2)\ni(0) and [ τ2\ni] = 1+∆2/12.\nIf for t→ ∞ theτ(1)\niand τ(2)\nilengths fully corre-\nlate again, then 1 should be recovered in this limit as\nwell. This is confirmed in Fig. SM23 where this time-\ndependent correlation is shown for various values of ∆ in\na system with L= 16 annealed following the protocol (6)\nfrom T0= 0.5 to zero temperature.\nFIG. SM23: The correlation of the τivariables according to\nEq. (SM30). L= 16 system annealed to zero temperature.\nB.6 The two-time correlation\nFinally, we have calculated the two-time correlation\nfunction, defined as\nC(t, tw) =1\nNNX\ni=1[⟨si(t)si(t+tw)⟩] (SM31)\nwhere si(t) =σi(t)τi(t) for the ∆-model and si(t) =σi(t)\nfor the EA one. As can be seen in Fig. SM24 the\nSWAP method induces a faster decay, compared to the\nsingle spin flip dynamics, in which the curves saturate\nrather quickly, consistent with plateau the Rpfound in\nFig. SM20.\nB.8 Measures of frustration\nIn Fig. SM29 we follow the time evolution of the in-\nstantaneous configuration of the system, parametrized as\nits projection on the ground state of the final couplings\nJ∗\nij=Jij(tmax). More precisely, each square in the im-\nages represents\nsi(t)σgs\ni=τi(t)σi(t)σgs\ni. (SM32)\nThe color code goes from −2 (light blue) to +2 (light\nred), A very negative value represents a long spin which20\nFIG. SM24: The two-time correlation of the sivariables, con-\nfronting the SWAP method ( pswap= 0.1) with the ∆ = 1 .5-\nmodel with the single-spin-flip kinetics of the 2DEA model,\nboth with sizes L= 512 and the waiting time values displayed\nin the key.\nis anti-aligned with the ground state, while a very pos-\nitive value represents a long spin which is aligned with\nthe ground state. Dark squares represent short spins. On\neach vertex of the (dual) lattice we place a symbol when\nthe corresponding plaquette of the original lattice is frus-\ntrated. The strength of the local frustration is quantified\nby\nfP(t) =Y\n⟨ij⟩∈PJij(t) =Y\n⟨ij⟩∈PJijτi(t)τj(t).(SM33)\nConcretely, on a square plaquette with site labels\n1,2,3,4,\nfP(t) =J12J23J34J41τ2\n1(t)τ2\n2(t)τ2\n3(t)τ2\n4(t).(SM34)\nThe sign, and whether the plaquette is frustrated or\nFIG. SM25: The number density of plaquettes with frus-\ntration fP<0, distinguished by the modulus being larger or\nsmaller than 1.\nnot, is decided by the factor J12J23J34J41while the\nmagnitude of the potential frustration is determined by\nτ2\n1(t)τ2\n2(t)τ2\n3(t)τ2\n4(t) which depends on time. Initially one\ncan imagine that the last four factors are independent\nandτ2\n1τ2\n2τ2\n3τ2\n4∼(1 + ∆2/12)4. Going back to the snap-\nshots, we used two kinds of symbols: a bullet for |fP(t)|greater than one, a triangle for |fP(t)|smaller than 0.1,\nrespectively. We called nthe density of each of these frus-\ntrated plaquettes, that is the number of the frustrated\nplaquettes of each kind divided by the total number of\nfrustrated plaquettes NFwhich is constant and approxi-\nmately equal to half the total number of plaquettes.\nA total frustration density, defined as the normalized\nsum of the local ones, reads\nf(t) =1\nNFX\nPfP(t) (SM35)\nwith the sum running over frustrated plaquettes.\nFIG. SM26: The magnitude of the total frustration f, de-\nfined in Eq. (SM35) in ∆-models with different values of ∆\nspecified in the key and L= 32 quenched to zero temperature\nand evolved with SWAP. The initial value is (1 + ∆2/12)4.\nIn the course of time we see the following.\n– The frustrated or unfrustrated nature of the pla-\nquettes does not change in time. When there is a\nsymbol (bullet or triangle) attached to them, they\ndo not disappear. They are not created elsewhere\neither.\n– The strength of the local frustration does vary in\ntime, so the symbols can change, for example, from\nbullet to triangle, or vice versa . The time depen-\ndence of the densities of frustrated plaquettes of\neach kind are shown in Fig. SM25. The number\ndensity of plaquettes with small frustration domi-\nnates over the ones with large frustration asymp-\ntotically.\n– In the course of time the symbols rearrange spa-\ntially. Rather large regions with small frustration\nare visible in the late snapshots.\n– The magnitude of the total frustration ftends to\ndiminish in time.\n– Initially, the color of the images is roughly ran-\ndomly distributed over the full palette. In the\ncourse of time, blue cells tend to disappear and the21\nimage becomes globally reddish. This means that\nthe system orders in the direction of the selected\nground state.\n– In the last images the crosses (weak frustration) are\nlocated in regions where the spins have short length\nand the bullets (strong frustration) are placed in\nregions where the spins have long length.\nFIG. SM27: Time evolution for the ratio of small-length ( τi≤\n1) and large-length (1 < τi≤2) spins with σinot aligned with\nthe ground state σgs\ni, i.e. σi(t)σgs\ni<0.\nB.7 Instantaneous configurations\nThe mechanism that renders the SWAP method effi-\ncient in the frustrated ∆-Model is clarified by inspectingsome snapshots, as we did for the ferromagnetic model.\nIn Fig. SM28, we show the actual configuration (first\nrow) and the overlap with the ground state of the sys-\ntem with the final Jijinteractions (second row). (In the\nmain text we explain how we obtain the σgs\niconfigura-\ntion.) We have performed a sub-critical quench at T= 0,\nforL= 128 using SWAP dynamics with pswap= 0.1.\nThe first row displays the spin configurations at four\ntimes after the quench. The images do not show much\nstructure apart from a slight tendency of long spins with\nthe same orientation to group locally. Still, and as ex-\npected, no structure with long-range spin ordering devel-\nops.\nOne can recognize the formation of domains in the\noverlap of the configurations with the ground state. In-\ndeed, the images shown in the second row are primarily\nred or yellow, that is, the system acquires a positive over-\nlap with the ground state all over space. The SWAP al-\ngorithm produces domain walls consisting mostly of short\nlength spins (green and yellow). This structure lowers the\nenergy barriers locally, making the spin-flip more feasible\nin these regions. The ratio of {τi}variables for which the\ncorresponding Ising spins do not match the ground state\n(i.e. σi(t)σgs\ni<0) goes to zero, see Fig. SM27, in the\ncourse of time. However, the decay rate depends strongly\non the length of the spins. Larger spins (1 < τi≤2) align\nfaster on the ground state directions compared to smaller\nones ( τi≤1), as can also be seen in Fig. SM27, support-\ning the previous explanation.22\nt=0t= 128t=2048t=32768\nsi\n-2 -1 0 1 2\nFIG. SM28: Instantaneous snapshots of the frustrated ∆-model with ∆ = 2, at four times indicated in the figure after a quench\ntoT= 0. First row: The spin siconfigurations. Second row: Overlap with the σ-ground state ( si(t)σgs\ni). The color scale\nbinned as indicated in the bar shows the lengths of the local spins.23\nFIG. SM29: Instantaneous configurations of the ∆-model obtained at subsequent (though not equally spaced) times after\na quench to T= 0 of four initial conditions (different rows) evolved with SWAP. Each square represents the instantaneous\nlocal overlap τi(t)σi(t)σgs\ni, see Eq. (SM32). The color code spans the interval [ −2 (light blue), 2 (light red)]. The frustrated\nplaquettes are indicated with bullets and triangles according to the local frustration fP(t) being greater or smaller than one,\nrespectively."    },    {        "title": "2207.10688v2.Probing_dynamics_of_a_two_dimensional_dipolar_spin_ensemble_using_single_qubit_sensor.pdf",        "content": "Probing dynamics of a two-dimensional dipolar spin\nensemble using single qubit sensor\nKristine Rezai1,3, Soonwon Choi2, Mikhail D. Lukin1, Alexander O. Sushkov3,4,5\n1Department of Physics, Harvard University, Cambridge, MA 02138, USA\n2Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02142, USA\n3Department of Physics, Boston University, Boston, MA 02215, USA\n4Department of Electrical and Computer Engineering, Boston University,\nBoston, MA 02215, USA\n5Photonics Center, Boston University, Boston, MA 02215, USA\n∗To whom correspondence should be addressed; e-mail: asu@bu.edu\nAbstract\nUnderstanding the thermalization dynamics of quantum many-body systems at\nthe microscopic level is among the central challenges of modern statistical physics.\nHere we experimentally investigate individual spin dynamics in a two-dimensional\nensemble of electron spins on the surface of a diamond crystal. We use a near-\nsurface NV center as a nanoscale magnetic sensor to probe correlation dynamics of\nindividual spins in a dipolar interacting surface spin ensemble. We observe that the\nrelaxation rate for each spin is significantly slower than the na¨ ıve expectation based\non independently estimated dipolar interaction strengths with nearest neighbors\nand is strongly correlated with the timescale of the local magnetic field fluctuation.\nWe show that this anomalously slow relaxation rate is due to the presence of strong\ndynamical disorder and present a quantitative explanation based on dynamic reso-\nnance counting. Finally, we use resonant spin-lock driving to control the effective\nstrength of the local magnetic fields and reveal the role of the dynamical disorder in\ndifferent regimes. Our work paves the way towards microscopic study and control\nof quantum thermalization in strongly interacting disordered spin ensembles.\n1arXiv:2207.10688v2  [quant-ph]  20 Jul 2023Quantum thermalization connects statistical physics with unitary quantum mechan-\nics. Recent technological developments in quantum information science have enabled\ndetailed studies of isolated quantum systems, revealing a variety of novel phenomena\nsuch as the role of entanglement in thermalization [1, 2, 3], the localization in the ab-\nsence of strong disorder [4], non-equilibrium phases [5, 6, 7, 3, 8, 9], and quantum many-\nbody scarring [10, 11, 12]. Despite this progress, one of the key open questions in this\nfield is the nature of microscopic relaxation dynamics in presence of dynamical disor-\nder and long-range interactions [13]. The majority of existing studies focus on ensemble\nmeasurements in systems dominated by static disorder [14, 15, 16, 17, 18, 19]. How-\never ensemble averaging can conceal important features of microscopic dynamical evolu-\ntion [20, 21], and in many noisy real-world quantum systems disorder is dynamic, exhibit-\ning non-trivial time-dependence. Another important factor is system dimensionality and\nits interplay with the distance scaling of interactions. The long-range 1 /r3dipolar spin\ninteraction implies that a two-dimensional system should be localized according to the\nsingle-particle Anderson model [22], but delocalized according to the many-body inter-\nacting treatment [23, 24, 25, 26, 27]. In addition to fundamental interest, understanding\nthe dynamics of two-dimensional systems is important for quantum sensing applications,\nsince such systems can be positioned in close proximity to a sensing target [28, 29].\nWe experimentally investigate spin transport dynamics of a two-dimensional ensemble\nof randomly-positioned spin-1/2 qubits with long-range magnetic dipolar interactions [29].\nNa¨ ıvely, one could expect that the spin exchange (flip-flop) component of dipolar interac-\ntions limits the lifetime of local polarization of spins. Our observation, however, reveals a\nsurprising finding that the spin lifetime is, in fact, significantly longer than independently-\nestimated interaction strengths. We attribute this anomalously-slow spin relaxation dy-\nnamics to the interplay between interactions and strong time-dependent local disorder,\ncreated by hyperfine fields of proximal nuclear spins. We control the effective strength of\nthis disorder and observe the corresponding scaling of the relaxation rate. We present a\ntheoretical model, based on dynamic resonance-counting arguments, which is in quantita-\ntive agreement with our experimental observations and reveals a universal scaling collapse\nof our data for different values of disorder strengths and correlation times.\nOur experimental platform is the ensemble of paramagnetic two-level systems on the\nsurface of a diamond crystal, fig. 1 (a) [30, 31, 32]. These surface spins are associated\nwith electron spin S= 1/2 impurities that are not optically active. Their exact nature\nis subject to ongoing investigation, but they are likely localized defect surface states\n[33, 34]. Due to faster decoherence of shallow NV centers [35, 36, 37, 38, 39], these surface\nspins have been considered to be deleterious and significant effort has gone into treating\nand engineering diamond surfaces to minimize their density [40]. However, with proper\nquantum control, they can be turned into a useful resource. The surface spins can be\ncoherently manipulated and measured, and can be used as the so-called quantum reporters\nthat probe and report the local magnetic environment. For example, by addressing a single\nsurface spin, it is possible to detect and localize proximal single proton nuclear spins on\n2the diamond surface under ambient conditions [28].\nThe dynamics of individual surface spins are measured by a single near-surface NV\ncenter, acting as a nanoscale sensor of magnetic fields created by the surface spins. NV\ncenters are addressed using a confocal microscopy setup, combined with radiofrequency\n(RF) spin drive fields that are delivered via a transmission line fabricated on a glass\ncoverslip. The surface spin transitions can be addressed with RF pulses delivered in the\nsame manner. The 2.87 GHz zero-field splitting of the NV center enables independent\naddressing of the NV spin and the surface spin transitions by using different resonant RF\ntones at a given bias magnetic field, aligned with the NV axis (z-axis), fig. 1(a), inset. The\nmagnetic dipole coupling between the NV center and the surface spin ensemble is charac-\nterized using a double electron-electron resonance (DEER) sequence, fig. 1(b). The NV\ncenter spin-echo decays on the timescale T(nv)\n2, but when a π-pulse flips the surface spins\nsimultaneously with the NV spin, the NV spin echo collapses on a timescale that depends\non the strength of the dipolar field created by the surface spins near the NV center. Be-\ncause the magnetic dipole interaction is long-range, the NV center is, in general, coupled\nto multiple surface spins, with coupling strengths dependent on locations of surface spins\non the diamond surface. The oscillations present in the data at short time (fig. 1(b) red\npoints) indicate that there is one “central” surface spin, whose dipolar coupling strength\nto the NV center kmaxdominates over other surface spins. Recent experimental studies\nhave shown that after certain surface treatments, some of the surface spins may be mobile,\nlikely changing their positions under green laser light illumination [41]. We have verified\nthat in our experiments the central surface spin remains in place, even when subjected\nto green laser light illumination of up to 50 µs [42]. In fact, the central spin position\nis stable for the experiments carried over the timescale of months. This is likely due to\nthe combination of small photo-ionization cross-section and steric protection by chemical\nsurface groups [41].\nOur experiments take place at room temperature, with the bias magnetic field on\nthe order of 1000 G. Therefore the surface spin ensemble is effectively at infinite spin\ntemperature. Nevertheless, we study the dynamics of an individual central surface spin\nSj, by measuring its spin autocorrelation functions. Quantum logic gates between the NV\ncenter and the central surface spin correlate the NV and the central spin evolution, via\ntheir magnetic-dipole interaction, fig. 2(a) [43]. Each measurement consists of a sequence\nof RF pulses, applied to the NV and surface spins, as well as NV optical spin polarization\nand readout steps. The RF pulse sequences include two probe segments (CNOT gates),\nin which the NV center probes the quantum state of the surface spin ensemble, separated\nby a time interval, in which this state can evolve with control pulses applied to the surface\nspins (fig. 2(a), grey box). The interpulse spacing tNVof the probe segments is tuned to\nthe timescale of the dipolar interaction between the NV center and the central surface\nspin so that the final NV center state is contingent on whether the z-projection of the\ncentral surface spin changes between the gates. Applying this method, the NV center\nacts as a probe of central surface spin autocorrelation functions ⟨Sx,y,z\nj(t)Sx,y,z\nj(0)⟩, where\n3the time tand the measured spin projection depend on the pulses applied during the\nevolution interval. We performed measurements on 11 separate NV center-surface spin\nsystems [42].\nThe Ramsey measurement probes fluctuations of the local magnetic field at the cen-\ntral surface spin site (fig. 2(b) blue). The Hahn echo sequence decouples the central\nsurface spin from low-frequency fluctuations, which extends the coherence time to T2\n(fig. 2(b) red). Further decoupling can be achieved with higher-order dynamical decou-\npling experiments, such as XY-4 (fig. 2(c) blue). However we observe that this does not\nfurther increase the coherence time, indicating the presence of a separate decoherence\nmechanism. The presence of dipolar interactions with other surface spins motivates the\nMREV-8 experiment, which decouples dipolar interactions. Indeed, we observe that the\ncoherence time in an MREV-8 experiment is extended, compared to the Hahn echo and\nXY-4 (fig. 2(c) red curve), showing that the dynamics of surface spins are strongly affected\nby dipolar interactions between them.\nIn order to understand our observations quantitatively, we consider the effective Hamil-\ntonian in the rotating frame of a single surface spin Sj:\nHj=ℏγeBz\nj(t)Sz\nj+X\niℏ2γ2\ne\nr3\nij(1−3 cos2θij)[Sz\niSz\nj−1\n4(S+\niS−\nj+S−\niS+\nj)], (1)\nwhere ℏis the reduced Planck constant, γe= 2π×2.80 MHz/G is the electron gyro-\nmagnetic ratio, Bz\nj(t) is the fluctuating magnetic field at the site of the surface spin, rij\nis the distance between the central surface spin jand a different surface spin i, and θij\nis the angle between the direction of the applied external magnetic field and the vector\nrij. In eq. (1) we do not include the dipolar interaction between the NV center and the\ncentral surface spin. By running experiments that vary the NV center spin state during\nthe surface spin evolution period, we find that this interaction does not affect central\nsurface spin evolution within our experimental uncertainty [42].\nThe strengths of the dipolar interaction terms in eq. (1) vary over different pairs of\nspins, due to their random positions on the surface. However the 1 /r3distance dependence\nand the 2D nature of the surface spin ensemble imply that it is the proximal surface\nspins that dominate the spin echo decoherence. We use the spin echo data to extract\nthe strength of the central spin interaction with these proximal spins: J1= 1/T2=\n(0.71±0.05)µs−1.\nLet us consider the origin of the fluctuating local magnetic field Bz\nj. All measurements\nwere preformed with the diamond surface submerged in deuterated glycerol. This reduced\nthe density of proton nuclear spins near the surface. However, as observed in several other\nstudies of near-surface NV centers, an intrinsic ∼1 nm thick layer of surface water and\nhydrocarbons contains a high density of proton nuclear spins [44, 45, 46]. The periodic\nfeatures that appear at odd multiples of proton Larmor period in the spin echo, XY-4,\nand MREV-8 data indicate that these protons are the dominant source of the local field\n4Bz\nj. We model the dynamics of Bz\njby parametrizing its power spectrum as a combination\nof two terms:\nV(ω) =Z∞\n−∞⟨Bz(t)Bz(0)⟩e−iωtdt= 2W2\nγ2\ne\u0012τ\nω2τ2+ 1+5\n9τ\n(ω−ωL)2τ2+ 1\u0013\n.(2)\nThe first term, centered at zero frequency, quantifies slow fluctuations with strength W\nand correlation time τ, due to proton spin projections along the bias magnetic field [42].\nThe second term quantifies fluctuations near the proton Larmor frequency, due to trans-\nverse proton spin projections (fig. 2(d)). We simultaneously fit the Ramsey, spin echo,\nXY-4, and MREV-8 experimental data shown in fig. 2, with J1,W, and τas fit parame-\nters [42]. For this NV-surface spin system, we extract W= (4.40 ±0.38) µs−1andτ=\n(14.6±4.2)µs. We note that the decoherence in the Ramsey sequence is significantly\nfaster than that of the spin echo, indicating that Wis greater than J1. The decoherence\nof the MREV-8 data is dominated by the low-frequency noise component in eq. (2).\nHaving characterized the local environment of the central surface spin, we study the\nspin flip-flop transport dynamics in this system by measuring the autocorrelation of the\nspin component along the applied magnetic field: ⟨Sz\nj(t)Sz\nj(0)⟩, fig. 3(a). The presence\nof the S+S−flip-flop terms in the dipolar interaction in eq. (1) motivates an expectation\nthat this single-spin autocorrelation would decay over the timescale Tz, which is on the\norder of 1 /J1=T2. In contrast, we consistently observe remarkably slow decay of the\nlongitudinal correlation, which persists over a timescale Tz, in this instance ∼30×slower\nthan the dipolar interaction timescale T2, fig. 2(e). We emphasize that these observations\nare especially unexpected for individual spin measurements, in contrast with experiments\nprobing a macroscopic spin ensemble (such as a bulk magnetic resonance measurement),\nwhere the total spin z-projection is unaffected by flip-flops within the ensemble.\nWe attribute such anomalously slow relaxation dynamics to the presence of strong\ndisorder. To quantify its effect, we consider a theory model based on an effective single-\nparticle resonance counting. In this model, each spin Sjexperiences a time-dependent\nenergy shift owing to the combination of two effective sources of disorder: i) the extrinsic\non-site disordered magnetic field Bz\nj(t) of typical strength ∼Wand ii) the intrinsic local\nfield of strength ∼J1arising from the Ising component of the dipolar interactions among\nsurface spins. These two components have distinct correlation times and strengths. To\nquantify the combined dynamical disorder, we introduce the effective disorder strength\nWe=p\nW2+J2\n1and the effective disorder correlation time τe, defined viap\n1/τe=\n(Wp\n1/τ+J1p\n1/Tz)/We[42]. We assume spins exchange their polarization via dipolar\ninteractions when a pair of spins i, jbecomes “resonant”, i.e. when the difference between\ntheir spin energy shift is smaller than the flip-flop interaction between them, fig. 3(d).\nOtherwise, energy conservation blocks flip-flops [18]. The spin relaxation dynamics is\nprobed by estimating the probability for a central spin to flip-flop with its neighbor as a\nfunction of time. This model predicts that for a 2D dipolar spin system, after averaging\nover random positioning of spins, the central spin autocorrelation ⟨Sz\nj(t)Sz\nj(0)⟩decays\n5with an approximate functional form exp ( −(t/Tz)2/3) [42], where Tz=κτeWe/J. The\nnumerical constant κis of order unity, and Jis the average dipolar interaction strength\nover the spin ensemble. We compare this prediction with our experimental data by scaling\nthe time axis of the autocorrelation data by independently estimated τeWe, and observe\nthat the data sets for the 7 different NV-central spin systems collapse towards a universal\ncurve, fig. 3(b). Plotting the individual relaxation times Tzversus the product τeWe\nreveals the linear relationship predicted by our resonance counting, with the best-fit value\nκ= (0.31 ±0.14) and J = (0.57 ±0.25) µs−1, consistent with the interaction strengths\nextracted from the spin echo and XY-4 data sets, fig. 3(c).\nTo further investigate the role of disorder, we control the magnitude of the effective\ndisorder by spin-lock driving, on resonance with the surface spin Larmor frequency. The\nresonant drive along the y-axis of the rotating frame adds to the Hamilonian (1) the\nadditional term Hd=ℏΩSy, where Ω is the drive Rabi frequency. We measure the\ndynamics of the central surface spin projection Sy\njalong the driving field, fig. 4(a). This\nis equivalent to T1ρmeasurement in NMR [47, 48]. The relaxation at small Ω is dominated\nby the local magnetic field noise Bz\nj(t), which can flip Sy\nj. In this regime, the relaxation\nrate calculated using the noise model in eq. (2) is consistent with our measurements\n(fig. 4(b), black line). At large drive strength Ω, spin flips due to Bz\nj(t) are suppressed, such\nthat the strength of the effective extrinsic disorder scales as W2/(√\n2Ω). Consequently,\nthe disorder experienced by individual spins is dominated by the Ising component of\ndipolar interactions along the dressed quantization axis Sy\nj. The strength of the latter is\nreduced to the half of the original Ising interaction along Sz\njin the undriven case [18].\nThus, systematic comparisons of T1ρ/2 and Tzenable us to understand the interplay\nbetween the extrinsic and intrinsic disorder fields. Shown in fig. 4(c), we find that the\nrelaxation time becomes relatively faster by suppressing the extrinsic disorder when Wis\nsufficiently large. However, for samples with small or intermediate W, the difference in T1ρ\nandTzis not substantial, suggesting that the Ising interaction constitutes the dominant\nsource of disorder in this regime. Interestingly, even in this regime, the relaxation rate\nis approximately a factor of 20 slower than the dipolar interaction scale J1. This implies\nthat, even in the absence of extrinsic on-site disorder, the intrinsic disorder associated\nwith random positioning of spins is sufficient to strongly suppress spin transport in 2D\nspin ensemble [42].\nA full many-body theoretical treatment of a random dipolar-interacting two-dimensional\nspin ensemble may be needed to quantitatively predict the relaxation timescale in the ab-\nsence of extrinsic disorder. A mean-field approach may also provide an approximate\nsolution [49]. It is natural to inquire if a many-body localized phase could be observed for\nthis system. While we do not observe localization directly within the accessible parameter\nrange, we note that theoretical analyses of localization physics typically consider quasi-\nstatic disorder models, where the value of on-site field strength does not change on the\ntimescale of a single measurement. Quasi-static disorder generally slows down relaxation\nas shown in our experiments, but our model demonstrates that time-dependent disorder\n6may speed it up, for a subset of values of Wandτ, by allowing previously off-resonant\nspins to become resonant. Achieving finer control over disorder parameters may allow\na detailed single-spin-resolution study of various aspects of localization physics in the\npresent system.\nOur observations open the door for in-depth explorations of quantum dynamics in\ntwo-dimensional spin systems. In particular, they indicate that systems of interacting\nspin qubits with long coherence times can be created and manipulated under ambient\nconditions at room temperature. Our approach can be used used for the controlled gen-\neration of entanglement among spins on the diamond surface with potential applications\nto quantum sensing and magnetic resonance imaging of single molecules.\nAcknowledgements:\nThe authors acknowledge discussions with Timo Gr¨ aßer, G¨ otz Uhrig, Elana Urbach,\nTamara Sumarac, Emma Rosenfeld, Bo Dwyer, Harry Zhou, and Oleg P. Sushkov. This\nwork was supported by the National Science Foundation grant PHY-2014094, NSF, CUA,\nARO MURI, and Moore Foundation.\nReferences\n[1] Kaufman, A. M. et al. Quantum thermalization through entangle-\nment in an isolated many-body system. Science 353, 794–800 (2016).\nURL https://science.sciencemag.org/content/353/6301/794https:\n//science.sciencemag.org/content/353/6301/794.abstract .\n[2] Lukin, A. et al. Probing entanglement in a many-body-localized system. Science\n364, 256–260 (2019). URL https://www-science-org.ezp-prod1.hul.harvard.\nedu/doi/abs/10.1126/science.aau0818 .1805.09819 .\n[3] Rispoli, M. et al. Quantum critical behaviour at the many-body localization transi-\ntion. Nature 2019 573:7774 573, 385–389 (2019). URL https://www.nature.com/\narticles/s41586-019-1527-2 .\n[4] Morong, W. et al. Observation of Stark many-body localization without disorder.\nNature — 599, 393 (2021). URL https://doi.org/10.1038/s41586-021-03988-0 .\n[5] Zhang, J. et al. Observation of a Discrete Time Crystal. Nature 543, 217 (2017).\n1609.08684 .\n[6] Choi, J. et al. state spin ensemble ( a ) ( b ) vi, 1–11 (2017).\n[7] Keesling, A. et al. Quantum Kibble–Zurek mechanism and critical dynamics on a\nprogrammable Rydberg simulator. Nature 2019 568:7751 568, 207–211 (2019). URL\nhttps://www.nature.com/articles/s41586-019-1070-1 .\n7[8] Ebadi, S. et al. Quantum phases of matter on a 256-atom programmable quantum\nsimulator. Nature 2021 595:7866 595, 227–232 (2021). URL https://www.nature.\ncom/articles/s41586-021-03582-4 .\n[9] Kyprianidis, A. et al. Observation of a prethermal discrete time crystal. Sci-\nence 372, 1192–1196 (2021). URL https://www-science-org.ezp-prod1.hul.\nharvard.edu/doi/abs/10.1126/science.abg8102 .2102.01695 .\n[10] Turner, C. J., Michailidis, A. A., Abanin, D. A., Serbyn, M. & Papi´ c, Z. Weak er-\ngodicity breaking from quantum many-body scars URL https://doi.org/10.1038/\ns41567-018-0137-5 .\n[11] Bluvstein, D. et al. Controlling quantum many-body dynamics in driven Rydberg\natom arrays. Science 371, 1355–1359 (2021). URL https://www-science-org.\nezp-prod1.hul.harvard.edu/doi/abs/10.1126/science.abg2530 .\n[12] Kao, W., Li, K. Y., Lin, K. Y., Gopalakrishnan, S. & Lev, B. L. Topological pumping\nof a 1D dipolar gas into strongly correlated prethermal states. Science 371, 296–300\n(2021). URL https://www-science-org.ezp-prod1.hul.harvard.edu/doi/abs/\n10.1126/science.abb4928 .\n[13] Cardellino, J. et al. The effect of spin transport on spin lifetime in nanoscale systems.\nNature Nanotechnology 9, 343–347 (2014).\n[14] Alvarez, G. A., Suter, D. & Kaiser, R. Localization-delocalization transition in the\ndynamics of dipolar-coupled nuclear spins. Science 349(2015).\n[15] Wei, K. X., Ramanathan, C. & Cappellaro, P. Exploring Localization in Nuclear\nSpin Chains. Physical Review Letters 120, 70501 (2018). URL https://link.aps.\norg/doi/10.1103/PhysRevLett.120.070501 .\n[16] Smith, J. et al. Many-body localization in a quantum simulator with programmable\nrandom disorder. Nature Physics 2016 12:10 12, 907–911 (2016). URL https:\n//www.nature.com/articles/nphys3783 .\n[17] Choi, J.-y. et al. Exploring the many-body localization transition in two dimensions.\nScience 352, 1547–1552 (2016). URL http://www.ncbi.nlm.nih.gov/pubmed/\n27339981 .\n[18] Kucsko, G. et al. Critical Thermalization of a Disordered Dipolar Spin System in\nDiamond. Physical Review Letters 121, 23601 (2018). URL https://link.aps.\norg/doi/10.1103/PhysRevLett.121.023601 .\n8[19] Schreiber, M. et al. Observation of many-body localization of interacting fermions\nin a quasi-random optical lattice (2015). URL http://arxiv.org/abs/1501.05661 .\n1501.05661 .\n[20] Fel’dman, E. B. & Lacelle, S. Cite as. J. Chem. Phys 104, 2000 (1996). URL\nhttps://doi.org/10.1063/1.470956 .\n[21] Dobrovitski, V. V., Feiguin, A. E., Awschalom, D. D. & Hanson, R. Decoherence\ndynamics of a single spin versus spin ensemble. Physical Review B - Condensed Mat-\nter and Materials Physics 77, 245212 (2008). URL https://journals-aps-org.\nezp-prod1.hul.harvard.edu/prb/abstract/10.1103/PhysRevB.77.245212 .\n[22] Anderson, P. W. Absence of Diffusion in Certain Random Lattices. Physical Review\n109, 1492–1505 (1958).\n[23] Burin, A. L. Energy delocalization in strongly disordered systems induced by the\nlong-range many-body interaction (2006). URL http://arxiv.org/abs/cond-mat/\n0611387 .0611387 .\n[24] Yao, N. Y. et al. Many-body localization in dipolar systems. Physical Review Letters\n113, 243002 (2014). URL https://journals.aps.org/prl/abstract/10.1103/\nPhysRevLett.113.243002 .\n[25] Gong, Z.-X., Foss-Feig, M., Brand˜ ao, F. G. S. L. & Gorshkov, A. V. Entanglement\nArea Laws for Long-Range Interacting Systems. Physical Review Letters 119, 50501\n(2017). URL http://link.aps.org/doi/10.1103/PhysRevLett.119.050501 .\n[26] Orioli, A. P. et al. Relaxation of an Isolated Dipolar-Interacting Rydberg Quantum\nSpin System. Physical Review Letters 120, 063601 (2018). 1703.05957 .\n[27] Gopalakrishnan, S., Agarwal, K., Demler, E. A., Huse, D. A. & Knap, M. Griffiths\neffects and slow dynamics in nearly many-body localized systems. Physical Review\nB93, 1–12 (2016). 1511.06389 .\n[28] Sushkov, A. O. et al. Magnetic resonance detection of individual proton\nspins using quantum reporters. Physical Review Letters 113, 197601 (2014).\nURL https://journals-aps-org.ezp-prod1.hul.harvard.edu/prl/abstract/\n10.1103/PhysRevLett.113.197601 .\n[29] Davis, E. J. et al. Probing many-body noise in a strongly interacting two-dimensional\ndipolar spin system (2021). URL http://arxiv.org/abs/2103.12742 .2103.12742 .\n[30] Grotz, B. et al. Sensing external spins with nitrogen-vacancy diamond. New Journal\nof Physics 13, 55004 (2011).\n9[31] Grinolds, M. S. et al. Subnanometre resolution in three-dimensional magnetic reso-\nnance imaging of individual dark spins. Nature Nanotechnology 9, 279–284 (2014).\nURL www.nature.com/naturenanotechnology .1401.2674 .\n[32] Tetienne, J. P. et al. Spin properties of dense near-surface ensembles of\nnitrogen-vacancy centers in diamond. Physical Review B 97, 085402 (2018).\nURL https://journals-aps-org.ezp-prod1.hul.harvard.edu/prb/abstract/\n10.1103/PhysRevB.97.085402 .1711.04429 .\n[33] Sangtawesin, S. et al. Origins of Diamond Surface Noise Probed by Correlating\nSingle-Spin Measurements with Surface Spectroscopy 031052 , 1–17 (2019).\n[34] Stacey, A. et al. Evidence for Primal sp ¡sup¿2¡/sup¿ Defects at the Diamond Surface:\nCandidates for Electron Trapping and Noise Sources. Advanced Materials Interfaces\n6, 1801449 (2019). URL http://doi.wiley.com/10.1002/admi.201801449 .\n[35] Rosskopf, T. et al. Investigation of Surface Magnetic Noise by Shallow Spins in\nDiamond 147602 , 1–5 (2014).\n[36] Romach, Y. et al. Spectroscopy of Surface-Induced Noise Using Shallow Spins in\nDiamond. Physical Review Letters 114, 17601 (2015). URL https://link.aps.\norg/doi/10.1103/PhysRevLett.114.017601 .\n[37] Myers, B. A. et al. Probing Surface Noise with Depth-Calibrated Spins in Diamond.\nPhysical Review Letters 113, 27602 (2014).\n[38] Myers, B. A., Ariyaratne, A. & Jayich, A. B. Double-Quantum Spin-Relaxation\nLimits to Coherence of Near-Surface Nitrogen-Vacancy Centers. Physical Review\nLetters 118, 197201 (2017).\n[39] Bluvstein, D., Zhang, Z., McLellan, C. A., Williams, N. R. & Jayich, A. C. Ex-\ntending the Quantum Coherence of a Near-Surface Qubit by Coherently Driv-\ning the Paramagnetic Surface Environment. Physical Review Letters 123, 146804\n(2019). URL https://journals-aps-org.ezp-prod1.hul.harvard.edu/prl/\nabstract/10.1103/PhysRevLett.123.146804 .1905.06405 .\n[40] F´ avaro de Oliveira, F. et al. Tailoring spin defects in diamond by lattice charging. Na-\nture Communications 8, 15409 (2017). URL http://www.nature.com/doifinder/\n10.1038/ncomms15409 .\n[41] Dwyer, B. L. et al. Probing spin dynamics on diamond surfaces using a single quan-\ntum sensor (2021). URL http://arxiv.org/abs/2103.12757 .2103.12757 .\n[42]Supplementary Information .\n10[43] Laraoui, A. et al. High-resolution correlation spectroscopy of 13C spins near a\nnitrogen-vacancy centre in diamond. Nature Communications 4, 1–7 (2013). URL\nwww.nature.com/naturecommunications .\n[44] Staudacher, T. et al. Probing molecular dynamics at the nanoscale via an individual\nparamagnetic centre. Nature Communications 6, 8527 (2015).\n[45] Loretz, M. et al. Single-proton spin detection by diamond magnetometry. Science\nscience.1259464—- (2014).\n[46] DeVience, S. J. et al. Nanoscale NMR spectroscopy and imaging of multiple nuclear\nspecies. Nature Nanotechnology 10, 129–134 (2015). URL http://www.nature.com/\narticles/nnano.2014.313 .\n[47] Abragam, A. The Principles of Nuclear Magnetism: The International Series of\nMonographs on Physics (1961).\n[48] Slichter, C. P. Principles of Magnetic Resonance , vol. 1 of Springer Series in Solid-\nState Sciences (Springer Berlin Heidelberg, Berlin, Heidelberg, 1978). URL http:\n//link.springer.com/10.1007/978-3-662-12784-1 .\n[49] Gr¨ aßer, T., Bleicker, P., Hering, D.-B., Yarmohammadi, M. & Uhrig, G. S. Dynamic\nmean-field theory for dense spin systems at infinite temperature (2021). URL https:\n//arxiv.org/abs/2107.07821v1 .2107.07821 .\n[50] Taylor, J. M. et al. High-sensitivity diamond magnetometer with nanoscale reso-\nlution. Nature Physics 4, 810–816 (2008). URL www.nature.com/naturephysics .\n0805.1367 .\n[51] Mamin, H. J. et al. Nanoscale nuclear magnetic resonance with a nitrogen-vacancy\nspin sensor. Science 339, 557–560 (2013).\n[52] Pham, L. M. et al. NMR technique for determining the depth of shallow nitrogen-\nvacancy centers in diamond. Physical Review B 93(2016). 1508.04191 .\n11surface spin\nproton spinSj\nNVBFigure 1: Characterization of a representative NV center - surface spin system. (a) The experimental platform\nconsists of a single, shallow NV center strongly coupled to one surface electron spin Sj. Other electron spins and\nproton nuclear spins reside on the surface of the diamond. An external magnet splits the electron and nuclear\nspin states and allows independent manipulation of NV center and surface spins at their corresponding magnetic\nresonance frequencies. Inset: NV center and surface spin level structure. NV center transition is addressed at\nangular frequency ωNVwhile surface spin transition is addressed at ωS. (b) NV center Hahn echo (blue) and\ndouble electron-electron resonance (DEER) (red) measurements. The oscillations present in the DEER data\nindicate that the magnetic dipole interactions of this NV center with the surface spin ensemble are dominated by\na central surface spin with the coupling strength kmax= (4.90 ±0.57) µs−1(red line fit). Inset: the DEER pulse\nsequence.\n12 \n \nFigure 2: Characterization of the local magnetic environment of a central surface spin. (a) Top: the correlation\nspectroscopy pulse sequence. In our experiments the NV center interpulse spacing is set such that the NV center\nis primarily sensitive to the central surface spin autocorrelation. Manipulating the surface spins during the\ngray shaded region allows us to measure different central surface spin correlation functions. The complete pulse\nsequences for all measurements are described in SI. Bottom: Quantum logic gate representation of the correlation\nspectroscopy pulse sequence. The two CNOT gates are applied, such that the final state of the NV center is\ncontingent on whether the z-projection of the central surface spin changed during the time between the gates. (b)\nSurface spin detuned Ramsey (blue) and Hahn echo (red) measurements of ⟨Sx\nj(t)Sx\nj(0)⟩transverse central spin\nautocorrelation. The Hahn echo decay time is T2= (1.41 ±0.11) µs and the Ramsey decay time is T∗\n2= (0.32 ±\n0.02) µs. Solid lines are fits using the model in eqs. (1,2). The oscillations in the Ramsey data are due to the 9.2\nMHz detuning from the surface spin Larmor frequency. (c) Surface spin XY-4 measurement (blue) and MREV-8\nmeasurement (red). The MREV-8 measurement is performed within a spin echo (see SI for full pulse sequences).\nSolid lines are fits using our model. The oscillations present in both measurements are due to proton nuclear spin\nprecession. (d) Nuclear spin bath noise power spectrum V( ω), with peaks at zero frequency and proton Larmor\nfrequency ωL. (e) The full characterization of relaxation timescales for one of the central surface spin systems,\nincluding the Ramsey, XY-4, MREV-8, and Szautocorrelation measurements.\n13Sj Si\nFigure 3: Measurements of central surface spin ⟨Sz\nj(t)Sz\nj(0)⟩autocorrelation. (a) Szautocorrelation measure-\nments for 7 different central surface spin systems. The black line is a stretched-exponential fit to one of the data\nsets, from which we extract the relaxation time Tz. (b) Szautocorrelation measurements with the delay time\nre-scaled by the product τeWeof the effective disorder correlation time and disorder width, independently mea-\nsured for each surface spin system. We observe a collapse of the 7 data sets to a universal stretched-exponential\ndecay curve, in agreement with our dynamical disorder model. (c) S zautocorrelation decay time plotted as a\nfunction of the product τeWe, with the linear fit shown by the black line. (d) A schematic of the spin hopping\nprocess in our model, showing a central surface spin Sjand another surface spin Si. Each spin state is shifted by\ntime-dependent on-site detuning δ(blue line), due to the local dynamical disorder created by nuclear spin bath\nnoise and Ising dipolar interaction terms. The spins can flip-flop if their detunings differ by less than the strength\nJof the dipolar interaction between them (grey band).\n14Figure 4: Evolution of central surface spin under spin-lock driving. (a) Measurements of ⟨Sy\nj(t)Sy\nj(0)⟩autocor-\nrelation under spin-lock driving for different drive strengths. Stronger driving results in better decoupling from\nthe nuclear spin bath magnetic noise and longer T1ρrelaxation time. For drive strength Ω ≳50µs−1, relaxation\nrate is independent of Ω. Lines show stretched exponential decay fits. Inset: radio frequency NV and surface\nspin pulses; green laser polarization and readout pulses applied before and after this sequence of RF pulses. (b)\nSpin-lock driving relaxation rate as a function of drive strength for the surface spin system presented in (a), along\nwith the corresponding decay rates for Hahn echo (1/ T2) and Szautocorrelation (1/ Tz) experiments, shown by\nthe gray bands. The point at Ω = 0 corresponds to the decay of the Ramsey coherence (1/ T2*). The mea-\nsurements performed with drive strength below 50 µs−1are consistent with direct relaxation due to the nuclear\nspin bath noise with spectrum V( ω). Relaxation rates for drive strengths Ω ≳50µs−1are independent of Ω and\nconsistently much slower than the dipolar interaction scale 1/ T2. (c) Comparison between different autocorrela-\ntion decay timescales for 4 different surface spin systems. Red circles, labeled T1ρ/2, show the decay time under\nspin-lock at strong driving. As the dynamical on-site disorder strength W increases, the Tzautocorrelation decay\ntime increases compared with the autocorrelation decay at strong driving with suppressed disorder ( T1ρ/2). Both\nof these decays are much slower (by a factor of ≈30) than the dipolar interaction scale, given by T2.\n15Supplemental Material for\n“Probing dynamics of a two-dimensional dipolar spin\nensemble using single qubit sensor”\nKristine Rezai1,3, Soonwon Choi2, Mikhail D. Lukin1, Alexander O. Sushkov3,4,5\n1Department of Physics, Harvard University, Cambridge, MA 02138, USA\n2Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02142, USA\n3Department of Physics, Boston University, Boston, MA 02215, USA\n4Department of Electrical and Computer Engineering, Boston University,\nBoston, MA 02215, USA\n5Photonics Center, Boston University, Boston, MA 02215, USA\n∗To whom correspondence should be addressed; e-mail: asu@bu.edu\n16Contents\nA Materials and Setup 18\nA.1 Diamond Sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18\nA.2 Optical Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18\nA.3 Radiofrequency Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18\nB Nuclear Spin Bath Noise Spectrum 18\nB.1 Wiener-Khinchin Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 19\nB.2 Calculate ⟨B2\n∥,rms⟩from Longitudinal Component of the Nuclear Magnetic\nMoment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20\nB.3 Calculate ⟨B2\n⊥,rms⟩from Transverse Component of the Nuclear Magnetic\nMoment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22\nB.4 Noise Spectrum Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 23\nC Characterizing NV Center - Surface Spin System 23\nC.1 NV Center Depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23\nC.2 The Double Electron-Electron Resonance (DEER) Sequence . . . . . . . . 24\nC.3 The Correlation Spectroscopy Pulse Sequence . . . . . . . . . . . . . . . . 25\nC.4 Surface Spin Pulse Sequences . . . . . . . . . . . . . . . . . . . . . . . . . 25\nC.5 Surface Spin Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26\nD Characterization of Surface Spin System 27\nD.1 Decoherence due to the Nuclear Spin Bath . . . . . . . . . . . . . . . . . . 27\nD.2 Decay due to Dipolar Interactions . . . . . . . . . . . . . . . . . . . . . . . 28\nD.3 Extraction of Surface Spin On-site Disorder Strength W . . . . . . . . . . 29\nD.4 Extraction of Proton Nuclear Spin Signal Correlation Time τ. . . . . . . . 29\nD.5 Extraction of Surface Spin Density . . . . . . . . . . . . . . . . . . . . . . 30\nD.6 Effects of Finite Pulse Duration . . . . . . . . . . . . . . . . . . . . . . . . 30\nD.7 NV Center’s Effects on Surface Spin Dynamics . . . . . . . . . . . . . . . . 30\nE Resonance Counting Theory 32\nE.1 Dynamic Disorder Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32\nE.2 Two Sources of Dynamic Disorder . . . . . . . . . . . . . . . . . . . . . . . 35\nF S zAutocorrelation Scaling 35\nG Spin-lock Driving Measurements (T 1ρ) 36\n17A Materials and Setup\nA.1 Diamond Sample\nNitrogen implantation was done on a12C enriched {001}diamond surface at an implantation energy of 3\nkeV. After annealing, the diamond was cleaned in a 3-acid mixture (equal volumes of concentrated H 2SO4,\nHNO 3, and HClO 4) for 3 hours. All measurements in the main text were performed with the diamond placed\nin deuterated glycerol (CDN Isotopes D-0302) and under ambient conditions.\nA.2 Optical Setup\nA home built scanning confocal microscope optically addresses and reads out single NV centers. The diamond\nwas placed NV side down on a RF transmission line, and an inverted Nikon CFI Apo TIRF objective (NA\n= 1.49) was positioned under the stripline. A piezoelectric scanner (Physik Instrumente P-721 PIFOC) was\nused to control the height of the objective, and a closed-loop scanning galvanometer (Thorlabs GVS012) was\nused to direct the lateral position of the laser beam focus. Optical excitation was provided by a 532 nm laser\n(Opto Engine MLL-FN-532-200mW) gated by an acousto-optic modulator (Isomet 1250C) in a double pass\nconfiguration. NV center fluorescence passes through a 842 nm short pass filter (Semrock FF01-842/SP-25),\na 594 nm long pass filter (Semrock BLP01-594R-25), and a 532 nm notch filter (Semrock NF03-532E-25)\nbefore collection by a single-photon counter (Excelitas Technologies SPCM-AQRH-14-FC). The single-photon\ncounter and acousto-optic modulator were gated using TTL pulses produced by a 500 MHz PulseBlasterESR-\nPRO pulse generator from Spincore Technologies. A typical photon counting rate of 20 kHz was observed,\nand a photon counter acquisition window of 900 ns was used.\nA.3 Radiofrequency Setup\nTwo Stanford Research SG384 signal generators were used to deliver RF tones to address the NV center\nand surface spin transitions. IQ modulation was used to control the x and y quadrature amplitude of the\nsignal generator driving the NV center. The output of the signal generator driving the surface spins is fed\ninto a 90 degree phase shifter (RF-Lambda RFHB01G04GVT). The two outputs of the 90 degree phase\nshifter were each fed into 180 degree phase shifters (Pulsar Microwave JSO-10-47/3S). Microwave switches\n(MiniCircuits ZASWA-2-50DR+), controlled by TTL pulses produced by the pulse generator, gated the RF\ndrives. The zero, 90 degree, 180 degree, and 270 degree phase shifted RF pulses that drive the surface spins\nwere combined using a SigaTek SP51400 combiner. The RF signals were power boosted by MiniCircuits\namplifiers ZHL-5W-422+, ZHL-30W-252-S+, and ZHL-16W-43-S+, combined using MCLI PS2-134, and fed\ninto a 50Ω-terminated RF transmission line with the diamond sample on top.\nB Nuclear Spin Bath Noise Spectrum\nThe presence of a 2D layer of proton nuclear spins on the diamond surface will produce a fluctuating magnetic\nfield at the site of the surface spins Bz(t) characterized by a magnetic power spectral density V(ω) where\n18Bz(t) =Bz\n⊥(t) +Bz\n∥(t). (3)\nWe define the z-axis as along the direction of the externally applied magnetic field, which we align along\nthe NV axis. We calculate the projection of the magnetic field due to sum of nuclear spin magnetic moments\nalong the direction of the applied external magnetic field. Bz\n⊥(t) is this projection due to components of the\nnuclear spin magnetic moment perpendicular to the direction of the external magnetic field, which rotate\nat the nuclear Larmor frequency, and Bz\n∥(t) is the sum of the contributions from the nuclear spin magnetic\nmoments parallel to the direction of the external magnetic field. From the expression for Bz(t), the noise\nspectrum V(ω) will have a low-frequency term due to slow time dependence in Bz\n∥(t) as well as higher\nfrequency components due to the precession of the transverse magnetic moment, captured by Bz\n⊥(t).\nWe can characterize the low frequency term in V(ω) following the treatment in [50]. We model the flip-flop\nrelaxation process of the proton nuclear spins as an exponentially correlated Gaussian fluctuating field Bz\n��(t)\nhaving correlation function ⟨Bz\n∥(t0+t)Bz\n∥(t0)⟩=⟨B2\n∥,rms⟩exp(−t/τ) where τis the longitudinal correlation\ntime of the nuclear spins and B∥,rmsis the rms value of the magnetic field’s longitudinal component projected\nalong the z-axis.\nB.1 Wiener-Khinchin Theorem\nUsing the correlation of the magnetic noise from the fluctuating nuclear spins, we construct the power\nspectrum for this noise using the Wiener-Khinchin Theorem. The Wiener-Khinchin theorem states that\ngiven a random process x(t) with autocorrelation function\nC(t) =⟨x(t0+t)x(t0)⟩, (4)\nthe spectral power density is the Fourier transform of the autocorrelation function\nV(ω) =Z∞\n−∞C(t)e−iωtdt. (5)\nFor our calculations, we use the following definitions of Fourier transform and inverse Fourier transform:\nX(ω) =Z∞\n−∞x(t)e−iωtdt (6)\nx(t) =1\n2πZ∞\n−∞X(ω)eiωtdω (7)\nUsing the Wiener-Khinchin theorem with the above definition of Fourier transform, we calculate the\nspectral power density for the low frequency noise associated with relaxation processes for proton nuclear\nspins as\nV1(ω) =Z∞\n−∞⟨B2\n∥,rms⟩e−|t|/τe−iωtdt= 2⟨B2\n∥,rms⟩τ\nω2τ2+ 1. (8)\n19We apply the same procedure to determine the noise spectrum V2(ω) due to proton nuclear spins precessing\nat their Larmor frequency ωL. We begin by writing the correlation of the magnetic field noise:\n⟨B⊥(t0+t)B⊥(t0)⟩=⟨B2\n⊥,rms⟩eiωLt−t/τ. (9)\nWe then apply the Wiener-Khinchin formula and take the Fourier transform of ⟨B⊥(t0+t)B⊥(t0)⟩in order\nto arrive at the power spectral density\nV2(ω) =Z∞\n−∞⟨B2\n⊥,rms⟩eiωL|t|−|t|/τe−iωtdt= 2⟨B2\n⊥,rms⟩τ\n(ω−ωL)2τ2+ 1. (10)\nWe combine the contribution of the nuclear spin bath’s component at zero frequency and at Larmor\nfrequency to write the final nuclear spin noise power spectrum as\nV(ω) = 2⟨B2\n∥,rms⟩τ\nω2τ2+ 1+ 2⟨B2\n⊥,rms⟩τ\n(ω−ωL)2τ2+ 1. (11)\n⟨B2\n∥,rms⟩and⟨B2\n⊥,rms⟩are geometrically related and we will next derive their relationship. For all following\ncalculations, we use CGS units.\nB.2 Calculate ⟨B2\n∥,rms⟩from Longitudinal Component of the Nuclear Magnetic\nMoment\nHere we wish to calculate the magnetic field at the site of surface spins from nuclear spins in the system. We\nfollow the treatment presented in [51]. An external field along the NV symmetry axis sets the longitudinal\nquantization axis ˆ q. The system is comprised of shallow electronic spins near the surface of diamond with\nproton nuclear spins on the diamond surface. All nuclear spins are unpolarized and produce a fluctuating\nmagnetic field at the site of each surface spin, and we are interested in calculating the RMS amplitude of the\ncomponent of this magnetic field along the quantization axis ˆ qof the surface spins. We begin by considering\nthe longitudinal component of the magnetic moment of nuclear spins. The component of magnetic field along\nˆ qfrom a single nuclear spin parallel or anti-parallel to ˆ qis\nBq(r) =ˆ q·B=ˆ q·3ˆ r(⃗ mn·ˆ r)−⃗ mn\nr3=mn[3(ˆ r·ˆ q)2−1\nr3] (12)\nwhere ris the vector from the nuclear spin to the surface electron spin and mnis the magnetic moment\nof the nuclear spin. The mean square field from a collection of spins is\nB2\n∥,rms=⟨[X\niBq(ri)]2⟩. (13)\nBy expanding the sum, we get\nB2\n∥,rms=⟨X\ni̸=jBq(ri)Bq(rj) +X\niB2\nq(ri)⟩. (14)\n20Since the nuclear spins are random and uncorrelated, this reduces to\nB2\n∥,rms=X\niB2\nq(ri). (15)\nWe can convert the sum to an integral by including the nuclear spin density ρn:\nB2\n∥,rms=ρnZ\nV[Bq(r)]2dV. (16)\nWe can define ˆ qas the unit vector along the [1,1,1] axis and define ˆ rin terms of its spherical coordinates:\nˆ q= [1,1,1]/√\n3 (17)\nˆ r= [sin( θ) cos( ϕ),sin(θ) sin( ϕ),cos(θ)] (18)\nwhere θis the polar angle measured from the normal of the diamond surface, and ϕis the aximuthal\nangle.\nWhen the diamond is immersed in deutarated glycerol, the simplest assumption to make is that the proton\nnuclear spins form a 2D layer at the diamond surface [44, 45, 46]. Thus, for the case of surface spins which\nreside a depth d below the 2D layer of proton nuclear spins, we can relate the distance from the surface spins\nto a nuclear spin plane through the depth dand polar angle θ\ncosθ=d\nr(19)\nsinθ=r\n1−d2\nr2(20)\nThe rms magnetic field can be calculated using\nB2\n∥,rms=m2\nnρnZ2π\n0Z∞\nd[(q\n1−d2\nr2cosϕ+q\n1−d2\nr2sinϕ+d/r)2−1\nr3]2rdrdϕ (21)\nB2\n∥,rms=m2\nnρn3π\n8d4(22)\nHowever, one must be careful when calculating the magnetic moment. The magnetic moment is defined\nas\n⃗ m=gµN⃗S (23)\nwhere gis the effective g-factor and µNis the nuclear magneton. We can define\n21⟨m⟩=gµN⟨S⟩ (24)\nm=gµNp\nS(S+ 1) = γNℏp\nS(S+ 1) (25)\nwhere γNis the nuclear spin gyromagnetic ratio.\nFor proton nuclear spins in the 2D plane of the diamod surface, the rms magnetic field on the surface spin\nz-axis from the longitudinal component of the magnetic field is given by\nB2\n∥,rms=3π\n8d4ρn(γNℏp\nS(S+ 1))2(26)\nfor surface spins with depth d below the proton layer.\nWhen we perform depth measurements, we place the diamond in undeuterated glycerol. For proton\nnuclear spins in the half space above the diamond surface, we can calculate the RMS magnetic field from the\ncomponent of the magnetic moment paralel or antiparallel to ˆ q:\nB2\nrms=m2\nnρnZπ/2\n0Z∞\nd/cosθZ2π\n0[(sinθcosϕ+ sin θsinϕ+ cos θ)2−1\nr3]2r2sinθdϕdrdθ (27)\nwhere d is the depth of the surface spin. Solving this integral yields\nB2\nrms=πm2\nnρn\n8d3(28)\nUsing the expression for magnetic moment, this becomes\nB2\nrms=πρn\n8d3(γNℏp\nS(S+ 1))2(29)\nB.3 Calculate ⟨B2\n⊥,rms⟩from Transverse Component of the Nuclear Magnetic\nMoment\nWe follow the treatment in [52] to calculate ⟨B2\n⊥,rms⟩due to the transverse component of the nuclear spin\nmagnetic moment.\nWe can calculate the rms magnetic field from the transverse component of the magnetic moment of proton\nnuclear spins in the half space above the diamond surface:\nB2\n⊥,rms=2(gµNp\nS(S+ 1))2\n3ρZ∞\n−∞dxZ∞\n−∞dyZ∞\nddz(x+y+z)2(r2−xy−yz−xz)\n(x2+y2+z2)5(30)\nB2\n⊥,rms=5πρ\n72d3(gµNp\nS(S+ 1))2(31)\n22Using this technique, we can calculate the rms magnetic field from a 2D layer of proton spins on the\ndiamond surface:\nB2\n⊥,rms=2(gµNp\nS(S+ 1))2\n3ρZ∞\n−∞dxZ∞\n−∞dy(x+y+d)2(r2−xy−ya−xa)\n(x2+y2+d2)5(32)\nB2\n⊥,rms=5πρ\n24d4(gµNp\nS(S+ 1))2(33)\nWe find that the projection of the rms magnetic field from the longitudinal component of the magnetic\nmoment on the surface spin z-axis is 3 /√\n5 times larger than the projection from the transverse component\nof the nuclear spin magnetic moment for both nuclear spins in the 2D plane as well as nuclear spins in the\nhalf space.\nB.4 Noise Spectrum Equation\nUsing the relationship previously derived between the transverse and longitudinal ⟨B2\nrms⟩for the given geom-\netry, we can write the expression for the nuclear spin bath noise spectrum as\nV(ω) = 2⟨B2\n∥,rms⟩(τ\nω2τ2+ 1+ +5\n9τ\n(ω−ωL)2τ2+ 1) (34)\nwhere\nB2\n∥,rms=3π\n8d4ρn(gµNp\nS(S+ 1))2(35)\nfor a system with proton nuclear spins residing on the 2D surface of the diamond and surface spins a\ndepth d below. We can make one further step and rewrite the power spectral density replacing ⟨B2\n∥,rms⟩with\nW2, which are related by γ2\ne, the electron gyromagnetic ratio. We recover the expression for the nuclear spin\nbath noise spectrum that we use in the main text:\nV(ω) = 2W2\nγ2\ne(τ\nω2τ2+ 1+ +5\n9τ\n(ω−ωL)2τ2+ 1). (36)\nC Characterizing NV Center - Surface Spin System\nC.1 NV Center Depth\nIn order to extract the depth of the NV centers used in this study, we placed the diamond in protonated\nglycerol and performed XY RF pulse sequences on the NV center to correlate the strength of the measured\nproton signal to the depth of the NV center. Following the treatment in [52], when performing an XY-64\npulse sequence on one NV center, an NV center depth of d NV= (2.92 ±0.24) nm was extracted. Depth\nmeasurements on other NV centers used in this study yielded similar results.\n23Figure S5: NV XY-64 measurement, with total sequence time centered around expected\nproton peak, performed in an external magnetic field of 735 G. Fit (blue curve) to Eq. 1\nin [52], extracting a proton Brmscorresponding to an NV center depth of d NV= (2.92 ±\n0.24) nm.\nC.2 The Double Electron-Electron Resonance (DEER) Sequence\nThe double elecron-electron resonance (DEER) sequence characterizes the strength of the dipolar interaction\nbetween the NV center and nearby surface electron spins. The dipolar interaction Hamiltonian between the\nNV center Jand surface spins Siis written as\nHNV−ss=X\niℏ2γ2\ne\nr3\nNV−i[J·Si−3(J·rNV−i)(Si·rNV−i)\nr2\nNV−i], (37)\nwhere rNV−iis the vector from the NV center to surface spin i. When we align the external magnetic\nfield along the NV center axis, we can make a secular approximation and only consider terms that commute\nwith Sz\niandJz:\nHNV−ss=X\niℏ2γ2\ne\nr3\nNV−i(1−3 cos2θi)Sz\niJz=X\niℏkiSz\niJz(38)\nwhere θiis the angle the vector rNV−imakes with the external magnetic field, and the coupling strength\nis\nki=ℏγ2\ne\nr3\nNV−i(1−3 cos2θi). (39)\nThe NV center |ms= 0⟩state after performing the DEER sequence with total sequence time tNVis\nSDEER =1\n2[1 + cos(X\nikiSz\nitNV)]. (40)\nAfter performing many averages with different initial surface spin projections, the final expected signal [28]\nis\n24SDEER =1\n2[1 +Y\nicos(kitNV/2)]. (41)\nFor systems of strong coupling between the NV center and one surface electron spin, oscillations at the\nlargest coupling strength are present at short time. For the data presented in the main text fig. 1 and fig.\n2, the NV spin-echo decays on a time scale TNV\n2= (2.67 ±0.13) µs and has a coupling strength of kmax=\n(4.90±0.57) µs−1to the strongest coupled surface spin.\nC.3 The Correlation Spectroscopy Pulse Sequence\nThe correlation spectroscopy pulse sequence (fig. 2a) measures the autocorrelation of the surface spin Sz\nstate⟨Sz(t0+t)Sz(t0)⟩[43]. After the initialπ\n2NV\nx−πNV\nyπSS−π\n2NV\nypulses, the NV phase accumulation ϕ1due\nto the surface spin Szstate is projected along the NV z-axis. If the NV state is read out at this time, the NV\npopulation would be ∼ ⟨sinϕ1⟩. By applying two sequences ofπ\n2NV\nx−πNV\nyπSS−π\n2NV\nypulses, the initial and\nfinal surface spin Szprojections are converted to population along the NV z-axis, with an amplitude related\nto⟨sinϕ1sinϕ2⟩where ϕ2is the phase the NV spin acquires during the finalπ\n2NV\nx−πNV\nyπSS−π\n2NV\nypulses.\nFor all correlation spectroscopy sequences that we perform, we normalize the result by the correlation\nspectroscopy contrast. We define the contrast as the NV spin population difference between when we run the\ncorrelation spectroscopy pulse sequence (fig. 2a) without any pulses applied during the grey shaded region\nand when we run the sequence with a pi pulse applied to the surface spins during this time. We normalize\nthe final signal between +1 and -1, plotting 4 ⟨Sz(t0+t)Sz(t0)⟩.\nC.4 Surface Spin Pulse Sequences\nIn this section we present the pulse sequences that we use in the main text for fig. 2. All of the following\nsequences are applied during the shaded region of the correlation spectroscopy pulse sequence (fig. 2 (a)).\nt\nt\ntt\n(a) (b) (c)\n(d)\nFigure S6: Pulse sequences applied to surface spins during the correlation spectroscopy\npulse sequence. (a) Surface spin Ramsey, (b) surface spin echo, (c) surface spin XY-4, and\n(d) surface spin MREV-8 measurement, which we apply sandwiched in the free precession\ninterval of the spin echo.\n25C.5 Surface Spin Stability\nRecent work has indicated that surface electron spins may not have stable positions but may be mobile on the\ndiamond surface, potentially during the application of green laser light [41]. We directly probe the stability\nof the strongest coupled surface spin by measuring its state under the application of green laser light. First,\nwe polarize the surface electron spin by driving both the surface spins and the NV center at the same drive\nstrength for a time set by the dipolar interaction between the NV center and the central surface spin. We\nfurther calibrate the Hartmann-Hahn polarization transfer by observing maximal polarization transfer when\nthe drive strength of the surface spins matches the drive strength of the NV center.\n \nlaser(a) (b)\nFigure S7: Hartmann-Hahn spin polarization transfer from the NV center to the strongest\ncoupled surface spin. (a) HH polarization transfer sequence and readout of final NV\ncenter state contingent on polarization of central surface spin. (b) Measurement result\nof sweeping the surface spin drive where the resulting polarization of the central surface\nspin is read out using the NV center.\nTo study the stability of the strongest coupled surface spin, we sweep the duration of the green laser\npulse after the polarization transfer. The strongest coupled surface spin is stable under green laser light\nillumination for at least 50 µs.\n \nlaser\ndt(a) (b)\nFigure S8: Surviving polarization of the central surface spin after green laser light illu-\nmination. (a) HH polarization transfer sequence and readout of final NV center state\ncontingent on polarization of central surface spin while green NV repolarization laser\npulse duration is swept. (b) Measurement result, normalized by the contrast of the Sz\nautocorrelation measurement of equivalent initial time delay.\n26D Characterization of Surface Spin System\nIn this section, we extract parameters relevant to central surface spin dynamics including the strength and\ncorrelation time of the on-site disorder from the nuclear spin bath, the dipolar interaction strength between\nnearby surface spins, and the density of surface spins.\nD.1 Decoherence due to the Nuclear Spin Bath\nIn a previous section, we derived the proton nuclear spin bath noise spectrum as a sum of Lorentzians with\ncomponents centered at zero and at proton Larmor period:\nV(ω) = 2W2\nγ2\ne(τ\nω2τ2+ 1+5\n9τ\n(ω−ωL)2τ2+ 1) (42)\nFor the pulse sequences applied to the surface spins, we calculate the expected signal from the nuclear\nspin bath. For a given sequence, the expected signal can be written as\nS(t) =e−χ(t)(43)\nwhere\nχ(t) =γ2\neZdω\n2πV(ω)\nω2F(ωt) (44)\nwhere F(ωt) is the filter function of the pulse sequence and tis the total sequence time. In order to\nsimplify calculations, we treat the peak at proton Larmor frequency ωLas a delta function\nV2(ω) = 2 π5\n9W2\nγ2\neδ(ω−ωL). (45)\nWe select a delta function with a factor of 2 πin front so that the integral of the Lorentzian is matched by\nthe integral of this delta function, and we numerically verify that for our given pulse sequences, the response\ndue to a narrow Lorentzian is consistent with the response due to a delta function. For the Ramsey pulse\nsequence with filter function FRamsey (ωt) = 2 sin2(ωt/2), the expected decay due to the nuclear spin bath is\nSRamsey (t) =e−(Wt/√\n2)2eW2t3/(6τ)e−25W2\n9ω2\nLsin2(ωLt/2). (46)\nThe filter function for the spin echo sequence is Fecho(ωt) = 8 sin4(ωt/4), and the expected signal decay\ndue to the nuclear spin bath is\nSecho(t) =e−W2t3\n12τe−40W2\n9ω2\nLsin4(ωLt/4). (47)\nThe XY-4 pulse sequence has the filter function FXY−4(ωt) = 128 sin6(ωt/16)(cos(3 ωt/16)+cos(5 ωt/16))2\nand the decay due to the nuclear spin bath is given by\n27SXY−4(t) =e−13W2t3\n4500τe−1285W2\n9ω2\nLsin6(ωLt/16)(cos(3 ωLt/16)+cos(5 ωLt/16))2\n. (48)\nThe MREV-8 sequence in an echo has a filter function given by\nFMREV (ωt) = 16(1+2 cos( ωt/12))2(sin(ωt/12)−sin(ωt/6)4(3−4 cos( ωt/24)+3 cos( ωt/12)−2 cos( ωt/8)+cos( ωt/6))\n(49)\nwith a resulting signal of\n−log(SMREV (t)) =49W2t3\n2592τ+ 165W2\n9ω2\nL(1 + 2 cos( ωLt/12))2(sin(ωLt/12)−sin(ωLt/6)4(3−4 cos( ωLt/24)+\n3 cos( ωLt/12)−2 cos( ωLt/8) + cos( ωLt/6)).\n(50)\nFor each of the pulse sequences, the resulting signal has two components: a decay envelope due to the\nnoise spectrum peak at zero frequency and a modulation due to the signal at proton Larmor frequency.\nD.2 Decay due to Dipolar Interactions\nWhile the central surface spin Ramsey, echo, and XY measurements will be affected by the nuclear spin bath,\nthe dipolar interations between the central surface spin and nearby surface spins will also cause decoherence.\nIf there were one other surface spin in the system interacting with the central surface spin with dipolar\ninteraction strength J1=ℏγ2\ne\nr3(1−3 cos2θ), the resulting signal for the Ramsey, echo, and XY sequences would\nbe\nSRamsey,XY −4,Echo(t) =1\n2[cos(3 J1t/4) + cos( J1t/4)]. (51)\nOn short time scales, the central surface spin is mostly sensitive to nearest neighbor dipolar interactions,\nso we approximate the decay profile as a stretched exponential with a power of 2, e−(t/T2)2, with decay time\nT2≈1/J1, which we numerically verify. If we assume that the decays due to dipolar interactions and the\nnuclear spin bath are independent processes, then we can express the overall Ramsey, echo, and XY-4 decays\nas the product of the stretched exponential decay due to dipolar interactions with the decay and modulation\ndue to the nuclear spin bath. Therefore, as an example, the fit function for the XY-4 measurement becomes\nSXY−4(t) = exp( −(t/T2)2) exp(−13W2t3\n4500τ) exp(−1285W2\n9ω2\nLsin6(ωLt/16)[cos(3 ωLt/16) + cos(5 ωLt/16)]2) (52)\nwhere ωLis the proton Larmor frequency. We can show that the dominant source of decoherence in the\necho and XY-4 sequences comes from dipolar interactions between surface spins by comparing the decay\nrate of the echo, XY-4, and MREV-8 measurements. If the nuclear spin bath were the primary source of\n28decoherence for the echo or XY-4 sequences, there would be an extension of the coherence time between\nthe echo and XY-4 measurement results. However, the echo and XY-4 measurements decay over the same\ntimescale. Moreover, after applying an MREV-8 sequence, which decouples surface spins from homonuclear\ndipolar interactions, the coherence time increases, suggesting that dipolar interactions between surface spins\nare the primary contributor to decoherence in echo and XY-4 measurements.\nFigure S9: Comparison of surface spin echo, XY-4, and MREV-8 measurements, showing\nextension of coherence time with MREV-8.\nD.3 Extraction of Surface Spin On-site Disorder Strength W\nIn the previous sections, we calculated the expected signals given dipolar interactions and the nuclear spin\nbath. For the experimentally relevant regime where the nuclear spin correlation time τis much longer than\nthe sequence duration, a detuned Ramsey sequence decay is given by\nSRamsey (t) = cos(∆ t)e−(t/T2)2e−(Wt/√\n2)2e−25W2\n9ω2\nLsin2(ωLt/2)(53)\nwhere ∆ is the detuning between the surface spin splitting ωSand the microwave drive frequency. By\nfitting the echo and XY-4 measurements, T2can be extracted, so Wis the only free parameter in a Ramsey\nfit.\nD.4 Extraction of Proton Nuclear Spin Signal Correlation Time τ\nIn a previous section, we found the MREV-8 decay to be given by\n−log(SMREV (t)) =49W2t3\n2592τ+ 165W2\n9ω2\nL(1 + 2 cos( ωLt/12))2(sin(ωLt/12)−sin(ωLt/6)4(3−4 cos( ωLt/24)+\n3 cos( ωLt/12)−2 cos( ωLt/8) + cos( ωLt/6)).\n(54)\nSince MREV-8 decouples surface spins from dipolar interactions with other surface spins, the decay\nenvelope is determined by the disorder width W and the nuclear spin correlation time τ. We extracted the\n29disorder width from the Ramsey decay so by fitting the MREV-8 decay, we extract the nuclear spin correlation\ntime τ.\nD.5 Extraction of Surface Spin Density\nWe extract the average surface spin density by comparing the results of numeric simulations to averaged\nexperimental data. We begin by averaging together all XY-4 data taken at an external magnetic field of 730\nG and fit the periodic features in the data, extracting a value of W of 4.13 ±0.24µs−1. Next we perform\nnumeric simulations with a central surface spin and five neighboring surface spins. We perform simulations\nfor different values of average separation between spins and average together the results for 500 averages with\nrandom spin positions for each value of average separation. We extract an average surface spin separation of\n8.4±1.3 nm. This corresponds to an average interaction strength J = J0n3/2= (0.57 ±0.25) µs−1.\n(a) (b)\nFigure S10: (a) Simulation results with averaged experimental data (white dots). (b) χ2\nvs average separation, where an average separation between surface spins of 8.4 ±1.3 nm\nis extracted.\nD.6 Effects of Finite Pulse Duration\nIn all measurements, pulses are applied with a 12.5 MHz drive strength (40 ns πpulse). The total drive\nduration for the Ramsey (0.04 µs), echo (0.08 µs), and XY-4 (0.2 µs) sequences are significantly shorter\nthan the timescale of the decay, but for the MREV-8 sequence (0.4 µs), we consider the effects of the finite\nduration of the drive on the decay. We do this by comparing the numeric simulation results of the MREV-8\npulse sequence with a nuclear spin bath for the case of finite pulse duration and infinitely fast pulses. We\nplot on the x-axis the total evolution time, which includes the evolution time during the pulses for the case\nof finite pulse duration. We can account for the finite pulse duration by adding the time of the drive to the\ntotal free precession time.\nD.7 NV Center’s Effects on Surface Spin Dynamics\nWith the ability polarize the central surface spin using the NV center, we can study the effect the NV center\nspin state has on surface spin dynamics. We can achieve this by first polarizing the central surface spin and\nthen preparing the NV center in a particular spin state. If the NV center is in the m s= 0 state, it will have\n30Figure S11: Simulation results for finite pulses versus infinitely fast pulses. For infinitely\nfast pulses, the x-axis is given by the total free precession time and for finite pulses, the\nx-axis corresponds to the free precession time between pulses plus the finite duration of\nthe pulses.\nno effect on the surface spins through the dipolar interaction. However, if the NV center is in either the m s\n= 1 or the m s= -1 state, it will affect the surface spins through the dipolar interaction. We can perform a\nsurface spin Ramsey experiment after polarizing one surface spin with the NV in either m s= 0 or m s= -1\nto observe the effect of the dipolar interaction of the NV center on the central surface spin. Note that the\ndrive pulses to the surface spins are detuned, which produce the oscillations.\n \nlasert\nFigure S12: Surface spin detuned Ramsey measurement with the NV center in m s= 0\nor m s= -1. The oscillations are due to detuned surface spin pulses, and we observe a\ndifference in oscillation frequency between the red and blue curves equal to the dipolar\ninteraction between the NV center and the central surface spin. Inset: pulse sequence with\nHartmann-Hahn polarization transfer followed by Ramsey sequence applied to surface\nspins and then NV center state readout pulses.\nThe difference in oscillation frequency is given by the dipolar interaction strength between the NV center\nand the central surface spin and acts to further detune the surface spins when the NV is in m s= -1.\nWe can further observe the effect of the NV center spin state on surface spin dynamics by measuring the\nsurface spin T zof the strongest coupled surface spin after polarization with the NV in either m s= 0 or m s\n31= -1.\n \nlasert\nFigure S13: Surface spin T zmeasurement after polarization with the NV center in either\nms= 0 or m s= -1. Measurement results indicate no difference in relaxation rate between\nthe two cases, to within measurement uncertainty. Inset: pulse sequence with Hartmann-\nHahn spin polarization following by a swept delay time before NV pulses read out the\nfinal surface spin state.\nWe observe that the dipolar interaction between the NV center and the surface spins does not affect the\ncentral surface spin relaxation. For our parameter ranges of NV - surface spin dipolar interaction strength,\nsurface spin - surface spin dipolar intreaction strength, and on-site nuclear spin disorder width, we can\nnumerically verify that the relaxation of the central surface spin is dominated by the effects of the nuclear\nspin disorder and surface spin - surface spin dipolar interactions. Thus, we can neglect the effect of the NV\ncenter in our resonance counting treatment.\nE Resonance Counting Theory\nIn this section, we present the single-particle resonance counting theory for the case of time-varying on-site\nfields at the sites of the surface spins. We follow the treatment presented in [18], which also treats the case\nof quasi-static on-site fields.\nE.1 Dynamic Disorder Model\nWe estimate the survival probability of a central spin based on simple resonance counting arguments. The\nprobability of a given spin being on resonance with the central spin is given byβJ0/r3\nW, where J0=ℏγ2\ne,J0/r3\nis the interaction strength of the central spin with a bath spin a distance raway, Wis the disorder width,\nandβis a dimensionless parameter that characterizes the resonance probability, of order unity. We consider\nthe case of time dependent disorder, where the value of on-site detuning δjchanges over time by uncorrelated\njumps at a rate γ= 1/τ. We consider the case where W≥nJ0>1/τ. As presented in [18], two spins jand\niare on resonance at time t if (1) |δj(t)−δi(t)|< βJ 0/r3at any time less than t (main text fig. 3(e)) and\n(2) the interaction occurs on a time-scale t >1\nβJ0/r3.\n32The probability of having a resonant pair is given by\nPres(t) = 1−e−βJ0/r3\nWt\nτ(1−βJ0/r3\nW). (55)\nFor a 2D spin system with density n, we can calculate the probability of a central spin having no resonance\nas\nP(t) = exp( −ZR(t)\nr02πnrP res(t)dr). (56)\nNote that R(t) = ( J0t)1/3. WhenβJ0/r3\nWis small (at large times), (1 −βJ0/r3\nW) can be rewritten as\nexp(−βJ0/r3\nW) and\nPres(t) = 1−e−βJ0/r3\nW(t\nτ+1). (57)\nWe can write the probability that the central spin has no resonance as\nP(t) = exp( −ZR(t)\nr02πnr(1−e−βJ0/r3\nW(t\nτ+1))dr). (58)\nTo simplify the expression, we can take the −logof each side:\n−log(P(t)) =ZR(t)\nr02πnr(1−e−βJ0/r3\nW(t\nτ+1)dr). (59)\nWe can next perform a change of variables to evaluate the integral. We define\nz=βJ0/r3\nW(t\nτ+ 1). (60)\nWe can rewrite this in terms of r as\nr= (βJ0/z\nW(t\nτ+ 1))1/3. (61)\nUsing the fact that 2 rdr=d(r2), we can rewrite the expression for P(t) in terms of z:\n−log(P(t)) =ZR(z,t)\nr(z,t)nπ(1−e−z)d(βJ0/z\nW(t\nτ+ 1))2/3(62)\nwhich becomes\n−log(P(t)) =Zz(r,t)\nz0(r0,t)nπ(1−e−z)(−2/3)z−5/3dz(βJ0\nW(t\nτ+ 1))2/3. (63)\nWe can then rewrite −log(P(t)) as the product of two terms:\n33−log(P(t)) =P1(t)P2(t) (64)\nP1(t) =Zz(r,t)\nz0(r0,t)π(1−e−z)(−2/3)z−5/3dz (65)\nP2(t) = (n3/2βJ0\nW(t\nτ+ 1))2/3(66)\nIn the limit t≫τ,P2(t) simplifies to\nP2(t) = (n3/2βJ0\nWt\nτ)2/3. (67)\nWe want to understand the time dependence in P1(t) so we define\nz0(r0, t) =βJ0/r3\n0\nW(t/τ+ 1) (68)\nz(R, t) =βJ0/R(t)3\nW(t/τ+ 1) (69)\nSince R(t) = ( J0t)1/3, in the limit t≫τ, z has weak dependence on time. We can numerically integrate\nP1(t) in order to understand the time dependence, and we plot P1(t) and P2(t) for comparison.\n(a) (b)\nFigure S14: Plots of (a) P1(t) and (b) P2(t) with r0= 2 nm, β= 1, τ= 15 µs,W= 3.77\nµs−1, and J0= 326.7 nm3/µs.P2(t) has a stronger dependence on time than P1(t) for\ntimes t > τ.\nFor times t > τ , there is some dependence on time in P1(t), but it is not as strong as the time dependence\ninP2(t). Therefore, we can write the probability of having a resonant pair as\nP(t) = exp( −α(n3/2βJ0\nWτ(t+τ))2/3) (70)\nwhere αis the average value of P1at times t > τ. For our parameter range, we set α= 5, and we\nhave numerically verified that P(t) in Eq. E16 is consistent with Eq. E4 numerically integrated for relevant\n34parameters. From this expression, we can see that changing Worτsimply amounts to changing the timescale\nof the decay. For times much longer than τ, we can further simplify P(t) to\nP(t) = exp( −α(n3/2βJ0\nWτt)2/3). (71)\nE.2 Two Sources of Dynamic Disorder\nIn this section we discuss the effects of having 2 sources of dynamic disorder with disorder widths W1and\nW2and correlation times τ1andτ2. We define an effective disorder width Weby adding the disorder widths\nfrom the 2 sources in quadrature:\nWe=q\nW2\n1+W2\n2. (72)\nWe can determine the effective correlation time by adding the correlation rates of the two disorder sources,\nweighted by the strength of each disorder source, as follows\np\n1/τe=W1p\n1/τ1+W2p\n1/τ2\nWe. (73)\nWe have numerically verified these relationships. For the central surface spin system, the nuclear spin\ndisorder has width W and correlation time τwhile the positional/interaction disorder has width J= 1/T2\nand correlation time Tz.\nF S zAutocorrelation Scaling\nFor the 7 systems with strong coupling between the NV center and a central surface spin presented in Fig. 3\nof the main text, we have independently measured and extracted T z, T2,τ, and W, and present those values\nbelow.\n(a) (b) (c)\nFigure S15: Dependence of T zon different system parameters. (a) T zvs T 2, (b) T zvsτ,\nand (c) T zvs W\n35G Spin-lock Driving Measurements (T 1ρ)\nIn order to control the strength of the nuclear spin bath disorder, we perform spin-lock driving measurements\non the surface spin system. In this section we will discuss our treatment and understanding of those results.\nWe begin by considering the Hamiltonian in the frame rotating with respect to the bare splitting ωS=γeBz\next.\nThe resulting Hamiltonian for the central surface spin Sjfrom the main text plus an additional driving term\nin the rotating fame is\nH′\nj=ℏγeBz\nj(t)Sz\nj+ ΩSy\nj+X\niℏ2γ2\ne\nr3\nij(1−3 cos2θij)[Sz\niSz\nj−1\n4(S+\niS−\nj+S−\niS+\nj)] (74)\nwhere Ω is the driving strength.\nWe consider the case of a two level system where spin operators are given by Sα=1\n2σα. One may further\nmove into a second interaction picture with respect to H1= ΩSy=1\n2Ωσy. Our spin operators are then\ntransformed by replacing Sαwith U†(t)SαU(t), where U(t) = exp( −iΩtσy/2). Our initial spin operators are\ntransformed as follows:\nSx(t)→cos(Ω t)Sx′+ sin(Ω t)Sz′(75)\nSy(t)→Sy′(76)\nSz(t)→cos(Ω t)Sz′−sin(Ω t)Sx′(77)\nwhere Sα′is the spin operator in the doubly rotated frame. The resulting dipolar Hamiltonian has half\nthe interaction strength as in the singly rotating frame:\nHdd\nj=X\nijℏ2γ2\ne\nr3\nij(1−3 cos2θij)[1\n4Sz′\niSz′\nj+1\n4Sx′\niSx′\nj−1\n2Sy′\niSy′\nj]. (78)\nIn the doubly rotating frame with respect to the drive, the Hamiltonian due to the on-site fields from the\nnuclear spin bath becomes\nHnuc\nj=ℏγeBz\nj(t)[cos(Ω t)Sz′\nj−sin(Ω t)Sx′\nj]. (79)\nIn this new basis, the energy eigenstates are split by the effective Rabi frequency of the drive Ω eff=√\nΩ2+δ2where δis the on-site detuning value, characterized by the power spectral density V(ω). Following\nthe treatment in [18], the effective detuning width at strong driving strength scales as Weff=W2√\n2Ω. We\ncan numerically calculate δvalues from the power spectral density V(ω) and calculate Ω efffor different\nvalues of drive strength Ω, verifying that at large drive strengths, the effective on-site disorder width scales\nasWeff=W2√\n2Ω.\nFor weaker drives, the presence of nuclear spin bath low frequency noise will directly drive transitions\nbetween the dressed states of the surface spins, split by drive strength Ω. We can treat the nuclear spin bath\nnoise spectrum as a perturbation and directly calculate the transition probability and relaxation rate given\nV(ω). First, let’s assume that the nuclear spin bath noise is monochromatic at frequency ω:\n36H′=ℏγeB0Szcos(ωt). (80)\nThe matrix element Qabis given by\nQab=−1\n2ℏγeB0. (81)\nWe can then calculate the transition probability as\nPab(t) =|Qab|2\nℏ2sin2[(Ω−ω)t/2]\n(Ω−ω)2(82)\nPab(t) =ℏ2γ2\neB2\n0/4\nℏ2sin2[(Ω−ω)t/2]\n(Ω−ω)2. (83)\nThe above expressions are all for monochromatic perturbations. In our system, B(t) can be described\nfrom a power spectrum V(ω), so we need to take this into account. The energy density in CGS units of an\nelectromagnetic wave is U=B2\n0\n8π. We can then rewrite the transition probability as\nPab(t) = 2 πUγ2\nesin2[(Ω−ω)t/2]\n(Ω−ω)2. (84)\nSince the central surface spin is exposed to magnetic noise at a range of frequencies, we rewrite U as\nρ(ω)dωwhere ρ(ω)dωis the energy density in the frequency range dω, and we can rewrite the transition\nprobability as\nPab= 2πγ2\neZ∞\n0ρ(ω)sin2[(Ω−ω)t/2]\n(Ω−ω)2dω. (85)\nSincesin2[(Ω−ω)t/2]\n(Ω−ω)2is sharply peaked around Ω, we can replace ρ(ω) by ρ(Ω) and take it outside of the\nintegral:\nPab= 2πγ2\neρ(Ω)Z∞\n0sin2[(Ω−ω)t/2]\n(Ω−ω)2dω. (86)\nWe can solve the integral by noting that\nZ∞\n−∞sin2x\nx2dx=π. (87)\nWe find that\nPab= 2πγ2\neρ(Ω)tπ/2. (88)\nWe find that the transition rate ( R=dP/dt ) is\n37Rab=π2γ2\neρ(Ω) (89)\nwhere ρ(Ω) is the energy in the field at Ω and directly corresponds to the value of the power spectral\ndensity S(ω) at that frequency.\n(a) (b) (c)\n1/T2\n1/T2\n1/Tz\n1/Tz\n1/Tz1/T2\nFigure S16: T 1ρrelaxation rate as a function of drive strength for the 3 other NV center\n- surface spin system data sets shown in the main text fig 4c. Black line represents\nthe predicted relaxation rate from extracting W from each measurement’s Ramsey decay\nand estimating τ. 1/T 2and 1/T zdecay rates shown by the gray bands, independently\nextracted for each central surface spin system.\nReferences\n[1] Kaufman, A. M. et al. Quantum thermalization through entangle-\nment in an isolated many-body system. Science 353, 794–800 (2016).\nURL https://science.sciencemag.org/content/353/6301/794https:\n//science.sciencemag.org/content/353/6301/794.abstract .\n[2] Lukin, A. et al. Probing entanglement in a many-body-localized system. Science\n364, 256–260 (2019). URL https://www-science-org.ezp-prod1.hul.harvard.\nedu/doi/abs/10.1126/science.aau0818 .1805.09819 .\n[3] Rispoli, M. et al. Quantum critical behaviour at the many-body localization transi-\ntion. Nature 2019 573:7774 573, 385–389 (2019). URL https://www.nature.com/\narticles/s41586-019-1527-2 .\n[4] Morong, W. et al. Observation of Stark many-body localization without disorder.\nNature — 599, 393 (2021). URL https://doi.org/10.1038/s41586-021-03988-0 .\n[5] Zhang, J. et al. Observation of a Discrete Time Crystal. Nature 543, 217 (2017).\n1609.08684 .\n[6] Choi, J. et al. state spin ensemble ( a ) ( b ) vi, 1–11 (2017).\n38[7] Keesling, A. et al. Quantum Kibble–Zurek mechanism and critical dynamics on a\nprogrammable Rydberg simulator. Nature 2019 568:7751 568, 207–211 (2019). URL\nhttps://www.nature.com/articles/s41586-019-1070-1 .\n[8] Ebadi, S. et al. Quantum phases of matter on a 256-atom programmable quantum\nsimulator. Nature 2021 595:7866 595, 227–232 (2021). URL https://www.nature.\ncom/articles/s41586-021-03582-4 .\n[9] Kyprianidis, A. et al. Observation of a prethermal discrete time crystal. Sci-\nence 372, 1192–1196 (2021). URL https://www-science-org.ezp-prod1.hul.\nharvard.edu/doi/abs/10.1126/science.abg8102 .2102.01695 .\n[10] Turner, C. J., Michailidis, A. A., Abanin, D. A., Serbyn, M. & Papi´ c, Z. Weak er-\ngodicity breaking from quantum many-body scars URL https://doi.org/10.1038/\ns41567-018-0137-5 .\n[11] Bluvstein, D. et al. Controlling quantum many-body dynamics in driven Rydberg\natom arrays. Science 371, 1355–1359 (2021). URL https://www-science-org.\nezp-prod1.hul.harvard.edu/doi/abs/10.1126/science.abg2530 .\n[12] Kao, W., Li, K. Y., Lin, K. Y., Gopalakrishnan, S. & Lev, B. L. Topological pumping\nof a 1D dipolar gas into strongly correlated prethermal states. Science 371, 296–300\n(2021). URL https://www-science-org.ezp-prod1.hul.harvard.edu/doi/abs/\n10.1126/science.abb4928 .\n[13] Cardellino, J. et al. The effect of spin transport on spin lifetime in nanoscale systems.\nNature Nanotechnology 9, 343–347 (2014).\n[14] Alvarez, G. A., Suter, D. & Kaiser, R. Localization-delocalization transition in the\ndynamics of dipolar-coupled nuclear spins. Science 349(2015).\n[15] Wei, K. X., Ramanathan, C. & Cappellaro, P. Exploring Localization in Nuclear\nSpin Chains. Physical Review Letters 120, 70501 (2018). URL https://link.aps.\norg/doi/10.1103/PhysRevLett.120.070501 .\n[16] Smith, J. et al. Many-body localization in a quantum simulator with programmable\nrandom disorder. Nature Physics 2016 12:10 12, 907–911 (2016). URL https:\n//www.nature.com/articles/nphys3783 .\n[17] Choi, J.-y. et al. Exploring the many-body localization transition in two dimensions.\nScience 352, 1547–1552 (2016). URL http://www.ncbi.nlm.nih.gov/pubmed/\n27339981 .\n39[18] Kucsko, G. et al. Critical Thermalization of a Disordered Dipolar Spin System in\nDiamond. Physical Review Letters 121, 23601 (2018). URL https://link.aps.\norg/doi/10.1103/PhysRevLett.121.023601 .\n[19] Schreiber, M. et al. Observation of many-body localization of interacting fermions\nin a quasi-random optical lattice (2015). URL http://arxiv.org/abs/1501.05661 .\n1501.05661 .\n[20] Fel’dman, E. B. & Lacelle, S. Cite as. J. Chem. Phys 104, 2000 (1996). URL\nhttps://doi.org/10.1063/1.470956 .\n[21] Dobrovitski, V. V., Feiguin, A. E., Awschalom, D. D. & Hanson, R. Decoherence\ndynamics of a single spin versus spin ensemble. Physical Review B - Condensed Mat-\nter and Materials Physics 77, 245212 (2008). URL https://journals-aps-org.\nezp-prod1.hul.harvard.edu/prb/abstract/10.1103/PhysRevB.77.245212 .\n[22] Anderson, P. W. Absence of Diffusion in Certain Random Lattices. Physical Review\n109, 1492–1505 (1958).\n[23] Burin, A. L. Energy delocalization in strongly disordered systems induced by the\nlong-range many-body interaction (2006). URL http://arxiv.org/abs/cond-mat/\n0611387 .0611387 .\n[24] Yao, N. Y. et al. Many-body localization in dipolar systems. Physical Review Letters\n113, 243002 (2014). URL https://journals.aps.org/prl/abstract/10.1103/\nPhysRevLett.113.243002 .\n[25] Gong, Z.-X., Foss-Feig, M., Brand˜ ao, F. G. S. L. & Gorshkov, A. V. Entanglement\nArea Laws for Long-Range Interacting Systems. Physical Review Letters 119, 50501\n(2017). URL http://link.aps.org/doi/10.1103/PhysRevLett.119.050501 .\n[26] Orioli, A. P. et al. Relaxation of an Isolated Dipolar-Interacting Rydberg Quantum\nSpin System. Physical Review Letters 120, 063601 (2018). 1703.05957 .\n[27] Gopalakrishnan, S., Agarwal, K., Demler, E. A., Huse, D. A. & Knap, M. Griffiths\neffects and slow dynamics in nearly many-body localized systems. Physical Review\nB93, 1–12 (2016). 1511.06389 .\n[28] Sushkov, A. O. et al. Magnetic resonance detection of individual proton\nspins using quantum reporters. Physical Review Letters 113, 197601 (2014).\nURL https://journals-aps-org.ezp-prod1.hul.harvard.edu/prl/abstract/\n10.1103/PhysRevLett.113.197601 .\n[29] Davis, E. J. et al. Probing many-body noise in a strongly interacting two-dimensional\ndipolar spin system (2021). URL http://arxiv.org/abs/2103.12742 .2103.12742 .\n40[30] Grotz, B. et al. Sensing external spins with nitrogen-vacancy diamond. New Journal\nof Physics 13, 55004 (2011).\n[31] Grinolds, M. S. et al. Subnanometre resolution in three-dimensional magnetic reso-\nnance imaging of individual dark spins. Nature Nanotechnology 9, 279–284 (2014).\nURL www.nature.com/naturenanotechnology .1401.2674 .\n[32] Tetienne, J. P. et al. Spin properties of dense near-surface ensembles of\nnitrogen-vacancy centers in diamond. Physical Review B 97, 085402 (2018).\nURL https://journals-aps-org.ezp-prod1.hul.harvard.edu/prb/abstract/\n10.1103/PhysRevB.97.085402 .1711.04429 .\n[33] Sangtawesin, S. et al. Origins of Diamond Surface Noise Probed by Correlating\nSingle-Spin Measurements with Surface Spectroscopy 031052 , 1–17 (2019).\n[34] Stacey, A. et al. Evidence for Primal sp ¡sup¿2¡/sup¿ Defects at the Diamond Surface:\nCandidates for Electron Trapping and Noise Sources. Advanced Materials Interfaces\n6, 1801449 (2019). URL http://doi.wiley.com/10.1002/admi.201801449 .\n[35] Rosskopf, T. et al. Investigation of Surface Magnetic Noise by Shallow Spins in\nDiamond 147602 , 1–5 (2014).\n[36] Romach, Y. et al. Spectroscopy of Surface-Induced Noise Using Shallow Spins in\nDiamond. Physical Review Letters 114, 17601 (2015). URL https://link.aps.\norg/doi/10.1103/PhysRevLett.114.017601 .\n[37] Myers, B. A. et al. Probing Surface Noise with Depth-Calibrated Spins in Diamond.\nPhysical Review Letters 113, 27602 (2014).\n[38] Myers, B. A., Ariyaratne, A. & Jayich, A. B. Double-Quantum Spin-Relaxation\nLimits to Coherence of Near-Surface Nitrogen-Vacancy Centers. Physical Review\nLetters 118, 197201 (2017).\n[39] Bluvstein, D., Zhang, Z., McLellan, C. A., Williams, N. R. & Jayich, A. C. Ex-\ntending the Quantum Coherence of a Near-Surface Qubit by Coherently Driv-\ning the Paramagnetic Surface Environment. Physical Review Letters 123, 146804\n(2019). URL https://journals-aps-org.ezp-prod1.hul.harvard.edu/prl/\nabstract/10.1103/PhysRevLett.123.146804 .1905.06405 .\n[40] F´ avaro de Oliveira, F. et al. Tailoring spin defects in diamond by lattice charging. Na-\nture Communications 8, 15409 (2017). URL http://www.nature.com/doifinder/\n10.1038/ncomms15409 .\n[41] Dwyer, B. L. et al. Probing spin dynamics on diamond surfaces using a single quan-\ntum sensor (2021). URL http://arxiv.org/abs/2103.12757 .2103.12757 .\n41[42]Supplementary Information .\n[43] Laraoui, A. et al. High-resolution correlation spectroscopy of 13C spins near a\nnitrogen-vacancy centre in diamond. Nature Communications 4, 1–7 (2013). URL\nwww.nature.com/naturecommunications .\n[44] Staudacher, T. et al. Probing molecular dynamics at the nanoscale via an individual\nparamagnetic centre. Nature Communications 6, 8527 (2015).\n[45] Loretz, M. et al. Single-proton spin detection by diamond magnetometry. Science\nscience.1259464—- (2014).\n[46] DeVience, S. J. et al. Nanoscale NMR spectroscopy and imaging of multiple nuclear\nspecies. Nature Nanotechnology 10, 129–134 (2015). URL http://www.nature.com/\narticles/nnano.2014.313 .\n[47] Abragam, A. The Principles of Nuclear Magnetism: The International Series of\nMonographs on Physics (1961).\n[48] Slichter, C. P. Principles of Magnetic Resonance , vol. 1 of Springer Series in Solid-\nState Sciences (Springer Berlin Heidelberg, Berlin, Heidelberg, 1978). URL http:\n//link.springer.com/10.1007/978-3-662-12784-1 .\n[49] Gr¨ aßer, T., Bleicker, P., Hering, D.-B., Yarmohammadi, M. & Uhrig, G. S. Dynamic\nmean-field theory for dense spin systems at infinite temperature (2021). URL https:\n//arxiv.org/abs/2107.07821v1 .2107.07821 .\n[50] Taylor, J. M. et al. High-sensitivity diamond magnetometer with nanoscale reso-\nlution. Nature Physics 4, 810–816 (2008). URL www.nature.com/naturephysics .\n0805.1367 .\n[51] Mamin, H. J. et al. Nanoscale nuclear magnetic resonance with a nitrogen-vacancy\nspin sensor. Science 339, 557–560 (2013).\n[52] Pham, L. M. et al. NMR technique for determining the depth of shallow nitrogen-\nvacancy centers in diamond. Physical Review B 93(2016). 1508.04191 .\n42"    },    {        "title": "2311.11308v2.Spin_squeezing_generated_by_the_anisotropic_central_spin_model.pdf",        "content": "Spin squeezing generated by the anisotropic central spin model\nLei Shao1and Libin Fu1,∗\n1Graduate School of China Academy of Engineering Physics, Beijing 100193, China\nSpin squeezing, as a crucial quantum resource, plays a pivotal role in quantum metrology, enabling us to\nachieve high-precision parameter estimation schemes. Here we investigate the spin squeezing and the quantum\nphase transition in an anisotropic central spin system. We find that this kind of central spin systems can be\nmapped to the anisotropic Lipkin-Meshkov-Glick model in the limit where the ratio of transition frequencies\nbetween the central spin and the spin bath tends towards infinity. This property can induce a one-axis twisting\ninteraction and provides a new possibility for generating spin squeezing. We separately consider generating\nspin-squeezedstatesviathegroundstateandthedynamicevolutionofthecentralspinmodel. Theresultsshow\nthat the spin squeezing parameter improves as the anisotropy parameter decreases, and its value scales with\nsystem size as 𝑁−2/3. Furthermore, we obtain the critical exponent of the quantum Fisher information around\nthe critical point by numerical simulation, and find its value tends to 4/3as the frequency ratio and the system\nsize approach infinity. This work offers a promising scheme for generating spin-squeezed state and paves the\nway for potential advancements in quantum sensing.\nI. INTRODUCTION\nAs the understanding of the quantum world continues to\ndeepen, quantum technology is ushering in in a series of\nrevolutionary changes in the real world. Notably, quantum\nprecision measurement is progressively gaining prominence,\nespecially playing a crucial role in critical areas such as bio-\nphysics [1–4], inertial sensors [5, 6], and measurement of\nphysicsconstants[7–9]. Intherealmofquantummetrology,a\ncoregoalistoimprovethemeasurementprecisionofparame-\nters of interest. As a result, a central focus of research lies in\nexploring how to utilize the nonclassical properties of quan-\ntum resources to achieve heightened precision. In the past\ndecades, quantum spin squeezing has garnered considerable\nattentionasanimportantconceptinthefieldofquantuminfor-\nmation,particularlywithinthedomainsofquantummetrology\nandquantumsensing. Duetoitscapabilitytoreducequantum\nfluctuations of a specific spin component, spin squeezing can\nenhance the precision of measurements, thus spin-squeezed\nstates are widely applied to quantum precision measurement,\nsuch as atomic clock [10–13], Ramsey spectroscopy [14–18]\nandgravitationalwaveinterferometers[19–21]. Furthermore,\ninvestigatingthespinsqueezingisalsohelpfultogainadeeper\nunderstanding of the correlations, entanglement, and quan-\ntum information processing among particles in the quantum\nworld [22–25].\nThe generation of spin squeezing involves two cate-\ngories [26], including nonlinear atomic-atomic collisions\nin Bose-Einstein condensates (BECs) [27–38] and atomic-\nphotoninteractions[39–55]. Theformerexploitsthenonlinear\neffects of atomic collisions to generate spin-squeezed states,\nwhile the latter mainly involves utilizing squeezing transfer\nfrom the optical field to spin ensemble [40, 48, 50, 53, 54],\nthe coherent feedback of the optical cavity [49, 55] or\nquantum nondemolition (QND) measurement of the output\nlight [45, 51, 52, 54].\nUtilizing bosonic atoms to generate spin squeezing can re-\n∗lbfu@gscaep.ac.cnsult in more losses due to collisions [56–58], hence it is es-\nsential to explore the use of spinful fermions to achieve spin\nsqueezing. Due to the similarity between the light-matter in-\nteractionandthespin-spininteraction,thecentralspinsystems\ncanalsoinduceone-axistwisting(OAT)interactionundercer-\ntainconditions,thusitcanberegardedasapromisingalterna-\ntivemodeltorealizespinsqueezing. Theschemeproposedin\n[59]alsoutilizesasimilarapproachtoachievespinsqueezing\nin a quantum dot composed of an electron and a large ensem-\nbleofnuclearspins. However,themechanismofinducingthe\neffective one-axis twisting interaction in central spin systems\nisstillambiguous,thus,weemploytheSchrieffer-Wolfftrans-\nformationtorigorouslyillustratethegenerationmechanismof\nOATinteractionandextendittotheanisotropiccase. Wefind\nthat when the ratio of the transition frequency of the central\nspin to that of the bath spin tends to infinity, the anisotropic\ncentral spin model can be mapped to the anisotropic Lipkin-\nMeshkov-Glick(LMG)model[60],wheretheOATinteraction\nin the Hamiltonian can dynamically generate spin squeezing .\nIn addition, there exists a quantum phase transition (QPT) in\nthisanisotropiccentralspinmodel,anditsquantumcriticality\ncan be viewed as a quantum resource for quantum metrology.\nThusweinvestigatethefinite-sizebehavioraroundthecritical\npoint and numerically calculate the critical exponent of the\nquantum Fisher information (QFI). It is found that when the\nfrequency ratio tends to infinity, the critical exponents of the\ncentral spin model become equal to those of the LMG model,\nhowever,asthefrequencyratiodecreases,thecriticalexponent\nwill decay to zero.\nThis article is organized as follows. In Sec. II, we give the\nderivationofthemappingbetweentheanisotropiccentralspin\nmodel and the anisotropic LMG model. In Sec. III, we inves-\ntigatethespinsqueezingandquantumphasetransitionsofthe\ncentralspinmodelunderbothisotropicandanisotropiccondi-\ntions. In Sec. IV, we discuss the quantum Fisher information\nanditscriticalexponentforthegroundstateintheanisotropic\ncentral spin model. Finally, we give a conclusion in Sec.V.arXiv:2311.11308v2  [quant-ph]  28 Nov 20232\nII. MODEL\nThe central spin model can be described as an ensemble\nof𝑁identical spin-1\n2particles with a total spin of 𝐼=𝑁/2\ninteractwithacentralspin-1\n2particle. Thismodelisdiscussed\nin the Ref. [61–67] and its Hamiltonian is written as (we set\nℏ=1)\n𝐻=Ω\n2𝜎(0)\n𝑧+𝜔\n2𝑁∑︁\n𝑘=1𝜎(𝑘)\n𝑧+𝑁∑︁\n𝑘=1\u0012𝐴𝑥\n2𝜎(𝑘)\n𝑥𝜎(0)\n𝑥\n+𝐴𝑦\n2𝜎(𝑘)\n𝑦𝜎(0)\n𝑦+𝐴𝑧\n2𝜎(𝑘)\n𝑧𝜎(0)\n𝑧\u0013\n,(1)\nwhereΩand𝜔arethetransitionfrequenciesofthecentralspin\nand bath spins induced by an external magnetic field, respec-\ntively. And𝐴𝑖(𝑖=𝑥,𝑦,𝑧)representsthestrengthofinteraction\nbetweenthecentralspinandbathspinsindifferentdirections.\nHere𝜎(0)\n𝑖representsthePaulioperatorofthecentralspinand\n𝜎(𝑘)\n𝑖(𝑘≠0,𝑖=𝑥,𝑦,𝑧) is the𝑖th Pauli operator of the bath\nspins. Centralspinmodeliswidelyusedtostudythespin-spin\ninteractions in quantum dots [64–66] and nitrogen-vacancy\ncenter [67]. Some intriguing physical phenomena, such as\ncollapse and revival [61, 62], superradiance effect [61], and\ndissipative phase transitions [68], emerge within this model.\nIn this article, we mainly focus on the anisotropic model, that\nis,thecouplingstrengthofthe 𝑋and𝑌directionsisdifferent.\nHere we set 𝐴𝑥=(1+𝜆)𝐴,𝐴𝑦=(1−𝜆)𝐴, and𝐴𝑧=0, and\nEq. (1) can be rewritten as\n𝐻=Ω\n2𝜎𝑧+𝜔𝐼𝑧+𝐴[(𝐼+𝜎−+𝐼−𝜎+)+𝜆(𝐼+𝜎++𝐼−𝜎−)],(2)\nwhere𝜎(0)\n𝑧≡𝜎𝑧,𝐼𝑧=1\n2Í𝑁\n𝑘=1𝜎(𝑘)\n𝑧, and𝜆is the anisotropy\nparameter. Note that the Hamiltonian in Eq. (2) commutes\nwith the parity operator Π= exp[𝑖𝜋(𝜎+𝜎−+𝐼𝑧+𝑁/2)], i.e.\n[Π,𝐻]=0, indicating that it possesses the 𝑍2symmetry.\nHowever, if 𝜆≠1then it can not be solved with a closed\nsubspace, which complicates further analysis of this model.\nTo this end, we apply a Schrieffer-Wolff transformation 𝑒𝑆\nwith𝑆=−𝑖(1+𝜆)/Ω𝐼𝑥𝜎𝑦+𝑖(1−𝜆)/Ω𝐼𝑦𝜎𝑥toEq.(2),and\nin the limit of 𝜂=Ω/𝜔→∞we obtain\n𝐻=Ω\n2𝜎𝑧+𝛾𝑧𝐼𝑧+1\n𝑁\u0010\n𝛾𝑥𝐼2\n𝑥+𝛾𝑦𝐼2\n𝑦\u0011\n𝜎𝑧,(3)\nwhere\n𝛾𝑥=˜𝑔2𝜔(1+𝜆)2\n4, 𝛾𝑦=˜𝑔2𝜔(1−𝜆)2\n4,(4)\n𝛾𝑧=𝜔−e𝑔2(1+𝜆)(1−𝜆)𝜔\n4𝑁, (5)\n˜𝑔=2𝐴√\n𝑁√\nΩ𝜔. (6)\nHere ˜𝑔in Eq. (6) is a dimensionless coupling strength pa-\nrameter. It can be seen that the Hamiltonian in Eq. (3) isblock-diagonal with respect to spin states and its low-energy\neffective Hamiltonian in the spin down subspace is\n𝐻↓=−Ω\n2+𝛾𝑧𝐼𝑧−1\n𝑁\u0010\n𝛾𝑥𝐼2\n𝑥+𝛾𝑦𝐼2\n𝑦\u0011\n. (7)\nOne can see that Eq. (7) is exactly the Hamiltonian of the\nanisotropic LMG model. In other words, there exists a map-\npingbetweentheanisotropicmodelandtheLMGmodelinthe\nlimit of𝜂→∞. The detailed derivation of the above process\nis presented in Appendix A. A Similar method for generating\nspin squeezing has been discussed in Ref. [59], where they\nprovided an approximate relationship. In contrast, we present\na more detailed mapping relationship and derivation process\nhere.\nIII. SPIN SQUEEZING AND QUATNUM PHASE\nTRANSITIONS IN CENTRAL SPIN MODEL\nInthissection,wewilldiscussthespinsqueezingandquan-\ntum phase transtion in the anisotropic central spin model. We\nwill begin by introducing two commonly used definitions of\nspinsqueezingproposedbyKitagawa et al.[69]andWineland\net al. [70], which are given by\n𝜉2\n𝑆=4min\u0000Δ2𝐼®𝑛⊥\u0001\n𝑁, (8)\nand\n𝜉2\n𝑅=𝑁min\u0000Δ2𝐼®𝑛⊥\u0001\n\f\f\fD\n®𝐼E\f\f\f2, (9)\nwhere®𝑛⊥is the direction perpendicular to the mean spin di-\nrectionD\n®𝐼E\nandΔ2𝐼®𝑛⊥is the variance of 𝐼®𝑛⊥=®𝐼·®𝑛⊥. And if\n𝜉2\n𝑆<1or𝜉2\n𝑅<1, it implies that this state is a spin-squeezed\nstate. In the following subsections, we will analyze the spin\nsqueezing in the isotropic ( 𝜆=0) and anisotropic ( 𝜆≠0)\ncases separately.\nA. Isotropic case\nFor𝜆=0, the Hamiltonian in Eq. (2) becomes\n𝐻=Ω\n2𝜎𝑧+𝜔𝐼𝑧+˜𝑔\n2√︂\nΩ𝜔\n𝑁(𝐼+𝜎−+𝐼−𝜎+).(10)\nWe can find that the above Hamiltonian remains invariant un-\nder the action of Π′=exp[𝑖𝜃(𝜎+𝜎−+𝐼𝑧+𝑁/2)], indicating\nthat it possesses 𝑈(1)symmetry. The analytical solution of\nisotropiccasehasbeendiscussedindetailinRef.[62,63]. First\nof all, we calculate the squeezing parameters for the ground3\nstateintheisotropiccase. Ref.[63]showsthatthereexistsaa\nnormal-to-superradiantphasetransitioninthelimitof 𝜂→∞\nand𝑁→∞, and the critical point is ˜𝑔𝑐=2/(1+𝜆)=2. A\ndetailed discussion is also provided in Ref. [63].\nFor further analysis, we introduce the Dicke state |𝑛⟩≡ \f\f𝑁\n2,−𝑁\n2+𝑛\u000b\n(𝑛∈[0,𝑁]), which is the eigenstate of the op-\nerators 𝑰2and𝐼𝑧. For𝜂≫1and𝑁≫1, the ground state is\n|↓,0⟩≡|↓⟩⊗\f\f𝑁\n2,−𝑁\n2\u000b\nwhen ˜𝑔<2,where|↓⟩(|↑⟩)istheeigen-\nstate of the operator 𝜎𝑧. Immediately, we obtain 𝜉2\n𝑆=𝜉2\n𝑅=1\nunder the condition of ˜𝑔 <2. For ˜𝑔 >2, the ground state is\ngiven by [63]\n|𝜓−(𝑛)⟩=e𝑃−\n↑,𝑛−1|↑,𝑛−1⟩+e𝑃−\n↓,𝑛|↓,𝑛⟩,(11)\nwhere\ne𝑃−\n↑,𝑛−1=eΩ−√︁\n1+eΩ2\n√︂\n2\u0010\n1+eΩ2\u0011\n−2eΩ√︁\n1+eΩ2,(12)\ne𝑃−\n↓,𝑛=1√︂\n2\u0010\n1+eΩ2\u0011\n−2eΩ√︁\n1+eΩ2,(13)\neΩ=Ω−𝜔\n˜𝑔√︁\n(𝑁−𝑛+1)𝑛Ω𝜔, (14)\nand\n𝑛=𝜂\n4\u0010\n˜𝑔2−˜𝑔−2\u0011\n. (15)\nHere𝑛inEq.(15)canberegardedastheexcitationnumberof\nthegroundstate. Utilizingtheaboveequations,wecanobtain\ntheexpressionofspinsqueezingparametersofthegroundstate\nfor𝜂≫1, which is\n𝜉2\n𝑆=−2\u0000𝑛−𝑁\n2\u00012\n𝑁+𝑁\n2+1, (16)\nand\n𝜉2\n𝑅=−𝑁\n2+𝑁2(𝑁+2)\n8\u0000𝑛−𝑁\n2\u00012. (17)\nThe derivation of the above equations is provided in Ap-\npendix B. The analytical results of Eq. (16) and Eq. (17) are\nshowninFig.1. Onecanseethatthesqueezingparameters 𝜉2\n𝑆\nand𝜉2\n𝑅remain 1when ˜𝑔<˜𝑔𝑐,andtheybegintocontinuously\nincreaseaftercrossingthecriticalpoint ˜𝑔𝑐,whichimpliesthat\nthe squeezing parameters can be used as an indicator to char-\nacterize the phase transition [26, 71]. Apart from this, we\ncan also find that the squeezing parameters of the central spin\nmodel tend to approach those of the LMG model as the fre-\nquencyratio 𝜂increases. AsshowninFig.1(b)andFig.1(d),\nthe two spin squeezing parameters are essentially consistent\nwhen the frequency ratio is 𝜂=105.\nHowever, from the above analysis, we can find that the\n0 2 4 6020406080100LMG model\ncentral spin model\n0 2 4 6020406080100LMG model\ncentral spin model\n0 2 4 60200040006000800010000LMG model\ncentral spin model\n0 2 4 60200040006000800010000LMG model\ncentral spin model(a) (b)\n(c) (d)FIG. 1. Spin squeezing parameters 𝜉2\n𝑆and𝜉2\n𝑅of the ground state\nfor the isotropic case as functions of ˜𝑔with𝑁=200. The first row\n[panels(a)-(b)]correspondstothecasesofspinsqueezingparameter\n𝜉2\n𝑆with𝜂=103and𝜂=105, respectively. The second row [pan-\nels (c)-(d)] corresponds to the cases of spin squeezing parameter 𝜉2\n𝑅\nwith𝜂=103and𝜂=105,respectively. Thesolidlinearetheresults\nof the LMG model and the dashed line are the results of the central\nspin model.\nground state of the central spin is not a spin-squeezed state\ndue to𝜉2\n𝑅≥𝜉2\n𝑆≥1. Subsequently, we will explore the\npossibility of generating spin-squeezed states through a dy-\nnamical approach. We choose |↓⟩⊗|𝜓cs⟩as the initial\nstate, where|𝜓cs⟩is the spin coherent state, i.e., |𝜓cs⟩=\n⊗𝑁\n𝑘=0\u0002\ncos(𝜃0/2)|↑⟩𝑘+𝑒𝑖𝜙0(𝜃0/2)|↓⟩𝑘\u0003\n, and the exact solu-\ntionofthisdynamicalprocesshasbeenprovidedin[62],which\nis\n\f\f𝜓𝑓(𝑡)\u000b\n=𝑁∑︁\n𝑚=0√︃\n𝐶𝑚\n𝑁𝑒−𝑖(−𝑁\n2+𝑚−1\n2)𝜔𝑡\u0012\ncos𝜃0\n2\u0013𝑚\u0012\nsin𝜃0\n2\u0013𝑁−𝑚\n×\u0010\n𝑃𝑚\n↓|↓,𝑚⟩+𝑃𝑚\n↑|↑,𝑚−1⟩\u0011\n, (18)\nwhere𝑃𝑚\n↓=𝑖Ωsin(Ω𝑚𝑡/2)/Ω𝑚+cos(Ω𝑚𝑡/2),𝑃𝑚\n↑=\n−𝑖2√𝑘𝑚𝐴sin(Ω𝑚𝑡/2)/Ω𝑚, andΩ𝑚=√︁\nΩ2+4𝑘𝑚𝐴2with\n𝑘𝑚=𝑚(𝑁−𝑚+1). Note that we set 𝜙0=0\nfor simplicity, and the mean spin direction is ®𝑛0=\n(sin𝜃cos𝜙,sin𝜃sin𝜙,cos𝜃),whilethedirectionsperpendic-\nular to it are given by ®𝑛1=(−sin𝜙,cos𝜙,0)and®𝑛2=\n(cos𝜃cos𝜙,cos𝜃sin𝜙,−sin𝜃). The spin squeezing parame-\nter is given by\n𝜉2\n𝑆=2\u0010\nC−√\nA2+B2\u0011\n𝑁, (19)4\n0 5 10 1500.20.40.60.81\n0 10 20 3000.20.40.60.81(b) (a)\nFIG. 2. Time evolution of the spin squeezing parameters 𝜉2\n𝑆\n[panel (a)] and 𝜉2\n𝑅[panel (b)] for different frequency ratios 𝜂. The\nparameters chosen here are 𝑁=200,𝜔=1and˜𝑔=2. The curves\nfrom top to bottom correspond to the spin squeezing parameter 𝜉2\n𝑆of the LMG model (blue solid line) and the central spin model with\nfrequency ratios 𝜂=103(yellow dotted line), 𝜂=104(red dash-dot\nline), and𝜂=105(purple dashed line), respectively. In (a), the bot-\ntomstraightlinecorrespondstotheoptimalspinsqueezingparameter\n𝜉𝑆,min(green dash-dot line).\nwhere A=\n𝐽®𝑛1−𝐽®𝑛2\u000b\n,B=\n𝐽®𝑛1𝐽®𝑛2−𝐽®𝑛2𝐽®𝑛1\u000b\n, and𝐶=D\n𝐽2\n®𝑛1+𝐽2\n®𝑛2E\n. The values of the above parameters only de-\npend on the following five parameters: ⟨𝐼𝑧⟩,\n𝐼2\n𝑧\u000b\n,⟨𝐼+⟩,\n𝐼2\n+\u000b\n,\n⟨𝐼+(2𝐼𝑧+1)⟩[72]. The specific expressions for the above\nparameters are presented in Appendix B.\nIn Fig. 2, we show the time evolution of spin squeezing\nparameters𝜉2\n𝑆and𝜉2\n𝑅of the central spin model with different\nfrequency ratios 𝜂. Here we choose 𝜃0=𝜋/2and𝑁is large\nenough (𝑁=200), and the optimal squeezing parameter for\nthe above model is given by\n𝜉2\n𝑆,min≃1\n2\u0012𝑁\n3\u0013−2\n3\n, (20)\nand the corresponding squeezing time is\n𝑡min≃4·31\n6𝑁1\n3\n˜𝑔2𝜔. (21)\nIn fact, for the OAT interaction, i.e., 𝜒𝐼2\n𝑧, it has the following\npower-law scalings: 𝜉2\n𝑆,min∼𝑁−2/3and𝜒𝑡min∼𝑁−2/3[69,\n72–74]. From Eq. (20) and Eq. (21), we can see that the\npower-law scalings of 𝜉2\n𝑆,minand𝑡min(here𝜒=˜𝑔2𝜔/4𝑁) are\nthe same as the one-axis twisting case when 𝜂is sufficiently\nlarge. On the other hand, a substantial detuning between the\ncentral spin and bath spins ( Ω/𝜔≫1) can induce an intra-\nspecies interaction, which is reported in Ref. [59, 75].\nB. Anisotropic case\nNow we consider the anisotropic case, and without loss of\ngeneralityweset 0<𝜆≤1. FortheHamiltonianinEq.(2),itis difficult to obtain its analytical solution. To further analyze\nit, we begin with the Hamiltonian in Eq. (7) ( 𝜂→∞) and\nemploy the mean-field approximation [71, 76] to obtain the\nfollowing Hamiltonian to obtain\n𝐻MF=⟨𝜓cs|𝐻|𝜓cs⟩\n=𝑁𝜔\n2cos𝜃0−˜𝑔2𝜔sin2𝜃0\n4\u0010\n1+𝜆2+2𝜆cos 2𝜙0\u0011\n,\n(22)\nwhere|𝜓cs⟩isthespincoherentstate. Inordertominimizethe\nenergy,wefindthefollowingconditions: (i)for ˜𝑔<2/(1+𝜆),\nwe have𝜃0=𝜋and𝜙0is arbitrary; (ii) for ˜𝑔 >2/(1+𝜆), we\nhave𝜃0=arccos(−𝜔/𝛾𝑥), and𝜙0=0,𝜋. Furthermore, we\nrotate the the spin operators around the 𝑦axis to align the 𝑧\naxiswiththedirectionofthesemiclassicalmagnetization. The\nrotated operators are expressed as ˜𝐼𝑥=cos𝜃0𝐼𝑥−sin𝜃0𝐼𝑧,\n˜𝐼𝑦=𝐼𝑦, and ˜𝐼𝑧=sin𝜃0𝐼𝑥+cos𝜃0𝐼𝑧, and the Hamiltonian\nbecomes\n𝐻↓=−𝛾𝑥\n𝑁\u0002\ncos2𝜃0˜𝐼2\n𝑥+sin2𝜃0˜𝐼2\n𝑧+sin𝜃0cos𝜃0\u0000˜𝐼𝑥˜𝐼𝑧+˜𝐼𝑧˜𝐼𝑥\u0001\u0003\n−𝛾𝑦\n𝑁𝐼2\n𝑦+𝛾𝑧\u0000−sin𝜃0˜𝐼𝑥+cos𝜃0˜𝐼𝑧\u0001. (23)\nThen we apply the Holstein-Primakoff transformation to the\nabove equation, i.e., ˜𝐼+=√\n𝑁𝑎,˜𝐼−=√\n𝑁𝑎†, and ˜𝐼𝑧=𝑁/2−\n𝑎†𝑎, and assume that 𝑁is sufficiently large (√︁\n𝑎†𝑎/𝑁≃1),\nthen Eq. (23) can be rewritten as\n𝐻↓=−𝛾𝑥\n2cos2𝜃0𝑥2−𝛾𝑦\n2𝑝2+\u0010\n𝛾𝑥sin2𝜃0−𝜔cos𝜃0\u0011\n𝑎†𝑎,\n(24)\nwhere𝑥=\u0000𝑎†+𝑎\u0001/√\n2and𝑝=𝑖\u0000𝑎†−𝑎\u0001/√\n2. In the above\nderivation,weutilize 𝜔sin𝜃0+𝛾𝑥sin𝜃0cos𝜃0=0andneglect\nthe constant term.\nSimilarly, the spin squeezing parameter differs before and\nafter the critical point. First of all, for ˜𝑔 < 2/(1+𝜆)and\n𝜃𝜃=𝜋, Eq. (24) becomes\n𝐻=𝜔𝑎†𝑎−1\n2\u0010\n𝛾𝑥𝑥2+𝛾𝑦𝑝2\u0011\n. (25)\nWe use a unitary transformation with a squeezed operator\nS(𝑟)=exp\u0002\n(𝑟/2)\u0000𝑎2−𝑎†2\u0001\u0003\nto Eq. (24) and obtain\n𝐻=√︃\n(𝜔−𝛾𝑥)\u0000𝜔−𝛾𝑦\u0001\u0012\n𝑎†𝑎+1\n2\u0013\n,(26)\nwhere𝑟=1\n4ln\u0002\n(𝜔−𝛾𝑥)/\u0000𝜔−𝛾𝑦\u0001\u0003\n. It is worth noting that\nthe ground state of the Hamiltonian in Eq. (7) satisfies that\n\b\n𝐼𝑥,𝐼𝑦\t\u000b\n=0,thusspinsqueezingparameterisgivenby[71]\n𝜉2\n𝑆=4min\b\n𝐼2\n𝑥\u000b\n,\n𝐼2\n𝑦\u000b\t\n𝑁. (27)\nUtilizing the above equations, we ultimately obtain the spin5\n5 5.5 6 6.5-2.6-2.4-2.2-2-1.8\n0 5 10 15 20 25 3000.250.50.7510 0.5 1 1.5 200.250.50.751(b) (a)\n(c)\nFIG. 3. (a) Spin squeezing parameter 𝜉2\n𝑆of the ground state for\nthe anisotropic case as functions of ˜𝑔with different frequency ratios\n𝜂. Here we choose 𝑁=200,𝜔=1and𝜆=1. The curves from\ntop to bottom correspond to the spin squeezing parameter 𝜉2\n𝑆of the\ncentralspinmodelwithfrequencyratios 𝜂=10(bluedash-dotline),\n𝜂=102(red dashed line), and 𝜂=103(yellow solid line), the\nLMG model (purple dotted line), and the analytical results given by\nEq. (28) and Eq. (30) (solid green line), respectively. (b) Optimal\nsqueezing parameters 𝜉2\n𝑆,minof the ground state as functions of ln𝑁\nwithdifferentfrequencyratios 𝜂. (c)Timeevolutionofspinsqueezing\nparameters for different anisotropy parameters. The curves from top\ntobottomcorrespondto 𝜆=1(bluecircleline), 𝜆=0.25(bluecircle\nline),𝜆=0.01(yellow point line), 𝜆=0(purple solid line).\nsqueezing parameter is given by\n𝜉2\n𝑆=\u0014˜𝑔2\n𝑐−˜𝑔2\n˜𝑔2𝑐−𝛾˜𝑔2\u00151\n2\n, (28)\nwhere𝛾=𝛾𝑦/𝛾𝑥. And for ˜𝑔 > 2/(1+𝜆)and𝜃0=\narccos(−𝜔/𝛾𝑥), then we have\n𝐻=𝜔√︄\n˜𝑔2𝜆\u0012𝛾𝑥\n𝜔−𝜔\n𝛾𝑥\u0013 \u0012\n𝑎†𝑎+1\n2\u0013\n−𝛾𝑥\n2,(29)\nand𝑟=1\n4ln\u0002\n(𝛾𝑥/𝜔−𝜔/𝛾𝑥)/˜𝑔2𝜆\u0003\n. The corresponding spin\nsqueezing parameter is\n𝜉2\n𝑆=\u00141\n˜𝑔2𝜆\u0012˜𝑔2\n˜𝑔2𝑐−˜𝑔2\n𝑐\n˜𝑔2\u0013\u00151\n2\n. (30)\nFrom Eq. (28) and Eq. (30), we can see that 𝜉2\n𝑆<1when\n˜𝑔 <2/√︁\n(1+𝜆)(1−𝜆), which implies that one can generate\nthe spin-squeezed state by preparing the ground state of the\nanisotropic central spin model with a large frequency ratio 𝜂.\nIn Fig. 3 (a), we present the spin squeezing parameter 𝜉2\n𝑆of\n0.98 0.99 1 1.01 1.0202468104(a) (b)FIG. 4. (a) QFI of the ground state as functions of ˜𝑔with different\nnumbers of bath spins 𝑁. Here we choose 𝜂=105,𝜔=1,𝜆=1\nand˜𝑔𝑐=1. ThecurvesfromtoptobottomcorrespondtotheQFIof\nthecentralspinmodelwith 𝑁=750(purpleline), 𝑁=1000(yellow\nline),and𝑁=1250(redline)and 𝑁=1250(blueline),respectively.\n(b) critical exponent 𝜇within various system size [𝑁1,𝑁2]. The\ncurves from top to bottom correspond to the cases of 𝜂=1000\n(blue line), 𝜂=10000(red line), and 𝜂=50000(yellow line),\nrespectively. Inaddition,thecaseoftheLMGmodelisalsopresented\nfor comparison (top blue line).\nthe ground state for the anisotropic case varies with ˜𝑔. It can\nbeseenthatthespinsqueezingparameter 𝜉2\n𝑆oftheanisotropic\ncentral spin model tends to that of the LMG model as the\nfrequency ratio 𝜂increases.\nInFig.3(a),itisevidentthattheoptimalsqueezingparam-\neter appears in the vicinity of the critical point. However, one\ncan see that the analytical expression of 𝜉2\n𝑆(solid green line)\ndecreasestozeroatthecriticalpoint,whichmeanstheresults\nof Eq. (28) and Eq. (30) fail when ˜𝑔approach ˜𝑔𝑐. In such a\ncase, it is necessary to consider the higher-order corrections.\nWe utilize numerical simulations to calculate the variation of\ntheoptimalspinsqueezingparameter 𝜉2\n𝑆,minwithrespectto 𝑁\nand find that 𝜉2\n𝑆,min∼𝑁−1/3in the limit of 𝜂→∞, which is\nconsistent with the result of the LMG model in Ref. [26, 77].\nThespinsqueezingparametersofthedynamicprocesswith\nvarious𝜆are also investigated by numerical simulation. In\nFig.3(c),wecomparethedynamicalevolutionofspinsqueez-\ningparameter 𝜉2\n𝑆withdifferentvaluesof 𝜆. Itisclearthatthe\nsmaller the value of 𝜆, the smaller the corresponding optimal\nsqueezingparameter. Inotherwords,for 𝜆=0,theassociated\noptimal squeezing parameter 𝜉2\n𝑆,minis minimal.\nIn summary, our analysis of the above results reveals that,\nunder the condition of fixed 𝑁, the most effective method\nfor generating the optimal spin-squeezed state is through the\ndynamic evolution of the isotropic central spin model. In this\nprocess,itiscrucialtomaketheratio 𝜂betweenthefrequency\nofthecentralspinandthatofthebathspinaslargeaspossible.6\nIV. QUANTUM FISHER INFORMATION AND\nFINITE-SIZE ANALYSIS\nQuantum Fisher information is a core concept in quantum\nmetrology, and the QFI of the ground state can be regard as\nan indicator to detect quantum phase transition even with-\nout knowing the information related to order parameter [78].\nTherefore, in this section, we will utilize QFI to analyze the\nfinite-sizebehavioroftheanisotropiccentralspinmodel. The\nfinite-size analysis of the isotropic case ( 𝜆=0) has been dis-\ncussedinRef.[63],thusherewewillfocusontheanisotropic\ncase (𝜆≠0).\nFor a pure state|𝜓(˜𝑔)⟩, the QFI with respect to parameter\n˜𝑔is give by\n𝐹(˜𝑔)=4\n𝜕˜𝑔𝜓|𝜕˜𝑔𝜓\u000b\n−4\f\f\n𝜕˜𝑔𝜓|𝜓\u000b\f\f2,(31)\nand its value is four times the fidelity susceptibility [78, 79].\nAccordingtotheconclusioninRef.[80],fortheLMGmodel,\nthe QFI of the ground state around the critical point presents\nscaling behavior as\n𝐹(˜𝑔m)∝𝑁𝜇, (32)\nwhere ˜𝑔mis the position of the maximal QFI, 𝑁is the size\nof the system, and 𝜇is the critical exponent with a specific\nvalue of𝜇=4/3. For the anisotropic central spin model, two\nconditionsarerequiredforaphasetransitiontooccur,namely,\nthe frequency ratio 𝜂and the size of the system 𝑁tend to\ninfinity. Nevertheless, when the frequency ratio 𝜂is finite,\nthe scaling behavior of 𝐹(˜𝑔m)in Eq. (32) is still unknown.\nTherefore, we utilize the exact diagonalization to obtain the\nground state and calculate the critical exponent 𝜇for a finite\n𝜂.\nInFig.4(a),wedepictthevariationoftheQFIoftheground\nstatefortheanisotropiccentralspinmodelwithdifferentnum-\nbers of bath spins. It is observed that as the 𝑁increases, the\nmaximum values of the QFI exhibits a progressively growing\ntrend, and their corresponding positions gradually approach\nthecriticalpoint. Thecriticalexponent 𝜇withindifferentsys-\ntemsizeranges[𝑁1,𝑁2]isprovidedinFig.4(b). Itisevident\nthat, as the frequency ratio 𝜂increases, the critical exponent\nconverges to 4/3when𝑁is sufficiently large, which corre-\nsponds precisely to the critical exponent of the LMG model\n(solid bule line) [80]. On the other hand, we find that if 𝜂\nis not sufficiently large, the critical exponent will rapidly de-\ncreasewithincreasing 𝑁untilitconvergestozero. Namely,if\n𝑁tends to infinity while 𝜂remains finite, the corresponding\ncritical exponent 𝜇will decay to zero, which implies that the\nquantumphasetransitionwillnotoccur. Hence,thecondition\nfor the phase transition to occur requires 𝜂and𝑁both tend\ntowards infinity.V. CONCLUSION\nInsummary,wehaveanalyzedthespinsqueezingandquan-\ntum phase transition in the anisotropic central spin model.\nUtilizing the Schrieffer-Wolff transformation, we analytically\nestablished the mapping between the anisotropic central spin\nmodel and the anisotropic LMG model. Meanwhile, it is\nshown that the substantial detuning between the central spin\nand bath spins can induce an intra-species nonlinear interac-\ntion. Inspiredbytheaboveresults,weinvestigatedthepotential\npathwaysforgeneratingspin-squeezedstateswithcentralspin\nsystems. Through comparisons, we found that the dynamics\ninitiated with a spin coherent state can generate the optimal\nspin-squeezedstate. Furthermore,whentheanisotropyparam-\neter𝜆iszero,theoptimalspinsqueezingparameterscaleswith\nthesystemsizeas 𝑁−2/3. Thecriticalexponent 𝜇ofthequan-\ntumFisherinformationaroundthecriticalpointhasalsobeen\nemployed to study the finite-size behavior in the anisotropic\ncentral spin model. Specifically, we obtained 𝜇=4/3as𝜂\nand𝑁approach infinity, however, when 𝜂is finite, it is found\nthat the value of 𝜇decreases to zero with the increase of 𝑁.\nThis work provides a new potential pathway for generating\nspin-squeezed states in practical physical systems, simultane-\nouslyofferingafreshperspectiveforourunderstandingofthe\nrelationship between spin squeezing and quantum criticality.\nACKNOWLEDGMENTS\nWe acknowledge Dr. Zhucheng Zhang for valuable sug-\ngestions. This work was supported by the National Natu-\nral Science Foundation of China (Grants No. 12088101 and\nNo. U2330401) and Science Challenge Project (Grant No.\nTZ2018005).\nAppendix A: DERIVATION OF MAPPING BEWTEEN\nCENTRAL SPIN MODEL AND LMG MODEL\nIn this section, we give the derivation of the mapping be-\ntweenthecentralspinmodelandtheLMGmodel. Firstofall,\nEq. (2) can be rewritten as\n𝐻=𝐻0+𝐴𝑉, (A1)\nwhere\n𝐻0=Ω\n2𝜎𝑧+𝜔𝐼𝑧, (A2)\n𝑉=(1+𝜆)𝐽𝑥𝜎𝑥+(1−𝜆)𝐽𝑦𝜎𝑦. (A3)\nNote that𝐻0is diagonal with respect to the spin subspace\nwhile𝑉isoff-diagonal. ThenweperformtheSchrieffer-Wolff7\ntransformation 𝑒𝑆on Eq. (A1) and obtain [81, 82]\n𝐻′=𝑒−𝑆𝐻𝑒𝑆=∞∑︁\n𝑘=01\n𝑘![𝐻,𝑆](𝑘), (A4)\nwhere\u0002\n𝐻,𝑆\u0003(𝑘)=\u0002\u0002\n𝐻,𝑆\u0003(𝑘−1),𝑆\u0003\n,\u0002\n𝐻,𝑆\u0003(0)=𝐻, and the\ngenerator𝑆is an anti-Hermitian and block-off-diagonal oper-\nator. In order to diagonalize the Hamiltonian 𝐻′, we need the\nblock-off-diagonal part of 𝐻′to be zero up to second order in\n𝐴, which leads to the follow relationship\n[𝐻0,𝑆]=−𝑉, (A5)\nandwefindtheexpressionof 𝑆satisfyingEq.(A5)isasfollows\n𝑆=−𝑖(1+𝜆)\nΩ𝐼𝑥𝜎𝑦+𝑖(1−𝜆)\nΩ𝐼𝑦𝜎𝑥.(A6)\nIn the limit of 𝜂→∞, the Hamiltonian in Eq. (A4) becomes\n𝐻′=𝐻0+𝐴2\n2[𝑉,𝑆]\n=Ω\n2𝜎𝑧+\u0014\n𝜔−𝐴2(1+𝜆)(1−𝜆)\nΩ\u0015\n𝐼𝑧\n+𝐴2\u0014(1+𝜆)2\nΩ𝐼2\n𝑥𝜎𝑧+(1−𝜆)2\nΩ𝐼2\n𝑦𝜎𝑧\u0015\n.(A7)\nFurthermore, we set ˜𝑔=2𝐴√︁\n𝑁/Ω𝜔,𝛾𝑥=˜𝑔2𝜔(1+𝜆)2/4,\n𝛾𝑦=˜𝑔2𝜔(1−𝜆)2/4, and finally we get the Hamiltonian in\nEq.(3),whichpreciselycorrespondstotheHamiltonianofthe\nanisotropic LMG model.\nAppendix B: DERIVATION OF SPIN SQUEEZING\nPARAMETER IN DYNAMICAL PROCESS\nFirst of all, we give the derivation of Eq. (16) and Eq. (17).\nForthegroundstateoftheLMGmodel,themeanspindirectionis always along the 𝑧-axis, thus we have ⟨𝐼𝑥⟩=\n𝐼𝑦\u000b\n=0, and\nthen we obtain [26]\nmin\u0010\nΔ2𝐼®𝑛⊥\u0011\n=1\n2\u0014\n𝐼2\n𝑥+𝐼2\n𝑦\u000b\n−√︃\u0000\n𝐼2𝑥−𝐼2𝑦\u000b\u00012+4Cov\u0000𝐼𝑥,𝐼𝑦\u00012\u0015\n=1\n2\n𝐼2\n𝑥+𝐼2\n𝑦\u000b\n, (B1)\nwhere Cov\u0000𝐼𝑥,𝐼𝑦\u0001=1\n2\n𝐼𝑥𝐼𝑦+𝐼𝑦𝐼𝑥\u000b\n.\nFor𝜂→ ∞, we make an approximation that e𝑃−\n↑,𝑛−1≃\n0,e𝑃−\n↓,𝑛≃1, and Eq. (B1) becomes\nmin\u0010\nΔ2𝐼®𝑛⊥\u0011\n=1\n2\n𝑰2−𝐼2\n𝑧\u000b\n=−1\n2\u0012\n𝑛−𝑁\n2\u00132\n+𝑁2\n8+𝑁\n4.(B2)\nUtilizing Eq. (B2) we can get Eq. (16) and Eq. (17).\nThen we give the derivation of the time evolution of spin\nsqueezing parameters 𝜉2\n𝑆and𝜉2\n𝑅. The initial state we choose\nisgiveninEq.(18),andundertheconditionof 𝜂≫1wehave\n⟨𝐼𝑧⟩=𝑁\n2cos𝜃0, (B3)\n\n𝐼2\n𝑧\u000b\n=1\n8𝑁[𝑁+1+(𝑁−1)cos 2𝜃0].(B4)\nThen we make the following approximations: Ω𝑛≃Ω,\n𝑃𝑛\n↓≃𝑒𝑖Ω𝑛𝑡\n2,𝑃𝑛\n↑≃0andΩ𝑛≃Ω+2𝑘𝑛𝐴2. And we obtain\n⟨𝐼+⟩≃𝑁−1∑︁\n𝑛=0𝐶𝑛\n𝑁−1𝑁\u0012\ncos2𝜃0\n2\u0013𝑛\u0012\nsin2𝜃0\n2\u0013𝑁−1−𝑛\n𝑒−𝑖(𝑁−2𝑛)𝐴2𝑡\nΩ𝑒𝑖𝜔𝑡cot𝜃0\n2\n=1\n2𝑁𝑒𝑖\u0010\n𝜔−𝐴2\nΩ\u0011\n𝑡\u0012\ncos𝐴2𝑡\nΩ+𝑖sin𝐴2𝑡\nΩcos𝜃0\u0013𝑁−1\nsin𝜃0, (B5)\n\n𝐼2\n+\u000b\n≃𝑒2𝑖𝜔𝑡𝑁−1∑︁\n𝑛=0\u0012\ncos2𝜃0\n2\u0013𝑛\u0012\nsin2𝜃0\n2\u0013𝑁−𝑛\n𝐶𝑛\n𝑁−2𝑁(𝑁−1)𝑒−𝑖2(𝑁−2𝑛−1)𝐴2𝑡\nΩ cot2𝜃0\n2\n=1\n4𝑒2𝑖\u0010\n𝜔−𝐴2\nΩ\u0011\n𝑡𝑁(𝑁−1)\u0014\ncos\u00122𝐴2𝑡\nΩ\u0013\n+𝑖cos𝜃0sin\u00122𝐴2𝑡\nΩ\u0013\u0015𝑁−2\nsin2𝜃0, (B6)8\n2⟨𝐼+𝐼𝑧⟩+⟨𝐼+⟩≃𝑁−1∑︁\n𝑛=0𝐶𝑛\n𝑁−1𝑁\u0012\ncos2𝜃0\n2\u0013𝑛\u0012\nsin2𝜃0\n2\u0013𝑁−𝑛\n𝑒−𝑖(𝑁−2𝑛)𝐴2𝑡\nΩ𝑒𝑖𝜔𝑡\u0014\n2\u0012\n−𝑁\n2+𝑛\u0013\n+1\u0015\ncot𝜃0\n2\n=1\n2𝑒𝑖\u0010\n𝜔−𝐴2\nΩ\u0011\n𝑡𝑁(𝑁−1)\u0014\ncos\u0012𝐴2𝑡\nΩ\u0013\n+𝑖cos𝜃0sin\u0012𝐴2𝑡\nΩ\u0013\u0015𝑁−2\u0014\ncos𝜃0cos\u0012𝐴2𝑡\nΩ\u0013\n+𝑖sin\u0012𝐴2𝑡\nΩ\u0013\u0015\nsin𝜃0.(B7)\nFor the sake of simplicity, we choose 𝜃0=𝜋/2, and the\ndirections perpendicular to the mean spin direction ®𝑛0are\n®𝑛1=(−sin𝜙,cos𝜙,0),®𝑛2=(0,0,1)and𝜙=𝜔𝑡. Therefore,\nthe spin squeezing parameter is given by [26]\n𝜉2\n𝑆=2\n𝑁\"D\n𝐼2\n®𝑛1+𝐼2\n®𝑛2E\n−√︂\u0010D\n𝐼2\n®𝑛1−𝐼2\n®𝑛2E\u00112\n+4Cov\u0000𝐼®𝑛1,𝐼®𝑛2\u00012#\n=2\n𝑁\u0010\nC−√︁\nA2+B2\u0011\n, (B8)where\nA=1\n8(𝑁−1)𝑁\"\n1−cos\u0012𝐴2𝑡\nΩ\u0013𝑁−2#\n, (B9)\nB=𝑁\n2(𝑁−1)cos\u0012𝐴2𝑡\nΩ\u0013𝑁−2\nsin\u0012𝐴2𝑡\nΩ\u0013\n,(B10)\nC=A+𝑁\n2. (B11)\nBy substituting Eq. (B9), Eq. (B10), and Eq. (B11) into\nEq. (19), we can obtain the analytical solution for the time\nevolution of the spin squeezing parameter 𝜉2\n𝑆.\n[1] M. A. Taylor and W. P. Bowen, Quantum metrology and its\napplicationinbiology,PhysicsReports 615,1(2016),quantum\nmetrology and its application in biology.\n[2] M. A. Taylor, J. Janousek, V. Daria, J. Knittel, B. Hage, H.-A.\nBachor, and W. P. Bowen, Biological measurement beyond the\nquantum limit, Nature Photonics 7, 229 (2013).\n[3] M. B. Nasr, D. P. Goode, N. Nguyen, G. Rong, L. Yang, B. M.\nReinhard, B. E. Saleh, and M. C. Teich, Quantum optical co-\nherence tomography of a biological sample, Optics Communi-\ncations 282, 1154 (2009).\n[4] P. A. Morris, R. S. Aspden, J. E. C. Bell, R. W. Boyd, and\nM. J. Padgett, Imaging with a small number of photons, Nature\nCommunications 6, 5913 (2015).\n[5] S. Qvarfort, A. Serafini, P. F. Barker, and S. Bose, Gravimetry\nthrough non-linear optomechanics, Nature Communications 9,\n3690 (2018).\n[6] S. Templier, P. Cheiney, Q. d’Armagnac de Castanet,\nB.Gouraud,H.Porte,F.Napolitano,P.Bouyer,B.Battelier,and\nB.Barrett,Trackingthevectoraccelerationwithahybridquan-\ntumaccelerometertriad,ScienceAdvances 8,eadd3854(2022),\nhttps://www.science.org/doi/pdf/10.1126/sciadv.add3854.\n[7] G. Lamporesi, A. Bertoldi, L. Cacciapuoti, M. Prevedelli, and\nG. M. Tino, Determination of the newtonian gravitational con-\nstant using atom interferometry, Phys. Rev. Lett. 100, 050801\n(2008).\n[8] S.-w. Chiow, S. Herrmann, S. Chu, and H. Müller, Noise-\nimmune conjugate large-area atom interferometers, Phys. Rev.\nLett. 103, 050402 (2009).\n[9] J. M. McGuirk, G. T. Foster, J. B. Fixler, M. J. Snadden, and\nM. A. Kasevich, Sensitive absolute-gravity gradiometry using\natom interferometry, Phys. Rev. A 65, 033608 (2002).\n[10] D. Oblak, P. G. Petrov, C. L. Garrido Alzar, W. Tittel, A. K.\nVershovski, J. K. Mikkelsen, J. L. Sørensen, and E. S. Polzik,\nQuantum-noise-limited interferometric measurement of atomic\nnoise: Towards spin squeezing on the cs clock transition, Phys.\nRev. A 71, 043807 (2005).\n[11] A. Sørensen and K. Mølmer, Spin-spin interaction and spinsqueezinginanopticallattice,Phys.Rev.Lett. 83,2274(1999).\n[12] D.Döring,G.McDonald,J.E.Debs,C.Figl,P.A.Altin,H.-A.\nBachor, N. P. Robins, and J. D. Close, Quantum-projection-\nnoise-limited interferometry with coherent atoms in a ramsey-\ntype setup, Phys. Rev. A 81, 043633 (2010).\n[13] A.André,A.S.Sørensen,andM.D.Lukin,Stabilityofatomic\nclocks based on entangled atoms, Phys. Rev. Lett. 92, 230801\n(2004).\n[14] G. S. Agarwal and M. O. Scully, Ramsey spectroscopy with\nnonclassical light sources, Phys. Rev. A 53, 467 (1996).\n[15] S.Chu,Atominterferometry,in Coherent atomic matter waves ,\neditedbyR.Kaiser,C.Westbrook,andF.David(SpringerBerlin\nHeidelberg, Berlin, Heidelberg, 2001) pp. 317–370.\n[16] G. Xu and D. J. Heinzen, State-selective rabi and ramsey mag-\nnetic resonance line shapes, Phys. Rev. A 59, R922 (1999).\n[17] V.Meyer,M.A.Rowe,D.Kielpinski,C.A.Sackett,W.M.Itano,\nC.Monroe,andD.J.Wineland,Experimentaldemonstrationof\nentanglement-enhancedrotationangleestimationusingtrapped\nions, Phys. Rev. Lett. 86, 5870 (2001).\n[18] G.A.Smith,S.Chaudhury,A.Silberfarb,I.H.Deutsch,andP.S.\nJessen,Continuousweakmeasurementandnonlineardynamics\nin a cold spin ensemble, Phys. Rev. Lett. 93, 163602 (2004).\n[19] D. Walls and P. Zoller, Enhanced sensitivity of a gravitational\nwave detector, Physics Letters A 85, 118 (1981).\n[20] J.A.DunninghamandK.Burnett,Sub-shot-noise-limitedmea-\nsurements with bose-einstein condensates, Phys. Rev. A 70,\n033601 (2004).\n[21] K. Goda, O. Miyakawa, E. E. Mikhailov, et al., A quantum-\nenhancedprototypegravitational-wavedetector,NaturePhysics\n4, 472 (2008).\n[22] X. Wang and B. C. Sanders, Spin squeezing and pairwise en-\ntanglement for symmetric multiqubit states, Phys. Rev. A 68,\n012101 (2003).\n[23] A.S.SørensenandK.Mølmer,Entanglementandextremespin\nsqueezing, Phys. Rev. Lett. 86, 4431 (2001).\n[24] G.Tóth,C.Knapp,O.Gühne,andH.J.Briegel,Spinsqueezing\nand entanglement, Phys. Rev. A 79, 042334 (2009).9\n[25] J. Estève, C. Gross, A. Weller, S. Giovanazzi, and M. K.\nOberthaler,Squeezingandentanglementinabose–einsteincon-\ndensate, Nature 455, 1216 (2008).\n[26] J. Ma, X. Wang, C. Sun, and F. Nori, Quantum spin squeezing,\nPhysics Reports 509, 89 (2011).\n[27] A. Sørensen, L.-M. Duan, J. I. Cirac, and P. Zoller, Many-\nparticle entanglement with bose–einstein condensates, Nature\n409, 63 (2001).\n[28] C. Orzel, A. K. Tuchman, M. L. Fenselau, M. Yasuda, and\nM.A.Kasevich,Squeezedstatesinabose-einsteincondensate,\nScience 291, 2386 (2001).\n[29] C. K. Law, H. T. Ng, and P. T. Leung, Coherent control of spin\nsqueezing, Phys. Rev. A 63, 055601 (2001).\n[30] S.D.JenkinsandT.A.B.Kennedy,Spinsqueezinginadriven\nbose-einstein condensate, Phys. Rev. A 66, 043621 (2002).\n[31] U. V. Poulsen and K. Mølmer, Positive-p simulations of spin\nsqueezing in a two-component bose condensate, Phys. Rev. A\n64, 013616 (2001).\n[32] D. Jaksch, J. I. Cirac, and P. Zoller, Dynamically turning off\ninteractions in a two-component condensate, Phys. Rev. A 65,\n033625 (2002).\n[33] A. Micheli, D. Jaksch, J. I. Cirac, and P. Zoller, Many-\nparticle entanglement in two-component bose-einstein conden-\nsates, Phys. Rev. A 67, 013607 (2003).\n[34] M.Jääskeläinen,W.Zhang,andP.Meystre,Limitstophaseres-\nolutioninmatter-waveinterferometry,Phys.Rev.A 70,063612\n(2004).\n[35] S. Choi and N. P. Bigelow, Quantum squeezing and entan-\nglement in a two-mode bose-einstein condensate with time-\ndependent josephson-like coupling, Phys. Rev. A 72, 033612\n(2005).\n[36] M.JääskeläinenandP.Meystre,Dynamicsofbose-einsteincon-\ndensates in double-well potentials, Phys. Rev. A 71, 043603\n(2005).\n[37] G.-R. Jin and S. W. Kim, Storage of spin squeezing in a\ntwo-component bose-einstein condensate, Phys. Rev. Lett. 99,\n170405 (2007).\n[38] G.-R.JinandS.W.Kim,Spinsqueezingandmaximal-squeezing\ntime, Phys. Rev. A 76, 043621 (2007).\n[39] D. J. Wineland, J. J. Bollinger, W. M. Itano, F. L. Moore, and\nD. J. Heinzen, Spin squeezing and reduced quantum noise in\nspectroscopy, Phys. Rev. A 46, R6797 (1992).\n[40] J. Hald, J. L. Sørensen, C. Schori, and E. S. Polzik, Spin\nsqueezed atoms: A macroscopic entangled ensemble created\nby light, Phys. Rev. Lett. 83, 1319 (1999).\n[41] B.Julsgaard,A.Kozhekin,andE.S.Polzik,Experimentallong-\nlivedentanglementoftwomacroscopicobjects,Nature 413,400\n(2001).\n[42] K.Hammerer,A.S.Sørensen,andE.S.Polzik,Quantuminter-\nface between light and atomic ensembles, Rev. Mod. Phys. 82,\n1041 (2010).\n[43] G.M.PalmaandP.L.Knight,Phase-sensitivepopulationdecay:\nThe two-atom dicke model in a broadband squeezed vacuum,\nPhys. Rev. A 39, 1962 (1989).\n[44] A.Banerjee,Generationofatomic-squeezedstatesinanoptical\ncavitywithaninjectedsqueezedvacuum,Phys.Rev.A 54,5327\n(1996).\n[45] A. Kuzmich, L. Mandel, J. Janis, Y. E. Young, R. Ejnisman,\nand N. P. Bigelow, Quantum nondemolition measurements of\ncollective atomic spin, Phys. Rev. A 60, 2346 (1999).\n[46] J. L. Sørensen, J. Hald, and E. S. Polzik, Quantum noise of an\natomicspinpolarizationmeasurement,Phys.Rev.Lett. 80,3487\n(1998).\n[47] L.Vernac,M.Pinard,andE.Giacobino,Spinsqueezingintwo-level systems, Phys. Rev. A 62, 063812 (2000).\n[48] A.DantanandM.Pinard,Quantum-statetransferbetweenfields\nand atoms in electromagnetically induced transparency, Phys.\nRev. A 69, 043810 (2004).\n[49] M. H. Schleier-Smith, I. D. Leroux, and V. Vuletić, Squeez-\ning the collective spin of a dilute atomic ensemble by cavity\nfeedback, Phys. Rev. A 81, 021804 (2010).\n[50] Z. Zhang, L. Shao, W. Lu, Y. Su, Y.-P. Wang, J. Liu, and\nX.Wang,Single-photon-triggeredspinsqueezingwithdecoher-\nencereductioninoptomechanicsviaphasematching,Phys.Rev.\nA104, 053517 (2021).\n[51] J. Geremia, J. K. Stockton, and H. Mabuchi,\nReal-time quantum feedback control of atomic\nspin-squeezing, Science 304, 270 (2004),\nhttps://www.science.org/doi/pdf/10.1126/science.1095374.\n[52] M.H.Schleier-Smith,I.D.Leroux,andV.Vuletić,Statesofan\nensembleoftwo-levelatomswithreducedquantumuncertainty,\nPhys. Rev. Lett. 104, 073604 (2010).\n[53] A. E. Kozhekin, K. Mølmer, and E. Polzik, Quantum memory\nfor light, Phys. Rev. A 62, 033809 (2000).\n[54] A. Kuzmich, L. Mandel, and N. P. Bigelow, Generation of\nspin squeezing via continuous quantum nondemolition mea-\nsurement, Phys. Rev. Lett. 85, 1594 (2000).\n[55] I. D. Leroux, M. H. Schleier-Smith, and V. Vuletić, Implemen-\ntationofcavitysqueezingofacollectiveatomicspin,Phys.Rev.\nLett. 104, 073602 (2010).\n[56] T. Hernández Yanes, M. Płodzień, M. Mackoit Sinkevičien ˙e,\nG.Žlabys,G.Juzeli ¯unas,andE.Witkowska,One-andtwo-axis\nsqueezingvialasercouplinginanatomicfermi-hubbardmodel,\nPhys. Rev. Lett. 129, 090403 (2022).\n[57] L.Pezzè,A.Smerzi,M.K.Oberthaler,R.Schmied,andP.Treut-\nlein,Quantummetrologywithnonclassicalstatesofatomicen-\nsembles, Rev. Mod. Phys. 90, 035005 (2018).\n[58] S. Kolkowitz, S. L. Bromley, T. Bothwell, M. L. Wall, G. E.\nMarti, A. P. Koller, X. Zhang, A. M. Rey, and J. Ye, Spin–\norbit-coupled fermions in an optical lattice clock, Nature 542,\n66 (2017).\n[59] M. S. Rudner, L. M. K. Vandersypen, V. Vuletić, and L. S.\nLevitov,Generatingentanglementandsqueezedstatesofnuclear\nspins in quantum dots, Phys. Rev. Lett. 107, 206806 (2011).\n[60] H. Lipkin, N. Meshkov, and A. Glick, Validity of many-body\napproximationmethodsforasolvablemodel: (i).exactsolutions\nand perturbation theory, Nuclear Physics 62, 188 (1965).\n[61] S. Dooley, F. McCrossan, D. Harland, M. J. Everitt, and T. P.\nSpiller,Collapseandrevivalandcatstateswithan 𝑛-spinsystem,\nPhys. Rev. A 87, 052323 (2013).\n[62] W.-B.He,S.Chesi,H.-Q.Lin,andX.-W.Guan,Exactquantum\ndynamicsofxxzcentralspinproblems,Phys.Rev.B 99,174308\n(2019).\n[63] L. Shao, R. Zhang, W. Lu, Z. Zhang, and X. Wang, Quantum\nphasetransitioninthe 𝑥𝑥𝑧centralspinmodel,Phys.Rev.A 107,\n013714 (2023).\n[64] A. V. Khaetskii, D. Loss, and L. Glazman, Electron spin deco-\nherence in quantum dots due to interaction with nuclei, Phys.\nRev. Lett. 88, 186802 (2002).\n[65] M. Bortz, S. Eggert, C. Schneider, R. Stübner, and J. Stolze,\nDynamicsanddecoherenceinthecentralspinmodelusingexact\nmethods, Phys. Rev. B 82, 161308 (2010).\n[66] M. Bortz and J. Stolze, Exact dynamics in the inhomogeneous\ncentral-spin model, Phys. Rev. B 76, 014304 (2007).\n[67] G.-Q.Liu,J.Xing,W.-L.Ma,P.Wang,C.-H.Li,H.C.Po,Y.-R.\nZhang,H.Fan,R.-B.Liu,andX.-Y.Pan,Single-shotreadoutof\na nuclear spin weakly coupled to a nitrogen-vacancy center at\nroom temperature, Phys. Rev. Lett. 118, 150504 (2017).10\n[68] E. M. Kessler, G. Giedke, A. Imamoglu, S. F. Yelin, M. D.\nLukin, and J. I. Cirac, Dissipative phase transition in a central\nspin system, Phys. Rev. A 86, 012116 (2012).\n[69] M. Kitagawa and M. Ueda, Squeezed spin states, Phys. Rev. A\n47, 5138 (1993).\n[70] D. J. Wineland, J. J. Bollinger, W. M. Itano, and D. J. Heinzen,\nSqueezed atomic states and projection noise in spectroscopy,\nPhys. Rev. A 50, 67 (1994).\n[71] J.MaandX.Wang,Fisherinformationandspinsqueezinginthe\nlipkin-meshkov-glick model, Phys. Rev. A 80, 012318 (2009).\n[72] G.-R. Jin, Y.-C. Liu, and W.-M. Liu, Spin squeezing in a gen-\neralized one-axis twisting model, New Journal of Physics 11,\n073049 (2009).\n[73] L.PezzéandA.Smerzi,Entanglement,nonlineardynamics,and\nthe heisenberg limit, Phys. Rev. Lett. 102, 100401 (2009).\n[74] L.-G. Huang, F. Chen, X. Li, Y. Li, R. Lü, and Y.-C. Liu,\nDynamic synthesis of heisenberg-limited spin squeezing, npj\nQuantum Information 7, 168 (2021).\n[75] L.-G. Huang, X. Zhang, Y. Wang, Z. Hua, Y. Tang, and Y.-C.\nLiu,Heisenberg-limitedspinsqueezingincoupledspinsystems,Phys. Rev. A 107, 042613 (2023).\n[76] S.DusuelandJ.Vidal,Continuousunitarytransformationsand\nfinite-sizescalingexponentsinthelipkin-meshkov-glickmodel,\nPhys. Rev. B 71, 224420 (2005).\n[77] J.Vidal,G.Palacios,andR.Mosseri,Entanglementinasecond-\norderquantumphasetransition,Phys.Rev.A 69,022107(2004).\n[78] S.-J.GU,Fidelityapproachtoquantumphasetransitions,Inter-\nnational Journal of Modern Physics B 24, 4371 (2010).\n[79] X.-Y. Chen, Y.-F. Xie, and Q.-H. Chen, Quantum criticality of\ntherabi-starkmodelatfinitefrequencyratios,Phys.Rev.A 102,\n063721 (2020).\n[80] H.-M. Kwok, W.-Q. Ning, S.-J. Gu, and H.-Q. Lin, Quantum\ncriticalityofthelipkin-meshkov-glickmodelintermsoffidelity\nsusceptibility, Phys. Rev. E 78, 032103 (2008).\n[81] M.-J. Hwang, R. Puebla, and M. B. Plenio, Quantum phase\ntransition and universal dynamics in the rabi model, Phys. Rev.\nLett. 115, 180404 (2015).\n[82] M.-J. Hwang and M. B. Plenio, Quantum phase transition in\nthefinitejaynes-cummingslatticesystems,Phys.Rev.Lett. 117,\n123602 (2016)."    },    {        "title": "2403.10465v1.Microscopic_understanding_of_NMR_signals_by_dynamic_mean_field_theory_for_spins.pdf",        "content": "Microscopic understanding of NMR signals by dynamic mean-field theory for spins\nTimo Gräßer,1,∗Thomas Hahn,1,2, †and Götz S. Uhrig1,‡\n1Condensed Matter Theory, TU Dortmund University,\nOtto-Hahn Straße 4, 44221 Dortmund, Germany\n2School of Physics and Astronomy, The University of Manchester, Manchester M13 9PL, United Kingdom\n(Dated: March 18, 2024)\nA recently developed dynamic mean-field theory for disordered spins (spinDMFT) is shown to\ncapture the spin dynamics of nuclear spins very well. The key quantities are the spin autocorrela-\ntions. In order to compute the free induction decay (FID), pair correlations are needed in addition.\nThey can be computed on spin clusters of moderate size which are coupled to the dynamic mean\nfields determined in a first step by spinDMFT. We dub this versatile approach non-local spinDMFT\n(nl-spinDMFT). It is a particular asset of nl-spinDMFT that one knows from where the contribu-\ntions to the FID stem. We illustrate the strengths of nl-spinDMFT in comparison to experimental\ndata for CaF 2. Furthermore, spinDMFT provides the dynamic mean-fields explaining the FID of\nthe nuclear spins in13C in adamantane up to some static noise. The spin Hahn echo in adamantane\nis free from effects of static noise and agrees excellently with the spinDMFT results without further\nfitting.\nI. INTRODUCTION\nNuclear magnetic resonance (NMR) has been success-\nfully employed for decades to obtain microscopic infor-\nmation about matter [1]. Experimental measurements of\nfree-induction decays (FIDs), spin echoes and more allow\nconclusions to be drawn about chemical compounds and\nmolecular distances in a sample. At the same time, a\nfully theoretical description of what is measured is rarely\npossible, since it requires the simulation of large numbers\nof nuclear spins. Conventional methods such as exact di-\nagonalization and Chebyshev polynomial expansion are\nrestricted to a couple of tens of spins [2, 3]. To capture\nrealistic system sizes, fundamental approximations have\nto be considered. For the computation of FIDs, many\ndifferent approximation schemes have been considered in\nthepast[4–9](seealsothereferencescitedinRef.[8]). Al-\nthoughsomeofthemarequiteaccuratefortheconsidered\nexample, they are mostly difficult to generalize and, thus,\nrarely used at present. Recently proposed strategies in-\nclude replacing all or many of the spins by classical ones\n[8–10], which is particularly successful in systems where\neachspinhasalargenumberofneighbors. Thepredictive\npower of such approaches can be questioned if quantum\neffects are more important, for instance, due to strong\nlocal quantum bonds or quadrupolar interactions [11].\nTurning to non-equilibrium NMR measurements makes\nthe situation even more complicated. A non-Markovian\nsimulation of spin echoes, for example, is rarely even at-\ntempted [12–14].\nThe concept of a “mean field” has been considered in\nphysics and chemistry for many decades. Static mean-\nfield theories are not known for their accuracy, but\n∗timo.graesser@tu-dortmund.de\n†thomas.hahn@student.manchester.ac.uk\n‡goetz.uhrig@tu-dortmund.dethey often provide us with a first qualitative picture\nof a complex many-body problem. Making the mean-\nfields dynamic immensely increases the accuracy, see\nthe fermionic DMFT developed in the nineties [15] and\nits various extensions, for example cellular or extended\nDMFT [16, 17]. The recently developed dynamic mean-\nfield theory for spins (spinDMFT) [18] can be seen as\nan analogue for spin systems. Yet, spinDMFT is derived\nonly for disordered spin systems. This makes it less gen-\neral, but perfectly tailored to NMR.\nThe main purpose of this article is to establish\nspinDMFT and its extensions in the NMR community\nas simulation tools. In doing so, we present an exten-\nsion of spinDMFT which we call non-local spinDMFT\n(nl-spinDMFT). The idea is to include the mean-field\nmoments computed by spinDMFT in the simulation of a\nlocal quantum cluster coupled to stochastic mean fields.\nThis approach allows for an efficient computation of non-\nlocalquantities and is related to the cluster-extension of\nspinDMFT (CspinDMFT) [19].\nWe emphasize two advantages of these approaches.\nThe first one is that the effective mean-field model main-\ntains a microscopic picture of the considered system.\nQuantities such as spin-spin correlations are directly\naccessible which gives us deeper insight into how the\nglobal NMR measurement is connected to microscopic\nspin physics. We will demonstrate this for the example\nof calcium fluoride, which is a common test substance for\nbenchmarking approximation schemes due to a relatively\nnoise-free FID in experiment [20]. The second advan-\ntage of spinDMFT is its high versatility. Treating several\nspins on the quantum level as it is done in CspinDMFT\ncan be very helpful in inhomogeneous systems [19]. For\nspins with S >1/2, a quadrupolar interaction can be di-\nrectly included in the effective mean-field model. More-\nover, external time-dependent fields and noises are easily\nincluded in spinDMFT andits extensions without in-\ncreasing the numerical effort much. Therefore, dynami-\ncal decoupling [21–27] can also be simulated within thesearXiv:2403.10465v1  [cond-mat.stat-mech]  15 Mar 20242\napproaches. We will demonstrate this in the second part\nof the article for the example of polycrystalline adaman-\ntane, which is another common test substance in NMR\n[1, 23–27].\nSec. II is devoted to a short overview of the considered\nsubstances and a brief derivation of spinDMFT. Subse-\nquently, we introduce nl-spinDMFT and use it to esti-\nmate the FID of calcium fluoride in Sec. III. In Sec. IV,\nwe consider the microscopic proton spin dynamics in\nadamantane using spinDMFT. The spin dynamics are\nused in Sec. V to compute the carbon FID and spin echo\nin adamantane.\nII. MODELS AND spinDMFT\nA. Calcium fluoride\nCalcium fluoride, CaF 2, is a crystal with calcium ions\non an fcc-lattice and fluoride ions ordered in an sc-lattice\nintertwined with the former. For NMR purposes, the\ncalcium ions can be neglected, since the abundance of\nnuclei with nonzero spin is very small, ≈0.13 %. A well-\nmeasurable NMR signal results from fluoride19Fwhich\nhas an abundance of 100 %and carries a nuclear spin of\nS=1/2.\nThe basis for an NMR experiment is a strong magnetic\nfield ⃗B, which polarizes the spin ensemble into the field\ndirection ⃗ nB:=⃗B/|⃗B|. The secular Hamiltonian for the\nhomonuclear dipole-dipole interaction is given by\nHhom=1\n4X\ni,jJij\u0000\n−Sx\niSx\nj−Sy\niSy\nj+ 2Sz\niSz\nj\u0001\n.(1)\nHere and henceforth, we set Jii= 0. Ifi̸=j, the secular\ncouplings read\nJij:=J⃗ rij(⃗ nB) =\u0010\n1−3 (⃗ nij·⃗ nB)2\u0011µ0\n4πγiγj\n|⃗ rij|3,(2a)\n⃗ rij:=⃗ ri−⃗ rj ⃗ nij:=⃗ rij/|⃗ rij|, (2b)\nwith ⃗ riand⃗ rjbeing the nuclear spin positions fixed by\nthe lattice and γiandγjbeing the gyromagnetic ratios.\nThe latter are equal in the homonuclear case.\nB. Adamantane\nThe adamantane molecule consists of 10 carbon and 16\nhydrogen atoms located on approximate spheres. Four of\nthe carbon atoms ( CA) are each bound to one hydrogen\natom ( HA) and six carbons ( CB) are each bound to two\nhydrogen atoms ( HB). The molecule is displayed in the\nleft panel of Fig. 1. We set all bond angles to the tetra-\nhedral angle α≈109.5◦and use the bond lengths [28]\ndCC= 1.540(2)Å, d CH= 1.112(4)Å.(3)\nrCArCB\nrHB\nrHAHA\nCA HB\nCB\nFIG. 1. Sketch of the adamantane molecule. The hydrogen\natoms are shown in dark blue ( HA) and light blue ( HB) and\nthecarbonatomsinblack( CA)andgray( CB). Weusecircles\nwith the corresponding colors to represent the radii of the\natoms with respect to the center of gravity of the molecule,\nsee Eq. (4). The circles for the two types of hydrogen are\nnearly indistinguishable.\nThe distances between the four types of atoms with re-\nspect to the CMS of the molecule are then given by\nrCA≈1.54Å, r CB≈1.78Å,(4a)\nrHA≈2.65Å, r HB≈2.58Å.(4b)\nConsidering these values and Fig. 1, we note that the two\nclasses of hydrogen and the two classes of carbon are each\nlocated nearly on the same shell. Hence, we consider only\na single type of hydrogen and a single type of carbon at\nthe averaged radii\nrH:=12rHA+ 4rHB\n16= 2.60(3)Å,(5a)\nrC:=6rCA+ 4rCB\n10= 1.68(12)Å (5b)\nhenceforth. As we will see later, this simplifies the appli-\ncation of spinDMFT, since only a single kind of mean-\nfield needs to be considered.\nIn line with the experiment in Ref. [23], we con-\nsider polycrystalline adamantane at room temperature.\nThe adamantane molecules form an fcc-lattice with\nafcc= 9.426(8)Å and four molecules per unit cell [29].\nWhile their centers of mass are fixed, the molecules con-\ntinuously change their orientations. These molecular ro-\ntations are isotropic and happen on a much shorter time\nscale than the spin dynamics so that motional narrowing\nhas to be considered for the couplings.\nAside from the proton spins, only the carbon isotopes\n13Care of interest for NMR, since they carry a spin of\nI=1/2in contrast to the spinless nuclei of12C. Gen-\nerally, we denote the1Hnuclear spins by Sand the13C\nnuclear spins by I. The natural abundance of13Cis\nabout 1.1 %. Since the carbon spins are so rare, we ne-\nglect any carbon-hydrogen interaction for the hydrogen\ndynamics and any carbon-carbon interaction in general.3\nThe homonuclear secular Hamiltonian is given by\nHhom=1\n4X\ni,jJijma\u0000\n−Sx\niSx\nj−Sy\niSy\nj+ 2Sz\niSz\nj\u0001\n.(6)\nThis is equivalent to Eq. (1), but with the couplings Jij\nreplaced by their motional average Jijma, see below. For\nthedynamicsof13C,wealsousetheheteronuclearHamil-\ntonian\nHhet=X\njJ0jmaIzSz\nj, (7)\nwhere, again, the averaged couplings have to be consid-\nered. The indices i∈ {1,2,3, . . .}label the proton spins\nand the carbon index is 0. As justified above, we consider\nthe13Cspins to be isolated from one another.\nThemotionalaveragingcausesanyintramolecularcou-\nplings to average out completely, i.e., motional narrowing\ntakes place. The averaged intermolecular couplings de-\npend on (i) the distances of the considered nuclei to the\ncenter of mass and (ii) the lattice vector between the\nmolecules to which the nuclei belong. For this reason, it\nis helpful to introduce double indices (mX):=i, where\nm∈ {1,2,3, . . .}denotes the molecule and X∈H,Cis\nthe nucleus type. The averaged couplings can be com-\nputed from\nJijma(⃗ nB):=J(m1X1),(m2X2)ma(⃗ nB) (8a)\n=ZdΩ1dΩ2\n(4π)2J⃗ am1,m2+rX1⃗ n(Ω1)−rX2⃗ n(Ω2)(⃗ nB).(8b)\nHere, ⃗ am1,m2is the lattice vector from center to\ncenter of the molecules and ⃗ n(Ω) = ⃗ n(θ, ϕ):=\n(sinϑcosφ,sinϑsinφ,cosϑ)⊤. Theradii rX1, rX2arethedis-\ntances from the nucleus to the center of its molecule, see\nEq. (5).\nAn exact many-body treatment of adamantane is even\nfurther beyond reach than for calcium fluoride. Due to\nthe motional averaging the intermolecular couplings are\nthe dominant ones. Note that a single molecule carries\nalready 16proton spins, so the number of couplings is\nimmense, so that brute-force calculations are out of ques-\ntion. On the other hand, this is an ideal condition for a\nmean-field approach. Due to the large number of inter-\naction partners, the mean-field average is dominant over\nfluctuations. In the following, we present a brief sum-\nmary of spinDMFT and show the generic results.\nC. Simulation of homonuclear dynamics by\nspinDMFT\nApplying spinDMFT to a system containing only a\nsingle kind of spins is straight-forward [18]. As an exam-\nple, we consider the homonuclear Hamiltonian in Eq. (1).\nDefining the operators of the local environment,\n⃗Vi:=X\njJijD⃗Sj, D =\n−1\n20 0\n0−1\n20\n0 0 1\n,(9)it can be rewritten according to\nHhom=1\n2X\ni⃗Si·⃗Vi. (10)\nThe first step of spinDMFT is to replace these opera-\ntors by classical time-dependent mean-fields ⃗Vi(t). Each\nmean-field summarizes the interaction of a large number\nof spins. Therefore, according to the central limit the-\norem, the mean-fields underlie a Gaussian distribution.\nReplacing the local-environment operators, we arrive at\nan effective single-site model\nHmf\ni(t) =⃗Si·⃗Vi(t). (11)\nThe quantum degrees of freedom of each spin are decou-\npled from those of the rest of the ensemble; hence, the\nspin index can be omitted henceforth. The mean-fields\ncan vary from site to site, even on average. However, for\nthe fluoride ions in CaF 2as well as for the protons in\nadamantane, translational invariance is a well justified\nassumption. Therefore, the distribution of each mean-\nfield is independent of the site iso that this index can be\nomitted as well. We consider a single site, i.e., a single\nspin and a single mean-field henceforth.\nSince the mean-field is Gaussian, only the first and\nthe second moment are required to fully determine it.\nIn the high-temperature regime of completely disordered\nspins, expectation values are evaluated with respect to\nthe disordered state ρ0∝ 1. The first moment is thus\nzero,\nVα(t)mf=⟨Vα(t)⟩= 0, (12)\nwhere α∈ {x, y, z}and the line with upper index “mf”\ndenotes the mean-field averaging. The second moments\ncan be related to quantum correlations of the original\nfield according to\nVα(t)Vα(0)mf=⟨Vα(t)Vα(0)⟩ (13a)\n=D2\nααJ2\nQ⟨Sα(t)Sα(0)⟩,(13b)\nVα(t)Vβ(0)mf= 0 α̸=β. (13c)\nwhere we defined the quadratic coupling constant\nJ2\nQ:=X\njJ2\nij. (14)\nNote that although the sum runs only over j, the to-\ntal expression does not depend on idue to translational\ninvariance. We stress that correlations between mean-\nfields at different sites are not relevant in spinDMFT due\nto the large coordination number [18]. For the derivation\nof Eq. (13b) we used that pair correlations are subdom-\ninant and can be neglected relative to the autocorrela-\ntions, that is,\nX\nj,k\nj̸=kJijJik⟨Sα\nj(t)Sα\nk(0)⟩ ≪X\njJ2\nij⟨Sα\nj(t)Sα\nj(0)⟩.(15)4\nThis is also justified by the large coordination number z.\nNote that zis typically defined by the number of nearest\nneighbors in the spin lattice. However, since the dipolar\ninteraction is long-range, the effective number of inter-\naction partners of each spin can be a lot higher than z.\nTherefore, we consider the effective coordination number\n[30]\nzeff:=J4\nQ\nJ4\nT, J4\nT:=X\njJ4\nij,(16)\nto be more relevant for the justification of the mean-field\napproach. For the sake of brevity, we sometimes use z\ninstead of zeffin our argumentation.\nEqs. (11), (13b) and (13c) form a closed self-\nconsistency problem which can be solved numerically.\nA comprehensive justification of spinDMFT and details\non the numerical implementation can be found in the\noriginal article [18]. In essence, one has to simulate\na single spin in a Gaussian noise, which can be done\nby Monte-Carlo simulation. The resulting spin autocor-\nrelations are transformed to mean-field moments using\nthe self-consistency conditions in Eqs. (13b) and (13c),\nwhich are then fed into the next simulation of a single\nspin in a Gaussian noise. This process is repeated until\nthe spin autocorrelations have converged which typically\ntakes only about 5 iterations.\nHenceforth, we will occasionally use the short-hands\ngαα\nij(t):=⟨Sα\ni(t)Sα\nj(0)⟩, (17a)\nGαα\nij(t):= 4gαα\nij(t). (17b)\nThe site indices are omitted if i=j. One should note\nthat the only remaining parameter in the self-consistency\nproblem is the quadratic coupling constant JQ. This con-\nstant sets the time scale. We present the generic auto-\ncorrelationresultsofspinDMFTforHamiltoniansoftype\n(1) in Fig. 2. The difference in the time scale between the\ntransverse ( Gxx) and longitudinal ( Gzz) autocorrelation\nis expected due to the anisotropy factor of 2 in the cou-\nplings. We find, that the decays are very well captured\nby the fits\nFxx(t) =exp \n−1\n2\u0012t\nσ\u00132!\n(18a)\nFzz(t) =exp\u0010\n−ρ\u0010p\nt2+κ2− |κ|\u0011\u0011\n(18b)\nwith\nσ= 1.769(1) J−1\nQ, (19a)\nρ= 0.2149(4) JQ, (19b)\nκ= 1.26(1) J−1\nQ. (19c)\nWe emphasize the qualitative difference in the temporal\ndependence of FxxandFzzfort→ ∞, Gaussian versus\nexponential. The spin autocorrelations yield a good es-\ntimate for the time scale of the spin dynamics. The key\nquantity is the quadratic coupling constant.\n0 5 10 15 20 25 30\nt(units ofJ−1\nQ)0.00.20.40.60.81.0GααGxx\nGzzFIG. 2. Generic spin autocorrelations computed by\nspinDMFT for dipolar spins in the rotating frame accord-\ning to Eq. (1). The quadratic coupling constant is defined\nin Eq. (14) and depends on the geometry of the spin system.\nThe dashed-dotted black lines correspond to the fit functions\nin Eq. (18).\nIII. SIMULATING FIDs WITH nl-spinDMFT\nWe stress that computing the FID with spinDMFT\nrepresents a challenging task, since the approach accesses\nthespindynamicsfromamicroscopicpointofview, while\nthe FID is a macroscopic quantity. Our strategy for com-\nputing the FID is to relate it to microscopic spin-spin\ncorrelations. This is explained in the first part of this\nsection. Then, we present an extension of spinDMFT,\ncalled nl-spinDMFT, and use it to evaluate the FID of\ncalcium fluoride.\nA. FID and microscopic correlations\nThe FID corresponds to a measurement of the trans-\nverse magnetization over time considering a slightly po-\nlarized initial state with the statistical operator [31]\nρ→:=1\nZehP\njSx\nj≈1\nZ\n 1+hX\njSx\nj\n(20)\nwhere h:=βγB,βis the inverse temperature, Zis the\npartitionsumand γisthegyromagneticratio. Thetrans-\nverse magnetization can be expressed by infinite temper-\nature spin correlations according to\n⟨Mx(t)⟩ρ→=AX\ni,j⟨Sx\ni(t)Sx\nj⟩, (21)\nwhere Ais time-independent prefactor, which is not rel-\nevant to us. The double sum can be divided into a sum5\nover autocorrelations and a sum over pair correlations\naccording to\nX\ni|{z}\n∝N⟨Sx\ni(t)Sx\ni⟩|{z}\n∝1+X\ni|{z}\n∝NX\nj̸=i|{z}\n∝z⟨Sx\ni(t)Sx\nj⟩\n|{z}\n≤∝1/z.(22)\nWe recall that a key aspect of spinDMFT is that pair\ncorrelations are suppressed by at least a factor1/z, where\nzis the coordination number. Often, this allows us to\nneglect them for the dynamics [18], e.g., in Eq. (15).\nWe stress that for this argument to work the presence\nof the couplings as factors in Eq. (15) is crucial. They\nassist in suppressing any contribution from pair correla-\ntions. In Eq. (22), however, the double sum for the pair\ncorrelations contains no couplings. If we consider only\nnearest-neighbor pair correlations, the double sum con-\ntains∝zNterms, where Nis the particle number. As\na consequence, the whole expression scales like Nwhich\nis the same as the scaling of the autocorrelation sum. It\nis the additional factor zwhich has the pair correlations\nplay a role in the FID although they are unimportant for\nthe microscopic spin dynamics. This is the key observa-\ntion in this section.\nIntuitively, one would expect the relevance of pair cor-\nrelations to decrease as the direct couplings of the two in-\nvolved spins decreases. We stress, however, that indirect\ncouplings, for example via a third spin, can also matter.\nIt is not a priori clear, if or how much pair correlations\nbecome irrelevant if we increase the distance of the two\ninvolved spins. However, we will see that the more indi-\nrect a coupling is, the more time a correlation needs to\nbuild up between the corresponding spins. To capture\nthe FID at short and moderate times, it is a promising\nstrategy to start with the pair correlations of close spins.\nIn Tab. I, we list and number pair correlations, sorted\nfirst by the distance between the pair of spins, and sec-\nond, if the distances are equal, by the coupling between\nthe spins.\nNotethatitisnotpossibletocomputepaircorrelations\nin spinDMFT, since they are excluded from the approach\nby construction. The extension, CspinDMFT, exactly\nsimulates a small number of spins, the so-called cluster,\nand regards the remainder of the lattice by mean-fields\n[19]. CspinDMFT can access these pair correlations, but\neach pair correlation would require an adapted cluster\nwhich would make the calculation numerically very ex-\npensive. In addition, the self-consistency in each cluster\nwould yield autocorrelations which differ from cluster to\ncluster. This is not systematic. Thus we adopt the fol-\nlowing two-step strategy instead.\nB. Overview of nl-spinDMFT\nIn step (1), we estimate the autocorrelations by a sin-\ngle self-consistent computation with spinDMFT. This is\nwelljustifiedforlarge zeff. Instep(2), weperformsimula-\ntions of a quantum cluster in a mean-field background toc⃗ r⊤\nc/asc|⃗ rc|/asc|Jc|/J0mc\n0 1\n1(0 0 -1) 1.0 2.02\n2(-1 0 0) 1.0 1.04\n3(-1 -1 0) 1.4142 0.35354\n4(-1 0 -1) 1.4142 0.17688\n5(-1 -1 -1) 1.732 0.08\n6(0 0 -2) 2.0 0.252\n7(-2 0 0) 2.0 0.1254\n8(-1 0 -2) 2.2361 0.12528\n9(-2 -1 0) 2.2361 0.08948\n10(-2 0 -1) 2.2361 0.03588\n11(-1 -1 -2) 2.4495 0.0688\n12(-2 -1 -1) 2.4495 0.03416\n13(-2 -2 0) 2.8284 0.04424\n14(-2 0 -2) 2.8284 0.02218\n15(0 0 -3) 3.0 0.07412\n16(-3 0 0) 3.0 0.0374\n17(-2 -2 -1) 3.0 0.02478\n18(-2 -1 -2) 3.0 0.012416\n19(-1 0 -3) 3.1623 0.05388\nTABLE I. List of correlations considered for the FID of cal-\ncium fluoride in [100]-direction ( z). The first entry ( c= 0) is\nthe autocorrelation and the other entries correspond to pair\ncorrelations. ⃗ rcis an exemplary distance vector between the\ntwo spins involved in the pair correlation and |Jc|/J0is the\nabsolute of their direct coupling with J0:=µ0γ2\nF/(4πa3\nsc). By\nmcwedenotethemultiplicityofthecorrelation, i.e., thenum-\nber how often this correlation occurs per spin.\ndetermine each pair correlation listed in Tab. I. Thereby,\nwe use the previously obtained autocorrelations as build-\ningblockstodeterminetherequiredmean-fieldmoments.\nThe simulations of step (2) are similar to CspinDMFT,\nbut we do not require the self-consistency anymore be-\ncause the mean-field moments are already well known\nfrom the upstream self-consistency in step (1). At the\nsame time, we are flexible to choose the quantum clus-\nter individually for each simulation in step (2). Hence,\nthe choice of the cluster can be optimized with respect\nto the pair correlation of interest. Finally, the FID is\nobtained by superposing all correlations obtained in the\nsecond step. Since the pair correlations and the FID are\nnon-local quantities, we refer to the procedure in second\nstep as non-local spinDMFT (nl-spinDMFT). The whole\nstrategy is summarized in Fig. 3. We provide more de-\ntails in the following.\nThe self-consistent calculation required for the first\nstep is discussed in Sec. IIC (spinDMFT). If the coor-\ndination number is high, spinDMFT is sufficient, but for\nexample in dimerized systems, one should simulate at\nleast a single dimer of spins using CspinDMFT. For cal-\ncium fluoride with magnetic field in [100]-direction, we\nhave z[100]\neff = 4.9, which is sufficiently large for using\nonly spinDMFT.\nIn the second step, we perform several simulations of a\nquantum cluster in a mean-field background. The clus-\nter is henceforth denoted by Γl, where lis the number\nof spins. The spins in the cluster interact quantum-6\nspin\nautocorrelations2nd mean-field\nmoments(1)\nself-consistency\n(spinDMFT)\n(2)\nnl-spinDMFT\nspin auto- and pair correlations2nd mean-field moments...\nc= 0\n c= 1\n c= 2\n...\nFree induction decay\nFIG. 3. Strategy to compute spin correlations and finally\nthe FID using dynamic mean-field theory. The dashed ar-\nrows correspond to simple, numerically cheap superpositions.\nThe solid arrows correspond to numerically more demanding\nMonte-Carlo simulations which are required to compute ex-\npectation values.\nmechanically with one another and we regard the re-\nmainder of the lattice by coupling each of the spins to\na dynamic Gaussian mean-field ⃗Vi. This is summarized\nby the Hamiltonian\nHmf\nΓl(t) =1\n2X\ni,j∈ΓlJij⃗S⊤\niD⃗Sj+X\ni∈Γl⃗Si·⃗Vi(t).(23)\nAs mentioned before, the second moments of the mean-\nfields are determined from the autocorrelations obtained\nin the first step. Generally, the second moments read\nVα\ni(t)Vα\nj(0)mf=D2\nααX\nk,m/∈ΓlJikJjm⟨Sα\nk(t)Sα\nm(0)⟩.(24)\nUsing translational invariance and the sorting scheme\nfrom Tab. I, this can be rewritten as\nVα\ni(t)Vα\nj(0)mf=D2\nαα∞X\nc=0J2\nc,ijGαα\nc(t),(25a)\nJ2\nc,ij:=X\nk,m/∈Γl\n⃗ rk−⃗ rm!∈{⃗ rc}JikJjl, (25b)\nwhere Gxx\ncis the pair correlation of type cand\nGxx\n0=Gxxis the autocorrelation. By {⃗ rc}we de-\nnote the set of distance vectors ⃗ rcthat correspond to thecorrelation with index c. The number of these distance\nvectors per site corresponds to the multiplicity mcwhich\nis also provided in Tab. I. For example, the set for c= 1\ncontains ⃗ rc=(0 0−1)⊤and⃗ rc=(0 0 1)⊤, which entails\nmc= 2. Of course, we can only insert those correlations\nthat have been computed in the first step, i.e., the auto-\ncorrelations, so that the summation in Eq. (25) reduces\nto\nVα\ni(t)Vα\nj(0)mf≈D2\nααJ2\n0,ij⟨Sα(t)Sα(0)⟩.(26)\nNote that this approximation is also justified by the co-\nordination number. For example, for i=j, the approxi-\nmation in Eq. (26) follows directly from Eq. (15).\nThe expression in Eq. (26) can be straightforwardly\ncomputedafterobtainingtheautocorrelationsinstep(1).\nThe resulting second mean-field moments are then fed\ninto the simulations of a spin cluster Γlin a mean-field\nbackground. For calcium fluoride, we consider cluster\nsizes of up to l= 9spins and access all pair correlations\nlisted in Tab. I. Next, we discuss how one can optimize\nthe choice of clusters for the computation of the pair\ncorrelations.\nC. Choice of the clusters\nClearly, for a pair correlation the two directly involved\nspins must be included in Γl. For a reasonable choice\nof the other spins, one has to consider that two spins\ncan become correlated indirectly, for example, via their\ncoupling to a third spin. Such processes can be only ac-\ncounted for, if the third spin is included in Γl, since back-\nactionsfromtheclusterontothemeanfieldsareexcluded\nby construction. For an optimal choice of the cluster, the\nmost important indirect couplings have to be identified.\nA selection criterion for the spins to be put into the\ncluster can be developed, for example, on the basis of\nthe fourth moment of the coupling constants\nJ4\nk,ij:=J2\nikJ2\nkj. (27)\nHere, iandjdenote the pair of spins of the pair correla-\ntion in question and kis some arbitrary third spin. The\ncoupling constant connects spin iandjvia two links,\nwhich captures an indirect process correlating the two\nspins with each other. We stress, however, that indirect\nprocessesinvolvingthreeormorelinkscanalsobeimpor-\ntant. Respecting several degrees of indirect correlations\nsimultaneously in a criterion forms a challenging task.\nFormally, we define an orderas the number of links\nbetweensite iandj. Weproposeastrategythataddsthe\npresumably most important spins order by order to the\ncluster and start with the lowest relevant order, n= 2.\nThe first order, n= 1, is automatically accounted for,\nsince we always include the spins iandjin the cluster.\nDipolar couplings are long-range so that, in principle,\nthere is an infinite number of processes for n≥2. To\ndevelop an appropriate criterion, we consider only the7\nLstrongest absolute couplings of a spin in the lattice.\nFor example, for calium fluoride with magnetic field in\n[100]-direction a plausible choice would be L= 6, which\nincludes only the nearest-neighbor couplings. We suggest\nthe following procedure for the n-th order:\n1. Determineallpathsfromsite itosite jwith nlinks.\nOnlythoselinksareallowed,thatcorrespondtoone\nof the Lstrongest couplings.\n2. Collect the sites of all paths in a set K. Add all\nspins in Kto the cluster, if this does not exceed\nthe desired cluster size l, and proceed with the next\norder. If adding all spins would exceed l, consider\nthe next step instead.\n3. Let fbe the number of spins that are yet missing\nin the cluster. Find all possibilities to distribute f\nspins on the sites in K. For each possibility, com-\npute\nJ2n\nij,n(Γl):=X\nk1,k2,...,k n−1∈ΓlJ2\nik1J2\nk1k2. . . J2\nkn−1kj.(28)\nHere, kican be any spin in the cluster with the f\nspins added (total size is l) and we consider only\nthe most important couplings. Accept the cluster\nΓlwith the highest value for J2(n+1)\nij,nand stop the\nprocedure.\nWe consider this strategy to be a good compromise be-\ntween lower and higher orders. Aside from the cluster\nsizelwhich is set by the available numerical resources,\nthe only free parameter is the number of considered cou-\nplings L.\nCarrying this procedure out for L= 6, we arrive at\nthe 20 clusters shown in Fig. 4. Note that the procedure\nhas also been used for determining the optimal cluster\nfor the autocorrelation c= 0(spin i=j).\nD. Results for calcium fluoride\nTo obtain the actual contribution of a pair correlation\nto the FID F(t), one has to multiply it with its multi-\nplicity mcin the lattice so that\nF(t) =∞X\nc=0mcGxx\nc(t). (29)\nIn practice, the sum needs to be truncated. We de-\nnote the corresponding truncation value by cmax. Theresults of the simulated FID for ascending cmaxare plot-\nted in Fig. 5 in comparison with experimental data from\nRef. [20]. In addition to the FID, we plot\nMz\ncmax(t) =cmaxX\nc=0mcGzz\nc(t), (30)\nwhich is exactly constant for cmax =∞, since the\nlongitudinal magnetization Mzis a conserved quantity.\nHence, the deviation of Mz(t)from 1is an excellent mea-\nsure for the good accuracy of the approximation in Fig. 5\n(e).\n0123\n4567\nz\ny x891011\n12131415\n16171819\nFIG. 4. Considered clusters in the simple cubic lattice for\nCaF 2for magnetic field in [100]-direction ( z) according to the\npresented strategy in the text. The numbering is adapted\nfrom Tab. I. The orange lines represent the pair correlations\naccessed by the corresponding cluster.8\n0.00.51.0F(a)\ncmax= 0(b)\ncmax= 1\n0 25 50 75\nt(units ofµs)0.00.51.0F(c)\ncmax= 3\n0 25 50 75\nt(units ofµs)(d)\ncmax= 5\n0 20 40 60 80\nt(units ofµs)−0.20.00.20.40.60.81.0\n(e)\ncmax= 19FID, experiment\nFID, nl-spinDMFT\nMz, nl-spinDMFT\nFIG. 5. Comparison of the FID in [100]-direction simulated with nl-spinDMFT with the experimental one measured in Ref. [20].\nThe clusters considered for the simulation are depicted in Fig. 4; they comprise l= 9sites and were chosen considering the\nL= 6strongest couplings. The different panels show the improved agreement as cmaxis increased, see Eq. (29) for clarity.\nIn addition to the FID, we plot the longitudinal magnetization from nl-spinDMFT, see Eq. (30). This quantity is rigorously\nconserved in the absence of any approximations. The experimental data were measured at 4.2 K. The lattice constant for the\nflourine ions at this temperature is about asc= 2.724Å.\nAs we increase the truncation cmax, the agreement\nbetween nl-spinDMFT and experiment improves as ex-\npected. Adding the contributions from 19 pair corre-\nlations leads to a remarkable agreement up to about\nt= 40 µs. Beyond this time, the amplitude of the os-\ncillation is slightly overestimated by nl-spinDMFT. We\ncan rule out the statistical error of the Monte-Carlo sim-\nulation as well as the error induced by time discretization\nas origin for the deviation, since they are more than one\norder of magnitude smaller. A probable reason for the\ndeviation is that the considered quantum clusters are not\nlarge enough to capture long-range pair correlations reli-\nably. The latter result from indirect processes involving\nmany spins and become, thus, important at later times.\nTo illustrate this, we present the individual transverse\nand longitudinal pair correlations in Figs. 6 and 7. Some\nof them are very close to zero and hence not shown. The\ntransverse correlations are mostly characterized by a sin-\ngle peak, which can be positive or negative. Depending\non the peak position and its sign we can group them phe-\nnomenologically into four classes on the considered time\ninterval. The longitudinal pair correlations on the other\nhand are all positive. Most of them increase monotoni-\ncally over the considered time interval. No. 1 and 2 rise\nthe fastest and slowly decay after their maximum value\nat about t= 40 µs, see panel (a) in Fig. 7. Note that we\napplied the same classification scheme as for the trans-\nverse pair correlations. The speed of the increase of the\nlongitudinal pair correlations decreases with c, i.e., with\nthe pair distance.\nHenceforth, we focus on the transverse pair correla-\ntions as they matter for the FID. The first peak is dueto 1st-order processes, i.e., due to the direct coupling be-\ntween the pair of spins of the corresponding pair corre-\nlation. We can verify this by analytically computing the\nexact pair correlations, while setting all couplings except\nthe direct one to zero yielding\nGxx\ndimer(t) =−sin\u0012Jt\n2\u0013\nsin\u0012Jt\n4\u0013\n,(31a)\nGzz\ndimer(t) = sin2\u0012Jt\n4\u0013\n, (31b)\nwhich is depicted by the dashed lines in Figs. 6 and 7 in\npanel (a), respectively. The agreement of these formulas\nwith the pair correlations of nl-spinDMFT is excellent at\nshorttimeswhichindicatesthatthedirectcouplingisthe\norigin of the first peak in the transverse correlations. The\nother peaks are likely to result from indirect couplings.\nOne can presume that the peak of the nth class results\nfrom processes of nth order according to the definition in\nSec. IIIC, that is, from processes with nlinks. This also\nexplains the alternating signs of the peaks: If the direct\ncoupling between two spins leads to anti-correlation ( −)\nat short times, the indirect coupling via two links leads\nto correlation ( +), the indirect coupling via three links\nagain to anti-correlation and so on. Note that, generally,\nweexpectall orderstoberelevantforallpaircorrelations\nto some extent. This can be seen well in pair correlation\nNo. 5 in panel (c), in which both the second and third\npeak are clearly visible. However, the obtained tendency\nis that the larger the distance between a pair of spins,\nthe higher is the order of the processes inducing the rele-\nvant peak in the pair correlation. This entails that more-\ndistant pair correlations tend to involve more links and,9\n−0.2−0.10.00.1mcGxx\nc(a)\n1\n2\n3 0.000.050.100.15 (b) 4\n6\n7\n0 50\nt(units ofµs)−0.06−0.04−0.020.000.02mcGxx\nc(c)\n5\n8\n10\n15\n0 50\nt(units ofµs)0.000.02(d) 11\n14\n19\nFIG. 6. Contributions of the transverse pair correlations com-\nputed by nl-spinDMFT. The numbering corresponds to the\none in Tab. I and Fig. 4. Some pair correlations are not\nshown as they are very close to zero (absolute value smaller\nthan 0.002on the whole interval). The orange bars corre-\nspond to the approximate peak positions which are briefly\ndiscussed in the main text.\nthus, tendtopopupatlatertimes. InApp.A,wediscuss\nthe convergence of the correlations upon increasing the\ncluster size l. We find that short-range pair correlations\nconverge much quicker than long-range pair correlations.\nThis is plausible considering the fact that the order of\nthe involved processes increases with the pair distance.\nFig. 5 could give the impression that the mean-field\napproachcorrespondstoashort-timeapproximation, but\nthis is not a valid view. In fact, nl-spinDMFT is a short-\nrange approximation and range and time correspond to\neach other in the FID. We emphasize that short-ranged\nquantities such as the autocorrelations are well captured\nby the approach even at long times . This is underlined\nby the consideration in App. A.\nThere are several ways to improve the accuracy of the\nsimulated FID at later times. An obvious step is to in-\ncrease the cluster size further; one or two more spins are\nrealistic. Besides this, one has to consider that the prime\nreason for inaccuracies in long-range pair correlations is\nthe lack of back actions in the mean-field approach. In\nclassical or hybrid quantum-classical approaches, such\nbackactionsaretreatedatleastonaclassicallevel[8–10].\nThe advantage of the expansion in Eq. (29) is that the\nsummed up quantities can all be computed separately. A\npromising strategy for very accurate FIDs could be to es-\ntimate any correlations below a properly chosen cmaxby\nnl-spinDMFT and to compute correlations for c > cmax\nby a fully classical or by a hybrid approach. We leave\nthese ideas for future research.\nIn App. B, we also compare nl-spinDMFT with the ex-\nperimentaldatafortheFIDsin[110]-and[111]-direction.\nThe observations are qualitatively the same as above.\nThe nl-spinDMFT is precise at short times and performs\n0.00.10.20.3mcGzz\nc(a)\n1\n2\n3 0.00.10.20.3\n(b) 4\n6\n7\n0 50\nt(units ofµs)0.000.05mcGzz\nc(c) 5\n8\n9\n10\n15\n0 50\nt(units ofµs)0.000.010.02(d) 11\n12\n14\n19FIG. 7. Same as Fig. 6, but for the longitudinal pair correla-\ntions.\nless well at later times, since long-range effects are not\naccounted for accurately. In fact, the deviations at later\ntimesareconsiderablyhigherthanforthe[100]-direction.\nWe assign this to the fact, that the effective coordination\nnumber is much larger for the [110]- and [111]-direction,\nz[110]\neff= 9.2andz[111]\neff= 22 .5. This may sound counter\nintuitive, because the mean-field approximation is justi-\nfied by a1/z-expansion. However, accessing pair corre-\nlations requires to simulate compact clusters, in which\nthe most important intermediate spins inducing indirect\ncorrelations are included. The larger the coordination\nnumber z, the larger is the number of important inter-\nmediate spins which complicates the choice of compact\nclusters enormously. If zis lower, the mean-field ap-\nproximation is, in principle, less well justified. However,\nthe agreement of the results with the experimental FID\nin [100]-direction together with the convergence plots in\nApp. A suggest that even for a smaller effective coordina-\ntion number, z[100]\neff= 4.9, the local dynamics is captured\naccurately by the mean-field approach. Therefore, we be-\nlieve that nl-spinDMFT can be a very powerful tool for\nthe computation of FIDs in systems with moderate co-\nordination numbers, for example, in dimerized systems\nsuch as crystalline silicon [9] or in two-dimensional sys-\ntems, see for example Ref. [19]. To access systems with\nveryhighcoordinationnumbers, onemaycombineitwith\napproaches that capture back actions better.\nIV. MICROSCOPIC PROTON SPIN DYNAMICS\nIN ADAMANTANE\nIn the previous section, we demonstrated how\nspinDMFT and its extensions can be employed to simu-\nlate the FID. To this end, we computed individual micro-\nscopic pair correlations by simulating clusters of 9 spins\nin a mean-field background. An important ingredient10\nwas the choice of compact clusters on the lattice. Such\na choice is not possible for adamantane, where the mo-\ntional averaging leads to exceptionally high coordination\nnumbers in the leading non-vanishing couplings to ad-\njacent molecules and beyond. In this scenario, at least\ntwo or three molecules would need to be included for\na reliable choice of clusters. This would entail simulat-\ning more than 30 proton spins which is numerically not\nfeasible. We conclude on the one hand that the proton\nFID of adamantane cannot be simulated by the described\nprocedure. On the other hand, the mean-field approxi-\nmation itself is excellently justified for the nuclear spins\nin adamantane, even more than for CaF 2because of the\nlarge number of interaction partners, i.e., large values\nofz. For this reason, we expect the autocorrelations of\nthe proton spins to be captured very well by spinDMFT.\nComputing them requires to perform a powder average,\nwhich is discussed in the following.\nToapplyspinDMFT,werequirethequadraticcoupling\nconstant for the proton spins. For a given direction ⃗ nB\nof the magnetic field, it can be computed by\nJ2\nQ,H(⃗ nB) =Nmol\nHX\nm>1\u0010\nJ(1H),(mH)ma(⃗ nB)\u00112\n,(32)\nwith the motional-averaged couplings provided in\nEq. (8b) and Nmol\nH= 16being the number of hydro-\ngen nuclei per molecule. To compute JQ,Hnumerically,\nwe rewrite the molecule sum as a double sum and define\na cutoff smax,\nJ2\nQ,H(⃗ nB)≈Nmol\nHsmaxX\ns=1X\nm\n|⃗ a1,m|=ds\u0010\nJ(1H),(mH)ma(⃗ nB)\u00112\n.(33)\nThe first sum runs over all “shells” of molecules around\nthe considered central molecule in the fcc lattice. The\nshells are defined by their distance to the central\nmolecule. Here, smaxis the index of the last shell that is\nincluded in the numerical calculation. The second sum\nruns over all molecules within the same shell, that means,\nwith the same distance dsto the central molecule. The\nlattice distances dsand the number of molecules per shell\nin the fcc lattice are extracted from Ref. [32]. For an\nanalytical estimate of the contributions from shells with\ns > smax, we assume for simplicity each nucleus to be at\nthe center of its molecule and introduce the continuum\nlimit to render the molecule sum tractable. Both approx-\nimations become more accurate as smaxis increased. We\nobtain\nJ2\nQ,H,rest≈Nmol\nHn0Z∞\ndsmax+1da a2Z\ndΩJ2(a⃗ n(Ω),⃗ nB)(34a)\n=64πNmol\nH\n15a3\nfcc\u0012µ0γ2\nH\n4π\u00132\n(dsmax+1)−3,(34b)\nwhere ⃗ n(Ω) = ⃗ n(θ, ϕ):=(sinϑcosφ,sinϑsinφ,cosϑ)⊤and\nn0:= 4/a3\nfccis the density of molecules. Note that dueto the continuum limit the lattice becomes completely\nisotropic so that the result is independent of ⃗ nB.\nThe coupling J(1H),(mH)mahas to be evaluated for each\nmolecule min each shell sfor different orientations ⃗ nB\nof the magnetic field. We compute the four-dimensional\nangular integrals in Eq. (8b) by Monte Carlo integra-\ntion. In total, we consider the first eight shells around\nthe central molecule, i.e. smax= 8, and approximate\ncontributions from higher shells using Eq. (34b). In\nspinDMFT, the orientation of the magnetic field itself\nis not important, but the resulting quadratic coupling\nJQis the key parameter. Therefore, we are not inter-\nested in the distribution of JQ,Hdepending of ⃗ nB, but\nrather in the distribution ρ(JQ,H), which is displayed in\nFig. 8. The powder-averaged value from our simulation\nisJQ,Hpa= 35 .038(7) kHz which agrees very well with\nthe experimentally obtained value, JQ,Hpa≈34 kHz\n[33] (Ref. [33] provides the second moment which dif-\nfers from the quadratic coupling constant by a factor\nofp\n3I(I+ 1)/4 =3/4). However, as ρ(JQ,H)is not\nstrongly peaked, but spreads over an interval of about\n8 kHz, thisaveragevalueisnotcompletelyrepresentative.\nApplying the powder average directly to the quadratic\ncoupling constant, that is, beforethe spinDMFT simu-\nlation is not fully justified. Instead, we should produce\nspinDMFT results for different values of JQ,Hand super-\npose the results according to the distribution ρ(JQ,H).\nNotethatthisdoesnotrequireadditionaleffort, sincethe\nsimulation results from spinDMFT, see Fig. 2, can sim-\nply be rescaled in order to correspond to a different JQ,H.\nThe powder-averaged autocorrelations of spinDMFT ob-\ntained in this way are plotted in Fig. 9 together with\nthe experimentally measured FID. It turns out that the\npowder average does not affect the shapes of the auto-\ncorrelations much.\nIn contrast to the FID, the transverse autocorrelation\ndoes not contain any oscillations. This is the same phe-\nnomenon as in calcium fluoride. The pure autocorrela-\ntion, see panel (a) of Fig. 5, does not oscillate, but after\nadding pair correlations the estimated FID contains min-\nima and maxima, see panel (e). As explained above, we\ncannot access the pair correlations in adamantane reli-\nably, since the required cluster sizes would be too large.\nHence, there is no direct comparison between spinDMFT\nand experiment concerning the proton spin dynamics.\nBut the situation is much more favorable for the dynam-\nics of the nuclear spins of13C, which will be considered\nin the next section.\nV. SIMULATING CARBON NMR SIGNALS IN\nADAMANTANE\nSince the13C-spins are rare, their dynamics is com-\npletely dominated by the interaction with the proton\nspins. The latter occur in much higher numbers and\ntheir gyromagnetic ratio is larger so that we can safely\nneglect any back action from the carbon spins onto the11\n30 32 34 36 38\nγH\nγiJQ,i(units of kHz)0.000.050.100.15ρH\nC\nFIG. 8. Distribution of the quadratic coupling constant\nγH/γiJQ,ifori∈ {H,C}. The dashed vertical lines corre-\nspond to the powder-average JQ,ipa(the distribution and av-\nerage for hydrogen and carbon is barely distinguishable). ρ\nis determined by organizing the resulting values for JQ,ifor\ndifferent field directions ⃗ nBinto bins. The horizontal error\nbars correspond to the bin widths. The vertical error bars\nare determined from the error bars of JQ,iwhich result from\nthe Monte Carlo sampling.\n0.0 0.1 0.2 0.3 0.4 0.5 0.6\nt(units of ms)0.00.20.40.60.81.0GααFID, experiment\nGxxpa\nGzzpa\nFIG. 9. Powder-averaged proton autocorrelations for\nadamantane computed by spinDMFT in comparison with the\nproton FID measured in Ref. [23]. The agreement is only ap-\nproximate because the pair correlations are missing.\nproton spins. In this case, a substitution of the proton\nfield in Eq. (7) by a Gaussian mean-field is very justified.\nThis leads to a simple spin-noise model according to the\nHamiltonian\nHC(t) =IzVz\nC(t). (35)\nThe mean-field has zero average and its second moments\nare given by\nVz\nC(t)Vz\nC(0)mf=J2\nQ,C⟨Sz(t)Sz(0)⟩(36)\nwith the longitudinal spin autocorrelation provided in\nFig. 2. Similar to Eq. (32), the quadratic coupling con-stant is given by\nJ2\nQ,C(⃗ nB) =Nmol\nHX\nm>1\u0010\nJ(1C),(mH)ma(⃗ nB)\u00112\n.(37)\nNote that we do not consider the powder average in\nEq. (36). Within each crystal, the carbon spins couple to\nan individual spin environment with fixed JQ,Cand fixed\nproton autocorrelation. Therefore, one has to compute\nany signals of the carbon spins individually for each crys-\ntal or, equivalently, for each JQ,Cand the corresponding\nproton autocorrelation. Averaging these signals leads to\nthe appropriate powder-average measured in experiment.\nUsing the same procedure as in the previous section,\nwe can estimate JQ,Cby Monte-Carlo integration. In\nour calculations it turns out that, although the carbon\nspins sit on a much smaller radius in the adamantane\nmolecule, their quadratic coupling constant is always ex-\nactly the same as for the proton spins except for a factor\nrγ:=γC/γH≈0.251due to the different gyromagnetic\nratios. This is true for any considered field direction ⃗ nB.\nTo underline this observation, we included the density\nρ(rγJQ,C)in Fig. 8, which is remarkably close to ρ(JQ,H).\nWe compute the transverse autocorrelation gener-\nated by the Hamiltonian in Eq. (35) for different val-\nues of JQ,Cand superpose the results according to\nρ(JQ,C)≈ρ(r−1\nγJQ,H)to obtain the powder average.\nThe resulting signal is shown in Fig. 10 together with\nthe experimental result for the FID. Note that since the\n13C-atoms have a low abundance, two individual carbon\nspinsseeatotallydifferentenvironment. Hence, paircor-\nrelations are not relevant for the corresponding FID and\nthe transverse autocorrelation of carbon should, in prin-\nciple, be equivalent to the FID. However, as it turns out,\nthe result from the noise model deviates a bit from the\nexperiment at moderate and large times. For a better\nunderstanding of this observation, we consider the spin\necho signal in the following.\nTo simulate a spin Hahn echo [34] within the spin noise\nmodel, we include an instantaneous pulse according to\nHpulse=πδ(t−τ/2)Iy(38)\nand measure the result of the transverse autocorrelation\natt=τ. The result of this is plotted versus τin Fig. 11.\nHere, the agreement between theory and experiment is\nvery good. We conclude that the dynamic noise result-\ning from the proton spins as captured by spinDMFT is\nperfectly accurate. Since a spin echo eliminates static\nnoise we attribute the discrepancy in the FID to some\nsmall additional static noise.\nTo underline this interpretation, we redo the computa-\ntion of the FID and include a weak static Gaussian noise.\nThis is done by adding a constant term W2to the second\nmean-field moment in Eq. (36), where Wis the standard\ndeviation of the noise. The best fit to the experimental\ndata is obtained for W= 0.29(6) kHz and plotted in or-\nange in Fig. 10. The agreement for the adapted model is\nremarkable and supports our assumption. We stress that12\n0.00 0.25 0.50 0.75 1.00 1.25 1.50\nt(units of ms)0.00.20.40.60.81.0Gxx\nexperiment\nspinDMFT\nspinDMFT + noise\nFIG. 10. Powder-averaged transverse autocorrelation for13C\nin adamantane from the spin-noise model in Eq. (35). Results\nwithout and with an additional static noise source are shown.\nThe carbon FID measured in Ref. [23] is plotted with black\ncircles.\n0.0 0.5 1.0 1.5\nt(units of ms)0.00.20.40.60.81.0Gxx\nexperiment\nspinDMFT\nFIG. 11. Powder-averaged spin echo for13Cin adamantane\nfrom the spin-noise model in Eq. (35). The experimental spin\necho measured in Ref. [23] is plotted with black circles.\nthe spin echo exactly reverts the effect of the noise in the\nconsidered model so that the result in Fig. 11 does not\nchange at all.\nIn addition, we analyzed how such an additional noise\nsource would affect the proton spin dynamics. To this\nend, we added the noise, amplified by the factor1/rγ, in\nthe spinDMFT simulation finding no visible difference in\nthe resulting autocorrelations relative to the ones shown\nin Fig. 9. The homonuclearly coupled proton spins are\nmuch more robust against disturbing fields. A candidate\nfor the origin of the inhomogeneous broadening is the\nchemical shift anisotropy. It is anisotropic in solids and\nbroadens the NMR spectra of powders [1].VI. CONCLUSION\nWe applied a recently developed dynamic mean-field\ntheoryforspins(spinDMFT)[18]toNMRexperimentsin\nordertoestablishitasaversatileandaccuratesimulation\ntool enabling quantitative understanding.\nInthefirstpart, weextendedspinDMFTtoaccessfree-\ninduction decays in homonuclear spin systems. This ap-\nproach is called non-local spinDMFT (nl-spinDMFT) be-\ncause it involves calculating non-local quantities that are\nnot directly accessible in spinDMFT. The nl-spinDMFT\nusesa prioricomputed mean-fields to efficiently simulate\na quantum cluster of about 10 spins in a mean-field back-\nground. The simulation allows us to compute pair cor-\nrelations, which contribute to the FID. This gives us an\nunderstanding of how the macroscopic FID results from\nmicroscopic spin dynamics. The extrema in the FID can\nbe traced back order by order to processes involving n\nspin-spin couplings.\nThe agreement of the method with the experimental\nFID in CaF 2[20] is excellent for the [100]-direction at\nshortandmoderatetimes. Smalldeviationsatlatertimes\nare found to be due to less well captured long-range pair\ncorrelations, which become relevant at later times. These\ndeviations are larger for the [110]- and [111]-direction\nbecause selecting compact clusters with at maximum 9\nspinsbecomesmoredifficultforhighercoordinationnum-\nbers. In these cases, it may be promising to combine\nthe approach with purely classical or hybrid quantum-\nclassical simulations [8–10]. Our simulations suggest that\nthe choice of the cluster is of great importance. Differ-\nent pair correlations are accessed by different clusters in\nnl-spinDMFT. For this reason, we propose to improve\nthe hybrid approach presented in Ref. [8, 9] by com-\nputing pair correlations separately considering individual\nadapted clusters.\nIn spin systems with moderate coordination numbers,\nfor example in two-dimensional systems, we expect an\nexcellent performance of nl-spinDMFT. Apart from pair\ncorrelations, nl-spinDMFT easily allows to compute ar-\nbitrary quantities on the quantum cluster, which are of\ninterest, for example, for experiments measuring multi-\nspin correlations [31].\nIn the second part, we demonstrated the versatility of\nthemean-fieldframeworkbyconsideringthespindynam-\nics in polycrystalline adamantane [23], where motional\nand powder averaging has to be taken into account. Us-\ning spinDMFT, we estimated the proton-spin autocorre-\nlations and used them to calculate the FID and the spin\necho of the rare13Cnuclear spins. While there is some\ndeviation between the numerical and experimental car-\nbon FID, the carbon spin echo is captured very well by\nthe simulation. This suggests the presence of inhomo-\ngeneous broadening in the experiment. We account for\nthis in an additional simulation by adding a weak static\nnoise, which leads to an excellent agreement with the\nexperimental FID.\nWe consider the mean-field framework described by13\nspinDMFT and nl-spinDMFT to be very versatile due to\nthe microscopic nature of the effective mean-field models.\nFor example, spins with S >1/2including quadrupolar\ninteractions can be easily considered in the simulations.\nExplicittimedependenciescanbeincludedinspinDMFT\nand its extensions without increasing the numerical effort\nmuch. This has been demonstrated for the spin echo in\nadamantane, but more complicated pulse sequences and\neven pulse imperfections can be considered as well.\nFurthermore, we point out that the nl-spinDMFT can\nbe enhanced by using CspinDMFT [19] in the first step\ninstead of spinDMFT. The second step remains essen-\ntially the same. This straigtfoward extension is indicated\nif there is are strong dominating spin interactions which\ncannot be treated as noise, for instance if dimers of spins\nwith strong intradimer, but weaker interdimer couplingneed to be dealt with. Also larger cluster are conceivable\nas an extension.\nIn conclusion, we strongly advocate the use of\nspinDMFT and its extensions for the prediction and un-\nderstanding of NMR signals.\nACKNOWLEDGMENTS\nWe acknowledge funding by the Deutsche Forschungs-\ngemeinschaft (DFG) in project UH90/14-1 as well as\nthe DFG for Basic Research in TRR 160. We also\nthank the TU Dortmund University for providing com-\nputing resources on the High Performance Computing\ncluster LiDO3, partially funded by the DFG in project\n271512359. We are grateful for the funding received by\nT.H. from the DAAD RISE Germany program.\n[1] M.H.Levitt, Spin Dynamics, Basics of Nuclear Magnetic\nResonance (John Wiley & Sons, Ltd, Chichester, 2005).\n[2] H. Tal-Ezer and R. Kosloff, An accurate and efficient\nscheme for propagating the time dependent Schrödinger\nequation, Journal of Chemical Physics 81, 3967 (1984).\n[3] A. Weiße, G. Wellein, A. Alvermann, and H. Fehske, The\nkernel polynomial method, Review of Modern Physics\n78, 275 (2006).\n[4] J. H. V. Vleck, The Dipolar Broadening of Magnetic\nResonance Lines in Crystals, Physical Review 74, 1168\n(1948).\n[5] I. J. Lowe and R. E. Norberg, Free-induction decays in\nsolids, Phys. Rev. 107, 46 (1957).\n[6] G. W. Parker and F. Lado, Calculation of NMR line\nshapes in calcium fluoride from modified moment expan-\nsions, Physical Review B 9, 22 (1974).\n[7] B. V. Fine, NMR Spin-Spin Relaxation as Kinetics in\nSpin Phase Space, Physical Review Letters 79, 4673\n(1997).\n[8] G. A. Starkov and B. V. Fine, Hybrid quantum-classical\nmethod for simulating high-temperature dynamics of\nnuclear spins in solids, Physical Review B 98, 214421\n(2018).\n[9] G. A. Starkov and B. V. Fine, Free induction decays in\nnuclear spin-1\n2lattices with a small number of interacting\nneighbors: The cases of silicon and fluorapatite, Physical\nReview B 101, 024428 (2020).\n[10] T. A. Elsayed and B. V. Fine, Effectiveness of classical\nspin simulations for describing NMR relaxation of quan-\ntum spins, Physical Review B 91, 094424 (2015).\n[11] M. M. Glazov, Electron & Nuclear Spin Dynamics in\nSemiconductor Nanostructures (Oxford University Press,\n2018).\n[12] B. V. Fine, Long-time behavior of spin echo, Physical\nReview Letters 94, 247601 (2005).\n[13] W. M. Witzel, R. de Sousa, and S. Das Sarma, Quantum\ntheory of spectral-diffusion-induced electron spin deco-\nherence, Physical Review B 72, 161306(R) (2005).\n[14] W. M. Witzel and S. Das Sarma, Quantum theory for\nelectron spin decoherence induced by nuclear spin dy-\nnamics in semiconductor quantum computer architec-tures: Spectral diffusion of localized electron spins in the\nnuclear solid-state environment, Physical Review B 74,\n035322 (2006).\n[15] A. Georges, G. Kotliar, W. Krauth, and M. J. Rozen-\nberg, Dynamical mean-field theory of strongly correlated\nfermion systems and the limit of infinite dimensions, Re-\nview of Modern Physics 68, 13 (1996).\n[16] G. Kotliar, S. Y. Savrasov, G. Pálsson, and G. Biroli,\nCellular dynamical mean field approach to strongly cor-\nrelated systems, Physical Review Letters 87, 186401\n(2001).\n[17] P. Sun and G. Kotliar, Extended dynamical mean-field\ntheory and GWmethod, Physical Review B 66, 085120\n(2002).\n[18] T.Gräßer, P.Bleicker, D.-B.Hering, M.Yarmohammadi,\nand G. S. Uhrig, Dynamic mean-field theory for dense\nspin systems at infinite temperature, Physical Review\nResearch3, 043168 (2021).\n[19] T.Gräßer, K.Rezai, A.O.Sushkov,andG.S.Uhrig,Un-\nderstanding the dynamics of randomly positioned dipo-\nlar spin ensembles, Physical Review Research 5, 043191\n(2023).\n[20] M. Engelsberg and I. J. Lowe, Free-inducation-decay\nmeasurements and determination of moments in CaF 2,\nPhysical Review B 10, 822 (1974).\n[21] G. S. Uhrig, Keeping a quantum bit alive by optimized\nπ-pulse sequences, Physical Review Letters 98, 100504\n(2007).\n[22] C. A. Ryan, J. S. Hodges, and D. G. Cory, Robust de-\ncoupling techniques to extend quantum coherence in di-\namond, Physical Review Letters 105, 200402 (2010).\n[23] G. A. Álvarez, A. Ajoy, X. Peng, and D. Suter, Perfor-\nmance comparison of dynamical decoupling sequences for\na qubit in a rapidly fluctuating spin bath, Physical Re-\nview A82, 042306 (2010).\n[24] A.Ajoy, G.A.Álvarez,andD.Suter,Optimalpulsespac-\ning for dynamical decoupling in the presence of a purely\ndephasing spin bath, Physical Review A 83, 032303\n(2011).\n[25] G. A. Álvarez and D. Suter, Measuring the spectrum of\ncolored noise by dynamical decoupling, Physical Review14\nLetters107, 230501 (2011).\n[26] A. M. Souza, G. A. Álvarez, and D. Suter, Robust dy-\nnamicaldecouplingforquantumcomputingandquantum\nmemory, Physical Review Letters 106, 240501 (2011).\n[27] A. M. Souza, G. A. Álvarez, and D. Suter, Robust dy-\nnamical decoupling, Phil. Trans. R. Soc. Lond. A 370,\n4748 (2012).\n[28] I. Hargittai and K. Hedberg, Electron-diffraction Inves-\ntigation of the Molecular Structures of Adamantane, J.\nChem. Soc. D , 1499 (1971).\n[29] R. C. Fort and P. v. R. Schleyer, Adamantane: Conse-\nquences of the diamondoid structure, Chemical Reviews\n64, 277 (1964).\n[30] In Ref. [18], we used the symbol z2forzeff, because we\ndiscussed also z1=J2\nL/J2\nQ, where JLis the linear cou-\npling sum.\n[31] H. Cho, T. D. Ladd, J. Baugh, D. G. Cory, and C. Ra-\nmanathan, Multispin dynamics of the solid-state NMR\nfree induction decay, Physical Review B 72, 054427\n(2005).\n[32] S. Burg, V. Stukalov, and E. Kogan, On the theory of\nindirect exchange in EuO, physica status solidi (b) 249,\n847 (2011).\n[33] K.-H. H. Kalman Tompa, Monika Bokor and P. Tompa,\nHydrogen skeleton, mobility and protein architecture, In-trinsically Disordered Proteins 1, e25767 (2013).\n[34] E. L. Hahn, Spin echoes, Physical Review 80, 580 (1950).\nAppendix A: Convergence of pair correlations in\ncalcium fluoride\nWith the help of the algorithm presented in Sec. IIIC,\nwe add the most important intermediate spins inducing\nindirect couplings order by order to the cluster. Com-\nparing the simulation results for different cluster sizes\nlallows us to track the convergence. We perform this\ncomparison for the autocorrelation and some selected rel-\nevant pair correlations. The result is shown in Fig. 12.\nAs can be seen in panel (e), the autocorrelation changes\nonly very slightly as lis increased. The result for l= 9\nis barely distinguishable from l= 1(spinDMFT), ex-\ncept for a tiny maximum at about 60µs. This is a\nperfect a posteriori justification for the applicability of\nspinDMFT. Such a quick convergence is also obtained\nfor the nearest-neighbor pair correlation c= 1, see panel\n(a). However, as the pair distance is increased, the con-\nvergence becomes slower and slower.\n−0.3−0.2−0.10.0mcGxx\nc(a)\nc= 10.000.050.100.15\n(b)\nc= 4\n0 50\nt(units ofµs)−0.06−0.04−0.020.00mcGxx\nc(c)\nc= 8\n0 50\nt(units ofµs)0.000.010.020.03(d)\nc= 19\n0 20 40 60 80\nt(units ofµs)0.00.20.40.60.81.0\n(e)\nc= 0l= 1\nl= 2\nl= 3\nl= 4\nl= 5\nl= 6\nl= 7\nl= 8\nl= 940 60 800.000.02\nFIG. 12. Convergence of the pair correlations from nl-spinDMFT in [100]-direction with ascending cluster size l. Panels (a)-(d)\nshow selected pair correlations with index caccording to Tab. I and panel (e) shows the autocorrelation. Pair correlations do\nnot exist for l= 1which is why the orange line can only be found in panel (e).\nOften, we obtain considerable leaps upon adding a sin-\ngle spin, because this spin allows the pair correlation to\nbuild up along an additional important path on the lat-\ntice. For c= 4, see panel (b), the third and fourth spin\nare very important. As can be seen in Fig. 13, these are\nthe common neighbors of the two spins directly involved\nin the pair correlation. The two indirect couplings via\nspin No. 3 and 4 are the main contributions to this pair\ncorrelation. The same behavior can be observed in paircorrelation No. 19, see panel (d) and Fig. 14 for the clus-\nters. Adding the sixth spin enables the first path with\nfour nearest-neighbor links between the two spins of the\npair correlation. Each spin that is added till l= 8en-\nables another fourth-order path and, thus, increases the\nfourth-order peak in the pair correlation. Remarkably,\nthe increase is always nearly the same indicating that\nthe effects from different paths roughly adds up linearly\nin the pair correlation. Studying such effects in detail15\n2345\n6789\nz\ny x\nFIG. 13. Obtained clusters for the pair correlation with index\nc= 4, depicted in orange, for different cluster sizes l.\n2345\n6789\nz\ny x\nFIG. 14. Obtained clusters for the pair correlation with index\nc= 19, depicted in orange, for different cluster sizes l.\nleads to a much better understanding of the microscopic\ncorrelations. Note that with a cluster size of l= 8spins,any paths with four nearest-neighbor links are included.\nHence, the effect of adding spin No. 9 is significantly\nsmaller. These observations are in perfect accordance\nwith our arguments in Secs. IIIC and IIID.\nThe direct understanding of the leaps in the conver-\ngence plots allows us to draw conclusions about the ac-\ncuracy of the computed pair correlations. We consider,\nfor instance, pair correlation No. 19, see Fig. 12 (d), to\nbe fairly well captured for l≥8, because all paths with\nfour nearest-neighbor links are included. Hence, the low-\nest, most important order is completely accounted for.\nThe clusters for l= 8andl= 9may be referred to as\ncompact. For several other pair correlations with com-\nparably large pair distances this is not the case, see for\nexample c= 11,12,17,18in Fig. 4. Here, the lowest or-\nders are only included partly for l= 9which makes the\nresults less reliable. We conclude that the larger the pair\ndistance, thelesscompactaretheclustersatgivencluster\nsize and, hence, the less accurate are the resulting pair\ncorrelations. Note that the absolute magnitude of the\npair correlations decreases as the pair distance increases\nso that it is justified to truncate at a certain range.\nAppendix B: FID of CaF 2in different directions\nThe results for external magnetic field in [110]- and\n[111]-direction are shown in Figs. 15 and 16. Note that\nthe number of most important couplings Lis chosen\nlarger than for the [100]-direction. This is because there\naremorecouplingsofsimilarstrength. Thetimeatwhich\nnl-spinDMFT becomes inaccurate is about the same as\nforthe[100]-direction(about 40µs). Inparticular, forthe\n[111]-direction the rather late minimum is not captured\ncompletely quantitatively. Also, the sum rule of the z-\ndirection (blue line) is not well fulfilled for the discussed\nreasons. The choice of compact clusters requires more\nand more spins as the coordination number is increased.\nSince we are limited to 9 spins, the FIDs in [110]- and\n[111]-direction are less well captured.16\n0.00.51.0F(a)\ncmax= 0(b)\ncmax= 1\n0 50 100\nt(units ofµs)0.00.51.0F(c)\ncmax= 3\n0 50 100\nt(units ofµs)(d)\ncmax= 5\n0 20 40 60 80 100 120\nt(units ofµs)−0.20.00.20.40.60.81.0\n(e)\ncmax= 19FID, experiment\nFID, nl-spinDMFT\nMz, nl-spinDMFT\nFIG. 15. Same as Fig. 5, but in [110]-direction and with L= 8.\n0.00.51.0F(a)\ncmax= 0(b)\ncmax= 1\n0 50 100 150\nt(units ofµs)0.00.51.0F(c)\ncmax= 3\n0 50 100 150\nt(units ofµs)(d)\ncmax= 5\n0 20 40 60 80 100 120 140\nt(units ofµs)−0.20.00.20.40.60.81.0\n(e)\ncmax= 19FID, experiment\nFID, nl-spinDMFT\nMz, nl-spinDMFT\nFIG. 16. Same as Fig. 5, but in [111]-direction and with L= 14."    },    {        "title": "1211.0124v1.Dynamically_generated_pure_spin_current_in_single_layer_graphene.pdf",        "content": "Dynamically-generated pure spin current in single-layer graphene  Zhenyao Tang1,#, Eiji Shikoh1,#, Hiroki Ago2, Kenji Kawahara2, Yuichiro Ando1, Teruya Shinjo1 and Masashi Shiraishi1,*  1. Graduate School of Engineering Science, Osaka Univ. Toyonaka 560-8531, Japan 2. Institute for Material Chemistry and Engineering, Kyushu Univ. Fukuoka 816-8508, Japan  # These two authors contributed equally to this work. *Corresponding author: shiraishi@ee.es.osaka-u.ac.jp  Abstract The conductance mismatch problem limits the spin-injection efficiency significantly, and spin-injection into graphene has been usually requiring high-quality tunnel barriers to circumvent the conductance mismatch. We introduce a novel approach, which enables generation of a pure spin current into single-layer graphene (SLG) free from electrical conductance mismatch by using dynamical spin injection. Experimental demonstration of spin-pumping-induced spin current generation and spin transport in SLG at room temperature was successfully achieved and the spin coherence was estimated to be 1.36 µm by using a conventional theoretical model based on Landau-Lifshitz-Gilbert equation. The spin coherence is proportional to the quality of SLG, which indicates that spin relaxation in SLG is governed by the Elliot-Yafet mechanism as was reported.       Electrical spin injection and generation of a pure spin current in graphene using non-local electrical technique has opened a new frontier in molecular spintronics [1-3]. A number of studies have been performed using this method, investigating spin drift [4,5], long spin coherence in bilayer graphene (BLG) [6,7] and robustness of spin polarization [8]. Since SLG possesses a linear band structure around the K and K’ points, the effective mass of spin carriers is quite small, which allows us to expect much better spin coherence than that observed in BLG. However, the experimentally observed coherence in SLG is still limited and shorter than that in theoretical predictions. In addition, the spin relaxation mechanisms in SLG and BLG are reported to be different, namely, Elliot-Yafet in SLG and D’yakonov-Perel in BLG [6]. Hence, there are still many issues in spin transport in graphene that need to be clarified, and the establishment of a novel technique for spin injection and generation of a pure spin current in graphene is strongly desired for discussing spin transport phenomena in graphene from a different and new standpoints. Here, we report on a novel approach to spin injection and generation of pure spin current in SLG enabling to circumvent the conductance mismatch problem [9], where dynamical spin injection without using electric current for generating a spin current is established. The estimated spin coherence at room temperature (RT) is 1.36 µm. This achievement provides a new platform for discussing spin transport physics in graphene.      The SLG used in this study was a large-area SLG grown by chemical vapor deposition (CVD). The SLG was transferred to an SiO2/Si substrate (SiO2 is 300 nm thick, and the details were described in elsewhere [10]). Figure 1a shows a schematic image of the sample used in this study. A Ni80Fe20 film (Py, 25 nm in thick and 900×300 µm2 in size) and a Pd wire (5 nm in thick and 1 µm in width) were fabricated using electron beam lithography and an evaporation method. An Al capping layer was evaporated onto the Py layer in order to prevent oxidation of the Py. The gap length between the Py and the Pd was ca. 970 nm. An external magnetic field was applied by changing the angle θ, as shown in Fig. 1(a) (θ is equal to 0 degree when the magnetic field is parallel to the SLG plane). Fig. 1(b) shows typical Raman spectra of the SLGs, and the Raman spectra were obtained by using a Raman spectrometer (Tokyo Instrument, Nanofinder30). We prepared two different qualities of the SLGs (sample #1 and #2). The quality of sample #1 is better than that of sample #2, since D-band can be observed only from sample #2 (see Fig. 1(b)). Furthermore, we clarified that they are SLGs by measuring transport characteristics (see supplemental information).      The principle of dynamical spin injection is as follows. Magnetization dynamics in a ferromagnet (the Py in this study) are described by the Landau-Lifsitz-Gilbert (LLG) equation as,                        dMdt=!Heff!M+\"M!dMdt,        (1) where γ, M, Heff, α and Ms are the gyro-magnetic ratio of the ferromagnet, the time-dependent magnetization of the ferromagnet, an external magnetic field, the Gilbert damping constant and saturation magnetization of the ferromagnet, respectively. The first and the second terms are the field term that describes magnetization precession and the damping term that describes damping torque. The Ferromagnetic resonance (FMR) occurs when a microwave (9.62 GHz in this study) is applied to the ferromagnet. The damping torque is suppressed by the applied microwave under the FMR, which induces pumping of spins into SLG due to spin angular momentum conservation. The pumped spins induce spin accumulation in the SLG, which allows generation of a pure spin current in the SLG (see supplemental information). The propagating spins are absorbed in the Pd wire, where a pure spin current (Js) is converted to a charge current (Jc) due to the inverse spin Hall effect (ISHE) [11] and an electromotive force at the Pd wire is generated. The ISHE is the reciprocal effect of the spin Hall effect, and the Jc is described as Jc~Js!!(σ : spin direction). Note that the sign of Jc is changed by varying the direction of spin (σ), namely, the sign of the electromotive force at the Pd wire becomes opposite, when the pure spin current is successfully generated and propagated in the SLG and σ is reversed by a static external magnetic field. Figure 1(c) shows the FMR spectra of the Py with and without the SLG/Pd. The line width of the spectrum from the sample with the SLG/Pd is larger than that without the SLG/Py, which is attributed to the modulation of α due to successful spin pumping into the SLG/Pd.       Figures 2(a)-2(c) show the FMR signals as a function of θ, where the FMR of the Py occurs in every condition. The electromotive force of the Pd wire is shown in Figs. 2(d)-2(f), where the electromotive force is observed when θ is set to be 0 and 180 degrees, whereas no signal was observed at θ = 90 degree. Since this finding is in accordance with the angular dependence of the electromotive force due to the ISHE (Jc~Js!!), the observed electromotive force is ascribed to the ISHE of the Pd, which is due to spin pumping into the SLG and achievement of spin transport of a dynamically generated pure spin current at room temperature. Here, note that there is no spurious effect with the observed symmetry and only the ISHE possesses the symmetry. For example, the anomalous Hall effect (AHE) signal can be included as a spurious signal, which may impede detection of the ISHE signals, but the AHE does not show such an external magnetic field dependence of the electromotive force. The following investigations also support the result that the ISHE signals were dominantly observed. The theoretical fitting was performed in order to separate the ISHE and the AHE signals in the observed electromotive force by using the following equation,V=VISHE!2(H\"HFMR)2+!2+VAHE\"2!(H\"HFMR)(H\"HFMR)2+!2+aH+b,where VISHE is the electromotive force, VAHE is the voltage due to the AHE, H is an external static magnetic field for the FMR, HFMR is the magnetic field where the FMR occurs, and Γ, a, b are fitting parameters. An example of the fitting is shown in Fig. 3(a), and VISHE and VAHE are estimated to be 1.16×10-5 V and 6.67×10-7 V, respectively. The contribution from the AHE to the electromotive force by the ISHE was revealed to be very weak. Figure 3(b) shows the microwave power dependence of the electromotive force at the Pd for the Py/SLG/Pd sample. The electromotive force at the Pd wire, VISHE, is nearly proportional to the microwave power, which indicates that the density of the generated spin current in the SLG proportionally increases with the applied microwave power [12]. Also, as shown in Fig. 3(c), the electromotive force proportionally increased with the microwave power, although the voltage due to the AHE was small enough. In fact, the ratio of the signal intensities by the ISHE and AHE at 200 mW was estimated to be 17, which indicates that the ISHE signal is dominant in the observed electromotive force and that the observed signal was mainly due to spin transport in the SLG.  Spin coherence in the experiment is estimated based on a conventional spin pumping theory [13]. The dynamical magnetization process induces spin pumping from the Py layer to the graphene layer and generates a spin current, js, as,           js=!2\"!4!02!\"!gr\"#1Ms2[M(t)$dM(t)dt]zdt.      (2) Here, gr!\"and ! are the real part of the mixing conductance [14] and the Dirac constant, respectively. Note that gr!\" is given by,           ),(32/PySLGPyBFsrWWgdMg−=↑↓ωµγπ             (3) where g,Bµand dF are the g-factor, the Bohr magneton and thickness of the Py layer, respectively, and, dF, WPy/SLG and WPy in this study were 25 nm, 3.10 mT and 2.57 mT, respectively. From Eqs. (1)-(3), the spin current density at the Py/SLG interface is obtained as, đ]4)4[(8]4)4(4[222222222ωγππαωγπγπγ+++=↑↓sssrsMMMhgjwhere h is the microwave magnetic field, set to 0.16 mT at a microwave power of 200 mW. As discussed above, the broadening of W in the Py/SLG compared with that in the Py film was attributed to spin pumping into the SLG and ↑↓rg in the Py/SLG layer was calculated to be 1.6×1019 m-2, and thus js was calculated to be 7.7×10-9 Jm-2. Here, half of the generated js can contribute to the electromotive force in the Pd electrode in our device geometry, since a pure spin current diffuses isotropically. The generated js decays by spin diffusion in SLG, js decays to js!exp(-d!) when spins diffuses to Pd wire. Furthermore, the electromotive force taking the spin relaxation in the Pd wire into account can be written as, sPdPdSLGSLGPdPdPdSHEISHEjedddwV)2()2/tanh(σσλλθ+=, in the simplest model. Here, w,Pdλ, dPd and σPd are the length of the Pd wire facing the Py (900 nm), the spin diffusion length (9 nm) [15], the thickness (5 nm) and the conductivity of the Pd, respectively, and dSLG and σSLG are the thickness (assumed to be ca. 0.3 nm) and conductivity of the SLG (measured to be ca. 3.10×106 S/m under the zero gate voltage application. See supplemental information). The spin-Hall angle in a Py/Pd junction, θSHE, has been reported to be 0.01 [12], which allows us to theoretically estimate the electromotive force in the Pd wire as 2.37×10-5 V if no spin relaxation occurred in the SLG. In contrast, the experimentally observed electromotive force was 1.16×10-5 V, this discrepancy is ascribed to dissipation of spin coherence during spin transport in the SLG (the decay of js), which can be described as an exponential damping dependence on the spin transport. From the above calculations, the spin coherent length in the SLG is estimated to be 1.36 µm. For comparison, we carried out the same experiments by using the other sample (sample #2, the gap length was measured to be 780 nm.), of which quality is not as good as that of sample #1 (the conductivity was measured to be 6.40×105 S/m under the zero gate voltage application). Figures 4(a)-4(f) show the result, and here again, the obvious ISHE signals and the inversion of the ISHE signals can be seen as the external magnetic field was reversed. However, the signals were comparatively weak and the estimated spin coherent length was ca. 460 nm, which is in agreement with the sample qualities and also with the reported spin relaxation mechanism in SLG, i.e., the Elliot-Yafet type. These observations also corroborates that our result is attributed to dynamical spin injection and spin transport in the SLGs. The spin coherent length in the CVD-grown SLG, which was estimated by using the dynamical method, is good and comparable to the previously reported value (1.1 µm) estimated by using an electrical method [16]. In contrast, the spin coherence of the SLG in this study is much better than that in p-Si (ca. 130 nm) in our previous study [17], which directly indicates the strong superiority of graphene for spin transport.       In summary, we successfully demonstrated dynamical spin injection, resulting in generation of spin current in SLG at room temperature, which enables generation of a pure spin current free from electrical conductance mismatch. The spin coherent length of CVD-grown SLG was 1.36 µm. This achievement provides a novel platform for discussing spin transport physics in SLG from a new viewpoint.  A part of this study was supported by the Japan Science and Technology Co. (JST), Global COE program of Osaka Univ., and the Japan Society for the Promotion of Science (JSPS). References 1. M. Ohishi, M. Shiraishi, R. Nouchi, T. Nozaki, T. Shinjo, Suzuki, Y. Jpn. J. Appl. Phys. 46, L605 (2007). 2. N. Tombros, C. Jozsa, M. Popinciuc, H.T. Jonkman, B.J. van Wees. Nature 448, 571 (2007). 3. S. Cho, Y-H. Chen, M.S. Fuhrer. Appl. Phys. Lett. 91, 123105 (2007). 4. C. Jozsa, M. Popinciuc, N. Tombros, H.T. Jonkman, B.J. van Wees. Phys. Rev. Lett. 100, 236603 (2008). 5. C. Jozsa, M. Popinciuc, N. Tombros, H.T. Jonkman, B.J. van Wees. Phys. Rev. B 79, 081402(R) (2009). 6. W. Han, R.K. Kawakami. Phys. Rev. Lett. 107, 047206 (2011).  7. T.-Y. Yang, J. Balakrishnan, F. Volmer, A. Avsar, M. Jaiswal, J. Samm, S.R. Ali, A. Pachoud, M. Zeng, M. Popinciuc, G. Guentherodt, B. Beschotenand, B. Ozyilmaz. Phys. Rev. Lett. 107, 47204 (2011). 8. M. Shiraishi, M. Ohishi, R. Nouchi, N. Mitoma, T. Nozaki, T. Shinjo, Y. Suzuki. Adv. Funct. Mater. 19, 3711 (2009). 9. A. Fert, H. Jaffres. Phys Rev. B 64, 184420 (2001). 10. C.M. Orofeo, H. Hibino, K. Kawahara, Y. Ogawa, M. Tsuji, K. Ikeda, S. Mizuno, H. Ago. Carbon 50, 2189 (2012). 11. E. Saitoh, M. Ueda, H. Miyajima, G. Tatara. Appl. Phys. Lett. 88, 182509 (2006). 12. K. Ando, E. Saitoh. J. Appl. Phys. 108, 113925 (2010) 13. Y. Tserkovnyak, A. Brataas, G. E.W. Bauer. Phys. Rev. Lett. 88, 117601 (2002). 14. Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, B. I. Halperin. Rev. Mod. Phys. 77, 1375 (2005). 15. J. Foros, G. Woltersdorf, B. Heinrich, A. Brataas. J. Appl. Phys. 97, 10A714 (2005). 16. A. Avsar, T.-Y. Yang, S. Bae, J. Balakrishnan, F. Volmer, M. Jaiswal, Z. Yi, S.R. Ali, G. Guentherodt, B. H. Hong, B. Beschoten, B. Ozyilmaz. Nano Lett. 11, 2363 (2011). 17. E. Shikoh, K. Ando, K. Kubo, E. Saitoh, T. Shinjo, M. Shiraishi, cond-mat arXiv 1107.0376. Figures and figure captions  \nSiO2 subSLGPyPdgapAlHВ\nSiO2 subSLGPyPdgapAlHВSiO2 subSLGPyPdgapAlHВ\n Figure 1(a). A schematic image of the SLG spin pumping sample. The CVD-grown SLG is transferred to the SiO2/Si substrate, and the Al/Py and the Pd electrodes are separately evaporated on the SLG. The angle of the external magnetic field is shown in the inset.    \n10001500200025003000  SLG #1  SLG #2IntensityRaman shift (cm-1)  \nDG2D\n Figure 1(b). Raman spectra of the SLG just after the synthesis. The black and red solid lines are data from sample #1 and #2, respectively. The typical symmetric Raman peaks of the G- and 2D-bands from SLG can be seen. The D-band can be hardly seen in sample #1, which shows the defect of sample #1 is less than that of sample #2.  \n-20020-101\n   \n  \nPyPy/SLGdI(H)/dH (a.u.)H-HFMR (mT) Figure 1(c). FMR spectra of the Py on the SiO2 substrate (a black solid line) and the Py/SLG on the SiO2 substrate (a red solid line, sample #1). An increase of the line width can be seen, which is attributed to a shift of the Gilbert damping constant, α, namely, spin pumping into the SLG.   -40-2002040H-HFMR (mT) \n   θ = 0°V (µV)5µV(d)-101(a)\n  dI(H)/dH (a.u.)(c)\n  \n \n-40-2002040 (f)\nH-HFMR (mT) \n   θ =180°5µV   \n (b)\n-40-2002040(e)\n H-HFMR (mT)\n   θ =90°5µV\n-40-2002040H-HFMR (mT) \n   θ = 0°V (µV)5µV(d)-101(a)\n  dI(H)/dH (a.u.)(c)\n  \n \n-40-2002040 (f)\nH-HFMR (mT) \n   θ =180°5µV   \n (b)\n-40-2002040(e)\n H-HFMR (mT)\n   θ =90°5µV Figure 2. Results on spin pumping and spin transport in sample #1. (a)-(c) Ferromagnetic resonance spectra of the Py under the external magnetic field at (a) 0 degree, (b) 90 degree and (c) 180 degree under the microwave power of 200 mW. (d)-(f) Electromotive forces from the Pd wire on the SLG when θ is set to be (d) 0, (e) 90 and (f) 180 degrees under the microwave power of 200 mW. Electromotive forces are observed at θ = 0 and 180 degrees but the polarity of the signals are opposite, which is in accordance with the symmetry of the inverse spin Hall effect. No signal can be observed when the θ  was set to be 90 degrees, which corroborates our observation originating from the spin pumping and propagation of a pure spin current in the SLG.   -40-2002040V (µV)\n \n  \nH-HFMR (mT)5µV\n Figure 3(a). Results of the analysis of the contribution from the electromotive force due to ISHE and AHE. The open circles are external magnetic field dependence of ΔV for sample #1, where ()()[]/2180-0===ΔθθVVV in order to eliminate the heat effect during measurement. The solid line is the fitting line, which is calculated by using Eq. (2).    -10010  V (µV)H-HFMR (mT) 200 mW 100 mW 50 mW 0 mW5µV\n Figure 3(b). Microwave power dependence of the electromotive forces in the Pd wire. A magnetic field was applied parallel to the film plane. A monotonical increase of the electromotive force can be seen.   0501001502000510 VISHE VAHE linear fit\n   V (µV)PMW (mW)  Figure 3(c). Microwave power (PMW) dependence of VISHE and VAHE measured for the Py/SLG/Pd sample. The contributions from the ISHE (VISHE) and anomalous Hall effect (VAHE) are plotted by black circles and red triangles, respectively. The dashed line shows the linear fit of the data for the VISHE.   -40-2002040θ = 180°   \n  5µVH-HFMR (mT)-40-2002040  θ =0 5µVV (µV)H-HFMR (mT)-40-2002040θ = 90° \nH-HFMR (mT)\n  5µV  dI(H)/dH (a.u.)\n   \n \n   (a)(b)(c)(d)(e)(f)\n-40-2002040θ = 180°   \n  5µVH-HFMR (mT)-40-2002040  θ =0 5µVV (µV)H-HFMR (mT)-40-2002040θ = 90° \nH-HFMR (mT)\n  5µV  dI(H)/dH (a.u.)\n   \n \n   (a)(b)(c)(d)(e)(f)  Figure 4. Results on spin pumping and spin transport in sample #2. (a)-(c) Ferromagnetic resonance of the Py under θ is set to be (a) 0 degree, (b) 90 degree and (c) 180 degree under the microwave power of 200 mW. (d)-(f) Electromotive forces from the Pd wire on the SLG when θ is set to be (d) 0, (e) 90 and (f) 180 degrees under the microwave power of 200 mW. Electromotive forces can be seen as expected, indicating successful dynamical spin injection. However, the signal was weak and noisy. The red solid lines in (d) and (f) show the fitting lines obtained by using Eq. (2).  Supplemental Materials for “Dynamically-generated pure spin current in single-layer graphene” by Tang et al.       In order to clarify that the graphene channel was definitely formed, we measured the field-effect transistor (FET) characteristics made of sample #1, and clarified that typical FET features were observed as shown in Fig. S1. Here, a slight distortion in the gate voltage dependence of source-drain current was observed, which often appears SLG-FETs with ferromagnetic contacts [S1]. This FET characteristic apparently indicates that the charge current flows in the SLG.      Figure S2 shows a schematic image of spin-pumping-induced generation of pure spin current and its isotropic diffusion in SLG. The isotropic spin diffusion is ascribed to the uniform electrochemical potential of spins in the SLG, since no electric field is applied to the SLG. The diffused pure spin current are adsorbed in the Pd wire and converted to an electric current due to ISHE.  [S1] R. Nouchi, M. Shiraishi, Y. Suzuki. Appl. Phys. Lett. 2008, 93, 152104.  -101-0.20.00.2\n  ID (mA)VDS (V)-20020400.10.20.3\n  ID (mA)VG (V) Figure S1. FET characteristics of SLG #1. The inset shows the the I-V curve of the source-drain current. The measurement was carried out at room temperature. SLG µ z Py\u0003Pd\u0003\nSpin Pumping \n Figure S1b. A schematic image of spin pumping into the SLG. Spin angular momentum is transformed from the Py to the SLG due to π-d coupling, which induces spin accumulation in the SLG (the accumulation induces separation of electrochemical potentials, µ, of the up- and down-spins). The accumulated spins diffuse in the SLG, resulting in a pure spin current. This pure spin current flows into the Pd electrode, where electromotive force arises due to the inverse spin Hall effect.  "    },    {        "title": "1105.4740v1.Scalable_Spin_Amplification_with_a_Gain_over_a_Hundred.pdf",        "content": "arXiv:1105.4740v1  [quant-ph]  24 May 2011Scalable Spin Amplification with a Gain over a Hundred\nMakoto Negoro, Kenichiro Tateishi, Akinori Kagawa, Masahiro Kitag awa1\n1Division of Advanced Electronics and Optical Science,\nGraduate School of Engineering Science, Osaka University,\nToyonaka, Osaka, 560-8531, Japan.\n(Dated: September 26, 2018)\nWe propose a scalable and practical implementation of spin a mplification which does not require\nindividual addressing nor a specially tailored spin networ k. We have demonstrated a gain of 140\nin a solid-state nuclear spin system of which the spin polari zation has been increased to 0.12 using\ndynamic nuclear polarization with photoexcited triplet el ectron spins. Spin amplification scalable\nto a higher gain opens the door to the single spin measurement for a readout of quantum computers\nas well as practical applications of nuclear magnetic reson ance (NMR) spectroscopy to infinitesimal\nsamples which have been concealed by thermal noise.\nPACS numbers: 03.67.-a, 76.60.-k, 76.70.Fz\nA magnetic moment of a single nuclear spin is so small\nthat its induction signal is buried under thermal noise.\nThe minimum detectable number of perfectly polarized\nnuclear spins is still in the millions with a cryogenic in-\nductive detection circuit. If information of a single nu-\nclear spin is quantum-logically copied to a large number\nof spins, a spin component can be amplified and hence\nits signal may be detected with sufficient signal-to-noise\nratio (SNR). This scheme is called spin amplification ,\nwhich can be realized by the quantum circuit shown in\nFig. 1(a) [1]. Although copy of an arbitrary, unknown\nstate itself is prohibited by the no-cloning theorem [2],\nthe information about whether the state is in |0/an}bracketri}htor|1/an}bracketri}ht\ncan be copied. Fig. 1(a) can be used for a readout of\nquantum computers.\nWe propose other potential applications of spin am-\nplification. To experimentally determine an arbitrary,\nunknown quantum state |ψ/an}bracketri}htof a spin, quantum state to-\nmography is used, in which the state |ψ/an}bracketri}htis repeatedly\nprepared and measured [3]. If the number of repetitions\nisNand the measurementis limited bythe thermal noise\nof the detection apparatus, then the SNR is increased by\na factor of√\nN. If the state is prepared and accumulated\nelsewhere to make Nidentical copies of |ψ/an}bracketri}ht, as shown in\nFig. 1(b), and measured at once, then the signal becomes\nNtimes as large and the SNR is increased by a factor of\nN, which is non-trivial because it is much more efficient\nthan√\nNattained by the simple repetition. This scheme\ncan also be regarded as amplification in a broader sense\nbecause it improves the SNR against the noise entering\nafterwards. The spin amplification (Fig. 1(b)) can be\napplied to quantum metrology which estimates a small\nunknown parameter φcontained in a quantum evolution\nUfrom the initial state |0/an}bracketri}htto the final state |ψ/an}bracketri}ht[4]. This\nprocedure is regarded as spectroscopy as well when the\nparameterφis given by an internal Hamiltonian of a spin\nsystem of interest. The minimum detectable value of φ\nis improved by a factor of Nas long as it is limited by\nthe thermal noise of the detection apparatus.0\n0\n0\n0U\n0\n0\n0\n0Ψ\n0\n0\n0\n0(a) (b)\n(d) (c)0\nΨS: \nI1:\nI2:\nI3:\nIm:S: \nI1:\nI2:\nI3:\nIm:\n0\n0\n0\n00S: \nI1:\nI2:\nI3:\nIm:U U U\nU U U U\nhetero- SD hetero- SD hetero- SD hetero- SD hetero- SD hetero- SD hetero- SD hetero- SD S: \nIb:\nI1:\nI2:\nIm:N (= m)\nNN (= m)\nNΨ\nΨ\nFIG. 1: (a) Spin amplification referred to by DiVincenzo [1].\n(b) Spin amplification of the S spin response signal using se-\nlective SWAPs, and (c) that using the hetero-SD process. (d)\nScalable version of (a). The I spins are initialized to |0/angbracketrightfor\nsimplicity, although they do not have to be in practice.\nIn the S-Imspin system (a rarely-existing S spin and\nmabundant I spins), the absolute maximum gain of spin\namplification shown in Figs. 1(a) and 1(b) is limited by\nm. To increase the maximum gain, mmust be increased.\nFigs. 1(a) and 1(b) are, in principle, scalable, but they\nhardly scale up to larger min practice because they re-\nquire individual addressing to all spins or a special spin\nnetwork with tailored couplings such as a spin chain and\na star topology [5–8]. Until now, a gain as low as four\nhas been demonstrated with nuclear spins in molecules\nin solution [8, 9]. Building a spin amplifier with a high\ngain, say 100, seems as daunting as building a 100-qubit\nquantum computer.\nIn this Letter, we propose a new scalable implementa-\ntion of spin amplification shown in Fig. 1(c), which uti-\nlizesspin diffusion [10] and magnetic-field cycling [11].\nThe total Hamiltonian of an S-Imnuclear spin system\nwith a dipolar coupling network, naturally existing in a2\nbulk solid, is given by\nH=ωSSZ+ωI/summationdisplay\niIiZ+/summationdisplay\ni,jHi,j+/summationdisplay\niHi,S,(1)\nwhereωSandωIare the resonance frequencies of the\nrare spin S and the abundant spins I. Hi,jandHi,Srep-\nresent the homonuclear and the heteronuclear dipolar in-\nteractions, respectively. The dipolar interaction, Hi,j,\nbetween the i-th spin I iand thej-th spin I j(or S for\nheteronuclear case, Hi,S) is given by Hi,j=HZZ\ni,j+HXY\ni,j\nwhereHZZ\ni,j=di,jIiZIjZ(for heteronuclear case, the\ndipolar coupling strength, di,j, is replaced with di,S) gen-\nerates mutually controlled rotations about zaxis and\nHXY\ni,j=−di,j(IiXIjX+IiYIjY)/2 generates fractional\npower of iSWAP . Both are always on between homonu-\nclearspinsregardlessofthestaticmagneticfieldstrength.\nBetween heteronuclearspins, the former is always on and\nthe latter is turned off in high static magnetic fields by\nthe Zeeman energy difference being much larger than the\ndipolar interaction strength ( |ωS−ωI| ≫ |di,S|). In low\nfields (|ωS−ωI|<∼|di,S|), the latter is turned on. Un-\nder free time evolution of the total Hamiltonian in the\nhigh field, the zcomponents of only the I spins are shuf-\nfled by simultaneous fractional powers of iSWAPs, and\nthis behavior is known as spin diffusion [10]. In the\nlow field, the S spin is also involved in the spin diffusion\namong the I spins via its coupling to at least one I spin:\nwe call it the heteronuclear-spin-involved spin diffusion\n(hetero-SD) process, where the total zspin component\nis preserved. The hetero-SD process can be switched on\nand off by magnetic-field cycling between the low and\nthe high fields [11–13]. Therefore the quantum circuit of\nFig. 1(c) can be easily realized by the simple repetition\nof synchronized field cycling and a radiofrequency (rf)\npulse irradiation for the quantum operation Uto the S\nspin alone. The process requires only a reasonably dense\nnetwork of dipolar-coupled spins without requiring the\ntailored coupling strength ( di,j), and therefore, is quite\ngeneral, robust, and practical in real samples which may\nhave some irregularities or randomness.\nBy exciting the zcomponent of the S spin with an\nrf pulse as Uand subsequently switching on the hetero-\nSD process for a sufficiently long duration, as shown in\nFig.1(c), theexcited zcomponentoftheSspinisdiffused\nin the reservoirof the abundant I spins. By repeating the\nsequence, the excitations are accumulated in the abun-\ndantIspins. Theinformationaboutwhetherthe Sspinis\ninvertedby Uis convertedinto the decreasein the zcom-\nponent of the total magnetization of mabundant I spins,\nwhich is proportional to the I spin polarization. Assum-\ning that the polarizationsofthe S spin and the I spinsare\nuniformly-mixed in the hetero-SD process, the I spin po-\nlarization after the Nth repetition of the U= NOT and\nthehetero-SD process decreases to ǫ0[(m−1)/(m+1)]N\nfrom the initial polarization ǫ0of the S and the I spins.m = 1599\nm = 299m = 799\naccumulation: G =   N ideal spin amplification: G = N 1000\n500\n0Gain G\n1000 500 0\nSteps N(for N  =     )G = N (1- e -1)\nm\n2\nFIG. 2: The top dashed line shows the gain Gof the SNR\nof ideal spin amplification (Figs. 1(a) and 1(b)) with respec t\nto the number of SWAP gates N. The solid lines show the\nexpected gain (Eq. (2)) of the proposed one (Fig. 1(c)) with\nrespect to the number of the hetero-SD process gates Nfor\ndifferent system sizes. The bottom dashed line shows the gain\nof the direct simple repetitive detection with respect to th e\nnumber of repetitions N.\nCompared with the difference between the directly ob-\nserved signal of the S spin of polarization ǫ0and that\nof the inverted S spin, the expected gain of the signal\nobtained from the polarization difference of mabundant\nI spins at the Nth step of spin amplification with and\nwithoutUis given by\nG=m/bracketleftbigg\nǫ0−ǫ0/parenleftBig\nm−1\nm+1/parenrightBigN/bracketrightbigg\nǫ0−(−ǫ0)=m\n2/bracketleftBigg\n1−/parenleftbiggm−1\nm+1/parenrightbiggN/bracketrightBigg\n.\n(2)\nIn an S-Imspin system, the gain of the spin amplifi-\ncation using the hetero-SD process increases with Nbut\nis saturated to m/2 forN≫m/2, as shown in Fig. 2.\nThe proposed implementation using the hetero-SD pro-\ncess gives better SNR improvement with respect to N\nthan the simple Nrepetitive detection. When mand\nNare increased simultaneously while keeping N=m/2,\nthe amplification gain scales up linearly as m(1−e−1)/2\n(the dotted line) with respect to m(≫1).\nThe quantum circuit of Fig. 1(d) is a scalable version\nof Fig. 1(a) and could be used for a readout of quantum\ncomputers [14]. While there is a functional difference be-\ntween Figs. 1(d) and 1(c), the most difficult part, the\nscalability of implementation to achieve a higher gain, is\ncommon in Figs. 1(d) and 1(c). In Fig. 1(d), the infor-\nmation about whether the state of a rare spin S is |0/an}bracketri}ht\nor|1/an}bracketri}htis non-destructively copied to a buffer spin I bby\nthe CNOT gate and accumulated in abundant I spins\nwithhetero-SD processes. The field cycling between the\nhigh field and the low field where the I-I bZeeman energy\ndifference is comparable to the I-I bcoupling energy and\nthe S-I bZeeman energy difference is much larger than\nthe S-I bcoupling energy enables the hetero-SD process\nin which the I bspin is involved but the S spin is not.3\nx240000Hyperpolarize 1H\nby DNPHyperpolarize 19 F\nby sample shuttlingrf pulse 1H detection\n4000 G100 G\n11 2 3 5\n2\n34\n5stepping motor\ncavity for DNPrf coil tuned\nat 1H and 19 F\nsample\nplates of \nelectromagnet(a)\n(c) (b)\nlaser\nmicro\nfield\nsweep1 µs\n10  µs\n- 35  G+35  G40  W20  ms by sample shuttlingxN\n598  nm \nwavehetero- SD process4\nFIG. 3: (a) An experimental procedure of spin amplification\nusing the hetero-SD process [15]. (b) A DNP sequence with\nISE [16]. (c) A schematic diagram of an experimental setup\nwhere the sample position at each stage is indicated.\nOur implementations, both Figs. 1(c) and 1(d), using the\nfield cycling do not require rf irradiation to the abundant\nI spins, of which a subtle imperfection causes the large\ndegradation of the spin amplification gain.\nIn this work, we have performed experiments to am-\nplify the response of a rare spin with respect to an rf\npulse and demonstrated a spectroscopic application in\nan S-I799spin system in a bulk solid with the quantum\ncircuit shown in Fig. 1(c). We used a sample of a single\ncrystal of naphthalene doubly-doped with ∼0.005 mol%\npentaceneand ∼1mol% 2-fluoronaphthalene. Ifthe sam-\nple is uniform, it is approximately regarded as the en-\nsemble of S-I799spin systems because a19F spin is sur-\nrounded by 7991H spins on average. The experimental\nprocedure is shown in Fig. 3(a) [15]. In the first stage\nof every experiment, we enhanced the1H spin polariza-\ntion to∼0.12 at 233 K by dynamic nuclear polarization\n(DNP) with photoexcited triplet electron spins of pen-\ntacene molecules [16–19], the sequence of the photoexci-\ntation and the integrated solid effect (ISE) [16] shown in\nFig. 3(b). The key stage in this experiment, the hetero-\nSDprocess,wasrealizedbyfieldcycling(4000G →100G\n→4000 G) using sample shuttling, as shown in Fig. 3(c).\nIn 100 G, the resonance frequency difference between the\n1H and the19F spins is of the same order as the strength\nof the nearest neighbor1H-19F dipolar coupling. After\nrepeating an rf pulse to19F spins as the quantum oper-\nationUand the sample shuttling Ntimes, we observed\nthe1H magnetization.\nWe investigated the behavior of the1H spin polar-\nization with respect to the number of steps Nof spin\namplification (Fig. 4(a)). In the case of U=Iwith\nno rf pulse, the1H spin polarization decayed with ηof\n∼99.91% per cycle, which gave close agreement with the\ndecay rate calculated from T1measured at each sample\nposition. In the case of an on-resonance πpulse (i.e.\nU= NOT), the result was close to the theoretical decay\nǫ0{(m−1)/(m+ 1)}NηN(the dashed line) calculated0 50 100 150 200\nSteps0 -200 200\nOffset (kHz)400 -400\nN0.12\n0.10\n0.08\n0.06Polarization (a) (b)\nwithout rf pulse \nwith on-resonance pulse \ntheoretical140  kHz45  kHzN=40\nN=200\n19 F spectrum directly observed \nFIG. 4: (a) The behaviors of the1H spin polarization with\nrespect to the number of steps Nof spin amplification with-\nout rf pulse ( U=I) and with on-resonance πpulses (U=\nNOT). The dashed line is the theoretical decay. (b) The cir-\ncles show the spin-amplified frequency response spectra ob-\ntained for N= 40 and 200. They were obtained with the rf\npulse with a peak amplitude of 45 kHz (the pulse length is ∼3\ntimes as long as that of 140 kHz). The squares show that for\nN=200 obtained with the rf pulse with a peak amplitude of\n140 kHz. Directly detected19F spectrum vertically magnified\nis also shown for comparison.\nfrom the decreased polarization in Eq. (2) and the decay\nη. The S spin response to Uis converted into the dif-\nference between the I spin polarization with and without\nU. The polarization difference was amplified 37 times\nforN= 40 and 136 times for N= 200, compared with\nthat forN= 1. The amplification gain anticipated from\nEq. (2) is 38 for N= 40 and 157 for N= 200. By tak-\ning into account the frequencies for19F and1H spins, the\nsignal obtained from the polarization difference of the1H\nspins with and without U= NOT was estimated to be\n∼140timesaslargeasthatofthe19Fspinwiththepolar-\nization 0.11. It is noteworthythat the 140-timeamplified\nsignal is more than ten times as large as the signal of the\nperfectly polarized19F spins. Furthermore, a factor of\n140 spin amplification in conjunction with the modern\nmagnetic resonance force microscopy (MRFM [20]) tech-\nnology, which is expected to detect a hundred nuclear\nspins in the near future [21], may open the door to the\ndetection of a faint signal from a single nuclear spin.\nAmplified responses to the rf pulses with various car-\nrier frequencies for N= 200 give the excitation spec-\ntrum of the19F spin, as represented by the squares in\nFig. 4(b). The spin-amplified response spectra with the\nweaker rf pulse (the circles) were narrower than that but\nstill broader than the directly observed19F NMR spec-\ntrum. The pulse intensity should be lowered in order\nto avoid power broadening. The depth of the spectral\nresponse was increased with N.\nAll spin-amplified1H NMR signals were reduced by a4\nfactor of a hundred on purpose to avoid saturation of the\ndetection circuit [15]. With the reduction, the directly\nobserved19F NMR signal was completely-concealed with\nthe thermal noise. The19F signal observed without the\nreduction is shown in Fig. 4(b). If the sample size is\n100 times as small, the19F signal can not be detected\neven without the reduction. On the other hand, the spin\namplified1H signals of the small sample can be detected\nwith the approximately same SNR as the present result\nwithout the reduction. Spin amplification will havea dis-\ntinct advantage over the simple repetitive detection in a\nsmalleroramorediluted sample. Ifwe usea samplewith\na lower 2-fluoronaphthalene concentration, we could get\na higher gain by increasing Nin the same manner. The\nmaximum achievable gain is inverselyproportionalto the\nconcentration. It has been reported that T1of the1H\nspins in naphthalene doped with ∼0.01 mol% pentacene\nin∼7 G at 77 K is ∼166 minutes [18], and the result im-\npliesthatT1ofasamplewithalower2-fluoronaphthalene\nconcentration at a lower temperature is longer. In addi-\ntion, there is room for speeding up field cycling [11, 22],\nand hence earning the number of steps Nwell before the\n1H polarization vanishes by T 1relaxation. Scalable spin\namplification will enable practical applications of NMR\nspectroscopy to rarely-existing nuclear spins in infinitesi-\nmal bulk samples which have been concealed by thermal\nnoise. For higher sensitivity and spectral resolution, the\nquantum operation Uand the NMR detection should be\nperformed in a higher magnetic field, which is realized by\nthe field-cycling system described in [23].\nIn the proposed implementation using field cycling for\nswitching the hetero-SD process, the nuclear species are\nmodestly restricted to those with the resonant frequency\nclose to that of the abundant1H spins. For other nuclear\nspecies to be involved in spin diffusion process, the dou-\nble resonance irradiation satisfying the Hartmann-Hahn\ncondition[24], asinthecrosspolarizationexperiment[25]\nmay be used, although it is subject to the rf pulse im-\nperfection and the much faster decay with T1ρ. State-\nof-the-art high fidelity pulse engineering [26, 27] may be\nrequired to implement high-gain spin amplification.\nThe demonstratedspin amplification(Fig. 1(c)) canbe\nstraightforwardly modified to Fig. 1(d). It can be real-\nized, for example, by doping 2-13C 2-fluoronaphthalene,\nin which the13C,19F, and1H spins play the roles of the\nS, the I b, and the I spins, respectively. Recently, opti-\ncally detected magnetic resonance (ODMR) [28, 29] has\naccomplished single-event state-readout of a single nu-\nclear spin [30], which is attracting attention in quantum\ninformationscience. The breakthroughofthe gainshown\nin this Letter is an important step for state-readout of a\nsingle nuclear spin in bulk solids with more versatile de-\ntection such as MRFM and the inductive detection.\nIn this Letter, we have proposed a scalable and practi-\ncal implementation of spin amplification and successfully\ndemonstrated a gain as large as 140. A faint magnetiza-tion ofa rarespin hasbeen quantum-logicallytransferred\nbeyond the molecular boundary to abundant spins in the\nbulk solid sample through spin diffusion process which\ndoes not requireindividual addressingnor a specially tai-\nlored spin network. The amplified signal obtained from\nthe abundant spins is more than ten times as large as\nthat of the completely polarized rare spins. A spectro-\nscopic application has been demonstrated by 140-time\namplified frequency response spectra of the rare spin. It\nwill enable practical applications of NMR spectroscopy\nto infinitesimal samples which have been concealed by\nthermal noise. A higher gain is possible in the sample\nwith a lower concentration of rare spins. Spin amplifica-\ntion scalable to a higher gain opens the door to a single\nspin measurement and a readout of spin-based quantum\ncomputers.\nWe thank Kazuyuki Takeda for fruitful discussions.\nThis work was supported by the CREST program of\nJST, MEXT Grant-in-Aid for Scientific Research on In-\nnovative Areas 21102004, and the Funding Program for\nWorld-Leading Innovative R&D on Science and Technol-\nogy (FIRST). M.N. and K.T. were also supported by the\nGlobal-COE Program of Osaka University.\n[1] D. P. DiVincenzo, Fortschr. Phys. 48, 771 (2000).\n[2] W. K. Wooters and W. H. Zurek, Nature (London) 299,\n802 (1982).\n[3] M. A. Nielsen and I. L. Chuang, Quantum Computa-\ntion and Quantum Information (Cambridge Univ. Press,\nCambridge, 2000).\n[4] N. F. Ramsey, Molecular Beams (Oxford Univ. Press,\nLondon, 1956).\n[5] C. A. Pe´ rez-Delgado, M. Mosca, P. Cappellaro, and\nD. G. Cory, Phys. Rev. Lett. 97, 100501 (2006).\n[6] A. Kay, Phys. Rev. Lett. 98, 010501 (2007).\n[7] J. A. Jones, et al., Science 324, 1166 (2009).\n[8] J.-S. Lee, T. Adams, and A. K. Khitrin, New J. Phys. 9,\n83 (2007).\n[9] P. Cappellaro, et al., Phys. Rev. Lett. 94, 020502 (2005).\n[10] N. Bloembergen, Physica 15, 386 (1949).\n[11] F. Noack, Prog. Nucl. Magn. Reson. Spectrosc. 18, 171\n(1986).\n[12] K. L. Ivanov, A. V. Yukovskaya, and H.-M. Vieth,\nJ. Chem. Phys. 128, 154701 (2008).\n[13] U. Bommerich, et al., Phys. Chem. Chem. Phys. 12,\n10309 (2010).\n[14] Fig. 1(d) is similar to the last scheme in Ref. [9] but is\nsimpler than that. The latter relies on multiple pulse se-\nquences to establish entanglement while the former does\nnot require any rf irradiation to the abundant I spins.\n[15] See supplemental material for experimental details.\n[16] A. Henstra, T.-S. Lin, J. Schmidt, and W. Th. Wencke-\nbach, Chem. Phys. Lett. 165, 6 (1990).\n[17] D. Stehlik and H.-M. Vieth, in Pulsed Magnetic Reso-\nnance NMR, ESR and Optics (Ed. D. M. S. Bagguley,)\n446-477 (Oxford Univ. Press, New York, 1992).\n[18] M. Iinuma, et al., Phys. Rev. Lett. 84, 171 (2000).5\n[19] K. Takeda, Triplet State Dynamic Nuclear Polarization\n(VDM Verlag, Saarbr¨ ucken, 2009).\n[20] D. Rugar, R. Budakian, H. J. Mamin, and B. W. Chui,\nNature (London) 430, 329 (2004).\n[21] M. Poggio and C. L. Degen, Nanotechnol. 21, 342001\n(2010).\n[22] A. Krahn, et al., Phys. Chem. Chem. Phys. 12, 5830\n(2010).\n[23] A. Kagawa, M. Negoro, K. Takeda, and M. Kitagawa,\nRev. Sci. Instrum. 80, 044705 (2009).\n[24] S. R. Hartmann and E. L. Hahn, Phys. Rev. 128, 2042\n(1962).\n[25] A. Pines, M. G. Gibby, and J. S. Waugh, J. Chem. Phys.\n59, 569 (1973).\n[26] N. Khaneja, et al., J. Magn. Reson. 172, 296 (2005).\n[27] C. A. Ryan, O. Moussa, J. Baugh, and R. Laflamme,\nPhys. Rev. Lett. 100, 140501 (2008).\n[28] J. Kohler, et al., Nature (London) 363, 242 (1993).\n[29] J. Wrachtrup, et al., Nature (London) 363, 244 (1993).\n[30] P. Neumann, et al., Science 329, 542 (2010).\nSupplemental Material\nNaphthalene was extensively purified by the zone\nmelting method and then a single crystal of naphtha-\nlene doubly-doped with ∼0.005 mol% pentacene and\n∼1 mol% 2-fluoronaphthalene was grown by the Bridg-\nmanmethod. Itwascutintoapiecewithaweightof ∼2.5\nmg and mounted in a cylindrical cavity (TE011, 12.08\nGHz)sothatthelongaxisofthepentacenemoleculeswas\nparallel to the static magnetic field [1, 2]. Supplementary\nFigure 1 is a detailed drawing of the experimental setup.\nDNP with photoexcited triplet electron spins was per-\nformed in the static field of 4000 G. We show the energy\ndiagram of pentacene in Supplementary Figure 2. 598\nnm laser irradiation was applied to the sample, so that\nthe pentacene molecules underwent intersystem cross-\ning from the excited singlet state to the excited triplet\nstate. The population distribution over the triplet sub-\nlevelswashighlybiasedinthisorientation[3]. Atthemo-\nment, the highnon-equilibriumelectronspin polarization\nwas transferred to the1H spins in pentacene molecules\nby ISE [4]. In the ISE, the microwave irradiation and\nthe field sweep were applied in such a way that the\ninhomogeneously-broadened electron spin packets were\nadiabatically swept over, and the Hartmann-Hahn con-\ndition was satisfied between the electron spins in the ro-\ntating frame and the1H spins in the laboratory frame\nat a certain moment during the adiabatic passage. The\nelectrons were returned to the ground state immediately\nafter the ISE. We accumulated the1H spin polarization\nby repeating the photoexcitation and the ISE at suffi-\nciently long intervals, during which the1H polarization\nlocalized in pentacene diffused to1H spins in the sample.\nIt has been reported that the1H polarization in a single\ncrystal of 0.018 mol% pentacene doped naphthalene has\nbeen enhanced to ∼0.7 in 3000 G at 105 K [5].The sample shuttling from the center of the cavity in\n4000 G to 13.5 centimeters above it in 100 G took ∼0.67\nseconds, and then, the sample stayed for 10 milliseconds\nand returned to an rf coil in 4000 G in ∼0.67 seconds.\nIn such a low field as 100 G, polarization transfer oc-\ncurs even between heteronuclear spins. This spin dy-\nnamicsmechanismhasbeenstudiedfortransferring para-\nhydrogen induced polarization (PHIP) into a molecule in\nsolution [6, 7]. After enhancing the1H polarization to\n∼0.12 by DNP and sample shuttling, the19F spin po-\nlarization measured by π/2 pulse excitation was ∼0.11,\nand therefore, the transfer efficiency of the polarization\nwas∼92%. The whole sample path was thermally insu-\nlated with a double-layered glass tube and cooled with\nnitrogen gas flow down to 233 K in order to suppress\nnaphthalene sublimation and reduce T 1relaxation. T1\nof the1H spins in 4000 G was ∼212 minutes and that\nin 100 G, which was the lowest field on the sample path,\nwas∼34 minutes at 233 K. Therefore, the total magne-\ntization was scarcely decreased by the sample shuttling.\nThe present sample system, which has very long T1and\ncan be hyperpolarized, is a good test-bed for spin ampli-\nfication experiments.\nWe used the Hermite 180◦pulse [8] with a peak am-\nplitude of 140 kHz shown in Supplementary Figure 3a as\nthe quantum operation U. As shown in Supplementary\nFigure 3b, its excitation range is broader than the inho-\nmogeneous broadening of the19F spin in the sample, and\nthe decrease in magnetization with the pulse at the offset\nof more than 300 kHz is less than 0.1%. We also used the\nHermite 180◦pulse with a peak amplitude of 45 kHz, of\nwhich the pulse length was ∼3 times as long as that of\n140 kHz.\nThe1H NMR signal was excited by a 5◦pulse and\nattenuated with -20 dB and, therefore, reduced by a fac-\ntor of more than a hundred on purpose to avoid sat-\nuration of the detection circuit. The1H and19F po-\nlarizations were determined by comparing them with\nthe1H signal of ethanol and the19F signal of 2,3,4-\ntrifluorobenz-aldehyde, respectively, in thermal equilib-\nrium at 4000 G and 233 K. The polarization is defined\nby (N|↑/angbracketright−N|↓/angbracketright)/(N|↑/angbracketright+N|↓/angbracketright), whereN|↑/angbracketrightandN|↓/angbracketrightare\nthe populations of up and down spins. The pulse shape,\nNMR detection, DNP sequence, and motor control were\nprogrammed by a homebuilt spectrometer [9, 10].\n[1] Abrahams, S. C., Robertson, J. M. & White, J. G. The\ncrystal and molecular structure of naphthalene. I. X-ray\nmeasurements. Acta Crystallogr. 2, 233-238 (1949).\n[2] van Strien, A. J. & Schmidt, J. An EPR study of the\ntriplet state of pentacene by electron spin-echo tech-\nniques and laser flash excitation. Chem. Phys. Lett. 70,\n513-517 (1980).\n[3] Sloop, D. J., Yu, H.-L., Lin, T.-S. & Weissman, S. I.6\nElectron-spin echoes of a photo-excited triplet - pen-\ntacene in para-terphenyl crystals. J. Chem. Phys. 75,\n3746-3757 (1981).\n[4] Henstra, A., Lin, T.-S., Schmidt, J. & Wenckebach,\nW. Th. High dynamic nuclear-polarization at room-\ntemperature. Chem. Phys. Lett. 165, 6-10 (1990).\n[5] Takeda, K., Takegoshi, K. & Terao, T. Dynamic nuclear\npolarization by electron spins in the photoexcited triplet\nstate: I.Attainmentofprotonpolarization of0.7at105K\nin naphthalene. J. Phys. Soc. Jpn. 73, 2313-2318 (2004).\n[6] Ivanov, K. L., Yukovskaya, A. V. & Vieth, H.-M. Coher-\nent transfer of hyperpolarization in coupled spin systems\nat variable magnetic field. J. Chem. Phys. 128, 154701\n(2008).\n[7] Bommerich, U. et al. Hyperpolarized19F-MRI:\nparahydrogen-induced polarization and field varia-\ntion enable19F-MRI at low spin density. Phys. Chem.\nChem. Phys. 12, 10309-10312 (2010).\n[8] Warren, W. S. Effects of arbitrary laser or NMR pulse\nshapes on population-inversion and coherence. J. Chem.\nPhys.81, 5437-5448 (1984).\n[9] Takeda, K. A highly integrated FPGA-based nuclear\nmagnetic resonance spectrometer. Rev. Sci. Instrum. 78,\n033103 (2007).\n[10] Takeda, K. OPENCORE NMR: Open-source core mod-\nules for implementing an integrated FPGA-based NMR\nspectrometer. J. Magn. Reson. 192, 218-229 (2008).\nrectangular\nwaveguidecopper disk\nfor matching\nsweep coilcollimator\nlensoptical\nfiber\ngoniometerquartz\ntubecooled nitrogen\n                gas\nsample\nB  field0\nTE011 cylindrical cavity= 4000  G= 100  G\nNMR coil\ntuned at 1H and 19 F13.5 cm sample position for \nhetero -SD  process : stepping motor \ndouble-layered \nglass tube string \nSupplementary Figure 1. Detailed drawing of the ex-\nperimental setup.\nT3\nT2\nT1S1\nS0598 nm\nlaserISC\nIC \ndark\ndecaymicrowave\nSupplementary Figure 2. Energy diagram of pen-\ntacene. ISC and IC mean intersystem crossing and in-\nternal conversion, respectively.a b1\n0\n-1 \n0 200 400200 3001\n0.99\n0.98\nOffset (kHz) Normalized amplitude 0\n0 10 20 50 100150\nTime ( µs) amplitude (kHz) \nSupplementary Figure 3. Hermite 180◦pulse. a , The\namplitude profile of the Hermite 180◦pulse of the peak\namplitude140kHz. b, Theexcitationfrequencyresponse\nto this Hermite 180◦pulse with respect to the carrier\nfrequency offset. The vertical axis shows the remaining\nlongitudinal magnetization after the pulse, which is nor-\nmalized with the initial magnetization."    },    {        "title": "1505.04301v1.Dynamics_of_a_macroscopic_spin_qubit_in_spin_orbit_coupled_Bose_Einstein_condensates.pdf",        "content": "arXiv:1505.04301v1  [quant-ph]  16 May 2015Dynamics of a macroscopic spin qubit in spin-orbit\ncoupled Bose-Einstein condensates\nSh Mardonov1,2,3, M Modugno4,5and E Ya Sherman1,4\n1Department of Physical Chemistry, The University of the Basque C ountry, 48080\nBilbao, Spain\n2The Samarkand Agriculture Institute, 140103 Samarkand, Uzbek istan\n3The Samarkand State University, 140104 Samarkand, Uzbekistan\n4IKERBASQUE Basque Foundation for Science, Bilbao, Spain\n5Department of Theoretical Physics and History of Science, Univer sity of the Basque\nCountry UPV/EHU, 48080 Bilbao, Spain\nE-mail:evgeny.sherman@ehu.eus\nAbstract. We consider a macroscopic spin qubit based on spin-orbit coupled Bos e-\nEinstein condensates, where, in addition to the spin-orbit coupling, spin dynamics\nstrongly depends on the interaction between particles. The evolut ion of the spin\nfor freely expanding, trapped, and externally driven condensate s is investigated. For\ncondensates oscillating at the frequency corresponding to the Ze eman splitting in the\nsynthetic magnetic field, the spin Rabi frequency does not depend on the interaction\nbetween the atoms since it produces only internal forces and does not change the\ntotal momentum. However, interactions and spin-orbit coupling br ing the system\ninto a mixed spin state, where the total spin is inside rather than on t he Bloch\nsphere. This greatly extends the available spin space making it three -dimensional, but\nimposes limitations on the reliable spin manipulation ofsuch a macroscop icqubit. The\nspin dynamics can be modified by introducing suitable spin-dependent initial phases,\ndetermined by the spin-orbit coupling, in the spinor wave function.\nPACS numbers: 03.75.Mn, 67.85.-d\nKeywords : Two-component Bose-Einstein condensate, spin-orbit coupling, spin\ndynamics\nSubmitted to: J. Phys. B: At. Mol. Opt. Phys.Dynamics of a macroscopic spin qubit in spin-orbit coupled B EC 2\n1. Introduction\nThe experimental realization of synthetic magnetic fields and spin-o rbit coupling (SOC)\n[1, 2] in Bose-Einstein condensates (BECs) of pseudospin-1/2 par ticles has provided\nnovel opportunities for visualizing unconventional phenomena in qu antum condensed\nmatter [3, 4]. More recently, also ultracold Fermi gases with synthe tic SOC have been\nproduced and studied [5, 6]. These achievements have motivated an d intense activity,\nand a rich variety of new phases and phenomena induced by the SOC h as been discussed\nboth theoretically and experimentally [7, 8, 9, 10, 11, 12, 13, 14, 15 , 16, 17, 18, 19, 20].\nRecently, it has also been experimentally demonstrated [3, 21] the a bility of a reliable\nmeasurement of coupled spin-coordinate motion.\nOne of the prospective applications of spin-orbit coupled Bose-Eins tein condensates\nconsists in the realization of macroscopic spin qubits [8]. A more detaile d analysis of\nquantum computation based on a two-component BEC was propose d in [22]. The\ngates for performing these operations can be produced by means of the SOC and of an\nexternal synthetic magnetic field. Due to the SOC, a periodic mecha nical motion of\nthe condensate drives the spin dynamics and can cause spin-flip tra nsitions at the Rabi\nfrequency depending on the SOC strength. This technique, known in semiconductor\nphysics as the electric dipole spin resonance, is well suitable for the m anipulation of\nqubits based on the spin of a single electron [23, 24, 25]. For the macr oscopic spin qubits\nbased onBose-Einstein condensate, the physics is different in at lea st two aspects. First,\na relative effect of the SOC compared to the kinetic energy can be mu ch stronger here\nthan in semiconductors. Second, the interaction between the bos ons can have a strong\neffect on the entire spin dynamics.\nHere we study how the spin evolution of a quasi one-dimensional Bos e-Einstein\ncondensate depends on the repulsive interaction between the par ticles and on the SOC\nstrength. The paper is organized as follows. In Section 2 we remind t he reader the\nground state properties of a quasi-one dimensional condensate a nd consider simple spin-\ndipole oscillations. In Section 3 we analyze, by means of the Gross-Pit aevskii approach,\nthedynamics of free, harmonically trapped, andmechanically driven macroscopic qubits\nbased on such a condensate. We assume that the periodic mechanic al driving resonates\nwith the Zeeman transition in the synthetic magnetic field and find diffe rent regimes of\nthe spin qubit operation in terms of the interaction between the ato ms, the driving\nfrequency and amplitude. We show that some control of the spin qu bit state can\nbe achieved by introducing phase factors, dependent on the SOC, in the spinor wave\nfunction. Conclusions will be given in Section 4.\n2. Ground state and spin-dipole oscillations\n2.1. Ground state energy and wave function\nBefore analyzing the spin qubit dynamics, we remind the reader how t o obtain the\nground state of an interacting BEC. In particular, we consider a ha rmonically trappedDynamics of a macroscopic spin qubit in spin-orbit coupled B EC 3\nquasi one-dimensional condensate, tightly bounded in the transv erse directions. The\nsystem can be described by the following effective Hamiltonian, where the interactions\nbetween the atoms are taken into account in the Gross-Pitaevskii form:\n/hatwideH0=/hatwidep2\n2M+Mω2\n0\n2x2+g1|ψ(x)|2. (1)\nHereψ(x) is the condensate wave function, Mis the particle mass, ω0is the frequency\nof the trap (with the corresponding oscillator length aho=/radicalbig\n/planckover2pi1/Mω0), andg1= 2as/planckover2pi1ω⊥\nis the effective one-dimensional interaction constant, with asbeing the scattering length\nof interacting particles, and ω⊥≫ω0being the transverse confinement frequency. For\nfurther calculations we put /planckover2pi1≡M≡1, and measure energy in units of ω0and length\nin units of aho, respectively. All the effects of the interaction are determined by the\ndimensionless parameter /tildewideg1N, where/tildewideg1= 2/tildewideas/tildewideω⊥, where/tildewideasis the scattering length in\nthe units of aho,/tildewideω⊥is the transverse confinement frequency in the units of ω0, andN\nis the number of particles. In physical units, for a condensate of87Rb and an axial\ntrapping frequency ω0= 2π×10 Hz, the unit of time corresponds to 0 .016 s, the unit\nof lengthahocorresponds to 3 .4µm, and the unit of speed ahoω0becomes 0.021 cm/s,\nrespectively. In addition, considering that as= 100aB,aBbeing the Bohr radius, in the\npresence of a transverse confinement with frequency ω⊥= 2π×100 Hz the dimensionless\ncoupling constant /tildewideg1turns out to be of the order of 10−3.\nIn order to find the BEC ground state we minimize the total energy in a properly\ntruncated harmonic oscillator basis. To design the wave function we take the real sum\nof even-order eigenfunctions\nψ0(x) =N1/2nmax/summationdisplay\nn=0C2nϕ2n(x). (2)\nHere\nϕ2n(x) =1/radicalbig\nπ1/2(2n)!22nH2n(x)exp/bracketleftbigg\n−x2\n2/bracketrightbigg\n, (3)\nwhereH2n(x) are the Hermite polynomials, and the normalization is fixed by requirin g\nthat\nnmax/summationdisplay\nn=0C2\n2n= 1. (4)\nThe coefficients C2nare determined by minimizing the total energy Etot, such that\nEmin= min\nC2n{Etot}, (5)\nwhere\nEtot=1\n2/integraldisplay/bracketleftBig\n(ψ′(x))2+x2ψ2(x)+/tildewideg1ψ4(x)/bracketrightBig\ndx, (6)\nand|C2nmax| ≪1.\nFormulas (5) and (6) yield the ground state energy, while the width o f the\ncondensate is defined as:\nwgs=/bracketleftbigg2\nN/integraldisplay\nx2|ψ0(x)|2dx/bracketrightbigg1/2\n. (7)Dynamics of a macroscopic spin qubit in spin-orbit coupled B EC 4\nFigure 1. (Color online) Ground-state probability density of the condensate obtained\nfrom (2)-(6) (blue solid line), compared with the Thomas-Fermi app roximation in (8)\n(red dashed line) for /tildewideg1N= 40.\nIn the non interacting limit, /tildewideg1= 0,ψ0(x) is the ground state of the harmonic\noscillator (nmax= 0), that is a Gaussian function with wgs= 1. In the opposite, strong\ncoupling limit /tildewideg1N≫1, the exact wave function (2) is well reproduced (see Figure 1)\nby the Thomas-Fermi expression\nψTF(x) =√\n3\n2√\nN\nw3/2\nTF/parenleftbig\nw2\nTF−x2/parenrightbig1/2;|x| ≤wTF, (8)\nwherewTF= (3/tildewideg1N/2)1/3.\nIn general, for a qualitative description of the ground state one ca n use instead of\nthe exact wave function (2), the Gaussian ansatz\nψG(x) =/parenleftbiggN\nπ1/2w/parenrightbigg1/2\nexp/bracketleftbigg\n−x2\n2w2/bracketrightbigg\n, (9)\nwhere the width wis single variational parameter for the energy minimization. Then\nthe total energy (6) becomes:\nEtot=N/bracketleftbigg1\n4/parenleftbigg\nw2+1\nw2/parenrightbigg\n+/tildewideg1N\n2(2π)1/2w/bracketrightbigg\n. (10)\nThe latter is minimized with respect to wby solving the equation\ndEtot\ndw=N/bracketleftbigg1\n2/parenleftbigg\nw−1\nw3/parenrightbigg\n−/tildewideg1N\n2(2π)1/2w2/bracketrightbigg\n= 0. (11)\nInthestrongcouplingregime, /tildewideg1N≫1, wehavew≫1sothat-toafirstapproximation\n- the kinetic term ∝1/w3in (11) can be neglected, yielding the following value for the\nwidth of the ground state:\n/tildewidewG=/parenleftbigg/tildewideg1N√\n2π/parenrightbigg1/3\n. (12)\nThe first order correction can be obtained by writing w=/tildewidew+ǫ(ǫ≪1), so that from\n(11) it follows:\nw=/parenleftbigg/tildewideg1N√\n2π/parenrightbigg1/3\n+√\n2π\n3/tildewideg1N. (13)Dynamics of a macroscopic spin qubit in spin-orbit coupled B EC 5\n0 10 20 30 40012345\ng/OverTilde\nl1NEmin,wgs\nFigure 2. (Color online) Ground state energy (black solid line) and condensate width\n(red dashed line) vs. the interaction parameter /tildewideg1N.\nBy substituting (13) in (10) we obtain that the leading term in the gro und state energy\nfor/tildewideg1N≫1 is:\nE[G]\nmin=3\n4N/parenleftbigg/tildewideg1N√\n2π/parenrightbigg2/3\n. (14)\nIn figure 2 we plot the ground state energy and the condensate wid th as a function\nof the interaction, as obtained numerically from (5) and (7), respe ctively. As expected,\nin the strong coupling regime /tildewideg1N≫1 both quantities nicely follow the behavior (not\nshown in the Figure) predicted both by the Gaussian approximation a nd by the exact\nsolution, namely Emin∝(/tildewideg1N)2/3andwgs∝(/tildewideg1N)1/3.\n2.2. Simple spin-dipole oscillations\nLet us now turn to the case of a condensate of pseudospin 1/2 ato ms. Here the system\nis described by a two-component spinor wave function Ψ = [ ψ↑(x,t),ψ↓(x,t)]T, still\nnormalized to the total number of particles N.The interaction energy (third term in\nthe functional (6)) now acquires the form (see, e.g. [9])\nEint=1\n2/tildewideg1/integraldisplay/bracketleftbig\n|ψ↑(x,t)|2+|ψ↓(x,t)|2/bracketrightbig2dx, (15)\nwhere, for simplicity and qualitative analysis, we neglect the depende nce of interatomic\ninteraction on the spin component ↑or↓and characterize all interactions by a single\nconstant/tildewideg1.\nHere we consider spin dipole oscillations, induced by a given small initial s ymmetric\ndisplacement of the two spin components ±ξ. For a qualitative understanding, we\nassume a negligible spin-orbit coupling and a Gaussian form of the wave function\npresented as\nΨG(x) =1√\n2/bracketleftBigg\nψG(x−ξ)\nψG(x+ξ)/bracketrightBigg\n, (16)\nwhereψGis given by (9), and ξ≪w. The corresponding energy is given by:\nE=E[G]\nmin+Esh, (17)Dynamics of a macroscopic spin qubit in spin-orbit coupled B EC 6\nwhereE[G]\nminis defined by (14) and Eshis the shift-dependent contribution:\nEsh=N\n2ξ2/parenleftbigg\n1−/tildewideg1N√\n2πw3/parenrightbigg\n. (18)\nThen, it follows that the corresponding oscillation frequency is\nωsh=/radicalBigg\n1−/tildewideg1N√\n2πw3. (19)\nFor strong interaction ( /tildewideg1N≫1) by substituting (13) in (19) we obtain:\nωsh≈/parenleftBigg√\n2π\n/tildewideg1N/parenrightBigg2/3\n. (20)\nTherefore, for strong interaction the frequency of the spin dipo le oscillations falls as\n(/tildewideg1N)−2/3, and this result is common for the Gaussian ansatz and for the exac t solution;\nit will be useful in the following section.\n3. Spin evolution and particles interaction\n3.1. Hamiltonian, spin density matrix, and purity\nTo consider the evolution of the driven quasi one dimensional pseud ospin-1/2 SOC\ncondensate we begin with the effective Hamiltonian\n/hatwideH=α/hatwideσz/hatwidep+/hatwidep2\n2+∆\n2/hatwideσx+1\n2(x−d(t))2+/tildewideg1|Ψ|2. (21)\nHereαis the SOC constant (see [11] and [12] for comprehensive review on t he SOC in\ncoldatomicgases), /hatwideσxand/hatwideσzarethePauli matrices, ∆isthesynthetic Zeemansplitting,\nandd(t) is the driven displacement of the harmonic trap center as can be ob tained by\na slow motion of the intersection region of laser beams trapping the c ondensate.\nThetwo-componentspinorwavefunctionΨisobtainedasasolutiono fthenonlinear\nSchr¨ odinger equation\ni∂Ψ\n∂t=/hatwideHΨ. (22)\nTo describe spin evolution we introduce the reduced density matrix\nρ(t)≡ |Ψ∝an}bracketri}ht∝an}bracketle{tΨ|=/bracketleftBigg\nρ11(t)ρ12(t)\nρ21(t)ρ22(t)/bracketrightBigg\n, (23)\nwhere we trace out the x−dependence by calculating integrals\nρ11(t) =/integraldisplay\n|ψ↑(x,t)|2dx, ρ22(t) =/integraldisplay\n|ψ↓(x,t)|2dx,\nρ12(t) =/integraldisplay\nψ∗\n↑(x,t)ψ↓(x,t)dx, ρ21(t) =ρ∗\n12(t), (24)\nand, as a result,\ntr(ρ(t))≡ρ11(t)+ρ22(t) =N. (25)Dynamics of a macroscopic spin qubit in spin-orbit coupled B EC 7\nFigure 3. (Color online) (a) Separation of a freely expanding condensate in tw o spin-\nup and spin-down components with opposite anomalous velocities. (b ) Oscillation of\nthe spin-up and spin-down components in the harmonic trap.\nAlthough the |Ψ∝an}bracketri}htstate is pure, integration in (24) produces ρ(t) formally describing a\nmixed state in the spin subspace. One can characterize the resultin g spin state purity\nby a parameter Pdefined as\nP=2\nN2/parenleftbigg\ntr/parenleftbig\nρ2/parenrightbig\n−N2\n2/parenrightbigg\n, (26)\nwhere 0≤P≤1,\ntr/parenleftbig\nρ2/parenrightbig\n=N2+2(|ρ12|2−ρ11ρ22), (27)\nand we omitted the explicit t−dependence for brevity. The system is in the pure state\nwhenP= 1,that is tr(ρ2) =N2withρ11ρ22=|ρ12|2. In the fully mixed state, where\ntr(ρ2) =N2/2, we have P= 0 with\nρ11=ρ22=N\n2, ρ 12= 0. (28)\nThe spin components defined by ∝an}bracketle{t/hatwideσi∝an}bracketri}ht ≡tr(/hatwideσiρ)/Nbecome\n∝an}bracketle{t/hatwideσx∝an}bracketri}ht=2\nNRe(ρ12),∝an}bracketle{t/hatwideσy∝an}bracketri}ht=−2\nNIm(ρ12),\n∝an}bracketle{t/hatwideσz∝an}bracketri}ht=2\nNρ11−1, (29)\nand the purity P=/summationtext3\ni=1∝an}bracketle{t/hatwideσi∝an}bracketri}ht2, which allows one to match the value of Pand the length\nof the spin vector inside the Bloch sphere. For a pure state/summationtext3\ni=1∝an}bracketle{t/hatwideσi∝an}bracketri}ht2= 1, and the\ntotal spin is on the Bloch sphere. Instead, for a fully mixed state/summationtext3\ni=1∝an}bracketle{t/hatwideσi∝an}bracketri}ht2= 0, and\nthe spin null.\n3.2. A simple condensate motion\nLet us suppose that a condensate of interacting spin-orbit couple d particles is located\nin a harmonic trap and characterized by an initial wave function\nΨ0(x,0) =1√\n2ψin(x)/bracketleftBigg\n1\n1/bracketrightBigg\n, (30)Dynamics of a macroscopic spin qubit in spin-orbit coupled B EC 8\n(a)\n024680.00.20.40.60.81.0\ntP/LParen1t/RParen1(b)\n0246810120.00.20.40.60.81.0\nt/LAngleBracket1Σ/Hat\nx/LParen1t/RParen1/RAngleBracket1\nFigure 4. (Color online) ( a) Purity and ( b) spin component as a function of time for\na condensate released from the trap, for α= 0.2. The lines correspond to /tildewideg1N= 0\n(black solid line; for the purity cf. (32)), /tildewideg1N= 10 (red dashed line), and /tildewideg1N= 20\n(blue dot-dashed line).\nwith the spin parallel to the x−axis.\nThe spin-orbit coupling modifies the commutator corresponding to t he velocity\noperator by introducing the spin-dependent contribution as:\n/hatwidev≡i/bracketleftbigg/hatwidep2\n2+α/hatwideσz/hatwidep,/hatwidex/bracketrightbigg\n=/hatwidep+α/hatwideσz. (31)\nThe effect of the spin-dependent anomalous velocity term on the co ndensate motion\nwas clearly observed experimentally in [3] as the spin-induced dipole os cillations and in\n[21] as the Zitterbewegung . Since the initial spin in (30) is parallel to the x-axis, the\nexpectation value of the velocity vanishes, ∝an}bracketle{t/hatwidev∝an}bracketri}ht= 0.\nFree and oscillating motion of the BEC is shown in figure 3(a) and figure 3(b),\nrespectively. When one switches off the trap, the condensate is se t free, and the two\nspin components start to move apart and the condensate splits up , see figure 3(a).\nEach spin-projected component broadens due to the Heisenberg momentum-coordinate\nuncertainty and interaction. The former effect is characterized b y a rate proportional\nto 1/wgs.At large/tildewideg1N,the width wgs∼(/tildewideg1N)1/3,and, as a result, the quantum\nmechanical broadening rate decreases as ( /tildewideg1N)−1/3.At the same time, the repulsion\nbetween the spin-polarized components accelerates the peak sep aration [26] and leads\nto the asymptotic separation velocity ∼(/tildewideg1N)1/2. This acceleration by repulsion leads\nto opposite time-dependent phase factors in ψ↑(x,t) andψ↓(x,t) in (24) and, therefore,\nresults in decreasing in |ρ12(t)|and in the purity. Thus, with the increase in the\ninteraction, the purity and the x−spin component asymptotically tend to zero faster,\nas demonstrated in figure 4. For a noninteracting condensate with the initial Gaussian\nwave function ψin∼exp(−x2/2w2) the purity can be written analytically as\nP0(t) = exp/bracketleftBigg\n−2/parenleftbiggαt\nw/parenrightbigg2/bracketrightBigg\n. (32)\nIn the presence of the trap (figure 3(b)), the anomalous velocity in (31) causes spin\ncomponents (spin-dipole) oscillations with a characteristic frequen cy of the oder of ωshDynamics of a macroscopic spin qubit in spin-orbit coupled B EC 9\n(a)\n0102030405060700.00.20.40.60.81.0\ntP/LParen1t/RParen1\n(b)\n0102030405060700.00.20.40.60.81.0\nt/LAngleBracket1Σ/Hat\nx/LParen1t/RParen1/RAngleBracket1\n(c)\n010203040506070/MinuΣ3/MinuΣ2/MinuΣ10123\nt/LAngleBracket1xΣ/Hat\nz/RAngleBracket1\nFigure 5. (Color online) (a) Purity, (b) spin component, and (c) spin dipole mom ent\nas a function of time for the system in the harmonic trap with α= 0.2,∆ = 0, d0= 0.\nThe different lines correspond to /tildewideg1N= 0 (black solid line), /tildewideg1N= 10 (red dashed\nline),/tildewideg1N= 20 (blue dot-dashed line), and /tildewideg1N= 60 (green dotted line).\nin (20). With the increase in the interatomic interaction, the freque ncyωshdecreases\nand, therefore, the amplitude of the oscillations arising due to the a nomalous velocity\n(∼α/ωsh) increases. As a result, the acceleration and separation of the sp in-projected\ncomponents increase, the off-diagonal components of the densit y matrix in (24) became\nsmaller, and the minimum in P(t) rapidly decreases to P(t)≪1 as shown by the exact\nnumerical results presented in figures 5(a) and (b) [27]. In figure 5 (c) we show the\ncorresponding evolution of spin density dipole moment\n∝an}bracketle{tx/hatwideσz∝an}bracketri}ht=1\nN/integraldisplay\nΨ†x/hatwideσzΨdx. (33)\nHere the oscillation frequency is a factor of two larger than that of the spin density\noscillation.\n3.3. Spin-qubit dynamics and the Rabi frequency\nTo manipulate the macroscopic spin qubit, the center of the trap is d riven harmonically\nat the frequency corresponding to the Zeeman splitting ∆ as\nd(t) =d0sin(t∆), (34)\nwhered0is an arbitrary amplitude and the corresponding spin rotation Rabi f requency\nΩRis defined as αd0∆.At ∆≪1,as will be considered here, for a noninteractingDynamics of a macroscopic spin qubit in spin-orbit coupled B EC 10\n(a)\n0 50 100 150 2000.00.20.40.60.81.0\ntP/LParen1t/RParen1(b)\n050100 150 200/MinuΣ1.0/MinuΣ0.50.00.51.0\nt/LAngleBracket1Σ/Hat\nx/LParen1t/RParen1/RAngleBracket1\nFigure 6. (Color online) ( a) Purity and ( b) spin component as a function of time for\na driven condensate with α= 0.1,∆ = 0.1, d0= 2. The lines correspond to /tildewideg1N= 0\n(black solid line), /tildewideg1N= 10 (red dashed line), and /tildewideg1N= 20 (blue dot-dashed line).\ncondensate and a very weak spin-orbit coupling, the spin componen t∝an}bracketle{t/hatwideσx(t)∝an}bracketri}htis expected\nto oscillate approximately as\n∝an}bracketle{t/hatwideσx(t)∝an}bracketri}ht= cos(Ω Rt). (35)\nThe corresponding spin-flip time Tsfis\nTsf=π\nΩR. (36)\nFigure 6 shows the time dependence of the purity and the spin of the condensate for\ngivenα,d0, and ∆ at different interatomic interactions. In figure 6(a) one can see that\nthe increase of /tildewideg1Nenhances the variation of the purity (cf. Fig 5(a)). This variation\nprevents a precise manipulation of the spin-qubit state in the conde nsate [28]. It follows\nfrom figure 6(b) that although increasing the interaction strongly modifies the spin\ndynamics, it roughly conserves the spin-flip time Tsf= 50π, see (36). To demonstrate\nthe role of the SOC coupling strength αand interatomic interaction at nominally the\nsame Rabi frequency Ω R,we calculated the spin dynamics presented in Figure 7. By\ncomparing Figures 6 and 7(a),(b) we conclude that the increase in th e SOC, at the\nsame Rabi frequency, causes an increase in the variation of the pu rity and of the spin\ncomponent. These results show that to achieve a required Rabi fr equency and a reliable\ncontrol of the spin, it is better to increase the driving amplitude d0rather than the spin-\norbit couping α.The increase in the SOC strength can result in losing the spin state\npurity and decreasing the spin length. Figure 7(c) shows the irregu lar spin evolution of\nthe condensate inside the Bloch sphere. In figure 7(a), for α= 0.2 and/tildewideg1N= 20,the\npurity decreases almost to zero, placing the spin close to the cente r of the Bloch sphere,\nas can be seen in figure 7(c). It follows from Figures 6(b) and 7(b) t hat in order to\nprotect pure macroscopic spin-qubit states, the Rabi frequenc y should be small. Then,\ntaking into account that the displacement of the spin-projected w ave packet is of the\norder ofα(/tildewideg1N)2/3and the packet width is of the order of ( /tildewideg1N)1/3, we conclude that\nforα/greaterorsimilar(/tildewideg1N)−1/3, the purity of the driven state tends to zero. As a result, the Rab i\nfrequency for the pure state evolution is strongly limited by the inte raction between the\nparticles and cannot greatly exceed d0∆/(/tildewideg1N)1/3.Dynamics of a macroscopic spin qubit in spin-orbit coupled B EC 11\n(a)\n0 50 100 150 2000.00.20.40.60.81.0\ntP/LParen1t/RParen1(b)\n050100 150 200/MinuΣ1.0/MinuΣ0.50.00.51.0\nt/LAngleBracket1Σ/Hat\nx/LParen1t/RParen1/RAngleBracket1\nFigure 7. (Color online) ( a) Purity, ( b) spin component, and ( c) spatial trajectory of\nthe spin inside the Bloch sphere for the driven BEC with α= 0.2,∆ = 0.1, d0= 1\nresulting in the same Rabi frequency as in Figure (6). In Figures ( a) and (b) the lines\ncorrespond to: /tildewideg1N= 0 (black solid line), /tildewideg1N= 10 (red dashed line), and /tildewideg1N= 20\n(blue dotted line). At /tildewideg1N= 0,the time dependence of ∝an}bracketle{tσx∝an}bracketri}htis rather accurately\ndescribed by cos(Ω Rt) formula, corresponding to a relatively small variation in the\npurity, 1 −P(t)≪1.With the increase in /tildewideg1N,the purity variation increases and the\nbehavior of ∝an}bracketle{tσx∝an}bracketri}htdeviates stronger from the conventional cos(Ω Rt) dependence. ( c)\nHere the interaction is fixed to /tildewideg1N= 20. The green and red vectors correspond to the\ninitial and final states of the spin, respectively. Here the final time is fixed to tfin=Tsf,\nsee (36).\nIn addition, it is interesting to note that for /tildewideg1N≫1,where the spin dipole\noscillates at the frequency of the order of ( /tildewideg1N)−2/3(as given by (20)), the perturbation\ndue to the trap motion is in the high-frequency limit already at ∆ ≥(/tildewideg1N)−2/3, having\na qualitative influence on the spin dynamics [29, 30, 31].\n3.4. Phase factors due to spin-orbit coupling\nThe above results show that the spin-dependent velocity in (31), a long with the\ninteratomic repulsion, results in decreasing the spin state purity an d produces irregular\nspin motion inside the Bloch sphere. To reduce the effect of these an omalous velocities\nand to prevent the resulting fast separation (with the relative velo city of 2α) of the spin\ncomponents, we compensate them by introducing coordinate-dep endent phases (similar\nto the Bragg factors) in the wave function [32]. To demonstrate th e effect of theseDynamics of a macroscopic spin qubit in spin-orbit coupled B EC 12\n(a)\n0 50 100 150 2000.00.20.40.60.81.0\ntP/LParen1t/RParen1(b)\n050100 150 200/MinuΣ1.0/MinuΣ0.50.00.51.0\nt/LAngleBracket1Σ/Hat\nx/LParen1t/RParen1/RAngleBracket1\nFigure 8. (Color online) ( a) Purity, ( b) spin component, and ( c) trajectory of the\nspin inside the Bloch sphere for a driven BEC with initial phases as in (37 ) and\nα= 0.2,∆ = 0.1, d0= 1.In Figures ( a) and (b) the black solid line is for /tildewideg1N= 0,\nthe red dashed line is for /tildewideg1N= 10, and the blue dotted line is for /tildewideg1N= 20. In Figure\n(c) the interaction is /tildewideg1N= 20. The green and red vectors correspond to the initial\nand final states of the spin, respectively ( tfin=Tsf). The initial spin state (a solid-line\ncircle with white filling) is inside the Bloch sphere since ∝an}bracketle{tσx(t= 0)∝an}bracketri}ht=/radicalbig\nP(0), and\nP(0)<1 due to the mixed character in the spin subspace of the state in (37 ).\nphase factors, we construct the initial spinor Ψ α(x,0) by a coordinate-dependent SU(2)\nrotation [33] of the state with ∝an}bracketle{tσx∝an}bracketri}ht= 1 in (30) as\nΨα(x,0) =e−iαx/hatwideσzΨ0(x,0) =ψin(x)√\n2/bracketleftBigg\ne−iαx\neiαx/bracketrightBigg\n. (37)\nThe expectation value of the velocity (31) at each component ψin(x)exp(±iαx) is zero,\nand, as a result, the α-induced separation of spin components vanishes, making, as can\nbe easily seen [33], the spinor (37) the stationary state of the spin- orbit coupled BEC\nin the Gross-Pitaevskii approximation.\nIn terms of the spin density matrix (24), the state (37) is mixed. Fo r a Gaussian\ncondensate with the width w, we get the following expression for the purity at t= 0\nP[G]\nα(0) = exp/bracketleftbig\n−2(αw)2/bracketrightbig\n. (38)Dynamics of a macroscopic spin qubit in spin-orbit coupled B EC 13\nInstead, in the case of a Thomas-Fermi wave function as in (8), in t he limitαwTF≫1\nthe initial purity behaves as\nP[TF]\nα(0)∼cos2(2αwTF)\n(αwTF)4. (39)\nBoth cases are characterized by a rapid decrease as αwTFis increased [25].\nIn the absence of external driving, the spin components and purit y of (37) state\nremain constant. With the driving, spinor components evolve with tim e and the\nobservables show evolution quantitatively different from that pres ented in Figure 7. In\nfigure 8 we show the analog of figure 7 for the initial state in (37), wit hψin(x) =ψ0(x)\ngiven by (2)-(5). By comparing these Figures one can see that the inclusion of the\nspin-dependent phases in (37) strongly reduces the oscillations in t hex−component of\nthe spin, making the spin trajectory more regular and decreasing t he variations in the\npurityP(t) compared to the initial state without these phase factors.\nA general effect of the interatomic interaction can be seen in both fi gures 7 and 8.\nNamely, for smaller interactions /tildewideg1N, the destructive role of the interatomic repulsion\non the spin state purity is reduced and the spin dynamics becomes mo re regular. As a\nresult, at smaller /tildewideg1Nthe spin trajectory is located closer to the Bloch sphere.\n4. Conclusions\nWe have considered the dynamics of freely expanding and harmonica lly driven\nmacroscopic spin qubits based on quasi one-dimensional spin-orbit coupled Bose-\nEinstein condensates in a synthetic Zeeman field. The resulting evolu tion strongly\ndepends in a nontrivial way on the spin-orbit coupling and interaction between the\nbosons. On one hand, spin-orbit coupling leads to the driven spin qub it dynamics. On\nthe other hand, it leads to a spin-dependent anomalous velocity cau sing spin splitting\nof the initial wave packet and reducing the purity of the spin state b y decreasing the\noff-diagonal components of the spin density matrix. This destruct ive influence of spin-\norbit coupling is enhanced by interatomic repulsion. The effects of th e repulsion can be\ninterpreted in terms of the increase in the spatial width of the cond ensate and the\ncorresponding decrease in the spin dipole oscillation frequency with t he interaction\nstrength. The joint influence of the repulsion and spin-orbit couplin g can spatially\nseparate and modify the spin components stronger than just the spin-orbit coupling\nand result in stronger irregularities in the spin dynamics. The spin-flip Rabi frequency\nremains, however, almost unchangedinthepresence oftheintera tomicinteractions since\nthey lead to only internal forces and do not change the condensat e momentum. As a\nresult, to preserve the evolution within a high-purity spin-qubit sta te, with the spin\nbeing always close to the Bloch sphere, the spin-orbit coupling should be weak and, due\nto this weakness, the spin-rotation Rabi frequency should be sma ll and spin rotation\nshould take a long time. The destructive effect of both the spin-orb it coupling and\ninteratomicrepulsiononthepurityofthespinstatecanbecontrolla blyandconsiderablyDynamics of a macroscopic spin qubit in spin-orbit coupled B EC 14\nreduced, although not completely removed, by introducing spin-de pendent Bragg-like\nphase factors in the initial spinor wave function.\nAcknowledgments\nThis work was supported by the University of Basque Country UPV/ EHU under\nprogram UFI 11/55, Spanish MEC (FIS2012-36673-C03-01 and FI S2012-36673-C03-\n03), and Grupos Consolidados UPV/EHU del Gobierno Vasco (IT-47 2-10). S.M.\nacknowledges EU-funded Erasmus Mundus Action 2 eASTANA, “evr oAsian Starter for\nthe Technical Academic Programme” (Agreement No. 2001-2571/ 001-001-EMA2).\nReferences\n[1] Lin Y-J, Compton R L, K Jim´ enez-Garcia, Porto J V and Spielman I B 2009Nature462628\n[2] Lin Y-J, Jim´ enez-Garc´ ıa K and Spielman I B 2011 Nature47183\n[3] Zhang J-Y, Ji S-C, Z Chen, Zhang L, Z-D Du, Yan B, G-S Pan, Zha o B, Deng Y-J, Zhai H, Chen\nS and Pan J-W 2012 Phys. Rev. Lett. 109115301\n[4] Ji S-C, Zhang J-Y, L Zhang, Du Z-D, Zheng W, Deng Y-J, Zhai H, Chen Sh and Pan J-W 2014\nNature Physics 10314\n[5] Wang P, Yu Z-Q, Fu Z, Miao J, Huang L, Chai Sh, Zhai H and Zhang J 2012Phys. Rev. Lett. 109\n095301\n[6] Cheuk L W, Sommer A T, Hadzibabic Z, Yefsah T, Bakr W S and Zwierle in M W 2012 Phys.\nRev. Lett. 109095302\n[7] Liu X-J, Borunda M F, Liu X and Sinova J 2009 Phys. Rev. Lett. 102046402\n[8] Stanescu T D, Anderson B and Galitski V 2008 Phys. Rev. A78023616\n[9] Li Y, Martone G I and Stringari S 2012 EPL9956008\nMartone G I, Li Y, Pitaevskii L P and Stringari S 2012 Phys. Rev. A86063621\n[10] Zhang Y, Mao L and Zhang Ch 2012 Phys. Rev. Lett. 108035302\n[11] Zhai H 2012 Int. J. Mod. Phys. B261230001\n[12] Galitski V and Spielman I B 2013 Nature49449\n[13] Zhang Y, Chen G and Zhang Ch 2013 Scientific Reports 31937\n[14] Achilleos V, Frantzeskakis D J, Kevrekidis P G and Pelinovsky D E 20 13Phys. Rev. Lett. 110\n264101\n[15] Ozawa T, Pitaevskii L P and Stringari S 2013 Phys. Rev. A87063610\n[16] Wilson R M, Anderson B M and Clark Ch W 2013 Phys. Rev. Lett. 111185303\n[17] L¨ u Q-Q and Sheehy D E 2013 Phys. Rev. A88043645\n[18] Xianlong G 2013 Phys. Rev. A87023628\n[19] Anderson B M, Spielman I B and Juzeliu˜ nas G 2013 Phys. Rev. Lett. 111125301\n[20] Dong L, Zhou L, Wu B, Ramachandhran B and Pu H 2014 Phys. Rev. A89011602\n[21] Qu Ch, Hamner Ch, Gong M, Zhang Ch and Engels P 2013 Phys. Rev. A88021604\n[22] Byrnes T, Wen K and Yamamoto Y 2012 Phys. Rev. A85040306\n[23] Nowack K C, Koppens F H L, Nazarov Yu V and Vandersypen L M K 2 007Science3181430\n[24] Rashba E I and Efros Al L 2003 Phys. Rev. Lett. 91126405\n[25] It is interesting to mention that an increase in spin-orbit coupling strength after a certain value\nleads to a less efficient spin driving as shown with perturbation theory approach by Li R, You\nJ Q, Sun C P and Nori F 2013 Phys. Rev. Lett. 111086805\n[26] This procedure models the von Neumann quantum spin measurem ent, see Sherman E Ya and\nSokolovski D 2014 New J. Phys. 16015013\n[27] An analysis of the spin-dipole oscillations based on the sum rules wa s presented in [9]Dynamics of a macroscopic spin qubit in spin-orbit coupled B EC 15\n[28] This low precision of the spin control can be seen as a general fe ature of the systems where\nexternal perturbation strongly drives the orbital motion. See, e .g. Khomitsky D V, Gulyaev L\nV and Sherman E Ya 2012 Phys. Rev. B85125312\n[29] A technique to study and engineer the high-frequency behavio r has been introduced in Bukov M,\nD’Alessio L and Polkovnikov A 2014 arXiv:1407.4803. Its application to t he spin dynamics of\nthe spin-orbit coupled BEC goes far beyond the scope of this paper , however.\n[30] Optimal control theory proposed by Budagosky J A and Castr o A 2015 The European Physical\nJournal B8815 can potentially be applied to the engineering of a d(t)−dependence more\ncomplicated than a harmonic oscillation.\n[31] Xiong B, Zheng J-H and Wang D-W 2014 arXiv:1410.8444 analyzed t he condensate driving in\nterms of the multichannel quantum interference.\n[32] These phase-related Josephson effect in the spin-orbit couple d BEC in a double-well potential\nhas been recently analyzed in Garcia-March M A, Mazzarella G, Dell’Ann a L, Juli´ a-D´ ıaz B,\nSalasnich L and Polls A 2014 Phys. Rev. A89063607\nCitro R and Naddeo A 2014 arXiv:1405.5356\n[33] Tokatly I V and Sherman E Ya 2010 Phys. Rev. B82161305"    },    {        "title": "0706.1514v1.Electron_and_hole_spin_dynamics_and_decoherence_in_quantum_dots.pdf",        "content": "arXiv:0706.1514v1  [cond-mat.mes-hall]  11 Jun 2007November 15, 2018 9:7 World Scientific Review Volume - 9in x 6i n decrev\nChapter 1\nElectron and hole spin dynamics and decoherence in\nquantum dots\nD. Klauser, D. V. Bulaev, W. A. Coish and Daniel Loss\nDepartment of Physics and Astronomy, University of Basel,\nKlingelbergstrasse 82, CH-4056 Basel, Switzerland\nIn this article we review our work on the dynamics and decoher ence\nof electron and hole spins in single and double quantum dots. The first\npart, on electron spins, focuses on decoherence induced via the hyperfine\ninteraction while the second part covers decoherence and re laxation of\nheavy-holespins duetospin-orbit interaction as well as th emanipulation\nof heavy-hole spin using electric dipole spin resonance.\n1.1. Introduction\nThe Loss-DiVincenzo proposal1to use the spin of a single electron con-\nfined in a quantum dot as a qubit for quantum computation has trigge red\nsignificant interest in the dynamics and control of single spins in quan tum\ndots. This has led to numerous exciting experimental achievements , among\nthem the realization of single electrons in single dots2,3as well as double\ndots,4–6the implementation of single-spin read out,7,8the demonstration\nof the√\nSWAP operation via pulsed exchange interaction9and the mea-\nsurement of single-spin ESR.10For a detailed account of the progress in\nimplementing the Loss-DiVincenzo proposal, see the extensive revie ws in\nRefs. 11 and 12.\nOn the theoretical side, one focus was, and still is, the investigatio n of\nthe decoherence induced by the nuclei in the host material via the h yperfine\ninteraction. The first part of this review article is devoted to the dis cussion\nof the rich spin dynamics that results from the hyperfine interactio n. We\nfirst give an introduction to hyperfine interaction in quantum dots ( Sec.\n1.2). Subsequently, we discuss dynamics under the influence of hyp erfine\ninteraction for the case of a single spin in a single dot (Sec. 1.3) and fo r a\ndouble dot with one electron in each dot (Sec. 1.4). To conclude the p art\n1November 15, 2018 9:7 World Scientific Review Volume - 9in x 6i n decrev\n2 D. Klauser et al.\nabout hyperfine interaction, we discuss the idea of narrowing the n uclear\nspin state in order to increase the spin coherence time (Sec. 1.5).\nThe second part of the article is devoted to the dynamics and the ma -\nnipulation of heavy-hole spins in quantum dots. The motivation to stu dy\nhole spins comes from the fact that the valence band has p-symmetry and\nthus the hyperfine interaction with lattice nuclei for holes is suppre ssed in\ncomparison to that of the conduction band electrons. As a conseq uence,\nthe main interest for hole spin dynamics is the relaxation and decoher ence\ndue to spin-orbit interaction and we discuss this in Sec. 1.6. The next task\ntowardsusing hole spins as qubits for quantum computation is ofcou rse the\ncoherent manipulation of single hole spins. A potentially powerful met hod\nto achieve coherent manipulation of spins is electric dipole spin resona nce\n(EDSR). An analysis of EDSR for heavy holes in quantum dots will be\npresented in Sec. 1.7.\n1.2. Hyperfine interaction for electrons in quantum dots\nIn this part of the article concerning electron spin decoherence we assume\nthat the orbital level spacing is much larger than the typical energ y scale of\nthe hyperfine interaction. This is the case in typical lateral quantu m dots\ncontaining single electrons and allows one to write an effective hyperfi ne\nHamiltonian Hhffor a single electron confined to the quantum-dot orbital\nground state ψ0\nHhf=h·S,h=Aν/summationdisplay\nk|ψ0(rk)|2Ik, (1.1)\nwhereSis the spin-1/2 operator for a single electron and Ikis the spin\noperator for the nuclear spin at lattice site k, whileνis the volume of the\ncrystal unit cell and Ais the hyperfine coupling strength. For GaAs, which\nis mostly used for the fabrication of lateral dots, the average hyp erfine\ncoupling strength weighted by the natural abundance of each isot ope is\nA≈90µeV.13In Fig. 1.1 the hyperfine coupling of the electron spin in\na lateral double quantum dot is illustrated. The electron spin dynamic s\nunderHhfhave been studied under various approximations and in different\nparameter regimes. For an extensive overview, see reviews in Refs . 11, 14\nand 15. Here, we briefly mention parts of this study before we focu s on a\nfew cases of special interest. The first analysis of electron spin dy namics\nunderHhfin this context showed that the long-time longitudinal spin-flip\nprobability is ∼1/p2N,16i.e., this probability is suppressed in the limitNovember 15, 2018 9:7 World Scientific Review Volume - 9in x 6i n decrev\nElectron and hole spin dynamics and decoherence in quantum d ots 3\nFig. 1.1. A double quantum dot. Top-gates are set to a voltage configuration that con-\nfines the electrons in the two-dimensional electron gas (gre en) to quantum dots (yellow).\nThe blue line indicates the envelope wave function of the ele ctron (blue arrow). The\nhyperfine interaction with a particular nuclear spins (red a rrows) is proportional to the\nenvelope wave function squared at the position of the nuclei . Thus the nuclear spins in\nthe center are drawn bigger since they couple stronger to the electron spin.\nof large nuclear spin polarization pand large number Nof nuclear spins\nin the dot. An exact solution for the case of full polarization ( p= 1)\ngives, for both transverse and longitudinal electron spin compone nts, a\nlong-time power law decay ∼1/t3/2by a fraction ∼1/Non a timescale of\nτ∼/planckover2pi1N/A∼1µs(for aGaAs dot with N∼105).17The fact that this exact\nsolution shows a non-exponential decay demonstrates the non-M arkovian\nbehavior of the nuclear spin bath. For non-fully polarized systems p <1\nand in the limit of large magnetic fields (or high polarization p≫1/√\nN),\nthe transverse electron spin undergoes a Gaussian decay17,18on a timescale\nτ∼/planckover2pi1√\nN/A/radicalbig\n1−p2(τ∼5nsfor GaAs with p≪1 andN∼105).19This\nGaussian decay can be reversed using a spin-echo sequence or by p reparing\nthe nuclear spin system in an eigenstate of hz.19\nSeveral methods to prepare the nuclear spin system have recent ly been\nsuggested20–22and we discuss one of these methods21in Sec. 1.5. Once\nthe nuclear spin system is prepared in an eigenstate, the electron s pin co-\nherence is on one hand limited by dynamics in the nuclear spin system\ndriven by the dipole-dipole interaction for which a worst case estimat e21\ngives∼100µsand on the other hand, even for an eigenstate of hz, there\nis decoherence due to the flip-flop dynamics which can be important a t\ntimes∼/planckover2pi1N/A∼1µs(or less, depending on the size of the electron Zee-\nman splitting). For the decay of nuclear spin polarization experiment s\nsuggest timescales up to tens of seconds23–26and hysteretic behavior of\nthe nuclear spin polarization with respect to the external magnetic fieldNovember 15, 2018 9:7 World Scientific Review Volume - 9in x 6i n decrev\n4 D. Klauser et al.\nhas been observed.27Further, measurements of the transport current in\nthe so-called spin-blockade regime28revealed hysteretic behavior with re-\nspect to the magnetic field23,29and bistable behavior in time,23which is\nattributed to a bistability in the nuclear spin polarization. Further, v ery\nrecent experiments suggest a strong dependence of the nuclear field corre-\nlation time depending on whether an electron is present in the dot or n ot\nand thus hyperfine mediated nuclear spin flips are a possible mechanis m for\nnuclear spin diffusion.24This last mechanism has been estimated to lead\nto fluctuations of nuclear spin polarization on a timescale of ∼100µs.21\n1.3. Single-electron spin decoherence\nIn this section we look in more detail at hyperfine-induced decohere nce for\na single spin in a quantum dot in the regime of large Zeeman splitting\nǫz=gµBbz(due to an externally applied magnetic field bz). Ifǫzis much\nlarger than σ= [Tr{ρI(hz−h0)2}]1/2, withh0= Tr{ρIhz}(whereρIis the\ndensity matrix of the nuclear spin system), we may neglect the tran sverse\ntermS⊥·h⊥and find that the Hamiltonian is simply\nH0= (ǫz+hz)Sz. (1.2)\nThisHamiltonianjustinducesprecessionaroundthe z-axiswithafrequency\nthat is determined by the eigenvalue hn\nzofhz, wherehz|n∝an}bracketri}ht=hn\nz|n∝an}bracketri}ht. For a\nlargebut finite number of nuclearspins ( N∼105for lateral GaAs dots) the\neigenvalues hn\nzare Gaussian distributed (due to the central limit theorem)\nwith meanh0and variance σ≈/planckover2pi1A/√\nN.19Calculatingthe dynamicsunder\nH0(which is valid up to a timescale of ∼ǫz/σ2∼1µs, where the transverse\nterms become relevant) leads to a Gaussian decay of the transver seelectron\nspin state |+∝an}bracketri}ht= (| ↑∝an}bracketri}ht+| ↓∝an}bracketri}ht)/√\n2:19\nC0\n++(t) =1√\n2πσ/integraldisplay∞\n−∞dhn\nze„\n−(hnz−h0)2\n2σ2«\n|∝an}bracketle{tn|⊗∝an}bracketle{t+|e(−iH0t)|+∝an}bracketri}ht⊗|n∝an}bracketri}ht|2\n=1\n2+1\n2e“\n−t2\n2τ2”\ncos[(ǫz+h0)t];τ=1\nσ=/radicalBigg\nN\n1−p22/planckover2pi1\nA.(1.3)\nHere again, pdenotes the polarization and for an unpolarized GaAs quan-\ntum dot with N∼105we findτ∼5ns. Applying an additional ac driving\nfield with amplitude balong thex-direction leads to single-spin ESR. As-\nsuming again that ǫz≫σ, we have the Hamiltonian\nHESR=H0+bcos(ωt)Sx. (1.4)November 15, 2018 9:7 World Scientific Review Volume - 9in x 6i n decrev\nElectron and hole spin dynamics and decoherence in quantum d ots 5\nIn a rotating-waveapproximation(which is valid for ( b/ǫz)2≪1) the decay\nof the driven Rabi oscillations is given by30\nCESR\n↑↑(t)∼1−C+/radicalbigg\nb\n8σ2tcos/parenleftbiggb\n2t+π\n4/parenrightbigg\n+O/parenleftbigg1\nt3/2/parenrightbigg\n,(1.5)\nfort/greaterorsimilarmax/parenleftbig\n1/σ,1/b,b/2σ2/parenrightbig\nandǫz+h0−ω= 0. Here,CESR\n↑↑(t) is defined\nin the same way as C0\n++(t) in Eq. (1.3). The time-independent constant\nis given by C= exp(b2/8σ2)erfc(b/√\n8σ)√\n2πb/8σ. The two interesting\nfeatures of the decay are the slow ( ∼1/√\nt) power law and the universal\nphase shift of π/4. The fact that the power law already becomes valid after\na short time τ∼15ns(forb≈σ) preserves the coherence over a long\ntime, which makes the Rabi oscillations visible even when the Rabi perio d\nis much longer than the timescale τ∼15nsfor transverse spin decay. Both\nthe universal phase shift and the non-exponential decay have re cently been\nobservedin experiment.30In order to take correctionsdue to the transverse\ntermsS⊥·h⊥into account, a more elaborate calculation is required. The\nHamiltonian with flip-flop terms (but without a driving field) takes the\nform\nHff=H0+1\n2(S+h−+S−h+). (1.6)\nIn Ref. 19 a systematic calculation taking into account these so-ca lled flip-\nflop terms was performed using a generalized master equation, valid in the\nlimit of large magnetic field or large polarization. This calculation shows\nthat even for an eigenstate of hz, for which the Gaussian decay in Eq. (1.3)\nvanishes, the electron spin undergoes nontrivial non-Markovian d ecay on a\ntimescale /planckover2pi1N/A∼10µs.\nOthercalculations31–33givemicrosecondtimescalesforthe electronspin\ndecoherence due to electron-nuclear spin flip-flops processes. T he results\nin Ref. 31 suggest that also the decoherence due to dynamics in the nu-\nclear spin system via electron mediated nuclear dipole-dipole interact ion\nis suppressed by a spin echo and thus that the spin-echo decay time may\nbe considerably different from the (not ensemble averaged) free- induction\ndecay.\n1.4. Singlet-triplet decoherence in a double quantum qot\nWe now move on to discuss hyperfine induced decoherence in a double\nquantum dot. The effective Hamiltonian in the subspace of one electr on\non each dot is best written in terms of the sum and difference of elect ronNovember 15, 2018 9:7 World Scientific Review Volume - 9in x 6i n decrev\n6 D. Klauser et al.\nspin and collective nuclear spin operators: S=S1+S2,δS=S1−S2and\nh=1\n2(h1+h2),δh=1\n2(h1−h2):\nHdd(t) =ǫzSz+h·S+δh·δS+J\n2S·S−J. (1.7)\nHere,Jis the Heisenbergexchangecouplingbetweenthe twoelectronspins .\nSimilar to the single-dot case, we assume that the Zeeman splitting is m uch\nlarger than ∝an}bracketle{tδh∝an}bracketri}htrmsand∝an}bracketle{thi∝an}bracketri}htrms, where∝an}bracketle{tO∝an}bracketri}htrms= [Tr{ρI(O −∝an}bracketle{tO∝an}bracketri}ht)2}]1/2is\nthe root-mean-square expectation value of the operator Owith respect to\nthe nuclearspin state ρI. Under these conditions the relevantspin Hamilto-\nnian becomes block diagonal with blocks labeled by the total electron spin\nprojection along the magnetic field Sz. In the subspace of Sz= 0 (singlet\n|S∝an}bracketri}ht, and triplet |T0∝an}bracketri}ht) the Hamiltonian can be written as21,34\nHsz0(t) =J\n2S·S+(δhz+δbz)δSz (1.8)\nHere,δbzis the inhomogeneity of the externally applied classical static\nmagnetic field with δbz≪ǫz, while the nuclear difference field δhzis\nGaussian distributed, as was hzin the single dot case. A full account\nof the rich pseudo-spin dynamics under Hsz0(t) can be found in Refs. 21\nand 34. Here we only discuss the most prominent features for Csz0\nSS(t),\nwhich gives the probability to find the singlet |S∝an}bracketri}ht, if the system was ini-\ntialized to |S∝an}bracketri}htatt= 0. The parameters that determine the dynamics\nare the exchange coupling J, the expectation value of the total differ-\nence fieldx0=δbz+δh0and the width of the difference field σδ(with\nδh0=∝an}bracketle{tψI|δhz|ψI∝an}bracketri}htandσδ=∝an}bracketle{tψI|(δhz−δh0)2|ψI∝an}bracketri}ht1/2). For the asymptotics\nonefinds thatthe singletprobabilitydoesnotdecayto zero,but go estoafi-\nnite, parameter-dependent value.34In the case of strong exchange coupling\n|J| ≫max(|x0|,σδ) the singlet only decays by a small fraction quadratic in\nσδ/Jorx0/J:\nCsz0\nSS(t→ ∞)∼/braceleftBigg\n1−2/parenleftbigσδ\nJ/parenrightbig2,|J| ≫σδ≫ |x0|,\n1−2/parenleftbigx0\nJ/parenrightbig2,|J| ≫ |x0| ≫σδ.(1.9)\nAt short times Csz0\nSS(t) undergoes a Gaussian decay on a timescale/radicalbig\nJ2+4x2\n0/4|x0|σδwhile at long times t≫ |J|/4σ2\nδwe have a power law\ndecay\nCsz0\nSS(t)∼Csz0\nSS(t→ ∞)+e−x2\n0\n2σ2\nδcos(|J|t+3π\n4)\n4σδ/radicalbig\n|J|t3\n2. (1.10)November 15, 2018 9:7 World Scientific Review Volume - 9in x 6i n decrev\nElectron and hole spin dynamics and decoherence in quantum d ots 7\nAs in the case of single-spin ESR, we again have a power-law decay, no w\nwith 1/t3/2and a universal phase shift, in this case: 3 π/4. Measurements35\nof the correlator Csz0\nSS(t) confirmed the parameterdependence ofthe satura-\ntion value and were consistent with the theoretical predictions con cerning\nthe decay. Using the same methods, one may also look at transvers e cor-\nrelators in the Sz= 0 subspace and find again power-law decays and a\nuniversal phase shift, albeit, with different decay power and differe nt value\nof the universal phase shift.21Looking at the short-time behavior of the\ntransverse correlators also allows one to analyze the fidelity of the√\nSWAP\ngate.21\n1.5. Nuclear spin state narrowing\nThe idea to prepare the nuclear spin system in order to prolong the e lectron\nspin coherence was put forward in Ref. 34. Specific methods for nu clear\nspin state narrowing have been described in Ref. 21 in the context o f\na double dot with oscillating exchange interaction, in Ref. 22 for phas e-\nestimation of a single (undriven) spin in a single dot and in an optical set up\nin Ref. 20. Here, we discuss narrowing for the case of a driven single spin\nin a single dot, for which the details are very similar to the treatment in\nRef. 21. The general idea behind state narrowing is that the evolut ion of\nthe electron spin system depends on the value of the nuclear field sin ce the\neffective Zeeman splitting is given by ǫz+hn\nz. This leads to a nuclear field\ndependent resonance condition ǫz+hn\nz−ω= 0 for ESR and thus measuring\ntheevolutionoftheelectronspinsystemdetermines hz\nnandthusthenuclear\nspin state.\nWe startfromthe Hamiltonian forsingle-spinESRasgivenin Eq. (1.4).\nThe electron spin is initialized to the | ↑∝an}bracketri}htstate at time t= 0 and evolves\nunderHesrup to a measurement performed at time tm. The probability\nto find| ↓∝an}bracketri}htfor a given eigenvalue hn\nzof the nuclear field operator ( hz|n∝an}bracketri}ht=\nhn\nz|n∝an}bracketri}ht) is then given by\nPn\n↓(t) =1\n2b2\nb2+4δ2n/bracketleftbigg\n1−cos/parenleftbiggt\n2/radicalbig\nb2+4δ2n/parenrightbigg/bracketrightbigg\n(1.11)\nwhereδn=ǫz+hn\nz−ωandbis the amplitude of the driving field. As\nmentioned above, in equilibrium we have a Gaussian distribution for the\neigenvalues hn\nz, i.e., for the diagonal elements of the nuclear spin density\nmatrixρI(hn\nz,0) =∝an}bracketle{tn|ρI|n∝an}bracketri}ht= exp/parenleftbig\n−(hn\nz−h0)2/2σ2/parenrightbig\n/√\n2πσ. Thus, av-\neraged over the nuclear distribution we have the probability P↓(t) to findNovember 15, 2018 9:7 World Scientific Review Volume - 9in x 6i n decrev\n8 D. Klauser et al.\nthe state | ↓∝an}bracketri}ht, i.e.,P↓(t) =/integraltext\ndhn\nzρI(hn\nz,0)Pn\n↓(t). After one measurement\nwith outcome | ↓∝an}bracketri}ht, we thus find for the diagonal of the nuclear spin density\nmatrix36\nρI(hn\nz,0)|↓/angbracketright−→ρ(1,↓)\nI(hn\nz,tm) =ρI(hn\nz,0)Pn\n↓(tm)\nP↓(tm).(1.12)\nAssuming now that the measurement is performed in such a way that it\ngives the time averaged value (i.e., with a time resolution less than 1 /b)\nwe have for the probability Pn\n↓of measurement result | ↓∝an}bracketri}htas a function of\nthe nuclear field eigenvalue Pn\n↓=1\n2b2\nb2+4δ2n. Thus, by performing a mea-\nsurement on the electron spin (with outcome | ↓∝an}bracketri}ht), the nuclear-spin density\nmatrix is multiplied by a Lorentzian with width bcentered around the hn\nz\nthat satisfies the resonance condition ǫz+hn\nz−ω= 0. This results in a\nnarrowed nuclear spin distribution, and thus an extension of the ele ctron\nspin coherence, if b<σ. In the case of measurement outcome | ↑∝an}bracketri}htwe find\nρI(hn\nz,0)|↑/angbracketright−→ρ(1,↑)\nI(hn\nz,tm) =ρI(hn\nz,0)1−Pn\n↓(tm)\n1−P↓(tm), (1.13)\ni.e., the Gaussian nuclear spin distribution is multiplied by one minus a\nLorentzian, thus reducing the probability for the nuclear field to ha ve a\nvalue matching the resonance condition ǫz+hn\nz−ω= 0. Due to the\nslow dynamics of the nuclear spin system (see discussion at the end o f\nSec. 1.2), many such measurements of the electron spin are possib le (with\nre-initialization of the electron spin between measurements). Unde r the\nassumption of a static nuclear field during Msuch initialization and mea-\nsurement cycles we find\nρI(hn\nz,0)−→ρ(M,α↓)(hn\nz) =1\nNρI(hn\nz,0)(Pn\n↓)α↓(1−Pn\n↓)M−α↓,(1.14)\nwhereα↓is the number of times the measurement outcome was | ↓∝an}bracketri}ht. The\nsimplestwaytonarrowistoperformsinglemeasurementswith b≪σ. Ifthe\noutcomeis | ↓∝an}bracketri}ht,narrowinghasbeenachieved. Otherwise,thenuclearsystem\nshould be allowedto re-equilibratebeforethe next measurement.37Inorder\nto achieve a systematic narrowing, one can envision adapting the dr iving\nfrequency (and thus the resonance condition) depending on the o utcome of\nthe previous measurements. Such an adaptive scheme is described in detail\nin Refs. 20 and 21. With this we conclude the part on hyperfine-induc ed\ndecoherence of electron spins in quantum dots and move on to the h eavy\nholes.November 15, 2018 9:7 World Scientific Review Volume - 9in x 6i n decrev\nElectron and hole spin dynamics and decoherence in quantum d ots 9\n1.6. Spin decoherence and relaxation for heavy holes\nNow we consider the spin coherence of heavy holes in quantum dots. The\ncontact hyperfine interaction between lattice nuclei and heavy-h ole spin is\nmuch weaker than that for electrons, since the valence band has psym-\nmetry. Thus (neglecting sphybridization) only the weaker anisotropic hy-\nperfine interaction is present. Therefore, the decoherence due to hyperfine\ninteraction is suppressed for heavy holes and in this section we focu s only\non the spin decoherence due to spin-orbit interactioninduced by he avy-hole\n- phonon coupling.\nFrom the two-band Kane model, the Hamiltonian for the valence band\nof III–V semiconductors is given by\nHbulk=HLK+ηJ·Ω+HZ, (1.15)\nwhereHLKis the Luttinger-Kohn Hamiltonian.38The second term is the\nDresselhaus spin-orbit coupling (due to bulk inversion asymmetry) f or the\nvalence band,39,40J= (Jx,Jy,Jz) are 4×4 matrices corresponding to spin\n3/2, Ωz=Pz(P2\nx−P2\ny), and Ω x, Ωyare given by cyclic permutations.\nThe last term in Eq. (1.15) HZ=−2κµBB·J−2qµBB·Jis the Zeeman\nterm for the valence band41(κandqare the Luttinger parameters41and\nJ= (J3\nx,J3\ny,J3\nz)).\nWe consider a [001]-grown two-dimensional system. In the case of a n\nasymmetric quantum well, due to structure inversion asymmetry alo ng\nthe growth direction, there is an additional spin-orbit term, the By chkov-\nRashba spin-orbit term, which, in the two-band model is given by42\nαRP×E·J, whereαRis the Bychkov-Rashbaspin-orbit coupling constant\nandEis an effective electric field along the growth direction. Due to con-\nfinement along the growth direction, the valence band splits into a he avy-\nholesubbandwith Jz=±3/2andalight-holesubbandwith Jz=±1/2(see\nFig. 1.2 and Ref. 40). If the splitting ∆ of heavy-hole and light-hole su b-\nbands is large,we describe the propertiesofheavy-holesand light- holessep-\narately, using only the 2 ×2 submatrices for the Jz=±3/2 andJz=±1/2\nstates, respectively. The heavy-hole submatrices have the prop erty that\n˜Jx=˜Jy= 0 and ˜Jz=3\n2σz. For such a system and low temperatures, only\nthe lowest heavy-hole subband is significantly occupied. In this case , we\nconsider heavy holes only. In the frameworkof perturbation theo ry,39using\nEq. (1.15) and taking into account the Zeeman energy and the Bych kov-\nRashba spin-orbit coupling term, the effective Hamiltonian for heavy holesNovember 15, 2018 9:7 World Scientific Review Volume - 9in x 6i n decrev\n10 D. Klauser et al.\nFig. 1.2. Band structure of a III-V semiconductor quantum we ll with the [001]-growth\ndirection, where Egis the band gap, ∆ is the splitting between the light- and heav y-hole\nsubbands due to quantum-well confinement, and ∆ SOis the splitting of the valence band\ndue to spin-orbit interaction.\nof a quantum dot with lateral confinement potential U(x,y) is given by\nH=1\n2m(P2\nx+P2\ny)+U(x,y)+HHH\nSO−1\n2g⊥µBB⊥σz,(1.16)\nwheremis the effective heavy-hole mass, P=p+|e|A(r)/c,A(r) =\n(−yB⊥/2,xB⊥/2,yBx−xBy),g⊥is the component of the g-factor tensor\nalong the growth direction, and\nHHH\nSO=iαP3\n−σ++βP−P+P−σ++γB−P2\n−σ++H.c. (1.17)\nis the spin-orbit coupling of heavy holes consisting of three contribu tions:\nthe Dresselhaus term ( β),40the Rashba term ( α),43and the last term\n(γ) combines two effects: orbital coupling via non-diagonal elements in\nthe Luttinger-Kohn Hamiltonian ( ∝P2\n±) and magnetic coupling via non-\ndiagonal elements in the Zeeman term ( ∝B±). This latter term repre-\nsents a new type of spin-orbit interaction which is unique for heavy h oles.44\nHere,α= 3γ0αR∝an}bracketle{tEz∝an}bracketri}ht/2m0∆,β=−3γ0η∝an}bracketle{tP2\nz∝an}bracketri}ht/2m0∆,γ= 3γ0κµB/m0∆,\nσ±= (σx±iσy)/2,P±=Px±iPy,B±=Bx±iBy,m0is the free electron\nmass,γ0is the Luttinger parameter,41∝an}bracketle{tEz∝an}bracketri}htis the averagedeffective electricNovember 15, 2018 9:7 World Scientific Review Volume - 9in x 6i n decrev\nElectron and hole spin dynamics and decoherence in quantum d ots 11\nfield along the growth direction of a quantum dot, and ∆ is the splitting of\nlight-hole and heavy-hole subbands. The splitting between heavy-h ole and\nlight-hole subbands ∆ ∼h−2, wherehis the quantum-dot height.\nThe spectrum of (1.16) for parabolic lateral confinement [ U(x,y) =\nmω2\n0(x2+y2)/2] and for vanishing spin-orbit interaction ( HHH\nSO= 0) is the\nFock-Darwin spectrum split by the Zeeman term.45,46From Eq. (1.17), it\ncan be seen that HHH\nSOleads to coupling of the two lowest states |0,±3/2∝an}bracketri}ht\nto the states with the opposite spin orientations and different orbit al mo-\nmenta|l,∓3/2∝an}bracketri}ht. Note that the three spin-orbit terms in Eq. (1.17) differ\nby symmetry in momentum space and hence mix different states resu lting\nin avoided crossings of the energy levels (see inset of Fig. 1.3). Due t o\nthis spin-orbit mixing of the heavy-hole states, the transitions bet ween the\nstates|0,±3/2∝an}bracketri}htwith emission or absorption of an acoustic phonon become\npossible and this is the main source of spin relaxation and decoherenc e for\nheavy-holes.40\nWe consider a single-particle quantum dot, in which a heavy hole can\noccupy one of the low-lying levels. In the following, we study the relax ation\nof ann-level system, the first n−1 levels have the same spin and the n-th\nlevel has the opposite spin orientation. In the framework of Bloch- Redfield\ntheory,47the Bloch equations for heavy-hole spin motion for such a system\nin the interaction representation are given by\n∝an}bracketle{t˙Sz∝an}bracketri}ht= (ST−∝an}bracketle{tSz∝an}bracketri}ht)/T1−R(t), (1.18)\n∝an}bracketle{t˙Sx∝an}bracketri}ht=−∝an}bracketle{tSx∝an}bracketri}ht/T2,∝an}bracketle{t˙Sy∝an}bracketri}ht=−∝an}bracketle{tSy∝an}bracketri}ht/T2, (1.19)\nwhereR(t) =Wn1ρnn(t)+/summationtextn−1\ni=1Wniρii(t),ρ(t) is the density matrix, Wij\nis the transition rate from state jto statei,STis a constant (which has\nthe value of ∝an}bracketle{tSz∝an}bracketri}htin thermodynamic equilibrium if R(t) = 0),\n1\nT1=Wn1+n−1/summationdisplay\ni=1Win,1\nT2=1\n2T1+1\n2n−1/summationdisplay\ni=2Wi1, (1.20)\nwherepuredephasing(dueto fluctuationsalong zdirection)isabsentin the\nspin decoherence time T2since the spectral function is superohmic. As can\nbe seen from Eq. (1.18), the spin motion has a complex dependence o n the\ndensity matrixand, in the generalcase, thereare n−1spin relaxationrates.\nHowever, in the case of low temperatures ( /planckover2pi1qsα≫kBT), when phonon\nabsorption becomes strongly suppressed, solving the master equ ation, we\nfind thatR(t)≈0. Therefore, there is only one spin relaxation time T1. In\nthis limit, the last sum in Eq. (1.20) is negligible and the spin decoherence\ntime saturates, i.e., T2= 2T1.November 15, 2018 9:7 World Scientific Review Volume - 9in x 6i n decrev\n12 D. Klauser et al.\n1    10  102103104105106107\n 0.3  0.5  0.7  0.9  1.1  1.3  1.5 1 / T1 (1/s)\nB⊥ (T)B|| = 0 T\nB|| = 0.5 T\nB|| = 3 T 0 0.1 0.2 0.3\n 0.3 0.7 1.1 1.5Energy (meV)\nB⊥ (T)\nFig. 1.3. Heavy hole spin relaxation rate 1 /T1in a GaAs quantum dot versus an applied\nperpendicular magnetic field B⊥(the height of the quantum dot is chosen to be h= 5nm,\nthe lateral size l0=p\n/planckover2pi1/mω0= 40 nm, κ= 1.2,γ0= 2.5,g⊥= 2.5,49and the\nother parameters are given in Ref.40). Inset: Energy differences of lowest excited levels\nwith respect to the ground state E0,+3/2. The second avoided crossing comes from the\nspin-orbit interaction and the in-plane magnetic field B/bardbl(3rd term in Eq. 1.17). The\nanticrossing gap is proportional to B/bardbl, implying that the coupling between corresponding\nstates can be controlled externally.\nNote that in contrast to electrons48there are no interference effects\nbetween different spin-orbit coupling terms, thus the total spin re laxation\nrate 1/T1is the sum of rates 1 /T1= 1/TD\n1+1/TBR\n1+1/T/bardbl\n1:40,44\n1\nTBR\n1∝α2ω7\nZ/parenleftbiggω3\n+\n3ω++ωZ−ω3\n−\n3ω−−ωZ/parenrightbigg2\n,\n1\nTD\n1∝β2ω3\nZ/parenleftbiggω+\nω++ωZ−ω−\nω−−ωZ/parenrightbigg2\n,\n1\nT/bardbl\n1∝γ2B2\n/bardblω5\nZ/parenleftbiggω2\n+\n2ω++ωZ+ω2\n−\n2ω−−ωZ/parenrightbigg2\n, (1.21)\nwhereω±=/radicalbig\nω2\n0+ω2c/4±ωc/2,ωZ=g⊥µBB⊥//planckover2pi1,B/bardbl=/radicalBig\nB2x+B2y. In\nFig. 1.3 the total spin relaxation rate 1 /T1is plotted as a function of per-November 15, 2018 9:7 World Scientific Review Volume - 9in x 6i n decrev\nElectron and hole spin dynamics and decoherence in quantum d ots 13\npendicular magnetic field B⊥. There are three peaks in the relaxation rate\ncurve atωZ=ω−,2ω−, and 3ω−, which are caused by strong spin mixing\nat the anticrossing points. In the inset, the first (third) avoided c rossing\nresulting from Dresselhaus (Rashba) spin-orbit coupling correspo nds to the\nfirst (third) peak of the spin relaxation curve in Fig. 1.3. At non-zer o in-\nplane magnetic fields ( B/bardbl), there is an additional peak which is due to an\nanticrossing between the energy levels E0,+3/2andE2,−3/2(see the second\navoided crossing in the inset). Note that the spin relaxation rate fo r heavy\nholes is comparable to that for electrons40,50due to the fact that spin-orbit\ncoupling of heavy holes is strongly suppressed for flat quantum dot s (see\nEq. (1.17)), as confirmed by a recent experiment.51\n1.7. Electric dipole spin resonance for heavy holes\nLet us now consider methods for the manipulation and detection of t he\nheavy-hole spin in quantum dots. For electrons in two-dimensional s truc-\ntures, an applied oscillating in-plane magnetic field couples spin-up and\nspin-down states via magnetic-dipole transitions and is commonly use d in\nelectron spin resonance, Rabi oscillation, and spin echo experiment s.10It\ncan be shown that magnetic-dipole transitions (∆ n= 0, ∆ℓ= 0, and\n∆s=±1)areforbiddenand, duetospin-orbitmixingofthestates |0,±3/2∝an}bracketri}ht\nwithiβ±\n1|1,∓3/2∝an}bracketri}ht, electric-dipole transitions (∆ n=±1, ∆ℓ=±1, and\n∆s= 0) are most likely to occur. Therefore, the heavy holes are affect ed\nby the oscillating electric field component and not by the magnetic one .\nWe consider a circularlypolarized electric field rotating in the XY-plane\nwith frequency ω:E(t) =E(sinωt,−cosωt,0). Therefore, the interac-\ntion of heavy holes with the electric field is described by the Hamiltonian\nHE(t) = (|e|E/mω)(cosωtPx+sinωtPy). The coupling between the states\n|±∝an}bracketri}htis given by ∝an}bracketle{t+|HE(t)|−∝an}bracketri}ht=HE\n+−=/parenleftbig\nHE\n−+/parenrightbig∗=dSOEe−iωt, where\ndSO= (|e|l/2ω)(β−\n1ω−+β+\n1ω+) (1.22)\nis an effective dipole moment of a heavy hole depending on Dresselhaus\nspin-orbit coupling constants, perpendicular magnetic field B⊥, lateral size\nof a quantum dot, and frequency ωof an rf electric field.\nIn the framework of the Bloch-Redfield theory47(taking into account\nalsooff-diagonalmatrixelements), theeffectivemasterequationf ortheden-\nsity matrix ρnmassumes the form of Bloch equations,47with the detuning\nof the rf field given by δrf=ωZ−ω. 2dSOE//planckover2pi1is the Larmor frequency,\nT1= 1/(W+−+W−+) the spin relaxation time ( Wnmis the transitionNovember 15, 2018 9:7 World Scientific Review Volume - 9in x 6i n decrev\n14 D. Klauser et al.\nrate from state mto staten),T2= 2T140the spin decoherence time, and\nρT\nz= (W+−−W−+)T1the equilibrium value of ρzwithout rf field.\nThe coupling energy between a heavy hole and an oscillating electric\nfield is given by\n∝an}bracketle{tHE(t)∝an}bracketri}ht= Tr(ρHE(t)) =−dSO·E(t), (1.23)\nwheredSO=dSO(iρ−+−iρ+−,ρ+−+ρ−+,0) is the dipole moment of a\nheavy hole. Therefore, the rf power P=−d∝an}bracketle{tHE(t)∝an}bracketri}ht/dt=−ωdSOEρ−\nabsorbed by a heavy-hole spin system in the stationary state is give n by52\nP=2ω(dSOE)2T2ρT\nz//planckover2pi1\n1+δ2\nrfT2\n2+(2dSOE//planckover2pi1)2T1T2. (1.24)\nIn Fig. 1.4, the dependence of Pon a perpendicular magnetic field B⊥\nand frequency ωof the oscillating electric field is plotted. The rf power P\nabsorbed by the system has three resonances and one resonant dip. The\nfirst resonance appears when the energy of rf radiation equals th e Zeeman\nenergy of heavy holes: Br,1\n⊥=/planckover2pi1ω/g⊥µB. The shape of this resonance (at\ncertainω) is given by P≈/planckover2pi1ωρT\nz/2/planckover2pi1[1+/planckover2pi12δ2\nrfT2/(2dSOE)2T1].\nFig. 1.4. Absorbed power P(meV/s) as a function of perpendicular magnetic field B⊥\nand rf frequency ω(T2= 2T1,E= 2.5 V/cm,B/bardbl= 1 T, and the other parameters are\nthe same as those in Fig. 1.3).\nIf the first and second resonances are well separated ( ω≪ω−), then\nthe absorbed power can be estimated as\nP≈2ω(dSOE)2ρT\nz//planckover2pi1δ2\nrfT2 (1.25)November 15, 2018 9:7 World Scientific Review Volume - 9in x 6i n decrev\nElectron and hole spin dynamics and decoherence in quantum d ots 15\nin the region of the second and third resonances and the resonant dip. The\nsecond resonance corresponds to an anticrossing of the levels E0,−3/2and\nE1,3/2(see the first avoided crossing in the inset of Fig. 1.3) at ω−=ωZ40\n(atB=Br,2\n⊥). At the anticrossing point, there is strong mixing of the\nspin-up and spin-down states and the dipole moment of a heavy-hole\nspin system is maximal dmax\nSO=|e|lωZ/2ωand equals half of the low-\nest electric dipole moment of a quantum dot ( |e|lω−/ω). Therefore, the\nheight of the second resonance is given by ( elωZE)2/2/planckover2pi1ωδ2\nrfT2. The res-\nonant dip appears at Bd\n⊥= (/planckover2pi1ω0/2g⊥µB)/radicalbig\n2m0/g⊥m, which corresponds\ntoβ−\n1ω−+β+\n1ω+= 0 and to zero dipole moment (see Eq. (1.22)). The\nthird resonance reflects the peak in the spin decoherence rate T−1\n2due to\nan applied in-plane magnetic field (see Fig. 1.3) at the second anticros s-\ning point (the second avoided crossing in inset of Fig. 1.3) at 2 ω−=ωZ\n(Br,3\n⊥= 4/planckover2pi1ω0/g⊥µB/radicalbig\n1+4m0/g⊥m). From the positions of the resonances\nwe can determine g⊥,m, andω0, from the shape and the height of those\nwe can extract information about the spin-orbit interaction const antsα,β,\nand spin-orbit interaction strength due to in-plane magnetic field (w hich\nis proportional to γ0κ/∆). Moreover, we can determine the dependence of\nthe spin relaxation and decoherence times on B⊥.\n1.8. Conclusions\nWe have discussed the rich dynamics of single electron spins in single an d\ndouble quantum dots due to hyperfine interaction with the nuclei. Ke y fea-\ntures are non-exponential decays of various kinds and a remarka ble univer-\nsalphase-shift. Further, we havestudied spin decoherenceand relaxationof\nheavyholesinquantumdotsduetospin-orbitcoupling. Thespinrelax ation\ntimeT1for heavy holes in flat quantum dots can be comparable to that for\nelectrons40,50as confirmed by experiment.51The spin decoherence time for\nheavy holes is given by T2= 2T1at low temperatures. There is strong spin\nmixing at energy-level crossings resulting in a non-monotonic depen dence\nT1(B). We have proposed a new method for manipulation of a heavy-hole\nspin in a quantum dot via rf electric fields. This method can be used for\ndetection of heavy-hole spin resonance signals, for spin manipulatio n, and\nfor determining important parameters of heavy holes.44\nAknowledgments: We aknowledge financial support from the Swiss\nNSF, the NCCR Nanoscience and JST ICORP.November 15, 2018 9:7 World Scientific Review Volume - 9in x 6i n decrev\n16 D. Klauser et al.\nReferences\n1. D. Loss and D. P. DiVincenzo, Quantum computation with qua ntum dots,\nPhys. Rev. A .57, 120–126, (1998).\n2. S. Tarucha, D. G. Austing, T. Honda, R. J. van der Hage, and L . P. Kouwen-\nhoven, Shell filling and spin effects in a few electron quantum dot,Phys. Rev.\nLett.77(17), 3613–3616, (1996).\n3. M. Ciorga, A. S. Sachrajda, P. Hawrylak, C. Gould, P. Zawad zki, S. Jul-\nlian, Y. Feng, and Z. Wasilewski, Addition spectrum of a late ral dot from\ncoulomb and spin-blockade spectroscopy, Phys. Rev. B .61(24), R16315–\nR16318, (2000).\n4. J. M. Elzerman, R. Hanson, J. S. Greidanus, L. H. Willems va n Beveren,\nS.De Franceschi, L. M. K.Vandersypen, S.Tarucha, andL. P. K ouwenhoven,\nFew-electron quantumdot circuit with integrated charge re ad out,Phys. Rev.\nB.67(16), 161308, (2003).\n5. T. Hayashi, T. Fujisawa, H. D. Cheong, Y. H. Jeong, and Y. Hi rayama,\nCoherent manipulation of electronic states in a double quan tum dot, Phys.\nRev. Lett. 91(22), 226804, (2003).\n6. J. R. Petta, A. C. Johnson, C. M. Marcus, M. P. Hanson, and A. C. Gossard,\nManipulation of a single charge in a double quantum dot, Phys. Rev. Lett.\n93(18), 186802, (2004).\n7. J. M. Elzerman, R. Hanson, L. H. Willems van Beveren, B. Wit kamp,\nL. M. K. Vandersypen, and L. P. Kouwenhoven, Single-shot rea d-out of an\nindividual electron spin in a quantum dot, Nature.430, 431–435, (2004).\n8. R. Hanson, L. H. vanBeveren, I. T. Vink, J. M. Elzerman, W. J . Naber, F. H.\nKoppens, L. P. Kouwenhoven, and L. M. Vandersypen, Single-s hot readout\nof electron spin states in a quantum dot using spin-dependen t tunnel rates,\nPhys. Rev. Lett. 94(19), 196802, (2005).\n9. J.R.Petta, A.C.Johnson, J.M.Taylor, E.A.Laird, A.Yaco by, M.D.Lukin,\nC. M. Marcus, M. P. Hanson, and A. C. Gossard, Coherent manipu lation of\ncoupled electron spins in semiconductor quantum dots, Science.309(5744),\n2180–2184, (2005).\n10. F. H. L. Koppens, C. Buizert, K. J. Tielrooij, I. T. Vink, K . C. Nowack,\nT. Meunier, L. P. Kouwenhoven, and L. M. K. Vandersypen, Driv en coherent\noscillations of a single electron spin in a quantum dot, Nature.442, 766–771,\n(2006).\n11. V. Cerletti, W. A. Coish, O. Gywat, and D. Loss, TUTORIAL: Recipes for\nspin-based quantum computing, Nanotechnology .16, 27, (2005).\n12. W. A. Coish and D. Loss, Quantum computing with spins in so lids,\nhttp://arXiv.org/cond-mat/0606550 . (2006).\n13. D. Paget, G. Lampel, B. Sapoval, and V. I. Safarov, Low fiel d electron-\nnuclear spin coupling in gallium arsenide under optical pum ping conditions,\nPhys. Rev. B .15, 5780–5796, (1977).\n14. J. Schliemann, A. Khaetskii, and D. Loss, TOPICAL REVIEW : Electron\nspin dynamics in quantum dots and related nanostructures du e to hyperfine\ninteraction with nuclei, J. Phys.: Condens. Matter .15, 1809 (Dec., 2003).November 15, 2018 9:7 World Scientific Review Volume - 9in x 6i n decrev\nElectron and hole spin dynamics and decoherence in quantum d ots 17\n15. J. M. Taylor, J. R. Petta, A. C. Johnson, A. Yacoby, C. M. Ma rcus, and\nM. D. Lukin, Relaxation, dephasing, and quantum control of e lectron spins\nin double quantum dots, http://arXiv.org/cond-mat/0602470 . (2006).\n16. G. Burkard, D. Loss, and D. P. DiVincenzo, Coupled quantu m dots as quan-\ntum gates, Phys. Rev. B .59, 2070–2078, (1999).\n17. A. V. Khaetskii, D. Loss, and L. Glazman, Electron spin de coherence in\nquantumdots due tointeraction with nuclei, Phys. Rev. Lett. 88(18), 186802,\n(2002).\n18. I. A. Merkulov, A. L. Efros, and M. Rosen, Electron spin re laxation by nuclei\nin semiconductor quantum dots, Phys. Rev. B .65(20), 205309, (2002).\n19. W. A. Coish and D. Loss, Hyperfine interaction in a quantum dot: Non-\nMarkovian electron spin dynamics, Phys. Rev. B .70(19), 195340, (2004).\n20. D. Stepanenko, G. Burkard, G. Giedke, and A. Imamoˇ glu, E nhancement of\nelectron spin coherence by optical preparation of nuclear s pins,Phys. Rev.\nLett.96(13), 136401, (2006).\n21. D. Klauser, W. A. Coish, and D. Loss, Nuclear spin state na rrowing via\ngate–controlled Rabi oscillations in a double quantum dot, Phys. Rev. B .74\n(20), 205302, (2006).\n22. G. Giedke, J. M. Taylor, D. D’Alessandro, M. D. Lukin, and A. Imamo˘ glu,\nQuantum measurement of a mesoscopic spin ensemble, Phys. Rev. A .74(3),\n032316, (2006).\n23. F. H. L. Koppens, J. A. Folk, J. M. Elzerman, R. Hanson, L. H . W. van\nBeveren, I. T. Vink, H. P. Tranitz, W. Wegscheider, L. P. Kouw enhoven, and\nL. M. K. Vandersypen, Control and detection of singlet-trip let mixing in a\nrandom nuclear field, Science.309, 1346–1350, (2005).\n24. P. Maletinsky, A. Badolato, and A. Imamoˇ glu, Dynamics o f quan-\ntum dot nuclear spin polarization controlled by a single ele ctron,\nhttp://arXiv.org/abs/0704.3684v1 . (2007).\n25. J. Baugh, Y. Kitamura, K. Ono, and S. Tarucha, Large nucle ar\nOverhauser fields detected in vertically-coupled double qu antum dots,\nhttp://arxiv.org/abs/0705.1104 . (2007).\n26. D. Reilly, C. M. Marcus, et al. Unpublished .\n27. P. Maletinsky, C. W. Lai, A. Badolato, and A. Imamoˇ glu, N onlinear dynam-\nics of quantum dot nuclear spins, Phys. Rev. B .75, 035409, (2007).\n28. K. Ono, D. G. Austing, Y. Tokura, and S. Tarucha, Current r ectification\nby pauli exclusion in a weakly coupled double quantum dot sys tem,Science.\n297, 1313–1317, (2002).\n29. K. Ono and S. Tarucha, Nuclear-spin-induced oscillator y current in spin-\nblockaded quantum dots, Phys. Rev. Lett. 92(25), 256803, (2004).\n30. F. H. L. Koppens, D. Klauser, W. A. Coish, K. C. Nowack, L. P . Kouwen-\nhoven, D. Loss, and L. M. K. Vandersypen, Universal phase shi ft and non-\nexponential decay of driven single-spin oscillations, http://arXiv.org/cond-\nmat/0703640 . (2007).\n31. W. Yao, R.-B. Liu, and L. J. Sham, Theory of electron spin d ecoherence by\ninteracting nuclear spins in a quantum dot, Phys. Rev. B .74(19), 195301,\n(2006).November 15, 2018 9:7 World Scientific Review Volume - 9in x 6i n decrev\n18 D. Klauser et al.\n32. N. Shenvi, R. de Sousa, and K. B. Whaley, Nonperturbative bounds on elec-\ntron spin coherence times induced by hyperfine interactions ,Phys. Rev. B .\n71(14), 144419, (2005).\n33. C. Deng and X. Hu, Analytical solution of electron spin de coherence through\nhyperfineinteraction inaquantumdot, Phys. Rev. B .73(24), 241303, (2006).\nErratum: Phys. Rev. B 74, 129902.\n34. W. A. Coish and D. Loss, Singlet-triplet decoherence due to nuclear spins in\na double quantum dot, Phys. Rev. B .72(12), 125337, (2005).\n35. E. A. Laird, J. R. Petta, A. C. Johnson, C. M. Marcus, A. Yac oby, M. P.\nHanson, and A. C. Gossard, Effect of exchange interaction on s pin dephasing\nin a double quantum dot, Phys. Rev. Lett. 97(5), 056801, (2006).\n36. A. Peres, Quantum Theory: Concepts and Methods . (Kluwer Academic Pub-\nlishers, 1993).\n37. D. Klauser, W. A. Coish, and D. Loss, Quantum-dotspin qub itandhyperfine\ninteraction, http://arXiv.org/cond-mat/0604252 . (2006).\n38. J. M. Luttinger and W. Kohn, Motion of electrons and holes in perturbed\nperiodic fields, Phys. Rev. 97(4), 869–883, (1955).\n39. B. P. Zakharchenya and F. Mier, Eds., Optical Orientation . (North-Holland,\nAmsterdam, 1984).\n40. D. V. Bulaev and D. Loss, Spin relaxation and decoherence of holes in quan-\ntum dots, Phys. Rev. Lett. 95, 076805, (2005).\n41. J. M. Luttinger, Quantum theory of cyclotron resonance i n semiconductors:\nGeneral theory, Phys. Rev. 102, 1030, (1956).\n42. R. Winkler, Rashba spin splitting in two-dimensional el ectron and hole sys-\ntems,Phys. Rev. B .62, 4245, (2000).\n43. R. Winkler, H. Noh, E. Tutuc, and M. Shayegan, Anomalous R ashba spin\nsplitting in two-dimensional hole systems, Phys. Rev. B .65, 155303, (2002).\n44. D. V. Bulaev and D. Loss, Electric dipole spin resonance f or heavy holes in\nquantum dots, Phys. Rev. Lett. 98, 097202, (2007).\n45. V. Fock, Bemerkung zur Quantelung des harmonischen Oszi llators im Mag-\nnetfeld,Z. Phys. 47, 446–448, (1928).\n46. C. G. Darwin, The diamagnetism of the free electron, Proc. Cambridge Phi-\nlos. Soc.27, 86–90, (1930).\n47. K. Blum, Density Matrix Theory and Applications . (Plenum, New York,\n1996).\n48. V. N. Golovach, A. Khaetskii, and D. Loss, Phonon-Induce d Decay of the\nElectron Spin in Quantum Dots, Phys. Rev. Lett. 93, 016601, (2004).\n49. H. W. van Kestern, E. C. Cosman, W. A. J. A. van der Poel, and C. T.\nFoxton, Fine structure of excitons in type-II GaAs/AlAs qua ntum wells,\nPhys. Rev. B .41, 5283, (1990).\n50. D. V. Bulaev and D. Loss, Spin relaxation and anticrossin g in quantum dots:\nRashba versus Dresselhaus spin-orbit coupling, Phys. Rev. B .71, 205324,\n(2005).\n51. D. Heiss, S. Schaeck, H. Huebl, M. Bichler, G. Abstreiter , J. J. Finley, D. V.\nBulaev, and D. Loss, Observation of extremely slow hole spin relaxation in\nself-assembled quantum dots, http://arxiv.org/abs/0705.1466 . (2007).November 15, 2018 9:7 World Scientific Review Volume - 9in x 6i n decrev\nElectron and hole spin dynamics and decoherence in quantum d ots 19\n52. A.Abragam, The Principles of Nuclear Magnetism . (OxfordUniversityPress,\nLondon, 1961)."    },    {        "title": "1103.3231v1.Spin_relaxation_in_Rashba_rings.pdf",        "content": "Spin relaxation in Rashba rings\nValeriy A. Slipko1, 2and Yuriy V. Pershin1,\u0003\n1Department of Physics and Astronomy and USC Nanocenter,\nUniversity of South Carolina, Columbia, SC 29208, USA\n2Department of Physics and Technology, V. N. Karazin Kharkov National University, Kharkov 61077, Ukraine\nSpinrelaxationdynamicsinringswithRashbaspin-orbitcouplingisinvestigatedusingspinkinetic\nequation. We find that the spin relaxation in rings occurs toward a persistent spin configuration\nwhose final shape depends on the initial spin polarization profile. As an example, it is shown that a\nhomogeneous parallel to the ring axis spin polarization transforms into a persistent crown-like spin\nstructure. It is demonstrated that the ring geometry introduces a geometrical contribution to the\nspin relaxation rate speeding up the transient dynamics. Moreover, we identify several persistent\nspin configurations as well as calculate the Green function of spin kinetic equation.\nPACS numbers: 72.25.Rb, 71.70.Ej, 71.10.-w\nINTRODUCTION\nThe understanding of spin relaxation dynamics in\nsemiconductor structures is an active area of research\nrelated to the field of spintronics [1, 2]. Previously, the\nspin relaxation dynamics has been investigated mainly in\ninfinite two-dimensional (2D) systems with D’yakonov-\nPerel’ [3, 4] spin relaxation mechanism (see, e.g., Refs.\n[4–11]). There are only several examples in the lit-\nerature where the effects of boundary conditions [12]\non D’yakonov-Perel’ spin relaxation have been explored.\nThese examples include investigations of spin relaxation\nin 2D channels [13, 14], 2D half-space [15], 2D systems\nwith antidots [16], small 2D systems [17], and finite-\nlength one-dimensional (1D) structures [18]. The main\nconclusion of all these studies is that the introduction\nof boundary conditions results in an increased spin life-\ntime. In particular, in Ref. [17] Lyubinskiy reports on\na strong suppression of spin relaxation in small 2D sys-\ntems. In Ref. [18] the present authors demonstrate that\na persistent spin helix spontaneously emerges in course\nof relaxation of homogeneous spin polarization. As real\nelectronic devices are always of a finite size, the under-\nstanding of spin relaxation dynamics in reduced geome-\ntries is of a crucial importance.\nIn this paper, we present our studies of spin relaxation\nin rings with Bychkov-Rashba [19] spin-orbit coupling.\nThe ring geometry is especially interesting since on the\none hand, the electron space motion in rings is confined\nto a limited space region and on the other hand, the\nring geometry does not require any boundary conditions\n(such as those derived in Ref. [12]) for 1D transport that\nnormally have to be used, for example, to describe spin\ndynamics in finite-length structures [18]. Starting with\na spin kinetic equation, we formulate a set of equations\nfor spin polarization components on the ring and solve\nthese equations in some particular cases. Specifically,\nwe find that a homogeneous spin polarization transforms\ninto a persistent crown-like spin structure schematically\nshown in Fig. 1. Moreover, we determine a set of non-trivial spin persistent states that are realized at some\nspecific sizes of the ring. Finally, we derive expression\nfor Green function that can be used to find evolution of\nspin polarization for any initial spin polarization profile.\nAll these results indicate an unusual character of spin\nrelaxation in finite size structures that is dramatically\ndifferent from the spin relaxation in the bulk.\nSPIN KINETIC EQUATION\nIn this section, we introduce the model and derive\nequations governing spin relaxation dynamics in rings.\nThe main results of this section are given by Eqs. (7)-(9).\nThederivationoftheseequationsisbasedonakineticap-\nproach. It is important to mention that Eqs. (7)-(9) are\nmore general than traditional spin drift-diffusion equa-\ntions [20–24]. In particular, Eqs. (7)-(9) are applicable\nin both diffusive and ballistic regimes at any value of spin\nrotation angle per mean free path.\nLet us consider spin-polarized electrons that can move\nalong a ring of a sufficiently large radius r\u001d~=pso\n00=t\nz\n∞→t\nFIG.1: (Coloronline)Schematicdiagramofspinrelaxationin\na ring. Initially homogeneous spin polarization in zdirection\n(along the ring axis) transforms into a persistent crown-like\nspin structure. Note that the spin polarization amplitude\ndecreases in this process according to Eq. (21).arXiv:1103.3231v1  [cond-mat.mes-hall]  16 Mar 20112\nthat small-radius corrections [25, 26] to spin-orbit Hamil-\ntonian [19] can be neglected. The spin-orbit Hamilto-\nnian is taken in the form HSO=\u000b(^\u001b\u0002^ p)\u0001ez. Here,\n^ p= (^px;^py)is the 2D electron momentum operator, m\nis the effective electron’s mass, ^\u001bis the Pauli-matrix vec-\ntor,\u000bis the spin-orbit coupling constant and ezis a unit\nvector along the ring axis. It can be shown that a mo-\ntion of an electron with a momentum palong a straight\ntrajectory is accompanied by a spin rotation with the\nangular velocity \np= \np\u0002ez=p, where \n = 2\u000bp=~.\nThe spin precession angle per unit length is given by\n\u0011= 2\u000bm~\u00001. The ring geometry, however, changes the\ncharacter of spin rotations since the spin rotation axis\nrotates as electron moves along the ring.\nOur consideration of spin dynamics in rings is based on\nthe kinetic equation for electron spin polarization (see,\ne.g., Ref. [17, 27]). In quasi-classic approximation such\nan equation can be written as\n\u0012@\n@t+p\nm\u0001r\u0013\nSp=\np\u0002Sp+StfSpg;(1)\nwhere Sp(r;t)is the vector of spin polarization of elec-\ntrons, andStfSpgis the collision integral describing elec-\ntron scattering processes. Eq. (1) describes change in\nspin polarization of electrons moving with momentum p.\nThe RHS of Eq. (1) includes two terms describing rota-\ntionofspinpolarizationwiththeangularvelocity \npand\nchange in spin polarization due to scattering processes.\nIn the\u001c-approximation [28] the collision integral is given\nby\nStfSpg=\u00001\n\u001c(Sp\u0000hSpi); (2)\nwhere the angle brackets denote averaging over direction\nof electron momentum. The collision integral (2) cor-\nresponds to the elastic scattering of electrons by strong\nscatterers with a characteristic time \u001cbetween the col-\nlisions. For 1D case the average spin polarization sim-\nplifies to the following expression hSpi= (S++S\u0000)=2,\nwhere S+andS\u0000are the spin polarizations of electrons\nmoving along the ring in the clockwise (with momentum\np=mve\u0012), and counterclockwise ( p=\u0000mve\u0012) direc-\ntions with the average velocity vconnected to the mean-\nfree path`by`=v\u001c. Thus, the kinetic equation (1) for\n1D ring of a radius rtakes the form of the system of two\nvector equations\n\u0012@\n@t+v\nr@\n@\u0012\u0013\nS+= \ner\u0002S+\u00001\n2\u001c(S+\u0000S\u0000);(3)\n\u0012@\n@t\u0000v\nr@\n@\u0012\u0013\nS\u0000=\u0000\ner\u0002S\u0000\u00001\n2\u001c(S\u0000\u0000S+):(4)\nThis system of equations should be complimented by the\ninitial conditions for spin polarizations S+(\u0012;t= 0)and\nS\u0000(\u0012;t= 0)(for clockwise and counterclockwise moving\nelectrons).Taking the sum and difference of Eqs. (3, 4) we easily\nobtain\n@S\n@t=\u0000v\nr@\u0001\n@\u0012+ \ner\u0002\u0001; (5)\n@\u0001\n@t=\u0000v\nr@S\n@\u0012+ \ner\u0002S\u0000\u0001\n\u001c; (6)\nwhere the following notations are used: S=S++S\u0000and\n\u0001=S+\u0000S\u0000. As we are mainly interested in finding\nthe total spin polarization S,\u0001can be eliminated from\nEqs. (5) and (6) via a simple transformation. The final\nequations for three components of electron spin polariza-\ntion in cylindrical coordinates, S=Srer+S\u0012e\u0012+Szez,\ncan be presented in the form\n@2Sr\n@t2+1\n\u001c@Sr\n@t=!2@2Sr\n@\u00122\u0000!2Sr\u00002!2@S\u0012\n@\u0012\u0000!\nSz;(7)\n@2S\u0012\n@t2+1\n\u001c@S\u0012\n@t=!2@2S\u0012\n@\u00122\u0000(!2+ \n2)S\u0012 (8)\n+2!2@Sr\n@\u0012+ 2!\n@Sz\n@\u0012;\n@2Sz\n@t2+1\n\u001c@Sz\n@t=!2@2Sz\n@\u00122\u0000\n2Sz\u0000!\nSr\u00002!\n@S\u0012\n@\u0012;(9)\nwhere!=v=ris the angular velocity.\nThe initial rate of change of spin polarization Smay be\ncalculated by using the initial condition for \u0001from Eq.\n(5). In particular, if polarizations of electrons moving\nclockwise and counterclockwise at the initial moment are\nthe same, i.e. \u0001(\u0012;t= 0) = 0 , then we find from Eq. (5)\nthat in this case\n\u0012@S\n@t\u0013\nt=0= 0: (10)\nThe diffusive limit Eqs. (7)-(9) is realized on time scales\nmuchlongerthan \u001c. Inthiscase, wecanneglect @2Si=@t2\nterms in these equations and notice that the resulting\nequations are of diffusion type with the diffusion coeffi-\ncientD=v2\u001c.\nRELAXATION OF HOMOGENEOUS SPIN\nPOLARIZATION\nThe most interesting and simple solution of Eqs. (7)-\n(9) describes the relaxation of the homogeneous spin po-\nlarization initially directed along z-axis (see Fig.1)\n(S)t=0=S0ez; (11)\nwhereS0is the initial spin polarization amplitude. Since\nthe initial condition (11) is symmetrical with respect to\nrotations about the ring axis, S\u0012= 0at anytime, and\nthe spin polarization components SrandSzshould not\ndepend on\u0012. Then Eq. (8) satisfies identically, and Eqs.3\n(7) and (9) can be simplified to the following relations\nd2Sr\ndt2+1\n\u001cdSr\ndt+!2Sr+!\nSz= 0; (12)\nd2Sz\ndt2+1\n\u001cdSz\ndt+ \n2Sz+!\nSr= 0:(13)\nSolving these simple equations with the initial conditions\n(10)and(11), wefindthattherelaxationofhomogeneous\nspin polarization initially directed along the ring’s axis is\ndescribed by\nSr(t) =\u0000!\nS0\n!2+ \n2\u0014\n1\u0000e\u0000t\n2\u001c\u0012\ncosh\u0014t+sinh\u0014t\n2\u001c\u0014\u0013\u0015\n;(14)\nSz(t) =!2S0\n!2+ \n2\u0014\n1 +\n2\n!2e\u0000t\n2\u001c\u0012\ncosh\u0014t+sinh\u0014t\n2\u001c\u0014\u0013\u0015\n;(15)\nwhere\u0014=q\n1\n4\u001c2\u0000(!2+ \n2). It should be emphasized\nthat the parameter \u0014can take both real and imaginary\nvalues depending on system parameters. In particular,\nwe can define a diffusive ( \u001cp\n!2+ \n2\u001c1, Im(\u0014) = 0)\nand ballistic ( \u001cp\n!2+ \n2\u001d1, Re(\u0014) = 0) limits of spin\ndynamics. According to these definitions, in the diffusive\nlimit the mean free path is much smaller than the ring\nradius,`\u001cr, andthespinprecessionanglepermeanfree\npath is small, \u0011`\u001c1. In the opposite ballistic limit, at\nleast one of the following inequalities should hold: `\u001dr,\n\u0011`\u001d1. Any of these two conditions for the ballistic limit\nresults in oscillations in spin relaxation dynamics. In\nthe diffusive limit, however, the spin polarization decays\nexponentially without any oscillations.\nEqs. (14) and (15) can be further simplified in\nthe diffusive and ballistic limits. In diffusive limit,\n\u001cp\n!2+ \n2\u001c1, Eqs. (14) and (15) take the form\nSr(t) =\u0000!\nS0\n!2+ \n2h\n1\u0000e\u0000\u001c(!2+\n2)ti\n;(16)\nSz(t) =!2S0\n!2+ \n2\u0014\n1 +\n2\n!2e\u0000\u001c(!2+\n2)t\u0015\n:(17)\nIt is clearly seen from the above expressions that in this\ncase the relaxation of spin polarization is characterized\nby the time (\u001c(!2+ \n2))\u00001, which is much longer than\n\u001c. In the opposite ballistic limit, when \u001cp\n!2+ \n2\u001d1,\nEqs. (14) and (15) can be reduced to\nSr(t) =\u0000!\nS0\n!2+ \n2h\n1\u0000e\u0000t\n2\u001ccos(p\n!2+ \n2t)i\n;(18)\nSz(t) =!2S0\n!2+ \n2\u0014\n1 +\n2\n!2e\u0000t\n2\u001ccos(p\n!2+ \n2t)\u0015\n:(19)\nConsequently, the time-dependence of spin polarization\ncomponents involves an exponential decay with a time\nconstant 2\u001cmodulated by oscillating functions. Fig. 2\nshows an example of spin relaxation dynamics in both\ndiffusive and ballistic limits.\n0 40 80 120 160 2000.00.20.40.60.81.0\n02468 1 0 1 2 1 40.00.20.40.60.81.0 Sr/S0\n Sz/S0\n Sr/S0\n Sz/S0\n Sr,z/S0\nt/τ\n(b)(a)\n  Sr,z/S0\nt/τFIG.2: (Coloronline)Relaxationofelectronspinpolarization\nin a ring in the (a) diffusive and (b) ballistic regimes. S\u0012= 0.\nThis plot was obtained using Eqs. (14) and (15) with the\nfollowing parameter values: \n=!=\u00000:5,\u001c\u0014= 0:45(in (a))\nand\u001c\u0014=i5(in (b)).\nThe most important feature of the solution given by\nEqs. (14) and (15) is that the electron spin polariza-\ntion does not decay completely to zero. At long times,\nit shapes into a persistent crown-like spin polarization\nstructure (schematically presented in Fig. 1). The long-\ntime values of spin polarization components are\nSr=\u0000!\n!2+ \n2S0; Sz=!2\n!2+ \n2S0:(20)\nThe above relations indicate that during the spin relax-\nationprocesstheamplitudeofspinpolarizationdecreases\nby the factor\nS\nS0=p\nS2r+S2z\nS0=!p\n!2+ \n2: (21)\nIn this persistent crown-like spin polarization structure\nthe angle between the ring’s axis and the direction of\nspin polarization does not depend on the magnitude of\nthe initial polarization S0and is equal to\ntan =Sr\nSz=\u0000\n!=\u0000\u0011r: (22)\nIt follows from Eqs. (21) and (22) that the geometric pa-\nrameters of persistent crown-like spin structure depend\n(besides the ring radius r) only on the ratio of the fre-\nquency of spin-orbit precession to the frequency of the\nelectron rotation, \n=!. Moreover, we note that in the\ncase of the opposite direction of the initial spin polariza-\ntion the asymptotic value of Srchanges to the opposite\none. In particular, if we consider the situation shown\nin Fig. 1 (realizable at a negative value of \u000b), then the4\nchangeS0!\u0000S0will result in the opposite directions of\nthe radial and zcomponents of spin polarization. Such\nan asymmetry can be considered as a result of the sym-\nmetry breaking by the spin-orbit interaction.\nPERSISTENT SPIN STATES\nThe crown-like persistent spin polarization structure\ndiscussed in the previous section is the simplest example\nof persistent spin states of spin kinetic equation (7)-(9).\nIn this section we find the complete set of the persistent\nspin states of Eqs. (7)-(9). For this purpose, we search\nspecific solutions of these equations in the complex form\nS(\u0012;t) =0\n@Sr(\u0012;t)\nS\u0012(\u0012;t)\nSz(\u0012;t)1\nA=0\n@~Sr\n~S\u0012\n~Sz1\nAein\u0012e\u0015t;(23)\nwherenis an integer number. The possible values of\n\u0015and corresponding amplitudes ~Sr,~S\u0012,~Szhave to be\nfound. Substituting Eq. (23) into Eqs. (7)-(9) we readily\nobtain a system of linear homogeneous equations. The\ncondition of consistency of these equations give rise for\nthe following values of \u0015:\n\u0015\u0006\n1=\u00001\n2\u001c\u0006r\n1\n4\u001c2\u0000n2!2; (24)\n\u0015\u0006\n2=\u00001\n2\u001c\u0006r\n1\n4\u001c2\u0000\u0010\nn!+p\n!2+ \n2\u00112\n;(25)\n\u0015\u0006\n3=\u00001\n2\u001c\u0006r\n1\n4\u001c2\u0000\u0010\nn!\u0000p\n!2+ \n2\u00112\n:(26)\nSolving the linear homogeneous equations for each value\nof\u0015(given by Eqs. (24)-(26)) we find corresponding\namplitudes\n~S1=0\nB@\u0000\np\n!2+\n2\n0\n!p\n!2+\n21\nCA;~S2=1p\n20\nB@!p\n!2+\n2\n\u0000i\n\np\n!2+\n21\nCA;\n~S3=1p\n20\nB@!p\n!2+\n2\ni\n\np\n!2+\n21\nCA: (27)\nIt is interesting to note that the vectors given by Eqs.\n(27) do not depend on \u001candn. They are orthogonal and\nnormalized(incomplexsense), soitiseasytoexpandany\nvector (for example, related to initial spin polarization)\nin this basis.\nA spin state is persistent when \u0015= 0(this follows\ndirectly from Eq. (23)). It can be seen from Eqs. (24)-\n(26) that there are three possibilities when the condition\n\u0015= 0is realized. The first possibility is when n= 0,\nin this case, \u0015\u0000\n1is equal to zero. The corresponding per-\nsistent spin polarization structure is given by S=A~S1,\nFIG. 3: (Color online) n= 2stationary state (as follows from\nEq. (29) at \u00120= 0).\nthat is\nSr=\u0000\np\n!2+ \n2A; S\u0012= 0; Sz=!p\n!2+ \n2A;(28)\nwhereAis an arbitrary constant describing the magni-\ntude of spin polarization in this state. It can be seen that\nthe persistent spin state given by Eq. (28) is realized in\nthe relaxation of homogeneous spin polarization initially\ndirected in zdirection (this problem was considered in\nthe previous section, see Eq. (20)).\nAs it is seen from Eqs. (25) and (26), two other pos-\nsibilities (\u0015\u0000\n2= 0or\u0015\u0000\n3= 0) may only be realized if\nn=\u0000q\n1 +\n2\n!2orn=q\n1 +\n2\n!2correspondingly. Since\nnis an integer number, these conditions are fulfilled only\nfor certain combinations of system parameters as dis-\ncussed below. By substituting Eqs. (27) for ~S2or~S3\ninto Eq. (23) and taking real or imaginary parts, we ob-\ntain the persistent spin polarization structures that can\nbe presented as\nSr=B\nnsinn(\u0012\u0000\u00120);S\u0012=Bcosn(\u0012\u0000\u00120);\nSz=Bp\nn2\u00001\nnsgn(\n) sinn(\u0012\u0000\u00120);(29)\nwheren=q\n1 +\n2\n!2= 2;3;4;:::,Band\u00120are arbitrary\nreal constants, which determine spin polarization ampli-\ntude and angular position of the structure respectively.\nFig. 3 shows schematically the persistent spin polariza-\ntion structure (29) for n= 2.\nIt should be noted that the persistent spin polarization\nstructures (Eq. (29)) exist only at certain relations be-\ntween the ring’s radius rand the strength of spin-orbit\ninteraction \u0011. Since \n=!=\u0011r, the conditions for persis-\ntent spin states are given by\nj\u0011jr=p\nn2\u00001; n= 2;3;4;::: (30)\nIn addition, we note that at r!1,nbecomes much\ngreater than 1. In this case in accordance with Eq. (30)\nn\u0019\u0011r\u001d1, and persistent spin polarization structure\n(29) becomes the usual plain spin helix [6] with wave\nvector\u0011.5\nGREEN FUNCTION\nLinear combinations of specific solutions (23) (with\namplitudes (27) and corresponding \u0015-s given by Eqs.\n(24)-(26)) of homogeneous linear Eqs. (7)-(9) are also\nsolutions of these equations. Therefore, the general solu-\ntion of the system (7)-(9) can be presented as a sum\nS(\u0012;t) =X\nn;\u0017;\u001bA\u001b\n\u0017(n)~S\u0017e\u0015\u001b\n\u0017(n)tein\u0012; (31)\nwhereA\u001b\n\u0017(n)are complex constants, \u0017= 1;2;3,\u001b=\n+;\u0000, andnruns over all integer values. In order to\nfind the constants A\u001b\n\u0017(n)we should specify the initial\nconditions for spin polarization\nS(\u0012;t)t=0\u0011S(\u0012;0);\u0012@S(\u0012;t)\n@t\u0013\nt=0\u0011_S(\u0012;0):(32)\nUsing initial conditions (32) we obtain\nA\u0006\n\u0017(n) =Z2\u0019\n0d\u00120\n2\u0019e\u0000in\u00120h~S\u0017;_S(\u00120;0)\u0000\u0015\u0007\n\u0017(n)S(\u00120;0)i\n\u0015\u0006\u0017(n)\u0000\u0015\u0007\u0017(n);(33)\nwhereh\u001e; i=\u001er r+\u001e\u0012 \u0012+\u001ez zis the inner product\nof amplitudes. It is easy to check that h~S\u0017;~S\u0016i=\u000e\u0017\u0016.\nThe Green function of spin kinetic equation can be\nobtained substituting Eq. (33) into Eq. (31). The Green\nfunction can be employed to find spin polarization at any\nmoment of time for any given initial conditions as\nS\u000b(\u0012;t) =\u0014@\n@t+1\n\u001c\u0015Z2\u0019\n0d\u00120G\u000b\f(\u0012\u0000\u00120;t)S\f(\u00120;0)\n+Z2\u0019\n0d\u00120G\u000b\f(\u0012\u0000\u00120;t)_S\f(\u00120;0);(34)\nwhere\u000bor\ftake the values r;\u0012;zand summation over\nrepeated indexes is implied.\nIn particular, the components of the Green function\n(it is actually 3x3 matrix) of the system (7)-(9) can be\nwritten as\nGrr(\u0012;t) =\n2G0(\u0012;t) +!2Gc(\u0012;t)\n!2+ \n2;(35)\nG\u0012r(\u0012;t) =\u0000!Gs(\u0012;t)p\n!2+ \n2; (36)\nGzr(\u0012;t) =\u0000!\n[G0(\u0012;t)\u0000Gc(\u0012;t)]\n!2+ \n2;(37)\nGr\u0012(\u0012;t) =\u0000G\u0012r(\u0012;t) (38)\nG\u0012\u0012(\u0012;t) =Gc(\u0012;t); (39)\n0 5 10 15 200.00.10.20.30.4\n0 1000 2000 30000.00.10.2 Sr\n Sθ\n Sz Sr\n Sθ\n Sz\n \n(b)t/τ\nt/τx5\n  Spin polarization (arb.units)x5(a) \n \n  FIG. 4: (Color online) Spin polarization components at\n\u0012=\u0019=4induced by a delta function excitation at \u0012= 0\n(S\f(\u0012;t= 0) = 0 ,_S\f(\u0012;t= 0) =\u000e(\u0012)\u000e\f;z, where\u000e(::)is the\ndelta function and \u000ei;jis the Kronecker delta). This plot was\nobtained using the parameter values \u001c\n =\u00000:1,\u001c!= 1(in\n(a)), and\u001c!= 0:1(in (b)).\nGz\u0012(\u0012;t) =\nGs(\u0012;t)p\n!2+ \n2; (40)\nGrz(\u0012;t) =Gzr(\u0012;t) (41)\nG\u0012z(\u0012;t) =\u0000Gz\u0012(\u0012;t); (42)\nGzz(\u0012;t) =!2G0(\u0012;t) + \n2Gc(\u0012;t)\n!2+ \n2;(43)\nwhere\nG0(\u0012;t) =e\u0000t\n2\u001c\n2\u0019n=+1X\nn=\u00001cosn\u0012sin\u0010\ntp\n!2n2\u00001=(2\u001c)2\u0011\np\n!2n2\u00001=(2\u001c)2;(44)\nGc(\u0012;t) =e\u0000t\n2\u001c\n4\u0019n=+1X\nn=\u00001cosn\u0012\u001asin \n+\nnt\n\n+n+sin \n\u0000\nnt\n\n\u0000n\u001b\n;(45)\nGs(\u0012;t) =\u0000e\u0000t\n2\u001c\n4\u0019n=+1X\nn=\u00001sinn\u0012\u001asin \n+\nnt\n\n+n\u0000sin \n\u0000\nnt\n\n\u0000n\u001b\n;(46)\n\n\u0006\nn=r\u0010\n!n\u0006p\n!2+ \n2\u00112\n\u00001=(2\u001c)2:(47)\nThe above equations can be further simplified. In fact,\nthesumsappearinginEqs. (44)-(46)canbecalculatedin6\nthe following way. First of all, we employ the well-known\nformula from the Bessel function theory\nsin\u0010\ntp\n!2\u0000q2\u0011\np\n!2\u0000q2=\n1\n2Z+1\n\u00001d\u0018ei!\u0018\b(t\u0000j\u0018j)I0\u0010\nqp\nt2\u0000\u00182\u0011\n;(48)\nwhere \b(t)is the Heaviside step function, and I0(t)is the\nmodified Bessel function of zero order. Substituting Eq.\n(48) into Eqs. (44)-(46) we perform summation in these\nequations taking into account that [29]\nn=+1X\nn=\u00001ein\u0018= 2\u0019n=+1X\nn=\u00001\u000e(\u0018\u00002\u0019n): (49)\nThen, the integrals in Eqs. (44)-(46) modified by Eq.\n(48) are trivially evaluated yielding the following results\nG0(\u0012;t) =e\u0000t\n2\u001c\n2!n=+1X\nn=\u00001I0 p\n!2t2\u0000(\u0012\u00002\u0019n)2\n2!\u001c!\n\u0002\b(!t\u0000j\u0012\u00002\u0019nj);(50)\nGc(\u0012;t) =e\u0000t\n2\u001c\n2!n=+1X\nn=\u00001I0 p\n!2t2\u0000(\u0012\u00002\u0019n)2\n2!\u001c!\n\u0002cos\"r\n1 +\n2\n!2(\u0012\u00002\u0019n)#\n\b(!t\u0000j\u0012\u00002\u0019nj);(51)\nGs(\u0012;t) =e\u0000t\n2\u001c\n2!n=+1X\nn=\u00001I0 p\n!2t2\u0000(\u0012\u00002\u0019n)2\n2!\u001c!\n\u0002sin\"r\n1 +\n2\n!2(\u0012\u00002\u0019n)#\n\b(!t\u0000j\u0012\u00002\u0019nj):(52)\nIn fact, all sums in Eqs. (50)-(52) are finite sums be-\ncause of the Heaviside functions \b(!t\u0000j\u0012\u00002\u0019nj). For\nanygivenvaluesoftime tandangle\u0012onlythetermswith\nnin the range (\u0012\u0000!t)=(2\u0019)< n < (\u0012+!t)=(2\u0019)give\nnon-zero contribution to the R.H.S. of Eqs. (50)-(52).\nFig. 4 shows an example of application of the Green\nfunction formalism to calculations of spin polarization\ndynamics. Specifically, we plot spin polarization compo-\nnents at\u0012=\u0019=4induced by spin polarization excitation\nat\u0012= 0. Such an excitation induces clockwise and coun-\nterclockwise propagating waves of spin polarization that\nfinally evolve into a steady homogeneous spin polariza-\ntion. Each time when clockwise and counterclockwise\npropagating waves of spin polarization reach \u0012=\u0019=4,\nthe spin polarization at this point changes in steps (see\nFig. 4(a)). The step amplitudes, however, decrease intime because of the diffusive component in spin trans-\nport. This type of behavior resembles a multiple echo.\nIn addition, in the diffusive limit shown 4(b) we did not\nobserve clear steps in spin polarization. At long times,\nS\u0012!0and the crown-like spin polarization configura-\ntion is formed.\nCONCLUSIONS\nSpinrelaxationinconfinedstructuresexhibitssomere-\nmarkable properties and behavior that are not found in\ninfinite 2D systems. In this paper we have studied the\nelectron spin relaxation in the ring. It has been found\nthat the homogeneous spin polarization along the ring\naxis transforms into a persistent crown-like spin struc-\nture. Moreover, a family of persistent spin states in the\nring has been identified. We have also derived the Green\nfunction of spin kinetic equation and used it to investi-\ngate the propagation of a point spin excitation.\n\u0003Electronic address: pershin@physics.sc.edu\n1I. Zutic, J. Fabian, and S. Das Sarma, Rev. Mod. Phys.\n76, 323 (2004).\n2M. Wu, J. Jiang, and M. Weng, Phys. Reports 439, 61\n(2010).\n3M. I. Dyakonov and V. I. Perel’, Sov. Phys. Solid State 13,\n3023 (1972).\n4M. I. Dyakonov and V. Y. Kachorovskii, Sov. Phys. Semi-\ncond.20, 110 (1986).\n5E. Y. Sherman, Appl. Phys. lett 82, 209 (2003).\n6Y. V. Pershin, Phys. Rev. B 71, 155317 (2005).\n7B. A. Bernevig, J. Orenstein, and S.-C. Zhang, Phys. Rev.\nLett.97, 236601 (2006).\n8M. Q. Weng, M. W. Wu, and H. L. Cui, J. Appl. Phys.\n103, 063714 (2008).\n9P. Kleinert and V. V. Bryksin, Phys. Rev. B 79, 045317\n(2009).\n10I. V. Tokatly and E. Y. Sherman, Ann. Phys. 325, 1104\n(2010).\n11Y. V. Pershin and V. A. Slipko, Phys. Rev. B 82, 125325\n(2010).\n12V. M. Galitski, A. A. Burkov, and S. Das Sarma, Phys.\nRev. B 74, 115331 (2006).\n13A. A. Kiselev and K. W. Kim, Phys. Rev. B 61, 13115\n(2000).\n14A. W. Holleitner, V. Sih, R. C. Myers, A. C. Gossard, and\nD. D. Awschalom, New J. Phys. 9, 342 (2007).\n15Y. V. Pershin, Phys. E 27, 77 (2005).\n16Y. V. Pershin and V. Privman, Phys. Rev. B 69, 073310\n(2004).\n17I. S. Lyubinskiy, JETP Lett. 83, 336 (2006).\n18V. A. Slipko and Y. V. Pershin, arXiv:1102.0039 (2011).\n19Y. Bychkov and E. Rashba, JETP Lett. 39, 78 (1984).\n20Z.G.YuandM.E.Flatté, Phys.Rev.B 66, 201202(2002).\n21A. A. Burkov, A. S. Núñez, and A. H. MacDonald, Phys.\nRev. B 70, 155308 (2004).\n22S. Saikin, J. Phys.: Condens. Matter 16, 5071 (2004).7\n23Y. V. Pershin, Phys. E 23, 226 (2004).\n24Y. V. Pershin and V. A. Slipko, arXiv:1007.0853v1 (2010).\n25F. E. Meijer, A. F. Morpurgo, and T. M. Klapwijk, Phys.\nRev. B 66, 033107 (2002).\n26B. Berche, C. Chatelain, and E. Medina, Eur. J. Phys. 31,\n1267 (2010).\n27I. S. Lyubinskiy and V. Y. Kachorovskii, Phys. Rev. B 73,041301 (2006).\n28L. P. Pitaevskii and E. M. Lifshitz, Physical Kinetics\n(Butterworth-Heinemann, 1981).\n29V. Vladimirov, Equations of Mathematical Physics (Mir,\n1985)."    }]
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